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+{"text":"\\section{Introduction}\n\n\nBL Lacertae is a well known source which has been used to define a class of active\ngalactic nuclei (AGNs) that, together with flat spectrum radio quasars, make up the highly\nvariable objects called blazars.  The BL Lacertae class is characterized by the absence or extreme\nweakness of  emission lines (with equivalent width in the rest frame of the host galaxy of $< 5$ \\AA),\nintense flux and spectral variability across the\ncomplete electromagnetic spectrum  on a wide variety of time-scales,\nand highly variable optical and radio polarization  (e.g., Wagner \\& Witzel 1995).  \nA relativistic plasma jet pointing close to our line of sight can account for the\nobserved properties of these objects.\nBL Lacertae is an optically bright blazar located at $z=0.0688 \\pm 0.0002$ (Miller \\&\nHawley 1977) hosted by a giant elliptical galaxy with R$=15.5$ (Scarpa et al.\\ 2000).\nBL Lacertae is an LBL (low frequency peaked blazar) as its low energy spectral component  peaks at\nmillimeter to micron wavelengths while the high energy spectral component peaks in the MeV-GeV  range.\nOn some occasions, BL Lacertae has shown broad $H \\alpha$ and $H  \\beta$ emission lines\nin its spectrum, raising the issue of its membership in its eponymous class (Vermeulen et al. 1995). \n\nBL Lacertae was observed by several multi-wavelength campaigns carried out by the Whole Earth Blazar Telescope \n(WEBT\/GASP; B\\\"ottcher et al. 2003; Villata et al.\\ 2009; Raiteri et al. 2009, 2010, 2013 and references therein).\nBL Lacertae is well known for its intense optical variability on short and intra-day time-scales\n(e.g.\\ Massaro et al.\\ 1998; Tosti et al.\\ 1999; Clements \\& Carini 2001; Hagen-Thorn et al.\\ 2004)\nand strong polarization variability (Marscher et al.\\ 2008; Gaur et al.\\ 2014 and references therein).\nExtensive light curves for BL Lacertae have been presented by many authors and hence a number\nof investigations have been carried out to search for the flux variations, spectral changes \nand any possible periodicities in the light curves (e.g.\\ Racine 1970; Speziali \\& Natali 1998; \nB{\\\"o}ttcher et al.\\ 2003; Fan et al.\\ 2001; Hu et al.\\ 2006).  Nesci et al.\\ (1998) found the \nsource to be variable with the amplitudes of flux\nvariations larger at shorter wavelengths. \nPapadakis et al.\\ (2003) studied the rise and decay time-scales of the source during the course of a single \nnight and found them to increase with decreasing frequency. They also studied the time-lags between the light\ncurves in different optical bands and found the B band to lead the I band by $\\sim$0.4 hours. Villata et al.\\ (2002) \ncarried out a campaign in 2000--2001 with exceptionally dense\ntemporal sampling, which was able to measure\nintra-night flux variations of this blazar. They found the optical spectrum to \nbe only weakly sensitive to the long term brightness trend and argued that this achromatic modulation of the flux base level \non long time-scales is due to variations of the jet Doppler factor. However, the short-term\n flux variations and especially the bluer-when-brighter trend indicate the importance of intrinsic  processes related to the jet \n emission mechanism (Raiteri et al. 2013; Agarwal \\& Gupta 2015).\n\nA key motivation of this study is to look for intra-day flux and spectral variations\nin optical bands during the active state of BL Lacertae in 2010--2012. We also studied interband BVRI time delays\non intra-day time-scales of BL Lacertae. As BL Lacertae is a very well known LBL and the optical bands are located\nabove the first peak of the spectral energy distribution, the fast intra-day variability (IDV) properties\ncan yield rather direct implications for the nature of  the acceleration and cooling mechanisms of the   \nrelativistic electron populations.\nOver the course of 3 years, we performed quasi-simultaneous optical multi-band photometric monitoring of this source\nfrom various telescopes in Bulgaria, Greece, India and the USA on intra-day time-scales.\n\nThe paper is organized as follows. In Section 2 we briefly describe the observations and data reductions. Section 3 discusses\nthe methods of quantifying variability. We present our results in Section 4.  Sections 5 \\& 6 contain a discussion and our conclusions,\nrespectively.\n\n\\section{Observations and Data Reduction}\n\nOur observations of BL Lacertae started on 10 June 2010 and ran through 26 October 2012.\n The entire observation log is presented in Table 1. The observations were carried out at \nseven telescopes in Bulgaria,  Greece, India and the USA.\nThe telescopes in Bulgaria, Greece and India are described in detail in Gaur et al.\\ (2012, Table 1) \nand the standard data reduction methods we used at each telescope are given in Section 3 of that paper,\nso we will not repeat them here. During our observations, typical seeing vary\nbetween 1--3 arcsec.\nIn our  observations of BL Lacertae, comparison stars are observed in the same field as the blazar \nand their magnitudes are taken from Villata et al.\\ (1998).\nWe used star C  for calibration as it has both magnitude and colour close to those of BL Lacertae \nduring our observations.\n\nAt the MDM Observatory on the south-west ridge of Kitt Peak, Arizona, USA,\ndata were taken for limited periods during the nights of 3, 4, 5, 6, 7 and 8\nJuly 2010 with the 1.3 m McGraw-Hill Telescope, using the Templeton CCD with\nB, V, R, and I filters. CCD parameters are described in Table 2. The standard data reduction was performed using \\texttt{IRAF}\n\\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which are operated\nby the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the\nNational Science Foundation.},\nincluding bias subtraction and flat-field division. Instrumental magnitudes of BL Lacertae\n plus four comparison stars in the field (Villata et al.\\ 1998) were extracted\nusing the \\texttt{IRAF} package \\texttt{DAOPHOT}\\footnote{Dominion Astrophysical\nObservatory Photometry software} with an aperture radius of 6 arcsec and\na sky annulus between 7.5 and 10 arcsec. \n\nThe host galaxy of BL Lacertae is relatively bright, so, in order to remove its contribution from the\nobserved magnitudes, we first dereddened the magnitudes using the Galactic extinction coefficient of Schlegel\net al.\\ (1998) and converted them into fluxes. We then subtracted the host galaxy contribution\nfrom the observed fluxes in R band  by considering different aperture radii used by different observatories for the\nextraction of BL Lacertae magnitudes, using Nilsson et al.\\ (2007). We inferred the host \ngalaxy contribution in B, V and I bands\nby adopting the elliptical galaxy colours of V$-$R $ = 0.61$, B$-$V $= 0.96$ and R$-$I $= 0.70$ \nfrom Fukugita et al.\\ (1995).\nFinally, we subtracted the host galaxy contribution in the B, V and I bands in order to avoid  host\ncontamination in the extraction of colour indices.\n\n\\section{Variability Detection Criterion}\n\n\\subsection{Power Enhanced F-test}\n\nThe F-test, as described by de Diego  (2010), provides a standard criterion for testing for the presence of intra-night\nvariability. The F-statistic is defined as the ratio of two given sample variances such as s$_Q ^{2}$ for the blazar instrumental light \ncurve measurements and s$_* ^{2}$ for that of the standard star, i.e.,\n\\begin{equation}\nF=\\frac {s_Q ^{2}}{s_* ^{2}} .\n\\end{equation}\nUsually, two comparison stars in the blazar field  are used to calculate $F_{1}$ and $F_{2}$, and evidence of variability is \nclaimed if both the F-tests simultaneously reject the null hypothesis at a specific significance level (usually 0.01 or 0.001) ( \nGaur et al. 2012 and references therein).\nIn this case, the number of degrees of freedom for each sample, $\\nu_Q$ and $\\nu_*$ will be the same, and equal to the number of\nmeasurements, $N$ minus 1 ($\\nu = N - 1$).\n\nRecently, de Diego (2014) called this procedure the ``Double Positive Test'' (DPT) and pointed out a problem with this \nprocedure that DPT has very low power and in practice its significance level can not be calculated. Also,\n  a large brightness difference or some variability in one of the stars may lead to an underestimation of the \nsource's variability with respect to the dimmer or less variable star.\nTo avoid this issue we employed the power-enhanced F-test using the approach of de Diego (2014) and de Diego et al. (2015). \nIt consists of increasing the number of degrees of freedom in the denominator of the F-distribution by stacking \nall the light curves of the standard stars and it consists\nin transforming the comparison star differential light curves to have the same photometric noise as if their magnitudes\n matched exactly the mean magnitude of the target object. The mean brightness of both the comparison star and the target\nobject are matched to ensure that the photometric errors are equal. Including \nmultiple standard stars reduces the possibility of false detections of intra-night variability that can be produced by \none single peculiar comparison star light curve. More details are provided in de Diego et al. 2015. In the analysis,\n we used three standard stars B, C and H in the field of BL Lacertae whose brightnesses are very close to \nthe brightness of BL Lacertae. Thus, for the $i^{th}$ observation of the \nlight curve of standard star $j$, for which we have $N_j$ data points, we calculate the square deviation as: \n\\begin{equation}\n    s_{j,i}^2 = (m_{j,i} - \\overline{m}_{j})^2.\n\\end{equation}\n\nBy stacking the results of all observations on the total of $k$ comparison stars, we can calculate the combined variance as:\n\\begin{equation}\n    s_c^2 = \\frac{1}{(\\sum_{j=1}^k N_j) - k} \\sum_{j=1}^k \\sum_{i=1}^{N_j} s_{j,i}^2.\n\\end{equation}\nThen, we compare this combined variance with the blazar light curve variance to obtain the $F$ value\nwith $\\nu_Q = N - 1$ degrees of freedom in the numerator and $\\nu_* = k (N - 1)$  degrees of freedom in the denominator. \nThis value is then compared with the $F^{(\\alpha)}_{\\nu_Q,\\nu_*}$ critical value, where $\\alpha$ is the significance \nlevel set for the test.\nThe smaller the $\\alpha$ value, the more improbable it is that the result is produced by chance. If $F$ is larger\nthan the critical value, the null hypothesis (no variability) is discarded. We have performed the\n$F$-test at the $\\alpha = 0.001$ level.\n\n\\subsection{Discrete Correlation Function (DCF)}\n\nTo estimate the variability time-scales in the observed light curves of BL Lacertae and to determine the cross-correlations\nbetween different optical bands, we used the Discrete Correlation Function (Edelson \\& Krolik 1988; Hovatta et al. 2007).\n\n \nThe first step is to calculate the unbinned correlation (UDCF) using the given time series by:\n\\begin{equation}\nUDCF_{ij} = {\\frac{(a(i) - \\bar{a})(b(j) - \\bar{b})}{\\sqrt{\\sigma_a^2 \\sigma_b^2}}}.\n\\end{equation}\nHere $a(i)$ and $b(j)$ are the individual points in two time series $a$ and $b$, respectively, $\\bar{a}$\nand $\\bar{b}$ are respectively the means of the time series, and $\\sigma_a^2$ and $\\sigma_b^2$ are their variances.\nThe correlation function is binned after calculation of the UDCF. The DCF can be calculated by averaging the \nUDCF values ($M$ in number) for each time delay\n\\( \\Delta t_{ij} = (t_{yj}-t_{xi}) \\) lying in the range  \\( \\tau - \\frac{\\Delta\\tau}{2} \\leq t_{ij} \n\\leq \\tau+ \\frac{\\Delta\\tau}{2} \\) via\n\n\\begin{equation}\nDCF(\\tau) = {\\frac{1}{n}} \\sum ~UDCF_{ij}(\\tau) .\n\\end{equation}\nwhere $\\tau$ is the centre of a time bin and $n$ are the number of points in each bin.\n DCF analysis is frequently used for finding the correlation and possible lags between multi-frequency\nAGN data. When the same data train is used, so $a$=$b$, there is obviously a peak at zero lag and is called Auto \nCorrelation Function (ACF), indicating that there \nis no time lag between the two but any other strong peaks in the ACF give indications of variability timescales. \n\n\n\n\\section{Results}\n\n\\subsection{Flux variations}\n\nWe extensively observed the source for a  total of 38 nights during 2010--2012.   During  \n26 of those nights, we observed the source quasi-simultaneously in the B, V, R and I bands, providing a total of 113 light \ncurves in B, V, R and I. These light curves are displayed in Figs.\\ 1 and 2.\nThe  lengths of the individual observations were usually between  2 and 6 hours.\nIt is clear from the figures that BL Lacertae was variable from day to day and also on hourly time-scales.\nWe searched for genuine flux variations on IDV time-scales and found 50 light curves to be variable in the whole set of filters\n during a total of 19 nights using the enhanced F-test. \nThe intranight variability amplitudes (in per cent) are given by Heidt \\& Wagner (1996):\n\\begin{eqnarray}\nAmp = 100\\times \\sqrt{{(A_{max}-A_{min}})^2 - 2\\sigma^2},\n\\end{eqnarray}\nwhere $A_{max}$ and $A_{min}$ are the maximum and minimum values in the calibrated light curves of the blazar, and $\\sigma$\nis the average measurement error. \nThe enhanced F-test values\nare presented in Table 1. \n\nThe duty cycle (in per cent) (DC) of BL Lacertae is computed following the definition of Romero et al.\\ (1999) that \nlater was used by many authors (e.g., Stalin et al.\\ 2004; Goyal et al.\\ 2012, and references therein),\n\n\\begin{equation} \nDC = 100\\frac{\\sum_{i=1}^n N_i(1\/\\Delta t_i)}{\\sum_{i=1}^n (1\/\\Delta t_i)}  ,\n\\label{eqno1} \n\\end{equation}\nwhere $\\Delta t_i = \\Delta t_{i,obs}(1+z)^{-1}$ is the duration of the\nmonitoring session of a source on the $i^{th}$ night, corrected for\nits cosmological redshift, $z$. Since for a given source the monitoring durations on different nights are not always equal, \nthe computation of the DC is weighted by the actual monitoring duration\n$\\Delta t_i$ on the $i^{th}$ night. $N_i$ is set equal  to 1 if intra-day variability \nwas detected by the F-test (given in Table 1),  otherwise $N_i$ = 0.\nWe found the duty cycle of BL Lacertae to be around 44 per cent. \n\n\\subsection{Amplitude of Variability}\n\nWhen the variability is substantial the amplitude of variability usually is greater in higher energy bands (B) and smaller \nin lower energy bands (R) in most of the observations.\nIt has been found by many investigators that amplitude of variability is greater at higher frequencies for\nBL Lacertaes (Ghisellini et al. 1997; Massaro et al. 1998; Bonning et al. 2012) however, the amplitude\nof variability is not systemaically larger at higher frequencies is also found on some occasions (Ghosh et al. 2000;\nRamirez et al. 2004). In few cases, we found the amplitude of variability in lower energy bands \nto be comparable to or greater than the amplitude of variability in higher energy bands. Still, because \nthe errors in the B band are higher in these cases, while they are often lower in the lower frequency band,  there are \nsome nights during which the statistical significance of the variations falls below our thresholds in \nB (also sometimes V), when  clear variability is detected in R and I bands.  Also, differences \nin the amplitude of variability in various optical bands change from one observation to another. \nThe highest fractional amplitude of variability was found to be $\\sim$38 per cent on 6 July 2010, where the  \nsource showed a $\\sim$0.3 mag change in less than 3.5 hours in V band. \\\\\n\nWe searched for possible correlations between the variability amplitude and the duration of the observation, as displayed\nin Fig.\\ 3 (left panel). We found a significant positive correlation ($\\rho=0.3306$ with $p=0.0190$, where $\\rho$ and $p$ are\n the Spearman Correlation coefficient and its p-value, respectively) between them, i.e., the variability amplitude \nincreases with duration of the observation.  We conclude that there is an enhanced likelihood of \nseeing higher amplitudes of variability in longer duration light curves, which is in agreement with Gupta \\& Joshi (2005) report\nfor a larger sample of blazars.\n\nNext we examined the possible correlation between the amplitude of variability with \nthe source flux (Fig.\\ 3, right panel). It can be seen that amplitude of variability decreases as the source flux increases \n($\\rho$=-0.4990 with $p$=0.0002). \nThis might be explained as, if the source attains a high flux state, the irregularities in the jet flow decreases \nparticularly if fewer \nnon-axisymmetric bubbles were carried outward in the relativistic magnetized jets (Gupta et al.\\ 2008).\nFor instance, the blazar 3C 279 was observed in 2006 during its outburst\/high state but did not show any genuine micro-variability, \nwhile other\nsources which were in their pre\/post outburst state showed significant micro-variability  (Gupta et al.\\ 2008). \nThis anti-correlation between variability amplitude and flux could also be explained by a two component\nmodel where in more active phases or in an outburst state, the slowly varying jet component rises and dominates on the emission\nfrom the more variable regions, i.e., shocks\/knots, and therefore reduces the fractional amplitude. However, there is \n scatter in the plot, some of which could be due to an observational bias, as some observations were longer than others. \nDue to this, the observed variability amplitude at similar luminosities could be quite different as the\n amplitude of variability increases with the duration of the observation.\n\n\\subsection{R-band Autocorrelations}\n\nWe searched for the presence of a characteristic time-scale of variability in all of the nights by auto-correlating the\n R-band measurements, for which we had the most data. In most of the nights, the\n auto-correlation function (ACF) shows a sharp maximum at nearly zero lag, as expected, and it stays positively \nauto-correlated with itself for time lags of somewhat less than 1 hour followed by dropping to negative values. \n Hence, we conclude that we did not find any evidence for  characteristic variability time-scales from this approach.\nOne example of R-band ACF is shown in the left panel of figure 4.\n\\begin{figure*}\n   \\centering\n\\includegraphics[width=4cm , angle=0]{fig1a.eps}\n\\includegraphics[width=4cm , angle=0]{fig1b.eps}\n\\includegraphics[width=4cm , angle=0]{fig1c.eps}\n\\includegraphics[width=4cm , angle=0]{fig1d.eps}\n\\includegraphics[width=4cm , angle=0]{fig1e.eps}\n\\includegraphics[width=4cm , angle=0]{fig1f.eps}\n\\includegraphics[width=4cm , angle=0]{fig1g.eps}\n\\includegraphics[width=4cm , angle=0]{fig1h.eps}\n\\includegraphics[width=4cm , angle=0]{fig1i.eps}\n\\includegraphics[width=4cm , angle=0]{fig1j.eps}\n\\includegraphics[width=4cm , angle=0]{fig1k.eps}\n\\includegraphics[width=4cm , angle=0]{fig1l.eps}\n\\includegraphics[width=4cm , angle=0]{fig1m.eps}\n\\includegraphics[width=4cm , angle=0]{fig1n.eps}\n\\includegraphics[width=4cm , angle=0]{fig1o.eps}\n\\includegraphics[width=4cm , angle=0]{fig1p.eps}\n\\includegraphics[width=4cm , angle=0]{fig1q.eps}\n\\includegraphics[width=4cm , angle=0]{fig1r.eps}\n\\includegraphics[width=4cm , angle=0]{fig1s.eps}\n\\includegraphics[width=4cm , angle=0]{fig1t.eps}\n\n\\caption{IDV light curves of BL Lacertae during 2010 and early 2011 in the  B (blue), V(green), R(red) and I(black) bands.  The\nX-axis is JD (2455000+), and the Y-axis is the calibrated magnitudes in each of the panels. The B, V and I \nbands are shifted by\narbitrary offsets with respect to R-band light curve. Observations from observatory A are represented by squares; those from\nC are represented by triangles; D  by filled circles; E by open circles and F by starred symbols.}\n \\end{figure*}\n\n\n\\begin{figure*}\n   \\centering\n\\includegraphics[width=4cm , angle=0]{fig2a.eps}\n\\includegraphics[width=4cm , angle=0]{fig2b.eps}\n\\includegraphics[width=4cm , angle=0]{fig2c.eps}\n\\includegraphics[width=4cm , angle=0]{fig2d.eps}\n\\includegraphics[width=4cm , angle=0]{fig2e.eps}\n\\includegraphics[width=4cm , angle=0]{fig2f.eps}\n\\includegraphics[width=4cm , angle=0]{fig2g.eps}\n\\includegraphics[width=4cm , angle=0]{fig2h.eps}\n\\includegraphics[width=4cm , angle=0]{fig2i.eps}\n\\includegraphics[width=4cm , angle=0]{fig2j.eps}\n\\includegraphics[width=4cm , angle=0]{fig2k.eps}\n\\includegraphics[width=4cm , angle=0]{fig2l.eps}\n\\includegraphics[width=4cm , angle=0]{fig2m.eps}\n\\includegraphics[width=4cm , angle=0]{fig2n.eps}\n\\includegraphics[width=4cm , angle=0]{fig2o.eps}\n\\includegraphics[width=4cm , angle=0]{fig2p.eps}\n\\includegraphics[width=4cm , angle=0]{fig2q.eps}\n\\includegraphics[width=4cm , angle=0]{fig2r.eps}\n\\includegraphics[width=4cm , angle=0]{fig2s.eps}\n\n\\caption{As in Fig.\\ 1 for August 2011 through October 2012.}\n\\end{figure*}\n\n\\subsection{Inter-band cross-correlations}\n\nWe computed the DCFs to determine the cross-correlations and time delays between the B and I, V and I, and R and I bands. \nThe time delays are expected between emission in different energy bands, as the flare usually begin at higher frequencies\nand then propagates to lower frequencies in the inhomogeneous jet model. Injected high energy electrons\nemit synchrotron radiation first at higher frequencies and then cool, emitting at progressively lower frequencies, resulting\nin time lag between high and low frequencies.\nThe DCFs between the light curves in all bands show close correlations among the various bands in the nights\nwhere genuine variability is present. For the nights in which no genuine variability is present, we normally found\nmuch weaker correlations between the bands ($<$0.4). As the peak of the DCFs are broad,  we fit the DCFs with\n Gaussian functions to determine the possible time delays; \nhowever, lags indicated by the DCFs are all consistent with zero. This is not surprising due to the closeness of the various optical \nbands we measured in frequency space, so if any lags are present they appear to be less than the resolution of our light curves.\nAn example of the DCF is shown in the right panel of fig.\\ 4.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm , angle=0]{fig3a.eps}\n\\includegraphics[width=8cm , angle=0]{fig3b.eps}\n\n\\caption{Dependence of amplitude of variability on duration (left panel) and flux (right panel) of the observations. \nHere, B band is represented by squares (blue), V by solid circles (green), R by triangles (red) and I by starred symbols (black) .}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm , angle=0]{fig4a.eps}\n\\includegraphics[width=8cm , angle=0]{fig4b.eps}\n\n\\caption{One example of Auto-correlation (R band) is shown in the left figure and DCF (R versus I) is shown in the right figure.}\n \\end{figure*}\n\n\n\n\\subsection{Colour Indices}\n\nWe next investigated the existence of spectral variations by studying the behaviour of colour variations with respect \nto the brightness of BL Lacertae. The colour indices (CIs) are calculated by combining almost simultaneous (within 8 minutes)\n B, V, R and I magnitudes to yield  CIs = B$-$V, V$-$R, R$-$I and B$-$I.  For a particular night, we studied the \ncolour variability only when both light curves were identified as variable. Our data have the \nadvantage of being almost simultaneous in various optical bands.  It is important to recall that the underlying host galaxy,\neffect of the accretion disk component and the Gravitational microlensing can also \nlead to apparent, but unreal, colour variations (Hawkins 2002). But, since Gravitational microlensing is important\non weeks to months time-scales and during our observations, BL Lacertae was in flaring state (Raiteri et al. 2013) where the Doppler\nboosting flux from the relativistic jet almost invariably swamps out the contribution of the accretion disk component, so we can\nrule out the contribution of these components. Also, the data we have used in calculating the colour indices are host \ngalaxy subtracted so we conclude that  our results indicates  \nvariability of the non-thermal continuum radiation. \n\nWe studied the variations of colour indices with respect to brightness in 13 of these observations. We fit all the \ncolour--magnitude diagrams with a linear model of the form CI = $m \\times $ mag $+$~c (where $m$ is the slope in the \nfit,  V magnitude is taken as mag and $c$ is its respective intercept). The Pearson correlation coefficient (r), \nits p-value (null hypothesis probability; we consider a confirmed colour index correlation with the V magnitude when p $<$0.01) \nalong with the slopes and intercepts are presented in Table 3. The significant positive correlations between the \ncolour index and magnitude along with a change of slope $>$3 $\\sigma$ indicates that the source exhibits a bluer-when-brighter trend. \n\nIn five of the nights, which are marked by asterisks in Table 3,  we found significant variations in colour indices at the  3-$\\sigma$ level.\nIn these observations, colour indices correlate with the source brightness and the overall correlation is positive, \nwhich indicates hardening of the spectrum as the source brightens. So, in these observations, BL Lacertae exhibits bluer-when-brighter\ntrend with different regression slopes. No significant negative correlations are found for the source.\nThe bluer-when-brighter tendency in BL Lacertae has been seen by several groups on long-term as well as short-term time-scales\n(e.g.\\ Papadakis et al.\\ 2003; Villata et al.\\ 2004; Stalin et al.\\ 2006; Larionov et al.\\ 2010; \nGu et al.\\ 2006; Gaur et al.\\ 2012; Wierzcholska et al.\\ 2015).\nVillata et al.\\ (2004) characterized the intra-day flares to be strong bluer-when-brighter chromatic events with a\nslope of $\\sim$0.4. In five observations, we found a bluer-when-brighter trend with a slope varying between 0.18--0.28 (in Table 3). \n\nIn the other eight observations, we did not find significant linear correlations between colour indices and magnitudes (Table 3). \nIn some of them significant colour variations are seen but are not well fitted by linear functions. \nDuring the observations on 6 July 2010 (Fig.\\ 1, fourth panel), we found a strong flare with magnitude variation of $\\sim$0.3\nin all the optical  bands and the behaviour of the colour magnitude diagram varies  according to the different flux states. \nBut, on intra-day timescales it is difficult to judge the variations of the colour indices with respect to different brightness states\n as the flare is on hours like timescales.  \nAlso, in other observations, i.e, 8 July 2010, 24 and 25 August 2011,  we saw small sub-flares superimposed on the long-term trend. \nIn these cases, the colour indices vary significantly within the individual observations, sometimes showing different branches in the \ncolour magnitude diagrams according to the flux states. Spectral steepening during the flux rise can be explained by the\npresence of two components, one variable with a flatter slope which dominates during the flaring states and  another one that is\nmore stable and contributes to the long term achromatic emission (Villata et al.\\ 2004). So, it could be possible that during these \nobservations, the superposition of many distinct new variable components lead to the overall weaking of the colour-magnitude \ncorrelations. \nBonning et al.\\ (2012) studied a sample of FSRQs and BL Lacertaes and found that FSRQs follow redder-when-brighter trends while BL Lacertaes\nshow no such trends. They found complicated behaviour of the blazars on colour-magnitude diagrams: hysteresis tracks, and\nacromatic flares which depart from the trend suggesting different jet components becoming important at different times.\nTherefore, our colour variability results show that the intra-night \nflares between 2010 and early 2012 are chromatic but do not always follow simple bluer-when-brighter trends. \n\n\n\\begin{table*}\n\\center\n\\caption{ Results of Intra-day Variability of BL Lacertae}\n\\setlength{\\tabcolsep}{0.03in}\n\\begin{tabular}{lccccccr@{\\hskip 0.3in}lcccccc} \\cline{1-7} \\cline{9-15}\nDate            &Telescope &Band       &$F_{enh}$    &$F_{c}$(0.001)   &$Amp\\%$   &Variable  & & Date            &Telescope &Band       &$F_{enh}$    &$F_{c}$(0.001)   &$Amp\\%$   &Variable   \\\\ \\cline{1-7} \\cline{9-15}\n10.06.2010           &A    &R        &1.018  &2.386   &-   & NV & & 24.08.2011           &D    &B        &4.035 &2.849    &13.55   &Var\\\\\n11.06.2010           &A    &R        &0.985  &2.008   &-   & NV & &                      &    &V         &7.389 &2.849    &11.36   &Var   \\\\\n12.06.2010           &A    &R         &1.034 &2.386   &-  &NV & &                      &    &R         &16.187&2.849    &11.12 &Var   \\\\\n14.06.2010           &A    &R         &1.054 &2.076     &-   &NV & &                     &    &I         &13.665&2.849    &9.98 &Var  \\\\\n                     &C    &R         &0.934 &2.033     &-    &NV & & 25.08.2011           &D   &B         &11.247&2.790   &30.14 &Var \\\\\n18.06.2010           &E    &R         &51.328&1.940      &6.63 &Var  & &                      &    &V         &56.7659&2.790   &26.30 &Var  \\\\\n19.06.2010           &E    &R         &1.861 &1.940      &-       &NV  & &                     &    &R         &63.145&2.790   &25.55 &Var\\\\\n20.06.2010           &E    &R         &21.725&1.930      &3.48&Var   & &                     &    &I         &17.878&2.790   &24.10 &Var\\\\\n21.06.2010           &E    &R         &1.227 &1.972     &-        &NV & & 22.09.2011           &D   &B         &1.354 &2.639   &-        &NV\\\\\n22.06.2010           &E    &R         &29.890 &1.94     &5.28 &Var & &                     &    &V         &1.565 &2.517   &-        &NV\\\\\n04.07.2010           &F    &B         &2.164 &2.281    &-        &NV  & &                     &    &R         &1.329 &2.517  &-  &NV\\\\\n                     &    &V          &2.155 &2.481    &-        &NV & &                     &    &I         &0.930 &2.517   &- &NV\\\\\n                     &    &R          &0.146 &2.236    &-        &NV & & 19.10.2011           &D   &R         &0.574 &3.239   &- &NV \\\\\n                     &    &I          &0.938 &2.281    &-        &NV & &                      &    &I         &0.841 &4.142   &-  &NV \\\\\n05.07.2010           &F    &B         &1.527 &2.305    &-        &NV & & 06.07.2012           &E   &V        &25.590&2.596   &6.18 &Var \\\\\n                     &    &V          &0.618 &2.281    &-        &NV & &                     &    &R         &21.632&2.596   &6.14  &Var \\\\\n                     &    &R          &0.109 &2.258    &-        &NV & &                     &    &I         &9.829  &2.596   &5.30 &Var \\\\\n                     &    &I          &1.052 &2.281    &-        &NV & & 10.07.2012           &E   &B         &16.174&2.983   &6.40 &Var\\\\\n06.07.2010           &F    &B         &32.4002&1.995    &35.85 &Var & &                     &    &V         &57.750&2.913   &6.27 &Var \\\\\n                     &    &V         &36.577 &1.995    &38.64 &Var & &                     &    &R         &28.700&2.913   &6.15  &Var\\\\\n                     &    &R          &45.931&2.047    &35.64        &Var & &                     &    &I         &11.878&2.913   &5.54 &Var\\\\\n                     &    &I          &55.757&1.995    &34.82 &Var & & 07.08.2012           &D   &B         &2.110 &3.753   &-        &NV \\\\\n07.07.2010           &F    &B         &2.307 &2.481    &-        &NV & &                     &    &V         &0.558 &3.932   &-        &NV\\\\\n                     &    &V          &2.284 &2.305    &-        &NV & &                     &    &R         &0.638 &3.932   &-        &NV\\\\\n                     &    &R          &1.611 &2.258    &-        &NV & &                     &    &I         &0.746 &3.932   &- &NV\\\\\n                     &    &I          &1.247 &2.258    &        &NV   & & 12.08.2012           &D    &B        &4.669 &3.598   &20.67 &Var  \\\\\n08.07.2010           &F    &B         &17.271&2.305    &14.49        &Var & &                     &    &V         &18.835 &3.598  &17.35 &Var \\\\\n                     &    &V          &3.824 &2.281    &13.76       &Var & &                     &    &R         &36.030&3.598   &14.52 &Var \\\\\n                     &    &R          &6.864 &2.281    &12.40       &Var & &                     &    &I         &24.429&3.753   &12.43 &Var  \\\\\n                     &    &I          &4.720 &2.331    &10.96 &Var  & & 15.08.2012           &D    &B        &0.670 &3.932   &-        &NV\\\\\n17.07.2011           &E   &B         &35.089&3.239    &8.23  &Var & &                     &    &V         &0.417 &3.932   &-        &NV\\\\\n                     &    &V          &90.629&3.239    &7.28 &Var   & &                     &    &R         &0.908 &3.932   &-  &NV\\\\\n                     &    &R          &50.931&3.239    &7.20 &Var   & &                     &    &I         &0.754 &3.932   &- &NV \\\\\n01.08.2011           &D   &B         &1.860 &3.932     &-       &NV & & 16.08.2012           &D    &B        &0.908 &3.463   &-        &NV\\\\\n                     &    &V         &2.774 &3.932     &-       &NV & &                     &    &V         &1.726 &3.463   &-        &NV\\\\\n                     &    &R         &2.094 &3.932     &- &NV  & &                     &    &R         &1.622 &3.463   &-        &NV\\\\\n                     &    &I         &1.862 &3.932     &- &NV   & &                     &    &I         &0.764 &3.463   &- &NV \\\\\n02.08.2011           &D   &B         &1.584 &6.195     &- &NV   & & 18.09.2012           &D    &B        &0.540 &3.345   &-        &NV\\\\\n                     &    &V         &4.858 &6.195     &- &NV   & &                     &    &V         &1.395  &3.345   &-        &NV\\\\\n                     &    &R         &5.322&7.077    &- &NV   & &                     &    &R         &1.034 &3.345   &-        &NV\\\\\n                     &    &I         &2.540 &7.077    &- &NV   & &                     &    &I         &0.678 &3.345   &-       &NV \\\\\n04.08.2011           &D   &B         &0.907 &2.849    &-       &NV  & & 08.10.2012           &C    &R        &3.007 &1.961    &7.89       &Var\\\\\n                     &    &V         &0.552 &2.983    &-  &NV   & & 13.10.2012           &C    &R        &6.686 &3.239    &11.76       &Var\\\\\n                     &    &R         &1.507 &2.983    &-  &NV   & & 17.10.2012           &D    &B        &6.863 &3.932    &24.58  &Var  \\\\\n                     &    &I         &1.346 &2.983    &- &NV   & &                     &    &V         &27.535 &3.932    &19.40 &Var  \\\\\n06.08.2011           &D   &B         &0.368 &2.736    &-       &NV & &                     &    &R         &42.842&3.932    &16.02  &Var   \\\\\n                     &    &V         &0.300 &2.736    &-       &NV & &                     &    &I         &33.239&3.932    &13.18   &Var   \\\\\n                     &    &R         &0.670 &2.790     &-       &NV & & 22.10.2012           &D    &B        &1.383   &2.686    &-       &NV  \\\\\n                     &    &I         &0.859 &2.849    &- &NV   & &                     &    &V         &1.462 &2.555    &-       &NV\\\\\n07.08.2011           &D   &B         &0.997 &3.239    &-       &NV & &                     &    &R         &10.608 &2.555    &13.80 &Var  \\\\\n                     &    &V         &7.058&3.239    &10.63  &Var   & &                     &    &I         &10.043 &2.555    &12.95  &Var   \\\\\n                     &    &R         &7.773 &3.239    &8.23  &Var    & & 23.10.2012           &D    &B        &0.606 &2.596    &       &NV\\\\\n                     &    &I         &2.046 &3.239    &      &NV    & &                      &    &V         &3.585 &2.481    &9.29       &Var\\\\\n23.08.2011           &D   &B         &0.730 &2.448    &-       &NV & &                     &    &R         &5.166 &2.517    &10.25  &Var   \\\\\n                     &    &V         &4.960 &2.448    &12.49  &Var    & &                     &    &I         &3.329 &2.481    &9.68  &Var  \\\\\n                     &    &R         &8.400  &2.448    &9.47  &Var   & & 26.10.2012           &C   &R         &4.330 &2.596    &9.42 &Var   \\\\\n                     &    &I         &5.092 &2.448    &9.69   &Var   \\\\\n\n\\cline{1-7} \\cline{9-15}\n\\end{tabular} \\\\\n\\footnotesize\n\\begin{tabbing}\nA: \\= 1.04 meter Samprnanand Telescope, ARIES, Nainital, India  \\=\n\\hspace{3em} \\=\nC: \\= 50\/70-cm Schmidt Telescope at National Astronomical \\\\\nD: \\> 60-cm Cassegrain Telescope at Astronomical Observatory \\> \\> \\> Observatory, Rozhen, Bulgaria   \\\\\n\\> Belogradchik, Bulgaria \\> \\>\nE: \\> 1.3-m Skinakas Observatory, Crete, Greece \\\\\nF: \\> 1.3 m McGraw-Hill Telescope, Arizona, USA\\\\\n$F_{enh}$:   \\> \\hspace{1em} Enhanced F-test values \\> \\>\n$F_{c}$(0.001): \\> \\hspace{3em} Critical Values of F Distribution at 0.1\\%; \\\\\nAmp:  \\> \\hspace{1em} Variability Amplitude \\> \\>\n Var \/ NV: \\> \\hspace{3em} Variable \/ Non-variable \\\\\n\\end{tabbing}\n\n\\end{table*}\n\n\n\\begin{table}\n\\caption{ Details of telescopes and instruments}\n\\textwidth=6.0in\n\\textheight=9.0in\n\\vspace*{0.2in}\n\\noindent\n\\begin{tabular}{ll} \\hline\nTelescope:        &1.3 m McGraw-Hill Telescope     \\\\\\hline\nChip size:   & $4064\\times4064$ pixels      \\\\\nPixel size:  &$15\\times15$ $\\mu$m           \\\\\nScale:       &0.315\\arcsec\/pixel             \\\\\nField:       & $21.3\\arcmin\\times21.3\\arcmin$   \\\\\nGain:        &2.2-2.4 $e^-$\/ADU                   \\\\\nRead Out Noise:         &5 $e^-$ rms         \\\\ \\hline\n\\end{tabular} \\\\\n\\noindent\n\\end{table}\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5cm , angle=0]{fig5a.eps}\n\\includegraphics[width=5cm , angle=0]{fig5b.eps}\n\\includegraphics[width=5cm , angle=0]{fig5c.eps}\n\\includegraphics[width=5cm , angle=0]{fig5d.eps}\n\\includegraphics[width=5cm , angle=0]{fig5e.eps}\n\\includegraphics[width=5cm , angle=0]{fig5f.eps}\n\\includegraphics[width=5cm , angle=0]{fig5g.eps}\n\\includegraphics[width=5cm , angle=0]{fig5h.eps}\n\\includegraphics[width=5cm , angle=0]{fig5i.eps}\n\\includegraphics[width=5cm , angle=0]{fig5j.eps}\n\\includegraphics[width=5cm , angle=0]{fig5k.eps}\n\\includegraphics[width=5cm , angle=0]{fig5l.eps}\n\\includegraphics[width=5cm , angle=0]{fig5m.eps}\n\\caption{Colour index versus magnitude diagrams for BL Lacertae. The solid line is the best fit on the observations.}\n \\end{figure*}\n\n\n\\begin{table}\n\\caption{ Results of linear fits to Colour-Flux Diagrams.}\n\\setlength{\\tabcolsep}{0.03in}\n\\begin{tabular}{lccccc} \\hline \\hline\n\nDate of          &Band             &r   &p    &c$\\pm$$\\Delta$c  &m$\\pm$$\\Delta$m     \\\\            \nObservation      &                 &    &        &Intercept   &Slope                 \\\\\\hline\n06.07.2010   &(B-V)                &-0.138 &0.339     & 1.038$\\pm$0.446     &-0.032$\\pm$0.032     \\\\\n             &(V-R)                &0.315  &0.026     &-1.055$\\pm$0.675     & 0.110$\\pm$0.048      \\\\\n             &(R-I)                &0.023  &0.873     & 0.459$\\pm$0.675     &0.008$\\pm$0.048       \\\\\n             &(B-I)                &0.316  &0.026     & 0.442$\\pm$0.534     &0.087$\\pm$0.038     \\\\\n08.07.2010   &(B-V)                &-0.106 &0.537     & 1.500$\\pm$1.455     &-0.066$\\pm$0.105      \\\\\n             &(V-R)                &0.205  &0.230     &-1.389$\\pm$1.540     &0.136$\\pm$0.111       \\\\\n             &(R-I)                &0.110  &0.545     &-0.626$\\pm$1.836     &0.085$\\pm$0.133       \\\\\n             &(B-I)                &0.220  &0.198     &-0.516$\\pm$1.643     &0.156$\\pm$0.119       \\\\\n17.07.2011   &(B-V)*               &0.629  &0.003     &-2.424$\\pm$0.879     &0.234$\\pm$0.068      \\\\\n             &(V-R)                &0.512  &0.021     &-1.150$\\pm$0.627     &0.123$\\pm$0.049      \\\\\n07.08.2011   &(V-R)                &0.193  &0.415     &-0.434$\\pm$1.058     &0.067$\\pm$0.081       \\\\\n23.08.2011   &(V-R)*               &0.714  &$<$0.001  &-3.423$\\pm$0.670     &0.289$\\pm$0.050      \\\\\n             &(R-I)                &0.280  &0.109     &-0.658$\\pm$0.779     &0.096$\\pm$0.058       \\\\\n24.08.2011   &(B-V)                &0.057  &0.788     &0.113$\\pm$1.910      &0.038$\\pm$0.141      \\\\\n             &(V-R)                &0.181  &0.388     &-0.805$\\pm$1.446     &0.094$\\pm$0.107       \\\\\n             &(R-I)                &-0.022 &0.917     &0.787$\\pm$1.219      &-0.009$\\pm$0.090      \\\\\n             &(B-I)                &0.264  &0.202     &0.095$\\pm$1.269      &0.123$\\pm$0.094       \\\\\n25.08.2011   &(B-V)                &0.087  &0.673     &0.235$\\pm$0.836      &0.027$\\pm$0.063       \\\\\n             &(V-R)                &0.345  &0.118     &-0.162$\\pm$0.366     &0.045$\\pm$0.028       \\\\\n             &(R-I)                &0.251  &0.217     &0.051$\\pm$0.439      &0.042$\\pm$0.033       \\\\\n             &(B-I)                &0.325  &0.105     &0.125$\\pm$0.894      &0.114$\\pm$0.067      \\\\\n06.07.2012   &(V-R)*               &0.530  &0.003     &-1.643$\\pm$0.641     &0.159$\\pm$0.048       \\\\\n             &(R-I)*               &0.506  &0.004     &-2.429$\\pm$0.994     &0.231$\\pm$0.074       \\\\\n10.07.2012   &(B-V)                &-0.084 &0.703     &1.019$\\pm$0.956      &-0.027$\\pm$0.071       \\\\\n             &(V-R)                &0.061  &0.781     & 0.332$\\pm$0.550     &0.011$\\pm$0.040       \\\\\n             &(R-I)               &0.487  &0.019     &-2.201$\\pm$1.125     &0.212$\\pm$0.083       \\\\\n             &(B-I)                &0.395  &0.062     &-0.850$\\pm$1.347     & 0.196$\\pm$0.099       \\\\\n12.08.2012   &(B-V)                &0.310  &0.243     &-2.208$\\pm$2.320     &0.205$\\pm$0.168       \\\\\n             &(V-R)                &0.101  &0.709     &0.052$\\pm$1.131      &0.031$\\pm$0.082       \\\\\n             &(R-I)                &0.581  &0.011     &-1.618$\\pm$0.861     &0.166$\\pm$0.062      \\\\\n             &(B-I)                &0.517  &0.040     &-3.774$\\pm$2.458     &0.402$\\pm$0.178     \\\\\n17.10.2012   &(B-V)                &-0.103 &0.716     &1.370$\\pm$1.927      &-0.051$\\pm$0.136       \\\\\n             &(V-R)*               &0.674  &0.006     &-1.526$\\pm$0.614     &0.142$\\pm$0.043     \\\\\n             &(R-I)*               &0.668  &0.006     &-1.945$\\pm$0.816     &0.186$\\pm$0.058      \\\\\n             &(B-I)                &0.542  &0.037     &-2.102$\\pm$1.694     &0.278$\\pm$0.120      \\\\\n22.10.2012   &(R-I)*               &0.482  &0.006     &-2.203$\\pm$0.970     & 0.210$\\pm$0.071      \\\\\n23.10.2012   &(V-R)                &0.217  &0.233     &-1.919$\\pm$1.973     & 0.170$\\pm$0.140       \\\\\n             &(R-I)                &0.163  &0.373    &-1.262$\\pm$2.147      & 0.137$\\pm$0.152       \\\\ \\hline\n\n\n\n\\end{tabular} \\\\\n$ $*: Significant variations are found in these observations.  \\\\\nr \\& p: Pearson Correlation Coefficient and its probability values respectively. \\\\\n\\end{table}\n\n\\section{Discussion}\n\nWe performed photometric monitoring of  BL Lacertae during the period 2010--2012 for a total of 38 nights \nin the B, V, R and I bands in order to study its flux and spectral variability. In 19 of those nights, we found \ngenuine IDV. The light curves often show gradual rises and decays,  sometimes with smaller \nsub-flares superimposed. No evidence for periodicity or other characteristic time scales was found.  \n We find the duty cycle of the source during this period to be $\\sim$44\\%.\nIn the earlier studies, it has been found that LBLs display stronger IDV than HBLs (high frequency peaked blazars)\nand the DC has been estimated to be $\\sim$70\\% for LBLs and $\\sim$30--50\\% for HBLs (Heidt \\& Wagner 1998; Romero et al. 2002;\nGopal-Krishna et al. 2003). Gopal-Krishna et al. (2011) studied a large sample of blazars and found that if variability\namplitude (Amp) $>$3\\% is considered, the DC is 22\\% for HBLs and 50\\% for LBLs. We found DC of $\\sim$44\\% for BL Lacertae \n(which is a well known LBL) is in accordance with the previous studies.\n\nIn the literature, there are various models which explains intra-day variability of blazars.  Intrinsic ones focus on the\n evolution of the electron energy density distribution of the relativistic paricles leading to a variable synchrotron\nemission, with shocks accelerating turbulent particles in the plasma jet which then cools by synchrotron emission (e.g., \nMarscher, Gear \\& Travis 1992; Marscher 2014; Calafut \\& Wiita 2015). \nExtrinsic ones involve  geometrical effects like swinging jets where the path of the relativistic moving blobs along the jet\ndeviated slightly from the line of sight, leading to a variable Doppler factor (e.g., Gopal-Krishna \\& Wiita 1992). The long term \nperiodic and acromatic BL Lacertae variability may be mostly explained by the geometrical scenarios where viewing angle\n variation can be due to the rotation of an inhomogeneous helical jet which causes variable Doppler boosting of the\n corresponding radiation (Villata et al 2002; Larionov et al. 2010 and references therein). As we are considering the \nfaster intra-night flux\nvariations that are associated with the colour variations, they are more likely to be associated with models\ninvolving shock propagating in a turbulent plasma jet. \n\nWhen variability is clearly detected, its amplitude is usually greater at higher frequencies, which is consistent with previous studies\n(Papadakis et al.\\ 2003; Hu et al.\\ 2006) and can be well explained by electrons that are accelerated at the shock front and then \nlose energy as they move away from the front. Higher-energy electrons lose energy faster through the production of synchrotron radiation, \n and are produced in a thin layer behind the shock front. In contrast, the lower-frequency emission is spread out over a larger volume \nbehind the shock front (Marscher \\& Gear 1985). This leads to time lags of the peak of the light curves toward lower frequencies and\namplitude of variability higher at higher frequencies which is clearly visible in the multi-frequency blazars light curves. \nDue to the closeness of the various optical bands, the starting time of a flare should be almost the same and hence \non short time-scales, it is difficult to detect the time lags between the optical bands.\nAlthough the  amplitude of variability is an inherent property of the source,  we had to examine whether\nit has any dependence on the duration of observation, and found a\nsignificant positive correlations between the observed amplitude of variability of the light curves and the duration of the observations.\nAs noted by Gupta \\& Joshi (2005),  on intra-day\ntimescales, the probability of seeing a significant intra-day variability generally increases if the source is continuously observed for\nlong durations. In our observations, duration of monitoring varies between 1.5--5.8 hours. Also, we found that the \namplitude of variability decreases as the source flux increases which can be explained as the source flux increases, the irregularities\nin the turbulent jet (Marscher 2014) decrease and the jet flow becomes more uniform leading  to a decrease in amplitude of variability. \n\nWe searched for the possible correlations between colour versus magnitude and found significant positive correlations\nbetween them in five of the observations out of total 13. So, BL Lacertae showed significant bluer-when-brighter trends\non these night with different regression slopes (in Table 3). This behaviour is very well known for BL Lacertaes \nand can be interpreted as resulting from rapid, impulsive injection\/acceleration of relativistic electrons, followed by\nsubsequent radiative cooling (e.g., B{\\\"o}ttcher \\& Chiang 2002). However, other observations do not show significant\n linear correlations and show complicated behaviour on colour-magnitude diagrams i.e different slopes according to the\ndifferent flux states or nearly zero slopes between colour-magnitude (Table 3).\nOf course it is possible that superposition of different spectral slopes from many variable components (standing shocks in\ndifferent parts of the jet) could lead to the overall weakening of the colour-magnitude correlations (Bonning et al.\\ 2012).\n\nHence, the behaviour of the colour--magnitude diagrams  provides us with indirect information on the amplitude \ndifference and time-lags between these bands as Dai et al.\\ (2011) performed simulations to confirm that both the \namplitude differences and time delays between variations at different wavelengths result in \na hardening of the spectrum during the flare rise. They showed that if there is a difference in amplitude \nin two light curves, it leads to the evolution of the object along a diagonal path in the colour-magnitude diagram.  \nIf there is a time-lag along with the amplitude change, counter-clockwise loop patterns on the colour index--magnitude \ndiagram arise (Dai et al.\\ 2011). However, any such lags are probably shorter than the sampling time of \nour observations, so we are not able to detect them through the DCFs we computed. \n\n\\section{Conclusions}\n \nOur conclusions are summarized as follows: \n\n\\begin{itemize}\n\\item During our observations in 2010-2012 BL Lacertae was highly variable in B, V, R and I bands. \nThe variations were well correlated in all four bands and were very smooth, with gradual rises\/decays.\nIn one of the observations, on  6 October 2010, we found a pronounced flare-like event, and the highest variability amplitude \nis found in the V band at $38$ per cent.\n\\item In the cases with significant variability, the amplitude of variability is highest in the highest energy band. \n\\item The amplitude of variability correlates positively with the duration of the observation and\n decreases as the flux of the source increases. \n\\item We searched for time delays between the B, V, R and I bands in our observations, but we did not find any\nsignificant lags. This implies that the variations are almost simultaneous in all of the bands and any time lags, if present, \nare less than our data sampling interval of $\\sim$8 minutes. \n\\item The flux variations are associated with spectral variations on intra-day time-scales. In  5 of the 13 observations, the\noptical spectrum showed the overall bluer-when-brighter trend which could well represent  highly variable jet emission.\n\\item The colour vs.\\ magnitude diagrams show different behaviours which could represent the contribution of different variable \ncomponents during the flaring states.\n\\item We conclude that the acceleration and cooling timescales\nare very short for these optical variations and hence dense optical observations with even shorter cadence and higher sensitivity are required\nto better characterize them.  \n\\end{itemize}\n  \n\nWe thank the referee for useful and constructive comments.\nThis research was partially supported by Scientific Research Fund of the Bulgarian Ministry of Education and Sciences under \ngrant DO 02-137 (BIn 13\/09). The Skinakas Observatory is a collaborative project of the University of Crete, the Foundation \nfor Research and Technology -- Hellas, and the Max-Planck-Institut f\\\"ur Extraterrestrische Physik. \nH.G. is sponsored by the Chinese Academy of Sciences Visiting Fellowship for Researchers from Developing Countries\n (grant No. 2014FFJB0005), and supported by the NSFC Research Fund for International Young Scientists (grant No. 11450110398).\nACG is partially supported by the Chinese Academy of Sciences Visiting Fellowship for Researchers\nfrom Developing Countries (grant no. 2014FFJA0004).\nMB acknowledges support by the South African Department of Science and Technology through the National Research Foundation \nunder NRF SARChI Chair grant no. 64789. MFG acknowledges support from the National Science Foundation\nof China (grant 11473054) and the Science and Technology Commission of Shanghai Municipality (14ZR1447100).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Free energy of a binary mixture on a membrane}\n\\subsection{Free energy for single species membranes}\nFor a membrane comprising of a single specie, the free energy of a given shape $S$ can be described via the Helfrich hamiltonian \\cite{Lipowsky1991}:\n\\begin{align}\nH_{el} = \\int_{S} d\\mu_S \\: [\\kappa_m (C- C_{sp})^2 + \\kappa_g G], \\tag{SM1} \\label{Hel1} \n\\end{align}where $C \\equiv (c_1 + c_2)\/2$ is the local mean curvature, $G = c_1 c_2$ is the local gaussian curvature, $c_{1,2}$ are the local principal curvatures, $C_{sp}$ is the intrinsic mean curvature of  the surface and $\\kappa_{m,g}$ are the mean and gaussian bending rigidities respectively. Furthermore, the Gauss-Bonnet theorem states that \\cite{Kamien02}:\n\\begin{equation}\n\\int_S d\\mu_S \\:G + \\oint_{\\partial S} dl \\:g_c = 2\\pi \\chi(S),  \\tag{SM2 }\\label{Hel2}\n\\end{equation}where $\\partial S$ stands for the boundary of $S$, $g_c$ for the local geodesic curvature of the boundary \\cite{Kamien02} and $\\chi(S)$ for the Euler characteristic of $S$. The key consequence is that, if the boundary and the topologies of the problem are fixed, minimizing Eq. \\eqref{Hel1} is equivalent to minimizing only the integral over the mean curvature $C$. In the absence of intrinsic curvature, the lowest energy solutions correspond to surfaces with exactly $C = 0$ at all points: such surfaces are called {\\it minimal} surfaces \\cite{MinimalSurfaces}. \nHere we focus on the triply periodic surfaces which are known to be formed by lipid mixtures in water \\cite{Templer1998,Angelov2003,Tyler2015}.\nWe use standard notations P, D, and G for the {\\it primitive}, {\\it Diamond}, and {\\it Gyroid} surfaces respectively. Since they are periodic, we characterize their topology with their Euler characteristic per unit cell ({\\it e.g.} in Eq. \\eqref{Hel2}). \n\n\\subsection{Free energy for a binary mixture on a minimal surface}\nIf instead of a single species, the membrane comprises two different species, then different allowed mixture configurations may have different topologies and one cannot disregard anymore the gaussian curvature contribution to the energy. The simplest extension of Eq. \\eqref{Hel1} to a binary mixture would then read:\n\\begin{align}\n\\tilde{H}^S_{el}(f_A) = \\int_{S} d\\mu_S(x) \\: [\\kappa_g^A \\sigma_A(x) + \\kappa_g^B(1-\\sigma_A(x))]G(x) \\tag{SM3} \\label{Hel3}\n\\end{align}where $x$ denotes a point on $S$, $d\\mu_S(x)$ is the area measure on $S$ at $x$, $G(x)$ is the gaussian curvature at point $x$, $f_A$ is the imposed area fraction of species $A$ (with $f_B = 1-f_A$), the field $\\sigma_{A}(x) \\in [0,1]$ is the {\\it mean} occupation number of species $A$ at $x$ and $\\kappa_g^{A,B}$ are the gaussian bending rigidities associated to the species $A$ and $B$ respectively. We then choose the free energy $\\tilde{H}^S_{el}(f_A = 0)$ as reference so that, in practice, we look at the free energy $H^S_{el}(f_A ) \\equiv \\tilde{H}^S_{el}(f_A) - \\tilde{H}^S_{el}(f_A = 0)$ which yields:\n\\begin{align}\nH^S_{el}(f_A) = \\delta \\kappa \\int_{S} d\\mu_S(x) \\:  \\sigma_A(x) G(x) \\tag{SM4} \\label{Hel4}\n\\end{align}where $\\delta \\kappa = \\kappa_g^A - \\kappa_g^B$. Eq. \\eqref{Hel4} is the starting point of our study.\n\n\n\\section{Weierstrass-Enneper representation}\n\\subsection{General formulation}\nIt can be shown that, locally, any minimal surface can be conformally mapped onto the complex plane via the Weierstrass-Enneper (W-E) representation \\cite{MinimalSurfaces}. More precisely, the W-E is a map from $\\mathbb{C}$ to $\\mathbb{R}^3$ which, to a point $(u,v)$ in an open subset of $\\mathbb{C}$, uniquely associates a point $(x(u,v), y(u,v), z(u,v))$ of $\\mathbb{R}^3$ that belongs to a minimal surface via:\n\n\\begin{align}\n&& x(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:f(w)(1-g(w)^2) \\right\\rbrace  \\tag{SM5}\\label{WE1} \\\\\n&& y(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:if(w)(1+g(w)^2) \\right\\rbrace  \\tag{SM6}\\label{WE2} \\\\\n&& z(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:2f(w)g(w) \\right\\rbrace  \\tag{SM7}\\label{WE3} \n\\end{align}where $g$ is a holomorphic function and $f$ is a meromorphic function such that $fg^2$ is analytic.\n\n\\vspace{2mm}\n\nThe gaussian curvature $G(x,y,z)$ at any point of a surface represented by Eqs. \\eqref{WE1}, \\eqref{WE2} and \\eqref{WE3} can be expressed as a function of $w=u+iv$ via:\n\n\\begin{equation}\nG(w) = - \\left(\\frac{4|g'(w)|}{|f(w)|(1+|g(w)|^2)^2} \\right)^2 .  \\tag{SM8}\\label{WE4} \n\\end{equation}\nThe negative sign is characteristic of minimal surfaces. Since the mean curvature $(c_1 + c_2)\/2 $ is zero everywhere, it implies that the gaussian curvature is always negative or zero. The W-E being a conformal map, it preserves the angles. Distances, however, are not conserved when mapping an infinitesimal segment from $\\mathbb{C}$ to $\\mathbb{R}^3$ and are scaled by a factor $\\Lambda(w)$ given by\n\\begin{equation}\n\\Lambda(w) = \\frac{|f(w)|(1+|g(w)|^2)}{2} \\tag{SM9} \\label{WE5}\n\\end{equation} \n\n\\subsection{Triply periodic Schwartz surfaces}\nThe three Schwartz surfaces {\\it P}, {\\it D} and {\\it G} can be obtained by choosing:\n\\begin{align}\n&& g(w) = w \\tag{SM10}\\label{WE6} \\\\\n&& f(w) = \\frac{e^{i \\theta_B}}{\\sqrt{w^8-14w^4+1}} \\tag{SM11}\\label{WE6bis}\n\\end{align}where $\\theta_B$ is the Bonnet angle such that $\\theta_B = 0 $ for the {\\it D} surface, $\\theta_B = \\pi\/2$ for the {\\it P} surface and $\\theta_B = \\mathrm{cotan}(K(1\/4)\/K(3\/4))$ for the {\\it G} surface. $K$ is a complete elliptic integral of the first kind. Since $e^{i \\theta_B}$ only changes the phase in the W-E representation, this means that the {\\it P}, {\\it D} and {\\it G} surfaces are simply related by an isometry called the Bonnet transformation and share many of their physical properties. \n\n\n\\subsection{Independence of the free energy on the member of the Bonnet family}\nIn particular, having the W-E map in mind, the whole integral in Eq. \\eqref{Hel4} can be thought of as an integral in the complex plane. Using the fact that $d\\mu_S(x(w)) =\\Lambda^2(w)dudv$, the Eq. \\eqref{Hel4} can be recast as:\n\\begin{equation}\nH^S_{el}(f_A) = \\delta \\kappa \\int_{\\mathcal{A}(S)} dudv \\: \\Lambda^2(w) \\: \\sigma_A(w) G(w) \\tag{SM12}\\label{WE8} \n\\end{equation}where $\\mathcal{A}(S)$ denotes the atlas used in $\\mathbb{C}$ to characterize $S$. It is worth noting that in the integrand of Eq. \\eqref{WE8}, the W-E functions $f$ and $g$ only appear via their complex modulus and therefore their contribution to the curvature energy would be unchanged by a phase factor. Thus, we concludethat the curvature energy of a binary mixture on a minimal surface is independent of which member of the Bonnet family is considered and, in particular, so is its ground state. In a similar fashion, a continuous model of the line tension contribution would read formally:\n\\begin{equation}\nH_{A-B}(f_A) = J \\int_{\\mathcal{I}(A-B)} d\\mu_l(x) \\tag{SM13}\\label{WE9}\n\\end{equation}where $\\mathcal{I}(A-B)$ denotes the set of points belonging to the $A-B$ interface on $S$ and $d\\mu_l(x)$ the length measure on $S$. Again, the integral can be thought as an integral on $\\mathbb{C}$ by virtue of the W-E map. Furthermore, the length measure on $S$ can be expressed in term of the length measure in $\\mathbb{C}$ via $d\\mu_l(x(w)) = \\Lambda(w) |dw|$. Eq. \\eqref{WE9} can thus be rewritten as:\n\\begin{equation}\nH_{A-B}(f_A) = J \\int_{WE^{-1}(\\mathcal{I}(A-B))} |dw| \\: \\Lambda(w) . \\tag{SM14}\\label{WE10}\n\\end{equation}\n\\begin{figure}\n\n \\vspace{2mm}\n  \\includegraphics[width=0.49\\columnwidth]{SMFig1a.jpg} \n   \\includegraphics[width=0.49\\columnwidth]{SMFig1b.jpg}\\\\\n    \\includegraphics[width=0.49\\columnwidth]{SMFig1c.jpg} \n     \\includegraphics[width=0.49\\columnwidth]{SMFig1d.jpg}  \n  \n  \\caption{\\label{fig1} {\\it Constructing the P-surface}. Top left: complex domain of the fundamental patch. Top right: fundamental patch in three dimensions. Bottom left: hexagonal patch ($\\Sigma$ in the main article) made of 12 fundamental patches. Bottom right: Full P-surface per cubic unit cell $S$ made of 8 patches $\\Sigma$.  }\n\\end{figure}\n\nAs before, the W-E functions $f$ and $g$ only contribute to the integral via their complex modulus and therefore $H_{A-B}$ is independent of the member of the Bonnet family under study.\nAs a consequence, the phenomenology of symmetry breaking and bridging-induced reentrant behaviour depicted in the phase diagrams of the P-surface in Fig. 2 of the main text hold in fact for all three Bonnet surfaces. The details of the phase diagram may slightly differ, however, as the cubic cells of the {\\it G} and {\\it D} surfaces do not contain the same surface area as the {\\it P} surface. This is outside the scope of this letter, and we will discuss these details in a separate publication.\n\n\\subsection{Angle of intersection between two geodesics}\nThe Euler characteristic of a compact domain $\\mathcal{D}$ of $A$ lipids that is not bridged to another domain on a neighbouring patch is $1$. Applying the Gauss-Bonnet theorem (Eq. \\eqref{Hel2}) to such a domain gives:\n\\begin{equation}\n\\int_{\\mathcal{D}} d\\mu_S(x) \\: G(x) + \\oint_{\\partial D} dl \\: g_c(x) = 2\\pi . \\tag{SM15} \\label{GB1} \n\\end{equation}Up to a factor the first term of the {\\it l.h.s} of Eq. \\eqref{GB1} is simply the curvature energy of the domain that we can denote $H^{\\Sigma_i}_{el}(f_i)$ in referring to notations introduced in the main text.\nBy using the fact that a patch has a 6-fold rotation symmetry, we can split the boundary integral of the geodesic curvature into 6 equivalent parts. If each piece of the boundary is almost everywhere a geodesic, then the second term of the {\\it l.h.s} of Eq. \\eqref{GB1} has zero integrand everywhere except at points where the geodesics meet. The total value of the contour integral becomes simply a sum over intersection angles that we call $\\theta$. We thus have:\n\\begin{equation}\n\\theta = \\frac{\\pi}{3} + \\frac{H^{\\Sigma_i}_{el}(f_A^i)}{6 |\\delta \\kappa|}. \\tag{SM16} \\label{GB2}\n\\end{equation}Finally, the actual interior angle $\\gamma$ between two geodesics making the hexagonal-like facetted domain is in fact the complementary angle of $\\theta$ and reads:\n\\begin{equation}\n\\gamma = \\frac{2\\pi}{3} - \\frac{H^{\\Sigma_i}_{el}(f_A^i)}{6 |\\delta \\kappa|}. \\tag{SM17} \\label{GB3}\n\\end{equation}\nNote that in the case where the curvature energy vanishes but the symmetry is still imposed, we retrieve the interior angle of a planar hexagon as expected.\n\n\\section{Numerical modelling of the P-surface}\n\\subsection{Fundamental patch}\nAll well behaved minimal surfaces admit a description in terms of a fundamental patch in $\\mathbb{R}^3$ that is repeated by using the symmetries of the surface. By the W-E representation, this fundamental patch is associated to a fundamental domain of the complex plane. For the {\\it P}, {\\it D} and {\\it G} surfaces, the fundamental domain is the set of complex points with positive real part bounded by the lines along the vectors $(1+i)\/\\sqrt{2}$ and $1$ and by the circle of radius $\\sqrt{2}$ whose center is located at the point $-(1+i)\/\\sqrt{2}$. \n\nThe top left of Fig. \\ref{fig1} shows the fundamental domain in $\\mathbb{C}$. The triangular tessellation is obtained by using the {\\it Surface Evolver} package \\cite{Brakke} and the images plus the management of the network structure have been performed with the {\\it Mathematica} software \\cite{Mathematica}. For the sake of illustration, Fig. \\ref{fig1} shows a coarse tessellation of the fundamental domain. The tessellations we used in the paper are typically 100 times finer. By using Eqs. \\eqref{WE1}-\\eqref{WE3} and \\eqref{WE6} and \\eqref{WE6bis}, we get a three dimensional realization of the fundamental patch that is represented in the top right of Fig. \\ref{fig1}. Then, following Ref. \\cite{Gandy-P}, we can generate first a full hexagonal patch of the P-surface ($\\Sigma$) by replicating and stitching together 12 fundamental patches as seen in the bottom left of Fig. \\ref{fig1}. The full cubic cell representation of the surface $S$ is then obtained by combining 8 such hexagonal patches with the right symmetry operations as illustrated on the bottom right of Fig. \\ref{fig1}. A similar procedure, albeit with different arrangements of the fundamental patches, can also be carried out for the D- and G-surfaces. \n\n\\subsection{Monte Carlo simulations}\nAs emphasized in Eq. \\eqref{WE8}, the curvature energy can be recast in terms of a sum over points on a euclidean (complex) plane of a curvature field that multiplies a scaling field. Moreover, the discretized surface $S$ on the bottom right of Fig. \\ref{fig1} is made of a network of cells $\\mathcal{N}(S)$ which are either an original version or a replica of a cell in the fundamental patch. Thus, the whole set of values of the curvature and scaling fields on the whole network is determined solely by that of the sub-network of cells in the fundamental patch. The particular topology of the P-surface (of genus 3 in a cubic cell) is then accounted for by the topology of the network {\\it i.e.} by assigning the right neighbours to each cell. If we add a species field $s$ into the picture such that $s_i= 1$ if cell $i \\:\\in \\: \\mathcal{N}(S)$ contains species $A$ and $s_i = 0$ otherwise, then the whole problem becomes that of paramagnetic spins on a network subject to an effective node-dependent magnetic field whose magnitude is $G(w)\\Lambda^2(w)\\Delta(w)$ and where $\\Delta(w)$ denotes the euclidean area of the triangular unit at point $w$ in the complex plane. Since the effective magnetic field is non uniform and non trivial, there is no simple explicit analytical expression for the thermodynamically favoured composition morphologies. By splitting the system into 8 equivalent patches, we could however suggest, as discussed in the main text, what would happen at low enough temperatures. In particular, a first approximation scheme neglecting the effect of curvature on the area measure and Taylor expanding the curvature field about its zero point $p_i$ suggested that the curvature energy $H^{\\Sigma_i}_{el}(f_A^i)$ in a patch $\\Sigma_i$, $i=1..8$, with an area fraction $f_A^i$ of $A$ lipids would go as $H^{\\Sigma_i}_{el}(f_A^i) \\sim (f_A^i)^{\\alpha}$ with $\\alpha = 2$. To test numerically this proposition, we performed Monte Carlo (MC) simulations of an Ising system whereby: \n\\begin{enumerate}\n\\item The total number of spins\/cells is fixed,\n\\item The total number of spins of value 1 is fixed,\n\\item A MC move consists then in:\n\\begin{enumerate}\n\\item  picking at random a cell among those which have $s=1$, \n\\item  picking at random a cell among those which have $s=0$,\n\\item  swap the cells spin values,\n\\item  accept the move with a probability satisfying the Metropolis criterion \\cite{Metropolis} $p_{acc} = \\min[1, e^{-\\beta \\Delta E}]$, where $\\Delta E = E_{final}-E_{initial}$ and the energy is in general given by Eqs. (4) and (5) of the main text article.\n\\end{enumerate}\n\\end{enumerate}\nIf we set the Ising parameter $J$ to zero, we can then probe the low energy curvature energy as a function of domain size for a single patch.\n\\begin{figure}\n\n \\vspace{2mm}\n  \\includegraphics[width=0.90\\columnwidth]{SMFig2bis.jpeg} \n  \n  \\caption{\\label{fig2} {\\it Curvature energy as a function of domain size}. The red triangles are MC data points for the energy $H^{\\Sigma_i}_{el}(f_A^i)$ of an $A$ domain on a patch $\\Sigma_i$. The lines correspond to the best fit to these data points with a scaling law behaviour $H^{\\Sigma_i}_{el}(f_A^i) \\sim (f_A^i)^{\\alpha}$. The solid red line corresponds to the best fit exponent value $\\alpha \\approx 1.83$, while the green dashed line is the best fit to the data with imposed $\\alpha = 2$ which shows quite good agreement with the data.}\n\\end{figure}As we see in Fig. \\ref{fig2}, the proposition that the curvature energy goes as a power low is in very good agreement with MC data. In addition, the approximation that $\\alpha = 2$ is in remarkably quite good agreement with the data.\n\n\\section{Validity of the minimal surface assumption}\nIn the current study, it is assumed that the underlying surface remains minimal during segregation of the lipids\/species. However, in general domain formation can induce local membrane deformation which in turn can destabilise the entire morphology of the membrane itself. In this section we will have a closer look at how phase separation induced bud formation may affect the conclusions drawn in the manuscript.\n\n\\subsection{Budding on flat multicomponent membranes}\nBased on the work by Lipowsky \\cite{Lipowsky1992} on flat membranes, domain formation may lead to a budding phenomenon driven by the line tension between the two coexisting demixed phases. Budding occurs when the line tension energy cost overcomes the bending energy penalty. \n\nConsider a membrane domain of area $A = \\pi L^2 = 2\\pi R^2 (1-\\cos\\theta)$, where $R$ is the radius of curvature of the deformed membrane domain, $\\theta$ is the contact angle of the domain with respect to the horizontal plane, and\n$L$ is the domain radius if there is no deformation to the membrane. \nWe will now consider the competition between two energy terms: a) line tension energy that increases with the perimeter of the domain, $2\\pi J R \\sin \\theta$; and (ii) bending energy which depends on the domain area and curvature, $2 \\kappa_m A \/ R^2$.\nHere $J$ is the line tension and $\\kappa_m$ is the mean curvature bending rigidity. We have also assumed that there is no mean spontaneous curvature. \n\nFig. \\ref{fig7} shows the total energy (normalised by the line tension energy for a flat domain) as a function of the reduced membrane mean curvature $L\/R $ that plays the role of an order parameter. $L\/R = 0$ corresponds to a flat membrane, i.e. no deformation. $L\/R = 2$ corresponds to a complete bud formation. As we see, there are three regimes depending on the membrane domain size $L$: (i) For $0 < L <  L^*= 4\\kappa_m\/J $, budding is unfavourable and $L\/R = 0$ is the global minimum configuration; (ii) For $ L^* \\le L <  L^o = 8\\kappa_m\/J $, budding is favourable but there is an energy barrier for its formation; (iii) Finally, only for $ L \\ge L^o$ that the energy barrier for bud formation disappears. Here the flat membrane geometry is completely unstable. \n\n\n\\vspace{2mm}\n\\begin{figure}[h]\n\n \\vspace{2mm}\n  \\includegraphics[width=0.99\\columnwidth]{SMFig7bis.jpeg} \n    \\caption{\\label{fig7} {\\it Sum of the line tension and curvature energies of a membrane domain}. Normalised energy curves as a function of reduced curvature $L\/R$ for given values of the domain size $L$. From top to bottom the domain size $L$ is increased from $L^*\/2$ to $L^o$.  }\n\\end{figure}\n\n\\subsection{Bud formation on a P-surface}\nSince the P-surface is a minimal surface with zero mean curvature everywhere, we will continue to assume that the spontaneous mean curvature is zero, as in the previous paragraph.\nThe presence of non-uniform gaussian curvature on the P-surface may alter the numerical prefactors in $L^*$ and $L^o$. \nHowever, since the domain formation occurs in the neighbourhood of very specific points on the P-surface {\\it i.e.} 8 zero-curvature points each at the centre of a patch in the cubic cell, to first approximation, it is reasonable to assume that the above results from \\cite{Lipowsky1992} to hold. As a result, we can approximate the two critical domain sizes as $L^* = 4\\kappa_m\/J$ and $L^o = 8 \\kappa_m\/J$.\nCorrespondingly, when $L < L^*$, we expect our assumption that the underlying surface remains minimal during phase separation to hold. Strong deviations to the results presented in the main text are only expected when $L \\ge L^*$.\n\n\\begin{figure}[h]\n\n \\vspace{2mm}\n  \\includegraphics[width=0.99\\columnwidth]{SMFig8.jpeg} \n  \n  \\caption{\\label{fig8} Modification of the phase diagram of Fig. 2(a) in the main manuscript resulting from budding instability for $\\kappa_m\/|\\delta \\kappa| = 1\/4$. The diagonally hashed region corresponds to a region where budding is preferable but there is an energy barrier for its formation, while the horizontally hashed region corresponds to a fully unstable bud formation. }\n\\end{figure}\n\nTo see which part of the phase diagram in Fig. 2 of the main manuscript is affected by the budding instability, we relate the area fraction $f_A$ occupied by the A-lipids to the size $L$ of the domains depending on which configuration $\\binom 8 k $ they are in. For the purpose of this analysis, we will focus on cases where there are no bridge formations between lipid $A$ domains. The total area occupied by the A-lipids on a cubic cell of the P-surface is $f_A \\mu_S(S)$. If the A-lipids are partitioned in $k$ patches among the 8 available, then the typical area per domain is $f_A \\mu_S(S)\/k \\approx \\pi L^2$. It follows that requiring full stability against budding ($L < L^*$) is equivalent to requiring $f_A \\mu_S(S)\/k < \\pi 16 (\\kappa_m\/J)^2$. If the cubic cell bounding the P-surface has sides of length $\\ell$, then $\\mu_S(S) = 24 \\ell^2 K(1\/4)\/K(3\/4)$, where $K(x)$ is the complete elliptic integral of the first kind \\cite{Gandy-P}. Thus, the stability criterion becomes\n\\begin{equation}\nf_A^*\\left(\\frac{J \\ell}{|\\delta \\kappa|}; k\\right) = \\left( \\frac{\\kappa_m}{|\\delta \\kappa|} \\right)^2 \\frac{2 k \\pi K(3\/4)}{3K(1\/4)} \\left(\\frac{|\\delta \\kappa|}{J \\ell}\\right)^2  \\tag{SM21} \\label{criticallines1}\n\\end{equation}\nfor configuration $\\binom 8 k $.\nAs we see in Eq. \\eqref{criticallines1}, the set of stability lines depends on the ratio $\\kappa_m \/|\\delta \\kappa|$ which constitutes an additional parameter in our modelling. We have observed that for values $\\kappa_m\/|\\delta \\kappa| > 0.4$, there is no intersection between any of the stability lines and the lipid repartition phases they correspond to within the parameter ranges of the present study, $0 \\le J \\ell \/ |\\delta \\kappa| \\le 2$. \n\nIt is worth emphasizing that experimental and numerical estimates of the ratio $\\kappa_m\/|\\delta \\kappa|$ for various lipid systems \\cite{Hu12} show that it is rarely below 1. As an example, consider the mixture of DOPC, sphingomyelin, and cholesterol forming coexisting $L_o$ and $L_d$ domains reported in references \\cite{Semrau,Baumgart}. Here the difference in Gaussian bending moduli $|\\delta \\kappa| \\simeq 3 \\times 10^{-19}$ J and the mean curvature bending modulus for the $L_o$ phase $\\kappa_m \\simeq 8 \\times 10^{-19}$ J, which gives us $\\kappa_m\/|\\delta \\kappa| \\simeq 8\/3$. Even if we use the mean curvature bending modulus for the $L_d$ phase $\\kappa_m \\simeq 2 \\times 10^{-19}$ J, which may be more appropriate when bridges are formed such that the lipid B domains are now surrounded by A (see Fig. 2(b) and (c) in the main text), we still have $\\kappa_m\/|\\delta \\kappa| \\simeq 2\/3 > 0.4$. Thus, this strongly suggests that our conclusions in the main text will hold even when taking into account segregation induced membrane deformation.\n\nWe can redo a similar calculation to that leading to Eq. \\eqref{criticallines1} to determine the phase boundaries beyond which the membrane is completely unstable against bud formation. Not surprisingly we find that they satisfy a very similar equation:\n\\begin{equation}\nf_A^o\\left(\\frac{J \\ell}{|\\delta \\kappa|}; k\\right) = \\left( \\frac{\\kappa_m}{|\\delta \\kappa|} \\right)^2 \\frac{4 k \\pi K(3\/4)}{3K(1\/4)} \\left(\\frac{|\\delta \\kappa|}{J \\ell}\\right)^2. \\tag{SM22} \\label{criticallines2}\n\\end{equation}\n\nTo illustrate how the phase diagram is modified when budding instability is taken into account, we show the results for $\\kappa_m \/| \\delta \\kappa| = 1\/4$ in Fig. \\ref{fig8}. The parameter regime susceptible to budding is for large area fraction of $A$ lipids, $f_A$, and large line tension, $J \\ell\/|\\delta \\kappa|$. The hashed regions correspond  to parameter regimes where budding is preferable. Energy barriers are present for budding to occur in the diagonally hashed region, while for the horizontally hashed region the P-surface is completely unstable against budding. \n\n\n\\bibliographystyle{apsrev4-1\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{SUPPLEMENTAL MATERIAL:\\\\ ``Quasiparticle lifetime of the repulsive Fermi polaron''}\n\\setcounter{page}{1}\n\\begin{center}\nHaydn S. Adlong,$^1$\nWeizhe Edward Liu,$^{1,2}$\nFrancesco Scazza,$^3$\nMatteo Zaccanti,$^3$\nNelson Darkwah Oppong,$^{4,5,6}$\nSimon~F\\\"olling,$^{4,5,6}$\nMeera M.~Parish,$^{1,2}$ and\nJesper~Levinsen$^{1,2}$, \\\\\n\\emph{\\small $^1$School of Physics and Astronomy, Monash University, Victoria 3800, Australia}\\\\\n\\emph{\\small $^2$ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia}\\\\\n\\emph{\\small $^3$Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO) and European Laboratory for Nonlinear Spectroscopy (LENS), 50019 Sesto Fiorentino, Italy}\\\\\n\\emph{\\small $^4$Ludwig-Maximilians-Universit{\\\"a}t, Schellingstra{\\ss}e 4, 80799 M{\\\"u}nchen, Germany}\\\\\n\\emph{\\small $^5$Max-Planck-Institut f{\\\"u}r Quantenoptik, Hans-Kopfermann-Stra{\\ss}e 1, 85748 Garching, Germany}\\\\\n\\emph{\\small $^6$Munich Center for Quantum Science and Technology (MCQST), Schellingstra{\\ss}e 4, 80799 M{\\\"u}nchen, Germany}\n\n\\end{center}\n\n\\section{Model and scattering parameters}\n\n\\subsection{Model}\n\nIn modelling the dynamics of impurities coupled to a Fermi sea, we use the following effective Hamiltonian\n\\begin{align} \\label{eq:effectiveHam}\n\\hat H=\\hat H_0+\\hat H_\\Omega+\\hat H_\\uparrow+\\hat H_\\downarrow,\n\\end{align}\nwhere\n\\begin{subequations}\n\\begin{align}\n    \\hat H_0&=\\sum_\\k(\\epsilon_{\\k}-\\mu) \\hat f_\\k^\\dag\\hat f_\\k, \\label{eq:supHam1}\\\\\n   \\hat H_\\Omega&=\\frac{\\Omega_0}{2} \\sum_{\\k} \\left(\n   \\hat c^\\dag_{\\k \\downarrow} \\hat c_{\\k \\uparrow} +\n   \\hat c^\\dag_{\\k \\uparrow} \\hat c_{\\k \\downarrow} \\right)+\\Delta\\omega\\,\\hat n_{\\downarrow}, \\label{eq:supHam2}\\\\\n    \\hat{H}_{\\sigma} &= \\sum_{\\k} \\left[\\epsilon_{\\k} \\hat\n                       c^\\dag_{\\k\\sigma} \\hat c_{\\k\\sigma} +\n                       ( \\epsilon_{\\k}\/2 + \\nu_{\\sigma}) \\hat d^\\dag_{\\k\\sigma} \\hat d_{\\k\\sigma}\\right]                 \n+g_{\\sigma}\\sum_{\\k, {\\bf q}} \\left( \\hat d^\\dag_{{\\bf q}\\sigma}\\hat c_{{\\bf q}\/2 - \\k,\\sigma}  \\hat f_{{\\bf q}\/2 + \\k}\n                       + \\hat f^\\dag_{{\\bf q}\/2 + \\k} \\hat c^\\dag_{{\\bf q}\/2\n                       -\\k, \\sigma}\\hat d_{{\\bf q}\\sigma} \\right).\\label{eq:supHam3}\n\\end{align}\n\\end{subequations}\nThe meaning of the various symbols is discussed in the main text.\n\n\\subsection{Relation between Hamiltonian operators and experimental atomic states}\n\nIn the 2D $^{173}$Yb experiment of Ref.~\\cite{Oppong2019}, the states of the atoms are defined by the electronic state, with ground state $^1S_0$ and long-lived excited state $^3P_0$ (i.e., the ``clock'' state), as well as the nuclear-spin state with $m_F \\in \\{-5\/2, -3\/2, \\ldots, +5\/2\\}$.\nThe majority $\\hat f_\\k^\\dag$ atoms exist in the $m_F=+5\/2$ ground state, which acts as a bath for the weakly interacting $\\hat c^\\dag_{\\k \\downarrow}$ and resonantly interacting $\\hat c^\\dag_{\\k \\uparrow}$ impurities in the $m_F=-3\/2$ ground state and $m_F=-5\/2$ `clock' state, respectively.\nInteractions between the $\\hat c^\\dag_{\\k \\uparrow}$ and $\\hat f_\\k^\\dag$ atoms are tunable through an orbital Feshbach resonance~\\cite{Zhang2015} as has been demonstrated experimentally~\\cite{Hofer2015,Pagano2015}.\n\nOn the other hand, the 3D experiment of Ref.~\\cite{Scazza2017} involves $^6$Li atoms in the three lowest Zeeman levels. The majority $\\hat f_\\k^\\dag$ atoms exist in the lowest Zeeman level, while the weakly interacting $\\hat c^\\dag_{\\k \\downarrow}$ and resonantly interacting $\\hat c^\\dag_{\\k \\uparrow}$ impurities occupy the second-lowest and third-lowest Zeeman levels respectively. Owing to two off-centered broad Feshbach resonances, the scattering between the  $\\hat c^\\dag_{\\k \\uparrow}$ and $\\hat f_\\k^\\dag$ atoms can be resonantly enhanced while only moderately increasing the comparatively weak interactions between the $\\hat c^\\dag_{\\k \\downarrow}$ and $\\hat f_\\k^\\dag$ atoms~\\cite{Scazza2017}.\n\n\n\n\\subsection{Scattering parameters}\n\nWithin the model in Eq.~\\eqref{eq:effectiveHam}, fermions of the same spin do not interact and the two spin states of the impurity are only coupled through the light-field. Ignoring for the moment the light-field (i.e., setting $\\Omega_0=0$), we calculate the vacuum spin-dependent impurity-fermion $T$ matrix, which characterizes the interactions between the majority fermions and a particular spin state of the impurity. %\nThis yields\n\\begin{align}\n    T_{\\sigma}(E)=\\left[\\frac{E - \\nu_\\sigma}{g^2_{\\sigma}}-\\sum_{\\k}^\\Lambda \\frac{1}{E-2 \\epsilon_{\\k}}\\right]^{-1}, %\n    \\label{eq:Tmatrix}\n\\end{align}\nwhere $E$ is the collision energy and $\\Lambda$ is the ultraviolet cutoff.\n\nTo proceed, we compare the scattering $T$ matrix with the 2D and 3D scattering amplitudes using $f_{2{\\rm D}\\sigma}(k)=mT_\\sigma(E)$ and $f_{3{\\rm D}\\sigma}(k)=-\\frac{m}{4\\pi}T_\\sigma(E)$, with $E=k^2\/m$. The scattering amplitudes at low energy are known to take the forms\n\\begin{subequations}\n\\begin{align} \n     f_{\\text{2D}\\sigma}(k) &\\simeq   \\frac{4 \\pi}{ - \\ln (k^2 a^2_{\\text{2D}\\sigma}) + R^2_{\\text{2D}\\sigma}k^2+i \\pi},\n     \\label{eq:f2D}\\\\\n     f_{\\text{3D}\\sigma}(k) &\\simeq -\\frac{1}{ a_{\\text{3D}\\sigma}^{-1} + R_{\\text{3D}\\sigma} k^2 + ik }.\n\\end{align}\n\\end{subequations}\nBy comparing Eq.~\\eqref{eq:Tmatrix} with the low-energy scattering amplitudes, we obtain the renormalized scattering parameters. In 2D, this results in\n\\begin{align}\n\\frac{\\nu_\\sigma+E_{2\\sigma}}{g^2_{\\sigma}} = \\sum_{\\k}^\\Lambda \\frac{1}{E_{2\\sigma}+2 \\epsilon_{\\k}},  \\qquad R_{\\text{2D}\\sigma}^2  = \\frac{4\\pi}{m^2 g^2_\\sigma},\n\\end{align}\nwhere $a_{{\\rm 2D}\\sigma}$ is the scattering length, $R_{2{\\rm D}\\sigma}$ is the effective range~\\cite{Kirk2017}, and $E_{2\\sigma}$ is the binding energy of the two-body bound state that exists for all interactions in 2D:\n\\begin{align}\n    E_{2 \\sigma} = \\frac{1}{m R^2_{\\text{2D}\\sigma}} W\\left( \\frac{R^2_{\\text{2D}\\sigma}}{a^2_{\\text{2D}\\sigma}} \\right),\n\\end{align}\nwith $W$ the Lambert $W$ function.\nIn 3D, we have\n\\begin{align}\n    \\frac{m}{4 \\pi a_{\\text{3D}\\sigma}} = - \\frac{\\nu_\\sigma}{g^2_\\sigma} + \\sum_{\\k}^\\Lambda \\frac{1}{ 2 \\epsilon_{\\k}}, \\qquad R_{\\text{3D} \\sigma} = \\frac{4 \\pi}{m^2 g_\\sigma^2},\n    \\label{eq:renorm3D}\n\\end{align}\nwhere $a_{{\\rm 3D}\\sigma}$ is the scattering length and $R_{3{\\rm D}\\sigma}$ is a range parameter~\\cite{Gurarie2007}. In this case, we only have a bound state when $a_{{\\rm 3D}\\sigma}>0$, with corresponding binding energy\n\\begin{align}\n    E_{3\\sigma}=\\frac{\\left[\\sqrt{1+4R_{3{\\rm D}\\sigma}\/a_{3{\\rm D}\\sigma}}-1\\right]^2}{4m{R_{3{\\rm D}\\sigma}^{2}}}.\n\\end{align}\nThrough this procedure, we have related the bare interaction parameters $g$, $\\Lambda$, and $\\nu$ to the physical parameters, the scattering length $a$ and effective range $R$, which characterises the relevant Feshbach resonance. For the broad Feshbach resonances in ${}^6$Li we use $R_{\\text{3D}\\sigma}=0$ for both spin states. For the orbital Feshbach resonance in quasi-2D, the description of the effective range is somewhat more complicated, as discussed in the following.\n\n\n\n\n\\subsection{Effective model for scattering at an orbital Feshbach resonance in the presence of confinement}\n\nWe now discuss how the scattering parameters, $a_{\\text{2D}\\sigma}$ and $R_{\\text{2D}\\sigma}$, of the $^{173}$Yb experiment are determined. The orbital Feshbach resonance is known to lead to a strongly energy dependent scattering~\\cite{Zhang2015}, which is well approximated by the introduction of a 3D effective range~\\cite{Xu2016}. Furthermore, quasi-2D confinement generally leads to a non-trivial energy dependence of the effective 2D scattering amplitude even for a broad Feshbach resonance~\\cite{Petrov2001}; for strong confinement, this energy dependence can be modelled by the introduction of a 2D effective range~\\cite{Levinsen2013}. Thus, we use the effective range in our model to provide the simplest possible description of both the orbital Feshbach resonance and the confinement. This greatly simplifies the numerical simulations of Rabi oscillations, since it drastically reduces the possible degrees of freedom in the problem.\n\nWe now describe the possible scattering channels in ${}^{173}$Yb atoms close to an orbital Feshbach resonance, following the analysis in Ref.~\\cite{Zhang2015}. We focus on the two electronic orbitals described above (denoted here by $\\ket{g}$ and $\\ket{e}$) as well as two particular nuclear spin states (denoted here by $\\ket{\\Downarrow}$ and $\\ket{\\Uparrow}$). These states form the open channel $\\ket{o} \\equiv \\ket{g \\Uparrow, e \\Downarrow}$ and the closed channel $\\ket{c} \\equiv \\ket{e \\Uparrow, g \\Downarrow}$, which are detuned by $\\delta = \\Delta \\mu B$, where $\\Delta \\mu = h\\times 554\\,\\mathrm{Hz}\/\\mathrm{G}$~\\cite{Oppong2019} is the differential Zeeman shift and $B$ is the magnetic field strength in Gauss.\nThe interactions in this system are not diagonal in the open- and closed-channel basis, but instead proceed via the triplet $(+)$ and singlet $(-)$ channels, with $\\ket{\\pm} \\equiv \\frac{1}{\\sqrt{2}}(\\ket{o} \\pm \\ket{c})$. Associated with the singlet and triplet interactions are the singlet and triplet scattering lengths $a_\\pm$~\\cite{Zhang2015} and effective ranges $r_\\pm$~\\cite{Hofer2015}. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{a2D_R2D_Figure.pdf}\n    \\caption{The open channel 2D scattering length (a) and 2D effective range (b) as a function of magnetic field strength (in Gauss). These parameters are used to create an effective 2D two-channel model description of scattering at an orbital Feshbach resonance in the presence of confinement. To match Ref.~\\cite{Oppong2019} we use $\\omega_z = 2\\pi \\times 37.1\\,\\mathrm{kHz}$.}\n    \\label{fig:a2DandR2D}\n\\end{figure}\n\nIn the experiment~\\cite{Oppong2019}, the ${}^{173}$Yb atoms were confined to move in a 2D plane with a harmonic potential $V(z) = \\frac{1}{2} m \\omega_z^2 z^2$ acting in the transverse direction. Reference~\\cite{Oppong2019} used this to extend the theoretical analysis of quasi-2D scattering in Refs.~\\cite{Petrov2001,Bloch2008mbp} to derive an effective scattering amplitude of the form in Eq.~\\eqref{eq:f2D}. For the weakly interacting $\\downarrow$ state, it was found that $\\ln (k_F a_{\\text{2D}\\downarrow}) \\simeq -4.9 (1)$~\\cite{Oppong2019}. In this case, the dependence on $R_{\\text{2D} \\downarrow}$ is strongly suppressed, and we simply take $R_{\\text{2D} \\downarrow}=0$. For the case of strong interactions, the open channel scattering amplitude was found to be given by~\\cite{Oppong2019}\n\\begin{align}\n    f_{\\text{q2D}}(E) &=2 \\sqrt{2 \\pi } \\frac{ l_z \\left(\\Tilde{a}^{-1}_- -  \\frac{m r_- E}{2} \\right)+ l_z \\left(\\Tilde{a}^{-1}_+ -  \\frac{m  r_+ E}{2} \\right)-2 \\mathcal{F}\\left(\\frac{-E + \\delta}{\\omega_z}\\right)}{ \\left\\{\\splitfrac{ -\\mathcal{F}\\left(\\frac{-E + \\delta}{\\omega_z}\\right) \\left[l_z (\\Tilde{a}^{-1}_- -  \\frac{m r_- E }{2})+l_z (\\Tilde{a}^{-1}_+ -  \\frac{m r_+ E }{2})-2\n   \\mathcal{F}\\left(-\\frac{E}{\\omega_z}\\right) \\right]}{-\\mathcal{F}\\left(-\\frac{E}{\\omega_z}\\right) \\left[l_z (\\Tilde{a}^{-1}_- -  \\frac{m r_- E }{2})+l_z (\\Tilde{a}^{-1}_+ -  \\frac{m r_+ E}{2}) \\right] +2 l_z^2 (\\Tilde{a}^{-1}_- - \\frac{m  r_- E}{2}) (\\Tilde{a}^{-1}_+ -  \\frac{ m r_+ E }{2}) } \\right\\}}, \\label{eq:fopen}\n\\end{align}\nwhere $l_z = 1\/\\sqrt{m \\omega_z}$ and $\\Tilde{a}_\\pm^{-1} = a^{-1}_\\pm - \\frac{r_\\pm}{4 l_z^2}  \\left( 1- \\delta\/\\omega_z \\right)$. Here, $\\mathcal{F}$ is a transcendental function defined by~\\cite{Bloch2008mbp}\n\\begin{align}\n    \\mathcal{F}(x) = \\int_0^\\infty \\frac{du}{\\sqrt{4 \\pi u^3}} \\left[ 1 - \\frac{e^{-xu}}{\\sqrt{(1-e^{-2u})\/2u}} \\right],\n\\end{align}\nand the energy is measured with respect to the quasi-2D zero point energy.\n\nTo extract an effective low-energy open-channel 2D scattering length and effective range for the strongly interacting spin-$\\uparrow$ impurity case, we perform a low energy expansion of  Eq.~\\eqref{eq:fopen} and compare it with the standard form of the low-energy scattering amplitude, Eq.~\\eqref{eq:f2D}. In the following, we use the notation $a_{\\text{2D}}\\equiv a_{\\text{2D}\\uparrow}$ and $R_{\\text{2D}}\\equiv R_{\\text{2D}\\uparrow}$, as in the main text. Assuming that $\\delta \\gg |E|$ (i.e., that we are not close to $B=0$) we find the 2D scattering length \\cite{Oppong2019}\n\\begin{align}\n    a_{\\text{2D}} = l_z \\sqrt{\\frac{\\pi}{D}} \\exp[-\\sqrt{2 \\pi} \\frac{ l_z^2 ( \\Tilde{a}_{-} \\Tilde{a}_{+})^{-1}- \\frac{1}{2}(l_z \\Tilde{a}_{-}^{-1}+ l_z \\Tilde{a}_{+}^{-1}) \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z} \\right) }{l_z \\Tilde{a}_{-}^{-1}+l_z \\Tilde{a}_{+}^{-1}-2 \\mathcal{F} \\left(\\frac{\\delta}{\\omega_z}\n   \\right)} ],\n\\end{align}\nwhere $D \\simeq 0.905$~\\cite{Petrov2001}. The 2D effective range takes the form\n\\begin{align}\n    \\left(\\frac{R_{\\text{2D}}}{l_z}\\right)^2 &=  \\ln 2 %\n    -\\frac{\\sqrt{2 \\pi } \\left\\{ l_z^{-1} r_+ \\left[ l_z \\Tilde{a}_-^{-1} - \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z}\n   \\right) \\right]^2 + l_z^{-1} r_- \\left[l_z \\Tilde{a}_+^{-1} - \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z}\n   \\right) \\right]^2  + \\left[l_z\\Tilde{a}_-^{-1} - l_z \\Tilde{a}_+^{-1} \\right]^2 \\mathcal{F}'\\left( \\frac{\\delta}{\\omega_z} \\right)\\right\\}}{ \\left(l_z \\Tilde{a}_-^{-1}+ l_z\\Tilde{a}_+^{-1} -2 \\mathcal{F}\\left( \\frac{\\delta}{\\omega_z} \\right) \\right)^2},\n\\end{align}\nwhere $\\mathcal{F}'$ is the first derivative of $\\mathcal{F}$. \n\nFigure~\\ref{fig:a2DandR2D} shows the extracted scattering parameters using the experimentally determined values for all parameters~\\cite{Oppong2019} (see also Ref.~\\cite{Hofer2015}). We see that indeed the change of the magnetic field provides a means to tune the interactions via $a_{{\\rm 2D}}$. On the other hand, the effective range is not strongly dependent on magnetic field --- indeed, the effective range is approximately constant ($k_F R_{\\text{2D}} \\sim 1$) within the magnetic field strengths of interest to the present work.\n\n\\jfl{\nIn general, we find that the 2D effective range is able to highly accurately capture the physics of the orbital Feshbach resonance and the confinement in the domain of interaction strengths used in Ref.~\\cite{Oppong2019}. This is shown in Fig.~\\ref{fig:fulltheoryvs2channel}, where we show the near perfect agreement between the full theory model \\cite{Oppong2019} and our effective model in calculating the energy of the Fermi polaron.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.35\\linewidth]{FullTheory_vs_2Channel.pdf}\n    \\caption{\\jfl{\n    The attractive (orange) and repulsive (blue) Fermi polaron energy as calculated through our effective two-channel model description of an orbital Feshbach resonance in the presence of confinement. The energies are highly consistent with the full theory calculation of the polaron energies (black) from Ref. \\cite{Oppong2019}. The curves are calculated at temperature $T\/T_F=0.16$.}}\n    \\label{fig:fulltheoryvs2channel}\n\\end{figure}\n\n\\section{Finite temperature variational approach for impurity dynamics}\nIn order to model the dynamics of the Rabi oscillations and the impurity spectral response, we use the finite-temperature Truncated Basis Method (TBM) developed in Refs.~\\cite{Parish2016,Liu2019}. The basic details of this method are discussed in the main text. Our ansatz for the impurity operator is\n\\begin{align} \\label{eq:OpApprox}\n    \\hat{c}(t)= &\n\\sum_{\\sigma}\\Bigg[\\alpha_{0}^\\sigma(t)\\hat\n  c_{{\\bf 0}\\sigma}+\\sum_\\k\\alpha_{\\k}^\\sigma(t)\\hat f^\\dag_\\k\\dh_{\\k\\sigma}\n+\\sum_{\\k,{\\bf q}} \\alpha_{\\k{\\bf q}}^\\sigma(t) \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\sigma}\\Bigg],\n\\end{align}\nwhere, for simplicity, we consider a vanishing total\nmomentum. As discussed in the main text, we may quantify the error incurred in the Heisenberg equation of motion by introducing the error operator $\\hat\\epsilon(t)\\equiv i\\partial_t \\hat{c}(t)- \\comm*{\\hat{c}(t)}{\\hat H}$ and the associated error quantity $\\Delta(t)\\equiv\\Tr[\\hat \\rho_0 \\hat\\epsilon(t) \\hat\\epsilon^\\dag(t)]$. Using the minimization condition $\\pdv*{\\Delta(t)}{\\dot\\alpha^{\\sigma*}_j(t)}=0$ with respect to the variational coefficients $\\{\\alpha_j\\}$,\nwe arrive at \n\\begin{subequations} \\label{eq:VariationalEqs}\n\\begin{align}\n    E  \\alpha^{\\uparrow}_{0} &= g_\\uparrow \\sum_{{\\bf q}} \\alpha^{\\uparrow}_{{\\bf q}} \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }]  + \\frac{\\Omega_0}{2} \\alpha^{\\downarrow}_{0}\\\\\n    (E - \\varepsilon_{{\\bf q} \\uparrow}) \\alpha^{\\uparrow}_{{\\bf q}} &= g_\\uparrow \\alpha^{\\uparrow}_{0} + g_\\uparrow \\sum_{\\k} \\alpha^{\\uparrow}_{\\k {\\bf q}} \\Tr[\\hat \\rho_0 \\hat f_{\\k}\\hat f^\\dag_{\\k}]\\\\\n    (E - \\varepsilon_{\\k {\\bf q}}) \\alpha^{\\uparrow}_{\\k {\\bf q}} &= g_\\uparrow \\alpha^{\\uparrow}_{{\\bf q}} + \\frac{\\Omega_0}{2} \\alpha^{\\downarrow}_{\\k {\\bf q}}\\\\\n    (E - \\Delta \\omega)  \\alpha^{\\downarrow}_{0} &=\n    g_\\downarrow \\sum_{{\\bf q}} \\alpha^{\\downarrow}_{{\\bf q}} \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }]  + \\frac{\\Omega_0}{2} \\alpha^{\\uparrow}_{0} \\\\\n    (E - \\varepsilon_{{\\bf q} \\downarrow} - \\Delta \\omega) \\alpha^{\\downarrow}_{{\\bf q}} &= g_\\downarrow \\alpha^{\\downarrow}_{0}  + g_\\downarrow \\sum_{\\k} \\alpha^{\\downarrow}_{\\k {\\bf q}} \\Tr[\\hat \\rho_0 \\hat f_{\\k}\\hat f^\\dag_{\\k}]\\\\\n    (E - \\varepsilon_{\\k {\\bf q}} - \\Delta \\omega) \\alpha^{\\downarrow}_{\\k {\\bf q}} &= g_\\downarrow \\alpha^{\\downarrow}_{{\\bf q}} + \\frac{\\Omega_0}{2} \\alpha^{\\uparrow}_{\\k {\\bf q}}.\n\\end{align}\n\\end{subequations}\nHere we have taken the stationary condition since our Hamiltonian is time-independent,\n\\begin{align}\n    \\alpha^\\sigma_{j}(t)=\\alpha^\\sigma_{j}(0)e^{-iEt}\\equiv\\alpha^\\sigma_{j}e^{-iEt},\n\\end{align}\nand defined $\\varepsilon_{{\\bf q} \\sigma} \\equiv  \\nu_{\\sigma}-\\epsilon_{{\\bf q}}\/2$ and $\\varepsilon_{\\k {\\bf q}} \\equiv \\epsilon_{{\\bf q} - \\k} + \\epsilon_{\\k} - \\epsilon_{{\\bf q}}$.\n\nEquation~\\eqref{eq:VariationalEqs} represents a set of linear integral equations that define the time dependence of the variational coefficients $\\{ \\alpha_j \\}$. It is quite similar to the corresponding set of equations derived in Refs.~\\cite{Parish2016,Liu2019}. However, \nhere we account for  temperature, initial-state interactions, and the Rabi coupling between the two impurity spin states. \nDiagonalizing Eq.~\\eqref{eq:VariationalEqs} yields eigenvectors $\\{ \\alpha_j^{(l)} \\}$ and corresponding eigenvalues $E_l$. In what follows it is useful to write the eigenvectors as the union of the spin-$\\uparrow$ and spin-$\\downarrow$ components, i.e., $\\{ \\alpha_j ^{(l)}\\} = \\{ \\alpha^{ \\uparrow (l)}_j \\} \\cup \\{ \\alpha^{ \\downarrow (l)}_j \\}.$\n\nThe solutions of Eq.~\\eqref{eq:VariationalEqs} allow us to obtain stationary impurity operators\n\\begin{align}\n    \\hat{\\phi}^{(l)} \\equiv \\sum_j \\alpha_j^{(l)} \\hat{O}_j,\n\\end{align}\nwhere the impurity basis operators $\\{ \\hat{O}_j \\}=\\{\\hat c_{{\\bf 0}\\uparrow}, \\hat f^\\dag_\\k\\dh_{\\k\\uparrow},  \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\uparrow},\\hat c_{{\\bf 0}\\downarrow}, \\hat f^\\dag_\\k\\dh_{\\k\\downarrow},  \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\downarrow}\\}$ are those introduced in Eq.~\\eqref{eq:OpApprox}. These all satisfy $\\Tr[\\hat\\rho_0\\hat{O}_j \\hat{O}^\\fix{\\dag}_k]=0$ when $j\\neq k$, since the trace is over medium-only states. This in turn allows us to normalize the stationary solutions according to\n\\begin{align}\n    \\Tr[\\hat \\rho_0 \\hat \\phi^{(l)}\\hat \\phi^{(m)\\dag}]=\\delta_{lm}.\n\\end{align}\n Using this, the impurity annihilation operator in Eq.~\\eqref{eq:OpApprox} can be expressed as\n\\begin{align}\n    \\hat{c}(t) = \\sum_l \\Tr[\\hat \\rho_0\\hat c(0)  \\Hat{\\phi}^{(l)\\dagger}] \\Hat{\\phi}^{(l)} e^{-i E_l t}=\\sum_l \\alpha_0^{\\fix{\\downarrow}(l)*} \\Hat{\\phi}^{(l)} e^{-i E_l t},\n\\end{align}\nwhere the initial impurity operator $\\hat c(0) = \\hat c_{{\\bf 0}\\downarrow}$. We take a bare impurity as the initial state even though there are initial-state interactions, since the simulations yield essentially the same result as when we use a weakly interacting polaron. We presume this is because the weakly interacting polaron forms quickly (on a time scale set by $a_{\\mathrm{3D}\\downarrow}^2$ in 3D) once we start the dynamics.\n\n\nFinally, we discuss the character of the medium-only states appearing in Eq.~\\eqref{eq:VariationalEqs}. Assuming that the interactions between the initial $\\downarrow$ impurity and the medium particles are negligible (as is the case in the experiments~\\cite{Scazza2017,Oppong2019}), we may take the medium states to be thermal eigenstates at temperature $T$. Therefore, we have\n\\begin{align}\n    \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }]\\equiv \\expval{\\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }}_\\beta = n_F(\\epsilon_{\\q})   \n\\end{align}\nwhere $\\beta$ is the inverse temperature and $n_F$ is the Fermi-Dirac distribution function:\n\\begin{align}\n    n_F(\\epsilon_{\\q})\\equiv \\Tr[\\hat \\rho_0 \\hat f^\\dag_{\\bf q} \\hat f_{\\bf q}]=\\frac1{e^{\\beta(\\epsilon_{\\q}-\\mu)}+1}.\n\\end{align}\nThe chemical potential $\\mu$ is related to the medium density $n$ via\n\\begin{align}\n    n=\\sum_{\\bf q} n_F(\\epsilon_{\\q})=\\begin{cases}\n    -\\left(\\frac{m}{2\\pi \\beta}\\right)^{3\/2}{\\rm Li}_{3\/2}(-e^{\\beta\\mu}) & \\rm(3D)\\\\[0.5em] \\frac{m}{2 \\pi \\beta} \\ln(1+ e^{\\beta \\mu}) & \n     \\mbox{(2D)}\\end{cases}\n\\end{align}\nwhere ${\\rm Li}$ is the polylogarithm. The density is related to the Fermi energy via\n\\begin{align}\n    E_F=\\frac{k_F^2}{2m}=\\begin{cases}\n    \\frac{(6\\pi^2n)^{2\/3}}{2m} & \\rm(3D)\\\\ \\frac{4\\pi n}{2m} & \\rm{(2D)}\\end{cases}\n\\end{align}\n\n\\subsection{Impurity spectral function}\n\nIn the limit of a weak Rabi coupling, we can obtain the spin-$\\uparrow$ impurity spectral function within linear response:\n\\begin{align}\n    A_\\uparrow(\\omega) \\simeq \\sum_l |\\alpha^{\\uparrow (l)}_0|^2 \\delta(\\omega - E_l),\n    \\label{eq:spec}\n\\end{align}\nwhere the variational equations in Eq.~\\eqref{eq:VariationalEqs} are solved at $\\Omega_0=0$.\nIn practice, the spin-$\\downarrow$ impurity interacts weakly with the medium, and thus the spectral function can be measured by driving transitions from the initial (nearly) non-interacting spin-$\\downarrow$ impurity state into the spin-$\\uparrow$ state. Indeed, this has been done in both experiments~\\cite{Scazza2017,Oppong2019}.\n\nSince the solution of Eq.~\\eqref{eq:VariationalEqs} are discrete, we convolve the resulting spectrum with a Gaussian to yield\n\\begin{align}\n    I(\\omega) &= \\sum_l |\\alpha^{\\uparrow (l)}_0|^2 g(\\omega - E_l).\n\\end{align}\nConvolution of the spectrum in this case has the added benefit of enabling one to approximately model the finite duration of the pulses used in experiment.\n\n\\subsection{Simulating Rabi oscillations}\n\nThe Rabi oscillations are defined by\n\\begin{align}\n{\\cal N}_\\downarrow(t)=\\expval{\\hat c(t)\\hat n_\\downarrow \\hat c^\\dag(t)}_\\beta.\n\\end{align}\nWe will take as our initial condition that $\\hat c(\\fix{t=0})=\\hat c_{{\\bf 0}\\downarrow}$, i.e., the impurity is initially in a bare spin-$\\downarrow$ state, which means that the impurity number is ${\\cal N}_\\downarrow+{\\cal N}_\\uparrow=1$ at all times. The Rabi oscillations are then given by\n\\begin{align}\n{\\cal N}_\\downarrow(t)& %\n\\simeq \\Trace[\\hat\\rho_0\\hat{c}_\\downarrow(t)\\hat n_\\downarrow\\hat{c}^\\dag_\\downarrow(t)]\n= \\sum_{j} \\fix{\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta} \\bigg|\\sum_l \\alpha_{0}^{\\downarrow(l)^*} e^{-i E_l t} \\alpha_j^{\\downarrow(l)} \\bigg|^2  .\\label{eq:RabiOscEq}\n\\end{align}\n\n\nWith the exception of detuning, all of the parameters used to define the Rabi oscillations are provided from the relevant experiment. The detuning in experiment is set to address the repulsive polaron peak, and to match this procedure, we use a calculated detuning such that the Rabi oscillations address the theoretically obtained spin-$\\uparrow$ repulsive polaron. Due to small but finite initial state interactions, this detuning must take into account the energy of the spin-$\\downarrow$ impurities. This is achieved by assuming the spin-$\\downarrow$ impurities exist as zero-momentum repulsive polarons with a narrow spectral width. Owing to this assumption, combined with thermal fluctuations, a finite density of impurities and experimental limitations, we place an uncertainty on the detuning (see Fig.~\\ref{fig:DetuningUncertaintyCalc}). This requires that we simulate Rabi oscillations over the range of possible detunings.\nThe parameters used in the simulations of the Rabi oscillations in Figs.~\\ref{fig:Rabi} and \\ref{fig:ResidueAndDamping} can be found in Table~\\ref{tab:params}.\n\nWe find that the inclusion of initial state interactions leads to a small reduction in the damping of the Rabi oscillations.\nThis can be understood from the fact that in the limit in which the spin states have equal interactions, spin-symmetry implies that the Rabi oscillations would be undamped and oscillate at the bare Rabi frequency.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\linewidth]{DetuningCalc.pdf}\n    \\caption{Example of the calculation of the uncertainty in the detuning of Rabi oscillations onto the repulsive polaron. We show the spectral function of the repulsive polaron in the 2D ${}^{173}$Yb experiment of Ref.~\\cite{Oppong2019} (solid line), with the dashed lines indicating the full-width half-maxima. The detuning is set %\n    between these half-maxima, leading to the uncertainty shown in Fig.~\\ref{fig:Rabi} in the main text. The shown spectral function is for $\\ln(1\/k_F a_{\\text{2D}}) = 0.41$, $k_F R_{\\text{2D}} = 0.69$ and $T\/T_F = 0.16$.}\n    \\label{fig:DetuningUncertaintyCalc}\n\\end{figure}\n\n\n\\begin{figure}[th]\n    \\centering\n    \\includegraphics[width=\\linewidth]{RabiSamples2D_FiniteOffset.pdf}\n    \\caption{\\jfl{Comparison of theory (blue solid lines) of Fig.~2(a)-(c) in the manuscript and the experimental data (black circles) with a constant offset of $0.2$ removed and normalized to the value at time $t=0$.\n    The light-gray circles correspond to the unmodified data in the main text.}}\n    \\label{fig:RabiSamples2D_finiteoffset}\n\\end{figure}\n\n\n\\jfl{\n\\subsection{Finite offset in the Rabi oscillation measurement of the 2D experiment}\nAs noted in the main text, there is a slight disagreement in the amplitude of the Rabi oscillations in the 2D experiment \\cite{Oppong2019} and our variational approach.\nThis disagreement can be explained with a constant offset $\\epsilon > 0$ of the $\\mathcal{N}_\\downarrow$ measurement in the experiment.\nIn our work, the relative population $\\mathcal{N}_\\downarrow\/(\\mathcal{N}_\\downarrow + \\mathcal{N}_\\uparrow)$ of the 2D experiment is inferred from the sole measurement of $\\mathcal{N}_\\downarrow(t)$ and\n\\begin{align}\n\\mathcal{N}_\\downarrow\\left\/(\\mathcal{N}_\\downarrow + \\mathcal{N}_\\uparrow)\\right. \\approx \\mathcal{N}_\\downarrow\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\mathcal{N}_\\uparrow(t = 0)]\\right. \\approx \\mathcal{N}_\\downarrow\/[\\mathcal{N}_\\downarrow(t = 0) + 0] = \\mathcal{N}_\\downarrow\\left\/\\mathcal{N}_\\downarrow(t = 0)\\right..\n\\end{align}\nThus, a constant offset in the measurement of $\\mathcal{N}_\\downarrow$  changes the relative population as\n\\begin{align}\n\\mathcal{N}_\\downarrow\\left\/\\mathcal{N}_\\downarrow(t = 0)\\right. \\rightarrow (\\mathcal{N}_\\downarrow + \\epsilon)\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right. = \\mathcal{N}_\\downarrow\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right. + \\epsilon\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right.,\n\\end{align}\nwhich directly shows how the mean of the relative population is artificially increased by $\\epsilon\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]$ and the amplitude is artificially reduced by the additional term $\\epsilon$ in the denominator.\n\nThe offset $\\epsilon$ in the experimental data originates from the detection method, for which the majority Fermi sea is removed, but a finite number of remaining majority atoms contributes a positive spurious signal to the measurement of $\\mathcal{N}_\\downarrow$.\nThis contribution is independent of $t$ and we estimate $\\epsilon\\lesssim 0.2\\, \\mathcal{N}_\\downarrow$ from a reevaluation of the existing data of Ref.~\\cite{Oppong2019}.\nIn Fig.~\\ref{fig:RabiSamples2D_finiteoffset}, we illustrate how the removal of such an offset considerably\nimproves the agreement between experiment and the variational approach.\nHowever, we have chosen not to subtract $\\epsilon$ from the data in Fig.~2 of the main text as we lack a precise number for each data set.}\n\n\\heavyrulewidth=.08em\n\\lightrulewidth=.05em\n\\cmidrulewidth=.03em\n\\belowrulesep=.65ex\n\\belowbottomsep=0pt\n\\aboverulesep=.4ex\n\\abovetopsep=0pt\n\\cmidrulesep=\\doublerulesep\n\\cmidrulekern=.5em\n\\defaultaddspace=.5em\n\n\\begin{table}[h]\n    \\caption{\\label{tab:params}%\n\t    Parameters used for obtaining the theoretical curves and simulations shown in Figs.~2~and~3, matching the experimental ones from Refs.~\\cite{Oppong2019} and \\cite{Scazza2017}. Experimental uncertainties are given in parenthesis were applicable. Note that the 3D effective range is always zero.\\bigskip}\n    \\begin{tabular}{l l c c c c c c c c c c c} \n        \\midrule\\midrule\n        && \\multicolumn{7}{c}{Fig.~2} & \\hspace{1.75em} & \\multicolumn{3}{c}{Fig.~3} \\\\ \\cmidrule{3-9} \\cmidrule{11-13}\n        && (a) & (b) & (c) & \\hspace{0.5em} & (d) & (e) & (f) & & (a), (c) & \\hspace{0.5em} & (b), (d) \\\\ \\midrule \n        Interaction parameter & $\\ln(1\/k_F a_{\\mathrm{2D}\\uparrow})$ \\hspace{0.75em} & 0.73(4) & 0.57(5) & 0.25(5) & & \\multicolumn{3}{c}{---} & & 0.07--0.91 & &  --- \\\\\n        & $\\ln(1\/k_F a_{\\mathrm{2D}\\downarrow})$ & \\multicolumn{3}{c}{4.9(1)} & & \\multicolumn{3}{c}{---} & & 4.9(1) & &  --- \\\\\n        & $1\/k_F a_{\\mathrm{3D}\\uparrow}$ & \\multicolumn{3}{c}{---} & & 2.63(4) & 1.27(2) & 0.22(1) & & --- & & 0.22--4.23 \\\\\n        & $1\/k_F a_{\\mathrm{3D}\\downarrow}$ & \\multicolumn{3}{c}{---} & & 9.20(15) & 6.94(11) & 5.03(8) & & --- & & 5.03--11.43 \\\\\n        Range parameter & $k_F R_{\\text{2D}\\uparrow}$ \\hspace{1.5em} & 0.71 & 0.67 & 0.68 & & \\multicolumn{3}{c}{---} & & 0.67--0.76 & &  --- \\\\\n        %\n        Rabi coupling & $\\Omega_0\/E_F$ & 0.95(11) & 0.96(11) & 0.94(11) && 0.68(1) & 0.69(1) & 0.67(1) & & 1.08 && 0.70 \\\\\n        Reduced temperature & $T\/T_F$ & \\multicolumn{3}{c}{0.16(4)} && \\multicolumn{3}{c}{0.13(2)} & & 0.16(4) && 0.13--0.14 \\\\\n        \\addlinespace[0.5em]\n        Rep. polaron energy & $E_{+ \\uparrow}\/E_F$ & 0.71 & 0.76 & 0.89 && 0.19 & 0.42 & 1.08 & & 0.64--1.02 && 0.11--1.08 \\\\\n         & $E_{+ \\downarrow}\/E_F$ & \\multicolumn{3}{c}{0.19} && 0.05 & 0.07 & 0.09 & & 0.19 && 0.04--0.09 \\\\\n        Detuning uncertainty & $\\delta E_+\/E_F$ & 0.24 & 0.31 & 0.47 && 0.02 & 0.05 & 0.42 & & 0.19--0.65 && 0.02--0.42 \\\\\n        \\midrule\\midrule\n    \\end{tabular}\n\\end{table}\n\n\n\\section{Green's function approach for quasiparticle properties}\nA key alternative to our study of the impurity dynamics and properties in the TBM is provided through a Green's function approach. It has been shown that a variational approach using a single particle-hole excitation is equivalent to a Green's function approach calculated with non-self consistent $T$ matrix theory --- see Ref.~\\cite{Combescot2007} for a zero-temperature treatment, or Ref.~\\cite{Liu2019} at finite temperature. %\n\nIn this section, we take $\\Omega_0=0$ in the variational equations~\\eqref{eq:VariationalEqs}, while we consider the more general case in the next section. This allows us to derive a finite temperature impurity self energy $\\Sigma_\\sigma(E)$ separately within each of the impurity subspaces. Solving these equations for the energy then yields the expression\n\\begin{align} \\label{eq:definingSelfE}\n    E =  \\sum_{{\\bf q}} n_F(\\epsilon_{\\q}) \\left[ \\frac{E - \\varepsilon_{{\\bf q} \\sigma}}{g_\\sigma^2} - \\sum_{\\k} \\frac{1 - n_F(\\epsilon_{\\k})}{E - \\varepsilon_{\\k {\\bf q}}} \\right]^{-1}.\n\\end{align}\nThe right hand side of this expression is precisely the impurity self energy at zero momentum using ladder diagrams at finite temperature~\\cite{Liu2019}. The self energy is then related to the impurity (single-particle) Green's function through Dyson's equation,\n\\begin{align} \\label{dyson}\n    G_\\sigma(E) = \\frac{1}{E - \\Sigma_\\sigma(E)}.\n\\end{align}\n\nThe relevant properties of the repulsive polaron can now be defined in terms of the impurity self energy. In particular, the repulsive polaron energy $E_{+\\sigma}$ is a (positive) solution to the implicit equation\n\\begin{align}\n    \\Re \\left[\\Sigma_\\sigma (E) \\right] = E.\n\\end{align}\nExpanding the Green's function around this pole, the repulsive polaron quasiparticle residue is\n\\begin{align}\n    Z_\\sigma = \\left( 1 - \\left. \\pdv{\\Re\\left(\\Sigma_\\sigma(E) \\right)}{E} \\right|_{E = E_{+\\sigma}} \\right)^{-1},\n\\end{align}\nand the quasiparticle width is\n\\begin{align} \\label{Eq:quasiparticlewidth}\n    \\Gamma_\\sigma = - Z_\\sigma \\Im \\left[ \\Sigma_\\sigma(E_{+\\sigma}) \\right].\n\\end{align}\nIn addition to these properties, the impurity spectral function in Eq.~\\eqref{eq:spec} is given by\n\\begin{align}\n    A_\\sigma(E) = -\\frac{1}{\\pi} \\Im[G_\\sigma(E)].\n\\end{align}\n\n\\subsection{Repulsive polaron width at weak interactions}\n\nHere we provide details of the calculation of the approximate width of the repulsive polaron peak at weak interactions, Eq.~\\eqref{eq:GammaPT}. To perform this calculation, we specialize to three dimensions, zero temperature, and to a broad Feshbach resonance, i.e., $R_{\\rm 3D}=0$. In fact, the arguments in the following are valid as long as the thermal wavelength exceeds the scattering length while $R_{\\rm 3D}\\lesssim 1\/(na_{\\rm 3D}^2)$. \n\nAssuming zero impurity momentum, the spin-$\\uparrow$ impurity self-energy in Eq.~\\eqref{eq:definingSelfE} is given by (for simplicity, in this section we suppress all spin indices as well as ``3D'' subscripts)\n\\begin{align}\n    \\Sigma(E) = \\sum_{\\vectorbold{q}} \\Theta(k_F - q) \\left[ \\frac{m}{4 \\pi a} - \\left( \\sum_{\\k} \\frac{1}{2 \\epsilon_{\\k}} + \\sum_{\\k} \\frac{1-\\Theta(k_F - k)}{E - \\varepsilon_{\\k \\vectorbold{q}}+i0} \\right) \\right]^{-1}.\n\\end{align}\nwhere $k\\equiv |\\k|$ and $q \\equiv |\\vectorbold{q}|$ and we take the limit of $R_{\\text{3D}}\\to 0$, which according to Eq.~\\eqref{eq:renorm3D} is equivalent to taking the limit of $\\nu,g\\to\\infty$ in such a way that\n\\begin{align}\n    \\frac{\\nu}{g^2} %\n    =   - \\frac{m }{4 \\pi a} + \\sum_{\\vb{k}}^\\Lambda \\frac{1}{2\\epsilon_{\\vb{k}} }.\n\\end{align}\nWe have also introduced a convergence factor $+i0$ which shifts the energy poles by an infinitesimal amount into the lower half of the complex plane.\n\nIn order to find the approximate form of the self-energy (and thereby the width) in the limit of weak interactions, we perturbatively expand the self-energy in scattering length (up to order $a^2$):\n\\begin{align}\n    \\Sigma(E) = \\sum_{\\vectorbold{q}} \\Theta(k_F - q) \\left[ \\frac{4 \\pi a}{m} + \\frac{16 \\pi^2 a^2}{m^2} \\left( \\sum_{\\k} \\frac{1}{2 \\epsilon_{\\k}} + \\sum_{\\k} \\frac{1-\\Theta(k_F - k)}{E - \\varepsilon_{\\k \\vectorbold{q}} + i 0} \\right) \\right].\n\\end{align}\nIn the limit of weak repulsive interactions, the repulsive polaron will have residue $Z \\simeq 1$ and the width is thus given by\n\\begin{align}\n    \\Gamma \\simeq - \\Im[\\Sigma(E_+)].\n\\end{align}\nWe can extract the imaginary component of the self-energy through the symbolic identity, which is valid for all real $\\alpha$:\n\\begin{align}\n    \\frac{1}{\\alpha + i 0} = \\mathscr{P} \\frac{1}{\\alpha} - i\\pi \\delta(\\alpha),\n\\end{align}\nwhere $\\mathscr{P}$ denotes the Cauchy principal value. We thus have,\n\\begin{align}\n    - \\Im[\\Sigma(E)] = \\frac{16 \\pi^3 a^2}{m^2 E_F} \\sum_{\\vectorbold{q}, \\k } \\Theta(k_F - q) (1 - \\Theta(k_F - k)) \\delta(E\/E_F - \\varepsilon_{ \\k \\vectorbold{q}}\/E_F).\n\\end{align}\nIn the thermodynamic limit, these sums reduce to integrals that can be solved analytically in spherical coordinates:\n\n\\begin{align}\n    - \\Im[\\Sigma(E)] = & \\frac{k_F^4 a^2}{16 \\pi m E_F^2} \\Bigg[ 4E^2  \\ln \\left(\\frac{2E_F}{\\sqrt{E_F (2   E+E_F)}+E_F}\\right) %\n    +3E^2-4 E E_F-2\n   E_F^2+2 (E+E_F)\n   \\sqrt{E_F (2 E+E_F)}\n    \\Bigg].\n\\end{align}\n\n\nImportantly, the imaginary part of the self energy is zero at zero energy, and we must therefore consider finite energy.\nIn the limit of weak interactions, the energy of the repulsive polaron is given by the mean-field approximation~\\cite{Bishop1973}:\n\\begin{align}\n    E_+ = \\frac{2 k_F^3 a}{3 \\pi m}.\n\\end{align}\nUsing this energy and only retaining terms of order $a^4$ (the lowest non-zero contribution) the width is given by\n\\begin{align} \\label{PertbExpDamp}\n    \\frac{\\Gamma}{E_F}= \\frac{8  }{9 \\pi^3} (k_F a)^4.\n\\end{align}\nSince we originally expanded the self-energy up to order $a^2$ and have ended with a result that is of order $a^4$ we justify this result numerically in Fig.~\\ref{fig:PertAndNumerical}. Furthermore, we have checked that all contributions with multiple particle-hole excitations vanish at order $a^4$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\linewidth]{LogLogNumericalVsPert.pdf}\n    \\caption{Comparison of the perturbative expression for the repulsive polaron quasiparticle width in Eq.~\\eqref{PertbExpDamp} against the numerically exact calculation from Eq.~\\eqref{Eq:quasiparticlewidth}.} %\n    \\label{fig:PertAndNumerical}\n\\end{figure}\n\n\n\n\\section{Approximate Rabi oscillations based on Green's functions}\n\nWe finally turn to the arguments that led to Eq.~\\eqref{eq:TheoryFit} in the main text, which allowed us to link the repulsive polaron width to the damping of the Rabi oscillations. This will allow us to extend the standard approximation for extracting polaron properties from impurity Rabi oscillations \\cite{Kohstall2012}. \nIt is useful to introduce a spectral decomposition of the Rabi oscillations, which is provided by the Fourier transform of Eq.~\\eqref{eq:RabiOscEq}: %\n\\begin{align} \\label{eq:RabiSpectrum}\n    \\mathcal{R}_\\downarrow (\\omega)\\equiv \\int dt\\,e^{i\\omega t}{\\cal N}_\\downarrow(t) \\simeq \\sum_{j, l,l'} \\fix{\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta}  \\, \\alpha^{\\downarrow (l)^*}_{0}  \\alpha^{\\downarrow(l)}_{j}\\alpha^{ \\downarrow(l')}_{0} \\alpha^{\\downarrow(l')^*}_{j} \\delta(\\omega - E_l + E_{l'}).\n\\end{align}\nIf the interactions are not too strong, the sum over intermediate states is dominated by the ``bare'' $\\alpha_0$ component. \\fix{In this case, $\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta \\simeq \\expval*{\\hat{O}_j \\hat{c}^\\dag_{{\\bf 0} \\downarrow} \\hat{c}_{{\\bf 0} \\downarrow} \\hat{O}^\\dag_j}_\\beta$} and thus\n\\begin{align} %\n    \\mathcal{R}_\\downarrow (\\omega) \\simeq \\sum_{l,l'} \\abs*{\\alpha^{\\downarrow (l)}_{0}}^2 \\abs*{\\alpha^{ \\downarrow(l')}_{0}}^2 %\n    \\delta(\\omega - E_l + E_{l'}).\n\\end{align}\nWe thus recognize the Rabi spectrum as developing according to a convolution of spectral functions \\textit{in the presence of} Rabi coupling:\n\\begin{align}\n{\\cal N}_\\downarrow(t)& \\simeq \n\\int d \\omega\\, d \\omega' \\, \\Tilde{A}_\\downarrow (\\omega) \\Tilde{A}_\\downarrow (\\omega') e^{-i (\\omega - \\omega')t}.\n\\end{align}\nHere \n\\begin{align}\n    \\Tilde{A}_\\downarrow (\\omega) = -\\frac{1}{\\pi} \\Im[\\Tilde{G}_\\downarrow (\\omega)],\n\\end{align}\nis calculated from the impurity Green's function including Rabi coupling.\n\n\nWe \\fix{can approximate} the Rabi coupled Green's function $\\tilde G$ via the relation\n\\begin{align} \\label{eq:ApproxGreenRelation}\n    \\Tilde{G}(\\omega) \\fix{\\simeq} \\mqty( G_\\uparrow^{-1}(\\omega) & \\Omega_0\/2 \\\\ \\Omega_0\/2 & G_\\downarrow^{-1}(\\omega))^{-1}\\fix{,}\n\\end{align}\n\\fix{where, for ease of notation,}\nwe define $\\Tilde{G}_\\downarrow (\\omega) \\equiv \\Tilde{G}_{22}(\\omega)$. \\fix{We point out that in using Eq.~\\eqref{eq:ApproxGreenRelation} to calculate $\\Tilde{G}_\\downarrow (\\omega)$, we are ignoring the coexistence of the spin-$\\downarrow$ impurity with excitations of the Fermi gas.}\nApproximating the decoupled Green's functions as\n\\begin{align} \n    G_\\sigma (\\omega) \\simeq \\frac{Z_\\sigma}{\\omega - E_{+\\sigma} - \\delta_{\\sigma \\downarrow} \\Delta \\omega + i \\Gamma_\\sigma},\n\\end{align}\nwe find that\n\\begin{align} \\label{eq:ApproxRabi}\n    {\\cal N}_\\downarrow(t) &\\simeq Z_\\downarrow^2 e^{- \\left(\\Gamma _{\\downarrow}+\\Gamma _{\\uparrow}\\right) t} \\left[\\frac{\\Gamma\n   _{\\uparrow}-\\Gamma _{\\downarrow} }{\\sqrt{\\Omega _0^2 Z_{\\downarrow}\n   Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n   _{\\downarrow}\\right)^2}} \\sin \\left(t \\sqrt{\\Omega _0^2 Z_{\\downarrow}\n   Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n   _{\\downarrow}\\right)^2}\\right) \\right. \\nonumber\\\\\n   &{}\\hspace{2em}+ \\left. \\left(1-\\frac{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}}{2\n   \\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}-2 \\left(\\Gamma _{\\uparrow}-\\Gamma\n   _{\\downarrow}\\right)^2}\\right) \\cos \\left(t \\sqrt{\\Omega _0^2 Z_{\\downarrow}\n   Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n   _{\\downarrow}\\right)^2}\\right)+\\frac{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}}{2\n   \\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}-2 \\left(\\Gamma _{\\uparrow}-\\Gamma\n   _{\\downarrow}\\right)^2}\\right].\n\\end{align}\nHere, we have taken $ \\Delta \\omega = E_{+\\uparrow} -E_{+\\downarrow}$\nfor simplicity (i.e., on resonance Rabi oscillations). In the cases of interest where $Z_\\downarrow$ is slightly below 1, the Rabi oscillations are not normalised. However, this is simply an artefact of our approximation of $G_\\downarrow(\\omega)$ and is overcome by dividing Eq.~\\eqref{eq:ApproxRabi} by $Z_\\downarrow^2$.\n\nEquation~\\eqref{eq:ApproxRabi} allows us to immediately identify the Rabi frequency and damping\n\\begin{align}\n    \\Omega &\\simeq \\sqrt{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}- \\left(\\Gamma _{\\uparrow}-\\Gamma_\\downarrow\\right)^2},\n   \\label{eq:OmegaRabi}\\\\\n     \\Gamma_R & \\simeq \\Gamma_\\downarrow + \\Gamma_\\uparrow.\n\\end{align}\nThese reduce to $\\Omega\\simeq \\sqrt{\\Omega _0^2 Z_{\\uparrow}-\\Gamma _{\\uparrow}^2}$ and $\\Gamma_R\\simeq \\Gamma_\\uparrow$ in the case of weak initial state interactions. Equation~\\eqref{eq:OmegaRabi} illustrates why a strong Rabi coupling is necessary in order to drive coherent oscillations once $\\Gamma_\\uparrow$ becomes appreciable, which is why the oscillations are strongly suppressed for the strongest repulsive interactions in the 2D case~(see Fig.~\\ref{fig:Rabi}). It also implies that we can estimate the quasiparticle residue by\n\\begin{align}\n    Z_\\uparrow \\simeq \\frac{ \\Omega ^2 + \\Gamma_\\uparrow^2}{\\Omega _0^2}.\n\\end{align}\nFinally, assuming weak initial state interactions such that $Z_\\downarrow=1$ and $\\Gamma_\\downarrow=0$, and taking $\\Omega_0 Z_\\uparrow\\gtrsim\\Gamma_\\uparrow$, we arrive at the form in Eq.~\\eqref{eq:MainApproxRabi} of the main text:\n\\begin{align} \n    {\\cal N}_\\downarrow(t) & \\simeq e^{- \\Gamma_\\uparrow t} \n    \\left[\\frac12+\\frac12 \\cos \\left(t \\sqrt{\\Omega_0^2\n   Z_\\uparrow-\\Gamma_\\uparrow^2}\\right)\\right].\n\\end{align}\n\n\nThe limiting feature of this effective model comes from our approximation of $\\hat n_\\downarrow \\simeq \\hat c_{{\\bf 0} \\downarrow}^\\dag \\hat c_{{\\bf 0} \\downarrow}$. For any $\\Gamma_\\uparrow, \\Gamma_\\downarrow >0$, this approximation will always lead to $\\mathcal{N}_\\downarrow(t) \\to 0$ for large $t$, which does not match the behaviour of Rabi oscillations in experiment or the TBM. However, at the intermediate times considered in Fig.~\\ref{fig:Rabi} it provides a good model of the actual oscillations.\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe non-decoupling of the relevant scales on a wide and continuous range of\nmagnitudes in many areas of physics has led to the invention (discovery) of\nthe renormalisation group (RG) \\cite{206}. Whereas they have been discovered\nin the framework of the perturbative (quantum field) theory, the RG\ntechniques tackle a nonperturbative physical phenomenon \\cite{425}.\nNonperturbative approaches are difficult to implement and to control, and\nduring a long time one has essentially carried on perturbative RG techniques\n(see, e.g., \\cite{4948}). Nowadays, the huge growth of the computing\ncapacity has greatly modified this behaviour pattern and, already since the\nbeginning of the ninety's, one has considered \\cite{4374} with a greater\nacuteness the exact RG equations (ERGEs) originally introduced by Wilson \n\\cite{Irvine}, Wegner and Houghton \\cite{414} in the seventy's and slightly\nreformulated by Polchinski \\cite{354} in the eighty's (for some reviews on\nthe ERGEs see \\cite{4595}).\n\nInitially, the ERGEs are integro-differential equations for the running\naction $S\\left[ \\phi ,t\\right] $ [assuming that $\\phi \\left( x\\right) $\ngenerically stands for some field with as many indices as necessary and $%\nt=-\\ln \\left( \\Lambda \/\\Lambda _{0}\\right) $ the logarithm of a running\nmomentum scale $\\Lambda $]. They have been extended to the running (average)\neffective action $\\Gamma \\left[ \\varphi ,t\\right] $ \\cite{4281,4374}. Such\ngeneral equations cannot be studied without the recourse to approximations\nor truncations. One of the most promising approximations is a systematic\nexpansion in powers of the derivative of the field (derivative expansion) \n\\cite{212} which yields a set of coupled nonlinear partial differential\nequations the number of which grows quickly with the order of the expansion.\nIn the simplest cases (e.g., for the scalar field), the determination of\nfixed points\\ (and of their stability) amounts to study ordinary\ndifferential equations (ODEs) with a two-point boundary value problem that\nmay be carried out numerically via a shooting (or a relaxation) method.\n\nA pure numerical study is in general not easy to implement and to control.\nFor example, in the shooting method, the discovery of the right adjustment\nof the parameters at the boundaries requires a good knowledge a priori of\ntheir orders of magnitude (initial guesses). It is thus interesting to\ndevelop concurrently some substitute analytical methods. A popular\nsubstitute to the ODEs of the derivative expansion is provided by an\nadditionnal expansion in powers of the field which yields a set of coupled\nalgebraic equations which may be solved analytically, at least with the help\nof a symbolic computation software. Various field expansions have been\nimplemented with more or less success \\cite{3478,3642,4192,3553}.\nUnfortunately, the methods proposed up to now, if they are easy to\nimplement, do not work in all cases and especially in the most famous and\nsimplest case of the Wilson-Polchinski ERGE \\cite{Irvine,354} (equation for\nthe running action $S\\left[ \\phi ,t\\right] $ with a smooth cutoff).\n\nThe object of this paper is to present two new substitute analytical methods\nfor studying ODEs which, at least in\\ the local potential approximation of\nthe derivative expansion (LPA), works for the Wilson-Polchinski ERGE. One of\nthe methods, recently proposed in \\cite{6110}, is a genuine analytical\napproximation scheme to two-point boundary value problems of ODEs. The other\nmethod is new. It is based on approximations of the solution looked for by\ngeneralized hypergeometric functions. It has a certain similarity with\nanother new and interesting method based on the representation of the\nsolution by Pad\\'{e} approximants just proposed in \\cite{6201} by P. Amore\nand F. M. Fernandez independantly from the present work. We illustrate the\neffectiveness of the two methods with the explicit consideration of two\nERGEs in the local potential approximation: the Wilson-Polchinski equation\nand the Litim optimized RG equation \\cite{5020} for the running effective\naction (named the Litim equation in the following).\\ Following a conjecture\nfirst stated in \\cite{5049,5902}, the equivalence of these two equations (in\nthe LPA) has been proven by Morris \\cite{5911} and recently been numerically\nillustrated \\cite{6137} with an unprecedented accuracy for the scalar field\nin three dimensions ($d=3$). This particular situation provides us with the\nopportunity of testing efficiently the various methods of study at hand.\n\nThe following of the paper is divided in five sections. In section \\ref%\n{Calculations}, we briefly present the direct numerical integration of the\nODEs for the scalar model\\ using the shooting method: determinations of the\nfixed point and the critical exponents for both the Wilson-Polchinski and\nLitim equations in the LPA (distinguishing between the even and odd\nsymmetries). A brief presentation of the currently used field expansion is\ngiven in section \\ref{WPE}. In section \\ref{BFG}, we analyse several aspects\nof the method of \\cite{6110} applying it to the study of the two equations.\nWe calculate this way the fixed point locations with high precision and\ncompare the results with the estimates obtained in section \\ref{Calculations}%\n. We show how the leading and the subleading critical exponents may be\nestimated using this recent method. In section \\ref{Ratios} we present a new\napproximate analytical method for ODEs which is based on the definition of\nthe generalized hypergeometric functions. We show that it is well adapted to\ntreat the Wilson-Polchinski case whereas the Litim case is less easily\ntreated. We relate these effects to the convergence properties of the series\nin powers of the field. Finally we summarize this work and conclude in\nsection \\ref{Conc}.\n\n\\section{Two-point boundary value problem in the LPA}\n\n\\label{Calculations}\n\nIn this section we briefly present the two-point boundary value problem to\nbe solved in the LPA of the ERGE. The Wilson-Polchinski equation is first\nchosen as a paradigm in section \\ref{WilPol}. The principal numerical\nresults obtained from the numerical integration of the ODE using the\nshooting method are given. In section (\\ref{Litim}), the Litim equation is\nalso studied.\n\n\\subsection{Wilson-Polchinski's flow equation for the scalar-field\\label%\n{WilPol}}\n\nThe original Wilson-Polchinski ERGE in the LPA expresses the evolution of\nthe potential $U\\left( \\phi ,t\\right) $ as varying the logarithm of the\nmomentum scale of reference $t=-\\ln \\left( \\Lambda \/\\Lambda _{0}\\right) $\n(with $\\phi \\in \n\\mathbb{R}\n$). In three dimensions, it reads:%\n\\begin{equation}\n\\dot{U}=U^{\\prime \\prime }-\\left( U^{\\prime }\\right) ^{2}-\\frac{1}{2}\\phi\nU^{\\prime }+3U\\,,  \\label{eq:LPAV}\n\\end{equation}%\nin which $\\dot{U}\\equiv \\partial U\\left( \\phi ,t\\right) \/\\partial t$, $%\nU^{\\prime }\\equiv \\partial U\\left( \\phi ,t\\right) \/\\partial \\phi $, $%\nU^{\\prime \\prime }\\equiv \\partial ^{2}U\\left( \\phi ,t\\right) \/\\partial \\phi\n^{2}$.\n\n\\subsubsection{Fixed point equation\\label{FP1}}\n\nThe fixed point equation corresponds to $\\dot{U}=0$. It is a second order\nODE for the function $U\\left( \\phi \\right) $:%\n\\begin{equation}\nU^{\\prime \\prime }-\\left( U^{\\prime }\\right) ^{2}-\\frac{1}{2}\\phi U^{\\prime\n}+3U=0\\,,  \\label{eq:FPF}\n\\end{equation}%\nthe solution of which (denoted $U^{\\ast }\\left( \\phi \\right) $ below)\ndepends on two integration constants which are fixed by two conditions. The\nfirst one comes from a property of symmetry assumed to be\\footnote{%\nThe other possibility $U^{\\ast }\\left( -\\phi \\right) =-U^{\\ast }\\left( \\phi\n\\right) $ gives only singular solutions at finite $\\phi $.} $U^{\\ast }\\left(\n-\\phi \\right) =U^{\\ast }\\left( \\phi \\right) $ which provides the following\ncondition at the origin for $U^{\\ast }\\left( \\phi \\right) $:%\n\\begin{equation}\nU^{\\ast \\prime }\\left( 0\\right) =0\\,.  \\label{eq:fori}\n\\end{equation}%\nThe second condition is the requirement that the solution we are interested\nin must be non singular in the entire range $\\phi \\in \\left[ 0,\\infty \\right[\n$. Actually, the general solution of (\\ref{eq:FPF}) involves a moving\nsingularity \\cite{2080} of the form:%\n\\begin{equation}\nU_{\\text{sing}}=-\\ln \\left\\vert \\phi _{0}-\\phi \\right\\vert \\,,\n\\label{eq:sing}\n\\end{equation}%\ndepending on the arbitrary constant $\\phi _{0}$. Pushing $\\phi _{0}$ to\ninfinity allows to get a non-singular potential since, in addition to the\ntwo trivial fixed points $U^{\\ast }\\equiv 0$ (Gaussian fixed point) and $%\nU^{\\ast }\\equiv -\\frac{1}{3}+\\frac{{\\phi }^{2}}{2}$ (high temperature fixed\npoint), eq.(\\ref{eq:FPF}) admits a non-singular solution which, for $\\phi\n\\rightarrow \\infty $, has the form:%\n\\begin{equation}\nU_{\\text{asy}}(\\phi )=\\frac{{\\phi }^{2}}{2}+b\\,{\\phi }^{\\frac{6}{5}}+\\frac{%\n18\\,b^{2}\\,{\\phi }^{\\frac{2}{5}}}{25}-\\frac{1}{3}+\\frac{108\\,b^{3}}{625\\,{%\n\\phi }^{\\frac{2}{5}}}+O\\left( \\phi ^{-4\/5}\\right) \\,,  \\label{eq:fasy}\n\\end{equation}%\nin which $b$ is the only remaining arbitrary integration constant. The non\ntrivial (Wilson-Fisher \\cite{439}) fixed point solution which we are\ninterested in must interpolate between eqs. (\\ref{eq:fori}) and (\\ref%\n{eq:fasy}). Imposing these conditions fixes uniquely the value $b^{\\ast }$\nof $b$ which corresponds to the fixed point solution we are looking for$.$\n\nWe have determined $b^{\\ast }$ by using the shooting method \\cite{5465}:\nstarting from a value $\\phi _{a}$ supposed to be large where the condition (%\n\\ref{eq:fasy}) is imposed (with a guess, or trying, value of $b\\simeq\nb^{\\ast }$), we integrate the differential equation (\\ref{eq:FPF}) toward\nthe origin where the condition (\\ref{eq:fori}) is checked (shooting to the\norigin), we adjust the value of $b$ to $b^{\\ast }$ so as the latter\ncondition\\ is satisfied with a required accuracy. A study of the stability\nof the estimate of $b^{\\ast }$ so obtained on varying the value $\\phi _{a}$\nprovides some information on the accuracy of the calculation.\n\nRather than (\\ref{eq:fasy}), it is more usual to characterize the fixed\npoint solution from its small field behaviour:\n\n\\begin{equation}\nU(\\phi )=k-\\frac{3\\,k\\,}{2}\\phi ^{2}+\\frac{k\\,\\left( 1+3\\,k\\right) \\,}{4}%\n\\phi ^{4}-\\frac{k\\,\\left( 1+3\\,k\\right) \\,\\left( 1+24\\,k\\right) \\,}{120}\\phi\n^{6}+O\\left( \\phi ^{8}\\right) \\,,  \\label{eq:Uori}\n\\end{equation}%\nand to provide the value of either of the two (related) quantities: \n\\begin{eqnarray}\nk^{\\ast } &=&U^{\\ast }\\left( 0\\right) \\,,  \\label{eq:kstarWP} \\\\\nr^{\\ast } &=&U^{\\ast \\prime \\prime }\\left( 0\\right) =-3k^{\\ast }\\,.\n\\label{eq:rstar}\n\\end{eqnarray}\n\nIn the shooting-to-origin method, the determination of $r^{\\ast }$ (or $%\nk^{\\ast }$)\\ is a byproduct of the adjustment of $b^{\\ast }$.\n\nThe adjustment of $b^{\\ast }$ may be bypassed by shooting \\emph{from} the\norigin toward $\\phi _{a}$, then $r^{\\ast }$ is adjusted in such a way as to\nreach the largest possible value of $\\phi _{a}$. In that case $b^{\\ast }$ is\na byproduct of the adjustment.\n\nBecause the boundary condition at $\\phi _{a}$ is under control, the shooting-%\n\\emph{to}-origin method provides a better determination of $r^{\\ast }$ than\nthe shooting-\\emph{from}-origin method. However, this latter method is more\nflexible and may easily yield a rough estimate on $r^{\\ast }$ which can be\nused as a guess in a more demanding management of the method. Notice that,\ndue to the increase of the number of adjustable parameters, this way of\ndetermining a guess is no longer possible in a study involving several\ncoupled EDOs. Consequently, the development of other methods as, for\nexample, those two presented below is useful to this purpose (see also \\cite%\n{6201}).\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$r^{\\ast }$ &  & $b^{\\ast }$ &  & $\\phi _{a}$ \\\\ \\hline\n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,022\\,0$} &  & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 296\\,3$} &  & $10$ \\\\ \n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,021\\,9$} &  & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 311\\,6$} &  & $20$ \\\\ \n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,021\\,9$} &  & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 316\\,2$} &  & $40$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ The fixed point parameter $r^{\\ast }$ is already well determined\nfor rather small values of $\\protect\\phi _{a}$ whereas $b^{\\ast }$ [fixed\npoint value of $b$ in (\\protect\\ref{eq:fasy})] still is not. }\n\\label{Table 1}\n\\end{table}\n\nTable \\ref{Table 1} displays the determinations of $r^{\\ast }$ and $b^{\\ast\n} $ for three values of $\\phi _{a}$. One may observe that a high accuracy on \n$r^{\\ast }$ is required to reach a yet small value of $\\phi _{a}$ whereas $%\nb^{\\ast }$ is only poorly determined. Obviously, considering higher values\nof $\\phi _{a}$ and\/or higher order terms in eq. (\\ref{eq:fasy}) allows to\nbetter determine $b^{\\ast }$, one more term in (\\ref{eq:fasy}) and $\\phi\n_{a}=1000$ yields:%\n\\begin{equation}\nb^{\\ast }=-2.318\\,29\\,,  \\label{eq:bstar}\n\\end{equation}%\nbut the estimate of $r^{\\ast }$ is not improved compared to the values given\nin table \\ref{Table 1} (the machine-precision was already reached). We\nfinally extract from table \\ref{Table 1} our best estimate of $r^{\\ast }$\n(or $k^{\\ast }$) as obtained from the study of the fixed point equation (\\ref%\n{eq:FPF}) alone:%\n\\begin{eqnarray}\nr^{\\ast } &=&-0.228\\,598\\,202\\,437\\,022\\pm 10^{-15}\\,\\,,  \\label{eq:rstar1}\n\\\\\nk^{\\ast } &=&0.076\\,199\\,400\\,812\\,340\\,7\\pm 10^{-16}\\,.  \\label{eq:kstar1}\n\\end{eqnarray}\n\nIndividually, these values do not define the potential function $U^{\\ast\n}\\left( \\phi \\right) $ the knowledge of which requires the numerical\nintegration explicitly performed in the shooting method.\n\n\\subsubsection{Eigenvalue equation}\n\nThe critical exponents are obtained by linearizing the flow equation (\\ref%\n{eq:LPAV}) near the fixed point solution $U^{\\ast }\\left( \\phi \\right) $. If\none inserts:%\n\\begin{equation*}\nU\\left( \\phi ,t\\right) =U^{\\ast }\\left( \\phi \\right) +\\epsilon \\,e^{\\lambda\nt}g\\left( \\phi \\right) \\,,\n\\end{equation*}%\ninto the flow equation and keeps the linear terms in $\\epsilon $, one\nobtains the eigenvalue equation:%\n\\begin{equation}\ng^{\\prime \\prime }-2\\,g^{\\prime }U^{\\ast \\prime }-\\frac{\\phi }{2}\\,g^{\\prime\n}+\\left( 3-\\lambda \\right) \\,g=0\\,.  \\label{eq:VPeq}\n\\end{equation}\n\nAgain it is a second order ODE the solutions of which are characterized by\ntwo integration constants.\n\nSince $U^{\\ast }\\left( \\phi \\right) $ is an even function of $\\phi $, eq. (%\n\\ref{eq:VPeq}) is invariant under a parity change. Then one of the\nintegration constants is fixed by looking for either an even or an odd\neigenfunction $g\\left( \\phi \\right) $ which implies either $g^{\\prime\n}\\left( 0\\right) =0$ (even) or $g\\left( 0\\right) =0$ (odd). The second\nintegration constant is fixed at will due to the arbitrariness of the\nnormalisation of an eigenfunction. Thus, assuming either $g\\left( 0\\right)\n=1 $ (even) or $g^{\\prime }\\left( 0\\right) =1$ (odd), the solutions of (\\ref%\n{eq:VPeq}) depend only on $\\lambda $ and on the fixed point parameter $%\nk^{\\ast }$. For example, these solutions have the following expansions about\nthe origin $\\phi =0$ :\n\n\\begin{eqnarray*}\n&&g_{\\text{even}}\\left( \\phi \\right) =1+\\frac{\\left( \\lambda -3\\right) }{2}%\n\\phi ^{2}\\left[ 1+\\frac{\\,\\,\\left( \\lambda -2-12\\,k^{\\ast }\\right) }{12}\\phi\n^{2}\\right] +O\\left( \\phi ^{6}\\right) \\,, \\\\\n&&g_{\\text{odd}}\\left( \\phi \\right) =\\phi +\\frac{\\,\\left( 2\\,\\lambda\n-5-12\\,k^{\\ast }\\right) }{12}\\phi ^{3}+O\\left( \\phi ^{5}\\right) \\,.\n\\end{eqnarray*}%\nWhen the fixed point solution $U^{\\ast }$ is known, the values of $\\lambda $\n[the only remaining unknown parameter in (\\ref{eq:VPeq})] are determined by\nlooking for the solutions which interpolate between either $g^{\\prime\n}\\left( 0\\right) =0$ (even) or $g\\left( 0\\right) =0$ (odd) and the regular\nsolution of (\\ref{eq:VPeq}) \\bigskip which, for $\\phi \\rightarrow \\infty $,\nis:%\n\\begin{equation}\ng_{\\text{asy}}(\\phi )=S_{0}{\\phi }^{\\frac{2\\,\\left( 3-\\lambda \\right) }{5}%\n}\\left\\{ 1+\\left( 3-\\lambda \\right) \\left[ \\frac{12\\,b^{\\ast }\\,}{25\\,{\\phi }%\n^{\\frac{4}{5}}}-\\frac{36\\,b^{\\ast 2}\\,\\left( 2\\,\\lambda -3\\right) }{625\\,{%\n\\phi }^{\\frac{8}{5}}}+\\frac{2\\,\\,\\left( 2\\,\\lambda -1\\right) }{125\\,{\\phi }%\n^{2}}+O\\left( {\\phi }^{-\\frac{12}{5}}\\right) \\right] \\right\\} \\,,\n\\label{eq:gasy}\n\\end{equation}%\nin which $b^{\\ast }$ is given by (\\ref{eq:bstar}). The value of $S_{0}$ is\nrelated to the choice of the normalisation of the eigenfunction at the\norigin, it is a byproduct of the adjustment in a shooting-\\emph{from}-origin\nprocedure.\n\nIn the even case, it is known that the first nontrivial positive eigenvalue $%\n\\lambda _{1}$ (there is also the trivial value $\\lambda _{0}=d=3$), is\nrelated to the critical exponent $\\nu $ which characterizes the Ising-like\ncritical scaling of the correlation length $\\xi $. One has $\\nu =1\/\\lambda\n_{1}$ and the first negative eigenvalue, $\\lambda _{2}$, is minus the\nIsing-like first correction-to-scaling exponent $\\omega _{1}$ ($\\omega\n_{1}=-\\lambda _{2}$) and so on.\n\nIn the odd case, the two first (positive) eigenvalues are trivial in the\nLPA. One has:%\n\\begin{eqnarray}\n\\breve{\\lambda}_{1} &=&\\frac{d+2-\\eta }{2}\\,,  \\label{eq:lambdac1} \\\\\n\\breve{\\lambda}_{2} &=&\\frac{d-2+\\eta }{2}\\,,  \\label{eq:lambdac2}\n\\end{eqnarray}%\nin which $\\eta $ is the critical exponent which governs the large distance\nbehaviour of the correlation functions\\ right at the critical point, it\nvanishes in the LPA. With the dimension $d=3$ and the approximation (LPA)\npresently considered, (\\ref{eq:lambdac1}) and (\\ref{eq:lambdac2}) reduce to $%\n\\breve{\\lambda}_{1}=2.5$ and $\\breve{\\lambda}_{2}=0.5$. Consequently the\nfirst non-trivial eigenvalue is negative and defines the subcritical\nexponent $\\theta _{5}=\\breve{\\omega}_{1}=-\\breve{\\lambda}_{3}$ sometimes\nconsidered to characterize the deviation of the critical behaviour of fluids\nfrom the pure Ising-like critical behaviour.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$\\nu $ &  & $b^{\\ast }$ &  & $\\phi _{a}$ \\\\ \\hline\n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,11$} &  & \\multicolumn{1}{l}{$%\n-2.\\,\\allowbreak 318\\,145$} &  & \\multicolumn{1}{l}{$12$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,80$} &  & \\multicolumn{1}{l}{$%\n-2.318\\,257$} &  & \\multicolumn{1}{l}{$22$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,65$} &  & \\multicolumn{1}{l}{$%\n-2.318\\,280$} &  & \\multicolumn{1}{l}{$32$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,65$} &  & \\multicolumn{1}{l}{$%\n-2.318\\,285$} &  & \\multicolumn{1}{l}{$40$} \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Values of the critical exponent $\\protect\\nu $ determined together\nwith $b^{\\ast }$ (and thus $r^{\\ast }$) whereas $\\protect\\phi _{a}$ is\nvaried. Compared to table \\protect\\ref{Table 1}, a better determination of $%\nb^{\\ast }$ is obtained [see the best value of $b^{\\ast }$ \\ given by eq. (%\n\\protect\\ref{eq:bstar})].}\n\\label{Table 2}\n\\end{table}\n\nTo determine the eigenvalues we use again the shooting-to-origin method with\nthe two equations (\\ref{eq:FPF}, \\ref{eq:VPeq}). However, in addition to $%\n\\lambda $, we leave also $b^{\\ast }$ adjustable instead of fixing it to the\nvalue given in (\\ref{eq:bstar}).\n\nIn the even case, the values we obtain\\ for $\\nu $ and $b^{\\ast }$ are shown\nin table \\ref{Table 2} for four values of $\\phi _{a}$. Comparing with the\nvalues displayed in table \\ref{Table 1} one observes a better convergence of \n$b^{\\ast }$ to the best value (\\ref{eq:bstar}) whereas $r^{\\ast }$ remains\nunchanged compared to (\\ref{eq:rstar1}). As for the best estimate of $\\nu $,\nit is:%\n\\begin{equation}\n\\nu _{\\mathrm{best}}=0.649\\,561\\,773\\,880\\pm 10^{-12}\\,,  \\label{eq:nubest}\n\\end{equation}%\nthat is to say:%\n\\begin{equation}\n\\lambda _{1\\mathrm{best}}=1.539\\,499\\,459\\,808\\pm 10^{-12}\\,.\n\\label{eq:l1best}\n\\end{equation}\n\nWe have proceeded similarly to determine the Ising-like subcritical exponent\nvalues displayed in table \\ref{Table 3}.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccccccccc}\n\\hline\\hline\n$\\omega _{1}$ &  & $\\omega _{2}$ &  & $\\omega _{3}$ &  & $\\omega _{4}$ &  & $%\n\\omega _{5}$ &  & $\\omega _{6}$ \\\\ \\hline\n$0.655\\,745\\,939\\,193$ &  & $3.180\\,006\\,512\\,059$ &  & $5.912\\,230\\,612$ & \n& $8.796\\,092\\,825$ &  & $11.798\\,087\\,66$ &  & $14.896\\,053\\,176$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Best estimates of the six first subcritical exponents for the\nIsing-like scalar model (i.e. even case), all digits are significant.}\n\\label{Table 3}\n\\end{table}\n\nIn the odd case, we obtain: \n\\begin{equation}\n\\breve{\\omega}_{1}=1.886\\,703\\,838\\,091\\pm 10^{-12}\\,.  \\label{eq:omegac}\n\\end{equation}\n\nTable \\ref{Table 4} displays the values of the other subcritical exponents\nof the same family as $\\breve{\\omega}$ but with a lower accuracy. Of course,\nthe values presently obtained are in agreement with the previous estimates \n\\cite{3491,6137}.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$\\breve{\\omega}_{2}$ &  & $\\breve{\\omega}_{3}$ &  & $\\breve{\\omega}_{4}$ \\\\ \n\\hline\n$4.524\\,390\\,734$ &  & $7.337\\,650\\,643$ &  & $10.283\\,900\\,73$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Best estimates of the odd-case subcritical exponents other than $%\n\\breve{\\protect\\omega}_{1}$ for the scalar model.}\n\\label{Table 4}\n\\end{table}\n\n\\subsection{ Litim's flow equation for the scalar field \\label{Litim}}\n\nFollowing a conjecture first stated in \\cite{5049,5902}, the equivalence in\nthe LPA between the Wilson-Polchinski flow (\\ref{eq:LPAV}) and the Litim\noptimized ERGE \\cite{5020} for the running effective action $\\Gamma \\left[\n\\varphi ,t\\right] $ has been proven by Morris \\cite{5911}. The Litim flow\nequation for the potential $V\\left( \\varphi ,t\\right) $ reads in three\ndimensions (compared to \\cite{5911} an unimportant shift $V\\rightarrow V-1\/3$\nis performed):%\n\\begin{equation}\n\\dot{V}=1-\\frac{1}{1+V^{\\prime \\prime }}-\\frac{\\varphi }{2}V^{\\prime }+3V\\,.\n\\label{eq:LPAVLitim}\n\\end{equation}\n\nIt is related to (\\ref{eq:LPAV}) via the following Legendre transformation:%\n\\begin{equation}\n\\left. \n\\begin{array}{l}\n\\left[ \\frac{1}{2}\\phi ^{2}-U\\left( \\phi ,t\\right) \\right] +\\left[ \\frac{1}{2%\n}\\varphi ^{2}+V\\left( \\varphi ,t\\right) \\right] =\\varphi \\phi \\\\ \n\\varphi =\\phi -U^{\\prime }\\left( \\phi ,t\\right)%\n\\end{array}%\n\\right\\} \\,.  \\label{eq:Legendre}\n\\end{equation}\n\nThe general solution of the fixed point equation ($\\dot{V}=0$) involves the\nfollowing moving \\textquotedblleft singularity\\textquotedblright\\ ($%\nV^{\\prime \\prime }$ is singular)\\ at the arbitrary point $\\varphi _{0}$:%\n\\begin{equation}\nV_{\\text{sing}}\\left( \\varphi \\right) =-\\frac{1}{3}+\\frac{4}{3\\sqrt{\\varphi\n_{0}}}\\left\\vert \\varphi _{0}-\\varphi \\right\\vert ^{3\/2}\\,.  \\label{eq:singL}\n\\end{equation}\n\n\\subsubsection{Fixed point solution}\n\nThe numerical study of the fixed point solution of (\\ref{eq:LPAVLitim})\nfollows the lines described in the preceding sections. This may be done\nindependently, but due to (\\ref{eq:Legendre}), one may already deduce from\nthe previous study the expected results. Similarly to (\\ref{eq:fasy}), the\nasymptotic behaviour of the non trivial fixed point potential is\ncharacterized by the integration constant $b_{L}$ in the following\nexpression [deduced from (\\ref{eq:LPAVLitim})]:%\n\\begin{equation}\nV_{\\mathrm{asy}}(\\varphi )=b_{L}\\,{\\varphi }^{6}-\\frac{1}{3}+\\frac{1}{%\n150\\,b_{L}\\,{\\varphi }^{4}}-\\frac{1}{6300\\,b_{L}^{2}\\,{\\varphi }^{8}}%\n+O\\left( {\\varphi }^{-12}\\right) \\,.  \\label{eq:asyv}\n\\end{equation}\n\nIt is easy to show from (\\ref{eq:fasy}) and (\\ref{eq:Legendre}) that the\nvalue $b_{L}^{\\ast }$ we are looking for is related to $b^{\\ast }$ as\nfollows:%\n\\begin{equation*}\nb_{L}^{\\ast }\\,=-\\frac{1}{6^{6}}\\left( \\frac{5}{b^{\\ast }}\\right) ^{5},\n\\end{equation*}%\nthen, from the previous result (\\ref{eq:bstar}) we get:%\n\\begin{equation}\nb_{L}^{\\ast }\\simeq 0.001\\,000\\,25\\,.  \\label{eq:bstarL}\n\\end{equation}\n\nSimilarly for the potential parameters \n\\begin{eqnarray*}\nk_{L}^{\\ast } &=&V^{\\ast }\\left( 0\\right) \\,, \\\\\nr_{L}^{\\ast } &=&V^{\\ast \\prime \\prime }\\left( 0\\right) \\,,\n\\end{eqnarray*}%\nwhich correspond to $b_{L}^{\\ast }$, they are related to the\nWilson-Polchinski counterparts $k^{\\ast }$ and $r^{\\ast }$ as follows:%\n\\begin{eqnarray}\nk_{L}^{\\ast } &=&k^{\\ast }\\,,  \\label{eq:kstarL0} \\\\\nr_{L}^{\\ast } &=&\\frac{r^{\\ast }}{1-r^{\\ast }}\\,.  \\label{eq:rstarL0}\n\\end{eqnarray}%\nThis latter relation, using (\\ref{eq:rstar1}), gives:%\n\\begin{equation}\nr_{L}^{\\ast }\\simeq -0.186\\,064\\,249\\,470\\,314\\pm 10^{-15}\\,.\n\\label{eq:rstarL}\n\\end{equation}\n\nAs precedingly, those values do not provide the potential function $V^{\\ast\n}\\left( \\varphi \\right) $ the knowledge of which requires an explicit\nnumerical integration.\n\n\\subsubsection{Eigenvalue equation}\n\nA linearization of the flow equation (\\ref{eq:LPAVLitim}) near the fixed\npoint solution $V^{\\ast }\\left( \\varphi \\right) $:%\n\\begin{equation*}\nV\\left( \\varphi ,t\\right) =V^{\\ast }\\left( \\varphi \\right) +\\epsilon\n\\,e^{\\lambda t}h\\left( \\varphi \\right) \\,,\n\\end{equation*}%\nprovides the Litim eigenvalue equation:%\n\\begin{equation}\n\\left( 3-\\lambda \\right) \\,h-\\frac{\\varphi \\,h^{\\prime }}{2}+\\frac{h^{\\prime\n\\prime }}{{\\left( 1+{V^{\\ast }}^{\\prime \\prime }\\right) }^{2}}=0\\,.\n\\label{eq:VPeqLitim}\n\\end{equation}\n\nTaking into account (\\ref{eq:asyv}), one can show that (\\ref{eq:VPeqLitim})\nadmits a regular solution which, for $\\varphi \\rightarrow \\infty $, has the\nform:%\n\\begin{equation}\nh_{\\mathrm{asy}}\\left( \\varphi \\right) =S_{1}{\\varphi }^{2\\,\\left( 3-\\lambda\n\\right) }\\left\\{ 1-\\left( \\lambda -3\\right) \\,\\left( 2\\,\\lambda -5\\right) %\n\\left[ \\frac{1}{2250\\,b_{L}^{\\ast 2}\\,{\\varphi }^{10}}-\\frac{1}{%\n47250\\,b_{L}^{\\ast 3}\\,{\\varphi }^{14}}+O\\left( \\varphi ^{-18}\\right) \\right]\n\\right\\} \\,,  \\label{eq:hasy}\n\\end{equation}%\nin which $b_{L}^{\\ast }$ is given by (\\ref{eq:bstarL}). In the following we\nmay set $S_{1}=1$ since the normalisation of the eigenfunction may be chosen\nat will.\n\nAs precedingly, we must distinguish between the odd and even eigenfunction $%\nh\\left( \\varphi \\right) $. The shooting method gives the same values as in\nthe Wilson-Polchinski case (see \\cite{5049,5252,5625,6137}) and we do not\npresent them again.\n\n\\section{Expansion in powers of the field}\n\n\\label{WPE}\n\nIn advanced studies of the derivative expansion \\cite{5469} or other\nefficient approximations of the ERGE \\cite{5903} and in the consideration of\ncomplex systems via the ERGEs \\cite{5677}, a supplementary truncation in\npowers of the field is currently used (see also \\cite{4595}). With a scalar\nfield, this expansion transforms the partial differential flow equations\ninto ODEs whereas the fixed point or eigenvalue ODEs are transformed into\nalgebraic equations. Provided auxiliary conditions are chosen, the latter\nequations are easy to solve analytically using a symbolic computation\nsoftware. Actually the auxiliary conditions currently chosen are extremely\nsimple: they consist in setting equal to zero the highest terms of the\nexpansion so as to get a balanced system of equations.\n\nA first kind of expansion, about the zero field --referred to as the\nexpansion I in the following, has been proposed by Margaritis et al \\cite%\n{3478} and applied to the LPA of Wegner-Houghton's ERGE \\cite{414} (the hard\ncutoff version of the Wilson-Polchinski equation). A second kind of\nexpansion, relative to the (running) minimum of the potential (expansion\nII), has been proposed by Tetradis and Wetterich \\cite{3642} and more\nparticularly presented by Alford \\cite{4192} using it, again, with the sharp\ncutoff version of the ERGE.\n\nIt is known that, for the Wegner-Houghton equation in the LPA, expansion I\ndoes not converge due to the presence of singularities in the complex plane\nof the expansion variable \\cite{3358}. Expansions I and II have been more\nconcretely studied and compared to each other by Aoki et al in \\cite{3553}\nwho also propose a variant to II (expansion III) by letting the expansion\npoint adjustable. They showed, again on the LPA of\\ the Wegner-Houghton\nequation, that expansion II is much more efficient than expansion I although\nit finally does not converge and expansion III is the most efficient one.\nExpansions II and III work well also on the ERGE expressed on the running\neffective action (effective average action, see the review by Berges et al\nin \\cite{4595}). The convergence of those expansions have also been studied\nin \\cite{5252} according to the regularisation scheme chosen and in\nparticular for the Litim equation (\\ref{eq:LPAVLitim}). In this latter study\nit is concluded that both expansions I and II seem to converge although II\nconverges faster than I.\n\nA striking fact emerges from those studies, the Wilson-Polchinski equation\nin the LPA, the simplest equation, is never studied using the field\nexpansion method. The reason is simple: none of the expansions currently\nused works in that case.\n\nActually the strategy of these methods, which consists in arbitrarily\nsetting equal to zero one coefficient for the expansion I and two for the\nexpansions II and III, is probably too simple. With regards to this kind of\nauxiliary conditions, the failure observed with the Wilson-Polchinski\nequation is not surprising and, most certainly, there should be many other\ncircumstances where such simple auxiliary conditions would not solve\ncorrectly the derivative expansion of an ERGE.\n\nIn the following sections we examine two alternative methods with more\nsophisticated auxiliary conditions. We show that they yield the correct\nsolution for the Wilson-Polchinski and its Legendre transformed (Litim)\nequations. Both methods are associated to expansion I (about the\nzero-field). The first one has recently been proposed in \\cite{6110} as a\nmethod to treat the two point boundary value problem of ODEs. It relies upon\nan efficient account for the large field behaviour of the solution looked\nfor. An attempt of accounting for this kind of behaviour within the field\nexpansion had already been done by Tetradis and Wetterich via their eq.\n(7.11) of \\cite{3642}. In the present work, a much more sophisticated\nprocedure is used. It relies upon the construction of an added auxiliary\ndifferential equation (ADE). We refer to it in the following as the ADE\nmethod. The second method is new. It relies upon the approximation of the\nsolution looked for by a generalized hypergeometric function. We refer to it\nin the following as the\\ hypergeometric function approximation (HFA) method.\n\n\\section{Auxiliary differential equation method}\n\n\\label{BFG}\n\nLet us first illustrate the auxiliary differential equation (ADE) method on\nthe search for the non trivial fixed point in the LPA for both the\nWilson-Polchinski equation (\\ref{eq:FPF}) and the Litim optimized equation (%\n\\ref{eq:LPAVLitim}). Since there are two boundaries (the origin and the\n\"point at\" infinity), we distinguish between two strategies.\n\n\\begin{itemize}\n\\item An expansion about the origin in the equations (small field expansion)\nand the account for the leading high field behaviour of the regular solution\nwhich we are looking for. This determines the value of $r^{\\ast }$ or $%\nr_{L}^{\\ast }$.\n\n\\item A change of variable $\\phi \\rightarrow 1\/\\phi $ or $\\varphi\n\\rightarrow 1\/\\varphi $ which reverses the problem: an expansion about\ninfinity (new origin) in the equations (high field expansion) and the\naccount for the leading small field behaviour of the regular solution which\nwe are looking for. This determines the value of $b^{\\ast }$ or $b_{L}^{\\ast\n}$.\n\\end{itemize}\n\n\\subsection{Wilson-Polchinski's fixed point}\n\n\\subsubsection{ Small field expansion and leading high field behaviour\\label%\n{WPsmallField}}\n\nFor practical and custom reasons\\footnote{%\nThe change $x=\\phi ^{2}$ is useful in practice to avoid some degeneracies\nobserved in \\cite{6110} when forming the auxiliary differential equation.\nTaking the derivative $f=U^{\\prime }$ is only a question of habit.}, instead\nof (\\ref{eq:FPF}) we consider the equation satisfied by the function $%\nw\\left( x\\right) $ related to the derivative of the potential $U^{\\prime\n}\\left( \\phi \\right) $ as follows:\n\n\\begin{equation}\nU^{\\prime }\\left( \\phi \\right) =\\phi \\,w\\left( \\phi ^{2}\\right) \\,,\n\\label{eq:chgt1}\n\\end{equation}%\nso that, with $x=\\phi ^{2}$, the fixed point equation (\\ref{eq:FPF}) reads:\n\n\\begin{equation}\n4\\,x\\,w^{\\prime \\prime }-2\\,{w}^{2}-4\\,x\\,w\\,w^{\\prime }+\\left( 6-\\,x\\right)\n\\,\\,w^{\\prime }+2\\,w=0\\,,  \\label{eq:FPw}\n\\end{equation}%\nin which a prime indicates a derivative with respect to $x$.\n\nThis second order ODE has a singular point at the origin and, by analyticity\nrequirement, the solution we are looking for depends on a single unknown\nintegration-constant (noted $r$ below).\n\nLet us first introduce the expansion I of Margaritis et al \\cite{3478}. The\nfunction $w\\left( x\\right) $ is expanded up to order $M$ in powers of $x$:%\n\\begin{equation}\nw_{M}\\left( x\\right) =r+\\sum\\limits_{n=1}^{M}a_{n}x^{n}\\,,  \\label{eq:wm}\n\\end{equation}%\nand inserted into the fixed point equation (\\ref{eq:FPw}).\n\nRequiring that (\\ref{eq:FPw}) be satisfied order by order in powers of $x$\nprovides an unbalanced system of $M$ algebraic equations with $M+1$ unknown\nquantities $\\left\\{ r,a_{1},\\cdots ,a_{M}\\right\\} $ [eq. (\\ref{eq:FPw}) is\nthen satisfied up to order $M-1$ in powers of $x$]. With a view to balancing\nthe system, $a_{M}=0$ is simply set equal to zero and if the solution\ninvolves a stable value $r_{M}^{\\ast }$ as $M$ grows, then it constitutes\nthe estimate at order $M$ of the fixed point location corresponding to\nexpansion I. As already mentioned, in the case of the Wilson-Polchinski\nequation (\\ref{eq:FPw}) under study, the method fails: all the values\nobtained for $r_{M}^{\\ast }$ are positive whatever the value of $M$ whereas\nthe correct value should be negative as shown in section \\ref{FP1}.\n\nIn the ADE method, the condition $a_{M}=0$ is not imposed. The previous\nalgebraic system is first solved in terms of the unknown parameter $r$ \\ so\nas to get the generic solution of (\\ref{eq:FPw}) at order $M$ in powers of $%\nx $:%\n\\begin{equation}\nw_{M}\\left( r;x\\right) =r+\\sum\\limits_{n=1}^{M}a_{n}\\left( r\\right) x^{n}\\,.\n\\label{eq:wmrx}\n\\end{equation}\n\nIn order to get a definite value for $r$, instead of arbitrarily imposing $%\na_{M}\\left( r\\right) =0$, an auxiliary condition is formed which explicitly\naccounts for the behaviour at large $\\phi $ given by (\\ref{eq:fasy}). With $%\nw\\left( x\\right) $, this behaviour corresponds to:%\n\\begin{eqnarray}\n&&w_{\\text{asy}}(x)\\underset{x\\rightarrow \\infty }{=}1\\,,  \\label{eq:asyw} \\\\\n&&w_{\\text{asy}}^{\\prime }(x)\\underset{x\\rightarrow \\infty }{=}0\\,.\n\\label{eq:asyw'}\n\\end{eqnarray}%\nThe auxiliary condition is obtained via the introduction of an auxiliary\ndifferential equation:\n\n\\begin{itemize}\n\\item Consider a first order differential equation for $w\\left( x\\right) $\nconstructed as a polynomial of degree $s$ (eventually incomplete) in powers\nof the pair $\\left( w,w^{\\prime }\\right) $: \n\\begin{equation}\nG_{1}+G_{2}\\,w+G_{3}\\,w^{\\prime }+G_{4}\\,w^{2}+G_{5}\\,w\\,w^{\\prime\n}+G_{6}w^{\\prime 2}+\\cdots +G_{n}\\,w^{s-q}\\,w^{\\prime q}=0\\,,\n\\label{eq:auxil1}\n\\end{equation}%\nin which, when the degree $s$ of the polynomial is saturated then $q=s$ and\nthe number $n$ of coefficients $G_{i}$ is equal to $\\left( s+1\\right) \\left(\ns+2\\right) \/2$, conversely when it is not\\ then $0\\leq q<s$ and $%\nn=s(s+1)\/2+q+1$.\n\n\\item The constant coefficients $G_{i}$ are then determined as functions of $%\nr$ by imposing that the solution $w_{M}\\left( r;x\\right) $ of (\\ref{eq:FPw})\npreviously determined for arbitrary $r$ at order $M$ in powers of $x$ be\nalso solution of (\\ref{eq:auxil1}) (at the same order $M$). Due to an\narbitrary normalisation which allows to fix, for example $G_{1}=1$, a simple\ncounting shows that the identification implies $M=n-1$. The resulting set $%\n\\{G_{i}\\left( r\\right) ;i=2,\\ldots ,n\\}$ is formed of rational functions of\nthe unknown parameter $r$. Hence, a new differential equation for $w\\left(\nx\\right) $ is obtained:%\n\\begin{equation}\n1+G_{2}\\left( r\\right) \\,w+G_{3}\\left( r\\right) \\,w^{\\prime }+\\,G_{4}\\left(\nr\\right) \\,w^{2}+G_{5}\\left( r\\right) \\,w\\,w^{\\prime }+G_{6}\\left( r\\right)\n\\,w^{\\prime 2}+\\cdots +G_{n}\\left( r\\right) \\,w^{s-q}\\,w^{\\prime q}=0\\,,\n\\label{eq:auxilw}\n\\end{equation}%\nwhich is satisfied by construction at order $M$ in powers of $x$ by (\\ref%\n{eq:wmrx}) which is already solution at the same order of (\\ref{eq:FPw}).\n\n\\item \\bigskip The last step is then to impose that the new equation (\\ref%\n{eq:auxilw}) be also satisfied when $x\\rightarrow \\infty $. Taking into\naccount (\\ref{eq:asyw}, \\ref{eq:asyw'}) it comes the final auxiliary\ncondition:%\n\\begin{equation}\n1+G_{2}\\left( r\\right) \\,+\\,G_{4}\\left( r\\right) +\\cdots +G_{s\\left(\ns+1\\right) \/2+1}\\left( r\\right) \\,=0\\,.  \\label{eq:auxilCond}\n\\end{equation}\n\\end{itemize}\n\nSolving this auxiliary condition for $r$ amounts to determining the roots of\na polynomial in $r$. As the order $M$ grows some root values appear to be\nstable. Those stable values are candidates for the fixed point solutions we\nare looking for. In a way similar to \\cite{6201}, the obtention of the\nauxiliary condition may be obtained without determinating explicitly the\ncoefficient functions $G_{i}\\left( r\\right) $. For this, it is sufficient to\nconsider the matrix $\\mathcal{F}$ of the homogeneous system of linear\nequations for all the $G_{i}$'s formed with eq (\\ref{eq:auxil1}) to which is\nadded its expression when $x\\rightarrow \\infty $. When the function $w\\left(\nx\\right) $ is replaced by the expansion (\\ref{eq:wmrx}) at the required\norder the matrix $\\mathcal{F}$ depends only on the coefficients $a_{n}\\left(\nr\\right) $ of the Taylor expansion (\\ref{eq:wmrx}) and the auxiliary\ncondition then finally reduces to: \n\\begin{equation}\n\\det \\mathcal{F}=0  \\label{eq:detF}\n\\end{equation}\n\nBefore going further, it is worthwhile indicating that a variant of the\nmethod which consists in remplacing $w^{\\prime }$ by $xw^{\\prime }$ in the\nauxiliary differential equation (\\ref{eq:auxil1}) has appeared more\nefficient [e.g., see figure \\ref{fig5}]).\n\nFigure (\\ref{fig2}) shows the distribution of all the real roots $r_{M}$ of (%\n\\ref{eq:auxilCond}) for the variant as the order $M$ varies up to 28. The\nthree expected fixed points encountered in section (\\ref{FP1}) are clearly\nevidenced by a threefold accumulation about the respective values $1$ (HT)$,$\n$0$ (Gaussian) and $r^{\\ast }$ (Wilson-Fisher). Although a huge accumulation\nof roots around the right value occurs, the approach to $r^{\\ast }$, which\nwe are interested in, may be followed step by step as the order $M$ grows. \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure1.eps}\n\\end{center}\n\\caption{Distribution of the real roots $r_{M}$ (open circles) of (\\protect\n\\ref{eq:auxilCond}) as function of the order $M$ of the Taylor series about\nthe origin (\\protect\\ref{eq:wm}) [with the ADE pair $\\left( w,xw^{\\prime\n}\\right) $]. A threefold accumulation occurs about the expected fixed\npoints: trivial high temperature ($r^{\\ast }=1$) and Gaussian ($r^{\\ast }=0$%\n)\\ fixed points and about the non-trivial Wilson-Fisher fixed point ($%\nr^{\\ast }\\simeq -0.2286$) [LPA, d=3].}\n\\label{fig2}\n\\end{figure}\n\n\\paragraph{Selection of the root}\n\nTo select the right value of the root corresponding to the nontrivial\nWilson-Fisher fixed point, the following procedure has been applied. We know\nthat the root of interest is negative and real, then we select the first\nnegative real root that appears at the smallest possible order. At the next\norder we choose the real root the closest to the previous choice and so on.\nWe obtain this way with $M=28$ the following excellent estimate: \n\\begin{equation}\nr^{\\ast }=-0.228\\,598\\,202\\,437\\,02\\,,  \\label{eq:rstar2}\n\\end{equation}%\nwhich coincides, up to the 14$^{th}$ digit, with the estimate (\\ref%\n{eq:rstar1}) obtained by the shooting method. Figure \\ref{fig5} shows the\naccuracy obtained on $r^{\\ast }$ by selecting the roots this way as $M$\nvaries. \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure2.eps}\n\\end{center}\n\\caption{Approximate number of accurate digits $N_{d}=-\\log \\left\\vert\n1-r_{M}\/r^{\\ast }\\right\\vert $ obtained on the selected roots $r_{M}$ as a\nfunction of $M$ and for two ADE pairs; the original $\\left( w,w^{\\prime\n}\\right) $: crosses, and the variant $\\left( w,xw^{\\prime }\\right) $: open\ncircles [$r^{\\ast }$ is given in eq. (\\protect\\ref{eq:rstar1})]. The highest\nvalues of $M$ in each case is limited by time computing. A better efficiency\nis obtained with the variant.}\n\\label{fig5}\n\\end{figure}\n\n\\subsubsection{Subleading high field behaviour\\label{SubWP}}\n\nEquations (\\ref{eq:asyw}, \\ref{eq:asyw'}) used in the preceding calculations\nexpress exclusively the limit of the solution $w^{\\ast }\\left( x\\right) $\nwhen $x\\rightarrow \\infty $, and we get the unique condition (\\ref%\n{eq:auxilCond}) to estimate $r^{\\ast }$. In fact there are higher correction\nterms to (\\ref{eq:asyw}, \\ref{eq:asyw'}) which vanish as $x\\rightarrow\n\\infty $ [the first of which correspond to those written in (\\ref{eq:fasy}%\n)]. Such subleading contributions may as well be imposed in (\\ref{eq:auxilw}%\n). In so doing, we require the auxiliary differential equation to be\nsatisfied not only at infinity but also in approaching this point.\nConsequently we obtain several auxiliary conditions similar to (\\ref%\n{eq:auxilCond}), each of them corresponding to the cancellation of the\ncoefficient of a given power of $x.$ We have used them to determine $r^{\\ast\n}$ again (the asymptotic constant $b$ factorizes in the first subleading\nconditions so obtained). The results are similar to those obtained\nprecedently with the leading conditions (\\ref{eq:asyw}, \\ref{eq:asyw'})\nalone. We have observed only a slight decrease in the accuracy: the higher\nthe subleading term considered the weaker the convergence to $r^{\\ast }$.\nThis shows the coherence of the ADE method: the auxiliary condition is not\nan isolated point condition, it emanates from a differential equation\nconstructed to be satisfied by the \\emph{function} looked for.\n\nWhen the order of the subleading contribution is high enough, the constant $%\nb $ no longer factorizes and the subleading auxiliary condition depends non\ntrivialy on the (non-independent) integration constants ($r$ and $b$)\ncharacterizing the fixed point solution. We have tried to determine the\nvalue $b^{\\ast }$ by imposing the individual vanishing of such contributions\nfor $r=r^{\\ast }$. Unfortunately, at the orders considered, the only\nknowledge of $r^{\\ast }$ suffices to satisfy the condition (whatever the\nvalue of $b)$. It is possible that considering much higher orders would\nallow us to get an estimate of $b^{\\ast }$ this way.\n\n\\subsubsection{ High field expansion and leading small field behaviour}\n\nWith the determination of $b^{\\ast }$ by the ADE method in view, let us\nperform the change of variable $x\\rightarrow y^{-5}$ and the following\nchange of function:%\n\\begin{equation}\nu\\left( y\\right) =y^{-2}\\left[ w\\left( y^{-5}\\right) -1\\right] \\,,\n\\label{eq:chgt2}\n\\end{equation}%\nso that, from (\\ref{eq:fasy}) and (\\ref{eq:chgt1}), $u^{\\ast }\\left(\ny\\right) $ has the following form for small $y:$%\n\\begin{equation}\nu^{\\ast }\\left( y\\right) =A^{\\ast }+\\frac{\\,1}{5}A^{\\ast 2}{y}^{2}-\\frac{1}{%\n25}A^{\\ast 3}{y}^{4}+O\\left( y^{5}\\right) \\,,  \\label{eq:ustary}\n\\end{equation}%\nwith \n\\begin{equation}\nA^{\\ast }=6b^{\\ast }\/5\\,.  \\label{eq:astar}\n\\end{equation}\n\nThe\\ fixed point differential equation (\\ref{eq:FPF}) is then transformed\ninto:\n\n\\begin{equation}\n-10\\,y\\,{u}^{2}+5\\,\\left( 5+2\\,y^{5}\\right) \\,u^{\\prime }-4\\,\\left(\ny^{4}-5\\,y^{2}\\,u^{\\prime }\\right) \\,u+4\\,y^{6}\\,u^{\\prime \\prime }=0\\,,\n\\label{eq:WFPasy}\n\\end{equation}%\nthe solution of which must satisfy the following condition, see (\\ref%\n{eq:ustary}):%\n\\begin{eqnarray*}\nu^{\\ast }\\left( 0\\right) &=&A^{\\ast }\\,, \\\\\nu^{\\ast \\prime }\\left( 0\\right) &=&0\\,,\n\\end{eqnarray*}%\nwith $A^{\\ast }$ to be determined so as, using (\\ref{eq:wm}, \\ref{eq:chgt2}%\n), to get at infinity:\n\n\\begin{equation*}\nu_{\\text{asy}}^{\\ast }\\left( y\\right) \\underset{y\\rightarrow \\infty }{=}0\\,.\n\\end{equation*}\n\nThe ADE method described in the preceding sections is used to determine the\nvalue of $A^{\\ast }$. Since there are some holes in the first terms of the\nseries (\\ref{eq:ustary}), the first significant estimates are obtained for\nvalues of $M$ higher than in section \\ref{WPsmallField}. Figure (\\ref{fig3})\nshows that the selected sequence of roots corresponding to $A^{\\ast }$\nconverges to $-2.73532$ whereas, according to (\\ref{eq:bstar}, \\ref{eq:astar}%\n), the right value expected from the shooting method is $-2.78195$. \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure3.eps}\n\\end{center}\n\\caption{ In the Wilson-Polchinski case, $A^{\\ast }$ does not converge\n(dots) to the right value $-2.78195$ (horizontal line) but to $-2.73532$.}\n\\label{fig3}\n\\end{figure}\n\nThis failure of the ADE method in determining correctly $A^{\\ast }$ is\npresumably due to the zero radius of convergence of the Taylor series of $%\nu^{\\ast }\\left( y\\right) $ about $y=0$. Actually, we have estimated this\nradius as the limit of the ratio of two consecutive terms and observed that\nit goes slowly but continuously to zero as the order $M$ increases. This\ncontrasts with the case of $w\\left( x\\right) $ for which the same procedure\nquickly tends to the following finite limit for the fixed point solution\ncorresponding to (\\ref{eq:rstar1}, \\ref{eq:rstar2}):%\n\\begin{equation}\nR_{WP}=5.721\\,67\\,.  \\label{eq:RWP}\n\\end{equation}\n\nNotice that, although the ADE method does not provide the right estimate of $%\nA^{\\ast }$ (or $b^{\\ast }$), it gives a value close enough to it to be used\nas a guess in the shooting method.\n\n\\subsection{Litim's fixed point}\n\n\\subsubsection{ Small field expansion and leading high field behaviour}\n\nFor convenience we perform the following change, compared to section \\ref%\n{Litim}:%\n\\begin{equation}\nV\\left( \\varphi \\right) =\\bar{w}\\left( \\varphi ^{2}\\right) -\\frac{1}{3}\\,,\n\\label{eq:Translation}\n\\end{equation}%\nso that the fixed point equation corresponding to (\\ref{eq:LPAVLitim}) reads\n(with $\\bar{x}=\\varphi ^{2}$):\n\n\\begin{equation}\n3\\,\\bar{w}-\\bar{x}\\,\\bar{w}^{\\prime }-\\frac{1}{1+2\\,\\bar{w}^{\\prime }+4\\,%\n\\bar{x}\\,\\bar{w}^{\\prime \\prime }}=0\\,.  \\label{eq:wbarxbar}\n\\end{equation}\n\nThe singularity at $\\bar{x}=0$ of this second order ODE allows us to look\nfor an analytic solution which satisfies, in terms of a single unknown\nparameter $\\bar{k}$, the following conditions at the origin:\n\n\\begin{eqnarray}\n\\bar{w}\\left( 0\\right) &=&\\bar{k}\\,,  \\label{eq:wbarori1} \\\\\n\\bar{w}^{\\prime }\\left( 0\\right) &=&\\frac{1}{6\\bar{k}}-\\frac{1}{2}\\,,\n\\label{eq:wbarori2}\n\\end{eqnarray}%\nwith $\\bar{k}$ adjusted to $\\bar{k}^{\\ast }$ so as to reach at infinity\n[from (\\ref{eq:asyv})]:%\n\\begin{equation}\n\\bar{w}_{\\mathrm{asy}}^{\\ast }(\\bar{x})=b_{L}^{\\ast }\\,{\\bar{x}}^{3}+\\frac{1%\n}{150\\,b_{L}^{\\ast }\\,{\\bar{x}}^{2}}-\\frac{1}{6300\\,b_{L}^{\\ast 2}\\,{\\bar{x}}%\n^{4}}+O\\left( {\\bar{x}}^{-6}\\right) \\,.  \\label{eq:asywbar}\n\\end{equation}\n\nThe expected value of $\\bar{k}^{\\ast }$ is related to $r_{L}^{\\ast }$ given\nin (\\ref{eq:rstarL}) as:%\n\\begin{equation*}\n\\,\\bar{k}^{\\ast }=\\frac{1}{3\\left( 1+\\,r_{L}^{\\ast }\\right) }\\,.\n\\end{equation*}\n\nIt is also related to $k^{\\ast }$ given in (\\ref{eq:kstar1}) via (\\ref%\n{eq:kstarL0}, \\ref{eq:Translation}) as $\\bar{k}^{\\ast }=k^{\\ast }+1\/3$.\nConsequently the estimation by the shooting method is: \n\\begin{equation}\n\\,\\bar{k}^{\\ast }=0.409\\,532\\,734\\,\\allowbreak 145\\,674\\pm 10^{-15}\\,.\n\\label{eq:kstar}\n\\end{equation}\n\nThe object of this section is thus to test whether the ADE method yields\nthat value of $\\,\\bar{k}^{\\ast }$ [and also that of $b_{L}^{\\ast }$ given in\n(\\ref{eq:bstarL})].\n\nContrary to the Wilson-Polchinski case, the asymptotic behaviour (\\ref%\n{eq:asywbar}) does not reach a finite value when $\\bar{x}\\rightarrow \\infty $%\n. But the third derivative of $\\bar{w}^{\\ast }$ does. Hence, since $%\nb_{L}^{\\ast }$ is still supposed unknown, the auxiliary first order\ndifferential equation (\\ref{eq:auxil1}) may be used with $w$ and $w^{\\prime\n} $ replaced respectively by $\\bar{w}^{(4)}$ and $\\bar{w}^{(5)}$ (where $%\n\\bar{w}^{(n)}$ stands for d$^{n}\\bar{w}\/$d$\\bar{x}^{n}$). Actually, both of\nthese\\ two derivatives go to zero as $\\bar{x}\\rightarrow \\infty $ so that\nfinally the auxiliary condition similar to (\\ref{eq:auxilCond}), but with\nanother normalisation of the $G_{i}$'s (e.g. $G_{2}=1$), reduces to:%\n\\begin{equation}\nG_{1}\\left( \\bar{k}\\right) =0\\,,  \\label{eq:auxilCondL}\n\\end{equation}%\nwhereas the function $\\bar{w}\\left( \\bar{x}\\right) $ is expanded up to order \n$M$ in powers of $\\bar{x}$ and inserted into (\\ref{eq:wbarxbar}) to get the\nsolution at this order as function of $\\bar{k}$:%\n\\begin{equation}\n\\bar{w}_{M}\\left( \\bar{k};\\bar{x}\\right) =\\bar{k}+\\sum\\limits_{n=1}^{M}a_{n}%\n\\left( \\bar{k}\\right) \\bar{x}^{n}\\,.  \\label{eq:wbarm}\n\\end{equation}\n\nSimilarly to the Wilson-Polchinski case, the complete set of real roots of (%\n\\ref{eq:auxilCondL}) shows accumulations about the expected fixed point\nvalues. However the selection process described previously fails in picking\nthe right value $\\bar{k}^{\\ast }$ (of the nontrivial fixed point) although\nit is present among the roots. Actually, for $M=14$ the selection gives $%\n0.409\\,627\\,819\\,729\\,71$ whereas a better value ($0.409\\,532\\,733\\,212\\,35$%\n) exists at the same order [compare with (\\ref{eq:kstar})]. The variant\nutilised in the preceding case which consists in replacing $\\bar{w}^{(5)}$\nby $\\bar{x}\\bar{w}^{(5)}$ does not circumvents this difficulty.\n\nIf instead of $\\left( \\bar{w}^{(4)},\\bar{w}^{(5)}\\right) $ as ADE pair, we\nconsider the combination $h=3\\,\\bar{w}-\\bar{x}\\,\\bar{w}^{\\prime }$ and its\nderivative $h^{\\prime }$ with respect to $\\bar{x}$ (or the variant $\\bar{x}%\nh^{\\prime }$ to save some time computing), then the new pair, according to (%\n\\ref{eq:asywbar}), vanishes also as $\\bar{x}\\rightarrow \\infty $ , and we\nobserve, this time, that the selection process works again. This way, at\norder $M=19$ the selection gives:%\n\\begin{equation*}\n\\bar{k}^{\\ast }=0.409\\,532\\,734\\,16\\,,\n\\end{equation*}%\na value which coincides with (\\ref{eq:kstar}) up to the 10$^{th}$ digit. No\ndoubt that considering higher values of $M$ would have improved the\naccuracy. We note that, as with Wilson-Polchinski's function, the radius of\nconvergence of the Taylor series of $\\bar{w}\\left( \\bar{x}\\right) $ about\nthe origin is finite, and is about:%\n\\begin{equation}\nR_{L}\\simeq 11.5\\,.  \\label{eq:radiusL}\n\\end{equation}\n\nLet us specify however that, contrary to the Wilson-Polchinski case, the\ntest of the ratio $a_{i}\/a_{i+1}$ of two consecutive terms\\ of the Taylor\nseries about the origin does not converge. We have obtained (\\ref{eq:radiusL}%\n) by explicitly performing a partial summation of the series and studying it\nas a function of $\\bar{x}$. Nevertheless, we have also observed that the\nratio $\\left\\vert a_{i}\/a_{i+3}\\right\\vert $ raised to the power $1\/3$,\nroughly converges to (\\ref{eq:radiusL}). This remark will have some\nimportance in section \\ref{GeoLitim}.\n\nSince expansions I and II work in the Litim case (see \\cite{5252}), we can\ncompare the ADE method with those two methods. Figure (\\ref{fig1}) shows the\nrespective accuracies obtained on $\\bar{k}^{\\ast }$ with the three methods\nas functions of the order $M$ of the field expansion. One sees that\nexpansion II and the ADE method provide better results than expansion I\n(which likely does not converge) and that the ADE method is most efficient\nthan expansion II (we have not studied expansion III). \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure6.eps}\n\\end{center}\n\\caption{Approximate number of accurate digits $N_{d}=-\\log \\left\\vert 1-%\n\\bar{k}_{M}\/\\bar{k}^{\\ast }\\right\\vert $ [with $\\bar{k}^{\\ast }$ given by (%\n\\protect\\ref{eq:kstar})] as functions of $M$ for the estimations of the\nLitim fixed point value of $\\bar{k}$ using three methods: expansion I (black\ndots), expansion II (crosses), and ADE (open circles). A better efficiency\nis obtained with the ADE method.}\n\\label{fig1}\n\\end{figure}\n\n\\subsubsection{Subleading high field behaviour}\n\nAs in the case of Wilson-Polchinski's equation, the subleading terms in (\\ref%\n{eq:asywbar}) may be used to impose the auxiliary condition not only at\ninfinity but also in approaching this point whatever the value of $\\bar{x}$.\nWe observe the same phenomenon as in section \\ref{SubWP}: the higher the\nsubleading term considered the weaker the convergence to $\\bar{k}^{\\ast }$\nwhereas $b_{L}^{\\ast }$ cannot be determined by imposing the individual\nvanishing of the subleading contributions for $\\bar{k}=\\bar{k}^{\\ast }$.\n\nHowever, the fact that the asymptotic behaviour (\\ref{eq:asywbar}) is an\ninteger power of $\\bar{x}$ provides us with the oportunity of determining $%\nb_{L}^{\\ast }$ from the knowledge of $\\bar{k}^{\\ast }$ as a boundary limit\n(a point condition). Actually, since $\\bar{w}^{(3)}\\rightarrow 6b_{L}^{\\ast\n}\\,$ when $\\bar{x}\\rightarrow \\infty $, we may choose $\\left( \\bar{w}^{(3)},%\n\\bar{w}^{(4)}\\right) $ as ADE pair [or the variant $\\left( \\bar{w}^{(3)},%\n\\bar{x}\\bar{w}^{(4)}\\right) $], and for $\\bar{k}$ fixed to $\\bar{k}^{\\ast }$\nsolve for $b_{L}^{\\ast }$ the resulting auxiliary condition at infinity. The\naccuracy on $b_{L}^{\\ast }$ obtained this way is not as large as in the case\nof $\\bar{k}^{\\ast }$, nevertheless, for $M=31$ we obtain the following\nestimation:%\n\\begin{equation}\nb_{L}^{\\ast }\\simeq 0.001\\,007\\,,  \\label{eq:bstarLsub}\n\\end{equation}%\nwhich is rather close to the shooting value (\\ref{eq:bstarL}). We indicate\nalso that rough estimates of $b_{L}^{\\ast }$ already sufficiently accurate\nto be used as guesses in the shooting method are obtained for small values\nof $M$, e.g.: $0.000\\,989$ for $M=11$ or even $0.0012$ for $M=5$.\n\n\\subsubsection{ High field expansion and leading small field behaviour}\n\nWith a view to determining $b_{L}^{\\ast }$ directly by the ADE method, we\ninvert the boundaries by changing the variable $\\bar{x}\\rightarrow \\bar{y}%\n^{-1}$ and by performing the following change of function:%\n\\begin{equation}\n\\bar{u}\\left( \\bar{y}\\right) ={\\bar{y}}^{3}\\bar{w}\\left( \\frac{1}{\\bar{y}}%\n\\right) \\,,  \\label{eq:ubar}\n\\end{equation}%\nso that, from (\\ref{eq:asywbar}), we deduce that $\\bar{u}^{\\ast }\\left( \\bar{%\ny}\\right) $ has the following form for small $\\bar{y}:$%\n\\begin{equation}\n\\bar{u}^{\\ast }\\left( \\bar{y}\\right) =b_{L}^{\\ast }\\,+\\frac{{\\bar{y}}^{5}}{%\n150\\,b_{L}^{\\ast }\\,}-\\frac{{\\bar{y}}^{7}}{6300\\,b_{L}^{\\ast 2}\\,}+O\\left( {%\n\\bar{y}}^{9}\\right) \\,.  \\label{eq:ubarstarori}\n\\end{equation}\n\nThe differential equation for $\\bar{u}\\left( \\bar{y}\\right) $ is:\n\n\\begin{equation*}\n\\bar{y}^{4}+18\\,\\bar{y}\\,\\left( \\bar{u}^{\\prime }\\right) ^{2}-\\bar{u}%\n^{\\prime }\\left[ 30\\,\\bar{u}+\\bar{y}^{2}\\,\\left( 1+4\\,\\bar{u}^{\\prime \\prime\n}\\right) \\right] =0\\,.\n\\end{equation*}%\n\\bigskip\n\nThe solution must satisfy the following condition at the origin $\\bar{y}=0$\n[see (\\ref{eq:ubarstarori})]:%\n\\begin{eqnarray*}\n\\bar{u}^{\\ast }\\left( 0\\right) &=&b_{L}^{\\ast }\\,, \\\\\n\\bar{u}^{\\ast \\prime }\\left( 0\\right) &=&0\\,,\n\\end{eqnarray*}%\nwith $b_{L}^{\\ast }$ to be determined so as, using (\\ref{eq:wbarori1}, \\ref%\n{eq:wbarori2}, \\ref{eq:ubar}), to get at infinity:\n\n\\begin{equation*}\n\\bar{u}_{\\text{asy}}^{\\ast }\\left( \\bar{y}\\right) =\\bar{k}^{\\ast }{\\bar{y}}%\n^{3}+\\left( \\frac{1}{6\\bar{k}^{\\ast }}-\\frac{1}{2}\\right) {\\bar{y}}%\n^{2}+O\\left( \\bar{y}\\right) \\,.\n\\end{equation*}\n\nAs previously, we use the ADE method with a view to determining the value of \n$b_{L}^{\\ast }\\,$. For this we consider, the pair $\\left( \\bar{u}^{(4)},\\bar{%\nu}^{(5)}\\right) $ which vanishes at infinity ($\\bar{y}\\rightarrow \\infty $).\nSince there are some holes in the first terms of the series about the\norigin, see (\\ref{eq:ubarstarori}), the first significant estimates are\nobtained for values of $M$ higher than with the original function $\\bar{w}%\n\\left( \\bar{x}\\right) $. Although the positive roots obtained for $b_{L}$\n(we know that $b_{L}^{\\ast }$ is positive)\\ have the right order of\nmagnitude compared to (\\ref{eq:bstarL}) the apparent convergent sequences do\nnot provide the right value. Again, as in the Wilson-Polchinski case, we\nthink that the failure of the ADE method is due to the (observed) zero\nradius of convergence of the Taylor series for $\\bar{u}\\left( \\bar{y}\\right) \n$ about the origin.\n\n\\subsection{Eigenvalue estimates}\n\nLet us consider the eigenvalue problem with the ADE method. This time two\ncoupled nonlinear ODEs have to be solved together (the fixed point equation\nand the linearisation of the flow in the vicinity of the fixed point). We\ncan solve these two equations together as the order of the field expansion $%\nM $ grows or consider separately the eigenvalue equation after having solved\nthe fixed point equation with some accuracy. With the aim to be short, we\npresent only the latter possibility which illustrates well the property of\nconvergence of the method.\n\n\\subsubsection{Wilson-Polchinski's eigenvalues}\n\n\\paragraph{ Small field expansion and leading high field behaviour}\n\nUsing a change of eigenfunction, $g\\rightarrow \\mathrm{v}$, similar to (\\ref%\n{eq:chgt1}) for the fixed point function, it comes:\n\n\\begin{itemize}\n\\item in the even case:%\n\\begin{equation*}\ng^{\\prime }\\left( \\phi \\right) =\\phi \\,\\mathrm{v}\\left( \\phi ^{2}\\right) \\,,\n\\end{equation*}\n\nand eq. (\\ref{eq:gasy}) yields the following behaviour at large $x=\\phi ^{2}$%\n:%\n\\begin{equation*}\n\\mathrm{v}_{\\text{asy}}\\left( x\\right) =\\frac{2\\left( 3-\\lambda \\right) }{5}%\nS_{0}\\,x^{-\\left( 2+\\lambda \\right) \/5}\\left[ 1+O\\left( x^{-2\/5}\\right) %\n\\right] \\,.\n\\end{equation*}\n\n\\item in the odd case:%\n\\begin{equation*}\ng^{\\prime }\\left( \\phi \\right) =\\,\\mathrm{v}\\left( \\phi ^{2}\\right) \\,,\n\\end{equation*}%\nand eq. (\\ref{eq:gasy}) gives:%\n\\begin{equation*}\n\\mathrm{v}_{\\text{asy}}\\left( x\\right) =\\frac{2\\left( 3-\\lambda \\right) }{5}%\nS_{0}\\,x^{\\left( 1-2\\lambda \\right) \/10}\\left[ 1+O\\left( x^{-2\/5}\\right) %\n\\right] \\,.\n\\end{equation*}\n\\end{itemize}\n\nThe arbitrariness of the global normalisation of the eigenfunctions allows\nto choose $\\mathrm{v}\\left( 0\\right) =1$ (even) and $\\mathrm{v}^{\\prime\n}\\left( 0\\right) =1$ (odd) corresponding respectively to some definite\nvalues of $S_{0}$. So defined, the functions $\\mathrm{v}\\left( x\\right) $\nand $\\mathrm{v}^{\\prime }\\left( x\\right) $ vanish at infinity provided that $%\n\\lambda >-2$ in the even case and $\\lambda >1\/2$ in the odd case. Hence one\ncould expect that, with the simple condition at infinity: $\\mathrm{v}=%\n\\mathrm{v}^{\\prime }=0$ imposed in the auxiliary differential equation, the\nADE procedure will, at best, allow the determination of exclusively the\nleading ($\\lambda _{1}=1\/\\nu $) and first subleading ($\\lambda _{2}=-\\omega\n_{1}$) eigenvalues in the even case and of only the trivial eigenvalue $%\n\\breve{\\lambda}_{1}=-\\breve{\\omega}_{1}$ in the odd case [see the values of\nthese quantities in eqs. (\\ref{eq:l1best}, \\ref{eq:omegac}) and tables (\\ref%\n{Table 3}, \\ref{Table 4})]. Actually it is better than that since, as $M$\ngrows, we observe\\ among the real roots of the auxiliary condition for $%\n\\lambda $ that a hierarchy of successive accumulations takes place about the\nright values of the leading and subsequent eigenvalues [see figure \\ref{fig4}%\n]. \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure4.eps}\n\\end{center}\n\\caption{Accumulations of real roots (open circles)\\ of the auxiliary\ncondition (\\protect\\ref{eq:auxilCond}) about eigenvalues as the order $M$\\\nof the series varies in the Wilson-Polchinski even case. From top to bottom: \n$\\protect\\lambda _{1}=1\/\\protect\\nu $ (second horizontal line), $\\protect%\n\\lambda _{2}=-\\protect\\omega _{1}$ (third h. line) and $\\protect\\lambda %\n_{3}=-\\protect\\omega _{2}$ (fourth h. line). A simple criterion of choice\nallows to determine their estimates at $M=20$, see the values in eqs. (%\n\\protect\\ref{eq:WPvpnu}--\\protect\\ref{eq:WPvpomega2}). An accumulation also\noccurs about the spurious value 5.8 (first h. line).}\n\\label{fig4}\n\\end{figure}\n\nWithin each of these accumulations of real roots, we have been able to\nfollow without ambiguity a convergent sequence to the right estimate. At\norder $M=20$ with the ADE pair $\\left( \\mathrm{v},\\mathrm{v}^{\\prime\n}\\right) $ supposed to vanish at infinity, and $r^{\\ast }$ fixed to the\nvalue given in (\\ref{eq:rstar2}), we have obtained the following estimates\nin the even case\n\n\\begin{eqnarray}\n\\nu &=&0.649\\,561\\,773\\,86\\,,  \\label{eq:WPvpnu} \\\\\n\\omega _{1} &=&0.655\\,745\\,92\\,,  \\label{eq:WPvpomega1} \\\\\n\\omega _{2} &=&3.178\\,,  \\label{eq:WPvpomega2}\n\\end{eqnarray}%\nwhere the number of digits has been truncated with regard to the accuracy of\nthe estimates obtained [by comparison with (\\ref{eq:nubest}) and table \\ref%\n{Table 3}]. We see that the accuracy decreases as the order of the\neigenvalue grows but also that we obtain an estimate of $\\omega _{2}$\nwhereas for that value $\\mathrm{v}$ does not vanish at infinity.\n\nThe same kind of observations stands in the odd case. We take the\nopportunity to indicate that choosing the ADE pair $\\left( f,f^{\\prime\n}\\right) $ with $f=$ $\\frac{\\left( 1-2\\lambda \\right) }{10}\\mathrm{v}$-$x%\n\\mathrm{v}^{\\prime }$ makes \\ $f$ vanish for $\\lambda >-3\/2$ and the\nprocedure gives a better accuracy on $\\breve{\\omega}_{1}$ than with the pair \n$\\left( \\mathrm{v},\\mathrm{v}^{\\prime }\\right) $. This way we obtain the\nfollowing estimate [at order $M=20,$ compare with (\\ref{eq:omegac})]%\n\\begin{equation*}\n\\breve{\\omega}_{1}=1.886\\,718\\,.\n\\end{equation*}\n\nWe have also noted the presence of accumulations of real roots about\nspurious positive values of order 5.8 in the even case and 3.77 in the odd\ncase.\n\n\\subsubsection{Litim's eigenvalues}\n\nThe determination using the ADE method of the eigenvalues from the Litim\nflow equation follows the same lines as previously for the Wilson-Polchinski\nflow equation. We limit ourselves in this section to a brief presentation of\nthe main differences encountered.\n\n\\paragraph{ Small field expansion and leading high field behaviour}\n\nCompared to (\\ref{eq:VPeqLitim}), we perform a change of eigenfunction, $%\nh\\rightarrow \\,\\mathrm{v}_{L}$, according to the symmetry considered:\n\n\\begin{itemize}\n\\item in the even case:%\n\\begin{equation*}\nh\\left( \\varphi \\right) =\\,\\mathrm{v}_{L}\\left( \\varphi ^{2}\\right) \\,,\n\\end{equation*}\n\nthen eq. (\\ref{eq:hasy}) yields the following behaviour at large $\\bar{x}%\n=\\varphi ^{2}$:%\n\\begin{equation*}\n\\mathrm{v}_{L\\text{asy}}\\left( \\bar{x}\\right) =S_{1}{\\bar{x}}^{\\,\\left(\n3-\\lambda \\right) }\\left[ 1+O\\left( \\bar{x}^{-5}\\right) \\right] \\,.\n\\end{equation*}\n\n\\item in the odd case:%\n\\begin{equation*}\nh\\left( \\varphi \\right) =\\varphi \\,\\mathrm{v}_{L}\\left( \\varphi ^{2}\\right)\n\\,,\n\\end{equation*}%\nand eq. (\\ref{eq:hasy}) gives:%\n\\begin{equation*}\n\\mathrm{v}_{L\\text{asy}}\\left( \\bar{x}\\right) =S_{1}{\\bar{x}}^{\\,\\left(\n5\/2-\\lambda \\right) }\\left[ 1+O\\left( \\bar{x}^{-5}\\right) \\right] \\,.\n\\end{equation*}\n\\end{itemize}\n\nSo defined, the two functions $\\mathrm{v}_{L}\\left( \\bar{x}\\right) $ and $%\n\\mathrm{v}_{L}^{\\prime }\\left( \\bar{x}\\right) $ vanish at infinity provided\nthat $\\lambda >3$ in the even case and $\\lambda >5\/2$ in the odd case\n(whereas the arbitrary global normalisation of the eigenfunctions allows to\nchoose $\\mathrm{v}_{L}\\left( 0\\right) =1$ (even) and $\\mathrm{v}_{L}^{\\prime\n}\\left( 0\\right) =1$ (odd) corresponding respectively to specific values of $%\nS_{1}$).\n\nAlthough it works, the original ADE pair $\\left( \\mathrm{v}_{L},\\mathrm{v}%\n_{L}^{\\prime }\\right) $ is not the most efficient choice to obtain estimates\nof the first nontrivial eigenvalues. A better choice appears to be the pairs \n$\\left( f\\left( \\bar{x}\\right) ,f^{\\prime }\\left( \\bar{x}\\right) \\right) $\nwith $f\\left( \\bar{x}\\right) =\\left( 3-\\lambda \\right) \\mathrm{v}_{L}\\left( \n\\bar{x}\\right) -\\bar{x}\\mathrm{v}_{L}^{\\prime }\\left( \\bar{x}\\right) $ in\nthe even case and $f\\left( \\bar{x}\\right) =$ $\\left( 5\/2-\\lambda \\right) \n\\mathrm{v}_{L}\\left( \\bar{x}\\right) -\\bar{x}\\mathrm{v}_{L}^{\\prime }\\left( \n\\bar{x}\\right) $ in the odd case (they correspond to eigenfunctions which\nvanish as $\\bar{x}\\rightarrow \\infty $ for more negative values of $\\lambda $%\n). With these choices and $\\bar{k}^{\\ast }=0.409\\,532\\,734\\,145\\,7$ we\nidentify immediately the trivial eigenvalues $\\lambda _{0}=3$ in the even\ncase and $\\breve{\\lambda}_{1}=2.5$, $\\breve{\\lambda}_{2}=0.5$ in the odd\ncase but also, for $M=20$, we obtain good estimates of the nontrivial\nleading and first subleading eigenvalues:%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =\\allowbreak 0.649\\,561\\,774,\\quad & \\omega _{1}=0.655\\,745\\,5,\\quad & \n\\omega _{2}=3.180\\,008,\\quad & \\omega _{3}=5.896, \\\\ \n\\breve{\\omega}_{1}=1.886\\,703\\,7, & \\breve{\\omega}_{2}=4.524\\,1\\,, &  & \n\\end{array}%\n\\end{equation*}%\nwhere the numbers of digits have been limited with respect to the estimated\naccuracy [compare with (\\ref{eq:nubest}), table \\ref{Table 3} (even) and (%\n\\ref{eq:omegac}), table \\ref{Table 4} (odd)]. For each eigenvalue, the\nsuccessive estimates may be followed unambiguously step by step when $M$\ngrows so that the right values may be easily selected following the rules\ndefined\\ precedently.\n\nWe notice also the presence of spurious convergences and especially in the\neven case to the value about 5.8 already encountered with the\nWilson-Polchinski case.\n\n\\section{Approximating by hypergeometric functions (HFA)}\n\n\\label{Ratios}\n\nThe ADE method is most certainly efficient in many cases but it is\nrelatively heavy regarding the computing time whereas the current methods,\nwhen they work, are lighter. In addition, none of these methods provides a\nglobal solution to the ODE studied: they yield an approximate value of the\nintegration constant but not a function as global approximation of the\nsolution looked for.\n\nWe propose in this section an alternative method which is lighter than the\nADE method and which provides a global approximation of the solution of\ninterest. This new method is based on the definition property of the\ngeneralized hypergeometric functions. Let us first review the definition and\nmain properties of these functions.\n\n\\subsection{Generalized hypergeometric functions\\label{Brief}}\n\nFor $x\\in \n\\mathbb{C}\n$, a series $S=\\sum_{n=0}^{\\infty }a_{n}x^{n}$ is hypergeometric (see for\nexample \\cite{6181}) if the ratio $a_{n+1}\/a_{n}$ is a rational function of $%\nn$, i.e.%\n\\begin{equation*}\n\\frac{a_{n+1}}{a_{n}}=\\frac{P\\left( n\\right) }{Q\\left( n\\right) }\\,,\n\\end{equation*}%\nfor some polynomials $P\\left( n\\right) $ and $Q\\left( n\\right) $.\n\nIf we factorize the polynomials, we can write:%\n\\begin{equation}\n\\frac{a_{n+1}}{a_{n}}=\\alpha _{0}\\frac{\\left( n+\\alpha _{1}\\right) \\left(\nn+\\alpha _{2}\\right) \\cdots \\left( n+\\alpha _{p}\\right) }{\\left( n+\\beta\n_{1}\\right) \\left( n+\\beta _{2}\\right) \\cdots \\left( n+\\beta _{q}\\right)\n\\left( n+1\\right) }\\,.  \\label{eq:PolyFac}\n\\end{equation}%\nThe factor $\\left( n+1\\right) $ in the denominator may or may not result\nfrom the factorization. If not, we add it along with the compensating factor\nin the numerator. Usually, the global factor $\\alpha _{0}$ is set equal to 1.\n\nIf the set $\\left\\{ \\alpha _{i}\\right\\} $ includes negative integers, then $%\nS $ degenerates into a polynomial in $x.$\n\nWhen it is not a polynomial, the series $S$ converges absolutely for all $x$\nif $p\\leq q$ and for $\\left\\vert x\\right\\vert <1\/\\left\\vert \\alpha\n_{0}\\right\\vert $ if $p=q+1$. It diverges for all $x\\neq 0$ if $p>q+1.$\n\nThe analytic continuation of the hypergeometric series $S$ with a non-zero\nradius of convergence is called a generalized hypergeometric function and is\nnoted:%\n\\begin{equation*}\n_{p}F_{q}\\left( \\alpha _{1},\\cdots ,\\alpha _{p};\\beta _{1},\\cdots ,\\beta\n_{q};\\alpha _{0}x\\right) =\\frac{1}{a_{0}}S\\,.\n\\end{equation*}\n\n$_{p}F_{q}\\left( x\\right) $ is a solution of the following differential\nequation (for $\\alpha _{0}=1$):%\n\\begin{equation}\n\\left[ \\theta \\left( \\theta +\\beta _{1}-1\\right) \\cdots \\left( \\theta +\\beta\n_{q}-1\\right) -x\\left( \\theta +\\alpha _{1}\\right) \\cdots \\left( \\theta\n+\\alpha _{p}\\right) \\right] \\,_{p}F_{q}\\left( x\\right) =0\\,,\n\\label{eq:EDOhyper}\n\\end{equation}%\nwhere%\n\\begin{equation*}\n\\theta =x\\frac{d}{dx}.\n\\end{equation*}\n\nWhen $p>2$ or $q>1$, the differential equation (\\ref{eq:EDOhyper}) is of\norder $\\max \\left( p,q+1\\right) >2$. It is of second order when $q=1$ and $%\np=0$, $1$ or $2$. It is of first order when $q=0$ and $p=1$\n\n$_{2}F_{1}$ is currently named the hypergeometric function. A number of\ngeneralized hypergeometric functions have also special names: $_{0}F_{1}$ is\ncalled confluent hypergeometric limit function and $_{1}F_{1}$ confluent\nhypergeometric function.\n\nIn the cases $p\\leq q$ for fixed $\\left\\{ \\alpha _{i}\\right\\} $ and $\\left\\{\n\\beta _{i}\\right\\} $, $_{p}F_{q}\\left( x\\right) $ is an entire function of $%\nx $ and has only one (essential) singular point at $x=\\infty $.\n\nFor $p=q+1$ and fixed $\\left\\{ \\alpha _{i}\\right\\} $ and $\\left\\{ \\beta\n_{i}\\right\\} $ in non-polynomial cases\\textbf{, }$_{p}F_{q}\\left( x\\right) $\ndoes not have pole nor essential singularity. It is a single-valued function\non the $x$-plane cut along the interval $\\left[ 1,\\infty \\right] $, i.e. it\nhas two branch points at $x=1$ and at $x=\\infty $.\n\nConsidered as a function of $\\left\\{ \\beta _{i};i=1,\\cdots ,q\\right\\} $, $%\n_{p}F_{q}\\left( x\\right) $ has an infinite set of singular points:\n\n\\begin{enumerate}\n\\item $\\beta _{i}=-m$, $m\\in \n\\mathbb{N}\n$ which are simple poles\n\n\\item $\\beta _{i}=\\infty $\\ which is an essential singular point (the point\nof accumulation of the poles).\n\\end{enumerate}\n\nAs a function of $\\left\\{ \\alpha _{i};i=1,\\cdots ,p\\right\\} $, $%\n_{p}F_{q}\\left( x\\right) $ has one essential singularity at each $\\alpha\n_{i}=\\infty $.\n\nThe elementary functions and several other important functions in\nmathematics and physics are expressible in terms of hypergeometric functions\n(for more detail see \\cite{6181}).\n\nThe wide spread of this family of functions suggests trying to represent the\nsolution of the ODEs presently of interest in this article, under the form\nof a generalized hypergeometric function.\n\n\\subsection{\\textbf{\\ }The HFA method\\label{HFA}}\n\nFor the sake of the introduction of the new method, let us first consider\nthe Wilson-Polchinski fixed point equation (\\ref{eq:FPw}) and the truncated\nexpansion (\\ref{eq:wmrx}) in which the coefficients $a_{n}\\left( r\\right) $ $%\n(n=1,\\cdots ,M)$ are already determined as function of $r$ via a generic\nsolution of (\\ref{eq:FPw}) truncated at order $M$ (in powers of $x)$. The\nquestion is again to construct an auxiliary condition to be imposed\\ with a\nview to determining the fixed point value $r^{\\ast }$. To this end, by\nanalogy with the generalized hypergeometric property definition recalled in\nsection \\ref{Brief}, we construct the ratio of two polynomials in $n$:%\n\\begin{equation}\n\\frac{P_{m_{1}}\\left( n\\right) }{Q_{m_{2}}\\left( n\\right) }=\\frac{%\n\\sum_{i=1}^{m_{1}}c_{i}\\,n^{i-1}}{\\sum_{i=1}^{m_{2}}d_{i}\\,n^{i-1}}\\,,\n\\label{eq:hyper0}\n\\end{equation}%\nso that $P_{m_{1}}\\left( n\\right) \/Q_{m_{2}}\\left( n\\right) $ \\ match the $%\nM-2$ ratios $a_{n+1}\\left( r\\right) \/a_{n}\\left( r\\right) $ \\ for $%\nn=1,\\cdots ,M-2$. Hence, accounting for the arbitrariness of the global\nnormalisation of (\\ref{eq:hyper0}), the complete determination of the two\nsets of coefficients $\\left\\{ c_{i};i=1,\\cdots ,m_{1}\\right\\} $ and $\\left\\{\nd_{i};i=1,\\cdots ,m_{2}\\right\\} $ as functions of $r$ implies $%\nm_{1}+m_{2}=M-1$.\\ Finally, the auxiliary condition on $r$ is obtained by\nrequiring that the last (still unused) ratio $a_{M}\\left( r\\right)\n\/a_{M-1}\\left( r\\right) $ satisfies again the $n$-dependency satisfied by\nits predecessors, namely that:%\n\\begin{equation}\n\\frac{\\sum_{i=1}^{m_{1}}c_{i}\\left( r\\right) \\left( M-1\\right) ^{i-1}}{%\n\\sum_{i=1}^{m_{2}}d_{i}\\left( r\\right) \\left( M-1\\right) ^{i-1}}=\\frac{%\na_{M}\\left( r\\right) }{a_{M-1}\\left( r\\right) }\\,.  \\label{eq:hyperauxil}\n\\end{equation}\n\nThe auxiliary condition so obtained is a polynomial in $r$, the roots of\nwhich are candidates to give an estimate at order $M$ of $r^{\\ast }$ (noted\nbelow $r_{M}^{\\ast }$). Notice that, to obtain faster this auxiliary\ncondition, one may avoid the calculation of the coefficients $c_{i}\\left(\nr\\right) $\\ and $d_{i}\\left( r\\right) $\\ by following the same\nconsiderations as those leading to (\\ref{eq:detF}) with the ADE method.\n\nAt this point, the method potentially reaches the same goal as the ADE and\nother preceding methods. However, according to section \\ref{Brief}, in\ndetermining the ratio of polynomials (\\ref{eq:hyper0}) we have also\nexplicitly constructed the function \n\\begin{equation}\nF_{M}\\left( x\\right) =r_{M}^{\\ast }\\cdot \\,_{m_{1}+1}F_{m_{2}}\\left( \\alpha\n_{1},\\cdots ,\\alpha _{m_{1}},1;\\beta _{1},\\cdots ,\\beta _{m_{2}};\\alpha\n_{0}x\\right) \\,,  \\label{eq:FM}\n\\end{equation}%\nin which $r_{M}^{\\ast }$ is the selected estimate of $r^{\\ast }$, the sets $%\n\\left\\{ -\\alpha _{i}\\right\\} $ and $\\left\\{ -\\beta _{i}\\right\\} $ are the\nroots of the two polynomials $P_{m_{1}}\\left( n\\right) $ and $%\nQ_{m_{2}}\\left( n\\right) $ when $r=r_{M}^{\\ast }$ whereas: \n\\begin{equation}\n\\alpha _{0}=\\frac{c_{m_{1}}\\left( r_{M}^{\\ast }\\right) }{d_{m_{2}}\\left(\nr_{M}^{\\ast }\\right) }\\,.  \\label{eq:alpha0}\n\\end{equation}\n\nNow, by construction, $F_{M}\\left( x\\right) $, has the same truncated series\nin $x$ as the solution of (\\ref{eq:FPw}) we are looking for. This function\nis thus a candidate for an approximate representation of this solution.\n\nIt is worth noticing that, contrary to the ADE method, the HFA method does\nnot make an explicit use of the conditions at infinity (large $x$) to\ndetermine $r^{\\ast }$. Only a local information, in the neighbourhood of the\norigin $x=0$, is explicitly employed.\n\nLet us apply the method to the two equations of interest in this paper.\n\n\\subsection{Wilson-Polchinski's equation}\n\n\\subsubsection{Fixed point}\n\nWe know that the absolute value of the ratio $a_{n}\\left( r^{\\ast }\\right)\n\/a_{n+1}\\left( r^{\\ast }\\right) $ has a definite value $R_{WP}$ [given by\neq. (\\ref{eq:RWP})] as $n\\rightarrow \\infty $. Consequently, we must\nconsider the ratio (\\ref{eq:hyper0}) with $m_{1}=m_{2}$ (this implies also\nthat $M$ be odd). In this circumstance, according to section \\ref{Brief},\nthe relevant hypergeometric functions have a branch cut on the positive real\naxis (as functions of $\\alpha _{0}x$). Consequently the analytic\ncontinuation to large positive values of $x$ is only possible if $\\alpha\n_{0}<0$. We note also that, according to (\\ref{eq:RWP}), $\\left\\vert \\alpha\n_{0}\\right\\vert $ should converge to $1\/R_{WP}=0.174774$. Finally by\nconsidering the large $x$ behaviour directly on (\\ref{eq:EDOhyper}), it is\neasy to convince oneself that the leading power is given by one of the\nparameters $\\left\\{ -\\alpha _{i}\\right\\} $, consequently we expect to\nobserve a stable convergent value among the $\\alpha _{i}$'s toward the\nopposite of the leading power at large $x$ of the solution looked for. For\nthis reason, instead of the function $w\\left( x\\right) $ of section \\ref%\n{WPsmallField} the limit of which is 1 as $x\\rightarrow \\infty $ [see (\\ref%\n{eq:asyw})], we have considered the translated function $w_{t}\\left(\nx\\right) =w\\left( x\\right) -1$ which, according to eqs (\\ref{eq:fasy}) and (%\n\\ref{eq:chgt1}, \\ref{eq:astar}), tends to $A^{\\ast }x^{-2\/5}$. In this case\nwe\\ thus expect to observe a stable value among the $\\alpha _{i}$'s about $%\n0.4$ with the eventual possibility of estimating $A^{\\ast }$.\n\nWhen looking at the roots of the auxiliary condition (\\ref{eq:hyperauxil})\nas $M$ varies, we obtain the same kind of accumulations about the expected\nfixed point value $r^{\\ast }$ as shown in figure \\ref{fig2} (with much less\npoints however). We can also easily select the right nontrivial solution\nusing the procedure described just above (\\ref{eq:rstar2}). We get precisely\nthis excellent estimate with $M=25$ and a reduced computing time compare to\nthe ADE method. Figure (\\ref{fig6}) shows the accuracies obtained on $%\nr^{\\ast }$ (crosses) compared to the ADE method (open circles). \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure5.eps}\n\\end{center}\n\\caption{Respective approximate number of digits (defined in the caption of\nfigure \\protect\\ref{fig5}) obtained for r$_{M}^{\\ast }$ with the HFA method\n(crosses) and the ADE method (open circles) for the Wilson-Polchinski fixed\npoint equation. Whereas at a given order $M$ the accuracy is similar, a\nsmaller time computing is necessary with the HFA method.}\n\\label{fig6}\n\\end{figure}\n\nFurthermore, the sets of parameters of the successive hypergeometric\nfunctions involve two stable quantities the values of which at $M=25$ are:%\n\\begin{eqnarray}\n\\alpha _{0} &=&-0.174\\,775\\,,  \\label{eq:alpha0estim} \\\\\n\\alpha _{1} &=&0.396\\,2\\,.  \\label{eq:alpha1estim}\n\\end{eqnarray}\n\nThose two results are quantitatively and qualitatively very close to the\nexpected values (respectively $-0.174774$ and $0.4$ as given just above).\n\nThis clearly shows that the hypergeometric function determined this way\nprovides us with a really correct (but approximate) global representation of\nthe fixed point function. This contrasts strongly with the numerical\nintegration of the ODE which, due to the presence of the moving singularity,\nnever provides us with such an approximate global representation of the\nsolution looked for.\n\nFrom (\\ref{eq:alpha1estim}) we have obtained a rough estimate of $A^{\\ast }$\n($=6b^{\\ast }\/5$) by a direct consideration of the value of the\ncorresponding function\\ $F_{M}\\left( x\\right) $ defined in (\\ref{eq:FM}) for\nsome relatively large value of $x$ and we obtain $A^{\\ast }\\simeq -2.6$ what\nis a sufficiently accurate estimate to serve as a guess in the shooting\nmethod.\n\nWe have also tried to determine, using the HFA method, the value $A^{\\ast }$\ndirectly from the \\textquotedblleft reverse side\\textquotedblright\\\ncorresponding to (\\ref{eq:WFPasy}). We have not improved the previous\\\nbiased estimate obtained by ADE (about $A^{\\ast }=-2.735$). We do not\nunderstand the significance of this coincidence. We recall, however, that\nthe radius of convergence of the Taylor series of $u^{\\ast }\\left( y\\right) $\nabout $y=0$ probably vanishes. This biased result shows again that the\nproperty of convergence of the Taylor series is crucial for the accuracy of\nthe two methods.\n\n\\subsubsection{Eigenvalues}\n\nWe have also applied the HFA method to the determination of the eigenvalues.\nWith $M=17$, we have easily and without ambiguity obtained the following\nexcellent estimates [compare with (\\ref{eq:nubest}, \\ref{eq:omegac}) and\ntables \\ref{Table 3} and \\ref{Table 4}]:%\n\\begin{equation*}\n\\begin{array}{lll}\n\\nu =\\allowbreak \\allowbreak 0.649\\,561\\,774\\,,\\quad & \\omega\n_{1}=0.655\\,745\\,939\\,3\\,,\\quad & \\omega _{2}=3.180\\,006\\,53\\,, \\\\ \n\\omega _{3}=5.912\\,229\\,4\\,,\\quad & \\omega _{4}=8.796\\,045\\,,\\quad & \\omega\n_{5}=11.800\\,4\\,, \\\\ \n\\breve{\\omega}_{1}=1.886\\,703\\,839\\,,\\quad & \\breve{\\omega}%\n_{2}=4.524\\,390\\,3\\,,\\quad & \\breve{\\omega}_{3}=7.337\\,635\\,.%\n\\end{array}%\n\\end{equation*}\n\nThese results show a greater efficiency than with the ADE method especially\nin the determination of the subleading eigenvalues.\n\nIt is worth indicating also that, surprisingly enough, we observe again\n(i.e. as with the ADE method) the presence of convergences to the same\nspurious eigenvalues: 5.8 and 3.8 in the even and odd cases respectively.\n\n\\subsection{Litim's equation\\label{GeoLitim}}\n\n\\subsubsection{Fixed point}\n\nApplying the HFA method with the ratio of two successive coefficients $%\na_{n}\\left( \\bar{k}\\right) $ provides again an accumulation of roots about\nthe right value of $\\bar{k}^{\\ast }$ given in (\\ref{eq:kstar}). However,\nthis time, we have encountered some difficulties in defining a process of\nselection of the right root. We obtain the following estimate for $M=21$:%\n\\begin{equation*}\n\\bar{k}^{\\ast }\\simeq 0.409\\,531\\,,\n\\end{equation*}%\nwhich is not bad [compare with (\\ref{eq:kstar})] but not as satisfactory as\nin the preceding Wilson-Polchinski's case.\n\nWith regard to the transformation (\\ref{eq:Legendre}) and the preceding\nsuccess of the HFA method, it is not amazing that the representation of the\nsolution in the Litim case be more complicated than in the Wilson-Polchinski\ncase.\n\nWe have already mentioned that, instead of the ratio of two successive terms\nof the series $a_{n}\\left( \\bar{k}\\right) $, it is a shifted ratio that\nroughly converges to the finite radius of convergence (\\ref{eq:radiusL}).\\\nAs a matter of fact, if we use the ratios%\n\\begin{equation*}\n\\frac{a_{n+3}\\left( \\bar{k}\\right) }{a_{n}\\left( \\bar{k}\\right) }\\,,\n\\end{equation*}%\ninstead of the ratio $a_{n+1}\/a_{n}$ without changing the procedure\\footnote{%\nNotice that the procedure does not define some generalized hypergeometric\nfunction of $\\bar{x}^{3}.$ This would have been obtained by considering\nseparately three series in the original series. Then a combination of three\ngeneralized hypergeometric functions would have represented the solution\nlooked for.} described in section \\ref{HFA}, then we get a better estimate\nfor $M=21$ [compare with (\\ref{eq:kstar})]:%\n\\begin{equation*}\n\\bar{k}^{\\ast }\\simeq 0.409\\,532\\,737\\,,\n\\end{equation*}%\nalthough the convergence properties are not substantially modified.\n\nBecause the case is apparently more complicated than precedently, we do not\npursued further the discussion of the global representation of the fixed\npoint solution by generalized hypergeometric functions.\n\n\\subsubsection{Eigenvalues}\n\nFor the eigenvalue problem, a similar difficulty occurs where the right\nvalues do not appear as clear convergent series of roots. At order $M=17$,\nwe get the following estimates:$\\allowbreak $%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =0.649\\,55\\,,\\quad & \\omega _{1}=0.657\\,6\\,,\\quad & \\omega\n_{2}=3.20\\,,\\quad & \\omega _{3}=5.8\\,, \\\\ \n\\breve{\\omega}_{1}=1.89\\,,\\quad & \\breve{\\omega}_{2}=4.5\\,. &  & \n\\end{array}%\n\\end{equation*}\n\nAs in the case of the fixed point determination, if instead of applying the\nmethod with the ratio of two successive terms of the series $a_{n}\\left( \n\\bar{k}\\right) $ we consider the ratios%\n\\begin{equation*}\n\\frac{a_{n+3}\\left( \\bar{k}\\right) }{a_{n}\\left( \\bar{k}\\right) }\\,,\n\\end{equation*}%\nthen we get better estimates for $M=19$:%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =0.649\\,561\\,774\\,,\\quad & \\omega _{1}=0.655\\,75\\,,\\quad & \\omega\n_{2}=3.180\\,7\\,,\\quad & \\omega _{3}=5.905\\,, \\\\ \n\\breve{\\omega}_{1}=1.886\\,71\\,,\\quad & \\breve{\\omega}_{2}=4.524\\,. &  & \n\\end{array}%\n\\end{equation*}%\nwhere the numbers of digits have been limited having regard to the estimated\naccuracies [compare with (\\ref{eq:nubest}), table \\ref{Table 3} (even) and (%\n\\ref{eq:omegac}), table \\ref{Table 4} (odd)].\n\n\\section{Summary and conclusions}\n\n\\label{Conc}\n\nWe have presented the details of a highly accurate determination of the\nfixed point and the eigenvalues for two equivalent ERGEs in the local\npotential approximation. First, we have made use of a standard numerical\n(shooting) method to integrate the ODEs concerned. Beyond the test of the\nequivalence between the two equations, already published in \\cite{6137}, the\nresulting numerics have been used to concretely test the efficiency of two\nnew approximate analytic methods for solving two point boundary value\nproblems of ODEs based on the expansion about the origin of the solution\nlooked for (field expansion).\n\nWe have considered explicitly those two methods applied to the study of the\ntwo equivalent ODEs. We have shown that they yield estimates as accurate as\nthose obtained with the shooting method provided that the Taylor series\nabout the origin of the function looked for has a non-zero radius of\nconvergence.\n\nThis is an important new result since, up to now, no such approximate\nanalytical method was known to work in the simplest case of the\nWilson-Polchinski equation. In the case of the Litim equation the two\nmethods converge better than the currently used expansions (usually referred\nto as I and II in the literature, see e.g. [\\cite{3553}]). Our results\nsupport concretely the conclusions of \\cite{5902} which indicated that the\nhigh field contributions were important in the Wilson-Polchinski case\nwhereas they were less important in the Litim case.\n\nThe first of the two methods relies upon the construction of an auxiliary\ndifferential equation (ADE) satisfied by the Taylor series at the origin and\nto which is imposed the condition of the second boundary (at infinity) \\cite%\n{6110}.\n\nThe second method (HFA) is new. It consists in defining a global\nrepresentation of the solution of the ODE via a generalized hypergeometric\nfunction. The HFA method provides the advantage of yielding a global\n(approximate) representation of the solution via an explicit hypergeometric\nfunction.\n\nIn both cases it is possible to obtain easily (with few terms in the field\nexpansion) rough estimates of the solution which may be used as guesses in a\nsubsequent shooting method.\n\nThe procedures may be applied to several coupled ODEs as shown in \\cite{6110}\nfor the ADE method. Hence, we hope that the present work will\\ make easier\nand more efficient future explicit (and ambitious) considerations of the\nderivative expansion of exact renormalisation group equations.\n\n\\section{Acknowledgements}\n\nWe thank D. Litim for comments on an earlier version of this article.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nIn the standard model (SM) of particle physics, \\ensuremath{C\\!P}\\xspace violation in the quark sector of weak interactions arises from a single\nirreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) that describes the mixing of quarks~\\cite{ref:CKM}. \nThe unitarity of the CKM \\mbox{matrix $V$} defines a unitarity triangle (UT) in the complex plane. \n\\ensuremath{C\\!P}\\xspace violation measurements and semileptonic decay rates (and other methods)\ncan be conveniently displayed and compared as constraints on the angles and sides, respectively, of this triangle.\nInconsistencies between all these (in general) precise and redundant constraints can be used to search for new physics (NP). \nAs today, there is an impressive overall agreement between all measurements~\\cite{ref:globalCKMfits}.\nAmong these the angle $\\gamma$, defined as the phase of $V_{ub}$ in the Wolfenstein parametrization~\\cite{ref:CKM},\nis particularly relevant since it is the only \\ensuremath{C\\!P}\\xspace-violating measurement that,\ntogether with the determination of the \\ensuremath{C\\!P}\\xspace-conserving magnitude of $V_{ub}$,\nselects a region of the UT apex independently of most types of NP, and thus constitutes a SM candle type of measurement.\nCurrent constraints, provided by the \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ and Belle experiments, make use of \n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays, and are still weak ($\\sim 15^\\circ$).\nNeutral \\ensuremath{B}\\xspace decays have also been proposed, although \ndo not yet provide significant constraints.\n\n\nThe angle $\\gamma$ from $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays is determined\nmeasuring the interference between the amplitudes $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{c}\\xspace$,\nwhen the neutral \\ensuremath{D}\\xspace meson is reconstructed in a final state accessible from both \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace decays. \nSince both amplitudes are tree level, the interference is unaffected by NP appearing in the loops, making the theoretical \ninterpretation of observables in terms of $\\gamma$ very clean. \nThe disadvantage is that the branching fractions of the involved decays are small due to CKM suppression ($10^{-5}-10^{-7}$), \nand the size of the interference, given by the ratio $r_\\ensuremath{B}\\xspace$ between the magnitudes of the $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{c}\\xspace$ amplitudes, \nis small due to further CKM and color suppressions ($\\sim 10\\%$). As a consequence, the measurements are\nstatistically limited and one has to combine complementary methods applied on the same \\ensuremath{B}\\xspace decay modes sharing the\nhadronic parameters ($r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$, i.e. the relative magnitude and phase of the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ transitions) and $\\gamma$, \nand use as many as possible different \\ensuremath{B}\\xspace decay modes to improve the overall sensitivity to $\\gamma$.\n\nIn this talk we present the most recent determinations of $\\gamma$ obtained by \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}, based on the full\ndata sample of charged \\ensuremath{B}\\xspace meson decays produced in $e^+e^- \\ensuremath{\\rightarrow}\\xspace \\Y4S \\ensuremath{\\rightarrow}\\xspace \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace$ and \nrecorded in the years 1999-2007, about $468\\times10^6$ \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace pairs.\nWe have studied $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays,\nwith the neutral \\ensuremath{D}\\xspace mesons reconstructed in a number of different final states: $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace \\KS h^+ h^-$, with $h=\\pi,\\ensuremath{K}\\xspace$\n(Dalitz plot method); $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{K^\\pm}\\xspace\\ensuremath{\\pi^\\mp}\\xspace$ (ADS method); \nand $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace f_{\\ensuremath{C\\!P}\\xspace}$, with $f_{\\ensuremath{C\\!P}\\xspace}$ a \\ensuremath{C\\!P}\\xspace-eigenstate (GLW method)~\\cite{ref:dalitz_ads_glw}.\n\n\n\n\n\n\nOne of the charged \\ensuremath{B}\\xspace mesons produced in the \\Y4S decay is fully reconstructed, with efficiencies ranging \nbetween 40\\% (for low-multiplicity decays with no neutrals) \nand 10\\% (for high-multiplicity decays with neutrals). The selection is optimized to maximize the statistical sensitivity. \nThe reconstruction efficiencies have substantially improved (20\\% to 60\\% relative) with respect to our previous measurements \nbased on $384\\times10^6$ \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace pairs, reflecting improvements in tracking and particle identification,\nand optimization of analysis procedures.\nSignal \\ensuremath{B}\\xspace decays are characterized by means of two nearly independent kinematic variables exploiting the constraint from the\nknown beam energies: \nthe beam-energy $\\mbox{$m_{\\rm ES}$}\\xspace \\equiv \\sqrt{E^{*2}_{\\rm beam}-|p^{*}_{\\ensuremath{B}\\xspace}|^2}$\nand the energy-difference $\\mbox{$\\Delta E$}\\xspace \\equiv E^*_\\ensuremath{B}\\xspace - E^*_{\\rm beam}$.\nSince the main source of background comes from \\ensuremath{q\\overline q}\\xspace continuum production, additional discrimination\nis achieved using multivariate analysis tools, from the combination (either a linear Fisher discriminant \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace,\nor a non-linear neural network $NN$) of several event-shape quantities. These variables distinguish between\nspherical \\BB events from more \njet-like continuum events and exploit the different angular correlations in the two event categories. \nThe signal is finally separated from background through \nunbinned maximum likelihood (UML) fits to the\n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ data using \\mbox{$m_{\\rm ES}$}\\xspace, \\mbox{$\\Delta E$}\\xspace, and \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace or $NN$. \n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{\\pi^\\pm}\\xspace$ decays, \nwhich are\nabout 12 times more abundant than $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$, have a similar topology but are \ndiscriminated by means of excellent pion and kaon identification provided by $dE\/dx$ and Cerenkov measurements,\nand show negligible \\ensuremath{C\\!P}\\xspace-violating effects ($r_\\ensuremath{B}\\xspace \\sim 1\\%$). Therefere, these decays provide powerful calibration\nand control samples for negative tests of \\ensuremath{C\\!P}\\xspace violation.\n\n\n\n\nIn the Dalitz plot (DP) method the amplitude for a \\ensuremath{\\Bub}\\xspace decay has for the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace$ transition the DP of the \\ensuremath{D^0}\\xspace decay, \nwhile for the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ transition the DP is the corresponding to the \\ensuremath{\\Dbar^0}\\xspace decay. If we assume no \\ensuremath{D}\\xspace mixing nor \\ensuremath{C\\!P}\\xspace violation \nin the \\ensuremath{D}\\xspace decay, and use as independent kinematic variables $s_\\pm=m^2(\\KS\\ensuremath{\\pi^\\pm}\\xspace)$,\nthen the two DPs are one rotated $90^\\circ$ to each other.\nThis is of critical importance since allows to determine directly from data the strong charm phase variation \nfor \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace, as well as well as the hadronic parameters $r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$, and the weak phase $\\gamma$, provided\nthat a \\ensuremath{D}\\xspace decay amplitude model is assumed. For \\ensuremath{\\Bu}\\xspace decays one has to interchange the \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace DPs, \nand change the sign of $\\gamma$. This results in an interference term proportional to our observables \n$x_\\pm \\equiv r_\\ensuremath{B}\\xspace \\cos(\\delta_\\ensuremath{B}\\xspace \\pm \\gamma)$ \nand\n$y_\\pm \\equiv r_\\ensuremath{B}\\xspace \\sin(\\delta_\\ensuremath{B}\\xspace \\pm \\gamma)$, i.e.\nthe real and imaginary parts of the ratio of $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace$ amplitudes for \\ensuremath{B^\\pm}\\xspace decays.\nWe reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$, $D^* \\ensuremath{K^\\pm}\\xspace$ with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace,D\\gamma$, and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ with $\\ensuremath{K^{*\\pm}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\KS\\ensuremath{\\pi^\\pm}\\xspace$ decays,\nfollowed by neutral \\ensuremath{D}\\xspace meson decays to the 3-body self-conjugate final states $\\KS h^+ h^-$, with $h=\\pi,\\ensuremath{K}\\xspace$. \nFrom the UML fit we determine the signal and background yields in each of the eight different final states for each \\ensuremath{B}\\xspace charge, \nalong with the \\ensuremath{C\\!P}\\xspace-violating \nparameters $x_\\pm$ and $y_\\pm$~\\cite{ref:babar_dalitzpub2010}.\nWe find 1507 \\ensuremath{B^\\pm}\\xspace signal candidates with $\\KS\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$, and 268 with $\\KS\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$. \nPrior to the \\ensuremath{C\\!P}\\xspace fit, we model the \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace decay amplitudes as a coherent sum of S-, P-, and D-waves,\nand determine their amplitudes and phases (along with other relevant parameters)\nrelative to the dominant two-body \\ensuremath{C\\!P}\\xspace-eigenstates $\\KS \\rho(770)$ (for $\\KS\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$) and $\\KS a_0(980)$ (for $\\KS\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$),\nusing a large ($\\approx 6.2\\times10^5$) and very pure ($\\approx 99\\%$) signal sample of flavor tagged \nneutral \\ensuremath{D}\\xspace mesons from $\\ensuremath{D^{*+}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{D^0}\\xspace\\ensuremath{\\pi^+}\\xspace$ decays produced in $e^+e^- \\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace\\ensuremath{\\overline c}\\xspace$ events~\\cite{ref:dmixing-kshh}.\n\\begin{figure}[htb!]\n\\begin{center}\n\\vskip-0.1cm\n\\begin{tabular}{ccc}\n\\includegraphics[width=0.32\\textwidth]{d0k_Dalitz-ubl-Contours-DzK-zoom.eps}&\n\\includegraphics[width=0.32\\textwidth]{d0k_Dalitz-ubl-Contours-DstK-zoom.eps} &\n\\includegraphics[width=0.32\\textwidth]{scan-gammat-Dalitz.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{fig:dalitz} $1\\sigma$ and $2\\sigma$ contours in the $(x_\\pm,y_\\pm)$ planes for\n(a) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ and (b) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^* \\ensuremath{K^\\pm}\\xspace$, for \\ensuremath{\\Bub}\\xspace (solid lines) and \\ensuremath{\\Bu}\\xspace (dotted lines) decays.\n(c) $1 - {\\rm CL}$ as a function of $\\gamma$ for $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{D}\\xspace\\ensuremath{K^\\pm}\\xspace, \\ensuremath{D^*}\\xspace\\ensuremath{K^\\pm}\\xspace, \\ensuremath{D}\\xspace\\ensuremath{K^{*\\pm}}\\xspace$ decays.\nThe dashed (upper) and dotted (lower) horizontal lines correspond to the $1\\sigma$ and $2\\sigma$ intervals, respectively.\n}\n\\end{figure}\nFrom the $(x_\\pm,y_\\pm)$ confidence regions for each of the 3 different \\ensuremath{B}\\xspace decay modes --Fig.~\\ref{fig:dalitz}.(a)(b)-- we determine, \nusing a frequentist procedure, $1\\sigma$ $[2\\sigma]$ intervals for $\\gamma$ --Fig.~\\ref{fig:dalitz}.(c)--. \nWe obtain $\\gamma~({\\rm mod}~180^\\circ) = (68\\pm14\\pm4\\pm3)^\\circ$ $[39^\\circ,98^\\circ]$, where the three uncertainties are statistical,\nexperimental systematic, and amplitude model systematic.\nWe also determine the hadronic parameters\n$r_\\ensuremath{B}\\xspace^{DK^\\pm}=(9.6\\pm2.9)\\%$ $[3.7,15.5]\\%$, \n$r_\\ensuremath{B}\\xspace^{D^*K^\\pm}=(13.3^{+4.2}_{-3.9})\\%$ $[4.9,21.5]\\%$, \n$\\kappa r_\\ensuremath{B}\\xspace^{DK^{*\\pm}}=(14.9^{+6.6}_{-6.2})\\%$ $[0,28.0]\\%$ ($\\kappa = 0.9\\pm0.1$ takes into account \nthe \\ensuremath{K^*}\\xspace intrinsic width),\nand the strong phases \n$\\delta_B^{DK^\\pm}$, \n$\\delta_B^{D^*K^\\pm}$, \nand $\\delta_B^{DK^{*\\pm}}$~\\cite{ref:babar_dalitzpub2010}. \nA $3.5\\sigma$ evidence of direct \\ensuremath{C\\!P}\\xspace violation ($\\gamma \\ne 0$) is found from the combination of the 3 channels,\nwhich corresponds to the significance of the separation between the $(x_+,y_+)$ and $(x_-,y_-)$ solutions in Fig.~\\ref{fig:dalitz}.(a)(b).\n\n\n\n\n\n\nIn the ADS method, we reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$, $D^* \\ensuremath{K^\\pm}\\xspace$ with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace,D\\gamma$, followed by \\ensuremath{D}\\xspace decays to both the doubly-Cabibbo-suppressed (DCS)\n\\ensuremath{D^0}\\xspace final state $K^+\\pi^-$ and the Cabibbo-favored (CF) $K^-\\pi^+$, which is used as normalization and control sample. \nFinal states with opposite-sign kaons arise \neither from the CKM favored \\ensuremath{B}\\xspace decay followed by the DCS \\ensuremath{D}\\xspace decay\nor from the CKM- and color-suppressed \\ensuremath{B}\\xspace decay followed by the CF \\ensuremath{D}\\xspace decay,\nproducing an interference which can be potentially large since the magnitudes of the interfering amplitudes are similar.\nHowever, their overall branching ratios are very small ($\\sim 10^{-7}$) and background suppression becomes crucial.\nThe UML fit directly determines the three branching fraction ratios $R_{ADS}$ between \\ensuremath{B}\\xspace decays with opposite-sign \nand same-sign kaons, and the three yields of \\ensuremath{B}\\xspace decays with same-sign kaons, using \\mbox{$m_{\\rm ES}$}\\xspace and $NN$.\nThe three \\ensuremath{C\\!P}\\xspace asymmetries $A_{ADS}$ are inferred from all these. \nWe obtain first indications of signals for the\n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^* \\ensuremath{K^\\pm}\\xspace$ (with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace$) opposite-sign modes --Fig.~\\ref{fig:ads}--, \nwith significances of $2.1\\sigma$ and $2.2\\sigma$, respectively~\\cite{ref:babar_ADS-DK-DstarK}.\nThe measured branching fraction ratios are \n$R_{ADS}^{DK}=(1.1\\pm0.5\\pm0.2)\\times10^{-2}$, \n$R_{ADS}^{[D\\ensuremath{\\pi^0}\\xspace]K}=(1.8\\pm0.9\\pm0.4)\\times10^{-2}$, and\n$R_{ADS}^{[D\\gamma]K}=(1.3\\pm1.4\\pm0.8)\\times10^{-2}$, and the \\ensuremath{C\\!P}\\xspace asymmetries are\n$A_{ADS}^{DK}=-0.86\\pm0.47^{+0.12}_{-0.16}$, \n$A_{ADS}^{[D\\ensuremath{\\pi^0}\\xspace]K}=0.77\\pm0.35\\pm0.12$, and\n$A_{ADS}^{[D\\gamma]K}=0.36\\pm0.94^{+0.25}_{-0.41}$.\nFrom these results and external measurements of the relative amplitude and phase of \\ensuremath{\\Dbar^0}\\xspace to \\ensuremath{D^0}\\xspace mesons \ndecaying into the $K^- \\ensuremath{\\pi^+}\\xspace$ final state~\\cite{ref:hfag} we infer, using a frequentist procedure similar to that used in\nthe DP method,\n$r_\\ensuremath{B}\\xspace^{DK^\\pm}=(9.5^{+5.1}_{-4.1})\\%$ $[0,16.7]\\%$, $r_\\ensuremath{B}\\xspace^{D^*K^\\pm}=(9.6^{+3.5}_{-5.1})\\%$ $[0,15.0]\\%$,\nand the strong phases $\\delta_B^{DK^\\pm}$, $\\delta_B^{D^*K^\\pm}$, in good agreement with those obtained with the DP technique.\n\\begin{figure}[htb]\n\\begin{center}\n\\vskip-0.1cm\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{fig6a.eps} &\n\\includegraphics[width=0.45\\textwidth]{fig6b.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{fig:ads}\nProjections on \\mbox{$m_{\\rm ES}$}\\xspace for (a) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{D}\\xspace\\ensuremath{K^\\pm}\\xspace$ and (b) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^*[\\ensuremath{D}\\xspace\\ensuremath{\\pi^0}\\xspace]\\ensuremath{K^\\pm}\\xspace$, $\\ensuremath{D}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^\\mp}\\xspace\\ensuremath{\\pi^\\pm}\\xspace$ \nopposite-sign decays, for ADS samples enriched in signal ($NN>0.94$). \nThe points with error bars are data while the curves represent the fit projections for\nsignal plus background (solid), the sum of all background components (dashed), and $q\\bar q$ background only (dotted).\n}\n\\end{figure}\n\n\n\n\n\n\n\nIn the GLW method, we reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays, followed by \\ensuremath{D}\\xspace decays to non-\\ensuremath{C\\!P}\\xspace ($\\ensuremath{D^0}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace$), \\ensuremath{C\\!P}\\xspace-even ($\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$, $\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$), \nand \\ensuremath{C\\!P}\\xspace-odd ($\\KS\\ensuremath{\\pi^0}\\xspace$, $\\KS\\phi$, $\\KS\\omega$) eigenstates. The partial decay rate charge asymmetries $A_{\\ensuremath{C\\!P}\\xspace\\pm}$ for\n\\ensuremath{C\\!P}\\xspace-even and \\ensuremath{C\\!P}\\xspace-odd \\ensuremath{D}\\xspace final states and the ratios $R_{\\ensuremath{C\\!P}\\xspace\\pm}$ of the charged-averaged \\ensuremath{B}\\xspace meson partial decay\nrates in \\ensuremath{C\\!P}\\xspace ($R_{K\/\\pi}^\\pm$) and non-\\ensuremath{C\\!P}\\xspace ($R_{K\/\\pi}$) decays (normalized to the corresponding $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{\\pi^\\pm}\\xspace$ decays, \nto reduce systematic uncertainties)\nprovide four observables from which the three \nunknowns $\\gamma$, $r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$ can be extracted (up to an 8-fold ambiguity for the phases). \nThe signal yields, expressed in terms of $A_{\\ensuremath{C\\!P}\\xspace\\pm}$, $R_{K\/\\pi}^\\pm$ and $R_{K\/\\pi}$ are extracted from UML fits \nto \\mbox{$m_{\\rm ES}$}\\xspace, \\mbox{$\\Delta E$}\\xspace, and \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace. We identify about 500 $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays with \\ensuremath{C\\!P}\\xspace-even \\ensuremath{D}\\xspace final states and a similar\namount for \\ensuremath{C\\!P}\\xspace-odd \\ensuremath{D}\\xspace final states, and measure~\\cite{ref:babar_GLW-DK} \n$A_{\\ensuremath{C\\!P}\\xspace+}=0.25\\pm0.06\\pm0.02$, \n$A_{\\ensuremath{C\\!P}\\xspace-}=-0.09\\pm0.07\\pm0.02$,\n$R_{\\ensuremath{C\\!P}\\xspace+}=1.18\\pm0.09\\pm0.05$, and\n$R_{\\ensuremath{C\\!P}\\xspace-}=1.07\\pm0.08\\pm0.04$. \nThe parameter $A_{\\ensuremath{C\\!P}\\xspace+}$ is different from zero with a significance of $3.6\\sigma$, and constitutes evidence for\ndirect \\ensuremath{C\\!P}\\xspace violation in $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays. \nThese results can be written in terms of the observables\n$x_\\pm$ using the relationship $x_\\pm = [R_{\\ensuremath{C\\!P}\\xspace+}(1\\mp A_{\\ensuremath{C\\!P}\\xspace+}) - R_{\\ensuremath{C\\!P}\\xspace-}(1\\mp A_{\\ensuremath{C\\!P}\\xspace-})]\/4$.\nExcluding the $D\\ensuremath{\\rightarrow}\\xspace\\KS\\phi$, $\\phi\\ensuremath{\\rightarrow}\\xspace K^+K^-$ channel to facilitate the combination with the DP method,\nwe find $x_+=-0.057\\pm0.039\\pm0.015$ and $x_-=0.132\\pm0.042\\pm0.018$, which are consistent (and of similar\nprecision) with the DP method.\nFrom these results and using a frequentist procedure similar to that used previously we infer\n$24\\% < r_\\ensuremath{B}\\xspace < 45\\%$ $[6,51]\\%$, and\nmod $180^\\circ$, \n$11^\\circ < \\gamma < 23^\\circ$ or $81^\\circ < \\gamma < 99^\\circ$ or $157^\\circ < \\gamma < 169^\\circ$\n$[7^\\circ,173^\\circ]$.\n\n\n\nWe have reported the recent progress in the determination of the CKM angle $\\gamma$, using the complete \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ data sample and three different\nand complementary methods (DP, ADS, and GLW). \nA coherent and consistent set of results on $\\gamma$ and the hadronic parameters characterizing the \\ensuremath{B}\\xspace decays has been\nobtained. The central value for $\\gamma$, around $70^\\circ$ with a precision around $15^\\circ$, is consistent with indirect determinations from\nCKM fits~\\cite{ref:globalCKMfits}. \nA proper average of all the three methods using the full \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ sample of $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D^{(*)} K^{\\pm}$, $D K^{*\\pm}$ decays is foreseen.\nWe obtain $x_- - x_+ = 0.175\\pm0.040$ by combining the $x_\\pm$ measurements from the DP and GLW methods for $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D K^{\\pm}$ decays,\nwhich is different from zero with a significance of $4.4\\sigma$,\nthus constitutes strong evidence for direct \\ensuremath{C\\!P}\\xspace violation in these charged \\ensuremath{B}\\xspace decays.\nFinally, we have the first sign of an ADS signal in $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D K^{\\pm}$ and  $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D^{(*)} K^{\\pm}$ decays.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{UNet and Attention UNet:}\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font).}\n\\label{tab:my-table1}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7151} & {\\color[HTML]{24292F} 0.7165} & {\\color[HTML]{24292F} 0.6674} & {\\color[HTML]{24292F} 0.8017} & {\\color[HTML]{24292F} 0.6753} & {\\color[HTML]{24292F} 0.6707} & {\\color[HTML]{24292F} 0.668} & {\\color[HTML]{24292F} 0.784} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7034} & {\\color[HTML]{24292F} 0.6933} & {\\color[HTML]{24292F} 0.699} & {\\color[HTML]{24292F} 0.718} & {\\color[HTML]{24292F} 0.7046} & {\\color[HTML]{24292F} 0.7035} & {\\color[HTML]{24292F} 0.7495} & {\\color[HTML]{24292F} 0.7475} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6963} & {\\color[HTML]{24292F} 0.6932} & {\\color[HTML]{24292F} 0.6873} & {\\color[HTML]{24292F} 0.7339} & {\\color[HTML]{24292F} 0.6781} & {\\color[HTML]{24292F} 0.6765} & {\\color[HTML]{24292F} 0.7423} & {\\color[HTML]{24292F} 0.7094} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7011} & {\\color[HTML]{24292F} 0.7037} & {\\color[HTML]{24292F} 0.714} & {\\color[HTML]{24292F} 0.7239} & {\\color[HTML]{24292F} 0.7032} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6684} & {\\color[HTML]{24292F} 0.8052} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.7231}} & {\\color[HTML]{24292F} \\textbf{0.7118}} & {\\color[HTML]{24292F} \\textbf{0.6763}} & {\\color[HTML]{24292F} \\textbf{0.7801}} & {\\color[HTML]{24292F} \\textbf{0.7248}} & {\\color[HTML]{24292F} \\textbf{0.7192}} & {\\color[HTML]{24292F} \\textbf{0.7105}} & {\\color[HTML]{24292F} \\textbf{0.8067}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.72} & {\\color[HTML]{24292F} 0.7217} & {\\color[HTML]{24292F} 0.7519} & {\\color[HTML]{24292F} 0.7185} & {\\color[HTML]{24292F} 0.7141} & {\\color[HTML]{24292F} 0.7068} & {\\color[HTML]{24292F} 0.6811} & {\\color[HTML]{24292F} 0.8166} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.709} & {\\color[HTML]{24292F} 0.7156} & {\\color[HTML]{24292F} 0.7195} & {\\color[HTML]{24292F} 0.7423} & {\\color[HTML]{24292F} 0.7027} & {\\color[HTML]{24292F} 0.7004} & {\\color[HTML]{24292F} 0.7043} & {\\color[HTML]{24292F} 0.7855} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.7127} & {\\color[HTML]{24292F} 0.7195} & {\\color[HTML]{24292F} 0.7012} & {\\color[HTML]{24292F} 0.7618} & {\\color[HTML]{24292F} 0.6643} & {\\color[HTML]{24292F} 0.6608} & {\\color[HTML]{24292F} 0.6444} & {\\color[HTML]{24292F} 0.7996} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6807} & {\\color[HTML]{24292F} 0.6838} & {\\color[HTML]{24292F} 0.6862} & {\\color[HTML]{24292F} 0.7167} & {\\color[HTML]{24292F} 0.6306} & {\\color[HTML]{24292F} 0.6296} & {\\color[HTML]{24292F} 0.6634} & {\\color[HTML]{24292F} 0.7316} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.6765} & {\\color[HTML]{24292F} 0.6813} & {\\color[HTML]{24292F} 0.6788} & {\\color[HTML]{24292F} 0.7183} & {\\color[HTML]{24292F} 0.6845} & {\\color[HTML]{24292F} 0.6855} & {\\color[HTML]{24292F} 0.8061} & {\\color[HTML]{24292F} 0.666} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6967} & {\\color[HTML]{24292F} 0.6244} & {\\color[HTML]{24292F} 0.8167} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.879} & {\\color[HTML]{24292F} 0.7981} & {\\color[HTML]{24292F} 0.6777} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.6112} & {\\color[HTML]{24292F} 0.6008} & {\\color[HTML]{24292F} 0.61} & {\\color[HTML]{24292F} 0.6325} & {\\color[HTML]{24292F} 0.6521} & {\\color[HTML]{24292F} 0.6545} & {\\color[HTML]{24292F} 0.8018} & {\\color[HTML]{24292F} 0.6245} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6954} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.6847} & {\\color[HTML]{24292F} 0.7387} & {\\color[HTML]{24292F} 0.6852} & {\\color[HTML]{24292F} 0.6986} & {\\color[HTML]{24292F} 0.7198} & {\\color[HTML]{24292F} 0.7462} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0289} & {\\color[HTML]{24292F} 0.0313} & {\\color[HTML]{24292F} 0.0373} & {\\color[HTML]{24292F} 0.0462} & {\\color[HTML]{24292F} 0.0260} & {\\color[HTML]{24292F} 0.0596} & {\\color[HTML]{24292F} 0.0561} & {\\color[HTML]{24292F} 0.0615} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7231} & {\\color[HTML]{24292F} 0.7217} & {\\color[HTML]{24292F} 0.7519} & {\\color[HTML]{24292F} 0.8167} & {\\color[HTML]{24292F} 0.7248} & {\\color[HTML]{24292F} 0.879} & {\\color[HTML]{24292F} 0.8061} & {\\color[HTML]{24292F} 0.8166} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6112} & {\\color[HTML]{24292F} 0.6008} & {\\color[HTML]{24292F} 0.61} & {\\color[HTML]{24292F} 0.6325} & {\\color[HTML]{24292F} 0.6306} & {\\color[HTML]{24292F} 0.6296} & {\\color[HTML]{24292F} 0.6444} & {\\color[HTML]{24292F} 0.6245} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: Attention UNet. Patch size: 128x128 pixels (best fold is reported in bold font).}\n\\label{tab:my-table2}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7654} & {\\color[HTML]{24292F} 0.7654} & {\\color[HTML]{24292F} 0.7833} & {\\color[HTML]{24292F} 0.8000} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.5959} & {\\color[HTML]{24292F} 0.8177} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7216} & {\\color[HTML]{24292F} 0.7216} & {\\color[HTML]{24292F} 0.7293} & {\\color[HTML]{24292F} 0.7890} & {\\color[HTML]{24292F} 0.6734} & {\\color[HTML]{24292F} 0.6734} & {\\color[HTML]{24292F} 0.6387} & {\\color[HTML]{24292F} 0.8258} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.7428} & {\\color[HTML]{24292F} 0.7422} & {\\color[HTML]{24292F} 0.7425} & {\\color[HTML]{24292F} 0.8108} & {\\color[HTML]{24292F} 0.6499} & {\\color[HTML]{24292F} 0.6499} & {\\color[HTML]{24292F} 0.6320} & {\\color[HTML]{24292F} 0.7883} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7646} & {\\color[HTML]{24292F} 0.7646} & {\\color[HTML]{24292F} 0.7876} & {\\color[HTML]{24292F} 0.7934} & {\\color[HTML]{24292F} 0.6851} & {\\color[HTML]{24292F} 0.6851} & {\\color[HTML]{24292F} 0.6582} & {\\color[HTML]{24292F} 0.7979} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_01} & {\\color[HTML]{24292F} 0.7272} & {\\color[HTML]{24292F} 0.7272} & {\\color[HTML]{24292F} 0.7916} & {\\color[HTML]{24292F} 0.7299} & {\\color[HTML]{24292F} 0.7128} & {\\color[HTML]{24292F} 0.7122} & {\\color[HTML]{24292F} 0.7454} & {\\color[HTML]{24292F} 0.7522} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.7335} & {\\color[HTML]{24292F} 0.7335} & {\\color[HTML]{24292F} 0.7593} & {\\color[HTML]{24292F} 0.7772} & {\\color[HTML]{24292F} 0.6909} & {\\color[HTML]{24292F} 0.6909} & {\\color[HTML]{24292F} 0.6836} & {\\color[HTML]{24292F} 0.7890} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.7658} & {\\color[HTML]{24292F} 0.7658} & {\\color[HTML]{24292F} 0.7946} & {\\color[HTML]{24292F} 0.7900} & {\\color[HTML]{24292F} 0.7184} & {\\color[HTML]{24292F} 0.7184} & {\\color[HTML]{24292F} 0.7109} & {\\color[HTML]{24292F} 0.8060} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_02\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.7159}} & {\\color[HTML]{24292F} \\textbf{0.7153}} & {\\color[HTML]{24292F} \\textbf{0.7565}} & {\\color[HTML]{24292F} \\textbf{0.7429}} & {\\color[HTML]{24292F} \\textbf{0.7263}} & {\\color[HTML]{24292F} \\textbf{0.7263}} & {\\color[HTML]{24292F} \\textbf{0.7950}} & {\\color[HTML]{24292F} \\textbf{0.7254}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.7809} & {\\color[HTML]{24292F} 0.7809} & {\\color[HTML]{24292F} 0.8232} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.7241} & {\\color[HTML]{24292F} 0.7488} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8180} & {\\color[HTML]{24292F} 0.8559} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6839} & {\\color[HTML]{24292F} 0.8008} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6843} & {\\color[HTML]{24292F} 0.7908} & {\\color[HTML]{24292F} 0.7140} & {\\color[HTML]{24292F} 0.7140} & {\\color[HTML]{24292F} 0.7781} & {\\color[HTML]{24292F} 0.7281} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.7857} & {\\color[HTML]{24292F} 0.7857} & {\\color[HTML]{24292F} 0.8297} & {\\color[HTML]{24292F} 0.7896} & {\\color[HTML]{24292F} 0.7024} & {\\color[HTML]{24292F} 0.7024} & {\\color[HTML]{24292F} 0.8435} & {\\color[HTML]{24292F} 0.6571} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.7516} & {\\color[HTML]{24292F} 0.7515} & {\\color[HTML]{24292F} 0.7750} & {\\color[HTML]{24292F} 0.7879} & {\\color[HTML]{24292F} 0.6920} & {\\color[HTML]{24292F} 0.6920} & {\\color[HTML]{24292F} 0.7074} & {\\color[HTML]{24292F} 0.7698} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0334} & {\\color[HTML]{24292F} 0.0335} & {\\color[HTML]{24292F} 0.0408} & {\\color[HTML]{24292F} 0.0301} & {\\color[HTML]{24292F} 0.0254} & {\\color[HTML]{24292F} 0.0254} & {\\color[HTML]{24292F} 0.0703} & {\\color[HTML]{24292F} 0.0469} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8297} & {\\color[HTML]{24292F} 0.8559} & {\\color[HTML]{24292F} 0.7263} & {\\color[HTML]{24292F} 0.7263} & {\\color[HTML]{24292F} 0.8435} & {\\color[HTML]{24292F} 0.8258} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6843} & {\\color[HTML]{24292F} 0.7299} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.5959} & {\\color[HTML]{24292F} 0.6571} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 256x256 pixels (best fold is reported in bold font).}\n\\label{tab:my-table3}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.707} & {\\color[HTML]{24292F} 0.7095} & {\\color[HTML]{24292F} 0.6557} & {\\color[HTML]{24292F} 0.7962} & {\\color[HTML]{24292F} 0.6423} & {\\color[HTML]{24292F} 0.6363} & {\\color[HTML]{24292F} 0.5839} & {\\color[HTML]{24292F} 0.8071} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.6509} & {\\color[HTML]{24292F} 0.6563} & {\\color[HTML]{24292F} 0.6744} & {\\color[HTML]{24292F} 0.6604} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.6743} & {\\color[HTML]{24292F} 0.6563} & {\\color[HTML]{24292F} 0.7767} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6663} & {\\color[HTML]{24292F} 0.6657} & {\\color[HTML]{24292F} 0.6725} & {\\color[HTML]{24292F} 0.6826} & {\\color[HTML]{24292F} 0.6368} & {\\color[HTML]{24292F} 0.6322} & {\\color[HTML]{24292F} 0.6356} & {\\color[HTML]{24292F} 0.7269} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_00}} & {\\color[HTML]{24292F} \\textbf{0.6939}} & {\\color[HTML]{24292F} \\textbf{0.7099}} & {\\color[HTML]{24292F} \\textbf{0.7041}} & {\\color[HTML]{24292F} \\textbf{0.7319}} & {\\color[HTML]{24292F} \\textbf{0.6963}} & {\\color[HTML]{24292F} \\textbf{0.6893}} & {\\color[HTML]{24292F} \\textbf{0.6632}} & {\\color[HTML]{24292F} \\textbf{0.7998}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_01} & {\\color[HTML]{24292F} 0.6411} & {\\color[HTML]{24292F} 0.642} & {\\color[HTML]{24292F} 0.6018} & {\\color[HTML]{24292F} 0.7157} & {\\color[HTML]{24292F} 0.6856} & {\\color[HTML]{24292F} 0.6785} & {\\color[HTML]{24292F} 0.6468} & {\\color[HTML]{24292F} 0.8008} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.6793} & {\\color[HTML]{24292F} 0.6829} & {\\color[HTML]{24292F} 0.6892} & {\\color[HTML]{24292F} 0.6979} & {\\color[HTML]{24292F} 0.6718} & {\\color[HTML]{24292F} 0.6654} & {\\color[HTML]{24292F} 0.6487} & {\\color[HTML]{24292F} 0.7732} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.6893} & {\\color[HTML]{24292F} 0.7077} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.6395} & {\\color[HTML]{24292F} 0.6352} & {\\color[HTML]{24292F} 0.6104} & {\\color[HTML]{24292F} 0.766} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.6911} & {\\color[HTML]{24292F} 0.698} & {\\color[HTML]{24292F} 0.6344} & {\\color[HTML]{24292F} 0.8027} & {\\color[HTML]{24292F} 0.621} & {\\color[HTML]{24292F} 0.6155} & {\\color[HTML]{24292F} 0.5748} & {\\color[HTML]{24292F} 0.7816} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6462} & {\\color[HTML]{24292F} 0.649} & {\\color[HTML]{24292F} 0.6593} & {\\color[HTML]{24292F} 0.6638} & {\\color[HTML]{24292F} 0.5949} & {\\color[HTML]{24292F} 0.595} & {\\color[HTML]{24292F} 0.6676} & {\\color[HTML]{24292F} 0.6185} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.6432} & {\\color[HTML]{24292F} 0.6529} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.7029} & {\\color[HTML]{24292F} 0.6507} & {\\color[HTML]{24292F} 0.65} & {\\color[HTML]{24292F} 0.7633} & {\\color[HTML]{24292F} 0.6242} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6747} & {\\color[HTML]{24292F} 0.6877} & {\\color[HTML]{24292F} 0.6581} & {\\color[HTML]{24292F} 0.7453} & {\\color[HTML]{24292F} 0.6507} & {\\color[HTML]{24292F} 0.6521} & {\\color[HTML]{24292F} 0.8146} & {\\color[HTML]{24292F} 0.5945} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.5391} & {\\color[HTML]{24292F} 0.5355} & {\\color[HTML]{24292F} 0.5558} & {\\color[HTML]{24292F} 0.5535} & {\\color[HTML]{24292F} 0.5834} & {\\color[HTML]{24292F} 0.5856} & {\\color[HTML]{24292F} 0.8118} & {\\color[HTML]{24292F} 0.5149} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.660} & {\\color[HTML]{24292F} 0.666} & {\\color[HTML]{24292F} 0.654} & {\\color[HTML]{24292F} 0.706} & {\\color[HTML]{24292F} 0.646} & {\\color[HTML]{24292F} 0.642} & {\\color[HTML]{24292F} 0.673} & {\\color[HTML]{24292F} 0.715} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.042} & {\\color[HTML]{24292F} 0.046} & {\\color[HTML]{24292F} 0.042} & {\\color[HTML]{24292F} 0.063} & {\\color[HTML]{24292F} 0.033} & {\\color[HTML]{24292F} 0.031} & {\\color[HTML]{24292F} 0.077} & {\\color[HTML]{24292F} 0.096} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.707} & {\\color[HTML]{24292F} 0.710} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.803} & {\\color[HTML]{24292F} 0.696} & {\\color[HTML]{24292F} 0.689} & {\\color[HTML]{24292F} 0.815} & {\\color[HTML]{24292F} 0.807} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.5391} & {\\color[HTML]{24292F} 0.5355} & {\\color[HTML]{24292F} 0.5558} & {\\color[HTML]{24292F} 0.5535} & {\\color[HTML]{24292F} 0.5834} & {\\color[HTML]{24292F} 0.5856} & {\\color[HTML]{24292F} 0.5748} & {\\color[HTML]{24292F} 0.5149} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: Attention UNet. Patch size: 256x256 pixels (best fold is reported in bold font).}\n\\label{tab:my-table4}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7288} & {\\color[HTML]{24292F} 0.7288} & {\\color[HTML]{24292F} 0.8318} & {\\color[HTML]{24292F} 0.6819} & {\\color[HTML]{24292F} 0.6439} & {\\color[HTML]{24292F} 0.6439} & {\\color[HTML]{24292F} 0.6312} & {\\color[HTML]{24292F} 0.7429} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.6698} & {\\color[HTML]{24292F} 0.6698} & {\\color[HTML]{24292F} 0.7152} & {\\color[HTML]{24292F} 0.6855} & {\\color[HTML]{24292F} 0.6038} & {\\color[HTML]{24292F} 0.6038} & {\\color[HTML]{24292F} 0.5860} & {\\color[HTML]{24292F} 0.7462} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6644} & {\\color[HTML]{24292F} 0.6644} & {\\color[HTML]{24292F} 0.6396} & {\\color[HTML]{24292F} 0.7920} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5132} & {\\color[HTML]{24292F} 0.7701} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7349} & {\\color[HTML]{24292F} 0.7349} & {\\color[HTML]{24292F} 0.8246} & {\\color[HTML]{24292F} 0.6971} & {\\color[HTML]{24292F} 0.6560} & {\\color[HTML]{24292F} 0.6560} & {\\color[HTML]{24292F} 0.6724} & {\\color[HTML]{24292F} 0.7162} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.6471}} & {\\color[HTML]{24292F} \\textbf{0.6471}} & {\\color[HTML]{24292F} \\textbf{0.6618}} & {\\color[HTML]{24292F} \\textbf{0.7102}} & {\\color[HTML]{24292F} \\textbf{0.6796}} & {\\color[HTML]{24292F} \\textbf{0.6790}} & {\\color[HTML]{24292F} \\textbf{0.6746}} & {\\color[HTML]{24292F} \\textbf{0.7599}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.6774} & {\\color[HTML]{24292F} 0.6774} & {\\color[HTML]{24292F} 0.6539} & {\\color[HTML]{24292F} 0.7842} & {\\color[HTML]{24292F} 0.6307} & {\\color[HTML]{24292F} 0.6307} & {\\color[HTML]{24292F} 0.5998} & {\\color[HTML]{24292F} 0.7847} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.7463} & {\\color[HTML]{24292F} 0.7463} & {\\color[HTML]{24292F} 0.8221} & {\\color[HTML]{24292F} 0.7172} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.6289} & {\\color[HTML]{24292F} 0.7087} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.6767} & {\\color[HTML]{24292F} 0.6761} & {\\color[HTML]{24292F} 0.7026} & {\\color[HTML]{24292F} 0.7199} & {\\color[HTML]{24292F} 0.6497} & {\\color[HTML]{24292F} 0.6497} & {\\color[HTML]{24292F} 0.6715} & {\\color[HTML]{24292F} 0.6983} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.7973} & {\\color[HTML]{24292F} 0.6619} & {\\color[HTML]{24292F} 0.6194} & {\\color[HTML]{24292F} 0.6194} & {\\color[HTML]{24292F} 0.6652} & {\\color[HTML]{24292F} 0.6538} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7662} & {\\color[HTML]{24292F} 0.8413} & {\\color[HTML]{24292F} 0.6413} & {\\color[HTML]{24292F} 0.6413} & {\\color[HTML]{24292F} 0.6032} & {\\color[HTML]{24292F} 0.7821} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.5635} & {\\color[HTML]{24292F} 0.8123} & {\\color[HTML]{24292F} 0.6731} & {\\color[HTML]{24292F} 0.6731} & {\\color[HTML]{24292F} 0.6997} & {\\color[HTML]{24292F} 0.7158} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.8009} & {\\color[HTML]{24292F} 0.6438} & {\\color[HTML]{24292F} 0.6169} & {\\color[HTML]{24292F} 0.6169} & {\\color[HTML]{24292F} 0.8428} & {\\color[HTML]{24292F} 0.5326} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.7316} & {\\color[HTML]{24292F} 0.7289} & {\\color[HTML]{24292F} 0.6342} & {\\color[HTML]{24292F} 0.6342} & {\\color[HTML]{24292F} 0.6490} & {\\color[HTML]{24292F} 0.7176} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0447} & {\\color[HTML]{24292F} 0.0447} & {\\color[HTML]{24292F} 0.0846} & {\\color[HTML]{24292F} 0.0606} & {\\color[HTML]{24292F} 0.0301} & {\\color[HTML]{24292F} 0.0300} & {\\color[HTML]{24292F} 0.0762} & {\\color[HTML]{24292F} 0.0667} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.8318} & {\\color[HTML]{24292F} 0.8413} & {\\color[HTML]{24292F} 0.6796} & {\\color[HTML]{24292F} 0.6790} & {\\color[HTML]{24292F} 0.8428} & {\\color[HTML]{24292F} 0.7847} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.5635} & {\\color[HTML]{24292F} 0.6438} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5132} & {\\color[HTML]{24292F} 0.5326} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\section{Scanner differences:}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font). Scanner: Hamamatsu NanoZoomer 2.0-RS.}\n\\label{tab:my-table5}\n\\resizebox{0.8\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7361} & {\\color[HTML]{24292F} 0.7354} & {\\color[HTML]{24292F} 0.693} & {\\color[HTML]{24292F} 0.8106} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7234} & {\\color[HTML]{24292F} 0.7284} & {\\color[HTML]{24292F} 0.7451} & {\\color[HTML]{24292F} 0.734} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_00\\_cv\\_02}} & {\\color[HTML]{24292F} \\textbf{0.7388}} & {\\color[HTML]{24292F} \\textbf{0.7338}} & {\\color[HTML]{24292F} \\textbf{0.7155}} & {\\color[HTML]{24292F} \\textbf{0.7744}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_03} & {\\color[HTML]{24292F} 0.7384} & {\\color[HTML]{24292F} 0.732} & {\\color[HTML]{24292F} 0.7063} & {\\color[HTML]{24292F} 0.7818} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.7342} & {\\color[HTML]{24292F} 0.7324} & {\\color[HTML]{24292F} 0.7150} & {\\color[HTML]{24292F} 0.7752} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0063} & {\\color[HTML]{24292F} 0.0026} & {\\color[HTML]{24292F} 0.0191} & {\\color[HTML]{24292F} 0.0274} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7388} & {\\color[HTML]{24292F} 0.7354} & {\\color[HTML]{24292F} 0.7451} & {\\color[HTML]{24292F} 0.8106} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.7234} & {\\color[HTML]{24292F} 0.7284} & {\\color[HTML]{24292F} 0.693} & {\\color[HTML]{24292F} 0.734} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font). Scanner: Hamamatsu NanoZoomer S60.}\n\\label{tab:my-table6}\n\\resizebox{0.8\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.6286} & {\\color[HTML]{24292F} 0.6429} & {\\color[HTML]{24292F} 0.6121} & {\\color[HTML]{24292F} 0.7143} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_00\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.6757}} & {\\color[HTML]{24292F} \\textbf{0.6695}} & {\\color[HTML]{24292F} \\textbf{0.6931}} & {\\color[HTML]{24292F} \\textbf{0.6855}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.617} & {\\color[HTML]{24292F} 0.6448} & {\\color[HTML]{24292F} 0.6668} & {\\color[HTML]{24292F} 0.6443} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_03} & {\\color[HTML]{24292F} 0.6167} & {\\color[HTML]{24292F} 0.6445} & {\\color[HTML]{24292F} 0.6653} & {\\color[HTML]{24292F} 0.6452} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6345} & {\\color[HTML]{24292F} 0.6504} & {\\color[HTML]{24292F} 0.6593} & {\\color[HTML]{24292F} 0.6723} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0243} & {\\color[HTML]{24292F} 0.0110} & {\\color[HTML]{24292F} 0.0294} & {\\color[HTML]{24292F} 0.0294} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.6757} & {\\color[HTML]{24292F} 0.6695} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.7143} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6167} & {\\color[HTML]{24292F} 0.6429} & {\\color[HTML]{24292F} 0.6121} & {\\color[HTML]{24292F} 0.6443} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\end{document}\n\\section{Introduction}\nAccumulations of Amyloid-$\\beta$ and tau protein aggregates, such as plaques in the brain gray matter, are well-known biomarkers of the neurodegenerative Alzheimer's disease (AD) \\cite{ben1}. Quantitative estimation of plaques is typically done by pathologists manually or semi-automatically, using proprietary black-box software from histopathological images of the brain -- a time and effort-intensive process prone to human observation variability and errors. \nIn recent times, deep learning (DL) based methods have shown promising results in digital pathology \\cite{jano1} and incredibly high accuracy segmentation of digital whole slide images \\cite{anant}. In \\cite{wurtz}, three different DL models were used to segment tau aggregates (tangles) and nuclei in postmortem brain Whole Slide Images (WSIs). The three models included a fully convolutional neural network (FCN), UNet, and Segnet, the latter achieving the highest accuracy in terms of IoU. In \\cite{signaevsky}, an FCN was trained on a dataset of 22 WSIs for semantic segmentation of tangle objects from postmortem brain WSIs. Their model can segment tangles of varying morphologies with high accuracy under diverse staining intensities. An FCN model was also used in \\cite{Vega2021} to classify morphologies of tau protein aggregates in the gray and white matter regions from 37 WSIs representing multiple degenerative diseases. In \\cite{manouskova2022}, tau aggregate analysis was done on a dataset of 6 WSIs with a combined classification-segmentation framework which achieved an F1 score of 81.3\\% and 75.8\\% on detection and segmentation tasks, respectively. \nSeveral domains in DL-based histopathological analysis of AD tauopathy remain unexplored. Firstly, most existing studies have used DL to segment tangles rather than plaques, which are harder to identify against the background gray matter due to their diffuse\/sparse appearance. Secondly, annotations of whole slide images are frequently affected by errors by human annotators. In such cases, a  DL preliminary model may be trained using weakly annotated data and used to assist the expert in refining annotations. Thirdly, contemporary tau segmentation studies do not consider context information, which is essential in segmenting plaques from brain WSIs as these emerge as sparse objects against an extended background of gray matter. Finally, DL models with explainability features have not yet been applied in tau segmentation from WSIs. This is a critical requirement for DL models used in clinical applications \\cite{explain1} \\cite{Yamamoto2019}. The DL models should not only be able to identify regions of interest precisely but also give clinicians and general users the knowledge about which image features the model found necessary that influenced its decision. \nBased on the above, a DL pipeline for the segmentation of plaque regions in brain WSIs is presented in our study. This pipeline uses context and explainability features with a UNet-based semantic segmentation model to identify plaque features from WSIs.\n\n\\section{Methodology}\n\\label{sec:methodology}\n\n\\subsection{Dataset characteristics}\n\\label{sec:data_characteristics}\nIn this work, we analyzed eight whole slide images containing histological sections from the frontal cortices of patients with AD, which were provided by the French national brain biobank Neuro-CEB. Signed informed consent for autopsy and histologic analysis was obtained in all cases from the patients or their family members. The present cohort represents a common heterogeneity of AD cases, including slides with variable tau pathology (e.g., different object densities), variable staining quality, and variable tissue preservation. Sections of the frontal lobe were stained with AT8 antibody to reveal phosphorylated tau pathology, using a standardized immunohistochemistry protocol. Obtained slides were scanned using two Hamamatsu slide scanners (NanoZoomer 2.0-RS and NanoZoomer s60 with 227 nm\/pixel and 221 nm\/pixel resolution, respectively) at 40x initial magnification. The slides were used for human-CNN iterative object annotation resulting in about 4000 annotated and expert-validated Neuritic plaques. The labels, extracted in an XML format, constitute the segmentation ground truth.\n\n\\subsection{Data preparation}\nFrom the WSIs, at 20x magnification, patches with two levels of context information were generated using an ROI-guided sampling method. The larger patches (256x256 pixels) capture a broader context containing object neighborhood and background pixels, whereas the smaller (128x 128 pixels) mainly focus on the plaque region without much context information. The amount of context present in each patch is quantified using a ratio of the area of annotated ROI to the total area of the patch. The plaque example in different patch sizes is shown in Fig~\\ref{fig:context} (note that the bigger patch has additional objects-plaques). In addition, two different normalizations are used and compared: Macenko~\\cite{macenko} and Vahadane~\\cite{vahadane2015normalisation} methods.\n\nA new scheme for data augmentation was implemented based on ROI-shifting to prevent the networks' bias from focussing on the center location of plaques in the patches. Accordingly, the annotated plaque ROIs are shifted to four corners of a patch, producing a four-fold augmentation of each patch containing an object. This augmentation aims to train the UNet models robustly in the presence of variable neighborhood context information, especially when closely-spaced plaque objects are present. An example of this augmentation is shown in Fig~\\ref{fig:ROI_aug}.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6] {images\/patchsize2}\n    \\caption{Example of plaque image for different levels of context.}\n    \\label{fig:context}\n\\end{figure}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.5] {images\/augmentations1}\n    \\caption{Example of ROI shifting augmentation.}\n    \\label{fig:ROI_aug}\n\\end{figure}\n\n\\subsection{Deep learning architecture for segmentation}\n\nIn order to segment the neuritic plaques, a UNet model adapted from \\cite{Ronnenbunet} is used with modifications for accommodating context information within the WSI patches during training and testing. The model architecture is modified to work with the two datasets containing different patch sizes -- i.e., $128\\times128$ (having low context information) and $256\\times256$ pixels (having more information about the plaque neighborhood). For the first dataset, the UNet architecture consists of 3 downsampling and 3 upsampling convolutional blocks, in addition to the convolutional middle block. For the $256\\times256$-patch-size dataset, we added a downsampling and upsampling convolutional block to the previous UNet model. For the downsampling block, we used a leaky ReLU activation function and ReLU for the upsampling block. In both blocks, we used batch-normalization following the suggestions in \\cite{Ioffe2015BatchNA} and \\cite{manouskova2022}. Dropout was used in each convolutional block with a probability of 0.5.\n\n\\subsection{Deep learning architecture for visual interpretation} \n\nIn addition to the segmentation, we focus on deriving locations within the patches where the DL model found significant features from the plaque objects. Therefore, we used an attention UNet described in \\cite{oktay2018attention}, which allows us to visualize the activated features at each iteration and evaluate qualitatively where the network focuses during training. The attention UNet architecture was also modified for the two different patch-size datasets following a configuration similar to the one described for the UNet.\n\n\\section{Experiments and results}\nData preparation and UNet experiments were executed on an 12-core Intel(R) Core i9-9920X @ 3.5GHz CPU with 128 GB RAM and two 12 GB RAM Nvidia GeForce RTX 2080 Ti GPUs. The attention UNet experiments run on a cluster (1 GPU Tesla V100S-PCIe-32GB, 12 CPU cores Intel(R) Xeon(R) Gold 6126 CPU @ 2.60GHz, and 80 GB of RAM). The average training and evaluation time of the UNet per epoch is approximately 2 minutes for the $128\\times 128$ patch-size database and 5 minutes for the $256\\times 256$ patch-size database. Meanwhile, for the attention UNet, approximately half the time is needed. On the other hand, data preprocessing takes 2 to 5 hours to process using parallel computation. Regarding memory consumption, we used at most 6 GB of GPU RAM for the larger patch dataset. In order to increase the performance, we cache the data and annotations first in CPU RAM and then move them to the GPU.\n\nWe randomly divided the 8 WSIs into 4 folds for the DL experiments. Then, we tested the network using a 4-fold cross-testing scheme, and with the remaining data from each test fold, we also performed a 3-fold cross-validation. In addition, we run a series of tests (using these folds) to select the loss function and the best optimizer for the UNet and attention UNet. We tested 4 loss functions (i.e., Focal loss, BCEwithLogits, Dice, and BCE-Dice loss) and 4 different optimizers (i.e., SGD, Adam, RMSProp, and AdaDelta). After the hyperparameter tuning, we obtained the best performance using the BCE-Dice loss with a 50\\% balance between Dice and BCE (Binary Cross Entropy) and the Adadelta optimizer with $\\rho = 0.9$ and a varying learning rate based on the evolution of the validation loss. Also, we implemented early stopping for training with a patience value of 15 epochs.\n\n\\subsection{Results from UNet architecture}\nThe segmentation evaluation metric used for all of the experiments regarding the UNet is the Dice score which is equivalent to the F1 score for binary segmentation problems. In the first experiment, the UNet model was trained with two datasets having different patch sizes: $128\\times 128$ and $256\\times 256$ pixels. The mean and standard deviations of the Dice coefficient for cross-validation and cross-testing are reported in Table~\\ref{tab:dice_results1}. The patches were previously normalized using the Macenko method and then separated in their corresponding fold for training, validation, and testing following the scheme described above. We observe a decrease in the Dice score for larger patches having additional environmental context from the neuritic plaque.\n\n\\begin{table}[ht]\n\\centering\n\\caption{UNet results (Dice score) for 4-fold cross testing and 3-fold cross validation for different patch sizes.}\n\\begin{tabular}[t]{|c|c|c|c|}\n\\hline\nPatch size & Normalization & Cross validation & Cross testing\\\\\n\\hline\n$128\\times128$ & Macenko & $ 0.6954 \\pm 0.0289 $ & $0.6852 \\pm 0.0260$\\\\\n$256\\times256$ & Macenko & $0.6600 \\pm 0.0420 $ & $0.6460 \\pm 0.0330$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:dice_results1}\n\\end{table}%\n\nAs described, the WSIs were acquired using two different scanners. Therefore, to study the impact of its properties, we divided the entire cohort into two independent datasets: 4 WSIs belonging to the NanoZoomer 2.0-RS and 4 WSIs scanned with the NanoZoomer s60. For both datasets, we only evaluate the performance of the DL architecture using 4-fold cross-validation and patches of $128\\times 128$ pixels size. Additionally, we normalize each dataset independently (i.e., using two reference patches: one for the NanoZoomer 2.0-RS and one for the NanoZoomer s60) using the Macenko method. The Dice score obtained using the images from the higher resolution Hamamatsu NanoZoomer S60 scanner was $0.6345 \\pm 0.0243$, whereas that from the NanoZoomer 2.0-RS was $0.7342 \\pm 0.0063$.\n\nWe also study the effect of normalization in the entire dataset (8 WSIs). We normalized the patches from the $128\\times 128$ dataset using Macenko and Vahadane methods, and we selected the best fold (i.e., highest Dice score in testing for the first experiment) to train, validate and test the UNet under different input color properties. Opposite to the results reported in~\\cite{manouskova2022}, the Dice score obtained was higher using the Macenko method (0.7248 in testing) than the Vahadane (0.7098 in testing), even in validation (0.72313 for Macenko and 0.6864 for Vahadane). For a full list of results, see supplementary material.\n\n\\subsection{Visual deep learning interpretation}\nThe attention UNet model was trained using the $128\\times 128$ and the $256\\times 256$ patch size dataset, and the results are summarized in Table~\\ref{tab:attunet_results1}. All images were normalized using the Macenko method, and we observed a similar trend as the UNet: better performance using patches containing less background information.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Attention UNet results (Dice score) for 4-fold cross testing and 3-fold cross validation for different patch sizes.}\n\\begin{tabular}[t]{|c|c|c|c|}\n\\hline\nPatch size & Normalization & Cross validation & Cross testing\\\\\n\\hline\n$128\\times128$ & Macenko & $ 0.7516 \\pm 0.0334 $ & $0.6920 \\pm 0.0254$\\\\\n$256\\times256$ & Macenko & $0.6931 \\pm 0.0447 $ & $0.6342 \\pm 0.0301$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:attunet_results1}\n\\end{table}%\n\nAn example segmentation result from the attention UNet model in a $128\\times128$ patch containing a plaque object and its corresponding ground-truth mask is shown in Fig~\\ref{fig:attunet1}. We observe that the attention UNet model finds significant activation features around the plaque object initially annotated by experts (see ground truth mask in Fig~\\ref{fig:attunet1}). We also notice that the loss at iteration 100 increases over iteration 1; however, we clearly distinguish the region of the object (dark red color). After 1000 iterations, the loss decreases 50\\% due to the fact that the Dice part of the BCE-Dice loss function influences the network into detecting a pattern very similar to the given ground truth.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.25] {images\/att_unet_res1}\n    \\caption{Global coherence of attention-UNet result with human annotation.}\n    \\label{fig:attunet1}\n\\end{figure}\n\nAnother result from attention UNet is in Fig~\\ref{fig:attunet2}. Here, the attention UNet focuses on 2 plaques initially annotated by a human expert. It also identifies strong activation features in regions with no ground truth annotations, which could indicate missed ROIs by human experts during the annotation process. Thus with the attention UNet, it is not only possible to segment the plaque objects but also to improve or refine the manual annotations by experts.\n\nWeak and imprecise annotations are frequently observed in histopathology arising from human or software errors. In such cases, deep learning attention maps could be useful to provide pathologists and biologists with refined annotations (e.g., precision on the boundaries of ROIs). An example is shown in Fig~\\ref{fig: attunet_vsexpert} where DL attention maps are closer to the shape of actual ROIs compared to human-made annotations.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.25] {images\/att_unet_res2}\n    \\caption{Focus progression using successive activation layers of attention-UNet.}\n     \\label{fig:attunet2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.7] {images\/attunet_compare}\n    \\caption{Improving human annotations using attention-based DL models.}\n     \\label{fig: attunet_vsexpert}\n\\end{figure}\n\n\\section{Discussion and conclusion}\nIn the presented work, we studied\/evaluated a number of factors that contribute to the segmentation of plaques from whole slide images using DL models. The key observations are the following: \n\\begin{enumerate}\n    \\item Use of biomarkers: the study in~\\cite{manouskova2022} uses the ALZ50 (used to discover compacted structures) biomarker, while our study uses the AT8 (majorly used in clinics, helps to discover all structures). We focus on AT8 in order to stay close to clinical protocols. The drawback is that this biomarker creates less compact structures meaning a slightly more difficult segmentation of the plaques, as our results support.\n    \\item Use of different modalities: using the AT8 biomarker, we analyzed 2 types of WSI scanners (see Section~\\ref{sec:data_characteristics}) with different resolutions. High-resolution scanners amplify the annotation errors (human-software). Accordingly, some results concerning the high-resolution scanners have been affected, generating lower dice scores. \n    \\item Context effect on results of DL models: We noticed that increasing the background information in the patches negatively affects the segmentation results, which can be explained by the imbalance between the foreground and background pixels. In future works, this will be addressed using adaptive loss functions to take advantage of context information around the ROIs.\n    \\item Attention maps: We observed that using the attention UNet model helps us see the weakness in the human-made annotations (see Fig~\\ref{fig: attunet_vsexpert}), generating precious insights about the segmentation DL protocol, which can be used to refine the annotations by improving the border of the detected objects. These refined patterns can be used for a morphology and topology pipeline toward a robust AD patient's stratification proof. In addition, quantitative results show better performance of the same UNet architecture with attention blocks.\n    \\item Comparison with state-of-the-art commercial software: We compared our WSI segmentation results with those generated by a commercial software. This software uses a UNet architecture with a VGG encoder which is different from our model. Our system outperforms this software (Dice score 0.63 for test), using the same WSI as the ones used in this paper. Besides, in this software, neither information about how patches are generated nor the type of normalization or pre-processing perfomed on the dataset is available.\n     \n\\end{enumerate}\nWhole slide histopathology images whose sizes range in giga-pixels often contain thousands of objects per image. As seen for plaques in this study, it becomes more challenging when the objects being annotated do not have clear boundaries separating them from their surrounding environments, which may give rise to errors in human-made annotations. We saw an example of how DL models with visual explanation properties can help pathologists refine the ROI identification process. Our future challenge is to create deep learning assistive tools that can improve human-made few and weak annotations, a generic problem of a wide range of biomedical applications. \n\n\\section*{Acknowlegements}\nThis research was supported by Mr Jean-Paul Baudecroux and The Big Brain Theory Program - Paris Brain Institute (ICM). The human samples were obtained from the Neuro-CEB brain bank (\\url{https:\/\/www.neuroceb.org\/en\/}) (BRIF Number 0033-00011),  partly  funded  by  the  patients' associations  ARSEP, ARSLA, \"Conna\u00eetre les Syndromes C\u00e9r\u00e9belleux\", France-DFT, France Parkinson and by Vaincre Alzheimer Fondation, to which we express our gratitude. We are also grateful to the patients and their families.\n\n\\bibliographystyle{splncs04}\n\n\\section{UNet and Attention UNet:}\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font).}\n\\label{tab:my-table1}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7151} & {\\color[HTML]{24292F} 0.7165} & {\\color[HTML]{24292F} 0.6674} & {\\color[HTML]{24292F} 0.8017} & {\\color[HTML]{24292F} 0.6753} & {\\color[HTML]{24292F} 0.6707} & {\\color[HTML]{24292F} 0.668} & {\\color[HTML]{24292F} 0.784} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7034} & {\\color[HTML]{24292F} 0.6933} & {\\color[HTML]{24292F} 0.699} & {\\color[HTML]{24292F} 0.718} & {\\color[HTML]{24292F} 0.7046} & {\\color[HTML]{24292F} 0.7035} & {\\color[HTML]{24292F} 0.7495} & {\\color[HTML]{24292F} 0.7475} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6963} & {\\color[HTML]{24292F} 0.6932} & {\\color[HTML]{24292F} 0.6873} & {\\color[HTML]{24292F} 0.7339} & {\\color[HTML]{24292F} 0.6781} & {\\color[HTML]{24292F} 0.6765} & {\\color[HTML]{24292F} 0.7423} & {\\color[HTML]{24292F} 0.7094} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7011} & {\\color[HTML]{24292F} 0.7037} & {\\color[HTML]{24292F} 0.714} & {\\color[HTML]{24292F} 0.7239} & {\\color[HTML]{24292F} 0.7032} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6684} & {\\color[HTML]{24292F} 0.8052} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.7231}} & {\\color[HTML]{24292F} \\textbf{0.7118}} & {\\color[HTML]{24292F} \\textbf{0.6763}} & {\\color[HTML]{24292F} \\textbf{0.7801}} & {\\color[HTML]{24292F} \\textbf{0.7248}} & {\\color[HTML]{24292F} \\textbf{0.7192}} & {\\color[HTML]{24292F} \\textbf{0.7105}} & {\\color[HTML]{24292F} \\textbf{0.8067}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.72} & {\\color[HTML]{24292F} 0.7217} & {\\color[HTML]{24292F} 0.7519} & {\\color[HTML]{24292F} 0.7185} & {\\color[HTML]{24292F} 0.7141} & {\\color[HTML]{24292F} 0.7068} & {\\color[HTML]{24292F} 0.6811} & {\\color[HTML]{24292F} 0.8166} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.709} & {\\color[HTML]{24292F} 0.7156} & {\\color[HTML]{24292F} 0.7195} & {\\color[HTML]{24292F} 0.7423} & {\\color[HTML]{24292F} 0.7027} & {\\color[HTML]{24292F} 0.7004} & {\\color[HTML]{24292F} 0.7043} & {\\color[HTML]{24292F} 0.7855} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.7127} & {\\color[HTML]{24292F} 0.7195} & {\\color[HTML]{24292F} 0.7012} & {\\color[HTML]{24292F} 0.7618} & {\\color[HTML]{24292F} 0.6643} & {\\color[HTML]{24292F} 0.6608} & {\\color[HTML]{24292F} 0.6444} & {\\color[HTML]{24292F} 0.7996} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6807} & {\\color[HTML]{24292F} 0.6838} & {\\color[HTML]{24292F} 0.6862} & {\\color[HTML]{24292F} 0.7167} & {\\color[HTML]{24292F} 0.6306} & {\\color[HTML]{24292F} 0.6296} & {\\color[HTML]{24292F} 0.6634} & {\\color[HTML]{24292F} 0.7316} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.6765} & {\\color[HTML]{24292F} 0.6813} & {\\color[HTML]{24292F} 0.6788} & {\\color[HTML]{24292F} 0.7183} & {\\color[HTML]{24292F} 0.6845} & {\\color[HTML]{24292F} 0.6855} & {\\color[HTML]{24292F} 0.8061} & {\\color[HTML]{24292F} 0.666} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6967} & {\\color[HTML]{24292F} 0.6244} & {\\color[HTML]{24292F} 0.8167} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.879} & {\\color[HTML]{24292F} 0.7981} & {\\color[HTML]{24292F} 0.6777} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.6112} & {\\color[HTML]{24292F} 0.6008} & {\\color[HTML]{24292F} 0.61} & {\\color[HTML]{24292F} 0.6325} & {\\color[HTML]{24292F} 0.6521} & {\\color[HTML]{24292F} 0.6545} & {\\color[HTML]{24292F} 0.8018} & {\\color[HTML]{24292F} 0.6245} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6954} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.6847} & {\\color[HTML]{24292F} 0.7387} & {\\color[HTML]{24292F} 0.6852} & {\\color[HTML]{24292F} 0.6986} & {\\color[HTML]{24292F} 0.7198} & {\\color[HTML]{24292F} 0.7462} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0289} & {\\color[HTML]{24292F} 0.0313} & {\\color[HTML]{24292F} 0.0373} & {\\color[HTML]{24292F} 0.0462} & {\\color[HTML]{24292F} 0.0260} & {\\color[HTML]{24292F} 0.0596} & {\\color[HTML]{24292F} 0.0561} & {\\color[HTML]{24292F} 0.0615} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7231} & {\\color[HTML]{24292F} 0.7217} & {\\color[HTML]{24292F} 0.7519} & {\\color[HTML]{24292F} 0.8167} & {\\color[HTML]{24292F} 0.7248} & {\\color[HTML]{24292F} 0.879} & {\\color[HTML]{24292F} 0.8061} & {\\color[HTML]{24292F} 0.8166} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6112} & {\\color[HTML]{24292F} 0.6008} & {\\color[HTML]{24292F} 0.61} & {\\color[HTML]{24292F} 0.6325} & {\\color[HTML]{24292F} 0.6306} & {\\color[HTML]{24292F} 0.6296} & {\\color[HTML]{24292F} 0.6444} & {\\color[HTML]{24292F} 0.6245} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: Attention UNet. Patch size: 128x128 pixels (best fold is reported in bold font).}\n\\label{tab:my-table2}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7654} & {\\color[HTML]{24292F} 0.7654} & {\\color[HTML]{24292F} 0.7833} & {\\color[HTML]{24292F} 0.8000} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.5959} & {\\color[HTML]{24292F} 0.8177} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7216} & {\\color[HTML]{24292F} 0.7216} & {\\color[HTML]{24292F} 0.7293} & {\\color[HTML]{24292F} 0.7890} & {\\color[HTML]{24292F} 0.6734} & {\\color[HTML]{24292F} 0.6734} & {\\color[HTML]{24292F} 0.6387} & {\\color[HTML]{24292F} 0.8258} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.7428} & {\\color[HTML]{24292F} 0.7422} & {\\color[HTML]{24292F} 0.7425} & {\\color[HTML]{24292F} 0.8108} & {\\color[HTML]{24292F} 0.6499} & {\\color[HTML]{24292F} 0.6499} & {\\color[HTML]{24292F} 0.6320} & {\\color[HTML]{24292F} 0.7883} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7646} & {\\color[HTML]{24292F} 0.7646} & {\\color[HTML]{24292F} 0.7876} & {\\color[HTML]{24292F} 0.7934} & {\\color[HTML]{24292F} 0.6851} & {\\color[HTML]{24292F} 0.6851} & {\\color[HTML]{24292F} 0.6582} & {\\color[HTML]{24292F} 0.7979} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_01} & {\\color[HTML]{24292F} 0.7272} & {\\color[HTML]{24292F} 0.7272} & {\\color[HTML]{24292F} 0.7916} & {\\color[HTML]{24292F} 0.7299} & {\\color[HTML]{24292F} 0.7128} & {\\color[HTML]{24292F} 0.7122} & {\\color[HTML]{24292F} 0.7454} & {\\color[HTML]{24292F} 0.7522} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.7335} & {\\color[HTML]{24292F} 0.7335} & {\\color[HTML]{24292F} 0.7593} & {\\color[HTML]{24292F} 0.7772} & {\\color[HTML]{24292F} 0.6909} & {\\color[HTML]{24292F} 0.6909} & {\\color[HTML]{24292F} 0.6836} & {\\color[HTML]{24292F} 0.7890} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.7658} & {\\color[HTML]{24292F} 0.7658} & {\\color[HTML]{24292F} 0.7946} & {\\color[HTML]{24292F} 0.7900} & {\\color[HTML]{24292F} 0.7184} & {\\color[HTML]{24292F} 0.7184} & {\\color[HTML]{24292F} 0.7109} & {\\color[HTML]{24292F} 0.8060} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_02\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.7159}} & {\\color[HTML]{24292F} \\textbf{0.7153}} & {\\color[HTML]{24292F} \\textbf{0.7565}} & {\\color[HTML]{24292F} \\textbf{0.7429}} & {\\color[HTML]{24292F} \\textbf{0.7263}} & {\\color[HTML]{24292F} \\textbf{0.7263}} & {\\color[HTML]{24292F} \\textbf{0.7950}} & {\\color[HTML]{24292F} \\textbf{0.7254}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.7809} & {\\color[HTML]{24292F} 0.7809} & {\\color[HTML]{24292F} 0.8232} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.6948} & {\\color[HTML]{24292F} 0.7241} & {\\color[HTML]{24292F} 0.7488} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8180} & {\\color[HTML]{24292F} 0.8559} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6959} & {\\color[HTML]{24292F} 0.6839} & {\\color[HTML]{24292F} 0.8008} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6843} & {\\color[HTML]{24292F} 0.7908} & {\\color[HTML]{24292F} 0.7140} & {\\color[HTML]{24292F} 0.7140} & {\\color[HTML]{24292F} 0.7781} & {\\color[HTML]{24292F} 0.7281} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.7857} & {\\color[HTML]{24292F} 0.7857} & {\\color[HTML]{24292F} 0.8297} & {\\color[HTML]{24292F} 0.7896} & {\\color[HTML]{24292F} 0.7024} & {\\color[HTML]{24292F} 0.7024} & {\\color[HTML]{24292F} 0.8435} & {\\color[HTML]{24292F} 0.6571} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.7516} & {\\color[HTML]{24292F} 0.7515} & {\\color[HTML]{24292F} 0.7750} & {\\color[HTML]{24292F} 0.7879} & {\\color[HTML]{24292F} 0.6920} & {\\color[HTML]{24292F} 0.6920} & {\\color[HTML]{24292F} 0.7074} & {\\color[HTML]{24292F} 0.7698} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0334} & {\\color[HTML]{24292F} 0.0335} & {\\color[HTML]{24292F} 0.0408} & {\\color[HTML]{24292F} 0.0301} & {\\color[HTML]{24292F} 0.0254} & {\\color[HTML]{24292F} 0.0254} & {\\color[HTML]{24292F} 0.0703} & {\\color[HTML]{24292F} 0.0469} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8193} & {\\color[HTML]{24292F} 0.8297} & {\\color[HTML]{24292F} 0.8559} & {\\color[HTML]{24292F} 0.7263} & {\\color[HTML]{24292F} 0.7263} & {\\color[HTML]{24292F} 0.8435} & {\\color[HTML]{24292F} 0.8258} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6962} & {\\color[HTML]{24292F} 0.6843} & {\\color[HTML]{24292F} 0.7299} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.6405} & {\\color[HTML]{24292F} 0.5959} & {\\color[HTML]{24292F} 0.6571} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 256x256 pixels (best fold is reported in bold font).}\n\\label{tab:my-table3}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.707} & {\\color[HTML]{24292F} 0.7095} & {\\color[HTML]{24292F} 0.6557} & {\\color[HTML]{24292F} 0.7962} & {\\color[HTML]{24292F} 0.6423} & {\\color[HTML]{24292F} 0.6363} & {\\color[HTML]{24292F} 0.5839} & {\\color[HTML]{24292F} 0.8071} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.6509} & {\\color[HTML]{24292F} 0.6563} & {\\color[HTML]{24292F} 0.6744} & {\\color[HTML]{24292F} 0.6604} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.6743} & {\\color[HTML]{24292F} 0.6563} & {\\color[HTML]{24292F} 0.7767} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6663} & {\\color[HTML]{24292F} 0.6657} & {\\color[HTML]{24292F} 0.6725} & {\\color[HTML]{24292F} 0.6826} & {\\color[HTML]{24292F} 0.6368} & {\\color[HTML]{24292F} 0.6322} & {\\color[HTML]{24292F} 0.6356} & {\\color[HTML]{24292F} 0.7269} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_00}} & {\\color[HTML]{24292F} \\textbf{0.6939}} & {\\color[HTML]{24292F} \\textbf{0.7099}} & {\\color[HTML]{24292F} \\textbf{0.7041}} & {\\color[HTML]{24292F} \\textbf{0.7319}} & {\\color[HTML]{24292F} \\textbf{0.6963}} & {\\color[HTML]{24292F} \\textbf{0.6893}} & {\\color[HTML]{24292F} \\textbf{0.6632}} & {\\color[HTML]{24292F} \\textbf{0.7998}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_01} & {\\color[HTML]{24292F} 0.6411} & {\\color[HTML]{24292F} 0.642} & {\\color[HTML]{24292F} 0.6018} & {\\color[HTML]{24292F} 0.7157} & {\\color[HTML]{24292F} 0.6856} & {\\color[HTML]{24292F} 0.6785} & {\\color[HTML]{24292F} 0.6468} & {\\color[HTML]{24292F} 0.8008} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.6793} & {\\color[HTML]{24292F} 0.6829} & {\\color[HTML]{24292F} 0.6892} & {\\color[HTML]{24292F} 0.6979} & {\\color[HTML]{24292F} 0.6718} & {\\color[HTML]{24292F} 0.6654} & {\\color[HTML]{24292F} 0.6487} & {\\color[HTML]{24292F} 0.7732} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.6893} & {\\color[HTML]{24292F} 0.7077} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.6395} & {\\color[HTML]{24292F} 0.6352} & {\\color[HTML]{24292F} 0.6104} & {\\color[HTML]{24292F} 0.766} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.6911} & {\\color[HTML]{24292F} 0.698} & {\\color[HTML]{24292F} 0.6344} & {\\color[HTML]{24292F} 0.8027} & {\\color[HTML]{24292F} 0.621} & {\\color[HTML]{24292F} 0.6155} & {\\color[HTML]{24292F} 0.5748} & {\\color[HTML]{24292F} 0.7816} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6462} & {\\color[HTML]{24292F} 0.649} & {\\color[HTML]{24292F} 0.6593} & {\\color[HTML]{24292F} 0.6638} & {\\color[HTML]{24292F} 0.5949} & {\\color[HTML]{24292F} 0.595} & {\\color[HTML]{24292F} 0.6676} & {\\color[HTML]{24292F} 0.6185} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.6432} & {\\color[HTML]{24292F} 0.6529} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.7029} & {\\color[HTML]{24292F} 0.6507} & {\\color[HTML]{24292F} 0.65} & {\\color[HTML]{24292F} 0.7633} & {\\color[HTML]{24292F} 0.6242} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6747} & {\\color[HTML]{24292F} 0.6877} & {\\color[HTML]{24292F} 0.6581} & {\\color[HTML]{24292F} 0.7453} & {\\color[HTML]{24292F} 0.6507} & {\\color[HTML]{24292F} 0.6521} & {\\color[HTML]{24292F} 0.8146} & {\\color[HTML]{24292F} 0.5945} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.5391} & {\\color[HTML]{24292F} 0.5355} & {\\color[HTML]{24292F} 0.5558} & {\\color[HTML]{24292F} 0.5535} & {\\color[HTML]{24292F} 0.5834} & {\\color[HTML]{24292F} 0.5856} & {\\color[HTML]{24292F} 0.8118} & {\\color[HTML]{24292F} 0.5149} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.660} & {\\color[HTML]{24292F} 0.666} & {\\color[HTML]{24292F} 0.654} & {\\color[HTML]{24292F} 0.706} & {\\color[HTML]{24292F} 0.646} & {\\color[HTML]{24292F} 0.642} & {\\color[HTML]{24292F} 0.673} & {\\color[HTML]{24292F} 0.715} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.042} & {\\color[HTML]{24292F} 0.046} & {\\color[HTML]{24292F} 0.042} & {\\color[HTML]{24292F} 0.063} & {\\color[HTML]{24292F} 0.033} & {\\color[HTML]{24292F} 0.031} & {\\color[HTML]{24292F} 0.077} & {\\color[HTML]{24292F} 0.096} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.707} & {\\color[HTML]{24292F} 0.710} & {\\color[HTML]{24292F} 0.716} & {\\color[HTML]{24292F} 0.803} & {\\color[HTML]{24292F} 0.696} & {\\color[HTML]{24292F} 0.689} & {\\color[HTML]{24292F} 0.815} & {\\color[HTML]{24292F} 0.807} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.5391} & {\\color[HTML]{24292F} 0.5355} & {\\color[HTML]{24292F} 0.5558} & {\\color[HTML]{24292F} 0.5535} & {\\color[HTML]{24292F} 0.5834} & {\\color[HTML]{24292F} 0.5856} & {\\color[HTML]{24292F} 0.5748} & {\\color[HTML]{24292F} 0.5149} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: Attention UNet. Patch size: 256x256 pixels (best fold is reported in bold font).}\n\\label{tab:my-table4}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} & {\\color[HTML]{24292F} \\textbf{test\\_dice}} & {\\color[HTML]{24292F} \\textbf{test\\_f1}} & {\\color[HTML]{24292F} \\textbf{test\\_recall}} & {\\color[HTML]{24292F} \\textbf{test\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7288} & {\\color[HTML]{24292F} 0.7288} & {\\color[HTML]{24292F} 0.8318} & {\\color[HTML]{24292F} 0.6819} & {\\color[HTML]{24292F} 0.6439} & {\\color[HTML]{24292F} 0.6439} & {\\color[HTML]{24292F} 0.6312} & {\\color[HTML]{24292F} 0.7429} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.6698} & {\\color[HTML]{24292F} 0.6698} & {\\color[HTML]{24292F} 0.7152} & {\\color[HTML]{24292F} 0.6855} & {\\color[HTML]{24292F} 0.6038} & {\\color[HTML]{24292F} 0.6038} & {\\color[HTML]{24292F} 0.5860} & {\\color[HTML]{24292F} 0.7462} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.6644} & {\\color[HTML]{24292F} 0.6644} & {\\color[HTML]{24292F} 0.6396} & {\\color[HTML]{24292F} 0.7920} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5132} & {\\color[HTML]{24292F} 0.7701} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_00} & {\\color[HTML]{24292F} 0.7349} & {\\color[HTML]{24292F} 0.7349} & {\\color[HTML]{24292F} 0.8246} & {\\color[HTML]{24292F} 0.6971} & {\\color[HTML]{24292F} 0.6560} & {\\color[HTML]{24292F} 0.6560} & {\\color[HTML]{24292F} 0.6724} & {\\color[HTML]{24292F} 0.7162} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_01\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.6471}} & {\\color[HTML]{24292F} \\textbf{0.6471}} & {\\color[HTML]{24292F} \\textbf{0.6618}} & {\\color[HTML]{24292F} \\textbf{0.7102}} & {\\color[HTML]{24292F} \\textbf{0.6796}} & {\\color[HTML]{24292F} \\textbf{0.6790}} & {\\color[HTML]{24292F} \\textbf{0.6746}} & {\\color[HTML]{24292F} \\textbf{0.7599}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_01\\_cv\\_02} & {\\color[HTML]{24292F} 0.6774} & {\\color[HTML]{24292F} 0.6774} & {\\color[HTML]{24292F} 0.6539} & {\\color[HTML]{24292F} 0.7842} & {\\color[HTML]{24292F} 0.6307} & {\\color[HTML]{24292F} 0.6307} & {\\color[HTML]{24292F} 0.5998} & {\\color[HTML]{24292F} 0.7847} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_00} & {\\color[HTML]{24292F} 0.7463} & {\\color[HTML]{24292F} 0.7463} & {\\color[HTML]{24292F} 0.8221} & {\\color[HTML]{24292F} 0.7172} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.6323} & {\\color[HTML]{24292F} 0.6289} & {\\color[HTML]{24292F} 0.7087} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_01} & {\\color[HTML]{24292F} 0.6767} & {\\color[HTML]{24292F} 0.6761} & {\\color[HTML]{24292F} 0.7026} & {\\color[HTML]{24292F} 0.7199} & {\\color[HTML]{24292F} 0.6497} & {\\color[HTML]{24292F} 0.6497} & {\\color[HTML]{24292F} 0.6715} & {\\color[HTML]{24292F} 0.6983} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_02\\_cv\\_02} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.6883} & {\\color[HTML]{24292F} 0.7973} & {\\color[HTML]{24292F} 0.6619} & {\\color[HTML]{24292F} 0.6194} & {\\color[HTML]{24292F} 0.6194} & {\\color[HTML]{24292F} 0.6652} & {\\color[HTML]{24292F} 0.6538} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_00} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7662} & {\\color[HTML]{24292F} 0.8413} & {\\color[HTML]{24292F} 0.6413} & {\\color[HTML]{24292F} 0.6413} & {\\color[HTML]{24292F} 0.6032} & {\\color[HTML]{24292F} 0.7821} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_01} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.5635} & {\\color[HTML]{24292F} 0.8123} & {\\color[HTML]{24292F} 0.6731} & {\\color[HTML]{24292F} 0.6731} & {\\color[HTML]{24292F} 0.6997} & {\\color[HTML]{24292F} 0.7158} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_03\\_cv\\_02} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.6783} & {\\color[HTML]{24292F} 0.8009} & {\\color[HTML]{24292F} 0.6438} & {\\color[HTML]{24292F} 0.6169} & {\\color[HTML]{24292F} 0.6169} & {\\color[HTML]{24292F} 0.8428} & {\\color[HTML]{24292F} 0.5326} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.7316} & {\\color[HTML]{24292F} 0.7289} & {\\color[HTML]{24292F} 0.6342} & {\\color[HTML]{24292F} 0.6342} & {\\color[HTML]{24292F} 0.6490} & {\\color[HTML]{24292F} 0.7176} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0447} & {\\color[HTML]{24292F} 0.0447} & {\\color[HTML]{24292F} 0.0846} & {\\color[HTML]{24292F} 0.0606} & {\\color[HTML]{24292F} 0.0301} & {\\color[HTML]{24292F} 0.0300} & {\\color[HTML]{24292F} 0.0762} & {\\color[HTML]{24292F} 0.0667} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.7855} & {\\color[HTML]{24292F} 0.8318} & {\\color[HTML]{24292F} 0.8413} & {\\color[HTML]{24292F} 0.6796} & {\\color[HTML]{24292F} 0.6790} & {\\color[HTML]{24292F} 0.8428} & {\\color[HTML]{24292F} 0.7847} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.6199} & {\\color[HTML]{24292F} 0.5635} & {\\color[HTML]{24292F} 0.6438} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5638} & {\\color[HTML]{24292F} 0.5132} & {\\color[HTML]{24292F} 0.5326} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\section{Scanner differences:}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font). Scanner: Hamamatsu NanoZoomer 2.0-RS.}\n\\label{tab:my-table5}\n\\resizebox{0.8\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.7361} & {\\color[HTML]{24292F} 0.7354} & {\\color[HTML]{24292F} 0.693} & {\\color[HTML]{24292F} 0.8106} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_01} & {\\color[HTML]{24292F} 0.7234} & {\\color[HTML]{24292F} 0.7284} & {\\color[HTML]{24292F} 0.7451} & {\\color[HTML]{24292F} 0.734} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_00\\_cv\\_02}} & {\\color[HTML]{24292F} \\textbf{0.7388}} & {\\color[HTML]{24292F} \\textbf{0.7338}} & {\\color[HTML]{24292F} \\textbf{0.7155}} & {\\color[HTML]{24292F} \\textbf{0.7744}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_03} & {\\color[HTML]{24292F} 0.7384} & {\\color[HTML]{24292F} 0.732} & {\\color[HTML]{24292F} 0.7063} & {\\color[HTML]{24292F} 0.7818} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.7342} & {\\color[HTML]{24292F} 0.7324} & {\\color[HTML]{24292F} 0.7150} & {\\color[HTML]{24292F} 0.7752} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0063} & {\\color[HTML]{24292F} 0.0026} & {\\color[HTML]{24292F} 0.0191} & {\\color[HTML]{24292F} 0.0274} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.7388} & {\\color[HTML]{24292F} 0.7354} & {\\color[HTML]{24292F} 0.7451} & {\\color[HTML]{24292F} 0.8106} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.7234} & {\\color[HTML]{24292F} 0.7284} & {\\color[HTML]{24292F} 0.693} & {\\color[HTML]{24292F} 0.734} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Architecture used: UNet. Patch size: 128x128 pixels (best fold is reported in bold font). Scanner: Hamamatsu NanoZoomer S60.}\n\\label{tab:my-table6}\n\\resizebox{0.8\\textwidth}{!}{%\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\color[HTML]{24292F} \\textbf{fold\\_name}} & {\\color[HTML]{24292F} \\textbf{dev\\_dice}} & {\\color[HTML]{24292F} \\textbf{dev\\_f1}} & {\\color[HTML]{24292F} \\textbf{dev\\_recall}} & {\\color[HTML]{24292F} \\textbf{dev\\_precision}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_00} & {\\color[HTML]{24292F} 0.6286} & {\\color[HTML]{24292F} 0.6429} & {\\color[HTML]{24292F} 0.6121} & {\\color[HTML]{24292F} 0.7143} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{test\\_00\\_cv\\_01}} & {\\color[HTML]{24292F} \\textbf{0.6757}} & {\\color[HTML]{24292F} \\textbf{0.6695}} & {\\color[HTML]{24292F} \\textbf{0.6931}} & {\\color[HTML]{24292F} \\textbf{0.6855}} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_02} & {\\color[HTML]{24292F} 0.617} & {\\color[HTML]{24292F} 0.6448} & {\\color[HTML]{24292F} 0.6668} & {\\color[HTML]{24292F} 0.6443} \\\\ \\hline\n{\\color[HTML]{24292F} test\\_00\\_cv\\_03} & {\\color[HTML]{24292F} 0.6167} & {\\color[HTML]{24292F} 0.6445} & {\\color[HTML]{24292F} 0.6653} & {\\color[HTML]{24292F} 0.6452} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{mean}} & {\\color[HTML]{24292F} 0.6345} & {\\color[HTML]{24292F} 0.6504} & {\\color[HTML]{24292F} 0.6593} & {\\color[HTML]{24292F} 0.6723} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{std}} & {\\color[HTML]{24292F} 0.0243} & {\\color[HTML]{24292F} 0.0110} & {\\color[HTML]{24292F} 0.0294} & {\\color[HTML]{24292F} 0.0294} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{max}} & {\\color[HTML]{24292F} 0.6757} & {\\color[HTML]{24292F} 0.6695} & {\\color[HTML]{24292F} 0.6931} & {\\color[HTML]{24292F} 0.7143} \\\\ \\hline\n{\\color[HTML]{24292F} \\textbf{min}} & {\\color[HTML]{24292F} 0.6167} & {\\color[HTML]{24292F} 0.6429} & {\\color[HTML]{24292F} 0.6121} & {\\color[HTML]{24292F} 0.6443} \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\end{document}\n\\section{Introduction}\nAccumulations of Amyloid-$\\beta$ and tau protein aggregates, such as plaques in the brain gray matter, are well-known biomarkers of the neurodegenerative Alzheimer's disease (AD) \\cite{ben1}. Quantitative estimation of plaques is typically done by pathologists manually or semi-automatically, using proprietary black-box software from histopathological images of the brain -- a time and effort-intensive process prone to human observation variability and errors. \nIn recent times, deep learning (DL) based methods have shown promising results in digital pathology \\cite{jano1} and incredibly high accuracy segmentation of digital whole slide images \\cite{anant}. In \\cite{wurtz}, three different DL models were used to segment tau aggregates (tangles) and nuclei in postmortem brain Whole Slide Images (WSIs). The three models included a fully convolutional neural network (FCN), UNet, and Segnet, the latter achieving the highest accuracy in terms of IoU. In \\cite{signaevsky}, an FCN was trained on a dataset of 22 WSIs for semantic segmentation of tangle objects from postmortem brain WSIs. Their model can segment tangles of varying morphologies with high accuracy under diverse staining intensities. An FCN model was also used in \\cite{Vega2021} to classify morphologies of tau protein aggregates in the gray and white matter regions from 37 WSIs representing multiple degenerative diseases. In \\cite{manouskova2022}, tau aggregate analysis was done on a dataset of 6 WSIs with a combined classification-segmentation framework which achieved an F1 score of 81.3\\% and 75.8\\% on detection and segmentation tasks, respectively. \nSeveral domains in DL-based histopathological analysis of AD tauopathy remain unexplored. Firstly, most existing studies have used DL to segment tangles rather than plaques, which are harder to identify against the background gray matter due to their diffuse\/sparse appearance. Secondly, annotations of whole slide images are frequently affected by errors by human annotators. In such cases, a  DL preliminary model may be trained using weakly annotated data and used to assist the expert in refining annotations. Thirdly, contemporary tau segmentation studies do not consider context information, which is essential in segmenting plaques from brain WSIs as these emerge as sparse objects against an extended background of gray matter. Finally, DL models with explainability features have not yet been applied in tau segmentation from WSIs. This is a critical requirement for DL models used in clinical applications \\cite{explain1} \\cite{Yamamoto2019}. The DL models should not only be able to identify regions of interest precisely but also give clinicians and general users the knowledge about which image features the model found necessary that influenced its decision. \nBased on the above, a DL pipeline for the segmentation of plaque regions in brain WSIs is presented in our study. This pipeline uses context and explainability features with a UNet-based semantic segmentation model to identify plaque features from WSIs.\n\n\\section{Methodology}\n\\label{sec:methodology}\n\n\\subsection{Dataset characteristics}\n\\label{sec:data_characteristics}\nIn this work, we analyzed eight whole slide images containing histological sections from the frontal cortices of patients with AD, which were provided by the French national brain biobank Neuro-CEB. Signed informed consent for autopsy and histologic analysis was obtained in all cases from the patients or their family members. The present cohort represents a common heterogeneity of AD cases, including slides with variable tau pathology (e.g., different object densities), variable staining quality, and variable tissue preservation. Sections of the frontal lobe were stained with AT8 antibody to reveal phosphorylated tau pathology, using a standardized immunohistochemistry protocol. Obtained slides were scanned using two Hamamatsu slide scanners (NanoZoomer 2.0-RS and NanoZoomer s60 with 227 nm\/pixel and 221 nm\/pixel resolution, respectively) at 40x initial magnification. The slides were used for human-CNN iterative object annotation resulting in about 4000 annotated and expert-validated Neuritic plaques. The labels, extracted in an XML format, constitute the segmentation ground truth.\n\n\\subsection{Data preparation}\nFrom the WSIs, at 20x magnification, patches with two levels of context information were generated using an ROI-guided sampling method. The larger patches (256x256 pixels) capture a broader context containing object neighborhood and background pixels, whereas the smaller (128x 128 pixels) mainly focus on the plaque region without much context information. The amount of context present in each patch is quantified using a ratio of the area of annotated ROI to the total area of the patch. The plaque example in different patch sizes is shown in Fig~\\ref{fig:context} (note that the bigger patch has additional objects-plaques). In addition, two different normalizations are used and compared: Macenko~\\cite{macenko} and Vahadane~\\cite{vahadane2015normalisation} methods.\n\nA new scheme for data augmentation was implemented based on ROI-shifting to prevent the networks' bias from focussing on the center location of plaques in the patches. Accordingly, the annotated plaque ROIs are shifted to four corners of a patch, producing a four-fold augmentation of each patch containing an object. This augmentation aims to train the UNet models robustly in the presence of variable neighborhood context information, especially when closely-spaced plaque objects are present. An example of this augmentation is shown in Fig~\\ref{fig:ROI_aug}.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6] {images\/patchsize2}\n    \\caption{Example of plaque image for different levels of context.}\n    \\label{fig:context}\n\\end{figure}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.5] {images\/augmentations1}\n    \\caption{Example of ROI shifting augmentation.}\n    \\label{fig:ROI_aug}\n\\end{figure}\n\n\\subsection{Deep learning architecture for segmentation}\n\nIn order to segment the neuritic plaques, a UNet model adapted from \\cite{Ronnenbunet} is used with modifications for accommodating context information within the WSI patches during training and testing. The model architecture is modified to work with the two datasets containing different patch sizes -- i.e., $128\\times128$ (having low context information) and $256\\times256$ pixels (having more information about the plaque neighborhood). For the first dataset, the UNet architecture consists of 3 downsampling and 3 upsampling convolutional blocks, in addition to the convolutional middle block. For the $256\\times256$-patch-size dataset, we added a downsampling and upsampling convolutional block to the previous UNet model. For the downsampling block, we used a leaky ReLU activation function and ReLU for the upsampling block. In both blocks, we used batch-normalization following the suggestions in \\cite{Ioffe2015BatchNA} and \\cite{manouskova2022}. Dropout was used in each convolutional block with a probability of 0.5.\n\n\\subsection{Deep learning architecture for visual interpretation} \n\nIn addition to the segmentation, we focus on deriving locations within the patches where the DL model found significant features from the plaque objects. Therefore, we used an attention UNet described in \\cite{oktay2018attention}, which allows us to visualize the activated features at each iteration and evaluate qualitatively where the network focuses during training. The attention UNet architecture was also modified for the two different patch-size datasets following a configuration similar to the one described for the UNet.\n\n\\section{Experiments and results}\nData preparation and UNet experiments were executed on an 12-core Intel(R) Core i9-9920X @ 3.5GHz CPU with 128 GB RAM and two 12 GB RAM Nvidia GeForce RTX 2080 Ti GPUs. The attention UNet experiments run on a cluster (1 GPU Tesla V100S-PCIe-32GB, 12 CPU cores Intel(R) Xeon(R) Gold 6126 CPU @ 2.60GHz, and 80 GB of RAM). The average training and evaluation time of the UNet per epoch is approximately 2 minutes for the $128\\times 128$ patch-size database and 5 minutes for the $256\\times 256$ patch-size database. Meanwhile, for the attention UNet, approximately half the time is needed. On the other hand, data preprocessing takes 2 to 5 hours to process using parallel computation. Regarding memory consumption, we used at most 6 GB of GPU RAM for the larger patch dataset. In order to increase the performance, we cache the data and annotations first in CPU RAM and then move them to the GPU.\n\nWe randomly divided the 8 WSIs into 4 folds for the DL experiments. Then, we tested the network using a 4-fold cross-testing scheme, and with the remaining data from each test fold, we also performed a 3-fold cross-validation. In addition, we run a series of tests (using these folds) to select the loss function and the best optimizer for the UNet and attention UNet. We tested 4 loss functions (i.e., Focal loss, BCEwithLogits, Dice, and BCE-Dice loss) and 4 different optimizers (i.e., SGD, Adam, RMSProp, and AdaDelta). After the hyperparameter tuning, we obtained the best performance using the BCE-Dice loss with a 50\\% balance between Dice and BCE (Binary Cross Entropy) and the Adadelta optimizer with $\\rho = 0.9$ and a varying learning rate based on the evolution of the validation loss. Also, we implemented early stopping for training with a patience value of 15 epochs.\n\n\\subsection{Results from UNet architecture}\nThe segmentation evaluation metric used for all of the experiments regarding the UNet is the Dice score which is equivalent to the F1 score for binary segmentation problems. In the first experiment, the UNet model was trained with two datasets having different patch sizes: $128\\times 128$ and $256\\times 256$ pixels. The mean and standard deviations of the Dice coefficient for cross-validation and cross-testing are reported in Table~\\ref{tab:dice_results1}. The patches were previously normalized using the Macenko method and then separated in their corresponding fold for training, validation, and testing following the scheme described above. We observe a decrease in the Dice score for larger patches having additional environmental context from the neuritic plaque.\n\n\\begin{table}[ht]\n\\centering\n\\caption{UNet results (Dice score) for 4-fold cross testing and 3-fold cross validation for different patch sizes.}\n\\begin{tabular}[t]{|c|c|c|c|}\n\\hline\nPatch size & Normalization & Cross validation & Cross testing\\\\\n\\hline\n$128\\times128$ & Macenko & $ 0.6954 \\pm 0.0289 $ & $0.6852 \\pm 0.0260$\\\\\n$256\\times256$ & Macenko & $0.6600 \\pm 0.0420 $ & $0.6460 \\pm 0.0330$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:dice_results1}\n\\end{table}%\n\nAs described, the WSIs were acquired using two different scanners. Therefore, to study the impact of its properties, we divided the entire cohort into two independent datasets: 4 WSIs belonging to the NanoZoomer 2.0-RS and 4 WSIs scanned with the NanoZoomer s60. For both datasets, we only evaluate the performance of the DL architecture using 4-fold cross-validation and patches of $128\\times 128$ pixels size. Additionally, we normalize each dataset independently (i.e., using two reference patches: one for the NanoZoomer 2.0-RS and one for the NanoZoomer s60) using the Macenko method. The Dice score obtained using the images from the higher resolution Hamamatsu NanoZoomer S60 scanner was $0.6345 \\pm 0.0243$, whereas that from the NanoZoomer 2.0-RS was $0.7342 \\pm 0.0063$.\n\nWe also study the effect of normalization in the entire dataset (8 WSIs). We normalized the patches from the $128\\times 128$ dataset using Macenko and Vahadane methods, and we selected the best fold (i.e., highest Dice score in testing for the first experiment) to train, validate and test the UNet under different input color properties. Opposite to the results reported in~\\cite{manouskova2022}, the Dice score obtained was higher using the Macenko method (0.7248 in testing) than the Vahadane (0.7098 in testing), even in validation (0.72313 for Macenko and 0.6864 for Vahadane). For a full list of results, see supplementary material.\n\n\\subsection{Visual deep learning interpretation}\nThe attention UNet model was trained using the $128\\times 128$ and the $256\\times 256$ patch size dataset, and the results are summarized in Table~\\ref{tab:attunet_results1}. All images were normalized using the Macenko method, and we observed a similar trend as the UNet: better performance using patches containing less background information.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Attention UNet results (Dice score) for 4-fold cross testing and 3-fold cross validation for different patch sizes.}\n\\begin{tabular}[t]{|c|c|c|c|}\n\\hline\nPatch size & Normalization & Cross validation & Cross testing\\\\\n\\hline\n$128\\times128$ & Macenko & $ 0.7516 \\pm 0.0334 $ & $0.6920 \\pm 0.0254$\\\\\n$256\\times256$ & Macenko & $0.6931 \\pm 0.0447 $ & $0.6342 \\pm 0.0301$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:attunet_results1}\n\\end{table}%\n\nAn example segmentation result from the attention UNet model in a $128\\times128$ patch containing a plaque object and its corresponding ground-truth mask is shown in Fig~\\ref{fig:attunet1}. We observe that the attention UNet model finds significant activation features around the plaque object initially annotated by experts (see ground truth mask in Fig~\\ref{fig:attunet1}). We also notice that the loss at iteration 100 increases over iteration 1; however, we clearly distinguish the region of the object (dark red color). After 1000 iterations, the loss decreases 50\\% due to the fact that the Dice part of the BCE-Dice loss function influences the network into detecting a pattern very similar to the given ground truth.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.25] {images\/att_unet_res1}\n    \\caption{Global coherence of attention-UNet result with human annotation.}\n    \\label{fig:attunet1}\n\\end{figure}\n\nAnother result from attention UNet is in Fig~\\ref{fig:attunet2}. Here, the attention UNet focuses on 2 plaques initially annotated by a human expert. It also identifies strong activation features in regions with no ground truth annotations, which could indicate missed ROIs by human experts during the annotation process. Thus with the attention UNet, it is not only possible to segment the plaque objects but also to improve or refine the manual annotations by experts.\n\nWeak and imprecise annotations are frequently observed in histopathology arising from human or software errors. In such cases, deep learning attention maps could be useful to provide pathologists and biologists with refined annotations (e.g., precision on the boundaries of ROIs). An example is shown in Fig~\\ref{fig: attunet_vsexpert} where DL attention maps are closer to the shape of actual ROIs compared to human-made annotations.\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.25] {images\/att_unet_res2}\n    \\caption{Focus progression using successive activation layers of attention-UNet.}\n     \\label{fig:attunet2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n    \\centering\n    \\includegraphics[scale=0.7] {images\/attunet_compare}\n    \\caption{Improving human annotations using attention-based DL models.}\n     \\label{fig: attunet_vsexpert}\n\\end{figure}\n\n\\section{Discussion and conclusion}\nIn the presented work, we studied\/evaluated a number of factors that contribute to the segmentation of plaques from whole slide images using DL models. The key observations are the following: \n\\begin{enumerate}\n    \\item Use of biomarkers: the study in~\\cite{manouskova2022} uses the ALZ50 (used to discover compacted structures) biomarker, while our study uses the AT8 (majorly used in clinics, helps to discover all structures). We focus on AT8 in order to stay close to clinical protocols. The drawback is that this biomarker creates less compact structures meaning a slightly more difficult segmentation of the plaques, as our results support.\n    \\item Use of different modalities: using the AT8 biomarker, we analyzed 2 types of WSI scanners (see Section~\\ref{sec:data_characteristics}) with different resolutions. High-resolution scanners amplify the annotation errors (human-software). Accordingly, some results concerning the high-resolution scanners have been affected, generating lower dice scores. \n    \\item Context effect on results of DL models: We noticed that increasing the background information in the patches negatively affects the segmentation results, which can be explained by the imbalance between the foreground and background pixels. In future works, this will be addressed using adaptive loss functions to take advantage of context information around the ROIs.\n    \\item Attention maps: We observed that using the attention UNet model helps us see the weakness in the human-made annotations (see Fig~\\ref{fig: attunet_vsexpert}), generating precious insights about the segmentation DL protocol, which can be used to refine the annotations by improving the border of the detected objects. These refined patterns can be used for a morphology and topology pipeline toward a robust AD patient's stratification proof. In addition, quantitative results show better performance of the same UNet architecture with attention blocks.\n    \\item Comparison with state-of-the-art commercial software: We compared our WSI segmentation results with those generated by a commercial software. This software uses a UNet architecture with a VGG encoder which is different from our model. Our system outperforms this software (Dice score 0.63 for test), using the same WSI as the ones used in this paper. Besides, in this software, neither information about how patches are generated nor the type of normalization or pre-processing perfomed on the dataset is available.\n     \n\\end{enumerate}\nWhole slide histopathology images whose sizes range in giga-pixels often contain thousands of objects per image. As seen for plaques in this study, it becomes more challenging when the objects being annotated do not have clear boundaries separating them from their surrounding environments, which may give rise to errors in human-made annotations. We saw an example of how DL models with visual explanation properties can help pathologists refine the ROI identification process. Our future challenge is to create deep learning assistive tools that can improve human-made few and weak annotations, a generic problem of a wide range of biomedical applications. \n\n\\section*{Acknowlegements}\nThis research was supported by Mr Jean-Paul Baudecroux and The Big Brain Theory Program - Paris Brain Institute (ICM). The human samples were obtained from the Neuro-CEB brain bank (\\url{https:\/\/www.neuroceb.org\/en\/}) (BRIF Number 0033-00011),  partly  funded  by  the  patients' associations  ARSEP, ARSLA, \"Conna\u00eetre les Syndromes C\u00e9r\u00e9belleux\", France-DFT, France Parkinson and by Vaincre Alzheimer Fondation, to which we express our gratitude. We are also grateful to the patients and their families.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec1}\n\nFlow and transport are the most fundamental phenomena in subsurface porous media associated with various physical processes, e.g., oil and gas flow in petroleum reservoir \\cite{terry2015applied}, CO$_2$ sequestration \\cite{zhang2015sequentially}, water pollution dispersion \\cite{bear2010modeling}, etc. The numerical simulation and analyses of flow and transport in subsurface porous media are highly demanded in practical engineering and mechanism studies. However, the simulation results are always subject to the influence of uncertainties, mainly stemming from the inherent spatial heterogeneity of media properties caused by complex geological processes \\cite{boschan2012scale}. It has been widely recognized that in natural subsurface porous media, most properties, such as permeability, porosity, etc., exhibit an uneven spatial distribution. For example, the hydraulic conductivity can span several orders of magnitude in an aquifer or reservoir. How to quantificationally identify the influence of uncertainties of porous media properties on the flow and transport behaviors in subsurface physical processes has been a research hot spot in recent years.\n\nTherefore, the uncertainty quantification is an essential task in the simulation of practical subsurface flows where porous media properties that unknown or partially known are taken as the input parameters. A possible way to deal with uncertainties of subsurface porous media is to treat porous media properties as random fields, then perform the stochastic simulation on the subsurface flow governing equations with random coefficients to evaluate the quality of interest (QoI). Among commonly-used stochastic simulation methods, e.g. Monte Carlo(MC) method, stochastic finite element method, stochastic collocation method, the MC method demonstrates apparent advantages such as it is a non-intrusive approach that only the realization of coefficients is needed while the original model code remains unchanged, and it is more easily to be implemented. In the standard MC method, the computer-generated (pseudo) random points are used, and in many cases, the computational efficiency is always unsatisfied for large-scale problems. The Quasi-Monte Carlo (QMC) method improve the demerit of MC by using deterministic quasi-random points. These points exhibit lower discrepancy and distribute more uniformly in the probability space. Moreover, to reduce the sample variance and further improve the computational efficiency, the multilevel Monte Carlo (MLMC) method was proposed and developed by Heinrich \\cite{Heinrich2001MultilevelMC} and Giles \\cite{giles2008multilevel}. It applies the control variates technique that a series of discretization is adopted with increasing resolution and computes the QoI on each of them, the success of which lies in the effective variance reduction sequentially.\n\nIt should be mentioned that in the particular case of subsurface flow with random coefficients, the problem is further aggravated where very detailed geological models are needed (a large number of cells) for an accurate description of the flow. To further alleviate the computational burden connected to the evaluation of random parameter effects on subsurface flow using the MLMC method, in this study, we exploit the similar hierarchies of MLMC and multigrid methods and proposed a full multigrid multilevel (quasi-) Monte Carlo (FMG-MLQMC) approach. In this proposed method, the solution on coarse mesh $Q_l^c$ can be obtained as a byproduct of the full multigrid solution on fine mesh $Q_l^f$ on each mesh level $l$, instead of directly solving the equations on the coarse mesh as the standard MLMC does. The proposed FMG-MLQMC method saves the computation of the $Q_l^c$. There have been works coupling the multigrid solver with the multilevel framework, see \\cite{kumar2017multigrid,robbe2018recycling} for example. However, the FMG-MLQMC method we proposed saves the computational cost without modifying the MLMC framework. We exploit the implementation method for upscaling the random coefficient from fine mesh to neighboring coarse mesh. Although in this study we only focus on the simple single-phase subsurface flow with random coefficients, the proposed approach can be applied and extended naturally to multiphase flow and transfer in porous media and any other flow and transport problems associated with uncertainty effect. \n\nThe rest of the paper is organized as follows: In Section \\ref{Sec2}, we give a brief description of the single-phase subsurface flow and then introduce the proposed full multigrid multilevel (quasi-) Monte Carlo method in detail. The methodology on the upscaling method of random coefficients from the fine mesh to the coarse mesh, which preserves the random structure, is presented as well. In Section \\ref{Sec3}, we verify the effectiveness (a smaller estimator variance and faster convergence rate) of the presented method by comparing with standard MLMC method in two numerical experiments. Finally, in Section \\ref{Sec4}, we report the concluding remarks of this work along with a brief discussion of future directions.\n\n\\section{Algorithms}\\label{Sec2}\n\\subsection{Model problem and MLMC method}\n\nIn this work, we consider the following elliptic problem, \n\n\\begin{equation} \\label{eq:elliptic_pde}\n    \\left\\{\n    \\begin{aligned} \n        -\\nabla \\cdot (k(\\bm{x}, \\omega) \\nabla u(\\bm{x},\\omega)) &= f(\\bm{x}) \\quad \\text{ in } D \\\\\n        u(\\bm{x},\\omega) &= g(\\bm{x}) \\quad \\text{ on } \\Gamma_D\\\\\n        \\frac{\\partial u(\\bm{x},\\omega)}{\\partial \\bm{n}} &= v(\\bm{x}) \\quad \\text{ on } \\Gamma_N\\\\\n    \\end{aligned}\n    \\right.\n\\end{equation}\nwhere $k(\\bm{x},\\omega)$ is the random, spatial-varying coefficient, $D$ is the computational domain, $\\omega$ is a sample from the probability triple $(\\Omega, \\mathcal{F}, P)$. $\\Gamma_D$ and $\\Gamma_N$ are Dirichlet and Neumann boundaries respectively. In single-phase flow context, Eq.\\ref{eq:elliptic_pde} corresponds the steady-state situation, when $g$ and $v$ prescribe the pressure and velocity of the fluid at the boundary, then the solution $u$ depicts the pressure in the domain $\\Omega$. \n\nIn this work, we address the random elliptic problem using the multi-level algorithm. Basically, the MLMC method employs a series of control variates, which are often the discretized models with increasing resolution levels. Here, we associate each level with one mesh with given resolution. The approximations of quantity of interest(QoI) on these levels are denoted as $Q_0,Q_1,\\cdots,Q_L$, see Figure \\ref{fig:chap2_MLMC}. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{MLMC.png}\n    \\caption{Multilevel Monte Carlo}\n    \\label{fig:chap2_MLMC}\n\\end{figure}\n\nWe are interested in the approximation $Q_L$ on the finest level $L$. The MLMC method not only computes the solution on level $L$ itself, but also calculates the solutions on all the preceding meshes. The expectation of such quantity can be expressed by the following telescoping formula,\n\\begin{equation} \\label{eq:Chap2_Telescoping_Formula}\n    \\bb{E}[Q_L] = \\bb{E}[Q_0] + \\sum_{l=1}^{L} (\\bb{E}[Q_l] - \\bb{E}[Q_{l-1}])\n\\end{equation}\n\nWe can approximate each expectation using the Monte Carlo approach as follows,\n\\begin{equation} \\label{eq:Chap2_Telescoping_Formula_approx}\n\\bb{E}[Q_L]\\approx \\frac{1}{N_0}\\sum_{i=1}^{N_0}Q_0(\\omega_{i,0})+\\sum_{l=1}^{L}\\frac{1}{N_l}[\\sum_{i=1}^{N_l}(Q_l(\\omega_{i,l})-Q_{l-1}(\\omega_{i,l})]\n\\end{equation}\n\nHere, we associate level $l$ with the $l$-th term in the telescoping formula \\eqref{eq:Chap2_Telescoping_Formula}. Notice that on each level $l$, we use the same samples $\\omega_{i,l}$ to calculate $Q_l$ and $Q_{l-1}$. Then $Q_l$ and $Q_{l-1}$ are likely to correlate well, and the variance of $Q_l - Q_{l-1}$ will be small, see Eq.\\ref{eq:Chap2_variance_small}. \n\n\\begin{equation}\\label{eq:Chap2_variance_small}\n\\begin{split}\nV_l = \\bb{V}[Q_l - Q_{l-1}] &= \\bb{V} [Q_l] + \\bb{V} [Q_{l-1}] - 2Cov(Q_l,Q_{l-1})\\\\\n& \\ll \\bb{V} [Q_l] + \\bb{V} [Q_{l-1}]\n\\end{split}\n\\end{equation}\nwhere we denote $V_l$ as the variance of the $Q_l - Q_{l-1}$ on level $l$. \n\nAlso notice that on different levels, independent samples are used so that the variance of the multilevel estimator $Q_L$ is the summation of the variance on each level.\\\\\n\\begin{equation} \\label{eq:chap2_variance_sum}\n    \\bb{V}[Q_L] = \\sum_{l=0}^{L} \\bb{V}[Q_l - Q_{l-1}] = \\sum_{l=0}^L V_l,\n\\end{equation}\nwhere we let $Q_{-1} = 0$. If we write $Y_0=Q_0$ and $Y_l=Q_l-Q_{l-1}$, then\n\\begin{equation*}\n\\bb{E}[Q_L]=\\sum_{l=0}^{L}\\bb{E}[Y_l]\n\\end{equation*}\nLet $\\hat{Y}_l$ be an unbiased estimator for $E[Y_l]$, \n\\begin{equation*}\n\\begin{aligned}\n\\hat{Y}_0 &=\\frac{1}{N_0}\\sum_{i=1}^{N_0}Q_0(\\omega_{i,0})\\\\\n\\hat{Y}_l &=\\frac{1}{N_l}[\\sum_{i=1}^{N_l}(Q_l(\\omega_{i,l})-Q_{l-1}(\\omega_{i,l})] \\qquad {l=1,2,3,\\cdots,L}\n\\end{aligned}\n\\end{equation*}\nthen the multilevel estimator becomes,\n\\begin{equation}\n\\hat{Q}^{ML}_L=\\sum_{l=0}^{L}\\hat{Y}_l    \n\\end{equation}\n\n\n\\subsection{MLMC Complexity Theory}\nLet $Q$ denote a quantity of interest, and $Q_l$ denote the corresponding numerical approximation on $l$-th mesh. If we assume that the weak error and the level variance decreases exponentially while the cost per sample on each level increases exponentially, there exist positive constants $\\alpha$, $\\beta$ and $\\gamma$ satisfying the following,\n\\begin{equation}\n    \\begin{split}\n        \\left \\lvert \\bb{E}[Q_l-Q] \\right\\rvert &= \\mathcal{O}(2^{-\\alpha l})\\\\\n        \\bb{V}[Y_l] &= \\mathcal{O}(2^{-\\beta l})\\\\\n        C_l &= \\mathcal{O}(2^{\\gamma l})\n    \\end{split}\n    \\label{eq:chap2_assumption}\n\\end{equation}\nwhere $C_l$ is the cost per sample on level $l$. With the mean square error less than a threshold,\n\n\\begin{equation}\n\\bb{E}[(\\sum_{l=0}^{L} \\hat{Y}_l - \\bb{E}[Q])^2] = \\sum_{l=0}^L N_l^{-1} V_l+(\\bb{E}[\\hat{Q}^{ML}_L-Q])^2 < \\epsilon ^{2}\\\\\n\\label{eq:chap2_mlmc_computation_goal}\n\\end{equation}\nthe total computational cost satisfies,\n\\[\nC=\\left\\{\n    \\begin{array}{ll}\n         \\mathcal{O}(\\epsilon^{-2}) & \\beta > \\gamma\\\\\n         \\mathcal{O}(\\epsilon^{-2} (\\log \\epsilon )^2) & \\beta = \\gamma\\\\\n         \\mathcal{O}(\\epsilon^{-2-(\\gamma-\\beta)\/\\alpha})  & \\beta < \\gamma\\\\\n    \\end{array}\n\\right.\n\\]\nas $\\epsilon \\to 0$. \n\n\n\n\\subsection{MLMC Algorithm} \\label{Chapter:MLMC_Algorithm}\nThis subsection gives a MLMC algorithm initially proposed by M.Giles\\cite{giles2008multilevel}.\\\\\n\n\\begin{algorithm}[H]\n\\SetAlgoLined\n Start with $L=2$, set the initial number of samples on level 0,1,2\\\\\n \\While{extra samples need to be evaluated}{\n  evaluate $Q_l(\\omega_{i,l})$ and $Q_{l-1}(\\omega_{i,l})$, for $\\{i,l:dN_l \\neq 0, i = 1,\\cdots,dN_l\\}$\\\\\n  update estimates for $V_l$, $l=0,\\cdots,L$\\\\\n  update optimal $N_l$, compute the number of extra samples $dN_l$\\\\\n  \\eIf{$|\\bb{E}[Q_L-Q]| \\approx \\frac{|\\bb{E}[Q_L-Q_{L-1}]|}{(2^{\\alpha}-1)}<\\frac{\\epsilon}{\\sqrt{2}}$}{\n   \\textbf{break}\n   }{\n   $L=L+1$ and initialize $N_L$\\;\n  }\n }\n \\caption{MLMC Algorithm}\n \\label{alg:MLMC}\n\\end{algorithm}\n\nIn Algorithm \\ref{alg:MLMC}, the variances $V_l$ are approximated by the sample variances on the run. The weak error $\\abs{\\bb{E}[Q_L-Q]}$ is approximated by Richardson extrapolation $\\frac{\\abs{\\bb{E}[Q_L-Q_{L-1}]}}{(2^{\\alpha}-1)}$. \n\nAnd here we consider a equal split of the estimator variance and the approximation error, i.e.,\n\\begin{align}\n    \\sum_{l=0}^L N_l^{-1} V_l &< \\epsilon^2\/2\\\\\n    \\bb({E}[\\hat{Q}^{ML}_L - Q])^2 &< \\epsilon^2\/2. \\label{eq:Chap_2_variance_constraint}\n\\end{align}\nHowever, it is possible to determine the split factor in an optimal way\\cite{Collier2015}. \n\nThe optimal number of samples $N_l$ can be obtained by solving a constrained optimization : minimizing the total computational cost subject to the constraint Eq.\\ref{eq:Chap_2_variance_constraint}. \n\nMLMC algorithm will work under the following three conditions. \n\n\\paragraph{Convergence} The sequence $Q_0, Q_1, \\cdots, Q_L, \\cdots$ converges. Otherwise, the telescoping equation \\eqref{eq:Chap2_Telescoping_Formula} does not yield a converging result. \n\\paragraph{Correlation} $Q_l$ and $Q_{l-1}$ are estimated using the same underlying random sample $\\omega_{i,l}$ in equation \\eqref{eq:Chap2_Telescoping_Formula_approx}, and are thus well correlated. In this case, the estimator variance is significantly reduced. \n\\paragraph{Consistency} The telescoping sum \\eqref{eq:Chap2_Telescoping_Formula} introduces no bias error. Notice that in the telescoping equation, the term $Q_{l-1}$ for $l=1,\\cdots,L$ appears twice. However, the two $Q_{l-1}$ may be evaluated differently. If we denote the $l$-th term in the telescoping equation $Q_{l}$ and $Q_{l-1}$ by $Q_{l}^{f}$ and $Q_{l}^{c}$, respectively, which denote the fine mesh solution and coarse mesh solution on level $l$, then the condition $\\bb{E}[Q_{l-1}^f] = \\bb{E}[Q_l^c]$ needs to be satisfied in order to introduce no bias error in equation \\eqref{eq:Chap2_Telescoping_Formula}. The expectation of fine mesh solution on level $l-1$ should be the same as that of the coarse mesh solution on level $l$.\n\n\n\\subsection{MLQMC Algorithm}\n\nThe QMC method can solve integration problems as well. In contrast to the MC method, the QMC method replaces random points with deterministic points. Figure \\ref{fig:MC_Lattice_Sobol} gives an example of Monte Carlo points, lattice rule, and Sobol' sequence. \n\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}{.33\\textwidth}\n        \\centering\n        \\includegraphics[width=0.97\\textwidth]{Monte_Carlo_Random_64.png}\n        \\caption{Monte Carlo}\n    \\end{subfigure}\n    \\begin{subfigure}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=0.97\\textwidth]{Lattice_Sequence_64.png}\n        \\caption{Lattice rule}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=0.97\\textwidth]{Sobol_Sequence_64.png}\n        \\caption{Sobol' sequence}\n    \\end{subfigure}\n    \\caption{An example of Monte Carlo points, Lattice rule and Sobol' sequence in $[0,1]\\times[0,1]$ domain.}\n    \\label{fig:MC_Lattice_Sobol}\n\\end{figure}\n\nThe QMC approximation is given by,\n\\[\n\\mathcal{I}_{QMC}^{N}=\\frac{1}{N} \\sum_{i=1}^{N} Q(t_i) \\approx \\bb{E}[Q].\n\\]\n\nNotice that the points $\\{t_i\\}_{i=1}^{N}$ are deterministic. However, deterministic points yield biased estimates. In this work, we use the randomized QMC: random shift and digital scramble for lattice rule and Sobol' sequence respectively. The interested readers may refer to \\cite{dick2013high} for the above mentioned two randomization techniques. \n\nThere have been numerous studies combining the MLMC and QMC methods\\cite{giles2009multilevel,giles2016combining,kuo2017multilevel}. \nWe follow the multilevel quasi-Monte Carlo (MLQMC) settings from these works and list the algorithm here.\n\n\\begin{algorithm}[H]\n\\SetAlgoLined\nStart with $L=2$, set initial number of samples $N_0$ on level 0,1,2\\\\\n\\While{extra samples need to be evaluated}{\n\tevaluate $Q_l(\\omega_{i,l})$ and $Q_{l-1}(\\omega_{i,l})$, for $\\{i,l:dN_l \\neq 0, i = 1,\\cdots,dN_l\\}$\\\\\n\tupdate estimates for $V_l$, $l=0,\\cdots,L$ and compute $\\bb{V}[Q]$\\\\\n\t\\eIf{$\\bb{V}[Q] > \\epsilon^2\/2$}{\n\t\tselect level $l$ such that $l=\\text{argmax} \\frac{\\bb{V}[Y_l]}{N_lC_l}$ and double $N_l$}{\n\t\t\\eIf{$\\bb{E}[|Q_L-Q|]\\approx\\frac{\\abs{\\bb{E}[Q_L-Q_{L-1}]}}{(2^{\\alpha}-1)}<\\frac{\\epsilon}{\\sqrt{2}}$}{\n\t\t\t\\textbf{break}\n\t\t}{\n\t\t\t$L=L+1$ and initialize $N_L$ \n\t\t}\n\t}\n}\n\\caption{MLQMC Algorithm}\n\\end{algorithm}\n\n\\subsection{Multigrid}\n\nThe multigrid method was originally introduced to solve elliptic boundary-value problems efficiently. It has since been developed to solve either linear or non-linear systems. Multigrid methods compute the solution on a sequence of grids. Figure. \\ref{fig:FMG-MLMC} gives an illustration of the full multigrid scheme\n\nWe observe that, when the full multigrid solver is applied to the MLMC problem, based on the same level hierarchies, the solution on the coarse mesh $Q^c_l$ can be obtained as a byproduct of the multigrid solution on fine mesh $Q^f_l$ on each level $l$. Thus, in our proposed FMG-MLMC method, we have saved the computation for $Q^c_l$. Also notice that at the red circles in Figure. \\ref{fig:FMG-MLMC} the solutions are exact, since they are the end point of each V-cycle. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.80\\textwidth]{FMG-MLMC.png} \n    \\caption{An illustration of Full-Multigrid-Multilevel Monte Carlo method.}\n    \\label{fig:FMG-MLMC}\n\\end{figure}\n\nRecall that $Q_l(\\omega_{i,l})$ and $Q_{l-1}(\\omega_{i,l})$ are evaluated by the same underlying random sample. In level $l$, we denote $K_l^f$ and $K_l^c$ as the coefficients of the fine and coarse models, respectively. Also recall the consistency condition, since we use the same numerical solver for $Q_l^c$ and $Q_{l-1}^f$, we only require that $K_l^c$ and $K_{l-1}^f$ follow the same distribution. \n\n$K_l^f$ can be generated by the matrix decomposition method, KL-expansion method or other random field generation methods, see \\cite{Liu2019} for example. A way to generate $K_l^c$ is to coarsen $K_l^f$. In order to prevent bias error, $K_l^c$ should satisfy the same distribution law as $K_l^f$. Figure \\ref{fig:chap3_coarsening} shows a way of coarsening. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{coarsen.jpg}\n    \\caption{Coarsening}\n    \\label{fig:chap3_coarsening}\n\\end{figure}\nIn this scheme, the value of the coefficient in the coarse grid is selected to be the corresponding block in the fine grid. We denote the blocks in fine grid and coarse grid by $k^f_{i,j}$ and $k^c_{I,J}$ respectively, then we have the following,\n\n\\begin{equation*}\n    k^c_{I,J} = k^f_{2I-1, 2J-1}.\n\\end{equation*}\n\nHere we give an algorithm to describe our proposed FMG-MLMC method. \n\n\\begin{algorithm}[H]\n\\SetAlgoLined\n Start with $L=2$, set the initial number of samples on level 0,1,2\\\\\n \\While{extra samples need to be evaluated}{\n  coarsen $K_l^f$ to obtain $K_l^c$\\\\\n  use the multigrid solver to compute realizations $Q_l(\\omega_{i,l})$, and obtain $Q_{l-1}(\\omega_{i,l})$ as a byproduct, for $\\{i,l:dN_l \\neq 0, i = 1,\\cdots,dN_l\\}$\\\\\n  update estimates for $V_l$, $l=0,\\cdots,L$\\\\\n  update optimal $N_l$, compute number of extra samples $dN_l$\\\\\n  \\eIf{$|\\bb{E}[Q_L-Q]| \\approx \\frac{|\\bb{E}[Q_L-Q_{L-1}]|}{(2^{\\alpha}-1)}<\\frac{\\epsilon}{\\sqrt{2}}$}{\n   \\textbf{break}\n   }{\n   $L=L+1$ and initialize $N_L$\\;\n  }\n }\n \\caption{FMG-MLMC Algorithm}\n\\end{algorithm}\n\nNotice that in our scheme, no correlation is introduced into each levels. Thus the equation \\eqref{eq:chap2_variance_sum} still holds true. \n\n\\section{Numerical Validation}\\label{Sec3}\n\\subsection{Problem Statement}\\label{Sec3.1}\nRecall the elliptic problem \\ref{eq:elliptic_pde} in Section 2, now we focus the physical domain $\\Omega=[0,1]^2$. In this work, we consider cases in two different boundary conditions and quantities of interest, as listed in Table \\ref{tab:Chap4_cases}. Case I is of the Dirichlet boundary type, with pointwise output quantity. Case II is of the mixed Dirichlet-Neumann boundary condition, whose output is the outflow at the east boundary.\n\n\\begin{table}[h]\n\\centering\n\\caption{Case Settings}\n\\begin{tabular}{ccc}\\hline\nCase & Boundary Condition & QoI \\\\ \\hline\n1 & $u\\mid_{\\partial W} = 100$, $u\\mid_{\\partial E} = 0$, $u\\mid_{\\partial N} = 50$, $u\\mid_{\\partial S} = 10$ & $u(0.5,0.5)$ \\\\\n2 & $u\\mid_{\\partial W} = 100$, $u\\mid_{\\partial E} = 0$, $\\frac{\\partial u}{\\partial n}\\mid_{\\partial N}=0$, $\\frac{\\partial u}{\\partial n}\\mid_{\\partial S}=0$ & $\\int_{\\partial E} -k\\nabla udx$ \\\\\\hline\n\\end{tabular}\n\\label{tab:Chap4_cases}\n\\end{table}\n\n\\null\n\\noindent (1) \\textbf{Discretization}\\\\\nThe governing equation (\\ref{eq:elliptic_pde}) is discretized by the finite-volume method on rectangular grids. On level $l$ the degree of freedom is $2^{l+2} \\times 2^{l+2}$.\\\\\n(2) \\noindent\\textbf{Random Fields}\\\\\nWe choose the Mat\\'ern covariance function \n\\begin{equation} \\label{eq:Matern Covariance}\nC_\\nu (d)=\\sigma^2 \\frac{2^{1-\\nu}}{\\Gamma(\\nu)} (\\sqrt{2\\nu} \\frac{d}{\\lambda})^{\\nu} K_\\nu (\\sqrt{2\\nu}\\frac{d}{\\lambda}),\n\\end{equation}\nwhere $d$ is the Euclidean distance of two points, $\\lambda$ is the correlation length, and $\\nu$ controls the smoothness of the field.\n\nThe parameters of Mat\\'ern covariance for four different random fields are given in Table \\ref{table:Random Field Parameters Setting}.\n\\begin{table}[h] \n\\centering\n\\caption{Random Field Parameters Settings}\n\\label{table:Random Field Parameters Setting}\n\\begin{tabular}{cc}\\hline\nRandom Field & Parameters \\\\ \\hline\n1 & $\\nu = 0.5, \\lambda = 0.5, \\sigma^2 = 1$ \\\\\n2 & $\\nu = 0.5, \\lambda = 1, \\sigma^2 = 1$\\\\\n3 & $\\nu = 1, \\lambda = 0.5, \\sigma^2 = 1$\\\\\n4 & $\\nu = 1, \\lambda = 1, \\sigma^2 = 1$\\\\\\hline\n\\end{tabular}\n\\end{table} \n\nThe random fields are generated using the KL-expansion method. The truncation term is determined when $99\\%$ of the variability is captured, meaning that\n\\[\n\\frac{\\sum_{i=1}^{N_{KL}} \\theta_i}{\\sum_{i=1}^{\\infty} \\theta_i} = 99\\%,\n\\]\nwhere the summation of all eigenvalues satisfies the following\n\\[\n\\sum_{i=1}^{\\infty} \\theta_i = \\sigma^2 meas(\\Omega) = \\sigma^2 \\int_{\\Omega} dx,\n\\]\nReaders of interest can see \\cite{ernst2009efficient} for examples. $\\Omega$ is the random field region. \\\\\n(3) \\noindent\\textbf{QMC Method}\\\\\nIn QMC method, the Lattice rule points are generated using the software from \\cite{kuo2016application}, Sobol' matrices from \\cite{joe2008constructing} are used to generate Sobol' sequences. In both cases, 24 randomizations are applied. The confidence intervals are obtained by 10 sets of randomization.\n\n\\subsection{Numerical results}\\label{Sec3.2}\n\nIn this subsection we present the numerical results of the two cases (Tab. \\ref{tab:Chap4_cases}) with four random field settings (Tab. \\ref{table:Random Field Parameters Setting}). In each figure, the first and second row corresponds the cases $\\nu = 0.5$ and $\\nu = 1.0$ respectively, while the first column and second column corresponds the cases $\\lambda = 0.5$ and $\\lambda = 1.0$. We will first give the results of the first case in the following. \n\nThe simulation starts by estimating the asymptotic rates $\\alpha, \\beta, \\gamma$ in the assumptions \\eqref{eq:chap2_assumption}. Figs. \\ref{fig:mlmc_1_var}, \\ref{fig:mlqmc_lattice_1_var} and \\ref{fig:mlqmc_sobol_1_var} plot the variance of the QoI $Q_l$ and $Y_l = Q_l - Q_{l-1}$ against level $l$. By comparison, the QMC method reduces in the variance not in the asymptotic variance convergence rate, but in the y-axis offsets. \n\n\\begin{figure}[H]\n    \\begin{center} \n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{MLMC_Case_I_0.5_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{MLMC_Case_I_0.5_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{MLMC_Case_I_1.0_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{MLMC_Case_I_1.0_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n   \n    \\caption{Variance of $Q_l$ and $Y_l$ for 4 random fields}\n    \\label{fig:mlmc_1_var}\n\\end{figure}\n\nFor the results of MLQMC-Lattice, the variance test (Fig. \\ref{fig:mlqmc_lattice_1_var}) is presented. \n\n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_I_0.5_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_I_0.5_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_I_1.0_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_I_1.0_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Variance of $Q_l$ and $Y_l$ for 4 random fields}\n    \\label{fig:mlqmc_lattice_1_var}\n\\end{figure}\n\nFor the results of MLQMC-Sobol', the variance (Fig. \\ref{fig:mlqmc_sobol_1_var}) is presented. \n\n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_I_0.5_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_I_0.5_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_I_1.0_0.5_Variance}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_I_1.0_1.0_Variance}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Variance of $Q_l$ and $Y_l$ for 4 random fields}\n    \\label{fig:mlqmc_sobol_1_var}\n\\end{figure}\n\nThe QMC method does not affect the expectation value nor the computational cost. Here we skip the comparison of $\\alpha$ and $\\gamma$, but present the final computational cost of the three methods, given $\\epsilon$. The results in 4 random fields are presented in Fig. \\ref{fig:case1_comparison}. \n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_I_0.5_0.5_Total_Cost}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_I_0.5_1.0_Total_Cost}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_I_1.0_0.5_Total_Cost}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_I_1.0_1.0_Total_Cost}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Computational complexity of the three methods; the asymptotic rates are marked on the plot}\n    \\label{fig:case1_comparison}\n\\end{figure}\n\nFor case II, we test the QMC convergence rate and then present the computational complexity. \n\nNext the results of MLQMC-Lattice, the convergence test (Fig. \\ref{fig:mlqmc_lattice_2_test}) variance of level estimator $Y_l$ against $N_l$ in each level. The offset between the lines reveals the variance decreases with the levels. The comparison shows Sobol' sequence's advantage in decreasing variance. The random parameter settings have small impact on the variances in this case. \n\n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_II_0.5_0.5_QMC_Test}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_II_0.5_1.0_QMC_Test}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_II_1.0_0.5_QMC_Test}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Lattice_Case_II_1.0_1.0_QMC_Test}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Variance test of MLQMC-Lattice for all levels. The variance of the estimator is plotted as a function of the number of samples $N_l$. The convergence rates $\\lambda$ are marked on the plot}\n    \\label{fig:mlqmc_lattice_2_test}\n\\end{figure}\n\nNext, the results of MLQMC-Sobol', the convergence test (Fig. \\ref{fig:mlqmc_sobol_2_test})\n\n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_II_0.5_0.5_QMC_Test}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_II_0.5_1.0_QMC_Test}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_II_1.0_0.5_QMC_Test}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Sobol_Case_II_1.0_1.0_QMC_Test}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Variance test of MLQMC-Sobol for all levels. The variance of the estimator is plotted as a function of the number of samples $N_l$. The convergence rates $\\lambda$ are marked on the plot}\n    \\label{fig:mlqmc_sobol_2_test}\n\\end{figure}\n\nFinally, we compare the computational cost of the three methods, given $\\epsilon$. The results in 4 random fields are presented in Fig. \\ref{fig:case2_comparison}. \n\\begin{figure}[H]\n    \\begin{center}\n    \\hfill\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_Sobol_Case_II_0.5_0.5_Total_Cost}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_Sobol_Case_II_0.5_1.0_Total_Cost}.png}\n       \n    \\end{minipage}\\hfill\n    \\null \\\\\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_Sobol_Case_II_1.0_0.5_Total_Cost}.png}\n       \n    \\end{minipage}\n    \\begin{minipage}{0.33\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{{Comparison_Sobol_Case_II_1.0_1.0_Total_Cost}.png}\n       \n    \\end{minipage}\\hfill\n    \\end{center}\n    \\caption{Computational complexity of the three methods}\n    \\label{fig:case2_comparison}\n\\end{figure}\n\n\\section{Conclusions}\\label{Sec4}\n\nIn this work, we combined the MLMC with a full multigrid method. We saved the computation for the coarse grid solution on each level without modifying MLMC hierarchy and introducing correlation between different levels. We applied the consistent coarsening approach such that no bias error was introduced in the telescoping formula \\eqref{eq:Chap2_Telescoping_Formula}. \n\nWe tested our FMG-MLQMC algorithm on 2-D elliptic PDE with random coefficients for two different types of boundary condition settings and QoIs. The random coefficients were modelled as lognormal random fields with the Mat\\'ern covariance function with various parameter settings. We observed that quasi-Monte Carlo approaches have better performance on smoother random fields and problems with more regularity. Also, the comparison of Monte Carlo and quasi-Monte Carlo methods (including Lattice rule and Sobol' sequence) revealed that QMC outperforms MC due to a smaller estimator variance, and Sobol' sequence performs slightly better than Lattice rule.\n\n\n\nOne future work could be the substitution of the geometric multigrid solver with the algebraic multigrid (AMG) solver. In the AMG scheme, the grids are not associated with physical meshes, rather, the grids are fully determined by the matrix entries algebraically. \n\nAnother work could be to extend the elliptic model to more sophisticated models, such as two-phase porous flow. The efficient sampling and fast simulation of multi-phase subsurface flow under heterogeneous media could produce practical values.\n\nFurther work could extend the multilevel model to a multiscale model, and multiscale meshes rather than geometric meshes would be used. In this case, one level could correspond to one scale, and sampling would be performed on each scale. The literature on multiscale modeling can be found in \\cite{jenny2003multi,efendiev2009multiscale}. \n\n\\section*{Acknowledgements}\nThe authors gratefully acknowledge the support from the National Natural Science Foundation of China (Nos. 51874262, 51904031) and the Research Funding from King Abdullah University of Science and Technology (KAUST) through the grants BAS\/1\/1351-01-01.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe notion of generalized geometry goes back to the work of Hitchin \\cite{Hi1} (see also \\cite{Hi3}).  In this context,\n Witt  \\cite{W} introduced a  new type of structures on a $7$-dimensional\nmanifold  $M$ in terms of a differential form of mixed degree,\nthus generalizing the classical notion of $G_2$-structure\ndetermined by a stable and positive $3$-form. Instead of studying\ngeometry on the tangent bundle $TM$ of the manifold,  one\nconsiders the   bundle   $TM \\oplus T^*M$ endowed with a natural\norientation and an  inner product  of signature $(7, 7)$,  where\n$T^*M$ denotes  the cotangent bundle of $M$. In this way,  if $M$\nis spin,  then the differential form of mixed type can be viewed\nas a $G_2 \\times G_2$-invariant spinor  $\\rho$ for the  bundle and\nit is  called  the structure form.\n\nThese  structures are called {\\it generalized $G_2$-structures}\nand they induce a Riemannian metric, a $2$-form $b$ (the\n$B$-field), two unit spinors $\\Psi_{\\pm}$   and a function $\\phi$\n(the dilaton).   By \\cite{W},  any  $G_2 \\times G_2$-invariant\nspinor   $\\rho$ is stable and has a canonical  expression by $\\rho\n= e^{-\\phi} e^{\\frac{b}{2}} \\wedge (\\Psi_+ \\otimes\n\\Psi_-)^{ev,od}$ in terms of the two spinors, the $B$-field and\nthe dilaton function. In the paper we will  restrict to  the case\nof constant dilaton, i.e. $\\phi = {\\mbox const}$, and trivial\n$B$-field.\n\nUp to a $B$-field transformation, a generalized $G_2$-structure is\nessentially a pair of $G_2$-structures.  If the two spinors\n$\\Psi_+$ and $\\Psi_-$ are linearly independent, then the\nintersection of  the two isotropy groups, both  isomorphic to\n$G_2$, determined  by the two spinors coincides with $SU(3)$.\nTherefore,  one can express the structure form in terms of  the\nform $\\alpha$ dual to the unit vector stabilized by $SU(3)$ and of\nthe forms $(\\omega, \\psi = \\psi_+ + i \\psi_-)$, associated  with\n$SU(3)$, where $\\omega$ is the fundamental form and  $\\psi$  is\nthe complex volume form. Assuming that the angle between $\\Psi_+$\nand $\\Psi_-$ is $\\frac {\\pi} {2}$, then it turns out that\n\\begin{equation} \\label{expression}\n\\begin{array}{l}\n\\rho = (\\Psi_+ \\otimes \\Psi_-)^{ev} = \\omega +  \\psi_+ \\wedge\n\\alpha -  \\frac{1}{6}  \\omega^3 \\wedge \\alpha,\\\\ [5pt] \\hat \\rho =\n(\\Psi_+ \\otimes \\Psi_-)^{od} = \\alpha -  \\psi_-  -  \\frac{1}{2}\n\\omega^2 \\wedge \\alpha,\n\\end{array}\n\\end{equation}\nwhere $\\hat \\rho$ is the companion of $\\rho$ and $\\omega^k$\ndenotes the $k$-power wedge of $\\omega$. In this paper we will\nconsider generalized $G_2$-structures defined by the previous\nstructure forms.  In this case, the two associated\n$G_2$-structures do not coincide.\n\n\nIf $H$ is a $3$-form  (not necessarily closed) on $M$, then one can\n consider two\ntypes of  generalized $G_2$-structures with respect to the\n$3$-form $H$: \\newline the {\\em strongly integrable} ones, i.e.\nthose associated to  a structure form $\\rho$  which satisfies\n\\begin{equation} \\label{strongnotclosed}\nd  \\rho  + H \\wedge \\rho =  d \\hat \\rho + H \\wedge \\hat \\rho  =0,\n\\end{equation}\n and the {\\em\nweakly integrable} ones, i.e. those defined by the condition $$ d\n\\rho  + H \\wedge \\rho = \\lambda \\hat \\rho, $$ where $\\lambda$ is a\nnon-zero constant. The  previous structures are said  of even or\nodd type according to the parity of  $\\rho$.\n\nNote that these definitions of integrability are slightly\ndifferent from the ones given  in \\cite{W}, where  the closure of\nthe $3$-form $H$ is assumed.\n\n\n\nIf $H$ is closed, then   the twisted operator $d_H   \\cdot =  d\n\\cdot + H \\wedge \\cdot$ defines a differential complex and if, in\naddition, $M$ is compact, then  the strongly integrable\ngeneralized $G_2$-structures can be interpreted  as critical\npoints of a  a certain  functional  \\cite[Theorem 4.1]{W}. In this\ncase  the underlying spinors $\\Psi_{\\pm}$ are parallel with\nrespect to the Levi-Civita connection and therefore there exist no\nnon-trivial compact examples  with such structures, i.e. there are\nonly the classical examples of manifolds with   holonomy contained\nin $G_2$. If $H$ is not closed, then we will show that compact\nexamples can be constructed starting from a $6$-dimensional\nmanifold endowed with an $SU(3)$-structure.\n\nIf $H$ is closed, then the weakly integrable generalized\n$G_2$-structures can be also viewed as critical points of a\nfunctional under a constraint, but they have no classical\ncounterpart.  The existence of weakly integrable generalized\n$G_2$-structures  with respect to a closed $3$-form $H$ on a\ncompact manifold  was  posed as    an open problem in \\cite{W}. We\n construct such  structures on a family of compact manifolds\nand we  relate them with $SU(3)$-structures in dimension $7$,\nwhere $SU(3)$ is identified with the subgroup $SU(3) \\times \\{ 1\n\\}$ of $SO(7)$.\n\n\n\n\n\nAfter reviewing the general theory of generalized $G_2$-structures, in\n section 3 we  construct  a family of  compact  $7$-dimensional manifolds endowed\n with a weakly integrable generalized $G_2$-structure with respect to\n a closed and non-zero $3$-form $H$ (Theorem \\ref{example}).\n The corresponding structure form is the odd type form  $\\hat \\rho$  given by  \\eqref{expression}. These manifolds are\n obtained as a compact quotients $M_{\\beta}$ by  uniform discrete subgroups (parametrized by  the  p-th roots of  unity $e^{i \\beta}$) of\n a semi-direct product $SU(2) \\ltimes\n \\H$, where  $\\H$\n denotes the quaternions. It turns out that these manifolds have an $SU(3)$-structure $(\\omega, \\eta, \\psi)$ such that\n\\begin{equation}\\label{hyposystem}\nd \\eta = \\lambda \\omega, \\quad d (\\eta \\wedge \\psi_{\\pm}) =0.\n\\end{equation}\nIn particular they are contact metric. The structures satisfying\nthe condition \\eqref{hyposystem} can arise on hypersurfaces of\n$8$-dimensional manifolds with an integrable $SU(4)$-structure and\nthey are the analogous of the \\lq \\lq hypo\\rq \\rq\n$SU(2)$-structures in dimension $5$ (see \\cite{CS2}). In the same\nvein of \\cite{Hi1}, we consider a family $(\\omega(t), \\eta(t),\n\\psi(t))$ of $SU(3)$-structures containing the $SU(3)$-structure\n$(\\omega, \\eta, \\psi)$ and the corresponding evolution equations.\nIn this way  in section 4 we\n show that on the product of $M_{\\beta}$ with an open interval\nthere exists a Riemannian metric with  discrete holonomy\ncontained  in $SU(4)$ (Theorem 4.1).\n\nStarting from a $6$-dimensional manifold $N$ endowed with an\n$SU(3)$-structure $(\\omega, g, \\psi)$, it is possible to define in\na natural way a generalized $G_2$-structure with  the structure\nform $\\rho$ of even type given by \\eqref{expression} on the\nRiemannian product $(M = N \\times S^1, h)$, with\n$$\nh = g + dt \\otimes dt\n$$\nand $\\alpha = dt$.\n In \\cite{W} an example of this type  with\na $6$-dimensional nilmanifold $N$ was considered in order to\nconstruct a compact manifold endowed  with a  strongly integrable\ngeneralized $G_2$-structure with respect to a non-closed $3$-form\n$H$.\n\n\nWe will prove in general  that if $N$  is a $6$-dimensional\nmanifold endowed with an $SU(3)$-structure $(\\omega, g, \\psi)$,\nthen the generalized $G_2$-structure defined by $\\rho$ on $N\n\\times S^1$ satisfies the conditions \\eqref{strongnotclosed}, for\na non-zero $3$-form $H$, if and only if\n\\begin{equation}\\label{symplectichalfflat}\nd \\omega =0,  \\quad  d \\psi_+  = - \\pi_2 \\wedge \\omega, \\quad d \\psi_- =0,\n\\end{equation}\nwhere the $2$-form $\\pi_2$ is the unique  non zero  component of\nthe intrinsic torsion (see Theorem \\ref{stronglyint}). We will\ncall $SU(3)$-structures which satisfy the previous conditions\nbelonging to the class ${\\mathcal W}_2^+$.  The $3$-form $H$ is\nrelated to\n the component $\\pi_2$ of the intrinsic torsion by $ H = \\pi_2\n \\wedge \\alpha$ and we\nwill show that $H$ will never be closed unless $\\pi_2 = 0$.\n\nIt has to be noted that, if $(\\omega, g,  \\psi)$ is in the class\n${\\mathcal W}_2^+$, then  the $SU(3)$-structure given by $(\\omega,\ng,  i \\psi)$ is symplectic half-flat (see \\cite{CS}), i.e. the\nfundamental form $\\omega$ and the real part of the  complex volume\nform are both closed.  The half-flat structures turn out to be\nuseful in the construction of metrics with holonomy group\ncontained in $G_2$ (see e.g. \\cite{Hi1,CS,CF}). Indeed, starting\nwith a half-flat structure on $N$, if certain evolution equations\nare satisfied, then there exists a Riemannian metric with holonomy\ncontained in $G_2$ on the product of the manifold $N$ with some\nopen interval. Examples of compact manifolds with symplectic\nhalf-flat structures have been given in \\cite{CT}, where invariant\nsymplectic half-flat structures on nilmanifolds are classified.\nOther examples are considered in \\cite{dBT}  where Lagrangian\nsubmanifolds are studied instead.\n\n\n\\medskip\n\n{\\em{Acknowledgements}}\nThe authors thank Simon Salamon, Frederik Witt   for useful comments and suggestions. They also thank the  Centro di Ricerca Matematica \\lq \\lq Ennio De Giorgi\\rq \\rq, Pisa,  for the warm hospitality.\n\n\n\n\n\\section{Generalized $G_2$ structures and spinors}\n\nIn this section we are going to recall some facts on generalized\n$G_2$-structures which have been studied by Jeschek  and Witt in\n\\cite{W,W2,JW} in the general case of $\\phi$ non-constant and\nnon-trivial $B$-field. In the next sections we will deal with the\ncase $\\phi = const$ and trivial $B$-field.\n\n\n Let $V$ be a\n$7$-dimensional real vector space and denote by $V^*$  the dual\nspace of $V$. Then $V \\oplus V^*$ has a natural  orientation and a\ninner product of signature $(7,7)$ defined by\n$$\n(v + \\xi, v + \\xi) = -  \\frac 12 \\xi(v), \\quad \\forall v \\in V, \\,\n\\xi \\in V^*.\n$$\nThe inner product determines a group coniugate to $SO(7,7)$ inside the linear group\n$GL(14)$. Since as $GL(7)$-space ${\\mathfrak {so}} (7,7) = End(V)\n\\oplus \\Lambda^2 V^* \\oplus \\Lambda^2 V$, any $b \\in \\Lambda^2\nV^*$ defines an element (called {\\em B-field}) in ${\\mathfrak\n{so}} (7,7)$. By exponentiating to $SO(7,7)$ the action of\n$\\Lambda^2 V^* \\subset {\\mathfrak {so}} (7,7)$\n$$\nv  \\to    v \\lrcorner  b,\n$$\n one gets an action on $V \\oplus V^*$, given by $ \\exp (b) (v \\oplus\n \\xi) = v \\oplus\n ( v  \\lrcorner  b+ \\xi)$.\n Then  $V \\oplus V^*$ acts  on $\\Lambda^* V^*$ by\n$$\n(v + \\xi) \\eta = \\iota (v) \\eta + \\xi \\wedge \\eta,\n$$\nand we have\n$$\n(v + \\xi)^2 \\eta = - (v + \\xi, v + \\xi) \\eta.\n$$\n Therefore $\\Lambda^* V^*$ can be viewed  as  a module over the  Clifford algebra of $V \\oplus V^*$. The space\n  $\\Lambda^*  V^*$,   as the spin\n representation of  $Spin(7, 7)$,    determines the splitting of  $\\Lambda^* V^* \\otimes (\\Lambda^7 V)^{\\frac 12}$\n $$\n \\begin{array} {l}\n S^+ = \\Lambda^{ev} V^* \\otimes (\\Lambda^7 V)^{\\frac 12}\\\\\n S^- = \\Lambda^{od} V^* \\otimes (\\Lambda^7 V)^{\\frac 12}\n \\end{array}\n $$\n into the sum of the two irreducible spin representations.\nBy considering $b \\in \\Lambda^2 V^*$, then one has the following\ninduced action on\n spinors given by\n $$\n \\exp (b) \\eta = (1 + b + \\frac 12 b \\wedge b + \\cdots) \\wedge \\eta = e^b \\wedge \\eta.\n $$\n   If $\\sigma$ is the Clifford algebra anti-automorphism defined by $\\sigma (\\gamma^p) = \\epsilon(p) \\gamma^p$, on any element of degree $p$, with\n $$\n  \\epsilon  (p) = \\left\\{  \\begin{array} {cll}\n 1 \\quad &{\\mbox{for}} &\\quad p \\equiv 0, 3 \\quad {\\mbox{mod}} \\, 4,\\\\\n - 1 \\quad &{\\mbox{for}}& \\quad p \\equiv 1, 2  \\quad {\\mbox{mod}} \\,  4,\n  \\end{array}\n  \\right.\n  $$\n  then $S^+$ and $S^-$ are totally isotropic with respect\n  to the symmetric  bilinear form $q(\\alpha, \\beta)$ defined as the\n  top degree component of $\\alpha \\wedge \\sigma (\\beta)$ (see \\cite{W}).\n\n\n\n A {\\it generalized $G_2$-structure} on  a $7$-dimensional manifold $M$ is a reduction from the structure group  $\\R^* \\times Spin(7,7)$ of\n  the bundle $TM  \\oplus  T^*M$ to $G_2 \\times G_2$.\n  Such a structure determines a generalized oriented metric structure\n  $(g, b)$, (i.e. a Riemannian  metric $g$, a {\\em B}-field $b$ and   an orientation on $V$) and a real scalar function  $\\phi$ (the {\\em dilaton}).\n    Therefore we get a pair of two $G_2$-structures associated with\n two unit spinors $\\Psi_{\\pm}$ in the irreducible spin representation\n  $\\Delta = \\R^8$ of $Spin(7)$. There is,  up to a scalar,  a unique invariant in $\\Lambda^{ev} V^* \\otimes \\Lambda^{od} V^*$, given by the  box operator\n $$\n \\Box_{\\rho}: \\Lambda^{ev,od} V^*  \\to  \\Lambda^{od,ev} V^*, \\quad \\tilde \\rho \\to e^{\\frac{b}{2}} \\wedge  \\ast_g \\sigma (e^{- \\frac{b}{2}}  \\wedge \\tilde \\rho).\n $$\n\n If $\\rho$ is a $G_2 \\times G_2$-invariant spinor, then  its  {\\it companion}   $\\hat \\rho = \\Box_{\\rho} \\rho$  is still a\n $G_2 \\times G_2$-invariant spinor.  To any $G_2 \\times G_2$-invariant spinor $\\rho$ one can associate a volume  form  $\\mathcal Q$ defined by\n\\begin{equation} \\label{volume}\n {\\mathcal Q}: \\rho \\to q (\\hat \\rho, \\rho).\n\\end{equation}\n Using the isomorphism $\\Delta \\otimes \\Delta \\cong \\Lambda^{ev,od}$, Witt  in \\cite[Proposition 2.4]{W} derived the following  normal form for\n $[\\Psi_+ \\otimes \\Psi_-]^{ev,od}$ in terms of a suitable orthonormal\n basis\n $(e^1, \\ldots, e^7)$, namely\n $$\n\\begin{array}{lcl}\n(\\Psi_+ \\otimes \\Psi_-)^{ev} & = & \\cos (\\theta) + \\sin( \\theta) (e^{12} + e^{34} + e^{56}) +\\\\[4pt]\n&& \\cos  (\\theta) (- e^{1367} - e^{1457} - e^{2357} + e^{2467} - e^{1234} - e^{1256} - e^{3456}) +\\\\[4pt]\n&& \\sin (\\theta )(e^{1357} - e^{1467} - e^{2367} - e^{2457}) - \\sin (\\theta) e^{123456},\\\\[4pt]\n (\\Psi_+ \\otimes \\Psi_-)^{odd} &=&  \\sin (\\theta)  e^7 + \\sin (\\theta )(-e^{136} - e^{145} - e^{235} + e^{246}) +\\\\[4pt]\n && \\cos  (\\theta) (-e^{127} - e^{347} - e^{567} - e^{135} + e^{146} + e^{236} + e^{245}) +\\\\[4pt]\n && \\sin( \\theta) (-e^{12347} - e^{12567} - e^{34567}) +\\cos (\\theta) e^{1234567}, \\end{array}$$\n where $\\theta$ is the angle between $\\Psi_+$ and $\\Psi_-$ and $e^{i \\ldots j}$ denotes the wedge product $e^i \\wedge \\ldots \\wedge e^j$.\n\n If  the spinors $\\Psi_+$ and $\\Psi_-$ are linearly independent, then\n (see Corollary 2.5 of \\cite{W})\n $$\n\\begin{array}{lcl}\n(\\Psi_+ \\otimes \\Psi_-)^{ev} & = &  \\cos (\\theta) + \\sin(\\theta) \\omega  - \\cos (\\theta) (\\psi_-  \\wedge \\alpha + \\frac 12 \\omega^2) \\\\[4pt]\n&& + \\sin (\\theta) \\psi_+ \\wedge \\alpha - \\frac{1}{6} \\sin (\\theta) \\omega^3,\\\\[5pt]\n(\\Psi_+ \\otimes \\Psi_-)^{od} & = & \\sin (\\theta) \\alpha -  \\cos (\\theta) (\\psi_+ + \\omega \\wedge \\alpha) - \\sin (\\theta) \\psi_- \\\\[5pt]\n&&- \\frac 12 \\sin (\\theta) \\omega^2 \\wedge \\alpha + \\cos  (\\theta)\n{\\mbox {vol}}_g,\n\\end{array}\n $$\n where $\\alpha$  denotes the dual of the unit vector in $V$, stabilized by $SU(3)$,\n $$\\omega = e^{12} + e^{34}+ e^{56}$$ is  the fundamental form and\n $\\psi_{\\pm}$ are  the real and imaginary parts respectively of the complex volume form\n $$\n \\psi = (e^1 + i e^2) \\wedge (e^3 + i e^4) \\wedge (e^5 + i e^6).\n $$\n A $G_2 \\times G_2$-invariant spinor $\\rho$ is stable in the sense of Hitchin (see \\cite{Hi3}), i.e. $\\rho$ lies in an open orbit under the\n  action of $\\R^+  \\times Spin(7,7)$.\n\n  By \\cite[Theorem 2.9]{W} the generalized $G_2$-structures are in $1-1$ correspondence with lines of spinors\n  $\\rho$ in $\\Lambda^{ev}$ (or $\\Lambda^{od} $) whose stabilizer under the action of $Spin(7,7)$ is isomorphic to $G_2 \\times G_2$.\n\n The spinor $\\rho$ is called the {\\it structure form} of the generalized $G_2$ structure and it can be uniquely written\n  (modulo a simultaneous change of sign for $\\Psi_+$ and $\\Psi_-$) as\n $$\\rho = e^{-\\phi}  (\\Psi_+ \\otimes \\Psi_-)^{ev}_b ,\n $$\nwhere $b$ is the $B$-field, $\\Psi_{\\pm} \\in \\Delta$ are two unit\nspinors, the function $\\phi$ is  the dilaton and the subscript $b$\ndenotes the wedge with the exponential $e^{\\frac{b}{2} }$.\n\nA {\\it {\\rm (}topological{\\rm )} generalized $G_2$-structure}  over $M$ is a topological $G_2 \\times G_2$-reduction of the $SO(7,7)$-principal\n bundle associated with $TM \\oplus T^* M$ and it  is characterized by a stable even or odd  spinor $\\rho$ which can be viewed as a form.\n This is equivalent to say  that there exists an $SO(7)$-principal fibre bundle which has two $G_2$-subbundles (or equivalently two $G_2{^\\pm}$-structures).\n\n\n In the sequel we will omit  topological when we will refer to a generalized $G_2$-structure.\n\n\nLet $H$ be a $3$-form and $\\lambda$ be a real, non-zero constant. A generalized $G_2$-structure $(M, \\rho)$  is called  {\\it strongly integrable} with respect to\n$H$ if\n$$\nd_H \\rho = 0, \\quad d_H \\hat \\rho =0,\n$$\nwhere $d_H \\cdot = d \\cdot + H \\wedge \\cdot$ is the twisted operator\nof $d$.  By \\cite{W} there are no non-trivial  compact examples  with\na strongly integrable generalized  $G_2$- structure with respect to a\nclosed $3$-form $H$.\n\nIf $$d_H \\rho = \\lambda  \\hat \\rho,$$ then the generalized\n$G_2$-structure is said  to be {\\it weakly  integrable} of {\\it\neven} or {\\it odd} type according to the parity of the form\n$\\rho$. The constant $\\lambda$ (called the {\\em Killing number})\nis the $0$-torsion form of the two underlying $G_2$-structures.\nIndeed, by Corollary 4.6 of \\cite{W}, there exist two unique\ndetermined linear connections $\\nabla^{\\pm}$, preserving the two\n$G_2^\\pm$-structures, with skew-symmetric torsion $\\pm T = \\frac\n12 db + H$. If the structure is of odd type, then\n$$\n\\begin{array}{l}\nd \\varphi_+ = \\frac{12}{7} \\lambda * \\varphi_+ + \\frac 32 d \\phi \\wedge \\varphi_+ - * T_{27}^+,\\\\[5pt]\nd * \\varphi_+ = 2 d \\phi \\wedge * \\varphi_+\n\\end{array}\n$$\nand\n$$\n\\begin{array}{l}\nd \\varphi_- = \\frac{12}{7} \\lambda * \\varphi_- + \\frac 32 d \\phi \\wedge \\varphi_- - * T_{27}^-,\\\\[5pt]\nd * \\varphi_- = 2 d \\phi \\wedge * \\varphi_-,\n\\end{array}\n$$\nwhere $T_{27}^\\pm$ denotes the component of $T$ into the $27$-dimensional irreducible $G_2^\\pm$-module\n$${\\Lambda^3_{27}}^\\pm = \\{ \\gamma \\in \\Lambda^3 \\, \\vert \\, \\gamma \\wedge \\varphi_+ = \\gamma \\wedge \\varphi_- =0 \\}.$$\nThis is equivalent to say that $e^{-\\phi} [\\Psi_+ \\otimes \\Psi_-]$ satisfies the generalized Killing and dilatino equation (see \\cite{W,GMPW}).\n\n\n In both cases there is a characterization in terms  of the two metric\n connections $\\nabla^{\\pm}$ with skew symmetric torsion $\\pm T$ (see\n \\cite[Theorem 4.3]{W}). Indeed, a generalized $G_2$-manifold  $(M,\n \\rho)$    is weakly integrable with respect to $H$ if and only if\n $$\n  \\nabla^{LC} \\Psi_{\\pm} \\pm \\frac 14 ( X \\lrcorner T) \\cdot \\Psi_{\\pm} =0,\\\\\n  $$\n where $\\nabla^{LC}$ is the Levi-Civita connection,  $X \\lrcorner$ denotes the contraction by $X$ and the following\n additional conditions are satisfied\n $$\n \\left (d \\phi \\pm \\frac 12 ( X \\lrcorner T)   \\pm \\lambda \\right) \\cdot \\Psi_{\\pm} =0, \\quad\n $$\n  if $\\rho$ is of even type\n  or\n $$\n \\left (d \\phi \\pm \\frac 12 ( X \\lrcorner T)  + \\lambda \\right) \\cdot \\Psi_{\\pm} =0, \\quad\n $$\n  if $\\rho$ is of odd type.\n  Taking $\\lambda = 0$ above equations yield strong integrability  with respect to $H$, instead.\n\n\nExamples  of  generalized $G_2$-structures  are given by  the {\\em\nstraight} generalized $G_2$-structures, i.e. structures defined by\none spinor $\\Psi = \\Psi_+ = \\Psi_-$. These structures are induced by\na classical $G_2$-structure $(M, \\varphi)$ and  are strongly\nintegrable with respect to a closed  $3$-form $T$ only if the\nholonomy  of the metric  associated with $\\varphi$ is contained in $G_2$.\n\nIf $H$ is closed, then it has to be noted that, in the compact case, the structure form  $\\rho$ of  a strongly integrable generalized $G_2$-structure corresponds to a critical point of a functional on stable forms. Indeed, since stability is an open condition, if $M$ is compact  then one can consider the functional\n$$\nV (\\rho) = \\int_M {\\mathcal Q}(\\rho),\n$$\nwhere $\\mathcal Q$ is defined as in \\eqref{volume}.  By \\cite[Theorem 4.1]{W} a $d_H$-closed stable form\n$\\rho$ is a critical point in its cohomology class if and only if $d_H \\hat \\rho =0$.\n\nAgain in the compact case  a $d_H$-exact form $\\hat \\rho \\in \\Lambda^{ev,od} (M)$  is  a  critical point  of the functional\n $V$ under some  constraint  if and only if\n $d_H \\rho =  \\lambda \\hat \\rho$, for a real non zero constant $\\lambda$.\n\n\n\\section{Compact examples of weakly integrable manifolds}\n\nIn this section we will construct   examples of compact manifolds endowed with a weakly integrable generalized $G_2$-structure with respect to a closed  $3$-form $H$.\n\n\n\nConsider the $7$-dimensional Lie algebra $\\mathfrak g$ with structure equations:\n$$\n\\left \\{ \\begin{array} {l}\nd e^1 = a e^{46},\\\\[3pt]\nd e^2 = - \\frac 12 a e^{36} - \\frac 12 a e^{45} +  \\frac 12 a e^{17},\\\\[3pt]\nd e^3 = - \\frac 12 a e^{15} +\\frac 12 a e^{26} -  \\frac 12 a e^{47},\\\\[3pt]\nd e^4 = -a e^{16},\\\\[3pt]\nd e^5 =  \\frac 12 a e^{13} -\\frac 12 a e^{24} -  \\frac 12 a e^{67},\\\\[3pt]\nd e^6 = a e^{14},\\\\[3pt]\nd e^7 =  -\\frac 12 a e^{12} -\\frac 12 a e^{34} -  \\frac 12 a e^{56},\n\\end{array}\n\\right.\n$$\nwhere $a$ is a real parameter different from zero.\n\nIt can be easily checked  that the Lie algebra $\\mathfrak g$ is not solvable since $[{\\mathfrak g},{\\mathfrak g}]  ={\\mathfrak g}$ and that  it is unimodular. We can also view  $\\mathfrak g$ as the semidirect sum\n$$\n{\\mathfrak g} = {\\mathfrak {su}} (2) \\oplus_{\\delta} \\R^4,\n$$\nwhere\n$$\n {\\mathfrak {su}} (2)  = {\\mbox {span}} <e_1, e_4, e_6>, \\quad   \\R^4 = {\\mbox {span}} <e_2, e_3, e_5, e_7>\n $$\n and $\\delta:   {\\mathfrak {su}} (2)  \\to {\\mathfrak {Der}} (\\R^4)$ is given by\n $$\n\\delta(e_1) = ad_{e_1} = \\left( \\begin{array}{cccc}   0&0&0&-\\frac 12 a\\\\\n 0&0&\\frac 12 a&0\\\\\n0&  - \\frac 12 a&0&0\\\\\n  \\frac 12 a &0&0&0 \\end{array} \\right),\n  $$\n    \\vskip 0.2cm\n $$\n  \\delta(e_4) = ad_{e_4} = \\left( \\begin{array}{cccc}   0&0&\\frac 12 a&0\\\\\n  0&0&0&\\frac 12 a\\\\\n  - \\frac 12 a&0&0&0\\\\\n  0&-  \\frac 12 a &0&0\\end{array} \\right),\n  $$\n  \\vskip 0.2cm\n $$\n \\delta(e_6) =  ad_{e_6} = \\left( \\begin{array}{cccc}   0&-\\frac 12 a&0&0\\\\\n  \\frac 12 a&0&0&0\\\\\n  0&0&0& \\frac 12 a\\\\\n  0&0&- \\frac 12 a &0 \\end{array} \\right).\n  $$\n If we identify  $\\R^4$ with  the space $\\H$  of quaternions, then\n $$\n ad_{e_1} = \\frac 12 a L_k, \\quad ad_{e_4} = \\frac 12 a L_{-j}, \\quad ad_{e_6} = \\frac 12 a L_{i},\n $$\n where $L_q$ denotes the left multiplication by the quaternion $q$.\n\nTherefore,  the product on the corresponding Lie group $G = SU(2)\n\\ltimes \\H$, for $a = 2$,  is given by\n$$\n(A, q) \\cdot (A', q') = (A A', Aq' + q),  \\quad A,A' \\in SU(2),\n\\quad q,q' \\in \\H,\n$$\nwhere  we identify $SU(2)$ with the group of quaternions of unit\nnorm.\n\n\\begin{theorem} \\label{example} The Lie group $G = SU(2) \\ltimes \\H$ admits  compact quotients  $M_{\\beta} = G\/  \\Gamma_{\\beta}$, with\n $e^{i \\beta}$ primitive p-th root of unity $(p$ prime$)$,  and  $M_{\\beta}$ has an invariant weakly integrable generalized $G_2$-structure with respect to a closed $3$-form $H$.\n\\end{theorem}\n\n\\begin{proof}\nConsider the discrete subgroup $\\Gamma_{\\beta} = <A_{\\beta} >\\ltimes \\Z^4$, where $<A_{\\beta} >$ is the subgroup of $SU(2)$ generated by\n$$\nA_{\\beta} = \\left( \\begin{array}{cc} e^{i \\beta} &0\\\\ 0&e^{-i \\beta}  \\end{array} \\right),\n$$\nwith $e^{i \\beta}$ primitive p-th root of unity and $p$ prime.\n\n Then one can check that $\\Gamma_\\beta$ is a closed subgroup of $G$.  Let $(A', q')$ be any point of\n$G$. Thus  $$ [ (A', q')] = \\{ (A_{\\beta}^m A', A_{\\beta}^m q' +\nr),  \\, m \\in \\Z \\, , r \\in \\Z^4 \\}\n$$\nis the equivalence class of $(A', q')$. In particular, $[ (A',\nq')] = [(A', q' + r)]$ and therefore the restriction of the\nprojection $\\pi: G \\to G\/\\Gamma_{\\beta}$  to  $SU(2) \\times\n[0,1]^4$ is surjective.\\newline Then the quotient $M_{\\beta} =\n(SU(2) \\ltimes \\H )\/ \\Gamma_{\\beta}$ is a compact manifold.\n\n\nConsider the invariant metric $g$ on $M_{\\beta}$ such that the basis $(e^1, \\ldots, e^7)$ is orthonormal\nand take the  generalized $G_2$ structure defined by the structure form of odd type\n$$\n\\rho = e^7 - e^{136}- e^{145} - e^{235}+ e^{246} -  e^{12347} - e^{12567} -  e^{34567},\n$$\nin terms of the basis $(e^1, \\ldots, e^7)$.\nThe\ncompanion  of $\\rho$ is\n$$\n\\hat \\rho = e^{12} + e^{34} + e^{56}+  e^{1357} -e^{1467} - e^{2367}- e^{2457} - e^{123456}.\n$$\nThen  the structure form $\\rho$ defines a weakly integrable\ngeneralized $G_2$-structure with   respect to  a closed  $3$-form\n$H$, i.e. $d_H \\rho = \\lambda \\hat \\rho$ ($\\lambda$ non-zero\nconstant),\n  if and only if\n\\begin{equation} \\label{weakeq}\n\\left\\{\n \\begin{array}{l}\n d e^7 = \\lambda \\omega,\\\\[5pt]\n d \\psi_- = (H - \\lambda \\psi_+) \\wedge e^7,\\\\[5pt]\n H \\wedge \\psi_- = - \\frac 13 \\lambda \\omega^3,\n \\end{array}\n \\right.\n \\end{equation}\n where $\\omega, \\psi_{\\pm}$ are given by\n\\begin{equation} \\label{definitionforms}\n\\left\\{\n\\begin{array}{lcl}\n\\omega &= & e^{12} + e^{34} + e^{56},\\\\[5pt]\n\\psi_+ &=& e^{135} - e^{146} - e^{236} - e^{245},\\\\[5pt]\n\\psi_- &=&  e^{136} + e^{145} + e^{235} - e^{246}.\n\\end{array}\n\\right.\n\\end{equation}\nThe equations \\eqref{weakeq} are satisfied with  $\\lambda = - \\frac 12 a$ and\n$$\n H=     - a e^{146}.\n$$\n\\end{proof}\n\nObserve that $H$ is also co-closed, i.e. $d*H =0$. Moreover, if $a\n\\leq 1$, $H$ is a calibration in the sense of \\cite{HL}.\n\n In this way we get  compact examples with a weakly\nintegrable  generalized  $G_2$-structure with respect to the closed\n$3$-form $H$.  The induced invariant  metric on $M_{\\beta}$ is not flat, since the inner product\n$$g = \\sum_{i = 1}^7 (e^i)^2$$\non the Lie algebra  $\\mathfrak g$  is not flat. Indeed,  the  Ricci\ntensor of $g$ is diagonal with respect to the orthonormal basis $(e_1,\n\\ldots, e_7)$ and its   non zero components are  given by:\n$$ Ric (e_1, e_1) = \\frac 12 a^2 =Ric (e_4, e_4) = Ric (e_6, e_6).\n$$\n\n\\section {Link with $SU(3)$-structures in dimension $7$ and\nevolution equations}\n\nIn this section we will relate the weakly integrable generalized\n$G_2$-structures constructed in the previous section with\n$SU(3)$-structures in dimension $7$.\n\n Since the $1$-form $\\eta = e^7$ is a contact\nform on the Lie algebra ${\\mathfrak g}$, then $M_{\\beta}$ is   a\ncontact metric manifold. Moreover, by  \\eqref{weakeq} $M_{\\beta}$\nhas an $SU(3)$-structure defined by $(\\omega, \\eta, \\psi = \\psi_+\n+ i \\psi_-)$ such that\n\\begin{equation} \\label{SU3hypo}\n\\left \\{ \\begin{array} {l}\nd \\omega =0,\\\\[5pt]\nd (\\psi_\\pm \\wedge \\eta) =0.\n\\end{array} \\right.\n\\end{equation}\nHere we identify $SU(3)$ as the subgroup $SU(3) \\times \\{1 \\}$ of\n$SO(7)$.\n\n\\smallskip\n\nNote that the $SU(3)$-structures $(\\omega, \\eta, \\psi = \\psi_+ + i\n\\psi_-)$ on $7$-dimensional manifolds for which $d \\omega =0$ and\n$d (\\psi_{\\pm}) =0$ where considered in \\cite{TV}. In this case\none cannot find any closed $3$-form $H$ such that conditions\n\\eqref{SU3hypo} are satisfied since $H$ has to be equal to\n$\\lambda \\psi_+$ and the third equation cannot hold. It would be\ninteresting to investigate if there are other $7$-dimensional\nexamples endowed with an $SU(3)$-structures which satisfy the\nconditions \\eqref{SU3hypo} and  giving rise to a weakly integrable\n$G_2$-structure with respect to a closed $3$-form $H$.\n\n\n In general, let ${\\iota}: M^7 \\to N^8$ be an embedding\nof a an oriented $7$-manifold $M^7$ into a $8$-manifold $N^8$ with\nunit normal vector $V$. Then an $SU(4)$-structure  $(\\tilde\n\\omega, \\tilde g, \\tilde \\psi)$ (or equivalently a special almost\nHermitian structure, see e.g. \\cite{Ca2}), where  $(\\tilde \\omega,\n\\tilde g)$ is a $U(4)$-structure and $\\tilde \\psi= \\tilde \\psi_+ +\ni \\tilde \\psi_-$ is  complex  $4$-form of unit norm, defines in a\nnatural way an $SU(3)$-structure  $(\\omega, \\eta, g, \\psi = \\psi_+\n+ i \\psi_-)$ on $M^7$ given by:\n$$\n\\eta = - V  \\lrcorner  \\tilde\\omega, \\quad\n\\omega = {\\iota}^* \\tilde \\omega, \\quad\ng = {\\iota}^* g, \\quad\n\\psi_+ =  - V  \\lrcorner   \\tilde\\psi_+, \\quad\n\\psi_- =   V \\lrcorner \\tilde\\psi_- .\n$$\nThen, if $\\gamma$ denotes the $1$-form dual to $V$, then  we have\n$$\n\\begin{array} {l}\n\\tilde \\omega =  \\omega +  \\eta \\wedge \\gamma,\\\\[5pt]\n\\tilde \\psi = (\\psi_+ + i \\psi_-) \\wedge (\\eta + i \\gamma).\n\\end{array}\n$$\nThe integrability of the $SU(4)$-structure $(\\tilde \\omega, \\tilde\ng, \\tilde \\psi)$ implies  conditions \\eqref{SU3hypo}, which can be\nviewed as the analogous of the equations defining the hypo\n$SU(2)$-structures in dimension 5 (see \\cite{CS}).\n\n\nVice versa, given an $SU(3)$-structure $(\\omega,  \\eta,   \\psi )$ on $M^7$, an $SU(4)$-structure on $M^7 \\times \\R$ is defined by\n\\begin{equation} \\label{SU4}\n\\begin{array} {l}\n\\tilde\\omega = \\omega + \\eta \\wedge dt,\\\\[5pt]\n\\tilde \\psi = \\psi \\wedge (\\eta + i dt),\n\\end{array}\n\\end{equation}\nwhere $t$ is a coordinate on $\\R$.\n\nIf the $SU(3)$-structure $(\\omega, \\eta,   \\psi )$ on $M^7$  belongs to a one-parameter family of $SU(3)$-structures   $(\\omega(t), \\eta(t),   \\psi(t) )$ satisfying the equations \\eqref{SU3hypo} and such that\n\\begin{equation} \\label{evolutions}\n\\left \\{ \\begin{array} {l}\n\\partial_t  \\omega(t) = - \\hat d \\eta(t),\\\\[5pt]\n\\partial_t (\\psi_+(t) \\wedge \\eta (t)) = \\hat d \\psi_-(t),\\\\[5pt]\n\\partial_t (\\psi_-(t) \\wedge \\eta (t)) =  -\\hat d \\psi_+(t),\n\\end{array} \\right.\n\\end{equation}\nfor all $t \\in (b, c)$, where $\\partial_t$ denotes the derivative\nwith respect to $t$ and $\\hat d$ is the exterior differential on\n$M^7$, then the $SU(4)$-structure given  by \\eqref{SU4} on $M^7\n\\times (b, c)$  is integrable, i.e. $\\tilde \\omega$ and $\\tilde\n\\psi$ are both closed. In particular, the associated Riemannian\nmetric  on $M^7 \\times (b, c)$ has holonomy contained in $SU(4)$\nand consequently it is Ricci-flat.\n\nFor the manifolds $M_{\\beta}$ a solution of the evolution\nequations \\eqref{evolutions} is given by\n$$\n\\begin{array}{l}\n\\omega(t) = u(t) v(t) (e^{12} + e^{34} + e^{56}),\\\\[5pt]\n\\psi_+(t) = u(t) v(t)^2 (e^{135} - e^{236} - e^{245}) - u(t)^3\ne^{146},\\\\[5pt]\n\\psi_-(t) = u(t)^2 v(t) (e^{136} + e^{145} - e^{246}) + v(t)^3\ne^{235},\\\\[5pt]\n\\eta(t) = \\frac {1}{v(t)^3} e^7,\n\\end{array}\n$$\nwhere $u(t), v(t)$ solve the system of ordinary differential\nequations\n$$\n\\left\\{ \\begin{array}{l} \\displaystyle\\frac {d}{dt} (u(t) v(t)) =\n\\frac 12 a\n\\displaystyle\\frac {1}{v(t)^3},\\\\[10pt]\n\\displaystyle\\frac {d}{dt} \\left( \\displaystyle\\frac {u(t)}{v(t)}\n\\right) = \\frac 12 a v(t)^3\\,,\n\\end{array} \\right.\n$$\nsuch that $u(0) = v(0) = 1$. The previous system is equivalent to\n\\begin{equation} \\label{odiffsystem}\n\\left\\{ \\begin{array}{l}  u'(t) = \\frac 14 a\n\\left(\\displaystyle\\frac {1} {v(t)^4} +\nv(t)^4\\right),\\\\[10pt]\nv'(t) = \\frac 14 a \\left(\\displaystyle\\frac {1} {u(t) v(t)^3} -\n\\displaystyle\\frac{v(t)^5}{u(t)}\\right).\n\\end{array}\n\\right.\n\\end{equation}\n\nThen, by the theorem on existence of solutions for a  system of\nordinary differential equations, one can show that on a open\ninterval $(b, c)$ containing $t =0$ the system \\eqref\n{odiffsystem} admits a unique solution $(u(t), v(t))$ satisfying\nthe initial condition $u(0) = v(0) = 1$. Actually, the solution is given by\n$$\nu(t) = 1 + \\frac 12 a t, \\quad v(t) =1.\n$$\n\nHence, we can prove   the following\n\n\\begin{theorem}\nOn the product of $M_{\\beta}$ with some open interval $(b, c)$\nthere exists a Riemannian metric with  discrete holonomy contained in\n$SU(4)$.\n\\end{theorem}\n\n\\begin{proof}\nThe basis of $1$-forms on the manifold $M_{\\beta} \\times (b,c)$ given by\n$$\n\\begin{array}{l}\nE^1 = (1 + \\frac 12 a t) e^1, \\, \\, E^2 =  e^2,  \\,\\, E^3 = (1 + \\frac 12 a t) e^3,\\,\\, E^4 = (1 + \\frac 12 a t) e^4, \\\\[10pt]\n E^5 =  e^5,\\,\\, E^6 = (1 + \\frac 12 a t) e^6,\\,\\, E^7 =  e^7,\\,\\, E^8 = dt\n \\end{array}\n$$\nis orthonormal with respect to the Riemannian metric with holonomy   contained in $SU(4)$. By a direct computation\nwe have that the non zero Levi-Civita connection 1-forms are given by\n$$\n\\begin{array} {l}\n\\theta^1_4 =  -\\theta^2_3 = \\theta^5_7 = \\theta^6_8 = \\displaystyle\\frac {a} {2 + at} E^6, \\\\[12pt]\n  \\theta^1_6 = -\\theta^2_5  = - \\theta^3_7=  -\\theta^4_8 =  \\displaystyle-\\frac {a} {2 + at} E^4,\\\\[12pt]\n    \\theta^1_8  =  - \\theta^2_7=   \\theta^3_5 = \\theta^4_6 =  \\displaystyle\\frac {a} {2 + at} E^1.\n  \\end{array}\n   $$\nTherefore, all  the curvature forms $\\Omega^i_j$ vanish and consequently the holonomy  algebra is trivial.\n\\end{proof}\n\n\\section{Strong integrability and $SU(3)$-structures in dimension 6}\nIn this section we are going to consider the structure form $\\rho$ of\neven type\n\\begin{equation} \\label{rho}\n\\rho = \\omega + \\psi_+ \\wedge\\alpha-\\frac{1}{6}\\omega^3\n\\end{equation}\non the product of a $6$-dimensional manifold $N$ endowed with  an $SU(3)$-structure cross $S^1$. We will investigate which type of\n$SU(3)$-structures give rise to a strongly integrable generalized\n$G_2$-structure with respect to a non-zero $3$-form.\n\nLet  $N$ be a $6$-dimensional manifold. An {\\it $SU(3)$-structure} on $N$ is determined by a Riemannian metric $g$, an orthogonal almost complex structure $J$ and a choice of a complex volume form  $\\psi = \\psi_+ + i \\psi_-$  of unit norm.  We will denote by $(\\omega, \\psi)$  an $SU(3)$-structure, where $\\omega$ is the fundamental form defined by\n$$\n\\omega(X, Y) = g (J X, Y),\n$$\nfor any pair of vector fields $X, Y$ on $N$.  Locally one may choose\nan   orthornormal basis $(e^1, \\ldots, e^6)$ of  the vector cotangent\nspace $T^*$  such that $\\omega$ and $\\psi_{\\pm}$ are given by\n\\eqref{definitionforms}.\n\nThese forms satisfy the following compatibility relations\n$$\n\\omega \\wedge \\psi_{\\pm} =0, \\quad \\psi_+ \\wedge \\psi_- = \\frac 23 \\omega^3.\n$$\nThe intrinsic torsion of the $SU(3)$-structure belongs to the space (see \\cite{CS})\n$$\nT^* \\otimes {\\mathfrak {su}} (3)^{\\perp} =  {\\mathcal W}_1 \\oplus  {\\mathcal W}_2 \\oplus  {\\mathcal W}_3 \\oplus  {\\mathcal W}_4  \\oplus  {\\mathcal W}_5,\n$$\n$ {\\mathfrak {su}} (3)^{\\perp} $ being the orthogonal complement of $ {\\mathfrak {su}} (3)$ in  ${\\mathfrak {so}} (6)$ and\n$$\n\\begin{array}{ll}\n {\\mathcal W}_1 =  {\\mathcal W}^+_1 \\oplus  {\\mathcal W}^-_1, &\\quad  {\\mathcal W}^{\\pm}_1 \\cong \\R,\\\\[5pt]\n  {\\mathcal W}_2 =  {\\mathcal W}^+_2 \\oplus  {\\mathcal W}^-_2, &\\quad  {\\mathcal W}^{\\pm}_2 \\cong {\\mathfrak {su}}(3),\\\\[5pt]\n  {\\mathcal W}_3 \\cong [\\![{\\mathrm S}^{2,0}]\\!],  &\\quad {\\mathcal W}_4  \\cong {\\mathcal W}_5 \\cong T^*,\n  \\end{array}\n  $$\n  where $[\\![{\\mathrm S}^{2,0}]\\!]$ denotes the real representation associated with  the space ${\\mathrm S}^{2,0}$ of  complex symmetric tensors of type $(2,0)$.\\newline\n  The components  of the intrinsic torsion of an $SU(3)$-structure can be expressed by (see e.g. \\cite{CS,BV})\n\\begin{equation} \\label{intrinsicforms}\n\\left\\{\n\\begin{array}{lll}\n d \\omega &=&  \\nu_0 \\, \\psi_+  +  \\alpha_0 \\,  \\psi_- +   \\nu_1  \\wedge \\omega+ \\nu_3,\\\\[5pt]\nd \\psi_+ &=&  \\frac23 \\alpha_0 \\, \\omega^2 + \\pi_1 \\wedge \\psi_+ -\\pi_2 \\wedge \\omega,\\\\[5pt]\nd \\psi_- &= & - \\frac 23 \\nu_0 \\, \\omega^2 +  J \\pi_1 \\wedge \\psi_+  - \\sigma_2 \\wedge   \\omega,\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\alpha_0 \\in {\\mathcal W}_1^+$, $\\pi_1 \\in {\\mathcal W}_5$, $\\pi_2 \\in {\\mathcal W}_2^+$, $\\nu_0 \\in {\\mathcal W}_1^-$,\n$ \\sigma_2 \\in  {\\mathcal W}_2^-$, $\\nu_1 \\in {\\mathcal W}_4$, $\\nu_3 \\in {\\mathcal W}_3$.\n\nBy definition, an $SU(3)$-structure is called {\\it integrable} if the intrinsic torsion vanishes. In this case $\\omega$ and $\\psi$ are both closed. Therefore, the intrinsic torsion measures the failure of the holonomy group of the Levi-Civita connection of $g$ to reduce to $SU(3)$.\n\nIf $(\\omega, \\psi)$ is in the class ${\\mathcal W}_2^+$, then  by using \\eqref{intrinsicforms} and taking into account  the conditions $d \\omega = d \\psi_- =0$,  we get  that  the components $\\nu_0,  \\alpha_0, \\sigma_2, \\nu_3, \\nu_1, \\pi_1$ vanish  and hence\n$$d \\psi_+ =  -\\pi_2 \\wedge \\omega, $$\nwith $\\pi_2$ belonging to the space\n\\begin{equation} \\label{spacepi2}\n\\begin{array}{lcl}\n{\\mathcal W}_2^+ &\\cong& \\{  \\gamma \\in  \\Lambda^2 \\quad  \\vert  \\quad \\gamma \\wedge \\psi_+=0, \\quad  * J \\gamma = -\n\\gamma \\wedge \\omega \\}\\\\[5pt]\n&=& \\{  \\gamma \\in  \\Lambda^2 \\quad \\vert \\quad J \\gamma = \\gamma, \\quad \\gamma \\wedge \\omega^2 =0 \\}.\n\\end{array}\n\\end{equation}\n\n\nBy \\cite{BV} the scalar curvature ${\\mbox {scal}} (g)$ of the metric $g$ is given by:\n$$\n{\\mbox {scal}}  (g)  = - \\frac 12 \\vert \\pi_2 \\vert^2 \\, .\n$$\n\n\\medskip\n\n\nLet $\\alpha$ be  a  closed 1-form on $S^1$.  Consider on  the\nproduct $N  \\times S^1$,   the  generalized $G_2$-structure\ndefined by  the structure form of even type $\\rho$ given by\n\\eqref{rho} with companion\n$$\n\\hat \\rho = \\alpha - \\psi_-  - \\frac12  \\omega^2 \\wedge \\alpha.\n$$\nWe have the following\n\\begin{theorem} \\label{stronglyint}\nLet $(N, \\omega, \\psi)$ be a $6$-dimensional manifold endowed with an\n $SU(3)$-structure. The structure form $\\rho$, given by \\eqref{rho},\n defines  a   strongly integrable generalized $G_2$-structure on $N \\times S^1$\n with respect to a  $3$-form $H$ $($ non necessarily closed$)$, i.e. $\\rho$ satisfies the conditions\n \\begin{equation} \\label{dequations} d_H \\rho  = d_H  \\hat \\rho =0 \\end{equation}\n if and only if\n $N$ is in the class ${\\mathcal W}_2^+$ and $H = \\pi_2 \\wedge \\alpha$.\n\\end{theorem}\n\n\\begin{proof} By \\eqref{dequations}  we get\n$$\n\\left\\{\n\\begin{array}{l}\nd \\omega + d (\\psi_+ \\wedge  \\alpha) - \\frac16 d(\\omega^3)  +\nH \\wedge \\omega + H \\wedge \\psi^+ \\wedge \\alpha =0,\\\\[5pt]\n d \\hat  \\rho   + H \\wedge \\hat \\rho = - d \\psi_-    - \\frac12 d(\\omega^2 \\wedge \\alpha) + H \\wedge \\alpha - H \\wedge \\psi_-  =0.\n\\end{array}\n\\right.\n$$\nThis is equivalent to say:\n\\begin{equation} \\label{strongconditions}\n\\left\\{\n\\begin{array}{l}\nd \\omega =0,\\\\[5pt]\nd (\\psi_+ \\wedge  \\alpha) = - H \\wedge \\omega,\\\\[5pt]\nH \\wedge \\psi_+  \\wedge \\alpha=0 \\\\[5pt]\nd \\psi_- =  H \\wedge \\alpha,\\\\[5pt]\nH \\wedge \\psi_- =0\\,.\n\\end{array}\n\\right.\n\\end{equation}\nHence, in particular\n$$\nd \\psi_-  = 0 , \\quad  H \\wedge \\alpha =0.\n$$\nIt follows that $H =  S \\wedge f \\alpha$, with $S$ a $2$-form on\n$N$ and $f$ a function on $S^1$. Since  $d \\omega =  0$, we obtain\n$$d \\psi^+ \\wedge \\alpha = - S \\wedge \\omega \\wedge f \\alpha,\n$$ we have that $f$ has to be a constant $k$ and  $$d \\psi_+ = - k S\n\\wedge \\omega,$$\n with $k S = \\pi_2$. Since $\\pi_2$ is a\n$(1,1)$-form, then $\\pi_2 \\wedge \\psi_{\\pm} =0$. Therefore,\nequations \\eqref{strongconditions} are satisfied if and only if\n$N$ belongs to the class ${\\mathcal W}_2^+$.\n\\end{proof}\n\nNote that $H$ is closed if and only if $d \\pi_2 =0$.\n\n\\smallskip\n\n\n\n\nHomogeneous examples of $6$-dimensional manifolds with a\n$SU(3)$-structure in the class ${\\mathcal W}_2^+$ are given in\n[8]. There it was proved that the  $6$-dimensional nilmanifolds\n$\\Gamma \\backslash G$ which  carry an invariant $SU(3)$-structures\nin the class ${\\mathcal W }_2^+$ are the torus, the $\\T^2$-bundle\nover $\\T^4$ and the $\\T^3$-bundle over $\\T^3$ associated with the\nfollowing nilpotent Lie algebras\n$$\n\\begin{array}{l}\n(0,0,0,0,0,0),\\\\[3pt]\n(0,0,0,0,12,13),\\\\[3pt]\n(0,0,0,12,13,23),\n\\end{array}\n$$\nwhere  the notation   $(0,0,0,0, 12,13)$ means that  the dual ${\\mathfrak g}^*$ of the Lie algebra ${\\mathfrak g}$ has a basis $(e^1, \\ldots, e^6)$ such that $d  e^i =0, i =1, \\ldots, 4$,\n $d e^5 = e^1 \\wedge e^2$ and $d e^6 = e^1 \\wedge e^3$.\n\nIn \\cite{W}  the $\\T^2$-bundle over $\\T^4$ has been considered and it has been  proved that\nit admits a $SU(3)$-structure in the class ${\\mathcal W}_2^+$.\\newline\nBy \\cite{dBT} the $\\T^3$-bundle over $\\T^3$ admits a family of $SU(3)$-structures in the class ${\\mathcal W}_2^+$ given by\n$$\n\\begin{array}{lcl}\n\\omega &=& e^{16} + \\mu e^{25} + (\\mu - 1) e^{34},\\\\[5pt]\n\\psi_+ &=&  (1 - \\mu) e^{124}  + \\mu e^{135}   - \\mu (\\mu - 1) e^{456}  - e^{236},\\\\[5pt]\n\\psi_- &=&- \\mu (1- \\mu) e^{145}  + (\\mu - 1) e^{246} + \\mu e^{356} + e^{123},\n\\end{array}\n$$\nwhere $\\mu$ is  a real number different from $0$ and $1$. Such  a family of $SU(3)$-structures belongs to  the  class ${\\mathcal W}_2^+$ with $$\n\\pi_2 = \\mu^2 e^{25} - (\\mu- 1)^2 e^{36} - e^{14},\n$$\nand  $d \\pi_2 \\neq 0$.\n\\newline\nManifolds in the class ${\\mathcal W}_2^+$ can be also obtained as\nhypersurfaces of $7$-dimensional manifolds with a $G_2$-structure.\nThe $\\T^2$-bundle over $\\T^4$ can be also be viewed as a\nhypersurface of a $7$-dimensional  manifold with a calibrated\n$G_2$-structure, i.e. such that the associated stable $3$-form is\nclosed. Indeed, if $(M, \\varphi)$ is a $7$-dimensional manifold\nwith a calibrated $G_2$-structure, then any hypersurface $\\iota: N\n\\hookrightarrow M$ with unit normal vector $\\nu$ such that the Lie\nderivative $L_{\\nu} \\varphi =0$ admits an $SU(3)$-structure\n$(\\omega, \\psi)$ in the class ${\\mathcal W}_2^+$ defined by\n$$\n\\begin{array}{l}\n\\omega = \\nu \\lrcorner \\varphi,\\\\[3pt]\n\\psi_+ =  \\nu \\lrcorner * \\varphi,\\\\[3pt]\n\\psi_- = \\iota^* \\varphi.\n\\end{array}\n$$\nFor general theory on  an oriented  hypersurface of a $7$-dimensional manifold endowed with a $G_2$-structure see \\cite{C}.\n\nIf we consider the 7-dimensional nilmanifold associated with the\nLie algebra (see \\cite{F})\n$$\n(0,0,0,-13,-23,0,0)\n$$\nand the hypersurface which is a maximal integral submanifold of the involutive  distribution defined by the $1$-form $e^6$, then one gets the\n $SU(3)$-structure considered above.\n\n Another example of hypersurface (non  nilmanifold) can be obtained by the  7-dimensional compact manifold $M = X \\times S^1$, where $X$ is the compact\n solvmanifold\n  considered by Nakamura (see \\cite{N}), associated with the solvable Lie algebra\n  $$\n  (0, 12 - 45, - 13 + 46, 0,15-24,-16+34,0)\n  $$\n  and endowed with the $G_2$-structure\n  $$\n  \\varphi = e^{147} + e^{357} - e^{267} + e^{136} + e^{125} + e^{234} - e^{456}.\n  $$\n  The compact hypersurface, maximal integral submanifold of the involutive  distribution defined by the $1$-form $e^7$, has\n an $SU(3)$-structure in the class ${\\mathcal W}_2^+$.\n\n\\medskip\n\n\n\nWe will show that,  if the $SU(3)$-structure is not integrable, then\nthe $2$-form $\\pi_2$ cannot be closed. Indeed,\n\n\n\\begin{prop}\\label{strongconstruction} Let $N$ be a $6$-dimensional\n  manifold endowed with an $SU(3)$-structure $(\\omega, \\psi)$ in the class ${\\mathcal W}_2^+$.\nIf    $\\pi_2$ is  closed,  then the $SU(3)$-structure is integrable. In particular, the associated Riemannian metric $g$ is Ricci flat.\n\\end{prop}\n\n\\begin{proof} As already remarked, $(\\omega, \\psi)$ is  in the class ${\\mathcal W}_2^+$ if and only if \\begin{equation} \\label{W_2^+}\nd \\psi_+ =  -\\pi_2 \\wedge \\omega\\,,\\quad\nd \\psi_- =  d \\omega = 0,\n\\end{equation}\nwith $\\pi_2$ satisfying the following relations\n$$\n\\begin{array}{ll}\n \\pi_2 \\wedge \\psi_- =0, \\quad & * J \\pi_2 = -\n\\pi_2 \\wedge \\omega\\\\[3pt]\nJ \\pi_2 = \\pi_2, \\quad &\\pi_2 \\wedge \\omega^2 =0.\n\\end{array}\n$$\nBy our assumption  that $\\pi_2$ is  closed,  \\eqref{W_2^+}  and the above definition of ${\\mathcal W}^+_2$ (see \\eqref{spacepi2}) we have\n$$\n0 = d (\\pi_2 \\wedge \\psi_+) =  \\pi_2  \\wedge d \\psi_+ = \\pi_2 \\wedge d \\psi_+ = \\vert \\pi_2 \\vert^2 * 1 \\,.\n$$\nThen  $\\pi_2 =0$ and we get   the result.\n\\end{proof}\n\nIn particular, as a consequence we have that  if  $(N, \\omega,\n\\psi)$ is  $6$-dimensional manifold endowed with a (not\nintegrable)\n $SU(3)$-structure in the class ${\\mathcal W}_2^+$, the  $3$-form $H = \\pi_2 \\wedge \\alpha$ on $N \\times S^1$ cannot be closed.\n\n\\begin{rem} {\\rm It has to be noted that, in view of Proposition \\ref{strongconstruction}, for $SU(3)$-manifolds in the\nclass ${\\mathcal W}_2^+$, the two conditions\n$$\nd\\pi_2=0\\quad \\hbox{\\rm and}\\quad d\\psi_+=0\n$$\nare equivalent.\\newline Furthermore, under the conditions of\nProposition \\ref{strongconstruction}, the holonomy group of the\nmetric on the manifold $N$ can be properly contained in $SU(3)$. Indeed, for example, if one\ntakes the  $6$-manifold  $N=M^4 \\times \\mathbb{T}^2$, where $(M^4$, $ \\omega_1, \\omega_2, \\omega_3)$\nis an hyper-K\\\"ahler manifold and\n$\\mathbb{T}^2$ is a $2$-dimensional torus, then an $SU(3)$-structure is defined by\n\\begin{eqnarray*}\n\\omega &=& \\omega_1 + e^5 \\wedge e^6,\\\\\n\\psi_+ &=& \\omega_2 \\wedge e^5 - \\omega_3 \\wedge e^6,\\\\\n\\psi_- &=& \\omega_2 \\wedge e^6 + \\omega_3 \\wedge e^5,\n\\end{eqnarray*}\nwhere $\\{e^5, e^6\\}$ is an orthonormal coframe on $\\mathbb{T}^2$. Since\n$$\nd \\omega_i =0\\,,\\,\\, i = 1,2,3\\,,\\quad \\quad d e^5 = d e^6 =0\\,,\n$$\nwe have\n$$\nd \\omega =0\\,,\\quad d \\psi_{\\pm} =0\\,.\n$$\nTherefore, the manifold $N$ endowed with the $SU(3)$-structure defined by $(\\omega,\\psi)$ belongs to the class ${\\mathcal W}_2^+$ and\nthe holonomy of the associated Riemannian metric is strictly contained in $SU(3)$, since the metric is a product.}\n\\end{rem}\n\\begin{rem}  {\\rm Consider on $N \\times \\R$ the generalized $G_2$-structure defined by  the structure form $\\rho$ given\nby \\eqref {rho} and let $H$ be a closed non-zero $3$-form.\nIf we drop  the condition $d_H \\hat \\rho =0$, then  the $SU(3)$-structure $(\\omega, \\psi)$ on $N$ has to be in the class\n${\\mathcal W _2}^+ \\oplus {\\mathcal W _2}^- \\oplus {\\mathcal W _5}$ with\n$$\nd \\psi_+  = \\pi_1  \\wedge \\psi_+  - \\pi_2 \\wedge \\omega = - S \\wedge \\omega,  \\quad dS =0.\n$$\n Indeed,\n$\\rho$ is $d_H$-closed  if and only if\n$$\n\\left\\{\n\\begin{array}{l}\nd \\omega=0\\,,\\\\[5pt]\nd  \\psi_+   \\wedge \\alpha =  - H  \\wedge \\omega\\,,\\\\[5pt]\nH \\wedge \\psi_+ \\wedge \\alpha =0\\,.\n\\end{array}\n\\right.\n$$\nSetting\n$$\nH = \\tilde H + S \\wedge \\alpha,\n$$\nwith $\\tilde H$  and $S$ a $3$-form  and a $2$-form respectively on $N$, then one gets the equivalent conditions:\n$$\n\\left\\{\n\\begin{array}{l}\nd \\omega =0,\\\\[5pt]\nd \\psi_+ = - S \\wedge \\omega\\,,\\\\[5pt]\n\\tilde H \\wedge \\psi_+ = \\tilde H \\wedge \\omega =0\\,,\\\\[5pt]\nd S  = d \\tilde H =0\\,.\n\\end{array}\n\\right.\n$$\nIn terms of the components of the intrinsic torsion one has that $\\nu_0, \\alpha_0,  \\nu_1, \\nu_3$ vanish and  $$\nd \\psi_+ = - S  \\wedge \\omega.\n$$\nIn contrast with the   case of $SU(3)$-manifolds in the class\n${\\mathcal W}_2^+$  (see Proposition \\ref{strongconstruction}),\n$6$-dimensional compact examples of this type  may exist, as\nshowed by the following\n\n\\begin{ex} {\\rm Consider the $6$-dimensional nilpotent Lie algebra $\\mathfrak l$  with structure equations\n$$\n(0,0,0,0,0,25)\n$$\nand  the  $SU(3)$-structure given by\n$$\n\\begin{array}{l}\n\\omega = e^{12} + e^{34} + e^{56},\\\\[3pt]\n\\psi = (e^1 + i e^2) \\wedge  (e^3 + i e^4) \\wedge  (e^5 + i e^6).\n\\end{array}\n$$\nLet  $H$ be  the   closed $3$-form\n$$\n\\begin{array} {lcl}\nH &= &- e^{457} + a_1 (e^{124} - e^{456}) + a_2 (e^{125} - e^{345})  - a_3 (e^{134}  - e^{156}) + a_4 e^{135} + \\\\[3pt]\n&& a_5 (e^{145} - e^{235}) + a_6 (e^{145} + e^{246}) + a_7 (e^{234} - e^{256}) + a_8 e^{245},\n\\end{array}\n$$\nwith $a_i \\in \\R$, $i = 1, \\ldots, 8$.\nThen $( \\omega, \\psi)$ induces a structure form  $\\rho$ on  a compact quotient of $L \\times  \\R$, where $L$ is the simply connected nilpotent Lie group\nwith Lie algebra $\\mathfrak l$, by a uniform discrete subgroup. A straightforward computation shows that  $d_H \\rho =0$ . }\n\\end {ex}\n\n}\n\\end{rem}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThe purpose of this paper is to point out\nsome connections between:\n\\begin{enumerate}\n\\item The monodromy of periodic linear\ndifferential equations;\n\\item The pentagram map, which\nwe studied in [{\\bf S1\\\/}] and [{\\bf S2\\\/}];\n\\item Dodgson's method of condensation for\ncomputing determinants;\n\\end{enumerate}\nWe discovered most of these connections through\ncomputer experimentation.\n\n\\subsection{Monodromy}\n\nConsider the second order O.D.E.\n\\begin{equation}\n\\label{diff2}\nf''(t)+\\frac{1}{2} q(t) f(t)=0.\n\\end{equation}\nHere $q(t)$ is $1$-periodic.\nIf $\\{f_1,f_2\\}$ is a basis for the solution space of\nEquation \\ref{diff2} then\nthere is some linear $T \\in SL_2(\\R)$ such\nthat $f_j(t+1)=T(f_j(t))$ for $j=1,2$.\nThe trace tr$(T)$, which is independent of\nbasis, is sometimes called the\n{\\it monodromy\\\/} of Equation\n\\ref{diff2}.\nThe ratio $f=f_1\/f_2$ gives a smooth map\nfrom $\\R$ into the projective line.\nHere $q$ is \ngiven by the {\\it Schwarzian derivative\\\/}:\n\\begin{equation}\nq=\\frac{f'''}{f'}-\\frac{3}{2} \\bigg(\n\\frac{f''}{f'}\\bigg)^2.\n\\end{equation}\n\nHere is a discrete analogue of Equation \\ref{diff2}.\nThe {\\it cross ratio\\\/} of $4$ points\n$a,b,c,d \\in \\R$ is given by\n\\begin{equation} \\label{cro}\nx(a,b,c,d)=\\frac{(a-c)(b-d)}{(a-b)(c-d)}.\n\\end{equation}\nA calculation shows that the quantity\n\\begin{equation}\n\\lim_{\\epsilon \\to 0} \\frac{1}{\\epsilon^2}\nx(f(t-3 \\epsilon),f(t-\\epsilon),f(t+\\epsilon),f(t+3 \\epsilon))\n\\end{equation}\nconverges to a multiple of $q$, when $f$ is sufficiently smooth.\nThus, the cross ratio is a \ndiscrete analogue of the Schwarzian derivative.\nSuppose  we have an infinite $n$-periodic sequence\n$...q_n,q_1,q_2,...,q_n,q_1,...$.\nWe can find points $...,f_1,f_2,f_3,...$ in \nthe projective line such that \n\\begin{equation}\n\\label{diff1}\nx(f_j,f_{j+1},f_{j+2},f_{j+3})=q_j \\hskip 30 pt \\forall j\n\\end{equation}\nThere will be a projective transformation\n$T$ such that $f_{j+n}=T(f_j)$ for all $j$.\nThe conjugacy class of $T$ only depends on $q$.    To obtain\na numerical invariant, we can lift $T$\nto $SL_2(\\R)$ and take its trace.  \nThis quantity is a rational function in\nthe variables $q_1,...,q_n$. \n\nA main focus of this paper is a discrete analogue\nfor the third order case.  This analogue involves\ninfinite polygons in the projective plane.  In\nanalogy to the cross ratio \nwe will define {\\it projective invariants\\\/} of\npolygons in \\S 3.1.\nWe begin with an\ninfinite sequence $...,x_1,x_2,...$ of projective invariants\nhaving period $2n$.  \nThese invariants determine, up to a projective\ntransformation, an infinite polygon which is\ninvariant under a projective transformation.\nWe call $P$ a {\\it twisted $n$-gon\\\/}.\nIn other words, we have a map $P: \\Z \\to \\R\\P^2$\nand a projective transformation $T$ such that\n$P(n+j)=T(P(n))$ for all $j$.\n\nThe monodromies $\\Omega_1$ and $\\Omega_2$ corresponding to $T$\nare rational functions of \nthe variables $x_1,...,x_{2n}$. \nLet $[\\cdot ]$ denote the floor function.\nIn \\S 2.1 we will define polynomials\n$O_1,...,O_{[n\/2]},O_n$ and\n$E_1,...,E_{[n\/2]},E_n$.  We call these\npolynomials the {\\it pentagram invariants\\\/}.\nWe will express the monodromies explicitly\nin terms of the pentagram invariants:\n\\begin{equation}\n\\label{main}\n\\Omega_1=\\frac{(\\sum_{k=0}^{[n\/2]} O_k)^3}{O_n^2E_n};  \\hskip 40 pt\n\\Omega_2=\\frac{(\\sum_{k=0}^{[n\/2]} E_k)^3}{E_n^2O_n}.\n\\end{equation}\n\n\\subsection{The Pentagram Map}\n\n\nRoughly, the {\\it pentagram map\\\/} is the map which\ntakes the polygon $P$ to the polygon $P'$,\nas indicated in Figre 1.  In \\S 4 we\nwill give a precise definition, which\nexpresses the pentagram map as a\ncomposition of two involutions\n$\\alpha_1$ and $\\alpha_2$.\n\n\\begin{center}\n\\psfig{file=Pix\/pix3.ps}\nFigure 1\n\\end{center}\n\nExpressed in our\nprojective invariant coordinates$-$the cross\nratio generalizations discussed in the previous\nsection$-$the pentagram map\nhas the form\n$\\alpha_1(x_1,...,x_{2n})=(x'_1,...,x'_{2n})$ and\n$\\alpha_2(x_1,...,x_{2n})=(x''_1,...,x''_{2n})$ where\n\\begin{eqnarray}\n\\label{basic}\nx_{2k-1}'=x_{2k} \\frac{1-x_{2k+1}x_{2k+2}}{1-x_{2k-3}x_{2k-2}};\n\\hskip 25 pt\nx_{2k}'=x_{2k-1}\\frac{1-x_{2k-3}x_{2k-2}}{1-x_{2k+1}x_{2k+2}};\n\\cr \\cr \\cr\nx_{2k+1}''=x_{2k}\\frac\n{1-x_{2k-2}x_{2k-1}}\n{1-x_{2k+2}x_{2k+3}}\n\\hskip 25 pt\nx_{2k}''=x_{2k+1}\\frac\n{1-x_{2k+2}x_{2k+3}}\n{1-x_{2k-2}x_{2k-1}}\n\\end{eqnarray}\nIn these formulas, the indices are taken mod $2n$.\nWe let $\\alpha=\\alpha_1 \\circ \\alpha_2$.\nIn general, $\\alpha$ has infinite order.\n\nIt turns out that the pentagram invariants are\ninvariant polynomials for the {\\it pentagram map\\\/},\nwhen it is expressed in suitable coordinates.\n\n\\begin{theorem} \n\\label{trans}\n$O_k \\circ \\alpha_j=E_k$ and\n$E_k \\circ \\alpha_j=O_k$ for\n$j=1,2$ and for all $k$.\n\\end{theorem}\n\n\\noindent\nIn \\S 2 we will give a completely algebraic proof of\nTheorem \\ref{trans}.  In \\S 3-4 we will give a\nmore conceptual proof which goes roughly as follows:\nThe pentagram map commutes with projective transformations\nand therefore must preserve the monodromies\n$\\Omega_1$ and $\\Omega_2$.\nIt follows from the general homogeneity\nproperties of Equation \\ref{basic} that\nthe pentagram map must preserve the properly\nweighted homogeneous pieces of the\nmonodromies, and these pieces are precisely\nthe pentagram invariants.   In \\S 6 we prove\n\n\\begin{theorem}\n\\label{precise}\nThe pentagram invariants are algebraically\nindependent, so that\n$\\alpha$ has at least $2[n\/2]+2$ algebraically independent\npolynomial invariants.\n\\end{theorem}\n\nWe conjecture that the pentagram invariants give the complete\nlist of invariants for the pentagram map, at least when it\nacts on the spaces of twisted $n$-gons.  We also\nconjecturethat the algebraic varieties cut out\nby the pentagram invariants are complex\ntori, after a suitable compactification.\nFinally we conjecture that the pentagram map acts\non these complex tori as a translation in\nthe natural flat metric. \n\n\\subsection{The Method of Condensation}\n\nLet $M$ be an $m \\times m$ matrix. Let\n$M_{NW}$ be the $(m-1) \\times (m-1)$ minor\nobtained by crossing off the last row and\ncolumn of $M$.  Here $N$ stands for ``north''\nand $W$ stands for ``west''.  We define the\nother three $(m-1) \\times (m-1)$ minors $M_{SW}$,\n$M_{NE}$ and $M_{SW}$ in the obvious way.\nFinally, we define\n$M_C$ to be the ``central''  $(m-2) \\times (m-2)$ minor\nobtained by crossing off all the extreme\nrows and columns of $M$.  Dodgson's\nidentity says  \\begin{equation}\n\\label{lc}\n\\det(M) \\det(M_C)=\\det(M_{NW}) \\det(M_{SE})-\n\\det(M_{SW}) \\det(M_{NE}).\n\\end{equation}\nAssuming that $\\det(M_C)$ is non-zero,\nEquation \\ref{lc} expresses $\\det(M)$ \nas a rational function of determinants\nof matrices of smaller size.  \nThis procedure can be iterated, expressing\nthe determinants of these smaller matrices\nas rational functions of determinants of still\nsmaller matrices.  And so on.\nThis method of computing matrices\nis called {\\it Dodgson's method of\ncondensation\\\/}.  See\n[{\\bf RR\\\/}] for a detailed discussion of\nthis method and the rational functions that arise.\n\nIn \\S 5 we will relate the pentagram map to\nthe method of condensation.   In some sense,\n{\\it the pentagram map\ncomputes determinants\\\/}. We exploit this\npoint of view to prove\n\n\\begin{theorem}\n\\label{hyper}\nSuppose that $P$ is a $4n$-gon whose sides\nare alternately parallel to the $x$ and\n$y$ axes.  Then (generically) the\n$(2n-2)$nd iterate of the pentagram map transforms\n$P$ into a polygon whose odd vertices are \nall collinear and whose even vertices are all\ncollinear.  \n\\end{theorem}\n\nThe surprise in Theorem \\ref{hyper} is that $P$ could\nhave trillions of sides.\nThe pentagram map goes about its business for\ntrillions of iterations and then the whole thing\ncollapses all at once into a polygon whose\nvertices lie on a pair of lines. \nTheorem \\ref{hyper} is closely related\nto the main result in [{\\bf S3\\\/}],\nwhich we proved by geometric methods.\n\n\\subsection{Paper Overview}\n\\begin{tabular}{ll} \n {\\bf \\S 2: The Invariants\\\/}& \\\\\n\\S 2.1: Basic Definitions \\\\\n\\S 2.2: Proof of Theorem \\ref{trans} \\\\ \n\n {\\bf \\S 3: Discrete Monodromy\\\/}& \\\\\n\\S 3.1: PolyPoints and PolyLines &  \\\\\n\\S 3.2: Constructing the PolyPoints from its Invariants & \\\\\n\\S 3.3: The Final Calculation & \\\\\n\n {\\bf \\S 4: The Pentagram\\\/}& \\\\\n\\S 4.1: Basic Definitions& \\\\\n\\S 4.2: The Pentagram Map in Coordinates & \\\\\n\\S 4.3: Second Proof of Theorem \\ref{trans} & \\\\\n\\S 4.4: Conic Sections \\\\\n\n {\\bf \\S 5: The Method of Condensation\\\/}& \\\\\n\\S 5.1: Octahedral Tilings & \\\\\n\\S 5.2: Picture of the Pentagram Map & \\\\\n\\S 5.3: Circulent Condensations & \\\\\n\\S 5.4: The Lifting Problem \\\\\n\\S 5.5: Degenerate Polygons \\\\\n\\S 5.6: Proof of Theorem \\ref{hyper}\\\\\n\n{\\bf \\S 6: Proof of Theorem \\ref{precise}\\\/} \\\\\n\\S 6.1: Proof modulo the Vanishing Lemma \\\\\n\\S 6.2: Proof of the Vanishing Lemma\n\\end{tabular}\n\n\\subsection{Acknowledgements}\n\nI would like to thank Peter Doyle, Bill Goldman,\nPat Hooper,\nFrancois Labourie, and John Millson for interesting\nconversations related to this work. \n\\newpage\n\n\\section{The Invariants}\n\n\\subsection{Basic Definitions}\n\nAll our definitions depend on a fixed\ninteger $n \\geq 3$.  We will sometimes\nsuppress $n$ from our notation.\nLet $Z=\\{1,2,3,...,2n\\}$.\nWe think of the elements of $Z$ as\nbeing ordered cyclically, so that\n$2n$ and $1$ are consecutive.\nAlso, in our notation all our indices\nare taken cyclically.\n\nWe say that an {\\it odd unit\\\/} of $Z$\nis a subset having one of the two\nforms:\n\\begin{enumerate}\n\\item $U=\\{j\\}$, where $j$ is odd.\n\\item $U=\\{k-1,k,k+1\\}$, where $k$ is even.\n\\end{enumerate}\nWe say that two odd units $U_1$ and $U_2$ are\n{\\it consecutive\\\/} if the set of odd\nnumbers in the union\n$U_1 \\cup U_2$ are consecutive.  For\ninstance $\\{1\\}$ and $\\{3,4,5\\}$ are\nconsecutive whereas\n$\\{1,2,3\\}$ and $\\{7,8,9\\}$ are not.\n\nWe say that an {\\it odd admissible subset\\\/} is\na nonempty subset $S \\subset X$ consisting of a\nfinite union of odd units, no two of which\nare consecutive.   We define the {\\it weight\\\/}\nof $S$ to be the number of odd units it contains.\nWe denote this quantity by $|S|$.   We define the\n{\\it sign\\\/} of $S$ to be the $+1$ is $S$ contains\nan even number of singleton units, and $-1$ if $S$ contains\nan odd number of singleton units.\nAs an example, the subset\n$$\\{1,5,6,7,11\\}=\\{1\\} \\cup \\{5,6,7\\} \\cup \\{11\\}$$\nis an odd admissible subset\nif $n \\geq 7$.  This subset has weight $3$ and sign $+1$.\nAs an exception to this rule, we call the\nset $\\{1,3,5,7,...,2n-1\\}$ odd admissible as well.\n\n\nEach odd admissible subset $S$ defines a\nmonomial $O_S \\in R$:\n\\begin{equation}\nO_S={\\rm sign\\\/}(S) \\prod_{j \\in S} x_j.\n\\end{equation}\nLet $O(k)$ denote the set of\nweight $k$ odd admissible subsets of $Z$.\nIf $n$ is even then $O(k)$ is nonempty iff\n$k \\in \\{1,2,...,n\/2,n\\}$.  If $n$ is odd then\n$O(k)$ is nonempty iff $k \\in \\{1,2,...,(n-1)\/2,n\\}$.\nWe define\n\\begin{equation}\nO_k=\\sum_{S \\in O(k)} O_S.\n\\end{equation}\nBy convention we set $O_0=1$.\n\nWe can make all the same definitions with the\nword {\\it even\\\/} replacing the word {\\it odd\\\/}.\nThis leads to the definition of the $E$ polymonials.\n\n\\subsection{Proof of Theorem \\ref{precise}}\n\nLet $\\alpha=\\alpha_1 \\circ \\alpha_2$ be as\nin the introduction.\nFor any rational function $f$, we define\n$\\alpha(f)=f \\circ \\alpha$.\n\nBy definition\n\\begin{equation}\nO_n=x_1x_3...x_{2n-1}; \\hskip 20 pt\nE_n=x_2x_4...x_{2n}.\n\\end{equation}\nIf is easy to see directly from\nEquation \\ref{basic} that \n$\\alpha_j(O_n)=E_n$ and\n$\\alpha_j(E_n)=O_n$.   When $n$ is even, we have\n\\begin{equation}\nO_{n\/2}=x_1x_5x_9...+x_3x_7x_{11}...; \\hskip 20 pt\nE_{n\/2}=x_2x_6x_{10}...+x_4x_8x_{12}....\n\\end{equation}\nOnce again, it is easy to see directly from\nEquation \\ref{basic} that\n$\\alpha_j(O_{n\/2})=E_{n\/2}$ and\n$\\alpha_j(E_{n\/2})=O_{n\/2}$.\nThe interesting cases, which we now consider,\nare when $k<n\/2$.  We will show that\n$\\alpha_1(O_k)=E_k$.  The other cases have\nsimilar derivations.\n\nBefore we treat the general case we consider an\nexample:  We have\n$$O_1=\\sum_{j=1}^n (-x_{2j+1}+x_{2j-1}x_{2j}x_{2j+1}).$$\nHere indices are taken mod $2n$.\nWe compute easily that  \\begin{equation}\n\\label{action}\n\\alpha_1(x_{2j+1}- x_{2j-1}x_{2j}x_{2j+1})=\nx_{2j+2}-x_{2k+2}x_{2j+3}x_{2j+4}.\n\\end{equation}\nTherefore\n$$\\alpha_1(O_1)=\n\\sum_{j=1}^n (-x_{2j+2}+x_{2k+2}x_{2j+3}x_{2j+4})=E_1.$$\nThis example suggests that the key to proving\nTheorem \\ref{precise} lies in partitioning\nour polynomials in the right way.\n\nRecall that $O(k)$ is the collection of weight $k$\nodd admissible sequences.   Let $O_s(k) \\subset O(k)$\nconsist of those sequences whose individual\nunits are singletons.  For instance\n$\\{1,5,9\\}$ is a member of $O_s(3)$ as long as $n \\geq 6$.\nWe call the odd units $\\{j\\}$ and $\\{j-2,j-1,j\\}$\n{\\it right partners\\\/}.   We say that a sequence\n$S' \\in O(k)$ is a {\\it right partner\\\/} of\na sequence $S \\in O_s(k)$ if every odd\nunit in $S$ has a right partner odd unit in $S'$ and\n{\\it vice versa\\\/}.  For instance,\nthe sequence $S=\\{1,7,19\\}$ and $S'=\\{1,5,6,7,17,18,19\\}$\nare right partners.   Also, $S$ is a right partner with\n$\\{1,5,6,7,19\\}$.\nFor any $S \\in O_s(k)$ let \n$S_R \\subset O(k)$ be the subcollection of right\npartners.   Every element of $O(k)$ has a unique\nright partner in $O_s(K)$.  Therefore, we have a\npartition\n$$O(k)=\\bigcup_{S \\in O_s(k)} S_R.$$\nCorrespondingly, we can write\n$$O_k=\\sum_{S \\in O_s(k)} RO_S; \\hskip 30 pt\nRO_S=\\sum_{S' \\in S_R} O_{S'}.$$\n\nWe can make all the same definitions, with\n{\\it even\\\/} replacing {\\it odd\\\/} and\n{\\it left\\\/} replacing {\\it right\\\/}.\nThus, we have a partition\n$$E(k)=\\bigcup_{S \\in E_s(k)} S_L.$$\nHere $S_L$ consists of the set of left\npartners of $S$.\nCorrespondingly we can write\n$$E_k=\\sum_{S \\in E_s(k)} LE_S; \\hskip 30 pt\nLE_S=\\sum_{S' \\in S_L} E_{S'}.$$\n\nBelow we will prove\n\\begin{lemma}\n\\label{seq}\nFor any sequence $S \\in S_s(k)$ there is a sequence\n$\\overline S \\in S_s(k)$ such that\n$\\alpha_1(RO_S)=LE_{\\overline S}$. \n\\end{lemma}\n\nSince\n$\\alpha_1$ is an involution\nthe assignment $S \\to \\overline S$ is a bijection.\nSumming over the individual terms we have\n$\\alpha(O_k)=E_k$.  Thus, Lemma \\ref{seq} implies\nTheorem \\ref{precise}.  \n\nTo prove Lemma \\ref{seq} we will decompose sequences\nin $S_s(k)$, which could be quite complicated,\ninto much simpler sequences. We say that an\nodd admissible sequence is {\\it tight\\\/} if\nit has the form\n$$\\{j,j+4,j+8,...,j+4a\\}.$$  As usual, these numbers are\ntaken mod $2n$.  \nGiven any $S \\in S_s(K)$ let $T_S$ denote the\nset of maximal tight subsequences of $S$.\nFor instance, if $S=\\{3,7,19,23,35\\}$ and $n=18$ then\n$T_S$ consists of the two sequences $\\{19,23\\}$\nand $\\{35,3,7\\}$.  (The second sequence is\ncongruent mod $36$ to $\\{35,39,43\\}$.)\nSince $k<n\/2$ every sequence in $S_s(k)$ decomposes\nin this way.\n\n\\begin{lemma}\n\\label{seq2}\nIf $S \\in S_s(k)$ then\n$RO_{S}=\\prod_{T \\in T_S} RO_T.$\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nLet $T_S=\\{T_1,...,T_h\\}$.\nNote that any number in $T_i$ is at least $5$ numbers\naway from any number in $T_j$.  Otherwise,\n$T_i \\cup T_j$ would be tight, contradicting\nmaximality.  From this observation we see that\n$T_1' \\cup ... \\cup T_h'$ is odd-admissible\nfor any choice of right partners\n$T_1', T_2',...,T_h'$ of\n$T_1,...,T_h$.  Conversely, any right\npartner of $S$ decomposes this way.\nTherefore\n$S_R$ is precisely the union of the\nsets of the form $T_1' \\cup ... \\cup T_h'$,\nwhere $T'_j \\in (T_j)_R$ is arbitrary.\nOur lemma follows from this and from the\ndistributive law.\n$\\spadesuit$  \\newline\n\n\\begin{lemma}\n\\label{seq3}\nFor any tight sequence $T$ there is a tight\nsequence $\\overline T$, having the same\nlength, such that\n$\\alpha_1(RO_T)=LE_{\\overline T}$. \n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nLet $P=RO_{T}$ and let $P'=\\alpha(P)$.\nCyclically relabelling we can assume that\n$T=\\{3,7,11,...,4a+3\\}$.\nThe set $T_R$ of right partners of $T$ consists of\n$T$ and the sequence\n$\\{1,2,3,7,11,...,4a+3\\}$.\nTherefore\n$$P=(1-x_1x_2)x_3x_7x_{11}...x_{4a+3}.$$\nWriting $x_j'=\\alpha(x_j)$ we have\n$$P'=(1-x'_1x'_2)x'_3x'_7x'_{11}...x'_{4a+3}.$$\nUsing Equation \\ref{action} we see that\n$(1-x'_1x'_2)x'_3=x_4(1-x_5x_6)$.  We also have\n$x_5x_6=x'_5x'_6$.\nTherefore\n$$P'=x_4(1-x'_5x'_6)x_7',x_{11}'...x'_{4a+3}.$$\nUsing Equation \\ref{action} we see that\n$(1-x'_5x'_6)x'_7=x_8(1-x_9x_{10})$.  We also have\n$x_9x_{10}=x'_9x'_{10}.$\nTherefore\n$$P'=x_4x_8(1-x'_9x'_{10})x_{11}',...,x'_{4a+3}.$$\nContinuing in this way we see that\n$$P'=x_4x_8...x_{4a+4}(1-x'_{4a+5}x'_{4a+6})=\nx_4x_8...x_{4a+4}(1-x_{4a+5}x_{4a+6}).$$\nThis last expression is\nexactly $LE_{\\overline T}$, where\n$\\overline T=\\{4,8,...,4a+4\\}$.\n$\\spadesuit$  \\newline\n\nLemma \\ref{seq} follows immediately from Lemma\n\\ref{seq2}, Lemma \\ref{seq3}, and the uniqueness of\nour decomposition.  As we mentioned before,\nTheorem \\ref{precise} follows from Lemma \\ref{seq}.\nThis completes our proof.\n\n\\newpage\n\n\\section{The Proof}\n\n\\section{Proof of Theorem \\ref{precise}}\n\\label{independence}\n\n\\subsection{Proof modulo the Vanishing Lemma}\n\nIn this chapter we prove that the\npentagram invariants,\nconstructed in \\S 2.1, are algebraically independent.\nWe take $n$ odd, so that there are $n+1$\ndihedral invariants.  The even case is very similar.\n\nGiven a polynomial map \n$f: \\C^{2n} \\to \\C$, let\n$\\nabla f=(\\partial f\/\\partial z_1,...,\\partial f\/\\partial z_{2n})$.\n\n\\begin{lemma}\n\\label{C}\nSuppose there is a $p \\in \\C^{2n}$ such\nthat $\\nabla f_1(p),...,\\nabla f_{n+1}(p)$ are\nlinearly independent over $\\C$.  Then\n$f_1,...,f_{n+1}$ are algebraically independent.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nFor the sake of contradiction assume that there is some\nnontrivial $F \\in \\Z[x_1,...,x_{n+1}]$ such\nthat $F(f_1,...,f_{n+1})=0$.  \nBy continuity, there is an open set $U$ such\nthat $\\nabla f_1(q),...,\\nabla f_{n+1}(q)$ are\nlinearly independent for any point $q \\in U$.\nSince $F$ is a nontrivial polynomial map there\nis some $q \\in U$ such that $dF(q) \\not = 0$.\nBy the chain rule, $\\nabla f_k \\in {\\rm ker\\\/}(dF)$\nfor all $k$.\nThese vectors span $\\C^{n+1}$, forcing\n$dF(q) =0$, a contradiction.\n$\\spadesuit$  \\newline\n\nLet $\\omega=\\exp(2 \\pi i\/ n)$. For $v \\in [1,(n-3)\/2]$\nlet $\\Lambda_v$ be the collection of all sequences\n$\\{s_1,...,s_v\\} \\subset \\{2,3,...,n-2\\}$ such that \n$s_j \\leq s_{j+1}+2$ for all $j$.   \nDefine\n\\begin{equation} \\label{exp}\n\\lambda_v=\\sum_{I \\in \\Lambda_v}\\omega^I; \\hskip 40 pt\n\\omega^I=\\prod_{j=1}^v \\omega^{s_j};\n\\hskip 15 pt I=(s_i,...,s_v).\n\\end{equation}\nIn the next section we will prove\nthat $\\lambda_v \\not = v$ for all\n$v \\in [1,(n-3)\/2]$.  In this section\nwe take this result, which we call\nthe Vanishing Lemma, for granted.\n\nWe have two embeddings of $\\C^n$ into $\\C^{2n}$.\nNamely \\begin{equation}\no(z_1,...,z_n)=(z_1,0,z_2,0,...,z_n,0); \\hskip 15 pt\ne(z_1,...,z_n)=(0,z_1,0,z_2,...,0,z_{n}).\n\\end{equation}\nDefine $H_k=O_k \\circ o=E_k \\circ e$.\nThe function $H_n$ is defined in $\\C^n$.\n\n\\begin{lemma}\n\\label{alg2}\n$\\nabla H_1(p),...,\\nabla H_{(n-1)\/2}(p), \\nabla H_n(p)$ are\nlinearly independent at\n$p=(\\omega,\\omega^2,...,\\omega^n)$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nSince $H_{i+1}$ is cyclically invariant, and homogeneous\nof degree $i+1$, the gradient $\\nabla H_i$ is homogeneous\nof degree $i$.  Furthermore, the $(j+1)$th entry of\n$\\nabla H_i$ is obtained from the $j$th entry by shifting\nthe indices of the variables by $1$.  These two\nfacts imply that\n$\\nabla H_{i+1}(p)=\\mu_i V_i,$ where\n$$\\mu_i=\\frac{\\partial H_{i}}{\\partial z_n}\\bigg|_p; \\hskip 30 pt\nV_i=(\\omega^i,\\omega^{2i},...,\\omega^{ni}).$$\nThe vectors $V_1,...,V_{\\frac{n-1}{2}},V_n$ are\ncertainly linearly independent over $\\C$.\nIt suffices to prove all the\n$\\mu_i$ are nonzero.\n\nAs $H_{1}=z_1+...+z_n$ and\n$H_{n}=z_1...z_n$, we have $\\mu_1=\\mu_n=1$.\nFor the intermediate values of $i$, the terms in $H_{i}$ have the form\n$(-1)^i z_{k_1},...,z_{k_i}$, where successive or\nrepeating indices are not allowed.  Note, in particular,\nthat $...z_1z_n...$ never occurs, because the notion of\nsuccession is reckoned cyclically.  The\nterms in $\\partial H_i\/\\partial z_n$ therefore have the form\n$$(-1)^iz_{k_1},...,z_{k_{i-1}}; \\hskip 30 pt\nk_{\\alpha} \\leq k_{\\alpha+1}+2 \\hskip 10 pt \\forall \\alpha.$$\nAlso, the terms $z_{n-1}$, $z_n$ and $z_1$ do not occur.\nThus, we see that $\\mu_i=\\pm \\lambda_{i-1}$.  Since\n$i \\leq (n-1)\/2$, the Vanishing Lemma says that $\\mu_i \\not = 0$.\n$\\spadesuit$  \\newline\n\n\\begin{lemma}\n\\label{limm}\nLet $p_t=(t\\omega,t\\omega,t\\omega^2,t\\omega^2,...\n       t\\omega^n,t\\omega^n)$.  Then\n$$\\lim_{t \\to 0} t^{1-k} \\nabla O_k(p_t)=\no(\\nabla H_k(p)); \\hskip 20 pt\n\\lim_{t \\to 0} t^{1-k} \\nabla E_k(p_t)=\ne(\\nabla H_k(p)).$$\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nWe will derive the first equation, the second being similar.\nGiven any point $q=(x_1,...,x_{2n})$ define \n\\begin{equation}\n\\label{hom}\nS_t(q)=(t^{-1}x_1,tx_2,t^{-1}x_3,tx_4,t^{-1}x_5,...,tx_{2n}).\n\\end{equation}\nWe have \\begin{equation}\n\\label{ind1}\n\\lim_{t \\to 0} S_{t}(p_t)=\n(\\omega,0,\\omega^2,0,...,\\omega^n,0)=o(p).\n\\end{equation}\nLet $\\partial_jO_k$ be the $j$th partial derivative\nof $O_k$.\nFor any point $q$ we have the general\nhomogeneity: \\begin{equation}\nO_k(S_t(q))=t^{-k}(O_k(q)). \\end{equation}\nHence\n\\begin{equation}\n\\label{ind2}\n\\partial_jO_k(S_{t}p_t)=t^{-k_j}\\partial_jO_k(p_t)\n\\hskip 20 pt k_j=k+(-1)^j.\n\\end{equation}\nCombining Equations \\ref{ind1} and \\ref{ind2} we have\n\\begin{equation}\n\\label{ind4}\n\\lim_{t \\to 0} t^{1-k} \\partial_jO_k(p_t)=\n\\left\\{\\matrix{\\partial_jO_k(o(p))&&&{\\rm j\\ odd\\\/} \\cr \\cr 0\n&&&{\\rm j\\ even\\\/}}\\right.\n\\end{equation}\nFinally, \\begin{equation}\n\\label{ind5}\n\\partial_{2j+1}(O_k(o(p))=o(\\partial_j H_k(p)).\n\\end{equation}\nEquations \\ref{ind4} and \\ref{ind5} together establish\nthe first equation.\n$\\spadesuit$  \\newline\n\nWe claim that the vectors\n$\\nabla O_1(p_t),...,\\nabla E_n(p_t)$ are\nindependent for some $t$.  Otherwise there are\nfunctions $a_{k,t}$ and $b_{k,t}$ such that\n\\begin{equation}\n\\label{ind7}\n\\sum a_{k,t} \\nabla O_k(p_t)+\n\\sum b_{k,t} \\nabla E_k(p_t)=0;\n\\hskip 30 pt\n\\max(|a_{k,t}|,...,|b_{n,t}|)=1.\n\\end{equation}\nWe let $t \\to 0$.\nTaking a subsequence, we can arrange that\n$\\lim a_{k,t}=a_{k,0}$ and\n$\\lim_{b,t}=b_{k,0}$ for all relevant $k$, with\nat least one limit being nonzero.\n\nWe multiply Equation \\ref{ind7} by $t^{1-k}$\nand take the limit using Lemma \\ref{limm} to\nobtain\n\\begin{equation}\n\\sum a_{k,0}\\ o(\\nabla H_k(p))+\n\\sum b_{k,0}\\ e(\\nabla H_k(p))=0.\n\\end{equation}\nThe subspaces $o(\\C^n)$ and\n$e(\\C^n)$ are orthogonal and the\nvectors $\\{\\nabla H_k(p)\\}$ are\nlinearly independent.  This is a contradiction.\nHence $\\nabla O_1(p_t),...,\\nabla E_n(p_t)$ are\nalgebraically independent for some value of $t$.\nLemma \\ref{C} now completes our algebraic\nindependence proof, modulo the Vanishing Lemma.\n\n\\subsection{Proof of The Vanishing Lemma}\n\\label{independence proof}\n\nWe begin with some algebraic preliminaries.\nSay that an {\\it adapted measure\\\/} is a \npositive measure $\\tau$, with integer sized\natoms, supported in the $n$th roots of unity.\n(Here $n$ is fixed, as above.)\nEach adapted measure $\\tau$ is encoded by a\nnon-decreasing sequence \n$I=\\{s_1,...,s_k\\}$.  The $j$th root\nof unity has $\\tau$-mass $m$ iff $j$ appears\n$m$ times in $I$.\nWe define $\\langle \\tau \\rangle=\\omega^I$, as in Equation \\ref{exp}.\nBy convention, the $\\langle \\emptyset \\rangle=1$.\nWe define the product\n$\\tau_1 \\cdot \\tau_2$ to be the measure\nobtained by adding $\\tau_1$ and $\\tau_2$ together.\n(We will reserve the $+$ symbol\nfor another purpose.)\nNote that \n$\\langle \\tau_1 \\cdot \\tau_2 \\rangle = \\langle \\tau_1\n\\rangle \\langle \\tau_2 \\rangle$.  \nFor $m \\in \\N$ we define $m \\tau=\n\\tau \\cdot ... \\cdot \\tau$, a total of $m$ times.\n\nLet $M$ denote the free abelian group\ngenerated by the adapted measures.  A\ntypical element of $M$ is a finite formal\nsum $\\sigma=\\sum m_i \\tau _i$.\nWe define the {\\it evaluation map\\\/}\n \\begin{equation}\n\\langle \\sigma \\rangle=\n\\sum \\langle \\tau_i \\rangle^{m_i}.\n\\end{equation}  We make $M$ into a ring\nusing the product rule\n\\begin{equation}\n\\big(\\sum_i m_i \\sigma_i\\big) \n\\big(\\sum_j n_j \\sigma_j\\big)=\n\\sum_{i,j} m_in_j( \\sigma_i \\cdot \\tau_j)\n\\end{equation}\nThe ring $M$ is the {\\it group ring\\\/} generated\nby the adapted measures.  The evaluation\nmap is a ring homomorphism from $M$ to $\\C$.\n\nIf $A \\subset S^1$ is an arc and $v \\in \\N$ is\nan integer, let $\\Psi(A,v) \\subset M$ denote the\nsum, taken over all adapted measures\nwhich have mass $v$ and are supported in $A$.\nUsing the notation from the Vanishing Lemma,\nlet $A_v \\subset S^1$ be the open arc,\ncontaining $-1$, whose endpoints are\n$\\omega^v$ and $\\omega^{-v}$.\n\n\\begin{lemma}[Centrally Symmetric Compression]\n$\\lambda_v=\\langle\\Psi(A_v,v)\\rangle.$ \n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nLet $\\Lambda_v$ be the set of sequences used to\ndefine $\\lambda_v$ in Equation \\ref{exp}.\nIf $I=(s_1,...,s_v) \\in \\Lambda_v$, then let\n$\\phi(I)$ be the summand of $\\Psi(A_v,v)$ indexed by\n$(s_1+v-1,s_2+v-3,s_2+v-5,...,s_{v-2}-v+5,s_{v-1}-v+3,s_v-v+1).$\nThis map is a bijection which\npreserves the total sum of the elements in $I$,\nso that $\\omega^I=\\langle \\phi(I) \\rangle$. \nThis lemma now follows from the definitions\nof the two relevant quantities.\n$\\spadesuit$  \\newline\n\nOur proof breaks down into two main cases, which we\ntreat in turn.\n\n\\subsubsection{Case 1: $v<n\/4$}\n\nWe say that a measure is {\\it sparse\\\/} if it\nassigns at most mass $1$ to any given point.\nFor any pair $(A,m)$ let $\\Psi'(A,m)$ denote the\nformal sum of sparse mass-$m$ measures, supported in\n$A$. Let $A^c=S^1-A$.\n\n\\begin{lemma} [Binomial Theorem]\n$\\langle\\Psi(A_v,v)\\rangle=\n\\langle\\Psi'(A_v^c,v)\\rangle$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nWe write $A=A_v$, $A^c=A^c_v$,\n$\\Psi=\\Psi(A_v,v)$, and\n$\\Psi'=\\Psi'(A^c_v,v)$.  \nLet $\\Phi_j$ be the\nformal sum of all mass $v$ measures whose\nsupport intersects $A^c$ in exactly $j$ points.\nNote that $\\Phi_0=\\Psi$ and $\\Phi_v=\\Psi'$.\nLet $\\Theta_j$ denote the formal sum of\nall mass $j$ adapted measures.  By symmetry\n$\\langle \\Theta_j \\rangle=0$ for $j>0$. \nLet $\\Delta_j=\\Psi_j(A^c,j)$ be the formal\nsum of sparse adapted measures of mass $j$ which\nare supported in $A^c$.  Note\nthat $\\Delta_v=\\Psi'$. \n\nSuppose that $k \\in \\{0,....,v-1\\}$.  If\n$j \\geq k$ and $\\tau$ is a summand of\n$\\Phi_j$ there are exactly $j$ choose $k$\nways to write \n$\\tau=\\tau_1 \\cdot \\tau_2$, where\n$\\tau_1 \\in \\Delta_k$ and\n$\\tau_2 \\in \\Theta_{v-k}.$\nThe point is that we can choose the support of\n$\\tau_1$ to be any $k$-element subset of\nthe $A^c$-support of $\\tau$. \nThis way of counting things gives the relation:\n\\begin{equation}\n\\Delta_k \\Theta_{v-k}=\\sum_{j=k}^v \n\\left(\\begin{array}{c} j \\\\k \\end{array} \\right) \\Phi_j,\n\\end{equation}\nfor $k=0,...,v-1$.\nCombining the previous equation with\na familiar corollary of the binomial theorem,\n\\begin{equation}\n\\sum_{k=0}^{v-1} (-1)^k \\Delta_k \\Theta_{v-k}=\n\\Phi_0 + (-1)^v \\Phi_v.\n\\end{equation}\nSince\n$\\langle \\Delta_k \\Theta_{v-k} \\rangle=\n\\langle \\Delta_k \\rangle \\langle \\Theta_{v-k} \\rangle=\n0.$ we have\n$\\langle \\Psi \\rangle=\n\\langle \\Phi_0 \\rangle= \\pm\n\\langle \\Phi_v \\rangle= \n\\langle \\Psi' \\rangle$.\n$\\spadesuit$  \\newline\n\nSince $v<n\/4$ we have\n$\\Re(z)>0$ for all $z \\in A_v$.\nWe will use induction \nto show that $\\langle \\Psi'(A_v,w) \\rangle>0$ for all $v,w \\geq 1$.\nLet $\\underline \\omega^v$ be the mass $1$ measure\nsupported on $\\omega^v$.  If $\\tau$ is a\nmass $w$ sparse measure supported in $A_v^c$ then\nthe support of $\\tau$ intersects\n$\\{\\omega^v,\\omega^{-v}\\}$ in $0$, $1$, or $2$\npoints.  Thus\n\\begin{equation}\n\\label{indu}\n\\Psi'(A_v^c,w)=\\left\\{\\matrix{\\Psi'(A^c_{v-1},w) \\cr + \\cr\n(\\underline \\omega^v+\\underline \\omega^{-v}) \n\\cdot \\Psi'(A^c_{v-1},w-1) \\cr + \\cr\n(\\underline \\omega^v \\cdot \\underline \\omega^{-v}) \\cdot \n\\Psi'(A^c_{v-1},w-2).}\\right\\}\n\\end{equation}\nAt least one term on the right is nontrivial.  From\n\\begin{equation}\n\\label{indu2}\n\\langle \\underline \\omega^v+\\underline \\omega^{-v}\\rangle=\n2 \\Re(\\omega^v)>0; \\hskip 30 pt\n\\langle \\underline \\omega^v \\cdot \\underline \\omega^{-v} \\rangle=1.\n\\end{equation} \nand induction, any nontrivial term on the\nright hand side of Equation \\ref{indu} evaluates\nto a positive number.   Therefore, the\nleft hand side evaluates to a positive\nnumber as well.\n\n\\subsubsection{Case 2: $v \\geq n\/4$}\n\nFor each integer $w \\in (0,n\/4]$ we choose an open arc $B_w$, \ninvariant under complex conjugation, such that\n$-1 \\in B_w$ and\nthere are exactly $w$ $n$th roots of\nunity contained in $B_w$.\nLet $\\Psi(w,k',k)$ denote the formal sum of\nadapted mass $k$ measures $\\mu$ such that\n$\\mu$ is supported in $B_w$ and\n$\\mu(B_w-B_{w-2}) \\leq k'$.\n\nOur goal is to show that\n$\\langle \\Psi(w,v,v) \\rangle \\not = 0$,\nwhere $w$ is the number of $n$th roots of\nunity in $A_v$.   \nWe order the triples $(w,k',k)$ lexicographically.\nWe will show inductively that\n$\\langle \\Psi(w,k',k) \\rangle>0$ if $k$ is even\nand $\\langle \\Psi(w,k',k)<0$ if $k$ is odd.\n(These sums are real, by symmetry.)\n\nIf $k=1$ then \n$\\langle \\Psi(w,k',k) \\rangle$ is the sum of numbers\nall of which have negative real part, so that\n$\\langle \\Psi(w,k',k) \\rangle<0$ in this case.\nAlso,\n$\\langle\\Psi(1,k,k) \\rangle=(-1)^k.$\nHenceforth we assume that $w \\geq 2$ and\n$k \\geq 2$.   Since $w \\geq 2$ there are\ntwo $n$th roots of unity\n$\\alpha_1$ and $\\alpha_2=\\overline \\alpha_1$\nin $B_w-B_{w-2}$.\n\nSuppose $w=2$.\nA simple counting argument gives\n$$\\Psi(w,k,k)=(\\underline \\alpha_1 + \\underline \\alpha_2) \\cdot\n\\Psi(v,k-1,k-1) + \\underline \\alpha_1 \\cdot \\underline \\alpha_2\n\\cdot \\Psi(v,k-2,k-2).$$\nNote that $\\alpha_1+\\alpha_2<0$.\nBy induction, both terms on the right have the\ndesired sign when evaluated.\nHenceforth we assume that $w \\geq 3$.\n\nSuppose that $k'=1$.\nA counting argument gives\n$$\\Psi(w,1,k)=\\Psi(w-2,k,k)+\n(\\underline \\alpha_1+\\underline \\alpha_2) \\cdot\n\\Psi(w-2,k-1,k-1)$$\nAgain, we note that $\\alpha_1+\\alpha_2<0$.\nSince\n$w \\geq 3$ both terms on the right have the \ndesired sign when evaluated.\n\nSuppose that $k'=2$.\nA counting argument gives\n$$\\Psi(w,2,k)=\\Psi(w-2,k,k)+\n(\\underline \\alpha_1+\\underline \\alpha_2) \\cdot \\Psi(1,k-1)+\n\\underline \\alpha_1 \\cdot \\underline \\alpha_2\n\\cdot \\Psi(w-2,k-2,k-2).$$\nBy induction, all terms on the right have\nthe desired sign when evaluated.\n\nSuppose that $k' \\geq 3$.\nA counting argument gives\n$$\n\\label{induct}\n\\Psi(w,k',k)= \\left\\{ \\matrix{\n\\Psi(w-2,k'-2,k) \\cr + \\cr\n(\\underline \\alpha_1 + \\underline \\alpha_2) \\cdot\n\\Psi(w,k'-1,k-1) \\cr + \\cr\n\\underline \\alpha_1 \\cdot \\underline \\alpha_2\n\\cdot \\Psi(w,k'-2,k'-2)} \\right\\} .\n$$\nBy induction, all three terms on the right\nhave the desired sign when evaluated.\n\nThis completes our proof.\n\n\n\\newpage\n\n\\section{Discrete Monodromy}\n\n\\subsection{PolyPoints and PolyLines}\n\\label{poly}\n\nAs in previous chapters we will fix some\npositive integer $n \\geq 3$.\n\nLet $\\P$ be the projective plane over the field\n$\\F$.   Say that a {\\it PolyPoint\\\/} is a \nbi-infinite sequence $A=\\{...A_{-3},A_1,A_5,..\\}.$ of\npoints in $\\P$.   (For technical reasons\nwe always index these points by integers\nhaving the same odd congruence mod $4$.)\nWe assume also that there\nis a projective transformation $T$ such\nthat $T(A_j)=A_{j+4n}$ for all $j \\in \\Z$.\nWe call $T$ the {\\it monodromy\\\/} of $A$.\n\nSay that a {\\it PolyLine\\\/} is a \nbi-infinite sequence $B=\\{...B_{-1},B_3,B_7,..\\}$ of\nlines in $\\P$. \nWe assume also that there\nis a projective transformation $T$ such\nthat $T(B_j)=B_{j+4n}$ for all $j \\in \\Z$.\nWe call $T$ the {\\it monodromy\\\/} of $B$.\n\nGiven two points $a,a' \\in \\P$ we let $(aa')$ be the\nline containing these two points.  Given two\nlines $b,b' \\in \\P$ we let $(bb')$ be the point\nof intersection of these two lines.  Every PolyPoint\n$A$ canonically determines a PolyLine $B$, by the\nrule $B_j=(A_{j-2}A_{j+2})$.   At the same time\nevery PolyLine $B$ determines a PolyPoint $A$ by the\nrule $A_j=(B_{j-2}B_{j+2})$.  In this case we call\n$A$ and $B$ {\\it associates\\\/}.   By construction\nassociates have the same monodromy.\n\nThe {\\it dual space\\\/} to $\\P$ is the space \nof lines in $\\P$.  This space, denoted by\n$\\P^*$, is isomorphic to $\\P$.\nIndeed $\\P^*$ is the projectivization of\nthe vector space dual to $\\F^3$.\nAny projective transformation $T: \\P \\to \\P$\nautomatically induces a projective transformation\n$T^*: \\P^* \\to \\P^*$, and {\\it vice versa\\\/}.\nAny point in $\\P$ canonically determines a\nline in $\\P^*$.  Likewise, points in $\\P^*$\ncanonically determine lines in $\\P$ and\nlines in $\\P^*$ canonically determine\npoints in $\\P$.  The two spaces are on\nan equal footing.\n\nGiven the PolyPoint $A$, we define\n$A^*$ to be the PolyPoint in $\\P^*$ whose\nlines are given by the associate $B$.\nIf the points of $A$ are indexed by\nnumbers congruent to $1$ mod $4$ then\nthe points of $A^*$ are indexed by\nnumbers congruent to $3$ mod $4$, and\n{\\it vice versa\\\/}.\nWe make the same definitions for\nPolyLines.\nBy construction\n$A^{**}=A$ and $B^{**}=B$.  If $T$ is\nthe common monodromy of $A$ and $B$ then\n$T^*$ is the common monodromy of\n$A^*$ and $B^*$.  We call $A^*$ and\n$B^*$ the {\\it duals\\\/} of $A$ and $B$.\n\nFor any projective transformation $T$,\nacting either on $\\P$ or $\\P^*$ we\ndefine\n\\begin{equation}\n\\Omega_1(T)=\\frac{{\\rm tr\\\/}^3(\\widetilde T)}{\\det(\\widetilde T)};\n\\hskip 30 pt\n\\Omega_2(T)=\\Omega_1(T^*).\n\\end{equation}\nHere $\\widetilde T$ is a linear transformation whose\nprojectivization is $T$.  That is, $\\widetilde T$ is a\n{\\it lift\\\/} of $T$.\nIt is easy to see that these quantities are\nindependent of lift.  Moreover,\n$\\Omega_j(T)$ only depends on the conjugacy\nclass of $T$.  Finally,\n$\\Omega_{3-j}(T^*)=\\Omega_j(T)$ for any projective\ntransformation.\n\nIf $T$ is the monodromy of $A$ we call\n$\\Omega_1(T)$ and $\\Omega_2(T)$ the\n{\\it monodromy invariants\\\/} of $A$.\nBy construction $A^*$ has the same\n{\\it set\\\/} of monodromy invariants as\n$A$, but their order is switched.\nThe same goes for $B$.   If $S$ is\nsome other projective transformation,\nthen $A$ and $S(A)$ have the same\nmonodromy invariants.  Likewise,\n$B$ and $S(B)$ have the same\nmonodromy invariants.\n\nWe now introduce our $2$-dimensional versions of the\ncross ratio.   If $j$ is\none of the indices for the points of $A$ we define\n\\begin{eqnarray}\n\\label{invt}\np_{(j+1)\/2}(A)=\nx(A_{j + 8}, A_{j + 4}, (B_{j + 6} B_{j - 2}),\n                            (B_{j + 6} B_{j - 6})) \\cr \\cr\nq_{(j-1)\/2}(A)=x(A_{j - 8}, A_{j - 4}, (B_{j - 6} B_{j + 2}),\n                            (B_{j - 6} B_{j + 6}))\n\\end{eqnarray}\nHere $x$ stands for the ordinary cross ratio,\nas in Equation \\ref{cro}.  \nIn the first equation, all $4$ points lie on the\nline $B_{j+6}$.  In the second equation, all\n$4$ points lie on $B_{j-6}$.   Conpare Figure 3 below.\nIf the points of $A$ are labelled by \nintegers congruent to $1$ mod $4$ then the\ninvariants of $A$ are\n$...q_0,p_1,q_2,p_3,...$.\nIf the points of $A$ are indexed by integers\ncongruent to $3$ mod $4$ then the invariants of\n$A$ are\n$...p_0,q_1,p_2,q_3,...$  In this chapter we will\nonly consider the case when the points of $A$ are\nindexed by integers congruent to $1$ mod $4$,\nthough in the next chapter we will consider both\ncases on an equal footing.\n\nWe can make all the same definitions for $B$, simply\nby interchanging the two roles of $A$ and $B$ in\nEquation \\ref{invt}.\nIt turns out that our invariants are not\njust invariant under projective transformations,\nbut also invariant under projective duality.\nPrecisely, we have\n\\begin{equation}\n\\label{dualinv}\np_j(A)=q_j(A^*); \\hskip 15 pt\nq_j(A)=p_j(A^*); \\hskip 15 pt\np_j(B)=q_j(B^*); \\hskip 15 pt\nq_j(B)=p_j(B^*)\n\\end{equation}\nfor all relevant indices.  To see this symmetry,\nwe will consider an example.\n\nSuppose that points of $A$ are labelled by\nintegers congruent to $1$ mod $4$.  The\nfirst half of Figure 3 highlights the\n$4$ points whose cross ratio is $p_3(A)$.\nThe second half shows the lines whose\ncross ratio is used to define $q_3(A^*)$.\nThe highlighted\npoints are exactly the intersection points of\nthe highlighted line with an auxilliary line.\nHence, the two cross ratios are the same.\n\n\n\\begin{center}\n\\psfig{file=Pix\/pix4.ps}\nFigure 3\n\\end{center}\n\n\nThis chapter is devoted to establising\nEquation \\ref{main}, which gives the formulas\nfor $\\Omega_1$ and $\\Omega_2$ in terms of\nour invariants.  Given the formula for\n$\\Omega_1$, the formula for $\\Omega_2$ follows\nfrom projective duality and from \nEquation \\ref{dualinv}.\nThus, to establish Equation \\ref{main} it\nsuffices to derive the equation for\n$\\Omega_1$.\n\n\n\\subsection{Constructing the PolyPoint from its Invariants}\n\\label{monodromy series}\n\nIn \\S 2 we constructed our polynomials from the variables\n$x_1,...,x_{2n}$.  In this section we are going to\nuse the alternate list of variables\n$p_1,q_2,p_3,q_4,...$.  The reason for the alternate\nnotation is that it is useful to distinguish the\neven and odd variables in our constructions.\nThe polynomials in \\S 2 are obtained from the ones\nhere using the substitution\n$p_i \\to x_i$ when $i$ is odd and\n$q_i \\to x_i$ when $i$ is even.\n\nSuppose that $p_1,q_2,p_3,q_4,...$ are given variables.\nWe seek an infinite PolyPoint $A$ such that\n\\begin{equation}\n\\label{mainn}\np_{2i-1}(A)=p_{2i-1}; \\hskip 15 pt\n  q_{2i}(A)=q_{2i}; \\hskip 15 pt i=1,2,3....\n\\end{equation}\nWhat we mean by Equation \\ref{mainn} is that we\nwish to specify the points of $A$ in such a\nway that the invariants we seek match a\nspecified list $p_1,q_2,p_3,...$.\nLikewise, we seek a formula for the associate $B$.\nFor our purposes we only need the formulas\nfor ``half'' of $A$ and ``half'' of $B$.\nThat is, we just need to know $A_{-3},A_1,A_5,...$ and \n$B_{-5},B_{-1},B_3,...$.\n\nHere we make the same definitions as in \\S 2.1, with respect\n$\\Z$ (the integers) rather than the finite set $Z$.\nTo each admissible\nsequence $S$ we associate a monomial $O_S$ in the formal\npower series ring\n$\\A=\\Z[[...p_1,q_2,p_3...]]$.\n(Again, under the substitution mentioned above,\nthe ring $\\A$ is identified with\n$\\Z[[...x_1,x_2,x_3,...]]$.) \nFor instance if $S=\\{1,2,3,9\\}$ then\n$O_S=-p_1q_2p_3q_9$.\nWe count the empty subset as both even and odd\nadmissible, and we define $O_{\\emptyset}=E_{\\emptyset}=1$.\nLet $O$ be the\nsum over all odd admissible sequences of finite\nweight.  Likewise\nlet $E$ be the sum over all even admissible sequences\nof finite weight. \nWe have $O, E \\subset \\A$.\nGiven a pair of odd integers, $(r,s)$  we\ndefine $O_r^s$  to be the polynomial\nobtained from $O$ by setting $p_j$\nequal to zero, for $j \\leq r$ and $j \\geq s$.\nWe make the same definitions, with\neven replacing odd.\n\nLet $A=\\{...A_{-3}, A_{1}, A_{5},...\\}$ and\n$B=\\{...B_{-5}, B_{-1}, B_3, ...\\}$, where\n(in homogeneous coordinates)\n\n\\begin{eqnarray}\n\\label{PA}\nA_{-3}=[0,1,0]; \\hskip 15 pt \n  A_{1}=[0,1,1]; \\hskip 15 pt \n  A_{5}=[1,1,1];  \\cr \\cr\nA_{4j+1}=[{O_1^{2j-1}},\n\\   {O_{-1}^{2j-1}+p_1 O_3^{2j-1}},\n\\   {O_{-1}^{2j-1}}]; \\hskip 15 pt j=2,4,6... \n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{PB}\nB_{-5}=[0,0,1]; \\hskip 15 pt \n  B_{-1}=[1,0,0]; \\hskip 15 pt \n  B_{3}=[0,1,-1]; \\hskip 15 pt \n  B_{7}=[1,-1,0]; \\cr \\cr\nB_{4j+3}=[{-E_2^{2j}+p_1q_2E_4^{2j}},\n\\   {E_0^{2j}},\n\\   {-E_0^{2j}+E_2^{2j}}]; \\hskip 15 pt j=2,4,6...\n\\end{eqnarray}\n\nIn \\S 5.2 we explicitly list out the first $7$ points of $A$.\nWe discovered these formulas as follows.  We normalized\nthe first few points of $A$ and then found the\nequations for successive points using the definitions\nof the invariants.  At some point we saw a pattern\nin the growing polynomials we were generating.\nThe algebraic proofs we give in this section are\nreally more like verifications.  We did everything\non the computer and simply converted our observations\ninto a proof.  \n\nThe basic tool for us is the following set\nof relations, which are easily derived.\n\n\\begin{eqnarray}\n\\label{relations}\nO_r^s=0 \\hskip 15 pt \\forall r>s; \\hskip 30 pt\n  O_{s-2}^s=O_s^s=1; \\cr\nE_r^s=0 \\hskip 15 pt \\forall r>s; \\hskip 30 pt\n  E_{s-2}^s=E_s^s=1; \\cr \\cr\nO_r^s=O_{r+2}^s\n-p_{r+2}O_{r+4}^s+\nP_{r+3} O_{r+6}^s; \\hskip 15pt r <s. \\cr \\cr\nE_r^s=E_{r+2}^s\n-q_{r+2}E_{r+4}^s+\nQ_{r+3}E_{r+6}^s. \\hskip 15 pt r<s. \\cr \\cr\nO_r^s=O_r^{s-2}-p_{s-2} O_r^{s-4}+P_{s-3} O_r^{s-6};\n\\hskip 15 pt r<s. \\cr \\cr\nE_r^s=E_r^{s-2}-q_{s-2} E_r^{s-4}+Q_{s-3} E_r^{s-6};\n\\hskip 15 pt r<s. \n\\end{eqnarray}\nHere we have set \\begin{equation} \\label{shorten}\nP_j=p_{j-1}q_jp_{j+1}; \\hskip 30 pt Q_j=q_{j-1}p_jq_{j+1}.\n\\end{equation}\n\n\nLet $\\cdot$ stand for the dot product, and let\n$\\times$ stand for the cross product.\n\n\\begin{lemma}\n\\label{id2}\nLet $k \\geq 2$ and $d \\geq 0$.  Then\n$$A_{4k+1} \\cdot B_{4k+3+4d}=\np_1q_2...q_{2k} E_{2k+2}^{2k+2d}.$$\n$$B_{4k+3} \\cdot A_{4k+5+4d}=\np_1q_2...q_{2k}p_{2k+1} O_{2k+3}^{2k+1+2d}.$$\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nWe will prove the first identity.  The second\none is very similar.\nWe use the notation \n$$(r,s)=O_{r}^{2k-1}E_s^{2k+2d}.$$\nThis notation should suggest to the reader that they plot the\nvarious {\\it points\\\/} $(r,s)$, given below, on a grid.\nThe result is a neat graphical representation of the\nalgebra we will be doing.\nUsing Equations \\ref{PA} and \\ref{PB} we have\n\\begin{equation}\n\\label{start}\nA_{4k+1} \\cdot B_{4k+3+4d}=\n(-1,2)-(1,2)+p_1(3,0)+p_1q_2(1,4). \\end{equation}\nThe basic relation\n$$O_{-1}^*-O_1^*=-p_1 O_3^*+p_1q_2p_3O_5^*,$$\nimplies that\n$$(-1,2) - (1,2) = -p_1(3,2)  + p_1q_2p_3(5,2).$$\nPlugging this into Equation \\ref{start} we have\n\\begin{equation}\n\\label{start1}\nA_{4k+1} \\cdot B_{4k+3+d}=p_1((3,0)-(3,2)+q_2(1,4)+q_2p_3(5,2).\n\\end{equation}\nThe basic relation\n$$E_0^*-E_2^*=-q_2 E_4^*+q_2p_3q_4E_6^*,$$\nimplies that\n$$(3,0)-(3,2)=-q_2(3,4)+q_2p_3q_4(5,4).$$\nPlugging this into Equation \\ref{start1} gives\n\\begin{equation} \\label{start2}\nA_{4k+1} \\cdot B_{4k+3+4d}=\np_1q_2((1,4)-(3,4)+p_3(5,2)+p_3q_4(3,6)).\n\\end{equation}\nnote that Equation \\ref{start2} has the same form\nas Equation \\ref{start} except that all the indices\nhave been shifted by $2$ and\na factor of $p_1q_2$ appears. \nThis process repeats until we reach:\n$$A_{4k+1} \\cdot B_{4k+3+4d}=\np_1q_2...p_{2k-3}q_{2k-2} X,$$\nwhere\n$$X=(2k-3,2k+4)-(2k-1,2k+4)+\np_{2k-1}(2k+3,2k)+p_{2k-1}q_{2k}(2k-1,2k+2)$$ \n$$=1-1+0+p_{2k-1}q_{2k}=p_{2k-1}q_{2k}.$$\nThe last two equations combine to give\nour identity.\n$\\spadesuit$  \\newline\n\n\n\\begin{lemma}\n\\label{id1}\nThe following identities hold for all\n$k \\geq 2$.\n\\begin{enumerate}\n\\item $A_{4k+1} \\times A_{4k+5}=\np_1...p_{2k-1} B_{4k+3}$. \n\\item $B_{4k+3} \\times B_{4k+7}=\nq_2...q_{2k} A_{4k+5}$. \n\\item $A_{4k+1} \\cdot B_{4k+7}=p_1q_2...p_{2k-1}q_{2k}.$ \n\\item $B_{4k+3} \\cdot A_{4k+9}=p_1q_2...q_{2k}p_{2k+1}.$ \n\\item $A_{4k+1} \\cdot B_{4k+11}=p_1q_2...p_{2k-1}q_{2k}.$\n\\item $B_{4k+3} \\cdot A_{4k+13}=p_1q_2...q_{2k}p_{2k+1}.$\n\\end{enumerate}\n\\end{lemma}\n\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nWe will derive these $6$ identities from the\ntwo identities of Lemma \\ref{id2}.\nTaking $d=1$ we obtain Identities 3 and 4 listed above.\nTaking $d=2$ we obtain Identities 5 and 6 listed above.\nTaking $d=0$, we see that \n$$A_{4k+1} \\cdot B_{4k+3}=\n  B_{4k+3} \\cdot A_{4k+5}=0.$$\nThus, we may write\n$$A_{4k+1} \\times A_{4k+5}=\\lambda_k B_{4k+3},$$ for\nsome $\\lambda_k$.\nAn easy calculation verifies that $\\lambda_2=p_1p_3.$\nSuppose, by induction, that $\\lambda_{k-1}=p_1...p_{2k-3}$.\nWe use the fact that\n$$(A_{4k+5} \\times A_{4k+1}) \\cdot A_{4k-3}=\n  (A_{4k+1} \\times A_{4k+3}) \\cdot A_{4k-5},$$\nand the already proven identities show that\n$\\lambda_{k}=p_1...p_{2k-1}$.  The case for the\n$B$'s is similar. $\\spadesuit$  \\newline\n\\newline\n\nIdentity 1 \nsays that $A_i$ is the intersection\npoint of the lines $B_{i-2}$ and $B_{i+2}$.\nIdentity 2 says that\n$B_{j}$ is the line determined by \n$A_{j-2}$ and $A_{j+2}$.   Hence $A$ and $B$ are\nassociates. \nIdentities 3 and 4\nsay that generically $A_i \\cdot B_{i-3} \\not =0$ and\n$B_j \\cdot A_{j-3} \\not =0$.\nThese statements also hold true\nin the few cases where some of the indices are negative,\nas may be verified by hand.  Thus, \n$A$ and $B$ are in general position for\ngeneric choices of variables.\n\nThe first few identities of Equation $(*)$ can be\nverified by hand.  We will show that\n$p_{2k+5}(P)=p_{2k+5}$ for $k \\geq 2$.  The case\nfor the $q$'s has a similar treatment.\nTo aid us in the computation, we recall some\nvector identities:\n\\begin{eqnarray}\nA \\times (B \\times C)=-(A \\cdot B)C+(A \\cdot C)B \\cr \\cr\n  (A \\times B) \\times (A \\times C)=((A\\times B) \\cdot C)A.\n\\end{eqnarray}\nIf $w_1,w_2,w_3,w_4$ are vectors lying in the same\n$2$-dimensional linear subspace of $\\F^3$ we define\n\\begin{equation}\n\\label{cr}\nX(w_1,w_2,w_3,w_4)=\\frac{(w_1 \\times w_2)*(w_3 \\times w_4)}\n        {(w_1 \\times w_3)*(w_2 \\times w_4)}.\n\\end{equation}\nThe operation $*$ means coordinate-wise multiplication.\n$X$ will be a vector of the form\n$(x,x,x)$.   The number $x$ is the classical\ncross ratio of the $4$ points in the projective\nplane $\\P$ represented\nby the vectors $w_1,w_2,w_3,w_4$.\n\nBy definition\n$$p_{2k+5}(P)= \nX(A_{4k+17},A_{4k+13},B_{4k+15} \\times B_{4k+7},\n\t\t    B_{4k+15} \\times B_{4k+3}),$$\n\nFirst:\n$$A_{4k+17} \\times A_{4k+13}=\n-p_1...p_{2k+5} B_{4k+15}.$$\n\nSecond:\n$$(B_{4k+15} \\times B_{4k+7}) \\times \n  (B_{4k+15} \\times B_{4k+3})=\n  ((B_{4k+15} \\times B_{4k+7}) \\cdot B_{4k+3}) B_{4k+15}=$$\n  $$((B_{4k+3} \\times B_{4k+7}) \\cdot B_{4k+15}) B_{4k+15}=\n    (q_2...q_{2k} A_{4k+5} \\cdot B_{4k+15}) B_{4k+15}=$$\n  $$(q_2...q_{2k})(p_1,q_2,...,p_{2k+1}q_{2k+2}) B_{4k+15}.$$ \n\nThird:\n$$A_{4k+17} \\times (B_{4k+15} \\times B_{4k+7})=$$\n$$(A_{4k+17} \\times B_{4k+7}) B_{4k+15}\n -(A_{4k+17} \\times B_{4k+15}) B_{4k+7}=$$\n$$-(p_1q_2...p_{2k+2}q_{2k+3}) B_{4k+15}.$$\n\nFourth:\n$$A_{4k+13} \\times (B_{4k+15} \\times B_{4k+3})=$$\n$$-(A_{4k+13} \\times B_{4k+15}) B_{4k+3}+\n  (A_{4k+13} \\times B_{4k+7}) B_{4k+15}=$$\n$$(p_1q_2...q_{2k}p_{2k+1}) B_{4k+15}.$$\n\nNotice that all the terms are multiples of\nthe same vector.   \nUsing the formula for $X$, we have:\n$$x=\\frac{(p_1,p_{2k+5})(q_2...q_{2k})(p_1q_2,..p_{2k+1}q_{2k+2})}\n\t {q_{2k+2} p_{2k+3}(p_1q_2,...,p_{2k+1})^2}=p_{2k+5}.$$\n\n\n\n\n\\subsection{The Final Calculation}\n\\label{monodromy}\n\nRecall that $T$ is the monodromy of $A$.\nWe will first compute a lift\nof $T$ to $GL_3(\\F)$.\nWe may interpret an element in $GL_3(\\F)$ as \na triple $\\widetilde T=(V_1,V_2,V_3)$ of vectors in $\\F^3$.\nThe linear action of $T$ is then given as follows:\nIf $W=[w_1,w_2,w_3]$, then  \\begin{equation}\n\\label{mon1}\n\\widetilde T(W)=w_1 V_1+w_2 V_2+ w_3 V_3.\n\\end{equation}\n\nConsider the element\n$\\widetilde T=(V_1,V_2,V_3),$\nwhere \\begin{eqnarray}\n\\label{mon2}\nV_1=p_1 A_{4n+5}-p_{2n+1} A_{4n+1}; \\cr \\cr\nV_2=p_{2n-1}q_{2n}p_{2n+1} A_{4n-3}; \\cr \\cr\nV_3= p_{2n+1}A_{4n+1}-\n       p_{2n-1}q_{2n}p_{2n+1} A_{4n-3}. \n\\end{eqnarray}\n\n\\begin{lemma}\n$\\widetilde T$ is a lift of $T$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nUsing Equations \\ref{mon1}, \\ref{mon2} and \\ref{PA} we compute\n$$\\widetilde T(A_{-3})=\\widetilde T[0,1,0]=\n                     p_{2n-1}q_{2n}p_{2n+1} A_{4n-3}.$$\n$$\\widetilde T(A_{1})=\\widetilde T[0,1,1]=\n                     p_{2n+1} A_{4n+1}.$$\n$$\\widetilde T(A_{5})=\\widetilde T[1,1,1]=\n                     p_1 A_{4n+5}.$$\n$$\\widetilde T(A_9)=\\widetilde T[1,1,1-p_1]=$$\n$$\\hskip 15 pt \np_1(A_{4n+5}-p_{2n+1} A_{4n+1}+p_{2n-1}q_{2n}p_{2n-1} A_{4n-3})=\np_1 A_{4n+9}.$$\nThis last equality follows Equation \\ref{relations},\napplied componentwise to our vectors.\nOur four computations show that the projectivization\nof $\\widetilde T$ has the same action on\nthe points $A_{-3}, A_{1}, A_{5}, A_9$\nas $T$ does.  These $4$ points are (for generic\nchoice of variables) in\ngeneral position, and projective transformations\nare determined by their action on $4$ general\nposition points. \nHence the projective action of $\\widetilde T$ coincides with\nthe action of $T$.\n$\\spadesuit$  \\newline\n\nNow we compute $\\Omega_1$.  We\nset \n$$\\widetilde O=\\sum_{i=0}^{[n\/2]} O_k.$$\nBefore we make the next calculation we note\nthat $p_{2n+1}=p_1$, under the assumption that\nthe invariants are $2n$-periodic.   Now for\nthe calculation:\n\\begin{eqnarray}\n\\label{trace}\n{\\rm tr\\\/}(T)=V_{11}+V_{22}+V_{33}=  \\cr \\cr\n(p_1 O_1^{2n+1}-p_{2n+1}O_1^{2n-1})+  \\cr \\cr\n(p_{2n-1}q_{2n}p_{2n+1}O_{-1}^{2n-3}+  \n  p_1p_{2n-1}q_{2n}p_{2n+1}O_{3}^{2n-3})+ \\cr \\cr\n(p_{2n+1}O_{-1}^{2n-1}- \np_{2n-1}q_{2n}p_{2n+1}O_{-1}^{2n-3})= \\cr \\cr\np_1\\big([O_1^{2n+1}]+[O_{-1}^{2n-1}-O_1^{2n-1}]\n      +[p_{2n-1}q_{2n}p_1 O_3^{2n-1}\\big]) \n=p_1\\widetilde O.\n\\end{eqnarray}\n\nIn the last line, we have bracketed terms\nso as to isolate the different kinds of\nterms in $\\widetilde O$.   The first expression\ndescribes the terms of $\\widetilde O$ which involve $p_1$,\nbut not $p_{2n-1}q_{2n}p_1$.  The second expression\ndescribes the terms of $\\widetilde O$ which do not involve\n$p_1$.  The third expression describes terms\nof $\\widetilde O$ which involve\n$p_{2n-1}q_{2n}p_1.$\n\nAgain using the fact that $p_{2n+1}=p_1$ we have\n\\begin{eqnarray}\n\\label{dett}\n\\det(T)=(V_1 \\times V_2) \\cdot V_3= \\cr \\cr\n(p_1)(p_{2n-1})(p_{2n-1}q_{2n}p_{2n+1})\n((A_{4n-5} \\times A_{4n-1}) \\cdot A_{4n+3})= \\cr \\cr\np_1^3(p_3...p_{2n+1})^2(q_2...q_{2n})= \np_1^3(p_1p_3...p_{2n-1})^2(q_2...q_{2n}).\n\\end{eqnarray}\nCombining Equations \\ref{trace} and\n\\ref{dett} we get the formula for\n$\\Omega_1$.  \n\n\\newpage\n\n\\section{The Pentagram}\n\\label{pentagram}\n\n\\subsection{Basic Definition}\n\nWe fix $n$, as in previous chapters. \nLet $P_1$ (respectively $P_2$) be the space of\nPolyPoints which have $2n$ periodic\ninvariant coordinates, and whose points\nare labelled by integers congruent to\n$1$ (respectively $3$) mod $4$.  Likewise we define\n$L_1$ and $L_2$ for PolyLines.\n\n\nDefine\n\\begin{equation}\nX=P_1 \\cup P_2 \\cup L_1 \\cup L_2.\n\\end{equation}\nSuppose that $A=\\{A_j\\}$ is a PolyPoint.   We define\n$\\delta_1(A)=\\{B_j\\}$ where\n\\begin{equation}\nB_j=(A_{j-2}A_{j+2}).\n\\end{equation}\n$\\delta_1(A)$ is just the associate of $A$.\nWe make the same definition for PolyLines.\n$\\delta_1$ is an involution of $X$ which\ninterchanges spaces $P_{1}$ and $L_{3}$ and \ninterchanges \n$P_{3}$ and $L_{1}$.\n\n  We define\n$\\delta_1(A)=\\{B'_j\\}$ where\n\\begin{equation}\nB_j=(A_{j-4}A_{j+4}).\n\\end{equation}\n$\\delta_2$ is an involution of $X$ which\ninterchanges spaces $P_{1}$ and $L_{1}$ and \ninterchanges \n$P_{3}$ and $L_{3}$.\nFigure 5.1 shows the action of $\\delta_2$ on a\nPolyPoint in $P_1$.\n\n\\begin{center}\n\\psfig{file=Pix\/pix5.ps}\nFigure 5.1\n\\end{center}\n\nWe define\n\\begin{equation}\n\\alpha_1=\\delta_1 \\circ \\delta_2 \\circ \\delta_1; \\hskip 30 pt\n\\alpha_2=\\delta_2.\n\\end{equation}\n\nWe call the pair $(\\alpha_1,\\alpha_2)$ the {\\it pentagram map\\\/}.\nBoth $\\alpha_1$ and $\\alpha_2$ are involutions.  Moreover,\nconjugation by $\\delta_1$ interchanges these two maps.\n\n\\subsection{The Pentagram Map in Coordinates}\n\\label{coord}\n\nThere is a ``forgetful map'' which takes the variables\n$p_j$ and $q_j$ and calls them $x_j$, regardless of\nthe letter.  There is also a map which takes a\npolygon and sends it to its invariant coordinates.\nComposing this coordinate map with the forgetful map\nwe get generically defined bijections from any\nof our $4$ spaces to $\\F^{2n}$.   In this way we\ncan work out the pentagram map in the same coordinates\nused in Equation \\ref{basic}.\n\nWe can exploit symmetry to reduce the amount\nof computing we have to do.  First, it is\na consequence of Equation \\ref{dualinv} and\nthe invariants of our coordinates under\nprojective duality that\n\\begin{enumerate}\n\\item The action of $\\alpha_j$ on $P_j$ is the same\nas the action of $\\alpha_j$ on $L_j$, and this\naction does not depend on $j$.\n\\item The action of $\\alpha_j$ on $P_{3-j}$ is the same\nas the action of $\\alpha_j$ on $L_{3-j}$, and this\naction does not depend on $j$.\n\\end{enumerate}\nWe will show that Equation \\ref{basic} describes\nthe action of $\\alpha_1$ and $\\alpha_2$ on \n$P_1 \\cup L_1$.  If we computed the action on\n$P_2 \\cup L_2$ we would have to interchange\n$\\alpha_1$ with $\\alpha_2$ to get the\nright formulas.  We will compute the action\nof $\\alpha_2$ on $P_1$.  The derivation for\n$\\alpha_1$ is similar and follows from symmetry.\nTo see that the formula in Equation \\ref{basic} is correct,\nwe just have to work out a single invariant of\n$\\alpha_2(P)$.  The formulas for the other\ninvariants are forced by the dihedral\nsymmetry of our constructions.\n\nWe will not do this calculation by hand.  However, we will\nset it up exactly, so that the inclined reader can type\neverything into a symbolic manipulator and push the\nbutton, as we did.  Let $A$ be as in\n\\S 4.  We will compute the invariant $q'_4=q_4(B')$, which\nis the variable $x$ in the expression $X=(x,x,x)$, where\n(as in Equation \\ref{cr})\n\\begin{equation}\nX=\nX(B'_1, B'_5, A'_3 \\times A'_{11},A'_3 \\times A'_{15}).\n\\end{equation}\nHere $A'$ is the associate of $B'$, so that\n\\begin{equation}\nA'_3=B'_1  \\times B'_5; \\hskip 15 pt\nA'_{11}=B'_{9} \\times B'_{13}; \\hskip 15 pt\nA'_{15}=B'_{13} \\times B'_{17}.\n\\end{equation}\nFinally, we have\n\\begin{equation}\nB'_1=A_{-3} \\times A_5; \\hskip 15 pt\nB'_5=A_1 \\times A_9; \\hskip 15 pt ... \\hskip 15 pt\nB'_{17}=A_{13} \\times A_{21}.\n\\end{equation}\n\nTo make our computation we need to know the \n$7$ points $A_{-3},...,A_{21}$. Once we have\nthese $7$ points, we just take a bunch of\ncross products.\n\nUsing the convention of Equation \\ref{shorten} we\nlist these points explicitly.\n$$A_{-3}=[0,1,0]; \\hskip 15 pt\nA_{1}=[0,1,1]; \\hskip 15 pt\nA_{5}=[1,1,1]; \\hskip 15 pt\nA_9=[1,1,1-p_1];$$\n\n$$A_{13}=[1-p_3,1-p_3+Q_2,1-p_1-p_3+Q_2];$$\n$$A_{17}=[1-p_3-p_5+Q_4,1-p_3-p_5+Q_2+Q_4,\n1-p_1-p_3-p_5+p_1p_5+Q_2+Q_4];$$\n\n$$\nA_{21}=[1-p_3-p_5-p_7+p_3p_7+Q_4+Q_6, $$\n$$1-p_3-p_7-p_9+Q_2+Q_4+Q_6+p_3p_7 -p_7Q_2,$$\n$$1-p_1-p_3-p_7+Q_2+Q_4+Q_6+p_1p_5+p_1p_7+p_3p_7\n-p_1Q_6 - p_7 Q_2]$$\nWe double checked these formulas using\nEquation \\ref{relations}.  \n\nWhen we plug\neverything into Mathematica and compute,\nwe get\n$$q'_4=p_5 \\frac{1-q_6p_7}{1-q_2p_3}.$$\nForgetting about the lettering, we have\n$$x'_4=x_5 \\frac{1-x_6x_7}{1-x_2x_3}.$$\nOne can see that this exactly matches the\nequation for $\\alpha_2$ given in\nEquation \\ref{basic}, for $j=2$.\nThe general case follows from dihedral symmetry.\n\n\\subsection{Second Proof of Theorem \\ref{precise}}\n\nSuppose $A$ is a PolyPoint,\nwith invariants $p_1,q_2,...,q_{2n}$, as above.\nLet $T$ be the projective transformation \nsuch that $T(A_{j+2n})=A_j$.  Let\n$\\Omega_1$ and $\\Omega_2$ be the two\nmonodromy invariants of $A$.\n\nWe will just consider $\\alpha_1$.  The proof\nfor $\\alpha_2$ is the same.\nLet $B'=\\alpha_1(A)$.  Our constructions\ncommute with projective transformations.  Hence\n$B'$ is also invariant under $T$.\nThis is to say that the dual of $B'$, which is\nanother PolyPoint, is\ninvariant under $T^*$, the dual of $T$.   Hence\n$\\alpha_1(\\Omega_j)=\\Omega_{3-j}$.\nIt now follows from\nEquation \\ref{main} that\n\\begin{equation}\n \\alpha_1 \\bigg(\\frac{(\\sum_{k=0}^{[n\/2]} O_k)^3}{O_n^2E_n} \\bigg)\n=\\frac{(\\sum_{k=0}^{[n\/2]} E_k)^3}{E_n^2O_n}.\n\\end{equation}\nOne can see easily from Equation \\ref{basic} that\n$\\alpha_1(O_n)=E_n$ and\n$\\alpha_1(E_n)=O_n$.  Therefore\n\\begin{equation}\n \\alpha_1\\bigg(\\sum_{k=0}^{[n\/2]} O_k\\bigg)=\n\\sum_{k=0}^{[n\/2]} E_k.\n\\end{equation}\n\nNow for the moment of truth.\nLet $S_t: \\F^{2n} \\to \\F^{2n}$ be as in\nEquation \\ref{hom}.  \nLooking at\nEquation \\ref{basic} we see that\n\\begin{equation}\n\\alpha_1 \\circ S_t=S_{t^{-1}} \\circ \\alpha_1.\n\\end{equation}\nAt the same time, we have\n\\begin{equation}\nO_k \\circ S_t=t^{-k} O_k;  \\hskip 20 pt\nE_k \\circ S_t=t^{k} E_k.\n\\end{equation}\nTherefore\n\\begin{equation}\n \\alpha_1\\bigg(\\sum_{k=0}^{[n\/2]} t^k O_k\\bigg)=\n\\sum_{k=0}^{[n\/2]} t^k E_k.\n\\end{equation}\nSince this last equation is true for all $t$, we must\nhave $\\alpha_1(O_k)=E_k$ for all $k$.\nSince $\\alpha_1$ is an involution,\n$\\alpha_1(E_k)=O_k$ for all $k$.  \nThis completes our proof.\n\n\\subsection{Conic Sections}\n\nIn this section we establish a technical result\nused in later sections.  Namely, if $A$\nis inscribed in a conic then $E_n(A)=O_n(A)$.\nWe continue using the notation established\nabove, and take $A \\in P_1$.\n\n\\begin{lemma}\n\\label{conic1}\n Suppose $A$ is inscribed in a conic.\nThen \n$$(1-q_{2k})(1-q_{2k-2}p_{2k-1})=\n(1-p_{2k-1})(1-q_{2k}p_{2k+1});$$ $$\n(1-p_{2k-1})q_{2k}(1-p_{2k+1})=\n(1-q_{2k-2})p_{2k-1}(1-q_{2k})$$\nfor $k=1,...,n$.  Here indices are\ntaken mod $2n$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nWe will derive the first identity.  The second\none is a fairly straightforward rearrangement\nof the first.   Figure 5.2 shows an application\nof Pascal's theorem.  If the $6$ points\n$A_{-3},A_1,A_5,A_9,A_{13},A_{17}$ lie on a\nconic section then the points\n$C_1,C_2,C_3$ lie on a line.\n\n\n\\begin{center}\n\\psfig{file=Pix\/pix52.ps}\nFigure 5.2\n\\end{center}\n\nIf we express the $C_j$ in homogeneous coordinates,\nwe have \\begin{eqnarray}\nC_1=(A_{-3} \\times A_{13}) \\times (A_1 \\times A_9) \\cr \\cr\nC_2=(A_{-3} \\times A_{17}) \\times (A_5 \\times A_9) \\cr \\cr\nC_3=(A_{5} \\times A_{13}) \\times (A_1 \\times A_{17})\n\\end{eqnarray}\n\nThe condition that the $C$s lie on the same line is\ngiven by\n\\begin{equation}\n\\label{det}\n\\det(C_1,C_2,C_3)=0.\n\\end{equation}\nIn other words, we arrange these vectors into\na $3 \\times 3$ matrix and set the determinant\nequal to $0$.  \n\nWhen we do this calculation on Mathematica, we see that\nEquation \\ref{det} holds if and only if\n$(1-q_4)(1-q_2p_3)=(1-p_3)(1-q_4p_5).$  If all points\nof $A$ lie on the same conic, then by symmetry the\nsame relation holds with the indices shifted\nby $2,4,6...$\nThis completes our proof.\n$\\spadesuit$  \\newline\n\n\\begin{corollary}\n\\label{later}\nIf $A$ is inscribed in a conic then\n$O_n(A)=E_n(A)$.\n\\end{corollary}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nTaking the product of the first equation over all\n$k$, and cancelling terms, we see that\n$\\prod (1-p_{2k-1})=\\prod (1-q_{2j})$.\nTaking the product of the second equation,\nover all $k$, we get\n$O_n (\\prod (1-q_{2k})^2=\nE_n (\\prod (1-p_{2k-1})^2$.\nUsing the first equation to cancel terms,\nwe are left with $O_n=E_n$.\n$\\spadesuit$  \\newline\n\n\n\n\\newpage\n\n\\section{The Method of Condensation}\n\n\n\\subsection{Octahedral Tilings}\n\nThere is a tiling $T$ of $\\R^3$ by octahedra, which\ncan be described as follows.  The vertices of\n$T$ have the form $(a,b,c) \\in \\Z^3$, subject\nto the constraints that the three coordinates\nare either all even or all odd.\nTwo vertices are joined by an edge if their\ndistance is exactly $\\sqrt 3$.  One of the\noctahedra in $T$ has the $6$ vertices\n$(0,0,0)$,\n$(\\pm 1, \\pm 1,1)$ and $(2,0,0)$.\nAll the other octahedra are translates of this one.\nIn our model octahedron, we call $(2,0,0)$ the\n{\\it top\\\/} and call $(0,0,0)$ the {\\it bottom\\\/}.\nWe call $(-1,1,1)$ the {\\it northwest vertex\\\/},\n$(1,1,1)$ the {\\it northeast vertex\\\/} and so forth.\nWe extend this definition to all octahedra\nusing translations.\n\nSuppose that $M$ is an $m \\times m$ matrix.\nDodgson's method of condensation involves the\nconnected square minors of $M$.  \nSuppose that\n\\begin{equation}\nH=\\{M_{ij}|\\ a_1 \\leq i \\leq a_2;\\ b_1 \\leq j \\leq b_2\\}\n\\end{equation}\nis such a minor.  We define\n$f(H)=(a,b,c)$ where\n\\begin{equation}\na=a_1+a_2; \\hskip 20 pt b=b_1+b_2; \\hskip 20 pt\nc=a_1-a_2=b_1-b_2.\n\\end{equation}\nFor instance, if $H$ is the singleton $M_{11}$ then\n$f(H)=(2,2,0)$.   If $H=M$ then\n$f(H)=(m+1,m+1,m-1)$.  We let $[M]$ denote the\nset of vertices of the tiling which have the\nform $f(H)$.  Note that $[M]$ looks like\na pyramid with square base.\n\nWe can label the point $f(H) \\in [M]$ by\n$\\det(H)$.   Dodgson's identity gives a\nsingle relation for each octahedron:\n\\begin{equation}\nV_t V_b= V_{nw} V_{se} - V_{sw} V_{ne}.\n\\end{equation}\nHere $V_t$ is the label of the top vertex,\n$V_b$ is the label of the bottom vertex, and\nso forth.\nTo compute $\\det(M)$, begins at the base of the\npyramid and computes successive layers using\nthe octahedron rule.  At the end, one arrives\nat the apex of the pyramid, with the final answer.\n\nWhat we do next works most gracefully when\n$\\det(M) \\not =0$.  We say that a labelling\nof a horizontal layer of $T$ is {\\it constant\\\/}\nif every vertex gets the same label.  \nWe say that a {\\it sandwich\ncondensation\\\/} is a labelling of all the layers\nof $T$ between two constant horizontal layers.\nHere is an example:\nWe label the layer $\\{z=0\\}$ by $1$'s.\nNext, we label the layer\n$\\{z=1\\}$ so as to be doubly periodic, with period\n$m$ in each direction.  In one $m \\times m$ block, we put the\nentries of $M$.\nThe second layer will be labelled\nby determinants of $2 \\times 2$ connected minors\nof $M$ and its cyclic permutations.\nBy Dodgson's identity, the third layer\nwill be labelled by the determinants of\nthe $3 \\times 3$ connected\nminors of $M$ and its cyclic permutations.\nAnd so forth.  The $m$th layer will be\nlabelled by the determinants of $M$ and\nits cyclic permutations.  But all\nthese determinants have the same value.  This\nis our sandwich.  It has {\\it width\\\/} $m$.\n\nTo consider a more generic situation,\nwe say that a {\\it condensation\\\/}\nis a labelling of the vertices of the\ntiling $T$, such that the labelling satisfies\nthe octahedron rule at every octahedron.\nThe labellings on two successive horizontal layers\ndetermine the condensation.  If these labels\nare chosen generically, in $\\C$ for example,\nthen we will not\nencounter singularities when trying to\npropagate the condensation to the other\nlayers, both above and below the two\ninitial ones.\n\n\\subsection{Picture of the Pentagram Map}\n\n\\begin{center}\n\\psfig{file=Pix\/pix1.ps}\nFigure 5.1\n\\end{center}\n\nFigure 5.1 shows a tiling of the\nplane by isosceles triangles.  Suppose one labels\nthe edges of the tiling with elements of a field,\nin a way which is invariant under a horizontal\ntranslation, subject to the compatibility rules\nindicated in the figure.    \n(These compatibility rules are supposed\nto hold for all configurations isometric to the\nones highlighted.)  We call such a labelling \na {\\it pentagram labelling\\\/}.\n\nGiven a pentagram labelling, the maps\n$\\alpha_1$ and $\\alpha_2$,\nthe two involutions from Equation \\ref{basic},\nexpress how one one deduces the labellings on a given\nrow from the labellings on the rows above or\nbelow it.  Thus, an orbit of the pentagram map\nis encoded by a pentagram labelling.\n\n\\subsection{Circulent Condensations} \n\nConsider the linear projection\n$\\pi: \\R^3 \\to \\R^2$ given by\n\\begin{equation} \\label{proj}\n\\pi(x,y,z)=(2x - y,z).\n\\end{equation}\nWe say that a condensation is {\\it circulent\\\/}\nto $\\pi$ if every vertex in a fiber\n$\\pi^{-1}(p)$ gets the same label.\nReferring again to Dodgson's method of\ncondensation, the circulent labellings\ncorrespond to certain circulent matrices.\n\n\\begin{center}\n\\psfig{file=Pix\/pix6.ps}\nFigure 5.2\n\\end{center}\n\nWe can use $\\pi$ to push a circulent condensation\ninto the plane, without losing any information.\n$\\pi$ maps the vertices of $T$ to the vertices\nof the tiling $\\tau$ shown in Figure 5.1.\nFigure 5.2 shows the image of a single octahedron under $\\pi$,\nsuperimposed onto $\\tau$.\nFigure 5.2 also shows the local rule satisfied\nby the image of an circulent condensation.\n\nWe can think of an circulent condensation as a\nlabelling of the vertices of $\\tau$ which\nsatisfies the local condition shown in Figure 5.2.\nFigure 5.3 shows how to convert from an\ncirculent condensation to a pentagram labelling.\n\n\\begin{center}\n\\psfig{file=Pix\/pix61.ps}\nFigure 5.3\n\\end{center}\n\nTo verify the first of the two compatibility\nrules for pentagram labellings we refer to\nFigure 5.4 and compute\n$$ab-c=\\frac{CD}{AF} \\frac{AG}{CE} - \\frac{BH}{EF}=\n\\frac{DG-BH}{EF}=1.$$\nThe last equality is the compatibility rule for the\ncirculent condensation.\n\n\\begin{center}\n\\psfig{file=Pix\/pix62.ps}\nFigure 5.4\n\\end{center}\n\nTo verify the second compatibility rule we refer to\nFigure 5.5 and compute\n$$wx=\\frac{AI}{EF} \\frac{EK}{GI}=\\frac{AK}{FG}=\n\\frac{AH}{CF} \\frac{CK}{GH}=yz.$$\n\n\\begin{center}\n\\psfig{file=Pix\/pix63.ps}\nFigure 5.5\n\\end{center}\n\n\\subsection{The Lifting Problem}\n\nAs usual, suppose that $n$ is fixed, as well as\na suitable base field.\nLet $C$ denote the space of circulent condensations\nwhich are periodic under a horizontal translation\nwhich shifts the labels by $n$.  Let $P$ denote the\nspace of pentagram labels which are periodic under\nthe same translation.  The translation given in\nthe previous section gives a map $\\psi: C \\to P$.\nIn this section we ask about the extent to which\n$\\psi$ is invertible.\n\nAn element of $C$ is determined by the values it\nattains on two successive horizontal rows.  \nAn element of $P$ is determined by the values\nit takes on the non-horizontal edges of a single\nrow.  In both cases one puts down $2n$ values\nbefore the period repeats.   That is, both\n$C$ and $P$ are $2n$ dimensional.  The map\n$\\psi$ is a quotient map and certainly not\n$1$-to-$1$.  For instance, all the scalar\nmultiples of a single element of $C$ get\nmapped to the same element of $P$.\nSince $C$ and $P$ have the same dimension and\n$\\psi$ is far from injective, $\\psi$ is\nalso far from surjective.\n\nTo give a simple and clean answer we will assume\nthat $n=4m$ for some $m \\in \\N$.  The other cases\nhave a somewhat messier analysis.  Given an\nelement $X=(x_1,...,x_{2n}) \\in P$ we have the\npentagram invariants defined in \\S 2.1.\n\n\\begin{theorem}\n\\label{lift}\nIf $n=4m$ than an element\n$X \\in P$ is contained in the image of\n$\\psi$ if and only if\n$O_{2m}(X)=E_{2m}(X)=2$ and $O_{n}(X)=E_{n}(X)=1$.\n\\end{theorem}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nFor $j=1,2,3,4$ we define\n\\begin{equation}\n\\label{mod4}\nf_j=\\prod_{i \\equiv j {\\rm mod \\\/} 4} x_i.\n\\end{equation}\nIt is not hard to see that\n$O_{2m}=f_1+f_3$.  Also $O_{n}=f_1f_3$.\nBy hypotheses, we have $f_1+f_3=2$ and\n$f_1f_3=1$.  This forces $f_1=f_3=1$.\nLikewise $f_2=f_4=1$.\nConversely, if $f_j=1$ for $j=1,2,3,4$ then\nwe have the hypothesis of this theorem.\nAll in all, the hypothesis of this theorem\nare equivalent to the hypothesis that\n$f_j=1$ for $j=1,2,3,4$.  We will work\nwith this latter hypothesis.\n\nFirst we prove necessity.  Suppose\n$X$ is in the image of $\\psi$.\nWe focus on a single row of our triangulation.\nLet $c_k$ be the label of the vertex which\nis incident to the edges labelled \n$x_{k-1\/2}$ and $x_{k+1\/2}$.   Here $k$ is a half-integer.\nWe have\n\\begin{equation}\nx_1=\\frac{c_{-3\/2} c_{7\/2}}{c_{-1\/2}c_{5\/2}}; \\hskip 15 pt\nx_5=\\frac{c_{5\/2} c_{15\/2}}{c_{7\/2}c_{13\/2}}; \\hskip 15 pt\nx_9=\\frac{c_{13\/2} c_{23\/2}}{c_{15\/2}c_{21\/2}} \\hskip 15 pt ...\n\\end{equation}\nWhen we compute $f_1$, each $c$-term appears exactly\nonce in the numerator and exactly once in the\ndenominator.\nHence $f_1=1$.  A similar argument shows that\n$f_2=f_3=f_4=1$.\n\nTo prove sufficiency, suppose that\n$f_1=f_2=f_3=f_4=1$. We claim that there exist\nlabels $r_1,r_2,...,r_{2n}$ such that\n\\begin{equation}\nx_j=\\frac{r_{j+2}}{r_{j-2}}.\n\\end{equation}\nTo see this, we pick $r_1$ arbitrarily.\nThen set, successively\n\\begin{equation}\nr_5=x_3r_1; \\hskip 15 pt\nr_9=x_7x_3r_1; \\hskip 15 pt\nr_{11}=x_{11}x_7x_3 r_1 \\hskip 15 pt...\n\\end{equation}\nSince $n \\equiv 0$ mod $4$ when we cycle through\none period we return back to the value\n$r_1 f_3=r_1$. \nThus, we can make a completely consistent\nchoice of $r$ for all indices congruent to\n$1$ mod $4$.  The same argument works for the\nother congruences.\n\nGiven the $r$'s, it suffices to find $c$'s such that\n\\begin{equation}\nr_j=\\frac{c_{j-1\/2}}{c_{j+1\/2}}.\n\\end{equation}\nIn our construction of the $r$'s we had a free\nchoice for each congruence mod $4$.\nThus, (in the generic case)\nwe can choose the $r$'s so that\n\\begin{equation}\n\\rho_j=\\prod_{i \\equiv j {\\rm mod\\\/} 4} r_i=1\n\\end{equation}\nfor $j=1,2,3,4$.\nFinding the $c$'s in terms of the $r$'s is exactly\nthe same problem as finding the $r$'s in terms of\nthe $x$'s.  The $\\rho$'s play the same role\nas the $f$'s.  Thus, we can find our $c$'s, which\ngives a preimage of $X$ in $C$.\n$\\spadesuit$  \\newline\n\n\\subsection{Degenerate polygons}\n\nWe fix $n=4m$.   To avoid trivial cases, we assume\nthat $m$ is large$-$say, $m \\geq 3$.  We work\nover $\\C$.  Everything we do is understood to\nbe defined for the generic example, but\nperhaps not for every example.  \nThe reader should insert this caveat before\nevery assertion.\nWe will work with PolyPoints\nand PolyPoints which are periodic mod $n$.\nBy this we mean that the points or lines\nthemselves are periodic, not just the\nprojective invariants.  We will call such\nobjects {\\it periodic PolyPoints\\\/} or\n{\\it periodic PolyLines\\\/}.  We take these\nto be elements of the spaces $P_1$ and $L_1$\nintroduced in \\S 5.1.\n\nWe say that the periodic PolyPoint $A=\\{...A_1,A_5,...\\}$ is\n{\\it degenerate\\\/} if $...A_1,A_9,A_{13}...$ lie\non the same line and if $...A_3,A_{11},A_{15},...$ all\nlie on the same line.  We make the dual definition\nfor PolyLines.  A polygon (in the ordinary sense)\nsatisfying the hypothesis of Theorem \\ref{hyper}\nnaturally determines a degenerate PolyLine.\nOne simply considers the lines considered by the\nedges of the polygon.\n\n\\begin{lemma}\n\\label{recip}\nA periodic element in $P_1$ is degenerate iff\nits projective invariants satisfy $q_jp_{j+1}=1$\nfor all $j$.  Dually, a periodic element in $L_1$\nis degenerate if and only if its projective\ninvariants satisfy $p_jq_{j+1}=1$ for all $j$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nBy duality it suffices to prove this result for\nPolyPoints.  Referring to the PolyPoint constructed\nin \\S 4, we compute\n$$\\det(A_{-1},A_5,A_{13})=p_1(1-q_2p_3).$$\nThe three points in question lie on the same\nline iff $q_2p_3=1$.  By symmetry, similar\nstatements hold for other triples of points.\n$\\spadesuit$  \\newline\n\n\n\\begin{lemma}\n\\label{closed}\nIf $A$ is a periodic $2n$-gon then\n$$(\\sum_{k=0}^{[n\/2]} O_k)^3=27 O_n^2E_n; \\hskip 15 pt\n(\\sum_{k=0}^{[n\/2]} E_k)^3=27 E_n^2O_n.$$\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nThis is a corollary of Equation \\ref{main}. If\n$A$ is $2n$-periodic then its monodromy\n$T$ is the identity.  Likewise,\n$T^*$ is the identity matrix.\nHence $\\Omega_1=\\Omega_2=27$.\n$\\spadesuit$  \\newline\n\nFor Theorem \\ref{hyper} we only\nneed the second statement of the next result.\nThe spaces $P_1$ and $L_1$ are defined in\n\\S 5.1.\n\n\\begin{lemma}\n\\label{van}\nA degenerate element in either $P_1$ or $L_1$\nhas pentagram invariants \\begin{enumerate}\n\\item $O_k=E_k=0$ for $k<2m$ and \n\\item $O_{2m}=E_{2m}=2$ and\n$O_n=E_n=1$.\n\\end{enumerate}\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nFirst suppose that $k<2m=n\/2$.  Using\nthe reasoning in the proof of Theorem \\ref{trans} we\ncan group\n$O_k$ into pieces having the form\n$\\prod \\mu$, where each $\\mu$ is a monomial\ntimes  $(1-q_jp_{j+1})$ for some $j$.\n(Compare Lemma \\ref{seq2}, but interchange\n$p$ with $q$.)\nBy Lemma \\ref{recip} each $\\mu$ is zero\nand hence $O_k=0$.  The same argument\nworks for $E_k$.\n\nFor the second item note that\n$O_nE_n=(p_2q_3)(p_4q_5)...=(1)(1)...=1.$\nElements in $P_1$ or $L_1$ are degenerate\nlimits of polygons which are either inscribed\nin, or superscribed about, a conic section.\nThus, Lemma \\ref{later} applies to these elements,\nby continuity.\nIt now follows from Lemma \\ref{later}\nthat $E_n^2=O_n^2=1$.   If $O_n=-1$, for any particular\nexample, then by continuity\n$O_n$ is identically $-1$.  However, we\ncan choose a $4n$-point, contained in $\\R^2$, which is essentially\nthe double cover of a $2n$-point.  The\ninvariants of such a $4n$-point would\nrepeat with period $2n$, forcing $O_n$ to be\na square of a real number.  This rules\nout the possibility that $O_n=-1$.  Hence\n$O_n=1$ for every example.  Likewise\n$E_n=1$ for every example.\nIf now follows from Corollary \\ref{closed},\nand from the vanishing of all the other\npentagram invariants, that\n$1+O_{2m}=3O_n=3$.  Hence $O_{2m}=2$.  Likewise\n$E_{2m}=2$.\n$\\spadesuit$  \\newline\n\n\\subsection{Proof of Theorem \\ref{hyper}}\n\nWe start with a degenerate PolyLine.  This\nPolyLine determines a labelling of a single\nrow of \nthe triangulation $\\tau$, considered in \\S 6.\nCombining Theorem\n\\ref{lift} and Lemma \\ref{van} we see that\nwe can find a circulent condensation $C$\nwhich translates into our pentagram labelling.\n\n\\begin{center}\n\\psfig{file=Pix\/pix65.ps}\nFigure 5.5\n\\end{center}\n\n\nReferring to the proof of Theorem \\ref{lift} we\ncan use the fact that $p_jq_{j+1}=1$ to arrange\nso that $r_jr_{j+1}=1$ for all $j$.  This allows\nto pick our $c$ labelling so that the row below the\nedge labels is identically $1$, as shown in\nFigure 5.5.  Figure 5.5 is supposed to be periodic\nin the horizontal direction.\n\nWe pull our $c$ labelling back to give a labelling of\n$T$, the tiling of $\\R^3$ by  octahedra.\nThe bottom row of dots pulls back to an infinite\nhorizontal plane of dots labelled by $1$.  The next layer up\npulls back to give the labelling of the next\nhorizontal plane, as shown in Figure 5.6.\nThis labelling is periodic with respect to\nhorizontal and vertical translation by $n$.\nEach $n \\times n$ block is a circulent matrix.\nFor generic choice of variables, this\ncirculent matrix will have nonzero determinant.\n\n\\begin{center}\n\\psfig{file=Pix\/pix66.ps}\nFigure 5.6\n\\end{center}\n\n\nWhen we develop the condensation upwards, we\narrive precisely at a sandwich condensation of\nwidth $n$.   Going back to the planar picture,\nwe see that the $n$th horizontal row of our\nvertex labelling is a constant labelling,\nas shown in Figure 5.7.\n\n\\begin{center}\n\\psfig{file=Pix\/pix67.ps}\nFigure 5.7\n\\end{center}\n\nSince the top later of vertices is all constant, the\nlayer of edges directly below it must have labels\n$s_1,t_2,s_3,t_4...$ where $t_2=1\/s_1$, $t_4=1\/s_3$, etc. \nBy Lemma \\ref{recip} the PolyPoint corresponding\nto this row is degenerate.  That is, the vertices\nof this PolyPoint lie on a pair of lines.\nTranslating this information back into the\nlanguage of the pentagram map, we see that\nit corresponds exactly to the statement\nof Theorem \\ref{hyper}.\n\n\n\\newpage\n\n\n\\section{Proof of Theorem \\ref{hyper}}\n\n\\subsection{Degenerate polygons}\n\nWe fix $n=4m$.   To avoid trivial cases, we assume\nthat $m$ is large$-$say, $m \\geq 3$.  We work\nover $\\C$.  Everything we do is understood to\nbe defined for the generic example, but\nperhaps not for every example.  \nThe reader should insert this caveat before\nevery assertion.\nWe will work with PolyPoints\nand PolyPoints which are periodic mod $n$.\nBy this we mean that the points or lines\nthemselves are periodic, not just the\nprojective invariants.  We will call such\nobjects {\\it periodic PolyPoints\\\/} or\n{\\it periodic PolyLines\\\/}.  We take these\nto be elements of the spaces $P_1$ and $L_1$\nintroduced in \\S 5.1.\n\nWe say that the periodic PolyPoint $A=\\{...A_1,A_5,...\\}$ is\n{\\it degenerate\\\/} if $...A_1,A_9,A_{13}...$ lie\non the same line and if $...A_3,A_{11},A_{15},...$ all\nlie on the same line.  We make the dual definition\nfor PolyLines.  A polygon (in the ordinary sense)\nsatisfying the hypothesis of Theorem \\ref{hyper}\nnaturally determines a degenerate PolyLine.\nOne simply considers the lines considered by the\nedges of the polygon.\n\n\\begin{lemma}\n\\label{recip}\nA periodic element in $P_1$ is degenerate iff\nits projective invariants satisfy $q_jp_{j+1}=1$\nfor all $j$.  Dually, a periodic element in $L_1$\nis degenerate if and only if its projective\ninvariants satisfy $p_jq_{j+1}=1$ for all $j$.\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nBy duality it suffices to prove this result for\nPolyPoints.  Referring to the PolyPoint constructed\nin \\S 4, we compute\n$$\\det(A_{-1},A_5,A_{13})=p_1(1-q_2p_3).$$\nThe three points in question lie on the same\nline iff $q_2p_3=1$.  By symmetry, similar\nstatements hold for other triples of points.\n$\\spadesuit$  \\newline\n\n\n\\begin{lemma}\n\\label{closed}\nIf $A$ is a periodic $2n$-gon then\n$$(\\sum_{k=0}^{[n\/2]} O_k)^3=27 O_n^2E_n; \\hskip 15 pt\n(\\sum_{k=0}^{[n\/2]} E_k)^3=27 E_n^2O_n.$$\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nThis is a corollary of Theorem\n\\ref{monodrom}. If\n$A$ is $2n$-periodic then its monodromy\n$T$ is the identity.  Likewise,\n$T^*$ is the identity matrix.\nHence $\\Omega_1=\\Omega_2=27$.\n$\\spadesuit$  \\newline\n\nFor Theorem \\ref{hyper} we only\nneed the second statement of the next result.\nThe spaces $P_1$ and $L_1$ are defined in\n\\S 5.1.\n\n\\begin{lemma}\n\\label{van}\nA degenerate element in either $P_1$ or $L_1$\nhas pentagram invariants \\begin{enumerate}\n\\item $O_k=E_k=0$ for $k<2m$ and \n\\item $O_{2m}=E_{2m}=2$ and\n$O_n=E_n=1$.\n\\end{enumerate}\n\\end{lemma}\n\n{\\bf {\\medskip}{\\noindent}Proof: }\nFirst suppose that $k<2m=n\/2$.  Using\nthe reasoning in the proof of Theorem \\ref{trans} we\ncan group\n$O_k$ into pieces having the form\n$\\prod \\mu$, where each $\\mu$ is a monomial\ntimes  $(1-q_jp_{j+1})$ for some $j$.\n(Compare Lemma \\ref{seq2}, but interchange\n$p$ with $q$.)\nBy Lemma \\ref{recip} each $\\mu$ is zero\nand hence $O_k=0$.  The same argument\nworks for $E_k$.\n\nWe now turn to the second item.\nNote that $$O_nE_n=(p_2q_3)(p_4q_5)...=(1)(1)...=1.$$\nIt now follows from Lemma \\ref{later}\nthat $E_n^2=O_n^2=1$.   If $O_n=-1$, for any particular\nexample, then by continuity\n$O_n$ is identically $-1$.  However, we\ncan choose a $4n$-point, contained in $\\R^2$, which is essentially\nthe double cover of a $2n$-point.  The\ninvariants of such a $4n$-point would\nrepeat with period $2n$, forcing $O_n$ to be\na square of a real number.  This rules\nout the possibility that $O_n=-1$.  Hence\n$O_n=1$ for every example.  Likewise\n$E_n=1$ for every example.\nIf now follows from Corollary \\ref{closed},\nand from the vanishing of all the other\npentagram invariants, that\n\\begin{equation} \\label{closed2}\n1+O_{2m}=3O_n=3; \\hskip 30 pt\n1+E_{2m}=3E_n=3. \\end{equation}\nHence $E_{2m}=O_{2m}=2$.\n$\\spadesuit$  \\newline\n\n\\subsection{The Main Argument}\n\nWe start with a degenerate PolyLine.  This\nPolyLine determines a labelling of a single\nrow of \nthe triangulation $\\tau$, considered in \\S 6.\nCombining Theorem\n\\ref{lift} and Lemma \\ref{van} we see that\nwe can find a circulent condensation $C$\nwhich translates into our pentagram labelling.\n\n\\begin{center}\n\\psfig{file=Pix\/pix65.ps}\nFigure 6.5\n\\end{center}\n\n\nReferring to the proof of Theorem \\ref{lift} we\ncan use the fact that $p_jq_{j+1}=1$ to arrange\nso that $r_jr_{j+1}=1$ for all $j$.  This allows\nto pick our $c$ labelling so that the row below the\nedge labels is identically $1$, as shown in\nFigure 6.5.  Figure 6.5 is supposed to be periodic\nin the horizontal direction.\n\nWe pull our $c$ labelling back to give a labelling of\n$T$, the tiling of $\\R^3$ by  octahedra.\nThe bottom row of dots pulls back to an infinite\nhorizontal plane of dots labelled by $1$.  The next layer up\npulls back to give the labelling of the next\nhorizontal plane, as shown in Figure 6.6.\nThis labelling is periodic with respect to\nhorizontal and vertical translation by $n$.\nEach $n \\times n$ block is a circulent matrix.\nFor generic choice of variables, this\ncirculent matrix will have nonzero determinant.\n\n\\begin{center}\n\\psfig{file=Pix\/pix66.ps}\nFigure 6.6\n\\end{center}\n\n\nWhen we develop the condensation upwards, we\narrive precisely at a sandwich condensation of\nwidth $n$.   Going back to the planar picture,\nwe see that the $n$th horizontal row of our\nvertex labelling is a constant labelling,\nas shown in Figure 6.7.\n\n\\begin{center}\n\\psfig{file=Pix\/pix67.ps}\nFigure 6.7\n\\end{center}\n\nSince the top later of vertices is all constant, the\nlayer of edges directly below it must have labels\n$s_1,t_2,s_3,t_4...$ where $t_2=1\/s_1$, $t_4=1\/s_3$, etc. \nBy Lemma \\ref{recip} the PolyPoint corresponding\nto this row is degenerate.  That is, the vertices\nof this PolyPoint lie on a pair of lines.\nTranslating this information back into the\nlanguage of the pentagram map, we see that\nit corresponds exactly to the statement\nof Theorem \\ref{hyper}.\n\n\\newpage\n\n\n\n\\section{References}\n\n[{\\bf RR\\\/}] D.P. Robbins and H. Rumsey, \\newline\n{\\it Determinants and Alternating Sign Matrices\\\/} \\newline\nAdvances in Mathematics {\\bf 62\\\/} (1986) \n\\newline\n\\newline\n[{\\bf S1\\\/}] R. Schwartz, {\\it The Pentagram Map\\\/} \\newline\nJournal of Experimental Mathematics {\\bf 1\\\/} (1992)  pp. 85-90\n\\newline\n\\newline\n[{\\bf S2\\\/}] R. Schwartz, {\\it Recurrence of the Pentagram Map\\\/}, \\newline\nJournal of Experimental Mathematics (2001)\n\\newline\n\\newline\n[{\\bf S3\\\/}] R. Schwartz, \\newline\n {\\it Desargues Theorem, Dynamics, and Hyperplane Arrangements\\\/}  \\newline\nGeometriae Dedicata (2001)\n\\newline\n\\newline\n[{\\bf W\\\/}] S. Wolfram,\n{\\it The Mathematica Book\\\/},\nFourth Edition,\nCambridge University Press (1999)\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-1}\n\n\nThe statement, that the assumption of non-interacting magnetic\nmolecules constitutes a paradigm, is of course exaggerated. In\nthe more specific scientific community which deals with magnetic\nmolecules for quantum computing the question of intermolecular\n(and other unwanted) interactions is of course of utmost\nimportance.\\cite{ARM:PRL07,CLT:PRL10,CGT:PRA14,PCM:NC14,LGB:PRL14,LKC:JPSJ14,YNS:PCCP15}\nAlso in low-dimensional magnetism \nthe question arises for instance in connection with dimerization\nor interchain as well as interlattice\ninteractions.\\cite{Ogu:PR64,TaY:JPSJ70,TYM:JPSJ70,OgB:JPSJ81,NTS:PRB92,KTH:Science02,SSS:PRB03,SRI:PRB05,YTH:PRL05,LWL:JACS07,NDF:DT08,BLT:ICA08,MMY:JACS09,AVA:PRB11}\nIntermolecular exchange is also carefully taken care of when\ndesigning magnetic multi-qubit devices. For these cases the\nintermolecular interactions are even made\nswitchable.\\cite{MMY:JACS09}\n\nIn this paper we would like to approach the problem from a\nsomewhat different perspective. We would like to ask how big\nintermolecular interactions have to be in order to modify dc\nmagnetic observables so drastically that the fingerprint of the\nunderlying molecular subunits is masked. We concentrate\nour investigations on molecules with an antiferromagnetic\nintramolecular coupling. For these cases the low-temperature\nmagnetization curves consist of clearly spaced\nsteps,\\cite{ShB:JAP02} which disappear with increasing\nantiferromagnetic intermolecular interaction. Such a behavior\nwas observed in several recent investigations, compare\ne.g. Refs.~\\onlinecite{SPK:PRB08,KKA:CC15,PPS:CPC15}, \nand interpreted in various ways. We don't offer a solution to\nspecific problems, but we would like to present\norder-of-magnitude calculations that show for which ratios of\ninter- and intramolecular interactions magnetization steps of\nfinite clusters disappear. We demonstrate that the space\ndimension of the embedding plays a strong role.\n\nThe evaluation of magnetic properties of interacting magnetic\nmolecules constitutes a massive quantum many-body problem, even\nif only spin Hamiltonians are considered. That's why only two to three\ndozens of spins $s=1\/2$ can be modeled numerically exactly by\ndiagonalizing a spin Hamiltonian even when symmetries are\nemployed.\\cite{DGP:IC93,GaP:GCI93,BCC:IC99,BeG:EPR,Tsu:group_theory,Wal:PRB00,Tsu:ICA08,BBO:PRB07,ScS:PRB09,ScS:IRPC10}\nBut for\nnon-frustrated spin systems quantum Monte\nCarlo\\cite{SaK:PRB91,San:PRB99,SyS:PRE02} (QMC) provides quasi\nexact thermodynamic observables. We therefore restrict our\ninvestigations to a series of such systems, which should be\nsufficient for the purpose of this paper. The clear advantage is\nthat we do not rely on mean field\napproximations.\\cite{TaY:JPSJ70,OgB:JPSJ81,Sch:PRL96} \n\nThe result of our investigations is that single molecule\nsignatures are washed out in the magnetization if the\nintermolecular interactions are stronger than about 10~\\% of the\nintramolecular interactions. The specific value depends on the\nspace dimension of the embedding, e.g. for intermolecular interactions\nin three dimensions a smaller intermolecular interaction is\nneeded to mask the molecular behavior than in lower dimensions.\nWe compare some of our results with the scenario of J-strain,\nthat sometimes is also taken into consideration for the\ninterpretation of experimental data.\n\nFinally, we would like to draw the readers attention to related\ninvestigations. An important related problem is given by the \ninfluence of interchain or interplane interactions on magnetic\nobservables and in particular ordering temperatures in\nantiferromagnetic systems. Such questions are also dealt with\nby means of QMC for instance in\nRefs.~\\onlinecite{SSS:PRB03,YTH:PRL05,BLT:ICA08}. \nRandom-exchange quantum Heisenberg antiferromagnets on a square\nlattice have been investigated by QMC in Ref.~\\onlinecite{LWL:PRB06}.\nThe influence\nof the embedding medium on ground state properties of a\nHeisenberg star system was investigated in\nRef.~\\onlinecite{RVK:JPAMG96}.\n\nThe article is organized as follows. In Section~\\xref{sec-2} the\ntheoretical framework is explained. Section~\\xref{sec-3} \npresents magnetization, susceptibility and specific heat for\ndimers, squares and cubes embedded in one, two and three\ndimensions, respectively, whereas Section~\\xref{sec-4} discusses\nthe magnetism of dimers with intermolecular interactions in one,\ntwo and three dimensions. \nIn Section~\\xref{sec-5} a comparison with the scenario of\nJ-strain is presented. The article closes with summary and\noutlook. \n\n\n\n\\section{Hamiltonian and calculational scheme}\n\\label{sec-2}\n\nThe investigated spin systems are modeled by a Heisenberg\nHamiltonian augmented with a Zeeman term, i. e.  \n\\begin{eqnarray}\n\\label{E-2-1}\n\\op{H}\n&=&\n-\n\\sum_{i<j}\\;\n{J}_{ij}\n\\op{\\vec{s}}_i \\cdot \\op{\\vec{s}}_j\n+\ng \\mu_B\\, B\\,\n\\sum_{i}\\;\n\\op{s}^z_i\n\\ .\n\\end{eqnarray}\nThermodynamic observables are evaluated by means of \nquantum Monte Carlo\\cite{SaK:PRB91,San:PRB99,SyS:PRE02} (QMC)\nusing the ALPS package.\\cite{ALPS:JMMM07,BCE:JSMTE11} The\nprefactors in \\fmref{E-2-1} are chosen according to the \nconvention used in ALPS, in particular a negative\nHeisenberg exchange corresponds to an antiferromagnetic\ninteraction. Without loss of generality \n$s=1\/2$ is chosen as spin quantum number, and the spectroscopic \nsplitting factor is taken as $g=2$ for all spins. \n\nIn our QMC calculations we choose $N=100$ for one-dimensional\nproblems, $N=10\\times 10$ for two-dimensional problems as well\nas $N=10\\times 10\\times 10$ for three-dimensional problems. In\nall cases periodic boundary conditions are applied. For\nmagnetization curves we use 100000 steps for thermalization and\nanother 100000 steps for sampling. In case of susceptibility and\nspecific heat functions of temperature at $B=0$ thermalization\nis done with 10000000 steps and sampling with 200000000\nsteps. Although we used $\\epsilon=0.1$ for the latter functions,\nconvergence was very slow at the lowest temperatures, which can\nbe seen in the upcoming plots. Fortunately, this does not alter\nour conclusions.\n\n\n\n\\section{Magnetic observables for one-, two-, and\n  three-dimensional intermolecular interactions}\n\\label{sec-3}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-1.pdf}\n\\caption{(Color online) Schematic structure of the\n  one-dimensional (a) and the two-dimensional bipartite lattice; solid bonds\n  depict interactions $J_1$, dashed bonds $J_2$.}   \n\\label{intermolecular-f-1}\n\\end{figure}\n\nIn the following magnetic observables are presented for small\nmagnetic units (molecules) that are investigated in\ndependence of the intermolecular interaction. These\ninvestigations have been performed for couplings in one, two and\nthree space dimensions. Our main concern is the magnetization\ncurve, since this curve usually exhibits a very strong\ndependence on whether and how subunits are magnetically\ncoupled.\\cite{ShB:JAP02} For small systems the magnetization\ncurve exhibits clear fingerprints of the specific unit, such as\nmagnetization steps.  \n\nWe follow two rationals when comparing the behavior in different\nspace dimensions. In the first investigation each spin interacts\nwith the same number of intramolecular $J_1$ and intermolecular\n$J_2$ bonds with its neighbors. This is realized by dimers in\none, squares in two and cubes in three dimensions. In addition,\nthe size of the singlet-triplet gap is the same for dimer and\nsquare and almost the same (80~\\%) for the cube. In a second\ninvestigation the molecular unit is kept fixed as dimers, and\nthe dimension of the embedding is varied. \n\n\n\n\n\n\\subsection{One-dimensional system}\n\\label{sec-3-1}\n\nOne-dimensional systems have been very thoroughly investigated\nover almost a century. The famous Bethe ansatz for the spin-1\/2\nHeisenberg chain and the resulting knowledge on observables as\nwell as the Haldane conjecture are cornerstones of this\nresearch.\\cite{Bet:ZP31,Hul:AMAFA38,Gri:PR64,Hal:PL83,Hal:PRL83,AfL:LMP86,AGS:JPA89,Klu:EPJB98}\nHere we focus on the question how magnetic observables develop\nwith an increasing intermolecular interaction $J_2$ between\ndimers that are coupled through $J_1$, compare\n\\figref{intermolecular-f-1}~(a). Both interactions are\nantiferromagnetic.  Since we deal with four quantities, $J_1$,\n$J_2$, $T$, and $B$, we\ndecided to assume some reasonable values throughout the article\nthat are common to \nmaterials in molecular magnetism, in particular we choose\n$J_1=-10$~K.\\footnote{The reader is of \n  course free to make all quantities dimensionless, e.g. by\n  dividing by $J_1$.}\nIn some sense this investigation touches earlier works on\ndimerized (or spin-Peierls) spin chains,\\cite{Hid:PRB92,RAH:PRB96,UhS:PRB96,KnU:00} \nwhich investigate similar structures as\n\\figref{intermolecular-f-1}~(a) but usually for a fixed ratio of\ninteractions $J_2\/J_1$ or some small interval of this ratio. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-2.pdf}\n\\caption{(Color online) Low-temperature magnetization of the\n  one-dimensional spin system shown in\n  \\figref{intermolecular-f-1}~(a) for various interdimer\n  couplings $J_2$ and $T=0.1$~K.}   \n\\label{intermolecular-f-2}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-3a.pdf}\n\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-3b.pdf}\n\\caption{(Color online) Zero-field susceptibility and specific\n  heat of the \n  one-dimensional spin system shown in\n  \\figref{intermolecular-f-1}~(a) for various interdimer\n  couplings $J_2$ and $B=0$. Fluctuations of $C$ at the lowest $T$\n  result from very slow and thus insufficient convergence.}   \n\\label{intermolecular-f-3}\n\\end{figure}\n\n\nFor vanishing interdimer interaction $J_2=0$ the magnetization\ncurve at low temperatures features one jump to saturation at the\nexternal field where the singlet and the lowest triplet level cross. \nThis simple magnetization curve is characteristic for the af\ndimer of two spins $s=1\/2$. The jump is rather stable against an\nincrease of $J_2$; even at $J_2\/J_1=0.5$ the curve still warps\naround the former jump. For $J_2\/J_1=1$ the limit of the\naf Heisenberg chain is reached, which results in a gapless\ncontinuous rise of the magnetization with increasing field. \n\nThe robustness of the dimer properties is also reflected in the\nsusceptibility as well as specific heat functions, here in zero\nfield. Both functions do not change their dimer character up to\nat least $J_2\/J_1=0.5$. For $J_2\/J_1=1$ they assume their\ncharacteristic behavior known from the Bethe ansatz: the system\nis gapless, and the specific heat shows Luttinger liquid\nbehavior, i.e. grows linearly with $T$ for low temperatures. \n\n\n\n\n\\subsection{Two-dimensional system}\n\\label{sec-3-2}\n\nFor the investigations in two dimensions we chose a square as\nthe unperturbed molecular system. The structure is shown in\n\\figref{intermolecular-f-1}~(b). A Heisenberg square with af\ninteraction shows two steps in the magnetization at fields $B_1$\nand $B_2$ where the lowest triplet crosses the singlet and where\nthe lowest pentet crosses the lowest triplet. This is clearly\nvisible in \\figref{intermolecular-f-4} (black solid curve).\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-4.pdf}\n\\caption{(Color online) Low-temperature magnetization of the\n  two-dimensional spin system shown in\n  \\figref{intermolecular-f-1}~(b) for various interdimer\n  couplings $J_2$ and $T=0.1$~K.}   \n\\label{intermolecular-f-4}\n\\end{figure}\n\nWith increasing ratio $J_2\/J_1$ the step-like structure is more\nquickly destroyed by the intermolecular interactions compared to\nthe one-dimensional case. For\n$J_2\/J_1=0.1$ the curve still warps around the former jumps, but\nfor $J_2\/J_1=0.5$ it is already almost continuous and in its\ncharacter not much different from the magnetization curve of the\naf square lattice.\\cite{Man:RMP91,Ric:PRB93,San:PRB97} This\nbehavior is not so clearly reflected by \nthe thermal functions. These are again rather stable against\nvariations of $J_2\/J_1$, only the susceptibility displays the\nchange by a very different rise at low-temperature. The specific\nheat does not display any feature since no ordering can occur in\none or two dimensions for non-zero temperature.\\cite{MeW:PRL66}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-5a.pdf}\n\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-5b.pdf}\n\\caption{(Color online) Zero-field susceptibility and specific heat of the \n  two-dimensional spin system shown in\n  \\figref{intermolecular-f-1}~(b) for various interdimer\n  couplings $J_2$ and $B=0$.}   \n\\label{intermolecular-f-5}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Three-dimensional system}\n\\label{sec-3-3}\n\nFor the three-dimensional case we choose a simple cubic lattice\nwhere the molecular unit is given by cubes as sketched in\n\\figref{intermolecular-f-6}. An isolated cube of spins $s=1\/2$\nand af bonds along the edges shows four magnetization steps at\nlow temperature which result from the successive level crossings\nof the lowest states with total spin $S=0, 1, 2, 3, 4$. The\nsimple cubic lattice with $J_2\/J_1=1$ on the other hand is a\nsystem with long range order at $T>0$, a property which the lower\ndimensional systems did not show in accordance with the theorem\nof Mermin and Wagner.\\cite{MeW:PRL66}\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-6.pdf}\n\\caption{Schematic structure of the\n  investigated three-dimensional bipartite lattice; solid bonds\n  depict interactions $J_1$, dashed bonds $J_2$.}   \n\\label{intermolecular-f-6}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-7.pdf}\n\\caption{(Color online) Low-temperature magnetization of the\n  three-dimensional spin system shown in\n  \\figref{intermolecular-f-6} for various interdimer\n  couplings $J_2$ and $T=0.1$~K.}   \n\\label{intermolecular-f-7}\n\\end{figure}\n\nLooking at the magnetization in \\figref{intermolecular-f-7} one\nimmediately realizes that already a rather small intermolecular\ninteraction of 10~\\% suffices to wash out the magnetization steps\nof the spin cube. It is important to keep in mind that the cube\nhas almost the same singlet-triplet gap as dimer and square, so\nthe effect is not thermal. We thus speculate that the\ndimensionality of the embedding structure, here three, is\nresponsible for the quick disappearance of the molecular\nfingerprints with increasing intermolecular interaction.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-8a.pdf}\n\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-8b.pdf}\n\\caption{(Color online) Zero-field susceptibility and specific heat of the\n  three-dimensional spin system shown in\n  \\figref{intermolecular-f-6} for various interdimer\n  couplings $J_2$ and $B=0$.}   \n\\label{intermolecular-f-8}\n\\end{figure}\n\nAlthough the magnetization is already drastically altered by\n10~\\% intermolecular interactions, the temperature dependence\nof the susceptibility does not show much deviation in this case,\ncompare \\figref{intermolecular-f-8}. The same holds for the\nspecific heat. These functions are modified only for larger\nintermolecular interactions in accord with the one- and\ntwo-dimensional cases. The peaks of the specific heat for\n$J_2\/J_1=0.5$ and $J_2\/J_1=1.0$ mark phase transitions to\nthree-dimensional ordered phases -- they correspond exactly to\nthose shown in Ref.~\\onlinecite{SSS:PRB03}. \n\n\n\n\\section{Dimers in various dimensions}\n\\label{sec-4}\n\nIn a second setup we kept the molecular unit fixed as a\ndimer and varied the dimension of the embedding. The\none-dimensional case remains the same. The two-dimensional case\ncan be derived from \\figref{intermolecular-f-1}~(b) by replacing\nall (thick) vertical $J_1$-bonds by (dashed) $J_2$-bonds. For\nthe three-dimensional case the two-dimensional lattices are stacked on top of\neach other with $J_2$-bonds in between. Thus each spin is\nconnected by one $J_1$-bond and one, three, and five $J_2$-bonds\nfor the one-, two-, and three-dimensional case, respectively.\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-9.pdf}\n\\caption{(Color online) Low-temperature magnetization of dimers\n  in one-, two and three-dimensional arrangements for\n  $J_2=1.0$~K and $T=0.1$~K.}    \n\\label{intermolecular-f-9}\n\\end{figure}\n\nFor the following investigation $J_2\/J_1=0.1$ as well as the\ntemperature were kept constant. As can be clearly seen in\n\\figref{intermolecular-f-9} the magnetization step is more\nstrongly washed out with increasing dimensionality. The\ninfluence on the temperature dependence of both susceptibility\nas well as specific heat is again weak, see\n\\figref{intermolecular-f-10}. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-10a.pdf}\n\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-10b.pdf}\n\\caption{(Color online) Zero-field susceptibility and specific\n  heat of dimers   in one-, two and three-dimensional\n  arrangements for   $J_2=1.0$~K and $T=0.1$~K.}   \n\\label{intermolecular-f-10}\n\\end{figure}\n\n\n\n\n\n\\section{Comparison with J-strain}\n\\label{sec-5}\n\nFinally, as a supplement to the presented investigations, we\nwould like to discuss the question whether a similar\nmodification of observables could stem from J-strain. The\nassumption of strain, for instance g-strain, is not unusual for\ninstance when modeling EPR lines. J-strain, i.e. a distribution of\n$J$ values about a mean was used in several theoretical models,\nsee e.g. Refs.~\\onlinecite{SPK:PRB08,SFF:JPCM:10,PPS:CPC15}. The\neffect of J-strain is rather similar to that of intermolecular\ninteractions: magnetization steps are smeared out, and\nsusceptibility as well as specific heat as functions of\ntemperature are not much altered. \n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-11.pdf}\n\\caption{(Color online) Low-temperature magnetization of dimers\n  in one-, two and three-dimensional arrangements for\n  $J_2=1.0$~K and $T=0.1$~K (dashes) compared to isolated dimers\n  with a J-strain of $\\Delta = 1.0, 3.0, 5.0$~K (solid curves),\n  respectively.}\n\\label{intermolecular-f-11}\n\\end{figure}\n\nIn the following we present an investigation in which\nindependent dimers with a flat distribution of $J_1$-values in the\ninterval $[\\bar{J}-\\Delta,\\bar{J}+\\Delta ]$ have been\nsimulated. $\\Delta$ was chosen such, that the saturation field\nfor the three cases discussed in section \\ref{sec-4} is met.\nFigure~\\xref{intermolecular-f-11} shows a comparison of the\nmagnetization of a single dimer (black solid curve), of dimers\nwith intermolecular interactions in one, two, and three space\ndimensions (dashed curves) as well as of dimers with J-strain\naccording to the flat distribution (solid colored curves). One\nimmediately realizes that the functional form of the\nmagnetization curve with J-strain is different from the behavior\nunder the influence of intermolecular interactions. Although the\nsaturation field is met by tuning $\\Delta$ appropriately, the\nonset of the magnetization curves happens already at smaller\nfields. In addition, at the field value where the magnetization\nstep happens for the unperturbed dimer, the magnetization curves\nof dimers with J-strain cross at half the step height whereas for\nintermolecular interactions the magnetization curves cross at a\nlower magnetization. Overall, the magnetization curves for\nJ-strain are symmetric about the crossing field value. This\nwould also hold if another (more realistic, but also symmetric\nabout $\\bar{J}$) Gaussian\ndistribution of $J_1$ values would have been taken. Intermolecular\ninteractions on the contrary seem to lead to magnetization\ncurves, that do not show any symmetry with respect to the\noriginal crossing field.\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-12a.pdf}\n\n\\includegraphics*[clip,width=60mm]{intermolecular-fig-12b.pdf}\n\\caption{(Color online) Zero-field susceptibility and specific\n  heat of dimers in one-, two and three-dimensional\n  arrangements for $J_2=1.0$~K and $T=0.1$~K (dashes) compared\n  to isolated dimers with a J-strain of $\\Delta = 1.0, 3.0,\n  5.0$~K (solid curves), respectively.}\n\\label{intermolecular-f-12}\n\\end{figure}\n\nFigure~\\xref{intermolecular-f-12} demonstrates that somewhat\ncontrary to the findings of section \\ref{sec-4} now the\nsusceptibility is only very weakly altered whereas the specific\nheat is more drastically modified especially for the case of the \nlargest J-strain.\n\n\n\n\n\\section{Summary and Outlook}\n\\label{sec-6}\n\nWe investigated the question how intermolecular interactions\ninfluence magnetic observables for small (molecular) magnetic\nunits. In particular we investigated for certain bipartite\nconfigurations how large the intermolecular interaction needs to\nbe compared to the intramolecular interaction in order to mask\nthe molecular behavior. It could be demonstrated that the\nvarious static magnetic observables reflect intermolecular\ninteractions differently: the low-temperature magnetization\nturned out to be most sensitive, since the appearance of\nmagnetization steps appears to be fragile. In addition\ndimensionality plays a role. With increasing space\ndimensionality of the intermolecular coupling the effect of\nmasking molecular properties happens for smaller intermolecular\ncoupling. Finally we discussed briefly whether similar\nmodifications of observables could be misinterpreted as\nJ-strain. We pointed out, that certain features of the\nobservables are different in the two scenarios, so that with\ngood quality of experimental data a discrimination should be\npossible. \n\n\n\\section*{Acknowledgment}\n\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG\nSCHN 615\/20-1). I would like to thank Arzhang Ardavan, Stephen\nBlundell, Marco Evangelisti, Andreas Honecker, Franziska\nKirschner, Hiroyuki Nojiri and Johannes Richter for valuable\ndiscussions.  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nNetworks have been introduced in  phylogenetics to generalize the tree paradigm, which permits to  \n represent only descent with modification (i.e. speciation events). Phylogenetic networks allow to model evolutionary scenarios including a larger class of evolutionary events, such as  recombinations,  lateral gene transfers and hybridizations. \n\nIn this paper, we shall focus on directed phylogenetic networks \\citep[see][for a short survey on the phylogenetic network paradigm also covering undirected phylogenetic networks]{survey}. Mathematically, such networks are, in the broadest sense, directed acyclic graphs with a single node with no incoming edges --the \\emph{root}-- representing the common ancestor of all the Operational Taxonomic Units (OTUs for short) under study, which are represented by the  nodes with no outgoing edges -- the \\emph{leaves}-- of the graph; internal nodes represent either (hypothetical) speciations or (hypothetical) reticulated events.\nUnfortunately, this definition is too broad, both for representing biologically-meaningful evolutionary scenarios, and for giving objects that can be efficiently handled.\n\nSo far, several restrictions on this general definition have been introduced in the literature. Some of them are based on biological considerations, while others have been introduced to artificially narrow the space of networks under study. \nThis led to the introduction of a panoply of different classes of phylogenetic networks, such as time-consistent networks \\citep{baroni2006}, regular networks \\citep{Baroni2005}, orchard networks \\citep{orchard}, galled trees \\citep{galled_trees} and galled networks \\citep{galled_net}, level-$k$ networks \\citep{vanIersel2011}, tree-sibling networks \\citep{Cardona2009}, tree-based networks \\citep{Francis2015} and LGT networks \\citep{cardona2015reconstruction}, just to name a few.\n\nIn this paper, we shall focus on binary tree-child networks (BTC networks, for short), which were introduced by \\cite{Cardona2009} and are one of the most studied classes of phylogenetic networks \\citep{VANIERSEL2010, vanIersel2014, Semple2016, Bordewich2018}. Mathematically, being tree-child means that every internal node is compelled to have at least a child node with a single incoming arc. Biologically, it boils down to say that every non-extant OTU has at least a direct descendant through mutation. \n\n\n\n\n The combinatorial study of phylogenetic networks is nowadays a challenging and  active field of research. Nevertheless, the problem of counting how many phylogenetic networks are in a given subclass  of networks is still open even for long-established classes. More precisely, this problem has been only recently solved for galled networks \\citep{COuntGalled}; for other classes, including tree-child networks, we only have  \n asymptotic results \\citep{McDiarmid2015,CountTC}. \n\nAssociated to the problem of counting networks, we find the problem of their ``injective'' generation, \n\n i.e. without having to check for isomorphism between pairs of constructed networks. \n \n The main result of this paper is a systematic way of recursively generating, with unicity, all BTC networks with a given number of leaves. This generation relies on a pair of reduction\/augmentation operations --both producing BTC networks-- where reductions decrease by one the number of leaves in a network, and augmentations increase it. The idea of using pairs of operation has already been used to deal either with other classes of phylogenetic networks \\citep{cardona2008distance, cardona2009metrics2}, or for BTC networks but without the unicity feature \\citep{orchard}.\n \n As interesting side product, this procedure gives a recursive formula providing an upper bound on the number of BTC networks. \n \n The paper is organized as follows. In Section~\\ref{sec:preliminaries}, we review the basic definitions that will be used throughout the paper. Section~\\ref{sec:reduction} is devoted to the reduction procedure, while  in Section~\\ref{sec:generation-networks} we introduce the augmentation operation and prove that any BTC network can be obtained, in an unique way, via a sequence of augmentation operations applied to the trivial network with one leaf. In Section~\\ref{sec:bounding-offspring}, we show how to relax the conditions for the applicability of the augmentation operation  to obtain a recursive formula providing an upper bound on the number of BTC networks. In Section~\\ref{sec:computations} we introduce the implementation of the algorithms presented in the paper, and some experimental results, including the exhaustive generation of all BTC networks with up to six leaves and an upper bound of their number up to ten leaves.\n  Finally, in Section~\\ref{sec:conclusions} we discuss how our reduction\/augmentation operations extend and generalize analogous operations for phylogenetic trees. \n\n \n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nThroughout this paper, a \\emph{tree node} in a directed graph is a node $u$ whose pair of degrees $d(u)=(\\indeg u,\\outdeg u)$ is $(1,0)$ for the \\emph{leaves}, $(0,2)$ for the \\emph{roots}, or $(1,2)$ for \\emph{internal} tree nodes; a \\emph{hybrid node} is a node $u$ with $d(u)=(2,1)$.\n\nA \\emph{binary phylogenetic network} over a set $X$ of taxa is a directed acyclic graph with a single root such that all its nodes are either tree nodes or hybrid nodes, and whose leaf set is bijectively labeled by the set $X$. In the following, we will implicitly identify every leaf with its label.\nA binary phylogenetic network is \\emph{tree-child} if every node either is a leaf or has at least one child that is a tree node \\citep{Cardona2009}; in particular, the single child of a hybrid node must be a tree node.\nWe will denote by $\\mathcal{BTC}_n$ the set of binary tree-child phylogenetic networks over the set $[n]=\\{1,\\dots,n\\}$.\n\n\nAn \\emph{elementary node} in a directed graph is a node $u$ with $d(u)=(1,1)$ or $d(u)=(0,1)$. An \\emph{elementary path} $p$ is a path $u_1,\\dots,u_k$ composed of  elementary nodes such that neither the single parent of $u_1$ (if it exists) nor the single child of $u_k$ are elementary. We  call these last two nodes respectively the \\emph{grantor} (if this node is well-defined) and \\emph{heir} of the nodes in the elementary path. In case of an elementary node, its grantor and heir are those of the nodes in the single elementary path that contains the given node. The \\emph{elimination} of an elementary path $p$ consists in deleting all nodes in $p$, together with their incident arcs, and adding an arc between the grantor and the heir of $p$ (provided that the grantor exists; otherwise, no arc is added). The elimination of an elementary node is defined as the elimination of the elementary path that contains the given node.\n\nGiven a node $u$, we can \\emph{split} it by adding a new node $\\tilde u$, an arc $(\\tilde u,u)$, and replacing every arc $(v,u)$ with $(v,\\tilde u)$. If $u$ is a tree node, then $\\tilde u$ is an elementary node whose heir is $u$, and the elimination of $\\tilde u$ recovers the original network. The successive splitting (say $k$ times) of a tree node $u$ generates an elementary path formed by $k$ nodes, whose heir is $u$, and whose elimination recovers the original network.\n\n\n\n\\section{Reduction of networks}\n\\label{sec:reduction}\n\nThe goal of this section is to define a reduction procedure on BTC networks that can be applied to any such network, and producing a  BTC network with one leaf less. By successive application of this procedure, any BTC network can thus be reduced to the trivial network with a single leaf.\n\nWe start by associating to each leaf $\\ell$ a path whose removal will produce the desired reduction (up to elementary paths).\n\nLet $\\ell$ be a leaf of a BTC network $N$. A \\emph{pre-TH-path} for $\\ell$ is a path $u_1,\\dots,u_r=\\ell$ such that: \n\\begin{enumerate}\n    \\item Each node $u_i$ in the path is a tree node.\n    \\item For each $i=1,\\dots,r-1$, the child of $u_i$ different from $u_{i+1}$, denoted by $v_{i}$, is a hybrid node.\n    \\item For each $i\\neq j$, we have that $v_i\\neq v_j$.\n\\end{enumerate}\nA \\emph{TH-path} is a maximal pre-TH-path, i.e. a pre-TH-path that cannot be further extended. Note that, since all nodes in a pre-TH-path $p$ are tree nodes, if $p$ can be extended by prepending one node, then this extension is unique. Hence, starting with the trivial pre-TH-path formed by the leaf $\\ell$ alone, and extending it by prepending the parent of the first node in the path as many times as possible, we obtain a TH-path that is unique by construction.\nLet $u_1,\\dots,u_r=\\ell$ be a TH-path; different possibilities may arise that make it maximal: (1) $u_1$ is the root of $N$; (2) the parent of $u_1$, call it $x$, is a hybrid node; (3) $x$ is a tree node whose both children are tree nodes; (4) $x$ is a parent of $v_i$ for some $i \\in [r-1]$. (We shall see in Lemma \\ref{lemma:noRoot} that the first case  cannot hold). \n\n\nFor each leaf $\\ell$, we denote by $\\THP(\\ell)$ its single TH-path and by $\\THP(\\ell)_1$ the first node of this path.\nNote that we allow the case $r=1$.\nIn this case, if we are not in a trivial BTC network (i.e.\\ a network consisting of a single node),  the parent of $\\ell$ is either a hybrid node, or a tree node whose two children are tree nodes.\n\n\n\n\n\n\\begin{lem}\\label{lemma:noRoot}\n  Let $N$ be a non-trivial BTC network  and let $\\ell$ be any of its leaves. Then, $\\THP(\\ell)_1$ cannot be the root of $N$.\n\\end{lem}\n\n\\begin{pf}\n  Let $u_1,\\dots,u_r=\\ell$ be the path $\\THP(\\ell)$ and assume for the sake of contradiction that $u_1$ is the root of $N$. For each $i=1,\\dots,r-1$, let $v_i$ be the hybrid node that is a child of $u_i$ and $x_i$ the parent of $v_i$ different from $u_i$;  recall that $x_i$ does not belong to $\\THP(\\ell)$ by the definition of a pre-TH-path. Since $u_1$ is the root of $N$, every node of $N$ either belongs to the path $\\THP(\\ell)$ or is descendant of a node in $\\{v_i\\mid i\\in[r-1]\\}$.\n  In particular, for each $i\\in[r-1]$, there exists some $\\sigma(i)\\in[r-1]$ such that\n \n  $x_i$ is descendant of $v_{\\sigma(i)}$, and since this node is descendant of $x_{\\sigma(i)}$, $x_i$ is descendant of $x_{\\sigma(i)}$. Hence, starting with $x_1$ we get a sequence\n  $x_1, x_{\\sigma(1)},  x_{\\sigma(\\sigma(1))}, \\dots$\n  where each node in the sequence is a descendant of the following one.\n  Since there is a finite number of nodes, at some point we find a repeated node, which means that $N$ contains a cycle and hence we have a contradiction.\n  }\\end{proof}\n\\end{pf}\n\nWe say that a leaf $\\ell$ is of \\emph{type} $T$ (resp. of \\emph{type} $H$) if the parent of $\\THP(\\ell)_1$ is a tree node (resp. a hybrid node). If $\\ell$ is of type $H$, we  indicate by $\\overline\\THP(\\ell)$ the path obtained by prepending to $\\THP(\\ell)$ the parent of $\\THP(\\ell)_1$.\nFor convenience, we let $\\overline\\THP(\\ell)=\\THP(\\ell)$ if $\\ell$ is of type $T$.\n\n\n\n\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}\n    \\node[treenode,label=left:$w_1$] (w) at (0,1) {};\n    \\node[treenode,label=left:{$u_0=u_1$}] (u1) at (0,0) {};\n    \\node[treenode,label=left:{$u_i$}] (u2) at (0,-1.5) {};\n    \\node[treenode,label=left:{$u_{r-1}$}] (u3) at (0,-3) {};\n    \\node[treenode, label=left:{$u_r=\\ell$}] (u4) at (0,-4) {};\n    \\node[hybnod\n    ] (v1) at ($(u1) +(-30:1)$) {};\n    \\node[hybnode,label=above:$v_i$] (v2) at ($(u2) +(-30:1)$) {};\n    \\node[hybnod\n    ] (v3) at ($(u3) +(-30:1)$) {};\n    %\n    \\coordinate (v1p) at ($(v1) +(30:0.5)$);\n    \\coordinate (v1c) at ($(v1) +(-90:0.5)$);\n    \\draw[->] (v1p)--(v1);\n    \\draw[->] (v1)--(v1c);\n    \\draw[->,dashed] (v1p) to\n    (v1c);\n    %\n    \\coordinate[label=right:{$x_i$}] (v2p) at ($(v2) +(30:0.5)$);\n    \\coordinate[label=right:{$y_i$}] (v2c) at ($(v2) +(-90:0.5)$);\n    \\draw[->] (v2p)--(v2);\n    \\draw[->] (v2)--(v2c);\n    \\draw[->,dashed] (v2p) to\n    (v2c);\n    %\n    \\coordinate (v3p) at ($(v3) +(30:0.5)$);\n    \\coordinate (v3c) at ($(v3) +(-90:0.5)$);\n    \\draw[->] (v3p)--(v3);\n    \\draw[->] (v3)--(v3c);\n    \\draw[->,dashed] (v3p) to\n    (v3c);\n    %\n    \\coordinate[label=right:$t_1$] (wc) at ($(w) +(-30:0.5)$) {};\n    \\coordinate[label=right:$z_1$] (wp) at (0,1.5) ;\n    \\draw[->] (wp)--(w);\\draw[->] (w)--(wc);\n    \\draw[->,dashed] (wp) to\n    (wc);\n    %\n    \\draw[->] (w)--(u1);\n    \\draw[path] (u1)--(u2); \n    \\draw[path] (u2)--(u3); \n    \\draw[->] (u3)--(u4);\n    \\draw[->] (u1)--(v1);\n    \\draw[->] (u2)--(v2);\n    \\draw[->] (u3)--(v3);\n    %\n    \\draw[dotted] ($(u4)+(-0.2,-0.2)$) rectangle ($(u1)+(0.2,0.2)$);\n  \\end{tikzpicture}\n  \\qquad\n  \\begin{tikzpicture}\n    \\node[hybnode,label=left:$u_0$] (u0) at (0,1) {};\n    \\node[treenode,label=left:{$u_1$}] (u1) at (0,0) {};\n    \\node[treenode,label=left:{$u_i$}] (u2) at (0,-1.5) {};\n    \\node[treenode,label=left:{$u_{r-1}$}] (u3) at (0,-3) {};\n    \\node[treenode, label=left:{$u_r=\\ell$}] (u4) at (0,-4) {};\n    \\node[hybnod\n    ] (v1) at ($(u1) +(-30:1)$) {};\n    \\node[hybnode,label=above:$v_i$] (v2) at ($(u2) +(-30:1)$) {};\n    \\node[hybnod\n    ] (v3) at ($(u3) +(-30:1)$) {};\n    %\n    \\coordinate (v1p) at ($(v1) +(30:0.5)$);\n    \\coordinate (v1c) at ($(v1) +(-90:0.5)$);\n    \\draw[->] (v1p)--(v1);\n    \\draw[->] (v1)--(v1c);\n    \\draw[->,dashed] (v1p) to\n    (v1c);\n    %\n    \\coordinate[label=right:{$x_i$}] (v2p) at ($(v2) +(30:0.5)$);\n    \\coordinate[label=right:{$y_i$}] (v2c) at ($(v2) +(-90:0.5)$);\n    \\draw[->] (v2p)--(v2);\n    \\draw[->] (v2)--(v2c);\n    \\draw[->,dashed] (v2p) to\n    (v2c);\n    %\n    \\coordinate (v3p) at ($(v3) +(30:0.5)$);\n    \\coordinate (v3c) at ($(v3) +(-90:0.5)$);\n    \\draw[->] (v3p)--(v3);\n    \\draw[->] (v3)--(v3c);\n    \\draw[->,dashed] (v3p) to\n    (v3c);\n    %\n   \n   \n   \n   \n   \n    %\n    \\draw[->] (w)--(u1);\n    \\draw[path] (u1)--(u2); \n    \\draw[path] (u2)--(u3); \n    \\draw[->] (u3)--(u4);\n    \\draw[->] (u1)--(v1);\n    \\draw[->] (u2)--(v2);\n    \\draw[->] (u3)--(v3);\n    %\n   \n   \n   \n   \n   \n   \n   \n   \n    \\node[treenode,label=left:$w_2$] (w') at ($(u0) + (30:1)$) {};\n    \\node[treenode,label=right:$w_1$] (w) at ($(u0) + (150:1)$) {};\n    \\draw[->] (w')--(u0);\n    \\draw[->] (w)--(u0);\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    %\n    \\coordinate[label=left:$z_1$] (wp) at ($(w) +(90:0.5)$);\n    \\coordinate[label=left:$t_1$] (wc) at ($(w) +(210:0.5)$);\n    \\draw[->] (wp)--(w);\n    \\draw[->] (w)--(wc);\n    \\draw[->,dashed] (wp) to\n    (wc);\n    %\n    \\coordinate[label=right:$z_2$] (w'p) at ($(w') +(90:0.5)$);\n    \\coordinate[label=right:$t_2$] (w'c) at ($(w') +(-30:0.5)$);\n    \\draw[->] (w'p)--(w');\n    \\draw[->] (w')--(w'c);\n    \\draw[->,dashed] (w'p) to\n    (w'c);\n    %\n   \n   \n   \n   \n   \n   \n   \n    \\draw[dotted] ($(u4)+(-0.2,-0.2)$) rectangle ($(u0)+(0.2,0.2)$);\n  \\end{tikzpicture}\n  \\caption{Sketches of\n  a $T$-reduction (left) and\n  a $H$-reduction (right). Tree nodes are represented by circles and hybrid nodes by squares.\n  The nodes inside the dotted box form $\\overline\\THP(\\ell)$ and will be removed, which will create elementary nodes that will be substituted by the dashed arcs.}\n  \\label{fig:T-red}\n\\end{figure}\n\n\\begin{defn}\n  \\label{def:reductions}\nLet $\\ell$ be a leaf in a BTC network $N$. We define the \\emph{reduction of $N$ with respect to $\\ell$} as the result of the following procedure:\n\\begin{enumerate}\n\\item Delete all nodes in $\\overline\\THP(\\ell)$ (together with any arc incident on them).\n\\item Eliminate all elementary nodes.\n\\end{enumerate}\nWe indicate this reduction by $R(N,\\ell)$. If we want to emphasize the type of the deleted leaf, we  indicate the reduction by $T(N,\\ell)$ and say it is a $T$-reduction if $\\ell$ is of type $T$, or by $H(N,\\ell)$ and say that it is a $H$-reduction if $\\ell$ is of type $H$. \n\\end{defn}\n\n\nTo ease of reading, we shall introduce some  notations:\n\\begin{defn}\n  \\label{defn:elementarynodes}\nLet $u_1,\\dots,u_r=\\ell$ be the path $\\THP(\\ell)$ and let $u_0$ be the first node in $\\overline\\THP(\\ell)$. For each $i \\in [r-1]$, $v_i$ is the hybrid child of $u_i$, $x_i$ the parent of $v_i$ different from $u_i$, and $y_i$ the single child of $v_i$. The parent(s) of $u_0$ is $w_1$ (are $w_1,w_2$); the node $w_j$ is always a tree node, $z_j$ is its parent (if it exists, since $w_j$ could be the root of $N$), and $t_j$ its child different from $u_0$. \n\\end{defn}\nIn the following, we shall use the notations given in the definition above, which are illustrated in Figure~\\ref{fig:T-red}.\n\n\\begin{rmk}\n  \\label{rmk:elementarynodes}\n  Since $N$ is tree-child, the nodes $y_i$ are always tree nodes, and so are $t_1$ and $t_2$ in case of an $H$-reduction. In case of a $T$-reduction, by definition of a TH-path, $t_1$ is either a tree node or coincides with one  of the hybrid nodes $v_i$.\nAlso, the removal of the arcs of the form $(u_i,v_i)$  and $(w_j,u_0)$ makes nodes $v_i$ and  $w_j$ elementary in $N\\setminus \\overline\\THP(\\ell)$, where $i \\in [r-1]$, and $j=1 $ for $T$-reductions and $j \\in [2]$ for $H$-reductions. Since no other arc is removed, no other node can be elementary. In order to find the heirs of nodes $v_i$ and  $w_j$, we must analyse under which circumstances two of these elementary nodes are adjacent in $N\\setminus \\overline\\THP(\\ell)$. \n\\begin{enumerate}\n    \\item If we had that two nodes $v_i$ and $v_j$ were connected by an arc in $N\\setminus \\overline\\THP(\\ell)$, then the single child of a hybrid node in $N$ would be also a hybrid. This contradicts the fact  that $N$ is tree-child.\n    \\item The existence of an arc $(v_i,w_j)$ would imply the existence of a cycle in $N$, which is impossible. \n    \\item Consider now the case of an arc $(w_j,v_i)$. In case of an $H$-reduction, it would imply that  both children of $w_j$ are hybrid nodes, which is impossible. However, such an arc can be present in a $T$-reduction: when $t_1$ is equal to $v_i$. In this last case, $w_1$ and $v_i$ form an elementary path in $N\\setminus \\overline\\THP(\\ell)$ and their common\n      heir is $y_i$.\n    \\item Finally, in case of an $H$-reduction, it can exist an arc between $w_1$ and $w_2$, say that the arc is $(w_1,w_2)$ (which implies, $t_1=w_2$, $z_2=w_1$). In this case, $w_1$ and $w_2$ form an elementary path in $N\\setminus \\overline\\THP(\\ell)$ and their common\n     \n      heir is $t_2$.\n\\end{enumerate}\nIn all other cases, the elementary nodes $v_i$ and $w_j$ are isolated, and their respective heirs are $y_i$ and $t_j$.\n\\end{rmk}\n\n\nWe study now what we call the \\emph{recovering data} of a reduction. This information will be used in the next section to recover the original network from its reduction.\n\n\\begin{defn}\n  The \\emph{recovering data} of the reduction $N'=R(N,\\ell)$ is the pair $(S_1,S_2)$, where:\n  \\begin{itemize}\n      \\item $S_1$ is the multiset of the nodes of $N'$ that are heirs of the nodes $w_j$. \n      The cardinality of $S_1$ (as a multiset) is either $1$ or $2$, depending on the type of the reduction, and will be denoted by $|S_1|$.\n     \n      \\item $S_2$ is the tuple $(y_1,\\dots,y_{r-1})$ of nodes of $N'$, which are the heirs of the nodes $v_i$. This tuple could be empty, corresponding to the case $r=1$.\n  \\end{itemize}\n \\end{defn}\n\nWe introduce now a set of conditions on multisets and tuples of nodes, and prove that the recovering data associated to any of the defined reductions satisfies them.\n\n\\begin{defn}\\label{dfn:feasible}\n  Given a BTC network $N'$ and a pair $(S_1,S_2)$ with\n    \\begin{itemize}\n      \\item $S_1$ a multiset of tree nodes of $N'$,\n      \\item $S_2=(y_1,\\dots,y_{r-1})$, with $r\\ge1$, an (eventually empty) tuple of $r-1$ tree nodes of $N'$,\n  \\end{itemize}\n  consider the following set of conditions:\n \n  \\begin{enumerate}\n      \\item For every $i,j\\in [r-1]$ with $i\\neq j$, the nodes $y_i$ and $y_j$ are different, and if they are siblings, then $y_i\\in S_1$ or $y_j\\in S_1$.\n      \\item For every $i\\in[r-1]$, if $y_i$ is the child of a hybrid node or has a hybrid sibling, then $y_i\\in S_1$.\n      \\item No node in $S_1$ is a proper descendant of any node in $S_2$.\n \n \n \n      \\item[4T.] $|S_1|=1$.\n \n \n \n      \\item[4H.]  $|S_2|=2$ and no node of $S_1$ appears in $S_2$.\n  \\end{enumerate}\n  We say that $(S_1,S_2)$ is \\emph{$T$-feasible} if it satisfies conditions 1, 2, 3, and 4T, and \\emph{$H$-feasible} if it satisfies conditions  1, 2, 3, and 4H. Finally,  we say that $(S_1,S_2)$ is \\emph{feasible} if it is either $T$-feasible or $H$-feasible.\n\\end{defn}\n\n\n\n\n\n\\begin{prop}\n  Let $N'=T(N,\\ell)$ be a $T$-reduction of a BTC network $N$. Then,  its recovering data  $(\\{\\tau_1\\},(y_1,\\dots,y_{r-1}))$ is $T$-feasible.\n\\end{prop}\n\n\\begin{pf}\nFirst, note that, by Remark \\ref{rmk:elementarynodes}, all nodes in  $(\\{\\tau_1\\},(y_1,\\dots,y_{r-1}))$ are tree nodes and  that Condition 4T holds trivially.\nNote also that $\\tau_1$ is equal to $y_i$ if $t_1=v_i$, or  to $t_1$ if this node is different from all the nodes $v_i$.\nWe now prove that Conditions 1, 2 and 3 hold: \n\\begin{enumerate}\n    \\item If $y_i=y_j$, then in $N$ we have $v_i=v_j$, which is impossible by definition of TH-path. If $y_i$ and $y_j$ are siblings in $N'$ but none of these nodes is equal to $\\tau_1$, then $v_i$ and $v_j$ are siblings in $N$, which implies that their common parent has two hybrid children, which is impossible in a BTC network.\n    \\item If $y_i$ is the child in $N'$ of a hybrid node and $\\tau_1\\neq y_i$, then in $N$ we have that $v_i$, which is a hybrid node, is the child of a hybrid node, which is impossible in a tree-child network. Analogously, if $y_i$ has a sibling in $N'$ which is a hybrid node, and $y_i\\neq \\tau_1$, then in $N$ we have that $v_i$ is sibling of another hybrid node, which is again impossible.\n    \\item The existence of  a non-trivial path in $N'$ from $y_i$ to $\\tau_1$ would, by construction, imply the existence of a path from $y_i$ to $w_1$ in $N$. Since there exists also a path in $N$ from $w_1$ to $y_i$, this would contradict the fact that $N$ is a DAG.\n    }\\end{proof}\n\\end{enumerate}\n\\end{pf}\n\n\n\n\n\n\\begin{prop}\n  Let $N'=H(N,\\ell)$ be an $H$-reduction of a BTC network $N$. Then,  its recovering data $(\\{\\tau_1,\\tau_2\\},(y_1,\\dots,y_{r-1}))$ is $H$-feasible.\n\\end{prop}\n\n\\begin{pf}\nAgain we have, by Remark \\ref{rmk:elementarynodes}, that all nodes in the recovering data are tree nodes. Additionally, by the same remark, we have that $|S_1|=2$ --and hence the first part of Condition 4H holds-- and if $(w_1,w_2)$ is an arc of $N$, then $S_1=\\{t_2,t_2\\}$, otherwise $S_1=\\{t_1,t_2\\}$ with $t_1\\neq t_2$.\nNote that Condition $3$ implies that Conditions 1 and 2 can be simplified as follows:\nfor all $i,j\\in [r-1]$ with $i\\neq j$, $y_i$ and $y_j$ are neither equal nor siblings, and for all $i\\in[r-1]$, $y_i$ is neither the child nor the sibling of a hybrid node.\n  \nConditions 1 and Conditions 2 and 3 in their simplified form follow using the same arguments as in the previous proposition. As for the condition 4H,\nthe nodes $\\tau_1$ and $\\tau_2$ are different from the nodes $y_i$ since the parents of $\\tau_1$ and $\\tau_2$ in $N$ are tree nodes, while the parent of each of the nodes $y_i$ is hybrid.\n    }\\end{proof}\n\\end{pf}\n\n\n\nThe following proposition is the main result of this section, since it shows that the reduction that we have defined, when applied to a BTC network, gives another BTC network with one leaf less. Hence,  successive applications of these reductions reduce any BTC network to the trivial BTC network.\n\n\\begin{prop}\n  Let $N$ be a BTC network over $X$ and $\\ell$ one of its leaves. Then, $R(N,\\ell)$ is a BTC network over $X\\setminus\\{\\ell\\}$.\n\\end{prop}\n\n\\begin{pf}\nFirst, it is easy to see that, since no new path is added, the resulting directed graph is still acyclic.\n\n  Then, we need to check that  $R(N,\\ell)$ is binary. To do so, we start noting that every node in $N\\setminus\\overline\\THP(\\ell)$ is either a tree node, a hybrid node, or an elementary node. Indeed, the removal of $\\overline\\THP(\\ell)$ (Phase 1 of Definition \\ref{def:reductions}) only affects the nodes adjacent to this path, that is the nodes $v_i$ and $w_i$,  which, as shown in Remark~\\ref{rmk:elementarynodes}, become elementary. The elimination of all elementary nodes (Phase 2 of Definition \\ref{def:reductions}) does not affect the indegree and outdegree of any other node, apart when the root $\\rho$ of $N\\setminus\\overline\\THP(\\ell)$ is elementary. In such a case, the heir of $\\rho$ becomes the new root. Hence, $R(N,\\ell)$  is binary and rooted.\n\n\n \n\n \n\n \n\n \n  Note also that the set of leaves of $R(N,\\ell)$ is $X\\setminus\\{\\ell\\}$, since in $N\\setminus\\overline\\THP(\\ell)$ no node becomes a leaf and the only leaf that is removed is $\\ell$.\n  \n  Finally, we need to prove that $R(N,\\ell)$ is tree-child. Note that, from what we have just said about how the reduction affects indegrees and outdegrees of the nodes that persist in the network, it follows that each hybrid node of $R(N,\\ell)$ is also a hybrid node of $N$, and that its parents in $R(N,\\ell)$ are the same as in $N$. It follows that no node in $R(N,\\ell)$ can have that all its children are hybrid, since this would imply that $N$ is not tree-child, a contradiction.\n  }\\end{proof}\n\\end{pf}\n\n\n\\begin{cor}\\label{cor:decomposition-of-network}\n  Let $N\\in\\mathcal{BTC}_n$ be a BTC network over $[n]$. Let $N_n=N$ and define recursively $N_{i}=R(N_{i+1},i+1)$ for each $i=n-1,n-2,\\dots,1$. Then, $N_i$ is a BTC network over $[i]$. In particular, $N_{1}$ is the trivial BTC network with its single node labeled by $1$.\n\\end{cor} \n\nWe finish this section with the computation of the number of tree nodes and hybrid nodes that the reduced network has, both in terms of the original network and of the reduction operation that has been applied. But before, we give an absolute bound on the number of these nodes in terms of the number of leaves. \n\n\\begin{lem}\\label{lem:number_nodes}\nLet $N$ be BTC network over $[n]$ with $t$ tree nodes and $h$ hybrid nodes. Then $t-h=2n-1$, $h\\le n-1$ and $t\\le 3n-2$.\n\\end{lem}\n\n\\begin{pf}\nThe equality $t-h=2n-1$ follows easily from the handshake lemma taking into account the number of roots, internal tree nodes, leaves and hybrid nodes in $N$, and their respective indegrees and outdegrees. The inequality  $h\\le n-1$ is shown in Proposition~1 in \\citep{Cardona2009}, and the last inequality is a simple consequence of the equality and the inequality already proved.\n  }\\end{proof}\n\\end{pf}\n\n\\begin{prop}\n  Let $N$ be a BTC network and $\\ell$ one of its leaves, and $N'=R(N,\\ell)$. Let $t,h$ (resp. $t',h'$) the number of tree nodes and hybrid nodes of $N$ (resp. of $N'$). Then\n  $$t'=t-|\\overline\\THP(\\ell)|-1,\\qquad h'=h-|\\overline\\THP(\\ell)|+1,$$\n  where $|\\overline\\THP(\\ell)|$ is the number of nodes in $\\overline\\THP(\\ell)$.\n\\end{prop}\n\n\\begin{pf}\n  Since the number of tree nodes and hybrid nodes are linked by the equality in Lemma~\\ref{lem:number_nodes}, it is enough to prove that $h'=h-|\\overline\\THP(\\ell)|+1$. From the discussion in Remark~\\ref{rmk:elementarynodes}, it is straightforward to see that the number of hybrid nodes in $N$ that are not in $N'$ is $r-1$ if $\\ell$ is of kind $T$, and $r$ otherwise. Hence, in both cases we have $h'=h-(|\\overline\\THP(\\ell)|-1)$ and the result follows. \n \n \n \n \n  }\\end{proof}\n\\end{pf}\n\n\\section{Generation of networks}\n\\label{sec:generation-networks}\n\nIn this section, we consider the problem of how to revert the reductions defined in the previous section, taking as input the reduced network and its recovering data. This will allow us to define a procedure that, starting with the trivial BTC network with one leaf, generates all the BTC networks with any number of leaves in an unique way. \n\nWe start by defining two augmentation procedures that take as input a BTC network and a feasible pair, and produce a BTC network with one leaf more.\n\n\\begin{defn}\n\\label{T-augmentation}\n  Let $N$ be a BTC network over $X$, $\\ell$ a label not in $X$, and  $(\\{\\tau_1\\},(y_1,\\dots,y_{r-1}))$ a $T$-feasible pair.\n  We apply the following operations to $N$:\n  \\begin{enumerate}\n  \\item Create a path of new nodes $u_1,\\dots,u_r$.\n  \\item Split the node $\\tau_1$ creating one elementary node $w_1$ and add an arc $(w_1, u_1)$.\n  \\item For each node $y_i$, split it introducing one elementary node $v_i$ and add an arc $(u_i,v_i)$.\n  \\item Label the node $u_r$  by $\\ell$.\n  \\end{enumerate}\n    We denote by $T^{-1}(N,\\ell;\\{\\tau_1\\},(y_1,\\dots,y_{r-1}))$ the resulting network and say that it has been obtained by an \\emph{augmentation operation of type} $T$.\n\\end{defn}\n\nNote that the order in which steps 2 and 3 are done is relevant in the case that $\\tau_1=y_i$ for some $i\\in[r-1]$. In such a case, two  nodes $w_1$ and $v_i$ are created, linked by an arc $(w_1,v_i)$.\n\n\\begin{prop}\\label{thm:correctness-T-augmentation}\n  Using  the notations of Definition~\\ref{T-augmentation}, the network \n  $$\\tilde N=T^{-1}(N,\\ell;\\{\\tau_1\\},(y_1,\\dots,y_{r-1}))$$ \n  is a  BTC network over $X\\cup\\{\\ell\\}$. Moreover, if $N$ has $h$ hybrid nodes, then $\\tilde N$ has $h+r-1$ hybrid nodes.\n\\end{prop}\n\n\\begin{pf}\nWe first check that the resulting directed graph is acyclic. Let us assume that $\\tilde N$ contains a cycle. If we define  $U_1=\\{u_1,\\dots,u_r\\}$ and $U_2=V(\\tilde N)\\setminus U_1$, we have that the only arcs connecting $U_1$ with $U_2$ are $(u_i,v_i)$ (with $i=1,\\dots,r-1$), and $(w_1,u_1)$ is the only arc connecting $U_2$ with $U_1$. The cycle can be contained neither inside $U_1$, since these nodes are linked by a single path, nor inside $U_2$, since otherwise $N$ would contain a cycle. Hence, the cycle must contain at least the arc $(w_1,u_1)$ and an arc $(u_i,v_i)$. This implies the existence of a path from $v_i$ to $w_1$ visiting only nodes in $U_2$, which in turn means that $N$ contains a path from $y_i$ to $\\tau_1$, against Condition 3 of Definition  \\ref{dfn:feasible}.\n\nNote that the nodes in $U_1$ are tree nodes by construction. Also by construction, the node $w_1$ is a tree node,  the nodes $v_i$ are hybrid nodes and $u_r$ is a leaf which is labelled with $\\ell$. Finally, the other nodes keep the same degrees they had in $N$ and hence $\\tilde N$ is a  binary phylogenetic network  over $X\\cup\\{\\ell\\}$ with $h+r-1$ hybrid nodes.\n\nSince $N$ is tree-child, in order to check that $\\tilde N$ is also tree-child, we only need to check the newly added  hybrid nodes, which are the parents of the nodes $v_i$. \n\nLet us first consider the case that $\\tau_1\\neq y_i$ for all $i\\in[r-1]$.\n \nFor each node $v_i$, its parents are $u_i$ and the parent $x_i$ of $y_i$ in $N$. The node $u_i$ is by construction a tree node whose other child is $u_{i+1}$, which, in turn, is a tree node. \nSince $\\tau_1\\neq y_i$, by Condition 2 of Definition \\ref{dfn:feasible},  $y_i$ can have neither a hybrid parent nor a hybrid sibling, and it cannot be a sibling of any other node $y_j$ with $j\\in[r-1]$. This latter restriction implies that  $y_i$ has the same sibling $\\tilde x_i$ in $N$ and $\\tilde N$. Thus both $x_i$ and $\\tilde x_i$ are not hybrid nodes, and the network is tree-child.\n\n\nLet us now consider the case that $\\tau_1=y_i$ for a single choice of $i\\in[r-1]$. The hybrid node $v_i$ in $\\tilde N$ has as parents the nodes $w_1$ and $u_i$, and these two nodes have as respective children $u_1$ and $u_{i+1}$, which are tree nodes. For each other node $v_j$ with $j\\neq i$ and such that $y_j$ is a not sibling of  $y_i$, the same argument as in the previous case proves that both parents of $v_j$ have a tree child. If $y_j$ is a sibling of $y_i$, it is easy to see that the parent of $v_j$ is still tree-child since it has $w_1$ as child.\n}\\end{proof}\n\\end{pf}\n\n\\begin{defn}\\label{dfn:H-augmentation}\n  Let $N$ be a BTC network over $X$, $\\ell$ a label not in $X$, and  $(\\{\\tau_1,\\tau_2\\},(y_1,\\dots,y_{r-1})$  a $H$-feasible pair. \n  We apply the following operations to $N$:\n  \\begin{enumerate}\n  \\item Create a path of new nodes $u_0,u_1,\\dots,u_r$.\n  \\item Split each of the nodes $\\tau_i$ introducing one elementary node $w_i$ and add an arc from $w_i$ to $u_0$. Note that, if $\\tau_1=\\tau_2$, two consecutive elementary nodes must be created.\n  \\item For each node $y_i$, split it introducing one elementary node $v_i$ and add an arc $(u_i,v_i)$.\n  \\item Label the node $u_r$  by $\\ell$.\n  \\end{enumerate}\nWe  denote by $H^{-1}(N,\\ell;\\{\\tau_1,\\tau_2\\},(y_1,\\dots,y_{r-1}))$ the resulting network and say that it has been obtained by an augmentation operation of type $H$.\n\\end{defn}\n\n\\begin{prop}\n  Using  the notations of Definition~\\ref{dfn:H-augmentation}, the network\n  $$\\tilde N=H^{-1}(N,\\ell;\\{\\tau_1,\\tau_2\\},(y_1,\\dots,y_{r-1}))$$ \n  is a BTC network over $X\\cup\\{\\ell\\}$. If $N$ has $h$ hybrid nodes, then $\\tilde N$ has $h+r$ hybrid nodes.\n\\end{prop}\n\n\\begin{pf}\n  The proof is completely analogous to that of Proposition~\\ref{thm:correctness-T-augmentation}, taking into account that one extra hybrid node is created.\n}\\end{proof}\n\\end{pf}\n\nGiven a BTC network over $X$, a label $\\ell\\notin X$ and a feasible pair $(S_1,S_2)$, in order to unify notations  we define the augmented network $R^{-1}(N,\\ell,S_1,S_2)$ as $T^{-1}(N,\\ell,S_1,S_2)$, if $|S_1|=1$, and as $H^{-1}(N,\\ell,S_1,S_2)$, if $|S_1|=2$. Also, we shall generically say that the \\emph{offspring} of a BTC network is the set of networks that can be obtained from it by means of augmentation operations.\n\nOur next goal is to prove that different augmentation operations applied to a same BTC network or different BTC networks over the same set of taxa  provide different networks. We start with the case of different networks.\n\n\n\n\\begin{prop}\\label{prp:different_nets}\n      Let $\\tilde N_1$ and $\\tilde N_2$ be two BTC networks, both obtained by one augmentation operation applied to two non-isomorphic BTC networks $N_1$ and $N_2$ over the same set of taxa $X$. Then $\\tilde N_1$ and $\\tilde N_2$ are not isomorphic.\n\\end{prop}\n\n\\begin{pf}\n    If $\\tilde N_1$ and $\\tilde N_2$ have different set of labels, then it is clear that they are not isomorphic. We can therefore assume that both augmentation operations introduced the same new leaf $\\ell$.\n    Suppose that $\\tilde N_1\\simeq \\tilde N_2$. Then $R(\\tilde N_1,\\ell)\\simeq R(N_2,\\ell)$. Now, from the definitions of the reductions and augmentations it is straightforward to check that $R(\\tilde N_i,\\ell)=N_i$\n    and we get that $N_1\\simeq N_2$, a contradiction.\n  }\\end{proof}\n\\end{pf}\n\nWe treat now the case of applying different augmentation operations to the same BTC network. But first, we give a technical lemma that will be useful in the proof of the proposition.\n\n\\begin{lem}\\label{lem:automorph}\n  Let $N$ be a BTC network. Then, the identity is the only automorphism (as a leaf-labeled directed graph) of $N$.\n\\end{lem}\n\n\n\\begin{pf}\n    Let $\\phi$ be any automorphism of $N$. Since $\\phi$ is an automorphism of directed graphs and sends each leaf to itself, it follows that $\\mu(u)=\\mu(\\phi(u))$ for each node $u$ of $N$, where $\\mu(u)$ is the $\\mu$-vector of $u$ as defined in \\citet{Cardona2009}. Then, by \\citep[Lemma~5c]{Cardona2009}, it follows that $u$ and $\\phi(u)$ are either equal, or one of them is the single child of the other one; according to our definition of BTC networks, this last possibility implies that one of them is a hybrid node and the other one is a tree node, which is impossible if $\\phi$ is an automorphism. Hence $\\phi(u)=u$ for every node $u$.\n    }\\end{proof}\n\\end{pf}\n\n\n\n\\begin{prop}\\label{thm:unicity-operation}\n  Let $\\tilde N_1$ and $\\tilde N_2$ be two BTC networks, both obtained by one augmentation operation applied to the same BTC network $N$. If either the kinds of operation or the feasible pairs used to construct $\\tilde N_1$ and $\\tilde N_2$ are different, then $\\tilde N_1$ and $\\tilde N_2$ are not isomorphic.\n\\end{prop}\n\n\\begin{pf}\n  Let us assume that $\\tilde N_1$ and $\\tilde N_2$ are isomorphic. Then, it is clear that they have the same set of labels, and exactly one of them, say $\\ell$, is not a label of $N$. Since $\\tilde N_1$ and $\\tilde N_2$ are isomorphic, the kind of $\\ell$ is the same in both networks, which implies that the kind of augmentation operations used to construct $\\tilde N_1$ and $\\tilde N_2$ are the same. Also, since $\\tilde N_1$ and $\\tilde N_2$ are isomorphic, the nodes in the respective recovering data of the reductions $R(\\tilde N_i,\\ell)$ must be linked by an isomorphism of phylogenetic networks. Therefore, and since by Lemma~\\ref{lem:automorph} BTC networks do not have a nontrivial automorphism, the respective recovering data must be equal.\n  }\\end{proof}\n\\end{pf}\n\nThe following proposition shows that the reduction procedure defined in the previous section can be reverted using the augmentation operations presented in this section.\n\n\\begin{prop}\\label{thm:reduction-augmentation}\n  Let $N$ be a BTC network and $\\ell$ a leaf of $N$. \n \n  Let $N'=R(N,\\ell)$,\n   $(S_1,S_2)$ its recovering data, and\n   \n    $\\tilde N=R^{-1}(N',\\ell,S_1,S_2)$.\n    Then, $N$ and $\\tilde N$ are isomorphic.\n\\end{prop}\n\\begin{pf}\n  It is straightforward to see that the operations $T^{-1}$ and $H^{-1}$ reverse the effects of $T$ and $H$, respectively. The only points worthy of attention correspond to the cases where the single node in $S_1$ appears in $S_2$ (for reductions\/augmentations of type $T$) or where there is a single node in $S_1$ with multiplicity two (for reductions\/augmentations of type $H$). In the first case, the augmentation process creates two elementary nodes, $w_1$ and $v_i$, connected by an arc $(w_1,v_i)$, which is the same situation as in $N$ after the removal of the nodes in $\\overline\\THP(\\ell)$. In the second case, two elementary nodes $\\tau_1$ and $\\tau_2$ are created, connected by an arc, once again the same situation as in $N$ after the removal of the nodes in $\\overline\\THP(\\ell)$.\n}\\end{proof}\n\\end{pf}\nA direct consequence of  the results in this section is the following theorem, which can be used to generate in an effective way all BTC networks over a set of taxa. See Figure~\\ref{fig:figuraJC} for an example.\n\\begin{thm}\n  Let $N\\in\\mathcal{BTC}_n$ be a BTC network over $[n]$. Then, $N$ can be constructed from the trivial network in $\\mathcal{BTC}_1$ (with one node labeled by $1$) by application of $n-1$ augmentation operations, where at each step $i$, the leaf $i+1$ is added. Moreover, these augmentation operations are unique.\n \n\\end{thm}\n\\begin{pf}\n  The existence is a direct consequence of Corollary~\\ref{cor:decomposition-of-network} and Proposition~\\ref{thm:reduction-augmentation}. Unicity comes from\n  Propositions~\\ref{prp:different_nets} and~\\ref{thm:unicity-operation}.\n}\\end{proof}\n\\end{pf}\n\nIt should be noted that very recently, other methods to generate all BTC networks over a set of taxa have been proposed \\citep{orchard}, but,  to our knowledge, this is the first time that the networks are generated with unicity. In previous attempts, an \nisomorphism check was needed after the generation phase.\n  %\n  %\n\\section{Bounding the number of networks}\n\\label{sec:bounding-offspring}\n\nIn this section, we shall first give bounds for the number of BTC networks that can be obtained from a given one by means of augmentation operations. This will be done by bounding the number of feasible pairs in such a network. Then, we shall find bounds for the number of BTC networks with a fixed number $n$ of leaves.\n\n\nLet $N$ be a BTC network over $[n]$ with $h$ hybrid nodes. From Lemma~\\ref{lem:number_nodes} we know that it has $t=2n+h-1$ tree nodes, and that $h\\le n-1$ and $t\\le 3n-2$.\nIn the following, we shall show how to \ncompute the number of pairs $(S_1,S_2)$ satisfying all conditions of Definition \\ref{dfn:feasible}, except for Condition 3, via an auxiliary problem. Note that this will only give an upper bound for the number of networks, since the pairs we find can  produce networks with cycles. \n\n\\begin{figure}\n  \\centering\n\n  \\begin{tikzpicture}\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[label=$N_1$] (N) at (0,-6.8) {};\n\\end{tikzpicture}\n\\qquad\n \\begin{tikzpicture}\n    \\node[treenode] (r) at (0.5,-6) {};\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[treenode,label=left:$2$] (l2) at (1,-7) {};\n    \\draw[->] (r)--(l1);\n    \\draw[->] (r)--(l2);\n    \\node[label=$N_2$] (N) at (0.5,-5.75) {};\n\\end{tikzpicture}\n \\qquad\n    \\vspace{0.5cm}\n \\begin{tikzpicture}\n    \\node[treenode] (r) at (1,-5) {};\n    \\node[treenode] (x) at (1.5,-6) {};\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[treenode,label=left:$2$] (l2) at (1,-7) {};\n    \\node[treenode,label=left:$3$] (l3) at (2,-7) {};\n    \\draw[->] (r)--(l1);\n    \\draw[->] (r)--(x);\n    \\draw[->] (x)--(l2);\n    \\draw[->] (x)--(l3);\n    \\node[label=$N_3$] (N) at (1,-4.75) {};\n\\end{tikzpicture}\n\n \\vspace{0.5cm}\n \\begin{tikzpicture}\n    \\node[treenode] (r) at (1,-4) {};\n    \\node[treenode,label=left:$a$] (a) at (1.5,-5) {};\n    \\node[treenode] (x) at (1.25,-6) {};\n    \\node[treenode,label=right:$b$] (b) at (1.75,-5.65) {};\n    \\node[hybnode] (h) at (3,-6.5) {};\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[treenode,label=left:$2$] (l2) at (1,-7) {};\n    \\node[treenode,label=left:$3$] (l3) at (2,-7) {};\n    \\node[treenode,label=left:$4$] (l4) at (3,-7) {};\n    \\draw[->] (r)--(l1);\n    \\draw[->] (r)--(a);\n    \\draw[->] (a)--(x);\n    \\draw[->] (a)--(b);\n    \\draw[->] (x)--(l2);\n    \\draw[->] (x)--(h);\n    \\draw[->] (b)--(l3);\n    \\draw[->] (b)--(h);\n    \\draw[->] (h)--(l4);\n    \\node[label=$N_4$] (N) at (0.3,-4.75) {};\n\\end{tikzpicture}\n\n  \\vspace{0.5cm}\n \\begin{tikzpicture}\n    \\node[treenode,label=left:$c$] (r) at (1,-3) {};\n    \\node[treenode,label=left:$a$] (a) at (1.5,-5) {};\n    \\node[treenode] (x) at (1.15,-6.25) {};\n    \\node[treenode,label=right:$b$] (b) at (1.75,-6) {};\n    \\node[hybnode] (h) at (3,-6.5) {};\n    \\node[treenode] (w) at (1.25,-3.5) {};\n    \\node[hybnode] (v1) at (1.6,-5.5) {};\n    \\node[hybnode] (v2) at (1.5,-4.5) {};\n    \\node[treenode] (u1) at (3,-3.75) {};\n    \\node[treenode] (u2) at (3.25,-4.25) {};\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[treenode,label=left:$2$] (l2) at (1,-7) {};\n    \\node[treenode,label=left:$3$] (l3) at (2,-7) {};\n    \\node[treenode,label=left:$4$] (l4) at (3,-7) {};\n    \\node[treenode,label=left:$5$] (l5) at (4,-7) {};\n    \\draw[->] (r)--(l1);\n    \\draw[->] (r)--(w);\n    \\draw[->] (w)--(u1);\n    \\draw[->] (w)--(v2);\n    \\draw[->] (u1)--(u2);\n    \\draw[->] (u1)--(v1);\n    \\draw[->] (v2)--(a);\n    \\draw[->] (a)--(x);\n    \\draw[->] (a)--(v1);\n    \\draw[->] (x)--(l2);\n    \\draw[->] (x)--(h);\n    \\draw[->] (v1)--(b);\n    \\draw[->] (b)--(l3);\n    \\draw[->] (b)--(h);\n    \\draw[->] (h)--(l4);\n    \\draw[->] (u2)--(v2);\n    \\draw[->] (u2)--(l5);\n    \\node[label=$N_5$] (N) at (0.3,-3.75) {};\n\\end{tikzpicture}\n\\qquad\n \\begin{tikzpicture}\n    \\node[treenode,label=left:$c$] (r) at (1,-3) {};\n    \\node[treenode] (a) at (1.5,-5) {};\n    \\node[treenode] (x) at (1.15,-6.25) {};\n    \\node[treenode] (b) at (1.75,-6) {};\n    \\node[hybnode] (h) at (3,-6.5) {};\n    \\node[treenode] (w) at (1.25,-3.5) {};\n    \\node[hybnode] (v1) at (1.6,-5.5) {};\n    \\node[hybnode] (v2) at (1.5,-4.5) {};\n    \\node[treenode] (u1) at (3,-3.75) {};\n    \\node[treenode] (u2) at (3.25,-4.25) {};\n  \n    \\node[hybnode] (u0) at (-1,-4) {};\n    \\node[treenode] (u_1) at (-1,-5) {};\n    \\node[treenode] (w1) at (1,-2) {};\n    \\node[treenode] (w2) at (1,-2.5) {};\n    \\node[hybnode] (v_1) at (0,-6) {};\n  \n    \\node[treenode,label=left:$6$] (l6) at (-1,-7) {};\n    \\node[treenode,label=left:$1$] (l1) at (0,-7) {};\n    \\node[treenode,label=left:$2$] (l2) at (1,-7) {};\n    \\node[treenode,label=left:$3$] (l3) at (2,-7) {};\n    \\node[treenode,label=left:$4$] (l4) at (3,-7) {};\n    \\node[treenode,label=left:$5$] (l5) at (4,-7) {};\n  \n    \\draw[->] (w1)--(w2);\n    \\draw[->] (w1)--(u0);\n    \\draw[->] (w2)--(u0);\n    \\draw[->] (w2)--(r);\n    \\draw[->] (u0)--(u_1);\n    \\draw[->] (u_1)--(l6);\n    \\draw[->] (u_1)--(v_1);\n    \\draw[->] (v_1)--(l1);\n    \\draw[->] (r)--(v_1);\n    \\draw[->] (r)--(w);\n    \\draw[->] (w)--(u1);\n    \\draw[->] (w)--(v2);\n    \\draw[->] (u1)--(u2);\n    \\draw[->] (u1)--(v1);\n    \\draw[->] (v2)--(a);\n    \\draw[->] (a)--(x);\n    \\draw[->] (a)--(v1);\n    \\draw[->] (x)--(l2);\n    \\draw[->] (x)--(h);\n    \\draw[->] (v1)--(b);\n    \\draw[->] (b)--(l3);\n    \\draw[->] (b)--(h);\n    \\draw[->] (h)--(l4);\n    \\draw[->] (u2)--(v2);\n    \\draw[->] (u2)--(l5);\n    \\node[label=$N_6$] (N) at (0,-2.5) {};\n\\end{tikzpicture}\n  \\caption{Example of a BTC network and the chain of augmentation operations that generate it. Namely: $N_2=T^{-1}(N_1,2;\\{1\\},\\emptyset)$, $N_3=T^{-1}(N_2,3;\\{2\\},\\emptyset)$,\n  $N_4=H^{-1}(N_3,4;\\{2,3\\},\\emptyset)$,\n  $N_5=T^{-1}(N_4,5;\\{a\\},(b,a))$, and\n  $N_6=H^{-1}(N_5,6;\\{c,c\\},(1))$.}\n  \\label{fig:figuraJC}\n\\end{figure}\n\n\n\\subsection{An auxiliary problem}\n\nLet $P(N,k)$ be the set of tuples of length $k$ of tree nodes of $N$ such that (1) no pair of them are equal or siblings, and (2) none of them has a hybrid parent or sibling. We indicate the number of such tuples as $p(N,k)=|P(N,k)|$, and since this number will only depend on $n$, $h$ and $k$, we indicate it also by $p(n,h,k)$. We consider the problem of computing $p(n,h,k)$.\n\n\nWe compute first how many tree nodes are there that do not have neither a hybrid parent nor a hybrid sibling. Since the single child of a hybrid node must be a tree node, there are $h$ tree nodes that have a hybrid parent. Note that each hybrid node has two siblings that must be tree nodes; also, a tree node cannot be sibling of two different hybrid nodes; hence, there are $2h$ tree nodes that have a hybrid sibling. Since there cannot be a tree node having the two properties (if it has a hybrid parent, then it does not have any hybrid sibling), there are $3h$ tree nodes that are either a child or a sibling of a hybrid node.\nThen, the number of tree nodes that do not have neither a hybrid parent nor a hybrid sibling is $t-3h=2n-2h-1$. Note that this set of nodes is composed by the root of the network and pairs of tree nodes that are siblings.\n\nConsider now the problem of counting the number of tuples $(y_1,\\dots,y_k)$ in this set that are neither equal nor siblings. We distinguish two cases:\n\\begin{itemize}\n\\item If none of the nodes $y_i$ is the root of $N$, we start having $2n-2h-2$ choices for $y_1$, and at each stage the number of choices decreases of two units. Hence, the number of choices is\n  $$\n\\begin{aligned}\n    p_0(n&,h,k)=\\\\\n    &=(2n-2h-2)(2n-2h-4)\\cdots(2n-2h-2k)\\\\\n    &=2^k(n-h-1)(n-h-2)\\cdots(n-h-k)\\\\\n    &=2^k\\frac{(n-h-1)!}{(n-h-k-1)!}.\n\\end{aligned}\n$$\n\\item If one of the nodes $y_i$ is the root of $N$, then the process of constructing an element in $P(N,k)$ can be described as first choosing at which position $i$ one puts the root, and then filling in the remaining $k-1$ positions with \na tuple of the set $P(N,k-1)$ such that none of the nodes is the root (which is what we have just computed). Hence, the number of possibilities is\n  $$p_1(n,h,k)=k 2^{k-1}\\frac{(n-h-1)!}{(n-h-k)!}.$$\n\\end{itemize}\nThen we get that\n$$\n\\begin{aligned}\np(n,h,k)&=p_0(n,h,k)+p_1(n,h,k)\\\\\n&=2^k\\frac{(n-h-1)!}{(n-h-k-1)!}+k 2^{k-1}\\frac{(n-h-1)!}{(n-h-k)!}\n\\end{aligned}\n$$\n\\subsection{Counting pairs satisfying Conditions 1, 2 and 4H}\n\\label{sec:counting-h-feasible}\nConsider pairs $(S_1,S_2)$ satisfying Conditions 1, 2 and 4H. \nRecall that, since condition 4H implies that $S_1$ and $S_2$ cannot have elements in common, Conditions 1 and 2 are simplified: no pair of nodes in $S_2$ can be siblings and none of them can either be the child of a hybrid node or have a hybrid sibling. Hence,\nthe problem is equivalent to finding a tuple $(y_1,\\dots,y_{r-1})$ in $P(N,r-1)$ and then either a tree node $\\tau_1$ or an unordered pair $\\{\\tau_1,\\tau_2\\}$ of different tree nodes, in either case disjoint from those in $(y_1,\\dots,y_{r-1})$. Once the tuple $(y_1,\\dots,y_{r-1})$ is fixed, the number of tree nodes available for choosing $\\tau_1$ and $\\tau_2$ is $t-r+1=2n+h-r$.\nHence, the number of possible pairs is $$F_H(n,h,r-1)=F_{H,1}(n,h,r-1)+F_{H,2}(n,h,r-1),$$ \nwhere\n$$\n\\begin{aligned}\nF_{H,1}(n,h,r-1)&=p(n,h,r-1)\\cdot (2n+h-r),\\\\%\\qquad\nF_{H,2}(n,h,r-1)&=p(n,h,r-1)\\cdot{}\\\\\n&\\qquad\\cdot\\frac12(2n+h-r)(2n+h-r-1).\n\\end{aligned}\n$$\n\n\\subsection{Counting pairs satisfying Conditions 1, 2 and 4T}\nThe problem now is to count the ways of choosing $S_1=\\{\\tau_1\\}$ and a tuple  $S_2=(y_1,\\dots,y_{r-1})$ satisfying Conditions 1, 2 and 4T. Now $\\tau_1$ can appear in $S_2$, and different possibilities arise, since it allows that one of the nodes in $S_2$ has a sibling in $S_2$, or that it has a hybrid parent or sibling. We consider, thus, these different possibilities:\n\\begin{itemize}\n\\item $\\tau_1\\neq y_i$ (for all $i$): This case is very similar to one considered in the previous paragraph, specifically the case where only a single node $\\tau_1$ had to be taken. The number of possible pairs is\n$$F_{T,1}(n,h,r-1)=p(n,h,r-1)\\cdot(2n+h-r).$$\n\\item $\\tau_1=y_i$ is a child or a sibling of a hybrid node: Choosing one of these pairs is equivalent to first choosing the position $i$, then filling the other $r-2$ positions with a tuple in $P(N,r-2)$, and then choosing a node that is a child or sibling of a hybrid node to be put in the position $i$. The number of ways to do this procedure is\n  $$F_{T,2}(n,h,r-1)=p(n,h,r-2)\\cdot (r-1)\\cdot 3h,$$\n  since each hybrid node has a single child and two siblings, and none of these $3h$ nodes appears twice, associated to two different hybrid nodes.\n\\item $\\tau_1=y_i$ is a sibling of some other node $y_j$ in $S_2$: In this case one has to choose the positions $i$ and $j$ where to put the pair of sibling nodes, fill the other $r-3$ positions with a tuple in $P(N,r-3)$, choose a pair of sibling tree nodes to take as $y_i$ and $y_j$, and finally set $\\tau_1=y_i$. The choice of $i$ and $j$ can be done in $(r-1)(r-2)$ different ways. The choice of the tuple of length $r-3$ can be done in $p(n,h,r-3)$ ways; $p_1(n,h,r-3)$ of them contain the root of $N$ (and $r-4$ tree nodes with a sibling tree node) and $p_0(n,h,r-3)$ do not contain the root (and contain $r-3$ tree nodes with a sibling tree node). Once this is done, the number of available pairs of sibling tree nodes is $n-h-1-(r-4)=n-h-r+3$, if the root of $N$ was chosen, or $n-h-1-(r-3)=n-h-r+2$ otherwise. \n  Hence, the total number of pairs is $F_{T,3}(n,h,r-1)=F_{T,3,A}(n,h,r-1)+F_{T,3,B}(n,h,r-1)$, corresponding to these two cases, with:\n    $$\n  \\begin{aligned}\n  F_{T,3,A}(n,h,r-1)&=(r-1)(r-2)\\cdot p_1(n,h,r-3)\\cdot{}\\\\\n  &\\qquad\\cdot(2n-2h-2r+6),\\\\\n  F_{T,3,B}(n,h,r-1)&=(r-1)(r-2)\\cdot p_0(n,h,r-3)\\cdot{}\\\\\n  &\\qquad\\cdot(2n-2h-2r+4).\n\\end{aligned}\n  $$\n  \n\\item $\\tau_1=y_i$ but none of the previous conditions hold: In this case one only has to take a tuple in $P(N,r-1)$ and choose which of the $r-1$ nodes to take as $\\tau_1$. The number of possible pairs is then\n  $$F_{T,4}(n,h,r-1)= p(n,h,r-1)\\cdot (r-1).$$\n\\end{itemize}\nNote that the four conditions above are mutually exclusive. Hence, the overall number of possible pairs $(S_1,S_2)$ is the sum of all numbers found:\n$$\n  \\begin{aligned}\nF_T(n,h,r-1)&=F_{T,1}(n,h,r-1)+F_{T,2}(n,h,r-1)+{}\\\\\n&\\qquad+F_{T,3}(n,h,r-1)+F_{T,4}(n,h,r-1). \n\\end{aligned}\n$$\n\n\\subsection{Bounds for the number of networks}\n\n\nEach network $N\\in\\mathcal{BTC}_n$ with $h$ hybrid nodes, appears as augmentation $R^{-1}(N',n,S_1,S_2)$\nof an unique network $N'\\in\\mathcal{BTC}_{n-1}$ with $h'$ hybrid nodes, where $S_2$ has length $r-1=h-h'$, if the augmentation is of type $T$, or $r-1=h-h'-1$ if it is of type $H$. If we call $B(n,h)$ the number of networks in $\\mathcal{BTC}_n$ with $h$ hybrid nodes, and since we have bounded the number of feasible pairs, we have that\n$$\\begin{aligned}\nB(n,h)&\\le\n\\sum_{h'=0}^h B(n-1,h')\\cdot F_T(n-1,h',h-h')+{}\\\\\n&\\qquad +\n\\sum_{h'=0}^{h-1} B(n-1,h')\\cdot F_H(n-1,h',h-h'-1)\n\\end{aligned}\n$$\n\n\nAlso, since the number of hybrid nodes in a BTC network with $n$ leaves is at most $n-1$, we have that\n$$|\\mathcal{BTC}_n|=\\sum_{h=0}^{n-1}B(n,h),$$\nand the expression above allows us to compute a bound for this number of networks. See section~\\ref{sec:computations} for an experiment with these bounds.\n\n\nThe asymptotic formula $|\\mathcal{BTC}_n|= 2^{2n \\log n+ O(n)}$ is given in \\citet{McDiarmid2015}, and both our experimental results in Section~\\ref{sec:computations} for $n\\le 7$ and the bounds that we have computed for $n\\le 10$ \nare coherent with this expression.\nHowever, the problem of finding a closed expression for the asymptotic behaviour of our bounds is still open.\n\n\n\\section{Computational experiments}\n\\label{sec:computations}\n\nThe algorithms in this paper have been implemented in python using the python library PhyloNetworks \\citep{pypi_phylonetworks}. This implementation, together with the sources for the experiments that we comment in this section can be downloaded from \\url{https:\/\/github.com\/bielcardona\/TCGenerators}. \n\n\\paragraph{Exhaustive and sequential construction of networks in $\\mathcal{BTC}_n$.}\nWe have implemented  both the exhaustive and sequential construction of BTC networks with $n$ leaves. \nThe number of such networks increases very rapidly, and hence the exhaustive construction is not feasible for $n\\ge 8$. For $n\\le 6$ we generated all the networks in $\\mathcal{BTC}_n$; see Table~\\ref{number-btc} for the number of such networks. For $n=7$ we could not find in a reasonable time all the networks (in our implementation on a cluster of 32 CPUs it would have taken 15 days). Instead, we took uniform samples of networks in $\\mathcal{BTC}_6$ and computed their respective offspring, and repeated this procedure until the average number of offsprings per network stabilized up to 4 digits; this allowed us to give an estimate for $|\\mathcal{BTC}_7|$. \n\n\n\\paragraph{Random construction of networks in $\\mathcal{BTC}_n$.}\nWe have implemented the following construction, that does not generates networks uniformly, but is the closest we could get to it. We start with the network $N_1$ with a single node labeled by $1$. At each stage $i=1,\\dots,n-1$, we explicitly find all feasible pairs inside $N_i$ and choose at random and uniformly one of them to generate the network $N_{i+1}$. This procedure generates all possible networks in $\\mathcal{BTC}_n$, but not uniformly, since different networks over the same set of taxa may have different number of feasible pairs.\n\n\\paragraph{Computation of bounds for $|\\mathcal{BTC}_n|$.}\nFinally, we have implemented the recursive computation for the upper bounds of $|\\mathcal{BTC}_n|$ using the bounds for the offsprings of BTC networks found in Section~\\ref{sec:bounding-offspring}. The results for $n$ up to $10$ are given in Table~\\ref{number-btc}, where it is observed that, at least for small values of $n$, the true number of networks and the upper bounds have the same order of magnitude.\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{rrr}\n    \\toprule\n    $n$&$|\\mathcal{BTC}_n|$& upper bound\\\\\\midrule\n1 & 1 & 1 \\\\\n2 & 3 & 3 \\\\\n3 & 66 & 85 \\\\\n4 & 4,059& 7,442 \\\\\n5 & 496,710& 1,317,098 \\\\\n6 &             101,833,875 & 387,405,870 \\\\\n7 & $\\simeq$ 31,500,000,000 & 169,781,857,790 \\\\\n8 & ? & 103,409,407,515,286 \\\\\n9 & ? & 83,400,205,845,281,275 \\\\\n10 & ? & 85,947,517,732,640,544,027 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Exact number of BTC networks over $[n]$ for $n=1,\\dots,6$, an estimate for $n=7$, and their upper bounds for $n\\le10$. }\n  \\label{number-btc}\n\\end{table}\n\n\\section{Conclusion\\label{sec:conclusions}}\n\nThe main result of this paper is a systematic way of recursively generating, with unicity, all BTC networks with a given number of leaves. This procedure relies on a pair of reduction\/augmentation operations that generalize  analogous operations for phylogenetic trees. \nIndeed, given a (rooted, binary) phylogenetic tree over $[n]$, we can obtain a phylogenetic tree over $[n-1]$  by deleting the leaf labeled by $n$ and removing the elementary node that this deletion generates. Conversely, given a tree $T$ over $[n-1]$ and one of its nodes $u$, we can construct a tree over $[n]$ by simply hanging a pendant leaf labeled by $n$ to the single incoming arc of $u$. Since different choices for $T$ and $u$ give different trees over $[n]$, this gives  a recursive procedure to generate, with unicity, all binary rooted phylogenetic trees over a given set of taxa: we start with the leaf labeled by $1$, then we add the leaf labeled by $2$, then the leaf labeled by $3$ in all possible ways, and so on.\nBiologically, we can think of this procedure as follows: Once the evolutionary history of a given set of OTUs is correctly established\\footnote{In practice, we can never be sure that we got the correct tree, but here we suppose we do.} and  modeled by a phylogenetic tree, extending this evolutionary history  to consider a `new''  OTU $n$ consists in finding where to place $n$ in the tree, i.e. finding the speciation event that leads to the diversification of $n$.\n\nUnfortunately, when working with classes of phylogenetic networks, the removal of a single leaf (and of all elementary nodes created by this removal) does not necessarily give a phylogenetic network within the same class. In the case of BTC networks, we were able to find the minimal set of nodes that one must remove so that, after their deletion and that of all elementary nodes created by this removal, one gets a BTC network with one leaf less. \nAs in the case of trees, given a BTC network over $[n-1]$ and some set of nodes with certain restrictions (i.e. the feasible pairs $S_1$ and $S_2$) we can construct a BTC network over $[n]$ leaves, in such a way that different choices for the BTC network or for the feasible pair give different BTC networks over $[n]$. Hence, we find a procedure to recursively generate all BTC networks over a given set of taxa.\nBiologically, we can think of this procedure as an extension of what can happen when adding a new OTU $n$ to a phylogenetic tree: here the diversification of $n$ can involve a reticulated event (when $n$ is added as hybrid node) and  the ancestors of $n$  participate to  $|S_2|$ reticulated events, which were\nimpossible to detect before the introduction of $n$. \n\n\n\n\n\n\\section*{Bibliography}\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe recent commissioning of several laser facilities of multi-petawatt (PW) power, including APRI \\cite{APRI}, CAEP \\cite{CAEP} and SULF \\cite{SULF}, as well as the decisive progress in the construction of the ELI laser pillars~\\cite{ELI,P3-ELI}, belonging to the same multi-PW class, open the door to laboratory studies of the interaction between extremely intense light and matter in the new, so far unexplored regimes.\nThe transition from sub-PW to multi-PW laser beams implies the extension of the experimentally available peak laser intensity from the currently accessible range $10^{20}\\div 10^{21}{\\rm W\/cm}^2$ to $10^{22}\\div 10^{23}{\\rm W\/cm}^2$.\nFurthermore, the currently developing projects of even more powerful, sub-exawatt lasers sources \\cite{XCELS}, promise reaching intensities up to $10^{25}\\div 10^{26}{\\rm W\/cm}^2$.\nThese technical achievements will allow, in particular, to study in laser laboratories a number of new phenomena of classical and quantum physics, including radiation-dominated dynamics of isolated charges and plasmas, laser initiation of cascades of elementary particles, excitation of extremely strong magnetic fields, production of electron-positron pairs, etc. \nThe respective research fields and the current achievements received a detailed description in a set of review articles~\\cite{bulanov-rmp09,dipiazza-rmp12,fedotov-cp15}.\n\nIn order to provide a reliable correspondence between experimental data on the interaction of ultra intense laser radiation with matter and theoretical predictions, a detailed characterization of this radiation is required.\nThis includes, in particular, knowing the peak laser intensity value in the focus with reasonable accuracy.\nFor laser-induced and assisted phenomena, which may proceed in the strong-field regime, the dependence of observables on the laser intensity appears to be highly nonlinear so that a relatively small variation in its peak value may cause a considerable change in the response of the laser-irradiated target.\nThus, an accurate determination of the focal intensity distribution and the intensity peak value is crucially important to correctly interpret experimental results and connect them to theory.\nAt ultrahigh intensities, no direct measurement of the intensity distribution in the focus is possible with cameras or other similar detectors.\nExtrapolation of measurements made in the low-power mode to the high-power regime is questionable because of the various nonlinear effects happening throughout the amplifying and focusing systems.\nTherefore, the only reliable methods for characterization of the laser focus at extreme intensities can be those based on the observation of effects of laser-matter interactions sensitive to the laser parameters of interest. \n\nAs far as the measurement of intensity is concerned, several essential constraints limit possible interactions scenarios. \nFirstly, the interaction must not make any significant effect on the electromagnetic field distribution in the focus.\nThis immediately excludes dense plasma and solids as possible targets.\nFor this reason only low-density atomic gases or particle beams can be used.\nSecondly, the response should be sensitive more to the local value of intensity than to its global distribution in space and time.\nRecently, several effects have been discussed and numerically examined as possible tools for the determination of laser peak intensities in the ultrahigh-power regime.\nIn Refs.~\\cite{ciappina-pra19,ciappina-book19} sequential tunneling ionization of multielectron atoms has been considered.\nIt was shown that, owing to the highly nonlinear dependence of the tunneling ionization rate on the electric field strength of the laser wave, the maximal charge state of a given atomic specie produced in the laser focus appears highly sensitive to the peak value of intensity.\nThe same value is much less sensitive to the focal distribution of intensity and to the pulse duration.\nThis method roots back to the experimental work \\cite{walker-pra01,walker-josa03,yamakawa-pra03,yamakawa-jmo03} where the commonly known analytic formulas for the rate of tunneling ionization \\cite{ppt-jetp66a,pp-jetp67,popov-usp04} have been quantitatively verified by observing tunneling ionization of rare gases at intensities $\\simeq 10^{19}{\\rm W\/cm}^2$.\nAnother proposal, based on the observation of an intensity-dependent shift in spectra of Thomson scattering, has been experimentally tested in~\\cite{he-oe19}. \nIn this approach, the wavelength shifts proportional to the laser intensity, appear to be in fair agreement with estimates of the peak intensity extracted from images of the focal area, obtained at reduced laser power. \nThe original approach, based on the measurement of electron radiation spectra, could be extended further by using heavier particles, e.g.~protons. In this way, the applicability range broadens and the scheme could be able to gauge intensities in the range of $10^{25}$ W\/cm$^{2}$.\nFurthermore, a recent theoretical analysis \\cite{marklund-arx19} of angular distributions of radiation emitted by an ultra relativistic electron beam interacting with an intense laser pulse also identified features which can be used for precise determination of the intensity value.\nFinally, ponderomotive scattering of relativistic electrons has been theoretically examined in view of a more general task of the focus characterization \\cite{mackenroth-arx19}.\nAlthough this scheme involves several additional parameters characterizeing the initial and the scattered electron beams, it has also been shown potentially capable to measure the peak value of intensity in a laser focus.\nThese four schemes utilize essentially different physical effects, assume different interaction scenarios and therefore can be considered as independent and complimentary approaches to {\\em in situ} diagnostics of ultrahigh laser intensities.\n\n\nIn this paper, we analyze the atomic diagnostics introduced in \\cite{ciappina-pra19,ciappina-book19}.\nOur purpose is to check the sensitivity of the method with respect to two essential factors present in experiment and in the theory.\nFirstly, we check the sensitivity of our theoretical scheme to the model of tunneling ionization used to determine ionization rates which enter the system of rate equations for the populations of ionic charge states.\nSecondly, we examine the focal volume effect on the distributions in ionic charge and check to which extent this volume effect smears out the sharp off-set used as an indicator of the peak laser intensity.\nOur results show that at ultrahigh laser intensities of interest, the barrier suppression affects the charge distributions and can reduce the accuracy of the measurement.\nWe show that its influence minimizes for ionization of He- and H-like ions, and therefore the atomic specie should be selected such to reach the complete ionization in the central part of the laser focus.\nSimultaneously, we show that the sharp dependance of the ion yield on the peak intensity value survives the focal averaging.\nThese results allow us to classify the suggested atomic diagnostics as a quantitatively reliable scheme which can be realized in practice.\n\nThe paper is organized as follows.\nIn the next section we discuss the interplay between the tunneling and the barrier suppression ionization (BSI) regimes. \nSection 3 presents an analysis of the focal averaging effect and, finally, the last section contains conclusions.\nAtomic units are used throughout unless otherwise stated.\n\n\n\n\\section{Barrier-suppresion vs tunneling ionization}\n\nIn order to calculate the populations of different charge states in a strong laser field, we solve a system of rate equations (for more details see~\\cite{ciappina-pra19}), using tunneling expressions for sequential ionization rates known as Perelomov-Popov-Terentiev (PPT) formulas \\cite{ppt-jetp66a,pp-jetp67,popov-usp04,poprz-jpb14}.\nWithin the PPT theory, the time-dependent tunneling rate for a bound state with ionization potential $I_p$, orbital and magnetic quantum numbers $l$ and $m$, and residual charge $Z$ ($Z=1$ for ionization of neutral atoms) is given by \n\\begin{equation}\nw_{\\rm TI}(\\nu,l,m;t)=C_{\\nu l}^2B_{lm}I_pF^{1+|m|-2\\nu}(t)\\exp\\bigg\\{-\\frac{2}{3F(t)}\\bigg\\}~.\n\\label{w-lm}\n\\end{equation}\nHere\n\\begin{equation}\n\\nu=\\frac{z}{\\sqrt{2I_p}}~,\n\\label{nu}\n\\end{equation}\nis the effective principal quantum number and the coefficients $B_{lm}$ and $C_{\\nu l}$ are \n\\begin{equation}\nB_{lm}=\\frac{(2l+1)(l+|m|)!}{2^{2|m|}|m|!(l-|m|)!}~,~~~\nC_{\\nu l}^2=\\frac{2^{2\\nu-2}}{\\nu\\Gamma(\\nu+l+1)\\Gamma(\\nu-l)}~.\n\\label{BC}\n\\end{equation}\nFinally, the time-dependent reduced field $F(t)$ defined as:\n\\begin{equation}\nF(t)=\\frac{\\sqrt{{\\bf E}_L^2(t)}}{(2I_p)^{3\/2}}~,~~~F\\equiv {\\rm max}\\; F(t)=\\frac{E_0}{(2I_p)^{3\/2}}~,\n\\label{Ft}\n\\end{equation}\nwhere ${\\bf E}_L(t)$ is the laser electric field with an amplitude value $E_0$.\n\nThe choice of quasistatic rates is well justified by small values of the Keldysh parameter $\\gamma=\\sqrt{2I_p}\\omega\/E_0$ \\cite{keldysh,popov-usp04,poprz-jpb14}, which fall into the interval $\\gamma\\simeq 10^{-2}\\div 10^{-3}$ for ionization potentials $I_p\\simeq 2\\div 40$ keV, laser intensities $10^{20}\\div 10^{24}{\\rm W\/cm}^2$ and laser frequencies $\\omega\\approx 0.05$ a.u., i.e. parameters considered in \\cite{ciappina-pra19}. \nThe tunneling ionization rates are asymptotically exact in the limit $F\\to 0$ and remain quantitatively accurate in the domain $F\\ll 1$ \\cite{popov-usp04,poprz-jpb14}. \n\nAt the same time, applicability of the tunneling approximation for quasistatic ionization rates can be considerably restricted from the side of high field strengths. \nPractically, for ground states of atoms and positive ions, the tunneling ionization rate (\\ref{w-lm}) deviates from the respective exact value by $\\sim 10\\%$ at $F\\approx 0.02...0.03$.\nWith a further growing of $F$ the tunneling formula quickly become quantitatively inaccurate.\nThe physical reason for that is in the suppression of the potential barrier the electron tunnels through.\nThis suppression leads to a violation of the WKB applicability conditions resulting in a considerable disagreement between the PPT and the exact ionization rates. \nIn Ref.\\cite{ciappina-pra19}, a criterion of fast ionization was formulated assuming that an initially populated atomic level is fully exhausted in a single optical cycle. \nThis takes place at $F\\simeq F_*\\approx 0.05$ (see Eqs.~(13) and (14) there) and leads to the following empirical rule: an atomic level with a given ionization potential can hardly survive fields $F>F_*$.\nTherefore if the tunneling rate (\\ref{w-lm}) remains applicable up to $F\\sim F_*$ the scheme of \\cite{ciappina-pra19} is expected to be quantitatively reliable, otherwise effects of barrier suppression have to be included.\n\nFollowing the 1D model of a hydrogen-like ion and neglecting the bound state Stark shift we may estimate the barrier-suppression (BS) field as $E_{\\rm BS}\\approx I_p^2\/4Z$ which gives for the corresponding reduced field\n\\begin{equation}\nF_{\\rm BS}=\\frac{E_{\\rm BS}}{(2I_p)^{3\/2}}\\approx\\frac{1}{16\\nu}~.\n\\label{FBS}\n\\end{equation}\nFor the ground state of hydrogen, $F_{\\rm BS}=1\/16\\approx 0.063>F_*$ while for ${\\rm Kr}^{29+}$ with $I_p=3381$ eV and $\\nu=1.84$, $F_{\\rm BS}\\approx 0.033<F_*$.\nThis means that, at intensities when fast ionization takes place, the actual ionization rates can in general considerably differ from their tunneling asymptotics.\n\nThis deviation can be accounted for numerically or by several approximate analytic methods (for a review of this long-standing problem, see e.g.~Chapter 3 of \\cite{popov-usp04}, recent publications \\cite{kost-pra18,kost-arx19} and references there).\nHere we use an empiric formula proposed by Tong and Lin \\cite{lin-jpb05}, which has been shown quite accurate in the intermediate domain $F\\simeq F_{\\rm BS}$.\nAccording to \\cite{lin-jpb05} the BS correction reduces to an additional factor in the rate:\n\\begin{equation}\nw_{\\rm BSI}(\\nu,l,m;t)=w_{\\rm TI}(\\nu,l,m;t)\\exp\\bigg(-2\\alpha\\nu^2F(t)\\bigg),\n\\label{WBS}\n\\end{equation}\nwhere the fitting numerical coefficient $\\alpha$ is chosen in the interval $6...9$ \\cite{lin-jpb05}.\nDepending on the values of $\\nu$ and $\\alpha$, the factor $\\exp\\bigg(-2\\alpha\\nu^2F\\bigg)$ can vary from $\\sim 2\\div 3$ to $\\simeq 10$, at $F\\simeq F_{\\rm BS}$.\nThus, the influence of the BS effect on the value ${\\cal I}_{\\rm s}$ of saturation intensity (i.e., intensity at which the given ionic state is generated with the probability close to unity, see a precise definition below) could be numerically significant.\nTaking into account that no accurate analytic theory of BSI exists and the presently known models gain quantitatively different predictions (see, e.g. comparisons in \\cite{kost-pra18}), there is a danger that the numerically found values of intensity ${\\cal I}_{\\rm s}$, may appear model-dependent.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=8cm]{Fig1-new.png}\n\\end{center}\n\\caption{(color online) Ionization probabilities $W=2\\pi w\/\\omega$ per single cycle of a laser with wavelength $1\\mu$m versus the peak reduced field $F$, calculated for ionization of ${\\rm Kr}^{29+}$ with the electronic configuration $1s^22s^22p^3$ and ionization potential $I_p=3381$ eV (black lines) and of $1s^1$ ${\\rm Ar}^{17+}$ with $I_p=4407$eV (red curves). Thick solid lines show the tunneling probability obtained from (\\ref{w-lm}), dashed lines -- the BSI probability determined by (\\ref{WBS}) with $\\alpha=6$ and thin solid lines -- that with $\\alpha=9$. The chosen interval of $F=0.03...0.08$ corresponds to the intensity interval $4.85\\cdot 10^{20}...3.45\\cdot 10^{21}{\\rm W\/cm}^2$ for Kr and $1.07\\cdot 10^{21}...7.64\\cdot 10^{21}{\\rm W\/cm}^2$ for Ar. Vertical dotted lines indicate the values of BS field (\\ref{FBS}) equal to $0.033$ and $0.062$ correspondingly. A gray rectangular area shows the interval where the total probability per cycle changes from 0.1 (lower boundary) to 1.0 (upper boundary). Fast ionization occurs predominantly when the reduced field value falls into this interval.}\n\\label{Fig1}\n\\end{figure}\n\nCurves of Fig.~1, where the total probability $W$ of ionization per laser cycle is shown versus the peak reduced laser field $F$, allow to qualitatively estimate the significance of the BS factor in (\\ref{WBS}).\nThe main message of Fig.~1 is the relative insignificance of this factor for atomic states with $\\nu\\simeq 1$.\nFor ions of rare gases this corresponds to ionization of their $1s$ shell, i.e. of He- and H-like ions.\nIn this case the value of (\\ref{FBS}) $F_{\\rm BS}\\approx 0.06>F_*\\approx 0.05$ so that, except of extremely short laser pulses which can not be realized with $\\mu{\\rm m}$-scale wavelengths, such shells have enough time to be ionized in the regime of tunneling before the filed amplitude enters the BS domain.\nConsequently, the BS correction is only expected to slightly enhance the saturation intensity ${\\cal I}_{\\rm s}$.\nIn contrast, the ionization probabilities for outer shells with $\\nu\\approx 2\\div 3$ demonstrate a stronger sensitivity to the BS regime, because the bound state lies higher and therefore $F_{\\rm BS}<F_*$.\nFor such levels, the tunneling ionization process may happen too slow to fully strip the level out before the barrier is suppressed by the growing laser field.\nThus, for $2s,~2p$ and higher shells the value of ${\\cal I}_{\\rm s}$ can be considerably enhanced by the BS effect. \n\nIn order to quantify the effect of BSI on the value of ${\\cal I}_{\\rm s}$ we numerically solve the system of rate equations with the tunneling (\\ref{w-lm}) and BS (\\ref{WBS}) ionization rates for  argon and krypton in the intensity interval $10^{19}\\div 10^{22}{\\rm W\/cm}^2$.\nThe laser pulse shape and duration are the same as in \\cite{ciappina-pra19}:\n\\begin{equation}\nE(t)=E_0\\cos(\\omega t)\\sin^{2}\\left(\\frac{\\omega t}{2 N}\\right)~.\n\\label{Et}\n\\end{equation}\nIn our simulations we use $\\omega=0.0455$ a.u. and $N=10$, that correspond to a laser wavelength $\\lambda=1$ $\\mu$m and a total pulse length $T\\sim33$ fs, respectively.\nThe systems of rate equations for populations $0\\le c_n\\le 1$ of ions $A^{n+}$ are taken from \\cite{ciappina-pra19}.\n\n\\begin{figure}\n\\begin{center}\n\\leftline{\\includegraphics[height=10cm]{Fig2-final.pdf}}\n\\end{center}\n\\caption{(color online) Populations $c_n$ of ionic states at the end of the laser pulse (\\ref{Et}) calculated by solving numerically the system of rate equations for argon (left column) and krypton (right column) using the tunneling (first row) and BS ionization rates with $\\alpha=6.0$ (second row) and $\\alpha=9.0$ (third row). The initial populations are set $c_{14}(0)=1.0$, $c_{15}(0)=...c_{18}(0)=0$ for argon and $c_{26}(0)=1.0$, $c_{27}(0)=...c_{36}(0)=0$ for krypton. Vertical lines indicate the saturation intensities ${\\cal I}_{\\rm s}$ defined such that the corresponding $c_n=0.9$, for ${\\rm Ar}^{16+}$ (blue lines), ${\\rm Ar}^{18+}$ (red lines) and ${\\rm Kr}^{34+}$ (red lines). Dashed and dotted lines indicate the values of ${\\cal I}_{\\rm s}$ for the tunneling and BS ionization mechanisms correspondingly.}\n\\label{Fig2}\n\\end{figure}\n\nFigure 2 shows the intensity-dependent populations at the end of the laser pulse calculated for ionization cascades developing from ${\\rm Ar}^{14+}$ and ${\\rm Kr}^{26+}$.\nThese initial states were chosen different from those for neutral atoms to greatly reduce the number of rate equations in the systems.\nPractical independence of ionization cascades for high charge states on the initial condition was demonstrated in \\cite{ciappina-pra19}.\nThe value of saturation intensity ${\\cal I}_{{\\rm s}n}$ is defined such that $c_n({\\cal I}_{{\\rm s}n})=0.9$.\nThe BS effect is clearly seen.\nIt mostly reduces to the overall shift of the distributions toward higher intensities.\nAs suggested by the analysis above, for ionization of He- and H-like ions ${\\rm Ar}^{16+}$ and ${\\rm Ar}^{17+}$ this shift appears smaller than for other states where $\\nu\\approx 2$.\nTo quantify the BS effect we introduce a relative shift of the saturation intensity\n\\begin{equation}\n\\delta({\\rm A}^{n+})=\\frac{{\\cal I}_{{\\rm s}n}(\\alpha=9)-{\\cal I}_{{\\rm s}n}(\\alpha=0)}{{\\cal I}_{{\\rm s}n}(\\alpha=9)+{\\cal I}_{{\\rm s}n}(\\alpha=0)}\n\\label{delta}\n\\end{equation}\nFor parameters of Fig.2, Eq.(\\ref{delta}) gives $\\delta({\\rm Ar}^{18+})\\approx 0.07$, $\\delta({\\rm Ar}^{16+})\\approx 0.16$ and $\\delta({\\rm Kr}^{34+})\\approx 0.30$.\nMoreover, for ${\\rm Ar}^{16+}$ and ${\\rm Kr}^{34+}$ the shift in the saturation intensity $\\Delta{\\cal I}_{\\rm s}$ is comparable to the value of intensity interval where the number of ions of given charge grows by approximately one order in magnitude, while for ${\\rm Ar}^{18+}$ this value is approximately 3 times smaller.\nThese observations are in favor of using $1s$ states for determination of the peak intensity value.\n\n\n\\section{Focal volume effect}\n\nThe ionic signal collected by a time-of-flight (TOF) detector presents a distribution in charge states averaged over the whole focal volume or its significant part.\nThis space averaging affects both the distribution shape and the intensity dependence of the yield for a given charge state.\nHere we focus on the latter and calculate the number of ions produced in the laser focus using results obtained in Ref.\\cite{ciappina-pra19} and in Section 2 above.\nTaking into account that the accuracy of the method appears the highest for ionization of He- and H-like ions, we restrict the analysis by this case and use the pulse form (\\ref{Et}) of the previous section.\nIonic states populations $c_n({\\cal I})$ at the end of the laser pulse are given by plots of Fig.2 (see also Figs.4-7 in \\cite{ciappina-pra19}).\nUsing these data, one can calculate the total number of ions produced in the focus by replacing ${\\cal I}\\to{\\cal I}(\\mathbf{r})$, choosing a model for the focal intensity distribution ${\\cal I}(\\mathbf{r})$ and integrating over the focal volume:\n\\begin{equation}\nN({\\rm A}^{n+})=n_0\\int c_n({\\cal I}(\\mathbf{r}))d^3r~.\n\\label{N}\n\\end{equation}\nBelow we assume the Gaussian beam\n\\begin{equation}\n{\\cal I}(\\rho,z)=\\frac{{\\cal I}_0w_0^2}{w^2(z)}\\exp\\bigg(-\\frac{2\\rho^2}{w^2(z)}\\bigg)~,~~~w(z)=w_0\\sqrt{1+\\frac{z^2}{z_R^2}}~,~~~z_R=\\frac{\\pi w_0^2}{\\lambda}\n\\label{gauss}\n\\end{equation}\nwith $w_0$ and $z_R$ being the beam waist and Rayleigh length correspondingly, and the wavelength is taken $\\lambda\\simeq 1\\mu{\\rm m}$.\n\nUsing the numerically found populations $c_n({\\cal I})$ we calculated the number of ions ${\\rm Ar}^{17+}$ and ${\\rm Ar}^{18+}$ from (\\ref{N}) as functions of the peak intensity value ${\\cal I}_0$ for the case when the tunneling theory of ionization applies $(\\alpha=0)$.\nThe initial gas concentration is taken $n_0=2\\cdot 10^{12}{\\rm cm}^{-3}$ corresponding to the pressure $3\\cdot 10^{-5}$Torr at room temperature.\nAt these parameters, the number of atoms in the central part of the focus $N_0\\approx 3\\cdot 10^3$.\nResults shown on Fig.3 demonstrate a steep growth in the total number of ions by two orders in magnitude when the peak intensity increases by approximately factor of two, between $3\\cdot 10^{21}$ and $5\\cdot 10^{21}{\\rm W\/cm}^2$ for ${\\rm Ar}^{18+}$.\nBecause of the relatively small difference in ionization potentials of the $1s^2$ and $1s$ states of argon, the number ${\\rm Ar}^{18+}$ ions can exceed that of ${\\rm Ar}^{17+}$ because of the volume effect, that happens at ${\\cal I}_0\\approx 7\\cdot 10^{21}{\\rm W\/cm}^2$.\nIon yields for ${\\rm Xe}^{53+}$ and ${\\rm Xe}^{54+}$ are shown on the inset.\nThey demonstrate a similar steep two orders of magnitude growth in the intensity interval $(1.5\\div 2.0)\\cdot 10^{24}{\\rm W\/cm}^2$.\nThe yields become equal at ${\\cal I}_0\\approx 6.5\\cdot 10^{24}{\\rm W\/cm}^2$ (not shown on the plot).\nThis steep increase in the number of ions results from the simultaneous action of the focal volume effect and of the increase of the coefficient $c_n$ as shown on Fig.2. \n\n\\begin{figure}\n\\begin{center}\n\\leftline{\\includegraphics[height=8cm]{Fig3-final.pdf}}\n\\end{center}\n\\caption{Total number of ions $N({\\rm A}^{n+})$ produced in a Gaussian laser focus with waist $w_0=3\\mu{\\rm m}$ and concentration of neutral atoms $n_0=2\\cdot 10^{12}{\\rm cm}^{-3}$. The main plot shows the intensity dependence of $N$ for ${\\rm Ar}^{17+}$ and ${\\rm Ar}^{18+}$; analogous dependences for ${\\rm Xe}^{53+}$ and ${\\rm Xe}^{54+}$ are shown on the inset. A gray arrow indicates the intensity ${\\cal I}_0\\approx 7\\cdot 10^{21}{\\rm W\/cm}^2$ where $N({\\rm Ar}^{17+})=N({\\rm Ar}^{18+})$.}\n\\label{Fig3}\n\\end{figure}\n\nBoth the initial steep part of the ion yield curve and the intersection point can be used for determination of the intensity value. \nThe main advantage of the first case is that the steep part of the yield curve only weakly depends on the focal spot shape. \nThis becomes clear when we note that for intensities only slightly exceeding the ionization threshold, the ion signal comes from a small vicinity of the focus center (i.e. the point where the intensity reaches its peak value) where the intensity distribution can be described by a parabolic shape.\nFrom the other side, if the effectiveness of a time of flight detector, which depends in particular on the collection angle, is not precisely known, a direct correspondence between the number of ions shown on Fig.3 and the number of counts cannot be established.\nThen the intensity value can only be located within the interval corresponding to the steep slope of the yield.\nThis imposes the lower limit on the measurement uncertainly equal to $\\Delta {\\cal I}_0\\approx 2\\cdot 10^{21}{\\rm W\/cm}^2$ for argon and $\\Delta {\\cal I}_0\\approx 0.5\\cdot 10^{24}{\\rm W\/cm}^2$ for xenon which gives a $50\\%$ and $30\\%$ accuracy correspondingly.\nThe second method based on the determination of the intersection point, $N(A^{n+})=N(A^{n+1})$, can potentially provide a higher accuracy.\nHowever the intersection of the yields happens when the laser intensity exceeds its threshold value already in the significant part of the focal spot.\nAs a result, the position of this point can considerably depend on the focal shape introducing an additional source of uncertainty. \n\n\n\\section{Conclusions}\n\nIn conclusion, we presented a numerical analysis on the scheme for experimental determination of the peak value of intensity in a focus on a multy-petawatt power laser beam, based on tunneling ionization of multielectron atoms \\cite{ciappina-pra19}.\nOur analysis shows that:\n\\begin{itemize}\n\\item {The saturation intensity value ${\\cal I}_s$ appears sensitive to the barrier-suppresion correction to the ionization rate which makes the whole algorithm of  intensity extraction dependent on the model of BS ionization.\nThis dependence is however not crucially strong and appears minimal for ionization of He- and H-like ions with effective principal quantum numbers $\\nu\\approx 1$.\nThus such ions should be used as intensity markers.}\n\\item {The highly nonlinear dependence of the ion yield on intensity at ${\\cal I}\\approx {\\cal I}_{s}$ survives the focal averaging procedure and the steep part of the yield curve can be used to measure intensity with accuracy $30\\div 50\\%$.}\n\\item{A simultaneous determination of the intersection point where $N(A^{n+})=N(A^{(n+1)+})$ can provide some information on the intensity distribution in the focal spot.}\n\\end{itemize}\nThese results can be used to design experiments on characterization of extremely intense electromagnetic pulses delivered by the future multi-PW and exawatt laser sources. \n\n\n\\section*{Acknowledgments}\nAuthors acknowledge useful discussions with S.V. Bulanov, G. Korn, D. Kumar and S. Weber.\nSVP acknowledges financial support of the Russian Foundation for Basic Research via Grant No.19-02-00643 and of the Ministry of Education\nand Science of the Russian Federation via Grant No.3.1659.2017\/4.6. This work was also supported by the project Advanced research using high intensity laser produced photons and particles (No. CZ.02.1.01\/0.0\/0.0\/16\\_019\/0000789) from European Regional Development Fund (ADONIS).\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion} \\label{sec:conclusion}\n\nThis paper addresses the\nquestion of determining what is interesting for a specific user during an interactive  \nexploration of a multidimensional cube. To that extent, the paper draws a\nparallel with De Bie's Forsied framework \\cite{DBLP:conf\/ida\/Bie13}, and defines a subjective interestingness measure for a query as a ratio between the surprise expressed \nthrough this query and its complexity.\nDefining such a measure raises $3$ main challenges: ($1$) how to model and learn the prior user belief as a probability distribution\nto model surprise?\n($2$) How to efficiently recompute this prior belief after each user query? And, ($3$) how to implement a realistic Subjective Interestingness measure that captures the complexity of each query?\n\nOur measure definition takes advantage of\nthe specificities  of Business Intelligence explorations of multidimensional data cubes.\nWe represent the prior knowledge of a specific user as a directed graph of query parts that relies: on former users' explorations, as a proxy of what the current user might find interesting, on the cube schema, that indicates how prior knowledge is structured, and finally on this specific users' past activity. \nFinally, the user belief is derived from this query parts graph as the stationary probability distribution of a PageRank algorithm.\nExperiments conducted on simulated realistic user explorations\nor on real user explorations\nshow that the observed belief distribution and subjective interestingness values are aligned with prior knowledge on these datasets.\nThis first work on the definition of a subjective interestingness measure in Business Intelligence shows that query parts offer a reasonable proxy\nto learn an appropriate model of user belief and determine what is interesting in an exploration.\n\nHowever, in De Bie's framework, the belief is expressed on the extension of the data and not on the intention of the way of characterizing the data subgroup. A first major extension to our work will consist in proposing a belief and subjective interestingness measure on the value of each cube cells.\nSecond, we plan to investigate how to define such an interestingness measure for less structured databases, like relational (non multidimensional) databases, or data lakes.\nFinally, we aim at studying how user belief modeling and subjective interestingness measures combine with\nhigher level intentional languages to query and learn from the data \\cite{DBLP:conf\/dolap\/VassiliadisM18}.\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{sec:intro}\nBusiness intelligence (BI) exploration can be seen as an iterative process that involves expressing and executing queries over multidimensional data (or cubes) and analyzing their results,  to ask more focused queries to reach a state of knowledge that allows to answer a business question at hand. This complex task can become tedious, and for this reason, several approaches have been proposed to facilitate the exploration by\npre-fetching data \\cite{DBLP:conf\/dawak\/Sapia00},\ndetecting interesting navigation paths \\cite{DBLP:journals\/vldb\/Sarawagi01},\nrecommending appropriate queries based on past interactions \\cite{DBLP:journals\/dss\/AligonGGMR15} or by modeling user intents \\cite{DBLP:conf\/caise\/DrushkuALMPD17}. \n\nIdeally, such systems should be able to measure to which extent a query would be interesting for a given user prior to any exploration. Indeed, as illustrated in \\cite{IDEA:2018} and first elicited in \\cite{DBLP:conf\/kdd\/SilberschatzT95} in the context of Explorative Data Mining (EDM), the interestingness of a pattern\ndepends on the problem at hand, and, most importantly, on the user that extracts the pattern.\nAn interestingness measure for such explorative tasks should  therefore be\ntailored for a specific user.\n\nFollowing the idea of subjective interestingness measures initiated and developed by De Bie \n\\cite{DBLP:conf\/ida\/Bie13}, our aim is to measure\nthe subjective interestingness of a query a expressed by a coherent set of query parts,\n based on the prior knowledge that the user has about the cube and the cost for the user to understand the query and its evaluation. \n\n\nIt is therefore crucial, before reaching the definition of such an interestingness measure for BI, to be able to transcribe, with an appropriate information-theoretic formalism, the prior user knowledge, also called belief, on the data. \nDe Bie proposes to represent this belief as a probability distribution over the set of data. However, \nit is clearly not possible to explicitly ask a user about the degree of belief in each element composing their knowledge prior to each query, let alone\nidentifying on which element of knowledge expressing this probability distribution. This motivates the investigation of approaches for automatically estimating the user's belief\nbased on their implicit feedback.\nLet us now consider the following example to illustrate the difficulty of estimating probabilities for the belief.\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[scale=0.5]{img\/SSB_session-new.png}\n    \\caption{Toy SSB benchmark session}\n    \\label{fig:ssb}\n\\end{figure}\n\n\n\n\\paragraph{Example} Consider the  exploration  over the schema of the  Star Sche\\-ma Benchmark  \\cite{DBLP:conf\/tpctc\/ONeilOCR09}, consisting of $3$ queries, as illustrated in Figure \\ref{fig:ssb}, and loosely inspired by session 3 of the SSB's workload. \nFor the sake of readability, only the relevant query parts (grouping set, filters and measures) are shown.\nThis  example showcases the short session initiated by a user that explores the cube looking for information on revenue some company makes in different locations. \nAssume we are interested in recommending a query to the user\nfor continuing the exploration. This recommendation should be both\nconnected to the most used query parts, so as not to loose focus,\nbut also should bring new, possibly unexpected information, so as\nnot to feed the user with already known or obvious information.\nA naive solution would be to use the set of all possible query parts as the set of data and to express the belief based on the frequency of each query part in the past user history.\nFrom the session in Figure \\ref{fig:ssb} it is possible to compute the number of occurrences of each query part (for instance, SUM REVENUE appears 3 times, \nCUSTOMER.CITY 2 times, while SUPPLIER.REGION=AMERICA appears only once, etc.).\nHowever, this simple representation raises major problems: first, the vector of user belief computed from the number of occurrences will mostly contain\nzero values because the majority of users will concentrate their exploration to a certain region of the data cube. Second, this belief would not give any probability to query parts such as CUSTOMER.NATION=CANADA, while if user knows about \\textit{AMERICA} and \\textit{USA}, she is likely to have a basic knowledge about sibling countries to \\textit{USA} in the dimension \\textit{CUSTOMER.NATION}. Finally, it may also be taken advantage\nof  other users' former explorations, as a proxy of what the current user might find interesting.\n\nThis example stresses the need for an approach to define the belief based on the users' past activity, as well as an information about how knowledge is structured, which, in the case of the data cube, can be found in the cube schema.\nWe note that while previous works already investigated surprising data in cubes (see e.g., \n\\cite{DBLP:journals\/vldb\/Sarawagi01,DBLP:journals\/ijbidm\/CariouCDGGK09}), to the best of our knowledge none of them did so by explicitly modeling a user's belief.\n\nAs a first step in this direction, our previous paper \\cite{DBLP:conf\/dolap\/ChansonCDLM19} tracks user belief in BI interactions for measuring the subjective interestingness of a set of queries executed on a data cube. This approach builds a model of the user's past explorations that is then used to infer the  belief of the user about the query parts being useful for the exploration. \nContrary to the context of pattern mining \n\\cite{DBLP:conf\/ida\/Bie13} where in general no metadata  information is available, the query parts that are employed in this model cannot be considered agnostically of the cube schema,\nthat the user typically knows. In this context, the method introduced in \n \\cite{DBLP:conf\/dolap\/ChansonCDLM19}\n takes advantage of the schema to infer what a user may or may not know based on what has been already visited and what is accessible from the previous queries.\nThe  belief of a user is defined as a probability distribution over the set of query parts coming from the log of past activities and the cube schema. This probability distribution is learned as the resulting stationary distribution of a modified topic-specific PageRank algorithm, where the underlying graph topology matrix is based on previous usage and the schema of the cube, and where a teleportation matrix that corresponds to a specific user model is introduced to \nensure connectivity.\nFinally, \\cite{DBLP:conf\/dolap\/ChansonCDLM19} takes advantage of the artificial exploration generator CubeLoad \\cite{DBLP:conf\/caise\/RizziG14} that mimics several prototypical user behaviors to evaluate qualitatively and quantitatively divergences in the estimated user belief.\n\n\nThe work presented here improves on that introduced in \\cite{DBLP:conf\/dolap\/ChansonCDLM19} with several major contributions, both at the methodological level and at the experimental level: \n\\begin{itemize}\n    \\item it refines the belief model by taking into account the filter values of the query parts. This is a strong bottleneck since it adds a lot of vertices in the graph used to compute user belief. To solve this problem, new rules to build the graph are introduced in Section \\ref{sec:PRbelief} ;\n    \n    \\item as stated before, this new belief model benefits from information from the cube schema, the past logs and an actual user profile  represented by her explorations. All the underlying operations are now defined as simple graph manipulation primitives and allow for an easier understanding of the whole process ; reproducibility of the experiments \n   \n    is permitted thanks to the shared code repository\\footnote{\\url{https:\/\/github.com\/AlexChanson\/IM-OLAP-Sessions}} ;\n    \n    \\item according to these new construction rules, the approach now results in only one strongly connected graph which avoids the need for\n    a teleportation matrix and ensures a cleaner convergence mechanism. Details of the simplified PageRank algorithm is provided in Section \\ref{sec:pr_section} ; \n    \n    \\item as in \\cite{DBLP:conf\/dolap\/ChansonCDLM19}, we introduce a real valued parameter $\\alpha \\in [0,1]$ that allows to give more or less importance to the user specific exploration bias in the computation of the belief distribution. This parameter is at the core of our experiments as it allows to reveal, when set close to $1$, significant differences in belief and subjective interestingness models ;\n    \n   \n    \n    \\item our approach now contains a simple yet efficient incremental mechanism that allows to track the evolution of the belief during an exploration  as described in Section \\ref{sec:inc_belief} ;\n    \n    \\item finally, this paper introduces a first formalization of a subjective interestingness measure for Business Intelligence explorations based on the belief distribution and on a simple measure of the complexity of  a query formed by several query parts, as described in Section \\ref{sec:subj_interest_dopan}.\n\\end{itemize}\n\nAs in \\cite{DBLP:conf\/dolap\/ChansonCDLM19}, one difficulty stands in the evaluation of our proposal, as there is no ground truth available. In this context, we propose several experiments:\n\\begin{itemize}\n    \\item an updated qualitative and a quantitative evaluation of the belief distributions for several simulated user profiles using CubeLoad generator \\cite{DBLP:conf\/caise\/RizziG14} and  new experiments on real data with the DOPAN workload \\cite{DjedainiDLMPV19} ;\n    \\item a set of novel experiments that a posteriori estimates the subjective interestingness of queries in simulated and real explorations from CubeLoad and the DOPAN workload.\n\\end{itemize}\n\nExperimental conclusions show that our approach to model user belief and subjective interestingness from a graph of query parts: ($1$) behaves as expected on both prototypical \nuser behaviors and real user explorations, \nand ($2$) indicates that query parts are a good proxy to infer user belief.\n\n\nThis paper is organized as follows: Section \\ref{sec:example} motivates\nthe use of user belief and subjective interestingness measure in the context of \nBI exploration.\nSection \\ref{sec:preliminaries} introduces the concepts used in our approach:  formal definitions  BI explorations, query parts and concepts related to Subjective Interestingness and PageRank algorithm. Section \\ref{sec:IM} introduces the graph based user belief model using past explorations and schema as inputs, and introduces a novel algorithm to deal with incremental belief estimation. Section \\ref{sec:subj_interest_dopan} introduces the Subjective Interestingness definition based on said model. Finally, Sections \\ref{sec:tests} and \\ref{sec:test_si_dopan} present our experiments to assess  our belief model and our Subjective Interestingness measure, both on artificially generated  explorations and on real user explorations. Finally, Section \\ref{sec:related} discusses related work and \nSection \\ref{sec:conclusion} concludes and draws perspectives.\n\n\n\n\n\n\n\\section{Inferring user belief from schema and log usage}\n\\label{sec:IM}\n\nThis section presents our first contribution and addresses the following questions: ($1$) what is user belief in BI exploration? ($2$) How to estimate it? ($3$) How to make it evolve during an exploration?\n\n\n\\subsection{What is user belief in BI?}\\label{sec:beliefBI}\n\nIdeally, in the context of BI exploration, the user belief would be a probability the user attaches to the statement \"I believe the value of this cell is exactly this one\". Modeling such a belief is one of our long term perspectives, as such this would raise several questions such as: ($1$) how to ensure efficiency while executing all the queries to update the belief over cells? ($2$) How to cope with the combinatorial complexity of expressing the belief over subsets of cells rather than a single cell? ($3$) How to update the belief distribution over cells when observing  aggregated values at higher level of granularity? \nAs an example, Sarawagi in her seminal work \\cite{DBLP:conf\/vldb\/Sarawagi00} \nto estimate user belief,\nrestricts her study to the sum measure paired with an assumption of uniform distribution to locally estimate belief on cells when exploring aggregates.\nSarawagi's work relies on the following assumptions: belief is only expressed\nover a limited set of cells (relating to those already explored) and the\ncube instance and past query answers are available to estimate this belief.\n\n\n\n\nIn a first methodological step towards this ambitious direction, \nwe use query parts as proxies to estimate this user belief.\nWe consider in this work that the user belief is the importance  the user attaches to the statement \"I believe this query part is relevant for my exploration\". \nIn some sense, we consider query parts as\npieces of knowledge about the data that\nreduce the set of possible values it may take from the original data space, inspired by the De Bie's view of explorative pattern mining \n\\cite{DBLP:conf\/kdd\/Bie11,DBLP:journals\/ml\/KontonasiosB15} and as illustrated in Figure \\ref{fig:querypart_asrestriction}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[scale=0.3]{img\/cube_belief.png}\n    \\caption{Aligned with De Bie's framework, query parts can be seen as restrictions to the original data space in the case of an OLAP cube exploration}\n    \\label{fig:querypart_asrestriction}\n\\end{figure}\n\nWe propose to define the user belief over the set of query parts for the following reasons. First, the set of query parts is measurable (and usually reasonable in size) and thus respects the formal constraints in the model of De Bie \\cite{DBLP:conf\/ida\/Bie13} to extend the belief to an interestingness measure. \nSecond, database instance or query answers may not be available,\ne.g., for privacy or confidentiality reasons, when query logs are anonymized. Finally, query parts provide a finer level to work with compared to queries.\nWorking at the query level would end-up with a very sparse representation of the data space, as the probability that two queries occur in the same exploration is much lower than the probability that two query parts appear in the same query or exploration. \nMoreover, when considering query parts, the most interesting ones for the user may appear in several consecutive queries and thus might have more prominent probability values.\n\nAs we cannot \"brain-dump\" the user, the belief is approximated by the importance of the available query parts. The challenge lies in a way to find this probability distribution over a possibly infinite or too large set of query parts even if we restrict to the attributes in a given schema.\nPractically, in order to avoid to deal with all these query parts, we restrict to those appearing  in a query log or in the schema, where only the active domain of the attributes is considered. \nSubjectiveness is ensured by the importance attached to the query parts appearing of the user's log of former explorations.\n\n\n\n\n\n\\subsection{Using PageRank as a belief function}\n\\label{sec:PRbelief}\nOnce restricted the set of query part\n, we still need to compute their relative importance expressed as a probability distribution for a specific user. As explained previously, this is done by a PageRank (PR) algorithm that computes the probability \nfor a user $u$ to end up on a query part $p$ when using the cube schema during the exploration, knowing past explorations by other users and knowing the profile of $u$.\nA naive assumption that could be made on this set of parts for an initial background distribution is that all parts not seen by the user are equally probable and those seen are as probable as often they appear in the user's log. This  would ignore many behaviors evidenced in the user explorations and connections of parts in  the schema of the cube. Our approach  incorporates those elements.\n\n\n\n\nGiven a database schema $D$ with query parts $P_D$,\nthe input to the PageRank algorithm is \na directed graph of query parts $G = \\tuple{P_D,E}$,\n computed by Algorithms \\ref{algo:graph_building_schema}, \n \\ref{algo:graph_building}\nand  \\ref{algo:eval}\nrespectively, as detailed in the following paragraphs.\n\n\nNote that compared to our preliminary work \\cite{DBLP:conf\/dolap\/ChansonCDLM19}, \nwe use a more elaborated technique for building the graph.\nIn particular, we now consider as vertices the filter values (i.e., members) of potential selection predicates instead of the hierarchy levels on which they apply.\nThis brings richer information about the data in our approach without changing the overall method described in \\cite{DBLP:conf\/dolap\/ChansonCDLM19}. The relationships between the selection predicates and their associated levels in the hierarchy are conserved but transcribed into the edges (see below and in Algorithm \\ref{algo:graph_building}).\n\n\\input{graph_building.tex}\n\\paragraph{Schema based construction rules} \nTo represent the global topology induced by a database schema $D$, a  graph is constructed as follows,:\n(i) there is an edge between any two consecutive levels of a hierarchy ;\n(ii) for any there is an edge between a member and its direct children in the hierarchy of this member ;\n(iii) finally, there is an edge between each member  and its\nlevel attribute.\nDetails about the implementation of these rules are provided in Algorithm \\ref{algo:graph_building_schema}.\n\n\\paragraph{Log usage construction rules} \nTo represent the activity of a user or group of users, a graph can be constructed as follows.\nThere is an edge $e$ from query part  $p_1$ to query part $p_2$ and an edge $e'$ from $p_2$ to $p_1$ if $p_1$ and $p_2$ appear together in the same query. There is also an edge $e$ from query part $p_1$ to query part $p_2$ if $p_1$ is in a query that precedes, in an exploration, another query where $p_2$ appears. As described in Algorithm \\ref{algo:graph_building}, those rules\n can be applied either to generate the graph of all users past queries or to produce the graphs of a specific user by restricting the log used as input. Note that  Algorithm \\ref{algo:graph_building} can either be used to update a pre-existing schema graph or, if the input graph is set to an empty graph, only build a new graph related to usage.\n \n\n \n\\paragraph{Introducing subjectivity}\nAlgorithms \\ref{algo:graph_building_schema} and \\ref{algo:graph_building} can be used to construct a graph that represents a general topology of the query space \n(Algorithm \\ref{algo:graph_building_schema})\nand transcribes important relationships established by the past users explorations (Algorithm \\ref{algo:graph_building}, called with a query log detailing the past activities of all users). This graph $G_t$,\ncalled the {\\em topology graph} from now on,\nis  however not subjective in any way. It has to be biased toward a specific user $U$, represented by the subset  of the query parts occurring in their sessions.\nTo this end, Algorithm \\ref{algo:graph_building} can be called\nover the query log of user $U$, and  defines $G_u = \\tuple{P_D, E_u}$ called the {\\em specific subjective user centered graph}.\n\n\n\n\\paragraph{Constructing the PageRank graph}\nOnce we have a graph representing the topology induced by the schema and the past logs graph, $G_t$, and a specific subjective user centered graph, $G_u$, we can aggregate them to produce the graph that will serve as an input for the PageRank algorithm described in Section \\ref{sec:pr_section}. To that aim, Algorithm \\ref{algo:merge_graph} introduces a real parameter $\\alpha \\in [0,1]$ that  allows to give more or less weight to graph $G_u$ compared to the topology graph $G_t$. Indeed, the topology graph is generally very large and the subjective user centered graph only modifies a small portion of it which may be barely noticeable in terms of belief distribution. In that sense, $\\alpha$ can be seen as a normalization factor, as it can be used to control the relative importance of the user's log against the general log and topology inherited from the schema.\n\n\\input{merge_graph_algo.tex}\n\n\n\n\n\n\n\nFinally, the probability distribution over the set of query parts $P_D$ is computed as the PageRank vector on the graph resulting from Algorithm \\ref{algo:merge_graph}. \nThis vector is obtained by an iterative approach that converges after a sufficient number of iterations.\n\n\n\n\n\n\\paragraph{About the connectivity of the graph} This  approach has the advantage to produce a graph with only one connected component under the weak assumption that the log contains at least one query for each measure (i.e. each measure must appear in the log at least once),\nwhich is a direct consequence of graph construction by Algorithms \\ref{algo:graph_building_schema} and \\ref{algo:graph_building}.\nThis property is crucial since it allows to simplify the Topic Specific PageRank algorithm used in \\cite{DBLP:conf\/dolap\/ChansonCDLM19} into a more conventional PageRank algorithm as introduced in Section \\ref{sec:pr_section}. Moreover, the construction of the\naggregated graph that represents the logs, the schema and a specific user allows to tune the system to give more weight to any of these aspects.\n\n\n\\subsection{Incrementality of belief}\\label{sec:inc_belief}\n\nContrary to most implementations of De Bie's framework, we don't enumerate possible patterns to recommend to the user. These implementations are able to update the beliefs assuming that each previous pattern has been seen and understood by the user without any user interaction. Instead, we recompute the estimated user beliefs as the user makes new queries which in turn will allow to recompute the Subjective Interestingness as described in Section \\ref{sec:subj_interest_dopan}. In our experiments, we develop an a-posteriori method which quantifies the subjective interest of queries at any point of the exploration according to the previous queries that we know were launched by the user.\n\nLet $\\tuple{s,u}$ be a session defined as a sequence $s$ of queries and the user $u$ that produced this session and let $G_s = \\tuple{V_s, E_s}$ be the \\textit{active session graph}. The latter is constructed \nwith \nAlgorithm \\ref{algo:graph_building}, {\\tt BuildLogGraph}, \nwith parameter $L$ \nrestricted to\nthe queries executed\nat some point of \nthe user session $s$. By applying  Algorithm \\ref{algo:graph_building} iteratively each time a new query is issued in session $s$,  with input parameter $G_s$ and a log restricted to the new query,  an updated version of $G_s$ is obtained.\n\n\n\n\nThen,  Algorithm \\ref{algo:merge_graph}\nis applied to aggregate\nthe updated graph  $G_s$  with the topology graph $G_t$ \nusing the {\\tt merge($G_t$, $G_s$, $\\alpha$)} method \nand the value  of  $\\alpha$ to control how much of the active session influences the computation of the PR.\n\n\n\n\n\nFinally, executing the PR algorithm on this aggregated graph leads to the updated belief distribution. Algorithm \\ref{algo:eval} shows how this incremental computation of belief is implemented to define the expected subjective interestingness measure.\n\n\n\n\\input{eval_algo.tex}\n\n\n\n\n\n\n\n\n\n\n\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{Preliminaries}\\label{sec:preliminaries}\n\nThis section introduces the basic definitions of our framework.\n\n\\subsection{BI explorations}\n\nOur work considers BI explorations,\ni.e., sequences of OLAP queries over a \ndatabase instance under a star schema, called datacube.\n\n\n\n\nLet $D$ be a database schema, $I$ an instance of $D$ and $Q$ the set of formal queries one can express over $D$. For simplicity, in this paper, we consider relational databases under star schemata, queried with multidimensional queries.\nLet $A$ be the set of attributes of the relations of $D$. Let $M \\subset A$ be a set of attributes defined on numerical domains called measures.\nLet \n$H = \\{h_1 , \\ldots, h_n \\}$ be a finite set of hierarchies, each characterized by (1) a subset $Lev(h_i)$ of attributes called levels and (2) a roll-up total order $\\succeq_{h_i}$ of $Lev(h_i)$. \nWe denote by $adom(A)$ the set of all constants appearing the instance $I$ of $D$ for attribute $A$.\nFor each hierarchy $h_i$, $Lev(h_i)$ includes\na top-most level $l_i^{ALL}$ such that $\\nexists l\\in Lev(h_i), l_i^{ALL} \\succeq_{h_i} l$. \nThis level only has one value called $all_i$, i.e., $adom(l_i^{ALL})=\\{all_i\\}$.\nFor any two consecutive levels $l^1_i, l^2_i$ of a hierarchy $h_i$,\nfunction $children(m)$ applied to $m\\in adom(l^1_i)$ returns\nthe set of values in $adom(l^2_i)$\nthat are direct children of $m$ according to $h_i$.\n\nTo simplify, we describe an OLAP query $q$ in $Q$ as a set of query parts.  Note that the term query parts can undergo different meanings. Coherent with our objective of taking into account both usage (i.e., previous explorations) and cube schema, our query part definitions encompasses both.\nWe rely on the definition of query part provided by \\cite{DBLP:conf\/caise\/RizziG14}, where the authors consider it is one\nconstituent of a \n\\begin{quote}\nmultidimensional query consisting of (i) a group-by (i.e., a set of hierarchy levels on which measure values are grouped); (ii) one or more measures whose values are returned (the aggregation operator used for each measure is defined by the multidimensional schema); and (iii) zero or more selection predicates, each operating on a hierarchy level.    \n\\end{quote}\n\nHowever, in our case, a query part is not necessarily  attached to a query already expressed by some user,  since we aim at considering also query parts that might be used in the future.\n\nFormally, a query part is either (i) a level $l$ of a hierarchy in $H$, (ii) a  measure in $M$, or the member $v$ of a simple Boolean predicate of the form  $l = v$, where $l$ is a level of a hierarchy in $H$, and $v$ is a constant in $adom(l)$. \nNote that each member $v$ identifies its level and hierarchy.\nGiven a database $D$  we call $P_D$ the set of query parts.\nIn what follows, queries are confounded with their sets of query parts, unless otherwise stated, and we assume a function $parts(q)$ that applied over a query $q$ returns the subset of $P_D$ \ncontaining its query parts.\n\n\n\n\n\n\n\nFinally, a BI exploration $s$  is a sequence $[q_1, \\ldots, q_p]$  of $p$ OLAP queries, and a log is a set of explorations.\n\n\n\n\n\n\n\\subsection{Interestingness for exploratory data mining} \\label{sec:DeBie}\n\nThe framework proposed by De Bie \\cite{DBLP:conf\/ida\/Bie13}, in the context of exploratory data mining, is based on the idea that the goal of any exploratory data mining task is to pick patterns that will result in the best updates of the user's knowledge or belief state, while presenting a minimal strain on the user's resources. \nIn De Bie's proposal, the belief is defined for each possible value for the data from the data space and can be approximated by a background distribution.\n\nAs a consequence, a general definition for this interestingness measure (IM) is a real-valued function of a background distribution, that represents the belief of a user, and a pattern, that is to say the artifact to be presented to the explorer. Given a set $\\Omega$, the data space, and a pattern $\\Omega'$ a subset of $\\Omega$, the belief is the probability P$(\\Omega')$ of the event  $x \\in \\Omega'$, i.e., the degree of belief the user attaches to a pattern characterized by $\\Omega'$ being present in the data $x$. In other words, if this probability is small, then the pattern is subjectively surprising for the explorer and thus interesting. In this sense, the IM is subjective in that it depends on the belief of the explorer. De Bie also proposes to weight this surprise by the complexity of the pattern $\\Omega'$ as follows:\n\\begin{equation}\\label{eq:IM_DeBie}\n    IM_{De Bie}(P,\\Omega') = \\frac{-log(P(\\Omega'))}{descComp(\\Omega')}\n\\end{equation}\nwhere $P$ represents the user belief, i.e., the background distribution of the pattern $\\Omega'$ over the set of data $x$ and $descComp(\\Omega')$ denotes the description complexity of a pattern $\\Omega'$.\n\nThe data mining process consists in extracting patterns and presenting first those that are subjectively interesting, and then refining the belief background distribution based the newly observed pattern $\\Omega'$.\nThe key to such modeling as proposed by De Bie lies in the definition  of the belief of each user for all possible patterns and how it should evolve based on new patterns explored during time. \n\n\n\\subsection{PageRank}\n\\label{sec:pr_section}\nInitially, the PageRank algorithm is designed to estimate the relative importance of web pages as a probability to end up on this web page after an infinite surf on the web \\cite{DBLP:journals\/cn\/BrinP12}. This algorithm is based on ergodic Markov Chains. A Markov Chain models a collection of states and transitions probabilities from a state to the next. At a particular time t, a random walker is assumed to be in a unique state of the chain. We are interested by the probability distribution over the states after some time. After an infinite amount of time, this distribution  is called the stationary distribution, if it exists. As the transitions probabilities themselves cannot change with time, these chains can be represented as simple directed graphs with the different possible states as nodes and the transitions as edges with their assigned probabilities. A Markov Chain can also be represented as an (N,N) matrix, where N is the number of possible states. A Markov Chain that is aperiodic and where all states are connected with all other states by a sequence of states whose transitional probabilities are not zero has the ergodic property. In other words, it is possible to reach any state from any initial state with enough time. This implies that a stationary distribution over the possible states exists and is unique after an infinite number of iterations, independently of the starting state.\n\nIn the classical Page Rank algorithm used for the web, the pages are the possible states of the Markov Chain and the hyperlinks are the transitions between states.\nThe transitional probabilities from a particular page are proportional to the number of hyperlinks targeting a common page. Thus, to rank the pages by popularity, one needs to find the stationary distribution over the pages. These are the probabilities of landing on a page after surfing for a long time, starting from any page.\nThe stationary distribution, called the PageRank vector $PR$, is the solution to the following equation:\n\\begin{equation}\n\\label{eq:PR}\nPR =  M \\times PR\n\\end{equation}\nwhere $M$ is the stochastic transition matrix of the graph of web pages hyperlinks.\n\n\nOur approach considers query parts as\nstates in a Markov chain and\nSection \\ref{sec:PRbelief} explains how\ntransitional probabilities are defined.\n\n\n\\section{Related Work} \\label{sec:related}\n\n\n\n\n\n\nOur work deals with subjective interestingness and how to define such a measure by learning a belief distribution from users' past activities in the context of Business Intelligence. This section  presents some interestingness measures, and how they have been used in the context of recommendation and exploratory data mining.\n\nDefining a good Interestingness measure has kept interested researchers for a long time in the context of data mining. Indeed, there exists numerous tasks, for example in pattern mining, for which it is critical to be able to filter out uninteresting patterns such as item sets or redundant rules, to control the complexity of the mining approaches and increase their usability. \n\nIn \\cite{Brijs2004DefiningIF, DBLP:journals\/csur\/GengH06}, the authors identify two main types of interestingness measures. Objective measures are based only on data, which corresponds to quality metrics such as generality, reliability, peculiarity, diversity and conciseness. For instance, directly measurable evaluation metrics such as support confidence, lift or chi-squared measures in the case of  association rules \\cite{BostonCollege-Alvarez-2003}. \n\nOn the contrary, subjective measures consider both the data and the user and characterize the patterns' surprise and novelty when compared to previous user knowledge or expected data distribution. The first work on the topic of subjective interestingness is certainly \\cite{DBLP:conf\/kdd\/SilberschatzT95} that is restricted to the pattern mining domain. In \\cite{DBLP:conf\/ida\/Bie13, IDEA:2018}, the author extends this notion to any explorative data mining task and represents interestingness as a ratio between information content and complexity of a discovered pattern being it an itemset, a cluster or a query evaluation result (see Section \\ref{sec:DeBie} for more formal details). In \\cite{IDEA:2018}, De Bie defines the subjective interestingness as a situation where a \\begin{quote}\n    ``user states expectations or beliefs formalized as a 'background distribution'. Any 'pattern' that contrasts with this and is easy to describe is subjectively interesting''.\n\\end{quote}\n\nThe authors in \\cite{DBLP:journals\/csur\/GengH06} consider also semantic measures of interestingness, based on the semantics\nand explanations of the patterns like utility and actionability. This latter property of actionability is not meaningful in our case where, as stated by De Bie \\cite{IDEA:2018}, we consider situations \\begin{quote}\n``where the user is interested in exploring without a clear anticipation of what to expect or what to do with the patterns found''.\n\\end{quote}\n\nDe Bie's framework \\cite{DBLP:conf\/ida\/Bie13} is usually used to model user belief about some \\textit{data}. The hypothesis is that the user has beliefs about all of the data and is interested by anything that is surprising according to her beliefs. This will usually be a piece of data whose properties greatly contradict the user's prior belief. This work is very similar to what Sarawagi did for multidimensional data exploration \\cite{DBLP:journals\/vldb\/Sarawagi01}. In her work, the user's previous observations about parts of the data is used to estimate the most probable cube instance in the user's brain. A dissimilarity to the actual cube instance data is then computed in order to recommend surprising subsets. Both approaches compute a kind of information gain conditioned by the knowledge of what has been seen by the user while he explored the data. This gives the system the ability to suggest the best action that will provide the most information out of the data to the user.\n\nModelling the probability over the user \\textit{intents} would be another subjective approach. This is well illustrated by the Bayesian Information Gain method, as shown in various works by Wanyu \\cite{DBLP:conf\/chi\/LiuDBR17} for instance. With this approach, the system makes hypothesis about the user's goal and test them by subjecting her to experiments. The goal is to find the experiment which would yield most information about the user's goal to the system, to help her reach it faster. Both approach seems quite complementary as De Bie's approach makes possible query recommendation through \\textit{interesting data discovery} and Wanyu's approach instead allows similar recommendation through \\textit{intent discovery}. \\nico{Interestingly, these intents are probability distribution over elements of knowledge, while other works have focused on capturing long or short term intents related to topics for data exploration such as \\cite{DBLP:conf\/caise\/DrushkuALMPD17}.}\n\nRecently, in \\cite{DBLP:conf\/icde\/PuolamakiOKLB18} the authors propose a data exploration study based on De Bie's FORSIED framework \\cite{foresied, IDEA:2018} that pairs a high level conjunctive query language to identify groups of data instances and expresses belief on some real-valued target attributes, based on location and spread patterns. This work is close to our proposal but expresses belief\non a summary of the data.\n\nIn general, most of recent work on De Bie's framework \\cite{DBLP:conf\/icde\/LijffijtKDPOB18, DBLP:journals\/ml\/LeeuwenBSM16} instantiate his framework to discover subjectively interesting pattern for different kind of data spaces. As a consequence, papers detail new pattern syntax as well as other statistics computed on the pattern extension. For example, in \\cite{DBLP:journals\/ml\/LeeuwenBSM16}, De Bie's uses his framework to find subjectively interesting dense graph patterns: each pattern is a set of nodes and the statistic is the average degree of the edges in the pattern's nodes. In \\cite{DBLP:conf\/icde\/LijffijtKDPOB18}, the pattern used are subgroups of instances described by conjunctions of categorical descriptive attributes, while the statistics are the mean and the covariance of the subgroup for any number of real-valued target attributes.\\\\\n\n\nIn the context of data cube exploration, to the best of our knowledge there is no final and consensual  interestingness measure or belief distribution elicitation method, while there exists  measures that are closely related. Measures have been defined as unexpectedness of skewness in navigation rules and navigation paths \\cite{KUMAR2008} and computed as a peculiarity measure of asymmetry in data distribution \\cite{KLEMETTINEN1999}. In \\cite{DBLP:conf\/sbbd\/FabrisF01}, the authors define interestingness measures in a data cube as a difference between expected and observed probability for each attribute-value pair and the the degree of correlation among two attributes. \nIn \\cite{DBLP:conf\/vldb\/Sarawagi00}, Sarawagi describes a method that profiles the exploration of a user, uses the Maximum Entropy principle and the Kullback-Leibler divergence as a subjective interestingness measure\nto recommend which unvisited parts of the cube can be the most surprising in a subsequent query.\n\n\nIn \\cite{DjedainiDLMPV19, DBLP:conf\/adbis\/DjedainiLMP17} the authors use supervised classification techniques to learn two interestingeness measures for OLAP queries: ($1$) focus, that indicates to what extent a query is well detailed and connected to other queries in the current exploration and ($2$) contribution that indicates to what extent a query contributes to the interest and quality of the exploration.\n\nFinally, interestingness and related principles have been studied in the context of recommendation but more widely used for evaluation rather than the recommendation itself \\cite{DBLP:journals\/tiis\/KaminskasB17}. Interestingness is reflected based on $4$ main criteria such as diversity, serendipity, novelty, and coverage, in addition to traditional accuracy measures. \n\nIn the context of OLAP query recommendation, several recommendation algorithms have been proposed that take into account the past history of queries of a user either based on a Markov model \\cite{DBLP:conf\/dawak\/Sapia00} or on extracted patterns \\cite{DBLP:journals\/dss\/AligonGGMR15}. Noticeably, \\cite{DBLP:journals\/dss\/AligonGGMR15} quantifies how distant is the recommendation from the current point of exploration to evaluate the interestingness of each candidate query recommendation.\n\n\n\n\\section{Our vision of User Centric Data Exploration}\n\\label{sec:example}\n\n\nThis section describes how the knowledge of a user belief, and by extension a subjective interestingness measure, could be used to improve the user's experience in the context of interactive data exploration. \nThis example highlights the main scientific challenges of such task, some of them being left as future work as the present paper exclusively focuses on a first expression of user belief and a derived subjective interestingness measure in the context of data cube exploration.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[scale=0.5]{img\/recommendation_interestingness}\n    \\caption{Envisioned use of belief and subjective interestingness measures in data exploration\n   \n    }\n    \\label{fig:reco_interestingness}\n\\end{figure*}\n\nIn our vision, illustrated in Figure \\ref{fig:reco_interestingness},\nhuman remains in the loop of data exploration,\ni.e., the exploration is not done fully automatically,\nbut we aim at making it less tedious.\nAll users, naive or expert,\nwilling to explore a dataset,\nexpress their information need \nthrough an exploration assistant.\nThis assistant is left with the task\nof deriving from the user's need the\nactual queries to evaluate over the\ndata source.\nThis exploration assistant communicates\nwith a belief processor that \nis responsible for the maintenance\nof the user's profile, i.e., a model\nof that user, in the sense that \nit includes an estimation  of the actual \nbelief unexpressed by the user.\nThis belief is manifold and concerns e.g., \nhypotheses on the value of the data,\nthe filters to use, how the answer should be\npresented, etc.\nThe belief processor  activates a series of \nsubjective interestingness measures that drives\nthe query generator for deriving \nand recommending the most\ninteresting queries for this user, in the sense that\nthey produce relevant, unexpected, diverse answers, avoiding \nundesirable artifacts such as  \nbiased or false discoveries, \nthe so-called cognitive bubble trap, etc.\nThese answers and recommendations are packaged \n(e.g., re-ranked, graphically represented)\nby the storytelling\nprocessor before being displayed to the user and\nsent to the belief processor for profile updating.\n\nNotably, thanks to the belief processor,\nonce enough diverse users are modeled, \nthe storytelling processor may \ncope with the  cold start problem of \ngenerating recommendation\nfor unknown users \n(the future user of Figure \\ref{fig:reco_interestingness}),\ne.g., by removing bias introduced by\ncommon beliefs.\n\n\n\nThe work presented in this paper is the first step in the implementation of this vision. We first concentrate on cube exploration, expressing the belief over query parts and deriving an incremental Subjective Interestingness measure on queries. Noticeably, all our definitions take advantage of the peculiarities of the data cube exploration context to be on par with what a human analyst would consider interesting.\n\n\n\\section{Exploration evaluation based on Subjective Interestingness}\n\\label{sec:test_si_dopan}\n\n\nWe use  Algorithm \\ref{algo:eval} to compute the subjective interestingness (SI) on the explorations of our two datasets. \nThe aim of these tests is to show how SI\nrelates to prototypical user behaviors (Cubeload use case) \nand to real user explorations  (DOPAN use case).\nWe start by describing the protocols for the two use cases and then\ncomment our experimental results.\n\n\\subsection{Experimental protocol}\n\nIn both use cases, we use $\\alpha=0.9$ for Algorithm \\ref{algo:eval}\nto compute the SI incrementally for each explorations, to better account\nfor the user (simulated or real) behavior.\nWe now explain the difference in protocol for the two uses cases.\n\n\n\\paragraph{CubeLoad} \nExplorations generated with CubeLoad correspond to prototypical\nbehaviors (profiles) of users navigating a datacube.\nFor our first experiment, we generate $50$ explorations for each different CubeLoad profile. \nWe first plot the accumulated number of unique query parts used at each moment of the exploration to understand how our complexity measure behaves.\nWe then run  Algorithm \\ref{algo:eval} \non each explorations to compute the subjective interestingness per query. For each  profile, we isolate the current explorations from the others generated. All the other explorations of the same profile will be used as a user past log. The results are finally aggregated per profile and query position in the exploration, to compute the mean and the standard deviation of SI. We display the results in the form of a line plot with error bars representing the standard deviation. The query position on the $x$ axis represents different moments of the exploration. Each line is a cubeload profile representing the mean behavior of all explorations generated using this profile. Our aim is to see how SI behaves along the explorations and allows to characterize the prototypical profiles.\n\n\n\n\\paragraph{DOPAN} For the DOPAN use case, we have at our disposal $32$ real user  explorations.\nWe will focus on the 22 explorations over the first cube of DOPAN, since SI cannot be compared across cubes with different schemas.\nContrary to the simple SSB cube, that one has 32 measures and 19 dimensions,  making it a much larger space to explore.\nOur protocol is a bit different than the one used for CubeLoad since we have not classified the explorations in how they follow a particular pattern.\nHowever, each exploration has been tagged by professors with labels B, C or D, depending on analyst's skills. Label D corresponds to good explorations, clearly following an information need, investigating it and containing coherent queries. Students producing such explorations are considered to have analysis skills. Contrarily, label B denotes those of the students that produced poor explorations, with less contributive queries, typically switching topics, with no clear information need. Label C corresponds to students that are learning analysis skills, but still produce middle-quality explorations.\n\nLike for the CubeLoad use case, we first plot the accumulated number of unique query parts used at each moment of the exploration.\nWe run Algorithm \\ref{algo:eval} on each exploration to compute SI per query, using, for the user history to build $G_s$, an exploration consisting of \nthe past queries of the exploration and, whenever possible, the queries of other explorations of that user. Note that users have done different number of explorations, each of different sizes, and therefore our protocol faithfully represent genuine user activities.\nWe use the same line plot as for CubeLoad to show the SI of each query of the explorations, to analyze how SI varies in the exploration and detects instantaneous analysis behavior. \nWe then group explorations per skill labels and plot the mean and confidence interval at 95\\% of SI for the 3 groups, again with line plot.\nWe finally compute the rank correlation between SI and the \nskill.\n\n\n\n\n\\subsection{Results}\n\n\\subsubsection{CubeLoad use case} \n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.23]{cubeload_results\/unique_query_parts_skill_index.png}\n    \\caption{Cumulated number of unique query parts by CubeLoad template for each query index in the explorations}\n    \\label{fig:cubeload-queryPartNumber}\n\\end{figure}\n\n\nFigure \\ref{fig:cubeload-queryPartNumber} shows the accumulated number of unique query parts used at each moment of the exploration. This measure increases monotonically since each newly generated query adds one or more parts, some being already seen and counted, allowing to rank the 4 profiles.\nAs expected, the slice and drill profile generates the more never encountered query parts since this pattern necessarily goes in one new direction, either by drilling or slicing.\nIt is followed by the slice all pattern, that only add new slices. \nThen comes the two profiles that are constrained by either a goal query (goal oriented) or a \"surprising\" query (explorative), and as such may not necessarily add never seen query parts since they are forced to stay around those queries. The goal oriented has a bit less new query\nparts since it is constrained all along the exploration, while explorative is only constrained half of it.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.25]{cubeload_results\/SI_by_profile_and_query_index.png}\n    \\caption{Subjective Interestingness for each cubeload  profile}\n    \\label{fig:cubeload_SI_profile}\n\\end{figure}\n\n\n\nAs can be observed in Figure \\ref{fig:cubeload_SI_profile}, SI is capable of discriminating the cubeload profiles, except for the \\textit{goal oriented} and \\textit{explorative} profiles that cannot be distinguished. This behavior is expected since both profiles \nexhibit changes to selection and group by attribute at each query. Both behaviors  will therefore generate quite large amounts of information and thus will hardly be  distinguished with SI.\n\nThe \\textit{slice all} profile shows the least amount of Subjective Interestingness because of the way its queries are generated. At each step, the query parts are very minimally altered, thus generating less surprise while keeping a constant \nquery complexity. The combination of these two behaviors cause the Subjective Interestingness to be almost constant across all queries of an exploration for this profile.\nThe \\textit{slice and drill} profile is clearly the profile generating the most subjectively interesting queries.\nCompared to the other profiles, slice and drill generates the greater variety of query parts since it systematically moves by using new query parts, without remaining at the same group by level (like \\textit{slice all}) and without  being\nattracted to some queries (like \\textit{explorative} or \\textit{goal oriented}). \nBeing both constrained, albeit not in the same way, explorative and goal oriented profiles exhibit an intermediate behavior and cannot be easily distinguished with SI.\n\n\n\n\n\\subsubsection{DOPAN use case}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.2]{dopan_results\/qp_so_far_skill_index.png}\n    \\caption{Cumulated number of unique query parts by skill for each query index in the explorations.}\n    \\label{fig:cumulated_query_parts}\n\\end{figure}\n\n\nFigure \\ref{fig:cumulated_query_parts} plots the cumulated number of unique query parts by skill (B, C or D) for each query in the explorations. \nClearly, B labelled explorations have more never seen new query parts, while D labelled explorations have less never seen query parts. The increase rate is also more pronounced for  B explorations, and  higher than for  D  explorations. This is explained by the fact that lower skill explorations exhibit a more explorative and erratic behaviour that periodically tries \nnew directions to explore. Manually reviewing the B explorations, we indeed have found that  their behavior is sometimes reminiscent of the slice and drill profile described in Figure \\ref{fig:template}.\nOn the contrary, D labelled explorations tend to produce less query parts, since these explorations exhibit the behavior\nof a user knowing how to formulate efficiently minimal queries to get to their objective. \n\nCategory C explorations have a mixed behavior with a similar behavior as category D at the beginning and then gradually converging to a situation where they produce as many query parts as the beginner explorations. \nInterestingly, category C also produces shorter explorations, as shown in Figure \\ref{fig:cumulated_query_parts},\nas these explorations were deemed to be executed by a \"user acquiring analytically skills\" ;  they will choose useful measures, use relevant selection predicates but their explorations might be cut short because they did not manage to answer a business questions or did not go the extra step to understand discrepancies in the data they find (e.g., by drilling down).\nFinally, Figure \\ref{fig:cumulated_query_parts} reflects with the cumulated number of query parts, the spread of each exploration skill profile in our query parts graph, which  directly impacts our Subjective Interestingness measure.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\hspace{-1cm}\n    \\includegraphics[scale=0.17]{dopan_results\/line_chart_skill.png}\n    \\includegraphics[scale=0.13]{dopan_results\/explorations_of_skill_B.png}\n\n    \\includegraphics[scale=0.15]{dopan_results\/explorations_of_skill_C.png}\n     \\includegraphics[scale=0.15]{dopan_results\/explorations_of_skill_D.png}\n    \\caption{Average and confidence interval of SI by skill for each query index in the explorations (top left), SI for all explorations of skill B (top right), C (bottom left) and D (bottom right), each color being an individual exploration.}\n    \\label{fig:query_SI_profile}\n\\end{figure}\n\n\n\n\nFigure  \\ref{fig:query_SI_profile} (top left) \nshows  the average SI with  confidence interval of each query of the explorations,  grouped by skill labels and presented by query position in the exploration.\nNoticeably, SI characterizes B labelled explorations with the highest average score and D labelled explorations with the lowest, while C labelled explorations are in between.\nHigh SI corroborates the explorative nature of B explorations, where users struggle to find their way in the multidimensional space by selecting unseen query parts somewhat erratically. On the contrary, low SI corroborates the more \"focused\" nature of D explorations, where users are more pragmatic in their choice of query part to express classical OLAP operations like roll-up or drill-down. Doing such choices, i.e., selecting query parts close to the ones already employed mechanically lowers SI, since the belief attached to those new parts is high and therefore their surprise is low.\nRegarding C explorations, they were labelled a such because they exhibit behaviors coming partly from unskilled explorations and partly from skilled ones. While SI is not conceived to discriminate user skills,\nit can still position those intermediate behaviors between the two extremes.\n\n\nFigure  \\ref{fig:query_SI_profile} (top right and bottom) shows the SI for the queries of each exploration, separated by skills.\nIt is immediately noticeable that there are sudden and short spikes in the SI. By looking at the specific queries where those spikes occur, \nwe could see that the users suddenly add large numbers of selection predicates. By design, our measure can\ndetect such behaviors since adding a high number of parts, even if they have a high probability of being selected, result in high SI due to how probabilities are used to compute surprise (see equation \\ref{eq:si_sum}) and because of the sudden increase in the complexity of the query. \nThe queries obtained with such a burst of new query parts are likely to be informative and correctly detected by our measure.\n\n\n\n\n\nFinally, we have validated these observations with  a rank correlation test between skills (B, C, D) and the Subjective Interestingness measure that outputs a score of $-0.28$. This result confirms that there is a correlation between the skill category and our Subjective Interestingness Measure: the lower the category, the higher the Subjective Interestingness.\n\n\n\n\n\n\n\n\n\n\\section{A first subjective interestingness measure for BI exploration}\n\\label{sec:subj_interest_dopan}\nIn this section we describe how a subjective interestingness measure can be defined\nfor interactive OLAP explorations.\n\n\n\\subsection{Definition of the measure}\nTo construct the Subjective Interestingness measure (denoted by $SI$ hereafter), we follow the same general principle established by De Bie \\cite{DBLP:conf\/ida\/Bie13} and the method presented in Section \\ref{sec:DeBie}. In this framework, $SI$ is the ratio of the surprise related to the observation of a pattern and the complexity to understand the pattern in the user point of view, recalled by the following general formulation of $SI$, for a user $u$ seeing a pattern $Q_t$:\n\n\n\\begin{equation}\\label{eq:si}\n    SI_u(Q_t) = \\frac{-log(belief_u^t(Q_t))}{complexity(Q_t)}\n\\end{equation}\n\n\nA query being composed of multiple query parts, \ncomputing the subjective interestingness of a query can\nbe done using the product of the individual query parts probabilities that compose it. We can therefore rewrite the equation above as:\n\n\\begin{equation}\\label{eq:si_sum}\n    SI_u(Q_t)=\\frac{-\\sum_{p \\in Q_t}\\log(belief_u^t(p))}{complexity(Q_t)}\n\\end{equation}\n\nIn the equation above, $Q_t$ is the  $t^{th}$ query of the exploration. $belief_u^t$ if a function dependent of the position $t$ for a specific user $u$. Note that we  assume  the probabilities of query parts to be independent after convergence of the stationary distribution to simplify computations and allow summation of the information content as described in Equation \\ref{eq:si_sum}. This assumption is reasonable since the PageRank vector is computed as a probability to reach a given query part after an infinite number of random walks in the graph. After convergence to the stationary distribution, the probability of going to another query part is independent of the previous query part, by definition of the PageRank vector. The probability of a particular sequence of query parts is independent of their order. So, the probability of the sequence of query parts constituting the query is the product of the individual probabilities of the query parts.\n\n\n\n\\subsection{Query complexity}\nComputing interestingness demands a measure that conveys the complexity of the queries. In De Bie's framework, this measure  reflects the difficulty of understanding the pattern. In our case, this \"pattern\" is actually the query. Previous work already explored such a metric, for instance  for the SQLShare workload  \\cite{Jain:2016:SRM:2882903.2882957}. In this paper, the authors use two complexity indicators separately to compare SQL queries originating from different datasets. Those two indicators are discriminant and allow to exhibit different behaviors when applied to the SQL workloads.\nThose two indicators are the number of distinct physical operators in the execution plan of a query in the one hand and the query length on the other hand.\nAs our work is done in the context of BI explorations, we consider in this work multidimensional queries, that may not be phrased in SQL. Indeed, in our experiment we work with queries phrased in MDX.\nWe therefore decided to use as complexity measure the\nnumber of query parts, since it is correlated to query length (which we have tested for the datasets we used),\nand considering that,  even if {\\it distinct operators} would be a finer measurement, it is not aligned with the spirit of this complexity measure that should be the complexity as  perceived by the user.\n\n\n\n\n\n\\section{Evaluation of the belief distribution}\n\\label{sec:tests}\n\n\nOur first experiments aim at showing that the belief probability distribution learned with our approach is coherent with what could be expected in realistic exploration situations. To settle such experiments, we envision two distinct use cases. First, we consider \nan ideal environment where all explorations are already categorized into several prototypical user profiles that could be used to bias our model. To this aim, we use the CubeLoad generator \\cite{DBLP:conf\/caise\/RizziG14} which  4  exploratory templates,  illustrated in Figure \\ref{fig:template},\n will serve as user profiles.\n\nSecond, we use real explorations over an open dataset, called the DOPAN  workload from now on,\nwhere users investigates about energy vulnerability in French R\\'{e}gion Centre Val de Loire. We expect these explorations to be more complex and potentially noisy because of queries that are more or less related to the task at hand. This dataset was used in our former work \\cite{DjedainiDLMPV19},\nwhere an expert classified  each users' exploration based on their skill expertise.\n\nFor each use case, several simulations are conducted to assess that our learned probability distributions behave differently and accordingly to what was expected.\nThis section first introduces the experimental protocol for each use case in Section \\ref{sec:xp_protocol}, proposes some hypothesis about the expected results in Section \\ref{sec:hypothesis} and finally describes and analyses the results in Section \\ref{sec:results}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=170pt]{img\/cubeload_templates.png}\n    \\caption{Exploration templates in CubeLoad (from \\cite{DBLP:conf\/caise\/RizziG14}): , {\\it ``seed queries in green, surprising queries in red''}.}\n    \\label{fig:template}\n\\end{figure}\n\n\n\\subsection{Experimental protocol}\\label{sec:xp_protocol}\n\n\\paragraph{Evaluation of quality} We will establish our results around two distinct evaluation methods:\n\\begin{enumerate}\n    \\item we run a quantitative evaluation that relies on a distance between two probability distributions: the goal is to estimate to which extent they are close and behave similarly. A classical choice could have been to use a Kullback-Leibler divergence, but here we prefer to use the discrete Hellinger distance that has the advantage of being symmetric and bounded in the interval $[0,1]$. The discrete Hellinger distance $H(P,Q)$ compares two discrete probability distributions $P = (p_1, \\ldots, p_k)$ and $Q = (q_1, \\ldots, q_k)$ as follows:\n\\begin{equation}\\label{eq:hell}\n    H(P,Q) = \\frac{1}{\\sqrt{2}}\\sqrt{\\sum_{i=1}^k (\\sqrt{p_i} - \\sqrt{q_i})^2}\n\\end{equation}\n    \n    \\item we run a qualitative evaluation based on a comparison of plots of average probability distributions presented in  decreasing order. Here, we do not look at a direct comparison of estimated probabilities for a given query part, but we are rather interested in the overall shape of the belief distribution and noticeably how the probability decreases and the long tail behavior\n\\end{enumerate}\n\n\\paragraph{Implementation} Our approach is implemented in Java using {\\tt jaxen} to read\ncube schemas and {\\tt Nd4j}\\footnote{\\url{https:\/\/deeplearning4j.org\/docs\/latest\/nd4j-overview}} for simple and efficient matrix computation. The code is open source and available in a {\\tt GitHub} public repository\\footnote{\\url{https:\/\/github.com\/AlexChanson\/IM-OLAP-Sessions}}.\n\n\\subsubsection{CubeLoad use case}\nWe generated a series of 50 explorations using the Cubeload generator \nover the schema of a cube constructed using the SSB benchmark \\cite{DBLP:conf\/tpctc\/ONeilOCR09},\nthat we split in 2 groups: the first 43 explorations are used to construct the topology graph, and the next 7 are taken from a single CubeLoad template, and are used to define the user profile.\nWe run 50 randomized samples to achieve a traditional cross-validation protocol.\n\n\\subsubsection{DOPAN use case} The DOPAN workload contains $32$ explorations in total, that have been authored by $10$ master students of a Business Intelligence program. DOPAN contains 3 data cubes with, respectively, 19, 14, and 27 dimensions, and 32, 20, and 58 measures. We expect this dataset to be more challenging than CubeLoad since real users are likely to be less predictable than simulated ones with potential erroneous queries during the exploration. Interestingly, users' explorations are categorised according to the skill expertise in $3$ groups: $B$ users are the less experienced, $C$ users show average skills while $D$ users are supposed to write the most appropriate queries. Noticeably, this dataset's noise and longer explorations can be explained by the behavior of OLAP tools, like Saiku, as they log a new query for each user action (including intermediate drag-and-drops).\n\n\n\n\\subsection{Hypothesis} \\label{sec:hypothesis}\n\n\\subsubsection{CubeLoad use case} We expect the 4 templates included in CubeLoad to behave differently.\nThe {\\it slice all} template is a local user model that only explores a small fraction of the data space.\nIt is thus expected that when comparing to a baseline distribution probability \nagnostic of any user specific graph, it will maximize this distance. In this case, only a few query parts concentrate most of the interactions with a higher probability, as all queries of the exploration share the same group by set and measure.\nSimilarly, as the {\\it slice all} template chooses one level in one hierarchy and then only varies the selection predicate, it is expected to show a larger standard deviation than the other templates from one exploration to the next.\n\n\nOn the contrary, the {\\it explorative} template simulates a broader exploration of the data space. This template should lead to minimizing its distance with a topology based distribution. In this case, it is expected that there are fewer very improbable query parts but that there are more higher probabilities on most query parts, because of the coverage of the data space by the template.\n\nThe {\\it goal-oriented} and {\\it slice-and-drill} templates are expected to be intermediate states between the two previous templates. Indeed, both models explore the data space more than {\\it slice-all}, but are a bit more constrained than {\\it explorative}.\n\n\n\n\n\\subsubsection{DOPAN use case}\n\nThe DOPAN use case is more complex since it deals with real explorations for which we do not know the profile of the users, contrary to CubeLoad. We expect these experiments to confirm the tendencies observed in the CubeLoad dataset, with an ability of our belief distribution to capture the knowledge of the users. However, we expect these results to be less contrasted than what is observed for CubeLoad for two reasons: ($1$) the cubes in DOPAN are more complex  in terms of schema ($2$) the users in the experiments were all trained in the same master program and thus should exhibit some common behviours\n, which may not help distinguishing from one profile to the next.\n\n\\subsection{Results}\\label{sec:results}\n\\subsubsection{CubeLoad use case}\nTable \\ref{tab:hellinger} represents the distance between:\n\\begin{itemize}\n\\item the PR vector computed over the topology graph $G_t$ (see Section \\ref{sec:IM}),\n\\item and the PR vector computed over the aggregated graph as produced by Algorithm \\ref{algo:merge_graph}, that merges the topology graph $G_t$ with the user specific graph $G_u$.\nIn order to bias our model, we gradually modify the parameter $\\alpha \\in [0,1]$ of the {\\tt merge} function in Algorithm \\ref{algo:merge_graph} to give more importance to $G_u$ compared to $G_t$\n\\end{itemize} \n\n\n\n\\begin{table}[ht]\n\\begin{center}\n{\\scriptsize{\n    \\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}\n  \\hline\n  User\/$\\alpha$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\\\\n  \\hline\nExplorative & 0.021 & 0.042 & 0.063 & 0.084 & 0.106 & 0.130 & 0.155 & 0.183 & 0.215\\\\\nGoal Oriented & 0.015 & 0.031 & 0.047 & 0.063 & 0.081 & 0.101 & 0.123 & 0.150 & 0.182\\\\\nSlice All & {\\bf 0.073} & {\\bf 0.127} & {\\bf 0.170} & {\\bf 0.209} & {\\bf 0.244} & {\\bf 0.279} & {\\bf 0.315} & {\\bf 0.350} & {\\bf 0.392}\\\\\nSlice and Drill & 0.022 & 0.044 & 0.066 & 0.089 & 0.114 & 0.139 & 0.167 & 0.200 & 0.236\\\\\n    \\hline\n\\end{tabular}\n}}\n    \\caption{Average Hellinger distance between the topology graph PR  and the user-specific graph PR, following the  templates of CubeLoad (std. dev. on the order of $10^{-3}$ no represented here)}\n    \\label{tab:hellinger}\n\\end{center}\n\\end{table}\n\nTable \\ref{tab:hellinger} details the\nHellinger distance values computed between the two PR's,\nfor each of the 4 CubeLoad templates, and values of parameter $\\alpha$ ranging from $0.1$ to $0.9$.\nThe first observation from Table \\ref{tab:hellinger} is that Cubeload profiles indeed differ in how close they are to the reference distribution. This means that different user activities can be characterized to correspond to different belief by our approach.\nWe can also observe that the distance between the resulting distributions is proportional to $\\alpha$ as expected. Indeed, if $\\alpha$ is very low, the biased distribution is very close to the PR topology distribution. The higher $\\alpha$, the more characteristics from the user profile are introduced in the transition matrix. Second, and as expected, we notice that the {\\it slice-all} profile bears the larger distance with the topology as it only explores a small portion of the possible space, while the other profiles seem to have a comparable behavior in terms of distance. It can be observed that {\\it goal-oriented} profile tends to generate the lowest distances to the PR that represents the topology graph. This can be explained by the fact that {\\it goal-oriented} is somehow the less constrained simulated user profile as it mainly performs a random walk in the topology graph to a destination query following the schema. In contrast other profiles and noticeably {\\it slice-all} and {\\it slice-and-drill} restrict more strongly their explorations of the graph to fixed patterns that may contradict probabilities of transitions as observed in most past usages and schema.\n\n\nFigures \\ref{fig:pr_probas02} and \\ref{fig:pr_probas08} represent, for two distinct values of parameter $\\alpha$, the average distribution of probabilities (and their standard deviation) for the $4$ user profiles and the PR distribution corresponding to the topology averaged on 200 tests. As expected, when $\\alpha = 0.2$ all distributions heavily tend to mimic the PR distribution. On the contrary, when $\\alpha = 0.8$ the difference brought by the user profile become clearly visible. The {\\it slice-all} profile tends to have a higher number of higher probabilities and then decreases with successive steps which are characteristic of this profile. Indeed, this profile explores all members at a given depth in a hierarchy and all the corresponding query parts are by construction almost equiprobable: only past usage may modify probability which translates in the observed decrease pattern. Then, as expected the {\\it slice-all} profile shows the largest standard deviation in our test, which can be explained by the variability of each exploration from this profile that explore each time a different hierarchy at a different level.\\\\\n\nSimilarly, {\\it slice-and-drill} profile exhibits some small but noticeable steps that can be explained by the fact that this profile alternatively navigate between members of a hierarchy at a given level and at some point drill down to query parts from lower level of the hierarchy and that are less likely to be used, as they are more specialized and thus less probable.\n\nThen, it is worth noticing that the PR plot also shows some steps which may traduce the presence of more strongly connected components inside the topology graph where query parts are equiprobable.\n\nFinally, {\\it explorative} and {\\it goal-oriented} profiles exhibits a more gradual decrease of the probabilities which is again expected as these approaches distributes more evenly their probabilities into more query parts because of their behavior. \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.055]{img\/dolap_alpha_02_newGraph.jpeg}\n    \\caption{Distribution of probabilities computed by our model for all $4$ user profiles when $\\alpha = 0.2$ (log scale). Each plot represents the probabilities of the query parts for one user profile in decreasing order}\n    \\label{fig:pr_probas02}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.055]{img\/dolap_alpha_08_newGraph.jpeg}\n    \\caption{Distribution of probabilities computed by our model for all $4$ user profiles when $\\alpha = 0.8$ (log scale). Each plot represents the probabilities of the query parts for one user profile in decreasing order.}\n    \\label{fig:pr_probas08}\n\\end{figure}\n\n\\subsubsection{DOPAN use case}\n\n\\begin{table}[ht]\n\\begin{center}\n{\\small{\n    \\begin{tabular}{|l|c|c|}\n  \\hline\n  User\/$\\alpha$ & 0.2 & 0.8 \\\\\n  \\hline\nUser 03 &0.00134 &0.0586 \\\\\nUser 04 &0.00417 &0.135 \\\\\nUser 05 &0.000565 &0.00244 \\\\\nUser 06 &0.00201 &0.0692 \\\\\nUser 07 &0.00560 &0.150 \\\\\nUser 09 &0.00423 &0.131 \\\\\nUser 10 &0.00367 &0.133 \\\\\nUser 12 &0.0000244 &0.00134 \\\\\nUser 14 &0.00567 &0.136 \\\\\nUser 16 &0.00530 &0.151 \\\\\n  \\hline\n\\end{tabular}\n}}\n    \\caption{Hellinger distance between PR and our biased PR with several user profiles on the DOPAN dataset}\n    \\label{tab:hellinger_dopan}\n\\end{center}\n\n\\end{table}\n\nSimilarly to Table \\ref{tab:hellinger}, Table \\ref{tab:hellinger_dopan} indicates the distances between the PR topology distribution and distributions obtained by biasing the topology graph with users explorations on the DOPAN cubes. For the sake of simplicity, Table \\ref{tab:hellinger_dopan} only presents the Hellinger distances computed for 2 distinct values of parameter $\\alpha$ that are either topology oriented ($\\alpha = 0.2$) or user oriented ($\\alpha = 0.8$).\n\nIt can be first noticed that the distance values are much smaller than in the case of CubeLoad explorations as expected. This is explained by the higher complexity of the DOPAN cubes combined with fewer explorations per user, in contrast to the CubeLoad experiments that were conducted with more explorations over a simpler SSB cube.\nThis indicates that our modeling of belief therefore correctly accounts for the complexity of the query space. \n\n\nNoticeably, two users exhibit very low distances to the reference belief. By manually reviewing those user logs we observed that they had very short explorations compared to the other users. As a consequence, a small user log will not change much the aggregated graph when mixed with the topology graph and therefore have almost no influence on the PR vector and on the computed distances.\nIn other words, \nthe probability to use a specific query part for a user with little experience is dictated by the schema and general user navigational habits.\n\n\\begin{figure}\n    \\begin{center}\n        \\includegraphics[scale=0.04]{img\/fig_dopan_Cube1MobProInd.jpg}    \n        \\caption{Distribution of probabilities over the first (and most used) cube of the DOPAN dataset. Query parts are ordered according to their decreasing probability value for each student. \\textit{Note that the plot has been limited to the first $3000$ queries for the sake of readability.}\n        \\label{fig:belief_dopan}\n    \\end{center}\n\\end{figure}\n\nIn Figure \\ref{fig:belief_dopan} we display in the same way as Figure \\ref{fig:pr_probas02} and Figure \\ref{fig:pr_probas08} the belief distribution of the user but here on the real explorations over the first cube of the Dopan dataset. Interestingly, we can observe that student $14$ behavior is reminiscent to the behavior exhibited by CubeLoad's \\textit{Slice All} profile. By reviewing the user's explorations we found that they change the selection predicate over several queries while keeping measures and group by set elements from previous queries. This behavior is  very similar to the one described by a \\textit{Slice All} pattern, and indicates that our modeling of belief \ncan help differentiating specific ways of exploring.\n\n\n\n\\section{How usage and recommendation impact belief}\\label{sec:tests3}\n\nThe first experiments reported in Section \\ref{sec:tests} show that our model of belief is able to transcribe different exploration templates based on the shape of their probability distribution functions, although $3$ templates, namely \\textit{Explorative}, \\textit{Goal Oriented} and \\textit{Slice And Drill}, remain very close in this respect.\n\nHowever, if our belief model is to be faithful to the user intent, as it is expressed at the query part level, any change in the accessed area of a cube should be traduced by a change in the assigned probability weights in our belief model. We propose to first simulate this diversity in the cube exploration by learning the belief model on different and independent log files generated by CubeLoad.\n\nSecond, as the long-term objective is to use belief-based interestingness measure as an input for tailoring personalized recommendations, we want to evaluate to which extent the output of a recommender system could impact the belief of a specific user. To this aim, we use a state-of-the-art OLAP recommender system \\cite{DBLP:journals\/dss\/AligonGGMR15} that is trained on a log file representative of a specific CubeLoad template and whose output recommendation is used to define a new belief distribution. It is important to notice that, at this step, we consider that all recommended queries are executed at the same time and that the belief can be computed based on the same prior knowledge from the user for all recommended queries at once. As explained earlier, dealing with the evolution of the user belief after each query is out of the scope of this paper.\n\nIn both cases, we need to define beforehand a \\textit{reference belief}  for each of the $4$ CubeLoad templates in our simulations. Then a divergence measure can traditionally be used to evaluate to which extent changing the template or changing the log file on which we learn the models impacts the belief distribution.\n\n\\subsection{Experimental protocol}\n\n\\paragraph{Evaluation scenarios} Figure \\ref{fig:protocol} presents the main evaluation protocol for the impact of usage and recommender system on the belief measure. The left-side of Figure \\ref{fig:protocol} illustrates how to generate the \\textit{reference belief}. Similarly to Section \\ref{sec:xp_protocol}, we use $50$ CubeLoad sessions among which $43$ are used to compute the topology graph of our PageRank (see \\textit{``Log usage construction rules''} in Section \\ref{sec:PRbelief}) and $7$ sessions issued from a single CubeLoad template are used to simulate a user and bias the topic-specific PageRank. \n\nThe right-side of Figure \\ref{fig:protocol} describes how we generate the \\textit{test belief} based on the  recommender system \\cite{DBLP:journals\/dss\/AligonGGMR15} and a simulated user of the recommender system. To do so, we generate a new \\textit{independent} log file from which $10$ random sessions from a single CubeLoad template are drawn. They are used to train the recommender system. In addition a separate session is drawn and truncated to be used as a seed for the recommender. In order to evaluate to which extent the diversity of queries affects our belief model, we also consider the case where the recommender system as well as the seed session for the recommender are drawn at random from the same log file as the one used for the reference belief. We call this scenario \\textit{identical} while the first one with distinct session files is named \\textit{independent} hereafter.\n\n\n\n\\begin{figure*}\n    \\begin{center}\n    \\includegraphics[scale=0.5]{img\/experimental_protocol.png}\n    \\caption{Evaluation protocol for the impact of usage and recommender system on the belief measure.}\n    \\label{fig:protocol}\n    \\end{center}\n\\end{figure*}\n\n\\paragraph{Evaluation of results} \nWe use the Hellinger distance as described in Section \\ref{sec:xp_protocol} to compare the reference and test belief distributions.\nWe generate 40 random logs (10 per profile)\nand we run them against randomly chosen seeds from the 4 profiles and against the 4 possible reference belief models. Results are   \naveraged \nand reported \nin Table \\ref{tab:identical} and \\ref{tab:independent}.\n\n\n\n\\subsection{Hypothesis}\n\nWe expect several observable results from these experiments. First, it is expected that the belief divergence between the recommended queries from the \\textit{test} and the queries used as a \\textit{re\\-fe\\-ren\\-ce} is much larger in case of \\textit{in\\-de\\-pen\\-dent} log files than in case of \\textit{identical} log files. As in both cases the recommender system is involved, this would mean that any observed difference would be due to the difference in log files and that our belief model while preserving the distribution probabilities of each CubeLoad template is able to reflect the differences in usage on query parts.\n\nSecond, we expect to observe differences between the $4$ CubeLoad templates that  could refine the observations of Section \\ref{sec:53}. Indeed, only based on the distribution it was not possible in our setting to distinguish clearly among the \\textit{Explorative}, \\textit{Goal Oriented} and \\textit{Slice and Drill} templates. We expect them to remain close but with slight divergence that would reflect the different abilities of each template to dig into new parts of the data cube. However, similarly to experiments in Section \\ref{sec:53} we expect \\textit{Slice All} to be significantly divergent from the other profiles. Indeed, it is known that this template exhibits tendencies to explore locally the cube by only navigating a dimension following siblings relations. \n\n\nFinally, if we consider our belief model as representative of the usage, we may be able to draw conclusions about the use of recommender systems with CubeLoad templates and to which extent the latter propose really distinguishable exploration patterns. To this aim, a comparison of divergence values when comparing, for each CubeLoad template, its reference to its recommended test on the same log file would indicate to which extent the template explores various portion of the cube. In other words, the more explorative a template is, the larger the divergence between a recommendation based on this template and its reference should be.\n\n\n\\subsection{Results}\n\nTables \\ref{tab:identical} and \\ref{tab:independent} present experimental values of Hellinger distance\nfor the \\textit{identical} and the \\textit{independent} scenarios and each type of recommendation based on the $4$ CubeLoad templates (lines) for each reference belief model (columns).\n\nIt can be seen that, as expected, the divergence is significantly lower when considering recommendations based on the same log file that was used for the reference belief computation. This shows that our belief method is sensitive to the actual queries that are involved in the building of the model despite the general trends observed for the probability distribution in Section \\ref{sec:53}.\n\n\\begin{table}[ht]\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{lcccc} \\hline\nTests$\\backslash$References & \\textit{Explorative}\t& \\textit{Goal Oriented} & \\textit{Slice All} & \\textit{Slice and Drill} \\\\ \\hline\n\\textit{Explorative} & \\textbf{0.64} & 0.60 & 0.47 & 0.60 \\\\\n\\textit{Goal Oriented} & 0.64 & \\textbf{0.61} & 0.46 & 0.60 \\\\\n\\textit{Slice All} & 0.67 & 0.63 & \\textbf{0.47} & 0.62 \\\\\n\\textit{Slice and Drill} & 0.63 & 0.60 & 0.43 & \\textbf{0.58} \\\\ \\hline\n\\end{tabular}\n\\caption\nAverage Hellinger distance values on 10 runs when log files are \\textbf{identical}. Lines represent test and columns represents reference belief.}\n\\label{tab:identical}\n\\end{center}\n\\end{table}\n\n\\begin{table}[ht]\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{lcccc} \\hline\nTests$\\backslash$References & \\textit{Explorative} & \\textit{Goal Oriented} & \\textit{Slice All} & \\textit{Slice and Drill} \\\\ \\hline\n\\textit{Explorative} & 0.85 & 0.86 & 0.81 & 0.86 \\\\\n\\textit{Goal Oriented} & 0.85 & 0.85 & 0.81 & 0.86 \\\\\n\\textit{Slice All} & 0.83 & 0.84 & 0.79 & 0.84 \\\\\n\\textit{Slice and Drill} & 0.86 & 0.86 & 0.82 & 0.87 \\\\ \\hline\n\\end{tabular}\n\\caption\nAverage Hellinger distance values on 10 runs when log files are \\textbf{independent}. Lines represent test and columns represents reference belief.}\n\\label{tab:independent}\n\\end{center}\n\\end{table}\n\n\nThen, results from Tables \\ref{tab:identical} and \\ref{tab:independent} corroborates that there exists differences between the CubeLoad templates but that are more or less difficult to observe depending on the CubeLoad template, as shown by the distribution of Section \\ref{sec:53}.\n\nIn the identical scenario, \\textit{Explorative} recommendation is the most distant to its corresponding \\textit{Explorative} reference (distance $= 0.64$) and the most distant to all other recommendations as shown in column \\textit{Explorative} from Table \\ref{tab:identical}. This\ncan be understood\nby the nature of the CubeLoad template which makes it difficult to model for a recommender system, which in turns implies a higher probability to access new portions of the cube and thus new query parts.\n\n\\textit{Goal oriented} column of Tables \\ref{tab:identical} and \\ref{tab:independent} presents not significantly different distance values whatever the recommendation profile is. This is expected as this template can generate several different paths in the cube to access the same region in the end. As a consequence, there is a variety in the accessed query parts which in turns tends to smooth the distance values.\n\n\\textit{Slice and Drill} exhibits significant differences and with a small distance to itself of $0.58$, which means that the recommendation that is produced reflects the CubeLoad template and stays in a localized region of the data cube with more common query parts between recommendations and the reference template.\n\n\n\n\nFinally, an interesting observation is that \\textit{Slice All} has a very low inner distance of $0.47$ in Table \\ref{tab:identical} which means that the exploration is very focused and that even a recommender system based on this template will reproduce this local exploration behaviour with no surprising query parts. This reflects a sort of cognitive bubble associated to this template, which can be detected with our approach of user belief estimation. \n\nThe latter is confirmed by the analysis of the line \\textit{Slice All} in Table \\ref{tab:identical} where the distance between the recommended \\textit{Slice All} and the other CubeLoad reference templates that are higher than the other distances for each of these templates. For example, in Table \\ref{tab:identical}, the distance of \\textit{Slice All} to \\textit{Explorative} is $0.67$ which is the largest distance for the column \\textit{Explorative}.\\\\\n\nThese first results bring new insights on our belief model, and to which extent it is coherent with the CubeLoad profiles. A more thoroughful analysis of these results is needed to conclude about differences on CubeLoad templates and is left as future work.\n\n\n\n\\subsubsection{DOPAN use case}\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.24]{dopan_results\/box_plot_SI_skill_and_user.png}\n    \\caption{Exploration Subjective Interestingness by skill level for users exploring the DOPAN cubes \\patrick{preciser que c'est par exploration et pas par user}}\n    \\label{fig:dopan_box_plot}\n\\end{figure}\n\nFrom Figure \\ref{fig:dopan_box_plot} we can observe that the explorations tagged with the lowest skill levels yield the most subjective information, which is expected because of the user weaker knowledge of the cubes. Indeed, if the user explores a larger part of the cubes, by construction, the probability of knowing the query parts attached to these portions of the cube increase and so the surprise and SI attached to these query decreases. This has been confirmed by a careful study of the logs where for some users, such as Student 4 whose skill went from ``B'' to ``D'' in the course of $5$ sessions over a few days. Overall only $2$ students were able to perform a session rated ``D'' (best score) on their first try and thus most users with ``D'' explorations will have already explored the cubes at least a few times. \n\\patrick{toujours pas clair pour moi}\n\n\nInterestingly, explorations with lower score (``B'') are not necessarily the shortest explorations in terms of number of queries. As a consequence, the observed difference in scores can mainly be explained by differences in the exploration pattern.\n\n\nAnalysis by an expert of all \"B\" skill sessions reveals a few insights. First, most sessions start with 10 or more near identical or identical queries where the user only changes {\\it measures}, then users will try to change the {\\it group by set} (one element at a time). Finally, at mid sessions,  users will start changing {\\it Group by set} elements combined with {\\it selection} predicates (very simple, one or two dimensions at most). This behavior looks similar to the Slice and Drill pattern described on Figure \\ref{fig:template} although noisier and more erratic at times\n\n\n\nAs shown in Figure \\ref{fig:dopan_box_plot}, the exploration Subjective Interestingness measure fails to fully discriminate between the medium and advanced users. \n\n\n\\patrick{il faudrait aussi analyser les 2 composantes de SI : le belief et le nombre de query parts}\n\nTo highlight the differences implied by the SI between the true skills, Figure \\ref{fig:dopan_violin} shows the distributions of the measure at the query level (and not by exploration) for each skill category, with the actual sample in white \n\\patrick{preciser: comment sont ordonnes les samples}\nas a KDE estimate alone can hide subtle details.  \n\n\\begin{figure}[ht]\n    \\centering\n    \\hspace{-1.5cm}\n    \\includegraphics[scale=0.38]{dopan_results\/violin_plot_SI_skills.png}\n    \\caption{Distributions of the query Subjective Interestingness by user skill level}\n    \\label{fig:dopan_violin}\n\\end{figure}\n\nIt is directly apparent that the distributions have different shapes in accordance to the skill level. The skewness of the distributions are very different (-0.68 for B 1.32 for C and 0.46 for D). Between ``B'' and ``C'', there is a great shift in skewness, from negative to positive. This can be explained by the fact that the user are gaining more knowledge of the cube as well as improving their exploration skills. A less skilled user will very easily find interesting patterns because of his limited knowledge of the cube at the beginning. This explain the negative skew of the distribution. For the second skill level, ``C'', their exploration skills are still improving. As they fail to find new interesting data with their exploration method, this effectively skews the distribution toward lower SI. Finally, the ``D'' skill levels shows two modes which seem to be the merging of the two previous behaviors. The lower mode is made of the queries who produced the least amount of interestingness because of the previous user's knowledge of the cube. The higher mode emphasize the fact that a skilled user will still find new ways to explore the cube, though generating a greater amount of Subjective Interestingness. \n\nMoreover, it is important to notice that using the query results or any data from cube were not needed to observe this differences and only depends on the sequences of query parts used in the explorations. This observation legitimates our choice of using the sole query parts as a proxy for the values of cells.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany papers in epidemiology proposing a mathematical model using dynamical systems, face the problem of parameter estimation. In general, some parameters are given, extracted from the literature, while the remaining unknown parameters are estimated by fitting the model to some observed data, usually by means of an optimization algorithm based on least squares or maximum likelihood methods.  Nevertheless, relatively few studies about the intrinsic property of a model to admit a unique set of parameters values for a given choice of measured variables. On the other hand, this a question that is well known in automatic control. Investigation of identifiability in mathematical epidemiology is relatively recent \n\\cite{MR3553947,MR2726194,MR1957979,MR2817815,MR2785878,Eisenberg:2013aa,MR2142487}.\nIndeed, to the best of our knowledge, the first paper considering the problem in a epidemic model studies identifiability of an intra-host model of HIV, and it has been published in 2003 in an Automatic Control journal \\cite{MR1957979}. It is also surprising that the observability and identifiability of the original Kermack-Mckendrick  model has not been more studied, since this system has been widely used to model an outbreak of an infection.\nThe  observability and identifiability of the classical model SIR, with demography and constant population, has been first studied in 2005 \\cite{MR2142487}.\n\nThe global health crisis of COVID-19 outbreak has led to a spectacular resurgence of interest in this type of models, but with specificities related to the detection and isolation of infected individuals \\cite{m-y-li-SIR}. This is why we revisit here the issues of identification and observability for an extended `SIQR' model \\cite{MR2003m:92074} for which such an analysis had not yet been performed (to the best of our knowledge). Once the question of identification has been settled, we also tackle the task of proposing a practical strategy for reconstructing the unique values of the parameters.\n\n\n\n\n\\section{The models}\n\nInspired by \\cite{MR2003m:92074,MR2428383}, we consider the classical SIR model (see for instance \\cite{MR3752193}), where $S$, $I$, $R$ denote the size of the populations of respectively susceptible, infected and recovered individuals, with an additive compartment where $Q$ denotes the size of the population of identified and isolated infectious individuals that have been removed from the infected population and placed in quarantine:\n\\begin{equation}\n\\label{model1}\n\\left\\{\\begin{array}{lll}\n\\dot S  & =  & -\\beta S\\frac{I}{N-Q} \\\\\n\\dot I & =  & \\beta S\\frac{I}{N-Q} - (\\rho+\\alpha) I\\\\\n\\dot Q  & =  & \\alpha I - \\rho Q\\\\ \n\\dot R  & = & \\rho I + \\rho Q\n\\end{array}\\right.\\,.\n\\end{equation}\nWhen the size of the total population $N$ is large and the size of the population placed in quarantine remains small compared to $N$ during the considered interval of time, one can consider a simplified model: \n\\begin{equation}\n\\label{model2}\n\\left\\{\\begin{array}{lll}\n\\dot S  & =  & -\\beta S\\frac{I}{N} \\\\\n\\dot I & =  & \\beta S\\frac{I}{N} - (\\rho+\\alpha) I\\\\\n\\dot Q  & =  & \\alpha I - \\rho Q\\\\ \n\\dot R  & = & \\rho I + \\rho Q\n\\end{array}\\right.\\,.\n\\end{equation}\nNote that for both models, one has\n\\[\nS(t)+I(t)+Q(t)+R(t)=N, \\quad \\forall t \\geq 0\\,.\n\\]\n\nThese models have three parameters: the infectivity parameter $\\beta$, the recovery rate $\\rho$, that we assume to be identical for the infected populations placed in quarantine or not, and  the rate of placement in quarantine $\\alpha$. Theses parameters are unknown but we assume the following hypothesis.\n\n\\smallskip\n\n\\begin{assum} \n\t\\label{assum1}\n\tThe reproduction number ${\\mathcal R}_0$ verifies\n\t\\[\n\t{\\mathcal R}_0:=\\frac{\\beta}{\\rho+\\alpha}>1\\,.\n\t\\]\n\\end{assum}\n\n\\smallskip\n\nThis assumption implies that the epidemic can spread in the population i.e.~at initial time with $S(0)=N-I(0)$ close to $N$ one has $\\dot I(0)>0$.\n\n\n\\section{The identification problem}\nWe assume that\n\\begin{itemize}\n\t\\item the flow $\\alpha I(t)$ of infected people placed in quarantine is known at any time $t\\geq 0$\n\t\\item the size $Q(t)$ of the population placed in quarantine is perfectly known at any time $t\\geq 0$\n\t\\item the size $N$ of the total population is known\n\t\\item at initial time $0$, one has $S(0)=N-\\varepsilon$, $I(0)=\\varepsilon$, $Q(0)=0$, $R(0)=0$ with $\\varepsilon \\in (0,N)$.\n\\end{itemize}\nWe consider then the observation function\n\\begin{equation}\n\\label{observation}\ny(t)=\\left[\\begin{array}{c} \ny_1(t)\\\\ \ny_2(t)\n\\end{array}\\right]:=\\left[\\begin{array}{c} \n\\alpha I(t)\\\\ \nQ(t)\n\\end{array}\\right]\n\\end{equation}\nand follow the usual definitions of identifiability and observability of systems \\cite{MR1482525,MR1408862}.\nHowever, note that when $Q=0$, the system is not infinitesimally identifiable: the knowledge of the outputs and all its derivative do not allow to determine formally $\\rho$. At $I=0$, the system is not identifiable neither. We adopt the following definition of identifiability for these models.\n\n\\smallskip\n\n\\begin{defi}\n\\label{defi}\n\tGiven $N>0$ and $\\varepsilon \\in (0,N)$, we shall say that system \\eqref{model1} resp.~\\eqref{model2} is identifiable for the observation \\eqref{observation} if there exists $t>0$ such that the map\n\t\\[\n\t\\left[\\begin{array}{c}\n\t\\alpha\\\\\n\t\\beta\\\\\n\t\\rho\\\\\n\t\\end{array}\\right]\\in \\left(\\mathbb{R}_{+}^\\star\\right)^3 \\quad \\longmapsto \\quad y(\\cdot)  \\in {\\mathcal C}^\\infty([0,t],\\mathbb{R}_+^2)\n\t\\]\n\tis injective, where $(S(\\cdot),I(\\cdot),Q(\\cdot),R(\\cdot))$ is solution of the Cauchy problem for the differential system \\eqref{model1} resp.~\\eqref{model2} with $S(0)=N-\\varepsilon$, $I(0)=\\varepsilon$, $Q(0)=0$ and $R(0)=0$.\n\tIf moreover the map\n\t\\begin{align*}\n\t\\left[\\begin{array}{c}\n\t\\alpha\\\\\n\t\\beta\\\\\n\t\\rho\\\\\n\t\\varepsilon\n\t\\end{array}\\right]\\in \\left(\\mathbb{R}_{+}^\\star\\right)^3\\times(0,N)\n\t \\longmapsto \\quad y(\\cdot)  \\in {\\mathcal C}^\\infty([0,t],\\mathbb{R}_+^2)\n\t\\end{align*}\n\tis injective, then the system \\eqref{model1} resp. \\eqref{model2} is identifiable and observable for the observation \\eqref{observation}.\n\n\\end{defi}\n\n\\section{Analysis of the first model}\n\n\\begin{prop}\n\tSystem \\eqref{model1} is identifiable and observable for the observation \\eqref{observation}, in the sense of Definition \\ref{defi}.\n\\end{prop}\n\n\\smallskip\n\n\\begin{proof}\nIt consists in showing that parameters and unmeasured variables $S$ and $I$ can be expressed as functions of the successive derivatives of the output vector $y$. As the variable $I$ cannot reach $0$ in finite time, we shall assume $I\\neq 0$ in the following.\n\n\\smallskip\n\nNote first that with $Q(0)=0$ one has $\\dot Q(0)>0$ and then $y_2(t)=Q(t)>0$ for any $t>0$. The dynamics of $Q$ gives directly the expression of the parameter $\\rho$ as:\n\\begin{equation}\n\\label{rho}\n\\rho= \\, \\dfrac{ y_1(t)-\\dot{y}_2(t)}{y_2(t)}\\,, \\quad t>0 .\n\\end{equation}\nPosit $h_1:=  \\dfrac{\\dot y_1 }{y_1}$. One has from the dynamics of $I$\n\\begin{equation}\\label{eq1}\nh_1 = \\dfrac{\\beta \\, S}{N-Q}-\\alpha- \\rho  .\n\\end{equation}\nand then\n\\begin{equation}\\label{eq2}\n(N-Q) \\, \\dot{h}_1=-\\dfrac{\\beta^2\\, S}{N-Q} \\, I + \\dfrac{\\beta \\, S}{N-Q}\\, \\dot Q.\n\\end{equation}\nUsing the equality $\\dfrac{\\beta \\, S}{N-Q}=\\alpha + h_1+\\rho$  from \\eqref{eq1}, one obtains\n\\begin{equation}\\label{eq3}\nh_2:= (N-y_2) \\, \\dot{h}_1 = (h_1+\\alpha +\\rho)\\, ( -\\beta \\,I+  \\dot Q).\n\\end{equation}\nLet us write the derivative of $h_2$:\n\\begin{align*}\n& \\dot h_2\u00a0= \\dot h_1\\, (-\\beta \\, I + \\dot Q)\\\\ \n& \\quad + (h_1+\\alpha+\\rho) \\, \\left [   -\\beta \\,  \\dfrac{\\beta \\, S}{N-Q}\\, I + \\beta \\, (\\alpha+\\rho) \\, I\u00a0 + \\ddot Q\\right ]\n\\end{align*}\nwhich can be also expressed as\n\\begin{align*}\n& \\dot h_2\u00a0= \\dot h_1\\, (-\\beta \\, I + \\dot Q)\\\\ \n& \\qquad + (h_1+\\alpha+\\rho) \\, \\left [   -\\beta \\, I  \\, (h_1+\\alpha+\\rho)+ \\beta \\, (\\alpha+\\rho) \\, I\u00a0 + \\ddot Q\\right ] \\\\\n& \\quad = \\dot h_1\\, (-\\beta \\, I + \\dot Q) + (h_1+\\alpha+\\rho)  \\, \\left [   -h_1\\, \\beta \\, I   + \\ddot Q\\right ] .\n\\end{align*}\nThen, using relation \\eqref{eq3}, one obtains the expression\n\\[\n\\dot h_2= \\dot h_1 \\,  (-\\beta \\, I + \\dot Q)+ \\dfrac{h_2}{(-\\beta\\, I +\\dot Q)}\\, \\left [   h_1\\, (-\\beta \\, I  + \\dot Q)  - h_1\\, \\dot Q +\\ddot Q\\right ]\n\\]\nwhich implies\n\\begin{align*}\n& (-\\beta \\, I  + \\dot Q) \\,  \\dot h_2 = \\dot h_1 \\,  (-\\beta \\, I + \\dot Q)^2\\\\\n& \\qquad + h_2\\, h_1 \\,  (-\\beta \\, I  + \\dot Q)+ h_2\\, ( - h_1\\, \\dot Q +\\ddot Q)\n\\end{align*}\nor equivalently the equation\n\\begin{equation*}\n\u00a0 \\dot h_1 \\,  (-\\beta \\, I + \\dot Q)^2+ (h_2\\, h_1 -\\dot h_2)  (-\\beta \\, I  + \\dot Q)+ h_2\\, ( - h_1\\, \\dot Q +\\ddot Q) =0\n\\end{equation*}\nto be fulfilled.\n\n\\medskip\n\nObserve that this last equation is a second order polynomial in the variable  $X=-\\beta \\, I + \\dot Q$.\nFrom \\eqref{eq2} and $\\mathcal R_0>1$ one has\n\\begin{equation}\n\\label{ineq1}\n\\left\\{\\dot h_1\\right\\}_{t=0} =\\dfrac{\\beta \\, \\varepsilon }{N }\\, (-\\beta +\\alpha)\\left(1-\\frac{\\varepsilon}{N}\\right) <0\n\\end{equation}\nand this allows us to show that one also has\n\\begin{equation}\n\\label{ineq2}\n\\left\\{h_2\\, ( - h_1\\, \\dot Q +\\ddot Q)\\right\\}_{t=0} =\\alpha \\,\u00a0\\beta \\, \\rho \\, \\varepsilon^2 (\\beta -\\alpha)\\left(1-\\frac{\\varepsilon}{N}\\right) >0\n\\end{equation}\nIndeed, one has \n\\begin{align*}\n&\\ddot Q = \\alpha \\beta \\dfrac{S\\, I}{N-Q}-\\alpha \\, (\\rho+\\alpha)\\, I -\\rho \\, \\dot Q\\\\\n& \\qquad \\Rightarrow \\left\\{\\ddot Q \\right\\}_{t=0}= \\alpha\\, \\varepsilon \\, \\left(\\beta\\left(1-\\frac{\\varepsilon}{N}\\right)-\\alpha-2 \\rho\\right)\n\\end{align*}\nWith $h_1(0) =\\beta\\left(1-\\frac{\\varepsilon}{N}\\right)-\\rho-\\alpha$, one obtains\n\\[\n\\begin{array}{lll}\n\\left\\{(\u00a0-h_1\\, \\dot Q + \\ddot Q)\\right\\}_{t=0} & = & \\alpha\\, \\varepsilon \\, (\\beta\\left(1-\\frac{\\varepsilon}{N}\\right)-\\alpha-2 \\rho)\\\\\n& & \\qquad - (\\beta\\left(1-\\frac{\\varepsilon}{N}\\right) -\\rho-\\alpha) \\, \\alpha \\, \\varepsilon\\\\\n& = & -\\alpha \u00a0\\rho \\,\\varepsilon <0 \n\\end{array}\n\\]\nand with $h_2(0)= \\beta \\,\\varepsilon\\,(-\\beta + \\alpha)\\left(1-\\frac{\\varepsilon}{N}\\right) $, one gets\n\\[\u00a0\n\\left\\{\\left(h_2\\, (\u00a0-h_1\\, \\dot Q + \\ddot Q)\\,\\right)\\right\\}_{t=0} =  \\alpha \\, \\beta \\, \\rho\\, \\varepsilon^2 (\\beta -\\alpha)\\left(1-\\frac{\\varepsilon}{N}\\right) >0 .\n\\]\nObserve also that one has $X(0)=-\\beta \\, I(0) + \\dot Q(t)=\\varepsilon\\, (-\\beta + \\alpha) <0$. Therefore, by continuity w.r.t.~$t$, we obtain that for $t>0$ small enough, $X$ is the unique negative solution of \n\\[\n\\dot h_1 \\,  X^2+ (h_2\\, h_1 -\\dot h_2)\\,  X + h_2\\, ( - h_1\\, \\dot Q +\\ddot Q) =0 .\n\\] \nthat is\n\\[\nX=\\frac{-2(h_2\\, h_1 -\\dot h_2)-\\sqrt{(h_2\\, h_1 -\\dot h_2)^2+4\\dot h_1 ( h_1\\, \\dot y_2 -\\ddot y_2)}}{2\\dot h_1}\\,.\n\\]\nThe parameter $\\alpha$ can be then obtained from equation \\eqref{eq2}\n\\[\n\\alpha= \\frac{(N-y_2)\\dot h_1}{X}-h_1-\\rho\n\\]\nwhere $\\rho$ is given by \\eqref{rho}. The initial condition $\\varepsilon$ is simply reconstructed by $\\varepsilon=y_1(0)\/\\alpha$ and finally one obtains the parameter $\\beta=\\alpha-X(0)\/\\varepsilon$.\n\\end{proof}\n\n\\section{Analysis of the simplified model}\n\n\\begin{prop}\n\tSystem \\eqref{model2} is identifiable and observable for the observation \\eqref{observation}, in the sense of Definition \\ref{defi}.\n\\end{prop}\n\n\\smallskip\n\n\\begin{proof}\n\tAs for model \\eqref{model1}, one can determine the parameter $\\rho$ from any positive time as\n\t\\[\n\t\\rho = \\frac{y_1(t)-\\dot y_2(t)}{y_2(t)} , \\quad t > 0 .\n\t\\]\n\tThen from the dynamics of $I$ one can write\n\t\\begin{equation}\n\t\\label{h1}\n\t\\frac{\\beta S(t)}{N}-\\alpha = h_1(t):=\\frac{\\dot y_1(t)}{y_1(t)} + \\rho , \\quad t >  0\n\t\\end{equation}\n\twhere $h_1$ is a known function. Differentiating $h_1$ with respect to the time gives\n\t\\begin{equation}\n\t\\label{h1dot}\n\t\\dot h_1(t)=-\\beta^2S(t)\\frac{I(t)}{N^2}=-\\frac{\\beta I(t)}{N}(h_1(t)+\\alpha)\n\t\\end{equation}\n\tand differentiating twice\n\t\\[\n\t\\ddot h_1(t)= -\\frac{\\beta}{N}\\left(\\frac{\\beta S(t)}{N}-\\rho-\\alpha\\right)I(t)(h_1(t)+\\alpha)-\\frac{\\beta I(t)}{N}\\dot h_1(t) .\n\t\\]\n\tWith the expression \\eqref{h1}, we rewrite this last equation as follows\n\t\\[\n\t\\ddot h_1(t)= -\\frac{\\beta I(t)}{N}(h_1(t)-\\rho)(h_1(t)+\\alpha)-\\frac{\\beta I(t)}{N}\\dot h_1(t)\n\t\\]\n\tand with expression \\eqref{h1dot}\n\t\\[\n\t\\ddot h_1(t)= -\\dot h_1(t)(h_1(t)-\\rho)-\\frac{\\beta I(t)}{N}\\dot h_1(t) .\n\t\\]\n\tOne obtains then the following expression for $\\beta I(t)$\n\t\\[\n\t\\beta I(t)=-N\\left(\\frac{\\ddot h_1(t)}{\\dot h_1(t)} +h_1(t)-\\rho\\right)\n\t\\]\n\t(note that this expression is well defined because $h_1(t)+\\alpha  > 0$ for any $t$ and thus $\\dot h_1(t)<0$).\n\t\n\t\\medskip\n\t\n\tFinally, from \\eqref{h1dot} one reconstructs the parameter\n\t\\[\n\t\\alpha=-\\frac{N\\dot h_1(t)}{\\beta I(t)}-h_1(t)\n\t\\]\n\tand then the parameter\n\t\\[\n\t\\beta = \\alpha\\frac{\\beta I(t)}{y_1(t)}\n\t\\]\n\tAt last, the initial condition is recovered as $\\varepsilon=y_1(0)\/\\alpha$.\n\\end{proof}\n\n\\section{Parameter estimation}\n\nThe former analyses have shown that models \\eqref{model1} and \\eqref{model2} are not infinitesimally identifiable at time $0$ when the initial state is $(N-\\epsilon,\\epsilon,0,0)$. One has to wait a short time $t>0$ to have $Q(t)>0$ and formally identify parameters. Thus, we expect a weak accuracy of the parameters estimation at the very beginning, that should improve with time while the state get away from this initial state and new measurements come. This is why we have opted for a dynamical estimation with the help of observers. The use of observers, although usually dedicated to state estimation (rather to parameters estimation) possesses the advantage to tune the speed of error decay. Moreover, the choice of a speed-accuracy compromise can be balanced thru simulations with synthetic data corrupted by noise, before facing real data. \n\nNote also that for large times, the solutions of \\eqref{model1} and \\eqref{model2} converge asymptotically to non-observable non-identifiable states when $I$ and $Q$ are both null. Consequently, we do not look precisely for results about asymptotic convergence of the error (as this is usually done in observers theory), but rather for an exponential decay of the error estimation during initial transients.\n\nIn this section, we shall consider the additional hypothesis\n\n\\smallskip\n\n\\begin{assum} \n\t\\label{assum2}\n\tOne has\n\t\\[\n\t\\alpha \\leq \\rho\n\t\\]\n\\end{assum}\nthat means that the rate of placement in quarantine is not larger than the recovery rate, which is often the case in epidemic regimes.\n\n\\medskip\n\nWe shall denote the elementary symmetric polynomials, where $X$ is a vector in $\\mathbb{R}^n$, as\n\\[\n\\sigma_k^n(X):=\\sum_{1\\leq i_1< \\cdots < i_k\\leq n}\\left( \\prod_{j=1}^k X_j\\right), \\quad i=1\\cdots n\n\\]\n\\begin{prop} Let $\\lambda$ and $\\mu$ be two positive vectors in $\\mathbb{R}^4$ and $\\mathbb{R}^3$ respectively. For $t>0$, consider the dynamical system\n\\begin{equation}\n\\label{full_observer}\n\\left\\{\\begin{array}{l}\n    \\dot {\\hat {z}}_1 = \\hat \\delta -\\hat\\rho\n    -K_1(\\hat{z_{1}} -\\log(y_1(t)),\\\\[2mm]\n   \\dot {\\hat {z}}_2  =  \\dfrac{y_1(t)}{y_2(t)} -\\hat\\rho -K_2(\\hat{z_{1}} -\\log(y_1(t)),\\\\[2mm]\n    \\dot {\\hat \\delta}= -K_3(\\hat{z_{1}} -\\log(y_1(t))), \n    \\\\[2mm]\n    \\dot {\\hat \\rho}= -K_4(\\hat{z_{1}} -\\log(y_1(t)))-(\\hat{z_{2}} -\\log(y_2(t))) ,  \\\\[2mm]\n    \\dot {\\hat{y}}_1 =  \\hat{v} y_1(t) -K_5 y_1(t)(\\hat{y_1}-y_1(t)), \\\\[2mm]\n \\dot {\\hat{v}} = -\\hat{k} \\dfrac{y_1(t)}{N}-K_6 y_1(t)(\\hat{y_1}-y_1(t)),\\\\[2mm]\n  \\dot {\\hat{k}} = - K_7Ny_1(t)(\\hat{y_1}-y_1(t)) \n\\end{array}\\right.\n\\end{equation}\nwith the gains vector\n\\[\nK=\\left[\\begin{array}{c}\n\\sigma_1^4(\\lambda)\\\\\n\\sigma_1^4(\\lambda)+\\sigma_3^4(\\lambda)\\\\\n-\\sigma_4^4(\\lambda)\\\\ \n-\\sigma_2^4(\\lambda)-\\sigma_4^4(\\lambda)-1\\\\\n\\sigma_1^3(\\mu)\\\\\n\\sigma_2^3(\\mu)\\\\\n-\\sigma_3^3(\\mu)\n\\end{array}\\right]\n\\]\nThen, the output vector\n\\begin{equation}\n    \\label{estimator}\n\\left[\\begin{array}{c}\n\\hat\\rho(t)\\\\[2mm]\n\\hat\\beta(t) = \\dfrac{1}{2}\\left(\\hat k(t)-\\sqrt{\\max(\\hat k(t)^2-4\\hat\\delta(t)\\hat k(t),0)}\\right)\\\\[2mm]\n\\hat \\alpha(t)=\\hat\\beta(t)-\\hat\\delta(t)\n\\end{array}\\right]\n\\end{equation}\nis an estimator of $[\\rho,\\beta,\\alpha]^\\top$, whose exponential decay of the error can be made as fast as desired keeping the number\n\\[\nl=\\min_{i,j} \\min(\\lambda_i,\\mu_j) .\n\\]\nlarge, as long as $S(t)\/N$ remains close to $1$. Moreover, the state $I$ is estimated with\n\\begin{equation}\n    \\label{Iestim}\n\\hat I(t) = \\frac{y_1(t)}{\\hat\\alpha(t)}\n\\end{equation}\n\\end{prop}\n\n\\medskip\n\n\\begin{proof}\nPosit\n\\begin{equation}\n    \\label{defdelta}\n\\delta:=\\beta-\\alpha .\n\\end{equation}\nAs far as $S\/N$ remains close to $1$, the size of the population $I$ is small compared to $S$, and the dynamics of the outputs $y_1=\\alpha I$ and $y_2=Q$ can be approximated by the linear dynamics\n\\begin{equation}\n\\label{ApproxLin1}\n\\left\\{\\begin{array}{l}\n    \\dot y_1 =  \\delta\\,y_1- \\rho y_1 \\\\[2mm]\n    \\dot y_2=   y_1-\\rho \\, y_2\\\\\n\\end{array}\\right.\n\\end{equation}\nFor $t>0$ and $i=1,2$, consider the new outputs $z_i(t)=\\log(y_i(t))$, whose dynamics is given by the system\n\\begin{equation}\n\\label{ApproxLin2}\n\\left\\{\\begin{array}{l}\n    \\dot z_1 =\\delta- \\rho  \\\\[2mm]\n    \\dot z_2=  \\exp{(z_1-z_2)}-\\rho \\\\\n\\end{array}\\right.\n\\end{equation}\nFor this sub-system with unknown parameters $\\delta$ and $\\rho$, we consider the following candidate observer in $\\mathbb{R}^4$:\n\\begin{equation}\n\\label{obs1}\n\\left\\{\\begin{array}{l}\n    \\dot {\\hat {z}}_1 = \\hat \\delta -\\hat\\rho\n    -K_1(\\hat{z_{1}} -z_{1}),\\\\[2mm]\n   \\dot {\\hat {z}}_2  =  \\exp{(z_1-z_2)} -\\hat\\rho -K_2(\\hat{z_{1}} -z_{1}),\\\\[2mm]\n    \\dot {\\hat \\delta}= -K_3(\\hat{z_{1}} -z_{1})\\\\[2mm]\n    \\dot {\\hat \\rho}= -K_4(\\hat{z_{1}}-z_1) -(\\hat{z_{2}} -z_{2}) ,   \n\\end{array}\\right.\n\\end{equation}\nThe dynamics of the error $e_1=(\\hat{z_{1}},\\hat{z_{2}},\\hat\\delta,\\hat\\rho)^\\top-(z_1,z_2,\\delta,\\rho)^\\top$ \nis given by the linear time invariant system $\\dot e_1 =M_1\\,e_1$ with \n\\[M_1= \n\\left(\\begin {array}{cccc} \n-K_1& 0 & 1 & -1\n\\\\[3mm] \n-K_2& 0 &0 & -1\\\\[3mm] \n-K_3 & 0 & 0 & 0\\\\[3mm] \n-K_4& -1 &  0 & 0\n\\end {array} \n\\right) \n\\]\nA calculation shows that with the choice\n\\[\\left\\{ \n\\begin{array}{l}\nK_{1}=\\lambda_1+\\lambda_2+\\lambda_3+\\lambda_4, \\\\[2mm]\nK_{2}=\\sum\\lambda_i\n+\\sum\\lambda_i\\lambda_j\\lambda_k \n,\\\\[2mm]\nK_{3}=-\\lambda_1\\,\\lambda_2\\,\\lambda_3\\,\\lambda_4\n{K_{42}}\\\\[2mm]\nK_{4}=-\\left(\\sum\\lambda_i\\lambda_j\n+ \\lambda_1\\lambda_2\\lambda_3\\lambda_4+1\n\\right),\n\\end{array}\\right.\n\\]\nthe spectrum of $M_1$ is $\\{-\\lambda_i, \\; i =1\\cdots 4\\}$. This shows that the first four equations of system \\eqref{full_observer} gives the reconstruction of the parameters $\\delta$ and $\\rho$ with an exponential decay of the error larger than $\\min_i \\lambda_i$.\n\n\\medskip\n\nPosit now\n\\begin{equation}\n \\label{defk}\n k:=\\frac{\\beta^2}{\\alpha} .\n\\end{equation}\nand consider the variable\n\\begin{equation}\n \\label{defv}\n v(t):=\\beta\\frac{S(t)}{N}-\\rho-\\alpha\n \\end{equation}\nAs far as $S(T)\/N$ remains close to $1$, one can write the approximation\n \\[\n \\dot v = -\\frac{\\beta^2}{\\alpha}\\frac{S(t)}{N^2}y_1(t) \\simeq -\\frac{\\beta^2}{\\alpha N}y_1(t)\n \\]\nThen, this amounts to approximate the dynamics of \\eqref{model1} or \\eqref{model2} in the $(y_1,v,k)$ coordinates by the following dynamical system\n\\[\n\\left\\{\\begin{array}{l}\n\\dot y_1 = v y_1\\\\[2mm]\n\\dot v = -\\dfrac{k}{N}y_1\n\\end{array}\\right.\n\\]\nwhere $k$ is an unknown parameter, for which we consider the following candidate observer in $\\mathbb{R}^3$\n\\begin{equation}\n\\label{obs2}\n\\left\\{\n\\begin{array}{l}\n \\dot {\\hat{y}}_1  =  \\hat{v} y_1 -K_5 y_1 (\\hat{y_1}-y_1) \\\\[3mm]\n \\dot {\\hat{v}}  = -\\hat{k} \\dfrac{y_1}{N}-K_6 y_1 (\\hat{y_1}-y_1)\\\\[3mm]\n  \\dot {\\hat{k}}  = -K_7N y_1(\\hat{y_1}-y_1) \n\\end{array}\n\\right.\n\\end{equation}\nwhose dynamics of the error $e_2=(\\hat{y_1},\\hat{v},\\hat{k})^\\top-(y_1,v,k)^\\top$ is given by \nthe non-autonomous linear system\n\\begin{equation}\n    \\label{dynerror2}\n    \\dot e_2 = y_1(t) M_2\\, e_2\n\\end{equation}\nwith\n\\[\nM_2=\\left(\\begin{array}{ccc}\n-K_5 & 1 & 0 \\\\[3mm]\n-K_6 & 0 & -\\dfrac{1}{N} \\\\[3mm]\n-K_7N & 0 & 0\n\\end{array}\\right)\\]\nOne can easily check that for the choice\n\\[\\left\\{\n\\begin{array}{l}\n K_5 =  \\mu_1 + \\mu_2 + \\mu_3, \\\\[3mm]\n K_6 = \\mu_1\\mu_2 + \\mu_1\\mu_3 + \\mu_2\\mu_3,\\\\[3mm]\n K_7 = -\\mu_1\\mu_2\\mu_3 ,\n\\end{array}\n\\right.\n\\]\nthe spectrum of $M_2$ is $\\{-\\mu_i , \\; i =1\\cdots 3\\}$. Then, from \\eqref{dynerror2}, we obtain the upper bound on the error decrease\n\\[\n|\\hat k(t)-k| \\leq \\exp\\left(-(\\min_j \\mu_j) \\int_0^T y_1(\\tau)d\\tau\\right)||e_2(0)|| \n\\]\nwhose exponential decay can be made as large as desired with large $l$.\n\n\\medskip\n\nFinally, from the reconstruction of parameters $\\delta$, $k$ by observers \\eqref{obs1}, \\eqref{obs2} and expressions \\eqref{defdelta} and \\eqref{defk}, the original parameters $\\alpha$, $\\beta$ are recovered as roots of \n\\begin{equation}\n    \\label{roots}\n\\beta^2-k\\beta+k\\delta=0 \\Rightarrow \\beta = \\frac{k\\pm\\sqrt{k^2-4k\\delta}}{2}\n\\end{equation}\nNote first that Assumption 1 implies $\\beta>\\alpha$ and thus $k^2-4k\\delta>0$. Moreover, one has $k>\\beta(1+\\frac{\\rho}{\\alpha})$, and by \nAssumption 2 one has $k>2\\beta$, which implies that only the smaller root of \\eqref{roots} is valid, leading to the expression \\eqref{estimator} of the estimator. \nNote that this expression preserves the exponential decay of the error obtained for $\\delta$, $k$ and $\\rho$.\n\\end{proof}\n\n\n\\medskip\n\n\nLet us make some comments about this observer. It consists in reconstructing functions of the parameters $\\delta$ and $k$ and not directly the parameters $\\alpha$, $\\beta$. There is an apparent redundancy of variables $\\hat z_1$ and $\\hat y_1$ in dynamics \\eqref{full_observer}, which reconstruct $\\log y_1$ and $y_1$. Indeed, this allows to decouple the observer into two sub-systems of dimensions $4$ and $3$, which avoids the use of two large correction gains compared to a full order observer. Finally, outputs of these two sub-systems are coupled in expression \\eqref{estimator} to reconstruct the original parameters.\n\n\n\\section{Numerical illustrations}\nThe proposed observer has been tested with synthetic data\nfor a population size $N=10^5$ with parameter values $\\alpha=0.07$, $\\beta=0.4$, $\\rho=0.1$ and initial condition\n$I(0)=10$, $Q(0)=5$, $R(0)=0$ over a time horizon of $10$ days (see Figure \\ref{fig-sansbruit}). The gains have been computed for\nthe choice of vectors $\\lambda=[1;1.5;2;2.5]$ and $\\mu=[1\/(13.10^3);1\/(15.10^3);1\/(19.10^3)]$. Note that vector $\\mu$ has been chosen quite small to avoid too large gains when multiplied by $N$ in the observer equations.\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=7cm]{IMAGES\/alphasansbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/betasansbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/rhosansbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/Isansbruit.pdf}\n\t\t\\caption{Observer simulation without measurement noise}\n\t\t\\label{fig-sansbruit}\n\t\\end{center}\n\\end{figure}\n\nThen, we have simulated a measurement noise with a centered Gaussian law of variance equal to $5\\%$ of the signal (see Figure \\ref{fig-avecbruit}).\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=7cm]{IMAGES\/alphaavecbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/betaavecbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/rhoavecbruit.pdf}\n\t\t\\includegraphics[width=7cm]{IMAGES\/Iavecbruit.pdf}\n\t\t\\caption{Observer simulation with measurement noise}\n\t\t\\label{fig-avecbruit}\n\t\\end{center}\n\\end{figure}\nBecause the expression \\eqref{Iestim} of the estimation of the state $I$ is not filtered, we have applied a moving average smoothing to the estimation $\\hat I$ (see Figure \\ref{figIlisse}).\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=6cm]{IMAGES\/Iavecbruitlisse.pdf}\n\t\t\\caption{Smoothed estimation of the infected population in presence of measurement noise}\n\\label{figIlisse}\n\t\\end{center}\n\\end{figure}\t\t\n\nFinally, these simulations show that the method allows to reconstruct the parameter values in few days in a quite accurate manner. The estimation of the size of the infected population $I$ allows then the use of the model for predictions of the epidemics.\n\n\n\\section{Conclusion}\nThis work shows that although the identifiability of the SIR-Q models has singularity points where measured variables are null, it is possible to design an observer with exponential decay of the estimation error during the first stage of the epidemics, and recover parameters in few days. Further investigations will concern real data of COVID epidemics provided by various territories.\n\n\\clearpage\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Estimation}\n\nDefine $\\boldsymbol{\\mu}(\\boldsymbol{\\theta})$ to be the marginals of the graphical model with parameters $\\boldsymbol{\\theta}$, which may be computed with the $\\text{MARGINAL-ORACLE}${}.\n\n\\subsection{Proximal Algorithm Derivation}\n\nOur goal is to solve the following optimization problem:\n$$ \\hat{\\boldsymbol{\\mu}} = \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} L(\\boldsymbol{\\mu}) $$\nwhere $L$ is some convex function such as $ \\norm{ \\matr{Q}_\\mathcal{C} \\boldsymbol{\\mu} - \\mathbf{y} } $.  \n\nUsing the mirror descent algorithm \\cite{beck2003mirror}, we can use the following update equation:\n$$\\boldsymbol{\\mu}^{t+1} = \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\nabla L(\\boldsymbol{\\mu}^t) + \\frac{1}{\\eta_t} D(\\boldsymbol{\\mu}, \\boldsymbol{\\mu}^t)$$\nWhere $D$ is a Bregman distance measure defined as \n$$D(\\boldsymbol{\\mu}, \\boldsymbol{\\mu}^t) = \\psi(\\boldsymbol{\\mu}) - \\psi(\\boldsymbol{\\mu}^t) - (\\boldsymbol{\\mu} - \\boldsymbol{\\mu}^t)^T \\nabla \\psi(\\boldsymbol{\\mu}^t)$$\nfor some strongly convex and continuously differentiable function $\\psi$.  Using $ \\psi = -H$ to be the negative entropy, we arrive at the following update equation:\n\\begin{align*}\n\\boldsymbol{\\mu}^{t+1} &= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\nabla L(\\boldsymbol{\\mu}^t) + \\frac{1}{\\eta_t} D(\\boldsymbol{\\mu}, \\boldsymbol{\\mu}^t) \\\\\n&= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\nabla L(\\boldsymbol{\\mu}^t) + \\frac{1}{\\eta_t} \\Big( -H(\\boldsymbol{\\mu}) + \\boldsymbol{\\mu}^T \\nabla H(\\boldsymbol{\\mu}^t) \\Big) \\\\\n&= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\Big(\\nabla L(\\boldsymbol{\\mu}^t) + \\frac{1}{\\eta_t} \\nabla H(\\boldsymbol{\\mu}^t) \\Big) - \\frac{1}{\\eta_t} H(\\boldsymbol{\\mu}) \\\\\n&= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\Big(\\eta_t \\nabla L(\\boldsymbol{\\mu}^t) + \\nabla H(\\boldsymbol{\\mu}^t) \\Big) - H(\\boldsymbol{\\mu}) \\\\\n&= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\Big(\\eta_t \\nabla L(\\boldsymbol{\\mu}^t) - \\boldsymbol{\\theta}^t \\Big) - H(\\boldsymbol{\\mu}) \\\\\n&= \\boldsymbol{\\mu} \\big( \\boldsymbol{\\theta}^t -\\eta_t \\nabla L(\\boldsymbol{\\mu}^t) \\big)\n\\end{align*}\nThe first four steps are simple algebraic manipulation of the mirror descent update equation.  The final two steps use the observation that $ \\nabla H(\\boldsymbol{\\mu}^t) = -\\boldsymbol{\\theta}^t $ and that marginal inference can be cast as the following optimization problem: \\cite{wainwright2008graphical,vilnis2015bethe}\n$$ \\boldsymbol{\\mu}(\\boldsymbol{\\theta}) = \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} -\\boldsymbol{\\mu}^T \\boldsymbol{\\theta} - H(\\boldsymbol{\\mu}) $$\n\nThus, optimization over the marginal polytope is reduced to computing the marginals of a graphical model with parameters $ \\boldsymbol{\\theta}^t -\\eta_t \\nabla L(\\boldsymbol{\\mu}^t) $, which can be accomplished using belief propagation or some other $\\text{MARGINAL-ORACLE}$.\n\n\\subsection{Accelerated Proximal Algorithm Derivation}\n\nThe derivation of the accelerated proximal algorithm is similar.  It is based on Algorithm 3 from \\cite{xiao2010dual}.  Applied to our setting, step 4 of that algorithm requires solving the following problem:\n\\begin{align*}\n\\vect{\\nu}^t &= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\vect{\\bar{g}} - \\frac{4 K}{t (t+1)} H(\\boldsymbol{\\mu}) \\\\\n&= \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\frac{t(t+1)}{4K} \\boldsymbol{\\mu}^T \\vect{\\bar{g}} - H(\\boldsymbol{\\mu}) \\\\\n&= \\boldsymbol{\\mu} \\Big(-\\frac{t (t+1)}{4 L} \\bar{\\vect{g}} \\Big) \\\\\n\\end{align*}\nwhich we solve by using the $\\text{MARGINAL-ORACLE}${}.\n\n\n\\subsection{Direct Optimization}\n\nIn preliminary experiments we also evaluated a direct method to solve the optimization problem.  For the direct method, we estimate the parameters $\\hat{\\boldsymbol{\\theta}}$ directly by reformulating the optimization problem and instead solving the unconstrained problem $ \\hat{\\boldsymbol{\\theta}} = \\argmin_{\\boldsymbol{\\theta}} L\\big( \\boldsymbol{\\mu}(\\boldsymbol{\\theta}) \\big)$\nTo evaluate the optimization objective, we use $\\text{MARGINAL-ORACLE}${} to compute $\\boldsymbol{\\mu}(\\boldsymbol{\\theta})$ and then compute the loss. For optimization, it has been observed that it is possible to backpropagate through marginal inference procedures (with or without automatic differentiation software) to compute their gradients~\\cite{eaton2009choosing,domke2013learning}. We apply automatic differentiation to the entire forward computation~\\cite{maclaurin2015autograd}, which includes $\\text{MARGINAL-ORACLE}${}, to compute the gradient of $L$. \n\n\nSince this is now an unconstrained optimization problem and we can compute the gradient of $L$, many optimization methods apply. In our experiments, we use $L_2$ loss, which is smooth, and apply the L-BFGS algorithm for optimization~\\cite{byrd1995limited}.\n\nHowever, despite its simplicity, there is a significant drawback to the direct algorithm. It is not, in general, convex with respect to $\\boldsymbol{\\theta}$. This may seem surprising since the original problem is convex, i.e., $L(\\boldsymbol{\\mu})$ is convex with respect to $\\boldsymbol{\\mu}$ and $\\mathcal{M}$ is convex. Also, the most well known problem of this form, maximum-likelihood estimation in graphical models, \\emph{is} convex with respect to $\\boldsymbol{\\theta}$~\\cite{wainwright2008graphical}; however, this relies on properties of exponential families that do not apply to other loss functions. One can verify for losses as simple as $L_2$ that the Hessian need not be positive definite. As a result, the direct algorithm is not guaranteed to converge to a global minimum of the original convex optimization problem $\\min_{\\boldsymbol{\\mu} \\in \\mathcal{M}} L(\\boldsymbol{\\mu})$. We did not observe convergence problems in our experiments, but it was not better in practice than the proximal algorithms, which is why it is not included in the paper.\n\n\n\\section{Inference} \\label{sec:inference}\n\n\\begin{algorithm}[t]\n    \\caption{Inference for Factored Queries} \\label{alg:inference}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Input:} Parameters $\\boldsymbol{\\theta}$, factored query matrix $\\matr{Q}$\n    \\STATE {\\bfseries Output:} Query answers $\\matr{Q}\\,\\mathbf{p}_{\\boldsymbol{\\theta}}$\n    \\STATE $\\psi = \\{ \\exp(\\boldsymbol{\\theta}_C) \\mid C \\in \\mathcal{C} \\} \\cup \\{ \\matr{Q}_i \\mid i \\in [d] \\} $\n    \\STATE $Z = $\\text{MARGINAL-ORACLE}$(\\boldsymbol{\\theta}$)\n    \\STATE {\\bfseries return} $\\text{VARIABLE-ELIM}(\\psi, \\mathcal{X}) \/ Z$\n\\end{algorithmic}\n\\end{algorithm}\n\nWe now discuss how to exploit our compact factored representation of $\\mathbf{p}_{\\boldsymbol{\\theta}}$ to answer new linear queries.\nWe give an efficient algorithm for answering \\emph{factored linear queries}.\n\\begin{definition}[Factored Query Matrix] \\label{def:kron}\n  A factored query matrix $\\matr{Q}$ has columns that are indexed by $\\vect{x}$ and rows that are indexed by vectors $ \\mathbf{z} \\in [r_1] \\times \\dots \\times [r_d]$. The total number of rows (queries) is $r = \\prod_{i=1}^d r_i$. The entries of $\\matr{Q}$ are given by\n  $\\matr{Q}(\\mathbf{z}, \\vect{x}) = \\prod_{i=1}^d \\matr{Q}_i(\\mathbf{z}_i, \\vect{x}_i)$,\n  where $\\matr{Q}_i \\in \\matr{R}^{r_i \\times n_i}$ is a specified factor for the $i$th attribute. The matrix $\\matr{Q}$ can be expressed as $\\matr{Q} = \\matr{Q}_1 \\otimes \\dots \\otimes \\matr{Q}_d $, where $\\otimes$ is the Kronecker product.\n\\end{definition}\n\nFactored query matrices are expressive enough to encode any conjunctive query (or a cartesian product of such queries), and more. There are a number of concrete examples that demonstrate the usefulness of answering queries of this form, including:\n\n\\begin{itemize}[leftmargin=*,topsep=0pt,itemsep=2pt,partopsep=1pt,parsep=1pt]\n\\item Computing the marginal $\\boldsymbol{\\mu}_C$ for any $C \\subseteq [d]$ (including unmeasured marginals)\n\\item Computing the multivariate CDF of $\\boldsymbol{\\mu}_C$ for any $C \\subseteq [d]$.\n\\item Answering range queries.\n\\item Compressing the distribution by transforming the domain.\n\\item Computing the (unnormalized) expected value of one variable conditioned on other variables.\n\\end{itemize}\n\nFor the first two examples, we could have used standard variable elimination to eliminate all variables except those in $C$. Existing algorithms are not able to handle the other examples without materializing $\\hat{\\mathbf{p}}$ (or a marginal that supports the queries). Thus, our algorithm generalizes variable elimination. A more comprehensive set of examples, and details on how to construct these query matrices are given in section \\ref{sec:factoredqueries}\n\nThe procedure for answering these queries is given in Algorithm~\\ref{alg:inference}, which can be understood as follows. For a particular $\\mathbf{z}$, write $f(\\mathbf{z}, \\vect{x}) = \\matr{Q}(\\mathbf{z}, \\vect{x})\\mathbf{p}_{\\boldsymbol{\\theta}}(\\vect{x}) = \\prod_i \\matr{Q}_i(\\mathbf{z}_i, \\vect{x}_i) \\mathbf{p}_{\\boldsymbol{\\theta}}(\\vect{x})$. This can be viewed as an augmented graphical model on the variables $\\mathbf{z}$ and $\\vect{x}$ where we have introduced new pairwise factors between each $(\\vect{x}_i, \\mathbf{z}_i)$ pair defined by the query matrix. Unlike a regular graphical model, the new factors can contain negative values. The query answers are obtained by multiplying $\\matr{Q}$ and $\\mathbf{p}$, which sums over $\\vect{x}$. The $\\mathbf{z}$th answer is given by:\n\\begin{align*}\n(\\matr{Q} \\mathbf{p}_{\\boldsymbol{\\theta}})(\\mathbf{z}) &= \\sum_{\\vect{x} \\in \\mathcal{X}} \\matr{Q}(\\mathbf{z},\\vect{x}) \\mathbf{p}_{\\boldsymbol{\\theta}}(\\vect{x}) \\\\\n&= \\frac{1}{Z} \\sum_{\\vect{x} \\in \\mathcal{X}} \\prod_{i=1}^d \\matr{Q}_i(\\mathbf{z}_i, \\vect{x}_i) \\prod_{C \\in \\mathcal{C}} \\exp[\\boldsymbol{\\theta}_C(\\vect{x}_C)]\n\\end{align*}\nThis can be understood as marginalizing over the $\\vect{x}$ variables in the augmented model $f(\\mathbf{z}, \\vect{x})$. The {\\sc Variable-Elim} routine referenced in the algorithm is standard variable elimination to perform this marginalization; it can handle negative values with no modification. We stress that, in practice, factor matrices $\\matr{Q}_i$ may have only one row ($r_i = 1$, e.g., for marginalization); hence the output size $r = \\prod_{i=1}^d r_i$ is not necessarily exponential in $d$.\n\n\\eat{\nAs shown in algorithm~\\ref{alg:inference}, we use variable elimination over an augmented model with factors from the original model combined with the $\\matr{Q}_i$ matrices interpreted as two variable factors.  The $\\matr{Q}_i$ factors introduce new variables $\\mathbf{z}_1, \\dots, \\mathbf{z}_d$.  The correctness of the algorithm can be seen by analyzing the following sum:\n\n$$(\\matr{Q} \\mathbf{p}_{\\boldsymbol{\\theta}})(\\mathbf{z}) = \\sum_{\\vect{x} \\in \\mathcal{X}} \\matr{Q}(\\mathbf{z},\\vect{x}) \\mathbf{p}_{\\boldsymbol{\\theta}}(\\vect{x}) = \\frac{1}{Z} \\sum_{\\vect{x} \\in \\mathcal{X}} \\prod_{i=1}^d \\matr{Q}_i(\\mathbf{z}_i, \\vect{x}_i) \\prod_{C \\in \\mathcal{C}} \\exp[\\boldsymbol{\\theta}_C(\\vect{x}_C)] $$\n\nOf course, evaluating the sum directly is intractable when $| \\mathcal{X} |$ is large, but this is exactly the type of sum that can be evaluated efficiently using variable elimination (when the factorization allows it).  \\ry{more details might be necessary here... Dan what are your thoughts?}\n}\n\n\\subsection{Factored Query Matrices} \\label{sec:factoredqueries}\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|l|c|c|l|l|}\n$\\matr{Q}_i$ & Requirements & Size & Definition $(\\forall a \\in [n_i])$ & Description \\\\\\hline\n$\\matr{I}$ & & $n_i \\times n_i $ & $\\matr{Q}_i(a,a) = 1$ & keep variable in \\\\\n$\\matr{1}$ & & $1 \\times n_i$ & $\\matr{Q}_i(1,a) = 1$ & marginalize variable out \\\\\n$\\vect{e}_j$ & $j \\in [n_i]$ & $1 \\times n_i$ & $\\matr{Q}_i(1,j) = 1$ & inject evidence \\\\\n$\\vect{e}_S$ & $S \\subseteq [n_i]$ & $1 \\times n_i$ & $\\matr{Q}_i(1,j) = 1 \\:\\forall j \\in S$ & inject evidence (disjuncts)\\\\\n$\\matr{P}$ & & $n_i \\times n_i$ & $\\matr{Q}_i(b,a) = 1 \\:\\forall b \\geq a$ & transform into CDF \\\\\n$\\matr{R}_f$ & $f : [n_i] \\rightarrow [r_i]$ & $r_i \\times n_i$ & $\\matr{Q}_i(f(a), a) = 1$ & compress domain \\\\\n$\\mathbf{E}$ & & $1 \\times n_i$ & $\\matr{Q}_i(1, a) = a$ & reduce to mean \\\\\n$\\mathbf{E}_k$ & $k \\geq 1$ & $ k \\times n_i$ & $\\matr{Q}_i(b,a) = a^b \\: \\forall b \\leq k$ & reduce to first k moments \\\\\\hline\n\\end{tabular}\n\\caption{Example factors in the factored query matrix} \\label{table:factored}\n\\end{table*}\n\n\nTable~\\ref{table:factored} gives some example ``building block'' factors that can be used to construct factored query matrices.  This is by no means an exhaustive list of possible factors but it provides the reader with evidence that answering these types of queries efficiently is practically useful.  The factored query matrix for computing the marginal $\\boldsymbol{\\mu}_C$ uses $\\matr{Q}_{i} = \\matr{I} $ for $ i \\in C $ and $ \\matr{Q}_i = \\matr{1}$ for $i \\notin C$.  Similarly, the factored query matrix for computing the multivariate CDF of $\\boldsymbol{\\mu}_C$ would simply use $\\matr{Q}_i = \\matr{P}$ for $i \\in C$.  A query matrix for compressing a distribution could be characterized by functions $ f_i : [n_i] \\rightarrow [2] $ or equivalently binary matrices $ \\matr{Q}_i = \\matr{R}_{f_i} \\in \\matr{R}^{2 \\times n_i} $.   The query matrix for computing the (unnormalized) expected value of variable $i$ conditioned on variable $j$ would use $ \\matr{Q}_i = \\mathbf{E} $ and $ \\matr{Q}_j = \\matr{I}$ (and $\\matr{Q}_k = \\matr{1}$ for all other $k$).  These are only a few examples; these building blocks can be combined arbitrarily to construct a wide variety of interesting query matrices.  \n\n\\section{Loss Functions}\n\n\\subsection{$L_1$ and $L_2$ losses}\n\nThe $L_1$ and $L_2$ loss functions have simple (sub)gradients.\n\\begin{align*}\n\\nabla L_1 (\\boldsymbol{\\mu}) &= \\matr{Q}_{\\mathcal{C}}^T \\text{sign}(\\matr{Q}_{\\mathcal{C}} \\boldsymbol{\\mu} - \\mathbf{y}) \\\\\n\\nabla L_2(\\boldsymbol{\\mu}) &= \\matr{Q}_{\\mathcal{C}}^T (\\matr{Q}_{\\mathcal{C}} \\boldsymbol{\\mu} - \\mathbf{y})\n\\end{align*}\n\n\\subsection{Linear measurements with unequal noise}\n\nWhen the privacy budget is not distributed evenly to the measurements in the we have to appropriately modify the loss functions, which assume that the noisy answers all have equal variance. \nIn order to do proper estimation and inference we have to account for this varying noise level in the loss function.  In section ~\\ref{sec:formulation} we claimed that $L(\\mathbf{p}) = \\norm{ \\matr{Q} \\mathbf{p} - \\mathbf{y} }$ makes sense as a loss function when the noise introduced to $\\mathbf{y}$ are iid.  Luckily, even if this assumption is not satisfied it is easy to correct.  Assume that $ y_i = \\vect{q}_i^T \\mathbf{p} + \\varepsilon_i $ where $ \\varepsilon_i \\sim Lap(b_i) $.  Then $ \\frac{1}{b_i} y_i = \\frac{1}{b_i} \\vect{q}_i^T \\mathbf{p} + \\frac{1}{b_i} \\varepsilon_i $ and $ \\frac{1}{b_i} \\varepsilon_i \\sim Lap(1) $.  Thus, we can replace the query matrix $ \\matr{Q} \\leftarrow \\mathbf{D} \\matr{Q}$ and the answer vector $ \\mathbf{y} \\leftarrow \\mathbf{D} \\mathbf{y} $ where $ \\mathbf{D} $ is the diagonal matrix defined by $\\mathbf{D}_{ii} = \\frac{1}{b_i}$.  All the new query answers have the same effective noise scale, and so the standard loss functions may be used.  This idea still applies if the noise on each query answer is sampled from a normal distribution as well (for $(\\epsilon, \\delta)$-differential privacy).    \n\n\\subsection{Dual Query Loss Function}\n\nAlgorithm \\ref{alg:dq} shows DualQuery applied to workloads defined over the marginals of the data.  There are five hyper-parameters, of which four must be specified and the remaining one can be determined from the others. \n\nThe first step of the algorithm computes the answers to the workload queries.  Then for $T$ time steps observations are made about the true data via samples from the distribution $Q^t$.  These observations are used to find a record $ \\vect{x} \\in \\mathcal{X} $ to add to the synthetic database.\n\n\\begin{algorithm}[H]\n    \\caption{Dual Query for marginals workloads} \\label{alg:dq}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Input:} $\\vect{X}$, the true data\n    \\STATE {\\bfseries Input:} ${\\matr{W}}_{\\mathcal{C}}$, workload queries\n    \\STATE {\\bfseries Input:} $(s, T, \\eta$, $\\epsilon$, $\\delta$), hyper-parameters\n    \\STATE {\\bfseries Output:} synthetic database of $T$ records\n    \\STATE $ \\mathbf{y} = {\\matr{W}}_C \\boldsymbol{\\mu}_{\\vect{X}} $\n    \\STATE $ Q^1 = \\text{uniform}({\\matr{W}}) $\n    \\FOR{$t = 1, \\dots, T$}\n    \\STATE sample $\\vect{q}^t_1, \\dots \\vect{q}^t_s$ from $Q^t$\n    \\STATE $ \\vect{x}^t = \\argmax_{\\vect{x} \\in \\mathcal{X}} \\sum_{i=1}^s \\vect{q}^t_i \\boldsymbol{\\mu} - \\vect{q}^t_i \\boldsymbol{\\mu}_{\\vect{x}} $\n    \\STATE $ Q^{t+1} = Q^t \\odot \\exp{(-\\eta*(\\mathbf{y} - {\\matr{W}}_C \\boldsymbol{\\mu}_{\\vect{x}^t}))} $\n    \\STATE normalize $Q^t$\n    \\ENDFOR\n    \\STATE {\\bfseries return} $(\\vect{x}^1, \\dots, \\vect{x}^T)$\n\\end{algorithmic}\n\\end{algorithm}\n\nAlgorithm \\ref{alg:dqloss} shows a procedure for computing the negative log likelihood (our loss function) of observing the DualQuery output, given some marginals.  Evaluating the log likelihood is fairly expensive, as it requires basically simulating the entire DualQuery algorithm.  Fortunately we do not have to run the most computationally expensive step within the procedure, which is finding $\\vect{x}^t$.  We differentiate this loss function using automatic differentiation \\cite{maclaurin2015autograd} for use within our estimation algorithms.\n\n\\begin{algorithm}[H]\n    \\caption{Dual Query Loss Function} \\label{alg:dqloss}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Input:} $\\boldsymbol{\\mu}$, marginals of the data\n    \\STATE {\\bfseries Input:} ${\\matr{W}}_{\\mathcal{C}}$, workload queries\n    \\STATE {\\bfseries Input:} cache, all relevant output from DualQuery \\\\\n        > $\\vect{q}^t_1, \\dots \\vect{q}^t_s$ - sampled queries at each time step \\\\\n        > $\\vect{x}^t$ - chosen record at each time step \\\\\n    \\STATE {\\bfseries Output:} $L(\\boldsymbol{\\mu})$, the negative log likelihood\n    \\STATE $ \\mathbf{y} = {\\matr{W}}_C \\boldsymbol{\\mu} $\n    \\STATE $ Q^1 = \\text{uniform}({\\matr{W}}) $\n    \\STATE loss $= 0$\n    \\FOR{$t = 1, \\dots, T$}\n    \\STATE  loss --$= \\sum_{i=1}^s \\log{(Q^t(\\vect{q}^t_i))} $\n    \\STATE $ Q^{t+1} = Q^t \\odot \\exp{(-\\eta*(\\mathbf{y} - {\\matr{W}}_C \\boldsymbol{\\mu}_{\\vect{x}^t}))} $\n    \\STATE normalize $Q^t$\n    \\ENDFOR\n    \\STATE {\\bfseries return} loss\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Additional Experiments}\n\n\\subsection{$L_1$ vs. $L_2$ Loss}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.33\\textwidth]{fig\/l1_vs_l2a}\n\\includegraphics[width=0.33\\textwidth]{fig\/l1_vs_l2b}\n\\includegraphics[width=0.33\\textwidth]{fig\/l1_vs_l2c}\n\\caption{$L_1$ minimization vs. $L_2$ minimization, evaluated on $L_1$ loss, $L_2$ loss, and workload error} \\label{fig:l1_vs_l2}\n\\end{figure*}\n\nIn Section ~\\ref{approach} we mentioned that minimizing $L_1$ loss is equivalent to maximizing likelihood for linear measurements with Laplace noise, but that $L_2$ loss is more commonly used in the literature.  In this experiment we compare these two estimators side-by-side.  Specifically, we consider the workload from Figure ~\\ref{fig:accuracy} and measurements chosen by HDMM with $\\epsilon = 1.0$.   As expected, performing $L_1$ minimization results in lower $L_1$ loss but higher $L_2$ loss, although the difference is quite small, especially for $L_1$ loss.  The difference is larger for $L_2$ loss.  Minimizing $L_2$ loss results in lower workload error, indicating that it generalizes better.  This is somewhat surprising given that $L_1$ minimization is maximizing likelihood.  Another interesting observation is that the workload error actually starts going up after about 200 iterations, suggesting that some form of over-fitting is occurring.  There minimum workload error achieved was $ 0.066 $ while the final workload error was $0.084$ --- a pretty meaningful difference.  Of course, in practice we cannot stop iterating when workload error starts increasing because evaluating it requires looking at the true data.\n\n\\section{Additional Details}\n\n\\subsection{Unknown Total}\n\nOur algorithms require $m$, the total number of records in the dataset is known or can be estimated.  Under a slightly different privacy definition where $\\textrm{nbrs}{(\\mathbf{X})}$ is the set of databases where a single record is added or removed (instead of modified), this total is a sensitive quantity which cannot be released exactly \\cite{Dwork14Algorithmic}.  Thus, the total is not known in this setting, but a good estimate can typically be obtained from the measurements taken, without spending additional privacy budget.  First observe that $ \\vect{1}^T \\boldsymbol{\\mu}_C = m$ is the total for an unnormalized database.  Now suppose we have measured $\\mathbf{y}_C = \\matr{Q}_C \\boldsymbol{\\mu}_C + \\mathbf{z}_C $.  Then as long as $ \\vect{1}^T $ is in the row-space of $\\matr{Q}_C$, $ m_C = \\vect{1}^T \\matr{Q}_C^+ \\mathbf{y}_C $ is an unbiased estimate for $m$ with variance $ Var(m_C) = Var(\\mathbf{y}_C) \\norm{\\vect{1}^T \\matr{Q}_C^+}_2^2 $. This is a direct consequence of Proposition 9 from \\cite{li2015matrix}.  We thus have multiple estimates for $m$ which we can combine using inverse variance weighting, resulting in the final estimate of $\\hat{m} = \\frac{\\sum_C m_C \/ Var(m_C)}{\\sum_C 1 \/ Var(m_C)} $, which we can use in place of $m$.\n\n\\subsection{Multiplicative Weights vs Entropic Mirror Descent} \\label{sec:mw_emd}\n\nRecall from Section \\ref{sec:methods} that the multiplicative weights update equation is:\n$$ \\hat{\\mathbf{p}} \\leftarrow \\hat{\\mathbf{p}} \\odot \\exp{(\\vect{q}_i (\\vect{q}_i^T \\hat{\\mathbf{p}} - y_i)) \/ 2m} \/ Z $$\nand the update is applied (possibly cyclically) for $i=1, \\dots, T$.  Now imagine taking all of the measurements and organizing them into a $ T \\times n$ matrix $\\matr{Q}$.  Then we can apply all the updates at once, instead of sequentially, and we end up with the following update equation.\n$$ \\hat{\\mathbf{p}} \\leftarrow \\hat{\\mathbf{p}} \\odot \\exp{(\\matr{Q}^T (\\matr{Q} \\hat{\\mathbf{p}} - \\mathbf{y}) \/ 2m)} \/ Z $$\nObserving that $ \\nabla L_2(\\hat{\\mathbf{p}}) = \\matr{Q}^T (\\matr{Q} \\hat{\\mathbf{p}} - \\mathbf{y}) $, this simplifies to:\n$$ \\hat{\\mathbf{p}} \\leftarrow \\hat{\\mathbf{p}} \\odot \\exp{(\\nabla L_2(\\hat{\\mathbf{p}}) \/ 2m)} \/ Z $$\nwhich is precisely the update equation for entropic mirror descent for minimizing $L_2(\\mathbf{p})$ over the probability simplex \\cite{beck2003mirror}.\n\n\\section{Background and Problem Statement} \\label{sec:background}\n\n\\def{\\cal A}{{\\cal A}}\n\\def\\mathbf{X}{\\mathbf{X}}\n\\def\\textrm{nbrs}{\\textrm{nbrs}}\n\n\\newcommand{\\compl}[1]{{#1}^{-}}\n\n\\textbf{Data.}\nOur input data represents a population of individuals, each contributing a single record \\(\\vect{x} = (x_1, \\ldots, x_d)\\) where \\(x_i\\) is the \\(i^{th}\\) attribute belonging to a discrete finite domain \\(\\mathcal{X}_i\\) of \\(n_i\\) possible values.  The full domain is \\(\\mathcal{X} = \\prod_{i=1}^d \\mathcal{X}_i\\) and its size \\(n = \\prod_{i=1}^d n_i\\) is exponential in the number of attributes. A dataset $\\mathbf{X}$ consists of $m$ such records \\(\\mathbf{X} = (\\vect{x}^{(1)}, \\ldots, \\vect{x}^{(m)})\\). We also consider a normalized contingency table representation $\\mathbf{p}$,\nwhich counts the fraction of the population with record equal to \\(\\vect{x}\\), for each \\(\\vect{x}\\) in the domain. That is, $\\mathbf{p}(\\vect{x}) = \\frac{1}{m}\\sum_{i=1}^m \\matr{I}\\{\\vect{x}^{(i)}  = \\vect{x}\\}, \\forall \\vect{x} \\in \\mathcal{X}$, where $\\matr{I}\\{ \\cdot \\}$ is an indicator function. Thus \\(\\mathbf{p}\\) is a probability vector in \\(\\matr{R}^{n}\\) with index set \\(\\mathcal{X}\\) (ordered lexicographically).  We write $\\mathbf{p} = \\mathbf{p}_{\\mathbf{X}}$ when it is important to denote the dependence on $\\mathbf{X}$.\n\n\\textbf{Queries, Marginals, and Measurements.}\nWe focus on the most common case of linear queries expressed over subsets of attributes. We will describe an extension to a generalized class of queries, including non-linear ones, in Section~\\ref{optimization-in-terms-of-marginals}.\nA {\\em linear query set} $f_{\\matr{Q}}(\\mathbf{X})$ is defined by a \\emph{query matrix} \\(\\matr{Q} \\in \\matr{R}^{r \\times n}\\) and has answer \\(f_{\\matr{Q}}(\\mathbf{X}) = \\matr{Q}\\, \\mathbf{p}_{\\mathbf{X}}\\). The $i$th row of $\\matr{Q}$, denoted $\\vect{q}_i^T$ represents a single scalar-valued query. In most cases we will refer unambiguously to the matrix $\\matr{Q}$, as opposed to $f_{\\matr{Q}}$, as the query set.\nWe often consider query sets that can be expressed on a {\\em marginal} (over a subset of attributes) of the probability vector $\\mathbf{p}$.\nLet \\(A \\subseteq [d]\\) identify a subset of attributes and, for $\\vect{x} \\in \\mathcal{X}$, let \\(\\vect{x}_A = (x_i)_{i \\in A}\\) be the sub-vector of \\(\\vect{x}\\) restricted to \\(A\\). \nThen the marginal probability vector (or simply ``marginal on A'') \\(\\boldsymbol{\\mu}_A\\), is defined by: \n\\[ \\boldsymbol{\\mu}_A(\\vect{x}_A) = \\frac{1}{m} \\sum_{i=1}^m \\matr{I}\\{ \\vect{x}^{(i)}_A = \\vect{x}_A\\}, \\quad \\forall \\vect{x}_A \\in \\mathcal{X}_A := \\prod_{i \\in A} \\mathcal{X}_i.\n\\]\nThe number of entries of the marginal is  $n_A := |\\mathcal{X}_A| = \\prod_{i \\in A} n_i$, which is exponential in $|A|$ but may be considerably smaller than $n$.\nNote that \\(\\boldsymbol{\\mu}_A(\\vect{x}_A)\\) is a linear function of \\(\\mathbf{p}\\), so there exists a matrix \\(\\matr{M}_A \\in \\matr{R}^{n_A \\times n}\\) such that \\(\\boldsymbol{\\mu}_A = \\matr{M}_A\\mathbf{p}\\).\nWhen a query set depends only on the marginal vector \\(\\boldsymbol{\\mu}_A\\), we call it a \\emph{marginal query set} written as \\(\\matr{Q}_A \\in \\matr{R}^{r_A \\times n_A}\\), and with answer \\(f_{\\matr{Q}_A}(\\mathbf{X}) = \\matr{Q}_A\\, \\boldsymbol{\\mu}_A\\). The marginal query set $\\matr{Q}_A$ is equivalent to the query set $\\matr{Q} = \\matr{Q}_A \\matr{M}_A$ on the full contingency table, since $\\matr{Q}_A \\boldsymbol{\\mu}_A = (\\matr{Q}_A \\matr{M}_A) \\mathbf{p}$. \nOne marginal query set asks for the marginal vector itself, in which case $\\matr{Q}_A = \\matr{I}_{n_A \\times n_A}$ (the identity matrix).\n\nIn our problem formulation, we consider measurements consisting of a collection of marginal query sets. Specifically, let \\(\\mathcal{C}\\) be a collection of \\emph{measurement sets}, where each $C \\in \\mathcal{C}$ is a subset of $[d]$.\\footnote{Later, these will comprise the cliques of a graphical model, as the notation suggests.}\nFor each measurement set $C \\in \\mathcal{C}$, we are given a marginal query set $\\matr{Q}_C$. \n The following notation is helpful to refer to combined measurements and their marginals.  Let \\(\\boldsymbol{\\mu} = (\\boldsymbol{\\mu}_C)_{C \\in \\mathcal{C}}\\) be the combined vector of marginals, and let $\\matr{Q}_{\\mathcal{C}}$ be the block-diagonal matrix with diagonal blocks $\\{\\matr{Q}_C\\}_{C \\in \\mathcal{C}}$, so that the entire set of query answers can be expressed as $\\matr{Q}_{\\mathcal{C}} \\boldsymbol{\\mu}$. Finally, let $\\matr{M}_{\\mathcal{C}}$ be the matrix that vertically concatenates the matrices $\\{\\matr{M}_C\\}_{C \\in \\mathcal{C}}$, so that $\\boldsymbol{\\mu} = \\matr{M}_{\\mathcal{C}} \\mathbf{p}$ and $\\matr{Q}_\\mathcal{C} \\boldsymbol{\\mu} = \\matr{Q}_\\mathcal{C} \\matr{M}_\\mathcal{C} \\mathbf{p}$. This shows that our measurements are equivalent to the combined query set $\\matr{Q} = \\matr{Q}_\\mathcal{C} \\matr{M}_\\mathcal{C}$ applied to the full table $\\mathbf{p}$. \n\n\n\n\\textbf{Differential privacy.}\nDifferential privacy protects individuals by bounding the impact any one individual can have on the output of an admissible  algorithm. This is formalized using the notion of neighboring datasets.\nLet $\\textrm{nbrs}(\\mathbf{X})$ denote the set of datasets formed by replacing any $\\vect{x}^{(i)} \\in \\mathbf{X}$ with an arbitrary new record $\\vect{x}'^{(i)} \\in \\mathcal{X}$.\n\n\\begin{definition}[Differential Privacy;~\\citeauthor{Dwork06Calibrating}, \\citeyear{Dwork06Calibrating}] \\label{def:dp}\nA randomized algorithm ${\\cal A}$ satisfies $(\\epsilon, \\delta)$-differential privacy if for any input $\\mathbf{X}$, any $\\mathbf{X}' \\in \\textrm{nbrs}(\\mathbf{X})$, and any subset of outputs $S \\subseteq \\textrm{Range}({\\cal A})$, \n$$ \\Pr[{\\cal A}(\\mathbf{X}) \\in S] \\leq \\exp(\\epsilon) \\Pr[{\\cal A}(\\mathbf{X}') \\in S] + \\delta$$\n\\end{definition}\n\nWhen $ \\delta = 0 $ we say $ {\\cal A} $ satisfies $\\epsilon$-differential privacy.\nDifferentially private answers to $f_{\\matr{Q}}$ are typically obtained with a noise-addition mechanism, such as the Laplace or Gaussian mechanism.\nFor $\\epsilon$-differential privacy, the noise added to the output of $f_{\\matr{Q}}$ is determined by the {\\em $L_1$ sensitivity} of $f_{\\matr{Q}}$, which, specialized to linear queries, is defined as\n$\\Delta_{\\matr{Q}} =  \\max_{\\mathbf{X}, \\mathbf{X}' \\in \\textrm{nbrs}(\\mathbf{X})} \\| \\matr{Q}\\, \\mathbf{p}_{\\mathbf{X}} - \\matr{Q}\\, \\mathbf{p}_{\\mathbf{X}'} \\|_1$. It is straightforward to show that $\\Delta_{\\matr{Q}}=\\frac{2}{m} \\norm{\\matr{Q}}_1$ where $\\norm{\\matr{Q}}_1$ is the maximum $L_1$ norm of the columns of $\\matr{Q}$. \n\n\n\\begin{definition}[Laplace Mechanism;~\\citeauthor{Dwork06Calibrating}, \\citeyear{Dwork06Calibrating}] \\label{def:laplace}\nGiven a query set $\\matr{Q} \\in \\matr{R}^{r \\times n}$ of $r$ linear queries, the Laplace mechanism is defined as $\\mathcal{L}(\\mathbf{X}) = \\matr{Q}\\, \\mathbf{p}_{\\mathbf{X}} + \\mathbf{z}$ where $\\mathbf{z} = (z_1, \\dots, z_r)$ and each $z_i$ is an i.i.d. random variable from $\\text{Laplace}(\\Delta_{\\matr{Q}}\/\\epsilon)$.\n\\end{definition}\n\nThe Laplace mechanism satisfies $\\epsilon$-differential privacy.  The sequential composition property implies that if we answer two query sets $\\matr{Q}_1$ and $\\matr{Q}_2$, under $\\epsilon_1$ and $\\epsilon_2$ differential privacy, respectively, then the combined answers are $(\\epsilon_1+\\epsilon_2)$-differentially private. The \\emph{post-processing} property of differential privacy~\\cite{Dwork14Algorithmic} asserts that an algorithm that accepts as input the output of an $\\epsilon$-differentially algorithm, but does not use the original protected data, is also $\\epsilon$-differentially private.  \n\n\n\\newcommand{\\target}{{\\matr{W}}}\n\n\\textbf{Problem Statement.}\nWe assume as given a collection \\(\\mathcal{C}\\) of \\emph{measurement sets}, and for each $C \\in \\mathcal{C}$: a marginal query set $\\matr{Q}_C$, a privacy parameter $\\epsilon_C$, and an $\\epsilon_C$-differentially private \\emph{measurement} \\(\\mathbf{y}_C = \\matr{Q}_C \\boldsymbol{\\mu}_C + \\text{Lap}(\\Delta_{\\matr{Q}_C}\/\\epsilon_C)\\).  The combined \\emph{measurements} are \\(\\mathbf{y} = (\\mathbf{y}_C)_{C \\in \\mathcal{C}}\\) which satisfy $\\epsilon$-differential privacy for $\\epsilon=\\sum_{C\\in\\mathcal{C}} \\epsilon_C$ by sequential composition.\nNote that there is no loss of generality in these assumptions; in the extreme case, there may be just a single measurement set $C = [d]$ consisting of all attributes. Formulating the problem this way will allow us to realize computational savings when measurements are not full-dimensional, which is common in practice. We also emphasize that the marginal query set $\\matr{Q}_C$ is often a complex set of linear queries expressed over measurement set $C$ (not simply a marginal).  Many past works~\\cite{li2015matrix,li2014data,qardaji2013understanding,Nikolov13Geometry,ding2011differentially,Xiao10Differential,li2010optimizing,hay2010boosting,Barak07Privacy} have shown that it is beneficial, in the presence of noise-addition for privacy, to measure carefully chosen query sets which balance sensitivity against efficient reconstruction of the workload queries.\n\n\nOur goal is: given $\\mathbf{y}$, derive answers to (possibly different) workload queries $\\target$. There are multiple possible motivations: $\\target$ may include new queries that were not part of the original measurements; or it is possible that $\\target$ is a subset of measurement queries, but we can obtain a more accurate answer by combining \\emph{all} of the available information to estimate \\(\\target \\mathbf{p}\\) as opposed to just using the noisy answer we got. We describe an extension to non-linear queries and more general linear queries in Section~\\ref{optimization-in-terms-of-marginals}; this will be applied to the DualQuery algorithm~\\cite{Gaboardi14Dual} in Section~\\ref{sec:methods}.\n\n\\section{Notation and setup}\\label{notation-and-setup}\n\n\\subsection{General}\\label{general}\n\n\\begin{itemize}\n\\item\n  \\(\\matr{I}\\{ \\cdot \\}\\) is indicator function.\n\\item\n  \\([m] = \\{1, 2, \\ldots, m\\}\\) for any integer \\(m\\)\n\\end{itemize}\n\n\\subsection{Individual data records}\\label{individual-data-records}\n\nSingle record \\(\\vect{x} = (x_1, \\ldots, x_d)\\)\n\\begin{itemize}\n\\item \\(x_i\\) is \\(i\\)th attribute\n\\item \\(x_i\\) belongs to domain \\(\\mathcal{X}_i\\), which is finite with \\(n_i\\) possible values\n\\item Overall domain is \\(\\mathcal{X} = \\prod_{i=1}^d \\mathcal{X}_i\\), which has size \\(n = \\prod_{i=1}^d n_i\\). This is exponential in the number of attributes \\(d\\).\n\\end{itemize}\n  \n\\subsection{Data set and full contingency\ntable}\\label{data-set-and-full-contingency-table}\n\n\\(m\\) individuals with records \\(\\vect{x}^{(1)}, \\ldots, \\vect{x}^{(m)}\\)\n\nData set \\(\\mathbf{X} = (\\vect{x}^{(1)}, \\ldots, \\vect{x}^{(m)})\\)\n\nContingency table \\(\\mathbf{p}\\) counts the number of individuals with record\nequal to \\(\\vect{x}\\) for each \\(\\vect{x}\\) in the domain. Definition:\n\\[\\mathbf{p}(\\vect{x}) = \\sum_{i=1}^m \\matr{I}\\{\\vect{x}^{(i)}  = \\vect{x}\\}, \\quad \\forall \\vect{x} \\in \\mathcal{X}\\]\n\n(Convention: use \\(\\mathbf{p}\\) to represent contingency tables because\ncontingency tables and their marginals behave much like probability\ntables and their marginals. The main difference is: contingency tables\nare non-negative integers and sum to \\(m\\); probabilities are\nnon-negative real numbers and sum to \\(1\\).)\n\nWe will consider \\(\\mathbf{p}\\) to be a vector in \\(\\matr{R}^{n}\\) with index set\n\\(\\mathcal{X}\\) (say, ordered lexicographically)\n\n\\subsection{Marginals}\\label{marginals}\n\nLet \\(A \\subseteq [d]\\) be a generic subset of attributes. Define:\n\n\\begin{itemize}\n\\item\n  \\(\\vect{x}_A = (x_i)_{i \\in A}\\) to be the subvector of \\(\\vect{x}\\) restricted to\n  attributes in \\(A\\)\n\\item\n  \\(\\mathcal{X}_A = \\prod_{i \\in A} \\mathcal{X}_i\\) be the restricted domain, with\n  size \\(n_A = \\prod_{i \\in A} n_i\\)\n\\item\n  Let \\(-A = [m]\\setminus A\\) be the complement of \\(A\\)\n\\end{itemize}\n\nDefine the marignal contingency table (or just ``marginal'') \\(\\boldsymbol{\\mu}_A\\)\nas \\[\n  \\boldsymbol{\\mu}_A(\\vect{x}_A) = \\sum_{i=1}^m \\matr{I}\\{ \\vect{x}^{(i)}_A = \\vect{x}_A\\}\n  \\]\n\nNote that \\(\\boldsymbol{\\mu}_A(\\vect{x}_A)\\) is a linear function of \\(\\mathbf{p}\\): \\[\n  \\boldsymbol{\\mu}_A(\\vect{x}_A) = \\sum_{\\vect{x}_{-A}\\in \\mathcal{X}_{-A}} \\mathbf{p}(\\vect{x}_{A}, \\vect{x}_{-A})\n  \\] so there exists a matrix \\(M_A \\in \\matr{R}^{n_A \\times n}\\) such that\n\\(\\boldsymbol{\\mu}_A = M_A\\mathbf{p}\\).\n\n\\textbf{(Q: Do you want to use bold or mathbb for matrices?)}\n\n\\subsection{Queries}\\label{queries}\n\nA linear query is a vector \\(\\vect{q}\\) with answer \\(\\vect{q}^T \\mathbf{p}\\).\n\nA set of \\(r\\) linear queries \\(\\vect{q}_1, \\ldots, \\vect{q}_r\\) can be arranged in\na matrix \\(Q \\in \\matr{R}^{r \\times n}\\) with \\(i\\)th row equal to \\(\\vect{q}_i\\).\n\nThe vector of answers to the queries is \\(Q \\mathbf{p}\\).\n\n\\(Q\\) is the set of \\emph{measurement queries}\n\n\\subsection{Queries over marginals}\\label{queries-over-marginals}\n\nA query may depend only on the marginal table \\(\\boldsymbol{\\mu}_A\\) for a subset\n\\(A\\) of attributes. In this case we write the query as\n\\(\\vect{q}_A \\in \\matr{R}^{n_A}\\) and the answer is \\(\\vect{q}_A^T \\boldsymbol{\\mu}_A\\).\n\n(If needed: write \\(\\vect{q}_{A,i}\\) if we need to index multiple queries.)\n\n(Optional: A query \\(\\vect{q}\\) can be written this way if it can be expressed\nas \\(\\vect{q} = M_A^T \\vect{q}_A\\). Then \\[\n  \\vect{q}^T \\mathbf{p} = \\vect{q}_A^T M_A \\mathbf{p} = \\vect{q}_A^T \\boldsymbol{\\mu}_A\n  \\] This means \\(\\vect{q}\\) is in the row space of \\(M_A\\). Additional\ninterpretation?)\n\n\\subsection{Measurement cliques}\\label{measurement-cliques}\n\nAssume each measurement query depends only on the marginal \\(\\boldsymbol{\\mu}_C\\)\nfor some \\(C \\in \\mathcal{C}\\), where \\(\\mathcal{C}\\) is a collection of cliques (subsets\nof attributes). We refer to the collection \\(\\mathcal{C}\\) as the\n\\emph{measurement cliques}.\n\nThere are \\(r_C\\) total queries for clique \\(C\\), given in the rows of\nthe matrix \\(Q_C \\in \\matr{R}^{r_C \\times n_C}\\).\n\nWe observe answers \\(\\mathbf{y}_C = Q_C \\boldsymbol{\\mu}_C + \\text{Lap}(\\Delta_C\/\\epsilon)\\)\nfor each \\(C\\). (Check this. Describe sensitivity \\(\\Delta_C\\).)\n\nLet \\(\\mathbf{y} = (\\mathbf{y}_C)_{C \\in \\mathcal{C}}\\) be the complete set of answers to\nmeasurement queries (the concatenation of the query answers from each\nclique). Let \\(\\boldsymbol{\\mu} = (\\boldsymbol{\\mu}_C)_{C \\in \\mathcal{C}}\\) be the complete set of\nmarginals for those cliques.\n\n(Optional: there is a matrix \\(M\\) such that \\(\\boldsymbol{\\mu} = M \\mathbf{p}\\). It is\nobtained by stacking the matrices \\(M_C\\) for all \\(C \\in \\mathcal{C}\\).)\n\n\\textbf{Observation}: measurements \\(\\mathbf{y}\\) depend only on \\(\\boldsymbol{\\mu}\\), which\nis much lower dimensional than \\(\\mathbf{p}\\).\n\n\\textbf{Terminology summary} (refine if needed):\n\n\\begin{itemize}\n\\item Measurement cliques \\(\\mathcal{C}\\)\n\\item Measurement marginals \\(\\boldsymbol{\\mu} = (\\boldsymbol{\\mu}_C)_{C \\in \\mathcal{C}}\\)\n\\item Measurement queries \\(Q = \\{Q_C\\}_{C \\in \\mathcal{C}}\\)\n\\item Measurements \\(\\mathbf{y} = (\\mathbf{y}_C)_{C \\in \\mathcal{C}}\\)\n\\end{itemize}\n\n\n\n\\section{Experiments} \\label{sec:experiments}\n\n\\ry{this section will need to be restructured quite a bit}\n\\ry{can we combine related work and experiments? As we will compare against related work}\n\\ry{also, plots are hard to read in black and white}\n\nNext we evaluate empirically the efficiency and accuracy of our proposed methods.  All experiments are performed on a quad-core Intel i7 3.6 GHz processor with 16 GB of RAM.  \n\nFor the efficiency experiments, we used synthetic data, allowing us to systematically vary the domain size and number of dimensions of the data and evaluate their impact.  Specifically, we consider a dataset with 1 million tuples and $d$ variables, each with domain size $n_i = 10$. We vary $d = 2 \\dots 16 $. We consider chain structured measurements, which include the $\\{ i, i+1 \\}$ marginals for $ 1 \\leq i \\leq d-1 $.  \n\nFor accuracy experiments, we use the Adult dataset obtained from the UCI machine learning repository \\cite{Dua:2017}, which contains 32,561 tuples and 11 discrete (or appropriately discretized) attributes: age, workclass, education, marital status, occupation, relationship to householder, race, sex, hours-per-week, country, and income>=50K.  The total domain size is $ 100 \\times 9 \\times 16 \\times 7 \\times 15 \\times 6 \\times 5 \\times 2 \\times 100 \\times 42 \\times 2 \\approx 7.6 \\times 10^{11} $.  \n\n\\begin{figure}[t]\n\\centering\n\\subcaptionbox{\\label{fig:chain} Model estimation from measurements}{\\includegraphics[width=0.45\\textwidth]{fig\/chain.pdf}}\n\\subcaptionbox{\\label{fig:inference} Target query inference}{\\includegraphics[width=0.45\\textwidth]{fig\/chain_inference.pdf}}\n\\caption{The runtime of model estimation and inference as dimensionality grows.} \\label{fig:efficiency}\n\\end{figure}\n\n\n\n\\textbf{Efficiency of estimation.}\nFig. \\ref{fig:chain} shows the runtime of our two algorithms for model estimation, Direct and Proximal, compared with two competing methods, LSQR and MW. LSQR \\cite{Zhang18Ektelo:} is an iterative algorithm for solving an ordinary least squares problem (without non-negativity constraints); it the most efficient method we are aware of that handles general linear measurements. Least-squares is a very common estimation method in the literature \\cite{hay2010boosting,Nikolov13Geometry,li2014data,qardaji2013understanding,ding2011differentially,Xiao10Differential,li2010optimizing}. We use it here to find the minimum-norm $d$-way contingency table which minimizes squared error given the chain-structured measurements.  \nMW uses multiplicative weights to repeatedly update an estimate of the full contingency table to reflect the noisy measurements \\cite{hardt2012simple}.  While the authors of  \\cite{hardt2012simple} describe an enhancement that uses a factored representation, it only applies when the measurements can be partitioned into groups that depend on disjoint subsets of variables.  That enhancement offers no benefit for chain-structured measurements, but would make MW efficient for measurements on disjoint sets of variables.  Nevertheless, our methods do not have such an extreme requirement in order to benefit from the efficiency of a factored representation.\n\n\n\nThe Direct and Proximal algorithms are both linear in the number of dimensions for chain structured measurements, and as shown in Fig. \\ref{fig:chain}, our algorithms are highly scalable, running in a matter of seconds up to $d=16$.  We were unable to run LSQR or MW beyond $d=7$ due to the memory requirements of jointly representing the measurements.  For these methods, the measurements are represented as a sparse matrix, which requires storing $(d-1) \\cdot 10^d$ values.  If the measurement queries were represented more compactly, these algorithms may be able to scale up to $d=8$ or $d=9$, but beyond that storing the full contingency table would be impossible on common hardware.\n\n\n\\textbf{Efficiency of inference.}\nWe now empirically evaluate the inference algorithm we described in Sec.~\\ref{inference} to answer factored queries with $\\matr{Q}_i \\in \\matr{R}^{2 \\times n_i}$.  The details of the queries encoded are not important for measuring efficiency, only the size of $\\matr{Q}_i$ matters.  However, we can interpret this query matrix as compressing the distribution (see supplemental material for more information).    \n\nWe compare our inference method (Algorithm~\\ref{alg:inference}) to a baseline algorithm that first computes the full contingency table (using the formula from definition~\\ref{def:graphical-model}) then answers the queries over the full contingency table.  As shown in Fig. ~\\ref{fig:inference}, the baseline breaks down after $ d = 7 $ due to memory limitations, but our algorithm easily scales to $d=16$.   \n\n\\textbf{Accuracy of inferred query estimates.}\nNext we demonstrate the accuracy improvements afforded by proper inference.  The measurement and target queries, for the Adult dataset, are listed in Fig. ~\\ref{table:key}.  There are six carefully chosen low-dimensional query matrices in the measurement set, in addition to the $11$ 1-way marginals (not listed).  The privacy budget of $ \\epsilon = 2.0$ is split equally among these $17$ measurement sets.  There are four additional target query matrices.  \nThe matrix $\\matr{P}$ in Fig. ~\\ref{table:key} contains range queries that characterize the cumulative distribution function, and the matrix $\\matr{A}$ is a low-sensitivity query matrix from which the answers to range queries can be accurately derived.\\cite{li2015matrix} These special query matrices are used for the continuous variables (age and hours-per-week) to demonstrate the important capability of our technology to handle arbitrary measurements of the marginals.\n\n\nWe evaluate accuracy of our inferred query answers in two settings.  First, we study accuracy on queries that are present in the measurement set (queries 1-6 in Fig.~\\ref{table:key}).   Our inferred answers are consistent and incorporate all available evidence.  We compare with a baseline method that simply reports the noisy measurement answer. Second, we study accuracy on queries {\\em not} present in the measurement set (queries 7-10 in Fig.~\\ref{table:key}).  \nSince the first baseline does not apply here, we compare with a different baseline that uses the 1-way marginal measurements and assumes independence among the variables involved in the queries.\nWe used the Proximal algorithm for estimating $\\mathbf{p}_{\\boldsymbol{\\theta}}$, running it for 2500 iterations (or about 25 seconds) using line search to select the step sizes $t_k$.\nWe evaluate the quality of a query estimate using the median $L_1$ error over 25 random trials.  The results are shown in Fig. ~\\ref{fig:adult1} and ~\\ref{fig:adult2}.\n\n\\begin{figure}\n\\centering\n\\captionsetup{justification=centering}\n\\subcaptionbox{\\label{table:key} Measurement queries (1-6) \\\\ \\hspace{7pt} and target queries (7-10)}{\n\\resizebox{0.31\\textwidth}{!}{%\n\\begin{tabular}{|c|l|l|}\n\\hline\n\\textbf{\\#} & \\textbf{Clique Variables}                                                      & \\textbf{Query Set} \\\\\\hhline{|=|=|=|}\n1           & workclass, education                                                           & $\\matr{I} \\otimes \\matr{I}$              \\\\\n2           & education, income                                                              & $\\matr{I} \\otimes \\matr{I}$              \\\\\n3           & age, marital status                                                            & $\\matr{A} \\otimes \\matr{I}$              \\\\\n4           & age, relationship                                                              & $\\matr{A} \\otimes \\matr{I}$              \\\\\n5           & \\begin{tabular}[c]{@{}c@{}}marital status, sex, \\\\ hours-per-week\\end{tabular} & $\\matr{I} \\otimes \\matr{I} \\otimes \\matr{A}$          \\\\\n6           & race, country                                                                  & $\\matr{I} \\otimes \\matr{I}$             \\\\ \\hhline{|=|=|=|}\n7           & workclass, income                                                              & $\\matr{I} \\otimes \\matr{I}$              \\\\\n8           & marital status, relationship                                                   & $\\matr{I} \\otimes \\matr{I}$            \\\\\n9           & relationship, hours-per-week                                                   & $\\matr{I} \\otimes \\matr{P}$              \\\\\n10          & occupation, sex                                                                & $\\matr{I} \\otimes \\matr{I}$             \\\\\\hline\n\\end{tabular}} \\vspace{3 mm}\n}\n\\subcaptionbox{\\label{fig:adult1} In-measurement \\\\ \\hspace{5pt} target queries}{\\includegraphics[width=0.33 \\textwidth]{fig\/adult1}}\n\\subcaptionbox{\\label{fig:adult2} Out-of-measurement \\\\ \\hspace{-10pt} target queries}{\\includegraphics[width=0.33 \\textwidth]{fig\/adult2}}\n\\caption{L1 Error from baseline estimates and our estimates.} \\label{fig:adult}\n\\end{figure}\n\nFor the in-measurement target queries, we found that our inferred query answers offered lower error than the corresponding baseline on all the measured 2-way and 3-way marginals, usually by a substantial amount.  In result not shown, it also improved the error on 9 out of 11 of the 1-way marginal estimates (the difference in error on the other 2 marginal estimates was negligible).\n\nFor the out-of-measurement target queries, our inferred query answers offered meaningful improvements to error for the first three measurements, and a small improvement for the fourth.  This is because the measurement queries contain more information about the first three target queries.  Note that the error of both methods is much worse for out-of-measurement queries because they can not be expressed as a linear combination of the measured queries.  Despite the high error, it is still useful to have better-than-baseline estimates of these quantities.  \n\nImprovements in accuracy stemming from inference have been observed before in the privacy literature (e.g. \\cite{hay2010boosting}), however existing inference algorithms are not capable of scaling to the large domain we considered in this experiment.\n\n\n\n\\section{Experimental evaluation} \\label{sec:experiments}\n\nIn this section, we measure the accuracy and scalability improvements enabled by probabilistic graphical-model (PGM) based estimation when it is incorporated into existing privacy mechanisms.  \n\n\\subsection{Adding PGM estimation to existing algorithms}\n\nWe run four algorithms: MWEM, PrivBayes, HDMM, and DualQuery, with and without our graphical model technology using a privacy budget of $ \\epsilon = 1.0 $ (and $\\delta = 0.001$ for DualQuery).  We run Algorithm \\ref{alg:proximal2} with line search for DualQuery and Algorithm \\ref{alg:proximal} for the other mechanisms, each for 10000 iterations.  We repeat each experiment five times and report the median workload error. Experiments are done on 2 cores of a single compute cluster node with 16 GB of RAM and 2.4 GHz processors.\n\n\nWe use a collection of four multi-dimensional datasets in our experiments, summarized in Table \\ref{table:datasets}.  Each dataset consists of a collection of categorical and numerical attributes (with the latter discretized into 100 bins).  Note the large domain of each dataset, which is the main property that makes efficient estimation challenging.  \n\n\\begin{table}[]\n\\caption{Datasets used in experiments along with the number of queries in the workload used with the dataset.} \\label{table:datasets}\n\\begin{tabular}{c|cccc}\n\\textbf{Dataset} & \\textbf{Records} & \\textbf{Attributes} & \\textbf{Domain} & \\textbf{Queries} \\\\\\hline\n\\textbf{Titanic} & 1304             & 9                   & 3e8  & 4851                \\\\\n\\textbf{Adult}   & 48842            & 15                  & 1e19 & 62876                \\\\\n\\textbf{Loans}   & 42535            & 48                  & 5e80 & 362201                \\\\\n\\textbf{Stroke}  & 19434           & 110                 & 4e104 & 17716               \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[t]\n  \\centering\n  \\begin{subfigure}[b]{13.1em}\n    \\centering\\includegraphics[width=13.15em]{fig\/privbayes} \n    \\caption{\\label{fig:privbayes} PrivBayes}\n  \\end{subfigure}%\n  \\hspace{-1.5em}\n  \\begin{subfigure}[b]{13.1em}\n    \\centering\\includegraphics[width=13.15em]{fig\/dualquery}\n    \\caption{\\label{fig:dq} DualQuery}\n  \\end{subfigure}%\n  \\hspace{-1.5em}\n  \\begin{subfigure}[b]{13.1em}\n    \\centering\\includegraphics[width=13.15em]{fig\/mwem} \n    \\caption{\\label{fig:mwem} MWEM}\n  \\end{subfigure}%\n  \\hspace{-1.5em}\n  \\begin{subfigure}[b]{13.1em}\n    \\centering\\includegraphics[width=13.15em]{fig\/hdmm} \n    \\caption{\\label{fig:hdmm} HDMM}\n  \\end{subfigure}%\n \\caption{Workload error of four mechanisms on four datasets, with and without our PGM estimation algorithm for $\\epsilon = 1.0$.} \\label{fig:accuracy}\n\\end{figure*}\n\n\nFor each dataset, we construct a workload of counting queries which is an extension of the set of three-way marginals. First, we randomly choose 15 subsets of attributes of size 3, $\\mathcal{C}$. For each subset $C \\in \\mathcal{C}$, if $C$ contains only categorical attributes, we define sub-workload ${\\matr{W}}_C$ to be a 3-way marginal.  However,  when $C$ contains any discretized numerical attributes, we replace the set of unit queries used in a marginal with the set of prefix range queries.  For example, if $C=\\langle$sex, education, income$\\rangle$ then the resulting subworkload ${\\matr{W}}_C$ would consist of all queries of the form:\n$\\mbox{sex}=x, \\mbox{education}=y, \\mbox{income} \\in [0,z]$ where $x,y,z$ range over the domains of the attributes, respectively.\nThe final workload is the union of the 15 three-way subworkloads defined above.\n\nWe measure the error on the workload queries as:\n$$ Error = \\frac{1}{| \\mathcal{C} |} \\sum_{C \\in \\mathcal{C}} \\frac{ \\norm{ {\\matr{W}}_{C} \\boldsymbol{\\mu}_C - {\\matr{W}}_{C} \\hat{\\boldsymbol{\\mu}}_C }_1}{2 \\norm{ {\\matr{W}}_C \\boldsymbol{\\mu}_C }_1} $$\nwhere the summand is related to the total variation distance (and is equal in the special case when ${\\matr{W}}_C = \\matr{I}$).  \n\n\\textbf{Improved accuracy.}\nPrivBayes and DualQuery are highly scalable algorithms supporting the large domains considered here.  Figures ~\\ref{fig:privbayes} and \\ref{fig:dq} show that incorporating PGM estimation significantly improves accuracy.\nFor PrivBayes, workload error is reduced by a factor of 6$\\times$ and 7$\\times$ on the Loans and Stroke datasets, respectively, and a modest $30\\%$ for Adult. For DualQuery, we also observe very significant error reductions of 1.2$\\times$, 1.8$\\times$, 3.5$\\times$, and 4.4$\\times$.\n\n\\textbf{Replacing infeasible estimation methods.}\nThe MWEM and HDMM algorithms fail to run on the datasets and workloads we consider because both require representations too large to maintain in memory.  However, incorporating PGM estimation makes these algorithms feasible. \n\nAs Figure \\ref{fig:mwem} shows, for the first three datasets, MWEM crashed before completing because it ran out of memory or timed out.  For example, on one run of the Adult dataset, the first three chosen queries were on the (race, native-country, income), (workclass, race, capital-gain), and (marital status, relationship, capital-gain) marginals.  Since these all overlap with respect to race and capital-gain, factored MW offers no benefit and the entire vector $\\mathbf{p}_{C}$ must be materialized over these attributes, which requires over 100 MB. After 5 iterations, the representation requires more than 2 GB, at which point it timed out.  Interestingly, MWEM was able to run on the stroke dataset, which has the largest domain and greatest number of attributes.  This is mainly because the workload did not contain as many queries involving common attributes.  In general, MWEM's representation will not explode as long as the workload (and therefore its measurements) consist solely of queries defined over low-dimensional marginals that do not have common attributes. Unfortunately this imposes a serious restriction on the workloads MWEM can support.\n\n\nAlthough the HDMM algorithm fails to run, for the purpose of comparison, we run a modified version of the algorithm (denoted HDMM+LLS) which uses local least squares independently over each measurement set instead of global least squares over the full data vector.  While scalable, Figure~\\ref{fig:hdmm} shows that this estimation is substantially worse than PGM estimation, especially on the titanic and loans dataset.  Incorporating PGM estimation offers error reductions of 6.6$\\times$, 3.2$\\times$, 27$\\times$, and 6.3$\\times$ on the four datasets.  These improvements primarily stem from non-negativity and global consistency. \n\n\\textbf{Varying epsilon.}\nWhile $\\epsilon$ is set to 1 in Figure ~\\ref{fig:accuracy}, in Figure ~\\ref{fig:sensitivity} we look at the impact of varying $\\epsilon$, for a fixed dataset and measurement set.  We use the Adult dataset and the measurements selected by HDMM, (which do not depend on $\\epsilon$).  The magnitude of the improvement offered by our PGM estimation algorithm increases as $\\epsilon$ decreases.  At $\\epsilon=0.3$ and below, the mechanism has virtually no utility without PGMs.  At the highest $\\epsilon$ of $10.0$, HDMM+LLS actually offers slightly lower error than HDMM+PGM on the workload, although both have very low error in an absolute sense.  The error of HDMM+PGM on the \\emph{measurements} is still better by more than a factor of three at this privacy level.  This behavior has been observed before in the low-dimensional setting, where the ordinary least squares estimator generalizes better than the non-negative least squares estimator for workloads with range queries \\cite{li2015matrix}.  \n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[b]{11.7em}\n    \\centering\\includegraphics[width=11.7em]{fig\/sensitivity} \\vspace{-1.5em}\n    \\caption{\\label{fig:sensitivity}\n\\end{subfigure}%\n\\begin{subfigure}[b]{11.7em}\n    \\centering\\includegraphics[width=11.7em]{fig\/scalability} \\vspace{-1.5em}\n    \\caption{\\label{fig:scalability}\n\\end{subfigure}%\n\\caption{ (a) Error of HDMM variants on Adult as a function of $\\epsilon$. (b) Scalability of estimation algorithms.} \n\\end{figure}\n\n\\subsection{The scalability of PGM estimation}\n\nWe now evaluate the scalability of our approach compared with two other general-purpose estimation techniques: multiplicative weights (MW; \\citeauthor{hardt2012simple}, \\citeyear{hardt2012simple}) and iterative ordinary least squares (LSMR; \\citeauthor{fong2011lsmr}, \\citeyear{fong2011lsmr}, \\citeauthor{Zhang18Ektelo:}, \\citeyear{Zhang18Ektelo:}).  We omit from comparison PrivBayes estimation and DualQuery estimation because they are special-purpose estimation methods that cannot handle arbitrary linear measurements.\nWe use synthetic data so that we can systematically vary the domain size and the number of attributes.  We measure the marginals for each triple of adjacent attributes --- i.e., $\\matr{Q}_{C} = \\matr{I}$ for all $C = (i, i+1, i+2) $ where $1 \\leq i \\leq d-2 $.  In Figure ~\\ref{fig:scalability}, we vary the number of attributes from $ 3 $ to $ 1000 $ (fixing the domain of each attribute, $| \\mathcal{X}_i |$ at 10), and plot the time per iteration of each of these estimation algorithms.  Both MW and LSMR fail to scale beyond datasets with $10$ attributes, as they both require materializing $\\mathbf{p}$ in vector form, while PGM easily scales to datasets with $1000$ attributes.\n\nThe domain size is the primary factor that determines scalability of the baseline methods.  However, the scalability of PGM primarily depends on the complexity of the measurements taken.  In the experiment above, the measurements were chosen to highlight a case where PGM estimation scales very well.  In general, when the graphical model implied by the measurements has high tree-width, our methods will have trouble scaling, as $\\text{MARGINAL-ORACLE}${} is computationally expensive.  In these situations, $\\text{MARGINAL-ORACLE}${} may be replaced with an approximate marginal inference algorithm, like loopy belief propagation \\cite{wainwright2008graphical}.\n\n\n\n\n\\section{Introduction} \\label{sec:intro}\n\nDifferential privacy \\cite{Dwork06Calibrating} has become the dominant standard for controlling the privacy loss incurred by individuals as a result of public data releases.  For complex data analysis tasks, error-optimal algorithms are not known and a poorly designed algorithm may result in much greater error than strictly necessary for privacy.  Thus, careful algorithm design, focused on reducing error, is an area of intense research in the privacy community. \n\n\n\nFor the private release of statistical queries, nearly all recent algorithms \\cite{zhang2017privbayes,li2015matrix,lee2015maximum,Proserpio12Calibrating,li2014data,qardaji2013understanding,Nikolov13Geometry,hardt2012simple,ding2011differentially,Xiao10Differential,li2010optimizing,hay2010boosting,hardt2010multiplicative,Hardt10Geometry,Barak07Privacy,Gupta11Privately,Thaler12Faster,Acs12Differentially,Zhang14Towards,Yaroslavtsev13Accurate,Cormode12Spatial,Qardaji12Differentially,mckenna2018optimizing} include steps within the algorithm where answers to queries are \\emph{inferred} from noisy answers to a set of  \\emph{measurement} queries already answered by the algorithm.\n \nInference is a critical component of privacy algorithms because: (i) it can reduce error when answering a query by combining evidence from multiple related measurements, (ii) it provides consistent query answers even when measurements are noisy and inconsistent, and (iii) it provides the above benefits without consuming the privacy-loss budget, since it is performed only on privately-computed measurements without re-using the protected data.  \n\n\nConsider a U.S. Census dataset, exemplified by the Adult table, which consists of 15 attributes including age, sex, race, income, education.  Given noisy answers to a set of measurement queries,\nour goal is to infer answers to one or more new queries.\nThe measurement queries might be expressed over each individual attribute (age), (sex), (race), etc., as well as selected combinations of attributes (age, income), (age, race, education), etc.  When inference is done properly, the estimate for a new query (e.g., counting the individuals with income>=50K, 10 years of education, and over 40 years old) will use many, or even all, available measurements. \n\n\n\nCurrent inference methods are limited in both scalability and generality.\nMost methods first estimate some model of the data and then answer new queries using the model.\nPerhaps the simplest model is a full contingency table, which stores a value for every element of the domain. When the measurements are linear queries (a common case, and our primary focus) least-squares \\cite{hay2010boosting,Nikolov13Geometry,li2014data,qardaji2013understanding,ding2011differentially,Xiao10Differential,li2010optimizing} and multiplicative-weight updates \\cite{hardt2010multiplicative,hardt2012simple} have both been used to estimate this model from the noisy measurements. New queries can then be answered by direct calculation. However, the size of the contingency table is the product of the domain sizes of each attribute, which means these methods break down for high-dimensional cases (or even a modest number of dimensions with large domains).  In the example above, the full contingency table would consist of $10^{19}$ entries.  To avoid this, factored models have been considered \\cite{hardt2012simple,zhang2017privbayes}. However, while scalable, these methods have other limitations including restricting the query class \\cite{hardt2012simple} or failing to properly account for (possibly varying) noise in measurements \\cite{zhang2017privbayes}. \n\n\n\n\n\n\nIn this work we show that graphical models provide a foundation for significantly improved inference. We propose to use a graphical model instead of a full contingency table as a model of the data distribution. Doing so avoids an intractable full materialization of the contingency table and retains the ability to answer a broad class of queries. We show that the graphical model representation corresponds to using a maximum entropy criterion to select a single data distribution among all distributions that minimize estimation loss. The structure of the graphical model is determined by the measurements, such that no information is lost relative to a full contingency table representation, but when each measurement is expressible over a low-dimensional marginal of the contingency table, as is common, the graphical model representation is much more compact. \n\nThis work is focused on developing a principled and general approach to inference in privacy algorithms. Our method is agnostic to the loss function used to estimate the data model and to the noise distribution used to achieve privacy.  We focus primarily on linear measurements, but also describe an extension to non-linear measurements\n\n\nWe assume throughout that the measurements are given, but we show our inference technique is versatile since it can be incorporated into many existing private query-answering algorithms that determine measurements in different ways.  For those existing algorithms that scale to high-dimensional data, our graphical-model based estimation method can substantially improve accuracy (with no cost to privacy).  Even more importantly, our estimation method can be added to some algorithms which fail to scale to high-dimensional data, allowing them to run efficiently in new settings.  We therefore believe our inference method can serve as a basic building block in the design of new privacy algorithms.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Use in Privacy Mechanisms} \\label{sec:methods}\n\nNext we describe how our estimation algorithms can improve the accuracy and\/or scalability of four state-of-the-art mechanisms: MWEM, PrivBayes, HDMM, and DualQuery.\n\n\\textbf{MWEM.} The multiplicative weights exponential mechanism \\cite{hardt2012simple} is an active-learning style algorithm that is designed to answer a workload of linear queries.  MWEM maintains an approximation of the data distribution and at each time step selects the worst approximated query $\\vect{q}_i^T$ from the workload via the exponential mechanism \\cite{mcsherry2007mechanism}. It then measures the query using the Laplace mechanism as $y_i = \\vect{q}_i^T \\mathbf{p} + z_i$ and then updates the approximate data distribution by incorporating the measured information using the multiplicative weights update rule.\nThe most basic version of MWEM represents the approximate data distribution in vector form, and updates it according to the following formula after each iteration:\n\\begin{equation} \\label{eq:mw}\n\\hat{\\mathbf{p}} \\leftarrow \\hat{\\mathbf{p}} \\odot \\exp{(-\\vect{q}_i (\\vect{q}_i^T \\hat{\\mathbf{p}} - y_i) \/ 2m)} \/ Z,\n\\end{equation}\nwhere $\\odot$ is elementwise multiplication and $Z$ is a normalization constant.\n\nIt is infeasible to represent $\\mathbf{p}$ explicitly for high-dimensional data, so this version of MWEM is only applicable to relatively low-dimensional data.  Hardt et al describe an enhanced version of MWEM, which we call \\emph{factored MWEM}, that is able to avoid materializing this vector explicitly, in the special case when the measured queries decompose\nover disjoint subsets of attributes.  \nIn that case, $\\mathbf{p}$ is represented implicitly as a product of independent distributions over smaller domains, i.e., $\\mathbf{p}(\\vect{x}) = \\prod_{C \\in \\mathcal{C}} \\mathbf{p}_C(\\vect{x}_C) $, and the update is done on one group at a time.\nHowever, this enhancement breaks down for measurements on overlapping subsets of attributes in high-dimensional data, so MWEM is still generally infeasible to run except on simple workloads.\n\nWe can replace the multiplicative weights update with a call to Algorithm \\ref{alg:proximal} using the standard $L_2$ loss function (on all measurements up to that point in the algorithm).  By doing so, we learn a compact graphical model representation of $\\hat{\\mathbf{p}}$, which avoids materializing the full $\\mathbf{p}$ vector even when the measured queries overlap in complicated ways.  This allows MWEM to scale better and run in settings where it was previously infeasible. We remark that Equation \\ref{eq:mw} is closely related to the update equation for entropic mirror descent \\cite{beck2003mirror}, suggesting that if the update equation is iterated until convergence, it solves the same $L_2$ minimization problem that we consider.  More details on this are given in Section ~\\ref{sec:mw_emd} of the supplement.\n\n\\textbf{PrivBayes.} PrivBayes \\cite{zhang2017privbayes} is a differentially private mechanism that generates synthetic data.  It first spends half the privacy budget to learn a Bayesian network structure that captures the dependencies in the data, and then uses the remaining privacy budget to measure the statistics---which are marginals---necessary to learn the Bayesian network parameters.  PrivBayes uses a heuristic of truncating negative entries of noisy measurements and normalizing to get conditional probability tables. It then samples a synthetic dataset of $m$ records from the Bayesian network from which consistent answers to workload queries can be derived.  While this is simple and efficient, the heuristic does not properly account for measurement noise and sampling may introduce unnecessary error.\n\nWe can replace the PrivBayes estimation and sampling step with a call to Algorithm ~\\ref{alg:proximal}, using an appropriate loss function (e.g. $L_1$ or $L_2$), to estimate a graphical model. \nThen we can answer new queries by performing graphical model inference (Section~\\ref{inference}), rather than using synthetic data. \n\n\n\n\\textbf{HDMM. } The high-dimensional matrix mechanism \\cite{mckenna2018optimizing} is designed to answer a workload of linear queries on multi-dimensional data. It selects the set of measurements that minimizes estimated error on the input workload.  The measurements are then answered using the Laplace mechanism, and inconsistencies resolved by solving an ordinary least squares problem of the form:\n$ \\hat{\\mathbf{p}} = \\argmin \\norm{ \\matr{Q} \\mathbf{p} - \\mathbf{y} }_2 $.\nSolving this least squares problem is the main bottleneck of HDMM, as it requires materializing the data vector even when $\\matr{Q}$ contains queries over the marginals of $\\mathbf{p}$.\n\nWe can replace the HDMM estimation procedure with  Algorithm~\\ref{alg:proximal}, using the same $L_2$ loss function.  If the workload contains queries over low-dimensional marginals of $\\mathbf{p}$, then $\\matr{Q}$ will contain measurements over the low-dimensional marginals too.\nThus, we replace the full ``probability'' vector $\\hat{\\mathbf{p}}$ with a graphical model $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$.\nAlso $\\hat{\\mathbf{p}}$ may contain negative values and need not sum to $1$ since HDMM solves an \\emph{ordinary} (unconstrained) least squares problem.\n\n\\textbf{DualQuery. } DualQuery \\cite{Gaboardi14Dual} is an iterative algorithm inspired by the same two-player game underlying MWEM. It generates synthetic data to approximate the true data on a workload of linear queries.  DualQuery maintains a distribution over the workload queries that depends on the true data so that poorly approximated queries have higher probability mass.  In each iteration, samples are drawn from the query distribution, which are proven to be differentially private.  The sampled queries are then used to find a single record from the data domain (without accessing the protected data), which is added to the synthetic database.\n\nThe measurements --- i.e., the random outcomes from the privacy mechanism --- are the queries sampled in each iteration.  Even though these are very different from the linear measurements we have primarily focused on, we can still express the log-likelihood as a function of $\\mathbf{p}$ and select $\\mathbf{p}$ to maximize the log-likelihood using Algorithm ~\\ref{alg:proximal2} or \\ref{alg:proximal}. The log-likelihood only depends on $\\mathbf{p}$ through the answers to the workload queries. If the workload can be expressed in terms of $\\boldsymbol{\\mu}$ instead, the log-likelihood can as well.\nThus, after running DualQuery, we can call Algorithm \\ref{alg:proximal2} with this custom loss function to estimate the data distribution, which we can use in place of the synthetic data produced by DualQuery.  The full details are given in the supplementary material.\n\n\n\\section{Related Work} \\label{sec:related}\n\nThe release of linear query answers has been extensively studied by the privacy community~\\cite{zhang2017privbayes,li2015matrix,Zhang14Towards,li2014data,Gaboardi14Dual,Yaroslavtsev13Accurate,qardaji2013understanding,Nikolov13Geometry,Thaler12Faster,hardt2012simple,Cormode12Spatial,Acs12Differentially,Gupta11Privately,ding2011differentially,Xiao10Differential,li2010optimizing,hay2010boosting,Hardt10Geometry,Barak07Privacy,mckenna2018optimizing,eugenio2018cipher}.  Early work using inference includes~\\cite{Barak07Privacy,hay2010boosting,williams2010probabilistic}, motivated by consistency as well as potential accuracy improvements. Inference has since been widely used in techniques for answering linear queries \\cite{lee2015maximum}.  These mechanisms often contain custom specialized inference algorithms that exploit properties of the measurements taken, and can be replaced by our algorithms. \n\n\\cite{williams2010probabilistic} introduce the problem of finding posterior distributions over model parameters from the output of differentially private algorithms. Their problem formulation requires a known model parameterization and a prior distribution over the parameter space.  Their approach requires approximating a high-dimensional integral, which they do either by Markov chain Monte Carlo, or by upper and lower bounds via the ``factored exponential mechanism''.  In the discrete data case, these bounds require summing over the data domain, which is just as hard as materializing $\\mathbf{p}$ and is not feasible for high-dimensional data.\n\n\n\\cite{bernstein2017differentially} consider the task of privately learning the parameters of an undirected graphical model. They do so by releasing noisy sufficient statistics using the Laplace mechanism, and then using an expectation maximization algorithm to learn model parameters from the noisy sufficient statistics. Their work shares some technical similarities with ours, but the aims are different. They have the explicit goal of learning a graphical model whose structure is specified in advance and used to determine the measurements. Our goal is to find a compact representation of some data distribution that minimizes a loss function where the measurements are determined externally; the graphical model structure is a by-product of the measurements made and the maximum entropy criterion.\n\n\\cite{chen2015differentially} consider the task of privately releasing synthetic data.  Their mechanism is similar to PrivBayes, but it uses undirected graphical models instead of Bayesian networks.  It finds a good model structure using a mutual information criteria, then measures the sufficient statistics of the model (which are marginals) and post-processes them to resolve inconsistencies.  This post-processing is based on a technique developed by \\cite{qardaji2014priview} that ensures all measured marginals are internally consistent, and may be improved with our methods.\n\n\n\n\n\t\n\n\\section{Algorithms for Estimation and Inference}\\label{approach}\n\n\n\n\nWhat principle can we follow to estimate answers to the workload query set? Prior work takes the approach of first using all available information to estimate a full contingency table \\(\\hat{\\mathbf{p}} \\approx \\mathbf{p}\\) and then using \\(\\hat{\\mathbf{p}}\\) to answer later queries~\\cite{hay2010boosting,li2010optimizing,ding2011differentially,qardaji2013understanding,lee2015maximum}.\nWe will call finding \\(\\hat{\\mathbf{p}}\\) \\emph{estimation}, and using \\(\\hat{\\mathbf{p}}\\) to answer new queries \\emph{inference}.\n\n\\subsection{Optimization Formulation} \\label{sec:formulation}\nThe standard framework for estimation and inference is:\n\\begin{align*}\n\\hat{\\mathbf{p}} &\\in \\argmin_{\\mathbf{p} \\in \\mathcal{S}} L(\\mathbf{p}), & \\text{(estimation)} \\\\\nf_{{\\matr{W}}}(\\mathbf{X}) &\\approx {\\matr{W}}\\, \\hat{\\mathbf{p}}. &\\text{(inference)}\n\\end{align*}\nHere $ \\mathcal{S} = \\big\\{ \\mathbf{p}: \\mathbf{p} \\geq 0, \\mathbf{1}^T \\mathbf{p} = 1\\big\\} $  is the probability simplex and $L(\\mathbf{p})$ is a loss function that measures how well $\\mathbf{p}$ explains the observed measurements. In past works, $L(\\mathbf{p}) = \\norm{ \\matr{Q} \\mathbf{p} - \\mathbf{y} }$ has been used as a loss function, where $\\matr{Q}$ is the measured query set and $ \\norm{ \\cdot } $ is either the $L_1$ norm or $L_2$ norm.  Minimizing the $L_1$ norm is equivalent to maximum likelihood estimation when the noise comes from the Laplace mechanism~\\cite{lee2015maximum}.\nMinimizing the $L_2$ norm is far more common in the literature however, and it is also the maximum likelihood estimator for Gaussian noise \\cite{hay2010boosting,Nikolov13Geometry,li2014data,qardaji2013understanding,ding2011differentially,Xiao10Differential,li2010optimizing, mckenna2018optimizing}. Our method supports both of these loss functions; we only require that $L$ is convex. Both of these loss functions are easily adapted to the situation where queries in $\\matr{Q}$ may be measured with differing degrees of noise.\nThe constraint $\\mathbf{p} \\in \\mathcal{S}$ may also be relaxed, which simplifies $L_2$ minimization; additionally, under different assumptions and an alternate version of privacy, the number of individuals may not be known.\nAll existing algorithms to solve these variations of the estimation problem suffer from the same problem: they do not scale to high dimensions since the size of $\\mathbf{p}$ is exponential in $d$ and we have to construct it explicitly as an intermediate step even if the inputs and outputs are small (e.g., all measurement queries are over low-dimensional marginals).  \n\n\n\n\\textbf{Optimization in Terms of Marginals.}\\label{optimization-in-terms-of-marginals}\nFor marginal query sets, a loss function will typically depend on $\\mathbf{p}$ only through its marginals $\\boldsymbol{\\mu}$. For example, when $\\matr{Q} = \\matr{Q}_\\mathcal{C} \\matr{M}_\\mathcal{C}$ we have $L(\\mathbf{p}) = \\| \\matr{Q}\\mathbf{p} - \\mathbf{y} \\| = \\| \\matr{Q}_\\mathcal{C} \\boldsymbol{\\mu} - \\mathbf{y}\\| = L(\\boldsymbol{\\mu})$ where we now write the loss function as $L(\\boldsymbol{\\mu})$. More generally, we will consider \\emph{any} loss function that only depends on the marginals. A very general case is when $L(\\boldsymbol{\\mu}) = -\\log p(\\mathbf{y} \\mid \\boldsymbol{\\mu})$ is the negative log-likelihood of \\emph{any} differentially private algorithm that produces output $\\mathbf{y}$ that depends only on the marginal vector $\\boldsymbol{\\mu}$ (see our treatment of DualQuery\\eat{, \\citeauthor{Gaboardi14Dual}, \\citeyear{Gaboardi14Dual},} in Section~\\ref{sec:methods})\n\nThe marginal vector $\\boldsymbol{\\mu}$ may be much lower dimensional than $\\mathbf{p}$. How can we take advantage of this fact? An ``obvious'' idea would be to modify the optimization to estimate only the marginals as $\\hat{\\boldsymbol{\\mu}} \\in \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} L(\\boldsymbol{\\mu})$, where \\(\\mathcal{M} = \\big\\{ \\boldsymbol{\\mu}: \\exists \\mathbf{p} \\in \\mathcal{S} \\text{ s.t. } \\matr{M}_\\mathcal{C} \\mathbf{p} = \\boldsymbol{\\mu}\\}\\) is the marginal polytope, which is the set of all valid marginals. There are two issues here. First, the marginal polytope has a complex combinatorial structure, and, although it is a convex set, it is generally not possible to enumerate its constraints for use with standard convex optimization algorithms. Note that this optimization problem is in fact a \\emph{generic} convex optimization problem over the marginal polytope, and as such it generalizes standard graphical model inference problems~\\cite{wainwright2008graphical}. Second, after finding $\\hat{\\boldsymbol{\\mu}}$ it is not clear how to answer new queries, unless they depend only on some measured marginal \\(\\boldsymbol{\\mu}_C\\).\n\n\n\n\n\\textbf{Graphical Model Representation.}\n\\label{factored-p-and-maximum-entropy-criterion}\nAfter finding an optimal $\\hat{\\boldsymbol{\\mu}}$ we want to answer new queries that do not necessarily depend directly on the measured marginals.  To do this we need to identify a distribution $\\hat{\\mathbf{p}}$ that has marginals $\\hat{\\boldsymbol{\\mu}}$, and we must have tractable representation of this distribution.  Also, since there may be many $\\hat{\\mathbf{p}}$ that give rise to the same marginals, we want a principled criteria to choose a single estimate, such as the principle of maximum entropy.  We accomplish these goals using undirected graphical models. \n\n\n\\begin{definition}[Graphical model]\n  Let \\(\\mathbf{p}_{\\boldsymbol{\\theta}}(\\vect{x}) = \\frac{1}{Z} \\exp \\big(\\sum_{C \\in \\mathcal{C}} \\boldsymbol{\\theta}_C(\\vect{x}_C) \\big)\\) be a normalized distribution, where $\\boldsymbol{\\theta}_C \\in \\matr{R}^{n_C}$. This distribution is a graphical model that factors over the measurement sets \\(\\mathcal{C}\\), which are the cliques of the graphical model. The vector \\(\\boldsymbol{\\theta} = (\\boldsymbol{\\theta}_C)_{C \\in \\mathcal{C}}\\) is the parameter vector.\n  \\label{def:graphical-model}\n\\end{definition}\n\n\\begin{theorem}[Maximum entropy \\cite{wainwright2008graphical}] \\label{theorem:maxent}\n  Given any $\\hat{\\boldsymbol{\\mu}}$ in the interior of $\\mathcal{M}$ there is a parameter vector \\(\\hat{\\boldsymbol{\\theta}}\\) such that the graphical model \\(\\mathbf{p}_{\\hat{\\boldsymbol{\\theta}}}(\\vect{x})\\) has maximum entropy among all $\\hat{\\mathbf{p}}(\\vect{x})$\nwith marginals \\(\\hat{\\boldsymbol{\\mu}}\\).\\footnote{If the marginals are on the boundary of $\\mathcal{M}$, e.g., if they contain zeros, there is a sequence of parameters $\\{\\boldsymbol{\\theta}^{(n)}\\}$ such that $\\mathbf{p}_{\\boldsymbol{\\theta}^{(n)}}(\\vect{x})$ converges to the maximum-entropy distribution\n   \n    as $n \\to \\infty$. See~\\cite{wainwright2008graphical}.\n   \n  }\n\n\\end{theorem}\nTheorem~\\ref{theorem:maxent} says that, after finding $\\hat{\\boldsymbol{\\mu}}$, we can obtain a factored representation of the maximum-entropy distribution with these marginals by finding the graphical model parameters $\\hat{\\boldsymbol{\\theta}}$.\nThis is the problem of learning in an graphical model, which is well understood~\\cite{wainwright2008graphical}.\n\n\n\n\\subsection{Estimation: optimizing over the marginal polytope}\\label{estimation}\n\n\\begin{algorithm}[tb]\n    \\caption{Proximal Estimation Algorithm} \\label{alg:proximal2}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Input:} Loss function $L(\\boldsymbol{\\mu})$ between $\\boldsymbol{\\mu}$ and $\\mathbf{y}$\n    \\STATE {\\bfseries Output:} Estimated data distribution $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$\n    \\STATE $\\boldsymbol{\\theta} = \\vect{0}$\n    \\FOR{$t=1, \\dots, T$}\n        \\STATE $\\boldsymbol{\\mu} = $\\text{MARGINAL-ORACLE}$(\\boldsymbol{\\theta})$\n        \\STATE $\\boldsymbol{\\theta} = \\boldsymbol{\\theta} - \\eta_t \\nabla L(\\boldsymbol{\\mu})$\n    \\ENDFOR\n    \\STATE {\\bfseries return} $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$\n\\end{algorithmic}\n\\end{algorithm}\n\nWe need algorithms to find \\(\\hat{\\boldsymbol{\\mu}}\\) and \\(\\hat{\\boldsymbol{\\theta}}\\).  We considered a variety of algorithms and present two of them here.  Both are proximal algorithms for solving convex problems with ``simple'' constraints \\cite{parikh2014proximal}.  Central to our algorithms is a subroutine $\\text{MARGINAL-ORACLE}${}, which is some black-box algorithm for computing the clique marginals $\\boldsymbol{\\mu}$ of a graphical model from the parameters $\\boldsymbol{\\theta}$.  This is the problem of \\emph{marginal inference} in a graphical model.  $\\text{MARGINAL-ORACLE}${} may be any marginal inference routine --- we use belief propagation on a junction tree. In the remainder of this section, we assume that the clique set $\\mathcal{C}$ are the cliques of a junction tree. This is without loss of generality, since we can enlarge cliques as needed until this property is satisfied.\n\nAlgorithm~\\ref{alg:proximal2} is a routine to find $\\hat{\\boldsymbol{\\mu}}$ by solving a convex optimization problem over the marginal polytope. Due to the special structure of the algorithm it also finds the parameters $\\hat{\\boldsymbol{\\theta}}$. Algorithm~\\ref{alg:proximal2} is inspired by the entropic mirror descent algorithm for solving convex optimization problems over the probability simplex \\cite{beck2003mirror}.  The iterates of the optimization are obtained by solving simpler optimization problems of the form: \\vspace{-1ex}\n\\begin{equation} \\label{eq:update}\n\\boldsymbol{\\mu}^{t+1} = \\argmin_{\\boldsymbol{\\mu} \\in \\mathcal{M}} \\boldsymbol{\\mu}^T \\nabla L(\\boldsymbol{\\mu}^t) + \\frac{1}{\\eta_t} D(\\boldsymbol{\\mu}, \\boldsymbol{\\mu}^t)\n\\end{equation}\nwhere $D$ is a Bregman divergence that is chosen to reflect the geometry of the marginal polytope.  Here we use the following Bregman divergence generated from the Shannon entropy: $ D(\\boldsymbol{\\mu}, \\boldsymbol{\\mu}^t) = -H(\\boldsymbol{\\mu}) + H(\\boldsymbol{\\mu}^t) + (\\boldsymbol{\\mu} - \\boldsymbol{\\mu}^t)^T \\nabla H(\\boldsymbol{\\mu}^t) $, where $H(\\boldsymbol{\\mu})$ is the Shannon entropy of the graphical model $\\mathbf{p}_{\\boldsymbol{\\theta}}$ with marginals $\\boldsymbol{\\mu}$. Since we assumed above that $\\boldsymbol{\\mu}$ are marginals of the cliques of a junction tree, the Shannon entropy is convex and easily computed  as a function of $\\boldsymbol{\\mu}$ alone~\\cite{wainwright2008graphical}.\\footnote{An alternative would be to use the Bethe entropy as in ~\\cite{vilnis2015bethe}. The Bethe entropy is convex and computable from $\\boldsymbol{\\mu}$ alone regardless of the model structure. Using Bethe entropy would lead to approximate marginal inference instead of exact marginal inference as the subproblems, which is an interesting direction for future work.}\n\nWith this divergence, the objective of the subproblem in Equation \\ref{eq:update} can be seen to be equal to a variational free energy, which is minimized by marginal inference in a graphical model.\nThe full derivation is provided in the supplement.  The implementation of Algorithm \\ref{alg:proximal2} is very simple --- it simply requires calling $\\text{MARGINAL-ORACLE}${} at each iteration.  Additionally, even though the algorithm is designed to find the optimal $\\boldsymbol{\\mu}$, it also returns the corresponding graphical model parameters $\\boldsymbol{\\theta}$ ``for free'' as a by-product of the optimization.  This is evident from Algorithm \\ref{alg:proximal2}: upon convergence, $\\boldsymbol{\\mu}$ is the vector of marginals of the graphical model with parameters $\\boldsymbol{\\theta}$. The variable $\\eta_t$ in this algorithm is a step size, which can be constant, decreasing, or found via line search.  This algorithm is an instance of mirror descent, and thus inherits its convergence guarantees.  It will converge for any convex loss function $L$ at a $O(1 \/ \\sqrt{t})$ rate,\\footnote{That is, $L(\\boldsymbol{\\mu}^t) - L(\\boldsymbol{\\mu}^*) \\in O(1 \/ \\sqrt{t})$.} even ones that are not smooth, such as the $L_1$ loss.\n\nWe now present a related algorithm which is based on the same principles as Algorithm \\ref{alg:proximal2} but has an improved $O(1\/t^2)$ convergence rate for convex loss functions with Lipchitz continuous gradients.  Algorithm~\\ref{alg:proximal} is based on Nesterov's accelerated dual averaging approach \\cite{nesterov2009primal,xiao2010dual,vilnis2015bethe}.  The per-iteration complexity is the same as Algorithm \\ref{alg:proximal2} as it requires calling the $\\text{MARGINAL-ORACLE}${} once, but this algorithm will converge in fewer iterations.  Algorithm \\ref{alg:proximal} has the advantage of not requiring a step size to be set, but it requires knowledge of the Lipchitz constant of $ \\nabla L$.  For the standard $L_2$ loss with linear measurements, this is equal to the largest eigenvalue of $ \\matr{Q}^T \\matr{Q} $.  The derivation of this algorithm appears in the supplement.\n\n\\begin{algorithm}[tb]\n    \\caption{Accelerated Proximal Estimation Algorithm} \\label{alg:proximal}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Input:} Loss function $L(\\boldsymbol{\\mu})$ between $\\boldsymbol{\\mu}$ and $\\mathbf{y}$\n    \\STATE {\\bfseries Output:} Estimated data distribution $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$\n    \\STATE $K = $ Lipchitz constant of $\\nabla L$\n    \\STATE $\\bar{\\vect{g}} = \\vect{0}$\n    \\STATE $\\vect{\\nu}, \\boldsymbol{\\mu} = $\\text{MARGINAL-ORACLE}$(\\vect{0})$\n    \\FOR{$t=1, \\dots, T$}\n        \\STATE $c = \\frac{2}{t+1}$\n        \\STATE $\\vect{\\omega} = (1-c) \\boldsymbol{\\mu} + c \\vect{\\nu}$\n        \\STATE $\\bar{\\vect{g}} = (1-c) \\bar{\\vect{g}} + c \\nabla L(\\vect{\\omega})$\n        \\STATE $\\boldsymbol{\\theta} = \\frac{-t (t+1)}{4 K} \\bar{\\vect{g}} $\n        \\STATE $\\vect{\\nu} = $\\text{MARGINAL-ORACLE}$(\\boldsymbol{\\theta})$\n        \\STATE $\\boldsymbol{\\mu} = (1-c) \\boldsymbol{\\mu} + c \\vect{\\nu}$\n    \\ENDFOR\n    \\STATE {\\bfseries return} graphical model $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$ with marginals $\\boldsymbol{\\mu}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n \n\\subsection{Inference}\\label{inference}\n\nOnce $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$ has been estimated, we need algorithms to answer new queries without materializing the full contingency table representation.  This corresponds to the problem of inference in a graphical model. If the new queries only depend on $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$ through its clique marginals $\\boldsymbol{\\mu}$, we can immediately answer them using $\\text{MARGINAL-ORACLE}${}, or by saving the final value of $\\boldsymbol{\\mu}$ from Algorithms~\\ref{alg:proximal2} or \\ref{alg:proximal}. If the new queries depend on some other marginals outside of the cliques of the graphical model, we instead use the variable elimination algorithm \\cite{koller2009probabilistic} to first compute the necessary marginal, and then answer the query.  In Section \\ref{sec:inference} of the supplement, we present a novel inference algorithm that is related to variable elimination but is faster for answering certain queries because it does not need to materialize full marginals if the query does not need them.  For more complicated downstream tasks, we can generate synthetic data by sampling from $\\hat{\\mathbf{p}}_{\\boldsymbol{\\theta}}$, although this should be avoided when possible as it introduces additional sampling error.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\\label{sec_intro}  \n\nAssociated absorbers are unique tools to probe the physical conditions of the gaseous environment in the immediate vicinity of the background quasar (QSOs). The abundance of heavy elements in these absorbers provides a direct measure of the star formation and chemical evolution in the center of galaxies hosting QSOs \\citep[]{Hamann97a}. Most importantly, good fraction of associated absorbers are believed to originate from the ejected material from the central engine of the QSOs \\citep[]{Gordon99}. These outflows are theoretically invoked to regulate the star formation of the host galaxies and growth of the supper massive black holes (SMBHs) at their centers \\citep[]{Silk98,King03,Bower06,Ostriker10}.   \n\n\nThere is no firm definition in the literature for an associated absorber. The absorbers with velocity spread of few $\\times$ 100 km~s$^{-1}$, which appear within few $\\times$ 1000 km~s$^{-1}$\\ from the emission redshift of the QSO, are generally defined to be associated absorbers \\citep{Hamann97a}. In addition to the proximity to the QSOs, associated absorbers are also characterized by (a) time variable absorption line strength \\citep[]{Barlow89,Barlow92,Hamann95,Srianand01,Hall11,Vivek12} (b) very high metallicity \\citep[e.g. near solar abundances;][]{Petitjean94a,Hamann97a} and high ionization parameter \\citep[e.g. log~U $\\gtrsim 0.0$;][]{Hamann98,Hamann00,Muzahid12b} (c) partial coverage of the continuum source  \\citep[e.g.][]{Barlow97,Srianand99,Ganguly99,Arav08} and (d) presence of excited fine structure lines \\citep[e.g.][]{Srianand00apm}. These above mentioned properties are unlikely to occur in intervening systems \\citep[however see][for a very special case]{Balashev11} and thus, they are believed to originate from gas very close to the QSO or a possible ejecta of the central engine. In the case of gas outflowing from the QSO, line driven radiative acceleration has often been suggested to be an important driving mechanisms however, only a handful of convincing evidences exists in the literature till date \\citep[see e.g.,][]{Arav94,Arav95,Srianand02}. \n \n\nBased on their line widths, the associated absorbers are broadly classified into two categories: (1) the broad absorption line (BAL) and (2) narrow absorption line (NAL) systems.\nBALs and NALs are predominantly detected through species (e.g., \\mbox{Mg\\,{\\sc ii}}, \\mbox{C\\,{\\sc iv}}, \\mbox{Si\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}\\ etc.) with low ionization potential (i.e. IP $\\lesssim 100$ eV) in the UV-optical regime. On the other hand, the soft X-ray spectra of $\\sim$ 40 -- 50\\% of the Seyfert galaxies and QSOs happen to show K-shell absorption edges of highly ionized oxygen \\citep[i.e., \\mbox{O\\,{\\sc vii}}, \\mbox{O\\,{\\sc viii}}\\ with IP $\\gtrsim 0.5$~keV;][]{Reynolds97a,George98,Crenshaw03}, known as X-ray ``warm absorbers\" (WAs). These X-ray WAs are often said to correlate with the presence of absorption in the UV regime \\citep[see e.g.,][]{Mathur94,Mathur95b,Mathur95a,Mathur98,Mathur99,Brandt00,Arav07}. \\citet{Telfer98} have argued that the BAL-like absorption seen towards SBS~1542$+$541 could be a potential X-ray WA candidate \\citep[see also][]{Hamann95}. However, QSOs known to have associated BAL absorption are generally found to be X-ray weak \\citep[]{Green95,Green96,Stalin11}. In few cases the physical conditions in the UV absorbers are shown to be incompatible with that of X-ray WAs \\citep[e.g.,][]{Srianand00apj,Hamann00}.  Therefore, although a unified picture of X-ray and UV associated absorbers is desirable, it is not clear whether there is any obvious connection between them. Even in cases, where simultaneous occurrences of the X-ray and UV absorption are seen, the absorbing gas need not be co-spatial. For example, envisaging a disk-wind model, \\cite{Murray95b} have shown that the X-ray absorption originates very close to the accretion disk whereas UV absorption predominantly occurs in the accelerated gas farther away.   \n\n\nThe study of the species with ionization potential intermediate between UV-optical and X-ray absorbers (i.e., few $\\times$ 100 eV) is important to understand the comprehensive nature of the ionization structure and thus the unified picture of QSO outflows detected in different wavebands. The resonant transitions of highly ionized species (e.g. \\mbox{Ne\\,{\\sc viii}}, \\nani, \\mgx, \\alel\\ and \\mbox{Si\\,{\\sc xii}}), that fall in the far-ultraviolet (FUV)  regime, are ideally suited for studying the intermediate ionization conditions of the associated absorbers. However, only a handful of absorbers showing some of these species have been reported till date. For example, the first tentative detection of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption was reported by \\citet{Korista92} towards Q~0226$-$1024. The three other tentative detections existing in literature are by \\citet{Petitjean96} towards HS~1700$+$6414, \\citet{Gupta05} towards 3C~48 and \\citet{Ganguly06} towards HE~0226$-$4110. We also note that a possible \\mbox{Ne\\,{\\sc viii}}\\ detection is reported in the composite {\\sl Far-Ultraviolet Spectroscopic Explorer} ($FUSE$) spectrum by \\citet{Scott04}. There are only six confirmed detections of \\mbox{Ne\\,{\\sc viii}}\\ absorption in associated absorbers reported till date [i.e., UM~675, \\citet{Hamann95}; SBS~1542$+$541, \\citet{Telfer98}; PG~0946$+$301, \\citet{Arav99a}; J2233$-$606, \\citet{Petitjean99}; 3C~288.1, \\citet{Hamann00}; HE~0238$-$1904, \\citet{Muzahid12b}]. Among these, both SBS~1542$+$541 and PG~0946$+$301 are BALQSOs. While multiphase photoionization models are generally used to explain most of these observations, \\citet{Muzahid12b} have shown that the models of collisional ionization equilibrium can also reproduce the observed column density ratios of high ions like \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}\\ and \\mgx. Therefore the collisional ionization could be an equally important ionizing mechanism in these absorbers.  \n\n\nAlthough the highly ionized UV absorbers are of prime interest to probe the ``missing link\" between UV and X-ray continuum absorbers, the low  rest frame wavelengths (i.e. $\\lambda_{\\rm rest}<$ 912 \\AA) of the diagnostic species (e.g. \\mbox{Ne\\,{\\sc viii}}, \\nani, \\mgx\\ etc.) make them difficult to detect. This is partly because of the Galactic Lyman-limit absorption in the spectra of low redshift sources and the Ly$\\alpha$\\ forest contamination in the spectra of high redshift sources. Hence the intermediate redshift (i.e. 0.5 $<$ $z_{\\rm em}$\\ $<$ 1.5) UV bright QSOs are ideal for this study. Note that such a study is only feasible with far-ultra-violet (FUV) sensitive space based telescopes like {\\sl Hubble Space Telescope} ($HST$). In this paper we present a sample of new class of associated absorbers detected through \\mbox{Ne\\,{\\sc viii}}\\ absorption in the FUV spectra of intermediate redshift QSOs obtained with the {\\sl Cosmic Origins Spectrograph} (COS) on board $HST$.   \n\n \nThis paper is organized as follows. In Section~2 we describe the observations and data reduction techniques for the sample of QSOs studied here. In Section~3 we discuss the effects of partial coverage in column density measurements and how we correct for it. Data sample and analysis of individual absorption systems are presented in Section~4. In Section~5 we explore ionization models for some of these systems detected with number of different ions. In Section~6 we discuss the overall properties of these absorbers. We summarize our main results in Section~7.  \nThroughout this paper we use flat $\\Lambda$CDM cosmology with ($\\Omega_{\\rm M}$, $\\Omega_{\\Lambda}$) = (0.27,0.73) and a Hubble parameter of $H_{0}$ = 71 km~s$^{-1}$ Mpc$^{-1}$. The solar relative abundances of heavy elements are taken from \\citet{Asplund09}.  \n\n\n\n\n\\input{table1.tex} \n\n\n\n\\section{Observations and data reduction}  \n\\label{sec_obs}   \n\n\nThe sample in which we searched for \\mbox{Ne\\,{\\sc viii}}\\ absorbers had the following selection criteria: (1) archived $HST$\/COS FUV spectra (G130M+G160M) of quasars which were public as of February 2012, (2) QSOs with emission redshift $z_{\\rm em} \\ge 0.45$ so that the the \\mbox{Ne\\,{\\sc viii}}\\ doublet transitions (770\\AA\\ and 780\\AA) are redshifted into the wavelength coverage of the COS G130M and G160M gratings, and (3) spectra with signal-to-noise ratio ($S\/N)$ per resolution element $>$10.     \nThe properties of COS and its in-flight operations are discussed by \\citet{Osterman11} and \\citet{Green12}. The data were retrieved from the $HST$ archive and reduced using the STScI {\\sc CalCOS} v.2.12 pipeline software. The reduced data were flux calibrated. The alignment and addition of the separate G130M and G160M exposures were done using the software developed by the COS team\\footnote{http:\/\/casa.colorado.edu\/$\\sim$danforth\/science\/cos\/costools.html}. The exposures were weighted by the integration time while coadding in flux units. The procedures followed for data reduction are described in greater detail in \\citet{Narayanan11}. \nThe unabsorbed QSO continuum is fitted using low-order polynomials interpolated between wavelength ranges devoid of strong absorption lines. We use the standard procedure that propagates the continuum placement uncertainty to the normalized flux. \n\nThe medium resolution ($R \\sim$ 20,000) with $S\/N \\ge 10$ COS data, covering 1150 -- 1800 \\AA\\ wavelength range, allow us to search for \\mbox{Ne\\,{\\sc viii}} $\\lambda\\lambda$770,780 doublets in the redshift range $\\sim$~0.45$-$1.31. Observational details of our final sample of 20 quasar sight lines are listed in Table~\\ref{tab:data}. Half of these sight lines were part of a blind survey to detect the warm-hot intergalactic gas (prop. ID 11741). Of the remaining, majority are from the COS-GTO program (prop. ID 11541) to probe the gas phases in the low redshift IGM and galaxy halos. For the QSO PKS~0405$-$123, we have combined spectra obtained under the GTO program of the COS science team from December 2009 and the $HST$ Early Release Observations (prop. ID 11508) of August 2009. While weak radio emission (i.e. flux density $\\le$ 1 mJy) is detected in most of the QSOs in our sample, only seven of them (called radio bright from now on) have radio flux density in excess of 50 mJy at 5 GHz.  \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=4.4cm,width=8.8cm,angle=00]{fc_fun.ps} \n}}\n}}\n\\caption{Demonstration of the effect of partial coverage in case of a heavily saturated \n({\\sl left}) and an unsaturated ({\\sl right}) \\mbox{Ne\\,{\\sc viii}}\\ lines. Partially covered saturated \nline will appear as flat bottom profile with nonzero flux. For a partially covered \nunsaturated line, column density measurement can lead to much lower value compared to the \ntrue value if we do not correct for the covering fraction.         \n} \n\\label{fc_fun} \n\\end{figure} \n\n\n\n\\section{Partial Coverage and Uncertainty in Column density measurement}  \n\\label{sec_parcov} \n\n\nBecause of the close physical association between the background QSO and the associated absorber, in many cases it so happens that the latter does not cover the former entirely. In such cases, the observed residual intensity at any frequency can be written as, \n\\begin{equation} \nI(\\nu) = I_{0}(\\nu)(1-f_{c})+f_{c} I_{0}(\\nu) {\\rm exp}[-\\tau(\\nu)]~ . \n\\label{eqn:covf1}\n\\end{equation} \nHere $I_{0}(\\nu)$ is the incident intensity, $\\tau(\\nu)$ is the true optical depth, and $f_{c}$ is the covering fraction. In the case of doublets with rest frame wavelengths $\\lambda_{1}$ \\& $\\lambda_{2}$ and oscillator strengths $f_{1}$ \\& $f_{2}$, the residual intensities $R_{1}$ and $R_{2}$ in the normalized spectra, at any velocity $v$ with respect to the line centroid are related by  \n\\begin{equation} \nR_{2}(v) = (1-f_{c})+f_{c}\\times \\left(\\frac{R_{1}(v)-1+f_{c}}{f_{c}}\\right)^{\\gamma},  \n\\label{eqn:covf2}\n\\end{equation}  \nwhere $\\gamma = f_{2}\\lambda_{2}\/f_{1}\\lambda_{1}$. The value of $\\gamma$ is very close to 2 for doublets  \\citep[see e.g.,][]{Srianand99,Petitjean99}. This equation in principle allows us to calculate the covering fraction of the absorbing gas.    \n\n\n\n\\begin{table}\n\\caption{List of important Extreme-UV (EUV) lines used in this paper$^{1}$.} \n\\centering  \n\\begin{tabular}{crrrcc} \n\\hline \n Ion & IP(1)$^{a}$ & IP(2)$^{b}$ & $\\lambda^{c}$ (\\AA) & $f_{\\rm osc}^{d}$ & log~$T_{\\rm max}^{e}$  \\\\   \n (1) &       (2)   &     (3)     &    (4)              &    (5)       & (6)       \\\\  \n\\hline \n\\mbox{O\\,{\\sc iv}}   &    54.93    &    77.41 &   787.7105 &  1.11$\\times10^{-1}$ & 5.20 \\\\ \n       &             &          &   608.3968 &  6.70$\\times10^{-2}$ &      \\\\ \n\\mbox{O\\,{\\sc v}}\\   &    77.41    &   113.90 &   629.7320 &  5.15$\\times10^{-1}$ & 5.40 \\\\   \n\\mbox{N\\,{\\sc iv}}\\  &    47.45    &    77.47 &   765.1467 &  6.16$\\times10^{-1}$ & 5.15 \\\\  \n\\mbox{Ne\\,{\\sc v}}\\ &    97.12    &   126.22 &   572.3380 &  7.74$\\times10^{-2}$ & 5.45 \\\\  \n\\mbox{Ne\\,{\\sc vi}}\\ &   126.22    &   157.93 &   558.5940 &  9.07$\\times10^{-2}$ & 5.65 \\\\ \n\\mbox{Ne\\,{\\sc viii}}\\  &   207.28    &   239.10 &   770.4089 &  1.03$\\times10^{-1}$ & 5.85 \\\\  \n       &             &          &   780.3240 &  5.05$\\times10^{-2}$ &      \\\\ \n\\arei\\ &   124.32    &   143.45 &   700.2450 &  3.85$\\times10^{-1}$ & 5.75 \\\\ \n       &             &          &   713.8100 &  1.88$\\times10^{-1}$ &      \\\\        \n\\nani\\ &   264.19    &   299.88 &   681.7190 &  9.24$\\times10^{-2}$ & 5.90 \\\\ \n       &             &          &   694.1460 &  4.54$\\times10^{-2}$ &      \\\\   \n\\mgx\\  &   328.24    &   367.54 &   609.7930 &  8.42$\\times10^{-2}$ & 6.05 \\\\ \n       &             &          &   624.9410 &  4.10$\\times10^{-2}$ &      \\\\   \n\\alel\\ &   399.37    &   442.08 &   550.0310 &  7.73$\\times10^{-2}$ & 6.15 \\\\ \n       &             &          &   568.1200 &  3.75$\\times10^{-2}$ &      \\\\  \n\\mbox{Si\\,{\\sc xii}}$^{\\dagger}$ & 476.08  &  523.52 &   499.4060 &  7.19$\\times10^{-2}$ & 6.35 \\\\ \n                  &         &         &   520.6650 &  3.45$\\times10^{-2}$ &      \\\\  \n\\hline \n\\end{tabular}\n~\\\\ \n\\flushleft  \n$^{1}$From \\citet{Verner94} \\\\ \n$^{a}$Creation ionization potential \\\\  \n$^{b}$Destruction ionization potential \\\\ \n$^{c}$Rest frame wavelength in \\AA \\\\ \n$^{d}$Oscillator strength  \\\\ \n$^{e}$Temperature at which collisional ionization fraction \\citep[]{Sutherland93} peaks \\\\ \n$^{\\dagger}$Not covered for any of the systems reported here  \n\\label{tab:atomic_data} \n\\end{table}    \n\n \n\n\n\nThe effects of partial coverage in case of a heavily saturated and an unsaturated line are shown in Fig.~\\ref{fc_fun}. In the left panel of the figure we plot synthetic profiles of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line (true line center optical depth $\\tau_0$= 21.0) with $N(\\mbox{Ne\\,{\\sc viii}})$ = 10$^{16}$ cm$^{-2}$ and $b$-parameter of 100 km~s$^{-1}$\\ for $f_c = 1.0$ (solid profile) and $f_c = 0.8$ (dashed profile). The heavy saturation in the profile with complete coverage suggests large optical depth (i.e. $e^{-\\tau(\\nu)}$ = 0). The dashed curve showing flat bottom profile but flux level not reaching to zero, clearly suggests a partial coverage scenario with $f_c = 1 - I(\\nu)\/I_{0}$. Evidently, presence of only one line is sufficient to compute $f_c$ in such a situation.     \nIn the right hand panel of Fig.~\\ref{fc_fun}, we show synthetic profiles of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line ($\\tau_0$= 2.1) with $N$ = 10$^{15}$ cm$^{-2}$ and $b$-parameter of 100 km~s$^{-1}$\\ for $f_c = 1.0$ (solid), $f_c = 0.8$ (short dashed) and  $f_c = 0.5$ (long dashed). For the same column density, profiles with different covering fraction look different. The line center becomes shallower for lower $f_c$ values. The line with a column density of 10$^{15}$ cm$^{-2}$ will appear as $N(\\mbox{Ne\\,{\\sc viii}})$ = 10$^{14.85}$ cm$^{-2}$ ($\\tau_0$= 1.5) and 10$^{14.60}$ cm$^{-2}$ ($\\tau_0$= 0.8) for covering fractions of $f_c =$ 0.8 and 0.5 respectively. Evidently, the observed optical depth in this case is degenerate between the true optical depth and the covering fraction. Therefore, unlike the saturated case, we need at least two lines from the same ground state to estimate the true column density. After estimating the covering fraction (either by flat bottom approximation or from doublet transitions) for a given species we recover the true optical depth by inverting Eq.~\\ref{eqn:covf1}. We then use the partial coverage corrected flux for Voigt profile fitting. Here we make an explicit assumption that the individual Voigt profile components in a blend all have same $f_c$ for a given ion, \nbut note that $f_c$ can be strongly dependent on the velocity along the absorption trough \\citep[]{Srianand99,Arav99b,Gabel05,Arav08}. In addition inhomogeneous absorption models were shown to produce good fits for the absorption troughs as well \\citep[]{Arav08,Borguet12a}. However, given the survey nature of this work and the limited $S\/N$ of the data, we deem it adequate to treat the absorber with simple covering fraction models and to reserve the use of more elaborate models for future investigations and high $S\/N$ observations.   \n\n\n\n\\section{Data Sample and Analysis}  \n\\label{sec_data} \n \n\\input{tab1.tex} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=9.2cm,angle=00]{outflow_HE0226_0.49253.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.49272 \ntowards HE~0226$-$4110. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.493) of the QSO. In case of \\mbox{H\\,{\\sc i}}, \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}, the smooth curves are the \nbest fitting Voigt profiles, overplotted on top of the data. The vertical ticks mark \nthe centroids of the individual Voigt profile components. Absorption features unrelated \nto this system are marked by the shaded regions.} \n\\label{vp_he0226} \n\\end{figure} \n\n\n\nFor the analysis presented in this paper, we concentrate on associated absorbers detected through \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780 doublets. In Table~\\ref{tab:atomic_data} we have summarised some of the important EUV lines used in this paper including \\mbox{Ne\\,{\\sc viii}}. Here we define, an associated absorbers as (a) those with ejection velocities $|v_{\\rm ej}|$ $\\lesssim$ 8000 km~s$^{-1}$\\ with respect to the QSO emission redshift \\citep[see e.g.][]{Fox08}, or (b) show clear signatures of partial coverage (see e.g. Section \\ref{sec_parcov}) even when having higher ejection velocities (i.e. $|v_{\\rm ej}|$~$\\ge 8000$ km~s$^{-1}$). Here, the ejection velocity $v_{\\rm ej}$ is defined as the velocity separation between the emission redshift of the QSO and the \\mbox{Ne\\,{\\sc viii}}\\ optical depth weighted redshift of the absorber. The $-ve$ sign in the ejection velocity is used whenever absorber redshift is less than the emission redshift of the QSO (i.e. $z_{\\rm abs}$\\ $\\le$ $z_{\\rm em}$). However, in subsequent discussions we will use the term ``higher velocity\" assuming modulus of the ejection velocity.   \n\n\nWe have searched for the \\mbox{Ne\\,{\\sc viii}}\\ doublets in the relevant spectral range by imposing the doublet matching criteria. For each identified coincidences we checked the consistency of the profile shape. However, we do not impose the condition of optical depth ratio consistency for the \\mbox{Ne\\,{\\sc viii}}\\ doublets, keeping in mind the effects of partial coverage as discussed in Section \\ref{sec_parcov}. We then checked for the presence of all other species at the redshift of the identified \\mbox{Ne\\,{\\sc viii}}\\ doublets.  We find the signatures of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption only in 8 out of 20 (40\\%) lines of sight. We have detected 12 associated \\mbox{Ne\\,{\\sc viii}}\\ absorption systems in total towards 8 lines of sight. Note that any continuous absorption comprised of single\/multiple component(s) are treated as system. Apart from the system detected towards PG~1206+459, all other systems are detected within $\\sim$8000 km~s$^{-1}$\\ with respect to the QSOs. Because of clear signature of partial coverage in the \\mbox{Ne\\,{\\sc viii}}\\ doublet we have included the system in our sample. Based on the number per unit redshift of \\mbox{Ne\\,{\\sc viii}}\\ absorbers \\citep{Narayanan09} we expect to detect only 2 \\mbox{Ne\\,{\\sc viii}}\\ absorbers from the intervening gas. Interestingly none of these \\mbox{Ne\\,{\\sc viii}}\\ absorption detected is towards the 7 radio bright QSOs. Although we search up to 8000 km~s$^{-1}$, 67 per cent of the absorbers are detected within 5000 km~s$^{-1}$\\ from the emission redshift of the QSO.   \n\nDetails of the sight lines and the \\mbox{Ne\\,{\\sc viii}}\\ absorbers are summarized in Table~\\ref{tab_list}. Apart for $z_{\\rm abs}$\\ = 0.94262 towards HB89~0107$-$025 (marked as ``Tentative\" in column \\#11 of Table~\\ref{tab_list}), all other associated system in our sample show at least one other species which indeed makes our \\mbox{Ne\\,{\\sc viii}}\\ identification robust. Next, we provide details of each individual \\mbox{Ne\\,{\\sc viii}}\\ systems detected in our sample.  \n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.49272 towards HE~0226$-$4110}  \n\\label{sec_discript_HE0226_0.49272}  \n\n\n\n \nThe ejection velocity of the absorber is only $\\sim -$56 km~s$^{-1}$. The velocity plot of this system clearly shows that the \\mbox{Ne\\,{\\sc viii}}\\ absorption is spread over $\\sim$~226 km~s$^{-1}$\\ (see Fig.~\\ref{vp_he0226}). However, as \\mbox{Ne\\,{\\sc viii}}\\ doublets occur in the extreme blue end of the COS spectrum, the $S\/N$ is not high. A tentative detection of \\mbox{Ne\\,{\\sc viii}}\\ in this system in the $FUSE$ data was reported earlier by \\citet{Ganguly06}. Here we confirm their detection. Apart from the weak \\mbox{Ne\\,{\\sc viii}}, other ions detected in the COS spectrum are \\mbox{C\\,{\\sc iii}}, \\mbox{O\\,{\\sc iii}}, \\mbox{N\\,{\\sc iv}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc vi}}\\ and possibly \\mbox{S\\,{\\sc v}}. The detection of \\mbox{O\\,{\\sc v}}\\ is also reported in the $FUSE$ spectrum by \\citet{Ganguly06} which is not covered by the COS spectrum.  \nAs the \\mbox{O\\,{\\sc vi}}~$\\lambda 1037$ line is severely blended (see Fig.~\\ref{vp_he0226}), we could not use \\mbox{O\\,{\\sc vi}}\\ doublets to estimate the \\mbox{O\\,{\\sc vi}}\\ covering fraction. \\mbox{Ne\\,{\\sc viii}}, on the other hand, is very weak. \\nani\\ and \\mgx\\ doublets as well as Ly$\\alpha$\\ line are not covered by the COS spectrum. Nevertheless, unblended profiles of Ly$\\beta$ and Ly$\\delta$ transitions are found to be consistent with covering fraction ($f_c$) being 1.0, suggesting complete occultation of the background source by the absorber. The measured column density is log~$N(\\mbox{H\\,{\\sc i}})$ [cm$^{-2}$] = 14.49$\\pm$0.01. The unblended \\mbox{O\\,{\\sc vi}}\\ $\\lambda 1031$ profile is fitted with four Voigt profile components. The total column density (i.e. the summed up column densities measured in four components) is log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] = 14.76$\\pm$0.13.  Because of the low $S\/N$ ratio we use the component structure of \\mbox{O\\,{\\sc vi}}\\ absorption to fit the \\mbox{Ne\\,{\\sc viii}}\\ doublets keeping the $b$-parameter tied with the corresponding \\mbox{O\\,{\\sc vi}}\\ component. The estimated total column densities of \\mbox{Ne\\,{\\sc viii}}\\ is log~$N(\\mbox{Ne\\,{\\sc viii}})$ [cm$^{-2}$] = 14.09$\\pm$0.19. The total column densities for \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc vi}}\\ as reported by \\citet{Ganguly06}, using apparent optical depth technique, (i.e. log~$N(\\mbox{Ne\\,{\\sc viii}})$ [cm$^{-2}$] = 14.25$\\pm$0.15 and log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] = 14.84$\\pm$0.08) are very similar to our measurements. The difference in profile between high ions (e.g. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}) and low ions (e.g. \\mbox{H\\,{\\sc i}}, \\mbox{C\\,{\\sc iii}}, \\mbox{O\\,{\\sc iii}}, etc.) is clearly evident from the system plot. Only the strongest \\mbox{O\\,{\\sc vi}}\\ component is accompanied  by these low ions. Such a difference in profiles possibly suggests multiphase nature of the absorbing gas. A detailed discussion on the absorbing system and the QSO properties can be found in \\citet{Ganguly06}, therefore we do not discuss this system in detail.     \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=9.0cm,angle=00]{outflow_HS1102_0.48432.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.48518 \ntowards HS~1102$+$3411. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.509) of the QSO. The smooth curves overplotted on top of the data are the \nbest fitting Voigt profiles. The vertical ticks mark the centroids of the individual \nVoigt profile components. Absorption features unrelated to this absorber are marked by \nthe shaded regions. \n}  \n\\label{vp_hs1102} \n\\end{figure} \n\n\n\\subsection{$z_{\\rm abs}$ = 0.48518 towards HS~1102$+$3441} \n\\label{sec_discript_HS1102_0.48518}  \n\n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -4768$ km~s$^{-1}$\\ and is \ndetected through \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ absorption, kinematically spread over $\\sim$700 km~s$^{-1}$\\ (see Fig.~\\ref{vp_hs1102}). The \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 is blended with unknown contaminants whereas \\mbox{O\\,{\\sc vi}}\\ $\\lambda 1031$ is found to be blended with Ly$\\alpha$\\ absorption of $z_{\\rm abs}$\\ = 0.26165 system. Unblended profiles of \\mbox{Ne\\,{\\sc viii}}\\ $\\lambda 780$ and \\mbox{O\\,{\\sc vi}}\\  $\\lambda 1037$ clearly show multicomponent structures with at least five components contributing to the absorption. Because of the blending we do not attempt to estimate the covering fraction for either of the detected ions. The Voigt profile fitting assuming complete coverage seems to give reasonably good fit to the unaffected pixels of the blended profiles. The estimated total column densities are log~$N(\\mbox{O\\,{\\sc vi}}) [{\\rm cm^{-2}}] = 15.00 \\pm 0.18$ and log~$N(\\mbox{Ne\\,{\\sc viii}})[{\\rm cm^{-2}}] = 15.22 \\pm 0.20$. Ly$\\alpha$\\ is not covered by the COS spectrum. Ly$\\beta$\\ and Ly$\\gamma$\\ lines are  contaminated. Nevertheless, we use the contaminated Ly$\\beta$\\ profile to put an upper limit on $N(\\mbox{H\\,{\\sc i}})$. Assuming component structure and  $b$-parameters similar to \\mbox{Ne\\,{\\sc viii}}, we find $N(\\mbox{H\\,{\\sc i}})< 10^{14.52}$~cm$^{-2}$. \n\n\n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.59795, 0.60406 \\& 0.60989 towards HE~0238$-$1904}  \n\\label{sec_discript_HE0238}  \n\nThese systems are detected at $v_{\\rm ej} \\sim -4500$ km~s$^{-1}$\\ away from the emission redshift of the QSO, in seven absorption components kinematically spread over $\\sim$~1800 km~s$^{-1}$. We have presented a detailed analysis of this absorber in an earlier paper \\citep[see][]{Muzahid12b}. \\mgx\\ lines from this system are severely affected by the Galactic H$_{2}$ absorption and we were able to measure $N(\\mgx)$ only in some of the components showing \\mbox{Ne\\,{\\sc viii}}\\ detection. The \\nani\\ doublets are not covered by the COS spectrum for this system. We looked at $FUSE$ LiF2a data covering the \\nani\\ doublets but do not find any clear signature of \\nani\\ absorption.           \n\n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{outflow_FBQS0751_0.91980.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.91983 \ntowards FBQS~0751$+$2919. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.915) of the QSO. The smooth curves overplotted on top of the data in the \n\\mbox{Ne\\,{\\sc viii}}\\ panel are the best fitting Voigt profiles. The vertical ticks mark the centroids \nof the individual Voigt profile components.} \n\\label{vp_fqbs0751} \n\\end{figure} \n\n\n\n\\subsection{$z_{\\rm abs}$ = 0.91983 towards FBQS~0751$+$2919} \n\\label{sec_discript_FBQS0751_0.91983}  \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim +598$ km~s$^{-1}$, suggesting $z_{\\rm abs}$~$>$~$z_{\\rm em}$. \\mbox{Ne\\,{\\sc viii}}\\ doublets in this system clearly show multicomponent structure spreads over $\\sim$~170~km~s$^{-1}$\\ (see Fig.~\\ref{vp_fqbs0751}). Apart from \\mbox{Ne\\,{\\sc viii}}, other ions detected in this system are \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ and \\mbox{S\\,{\\sc v}}. \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 seems to be mildly blended in both the wings. The optical depth ratios in the core pixels of \\mbox{Ne\\,{\\sc viii}}\\ absorption are consistent with $f_c$=1.0. The Voigt profile fitting of the \\mbox{Ne\\,{\\sc viii}}\\ doublets leads to a total column density of log~$N(\\mbox{Ne\\,{\\sc viii}}) [{\\rm cm^{-2}}] = 14.59 \\pm 0.09$.  For this system \\mbox{O\\,{\\sc vi}}\\ lines are not covered by the COS spectrum. The clear non-detection of \\nani\\ $\\lambda681$ transition is consistent with log~$N(\\nani)[{\\rm cm^{-2}}]<$ 13.94 at 3$\\sigma$ confidence level. The expected positions of both the members of \\mgx\\ doublet are heavily blended and hence we do not have any estimate on $N(\\mgx)$. Very high order Lyman series lines (i.e. with $\\lambda_{\\rm rest}<$930 \\AA) are covered by the COS spectrum where we do not find any clear signature of \\mbox{H\\,{\\sc i}}\\ absorption. Non-detection of Ly$-9$ transition is consistent with $N(\\mbox{H\\,{\\sc i}})<10^{14.34}$~cm$^{-2}$. \n\n \n\\subsection{$z_{\\rm abs}$ = 0.93287 towards PG~1407$+$265}   \n\\label{sec_discript_PG1407_0.93287}  \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{outflow_PG1407_0.93295.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 0.93287 \ntowards PG~1407$+$265. The zero velocity corresponds to the emission redshift ($z_{\\rm em}$\\ = 0.940) \nof the QSO. The smooth curves overplotted on top of the data are the best fitting Voigt \nprofiles. The vertical tick marks the centroids of the individual Voigt profile components.} \n\\label{vplot_PG1407} \n\\end{figure} \nThe ejection velocity of this system is $v_{\\rm ej} \\sim -1103$ km~s$^{-1}$. \\mbox{Ne\\,{\\sc viii}}\\ absorption has two components spread over $\\sim$~100~km~s$^{-1}$\\ (see Fig.~\\ref{vplot_PG1407}). \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line is found to be contaminated in both the wings. Nevertheless, the unblended core pixels are consistent with covering fraction $f_c = 1.0$. We estimate log~$N(\\mbox{Ne\\,{\\sc viii}})[{\\rm cm^{-2}}] = 14.36\\pm0.22$. \\mgx\\ doublet is fitted with two components slightly off-centered with respect to the \\mbox{Ne\\,{\\sc viii}}\\ components. Estimated total column density of \\mgx\\ absorption is log~$N(\\mgx)[{\\rm cm^{-2}}] = 14.29\\pm0.18$. The non-detection of \\nani\\ $\\lambda 681$ transition is consistent with log~$N(\\nani)[{\\rm cm^{-2}}]<13.60$ at 3$\\sigma$ confidence level. \\mbox{O\\,{\\sc vi}}\\ doublets are not covered by the COS spectrum. \nVery high order Lyman series lines (i.e. with $\\lambda_{\\rm rest}<$ 930 \\AA) are covered by the COS spectrum where we do not find any clear signature of \\mbox{H\\,{\\sc i}}\\ absorption. In addition, no convincing Ly$\\beta$\\ (or Ly$\\gamma$) absorption is seen in archival $HST$\/FOS spectrum, obtained with the G190H grating. We note that the non-detection of Ly$\\beta$\\ is consistent with $N(\\mbox{H\\,{\\sc i}})<10^{13.71}$~cm$^{-2}$.    \n\n   \n\n\\subsection{$z_{\\rm abs}$ = 0.94262 towards HB89~0107--025}  \n\\label{sec_discript_HB890107_0.94262}  \n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -2057$~km~s$^{-1}$\\ and is detected only through \\mbox{Ne\\,{\\sc viii}}\\ absorption spread over $\\sim 120$~km~s$^{-1}$\\ (see Fig.~\\ref{aod_hb890107}). The covering fraction of \\mbox{Ne\\,{\\sc viii}}\\ is consistent with $f_c = 1$ within the continuum placement uncertainty. We measure log~$N(\\mbox{Ne\\,{\\sc viii}})$[cm$^{-2}$] = $14.27\\pm0.04$, whereas the non-detection of \\nani~$\\lambda 681$ transition in the COS spectrum is consistent with $N(\\nani) < 10^{13.75}$~cm$^{-2}$ at 3$\\sigma$ confidence level. The expected positions of \\mgx\\ doublets are contaminated and thus we cannot confirm its presence. In addition, we do not detect any other ion in the COS spectrum, corresponding to this system. \\mbox{O\\,{\\sc vi}}\\ is not covered by COS and we do not find any signature of \\mbox{O\\,{\\sc vi}}\\ lines  in the FOS\/G190H spectra. Therefore, we treat this system as tentative one. The non-detection of Ly$\\beta$\\ in FOS\/190H spectra is consistent with $N(\\mbox{H\\,{\\sc i}})< 10^{14.75}$~cm$^{-2}$.   \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{HB890107_aod.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorber at $z_{\\rm abs}$\\ = 0.94262 \ntowards HB89~0107--025. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 0.956) of the QSO. The smooth curves overplotted on top of the data are \nthe best fitting Voigt profiles. The vertical tick marks the line centroid. The \napparent column density profiles of \\mbox{Ne\\,{\\sc viii}}\\ doublets \n[in units of 10$^{12}$ cm$^{-2}$(km~s$^{-1}$)$^{-1}$] are plotted in the top panel.}  \n\\label{aod_hb890107} \n\\end{figure} \n\n\n\n\\subsection{$z_{\\rm abs}$ = 1.02854 towards PG~1206$+$459} \n\\label{sec_discript_PG1206_1.02854}  \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=10.4cm,width=9.4cm,angle=00]{PG1206_1.02863.ps} \n}}\n}}\n\\caption{Velocity plot of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorption system at $z_{\\rm abs}$\\ = 1.02854 \ntowards PG~1206$+$459. The zero velocity corresponds to the emission redshift \n($z_{\\rm em}$\\ = 1.163) of the QSO. The smooth curves overplotted on top of the data are the \nbest fitting Voigt profiles after correcting for the effect of partial coverage. \nThe vertical ticks mark the centroids of the individual Voigt profile components.   \nLy$\\alpha$\\ and \\mbox{N\\,{\\sc v}}\\ are from STIS E230M spectrum. Absorption lines unrelated to \nthis system are marked by the shaded regions.} \n\\label{vp_pg1206} \n\\end{figure} \n\n\nThis is the highest ejection velocity associated system detected in our sample, with $v_{\\rm ej}$ $\\sim -19,228$ km~s$^{-1}$, and \\mbox{Ne\\,{\\sc viii}}\\ absorption is spread over $\\sim360$~km~s$^{-1}$. This system is part of our sample, despite having large ejection velocity, as it shows clear signature of partial coverage. In Fig.~\\ref{vp_pg1206} we show absorption profiles of different species as a function of their outflow velocity with respect to the QSO emission redshift ($z_{\\rm em}$\\ = 1.214). The highly ionized species like \\arei$\\lambda\\lambda$700,713; \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780; \\nani$\\lambda\\lambda$681,694 and \\mgx$\\lambda\\lambda$609,624, originating from this absorber are detected in the COS spectrum. In addition, we also detect species like \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}, \\mbox{N\\,{\\sc iv}}\\ in COS and \\mbox{H\\,{\\sc i}}, and \\mbox{N\\,{\\sc v}}\\ in the $HST$\/STIS E230M spectrum. The STIS spectrum does not cover Ly$\\beta$, \\mbox{C\\,{\\sc iii}}\\ or \\mbox{C\\,{\\sc iv}}\\ lines. However, expected wavelength range of \\mbox{Si\\,{\\sc iv}}, \\mbox{Si\\,{\\sc iii}}~$\\lambda$1206, \\mbox{Si\\,{\\sc ii}}~$\\lambda$1260 and \\mbox{C\\,{\\sc ii}}~$\\lambda$1334 lines are covered in the STIS data, but we do not detect any of these species. The profiles of \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ doublets are flat over $\\sim$~300 km~s$^{-1}$, indicating partial coverage and heavy saturation of these lines. The flat bottom assumption (see Section~\\ref{sec_parcov}) gives the covering fractions for \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}, and \\mbox{Ne\\,{\\sc viii}}\\ as $f_c$ = 0.21, 0.40, 0.32 and 0.59 respectively. Unlike these species, the doublets of \\mgx\\ absorption are unsaturated. The uncontaminated profile of \\mgx\\ $\\lambda 609$ clearly shows component structure with at least two components contributing to the absorption. For the subsequent discussions on this system (see section~\\ref{sec_phot_model_pg1206}) we will refer the higher and lower velocity components (i.e. blue and red) as component-1 and component-2 respectively. The blue wing of the \\mgx\\ $\\lambda 624$ is blended with S~{\\sc iv} $\\lambda 657$ line from $z_{\\rm abs}$\\ = 0.9275. The core pixels of \\mgx\\ $\\lambda 624$ which are not affected by this blending are consistent with $f_c = 0.68$. For the singlet transition of \\mbox{O\\,{\\sc v}}, we have taken $f_c$ = 0.59 which seems to be consistent with (nearly) flat bottom seen in the profile. We note that, \\mbox{O\\,{\\sc v}}\\ profile is unusually broad which could possibly due to unknown contamination. Therefore, the actual $f_c$ for \\mbox{O\\,{\\sc v}}\\ could be even less. Because of the weak line strength we do not attempt to estimate $f_c$ for \\nani, instead we use \\mbox{Ne\\,{\\sc viii}}\\ covering fraction for fitting. We  note that unlike \\mbox{Ne\\,{\\sc viii}}, profiles of \\nani\\ are not saturated.    \n\n\nIn Fig.~\\ref{IP_fc}, we have plotted the covering fractions ($f_c$) of different species detected in this system as a function of ionization potentials. It is clear from the figure that we have two sets of covering fractions for this system. The species with high ionization potentials (i.e. \\mbox{Ne\\,{\\sc viii}}, \\mgx) are showing covering fraction $f_c \\gtrsim 0.6$, whereas, the low ionization species (i.e. \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}, \\mbox{N\\,{\\sc v}}) show $f_c \\lesssim 0.4$. Ionization potential dependent covering fraction have already been reported by \\citet[]{Telfer98,Muzahid12b}, where, the idea of multiphase structure of the absorbing gas has been put forward, with different species having different projected area. In view of this, the covering fraction of \\nani\\ should be similar to that of \\mbox{Ne\\,{\\sc viii}}, as they have ionization potentials of the same order. This also justifies our use of \\mbox{Ne\\,{\\sc viii}}\\ covering fraction for the fitting of \\nani\\ doublets. Such an assumption indeed gives good fit to \\nani\\ doublets.    \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=4.4cm,width=8.4cm,angle=00]{IP_fc.ps} \n}}\n}}\n\\caption{Covering fractions of different species detected in $z_{\\rm abs}$\\ = 1.02854 \ntowards PG~1206$+$459 as a function of ionization potential. The energy range \nbetween the creation and destruction ionization potentials of a given species \nare shown by the two stars connected by a solid line.      \n} \n\\label{IP_fc} \n\\end{figure} \n\n\n\n\\begin{table}\n\\caption{Partial coverage corrected Voigt profile fit parameters for the absorber at $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459.}  \n\\begin{tabular}{ccccc} \n\\hline \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) & $f_c$ \\\\   \n  (1)    &    (2)               & (3)   &  (4)       & (5)         \\\\ \n\\hline \n\\hline \n $-$19290  &   \\mgx\\   &   65 $\\pm$ 4 &   15.31 $\\pm$ 0.06 & 0.68 ($db$) \\\\\n $-$19314  &  \\nani\\   &   93 $\\pm$12 &   14.93 $\\pm$ 0.05 & 0.59 ($aa$) \\\\ \n $-$19285  &   \\mbox{Ne\\,{\\sc viii}}\\   &   83 $\\pm$ 4 &   $>$15.90 $\\pm$ 0.06 & 0.59 ($fb$) \\\\\n $-$19285  &  \\arei\\   &   83 $\\pm$ 0 & $<$14.19 $\\pm$ 0.02 & 0.59 ($aa$) \\\\ \n $-$19285  &    \\mbox{O\\,{\\sc v}}\\   &  141 $\\pm$ 5 &   $>$14.92 $\\pm$ 0.03 & 0.59 ($fb$) \\\\\n $-$19339  &    \\mbox{N\\,{\\sc v}}\\   &   72 $\\pm$18 &   $>$15.38 $\\pm$ 0.40 & 0.32 ($fb$) \\\\\n $-$19285  &   \\mbox{O\\,{\\sc iv}}\\   &   57 $\\pm$11 &   $>$15.85 $\\pm$ 0.23 & 0.21 ($fb$) \\\\\n $-$19272  &   \\mbox{N\\,{\\sc iv}}\\   &   94 $\\pm$17 &   $>$14.77 $\\pm$ 0.18 & 0.40 ($fb$) \\\\ \n $-$19285  &    \\mbox{H\\,{\\sc i}}\\   &   83 $\\pm$ 0 &   $\\sim$13.78      & 0.59 ($aa$) \\\\ \n $-$19285  &    \\mbox{H\\,{\\sc i}}\\   &   83 $\\pm$ 0 &   $\\sim$14.42      & 0.30 ($aa$) \\\\ \n\\hline                                                                           \n $-$19163  &   \\mgx\\  &   88 $\\pm$ 9 &   15.28 $\\pm$ 0.06 & 0.68 ($db$) \\\\\n $-$19150  &   \\nani\\ &   70 $\\pm$11 &   14.68 $\\pm$ 0.08 & 0.59 ($aa$) \\\\ \n $-$19140  &    \\mbox{Ne\\,{\\sc viii}}\\ &   68 $\\pm$ 5 &   $>$15.69 $\\pm$ 0.11 & 0.59 ($fb$) \\\\\n $-$19140  &   \\arei\\ &   68 $\\pm$ 0 & $<$13.66 $\\pm$ 0.07 & 0.59 ($aa$) \\\\ \n $-$19140  &     \\mbox{O\\,{\\sc v}}\\ &  113 $\\pm$11 &   $>$14.97 $\\pm$ 0.05 & 0.59 ($fb$) \\\\\n $-$19202  &   \\mbox{N\\,{\\sc v}}\\   &   70 $\\pm$26 &   $>$15.21 $\\pm$ 0.38 & 0.32 ($fb$) \\\\\n $-$19140  &    \\mbox{O\\,{\\sc iv}}\\ &   53 $\\pm$12 &   $>$15.40 $\\pm$ 0.13 & 0.21 ($fb$) \\\\\n $-$19140  &    \\mbox{N\\,{\\sc iv}}\\ &   46 $\\pm$15 &   $>$14.28 $\\pm$ 0.36 & 0.40 ($fb$) \\\\ \n $-$19140  &    \\mbox{H\\,{\\sc i}}\\  &   68 $\\pm$ 0 &   $\\sim$13.61      & 0.59 ($aa$) \\\\\n $-$19140  &    \\mbox{H\\,{\\sc i}}\\  &   68 $\\pm$ 0 &   $\\sim$14.01      & 0.30 ($aa$) \\\\\n\\hline \n\\hline \n\\end{tabular}\n\\vfill ~\\\\ \nNote -- Listed errors on all the quantities in this paper only include the statistical \nerrors. For $v_{\\rm ej}$ the COS calibration uncertainty is $\\sim \\pm$~10 km~s$^{-1}$.  In addition, \nthe uncertainty in the COS LSF introduces errors of at least 1 to 3 km~s$^{-1}$\\ in these profile \nfit line widths. Zero error implies that the parameter was tied\/fixed during fitting. \nCovering fraction, $f_c$ used to estimate the column density is given in column~5. Method used \nto compute $f_c$ is mentioned in parenthesis. ``$fb$\"-- from flat bottom profile, ``$db$\"-- from \ndoublets, ``$aa$\"-- physically motivated assumed value. Note that the column density estimated \nfrom the flat bottom profile (i.e. ``$fb$\") should be taken as lower limit.         \n\\label{tab_pg1206}  \n\\end{table}  \n\n\nParameters estimated through Voigt profile fitting after correcting for partial coverage are given in Table~\\ref{tab_pg1206}. We treat column densities of all the species as lower limits in the case of ions showing flat bottom profiles. We note that, both the members of \\arei\\ doublets are partially blended by unknown contaminants which made the covering fraction estimation impossible. However, since the ionization potentials (creation$+$destruction) of \\arei\\ are comparable to those of \\mbox{O\\,{\\sc v}}, we take $f_c = 0.59$. We note that, because of blend $N(\\arei)$  should be taken as upper limit.  \n\n\nAt the expected position of \\mbox{O\\,{\\sc vi}}, some absorption is seen in the low resolution $HST\/$FOS G190H spectrum. However, due to severe blending in both members of the doublets, we do not attempt to estimate the covering fraction. Estimated conservative upper limit on \\mbox{O\\,{\\sc vi}}\\ column density is log~$N(\\mbox{O\\,{\\sc vi}})$ [cm$^{-2}$] $<$ 14.80, assuming $f_c$ = 0.59. In addition, we do not detect any clear signature of Ly$\\beta$\\ absorption in G190H spectrum. Weak Ly$\\alpha$\\ absorption line, seen in STIS\/E230M spectrum, is fitted with two different values of covering fractions (i.e. $f_c$ = 0.59 and 0.30; see Table~\\ref{tab_pg1206}), in order to estimate the maximum \\mbox{H\\,{\\sc i}}\\ content associated with the high and low ionization phases. However, in both the phases $N(\\mbox{H\\,{\\sc i}})$ found to be $< 10^{14.5}$ cm$^{-2}$. All these suggest a very little neutral hydrogen content in this absorber. \n \n \n\n\\subsection{$z_{\\rm abs}$ = 1.21534 towards PG~1338$+$416}  \n\\label{sec_discript_PG1338_1.21534}  \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim +181$~km~s$^{-1}$\\ suggesting $z_{\\rm abs}$~$>$~$z_{\\rm em}$. This absorber (see the rightmost panel of Fig.~\\ref{vp_pg1338}) is primarily detected through the presence of \\mbox{O\\,{\\sc vi}}\\ doublets in FOS\/G270H spectrum and subsequently confirmed with various other low ionization species (e.g. \\mbox{O\\,{\\sc iii}}, \\mbox{N\\,{\\sc iv}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ etc.) in COS spectrum. Weak absorption from high ionization species like \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ are also detected. However, \\mbox{Ne\\,{\\sc viii}}~$\\lambda780$ profile is blended with strong Ly$\\beta$\\ absorption from $z_{\\rm abs}$\\ = 0.6863 system. The \\mgx\\ $\\lambda 609$ line is heavily blended, possibly with low redshift Ly$\\alpha$\\ line and hence not shown in the figure. The non-detection of \\nani\\ $\\lambda 681$ is consistent with log~$N(\\nani)[{\\rm cm^{-2}}] < 14.18$ at 3$\\sigma$ confidence level. \\mbox{O\\,{\\sc iv}}~$\\lambda608$ line is severely blended with \\mgx~$\\lambda624$ line from $z_{\\rm abs}$\\ = 1.15456. \\mbox{O\\,{\\sc iii}}~$\\lambda702$ line is partially blended with unknown contaminants.      \nThe uncontaminated low ionization species (i.e. \\mbox{N\\,{\\sc iv}}~$\\lambda765$, \\mbox{O\\,{\\sc iv}}~$\\lambda787$) clearly show multicomponent structure. In both cases, at least two Voigt profile components (shown by vertical dashed lines) are required to get best fitted $\\chi^{2}$ close to 1. Unlike low ionization species, \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 absorption shows smooth and\/or broad profile which is well fitted by a single component. Due to poor spectral resolution, all the ions detected in FOS can be fitted with a single component.  \n\n\nWe do not find a clear signature of partial coverage in any line. For example, \\mbox{N\\,{\\sc v}}\\ and \\mbox{O\\,{\\sc vi}}\\ doublets are well fitted with $f_c$ = 1.0 and do not show non-zero flat bottom profiles. The Ly$\\alpha$\\ and (weak) Ly$\\beta$\\ absorption are also consistent with complete coverage of the background source by the absorber. The Voigt profile fit parameters for this absorber are given in Table~\\ref{tab3_pg1338}. We would like to mention here that, because of blending in \\mbox{O\\,{\\sc iii}}\\ line and saturation in \\mbox{O\\,{\\sc v}}\\ line, $N(\\mbox{O\\,{\\sc iii}})$ and $N(\\mbox{O\\,{\\sc v}})$ should be taken as upper and lower limits respectively. We will use these bounds in section~\\ref{sec_phot_model2_pg1338}, where we discuss the photoionization modelling of this system. In passing, we note that \\mbox{H\\,{\\sc i}}\\ and \\mbox{O\\,{\\sc vi}}\\ line centroids are offset by $\\sim$40 km~s$^{-1}$. This could be  a signature of multiphase gas. We also note that, \\mbox{H\\,{\\sc i}}\\ and \\mbox{C\\,{\\sc iii}}\\ line centroids are offset by $\\sim$60 km~s$^{-1}$. However, as they are detected in spectra taken with two different instruments (i.e. FOS G270H and G190H), such an offset could also be attributed to the systematic uncertainties.     \n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Voigt profile fit parameters for the absorber at $z_{\\rm abs}$\\ = 1.21534 towards \nPG~1338$+$416 using $f_c = 1$ for all the species.} \n\\begin{tabular}{rccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) \\\\   \n  (1)    &    (2)               & (3)   &  (4)                 \\\\ \n\\hline \n\\hline \n  $+$81  &   \\mbox{H\\,{\\sc i}}\\  &  137 $\\pm$14 &  14.04 $\\pm$ 0.04  \\\\\n $+$101  &   \\mbox{N\\,{\\sc v}}\\  &  144 $\\pm$15 &  14.36 $\\pm$ 0.04  \\\\\n $+$121  &   \\mbox{O\\,{\\sc vi}}\\  &  158 $\\pm$49 &  14.74 $\\pm$ 0.12  \\\\\n $+$126  &  \\mbox{N\\,{\\sc iv}}\\  &   44 $\\pm$ 2 &  14.25 $\\pm$ 0.03  \\\\\n $+$126  &  \\mbox{O\\,{\\sc iv}}\\  &   44 $\\pm$ 2 &  15.08 $\\pm$ 0.03  \\\\\n $+$126  &  \\mbox{O\\,{\\sc iii}}   &   44 $\\pm$ 0 &  14.82 $\\pm$ 0.01  \\\\\n $+$136  &   \\mbox{O\\,{\\sc v}}\\  &   45 $\\pm$ 3 &  14.78 $\\pm$ 0.05  \\\\\n $+$139  &   \\mbox{C\\,{\\sc iii}}\\  &  105 $\\pm$17 &  13.91 $\\pm$ 0.06  \\\\\n $+$181  &  \\mbox{Ne\\,{\\sc viii}}\\  &   91 $\\pm$12 &  14.42 $\\pm$ 0.05  \\\\\n $+$181  &  \\mgx\\  &   91 $\\pm$ 0 &  14.62 $\\pm$ 0.06  \\\\\n $+$214  &  \\mbox{N\\,{\\sc iv}}\\  &   40 $\\pm$ 5 &  13.80 $\\pm$ 0.06  \\\\\n $+$214  &  \\mbox{O\\,{\\sc iv}}\\  &   40 $\\pm$ 5 &  14.53 $\\pm$ 0.07  \\\\\n $+$214  &  \\mbox{O\\,{\\sc iii}}\\  &   40 $\\pm$ 0 &  14.26 $\\pm$ 0.03  \\\\\n $+$223  &   \\mbox{O\\,{\\sc v}}\\  &   32 $\\pm$ 3 &  14.55 $\\pm$ 0.06  \\\\\n\\hline \n\\hline \n\\end{tabular}\n\\label{tab3_pg1338}  \n\\end{center}\n\\end{table}  \n\n\n\n\n\\begin{figure*} \n\\begin{sideways}\n\\begin{minipage}{24cm} \n\\centerline{\\vbox{\n\\centerline{\\hbox{\n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow2_PG1338.ps} \n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow3_PG1338.ps} \n\\includegraphics[height=11.0cm,width=8.0cm,angle=00]{outflow1_PG1338.ps} \n}}\n}} \n\\caption{Velocity plot of the three associated \\mbox{Ne\\,{\\sc viii}}\\ systems detected towards PG~1338$+$416.  \nThe zero velocity corresponds to the emission redshift ($z_{\\rm em}$\\ = 1.214) of the QSO. The \nsmooth curves overplotted on top of the data are the best fitting Voigt profiles after \ncorrecting for the partial coverage whenever needed. The shaded regions mark the contamination \ndue to unrelated absorption to the system of interest. The vertical dashed lines mark the \npositions of individual Voigt profile components. \n{\\sl Left :} The system at $z_{\\rm abs}$\\ = 1.15456. {\\sl Middle :} The system at $z_{\\rm abs}$\\ = 1.16420. \nThe smooth dashed curves are not fit to the data, but the synthetic profiles corresponding \nto the maximum allowed column density assuming complete coverage (see text). \n{\\sl Right :} The system at $z_{\\rm abs}$\\ = 1.21534. Apart from \\mbox{C\\,{\\sc iii}}\\ all other species plotted in \nthe left hand sub-panel are from  FOS\/G270H spectrum whereas \\mbox{C\\,{\\sc iii}}\\ is from FOS\/G190H spectrum.  \nAll the species plotted in the right hand sub-panel are from COS spectrum. Two components are \nclearly seen in \\mbox{O\\,{\\sc iii}}, \\mbox{O\\,{\\sc iv}}, \\mbox{N\\,{\\sc iv}}\\ and \\mbox{O\\,{\\sc v}}\\ absorption (shown by two vertical dashed lines).  \nIn all other cases single component is needed as shown by (green) solid tick.   \n} \n\\label{vp_pg1338} \n\\end{minipage}\n\\end{sideways} \n\\end{figure*}\n  \n\n\\subsection{$z_{\\rm abs}$ = 1.16420 towards PG~1338$+$416}   \n\\label{sec_discript_PG1338_1.16420}  \n\n\nThe ejection velocity of this system is $v_{\\rm ej} \\sim -6818$~km~s$^{-1}$. The velocity plot of this systems is shown in the middle panel of Fig.~\\ref{vp_pg1338}. Apart from \\mbox{O\\,{\\sc v}}\\ and \\nani~$\\lambda 681$, all other detected ions in this system show complex blend in their profiles. The overall similarity in profiles of various ions clearly assures their presence. We do not find any other contamination in \\nani~$\\lambda 681$ absorption and it shows very similar profile like \\mbox{O\\,{\\sc v}}. Therefore, we believe \\nani\\ detection is robust, although the blue wing of \\nani~$\\lambda 694$ line is severely blended. \\mbox{Ne\\,{\\sc vi}}~$\\lambda 558$ absorption falls near the Galactic Ly$\\alpha$ absorption and hence the continuum around this absorption is not well constrained. Since \\mbox{O\\,{\\sc v}}\\ is singlet transition and \\nani~$\\lambda 694$ is blended, we did not estimate covering fraction for any of these ions. We assume $f_c = 1$ to get a lower limit on column densities. \nAt least four Voigt profile components are required to fit the unblended \\mbox{O\\,{\\sc v}}\\ and \\nani~$\\lambda 681$ profiles. The \\nani\\ to \\mbox{O\\,{\\sc v}}\\ column density ratio in all four components are consistent within factor $\\sim$~2 (e.g. log~$N(\\nani)\/N(\\mbox{O\\,{\\sc v}})$ = 0.27$\\pm$0.21). Because of contamination in the case of \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ lines and poorly constrained continuum in the case of \\mbox{Ne\\,{\\sc vi}}~$\\lambda 558$ line we do not perform Voigt profile fitting for these absorption. Instead, we check the consistency of synthetic profiles generated using the component structure and the $b$-parameters similar to \\mbox{O\\,{\\sc v}}\\ line, assuming $f_c = 1.0$. The synthetic profiles are shown in smooth dashed curves on top of data, in the middle panel of Fig.~\\ref{vp_pg1338}. The highest optical depth pixels in \\mbox{Ne\\,{\\sc viii}}\\ doublets are roughly consistent with $f_c \\gtrsim 0.8$. The line measurements for this system are presented in Table~\\ref{tab2_pg1338}.   \nThe low resolution FOS\/G190H spectrum shows  absorption in the expected position of \\mbox{O\\,{\\sc vi}}. However, contamination of \\mbox{O\\,{\\sc vi}}\\ lines from $z_{\\rm abs}$\\ = 1.15456 absorber do not allow any reliable column density estimation. Ly$\\alpha$\\ and Ly$\\beta$\\ absorption from this absorber are covered by the G270H and G190H spectra respectively. However, Ly$\\beta$\\ is found to be stronger than Ly$\\alpha$, suggesting a possible contamination in Ly$\\beta$. Ly$\\alpha$, on the other hand, is contaminated with Galactic Fe~{\\sc ii} lines. Therefore, we do not present any measurement for \\mbox{H\\,{\\sc i}}\\ in Table~\\ref{tab2_pg1338}. Due to poorly constrained $f_c$, the column density measurements are highly uncertain and hence we do not discuss the ionization modelling for this system, in spite of the presence of \\nani.  \n\n\\subsection{$z_{\\rm abs}$ = 1.15456 towards PG~1338$+$416}  \n\\label{sec_discript_PG1338_1.15456}  \n\n\n\\begin{table}\n\\begin{center} \n\\caption{Voigt profile fit parameters for $z_{\\rm abs}$\\ = 1.16420 towards PG~1338$+$416 \nassuming $f_c = 1$ for all the species.}  \n\\begin{tabular}{cccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$)   \\\\   \n  (1)    &    (2)               & (3)   &  (4)                   \\\\ \n\\hline \n\\hline \n$-7022$    & \\nani\\   &   100$\\pm$ 13 &     $>$ 14.50 $\\pm$ 0.05  \\\\\n           &    \\mbox{O\\,{\\sc v}}\\  &   40 $\\pm$  3 &     $>$ 14.03 $\\pm$ 0.02  \\\\\n           &    \\mgx\\ &   40          &           $\\sim$   14.67   \\\\\n           &    \\mbox{Ne\\,{\\sc viii}}\\ &   40          &           $\\sim$   14.94   \\\\\n           &   \\mbox{Ne\\,{\\sc vi}}\\ &   40          &           $\\sim$   14.68   \\\\ \n\\hline \n$-6932$\t   & \\nani\\   &   34 $\\pm$ 11 &      $>$ 13.91 $\\pm$ 0.15  \\\\  \t\n           &    \\mbox{O\\,{\\sc v}}\\  &\t  27 $\\pm$  4 &      $>$ 13.68 $\\pm$ 0.07  \\\\      \n           &    \\mgx\\ &\t  27          &           $\\sim$   14.67   \\\\      \n\t   &    \\mbox{Ne\\,{\\sc viii}}\\ &\t  27          &           $\\sim$   14.50   \\\\      \n           &   \\mbox{Ne\\,{\\sc vi}}\\ &\t  27          &           $\\sim$   14.14   \\\\      \n\\hline \n$-6832$\t   & \\nani\\   &  74 $\\pm$ 7   &      $>$ 14.65 $\\pm$ 0.04  \\\\ \t \n           &    \\mbox{O\\,{\\sc v}}\\  &  66 $\\pm$ 4   &      $>$ 14.17 $\\pm$ 0.02  \\\\ \n           &    \\mgx\\ &  66           & $\\sim$   15.20   \\\\ \n\t   &    \\mbox{Ne\\,{\\sc viii}}\\ &  66           & $\\sim$   15.14   \\\\ \n           &   \\mbox{Ne\\,{\\sc vi}}\\ &  66           & $\\sim$   14.87   \\\\  \n\\hline \n$-6613$\t   & \\nani\\   &  86 $\\pm$ 14  &     $>$  14.40 $\\pm$ 0.06  \\\\  \n           &    \\mbox{O\\,{\\sc v}}\\  &  57 $\\pm$  3  &     $>$  14.04 $\\pm$ 0.02  \\\\ \n           &    \\mgx\\ &  57           & $\\sim$   15.09   \\\\ \n\t   &    \\mbox{Ne\\,{\\sc viii}}\\ &  57           & $\\sim$   14.91   \\\\ \n           &   \\mbox{Ne\\,{\\sc vi}}\\ &  57           & $\\sim$   14.75   \\\\ \n\\hline \n\\hline \n\\end{tabular}\n\\label{tab2_pg1338}  \n\\end{center} \n\\end{table}  \n\n \nThe ejection velocity of this system is $v_{\\rm ej} \\sim -8156$~km~s$^{-1}$, with \\mbox{Ne\\,{\\sc viii}}\\ absorption spread over $\\sim 340$~km~s$^{-1}$. The profiles of different species originating from this system are plotted as a function of outflow velocity in the leftmost panel of Fig.~\\ref{vp_pg1338}. This is the highest \\mgx\\ column density system in our sample. The core pixels of \\mgx\\ doublets are free from any blend and clearly show broad multicomponent structure with at least two components contributing to the absorption. We will refer to the highest velocity component as component-1 and the other as component-2\\ in subsequent discussions regarding this system (e.g. in section~\\ref{sec_phot_model1_pg1338}). There is only a mild contamination from Ly$\\gamma$ of $z_{\\rm abs}$\\ = 0.3488 absorber in the blue wing of the \\mgx~$\\lambda609$ line as shown by shaded region. The red wing of the \\mgx~$\\lambda624$, on the other hand, is blended with \\mbox{O\\,{\\sc iv}}~$\\lambda608$ transitions from another associated absorber (i.e. $z_{\\rm abs}$\\ = 1.21534) along this sight line. We use the uncontaminated core pixels (i.e., between $-8350 < v~(\\rm km s^{-1}) < -8150$) of \\mgx\\ doublets and estimate the covering fraction $f_c = 0.8\\pm0.1$ (see Fig.~\\ref{covf_pg1338}). \\nani~$\\lambda681$ line is completely free from any contamination and shows remarkable similarity with \\mgx\\ profiles. This possibly means \\mgx\\ and \\nani\\ are originating from the same phase of the absorbing gas. The red wing of \\nani~$\\lambda694$ line, however, is blended by Ly$-9$ transition from a previously known DLA at $z_{\\rm abs}$\\ = 0.6214 \\citep[]{Rao06}. Therefore we use covering fraction for \\nani\\ similar to that of \\mgx. We note that such an assumption gives remarkably good fit to \\nani\\ doublets. \n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.8cm,angle=00]{covf_pg1338.ps} \n}}\n}}\n\\caption{Profiles of \\mgx\\ doublet for the system at $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416 \nare shown in the two bottom panels. Corresponding apparent column density distributions \n[in units of 10$^{13}$ cm$^{-2}$ (km s$^{-1}$)$^{-1}$] are shown in the second panel from \nthe top. The covering fraction distribution is shown in the topmost panel. The dashed line \nindicates the median value of covering fraction $f_c = 0.8 \\pm 0.1$, as measured in the core \npixels. The shaded regions show the velocity range affected by unrelated absorption.\n} \n\\label{covf_pg1338} \n\\end{figure} \n\n\nBoth transitions of \\mbox{Ne\\,{\\sc viii}}\\ doublet show very strong, albeit blended, absorption with flat bottom profiles consistent with $f_c = 0.8$. The strong uncontaminated \\mbox{O\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc vi}}\\ lines also show flat bottom profiles. The covering fraction in these two cases, as calculated from the flat bottom, are very similar and lower (i.e. $f_c$ = 0.67) than that of very highly ionized species (i.e. \\mbox{Ne\\,{\\sc viii}}, \\mgx). Clearly, like the previous case (i.e. $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459), here also we find two sets of covering fraction for the detected species suggesting ionization potential dependent phase separation of the absorbing gas. \\mbox{Ne\\,{\\sc v}}\\ line seen in this absorber is unsaturated and shows two possible velocity components. However, due to severe blending in both the wings of \\mbox{Ne\\,{\\sc v}}\\ absorption, we only estimate the upper limit on $N(\\mbox{Ne\\,{\\sc v}})$ assuming $f_c$ and $b$-parameters similar to those of \\mbox{O\\,{\\sc v}}\\ line. In section~\\ref{sec_phot_model}, we will show that, under photoionization equilibrium \\mbox{O\\,{\\sc v}}\\ and \\mbox{Ne\\,{\\sc v}}\\ trace each other for the whole range of ionization parameters. Therefore, using the \\mbox{O\\,{\\sc v}}\\ covering fraction for \\mbox{Ne\\,{\\sc v}}\\ is legitimate.  We also estimate upper limits on the weak absorption seen in the expected position of \\alel~$\\lambda550$ transition assuming $f_c$ and $b$-parameters similar to those of \\mgx, as they have ionization potentials of similar order. However, as both the wings of \\alel~$\\lambda550$ line is blended the measured column density is merely a upper limit. The other member of \\alel\\ doublet with $\\lambda_{\\rm rest} = 568$~\\AA, is severely affected by the Galactic Ly$\\alpha$\\ absorption and complex blend. In addition, we do not detect any clear signature of Ly$\\alpha$\\ absorption, in the FOS\/G270H spectrum. Some absorption is seen in the expected positions of \\mbox{O\\,{\\sc vi}}\\ doublets in the FOS\/G190H spectrum. However, the contamination of \\mbox{O\\,{\\sc vi}}\\ lines from $z_{\\rm abs}$\\ = 1.16420 absorber and the poor data quality prevent us from any reliable column density estimations.   \nThe partial coverage corrected Voigt profile fit parameters are given in Table~\\ref{tab1_pg1338}. In the case of non-detections (i.e. \\mbox{H\\,{\\sc i}}, \\mbox{O\\,{\\sc iv}}\\ and \\arei), we present 3$\\sigma$ upper limits on column densities as estimated from the error in the continuum. \n\n\\section{Ionization models} \n\\label{sec_phot_model} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{sed_mf.ps} \n}}\n}}\n\\caption{Typical shapes of spectral energy distributions of an AGN with arbitrary \nnormalization. The solid line gives the spectrum by \\citet{Mathews87} whereas the \ndotted curve is generated assuming a blackbody with temperature \n$T_{\\rm BB} \\sim 1.5\\times10^{5}$~K and power laws with typical slopes, \n$\\alpha_{\\rm uv} = -0.5$, $\\alpha_{\\rm x} = -0.7$ and $\\alpha_{\\rm ox} = -1.8$ \n(see text). The vertical lines with different line styles mark the frequency \ncorresponding to the ionization potential of the species mentioned in the plot.  \n} \n\\label{sed_mf} \n\\end{figure} \n \n\\begin{table}\n\\caption{Partial coverage corrected Voigt profile fit parameters for $z_{\\rm abs}$\\ = \n1.15456 towards PG~1338$+$416.}  \n\\begin{tabular}{ccccc} \n\\hline \n $v_{\\rm ej}$(km~s$^{-1}$) & Ion & $b$(km~s$^{-1}$) & log~$N$(cm$^{-2}$) & $f_c^{a}$ \\\\   \n  (1)    &    (2)               & (3)   &  (4)       & (5)         \\\\  \n\\hline \n\\hline \n$-$8195   & \\nani\\  &  165 $\\pm$ 12   & 15.05 $\\pm$ 0.03 & 0.80 ($aa$) \\\\ \n          & \\mgx\\   &  165 $\\pm$  7   & 15.62 $\\pm$ 0.02 & 0.80 ($db$) \\\\ \n\t  & \\alel\\  &        165      & $\\le$14.77 $\\pm$ 0.07 & 0.80 ($aa$) \\\\  \n          & \\mbox{Ne\\,{\\sc viii}}\\   &   89 $\\pm$ 35   & $>$15.94 $\\pm$ 0.41 & 0.80 ($fb$) \\\\ \n          & \\mbox{O\\,{\\sc v}}\\    &   91 $\\pm$  6   & $>$14.97 $\\pm$ 0.06 & 0.67 ($fb$) \\\\ \n          & \\mbox{Ne\\,{\\sc vi}}\\  &   87 $\\pm$  5   & $>$15.57 $\\pm$ 0.04 & 0.67 ($fb$) \\\\ \n\t  & \\mbox{O\\,{\\sc iv}}\\   &         91      & $\\le$ 13.69      & 0.67 ($aa$) \\\\  \n\t  & \\mbox{Ne\\,{\\sc v}}\\  &         91      & $\\le$ 15.26      & 0.67 ($aa$) \\\\ \n\t  & \\mbox{Ne\\,{\\sc iv}}\\  &         91      & $\\le$ 13.66      & 0.67 ($aa$) \\\\ \n          & \\arei\\  &         91      & $\\le$ 13.94      & 0.67 ($aa$) \\\\ \n\t  & \\mbox{H\\,{\\sc i}}\\    &        165      & $\\le$ 13.64      & 0.67 ($aa$) \\\\   \n\t  & \\mbox{H\\,{\\sc i}}\\    &        165      & $\\le$ 13.56      & 0.80 ($aa$) \\\\   \n\\hline                                                   \n$-$8055   & \\nani\\  &   90 $\\pm$  8   & 14.82 $\\pm$ 0.04 & 0.80 ($aa$) \\\\  \n          & \\mgx\\   &   86 $\\pm$  5   & 15.33 $\\pm$ 0.03 & 0.80 ($db$) \\\\  \n\t  & \\alel\\  &        86       & $\\le$14.66 $\\pm$ 0.06 & 0.80 ($aa$) \\\\  \n          & \\mbox{Ne\\,{\\sc viii}}\\   &   62 $\\pm$  6   & $>$15.45 $\\pm$ 0.12 & 0.80 ($fb$) \\\\    \n          & \\mbox{O\\,{\\sc v}}\\    &   65 $\\pm$  5   & $>$14.85 $\\pm$ 0.08 & 0.67 ($fb$) \\\\ \n          & \\mbox{Ne\\,{\\sc vi}}\\  &   66 $\\pm$  5   & $>$15.60 $\\pm$ 0.07 & 0.67 ($fb$) \\\\  \n\t  & \\mbox{O\\,{\\sc iv}}\\   &         65      & $\\le$ 13.90      & 0.67 ($aa$) \\\\    \n\t  & \\mbox{Ne\\,{\\sc v}}\\  &         65      & $\\le$ 14.87      & 0.67 ($aa$) \\\\ \n\t  & \\mbox{Ne\\,{\\sc iv}}\\  &         65      & $\\le$ 13.10      & 0.67 ($aa$) \\\\ \n\t  & \\arei\\  &         65      & $\\le$ 13.30      & 0.67 ($aa$) \\\\ \n\t  & \\mbox{H\\,{\\sc i}}\\    &         91      & $\\le$ 13.50      & 0.67 ($aa$) \\\\   \n\t  & \\mbox{H\\,{\\sc i}}\\    &         91      & $\\le$ 13.42      & 0.80 ($aa$) \\\\   \n\\hline  \n\\hline \n\\end{tabular}\n\\vfill ~\\\\ \nTable Note -- $^{a}$Same as Table~\\ref{tab_pg1206}\n\\label{tab1_pg1338}  \n\\end{table}  \n\n\n\\begin{figure*} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig1.ps} \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig2.ps} \n}}\n\\centerline{\\hbox{ \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig3.ps} \n\\includegraphics[height=6.4cm,width=8.4cm,angle=00]{fig4.ps} \n}}\n}} \n\\caption{Results of photoionization model calculation in optically thin condition with  \n$N(\\mbox{H\\,{\\sc i}})$ = 10$^{14}$ cm$^{-2}$, incident ionizing continuum given by \\citet{Mathews87}, \nand with solar metallicity $(Z = Z_{\\odot})$. In each panel column densities of various \nspecies (bottom) and their ratios (top) are plotted as a function of ionization parameter. \nA species will be detectable for the ionization parameter range in which it has column density \n$N\\gtrsim10^{13}$~cm$^{-2}$. In the detectable range if two species show constant ratio (i.e. \ninsensitive to ionization parameter), they are likely to originate from the same phase of \nthe absorbing gas. Such a pair of ions is good to estimate relative abundance of the elements. \nIonization parameter should be estimated from the pair of ions whose ratio is sensitive to the \nlog~U.      \nNote that, according to the ionization potentials, different species are grouped and plotted in \ndifferent panels for convenience. The notch seen in the $N(\\mgx)\/N(\\nani)$ ratio in panel-(D) is \nan artifact created by {\\sc cloudy} in the low column density limit of $N(\\mgx)$.  \n}   \n\\label{cloudy1} \n\\end{figure*} \n\n\nIn this section, we try to determine the ionization structure and the physical conditions in the outflowing gas with the help of photoionization equilibrium models using {\\sc cloudy  v(07.02)} \\citep[first described in][]{Ferland98}. First, we describe results of a general photoionization model to understand the variation of column densities of different high and low ions and their ratios over a wide range in ionization parameter. We then describe more detailed models (both PI and CI) only for those individual absorbers showing absorption lines from several ions with adequate column density measurements. \n\n\nOur photoionization models assume the absorbing gas to be an optically thin (i.e. stopping H~{\\sc i} column density of 10$^{14}$ cm$^{-2}$ as measured in most cases) plane parallel slab with solar metallicity and relative solar abundances, illuminated by the AGN spectrum. To draw some general conclusions we use the mean spectrum of \\citet{Mathews87} (hereafter MF87, see solid curve in Fig.~\\ref{sed_mf}). However, it is well known that the results of photoionization modeling is very sensitive to the shape of the ionizing radiation. In order to minimize the uncertainties, while modelling individual absorbers, we use the QSO SED of the form:  \n\\begin{equation} \nf_{\\nu} = \\nu^{\\alpha_{\\rm uv}} {\\rm exp} (-h\\nu\/kT_{\\rm BB}){\\rm exp}(-kT_{\\rm IR}\/h\\nu)+B\\nu^{\\alpha_{\\rm x}},  \n\\label{agn_cont}\n\\end{equation} \nwhile discussing individual systems. Here, $T_{\\rm BB}$, ${\\alpha_{\\rm uv}}$ and ${\\alpha_{\\rm x}}$ are disk black body temperature, UV spectral index and X-ray spectral index respectively. The normalization constant $B$ is fixed using the optical-to-X-ray powerlaw slope $\\alpha_{\\rm ox}$ and we use  $kT_{\\rm IR}$ = 0.01 Rydberg. We use the SED defined by the Eq.~\\ref{agn_cont} with appropriate values for the parameters (based on available observations) when we discuss the photoionization models of individual absorbers. \n\nThe model predictions for MF87 incident continuum are plotted in Fig.~\\ref{cloudy1}. In the bottom of each panel, we plot the column densities of different species having similar ionization potential, as a function of ionization parameter. For the sensitivity of our COS spectra, we find that the column density of individual species has to be $\\ge10^{13}$ cm$^{-2}$ to produce detectable absorption lines which are as broad as $\\sim$~100 km~s$^{-1}$.  \n\n\nIn panel {\\bf (A)} of Fig.~\\ref{cloudy1}, we plot the model predictions for the species \\mbox{N\\,{\\sc iv}}\\ (I.P = 47.5 eV), \\mbox{C\\,{\\sc iv}}\\ (I.P = 47.9 eV), \\mbox{O\\,{\\sc iv}}\\ (I.P = 54.9 eV) and \\mbox{Ne\\,{\\sc iv}}\\ (I.P = 63.5 eV). Among all these species, \\mbox{O\\,{\\sc iv}}\\ seems to be the dominant in the range $-2.0\\le$~log~U~$\\le 0.0$ and apart from \\mbox{C\\,{\\sc iv}}\\ all of them showing peak round log~U~$\\sim$~$-1.0$ (see bottom panel). \\mbox{C\\,{\\sc iv}}, however, shows relatively flat distribution over the above mentioned ionization parameter range. For log~U~$>0.0$, column densities of almost all these species become $<10^{13}$ cm$^{-2}$ and hence, they will not be detectable. \nFrom the top panel it is clear that the ionization parameter range where all the species are detectable (i.e., $-2.0\\le$~log~U~$\\le 0.0$), $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{N\\,{\\sc iv}})$ and $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{Ne\\,{\\sc iv}})$ ratios show remarkable constancy. The ratios where $N(\\mbox{C\\,{\\sc iv}})$ is involved [i.e. $N(\\mbox{Ne\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iv}})$ and $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iv}})$], on the other hand, show similar constancy for $-2.0 \\le$ log~U $\\le -1.0$ and fall by a factor of $\\ge$ 0.84 dex in the range $-1.0 \\le$ log~U $\\le 0.0$. \n\n\nIn panel {\\bf (B)} we plot the model predictions for the species \\mbox{S\\,{\\sc vi}}\\ (I.P = 72.7 eV), \\mbox{O\\,{\\sc v}}\\ (I.P = 77.4 eV), \\mbox{N\\,{\\sc v}}\\ (I.P = 77.5 eV) and \\mbox{Ne\\,{\\sc v}}\\ (I.P = 97.1 eV). From the bottom panel, it is apparent that \\mbox{O\\,{\\sc v}}\\ is the dominant species for the whole range in ionization parameters. In addition, all of them show roughly similar $N$ distribution with a peak around log~U $\\sim -0.5$. We also find that most of these species are detectable in the range $-1.5 \\le$ log~U $\\le 0.5$. From the top panel, it is interesting to note that, apart from $N(\\mbox{O\\,{\\sc v}})\/N(\\mbox{S\\,{\\sc vi}})$ ratio, all other ratios are exceptionally constant over the ionization parameter range where these species are detectable (i.e. $-1.5 \\le$ log~U $\\le 0.5$).                   \n\n\nIn panel {\\bf (C)} we plot the model predictions for the species \\mbox{O\\,{\\sc v}}\\ (I.P = 77.4 eV), \\mbox{O\\,{\\sc vi}}\\ (I.P = 113.9 eV), \\arei\\ (I.P = 124.3 eV) and \\mbox{Ne\\,{\\sc vi}}\\ (I.P = 126.2 eV). From the bottom panel, it is evident that, apart from \\arei\\ all other species are detectable roughly in the range $-1.0\\le$ log~U $\\le 1.0$. In addition, \\mbox{O\\,{\\sc vi}}\\ is found to be the dominant species in this ionization parameter range. \\arei, on the other hand, is detectable in a very narrow range in ionization parameter around log~U $\\sim 0.0$, where all these species show peak column densities. \nFrom the top panel, in is interesting to note that the $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc v}})$ ratio keeps on increasing with the increase of ionization parameter whereas $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{Ne\\,{\\sc vi}})$ ratio remains constant for the entire range in log~U (i.e. $-2.0 \\le$ log~U $\\le 1.0$). The $N(\\mbox{Ne\\,{\\sc vi}})\/N(\\arei)$ ratio also remains constant in the range $-1.0 \\le$ log~U $\\le 1.0$. $N(\\mbox{O\\,{\\sc vi}})\/N(\\arei)$ ratio, on the contrary, varies by a factor of $\\gtrsim$ 6\\ in the same ionization parameter range. \n\n\nIn panel {\\bf (D)}, we plot the model predictions for the high ionization species e.g., \\mbox{O\\,{\\sc vi}}\\ (I.P = 113.9 eV), \\mbox{Ne\\,{\\sc viii}}\\ (I.P = 207.3 eV), \\nani\\ (I.P = 264.2 eV) and \\mgx\\ (I.P = 328.2 eV). From the bottom panel, we note that \\nani\\ and \\mgx\\ are detectable only for log~U~$\\gtrsim$~0.5 and their column densities show peak at log~U $\\sim$~1.4. \\mbox{Ne\\,{\\sc viii}}, on the other hand, shows peak at log~U $\\sim$~1.0 and $N(\\mbox{Ne\\,{\\sc viii}})>10^{13}$ cm$^{-2}$ for log~U~$\\gtrsim -0.5$. \\mbox{O\\,{\\sc vi}}, in contrast, shows relative flat distribution and is detectable for the entire range in ionization parameter (e.g. $-1.5 \\le$ log~U $\\le 1.8$). \nIt is interesting to note that the ratios plotted in the top panel show smooth variation over the whole range in ionization parameter. For example, $N(\\mgx)\/N(\\nani)$ ratio varies by a factor $\\sim$ 3\\ in the range $0.0 \\le$ log~U $\\le 1.0$ and by a factor $\\sim$ 5\\ in the range 1.0 $\\le$ log~U $\\le$ 2.0. Note that notch seen in $N(\\mgx)\/N(\\nani)$ ratio around log~U = $-1.2$ is not real but a numerical artifact  where $N(\\mgx)$ become \nnegligibly small. \n\n\nThe above analysis clearly provides the rough range in the ionization parameter where species with similar ionization potentials are most likely to originate from the same phase of the absorber. In this U range the ratios of such ionic column densities are also useful in constraining the relative abundances of the heavy elements. On a different note, we wish to point out here that all these species originating from same phase (or density) will have similar projected area and hence they will show very similar covering fractions.    \nThe ratios of very highly ionized species (i.e. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, \\nani\\ and \\mgx\\ that are the main focus of this work) show smooth variation over ionization parameter. These ratios are sensitive probes of the ionization parameter provided these\nspecies originate from the same phase of the absorbing gas. The nature of absorption profiles (e.g. velocity alignment, line spread, component structure etc.) can  be used to decide whether these species originate from the same phase of the absorbing gas.    \n\n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{CIE_ratio.ps} \n}}\n}}\n\\caption{ {\\sl Bottom :} Column densities of various high ionization species \nas a function of gas temperature under collisional ionization equilibrium \n\\citep[]{Sutherland93}. Column densities are calculated for $N(\\mbox{H\\,{\\sc i}}) = 10^{14}$ \ncm$^{-2}$, assuming solar metallicity.     \n{\\sl Top :} Column density ratios are plotted as a function of gas temperature.}    \n\\label{CIE_ratio} \n\\end{figure} \n \n\\citet{Muzahid12b} have shown that the near constancy of $N$(O~{\\sc vi})\/$N$(Ne~{\\sc viii}) between different components in the associated absorber towards HE~0238--1904 can be explained if collisional excitation plays an important role. Therefore, we  now consider the collisional ionization equilibrium (CIE) model \\citep[]{Sutherland93}. The model predicted column densities of high ionization species discussed in the panel-{\\bf (D)} of  Fig.~\\ref{cloudy1} are plotted as a function of gas temperature, in the lower panel of Fig.~\\ref{CIE_ratio}. The column densities are calculated for $N(\\mbox{H\\,{\\sc i}})$ = 10$^{14}$ cm$^{-2}$ and $Z = Z_{\\odot}$, typically seen in most of the cases in our sample. It is clear from the figure that for log~$T >$ 6.0, all these high ionization species become fairly insensitive to the gas temperature. This fact is also manifested in the column density ratios, plotted in the top panel.  \n\nIn what follows we provide detailed models for some individual systems (specially the ones that show \\nani) in the framework of photoionization and CIE models.\n\n\n\n\\subsection{Models for the system $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459}\n\\label{sec_phot_model_pg1206} \n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=8.4cm,width=8.8cm,angle=00]{model1_pg1206.ps} \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459.  \n{\\sc cloudy} predicted column density ratios of various high ionization species as \na function of ionization parameter are plotted in different panels. The horizontal \ndashed line in the bottom  panel indicates the measured value of $N(\\mgx)\/N(\\nani)$ \nin component-1. In all other cases the horizontal dashed line marks the upper\/lower \nlimit on the ratio as shown by an arrow.    \nThe dotted (green) curves are the model prediction in case of $\\rm Na$ is overabundant by \na factor of 7 relative to $\\rm Mg$ and\/or $\\rm Ne$. The dotted vertical line represents a \npossible solution for the ionization parameter. \n} \n\\label{model_pg1206} \n\\end{figure} \n\n\n\\begin{figure*} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1206_CIE.ps} \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1338_CIE.ps} \n}}\n}}\n\\caption{{\\sl Left:} CIE model for the system at $z_{\\rm abs}$\\ = 1.02854 towards PG~1206$+$459. \n{\\sl Right:} CIE model for the system at $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. \nIn each panel model predicted column density ratios of various high ionization species \nare plotted as a function of gas temperature. In the bottommost panels the horizontal dotted \nline represents the measured $N(\\mgx)\/N(\\nani)$ ratios in component-1 and component-2. In all \nother panels horizontal dotted lines followed by an arrow  mark the observed upper\/lower limits \non the plotted ratios.\n} \n\\label{CIE_model} \n\\end{figure*} \n\n\n\n\nHere we discuss  the physical conditions in the $z_{\\rm abs}$\\ = 1.02854 system towards PG~1206$+$459. The ionizing background is characterized by the Eq.~\\ref{agn_cont} with  $T_{\\rm BB} \\sim 1.5 \\times 10^{5}$~K, $\\alpha_{ox} = -1.7$, $\\alpha_{x} = -0.7$ and   $\\alpha_{uv} = -0.5$. \nThe value of $\\alpha_{ox}$ has been calculated assuming power law shape of X-ray spectrum with photon index, $\\Gamma_{\\rm 0.3-12 keV} = -1.74\\pm0.09$ (or $\\alpha_{x} \\sim -0.7$) and normalization at 1keV is $A_{\\rm PL} = (2.4\\pm0.2)\\times10^{-4}$ $\\rm photons~cm^{-2} keV^{-1} s^{-1}$, as estimated for this source by \\citet{Piconcelli05}. Using the black hole mass, M$_{\\rm BH} = 1.0 \\times10^{9}$ M$_{\\odot}$, and $\\rm L_{Bol}\/L_{Edd} = 0.84$ from \\citet{Chand10}, the inner disk temperature ($T_{\\rm BB}$) is found to be very similar to the value used here. \n\nFrom Fig.~\\ref{vp_pg1206} (and Table~\\ref{tab_pg1206}) we notice that the covering fractions for \\nani, \\mbox{Ne\\,{\\sc viii}}, \\mbox{O\\,{\\sc v}}, \\mgx\\ and \\arei\\ are similar. It is clear that \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc v}}\\ column density estimation are lower limits as they are affected by saturation effects. The photoionization model predictions for the above mentioned SED is given in Fig.~\\ref{model_pg1206}. The horizontal dashed line in each panel represents the observed values for the component 1 (i.e., $v_{\\rm ej} \\sim -19,250$ km~s$^{-1}$ component in Table~\\ref{tab_pg1206}).  The upper limit on the observed $N(\\mgx)\/(\\mbox{Ne\\,{\\sc viii}})$ ratio, suggests log~U~$\\le$~1.0. The lower limit on the observed $N(\\mgx)\/(\\arei)$ ratio, on the other hand suggests log~U~$\\ge$~0.7. We notice that in this ionization parameter range (i.e. 0.7 $\\le$ log~U $\\le$ 1.0) the model over-predicts the observed $N(\\mgx)\/N(\\nani)$ ratio. The observed column density ratios involving \\nani\\ can be reproduced by the models if we assume ${\\rm Na}$ is enhanced by a factor of 0.85 dex with respect to ${\\rm Mg}$ and\/or ${\\rm Ne}$ [see the dotted (green) curves in Fig.~\\ref{model_pg1206}].\nWe estimate the upper limit for  log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 13.78 using $f_c = 0.59$ (see Table~\\ref{tab_pg1206}). For this model predicts log~$N$(\\mgx) (cm$^{-2}$) = 15.32, which is very close to the observed value implying the metallicity of the gas phase producing \\mbox{Ne\\,{\\sc viii}}\\ and \\mgx\\ is higher than solar.\nAmong the other species detected in this component only \\mbox{O\\,{\\sc iv}}\\ column density and covering fraction are well measured. We find $N$(\\mbox{O\\,{\\sc iv}}) predicted by our model for log~U$\\sim 1$ is a factor 25 times smaller than what is observed. This confirms that \\mbox{O\\,{\\sc iv}}\\ is originating from a distinctly different phase as suggested by the low covering fraction as well. The observed column density of \\mbox{O\\,{\\sc iv}}\\ for solar metallicity and log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 14.42 (for the similar covering fraction measured for \\mbox{O\\,{\\sc iv}}) we find log~U $\\sim 0$. This ionization parameter also produces the correct value of observed $N(\\mbox{N\\,{\\sc iv}})$. If both these phases are at the same distance from the QSO then we can conclude that there is a factor ten change in the density along the transverse direction for the absorbing gas. \n\n \nIn the case of component-2 (i.e., $v_{\\rm ej}\\sim -19,150$ km~s$^{-1}$\\ component in Table~\\ref{tab_pg1206}), the upper limit on observed $N(\\mgx)\/(\\mbox{Ne\\,{\\sc viii}})$ ratio, suggests log~U~$\\le$~1.0. The lower limit on observed $N(\\mgx)\/(\\arei)$ ratio,  on the other hand suggests log~U~$\\ge$~0.8. As in the case of component-1 for this ionization parameter range the photoionization model over predicts the observed $N(\\mgx)\/N(\\nani)$.  We find that the observed column density ratios involving \\nani\\ can be reproduced by the model if we assume $\\rm Na$ is enhanced by a factor of 0.60 dex with respect to $\\rm Mg$ and\/or $\\rm Ne$. Like in the previous case the model that reproduces the high ions under-predicts the \\mbox{O\\,{\\sc iv}}\\ column density. We also find \\mbox{O\\,{\\sc iv}}\\ is originating from a phase that is up to a factor 10 lower density if both phases are at same distance from the QSO. \n\nIn both the components \\mbox{N\\,{\\sc v}}\\ absorption is detected. The measured column densities are consistent with an ionization parameter intermediate between the gas traced by \\mbox{N\\,{\\sc iv}}\/\\mbox{O\\,{\\sc iv}} and \\mgx. All this suggests that the outflow having smooth density gradients in the transverse direction.\n\nFor log~U=1, the inferred total column density of system is $N(\\rm H)$ = 4.7~$\\times$~10$^{20}$ cm$^{-2}$ (when log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) = 14.0), whereas, \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ column densities are $N(\\mbox{O\\,{\\sc vii}})$~ = 1.0~$\\times$~10$^{17}$ cm$^{-2}$ and $N(\\mbox{O\\,{\\sc viii}})$~ = 9.5~$\\times$~10$^{16} \\rm cm^{-2}$, suggesting continuum optical depths of \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ are much less than 0.1. Therefore, this system may not be a potential X-ray WA candidate. \n \n\nIn the left hand panel of Fig.~\\ref{CIE_model}, the column density ratios of  various high ionization species predicted by the CIE models, are plotted as a function of gas temperature. The horizontal dashed lines followed by arrows, in each sub-panel except for the bottom one, indicate the upper limit on the column density ratios measured in component-1 and component-2. The measured values of $N(\\mgx)\/N(\\nani)$ ratio, shown in the bottom panel, are found to be very similar for both the components which corresponds to a temperature of log~$T \\sim 5.9$. Note that the upper limits on $N(\\mgx)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratios observed in both the components suggests log~$T \\lesssim 5.9$. The observed value of $N(\\mbox{O\\,{\\sc v}})$, on the other hand, suggests a temperature $T\\sim10^{5.8}$~K. On the other hand we notice that low ionization species like O~{\\sc iv} and N~{\\sc iv} require $T\\sim10^{5.2}$ K. In order for the two phases to be in pressure equilibrium the density of the low ionization phase needs to be a factor 4 higher. We next run {\\sc cloudy} model keeping the gas temperature to be constant at $T\\sim10^{5.8}$~K and found that the ionization parameter of the gas log~U$\\le-2$ so that the ratio of \\mgx\\ and \\mbox{Ne\\,{\\sc viii}}\\ are not affected by the QSO radiation. Given the luminosity of the QSO this corresponds to a radial separation of $\\gtrsim 2600\/\\sqrt{(n_{\\rm H}\/10^5)}$~pc between the absorbing gas and the QSO, so that the ionization state can be dominated by collisions.  \n\n\n\n\n\\subsection{Models for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416} \n\\label{sec_phot_model1_pg1338} \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=9.2cm,width=9.2cm,angle=00]{PG1338_mod.ps} \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. \n{\\sc cloudy} predicted column density ratios of different ions are plotted as a \nfunction of ionization parameter in different panels. The horizontal (red) dashed line \nfollowed by an arrow in each panel represents the observed upper\/lower limit on the \nplotted ratio as measured in component-1. \n{\\sl Left :} Ratios of different ionization states of neon are plotted. \n{\\sl Right :} Same as {\\sl Left} but for different ionization states of different elements. \nThe dotted (green) curves are the model prediction in case of $\\rm Na$ is overabundant by a \nfactor of 5 relative to $\\rm Mg$ and\/or $\\rm Ne$. The shaded region represents the allowed \nrange in ionization parameter as suggested by various ionic ratios.             \n} \n\\label{phot_pg1338} \n\\end{figure} \n \n\n\nHere we discuss the photoionization model for the system $z_{\\rm abs}$\\ = 1.15456 towards PG~1338$+$416. For SED we use $T_{\\rm BB}\\sim1.0\\times10^{5}$~K, $\\alpha_{\\rm x} = -1.5$, $\\alpha_{\\rm uv} = -0.5$ and $\\alpha_{\\rm ox} = -1.8$ \\citep[from][]{Anderson07}. From the \\mbox{Mg\\,{\\sc ii}}\\ emission line width \\citet{Chand10} have estimated the black hole mass for this source to be log~M$_{\\rm BH}$\/M$_{\\odot} \\sim$ 8.96 and 9.47 using the method by \\citet{McLure04} and \\citet{Dietrich09} respectively. They also find $L_{\\rm bol}\/L_{\\rm Edd} = 0.34$ for this source. Using these we calculate the inner disk temperature for this QSO to be $T_{\\rm BB} \\sim 1.2 \\times 10^{5}$~K and $9.1 \\times 10^{4}$~K for log~M$_{\\rm BH}$\/M$_{\\odot} \\sim$ 8.96 and 9.47 respectively. This is close to what we use to generate the SED.  \n\n\nIn Fig.~\\ref{phot_pg1338} we show  the results of our photoionization model. In the left hand panel of the figure we have shown the column density ratios of different ionization states of neon. All these ratios gives lower limits on ionization parameter. The best constraint comes from $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{Ne\\,{\\sc v}})$ ratio, which suggests log~U~$\\ge$~0.2. All other ratios are consistent with this lower limit. In the right hand panel we have plotted ionic ratios of different species of different elements which can provide useful constraints on the ionization parameter (see section \\ref{sec_phot_model}).  The observed upper limit on $N(\\mgx)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratio suggests log~U~$\\le$~0.8. Hence the physically allowed range in ionization parameter becomes $0.2 \\le$~log~U~$\\le 0.8$, as marked by the shaded region. We note that the observed limits on $N(\\mgx)\/N(\\alel)$ and\/or $N(\\mgx)\/N(\\mbox{O\\,{\\sc v}})$ are also consistent with this range. However, it is apparent from the right-bottom panel, that our model cannot reproduce the observed $N(\\mgx)\/N(\\nani)$ ratio for the whole range in ionization parameter, where the individual species (i.e. \\nani\\ and \\mgx) are detectable. Similarly, $N(\\nani)\/N(\\mbox{Ne\\,{\\sc viii}})$ ratio also suggests a very high log~U, which is not in the allowed range of ionization parameter (i.e. shaded region). Like the previous case (see section~\\ref{sec_phot_model_pg1206}), such a  discrepancy can be easily avoided if $\\rm Na$ is overabundant by factor of $\\sim$~5--6, as can be seen from the dotted (green) curves in the figure. \n\n\nAssuming log~U = 0.5 and using the estimated upper limit on $N(\\mbox{H\\,{\\sc i}})$ in the component-1 [i.e. log~$N(\\mbox{H\\,{\\sc i}})$ (cm$^{-2}$) $\\le$ 13.64; see Table~\\ref{tab1_pg1338}], we estimate the metallicity of the gas to be $\\gtrsim$ 10 $Z_{\\odot}$. The total column density of the system at log~U~$\\sim$~0.5 is log~$N(\\rm H)$ (cm$^{-2}$) = 20.09. Predicted column densities of \\mbox{O\\,{\\sc vii}}\\ and \\mbox{O\\,{\\sc viii}}\\ are log $N$ (cm$^{-2}$) = 17.47 and 17.02 respectively. Continuum optical depth of oxygen corresponding to these values is again much less than 0.1, suggesting that the system may not be a potential X-ray WA candidate.   \n\n\nFrom the right hand panel of Fig.~\\ref{CIE_model}, we can conclude that the observed ratios and limits of high ions can be explained if the gas temperature is $T\\sim 10^{5.9}$ K without the enhancement of $\\rm Na$ as required by the photoionization models. However, in order for the QSO radiation field to not affect the ionization state of the absorbing gas the ionization parameter has to be log~U~$\\le-1.0$. For the inferred luminosity this corresponds to a separation of $\\gtrsim400\/\\sqrt{(n_{\\rm H}\/10^5)}$~pc of the absorbing cloud from the QSO.  \n\n\n\n\\subsection{Model for the system $z_{\\rm abs}$\\ = 1.21534 towards PG~1338$+$416} \n\\label{sec_phot_model2_pg1338} \n\n \nIn this section we discuss the photoionization model for $z_{\\rm abs}$\\ = 1.21534 absorber towards PG~1338$+$416. This system has $z_{\\rm abs}$\\ very similar to $z_{\\rm em}$\\ = 2.2145$\\pm$0.0019. This is the only associated \\mbox{Ne\\,{\\sc viii}}\\ system along the line of sight without detectable \\nani\\ absorption. Unlike this system the other two have large outflow velocities and show signatures of partial coverage. The column density of \\mbox{Ne\\,{\\sc viii}}\\ is also high in the other two systems.\n\n\nIn section~\\ref{sec_discript_PG1338_1.21534}, we have seen that the low ionization species, detected in COS, originating from this system show 2 possible components. However, because of the poor spectral resolution, species detected in $HST\/$FOS spectra can be well fitted by a single Voigt profile component. Because of this disparity in the data quality, we use the total column densities (i.e. summed up component column densities), for the photoionization model. We run {\\sc cloudy} with same set of parameters as described in section~\\ref{sec_phot_model1_pg1338}. The results of our photoionization model are shown in Fig.~\\ref{fig:phot_1.21534_pg1338}. \n\n\nThe column density ratios of different ionization states of same element are very important diagnostics of ionization parameter. Therefore, we make use of simultaneous presence of \\mbox{O\\,{\\sc iii}}, \\mbox{O\\,{\\sc iv}}, \\mbox{O\\,{\\sc v}}\\ and \\mbox{O\\,{\\sc vi}}\\ lines of oxygen and \\mbox{N\\,{\\sc iv}}\\ and \\mbox{N\\,{\\sc v}}\\ lines of nitrogen to estimate the ionization parameter of the gas. Since we treat $N(\\mbox{O\\,{\\sc iii}})$ and $N(\\mbox{O\\,{\\sc v}})$ as upper and lower limits (see discussions in section~\\ref{sec_discript_PG1338_1.21534}), the ionization parameter is primarily decided by  $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc iv}})$ and $N(\\mbox{N\\,{\\sc v}})\/N(\\mbox{N\\,{\\sc iv}})$ ratios. It is clear from Fig.~\\ref{fig:phot_1.21534_pg1338} that, both the ratios are remarkably consistent with log~U $\\sim -1.0$. We also note that, the upper limit on $N(\\mbox{O\\,{\\sc vi}})\/N(\\mbox{O\\,{\\sc v}})$ and lower limits on $N(\\mbox{O\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iii}})$ ratios are also suggestive of such an ionization parameter. Using the ionization fractions at log~U $\\sim -1.0$, we find that the metallicity of the gas to be near solar, e.g. log~$Z\/Z_{\\odot}$ = 0.40$^{+0.90}_{-0.25}$. \n\n\n\\begin{figure} \n\\centerline{\n\\vbox{\n\\centerline{\\hbox{ \n\\includegraphics[height=7.2cm,width=8.8cm,angle=00]{phot_1.21534.ps}  \n}}\n}}\n\\caption{Photoionization model for the system $z_{\\rm abs}$\\ = 1.21534 towards PG~1338$+$416. \n{\\sc cloudy} predicted column density ratios of different ions are plotted as a function \nof ionization parameter in different panels. The horizontal dashed line in the bottom two \npanels mark the measured ionic ratios. The dashed lines with arrows in the top two panels \nshow measured limits on the ionic ratios. The vertical dotted line at log~U = $-1$ marks \na possible solution for the ionization parameter.   \n} \n\\label{fig:phot_1.21534_pg1338} \n\\end{figure} \n \n\n\n\n\nIn the section~\\ref{sec_phot_model}, we have seen that the species \\mbox{N\\,{\\sc iv}}\\ and \\mbox{O\\,{\\sc iv}}\\ traces each other for a wide range in ionization parameter. Therefore, $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio is a sensitive probe of the relative abundances. From the observed $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio we find that nitrogen is overabundant compared to oxygen by a factor of 0.93 dex (i.e., $\\rm [N\/O] = 0.07$). Furthermore, nitrogen is found to be overabundant compared to carbon by a factor of 0.89 dex (i.e. $\\rm [N\/C] = 0.29$), from the measured  $N(\\mbox{N\\,{\\sc iv}})\/N(\\mbox{C\\,{\\sc iii}})$ ratio. Since \\mbox{O\\,{\\sc iii}}\\ line is blended, we use $N(\\mbox{C\\,{\\sc iii}})\/N(\\mbox{O\\,{\\sc iv}})$ ratio to estimate $\\rm [C\/O]$ and found that the carbon and oxygen roughly follow solar abundance pattern. For example, estimated $\\rm [C\/O] = -0.22$, whereas, in sun $\\rm (C\/O) = -0.26$ \\citep[]{Asplund09}. Such an enhanced nitrogen abundance is seen in high redshift ($z\\ge2.0$) QSOs \\citep[]{Hamann92,Korista96,Petitjean99}. These authors suggested a rapid star formation scenario which produces a super solar metallicity in order to boost the nitrogen abundance through enhanced secondary production in massive stars.  We would like to mention that, with the estimated ionization parameter and metallicity, neither \\mbox{Ne\\,{\\sc viii}}\\ nor \\mgx\\ would be detectable [e.g., reproduced log~$N(\\mbox{Ne\\,{\\sc viii}})$ (cm$^{-2}$) $\\ll$ 14.0 at log~U = $-1.0$].     \n\n\nThe observed $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$ ratio require a different phase with fairly high ionization parameter (i.e., log~U~$\\sim$~1.3). If we assume  most of the \\mbox{O\\,{\\sc v}}\\ originate from \\mbox{Ne\\,{\\sc viii}}\\ phase then we get log~U $\\ge$~0.8. In equality in this case is because some part of  \\mbox{O\\,{\\sc v}}\\ will originate from \\mbox{O\\,{\\sc iv}}\\ phase. If we use $N$(\\mbox{O\\,{\\sc vi}})\/$N$(\\mbox{Ne\\,{\\sc viii}}) ratio then we get log~U $\\ge$~0.9. Thus one can conclude that the \\mbox{Ne\\,{\\sc viii}}\\ absorption is originating from a gas having log~U~$\\sim$ 1 (as we have seen in the other cases discussed above).\nIf we assume the \\mbox{Ne\\,{\\sc viii}}\\ phase has same metallicity as the low ionization phase discussed above then we can conclude that \\mbox{H\\,{\\sc i}}\\ associated with \\mbox{Ne\\,{\\sc viii}}\\ is $\\le 10^{12}$ cm$^{-2}$. We can conclude that the low hydrogen column density in this component  is the reason for the lack of \\nani\\ absorption in this system. \n\n\nLike in the previous cases our model suggests that the absorbing gas will not have sufficient optical depth to be a X-ray warm absorber. \n\n\n\n\n\\begin{table*}\n\\caption{Summary of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers from literature (only secure detections are listed here)}  \n\\centering  \n\\begin{tabular}{crrcccccccc} \n\\hline \n    &      &        &     \\multicolumn {5}{c} {log~$N$~(cm$^{-2}$)} & \\\\ \\cline{4-8}   \nQSO & $z_{\\rm em}$ & $v_{\\rm ej}$(km~s$^{-1}$) &  \\mbox{O\\,{\\sc vi}}  & \\mbox{Ne\\,{\\sc viii}} & \\mgx &  \\mbox{H\\,{\\sc i}} & $\\rm H$ & Type &  QSO Type &  Reference \\\\ \\hline   \n (1)  &     (2)    &     (3)     &    (4)      &    (5)    &  (6)    &   (7)   &    (8)       &    (9)   & (10)  & (11)  \\\\   \n\\hline \nUM~675         & 2.150 & $-$1500  & 15.5  &  15.4  &  ....  &  14.8  &  20.0  &  NAL  &  RQ$^{b}$  &  \\citet{Hamann95} \\\\ \nSBS~1542$+$541 & 2.631 & $-$11360 & 15.8  &  16.0  &  15.9  &  14.9  &  22.7  &  BAL  &  RQ        &  \\citet{Telfer98} \\\\ \nJ~2233$-$606   & 2.240 & $-$3900  & 15.4  &  15.1  &  ....  &  14.0  &  22.0  &  NAL  &  ...       &  \\citet{Petitjean99} \\\\ \nPG~0946$+$301  & 1.221 & $-$10000 & 16.6  &  16.7  &  16.6  &  15.3  &  ....  &  BAL  &  RQ        &  \\citet{Arav99a} \\\\ \n3C~288.1       & 0.965 & $+$250   & 15.8  &  15.4  &  15.0  &  15.8  &  20.2  &  NAL  &  RL$^{a}$  &  \\citet{Hamann00} \\\\ \n\\hline \n\\hline \n\\end{tabular}\n~\\\\ \nTable Note -- $^{a}$ Radio Loud; ~~$^{b}$ Radio Quiet \n\\label{tab:summary} \n\\end{table*}  \n\n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=8.0cm,width=8.0cm,angle=00]{b_dist.ps} \n}}\n}}\n\\caption{ {\\sl Bottom:} The distribution of Doppler parameter as measured in individual \n\\mbox{Ne\\,{\\sc viii}}\\ components. {\\sl Top:} Number of \\mbox{Ne\\,{\\sc viii}}\\ components against the spread of \\mbox{Ne\\,{\\sc viii}}\\ \nabsorption in each system.      \n} \n\\label{b_dist} \n\\end{figure} \n \n\\section{Discussions} \n\\label{sec_diss}\n\nIn this section, we try to draw a broad physical picture of the associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers. \n\n\n\\subsection{Incidence of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers} \n\n \nThe fraction of AGNs that show associated absorption is important for understanding the global covering fraction and the overall geometry of the absorbing gas \\citep[]{Crenshaw03,Ganguly08}. In low redshift Seyfert galaxies, surveys in the UV \\citep[]{Crenshaw99}, FUV \\citep[]{Kriss02}, and X-rays \\citep[]{Reynolds97a} found $\\sim$ 50 -- 70\\% incidence of associated absorbers. For quasars, the fraction of occurrence has been found to be somewhat lower. For example \\citet[]{Ganguly01} found signature of associated \\mbox{C\\,{\\sc iv}}\\ absorption in $\\sim$~25\\% of QSOs. On the other hand, \\citep[]{Dai08} have found occurrence of BAL in $\\sim$~40\\% cases. However, we note that depending upon the selection criteria (e.g. cutoff velocity, rest frame equivalent width etc.) these numbers could be very different. An exposition on the incidence of different forms of the associated absorbers can be found in \\citet{Ganguly08}. We have found 12  associated \\mbox{Ne\\,{\\sc viii}}\\ systems in 8 out of 20 QSOs in our sample while only 2 is expected based on the statistics of intervening systems. Even if we restrict ourself to $|v_{\\rm ej}|$ up to 5000 km~s$^{-1}$\\ instead of 8000 km~s$^{-1}$, we have 8 associated systems which is factor 4 higher compared to what is expected from statistics of intervening systems. Such an enhanced occurrence of associated absorbers have also been noticed in the case of high-$z$ \\citep[]{Fox08} and low-$z$ \\citep[]{Tripp08} \\mbox{O\\,{\\sc vi}}\\ absorbers.     \nThe incidence of associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers in our sample is $\\sim$40\\% ($\\sim$35\\% if we do not include the tentative system towards HB89 0107$-$025 or restrict to systems with $|v_{\\rm ej}|<$ 5000 km~s$^{-1}$). It is also interesting to note only 5\/12 systems along 3\/20 sightlines show signature of partial coverage. Therefore the incidence of partially covered associated \\mbox{Ne\\,{\\sc viii}}\\ absorber is 15\\%. No associated \\mbox{Ne\\,{\\sc viii}}\\ system is detected towards 7 radio bright QSOs in our sample. There are 5 \\mbox{Ne\\,{\\sc viii}}\\ absorption reported in the literature (see Table~\\ref{tab:summary}) and only one of them ($z_{\\rm abs}$ = 0.965 towards 3C~288.1) is towards radio bright QSO. Confirming the high detection rate of associated \\mbox{Ne\\,{\\sc viii}}\\ systems and relatively less incidence rate towards radio bright QSOs is very important to understand the possible influences of radio jets.   \n\n\n\n \n\\subsection{Line broadening}  \nIn the bottom panel of Fig.~\\ref{b_dist} we show the distribution of Doppler parameter as measured in individual \\mbox{Ne\\,{\\sc viii}}\\ components. The median value of $b(\\mbox{Ne\\,{\\sc viii}})$ is $\\sim$~58.7 km~s$^{-1}$. The upper limit on temperature corresponding to this value is 10$^{6.6}$~K. Under CIE, even \\mbox{Ne\\,{\\sc viii}}\\ will not be a dominant species at such high temperatures. The collisional ionization fraction of \\mbox{Ne\\,{\\sc viii}}\\ becomes only $\\sim 3\\times10^{-3}$ at $T \\sim 10^{6.6}$~K. Therefore the width of individual Voigt profile components are most probably dominated by non-thermal motions. We note that $b(\\mbox{Ne\\,{\\sc viii}}) \\sim$~22~km~s$^{-1}$\\ corresponds to a temperature of 10$^{5.8}$~K, at which $N(\\mbox{Ne\\,{\\sc viii}})$ peaks under CIE (see bottom panel of Fig.~\\ref{CIE_ratio}). This indeed suggests that, based on the observed $b$-values of \\mbox{Ne\\,{\\sc viii}}, we cannot rule out the possibility of gas temperature being 6--7$\\times$~10$^{5}$~K at which collisional ionization becomes important.  \nNote that we use minimum number of Voigt profile components needed to have a reduced $\\chi^2\\sim 1$. The discussions presented above are based on $b$-parameters derived this way. While we can not rule out each of our Voigt profile component being made of a blended large number of components, our analysis suggests that the observed line profiles allow for the gas temperature being higher than the typical photoionization equilibrium temperature.     \nIn the top panel of Fig.~\\ref{b_dist} we have plotted the number of components required to fit \\mbox{Ne\\,{\\sc viii}}\\ absorption against the velocity spread of the line. Lack of any significant correlation between these two suggests that the line spread may not dominated by the presence of multiple number of narrow components \\citep[as seen in the case of high redshift \\mbox{O\\,{\\sc vi}}\\ absorbers, e.g.][]{Muzahid12a}  but the line spread is related to the large scale velocity field. Further, $\\delta v (\\mbox{Ne\\,{\\sc viii}})$ lies roughly between 100 -- 800~km~s$^{-1}$\\ suggesting that these absorbers are intermediate of BAL and NAL. This type of associated absorbers are also known as mini-BAL. \n    \n\n\n\n\\subsection{Ejection velocities and correlations} \nThe ejection velocity is defined as the velocity separation between the emission redshift of the QSO and the \\mbox{Ne\\,{\\sc viii}}\\ optical depth weighted redshift of the absorber. The distribution of ejection velocities in our sample are shown in panel {\\bf (A)} of Fig.~\\ref{beta_dist}. Clearly most of these associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers are detected within $-5000$ km~s$^{-1}$\\ from the emission redshift of the QSO.  The highest velocity absorber is detected at a ejection velocity of $\\sim-19,000$ km~s$^{-1}$. In panel {\\bf (B)} we have plotted covering fraction corrected total column densities of \\mbox{Ne\\,{\\sc viii}}\\ in our sample ({\\sl stars}) as a function of ejection velocity. The hexagons in this panel are from literature (see Table~\\ref{tab:summary}). The overall sample shows a possible correlation between $N(\\mbox{Ne\\,{\\sc viii}})$ and $v_{\\rm ej}$. If we consider all the limits as detections we find a 2.1$\\sigma$ correlation for the systems in our sample. When we consider the measurements from the literature the significance of the correlation increase to 2.7$\\sigma$. However, we note that the top two ejection velocity systems from the literature (filled hexagons) are BAL in nature. The (green) arrows in the bottom, identify the systems with \\nani\\ detection. It is apparent that these are the ones having top three ejection velocities with $|v_{\\rm ej}| > 5000$ km~s$^{-1}$, in our sample. Interestingly, we note that the only possible \\nani\\ detection was reported before, by \\citet{Arav99a} towards BALQSO PG~0946$+$301 where the system has an ejection velocity of $-10,000$ km~s$^{-1}$, which is consistent with the trend seen in our sample.   \n \n  \nIn panel {\\bf (C)} we have plotted the Lyman continuum luminosity (i.e. $L_{912\\rm \\AA}$ in ergs s$^{-1}$ Hz$^{-1}$) of the sources in our sample as a function of $v_{\\rm ej}$. We do not find any obvious correlation between them. However, the highest velocity system, which also show \\nani\\ absorption, originates from the highest UV luminosity source. Here we note that, the sources with higher UV luminosities are found to be the ones with higher outflow velocities in the sample of SDSS BALQSOs \\citep[see e.g.][]{Gibson09}. \nThe estimated \\mbox{Ne\\,{\\sc viii}}\\ covering fractions in different systems in our sample are plotted against the ejection velocity in panel {\\bf (D)}. It is to be noted that majority of the systems at smaller ejection velocities show nearly 100\\% coverage of the background source whereas the systems with higher ejection velocity tend to have lower covering fractions. In panel {\\bf (E)} the line spreads of \\mbox{Ne\\,{\\sc viii}}\\ absorption in each system are plotted as function of $v_{\\rm ej}$. A mild 2$\\sigma$ level correlation is seen between $\\delta v(\\mbox{Ne\\,{\\sc viii}})$ and $v_{\\rm ej}$, suggesting systems with higher outflow velocity are likely to show wider spread. \n\n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=10.0cm,width=8.6cm,angle=00]{beta_dist.ps} \n}}\n}}\n\\caption{{\\bf (A):} The distribution of ejection velocity for the associated \\mbox{Ne\\,{\\sc viii}}\\ \nabsorbers presented in this paper. {\\bf (B):} The \\mbox{Ne\\,{\\sc viii}}\\ column density as a function \nof ejection velocity. The hexagons are from literature listed in Table~\\ref{tab:summary}. \nThe filled hexagons represent BALQSOs. The (green) upward arrows mark the systems where \nwe detect \\nani. {\\bf (C):} $L_{912\\AA}$ as a function of $v_{\\rm ej}$. {\\bf (D):} \\mbox{Ne\\,{\\sc viii}}\\ \ncovering fractions in individual systems against $v_{\\rm ej}$. {\\bf (E):} The line spread \nof \\mbox{Ne\\,{\\sc viii}}\\ absorbers against $v_{\\rm ej}$.    \n} \n\\label{beta_dist} \n\\end{figure} \n\n\n\\begin{table}\n\\vfill \n\\caption{List of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers that exist in literature}  \n\\begin{tabular}{cccccc} \n\\hline \n      &       &       & \\multicolumn{2}{c}{log~$N$ (cm$^{-2}$)} & \\\\ \\cline{4-5}\n  QSO & $z_{\\rm em}$\\ & $z_{\\rm abs}$ &       \\mbox{Ne\\,{\\sc viii}}    &       \\mbox{O\\,{\\sc vi}}      &  Ref.$^{a}$   \\\\  \\hline  \n  (1) &  (2)  &  (3)  &       (4)     &       (5)      &    (6)        \\\\ \n\\hline \n\\hline \nPG~1148$+$549    &  0.9754  &  0.6838  &  13.95 &  14.52  &     1     \\\\  \nPG~1148$+$549    &  0.9754  &  0.7015  &  13.86 &  14.37  &     1     \\\\ \nPG~1148$+$549    &  0.9754  &  0.7248  &  13.81 &  13.86  &     1     \\\\ \nPKS~0405$-$123   &  0.5726  &  0.4951  &  13.96 &  14.41  &     2     \\\\ \n3C~263\t         &   0.646  &  0.3257  &  13.98 &  13.98  &     3     \\\\ \nHE~0226$-$4110   &   0.495  &  0.2070  &  13.89 &  14.37  &     4     \\\\ \n\\hline  \n\\hline \n\\end{tabular} \n~\\\\ ~\\\\  \nNote-- $^{a}$Reference (1) \\citet{Meiring12}; (2) \\citet{Narayanan11}; (3) \n\\citet{Narayanan09,Narayanan12} (4) \\citet{Savage05a}   \n\\label{ne8_int}  \n\\end{table} \n\n\n\\subsection{Distribution of column densities} \n\n\nIn Fig.~\\ref{analysis1}, we show the column density distributions of the \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, and \\mgx, as measured in intervening and associated \\mbox{Ne\\,{\\sc viii}}\\ absorbers in our sample and from the existing literature (i.e. using Table~\\ref{tab_list}, \\ref{tab:summary} and \\ref{ne8_int}). The (blue) 120$^\\circ$ and (red) 60$^\\circ$ hashed histograms show the distributions corresponding to the intervening \\mbox{Ne\\,{\\sc viii}}\\ systems (i.e. from Table~\\ref{ne8_int}) and the associated \\mbox{Ne\\,{\\sc viii}}\\ systems from this paper (i.e. from Table~\\ref{tab_list}) respectively. The histograms clearly show that the column densities of \\mbox{O\\,{\\sc vi}}\\ and \\mbox{Ne\\,{\\sc viii}}\\ are systematically higher in case of associated absorbers compared to those of intervening absorbers. For example, the median values of log~$N(\\mbox{Ne\\,{\\sc viii}})$ ($\\rm cm^{-2}$) are 13.95$\\pm$0.10 and 15.40$\\pm$0.78 for the intervening and the associated absorbers respectively. The median values of log~$N(\\mbox{O\\,{\\sc vi}})$ ($\\rm cm^{-2}$), on the other hand, are 14.37$\\pm$0.27 and 15.40$\\pm$0.58 for intervening and associated absorbers respectively. However we note that, both at high and low redshifts, \\mbox{O\\,{\\sc vi}}\\ absorbers do not show any compelling evidence of having different column density distribution for intervening and associated systems \\citep[see e.g.][]{Tripp08,Fox08}. The apparent discrepancy is mainly because of the fact that we consider \\mbox{O\\,{\\sc vi}}\\ column densities measured in the \\mbox{Ne\\,{\\sc viii}}\\ absorbers, both in the cases of intervening and associated systems. To our knowledge no \\mgx\\ absorption has ever been reported in intervening systems. The median value of log~$N(\\mgx)$ ($\\rm cm^{-2}$) in associated systems turns out to be 15.47$\\pm$0.66. The median values of \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, and \\mgx\\ in our sample are very similar. But we caution here that we have assumed all the upper\/lower limits as measurements.  \n\n\nIt is evident from the middle panel of Fig.~\\ref{analysis1} that all the associated absorbers show $N(\\mbox{Ne\\,{\\sc viii}}) > 10^{14} \\rm cm^{-2}$. This could primarily be due to the fact that we are not sensitive enough to detect a broad line with $N(\\mbox{Ne\\,{\\sc viii}}) < 10^{14} \\rm cm^{-2}$ in the COS spectra used here. For example, for a typical $S\/N$ ratio of $\\sim 10$, the 5$\\sigma$ upper limit for non-detection of \\mbox{Ne\\,{\\sc viii}}$\\lambda$770 line is log~$N(\\mbox{Ne\\,{\\sc viii}})~(\\rm cm^{-2}) < 13.66$ for $b$-parameter of 100 km~s$^{-1}$. Here the assumed $b$ value (i.e. 100 km~s$^{-1}$) is typical for mini-BAL system. The previously reported associated \\mbox{Ne\\,{\\sc viii}}\\ systems (see e.g. Table~\\ref{tab:summary}) are all showing $N(\\mbox{Ne\\,{\\sc viii}})>$ 10$^{15}$ cm$^{-2}$.  \n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=4.4cm,width=8.4cm,angle=00]{ana2.ps} \n}}\n}}\n\\caption{Distribution of total column densities of \\mbox{O\\,{\\sc vi}}\\ (left), \\mbox{Ne\\,{\\sc viii}}\\ (middle) and \n\\mgx\\ (right) in all the associated \\mbox{Ne\\,{\\sc viii}}\\ systems (i.e. using Table~\\ref{tab_list} \\& \n\\ref{tab:summary}). The 60$^\\circ$ hashed (red) histogram shows the data points from this \npaper (i.e. using Table~\\ref{tab_list} only) whereas as 120$^\\circ$ hashed (blue) \nhistograms are for measurements in intervening systems as given in Table~\\ref{ne8_int}.  \n} \n\\label{analysis1} \n\\end{figure} \n\n\n\n\\subsection{Column density ratios and ionization state} \nIn different sub-panels of Fig.~\\ref{analysis2}, various column density ratios are plotted as a function of log~$N(\\mbox{Ne\\,{\\sc viii}})$. The {\\sl stars} and the {\\sl circles} in the bottom panel of Fig.~\\ref{analysis2} are representing the $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratios in associated and intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers respectively. A Spearman rank correlation analysis shows ($\\rho_s = 0.75$) a 2.3$\\sigma$ level correlation between \\mbox{Ne\\,{\\sc viii}}\\ and \\mbox{O\\,{\\sc vi}}\\ column densities in associated absorbers. When we include the intervening absorbers in the analysis, the correlation becomes even tighter (e.g. $\\rho_s = 0.91$ and $\\rho_s\/\\sigma$ = 3.5).  \nThe median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratio for the associated absorbers is 0.11$\\pm$0.50. Under photoionization equilibrium it corresponds to the ionization parameter log~U = 0.4$\\pm$0.2. Under collisional ionization equilibrium the above ratio is reproduced when T $\\sim$ $10^{5.8}$ K and log~U~$\\le-$2. Based on the present data we are not in a position to disentangle among different ionization mechanisms. However, detection of absorption line variability and its relationship to the continuum variation will enable us to distinguish between the two alternatives. For a flat SED, the ionization parameter, density ($n_{\\rm H}$) and distance between the absorber and the QSO are related by,  \n\\begin{equation} \n{\\rm log} \\left(\\frac{n_{\\rm H}}{10^{5} \\rm \/cc}\\right) \n= {\\rm log}~L_{912\\rm\\AA}^{30} -{\\rm log} \\left( \\frac{r}{100 \\rm pc}\\right)^{2} - {\\rm log~U} -1.25      \n\\label{eqn:nH_rC}\n\\end{equation} \nwhere, log~$L_{912\\rm\\AA}^{30}$ is the monochromatic luminosity of the QSO at the Lyman continuum in units of 10$^{30}$ erg s$^{-1}$ Hz$^{-1}$.  The density estimation using absorption line variability or fine-structure excitations will enable us to get the location of the absorbing gas with respect to the central engine. This will allow us to estimate the kinetic luminosity of the outflow which is very crucial for probing the AGN feedback \\citep[e.g.,][]{Moe09,Dunn10,Bautista10,Borguet12b}.  \n\n\nWe have mentioned earlier that the intervening absorbers are showing systematically lower values of $N(\\mbox{Ne\\,{\\sc viii}})$. But, as far as $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratio is concerned there is very little difference between associated and intervening absorbers. For example, the median values of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ ratios in intervening and associated absorbers are, $-$0.45$\\pm$0.50 and 0.11$\\pm$0.50 respectively, consistent within 1$\\sigma$ level. Naively this implies a large ionization parameter even for the intervening systems. In case of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers models of collisional ionization are generally proposed, as photoionization by the extragalactic UV background \\citep[]{Haardt96} requires unusually large cloud sizes \\citep[]{Savage05a,Narayanan09,Narayanan11}. Thus similar ratios seen between associated and intervening systems and between different components in an associated system as in the case of \\citet{Muzahid12b} favour collisional ionization in the associated absorbers as well.\n\n\n\nFurther, we notice a strong correlation between $N(\\mbox{Ne\\,{\\sc viii}})$ and $N(\\mgx)$ (i.e. $\\rho_s$ = 0.90 and $\\rho_s\/\\sigma$ = 2.4). The median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$ is found to be 0.11$\\pm$0.36 (see middle panel of Fig.~\\ref{analysis2}). These observed ratios correspond to a very narrow range in gas temperature (i.e.~$T \\sim 10^{5.95\\pm0.03}$~K) under CIE or very narrow range in ionization parameter (i.e. log~U $\\sim0.8\\pm0.2$) under photoionization [see panel~(D) of Fig.~\\ref{cloudy1}]. Given the high ionization parameter and low neutral hydrogen column density (i.e. $< 10^{14.5}$~cm$^{-2}$ in most of the cases), the predicted total hydrogen column density is too low (i.e. $N({\\rm H}) < 10^{20.5}$~cm$^{-2}$) to produce significant continuum optical depth in the soft X-ray regime.       \n \nIn the top panel of Fig.~\\ref{analysis2} we show $N(\\mbox{Ne\\,{\\sc viii}})\/N(\\nani)$ ratio as a function of $N(\\mbox{Ne\\,{\\sc viii}})$ in logarithmic scale. It is interesting to note that the three systems, where we detect \\nani\\ absorption (i.e. solid hexagons in the plot), are all showing log~$N(\\mbox{Ne\\,{\\sc viii}}) \\gtrsim$ 15.60. The solid (green) triangles in this panel represents the tentative detection of \\nani\\ reported by \\citet{Arav99a}, which also shows log~$N(\\mbox{Ne\\,{\\sc viii}})>$15.60. We notice that log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mgx)$  and  log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ are roughly similar between the systems with and without detectable \\nani\\ absorption. This clearly means that the lack of \\nani\\ detection can be attributed to low $N\\rm (H)$.  \n\n\\begin{figure} \n\\centerline{\\hbox{ \n\\centerline{\\vbox{\n\\includegraphics[height=8.4cm,width=8.4cm,angle=00]{ana4.ps} \n}}\n}}\n\\caption{Column density ratios (\\mbox{Ne\\,{\\sc viii}}\/\\mbox{O\\,{\\sc vi}}, {\\sl bottom}; \\mbox{Ne\\,{\\sc viii}}\/\\mgx, {\\sl middle};  \n\t\\mbox{Ne\\,{\\sc viii}}\/\\nani, {\\sl top}) as a function of $N(\\mbox{Ne\\,{\\sc viii}})$. The open triangles \n\tand (blue) filled circles are from Table~\\ref{tab:summary} and Table~\\ref{ne8_int}  \n\trespectively. In the top panel all the points apart from (green) triangle is from this \n\tpaper. The solid hexagon indicate \\nani\\ detections. The solid (green) triangle represents  \n\tthe tentative \\nani\\ detection by \\citet{Arav99a}. The mean values and \n\tcorresponding scatters, only for data points from our sample, are shown in each panel by \n\thorizontal dashed and dotted lines respectively.   \n\tHere we assumed all the limits as measurements.      \n} \n\\label{analysis2} \n\\end{figure} \n\n\\subsection{Multiple phases in \\nani\\ absorbers} \n\nWe report secure detections of \\nani\\ absorption in three associated \\mbox{Ne\\,{\\sc viii}}\\ systems for the first time. These systems show signatures of multiple component structure. Photoionization models with log~U $\\sim$ 1 explain the \\nani\\ phase of the absorbers. However these models require $\\rm Na$ abundance being enhanced by a factor of 4--7 with respect to $\\rm Mg$. Standard chemical evolution models do not predict such large enhancement of $\\rm Na$ over $\\rm Mg$ \\citep[see Fig. 18 of][and Fig. 6 of \\citet{Venn04}]{Timmes95}. The photoionization models also suggest a typical density of the absorbing region varying by up to a factor 10 along the transverse direction. As photoionization predicts roughly same temperature for the range of ionization parameters probed by low and high ions, different phases cannot be in pressure equilibrium. On the contrary, if collisional excitations are important then one may not need an enhancement of $\\rm Na$, provided the gas temperature $T = 10^{5.9}$ K. In the case of CIE the low ionization phase requires $T\\sim 10^{5.2}$ K. Therefore, a factor $\\sim$~5 density difference between two phases is needed for the gas to be in pressure equilibrium. In the case of CIE, absorbing gas has to be far away from the QSO for the gas to be unaffected by the QSO radiation. Therefore it is important to identify the source of energy that maintains the high temperature of the gas. Probing the optical depth variability and presence of fine-structure transitions with new $HST\/$COS observations will allow us to make good progress in this direction. \n\nAt last, we note that the element $\\rm Na$ has not been incorporated in the non-equilibrium collisional ionization calculations so far. For the metallicity as measured in our sample, non-equilibrium effects would be important and  can provide more realistic models of \\nani\\ absorbers. Therefore, inclusion of $\\rm Na$ in non-equilibrium calculations will be very useful. \n                \n\n\n\n\\section{Summary \\& Conclusions}  \n\\label{con} \n\nWe present a sample of new class of associated absorbers, detected through \\mbox{Ne\\,{\\sc viii}}$\\lambda\\lambda$770,780 absorption, in $HST\/$COS spectra of intermediate redshift (0.45~$\\le z \\le$~1.21) quasars. We searched for \\mbox{Ne\\,{\\sc viii}}\\ absorption in the public $HST\/$COS archive of QSOs with $S\/N \\ge 10$ and emission redshift $z_{\\rm em}$~$ > 0.45$. There were total 20 QSO sight lines in the $HST\/$COS archive before February 2012, satisfying these criteria. Seven of these QSOs are radio bright. The signatures of associated \\mbox{Ne\\,{\\sc viii}}\\ absorption are seen in 40\\% (i.e. 8 out of 20) of the lines of sight, with 10 secured and 2 tentative \\mbox{Ne\\,{\\sc viii}}\\ systems detected in total. None of them are towards radio bright QSOs. The associated absorbers detected towards QSO HE~0226$-$4110 and QSO HE~0238$-$1904 were previously reported by \\citet{Ganguly06} and \\citet{Muzahid12b} respectively. Here we summarize our main results.  \n\n\n\\vskip 0.2cm \\noindent {\\bf (1)}   \nMajority of the \\mbox{Ne\\,{\\sc viii}}\\ absorbers are detected with outflow velocities $\\lesssim$5000 km~s$^{-1}$. The highest velocity system shows $|v_{\\rm ej}| \\sim 19,000$~km~s$^{-1}$. Medium resolution COS spectra allow us to probe the component structure of \\mbox{Ne\\,{\\sc viii}}\\ absorption in most of the systems. The line spread of \\mbox{Ne\\,{\\sc viii}}\\ absorption is found to be in the range 100~$\\le \\delta v (\\rm km~s^{-1}) \\le$~1000, suggesting that these absorbers are most likely  mini-BALs. The Doppler parameters measured in individual components (with median 58.7$\\pm$31.7 km~s$^{-1}$) indicates domination of non-thermal motions.             \n\n\n\\vskip 0.2cm \\noindent {\\bf (2)}   \nWe detect \\mgx\\ absorption in 7 of 8 \\mbox{Ne\\,{\\sc viii}}\\ systems when the lines are not blended and are covered by the observations. Moreover, we report first secure detections of \\nani\\ absorption in three highest velocity systems in our sample. All three \\nani\\ systems show high $N(\\mbox{Ne\\,{\\sc viii}})$ (i.e.$ > 10^{15.6}$ cm$^{-2}$). The measurements and\/or limits on the column densities of different ions, detected in these \\nani\\ absorbers, require very high ionization parameter (i.e. log~U $\\ge 0.5$) and high metallicity (i.e. $Z \\ge Z_{\\odot}$) when we consider single phase photoionization models. However, ionization potential dependent covering fraction seen in these absorbers suggests kinematic coincidence of multiphase gas with higher ionization species having higher projected area. Given the high value of ionization parameter (log~U) and observed low $N(\\mbox{H\\,{\\sc i}})$, the model predicted $N(\\rm H)$ is too low (i.e. $<10^{20.5}$~cm$^{-2}$) to produce any significant continuum optical depth in the soft X-ray regime. The observed $N(\\mgx)\/N(\\nani)$ ratios, under single phase photoionization scenario, require a factor $\\gtrsim 5$ enhancement of $\\rm Na$ abundance with respect to $\\rm Mg$. However, such enhancement is not required in CIE models provided gas temperature is $T \\ge 10^{5.9}$~K. In the case of CIE, the low ions require a different phase with temperature $T \\sim 10^{5.2}$ K suggesting a factor of $\\sim 5$ difference in density between two gas phases to be in pressure equilibrium.             \n\n\n\\vskip 0.2cm \\noindent {\\bf (3)} \nWe notice a very narrow range in the column density ratios of high ions (i.e. \\mbox{O\\,{\\sc vi}}, \\mbox{Ne\\,{\\sc viii}}, \\mgx\\ etc.). This suggests a narrow range in ionization parameter (temperature) under photoionization (CIE). The median value of log~$N(\\mbox{Ne\\,{\\sc viii}})\/N(\\mbox{O\\,{\\sc vi}})$ $= 0.11\\pm0.50$ as measured in our sample is comparable to that measured in the intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers within the measurement uncertainties. In case of intervening \\mbox{Ne\\,{\\sc viii}}\\ absorbers collisional ionization is generally proposed, as photoionization by the extragalactic UV background requires unusually large cloud sizes. Indeed, CIE can play an important role in deciding the ionization structure of the absorbing gas in our sample as well. However, for CIE to be dominant, gas cloud has to be far away from the QSO. In that case it is crucial to understand sources of thermal and mechanical energy and the stability of the absorber. Variability study with repeated $HST\/$COS observation is needed to make further progress on these issues.     \n\n\n\n\\section{acknowledgment} \n  \nWe thank anonymous referee for useful comments. We appreciate the efforts of the people involved with the design and construction of COS and its deployment on the $HST$. Thanks are also extended to the people responsible for determining the orbital performance of COS and developing the {\\sc calcos} data processing pipeline. We thankfully acknowledge Dr. Jane Charlton for providing the STIS E230M spectrum of PG~1206$+$459. We thank Dr. Gulab C. Dewangan and Dr. Durgesh Tripathi for useful discussions. SM thanks Sibasish Laha for useful discussions on {\\sc cloudy} modelling. SM also thanks CSIR for providing support for this work. RS wish to thank Indo-French Centre for the Promotion of Advanced Research under the programme No. 4304--2. NA acknowledge support from NASA STScI grants AR-12653.  \n\n\\def\\aj{AJ}%\n\\def\\actaa{Acta Astron.}%\n\\def\\araa{ARA\\&A}%\n\\def\\apj{ApJ}%\n\\def\\apjl{ApJ}%\n\\def\\apjs{ApJS}%\n\\def\\ao{Appl.~Opt.}%\n\\def\\apss{Ap\\&SS}%\n\\def\\aap{A\\&A}%\n\\def\\aapr{A\\&A~Rev.}%\n\\def\\aaps{A\\&AS}%\n\\def\\azh{AZh}%\n\\def\\baas{BAAS}%\n\\def\\bac{Bull. astr. Inst. Czechosl.}%\n\\def\\caa{Chinese Astron. Astrophys.}%\n\\def\\cjaa{Chinese J. Astron. Astrophys.}%\n\\def\\icarus{Icarus}%\n\\def\\jcap{J. Cosmology Astropart. Phys.}%\n\\def\\jrasc{JRASC}%\n\\def\\mnras{MNRAS}%\n\\def\\memras{MmRAS}%\n\\def\\na{New A}%\n\\def\\nar{New A Rev.}%\n\\def\\pasa{PASA}%\n\\def\\pra{Phys.~Rev.~A}%\n\\def\\prb{Phys.~Rev.~B}%\n\\def\\prc{Phys.~Rev.~C}%\n\\def\\prd{Phys.~Rev.~D}%\n\\def\\pre{Phys.~Rev.~E}%\n\\def\\prl{Phys.~Rev.~Lett.}%\n\\def\\pasp{PASP}%\n\\def\\pasj{PASJ}%\n\\def\\qjras{QJRAS}%\n\\def\\rmxaa{Rev. Mexicana Astron. Astrofis.}%\n\\def\\skytel{S\\&T}%\n\\def\\solphys{Sol.~Phys.}%\n\\def\\sovast{Soviet~Ast.}%\n\\def\\ssr{Space~Sci.~Rev.}%\n\\def\\zap{ZAp}%\n\\def\\nat{Nature}%\n\\def\\iaucirc{IAU~Circ.}%\n\\def\\aplett{Astrophys.~Lett.}%\n\\def\\apspr{Astrophys.~Space~Phys.~Res.}%\n\\def\\bain{Bull.~Astron.~Inst.~Netherlands}%\n\\def\\fcp{Fund.~Cosmic~Phys.}%\n\\def\\gca{Geochim.~Cosmochim.~Acta}%\n\\def\\grl{Geophys.~Res.~Lett.}%\n\\def\\jcp{J.~Chem.~Phys.}%\n\\def\\jgr{J.~Geophys.~Res.}%\n\\def\\jqsrt{J.~Quant.~Spec.~Radiat.~Transf.}%\n\\def\\memsai{Mem.~Soc.~Astron.~Italiana}%\n\\def\\nphysa{Nucl.~Phys.~A}%\n\\def\\physrep{Phys.~Rep.}%\n\\def\\physscr{Phys.~Scr}%\n\\def\\planss{Planet.~Space~Sci.}%\n\\def\\procspie{Proc.~SPIE}%\n\\let\\astap=\\aap\n\\let\\apjlett=\\apjl\n\\let\\apjsupp=\\apjs\n\\let\\applopt=\\ao\n\\bibliographystyle{mn}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ INTRODUCTION }\n In the presently known energy range the spectrum of the\n elementary particles is {\\em ``chiral''} in the sense that no\n explicit fermion mass terms are allowed by the symmetry.\n Fermion masses, as well as all other masses, are entirely\n generated by spontaneous symmetry breaking, due to the nonzero\n vacuum expectation value $v_R \\simeq 250$ GeV of the Higgs-boson\n field.\n One of the consequences is the V-A structure of weak interactions,\n resulting in the breaking of space-reflection (parity) symmetry.\n\n A natural question is whether ``chirality'' and the accompanying\n parity breaking is perhaps only a low-energy phenomenon, and at\n high energy the space-reflection symmetry is restored by the\n existence of opposite chirality {\\em ``mirror fermions''}\n \\cite{LEEYAN}.\n If the presently known (almost complete) three fermion families\n were duplicated at the electroweak energy scale, in the range\n 100-1000 GeV, by three mirror fermion families with opposite\n chiralities and hence V+A couplings to the weak gauge vector\n bosons \\cite{MIRFAM}, then the whole fermion spectrum would be\n ``vectorlike''.\n This would very much simplify the nonperturbative lattice formulation\n of the Standard Model \\cite{CHFER,TSUKPR}.\n Of course, since no effects of the mirror fermions are experimentally\n observed up to now, first one has to ask whether the limits implied\n by the presently known experimental data allow their existence at all.\n\n\\section{ MIRROR FERMION PHENOMENOLOGY }\n The mirror partners of fermions have the same\n $\\rm SU(3) \\otimes SU(2)_L \\otimes U(1)_Y$ quantum numbers but\n opposite chiralities.\n For instance, the right-handed chiral components of mirror leptons\n form a doublet with $Y=-1$ with respect to\n $\\rm SU(2)_L \\otimes U(1)_Y$.\n Such particles appear in several extensions of the minimal Standard\n Model, for instance, in grand unified theories with large groups\n as O(16) or SU(15) \\cite{MAAROO}.\n In general mirror fermion models there might be some other quantum\n numbers which are different for fermions and mirror fermions, and the\n set of representations containing the mirror fermions may also be\n different.\n The lattice formulation of the Standard Model suggests a simple\n doubling of the fermion spectrum \\cite{NIENIN}, resulting in three\n mirror pairs of fermion families \\cite{MIRFAM}.\n\n\\subsection{ Present limits }\n The direct pair production of mirror fermions is not observed at LEP.\n This puts a lower limit on their masses of about 45 GeV.\n Heavier mirror fermions could be produced via their mixing to\n ordinary fermions.\n This implies some constraints on the mixing angles versus the masses.\n\n In order to discuss the mixing schemes, let us first consider the\n simplest case of a single fermion ($\\psi$) mirror fermion ($\\chi$)\n pair.\n The mass matrix on the $(\\overline{\\psi}_R,\\overline{\\psi}_L,\n \\overline{\\chi}_R,\\overline{\\chi}_L) \\otimes\n (\\psi_L,\\psi_R,\\chi_L,\\chi_R)$ basis is\n\\begin{equation} \\label{eq01}\nM = \\left( \\begin{array}{cccc}\n\\mu_\\psi  &  0  &  \\mu_R  &  0       \\\\\n0  &  \\mu_\\psi  &  0  &  \\mu_L       \\\\\n\\mu_L  &  0  &  \\mu_\\chi  &  0       \\\\\n0  &  \\mu_R  &  0  &  \\mu_\\chi\n\\end{array} \\right) \\ .\n\\end{equation}\n Here $\\mu_{(L,R)}$ are the fermion mirror fermion mixing mass\n parameters, and the diagonal elements are produced by spontaneous\n symmetry breaking:\n\\begin{equation} \\label{eq02}\n \\mu_\\psi=G_{R\\psi}v_R \\ , \\hspace{2em} \\mu_\\chi=G_{R\\chi}v_R \\ ,\n\\end{equation}\n with the renormalized Yukawa-couplings $G_{R\\psi}$, $G_{R\\chi}$.\n\n For $\\mu_R \\ne \\mu_L$ the mass matrix $M$ in (\\ref{eq01})  is not\n symmetric, hence one has to diagonalize $M^T M$ by\n $O^T_{(LR)} M^T M O_{(LR)}$, and\n $M M^T$ by $O^T_{(RL)} M M^T O_{(RL)}$, where\n$$\nO_{(LR)} =\n$$\n$$\n           \\left( \\begin{array}{cccc}\n\\cos\\alpha_L  &  0  &  \\sin\\alpha_L  &  0   \\\\\n0  &  \\cos\\alpha_R  &  0  &  \\sin\\alpha_R   \\\\\n-\\sin\\alpha_L  &  0  &  \\cos\\alpha_L  &  0  \\\\\n0  &  -\\sin\\alpha_R  &  0  &  \\cos\\alpha_R\n\\end{array} \\right) \\  ,\n$$\n$$\nO_{(RL)} =\n$$\n\\begin{equation} \\label{eq03}\n           \\left( \\begin{array}{cccc}\n\\cos\\alpha_R  &  0  &  \\sin\\alpha_R  &  0   \\\\\n0  &  \\cos\\alpha_L  &  0  &  \\sin\\alpha_L   \\\\\n-\\sin\\alpha_R  &  0  &  \\cos\\alpha_R  &  0  \\\\\n0  &  -\\sin\\alpha_L  &  0  &  \\cos\\alpha_L\n\\end{array} \\right) \\ .\n\\end{equation}\n The rotation angles of the left-handed, respectively, right-handed\n components satisfy\n$$\n\\tan(2\\alpha_L) = \\frac{2(\\mu_\\chi \\mu_L + \\mu_\\psi \\mu_R)}\n{\\mu_\\chi^2 + \\mu_R^2 - \\mu_\\psi^2 - \\mu_L^2} \\ ,\n$$\n\\begin{equation} \\label{eq04}\n\\tan(2\\alpha_R) = \\frac{2(\\mu_\\chi \\mu_R + \\mu_\\psi \\mu_L)}\n{\\mu_\\chi^2 + \\mu_L^2 - \\mu_\\psi^2 - \\mu_R^2} \\ ,\n\\end{equation}\n and the two (positive) mass-squared eigenvalues are given by\n$$\n\\mu_{1,2}^2 = \\half \\left\\{\n\\mu_\\chi^2 + \\mu_\\psi^2 + \\mu_L^2 + \\mu_R^2\n\\right.\n$$\n$$\n\\mp \\left[ (\\mu_\\chi^2 - \\mu_\\psi^2)^2 + (\\mu_L^2 - \\mu_R^2)^2\n\\right.\n$$\n\\begin{equation} \\label{eq05}\n\\left.\\left.\n+ 2(\\mu_\\chi^2 + \\mu_\\psi^2) (\\mu_L^2 + \\mu_R^2)\n+ 8\\mu_\\chi \\mu_\\psi \\mu_L \\mu_R \\right]^\\half \\right\\}  .\n\\end{equation}\n The mass matrix itself is diagonalized by\n$$\nO^T_{(RL)} M O_{(LR)} = O^T_{(LR)} M^T O_{(RL)}\n$$\n\\begin{equation} \\label{eq06}\n= \\left( \\begin{array}{cccc}\n\\mu_{1}  &  0  &  0  &  0   \\\\\n0  &  \\mu_{1}  &  0  &  0   \\\\\n0  &  0  &  \\mu_{2}  &  0   \\\\\n0  &  0  &  0  &  \\mu_{2}\n\\end{array} \\right) \\ .\n\\end{equation}\n This shows that for $\\mu_\\psi,\\mu_L,\\mu_R \\ll \\mu_\\chi$ there\n is a light state with mass $\\mu_{1}=O(\\mu_\\psi,\\mu_L,\\mu_R)$ and a\n heavy state with mass $\\mu_{2}=O(\\mu_\\chi)$.\n In general, both the light and heavy states are mixtures of\n the original fermion and mirror fermion.\n According to (\\ref{eq04}), for $\\mu_L \\ne \\mu_R$ the\n fermion-mirror-fermion mixing angle in the left-handed sector is\n different from the one in the right-handed sector.\n\n In case of three mirror pairs of fermion families the diagonalization\n of the mass matrix is in principle similar but, of course, more\n complicated.\n A particular class of mixing schemes will be discussed in the next\n subsection.\n The mirror fermions can be produced through their mixing to ordinary\n fermions.\n The upper limits on the absolute value of mixing angles depend on the\n mixing scheme.\n\n Indirect limits on the existence of heavy mirror fermions can also be\n deduced from the absence of observed effects in 1-loop radiative\n corrections \\cite{PESTAK,ALTBAR}, because of the non-decoupling of\n heavy fermions.\n The question of non-decoupling in higher loop orders is, however,\n open.\n In fact, one of the goals of nonperturbative lattice studies is to\n investigate this in the nonperturbative regime of couplings.\n\n\\subsection{ Mixing schemes }\n The strongest constraints on mixing angles between ordinary\n fermions and mirror fermions arise from the\n conservation of $e$-, $\\mu$- and $\\tau$- lepton numbers and from\n the absence of flavour changing neutral currents \\cite{MAAROO}.\n In a particular scheme these constraints can be avoided \\cite{MIRFAM}.\n In this {\\em ``monogamous mixing''} scheme the structure of the\n mass matrix is such that there is a one-to-one correspondence between\n fermions and mirror fermions, due to the fact that the family\n structure of the mass matrix for mirror fermions is closely related\n to the one for ordinary fermions.\n\n Let us denote doublet indices by $A=1,2$, colour indices by\n $c=1,2,3$ in such a way that the leptons belong to the fourth value of\n colour $c=4$, and family indices by $K=1,2,3$.\n In general the entries of the mass matrix for three mirror pairs\n of fermion families are diagonal in isospin and colour, hence they\n have the form\n$$\n\\mu_{(\\psi,\\chi);A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(A_1c_1)}_{(\\psi,\\chi);K_2K_1}  \\ ,\n$$\n$$\n\\mu_{L;A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(c_1)}_{L;K_2K_1}  \\ ,\n$$\n\\begin{equation} \\label{eq07}\n\\mu_{R;A_2c_2K_2,A_1c_1K_1} = \\delta_{A_2A_1}\n\\delta_{c_2c_1} \\mu^{(A_1c_1)}_{R;K_2K_1}  \\ .\n\\end{equation}\n The diagonalization of the mass matrix can be achieved for given\n indices $A$ and $c$ by two $6 \\otimes 6$ unitary matrices $F_L^{(Ac)}$\n and $F_R^{(Ac)}$ acting, respectively, on the L-handed and R-handed\n subspaces:\n$$\nF_L^{(Ac)\\dagger}(M^\\dagger M)_L F_L^{(Ac)} \\ ,\n$$\n\\begin{equation} \\label{eq08}\nF_R^{(Ac)\\dagger}(M^\\dagger M)_R F_R^{(Ac)} \\ .\n\\end{equation}\n\n The main assumption of the ``monogamous'' mixing scheme is that\n in the family space $\\mu_\\psi,\\mu_\\chi,\\mu_L,\\mu_R$ are hermitian\n and simultaneously diagonalizable, that is\n\\begin{equation} \\label{eq09}\nF_L^{(Ac)} = F_R^{(Ac)} =  \\left(\n\\begin{array}{cc}\nF^{(Ac)}  &  0  \\\\  0  &  F^{(Ac)}\n\\end{array} \\right)  \\ ,\n\\end{equation}\n where the block matrix is in $(\\psi,\\chi)$-space.\n The Kobayashi-Maskawa matrix of quarks is given by\n\\begin{equation} \\label{eq10}\nC \\equiv F^{(2c)\\dagger} F^{(1c)} \\ ,\n\\end{equation}\n independently for $c=1,2,3$.\n The corresponding matrix with $c=4$ and $A=1 \\leftrightarrow 2$\n describes the mixing of neutrinos, if the Dirac-mass of the neutrinos\n is nonzero.\n (Majorana masses of the neutrinos are not considered here, but\n in principle, they can also be introduced.)\n\n A simple example for the ``monogamous'' mixing is the following:\n$$\n\\mu^{(Ac)}_{\\chi;K_2K_1} = \\lambda^{(Ac)}_\\chi\n\\mu^{(Ac)}_{\\psi;K_2K_1} + \\delta_{K_2K_1}\\Delta^{(Ac)} \\ ,\n$$\n$$\n\\mu^{(c)}_{L;K_2K_1} = \\delta_{K_2K_1}\\delta^{(c)}_L \\ ,\n$$\n\\begin{equation} \\label{eq11}\n\\mu^{(Ac)}_{R;K_2K_1} = \\lambda^{(Ac)}_R\n\\mu^{(Ac)}_{\\psi;K_2K_1} + \\delta_{K_2K_1}\\delta^{(Ac)}_R \\ ,\n\\end{equation}\n where $\\lambda_\\chi^{(Ac)},\\; \\Delta^{(Ac)},\\; \\delta_L^{(c)},\\;\n \\lambda_R^{(Ac)},\\; \\delta_R^{(Ac)}$ do not depend on the family\n index.\n\n The general case can be parametrized by the eigenvalues\n $\\mu^{(AcK)}_\\psi,\\; \\mu^{(AcK)}_\\chi,\\; \\mu^{(AcK)}_R,\\;\n \\mu^{(cK)}_L$ and matrices $F^{(Ac)}$:\n$$\n\\mu^{(Ac)}_{(\\psi,\\chi,R);K_2K_1} = \\sum_K\nF^{(Ac)}_{K_2K} \\mu^{(AcK)}_{(\\psi,\\chi,R)} F^{(Ac)\\dagger}_{KK_1} \\ ,\n$$\n\\begin{equation} \\label{eq12}\n\\mu^{(c)}_{L;K_2K_1} = \\sum_K\nF^{(Ac)}_{K_2K} \\mu^{(cK)}_L F^{(Ac)\\dagger}_{KK_1} \\ .\n\\end{equation}\n Here in the second line the left hand side has to be independent of\n the value of $A=1,2$.\n\n The full diagonalization of the mass matrix on the\n $(\\psi_L,\\psi_R,\\chi_L,\\chi_R)$ basis of all three family pairs is\n achieved by the $96 \\otimes 96$ matrix\n$$\n{\\cal O}^{(LR)}_{A^\\prime c^\\prime K^\\prime,AcK}\n= \\delta_{A^\\prime A}\\delta_{c^\\prime c}\nF^{(Ac)}_{K^\\prime K}\n$$\n$$\n\\cdot \\left(\n\\begin{array}{cc}\n \\cos\\alpha^{(AcK)}_L  &  0  \\\\\n 0  &  \\cos\\alpha^{(AcK)}_R  \\\\\n-\\sin\\alpha^{(AcK)}_L  &  0  \\\\\n 0  & -\\sin\\alpha^{(AcK)}_R\n\\end{array}  \\right.\n$$\n\\begin{equation} \\label{eq13}\n\\left.\n\\begin{array}{cc}\n \\sin\\alpha^{(AcK)}_L  &  0  \\\\\n 0  &  \\sin\\alpha^{(AcK)}_R  \\\\\n \\cos\\alpha^{(AcK)}_L  &  0  \\\\\n 0  &  \\cos\\alpha^{(AcK)}_R\n\\end{array}  \\right) \\ .\n\\end{equation}\n $M^\\dagger M$ is diagonalized by\n\\begin{equation} \\label{eq14}\n{\\cal O}^{(LR)\\dagger} M^\\dagger M {\\cal O}^{(LR)} \\ ,\n\\end{equation}\n and $M M^\\dagger$ by\n\\begin{equation} \\label{eq15}\n{\\cal O}^{(RL)\\dagger} M M^\\dagger {\\cal O}^{(RL)} \\ ,\n\\end{equation}\n where ${\\cal O}^{(RL)}$ is obtained from ${\\cal O}^{(LR)}$ by\n $\\alpha_L \\leftrightarrow \\alpha_R$.\n\n In case of $\\mu_R=\\mu_L$, which happens for instance in (\\ref{eq11})\n if $\\lambda_R=0$ and $\\delta_R=\\delta_L$, the left-handed and\n right-handed mixing angles are the same:\n\\begin{equation} \\label{eq16}\n\\alpha^{(AcK)} \\equiv \\alpha_L^{(AcK)} =\n\\alpha_R^{(AcK)} \\ .\n\\end{equation}\n In Ref.~\\cite{MIRFAM} only this special case was considered.\n The importance of the left-right-asymmetric mixing was pointed out in\n Ref.~\\cite{CSIFOD}, where the constraints arising from the measured\n values of anomalous magnetic moments were determined.\n It turned out that for the electron and muon the upper limit is\n\\begin{equation} \\label{eq17}\n|\\alpha_L \\alpha_R| \\le 0.0004 \\ ,\n\\end{equation}\n which is much stronger than the limits obtained from all other\n data \\cite{LANLON}:\n\\begin{equation} \\label{eq18}\n\\alpha_L^2,\\; \\alpha_R^2 \\le 0.02 \\ .\n\\end{equation}\n In case of the L-R asymmetric mixing the constraint (\\ref{eq17})\n can be satisfied, if either the left- or right-handed mixing exactly\n vanishes (or is very small): $\\alpha_L=0$ or $\\alpha_R=0$.\n\n\\subsection{ Future colliders }\n The hypothetical mirror fermions can be discovered at the next\n generation of high energy colliders.\n At HERA the first family mirror fermions can be produced via mixing\n to ordinary fermions up to masses of about 200 GeV, if the mixing\n angles are close to their present upper limits \\cite{CSIMON,HERA}.\n At SSC and LHC mirror lepton pair production can be observed up to\n masses of a few hundred GeV \\cite{CSIKOR}.\n This has the advantage of being essentially independent of the small\n mixing.\n At a high energy $e^+e^-$ collider, e.~g. LEP-200 or NLC,\n mirror fermions can be pair produced and easily identified up to\n roughly half of the total energy, and also produced via mixing\n almost up to the total energy \\cite{NLC}.\n\n\\section{ LATTICE SIMULATION OF YUKAWA MODELS }\n\n\\subsection{ Lattice actions }\n The lattice formulation of the electroweak Standard Model is\n difficult because of the doubler fermions \\cite{NIENIN}.\n In fact, at present no completely satisfactory formulation is known\n \\cite{TSUKPR}: if one insists on explicit chiral gauge invariance,\n then mirror fermion fields have to be introduced \\cite{CHFER},\n otherwise one has to fix the gauge as in the ``Rome-approach''\n \\cite{ROMA2}.\n\n The situation is different if the\n $\\rm SU(3)_{colour} \\otimes U(1)_{hypercharge}$\n gauge couplings are neglected.\n In this case, as a consequence of the pseudo-reality of SU(2)\n representations, mirror fermions can be transformed to normal fermions\n by charge conjugation.\n This allows the gauge invariant lattice formulation of $\\rm SU(2)_L$\n symmetric models with an even number of fermion doublets.\n For simplicity, let us also neglect here the $\\rm SU(2)_L$ gauge\n interaction, and consider a chiral Yukawa-model of two fermion\n doublets.\n A global $\\rm SU(2)_L \\otimes U(1)$ symmetric chiral Yukawa-model\n can be formulated by the lattice action\n\\begin{equation} \\label{eq19}\nS = S_{scalar}+ S_{fermion} \\ ,\n\\end{equation}\n where the pure scalar part in terms of the $2 \\otimes 2$ matrix\n Higgs-boson field $\\varphi$ is\n$$\nS_{scalar} = \\frac{1}{4}\\sum_x \\left\\{\nm_0^2 {\\rm Tr\\,}(\\varphi^\\dagger_x\\varphi_x)\n+ \\lambda \\left[ {\\rm Tr\\,}(\\varphi^\\dagger_x\\varphi_x) \\right]^2\n\\right.\n$$\n\\begin{equation} \\label{eq20}\n\\left.\n+  \\sum_\\mu\n[ {\\rm Tr\\,}  (\\varphi^\\dagger_x \\varphi_x)\n- {\\rm Tr\\,}  (\\varphi^\\dagger_{x+\\hat{\\mu}}\\varphi_x) ]\n\\right\\} \\ ,\n\\end{equation}\n and the fermionic part with two doublet fields $\\psi_{1,2}$ is\n$$\nS_{fermion} = \\sum_x \\left\\{ \\frac{\\mu_0}{2} \\left[\n  (\\psi^T_{2x}\\epsilon C \\psi_{1x}) - (\\psi^T_{1x}\\epsilon C \\psi_{2x})\n\\right.\\right.\n$$\n$$\n\\left.\n+ (\\overline{\\psi}_{2x}\\epsilon C \\overline{\\psi}^T_{1x})\n- (\\overline{\\psi}_{1x}\\epsilon C \\overline{\\psi}^T_{2x}) \\right]\n$$\n$$\n- \\half \\sum_\\mu \\left[\n  (\\overline{\\psi}_{1 x+\\hat{\\mu}} \\gamma_\\mu \\psi_{1x})\n+ (\\overline{\\psi}_{2 x+\\hat{\\mu}} \\gamma_\\mu \\psi_{2x})\n\\right.\n$$\n$$\n- \\frac{r}{2} \\left(\n  (\\psi^T_{2x}\\epsilon C \\psi_{1x})\n- (\\psi^T_{2 x+\\hat{\\mu}}\\epsilon C \\psi_{1x})\n\\right.\n$$\n$$\n- (\\psi^T_{1x}\\epsilon C \\psi_{2x})\n+ (\\psi^T_{1 x+\\hat{\\mu}}\\epsilon C \\psi_{2x})\n$$\n$$\n+ (\\overline{\\psi}_{2x}\\epsilon C \\overline{\\psi}^T_{1x})\n- (\\overline{\\psi}_{2 x+\\hat{\\mu}}\\epsilon C \\overline{\\psi}^T_{1x})\n$$\n$$\n\\left.\\left.\n- (\\overline{\\psi}_{1x}\\epsilon C \\overline{\\psi}^T_{2x})\n+ (\\overline{\\psi}_{1 x+\\hat{\\mu}}\\epsilon C \\overline{\\psi}^T_{2x})\n\\right)\\right]\n$$\n$$\n+ (\\overline{\\psi}_{1Rx} G_1\\varphi^+_x \\psi_{1Lx})\n+ (\\overline{\\psi}_{1Lx} \\varphi_x G_1  \\psi_{1Rx})\n$$\n\\begin{equation} \\label{eq21}\n\\left.\n+ (\\overline{\\psi}_{2Rx} G_2\\varphi^+_x \\psi_{2Lx})\n+ (\\overline{\\psi}_{2Lx} \\varphi_x G_2  \\psi_{2Rx}) \\right\\} \\ .\n\\end{equation}\n Here the summations $\\sum_\\mu$ always go over eight directions of the\n neighbouring sites, $C$ is the Dirac matrix for charge conjugation and\n $\\epsilon = i\\tau_2$ is the antisymmetric unit matrix in isospin space.\n In the scalar part of the action $m_0^2$ is the bare mass squared and\n $\\lambda$ the bare quartic coupling.\n In the fermionic part $\\mu_0$ is an off-diagonal Majorana mass term,\n $r$ is the Wilson parameter for removing lattice fermion doublers,\n which is usually chosen to be $r=1$, and $G_{(1,2)}$ are diagonal $2\n \\otimes 2$ matrices in isospin space for the bare Yukawa-couplings:\n\\begin{equation} \\label{eq22}\nG_{(1,2)} \\equiv \\left(\n\\begin{array}{cc}\nG_{(1,2)u}  &  0  \\\\  0  &  G_{(1,2)d}\n\\end{array}  \\right)  \\ .\n\\end{equation}\n\n The global $\\rm SU(2)_L$ symmetry is acting on the fields as\n$$\n\\varphi_x^\\prime = U_L^{-1}\\varphi_x \\ ,\n$$\n$$\n\\psi_{(1,2)Lx}^\\prime = U_L^{-1}\\psi_{(1,2)Lx} \\ ,\n$$\n\\begin{equation} \\label{eq23}\n\\overline{\\psi}_{(1,2)Lx}^\\prime =\n\\overline{\\psi}_{(1,2)Lx} U_L \\ .\n\\end{equation}\n The right-handed components of the fermion fields $\\psi_{(1,2)Rx}$ and\n $\\overline{\\psi}_{(1,2)Rx}$ are, of course, invariant.\n\n The global U(1) symmetry corresponds to the conservation of the\n difference of the fermion number of $\\psi_1$ minus the fermion\n number of $\\psi_2$.\n (This means that if it is identified by the hypercharge $\\rm U(1)_Y$,\n then $\\psi_1$ and $\\psi_2$ have opposite hypercharges.)\n\n An equivalent form of the fermionic part of the above action is\n obtained, if one introduces the mirror fermion fields by charge\n conjugation:\n\\begin{equation} \\label{eq24}\n\\chi_x \\equiv \\epsilon^{-1} C \\overline{\\psi}_{2x}^T \\ ,  \\hspace{1em}\n\\overline{\\chi}_x \\equiv \\psi_{2x}^T \\epsilon C  \\ .\n\\end{equation}\n Since under charge conjugation the left- and right-handed components\n are interchanged, and $\\epsilon^{-1}U_L^T\\epsilon=U_L^{-1}$, under\n $\\rm SU(2)_L$ transformations we have\n\\begin{equation} \\label{eq25}\n\\chi_{Rx}^\\prime = U_L^{-1}\\chi_{Rx} \\ ,  \\hspace{1em}\n\\overline{\\chi}_{Rx}^\\prime = \\overline{\\chi}_{Rx} U_L \\ ,\n\\end{equation}\n and now the left-handed components $\\chi_{Lx}$ and\n $\\overline{\\chi}_{Lx}$ are invariant.\n Omitting the index on $\\psi_1$, one gets the action in terms of the\n mirror pair of fermion fields \\cite{CHFER}\n$$\nS_{fermion} = \\sum_x \\left\\{\n  \\mu_0 \\left[ (\\overline{\\chi}_x\\psi_x)\n+ (\\overline{\\psi}_x\\chi_x) \\right]\n\\right.\n$$\n$$\n- \\half \\sum_\\mu \\left[\n  (\\overline{\\psi}_{x+\\hat{\\mu}}\\gamma_\\mu\\psi_x)\n+ (\\overline{\\chi}_{x+\\hat{\\mu}}\\gamma_\\mu\\chi_x)\n\\right.\n$$\n$$\n- r \\left(\n  (\\overline{\\chi}_x\\psi_x)\n- (\\overline{\\chi}_{x+\\hat{\\mu}}\\psi_x)\n\\right.\n$$\n$$\n\\left.\\left.\n+ (\\overline{\\psi}_x\\chi_x)\n- (\\overline{\\psi}_{x+\\hat{\\mu}}\\chi_x) \\right) \\right]\n$$\n$$\n+ (\\overline{\\psi}_{Rx} G_\\psi\\varphi^\\dagger_x \\psi_{Lx})\n+ (\\overline{\\psi}_{Lx} \\varphi_x G_\\psi  \\psi_{Rx})\n$$\n\\begin{equation} \\label{eq26}\n\\left.\n+ (\\overline{\\chi}_{Lx} G_\\chi\\varphi^\\dagger_x \\chi_{Rx})\n+ (\\overline{\\chi}_{Rx} \\varphi_x G_\\chi  \\chi_{Lx}) \\right\\} \\ .\n\\end{equation}\n The Yukawa-coupling of the fermion doublet is denoted here by\n $G_\\psi = G_1$, and the Yukawa-coupling of the mirror fermion\n doublet is $G_\\chi = \\epsilon^{-1}G_2 \\epsilon$.\n This means that in $G_\\chi$ the isospin components are interchanged.\n The mass term proportional to $\\mu_0$ and the off-diagonal\n Wilson term multiplied by $r$ look in the second form (\\ref{eq26}) not\n Majorana-like but Dirac-like.\n\n Note that if the doublets are degenerate, that is the Yukawa-couplings\n are proportional to the unit matrix in isospin space, then the\n $\\rm SU(2)_L \\otimes U(1)$ symmetry is enlarged\n to $\\rm SU(2)_L \\otimes SU(2)_R$ defined by\n$$\n\\varphi_x^\\prime = U_L^{-1}\\varphi_x U_R \\ ,\n$$\n$$\n\\psi_{(L,R)x}^\\prime = U_{(L,R)}^{-1}\\psi_{(L,R)x} \\ ,\n$$\n$$\n\\overline{\\psi}_{(L,R)x}^\\prime =\n\\overline{\\psi}_{(L,R)x} U_{(L,R)} \\ ,\n$$\n$$\n\\chi_{(R,L)x}^\\prime = U_{(L,R)}^{-1}\\chi_{(R,L)x} \\ ,\n$$\n\\begin{equation} \\label{eq27}\n\\overline{\\chi}_{(R,L)x}^\\prime =\n\\overline{\\chi}_{(R,L)x} U_{(L,R)} \\ .\n\\end{equation}\n\n\\subsection{ Chirality and decoupling }\n An important property of the lattice action in the previous\n subsection is the possibility of decoupling half of the fermions from\n the interacting sector \\cite{ROMA1}.\n Let us formulate this in the mirror fermion langauge corresponding\n to (\\ref{eq26}).\n For vanishing Yukawa-coupling of the mirror fermion doublet $G_\\chi=0$\n and fermion mirror fermion mixing mass $\\mu_0=0$ the action is\n invariant with respect to the Golterman-Petcher shift symmetry\n\\begin{equation} \\label{eq28}\n\\chi_x \\rightarrow \\chi_x + \\epsilon \\ ,  \\hspace{1em}\n\\overline{\\chi}_x \\rightarrow \\overline{\\chi}_x + \\overline{\\epsilon} \\ .\n\\end{equation}\n This implies \\cite{GOLPET,LINWIT} that all higher vertex functions\n containing the $\\chi$-field identically vanish, and the $\\chi$-$\\chi$\n and $\\chi$-$\\psi$ components of the inverse propagator are equal to\n the corresponding components of the free inverse propagator:\n\\begin{equation} \\label{eq29}\n\\tilde{\\Gamma}_{\\psi\\chi} = \\mu_0 + \\frac{r}{2} \\hat{p}^2 \\ ,\n\\hspace{1em}\n\\tilde{\\Gamma}_{\\chi\\chi} = i\\gamma \\cdot \\bar{p} \\ ,\n\\end{equation}\n where, as usual, $\\bar{p}_\\mu \\equiv \\sin p_\\mu$ and\n $\\hat{p}_\\mu \\equiv 2\\sin \\half p_\\mu$.\n\n The consequence of (\\ref{eq29}) is that the fermion mirror fermion\n mixing mass $\\mu_0$ is not renormalized by the Yukawa-interaction\n of the $\\psi$-field.\n This is very useful in numerical simulations, because the\n corresponding bare parameter (usually the fermionic hopping parameter\n $K \\equiv (2\\mu_0+8r)^{-1}$) is fixed, and the number of bare\n parameters to be tuned is less.\n\n There is also another possible interpretation of the fermion\n decoupling.\n Namely, interchanging the r\\^oles of $\\psi$ and $\\chi$, in the case\n of $G_\\psi=\\mu_0=0$ the ordinary fermions are decoupled.\n This decoupling scenario is in fact a rather good approximation to the\n situation in phenomenological models with mirror fermions discussed in\n the previous section.\n This is due to the fact that all known physical fermions have very\n small Yukawa-couplings.\n The only fermion states with strong Yukawa-coupling would be the\n members of the mirror families, if they would exist.\n In fact, the smallness of the known fermion masses on the electroweak\n scale could then be explained by the approximate validity of the\n Golterman-Petcher shift symmetry.\n {\\em Low energy chirality would be the consequence of the\n approximate decoupling of light fermions.}\n In this sense the mirror fermion model is natural, because the\n smallness of some of its parameters is connected to an approximate\n symmetry \\cite{THOOFT}.\n\n Let us shortly discuss the form of the broken Golterman-Petcher\n identities.\n They are broken in general by the small Yukawa-coupling $G_\\chi$, by\n small mixing mass $\\mu_0$, and by the small\n $\\rm SU(3) \\otimes SU(2) \\otimes U(1)$ gauge couplings.\n For definiteness, let us consider here the case of $G_\\chi \\simeq 0$.\n Consider the generating function of the connected Green's functions\n $W[\\eta,\\overline{\\eta},\\zeta,\\overline{\\zeta},j]$, where the\n external sources $\\eta,\\zeta,j$ belong, respectively, to\n $\\psi,\\chi,\\varphi$.\n The identities obtained by shifting the $\\chi$- and\n $\\overline{\\chi}$-fields are\n$$\n\\overline{\\zeta}_x\n+ \\half \\sum_{\\mu=1}^4 (\\Delta^f_\\mu + \\Delta^b_\\mu)\n\\frac{\\partial W}{\\partial\\zeta_x}\\gamma_\\mu\n$$\n$$\n+ \\frac{r}{2} \\sum_{\\mu=1}^4 \\Delta^f_\\mu\\Delta^b_\\mu\n\\frac{\\partial W}{\\partial\\eta_x}\n= \\mu_0 \\frac{\\partial W}{\\partial\\eta_x}\n$$\n$$\n+ G_\\chi \\left(\n\\frac{\\partial^2 W}{\\partial j_{Rx}\\partial\\zeta_x}\n+ \\frac{\\partial W}{\\partial j_{Rx}}\n\\frac{\\partial W}{\\partial\\zeta_x} \\right) \\Gamma^\\dagger_R  \\ ,\n$$\n$$\n- \\zeta_x\n- \\half \\sum_{\\mu=1}^4 \\gamma_\\mu (\\Delta^f_\\mu + \\Delta^b_\\mu)\n\\frac{\\partial W}{\\partial\\overline{\\zeta}_x}\n$$\n$$\n+ \\frac{r}{2} \\sum_{\\mu=1}^4 \\Delta^f_\\mu\\Delta^b_\\mu\n\\frac{\\partial W}{\\partial\\overline{\\eta}_x}\n= \\mu_0 \\frac{\\partial W}{\\partial\\overline{\\eta}_x}\n$$\n\\begin{equation} \\label{eq30}\n+ G_\\chi \\left(\n\\frac{\\partial^2 W}{\\partial j_{Rx}\\partial\\overline{\\zeta}_x}\n+ \\frac{\\partial W}{\\partial j_{Rx}}\n\\frac{\\partial W}{\\partial\\overline{\\zeta}_x} \\right) \\Gamma^\\dagger_R\n\\ .\n\\end{equation}\n Here $\\Delta^f_\\mu$ and $\\Delta^b_\\mu$ denote, respectively, forward\n and backward lattice derivatives, and real scalar field components\n $\\phi_R$ $(R=0,1,2,3)$ are introduced by\n$$\n\\half \\left[ \\varphi_x (1+\\gamma_5)\n+ \\varphi^\\dagger_x (1-\\gamma_5) \\right]\n$$\n\\begin{equation} \\label{eq31}\n= \\phi_{0x} + i\\gamma_5 \\tau_r\\phi_{rx} \\equiv \\Gamma_R\\phi_{Rx} \\ .\n\\end{equation}\n\n Let us define the composite fermion field $\\Psi_x$ by\n\\begin{equation} \\label{eq32}\n\\Gamma_R\\phi_{Rx}\\chi_x\n= \\varphi_x\\chi_{Lx} + \\varphi^\\dagger_x\\chi_{Rx} \\equiv \\Psi_x \\ .\n\\end{equation}\n The mixed $\\psi$-$\\Psi$ two point function is\n\\begin{equation} \\label{eq33}\n\\langle \\psi_y \\Psi_x \\rangle \\equiv \\frac{1}{\\Omega}\n\\sum_k e^{ik \\cdot (y-x)} \\tilde{\\Delta}^{\\psi\\Psi}_k \\ .\n\\end{equation}\n Taking derivatives at zero sources, with the notation\n $\\langle \\phi_{0x} \\rangle \\equiv v$ for the vacuum expectation\n value one obtains, for instance,\n$$\n0 = (\\mu_0 + \\frac{r}{2} \\hat{k}^2) \\tilde{\\Delta}^{\\psi\\psi}_k\n+ \\tilde{\\Delta}^{\\psi\\chi}_k (i\\gamma \\cdot \\bar{k} + G_\\chi v)\n+ G_\\chi \\tilde{\\Delta}^{\\psi\\Psi}_k \\ ,\n$$\n$$\n1 = (\\mu_0 + \\frac{r}{2} \\hat{k}^2) \\tilde{\\Delta}^{\\chi\\psi}_k\n$$\n\\begin{equation} \\label{eq34}\n+ \\tilde{\\Delta}^{\\chi\\chi}_k (i\\gamma \\cdot \\bar{k} + G_\\chi v)\n+ G_\\chi \\tilde{\\Delta}^{\\chi\\Psi}_k \\ .\n\\end{equation}\n For $G_\\chi=0$ this is equivalent to (\\ref{eq29}).\n The case of small gauge couplings can be treated similarly.\n\n\\subsection{ A simple $SU(2)_L \\otimes SU(2)_R$ model }\n The lattice actions of Yukawa-models in the form (\\ref{eq21}) or\n (\\ref{eq26}) can be used for numerical simulation studies of chiral\n Yukawa-models.\n In order to apply Monte Carlo simulation methods one needs, however,\n a fermion determinant which is positive.\n For instance, in the Hybrid Monte Carlo algorithm \\cite{DKPR} one\n has to duplicate the number of fermionic degrees of freedom.\n Let us denote the fermion matrix corresponding to (\\ref{eq26}) by\n $Q$, then the replica fermions have $Q^\\dagger$, and the fermion\n determinant is $\\det(QQ^\\dagger)$, which is positive.\n Due to the adjoint, for the replica fermions $\\psi_x$ describes\n a mirror fermion doublet and $\\chi_x$ an ordinary fermion doublet.\n By charge conjugation as in (\\ref{eq24}) one can transform the\n $\\psi$-field of replica fermions to an ordinary doublet, and\n the $\\chi$-field of replica fermions to a mirror doublet.\n In this way one can consider two doublets described by the\n $\\psi$-fields and two mirror doublets described by the $\\chi$-fields.\n For simplicity, let us consider only degenerate doublets, that is,\n let the Yukawa-couplings $G_{(\\psi,\\chi)}$ be proportional to the\n unit matrix in isospin space.\n In this case the Hybrid Monte Carlo simulation describes two\n equal mass degenerate doublets plus two equal mass degenerate mirror\n doublets with exact global $\\rm SU(2)_L \\otimes SU(2)_R$ symmetry.\n In the phase with spontaneously broken symmetry the mass of the\n doublets is proportional to $G_\\psi$, and the mass of the mirror\n doublets to $G_\\chi$.\n\n As it was noted in the previous subsection, the limit $G_\\chi=\\mu_0=0$\n is particularly interesting, because it has less bare parameters to\n tune.\n In this case the $\\chi$-fields are exactly decoupled, and one is\n left in Hybrid Monte Carlo with two degenerate fermion doublets\n described by the $\\psi$-fields.\n This is the simplest model of heavy fermion doublets one can simulate\n by present day fermionic simulation techniques \\cite{SU2XSU2}.\n Besides the two bare parameters of the pure scalar sector\n $(m_0^2,\\lambda)$ there is only one additional bare parameter\n ($G_\\psi$) in the fermionic part of the action:\n$$\nS_{fermion} = \\sum_x \\left\\{ -\\half \\sum_\\mu \\left[\n  (\\overline{\\psi}_{x+\\hat{\\mu}}\\gamma_\\mu\\psi_x)\n\\right.\\right.\n$$\n$$\n+ (\\overline{\\chi}_{x+\\hat{\\mu}}\\gamma_\\mu\\chi_x)\n$$\n$$\n\\left. - \\left(\n  (\\overline{\\chi}_x\\psi_x)\n- (\\overline{\\chi}_{x+\\hat{\\mu}}\\psi_x)\n+ (\\overline{\\psi}_x\\chi_x)\n- (\\overline{\\psi}_{x+\\hat{\\mu}}\\chi_x) \\right) \\right]\n$$\n\\begin{equation} \\label{eq35}\n\\left. + G_\\psi \\left[\n  (\\overline{\\psi}_{Rx} \\varphi^+_x \\psi_{Lx})\n+ (\\overline{\\psi}_{Lx} \\varphi_x \\psi_{Rx}) \\right] \\right\\} \\ .\n\\end{equation}\n To have the smallest possible number of parameters is very important,\n in order to keep parameter tuning as easy as possible.\n\n By a further duplication of the fermion fields one can also\n simulate four degenerate fermion doublets, which correspond to\n a heavy degenerate family.\n The $\\rm SU(4)_{Pati-Salam}$ symmetry \\cite{PATSAL} of the four\n $\\psi$-doublets, including $\\rm SU(3)_{colour}$ for the quarks,\n is exact in the continuum limit, but for finite lattice spacings\n it is broken by the off-diagonal Wilson-terms which mix the $\\psi$- and\n $\\chi$-fields.\n Note, however, that there is an exact SU(4) symmetry also at finite\n lattice spacings, if one transforms the $\\psi$- and $\\chi$-fields\n simultaneously.\n Of course, the $\\chi$'s are mirror fermion fields, which mix with the\n $\\psi$'s through the nonzero $\\tilde{\\Gamma}_{\\psi\\chi}$ in\n (\\ref{eq29}).\n This mixing goes to zero only in the continuum limit.\n The decoupling in the continuum limit is exact in the Yukawa-model,\n but for nonzero gauge couplings decoupling the $\\chi$'s in a gauge\n invariant way does not work.\n\n Another way of simulating a heavy degenerate fermion family with\n only two pairs of $(\\psi,\\chi)$-doublet fields is to choose\n $G_\\chi = \\pm G_\\psi$ (the opposite sign is preferred by the study\n of the $K=0$ limit \\cite{LIMAMO}).\n Since without $\\rm SU(3) \\otimes U(1)$ gauge fields the mirror\n fermion doublets are equivalent to ordinary fermion doublets, this\n describes the same model in the continuum limit as the one with\n twice as much fields and decoupling.\n In this case, however, the fermion hopping parameter $K$ has to be\n tuned, too, which can be worse than having more field components per\n lattice sites.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig1}\n Phase structure of the $\\rm SU(2)_L\\otimes SU(2)_R$ symmetric Yukawa\n model at $\\lambda=\\infty$ and $G_\\chi=\\mu_0=0$ in the\n ($G_\\psi,\\,\\kappa$)-plane.\n Open circles denote points in the PM~phase, crosses represent\n points in the FM~phase.\n The points in the AFM and FI~phases are denoted by full circles and\n open squares, respectively.\n The dashed lines labelled~R,S,T each show the range of~$\\kappa$ used\n for a systematic scan of renormalized parameters at fixed~$G_\\psi$.\n The crosses along those lines denote the kappa values where the\n minimum scalar mass in the broken phase is encountered.\n Solid lines connect the critical values for $\\kappa$ estimated from\n the behaviour of the magnetization on $4^3\\cdot8$.\n Dashed lines around the FI~phase show the expected continuation of the\n critical lines. }\n\\end{figure}\n\n\\subsection{ Phase structure }\n The first step in a recent numerical simulation of the\n $\\rm SU(2)_L \\otimes SU(2)_R$ symmetric Yukawa-model with $N_f=2$\n fermion doublets in the decoupling limit $G_\\chi=0$ \\cite{SU2XSU2}\n was to check the phase structure at infinite bare quartic coupling\n $\\lambda=\\infty$.\n On the basis of experience in several different lattice Yukawa models\n \\cite{GOLREV,SHIREV,LIMOWI}, this is expected to possess several phase\n transitions between the {\\it ``ferromagnetic'' (FM)}, {\\it\n ``antiferromagnetic'' (AFM)}, {\\it ``paramagnetic'' (PM)} and {\\it\n ``ferrimagnetic'' (FI)} phases.\n The resulting picture in the ($G_\\psi,\\kappa$)-plane is shown in\n fig.~\\ref{fig1}.\n ($\\kappa \\equiv (1-2\\lambda)\/(m_0^2+8)$ is the bare parameter which\n is usually taken in numerical simulations instead of $m_0^2$.)\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig2}\n The cut-off dependent allowed region in the ($G_{R\\psi}^2,g_R$) plane\n for cut-off values equal to some multiples of the Higgs-boson mass\n $m_\\phi \\equiv m_{R\\sigma}$.\n In the Yukawa-model describing two degenerate heavy fermion doublets\n without gauge couplings the perturbative 1-loop $\\beta$-functions are\n assumed. }\n\\end{figure}\n\n\\subsection{ Allowed region in renormalized couplings }\n An important question for numerical simulations is the determination\n of the nonperturbative cut-off dependent {\\em allowed region} in the\n space of renormalized quartic and Yukawa-couplings.\n If the continuum limit of Yukawa-models is trivial, then there\n are cut-off dependent upper bounds on both the renormalized quartic\n and Yukawa-couplings, which tend to zero in the continuum limit.\n In pure $\\phi^4$ models the upper bound is qualitatively well\n described by the 1-loop perturbative $\\beta$-function, if the\n Landau-pole in the renormalization group equations is assumed to\n occur at the scale of the cut-off.\n The same might be true for scalar-fermion models with Yukawa-couplings.\n For instance, in the model with $\\rm SU(2)_L \\otimes SU(2)_R$ symmetry\n and $N_f$ degenerate fermion doublets the 1-loop $\\beta$-functions for\n the quartic ($g_R$) and Yukawa- ($G_{R\\psi}$) couplings are:\n$$\n\\beta_{g_R} = \\frac{1}{16\\pi^2} \\left(\n4g_R^2 + 16N_fg_RG_{R\\psi}^2 - 96N_fG_{R\\psi}^4 \\right) \\ ,\n$$\n\\begin{equation} \\label{eq36}\n\\beta_{G_{R\\psi}} = \\frac{1}{16\\pi^2} \\cdot\n4N_fG_{R\\psi}^3 \\ .\n\\end{equation}\n\n Since in the region where $G_{R\\psi}^2 \\gg g_R$ the 1-loop\n $\\beta$-function of the quartic coupling $\\beta_{g_R}$ is negative,\n besides the upper bounds there is also a lower bound on $g_R$ for fixed\n $G_{R\\psi}$, which is called in the literature {\\it vacuum stability\n bound} \\cite{SHER}.\n On the lattice, if one assumes the qualitative behaviour of the 1-loop\n $\\beta$-function to be valid also nonperturbatively, the vacuum\n stability lower bound occurs at zero bare quartic coupling $\\lambda=0$,\n whereas the upper bound at $\\lambda=\\infty$ \\cite{LMMW,LMMWTA,SHENTA}.\n Negative $\\lambda$-values are excluded, because there the path\n integral is divergent.\n For the 1-loop $\\beta$-functions in (\\ref{eq36}) with $N_f=2$\n the bounds in the plane of ($G_{R\\psi}^2,g_R$) are shown in\n fig.~\\ref{fig2}.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig3}\n The fermion mass in lattice units $\\mu_{R\\psi}$ plotted versus\n $\\kappa$ for $G_\\psi=0.3$ (triangles) and $G_\\psi=0,6$ (squares) on\n lattices of size $4^3\\cdot8$ (open symbols) and $6^3\\cdot12$\n (filled-in symbols).\n Errorbars are omitted when the variation is of the size of the symbols.\n It is seen that larger bare couplings $G_\\psi$ in general yield larger\n fermion masses. }\n\\end{figure}\n\n These curves have to be confronted with the results of the numerical\n simulations.\n It turned out (see fig.~\\ref{fig3}) that on $4^3 \\cdot 8$ and\n $6^3 \\cdot 12$ lattices the fermion mass in lattice units tends to\n zero, if one is approaching the FM-PM phase transition from above\n (i.~e. from the FM-side).\n This is as expected in the continuum limit on an infinite lattice.\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig4}\n The scalar mass in lattice units $m_{R\\sigma}$ plotted versus $\\kappa$.\n The explanation of symbols is similar to the previous figure.\n When approaching the phase transition the scalar masses increase again\n after going through a minimum. }\n\\end{figure}\n\\begin{figure}[tb]\n\\vspace{9cm}\n\\caption{ \\label{fig5}\n The obtained numerical estimate of the upper bound on the renormalized\n quartic coupling $g_R$ (or Higgs-boson mass $m_{R\\sigma}$) as a\n function of the renormalized Yukawa-coupling squared $G_{R\\psi}^2$.\n Two points with the smaller errors are on $6^3 \\cdot 12$, the\n third one on $8^3 \\cdot 16$ lattice.\n The lines are the upper bounds according to 1-loop perturbation\n theory at a cut-off $\\Lambda \\simeq 3m_{R\\sigma}$ and $4m_{R\\sigma}$,\n respectively. }\n\\end{figure}\n\n The behaviour of the Higgs-boson mass $m_{R\\sigma}$ on finite lattices\n is more involved.\n This is because the decrease of $m_{R\\sigma}$ is stopped by a minimum,\n and instead of a further decrease there is an increase (see\n fig.~\\ref{fig4}).\n This can be understood as a finite size effect.\n The value at the minimum is smaller on larger lattices.\n Nevertheless, for increasing bare Yukawa-coupling $G_\\psi$ the\n required lattice size is growing.\n Reasonably small masses $m_{R\\sigma} \\simeq 0.5-0.7$ can be\n achieved at $G_\\psi=0.3$ on $6^3 \\cdot 12$, at $G_\\psi=0.6$ on\n $8^3 \\cdot 16$ lattices \\cite{SU2XSU2}.\n At $G_\\psi=1.0$ presumably lattices with spatial extension of\n at least $16^3$ are necessary.\n Taking the values of the Higgs-boson mass at $\\kappa$-values above the\n minimum, one obtains for the upper limit on the renormalized quartic\n coupling the estimates in fig.~\\ref{fig5}.\n These agree within errors with the 1-loop estimates, although the\n values of the renormalized couplings are close to the tree\n unitarity limits.\n The continuation of the upper bound on $g_R$ towards larger\n Yukawa-couplings can only be obtained on larger lattices.\n Particularly interesting is the behaviour in the vicinity of\n the FM-FI phase transition near $G_\\psi=1.0-1.5$.\n A further interesting question is the phase structure near\n $\\lambda \\simeq 0$, which has an influence on the vacuum stability\n lower bound.\n\n\\begin{acknowledge}\n It is a pleasure to thank the organizers for giving me the opportunity\n to participate in this workshop.\n I benefited a lot from the discussions with the participants in a\n very pleasant atmosphere.\n\\end{acknowledge}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHarmonic analysis on a Riemannian symmetric space $G\/K$ of the noncompact type \nis by now well developed (cf. \\cite{Hel2}). \nA  natural extension is \nto study harmonic analysis on homogeneous vector bundles over $G\/K$. \nOne of fundamental problems \nin harmonic analysis is to establish the Plancherel theorem. \nHarish-Chandra establishes a general theory of \n the Eisenstein integrals and \nthe Plancherel theorem for noncompact real semisimple Lie groups (cf. \\cite{HC76,Kn0,Wal,War}). \nThe Plancherel theorem on a homogeneous vector bundle over $G\/K$ \nassociated with an irreducible representation $\\pi$ of $K$ follows from \nHarish-Chandra's result by restricting the Plancherel measure to \n$K$-finite functions of type $\\pi$. But \nit is a highly nontrivial and important problem to determine the Plancherel measure \non the associated vector bundle as \nexplicitly as in the case of the trivial $K$-type. \nThere are several studies in this direction (cf. \\cite{Camporesi2, CP, vDP, \nFJ, Hec, OS, SPlancherel, Shyperbolic}). \n\nIn our previous paper \\cite{OS}, we study elementary \nspherical functions on $G$ with a \nsmall $K$-type $\\pi$ (in the sense of Wallach \\cite[\\S 11.3]{Wal}). \nNamely, we identify elementary spherical functions with the Heckman-Opdam \nhypergeometric function (cf. \\cite{Hec, Op:lecture}) and apply the \ninversion formula and the Plancherel formula \nfor the hypergeometric Fourier transform (\\cite{Op:Cherednik}) \nto obtain the inversion formula and the Plancherel formula for the \n$\\pi$-spherical transform. But there is an exception in \\cite{OS}. \nNamely, for a certain small $K$-type of a noncompact Lie group \nof type $G_2$, elementary spherical functions can not be expressed by \nthe Heckman-Opdam hypergeometric function. \n\nIn this paper we give a complete treatment of harmonic analysis of \n$\\pi$-spherical transform for each small $K$-type $\\pi$ of  \n $G_2$. Namely, we give an explicit formula for the Harish-Chandra $c$-function \n$c^{\\pi}(\\lambda)$ \nand determine the Plancherel measure explicitly. The most continuous part \nof the Plancherel measure\n is $|c^\\pi(\\lambda)|^{-2}d\\lambda$ on $\\sqrt{-1}\\mathfrak{a}^*$ and the other \n spectra with supports of lower dimensions are given explicitly by \n using residue calculus.  \nAs indicated by Oshima \\cite{Oshima81a} and as was done for one-dimensional $K$-types \nby the second author \\cite{SPlancherel}, we could prove the inversion formula for the \n$\\pi$-spherical tranform in the case of $G_2$ by \n extending Rosenberg's method of a proof of the inversion formula in the case of \n the trivial $K$-type (\\cite{R}). Instead of doing this, \n we utilize general results on the Plancherel theorem and residue calculus on \n $G$ due to Harish-Chandra and Arthur (cf. \\cite{HC76, A, Kn0,Wal}) \n and devote ourselves to the determination of the Plancherel measure. \n\nThis paper is organized as follows. In Section~2 we give general results \nfor elementary $\\pi$-spherical functions, the \nHarish-Chandra $c$-function, the inversion formula for $\\pi$-spherical transform \nwith respect to a small $K$-type $\\pi$ \non a noncompact real semisimple Lie group of finite center. \n \n In Section~3 we study the case of $G_2$. We give an  explicit formula of \nthe $c$-function (Theorem~\\ref{thm:cfg2}), \nthe  inversion formula, and the Plancherel formula (Theorem~\\ref{thm:main}, \n Corollary~\\ref{cor:main}) for each small $K$-type. \nIn particular, they cover the small $K$-type that is not treated in \\cite{OS}. \n\n\\section{Elementary spherical functions \nfor small $K$-types}\n\n\\subsection{Notation}\nLet $\\mathbb{N}$ denote the set of the nonnegative integers. \nLet $G$ be a non-compact connected real semisimple Lie group of \nfinite center and $K$ a maximal compact subgroup of $G$. \nLet $e$ denote the identity element of $G$. \nLie algebras of Lie groups $G,\\,K$, etc. are denoted by the corresponding German letter  $\\mathfrak{g},\\,\\mathfrak{k}$, etc.  \nLet $\\mathfrak{g}=\\mathfrak{k}+\\mathfrak{p}$ be \nthe Cartan decomposition and $\\mathfrak{a}$ a maximal abelian subspace \nof $\\mathfrak{p}$. Let $\\varSigma$ denote the root system for $(\\mathfrak{g},\\mathfrak{a})$. \nFor $\\alpha\\in\\varSigma$, let $\\mathfrak{g}_\\alpha$ denote the corresponding \nroot space and $\\boldsymbol{m}_\\alpha=\\dim \\mathfrak{g}_\\alpha$. \nFix a \npositive system $\\varSigma^+\\subset \\varSigma$ and let \n$\\varPi=\\{\\alpha_1,\\dots,\\alpha_r\\}$ denote the set of simple roots in $\\varSigma^+$. \nDefine $\\mathfrak{n}=\\sum_{\\alpha\\in\\varSigma^+}\\mathfrak{g}_\\alpha$ and \n$N=\\exp \\mathfrak{n}$. Then we have the Iwasawa decomposition $G=K\\exp\\mathfrak{a}\\,N$. \nDefine $\\rho=\\frac12\\sum_{\\alpha\\in\\varSigma^+}\\boldsymbol{m}_\\alpha \\alpha$. \n\nLet $W$ denote the Weyl group of $\\varSigma$ and \n $s_i$ the reflection across $\\alpha_i^\\perp$ ($1\\leq i\\leq r$). \nWe have $W\\simeq M'\/M$, where $M'$ (resp. $M$) is the normalizer (resp. centralizer) \nof $\\mathfrak{a}$ in $K$. \n\nDefine\n\\begin{align*}\n& \\mathfrak{a}_+\n=\\{H\\in\\mathfrak{a}\\,|\\,\\alpha(H)>0 \\text{ for all } \\alpha\\in\\varSigma^+\\},\n\\\\\n& \\mathfrak{a}_\\text{reg}\n=\\{H\\in\\mathfrak{a}\\,|\\,\\alpha(H)\\not=0 \\text{ for all } \\alpha\\in\\varSigma^+\\}.\n\\end{align*}\nWe have the Cartan decomposition $G=K\\exp \\overline{\\mathfrak{a}_+}\\,K$. \n\nLet $\\langle\\,\\,,\\,\\,\\rangle$ denote the inner product on $\\mathfrak{a}^*$ \ninduced by the Killing form on $\\mathfrak{g}$ and $||\\,\\,||$ the corresponding norm. \nDefine\n\\[\n\\mathfrak{a}_+^*\n=\\{\\lambda\\in\\mathfrak{a}^*\\,|\\,\\langle\\lambda,\\alpha\\rangle>0 \\text{ for all } \\alpha\\in \\varSigma^+\\}.\n\\]\n\n\\subsection{Elementary $\\pi$-spherical function}\n\nIn this subsection, we review elementary $\\pi$-spherical functions for small $K$-types \naccording to \\cite{OS}. \n\nLet $(\\pi,V)$ be a small $K$-type, that is, $\\pi|_M$ is irreducible. \nWe call an $\\text{End}_\\mathbb{C} V$-valued function $f$ on $G$ satisfying \n\\[\nf(k_1gk_2)=\\pi(k_2^{-1})f(g)\\pi(k_1^{-1})\\quad (k_1,\\,k_2\\in K,\\,g\\in G)\n\\]\na $\\pi$-spherical function. \n\nLet $\\boldsymbol{D}^\\pi$ denote the algebra of the invariant differential operators on the homogeneous \nvector bundle over $G\/K$ associated with $\\pi$. \nLet $U(\\mathfrak{g}_\\mathbb{C})$ denote the universal enveloping algebra of \n$\\mathfrak{g}_\\mathbb{C}=\\mathfrak{g}\\otimes_\\mathbb{R}\\mathbb{C}$ and \n$U(\\mathfrak{g}_\\mathbb{C})^K$ the set of the \n$\\text{Ad}(K)$-invariant elements in $U(\\mathfrak{g}_\\mathbb{C})$. \nLet $J_{\\pi^*}=\\text{ker}\\,\\pi^*$ in $U(\\mathfrak{k}_\\mathbb{C})$. \nWe have \n\\[\n\\boldsymbol{D}^\\pi\\simeq U(\\mathfrak{g}_\\mathbb{C})^K\/\nU(\\mathfrak{g}_\\mathbb{C})^K\\cap U(\\mathfrak{g}_\\mathbb{C})J_{\\pi^*}.\n\\] \n\nLet $S(\\mathfrak{a}_\\mathbb{C})$ denote the symmetric algebra \nof $\\mathfrak{a}_\\mathbb{C}=\\mathfrak{a}\\otimes_\\mathbb{R}\\mathbb{C}$ and \n$S(\\mathfrak{a}_\\mathbb{C})^W$ the set of the $W$-invariant elements in $S(\\mathfrak{a}_\\mathbb{C})$. \nThere exists an algebra homomorphism \n\\[\n\\gamma^\\pi:U(\\mathfrak{g}_\\mathbb{C})^K\\rightarrow S(\\mathfrak{a}_\\mathbb{C})^W\n\\]\nwith the kernel $U(\\mathfrak{g}_\\mathbb{C})^K\\cap U(\\mathfrak{g}_\\mathbb{C})\nJ_{\\pi^*}$ (cf. \\cite[Lemma~11.3.2, Lemma~11.3.3]{Wal}). \nNotice that the homomorphism $\\gamma^\\pi$ is independent of the choice of \n$\\varSigma^+$. \nThus we have the generalized Harish-Chandra isomorphism \n$\\gamma^\\pi:\\boldsymbol{D}^\\pi\\xrightarrow{\\smash[b]{\\lower 0.7ex\\hbox{$\\sim$}}} S(\\mathfrak{a}_\\mathbb{C})^W$. \nTherefore, any algebra homomorphism \nfrom $\\boldsymbol{D}^\\pi$ to $\\mathbb{C}$ is of the form $D\\mapsto \\gamma^\\pi(D)(\\lambda)\\,\\,(D\\in\\boldsymbol{D}^\\pi)$ \nfor some $\\lambda\\in \\lower0.8ex\\hbox{$W$}\\backslash \\mathfrak a_{\\mathbb C}^*$. \n\n\nFor  $\\lambda\\in \\lower0.8ex\\hbox{$W$}\\backslash \\mathfrak a_{\\mathbb C}^*$ \nthere exists a unique \nsmooth $\\pi$-spherical \nfunction $f=\\phi_\\lambda^\\pi$ satisfying $f(e)=\\text{id}_V$ and \n$Df=\\gamma^\\pi(D)(\\lambda)f\\,\\,\\,(D\\in\\boldsymbol{D}^\\pi)$ (cf. \\cite[Theorem~1.4]{OS}). \nWe call $\\phi_\\lambda^\\pi$ \nthe  \nelementary $\\pi$-spherical function. \nSince $\\boldsymbol{D}^\\pi$ contains an elliptic operator, $\\phi_\\lambda^\\pi$ is \nreal analytic. Moreover, \nit has an integral representation \n\\begin{equation}\n\\phi^\\pi_{\\lambda}(g)  =\n\\int_Ke^{(\\lambda-\\rho)(H(gk))}\\pi(k\\kappa(gk)^{-1})dk.  \\label{eq:eisenstein}\n\\end{equation}\nHere given $x\\in G$, define $\\kappa(x)\\in K$ and $H(x)\\in\\mathfrak{a}$ by $x\\in \\kappa(x)e^{H(x)}N$. \nNotice that $\\phi_\\lambda^\\pi$ is independent of the choice of \n$\\varSigma^+$, though the right hand side of \\eqref{eq:eisenstein} depends on \n$\\varSigma^+$ at first glance. \nMoreover, $\\phi_\\lambda^\\pi$ depends \nholomorphically on $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$. \n\nFormula \\eqref{eq:eisenstein} is a special case of the integral representations of elementary spherical functions (or more generally the \n{Eisenstein integral}s) given by Harish-Chandra (cf. \\cite[\\S 6.2.2, \\S 9.1.5]{War}, \\cite[(42)]{Camporesi2}, \\cite[(14.20)]{Kn0}).\n\n\\subsection{Harish-Chandra series}\n\nIn this subsection, we review the Harish-Chandra expansion of the \nelementary spherical function according to \\cite[\\S~9.1]{War}. \nWe assume $(\\pi,V)$ is a small $K$-type. \n\nLet $C^\\infty(G,\\pi,\\pi)$ denote the space of the \nsmooth $\\pi$-spherical functions. \nIf $f\\in C^\\infty(G,\\pi,\\pi)$ then $f|_A$ takes values in $\\text{End}_M V\\simeq \\mathbb{C}$.\nHence we regard \n$\\varUpsilon^\\pi(f):=f|_A \\circ\\exp$ as a scalar valued function on $\\mathfrak{a}$. \nLet $C^\\infty(\\mathfrak{a})^W$ denote the space of the $W$-invariant smooth functions on $\\mathfrak{a}$.\nThe restriction map \n$\\varUpsilon^\\pi$ gives an isomorphism $C^\\infty(G,\\pi,\\pi)\\xrightarrow{\\smash[b]{\\lower 0.7ex\\hbox{$\\sim$}}}\nC^\\infty(\\mathfrak{a})^W$ (\\cite[Theorem~1.5]{OS}). \n\nLet $\\mathscr R$ be the unital algebra of functions on $\\mathfrak a_{\\mathrm{reg}}$\ngenerated by $(1\\pm e^{\\alpha})^{-1}$ ($\\alpha\\in\\varSigma^+$).\nFor any $D \\in U(\\mathfrak g_{\\mathbb C})^K$\nthere exists a unique $W$-invariant \ndifferential operator $\\varDelta^\\pi(D) \\in \\mathscr R \\otimes S(\\mathfrak a_{\\mathbb C})$\nsuch that for any $f \\in C^\\infty(G,\\pi,\\pi)$\n\\[\n\\varUpsilon^\\pi(Df)\n=\\varDelta^\\pi(D)\\varUpsilon^\\pi(f)\n\\]\non $\\mathfrak a_{\\mathrm{reg}}$ (\\cite[Proposition 3.10]{OS}). \nWe call $\\varDelta^\\pi(D)$ the $\\pi$-radial part of $D$. The function \n$\\Phi=\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)$ satisfies differential equations\n\\begin{equation}\\label{eqn:derad}\n\\varDelta^\\pi(D)\\Phi=\\gamma^\\pi(D)(\\lambda)\\Phi\\quad \n(D\\in U(\\mathfrak{g}_\\mathbb{C})^K).\n\\end{equation}\n\nLet $\\mathbb{N}\\varSigma^+$ denote the set of $\\mu\\in\\mathfrak{a}^*$ of the form \n$\\mu=n_1\\alpha_1+\\cdots +n_r\\alpha_r\\,\\,(n_i\\in\\mathbb{N})$. \nFor $\\mu\\in\\mathbb{N}\\varSigma^+\\setminus\\{0\\}$, let \n$\\sigma_\\mu$ denote the hyperplane\n\\[\n\\sigma_\\mu=\\{\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*\\,|\\,\n\\langle 2\\lambda-\\mu,\\mu\\rangle=0\\}.\n\\]\nIf $\\lambda\\not\\in\\sigma_\\mu$ for any $\\mu\\in\\mathbb{N}\\varSigma^+\\setminus\\{0\\}$, \nthen there exists a unique convergent series solution\n\\begin{equation}\\label{eqn:hcs1}\n\\Phi_\\lambda(H)=e^{(\\lambda-\\rho)(H)}\\sum_{\\mu\\in\\mathbb{N}\\varSigma^+}\n\\Gamma_\\mu(\\lambda)e^{-\\mu(H)}\n\\quad (H\\in \\mathfrak{a}_+)\n\\end{equation}\nof \\eqref{eqn:derad} \nwith $\\Gamma_\\mu(\\lambda)\\in\\mathbb{C}$ and $\\Gamma_0(\\lambda)=1$. \nThis is a special case of \\cite[Theorem~9.1.4.1]{War}. \n\nBy using differential equations \\eqref{eqn:derad}, apparent singularities of $\\Phi_\\lambda(H)$ \nas a function of $\\lambda$ is removable unless $\\mu=n\\alpha$ for some $n\\in \\mathbb{Z}_{>0}$ and \n$\\alpha\\in\\varSigma^+$ (cf. \\cite[Corollary~6.3]{A}, see also \n\\cite[Lemma~6.5]{Op:lecture} and \\cite[Proposition~7.5]{Hec}). \nFor $\\mu=n\\alpha$, $\\lambda\\not\\in \\sigma_{\\mu}$ if and only if $\\langle\\lambda,\\alpha^\\vee\\rangle\n\\not=n$. Here $\\alpha^\\vee =2\\alpha\/\\langle\\alpha,\\alpha\\rangle$. \nThus $\\Phi_\\lambda$ is defined if $\\langle\\lambda,\\alpha^\\vee\\rangle\\not\\in \\mathbb{Z}_{>0}$ for \nall $\\alpha\\in\\varSigma^+$. \n\nIf \n$\\langle\\lambda,\\alpha^\\vee\\rangle\\not\\in \\mathbb{Z}$ for all $\\alpha\\in\\varSigma^+$, then \n$\\{\\Phi_{w\\lambda}\\,|\\,w\\in W\\}$ forms a basis of the solution space \nof \\eqref{eqn:derad} on $\\mathfrak{a}_+$. Thus  \n$\\varUpsilon^\\pi(\\phi_\\lambda^\\pi)$ is a linear combination of \n$\\Phi_{w\\lambda}\\,\\,(w\\in W)$. \nSince $\\phi_{w\\lambda}^\\pi=\\phi_\\lambda^\\pi$, there exists a constant \n$c^\\pi(\\lambda)$ such that\n\\begin{equation}\\label{eqn:hcs2}\n\\varUpsilon^\\pi(\\phi_\\lambda^\\pi)(H)\n=\\sum_{w\\in W}c^\\pi(w\\lambda)\\Phi_{w\\lambda}(H)\n\\quad (H\\in \\mathfrak{a}_+).\n\\end{equation}\n\n\\subsection{Harish-Chandra ${c}$-function}\n\nIn this subsection, \nwe review \nthe Harish-Chandra $c$-function. We refer to \\cite{Schiffman}, \n \\cite[\\S 9.1.6]{War}, \\cite[Chapter 8]{Wallach73}, \n and \\cite[\\S 5]{Sekiguchi} for details. \n\nLet $H\\in\\mathfrak{a}_+$ and  \n$\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$ satisfying $\\text{Re}\\,\\lambda\\in \\mathfrak{a}_+^*$. \nThe leading \ncoefficient $c^\\pi(\\lambda)$ \nof $\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)$ at infinity in $A_+=\\exp\\mathfrak{a}_+$ is \ngiven by the Harish-Chandra ${c}$-function \n(cf. \\cite[Theorem 9.1.6.1]{War}, \\cite[Theorem~14.7, (14.29)]{Kn0}):\n\\begin{align}\n& \\lim_{t\\to\\infty}e^{t(-\\lambda+\\rho)(H)}\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)(e^{tH})={c}^\\pi(\\lambda), \n\\label{eq:limcf1} \\\\\n& {c}^\\pi(\\lambda)=\\int_{\\bar{N}}e^{-(\\lambda+\\rho)(H(\\bar{n}))}\\pi(\\kappa(\\bar{n}))d\\bar{n}, \n\\label{eq:cfgroup}\n\\end{align}\nwhere the Haar measure on $\\bar{N}$ is normalized so that \n\\[\n\\int_{\\bar{N}}e^{-2\\rho(H(\\bar{n}))}d\\bar{n}=1.\n\\]\nThe integral (\\ref{eq:cfgroup}) \nconverges absolutely for $\\text{Re}\\,\\lambda\\in \\mathfrak{a}_+^*$ and extends to a meromorphic function \non $\\mathfrak{a}_\\mathbb{C}^*$. \nNotice that ${c}^\\pi(\\lambda)\\in \\text{End}_M V\\simeq \\mathbb{C}$. \n\nDefine \n\\[\n\\varSigma_0=\\{\\alpha\\in\\varSigma\\,|\\,\\tfrac12\\alpha\\not\\in\\varSigma\\}\n\\]\nand $\\varSigma_0^+=\\varSigma_0\\cap \\varSigma^+$. \nFor $\\alpha\\in\\varSigma_0^+$ let $\\mathfrak{g}_{(\\alpha)}$ denote the Lie subalgebra of $\\mathfrak{g}$ \ngenerated by $\\mathfrak{g}_\\alpha$ and $\\mathfrak{g}_{-\\alpha}$. \nPut $\\mathfrak{k}_\\alpha=\\mathfrak{k}\\cap \\mathfrak{g}_{(\\alpha)}$, $\\mathfrak{p}_\\alpha=\\mathfrak{p}\\cap \\mathfrak{g}_{(\\alpha)}$, \n $\\mathfrak{a}_\\alpha=\\mathfrak{a}\\cap \\mathfrak{g}_{(\\alpha)}$, $\\mathfrak{n}_\\alpha=\\mathfrak{g}_\\alpha+\\mathfrak{g}_{2\\alpha}$, and \n$\\bar{\\mathfrak{n}}_{\\alpha}=\\theta\\mathfrak{n}_\\alpha$. \nFor $\\alpha\\in \\varSigma_0^+$, let $G_\\alpha,\\,K_\\alpha,\\,A_\\alpha,\\,N_\\alpha$, and $\\bar{N}_{\\alpha}$ denote the analytic subgroups of $G$ \ncorresponding to $\\mathfrak{g}_{(\\alpha)},\\,\\mathfrak{k}_\\alpha,\\,\n\\mathfrak{a}_\\alpha,\\,\\mathfrak{n}_\\alpha$, and $\\bar{\\mathfrak{n}}_{\\alpha}$, respectively. \nWe have the Iwasawa decomposition $G_\\alpha=K_\\alpha A_\\alpha N_\\alpha$. Put \n$\\rho_\\alpha=\\frac12(\\boldsymbol{m}_\\alpha+2\\boldsymbol{m}_{2\\alpha})\\alpha$. \nLet $d\\bar{n}_\\alpha$ denote the Haar measure on $\\bar{N}_\\alpha$ \nnormalized so that \n\\[\n\\int_{\\bar{N}_\\alpha}e^{-2\\rho_\\alpha(H(\\bar{n}_\\alpha))}d\\bar{n}_\\alpha=1. \n\\]\n\nLet $w^*\\in W$ be the element such that $w^*(\\varSigma^+)=-\\varSigma^+$. \nLet $w^*=s_{i_m}\\cdots s_{i_2}s_{i_1}$ be a reduced expression, where $m$ denotes \nthe length of $w^*$ and $1\\leq i_k\\leq r\\,\\,(1\\leq k\\leq m)$. Put \n$\\beta_k=s_{i_1}\\cdots s_{i_{k-1}}\\alpha_{i_k}$. Then we have \n$\\varSigma_0^+=\\{\\beta_1,\\beta_2,\\dots,\\beta_m\\}$ (cf. \\cite[Ch. IV Corollary 6.11]{Hel2}). \nWe have the decomposition $\\bar{N}=\\bar{N}_{\\beta_1}\\cdots \\bar{N}_{\\beta_m}$,  \nthe product map being a diffeomorphism. \nMoreover, there exists a positive constant $c_0$ such that \n\\begin{equation}\\label{eqn:normalize}\nd\\bar{n}=c_0\\, d\\bar{n}_{\\beta_1}\\cdots d\\bar{n}_{\\beta_m}.\n\\end{equation}\n\nFor $\\alpha\\in\\varSigma_0^+$ define\n\\[\nc_\\alpha^\\pi(\\lambda)=\\int_{\\bar{N}_\\alpha}e^{-(\\lambda+\\rho_\\alpha)(H(\\bar{n}_\\alpha))}\n\\pi(\\kappa(\\bar{n}_\\alpha))d\\bar{n}_\\alpha.\n\\]\n\nWe have the following product formula (\\cite[Theorem 1.2]{Schiffman}, \n\\cite[\\S  8.11.6]{Wallach73}, \\cite[\\S 9.1.6]{War}, \\cite[Theorem 5.1]{Sekiguchi}).\n\n\\begin{thm}\\label{thm:prod}\n$c^\\pi(\\lambda)=c_0 \\,c_{\\beta_1}^\\pi\\!(\\lambda)\\cdots c_{\\beta_m}^\\pi\\!(\\lambda)$.\n\\end{thm}\n\nFor the case of the trivial $K$-type $\\pi=\\text{triv}$, \n$c_{\\beta_i}^\\text{triv}$ can be written\n explicitly by the classical Gamma function and we have \n the Gindikin and Karpelevi\\v{c} product formula for ${c}^\\text{triv}(\\lambda)$ \n (cf. \\cite{GK}, see also \\cite[Ch IV, \\S 6]{Hel2} and \\cite[\\S 9.1.7]{War}). \n Note that the constant $c_0$ in \n \\eqref{eqn:normalize} is determined explicitly by the \n Gindikin and Karpelevi\\v{c} formula for $c^{\\text{triv}}(\\lambda)$. \n \nThe ${c}$-function for a one-dimensional $K$-type $\\pi$ \nof a group $G$ of Hermitian type is also given \nexplicitly by the Gamma function (\\cite{Sc}, \\cite{S:eigen}). \n\nIn \\cite{OS} we give an explicit formula  \nof $c^\\pi(\\lambda)$ for each \nsimple $G$ and each small $K$-type $\\pi$, with one exception for $G$ of type $G_2$ \nand a certain small $K$-type $\\pi$. \nThe method we use is to relate the $\\pi$-elementary spherical \nfunction $\\phi_\\lambda^\\pi$ with \nthe Heckman-Opdam hypergeometric function, \ninstead of computing the integral (\\ref{eq:cfgroup}) by using Theorem~\\ref{thm:prod}. \nHeckman~\\cite[Chapter 5]{Hec} gives in this way an explicit formula of ${c}^\\pi(\\lambda)$ \nfor a one-dimensional $K$-type $\\pi$ when the group $G$ is  of Hermitian type. \n\nIn \\S~\\ref{subsec:g2} we give an explicit formula of $c^\\pi(\\lambda)$ \nfor $G$ of type $G_2$ and each small $K$-type $\\pi$ by using Theorem~\\ref{thm:prod}. \n\n\\subsection{$\\pi$-spherical transform}\n\nLet $dH$ denote the Euclidean measure on $\\mathfrak{a}$ with respect to the Killing form. \nDefine $\\delta_{G\/K}=\\prod_{\\alpha \\in \\varSigma^+}\n|2\\sinh \\alpha|^{\\boldsymbol m_\\alpha}$. \nWe normalize the Haar measure $dg$ on $G$ so that\n\\begin{equation*}\n\\int_G \\psi(g)dg=\\frac1{\\#W}\\int_{\\mathfrak a} \\psi(e^H)\\, \\delta_{G\/K}(H)dH\n\\end{equation*}\nfor any compactly supported continuous $K$-bi-invariant function $\\psi$ on $G$\n(cf.~\\cite[Ch.~I, Theorem 5.8]{Hel2}). \n\nLet $C^\\infty_c(G,\\pi,\\pi)$ be the subspace of $C^\\infty(G,\\pi,\\pi)$\nconsisting of the compactly supported smooth $\\pi$-spherical functions. \nThe $\\pi$-spherical transform of $f\\in C_c^\\infty(G,\\pi,\\pi)$ is the $\\text{End}_M V\\simeq \\mathbb{C}$-valued \nfunction ${f}^\\wedge$ on $\\mathfrak{a}_\\mathbb{C}^*$ defined by\n\\begin{equation}\n{f}^\\wedge(\\lambda)=\\int_G \\phi_\\lambda^\\pi(g^{-1})f(g)dg.\n\\end{equation}\nThe $\\pi$-spherical transform $f\\mapsto f^\\wedge(\\lambda)$ is a homomorphism from the commutative convolution \nalgebra $C_c^\\infty(G,\\pi,\\pi)$ to $\\mathbb{C}$ (cf. \\cite{Camporesi2}). \nIt \n is a special case of the Fourier transform given by  \nArthur~\\cite{A} (see also \\cite[\\S 3]{BS3}). %\nBy the identification $C_c^\\infty(G,\\pi,\\pi)\\simeq C_c^\\infty(\\mathfrak{a})^W$, \nthe \n$\\pi$-spherical transform $f^\\wedge$ of $f\\in C_c^\\infty(\\mathfrak{a})^W$ is given by  \n\\begin{equation}\n{f}^\\wedge(\\lambda)=\\frac{1}{\\# W}\\int_{\\mathfrak{a}}f(H)\n\\varUpsilon^\\pi(\\phi_{-\\lambda}^\\pi)(H)\\delta_{G\/K}(H)\\,dH. \n\\end{equation}\nWe normalize the Haar measure $d\\lambda$ on $\\sqrt{-1}\\mathfrak{a}^*$\n so that the Euclidean Fourier transform and its inversion are given by \n\\[\n\\tilde{f}(\\lambda)=\\int_{\\mathfrak{a}}f(H)e^{-\\lambda(H)}dH, \\qquad\nf(H)=\\int_{\\sqrt{-1}\\mathfrak{a}^*} \\tilde{f}(\\lambda)e^{\\lambda(H)}d\\lambda. \n\\]\n\nLet $\\eta_1$ be a point in $-\\overline{\\mathfrak{a}_+^*}$ such \nthat $c^\\pi(-\\lambda)^{-1}$ is a regular \nfunction of $\\lambda$ for $\\text{Re}\\,\\lambda\\in \\eta_1-\\overline{\\mathfrak{a}_+^*}$. \nThe existence of such $\\eta_1$ follows from an \nexplicit formula of the $c$-function for each small $K$-type, \nwhich is determined by \\cite{OS} and \\S\\ref{sec:g2} for $G_2$.\nIt also follows from a general result on the Harish-Chandra $c$-function \ndue to Cohn \\cite{Cohn}. \n\nLet $F=f^\\wedge$. \nFollowing \\cite[Chapter II, \\S 1]{A}, \ndefine the function $F^\\vee(H)$ on $\\mathfrak{a}_+$ by\n\\begin{equation}\\label{eqn:invft}\nF^\\vee(H)=\n\\int_{\\eta_1+\\sqrt{-1}\\mathfrak{a}^*} F(\\lambda)\\Phi\n_\\lambda(H)c^\\pi(-\\lambda)^{-1}d\\lambda.\n\\end{equation}\nThe integral \\eqref{eqn:invft} converges and is independent of $\\eta_1$ \n(cf. \\cite[Chapter II, \\S 1]{A}). \n$F^\\vee$ defined above coincides with that given by Arthur, \nbecause  $\\phi_\\lambda^\\pi$ is $W$-invariant in $\\lambda$ and the \nHarish-Chandra $\\mu$-function associate with a minimal parabolic subgroup \nin our case is given by \n$c^\\pi(\\lambda)^{-1}c^\\pi(-\\lambda)^{-1}$ (cf. \\cite[\\S~10.5]{Wal}). \nThe following theorem is a special case of \\cite[Chapter III, Theorem 3.2]{A}.\nIt is also a special case of \\cite[Theorem 1.1]{BS2}, since $G$ is a \nsemisimple \nsymmetric space for $G\\times G$ (cf. \\cite{BS3}). \n\n\n\n\\begin{thm}\\label{thm:inv}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$ we have\n\\[\nf(H)=({f}^\\wedge)^\\vee(H)\\quad (H\\in\\mathfrak{a}_+).\n\\]\n\\end{thm}\n\nIf $c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$, \nthen we can take $\\eta_1=0$ and by \\eqref{eqn:hcs2} and $W$-invariance of \n$c^\\pi(\\lambda)^{-1}c^\\pi(-\\lambda)^{-1}$ in \n$\\lambda\\in\\sqrt{-1}\\mathfrak{a}^*$, we have \n\\begin{equation}\\label{eqn:invcont}\nf(H)\n=\n\\frac{1}{\\# W}\\int_{\\sqrt{-1}\\mathfrak{a}^*}\nf^\\wedge (\\lambda)\\varUpsilon^\\pi(\\phi^\\pi_\\lambda)(H)\n|c^\\pi(\\lambda)|^{-2}d\\lambda\\quad (H\\in\\mathfrak{a}^*). \n\\end{equation}\n\nIn \\cite[Corollary 7.6]{OS} \nwe prove the formula \\eqref{eqn:invcont} by using the inversion formula \nof the hypergeometric Fourier transform due to Opdam~\\cite{Op:Cherednik},\nunder the assumption that $\\pi$ is a small $K$-type of\na real simple $G$ which is not in the following list:\n\n\\smallskip\n\\noindent\n(1) $\\mathfrak{g}=\\mathfrak{sp}(p,1),\\,\\pi=\\pi_n\\circ \\text{pr}_2$ \n($\\pi_n$ : $n$-dimensional irred. rep. of $\\text{Sp}(1))$,  \n\\\\\n(2) $\\mathfrak{g}=\\mathfrak{so}(2r,1)$,\\\\ \n\\phantom{(3)} $\\pi=\\pi_s^{\\pm}$ : \nirred. rep. of $\\text{Spin}(2r)$ with h.w. $(s\/2,\\cdots s\/2,\\pm s\/2)\\,\\,\n(s\\in \\mathbb{N})$,\\\\\n(3) $\\mathfrak{g}=\\mathfrak{so}(p,q)\\quad (p>q\\geq 3,\\,\\,\\text{$p$ : even, \n$q$ : odd)}$,\\\\\n\\phantom{(3)} $\\pi=\\sigma\\circ\\text{pr}_1\\,\\,(\\sigma$ : one of half spin representation of $\\text{Spin}(p)$),\\\\\n(4) $\\mathfrak{g}$\\,:\\,Hermitian type, $\\pi$ : one dimensional $K$-type, \n\\\\\n(5) $\\mathfrak{g}=\\mathfrak{g}_2$, $\\pi=\\pi_2$ (see \\S \\ref{sec:g2} for the definition).\n\n\\medskip\nThough the case (3) is not covered by \\cite[Corollary 7.6]{OS}, \nthe formula \\eqref{eqn:invcont} holds in this case, since \n$c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$ as we mention in \nthe final part of \\cite{OS}. \n\nIf the parameter of the small $K$-type is \n``large enough'' in the cases (1), (2), and (4), \nthen $c^\\pi(-\\lambda)^{-1}$ has singularities in \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$ and \nwe must take account of residues during the contour \nshift $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*\\to \\sqrt{-1}\\mathfrak{a}^*$. \nThe most continuous part of the spectrum is given by the right hand side \nof \\eqref{eqn:invcont}. In addition, there are spectra with low dimensional \nsupports. The residue calculus in the case (4) is done by \\cite{SPlancherel}. \nFor the cases (1) and (2), $\\dim\\mathfrak{a}=1$ and the residue calculus is \neasy to proceed. Also these cases are covered by the inversion formula \nof the Jacobi transform (cf. \\cite[Appendix 1]{FJ}, \\cite{Koornwinder}). \nSee also \\cite{vDP} and \\cite{Shyperbolic} for the case (1). \n\nWe will discuss the case (5) in the next section. \n\n\\section{The case of $G_2$}\n\\label{sec:g2}\n\n\\subsection{Notation and preliminary results}\\label{subsec:g2prem}\nLet $\\mathfrak{g}$ be the simple split real Lie algebra of type $G_2$ and $G$ the connected simply connected  \nLie group with the Lie algebra $\\mathfrak{g}$. \n$G$ is the double cover of the adjoint group of $\\mathfrak{g}$. \nLet $K$ be a maximal compact subgroup of $G$ and \n$\\mathfrak{k}$ the Lie algebra of $K$. Then \n$K\\simeq \\SU(2)\\times \\SU(2)$ and $\\mathfrak{k}\\simeq \\mathfrak{su}(2)\\oplus \\mathfrak{su}(2)$. \n\nLet $\\mathfrak{t}$ be a maximal abelian subalgebra of $\\mathfrak{k}$. \nThen $\\mathfrak{t}$ is a Cartan subalgebra \nof $\\mathfrak{g}$. Let $\\Delta$ and $\\Delta_K$ denote the root system for $(\\mathfrak{g}_\\mathbb{C},\\mathfrak{t}_\\mathbb{C})$ \nand $(\\mathfrak{k}_\\mathbb{C},\\mathfrak{t}_\\mathbb{C})$, respectively. \nThen $\\Delta$ is a root system of type $G_2$. \nWe choose a positive system $\\Delta^+\\subset \\Delta$\nso that its base contains a short compact root $\\beta_1$.\nThe other simple root, say $\\beta_2$, is a long noncompact root.\nIf we put $\\Delta_K^+=\\Delta_K\\cap \\Delta^+$ then\n\\begin{align*}\n& \\Delta^+=\\{\\beta_1,\\beta_2,\\beta_1+\\beta_2,2\\beta_1+\\beta_2,3\\beta_1+\\beta_2,3\\beta_1+2\\beta_2\\},\\\\\n& \\Delta_K^+=\\{\\beta_1,3\\beta_1+2\\beta_2\\}.\n\\end{align*}\nWe fix an inner product on $\\sqrt{-1}\\mathfrak{t}^*$ such that \n$(\\beta_1,\\beta_1)=2$. Then $(\\beta_2,\\beta_2)=6$ and $(\\beta_1,\\beta_2)=-3$.\nWe let $K=K_1\\times K_2$ with $K_i\\simeq \\SU(2)\\,\\,(i=1,2)$\nassuming that\n$\\beta_1$ (resp. $3\\beta_1+2\\beta_2$) is a \nroot for $((\\mathfrak{k}_1)_\\mathbb{C}, \n(\\mathfrak{t}\\cap\\mathfrak{k}_1)_\\mathbb{C})$ (resp. \n$((\\mathfrak{k}_2)_\\mathbb{C}, \n(\\mathfrak{t}\\cap\\mathfrak{k}_2)_\\mathbb{C})$).\nLet  \n$\\pr_i$ denote the projection of $K$ to $K_i\\,\\,(i=1,2)$. \n\nThe classification of the small $K$-types for $G$ is given as follows:\n\\begin{thm}[{\\cite[Theorem 1]{SWL}}]\\label{thm:G2}\nThe non-trivial small $K$-types are $\\pi_1:=\\sigma\\circ\\pr_1$ and  $\\pi_2:=\\sigma\\circ\\pr_2$. \nHere $\\sigma$ is the two-dimensional irreducible representation of $\\SU(2)$. \n\\end{thm}\n\nA discrete series representation of $G$ is an irreducible representation of $G$ realized as a closed \n$G$-invariant subspace of the \nleft regular representation on $L^2(G)$. \n\n\\begin{lem}\nLet $G$ be as above.\nThen no small $K$-type appears in any discrete series representation of $G$.\n\\end{lem}\n\\begin{proof}\n\n\nIf $\\pi$ is the trivial $K$-type or $\\pi=\\pi_1$, then \nit follows from the Plancherel formula for the \n$\\pi$-spherical functions (cf. \\cite{Hel2}, \\cite{OS}) \nthat there are no discrete series representations having \nthe $K$-type $\\pi$. \n \nNext let us discuss the case of $\\pi_2$.\nThe positive open chamber $(\\sqrt{-1}\\mathfrak{t}^*)^+$ for $\\Delta_K^+$\ncontains the following three open chambers for $\\Delta$:\n\\begin{align*}\n(\\sqrt{-1}\\mathfrak{t}^*)^+_1&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_2)>0\\},\\\\\n(\\sqrt{-1}\\mathfrak{t}^*)^+_2&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_2)<0\\text{ and }(\\lambda,\\beta_1+\\beta_2)>0\\},\\\\\n(\\sqrt{-1}\\mathfrak{t}^*)^+_3&:=\\{\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+\\,|\\, (\\lambda,\\beta_1+\\beta_2)<0\\}.\n\\end{align*}\nLet $\\Delta_i^+$ be the corresponding positive systems ($i=1,2,3$).\nNote that $\\Delta_1^+=\\Delta^+$. \nIf we put $\\delta_i=\\frac12\\sum_{\\beta\\in\\Delta^+_i}\\beta$ then\n\\[\n\\delta_1=5\\beta_1+3\\beta_2,\\quad\n\\delta_2=5\\beta_1+2\\beta_2,\\quad\n\\delta_3=4\\beta_1+\\beta_2.\n\\]\nOn the other hand,\n$\\delta_K:=\\frac12\\sum_{\\beta\\in\\Delta_K^+}\\beta=2\\beta_1+\\beta_2.$\nNow suppose $\\pi_2$ appears in a discrete series representation with Harish-Chandra parameter \n$\\lambda\\in \\sqrt{-1}\\mathfrak{t}^*$.\nWe may assume $\\lambda\\in (\\sqrt{-1}\\mathfrak{t}^*)^+_i$\nfor $i=1,2$, or $3$.\nSince the highest weight of $\\pi_2$ is $\\frac32\\beta_1+\\beta_2$,\nit follows from \\cite[Theorem~8.5]{AS} that \n\\[\n\\frac32\\beta_1+\\beta_2=\\lambda+\\delta_i-2\\delta_K+\\sum_{\\beta\\in\\Delta^+_i}\nn_\\beta \\beta\\quad \\text{for some }n_\\beta\\in\\mathbb{N}. \n\\]\nIf $i=1$ then this reduces to\n\\[\n\\lambda=\\left(\\frac12-c_1\\right)\\beta_1-c_2\\beta_2\\quad\\text{for some }c_1,\\,c_2\\in\\mathbb{N}.\n\\]\nSince $(\\lambda,\\beta_j)>0\\,\\,(j=1,2)$, we have $1-2c_1+3c_2>0$ and $-\\frac32+3c_1-6c_2>0$, \nwhich are impossible.\nIf $i=2$ or $3$ then we can also deduce a contradiction in a similar way.\n\\end{proof}\n\n\\subsection{Harish-Chandra ${c}$-function for $G_2$\n}\n\\label{subsec:g2}\n\nThe restricted root system $\\varSigma=\\varSigma(\\mathfrak{g},\\mathfrak{a})$ is \na root system of type $G_2$. \nFor all $\\alpha\\in\\varSigma$ we have \n$\\mathfrak{g}_{(\\alpha)}\\simeq \\mathfrak{sl}(2,\\mathbb{R})$, since \n$\\boldsymbol{m}_\\alpha=\\dim\\mathfrak{g}_\\alpha=1$ and $2\\alpha\\not\\in\\varSigma$. \n\nWe recall the ${c}$-function for $\\mathfrak{g}=\\mathfrak{sl}(2,\\mathbb{R})$. \nPut \n\\[\nh=\\begin{pmatrix} 1 & 0 \\\\ 0 & -1\\end{pmatrix},\\quad \ne=\\begin{pmatrix} 0 & 1 \\\\ 0 & 0\\end{pmatrix},\\quad \nf=\\begin{pmatrix} 0 & 0 \\\\ 1 & 0\\end{pmatrix}. \n\\]\nThen $\\{h,e,f\\}$ is a basis of $\\mathfrak{sl}(2,\\mathbb{R})$ and also \nforms an $\\mathfrak{sl}_2$-triple. \nWe put $\\mathfrak{a}=\\mathbb{R}h$ and $\\mu=\\lambda(h)$ for $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$. \nBy \\cite[Remark 7.3]{Sc}, \nthe ${c}$-function  for $\\mathfrak{sl}(2,\\mathbb{R})$ with a one-dimensional \n$\\mathfrak{k}$-type $\\pi$ of the weight $\\nu\\in\\mathbb{Q}$ for $\\sqrt{-1}(e-f)$ is given by \n\\begin{equation}\\label{eq:cfsl2}\n{c}^\\pi(\\lambda)=\\frac{2^{1-\\mu}\\varGamma(\\mu)}{\\varGamma(\\frac12(\\mu+1+\\nu))\n\\varGamma(\\frac12(\\mu+1-\\nu))}.\n\\end{equation}\n\nNow we come back to the case of $G_2$. For $\\alpha\\in\\varSigma^+$ \nchoose $e_\\alpha\\in \\mathfrak{g}_{\\alpha}$ so \nthat $\\{\\alpha^\\vee,\\,e_\\alpha,\\,-\\theta e_\\alpha\\}$ is an $\\mathfrak{sl}_2$-triple. \nPut $t_\\alpha:=e_\\alpha+\\theta e_\\alpha\\in \\sqrt{-1}\\mathfrak{k}_\\alpha$.  \nIf $\\alpha\\in \\varSigma^+$ is a long root, then the \npossible weights of $t_\\alpha$ for \n$\\pi_i\\,(i=1,2)$ are $\\pm \\frac12$ by \\cite[Lemma 4.2]{SWL}. \nIf $\\alpha\\in \\varSigma^+$ is a short root, then the possible weights of $t_\\alpha$  \nfor $\\pi_1$ (resp. $\\pi_2$) are $\\pm\\frac12$ \n(resp. $\\pm\\frac32$) by \\cite[Lemma 4.3]{SWL}. \nSince \\eqref{eq:cfsl2} remains unchanged if we replace $\\nu$ by $-\\nu$, \n$c_\\alpha^{\\pi_i}(\\lambda)$ is a scalar operator for each $\\alpha\\in \\varSigma$ and $i=1,\\,2$. \n\nLet $\\varSigma_\\text{long}^+$ and $\\varSigma_\\text{short}^+$ denote \nthe sets of the long and short positive roots, respectively. \nDefine $\\lambda_\\alpha=\\langle\\lambda,\\alpha^\\vee\\rangle$ \nfor $\\lambda\\in\\mathfrak{a}_\\mathbb{C}^*$ and \n$\\alpha\\in\\varSigma^+$. \nIt follows from Theorem~\\ref{thm:prod}, \n\\eqref{eq:cfsl2}, and the proof of \\cite[Ch. IV, Theorem~6.13]{Hel2} \nthat\n\\begin{align*}\n{c}^{\\text{triv}}(\\lambda) & \n=\nc_0\\!\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{2^{1-\\lambda_\\alpha}\n\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)^2}\n=\nc_0\\!\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{\\pi^{-\\frac12}\n\\varGamma(\\frac12\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_1}(\\lambda) & =c_0\\!\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{2^{1-\\lambda_\\alpha}\\varGamma(\\lambda_\\alpha)}{\\varGamma(\\frac12(\\lambda_\\alpha+\\frac32))\n\\varGamma(\\frac12(\\lambda_\\alpha+\\frac12))} \n  =c_0\\!\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{2^{\\frac12}\\pi^{-\\frac12}\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_2}(\\lambda) & \n=c_0\\!\\!\n\\prod_{\\alpha\\in\\varSigma_\\text{long}^+} \\!\\!\n\\frac{2^{1-\\lambda_\\alpha}\\varGamma(\\lambda_\\alpha)}{\\varGamma(\\frac12(\\lambda_\\alpha+\\frac32))\n\\varGamma(\\frac12(\\lambda_\\alpha+\\frac12))} \n\\prod_{\\beta\\in\\varSigma_\\text{short}^+} \\!\\!\n\\frac{2^{1-\\lambda_\\beta}\\varGamma(\\lambda_\\beta)}{\\varGamma(\\frac12(\\lambda_\\beta+\\frac52))\n\\varGamma(\\frac12(\\lambda_\\beta-\\frac12))} \\\\\n&=c_0\\!\\!\n\\prod_{\\alpha\\in\\varSigma_\\text{long}^+} \\!\\!\n\\frac{2^{\\frac12}\\pi^{-\\frac12}\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}\n\\prod_{\\beta\\in\\varSigma_\\text{short}^+} \\!\\!\n \\frac{2^{\\frac12}\\pi^{-\\frac12}\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\end{align*}\n\nThe value of the constant $c_0$ is determined by $c^{\\text{triv}}(\\rho)=1$. \nWe have $c_0=2\\pi^2$ by direct computation. \nThus we have the following \ntheorem. \n\n\\begin{thm}\\label{thm:cfg2}\n\\begin{align*}\n{c}^{\\text{\\rm triv}}(\\lambda) & \n=\n\\frac{2}{\\pi}\n\\prod_{\\alpha\\in\\varSigma^+}\\frac{\n\\varGamma(\\frac12\\lambda_\\alpha)}\n{\\varGamma(\\frac12\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_1}(\\lambda) & \n  =\\frac{16}{\\pi}\\prod_{\\alpha\\in\\varSigma^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\\\\n{c}^{\\pi_2}(\\lambda) \n&=\\frac{16}{\\pi}\n\\prod_{\\alpha\\in\\varSigma_\\text{\\rm long}^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}\n\\prod_{\\beta\\in\\varSigma_\\text{\\rm short}^+} \n \\frac{\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\end{align*}\n\\end{thm}\n\nThe formula for $c^{\\text{triv}}(\\lambda)$ in Theorem~\\ref{thm:cfg2} is a special case of the \nGindikin-Karpelevi\\v{c} formula (cf. \\cite{GK}, \\cite[Ch.~IV, Theorem~6.13]{Hel2}). \nThe formula for $c^{\\pi_1}(\\lambda)$ \nis given in \\cite{OS} by use of a different method. \nThe formula for $c^{\\pi_2}(\\lambda)$ is new. \n\n\\subsection{$\\pi$-spherical transform}\n\nAn inversion formula for the $\\pi$-spherical transform is given by \nTheorem~\\ref{thm:inv}. We must shift the contour of integration from \n$\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ to $\\sqrt{-1}\\mathfrak{a}^*$ and get a \nglobally defined function on $\\mathfrak{a}$. \nWe refer to \\cite[Ch. II, Ch. III]{A} for the general residue scheme \n(see also \\cite{Oshima81a,BS,BS2,BS2000,BS3,BS4}). \n\nFor $\\pi=\\text{triv}$ and \n$\\pi_1$, $c^\\pi(-\\lambda)^{-1}$ is a regular function of $\\lambda$ for \n$\\text{Re}\\,\\lambda\\in -\\overline{\\mathfrak{a}_+^*}$, hence\n the inversion formula is given by \\eqref{eqn:invcont} for these \nsmall $K$-types (cf. \\cite[Ch~IV, Theorem~7.5]{Hel2}, \\cite[Corollary~7.6]{OS}). \n\nFor $\\pi=\\pi_2$, there appear singularities during the contour shift and \nwe must take account of residues. \nThe function $c^{\\pi_2}(-\\lambda)^{-1}$ for $\\text{Re}\\,\\lambda\\in\n\\mathfrak{a}_+^*$ \nhas singularities along \nlines $\\lambda_\\beta=-\\frac12\\,\\,(\\beta\\in\\varSigma^+_{\\text{short}})$. \nFigure~1 illustrates singular lines \n$\\lambda_{\\alpha_1}=-\\frac12,\\,\\lambda_{\\alpha_1+\\alpha_2}=-\\frac12$, and \n$\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12$ \n (dashed) for $c^{\\pi_2}(-\\lambda)^{-1}$ \nand $-\\overline{\\mathfrak{a}_+^*}$ \n(shaded region). \nHere $\\alpha_1$ and $\\alpha_2$ are the simple roots of $\\varSigma^+$ ($||\\alpha_1||<||\\alpha_2||$).\nThese singular lines divide $-\\overline{\\mathfrak{a}_+^*}$ into \nthe following four regions (indicated in Figure 1):\n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=0 105 0 105,width=9.2cm]{shift2b.pdf}\n\\caption{singular lines}\n\\end{center}\n\\end{figure}\n\\begin{align*}\n& \\text{\\emph{I}} \\,:\\, \\lambda_{\\alpha_1}<-\\frac12,\\quad \\lambda_{\\alpha_2}\\leq 0, \\\\\n& \\text{\\emph{II}} \\,:\\, -\\frac12<\\lambda_{\\alpha_1}\\leq 0,\\quad \\lambda_{\\alpha_1+\\alpha_2}<-\\frac12, \\\\\n& \\text{\\emph{III}} \\,:\\, -\\frac12<\\lambda_{\\alpha_1+\\alpha_2},\\quad \n\\lambda_{2\\alpha_1+\\alpha_2}<-\\frac12,\\quad \\lambda_{\\alpha_2}\\leq 0, \\\\\n& \\text{\\emph{IV}} \\,:\\,\\lambda_{2\\alpha_1+\\alpha_2}>-\\frac12,\\quad \\lambda_{\\alpha_1}\\leq 0,\\quad \n\\lambda_{\\alpha_2}\\leq 0. \n\\end{align*}\n\nFirst $\\eta_1\\in-\\mathfrak{a}_+^*$ in \\eqref{eqn:invft} \nlies in the region \\emph{I}. \nWe choose $\\eta_2,\\,\\eta_3$, and $\\eta_4$ \nin the regions \\emph{II,\\,III}, and \\emph{IV}, respectively. We may take $\\eta_4=0$. \nWe shift the contour of \nintegration from $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ to $\\eta_2+\\sqrt{-1}\\mathfrak{a}^*$ \nand so on, and finally to $\\eta_4+\\sqrt{-1}\\mathfrak{a}^*=\n\\sqrt{-1}\\mathfrak{a}^*$, picking up residues. \nDefine\n\\[\nF^\\vee_i(H)\n=\\int_{\\eta_i+\\sqrt{-1}\\mathfrak{a}^*}f^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}d\\lambda\\quad (1\\leq i\\leq 4).\n\\]\n\nWe regard \n$(\\lambda_{\\alpha_1},\\lambda_{3\\alpha_1+2\\alpha_2})\\in\\mathbb{C}^2$ as \na coordinate on $\\mathfrak{a}_\\mathbb{C}^*$. \nDefine \n\\begin{equation}\nc_1=\\frac{||\\alpha_1||\\,||3\\alpha_1+2\\alpha_2||}{4}.\n\\end{equation}\nFor $\\eta\\in\\mathfrak{a}^*$, the normalized measure $d\\lambda$ on $\\eta+\\sqrt{-1}\\mathfrak{a}^*$ is given by\n\\[\nd\\lambda=\\frac{c_1}{(2\\pi\\sqrt{-1})^2}d\\lambda_{\\alpha_1}d\\lambda_{3\\alpha_1+2\\alpha_2}.\n\\]\n\nFirst we change the contour of integration of $F^\\vee(H)=F^\\vee_1(H)$ from $\\eta_1+\\sqrt{-1}\\mathfrak{a}^*$ \nto $\\eta_2+\\sqrt{-1}\\mathfrak{a}^*$ with respect to the integration in the variable \n$\\lambda_{\\alpha_1}$. \nBy the explicit formula of $c^{\\pi_2}(\\lambda)$ in Theorem~\\ref{thm:cfg2}, \n$f^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}$ has a possible simple pole during the change of \nthe contour coming from the factor \n$(-\\lambda_{\\alpha_1}-\\frac12)^{-1}$ of $c^{\\pi_2}_{\\alpha_1}(-\\lambda)^{-1}$. \nThus the difference $F^\\vee_1(H)-F^\\vee_2(H)$ is\n\\[\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\mu+\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+2\\alpha_2}\n\\]\nfor some $\\mu$ with $\\mu_{\\alpha_1}=-\\frac12,\\,\\mu_{\\alpha_2}<0$. \nNext we move $\\mu_{3\\alpha_1+2\\alpha_2}$ to $0$ along the \nline $\\lambda_{\\alpha_1}=-\\frac12$. \nSingularities coming from $(-\\lambda_\\beta-\\frac12)^{-1}\n\\,\\,(\\beta=\\alpha_1+\\alpha_2,\\,2\\alpha_1+\\alpha_2)$ are on the walls and \nthey are canceled by $\\varGamma(-\\lambda_\\alpha)^{-1}\\,\\,\n(\\alpha=\\alpha_2,\\,\\alpha_1+\\alpha_2, \\text{respectively})$. Thus \nthe integrand \n$\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)$ is regular for $\\lambda_{3\\alpha_1+2\\alpha_2}\\leq 0$.  \nHence we have \n\\[\nF^\\vee_1(H)-F^\\vee_2(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+2\\alpha_2}.\n\\]\nSimilarly, we have\n\\begin{align*}\n& F^\\vee_2(H)-F^\\vee_3(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{3\\alpha_1+\\alpha_2}, \\\\\n& F^\\vee_3(H)-F^\\vee_4(H)=\n-\\frac{c_1}{2\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{\\lambda}(H)\nc^{\\pi_2}(-\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2}.\n\\end{align*}\nBy summing up and changing variables, we have\n\\begin{align*}\nF_1^\\vee & (H)-F_4^\\vee (H) \\\\ & =-\\frac{c_1}{2\\pi\\sqrt{-1}}\n\\!\\!\n\\sum_{w\\in\\{e,s_1,s_2 s_1\\}}\\!\n\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{w\\lambda}(H)\nc^{\\pi_2}(-w\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2}. \n\\end{align*}\nLet $W^{2\\alpha_1+\\alpha_2}=\\{e,s_1,s_2,s_1s_2,s_2s_1,s_2 s_1 s_2\\}$. \nBy changing variables, we have\n\\begin{align}\nF_1^\\vee & (H)  -F_4^\\vee (H) \\label{eqn:invtemp}\n \\\\ & =-\\frac{c_1}{4\\pi\\sqrt{-1}}\\!\\!\\!\n\\sum_{w\\in W^{2\\alpha_1+\\alpha_2}}\n\\int_{\\sqrt{-1}\\mathbb{R}}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\!\\left(\nf^\\wedge (\\lambda)\\Phi_{w\\lambda}(H)\nc^{\\pi_2}(-w\\lambda)^{-1}\\right)\\!\nd\\lambda_{\\alpha_2} . \\notag\n\\end{align}\n\nSince $W^{2\\alpha_1+\\alpha_2}=\\{w\\in W\\,|\\,w(2\\alpha_1+\\alpha_2)\\in\\varSigma^+\\}$, $c^{\\pi_2}(w\\lambda)|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}=0$ for \nany $w\\in W\\setminus W^{2\\alpha_1+\\alpha_2}$ by Theorem~\\ref{thm:cfg2}. \nNotice that the Harish-Chandra expansion \\eqref{eqn:hcs2} is valid for \n$\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12,\\,\n\\lambda_{\\alpha_2}\\in\\sqrt{-1}\\mathbb{R}\\setminus\\{0\\}$. \nHence \n\\begin{equation}\\label{eqn:hcstemp}\n\\varUpsilon^{\\pi_2}(\\phi_\\lambda^{\\pi_2})|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n=\\sum_{w\\in W^{2\\alpha_1+\\alpha_2}}c^{\\pi_2}(w\\lambda)\\Phi_{w\\lambda}|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n\\end{equation}\nfor $\\lambda_{\\alpha_2}\\in\\sqrt{-1}\\mathbb{R}\\setminus\\{0\\}$. \n\nWe write the $c$-function $c^{\\pi_2}(\\lambda)$ in Theorem~\\ref{thm:cfg2} as \n\\[\nc^{\\pi_2}(\\lambda)=\\frac{16}{\\pi}c_l(\\lambda)c_s(\\lambda)\n\\]\nwith\n\\[\nc_l(\\lambda)=\n\\prod_{\\alpha\\in\\varSigma_\\text{\\rm long}^+} \n\\frac{\\varGamma(\\lambda_\\alpha)}\n{\\varGamma(\\lambda_\\alpha+\\frac12)}, \n\\qquad \nc_s(\\lambda)=\n\\prod_{\\beta\\in\\varSigma_\\text{\\rm short}^+} \n \\frac{\\left(\\lambda_\\beta-\\frac12\\right)\\varGamma(\\lambda_\\beta)}\n{\\varGamma(\\lambda_\\beta+\\frac32)} .\n\\]  \nNotice that the functions  \n$(c_l(\\lambda)c_l(-\\lambda))^{-1}$ and $(c_s(\\lambda)c_s(-\\lambda))^{-1}$ \nare $W$-invariant.\n\\begin{lem}\nWe have\n\\begin{equation}\\label{eqn:restemp0}\n(c_l(\\lambda)c_l(-\\lambda))^{-1}|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n=\\frac{(4\\lambda_{\\alpha_2}^3-\\lambda_{\\alpha_2})\\sin\\pi\\lambda_{\\alpha_2}}\n{16\\cos\\pi\\lambda_{\\alpha_2}}\n\\end{equation}\nand \n\\begin{equation}\\label{eqn:restemp}\nc_s(w\\lambda)^{-1}|_\n{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}(c_s(-w\\lambda)^{-1})\n=\\frac{36\\lambda_{\\alpha_2}^2-1}{32\\pi}\n\\end{equation}\nfor any $w\\in W^{2\\alpha_1+\\alpha_2}$.\n\\end{lem}\n\\begin{proof}\nWe show only \\eqref{eqn:restemp} because \\eqref{eqn:restemp0} can be deduced in a similar way.\nSince the left hand side of \\eqref{eqn:restemp} is the residue of  \nthe $(c_s(\\lambda)c_s(-\\lambda))^{-1}$ \nas a function of $\\lambda_{2\\alpha_1+\\alpha_2}$ \nat $\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12$,\nit suffices to show \\eqref{eqn:restemp} for $w=1$.\nBy elementary calculation we have\n\\[\n \\frac{\\left(z-\\frac12\\right)\\varGamma(z)}\n{\\varGamma(z+\\frac32)}\n\\cdot\n \\frac{\\left(-z-\\frac12\\right)\\varGamma(-z)}\n{\\varGamma(-z+\\frac32)}\n=-\\frac{\\cos\\pi z}{z\\sin\\pi z}\\quad(z\\in{\\mathbb C}).\n\\]\nUsing\n$\\lambda_{\\alpha_1}=\\frac12\\lambda_{2\\alpha_1+\\alpha_2}-\\frac32\\lambda_{\\alpha_2}$\nand\n$\\lambda_{\\alpha_1+\\alpha_2}=\\frac12\\lambda_{2\\alpha_1+\\alpha_2}+\\frac32\\lambda_{\\alpha_2}$\nwe calculate\n\\begin{align*}\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}&((c_s(\\lambda)c_s(-\\lambda))^{-1})\\\\\n&=\\biggl.\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\biggr|_{z=-\\frac14-\\frac32\\lambda_{\\alpha_2}}\n\\biggl.\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\biggr|_{z=-\\frac14+\\frac32\\lambda_{\\alpha_2}}\n\\Res_{z=-\\frac12}\\biggl(-\\frac{z\\sin\\pi z}{\\cos\\pi z}\\biggr)\\\\\n&=\n\\frac{1-36\\lambda_{\\alpha_2}^2}{16}\n\\cdot\\frac{\\sin\\pi\\Bigl(-\\frac14-\\frac32\\lambda_{\\alpha_2}\\Bigr)\n\\sin\\pi\\Bigl(-\\frac14+\\frac32\\lambda_{\\alpha_2}\\Bigr)}%\n{\\cos\\pi\\Bigl(-\\frac14-\\frac32\\lambda_{\\alpha_2}\\Bigr)\n\\cos\\pi\\Bigl(-\\frac14+\\frac32\\lambda_{\\alpha_2}\\Bigr)}\n\\cdot\\biggl(-\\frac1{2\\pi}\\biggr).\n\\end{align*}\nIn the final expression the second factor reduces to $1$.\n\\end{proof}\n\nThus we have the following inversion formula for $\\pi_2$-spherical transform. \n\\begin{thm}\\label{thm:main}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$, we have\n\\begin{align*}\nf(H)  =  \\frac{1}{12} & \\int_{\\sqrt{-1}\\mathfrak{a}^*}\n  f^\\wedge(\\lambda)\\varUpsilon^{\\pi_2}\n  (\\phi^{\\pi_2}_\\lambda)(H)|c^{\\pi_2}(\\lambda)|^{-2}d\\lambda \\\\\n  &\n  -\\frac{c_1}{4\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n(f^\\wedge(\\lambda)\\varUpsilon^{\\pi_2}(\\phi^{\\pi_2}_\\lambda(H)))\n|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\\,\np(\\lambda_{\\alpha_2})d\\lambda_{\\alpha_2}\n\\end{align*}\nfor $H\\in \\mathfrak{a}$, \nwhere \n\\begin{align*}\np(\\lambda_{\\alpha_2}) & =\n\\Res_{\n\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}\n(c^{\\pi_2}(\\lambda)^{-1}c^{\\pi_2}(-\\lambda)^{-1})\n=\\frac{\\pi(36\\lambda_{\\alpha_2}^2-1)\n(4\\lambda_{\\alpha_2}^3-\\lambda_{\\alpha_2})\\sin\\pi\\lambda_{\\alpha_2}}\n {2^{17}\\cos\\pi\\lambda_{\\alpha_2}}\\\\\n& =-2^{-17}\\pi\\Bigl(36\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)^2+1\\Bigr)\n\\Bigl(4\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)^2+1\\Bigr)\n\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)\n\\tanh \\pi\\left(\\tfrac{\\lambda_{\\alpha_2}}{\\sqrt{-1}}\\right)\n.\n\\end{align*}\n\\end{thm}\n\nThe Plancherel formula follows from Theorem~\\ref{thm:main} by a \nstandard argument as in the case \nof $\\pi=\\text{triv}$ (cf. the proof of \\cite[Theorem~6.4.2]{GV} and \n\\cite[Ch~IV Theorem~7.5]{Hel2}). \n\n\\begin{cor}\\label{cor:main}\nFor $f\\in C_c^\\infty(\\mathfrak{a})^W$, we have\n\\begin{align*}\n\\frac{1}{12}\n\\int_\\mathfrak{a}|f(H)|^2\\delta_{G\/K} & (H)dH  =  \n\\frac{1}{12}  \\int_{\\sqrt{-1}\\mathfrak{a}^*}\n  |f^\\wedge(\\lambda)|^2|c^{\\pi_2}(\\lambda)|^{-2}d\\lambda \\\\\n - & \\frac{c_1}{4\\pi\\sqrt{-1}}\\int_{\\sqrt{-1}\\mathbb{R}}\n|f^\\wedge(\\lambda)|_{\\lambda_{2\\alpha_1+\\alpha_2}=-\\frac12}|^2\\,\np(\\lambda_{\\alpha_2})d\\lambda_{\\alpha_2}\n\\end{align*}\n\\end{cor}\n\nAs we see in \\S~\\ref{subsec:g2prem}, no discrete spectrum appears in \nthe inversion formula and the Plancherel formula. \nIn addition to the most continuous spectrum, there \nis a contribution of a principal series representation associated with a \nmaximal parabolic subgroup whose Levi part corresponds to a short restricted \nroot. \n\n\\section*{Acknowledgement}\nThe first author was supported by JSPS KAKENHI Grant Number 18K03346. \nThe authors \n thank anonymous reviewers for careful reading our manuscript and for giving useful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n \nOnline learning is a general model for sequential prediction. Within\nthat framework, the setting of prediction with expert advice has\nreceived widespread attention \\citep{LittlestoneWarmuth1994,\n  CesaBianchiLugosi2006,CesaBianchiMansourStoltz2007}.  In this\nsetting, the algorithm maintains a distribution over a set of experts,\nor selects an expert from an implicitly maintained distribution.  At\neach round, the loss assigned to each expert is revealed. The\nalgorithm incurs the expected loss over the experts and then updates\nits distribution on the set of experts.  Its objective is to minimize\nits expected regret, that is the difference between its cumulative\nloss and that of the best expert in hindsight.\n\nHowever, this benchmark is only significant when the best expert in\nhindsight is expected to perform well. When that is not the case, then\nthe learner may still play poorly. As an example, it may be that no\nsingle baseball team has performed well over all seasons in the past\nfew years. Instead, different teams may have dominated over different\ntime periods. This has led to a definition of regret against the best\nsequence of experts with $k$ shifts in the seminal work of\n\\cite{HerbsterWarmuth1998} on \\emph{tracking the best expert}.  The\nauthors showed that there exists an efficient online learning\nalgorithm for this setting with favorable regret guarantees.\n\nThis work has subsequently been improved to account for broader expert\nclasses \\citep{GyorgyLinderLugosi2012}, to deal with unknown\nparameters \\citep{MonteleoniJaakkola2003}, and has been further\ngeneralized \\citep{CesaBianchiGaillardLugosiStoltz2012, Vovk1999}.\nAnother approach for handling dynamic environments has consisted of\ndesigning algorithms that guarantee small regret over any subinterval\nduring the course of play. This notion, coined as \\emph{adaptive\n  regret} by \\cite{HazanSeshadhri2009}, has been subsequently\nstrengthened and generalized \\citep{DanielyGonenShalevShwartz2015,\n  AdamskiyKoolenChernovVovk2012}.  Remarkably, it was shown by\n\\cite{AdamskiyKoolenChernovVovk2012} that the algorithm designed by\n\\cite{HerbsterWarmuth1998} is also optimal for adaptive regret.\n\\cite{KoolenDeRooij2013} described a Bayesian framework for online\nlearning where the learner samples from a distribution of expert\nsequences and predicts according to the prediction of that expert\nsequence. They showed how the algorithms designed for $k$-shifting\nregret, e.g.\\ \\citep{HerbsterWarmuth1998, MonteleoniJaakkola2003}, can\nbe interpreted as specific priors in this formulation.  There has also\nbeen work deriving guarantees in the bandit setting when the losses\nare stochastic \\citep{BesbesGurZeevi2014, WeiHongLu2016}.\n\n\\begin{figure*}[t]\n\\vskip -.15in\n\\centering\n\\begin{tabular}{ccc}\n\\includegraphics[scale=0.4]{kshift} &\n  {\\includegraphics[scale=0.4]{wshift}} &\n  \\includegraphics[scale=0.4]{hierarchy} \\\\ \n  (i) &\n  (ii) &\n  (iii) \\\\ \n\\end{tabular}\n\\caption{WFAs representing sequences of experts in\n  $\\Sigma = \\set{a, b, c}$. (i) $\\sC_\\text{$k$-shift}$ with\n  $k = 2$ shifts, all weights are equal to one and not indicated; (ii)\n  $\\sC_\\text{weighted-shift}$ with $\\alpha, \\beta, \\gamma \\in [0, 1]$;\n  (iii) $\\sC_\\text{hierarchy}$ a hierarchical family of expert \n  sequences: the learner must select expert $a$ from the start, can only shift \n  onto $b$ once, and can only shift onto $c$ twice. }\n\\label{fig:kshift}\n\\vskip -.15in\n\\end{figure*}\n\nThe general problem of online convex optimization in the presence of\nnon-stationary environments has also been studied by many\nresearchers. One perspective has been the design of algorithms that\nmaintain a guarantee against sequences that do not vary too much\n\\citep{MokhtariShahrampourJadbabaieRibeiro2016,\n  ShahrampourJadbabaie2016, JadbabaieRakhlinShahrampourSridharan2015,\n  BesbesGurZeevi2015}.  Another assumes that the learner has access to\na dynamical model that is able to capture the benchmark sequence\n\\citep{HallWillett2013}. \\cite{GyorgySzepesvari2016} reinterpreted the\nframework of \\cite{HallWillett2013} to recover and extend the results\nof \\cite{HerbsterWarmuth1998}.\n\nIn this paper, we generalize the framework just described to the case\nwhere the learner's cumulative loss is compared to that of sequences\naccepted by a weighted finite automaton (WFA). This strictly\ngeneralizes the notion of $k$-shifting regret, since $k$-shifting\nsequences can be represented by an automaton (see\nFigure~\\ref{fig:kshift}), and further extends it to a notion of\n\\emph{weighted regret} which takes into consideration the sequence\nweights. Our framework covers a very rich class of competitor classes,\nincluding WFAs learned from past observations.  \n\nOur contributions are mainly algorithmic but also include\nseveral theoretical results and guarantees.  We first describe an\nefficient online algorithm using automata operations that achieves\nboth favorable weighted regret and unweighted regret\n(Section~\\ref{sec:AWM}).  Next, we present and analyze more efficient\nsolutions based on an approximation of the WFA representing the set of competitor\nsequences (Section~\\ref{sec:approx}), including a specific analysis of\napproximations via $n$-gram models both when minimizing the\n$\\infty$-R\\'enyi divergence and the relative entropy.  Finally, we\nextend the results above to the sleeping expert setting\n\\citep{FreundSchapireSingerWarmuth1997}, where the learner may not\nhave access to advice from every expert at each round\n(Section~\\ref{sec:sleep}).\n\n\\ignore{\nIn this paper, we significantly generalize the framework just\ndescribed and consider prediction with expert advice in a setting\nwhere the learner's cumulative loss is compared against that of\nsequences represented by an \\emph{arbitrary weighted family of\n  sequences}. We model this family using a weighted finite automaton\n(WFA). This strictly generalizes the notion of $k$-shifting regret and\nextends it to the notion of regret against a WFA.\n\nMeasuring regret against an automaton is both natural and flexible. In\nfact, it may often be sensible to \\emph{learn} the set of competitor\nsequences using data before competing against it. For instance, the\ncompetitor automaton could be a language model trained over best\nsequences of baseball teams in the past. Moreover, the competitor\nautomaton could be learned and reset incrementally. After each epoch,\nwe could choose to learn a new competitor model and seek to perform\nwell against that.\n\nWe show that not only it is possible to achieve favorable regret\nagainst a WFA but that there exist \\emph{computationally efficient}\nalgorithms to achieve that.  We give a series of algorithms for this\nproblem. Our first algorithm (Section~\\ref{sec:autalg}) is an\nautomata-based algorithm extending weighted-majority and using\nautomata operations such as composition and shortest-distance; its\ncomputational cost is exponentially better than that of a na\\\"{i}ve\nmethod.\n\nWe further present efficient algorithms based on a compact\napproximation of the competitor automaton\n(Section~\\ref{sec:min-Renyi}), in particular efficient $n$-gram models\nobtained by minimizing the R\\'enyi divergence, and present an\nextensive study of the approximation properties of such models. We\nalso show how existing algorithms for minimizing $k$-shifting regret\ncan be recovered by learning a Maximum-Likelihood bigram language\nmodel over the $k$-shifting competitor automaton. To the best of our\nknowledge, this is the first instance of recovering the algorithms of\n\\cite{HerbsterWarmuth1998} by way of solely focusing on minimizing the\n$k$-shifting regret.  Since approximating the competitor automaton is\nsubject to a trade-off between computational efficiency and\napproximation accuracy, we also design a model selection algorithm\nadapted to this problem.\n\nWe further improve that algorithm by using the notion of failure\ntransitions ($\\phi$-transitions) for a more compact and therefore more\nefficient automata representation.  Here, we design a new algorithm\n(Appendix~\\ref{app:phiaut}) that can convert any weighted finite\nautomaton into a weighted finite automaton with $\\phi$-transitions\n($\\phi$-WFA).  We then extend the classical composition and\nshortest-distance algorithms for WFAs to the setting of\n$\\phi$-WFAs. The shortest-distance algorithm is designed by extending\nthe probability semiring structure of the $\\phi$-WFA to that of a\nring. We show that if the number of consecutive $\\phi$-transitions is\nnot too large, then these algorithms have a computational complexity\nthat is comparable to those for standard WFAs. At the same time, our\nconversion algorithm can dramatically reduce the size of a WFA.\n\nFinally, we extend the results above to the sleeping expert setting\n\\citep{FreundSchapireSingerWarmuth1997}, where the learner may not\nhave access to advice from every expert at each round\n(Section~\\ref{sec:sleep}).  \\ignore{Finally, we extend the ideas for\nprediction with expert advice to online convex optimization and a\ngeneral mirror descent setting (Section~\\ref{sec:oco}). Here, we\ndescribe a related framework that parallels the previous discussion\nand also recovers existing algorithms for $k$-shifting regret.}\n}\n\n\\section{Learning setup}\n\\label{sec:setup}\n\nWe consider the setting of prediction with expert advice over\n$T \\in \\Nset$ rounds. Let $\\Sigma = \\set{a_1, \\ldots, a_N}$ denote a set\nof $N$ experts.  At each round $t \\in [T]$, an algorithm $\\cA$ specifies\na probability distribution $\\sfp_t$ over $\\Sigma$, samples an expert\n$i_t$ from $\\sfp_t$, receives the vector of losses of all experts\n$\\bfl_t \\in [0, 1]^N$, and incurs the specific loss $l_t[i_t]$.  A\ncommonly adopted goal for the algorithm is to minimize its static\n(expected) regret $\\Reg_T(\\cA, \\Sigma)$, that is the difference\nbetween its cumulative expected loss and that of the best expert in\nhindsight:\n\\begin{align}\n\\label{eq:staticregret}\n\\Reg_T(\\cA, \\Sigma) = \n\\max_{x \\in \\Sigma} \\sum_{t = 1}^T \\sfp_t \\cdot \\bfl_t - \\sum_{t = 1}^T l_t[x].\n\\end{align}\nHere, we will consider an alternative benchmark, typically more\ndemanding, where the cumulative loss of the algorithm is compared\nagainst the loss of the best sequence of experts\n$\\bx \\in \\Sigma^T$ among those accepted by a weighted\nfinite automaton (WFA) $\\sC$ over the semiring\n$(\\Rset_+ \\cup \\set{+\\infty}, +, \\times, 0, 1)$.\\footnote{Thus, the\n  weights in $\\sC$ are non-negative; the weight of a path is\n  obtained by multiplying the transition weights along that path and\n  the weight assigned to a sequence is obtained by summing the weights\n  of all accepting paths labeled with that sequence.} The sequences\n$\\bx$ accepted by $\\sC$ are those which are assigned a positive\nvalue by $\\sC$, $\\sC(\\bx) > 0$, which we will assume to be\nnon-empty. We will denote by $K \\geq 1$ the cardinality of that set.\n\nWe will take into account the probability distribution $\\sfq$ defined\nby the weights assigned by $\\sC$ to sequences of length $T$:\n$\\sfq(\\bx) = \\frac{\\sC(\\bx)}{\\sum_{\\bx \\in \\Sigma^T}\n  \\sC(\\bx)}$. This leads to the following definition of\n\\emph{weighted regret} at time $T$ given a WFA $\\sC$:\n\\begin{align}\n\\label{eq:weightedregret}\n& \\Reg_T(\\cA, \\sC) \\\\\\nonumber\n& = \\max_{\\substack{\\bx \\in \\Sigma^T\\\\ \\sC(\\bx) > 0}} \\set[\\Bigg]{ \\sum_{t = 1}^T \\sfp_t\n\\cdot \\bfl_t -  \\sum_{t = 1}^T l_t[\\bx[t]] \n  + \\log [\\sfq(\\bx) K ] },\n\\end{align}\nwhere $\\bx[t]$ denotes the $t$th symbol of $\\bx$.  The presence of the\nfactor $K$ only affects the regret definition by a constant additive\nterm $\\log K$ and is only intended to make the last term\nvanish when the probability distribution $\\sfq$ is uniform, i.e.\n$\\sfq(\\bx) = \\frac{1}{K}$ for all $\\bx$.  The last term in the\nweighted regret definition can be interpreted as follows: for a given\nvalue of an expert sequence loss $\\sum_{t = 1}^T l_t[\\bx[t]]$, the\nregret is larger for sequences $\\bx$ with a larger probability\n$\\sfq(\\bx)$.  Thus, with this definition of regret, the learning\nalgorithm is pressed to achieve a small cumulative loss compared \nto expert sequences with small loss and high\nprobability. Notice that when $\\sC$ accepts only constant sequences,\nthat is sequences $\\bx$ with $\\bx[1] = \\ldots = \\bx[T]$ and assigns\nthe same weight to them, then the notion of weighted regret coincides\nwith that of static regret (Formula~\\ref{eq:staticregret}).\n\nWe also define the \\emph{unweighted regret} $\\Reg_T^0(\\cA, \\sC)$ of\nalgorithm $\\cA$ at time $T$ given the WFA $\\sC$ as:\n\\begin{align}\n\\label{eq:unweightedregret}\n \\Reg_T^0(\\cA, \\sC) \n= \\max_{\\substack{\\bx \\in \\Sigma^T\\\\ \\sC(\\bx) > 0}} \\set[\\Bigg]{\n    \\sum_{t = 1}^T \\sfp_t \\cdot \\bfl_t - \\sum_{t = 1}^T l_t[\\bx[t]] }.\n\\end{align}\nThe weights of the WFA $\\sC$ play no role in this notion of regret.\nWhen $\\sC$ has uniform weights, then the unweighted regret and\nweighted regret coincide.\n\nAs an example, the sequences of experts with $k$ shifts studied by\n\\cite{HerbsterWarmuth1998} can be represented by the WFA\n$\\sC_\\text{$k$-shift}$ of\nFigure~\\ref{fig:kshift}(i). Figure~\\ref{fig:kshift}(ii) shows an\nalternative weighted model of shifting experts, and\nFigure~\\ref{fig:kshift}(iii) shows a hierarchical family of expert\nsequences.\n\n\\section{Automata Weighted-Majority algorithm}\n\\label{sec:AWM}\n\nIn this section, we describe a simple algorithm, \\emph{Automata\n  Weighted-Majority} (\\AWM), that can be viewed as an enhancement of\nthe weighted-majority algorithm \\citep{LittlestoneWarmuth1994} to the\nsetting of experts paths represented by a WFA. \\footnote{This\n  algorithm is in fact closer to the EXP4 algorithm\n  \\citep{AuerCesaBianchiFreundSchapire2002}.  However, EXP4 is\n  designed for the bandit setting, so we use the weighted-majority\n  naming convention.} We will show that it benefits from favorable\nweighted and unweighted regret guarantees.\n\nAs with standard weighted-majority, \\AWM\\ maintains a distribution\n$\\sfq_t$ over the set of expert sequences $\\bx \\in \\Sigma^T$ accepted\nby $\\sC$ at any time $t$ and admits a learning parameter $\\eta >\n0$. The initial distribution $\\sfq_1$ is defined in terms of the\ndistribution $\\sfq$ induced by $\\sC$ over $\\Sigma^T$, and\n$\\sfq_{t + 1}$ is defined from $\\sfq_t$ via an exponential update:\nfor all $\\bx \\in \\Sigma^T, t \\geq 1$,\n\\begin{align}\n  & \\sfq_1[\\bx] = \\frac{\\sfq[\\bx]^\\eta}{\\sum_{\\bx \\in \\Sigma^T}\n  \\sfq[\\bx]^\\eta}, \\nonumber \\\\\n  & \\sfq_{t + 1}[\\bx] = \\frac{e^{-\\eta \\, l_t[\\bx[t]]}\\sfq_t[\\bx]}{\\sum_{\\bx\n    \\in \\Sigma^T} e^{-\\eta \\, l_t[\\bx[t]]}\\sfq_t[\\bx]},\n\\end{align}\nwhere we denote by $\\bx[t] \\in \\Sigma$ the $t$th symbol in $\\bx$.\n$\\sfq_t$ induces a distribution $\\sfp_t$ over the expert set $\\Sigma$\ndefined for all $a \\in \\Sigma$ by\n\\begin{equation}\n\\sfp_t[a] = \\frac{\\sum_{\\bx \\in \\Sigma^T} \\sfq_t[\\bx] 1_{\\bx[t] =\n    a}}{\\sum_{a \\in \\Sigma} \\sum_{\\bx \\in \\Sigma^T} \\sfq_t[\\bx] 1_{\\bx[t] = a}}.\n\\end{equation}\nThus, $\\sfp_t[a]$ is obtained by summing up the $\\sfq_t$-weights of\nall sequences with the $t$th symbol equal to $a$ and normalization.\nThe distributions $\\sfp_t$ define the \\AWM\\ algorithm. Note that the\nalgorithm cannot be viewed as weighted-majority with $\\sfq$-priors on\nexpert sequences as $\\sfq_1$ is defined in terms of $\\sfq^\\eta$.\n\nThe following regret guarantees hold for \\AWM.\n\n\\begin{theorem}\n\\label{th:awm}\nLet $\\sfq$ denote the probability distribution over expert sequences\nof length $T$ defined by $\\sC$ and let $K$ denote the cardinality of\nits support. Then, the following upper bound holds for the weighted\nregret of \\AWM:\n\\begin{align*}\n  \\Reg_T(\\AWM, \\sC) \n  & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\bigg[ K^\\eta \\sum_{\\bx\n    \\in \\Sigma^T} \\sfq[\\bx]^\\eta \\bigg] \\\\\n    &\\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K. \n\\end{align*}\nFurthermore, when $K \\geq 2$, for any $T > 0$, there exists\n$\\eta^* > 0$, decreasing as a function of $T$, such that:\n\\begin{align*}\n    \\Reg_T(\\AWM, \\sC) \n    & \\leq \\sqrt{\\frac{T H_{\\eta^*}(\\sfq)}{2}} - H_{\\eta^*}(\\sfq) + \\log\n    K,\n\\end{align*}\nwhere\n$H_\\eta(\\sfq) = \\frac{1}{1 - \\eta} \\log\\left(\\sum_{\\bx \\in \\Sigma^T}\n  \\sfq[\\bx]^\\eta \\right)$ is the $\\eta$-R\\'{e}nyi entropy of $\\sfq$.  The\nunweighted regret of \\AWM\\ can be upper-bounded as follows:\n\\begin{align*}\n  \\Reg_T^0(\\AWM, \\sC) \n  & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K.\n\\end{align*}\n\n\\end{theorem}\n\nThe proof is an extension of the standard proof for the\nweighted-majority algorithm and is given in Appendix~\\ref{app:awm}.\nThe bound in terms of the R\\'enyi entropy shows that the regret\nguarantee can be substantially more favorable than standard bounds of\nthe form $O(\\sqrt{T \\log K})$, depending on the properties of the\ndistribution $\\sfq$. First, since the $\\eta$-R\\'enyi entropy is\nnon-increasing in $\\eta$ \\citep{VanErvenHarremos2014}, we have\n$H_{\\eta^*}(\\sfq) \\leq H_0(\\sfq) = \\log(|\\supp(\\sfq)|) \\leq \\log\nK$. Second, if the distribution $\\sfq$ is concentrated on a subset\n$\\Delta$ with a small cardinality, $|\\Delta| \\ll K$, that is\n$\\sum_{\\bx \\not \\in \\Delta} \\sfq[\\bx]^{\\eta^*} < \\e (1-\\eta^*)\n\\sum_{\\bx \\in \\Delta} \\sfq[\\bx]^{\\eta^*}$ for some $\\e > 0$ and for\n$\\eta^* < 1$, then, by Jensen's inequality, the following upper bound\nholds:\n\\begin{align*}\n  H_\\eta^*(\\sfq)\n\\ignore{\n  & \\leq \\frac{1}{1 - \\eta^*} \\log \\bigg( \\sum_{\\bx \\in \\Delta}\n    \\sfq[\\bx]^{\\eta^*\\!} + \\e (1 - \\eta^*) \\sum_{\\bx \\in \\Delta}\n  \\sfq[\\bx] \\bigg) \\\\\n}\n  & \\leq \\frac{1}{1 - \\eta^*} \\log \\bigg(\\sum_{\\bx \\in \\Delta}\n    \\sfq[\\bx]^{\\eta^*\\!}\\bigg) + \\e \\\\ \n    &\\leq \\frac{1}{1 - \\eta^*} \\log \\bigg(|\\Delta| \\bigg(\\frac{1}{|\\Delta|}\\sum_{\\bx \\in \\Delta}\n    \\sfq[\\bx]\\bigg)^{\\eta^*\\!}\\bigg) + \\e \\\\ \n   \n    &\\leq \\log(|\\Delta|) + \\e. \n\\end{align*}\n\n{\\bf Efficient algorithm}. We now present an efficient computation\nof the distributions $\\sfp_t$.  Algorithm~\\ref{alg:awm} gives the\npseudocode of our algorithm.  We will assume throughout that $\\sC$ is\ndeterministic, that is it admits a single initial state and no two\ntransitions leaving the same state share the same label.  We can\nefficiently compute a WFA accepting the set of sequences of length $T$\naccepted by $\\sC$ by using the standard intersection algorithm for\nWFAs (see Appendix~\\ref{app:intersection} for more detail on this\nalgorithm). Let $\\sS_T$ be a deterministic WFA accepting the\nset of sequences of length $T$ and assigning weight one to each (see\nFigure~\\ref{fig:S_T}). Then, the intersection of $\\sC$ and $\\sS_T$ is\na WFA denoted by $\\sC \\cap \\sS_T$ which, by definition, assigns to\neach sequence $\\bx \\in \\Sigma^T$ the weight\n\\begin{equation}\n  (\\sC \\cap \\sS_T) (\\bx) = \\sC(\\bx) \\times \\sS_T(\\bx) = \\sC(\\bx),\n\\end{equation}\nand assigns weight zero to all other sequences. Furthermore, the WFA\n$\\sB = (\\sC \\cap \\sS_T)$ returned by the intersection algorithm is\ndeterministic since both $\\sC$ and $\\sS_T$ are deterministic.  Next,\nwe replace each transition weight of $\\sB$ by its $\\eta$-power. Since\n$\\sB$ is deterministic, this results in a WFA that we denote by\n$\\sB^\\eta$ and that associates to each sequence $\\bx$ the weight\n$\\sC[\\bx]^\\eta$. Normalizing $\\sB^\\eta$ results in a WFA $\\sA$\nassigning weight\n$\\sA[\\bx] = \\frac{\\sB[\\bx]^\\eta}{\\sum_x \\sB[\\bx]^\\eta} = \\sfq_1[x]$ to\nany $\\bx \\in \\Sigma^T$.  This normalization can be achieved in time\nthat is linear in the size of the WFA $\\sB^\\eta$ using the\n\\textsc{Weight-Pushing} algorithm \\citep{Mohri1997bis,Mohri2009}. For\ncompleteness, we describe this algorithm in\nAppendix~\\ref{app:weightpush}. Note that since $\\sB^\\eta$ is acyclic,\nits size is in $\\cO(|E_\\sA|)$.\\footnote{The \\textsc{Weight-Pushing}\n  algorithm has been used in many other contexts to make a directed\n  weighted graph stochastic. This includes network normalization in\n  speech recognition \\citep{MohriRiley2001}, and online learning with\n  large expert sets\n  \\citep{TakimotoWarmuth2003,CortesKuznetsovMohriWarmuth2015}, where\n  the resulting stochastic graph enables efficient sampling.  The\n  problem setting, algorithms and objectives in the last two\n  references are completely distinct from ours. In particular, (a) in\n  those, each path of the graph represents a single expert, while in\n  our case each path is a sequence of experts; (b) in those,\n  weight-pushing is applied at every round, while in our case it is\n  used once at the start of the algorithm; (c) the regret is with\n  respect to a static expert, while in our case it is with respect to\n  a WFA of expert sequences.}  We will denote by $\\sA$ the resulting\nWFA.\n\nFor any state $u$ of $\\sA$, we will denote by $\\bbeta[u]$ the sum of\nthe weights of all paths from $u$ to a final state. The vector\n$\\bbeta$ can be computed in time that is linear in the number of states and\ntransitions of $\\sA$ using a simple \\emph{single-source\n  shortest-distance} algorithm in the semiring\n$(\\Rset_+ \\cup \\set{+\\infty}, +, \\times, 0, 1)$ \\citep{Mohri2009}, or\nthe forward-backward algorithm. We call this subroutine \\textsc{BwdDist} in \nthe pseudocode.\n\n\\begin{algorithm2e}[t]\n  \\TitleOfAlgo{\\AWM($\\sC$, $\\eta$)}\n  $\\sB \\gets \\sC \\cap \\sS_T$ \\\\\n  $\\sA \\gets \\textsc{Weight-Pushing}(\\sB^\\eta)$ \\\\\n  $\\bbeta \\gets \\textsc{BwdDist}(\\sA)$ \\\\\n  $\\balpha \\gets 0$; $\\balpha[I_\\sA] \\gets 1$ \\\\ \n  \\ForEach{$e \\in E_{\\sA}^{0 \\to 1}$}{\n      $\\sfp_1[\\lab[e]] \\gets \\weight[e]$.\n    }\n \\For{$t \\gets 1$ \\KwTo $T$}{\n    $i_t \\gets $\\textsc{Sample}($\\sfp_t$); \\textsc{Play}($i_t$); \\textsc{Receive}($\\bfl_t$)\\\\\n    $Z \\gets 0$; $\\bw \\gets 0$\\\\ \n    \\ForEach{$e \\in E_{\\sA}^{t \\to t + 1}$}{\n      $\\weight[e] \\gets \\weight[e] \\, e^{-\\eta l_t[\\lab[e]]}$ \\\\\n      $\\bw[\\lab[e]] \\gets \\bw[\\lab[e]] + \\balpha[\\src[e]] \\, \\weight[e] \\, \\bbeta[\\dest[e]]$\\\\\n      $Z \\gets Z + \\bw[\\lab[e]]$\\\\\n      $\\balpha[\\dest[e]] \\gets \\balpha[\\dest[e]] + \\balpha[\\src[e]] \\,\n      \\weight[e] $\n    }\n    $\\sfp_{t + 1} \\gets \\frac{\\bw}{Z}$ \\\\ \n  }\n\\caption{\\textsc{AutomataWeightedMajority}(AWM).} \n\\label{alg:awm}\n\\end{algorithm2e}\n\n\\begin{figure}[t]\n\\vskip -.15in\n\\centering\n\\includegraphics[scale=0.55]{S_T}\n\\caption{WFA $\\sS_T$, for $\\Sigma = \\set{a, b, c}$ and $T = 3$.}\n\\label{fig:S_T}\n\\vskip .1in\n\\centering\n\\includegraphics[scale=0.65]{awm}\n\\caption{Illustration of algorithm \\AWM.}\n\\label{fig:awm}\n\\vskip -.15in\n\\end{figure}\n \nWe will denote by $Q_t$ the set of states in $\\sA$ that can be reached\nby sequences of length $t$ and by $E_{\\sA}^{t \\to t + 1}$ the set of\ntransitions from a state in $Q_t$ to a state in $Q_{t + 1}$.  For each\ntransition $e$, let $\\src[e]$ denote its source state, $\\dest[e]$ its\ndestination state, $\\lab[e] \\in \\Sigma$ its label, and\n$\\weight[e] \\geq 0$ its weight.  Since $\\sA$ is normalized, the expert\nprobabilities $\\sfp_1[a]$ for $a \\in \\Sigma$ can be read off the\ntransitions leaving the initial state: $\\sfp_1[a]$ is the weight of\nthe transition in $E_{\\sA}^{0 \\to 1}$ labeled with $a$.\n\nLet $\\balpha_t[u]$ denote the \\emph{forward weights}, that is the sum\nof the weights of all paths from the initial state to state $u$ just\nbefore the $t$th round.  At round $t$, the weight of each transition\n$e$ in $E_{\\sA}^{t \\to t + 1}$ is multiplied by\n$e^{-\\eta l_t[\\lab[e]]}$.  This results in new forward weights\n$\\balpha_{t + 1}[u]$ at the end of the $t$-th iteration.  $\\balpha_{t + 1}$\ncan be straightforwardly derived from $\\balpha_t$ since for\n$u \\in Q_{t + 1}$, $\\balpha_{t + 1}[u]$ is given by\n$\\balpha_{t + 1}[u] = \\sum_{e\\colon \\dest[e] = u} \\balpha_t[\\src[e]]\n\\weight[e]$. \n\nObserve that for any $t \\in [T]$ and $\\bx$, $\\sfq_t[\\bx]$ can be\nwritten as follows by unwrapping its recursive update definition:\n$$\\sfq_t[\\bx] = \\frac{e^{-\\eta \\sum_{s = 1}^{t - 1} l_s[\\bx[s]]}\n  \\sfq_1[\\bx]}{\\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^{t - 1}\n    l_s[\\bx[s]]} \\sfq_1[\\bx]}.$$\n    In view of that, for any\n$a \\in \\Sigma$, $\\sfp_{t + 1}[a]$ can be written as follows:\n\\begin{align*}\n\\sfp_{t + 1}[a] \n& = \\frac{\\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^t\n    l_s[\\bx[s]]} \\sfq_1[\\bx] 1_{\\bx[t] =\n    a}}{\\sum_{a \\in \\Sigma} \\sum_{\\bx \\in \\Sigma^T} e^{-\\eta \\sum_{s = 1}^t\n    l_s[\\bx[s]]} \\sfq_1[\\bx] 1_{\\bx[t] =\n    a}}.\n\\end{align*}\nSince the WFA $\\sA$ is deterministic, for any $\\bx$ accepted by $\\sA$\nthere is a unique accepting path $\\pi$ in $\\sA$ labeled with\n$\\bx$. The numerator of the expression of $\\sfp_{t + 1}[a]$ is then\nthe sum of the weights of all paths in $\\sA$ with the $t$th symbol $a$\nat the end of $t$th iteration. This can be expressed as the sum over\nall transitions $e$ in $E_{\\sA}^{t \\to t + 1}$ with label $a$ of the\ntotal \\emph{flow} through $e$, that is the sum of the weights of all\naccepting path going through $e$:\n$\\balpha_t[\\src[e]] \\, \\weight[e] \\, \\bbeta[\\dest[e]]$ (see\nFigure~\\ref{fig:awm}).  This is precisely the formula \ndetermining $\\sfp_{t + 1}$ in the pseudocode, where $Z$ is the\nnormalization factor.\n\nThe \\AWM\\ algorithm is closely related to the Expert Hidden Markov\nModel of \\cite{KoolenDeRooij2013} given for the log loss. It can be\nviewed as a generalization of that algorithm to arbitrary loss\nfunctions. A key difference between our setup and the perspective\nadopted by \\cite{KoolenDeRooij2013} is that they assume a Bayesian\nsetting where a prior distribution over expert sequences is given and\nmust be used. We assume the existence of a competitor automaton\n$\\sC$, but do not necessarily need to sample from it for making\npredictions. This will be crucial in the next section, where we use a\ndifferent WFA than $\\sC$ to improve computational efficiency while\npreserving regret performance. Also,  \nthe prior distribution in \\citep{KoolenDeRooij2013} would be over\n$\\sC_T$ (for a large $T$) and not $\\sC$.\n\nThe computational complexity of \\AWM\\ at each round $t$ is\n$\\cO\\big(|E_{\\sA}^{t \\to t + 1}|\\big)$, that is the time to update the\nweights of the transitions in $E_{\\sA}^{t \\to t + 1}$ and to\nincrementally compute $\\balpha$ for states reached by paths of length\n$t + 1$. The total computational cost of the algorithm is thus\n$\\cO\\big (\\sum_{t = 1}^T |E_{\\sA}^{t \\to t + 1}| \\big) =\n\\cO(|E_\\sA|)$, where $E_\\sA$ is the set of transitions of\n$\\sA$.\\footnote{By the discussion above and\n  Appendix~\\ref{app:intersection}, the total complexity of the\n  intersection and weight-pushing operations is also in\n  $\\cO(|E_\\sA|)$, so that they do not add any additional\n  cost. Moreover, these two operations need only be carried out once\n  and can be performed offline.}  Note that $\\sA$ and $\\sC \\cap \\sS_T$\nadmit the same topology, thus the total complexity of the algorithm\ndepends on the number of transitions of the intersection WFA\n$\\sC \\cap \\sS_T$, which is at most $|\\sC| NT$. This can be\nsubstantially more favorable than a na\\\"{i}ve algorithm, whose\nworst-case complexity is exponential in $T$.\n\nWhen the number of transitions of the intersection WFA\n$\\sC \\cap \\sS_T$ is not too large compared to the number of experts\n$N$, the \\AWM\\ algorithm is quite efficient.  However, it is natural\nto ask whether one can design efficient algorithms even if the number\nof transitions $E_{\\sA}^{t \\to t + 1}$ to process per round is large\n(which may be the case even for a \\emph{minimized} WFA\n$\\sC \\cap \\sS_T$ \\citep{Mohri2009}).\n\n\n\nWe will give two sets\nof solutions to derive a more efficient algorithm, which can be\ncombined for further efficiency. In the next section, we\ndiscuss a solution that consists of using an\napproximate WFA with a smaller number of transitions.\nIn Appendix~\\ref{app:phiaut}, we show\nthat the notion of \\emph{failure transition}, originally used in the\ndesign of string-matching algorithms and recently employed for \nparameter estimation in backoff $n$-gram language models\n\\citep{RoarkAllauzenRiley2013}, can be used to derive a more compact representation\nof the WFA $\\sC \\cap \\sS_T$, thereby resulting in a significantly\nmore efficient online learning algorithm that still admits compelling regret\nguarantees.\n\n\\section{Approximation algorithms}\n\\label{sec:approx}\n\nIn this section, we present approximation algorithms for the problem\nof online learning against a weighted sequence of experts represented\nby a WFA $\\sC$.  Rather than using the intersection WFA\n$\\sC_T = \\sC \\cap \\sS_T$, we will assume that \\AWM\\ is run with an\napproximate WFA $\\h \\sC_T$. The main motivation for doing so is that\nthe algorithm can be substantially more efficient if $\\h \\sC_T$ admits\nsignificantly fewer transitions than $\\sC_T$. Of course, this comes at\nthe price of a somewhat weaker regret guarantee that we now analyze in\ndetail.\n\\ignore{In Appendix~\\ref{app:timeindepapprox}, we will also discuss\nhow to perform approximations directly on the WFA $\\sC$.}\n\n\\subsection{Effect of WFA approximation}\n\\label{sec:WFAapprox}\n\nWe first analyze the effect of automata approximation on the regret of \\AWM.\nAs in the previous section, we denote by $\\sfq$ the distribution\ndefined by $\\sC_T$ over sequences of length $T$. We will similarly\ndenote by $\\h \\sfq$ the distribution defined by $\\h \\sC_T$ over the\nsame set. The effect of the WFA approximation on the regret can be\nnaturally expressed in terms of the $\\infty$-R\\'{e}nyi divergence\n$D_\\infty(\\sfq \\| \\h \\sfq)$ between the distributions $\\sfq$ and\n$\\h \\sfq$, which is defined by\n$D_\\infty(\\sfq \\| \\h \\sfq) = \\sup_{\\bx \\in \\Sigma^T} \\log [\n\\sfq(\\bx)\/\\h \\sfq(\\bx) ]$\\ignore{ \\citep{Renyi1961}}.\n\n\\begin{theorem}\n\\label{th:WFAapprox}\nThe weighted regret of the \\AWM\\ algorithm with respect to the WFA\n$\\sC$ when run with $\\h \\sC_T$ instead of $\\sC_T$ can be\nupper bounded as follows:\n\\begin{align*}\n  \\Reg_T(\\cA, \\sC) \n  & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\Big[K^\\eta \\sum_{\\bx} \\h\n  \\sfq[\\bx]^\\eta \\Big] + D_\\infty(\\sfq \\| \\h \\sfq) \\\\ \n  & \\leq \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log K + D_\\infty(\\sfq \\| \\h \\sfq). \n\\end{align*}\nIts unweighted regret can be upper bounded as follows:\n\\begin{align*}\n  \\Reg_T^0(\\cA, \\sC) &\\leq \\max_{\\sC(\\bx) > 0} \\frac{\\eta T}{8} + \\frac{1}{\\eta} \\log \\bigg[\n    \\frac{1}{\\sfq[\\bx]}\\bigg] + \\frac{1}{\\eta}\nD_\\infty(\\sfq \\| \\h \\sfq).\n\\end{align*}\n\\end{theorem}\nThe proof is given in Appendix~\\ref{app:WFAapprox}.\nTheorem~\\ref{th:WFAapprox} shows that the extra cost of using an\napproximate WFA $\\h \\sC_T$ instead of $\\sC_T$ is\n$D_\\infty(\\sfq \\| \\h \\sfq)$ for the weighted regret and similarly\n$\\frac{1}{\\eta} D_\\infty(\\sfq \\| \\h \\sfq)$ for the unweighted regret.\nThe bound is tight since the best sequence in hindsight in the regret\ndefinition may also be the one maximizing the log-ratio.\n\nThe theorem suggests a general algorithm for selecting an approximate\nWFA $\\h \\sC$ out of a family $\\cC$ of WFAs with a relatively small\nnumber of transitions. This consists of choosing $\\h \\sC$ to\nminimize the R\\'enyi divergence as defined by the following program:\n\\begin{equation}\n\\label{eq:autapprox}\n\\min_{\\h \\sC \\in \\cC} D_\\infty( \\sfq \\| \\h \\sfq),\n\\end{equation}\nwhere $\\h q$ is the distribution induced by $\\h \\sC$ over $\\Sigma^T$\n(the one obtained by computing $\\h \\sC_T = \\h \\sC \\cap \\sS_T$ and\nnormalizing the weights). The theorem ensures that the solution\nbenefits from the most favorable regret guarantee among the WFAs in\n$\\cC$.  When the set of distributions associated to $\\cC$ is convex,\nthen the set of distributions defined over $\\Sigma^T$ is also\nconvex. This is then a convex optimization problem, since\n$\\h \\sfq \\mapsto \\log(\\sfq\/\\h \\sfq)$ is a convex function and the\nsupremum of convex functions is convex.\n\nThe choice of the family $\\cC$ is subject to a trade-off: approximation\naccuracy versus computational efficiency of using WFAs in $\\cC$.\nThis raises a model selection question for which we discuss in\ndetail a solution in Section~\\ref{sec:min-Renyi}:\ngiven a sequence of families $(\\cC_n)_{n \\in \\bN}$ with growing\ncomplexity and computational cost, the problem consists of selecting\nthe best $n$.\n\nIn the following, we will consider the case where the family $\\cC$ of\nweighted automata is that of \\emph{$n$-gram models}, for which we can\nupper bound the computational complexity.\n\n\n\\subsection{Minimum R\\'{e}nyi divergence $n$-gram models}\n\\label{sec:min-Renyi}\n\n\n\nLet $\\Sigma^{\\leq n - 1}$ denote the set of sequences of length at\nmost $n - 1$.  An $n$-gram language model is a Markovian model of\norder $(n - 1)$ defined over $\\Sigma^*$, which can be compactly\nrepresented by a WFA with each state identified with a sequence\n$\\bx \\in \\Sigma^{\\leq n - 1}$, thereby encoding the sequence\n\\emph{just read} to reach that state.  The WFA admits a transition\nfrom state $(\\bx[1] \\cdots \\bx[n - 1])$ to state\n$(\\bx[2] \\cdots \\bx[n - 1] a)$ with weight\n$\\sfw \\big[a \\, | \\, \\bx[1] \\cdots \\bx[n - 1] \\big]$, for any\n$a \\in \\Sigma$, and, for any $k \\leq n - 1$, a transition from state\n$(\\bx[1] \\cdots \\bx[k - 1])$ to state $(\\bx[1] \\cdots \\bx[k - 1] a)$\nwith weight $\\sfw \\big[ a \\, | \\, \\bx[1] \\cdots \\bx[k - 1] \\big]$, for\nany $a \\in \\Sigma$. It admits a unique initial state which is the one\nlabeled with the empty string $\\e$ (sequence of length zero) and all\nits states are final. The WFA is stochastic, that is outgoing\ntransition weights sum to one at every state: thus,\n$\\sum_{a \\in \\Sigma} \\sfw[a | \\bx ] = 1$ for all\n$\\bx \\in \\Sigma^{\\leq n - 1}$. Notice that this WFA is also\ndeterministic since it admits a unique initial state and no two\ntransitions with the same label leaving any\nstate. Figure~\\ref{fig:bigram} illustrates this definition in the case\nof a simple bigram model.\\ignore{\\footnote{Notice that $n$-gram models of this\n  form are commonly used in language and speech processing. In these\n  applications, the models are typically \\emph{smoothed} since they\n  are trained on a finite sample. In our case, smoothing will not be\n  needed since we can train directly on $\\sC_T$, which we interpret as\n  the full language.}}\n\nNote that the transition weights $\\sfw[a | \\bx]$, with $a \\in \\Sigma$\nand $\\bx \\in \\cup_{k \\leq n - 1} \\Sigma^k$ fully specify an $n$-gram\nmodel. Since for a fixed $\\bx \\in \\cup_{k \\leq n - 1} \\Sigma^k$,\n$\\sfw[\\cdot | \\bx]$ is an element of the simplex, an $n$-gram model\ncan be viewed as an element of the product of\n$\\sum_{k = 0}^{n - 1} \\Sigma^k$ simplices, a convex set. We will denote\nby $\\cW_n$ the family of all $n$-gram models.\n\n\\begin{figure}[t]\n\\vskip -.15in\n\\centering\n\\includegraphics[scale=0.5]{nbigram} \n  \\caption{A bigram language \n  model over the alphabet $\\Sigma =\\set{a, b, c}$.\n }\n\\label{fig:bigram}\n\\vskip -.15in\n\\end{figure}\n\nOne key advantage of $n$-gram models in this context is that the\nper-iteration complexity can be bounded in terms of the number of\nsymbols. Since an $n$-gram model has at most $|\\Sigma|^{n - 1}$\nstates, its per-iteration computational cost is in\n$\\cO\\big(|\\Sigma|^{n}\\big)$ as each state can take one of $|\\Sigma|$\npossible transitions. For $n$ small, this can be very advantageous\ncompared to the original $\\sC_T$, since in general the maximum\nout-degree of states reached by sequences of length $t$ in the latter\ncan be very large. For instance, the automaton $\\sC_\\text{weighted-shift}$ \nin Figure~\\ref{fig:kshift} (ii) can itself be viewed as a bigram model and \nadmits efficient computation.   \n\nFor $n$-gram models, our approximation algorithm\n(Problem~\\ref{eq:autapprox}) can be written as follows:\n\\begin{equation}\n\\label{eq:minrenyingram}\n\\min_{\\sfw \\in \\cW_n} D_\\infty(\\sfq \\| \\sfq_{\\sfw})\n=  \\min_{\\sfw \\in \\cW_n} \\sup_{\\bx \\in \\Sigma^T} \n\\log\\bigg[ \\frac{\\sfq[\\bx]}{\\sfq_{\\sfw}[\\bx]} \\bigg],\n\\end{equation}\nwhere $\\sfq_{\\sfw}$ is the distribution induced by the $n$-gram model\n$\\sfw$ on sequences in $\\Sigma^T$. By definition of the $n$-gram\nmodel, for any $\\bx \\in \\Sigma^T$, $\\sfq_{\\sfw}[\\bx]$ is given by the\nfollowing:\n\\begin{equation*}\n\\sfq_{\\sfw}[\\bx] = \\prod_{t = 1}^T \\sfw \\big[\\bx[t] \\big| \\bx_{\\max(t - n +\n  1, 1)}^{t - 1} \\big],\n\\end{equation*}\nsince the weights of sequences of any fixed\nlength sum to one in an $n$-gram model. Problem~\\ref{eq:minrenyingram} is a convex\noptimization problem over $\\cW_n$. The problem can be solved\nusing as an an extension of the Exponentiated Gradient (EG) algorithm\nof \\cite{KivinenWarmuth1997}, which we will refer to as\n\\textsc{Prod-EG}. The pseudocode of \\textsc{Prod-EG}, a general convergence\nguarantee, and its convergence guarantee in\nthe specific case of $n$-gram models are given in detail in\nAppendix~\\ref{app:Prod-EG} as Algorithm~\\ref{alg:prodeg}, Theorem~\\ref{th:prodeg},\nand Corollary~\\ref{cor:prodegngram} respectively.\n\n{\\bf Model selection}. In practice, we seek an $n$-gram model that\nbalances the tradeoff between approximation error and computational\ncost. Assume that we are given a maximum per-iteration computational\nbudget $B$. We therefore wish to determine an $n$-gram approximation\nmodel affordable within our budget and with the most favorable regret\nguarantee.  Let $F(\\sfq, \\sfq_\\sfw)$ denote the objective function of\nProblem~\\eqref{eq:minrenyingram}:\n$F(\\sfq, \\sfq_\\sfw) = D_\\infty(\\sfq \\| \\sfq_{\\sfw})$.  By the\nconvergence guarantee of Corollary~\\ref{cor:prodegngram}, if\n$\\sfq_\\sfw$ is the $n$-gram model returned by \\textsc{Prod-EG} after\n$\\tau$ iterations, we can write\n$F(\\sfq, \\sfq_\\sfw) - F(\\sfq, \\sfq_{\\sfw^*}) \\leq \\Delta(\\tau, n)$,\nwhere $\\sfw^*$ is the $n$-gram model minimizing\nProblem~\\eqref{eq:minrenyingram} over $\\cW_n$ and $\\Delta(\\tau, n)$\nthe upper bound given by Corollary~\\ref{cor:prodegngram}.  Thus, if\n$F(\\sfq, \\sfq_\\sfw) - \\Delta(\\tau, n) > \\sqrt{T}$ for some $n$, then,\neven the optimal $n$-gram model for this $n$ will cause an increase in\nthe regret.\n\nLet $n^*$ be the smallest $n$ such that\n$F(\\sfq, \\sfq_\\sfw) - \\Delta(\\tau, n) \\leq \\sqrt{T}$ (or the smallest\nvalue that exceeds our budget).  We can find this value in $\\log(n^*)$\ntime using a two-stage process. In the first stage, we double $n$\nafter every violation until we find an upper bound on $n^*$, which we\ndenote by $n_{\\text{max}}$. In the second stage, we perform a binary\nsearch within $[1, n_{\\text{max}}]$ to determine $n^*$. Each stage\ntakes $\\log(n^*)$ iterations, and each iteration is the cost of\nrunning \\textsc{Prod-EG} for that specific value of $n$.  Thus, the\noverall complexity of the algorithm is\n$\\cO\\left(\\log(n^*) \\, \\text{Cost}(\\textsc{Prod-EG})\\right)$, where\n$\\text{Cost}(\\textsc{Prod-EG})$ is the cost of a call to\n\\textsc{Prod-EG}. The full pseudocode of this algorithm, \n\\textsc{$n$-GramModelSelect}, is presented as Algorithm~\\ref{alg:ngramselect},\nwhere $\\sfu_n$ denotes the\nuniform $n$-gram model and $\\textsc{Prod-EG-Update}(\\sfq_\\sfw, \\cW_n)$\ndenotes one update made by \\textsc{Prod-EG} when optimizing over\n$\\cW_n$.\n\n\\begin{algorithm2e}[t]\n  \\TitleOfAlgo{\\textsc{$n$-GramModelSelect}($\\sfq$, $\\tau$, $B$)}\n  $n \\gets 1$;\n  $\\sfq_\\sfw \\gets \\sfq_{\\sfu_n}$;\n  $s \\gets 0$ \\\\\n  \\While{$s \\leq \\tau$}{\n    $\\sfq_\\sfw \\gets \\textsc{Prod-EG-Update}(\\sfq_\\sfw, \\cW_n)$  \\\\\n    $s \\gets s + 1$ \\\\\n    \\If{$F(\\sfq, \\sfq_\\sfw) - \\Delta(s, n) > \\sqrt{T}$ and $|\\Sigma|^n \\leq B$}{\n      $n \\gets 2n$;\n      $s \\gets 0$;\n      $\\sfq_\\sfw \\gets \\sfq_{\\sfu_n}$\n    }\n  }\n  $n_{\\max} \\gets n$.\\\\\n  $\\sfq_\\sfw \\gets \\textsc{BinarySearch}([1, n_\\text{max}], F(\\sfq, \\sfq_\\sfw) -\n  \\Delta(\\tau, n) \\leq \\sqrt{T})$\\\\\n  \\RETURN{$\\sfq_\\sfw$}\n\\caption{\\textsc{$n$-GramModelSelect}.}\n\\label{alg:ngramselect}\n\\end{algorithm2e}\n\n\n\nIn the simple case of a unigram automaton model over two symbols and when the\ndistribution $\\sfq$ defined by the intersection WFA $\\sC_T$ is uniform,\nwe can give an explicit form of the solution of Problem~\\ref{eq:minrenyingram}.\nThe solution is obtained from the paths with the\nsmallest number of occurrences of each symbol, which can be\nstraightforwardly found via a shortest-path algorithm in linear time.\n\n\\begin{theorem}\n\\label{th:inftyrd1gram}\nAssume that $\\sC_T$ admits uniform weights over all paths and\n$\\Sigma = \\set{a_1, a_2}$.  For $j \\in \\set{1, 2}$, let $n(a_j)$ be the smallest\nnumber of occurrences of $a_j$ in a path of $\\sC_T$.  For any $j \\in\n\\set{1, 2}$, define\n\\begin{equation*}\n\\sfq[a_j] = \\frac{\\max\\left\\{1, \\frac{n(a_j)}{T -\n      n(a_j)}\\right\\}}{1+\\max\\left\\{1, \\frac{n(a_j)}{T -\n      n(a_j)}\\right\\}}.\n\\end{equation*}\n\nThen, the unigram model $\\sfw \\in \\cW_1$ solution of\n$\\infty$-R\\'{e}nyi divergence optimization problem is defined by\n$\\sfw[a_{j^*}] = \\sfq[a_{j*}]$, $\\sfw[a_{j'}] = 1 - \\sfw[a_{j^*}]$,\nwith\n$j^* = \\argmax_{j \\in \\set{1, 2}} \\ n(a_j) \\log \\sfq[a_j] + \\left[T -\n  n(a_j) \\right] \\log\\big[ 1 - \\sfq[a_j] \\big]$.\n\\end{theorem}\n\nThe proof of this result is provided in Appendix~\\ref{app:unigram}.\n\nTheorem~\\ref{th:inftyrd1gram} shows that the solutions of the\n$\\infty$-R\\'enyi divergence optimization are based on the $n$-gram\ncounts of sequences in $\\sC_T$ with ``high entropy''. This can be\nvery different from the maximum likelihood solutions, which are based\non the average $n$-gram counts.  For instance, suppose we are under\nthe assumptions of Theorem~\\ref{th:inftyrd1gram}, and specifically,\nassume that there are $T$ sequences in $\\sC_T$. Assume that one of\nthe sequences has $\\left(\\frac{1}{2} + \\gamma\\right)T$ occurrences of\n$a_1$ for some small $\\gamma > 0$ and that the other $T-1$ sequences\nhave $T-1$ occurrences of $a_1$. Then,\n$n(a_1) = \\left(\\frac{1}{2} + \\gamma\\right)T$, and the solution\nof the $\\infty$-R\\'enyi divergence optimization problem is given by\n$\\sfq_\\infty(a_1) = \\frac{1 + 2\\gamma}{2}$ and\n$\\sfq_\\infty(a_2) = \\frac{1 - 2\\gamma}{2}$. On the other hand,\nthe maximum-likelihood solution would be\n$\\sfq_1(a_1) = 1 + \\frac{\\gamma}{T} - \\frac{3}{2T} + \\frac{1}{T^2} \\approx 1$\nand $\\sfq_1(a_2) = \\frac{3}{2T} - \\frac{\\gamma}{T} - \\frac{1}{T^2} \\approx 0$\nfor large $T$.\n\n\n\n\n\\subsection{Maximum-Likelihood $n$-gram models}\n\\label{subsec:mlapprox}\n\nA standard method for learning $n$-gram models is via\nMaximum-Likelihood, which is equivalent to minimizing the\nrelatively entropy between the target distribution $\\sfq$ and the\nlanguage model, that is via\n\\begin{align}\n  \\label{eq:mlngram}\n\\min_{\\sfw \\in \\cW_n} D(\\sfq \\| \\sfq_{\\sfw}),\n\\end{align}\nwhere, $D(\\sfq \\| \\sfq_{\\sfw})$ denotes the relative entropy,\n$D(\\sfq \\| \\sfq_{\\sfw}) = \\sum_\\bx \\sfq[\\bx]\n\\log\\Big[\\frac{\\sfq[\\bx]}{\\sfq_\\sfw[\\bx]} \\Big]$.  Maximum likelihood\n$n$-gram solutions are simple. For standard text data, the weight of each\ntransition is the frequency of appearance of the corresponding\n$n$-gram in the text. For a probabilistic $\\sC_T$, the weight can be\nsimilarly obtained from the expected count of the $n$-gram in the\npaths of $\\sC_T$, where the expectation is taken over the probability\ndistribution defined by $\\sC_T$ and can be computed efficiently\n\\citep{AllauzenMohriRoark2003}. In general, the solution of this\noptimization problem does not benefit from the guarantee of\nTheorem~\\ref{th:WFAapprox} since the $\\infty$-R\\'{e}nyi divergence is\nan upper bound on the relative entropy.\\ignore{\\footnote{The relative entropy\n  also coincides with the R\\'{e}nyi divergence of order $1$ and the\n  R\\'{e}nyi divergence is non-decreasing function of the order.}}\nHowever, in some cases, maximum likelihood solutions do benefit from\nfavorable regret guarantees.  In particular, as shown by the following\ntheorem, remarkably, the maximum-likelihood bigram approximation to\nthe $k$-shifting automaton coincides with the \\textsc{Fixed-Share}\nalgorithm of \\cite{HerbsterWarmuth1998} and benefits from a constant\napproximation error. Thus, we can view and motivate the design of the\n\\textsc{Fixed-Share} algorithm as that of a bigram approximation of\nthe desired competitor automaton, which represents the family of\n$k$-shifting sequences.\n\n\n\\begin{theorem}\n\\label{th:bigramkshift}\nLet $\\sC_T$ be the $k$-shifting automaton for some $k$.  Then, the\nbigram model $\\sfw_2$ obtained by minimizing relative entropy is\ndefined for all $a_1, a_2 \\in \\Sigma$ by\n\\begin{align*}\n\\sfp_{\\sfw_2}[a_1a_2]\n\\! = \\! \\frac{\\big[ 1 - \\frac{k}{(T - 1)}\\big] \\, 1_{a_1  = a_2} \n+ \\big[ \\frac{k}{(T - 1)(N - 1)} \\big] \\, 1_{a_1 \\neq a_2}}{N} .\n\\end{align*}\nMoreover, its approximation error can be bounded by a constant\n(independent of $T$):\n\\begin{equation*}\n    D_\\infty(\\sfq \\| \\sfq_{\\sfw_2}) \\leq  - \\log \\big[ 1 - 2e^{-\\frac{1}{12k}} \\big].\n\\end{equation*}\n\\end{theorem}\nThe proof of the theorem as well as other details about\nMaximum-Likelihood are given in Appendix~\\ref{app:mlapprox}.\nThe proof technique is illustrative\nbecause it reveals that the maximum likelihood $n$-gram model has low\napproximation error whenever (1) the model's distribution is\nproportional to the distribution of $\\sC_T$ on $\\sC_T$'s support and\n(2) most of the model's mass lies on the support of $\\sC_T$. When the\nautomaton $\\sC_T$ has uniform weights, then condition (1) is satisfied\nwhen the $n$-gram model is uniform on $\\sC_T$. This is true whenever\nall sequences in $\\sC_T$ have the same set of $n$-gram counts, and\nevery permutation of symbols over these counts is a sequence that lies\nin $\\sC_T$, which is the case for the $k$-shifting\nautomaton. Condition (2) is satisfied when $n$ is large enough, which\nnecessarily exists since the distribution is exact for $n = T$. On the\nother hand, note that a unigram approximation would have satisfied\ncondition (1) but not condition (2) for the $k$-shifting automaton.\n\nTo the best of our knowledge, this is the first framework that\nmotivates the design of \\textsc{Fixed-Share} with a focus on\nminimizing tracking regret. Other works that have recovered\n\\textsc{Fixed-Share} (e.g. \\citep{CesaBianchiGaillardLugosiStoltz2012,\n  KoolenDeRooij2013, GyorgySzepesvari2016}) have generally viewed the\nalgorithm itself as the main focus.\n\nOur derivation of \\textsc{Fixed-Share} also allows us to naturally\ngeneralize the setting of standard $k$-shifting experts to\n$k$-shifting experts with non-uniform weights. Specifically, consider\nthe case where $\\sC_T$ is an automaton accepting up to $k$-shifts but\nwhere the shifts now occur with probability\n$\\sfq[a_2 | a_1, a_1 \\neq a_2] \\neq \\frac{1}{N-1} 1_{\\{a_2 \\neq\n  a_1\\}}$.  Since the bigram approximation will remain exact on\n$\\sC_T$, we recover the exact same guarantee as in\nTheorem~\\ref{th:bigramkshift}.\n\nMaximum likelihood $n$-gram models can further benefit from our use of\nfailure transitions and the $\\phi$-conversion algorithm presented in\nAppendix~\\ref{app:phiaut}. This can reduce the size of the automaton\nand often dramatically improve its computational efficiency without\naffecting its accuracy.\n\n\\section{Extension to sleeping experts}\n\\label{sec:sleep}\n\nIn many real-world applications, it may be natural for some experts to\nabstain from making predictions on some of the rounds. For instance,\nin a bag-of-words model for document classification, the presence of a\nfeature or subset of features in a document can be interpreted as an\nexpert that is awake. This extension of standard prediction with\nexpert advice is also known as the \\emph{sleeping experts framework}\n\\citep{FreundSchapireSingerWarmuth1997}.  The experts are said to be\nasleep when they are inactive and awake when they are active and available\nto be selected. This framework is distinct from the\npermutation-based definitions adopted in the studies in \\citep{KleinbergNiculescuMizilSharma2010, KanadeMcMahanBryan2009, KanadeSteinke2014}.  \n\nFormally, at each round $t$, the adversary chooses an awake set\n$A_t \\subseteq \\Sigma$ from which the learner is allowed to query an\nexpert.  The algorithm then (randomly) chooses an expert $i_t$ from\n$A_t$, receives a loss vector $l_t \\in [0,1]^{|\\Sigma|}$ supported on\n$A_t$ and incurs loss $l_t[i_t]$. Since some experts may not be\navailable in some rounds, it is not reasonable to compare the loss\nagainst that of the best static expert or sequence of experts. In\n\\citep{FreundSchapireSingerWarmuth1997}, the comparison is made\nagainst the best fixed mixture of experts normalized at each round\nover the awake set:\n$\\min_{\\sfu \\in \\Delta_N}\\sum_{t = 1}^T \\frac{\\sum_{a \\in A_t} \\sfu[a] l_t[a]\n}{\\sum_{a' \\in A_t} \\sfu[a']}$, where $\\Delta_N$ is the $(N-1)$-dimensional \nsimplex.\n\nWe extend the notion of sleeping experts to the path setting, so that\ninstead of comparing against fixed mixtures over experts, we compare\nagainst fixed mixtures over the family of expert sequences. With some\nabuse of notation, let $A_t$ also represent the automaton accepting\nall paths of length $T$ whose $t$-th transition has label in $A_t$.\nThus, we want to design an algorithm that performs well with respect\nto the following quantity:\n$$\\min_{\\sfu \\in \\Delta_K}\\sum_{t = 1}^T \\frac{\\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] l_t[\\bx[t]] }{\\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx]},$$\nwhere $K$ is the number of accepting paths of $\\sC_T$.\n\nThis motivates the design of \\textsc{AwakeAWM}, a \npath-based weighted majority algorithm that generalizes the algorithms\nin \\citep{FreundSchapireSingerWarmuth1997} to arbitrary families of\nexpert sequences.  Like \\AWM, \\textsc{AwakeAWM} maintains a set of\nweights over all the paths in the input automaton. At each round $t$,\nthe algorithm performs a weighted majority-type update. However, it\nnormalizes the weights so that the total weight of the awake set\nremains unchanged. This prevents the algorithm from ``overfitting'' to\nexperts that have been asleep for many rounds.  The pseudocode of this\nalgorithm and the proof of its accompanying guarantee,\nTheorem~\\ref{th:awakeawm}, \nare provided in Appendix~\\ref{app:sleep}.\n\n\\begin{theorem}[Regret Bound for \\textsc{AwakeAWM}]\n  \\label{th:awakeawm}\n  Let $K$ denote the number of accepting paths of $\\sC_T = \\sC \\cap \\sS_T$,\n  and for each $t \\in [T]$, let $A_t\\subseteq \\Sigma$ denote the set of experts \n  that are awake at time $t$.\n  Then for any distribution $\\sfu\\in \\Delta_K$, \\textsc{AwakeAWM} admits\n  the following unweighted regret guarantee:\n  \\begin{align*}\n    &\\sum_{t = 1}^T \\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] \\E_{a \\sim \\sfp_t^{A_t}} [l_t[a]] - \n    \\sum_{t=1}^T \\sum_{\\bx \\in \\sC_T \\cap A_t} \\sfu[\\bx] l_t[\\bx[t]] \\\\\n    &\\leq \\frac{\\eta}{8} \\sum_{t = 1}^T \\sfu(A_t) + \\frac{1}{\\eta} \\log(K). \n  \\end{align*}\n\\end{theorem}\n\nAs with \\textsc{AWM}, \\textsc{AwakeAWM} is an efficient algorithm\nwith a total computational cost that is linear in the number of transitions \nof $\\sA$ (or equivalently, $\\sC_T$). \nMoreover, as in the non-sleeping expert setting, we can further improve the\ncomputational  complexity by applying $\\phi$-conversion to arrive at a \nor $n$-gram approximation and then $\\phi$-conversion.\nAll other improvements in the sleeping expert setting will similarly mirror\nthose for the non-sleeping expert algorithms.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe studied a general framework of online learning against a competitor\nclass represented by a WFA and presented a number of algorithmic\nsolutions for this problem achieving sublinear regret guarantees using\nautomata approximation and failure transitions.  We also extended our\nalgorithms and results to the sleeping experts framework\n(Section~\\ref{sec:sleep}).\\ignore{and to the online convex optimization\nsetting (Appendix~\\ref{app:oco}).}  Our results can be\nstraightforwardly extended to the adversarial bandit scenario using\nstandard surrogate losses based on importance weighting techniques and\nto the case where more complex formal language families such as\n(probabilistic) context-free languages over expert sequences are\nconsidered.\n\n\\ignore{\nWe gave a series of algorithms for this problem, including an\nautomata-based algorithm extending weighted-majority whose\ncomputational cost at round $t$ depends on the total number of\ntransitions leaving the states of the competitor automaton reachable\nat time $t$, which substantially improves upon a na\\\"ive algorithm\nbased on path updates.  We used the notion of failure transitions to\nprovide a compact representation of the competitor automaton or its\nintersection with the set of strings of length $t$, thereby resulting\nin significant efficiency improvements. This required the\nintroduction of new failure-transition-based composition and\nshortest-distance algorithms that could be of independent interest.\n\nWe further gave an extensive study of algorithms based on a compact\napproximation of the competitor automata.  We showed that the key\nquantity arising when using an approximate weighted automaton is the\nR\\'enyi divergence of the original and approximate automata. We\npresented a specific study of approximations based on $n$-gram models\nby minimizing the R\\'enyi divergence and studied the properties of\nmaximum likelihood $n$-gram models. We pointed out the efficiency\nbenefits of such approximations and provides guarantees on the\napproximations and the regret.  We also extended our algorithms and\nresults to the framework of sleeping experts.  We further described the\nextension of the approximation methods to online convex optimization\nand a general mirror descent setting.\n\nOur description of this general (weighted) regret minimization\nframework and the design of algorithms based on automata provides a\nunifying view of many similar problems and leads to general\nalgorithmic solutions applicable to a wide variety of problems with\ndifferent competitor class automata.  In general, automata \nlead to a more general and cleaner analysis.  \nAn alternative approximation method consists of directly\nminimizing the competitor class automaton before intersection\nwith the set of strings of length $t$. We have also studied\nthat method, presented guarantees for its success, and illustrated \nthe approach in a special case.\n\nNote that, instead of automata and regular languages, we could have\nconsidered more complex formal language families such as\n(probabilistic) context-free languages over expert sequences.\nHowever, more complex languages can be handled in a similar way since\nthe intersection with $\\sS_T$ would be a finite language.  The method\nbased on a direct approximation would require approximating a\nprobabilistic context-free language using weighted automata, a problem\nthat has been extensively studied in the past\n\\citep{PereiraWright1991, Nederhof2000, MohriNederhof2001}.\n\n}\n\n\n\\newpage\n\\bibliographystyle{abbrvnat} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{sec:intro}\n\\footnotetext{\\footnotemark[1]Work was done while Gunnar was an intern at AI2.}\n\nConsider the video shown in Figure~\\ref{fig:teaser1}: A man walks through a doorway, stands at a table, holds a cup, pours something into it, drinks it, puts the cup on the table, and finally walks away. Despite depicting a simple activity, the video involves a rich interplay of a sequence of actions with underlying goals and intentions. For example, the man stands at the table `to take a cup', he holds the cup `to drink from it', etc. Thorough understanding of videos requires us to model such interplay between activities as well as to reason over extensive time scales and multiple aspects of actions (objects, scenes, etc). \n\nMost contemporary deep learning based methods have treated the problem of video understanding as that of only appearance and motion (trajectory) modeling~\\cite{Simonyan13,Tran15,Georgia15,Snoek16}. While this has fostered interesting progress in this domain, these methods still struggle to outperform models based on hand-crafted features, such as Dense Trajectories~\\cite{WangIDT13}. \nWhy such a disconnect? We argue that video understanding requires going beyond appearance modeling, and necessitates reasoning about the activity sequence as well as higher-level constructs such as intentions. The recent emergence of large-scale datasets containing rich sequences of realistic activities~\\cite{charades,yeung2015every,weinzaepfel2016towards} comes at a perfect time facilitating us to explore such complex reasoning. \n\nBut what is the right way to model and reason about temporal relations and goal-driven behaviour? Over the last couple of decades, graphical models such as Conditional Random Fields (CRFs) have been the prime vehicles for structured reasoning. \nTherefore, one possible alternative is to use ConvNet-based approaches~\\cite{alexnet12} to provide features for a CRF training algorithm. \nAlternatively, it has been shown that integrating CRFs with ConvNet architectures and training them in an end-to-end manner provides substantial improvements in tasks such as segmentation and situation recognition~\\cite{raqueldense2016,ChenSchwingICML2015,yatskarsituation}. \n\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{teaser2f.pdf}\n\\caption{Understanding human activities in videos requires jointly reasoning about multiple aspects of activities, such as `what is happening', `how', and `why'. In this paper, we present an end-to-end deep structured model over time trained in a stochastic fashion. The model captures rich semantic aspects of activities, including \\emph{Intent} (why), \\emph{Category} (what), \\emph{Object} (how). The figure shows video frames and annotations used in training from the {\\em Charades}~\\cite{charades} dataset. }\n\\label{fig:teaser1}\n\\end{figure}\n\n\nInspired by these advances, we present a deep-structured model that can reason temporally about multiple aspects of activities. For each frame, our model infers the activity category, object, action, progress, and scene using a CRF, where the potentials are predicted by a jointly end-to-end trained ConvNet over all predictions in all frames. This CRF has a latent node for the intent of the actor in the video and pair-wise relationships between all individual frame predictions. \n\nWhile our model is intuitive, training it in an end-to-end manner is a non-trivial task. Particularly, end-to-end learning requires computing likelihoods for individual frames and doing joint inference about all connected frames with a CRF training algorithm. \nThis is in stark contrast with the standard stochastic gradient descent (SGD) training algorithm (backprop) for deep networks, where we require mini-batches with a large number of independent and uncorrelated samples, not just a few whole videos.\nIn order to handle this effectively: (1) we relax the Markov assumption and choose a fully-connected temporal model, such that each frame's prediction is influenced by all other frames, and (2) we propose an asynchronous method for training fully-connected structured models for videos. Specifically, this structure allows for an implementation where the influence (messages) from other frames are approximated by emphasizing influence from frames computed in recent iterations. They are more accurate, and show advantage over being limited to only neighboring frames. In addition to being more suitable for stochastic training, fully-connected models have shown increased performance on various tasks~\\cite{densecrfsegmentation,raqueldense2016}.\n\nIn summary, our key contributions are: (a) a deep CRF based model for structured understanding and comprehensive reasoning of videos in terms of multiple aspects, such as action sequences, objects, and even intentions; (b) an asynchronous training framework for expressive temporal CRFs that is suitable for end-to-end training of deep networks; and, (c) substantial improvements over state-of-the-art, increasing performance from 17.2\\% mAP to 22.4\\% mAP on the challenging Charades~\\cite{charades} benchmark.\n\n\\section{Related Work}\n\nUnderstanding activities and actions has an extensive history~\\cite{poppe2010survey,weinland2011survey,STIP05,HOG3D,HOF,MBH06,Matikainen09,WangIDT13,Peng14,Lan15}. Interestingly, \nanalyzing actions by their appearance has gone through multiple iterations. Early success was with hand-crafted representations such as Space Time Interest Points (STIP)~\\cite{STIP05}, 3D Histogram of Gradient (HOG3D)~\\cite{HOG3D}, Histogram of Optical Flow (HOF)~\\cite{HOF}, and Motion Boundary Histogram~\\cite{MBH06}. These methods capture and analyze local properties of the visual-temporal datastream. In the past years, the most prominent hand-crafted representations have been from the family of trajectory based approaches~\\cite{Matikainen09,WangIDT13,Peng14,Lan15}, where the Improved Dense Trajectories (IDT)~\\cite{WangIDT13} representation is in fact on par with state-of-the-art on multiple recent datasets~\\cite{THUMOS15,charades}. \n\nRecently there has been a push towards mid-level representations of video~\\cite{Corso12,YaleS13,ArpitJ13,lan2015iccv}, that capture beyond local properties. However, these approaches still used hand-crafted features. With the advent of deep learning, learning representations from data has been extensively studied~\\cite{3DCNN,Karpathy14,2stream14,TDD15,Taylor10,Tran15,Le11,Georgia15,Xu15,Vondrick16_repr,scnn_shou_wang_chang_cvpr16,Souza16}. Of these, one of the most popular frameworks has been the approach of Simonyan et al.~\\cite{2stream14}, who introduced the idea of training separate color and optical flow networks to capture local properties of the video. \n\nMany of those approaches were designed for short clips of individual activities and hence do not generalize well to realistic sequences of activities. \nCapturing the whole information of the video in terms of temporal evolution of the video stream has been the focus of some recent approaches~\\cite{KevinT12,Basura15,Izadinia12,Rohrbach12,Sun13,pirsiavash2014parsing}.\nMoving towards more expressive deep networks such as LSTM has become a popular method for encoding such temporal information~\\cite{Srivastava15,Donahue15,cnnlstm,Sun_2015_ICCV,Wang_Transformation,sigurdsson2016learning,yeung2015end}. Interestingly, while those models move towards more complete understanding of the full video stream, they have yet to significantly outperform local methods~\\cite{2stream14} on standard benchmarks.\n\nA different direction in understanding comes from reasoning about the complete video stream in a complementary direction --- Structure. Understanding activities in a human-centric fashion encodes our particular experiences with the visual world. Understanding activities with emphasis on objects has been a particularly fruitful direction~\\cite{li2007and,ryoo2007hierarchical,gupta2009observing,prest2012weakly,Vondrick16_actns}. In a similar vein, some works have also tried modeling activities as transformations~\\cite{Wang_Transformation} or state changes~\\cite{Fathi13}. Recently, there has been significant progress in modelling the complete human-centric aspect, where image recognition is phrased in terms of objects and their roles~\\cite{yatskarsituation,VSRL_gupta15}. Moving beyond appearance and reasoning about the state of {\\em agents} in the images requires understanding human intentions~\\cite{kitani2012activity,pirsiavash2014inferring}. This ability to understand people in terms of beliefs and intents has been traditionally studied in psychology as the Theory of mind~\\cite{premack1978does}.\n\nHow to exactly model structure of the visual and temporal world has been the pursuit of numerous fields. Of particular interest is work that combines the representative power of deep networks with structured modelling. Training such models is often cumbersome due to the differences in jointly training deep networks (stochastic sampling) and sequential models (consecutive samples)~\\cite{mnih2013playing,raqueldense2016}.\nIn this work, we focus on fully-connected random fields, that have been popular in image segmentation~\\cite{densecrfsegmentation}, where image filtering was used for efficient message passing, and later extended to use CNN potentials~\\cite{schwing2015fully}.\n\n\n\\section{Proposed Method}\n\\label{sect:app}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{model5.pdf}\n\\caption{An overview of our structured model. The semantic part captures \\emph{object}, \\emph{action}, etc. at each frame, and temporal aspects captures those over time. On the left side, we show how for each timepoint in the video, a Two-Stream Network predicts the potentials. Our model jointly reasons about multiple aspects of activities in all video frames. The \\emph{Intent} captures groups of activities of the person throughout the whole sequence of activities, and fine-grained temporal reasoning is through fully-connected temporal connections.}\n\\label{fig:model}\n\\end{figure*}\n\nGiven a video with multiple activities, our goal is to understand the video in terms of activities. Understanding activities requires reasoning about objects being interacted with, the place where the interaction is happening, what happened before and what happens after this current action and even the intent of the actor in the video. We incorporate all these by formulating a deep Conditional Random Field (CRF) over different aspects of the activity over time. That is, a video can be interpreted as a graphical model, where the components of the activity in each frame are nodes in the graph, and the model potentials are the edges in the graph. \n\nIn particular, we create a CRF which predicts activity, object, etc., for every frame in the video. For reasoning about time, we create a \\emph{fully-connected temporal CRF}, referred as Asynchronous Temporal Field in the text. That is, unlike a linear-chain CRF for temporal modelling (the discriminative counterpart to Hidden Markov Models), each node depends on the state of every other node in the graph. We incorporate intention as another latent variable which is connected to all the action nodes. \nThis is an unobserved variable that influences the sequence of activities. This variable is the common underlying factor that guides and better explains the sequence of actions an agent takes. Analysis of what structure this latent variable learns is presented in the experiments.\nOur model has three advantages: (1) it addresses the problem of long-term interactions; (2) it incorporates reasoning about multiple parts of the activity, such as objects and intent; and (3) more interestingly, as we will see, it allows for efficient end-to-end training in an asynchronous stochastic fashion.\n\n\\subsection{Architecture}\nIn this work we encode multiple components of an activity. Each video with $T$ frames is represented as $\\{X_1, \\dots, X_T, I\\}$ where $X_t$ is a set of frame-level random variables for time step $t$ and $I$ is an unobserved random variable that represent global intent in the entire video. We can further write $X_t = \\{C_t,O_t,A_t,P_t,S_t\\}$, where $C$ is the activity category (e.g., `drinking from cup'), $O$ corresponds to the object (e.g., `cup'), $A$ represents the action (e.g., `drink'), $P$ represents the progress of the activity \\{start, middle, end\\}, and $S$ represents the scene (e.g. `Dining Room'). \nFor clarity in the following derivation we will refer to all the associated variables of $X_t$ as a single random variable $X_t$. A more detailed description of the CRF is presented in the appendix.\n\nMathematically we consider a random field $\\{X,I\\}$ over all the random variables in our model ($\\{X_1, \\dots, X_T, I\\}$). Given an input video $V{=}\\{V_1,\\dots,V_T\\}$, where $V_t$ is a video frame, our goal is to estimate the maximum a posteriori labeling of the random field by marginalizing over the intent $I$. This can be written as:\n\\begin{equation}\n\\mathbf{x}^* = \\mathrm{arg}\\,\\max_x \\sum_I P(x,I|V).\n\\end{equation}\nFor clarity in notation, we will drop the conditioning on $V$ and write $P(X,I)$. We can define $P(X,I)$ using Gibbs distribution as: $P(X,I) {=} \\frac{1}{Z(\\mathbf{V})} \\exp \\left( -E(x,I) \\right)$ where $E(x,I)$ is the Gibbs energy over $x$. In our CRF, we model all unary and pairwise cliques between all frames $\\{X_1,\\dots,X_T\\}$ and the intent $I$. The Gibbs energy is:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\nE(\\mathbf{x},I) = \\underbrace{\\sum_i \\phi_\\mathcal{X}(x_i)}_{\\mathrm{Semantic}} + \\underbrace{\\sum_{i} \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,I) + \\sum_{\\substack{i,j\\\\i \\neq j}} \\phi_{\\mathcal{X}\\!\\mathcal{X}}(x_i,x_j)}_{\\mathrm{Temporal}},\n\\end{align}\n\\endgroup\nwhere $\\phi_{\\mathcal{X}\\!\\mathcal{X}}(x_i,x_j)$ is the potential between frame $i$ and frame $j$, and $\\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,I)$ is the potential between frame $i$ and the intent. For notational clarity $\\phi_\\mathcal{X}(x_i)$ incorporates all unary and pairwise potentials for ${C_t,O_t,A_t,P_t,S_t}$. \nThe model is best understood in terms of two aspects: Semantic aspect, which incorporates the local variables in each frame (${C_t,O_t,A_t,P_t,S_t}$); and Temporal aspect, which incorporates interactions among frames and the intent $I$. This is visualized in Figure~\\ref{fig:model}. \nWe will now explain the semantic, and temporal potentials.\n\n\\myparagraph{Semantic aspect}\nThe frame potential $\\phi_\\mathcal{X}(x_i)$ incorporates the interplay between activity category, object, action, progress and scene, and could be written explicitly as $\\phi_\\mathcal{X}(C_t,O_t,A_t,P_t,S_t)$. In practice this potential is composed of unary, pairwise, and tertiary potentials directly predicted by a CNN. We found predicting only the following terms to be sufficient without introducing too many additional parameters:\n$\\phi_\\mathcal{X}(C_t,O_t,A_t,P_t,S_t){=}\\phi(O_t,P_t){+}\\phi(A_t,P_t){+}\\phi(O_t,S_t)+ \\phi(C_t,O_t,A_t,P_t)$ where we only model the assignments seen in the training set, and assume others are not possible.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{messages2.pdf}\n\\caption{Illustration of the learning algorithm, and the message passing structure. Each timepoint that has been processed has a message (Blue highlights messages that have recently been computed). The loss receives a combination of those messages, uses those to construct new messages, and updates the network. }\n\\label{fig:learning}\n\\end{figure}\n\n\\myparagraph{Temporal aspect}\nThe temporal aspect of the model is both in terms of the frame-intent potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{I}}}(x_i,I)$ and frame-frame potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j)$. The frame-intent potentials are predicted with a CNN from video frames (pixels and motion). The pairwise potentials $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j)$ for two time points $i$ and $j$ in our model have the form:\n\\begin{align}\n\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,x_j) = \\mu(x_i,x_j)\\sum_{m} w^{(m)}k^{(m)}(v_i,v_j),\n\\end{align}\nwhere $\\mu$ models the asymmetric affinity between frames, $w$ are kernel weights, and each $k^{(m)}$ is a Gaussian kernel that depends on the videoframes $v_i$ and $v_j$. In this work we use a single kernel that prioritises short-term interactions:\n\\begin{align}\n\\label{eq:sigma}\nk(v_i,v_j) = \\exp \\left( - \\frac{(j-i)^2}{2\\sigma^2}\\right)\n\\end{align}\nThe parameters of the general asymmetric compatibility function $\\mu(x_i,x_j)$ are learned from the data, and $\\sigma$ is a hyper-parameter chosen by cross-validation.\n\n\\subsection{Inference}\n\nWhile it is possible to enumerate all variable configurations in a single frame, doing so for multiple frames and their interactions is intractable. Our algorithm uses a structured variational approximation to approximate the full probability distribution. In particular, we use a mean-field approximation to make inference and learning tractable. With this approximation, we can do inference by keeping track of message between frames, and asynchronously train one frame at a time (in a mini-batch fashion). \n\nMore formally, instead of computing the exact distribution $P(X,I)$ presented above, the structured variational approximation finds the distribution $Q(X,I)$ among a given family of distributions that best fits the exact distribution in terms of KL-divergence. By choosing a family of tractable distributions, it is possible to make inference involving the ideal distribution tractable. Here we use $Q(X,I)=Q_{\\mathcal{I}}(I) \\prod_i Q_i(x_i)$, the structured mean-field approximation. Minimizing the KL-divergence between those two distributions yields the following iterative update equation:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\nQ_i(x_i) \\propto \\exp \\bigg\\{ & \\phi_\\mathcal{X}(x_i) + \\mathrm{E}_{U\\sim Q_{\\mathcal{I}}} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_i,U) \\right] \\nonumber \\\\ \n&+ \\sum_{j > i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(x_i,U_j) \\right] \\bigg\\} \\nonumber \\\\\n&+ \\sum_{j < i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{{\\mathcal{X}\\!\\mathcal{X}}}(U_j,x_i) \\right] \\bigg\\} \\\\\nQ_{\\mathcal{I}}(I) \\propto \\exp \\bigg\\{ & \\sum_j \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}}(U_j,I) \\right] \\bigg\\}\n\\label{eq:Q}\n\\end{align}\n\\endgroup\nwhere $Q_i$ is marginal distribution with respect to each of the frames, and $Q_{\\mathcal{I}}$ is the marginal with respect to the intent. An algorithmic implementation of this equation is as presented in Algorithm~\\ref{alg:inference}.\n\n\\begin{algorithm}\n  \\caption{Inference for Asynchronous Temporal Fields\n    \\label{alg:inference}}\n    {\\footnotesize\n  \\begin{algorithmic}[1]\n      \\State Initialize Q \\Comment{Uniform distribution}\n      \\While{$\\textrm{not converged}$}\n        \\State Visit frame $i$\n        \\State Get $\\sum_{j > i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (x_i,U_j) \\right]$ \n        \\State Get $\\sum_{j < i} \\mathrm{E}_{U_j\\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (U_j,x_i) \\right]$ \n        \\State Get $\\sum_{j} \\mathrm{E}_{U_j \\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}} (U_j,I) \\right]$ \n        \\While{$\\textrm{not converged}$}\n          \\State Update $Q_i$ and $Q_{\\mathcal{I}}$ using Eq.~\\ref{eq:Q}\n        \\EndWhile\n        \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (x,U) \\right]$\n        \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{X}} (U,x) \\right]$\n        \\State Send $\\mathrm{E}_{U \\sim Q_i} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}} (U,I) \\right]$\n      \\EndWhile\n  \\end{algorithmic}\n  }\n\\end{algorithm}\n\n\n\\noindent Here `Get' and `Send' refer to the message server, and $f(x)$ is a message used later by frames in the same video. The term message server is used for a central process that keeps track of what node in what video sent what message, and distributes them accordingly when requested. In practice, this could be implemented in a multi-machine setup.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{messages.pdf}\n\\caption{Evolution of prediction with increasing messages passes. The first row shows the initial prediction for the category {\\em tidying with a broom} without any message passing, where darker colors correspond to higher likelihood, blue is then an increase in likelihood, and brown decrease. In the first message pass, the confidence of high predictions gets spread around, and eventually increases the confidence of the whole prediction.}\n\\label{fig:messages}\n\\end{figure}\n\n\n\\subsection{Learning}\n\nTraining a deep CRF model requires calculating derivatives of the objective in terms of each of the potentials in the model, which in turn requires inference of $P(X,I|V)$. The network is trained to maximize the log-likelihood of the data $l(X) = \\log \\sum_I P(x,I|V)$. \nThe goal is to update the parameters of the model, for which we need gradients with respect to the parameters. Similar to SGD, we find the gradient with respect to one part of the parameters at a time, specifically with respect to one potential in one frame. That is, $\\phi_\\mathcal{X}^i(\\hat{x})$ instead of $\\phi_\\mathcal{X}(\\hat{x})$.\nThe partial derivatives of this loss with respect to each of the potentials are as follows:\n\\begingroup\\makeatletter\\def\\f@size{8}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align}\n\\frac{\\partial l(X)}{\\partial \\phi_\\mathcal{X}^i(\\hat{x})} &=\n\\mathbf{1}_{x=\\hat{x}} - Q_i(\\hat{x}) \\label{eq:gradients1}\\\\ \n\\frac{\\partial l(X)}{\\partial \\phi_{\\mathcal{X}\\!\\mathcal{I}}^i(\\hat{x},\\hat{I})} &= \\frac{\\exp \\sum_j \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_j,\\hat{I})}{\\sum_I \\exp \\sum_j  \\phi_{\\mathcal{X}\\!\\mathcal{I}}(x_j,I)}\\mathbf{1}_{x=\\hat{x}} - Q_i(\\hat{x}) Q_{\\mathcal{I}}(\\hat{I}) \\label{eq:gradients2}\\\\\n\\frac{\\partial l(X)}{\\partial \\mu^i(a,b)} &= \n\\sum_{j>i} \\mathbf{1}_{x=a} k(v_i,v_j) - Q_i(\\hat{x}) \\sum_{j>i} Q_{\\mathcal{I}}(b) k(v_i,v_j) \\nonumber \\\\\n&+ \\sum_{j<i} \\mathbf{1}_{x=b} k(v_j,v_i) - Q_i(\\hat{x}) \\sum_{j<i} Q_{\\mathcal{I}}(a) k(v_i,v_j)\n\\label{eq:gradients3}\n\\end{align}\n\\endgroup\nwhere $\\phi_\\mathcal{X}^i(\\hat{x})$ and $\\phi_{\\mathcal{X}\\!\\mathcal{I}}^i(\\hat{x},\\hat{I})$ is the frame and frame-intent potentials of frame $i$, and we use $\\hat{x}$ to distinguish between the labels and variables the derivative is taken with respect to. $\\mu^i(a,b)$ are the parameters of the asymmetric affinity kernel with respect to frame $i$, and $\\mathbf{1}_{x=\\hat{x}}$ is a indicator variable that has the value one if the ground truth label corresponds to the variable. Complete derivation is presented in the appendix. These gradients are used to update the underlying CNN model. These update equations lead to the learning procedure presented in Algorithm~\\ref{alg:learning}.\n\\algnewcommand\\algorithmicforeach{\\textbf{for each}}\n\\algdef{S}[FOR]{ForEach}[1]{\\algorithmicforeach\\ #1\\ \\algorithmicdo}\n\\begin{algorithm}\n  \\caption{Learning for Asynchronous Temporal Fields\n    \\label{alg:learning}}\n    {\\footnotesize\n  \\begin{algorithmic}[1]\n      \\State Given videos $\\mathcal{V}$ \n      \\While{$\\textrm{not converged}$}\n        \\ForEach{example in mini-batch}\n          \\State Sample frame $v \\in \\mathbf{V}\\subseteq \\mathcal{V}$\n          \\State Get incoming messages\n          \\State Update $Q_i$ and $Q_{\\mathcal{I}}$\n          \\State Find gradients with Eq.~\\ref{eq:gradients1}-\\ref{eq:gradients3}\n          \\State Backprop gradients through CNN\n          \\State Send outgoing messages\n        \\EndFor\n      \\EndWhile\n  \\end{algorithmic}\n  }\n\\end{algorithm}\n\nFigure~\\ref{fig:learning} graphically illustrates the learning procedure. Since the videos are repeatedly visited throughout the training process, we do not have to run multiple message passes to calculate each partial gradient. This shares ideas with contrastive divergence~\\cite{hinton2002training,salakhutdinov2009deep}.\nGiven a single video at test time, we visualize in Figure~\\ref{fig:messages} how the predictions changes as the distribution converges with multiple messages passes.\n\n\\myparagraph{Message Passing}\nThe key thing to note is all the incoming messages are of the form $M(z) {=} \\sum_j f_j(z)$ where $f_j$ is some function from node $j$; for e.g., $M(z) = \\sum_j \\mathrm{E}_{U_j \\sim Q_j} \\left[ \\phi_{\\mathcal{X}\\!\\mathcal{I}} (U_j,z) \\right] = \\sum_j f_j(z)$ from Algorithm~\\ref{alg:inference}. We use the following approximation during training:\n\\begin{align}\nM(z) {\\approx} \\frac{h}{\\sum_j d^j} \\sum_j d^j f_{J(j)}(z),\n\\end{align}\nwhere $d\\in[0,1]$ is a discount factor, $h$ is a hyperparameter, and $J(\\cdot)$ is an ordering of the messages in that video based on the iteration in which the message was computed. The messages are a weighted combination of stored messages.\n\n\\section{Experimental Results and Analysis}\n\nWe analyzed the efficacy of our model on the challenging tasks of video activity classification and temporal localization. In addition, we investigated the different parts of the model, and will demonstrate how they operate together.\n\n\n\\myparagraph{Dataset}\nRecent years have witnessed an emergence of large-scale datasets containing sequences of common daily activities~\\cite{charades,yeung2015every,weinzaepfel2016towards}. For our evaluation, we chose the {\\em Charades} dataset~\\cite{charades}. This dataset is a challenging benchmark containing 9,848 videos across 157 action classes with 66,500 annotated activities, including nouns (objects), verbs (actions), and scenes. A unique feature of this dataset is the presence of complex co-occurrences of realistic human-generated activities making it a perfect test-bed for our analysis. We evaluate video classification using the evaluation criteria and code from~\\cite{charades}. Temporal localization is evaluated in terms of per-frame classification using the provided temporal annotations. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{vgg16.pdf}\n\\caption{The VGG-16 variant predicts the potentials for both RGB and Flow. The network predicts the values of all potentials except one (in this figure we group the frame potentials $\\phi_\\mathcal{X}$ into one layer for clarity). The model is trained end-to-end by passing gradients from the Asynchronous Temporal Field through the network.}\n\\label{fig:CNN}\n\\end{figure}\n\n\n\\myparagraph{Implementation details}\nWe use a VGG16 network~\\cite{Simonyan15} with additional layers to predict the model potentials (Figure~\\ref{fig:CNN}). We train both a network on RGB frames, and stacks of optical flow images, following the two-stream architecture~\\cite{2stream14}. The main challenge in training the network is the increase in the output layer size. For the larger potentials, we used the following structure to go from fc7 to $\\phi_{{\\mathcal{X}\\!\\mathcal{I}}}$: Linear layer (4096 to 100), ReLU, Dropout, Linear layer (100 to the potential values). \n\nThe input to the RGB network is an image of size $224{\\times}224{\\times}3$ where we crop random location, size, and aspect ratio. We use data augmentation with color jitter and PCA lighting noise. The RGB network was pretrained on ImageNet. The input to the Flow network is a stack of 10 consecutive optical flow frames at 24 FPS starting with the current frame. Since each optical flow has two channels, the input size is $224{\\times}224{\\times}20$ as in~\\cite{2stream14}. The Flow network was pretrained on UCF101~\\cite{UCF101} as in Sigurdsson et al.~\\cite{charades}, and random cropped in the same way as RGB. \n\nWe follow the training setup in Charades~\\cite{charades} and consider a frame to have one activity label at a time. Even so, our method is still able to reason about other activities in the video. Convergence of the model is evaluated using the approximate distribution $Q_i(X)$ at each frame. The Charades dataset has the property that scenes were chosen at random for each sequence of activities. For this reason, we found reasoning about scenes to reduce the performance, and the weight of that term was lowered in the model.\n\nTo obtain annotations for action progress $p_t$, we split each activity annotation into three equally sized parts. All layers of the network are trained with a batch size of $240$ and a learning rate of $10^{-3}$ (RGB), $10^{-5}$ (Flow). Learning rate was reduced by a factor of 10 every 30k iterations for RGB, and every 140k iterations for Flow. The value of the message decay parameter $d$ was set to $d=0.9$, and the standard deviation $\\sigma$ in (\\ref{eq:sigma}) was set to $6.25$ sec (150 frames).\n\nFor testing, we sampled 25 equally spaced frames from the video and synchronously pass messages between the frames until convergence (10 message passes). The predictions of the RGB and Flow networks are combined in a probabilistic fashion by multiplying their probabilistic predictions for each class. More implementation details may be found in the appendix. The networks were implemented in Torch, and the code is available on project page.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{convergence.pdf}\n\\caption{Convergence of our method compared to other methods that capture temporal structure. Our asynchronous training method contains more diverse batches, has faster and more stable convergence, and reaches higher accuracy on the test set.}\n\\label{fig:convergence}\n\\end{figure}\n\n\\myparagraph{Diverse batches}\nAs highlighted in Section~\\ref{sec:intro}, the standard way of sampling batches for temporal models results in high correlation between data points leading to a breakdown of the SGD. To understand the importance of having many diverse examples from multiple videos, we compare the convergence of our method to two alternatives using homogeneous batches: CNN+LSTM from Ng et al.~\\cite{cnnlstm}, and a synchronous version of our method, where each batch contains full videos (only three videos fit into each mini-batch). We do synchronous message passing until convergence before calculating gradients for backprop. Figure~\\ref{fig:convergence} shows that our asynchronous training method, containing more diverse training batches, has faster and more stable convergence. \n\n\\subsection{Video Classification}\n\n\nGiven a video, the task here is to verify whether it contains one or several of the $157$ activity categories. Classification accuracy is measured with the standard mean average precision (mAP) criterion, where a prediction is given for each video. This task has been shown to be highly challenging, with the state-of-the-art non-ensemble methods reaching an mAP of only $17.2\\%$, particularly as each video in this dataset has a sequence of multiple fine-grained activities with a real-world long-tailed activity distribution. \n\nWe trained our models using the provided training split following the procedure outlined in Section~\\ref{sect:app}. To make predictions for the whole video, we marginalize out everything except the activity category for 25 equidistant frames in the video. The score for each activity category is the maximum across all frames following the setup from~\\cite{charades}. In our analysis, we include the provided non-ensemble baselines from~\\cite{charades} as well as the following additional baselines:\n\n{\\em Two-Stream++.} We reimplemented the network described in~\\cite{charades}, which follows Simonyan et al.~\\cite{Simonyan15}, with the same parameters. We added data augmentation and fine-tuned all layers of the network. The performance of only the RGB stream is included ({\\em RGB++}). We also consider {\\em Two-Stream Extended} which is the same network, but the Flow network was trained for 25 times more iterations than the RGB network (two weeks of computation on a Titan X GPU). Combined with the augmentation, we found this to non-trivially increase the accuracy.\n\n{\\em Two-Stream+LSTM.} We followed the method outlined in~\\cite{cnnlstm} to jointly train a LSTM on top of the two-stream network. We trained both an RGB and an Optical Flow network using the same setup from~\\cite{charades}. The trained networks from Two-Stream++ were used to initialize the models.\n\n\\begin{table}[tb]\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{lcclc} \\toprule\nApproach & mAP & \\hspace{2em} &Approach & mAP \\\\ \\hline \\rule{0pt}{3ex}%\nRandom~\\cite{charades} & 5.9 & & RGB++ & 15.6 \\\\\nC3D~\\cite{Tran15} & 10.9 & &Two-Stream++ & 16.8\\\\\nAlexNet~\\cite{alexnet12} & 11.3 & &Two-Stream+LSTM & 17.8 \\\\\nIDT~\\cite{WangIDT13} & 17.2 & &Two-Stream Extended & 18.6\\\\ \nTwo-Stream~\\cite{Simonyan13} & 14.3 & & Ours (RGB Only) & 18.3 \\\\\n & & & Ours & {\\bf 22.4}  \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{Video classification results on Charades~\\cite{charades}. The left shows the published baselines from~\\cite{charades} and the right show additional new baselines. Our proposed approach outperforms all competing methods on this dataset.}\n\\label{tbl:classification}\n\\end{table}\n\nTable~\\ref{tbl:classification} displays the accuracy obtained by our method along with the baselines. Our proposed approach obtains an mAP of $22.4\\%$ substantially outperforming the Two-stream Extended baseline at $18.6\\%$ mAP, and the IDT baseline at $17.2\\%$. Our method reasons over significantly larger timescales and multiple aspects of the activities. To ascertain this, we highlight in Figure~\\ref{fig:top10bottom10}, the activity classes with the highest positive and negative difference between our method and the Two-Stream network.\nIt is interesting to note that two of those activities are {\\em opening} and {\\em closing} a refrigerator, that arguably have a significant causal structure (an {\\em open} refrigerator was {\\em opened} at some point), which our model harnesses to significantly increase the accuracy.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{top10bottom10.pdf}\n\\caption{The classes with the highest positive and negative difference between our method and Two-Stream (no structure). Our method does better on many classes, without doing much worse on any. In particular, activities that have temporal structure, such as Opening\/Closing a refrigerator have significantly higher performance, since our model can reason jointly about those.}\n\\label{fig:top10bottom10}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{ablationsplot.pdf}\n\\caption{Ablation analysis for our proposed model. Y-axis is video classification mAP \\%. Each factor helps in improving the overall model performance. $\\phi(P)$ indicates dropping the `progress' term within the semantic factor $\\phi_{\\mathcal{X}}$ .}\n\\label{tbl:ablation}\n\\end{figure}\n\n\\myparagraph{Ablation studies} To study the contribution of different model parts, we also train ablated versions of our model separately choosing the best hyperparameters for each version. In addition to our model with only RGB or Flow, we also consider dropping $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}$ (i.e., no sequential information), $\\phi_{{\\mathcal{X}\\!\\mathcal{I}}}$ (i.e., no intent), both (i.e., only semantic information), and further dropping $\\phi_{\\mathcal{X}}$ (i.e., dropping all structure). Figure~\\ref{tbl:ablation} shows that semantic reasoning improves over the baseline. Further, while both $\\phi_{{\\mathcal{X}\\!\\mathcal{I}}}$ and $\\phi_{{\\mathcal{X}\\!\\mathcal{X}}}$ capture temporal information, they are complementary. \n\n\n\\begin{figure*}[th]\n\\centering\n\\includegraphics[width=1\\linewidth]{examples.pdf}\n\\caption{Model predictions for a sample video. We see the interplay between categories, objects and actions over time. For example, model becomes confident about the action \\emph{sit} early, which aids the understanding of \\emph{Sitting in a chair} once the chair becomes visible, and helps predicting {\\em Reading a book}. Darker colors represent higher likelihood, and we average predictions to correspond to each frame.}\n\\label{fig:examples}\n\\end{figure*}\n\n\\subsection{Temporal Localization}\n\nTo measure the ability of the methods to temporally localize and understand when exactly activities happen, we adapt the benchmark of~\\cite{charades} to evaluate with the same mAP metric but on individual frames. That is, instead of having a single prediction per video, evaluation is now split into 25 equidistant timepoints having zero or more activities, and the models make a prediction for each of those\\footnote{This evaluation code has been included as a part of the Charades dataset (\\myurl{allenai.org\/plato\/charades\/}).}.\nWe find this way of evaluating localization robust to annotation ambiguity, and informative for challenging datasets.\nAll hyperparameters were kept equal between localization and classification experiments. All baselines are run on 75 frames across the video, and then every third frame selected for a total of 25 frames. We also considered methods with \\emph{post-processing} where the model predictions for the 75 frames are averaged across 30 frames to obtain more spatial consistency, and then 25 frames selected as before.\n\\begin{table}[tb]\n\\begin{center}\n\\setlength{\\tabcolsep}{8pt}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{rcccccc} \\toprule\n & \\multirow{ 2}{*}{Random} & \\multirow{ 2}{*}{RGB} & \\multirow{ 2}{*}{Two-Stream++} & Two-Stream & Two-Stream & \\multirow{ 2}{*}{Ours}  \\\\ \n & & & &  +LSTM & Extended & \\\\ \\hline \\rule{0pt}{3ex}\n\\emph{Standard} & 2.42 & 7.89 & 8.94 & 9.60 & 9.37 & {\\bf 9.69} \\\\\n\\emph{Post-processing} & 2.42 & 9.05 & 10.9 & 10.4 & 11.6 & {\\bf 12.8} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{Temporal localization results (mAP \\%) on the Charades~\\cite{charades} dataset. Our proposed method outperforms the LSTM model, and is also more tractable to train at a large-scale.}\n\\label{tbl:frameclassification}\n\\end{table}\n\nTable~\\ref{tbl:frameclassification} shows that our method outperforms the alternatives, including the LSTM model which has been shown to be a powerful temporal modeling tool, but challenging to train on top of a two-stream network due to correlations between consecutive samples. These results demonstrate the our method is tractable way of training end-to-end structured models to understand activities. Interestingly, our method still benefits from adding post-processing, significantly more than the LSTM baseline, likely since our method is reasoning on larger time-scales. This suggests that our model could further benefit from joint training with additional kernels in the temporal term.\n\n\n\n\\myparagraph{Qualitative visualization}\nA key advantage of our model is the structured understanding of videos in terms of multiple aspects, such as action sequences, objects, and even intentions. \nTo visualize this, we display predictions over time in Figure~\\ref{fig:examples} for the three most confident activity categories, two most confident actions, and the most confident object. More examples are presented in the Appendix.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{goals85.pdf}\n\\caption{To visualize the learned intent, we cluster videos based on intent. In \\emph{Cluster 1}, the model captures the intent of get up from lying down. In \\emph{Cluster 2}, folding clothes is followed by putting them away, and \\emph{Cluster 3} shows cleaning with a broom\/vacuum\/towel, followed by picking up things.}\n\\label{fig:goals}\n\\end{figure}\n\n\\myparagraph{Interpretation of Intent}\nIn our model, the intent $I$ is a continuous distribution over the latent variables. To get an insight into how our model learns the intent, we ran a simple experiment that clustered videos in the dataset that have the most similar inferred intent distributions. The first cluster in Figure~\\ref{fig:goals} shows the model captures the simple intent that the person intends to get up from lying down. In the videos, these actions are 10-20 seconds apart, demonstrating that the intent helps reason over large time scales.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{finaltsne.pdf}\n\\caption{t-SNE visualization for the learned intent. Each point corresponds to a video. In a) it is colored based on its activity shared by the most of the 10 nearest neighbors (each video has multiple actions). In b) videos with 6 annotated intent types are emphasized with larger points colored by the type.}\n\\label{fig:intentannotations}\n\\end{figure}\n\nIn order to further analyze the `intent' variable, we plot the t-SNE embedding of the intent variable for the videos in the test set. We see that there is clear clustering of similar videos in Fig.~\\ref{fig:intentannotations}a. We also annotated 10 types of intent (100 videos total). More details are presented in the Appendix. We observe that the intent representation preserves some of the intent types in Fig.~\\ref{fig:intentannotations}b. Quantitatively, even without mitigating outliers, the average distance (in $10^{-3}$) between pairs of videos within an intent type was $6.02$ compared to $7.25$ ($\\sigma{=}1.06$) for any points, and the difference is significant for 5 of 10 intent types (p=$0.1$). This tentatively suggest that the intent captures interesting structure in the data, and we hope this will encourage future work.\n\n\n\\section{Conclusion}\n\nIn this paper, we have presented a deep-structured model using a fully-connected temporal CRF that not only models semantic aspects of activities but also reasons about long-term temporal relations. We also presented an asynchronous stochastic inference algorithm that circumvents a key bottleneck in the large-scale end-to-end model learning. Using our proposed method, we have demonstrated impressive activity classification and temporal localization results on a challenging dataset of realistic activities. \n\n{\\footnotesize\n\\noindent {\\bf Acknowledgements:} This work was partly supported by ONR MURI N00014-16-1-2007, ONR N00014-13-1-0720, NSF IIS-1338054, NSF-1652052, NRI-1637479, Intel via the Intel Science and Technology Center for Visual Cloud Systems, Allen Distinguished Investigator Award, gifts from Google, and the Allen Institute for Artificial Intelligence. The authors would like to thank Mark Yatskar for lending his expertise on deep CRFs, and Olga Russakovsky, Christoph Dann, and the anonymous reviewers for their invaluable suggestions and advice.}\n\n\\balance\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nPre-trained language models (LMs, \\citealp{gpt2, t5}) are the backbone of contemporary task-oriented dialog (TOD) models \\cite{peng-etal-2020-soloist, yang-etal-2021-ubar}. However, the models are pre-trained on large generic corpora so that they do not contain task-specific knowledge. \nPrevious work primarily suggests further pre-training or fine-tuning the LMs on in-domain data for adaptation \\cite{wu-etal-2020-tod, simpletod}, but it cannot consider information above the surface level. This makes it challenging for downstream tasks especially in the few-shot learning setting because mapping representation to the output space and encoding knowledge into the model parameters are entangled, while the latter may require more training data.\nSome more recent research proposes to incorporate external knowledge for response generation tasks \\cite{dinan-etal-2019-wizard, shuster-etal-2022-language, chen-etal-2022-ketod, komeili-etal-2022-internet}, but it is not clear how to utilize such information for language understanding. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{kg_nlu_model.pdf}\n\\vspace{-1em}\n\\caption{Model architecture for our proposed knowledge-grounded DST. The encoder first encodes the query and knowledge into representations, and we find the top-k most relevant knowledge elements to the context in step 2. We flatten the retrieved elements in step 3 and append to the query context as the input to the encoder-decoder model. The retrieved elements serve as a prior for DST.}\n\\label{fig:model}\n\\vspace{-1em}\n\\end{figure}\n\nIn TOD settings, because the API call structure is restricted to certain intents, slots, and values, the schema is often provided. For example, in a flight booking system, queries like departure location and airlines are pre-defined. Users, even though not bounded directly by what they can say to agents, have a limited and predictable vocabulary set to some extent. If the schema information is utilized, a model does not need to learn that \"San Francisco\" represents a departure place, rather than a general city name from the LM. This is particularly important for new information, such as movie titles or locations that do not appear in the LM training corpus.\nSimilar to human annotators, grounding a dialog model on such knowledge makes it easier and more accurate in understanding conversations.\n\nIn this paper, we investigate knowledge-grounded understanding for dialog state tracking (DST). In addition to using structured knowledge such as the ontology of slot type-value pairs, we also consider unstructured knowledge from the raw training data. We train a TOD model to query relevant knowledge for each turn in the context, and leverage the retrieved knowledge to predict dialog state. \nWe evaluate our method on MultiWOZ \\cite{budzianowski-etal-2018-multiwoz} for both the full-data and few-shot settings, and show superior performance compared to previous methods.  \n\n\\section{Related Work}\n\n\\subsection{Knowledge grounding}\n\nTo relax the requirement of encoding knowledge of the whole world into model parameters,\none direction \nis to disentangle knowledge representation from LMs. Most of these methods are applied to knowledge-intensive text generation tasks such as open-domain question answering \\cite{lee-etal-2019-latent, \nkarpukhin-etal-2020-dense, guu-etal-2020-retrieval, \nlewis-etal-2020-retrieval, boregeaud-etal-2021-improving}, and response generation with factual information \\cite{dinan-etal-2019-wizard, \nkomeili-etal-2022-internet, lamda, \nkim-etal-2020-beyond, thulke-etal-2021-efficient, chen-etal-2022-ketod}. \nSimilarly, some work also considers retrieving information to serve as a reference to refine the model generation process \\cite{weston-etal-2018-retrieve, gonzalez-etal-2019-retrievalbased, khandelwal-etal-2021-nearest, zhang-etal-2021-jointRetrieval}.\nDifferent from these approaches, our method focuses on learning and utilizing available domain-relevant knowledge for language understanding tasks. Moreover, we propose to leverage knowledge of various formats.\n\n \n\\subsection{Knowledge guided dialog understanding}\n\nEncoding domain schema into model parameters \\cite{simpletod, madotto-etal-2020-learning} may not be efficient for unseen domains and tasks where the ontology can be different. \nOne line of research \\cite{ren-etal-2018-towards, wu-etal-2019-transferable,  zhou-small-2019-multi, rastogi-etal-2020-towards, du-etal-2021-qa, lee-etal-2021-dialogue}\nleverages question-answering techniques to predict values for each slot, or prepend all slot-value information to the context \\cite{zhao-etal-2022-description}.\nHowever, this method is not scalable when the number of slot-value pairs is large, especially in multi-domain TOD systems. In addition, probably due to blurry attention over long context \\cite{fan-etal-2021-augmenting}, \\citet{lee-etal-2021-dialogue} find that adding potential slot values does not improve the model performance. In contrast, retrieving only relevant schema effectively solves the scalability problem by specifying the knowledge with a fixed length.\n\nAlternatively, instead of structured schema knowledge, recent research proposes to use hand-crafted demonstrations as prompts \\cite{gupta-etal-2022-show} or find similar examples to guide understanding tasks \\cite{yu-etal-2021-shot, pasupat-etal-2021-controllable, yao-etal-2021-tlm} such as conversational semantic parsing. However, one turn can contain multiple dialog states so that retrieved examples from previous methods may not be sufficient to provide required evidence. Furthermore, our method can be applied to unify different forms of knowledge including structured and unstructured ones.\n\n\n\n\\section{Methodology}\nOur proposed method is illustrated in Figure \\ref{fig:model}. Given the context $\\mathbf{x}$, we first retrieve $\\mathbf{k}$ relevant knowledge entries $\\mathbf{e}$ by the similarity between $\\operatorname{Enc}(\\mathbf{x})$ and $\\operatorname{Enc}(\\mathbf{e})$ using an encoder $\\operatorname{Enc}$. Then we integrate the retrieved entries $e_1, e_2, ..., e_k$ with the original context to form $\\mathbf{x'}$, where $\\mathbf{x'}$ is used as the input for the target DST task.\n\n\\paragraph{Knowledge retrieval}\nDifferent from previous work (such as question answering) where there is only one ground-truth knowledge for each query, \nmultiple entries of the form slot-value pairs may exist in the ontology base that match the conversation context.\nImportantly, unlike passage retrieval where the query (e.g., a sentence) and the target (e.g., another sentence or passage) are similar to the pre-training corpus, structured knowledge such as schema pairs may have different representation distribution. Thus, an off-the-shelf encoder may retrieve noisy elements and degrade final performance, especially when training with the target task optimized on DST generation. Moreover, non-parametric retrieval methods such as TF-IDF and BM25 \\cite{robertson-zaragoza-2009-bm25} rely on lexical overlapping, which could be detrimental when entries in schemas contain high word overlapping (e.g., same value for different slots).\n\nWe therefore train our knowledge retriever to promote similar representations between a query and its ground truth knowledge. We started with optimizing the marginal likelihood over all positive knowledge entries, but found that it resulted in peaky distribution centered around specific elements in our preliminary studies. Instead,\nwe minimize binary cross-entropy with contrastive loss:\n\n\\begin{small}\n\\begin{equation}\n\\small{\n\\begin{split}\n    \\mathcal{L} &=\n    -\\sum_{i=1}^{i=n}y_i\\cdot \n    (log(sim(\\operatorname{Enc}(\\mathbf{x}),\\operatorname{Enc}(e_i))) \\\\ \n    &+(1 - y_i)\\cdot log(1 - sim(\\operatorname{Enc}(\\mathbf{x}),\\operatorname{Enc}(e_i))))\n\\end{split}\n}\n\\end{equation}\n\\end{small}\n\n\\noindent where $y_i$ is 1 if $e_i$ appears in the target dialog state and otherwise 0. \nIn our model, we use the same encoder $\\operatorname{Enc}$ for both the context and the knowledge, and $\\operatorname{Enc}$ is also used for the target DST task. \n$sim$ defines the retrieval score,\ncomputed as the dot product between representations of the first token from the last layer\\footnote{Other methods such as ColBERT \\cite{khattab-etal-2020-colbert} are also applicable. In our preliminary experiments, we found that dot product is an effective measure, corroborating findings from \\citet{ni-etal-2021-large}}.\n\n\\paragraph{Knowledge integration}\nOnce most relevant knowledge elements are retrieved by the model, this extra information can serve as a strong inductive bias to the downstream, knowledge-sensitive tasks. One common approach for knowledge integration is fusion-in-decoder \\cite{izacard-grave-2021-distilling}. Although efficient, it has been shown that retrieved information is likely to be ignored by a pre-trained model \\cite{shuster-etal-2022-language}. Hence, we concatenate retrieved knowledge with the context $\\mathbf{x'} = e_k, e_{k-1}, ..., e_1, \\mathbf{x}$, where the entries are ordered from the least similar ($e_k$) to the most similar ($e_1$). The similarity can also be considered as the confidence an element $e_i$ is relevant to the current context. We take the $\\mathbf{x'}$ as the context to the DST task. Therefore, our method is unified for knowledge of any format, and a bounded number of elements \ncan solve the problem of memory constraint in previous research \\cite{zhao-etal-2022-description}.\n\n\\section{Experiments and Results}\n\\subsection{Experiments and baselines}\nWe conduct experiments on MultiWOZ 2.4 \\cite{budzianowski-etal-2018-multiwoz, ye-etal-2021-multiwoz} for DST in both full-shot and few-shot (1\\%) learning settings. For all experiments, we use T5-base and T5-XXL encoder-decoder models \\cite{t5} as the initial checkpoints. \nWe use the publicly available T5 checkpoints \\footnote{\\url{2https:\/\/github.com\/google-research\/ text-to-text-transfer-transformer\/blob\/ main\/released_checkpoints.md}} for our experiments. T5-base has 250 million parameters, and T5-XXL has 11B parameters. We train all models on 64 (for T5-base) and 128 (for T5-XXL) TPU v3 chips \\cite{jouppi-etal-2017-in}. For fine-tuning, we set a learning rate of 1e-4 and a batch size of 32. We set the input and output sequence length to 1024 and 512 tokens. We train all models for 200k steps and report the performance on the test set from checkpoints achieving the best results on the development set. When multitask training on both retrieval and DST, we set 0.1 weight to the retrieval loss and 1 weight to the DST loss since it is relatively faster to converge for retrieval. We also experimented with 0.01, 0.05, 0.5, 1 for the retrieval loss weight, and found that 0.1 performs the best.\n\nWe compare with two baselines, seq2seq and D3ST. seq2seq takes the context as input, and predicts a sequence of linearized dialog state for each turn. Similarly, D3ST \\cite{zhao-etal-2022-description} adds descriptions of each slot with potentially values as the prompt and predicts dialog states as multiple choice. Both baselines use the same T5 initial checkpoints. \nWe report averaged joint goal accuracy (JGA) across three random seeds.\n\nFor our proposed method, we consider slot type (type), slot type and value (type+value), and training data (training) as knowledge sources. \nSpecifically, for type, we consider all slot types (35 in total such as ``hotel-parking'') as the knowledge base and retrieve corresponding top ten elements. For type+value, we consider each combination of types and their values in the form of ``type: value'' (1858 in total such as ``hotel-parking: don't care'') as the knowledge elements. Because there are more elements, we consider top 30 in our experiments for retrieval to achieve higher recall (with analysis later in Section \\ref{analysis}). For training data, because of memory concerns,  we randomly sample 500 training examples as the knowledge base and we consider the ground truth training example as the one with the highest F1 overlapping in the dialog slot types. We only consider top-1 due to the length constraint. For each knowledge source, we train retrieval together with the DST generator using the same model parameters. \n\n\\subsection{Results}\n\\input{results}\n\nTable \\ref{results} shows DST results produced by different methods. Compared to the seq2seq baseline and D3ST, grounding on relevant knowledge by retrieval achieves better JGA by a large margin especially in the few-shot learning setting ($> 4\\%$ absolute value). In the full data setting, our method performs on par with D3ST mostly due to that with more training data, the model can encode knowledge into parameters rather than relying on a separate, disentangled knowledge base. However, our method is not limited by the sequence length when we can specifically choose the number of retrieved elements regardless of the ontology size. \n\nWhen comparing among different knowledge formats, type+value performs better than retrieving type only despite that retrieving is a harder task. As shown by recall\\footnote{Precision is determined by the number of retrieved elements we set, whereas recall measures the percentage of ground-truth knowledge elements being correctly retrieved. Therefore, recall is more informative.}, with a large pre-trained model (XXL), recall for retrieving type only can achieve perfect scores ($>99\\%$), but recall for type+value can only be 48\\% in the few-shot and close to $70\\%$ in the full-data setting. This indicates that \nthe model can denoise distracting elements and make use \nof relevant knowledge as a positive inductive bias. Meanwhile, retrieving training data is similar to utilizing prompts \\cite{gupta-etal-2022-show}, but the worse performance compared to other knowledge formats suggests that selecting top-1 element is not optimal despite the relatively high recall. This is mostly due to that the retrieval results are noisy, as the small set of examples may contain slot types or values that are different from the ground truth. It is even less likely to find an example with exactly the same dialog state when the context is long. We leave further investigation by separating knowledge memory to support different knowledge sizes and external knowledge to future work.\n\n\\subsection{Analysis}\n\\label{analysis}\nWe study the relationship between retrieval and JGA in this section, and provide error analysis. \nWe also analyze the detailed comparison between our method and D3ST.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{recall_jga.pdf}\n\\caption{JGA with controlled retrieval recall from sanity check experimented with T5-XXL on the full-data setting. Results show that similar to our findings, even noisy retrieval improves model performance on DST.}\n\\label{fig:jga_recall}\n\\end{figure}\n\n\\paragraph{Relationship between retrieval quality and JGA}\nTo understand the relationship between retrieval and the downstream task, we show JGA corresponding to recall in a controlled sanity check. Specifically, we randomly sample ground truth slot type-value pairs to match a target recall score and replace the rest dialog states with pairs uniformly sampled from the whole ontology (excluding the ground truth) without replacement as negative examples. Results (detailed in Figure \\ref{fig:jga_recall}) show that with T5-XXL on the full-data setting, $50\\%$ recall can significantly improve the model performance (83.75 JGA) while $90\\%$ recall can result in 91.24 JGA. This suggests that a high recall for retrieval is critical to JGA, while the model remains robust against noisy retrieval results. It also indicates that a better retrieval method (such as an external one \\citealt{lazaridou-etal-2022-internet}) may achieve better performance.\nOn the other hand, if we consider DST as a multi-class classification task with a retrieval module only, the model has to pick relevant elements from top-k, which is non-trivial.\n\nWe also consider separating retrieval from DST, i.e., train the model for retrieval first and then on DST. Results show that although the model can achieve \n$97.38\\%$ recall, JGA actually drops to $70.33$ on the full-data setting with XXL. We conjecture the main reason to be that different from freezing retrieval index in previous question-answering work, knowledge such as ontology or training data are more homogeneous and thus being more sensitive.\nThis result is similar to our findings when training the two tasks jointly: retrieval metrics keep improving while JGA may drop with higher retrieval, even if we decrease the retrieval loss weight.\n\nWhen we optimize separate parameters (i.e., two additional layers) for retrieval instead of the whole model, we observe slightly lower performance on JGA ($54.76$ compared to $55.32$ on $1\\%$ data) and lower retrieval recall ($36$ compared to close to $46$).\nLastly, compared to top-30 with a JGA of $75.47$, we observe an absolute drop of $0.40$ for top-20, and $3.25$ for top-10. This indicates that compared to noise in precision, retrieving ground-truth elements for recall is more critical to JGA.\n\n\\paragraph{Comparison to D3ST}\nD3ST decodes the sequence of dialog state based on the order of slot types provided in the prompt by data pre-processing. In comparison, the order of retrieved elements varies while the order of dialog state depends on the ground-truth annotation. In other words, similar to the seq2seq baseline, our method requires learning the annotation order for DST prediction. This makes it more challenging to train, especially when there are similar knowledge elements retrieved. This can be justified by the slightly lower JGA with the full data setting.  On the other hand, D3ST can be considered as a special setting of our grounding method where all knowledge elements are provided, and the DST generation model needs to implicitly detect relevant information and decode accordingly. We conjecture that the better performance on the few-shot setting over D3ST is due to that retrieving target elements while filtering noisy ones is easier than selecting corresponding knowledge, which can be shown from the high recall scores compared to lower JGA for D3ST. One future direction is to combine the benefits of the two worlds by utilizing the retrieved knowledge without length restriction.\n\n\n\\paragraph{Error analysis}\nWe found qualitatively that instead of ignoring retrieved elements as shown in previous research, the model does attend to retrieved slot-value pairs when decoding dialog states. The main errors are from noisy retrieval, where a very similar elements with a higher rank (thus closer to the context in $\\mathbf{x'}$) than the ground truth knowledge may either stop the model from generating more states (i.e., missing target dialog states) or signal the model to generate the wrong elements directly. On the other hand, the model always predicts correctly if the ground truth are the most confident retrieved elements. To deal with the influence of attending only at the nearest few elements (which have the highest retrieval scores), we also experimented with randomly shuffling the retrieved knowledge but this results in lower scores ($71.0$ compared to $75.5$) because the model needs to denoise from potential top-k elements without any additional information.\n\n\\section{Conclusion}\nIn this paper, we propose to disentangle domain knowledge and encode knowledge as a prior to dialog state tracking. Compared to previous research of grounding on knowledge for factual generation, our method can be applied to multiple sources of knowledge in the task-oriented dialog understanding setting. We conduct experiments on the MultiWOZ dataset and show superior performance especially in the few-shot learning setting. We plan to apply our method on more general natural language understanding tasks in the future.\n\n\\section{Limitations}\nIn the experiments, we show model improvements over strong baselines. Despite the simplicity of the method, we acknowledge that the domain ontology is not always available since \nknowledge (e.g., non-categorical slots) may not be a closed set,\nsuch as type+value in DST. However, this limitation can be lifted in two ways. Firstly, as shown in our experiments, retrieving slot type alone can also improve the model performance, which indicates that we may choose a knowledge base mixing type and type+value when the assumption that all values are predefined does not hold. Moreover, in most DST applications, the schema is specified before data collection and model training, where all target types and values need to match a database for information lookup. If the schema is unavailable, we may consider schema induction \\cite{hudecek-etal-2021-discovering, yu-etal-2022-unsupervised} where we can build the schema before DST. We plan to investigate these directions in our future work.  \n\n\\section*{Acknowledgements}\nWe thank Abhinav Rastogi from Google Research, and anonymous reviewers for their constructive suggestions.\n\n\n\n\\section{Appendix}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHybrid   inflation~\\cite{Linde},   along   with   its   supersymmetric\nrealizations~\\cite{CLLSW,DSS,Halyo,CP},   remains  one  of   the  most\npredictive and  potentially testable scenarios of  inflation that have\nbeen suggested so far.  Hybrid inflation is predictive and testable, in the  sense that the inflaton dynamics  is mainly governed\nby  a  few  renormalizable   operators  which  might  have  observable\nimplications for laboratory experiments. In such a scenario, inflation\nterminates  through  the   so-called  waterfall  mechanism,  which  is\ntriggered, when  the inflaton field $\\phi$ passes  below some critical\nvalue  $\\phi_c$. From  that point  on, another  field $X$,  called the\nwaterfall field, held fixed at origin initially, quickly rolls down to\nits true  vacuum expectation value~(VEV) and  drastically modifies the\nslow-roll form of the $\\phi$-potential, thereby ending inflation.\n\nIn  supersymmetric   theories,  the   required  form  of   the  hybrid\ninflationary potential may originate  from either the $F$-terms of the\nsuperpotential     or    from    a     large    Fayet--Iliopoulos~(FI)\n$D$-term~\\cite{FI},  usually  induced  by  some anomalous  local  U(1)\nsymmetry within the context of  string theories.  In both the $F$- and\n$D$-term hybrid  inflation, the slow-roll  slope of the  potential may\ncome either from  supergravity (SUGRA) corrections~\\cite{CLLSW} and\/or\nfrom radiative effects~\\cite{DSS,Halyo,CP}.\n\nRecently, a new supersymmetric  hybrid inflationary model was proposed\nin~\\cite{GP} and studied in  detail in~\\cite{GPP}.  The model realizes\n$F$-term   hybrid   inflation   and   includes  a   subdominant   {\\em\nnon-anomalous}  FI  $D$-term  that  arises  from  the  U(1)$_X$  gauge\nsymmetry of  the waterfall  sector. It has  therefore been  called the\n$F_D$-term  model of  hybrid inflation,  or in  short,  the $F_D$-term\nmodel.  The  $F_D$-term model can naturally  accommodate the currently\nfavoured    red-tilted   spectrum    with    $n_{\\rm   s}-1    \\approx\n-0.037$~\\cite{Dunkley:2008ie},  along  with the  actual  value of  the\npower  spectrum   of  curvature  perturbations,   $P_{\\cal  R}  \\simeq\n4.86\\times   10^{-5}$~\\cite{WMAP3},  and   the   required  number   of\n$e$-folds, ${\\cal N}_e \\approx 55$~\\cite{review}.\n\nThe presence  of the FI term  in the $F_D$-term model  is necessary to\napproximately break  a $D$-parity  that governs the  waterfall sector.\nThe approximate breaking  of the $D$-parity gives rise  to late decays\nof the  superheavy waterfall-sector  particles that are  produced just\nafter         inflation          during         the         preheating\nepoch~\\cite{PREHEATING,GBRM}. These waterfall particles have masses of\nthe  Grand Unified  Theory (GUT)  scale  and can  dominate the  energy\ndensity of  the Universe, provided  the inflaton coupling  $\\kappa$ to\nthe waterfall sector is not too suppressed, i.e.~for values of $\\kappa\n\\stackrel{>}{{}_\\sim}  10^{-3}$.    Then,  the  late   decays  of  the\nGUT-scale  waterfall particles  produce an  enormous entropy  that can\nreduce  the  gravitino abundance  $Y_{\\widetilde{G}}$  well below  the\nlimits     imposed    by     big    bang     nucleosynthesis    (BBN),\ni.e.~$Y_{\\widetilde{G}}                           \\stackrel{<}{{}_\\sim}\n10^{-15}$~\\cite{gravitino}.   In   this  way,  the   $F_D$-term  model\nprovides   a   viable   solution   to  the   gravitino   overabundance\nproblem~\\cite{GPP},  without  the  need  to unnaturally  suppress  all\nrenormalizable  inflaton couplings  to  the particles  of the  Minimal\nSupersymmetric  Standard  Model  (MSSM)  sector, below  the  $10^{-6}$\nlevel.\n\nAnother  interesting  feature of  the  $F_D$-term  model  is that  the\n$\\mu$-parameter  of  the MSSM  can  be  generated  effectively by  the\nsuperpotential    operator    $\\lambda    \\widehat{S}    \\widehat{H}_u\n\\widehat{H}_d$,  when  the scalar  component  of  the inflaton  chiral\nmultiplet $\\widehat{S}$ receives a  non-zero VEV after the spontaneous\nsymmetry  breaking  (SSB)  of  the  local  U(1)$_X$  symmetry  of  the\nwaterfall   sector~\\cite{DLS}.   Moreover,  the   inflaton  superfield\n$\\widehat{S}$     couples     to     the     right-handed     neutrino\nsuperfields~$\\widehat{N}_{1,2,3}$,  via  the  superpotential  coupling\n$\\frac{1}{2}\\rho_{ij} \\widehat{S}  \\widehat{N}_i\\widehat{N}_j$, with $i,j=1,2,3$.\nHence, the inflaton VEV will produce an effective Majorana mass matrix\nas  well~\\cite{Francesca,GP}.  As a  consequence, the  resulting heavy\nMajorana neutrinos  are expected  to have masses  of order  $\\mu$.  If\n$\\rho_{ij}$ is approximately  SO(3) symmetric, i.e.~$\\rho_{ij} \\approx\n\\rho\\,  {\\bf 1}_3$,  a  possible explanation  of  the observed  baryon\nasymmetry  in   the  Universe  (BAU)   may  be  obtained   by  thermal\nelectroweak-scale resonant  leptogenesis, in a way  independent of any\npre-existing lepton- or baryon-number asymmetry~\\cite{PU2}.\n\nEven though the $F_D$-term model violates explicitly the lepton number\n($L$)  by  $\\Delta  L  =  2$ superpotential  operators,  it  conserves\n$R$-parity.   Hence,  the  lightest  supersymmetric particle  will  be\nstable and  so will  potentially qualify as  a candidate for  the cold\ndark matter (DM) in  the Universe.  Most interestingly, the $F_D$-term\nmodel provides a new candidate for  the cold DM.  This is the lightest\nright-handed  sneutrino  (LRHS),   which  may  possess  thermal  relic\nabundance~\\cite{GPP} for relatively large values of the aforementioned\nsuperpotential  couplings $\\lambda$  and $\\rho$,  i.e.~for $\\lambda,\\,\n\\rho \\stackrel{>}{{}_\\sim}  10^{-2}$.  This should  be contrasted with\nwhat is  happening in  standard seesaw extensions  of the  MSSM, where\n$\\widehat{N}_{1,2,3}$  have only  bare Majorana  masses.   Because the\nsmall neutrino Yukawa couplings  are the only possible interactions of\nsneutrinos with matter in these models, purely right-handed sneutrinos\nturn out to be non-thermal and  tend to overclose the Universe by many\norders of magnitude~\\cite{GGP,LMN}.  It is therefore difficult for the\nLRHS to  be a thermal  DM in seesaw  extensions of the MSSM  with bare\nMajorana masses.\n\nIn  this  paper  we analyze  in  detail  the  relic abundance  of  the\nright-handed  sneutrinos  in the  supersymmetric  $F_D$-term model  of\nhybrid  inflation.   In  this  model,  the $F$-term  of  the  inflaton\nsuperfield,   $F_S$,  gives   rise  to   the  new   quartic  coupling,\n$\\frac{1}{2}\\lambda\\rho\\,  \\widetilde{N}^*_i \\widetilde{N}^*_i  H_u H_d$,  in the\nscalar   potential,  which   involves   the  right-handed   sneutrinos\n$\\widetilde{N}_{1,2,3}$   and   the   Higgs  doublets~$H_{u,d}$.    As\nmentioned  above, unless the  couplings $\\lambda$  and $\\rho$  are too\nsmall,  the  new  quartic  coupling  will be  sufficiently  strong  to\nthermalize  the  sneutrinos  and  make  them  annihilate  to  a  level\ncompatible  with   the  current  cosmic   microwave  background  (CMB)\ndata~\\cite{Dunkley:2008ie},  from  which   the  DM  component  of  the\nUniverse was found to be\n\\begin{equation}\n  \\label{OmegaDM}\n\\Omega_{\\rm   DM}\\,h^2\\ =\\ 0.1099\\: \\pm\\: 0.0062\\; .\n\\end{equation}\nThe central goal  of our analysis is to  delineate the parameter space\nwithin  the context of  minimal supergravity  (mSUGRA), for  which the\nLRHS is the {\\em thermal} DM. In addition, we consider the constraints\nobtained  by  WMAP  observations  related to  cosmological  inflation.\nFinally,   we   present   numerical   estimates  of   the   scattering\ncross-section of the LRHS with  nuclei that will be relevant to direct\nDM searches in present and future experiments.\n\n\nAfter  this  introduction,  the  paper  is organized  as  follows:  in\nSection~\\ref{FDmodel} we present the basic structure of the $F_D$-term\nmodel and  briefly review the solution to  the gravitino overabundance\nproblem.   Moreover, in  the same  section we  derive  the constraints\nimposed on  the theoretical parameters by  cosmological inflation.  In\nSection~\\ref{CDM} we  perform a detailed study of  the relic abundance\nof the LRHS and  offer numerical estimates of representative scenarios\nwithin the  mSUGRA framework. We also present  numerical estimates for\nthe scattering cross-section of  the LRHS with the nucleon, indicating\nthe  presently achieved  and  future sensitivity  of  the current  and\nprojected  experiments for  DM  searches, such as CDM-II, SuperCDMS and Xenon1T.  Finally,  we summarize  our\nconclusions in Section~\\ref{conclusions}.\n\n\\bigskip\n\n\\setcounter{equation}{0}\n\\section{The {\\boldmath $F_D$}-Term Model of Hybrid Inflation}\\label{FDmodel}\n\nIn this section we first outline the basic structure of the $F_D$-term\nmodel of  hybrid inflation. Then  we briefly review how  the gravitino\nabundance  can be  solved  within the  $F_D$-term  model. Finally,  we\npresent the constraints on the theoretical parameters that are imposed\nby CMB data pertinent to  inflation. A more detailed discussion of all\nthe above issues may be found in~\\cite{GPP}.\n\n\n\\subsection{The Model}\n\nThe $F_D$-term model may be defined through the superpotential\n\\begin{equation}\n\\label{Wmodel}\n\tW\\ =\\  \\kappa\\,\\widehat{S}\\,\n        \\Big(\\widehat{X}_1\\widehat{X}_2\\:-\\:M^2\\Big)\\ \n\t+\\ \\lambda\\,\\widehat{S} \\widehat{H}_u  \\widehat{H}_d\\ \n\t+\\ \\frac{\\rho_{ij}}{2}\\,\\widehat{S}\\, \\widehat{N}_i\\widehat{N}_j\\ \n\t+\\ h^{\\nu}_{ij} \\widehat{L}_i \\widehat{H}_u\\widehat{N}_j\n \t+\\ W_{\\rm MSSM}^{(\\mu = 0)}\\; ,\n\\end{equation}\nwhere  $\\widehat{S}$  is  the  gauge-singlet inflaton  superfield  and\n$\\widehat{X}_{1,2}$  is  a  chiral  multiplet pair  of  the  so-called\nwaterfall fields which have  opposite charges under the U(1)$_X$ gauge\ngroup,  i.e.~$Q (\\widehat{X}_1)=-Q  (\\widehat{X}_2)=1$.   In addition,\n$W_{\\rm MSSM}^{(\\mu  = 0)}$ indicates the  MSSM superpotential without\nthe $\\mu$-term,\n\\begin{equation} \n\tW_{\\rm MSSM}^{(\\mu=0)}\\ =\\\n\th^u_{ij}\\,\\widehat{Q}_i\\widehat{H}_u\\widehat{U}_j\\: \n\t+\\: h^d_{ij}\\,\\widehat{H}_d\\widehat{Q}_i\\widehat{D}_j\\: \n\t+\\: h_l\\, \\widehat{H}_d\\widehat{L}_l\\widehat{E}_l \\; .\n\\end{equation}\nWithin the  SUGRA framework, the  sector of soft  supersymmetry (SUSY)\nbreaking (SSB) derived from~(\\ref{Wmodel}) is given by\n\\begin{equation}\\label{Lsoft}\n\t-\\,{\\cal L}_{\\rm soft} = \n\t  M^2_{\\tilde S} S^*S\n\t+ M^2_{\\tilde N} N_i^* N_i\n\t+ \\Big(\\kappa A_\\kappa\\, S X_1X_2\\: \n\t+ \\lambda A_\\lambda S H_u H_d\\: \\: \n\t+ \\frac{\\rho}{2}\\, A_\\rho\\, S\n\t    \\widetilde{N}_i\\widetilde{N}_i\\:\n\t- \\kappa a_S M^2 S \\: \\ +\\ {\\rm  H.c.}\\,\\Big)\\,,\n\\end{equation}\nwhere  $M_{\\tilde S}$,  $M_{\\tilde N}$,  $A_{\\kappa,\\lambda,\\rho}$ and\n$a_S$ are soft SUSY-breaking mass parameters that are all typically of\norder  $M_{\\rm SUSY}~\\sim~1$~TeV.  In addition,  the  $F_D$-term model\ncontains a FI $D$-term, $-\\frac{1}2 g m^2_{\\rm FI} D$, associated with\nthe U(1)$_X$ gauge symmetry of  the waterfall sector. The latter gives\nrise to the $D$-term potential\n\\begin{equation}\n  \\label{Dterm}\nV_D\\ =\\ \\frac{g^2}{8}\\; \\Big( |X_1|^2\\: -\\: |X_2|^2\\: -\\: m^2_{\\rm\n  FI}\\Big)^2\\; ,\n\\end{equation}\nwhere  $g$  is the  U(1)$_X$  gauge-coupling  constant.   The FI  mass\nparameter   $m_{\\rm  FI}$   is   subdominant  with   respect  to   the\nsuperpotential     tadpole     mass     $M$,    i.e.~$m_{\\rm     FI}\/M\n\\stackrel{<}{{}_\\sim} 10^{-5}$.\n\nAn interesting feature of the $F_D$-term model is the generation of an\neffective $\\mu$-term  of the required  order $M_{\\rm SUSY}$  after the\nSSB of  U(1)$_X$. To see  this, let us  neglect the VEVs  of $H_{u,d}$\nnext to  the large  VEVs of the  waterfall fields  $X_{1,2}$: $\\langle\nX_{1,2}\\rangle = M$. To a good  approximation, the VEV of $S$ may then\nbe determined by the following part of the potential:\n\\begin{equation}\n  \\label{VS}\nV_S\\ =\\ |F_{X_1}|^2\\: +\\: |F_{X_2}|^2\\: +\\: M^2_S\\,S^*S\\: +\\:\n        \\Big[\\, \\kappa\\,M^2 (A_\\kappa - a_S)\\, S\\ +\\ {\\rm H.c.}\\Big]\\; ,\n\\end{equation}\nwhere we have set the waterfall fields $X_{1,2}$ to their actual VEVs.\nSubstituting the $F$-terms of the waterfall fields,\n\\begin{equation}\n  \\label{FX12}\nF_{X_{1,2}}\\ =\\ \\kappa S\\, \\langle X_{2(1)}\\rangle\\ =\\ \\kappa M\\, S\\; ,\n\\end{equation}\ninto~(\\ref{VS}), we obtain \n\\begin{equation}\n  \\label{VStad}\nV_S\\ =\\ \\Big( 2\\kappa^2 M^2\\: +\\: M^2_S \\Big)\\, S^*S\\: +\\: \n \\Big[\\, \\kappa\\,M^2 (A_\\kappa - a_S)\\, S\\ +\\ {\\rm H.c.}\\Big]\\; .\n\\end{equation}\nIt  is then  not difficult  to derive  from~(\\ref{VStad}) that  at the\npresent  epoch of the  Universe, the  inflaton field,  $S$,  acquires the\nnon-zero VEV\n\\begin{equation}\n  \\label{VEVofS}\n\\langle S \\rangle\\ =\\ \n\\frac{1}{2\\kappa}\\, |A_\\kappa - a_S|\\: +\\: {\\cal O}(M^2_{\\rm\n  SUSY}\/M)\\; ,\n\\end{equation}\nin  the  phase  convention  that  $\\langle  S  \\rangle$  is  positive.\nEquation~(\\ref{VEVofS}) implies the effective $\\mu$-term\n\\begin{equation}\n  \\label{mu}\n\t\\mu\\ =\\  \n\t\\lambda\\, \\langle S  \\rangle\\ \n\t\\approx\\ \\frac{\\lambda}{2\\kappa}\\, |A_\\kappa - a_S|\\ .\n\\end{equation}\nIf  $\\lambda \\sim  \\kappa$, the  size of  $\\mu$-parameter is  of order\n$M_{\\rm  SUSY}$,  as  required  for  a  successful  electroweak  Higgs\nmechanism.\n\n\nIn  addition to the  generation of  an effective  $\\mu$-parameter, the\nthird term in~(\\ref{Wmodel}), $\\frac{1}{2}\\, \\rho_{ij}\\, \\widehat{S}\\,\n\\widehat{N}_i   \\widehat{N}_j$,    gives   rise   to    an   effective\nlepton-number-violating Majorana mass  matrix, i.e.~$M_N = \\rho_{ij}\\,\nv_S$.  If we assume that $\\rho_{ij}$ is approximately SO(3) symmetric,\ni.e.~$\\rho_{ij}  \\approx  \\rho\\;  {\\bf  1}_3$, one  obtains  3  nearly\ndegenerate right-handed neutrinos $N_{1,2,3}$, with mass\n\\begin{equation}\\label{mN}\n\tm_N\\ =\\ \\rho\\, v_S\\ .\n\\end{equation}\nIf  the  couplings  $\\lambda$  and  $\\rho$ are  comparable,  then  the\n$\\mu$-parameter will  set the  scale for the  SO(3)-symmetric Majorana\nmass $m_N$, i.e.~$m_N \\sim  \\mu$~\\cite{GP}.  Evidently, this will lead\nto a scenario where the singlet neutrinos $N_{1,2,3}$ have TeV or\nelectroweak-scale masses.   This opens up the  possibility of directly\ndetecting these singlet Majorana neutrinos through their lepton-number\nviolating     signatures     at     the     LHC~\\cite{NprodLHC}     or\nILC~\\cite{NprodILC}.   Furthermore, in  the $F_D$-term  model  the BAU\ncould   be    explained   by   thermal    electroweak-scale   resonant\nleptogenesis~\\cite{PU2}.\n\n\n\n\\subsection{Solution to the Gravitino Overabundance Problem}\n\nThe FI mass  term $m_{\\rm FI}$ plays a key role  in providing a viable\nsolution  to the  gravitino  overabundance problem  in the  $F_D$-term\nmodel,  without the  need  to unnaturally  suppress  all the  inflaton\ncouplings    $\\kappa$,     $\\lambda$    and    $\\rho$     below    the\n$10^{-6}$~level~\\cite{GP,GPP}.\n\nIn detail, the presence of  $m_{\\rm FI}$ explicitly breaks an unwanted\ndiscrete symmetry that arises from the permutation of the waterfall fields:\n$\\widehat{X}_1  \\leftrightarrow \\widehat{X}_2$.   If $m_{\\rm  FI}$ was\nabsent, the permutation symmetry would remain exact even after the SSB\nof the U(1)$_X$.  This would  act like parity and was therefore termed\n$D$-parity in~\\cite{GP}.  As a consequence of $D$-parity conservation,\nthe $D$-odd waterfall particles of  mass $g M$ would have been stable,\nand     if    abundantly     produced     during    the     preheating\nepoch~\\cite{PREHEATING,GBRM},  they could  overclose  the Universe  at\nlate times.\n\nTo avoid this undesirable situation, we introduce a small but non-zero\nFI term  $m_{\\rm FI}$. In  this case, the $D$-odd  waterfall particles\nwill have forbidden decays  to two $D$-even inflaton-related fields of\nmass $\\kappa  M$, induced by the  FI term. To  kinematically allow for\nsuch decays, we assume that $\\kappa  < g\/2$, where $g$ is the\nvalue of  the U(1)$_X$ coupling constant  at the GUT  scale.  The late\ndecays  of the  $D$-odd waterfall  fields will  then reheat  again the\nUniverse at  temperature $T_g$, and  so release enormous  entropy that\nmight    be   sufficient   to    reduce   the    gravitino   abundance\n$Y_{\\widetilde{G}}$ below the BBN  limits.  More explicitly, after the\nUniverse  passes  through  a  second reheating  phase,  the  gravitino\nabundance may be estimated by~\\cite{GPP}:\n\\begin{equation}\n  \\label{YGtilde}\nY_{\\widetilde{G}}\\ \\approx \\ \\frac{7.6\\times 10^{-11}}{\\kappa g}\\\n\\Bigg( \\frac{T_g}{10^{10}~{\\rm GeV}}\\Bigg)\\; ,\n\\end{equation}\nHence, for  second reheat temperatures  $T_g \\sim 1$~TeV  and inflaton\ncouplings $\\kappa  \\stackrel{>}{{}_\\sim} 10^{-2}$, the strict\nconstraint $Y_{\\widetilde{G}} \\stackrel{<}{{}_\\sim} 10^{-15}$, for \\(m_{\\tilde G}\\lesssim 500\\)~GeV, can be\ncomfortably met.\n\nTo  determine the  second reheat  temperature  $T_g$, we  may use  the\nstandard freeze-out condition $\\Gamma_g = H(T_g)$, where\n\\begin{equation}\n  \\label{Gg}\n\\Gamma_g\\ =\\ \\frac{g^4}{128\\pi}\\ \\frac{m^4_{\\rm FI}}{M^3}\n\\end{equation}\nis the  decay rate of the  $D$-odd particles and \n\\begin{equation}\n  \\label{Hrad}\nH(T)\\ =\\ \\Bigg( \\frac{\\pi^2 g_*}{90}\\Bigg)^{1\/2}\\, \\frac{T^2}{m_{\\rm Pl}}\n\\end{equation}\nis the  Hubble expansion parameter  in the radiation dominated  era of\nthe Universe and  $m_{\\rm Pl} = 2.4\\times 10^{18}$~GeV  is the reduced\nPlanck mass. In  particular, for a fixed given value  of~$T_g$, we may\ninfer the required size of the FI mass term $m_{\\rm FI}$~\\cite{GPP}:\n\\begin{equation}\n  \\label{mFI}\n\\frac{m_{\\rm FI}}{M}\\ \\approx\\ 8.4 \\times 10^{-4}\\times \\Bigg(\n\\frac{0.5}{g}\\Bigg)^{3\/4} \\Bigg(\\frac{T_g}{10^9~{\\rm\nGeV}}\\Bigg)^{1\/2}\\, \\Bigg( \\frac{10^{16}~{\\rm\nGeV}}{M}\\Bigg)^{1\/4}\\; .\n\\end{equation}\nAs can  be seen from~(\\ref{mFI}), for  $T_g \\sim 1$~TeV,  it should be\n$m_{\\rm FI}\/M \\sim 10^{-6}$, so the FI mass term $m_{\\rm FI}$ needs be\nmuch smaller  than $M$. Detailed  discussion of how such  an hierarchy\ncan  be naturally  achieved within  the SUGRA  framework may  be found\nin~\\cite{GPP}.\n\n\n\\subsection{Constraints from Cosmological Inflation}\n\nHere we recall the constraints derived in~\\cite{GPP} on the $F_D$-term\nmodel  from   cosmological  inflation.   In  fact,  there   are  three\nconstraints that need to be considered.\n\nThe  first  constraint arises  from  the  requirement  of solving  the\nhorizon  and flatness  problems  of the  standard Big-Bang  Cosmology.\nAccording to  the inflationary paradigm, these  problems may naturally\nbe solved, if our observable  Universe had an accelerated expansion of\na  number of 50--60  $e$-folds.  In  the slow-roll  approximation, the\nnumber of $e$-folds, ${\\cal N}_e$, may be calculated by~\\cite{review}\n\\begin{equation}\n  \\label{Nefold}\n{\\cal N}_e\\ =\\ \\frac{1}{m^2_{\\rm Pl}}\\; \\int_{\\phi_{\\rm\n    end}}^{\\phi_{\\rm exit}}\\, d\\phi\\: \\frac{V_{\\rm inf}}{V'_{\\rm inf}}\\\n    \\simeq\\ 55\\; ,\n\\end{equation}\nwhere $\\phi=\\sqrt{2}\\, {\\rm Re}\\,S$  is the inflaton field and $V_{\\rm inf}$\nis the $F_D$-term inflaton potential  that can be found in Section 2.1\nof~\\cite{GPP}. We  will always denote differentiation  with respect to\n$\\phi$ with a prime on  $V_{\\rm inf}$.  Moreover, $\\phi_{\\rm exit}$ is\nthe value of $\\phi$, when our present horizon scale exited inflation's\nhorizon, whilst $\\phi_{\\rm end}$ is its value at the end of inflation.\nSpecifically, the field value  $\\phi_{\\rm end}$ may be determined from\nthe condition:\n\\begin{equation}\n  \\label{slow}\n{\\sf max}\\{\\epsilon(\\phi_{\\rm end}),|\\eta(\\phi_{\\rm\nend})|\\}\\ =\\ 1\\, ,\n\\end{equation}\nwith\n\\begin{equation}\n  \\label{epseta}\n\\epsilon\\ =\\ \\frac{m_{\\rm Pl}^2}{2}\\ \\left(\n\\frac{V'_{\\rm inf}}{V_{\\rm inf}}\\right)^2\\,,\\qquad\n\\eta\\ =\\  m_{\\rm Pl}^2\\ \\frac{V''_{\\rm inf}}{V_{\\rm inf}}\\ .\n\\end{equation}\n\nThe other  two inflationary constraints come from  the so-called power\nspectrum  $P_{\\cal R}$  of  curvature perturbations  and the  spectral\nindex $n_s$.   The square root  of the power  spectrum, $P^{1\/2}_{\\cal\nR}$, is given by\n\\begin{equation}\n    \\label{PR}\nP^{1\/2}_{\\cal R}\\ =\\ \\frac{1}{2\\sqrt{3}\\, \\pi m^3_{\\rm Pl}}\\;\n\\frac{V_{\\rm inf}^{3\/2}(\\phi_{\\rm exit})}{|V'_{\\rm inf}(\\phi_{\\rm\nexit})|}\\ .\n\\end{equation}\nThis prediction must be compared  with the result obtained by a 3-years\nWMAP analysis of CMB data~\\cite{WMAP3},\n\\begin{equation}\n  \\label{Pr}\nP^{1\/2}_{\\cal R}\\ \\simeq\\ 4.86\\times 10^{-5}\\, .\n\\end{equation}\nMoreover, in  the slow-roll approximation, the  spectral index $n_{\\rm\ns}$ is given by~\\cite{review}\n\\begin{equation}\n  \\label{nS}\nn_{\\rm s}\\ =\\ 1-6\\epsilon(\\phi_{\\rm exit})\\ +\\ 2\\eta(\\phi_{\\rm exit})\\\n\\simeq\\ 1\\ +\\ 2\\eta(\\phi_{\\rm exit}),\n\\end{equation}\nwhere the parameter $\\epsilon$  is negligible in the $F_D$-term model.\nRecently, after analysing its data collected in the last 5 years, WMAP\nhas reported the value for the spectral index~\\cite{Dunkley:2008ie}:\n\\begin{equation}\n  \\label{nswmap}\nn_{\\rm  s}\\: -\\: 1\\ =\\ -0.037_{-0.015}^{+0.014}\\ .\n\\end{equation}\nThis result  slightly favours a red-tilted spectrum  and is consistent\nwith scale invariance at the 2.64~$\\sigma$ confidence level.\n\nGiven   the   three   constraints   (\\ref{Nefold}),   (\\ref{Pr})   and\n(\\ref{nswmap}), and  assuming that  all inflaton couplings  are equal,\ni.e.~$\\kappa =  \\lambda = \\rho$,  one obtains within mSUGRA  the upper\nbound~\\cite{GPP}\n\\begin{equation}\n  \\label{mSUGRA}\n\\kappa\\ \\stackrel{<}{{}_\\sim}\\ 2\\times 10^{-2}\\; .\n\\end{equation}\nOn the  other hand,  the inflationary  scale $M$ is  close to  the GUT\nscale,  i.e.~$M \\sim  10^{16}$~GeV,  when $\\kappa$  reaches its  upper\nbound imposed  by inflation.  For  an inflaton sector that  realizes a\nnext-to-minimal K\\\"ahler potential with a negative Hubble-induced mass\nterm  for  $S$~\\cite{hilltop}, the  upper  limit  on  $\\kappa$ may  be\nslightly relaxed to~\\cite{GPP}\n\\begin{equation}\n  \\label{nmSUGRA}\n\\kappa\\ \\stackrel{<}{{}_\\sim}\\ 3.2\\times 10^{-2}\\; ,\n\\end{equation}\nwhilst $M$ decreases to $M \\simeq 0.5 \\times 10^{16}$~GeV.\n\nIt is important to  properly translate the upper bounds~(\\ref{mSUGRA})\nand~(\\ref{nmSUGRA}) on $\\kappa$ obtained at the inflationary scale $M$\ninto  the  respective  ones  on  $\\lambda$ and  $\\rho$  for  the  soft\nSUSY-breaking scale~$M_{\\rm SUSY}$. As  we will see more explicitly in\nthe next  section, it  is the product  $\\lambda\\rho$ evaluated  at the\nscale $M_{\\rm SUSY}$ that controls the strength of annihilation of the\nLRHSs into the Higgs fields  and other SM particles.  Even though the\nrenormalization  group (RG) evolution  of $\\rho$  from $M$  to $M_{\\rm\nSUSY}$ may be ignored, as~$\\rho (M) \\approx \\rho (M_{\\rm SUSY})$, this\nis  not the  case for  the coupling  $\\lambda$.  Neglecting  gauge and\nsmall  Yukawa  couplings  of  order  $10^{-1}$, the  RG  equation  for\n$\\lambda$ is given by~\\cite{BS}\n\\begin{equation}\n  \\label{RGlambda}\n16\\pi^2\\, \\frac{d\\lambda}{dt}\\ =\\ \\lambda\\, \\Bigg(\\, \\frac{3}{2}\\,\nh^2_t\\: +\\: \\frac{3}{2}\\: h^2_b\\,\\Bigg)\\; ,\n\\end{equation}\nwhere $t = \\ln (Q^2\/M_{\\rm  SUSY})$. Assuming that the RG evolution is\ndominated  by the  top-quark Yukawa  coupling $h_t$,  the  solution to\n(\\ref{RGlambda}) is easily found to be\n\\begin{equation}\n  \\label{RGsolution}\n\\lambda (M_{\\rm SUSY})\\ =\\ \\lambda (M)\\, \n\\Bigg(\\frac{M_{\\rm SUSY}}{M}\\Bigg)^{3h^2_t\/(16\\pi^2)}\\ \\approx \\\n0.57\\times \\lambda (M)\\; .\n\\end{equation}\nTo obtain the last  result in~(\\ref{RGsolution}), we assumed that $h_t\n\\approx 1$ and $M_{\\rm SUSY}\/M \\sim 10^{-13}$. Then, starting with the\nboundary condition  $\\lambda = \\kappa$ at the  inflationary scale $M$,\nthe  RG  running~(\\ref{RGsolution})  of  $\\lambda$ implies  the  upper\nlimits:\n\\begin{equation}\n  \\label{Ulambda}\n\\lambda (M_{\\rm SUSY})\\ \\stackrel{<}{{}_\\sim}\\ 1.14\\times\n  10^{-2}\\,,\\qquad \\lambda (M_{\\rm SUSY})\\ \\stackrel{<}{{}_\\sim}\\ \n1.82\\times 10^{-2}\\; ,\n\\end{equation}\nfor an inflaton  sector with a minimal and  a next-to-minimal K\\\"ahler\npotential, respectively.\n\n\nIn  addition   to  constraints  from  inflation,  one   may  also  get\nconstraints on the  size of $M$ from cosmic strings  that arise due to\nthe SSB  of the local U(1)$_X$  symmetry.  For values  of $\\kappa \\sim\n10^{-2}$ of  our interest, this implies that  one must have~\\cite{GPP}\n$M \\stackrel{<}{{}_\\sim} 0.5\\times  10^{16}$~GeV.  This constraint may\nbe a  bit restrictive for the  mSUGRA model, but it  can be completely\navoided if  the waterfall sector realizes an  SU(2)$_X$ gauge symmetry\ninstead   of   U(1)$_X$,    whose   SSB   generates   no   topological\ndefects~\\cite{GPP}.  Consequently, we will conservatively consider the\nlimits  stated in~(\\ref{mSUGRA}),  (\\ref{nmSUGRA}) and~(\\ref{Ulambda})\nwhen implementing  inflationary constraints on the  relic abundance of\nthe LRHS in the next section.\n\n \n\n\\bigskip\n\n\\setcounter{equation}{0}\n\\section{Right-Handed Sneutrino as Thermal Dark Matter}\\label{CDM}\n\nIn the  $F_D$-term hybrid model  $R$-parity is conserved,  even though\nthe lepton number $L$, as well  as $B-L$, are explicitly broken by the\nMajorana   operator    $\\frac{1}{2}   \\rho   \\widehat{S}   \\widehat{N}\n\\widehat{N}$. We note that all  superpotential couplings either  conserve the $B-L$\nnumber or break it by even  number of units.  Since $R$-parity of each\nsuperpotential operator is determined to  be $R = (-1)^{3(B-L)} = +1$,\nthe $F_D$-term  hybrid model conserves $R$-parity.   As a consequence,\nthe LSP  of the spectrum is stable  and can be a  viable candidate for\nCold Dark  Matter~(CDM). As  an extension of  the MSSM, our  model can\naccommodate  the standard  SUSY CDM  candidates, such  as  the lightest\nneutralino. Because  of the connection between the  Higgs and neutrino\nsectors, on  the one  hand, and  inflation, on the  other, it  is very\ninteresting  to  explore  the  possibility of  having  a  right-handed\nsneutrino as LSP in order to solve the CDM problem.  As we will see in\nSection 3.3, this renders  the $F_D$-term model much more constrained,\nleading to sharp predictions for scattering cross-sections relevant to\nexperiments of direct searches for~CDM.\n\n\n\\subsection{Sneutrino Mass Spectrum}\n\nBefore  calculating the  sneutrino relic  abundance in  our  model, we\nfirst observe that light  right-handed sneutrinos may easily appear in\nthe spectrum.  Ignoring the terms proportional to the small neutrino-Yukawa  couplings, the\n\\(6\\times    6\\)   right-handed    sneutrino   mass    matrix   ${\\cal\nM}^2_{\\widetilde  N}$ is  given in  the weak  basis $(\\widetilde{N}_{1,2,3},\n\\widetilde{N}^*_{1,2,3})$ by\n\\begin{equation}\n  \\label{eq:SneutrinoMassMatrix}\n\t{\\cal M}^2_{\\widetilde N} = \n\t\\frac{1}{2} \n\t\\left(\\begin{array}{cc}\n\t\t\\rho^2 v^2_S      + M^2_{\\widetilde  N} & \n\t\t\\rho A_\\rho v_S   + \\rho\\lambda v_u v_d \\\\\n\t\t\\rho A^*_\\rho v_S + \\rho\\lambda v_u v_d &\n\t\t\\rho^2 v^2_S      + M^2_{\\widetilde  N}\n\t\\end{array} \\right),\n\\end{equation}\nwhere  $v_S   =  \\langle  S  \\rangle$,  $v_{u,d}   =  \\langle  H_{u,d}\n\\rangle$. Moreover, $M^2_{\\widetilde  N}$ is the soft  SUSY-breaking mass matrix\nassociated with  the sneutrino fields and \\(A_\\rho\\)  is the sneutrino\ntrilinear coupling  matrix. In general,  \\({\\cal M}^2_{\\widetilde N}\\)\nis diagonalized by a unitary matrix \\(U_{\\widetilde N}\\) such that\n\\begin{equation}\\label{eq:SneutrinoDiag}\n\tU_{\\widetilde N}^\\dagger  \\,\n\t{\\cal M}^2_{\\widetilde N} \\,\n\tU_{\\widetilde N} =\n\t\\textrm{diag}\n\t\\left(m_{\\tilde N_1}^2,m_{\\tilde N_2}^2,\\dots,m_{\\tilde N_6}^2\\right),\n\\end{equation}\nwhere the sneutrino masses are ordered, such that \\(m_{\\tilde N_1} < m_{\\tilde N_2}< \\dots < m_{\\tilde N_6}\\). Neglecting the possible\nflavor   structure   contained   in   the   \\(3\\times   3\\)   matrices\n\\(M^2_{\\widetilde  N}\\) and  \\(A_\\rho\\), the  sneutrino  spectrum will\nthen consist of 3 light (heavy) right-handed sneutrinos with masses\n\\begin{equation}\\label{eq:SneutrinoMasses}\n\tm^2_{\\tilde N_{L(H)}} = \n\t     \\rho^2 v^2_S \n\t+    M^2_{\\widetilde N}  \n\t-(+) \\left|\\rho A_\\rho v_S + \\rho\\lambda v_u v_d\\right|.\n\\end{equation}\nAll    mass   terms    in   (\\ref{eq:SneutrinoMasses})    are   ${\\cal\nO}(100$--1000)~GeV,  so  a  proper  choice  of  model  parameters  can\naccommodate  a LRHS  to act  as  LSP.  Unless  the trilinear  coupling\n\\(A_\\rho\\) is small compared  to \\(\\mu\\), the off-diagonal elements in\n(\\ref{eq:SneutrinoMassMatrix}) will  induce a sizeable  mixing between\nthe  heavy and  light right-handed  sneutrino states,  suppressing the\nlight masses to  values smaller than \\((\\mu^2+M_{\\tilde N}^2)^{1\/2}\\).\nThis will be  demonstrated in our discussion of  the numerical results\nin Section~3.3,  where the  $F_D$-term model is  embedded within the mSUGRA\nframework.\n\n\\subsection{Sneutrino Annihilation and Relic Density}\n\nRight-handed sneutrinos  as CDM  were considered in~\\cite{GGP}  in the\ncontext   of   the  MSSM   with   right-handed  neutrino   superfields\n$\\widehat{N}_i$  and  bare  Majorana  masses  $M^N_{ij}  \\widehat{N}_i\n\\widehat{N}_j$.   This   analysis  shows  that   thermal  right-handed\nsneutrinos  have  rather  high  relic abundances  and  will  generally\noverclose  the Universe.   The reason  is  that because  of the  small\nneutrino Yukawa couplings $h^\\nu_{ij}$,  the self- and co-annihilation\ninteractions of the sneutrino LSP with itself and other MSSM particles\nare rather weak.  These weak  processes do not allow the sneutrino LSP\nto  stay long  enough  in thermal  equilibrium  before its  freeze-out\ntemperature, such that its number density gets reduced to the observed\nvalue   $\\Omega_{\\rm  DM}   h^2   \\approx  0.11$~\\cite{Dunkley:2008ie}\n[cf. (\\ref{OmegaDM})].  In fact, the predicted values for $\\Omega_{\\rm\nDM}  h^2$ turn  out  to be  many  orders of  magnitude larger  than~1.\nInstead, right-handed  sneutrinos can be viable  thermal DM candidates in the MSSM if\nthey  significantly mix  with  left-handed sneutrinos,  either by increasing the \nSUSY-breaking  trilinear couplings~\\cite{Hooper:2004dc}~\\footnote{For an earlier discussion, see also\nthe paper by N.   Arkani-Hamed {\\em et al.}  in~\\cite{Francesca}.}, or by lowering the right-handed neutrino mass scale~\\cite{Arina:2008bb}. \nAlternatively, right-handed sneutrinos may become thermal DM by introducing a new U(1)' gauge coupling to make the self-annihilation interaction sufficiently strong~\\cite{LMN}. Recently, there has been a paper~\\cite{Cerdeno} discussing the possibility of right-handed sneutrinos as DM in an extended version of the next-to-minimal supersymmetric standard model.\n\nIn the $F_D$-term hybrid model, a novel possibility opens up.\nAs was first observed in~\\cite{GPP}, there   exists    a   new   quartic   coupling    described   by   the\nLagrangian~\\footnote{The    implications   of    a   generic\nsinglet-Higgs  quartic coupling  for the  CDM abundance  and detection\nwere studied  before in~\\cite{Silveira:1985rk,mcdonald,pospelov}, within a simple\nnon-SUSY model.}\n\\begin{equation}\\label{Llsp}\n\t{\\cal L}^{\\rm LSP}_{\\rm int} =\n\t\\frac{1}{2}\\lambda\\rho \\tilde N^*_i \\tilde N^*_i H_uH_d \n\t+{\\rm H.c.}\\; .\n\\end{equation}\nThis quartic coupling between right-handed sneutrinos and Higgs fields\nresults from  the $F$-term of  the inflaton field  $F_S$: $\\frac{1}{2}\n\\rho \\widehat{N}_i \\widehat{N}_i + \\lambda \\widehat{H}_u \\widehat{H}_d\n\\subset  F_S$.   If strong  enough,  the interaction~(\\ref{Llsp})  can\nthermalize  the  sneutrinos  and  make  them  annihilate  to  a  level\ncompatible  with the current  CMB  data  via the  processes  depicted  in\nFigure~\\ref{fig:graphs}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip,width=0.90\\textwidth]{plots\/graphs.eps}\n\\caption{Feynman graphs related to sneutrino annihilation.}\n\\label{fig:graphs}\n\\end{figure}\n\nFor sneutrino masses of our  interest, the most relevant processes are\nthe  off-resonant  pair-production  of  $W$ bosons  and  the  on-shell\npair-production  of light  Higgs bosons.  An initial  estimate  of the\nprocess  \\(\\tilde N\\tilde N  \\to \\langle  H_u\\rangle H_d  \\to W^+W^-\\)\nfor \\(m_{\\tilde N} > m_W\\) yields\n\\begin{equation}\n  \\label{estimate1}\n\t\\Omega_{\\rm DM}\\, h^2\\ \\approx\\\n\t\\left(\\frac{10^{-4}}{\\rho^2\\lambda^2}\\right)\n\t\\left(\\frac{\\tan\\beta\\, m_H}{g_W\\, m_W}\\right)^2\\; .\n\\end{equation}\nIn  order  to  obtain  an  acceptable CDM  density,  relatively  large\ncouplings \\(\\rho\\) and  \\(\\lambda\\) are needed, \\(\\rho\\lambda \\gtrsim\n0.1\\).  However, these  large values for $\\lambda$ and  $\\rho$ are not\ncompatible with  the constraints  derived by inflation.  \n\nThe situation  differs for sneutrino masses $m_{\\widetilde  N} < m_W$,\nin  large $\\tan\\beta$ scenarios,  in which  light Higgs  bosons couple\nappreciably  to  $b$-quarks~\\cite{Sabine}.    In  particular,  in  the\nkinematic region $m_{H_1}  \\approx 2 m_{\\widetilde{N}_{1}}$, the\nself-annihilation process  $\\widetilde{N}_{1} \\widetilde{N}_{1} \\to \\langle  H_u \\rangle H_d \\to b\\bar{b}$  becomes resonant, and\nthe above estimate modifies to\n\\begin{equation}\n  \\label{estimate2}\n\t\\Omega_{\\rm DM} h^2 \\approx 10^{-4} \\times B^{-1}(H_1 \\to\n\t\\widetilde N_1 \\widetilde N_1) \\times\n\t\\left(\\frac{m_{H_1}}{100~{\\rm GeV}}\\right)^2.\n\\end{equation}\nConsequently,  if  the couplings  $\\lambda,\\rho$  are  not too  small,\ne.g.~$\\lambda\\rho  \\gtrsim  10^{-3}$,  the  right-handed  sneutrino\n$\\widetilde{N}_1$  can   now  efficiently  annihilate   via  a  resonant\n$H_1$-boson  into  pairs of  $b$-quarks,  thus  obtaining  a relic  DM\ndensity compatible with the observed value (\\ref{OmegaDM}).\n\nWe   will  now   show  that   the   naive  estimates~(\\ref{estimate1})\nand~(\\ref{estimate2}) presented in~\\cite{GPP}  are in a fairly  good \nagreement  with  a complete  calculation  of  all relevant  sneutrino\nannihilation processes displayed  in Figure~\\ref{fig:graphs}.  To this\nend,  we use  the short-hand  notation \\(M_{XY}  =  M(\\tilde N_a\\tilde\nN_b\\to  XY)\\)  to  denote  the  individual  matrix  elements  for  the\nannihilation  of sneutrinos  \\(\\tilde N_a\\)  and \\(\\tilde  N_b\\).  The\ncontributing   processes    may   be   listed    as   follows   (\\(c_w\n=\\cos\\theta_w\\), \\(v=2m_W\/g_w\\)):\n\\begin{itemize}\n\\item[{\\bf (i)}]  \\(\\tilde N_a  \\tilde N_b \\longrightarrow  H^+ H^-\\),\n  via contact quartic interaction and $s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{H^+H^-}\\ =\\\n\tg_{\\tilde N_a \\tilde N_bH^+H^-}\n\t- v^2 \\sum_{k=1}^3\\:\n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k}g_{H_k H^+H^-}}\n\t{s-m_{H_k}^2+im_{H_k}\\Gamma_{H_k}}\\ ; \n\\end{equation}\n\n\\item[{\\bf (ii)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow W^+ W^-\\), via\n$s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{W^+W^-}\\ =\\\n\tg_w m_W v\\, \n\t\\Bigg[\\, 2\\: +\\: \\Bigg(1-\\frac{s}{2m_W^2}\\Bigg)^2\\,\\Bigg]^{1\/2}\\:\n\t\\sum_{k=1}^3\\:\n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k}g_{H_k VV}}\n\t{s-m_{H_k}^2 + i m_{H_k}\\Gamma_{H_k}}\\ ;\n\\end{equation}\n\n\\item[{\\bf (iii)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow ZZ\\), via\n$s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{ZZ}\\ =\\\n\t\\frac{g_w m_W v}{2c_w}\\, \n\t\\Bigg[\\, 2\\: +\\: \\Bigg(1-\\frac{s}{2m_Z^2}\\Bigg)^2\\,\\Bigg]^{1\/2}\\:\n\t\\sum_{k=1}^3\n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k} g_{H_k VV}}\n\t{s - m_{H_k}^2 + im_{H_k}\\Gamma_{H_k}}\\ ;\n\\end{equation}\n\n\\item[{\\bf (iv)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow f \\bar f\\), via\n$s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{f_\\alpha \\bar f_\\alpha}\\ =\\\n\tv\\sqrt{2s}\\,\n\t\\Bigg[\\,\n\t\t|A_S|^2\\Bigg(1-\\frac{4m_\\alpha^2}{s}\\Bigg)\\: +\\: |A_P|^2\\,\n\t\\Bigg]^{1\/2}\\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\tA_{S\/P}\\ =\\ \n\t\\sum_{k=1}^3\\:  \n   \\frac{\n\t\tg_{\\tilde N_a \\tilde N_b H_k}\n\t\tg_{f_\\alpha}g^{S\/P}_{H_k \\bar f_\\alpha f_\\alpha}\n\t}\n\t{s-m_{H_k}^2+im_{H_k}\\Gamma_{H_k}}\\ ;\n\\end{equation}\n\n\\item[{\\bf (v)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow H_i H_j\\), via\n  contact quartic interaction, $s$-channel Higgs exchange and\n  $t\/u$-channel sneutrino exchange:\n\\begin{eqnarray}\n\tM_{H_i H_j} \\!&=&\\!\n\tg_{N_aN_bH_iH_j}\\: \n\t-\\: v^2\\,\n\t\\sum_{k=1}^3\\: \n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k}g_{H_i H_j H_k}}\n\t{s-m_{H_k}^2+im_{H_k}\\Gamma_{H_k}} \\nonumber\\\\\n\t&-& v^2\\,\n\t\\sum_{c=1}^6\\: \n        \\frac{g_{\\tilde N_a \\tilde N_c H_i}\n\t\t\t\t\tg_{\\tilde N_b \\tilde N_c H_j}}\n\t{t-m_{\\tilde{N}_c}^2}\\ \n\t-\\ v^2\\,\n\t\\sum_{c=1}^6\n\t\\frac{g_{\\tilde N_a \\tilde N_c H_j}\n\t\t\tg_{\\tilde N_b \\tilde N_c H_i}}\n\t{u-m_{\\tilde{N}_c}^2}\\ ;\n\\end{eqnarray}\n\n\\item[{\\bf (vi)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow H^+ W^-\\),\nvia $s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{H^+W^-}\\ =\\\n\t\\frac{g_w v}{2}\\,\n\t\\Bigg[\\,\n\t\t\\frac{s^2}{4m_W^2}\n\t\t\\Bigg(1-\\frac{m_W^2+m_{H^+}^2}{s}\\Bigg)^2\\: -\\: m_{H^+}^2\n\t\\Bigg]^{1\/2}\n\t\\sum_{k=1}^3\n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k}\n\t\t\tg_{H_k H^+ W^-}}\n\t{s-m_{H_k}^2+im_{H_k}\\Gamma_{H_k}}\\ ;\n\\end{equation}\n\n\\item[{\\bf (vii)}] \\(\\tilde N_a \\tilde N_b \\longrightarrow H_i Z\\),\nvia $s$-channel Higgs exchange:\n\\begin{equation}\n\tM_{H_i Z}\\ =\\\n\t\\frac{g_w v}{4c_s}\\,\n\t\\Bigg[\\,\n\t\t\\frac{s^2}{4m_Z^2}\n\t\t\\Bigg(1-\\frac{m_Z^2+m_{H_i}^2}{s}\\Bigg)^2\\: -\\: m_{H_i}^2\n\t\\Bigg]^{1\/2}\n\t\\sum_{k=1}^3\n\t\\frac{g_{\\tilde N_a \\tilde N_b H_k}g_{H_k H_i Z}}\n\t{s-m_{H_k}^2+im_{H_k}\\Gamma_{H_k}}\\ .\n\\end{equation}\n\\end{itemize}\nIn the  above, the effective  sneutrino-to-Higgs couplings \\(g_{\\tilde\nN_a \\tilde  N_b H^+H^-}\\), \\(g_{\\tilde  N_a \\tilde N_b H_j  H_j}\\) and\n\\(g_{\\tilde  N_a \\tilde  N_c H_i}\\)  that arise  from  the interaction\nLagrangian~(\\ref{Llsp})    are    given    by    (\\(c_\\beta=\\cos\\beta,\ns_\\beta=\\sin\\beta\\))\n\\begin{eqnarray}\n\tg_{\\tilde N_a \\tilde N_b H^+H^-} \\!&=&\\! \n\t\\frac{\\lambda\\rho}{2}\\: c_\\beta s_\\beta\\delta_{ab}, \\\\\n\tg_{\\tilde N_a \\tilde N_b H_i H_j} &=&\n\t\\frac{\\lambda\\rho}{2}\\:  \n\t\\frac{\\delta_{ab}}{1+\\delta_{ij}}\n\t\\left[\\left(\n\t\t  O_{\\phi_u i} O_{\\phi_d j} \n\t\t+ O_{a i}      O_{\\phi_u j}s_\\beta\n\t\t+ O_{a i}      O_{\\phi_d j}c_\\beta\n\t\t- O_{a i}      O_{a j}s_\\beta c_\\beta\n\t\\right)\\right. \\nonumber\\\\\n\t&&\\qquad\\qquad\\quad+ \\left.(i \\leftrightarrow j)\n\t\\right], \\\\\n\tg_{\\tilde N_a \\tilde N_b H_i}  \\!&=&\\! \n\t\\frac{\\lambda\\rho}{2}\\:\n\t\\left(\n\t\tO_{\\phi_d i}s_\\beta + O_{\\phi_u i}c_\\beta\n\t\\right)\\delta_{ab}\\; ,\n\\end{eqnarray}\nwhere $O$ is  the $3\\times 3$ Higgs-boson mixing  matrix, defined such\nthat \n\\begin{equation}\n(\\phi_d,\\, \\phi_u,\\, a)^T\\ =\\ O \\: (H_1,\\, H_2,\\, H_3)^T\\; .\n\\end{equation}\nFor the  effective Higgs-boson  couplings \\(g_{H_k H_i  Z}\\), \\(g_{H_k\nH^+ W^-}\\),  \\(g_{H_i H_j  H_k}\\), \\(g_{H_k H^+H^-}\\),  \\(g_{H_k VV}\\)\nand   \\(g_{f_\\alpha}\\,  g^{S\/P}_{H_k   f_\\alpha  \\bar   f_\\alpha  }\\),\nincluding $O$, the Higgs-boson  masses $m_{H_{1,2,3}}$ and their decay\nwidths $\\Gamma_{H_{1,2,3}}$,  we follow the  notations and conventions\nof~\\cite{Lee:2003nta,Lee:2007gn}  and calculate them  by means  of the\ncomputational   package  {\\tt CPsuperH}.   \n\nThe   total  annihilation   cross-section  \\(\\sigma_{ab}=\\sigma(\\tilde\nN_a\\tilde N_b\\to \\textrm{all})\\) may then be conveniently expressed as\nthe sum of all channels,\n\\begin{equation}\n\t\\sigma_{ab}\\ =\\\n\t\\sigma_{H^+H^-}+\n\t\\sigma_{W^+W^-}+\n\t\\sigma_{ZZ}+\n\t\\sigma_{H^+W^-}+\\sigma_{H^-W^+}+\n\t\\sum_{i=1}^3        \\sigma_{H_i Z} +\n\t\\sum_{i,j=1}^3     \\sigma_{H_i H_j} + \n\t\\sum_{f=\\tau,b,t} \\sigma_{f \\bar f}\\; .\n\\end{equation}\nThe individual cross sections \\(\\sigma_{XY}\\) are defined by\n\\begin{equation}\n\t\\sigma_{XY} \\ = \\ \n\t\\frac{1}{1+\\delta_{XY}}\\:\n\t\\frac{1}{16\\pi \\lambda(s,m_{\\tilde N_a}^2,m_{\\tilde N_b}^2)}\\:\n\t\\int_{t^-}^{t^+} dt\\, |M_{XY}|^2,\n\\end{equation}\nwith\n\\begin{eqnarray}\n\tt^\\pm \\!&=&\\!  \n\tm_X^2 + m_{\\tilde N_a}^2 - \n\t\\frac{1}{2s}\n\t\\left(\n\t\t(s+m_{\\tilde N_a}^2-m_{\\tilde N_b}^2)(s+m_X^2-m_Y^2)\n\t\\right. \\nonumber\\\\\n\t\t&& \\qquad\\qquad\\qquad\\qquad\n\t\\left.\\mp \\lambda^{1\/2}(s,m_{\\tilde N_a}^2,m_{\\tilde N_b}^2)\n\t\t    \\lambda^{1\/2}(s,m_X^2,m_Y^2)\\right)\\;,\\qquad\\quad \\\\\n\t\\lambda(a,b,c) \\!&=&\\! (a-b-c)^2-4bc.\n\\end{eqnarray}\nIn     order     to     calculate     the    relic     density,     we\nfollow~\\cite{Griest:1990kh}   and  use   an   effective  cross-section\naveraged over all initial sneutrino channels,\n\\begin{eqnarray}\n  \\label{eq:EffectiveCrossSection}\n\t\\sigma_{\\rm eff} \\! &=&\\!\n\t\\sum_{a,b=1}^6\t\\sigma_{ab} \\frac{g_a g_b}{g_{\\rm eff}^2}\n\t\t(1+\\Delta_a)^{3\/2}\n\t\t(1+\\Delta_b)^{3\/2}\\;\n\\exp \\big[ -x(\\Delta_a+\\Delta_b)\\big]\\; ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\ng_{\\rm eff} \\ = \\\n \\sum_{a=1}^6 g_a(1+\\Delta_a)^{3\/2}e^{-x\\Delta_a}\\; , \\qquad\n\\Delta_a \\ =\\ \\frac{m_{\\tilde N_a}-m_{\\tilde N_1}}{m_{\\tilde N_1}}\\ .\n\\end{equation}\nIn~(\\ref{eq:EffectiveCrossSection}),   both   the   effects   of   LSP\nself-annihilation and co-annihilation  with the heavier sneutrinos are\nincluded\\footnote{Note    that    co-annihilation    effects    become\nsignificant, only if the  mass differences with the heavier sneutrinos\nare smaller or comparable to the LSP freeze-out temperature, i.e,~when\n\\(m_{\\tilde  N_a}-m_{\\tilde   N_b}\\stackrel{<}{{}_\\sim}  T_f\\).}.   In\nterms of the effective cross-section~(\\ref{eq:EffectiveCrossSection}),\nthe thermally-averaged effective cross-section may be calculated as\n\\begin{equation}\\label{eq:sigmavEffective}\n\t\\langle\\sigma v\\rangle\\ =\\ \\frac{x^{3\/2}}{2\\pi^{3\/2}}\n\t\\int_0^\\infty dv\\, v^2 (\\sigma_{\\rm eff} v)\\, e^{-x v^2\/4}\\ ,\n\\end{equation}\nwhere the integrand is expressed in terms of the relative velocity \\(v\\), such that\n\\begin{equation}\n\ts \\ = \\ \\frac{4m_{\\tilde N_1}^2}{1-v^2\/4} \\ . \n\\end{equation}\nFrom the expression (\\ref{eq:sigmavEffective}), we may determine the freeze-out temperature\n \\(x_f= m_{\\tilde N_1}\/T_f\\) by iteratively solving the equation\n\\begin{equation}\n\tx_f\\ =\\ \n\t\\ln\\Bigg(\n\\frac{0.038 g_{\\rm eff}\\, M_{\\rm Pl}\\, m_{\\tilde N_1} \\langle\\sigma v\\rangle}\n\t{g_*^{1\/2}x_f^{1\/2}}\\Bigg)\\; , \n\\end{equation}\nwhere  \\(  M_{Pl}=1.22\\times  10^{19}\\)~GeV  is the  Planck  mass  and\n\\(g_*\\)  is the  total  number of  effective  relativistic degrees  of\nfreedom at  the temperature  of the LSP  freeze-out.  The  present day\nrelic density is then given by\n\\begin{equation}\\label{eq:RelicDensity}\n\t\\Omega_{\\rm DM}\\, h^2\\ \\approx\\ \n\t\\frac{1.07\\times 10^9~\\textrm{GeV}^{-1}}{J\\, g_*^{1\/2} m_{Pl}}\n\\end{equation}\nwhere $J$ is the  post freeze-out annihilation efficiency factor given\nby\n\\begin{equation}\n\tJ\\ =\\ \\int_0^\\infty dv\\, v (\\sigma_{eff}v)\\, \n         \\textrm{erfc}(v\\sqrt{x_f}\/2).\n\\end{equation}\nIn our  numerical estimates,  we neglect the  flavor structure  of the\nright-handed  sneutrinos  and   treat  the  three  light  right-handed\nsneutrinos \\(\\tilde N_{1,2,3}\\) as being  essentially degenerate~\\footnote{Note that the second and third right-handed sneutrinos \\(\\tilde N_{2,3}\\) will decay to the LRHS \\(\\tilde N_1\\) through the processes~\\(\\tilde N_{2,3}\\to\\tilde N_1 \\gamma, \\tilde N_1 \\nu\\bar\\nu\\). We do not address potential problems for BBN from the late decays of~\\(\\tilde N_{2,3}\\), since their rates strongly depend on the flavor structure of \\(\\rho_{ij}\\) and the Yukawa couplings \\(h^\\nu_{ij}\\) [cf. (\\ref{Wmodel})] and on the details of the model in general.}.   Since all  three light\nsneutrinos  will contribute to  the relic  density, we must therefore\nmultiply~(\\(\\ref{eq:RelicDensity}\\)) by 3  to obtain the final relic\nDM abundance.\n\n\\subsection{Numerical Results}\n\nThe numerical analysis is separated  in two parts: in the first part, we perform\na scan  over the mSUGRA parameter  space to  calculate the\nsupersymmetric particle  spectrum and identify regions  where the LRHS\ncan be a possible candidate for CDM.  In the second part, we specify two mSUGRA  scenarios and calculate the  constraints on the\neffective sneutrino annihilation coupling \\(\\lambda\\rho\\) by requiring\na sneutrino relic density of \\(\\Omega_{\\rm DM} h^2=0.11\\).\n\nIn  Figure~\\ref{fig:scans}   we  plot  the   lightest  sneutrino  mass\n\\(m_{\\tilde  N_1}\\)   as  contours  in  the   mSUGRA  parameter  plane\n(\\(m_0,m_{1\/2}\\)), for two different  values of \\(\\tan\\beta = 10\\) (left)\nand  30  (right).  In both plots of Figure~\\ref{fig:scans}, we set \\(A_0=300\\)~GeV  and  \\(\\mu>0\\). For  the\ninflaton   couplings  \\(\\lambda,\\rho\\)   required  to   calculate  the\nsneutrino masses~(\\ref{eq:SneutrinoMasses}), we simply choose\n\\begin{equation}\n\t\\lambda=\\rho=10^{-2},\n\\end{equation}\nin accordance with the bounds (\\ref{Ulambda}) derived from inflation.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip,width=0.49\\textwidth]{plots\/scan.eps}\n\\includegraphics[clip,width=0.49\\textwidth]{plots\/scan2.eps}\n\\caption{\\it Allowed  $(m_0,\\,  m_{1\/2})$  parameter  space for  a  mSUGRA\nscenario  with  \\(A_0=300\\)~GeV,  \\(\\textrm{sign}\\mu=+\\), \\(\\lambda  =\n\\rho  = 10^{-2}\\)  and \\(\\tan\\beta=10\\)  (left panel)  and  30 (right\npanel).  The black contours show  the predicted LRHS mass, while the\nsneutrino \\(\\tilde N_1\\)\/neutralino \\(\\tilde\\chi^0_1\\)\/stau \\(\\tilde\\tau_1\\) LSP is  given by the blue\/green\/orange area.\nThe  red area is  excluded by  direct SUSY  mass searches.   The white\ncontour  is  defined by  the  condition \\(m_{\\tilde  N_1}=m_{H_1}\/2\\),\nallowing  for   rapid  sneutrino  annihilation   via  the  $H_1$-boson\nresonance.}\\label{fig:scans}\n\\end{figure}\nThe  coloured areas in  Figure~\\ref{fig:scans} denote  the LSP  in the\ngiven  parameter region: sneutrino  \\(\\tilde N_1\\)  (blue), neutralino\n\\(\\tilde\\chi^0_1\\) (green) or  stau \\(\\tilde\\tau_1\\) (orange). The red\narea on the bottom\/left is  excluded by direct searches for SUSY particles. Specifically, the following experimental mass limits are used~\\cite{PDG08}:\n\\begin{eqnarray} \n\tm_{\\tilde\\chi^-_1} \\!&>&\\! 104\\textrm{ GeV}\\; , \\nonumber\\\\\n\tm_{\\tilde q}       \\!&>&\\! 375\\textrm{ GeV}\\; , \\nonumber\\\\\n\tm_{\\tilde g}       \\!&>&\\! 289\\textrm{ GeV}\\; , \\\\\n\tm_{\\tilde \\ell}    \\!&>&\\!  95\\textrm{ GeV}\\; , \\nonumber\\\\\n\tm_{\\tilde\\nu_L}      \\!&>&\\! 130\\textrm{ GeV}\\; . \\nonumber\n\\end{eqnarray}\nFigure~\\ref{fig:scans} was determined by appropriately using universal\nsoft SUSY-breaking parameters at the GUT scale according to the mSUGRA\nscheme and then solving the  MSSM RG equations down to the electroweak\nscale.  In  this respect,  our computation was  aided by  the software\npackage SPheno \\cite{Porod:2003um}.  We  neglect the RG running of the\nsneutrino parameters \\(M_{\\tilde N}^2\\) and \\(A_\\rho\\) which enter the\nsneutrino  mass  matrix~(\\ref{eq:SneutrinoMassMatrix}),  and  identify\nthem  directly with \\(m_0^2\\)  and \\(A_0\\),  respectively.  This  is a\nreasonable approximation as  their RG evolution is only  driven by the\nsmall couplings \\(\\lambda\\) and \\(\\rho\\). The Higgs coupling parameter\n\\(\\mu\\)   is  then   calculated  consistently   by   requiring  proper\nelectroweak symmetry breaking. In  the \\(F_D\\)-term model, the \\(\\mu\\)\nterm  originates  from  the  VEV  of the  inflaton  (\\ref{mu}).   This\nimmediately allows us to calculate  both the inflaton VEV, \\(\\langle S\n\\rangle=\\mu\/\\lambda\\),  and   the  mass  scale   of  the  right-handed\nneutrinos,         \\(m_N=\\rho\\langle        S         \\rangle        =\n\\frac{\\rho}{\\lambda}\\mu\\)~(\\ref{Wmodel}). For the \\(\\tilde\nN_1\\) LSP region of interest and with our choice \\(\\lambda=\\rho=10^{-2}\\), \\(\\mu\\)\nand  \\(m_N\\)  are  equal  and  of  order  300~GeV.   The  mass \\(m_{\\tilde N_1}\\) of the LRHS as  LSP  ranges  between  20--100 GeV.  This \nallows  for a rapid  annihilation of \\(\\tilde N_1\\) via  the Higgs  resonance, \\(m_{\\tilde\nN_1}=m_{H_1}\/2\\approx   57\\)~GeV,   along   the   white   contour   in\nFigure~\\ref{fig:scans}.\n\nThe \\(F_D\\)-term model puts strong constraints on the mSUGRA parameter\nspace, when requiring  a sneutrino LSP and taking  into account bounds\nfrom  inflation.    As can be seen  in   Figure~\\ref{fig:scans},  the\nconnection between LRHS mass \\(\\tilde N_1\\) and \\(\\mu\\) generally points towards\na low-energy SUSY  spectrum. This coincidentally includes the \\(H_1\\)-boson funnel region, where \\(m_{H_1}\\approx 2m_{\\tilde\nN_1}\\).  On the\nother  hand,  very large and small values  for  \\(A_0\\)  and \\(\\tan\\beta\\)  are\ndisfavoured  as  they  generally   exclude  a  sneutrino  LSP.   The above correlations may  be somewhat relaxed if non-universal\ninflaton couplings \\(\\lambda\\) and \\(\\rho\\) are considered.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip,width=0.90\\textwidth]{plots\/CDM_scenario1.eps}\n\\caption{\\it  Effective  annihilation  coupling \\(\\lambda\\rho\\)  as  a\nfunction of the  mass of the LRHS \\(m_{\\tilde  N_1}\\) for the observed\nrelic density \\(\\Omega_{\\rm DM}  h^2=0.11\\) (blue curve) in the mSUGRA\nScenario~I~(\\ref{ScenI}).  The actual  sneutrino and neutralino masses in the scenario\nare indicated by vertical lines.  The red curves denote the upper bound on\n\\(\\lambda\\rho\\) as obtained by the  CDMS-II experiment and as expected by\nthe projected sensitivities of SuperCDMS and Xenon1T.}\n\\label{fig:CDMscenarioI}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[clip,width=0.90\\textwidth]{plots\/CDM_scenario2.eps}\n\\caption{\\it As  in Figure~\\ref{fig:CDMscenarioI}, but  for the mSUGRA\nScenario II~(\\ref{ScenII}).}\n\\label{fig:CDMscenarioII}\n\\end{figure}\n\n\nIn  order to  compute  the  sneutrino relic  density  and analyze  the\nconstraints  on the  effective annihilation  coupling \\(\\lambda\\rho\\),\nthe following two mSUGRA scenarios have been selected:\n\n\\begin{itemize}\n\n\\item Scenario I:\n\\begin{equation}\n  \\label{ScenI}\nm_0=70\\textrm{ GeV},\\ m_{1\/2}=243\\textrm{ GeV},\\\nA_0=300\\textrm{ GeV},\\ \\tan\\beta=10,\\ \\mu=303\\textrm{ GeV}\\; .\n\\end{equation}\n\n\\item Scenario II: \n\\begin{equation}\n  \\label{ScenII}\nm_0=125\\textrm{ GeV},\\ m_{1\/2}=212\\textrm{ GeV}, \nA_0=300\\textrm{ GeV},\\ \\tan\\beta=30,\\ \\mu=263\\textrm{ GeV}\\; .\n\\end{equation}\n\n\\end{itemize} \n\nIn  addition, we  keep the  LRHS  mass as  a free  parameter. The  effective annihilation  coupling\n\\(\\lambda\\rho\\)  (\\ref{Llsp}) is then consistently calculated so as to obtain  a  sneutrino relic  density\n\\(\\Omega_{\\rm DM}\\, h^2=0.11\\), consistent with observation.  Furthermore,  we assume that the mass\nsplitting  between  the light  and  heavy  right-handed sneutrinos  is\nsufficiently large so that co-annihilation can be safely ignored. This\nis valid as  long as there is a sizeable mixing  between the light and\nheavy right-handed  sneutrino states, which is certainly  true for the\nmass  range \\(m_{\\tilde  N_1}< m_{\\tilde\\chi^0_1}\\)  of  our interest.\nAll other MSSM parameters and masses were calculated within the mSUGRA\nframework.  Numerical  estimates  of  the allowed  parameters  in  the\n$(m_{\\tilde{N}_1},\\lambda\\rho)$-plane are shown for Scenarios~I and~II\nin  Figures~\\ref{fig:CDMscenarioI} and Figure~\\ref{fig:CDMscenarioII},\nrespectively.\n\nAs  we  have  seen  in Section~\\ref{FDmodel}.3,  the  requirement  for\nsuccessful inflation puts upper  limits on the couplings $\\lambda$ and\n$\\rho$. Given~(\\ref{mSUGRA}),  (\\ref{nmSUGRA}) and \\ref{Ulambda}), the\nupper limits on the product  $\\lambda\\rho$ for an inflaton sector with\na minimal and next-to-minimal K\\\"ahler potential may easily be deduced\nto be\n\\begin{equation}\n  \\label{Inflimits}\n\\lambda\\rho\\ \\stackrel{<}{{}_\\sim}\\ 2.3\\times 10^{-4}\\,,\\qquad\n\\lambda\\rho\\ \\stackrel{<}{{}_\\sim}\\ 5.8\\times 10^{-4}\\,,\n\\end{equation}\nrespectively, at the soft  SUSY-breaking scale $M_{\\rm SUSY}$.  On the\nother hand, Figures~\\ref{fig:CDMscenarioI} and~\\ref{fig:CDMscenarioII}\nshow that it should be\n\\begin{equation}\n  \\label{CDMlimits}\n\\lambda\\rho\\ \\stackrel{>}{{}_\\sim}\\ 2\\times 10^{-4}\\; ,\n\\end{equation}\nin  order  to account  for  the observed  DM  relic  abundance in  the\n$H_1$-boson funnel region,  where \\(m_{\\tilde N_1}\\approx m_{H_1}\/2\\). Larger\nvalues of \\(\\tan\\beta\\)  do suppress the coupling required  to get the\nobserved  relic  density,  but  not  to a  level  compatible  with  the\ninflationary constraints~(\\ref{Inflimits}).  In  general, we find that\nLRHS masses larger than about 100~GeV are not possible within a mSUGRA\nrealization of the $F_D$-term model.  This is indicated by the\nvalue of the neutralino mass in the given mSUGRA scenario as displayed by vertical lines in Figures~\\ref{fig:CDMscenarioI} and \\ref{fig:CDMscenarioII}.\n\nFurther  constraints on the  $(m_{\\tilde{N}_1},\\lambda\\rho)$-plane may\nbe obtained by taking into  account the limits from direct searches of\nexperiments  which  look for  scattering  between Weakly  Interacting\nMassive Particles (WIMPs) and nuclei. Specifically, a WIMP, such as the\nLRHS, can directly be detected through its elastic scattering with a\nnucleus. In our case, the  relevant scattering process is \\(\\tilde N_1\n+{}^A_Z X \\to  \\tilde N_1 + {}^A_Z X\\) and  proceeds via a Higgs-boson\n$t$-channel  exchange.   Its   cross-section  may  well  be  estimated\nby~\\cite{pospelov}\n\\begin{equation}\n\t\\sigma_{\\rm el}^{\\rm nucleus}\\ \\approx\\\n\t\\frac{(1\/2\\lambda\\rho)^2 v^2 |M_X|^2}{\\pi}\\\n\t\\frac{m_{\\rm red}^2}{m_{\\tilde N_1}^2 m_{H_1}^4}\\ ,\n\\end{equation}\nwhere $m_{\\rm  red}$ is the  reduced mass of the  LRHS-nucleus system,\ni.e.\n\\begin{equation}\n\tm_{\\rm red}\\ =\\\n\t\\frac{m_{\\tilde N_1}m_{X}}{m_{\\tilde N_1}+m_X},\n\\end{equation}\nand \\(M_X\\) is the nuclear matrix element. For comparison purposes, we\nexpress  our results  in terms  of the  \\emph{nucleon}  cross section.\nAssuming the nucleus to be composed of \\(A\\) independent nucleons, the\nnuclear  cross  sections  then  simply scale  quadratically  with  the\nnucleon  number  \\(A\\)  and  the reduced  masses:  \\(m^2_{\\rm  red}(p)\n\\sigma_{\\rm   el}^{\\rm  nucleus}   =  A^2   m^2_{\\rm   red}({}^A_Z  X)\n\\sigma_{\\rm el}^{\\rm nucleon}\\).   The nucleon matrix element \\(M_{\\rm\nnucleon}  \\sim  10^{-3}\\) is  mostly  sensitive  to the  strange-quark\nYukawa coupling.  An adequate estimate of the elastic scattering cross\nsection  $\\sigma_{\\rm el}^{\\rm nucleon}$  of a  right-handed sneutrino\nwith a nucleon yields~\\cite{pospelov}\n\\begin{equation}\n  \\label{eq:elastic}\n\t\\sigma_{\\rm el}^{\\rm nucleon}\\ \\approx\\\n\\left(5\\times 10^{-50}~\\textrm{ cm}^2\\right)\\,\n\t\\left(\\frac{\\lambda\\rho}{10^{-4}}\\right)^2\\,\n\t\\left(\\frac{100\\textrm{ GeV}}{m_{H_1}}\\right)^4\\,\n\t\\left(\\frac{ 50\\textrm{ GeV}}{m_{\\tilde N_1}}\\right)^2\\; .\n\\end{equation}\nThe  upper limits  on  \\(\\lambda\\rho\\) are  derived  by comparing  the\nestimate~(\\ref{eq:elastic})   with   the    current   bound   on   the\nspin-independent nucleon cross section from the CDMS-II experiment and\nthe       expected  sensitivities    of       the      SuperCDMS\nextension~\\cite{Ogburn:2006hk} and the Xenon1T experiment~\\cite{Aprile:2006nz}.    These   limits   are  included   in\nFigures~\\ref{fig:CDMscenarioI}    and~\\ref{fig:CDMscenarioII}.     The\ncurrent    bound    already    excludes    large    parts    of    the\n$(m_{\\tilde{N}_1},\\lambda\\rho)$-parameter   plane,   except   of   the\nHiggs-boson  funnel  regions.   In   the  near  future,  the  upgraded\nexperiment SuperCDMS  will cover a large part of the parameter space,  but it\nwill  leave  open  the  lightest  Higgs-boson  pole  region  which  is\ntheoretically favoured by inflation  within the mSUGRA framework.  The\nproposed   Xenon1T  experiment is   expected  to\nfurther narrow  down this uncovered parameter range  of the $F_D$-term\nmodel.\n\nDark Matter may also be indirectly searched for through the detection of its final annihilation products, such as photons, positrons, anti-protons or neutrinos. The dominant channel of the LRHS annihilation in the Higgs funnel is determined by an effective scalar coupling with a $b\\bar b$ pair, which is approximately independent of the relative velocity of the annihiliating sneutrinos. Rates at low temperatures resulting in gamma-ray or charged particle fluxes are therefore not suppressed compared to the rates at the freeze-out temperature responsible for the LRHS relic density. There are several signals that could be explained as an observation of DM annihilation but, as of now, do not provide a consistent picture interpretable by a single DM candidate and model. For example, the excess in the diffuse galactic gamma ray spectrum measured by the EGRET detector may be interpreted by a 50-100~GeV WIMP, as given by the LRHS in our model, whereas the 511~keV line observed by the INTEGRAL satellite would hint at an MeV DM particle~\\cite{Taoso:2007qk,Hooper:2007vy}. Upcoming projects such as the GLAST and PAMELA satellites will have higher sensitivities, probe new energy ranges and should provide a clarification of the observational status.  \nHigh-energy neutrinos as annihilation products are expected and can be searched for in the Sun and the Earth, as WIMPs can accumulate in their centre. For the LHRS there is no spin-dependent coupling to nuclei, and its capture rate along with the produced neutrino flux is suppressed. In addition, for an annihilation via the Higgs resonance, the effective annihilation coupling required to get the correct relic density is very small. The LRHS is therefore not expected to be within the reach of high-energy neutrino telescopes~\\cite{Beltran:2008xg}, such as IceCube~\\cite{Rott:2007zz}.\n\n\\bigskip\n\n\\setcounter{equation}{0}\n\\section{Conclusions}\\label{conclusions}\n\nWe  have  analyzed in  detail  the  relic  abundance of  the  lightest\nright-handed sneutrinos (LRHS)  in the supersymmetric $F_D$-term model\nof hybrid inflation.  The  inflationary potential of the model results\nfrom  the  $F$-term  of  the inflaton  multiplet  $\\widehat{S}$.   The\n$F_D$-term  model also includes  a subdominant  non-anomalous $D$-term\ngenerated from  the local U(1)$_X$  symmetry of the  waterfall sector,\nwhich does not affect the  inflaton dynamics.  As was mentioned in the\nintroduction and further discussed in Section~\\ref{FDmodel}, the model\nadequately  fits the  current CMB  data  of inflation  and provides  a\nnatural  solution to  the so-called  gravitino  overabundance problem,\nwithout   resorting   to   an   excessive  suppression   of   possible\nrenormalizable  couplings  of  the  inflaton to  the  MSSM  particles.\nFinally, the  $F_D$-term model closely relates  the $\\mu$-parameter of\nthe MSSM to an SO(3) symmetric  Majorana mass $m_N$ through the VEV of\nthe inflaton  field.  If $\\lambda  \\sim \\rho$, this implies  that $\\mu\n\\sim m_N$, so the  model may naturally predict lepton-number violation\nat  the electroweak  scale and  potentially  account for  the BAU  via\nthermal resonant leptogenesis.\n\n\nIn spite of the  explicit lepton-number violation through the Majorana\nterm  $\\frac{1}{2}\\, \\rho\\, \\widehat{S}  \\widehat{N}_i \\widehat{N}_i$,\nthe $F_D$-term  hybrid model conserves  $R$-parity.  Consequently, the\nLSP of the spectrum is stable and so qualifies as candidate to address\nthe CDM problem. The new aspect of the $F_D$-term hybrid model is that\nthermal right-handed sneutrinos emerge as new candidates to solve this\nproblem,  by  virtue of  the  quartic coupling:  $\\frac{1}{2}\\,\\lambda\n\\rho\\,  \\widetilde{N}^*_i \\widetilde{N}^*_i  H_u H_d\\  +\\  {\\rm H.c.}$\nThis  new quartic  coupling results  in the  Higgs potential  from the\n$F$-terms of  the inflaton field,  and it is  not present in  the more\noften-discussed  extension of  the MSSM,  where  right-handed neutrino\nsuperfields have  bare Majorana  masses. Provided that  the couplings\n$\\lambda$  and  $\\rho$  are   not  too  small,  e.g.~$\\lambda,\\,  \\rho\n\\stackrel{>}{{}_\\sim} 10^{-2}$, the  LRHS $\\widetilde{N}_{\\rm\nLSP}$ as LSP can   efficiently  annihilate  via   the  lightest  Higgs-boson\nresonance  $H_1$ into  pairs of  $b$-quarks, in  the  kinematic region\n$m_{H_1}  \\approx 2 m_{\\widetilde{N}_{\\rm  LSP}}$, and  so drastically\nreduce its relic density  to the observed value: $\\Omega_{\\rm DM}\\,h^2\n\\approx 0.11$. \n\nExperiments, such  as CDMS-II, SuperCDMS  and Xenon1T, which  look for\nsignatures  of WIMPs  through their  elastic scattering  with nuclei,\nwill  significantly  constrain  the  allowed parameter  space  of  the\n$F_D$-term  model. They  will  exclude most  of  the parameter  space,\nexcept possibly of  a narrow region close to  the lightest $H_1$-boson\nresonance, where $m_{H_1}  \\approx 2 m_{\\widetilde{N}_{\\rm LSP}}$.  It\nmight seem that to obtain  this particular relation between the masses\nof the $H_1$ boson and $\\widetilde{N}_{\\rm LSP}$, a severe tuning of the\nmodel parameters is required. However, it is worth stressing here that\nsuch a mass relation may  easily be achieved within a mSUGRA framework\nof the $F_D$-term model that successfully realizes hybrid inflation.\n\n\nThe  LRHS scenario  of  CDM requires  relatively  large $\\lambda$  and\n$\\rho$  couplings that could,  in principle,  make Higgs  bosons decay\ninvisibly,   e.g.~$H\\to\\widetilde{N}_{\\rm   LSP}\\,  \\widetilde{N}_{\\rm\nLSP}$.  Also, right-handed sneutrinos  could be present in the cascade\ndecays  of   the  heavier  supersymmetric   particles.   The  collider\nphenomenology  of such a  CDM scenario  lies beyond  the scope  of the\npresent  article.   Instead,  we   note  that  the  $F_D$-term  hybrid\ninflationary model  can give rise  to rich phenomenology which  can be\nprobed at  high-energy colliders~\\cite{NprodLHC,NprodILC}, as  well as\nin low-energy experiments of lepton flavour and number violation, such\nas  $0\\nu\\beta\\beta$ decay,  $\\mu\\to e\\gamma$~\\cite{CL},  $\\mu\\to eee$\nand $\\mu\\to e$ conversion in nuclei~\\cite{IP,LFVN}. It would therefore\nbe  very interesting to  systematically analyze  possible correlations\nbetween predictions for  cosmological and phenomenological observables\nin the $F_D$-term model.\n\n\n\\subsection*{Acknowledgements}\nThis work is supported in part by the PPARC research grant: PP\/D000157\/1.\n\n\n\n\\newpage \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intr}\nVideo content understanding is one of the widely researched areas in computer vision with several applications ranging from automated surveillance to robotics, human computer interaction, video indexing and retrieval, \\etc., to name a few. Egocentric action recognition is a particularly challenging sub-task of video content understanding. Egocentric videos are captured using wearable cameras and are often characterized by the presence of a cluttered environment containing several objects and egomotion caused by movement of the camera wearer. Recognizing the action present in a video requires extraction of fine-grained spatio-temporal features that can discriminate one action from another. EPIC-Kitchens-100~\\cite{epic} is the largest egocentric action recognition dataset with ~90K video segments composed of 97 verb and 300 noun categories. The verb and noun labels of a video segment is combined to form its action label.\n\nTo participate in the challenge, we used two different video action recognition models composed of entirely different building blocks and feature aggregation strategy.\n\\begin{itemize}\n    \\item GSF\\cite{gsf}: A plug and play module that can transform 2D CNNs into a high performing spatio-temporal feature extractor;\n    \\item XViT\\cite{xvit}: A convolution free transformer based architecture for efficient video representation learning\n\\end{itemize}\n\nGate-Shift-Fuse (GSF) is a CNN based architecture that captures local relationship which introduces an inductive bias about the 3D structure of video frames within a small spatio-temporal receptive field.\nOn the other hand XViT, a transformer based model, captures global information and learns geometric relationship between the pixels. While GSF relies on the inductive bias owing to the locally connected convolution layers for feature extraction, XViT disregards any prior about the data and learns the relevant patterns in it that are suitable for addressing the end task. \nThus the two models used in the challenge extract different features and are highly complementary to each other. We deployed an ensemble of the two model families to participate in the challenge. The final score is obtained by averaging the prediction scores from the individual members in the ensemble. \n\n\n\n\\begin{table*}[t]\n\t\\centering\n\t\\begin{tabular}{c|c|c|c|c|c}\n\t\tMethod & Backbone & Pre-training & Verb & Noun & Action \\\\ \\hline \\hline\n\t\t\\multicolumn{6}{c}{Validation set} \\\\ \\hline\n\t\t\\multirow{3}{*}{GSF} & IncV3 & Kinetics400 & 68.89 (90) & 51.42 (75.49) & 43.11 (64.19)\n\t\t\\\\ \\cline{2-6}\n\t\t& Res-50 & Kinetics400 & 68.88 (90.44) & 52.73 (76.37) & 43.84 (64.95 \\\\ \\cline{2-6}\n\t\t& Res-101 & ImageNet & 69.06 (90.33) & 53.18 (75.81) & 44.48 (64.68) \\\\ \\hline\n\t\tXViT & ViT-B\/16 & Kinetics400 & 68 (90.08) & 55.63 (78.86) & 44.91 (65.97) \\\\ \\hline\n\t\tEnsemble & \\multicolumn{2}{c|}{-} & 70.86 (91.67) & 56.7 (79.9) & 46.88 (68.18) \\\\ \\hline \\hline\n\t\t\\multicolumn{6}{c}{Test set} \\\\ \\hline\n\t\tEnsemble & \\multicolumn{2}{c|}{-} & 68.16 (90.01) & 55.49 (78.98) & 44.82 (65.45) \\\\ \\hline\n\t\\end{tabular}\\vspace{.1in}\n\t\\caption{Performance of the models on the validation set (top) and test set (bottom) of EPIC-Kitchens 100 dataset. Ensemble score is generated by averaging the scores of individual models.}\n\t\\label{tab:ensemble}\n\\end{table*}\n\n\\section{Models} \\label{sec:models}\n\nWe describe details of the two model families in this section.\n\n\\subsection{GSF}\n\nGSF, an extension of GSM~\\cite{gsm}, is a light weight feature encoding module capable of converting a 2D CNN into an efficient and effective spatio-temporal feature extractor. The output features from a spatial convolution layer of the 2D backbone is first applied to a gating module, composed of a light-weight 3D convolution kernel,\nto generate grouped spatial gating. The spatial gating is then applied to the input features to obtain group-gated features and residual. Forward and backward shifting in time is then applied to the group-gated features. In GSM, the time-shifted features are combined with the residual using addition operation. GSF extends this simple fusion with a data dependent weighted channel fusion mechanism using a convolution layer. The resulting spatio-temporal features are then propagated to the next layer of the backbone CNN for further processing. \n\n\n\n\\subsection{XViT}\nVision transformers~\\cite{vit} can be extended for video recognition by extending the self attention mechanism between tokens within a frame to tokens from other frames as well. However, this will increase the complexity quadratically with the increase in the number of frames. To make the model tractable XViT~\\cite{xvit} proposes efficient space-time mixing attention as follows.\n\nLet $\\mathbf{q}_{s,t}\\in \\mathbf{R}^{1\\times d_h}$, $\\mathbf{k}_{s,t}\\in \\mathbf{R}^{1\\times d_h}$ and $\\mathbf{v}_{s,t}\\in \\mathbf{R}^{1\\times d_h}$ be the query, key and value at a spatial location $s$ and temporal location $t$. Then the self-attention $\\mathbf{y}_{s,t}$ is computed as \n\\begin{equation}\n    \\mathbf{y}_{s,t} = \\sum_{s'=0}^{S-1} \\textrm{softmax}\\{(\\mathbf{q}_{s,t} \\cdot \\tilde{\\mathbf{k}}_{s',-t_w:t_w})\/\\sqrt{d_h}\\} \\tilde{\\mathbf{v}}_{s',-t_w:t_w}  \\label{eqn:self-attn}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\tilde{\\mathbf{k}}_{s',-t_w:t_w} = [\\mathbf{k}_{s',t-t_w}(d_h^{t-t_w}), \\dots, \\mathbf{k}_{s',t+t_w}(d_h^{t+t_w})] \\\\\n\\tilde{\\mathbf{v}}_{s',-t_w:t_w} = [\\mathbf{v}_{s',t-t_w}(d_h^{t-t_w}), \\dots, \\mathbf{v}_{s',t+t_w}(d_h^{t+t_w})]\n\\end{eqnarray}\nwhere, $\\mathbf{k}_{s',t'}(d_h^{t'})$ and  $\\mathbf{v}_{s',t'}(d_h^{t'})$ denotes the operator for indexing the $d_h^{t'}$ channels  from $\\mathbf{k}_{s',t'}$ and $\\mathbf{v}_{s',t'}$, respectively.\n\nThe video transformer model used in this challenge is constructed by replacing the self-attention in \\cite{vit} with Eqn.~\\ref{eqn:self-attn}\n\n\n\n\n\\section{Experiments} \\label{sec:expts}\nWe describe the implementation details of the two model families along with their training and testing settings in this section.\n\\subsection{Implementation Details} \\label{sec:impl_det}\n\\noindent \\textbf{GSF.} Gate-Shift-Fuse Networks are instantiated by plugging in GSF to the backbone layers of a 2D CNN. For the challenge, we instantiated three different models by changing the backbone CNNs. This includes InceptionV3, ResNet50 and ResNet101. The GSF variant of InceptionV3 and ResNet50 are first pre-trained on Kinetics400 dataset while for ResNet101, we used the ImageNet pretrained weights and directly trained the model on EPIC-Kitchens-100 dataset.\n\n\\noindent \\textbf{XViT.} Backbone used is the base architecture ViT-B\/16 from \\cite{vit} with 12 transformer layers each with 12 attention heads and an embedding dimension of 768. Each frame from the video is first divided into non-overlapping patches of size 16$\\times$16 and are then applied to a linear layer for vectorization. The temporal window $t_w$ is set as 1.\n\n\\noindent \\textbf{Training.} We trained all our models using SGD with momentum (0.9) and a cosine scheduler with linear warmup. The base learning rate for GSF models are set as 0.01 for a batch size of 32 while XViT is trained with a base learning rate of 0.05 and a batch size of 128. GSF models are trained for 60 epochs and XViT is trained for 50 epochs. \n16 frames uniformly sampled from the input video clip are applied as input to all the models. We also applied temporal jittering during training, as done in \\cite{tsn}. All models are trained in a multi-task classification setting using three classification layers to predict verb, noun and action labels. We generated the action labels by combining the verb and noun label of the video provided with the dataset to obtain a total of 3806 action categories in the training set. More details regarding training can be found in \\cite{gsf} and \\cite{xvit}.\n\n\\noindent \\textbf{Testing.} We sample 2 clips consisting of 16 frames during testing. From each frame, 3 spatial crops are generated. Thus, from each video, we generate 6 clips. The prediction score from each of the 6 clips are averaged to obtain the video prediction.\n\n\n\n\\subsection{Results}\n\nTab.~\\ref{tab:ensemble} lists the performance of the various models used for the challenge. The top part of the table shows the results on the validation set. From the validation set results, one can see that GSF is strong on verb prediction while XViT results in a better performance on noun prediction. This shows that GSF is a powerful model for temporal reasoning. On the other hand, the presence of global spatial receptive field of XViT enables it to perform as a strong object recognition model. Combining the prediction scores obtained from both model families improves the performance considerably, showing their complementarity in extracting spatio-temporal features. The bottom part of the table shows the performance on the test set, which is visible on the leaderboard. Note that all model developments have been done on the validation set and evaluation of individual models is not done on the test set to tune the models' performance. This shows that our ensemble generalizes well to the test data.\n\n\n\n\n\\section{Conclusion} \\label{sec:concl}\n\nIn this report, we summarized the details of the two model families used for participating in the EPIC-Kitchens-100 Action Recognition Challenge 2021. The improved performance of the ensemble consisting of the two model families shows that the two models are complementary to each other. This resulted in achieving a top-3 action recognition performance on the leaderboard.\n\n\n\\section*{Acknowledgements}\nFBK gratefully acknowledge the support from Amazon AWS Machine Learning Research Awards (MLRA).\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Vela pulsar (PSR B0833-45) is the strongest $\\gamma$-ray source in the \nsky, but it is one of the most difficult pulsars to detect at X-ray energies.\nThis is in part because it is embedded in a very bright X-ray synchrotron nebula\nproviding a large unpulsed background, but also because its pulsed X-ray emission \nis comparatively weak.  The first detection of pulsed emission at X-ray energies was \nmade by ROSAT in the 0.1 - 2 keV band (Ogelman 1993), and the spectrum\nis consistent with a blackbody.  We detected the pulsar for the first \ntime in hard X-rays (2 - 30 keV) during a 93 ks RXTE Cycle 1 observation (Strickman,\nHarding \\& De Jager 1999 [SHD99]) and the pulse profile shows two peaks.  The first\nRXTE peak is closely aligned with the first EGRET $\\gamma$-ray peak, but the second\npeak has an energy-dependent phase and is aligned with the second EGRET peak\nonly at the highest energy (16 - 30 keV).  In our lowest energy RXTE band (2 - 8 keV)\nthe second peak is roughly aligned with the second peak of the optical profile\n(Gouiffes 1998).  The average pulse spectrum joins smoothly with the high-energy\nspectrum of OSSE, COMPTEL and EGRET, although the spectrum of the first peak is\nsignificantly harder than that of the second peak, which appeared to be consistent\nwith an extrapolation to the optical flux points.  We (SHD99) suggested that the\nsecond RXTE peak was a blend of separate hard and soft components.  In this \npaper, we report preliminary results of our analysis of a 300 ks RXTE Cycle 3  \nobservation which confirms this picture.  \n\n\\section{Cycle 3 RXTE Observations and Analysis}\n\n\\begin{figure}\n\\centerline{\\psfig{file=NewBBlcs.ps,height=13.5cm}}\n\\vskip -0.7 truecm\n\\caption[]{Light curves of Cycle 3 RXTE pulsed emission in three bands, shown together\nwith light curves in EGRET (Kanbach et al. 1994), ROSAT\n(Ogelman 1993) and optical (Gouiffes 1998) bands.}\n\\end{figure}\n\nThe Cycle 3 observations were carried out during April\/May and July\/August 1999 with a\ngood exposure of 274 ks.  For this analysis we epoch-folded the PCA data in GoodXenon \nevent-by-event mode at the pulsar period using the Princeton Pulsar Database\n(Arzoumian et al. 1992), to obtain the energy-dependent light curves which we\nhave summed into three broad energy bands shown in Figure 1.\nTo separate the various possible components of the light curve and compute individual\ncomponent spectra, we fit a five peak\nsinusoid model with peaks of the form\n\\begin{eqnarray}\nC(i) & = & A(i) |\\cos[{\\pi\\over 2}(\\phi - \\phi_{_0}(i))]|^{\\xi(i)}, \\\\\n\\nonumber \\\\\n\\xi(i) & = & -{0.693\\over \\log[\\cos({\\pi\\over 2}W(i))]} \\nonumber\n\\end{eqnarray}\nto the data in 94 energy channels (roughly spanning 2 - 30 keV), where $A(i)$, \n$\\phi_0(i)$ and $W(i)$ are the amplitude, center phase and width of peak $i$\nrespectively.  The model also includes a constant background level.  \nWe found that $\\phi_0(i)$ and $W(i)$ did not vary significantly with energy,\nexcept for the phase of Peak 3 which has a modest variation (see Table 1),\nso they were fixed while the\nvalues of $A(i)$ as a function of energy were determined. The counts for each\npeak were then found by integrating each sinusoid curve over phase.  We could\nthen obtain the photon spectrum of each peak separately by fitting a power law with\nphotoelectric absorption of fixed column density $10^{20}\\,\\rm cm^{-2}$.\n\n\\begin{table}\n\\caption{Center Phase and Spectrum of Light Curve Peaks} \\label{tbl-1}\n\\begin{center}\n\\begin{tabular}{lllr}\n\\tableline\nPeak & $\\phi_0$ & Photon index & $\\chi^2$ (92 dof) \\\\\n\\tableline\n1 & $0.117 \\pm 0.001$ & $0.8892\\pm 0.0923$ & 92.8\\\\\n2 & $0.463 \\pm 0.006$ & $1.848 \\pm 0.244$ & 132.6\\\\\n3 & $0.55  \\pm 0.008$ (0.50-0.6)& $1.462 \\pm 0.132$ & 99.8\\\\\n4 & $0.87 \\pm 0.02$ & $1.463 \\pm 0.445$ & 119.9\\\\\n5 & $1.006 \\pm 0.004$ & $2.057 \\pm 0.312$ & 127.9\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vskip -1.0 truecm\n\\end{table}\n\n\\section{Results}\n\nThe RXTE light curves in Figure 1 show a narrow peak (Peak 1) at the phase of the EGRET\nfirst peak and a second peak that is now clearly seen to be a blend of two\ncomponents.  The first of these components, which we call Peak 2, becomes dominant\nin the 2 - 8 keV band and is in phase with the optical second peak.  The second\ncomponent, Peak 3, is harder and is in phase with the EGRET second peak.  There are\ntwo other statistically significant peaks that appear in the RXTE light curve:\na peak (Peak 5) at the radio phase (0.0) and a weaker peak (Peak4) \nleading the radio peak.\nNote that there is also a peak in the optical light curve at the radio phase.\nThe energy spectra have been computed using the sinusoid model fits to each peak.\nThe resulting power law spectral indices and goodness of fit for each peak\nare listed in Table 1.  \nThe peaks in phase with the EGRET peaks (Peak 1 and Peak 3) both have hard, but\nsignificantly different, spectra.  The spectra of Peak 2 and Peak 5 are much softer\nand both their flux levels and indices are consistent within the uncertainties.\nInterestingly, an extrapolation of the Peak 2 spectrum falls near the optical \nflux points (Nasuti et al. 1997). \n\n\\section{Discussion}\n\nThese Cycle 3 observations have thus independently confirmed the multicomponent \nnature of pulsed emission from the Vela pulsar in the energy range 2 - 30 keV suggested \nby the Cycle 1 observations.  With the improved statistics of the Cycle 3 data, we \nhave been able to separate the broad second peak in the RXTE X-ray light curve into \nsoft (Peak 2) and hard (Peak3) spectral components which maintain their phase integrity throughout the RXTE energy range.  In addition, we have discovered a new feature in\nthe RXTE light curve: a peak (Peak 5) at the phase of the radio pulse with an extremely \nsoft spectrum.  There is, in addition, significant emission leading the radio phase \n(Peak 4).  Peaks 1 and 3 make up the hard spectral component \nwhose light curve peaks are in phase with those of the gamma-ray \nlight curve, and whose spectrum smoothly connects to the 100 keV - 5 GeV spectrum.\nPeaks 2 and 5, whose phases match those of the second optical and radio pulses \nrespectively,\nmake up the soft component.  Their spectra, consistent with each other in both flux\nand spectral index, extrapolate to the optical flux points.\n\nAlthough the RXTE hard component spectrum connects to the $\\gamma$-ray spectrum, \nthe X-ray spectrum is harder, requiring a break around 100 keV.  Such a break at\nthe local cyclotron energy, blueshifted by the parallel momentum of the pairs, \nis predicted by the polar cap cascade model (Harding \\& \nDaugherty 1999).  The RXTE soft component may be either inverse Compton scattering\nradiation of pairs in polar cap cascades (Zhang \\& Harding 1999) or synchrotron\nradiation of backflowing particles from the outer gap (Cheng \\& Zhang 1999). \n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe field equation\n\\[\n\\triangle u-\\partial_{y}^{2}u=0\n\\]\nin Minkowski spacetime is of central physical importance, as it describes the\npropagation of many of the physical quantities described by field theories,\nincluding the components of the electromagnetic field in a vacuum. Its\ngeneralization to a theory which has multiple times is an ultrahyperbolic\nequation. The study of these equations provides a useful window onto the\nmathematical status of physical theories involving multiple times, and perhaps\nmore importantly, provides insight into the extent to which the ordinary\nconcepts of causality and determinism survive the transition to multiple time dimensions.\n\nConsideration of theories with multiple times has been relatively rare because\nit is widely believed that they are inherently unstable, and thus are not\ndeterministic in a physically meaningful sense. Certain significant developments in\ntheoretical physics, notably string theory, require additional dimensions, and\nin most work to date\\footnote[1]{Exceptions include the work of Tegmark\n(1997), Hull (1999), Hull \\& Khuri (2000), and Bars (2001).} the signature for\nthe extra dimensions is spatial, reflecting in part this concern with\ninstability. Motivated by this, the purpose of the present paper is to\nreconsider the questions of stability, uniqueness and determinism of the\ninitial value problem in the presence of multiple time dimensions. We take the\nmodel field equation in this setting to be the simple generalizations of the\nwave equation to multiple times, the ultrahyperbolic equation. We find\nthat the issue of stability and uniqueness for the Cauchy problem can be\naddressed by imposing nonlocal constraints that arise naturally from the field equations.\n\nIt may be thought reasonable to go beyond the traditional Cauchy problem, and\ngive initial data on hypersurfaces of higher codimension.  We show\nthat under the above constraints one can preserve\nstability in this setting, but uniqueness is lost, and thus\ndeterminism. Indeed, one may specify an arbitrary finite number of normal\nderivatives of the solution on the higher codimension hypersurface, and insist\nupon smooth solutions, yet still fail to achieve uniqueness. In contrast to\nthis, we conclude with a result that essentially recovers and generalizes a\ntheorem of Courant, which shows that the values of a solution in an\narbitrarily small neighborhood of the initial hypersurface are sufficient to\ndetermine the solution uniquely. In related and prior work, Woodhouse\n(1992) studied the case of two space and two time dimensions with \ninitial data on a spacelike hypersurface (thus of codimension 2), \nusing the Penrose twistor transform in the real case. He also recovered\nthe uniqueness result of Courant and its implicit constraints\non well-posed inital data for the Cauchy problem. Our work\nprovides a rigorous analytic alternative for his solution\nmethod, which is not restricted to this choice of space and\ntime dimensions. We remark that none of these results rely\nupon properties of analyticity of the data or the solution.\n\nTo fix our notation, the wave equation in $d_{1}$-many space \ndimensions and one time dimension is\n\\begin{equation}\n\\triangle_{x}u-\\partial_{y}^{2}u:=%\n{\\displaystyle\\sum\\limits_{j=1}^{d_{1}}}\n\\partial_{x_{j}}^{2}u-\\partial_{y}^{2}u=0\\text{ .}%\n\\end{equation}\nThe standard Cauchy problem is posed on $N=\\left\\{  (x,y)\\in\\mathbb{R}%\n_{x}^{d_{1}}\\times\\mathbb{R}_{y}^{1}:y=0\\right\\}  $, a spacelike codimension\none linear hypersurface, for initial data\n\\[\nu(x,0)=f(x)\\text{, }\\partial_{y}u(x,0)=g(x)\\text{.}%\n\\]\nA nonstandard Cauchy problem is posed for a linear hypersurface of mixed\nsignature $N=\\left\\{  (x,y):x_{1}=0\\right\\}  \\subseteq\\mathbb{R}_{x}^{d_{1}%\n}\\times\\mathbb{R}_{y}^{1}$, namely\n\\[\nu(0,x^{\\prime},y)=f(x^{\\prime},y)\\text{, }\\partial_{x_{1}}u(0,x^{\\prime\n},y)=g(x^{\\prime},y)\\text{,}%\n\\]\nwhere the notation is that $x=(x_{1},x^{\\prime})\\in\\mathbb{R}^{d_{1}}$.\nCourant (1962) calls this the non-spacelike Cauchy problem, but to avoid\nconfusion with the non-characteristic Cauchy problem, we call it a Cauchy\nproblem of \\emph{mixed signature}.\n\nAn ultrahyperbolic equation has the form\n\\begin{equation}\n\\triangle_{x}u-\\triangle_{y}u:=%\n{\\displaystyle\\sum\\limits_{j=1}^{d_{1}}}\n\\partial_{x_{j}}^{2}u-%\n{\\displaystyle\\sum\\limits_{j=1}^{d_{2}}}\n\\partial_{y_{j}}^{2}u=0\\text{ ,} \\label{ultra}%\n\\end{equation}\nwhere $x\\in\\mathbb{R}^{d_{1}}$ are considered to be the spacelike variables\nand $y\\in\\mathbb{R}^{d_{2}}$ are timelike. The Cauchy problem considers\ninitial data posed on a linear hypersurface of codimension one. Choosing\n$y_{1}$ as the direction of evolution, Cauchy data consist of\n\\[\nu(x,0,y^{\\prime})=f(x,y^{\\prime})\\text{, }\\partial_{y_{1}}u(x,0,y^{\\prime\n})=g(x,y^{\\prime})\n\\]\non the hypersurface $N=\\left\\{  (x,y)\\in\\mathbb{R}_{x}^{d_{1}}\\times\n\\mathbb{R}_{y}^{d_{2}}:y_{1}=0\\right\\}  $.\n\nThe initial value problem on a \\textit{higher} codimension hypersurface $M$\ncould take various forms. A natural problem from the perspective of theories\nwith multiple times is to consider the spacelike hypersurface $M =\\left\\{\n(x,y)\\in\\mathbb{R}_{x}^{d_{1}}\\times\\mathbb{R}_{y}^{d_{2}}:y=0\\right\\}  $ of\ncodimension $d_{2}$. Alternatively, one may consider more general $M =\\{\n(x,y)\\in\\mathbb{R}_{x}^{d_{1}}\\times\\mathbb{R}_{y}^{d_{2}}:x_{p_{1}+1} =\n\\dots= x_{d_{1}}=0$, $y_{P_{2}+1} = \\dots=y_{d_{2}-1} = 0\\} $ where $0\\leq\np_{1} \\leq d_{1}$ and $0 \\leq p_{2} \\leq d_{2}-1 $. There is in either case a\nquestion as to how much data, and what constraints, are to be considered on\n$M$. Some of the options are to (i) give the zeroth and first normal\nderivatives of $u$ on $M$, (ii) give some finite number of derivatives of $u$\non $M$ which are compatible with the constraint imposed by the ultrahyperbolic\nequation, or (iii) specify infinitely many derivatives of $u$ on $M$. In this\npaper we consider the first two of these cases.\n\nAn outline of the results of this paper is as follows. In section 2 we use\nFourier methods to show that the Cauchy problem for the ultrahyperbolic\nequation \\eqref{ultra} is ill-posed in general but well-posed on Sobolev\nspaces $H^{m}$ if an explicit nonlocal constraint is imposed upon the Cauchy\ndata. This applies as well to the wave equation with Cauchy data on a mixed\nsignature hypersurface.\\ In section 3 we consider the initial value problem\nfor data given on higher codimension hypersurfaces, and we find that solutions\nare highly nonunique for the initial value problems of type (i) and (ii)\nabove, even among $H^{m}$ smooth solutions and with the imposition of the\nconstraint given in section~2. \\ In particular, for theories with multiple\ntimes that can be transformed to the form of equation (\\ref{ultra}), data\nposed on the hypersurface $M=\\left\\{  y=0\\right\\}  $ do not uniquely determine\nthe solution at any other point in time $y\\in\\mathbb{R}^{d_{2}}\\backslash\n\\{0\\}$. The extension problem for higher numbers of derivatives is treated by\nthe same method as case (i) of zeroth and first normal derivatives. Regarding\ncase (iii), in which one specifies infinitely many derivatives on the initial\nhypersurface $M$, we do not have an answer. We do show in section 4 that among\nsmooth solutions, data in an arbitrarily small ellipsoidal neighborhood of a\ndisk in $M$ uniquely determine the data in the envelope of its light cones.\nThis is analogous to a result in Courant (1962) that is derived from\nAsgeirsson's mean value theorem.\n\n\\section{The Cauchy problem}\n\nLet $x\\in\\mathbb{R}^{d_{1}}$ and $y\\in\\mathbb{R}^{d_{2}}$ be the Cartesian\ncoordinates of space-time, denote $y=(y_{1},y^{\\prime})$ and consider the\nCauchy problem of evolution in the coordinate $y_{1}$. The Cauchy problem of\nmixed signature that we address is posed as\n\\begin{equation}\n\\partial_{y_{1}}^{2}u=\\triangle_{x}u-\\triangle_{y^{\\prime}}u\\text{ ,}\n\\label{ultra_cauchy}%\n\\end{equation}\nwith Cauchy data $u(x,0,y^{\\prime})=u_{0}(x,y^{\\prime})$ and $\\partial_{y_{1}%\n}$ $u(x,0,y^{\\prime})=u_{1}(x,y^{\\prime})$. The standard Sobolev spaces\n$H^{m}$ of functions of the variables $(x,y^{\\prime})$ are defined as closures\nof $C_{0}^{\\infty}(\\mathbb{R}^{d_{1}}\\times\\mathbb{R}^{d_{2}-1})$ with respect\nto the norms\n\\[\n\\left\\Vert f\\right\\Vert _{m}^{2}=\n{\\displaystyle\\sum\\limits_{\\left\\vert \\alpha\\right\\vert +\\left\\vert\n\\beta\\right\\vert \\leq m}}\n{\\displaystyle\\int}\n\\left\\vert \\partial_{x}^{\\alpha}\\partial_{y^{\\prime}}^{\\beta}f(x,y^{\\prime\n})\\right\\vert ^{2}dxdy^{\\prime}\\text{ .}\n\\]\nAdditionally, there is an energy functional, of indefinite sign, that is\nassociated with equation (\\ref{ultra_cauchy}), namely%\n\\[\nE(u):=\\frac{1}{2}%\n{\\displaystyle\\iint}\n\\left\\vert \\partial_{y_{1}}u\\right\\vert ^{2}+\\left\\vert \\nabla_{x}u\\right\\vert\n^{2}-\\left\\vert \\nabla_{y^{\\prime}}u\\right\\vert ^{2}dxdy^{\\prime}\\text{ .}\n\\]\n\n\n\\begin{theorem}\n\\label{thm1}\nSuppose that the evolution mapping $y_{1}\\rightarrow\\binom{u(x,y_{1}%\n,y^{\\prime})}{\\partial_{y_{1}}u(x,y_{1},y^{\\prime})}$ is in $C^{1}%\n(\\mathbb{R}_{y_{1}}:H^{1}\\times H^{0})$. Then the energy is conserved along a\nsolution $u(\\cdot,y_{1},\\cdot)$:%\n\\[\nE(u(\\cdot,y_{1},\\cdot))=E(u(\\cdot,0,\\cdot))\\text{.}\n\\]\n\\end{theorem}\n\n\\begin{proof}\nGiven $\\binom{u(x,y_{1},y^{\\prime})}{\\partial_{y_{1}}u(x,y_{1},y^{\\prime})}\\in\nC^{1}$, the following calculation is justified:%\n\\[%\n\\begin{array}\n[c]{cl}%\n\\partial_{y_{1}}E(u) & =%\n{\\displaystyle\\iint}\n(\\partial_{y_{1}}u\\cdot\\partial_{y_{1}}^{2}u+\\nabla_{x}u\\cdot\\nabla\n_{x}\\partial_{y_{1}}u-\\nabla_{y^{\\prime}}u\\cdot\\nabla_{y^{\\prime}}%\n\\partial_{y_{1}}u)dxdy^{\\prime}\\\\\n& =%\n{\\displaystyle\\iint}\n\\partial_{y_{1}}u(\\partial_{y_{1}}^{2}u+\\triangle_{x}u+\\triangle_{y^{\\prime}%\n}u)dxdy^{\\prime}\\\\\n& =0\\text{ .}%\n\\end{array}\n\\]\n\\end{proof}\n\n\nThe key issue is that the Cauchy problem above for equation\n(\\ref{ultra_cauchy}) is ill-posed for $d_{2}\\geq2$ and solutions are \\emph{not} in\ngeneral in $C^{1}(\\mathbb{R}_{y_{1}}:H^{1}\\times H^{0})$. \\ The energy is\nindefinite and in particular not bounded below, hence it does not in general\ndefine an energy norm with which to control the Sobolev norms of solutions of\nthe evolution equations.\n\nTo move to the next level of analysis, we give a Fourier synthesis of the\nevolution operator for the Cauchy problem of mixed signature. \\ Given\n$\\binom{u_{0}}{u_{1}}\\in H^{m+1}\\times H^{m}$, consider the Fourier space\nvariables $(x,y^{\\prime})\\rightarrow(\\xi,\\eta^{\\prime})$ and define the\nFourier transform in the standard way,\n\\[\n\\binom{\\hat{u}_{0}(\\xi,\\eta^{\\prime})}{\\hat{u}_{1}(\\xi,\\eta^{\\prime})}%\n=\\frac{1}{\\sqrt{2\\pi}^{d}}%\n{\\displaystyle\\iint}\ne^{-i\\xi\\cdot x}e^{-i\\eta^{\\prime}\\cdot y^{\\prime}}\\binom{u_{0}(x,y^{\\prime}%\n)}{u_{1}(x,y^{\\prime})}dxdy^{\\prime}%\n\\]\nwhere $d=(d_{1}+d_{2}-1)$. On a formal level equation (\\ref{ultra_cauchy})\nunder Fourier transform will read\n\\[\n\\partial_{y_{1}}^{2}\\hat{u}=(-\\left\\vert \\xi\\right\\vert ^{2}+\\left\\vert\n\\eta^{\\prime}\\right\\vert ^{2})\\hat{u}\\text{ ,}%\n\\]\ngiving rise to the expression for the propagator, $\\exp(y_{1}\\sqrt\n{\\triangle_{x}-\\triangle_{y^{\\prime}}})$. The solution thus reads%\n\n\\[\n\\hat{u}(\\xi,y_{1},\\eta^{\\prime})=\\cos(\\sqrt{\\left\\vert \\xi\\right\\vert\n^{2}-\\left\\vert \\eta^{\\prime}\\right\\vert ^{2}}y_{1})\\hat{u}_{0}(\\xi\n,\\eta^{\\prime})+\\frac{\\sin(\\sqrt{\\left\\vert \\xi\\right\\vert ^{2}-\\left\\vert\n\\eta^{\\prime}\\right\\vert ^{2}}y_{1})}{\\sqrt{\\left\\vert \\xi\\right\\vert\n^{2}-\\left\\vert \\eta^{\\prime}\\right\\vert ^{2}}}\\hat{u}_{1}(\\xi,\\eta^{\\prime\n})\\text{ }%\n\\]\nfor $\\left\\vert \\eta^{\\prime}\\right\\vert \\leq\\left\\vert \\xi\\right\\vert $,\nwhile%\n\\[\n\\hat{u}(\\xi,y_{1},\\eta^{\\prime})=\\cosh(\\sqrt{\\left\\vert \\eta^{\\prime\n}\\right\\vert ^{2}-\\left\\vert \\xi\\right\\vert ^{2}}y_{1})\\hat{u}_{0}(\\xi\n,\\eta^{\\prime})+\\frac{\\sinh(\\sqrt{\\left\\vert \\eta^{\\prime}\\right\\vert\n^{2}-\\left\\vert \\xi\\right\\vert ^{2}}y_{1})}{\\sqrt{\\left\\vert \\eta^{\\prime\n}\\right\\vert ^{2}-\\left\\vert \\xi\\right\\vert ^{2}}}\\hat{u}_{1}(\\xi,\\eta\n^{\\prime})\\text{ }%\n\\]\nfor $\\left\\vert \\xi\\right\\vert <\\left\\vert \\eta^{\\prime}\\right\\vert $. That\nis, the dispersion relation\n\\begin{equation}\n\\omega(\\xi,\\eta^{\\prime})=\\sqrt{\\left\\vert \\xi\\right\\vert ^{2}-\\left\\vert\n\\eta^{\\prime}\\right\\vert ^{2}} \\label{dispersion}%\n\\end{equation}\nholds in the Fourier space region $\\left\\{  \\left\\vert \\eta^{\\prime\n}\\right\\vert \\leq\\left\\vert \\xi\\right\\vert \\right\\}  $, while in the\ncomplementary region the evolution of a Fourier mode is described by the\nLyapunov exponent\n\\begin{equation}\n\\lambda(\\xi,\\eta^{\\prime})=\\sqrt{\\left\\vert \\eta^{\\prime}\\right\\vert\n^{2}-\\left\\vert \\xi\\right\\vert ^{2}}. \\label{Lyapunov}%\n\\end{equation}\n\n\nWhen the propagator is applied to data $\\binom{u_{0}}{u_{1}}$ which is\nanalytic, this solution exists for at least short time; for analytic data of\nexponential type, the solution is global. \\ However, it is clear that general\ninitial data in $H^{m+1}\\times H^{m}$ do not even give rise to solutions which\nare tempered distributions for any nonzero $y_{1}$.\n\nOn the other hand, imposing a constraint on the initial data, the solution\nprocess is well defined. The fact that some constraint is necessary is indeed\nevident from the Asgeirsson mean value theorem, and its consequences, as\ndiscussed in Courant (1962). The form of this nonlocal constraint is evident\nfrom the Fourier synthesis, as we shall now see.\n\nDefine a phase space $X$ using an energy norm adapted to the propagator of\nequation (\\ref{ultra_cauchy}). Using the definition of the dispersion relation\n\\eqref{dispersion} and the Lyapunov exponent \\eqref{Lyapunov}, and the\nPlancherel identity, set $v=\\binom{v_{0}}{v_{1}}$ and\n\\begin{align*}\n\\left\\Vert v\\right\\Vert _{X}^{2}:=  &  \\iint\\limits_{\\left\\{  \\left\\vert\n\\eta^{\\prime}\\right\\vert <\\left\\vert \\xi\\right\\vert \\right\\}  }\\omega^{2}%\n(\\xi,\\eta^{\\prime})\\left\\vert \\hat{v}_{0}(\\xi,\\eta^{\\prime})\\right\\vert\n^{2}d\\xi d\\eta^{\\prime}\\\\\n&  +\\iint\\limits_{\\left\\{  \\left\\vert \\xi\\right\\vert \\leq\\left\\vert\n\\eta^{\\prime}\\right\\vert \\right\\}  }\\lambda^{2}(\\xi,\\eta^{\\prime})\\left\\vert\n\\hat{v}_{0}(\\xi,\\eta^{\\prime})\\right\\vert ^{2}d\\xi d\\eta^{\\prime}\\\\\n&  +\\iint\\left\\vert \\hat{v}_{1}(\\xi,\\eta^{\\prime})\\right\\vert ^{2}d\\xi\nd\\eta^{\\prime}~\\text{.}%\n\\end{align*}\nThis is a norm, unlike the actual energy associated with the equation\n(\\ref{ultra_cauchy}), and can be used to control solutions when the propagator\nis restricted to the appropriate stable and\/or unstable subspaces of $X$.\nDefine%\n\\begin{align}\nX^{S}  &  =\\left\\{  v=\\binom{v_{0}}{v_{1}}\\in X:\\ \\frac{1}{2}\\left(  \\hat\n{v}_{0}(\\xi,\\eta^{\\prime})+\\frac{\\hat{v}_{1}(\\xi,\\eta^{\\prime})}{\\lambda\n(\\xi,\\eta^{\\prime})}\\right)  =0\\text{ for }\\left\\vert \\xi\\right\\vert\n<\\left\\vert \\eta^{\\prime}\\right\\vert \\right\\} \\\\\nX^{U}  &  =\\left\\{  v\\in X:\\ \\frac{1}{2}\\left(  \\hat{v}_{0}(\\xi,\\eta^{\\prime\n})-\\frac{\\hat{v}_{1}(\\xi,\\eta^{\\prime})}{\\lambda(\\xi,\\eta^{\\prime})}\\right)\n=0\\text{ for }\\left\\vert \\xi\\right\\vert <\\left\\vert \\eta^{\\prime}\\right\\vert\n\\right\\}\n\\end{align}\nand%\n\\begin{align*}\nX^{C}  &  =\\left\\{  v\\in X:\\mathrm{supp}\\binom{\\hat{v}_{0}}{\\hat{v}_{1}}%\n(\\xi,\\eta^{\\prime})\\subseteq\\text{ }\\left\\{  \\left\\vert \\xi\\right\\vert\n>\\left\\vert \\eta^{\\prime}\\right\\vert \\right\\}  \\right\\} \\\\\n&  =X^{S}\\cap X^{U}\\text{.}%\n\\end{align*}\nThe subspace $X^{S}$ corresponds to the center stable subspace for evolution\nin $y_{1}\\in\\mathbb{R}^{+}$, the subspace $X^{U}$ corresponds to the center\nunstable subspace, and $X^{C}$ is the center subspace. This nomenclature is\nsupported by the following theorem.\n\n\\begin{theorem}\n\\label{thm2}\nFor $\\binom{u_{0}}{u_{1}}\\in X^{S}$, the Cauchy problem of mixed signature\nfor equation (\\ref{ultra_cauchy}) has a unique solution in $X$ for all $y_{1}\n\\in\\mathbb{R}^{+}$. For $\\binom{u_{0}}{u_{1}}\\in X^{U}$ the problem has a\nunique solution for all $y_{1}\\in\\mathbb{R}^{-}$, and whenever $\\binom{u_{0}\n}{u_{1}}\\in X^{C}$ the solution exists globally in $y_{1}\\in\\mathbb{R}$. In\neach of these cases, the map $y_{1}\\rightarrow u(x,y_{1},y^{\\prime})$ is $C^{1}$.\n\\end{theorem}\n\n\nDenote the propagators by $\\Phi^{S},\\Phi^{U}$ and $\\Phi^{C}$ for data in the\nrespective subspaces. \\ These solutions are continuous with respect to their\nCauchy data taken in the respective subspaces. This is the result of the next theorem.\n\n\\begin{theorem}\n\\label{thm3}\nGiven two phase space points $u=\\binom{u_{0}}{u_{1}},v=\\binom{v_{0}}{v_{1}}\\in\nX^{S}$, then for $y_{1}\\geq0,$\n\\begin{equation}\n\\left\\Vert \\Phi_{y_{1}}^{S}(u)-\\Phi_{y_{1}}^{S}(v)\\right\\Vert _{X}^{2}\n\\leq\\left\\Vert u-v\\right\\Vert _{X}^{2} ~. \\label{boundS}\n\\end{equation}\nThe analogous estimate holds for $u,v\\in X^{U}$, for $y_{1}\\leq0$:\n\\begin{equation}\n\\left\\Vert \\Phi_{y_{1}}^{U}(u)-\\Phi_{y_{1}}^{U}(v)\\right\\Vert _{X}^{2}\n\\leq\\left\\Vert u-v\\right\\Vert _{X}^{2} ~. \\label{boundU}\n\\end{equation}\nFor $u\\in X^{C}$, $\\Phi_{y_{1}}^{C}=\\Phi_{y_{1}}^{S}$ for $y_{1}\\geq0$ and\n$\\Phi_{y_{1}}^{C}=\\Phi_{y_{1}}^{U}$ for $y_{1}\\leq0$, and equality holds in\nboth (\\ref{boundS}) and (\\ref{boundU}).\n\\end{theorem}\n\n\n\\begin{proof}\nIt suffices in Theorem \\ref{thm3} to prove the first statement. In $X^{S}$ the solution\nhas two components, distinguished by their Fourier support. Consider first\n$\\binom{u_{0}}{u_{1}}$ such that $\\mathrm{supp}(\\hat{u}_{0},\\hat{u}%\n_{1})\\subseteq\\left\\{  \\left\\vert \\xi\\right\\vert \\geq\\left\\vert \\eta^{\\prime\n}\\right\\vert \\right\\}  := R_{1} $, which gives the center component of the\nevolution. The propagator is expressed%\n\\[\n\\mathcal{F}\\Phi_{y_{1}}^{S}\\binom{u_{0}}{u_{1}}=\\left(\n\\begin{array}\n[c]{cc}%\n\\cos(\\omega y_{1}) & \\frac{\\sin(\\omega y_{1})}{\\omega}\\\\\n-\\omega\\sin(\\omega y_{1}) & \\cos(\\omega y_{1})\n\\end{array}\n\\right)  \\binom{\\hat{u}_{0}}{\\hat{u}_{1}}%\n\\]\nwhere $\\omega=\\omega(\\xi,\\eta^{\\prime})$\nis the dispersion relation \\eqref{dispersion}.\nEvaluating this in the energy norm,%\n\\begin{align}\n\\left\\Vert \\Phi_{y_{1}}^{S}\\binom{u_{0}}{u_{1}}\\right\\Vert _{X}^{2}  &\n=\\iint\\left\\vert \\cos(\\omega y_{1})\\hat{u}_{0}+\\frac{\\sin(\\omega y_{1}%\n)}{\\omega}\\hat{u}_{1}\\right\\vert ^{2}\\omega^{2}\\label{E conserve}\\\\\n&  +\\left\\vert -\\omega\\sin(\\omega y_{1})\\hat{u}_{0}+\\cos(\\omega y_{1})\\hat\n{u}_{1}\\right\\vert ^{2}d\\xi d\\eta^{\\prime}\\nonumber\\\\\n&  =\\iint(\\left\\vert \\hat{u}_{0}\\right\\vert ^{2}\\omega^{2}+\\left\\vert \\hat\n{u}_{1}\\right\\vert ^{2})d\\xi d\\eta^{\\prime}\\nonumber\\\\\n&  =\\left\\Vert \\dbinom{u_{0}}{u_{1}}\\right\\Vert _{X}^{2}\\text{ .}\\nonumber\n\\end{align}\nThe propagator on the complementary space is more sensitive. Let us suppose\nthat $\\mathrm{supp}(\\hat{u}_{0},\\hat{u}_{1})\\subseteq\\left\\{  \\left\\vert\n\\eta^{\\prime}\\right\\vert >\\left\\vert \\xi\\right\\vert \\right\\}  $, then\n$\\lambda(\\xi,\\eta^{\\prime})>0$ and we express the propagator in terms of its\nFourier transform as\n\\begin{align*}\n\\mathcal{F}\\Phi_{y_{1}}^{S}\\binom{u_{0}}{u_{1}}  &  =\\left(\n\\begin{array}\n[c]{cc}%\n\\cosh(\\lambda y_{1}) & \\frac{\\sinh(\\lambda y_{1})}{\\lambda}\\\\\n\\lambda\\sinh(\\lambda y_{1}) & \\cosh(\\lambda y_{1})\n\\end{array}\n\\right)  \\binom{\\hat{u}_{0}}{\\hat{u}_{1}}\\\\\n&  =\\frac{e^{\\lambda y_{1}}}{2}\\left(\n\\begin{array}\n[c]{cc}%\n1 & \\frac{1}{\\lambda}\\\\\n\\lambda & 1\n\\end{array}\n\\right)  \\binom{\\hat{u}_{0}}{\\hat{u}_{1}}+\\frac{e^{-\\lambda y_{1}}}{2}\\left(\n\\begin{array}\n[c]{cc}%\n1 & -\\frac{1}{\\lambda}\\\\\n-\\lambda & 1\n\\end{array}\n\\right)  \\binom{\\hat{u}_{0}}{\\hat{u}_{1}}\\text{ .}%\n\\end{align*}\nThe subspace $X^{S}$ consists of precisely those data which lie in\nthe null space of the first term; this is the expression of the constraint\n\\begin{equation}\n\\lambda\\hat{u}_{0}(\\xi,\\eta^{\\prime})+\\hat{u}_{1}(\\xi,\\eta^{\\prime})=0\n\\label{Constraint}%\n\\end{equation}\nMeasuring the remaining term in energy norm, we find\n\\begin{align*}\n\\left\\Vert \\Phi_{y_{1}}^{S}\\binom{u_{0}}{u_{1}}\\right\\Vert _{X}^{2}  &\n=\\iint\\frac{e^{-2\\lambda y_{1}}}{4}\\left[  \\left\\vert \\hat{u}_{0}-\\frac\n{\\hat{u}_{1}}{\\lambda}\\right\\vert ^{2}\\lambda^{2}+\\left\\vert -\\lambda\\hat\n{u}_{0}+\\hat{u}_{1}\\right\\vert ^{2}\\right]  d\\xi d\\eta^{\\prime}\\\\\n&  \\leq\\iint e^{-2\\lambda y_{1}}\\left(  \\left\\vert \\hat{u}_{0}\\right\\vert\n^{2}\\lambda^{2}+\\left\\vert \\hat{u}_{1}\\right\\vert ^{2}\\right)  d\\xi\nd\\eta^{\\prime}\\text{ .}%\n\\end{align*}\nSince we consider the propagator $\\Phi_{y_{1}}^{S}$ for $y_{1}\\geq0$, the\nexponent $-\\lambda y_{1}$ is negative, and therefore%\n\\[\n\\left\\Vert \\Phi_{y_{1}}^{S}\\binom{u_{0}}{u_{1}}\\right\\Vert _{X}^{2}%\n\\leq\\left\\Vert \\binom{u_{0}}{u_{1}}\\right\\Vert _{X}^{2}%\n\\]\nfor $u=\\dbinom{u_{0}}{u_{1}}\\in X^{S}$. For general data in $X^{S}$, one\ndecomposes it into its components with support in $\\left\\{  \\left\\vert\n\\xi\\right\\vert \\geq\\left\\vert \\eta^{\\prime}\\right\\vert \\right\\}  $ for which\nwe use (\\ref{E conserve}), and its component supported in $\\left\\{  \\left\\vert\n\\eta^{\\prime}\\right\\vert >\\left\\vert \\xi\\right\\vert \\right\\}  $, which in\naddition satisfies the constraint (\\ref{Constraint}). Therefore on $X^{S}$%\n\\[\n\\left\\Vert \\Phi_{y_{1}}^{S}(u)\\right\\Vert _{X}^{2}\\leq\\left\\Vert u\\right\\Vert\n_{X}^{2}\\text{ .}%\n\\]\nBounded operators on $X^{S}$ are continuous with respect to $u\\in X^{S}$, and\nit is easy to see that the solution behaves continuously with respect to\n$y_{1}\\geq0$ as well. The case for the subspace $X^{U}$ is proved by the same\narguments, after reversing time $y_{1}\\longrightarrow-y_{1}$. This proves\nTheorems \\ref{thm2} and \\ref{thm3}. We remark that on the center subspace $X^{C}$, which yields global solutions, both\nconstraints are imposed%\n\\begin{equation}\n\\lambda\\hat{u}_{0}(\\xi,\\eta^{\\prime})\\pm\\hat{u}_{1}(\\xi,\\eta^{\\prime})=0\\text{\n,}%\n\\end{equation}\nimplying that $\\hat{u}_{0}(\\xi,\\eta^{\\prime})=0=\\hat{u}_{1}(\\xi,\\eta^{\\prime\n})$ for all $\\left\\{  \\left\\vert \\xi\\right\\vert \\leq\\left\\vert \\eta^{\\prime\n}\\right\\vert \\right\\}  $.\n\\end{proof}\n\nThe proof extends to the initial value problem posed in higher energy spaces,\ndefined by%\n\\[\n\\left\\Vert v\\right\\Vert _{X^{m}}^{2}:=\\sum\\limits_{\\left\\vert \\alpha\n\\right\\vert +\\left\\vert \\beta\\right\\vert \\leq m}\\left\\Vert \\partial\n_{x}^{\\alpha}\\partial_{y^{\\prime}}^{\\beta}v\\right\\Vert _{X}^{2}\\text{ .}\n\\]\nWe then have\n\n\\begin{corollary}\nThe higher energy space $X^{m}$ decomposes into three subspaces, $X^{m,S},$\n$X^{m,U}$ and $X^{m,C}=X^{m,S}\\cap X^{m,U}$ such that for $u,v\\in X^{m,S}$ and\n$y_{1}\\geq0$%\n\\[\n\\left\\Vert \\Phi_{y_{1}}^{S}(u)-\\Phi_{y_{1}}^{S}(v)\\right\\Vert _{X^{m}}^{2}%\n\\leq\\left\\Vert u-v\\right\\Vert _{X^{m}}^{2}\\text{ ,}\n\\]\nwhile for $y_{1}\\leq0$ and $u,v\\in X^{m,U}$,%\n\\[\n\\left\\Vert \\Phi_{y_{1}}^{U}(u)-\\Phi_{y_{1}}^{U}(v)\\right\\Vert _{X^{m}}^{2}%\n\\leq\\left\\Vert u-v\\right\\Vert _{X^{m}}^{2}\\text{ .}\n\\]\nFor $u,v\\in X^{m,C}$ both estimates hold, and a global solution exists which\nhas properties of higher Sobolev regularity. When $m>((d_{1}+(d_{2}-1))\/2)+2$\nthen such solutions are known to be classical $C^{2}$ solutions by the Sobolev\nembedding theorem.\n\\end{corollary}\n\nIt is natural to estimate solutions with respect to the energy norm; indeed,\nit \\emph{is }the energy when restricted to the center subspace $X^{C}$. Thus\nthe problem is well-posed in the following sense:\\ data in $X^{S}$\ncontinuously propagates to all $y_{1}\\in\\mathbb{R}^{+}$, data in $X^{U}$\ncontinuously propagates to all times $y_{1}\\in\\mathbb{R}^{-}$, and data in\n$X^{C}$, which constitute an infinite-dimensional Hilbert space, are defined\nglobally in time. In the case of the ordinary wave equation ($d_{1}=1$),\nsolutions in $X^{C}$ correspond to the full energy space $H^{1} \\times L^{2}$.\n\n\\section{The initial value problem in higher codimension}\n\nIn the presence of multiple time dimensions, spacelike hypersurfaces are\nnecessarily of higher codimension. Therefore one might consider the initial\nvalue problem with data posed on a hypersurface of codimension greater than or\nequal to two. Such problems are generally ill-posed. Indeed, we show that\nsolutions can be singular for standard classes of data. Moreover, even\nimposing the constraint discussed in section 2, which is the requirement of\nglobal existence, smooth solutions are highly non-unique. The purpose of this\nsection is to study the extension problem of data posed on a non-degenerate\nhigher codimension hypersurface $M$ to Cauchy data on a codimension one\nhypersurface $N$. There is a lot of freedom in choosing this extension, even\nunder the constraint equation (\\ref{Constraint}) on the resulting Cauchy data.\nOther extensions can be chosen to fail to satisfy the constraint. Thus the\ninitial value problem fails to be well-posed in several ways: resulting\nsolutions may be singular, or they may be selected to satisfy the constraint\nand be regular for all $y_{1}\\in\\mathbb{R}$, however they will not be unique.\n\n\\subsection{Codimension 2 to codimension 1 in $\\mathbb{R}^{3}$}\n\nOur analysis is illustrated in the example case of $M=\\{y_{1}=y_{2}=0\\}$ and\n$N=\\{y_{1}=0\\}$ subspaces of $\\mathbb{R}^{3}$. We suppose that initial data\nfor a solution $u(x,y)$ is given on $M$ in the form\n\\[\nw(x_{1})=(w_{0}(x_{1}),w_{10}(x_{1}),w_{01}(x_{1}))\n\\]\nwhere $w_{0}(x_{1})=u(x_{1},0)$, $w_{10}(x_{1})=\\partial_{y_{1}}u(x_{1},0)$\nand $w_{01}(x_{1})=\\partial_{y_{2}}u(x_{1},0)$, corresponding to the values of\nthe solution and its normal derivatives on $M$. The object is to extend\n$w(x_{1})$ to Cauchy data $(u_{0}(x_{1},y_{2}),u_{1}(x_{1},y_{2})$ on $N$\nwhich satisfies%\n\\begin{align*}\nu_{0}(x_{1},0)  &  =w_{0}(x_{1})\\\\\nu_{1}(x_{1},0)  &  =w_{10}(x_{1})\n\\end{align*}\nand the compatibility condition%\n\\[\n\\partial_{y_{2}}u_{0}(x_{1},0)=w_{01}(x_{1})\\text{.}%\n\\]\nWe give extensions which satisfy the constraint (\\ref{Constraint}), therefore\ngiving rise to global solutions in $y_{1}\\in\\mathbb{R}$. Such extensions are\nnonunique. Additionally, there are extensions which fail to satisfy the\nconstraint, lying in $X^{S}\\backslash X^{C}$ or $X^{U}\\backslash X^{C}$ or neither.\n\n\\begin{definition}\nThe extension operator is given by%\n\\[\nE(w)(x_{1},y_{2}):=\\frac{1}{2\\pi}\\iint e^{i(\\xi x_{1}+\\eta^{\\prime}y_{2})}\n\\hat{w}(\\xi)\\chi(\\xi,\\eta^{\\prime})d\\eta^{\\prime}d\\xi\n\\]\nwhere the kernel function $\\chi(\\xi,\\eta^{\\prime})$ is chosen such that for\nall $\\xi$,%\n\\[\n\\frac{1}{\\sqrt{2\\pi}}\\int\\chi(\\xi,\\eta^{\\prime})d\\eta^{\\prime}=1\\text{.}\n\\]\nIn order to satisfy the constraint, we ask additionally that\n$\\mathrm{supp(}\n\\chi(\\xi,\\eta^{\\prime}))\\subseteq\\{\\left\\vert \\eta^{\\prime}\\right\\vert\n<\\left\\vert \\xi\\right\\vert \\}$. A reasonable choice is to take%\n\\[\n\\chi(\\xi,\\eta^{\\prime})=\\psi(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac\n{1}{\\left\\vert \\xi\\right\\vert }\\text{,}%\n\\]\nfor $\\psi(\\theta)\\in C_{0}^{\\infty}([-1,1])$, $\\psi(\\theta)\\geq0$ even, and\n\\begin{equation}\n\\frac{1}{\\sqrt{2\\pi}}\\int_{-1}^{1}\\psi(\\theta)d\\theta=1. \\label{Normalization}%\n\\end{equation}\n\\end{definition}\n\n\\begin{theorem}\n\\label{thm4}\nThe extension operator $E$ is a bounded operator on the following space of\nfunctions:\n\\begin{eqnarray*}\n  &&  E:\\dot{H}^{-1\/2}(M)\\rightarrow L^{2}(N)  \\\\\n  &&  \\quad \\ (\\dot{H}^{-1\/2}\\cap H^{m-1\/2})(M)\\rightarrow H^{m}(N) ~.\n\\end{eqnarray*}\nIn addition, when $w\\in\\dot{H}^{-3\/2}(M)$ then $y_{2}E(w)\\in L^{2}(N)$ and\nfurthermore\n\\begin{eqnarray*}\n  && y_{2}E:\\dot{H}^{m-3\/2}(M)\\rightarrow H^{m}(N) \\text{.}\n\\end{eqnarray*}\n\\end{theorem}\n\n\nUsing the extension operator, we generate constraint-satisfying Cauchy data on\n$N$ from initial data on $M$ as follows:%\n\\begin{align*}\nu_{0}(x_{1},y_{2}) &  :=E(w_{0})(x_{1},y_{2})+y_{2}E(w_{01})(x_{1},y_{2})\\\\\nu_{1}(x_{1},y_{2}) &  :=E(w_{10})(x_{1},y_{2})\\text{.}%\n\\end{align*}\nChecking that this is a legitimate choice, we have\n\\begin{align*}\nu_{1}(x_{1},0) &  =\\frac{1}{2\\pi}\\iint e^{i\\xi x_{1}}\\hat{w}_{10}(\\xi)\\chi\n(\\xi,\\eta^{\\prime})d\\xi d\\eta^{\\prime}\\\\\n&  =\\frac{1}{\\sqrt{2\\pi}}\\int e^{i\\xi x_{1}}\\hat{w}_{10}(\\xi)\\left[  \\frac\n{1}{\\sqrt{2\\pi}}\\int\\psi(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac\n{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right]  d\\xi\\\\\n&  =\\frac{1}{\\sqrt{2\\pi}}\\int e^{i\\xi x_{1}}\\hat{w}_{10}(\\xi)\\left[  \\frac\n{1}{\\sqrt{2\\pi}}\\int\\psi(\\theta)d\\theta\\right]  d\\xi\\\\\n&  =w_{10}(x_{1})\n\\end{align*}\nbecause of the normalization (eqn \\ref{Normalization}) of $\\psi$.\\ Similarly,%\n\\[\nu_{0}(x_{1},0)=E(w_{0})(x_{1},0)=w_{0}(x_{1}).\n\\]\nThe compatibility condition is satisfied, since%\n\\begin{align*}\n\\partial_{y_{2}}u_{0}(x_{1},0) &  =\\partial_{y_{2}}E(w_{0})(x_{1}%\n,0)+E(w_{0})(x_{1},0)\\\\\n&  =\\partial_{y_{2}}E(w_{0})(x_{1},0)+w_{01}(x_{1})\\text{.}%\n\\end{align*}\nThe first term of the RHS vanishes because\n\\begin{align*}\n\\partial_{y_{2}}E(w_{0})(x_{1},0) &  =\\frac{1}{2\\pi}\\iint e^{i\\xi x_{1}}%\ni\\eta^{\\prime}\\hat{w}_{0}(\\xi)\\chi(\\xi,\\eta^{\\prime})d\\xi d\\eta^{\\prime}\\\\\n&  =\\frac{1}{\\sqrt{2\\pi}}\\int ie^{i\\xi x_{1}}\\hat{w}_{0}(\\xi)\\left[  \\frac\n{1}{\\sqrt{2\\pi}}\\int\\eta^{\\prime}\\psi(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert\n)\\frac{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right]  d\\xi\\\\\n&  =0\\text{,}%\n\\end{align*}\nwhere we have used that $\\int\\theta\\psi(\\theta)d\\theta=0$ because $\\psi\n(\\cdot)$ has been chosen to be even$.$\n\nThe pair of functions $(u_{0}(x_{1},y_{2}),u_{1}(x_{1},y_{2}))$ gives Cauchy\ndata for the codimension one problem that is discussed in Section 2. Because\nof the properties of the extension, it satisfies the constraint conditions of\n$X^{C}$ for solutions which are globally defined in $y_{1}$. In order to apply\nthe existence theorem, the energy norm of this Cauchy data must be finite.\n\n\\begin{theorem}\n\\label{thm5}\nSuppose that $w_{0}\\in\\dot{H}^{1\/2}(M)$, $w_{01}\\in\\dot{H}^{-1\/2}$ and\n$w_{10}\\in\\dot{H}^{-1\/2}$. Then the energy norm of the extension\n$u_{0}=E(w_{0})(x_{1},y_{2})+y_{2}E(w_{01})(x_{1},y_{2})$, $u_{1}(x_{1}%\n,y_{2})=E(w_{10})(x_{1},y_{2})$ is finite:%\n\\[\n\\left\\Vert (u_{0},u_{1})\\right\\Vert _{X^{C}}^{2}\\leq C(\\left\\Vert\nw_{0}\\right\\Vert _{\\dot{H}^{1\/2}}^{2}+\\left\\Vert w_{01}\\right\\Vert _{\\dot\n{H}^{-1\/2}}^{2}+\\left\\Vert w_{10}\\right\\Vert _{\\dot{H}^{-1\/2}}^{2})\\text{.}\n\\]\nAdditionally, the higher energy norms with which one defines the $X^{m}$\ntopology for $(u_{0},u_{1})$ are also bounded by this extension process,\nnamely\n\\[\n\\left\\Vert (u_{0},u_{1})\\right\\Vert _{X^{m,C}}^{2}\\leq C_{m}(\\left\\Vert\nw_{0}\\right\\Vert _{\\dot{H}^{m+1\/2}}^{2}+\\left\\Vert w_{01}\\right\\Vert _{\\dot\n{H}^{m-1\/2}}^{2}+\\left\\Vert w_{10}\\right\\Vert _{\\dot{H}^{m-1\/2}}^{2})\\text{.}\n\\]\n\\end{theorem}\n\n\n\\begin{proof}\n(of Theorem \\ref{thm4}): Using the Plancherel identity, the $L^{2}(N)$ norm of $E(w)$\nis%\n\\begin{align*}\n\\left\\Vert E(w)\\right\\Vert _{L^{2}(N)}^{2}  &  =\\iint\\left\\vert \\hat{w}%\n(\\xi)\\right\\vert ^{2}\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert\n)\\frac{1}{\\left\\vert \\xi\\right\\vert ^{2}}d\\eta^{\\prime}d\\xi\\\\\n&  =\\int\\frac{1}{\\left\\vert \\xi\\right\\vert }\\left\\vert \\hat{w}(\\xi)\\right\\vert\n^{2}\\left(  \\int\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac\n{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right)  d\\xi\\\\\n&  =\\left\\Vert \\psi\\right\\Vert _{L^{2}[-1,1]}^{2}\\left\\Vert w\\right\\Vert\n_{\\dot{H}^{-1\/2}(M)}^{2}\\text{,}%\n\\end{align*}\nsince $\\theta=\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert $ and\n\\[\n\\int\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert\n\\xi\\right\\vert }d\\eta^{\\prime}=\\int\\limits_{-1}^{1}\\psi^{2}(\\theta\n)d\\theta\\text{.}%\n\\]\nThe identity extends to the Sobolev space $H^{m}(N)$; it suffices to calculate\n$\\left\\Vert \\partial_{x_{1}}^{m}E(w)\\right\\Vert _{L^{2}}$ and $\\left\\Vert\n\\partial_{y_{2}}^{m}E(w)\\right\\Vert _{L^{2}}$:\n\\begin{align*}\n\\left\\Vert \\partial_{x_{1}}^{m}E(w)\\right\\Vert _{L^{2}(N)}  &  =\\iint\n\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\left\\vert \\xi\\right\\vert ^{2m}\\psi\n^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{2}}d\\eta^{\\prime}d\\xi     \\\\\n&  =\\int\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\left\\vert \\xi\\right\\vert\n^{2m-1}\\left(  \\int\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac\n{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right)  d\\xi\\\\\n&  =\\left\\Vert \\psi\\right\\Vert _{L^{2}[-1,1]}^{2}\\left\\Vert w\\right\\Vert\n_{\\dot{H}^{m-1\/2}(M)}^{2}\\text{.}%\n\\end{align*}\nThe second quantity is similar:%\n\\begin{align*}\n\\left\\Vert \\partial_{y_{2}}^{m}E(w)\\right\\Vert _{L^{2}(N)}  &  =\\iint\n\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\left\\vert \\xi\\right\\vert ^{2m}\\psi\n^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{2}}\\left\\vert \\eta^{\\prime}\\right\\vert ^{2m}d\\eta^{\\prime\n}d\\xi\\\\\n&  =\\int\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\left\\vert \\xi\\right\\vert\n^{2m-1}\\left(  \\int\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert\n)\\left\\vert \\frac{\\eta^{\\prime}}{\\left\\vert \\xi\\right\\vert }\\right\\vert\n^{2m}\\frac{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right)  d\\xi\\\\\n&  =C_{m}\\left\\Vert w\\right\\Vert _{\\dot{H}^{m-1\/2}(M)}^{2}\\text{,}%\n\\end{align*}\nwhere $C_{m}=\\int\\limits_{-1}^{1}\\theta^{2m}\\psi^{2}(\\theta)d\\theta$. The\nthird and fourth estimates of the theorem involve $y_{2}E(w)$, whose Fourier\ntransform is%\n\\[\n\\hat{w}(\\xi)\\frac{1}{i}\\partial_{\\eta^{\\prime}}\\chi(\\eta^{\\prime}\/\\left\\vert\n\\xi\\right\\vert )\\text{.}%\n\\]\nMeasuring the $L^{2}$ norm of $y_{2}E(w)$,\n\\begin{align*}\n\\left\\Vert y_{2}E(w)\\right\\Vert _{L^{2}(N)}^{2}  &  =\\iint\\left\\vert \\hat\n{w}(\\xi)\\right\\vert ^{2}\\left\\vert \\frac{1}{i}\\partial_{\\theta}\\psi\n(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert \\xi\\right\\vert\n^{2}}\\right\\vert ^{2}d\\eta^{\\prime}d\\xi\\\\\n&  =\\int\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{3}}\\left(  \\int\\left\\vert \\partial_{\\theta}\\psi(\\eta^{\\prime\n}\/\\left\\vert \\xi\\right\\vert )\\right\\vert ^{2}\\frac{1}{\\left\\vert\n\\xi\\right\\vert }d\\eta^{\\prime}\\right)  d\\xi\\\\\n&  =\\int\\left\\vert \\hat{w}(\\xi)\\right\\vert ^{2}\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{3}}d\\xi\\left(  \\int\\limits_{-1}^{1}\\left\\vert \\partial\n_{\\theta}\\psi\\right\\vert ^{2}d\\theta\\right) \\\\\n&  =C\\left\\Vert w\\right\\Vert _{\\dot{H}^{-3\/2}(M)}^{2}%\n\\end{align*}\nwith $C=\\int\\limits_{-1}^{1}\\left\\vert \\partial_{\\theta}\\psi\\right\\vert\n^{2}d\\theta$. The calculations of the $H^{m}$ norms of $y_{2}E(w)$ are similar.\n\\end{proof}\n\n\\begin{proof}\n(of Theorem \\ref{thm5}): Given initial data $(w_{0},w_{01},w_{10})(x_{1})$ we are to\ngive conditions under which the energy norm of the extension $(u_{0},u_{1})$\nis finite. First of all, the contribution to the energy given by $u_{1}$ is\nsimply $\\frac{1}{2}\\left\\Vert u_{1}\\right\\Vert _{L^{2}}^{2}$, hence by Theorem \\ref{thm4}\nit is bounded by $C\\left\\Vert w_{10}\\right\\Vert _{\\dot{H}^{-1\/2}}^{2}$.\nThere are two contributions from $u_{0}$, which can be expressed using the\nPlancherel identity:%\n\\[\n\\iint \\omega^{2}(\\xi,\\eta^{\\prime})\\left\\vert \\hat{w}_{0}(\\xi)\\right\\vert ^{2}%\n\\chi^{2}(\\xi,\\eta^{\\prime})d\\eta^{\\prime}d\\xi+\\iint \\omega^{2}(\\xi,\\eta^{\\prime\n})\\left\\vert \\hat{w}_{01}(\\xi)\\right\\vert ^{2}\\left\\vert \\partial\n_{\\eta^{\\prime}}\\chi^{2}(\\xi,\\eta^{\\prime})\\right\\vert ^{2}d\\eta^{\\prime}%\nd\\xi\\text{ .}%\n\\]\nUsing that $\\chi(\\xi,\\eta^{\\prime})=\\psi(\\eta^{\\prime}\/\\left\\vert\n\\xi\\right\\vert )\\frac{1}{\\left\\vert \\xi\\right\\vert }$, we estimate these two\nintegrals:%\n\\begin{align*}\n&  \\iint\\limits_{\\{\\left\\vert \\eta^{\\prime}\\right\\vert <\\left\\vert\n\\xi\\right\\vert \\}}\\left(  \\left\\vert \\xi\\right\\vert ^{2}-\\left\\vert\n\\eta^{\\prime}\\right\\vert ^{2}\\right)  \\left\\vert \\hat{w}_{0}(\\xi)\\right\\vert\n^{2}\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{2}}d\\eta^{\\prime}d\\xi\\text{ }\\\\\n&  =\\int\\left\\vert \\hat{w}_{0}(\\xi)\\right\\vert ^{2}\\left[  \\int\\left(\n\\left\\vert \\xi\\right\\vert -\\frac{\\left\\vert \\eta^{\\prime}\\right\\vert ^{2}%\n}{\\left\\vert \\xi\\right\\vert }\\right)  \\psi^{2}(\\eta^{\\prime}\/\\left\\vert\n\\xi\\right\\vert )\\frac{1}{\\left\\vert \\xi\\right\\vert }d\\eta^{\\prime}\\right]\nd\\xi\\text{ }\\\\\n&  \\leq C\\left\\Vert w_{0}\\right\\Vert _{\\dot{H}^{1\/2}}^{2}\\text{.}%\n\\end{align*}%\n\\begin{align*}\n&  \\iint\\limits_{\\{\\left\\vert \\eta^{\\prime}\\right\\vert <\\left\\vert\n\\xi\\right\\vert \\}}\\left(  \\left\\vert \\xi\\right\\vert ^{2}-\\left\\vert\n\\eta^{\\prime}\\right\\vert ^{2}\\right)  \\left\\vert \\hat{w}_{01}(\\xi)\\right\\vert\n^{2}\\psi^{2}(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert\n\\xi\\right\\vert ^{2}}d\\eta^{\\prime}d\\xi\\text{ }\\\\\n&  =\\int\\left\\vert \\hat{w}_{01}(\\xi)\\right\\vert ^{2}\\left[  \\int_{\\{\\left\\vert\n\\eta^{\\prime}\\right\\vert <\\left\\vert \\xi\\right\\vert \\}}\\left(  \\frac\n{1}{\\left\\vert \\xi\\right\\vert }-\\frac{\\left\\vert \\eta^{\\prime}\\right\\vert\n^{2}}{\\left\\vert \\xi\\right\\vert ^{3}}\\right)  \\psi^{2}(\\eta^{\\prime\n}\/\\left\\vert \\xi\\right\\vert )\\frac{1}{\\left\\vert \\xi\\right\\vert }d\\eta\n^{\\prime}\\right]  d\\xi\\text{ }\\\\\n&  \\leq C\\left\\Vert w_{01}\\right\\Vert _{\\dot{H}^{-1\/2}}^{2}\\text{.}%\n\\end{align*}\n\\end{proof}\n\n\\subsection{The extension problem for general spacelike data}\n\nWe now consider the general problem of initial data given on a maximal\nspacelike hypersurface of dimension $d_{1}$, extending it to Cauchy data on a\ncodimension one hypersurface. That is, for $(x,y)\\in\\mathbb{R}_{x}^{d_{1}}\n\\times\\mathbb{R}_{y}^{d_{2}}$,\n\\[\nM=\\left\\{  y=0\\right\\}  \\subseteq N=\\{y_{1}=0\\}\\text{.}\n\\]\nInitial data on $M$ take the form $w(x)=(w_{0}(x),w_{\\alpha}(x))$ where a\nsolution $u(x,y)$ of the field equation (\\ref{ultra}) is asked to satisfy\n\\[\nu(x,y)=w_{0}(x)\n\\]\nwith its first derivatives normal to $M$ satisfying%\n\\[\n\\partial_{y}^{\\alpha}u(x,0)=w_{\\alpha}(x)\n\\]\nwhere $\\alpha\\in\\mathbb{N}^{d_{2}}$ is the multi-index $\\alpha=(\\alpha\n_{1},...,\\alpha_{d_{2}}),$ $\\left\\vert \\alpha\\right\\vert =1$, such that only\none $\\alpha_{j}=1$ and the rest are zero. The object is to extend $w(x)$ to\nCauchy data on $N$ while satisfying the constraints (\\ref{Constraint}) to be\nin $X^{C}$. This Cauchy data satisfies%\n\\begin{align*}\nu_{0}(x,0)  &  =w_{0}(x)\\\\\nu_{\\alpha^{\\prime}}(x,0)  &  =w_{0\\alpha^{\\prime}}(x)\n\\end{align*}\nfor $\\alpha^{\\prime}=(\\alpha_{2},...,\\alpha_{d_{2}})$ and the first\nderivatives normal to $N$ satisfy%\n\\[\n\\partial_{y_{1}}u(x,0)=w_{10}(x)\\text{ .}%\n\\]\n\n\nFollowing the construction given in section 3.1, define an extension operator%\n\\[\nE(w)(x,y^{\\prime}):=\\frac{1}{\\sqrt{2\\pi}^{d_{1}+d_{2}-1}}\\iint\\hat{w}(\\xi\n)\\chi(\\xi,\\eta^{\\prime})e^{i\\xi\\cdot x}e^{i\\eta^{\\prime}\\cdot y^{\\prime}}d\\xi\nd\\eta^{\\prime}%\n\\]\nwhere the kernel function is even in $\\eta$ and satisfies%\n\\[\n\\frac{1}{\\sqrt{2\\pi}^{d_{2}-1}}\\int\\chi(\\xi,\\eta^{\\prime})d\\eta^{\\prime\n}=1\\text{ .}%\n\\]\nTo satisfy the constraint that $E(w)\\in X^{C}$, we ask that $\\mathrm{supp(}%\n\\chi(\\xi,\\eta^{\\prime}))\\subseteq\\{(\\xi,\\eta^{\\prime}):\\left\\vert \\eta\n^{\\prime}\\right\\vert <\\left\\vert \\xi\\right\\vert \\}$. Such kernel functions are\nreadily constructed (they are far from being uniquely determined). For\nexample, a variant of our construction of section 3.1 is based on choice of a\n$C_{0}^{\\infty}$ function $\\psi(\\theta)\\geq0$, with $\\mathrm{supp}%\n(\\psi)\\subseteq B_{1}(0)$, the ball of radius one. Then define\n\\[\n\\chi(\\xi,\\eta^{\\prime})=\\psi(\\eta^{\\prime}\/\\left\\vert \\xi\\right\\vert )\\frac\n{1}{\\left\\vert \\xi\\right\\vert ^{d_{2}-1}}~\\text{.}%\n\\]\nWe note that $\\chi$ is even in $\\eta$ if $\\psi(\\theta)$ is even, and that%\n\\begin{align*}\n\\frac{1}{\\sqrt{2\\pi}^{d_{2}-1}}\\int\\chi(\\xi,\\eta^{\\prime})d\\eta^{\\prime}  &\n=\\frac{1}{\\sqrt{2\\pi}^{d_{2}-1}}\\int\\psi(\\eta^{\\prime}\/\\left\\vert\n\\xi\\right\\vert )\\frac{1}{\\left\\vert \\xi\\right\\vert ^{d_{2}-1}}d\\eta^{\\prime}\\\\\n&  =\\frac{1}{\\sqrt{2\\pi}^{d_{2}-1}}\\int\\psi(\\theta)d\\theta\\text{ .}%\n\\end{align*}\nThis is normalized to one by choice of $\\psi$.\n\n\\begin{theorem}\n\\label{thm6}\nThe extension operator $E$ is bounded on the following function spaces:\n\\begin{eqnarray*}\n  && E:\\dot{H}^{\\frac{1-d_{2}}{2}}(M)\\rightarrow L^{2}(N)  \\\\\n  && \\quad \\ \\dot{H}^{\\frac{1-d_{2}}{2}}(M)\\cap\n  H^{m+\\frac{1-d_{2}}{2}}(M)  \\rightarrow H^{m}(N) ~,\n\\end{eqnarray*}\nwith $m$ the exponent of Sobolev regularity, and\n\\begin{eqnarray*}\n  && y^{\\prime}E:\\dot{H}^{\\frac{-(1+d_{2})}{2}}(M)\\rightarrow L^{2}(N)\\\\\n\\dot{H}^{\\frac{-(1+d_{2})}{2}}(M)\\cap H^{m-\\frac{1+d_{2}}{2}}(M)\\rightarrow\nH^{m}(N)\\text{ .}\n\\end{eqnarray*}\n\\end{theorem}\n\nUsing the extension operator $E$, the vector function $w(x)=(w_{0}%\n(x),w_{\\alpha}(x))$ extends to Cauchy data on $N$ as follows:%\n\\begin{align*}\nu_{0}(x,y^{\\prime}) &  :=E(w_{0})(x,y^{\\prime})+\\sum\\limits_{\\left\\vert\n\\alpha^{\\prime}\\right\\vert =1}y^{\\alpha^{\\prime}}\\cdot E(w_{0\\alpha^{\\prime}%\n})(x,y^{\\prime})\\\\\nu_{1}(x,y^{\\prime}) &  :=E(w_{10})(x,y^{\\prime})\\text{ .}%\n\\end{align*}\nThis is seen to extend the initial data $w(x)$ in the required way, and in\naddition it satisfies the constraint that $(u_{0},u_{1})\\in X^{C}$. However,\nmeasuring the functions $(u_{0},u_{1})$ in the energy norm is more appropriate\nfor the Cauchy problem, hence we also state estimates in this setting.\n\n\\begin{theorem}\n\\label{thm7}\nGiven $w_{0}\\in\\dot{H}(M)$ and $w_{\\alpha}\\in\\dot{H}(M)$, the energy norm of\nthe extension\n\\[\n(u_{0},u_{1})=(E(w_{0})+y^{\\alpha^{\\prime}}\\cdot E(w_{(0,\\alpha^{\\prime}%\n)}),E(w_{10}))\n\\]\nis finite; indeed\n\\[\n\\left\\Vert (u_{0},u_{1})\\right\\Vert _{X^{C}}^{2}\\leq C(\\left\\Vert\nw_{0}\\right\\Vert _{\\dot{H}^{\\frac{3-d_{2}}{2}}}^{2}+\\left\\Vert w_{0\\alpha\n^{\\prime}}\\right\\Vert _{\\dot{H}^{\\frac{1-d_{2}}{2}}}^{2}+\\left\\Vert\nw_{10}\\right\\Vert _{\\dot{H}^{\\frac{1-d_{2}}{2}}}^{2})  ~.\n\\]\n\\end{theorem}\n\nThe proofs of Theorems \\ref{thm6} and \\ref{thm7} are similar to the proofs of Theorems \\ref{thm4} and \\ref{thm5}, to which we refer the reader.\n\n\\subsection{The extension problem for mixed spacelike and timelike data}\n\nAs a final case, we consider the extension problem for initial data on a lower dimensional hypersurface $M$ of \\emph{mixed} signature. Given zero'th and first normal derivatives of a solution $u(x,y)$\non $M$, the object is to extend this data to the codimension one hypersurface\n$N=\\{y_{1}=0\\}$ in such a way that the constraint for well-posedness is\nsatisfied. This is not always possible for arbitrary data $w=(w_{0},w_{\\alpha\n})$ posed on $M$, due to analogous lower dimensional constraints on $M$. But\nit is possible, along with attendant Sobolev bounds on the extended functions,\nin most cases. This situation will be explained below.\n\nTo set the notation, we consider spacelike and timelike coordinates on $M$ to\nbe $(\\tilde{x},\\tilde{y})\\in\\mathbb{R}^{p_{1}}\\times\\mathbb{R}^{p_{2}}$, with\ntheir Fourier transform variables denoted $(\\tilde{\\xi},\\tilde{\\eta}%\n)\\in\\mathbb{R}^{p_{1}}\\times\\mathbb{R}^{p_{2}}$. The complementary variables\nwill be denoted $(x^{\\prime\\prime},y^{\\prime\\prime})\\in\\mathbb{R}^{d_{1}%\n-p_{1}}\\times\\mathbb{R}^{d_{2}-p_{2}-1}$ and $(\\xi^{\\prime\\prime},\\eta\n^{\\prime\\prime})\\in\\mathbb{R}^{d_{1}-p_{1}}\\times\\mathbb{R}^{d_{2}-p_{2}-1}$,\nso that coordinates on $N$ are $(x,y^{\\prime})=(\\tilde{x},x^{\\prime\\prime\n},\\tilde{y},y^{\\prime\\prime})$. The evolution variable remains $y_{1}$.\n\nInitial data for a solution $u(x,y)$ is given on $N$, which is expressed in\nthe form $(u,\\partial_{x^{\\prime\\prime}}^{\\alpha^{\\prime\\prime}}%\nu,\\partial_{y_{1}}^{\\beta_{1}}u,\\partial_{y^{\\prime\\prime}}^{\\beta\n^{\\prime\\prime}}u)(\\tilde{x},\\tilde{y},0,0)=(w_{0},w_{\\alpha^{\\prime\\prime}%\n},w_{\\beta_{1}},w_{\\beta^{\\prime\\prime}})(\\tilde{x},\\tilde{y})$, where\n$\\alpha^{\\prime\\prime}=(\\alpha_{p_{1}+1},...,\\alpha_{d_{1}})$, $\\beta\n^{\\prime\\prime}=(\\beta_{p_{2}+1},...,\\beta_{d_{2}})$ are multi-indices such\nthat $\\left\\vert \\alpha^{\\prime\\prime}\\right\\vert +\\left\\vert \\beta\n^{\\prime\\prime}\\right\\vert +\\left\\vert \\beta_{1}\\right\\vert =1$. The idea is\nthe same as in sections 3.1 and 3.2, namely to extend $(w_{0},w_{\\alpha\n^{\\prime\\prime}},w_{\\beta_{1}},w_{\\beta^{\\prime\\prime}})$ to\nconstraint-satisfying Cauchy data on $N$ in such a way that a solution\n$u(x,y)=u(\\tilde{x},x^{\\prime\\prime},\\tilde{y},y^{\\prime\\prime})$ to the field\nequation (\\ref{ultra}) satisfies%\n\\[\nu(\\tilde{x},0,0,\\tilde{y},0)=w_{0}(\\tilde{x},\\tilde{y})\n\\]\nand%\n\\[\n\\partial_{y_{1}}u(\\tilde{x},0,0,\\tilde{y},0)=w_{0\\beta_{1}}(\\tilde{x}%\n,\\tilde{y})~,\n\\]\nas well as the compatibility conditions%\n\\begin{align*}\n\\partial_{x^{\\prime\\prime}}^{\\alpha^{\\prime\\prime}}u(\\tilde{x},0,0,\\tilde\n{y},0) &  =w_{(\\alpha^{\\prime\\prime},0)}(\\tilde{x},\\tilde{y})\\\\\n\\partial_{y^{\\prime\\prime}}^{\\beta^{\\prime\\prime}}u(\\tilde{x},0,0,\\tilde{y},0)\n&  =w_{(0,\\beta^{\\prime\\prime})}(\\tilde{x},\\tilde{y})\n\\end{align*}\nThe existence of such an extension follows as in Theorems \\ref{thm6} and \\ref{thm7} from the\nconstruction of an extension operator $E$ with certain boundedness properties\non appropriate Sobolev spaces. We will focus our analysis therefore on the\nextension operators.\n\nAgain following section 3.1, define an extension operator\n\\[\nE(w)(x,y^{\\prime})=\\frac{1}{\\sqrt{2\\pi}^{d_{1}+d_{2}-1}}\\iint\\chi(\\tilde{\\xi\n},\\xi^{\\prime\\prime},\\tilde{\\eta},\\eta^{\\prime\\prime})d\\xi^{\\prime\\prime}%\nd\\eta^{\\prime\\prime}=1\\text{.}%\n\\]\nFurthermore, to satisfy the constraint that $E(w)\\in X^{C}$ for arbitrary data\n$w$, we ask that\n\\[\n\\mathrm{supp(\\chi(\\xi,\\eta}^{\\prime}))\\subseteq\\left\\{  (\\xi,\\eta^{\\prime\n}):\\left\\vert \\eta^{\\prime}\\right\\vert ^{2}<\\left\\vert \\xi\\right\\vert\n^{2}\\right\\}  :=R_{1}\\text{.}%\n\\]\nThese two conditions are always satisfiable, except in the case \n$\\xi^{\\prime\\prime}=\\left\\{  0\\right\\}  $, meaning that $d_{1}=p_{1}$ and the\nextension subspace $\\left\\{  (\\xi^{\\prime\\prime},\\eta^{\\prime\\prime})\\right\\}\n$ is purely timelike.\n\nIt is to be expected that the constraint induces a restriction on the data\n$w(\\tilde{\\xi},\\tilde{\\eta})$ in the vicinity of the ``lightcone'' $\\left\\{\n\\left\\vert \\tilde{\\xi}\\right\\vert =\\left\\vert \\tilde{\\eta}\\right\\vert\n\\right\\}  \\subseteq\\hat{M}$. Subdivide $\\hat{M}$ into two sets,%\n\\begin{align*}\n\\tilde{R}_{1}  &  :=\\left\\{  (\\tilde{\\xi},\\tilde{\\eta})\\in\\hat{M}:\\left\\vert\n\\tilde{\\eta}\\right\\vert \\leq\\left\\vert \\tilde{\\xi}\\right\\vert \\right\\} \\\\\n\\tilde{R}_{2}  &  :=\\left\\{  (\\tilde{\\xi},\\tilde{\\eta})\\in\\hat{M}:\\left\\vert\n\\tilde{\\eta}\\right\\vert >\\left\\vert \\tilde{\\xi}\\right\\vert \\right\\}  \\text{.}%\n\\end{align*}\nThe orthogonal projections onto functions supported in $\\tilde{R}_{1}$,\n$\\tilde{R}_{2}$ respectively, are denoted $\\pi_{1}$ and $\\pi_{2}$. We use\nstandard Sobolev spaces to quantify data supported in $\\tilde{R}_{1}$, namely\n\\[\nH^{r}=\\left\\{  w(\\tilde{x},\\tilde{y})\\in{\\mathrm{range}}(\\pi_{1}):\\left\\Vert\nw\\right\\Vert _{H^{r}}^{2}=\\iint\\limits_{\\tilde{R}_{1}}\\left\\vert \\hat\n{w}(\\tilde{\\xi},\\tilde{\\eta})\\right\\vert ^{2}(\\left\\vert \\tilde{\\xi\n}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{n}d\\tilde{\\xi\n}d\\tilde{\\eta}<+\\infty\\right\\}  ~\\text{.}%\n\\]\nOver $\\tilde{R}_{2}$ we use a modified form of Sobolev norm which is given by\n\\[\nK^{r}=\\left\\{  w(\\tilde{x},\\tilde{y})\\in{\\mathrm{range}}(\\pi_{2}):\\left\\Vert\nw\\right\\Vert _{K^{r}}^{2}=\\iint\\limits_{\\tilde{R}_{2}}\\left\\vert \\hat\n{w}(\\tilde{\\xi},\\tilde{\\eta})\\right\\vert ^{2}\\frac{(\\left\\vert \\tilde{\\xi\n}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{r}}{(\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}d\\tilde{\\xi}d\\tilde{\\eta}<+\\infty\\right\\}  \\text{.}%\n\\]\nwhere\n\\[\ne_{0}:=d_{1}+d_{2}-(p_{1}+p_{2})-1.\n\\]\nWe note that in the case where $d_{1}=p_{1}$, $\\tilde{R}_{1}=\\left\\{\n0\\right\\}  $ and $K^{n}=H^{r-\\frac{r}{2}(d_{2}-p_{2}-1)}$. More generally,\ndefine%\n\\[\nK_{s}^{r}=\\left\\{  w(\\tilde{x},\\tilde{y})\\in{\\mathrm{range}}(\\pi\n_{2}):\\left\\Vert w\\right\\Vert _{K_{s}^{r}}^{2}=\\iint\\limits_{\\tilde{R}_{2}%\n}\\left\\vert \\hat{w}(\\tilde{\\xi},\\tilde{\\eta})\\right\\vert ^{2}\\frac{(\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{r}%\n}{(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert\n^{2})^{\\frac{1}{2}e_{0}+s}}d\\tilde{\\xi}d\\tilde{\\eta}<+\\infty\\right\\}\n\\]\nDecompose an arbitrary function $w=\\pi_{1}w+\\pi_{2}w$, so that its components possess Fourier support in $\\tilde{R}_{1}$ and $\\tilde{R}_{2}$ respectively.\n\n\\begin{theorem}\n\\label{thm8}\nIf $d_{1}>p_{1}$ then there is a choice of kernel $\\chi$ (indeed there are\nmany such choices) such that $u=E(w)$ satisfies%\n\\[\n\\left\\Vert u\\right\\Vert _{L^{2}}^{2}\\leq C(\\left\\Vert \\pi_{1}w\\right\\Vert\n_{H^{-\\frac{1}{2}(e_{0})}}^{2}+\\left\\Vert \\pi_{2}w\\right\\Vert _{K^{0}}%\n^{2})\\text{.}%\n\\]\nHigher Sobolev norms of $u=E(w)$ are bounded as follows%\n\\[\n\\left\\Vert u\\right\\Vert _{H^{r}}^{2}\\leq C_{r}(\\left\\Vert \\pi_{1}w\\right\\Vert\n_{H^{r-\\frac{1}{2}(e_{0})}}^{2}+\\left\\Vert \\pi_{2}w\\right\\Vert _{K^{r}}%\n^{2}\\text{.}%\n\\]\nIn case $d_{1}=p_{1}$, it is not possible to extend arbitrary data to a\nfunction $u=E(w)$ which satisfies the constraint $\\mathrm{supp}(\\hat{u}%\n(\\xi,\\eta^{\\prime}))\\subseteq R_{1}$. However, if initially $\\mathrm{supp}%\n(\\hat{w}(\\xi,\\eta^{\\prime}))\\subseteq\\tilde{R}_{1}$ (i.e., $w=\\pi_{2}w$), then\nsuch an extension is possible, and we have, for $u=E(w)$,%\n\\[\n\\left\\Vert u\\right\\Vert _{L^{2}}^{2}\\leq C\\left\\Vert w\\right\\Vert _{K^{0}}%\n^{2}~\\text{,}%\n\\]%\n\\[\n\\left\\Vert u\\right\\Vert _{H^{r}}^{2}\\leq C_{r}\\left\\Vert w\\right\\Vert _{K^{r}%\n}^{2}~\\text{.}%\n\\]\n\\end{theorem}\n\n\n\\begin{proof}\nThe proof of Theorem \\ref{thm8} depends upon the construction of a kernel $\\chi\n(\\tilde{\\xi},\\tilde{\\eta},\\xi^{\\prime\\prime},\\eta^{\\prime\\prime})$ with\nsatisfactory properties. This construction is slightly different in the two\ndifferent regions of Fourier space%\n\\[\n\\tilde{R}_{1}:=\\left\\{  (\\tilde{\\xi},\\tilde{\\eta}):\\left\\vert \\tilde{\\eta\n}\\right\\vert \\leq\\left\\vert \\tilde{\\xi}\\right\\vert \\right\\}  \\text{ and\n}\\tilde{R}_{2}:=\\left\\{  (\\tilde{\\xi},\\tilde{\\eta}):\\left\\vert \\tilde{\\eta\n}\\right\\vert >\\left\\vert \\tilde{\\xi}\\right\\vert \\right\\}\n\\]\nwhere we note that the region $\\tilde{R}_{2}$ contains the data which would\nlead to an ill-posed initial value problem if $M$ were considered itself as a\ncodimension one hypersurface.\n\nTo extend data posed on region $\\tilde{R}_{1}$, define\n\\[\n\\chi_{1}(\\tilde{\\xi},\\tilde{\\eta},\\xi^{\\prime\\prime},\\eta^{\\prime\\prime\n}):= \\psi_{1}\\Bigl(\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert\n^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac\n{\\eta^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}\\text{,}%\n\\]\nwhere $\\psi_{1}(\\theta_{1},\\theta_{2})$ is a $C_{0}^{\\infty}$ function of\n$(d_{1}-p_{1})\\times(d_{2}-p_{2}-1)$ variables, respectively, with support in\nthe set $\\left\\vert \\theta_{2}\\right\\vert <\\left\\vert \\theta_{1}\\right\\vert $.\nTherefore $\\chi_{1}$ has support in the set\n\\[\n\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}}\\geq\\frac{\\eta^{\\prime\\prime}%\n}{(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2})^{\\frac{1}{2}}}\n\\]\nimplying that\n\\[\n\\left\\vert \\eta^{\\prime}\\right\\vert ^{2}=\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2}+\\left\\vert \\eta^{^{\\prime\\prime}}\\right\\vert ^{2}<\\left\\vert \\tilde{\\xi\n}\\right\\vert ^{2}+\\left\\vert \\xi^{\\prime\\prime}\\right\\vert ^{2}=\\left\\vert\n\\xi\\right\\vert ^{2}\\text{.}\n\\]\nThis is the appropriate region of support from functions $v=E(w)$ to lie in\nthe constraint-satisfying subspace of $L^{2}(N)$. In order that $E$ be an\nextension operator, we furthermore require that\n\\begin{align*}\n\\sqrt{2\\pi}^{d_{1}+d_{2}-1} &  =\\iint\\chi_{1}(\\tilde{\\xi},\\tilde{\\eta}%\n,\\xi^{\\prime\\prime},\\eta^{\\prime\\prime})d\\xi^{\\prime\\prime}d\\eta^{\\prime\n\\prime}\\\\\n&  =\\iint\\psi_{1}\\Bigl(\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert\n^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac\n{\\eta^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}d\\xi^{\\prime\\prime}d\\eta^{\\prime\\prime}\\\\\n&  =\\iint\\psi_{1}(\\theta_{1},\\theta_{2})d\\theta_{1}d\\theta_{2}\\text{.}%\n\\end{align*}\nAsking that this latter integral equal the normalizing constant $\\sqrt{2\\pi\n}^{d_{1}+d_{2}-1}$, and asking for $\\psi_{1}$ to be even in its variables\n$(\\theta_{1},\\theta_{2})$ gives an acceptable kernel for the extension\noperator. We note again that this choice of kernel is highly nonunique.\n\nOn the region $\\tilde{R}_{2}=\\left\\{  (\\tilde{\\xi},\\tilde{\\eta})\\in\\hat\n{M}:\\left\\vert \\tilde{\\eta}\\right\\vert >\\left\\vert \\tilde{\\xi}\\right\\vert\n\\right\\}  $, we can also attempt a construction of our extension operator.\nBy itself, this region would give rise to data in $L^{2}(M)$ for which the Cauchy problem\nof mixed type is ill-posed. The extension operator will nonetheless come up\nwith data $u=E(w)$ for which the well-posedness constraint is satisfied, if\nthis is possible. That is, as long as $d_{1}>p_{1}$, so that $\\left\\{\n\\xi^{\\prime\\prime}\\right\\}  $ is not restricted to the zero-dimensional vector\nspace, extensions can be found in a way that the default in satisfying the\nconstraint caused by the fact that $\\left\\vert \\tilde{\\xi}\\right\\vert\n<\\left\\vert \\tilde{\\eta}\\right\\vert $ can be made up with a choice of large\n$\\left\\vert \\xi^{\\prime\\prime}\\right\\vert $.\nIn practice, we will build $\\chi_{2}(\\tilde{\\xi},\\tilde{\\eta},\\xi\n^{\\prime\\prime}, \\eta^{\\prime\\prime})$ so that its support is in the regions\n\\[\n\\left\\{  \\left\\vert \\tilde{\\eta}\\right\\vert >\\left\\vert \\tilde{\\xi}\\right\\vert\n\\right\\}  = \\tilde{R}_{2}%\n\\]\nas well as\n\\[\n\\left\\{  \\left\\vert \\tilde{\\eta}\\right\\vert ^{2}+\\left\\vert \\eta^{\\prime\n\\prime}\\right\\vert ^{2}<\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert\n\\xi^{\\prime\\prime}\\right\\vert ^{2}\\right\\}  ;\n\\]\nimplying that $0\\leq(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2})+(\\left\\vert \\eta^{\\prime\\prime}\\right\\vert\n^{2}+\\left\\vert \\xi^{\\prime\\prime}\\right\\vert ^{2})$. Thus we require\n$d_{1}>p_{1}$.\nFollowing the above examples, assume that $d_{1}>p_{1}$ and set%\n\\[\n\\chi_{2}(\\tilde{\\xi},\\tilde{\\eta},\\xi^{\\prime\\prime},\\eta^{\\prime\\prime\n}):=\\psi_{2}\\Bigl(\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac{\\eta\n^{\\prime\\prime}}{(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}%\n\\]\nfor $(\\tilde{\\xi},\\tilde{\\eta})\\in\\tilde{R}_{2}$. Let $\\psi_{2}(\\theta\n_{1},\\theta_{2})$ be a $C_{0}^{\\infty}$ function of $e_{0}=d_{1}+d_{2}%\n-(p_{1}+p_{2})-1$ variables, as before and require that%\n\\begin{align*}\n\\int\\psi_{2}(\\theta_{1},\\theta_{2})d\\theta_{1}d\\theta_{2} & =\\int\\psi_{2}\n\\Bigl(\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac{\\eta\n^{\\prime\\prime}}{(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}d\\xi^{\\prime\\prime}d\\eta^{\\prime\\prime}\\\\\n&  =\\sqrt{2\\pi}^{e_{0}}\\text{.}%\n\\end{align*}\nFurthermore, ask that $\\psi(\\theta_{1},\\theta_{2})$ be even in $(\\theta\n_{1},\\theta_{2})$. Finally ask that the support of $\\psi(\\theta_{1},\\theta\n_{2})$ be in the set%\n\\[\n\\left\\{  (\\theta_{1},\\theta_{2}):\\theta_{1}^{2}-\\theta_{2}^{2}>1\\right\\}\n\\text{.}%\n\\]\nSuch requirements are satisfied by many possible choices of $\\psi$. In doing\nso, we arrive at a satisfactory kernel of an extension operator $E$ with the\nproperty that all functions $u=E(w)$ in its range have Fourier support\nsatisfying $\\mathrm{supp}(\\hat{u})\\leq\\left\\{  \\left\\vert \\eta^{\\prime\n}\\right\\vert ^{2}<\\left\\vert \\xi\\right\\vert ^{2}\\right\\}  $.\nThe singularities introduced at the boundaries of the lightcone $\\left\\{\n\\left\\vert \\tilde{\\eta}\\right\\vert =\\left\\vert \\tilde{\\xi}\\right\\vert\n\\right\\}  \\subseteq\\hat{M}$ by the kernel $\\chi_{2}$ impose more severe\nconstraints on the functions $w$ that are permitted in the domain of the\noperator $E$; this is the origin of the somewhat unusual requirements on\nfunctions $w(\\tilde{x},\\tilde{y})$ from which we can reasonably draw our data.\nThe Sobolev estimates of the proof are similar to those of Theorems \\ref{thm6} and \\ref{thm7} and we leave the details to the reader.\n\\end{proof}\n\nFinally, we show that a sufficiently large class of data $(w_{0}%\n,w_{(\\alpha^{\\prime\\prime},0)},w_{(0,\\beta^{\\prime\\prime})},w_{\\beta_{1}})$ on\n$M$ extends to Cauchy data on the hypersurface $N$ which is both of finite\nenergy and satisfies the constraint. This extension is given by\n\\begin{align}\nu_{0}(x,y^{\\prime}) &  =E(w_{0})(x,y^{\\prime})+\\sum\\limits_{\\left\\vert\n\\alpha^{\\prime\\prime}\\right\\vert =1}x^{\\prime\\prime\\alpha^{\\prime\\prime}%\n}E(w_{(\\alpha^{\\prime\\prime},0)})(x,y^{\\prime})+\\sum\\limits_{\\left\\vert\n\\beta^{\\prime\\prime}\\right\\vert =1}y^{\\prime\\prime\\beta^{\\prime\\prime}%\n}E(w_{(0,\\beta^{\\prime\\prime})})(x,y^{\\prime})\\label{Extension}\\\\\nu_{1}(x,y^{\\prime}) &  =E(w_{(0,\\beta_{1})})(x,y^{\\prime})\\text{.}\\nonumber\n\\end{align}\nBy design, this Cauchy data satisfies the constraint, that is, $(u_{0}%\n,u_{1})\\in X^{C}$, the center manifold. As before, its restriction to $M$\nreduces to the data $(w_{0},w_{(\\alpha^{\\prime\\prime},0)},w_{(0,\\beta\n^{\\prime\\prime})})(\\tilde{x},\\tilde{y},0)$. The only remaining task is to show\nthat its energy norm is finite. Recall that in this context the energy norm\nis\n\\begin{align*}\nH(u_{0},u_{1}) &  =\\frac{1}{2}\\iint\\limits_{N}\\left\\vert u_{1}\\right\\vert\n^{2}+\\left\\vert \\nabla_{x}u_{0}\\right\\vert ^{2}-\\left\\vert \\nabla_{y^{\\prime}%\n}u_{0}\\right\\vert ^{2}dxdy^{\\prime}\\\\\n&  =\\frac{1}{2}\\iint\\limits_{N}\\left\\vert \\hat{u}_{1}(\\xi,\\eta^{\\prime\n})\\right\\vert ^{2}+(\\left\\vert \\xi\\right\\vert ^{2}-\\left\\vert \\eta^{\\prime\n}\\right\\vert ^{2})\\hat{u}_{0}(\\xi,\\eta^{\\prime})d\\xi d\\eta^{\\prime}\\text{.}%\n\\end{align*}\nTo show that this energy is finite for the extension (\\ref{Extension}), we use\nthe results of Theorem \\ref{thm8}.\n\n\\begin{theorem}\n\\label{thm9}\nGiven data $(w_{0},w_{(\\alpha^{\\prime\\prime},0)},w_{(0,\\beta^{\\prime\\prime}%\n)},w_{\\beta_{1}})$ on $M$ with $\\left\\vert \\alpha^{\\prime\\prime}\\right\\vert\n=\\left\\vert \\beta^{\\prime\\prime}\\right\\vert =1$, suppose that\n\\begin{align}\n&  \\left\\Vert \\pi_{1}w_{0}\\right\\Vert _{H^{e_{0}+1}}+\\sum\\limits_{\\left\\vert\n\\alpha^{\\prime\\prime}\\right\\vert =1}\\left\\Vert \\pi_{1}w_{(\\alpha^{\\prime\n\\prime},0)}\\right\\Vert _{H^{e_{0}+1}}+\\sum\\limits_{\\left\\vert \\beta\n^{\\prime\\prime}\\right\\vert =1}\\left\\Vert \\pi_{1}w_{(0,\\beta^{\\prime\\prime}%\n)}\\right\\Vert _{H^{e_{0}+1}}<+\\infty\\\\\n&  \\left\\Vert \\pi_{2}w_{0}\\right\\Vert _{K^{1}}+\\sum\\limits_{\\left\\vert\n\\alpha^{\\prime\\prime}\\right\\vert =1}\\left\\Vert \\pi_{2}w_{(\\alpha^{\\prime\n\\prime},0)}\\right\\Vert _{K^{1}}+\\sum\\limits_{\\left\\vert \\beta^{\\prime\\prime\n}\\right\\vert =1}\\left\\Vert \\pi_{2}w_{(0,\\beta^{\\prime\\prime})}\\right\\Vert\n_{K^{1}}<+\\infty\n\\end{align}\nand%\n\\[\n\\left\\Vert \\pi_{1}w_{\\beta_{1}}\\right\\Vert _{H^{e_{0}}}+\\left\\Vert \\pi\n_{2}w_{\\beta_{1}}\\right\\Vert _{K^{0}}+\\infty\\text{.}%\n\\]\nThen the extension $(u_{0},u_{1})$ given by expression (\\ref{Extension}) has\nfinite energy and lies in the center subspace $X^{C}$. If $d_{1}=p_{1}$, then\nwe have to ask that $\\pi_{2}w_{\\gamma}=0$ in the above statement, for all\nmulti-indices $\\gamma$ in question.\n\\end{theorem}\n\n\n\\begin{proof}\nEstimates on the contributions of $w_{0}$ to $u_{0}$ follow immediately from\nTheorem \\ref{thm8}, as do the estimates for $u_{1}=E(w_{\\beta_{1}})$. Therefore we\nonly have to consider contributions in one of the two possible forms:%\n\\[%\n\\begin{array}\n[c]{cc}%\nx^{\\prime\\prime\\alpha^{\\prime\\prime}}E(w_{(\\alpha^{\\prime\\prime},0)}) &\n\\qquad\\left\\vert \\alpha^{\\prime\\prime}\\right\\vert =1\n\\end{array}\n\\]\nor%\n\\[%\n\\begin{array}\n[c]{cc}%\ny^{\\prime\\prime\\beta^{\\prime\\prime}}E(w_{(0,\\beta^{\\prime\\prime})}) &\n\\qquad\\left\\vert \\beta^{\\prime\\prime}\\right\\vert =1\n\\end{array}\n\\text{.}%\n\\]\nThe energy norm includes the quantities $\\left\\Vert x^{\\prime\\prime\n\\alpha^{\\prime\\prime}}E(w_{(\\alpha^{\\prime\\prime},0)})\\right\\Vert _{H^{1}}$\nand $\\left\\Vert y^{\\prime\\prime\\beta^{\\prime\\prime}}E(w_{(0,\\beta\n^{\\prime\\prime})})\\right\\Vert _{H^{1}}$; since the estimates are similar we\nwill give a sketch of one of them.\n\\begin{align*}\n& \\left\\Vert x^{\\prime\\prime\\alpha^{\\prime\\prime}}E(w_{(\\alpha^{\\prime\\prime\n},0)})\\right\\Vert _{H^{1}}^{2} =\\left\\Vert \\frac{1}{i}\\partial_{\\xi\n^{\\prime\\prime}}^{a^{\\prime\\prime}}\\widehat{E(w_{(\\alpha^{\\prime\\prime},0)}\n)}(\\left\\vert \\xi\\right\\vert ^{2}+\\left\\vert \\eta^{\\prime}\\right\\vert\n^{2})^{\\frac{1}{2}}\\right\\Vert _{L^{2}}^{2}  \\\\\n&\n\\leq\\iiiint\\Bigl[\\partial_{\\xi^{\\prime\\prime}}^{a^{\\prime\\prime}}\\psi_{1}\n\\Bigl(\\frac{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert\n^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac\n{\\eta^{\\prime\\prime}}{(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert\n\\tilde{\\eta}\\right\\vert ^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert ^{2})^{\\frac\n{1}{2}e_{0}}}\\Bigr]^{2}\\left\\vert \\widehat{\\pi_{1}w_{(\\alpha^{\\prime\\prime\n},0)}}(\\tilde{\\xi},\\tilde{\\eta})\\right\\vert ^{2}  \\\\\n&  +\\Bigl[\\partial_{\\xi^{\\prime\\prime}}^{a^{\\prime\\prime}}\\psi_{2}\\Bigl(\\frac\n{\\xi^{\\prime\\prime}}{(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}}},\\frac{\\eta^{\\prime\\prime}\n}{(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert\n^{2})^{\\frac{1}{2}}}\\Bigr)\\cdot\\frac{1}{(\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2}-\\left\\vert \\tilde{\\xi}\\right\\vert ^{2})^{\\frac{1}{2}e_{0}}}\n\\Bigr]^{2}\\left\\vert \\widehat{\\pi_{2}w_{(\\alpha^{\\prime\\prime},0)}}(\\tilde\n{\\xi},\\tilde{\\eta})\\right\\vert ^{2}d\\tilde{\\xi}d\\tilde{\\eta}d\\xi^{\\prime\n\\prime}d\\eta^{\\prime\\prime}\\text{.}\n\\end{align*}\nThe $\\xi^{\\prime\\prime}$-derivative introduces one extra factor of\n$(\\left\\vert \\tilde{\\xi}\\right\\vert ^{2}+\\left\\vert \\tilde{\\eta}\\right\\vert\n^{2})$, respectively $(\\left\\vert \\tilde{\\eta}\\right\\vert ^{2}-\\left\\vert\n\\tilde{\\xi}\\right\\vert ^{2})$, into the denominator. The integral over\n$(\\xi^{\\prime\\prime},\\eta^{\\prime\\prime})$ gives a constant, depending upon\n$\\psi_{1}$ and $\\psi_{2}$, as a bound, while the resulting integral over the\nvariable $(\\tilde{\\xi},\\tilde{\\eta})$ is bounded by the $H^{1-e_{0}}$ norm\n(respectively, the $K_{1}^{1}~$norm) of $w_{(\\alpha^{\\prime\\prime},0)}$. This\nfinishes the proof.\n\\end{proof}\n\n\\section{Failure of uniqueness in higher codimension}\n\n\\label{Sec:FailureUniqueness}\n\nThe question addressed in this section is the uniqueness of solutions with\nprescribed initial data on a hypersurface $M$ of codimension greater than one.\nThis is a nontrivial issue if one requires that solutions exist globally in\nspace-time, which has been the focus of the analysis in the preceding sections.\nIn section 3 we showed that initial data consisting of the values of the\nsolution $u(x,y)$ and its first normal derivatives on $M$, through a procedure\nof extension, give rise to constraint-satisfying Cauchy data on a\ncodimension-one hypersurface $N$. These extensions are highly nonunique, and\ntherefore so are the resulting global solutions.\n\nWe now raise the question whether prescribing an arbitrarily large but finite\nnumber of normal derivatives on $M$, as well as insisting upon global\nsolutions, would remedy the nonuniqueness. This data should satisfy the\ncompatibility conditions implied by the commuting of mixed partial derivatives\nand by equation \\eqref{ultra}. Given Courant's classic result (1962) in the\ncase of purely timelike $M$, that data given in any $\\varepsilon$-tubular\nneighborhood of $M$ within $N$ determine solutions uniquely in the $C^{2}$\ncategory, one might think that specifying additional data for $u(x,y)$ on $M$\nwould suffice. In fact, if one specifies any finite number of derivatives of\n$u$ on $M$ it does not.\n\n\\begin{theorem}\n\\label{thm10}\nGiven $k$, there exist constraint-satisfying data $u_{0},u_{1}$ on $N$ which\nvanish to order $k$ on $M$.\n\\end{theorem}\n\n\n\\noindent Therefore, there exists a globally defined solution $u(x,y)$ which\nhas initial data $u(x,y)=u_{0}$, $\\partial_{y_{1}}u(x,y)=u_{1}$ on $N$, which\nvanishes to order $k$ on $M$. Hence any other solution $v(x,y)$ which takes on\nspecified data on $M$ up to $k$-many derivatives may be changed by adding this\nsolution $u$ to it, without changing its initial data.\n\n\\begin{proof}\nWe follow a construction that was used for the extension operators of section\n$3$. Let $\\chi_{3}(\\xi,\\eta^{\\prime})$ be a Schwartz class function with\nsupport in the set $\\left\\{  \\left\\vert \\eta^{\\prime}\\right\\vert\n^{2}<\\left\\vert \\xi\\right\\vert ^{2}\\right\\}  \\subseteq\\hat{N}$. Its Fourier\nrestriction to $\\hat{M}$, given by\n\\[\n\\iint\\chi_{3}(\\tilde{\\xi},\\tilde{\\eta},\\xi^{\\prime\\prime},\\eta^{\\prime\\prime\n})d\\xi^{\\prime\\prime}d\\eta^{\\prime\\prime}=\\mu(\\tilde{\\xi},\\tilde{\\eta})\n\\]\nis in Schwartz class in $\\hat{M}$. Because of the support of $\\chi_{3}$,\n\\[\nv(x,y^{\\prime})=(\\mathcal{F}^{-1}\\chi_{3})(x,y^{\\prime})\n\\]\nsatisfies the constraint. While $v$ may be nonzero on $M$, as may its\nderivatives, it is the case that for homogeneous polynomials $p_{k+1}%\n(x^{\\prime\\prime},y^{\\prime\\prime})$ of degree $k+1$, the function\n$p_{k+1}(x^{\\prime\\prime},y^{\\prime\\prime})v(x,y^{\\prime})$ on $N$ vanishes on\n$M$ to at least order $k$. Furthermore $p_{k+1}v$ satisfies the constraint.\nIndeed,%\n\\[\n(\\mathcal{F}p_{k+1}v)(\\xi,\\eta^{\\prime})=p_{k+1}(\\frac{1}{i}\\partial\n_{\\xi^{\\prime\\prime}},\\frac{1}{i}\\partial_{\\eta^{\\prime\\prime}})\\chi_{3}%\n(\\xi,\\eta^{\\prime})\\text{,}%\n\\]\nand differential operators do not affect the support.\nSet data $u_{0}(x,y^{\\prime})=(p_{k+1}v)(x,y^{\\prime})$ and $u_{1}=0$, and\nsolve equation (\\ref{ultra}). Because this data satisfies the constraint, the\nsolution $u(x,y)$ is global. Because of the properties of the initial data,\nall $x$ and $y^{\\prime}$ derivatives of $u(x,y)$ vanish on $M$. Because\n$u_{1}=0$ and $u$ itself satisfies equation (\\ref{ultra}), all $y_{1}$\nderivatives up to order $k$ as well as any mixed derivatives also vanish.\n\\end{proof}\n\n\n\\section{A variant of a uniqueness theorem of Courant}\n\nCourant (1962) gives a uniqueness result for the ultrahyperbolic equation with\ndata posed on a hypersurface of mixed signature, which in our notation states\nthat, among $C^{2}$ solutions, initial values of $u(x,0,y^{\\prime})$ and\n$\\partial_{y_{1}}u(x,0,y^{\\prime})$ prescribed in the set in the Cauchy\nhypersurface $M$ given by\n\\begin{equation}\n\\sum_{\\ell=1}^{d_{1}}(x_{\\ell}-x_{\\ell}^{0})^{2}\\leq a^{2}~,\\qquad\\sum\n_{\\ell=2}^{d_{2}}(y_{\\ell}-y_{\\ell}^{0})^{2}\\leq\\varepsilon^{2}%\n\\end{equation}\nwill determine \\textit{a priori} the values of the data on the larger set\n\\begin{equation}\n\\left\\{  (x,y^{\\prime})\\in M\\ :\\ \\sqrt{\\sum_{\\ell=1}^{d_{1}}(x_{\\ell}-x_{\\ell\n}^{0})^{2}}+\\sqrt{\\sum_{\\ell=2}^{d_{2}}(y_{\\ell}-y_{\\ell}^{0})^{2}}\\leq\na\\right\\}  ~.\\label{Eqn:CourantStatement1}%\n\\end{equation}\nFurthermore the solution is determined uniquely in the space-time region\n\\begin{equation}\n\\left\\{  (x,y)\\in{\\mathbb{R}}^{d_{1}+d_{2}}\\ :\\ \\sqrt{\\sum_{\\ell=1}^{d_{1}%\n}(x_{\\ell}-x_{\\ell}^{0})^{2}}+\\sqrt{\\sum_{\\ell=1}^{d_{2}}(y_{\\ell}-y_{\\ell\n}^{0})^{2}}\\leq a\\right\\}  ~.\\label{Eqn:CourantStatement2}%\n\\end{equation}\nCourant's proof of this fact uses the Asgeirsson mean value theorem in a\nfundamental way.\n\nThe key implication from our point of view is that data on an arbitrarily\nsmall cylindrical subset of $M$, plus the stipulation of $C^{2}$ regularity,\ndetermine the data and indeed the solution on much larger sets of $M$ and of\nspace-time, respectively. In turn, knowledge of the data in a small cylinder\ndetermines the values of all of its derivatives on $N=\\{(x,y)\\ :\\ y=0\\}$ (if\nthe data are smooth). This contrasts to the case discussed in\nsection~\\ref{Sec:FailureUniqueness}, in which it is shown that specification\nof a possibly large but finite number of derivatives does not lead to unique\nsolutions, even when the constraint is imposed and the resulting solutions are\nglobally defined and smooth.\n\nIn this section we give a version of the above theorem of Courant, for data\nposed in ellipsoidal domains in the Cauchy hypersurface $M$, which are\nlocalized near the $\\{y^{\\prime}=0\\}$ coordinate axis (or any translate\nthereof). Our proof of this result is based on the Holmgren--John theorem\n(John 1982), and therefore remains true under perturbations to the equation.\nThus it is a robust generalization of the Courant result, which being based on\nAsgeirsson's theorem is true only for precisely the ultrahyperbolic equation.\n\n\\begin{theorem}\n\\label{thm11}\nLet $\\varepsilon >0$ and define the ellipsoid\n$Z_\\varepsilon \\subseteq M$ by\n\\begin{equation}\nZ_\\varepsilon = \\{(x,y) \\ : \\ y_1 = 0 \\ ,\n|x|^2 +\\frac{|y'|^2}{\\varepsilon^2} < 1 \\} ~, \\qquad\n0 < \\varepsilon \\leq 1~ .\n\\end{equation}\nA $C^2$ solution to \\eqref{ultra} whose Cauchy data vanishes on\n$Z_\\varepsilon$ must necessarily vanish on the set\n\\[\nD = \\{ (x,y) \\in {\\mathbb R}^{d_1+d_2} \\ : \\ |x| + |y| < 1 \\} ~\n\\]\nand in particular its Cauchy data along with all derivatives\nmust vanish on the subset  of the Cauchy hypersurface given by\n$\\{ (x,y') \\in M \\ : \\ |x| + |y'| < 1 \\}$.\n\\end{theorem}\n\n\n\\begin{proof}\nDefine $R_\\varepsilon(w)$ to be the cone over $Z_\\varepsilon$ with\nvertex $v = (0,w_1,w') \\in \\{(x,y) \\ : \\ x=0 \\}$. We will show\nthat for any $w = (w_1,w')$ with $|w| \\leq 1$ (namely the unit sphere\nin ${\\mathbb R}^{d_2}$), the region between the cone\n$R_\\varepsilon(w)$ and the ellipsoid $Z_\\varepsilon$ is a region\nof determinacy for the ultrahyperbolic equation. The closure of the\nenvelope of such ellipsoidal cones includes the region $D$; in fact it\nis slightly larger. The result will follow accordingly.\n\nFor a given $R_\\varepsilon(w)$, the Holmgren--John theorem is based\nupon the construction of an analytic family of noncharacteristic\nhypersurfaces $S_\\lambda$ with which to sweep the region between\n$Z_\\varepsilon$ and $R_\\varepsilon(w)$. Taking the case of the vertex\n$v = (0,w)$ with $w=e_1:=(1,0)$, define\n\\[\nS_\\lambda := \\{ (x,y) \\ : \\ (1-y_1)^2 - \\bigl( |x|^2 +\n\\frac{|y'|^2}{\\varepsilon^2}\\bigr) = - \\lambda \\}\n\\]\nwith $-1 \\leq \\lambda \\leq 0$. The normal to $S_\\lambda$ is\n$N_\\lambda = -2(x,(1-y_1),y'\/\\varepsilon^2)^T$, so that the\ncharacteristic form calculated on $N_\\lambda$ is\n\\[\n\\frac{1}{4} N_\\lambda^T\n\\begin{pmatrix} -I_{d_1\\times d_1} & 0 \\\\\n0 & I_{d_2\\times d_2}\n\\end{pmatrix} N_\\lambda\n= \\frac{1}{4} \\bigl( -|x|^2 + (1-y_1)^2 + |y'|^2\/\\varepsilon^2\n\\bigr) ~.\n\\]\nTaking into account that $(x,y_1,y') \\in S_\\lambda$ and solving\nfor $(1-y_1)^2$,\n\\[\n\\frac{1}{4} N_\\lambda^T\n\\begin{pmatrix} -I_{d_1\\times d_1} & 0 \\\\\n0 & I_{d_2\\times d_2}\n\\end{pmatrix} N_\\lambda\n= \\frac{1}{4} \\Bigl( \\bigl(\\frac{1+\\varepsilon^2}{\\varepsilon^4}\\bigr)\n|y'|^2 - \\lambda \\Bigr) ~.\n\\]\nRecalling that $\\lambda < 0$ (except in the limiting case\n$S_\\lambda \\to R_\\varepsilon$) observe that this family of hyperboloids constitutes\na noncharacteristic analytic family which sweeps the region between\n$Z_\\varepsilon$ and $R_\\varepsilon(e_1)$. Thus the Holmgren--John\nuniqueness theorem applies, and this region is a region of determinacy\nfor the ultrahyperbolic equation \\eqref{ultra}.\n\nWe have already achieved the analogue of the statement\n\\eqref{Eqn:CourantStatement2} of Courant. Namely, given\nthe values of a $C^2$ solution $u(x,y)$\nto \\eqref{ultra} in the space-time ellipsoid\n\\[\nW_\\varepsilon := \\{(x,y) \\ : \\\n|x|^2 +\\frac{|y|^2}{\\varepsilon^2} < 1 \\} ~,\n\\]\nwe may slice it with a hyperplane which contains the $x$-coordinate\naxes but which is otherwise arbitrarily oriented in $y$, to determine\na possible $Z_\\varepsilon$,\nwhich in turn determines the solution over the larger conical region\n$R_\\varepsilon$ with base $Z_\\varepsilon$. All of these regions have\nbeen shown to be domains of determinacy. Their union contains the\nset $D = \\{ (x,y) \\ : \\ |x| + |y| < 1 \\}$. Therefore if a\nsolution vanishes in $W_\\varepsilon$ it must also vanish in $D$.\n\nReturning to the problem of the domain of determinacy of the set\n$Z_\\varepsilon \\subseteq M$, we\ngeneralize the above construction to any $w \\in {\\mathbb R}^{d_2}$ with\n$|w| = 1$. Let $w = Re_1$ for $e_1 = (1,0\\dots)$, where $R$ is an\northogonal matrix. Changing variables to $z = Ry$\nand using a symmetric matrix $Q$ of signature $(-,+ \\dots)$, an\nanalytic family of hyperboloids is given by\n\\[\nS_\\lambda(w) := \\{ (x,z) \\ : \\\n|x|^2 + \\langle (z-e_1), Q (z-e_1) \\rangle = \\lambda \\}~,\n\\]\nwhere the Euclidean inner product is given by $\\langle \\cdot,\n\\cdot\\rangle$.\nThe matrix $Q$ is to be chosen so that the intersections of\nthe hyperboloids $S_\\lambda(w)$ with the hypersurface $M$ lie\nin $Z_\\varepsilon$, and sweeps it as $\\lambda$ is varied.\n\nAt this point we may assume without loss of generality that\n$w=(w_1,w') = (w_1,w_2, 0 \\dots)$, whereupon $Q$\nmay be chosen such that\n\\[\nQ = \\begin{pmatrix}  Q_2 & 0 \\\\\n0 & \\frac{1}{\\varepsilon^2} I'' \\end{pmatrix}\n~,\n\\qquad Q_2^T = Q_2 ~,\n\\]\nfor $Q_2$ a $2\\times 2$ symmetric matrix with signature $(-,+)$.\nFurthermore, the above rotation is then set to be\n\\[\nR = \\begin{pmatrix}  R_2 & 0 \\\\\n0 &   I'' \\end{pmatrix} ~, \\qquad\nR_2 = \\begin{pmatrix} \\cos(\\theta) & \\sin(\\theta) \\\\\n-\\sin(\\theta) & \\cos(\\theta) \\end{pmatrix} ~.\n\\]\nIn $y-$coordinates the hyperboloid family is expressed\n\\[\nS_\\lambda(w) := \\{ (x,y) \\ : \\\n|x|^2 + \\langle (y-w), R^T Q R(y-w) \\rangle = \\lambda \\}~,\n\\]\nand the stipulation is that $S_0(w)$ should intersect the hypersurface\n$M$ in the original ellipsoid $Z_\\varepsilon$. This imposes the\ncondition that\n\\[\n|x|^2 + \\langle (x,0,y'), R^T Q R(x,0,y') \\rangle\n:= |x|^2 + \\langle (x,0,y'), B (x,0,y') \\rangle\n= |x|^2 + \\frac{1}{\\varepsilon^2}|y'|^2 ~,\n\\]\nwhere $B_2$ is the upper left-hand $2\\times 2$ block of the matrix\n$B$. Therefore one finds the matrix elements of $B_2$\n\\[\nb_{11} = -\\frac{\\varepsilon^2 - \\sin^2(\\theta)}{\\varepsilon\n\\cos^2(\\theta)} ~, \\quad\nb_{12} = -\\frac{\\tan(\\theta)}{\\varepsilon^2} ~, \\quad\nb_{22} = \\frac{1}{\\varepsilon^2} ~,\n\\]\nand furthermore, the $2\\times 2$ matrix $Q_2$ is\n\\begin{equation}\\label{Eqn:Q2}\nQ_2 = \\begin{pmatrix} -1 & \\tan(\\theta) \\\\\n\\tan(\\theta) & \\frac{1}{\\varepsilon^2}a \\end{pmatrix} ~,\n\\end{equation}\nwhere $a = a(\\varepsilon,\\theta) = (1 + (1-\\varepsilon^2)\\tan^2(\\theta))$.\nCalculating the characteristic form on the hyperboloids $S_\\lambda(w)$,\nwe compute the normal $N_\\lambda(w)$ as\n\\[\n-\\frac{1}{2} N_\\lambda(w) = (x, Q(z-e_1))^T ~.\n\\]\nNoting that the characteristic form is invariant under rotations\n$R$ as above, which leave the coordinate subspaces\n${\\mathbb R}^{d_1}_x$ and ${\\mathbb R}^{d_2}_y$ invariant,\nwe find that\n\\[\n\\frac{1}{4} N_\\lambda(w)^T \\begin{pmatrix} -I_{d_1\\times d_1} & 0 \\\\\n0 & I_{d_2\\times d_2} \\end{pmatrix} N_\\lambda(w)\n= -|x|^2 + \\langle (z-e_1), Q^2 (z-e_1)\\rangle ~.\n\\]\nThis is evaluated on the hyperboloid $S_\\lambda(w)$, on which\n\\[\n|x|^2 + \\langle (z-e_1), Q(z-e_1)\\rangle = \\lambda ~.\n\\]\nSolving for $|x|^2$, we find\n\\[\n\\frac{1}{4} N_\\lambda(w)^T \\begin{pmatrix} -I_{d_1\\times d_1} & 0 \\\\\n0 & I_{d_2\\times d_2} \\end{pmatrix} N_\\lambda(w)\n= \\langle (z-e_1), [Q^2 + Q](z-e_1)\\rangle - \\lambda ~.\n\\]\nSpecifically, the matrix $[Q^2 + Q]$ is\n\\[\n[Q^2 + Q] = \\begin{pmatrix} Q_2^2 + Q_2 & 0 \\\\\n0 & \\bigl( \\frac{1+\\varepsilon^2}{\\varepsilon^4} \\bigr)\nI'' \\end{pmatrix} ~.\n\\]\nUsing the form \\eqref{Eqn:Q2} for $Q_2$, one calculates\n\\[\n[Q_2^2 + Q_2] = \\begin{pmatrix}\n\\tan^2(\\theta) & \\frac{a}{\\varepsilon^2}\\tan(\\theta) \\\\\n\\frac{a}{\\varepsilon^2}\\tan(\\theta) &\n\\frac{a^2}{\\varepsilon^4} + \\tan^2(\\theta)\n\\end{pmatrix} ~.\n\\]\nIt is easily verified that this is positive definite. Recalling that\n$\\lambda \\leq 0$ in the definition of the analytic families of\nhyperboloids, it follows that $S_\\lambda(w)$ are all noncharacteristic,\nand hence the Holmgren--John theorem applies, thus completing the\nargument.\n\\end{proof}\n\n\\begin{acknowledgements}\n\\textbf{Acknowledgements:}\nThe research of the first author was partially supported by the Canada Research\nChairs Program and NSERC grant \\#238452-06.  The research of the\nsecond author was partially supported by a grant from SSHRC.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe problem of inverse inference has been for a long time one of the\nmain issues in neural networks\nanalysis~\\cite{Bialek2006,Bialek2007,Bolle2009}.  Given a number of\nstimuli, one measures the activity of some local components, such as\nspike trains of the neurons, to identify which connections between\nthem, i.e. which synapses, are active, and, possibly, their\nstrength~\\cite{Amit,KappenRodriguez1997}.  More recently the problem\nof inverse inference has manifested itself in other branches of\nbiology. For example in structural biology the problem comes down to\ndetermine the probability distribution of amino acid strings by\nobserving the way in which proteins naturally\nfold~\\cite{Socolich.etal-2005}, or, in systems biology it consists in\nrecovering structural details of protein-protein interactions from\nprimary sequence information of gene regulatory\nnetworks~\\cite{Szallasi-1999,Weigt-PNAS2009}.  A solution to these\ntypes of problems usually comes about in the following way: one\nproposes a model capturing the characteristics of the network under\nstudy, and eventually develops methods to retrieve the structural\ncharacteristics, which can allow to disprove the underlying\nhypothesis depending on whether the findings are consistent with the observed\nbehavior.\n\nThe pairwise Ising spin glass has been widely used as a starting point\nfor analyzing the above problems.  The fact that these problems have a\nnumber of common features has allowed to develop several algorithms\naddressing the corresponding inverse inference\nproblem~\\cite{SessakMonasson2008,MezardMora2008,Roudi-Hertz2009,Roudi-Aurell2009}.\nHowever, as more accurate methods come along, new issues regarding the\nvalidation of the underlying hypothesis have been raised.  For\ninstance, it was pointed out in \\cite{Roudi-Nirenberg2009} that by\nonly observing a subset of the nodes composing a neural network,\nreconstructing the original couplings according to a pairwise model\ndoes not necessarily lead to any added information as there exists a\none-to-one relation between the couplings and the (normalized) second\norder correlation coefficient for a certain fraction of hidden\nvariables onward. Thus, in this case one needs to consider higher\norder couplings.  Moreover, binning spikes of the neurons, and\naccordingly representing their status by Ising-like variables, can\noversimplify the actual observations. Similarly, determining the\npossible states in protein folding needs at least Potts-like\nvariables.  Note also that while for protein networks it is natural to\nassume a pairwise interaction model, in neural networks this is not\nnecessarily the case and though in theory any network can always be\ntransformed into one which contains only pairwise\ninteractions~\\cite{YedidiaFreemanWeiss-2002}, the practical way to go\nabout this might not always be obvious.\n\nIn this paper, we do not address the interesting cases of the presence\nof hidden variables, of higher order interactions or of Potts-like\nvariables.  Rather, we question a more basic issue: given a situation\nwhere the pairwise Ising spin glass correctly describes the structure of\nthe network under study, how much information can be gathered from an\nexperimental input about the values of the couplings of the model?  In\nother words, given all two-point correlations, or equivalently, given\nthe susceptibilities, up to which point can we say something about the\nreconstructed couplings?  Does this information allow us to\nreconstruct the original model or are there intrinsic uncertainties to\nthis inverse inference problem? How does the quality of the\nreconstruction depend on the original distribution of couplings or on\nthe size of the system?  The main question can be phrased as to\nwhat amount do the statistical errors that affect the measurements\ninhibit us to reconstruct the original model: if the original\nobservations are incomplete or noisy, up to which point does it make\nsense to try and reconstruct the data?  Ideally, one would like to\nanswer the above questions from a theoretical viewpoint. Here,\nhowever, we start by numerically investigating several of the above\nproblems by means of a message  passing algorithm, first introduced\nin~\\cite{MezardMora2008}, which is currently among those delivering\nthe best results~\\cite{SessakMonasson2008}.  We will use this\nalgorithm to analyze the reconstruction of various types of networks\ngiven their first- and second-order local, possibly noisy,\nobservables.\n\nWe want to analyze some basic features of the\nreconstruction process, by spotting some relevant weakness and by\ntrying to focus on systematic trends that can be relevant in\nexploiting this approach and, consequently, in trying to devise improvements \nthat could lead to better performances. We consider these result as a\ntoolbox spelling and clarify a number of facts that can be useful\nfor a better understanding of and improving this class of methods.\n\nIn Section~\\ref{S:MET} we introduce the message passing method, and\ndefine some relevant quantities.  We describe in detail the iterative\nrules in presence of a memory term, that allows the convergence to a\nfixed point.  In Section~\\ref{S:DIS} we analyze the reconstruction\nprocedure for different distribution of the couplings: we look at\nbinary random couplings and at Gaussian couplings.  In\nSection~\\ref{S:SYN} we introduce synthetic random errors, by modifying\nrandomly exact values for the susceptibility, and we try to understand\nhow a larger incertitude affects the quality of the reconstruction of\nthe couplings. We analyze both the case of an additive error and the\none of a multiplicative error.  In Section~\\ref{S:MC} we analyze data\nobtained by a Monte Carlo simulation, and we study the quality of the\nreconstruction as a function of the accuracy of the measurements. We\ndraw our conclusions in Section~\\ref{S:CON}.\n\n\\section{Susceptibility propagation and the inverse Ising spin glass\\label{S:MET}}\n\nWhile message passing algorithms have been widely used to solve the\ndirect inference problem where the characteristics of the underlying\nnetwork are given and one wants to derive experimentally observable\nquantities~\\cite{MezardParisi2000}, their adaptation to tackle the\ninverse problem is relatively recent.  Here we consider the inverse\nIsing spin glass, which assumes that the basic constituent agents of the\nnetwork interact only in a pairwise, symmetric way with the other\nagents.  In other words, we assume the problem is described by the\nfollowing partition function:\n\\begin{equation}\nZ=\\sum_{\\mathbf{\\sigma}}\\exp \\left[-\\frac{1}{T} \n\\left(\\sum_{i=1}^N h_i \\sigma_i + \\sum_{i<j} J_{ij} \\sigma_i \\sigma_j \\right)\n\\right]\\;,\n\\label{eq:part_fction}\n\\end{equation}\nwhere the $\\sigma_i$ can take the values $\\pm 1$, $h_i$ are\n(potentially site dependent) magnetic fields, the couplings $J_{ij}$\nare quenched random variables that can have both a positive and a\nnegative sign and are distributed under a probability function that we\nwill discuss in the following, and $T$ is a ``temperature'' governing\nthe behavior of the system.  We can now define the inverse Ising spin\nglass problem by considering as given (by a, potentially numerical,\nexperiment) the local magnetizations and the susceptibilities, and by\ntrying to compute the $N$ local fields $h_i$ and the $N(N-1)\/2$\ncouplings $J_{ij}$.\n\nM\\'ezard and Mora have recently introduced a message passing\nprocedure, the so called {\\it susceptibility\n  propagation}~\\cite{MezardMora2008}, that is able, in appropriate\nconditions, to solve this problem.  As for the direct problem, it is a\ndistributed algorithm where messages are exchanged between any two\npairs of nodes. In case of the inverse problem, the content of some of\nthese messages concerns the probability distribution of the local\nfield of a vertex, while other messages contain information about the\ndistribution of the couplings between two vertexes.\n\nMore specifically in the inverse problem one defines four types of\nmessages, $h_{i \\to j}$, $u_{i \\to j}$, $v_{i \\to j,k}$ and $g_{i \\to\n  j,k}$.  The messages $h_{i \\to j}$ and $u_{i \\to j}$ are exchanged\nbetween couples of nodes, while the messages $v_{i \\to j,k}$ and $g_{i\n  \\to j,k}$ are exchanged between triples of nodes.  The\nsusceptibility propagation algorithm starts by assigning a random\nvalue to each of these messages and setting all the ``estimated\ncouplings'' $J_{ij}$ to zero. These quantities are updated iteratively\naccording to the following rules (where the $m_i$ and the $\\chi_{ij}$\nare the ``experimental'' inputs):\n\\begin{eqnarray}\nh_{i \\to j} & = & \\rm{arctanh}\\left( m_i\\right) -u_{j \\to i} \n\\label{eq:update_rule_h}\\\\\ng_{i \\to j,k} & = &\\frac{\\delta_{i,k}}{T} + \n\\sum_{l \\in \\partial i\\setminus j} v_{l \\to i,k} \\label{eq:update_rule_g}\\\\\n\\tanh \\left(\\frac{J_{ij}}{T}\\right) & = & \n\\varepsilon \\left[ \\frac{\\tilde{C}_{ij} \n- \\tanh \\left(h_{i \\to j}\\right) \n\\tanh \\left(h_{j \\to i}\\right) }\n{1- \\tilde{C}_{ij}\\tanh \\left(h_{i \\to j}\\right) \n\\tanh \\left(h_{j \\to i} \\right)}\\right]\\\\ \n&+& \\left(1-\\varepsilon\\right) \n\\tanh \\left(\\frac{J_{ij}}{T}\\right) \\label{eq:update_rule_j}\\\\\n\\tanh \\left(u_{k \\to i}\\right) & = & \n\\tanh \\left(\\frac{J_{ik}}{T}\\right) \\tanh \\left(h_{k \\to i}\\right) \n\\label{eq:update_rule_u}\\\\\nv_{l \\to i,k} & = & g_{l \\to i,k} \n\\tanh \\left(\\frac{J_{il}}{T} \\right)\n\\frac{1 - \\tanh^2 \\left(h_{l \\to i}\\right)}\n{1 - \\tanh^2 \\left(u_{l \\to i}\\right)}\\;,\n\\label{eq:update_rule_v}\n\\end{eqnarray} \nwhere\n\\begin{equation}\n\\nonumber\n\\tilde{C}_{ij}  \\equiv \\frac{\\chi_{ij}-g_{i \\to j,j}(1-m_i^2)}{g_{j\\to i,j}} \n+ m_i m_j \\label{eq:update_rule_c}\\;.\n\\end{equation}\nFor a detailed discussion of these equations (with $\\varepsilon=0$),\nsee~\\cite{Mora2007}.  If this set of equations converges to a fixed\npoint, the local fields are reconstructed as\n\\begin{equation}\nh_i  = T \\left( \\rm{arctanh}\\left( m_i\\right) \n- \\sum_{k \\in \\partial i} u_{k \\to i}\\right)\\;.\n\\label{eq:local_fields}\n\\end{equation}\nThe couplings can be read directly from equation\n(\\ref{eq:update_rule_j}).\n\nThe additional external parameter~\\cite{MezardMoraPC}\n$\\varepsilon\\in]0,1[$ that we have introduced in the update rule\n(\\ref{eq:update_rule_j}) is a memory term: we keep, when computing\n the new value of a coupling, part of the old value, and only modify\n the coupling partially. This is a possible way out to the fact that\n for $\\varepsilon=0$ the iteration typically fails since it ends up to\n propose an absolute value larger than one for updating\n $\\tanh\\left(J_{ij}\/T\\right)$. Clearly this slows down the\n convergence: in our scheme we had to use very small values of\n $\\varepsilon$, but this eventually guaranteed that the susceptibility\n propagation algorithm was eventually leading in most part of the\n cases to a fixed point solution (as we discuss in the following\n section).  We will discuss in some more detail the role of the\n ``temperature'' $T$ that appears in an explicit form in our iterative\n equations: if one thinks the local susceptibilities to be connected\n to spin-spin correlation functions according to the usual relation\n $\\chi_{ij}=\\beta_P \\langle \\sigma_i \\sigma_j\\rangle$ (where $\\beta_P$\n is the ``physical'' value of the inverse temperature) the parameter\n $T$ is naturally related to $\\beta_P^{-1}$, but one has indeed some\n more freedom in tuning $T$.\n\nIn this note we will consider the case where all local fields are zero\nand we will focus on how to reconstruct the couplings when the local\nsusceptibilities are given. One way to evaluate how close the\nreconstructed couplings are to the original ones is based on computing the\nKullback-Leibler distance, which measures the weighted difference\nbetween the exact and the reconstructed\ndistribution~\\cite{KappenRodriguez1997,Roudi-Nirenberg2009}.  An exact\ncomputation of the Kullback-Leibler distance involves a sum over all\nthe states of the system, and can only be performed for rather small\nsystems. We have used for our comparison a simpler indicator, that is\nappropriate for our goal:\nwe\nconsider the ratio of the\naverage of the squared difference of the\nreconstructed couplings\n$J_{ij}^r$ and the original couplings $J_{ij}$, and the variance\n$\\sigma$ of the distribution of the exact couplings,\n\\begin{equation}\n\\Delta=\\frac{\\sqrt{ \\left\\langle \\frac{\\sum_{i<j}(J_{ij}^{r}-J_{ij})^2\n    }{N(N-1)\/2}\\right\\rangle}} {\\sigma}\\;.\n\\label{eq:error}\n\\end{equation}\nIn the following we always consider coupling distributions with zero\nmean and variance $\\sigma=\\tilde{J}\/\\sqrt{N}$, and we will set\n$\\tilde{J}\\equiv 1$. In the case of fully connected graphs, i.e. the\nusual mean field theory of spin glasses (Sherrington-Kirkpatrick\nmodel), the spin glass phase sets in at\n$T_c = 1$~\\cite{SherringtonKirkpatrick1975}.\nUsing a different terminology~\\cite{KappenRodriguez1997,SessakMonasson2008}, \none fixes the temperature (that disappears from the iteration scheme) \n$T \\equiv 1$ and looks at $\\tilde{J}^{-1\/2}$, in which case the\ncritical transition on the complete graph occurs at $\\tilde{J}=1$.\nConsidering explicitly\na ``temperature'' \nhas the advantage that it provides direct connections to\nthe (direct) studies of spin glasses in statistical mechanics, and\nit can allow  \nto use the $T$ parameter to modify the iterative procedure.  We\nhave always averaged the quantity $\\Delta$ defined in\nEq.~(\\ref{eq:error}) over either a hundred (for small values of $N$)\nor twenty (for larger values $N$) different distributions of couplings\n(both in the case of binary couplings and in the case of Gaussian\ncouplings).  As we will describe in detail in the following we have\nanalyzed both cases where the input to our inverse reconstruction\nprocedure was based on exact values of the observables ($m$ and\n$\\chi$) computed using an exact enumeration in small systems and cases\nwhere they were computed by Monte Carlo simulations (where the\nstatistical accuracy of the time series was kept under control and is\none of the relevant issues we have investigated).\n\n\\section{The quality of the reconstruction as a function of temperature\n  and of the coupling distribution\\label{S:DIS}}\n\nThe first issue that we have analyzed is the quality of the\nreconstruction as a function of the temperature $T$ of the Ising spin glass\nand of the distribution of the couplings. We consider the two, somehow\nextreme cases, of binary couplings, where $J_{ij}=\\pm 1$ with\nprobability $\\frac12$, and of Gaussian couplings, where the $J$ have\nzero average and unitary variance. We obtain the susceptibilities that\nserve as input to the inverse reconstruction by exact enumeration. Thus, in this section we are dealing with inputs that are exact values (but for\nthe finite precision of the input words, see later). All the results\nare sample averages, as described in the former section. \n\nIn Fig.~\\ref{fig:exact_distributiondependence} we show a plot of the\naverage error (\\ref{eq:error}) of the reconstruction of the couplings\nas a function of the temperature for the two different distributions\nof the couplings and for different system sizes (in the main plot of\nthe figure we used variables of $12$ byte).  With our normalization\n$T=1$ is $T_c$, the critical temperature of the statistical model (the\nmean field, Sherrington-Kirkpatrick spin glass).\n\nIn a large temperature interval the quality of the reconstruction\nessentially only depends on the temperature $T$, and not on $N$, nor on\nthe details of the distribution of the couplings: when increasing $T$\nthe quality improves.  Close to $T_c$ the error becomes large. Still\neven for temperatures of the same order of magnitude as \n$T_c$ (at least down to $T=2$, say) our procedure allows to get precise hints\nabout the values of the individual couplings.\n\nIn the lower inset of the figure we enlarge the left part of the main\nplot, for temperatures in the range going from $2$ up to $12$. This \ndemonstrates even more clearly that the error in the reconstruction does not\ndepend much on the type of distribution we considered, nor on the size\nof the system. In other words, it is no problem for our algorithm\nto distinguish the two different cases where couplings can only take a\nfew values (here two) and where they come from a continuous\ndistribution: this can have some relevance when deciding how to\nanalyze and interpret experimental data.  Also $\\Delta$ is an average\nof the systematic error of the reconstruction of one single coupling:\nthis means that the algorithm is essentially more precise for\nincreasing $N$, as is well known for MP-algorithms and was already\nobserved in~\\cite{MezardMora2008}.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[angle=270, width=0.9\\textwidth]{coupling_DiffDistribJSK_h0.0_J1.0.eps}\n\\caption{\\label{fig:exact_distributiondependence} The reconstruction\n  error $\\Delta$ as a function of $T$ for fully connected graphs with\n  binary (full lines) or Gaussian distributed couplings (dotted\n  lines): here variables occupy $12$ bytes. \n  We considered two\n  different sizes of the graphs: $N=10$ ($+$) and $N=20$\n  ($\\square$). The $x$ and $y$ scales of the insets can be easily\n  deduced from the main plot. The lower inset contains a detail of the\n  same plot for $T \\in [2,12]$. The upper inset demonstrates the\n  dependence of $\\Delta(T)$ on the precision of the variables used in\n  implementing the susceptibility propagation algorithm that allows\n  reconstructing the binary distributed couplings. From top to bottom,\n  we used $4$, $8$ and $12$ byte variables, respectively. }\n\\end{figure}\n\nThese facts have strong implications about the scope that the pairwise\nstatistical model we are using can enjoy: there are situations, in\nother words, where we cannot gather any useful information by trying\nto determine the couplings $J_{ij}$. If all the correlation\nfunctions among the constituent agents of a system are known, and we\nknow that they only interact in a pairwise manner, we are able to\nreconstruct each single coupling if the system is at a relatively high\ntemperature, or if, in other words, the corresponding distributions of\nthe couplings have large variance. The fact that for decreasing\ntemperature the reconstruction becomes less precise, is most likely\ndue to a lack of information on how to capture the complex structure\nof the solution space near and below the critical temperature\n$T_c=1$. Unfortunately, all algorithms currently available to tackle\nthe inverse problem suffer this same symptom.  In the present case of\nthe susceptibility propagation algorithm the increasingly complex\nstructure of the phase space also manifests itself as a non-convergent\nbehavior of the algorithm.  While for larger temperatures, the\nalgorithm converges to a fixed point, as we approach the critical\npoint, the average change in the messages\n(\\ref{eq:update_rule_h})-(\\ref{eq:update_rule_v}) between two\nconsecutive updates gradually converges to some increasing positive\nvalue until they, eventually, no longer converge. \nA better reconstruction \nof the couplings defeating this critical point could use\nadditional information on the network, or a more advanced \nalgorithm which take this extra complexity into account, such as the survey\npropagation algorithms~\\cite{SP}.\n\nFinally, Fig.~\\ref{fig:exact_distributiondependence} evidences another \nimportant feature, regarding the\nnumerical precision needed for an accurate reconstruction. If the\ncorrelation functions or the susceptibilities have been obtained at\nhigh temperatures, and we are in the ideal case where at least in\nprinciple we know them exactly, the quality of the reconstruction will\nbe essentially limited by the precision with which we represent them\nand that we use in implementing the iterative scheme.  We analyze this\nphenomenon by finding the solution using variables of different sizes\n(respectively $4$, $8$ and $12$ bytes) and we plot the results in the\nupper inset of Fig.~\\ref{fig:exact_distributiondependence}. The\nsituation is quite clear: selecting a level of precision $p$ sets a\nmaximum temperature value $T^*(p)$ that would allow to improve the\nreconstruction. For $T>T^*(p)$ the information we gather is hidden by\nthe insufficient precision $p$, and the reconstruction quality does\nnot improve.  All the results we will discuss in the following have been\nobtained with $12$ bytes wide variables.\n\n\\section{Synthetic noisy data\\label{S:SYN}}\n\nThe susceptibilities (or the correlation functions) one obtains as the\noutput of an experiment are far from exact. The error can either be\ndue to the limitations of the experimental set-up, thus imposing some\nabsolute error on the measured date, or to statistical \nfluctuations, that can originate from \na number of different causes.  In case the\nobservables are averages of successive experiments, the two-point\ncorrelation functions are limited by some relative errors which can\npossibly be improved by performing more experiments. We will discuss\nhere how these different types of error can affect the reconstructions\nof the couplings.\n\nThe case of an error that is on average constant in magnitude\n(independently from the size of the observable we are considering) and\nthe one where its ratio to the signal is constant in magnitude are\nindeed very different. In the case of an error that is constant on\naverage small correlations functions will not give any significant\namount of information: if, for example, for a model endowed with a\nEuclidean distance $d$ we expect an exponential decay\nwith $d$, only the first, larger contributions, will be of use in our\nreconstruction, while the smaller ones will be completely hidden by\nthe noise.\n\nWe start again from exact values of the susceptibility that we compute\nby exact enumeration, summing all the contributions of the $2^N$ spin\nconfigurations.  We simulate the presence of an absolute error by\nincluding an additive noise term to the exact observables: the\nsusceptibilities $\\chi_{ij}^{A}$ used for reconstruction are given\nhere by the exact susceptibilities $\\chi_{ij}$ with the addition of a\nnoise term $r_{\\eta}$, uniformly drawn from the interval\n$[-\\eta,+\\eta]$, with $\\eta > 0$: $\\chi_{ij}^{A}=\\chi_{ij}+r_{\\eta}\\;,\\;\\;\n\\forall\\; ij$.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[angle=270, width=0.9\\textwidth]\n{coupling_sqrtJbimSK_h0.0_J1.0_ADDITIVEnoise.eps}\n\\caption{\\label{fig:ADDnoise} $\\Delta$ as a function of $T$. Here the\n  susceptibilities used as input for the \n  reconstruction of the couplings are only\n  approximate due to an additive noise. From bottom to top:\n  $\\eta=10^{-8}, 10^{-7}, 10^{-6}, 10^{-5}$ for $N=10$ ($+$) and\n  $N=20$ ($\\square$).}\n\\end{figure}\n\nWe show our results for the reconstructed couplings in\nFig.~\\ref{fig:ADDnoise}.  We show the values obtained for different\nchoices of $\\eta$.  The effect of this random noise is irrelevant for the\nlow $T$ range where the reconstruction is possible, but becomes large\nwhen $T$ increases. The larger is $\\eta$, the smaller is the $T$ range\nwhere the reconstruction becomes unreliable. In presence of this kind\nof noise the quality of the reconstruction worsens when $T$ increases:\nthis can be an interesting observation when trying to optimize a\nreconstruction scheme of experimental data.\nIt is interesting to note that the error on the reconstructed coupling\nis several orders of magnitude larger than the additive error on the\nsusceptibilities.  This is due to the fact that the order of magnitude\nof the susceptibilities is large at small temperatures, and much\nsmaller at high temperatures. Therefore, especially at high\ntemperatures, additive noise terms are very damaging.\n\nIn Fig.~\\ref{fig:MULnoise} we show the effect of a multiplicative\nnoise term on the susceptibilities. More precisely, these\nreconstructed couplings are computed starting from the approximated\nsusceptibilities $\\chi_{ij}^{M}$, which were obtained from the\noriginal susceptibilities by multiplying them by a factor\n$r_{\\epsilon}$, which was drawn uniformly from the interval\n$[1-\\epsilon,1+\\epsilon]$, with $\\epsilon>0$:\n$\\chi_{ij}^{'}=r_{\\epsilon}\\chi_{ij}\\;,\\;\\;\\forall\\; ij$.  \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[angle=270, width=0.9\\textwidth]\n{coupling_sqrtJbimSK_h0.0_J1.0_MULTIPLICATIVEnoise.eps}\n\\caption{\\label{fig:MULnoise} $\\Delta$ as a function of $T$. Here the\n  susceptibilities used as input to the reconstruction scheme are only\n  approximate due to the presence of a multiplicative noise. From\n  bottom to top: $\\epsilon=10^{-8}, 10^{-7}, 10^{-6}, 10^{-5},\n  10^{-4}, 10^{-3}$ for $N=10$ ($+$) and $N=20$ ($\\square$).}\n\\end{figure}\n\nAgain, the error on the observables influences severely the\nreconstruction of the couplings at higher temperatures. Here there is\na clearer threshold effect than in the former case, and there is a\nclear dependence of the ``breaking point'' $T^*(\\eta)$ over\n$\\eta$. The situation is very similar to the one that we have\ndiscussed in the previous section, where we were using ``short''\nvariables with a finite, small width (down to four bytes).\n\n\\section{Monte Carlo noisy data\\label{S:MC}}\n\nA Monte Carlo numerical experiment is one of the best proxy for a real\nexperiment. One gets sets of data that are asymptotically distributed\naccording to a certain probability function. These data are affected\nby statistical errors, as it would happen in an experiment.  We\nanalyze here how the reconstruction works when starting from Monte Carlo\ndata obtained under variable accuracy requirements: this is an issue\nof paramount interest, since we need to know if a given real\nexperiment, with a given level of accuracy, will give information\nthat can lead to a useful coupling reconstruction.\n\nSo here we do not start from exact data, but from data obtained by a \nusual Monte Carlo simulation, with a local, accept-reject Metropolis\nupdating scheme, and we use our inverse algorithm to get the couplings\nfrom these data.  We first lead the system to equilibrium (and discard\ndata obtained during this thermalization phase of the simulation), and\neventually collect data for a number of Monte Carlo steps.  We show in\nFig.~\\ref{fig:MCnoise} the error $\\Delta$ on the reconstruction of the\ncouplings, given the values $\\chi_{ij}^{MC}$ of the susceptibilities,\nobtained by sampling the solution space with a Monte Carlo Markov\nChain.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[angle=270, width=0.9\\textwidth]\n{DETAILcoupling_sqrtJbimSK_h0.0_J1.0_N16_MCvsEXACT.eps}\n\\caption{\\label{fig:MCnoise} The error $\\Delta$ as a \n  function of $T$ for \n  fully connected graphs of size $N=16$ ($\\circ$)\n  and $N=128$ ($\\Diamond$), with binary couplings. \n  The different curves represent\n  reconstructions starting from the exact susceptibilities for the\n  $N=16$ system (continuous line),\n  and starting from approximations of the susceptibilities\n  generated from $10^4$, $10^5$, $10^6$, $10^7$ and $10^9$ MC data\n  (from top to bottom) for $N=16$, and $10^5$, $10^6$ and $10^7$ MC\n  data (from top to bottom) for $N=128$.}\n\\end{figure}\n\nGiven a fixed number of Monte Carlo measurements the error on the\nsusceptibilities increases with the temperature, resulting in a less\nprecise reconstruction of the couplings.  However, by increasing the\nduration of the experiments, i.e. increasing the number of independent\nobservations, the relative error can be drastically reduced as can be\nseen from Fig.~\\ref{fig:MCnoise}. The pattern is, as one would have\nexpected, very similar to the one of a statistical error of constant\naverage size. Indeed, this is exactly what happens here, where we\nestimate all correlation functions by adding numbers of order one\n(the individual values of the correlations, that can be $\\pm 1$).\nFor each level of the error the reconstruction works as if correlations\nwere exact up to a given $T$ value, beyond which its accuracy does not\nincrease anymore with $T$, but, on the contrary, it starts decreasing\nwith $T$. \n\nWe also analyzed the low-temperature limit down to which the couplings\ncan be reconstructed starting from approximated two-point\ncorrelations. While the exact correlations in general allow to\nreconstruct the couplings down to a temperature as low as $T=1.7$, no\nsolution could be found, for example, starting from susceptibilities\nobtained from only $10^4$ independent MC data at this same\ntemperature: the susceptibility propagation algorithm can be\nadditionally limited by an inaccurate original data set.\n\nWe have also tried to understand how the performance of the \nsusceptibility propagation algorithm varies when we increase the\nnumber of elements of the system. In the Monte Carlo case we have\nstudied the two cases $N=16$ and $N=128$, where the second system is\neight times larger than the first one: we show both sets of data in\nFig.~\\ref{fig:MCnoise}.  Larger size systems require more experiments\nto get a reconstruction of the same quality than for the smaller systems:\nthe $N=128$ curves overlap with \n$N=16$ curves obtained with ten times less statistics. \nAfter assuming this rescaling\nour data clearly show that the reconstruction procedure also works\nvery well even when we heavily increase the volume of the system.  Let\nus look carefully at ``low'' values of $T$.  For example, when starting \nfrom observables with a good precision, at $T=2$ the\n$N=128$ reconstruction clearly improves in quality with respect to those \nfor $N=16$ (the\nsame phenomenon can already be observed, on a smaller scale, in\nFig.~\\ref{fig:exact_distributiondependence} when comparing $N=10$ and\n$N=20$). Reconstruction on large systems sizes is possible and\nreliable even if the ``temperature'' of the system is not so far from\ncriticality, which certainly is good and useful news.\n\n\\section{Conclusions\\label{S:CON}}\n\nWe have analyzed a number of features of the inverse Ising spin glass\nproblem, by using the susceptibility propagation algorithm, first\nintroduced in~\\cite{MezardMora2008}. In a very large temperature\nwindow, this algorithm is able to reconstruct the individual couplings\nand, consequently, their overall distributions with a remarkable\nprecision.  If the system is ``at high temperature'' (or, in other,\nmaybe more physical terms), if the (zero average) disorder does not\nfluctuate too much, the quality of the reconstruction is basically\nonly limited by the precision under which the experimental input is\nknown, and by the precision used when implementing the susceptibility\npropagation algorithm.\n\nFor smaller temperatures approaching the critical temperature, or\nequivalently, for distributions of the couplings characterized by a\nlarge variance, the reconstruction is less accurate and eventually the\nalgorithm fails to find any solutions to the problem. This is \ndue to the fact that it does not take the possibility of multiple states into account, which is exactly what happens in the spin-glass phase.\n\nAll algorithms currently available suffer this same problem. However,\nthe message passing algorithm used in this paper could possibly be\nimproved by using the probability distributions of the observables as\nbasic working ingredients, rather than the observables themselves to\nobtain a type of survey propagation algorithm for which the exchanged\nmessages do not contain information on the couplings, but rather on\nthe probability distribution of each individual coupling.\nFurthermore, the nature of the susceptibility propagation algorithm\nsuggests it could be easily adapted to include the case of Potts-like\nvariables allowing to treat problems in structural\nbiology~\\cite{Weigt-PNAS2009}.\n\nWhile the overall reconstruction of the pairwise model is quite\nprecise in case the original data set is accurate, the results can\ndeteriorate fast if data are affected by a statistical error.  The\nnumber of experiments that have to be used to obtain the average\ntwo-point correlations needs to be increasingly large for increasing\nsample size. Also, at large temperatures, where the values of the\nsusceptibilities are small, this error on the reconstruction of the\ncouplings becomes more pronounced. For the same reason, an absolute\nerror on the two-point correlations is increasingly damaging at higher\ntemperatures.\n\nAll together we feel that our conclusion lead to an optimistic\nscenario. Even in presence of a large statistical or systematic \nignorance the reconstruction is possible and can be effective. Large\nsamples still allow for a good reconstruction quality, under the\ncondition that the statistical inaccuracy that affects the data\nis lowered down to a reasonable level.\n\n\\section*{Acknowledgments\\label{S:ACK}}\nWe thank Thierry Mora for describing us the use of the\n$\\varepsilon$ term in this context. We acknowledge interesting\nconversations with Federico Ricci-Tersenghi. \n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Acknowledgements}\n\nWe thank Noga Alon for referring us to his paper \\cite{DBLP:conf\/stoc\/AlonMS12} on construction of Ruzsa-Szemer\u00e9di graphs and discussing its implications which were extremely insightful. We are in addition thankful to Hamed Saleh for fruitful discussions and also to anonymous STOC reviewers for helpful suggestions.\n\\section{The Algorithm and Basic Definitions}\\label{sec:analysissetup}\n\nThe algorithm that we analyze is formally stated as Algorithm~\\ref{alg:sampling}.\n\n\\begin{tboxalg}[ (\\cite{soda19})]{A sampling-based non-adaptive algorithm for stochastic matching.}\\label{alg:sampling}\n\t\\textbf{Parameter:} $R$, which controls the maximum degree of $Q$.\n\t\\vspace{0.1cm}\n\t\n\tTake $R$ realizations $\\mc{G}_1, \\ldots, \\mc{G}_R$ of $G$ independently where each realization $\\mc{G}_i$ includes each edge $e$ independently with probability $p_e$. Return subgraph $Q = \\MM{\\mc{G}_1} \\cup \\ldots \\cup \\MM{\\mc{G}_R}$.\n\\end{tboxalg}\n\nIn the algorithm above, $\\MM{\\mc{G}_i}$ returns a maximum matching of $\\mc{G}_i$. It will be convenient for the analysis to assume $\\MM{\\cdot}$ is a deterministic maximum matching algorithm.\n\nIn order to analyze Algorithm~\\ref{alg:sampling}, we will make the following assumption which will simplify many of our arguments.\n\n\\begin{assumption}\\label{ass:optlarge}\n\t$\\opt \\geq 0.1\\epsilon n$.\n\\end{assumption}\n\nAssumption~\\ref{ass:optlarge} comes w.l.o.g. due to a reduction of Assadi~\\etal{}~\\cite{AKL16}. The reduction is roughly as follows: If $n \\gg \\opt$, randomly put nodes of $G$ into $O(\\frac{\\opt}{\\epsilon})$ buckets and contract the nodes within each bucket. The resulting graph will have only $O(\\frac{\\opt}{\\epsilon})$ nodes but its expected maximum realized matching will be as large as $(1-O(\\epsilon))\\opt$. Solving this modified graph will then solve the original graph $G$ as well. We provide further details in Appendix~\\ref{app:optlarge} and note that for the reduction to work, it is important that our algorithm can handle different edge realization probabilities.\n\n\n\\subsection{A Crucial\/Non-crucial Decomposition}\n\nFor each edge $e$ define $q_e := \\Pr[e \\in \\MM{\\mc{G}}]$ where $\\MM{\\cdot}$ is the same matching algorithm used in Algorithm~\\ref{alg:sampling}. Since we assumed $\\MM{\\cdot}$ is deterministic, the probability is taken only over the randomization of the realization $\\mc{G}$. Having this definition, for any vertex $v$ we denote $q_v := \\sum_{e \\ni v} q_e$ and for any subset $E' \\subseteq E$ denote $q(E') := \\sum_{e \\in E'} q_e$. The following statements immediately follow from the definition:\n\n\\begin{observation}\n\t$q(E) = \\opt$.\n\\end{observation}\n\\begin{observation}\\label{obs:qv}\n\tFor any vertex $v$, $q_v$ denotes the probability that $v$ is matched in $\\MM{\\mc{G}}$.\n\\end{observation}\n\nWe will fix two thresholds $0 < \\tau_- < \\tau_+ < 1$ that both depend only on $\\epsilon$ and $p$. Next, for any edge $e$, we say $e$ is {\\em crucial} if $q_e \\geq \\tau_+$,  {\\em non-crucial} if $q_e \\leq \\tau_-$, and {\\em ignored} if $q_e \\in (\\tau_-, \\tau_+)$. We denote the crucial edges by $C := \\{ e \\in E \\mid \\text{$e$ is crucial} \\}$, and the non-crucial edges by $N := \\{ e \\in E \\mid \\text{$e$ is non-crucial} \\}$. Furthermore, we denote their realizations by $\\mc{C} := C \\cap \\mc{E}$ and $\\mc{N} := N \\cap \\mc{E}$. When confusion is impossible, we may use $C$ to denote graph $(V, C)$ instead of merely the edge-subset. The same also naturally generalizes to $N$, $\\mc{C}$, and $\\mc{N}$. We will further use $\\Delta_C$ to denote the maximum degree in graph $C$. Moreover, for any vertex $v$ we use $c_v$ (resp. $n_v$) to  denote the probability that $v$ is matched via a crucial (resp. non-crucial) edge in $\\MM{\\mc{G}}$.\n\n\\begin{observation}\\label{obs:crucialdegree}\n\t$\\Delta_C \\leq 1\/\\tau_+$.\n\\end{observation}\n\\begin{proof}\n\tEach edge $e \\in C$ has $q_e \\geq \\tau_+$ by definition. Thus, if there is a vertex $v$ of degree larger than $1\/\\tau_+$ in $C$, then it should hold that $q_v > 1\/\\tau_+ \\times \\tau_+ = 1$ which contradicts Observation~\\ref{obs:qv}.\n\\end{proof}\n\n\\subsection{Setting the Thresholds $\\tau_-$ and $\\tau_+$}\n\nTo describe how we set the values of $\\tau_-$ and $\\tau_+$, we state a lemma that we prove in Section~\\ref{sec:proofs}.\n\n\n\\begin{lemma}\\label{lem:gap}\n\tFix any arbitrary function $f(x)$ such that $0 < f(x) < x$ for any $0 < x < 1$. There is a choice of $0 < \\tau_- < \\tau_+ < 1$ such that: (1) $\\tau_- = f(\\tau_+)$. (2) $q(N) + q(C) \\geq (1-\\epsilon)\\opt$. (3) Both $\\tau_-$ and $\\tau_+$ depend only on $\\epsilon$ and $p$. And finally, (4) $\\tau_+ \\leq (\\epsilon p)^{50}$.\n\\end{lemma}\n\nThe lemma above essentially shows that we can have any desirably large gap between $\\tau_+$ and $\\tau_-$ and still ensure that $q(N)+q(C) \\geq (1-\\epsilon)\\opt$. That is, the ignored edges in expectation constitute at most $\\epsilon \\opt$ edges of  $\\MM{\\mc{G}}$. While this may sound counter-intuitive, it follows roughly speaking from the fact that by iteratively reducing the threshold $\\tau_+$ by a sufficient amount, all the previously ignored edges become crucial. Thus it cannot continue to hold that there are still a significant mass of the matching on the ignored edges after sufficiently many iterations. See Section~\\ref{sec:proofs} for the proof.\n\nHaving Lemma~\\ref{lem:gap}, we set our thresholds and the parameter $R$ of Algorithm~\\ref{alg:sampling} as follows:\n\n\\begin{highlighttechnical}\n\t\\textbf{Setting $\\tau_-, \\tau_+,$ and $R$:}\n\t\n\t\\vspace{0.4cm}\n\t\n\tDefine function $f(x) := x^{10g(x)}$ where $g(x) := \\epsilon^{-20}\\log\\frac{1}{x}$.\n\t\n\t\\smallskip\n\t\n\tWe plug this function $f$ into Lemma~\\ref{lem:gap} and define $\\tau_-$ and $\\tau_+$ accordingly. We also set $R = \\frac{1}{2\\tau_-}$.\n\\end{highlighttechnical}\n\nNote that function $f$ as defined above satisfies $0 < f(x) < x$ for any $0 < x < 1$ since clearly $g(x) \\geq 1$ so long as $0 < x < 1$. Therefore, we can indeed plug $f$ into Lemma~\\ref{lem:gap}. This results in the following properties:\n\n\\begin{corollary}\\label{cor:thresholds}\n\tIt holds that: (1) $\\tau_- = (\\tau_+)^{10g}$ where $g = \\epsilon^{-20}\\log\\frac{1}{\\tau_+}$. (2) $q(N) + q(C) \\geq (1-\\epsilon)\\opt$. (3) Both $\\tau_-$ and $\\tau_+$ depend only on $\\epsilon$ and $p$ and thus $R=O_{\\epsilon, p}(1)$. (4) $\\tau_- < \\tau_+ \\leq (\\epsilon p)^{50}$.\n\\end{corollary}\n\nThe next lemma shows that $R$ is set such that Algorithm~\\ref{alg:sampling}  samples (almost) all crucial edges.\n\n\\begin{observation}\\label{obs:samplealmostallcrucial}\n\tFor every edge $e \\in C$, $\\Pr[e \\in Q] \\geq 1-\\epsilon$.\n\\end{observation}\n\\begin{proof}\n\tNote that $e \\in Q$ if there is at least one $i \\in [R]$ where $e \\in \\MM{\\mc{G}_i}$. The probability that $e \\in \\MM{\\mc{G}_i}$ for any fixed $i$ is precisely $q_e$. Since realizations $\\mc{G}_1, \\ldots, \\mc{G}_R$ are independent, it holds that $\\Pr[e \\not\\in Q] = (1-q_e)^{R}$. On the other hand $q_e \\geq \\tau_+$ since $e$ is crucial. Also $R = \\frac{1}{2\\tau_-} > \\ln \\epsilon^{-1}\/\\tau_+$ where the latter inequality follows easily from Corrolary~\\ref{cor:thresholds} part (1). Combining all of these gives: \n\t$$\n\t\\Pr[e \\not\\in Q] = (1-q_e)^R < (1-\\tau_+)^{\\ln \\epsilon^{-1}\/\\tau_+} < e^{-\\ln \\epsilon^{-1}} = \\epsilon.\n\t$$\n\tTherefore indeed $\\Pr[e \\in Q] \\geq 1-\\epsilon$.\n\\end{proof}\n\n\\subsection{The Vertex-Independent Matching Lemma}\n\nAs discussed before, a key technical contribution of this work that allows getting an arbitrary good approximation-factor is a ``vertex-independent matching'' lemma that we state here. The proof of this lemma is involved and thus we defer it to Section~\\ref{sec:independentmatching}. In Section~\\ref{sec:analysisviavertexindependent}, we show how Lemma~\\ref{lem:independentmatching} can be used to analyze Algorithm~\\ref{alg:sampling} and prove Theorem~\\ref{thm:main}.\n\n\n\\newcommand{\\independentmatching}[0]{There is a randomized algorithm that constructs an integral matching $Z$ of $\\mc{C}$ (the realized subgraph of $C$) such that defining $X_v$ as the indicator random variable for $v \\in V(Z)$, we get:\n\t\\begin{enumerate}[itemsep=0.2pt,topsep=5pt]\n\t\t\\item $\\E[|Z|] \\geq q(C) - 30\\epsilon\\opt$.\n\t\t\\item For every vertex $v$, $\\Pr[X_v] \\leq \\max\\{c_v - \\epsilon^2, 0\\}$, where recall that $c_v$ is the probability that vertex $v$ is matched via a crucial edge in $\\MM{\\mc{G}}$.\n\t\t\\item The matching $Z$ is independent of the realization of non-crucial edges in $G$.\n\t\t\\item Let $\\lambda := \\epsilon^{-20}\\log\\Delta_C$. For every $k$ and every $\\{v_1, v_2, \\ldots, v_k \\} \\subseteq V$ such that $d_C(v_i, v_j) \\geq \\lambda$ for all $v_i \\not= v_j$, random variables $X_{v_1}, \\ldots, X_{v_k}$ are independent.\n\t\\end{enumerate}\nWe emphasize that $\\E[|Z|]$ and $X_v$ are both defined with respect to the randomizations in both the realization of $C$, and the randomization of the algorithm in constructing $Z$.\n\t}\n\\begin{lemma}[Vertex-Independent Matching Lemma]\\label{lem:independentmatching}\n\t\\independentmatching{}\n\\end{lemma}\n\n\\begin{observation}\\label{obs:ggtlambda}\n\tLet $g$ be as defined in Corollary~\\ref{cor:thresholds} and $\\lambda$ be as defined in Lemma~\\ref{lem:independentmatching}. Then it holds that $g \\geq \\lambda$.\n\\end{observation}\n\\begin{proof}\n\tSince $\\lambda = \\epsilon^{-20}\\log \\Delta_C$ by definition and $\\Delta_C \\leq 1\/\\tau_+$ by Observation~\\ref{obs:crucialdegree}, we get that $\\lambda \\leq \\epsilon^{-20}\\log \\frac{1}{\\tau_+}$. On the other hand $g = \\epsilon^{-20}\\log \\frac{1}{\\tau_+}$. Therefore, $g \\geq \\lambda$.\n\\end{proof}\n\\section{Concentration of the Maximum Realized Matching's Size}\\label{sec:concentration}\n\nIn this section, we prove that random variable $\\mu(\\mc{G})$, i.e. the size of the maximum realized matching of $G$, is highly concentrated around its mean $\\E[\\mu(\\mc{G})] = \\opt$. A similar concentration bound was previously proved also in the works of \\cite{DBLP:conf\/soda\/BlumCHPPV17,DBLP:conf\/soda\/AssadiBBMS19}. Nonetheless, we provide the full proof in this section for the sake of self-containment.\n\n\\begin{lemma}\\label{lem:concentration}\n\tFor every $0 < t \\leq \\opt$, $\\Pr[|\\mu(\\mc{G}) - \\opt| \\geq t] \\leq \\exp\\left(-\\frac{t^2}{2\\opt + 2t\/3}\\right) < \\exp\\left(-\\frac{t^2}{3\\opt}\\right)$.\n\\end{lemma}\n\n\\begin{corollary}\\label{cor:highprobability}\n\tLet $Q$ be a subgraph of $G$ obtained via a deterministic algorithm and suppose that $\\opt = \\omega(1)$. If $\\E[\\mu(\\mc{Q})]\/\\E[\\mu(\\mc{G})] \\geq \\alpha$ then with high probability $\\mu(\\mc{Q})\/\\mu(\\mc{G}) \\geq (1-o(1))\\alpha$.\n\\end{corollary}\n\\begin{proof}\n\tLemma~\\ref{lem:concentration} implies that w.h.p. $\\mu(\\mc{Q}) = (1\\pm o(1))\\E[\\mu(\\mc{Q})]$ and $\\mu(\\mc{G}) = (1\\pm o(1))\\E[\\mu(\\mc{G})]$. Therefore, w.h.p. $\\mu(\\mc{Q})\/\\mu(\\mc{G}) = (1\\pm o(1)) \\E[\\mu(\\mc{Q})]\/\\E[\\mu(\\mc{G})] \\geq (1-o(1))\\alpha$.\n\\end{proof}\n\nWe note that our construction of subgraph $Q$ in Algorithm~\\ref{alg:sampling} is randomized, thus the corollary above cannot be used as a black-box to imply a high probability bound. However, we remark that a similar proof to that of Lemma~\\ref{lem:concentration} which we give below,  proves $\\mu(\\mc{Q})$ in our algorithm is concentrated around its mean even considering the randomization of Algorithm~\\ref{alg:sampling}. Therefore, our algorithm also guarantees a high probability bound for the approximation-factor.\n\n\nIn order to prove this lemma, we use the concentration of ``self-bounding'' functions. See Sections~3.3 and 6.7 of book \\cite{DBLP:books\/daglib\/0035704} by Boucheron, Lugosi and Massart for a thorough discussion on this concentration inequality and its proof.\n\n\\begin{definition}[{\\cite[Section~6.7]{DBLP:books\/daglib\/0035704}}]\\label{def:selfbounding}\n\tA function $f: \\mc{X}^m \\to \\mathbb{R}$ is ``self-bounding'' if for every $i \\in [m]$ there is a function $f_i: \\mc{X}^{m-1} \\to \\mathbb{R}$ such that for all $x=(x_1, \\ldots, x_m) \\in \\mathcal{X}^m$,\n\t\\begin{enumerate}\n\t\t\\item $0 \\leq f(x) - f_i(x^{(i)})\\leq 1$ for all $i \\in [m]$, and\n\t\t\\item $\\sum_{i=1}^m (f(x)-f_i(x^{(i)})) \\leq f(x)$,\n\t\\end{enumerate}\n\twhere $x^{(i)} = (x_1, \\ldots, x_{i-1}, x_{i+1}, \\ldots, x_n)$.\n\\end{definition}\n\n\\begin{lemma}[{\\cite[Theorem~6.12]{DBLP:books\/daglib\/0035704}}]\\label{lem:selfbounding}\n\tIf $X_1, \\ldots, X_m$ are independent random variables taking values in $\\mathcal{X}$ and $Z = f(X_1, \\ldots, X_m)$ is self-bounding, then for every $0 < t \\leq \\E Z$,\n\t$$\n\t\t\\Pr[|Z - \\E Z| \\geq t] \\leq \\exp \\left(- \\frac{t^2}{2\\E Z + 2t\/3} \\right).\n\t$$\n\\end{lemma}\n\nHaving this inequality, Lemma~\\ref{lem:concentration} follows as follows.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:concentration}]\n\tLet $X_e$ for each edge $e$ in graph $G$ be the indicator of the event that $e$ is realized. We can use vector $X = (X_{e_1}, \\ldots, X_{e_m})$ to represent a realization of $G$ where $e_1, \\ldots, e_m$ are all edges in $G$. With a slight abuse of notation, we use $\\mu(X)$ to denote the size of the maximum matching in realization $X$. We first prove that function $\\mu(X)$ is self-bounding. For each $i \\in [m]$, define\n\t$$\n\t\t\\mu_i(X^{(i)}) = \\mu(X_{e_1}, \\ldots, X_{e_{i-1}}, 0, X_{e_{i+1}}, \\ldots, X_{e_m}).\n\t$$\n\tIn words, $\\mu_i(X^{(i)})$ is the maximum matching size in realization $X$ if we regard edge $e_i$ as unrealized. We need to show that the two conditions of Definition~\\ref{def:selfbounding} hold. First, we have to show that\n\t$$\n\t\t0 \\leq \\mu(X) - \\mu_i(X^{(i)}) \\leq 1 \\qquad \\text{for all $i \\in [m]$ and all realizations $X$.}\n\t$$\n\tObserve that removing a realized edge cannot increase the maximum realized matching size, thus clearly $\\mu(X) - \\mu(X^{(i)}) \\geq 0$. Moreover, removing each edge decreases the maximum matching size by at most 1. Thus $\\mu(X) - \\mu(X^{(i)}) \\leq 1$ proving the first condition. For the second condition, we have to show that\n\t$$\n\t\t\\sum_{i=1}^m \\left(\\mu(X) - \\mu_i(X^{(i)}) \\right) \\leq \\mu(X).\n\t$$\n\tTo see this, fix a maximum realized matching $M$ in realization $X$. For any edge $e_i$ outside this matching, we have $\\mu(X) - \\mu_i(X^{(i)}) = 0$. For the rest, as discussed above $\\mu(X) - \\mu_i(X^{(i)}) \\leq 1$. Therefore indeed $\\sum_{i=1}^m \\left(\\mu(X) - \\mu_i(X^{(i)}) \\right) \\leq |M| = \\mu(X)$.\n\t\n\tWe proved that $\\mu(X)$ is self-bounding. Since the edges are realized independently, we can plug this into Lemma~\\ref{lem:selfbounding} and immediately obtain Lemma~\\ref{lem:concentration}.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Construction of an Expected Fractional Matching $x$ on $\\mc{Q}$}\\label{sec:constructfractional}\n\nIn this section, we describe an algorithm that constructs an ``expected fractional matching'' $x$ on $\\mc{Q}$ as defined below.\n\n\\begin{definition}\\label{def:expfractional}\n\tLet $\\mathcal{A}$ be a random process that assigns a fractional value $x_e \\in [0, 1]$ to each edge $e$ of a graph $G(V, E)$. We say $x$ is an expected fractional matching if:\n\t\\begin{enumerate}[itemsep=0pt,topsep=5pt]\n\t\t\\item For each vertex $v$, defining $x_v := \\sum_{e \\ni v} x_e$ we have $\\E[x_v] \\leq 1$.\n\t\t\\item For all subsets $U \\subseteq V$ with $|U|\\leq 1\/\\epsilon$, $x(U) \\leq \\lfloor \\frac{|U|}{2} \\rfloor$ with probability 1.\n\t\\end{enumerate}\n\\end{definition}\n\nWe emphasize that the definition only requires $\\E[x_v] \\leq 1$, thus depending on the coin tosses of the process, it may occur that $x_v > 1$, violating the constraints of a normal fractional matching. We will later argue that in our construction, the values of $x_v$'s are sufficiently concentrated around their mean and thus we can turn our expected fractional matching to an actual fractional matching of (almost) the same size.\n\nAs described before, we construct an expected fractional matching $x$ on the edges of graph $\\mc{Q}$. Note that here the graph $\\mc{Q}$ itself is also stochastic. In the construction, we treat crucial and non-crucial edges completely differently.\n\n\\smparagraph{Crucial edges.} On the crucial edges, we first construct an integral matching $Z$ using the algorithm of Lemma~\\ref{lem:independentmatching}. Once we have $Z$, we define $x$ on crucial edges as follows.\n\n\\begin{highlighttechnical}\n\\vspace{-0.4cm}\n\\begin{flalign}\\label{eq:crucialxe}\n\t\\text{For every crucial edge $e$, } \\qquad\\qquad x_e := \\begin{cases}\n    1,& \\text{if $e \\in Z$ and $e \\in Q$,}\\\\\n    0,              & \\text{otherwise}.\n    \\end{cases}\n\\end{flalign}\n\\end{highlighttechnical}\n\nNote from Observation~\\ref{obs:samplealmostallcrucial} that each crucial edges belong to $Q$ with probability at least $1-\\epsilon$. Therefore the construction above (roughly speaking) sets $x_e = 1$ for most of the edges $e$ in $Z$.\n\n\\smparagraph{Non-crucial edges.} For defining $x$ on the non-crucial edges, we start with a number of useful definitions. For any edge $e$, define $t_e$ to be the number of matchings $\\MM{\\mc{G}_1}, \\ldots, \\MM{\\mc{G}_R}$ that include $e$. Then based on that, define \n\\begin{equation}\\label{eq:deff}\n\tf_e := \\begin{cases}\n\t\t\\frac{t_e}{R}, & \\text{if $\\frac{t_e}{R} \\leq \\frac{1}{\\sqrt{\\epsilon R}}$ and $e$ is non-crucial,}\\\\\n\t\t0, & \\text{otherwise.}\n\t\\end{cases}\n\\end{equation}\nNote that $f_e$ is a random variable of only the randomization of Algorithm~\\ref{alg:sampling}, i.e. it is independent of the realization. Also note that $f_e$ is desirably non-zero only on the edges that belong to graph $Q$. Having defined $f_e$, we define $x_e$ on the non-crucial edges as follows.\n\\begin{highlighttechnical}\nFor every non-crucial edge $e$, define\n\\begin{flalign}\\label{eq:noncrucialxe}\n\tx_e = \\begin{cases}\n    \\frac{f_e}{p_e(1-\\Pr[X_v])(1-\\Pr[X_u])}, & \\text{if $e$ is realized, $u, v \\not\\in V(Z)$, and $d_C(u, v) \\geq \\lambda$,}\\\\\n    0,              & \\text{otherwise}.\n    \\end{cases}\n\\end{flalign}\n\\end{highlighttechnical}\n\nWe note that $\\lambda$ in the definition above is the number defined in Lemma~\\ref{lem:independentmatching} and that $X_v$ is the indicator random variable for the event $v \\in V(Z)$.\n\nBefore concluding this section, let $f_v := \\sum_{e \\in N : v \\in e} f_e$ for each vertex $v$. We note the following properties of $f$, which can be derived directly from the definition above. The proof is given in Section~\\ref{sec:proofs}.\n\n\\begin{claim}\\label{cl:frange}\n\tIt holds that:\n\t\\begin{enumerate}\n\t\t\\item For every non-crucial edge $e$, $\\E[f_e] \\leq q_e$.\n\t\t\\item For every non-crucial edge $e$, $\\E[f_e] \\geq (1-\\epsilon)q_e$.\n\t\t\\item For every vertex $v$, it always holds that $\\sum_{e \\ni v} f_e \\leq 1$.\n\t\t\\item For every vertex $v$, $\\Pr[f_v > n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$, where recall that $n_v$ is the probability that $v$ is matched via a non-crucial edge in $\\MM{\\mc{G}}$.\n\t\\end{enumerate}\n\\end{claim}\n\nConsider a non-crucial edge $\\{u, v\\}$ between two nodes $u$ and $v$ with $d_C(u, v) \\geq \\lambda$. The probability that $x_e$ is non-zero is $p_e(1-\\Pr[X_v])(1-\\Pr[X_u])$: Both $u$ and $v$ should be unmatched in $Z$ and $e$ should be realized, and further all these events are independent. This intuitively explains why we set $x_e = \\frac{f_e}{p_e(1-\\Pr[X_v])(1-\\Pr[X_u])}$ if all these conditions hold: We want the denominator to cancel out with this probability so that we get $\\E[x_e] = f_e$. We will formalize this intuition in Section~\\ref{sec:analysisfractional} where we prove the expected size of $x$ is large.\n\n\\subsection{Validity of $x$}\\label{sec:validityofx}\n\nIn this section, we prove that $x$ is indeed an expected fractional matching of $\\mc{Q}$.\n\nFirst, we prove that $x$ is non-zero only on the edges of $\\mc{Q}$. This simply follows from the construction of $x$.\n\n\\begin{claim}\n\tAny edge $e$ with $x_e > 0$ belongs to $\\mc{Q}$. That is, $x$ is only non-zero on the set of edges queried by Algorithm~\\ref{alg:sampling} that are also realized.\n\\end{claim}\n\\begin{proof}\n\tFor any crucial edge $e$, we either have $x_e = 1$ or $x_e = 0$. By definition, if $x_e = 1$ then $e \\in Z \\cap Q$. By Lemma~\\ref{lem:independentmatching}, $Z$ is a matching of {\\em realized} crucial edges, i.e. $e \\in Z$ implies $e \\in \\mc{E}$. Therefore, $e \\in Z \\cap Q$ implies $e \\in \\mc{E} \\cap Q = \\mc{Q}$ as desired.\n\t\n\tFor any non-crucial edge $e$, if $e \\not\\in Q$, then $f_e = 0$ by definition of $f_e$. Therefore, if $x_e > 0$, then $f_e > 0$ which implies $e \\in Q$. Moreover, by (\\ref{eq:noncrucialxe}), $x_e > 0$ implies $e$ is realized. Combining these two, we get that if $x_e>0$ then $e \\in \\mc{Q}$.\n\\end{proof}\n\nNext, we prove condition (1) of Definition~\\ref{def:expfractional}.\n\n\\begin{claim}\\label{cl:expxvlt1}\n\tFor every vertex $v$, $\\E[x_v] \\leq 1$.\n\\end{claim}\n\\begin{proof}\n\tSuppose at first that there is an edge $e$ incident to $v$ that belongs to matching $Z$. Then we either have $x_e = 1$ or $x_e = 0$ (depending on whether $e \\in Q$ or not). For all other edges $e'$ connected to $v$ (crucial or non-crucial) we have $x_{e'} = 0$ by (\\ref{eq:crucialxe}) and (\\ref{eq:noncrucialxe}). Therefore if such edge $e$ exists, we indeed have $x_v \\leq 1$. For the rest of the proof, we condition on the event that no such edge $e$ exists, i.e. $v \\not\\in V(Z)$ and prove the claim.\n\t\n\tLet $u_1, u_2, \\ldots, u_r$ be neighbors of $v$ in graph $G$ such that for all $i \\in [r]$: (1) edge $\\{v, u_i\\}$ is non-crucial, (2) $d_C(v, u_i) \\geq \\lambda$. Let $e_i := \\{v, u_i\\}$; we  claim that conditioned on $v \\not\\in V(Z)$, we have\n\t\\begin{equation}\\label{eq:21410238471098412}\n\tx_v = x_{e_1} + x_{e_2} + \\ldots + x_{e_r}.\n\t\\end{equation}\n\tTo see this, fix an edge $e = \\{v, u\\}$ for some $u \\not\\in \\{u_1, \\ldots, u_r\\}$. We show that $x_e = 0$, which suffices to prove (\\ref{eq:21410238471098412}). First if $e$ is crucial, then $e \\not\\in Z$ given that $v \\not\\in V(Z)$; thus according to (\\ref{eq:crucialxe}) we set $x_e = 0$. Moreover, if $e$ is non-crucial, the assumption $u \\not\\in \\{u_1, \\ldots, u_r\\}$ implies $d_C(v, u) < \\lambda$ by definition of the set. In this case also, we set $x_e = 0$ according to (\\ref{eq:noncrucialxe}); concluding the proof of (\\ref{eq:21410238471098412}).\n\t\n\tBy linearity of expectation applied to (\\ref{eq:21410238471098412}), we get\n\t\\begin{equation}\\label{eq:612903247}\n\t\\E[x_v \\mid v \\not\\in V(Z)] = \\sum_{i=1}^r \\E[x_{e_i} \\mid v \\not\\in V(Z)].\n\t\\end{equation}\n\tMoreover, for any arbitrary $i \\in [r]$ we have\n\t\\begin{align}\n\t\t\\nonumber\\E[x_{e_i} \\mid v\\not\\in V(Z)] &= \\Pr[u_i \\not\\in V(Z), e_i \\text{ realized} \\mid v \\not\\in V(Z)] \\times \\frac{\\E[f_{e_i}]}{p_{e_i}(1-\\Pr[X_v])(1-\\Pr[X_{u_i}])}\\\\\n\t\t&= p_{e_i}(1-\\Pr[X_{u_i}]) \\times \\frac{\\E[f_{e_i}]}{p_{e_i}(1-\\Pr[X_v])(1-\\Pr[X_{u_i}])} = \\frac{\\E[f_{e_i}]}{1-\\Pr[X_v]}.\\label{eq:7128312098}\n\t\\end{align}\n\tThe second equality above follows from the fact that the event of $e_i$ being realized is independent of $u_i$ or $v$ being in $V(Z)$, as indicated by Lemma~\\ref{lem:independentmatching} part 3; and also the fact that $u_i \\not\\in V(Z)$ and $v \\not\\in V(Z)$ are also independent from each other due to Lemma~\\ref{lem:independentmatching} part 4 combined with the assumption that $d_C(u_i, v) \\geq \\lambda$. We also note that we have used $\\E[f_{e_i}]$ instead of $\\E[f_{e_i} \\mid v\\not\\in V(Z)]$ in the equation above since $f_{e_i}$ is only a random variable of the randomization used in Algorithm~\\ref{alg:sampling} whereas the matching $Z$ is constructed in Lemma~\\ref{lem:independentmatching} independent of the outcome of Algorithm~\\ref{alg:sampling}.\n\t\n\tCombining (\\ref{eq:612903247}) and (\\ref{eq:7128312098}) we get\n\t\\begin{equation}\\label{eq:421610986201363}\n\t\t\\E[x_v \\mid v\\not\\in V(Z)] = \\sum_{i = 1}^r \\frac{\\E[f_{e_i}]}{1-\\Pr[X_v]} = \\frac{1}{1-\\Pr[X_v]}\\sum_{i=1}^r\\E[f_{e_i}].\n\t\\end{equation}\t\n\tFrom Claim~\\ref{cl:frange} part 1, we know $\\E[f_{e_i}] \\leq q_{e_i}$. Replacing this into the equality above, we get\n\t$$\n\t\t\\E[x_v \\mid v\\not\\in V(Z)] \\leq \\frac{1}{1-\\Pr[X_v]} \\sum_{i=1}^r q_{e_i} \\leq \\frac{n_v}{1-\\Pr[X_v]}.\n\t$$\n\t\n\tLemma~\\ref{lem:independentmatching} part (2) guarantees that $\\Pr[X_v] < c_v$ which implies $1-\\Pr[X_v] > 1-c_v$. On the other hand, $c_v + n_v$ is upper bounded by the probability that $v$ is matched in $\\opt$, thus $c_v + n_v \\leq 1$, implying $n_v \\leq 1-c_v$. These, combined with the equation above, gives\n\t$$\n\t\t\\E[x_v \\mid v\\not\\in V(Z)] \\leq \\frac{n_v}{1-\\Pr[X_v]} \\leq \\frac{1-c_v}{1-c_v} = 1.\n\t$$\n\tRecalling also that $\\E[x_v \\mid v\\in V(Z)] \\leq 1$ as described at the start of the proof, this concludes the proof of the claim that $\\E[x_v] \\leq 1$.\n\\end{proof}\n\nNext, we show that condition (2) of Definition~\\ref{def:expfractional} also holds for our construction.\n\n\\begin{claim}\\label{cl:xblossom}\n\tFor all subsets $U \\subseteq V$ with $|U|\\leq 1\/\\epsilon$, $x(U) \\leq \\lfloor \\frac{|U|}{2} \\rfloor$ with probability 1.\n\\end{claim}\n\\begin{proof}\n\tBy definition of $x$, the value of $x_e$ on crucial edges is either 1 or 0. Moreover, the definition also implies that if a vertex $v$ is incident to a crucial edge $e$ with $x_e = 1$, for all other edges $e'$ incident to $v$ we have $x_{e'} = 0$. Call all such vertices {\\em integrally matched}. Fix a subset $U$ and let $U'$ be the subset of $U$ excluding its integrally matched vertices. One can easily confirm that if $x(U) > \\lfloor |U|\/2 \\rfloor$, then also $x(U') > \\lfloor |U'|\/2 \\rfloor$. Therefore, either the claim holds, or there should exist a subset with no integrally matched vertices that violates it. Let $U$ be the smallest such subset and observe that $|U| \\leq 1\/\\epsilon$ (otherwise $U$ does not contradict the claim's statement).\t\n\t\n\tSince $U$ has no integrally matched vertex, for every crucial edge $e$ inside $U$ we have $x_e = 0$ and for every non-crucial edge $e$ inside $U$ by definition (\\ref{eq:noncrucialxe}) we have\n\t$\n\t\tx_e \\leq \\frac{f_e}{p_e (1-\\Pr[X_u]) (1-\\Pr[X_v])}.\n\t$\n\tBy definition of $f_e$, it holds that $f_e \\leq 1\/\\sqrt{\\epsilon R}$ and by Lemma~\\ref{lem:independentmatching} part 2, $\\Pr[X_u], \\Pr[X_v] \\leq 1-\\epsilon^2$. Replacing these into the bound above, we get\n\t$\n\t\tx_e \\leq \\frac{1}{p \\times \\epsilon^2 \\times \\epsilon^2 \\sqrt{\\epsilon R}}.\n\t$ Noting from Corollary~\\ref{cor:thresholds} part 4 that $\\tau_- < (\\epsilon p)^{50}$ and that $R = 2\/\\tau_-$, we get $R > 2\/(\\epsilon p)^{50}$. Replacing this into the previous upper bound on $x_e$, we get that $x_e$ is much smaller than say $\\epsilon^3$.\n\t\n\tNow since $|U| \\leq 1\/\\epsilon$ there are at most $\\binom{|U|}{2} < 1\/\\epsilon^2$ edges $e$ inside $U$ that can have non-zero $x_e$. For each of these, as discussed above $x_e < \\epsilon^3$. Thus we have $x(U) < \\epsilon^3 \\times 1\/\\epsilon^2 < 1$ which cannot be larger than $\\lfloor |U|\/2 \\rfloor$ if $|U| \\geq 2$ (if $|U| \\leq 1$, then there are no edges with both endpoints in $U$ and thus clearly $x(U) = 0$). This contradicts the assumption that $x(U) > \\lfloor |U|\/2 \\rfloor$, implying that there is no such subset. \n\\end{proof}\n\n\\subsection{The Expected Size of $x$}\\label{sec:analysisfractional}\n\nIn this section we prove the following.\n\n\\begin{lemma}\\label{lem:sizeofx}\n\tIt holds that $\\E\\left[|x| \\right] \\geq (1-34\\epsilon)\\opt$.\n\\end{lemma}\n\nWe start by analyzing the size of $x$ on the crucial edges. This is a simple consequence of Lemma~\\ref{lem:independentmatching} part 1 which guarantees $\\E[Z]\\geq q(C)-30\\epsilon \\opt$ and Observation~\\ref{obs:crucialdegree} which guarantees each crucial edge belongs to $Q$ with probability at least $1-\\epsilon$.\n\n\\begin{claim}\\label{cl:sizeofxcrucial}\n\tIt holds that $\\E\\left[\\sum_{e \\in C} x_e \\right] \\geq q(C)-31\\epsilon \\opt$.\n\\end{claim}\n\\begin{proof}\n\tDenoting $x(C) = \\sum_{e \\in C} x_e$, we have\n\t\\begin{equation*}\n\t\t\\E[x(C)] = \\E \\Big[ \\sum_{e \\in C} x_e \\Big] = \\sum_{e \\in C} \\E[x_e] = \\sum_{e \\in C} \\Pr[e \\in Q \\text{ and } e \\in Z].\n\t\\end{equation*}\n\tObserve that $Z$ and $Q$ are picked independently as Lemma~\\ref{lem:independentmatching} is essentially unaware of $Q$. Therefore, for any crucial edge $e$ we get \n\t$$\n\t\\Pr[e \\in Q \\text{ and } e \\in Z] = \\Pr[e \\in Q] \\times \\Pr[e \\in Z] \\geq (1-\\epsilon)\\Pr[e \\in Z],\n\t$$\n\twhere the latter inequality comes from Observation~\\ref{obs:samplealmostallcrucial}. Replacing this to the equality above gives\n\t$$\n\t\\E[x(C)] \\geq (1-\\epsilon)\\sum_{e \\in C} \\Pr[e \\in Z] = (1-\\epsilon)\\E[|Z|] \\stackrel{\\text{Lemma~\\ref{lem:independentmatching} part 1}}{\\geq} (1-\\epsilon) (q(C)-30\\epsilon \\opt) \\geq q(C) - 31\\epsilon \\opt,\n\t$$\n\tcompleting the proof of the claim.\n\\end{proof}\n\nTo analyze the size of $x$ on the non-crucial edges, we first define $N'$ to be the subset of non-crucial edges $\\{u, v\\}$ such that $d_C(u, v) \\geq \\lambda$ and define $q(N') := \\sum_{e \\in N'} q_e$ and $x(N') := \\sum_{e \\in N'} x(N')$. Definition of $N'$ is useful since recall from (\\ref{eq:noncrucialxe}) that for any $\\{u, v\\} \\in N$ with $d_C(u, v) < \\lambda$ (i.e. $\\{u, v\\} \\not\\in N'$) we set $x_e = 0$. Therefore only the edges in $N$ that also belong to $N'$ have non-zero $x_e$, implying $x(N) = x(N')$.\n\n\\begin{claim}\\label{cl:nplarge}\n\tIt holds that $q(N') \\geq q(N)-\\epsilon q(C)$.\n\\end{claim}\n\\begin{proof}\n\tFor any edge $e = \\{u, v\\}$ in $N \\setminus N'$, we choose an arbitrary shortest path $P$ between $u$ and $v$ in graph $C$ and charge the edges of this path. Note that by definition of $N'$, such path between $u$ and $v$ exists and has size less than $\\lambda$. Now, take a crucial edge $f$. We denote by $\\Phi(f)$ the set of edges in $N \\setminus N'$ for which we charge a path containing $f$. Below, we argue that\n\t\\begin{equation}\\label{eq:612398273497}\n\t\t|\\Phi(f)| \\leq 4(1\/\\tau_+)^{2\\lambda} \\qquad \\forall f \\in C.\n\t\\end{equation}\n\t\n\tFix a crucial edge $f$ and an edge $\\{u, v\\} \\in \\Phi(f)$. As discussed above, there should be a path of length less than $\\lambda$ between $u$ and $v$ in graph $C$ that passes through $f$. This means that $d_C(u, f) < \\lambda$ and $d_C(v, f) < \\lambda$. Therefore, both $u$ and $v$ are at distance at most $\\lambda$ from $f$ in graph $C$. \n\t\n\tObserve that there are at most $2(\\Delta_C)^{\\lambda}$ vertices in the $\\lambda$-neighborhood of $f$ in graph $C$. Thus, there are at most $2(\\Delta_C)^{\\lambda} \\times 2(\\Delta_C)^{\\lambda} = 4(\\Delta_C)^{2\\lambda}$ pairs of vertices that can potentially charge $f$, proving $|\\Phi(f)| \\leq 4(\\Delta_C)^{2\\lambda} \\leq 4(1\/\\tau_+)^{2\\lambda}$ where the latter inequality comes from Observation~\\ref{obs:crucialdegree} that $\\Delta_C \\leq 1\/\\tau_+$. This concludes the proof of (\\ref{eq:612398273497}).\n\t\n\tAs discussed above, each edge $e \\in N \\setminus N'$ charges a path in $C$, thus belongs to $\\Phi(f)$ of at least one crucial edge $f$. Therefore, we get\n\t\\begin{equation}\\label{eq:10234817293478}\n\t\t|N \\setminus N'| \\leq \\sum_{f \\in C} \\Phi(f).\n\t\\end{equation}\n\tEvery edge $e$ in $N \\setminus N'$ is non-crucial, i.e. $q_e \\leq \\tau_-$. Thus:\n\t\\begin{equation}\\label{eq:7873241712304912348}\n\t\\sum_{e \\in N \\setminus N'} q_e \\leq \\tau_-|N \\setminus N'| \\stackrel{(\\ref{eq:10234817293478})}{\\leq} \\tau_- \\sum_{f \\in C} \\Phi(f) \\stackrel{(\\ref{eq:612398273497})}{\\leq} 4\\tau_- |C|(1\/\\tau_+)^{2\\lambda} \\leq 4\\tau_- q(C)(1\/\\tau_+)^{2\\lambda+1},\n\t\\end{equation}\n\twhere the last inequality comes from the fact that $q(C) \\geq |C| \\tau_+$ as for every edge $e \\in C$, $q_e \\geq \\tau_+$.\n\t\n\tFrom Corollary~\\ref{cor:thresholds} we have $\\tau_- = (\\tau_+)^{10g}$ and we have $g \\geq \\lambda > 1$ by Observation~\\ref{obs:ggtlambda}. Thus:\n\t$$\n\t 4\\tau_- (1\/\\tau_+)^{2\\lambda+1} = 4 (\\tau_+)^{10g} (1\/\\tau_+)^{2\\lambda+1} = 4 (\\tau_+)^{10g - (2\\lambda - 1)} < 4 \\tau_+ < \\epsilon.\n\t$$\n\tReplacing it into inequality (\\ref{eq:7873241712304912348}), we get\n\t$$\n\t\\sum_{e \\in N \\setminus N'} q_e \\leq \\epsilon q(C).\n\t$$\n\tThis concludes the proof since\n\t$$\n\t\tq(N') = \\sum_{e \\in N'} q_e = \\sum_{e \\in N \\setminus (N \\setminus N')} q_e \\geq \\sum_{e \\in N}q_e - \\sum_{e \\in N \\setminus N'} q_e \\geq q(N) - \\epsilon q(C)\n\t$$\n\tas it is desired.\n\\end{proof}\n\n\n\\begin{claim}\\label{cl:xnpgtqnp}\n\tIt holds that $\\E[x(N')] \\geq (1-\\epsilon) q(N')$.\n\\end{claim}\n\\begin{proof}\n\tBy linearity of expectation, we have \n\t\\begin{equation}\\label{eq:16234421340}\n\t\\E[x(N')] = \\E \\Big[ \\sum_{e \\in N'} x_e \\Big] = \\sum_{e \\in N'} \\E[x_e].\n\t\\end{equation}\n\tWe emphasize that the expectation here is taken over the randomization in Algorithm~\\ref{alg:sampling}, the randomization in matching $Z$, and the randomization in realization of non-crucial edges. Specifically, we write $\\E_{\\alg, Z, \\mc{N}}[x_e]$ to emphasize on this.\n\t\n\tThe randomization of Algorithm~\\ref{alg:sampling} determines the value of $f_e$ which is used in defining $x_e$. Let us first condition on $f_e$ and compute $\\E_{Z, \\mc{N}}[x_e \\mid f_e]$. We have\n\t\\begin{equation}\\label{eq:89123}\n\t\t\\E_{Z, \\mc{N}}[x_e \\mid f_e] = \\Pr[e \\in \\mc{E} \\text{ and } u, v \\not\\in V(Z) \\mid f_e] \\times  \\frac{f_e}{p_e(1-\\Pr[X_u])(1-\\Pr[X_v])}.\n\t\\end{equation}\n\tWe claim that \n\t\\begin{equation}\\label{eq:5123674182374}\n\t\t\\Pr[e \\in \\mc{E} \\text{ and } u, v \\not\\in V(Z) \\mid f_e] = p_e(1-\\Pr[X_u])(1-\\Pr[X_v]).\n\t\\end{equation}\n\tTo see this, first observe that the value of $f_e$ is determined solely by the random realizations taken by Algorithm~\\ref{alg:sampling}. In particular, the events $e \\in \\mc{E}$, and $u, v \\not\\in V(Z)$ are completely independent of the outcome of Algorithm~\\ref{alg:sampling}. This allows us to remove the condition on $f_e$ from the left hand side of (\\ref{eq:5123674182374}). Moreover, by Lemma~\\ref{lem:independentmatching} part 3, the matching $Z$ is chosen independently from the realization of non-crucial edges, thus events $e \\in \\mc{E}$ and $u, v \\not\\in V(Z)$ are independent. \tFinally, the assumption that $e \\in N'$, by definition of $N'$, implies that $d_C(u, v) \\geq \\lambda$. Therefore, by Lemma~\\ref{lem:independentmatching} part 4, events $v \\in V(Z)$ and $u \\in V(Z)$ (and for that matter their complements) are independent. Thus, indeed:\n\t\\begin{align*}\n\t\\Pr[e \\in \\mc{E} \\text{ and } u, v \\not\\in V(Z) \\mid f_e] &= \\Pr[e \\in \\mc{E}] \\times \\Pr[v \\not\\in V(Z)] \\times \\Pr[u \\not\\in V(Z)]\\\\\n\t&= p_e(1-\\Pr[X_u])(1-\\Pr[X_v]).\n\t\\end{align*}\n\tReplacing (\\ref{eq:5123674182374}) into (\\ref{eq:89123}) we get\n\t\\begin{equation*}\n\t\t\\E_{Z, \\mc{N}}[x_e \\mid f_e] = p_e(1-\\Pr[X_u])(1-\\Pr[X_v]) \\times \\frac{f_e}{p_e(1-\\Pr[X_u])(1-\\Pr[X_v])} = f_e.\n\t\\end{equation*}\n\tTaking expectation over $\\alg$ from both sides, we get\n\t\\begin{equation}\\label{eq:72313409}\n\t\\E_{\\alg}[\\E_{Z, \\mc{N}}[x_e \\mid f_e]] = \\E_{\\alg}[f_e].\n\t\\end{equation}\n\tThe left hand side equals $\\E_{\\alg, Z, \\mc{N}}[x_e]$. For the right hand side, by Claim~\\ref{cl:frange} we have $\\E[f_e] \\geq (1-\\epsilon)q_e$. \tReplacing both the left hand side and right hand side of (\\ref{eq:72313409}) by these bounds, we get\n\t\\begin{equation}\n\t\t\\E_{\\alg, Z, \\mc{N}}[x_e] \\geq (1-\\epsilon) q_e.\n\t\\end{equation}\n\tCombining this with (\\ref{eq:16234421340}) we get\n\t\\begin{equation*}\n\t\\E[x(N')] = \\sum_{e \\in N'} \\E[x_e] \\geq (1-\\epsilon) \\sum_{e \\in N'} q_e = (1-\\epsilon) q(N'),\n\t\\end{equation*}\n\tcompleting the proof.\n\\end{proof}\n\nWe are now ready to prove Lemma~\\ref{lem:sizeofx}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:sizeofx}]\n\tWe have\n\t$$\n\t\\E\\Big[\\sum_{e}x_e\\Big] = \\E\\Big[\\sum_{e \\in C} x_e\\Big] + \\E\\Big[\\sum_{e \\in N} x_e\\Big] \\stackrel{\\text{Claim~\\ref{cl:sizeofxcrucial}}}{\\geq} q(C) - 31\\epsilon \\opt + \\E\\Big[\\sum_{e \\in N} x_e\\Big].\n\t$$\n\tAlso note that for $e \\in N$, $x_e \\not= 0$ iff $e \\in N'$ by construction of $x$. Thus,\n\t$$\n\t\\E\\Big[\\sum_{e \\in N} x_e\\Big] = \t\\E\\Big[\\sum_{e \\in N'} x_e\\Big] = \\E[x(N')] \\stackrel{\\text{Claim~\\ref{cl:xnpgtqnp}}}{\\geq} (1-\\epsilon)q(N') \\stackrel{\\text{Claim~\\ref{cl:nplarge}}}{\\geq} (1-\\epsilon)(q(N)-\\epsilon q(C)).\n\t$$\n\tCombining the two equations above, we get\n\t\\begin{align*}\n\t\t\\E\\Big[\\sum_{e}x_e\\Big] &\\geq q(C) - 31\\epsilon \\opt + (1-\\epsilon)(q(N)-\\epsilon q(C)) > q(C) + q(N) - 33\\epsilon \\opt\\\\\n\t\t&\\stackrel{\\text{Lemma~\\ref{lem:gap} part (2)}}{\\geq} (1-\\epsilon)\\opt - 33\\epsilon \\opt \\geq (1-34\\epsilon)\\opt,\n\t\\end{align*}\n\tconcluding the proof.\n\\end{proof}\n\\subsection{From the Expected Fractional Matching to an Actual Fractional Matching}\\label{sec:turntofracmatching}\n\nWe showed that $x$ is an expected fractional matching satisfying $\\E[x_v] \\leq 1$ for every vertex $v$. However, as mentioned before, there is still a possibility that $x_v > 1$ depending on the coin tosses of the algorithms and the realization. This should never occur in a valid fractional matching. Thus, we define the following scaled fractional matching $y$ based on $x$ which decreases the fractional matching around vertices that deviate significantly from their expectation to 0.\n\n\\begin{equation}\\label{eq:defy}\n\t\\text{For any edge $e=\\{u, v\\}$,} \\qquad\\qquad y_e = \\begin{cases}\n\t\t\tx_e\/(1+\\epsilon) & \\text{if $x_v, x_u \\leq 1+\\epsilon$,}\\\\\n\t\t\t0 & \\text{otherwise.}\n\t\\end{cases}\n\\end{equation}\n\n\\begin{observation}\\label{obs:yblossom}\n\tBy definition above, $y$ is a valid fractional matching, i.e. $y_v \\leq 1$ for all $v \\in V$. In addition, since $y_e \\leq x_e$ for all edges $e$, Claim~\\ref{cl:xblossom} implies that for all $U \\subseteq V$ with $|U| \\leq 1\/\\epsilon$, $y(X) \\leq \\lfloor \\frac{|U|}{2} \\rfloor$. That is, $y$ also satisfies all blossom inequalities of size up to $1\/\\epsilon$.\n\\end{observation}\n\nIt remains to prove that while turning the expected fractional matching $x$ into an actual fractional matching $y$, we don't significantly hurt the matching's size. We address this in the lemma below.\n\n\\begin{lemma}\\label{lem:ylarge}\n\t$\\E[|y|] \\geq (1-55\\epsilon)\\opt$.\n\\end{lemma}\n\nThe main ingredient in proving Lemma~\\ref{lem:ylarge} is the following claim.\n\n\\begin{claim}\\label{cl:6123719801923}\n\tFor every vertex $v$, $\\Pr[x_v > 1+\\epsilon] \\leq \\epsilon^6p$.\n\\end{claim}\n\nLet us first see how Claim~\\ref{cl:6123719801923} suffices to prove Lemma~\\ref{lem:ylarge} and then prove it.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:ylarge}]\n\tWe have\n\t\\begin{flalign*}\n\t\t\\sum_{e} y_e &= \\sum_{e = \\{u, v\\}} \\mathbbm{1}(x_u \\leq 1+\\epsilon \\text{ and } x_v \\leq 1+\\epsilon) \\frac{x_e}{1+\\epsilon} && \\text{By definition of $y_e$ in (\\ref{eq:defy}).}\\\\\n\t\t&\\geq \\sum_{e = \\{u, v\\}} (1-\\mathbbm{1}(x_u > 1+\\epsilon)-\\mathbbm{1}(x_v > 1+\\epsilon)) \\frac{x_e}{1+\\epsilon} && \\text{Union bound.}\\\\\n\t\t&= \\sum_{e} \\frac{x_e}{1+\\epsilon} - 2\\sum_{v : x_v > 1+\\epsilon} \\sum_{e \\ni v} \\frac{x_e}{1+\\epsilon} = \\sum_{e} \\frac{x_e}{1+\\epsilon} - 2\\sum_{v : x_v > 1+\\epsilon} \\frac{x_v}{1+\\epsilon}.\n\t\\end{flalign*}\n\tTaking expectation from both sides, we get\n\t\\begin{flalign}\n\t\t\\nonumber\\E\\Big[ \\sum_e y_e \\Big] &\\geq \\E\\Big[\\sum_{e} \\frac{x_e}{1+\\epsilon} - 2\\sum_{v : x_v > 1+\\epsilon} \\frac{x_v}{1+\\epsilon}\\Big] = \\frac{1}{1+\\epsilon}\\left(\\E\\Big[\\sum_{e} x_e\\Big] - 2\\E\\Big[\\sum_{v : x_v > 1+\\epsilon} x_v \\Big]\\right)\\\\\n\t\t\\nonumber&\\geq \\frac{1}{1+\\epsilon}\\left((1-34\\epsilon)\\opt - 2\\E\\Big[\\sum_{v : x_v > 1+\\epsilon} x_v \\Big]\\right) \\qquad\\qquad \\text{By Lemma~\\ref{lem:sizeofx}.}\\\\\n\t\t\\nonumber &\\geq (1-35\\epsilon)\\opt - 2\\sum_{v} \\Pr[x_v > 1+\\epsilon]\\E[x_v \\mid x_v > 1+\\epsilon]\\\\\n\t\t&\\geq (1-35\\epsilon)\\opt - 2\\sum_{v} \\epsilon^6 p \\E[x_v \\mid x_v > 1+\\epsilon] \\qquad\\qquad \\text{By Claim~\\ref{cl:6123719801923}.}\\label{eq:98912308}\n\t\\end{flalign}\n\tWe will soon prove that for every vertex $v$, it \\underline{deterministically} holds that $x_v \\leq \\frac{1}{p\\epsilon^4}$. Replacing this into the last inequality above, gives the desired bound that\n\t\\begin{flalign*}\n\t\t\\E\\Big[\\sum_e y_e \\Big] &\\geq (1-35\\epsilon)\\opt - 2\\sum_{v} \\epsilon^6 p \\frac{1}{p \\epsilon^4} \\geq (1-35\\epsilon)\\opt - 2\\epsilon^2 n \\stackrel{\\text{Assumption~\\ref{ass:optlarge}}}{\\geq} (1-35\\epsilon)\\opt - 20\\epsilon \\opt \\\\\n\t\t&= (1-55\\epsilon)\\opt.\n\t\\end{flalign*}\n\tNow let's see why $x_v \\leq \\frac{1}{p\\epsilon^4}$. Observe from the definition of $x$ that if $v \\in V(Z)$ then $x_v \\leq 1$ and otherwise\n\t$$\n\t\tx_v = \\sum_{e = \\{v, u\\}} x_e \\leq \\sum_{e = \\{v, u\\}} \\frac{f_e}{p(1-\\Pr[X_u])(1-\\Pr[X_v])} \\leq \\frac{1}{p \\epsilon^4} \\sum_{e=\\{v, u\\}} f_e.\n\t$$\n\tThe last inequality above comes from the fact that for every vertex $w$, $\\Pr[X_w] \\leq 1-\\epsilon^2$ due to Lemma~\\ref{lem:independentmatching} part 2, which means $1-\\Pr[X_w] \\geq \\epsilon^2$. \n\t\n\tNow recall from Claim~\\ref{cl:frange} part 3 that $\\sum_{e \\ni v} f_e \\leq 1$. Thus we get our desired upper bound that $x_v \\leq \\frac{1}{p\\epsilon^4}$.\t As described above, this completes the proof that $\\E[\\sum_e y_e] \\geq (1-55\\epsilon)\\opt$.\n\\end{proof}\n\nWe now turn to prove Claim~\\ref{cl:6123719801923} that $\\Pr[x_v > 1+\\epsilon] \\leq \\epsilon^6 p$ for all $v$.\n\n\\newcommand{\\eventvz}[0]{\\ensuremath{A}}\n\n\\begin{proof}[Proof of Claim~\\ref{cl:6123719801923}]\n\tIf an edge incident to $v$ belongs to matching $Z$, i.e. if $X_v = 1$ (as defined in Lemma~\\ref{lem:independentmatching}), then one can confirm easily from the definition of $x$ in (\\ref{eq:crucialxe}) and (\\ref{eq:noncrucialxe}) that either $x_v = 1$ or $x_v = 0$, implying that $\\Pr[x_v > 1+\\epsilon \\mid X_v = 1] = 0$. As such, for the rest of the proof, we simply condition on the event that $X_v = 0$.\n\t\n\tSimilar to the proof of Claim~\\ref{cl:expxvlt1} let $u_1, u_2, \\ldots, u_r$ be the neighbors of $v$ such that for each $i \\in [r]$, (1) edge $e_i = \\{v, u_i\\}$ is non-crucial, and (2) $d_C(v, u_i) \\geq \\lambda$. Recall from (\\ref{eq:21410238471098412}) that given event $X_v = 0$, it holds that\n\t\t\\begin{equation*}\n\t\t\tx_v = x_{e_1} + x_{e_2} + \\ldots + x_{e_r}.\n\t\t\\end{equation*}\n\t\n\t\tLet $f'_v := \\sum_{i=1}^r f_{e_i}$ and note that $f'_v \\leq f_v$ since $f_v$ is sum of $f_e$ of all non-crucial edges $e$ connected to $v$. Claim~\\ref{cl:frange} part 4 proves that $\\Pr[f_v \\geq n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$. Therefore, it also holds that $\\Pr[f'_v \\geq n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$ since $f'_v \\leq f_v$. For the rest of the proof, we regard $f_{e_i}$'s as (adversarially) fixed with the only assumption that $f'_v < n_v + 0.1\\epsilon$ which happens with probability at least $1 - (\\epsilon p)^{10}$. We denote this event, as well as the event that $X_v = 0$, by $\\eventvz$ and prove\n\t\t\\begin{equation}\\label{eq:980989825627}\n\t\t\\Pr[x_v > 1 + \\epsilon \\mid \\eventvz] \\leq 0.5\\epsilon^6p,\n\t\t\\end{equation}\n\t\twhich clearly is sufficient for proving the claim.\n\t\n\t\tWe do this by proving a concentration bound using the second moment method. Consider the variance of $x_v$ conditioned on $\\eventvz$:\n\t\t\\begin{flalign*}\n\t\t\t\\Var[x_v \\mid \\eventvz] = \\sum_{i=1}^r \\sum_{j=1}^r \\Cov(x_{e_i}, x_{e_j} \\mid \\eventvz).\n\t\t\\end{flalign*}\n\t\tNow that $f_e$'s are fixed, $x_v$ is only a random variable of (1) the randomization used in Lemma~\\ref{lem:independentmatching} for obtaining matching $Z$, and (2) the realization of non-crucial edges.\n\t\t\n\t\tIn what follows we identify a condition under which covariance of $x_{e_i}$ and $x_{e_j}$ becomes $0$. We will use this later to upper bound $\\Var[x_v \\mid \\eventvz]$. \n\t\t\n\t\t\\begin{observation}\\label{obs:cov0}\n\t\t\tLet $i, j \\in [r]$ be such that $d_C(u_i, u_j) \\geq \\lambda$. Then $\\Cov(x_{e_i}, x_{e_j} \\mid \\eventvz) = 0$.\n\t\t\\end{observation}\n\t\t\\begin{proof}\n\t\t\tWe already had $d_C(v, u_i) \\geq \\lambda$ and $d_C(v, u_j) \\geq \\lambda$ by definition of $u_i, u_j$. Combined with assumption $d_C(u_i, u_j) \\geq \\lambda$ and using Lemma~\\ref{lem:independentmatching} part 4, we get that $X_v, X_{u_i}, X_{u_j}$ are independent. Realization of $e_i$ and $e_j$ are also independent even given $\\eventvz$. This is because these are non-crucial edges and thus are realized independently from $Z$ (according to Lemma~\\ref{lem:independentmatching} part 3) or the values of $f$ which are derived from Algorithm~\\ref{alg:sampling}.\n\t\t\t\n\t\t\tBy definition (\\ref{eq:noncrucialxe}), the value of $x_{e_i}$ conditioned on $\\eventvz$ is fully determined once we know $X_{u_i}$ and whether $e_i$ is realized. Similarly, the value of $x_{e_j}$ conditioned on $\\eventvz$ is fully determined once we know $X_{u_j}$ and whether $e_j$ is realized. These, as discussed above, are independent. Hence $x_{e_i}$ and $x_{e_j}$, conditioned on $\\eventvz$, are independent and thus their covariance is 0.\n\t\t\\end{proof}\n\t\t\n\t\tNow consider two vertices $u_i$ and $u_j$ (possibly $u_i = u_j$) where $d_C(u_i, u_j) < \\lambda$. Here, the covariance may not be 0. But we still can upper bound it as follows:\n\t\t\\begin{flalign}\n\t\t\\nonumber \\Cov(x_{e_i}x_{e_j} \\mid \\eventvz) &= \\E[x_{e_i}x_{e_j} \\mid \\eventvz] - \\E[x_{e_i} \\mid A]\\E[x_{e_j} \\mid A] \\leq \\E[x_{e_i}x_{e_j} \\mid \\eventvz]\\\\\n\t\t\\nonumber &\\leq \\frac{f_{e_i}}{p(1-\\Pr[X_v])(1-\\Pr[X_{u_i}])} \\times \\frac{f_{e_j}}{p(1-\\Pr[X_v])(1-\\Pr[X_{u_j}])}\\\\\n\t\t&\\leq \\frac{f_{e_i}f_{e_j}}{p^2 \\epsilon^8},\\label{eq:3819123987}\n\t\t\\end{flalign}\n\t\twhere the last inequality follows from Lemma~\\ref{lem:independentmatching} part 2 that states for all vertices $w$, $\\Pr[X_w] < 1-\\epsilon^2$ and thus $1-\\Pr[X_w] \\geq \\epsilon^2$.\n\t\t\n\t\tNow, for each $i \\in [r]$, let $D_i := \\{j : d_C(u_i, u_j) < \\lambda \\}$. Since $C$ is a graph of max degree $\\Delta_C$, the $\\lambda-1$ neighborhood of each vertex $u_i$ in $C$ includes $\\leq (\\Delta_C)^{\\lambda-1}$ vertices. Thus:\n\t\t\\begin{equation}\\label{eq:87193107123897}\n\t\t\t|D_i| \\leq (\\Delta_C)^{\\lambda-1} \\qquad\\qquad \\text{for every $i \\in [r]$.}\n\t\t\\end{equation}\n\t\tHaving these, we obtain that\n\t\t\\begin{flalign*}\n\t\t\t\\Var[x_v \\mid \\eventvz] &= \\sum_{i=1}^r \\sum_{i=1}^r \\Cov(x_{e_i}, x_{e_j} \\mid \\eventvz) \\stackrel{\\text{Obs~\\ref{obs:cov0}}}{=} \\sum_{i = 1}^r \\sum_{j \\in D_i} \\Cov(x_{e_i}, x_{e_j} \\mid \\eventvz) \\stackrel{(\\ref{eq:3819123987})}{\\leq} \\sum_{i = 1}^r \\sum_{j \\in D_i} \\frac{f_{e_i}f_{e_j}}{p^2\\epsilon^8} \\\\\n\t\t\t& = \\frac{1}{p^2 \\epsilon^8}\\sum_{i = 1}^r \\Big(f_{e_i}\\sum_{j \\in D_i} f_{e_j}\t\\Big) \\stackrel{f_{e_j} \\leq \\frac{1}{\\sqrt{\\epsilon R}} \\text{ by (\\ref{eq:deff})}}{\\leq} \\frac{1}{p^2\\epsilon^8} \\sum_{i = 1}^r \\Big(f_{e_i} |D_i| \\frac{1}{\\sqrt{\\epsilon R}}\t\\Big)\\\\\n\t\t\t& \\stackrel{(\\ref{eq:87193107123897})}{\\leq} \\frac{(\\Delta_C)^{\\lambda-1}}{p^2 \\epsilon^8\\sqrt{\\epsilon R}} \\sum_{i = 1}^r f_{e_i} \\stackrel{\\text{Claim~\\ref{cl:frange} part 3}}{\\leq} \\frac{(\\Delta_C)^{\\lambda-1}}{p^2 \\epsilon^8\\sqrt{\\epsilon R}} \\stackrel{\\text{Obs~\\ref{obs:crucialdegree}}}{\\leq} \\frac{(1\/\\tau_+)^{\\lambda-1}}{p^2\\epsilon^{8.5}\\sqrt{R}}.\n\t\t\\end{flalign*}\n\t\tReplacing $R$ with $\\frac{1}{2\\tau_-}$ and noting that $\\tau_- = (1\/\\tau_+)^{10 g}$, we get that\n\t\t\\begin{flalign*}\n\t\t\\Var[x_v \\mid A] \\leq \\frac{2(1\/\\tau_+)^{\\lambda}}{p^2 \\epsilon^{8.5}(1\/\\tau_+)^{10g}} &= \\frac{2}{p^2 \\epsilon^{8.5}}(\\tau_+)^{10g - \\lambda}\\\\\n\t\t&< \\frac{2\\tau_+}{p^2 \\epsilon^{8.5}} && \\text{By Observation~\\ref{obs:ggtlambda} $g \\geq \\lambda > 1$ and $\\tau_+ < 1$.}\\\\\n\t\t&< \\frac{2 (\\epsilon p)^{50}}{p^2 \\epsilon^{8.5}} && \\text{Corrolary~\\ref{cor:thresholds} part 4.}\\\\\n\t\t&= 2 \\epsilon^{41.5} p^{48} < 0.1 \\epsilon^8 p.\n\t\t\\end{flalign*}\n\t\tWith this upper bound on the variance, we can use Chebyshev's inequality to get\n\t\t\\begin{equation}\\label{eq:cheb18023}\n\t\t\t\\Pr\\Big[|x_v - \\E[x_v \\mid \\eventvz]| > 0.5\\epsilon \\,\\Big\\vert\\, \\eventvz\\Big] \\leq \\frac{\\Var[x_v \\mid \\eventvz]}{(0.5\\epsilon)^2} \\leq \\frac{0.1 \\epsilon^8 p}{0.25 \\epsilon^2} < 0.5\\epsilon^6 p.\n\t\t\\end{equation}\n\t\tNext, recall from (\\ref{eq:421610986201363}) in the proof of Claim~\\ref{cl:expxvlt1} that $\\E[x_v \\mid v\\not\\in V(Z)] \\leq \\frac{\\sum_{i=1}^r \\E[f_{e_i}]}{1-\\Pr[X_v]} = \\frac{f'_v}{1-\\Pr[X_v]}$. Event $\\eventvz$ in addition to $v \\not\\in V(Z)$ also fixes the value of $f'_v$. But recall that event $\\eventvz$ (as we defined it) guarantees $f'_v \\leq n_v + 0.5\\epsilon$. Therefore, we get\n\t\t\\begin{equation}\\label{eq:74777748123}\n\t\t\t\\E[x_v \\mid \\eventvz] \\leq \\frac{n_v + 0.5\\epsilon}{1-\\Pr[X_v]} \\stackrel{\\Pr[X_v] < c_v}{\\leq} \\frac{n_v + 0.5\\epsilon}{1-c_v} \\stackrel{n_v \\leq 1-c_v}{\\leq} \\frac{1-c_v+0.5\\epsilon}{1-c_v} \\leq 1 + 0.5\\epsilon.\n\t\t\\end{equation}\n\t\tCombining (\\ref{eq:cheb18023}) and (\\ref{eq:74777748123}) we get the claimed inequality of (\\ref{eq:980989825627}) that \n\t\t$$\n\t\t\\Pr[x_v > 1+\\epsilon \\mid \\eventvz] \\leq \\Pr[|x_v - \\E[x_v \\mid A]| > 0.5\\epsilon \\mid \\eventvz] \\leq 0.5\\epsilon^6p,\n\t\t$$\n\t\twhich as described before suffices to prove $\\Pr[x_v > 1+\\epsilon] \\leq \\epsilon^6p$.\n\\end{proof}\n\n\n\n\\subsection{Lemma~\\ref{lem:independentmatching} Property 1: The Matching's Size}\\label{sec:p1}\n\\input{matchingsize}\n\n\n\\subsection{Lemma~\\ref{lem:independentmatching} Property 2: Matching Probabilities}\\label{sec:p2}\n\\input{matchingprobs}\n\n\\subsection{Lemma~\\ref{lem:independentmatching} Property 4: Matching Independence}\\label{sec:p4}\n\\input{matchingindependence}\n\n\\section{Proof of the Vertex-Independent Matching Lemma}\\label{sec:independentmatching}\n\n\\newcommand{\\neighbors}[1]{\\ensuremath{\\mathsf{Neighbors}(#1)}}\n\\newcommand{\\isrealized}[1]{\\ensuremath{\\mathsf{IsRealized}(#1)}}\n\\newcommand{\\dependent}[0]{\\ensuremath{\\mathcal{D}}}\n\nIn this section we turn to prove Lemma~\\ref{lem:independentmatching} restated below.\n\n\\restate{Lemma~\\ref{lem:independentmatching}}{\n\t\\independentmatching{}\n}\n\n\\subsection{Overview of the Algorithm}\\label{sec:crucialoverview}\n\\input{ind-intuitions}\n\n\\subsection{The Formal Algorithm}\\label{sec:formalcrucialalg}\n\\input{ind-algorithm}\n\n\n\n\\section{Introduction}\\label{sec:intro}\n{\n\nWe study the following {\\em stochastic matching} problem. An arbitrary graph $G=(V, E)$ is given, then each edge $e \\in E$ is  retained (or to be consistent with the literature {\\em realized}) independently with some given probability $p \\in (0, 1]$. The goal is to pick a subgraph $Q$ of $G$ without knowing the edge realizations such that:\n\\begin{enumerate}[itemsep=0pt,topsep=5pt]\n\t\\item The expected size of the maximum matching among the realized edges of $Q$ approximates the expected size of the maximum matching among the realized edges in $G$.\n\t\\item The maximum degree in $Q$ is bounded by a function that may depend on $p^{-1}$ but must be independent of the size of $G$.\\footnote{In this paper, we solve a generalization of this problem where each edge $e$ has its own  realization probability $p_e$ and the degree of $Q$ can be proportional to $p = \\min_e p_e$. See Section~\\ref{sec:prelim} for the formal setting.}\n\\end{enumerate}\nIt would be useful to think of $p$ as some constant whereas $n := |V| \\to \\infty$. Then the second condition translates to $Q$ having $O(1)$ maximum degree. In other words, the subgraph $Q$ should provide a good approximation while having $O(n)$ edges, in contrast to $G$ which may have up to $\\Omega(n^2)$ edges.\n\n\\smparagraph{Applications.} The setting is mainly motivated by applications in which the process of determining an edge realization (referred to as {\\em querying} the edge) is considered time consuming or expensive. For such applications, one can instead of querying every edge of $G$, only query the edges of its much sparser subgraph $Q$ and still find a large realized matching in $G$. Kidney exchange and online labor markets are major examples of such applications. For more details on the role of the stochastic matching problem in these applications, see \\cite{arXivblumetal,blumetal,AKL16,AKL17,BR18} (particularly \\cite[Section~1.2]{arXivblumetal}) for kidney exchange and \\cite{BR18,soda19,sagt19} for online labor markets. Another natural application of the model is that this subgraph $Q$ can be used as a {\\em matching sparsifier} for $G$ which approximately preserves its maximum matching size under random edge failures \\cite{sosa19}.\n\n\\smparagraph{Related work.} The problem has received significant attention \\cite{blumetal,AKL16,AKL17,YM18,BR18,soda19,sosa19,sagt19} after the pioneering work of Blum~\\etal{}~\\cite{blumetal} who proved that it admits a $(\\frac{1}{2}-\\epsilon)$-approximation.  Earlier follow-up works revolved around the prevalent half-approximation barrier until it was first broken by Assadi~\\etal{}~\\cite{AKL16}. This was followed by a $0.6568$-approximation by Behnezhad~\\etal{}~\\cite{soda19} and eventually a $(\\frac{2}{3}-\\epsilon)$-approximation by Assadi and Bernstein \\cite{sosa19} which is the state-of-the-art. See also \\cite{YM18,BR18,soda19,YM19} for various natural generalizations of the problem.\n\n\\smparagraph{Our result.} In this work, we improve the approximation-factor all the way up to $(1-\\epsilon)$:\n\n\\begin{highlighttechnical}\n\t\\begin{theorem}\\label{thm:main}\n\t\tFor any $\\epsilon > 0$, there is an algorithm that picks an $O_{\\epsilon, p}(1)$-degree subgraph $Q$ of $G$ such that the expected size of the maximum realized matching in $Q$ is at least $(1-\\epsilon)$ times the expected size of the maximum realized matching in $G$.\n\t\\end{theorem}\n\\end{highlighttechnical}\n\nTo get a $(1-\\epsilon)$-approximation, the dependence of the maximum degree of $Q$ on both $\\epsilon$ and $p$ is necessary. Particularly, a simple lower bound shows that even when $G$ is a clique, to avoid too many singleton vertices in a realization of $Q$, the maximum degree in $Q$ must be $\\Omega(\\frac{\\ln \\epsilon^{-1}}{p})$ \\cite{AKL16}. The same lower bound also shows that a $(1-o(1))$ approximation is not achievable unless the maximum degree of $Q$ is $\\omega(1)$, meaning that our approximation-factor is essentially the best one can hope for.\n\n\\begin{remark}\n\tThe $O_{\\epsilon, p}(1)$ term in Theorem~\\ref{thm:main} is in the order \n\t$\\exp\\left(\\exp\\left(\\exp\\left(O\\left(\\epsilon^{-1} \\right)\\right) \\times \\log \\log p^{-1}\\right)\\right)$. We do not believe this dependence is optimal and leave it as an open problem to improve it. Particularly, we conjecture that the same algorithm that is analyzed in this work (see Algorithm~\\ref{alg:sampling}) should obtain up to  $(1-\\epsilon)$-approximation even by picking only a $\\poly(1\/\\epsilon p)$-degree subgraph.\n\\end{remark}\n\n\\smparagraph{The algorithm.} Many different constructions of $Q$ have been studied in the literature. A well-studied algorithm first considered by Blum~\\etal{}~\\cite{blumetal} which was further analyzed (module minor differences and generalizations) in the subsequent works of \\cite{AKL16,AKL17,BR18,YM18,YM19} is as follows: Iteratively pick a maximum matching $M_i$ from $G$, remove its edges, and finally let $Q = M_1 \\cup \\ldots \\cup M_R$ for some parameter $R$ that controls the maximum degree in $Q$. Despite the positive results proved for this algorithm, it was already shown in \\cite{blumetal} that its approximation-factor is not better than $5\/6$. Thus to obtain $(1-\\epsilon)$-approximation, one has to use a different algorithm.\n\nWe focus on an algorithm proposed previously by Behnezhad~\\etal{}~\\cite{soda19}, which they proved obtains at least a $0.6568$-approximation. The algorithm is equally simple, but subtly different: Draw $R$ independent realizations $\\mc{G}_1, \\ldots, \\mc{G}_R$ of $G$ and let $Q = \\MM{\\mc{G}_1} \\cup \\ldots \\cup \\MM{\\mc{G}_R}$ where $\\MM{\\mc{G}_i}$ is a maximum matching of $\\mc{G}_i$. Our main result is obtained via providing a different analysis of this algorithm. Within the next two paragraphs, we discuss how our analysis differs substantially from the previous approaches and in particular from the analysis of \\cite{soda19}.\n\n\\smparagraph{The analysis and the Ruzsa-Szemer\u00e9di barrier.} A major barrier to overcome in order to prove existence of a $(1-\\epsilon)$-approximate subgraph was already discussed in the work of Assadi, Khanna, and Li \\cite[Section~6]{AKL16} based on Ruzsa-Szemer\u00e9di graphs \\cite{ruzsa1978triple,DBLP:conf\/stoc\/FischerLNRRS02,DBLP:conf\/soda\/GoelKK12,DBLP:conf\/stoc\/AlonMS12} which we henceforth call the ``RS-barrier''. Consider an extension of the stochastic matching setting where the realization of edges in a single a-priori known matching $M$ of $G$ can be correlated while other edges are still realized independently. An implication of the RS-barrier is that in this extended model, no algorithm can obtain $(1-\\epsilon)$-approximation (or even beat $\\frac{2}{3}$-approximation\\footnote{The original proof of \\cite{AKL16} rules out  $>\\frac{6}{7}$-approximation. A similar instance can rule out $\\frac{2}{3}$-approximation using a more efficient construction of RS-graphs  \\cite{DBLP:conf\/soda\/GoelKK12} and allowing a subset of edges of $G$ to have realization probability 1.}) unless $Q$ has maximum degree  $n^{\\Omega(1\/\\log\\log n)} = \\omega(\\polylog n)$. Put differently, this proves that in order to beat $\\frac{2}{3}$-approximation, the analysis has to use the fact that {\\em every} edge around a vertex is realized independently. This explains why the previous arguments were short of bypassing $\\frac{2}{3}$-approximation: They can all (to our knowledge) be adapted to tolerate  adversarial realization of one edge per vertex.\n\n\\smparagraph{``Vertex-independent matchings'' to the rescue.} We overview our analysis soon in Section~\\ref{sec:techniques}. However, here we briefly mention our key analytical tool in bypassing the RS-barrier. It is an algorithm (Lemma~\\ref{lem:independentmatching}) for constructing a matching $Z$ on the realized {\\em crucial} edges (roughly, an edge is crucial if it has a sufficiently high probability of being part of an optimal realized matching). The algorithm constructs $Z$ such that among some other useful properties, it guarantees that each vertex is matched independently from all but $O(1)$ other vertices. Here the independence is with regards to both the randomization of the algorithm in constructing $Z$, and also importantly  \\underline{the edge realizations of $G$}. This independence property is the key that separates the stochastic matching model from the extended model of the RS-barrier: Due to the added correlations in the edge realizations, such vertex-independent matchings essentially do not exist in the model of the RS-barrier. Using this independence, we show that $Z$ can be well-augmented by the rest of the realized edges in $Q$. See Section~\\ref{sec:techniques} for a more detailed overview of our analysis and how the independence property helps.\n\nOur method of bypassing the RS-barrier via vertex-independent matchings sheds more light on the limitations imposed by Ruzsa-Szemer\u00e9di type graphs. These graphs are known to be notoriously hard examples in various other areas such as property testing, streaming algorithms, communication complexity, and additive combinatorics among others \\cite{DBLP:conf\/soda\/Kapralov13,DBLP:conf\/soda\/GoelKK12,DBLP:conf\/stoc\/AlonMS12,ruzsa1978triple,DBLP:conf\/stoc\/FischerLNRRS02,gowers2001some}. As such, we believe that this method may find applications beyond the stochastic matching problem.\n\n\\smparagraph{Organization of the paper.} In Section~\\ref{sec:techniques} we provide an informal overview of our  analysis. In Section~\\ref{sec:prelim} we formally state the problem and the notations used throughout the paper. In Section~\\ref{sec:analysissetup} we describe the algorithm and basic definitions that we will use throughout the analysis. In Section~\\ref{sec:analysisviavertexindependent} we prove how the vertex-independent matching lemma leads to a $(1-\\epsilon)$-approximation and in Section~\\ref{sec:independentmatching}, we prove the vertex-independent matching lemma. Finally, Section~\\ref{sec:proofs} contains the proofs of (less important) statements that are deferred.\n\n}\n\n\n\n\n\n\\section{On Generality of Assumption~\\ref{ass:optlarge}}\\label{app:optlarge}\n\nIn this section, we prove that Assumption~\\ref{ass:optlarge} comes without loss of generality. Precisely, we show that solving the problem for any input graph $G$ can be reduced to solving it for a graph $H$ with $O(\\opt\/\\epsilon)$ vertices and $\\E[\\mu(\\mc{H})] \\geq (1-\\epsilon)\\opt$ where $\\mc{H}$ is a realization $H$. To do this, we use a ``vertex sparsification'' idea of  Assadi~\\etal{}~\\cite{AKL16}. Our reduction is slightly different since we do not want parallel edges in the graph, but the main idea is essentially the same. It is also worth noting that for the reduction to work, it is crucial that our algorithm works for different edge realization probabilities. We provide the full proof for completeness.\n\nWe note that throughout the proof we may assume that $\\opt$ is larger than constant $3\\epsilon^{-3}$ and remark that the problem otherwise is trivial.\n\n\\smparagraph{Construction of $H$ from $G$.} We construct graph $H=(U, F)$ as follows. For $k = \\frac{8\\opt}{\\epsilon}$, define $k$ {\\em buckets} $U = \\{u_1, \\ldots, u_k\\}$. Each of these buckets $u_i$ will correspond to a node in $H$. Assign each vertex $v$ of graph $G$ to a bucket $b(v) \\in \\{u_1, \\ldots, u_k\\}$ picked independently and uniformly at random. Then for any edge $\\{v_1, v_2\\}$ in graph $G$, we add an edge  $\\{b(v_1), b(v_2)\\}$ to $F$. Finally, we turn $H$ into a simple graph by removing self-loops and merging parallel edges.\n\nNow we need to set the realization probability $p_e$ of every edge $e \\in F$ as well. For any $e \\in F$, let us denote by $E(e)$ the set of edges in the original graph $G$ that are mapped to $e$. We set\n$$\n\tp_e := 1 - \\prod_{e' \\in E(e)} (1-p_{e'}).\n$$\nWe note that $p_e$ is defined such that it precisely equals to the probability that at least one edge in $E(e)$ is realized.\n\n\\begin{claim}\\label{cl:largeinH}\n\tFix any matching $M$ in $G$ satisfying $|M| \\leq 2\\opt$. Then $\\E[\\mu(H)] \\geq (1-\\epsilon)|M|$ where the expectation is taken over the randomization of the algorithm in constructing $H$. \n\\end{claim}\n\\begin{proof}\n\tLet $V(M)$ be the vertex-set of matching $M$ in graph $G$ and define \n\t$$X := \\{v \\in V(M) \\mid \\exists u \\in V(M) \\text{ s.t. } v\\not= u \\text{ and } b(v)=b(u)\\},\n\t$$\n\twhich is the set of vertices in $V(M)$ whose bucket is not unique with regards to others in $V(M)$.\n\t\n\tWe first claim that $\\mu(H) \\geq |M| - |X|$. Call an edge $\\{u, v\\} \\in M$ {\\em good} if $u \\not\\in X$, $v \\not\\in X$, and {\\em bad} otherwise. Each bad edge has at least one endpoint in $X$, thus there are at least $|M| - |X|$ good edges in $M$. One can easily confirm that the set of corresponding edges of all good edges in $M$ forms a matching in $H$. Thus $\\mu(H) \\geq |M|-|X|$.\n\t\n\tTo conclude, we prove that $\\E[|X|] \\leq \\epsilon |M|$ which proves $\\E[\\mu(H)] \\geq |M| - \\epsilon |M| = (1-\\epsilon)|M|$. To see why $\\E[|X|] \\leq \\epsilon |M|$, fix any vertex $v \\in V(M)$ and suppose that we have adversarially fixed the bucket $b(u)$ of all other vertices $u \\in V(M)$. Since the bucket of $v$ is picked uniformly at random from $10\\opt\/\\epsilon$ buckets and $|V(M)| \\leq 2|M| \\leq 4\\opt$, the probability of $v$ choosing a bucket already chosen by another vertex in $V(M)$ would be $\\leq \\frac{4\\opt}{8\\opt\/\\epsilon} \\leq \\epsilon\/2$. By linearity of expectation over $2|M|$ vertices in $V(M)$, we get $\\E[|X|] \\leq \\epsilon |M|$, concluding the proof.\n\\end{proof}\n\n\\begin{claim}\\label{cl:713713}\n\tIt holds that $\\E[\\mu(\\mc{H})] \\geq (1-3\\epsilon)\\opt$. Here the expectation is taken over both the randomization in construction of $H$ and the randomization in realization $\\mc{H}$ of $H$.\n\\end{claim}\n\\begin{proof}\n\tWe first map each realization $\\mc{G}$ of $G$ to a realization $\\mc{H}$ of $H$. To do so, we say an edge $e \\in F$ is realized in $\\mc{H}$ if and only if at least one edge $e' \\in E(e)$ is realized in $\\mc{G}$. We argue that this mapping preserves independence of edge realizations in $H$ and their realization probabilities. First, since for any two edges $e_1, e_2 \\in F$ it holds that $E(e_1) \\cap E(e_2) = \\emptyset$, realization of an edge $e \\in F$ gives no information regarding realization of other edges. Moreover, observe that each edge $e \\in F$ will be precisely realized with probability $p_e$ as discussed above in defining $p_e$.\n\t\n\tLet $M$ be the maximum realized matching of $G$. By Lemma~\\ref{lem:concentration}, $\\Pr[||M|-\\opt| \\geq \\epsilon \\opt] < \\exp(-\\frac{(\\epsilon \\opt)^2}{3\\opt}) = \\exp( - \\frac{\\epsilon^2 \\opt}{3}) < \\epsilon$ where the last inequality follows from assumption $\\opt > 3\\epsilon^{-3}$. This means that with probability at least $1-\\epsilon$, $|M| \\in [(1-\\epsilon)\\opt, (1+\\epsilon)\\opt]$. Let us suppose that this event holds and denote it by $A$. Note that event $A$ is only with regards to realization of $G$ and reveals no information about the algorithm to construct $H$. Now plugging matching $M$ into Claim~\\ref{cl:largeinH}, we get that $\\E[\\mu(\\mc{H}) \\mid A] \\geq (1-\\epsilon)|M| \\geq (1-\\epsilon)(1-\\epsilon)\\opt \\geq (1-2\\epsilon)\\opt$. Incorporating also the probability that event $A$ holds, which as described is at least $1-\\epsilon$, we get $\\E[\\mu(\\mc{H})] \\geq (1-\\epsilon)(1-2\\epsilon)\\opt \\geq (1-3\\epsilon)\\opt$, concluding the proof.\n\\end{proof}\n\n\\smparagraph{The reduction.} We are now ready to give the full reduction. Suppose we are given $n$-vertex graph $G$ with $\\opt = \\E[\\mu(\\mc{G})]$ and assume that $\\opt < 0.1 \\epsilon n$ (otherwise Assumption~\\ref{ass:optlarge} holds). We first construct graph $H$ as described. Note that $H$ has at most $n' = \\frac{8\\opt}{\\epsilon}$ nodes by the construction and that $\\E[\\mu(\\mc{H})] \\geq (1-3\\epsilon)\\opt$ by Claim~\\ref{cl:713713}. Replacing $\\opt$ with $\\epsilon n'\/8$, we get $\\E[\\mu(\\mc{H})] \\geq (1-3\\epsilon)\\frac{\\epsilon n'}{8}$. Assuming $\\epsilon < 0.05$ (recall that we can assume $\\epsilon$ to be smaller than any needed constant), this implies $\\E[\\mu(\\mc{H})] \\geq \\frac{\\epsilon n'}{10}$ and thus Assumption~\\ref{ass:optlarge} holds for graph $H$.\n\nLet $Q$ be the result of running Algorithm~\\ref{alg:sampling} on graph $H$. Since Assumption~\\ref{ass:optlarge} holds for $H$, it leads to a $(1-\\epsilon)$-approximation. That is, we get $\\E[\\mu(\\mc{Q})] \\geq (1-\\Omega(\\epsilon))\\E[\\mu(\\mc{H})]$. We use this subgraph $Q$ to pick a bounded-degree subgraph $Q'$ of $G$ that provides a $(1-\\epsilon)$-approximation: For each edge $e \\in Q$, let us {\\em pick} $\\min\\{p^{-1} \\log \\epsilon^{-1}, |E(e)|\\}$ arbitrary edges from $E(e)$ and put them in $Q'$. We argue that this subgraph $Q'$ has maximum degree $O_{\\epsilon, p}(1)$ and that $\\E[\\mu(\\mc{Q}')] \\geq (1-\\Omega(\\epsilon))\\opt$.\n\n\\begin{claim}\n\t$Q'$ has maximum degree $O_{\\epsilon, p}(1)$.\t\n\\end{claim}\n\\begin{proof}\n\tObserve that an edge $e'$ incident to a vertex $v \\in V$ is in $Q'$ only if its corresponding edge $e$ in graph $H$ is in $Q$. Since $e$ corresponds to $e'$,  it should be incident to $b(v)$ of $v$ by the construction of $H$. Moreover, since $b(v)$ has maximum degree $O_{\\epsilon, p}(1)$ in $Q$ and that for each edge incident to $b(v)$ in $Q$, we put at most $O(p^{-1}\\log \\epsilon^{-1})$ edges in $Q'$, the degree of $v$ in $Q'$ is bounded by $O_{\\epsilon, p}(1) \\times O(p^{-1}\\log \\epsilon^{-1}) = O_{\\epsilon, p}(1)$. This bounds the maximum degree of $Q'$ by $O_{\\epsilon, p}(1)$.\n\\end{proof}\n\n\\begin{claim}\n\t$\\E[\\mu(\\mc{Q}')] \\geq (1-\\Omega(\\epsilon))\\opt$.\n\\end{claim}\n\\begin{proof}\n\tFor any edge $e \\in Q$, define $p'_e$ to be the probability that at least one of the edges in $G$ picked for $e$ is realized. We first argue that $p'_e \\geq (1-\\epsilon)p_e$. To see this, note that if $|E(e)| \\leq p^{-1}\\log \\epsilon^{-1}$, then all the edges in $E(e)$ will be picked. Thus by definition of $p_e$ we have $p'_e = p_e$. On the other hand, if $|E(e)| > p^{-1}\\log \\epsilon^{-1}$, we pick exactly $p^{-1}\\log \\epsilon^{-1}$ edges for $e$. Since each of these edges has realization probability at least $p$, the probability that at least one of them is realized is at least\n$$\n\t1-(1-p)^{p^{-1}\\log \\epsilon^{-1}} \\geq 1-\\epsilon \\geq (1-\\epsilon)p_e.\n$$\n\nNow let $M$ be any matching in $Q$. For each edge $e \\in M$, choose one arbitrary edge in $E(e)$. From the construction of $H$ from $G$, one can confirm that the set of these chosen edges will form a matching of size $|M|$ in $G$. This concludes the proof: For each edge $e \\in Q$, there is a probability at least $(1-\\epsilon)p_e$ that one picked edge in $Q'$ is realized, thus $\\E[\\mu(\\mc{Q}')] \\geq (1-\\epsilon)\\E[\\mu(\\mc{Q})]$. As it was previously shown that $\\E[\\mu(\\mc{Q})] \\geq (1-\\Omega(\\epsilon))\\opt$, we conclude that  $\\E[\\mu(\\mc{Q}')] \\geq (1-\\Omega(\\epsilon))\\opt$.\n\\end{proof}\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\n\\paragraph{General notations.} We denote the maximum matching size of any graph $G$ by $\\mu(G)$. For a matching $M$, we use $V(M)$ to denote the set of vertices matched in $M$. For any two nodes $u$ and $v$ in a graph $G$, we use $d_G(u, v)$ to denote their distance, i.e. the number of edges in their shortest path. Furthermore, the distance $d_G(u, e)$ between an edge $e$ and a node $u$ is the minimum distance between an endpoint of $e$ and $u$. We use $\\mathbbm{1}(A)$ as the {\\em indicator} of an event $A$, i.e. $\\mathbbm{1}(A) = 1$ if event $A$ occurs and $\\mathbbm{1}(A) = 0$ otherwise. Also, we may use $[k] := \\{1, 2, \\ldots, k\\}$ for any integer $k \\geq 1$.\n\nThroughout the paper, we define various functions of form $x : E \\to [0, 1]$ that map each edge $e \\in E$ to a real number in $[0, 1]$. Having such function $x$, for any vertex $v$ we define $x_v := \\sum_{e \\ni v} x_e$, for any edge subset $F$ we define $x(F) := \\sum_{e \\in F} q_e$, and for any vertex subset $U$ we define $x(U) := \\sum_{e=\\{u, v\\}: u, v \\in U} x_e$. We also denote $|x| = \\sum_e x_e$.\n\n\\smparagraph{The setting.} We consider a generalized variant of the standard stochastic matching problem studied in the literature where each edge $e$ has a realization probability $p_e$ that may be different from that of other edges. We then let $p = \\min_e p_e$, which is the parameter the degree of subgraph $Q$ can depend on. This generalization will actually help in solving the original model of the literature defined in Section~\\ref{sec:intro} which coincides with the case where $p_e = p$ for every edge $e$.\n\nWe denote realizations by script font; for instance, we use $\\mc{G}=(V, \\mc{E})$ to denote the realized subgraph of the input graph $G$, which includes each edge $e$ independently with probability $p_e$. Similarly, we use $\\mc{Q}$ to denote the realized subgraph of $Q$. The same notation also naturally extends to denote realization of other subgraphs of $G$ that we may later define.\n\nAs discussed in Section~\\ref{sec:intro}, the goal is to pick a sparse subgraph $Q$ of $G$ such that the ratio $\\E[\\mu(\\mc{Q})]\/\\E[\\mu(\\mc{G})]$, known as the approximation-factor, is large. Here  the expectations are taken over the realizations $\\mc{Q}$ and $\\mc{G}$, and possibly the randomization of the algorithm in constructing subgraph $Q$. For brevity, we use $\\opt$ to denote  $\\E[\\mu(\\mc{G})]$. Note that $\\opt$ is just a number.\n\nWe note that the {\\em expected} approximation-factor defined above can automatically be turned into {\\em high-probability} due to a simple concentration bound. See Appendix~\\ref{sec:concentration}.\n\n\n\\section{Deferred Proofs}\\label{sec:proofs}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:gap}]\n\tLet $t_0 = (\\epsilon p)^{50}$ and for any $i \\geq 1$ let $t_i = f(t_{i-1})$. Note that $t_0 > t_1 > t_2 > \\ldots$ by the assumption of the lemma that $0 < f(x) < x$ for all $0 < x < 1$. For any $i \\geq 1$ define $q_i = \\sum_{e \\in E: q_e \\in (t_i, t_{i-1}]} q_e$ and let $j$ be the smallest number where $q_j \\leq \\epsilon \\opt$. We will soon prove existence of such $j$ and also prove that $j = O(1\/\\epsilon)$. We claim that setting $\\tau_+ = t_{j-1}$ and $\\tau_- = t_j$ satisfies the conditions of the lemma.\n\t\n\t\\textbf{Condition} (1): This condition holds trivially since $\\tau_- = t_{j} = f(t_{j-1}) = f(\\tau_+)$. \n\t\n\t\\textbf{Condition} (2): Let us define $X := \\{ e \\mid \\tau_- < q_e < \\tau_+ \\}$. Recall that crucial and non-crucial edges are defined based on $\\tau_+$ and $\\tau_-$. That is, an edge $e$ is crucial (i.e. $e \\in C$) if $q_e \\geq \\tau_+$, and is non-crucial (i.e. $e \\in N$) if $q_e \\leq \\tau_-$. This implies that the remaining edges that are neither crucial nor non-crucial belong to $X$. Therefore,\n\t$$\n\t\t\\opt = q(E) = q(C) + q(N) + q(X).\n\t$$\n\tTo obtain $q(N)+q(C) \\geq (1-\\epsilon)\\opt$ it thus suffices to show $q(X) \\leq \\epsilon \\opt$. Noting that $\\tau_+ = t_{j-1}$ and $\\tau_- = t_j$ and also noting the definition of $q_j$ above, we get $q(X) \\leq q_j$. Recall that we chose $j$ such that $q_j \\leq \\epsilon \\opt$. Therefore we indeed get that $q(X) \\leq \\epsilon \\opt$.\n\t\n\t\\textbf{Condition} (3): We defined $t_0 = (\\epsilon p)^{50}$ and recursively defined $t_i = f(t_{i-1})$. Since $f(\\cdot)$ is only a function of its input, we get via a simple induction that both $t_j$ and $t_{j-1}$ are also functions of only $\\epsilon$ and $p$. (Recall that $j = O(1\/\\epsilon)$.)\n\t\n\t\\textbf{Condition} (4): We defined $t_0 = (\\epsilon p)^{50}$ and recall that we showed $t_0 > t_1 > t_2  > \\ldots$; this implies clearly that $\\tau_+ = t_{j-1} \\leq (\\epsilon p)^{50}$.\n\t\n\t\\textbf{Existence of $j$.} It only remains to prove that there exists a choice of $j$ satisfying $q_j \\leq \\epsilon \\opt$ and that this $j$ is not too large. Precisely, we show that $j = O(1\/\\epsilon)$. Since intervals $(t_1, t_0], (t_2, t_1], (t_3, t_2], \\ldots$ are disjoint, it holds that for each edge $e$ there is at most one $i$ for which $q_e \\in (t_i, t_{i-1}]$. This means that $\\sum_{i=1}^\\infty q_i \\leq \\sum_{e \\in E} q_e = \\opt$.  It thus has to hold that $j \\leq \\lceil 1\/\\epsilon \\rceil + 1$ or otherwise \n\t$$\n\t\\sum_{i=1}^{j-1} q_i \\geq \\sum_{i=1}^{\\lceil 1\/\\epsilon \\rceil + 1} \\epsilon \\opt = (\\lceil 1\/\\epsilon \\rceil + 1) \\epsilon \\opt >  \\opt\n\t$$ contradicting the previous statement. This concludes the proof of the lemma.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Claim~\\ref{cl:frange}]\n\tWe prove parts 1-3 one by one.\n\t\n\t\\smparagraph{Part 1.} The upper bound $\\E[f_e] \\leq q_e$ is simple to prove. Consider random variable $f'_{e} = t_e\/R$ and note that $f'_e \\geq f_e$. We have\n\t$$\n\t\\E[f'_e] = \\E\\left[\\frac{t_e}{R}\\right] = \\frac{1}{R} \\E[t_e] = \\frac{1}{R} \\left(\\sum_{i=1}^R \\Pr[e \\in \\MM{\\mc{G}_i}]\\right) = \\frac{1}{R} (R \\times \\Pr[e \\in \\MM{\\mc{G}_1}]) = q_e.\n\t$$\n\tSince $f_e \\leq f'_e$, we get $\\E[f_e]\\leq\\E[f'_e] = q_e$, concluding the proof of part 1.\n\t\n\t\\smparagraph{Part 2.} Next we turn to prove the lower bound $\\E[f_e] \\geq (1-\\epsilon)q_e$. Let $X_i$ be the indicator random variable for $e \\in \\MM{\\mc{G}_i}$. We have $t_e = X_1 + \\ldots + X_R$, $\\E[X_i] = q_e$, and $\\E[t_e] = Rq_e$. Note also that the $X_i$'s are independent since graphs $\\mc{G}_1, \\ldots, \\mc{G}_R$ are drawn independently. Therefore, $\\Var[t_e] = \\sum_{i=1}^R \\Var[X_i] = R(q_e - q_e^2)$. \n\t\n\tNoting that $R = 0.5\/\\tau_-$ and that $q_e < \\tau_-$ since $e$ is non-crucial, we get $R q_e < 1$. This means that if $t_e \\geq a + 1$, then $|t_e - Rq_e| \\geq a$; which implies $\\Pr[t_e \\geq a + 1] \\leq \\Pr[|t_e - Rq_e| \\geq a]$. Therefore by setting $a = \\sqrt{R\/\\epsilon}$ and also using Chebyshev's inequality, we get\n\t\\begin{equation}\\label{eq:1234123489172346}\n\t\t\\Pr\\left[t_e \\geq \\sqrt{R\/\\epsilon}+1\\right] \\leq  \\Pr\\left[|t_e - \\E[t_e]| \\geq \\sqrt{R\/\\epsilon}\\right] \\leq \\frac{\\Var[t_e]}{(\\sqrt{R\/\\epsilon})^2} = \\frac{R(q_e-q_e^2)}{(\\sqrt{R\/\\epsilon})^2} = \\epsilon (q_e - q_e^2) \\leq \\epsilon q_e.\n\t\\end{equation}\n\tFinally, we have\n\t\\begin{align*}\n\t\\E\\left[\\frac{t_e}{R}\\right] &= \\underbrace{\\Pr\\left[\\frac{t_e}{R} \\leq \\frac{1}{\\sqrt{\\epsilon R}}\\right] \\E\\left[\\frac{t_e}{R} \\mid \\frac{t_e}{R} \\leq \\frac{1}{\\sqrt{\\epsilon R}} \\right]}_{=\\E[f_e]} + \\Pr\\left[\\frac{t_e}{R} > \\frac{1}{\\sqrt{\\epsilon R}}\\right] \\underbrace{\\E\\left[\\frac{t_e}{R} \\mid \\frac{t_e}{R} > \\frac{1}{\\sqrt{\\epsilon R}} \\right]}_{\\leq 1 \\text{ since by definition, $t_e \\leq R$.}}\n\t\\end{align*}\n\tRearranging the terms and replacing the bounds specified, we get\n\t$$\n\t\\E[f_e] \\geq \\E\\left[\\frac{t_e}{R}\\right] - \\Pr\\left[\\frac{t_e}{R} > \\frac{1}{\\sqrt{\\epsilon R}}\\right] = \\frac{1}{R}\\E\\left[t_e\\right] - \\Pr\\left[t_e \\geq \\sqrt{R\/\\epsilon} + 1 \\right] \\stackrel{(\\ref{eq:1234123489172346})}{\\geq} \\frac{1}{R} \\times R q_e - \\epsilon q_e = (1-\\epsilon)q_e,\n\t$$\n\tconcluding the proof of part 2.\n\t\n\t\\smparagraph{Part 3.} Note that $f_e \\leq t_e\/R$ by definition. Thus, we have $\\sum_{e \\ni v} f_e \\leq \\sum_{e \\ni v} t_e\/R = R^{-1} \\sum_{e \\ni v} t_e$. Since each $\\MM{\\mc{G}_i}$ includes at most one incident edge of $v$ for being a matching, it holds that $\\sum_{e \\ni v} t_e \\leq R$, thus indeed $\\sum_{e \\ni v} f_e \\leq R^{-1} R = 1$.\n\t\n\t\\smparagraph{Part 4.} Let $X_i$ be the event that $v$ is matched in $\\MM{\\mc{G}_i}$ via a non-crucial edge and define $X := \\sum_{i=1}^R X_i$. Furthermore, define for each edge $e$,\n\t$$\n\t\tf'_e := \\begin{cases}\n\t\t\\frac{t_e}{R}, & \\text{if $e$ is non-crucial,}\\\\\n\t\t0, & \\text{otherwise.}\n\t\\end{cases}\n\t$$\n\tNote that $f'_e$ is very similar to the value of $f_e$ except for the case where $t_e\/R > 1\/\\sqrt{\\epsilon R}$. In this case, $f_e = 0$ but $f'_e$ remains to be the ratio $t_e\/R$. This implies that $f'_e \\geq f_e$. Now let $f'_v = \\sum_{e \\ni v} f'_e$. Since $f_e \\leq f'_e$ for all edges, we have $f_v \\leq f'_v$. Therefore, instead of proving $\\Pr[f_v > n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$, it suffices to prove $\\Pr[f'_v > n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$.\n\t\n\t\n\tIt holds from the definition that\n\t$$\n\tf'_v = \\sum_{e: e \\in N, v \\in e} \\frac{t_e}{R} = \\frac{1}{R} \\sum_{e: e \\in N, v \\in e} t_e = \\frac{1}{R} \\times (X_1 + \\ldots + X_R) = X\/R.\n\t$$\n\tReplacing this into $\\Pr[f'_v > n_v + 0.1\\epsilon] \\leq (\\epsilon p)^{10}$, we thus have to prove \n\t$\n\t\t\\Pr\\left[X\/R > n_v + 0.1\\epsilon \\right] \\leq (\\epsilon p)^{10},\n\t$\n\tor equivalently:\n\t$$\n\t\t\\Pr[X > R n_v + 0.1 R \\epsilon] \\leq (\\epsilon p)^{10}.\n\t$$\n\tTo prove this we use a concentration bound on $X$. Note  that the $X_i$'s are independent since graphs $\\mc{G}_1, \\ldots, \\mc{G}_R$ are drawn independently. Moreover, for each $i \\in [R]$, we have $\\E[X_i] = n_v$ since recall $X_i = 1$ iff $v$ is matched via a non-crucial edge in $\\MM{\\mc{G}_i}$ and this has probability $\\sum_{e: e\\in N, v \\in e} q_e = n_v$. Thus $\\E[X] = Rn_v$. While we can use Chernoff's bound here since all $X_i$'s are independent, even the second-moment method is enough for our desired inequality. The variance of $X$ can be bounded as follows:\n\t$$\n\t\\Var[X] = \\sum_{i=1}^R \\Var[X_i] = \\sum_{i=1}^R E[X_i^2] - \\E[X_i]^2 = R (n_v - n_v^2).\n\t$$\n\tBy Chebyshev's inequality, we get\n\t$$\n\t\t\\Pr[X > Rn_v + 0.1R\\epsilon] \\leq \\frac{R(n_v - n_v^2)}{(0.1 R \\epsilon)^2} = \\frac{100(n_v - n_v^2)}{R \\epsilon^2} \\leq \\frac{100}{R\\epsilon^2}.\n\t$$\n\tSince $R = 1\/2\\tau_-$ and $\\tau_- < (\\epsilon p)^{50}$ by Corrolary~\\ref{cor:thresholds}, we get \n\t$$\n\t\\Pr[X > Rn_v + R\\epsilon] \\leq \\frac{100}{R\\epsilon^2} < \\frac{200(\\epsilon p)^{50}}{\\epsilon^2} < (\\epsilon p)^{10},\n\t$$\n\twhich as described above concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Observation~\\ref{obs:samedist}]\nFirst note that realizations $\\mc{C}_1, \\ldots, \\mc{C}_\\alpha$ are all drawn precisely from the same distribution that realization $\\mc{C} = \\mc{C}_0$ is drawn from. Thus due to symmetry, matchings $M_0, \\ldots, M_\\alpha$ are all derived from the same distribution. Matchings $M'_0, \\ldots, M'_\\alpha$ are then the result of applying the augmenting-hyperwalks $I$ found by $\\apxMIS{H, \\epsilon}$ on graph $H$. Construction of graph $H$ is symmetrical w.r.t. matchings $M_0, \\ldots, M_\\alpha$. The only remaining component of the algorithm where this symmetry may break is in algorithm $\\apxMIS{H, \\epsilon}$ that may be biased towards picking augmenting-hyperwalks depending on which matching $M_i$ they would augment. This can be avoided by using an algorithm for $\\apxMIS{H, \\epsilon}$ that is oblivious to the indices of matchings $M_0, \\ldots, M_\\alpha$ used to construct graph $H$. That is, suppose e.g. that we pick the ID of nodes in $H$ randomly before feeding it into $\\apxMIS{H, \\epsilon}$. This guarantees that the obtained matchings $M'_0, \\ldots, M'_\\alpha$ will all have the same distribution due to their symmetry.\n\\end{proof}\n\n\\section{The Analysis via the Vertex-Independent Matching Lemma}\\label{sec:analysisviavertexindependent}\n\nIn this section, given correctness of Lemma~\\ref{lem:independentmatching}, we prove Theorem~\\ref{thm:main}. In what follows we give the outline of the proof by referring to the needed lemmas that will be proved in subsequent Sections~\\ref{sec:constructfractional}, \\ref{sec:validityofx}, \\ref{sec:analysisfractional}, and \\ref{sec:turntofracmatching}.\n\n\\smparagraph{Proof Outline for Theorem~\\ref{thm:main}.} \n\tLet $Q$ be the output of by Algorithm~\\ref{alg:sampling} where parameter $R$ is set as described above. We show that one can construct a matching of expected size at least $(1-56\\epsilon)\\opt$ on the realized subgraph $\\mc{Q}$ of $Q$. This implies that $\\E[\\mu(\\mc{Q})] \\geq (1-56\\epsilon)\\opt = (1-56\\epsilon)\\E[\\mu(\\mc{G})]$. In other words, this proves that the approximation-factor of the algorithm is at least $(1-56\\epsilon)$. (Note this is equivalent to $(1-\\epsilon)$ approximation  since one can choose $\\epsilon$ to be any desirably small constant.)\n\t\n\tIn order to construct a matching of expected size at least $(1-56\\epsilon)\\opt$ on $\\mc{Q}$, we first describe how to construct an ``expected fractional matching'' (see Definition~\\ref{def:expfractional}) $x$ on $\\mc{Q}$ in Sections~\\ref{sec:constructfractional}, \\ref{sec:validityofx}, and \\ref{sec:analysisfractional}. Later on, we show in Section~\\ref{sec:turntofracmatching} how to turn $x$ into a  fractional matching $y$ on $\\mc{Q}$ such that $\\E[|y|] \\geq (1-55\\epsilon)\\opt$ (see Lemma~\\ref{lem:ylarge}). Finally, to turn $y$ into an {\\em integral} matching, we show (Observation~\\ref{obs:yblossom}) that the so called ``blossom inequalities'' of size up to $1\/\\epsilon$ also hold for $y$. That is, we show that for all vertex subsets $U \\subseteq V$ with $|U| \\leq 1\/\\epsilon$, we have $y(U) \\leq \\lfloor \\frac{|U|}{2}\\rfloor$. By Edmond's celebrated theorem \\cite{edmonds1965maximum,schrijver2003combinatorial} on the matching polytope, this means that there is an integral matching of size at least $\\frac{1}{1+\\epsilon}|y| \\geq (1-\\epsilon)|y|$ in \\mc{Q}. As described, $\\E[|y|] \\geq (1-55\\epsilon)\\opt$, thus indeed $\\E[\\mu(\\mc{Q})] \\geq (1-\\epsilon)(1-55\\epsilon)\\opt \\geq (1-56\\epsilon)\\opt$ as desired.\n \n\\input{expfracmatching}\n\n\\input{exptoactualfracmatching}\n\\section{Our Techniques}\\label{sec:techniques}\n\n\nAs previously described, we consider the following algorithm for constructing subgraph $Q$ (see also Algorithm~\\ref{alg:sampling}): Draw $R$ realizations $\\mc{G}_1, \\ldots, \\mc{G}_R$ of graph $G$, then pick a matching $\\MM{\\mc{G}_i}$ from each realization, and finally set $Q = \\MM{\\mc{G}_1} \\cup \\ldots \\cup \\MM{\\mc{G}_R}$. In this section, we give an informal overview of our analysis for this algorithm. \n\nNote that these realizations $\\mc{G}_i$ are part of the randomization of the algorithm and may be very different from the actual realization $\\mc{G}$ of $G$. In fact, in expectation, only $p$ fraction of the edges of each matching $\\MM{\\mc{G}_i}$ are realized in $\\mc{G}$. Thus, we have to argue that the realized edges of these matchings can be used to augment each other and form a large matching in the realized subgraph $\\mc{Q}$ of $Q$. In order to do this, we will give a ``procedure'' to construct a matching in $\\mc{Q}$. To get a handle on the dependencies involved, the procedure carefully decides how the realization of edges in $Q$ are revealed and which are chosen to be in the matching. We emphasize that this procedure is merely an analytical tool for analyzing the approximation-factor. Thus, no matter how intricate it is, the algorithm for constructing $Q$ remains to be the simple Algorithm~\\ref{alg:sampling} described above.\n\n\\smparagraph{A crucial\/non-crucial decomposition.} Similar to  \\cite{soda19} (and also implicitly \\cite{AKL17}), we consider a partitioning of the edges of $G$ into what we call {\\em crucial} and {\\em non-crucial} edges. For each edge $e$, define $q_e := \\Pr[e \\in \\MM{\\mc{G}}]$ where $\\MM{\\cdot}$ is the same matching algorithm used to construct $Q$. We further assume that $\\MM{\\cdot}$ is deterministic, so the probability is taken only over the realization $\\mc{G}$. For two thresholds $0 < \\tau_- < \\tau_+ < 1$ that we fix later, we define:\n\\begin{itemize}[itemsep=0pt]\n\t\\item The crucial edges as $C := \\{ e \\in E \\mid q_e \\geq \\tau_+\\}$.\n\t\\item The non-crucial edges as $N = \\{ e \\in E \\mid q_e \\leq \\tau_-\\}$.\n\\end{itemize}\nNote that in the decomposition above edges $e$ with $q_e \\in (\\tau_-, \\tau_+)$ are neither crucial nor non-crucial. We will essentially ``ignore'' these edges in the analysis but ensure that we choose $\\tau_-$ and $\\tau_+$ such that there are few ignored edges. \n\nIn our procedure to construct a matching on $\\mc{Q}$, we treat crucial and non-crucial edges differently. We start with the crucial edges and (in Lemma~\\ref{lem:independentmatching}) construct a matching $Z$ on them whose expected size is (almost) as large as the expected number of crucial edges in the optimal maximum realized matching of $G$. We then show that this matching $Z$ can be augmented via the non-crucial edges to eventually form a matching whose expected size is arbitrarily close to $\\opt := \\E[|\\MM{\\mc{G}}|]$.\n\n\\smparagraph{The procedure for crucial edges.} In addition to the lower bound on the expected size of $Z$, we make sure that no vertex tends to be ``over-matched'' in $Z$. More formally, the probability of any vertex $v$ being matched in $Z$ should not be larger than the probability that $v$ is matched via a crucial edge in $\\MM{\\mc{G}}$. Both of these conditions can actually be satisfied by a very simple randomized procedure: Reveal the whole realization $\\mc{C}$ of $C$, also draw a random realization $\\mc{N}'$ of the non-crucial edges, and let $Z$ be the crucial edges in matching $\\MM{\\mc{C} \\cup \\mc{N}'}$. \n\nUnfortunately, the matching constructed via the above-mentioned procedure is hard to augment via the non-crucial edges as we have no control over the correlations. To get around this, we need an extra  ``independence'' property. Let $X_v$ be the indicator of the event that vertex $v$ is matched in $Z$. The independence property requires random variables $X_{v_1}, X_{v_2}, \\ldots, X_{v_n}$ to be (almost) independent where $\\{v_1, \\ldots, v_n\\}$ is the vertex-set of $G$. Clearly, perfect independence cannot be achieved: Given the event that a vertex $v$ is matched in $Z$, we derive that at least one of its neighbors in $C$ is also matched. What we prove can be achieved, though, is that each $X_{v}$ is independent from $X_u$ of vertices $u$ outside a small local neighborhood of $v$ in graph $C$. (See Lemma~\\ref{lem:independentmatching} part~4 for the formal statement.)\n\nIn order to satisfy the independence property described above, we will not reveal the whole realization $\\mc{C}$ outright and then construct $Z$ based on it as it was done in the simple procedure described above. Instead, we present a different algorithm (Algorithm~\\ref{alg:crucial}) for constructing this matching $Z$. To prove the independence property, we show that this algorithm can be simulated locally. In other words, for each vertex $v$, the value of $X_v$ can be determined uniquely by having the realization of edges in a small local neighborhood of $v$. Thus, if two vertices $u$ and $v$ are sufficiently far from each other in graph $C$, then $X_v$ and $X_u$ would be independent. \n\n\\smparagraph{Augmenting $Z$ via non-crucial edges.} We noted above that $\\E[|Z|]$ is (almost) as large as the expected number of crucial edges in $\\MM{\\mc{G}}$. Therefore, in order to construct a matching of $\\mc{Q}$ with expected size arbitrarily close to $\\opt$, we have to augment $Z$ via the non-crucial edges. To do this, we only use non-crucial edges $\\{u, v\\}$ in $Q$ such that $X_u$ and $X_v$ are independent. Describing how exactly we construct the matching on these non-crucial edges requires a number of definitions which we give in Section~\\ref{sec:constructfractional}. However, to convey the key intuition, here we only mention how and why the independence of $X_u$ and $X_v$ plays an important role in using a non-crucial edge $e = \\{u, v\\}$ to augment $Z$. Suppose that $\\Pr[X_u] = \\Pr[X_v] = 1\/2$. Note that it is only when {\\em both} $u$ and $v$ are unmatched in $Z$ that we can use edge $e$ to augment $Z$. If $X_u$ and $X_v$ are independent, there is a relatively large probability $(1-\\Pr[X_u])(1-\\Pr[X_v]) = \\frac{1}{4}$ that this occurs. However, if $X_u$ and $X_v$ can be correlated, it may be the case that with probability half  $X_u = 1$ and $X_v = 0$, and with probability half $X_u = 0$ and $X_v = 1$. In this case, the probability of both $u$ and $v$ being unmatched in $Z$ would be zero and thus we would never be able to use $e$ to augment $Z$. We remark that this is precisely the type of correlation introduced in the RS-barrier of \\cite{AKL16} which the independence property allows us to bypass.\n\n\n\n\n\\section{Approximate MIS}\\label{app:weakmis}\n\nIn this section we describe how Lemma~\\ref{lem:apxMIS} can be derived as a corollary of the algorithm of \\cite{DBLP:conf\/soda\/Ghaffari19}. Theorem~1.1 of \\cite{DBLP:conf\/soda\/Ghaffari19} gives a randomized \\local{} independent-set (IS) algorithm which guarantees that for each node $v$, the probability that $v$ ``has not made its decision'' after $O(\\log \\deg(v) + \\log \\frac{1}{\\delta})$ rounds is at most $\\delta$. The decision of $v$ is finalized if it is in the IS or it has a neighbor that is in the IS (implying that $v$ cannot be in the IS). \n\nTo achieve Lemma~\\ref{lem:apxMIS} we set $\\delta = \\frac{\\epsilon}{10\\Delta}$. Let $I$ denote the independent set returned by the algorithm after $O(\\log \\deg(v) + \\log \\frac{10\\Delta}{\\epsilon}) = O(\\log \\frac{\\Delta}{\\epsilon})$ rounds and let $U$ and $D$ respectively denote the set of undecided and decided vertices. We have\n$$\n\\E[|U|] = \\E\\Big[ \\sum_{v} \\mathbbm{1}(\\text{$v$ is undecided}) \\Big] = \\sum_v \\Pr[\\text{$v$ is undecided}] \\leq \\sum_v \\frac{\\epsilon}{10\\Delta} = \\frac{\\epsilon}{10\\Delta}n,\n$$\nand thus $\\E[|D|] = n - \\E[|U|] \\geq (1-\\frac{\\epsilon}{10\\Delta})n \\geq 0.9n$. There is at least one IS node among the at most $\\Delta + 1$ inclusive neighbors of any decided vertex; thus $\\E[|I|] \\geq \\frac{\\E[|D|]}{\\Delta+1} \\geq \\frac{0.9n}{\\Delta+1} \\geq \\frac{0.9n}{2\\Delta} = 0.45 \\frac{n}{\\Delta}$. On the other hand, let $I'$ be the MIS obtained by greedily adding the undecided nodes to $I$ until they form an MIS. We have $|I'| \\leq |I| + |U|$. Therefore, we indeed get that\n$$\n\\frac{\\E[|I|]}{\\E[|I'|]} \\geq \\frac{\\E[|I|]}{\\E[|I|] + \\E[|U|]} \\geq \\frac{0.45\\frac{n}{\\Delta}}{0.45\\frac{n}{\\Delta} + \\frac{\\epsilon}{10\\Delta}n} = \\frac{0.45\\frac{n}{\\Delta}}{(0.45 + 0.1 \\epsilon) \\frac{n}{\\Delta}} = \\frac{0.45}{0.45 + 0.1 \\epsilon} > 1-\\epsilon,\n$$ \nconcluding the proof.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe active galactic nucleus (AGN) is one of the most luminous class of objects \nin the Universe, whose huge radiative energy is released through the mass \naccretion onto the supermassive black hole (SMBH). The mass of SMBHs \n($M_{\\rm BH}$) is tightly correlated with the mass or the stellar velocity \ndispersion of their host galaxies \\citep[e.g.,][]{1998AJ....115.2285M, \n2003ApJ...589L..21M,2013ARA&A..51..511K, 2015ApJ...813...82R}, \nimplying that SMBHs and \ngalaxies have evolved with closely interacting in each other (the so-called \nco-evolution of SMBHs and galaxies). However, the physics behind the \nco-evolution is still unclear. To understand the total picture of the \nco-evolution, examining the scaling relations for AGNs in the early phase of the \nco-evolution is an interesting approach since different theoretical models predict \ndifferent redshift dependences of scaling relations \\citep[e.g.,][]\n{2003ApJ...583...85K, 2010MNRAS.405...29L}. One simple strategy to explore \nthe early phase of the co-evolution is measuring the scaling relations at high \nredshifts, where the typical age of AGNs is much younger than low-redshift \nAGNs. Many attempts have been made for measuring the scaling relations for \nhigh-redshift AGNs \\citep[e.g.,][]{2008A&A...478..311S, 2010ApJ...714..699W, \n2013A&A...559A..29C}, and a higher $M_{\\rm BH}$ with respect to the mass \nor velocity dispersion of host galaxies has been sometimes reported \n\\citep[e.g.,][]{2010ApJ...714..699W}. On the other hand, there are some reports \nclaiming that such a possible evolution in the scaling relation is a result of \nobservational bias through the sample selection \\citep[e.g.,][]\n{2011A&A...535A..87S}. Measuring the properties of AGN host galaxies at high \nredshift is generally very challenging, that prevents us from assessing the scaling\nrelations at high redshifts.\n\nAnother possible approach to study the early phase of the co-evolution is \nfocusing on young AGNs at low redshifts, where detailed observations are much \neasier than high redshifts. In this context, low-metallicity (i.e., chemically young) \nAGNs in the low-redshift Universe are particularly interesting. However, the \ntypical metallicity of AGNs inferred for broad-line regions (BLRs) and narrow-line\nregions (NLRs) is high ($Z \\gtrsim 2 Z_{\\odot}$; e.g., \n\\citealt{2006A&A...447..157N, 2009A&A...503..721M}) and low-metallicity AGNs\nare very rare \\citep[e.g.,][]{2008ApJ...687..133I}. \\citet{2006MNRAS.371.1559G}\nproposed a method to search for AGNs with a low-metallicity NLR, that utilizes \nan optical emission-line diagnostic diagram which consists of the flux ratios of\n[N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 and\n[O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861. This diagnostic diagram\nwas originally investigated for classifying emission-line galaxies into star-forming\ngalaxies and Seyfert 2 galaxies (BPT diagram, \\citealt{1981PASP...93....5B}).\n\\citet[hereafter Ke01]{2001ApJ...556..121K} established the ``maximum'' \nstarburst line in the BPT diagram by combining stellar population synthesis \nmodels and photoionization models. On the other hand, \\citet[hereafter Ka03]\n{2003MNRAS.346.1055K} derived empirical classification criteria for star-forming \ngalaxies while \\citet[hereafter Ke06]{2006MNRAS.372..961K} derived empirical \nclassification criteria for low-ionization nuclear emission-line regions (LINERs;\n\\citealt{1980A&A....87..152H}), using emission-line data taken from the database \nof Sloan Digital Sky Survey (SDSS; \\citealt{2000AJ....120.1579Y}).\n\n\\citet{2006MNRAS.371.1559G} pointed out that AGNs with a low-metallicity \nNLRs (i.e., characterized by the solar or sub-solar metallicity) should have a flux ratio of \n[O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861 as high as usual AGNs\n($\\sim 10^{0.5}-10^1$) but have an intermediate flux ratio of\n[N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 between usual AGNs and\nlow-mass (i.e., low-metallicity) star-forming galaxies ($\\sim 10^{-1}-10^{-0.5}$). \nThis is because the nitrogen relative abundance is in proportion to the metallicity \ndue to its nature as a secondary element \\citep[e.g.,][]{1998ApJ...497L...1V}. In \nthe BPT diagram, there are only few objects located at the region characterized \nby a high flux ratio of [O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861 and \nan intermediate flux ratio of [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563\n(hereafter ``BPT valley''; see Figure~\\ref{BPT_diagram}). \n\\citet{2006MNRAS.371.1559G} specifically focused on AGNs with a low-mass\nhost galaxy (i.e., $M_{\\rm host} < 10^{10} M_\\odot$), and then they selected \nlow-metallicity AGNs using another diagnostic\ndiagram that consists of [N~{\\sc ii}]$\\lambda$6584\/[O~{\\sc ii}]$\\lambda$3727 and\n[O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 flux ratios. However, it is not\nclear whether low-metallicity AGNs should be always found in a sample of AGNs with a\nlow-mass host galaxy. Also, the method adopted by \\citet{2006MNRAS.371.1559G}\nrequires a wide wavelength coverage ($\\lambda_{\\rm rest} \\sim 3700-6600$ \\AA), \nthat is not convenient for future applications to expand the search of low-metallicity\nAGNs toward the high-redshift Universe. \n\nTherefore, we focus on BPT-valley selection (requiring a moderately narrow \nwavelength coverage; $\\lambda_{\\rm rest} \\sim 4800-6600$ \\AA) to select \nlow-metallicity AGNs without any host-mass cut.\nHowever, there is a potentially serious problem in the BPT-valley selection for \nidentifying low-metallicity AGNs. \nThat is, not only low-metallicity AGNs are located in the BPT valley. \nAs \\citet{2013ApJ...774..100K} showed, star-forming galaxies with \na very hard radiation field or high-density H~{\\sc ii} regions are expected \nto be seen in the BPT valley (see also, e.g., \\citealt[][]{2014ApJ...787..120S}). \nAlso, star-forming galaxies with a high ionization parameter \n\\citep[e.g.,][]{2014ApJ...795..165S, 2015PASJ...67...80H}, a high nitrogen-to-oxygen \nabundance ratio (N\/O; e.g., \\citealt[][]{2014ApJ...785..153M, 2015ApJ...801...88S, 2015PASJ...67..102Y, \n2016arXiv160503436K}), \nor shocks \\citep[e.g.,][]{2014ApJ...781...21N} are also expected to be seen in the BPT valley. \nNot only star-forming galaxies, AGNs with a high electron density or high ionization parameter \n(i.e., not characterized by a low metallicity) could be also seen in the BPT valley \n\\citep[e.g.,][]{2001ApJ...546..744N}. \nTherefore, it is not completely clear whether the BPT-valley objects are really low-metallicity AGNs \nand whether the BPT diagram is a useful tool to search for low-metallicity AGNs. \nThis problem prevents us from selecting chemically-young AGNs observationally.\n\nIn this paper, we investigate the optical spectra of BPT-valley objects for \nexamining whether most of emission-line galaxies at the BPT valley are really \nlow-metallicity AGNs or not. Through this examination, it will be tested whether \nthe optical BPT diagram is an efficient and appropriate method to search for \nlow-metallicity AGNs. In Section 2, we present our selection procedure of the \nBPT-valley sample. In Section 3, we show how we identify BPT-valley AGNs to \navoid contaminating star-forming galaxies at the BPT valley. In Section 4, we \ninvestigate gas properties of the selected BPT-valley AGNs such as electron \ndensity and ionization parameter, for examining whether the BPT-valley AGNs\nare characterized by a low metallicity or not. \nIn Section 5, we disccus physical properties of the BPT-valley AGNs.\nSection 6 describes the summary of this work.\n\n\n\\section{Sample}\n\nIn order to select the BPT-valley objects, we use Max-Planck-Institute for \nAstrophysics (MPA)-Johns Hopkins University (JHU) SDSS Data Release 7 \n(\\citealt{2009ApJS..182..543A}) galaxy catalog\\footnote[1]\n{http:\/\/www.mpa-garching.mpg.de\/SDSS\/}. \nThe MPA-JHU DR7 catalog of spectral measurements contains various\nspectral properties such as emission-line fluxes and their errors, based on\nthe analysis for 927,552 objects without showing dominant broad Balmer lines \n(i.e., star-forming galaxies, composite galaxies, LINERs, and type-2 Seyfert galaxies) \nin the SDSS DR7.  \nOur sample selection is based on the following procedure (the flow chart of our \nsample selection process is shown in Figure~\\ref{flow_chart}).\n\nFirst, we select the initial sample according to the following criteria. \nWe require the reliable redshift measurement (i.e., $z_{\\rm warning} = 0$) and also $z>$ 0.02. \nThis redshift limit is required to cover [O~{\\sc ii}]$\\lambda$3727.\nThis results in 906,761 galaxies.\nThen we require a signal-to-noise ratio (S\/N) $>$ 3 for some key emission lines, i.e., \nH$\\beta$$\\lambda$4861, [O~{\\sc iii}]$\\lambda$5007, \n[O~{\\sc i}]$\\lambda$6300, H$\\alpha$$\\lambda$6563, \n[N~{\\sc ii}]$\\lambda$6584 and [S~{\\sc ii}]$\\lambda \\lambda$6717, 31 (212,866 galaxies).\n\nNext, we classify these 212,866 galaxies and extract the BPT-valley sample \naccording to the following steps. \n\\begin{enumerate}\n\\item \n   Using the \\citetalias{2003MNRAS.346.1055K} empirical line,\n   \\begin{eqnarray}\n   \\log \\left( \\frac{\\rm [O\\ {\\scriptstyle III}]}{\\rm H\\beta} \\right) \n   > \\frac{0.61}{\\rm log ([N\\ {\\scriptstyle II}]\/H\\alpha) -0.05}+1.3,\n   \\end{eqnarray}\n   for removing usual star-forming galaxies (56,217 galaxies).\n\\item \n   Using the \\citetalias{2001ApJ...556..121K} maximum starburst line,\n   \\begin{eqnarray}\n   \\log \\left( \\frac{\\rm [O\\ {\\scriptstyle III}]}{\\rm H\\beta} \\right) > \n   \\frac{0.61}{\\rm log ([N\\ {\\scriptstyle II}]\/H\\alpha) -0.47}+1.19,\n   \\end{eqnarray}\n   for removing so-called composite galaxies (22,865 galaxies).\n\\item \n   Using the \\citetalias{2006MNRAS.372..961K} empirical criterion,\n   \\begin{eqnarray}\n   \\log \\left( \\frac{\\rm [O\\ {\\scriptstyle III}]}{\\rm H\\beta} \\right) > \n   1.36\\log \\left( \\frac{\\rm [O\\ {\\scriptstyle I}]}{\\rm H\\alpha} \\right) + 1.4, \n   \\end{eqnarray}\n   for obtaining Seyfert sample by removing LINERs (14,253 galaxies).\n\\item  \n   Adopting the following criterion,\n   \\begin{eqnarray}\n   \\log \\left( \\frac{\\rm [N\\ {\\scriptstyle II}]}{\\rm H\\alpha} \\right) < -0.5,\n   \\label{BPT_valley}\n   \\end{eqnarray}\n   for finally selecting the BPT-valley sample (71 galaxies).\n\\end{enumerate}\nNote that 1 object in the 71 BPT-valley objects was observed twice and \nduplicated in the final sample, i.e., the final BPT-valley sample consists of 70 objects. \nThe BPT-valley criterion (Equation~\\ref{BPT_valley}) is determined empirically, \nby taking account of the frequency distribution of the [N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$ \nflux ratio of Seyfert galaxies. \nFigure~\\ref{NII_Ha} shows the [N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$ frequency distribution of \nSeyfert galaxies, \nwhere the average and standard deviation of the logarithmic [N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$ \nflux ratio are $-0.058$ and $0.145$, respectively.\nAccordingly, the $3\\ \\sigma$ bounding from the average value is $-0.493$, \nand therefore we adopt the threshold to categorize BPT-valley objects as described \nby Equation~\\ref{BPT_valley}.\nFigure~\\ref{BPT_diagram} shows the finally selected 70 BPT-valley objects \nin the BPT diagram that consists of\n[N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 versus \n[O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861.\nTable 1 shows the basic properties of the selected BPT-valley objects.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8.5cm]{figure_1.eps}\n \\caption{Flow chart of the selection of BPT-valley objects. Numbers shown in the chart denote \n the numbers of objects at each selection stage. \n The number shown in the parenthesis denotes the number of objects after removing the duplication.}\n \\label{flow_chart} \n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8.5cm]{figure_2.eps}\n \\caption{\n Histogram of the [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 flux ratio of Seyfert galaxies.\n Dashed line denotes the average of log ([N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$) $= -0.058$, \n while dotted line denotes the threshold of log ([N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$) $= -0.5$ \n to select BPT-valley objects.\n }\n \\label{NII_Ha} \n\\end{figure}\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8.5cm]{figure_3s.eps}\n \\caption{The BPT diagram ([N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 versus \n [O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861), showing the BPT-valley sample \n (red square) among the SDSS DR7 emission-line objects. \n The green solid line is the Ke01 extreme starburst criterion, \n while the red solid line denotes the criterion for separating star-forming galaxies and \n composite galaxies (Ka03). \n The violet solid line is the BPT valley criterion which is defined in this paper.\n The numbers of various galaxy populations are shown in the parenthesis \n in the lower-left box.}\n \\label{BPT_diagram} \n\\end{figure}\n\n\n\n\\begin{deluxetable*}{ccrrrrrrrrc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{The BPT-valley sample}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{ID} & \\colhead{SDSS Name} & \\colhead{Plate} & \\colhead{MJD} & \\colhead{Fiber} &\n\\colhead{$z$} & \\colhead{H$\\beta$$\\lambda$4861} & \\colhead{[O~{\\sc iii}]$\\lambda$5007} &\n\\colhead{[O~{\\sc i}]$\\lambda$6300} & \\colhead{H$\\alpha$$\\lambda$6563} &\n\\colhead{[N~{\\sc ii}]$\\lambda$6584}\\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{(5)} & \n\\colhead{(6)} & \\colhead{(7)} & \\colhead{(8)} & \\colhead{(9)} & \\colhead{(10)} & \n\\colhead{(11)}\n}\n\\startdata\n1.......... & SDSS J102310.97$-$002810.8 & 0272 & 51941 & 0238 & 0.11274 & 253.97  & 2279.29 & 125.19 & 881.10   & 235.20  \\\\\n2.......... & SDSS J111006.26$-$010116.5 & 0278 & 51900 & 0096 & 0.10949 & 382.92  & 1837.07 & 129.44 & 1484.61  & 438.76  \\\\\n3.......... & SDSS J230321.73+011056.4 & 0380 & 51792 & 0565 & 0.18136 & 235.38  & 1692.71 & 12.29  & 816.35   & 247.33  \\\\\n4.......... & SDSS J024825.26$-$002541.4 & 0409 & 51871 & 0150 & 0.02467 & 10.35   & 41.33   & 4.71   & 47.65    & 14.92   \\\\\n5.......... & SDSS J073506.37+393300.8 & 0432 & 51884 & 0316 & 0.03479 & 29.46   & 163.57  & 8.55   & 109.51   & 21.85   \\\\\n6.......... & SDSS J023310.78$-$074813.4 & 0455 & 51909 & 0388 & 0.03097 & 97.71   & 707.20  & 12.83  & 433.29   & 104.88  \\\\\n7.......... & SDSS J092907.78+002637.3 & 0475 & 51965 & 0205 & 0.11732 & 259.86  & 2602.18 & 56.12  & 1173.36  & 274.70  \\\\\n8.......... & SDSS J090613.76+561015.1 & 0483 & 51902 & 0016 & 0.04668 & 188.36  & 1144.93 & 73.96  & 621.46   & 189.46  \\\\\n9.......... & SDSS J144328.78+044022.0 & 0587 & 52026 & 0374 & 0.11411 & 36.11   & 300.13  & 3.37   & 137.53   & 32.98   \\\\\n10......... & SDSS J114825.71+643545.0 & 0598 & 52316 & 0189 & 0.04169 & 349.08  & 1347.75 & 60.17  & 1302.50  & 392.24  \\\\\n11......... & SDSS J213439.57$-$071641.9 & 0641 & 52199 & 0487 & 0.06377 & 215.60  & 2079.35 & 78.83  & 800.93   & 182.94  \\\\\n12......... & SDSS J001050.35$-$010257.4 & 0686 & 52519 & 0020 & 0.11299 & 223.89  & 1669.57 & 109.74 & 940.89   & 275.90  \\\\\n13......... & SDSS J124738.52+621243.1 & 0781 & 52373 & 0076 & 0.12112 & 171.82  & 915.80  & 33.03  & 697.15   & 162.91  \\\\\n14......... & SDSS J131659.37+035319.9 & 0851 & 52376 & 0219 & 0.04541 & 127.05  & 1399.96 & 80.91  & 698.40   & 136.73  \\\\\n15......... & SDSS J115908.55+525823.1 & 0881 & 52368 & 0623 & 0.06644 & 392.77  & 2812.16 & 52.13  & 1218.45  & 379.53  \\\\\n16......... & SDSS J104632.21+543559.7 & 0906 & 52368 & 0169 & 0.14475 & 83.71   & 938.64  & 19.52  & 375.92   & 101.46  \\\\\n17......... & SDSS J104600.36+061632.0 & 1000 & 52643 & 0035 & 0.18447 & 146.67  & 637.74  & 31.28  & 586.86   & 177.72  \\\\\n18......... & SDSS J101945.65+520608.6 & 1008 & 52707 & 0378 & 0.06492 & 29.60   & 115.17  & 6.75   & 92.29    & 28.00   \\\\\n19......... & SDSS J205111.11+000913.2 & 1023 & 52818 & 0393 & 0.06644 & 7.37    & 44.11   & 4.92   & 37.11    & 9.68    \\\\\n20......... & SDSS J214930.43+010509.4 & 1031 & 53172 & 0369 & 0.11399 & 28.14   & 210.35  & 3.43   & 109.29   & 31.13   \\\\\n21......... & SDSS J081653.27+285423.1 & 1206 & 52670 & 0530 & 0.05740 & 246.45  & 1610.19 & 131.99 & 1145.34  & 203.60  \\\\\n22......... & SDSS J095319.42+422912.2 & 1217 & 52672 & 0349 & 0.22343 & 207.67  & 1835.79 & 83.29  & 698.87   & 115.09  \\\\\n23......... & SDSS J110504.94+101623.5 & 1221 & 52751 & 0359 & 0.02076 & 104.02  & 466.39  & 29.14  & 481.20   & 151.62  \\\\\n24......... & SDSS J114440.53+102429.3 & 1226 & 52734 & 0435 & 0.12688 & 51.67   & 304.73  & 39.93  & 241.38   & 67.83   \\\\\n25......... & SDSS J124110.10+104143.7 & 1233 & 52734 & 0611 & 0.15613 & 214.49  & 1004.84 & 29.37  & 703.65   & 188.71  \\\\\n26......... & SDSS J092620.42+352250.3 & 1274 & 52995 & 0147 & 0.24729 & 118.98  & 868.18  & 81.24  & 513.89   & 75.74   \\\\\n27......... & SDSS J131756.07+491531.3 & 1282 & 52759 & 0390 & 0.09231 & 215.30  & 2050.51 & 105.88 & 711.65   & 176.30  \\\\\n28......... & SDSS J090107.41+085459.2 & 1300 & 52973 & 0335 & 0.08380 & 172.61  & 1116.69 & 153.30 & 822.23   & 210.37  \\\\\n29......... & SDSS J120134.05+581421.1 & 1313 & 52790 & 0527 & 0.04636 & 28.83   & 136.20  & 5.78   & 87.43    & 20.49   \\\\\n30......... & SDSS J152723.47+334919.1 & 1354 & 52814 & 0044 & 0.09116 & 44.63   & 173.46  & 24.18  & 254.04   & 79.05   \\\\\n31......... & SDSS J112314.89+431208.7 & 1365 & 53062 & 0119 & 0.08005 & 56.52   & 226.87  & 33.20  & 183.08   & 52.19   \\\\\n32......... & SDSS J152328.09+313655.6 & 1387 & 53118 & 0210 & 0.06850 & 382.47  & 1707.84 & 67.32  & 1300.98  & 322.89  \\\\\n33......... & SDSS J120900.89+422830.9 & 1448 & 53120 & 0075 & 0.02364 & 100.07  & 614.24  & 24.50  & 320.70   & 61.56   \\\\\n34......... & SDSS J121839.40+470627.6 & 1451 & 53117 & 0190 & 0.09389 & 478.35  & 5046.81 & 101.68 & 1861.68  & 380.45  \\\\\n35......... & SDSS J005231.29$-$011525.2 & 1496 & 52883 & 0089 & 0.13485 & 251.99  & 2400.52 & 55.68  & 826.34   & 254.44  \\\\\n36......... & SDSS J011341.11+010608.5 & 1499 & 53001 & 0522 & 0.28090 & 191.99  & 2118.06 & 38.27  & 779.72   & 162.79  \\\\\n37......... & SDSS J001901.52+003931.8 & 1542 & 53734 & 0375 & 0.09669 & 58.06   & 296.69  & 15.99  & 242.69   & 58.75   \\\\\n38......... & SDSS J034019.39+002530.6 & 1632 & 52996 & 0467 & 0.35296 & 40.58   & 375.40  & 10.78  & 167.65   & 46.21   \\\\\n39......... & SDSS J032224.64+401119.8 & 1666 & 52991 & 0048 & 0.02608 & 121.08  & 1388.24 & 50.70  & 428.76   & 130.77  \\\\\n40......... & SDSS J135855.82+493414.1 & 1670 & 53438 & 0061 & 0.11592 & 56.67   & 385.56  & 14.74  & 189.76   & 50.29   \\\\\n41......... & SDSS J160452.78+344540.4 & 1682 & 53173 & 0201 & 0.05493 & 87.33   & 437.20  & 37.34  & 364.99   & 111.94  \\\\\n42......... & SDSS J132011.71+125940.9 & 1698 & 53146 & 0327 & 0.11398 & 25.43   & 174.93  & 4.92   & 92.99    & 24.68   \\\\\n43......... & SDSS J143523.42+100704.1 & 1711 & 53535 & 0306 & 0.03122 & 128.18  & 530.61  & 15.31  & 473.30   & 123.16  \\\\\n44......... & SDSS J072637.94+394557.8 & 1733 & 53047 & 0326 & 0.11141 & 505.82  & 3357.26 & 23.23  & 1744.03  & 120.42  \\\\\n45......... & SDSS J095914.76+125916.4 & 1744 & 53055 & 0385 & 0.03432 & 1298.59 & 8418.94 & 361.09 & 4396.86  & 1088.11 \\\\\n46......... & SDSS J113714.22+145917.2 & 1755 & 53386 & 0463 & 0.03484 & 74.19   & 364.47  & 25.99  & 276.40   & 77.15   \\\\\n47......... & SDSS J120847.79+135906.7 & 1764 & 53467 & 0013 & 0.29030 & 136.33  & 659.31  & 28.11  & 554.94   & 137.92  \\\\\n48......... & SDSS J135429.05+132757.2 & 1777 & 53857 & 0076 & 0.06332 & 312.80  & 3422.56 & 111.50 & 1005.37  & 306.94  \\\\\n49......... & SDSS J130431.99+061616.7 & 1794 & 54504 & 0046 & 0.06283 & 184.84  & 1202.46 & 41.38  & 702.29   & 217.24  \\\\\n50......... & SDSS J134316.52+101440.1 & 1804 & 53886 & 0433 & 0.08132 & 186.67  & 960.51  & 170.90 & 635.68   & 198.21  \\\\\n51......... & SDSS J160032.89+052608.8 & 1822 & 53172 & 0012 & 0.11653 & 281.02  & 1630.88 & 69.22  & 1204.67  & 280.09  \\\\\n52......... & SDSS J081212.84+541539.8 & 1871 & 53384 & 0060 & 0.04417 & 93.10   & 809.73  & 34.28  & 326.93   & 94.82   \\\\\n53......... & SDSS J084038.99+245101.6 & 1931 & 53358 & 0396 & 0.04334 & 137.37  & 770.11  & 39.75  & 579.80   & 151.50  \\\\\n54......... & SDSS J122451.88+360535.4 & 2003 & 53442 & 0112 & 0.15094 & 25.95   & 148.22  & 11.11  & 126.62   & 35.77   \\\\\n55......... & SDSS J134237.37+273251.3 & 2017 & 53474 & 0127 & 0.04947 & 12.67   & 97.69   & 10.27  & 60.02    & 17.32   \\\\\n56......... & SDSS J140952.03+244334.6 & 2128 & 53800 & 0358 & 0.05215 & 45.42   & 220.76  & 12.29  & 198.30   & 41.19   \\\\\n57......... & SDSS J142535.21+314027.1 & 2129 & 54252 & 0618 & 0.03324 & 91.61   & 362.11  & 42.31  & 323.51   & 95.49   \\\\\n58......... & SDSS J145505.97+211121.1 & 2148 & 54526 & 0122 & 0.06751 & 82.30   & 441.80  & 12.31  & 437.94   & 126.61  \\\\\n59......... & SDSS J083200.51+191205.8 & 2275 & 53709 & 0472 & 0.03753 & 549.64  & 6069.28 & 42.46  & 15422.57 & 419.58  \\\\\n60......... & SDSS J103731.01+280626.9 & 2356 & 53786 & 0468 & 0.04263 & 54.57   & 447.10  & 24.55  & 216.95   & 65.48   \\\\\n61......... & SDSS J104403.52+282628.3 & 2356 & 53786 & 0618 & 0.16286 & 225.46  & 1047.00 & 17.43  & 794.20   & 193.30  \\\\\n62......... & SDSS J104724.40+204433.5 & 2478 & 54097 & 0541 & 0.26515 & 102.51  & 751.70  & 37.17  & 391.58   & 66.16   \\\\\n63......... & SDSS J160635.22+142201.9 & 2524 & 54568 & 0498 & 0.03245 & 162.66  & 621.83  & 41.72  & 517.08   & 160.44  \\\\\n64......... & SDSS J171901.28+643830.8 & 2561 & 54597 & 0345 & 0.08954 & 152.42  & 709.97  & 21.69  & 586.95   & 174.32  \\\\\n65......... & SDSS J084658.44+111457.5 & 2574 & 54084 & 0382 & 0.06296 & 130.82  & 638.15  & 41.04  & 557.91   & 161.29  \\\\\n66......... & SDSS J095745.49+152350.6 & 2584 & 54153 & 0442 & 0.05183 & 96.42   & 702.55  & 24.58  & 514.46   & 117.37  \\\\\n67......... & SDSS J133014.91+242153.9 & 2665 & 54232 & 0388 & 0.07151 & 50.49   & 244.84  & 15.74  & 284.46   & 76.98   \\\\\n68......... & SDSS J135007.07+164227.2 & 2742 & 54233 & 0551 & 0.13043 & 99.01   & 903.05  & 38.82  & 495.52   & 137.00  \\\\\n69......... & SDSS J153941.67+171421.9 & 2795 & 54563 & 0509 & 0.04583 & 157.66  & 758.85  & 14.76  & 500.33   & 119.40  \\\\\n70......... & SDSS J143730.46+620649.4 & 2947 & 54533 & 0227 & 0.21862 & 66.97  & 290.54  & 22.39  & 219.25   & 65.39   \n\n\\enddata\n\\tablecomments{Col. (1): Identification number assigned in this paper. \nCol. (2): Object name. \nCol. (3)--(5): Plate-MJD-Fiber ID in the SDSS observation for analyzed spectra. \nCol. (6): Redshift measured by the SDSS pipeline.\nCol. (7)--(11): Emission-line fluxes in units of $10^{-17}$ $\\rm erg\\ s^{-1}\\ cm^{-2}$.\n}\n\\end{deluxetable*}\n\n\n\n\\section{Selection of secure-AGN sample}\n\nAs described in Section 1, the BPT-valley sample potentially includes star-forming galaxies with \nspecial gas properties, not only AGNs. \nThus we first select objects showing secure evidence of the AGN from the BPT-valley sample. \nSpecifically, we regard objects showing at least one of the following two features \nin their SDSS spectra as secure AGNs; (1) a broad H$\\alpha$$\\lambda$6563 emission, \nand (2) a He~{\\sc ii}$\\lambda 4686$ emission line. \nDetails of the selection procedure of secure AGNs are given below.\n\n\n\\subsection{Broad H$\\alpha$$\\lambda$6563 emission line}\n\nThe velocity profile of recombination lines is a powerful tool to examine \nthe presence of AGNs, since star-forming galaxies never show a velocity \nwidth wider than $\\sim$1000 km~s$^{-1}$ in full-width at half maximum (FWHM). \nGenerally the optical spectra of type-1 AGNs show broad permitted lines whose \nvelocity width is $\\gtrsim$ 2000 km~s$^{-1}$ emitted from BLRs. \nThe origin of recombination lines with $\\rm FWHM \\sim 1000 - 2000$ km~s$^{-1}$ is \nnot very clear, since such lines may arise at BLRs in so-called narrow-line \nSeyfert 1 galaxies (NLS1s; e.g., Osterbrock \\& Pogge 1985) or \nat NLRs in type-2 AGNs with a relatively large velocity width \n(such as NGC 1068 and NGC 1275; see, e.g., \\citealt{1984ApJ...281..525H, 2000ApJ...532L.101C}). \nHowever, in either case, the detection of recombination lines with \n$\\rm FWHM > 1000$ km~s$^{-1}$ strongly suggests the presence of AGNs. \nTherefore we search for the broad H$\\alpha$$\\lambda$6563 component \nin the optical spectrum of the BPT-valley objects. Here we do not search for \nthe broad component of the H$\\beta$$\\lambda$4861 emission, \nsince it is intrinsically fainter than that of the H$\\alpha$$\\lambda$6563 emission \nand it is sometimes affected significantly by the Fe~{\\sc ii} multiplet emission\n\\citep[e.g.,][]{2001AJ....122..549V}.\n\nWe use an IRAF routine {\\tt specfit} \\citep{1994ASPC...61..437K} to find the broad \nH$\\alpha$$\\lambda$6563 component.\nSpecifically, we fit the SDSS optical spectrum of the BPT-valley objects \nin the range of $\\lambda_{\\rm rest} = 6200-6800\\ \\rm \\AA$ with and \nwithout the broad H$\\alpha$$\\lambda$6563 component, and examine \nwhether the addition of the broad component improves the spectral fit significantly. \nThe details of the fitting procedure are as follows. \nFirst, we fit the optical spectrum with a linear continuum component and \nsingle-Gaussian emission-line components for [O~{\\sc i}]$\\lambda$6300, \n[O~{\\sc i}]$\\lambda$6363, [N~{\\sc ii}]$\\lambda$6548, H$\\alpha$$\\lambda$6563, \n[N~{\\sc ii}]$\\lambda$6584, [S~{\\sc ii}]$\\lambda$6717, and [S~{\\sc ii}]$\\lambda$6731 \n(hereafter ``nobroad fitting''). \nHere we assume that the velocity width of all emission lines is the same, \nand the relative separation of the emission lines is fixed to be the same as \nthat of their laboratory wavelengths. \nThe flux ratios of [O~{\\sc i}]$\\lambda$6300 to [O~{\\sc i}]$\\lambda$6363 and \n[N~{\\sc ii}]$\\lambda$6584 to [N~{\\sc ii}]$\\lambda$6548 are fixed to be 3.00 and 2.96 \nrespectively \\citep{1983IAUS..103..143M}, and the flux ratios among the remaining \nemission lines are kept to be free. \nThen, we add a broad component for the H$\\alpha$$\\lambda$6563 emission to \nthe nobroad fit, where the flux, wavelength center and width of this additional \ncomponent are kept to be free (hereafter ``broad fitting''). \nHere we recognize that the additional broad component significantly improves \nthe fit by the following criterion:\n\\begin{eqnarray}\n\\frac{\\tilde{\\chi}^2_{\\rm nobroad}-\\tilde{\\chi}^2_{\\rm broad}}{\\tilde{\\chi}^2_{\\rm nobroad}}>0.4,\n\\label{chi}\n\\end{eqnarray}\nwhere $\\tilde{\\chi}^2_{\\rm nobroad}$ and $\\tilde{\\chi}^2_{\\rm broad}$ are the reduced \nchi-square of the nobroad fitting and broad fitting, respectively. \nNote that the threshold, 0.4, is determined empirically, so that the result \nbecomes consistent with the visual inspection. \nAs a result, 13 BPT-valley objects with a broad component are identified from the 70 BPT-valley objects.\nFigures~\\ref{broad_HeII_1} and~\\ref{broad_noHeII} show the SDSS spectrum with the best-fit result \nfor the BPT-valley objects with a broad H$\\alpha$$\\lambda$6563 component. \nFigure~\\ref{ID_48} shows an example of objects (ID = 48) whose fitting result does not satisfy the \ncriterion defined by Equation~\\ref{chi} (for this case, the improvement of the fit is slightly less than \nthe threshold, 0.32).\nNote that we regard object ID = 8 as an object with a broad H$\\alpha$ component, \nthough the FWHM of the broad H$\\alpha$ component is less than \n$1000\\ {\\rm km\\ s^{-1}}$ (Figure~\\ref{ID_8}). \nThis is because this object shows [Fe~{\\sc vii}]$\\lambda$6087 and [Fe~{\\sc x}]$\\lambda$6374 \nlines, that are seen only when the AGN presents. Note that such high-ionization forbidden \nemission lines are preferentially seen in type-1 AGNs \n\\citep[e.g.,][]{1998ApJ...497L...9M, 2000AJ....119.2605N}. \nNote that the [Fe~{\\sc vii}]$\\lambda$6087 line is seen in 8 objects \nwhile [Fe~{\\sc x}]$\\lambda$6374 line is seen in 3 objects \n(including ID = 8, note that 2 objects in addition to ID = 8 show both [Fe~{\\sc vii}]$\\lambda$6087 \nand [Fe~{\\sc x}]$\\lambda$6374). \nThe spectral properties of the BPT-valley objects with a broad H$\\alpha$$\\lambda$6583 \ncomponent are summarized in Table 2.\nOnly 1 BPT-valley object (ID = 25) shows the broad H$\\beta$ component among  \nthe 13 BPT-valley objects showing a broad H$\\alpha$ component (see Figure~\\ref{broad_HeII_1}).\n\n\\subsection{He~{\\sc ii}$\\lambda$4686 emission line}\nThe presence of a He~{\\sc ii}$\\lambda4686$ emission line indicates the existence of \nthe hard ionizing radiation since the ionization potential for \n$\\rm He^{+}$ is 54.4 eV. \nThis hard radiation is naturally produced by AGNs. \nTherefore, the He~{\\sc ii}$\\lambda4686$ emission line is a good indicator of AGNs. \nWe examine whether the SDSS optical spectrum of the BPT-valley objects show \nthe He~{\\sc ii}$\\lambda$4686 line by the visual inspection, \nsince the He~{\\sc ii}$\\lambda$4686 information is not given in the MPA-JHU database. \nAs a result, 38 BPT-valley objects with the He~{\\sc ii} emission line are identified from \nthe 70 BPT-valley objects. \nSome of the SDSS spectra of BPT-valley objects with the He~{\\sc ii} detection are shown \nin Figures~\\ref{broad_HeII_1}, while those without the He~{\\sc ii} detection are shown \nin Figure~\\ref{broad_noHeII}.\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=17cm]{figure_4.eps}\n \\vspace{2mm}\n \\caption{Spectra of the BPT-valley objects showing a broad H$\\alpha$ component and He~{\\sc ii} emission. \n Best-fit models are plotted in red, while the narrow-H$\\alpha$+[N~{\\sc ii}] Gaussian components, \n broad H$\\alpha$ component, and continuum are plotted in green, violet, and orange, respectively. \n Residuals are plotted in blue. Reduced chi-square values are given at the upper-right side in the \n right panels (the value before adding the broad $\\alpha$ component is given in the parenthesis).\n }\n \\label{broad_HeII_1} \n \\end{figure*}\n\n\\setcounter{figure}{3}\n\\begin{figure*}[t]\n \\centering\n  \\includegraphics[width=17cm]{figure_5.eps}\n\\caption{(Continued)}\n \\label{broad_HeII_2} \n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=17.cm]{figure_6.eps}\n \\vspace{2mm}\n \\caption{Same as Figure~\\ref{broad_HeII_1} but for objects showing the broad H$\\alpha$ emission \n but without the He~{\\sc ii} line.}\n \\label{broad_noHeII} \n \\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=17.cm]{figure_7.eps}\n \\vspace{2mm}\n \\caption{\n Same as Figure~\\ref{broad_HeII_1} but for an example of objects whose fitting result \n does not satisfy Equation~\\ref{chi}.\n }\n \\label{ID_48} \n \\end{figure*}\n\n\\begin{figure}[h]\n \\centering\n \\hspace{0cm}\n \\vspace{0.cm}\n \\includegraphics[width=8.5cm]{figure_8.eps}\n \\vspace{0cm}\n \\caption{Same as Figure~\\ref{broad_HeII_1}, but for objects ID = 8, whose FWHM of broad \n H$\\alpha$ component is less than $1500\\ {\\rm km\\ s^{-1}}$ (see Table 2). \n High-ionization forbidden emission lines of [Fe~{\\sc vii}]$\\lambda$6087 and \n [Fe~{\\sc x}]$\\lambda$6374 are clearly detected.\n Note that fitting range is from 6200 to 6800 $\\rm \\AA$ in the restframe.\n Continuum fitting is extrapolated below 6200 $\\rm \\AA$.}\n \\label{ID_8} \n\\end{figure}\n\n\\subsection{Classification result of the BPT-valley sample}\n\n The results of the classification of the BPT-valley objects are summarized in Table 3. \n Among the 70 BPT-valley objects, 8 objects show both broad H$\\alpha$ component and \n He~{\\sc ii} emission line, that are now confirmed to be AGNs. \n There are 5 objects showing the broad H$\\alpha$ component but without He~{\\sc ii} emission line, \n that are also regarded as AGNs. \n The non-detection of the He~{\\sc ii} line is likely due to insufficient signal-to-noise ratio, \n since the He~{\\sc ii} line is very weak. In addition, 30 objects show the He~{\\sc ii} line \n but without broad H$\\alpha$ component, that are thought to be typical type-2 AGNs. \n Here we should mention that the stellar absorption lines (mainly H$\\alpha$) are not \n considered in our fitting procedure. \n Though the stellar H$\\alpha$ absorption line could impact the narrow component of the H$\\alpha$ \n emission, the absorption effect is negligible for examining the presence of the broad H$\\alpha$ \n component. \n This is because the equivalent width of the detected broad H$\\alpha$ component is higher than \n $20\\ {\\rm \\AA}$ (the median value of $\\rm EW_{rest}(H\\alpha)_b$ is $44.74\\ {\\rm \\AA}$, Table 2) \n while the typical equivalent width of the stellar H$\\alpha$ absorption is $\\sim 2-3\\ {\\rm \\AA}$ \n in nearby galaxies \\citep[e.g.,][]{1997ApJS..112..315H}.\n Note that the detected He~{\\sc ii} line is not caused by Wolf-Rayet stars, \n because the typical velocity width of the detected He~{\\sc ii} line is not \n broad ($\\lesssim 1000\\ \\rm km\\ s^{-1}$). \n Therefore, at least 43 among the BPT-valley objects are regarded as AGNs. \n There may be some additional AGNs in the remaining 27 objects, possibly \n owing to insufficient S\/N to detect any AGN indicators in their spectra. \n Instead, some of those 27 objects could be non-AGNs, i.e., star-forming \n galaxies with a relatively high N\/O ratio or fast shocks. \n We do not discuss further about those 27 objects since the main interests \n of this work are on the BPT-valley AGN sample. \n Accordingly, we conclude that at least 43 objects of the BPT-valley sample \n (or $\\sim$ 60\\%, but probably more) are confirmed to be AGNs.\n  \n As described in Section 3.1, at least one of the [Fe~{\\sc vii}]$\\lambda$6087 and \n [Fe~{\\sc x}]$\\lambda$6374 lines are seen in 9 BPT-valley objects. \n Interestingly, a large fraction of objects showing both the broad H$\\alpha$ component and He~{\\sc ii} \n emission show such high-ionization iron lines (5 among 8 objects). \n On the other hand, objects showing neither the broad H$\\alpha$ component nor He~{\\sc ii} emission \n never shows those high-ionization iron lines. \n Then, a few objects in the remaining two classes show high-ionization iron lines (4 among 35 objects). \n This may infer that our classification is well tracing the presence of the AGN, \n but the absence of high-ionization iron lines could be simply due to a low S\/N ratio of the spectra. \n \n Figure~\\ref{BPT_classification} shows how various populations of galaxies classified in this work \n are populated in the BPT diagram. \n There are no significant segregation except for two BPT-valley objects whose \n [N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$ flux ratio is very low, $< 0.1$. \n Both of these two galaxies show no broad H$\\alpha$ component nor He~{\\sc ii} line, \n which is consistent with the idea that these two objects are not low-metallicity AGNs \n but somewhat extreme low-metallicity galaxies, characterized probably by a very \n high ionization parameter and\/or very hard ionization radiation.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=8.5cm]{figure_9s.eps}\n\\caption{The BPT diagram ([N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 versus \n [O~{\\sc iii}]$\\lambda$5007\/H$\\beta$$\\lambda$4861) with the classification result of the \n BPT valley sample among the SDSS DR7 emission-line objects. \n The numbers of various galaxy populations are shown in the parenthesis in the lower-right box.\n}\n\\label{BPT_classification} \n\\end{figure}\n\n\n\\begin{deluxetable*}{lrrrrr}\n\\tablewidth{0pt}\n\\tablecolumns{6} \n\\tablecaption{Broad-line AGNs in the BPT valley}\n\\tablehead{\n\\colhead{ID} & \\colhead{$f(\\rm H\\alpha)_n$} & \\colhead{$f(\\rm H\\alpha)_b$} & \\colhead{$\\rm FWHM_{H\\alpha}$} & \\colhead{$\\rm FWHM_{[S\\ {\\scriptscriptstyle II}]}$} & \\colhead{$\\rm EW_{\\rm rest}(H\\alpha)_b$}\\\\\n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{(5)} & \\colhead{(6)}\n}\n\\startdata\n\\multicolumn{6}{c}{broad H$\\alpha$ and He~{\\sc ii}}\\\\\n\\tableline\n3......... & 991.96 & 1834.96 & 7162.24 & 245.05 & 88.10\\\\\n8......... & 414.20 & 309.70 & 875.81\\tablenotemark{1} & 247.31 & 27.02\\\\\n12........ & 524.87 & 830.66 & 1682.70 & 324.79 & 57.25\\\\\n13........ & 748.78 & 728.55 & 2033.36 & 223.68 & 72.83\\\\\n25......... & 607.56 & 930.70 & 1974.28 & 248.66 & 159.88\\\\\n34........ & 2072.41 & 797.10 & 2083.37 & 377.39 & 33.06\\\\\n65........ & 573.27 & 761.34 & 4830.02 & 263.52 & 44.61\\\\\n66........ & 448.38 & 768.76 & 2430.79 & 221.86 & 47.03\\\\\n\\tableline\n\\multicolumn{6}{c}{broad H$\\alpha$ and noHe~{\\sc ii}}\\\\\n\\tableline\n\\dataset\n17........ & 719.07 & 392.29 & 3396.57  & 270.94 & 44.74\\\\\n21........ & 1014.89 & 1145.11 & 2440.19 & 337.97 & 44.13\\\\\n47........ & 515.66 & 726.01 & 3569.90 & 273.99 & 246.77\\\\\n58........ & 390.35 & 801.51 & 4347.42 & 307.51 & 28.83\\\\\n67........ & 276.51 & 335.10 & 2372.32 & 276.41 & 25.14    \n\\enddata\n\\tablenotetext{1}{Classified as an object with a broad H$\\alpha$ component through the FWHM \nof the additional H$\\alpha$ component is less than 1000 km s$^{-1}$ (see the main text).}\n\\tablecomments{\nCol. (1): Identification number assigned in this paper. \nCol. (2): Flux of the nallow component of H$\\alpha$ in units of $10^{-17}$ $\\rm erg\\ s^{-1}\\ cm^{-2}$.\nCol. (3): Flux of the broad component of H$\\alpha$ in units of $10^{-17}$ $\\rm erg\\ s^{-1}\\ cm^{-2}$.\nCol. (4): FWHM of the broad component of H$\\alpha$ in units of km $\\rm s^{-1}$.\nCol. (5): FWHM of the [S~{\\sc ii}]$\\lambda$6717 (i.e., narrow component) in units of km $\\rm s^{-1}$.\nCol. (6): Rest-frame equivalent width of the broad component of H$\\alpha$ in units of $\\rm \\AA$.\n}\n\n\\end{deluxetable*}\n\n\\begin{deluxetable}{lcc}\n\\tablecolumns{3} \n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pc}\n\\tablecaption{Classification result of the BPT-valley sample}\n\\tablehead{ \n\\colhead{} & \\colhead{broad} & \\colhead{nobroad} \n}\n\\startdata\n    He II  & 8 & 30 \\\\\n    noHe II   & 5 & 27 \n\\enddata\n\\end{deluxetable}\n\n\n\\section{Selection of Low-Metallicity AGNs}\n\nThe 43 BPT-valley objects confirmed to be AGNs are not necessarily low-metallicity AGNs, \nbecause AGNs with a very high electron density or very high ionization parameter are \nalso expected to be populated in the BPT valley as mentioned in Section 1. \nMore specifically, the [N~{\\sc ii}]$\\lambda$6584 emission in AGNs with a density higher than \nthe critical density of the [N~{\\sc ii}]$\\lambda$6584 transition ($\\sim$8.7 $\\times 10^4$ cm$^{-3}$) \nis significantly suppressed due to the collisional de-excitation effect. \nOn the other hand, a very high ionization parameter results in a higher relative ionic abundance of \nN$^{2+}$ (i.e., a lower relative ionic abundance of N$^{+}$), that results in a weaker \n[N~{\\sc ii}]$\\lambda$6584 emission.\nTherefore, in this section, we examine whether the 43 BPT-valley AGNs are characterized \nby a very high electron density or very high ionization parameter or not, and test whether \nthe AGNs in the BPT-valley are characterized by low-metallicity gas or not.\n\n\\subsection{Electron density}\nThe emission-line flux ratios of [S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 and \n[O~{\\sc ii}]$\\lambda$3729\/$\\lambda$3726 are famous good indicators of the electron density \n\\citep[e.g.,][]{1989agna.book.....O}.\nIn this work, we use the [S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 line ratio to estimate electron density, \nbecause the wavelength separation of the [O~{\\sc ii}] doublet is too small to be well resolved with \nthe SDSS spectral resolution.\nWe use an IRAF routine {\\tt temden} for deriving the electron density from the \n[S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 ratio, by assuming the electron temperature of 10,000 K. \nHere we derive the electron density whose [S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 ratio is \nwithin the range of 0.5--1.4. \nNo BPT-valley objects show the [S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 ratio lower than 0.5 \n(i.e., the high-density limit) while 11 among the 70 BPT-valley objects show the flux ratio \nhigher than 1.4 (i.e., the low-density limit). \nAmong 14,252 Seyfert sample, only 12 objects show the [S~{\\sc ii}]$\\lambda$6716\/$\\lambda$6731 \nratio lower than 0.5 while 2,880 objects show the flux ratio higher than 1.4.\n\nFigure~\\ref{SII_electron} shows the frequency distribution of the inferred gas density for objects whose \n[S~{\\sc ii}]$\\lambda$6717\/$\\lambda$6731 ratio is within the range of 0.5--1.4; i.e.,  41 BPT-valley AGNs \n(showing a broad H$\\alpha$ component and\/or He~{\\sc ii} emission), 59 BPT-valley objects \n(including objects without any AGN signatures), and 11,360 Seyfert galaxies. \nHere we show the histograms for both BPT-valley AGNs and BPT-valley objects, because some of \nBPT-valley objects without any AGN signatures could be also AGNs (see Section 3.3). \nThe median density of the BPT-valley AGNs, BPT-valley objects, and Seyfert galaxies are \n210 cm$^{-3}$, 210 cm$^{-3}$, and 270 cm$^{-3}$, respectively. \nIn order to investigate whether the frequency distribution of the gas density is statistically different \namong the samples, we apply the Kolmogorov-Smirnov (K-S) statistical test with a null hypothesis that \nthe frequency distribution of the gas density of two classes of objects comes from \nthe same underlying population. \nThe derived K-S probability for the BPT-valley AGNs and Seyferts is 0.207, while that for \nthe BPT-valley objects and Seyferts is 0.146. \nThese results strongly suggest that the BPT-valley sample is not characterized by the higher gas density \nwith respect to the Seyfert sample.\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=8.5cm]{figure_10.eps}\n \\caption{Histograms of the electron density of the BPT-valley AGN (filled green), \n BPT-valley objects (open red), and Seyfert galaxies (open gray), normalized by their peak count.\n Dashed lines denote the range of the electron density measurable through the [S~{\\sc ii}] doublet ratio.} \n \\label{SII_electron} \n\\end{figure}\n\n\\subsection{Ionization parameter}\nThe ionization palameter is the ratio of the number density of hydrogen-ionizing photons \nto that of Hydrogen atoms. \nIn order to investigate the ionization parameter, the [O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 \nflux ratio is a useful indicator because this ratio does not suffer significantly from chemical properties of \nthe gas in both AGNs and star-forming galaxies \n\\citep[see, e.g.,][]{1997A&A...323...31K, 2002ApJ...567...73N, 2014MNRAS.442..900N}.\nNote that this flux ratio is sensitive also to the gas density if the density is higher than \nthe critical density of [O~{\\sc ii}] ($\\sim$$10^{3.5}$ cm$^{-3}$), \nbut the typical density of NLRs inferred from the [S~{\\sc ii}] doublet ratio is much lower than \nthat as described in Section 4.1.\nThough the dust reddening is not corrected to study the BPT diagram due to the small wavelength \nseparation of emission-line pairs used for the BPT diagram (Section 2), we should correct for the reddening \neffect to investigate the [O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ flux ratio. \nFor this correction, we assume $R_V = A_V\/E(B-V) = 3.1$ and the intrinsic flux ratio \nof H$\\alpha$$\\lambda6584$\/H$\\beta$$\\lambda4861$ = 3.1, and adopt the reddening curve \nof \\cite{1989ApJ...345..245C}.\n\nFigure~\\ref{OIII_OII} shows the histogram of the [O~{\\sc iii}]$\\lambda$5007\/\n[O~{\\sc ii}]$\\lambda$3727 line \nratio of the BPT-valley AGNs, BPT-valley objects, and Seyferts, with S\/N([O~{\\sc ii}]$\\lambda$3727) $>$ 3.\nHere it should be noted that the BPT-valley objects show \nlog([O~{\\sc iii}]$\\lambda5007$\/H$\\beta$$\\lambda4861$) $>$ 0.5 by definition, \nwhile the Seyfert galaxies could have much lower [O~{\\sc iii}]$\\lambda5007$\/H$\\beta$$\\lambda4861$ flux ratios \ndown to $\\sim -0.2$. \nThis may introduce a selection effect in the sense that strong [O~{\\sc iii}] emitters could be selectively \nincluded in the BPT-valley sample. \nTherefore, for reducing this selection effect, only objects with \nlog([O~{\\sc iii}]$\\lambda5007$\/H$\\beta$$\\lambda4861$) $>$ 0.5 are examined for assessing \nthe ionization parameter. \nAfter adopting this additional criterion, the numbers of the BPT-valley AGNs, BPT-valley objects, and \nSeyferts examined in Figure~\\ref{OIII_OII} are 42, 69, and 8,500, respectively. \nThis figure shows that the BPT-valley samples seem to show systematically higher \n[O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ flux ratios than the Seyfert sample. \nThe median values of the logarithmic [O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ flux ratios of \nthe BPT-valley AGNs, \nBPT-valley objects, and Seyferts are 0.67, 0.65, and 0.46, respectively. \nIn order to investigate whether or not the distributions of the \n[O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 line ratio are \nstatistically different between BPT-valley sample and Seyfert sample, we apply the K-S statistical test. \nThe K-S probability that the underlying distribution of these two\ndistributions is the same is $3.925\\times 10^{-6}$ for the BPT-valley AGNs and Seyferts, \nwhile $1.803 \\times 10^{-5}$ for the BPT-valley objects and Seyferts. \nThese results suggest that the BPT-valley samples have statistically higher \n[O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ flux ratios, i.e., the ionization parameter, \nthan the Seyfert sample. \nNote that it is well known that low-metallicity galaxies are generally characterized by a relatively \nhigh ionization parameter, at least for star-forming galaxies \\citep[e.g.,][]{2006A&A...459...85N}. \nIt may be interesting that the BPT-valley objects show a clear edge at the lower side of the \n[O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ distribution in Figure~\\ref{OIII_OII}.\nHowever, probably this feature is not statistically significant, because the number of BPT-valley objects \nis not enough to discuss the tail of the frequency distribution of the \n[O~{\\sc iii}]$\\lambda5007$\/[O~{\\sc ii}]$\\lambda3727$ flux ratio.\n\nIn the next subsection, we will examine whether or not this difference in the ionization parameter can be \nresponsible for the lower [N~{\\sc ii}]$\\lambda6584$\/H$\\alpha$$\\lambda6563$ ratio observed in the \nBPT-valley samples with respect to the Seyfert sample.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8.5cm]{figure_11.eps}\n \\caption{Same as Figure~\\ref{SII_electron} but for the [O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 \n flux ratio.}\n \\label{OIII_OII} \n\\end{figure}\n\n\\subsection{Model calculations}\nAs shown in Section 4.2, the ionization parameter of the BPT valley sample is higher than that of \nthe Seyfert sample. \nSince it is interesting to examine either the BPT-valley AGNs are characterized by a low metallicity or \na high ionization parameter, we perform photoionization model calculations.\n\nWe perform photoionization model calculations for simulating the NLR of AGNs, \nusing the code CLOUDY version 13.03 \\citep{1998PASP..110..761F}. \nHere the main parameters for CLOUDY calculations are as follows \n\\citep[see][for more details]{2001ApJ...546..744N}:\n\\begin{enumerate}\n \\item The hydrogen density of the cloud ($n_{\\rm H}$).\n \\item The ionization parameter ($U$).\n \\item The chemical composition of the gas.\n \\item The shape of the input SED.\n\\end{enumerate}\nWe calculate photoionization models covering the following ranges of parameters:\n$10^1\\ {\\rm cm^{-3}} \\leq n_{\\rm H} \\leq 10^6\\ {\\rm cm^{-3}}$ and\n$10^{-4} \\leq U \\leq 10^{-1}$. \nWe set the gas-phase elemental abundance ratios to be the solar ones. \nThe adopted solar abundances relative to hydrogen are taken from \\cite{1989AIPC..183....1G} \nwith extensions by \\citet{1993oee..conf...15G}.\nThe adopted metallicity (i.e., the solar one) is not typical for usual Seyfert galaxies \n(whose NLR metallicity is generally higher than the solar metallicity), possibly nor BPT-valley AGNs \n(that could have sub-solar metallicity). \nHowever, as described below, it is useful to fix the metallicity to examine whether the ionization \nparameter alone can account for the difference in the emission-line flux ratios between \nBPT-valley objects and Seyferts. \nFor the input SED, we adopt the following one:\n\\begin{eqnarray}\nf_{\\nu}={\\nu}^{\\rm \\alpha_{UV}} \\exp \\left( -\\frac{h\\nu}{kT_{\\rm BB}}\\right) \\exp \\left( -\\frac{kT_{\\rm IR}}{h\\nu}\\right) \n+ a{\\nu}^{\\alpha_{\\rm  X}}\n\\end{eqnarray}\nas a typical spectrum of AGNs (see \\citealt{1996hbic.book.....F}). \n$kT_{\\rm IR}$ is the infrared cutoff of the big-blue bump, and we adopt \n$kT_{\\rm IR}=0.01$ ryd \\citep[see][]{1996hbic.book.....F}. \n$\\alpha_{\\rm UV}$ is the slope of the low-energy side of the big-blue bump.\nWe adopt $\\alpha_{\\rm UV} = 0.5$, which is typical for AGNs \n\\citep{1996hbic.book.....F}. \n$\\alpha_{\\rm ox}$ is the UV--to--X-ray spectral slope, which determines the parameter $a$ in equation (6).\nWe adopt $\\alpha_{\\rm ox}=-1.35$, which is the average value of nearby Seyfert 1 galaxies\n\\citep[see][]{1993A&A...274..105W}.\n$\\alpha_{\\rm x}$ is the X-ray slope, and we adopt $\\alpha_{\\rm x}=-0.85$ (see \\citealt{2001ApJ...546..744N}).\n$T_{\\rm BB}$ is the characterizing the shape of the big-blue bump, and we adopt 490,000 K \n(see \\citealt{2001ApJ...546..744N}).\nThe calculations end at the depth where the temperature falls to 3,000 K, \nbelow which gas does not contribute significantly to the flux of optical emission lines.\n\nFigure~\\ref{cloudy_1} shows the results of the photoionization model calculations, overlaid \non the BPT diagram. \nThough the density effect is not significant in the range of \n$10^1$ cm$^{-3}$ $<$ $n_{\\rm H}$ $<$ $10^5$ cm$^{-3}$, \nwe can see the effect of the collisional de-excitation at $n_{\\rm H}$ $>$ $10^4$ cm$^{-3}$. \nHowever, this figure suggests that the difference in the [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6583 \nflux ratio is more easily explained by the difference in the ionization parameter rather than by the difference \nin the gas density. \nMore specifically, a higher ionization parameter by 0.5--1 dex in the BPT-valley objects with respect to \nthe Seyfert sample is required to explain the lower [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6583 \nflux ratio of the BPT-valley objects. \n\nFor examining whether the BPT-valley objects have a higher ionization parameter than the Seyfert sample, \nwe investigate another diagnostic diagram that consists of the emission-line flux ratios of \n[O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 and [O~{\\sc i}]$\\lambda$6300\/[O~{\\sc iii}]$\\lambda$5007 \n(Figure~\\ref{cloudy_2}). \nThis diagram is useful to examine the effect of ionization parameter without suffering from \nthe metallicity effect, because only oxygen lines are used and thus less sensitive to the metallicity. \nFigure~\\ref{cloudy_2} shows that the BPT-valley sample and Seyfert sample have a similar gas density, \nthat is consistent with our analysis presented in Section 4.1. \nMore interestingly, Figure~\\ref{cloudy_2} shows that the BPT-valley sample shows a systematically \nhigher ionization parameter than the Seyfert sample, but the inferred difference in the ionization \nparameters is only less than 0.5 dex. \nThis strongly suggests that the lower [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 flux ratio \nin the BPT-valley sample with respect to the Seyfert sample is not explained by the ionization parameter \n(nor the gas density, as described in Section 4.1). \nTherefore we conclude that the BPT-valley AGNs are characterized by a systematlcally lower metallicity \nthan the Seyfert sample, as originally proposed by \\citet{2006MNRAS.371.1559G}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8.5cm]{figure_12s.eps}\n \\caption{Same as Figure~\\ref{BPT_classification} (without the inset panel in \n Figure~\\ref{BPT_classification}) but grids of photoionization models are overlaid. \n Different colors of lines denote different parameters adopted in the calculations, \n as shown in the inset panels.\n }\n \\label{cloudy_1} \n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=8.5cm]{figure_13.eps}\n \\caption{Diagnostic diagram of [O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 versus \n [O~{\\sc i}]$\\lambda$6300\/[O~{\\sc iii}]$\\lambda$5007. \n The symbols and lines are the same as in Figure~\\ref{cloudy_1}. \n Note that only the BPT valley objects with S\/N $>$ 5 of the [O~{\\sc ii}]$\\lambda$3727, \n [O~{\\sc iii}]$\\lambda$5007 and [O~{\\sc i}]$\\lambda$6300 line are plotted.}\n \\label{cloudy_2} \n\\end{figure}\n\n\\section{Disccusions}\nAs mentioned Section 1, low-metallicity AGNs are interesting to study the early phase of \nthe AGN evolution.\nHowever low-metallicity AGNs are very rare, so that little has been reported on\nphysical property of low-metallicity AGNs. \nIn this section, we present some basic properties of BPT-valley objects which are expected \nto be low-metallicity AGNs.\n\n\\subsection{Stellar mass}\nNaively it is expected that the stellar mass of low-metallicity AGNs is expected to be \nrelatively low, as suggested by the mass-metallicity relation seen in star-forming galaxies \n\\citep[e.g.,][]{2004ApJ...613..898T, 2006ApJ...647..970L}.\nAccordingly \\citet[][]{2006MNRAS.371.1559G} introduced a mass criterion \n(i.e., $M_{*} < 10^{10}\\ {\\rm M_{\\odot}}$) to select low-metallicity AGNs. \nHowever, it is not clarified whether low-metallicity AGNs should be always found in \na sample of AGNs with a low-mass host galaxy. \nTherefore, in this paper, we select low-metallicity AGNs without stellar-mass cut and \ninvestigate the mass distribution of host galaxies of low-metallicity AGNs. \nHere the stellar mass has been measured and given in the MPA-JHU DR7 catalog \n\\cite[see also][]{2003MNRAS.341...33K}.\nAmong the 43 BPT-valley AGNs and 70 BPT-valley objects, the host mass is \navailable for 39 and 64 objects, respectively. \nFigure~\\ref{mass} shows the histogram of the stellar mass of the 39 BPT-valley AGNs, \n64 BPT-valley objects and 13,662 Seyferts.\nThe median of the stellar mass of the BPT-valley AGNs, BPT-valley objects \nand Seyferts are $10^{10.15}\\ {\\rm M_{\\odot}}$, $10^{10.07}\\ {\\rm M_{\\odot}}$ and \n$10^{10.77}\\ {\\rm M_{\\odot}}$, respectively. \nThis result clearly shows that the stellar mass of the BPT-valley AGNs is systematically lower \nthan that of Seyferts. \nHowever, interestingly, a substantial fraction of the BPT-valley AGN (23 among 39 objects) \nare actually hosted by galaxies with $M_{*} > 10^{10}\\ {\\rm M_{\\odot}}$, \nsuggesting that low-metallicity AGNs are not necessarily hosted by low-mass galaxies. \nNote that such low-metallicity AGNs with a relatively massive host galaxy cannot be selected \nby the criteria of \\citet[][]{2006MNRAS.371.1559G} due to the mass criterion of \n$M_{*} < 10^{10}\\ {\\rm M_{\\odot}}$. \nSuch low-metallicity AGNs hosted by a relatively massive host galaxy may be realized by \ntaking into account of the inflow of low-metallicity gas from the surrounding environment  \n\\citep[e.g.,][]{2011A&A...535A..72H}. \n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=8.5cm]{figure_14.eps}\n \\caption{\n Same as Figure~\\ref{SII_electron} but for the stellar mass.\n }\n \\label{mass} \n\\end{figure}\n\n\n\n\n\\subsection{Electron temperature}\nConsidering the effect of the metal cooling, \nlow-metallicity AGNs are expected to be characterized by the higher electron temperature.\nHence we investigate the [O~{\\sc iii}]$\\lambda \\lambda(4959+5007)$\/[O~{\\sc iii}]$\\lambda4363$ line \nratio which is very sensitive to the gas temperature.\nHere it should be mentioned that, [O~{\\sc iii}]$\\lambda \\lambda(4959+5007)$\/[O~{\\sc iii}]$\\lambda4363$ line \nratio also depends on the electron density \\citep[see, e.g.,][]{2001ApJ...549..155N}. \nTherefore we investigate the  [O~{\\sc iii}]$\\lambda \\lambda(4959+5007)$\/[O~{\\sc iii}]$\\lambda4363$ and \n[S~{\\sc ii}]$\\lambda6717$\/[S~{\\sc ii}]$\\lambda6731$ line ratios simultaneously in Figure~\\ref{temperature_density}. \nHere this figure shows the emission-line flux ratios of BPT-valley objects and Seyferts but only for \nobjects with a significant detection of the [O~{\\sc iii}]$\\lambda4363$ line (S\/N $> 3$).\nAs described in Section 4.2, only objects with log ([O~{\\sc iii}]$\\lambda$5007\/H$\\beta$) $> 0.5$ are used \n(that results in 9,043 Seyferts and 70 BPT-valley objects).\nNote that [O~{\\sc iii}]$\\lambda \\lambda(4959+5007)$\/[O~{\\sc iii}]$\\lambda4363$ line ratio is corrected \nfor the reddening effect in the same way as Section 4.2.\nThe median values of log ([S~{\\sc ii}]$\\lambda6717$\/[S~{\\sc ii}]$\\lambda6731$) of the BPT-valley AGNs, \nBPT-valley objects and Seyferts with a [O~{\\sc iii}]$\\lambda4363$ detection are 0.088, 0.088 and 0.055, \nrespectively.\nTherefore the electron density of the BPT-valley sample is slightly higher than that of Seyferts \nas already mentioned in Section 4.1. \nThe median of log ([O~{\\sc iii}]$\\lambda \\lambda(4959+5007)$\/[O~{\\sc iii}]$\\lambda4363$) of \nthe BPT-valley AGN, BPT-valley objects and Seyferts are 1.77, 1.77 and 1.79, respectively.\nThis result suggests that the electron temperature of the BPT-valley objects is not significantly higher \nthan that of Seyferts. \nHowever, the fraction of objects showing a significant (S\/N $> 3$) [O~{\\sc iii}]$\\lambda$4363 emission \nis very different between the Seyferts and BPT-valley objects. \nMore specifically, 44 among the 70 BPT-valley objects ($\\sim 63\\ \\%$) show the [O~{\\sc ii}]$\\lambda$4363 \nemission while only 1,516 among 9,043 Seyferts ($\\sim 17\\ \\%$) show the [O~{\\sc iii}]$\\lambda4363$ line. \nThis difference infers that generally the gas temperature of the NLR in BPT-velley objects tends to be \nso high that the [O~{\\sc iii}] $\\lambda$4363 line is detected in most cases, \nwhile the typical gas temperature of the NLR in Seyferts may be lower than that in BPT-valley objects \nand only highly biased objects with a relatively high temperature in the Seyfert sample show the \n[O~{\\sc iii}]$\\lambda4363$ line. \nThis result is consistent to our expectation that the BPT-valley objects is actually characterized by a \nrelatively high gas temperature, due to the low gas metallicity. \n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=8.5cm]{figure_15.eps}\n \\caption{\n Diagnostic diagram of [O~{\\sc iii}]$\\lambda$$\\lambda$(4959+5007)\/\n [O~{\\sc iii}]$\\lambda$4363 versus [S~{\\sc ii}]$\\lambda$6717\/[S~{\\sc ii}]$\\lambda$6731. \n The symbols are the same as in Figure~\\ref{BPT_classification}. \n Note that only the BPT valley objects with S\/N $>$ 3 of the [O~{\\sc iii}]$\\lambda$4363 line are plotted. \n }\n \\label{temperature_density} \n\\end{figure}\n\n\n\\section{Conclusions}\n\nIn this paper, we focus on low-metallicity AGNs ($Z_{\\rm NLR}$ $\\lesssim$ $1\\ Z_{\\odot}$) \nwhich are very rare but important since they are in the early phase of the galaxy-SMBH co-evolution.\nSpecifically, in this work it is examined whether the BPT-valley selection is an effective and reliable \nway to identify low-metallicity AGNs, as proposed by \\citet{2006MNRAS.371.1559G}. \nThe main results are as follows:\n\\begin{itemize}\n \\item We select 70 BPT valley sample which expected low metallicity AGN from \n14,253 Seyfert galaxies of MPA-JHU SDSS DR7 galaxy catalog.\n \\item Out of 70 BPT-valley objects, 43 objects show clear evidence of the AGN based on \n the detection of the broad H$\\alpha$ component and\/or He~{\\sc ii}$\\lambda$4686 emission.\n  \\item The typical gas density of the BPT-valley sample ($\\sim$210 cm$^{-3}$) is not higher than that of \n  the Seyfert sample ($\\sim$270 cm$^{-3}$), suggesting that the lower \n  [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 ratio in the BPT-valley AGNs with respect to \n  the Seyfert sample is not caused by the collisional de-excitation effect. \n  \\item The higher [O~{\\sc iii}]$\\lambda$5007\/[O~{\\sc ii}]$\\lambda$3727 ratio in the BPT-valley sample \n  ($\\sim$4.5) with respect to that in the Seyfert sample ($\\sim$2.9) suggests a typically higher \n  ionization parameter of the BPT-valley sample; however, photoionization models suggest that \n  the inferred difference in the ionization parameter between the BPT-valley sample and \n  Seyfert sample is not enough to explain the observed lower \n  [N~{\\sc ii}]$\\lambda$6584\/H$\\alpha$$\\lambda$6563 ratio of the BPT-valley sample.  \n  \\item The BPT-valley selection for identifying low-metallicity AGNs is thus confirmed to be a useful method; \n  in our analysis, more than 60\\% of the BPT-valley sample are low-metallicity AGNs \n  ($Z_{\\rm NLR}$ $\\lesssim$ $1\\ Z_{\\odot}$). \n\n\\end{itemize}\n\n\n\\acknowledgments\n\nWe would like to thank the anonymous referee for her\/his\ncareful reading this paper and useful suggestions, and also\nMasaru Kajisawa and Kazuyuki Ogura for their useful comments.\nTN is financially supported by JSPS grants Nos. 25707010, 16H01101, and 16H03958. \nKM is also supported by JSPS grant No. 14J01811. \nFunding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http:\/\/www.sdss.org\/.\nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nEven though General Relativity (GR) is extensively tested in the weak-field regime, it is only recently that we have started constraining it in the strong field regime. Gravitational-wave observations \\cite{Abbott:2016blz,TheLIGOScientific:2017qsa,Abbott:2020niy} will soon be rising to the hundreds, providing us with enough data to accurately confront many of the proposed strong gravity GR deviations. Increased precision in observations will allow us to determine whether compact objects, which are associated with extremely large curvatures, have different properties than predicted by GR. \n\nThe phenomenon of \\textit{spontaneous scalarization} provides perhaps the most promising framework, in which we can investigate the manifestation of a strong gravity process that remains dormant in low curvature regimes. Spontaneous scalarization was initially proposed in the case of neutron stars by Damour and Esposito-Far\\`ese (DEF) \\cite{Damour:1992kf,Damour:1993hw}. According to it, a scalar field coupled to gravity in a suitable manner, might acquire a non-trivial structure only in the strong field regime of neutron stars, while remaining trivial and undetected in the weak field regime. In the DEF model black holes do not exhibit scalarization unless it is induced by matter in their vicinity \\cite{Hawking:1972qk,Sotiriou:2011dz,Cardoso:2013opa,Cardoso:2013fwa,Palenzuela:2013hsa}. However, recently, a different class of models in which there is scalarization of both black holes and neutron stars has been receiving a lot of attention: scalar-Gauss-Bonnet theories (\\textit{e.g.}  \\cite{Silva:2017uqg,Doneva:2017bvd,Doneva:2017duq}).\n\nScalarization of both black holes and neutron stars has been scrutinized in various works concerning many different modifications (bare mass, self-interactions, different field content, \\textit{etc} \\cite{Ramazanoglu:2016kul,Blazquez-Salcedo:2018jnn,Macedo:2019sem,Herdeiro:2018wub,Ramazanoglu:2017xbl,Ramazanoglu:2018hwk}). Scalarization can be thought of as triggered by a curvature-induced tachyonic instability of the scalar field. In more recent works, it has been shown that this instability can be triggered by spin \\cite{Dima:2020yac} and lead to black holes that are scalarized only when rapidly rotating \\cite{Herdeiro:2020wei,Berti:2020kgk}. It should be noted that scalarization models differ from certain hairy black hole models ({\\em e.g.}~\\cite{Sotiriou:2013qea,Sotiriou:2014pfa,Antoniou:2017acq,Antoniou:2017hxj}) in that, in the latter all black holes carry a non-trivial scalar configuration, whereas in the former only black holes with certain mass or spin characteristics deviate from the Kerr metric.\n\nThe onset of the tachyonic instability that triggers scalarization is controlled by linear terms (although Ref.~\\cite{Doneva:2021tvn} also examined what happens if linear terms are absent from the potential) but eventually this instability is quenched by non-linearities, which control the end-state.\nIn \\cite{Andreou:2019ikc}, all terms that can affect the onset of the instability in the framework of Horndeski theory were listed. However, one of these terms, namely the coupling to the Ricci scalar, has not received much attention in many of the aforementioned works. This is mostly due to the fact that,\nin the black-hole scenario, the onset of scalarization is only controlled by the Gauss-Bonnet invariant, since the Ricci scalar evaluates to zero for GR black holes. Nonetheless, including the Ricci term does  seem to provide us with several advantages. To begin with, as discussed in \\cite{Antoniou:2020nax}, the Ricci term is crucial if one wants to retrieve a late-time attractor to GR in a cosmological scenario. Additionally, it was shown in \\cite{Ventagli:2020rnx} that the Ricci term can help in suppressing the scalarization of neutron stars, which would otherwise tend to place significant constraints. Finally, Ref.~\\cite{Antoniou:2021zoy} showed that this term has very interesting effects on the properties of scalarized black holes. Even though the Ricci coupling does not affect the onset of black hole scalarization (being zero in a GR black hole background), it affects the properties of the scalarized solutions and, consequently, observables. For certain values of the Ricci coupling ---~which happen to be consistent with the ones associated with a late-time attractor behaviour~--- the presence of this operator is expected to render black holes radially stable, without the need to introduce self-interaction terms.\n\nFor the reasons presented above, it is of great interest to examine how the combination of Ricci and Gauss-Bonnet couplings affects neutron star properties. We present the analytic and numerical setup of our study in Sec.~\\ref{sec:setup}. The numerical results are presented in Sec.~\\ref{sec:results}. In Sec.~\\ref{Sec:parameterSpace}, we determine over which region of the parameter space scalarized solutions exist, for three different stellar scenarios. In Sec.~\\ref{Sec:betaNeg} and \\ref{Sec:betaPos}, we discuss the properties of the scalarized solutions, in particular their scalar charges and masses. Section \\ref{Sec:instabilityLines} investigates in more detail the solutions that always exist near the scalarization thresholds, while Sec.~\\ref{Sec:EffMass} explains how, already at the level of the GR solution, a given scalar profile may be favored. We conclude with a discussion in Sec.~\\ref{sec:discussion}.\n\n\n\n\n\n\n\\section{SETUP}\n\\label{sec:setup}\n\nIt has been shown in \\cite{Andreou:2019ikc} that, in the framework of Horndeski theories, the minimal action containing all the terms that can affect the onset of a tachyonic instability is\n\\begin{equation}\n\\begin{split}\\label{eq:ActionCaseI}\nS&=\\int\\mathrm{d}^4x\\sqrt{-g}\\bigg\\{\\dfrac{R}{2\\kappa}+X+ \\gamma\\, G^{\\mu\\nu}\\nabla_\\mu\\phi\\,\\nabla_\\nu\\phi\n\\\\\n&\\quad -\\left(m_\\phi^2+\\dfrac{\\beta}{2} R-\\alpha\\mathscr{G}\\right)\\dfrac{\\phi^2}{2}\\bigg\\}\n +S_\\mathrm{M},\n\\end{split}\n\\end{equation}\nwhere $X=-\\nabla_\\mu\\phi\\nabla^\\mu\\phi\/2$, $\\kappa=8\\pi G\/c^4$ and $\\mathscr{G}$ is the Gauss-Bonnet invariant\n\\begin{equation}\n\\mathscr{G}=R^2-4R_{\\mu\\nu}R^{\\mu\\nu}+R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}.\n\\end{equation}\n $S_\\text{M}$ is the matter action, where matter is assumed to couple minimally to the metric; in other words, we are working in the so-called Jordan frame. $m_\\phi$ is the bare mass of the scalar field, and $\\alpha$, $\\beta$ and $\\gamma$ parametrize the deviations from GR. Note that $\\beta$ is dimensionless, whereas $\\gamma$ and $\\alpha$ have the dimension of a length squared. $\\beta$ is defined such that it matches the notation of the (linearized) DEF model (see~\\cite{Andreou:2019ikc} for a detailed discussion on the relation to the original DEF model). For the purpose of this paper we set $\\gamma=0$ and $m_\\phi=0$. If a bare mass is included it needs to be tuned to rather small values else it can prevent scalarization altogether \\cite{Ramazanoglu:2016kul,Ventagli:2020rnx}, while $\\gamma$ has a very limited effect on the threshold of tachyonic scalarization \\cite{Ventagli:2020rnx}. Note that by setting these two parameters to zero, we retrieve the action studied in \\cite{Antoniou:2021zoy} in the context of spontaneously scalarized black holes. The modified Einstein equation is\n\n \\begin{equation}\\label{eq:grav_eq}\n    G_{\\mu\\nu}=\\kappa T^\\phi_{\\mu\\nu}+\\kappa T^\\text{M}_{\\mu\\nu},\n \\end{equation}\n\nwhere \n\\begin{equation}\n\\begin{split}\\label{eq:StressEnScal}\nT^{\\phi}_{\\mu\\nu} & =\\nabla_\\mu\\nabla_\\nu\\phi-\\frac{1}{2}g_{\\mu\\nu}\\nabla_\\lambda\\phi\\nabla^\\lambda\\phi\\\\\n& +\\frac{1}{2}\\beta\\left(G_{\\mu\\nu}-\\nabla_\\mu\\nabla_\\nu+g_{\\mu\\nu}\\nabla_\\lambda\\nabla^\\lambda \\right)\\phi^2\\\\\n& +2\\alpha\\big[R\\big(\\nabla_\\mu\\nabla_\\nu-g_{\\mu\\nu}\\nabla_\\lambda\\nabla^\\lambda\\big)\\phi^2\\\\\n& +2\\big(R_{\\mu\\nu}\\nabla_\\lambda\\nabla^\\lambda-2R_{(\\mu\\lambda}\\nabla_{\\nu)}\\nabla^\\lambda\\\\\n& +4g_{\\mu\\nu}R_{\\lambda\\sigma}\\nabla^\\lambda\\nabla^\\sigma\\big)\\phi^2-2R_{\\mu\\lambda\\nu\\sigma}\\nabla^\\lambda\\nabla^\\sigma\\phi^2\\big]\n\\end{split}\n\\end{equation}\ncomes from the variation of the $\\phi$-dependent part of the action with respect to the metric, and $T^\\mathrm{M}_{\\mu\\nu}=-(2\/\\sqrt{-g})(\\delta S_\\mathrm{M}\/\\delta g^{\\mu\\nu})$ is the matter stress-energy tensor. The scalar field equation reads\n\\begin{equation}\\label{eq:scal_eq}\n    \\DAlembert \\phi =m_\\text{eff}^2\\phi,\n\\end{equation}\nwhere the effective scalar mass is given by\n\\begin{equation}\\label{eq:eff_masss}\n    m_\\text{eff}^2=\\frac{\\beta}{2}R-\\alpha \\mathscr{G}.\n\\end{equation}\nA configuration with a sufficiently\\footnote{Any negative effective mass squared will cause an instability in Minkowski spacetime, but a curved spacetime is destabilized only if a certain threshold is exceeded.} negative effective mass squared will suffer from a tachyonic instability, triggering spontaneous scalarization. For the purpose of this paper, we restrict our analysis to static and spherically symmetric spacetimes:\n\\begin{equation}\\label{eq:metric}\n\\text{d}s^2= - e^{\\Gamma(r)}\\text{d}t^2+e^{\\Lambda(r)}\\text{d}r^2+r^2 \\text{d}\\Omega^2,\n\\end{equation}\nand we assume matter to be described by a perfect fluid with $T^\\text{M}_{\\mu\\nu}=(\\epsilon+p)u_\\mu u_\\nu+p\\,g_{\\mu\\nu}$, where $\\epsilon$, $p$ and $u_\\mu$ are respectively the energy density, the pressure and the 4-velocity of the fluid. The pressure is  directly related to the energy density through the equation of state. The field equations then take the form of coupled ordinary differential equations for $\\Gamma$, $\\Lambda$, $\\epsilon$ and $\\phi$, see Appendix. We can solve algebraically the $(rr)$ component of the modified Einstein equation for $e^\\Lambda$. The result is\n\\begin{equation}\\label{eq:ExpLambda}\ne^\\Lambda=\\frac{-B+\\delta\\sqrt{B^2-4\\,A\\,C}}{4 A},\\,\\,\\delta=\\pm 1\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n& A=1+\\kappa\\,r^2p-\\frac{1}{2}\\,\\beta\\kappa\\phi^2,\\\\\n& B=-2+\\beta\\kappa\\,\\phi^2-2\\,r\\Gamma'+r\\beta\\kappa\\,\\phi^2\\Gamma'+4\\,r\\beta\\kappa\\,\\phi\\phi'\\\\\n&\\qquad -8\\,\\alpha\\kappa\\,\\phi\\Gamma'\\phi'+r^2\\beta\\kappa\\phi\\Gamma'\\phi'+\\kappa\\,r^2\\phi'^2,\\\\\n& C=48\\,\\alpha\\kappa\\,\\phi\\,\\Gamma'\\phi'.\n\\end{split}\n\\end{equation}\nFor the $\\delta=-1$ branch of solutions we do not retrieve GR in the limit $\\alpha\\to 0$ and $\\beta \\to 0$, henceforth we will assume $\\delta=1$. By substituting Eq.~\\eqref{eq:ExpLambda} in the remaining differential equations, we can reduce our problem to an integration in three variables: $\\Gamma$, $\\phi$ and $\\epsilon$. \n\n\\subsection{Expansion for $r\\to 0$}\\label{Sec:Exp0}\nClose to the center of the star, we can perform an analytic expansion of the form \n\\begin{equation}\\label{eq:smallr}\nf(r)=\\sum_{n=0}^\\infty f_n r^n\n\\end{equation}\nfor the functions $\\Gamma$, $\\Lambda$, $\\epsilon$, $p$ and $\\phi$.\nPlugging these expansions in the field equations, we can solve order by order to determine the boundary conditions at the origin. At this point, there are essentially three quantities that one can freely fix: the central density $\\epsilon_0$, the value of the scalar field at the center $\\phi_0$, and the value of the time component of the metric at the center, determined by $\\Gamma_0$. On the other hand, $\\Lambda_0$ has to vanish in order to avoid a conical singularity at the center, while $p_0$ is directly related to $\\epsilon_0$ by the equation of state. All higher order quantities $\\{\\Gamma_i, ...,\\phi_i\\}$, $i\\geq1$ can be determined in terms of the three quantities $\\{\\epsilon_0,\\Gamma_0,\\phi_0\\}$. We will require that spacetime is asymptotically flat, with a trivial scalar field at spatial infinity, which fixes uniquely $\\Gamma_0$ and $\\phi_0$, or rather restricts $\\phi_0$ to a discrete set of values, each corresponding to a different mode; technically, these values are found through a numerical shooting method. Therefore, for given parameters $\\alpha$ and $\\beta$, a solution is eventually fully determined by the central density $\\epsilon_0$. Different choices of $\\epsilon_0$ will translate into different masses. \n\nWe must underline the difference with the black hole case, studied in \\cite{Antoniou:2021zoy}. For black holes, the equations are scale invariant up to a redefinition of the couplings. Practically, this means that it is enough to explore the full space of parameters $\\alpha$ and $\\beta$ for a \\textit{fixed} mass. One can then deduce all solutions, of arbitrary mass, by an appropriate rescaling. For neutron stars this scaling symmetry is broken by the equation of state that relates $p$ and $\\epsilon$. Therefore, one \\textit{a priori} has to explore a 3-dimensional space of parameters ($\\epsilon_0$, $\\alpha$ and $\\beta$) in the case of neutron stars. In order to keep this exploration tractable, as it was already done in \\cite{Ventagli:2020rnx}, we will focus our study on a selection of central densities and equations of state. We pick these in order to cover very diverse solutions, typically corresponding to the lightest\/heaviest observed stars in general relativity. We then explore a wide range of the $(\\alpha,\\beta)$ parameter space for these fixed densities and equations of state.\n\nTo complete this section, let us note that solving order by order the field equations for the higher order coefficients in the expansion \\eqref{eq:smallr} does not always yield solutions. All first order coefficients in this expansion have to vanish; one can express $\\Gamma_2$, $\\epsilon_2$, $p_2$ and $\\phi_2$ in terms of $\\Lambda_2$; however, $\\Lambda_2$ itself is determined by the following equation:\n\\begin{widetext}\n\\begin{equation}\\label{eq:Lambda2}\n\\begin{split}\n& \\Lambda_2^4(512\\,\\alpha^3\\kappa\\,\\phi_0^2-256\\,\\alpha^3\\beta\\kappa^2\\phi_0^4)+\\Lambda_2^3(512\\,p_0\\alpha^3\\kappa^2\\phi_0^2-64\\,\\alpha^2\\beta\\kappa\\phi_0^2+32\\,\\alpha^2\\beta^2\\kappa^2\\phi_0^4)+\\Lambda_2^2(12\\,\\alpha\\beta^3\\kappa^2\\phi_0^4-24\\,\\alpha\\beta^2\\kappa\\phi_0^2\\\\\n& -192\\,p_0\\alpha^2\\beta\\kappa^2\\phi_0^2)+\\Lambda_2\\left(2\\,\\beta-\\frac{16}{3}\\alpha\\epsilon_0\\kappa-2\\,\\beta^2\\kappa\\,\\phi_0^2+3\\,\\beta^3\\kappa\\,\\phi_0^2+24\\,p_0\\alpha\\beta^2\\kappa^2\\phi_0^2+\\frac{8}{3}\\alpha\\beta\\epsilon_0\\kappa^2\\phi_0^2+\\frac{16}{3}\\alpha\\beta^2\\epsilon_0\\kappa^2\\phi_0^2\\right.\\\\\n&\\left.+\\frac{1}{2}\\beta^3\\kappa^2\\phi_0^4-\\frac{3}{2}\\beta^4\\kappa^2\\phi_0^4\\right)-\\frac{2}{3}\\beta\\epsilon_0\\kappa+\\frac{16}{9}\\alpha\\epsilon_0^2\\kappa^2-p_0\\beta^3\\kappa^2\\phi_0^2+\\frac{1}{3}\\beta^2\\epsilon_0\\kappa^2\\phi_0^2-\\frac{2}{3}\\beta^3\\epsilon_0\\kappa^2\\phi_0^2=0.\n\\end{split}\n\\end{equation}\n\\end{widetext}\nEquation \\eqref{eq:Lambda2} is a fourth order equation in $\\Lambda_2$. Such an equation does not necessarily possess real solutions. Therefore, for any choice of parameters $(\\alpha,\\beta)$ and initial values $(\\epsilon_0,\\phi_0)$, we need to check that a real solution to Eq.~\\eqref{eq:Lambda2} exists. In particular, we need to check this when implementing the shooting method that will allow us to find the values of $\\phi_0$ such that the scalar field is trivial at spatial infinity. Such values might actually not exist in the domain where Eq.~\\eqref{eq:Lambda2} possesses real solutions.\nIn practice, we make sure that each choice of parameters that we consider guarantees not only that Eq.~\\eqref{eq:Lambda2} has a positive\\footnote{An acceptable solution to Eq.~\\eqref{eq:Lambda2} must be positive, otherwise $g_{rr}$ diverges at a finite radius, and consequently the pressure and the energy density diverge as well.} real solution, but that such a solution is connected to the GR one. We discard all other parameter combinations that do not respect such criteria.\n\n\\subsection{Expansion at spatial infinity}\n\nWe now analyze the asymptotic behaviour of the solutions at spatial infinity. This time, we expand the metric and scalar functions in inverse powers of $r$, and solve order by order.\nWe impose that the asymptotic value of the scalar field vanishes, that is $\\phi(r\\to\\infty)\\equiv\\phi_\\infty=0$, and that $\\Gamma(r\\to\\infty)=0$. The asymptotic solution then reads\n\\begin{widetext}\n\\begin{align}\n\\begin{split}\\label{eq:Asymptotic1}\ne^{-\\Lambda}&= 1-\\frac{2M}{r}+\\frac{1}{2}\\frac{Q^2\\kappa}{r^{2}}(1-2\\,\\beta\\kappa)+\\frac{1}{2}\\frac{MQ^2\\kappa}{r^{3}}(1-3\\,\\beta)+\\frac{1}{12}\\frac{Q^2\\kappa}{r^{4}}\\left[ M^2(8-28\\,\\beta)+Q^2\\beta\\kappa(1-5\\,\\beta+12\\,\\beta^2)\\right]\\\\\n& \\quad +\\frac{1}{48}\\frac{MQ^2\\kappa}{r^{5}}\\left[ 768\\,\\alpha+8\\,M^2(6-23\\,\\beta)-Q^2\\kappa(1-18\\,\\beta+77\\,\\beta^2-156\\beta^3)\\right]+O(r^{-6}),\n\\end{split}\n\\\\\n\\begin{split}\\label{eq:Asymptotic2}\ne^\\Gamma&=1-\\frac{2M}{r}+\\frac{1}{2}\\frac{Q^2\\beta\\kappa}{r^2}+\\frac{1}{6}\\frac{MQ^2\\kappa}{r^3}(1-3\\,\\beta)+\\frac{1}{r^4}\\left[ 4\\,M^4-\\frac{1}{3}M^2Q^2\\kappa(1+3\\,\\beta)+\\frac{1}{8}Q^4\\beta^2\\kappa^2 \\right]\\\\\n& \\quad -\\frac{1}{r^5}\\left\\{ 8\\,M^5-\\frac{1}{30}M^3Q^2\\kappa(58-75\\beta)-\\frac{1}{80}M Q^2\\kappa\\left[ 512\\,\\alpha-Q^2\\kappa(3+10\\,\\beta-85\\,\\beta^2+60\\,\\beta^3) \\right] \\right\\}+O(r^{-6}),\n\\end{split}\n\\\\\n\\begin{split}\\label{eq:Asymptotic3}\n\\phi&= \\frac{Q}{r}+\\frac{MQ}{r^2}+\\frac{1}{12}\\frac{Q}{r^3}\\left[ 16\\,M^2-Q^2\\kappa(1-2\\,\\beta+3\\,\\beta^2) \\right]+\\frac{1}{r^4}\\left[ 2\\,M^3Q-\\frac{1}{12}MQ^3\\kappa(4-9\\,\\beta+9\\,\\beta^2) \\right]\\\\\n& \\quad +\\frac{1}{480}\\frac{Q}{r^5}\\big\\{Q^4\\kappa^2(9-40\\,\\beta+86\\,\\beta^2-144\\,\\beta^3+117\\,\\beta^4) -8M^2\\left[ 144\\,\\alpha+Q^2\\kappa(58-140\\,\\beta+105\\,\\beta^2) \\right] \\\\\n& \\quad + 1536\\,M^4 \\big\\} +O(r^{-6}).\\vphantom{\\dfrac{Q}{r}}\n\\end{split}\n\\end{align}\n\\end{widetext}\nwhere $M$ and $Q$ are free. We identify $M$ as the ADM mass and $Q$ as the scalar charge, in the sense that it dictates the fall-off of the scalar field far away. As one can see from Eqs.~\\eqref{eq:Asymptotic1}--\\eqref{eq:Asymptotic3}, the contribution from the Ricci coupling dominates the asymptotic behaviour of the solutions over the Gauss-Bonnet coupling. Indeed, terms proportional to $\\beta$ enter the expansion already at order $r^{-2}$, whereas $\\alpha$-dependent terms arise only at order $r^{-5}$. This expansion is in fact entangled with the boundary conditions at the center of the star, as we already mentioned. For fixed parameters $\\alpha$ and $\\beta$, the freedom in $M$ directly relates to the freedom in the central density $\\epsilon_0$. On the other hand, the fact that only discrete values of $\\phi_0$ yield a vanishing scalar field at infinity means that the scalar profile is actually fixed once a central density (or a mass) is chosen. Therefore, $Q$ is fixed as a function of $M$, and does not constitute a free charge; this is sometimes referred to as secondary hair.\n\nThe scalar charge constitutes probably the most direct channel to test the theory through observations. Indeed, binaries of compact objects endowed with an asymmetric charge will emit dipolar radiation. This enhances the gravitational-wave emission of such systems: in a Post-Newtonian (PN) expansion, dipolar radiation contributes to the energy flux at order -1PN with respect to the usual quadrupolar GR flux. Generically, this dipolar emission is controlled by the sensitivities of the compact objects, defined as\\footnote{The factor of $1\/\\sqrt{4\\pi}$ is added to match the standard definition of the sensitivity in the literature, where a different normalization for the scalar field is generally used.}\n\\begin{equation}\n    \\alpha_I=\\dfrac{1}{\\sqrt{4\\pi}}\\,\\dfrac{\\partial\\text{ln}M_I}{\\partial\\phi_0},\n\\end{equation} \n$M_I$ being the mass of the component $I$, and $\\phi_0$ the value of the scalar field at infinity. The observation of various binary pulsars, notably the PSR~J1738+0333 system, allows one to set the following constraint:\n\\begin{equation}\n   | \\alpha_A-\\alpha_B|\\lesssim2\\times10^{-3},\n      \\label{eq:DEFbound}\n\\end{equation}\nwhere $A$ and $B$ label the two components of the system \\cite{Shao:2017gwu,Wex:2020ald}. We can then relate the sensitivity to the scalar charge $Q$, using the generic arguments of \\cite{Damour:1992we}. We have\n\\begin{equation}\n   Q_I=-\\dfrac{1}{4\\pi}\\,\\dfrac{\\partial M_I}{\\partial\\phi_0}.\n   \\label{eq:DEFcharge}\n\\end{equation} \nIf there is no accidental coincidence in the charge of the two components of the binary, Eqs.~\\eqref{eq:DEFbound}-\\eqref{eq:DEFcharge} translate as\n\\begin{equation}\n\\left|\\dfrac{Q}{M}\\right|\\lesssim6\\times10^{-4}\n\\label{eq:boundQ}\n\\end{equation}\nfor the solutions we consider. Only solutions satisfying this bound on the charge to mass ratio are relevant. It is however a non-trivial task to map this bound onto the parameters of the Lagrangian \\eqref{eq:ActionCaseI}. We will do so by exploring the parameter space in Sec.~\\ref{sec:results}.\n\n\\subsection{Numerical implementation}\n\nWe solve the system of three differential equations for the three independent functions $\\Gamma$, $\\phi$ and $\\epsilon$ by starting our integration from $r_0=10^{-5}~\\text{km}$. We fix the parameters of the theory $\\alpha$ and $\\beta$, and the central density $\\epsilon_0$, typically to values of order $10^{17}$~kg\/m$^3$. Then, we give an initial guess for $\\phi_0$, and determine boundary conditions as explained in Sec.~\\ref{Sec:Exp0}. The integration will generically give a solution; however, we also demand that the scalar field vanishes at infinity, that is $\\phi_\\infty=0$. Only a discrete set of $\\phi_0$ values will yield $\\phi_\\infty=0$. Each value corresponds to a different number of nodes of the scalar field in the radial direction. In practice, we integrate up to distances $r_\\text{max}=300\\, \\text{km}$ and we implement a shooting method to select the solutions with $\\phi_\\infty=0$. Generally, we use Mathematica's built-in function FindRoot.\n\nHowever, in some cases FindRoot fails to find the right solutions, even if one gives it a limited range $(\\phi_{0,\\:\\text{min}},\\phi_{0,\\:\\text{max}})$ where to look for. When this happens, we resort to  bisection instead. In this latter case, we require that $\\phi(r_\\text{max})\/\\phi_0 \\leq 10^{-2}$. \n\nAt each stage of the shooting method, we must check that Eq.~\\eqref{eq:Lambda2} gives a real positive solution for $\\Lambda_2$ that is connected to the GR solution. In some cases, we reach the limit of the region of the parameter space where these criteria are fulfilled before reaching $\\phi_\\infty=0$. When this is the case, there is no solution associated to the given choice of $\\alpha$, $\\beta$ and $\\epsilon_0$. Note also that, given a set of $\\alpha$, $\\beta$ and $\\epsilon_0$, there is a maximum number of nodes that the solution can have, consequently a maximum number of suitable choices of $\\phi_0$ (typically up to three modes in the regions we explore). Solutions with more nodes are encountered only for higher values of the parameters $\\alpha$ and $\\beta$, or at higher curvatures (that is, at higher $\\epsilon_0$).\n\nGiven a solution, we extract the value of the ADM mass $M$ and the scalar charge $Q$, as defined in the asymptotic expansion \\eqref{eq:Asymptotic1}--\\eqref{eq:Asymptotic3}. We then have\n\\begin{equation}\n    \\begin{split}\n    & M = -\\left(\\frac{1}{2}r^2\\Lambda'\\,e^{-\\Lambda}\\right) \\bigg|_{r_\\text{max}},\\\\\n    & Q = -\\left(r^2 \\phi'\\right)\\big|_{r_\\text{max}}.\n    \\end{split}\n\\end{equation}\n\n\n\\section{NUMERICAL RESULTS}\n\\label{sec:results}\n\n\\subsection{Existence regions of scalarized solutions}\n\\label{Sec:parameterSpace}\n\nIn this section, we will study the regions where scalarized solutions exist in the $(\\alpha,\\beta)$ parameter space. We analyze three different neutron star scenarios, which correspond to the three cases studied in \\cite{Ventagli:2020rnx}.\n\n\\subsubsection{Light star with SLy EOS}\n\\label{sec:lightSLy}\n\nFirst, we consider a neutron star described by the SLy equation of state \\cite{Haensel:2004nu}, with a central energy density of $\\epsilon_0=8.1\\times 10^{17}~\\text{kg}\/\\text{m}^3$, so that its gravitational mass in GR is $M_{\\text{GR}}=1.12 M_\\odot$. The results are summarized in Fig.~\\ref{fig:Sly112}, where we relate our new results to the previous study of the scalarization thresholds \\cite{Ventagli:2020rnx}.\n\\begin{figure}[ht]\n\t\\includegraphics[width=1\\linewidth]{Sly112n1.pdf}%\n\t\\caption{Regions of existence of scalarized solutions in the $(\\alpha,\\beta)$ space, for the SLy EOS with $\\epsilon_0=8.1\\times 10^{17}~\\text{kg}\/\\text{m}^3$. The red (respectively blue) region is the region where scalarized solutions with 0 (respectively 1) node exist. We superimposed the grey contours obtained in Ref.~\\cite{Ventagli:2020rnx}, which represent the lines beyond which GR solutions with the same density are unstable to scalar perturbations with 0, 1, 2, \\textit{etc} nodes. We see that the region where there exist scalarized solutions with $n$ nodes is included in the region where the GR solutions are unstable to scalar perturbations with $n$ nodes, but much smaller. The dashed boundary for the blue region corresponds to a breakdown of the integration inside the star. In GR, a star with this choice of $\\epsilon_0$ and EOS has a light mass, $M_\\text{GR}=1.12 M_\\odot$.}\n\t\\label{fig:Sly112}\n\\end{figure}\nThe white area corresponds to the region of the parameter space where the GR solution is stable. When cranking up the parameters $\\alpha$ or $\\beta$, a new unstable mode appears every time one crosses a black line. The first mode has 0 nodes, the second 1 node, \\textit{etc}. We will refer to these black lines as \\textit{instability lines}. Any point in the parameter space that lies within some grey region corresponds to a configuration where the GR solution is unstable. \nThe red (respectively blue) area corresponds to the region where scalarized solutions with $n=0$ (respectively $n=1$) nodes exist. We do not include the equivalent regions for higher $n$, to not complicate further the analysis. The region where a scalarized solution does exist\nis considerably reduced with respect to the region where the GR solution is unstable.\n\nOne of our main results is that the parameters $(\\alpha,\\beta)$ corresponding to the grey areas that are not covered by the colored regions must be excluded. Indeed, there, scalarized solutions do not exist while the GR solution itself is unstable. Therefore, neutron stars in these theories, when they reach a critical mass, will be affected by a tachyonic instability, but there does not exist a fixed point (a static scalarized solution) where the growth could halt. This would imply that neutron stars with this mass and EOS do not exist for the corresponding parameters  of the theory \\eqref{eq:ActionCaseI}. Considering that the properties of the scalarized star are sensitive to nonlinearities, adding further nonlinear interaction terms to the action, \\textit{e.g.} self-interactions in a scalar potential, as was proposed in \\cite{Macedo:2019sem}, or non-linear terms in the coupling functions \\cite{Doneva:2017bvd,Silva:2018qhn}, can potentially change this result.\n\nIn Fig.~\\ref{fig:Sly112}, the regions where scalarized solutions exist are delimited by \\textit{existence lines}, represented by a curve of the respective color. The plain lines correspond to boundaries beyond which it is no longer possible to find a value of $\\phi_0$ that allows a suitable solution to Eq.~\\eqref{eq:Lambda2}, while providing $\\phi_\\infty=0$. Beyond dashed lines, on the other hand, nothing special occurs at the center of the star, but the numerical integration breaks down at a finite radius inside the star. We do not know whether, when crossing these dashed lines, our integration is affected by numerical problems, or whether the divergence corresponds to an actual singularity of the solutions. \nIt could be that this singularity emerges as an artifact of the method we employ. Indeed, in our analysis, we keep the central density $\\epsilon_0$ fixed while pushing the couplings $\\alpha$ and $\\beta$ to larger and larger values. However, for each couple of parameters $(\\alpha,\\beta)$, there probably exists a maximal central density beyond which star solutions do not exist, or equivalently it becomes impossible to sustain such a high central density. The dashed line could correspond to this saturation, where we try to push all the parameters beyond values that can actually be sustained by the model.\n\nA surprising feature, which is not visible in Fig.~\\ref{fig:Sly112}, is that scalarized solutions always exist in a very narrow range along the instability lines. For example, when crossing the black instability line that delimitates the white region where the GR solution is stable, from the light-grey region where it is unstable against $n=0$ scalar perturbations, there exists a very narrow band (within the grey region) where scalarized solutions with zero node exist. We observed similar behaviours along each instability line, also in the scenarios discussed in the next paragraphs. We further investigate these particular solutions in Sec.~\\ref{Sec:instabilityLines}.\n\n\\subsubsection{Light star with MPA1 EOS}\n\nWe next consider a stellar model described by the MPA1 equation of state~\\cite{Gungor:2011vq}. We choose a central energy density of $\\epsilon_0=6.3\\times 10^{17}\\,\\text{kg}\/\\text{m}^3$, such that it corresponds to the same GR mass as in the previous case, that is $M_{\\text{GR}}=1.12 M_\\odot$. We report the results in Fig.~\\ref{fig:MPA1}.\n\\begin{figure}[ht]\n\t\\includegraphics[width=1\\linewidth]{MPA1n1.pdf}%\n\t\\caption{Regions of existence of scalarized solutions in the $(\\alpha,\\beta)$ space, for the MPA1 EOS with $\\epsilon_0=6.3\\times 10^{17}~\\text{kg}\/\\text{m}^3$. The conventions are the same as in Fig.~\\ref{fig:Sly112}.  In GR, a star with this choice of $\\epsilon_0$ and EOS is again light, with $M_\\text{GR}=1.12 M_\\odot$.}\n\t\\label{fig:MPA1}\n\\end{figure}\nAs one can see, changing the EOS has only mild effects on the region of existence of scalarized solutions. The analysis of the parameter space is qualitatively the same as for the SLy EOS. The main difference is that, for the range of parameters we considered, no numerical divergences (associated with dashed lines) appear with the MPA1 EOS.\n\n\\subsubsection{Heavy star with SLy EOS}\n\nLast, we consider a denser neutron star described by the SLy EOS, with $\\epsilon_0=3.4\\times 10^{18}\\,\\text{kg}\/\\text{m}^3$. It corresponds to an increased mass in GR of $M_{\\text{GR}}=2.04 M_\\odot$. The results are shown in Fig.~\\ref{fig:SLy204}.\n\\begin{figure}[ht]\n\t\\subfloat{\\includegraphics[width=\\linewidth]{SLy204.pdf}%\n\t}\n\t\\\\\n\t\t\\subfloat{%\n\t\\includegraphics[width=\\linewidth]{SLy204Zoom.pdf}%\n\t}\n\t\\caption{Regions of existence of scalarized solutions in the $(\\alpha,\\beta)$ space, for the SLy EOS with $\\epsilon_0=3.4\\times 10^{18}~\\text{kg}\/\\text{m}^3$. The conventions are the same as in Fig.~\\ref{fig:Sly112}. In GR, a star with this choice of $\\epsilon_0$ and EOS is the heaviest possible, $M_\\text{GR}=2.04 M_\\odot$. The bottom panel is simply a zoom of the upper one.}\n\t\\label{fig:SLy204}\n\\end{figure}\nIn this case, positive values of $\\beta$ can also lead to scalarized solutions. Already in \\cite{Mendes:2014ufa,Palenzuela:2015ima,Mendes:2016fby,Ventagli:2020rnx}, it was shown that, in GR, dense neutron possess a negative Ricci scalar towards the center, which allows for scalarization to be triggered even when $\\beta>0$. As before, a dashed line signals the appearance of divergences, which in this case show up already for the $n=0$ node.\n\nIn the lower panel of Fig.~\\ref{fig:SLy204}, we zoomed on the region of small couplings, in order to understand better what happens for natural values of the Ricci coupling $\\beta$. In the absence of the Gauss-Bonnet coupling, scalarization can occur either if $\\beta<-8.55$, or $\\beta>11.5$. Let us concentrate on the $\\beta>0$ scenario, which is motivated by the results of Ref.~\\cite{Antoniou:2020nax}, where it was shown that positive values of $\\beta$ make GR a cosmological attractor. We remind that black hole scalarization (at least for non-rotating black holes) occurs for $\\alpha>0$. Hence, we see that there exists an interesting region in the $\\alpha>0,~\\beta>0$ quadrant where even very compact stars do not scalarize, while black holes do. Such models can therefore \\textit{a priori} pass all binary pulsar tests, while being testable with black hole observations. On the other hand, for $\\beta\\gtrsim11.5$, the red region where GR solutions are replaced by scalarized solutions spreads very fast in the $\\alpha$ direction, and one has to be careful, when considering black hole scalarization, that such models are not already excluded by neutron star observations.\n\nSo far, we established the regions where scalarized solutions exist in the parameter space. In the next two sections, we will discuss the properties of these solutions, in particular their scalar charge and their mass. We separate this study into two cases: $\\beta<0$ (Sec.~\\ref{Sec:betaNeg}) and $\\beta>0$ (Sec.~\\ref{Sec:betaPos}); indeed, these two situations have different motivations and observational interests.\n\n\\subsection{Mass and scalar charge of the $\\beta<0$ solutions}\n\\label{Sec:betaNeg}\n\nWe now focus on the scenario where $\\beta<0$. This corresponds to the original situation studied by Damour and Esposito-Far\u00e8se. Typically, scalarized solutions with $\\beta<0$ and $\\alpha=0$ are extremely constrained by binary pulsar observations \\cite{Freire:2012mg,Antoniadis:2013pzd,Shao:2017gwu}. A particular motivation to study solutions with $\\beta<0$ is therefore to determine whether the addition of a non-zero Gauss-Bonnet coupling can improve their properties. We will consider three different choices of the Ricci coupling: $\\beta=-5.5,-10$ and $-100$. The two first choices are relevant astrophysically: $\\beta=-5.5$ is approximately the value where scalarization is triggered for small Gauss-Bonnet couplings, while $\\beta=-10$ corresponds to a region where neutron stars are scalarized, but with rather small deviations with respect to GR. The third choice, $\\beta=-100$, is certainly disfavored observationally, but it will allow us to illustrate an interesting behaviour concerning different scalar modes.\n\nLet us start with the comparison between the cases $\\beta=-5.5$ and $-10$. The results are summarized in Fig.~\\ref{fig:smallCoup}.\n\\begin{figure*}[ht]\n\\begin{center}\n\t\\subfloat{%\n\t\\includegraphics[width=0.4\\linewidth]{dMalpha1v1N.pdf}%\n\t}\n\t\\subfloat{%\n\t\\includegraphics[width=0.4\\linewidth]{dMalpha2v1N.pdf}%\n\t}\n\t\\\\\n\t\\subfloat{%\n\t\\includegraphics[width=0.4\\linewidth]{QMalpha1v1.pdf}%\n\t}\n\t\\subfloat{%\n\t\\includegraphics[width=0.4\\linewidth]{QMalpha2v1N.pdf}%\n\t}\n\t\\caption{Mass difference and scalar charge of scalarized solutions for $\\beta<0$. The two left (respectively right) panels show how these quantities evolve when varying $\\alpha$ at fixed $\\beta=-5.5$ (respectively $-10$). The scalar charge $Q$ (bottom panels) is normalized to the total mass of the solutions, $M$. For all curves, the mass difference $\\delta M$ (upper panels) is computed with respect to a GR star with the same central density and EOS. Plain curves correspond to a GR mass of $1.12~M_\\odot$, using the SLy EOS; dashed curves to the same GR mass, but the MPA1 EOS; and dotted-dashed curves to a GR mass of $2.04~M_\\odot$, using the SLy EOS. In this region of the parameter space, only solutions with 0 nodes for the scalar field exist. A generic feature of lighter stars (plain and dashed curves), is that the charge decreases when $\\alpha$ increases, \\textit{a priori} offering a way to evade the stringent bound of Eq.~\\eqref{eq:boundQ} when increasing $\\alpha$. However, it is only for values of $\\beta$ very close to the DEF threshold ($\\beta=-5.5$) that we can obtain scalar charges compatible with observations.\n\t}\n\t\\label{fig:smallCoup}\n\\end{center}\n\\end{figure*}\nThis figure shows two properties of scalarized stars. First, the mass default (or excess) of scalarized stars with respect to GR stars with the same central density and EOS: \n$\\delta M=M-M_{GR}$. \nSecond, the scalar charge of the scalarized solutions, $Q$. We compare the results for the three different stellar models considered in Sec.~\\ref{Sec:parameterSpace}, for the two values of $\\beta$. \nAll curves extend only over a finite range of $\\alpha$. Indeed, passed a certain value of $\\alpha$, we exit the red region on the $\\beta<0$ side of Figs.~\\ref{fig:Sly112}, \\ref{fig:MPA1} and \\ref{fig:SLy204} (moving vertically, since $\\beta$ is fixed to $-5.5$ or $-10$). Scalarized solutions do not exist outside of this region. \n\nFigure \\ref{fig:smallCoup} shows that the choice of EOS does not affect much the properties of the scalarized solutions.\nHowever, increasing the density drastically modifies these properties. In particular, at higher densities, there exist solutions with $\\delta M>0$. This can appear problematic at first. Indeed, one expects that, in a scalarization process, energy is stored in the scalar field distribution. Hence, the ADM mass, that constitutes a measure of the gravitational energy, should decrease in the process. \nHowever, we stress that we are not studying a dynamical process. Indeed, the stars for which we are computing the mass difference $\\delta M$ have, by construction, the same central energy density $\\epsilon_0$. In the scalarization process of a GR neutron star, the central energy density will not remain fixed. Hence, our results do not necessarily mean that a star will gain mass when undergoing scalarization.\n\nPerhaps more interestingly for observations, Fig.~\\ref{fig:smallCoup} also shows the behaviour of the scalar charge. For the light neutron stars, the scalar charge always decreases when $\\alpha$ increases. Therefore, the constraint on the scalar charge, Eq.~\\eqref{eq:boundQ}, disfavors the solutions with $\\alpha<0$ with respect to standard DEF ($\\alpha=0$) solutions. On the contrary, one could hope that a positive Gauss-Bonnet coupling could help evade these constraints even for $\\beta<-5.5$, by quenching the charge. Effectively, there will be a direction in the $\\alpha>0$ and $\\beta<0$ quadrant where the effects of the two operators, Ricci and Gauss-Bonnet, combine to yield a small scalar charge.\nThis interesting possibility is moderated by what happens in the case of denser stars (dotted-dashed line in Fig.~\\ref{fig:smallCoup}). For large negative values of the Ricci coupling ($\\beta=-10$), the scalar charge does not have a monotonic behaviour with $\\alpha$. In particular, as shown in the bottom-right panel of Fig.~\\ref{fig:smallCoup}, $Q$ starts increasing for positive values of $\\alpha$. Even at the point where $Q$ is minimal, its value ($Q\/M\\simeq8\\times10^{-3}$) already exceeds the bound of Eq.~\\eqref{eq:boundQ}. Therefore, it is only for values of $\\beta$ that are very close to the DEF threshold $\\beta\\simeq-5.5$, that the addition of the Gauss-Bonnet coupling can help to reduce the scalar charge, and to pass the stringent binary pulsar tests.\n\nTo conclude the study of the $\\beta<0$ region, we consider a significantly more negative Ricci coupling, namely $\\beta=-100$. To illustrate what happens at these large negative values of $\\beta$, it is enough to consider one scenario, for example the one of lighter neutron stars with the SLy EOS. For such negative values of $\\beta$, there exist several scalarized solutions, with different number of nodes. We can then compare the mass difference of these solutions between each other. Figure \\ref{fig:betaNeg100} shows that, for $\\alpha>\\alpha_\\text{c}\\simeq350\\, \\text{km}^2$, scalarized solutions with 1 node become lighter than scalarized solutions with 0 node.\n\\begin{figure}[ht]\n\t\\includegraphics[width=0.75\\linewidth]{dMalpha6v1N.pdf}\n\t\\caption{Mass difference $\\delta M$ vs $\\alpha$ at $\\beta=-100$. The EOS considered here is the SLy one, with $\\epsilon_0=8.1\\times 10^{17}\\,\\text{kg}\/\\text{m}^3$, which in GR corresponds to $M_\\text{GR}=1.12~M_\\odot$. The color and dashing conventions is the same as in Fig.~\\ref{fig:smallCoup}. We have more modes in this region of parameter space, that we represent as dotted-dashed (for $n=1$ node) and dashed (for $n=2$ nodes) curves. For $\\alpha\\gtrsim350\\, \\text{km}^2$, solutions with 1 node start having a smaller mass than solutions with 0 node, which can indicate that solutions with 1 node are more energetically favored.}\n\t\\label{fig:betaNeg100}\n\\end{figure}\nThis is a hint that, for $\\alpha>\\alpha_c$, the one node solution will be preferred energetically to the zero node solution. We cannot conclude definitively on this issue, as the ADM mass does not take into account the energy stored in the scalar distribution (which is non-zero for the two scalarized solutions). However, in the regime where this inversion happens, the mass difference with respect to GR, $\\delta M$, is rather small. If our interpretation in terms of energetic preference is correct, the transition from a preferred solution with zero node to a solution with one node is interesting. Indeed, the scalarized solution with zero node is associated with the fundamental mode of the GR background instability. At the perturbative level, all the other modes of instability have higher energies. It would then be natural to expect that, at the non-linear level of scalarized solutions, this energy hierarchy is respected. This is the case up to $\\alpha=\\alpha_\\text{c}$, but not anymore beyond. In Sec.~\\ref{Sec:EffMass}, we provide a putative explanation for this inversion: that for $\\alpha>\\alpha_\\text{c}$, the profile of the effective mass over the GR background tends to favor the growth of scalar field solutions with one node, rather than zero.\n\n\\subsection{Mass and scalar charge of the $\\beta>0$ solutions}\\label{Sec:betaPos}\n\nWe now consider the case of positive $\\beta$. Such solutions are less constrained by observations than their $\\beta<0$ counterparts. They are also very interesting from a cosmological perspective, where $\\beta>0$ allows a consistent history throughout different epochs \\cite{Antoniou:2020nax}. We have seen in Sec.~\\ref{Sec:parameterSpace} that, among the three different possible neutron star configurations we focus on, only the denser one leads to scalarized solutions for $\\beta>0$. In Fig.~\\ref{fig:50M204}, we show the mass difference $\\delta M$ and scalar charge $Q$ as functions of $\\alpha$ when $\\beta=50$.\n    \\begin{figure}[ht]\n\t\\subfloat{\\includegraphics[width=0.75\\linewidth]{dMalpha4v1N.pdf}}\n\t\\\\\n\t\\subfloat{\\includegraphics[width=0.75\\linewidth]{QMalpha4v1N.pdf}}      \n\t\\caption{Mass difference and scalar charge of scalarized solutions for $\\beta>0$ ($\\beta=50$ here). Among the three neutron star scenarios that we considered throughout the paper, only the heavier star ($\\epsilon_0=5.51 \\times 10^{-3}$~kg\/m$^3$, $M_\\text{GR}=2.04~M_\\odot$, SLy EOS) possesses some scalarized solutions in this region. The dashing convention is the same as in Fig.~\\ref{fig:betaNeg100}. Solutions that correspond to the interval of $\\alpha$ centered on 0 are interesting observationally, as they yield very small scalar charges, compatible with Eq.~\\eqref{eq:boundQ}.}\n\t\\label{fig:50M204}\n\\end{figure}\nNote that scalarized solutions with zero node exist over two disconnected ranges of $\\alpha$ ($-44~\\text{km}^2<\\alpha<57~\\text{km}^2$ and $174~\\text{km}^2<\\alpha<522~\\text{km}^2$). In the gap, GR solutions are stable and no scalarized solutions exist. This is obvious from Fig.~\\ref{fig:SLy204}, taking a cut along the vertical line $\\beta=50$. \n    \nOver the first interval, $\\alpha$ is rather small and the scalarization process is dominated by the negative Ricci scalar. For strictly vanishing $\\alpha$, the scalarization phenomenon with $\\beta>0$ has already been examined in \\cite{Mendes:2014ufa,Palenzuela:2015ima,Mendes:2016fby}. Here, we find that, in the interval of small values of $\\alpha$, the scalar charges of the $n=0$ solutions (as well as of the $n=1$ solutions) are very small. Typically, $Q\/M \\simeq 10^{-4}-10^{-5}$, compatible with Eq.~\\eqref{eq:boundQ}. Hence, all solutions  with $\\beta>0$ and rather small values of $\\alpha$ are interesting observationally: they display either no scalarization effects for neutron stars (for $\\beta\\lesssim 11.51$) or very mild scalar charges (for $\\beta\\gtrsim 11.51$). At the same time, they allow for a consistent cosmological history; finally, together with positive values of $\\alpha$, they will generically give rise to black hole scalarization, as studied in detail in \\cite{Antoniou:2021zoy}. In this region of parameter space, we can therefore hope to discover scalarization effects in the future gravitational-wave signals of binary black holes, that are either absent or suppressed in the case of neutron stars.\n    \nOver the second interval ($174~\\text{km}^2<\\alpha<522~\\text{km}^2$), the contribution of the Gauss-Bonnet invariant tends to dominate, and the scalar charges are more significant, as one can immediately notice in Fig.~\\ref{fig:SLy204}. Such setups are not compatible with Eq.~\\eqref{eq:boundQ}, and therefore less interesting phenomenologically.\n\n\\subsection{Scalarized solutions along the instability lines}\\label{Sec:instabilityLines}\n\nAs we mentioned at the end of Sec.~\\ref{sec:lightSLy}, a generic feature that is not observable in Figs.~\\ref{fig:Sly112}, \\ref{fig:MPA1} and \\ref{fig:SLy204}, is that scalarized solutions are present in a tiny band close to each instability line.\nLet us illustrate this with the light star model (with SLy EOS), that is the one which corresponds to Fig.~\\ref{fig:Sly112}. For simplicity, we also restrict our study to solutions with $\\beta=0$ (\\textit{i.e.}, we take a cut along the vertical axis in Fig.~\\ref{fig:Sly112}). The characteristics of the solutions are shown in Fig.~\\ref{fig:beta0}.\n\\begin{figure}[ht]\n\t\\subfloat{\\includegraphics[width=0.75\\linewidth]{dMalpha5v1N.pdf}}\n\t\\\\\n\t\\subfloat{\\includegraphics[width=0.75\\linewidth]{QMalpha5v1N.pdf}}      \n\t\\caption{Mass difference and scalar charge of the scalarized solutions along the instability lines, for $\\beta=0$. The scenario considered here corresponds to $\\epsilon_0=8.1\\times 10^{17}~\\text{kg}\/\\text{m}^3$ ($M_\\text{GR}=1.12M_\\odot$) together with the SLy EOS. Solutions with zero node acquire a significant charge and mass difference, and are apparently disconnected from GR when they appear while increasing $\\alpha$ towards positive values. Solutions with $n=1$ nodes are very close to GR, with a small charge and mass difference. Since they extend only over a small range of $Q$ and $\\delta M$, they are difficult to spot. They lie at the upper left (respectively lower left) of the top (respectively bottom) panel.\n\t}\n\t\\label{fig:beta0}\n\\end{figure}\nScalarized solutions with zero nodes (the ones lying close to the $n=0$ instability line of the GR solution) have a characteristic mass difference and scalar charge which is not particularly small. It is of the same order as for the solutions we previously examined (Figs.~\\ref{fig:smallCoup}--\\ref{fig:50M204}). They also exhibit a surprising behaviour: when increasing $\\alpha$ progressively from 0 towards positive values, the mass and scalar charge suddenly deviate from GR, instead of being smoothly connected; further increasing $\\alpha$, $\\delta M$ and $Q$ then tend to decrease. This behaviour is significantly different from what we could observe in Figs.~\\ref{fig:smallCoup}--\\ref{fig:50M204}.\n \n Solutions with more nodes ($n=1$, 2, 3...) exhibit a clear feature: they deviate very slightly from GR in terms of mass, and acquire only a small scalar charge (typically $\\delta M < 10^{-2}$ and $Q\/M < 10^{-4}$). We verified this behaviour for all higher nodes admitted; however, for simplicity, in Fig.~\\ref{fig:beta0} we show only the case $n=1$. This feature can be understood as follows; close to some instability line (on the unstable side), an unstable mode of the effective potential associated with the GR solution has just appeared. A very small deformation of the potential can therefore easily restore the equilibrium. This deformation can be caused by the back-reaction of the scalar onto the metric: the instability is triggered, the scalar field starts growing, but it immediately back-reacts on the potential, making it shallower and suppressing the instability. Clearly, such a behaviour can only happen close to instability lines, where a specific mode is on the edge of stability.\n\n\\subsection{Predicting the scalar profile of scalarized stars from GR solutions}\\label{Sec:EffMass}\n\nWe will conclude this study by arguing that, already at the perturbative level of the GR solution, we can identify an influence on the profile of the scalar field in the fully scalarized solution. To this end, let us focus on the effective mass given in Eq.~\\eqref{eq:eff_masss}, $ m_\\text{eff}^2=\\beta R\/2-\\alpha \\mathscr{G}$. This is a radially dependent quantity, and the scalar field is most likely to grow at radii where $m_\\text{eff}^2$ is most negative. In particular, it is natural to expect that, if $m_\\text{eff}^2$ has a minimum at $r=0$, this will favor a monotonic profile for the scalar field, and hence an $n=0$ type of solution. On the contrary, if $m_\\text{eff}^2$ has a minimum at $r>0$, this favors a peaked profile for the scalar field, which is more common in $n\\geq1$ solutions. Let us illustrate this with a concrete example. We will\nconsider the scenario that corresponds to $M_\\text{GR}=1.12 M_\\odot$, together with the SLy EOS, and two choices of $\\beta$: $\\beta=-10$ and $\\beta=-100$. In the first case, only solutions with 0 node exist; in the second case, we can construct solutions with 0 or 1 node.\n\nWe first focus on the case $\\beta=-10$. The Ricci scalar is everywhere positive over the background we consider, with a maximum at $r=0$; hence, $\\beta R$ contributes negatively to the squared mass, favouring the growth of the scalar field close to the center. The Gauss-Bonnet scalar, on the other hand, is negative in the central region of the star, and becomes positive towards the surface. Therefore, $-\\alpha\\mathscr{G}$ reinforces the effect of $\\beta R$ if $\\alpha<0$, while couterbalancing it if $\\alpha>0$. This is illustrated in the top panel of Fig.~\\ref{fig:EffMass2}.\n\\begin{figure}[ht]\n\t\\subfloat{\\includegraphics[width=0.7\\linewidth]{mEff2.pdf}}\n\t\\\\%\n\t\\subfloat{\\includegraphics[width=0.7\\linewidth]{phi2N.pdf}}\n\t\\caption{Upper panel: radial profile of the effective mass squared over the GR background, using the SLy EOS and a central density $\\epsilon_0=8.1\\times 10^{17}\\,\\text{kg}\/\\text{m}^3$ (yielding $M_{\\text{GR}}=1.12 M_{\\odot}$), for $\\beta=-10$ and $\\alpha=\\pm200$~km$^2$; Lower panel: radial profile of the scalar field, this time in the fully scalarized solution with the same EOS, central density, and Lagrangian parameters. The radial coordinate is normalized by $R_\\text{s}$, the radius of the star surface. In the lower panel, the scalar field is normalized to its central value for $\\alpha=-200\\, \\text{km}^2$. When the minimum of $m_\\text{eff}^2$ is shifted to $r>0$, so is the peak of $\\phi$.}\n\t\\label{fig:EffMass2}\n\\end{figure}\nThe bottom panel shows the scalar profile of the fully scalarized solutions associated with the same parameters. In this range of parameters, only solutions with 0 node are allowed (as one can check in Fig.~\\ref{fig:Sly112}); hence, pushing the minimum of $m_\\text{eff}^2$ away from the center cannot favour $n=1$ solutions, which do not exist. Still, we notice that positive $\\alpha$ values, which have the effect of displacing the minimum of $m_\\text{eff}^2$ to $r>0$, also displace the peak of the scalar field to $r>0$. The peak of the scalar field is located approximately at the minimum of $m_\\text{eff}^2$. Again, one must be careful in the comparison of the two panels, as one of them corresponds to a GR star while the other one corresponds to a scalarized star. However, our analysis seems to capture what happens during the transition from the GR to the scalarized branch.\n\nTo illustrate better the transition between $n=0$ and $n=1$ solutions, let us now consider the case $\\beta=-100$. \nThe qualitative discussion about the effect of $\\beta R$ and $-\\alpha\\mathscr{G}$ over the effective mass is exactly the same as in the previous case. We will therefore consider again a large negative and a large positive value of $\\alpha$, as well as an intermediate one: $\\alpha=-2000, \\,350$ and 1500~km$^2$. Note that the intermediate value corresponds to $\\alpha_\\text{c}$ in Sec.~\\ref{Sec:betaNeg}, the critical value at which scalarized stars with $n=0$ node become more massive (and hence probably less stable) than those with $n=1$ node. We show the results in Fig.~\\ref{fig:EffMass3}.\n\\begin{figure}[ht]\n\t\\subfloat{\\includegraphics[width=0.735\\linewidth]{mEff3.pdf}}\n\t\t\\\\%\n\t\\subfloat{\\includegraphics[width=0.735\\linewidth]{phi3n0N.pdf}}\n\t    \\\\%\n\t\\subfloat{\\includegraphics[width=0.735\\linewidth]{phi3n1N.pdf}}\n\t\\caption{Upper panel: radial profile of the effective mass squared over the GR background, using the SLy EOS and a central density $\\epsilon_0=8.1\\times 10^{17}\\,\\text{kg}\/\\text{m}^3$ (yielding $M_{\\text{GR}}=1.12 M_{\\odot}$), for $\\beta=-100$ and $\\alpha=-200$, 350 or 1500~km$^2$; Center (respectively lower) panel: radial profile of the scalar field solution with 0 (respectively 1) node in the fully scalarized solution with the same EOS, central density, and Lagrangian parameters. The normalization is similar to the one of Fig.~\\ref{fig:EffMass2}. When increasing $\\alpha$, the minimum of $m_\\text{eff}^2$ is progressively shifted from $r=0$ to a finite radius, alternatively favoring the growth of $n=0$ and $n=1$ solutions.}\n\t\\label{fig:EffMass3}\n\\end{figure}\nThe top panel shows the profile of the effective scalar mass. It behaves exactly as in the case $\\beta=-10$, with a minimum at $r=0$ for negative values of $\\alpha$, which is progressively shifted to larger radii when we increase $\\alpha$. For the parameters we chose, this time, both solutions with zero and one node exist. In the center (respectively bottom)  panel of Fig.~\\ref{fig:EffMass3}, we show the $n=0$ (respectively $n=1$) solutions. In Sec.~\\ref{Sec:betaNeg}, we stated that for $\\alpha<\\alpha_c$ we expected that the zero node solution will be energetically preferred over the one node solution, and vice-versa for $\\alpha>\\alpha_c$. The profiles of the effective mass squared give a complementary argument that strengthens this expectation. Indeed, for $\\alpha=-2000\\,\\text{km}^2\\ll\\alpha_c$ the shape of $m_\\text{eff}^2$ favours a scalar solution with a maximum at the center of the star, which decays monotonically with $r$, \\textit{i.e.} a $n=0$ solution. For $\\alpha=1500\\,\\text{km}^2\\gg\\alpha_c$, the tachyonic instability is still triggered inside the star, but away from the center. Thus, we expect that a solution with one node will be favoured. The transition between a minimum at $r=0$ and $r>0$ indeed seems to occur around $\\alpha_\\text{c}$.\n\n\n\\section{Conclusions}\n\\label{sec:discussion}\n\nWe have explored scalarized neutron stars when couplings between the scalar field and both the Ricci and the Gauss-Bonnet invariants are present. This completes the analysis initiated in~\\cite{Andreou:2019ikc,Ventagli:2020rnx}, where all the terms contributing to the onset of scalarization were identified, and continued in~\\cite{Antoniou:2021zoy} with the study of scalarized black holes in this minimal setup.\n\n\nWe have identified the regions of parameter space where solutions exist, considering three different stellar scenarios which correspond to different central densities and EOS. Although we have considered only a limited number of different central densities, we have selected the ones that correspond to the lowest\/largest neutron star mass in GR, in order to cover very different setups. The regions where scalarized solutions exist are systematically smaller than the ones where the GR branch is tachyonically unstable. The complementary regions, where the GR solution is unstable while no scalarized solution exists, should be excluded.\n\nWe then investigated in detail the physical characteristics of the scalarized solutions. In general, large parameters ($|\\beta|\\gg1$ or $|\\alpha|\\gg L^2$, where $L\\simeq10$~km is the typical curvature scale) lead to scalar charges that would be in conflict with binary pulsar constraints. However, it is interesting to notice that solutions with $\\beta>0$ and reasonably small $\\alpha$ (typically $|\\alpha|\\lesssim 50$~km$^2$) lead either to stable GR configurations, or to scalarized stars with small charges. Remarkably, this is the region of the $(\\alpha,\\beta)$ parameter space for which GR is a cosmological attractor \\cite{Antoniou:2020nax} and black holes scalarization can take place \\cite{Antoniou:2021zoy}. Therefore, it is possible to construct scalarization models that are consistent with current observations, while still having interesting strong field phenomenology. It's worth noting that future gravitational-wave observations, such as for instance the observations of extreme mass ratio inspirals by LISA~\\cite{Maselli:2020zgv, Maselli:2021men}, will reach the precision to measure small scalar charges for neutron stars and black holes.\n\nWe have also discovered that scalarized solutions systematically exist near the thresholds that delimit the stability of the GR solutions, and provided a putative explanation for this. Finally, we have shown that the profile of the effective mass at the GR level can foster the growth of certain modes with respect to others.\n\nAn obvious continuation of the present work is the stability analysis of the scalarized solutions, both the neutron stars presented here and the black holes investigated in~\\cite{Antoniou:2021zoy}.\nIt will also be interesting to combine the bounds coming from neutron star and black hole observations with the theoretical constraints that relate to the requirement that scalarization models have a well-posed initial value problem \\cite{Ripley:2020vpk}. So far, the combined theory with both Ricci and Gauss-Bonnet couplings has not been studied in detail from the initial value problem perspective. Finally, rotation is known to have important effects on black hole scalarization with a Gauss-Bonnet coupling, either quenching it (for $\\alpha>0$ \\cite{Cunha:2019dwb,Collodel:2019kkx}) or triggering it (for $\\alpha<0$ \\cite{Dima:2020yac}). The effect of rotation on neutron star scalarization was investigated in the framework of the DEF model \\cite{Doneva:2013qva}. It would be interesting to extend this analysis to coupled Ricci\/Gauss-Bonnet couplings, or pure Gauss-Bonnet ones.\n\n\n\\begin{acknowledgments}\n G.A. acknowledges partial support from\nthe Onassis Foundation.\nThis project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101007855.\nA.L. thanks FCT for financial support through Project~No.~UIDB\/00099\/2020.\nA.L. acknowledges financial support provided by FCT\/Portugal through grants PTDC\/MAT-APL\/30043\/2017 and PTDC\/FIS-AST\/7002\/2020.\nT.P.S. acknowledges partial support from the STFC Consolidated Grants No. ST\/T000732\/1 and No. ST\/V005596\/1. \nWe also acknowledge  networking support by the GWverse COST Action\nCA16104, ``Black holes, gravitational waves and fundamental physics.''\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\nIn {\\it Harry Potter and the Deathly Hallows} \\cite{rowling:HarryPotter:2007},\nJ.~K.~Rowling tells about the origin of the famous piece of cloaking tissue that\nallows Harry Potter, the renowned young wizard apprentice, to hide from unwanted company.\nThis is just an example taken from modern literature of how the idea of optical cloaking and\ninvisibility, the capability to hide objects from sight (people in this case), has permeated popular culture\nin recent times, although there are many more examples.\nFor instance, as is well known, a number of optical illusions in fashion during the XIXth and XXth centuries were\nbased on the reflection properties of mirrors and window glasses \\cite{steinmeyer-bk,zepf:physteach:2004}.\nIt is perhaps because of this capability to both fascinate and amaze at the same time, that the scientific\ncommunity has also shown much interest for the cloaking phenomenon and its possible applications\n\\cite{azanna:optica:2018}, from the stealth technology (widely used in the military\nindustry in aircrafts or warships, for example) to the design and fabrication of metamaterials\n\\cite{engheta:PRE:2005,engheta:PRE:2006,smith:Science:2006,shalaev:NatPhot:2007}.\n\nIn general, (optical) cloaking conditions are related to a particular geometrical design of the\nobject to be hidden or the cloaking device that is going to be used to hide the object.\nConversely, such a design strongly depends on the level at which cloaking is required.\nThis can be done either at the highly sophisticated level of electromagnetic optics\nor at the more elementary level of geometrical optics.\nAt the level of electromagnetic optics, this implies fabricating specific arrangements\nthat produce a direct, local influence on the phase of the incident light.\nThis is the basic principle behind the possibility to produce cloaking metamaterials, of much\ninterest at present, although out of the scope here.\nNevertheless, just to provide a quick glimpse on the issue, it is worth mentioning that\nin these new materials cloaking emerges from the collective behavior of repetitive\npatterns of assemblies of tiny units, smaller than the typical wavelengths they intend to\ninfluence, instead of the material itself that such units are made of.\nBy means of a specific design of such assemblies, the electric and magnetic properties\nof the material can be controlled, achieving narrow electromagnetic spectral bands,\ninfinite phase velocities, or negative refractive indexes.\nAt a pedagogical level, there have been some proposals to introduce the physics of\nmetamaterials in simple terms with the aid of inexpensive experimental setups, based\non elements available in any undergraduate teaching laboratory and the use of\nmicrowaves \\cite{marques:AJP:2011,fleming:AJP:2017} or sound waves \\cite{gennaro:AJP:2016}.\nOther authors have approached the issue, particularly cloaking properties, from a more\ntheoretical perspective, through easy-to-implement numerical tools \\cite{thompson:AJP:2008} or with\njust some elementary theory \\cite{longhi:AJP:2017}, also appropriate at the undergraduate level.\n\nAt the level of geometrical optics, on the other hand, the cloaking effect can\nbe achieved in a far simpler fashion, involving either multiple reflections \\cite{choi:ApplOpt:2014,howell:video}\nor the imaging properties of lens systems \\cite{choi:OptExp:2014,choi:OptExp:2015,howell:video2}.\nIn either case, the working principle is the same: the light coming from an object is redirected in a way\nthat it reaches the observer's eyes without being affected by the presence of an interposed element,\nwhich remains unnoticeable for the observer.\nThis is the case, for instance, of the optical cloaking device (OCD) proposed in Ref.~\\cite{choi:OptExp:2014},\nwhich is based on the very basic contracting\/expanding property of a two-lens system.\nMore specifically, the waist of an incident light beam coming from the object is reduced\nby a telescope-like two-lens configuration and, then, after the light has traveled a certain\ndistance, the inverted version of the same two-lens configuration expands again the beam waist\nto its initial size, which is eventually what the observer is looking at.\nIn the middle part of the OCD, between the reducer and the expander two-lens systems, the narrowness\nof the beam allows to accommodate around it whatever object we wish to hide, because its presence\nwill not be noticed by the observer.\nA nice video demonstration of the experiment can be seen in \\cite{howell:video2}.\nThe same experiment can also be performed with standard lenses, available in any teaching\noptics laboratory.\nAnalogous results can also be obtained with cheaper versions, such as the one specially prepared\nin our teaching laboratory for a popular science TV show \\cite{orbita-laika}.\n\nBecause of its simplicity, it is clear that for an optics instructor the aforementioned\nexperiment is very appealing at an elementary teaching level, as a way to introduce the concept\nof optical cloaking in the classroom in simple terms: just a simple arrangement of a few lenses\nand basic theory illustrates very nicely an application of the principles of geometrical optics\nin paraxial form, beyond other conventional examples that have been used in optics\nlaboratories for decades to demonstrate the functioning of telescopes, microscopes,\nphoto-cameras, the eye, or other optical instruments.\nFor undergraduate students, on the other hand, these experiments would constitute a beneficial\nfirst-approach to the phenomenon, since it is introduced at a very basic level, without getting into\nvery sophisticated mathematics or the use of specific optics software.\nNow, the question here is how to design an experiment in a way that can be quantitatively meaningful,\nis easy to be described and analyzed theoretically, and, very important, does not require expensive\n(professional) laboratory material, which is what one usually has at hand in typical teaching\nlaboratories.\nThe setup proposed in \\cite{choi:OptExp:2014,choi:OptExp:2015,howell:video2} produces\nvery spectacular results (just what everyone is expecting from this kind of experiences), but our\nexperience in the laboratory with this kind of OCD is that students lose track very quickly of the\nphysics behind it, because eventually all is based on apparatent sizes, semi-qualitative analyses\nand, of course, the funny cloaking effect when they interpose objects (or even their hands and\nfaces).\nHence, instead, we considered an observer-independent alternative setup, more appropriate\nto conduct quantitative measurements and to study the physics involved (and less prone to\ndistractions), where cloaking is not directly observed by looking through, but is analyzed in terms\nof the effects the hidden object makes on the image of the observed object.\n\nHere, we report on theory and measurements performed with the above mentioned OCD.\nThe work combines a preliminary theoretical\nanalysis with a subsequent experimental development of a simple OCD\nThe theoretical analysis is based on the transfer matrix method applied to optics \\cite{pedrotti-bk},\nwhich allows the student to understand the physics of the optical cloaking phenomenon by\ninvestigating imaging through a lens system in a relatively simple fashion (no high knowledge\non the issue is actually required, but just simple matrix algebra).\nThis analysis allows to determine optimal cloaking conditions.\nAccordingly, the simpler, optimal configuration consists of two sets of lenses with different\nfocal lengths arranged in the form of two Kepler-type telescopes faced by their eyepieces.\nThe analysis also provides the distance that should be considered between those ``eyepieces''.\nWith these data, the device is then mounted with cheap resources (as we have confirmed,\nthere is no need for high-quality, expensive lenses to observe the phenomenon) available in\nany teaching laboratory.\nIn this regard, we have noticed that, even with the lowest-quality lenses available in our laboratory\n(formerly used to study the effects of aberrations), the amount of cloaking obtained is already\nremarkable.\nFurthermore, the presence of spherical and chromatic aberrations is taken advantageously\nto determine unambiguously the limits of our optimal cloaking conditions (beyond them,\naberration effects start dominating very quickly the image observed).\n\nThe work has been organized as follows.\nTo be self-contained, in Sec.~\\ref{sec2} the basic theory on the transfer matrix method is\nintroduced, which is based on Gaussian paraxial optics (small-angle approximation) and\ntherefore does not require expert-level knowledge.\nAn analysis and discussion of the application of this approach to systems with increasing\nnumber of lenses is also presented in order to settle down the theory that the experiment is\nbased on.\nThis approach is applied to the analysis of cloaking with systems with increasing number of\nlenses.\nThe experimental setup considered here is described in Sec.~\\ref{sec3}, discussing some of\nthe main aspects that have been taken into consideration for its implementation as well as\nthe main outputs.\nTo conclude, a series of final remarks are summarized in Sec.~\\ref{sec4}.\n\n\n\\section{Theoretical analysis}\n\\label{sec2}\n\nThe OCD implemented in Ref.~\\cite{choi:OptExp:2014} worked in a very simple manner:\nwhen looking through the OCD, any object observed behind it should be seen as if there\nwas nothing between such an object and our eyes, even if we introduce another object\ninside the OCD.\nThis behavior can be summarized in terms of the following two conditions:\n\\begin{enumerate}\n \\item The image of the observed object must be direct, virtual, and with unitary\n magnification.\n\n \\item The OCD has some extension, henceforth denoted with $L$, so that the hidden object\n can be accommodated somewhere inside it.\n\\end{enumerate}\nAccordingly, the rays leaving the object will reach our eyes with a minor influence\nfrom the hidden object or the OCD itself.\n\nThe variant here proposed works in a similar way, although instead of a virtual image of a\nbackground object, we are going to focus on the real image produced by an illuminated object.\nThat is, instead of analyzing cloaking by direct observation, we are going to analyze it by\nobserving a projected image, although the theoretical analysis is equally suitable to both.\nConsequently, for optimal cloaking conditions, such an image should be unaffected by the\npresence of the OCD itself or an object hidden inside it.\n\nIt should also be mentioned that the above condition (i) is not essential regarding\ncloaking, unless the OCD is also required to be hidden from sight, i.e., we do not wish\nto notice the presence of the OCD, but only the background as it appears when there is\nnothing in front of it (along the direct line of sight).\nThis is worth mentioning, because it enables the possibility to construct alternative\nOCDs in a way that, even if their configuration is not exactly the same as the one\nreported in \\cite{choi:OptExp:2014}, still the cloaking phenomenon can be observed.\n\n\n\\subsection{Elementary aspects of the transfer matrix method}\n\\label{sec21}\n\nIn Gaussian paraxial optics, when dealing with simple lens systems, imaging is typically determined\nby means of ray tracing.\nAn efficient way to tackle the issue when the number of optical elements increases (which, from a\npractical point of view, essentially means considering more than two or three lenses) is by making\nuse of the so-called transfer matrix method \\cite{pedrotti-bk}.\nThis is an easy-to-handle input\/output method based on the linear relationship between the object\n(input) and its conjugate image (output) independently of the number of optical elements (lenses\nand mirrors) accommodated between both.\nSuch a relationship is given in terms of the so-called $ABCD$ matrix,\n\\begin{equation}\n \\mathbb{M} =\n \\left( \\begin{array}{cc}\n  A & B \\\\ C & D\n  \\end{array} \\right) ,\n \\label{eqA}\n\\end{equation}\nwhere each element is directly related with a property of the optical system itself, if the matrix\nis computed between its two boundary surfaces, or the imaging process, if it is defined from the\nobject plane to the image one.\n\nTo better understand this basic concept, consider an object point $P_O$ and its conjugate image\npoint $P_I$.\nThe point $P_O$ is at a height $h_O$ off the optical axis and $P_I$ is at $h_I$.\nBoth points can be joined by a swarm of rays, all employing the same time in going from one to the\nother, according to Fermat's least time principle.\nLet us consider one of such rays.\nThis ray leaves $P_O$ at an angle $\\alpha_O$ with respect to the direction of the optical axis,\nand reaches $P_I$ with an angle $\\alpha_I$ (also with respect to the optical axis).\nAlthough it is not shown here (but it is not difficult to prove either), $h$ and $\\alpha$ are\nthe only two parameters we need to characterized the imaging process in paraxial optics.\nThe relationship between the input (object) properties, $(h_O,\\alpha_O)$, and the output (image)\nones, $(h_I,\\alpha_I)$, is described by a linear matrix transformation, $\\mathbb{M}$, which\ntransfers the former to the latter.\nIf these properties are recast in vector form, we have\n\\begin{equation}\n {\\bf p}_I = \\left( \\begin{array}{c} h_I \\\\ \\alpha_I \\end{array} \\right)\n  = \\left( \\begin{array}{cc} A & B \\\\ C & D \\end{array} \\right)\n \\left( \\begin{array}{c} h_O \\\\ \\alpha_O \\end{array} \\right)\n  = \\mathbb{M} {\\bf p}_O .\n \\label{eqC}\n\\end{equation}\n\nAccording to Eq.~(\\ref{eqC}), the height and inclination of the image point are given,\nrespectively, by\n\\begin{eqnarray}\n h_I & = & A h_O + B \\alpha_O ,\n \\label{eqE} \\\\\n \\alpha_I & = & C h_O + D \\alpha_O ,\n \\label{eqF}\n\\end{eqnarray}\nfrom which it is readily seen that $A$ and $D$ are dimensionless parameters,\nwhile $B$ and $D$ have length and inverse-length dimensions, respectively.\nThe dimensionality of these matrix elements can easily be understood by noting that\n$A$ is related to the linear magnification of the image with respect to the object\n(perpendicularly measured from the optical axis, i.e., the ratio $h_I\/h_O$), while $D$\nis related to the angular magnification, which describes the apparent size with respect\nto the object (i.e., $\\alpha_I\/\\alpha_O$).\nRegarding the elements $C$ and $B$, they are associated with the positions of the first\nand second focal planes of the optical system, $h_I\/\\alpha_O$ and $h_O\/\\alpha_I$,\nrespectively, taking its input and output planes as a reference.\n\n\n\\subsection{Transfer matrix for an $N$-lens system}\n\\label{sec22}\n\nIn the particular case we are going to deal with here, only two types of matrices are needed, namely a matrix\ndescribing the passage of light through a single thin lens, which in paraxial form reads as\n\\begin{equation}\n \\mathbb{L} =\n \\left( \\begin{array}{cc}\n  1 & 0 \\\\ - 1\/f & 1\n  \\end{array} \\right) ,\n \\label{eq1}\n\\end{equation}\nwith $f$ being the lens effective focal length, and a translation matrix,\n\\begin{equation}\n \\mathbb{T} =\n \\left( \\begin{array}{cc}\n  1 & d \\\\ 0 & 1\n  \\end{array} \\right) ,\n \\label{eq2}\n\\end{equation}\naccounting for the transit of ray bundles through an empty space of length $d$\n(this can be the space between two consecutive lenses, or just the distance between\nthe object and the first lens and the distance from the last lens to the image, as will\nbe seen below).\n\nFrom the above considerations, (i) and (ii), an ideal OCD should behave analogously to a single\ntranslation matrix with $d = L$, i.e.,\n\\begin{eqnarray}\n h_I & = & h_O + L \\alpha_O ,\n \\label{eqEocd} \\\\\n \\alpha_I & = & \\alpha_O ,\n \\label{eqFocd}\n\\end{eqnarray}\nso that the image has the same size and orientation as the object when looked from any\ndirection (that is, any $\\alpha_O \\neq 0$).\nThus, let us consider a system of $N$ lenses with their centers aligned along the system\noptical axis.\nIn this system, $f_n$ is the effective focal length of the $n$th lens and $d_{n-1}$ denotes\nthe distance between the $n$th and $(n-1)$th lenses, with $d_0 \\equiv 0$.\nImaging in this system is determined by the matrix product\n\\begin{equation}\n \\mathbb{M}_N = \\mathbb{L}_N \\mathbb{T}_{N-1} \\mathbb{L}_{N-1}\n  \\cdots \\mathbb{L}_1 \\mathbb{T}_2 \\mathbb{L}_1\n  = \\overleftarrow{\\Pi}_{n=1}^N \\mathbb{S}_n ,\n \\label{eq3}\n\\end{equation}\nwith\n\\begin{equation}\n \\mathbb{S}_n \\equiv \\mathbb{L}_n \\mathbb{T}_{n-1} ,\n \\label{eq4}\n\\end{equation}\nand where the arrow over the product symbol ($\\Pi_n$) denotes that each new\nproduct element $n$ has to be added to the left instead of to the right.\nNotice that for $n=1$, we have $\\mathbb{T}_0 = \\mathbb{I}$, since $d_0 = 0$.\n\nWhen the explicit form of the matrices (\\ref{eq1}) and (\\ref{eq2}) is\nsubstituted into (\\ref{eq4}), with $f_n$ and $d_{n-1}$ instead of $f$ and $L$,\nrespectively, the product matrix (\\ref{eq3}) reads as\n\\begin{equation}\n \\mathbb{M}_N =\n  \\overleftarrow{\\Pi}_{n=1}^N\n  \\left( \\begin{array}{cc}\n  1 & d_{n-1} \\\\ - 1\/f_n & 1 - d_{n-1}\/f_n\n  \\end{array} \\right) ,\n \\label{eq5}\n\\end{equation}\nwhich is of the form\n\\begin{equation}\n \\mathbb{M}_N =\n \\left( \\begin{array}{cc}\n  A_N & B_N \\\\ C_N & D_N\n  \\end{array} \\right) .\n \\label{eq6}\n\\end{equation}\nThis matrix is to be compared with the total translation matrix that represents\nthe ideal OCD, i.e.,\n\\begin{equation}\n \\mathbb{M}_N =\n \\left( \\begin{array}{cc}\n  1 & L \\\\ 0 & 1\n  \\end{array} \\right) ,\n \\label{eq7}\n\\end{equation}\nwith $L = \\sum_{n=1}^N d_{n-1}$.\nBy comparing matrices (\\ref{eq6}) and (\\ref{eq7}) element by element, we obtain the\nset of equations\n\\begin{eqnarray}\n A_N & = & 1 ,\n \\label{eq8a}\n \\\\\n B_N & = & L ,\n \\label{eq8b}\n \\\\\n C_N & = & 0 ,\n \\label{eq8c}\n \\\\\n D_N & = & 1 ,\n \\label{eq8d}\n\\end{eqnarray}\nwhich are used to design the OCD.\nIn compliance with Eqs.~(\\ref{eqEocd}) and (\\ref{eqFocd}),\nThe fact that $A_N$ and $D_N$ are both unitary means that the image\nproduced by the device of an object located to its right must also have\nunitary lateral and angular magnification (image equal to object) even if\nthere is a cloaked object inside it.\nThat is, the picture collected by lens 1 is directly transferred to lens $N$,\nwithout any further optical operation, as it is inferred from the fact that\n$B_N = L$.\nMoreover, like a telescope, the system is afocal, since $C_N = 0$.\n\nNext we are going to consider systems with $N$ ranging from 2 to 4 in order\nto better understand how the cloaking property works.\nThe case $N=1$ is not considered, because it is the trivial one given by (\\ref{eq1})\nwhen instead of a lens we have a very thin plate (with negligible thickness).\nObviously, this can never be an OCD, because there is no room to hide an object\ninside it.\nWe thus need to take into account that the conditions specified by Eqs.~(\\ref{eq8a})\nto (\\ref{eq8d}) are necessary for cloaking, but not sufficient.\n\n\n\\subsection{Two-lens system}\n\\label{sec23}\n\nFor a two-lens system, after proceeding with the product of matrices, the conditions\ngiven by Eqs.~(\\ref{eq8a}) to (\\ref{eq8d}) read as\n\\begin{eqnarray}\n A_2 & = & 1 - \\frac{d_1}{f_1} ,\n \\label{eq9a}\n \\\\\n B_2 & = & d_1 ,\n \\label{eq9b}\n \\\\\n C_2 & = & - \\frac{1}{f_1} - \\frac{1}{f_2} + \\frac{d_1}{f_1 f_2} ,\n \\label{eq9c}\n \\\\\n D_2 & = & 1 - \\frac{d_1}{f_2} .\n \\label{eq9d}\n\\end{eqnarray}\nThis system is analogous to a thick lens with thickness $t = d_1$ and refractive\nindex $n_L = 1$.\nThe term $-C_2$ provides us with the equivalent power of the system, also known as Gullstrand's\nequation \\cite{milton-bk}, from which the system focal length is readily obtained: $f = -1\/C_2$.\n\nIf we apply the condition (\\ref{eq8c}) to the Eq.~(\\ref{eq9c}), we\nfind\n\\begin{equation}\n f_1 + f_2 = d_1 ,\n \\label{eq10}\n\\end{equation}\ni.e., for the system to be afocal, the distance between the two lenses must be equal to\nthe sum of their respective focal lengths.\nThis is precisely the condition that makes afocal a telescope.\nHowever, contrarily to a telescope, where we are interested in large magnification factors,\nin the case of the OCD we are looking for a unitary magnification.\nThus, in order to ensure that the magnification elements (\\ref{eq9a}) and (\\ref{eq9d}) are\nnearly unitary, both focal lengths, $f_1$ and $f_2$, must be much larger than $d_1$, so that\n$d_1\/f_1 \\ll 1$ and $d_1\/f_2 \\ll 1$.\nWhen this condition is satisfied, we find that, although there is room to place an object\ninside this two-lens device, the situation is again analogous to the above single-lens\ncase: cloaking conditions lead to a non-cloaking device formed by two plane-parallel\nplates.\nTherefore, this is another example where conditions (\\ref{eq8a}) to (\\ref{eq8d}) are\nnecessary for cloaking, but not sufficient.\n\n\n\\subsection{Three-lens system}\n\\label{sec24}\n\nIn the case of three lenses, the elements of the matrix $\\mathbb{M}_3$ describing\nthe system read as\n\\begin{eqnarray}\n A_3 & = & 1 - \\frac{L}{f_1} - \\left( 1 - \\frac{d_1}{f_1} \\right) \\frac{d_2}{f_2} ,\n \\label{eq11a}\n \\\\\n B_3 & = & L - \\frac{d_1 d_2}{f_2} ,\n \\label{eq11b}\n \\\\\n C_3 & = & - \\frac{1}{f_1} - \\frac{1}{f_2} - \\frac{1}{f_3}\n           + \\left( \\frac{d_1}{f_1} + \\frac{d_2}{f_3} \\right) \\frac{1}{f_2}\n \\nonumber \\\\ & &\n           + \\left( L - \\frac{d_1 d_2}{f_2} \\right) \\frac{1}{f_1 f_3} ,\n \\label{eq11c}\n \\\\\n D_3 & = & 1 - \\frac{L}{f_3} - \\left( 1 - \\frac{d_2}{f_3} \\right) \\frac{d_1}{f_2} ,\n \\label{eq11d}\n\\end{eqnarray}\nwith $L = d_1 + d_2$.\nAs before, we find that $f_2$ must be very large in order that the condition (\\ref{eq8b})\nsatisfies, which means that the second lens essentially behaves as a thin glass layer that\ndoes not affect all the other components of the system.\nActually, this leads to a two-lens system analogous to the previous one, for which the\nsame equations hold after replacing $f_2$ in Eqs.~(\\ref{eq9a})--(\\ref{eq9d}) by $f_3$.\n\nThere is, however, an alternative non-trivial solution.\nBy inspecting the magnification terms, $A_3$ and $D_3$, we notice that they display\nsome symmetry when $f_1$ and $f_3$ are exchanged, and also when the same is done with\n$d_1$ and $d_2$.\nIf we apply conditions (\\ref{eq8a}) and (\\ref{eq8d}), we obtain the following relation\n\\begin{equation}\n \\frac{f_1}{f_3} = \\frac{d_1}{d_2} .\n \\label{eq12}\n\\end{equation}\nThis means that, if the cloaking device is designed with inversion symmetry (it should\nnot matter whether we look through the front or through the back), then we can consider\nas a convenient working hypothesis that $d_1 = d_2 = L\/2$, which leads to $f_1 = f_3 = f$.\nMaking the corresponding substitutions in either Eq.~(\\ref{eq11a}) or Eq.~(\\ref{eq11d}),\nwith the cloaking conditions (\\ref{eq8a}) or (\\ref{eq8d}), and solving for $f_2$, we obtain\na non-vanishing value for this focal length:\n\\begin{equation}\n f_2 = \\frac{L - 2f}{4} .\n \\label{eq13}\n\\end{equation}\nIt can readily be noticed that, if this condition is substituted into (\\ref{eq11c}),\nthis matrix element vanishes.\nThe resulting matrix then reads as\n\\begin{equation}\n \\mathbb{M}_3 =\n \\left( \\begin{array}{cc}\n  1 & L\/(1 - L\/2f) \\\\ 0 & 1\n  \\end{array} \\right) ,\n \\label{eq14}\n\\end{equation}\nwhich, for $2f \\gg L$, can be recast as\n\\begin{equation}\n \\mathbb{M}_3 \\approx\n \\left( \\begin{array}{cc}\n  1 & L \\left( 1 + L\/2f \\right) \\\\ 0 & 1\n  \\end{array} \\right) .\n \\label{eq15}\n\\end{equation}\nIf the term $L\/2f$ can be neglected ($L \\ll f$), then the matrix (\\ref{eq15}) acquires\nthe form of (\\ref{eq7}) and, in principle, this condition might allow cloaking.\nMoreover, when applied to Eq.~(\\ref{eq13}), this assumption implies that $f_2$ must\nbe a negative lens, since $f_2 \\approx - f\/2$, unlike $f_1$ and $f_3$, which are both\npositive (convergent).\nTherefore, we find that the first non-trivial OCD can be achieved by playing with\nlenses with relatively large focal lengths and different vergence.\n\n\n\\subsection{Four-lens system}\n\\label{sec25}\n\nLet us now consider a setup consisting of four lenses.\nProceeding as before, the elements of the corresponding $\\mathbb{M}_4$\nmatrix read as\n\\begin{eqnarray}\n\\fl A_4 & = & 1 - \\frac{L}{f_1} - \\left( 1 - \\frac{d_1}{f_1} \\right)\n     \\left( d_2 + d_3 - \\frac{d_2 d_3}{f_3} \\right) \\frac{1}{f_2}\n   - \\left[ 1 - \\frac{(L - d_3)}{f_1} \\right] \\frac{d_3}{f_3} ,\n \\label{eq16a}\n \\\\\n\\fl B_4 & = & L - \\frac{\\left( L - d_1 \\right) d_1}{f_2}\n   - \\frac{\\left( L - d_3 \\right) d_3}{f_3} + \\frac{d_1 d_2 d_3}{f_2 f_3} ,\n \\label{eq16b}\n \\\\\n\\fl C_4 & = & - \\frac{1}{f_1} - \\frac{1}{f_2} - \\frac{1}{f_3} - \\frac{1}{f_4} + \\frac{L}{f_1 f_4}\n   + \\left( \\frac{1}{f_2} + \\frac{1}{f_3} \\right) \\left( \\frac{d_1}{f_1} + \\frac{d_3}{f_4} \\right)\n \\nonumber \\\\\n\\fl & & + \\left( \\frac{1}{f_1 f_3} + \\frac{1}{f_2 f_4} + \\frac{1}{f_2 f_3} \\right) d_2\n   - \\left[ \\frac{\\left( L - d_1 \\right) d_1}{f_2} + \\frac{\\left( L - d_3 \\right) d_3}{f_3} \\right]\n     \\frac{1}{f_1 f_4}\n \\nonumber \\\\\n\\fl & &  - \\left( \\frac{d_1}{f_1} + \\frac{d_3}{f_4} - \\frac{d_1 d_3}{f_1 f_4} \\right) \\frac{d_2}{f_2 f_3} ,\n \\label{eq16c}\n \\\\\n\\fl D_4 & = & 1 - \\frac{L}{f_4} - \\left( 1 - \\frac{d_3}{f_4} \\right)\n     \\left( d_1 + d_2 - \\frac{d_1 d_2}{f_2} \\right) \\frac{1}{f_3}\n   - \\left[ 1 - \\frac{(L - d_1)}{f_4} \\right] \\frac{d_1}{f_2} ,\n \\label{eq16d}\n\\end{eqnarray}\nwith $L = d_1 + d_2 + d_3$.\nAgain here we can notice a certain symmetry by exchange of indices (both in focal lengths\nand in inter-lens distances) that can be advantageously considered in order to determine\noptimal cloaking conditions.\n\nThus, as before, let us assume the OCD satisfies inversion symmetry.\nThis means that the focal lengths of the outermost lenses is equal ($f_1 = f_4 = f_\\alpha$),\nand the same holds for the innermost lenses ($f_2 = f_3 = f_\\beta$).\nMoreover, in order to preserve such symmetry, it is also required that the distances $d_1$\nand $d_3$ are equal.\nSo, from now on, $d_1 = d_3 = d_\\alpha$ and $d_2 = d_\\beta$.\nWith this, the matrix elements specified by Eqs.~(\\ref{eq16a}) to (\\ref{eq16d}) can be recast as\n\\begin{eqnarray}\n\\fl A_4 & = & 1 - \\left( \\frac{1}{f_\\alpha} + \\frac{1}{f_\\beta} \\right) L\n   + \\frac{2 d_\\alpha (L - d_\\alpha)}{f_\\alpha f_\\beta}\n   + \\frac{d_\\alpha d_\\beta}{f_\\beta^2} - \\frac{d_\\alpha^2 d_\\beta}{f_\\alpha f_\\beta^2} ,\n \\label{eq17a}\n \\\\\n\\fl B_4 & = & L - \\left( L - d_\\alpha \\right) \\frac{2 d_\\alpha}{f_\\beta} + \\frac{d_\\alpha^2 d_\\beta}{f_\\beta^2} ,\n \\label{eq17b}\n \\\\\n\\fl C_4 & = & - \\frac{2}{f_\\alpha} - \\frac{2}{f_\\beta} + \\frac{L}{f_\\alpha^2} + \\frac{d_\\beta}{f_\\beta^2}\n   + \\frac{2 L}{f_\\alpha f_\\beta} - \\frac{2 \\left( L - d_\\alpha \\right) d_\\alpha}{f_\\alpha^2 f_\\beta}\n   - \\frac{2 d_\\alpha d_\\beta}{f_\\alpha f_\\beta^2}\n   + \\frac{d_\\alpha^2 d_\\beta}{f_\\alpha^2 f_\\beta^2} ,\n \\label{eq17c}\n \\\\\n\\fl D_4 & = & 1 - \\left( \\frac{1}{f_\\alpha} + \\frac{1}{f_\\beta} \\right) L\n   + \\frac{2 d_\\alpha (L - d_\\alpha)}{f_\\alpha f_\\beta}\n   + \\frac{d_\\alpha d_\\beta}{f_\\beta^2} - \\frac{d_\\alpha^2 d_\\beta}{f_\\alpha f_\\beta^2} .\n \\label{eq17d}\n\\end{eqnarray}\nFrom the application of (\\ref{eq8b}) to (\\ref{eq17b}), we obtain\n\\begin{equation}\n f_\\beta = \\frac{d_\\alpha d_\\beta}{2(L - d_\\alpha)} ,\n \\label{eq18}\n\\end{equation}\nwhich avoids making further assumptions on the relative size of $f_\\beta$ with respect to $L$,\nas in the previous cases, and hence allows some freedom of choice.\nWith this result and applying (\\ref{eq8a}) to (\\ref{eq17a}), we find the value of other focal length,\n\\begin{equation}\n f_\\alpha = \\frac{d_\\alpha L}{2(L - d_\\alpha)} .\n \\label{eq19}\n\\end{equation}\nAs it can easily be noticed, the addition of these two focal lengths satisfies the relation\n\\begin{equation}\n d_\\alpha = f_\\alpha + f_\\beta .\n \\label{eq20}\n\\end{equation}\nThis means that the configuration of the OCD is such that the set of lenses 1 and 2, on the\none hand, and the set of lenses 3 and 4, on the other hand, form each a telescope, one in\nfront of the other.\nWithin this configuration, lenses 1 and 4 play the role of the objective, while lenses 2 and 3\nwould be the eyepieces, since $f_\\alpha > f_\\beta$, as it is inferred from the relation\n\\begin{equation}\n \\frac{f_\\alpha}{f_\\beta} = \\frac{L}{d_\\beta} .\n \\label{eq21}\n\\end{equation}\nThis relation also gives the magnification of each telescope.\nThus, we can see that the magnified image of a given object allocated before the first telescope\nis reversed by the other, so that the total magnification becomes unitary.\nThis is precisely the counterpart in geometrical optics of the invisibility recipe in terms\nof transfer matrices found by S\\'anchez-Soto and coworkers for general electromagnetic fields\n\\cite{sanchezsoto:EJP:2008,sanchezsoto:PhysRep:2012}.\nFurthermore, it should also be noticed that the condition leading to the cancelation of the\nelement $C_4$, in compliance with the functional form (\\ref{eq7}), is precisely (\\ref{eq20}),\nwhich can easily be shown by direct substitution.\n\nTaking into account that $L = 2d_\\alpha + d_\\beta$ in (\\ref{eq21}) and the value of $d_\\alpha$,\ngiven by (\\ref{eq20}), we can now obtain the value for $d_\\beta$, which reads as\n\\begin{equation}\n d_\\beta = \\frac{2 \\left( f_\\alpha + f_\\beta \\right) f_\\beta}{f_\\alpha - f_\\beta} .\n \\label{eq22}\n\\end{equation}\nThis provides us with an exact solution (condition) for optical cloaking if we have two\nsets of two lenses with focal lengths $f_\\alpha$ and $f_\\beta$.\nThe total size of the cloaking device will be\n\\begin{equation}\n L = 2 d_\\alpha + d_\\beta\n   = \\frac{2 \\left( f_\\alpha + f_\\beta \\right) f_\\alpha}{f_\\alpha - f_\\beta} .\n \\label{eq23}\n\\end{equation}\n\n\n\\section{Experimental implementation}\n\\label{sec3}\n\n\n\\subsection{General aspects}\n\\label{sec31}\n\nIn the analysis presented in the previous section, cloaking has been investigated within an ideal scenario based on the\nfollowing assumptions:\n\\begin{itemize}\n \\item Paraxial conditions are always guaranteed.\n \\item Lenses are aberration-free.\n \\item Lack of aperture effects associated with the diameter of the lenses considered.\n\\end{itemize}\nObviously, these are ideal conditions that simplify the theoretical analysis, as we have seen above, but that have to be taken\ninto account when considering sets of standard lenses, as it is the case here, where we actually were not so much concerned\nabout getting a high degree of cloaking as constructing a relatively simple and cheap device that would allow us to study this\nphenomenon.\nIn any case, all these handicaps are advantageous from a teaching perspective, since they can be used to better characterize\nthe cloaking conditions.\nFurthermore, this is the reason why a projective OCD has been chosen to the detriment of a more appealing direct-sight OCD.\n\nThe so-called projective OCD prepared here is based on some basic optical properties.\nConsider we illuminate a transparent slide and project its image on a somehow distant wall (or projection screen if the optical\nbench is long enough).\nWhen the OCD is inserted between the object and its image, if it has been properly implemented, its presence should not\naffect too much the image (or not, at least, to a great extent), except for a reduction of luminosity due to the many lenses\ninvolved in the setup.\nMoreover, if an additional object is inserted inside the OCD (the ``hidden object''), its presence should not either affect\nthe image projected on the wall.\nIn spite of the difference in its performance with respect to a direct-sight OCD, notice that the theory introduced in the\nprevious section is still applicable, since the operation principle is the same (i.e., it does not matter whether we look through\nthe OCD or we make the light from an object to cross it and form an image beyond it).\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.85\\textwidth]{figure1.png}\n \\caption{\\label{Fig1}\n  Full OCD experimental setup used here.\n  In order to better appreciate the cloaking effect, a ``twin'' imaging system is accommodated side by side,\n  so that the images produced by both are compared.\n  As it can be seen, the setup consists of four basic elements: a light source with the object (a transparent\n  slide with different shapes depicted), the projective lens system (see Sec.~\\ref{sec32}), the OCD itself,\n  and a series of diaphragms with variable diameter.}\n\\end{figure}\n\nA photograph of the full experimental setup implemented here to investigate optical cloaking, as it has been used\nduring the different experiments carried out, is displayed in Fig.~\\ref{Fig1}.\nIt consists of two standard optical benches with a length of about one meter and a half, a set of simple teaching lenses\nwith different focal lengths, several iris-type diaphragms, and two halogen lamps connected to 6~V\/12~V DC power\nsuppliers.\nThe setup essentially consists of the object light sources (surrounded by a blue dashed line), the projective system\n(orange dashed line), and the OCD (embraced by a white dashed line).\nThe hidden objects (plates with iris-type diaphragms) are all at display (their positions denoted with green arrows),\nalthough depending on the experiment performed they could all be mounted, or only one of them.\n\nWe have considered two optical benches, because the OCD is going to be mounted in one of them, while the other\nis just an idler partner.\nThe purpose of the idler bench is to produce a reference image that is not affected by any of the effects due to\neither the OCD or the hidden object.\nThese two benches were aligned side-by-side very close together, such that the image produced with the OCD\ncould be easily compared with the idler one by direct sight (it was difficult, though, to get an optimal photograph\nof them, as it can be inferred from Fig.~\\ref{Fig4}).\nThus, when the OCD is not mounted, the two optical systems produce exactly the same image (see Sec.~\\ref{sec32});\nwhen the OCD is mounted, the presence of the idler image allows us to determine to what extent cloaking is achieved\nand its quality (particularly, to detect any change in the size of the image, presence of aberrations or decrease in the\nluminosity).\n\nThe lenses considered have focal lengths of $+5$~cm, $+10$~cm, and $+20$~cm, all of them with a diameter\nof 3.5~cm and mounted on opaque $10\\times 11$~cm$^2$ square frame plates.\nThese lenses are used to construct both the projective system and the OCD (see below).\nAs mentioned above, the role of hidden object is played here by a series of iris-type diaphragms mounted on plates with\nthe same features as those for the lenses.\nThese diaphragms have all of them maximum and minimum diameters of 3~cm and 0.1~cm, respectively.\nThey allow us to determine the maximum area that can be covered inside the OCD, while still observing the image\nwithout much distortion (aberration effects) or remarkable loses of luminosity.\nOr, in other words, the regions around the transferred bundle of rays where an object can be hidden without noticing\nits presence.\nIt is worth stressing the fact that, because of the negligible thickness of the stop blades (and even the frame\nwhere they are accommodated), this working method is ideal to determine the optimal distances or longitudinal ranges\n(alongside the bench) where neither the position or the diameter of the diaphragm (or diaphragms) affect too much\nthe projected image, that is, the tolerance ranges of the OCD to hide object (see discussions in this respect\nin Sec.~\\ref{sec32} regarding the different experiments carried out and the corresponding tolerance ranges found).\n\n\n\\subsection{The projective imaging system}\n\\label{sec32}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{figure2.png}\n \\caption{\\label{Fig2}\n  Ray-tracing diagram showing the effects on imaging due to size limitation of the diameter of the lenses\n  used in the projective system.\n  As it can be seen, due to such limitation, not every point of the illuminated object has an image.\n  The original object is denoted with the shaded blue arrow on the left and its expected image with the inverted\n  shaded blue arrow on the right.\n  The final object and image are represented with the dark blue arrow (left and right, respectively).\n  Limiting rays for the on-axis object point are displayed with orange dashed line, while for off-axis points are denoted\n  green and read dashed lines (arising from the top and bottom parts of the object, respectively).}\n\\end{figure}\n\nPrevious to the practical implementation of the OCD, it is necessary to prepare the projective system that ensures a real\nimage with unitary linear magnification.\nTo that end, we have considered two lenses with the same focal length, $f = + 10$~cm.\nOne lens is positioned approximately at the focal distance away from the object (10~cm), while the other is at about the\nsame distance from the wall.\nThis particular arrangement, where both object and image are accommodated at the focal planes of the lenses\n(front and rear, respectively), ensures the production of a real image with the same size as the object, although\ninverted.\nThis can easily be seen from the ray diagram displayed in Fig.~\\ref{Fig2}, by inspecting the green and red rays (dashed\nlines) leaving, respectively, the top and bottom points of the object (dark blue arrow on the left).\nNevertheless, in terms of the transfer matrix method, if we construct the matrix from the object plane to the image one,\nwe have:\n\\begin{eqnarray}\n\\fl\n \\mathbb{M}_{IO} & = &\n  \\left( \\begin{array}{cc} 1 & f \\\\ 0 & 1 \\end{array} \\right)\n  \\left( \\begin{array}{cc} 1 & 0 \\\\ - 1\/f & 1 \\end{array} \\right)\n  \\left( \\begin{array}{cc} 1 & L \\\\ 0 & 1 \\end{array} \\right)\n  \\left( \\begin{array}{cc} 1 & 0 \\\\ - 1\/f & 1 \\end{array} \\right)\n  \\left( \\begin{array}{cc} 1 & f \\\\ 0 & 1 \\end{array} \\right) \\nonumber \\\\\n\\fl\n & = & \\left( \\begin{array}{cc} - 1 & 0 \\\\ - 2\/f + L\/f^2 & - 1 \\end{array} \\right) ,\n \\label{eqIO}\n\\end{eqnarray}\nwhich is a symmetric matrix (it displays the same functional form going from the object $O$ to the image $I$,\nas in the other way around).\nAs it can be noticed, the lateral magnification (element $A$) and the angular one (element $D$) are both equal\nto $-1$, which denotes the fact that the image is inverted with respect to the object, although the size is the\nsame.\nRegarding the element $C$, it corresponds to the equivalent power for two identical lenses separated a distance\n$L$, according to Gullstrand's equation \\cite{milton-bk}, while a vanishing element $B$ denotes the fact that\nthe input plane corresponds to the front (object) focal plane of the system and the output plane to the rear\n(image) focal plane.\n\nIn Fig.~\\ref{Fig2} it can also be noticed that the full object $O$, denoted with the shaded blue arrow on the left,\nis not going to produce an image $I$ (shaded blue arrow on the right).\nThis arises as a consequence of the above commented effect of the size-limitation of the lenses, which prevents rays\ncoming from any point on the object to pass through the two-lens system and produce a full image.\nTypically, ray tracing in paraxial optics assumes that extension of the object is relatively small compared to the diameter\nof the lens, by virtue of which sine and tangent functions can be approximated by the value of their arguments.\nIn realistic optical systems, like the one we are dealing with here, where the object is a rectangular transparent slide\nof several centimeters wide and high, while the diameter of the lenses is smaller, the approximation works fine only for\nobject points off the system optical axis but still close to it; as object points become more and more off the optical\naxis, the approximation breaks and additional considerations are required in order to explain or determine the imaging\nprocess.\nIn principle, this leads to introduce some more advanced technical knowledge on aperture and field stops.\nHowever, in order to keep the discussion here at the simplest level, which is one of the main purposes of the work,\nwe are going to further exploit the diagram of Fig.~\\ref{Fig2} and extract such an information directly from it.\n\nThus, consider again the object denoted with the left shaded blue arrow.\nAny bundle of rays leaving any point along this object will be able to pass through the front lens.\nIf the object is accommodated on the front focal plane of the lens, then ray bundles leaving the same point will\nemerge parallel from the lens.\nIn the case of the on-axis object point, this is illustrated by the two orange dashed lines.\nAs it can be seen, after reaching the rear lens, the corresponding bundle of parallel rays will merge into the on-axis\nimage point, at the back focal plane of such a lens.\nNow, if such ray bundles are also required to pass through a second lens, namely the rear lens here, this constitutes\na severe restriction, because not all the parallel ray bundles leaving the front lens will be able to reach totally or even\npartially the rear lens.\nThere will be ray bundles with such an inclination that, after having travel the distance $L$, will fall out of the diameter\nof the rear lens.\nFor example, in the diagram of Fig.~\\ref{Fig2} we notice that, compared to the on-axis object point, only a half of the\nray bundles leaving the front lens can reach the rear one if such rays come either from the top of the dark blue arrow\n(see green dashed lines) or the bottom (red dashed lines).\nWhen these bundles cross the rear lens, the merge respectively into the bottom and top off-axis image points, denoted\nwith the dark blue arrow on the right.\nIt is clear that, because only half of the initial bundle that penetrated the front lens is going to reach the focal plane of\nthe rear lens, the luminosity of the image in those points will be lesser than closer to the optical axis.\nThe same can be applied to object points further away from the optical axis, with a relatively quick loss of luminosity in\nthe corresponding image points.\n\nIn our particular case, taking also into account the points for which the ray bundles reaching the rear lens reduce to\na half, and considering that the lens diameter of 3.5~cm and a distance between both lenses $L = 84.7$~cm, a simple\ntrigonometry-based calculation renders an estimate for the size of the image of about 0.41~cm.\nThis is the same to say that only an effective circular spot in the object with such diameter is going to form a clear image,\neven if the illuminated area of the object is much larger.\nNonetheless, in practice we have noted that the image is a bit larger, namely a spot of about 0.7~cm (see Fig.~\\ref{Fig5}),\nwhich means that ray bundles coming from upper or lower points in the object are also going to contribute, although with\nsmaller luminosity and importantly affected by spherical and chromatic aberrations.\nFor instance, if we consider the limit of the off-axis object points for which only one of the corresponding outgoing rays is\ngoing to pass through both lenses, we find an estimate of the spot diameter of about 0.83~cm, although the borders of\nsuch a spot will be relatively dark.\nBy averaging with the previous value, we obtain a spot size of 0.62~cm, which is closer to the value observed experimentally.\nThis means that even object points contributing with about less than a quarter of the ray bundle that passes through the front\nlens are going to be significant.\nIn any case, these values are fine, because what we have used as a test object\/image is a picture of two parallel straight\nsegments of 0.2~cm length.\nNotice that with a smaller OCD, $L$ would also be smaller and therefore the projected image would be larger.\n\nRegarding this projective system, it is also worth mentioning that, if the light source and the two projective lenses are removed,\nand in the place of the original image on the wall we put a picture, when looking through the OCD we can see (although\naffected by some amount of aberration) the image of a such picture with exactly the same size and orientation.\nThis experiment was performed in order to confirm the conditions (i) and (ii) by direct sight, although the result was not\nas spectacular as in Ref.~\\cite{choi:OptExp:2014} and it was not possible to obtain any good quality photograph to be\nreported here.\nFurthermore, also notice that analogous size-limitation effects are going to be associated with the lenses of the OCD itself,\nalthough in a minor proportion, as it will be seen.\n\n\n\\subsection{Experiments with the projective OCD}\n\\label{sec33}\n\nFigure~\\ref{Fig3} shows a top view (a) and a side view (b) of the full experimental setup\nused here.\nThe top view in panel (a) shows how the OCD almost occupies the full length between the lenses\nof the projective system (compare the lower bench, where the OCD is mounted, with the idler,\nwhich is upper one), as well as the proximity between both benches.\nThe side view, in panel (b), gives an idea of the side-by-side alignment, very beneficial for\na direct comparison of the images $I$ (with the OCD) and $I'$ (idler).\n\nThe OCD implemented can be better seen in Fig.~\\ref{Fig3}, where a top view (a) and a side view (b) are shown.\nFor an easier identification of the different elements, the projective system lenses are denoted with $\\ell_O$ for\nthe front lens and $\\ell_I$ for the rear one (the same for the idler companion, but with primes).\nThe lenses constituting the OCD are denoted as $\\ell_i$, with $i = 1, 2, 3, 4$.\nThis criterion follows the same labeling used in the theoretical section, where the closest lens to the observer\nis precisely the one closer to image in our case here.\nAccordingly, we have also labeled the three spaces generated by every two consecutive lenses as 1, 2 and 3,\nincreasing from the image to the object.\nHence, the distances spanned by these spaces are $d_1$, $d_2$ and $d_3$, with the total length of the OCD being\n$L = d_1 + d_2 + d_3$.\nAs for the diaphragms, to keep a consistent notation, they are referred to as $A_1$, $A_2$ and $A_3$ ($A$ for\naperture), which makes reference to the space where they are accommodated, although they all are identical.\nBecause of the decrease of luminosity in the image when the OCD is introduced, in the corresponding setup\na power of 12~V is supplied to the halogen bulb, although only 6~V were required in the idler companion,\notherwise the image spot was too bright under darkness conditions, which were used for a better performance\nof the experiment (the photos of Figs.~\\ref{Fig1} and \\ref{Fig3} were taken with daylight conditions).\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{figure3.png}\n \\caption{\\label{Fig3}\n  Top view (a) and side view (b) of experimental setup used as OCD, where $\\ell_i$ ($i = 1, 2, 3, 4$) denote\n  the positions of the lenses used and $A_j$ denotes the positions of three ($j = 1, 2, 3$) iris-type diaphragms.\n  The distances between lenses $d_k$ (with $k = 1, 2, 3$) and the total length $L$ of the device are also shown.\n  Notice in panel (a) that both projective systems, test and OCD, are replicas one another.}\n\\end{figure}\n\nFigure~\\ref{Fig4} illustrates by means of a simple ray diagram the working principle of the OCD:\nthe rays leaving the on-axis object point (orange solid lines) pass through the device as if it\nwas not there (dashed lines).\nThis transit takes place by diverting the rays incident onto the OCD in the way indicated by\nthe red solid lines.\nNotice that this ray diversion is theoretically described by the matrix found in Sec.~\\ref{sec25}, which would\nreplay the central transit matrix in Eq.~(\\ref{eqIO}), in compliance with the fact that the effect of the four-lens\ntransfer matrix should be equivalent to having nothing along the path $L$ pursued by the incoming object rays.\nSpecifically, the lenses selected to built the OCD have focal lengths of $+20$~cm (for $\\ell_1$ and $\\ell_4$)\nand $+5$~cm ($\\ell_2$ and $\\ell_3$).\nAccordingly, from Eq.~(\\ref{eq20}), the distance between them is $d_1 = d_3 = 25$~cm, while\n$d_2 \\approx 16.7$~cm, from Eq.~(\\ref{eq22}).\nThe OCD was mounted in such a way that $\\ell_4$ was at 5.4~cm from $\\ell_O$ and $\\ell_1$ at 12.6~cm from $\\ell_I$.\nAs mentioned above, cloaking conditions have been investigated by using iris-type diaphragms.\nIt is clear that, as the diaphragm diameter is decreased, the bundle of rays will also decrease, which have\nan observable effect on the image.\nTherefore, by conveniently choosing a relatively narrow aperture that still allows the full bundle\nof rays to pass through, it is possible to determine a range of positions of the top along which its\npresence will not alter the image.\nIn other words, the presence of the diaphragm will be cloaked unless its diameter is so small that it starts\naffecting the projected image.\nSeveral experiments were carried out with analogous results.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{figure4.png}\n \\caption{\\label{Fig4}\n  Ray diagram illustrating the working principle of the OCD implemented here: the rays\n  going from the object to the image (orange solid lines) pass through the device as if\n  it was not there (orange dashed lines) by deviating them (red solid lines).}\n\\end{figure}\n\nThe first experiment performed consisted in determining the optimal cloaking conditions\nfor each diaphragm individually considered in the setup.\nNotice that, if the diaphragm diameter is reduced in an important amount, by moving it\nalong the corresponding section of the OCD we should obtain a spatial range for which\nthe projected image is not affected.\nThis range will be the optimal cloaking region, that is, an observer at the position of\nthe projected image will not be able to perceive the presence of any object accommodated\nwithin such a range beyond the boundary defined by the corresponding iris diameter.\nWith this in mind, we selected an aperture of 0.5~cm for all three diaphragms, which\ncorresponds to about a 2.8\\% of their maximum area when they are fully open.\nAccordingly, we have found the following optimal distances (tolerance ranges):\n\\begin{itemize}\n \\item For $A_1$, with a range going from 11.7~cm to 13.2~cm measured from $L_1$, no important\n effects were observed in the projected image, such as loss of luminosity or appearance of\n chromatic aberrations (in the form of light color rings surrounding the image).\n This range lies around the center of the section (at 12.5~cm from $L_1$).\n\n \\item For $S_2$, the range goes from 6.7~cm to 8.9~cm, measured from $L_2$, which is also\n around the center of the section ($\\approx 8.4$~cm from $L2$).\n\n \\item For $S_3$, the range was between 9.7~cm and 11.6~cm, measured from $L_3$, closer\n (although still below) the center of the section (12.5~cm from $L_3$).\n\\end{itemize}\nA photograph of what can be seen projected onto the wall during the performance of\nthe experiment is displayed in Fig.~\\ref{Fig5}.\nThe two images, the one produced with the OCD (left) and the idler one (right), are shown\ntogether for comparison in panel (a).\nDue to the small size of the illuminated spots, the distance between them seem to be\nrelatively large, although there are only 10~cm between their centers.\nActually, the distortion observed (the kind of oval shape that they display) is due to\nthe perspective introduced by the camera.\nIn panel (b) we show only the image produce with OCD, where the colored halo due to the\nincipient effects of the chromatic aberration can be seen.\nNevertheless, it is worth mentioning that the effect is much stronger than it is\nactually, because of the treatment of the image, which require a high contrast for\na better visualization.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.95\\textwidth]{figure5.png}\n \\caption{\\label{Fig5}\n  (a) Photograph of the illuminated spots that can be observed during the performance of\n  the experiments reported here.\n  The left spot corresponds to the bench with the OCD, while the right spot is the idler one.\n  Although they are circular, here the present an oval shaped due to the perspective introduced\n  by the camera.\n  (b) Enlargement of the left spot showing the colored halo produced by chromatic aberration\n  in the limit of the optimal cloaking region (it appears much more enhanced that it is\n  actually, because the photograph has received a high-contrast treatment in order to better\n  visualize it).}\n\\end{figure}\n\nOnce cloaking ranges are set on each section of the OCD, one might wonder about the maximum\ncloaking achievable with this device.\nBy inspecting Fig.~\\ref{Fig4}, given that the ray bundle crossing the OCD is going to get\nnarrower between $\\ell_2$ and $\\ell_3$, we are going to focus on this section.\nBy moving the diaphragm $A_2$ from one of these lenses to the other and making narrower its\ndiameter, we finally found such optimal cloaking conditions at 7.7~cm from $L_2$ (near the\ncentral value found in the previous experiment), for a diameter of 0.2~cm, i.e., the diaphragm\nopening is only about 0.4\\% of its maxima opening.\nFor these conditions, we noticed that:\n\\begin{itemize}\n \\item If the diameter was decreased at this position, there was a remarkable quick\n loss of luminosity in the projected image.\n\n \\item If the diaphragm was slightly displaced backwards, towards $L_2$, vignetting-related\n effects became remarkable.\n\n \\item If the diaphragm was slightly displaced forward, towards $L_3$, the effects of chromatic\n aberrations also appeared immediately (blue spots inside the image).\n\\end{itemize}\n\nOne might also wonder why instead of considering the central section of the OCD, the first or\nthird sections were not considered in the regions where the rays cross the foci of its lenses.\nThe reason is very simple: although there is a minimum (zero) waste there for incident rays\nthat are parallel to the optical axis, rays incident with any other inclination will not\npass through such points, which constitutes an important inconvenience.\nNevertheless, this led us to consider a third experiment to test the robustness of the cloaking\ncondition rendered by the previous experiment.\nThat is, without changing either the diameter or the position of the diaphragm $A_2$, we decided\nto determine where inside the other sections of the OCD an object could be hidden without noticing\nits presence.\nThus, we selected a diameter of 0.4~cm for the other two diaphragms, namely $A_1$ and $A_3$.\nWith this diameter we ensured a reduced loss of luminosity in the projected image when all three\ndiaphragms were used.\nThe optimal positions for these diaphragms were 12.1~cm for $A_1$, measured from $\\ell_1$, and\n10.7~cm for $A_3$, measured from $\\ell_3$.\nNotice that none of them is close to the corresponding focus, but they are closer to the center\nof the corresponding section of the OCD.\nFurthermore, also notice that the two positions lie close to the respective central values of\nthe ranges found in the first experiment.\n\n\n\\section{Concluding remarks}\n\\label{sec4}\n\nThe main purpose of this work has been to provide startup tools to introduce the\noptical cloaking phenomenon in the classroom both at the theory level and also at\nthe level of teaching experiment.\nThis has been done by exploring the performance of what we have called a projective\noptical cloaking device (OCD), where instead of directly observing through the device\nitself, as it is the case in Ref.~\\cite{choi:OptExp:2014}, the phenomenon is studied\nand analyzed by inspecting the projected image of an illuminated object.\nThe main advantage of this setup is that it can be built with material that is currently\navailable in any optics teaching laboratory (organic teaching lenses), without the\nnecessity to rely on high-quality material, such as good quality lenses (glass research\nlenses).\nBy means of a series of experiments (these are just some examples, but many others\ncan be devised) has served to detect the phenomenon and also to determine optimal\ncloaking regimes for the setup considered.\nIn this regard, we have advantageously used the fact that the lenses were not\naberration-free, which has allowed us establishing appropriate boundaries for the\ncloaking regimes.\nThese regimes have been found to happen in all three sections of the OCD, with\ncloaking ranges of about 2~cm, approximately, within each section, although in\nprinciple one would expect to detect cloaking only along the central section, where\nthe incident ray bundle gets narrower, as it is illustrated in Fig.~\\ref{Fig3}.\nIn this sense, we have also seen how putting a number of lenses one after the other\nbecomes very important regarding cloaking, not only because of a remarkable reduction of\nluminosity, but also because of aperture issues, exemplified by means of Fig.~\\ref{Fig2}.\n\nThe OCD here has been built taking into account a theoretical analysis based on the transfer\nmatrix formulation of Gaussian paraxial optics.\nThe main reason why we have chosen this method instead of more conventional ones\nbased on ray tracing is because it stresses in a nice manner the input\/ouput relationship\nenabled by the system analyzed.\nAlthough the transfer matrix method is not widely known, it is worth introducing at this\nlevel, because the construction of the system matrix allows to get a general view of\nthe path followed by the rays, since they depart from the object until they reach the\nprojection wall where the image is formed, without restricting ourselves to a limited set\nof rays.\nNevertheless, as it can be seen, it is not necessary a high knowledge on the method,\nbut just a few aspects; the rest is just standard matrix algebra.\nOn the other hand, the appealing feature of ray tracing constructions, however, is not lost\neither, because they have also been used here, in particular to illustrate the passage of rays\nthrough the projective system or the OCD, as shown respectively in Figs.~\\ref{Fig2} and\n\\ref{Fig3}.\nIn general terms, we have found that, in the same way that the experimental OCD is\nsimple, the theory here considered has also be presented at a simple level in order to\nmake it suitable for undergraduate optics courses.\nIn this regard, the introduction of more sophisticated, on-purpose software typically used\nin this kind of analysis either to proceed with the ray tracing or to compute and solve matrices\nhas been skipped, because the main idea is to tackle the problem just with simple tools.\nFor the same reason, a more refined and explicit analysis of the problem based on the role\nof lenses as aperture and field stops has also been omitted, because it already requires\nsome more advanced knowledge on the issue, which are not really necessary at this stage\nto explain imaging, as we have shown.\n\n\n\\ack\n\nFinancial support from the Spanish MINECO (Grant No.\\ FIS2016-76110-P)\nis acknowledged.\n\n\n\\section*{References}\n\n\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\vspace{-0.05in}\nSemantic segmentation aims at assigning semantic labels to every pixel of an image. Leveraging on CNNs~\\cite{he2016deep,Hu_2018_CVPR,ILSVRC15,simonyan2014very,szegedy2015going}, significant progress has been reported for this fundamental task~\\cite{chen2016deeplab,Chen_2018_ECCV,long2015fully,Peng_2017_CVPR}. One drawback of the existing approaches, nevertheless, is the requirement of large quantities of pixel-level annotations, such as in VOC \\cite{everingham2010pascal}, COCO \\cite{lin2014microsoft} and Cityscapes \\cite{Cordts2016Cityscapes} datasets, for model training. Labeling of semantics at pixel-level is cost expensive and time consuming.\nFor example, the Cityscapes dataset is composed of 5,000 high-quality pixel-wise annotated images, and the annotation on a single image is reported to take more than 1.5 hours.\n\nAn alternative is by utilizing synthetic data, which is largely available in 3D engines (e.g., SYNTHIA \\cite{ros2016synthia}) and 3D computer games (e.g., GTA5 \\cite{GTA5_richter2016playing}). The ground-truth semantics of these data can be automatically generated without manual labeling. Nevertheless, in the case where the synthetic data is different from the real images, the domain gap might be difficult to bridge. Unsupervised domain adaptation is generally regarded as an appealing way to address the problem of domain gap. The existing approaches include narrowing the gap by transferring images across domains \\cite{dundar2018domain,murez2018image,wu2018dcan} and learning domain-invariant representation via adversarial mechanism \\cite{Du_2019_ICCV,luo2019taking,Vu_2019_CVPR}.\n\n\\begin{figure*}[!tb]\n\\vspace{-0.05in}\n   \\centering {\\includegraphics[width=0.98\\textwidth]{intro.pdf}}\n   \\vspace{-0.1in}\n   \\caption{\\small The examples of (a) predictions on two domains by fully convolutional networks trained on synthetic data; (b)$\\sim$(d) the three evaluation criteria we studied, i.e., patch-based consistency, cluster-based consistency and spatial logic.}\n   \\label{fig:intro}\n   \\vspace{-0.25in}\n\\end{figure*}\n\nIn this paper, we consider model overfitting in source domain as the major cause of domain mismatch. As shown in Figure \\ref{fig:intro}(a), although Fully Convolutional Networks (FCN) perfectly segment the synthetic image by correct labeling of pixels, directly deploying this model for real image yields poor results. Instead of leveraging training samples in the target domain for model fine-tuning, this paper explores label-free constraints to alleviate the problem of model overfitting. These constraints are intrinsic and generic in the context of semantic segmentation. Figure \\ref{fig:intro}(b)$\\sim$(d) illustrate three label-free constraints being investigated. The first two constraints, namely patch-based and cluster-based consistencies guide the segmentation based on the prediction consistency among the pixels in an image patch and among the clusters of patches sharing similar visual properties, respectively. The last criterion, namely spatial logic, contextualizes the prediction of labels based on spatial relation between image patches. Based on these criteria, we propose a novel Regularizer of Prediction Transfer (RPT) for transferring the model trained on synthetic data for semantic segmentation of real images.\n\nThe main contribution of this paper is on the exploration of label-free data-driven constraints for transferring of model to bridge domain gap. These constraints are imposed as regularizers during training to transfer an overfitted source model for proper labeling of pixels in the target domain. Specifically, at the lowest level of regularization, majority voting is performed to derive a dominative category for each image patch. The dominative category serves as a local cue for pixels with low prediction confidence to adjust their label prediction during training. The patch-level regularization is then extended to a higher level of regularization to explore cluster-level and context-level prediction consistency.\nDespite its simplicity, the three regularizers, when jointed optimized in a fully convolutional network with adversarial learning, show impressive performances by outperforming several state-of-the-art methods, when transferring the models trained on GTA5 and SYNTHIA for semantic segmentation on the Cityscapes dataset.\n\n\\vspace{-0.05in}\n\\section{Related Work}\n\\vspace{-0.05in}\n\\textbf{CNN Based Semantic Segmentation.} As one of the most challenging computer vision task, semantic segmentation has received intensive research attention. With the surge of deep learning and convolutional neural networks (CNNs), Fully Convolutional Network (FCN)~\\cite{long2015fully} successfully serves as an effective approach that employs CNNs to perform dense semantic prediction. Following FCN, various schemes, ranging from multi-path feature aggregation and refinement~\\cite{ghiasi2016laplacian,Lin:2017:RefineNet,Peng_2017_CVPR,Pohlen_2017_CVPR,zhang2019customizable,Zhao_2018_ECCV} to multi-scale context extraction and integration~\\cite{chen2018searching,chen2016deeplab,He_2019_ICCV,qiu2017learning,yang2018denseaspp,Zhang_2018_CVPR_context,zhao2017pspnet}, have been developed and achieved great success in leveraging contextual information for semantic segmentation. Post-processing techniques, such as CRF~\\cite{chen2016deeplab} and MRF~\\cite{liu2018deep}, could further be applied to take the spatial consistency of labels into account and improve the predictions from FCNs. Considering that such methods typically rely on the datasets with pixel-level annotations which are extremely expensive and laborious to collect, researchers have also strived to utilize a weaker form of annotation, such as image-level tags \\cite{papandreou2015weakly,pinheiro2015image}, bounding boxes~\\cite{dai2015boxsup}, scribbles~\\cite{bearman2016s} and statistics \\cite{pathak2015constrained}, for semantic segmentation. The development of computer graphics techniques provides an alternative approach that exploits synthetic data with free annotations. This work aims to study the methods of applying the semantic segmentation model learnt on the computer-generated synthetic data to unlabeled real data.\n\n\\textbf{Domain Adaptation of Semantic Segmentation.}\nTo alleviate the issues of expensive labeling efforts in collecting pixel-level annotations, domain adaptation is studied for semantic segmentation. FCNWild~\\cite{BDDS_hoffman2016fcns}, which is one of the early works, attempts to align the features in different domains from both global and local aspects by adversarial training. Curriculum~\\cite{zhang2017curriculum} proposes a curriculum-style learning approach to bridge the domain gap between synthetic and real data. Later on, similar to domain adaptation in image recognition and object detection \\cite{cai2019exploring,pan2019transferrable,yao2015semi}, visual appearance-level and\/or representation-level adaptation are exploited in~\\cite{dundar2018domain,murez2018image,Tsai_2018_CVPR,Zhang_2018_CVPR} for this task. \\cite{dundar2018domain,murez2018image} perform an image-to-image translation that transfers the synthetic images to the real domain in the appearance-level. From the perspective of the representation-level adaptation, AdaSegNet~\\cite{Tsai_2018_CVPR} proposes to apply adversarial learning on segmentation maps for adapting structured output space.\nFCAN~\\cite{Zhang_2018_CVPR} employs the two levels of adaptation simultaneously, in which the appearance gap between synthetic and real images is minimized and the network is encouraged to learn domain-invariant representations.\nThere have been several other strategies~\\cite{chang2019all,chen2019learning,chen2018road,pmlr-v80-hoffman18a,iqbal2019mlsl,li2019bidirectional,zou2018unsupervised}, being performed for cross-domain semantic segmentation.\nFor example, ROAD~\\cite{chen2018road} devises a target guided distillation module and a spatial-aware adaptation module for real style and distribution orientation. Labels from the source domain are transferred to the target domain as the additional supervision in CyCADA~\\cite{pmlr-v80-hoffman18a}. Depth maps which are available in virtual 3D environments are utilized as geometric information to reduce domain shift in ~\\cite{chen2019learning}. \\cite{iqbal2019mlsl,li2019bidirectional,zou2018unsupervised} treat target predictions as the guide for learning a model applicable to the images in target domain by self-supervised learning. \\cite{chang2019all} proposes a domain invariant structure extraction framework that decouples the structure and texture representations of images and improves the performance of segmentation.\n\n\\textbf{Summary.} Most of the aforementioned approaches mainly investigate the problem of domain adaptation for semantic segmentation through bridging the domain gap during training. Our work is different in the way that we seek the additional regularization for the prediction in target domain based on the intrinsic and generic properties of semantic segmentation task. Such solution formulates an innovative and promising research direction for this task.\n\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.478\\textwidth]{patch.pdf}}\n   \\caption{\\small Example of pixels to be unpunished (a) or punished (b) in optimization. (a) For the unpunished cases, some pixels are very confident in the class differed from the dominative category. (b) For the punished cases, most pixels inside the region predict relatively high probabilities for the dominative category.}\n   \\label{fig:patch}\n   \\vspace{-0.15in}\n\\end{figure}\n\\section{Regularizer of Prediction Transfer}\nWe start by introducing the Regularizer of Prediction Transfer (RPT) for semantic segmentation.\nThree criteria are defined to assess the quality of segmentation. The result of assessment is leveraged to guide the transfer of a learnt model in the source domain for semantic segmentation in the target domain.\n\n\\subsection{Patch-based Consistency}\nThe idea is to enforce all pixels in a patch to be consistent in the prediction of semantic labels. Here, a patch is defined as a superpixel that groups neighboring pixels with similar visual appearance. We employ Simple Linear Iterative Clustering (SLIC)~\\cite{achanta2012slic}, which is both speed and memory efficient in the generation of superpixels by adopting k-means algorithm.\nGiven one image from target domain $x_t$, SLIC splits the image into $N$ superpixels $\\{S_i|i=1,...,N\\}$. Each superpixel $S_i=\\{p^j_i|j=1,...,M_i\\}$ is composed of $M_i$ adjacent pixels with similar appearance.\nWe assume that all or the majority of pixels will be annotated with the same semantic labels. Here, the dominative category $\\hat{y}_i$ of a superpixel is defined as the most number of predicted labels among all the pixels in this superpixel.\n\nAs SLIC considers only visual cue, a superpixel usually contains multiple regions of different semantic labels. Simply involving all pixels in network optimization can run into the risk of skew optimization. To address this problem, a subset of pixels is masked out from patch-based regularization.\nSpecifically, in superpixel $S_i$, pixels $p_i^j\\in S_i$ are clustered into two groups depending on the predicted probability of the dominative category $\\hat{y}_i$: (a) $P_{seg}(\\hat{y}_i| p^j_i)<=\\lambda_{pc}$ means that the probability is less than or equal to a pre-defined threshold $\\lambda_{pc}$. In other words, the pixel $p_i^j$ is predicted with labels different from the dominative category with relatively high probability. This group of pixels should be exempted from regularization. (b) $P_{seg}(\\hat{y}_i| p^j_i)>\\lambda_{pc}$ represents that $p_i^j$ has relatively higher confidence to be predicted as the dominative category. In this case, the dominative $\\hat{y}_i$ is leveraged as a cue to guide the prediction of these pixels. To the end, the loss item for patch-based consistency regularization of a target image $x_t$ is formulated as:\n\\begin{equation}\\label{eq:pc}\n\\begin{aligned}\n\\mathcal{L}_{pc}(x_t)=- \\sum_{i, j} I_{(P_{seg}(\\hat{y}_i| p^j_i)>\\lambda_{pc})} log P_{seg}(\\hat{y}_i| p^j_i)\n\\end{aligned}~~,\n\\end{equation}\nwhere $I_{(\\cdot)}$ is an indicator function to selectively mask out pixels from optimization by thresholding. Figure \\ref{fig:patch} shows examples of superpixels that are masked out (i.e., unpunished) and involved (i.e., punished) for optimization.\n\n\n\\subsection{Cluster-based Consistency}\n\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.40\\textwidth]{cluster.pdf}}\n   \\caption{\\small Feature space visualization of seven superpixel clusters using t-SNE. The dominative category is given for each cluster.}\n   \\label{fig:cluster}\n   \\vspace{-0.15in}\n\\end{figure}\nIn addition to patch, we also enforce the consistency of label prediction among the clusters of patches that are visually similar. Specifically, cluster-level regularization imposes a constraint that the superpixels with similar visual properties should predict the cluster dominative category as their label. To this end, superpixels are further grouped into clusters. The feature representation of a superpixel is extracted through ResNet-101~\\cite{he2016deep}, which is pre-trained on ImageNet dataset~\\cite{ILSVRC15}. The feature vector utilized for clustering is generated by averagely pooling the feature maps of the superpixel region from $res5c$ layer.\nAll the superpixels from target domain images are grouped into $K=2048$ clusters by k-means algorithm. The cluster-level dominative category $\\tilde{y}_k$ is determined by majority voting among the superpixels within a cluster. Figure \\ref{fig:cluster} visualizes seven examples of clusters and the corresponding dominative categories by t-SNE \\cite{maaten:JMLR08}. As clustering is imperfect, it is expected that some superpixels will be incorrectly grouped. Denote $P_{seg}(\\tilde{y}_k| p^j_i)$, where $p^j_i \\in S_i \\in C_k$, as the probability of predicting cluster-level dominative category as label for pixel $p^j_i$. Similar to patch-based consistency regularization, pixels with low confidence on the cluster-level category will not be punished during network optimization. Thus, the loss item of cluster-based consistency regularization for a target image $x_t$ is defined as:\n\\begin{equation}\\label{eq:pc}\n\\begin{aligned}\n\\mathcal{L}_{cc}(x_t)=- \\sum_{i, j, S_i \\in C_k} I_{(P_{seg}(\\tilde{y}_k| p^j_i)>\\lambda_{cc})} log P_{seg}(\\tilde{y}_k| p^j_i)\n\\end{aligned}~~,\n\\end{equation}\nwhere $\\lambda_{cc}$ is a pre-defined threshold to gate whether a pixel should be masked out from regularization.\n\n\n\\subsection{Spatial Logic}\nA useful cue to leverage for target-domain segmentation is the spatial relation between semantic labels. For instance, a superpixel of category \\emph{sky} is likely on the top of another superpixel labeled with \\emph{building} or \\emph{road}, and not vice versa. These relations are expected to be invariant across the source and target domains. The supportive hypothesis behind is introduced in \\cite{chang2019all} that the high-level structure information of an image is informative for semantic segmentation and can be readily shared across domains. As such, the motivation of spatial logic is to preserve the spatial relations learnt in source domain to target domain.\n\nFormally, we exploit the LSTM encoder-decoder architecture to learn the vertical relation between superpixels, as shown in Figure \\ref{fig:spatial}. The main goal of this architecture is to speculate the category of the masked segment in the sequence according to context information. Then, the produced probability can be used to evaluate the logical validity of the predicted category in the masked segment. Suppose we have a prediction sequence $\\mathcal{Y}$, where $\\mathcal{Y}=\\{\\mathbf{y}_1,\\mathbf{y}_2,...,\\mathbf{y}_{T-1},\\mathbf{y}_{T}\\}$ including $T$ superpixel predictions sliced from one column of prediction map. Let $\\mathbf{y}_{t} \\in \\mathbb{R}^{C+1}$ denote the one-hot vector of the $t$-th prediction in the sequence, and the dimension of $\\mathbf{y}_{t}$, i.e., $C+1$, is the number of semantic categories plus one symbol as an identification of masked prediction. The masked prediction sequence $\\hat{\\mathcal{Y}}$, which is fed into the LSTM encoder, is generated by masking a segment of consecutive predictions with the identical semantic category in the original sequence $\\mathcal{Y}$. The LSTM encoder embeds the masked prediction sequence $\\hat{\\mathcal{Y}}$ into a sequence representation. The LSTM decoder, which is attached on the top of the encoder, then speculates the categories of the masked segment and reconstructs the original sequence $\\mathcal{Y}$. To learn the aforementioned spatial logic, the encoder-decoder architecture is optimized with the cross-entropy loss supervised by the label from source domain.\n\nNext, the optimized model can be utilized to estimate the validity of each prediction from the view of spatial logic. For the target image $x_t$, we first slice the prediction map to several columns consisting of vertically neighbored superpixels. The patch-level dominative categories of the superpixels in the column are organized into a prediction sequence. For the superpixel $S_i$ in the column, the spatial logical probability $P_{logic}(\\hat{y}_i| S_i)$ is measured by the LSTM encoder-decoder only when the prediction of this superpixel is masked in the input sequence. Once this probability is lower than the threshold $\\lambda_{sl}$, we consider this prediction to be illogical and punish the prediction of $\\hat{y}_i$ by the segmentation network. The loss of spatial logic regularization is computed as:\n\\begin{equation}\\label{eq:pc}\n\\begin{aligned}\n\\mathcal{L}_{sl}(x_t)=\\sum_{i, j} I_{(P_{logic}(\\hat{y}_i| S_i)<\\lambda_{sl})} log P_{seg}(\\hat{y}_i| p^j_i)\n\\end{aligned}~~,\n\\end{equation}\nwhere $P_{logic}(\\cdot)$ denotes the prediction from LSTM encoder-decoder architecture.\n\n\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.45\\textwidth]{spatial.pdf}}\n   \\caption{\\small The LSTM encoder-decoder architecture to learn the spatial logic in the prediction map.}\n   \\label{fig:spatial}\n   \\vspace{-0.15in}\n\\end{figure}\n\n\n\\section{Semantic Segmentation with RPT}\nThe proposed Regularizer of Prediction Transfer (RPT) can be easily integrated into most of the existing frameworks for domain adaptation of semantic segmentation. Here, we choose the widely adopted framework based on adversarial learning as shown in Figure \\ref{fig:framework}. The principle in this framework is equivalent to guiding the semantic segmentation in both domains by fooling a domain discriminator $D$ with the learnt source and target representations. Formally, given the training set $\\mathcal{X}_{s}=\\{x_s^{i}|i=1,\\dots,N_s\\}$ in source domain and $\\mathcal{X}_{t}=\\{x_t^{i}|i=1,\\dots,N_t\\}$ in target domain, the adversarial loss $\\mathcal{L}_{adv}$ is the average classification loss, which is formulated as:\n\\begin{equation}\n   \\label{eq:adv}\n   \\begin{aligned}\n   \\mathcal{L}_{adv}(\\mathcal{X}_{s},\\mathcal{X}_{t})= \\mathop{-E}\\limits_{x_t \\sim \\mathcal{X}_t}[log(D(x_t))]\\mathop{-E}\\limits_{x_s \\sim \\mathcal{X}_s}[log(1 - D(x_s)]\n   \\end{aligned}~~.\n\\end{equation}\nwhere $\\mathop{E}$ denotes the expectation over the image set. The discriminator $D$ will attempt to minimize this loss by differentiating between source and target representations, and the shared Fully Convolutional Network (FCN) is learnt to fool the domain discriminator.\nConsidering that the image region corresponding to the receptive field of each spatial unit in the final feature map is treated as an individual instance during semantic segmentation, the representations of such instances are expected to be invariant across domains.\nThus we employ a fully convolutional domain discriminator whose outputs are the domain prediction of each image region corresponding to the spatial unit in the feature map.\n\nSince training labels are available in the source domain, the loss function is based on the pixel-level classification loss $\\mathcal{L}_{seg}$. In contrast, due to the absence of training labels, the loss function in the target domain is defined based upon the following three regularizers:\n\\begin{equation}\n   \\label{eq:rpt}\n   \\small\n   \\begin{aligned}\n   \\mathcal{L}_{rpt}(\\mathcal{X}_{t})= \\mathop{E}\\limits_{x_t \\sim \\mathcal{X}_t}[\\mathcal{L}_{cc}(x_t)+\\mathcal{L}_{pc}(x_t)+\\mathcal{L}_{sl}(x_t)]\n   \\end{aligned}~~.\n\\end{equation}\nHere, we empirically treat each loss in RPT equally. Thus, the overall objective of the segmentation framework integrates $\\mathcal{L}_{adv}$, $\\mathcal{L}_{seg}$ and $\\mathcal{L}_{rpt}$ as:\n\\begin{equation}\n   \\label{eq:all}\n   \\small\n   \\begin{aligned}\n   \\mathop{\\min}_{FCN}\\{-\\varepsilon \\mathop{\\min}_{D}\\mathcal{L}_{adv}(\\mathcal{X}_s, \\mathcal{X}_t) + \\mathcal{L}_{seg}(\\mathcal{X}_s) + \\mathcal{L}_{rpt}(\\mathcal{X}_t)\\}\n   \\end{aligned}~~,\n\\end{equation}\nwhere $\\varepsilon=0.1$ is the trade-off parameter to align the scale of different losses.\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.45\\textwidth]{framework.pdf}}\n   \\vspace{-0.05in}\n   \\caption{\\small The adversarial-based semantic segmentation adaptation framework with RPT. The shared FCN is learnt with adversarial loss for domain-invariant representations across two domains. The predictions on source domain are optimized by supervised label, while the target domain predictions are regularized by RPT loss.}\n   \\label{fig:framework}\n   \\vspace{-0.15in}\n\\end{figure}\n\n\\section{Implementation} \\label{sec:imp}\n\\textbf{Training strategy.} Our proposed network is implemented in Caffe~\\cite{jia2014caffe} framework and the weights are trained by SGD optimizer. We employ dilated FCN~\\cite{chen2016deeplab} originated from the ImageNet pre-trained ResNet-101 as our backbone followed by a PSP module~\\cite{zhao2017pspnet}, unless otherwise stated. The domain discriminator for adversarial learning is borrowed from FCAN~\\cite{Zhang_2018_CVPR}. During the training stage, images are randomly cropped to $713\\times713$ due to the limitation of GPU memory. Both random horizontal flipping and image resizing are utilized for data augmentation. To make the training process stable, we pre-train the FCN on data from the source domain with annotations. At the stage of pre-training, the ``poly'' policy whose power is fixed to 0.9 is adopted with the initial learning rate 0.001. Momentum and weight decay are 0.9 and 0.0005 respectively. Each mini-batch has 8 samples and maximum training iterations is set as 30K. With the source domain pre-trained weights, we perform the domain adaptation by finetuning the whole adaptation framework which is equipped with our proposed RPT. The initial learning rate is 0.0001 and the total training iteration is 10K. Other training hyper-parameters remain unchanged.\nFollowing \\cite{lian2019constructing}, we randomly selected 500 images from the official training set of Cityscapes as a general validation set. The hyper-parameters ($\\lambda_{pc}=\\lambda_{cc}=\\lambda_{sl}=0.25$, $\\varepsilon=0.1$) are all determined on this set.\n\n\\textbf{Complexity of superpixel.}\nRPT highly relies on the quality of superpixel extraction. For robustness, superpixels with complex content ideally should be excluded from model training. The term ``complex'' refers to the distribution of semantic labels in a superpixel. In our case, we measure complexity based on the proportion of pixels being predicted with the dominative category over the number of pixels in a superpixel. A larger value implies consistency in prediction and hence safer to involve the corresponding superpixel in regularizations. Empirically, RPT only regularizes the top-50\\% of superpixels. The empirical choice will be further validated in the next section.\n\n\\textbf{State update of RPT.}\nDuring network optimization, the segmentation prediction $P_{seg}$, superpixel dominative category $\\hat{y}_i$ and cluster dominative category $\\tilde{y}_k$ change gradually. Iteratively updating these ``states'' is computationally expensive because reassigning the categories to superpixel and cluster (e.g., $\\hat{y}_i$ and $\\tilde{y}_k$) requires the semantic predictions collected from the whole training set of the target domain. Considering these predictions only change slightly during training, we first calculate these states before the optimization (without regularization) and fix these states at the beginning of iterations. Then, we will update the predictions or states for $N_{su}$ times evenly during training.\n\n\\vspace{-0.1in}\n\\section{Experiments}\n\\vspace{-0.05in}\n\\subsection{Datasets}\n\\vspace{-0.05in}\nThe experiments are conducted on GTA5~\\cite{GTA5_richter2016playing}, SYNTHIA~\\cite{ros2016synthia} and Cityscapes~\\cite{Cordts2016Cityscapes} datasets. The proposed RPT is trained on GTA5 and SYNTHIA (source domain) and Cityscapes (target domain). GTA5 is composed of 24,966 synthetic images of size $1914 \\times 1052$. These images are generated by Grand Theft Auto V (GTA5), a modern computer game, to render city scenes. The pixels of these images are annotated with 19 classes that are compatible with the labels in Cityscapes. Similarly, SYNTHIA consists of synthetic images of urban scenes with resolutions of $1280 \\times 760$. Following~\\cite{chang2019all,chen2019learning,hong2018conditional,li2019bidirectional,Tsai_2018_CVPR}, we use the subset, SYNTHIA-RAND-CITYSCAPES, which has 9,400 images being annotated with labels consistent with Cityscapes for experiments. Cityscapes is composed of 5,000 images of resolution $2048 \\times 1024$. These images are split into three subsets of sizes 2,975, 500 and 1,525 for training, validation and testing, respectively. The pixels of these images are annotated with 19 classes. In the experiments, the training subset is treated as the target-domain training data, where the pixel-level annotation is assumed unknown to RPT. On the other hand, the target-domain testing data is from validation subset. The same setting is also exploited in~\\cite{chang2019all,li2019bidirectional,Tsai_2018_CVPR}.\n\nTo this end, the performance of RPT is assessed by treating GTA5 as source domain and Cityscapes as target domain (i.e., GTA5~$\\to$~Cityscapes), and similarly, SYNTHIA~$\\to$~Cityscapes. The metrics are per class Intersection over Union (IoU) and mean IoU over all the classes.\n\n\\begin{table}\n   \\centering\n   \\small\n   \\caption{\\small RPT performances in terms of mean IoU for domain adaptation of semantic segmentation on GTA5~$\\to$~Cityscapes.}\n   \\begin{tabular}{l|c@{~~}c@{~~}c|c@{~~}c@{~~}c} \\hline\n      \\multirow{2}{*}{\\textbf{Method}}  & \\multicolumn{3}{c|}{\\textbf{ResNet-50}}& \\multicolumn{3}{c}{\\textbf{ResNet-101}}\\\\\n      & FCN & +ABN & +ADV & FCN & +ABN & +ADV \\\\ \\hline\n      baseline & 30.1 & 35.7 & 45.7 & 32.3 & 39.1 & 47.2 \\\\ \\hline\n      \\textbf{RPT$^{1}$} & 33.0 & 39.3 & 48.7 & 36.1 & 42.9 & 50.4 \\\\\n      \\textbf{RPT$^{2}$} &  33.4 & 39.9 & 50.0 & 37.9 & 44.2 & 51.7\\\\\n      \\textbf{RPT$^{3}$} & 33.5 & 40.0 & 50.0 & 39.1 & 44.6 & 52.6\\\\ \\hline\n   \\end{tabular}\n   \\label{tab:effectiveness}\n   \\vspace{-0.15in}\n\\end{table}\n\n\\begin{figure}[!tb]\n   \\centering\n   \\subfigure[State updating]{\n      \\label{fig:curve:a}\n      \\includegraphics[width=0.22\\textwidth]{exp_iter.pdf}}\n   \\subfigure[Filtering complex superpixels]{\n      \\label{fig:curve:b}\n      \\includegraphics[width=0.23\\textwidth]{exp_complex.pdf}}\n   \\caption{\\small Two analysis experiments of (a) the effectiveness of state updating during training of RPT$^{3}$; (b) the percentage of filtered complex superpixels of RPT$^{1}$.}\n   \\label{fig:curve}\n   \\vspace{-0.15in}\n\\end{figure}\n\\subsection{Evaluation of RPT}\nRPT is experimented on top of six different network architectures derived from \\textbf{FCN} which leverages on either \\textbf{ResNet-50} or \\textbf{ResNet-101} as the backbone network. Specially, we adopt Adaptive Batch Normalization (\\textbf{ABN}) to replace the mean and variance of BN in the original version of FCN, resulting in a variant of network named FCN+ABN. Note that the BN layer is first learnt in source domain and then replaced by ABN when being applied to the target domain. In addition, leveraging on the adversarial training (\\textbf{ADV}), another variant, FCN+ABN+ADV, is trained to learn domain-invariant representations.\n\nWe first verify the impact of $N_{su}$, the number of state updating, in RPT. Table~\\ref{tab:effectiveness} summarizes the impact on six variants of network for domain adaptation on GTA5~$\\to$~Cityscapes.\nAll the networks are pre-trained on ImageNet dataset and then injected with RPT.\nThe superscript, RPT$^{n}$, refers to the number of times for state updating (see Table~\\ref{tab:effectiveness} for exact number).\nThe baselines are obtained by performing domain adaptation of semantic segmentation on the use of the corresponding network architectures, but without RPT.\nOverall, RPT improves the baseline without regularization. The improvement is consistently observed across the variants of networks, and proportional to the number of state updating at the expense of computation cost. RPT$^{3}$ achieves the best performance (mIoU = 52.6\\%) and with 5.4\\% improvement over the baseline of the same network (FCN+ABN+ADV). Figure \\ref{fig:curve:a} shows the performance changes in terms of mIoU during training over different times of state updating. The training starts with model learning in source domain. State updating, such as the assignment of dominative categories at superpixel and cluster levels, is then performed three times evenly during the training process in the target domain.\nDespite dropping in performance at the start of training after each state updating, mIoU gradually improves and eventually converges to a higher value than the previous round.\nFigure \\ref{fig:curve:b} shows the performance trend when the percentage of complex superpixels being excluded from learning gradually increases. As shown, the value mIoU constantly increases till reaching the level when 50\\% of superpixels are filtered. In the remaining experiments, we fix the setting of RPT to involve 50\\% of superpixels in regularization.\n\n\\begin{table}\n      \\centering\n      \\small\n      \\caption{\\small Contribution of each design in RPT for domain adaptation of semantic segmentation on GTA5~$\\to$~Cityscapes.}\n      \\begin{tabular}{l|c@{~}c@{~}c@{~}c@{~}c@{~}c|c} \\hline\n         \\textbf{Method} & \\textbf{ABN} & \\textbf{ADV} & \\textbf{PCR} & \\textbf{CCR}  & \\textbf{SLR} & \\textbf{SU} & \\textbf{mIoU} \\\\\\hline\n         FCN           &              &              &               &               &          &     & 32.3          \\\\\n         +ABN         & $\\surd$      &              &               &               &          &    & 39.1           \\\\\n         FCN$_{adv}$ (+ADV)        & $\\surd$      & $\\surd$      &               &               &          &    &  47.2          \\\\ \\hline\n         +{PCR}         & $\\surd$      & $\\surd$      & $\\surd$       &               &          &    & 49.0          \\\\\n         +{CCR}        & $\\surd$      & $\\surd$      & $\\surd$       & $\\surd$       &          &     & 49.6          \\\\\n         \\textbf{RPT$^{1}$} (+{SLR})         & $\\surd$      & $\\surd$      & $\\surd$       & $\\surd$       & $\\surd$  &    &  50.4          \\\\\n         \\textbf{RPT$^{3}$}           & $\\surd$      & $\\surd$      & $\\surd$       & $\\surd$       & $\\surd$  &  $\\surd$ &   52.6          \\\\\\hline\n      \\end{tabular}\n      \\label{tab:contribution}\n   \\vspace{-0.15in}\n\\end{table}\n\n\\begin{table*}\n   \\centering\n   \\footnotesize\n   \\caption{\\small Comparisons with the state-of-the-art unsupervised domain adaptation methods on GTA5~$\\to$~Cityscapes adaptation. Please note that the baseline methods are divided into five groups: (1) representation-level domain adaptation by adversarial learning \\cite{chen2018road,Du_2019_ICCV,pmlr-v80-hoffman18a,hong2018conditional,luo2019taking,sankaranarayanan2018learning,Tsai_2018_CVPR}; (2) appearance-level domain adaptation by image translation \\cite{dundar2018domain,murez2018image}; (3) appearance-level + representation-level adaptation~\\cite{chang2019all,wu2018dcan,Zhang_2018_CVPR}; (4) self-learning \\cite{iqbal2019mlsl,lian2019constructing,zhang2018fully,zou2018unsupervised}; (5) others \\cite{Chen_2019_ICCV,li2019bidirectional,saleh2018effective,zhang2017curriculum,zhu2018penalizing}.}\n   \\begin{tabular}{l@{~}|@{~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~}|@{~}c@{~~}} \\hline\n      Method                                        & road          & sdwlk         & bldng         & wall          & fence         & pole          & light         & sign          & vgttn         & trrn          & sky           & person        & rider         & car           & truck         & bus           & train         & mcycl         & bcycl         & mIoU          \\\\ \\hline\n      FCNWild~\\cite{BDDS_hoffman2016fcns}          & 70.4 & 32.4  & 62.1  & 14.9 & 5.4   & 10.9 & 14.2  & 2.7  & 79.2  & 21.3 & 64.6 & 44.1   & 4.2   & 70.4 & 8.0   & 7.3  & 0.0   & 3.5   & 0.0   & 27.1 \\\\\n      Learning~\\cite{sankaranarayanan2018learning} & 88.0 & 30.5  & 78.6  & 25.2 & 23.5  & 16.7 & 23.5  & 11.6 & 78.7  & 27.2 & 71.9 & 51.3   & 19.5  & 80.4 & 19.8  & 18.3 & 0.9   & 20.8  & 18.4  & 37.1 \\\\\n      ROAD~\\cite{chen2018road}                     & 76.3 & 36.1  & 69.6  & 28.6 & 22.4  & 28.6 & 29.3  & 14.8 & 82.3  & 35.3 & 72.9 & 54.4   & 17.8  & 78.9 & 27.7  & 30.3 & 4.0   & 24.9  & 12.6  & 39.4 \\\\\n      CyCADA~\\cite{pmlr-v80-hoffman18a}            & 79.1 & 33.1  & 77.9  & 23.4 & 17.3  & 32.1 & 33.3  & 31.8 & 81.5  & 26.7 & 69.0 & 62.8   & 14.7  & 74.5 & 20.9  & 25.6 & 6.9   & 18.8  & 20.4  & 39.5 \\\\\n      AdaptSegNet~\\cite{Tsai_2018_CVPR}            & 86.5 & 36.0  & 79.9  & 23.4 & 23.3  & 23.9 & 35.2  & 14.8 & 83.4  & 33.3 & 75.6 & 58.5   & 27.6  & 73.7 & 32.5  & 35.4 & 3.9   & 30.1  & 28.1  & 42.4 \\\\\n      CLAN~\\cite{luo2019taking}                    & 87.0 & 27.1  & 79.6  & 27.3 & 23.3  & 28.3 & 35.5  & 24.2 & 83.6  & 27.4 & 74.2 & 58.6   & 28.0  & 76.2 & 33.1  & 36.7 & 6.7   & 31.9  & 31.4  & 43.2 \\\\\n      Conditional~\\cite{hong2018conditional}       & 89.2 & 49.0  & 70.7  & 13.5 & 10.9  & 38.5 & 29.4  & 33.7 & 77.9  & 37.6 & 65.8 & \\textbf{75.1}   & \\textbf{32.4}  & 77.8 & \\textbf{39.2}  & 45.2 & 0.0   & 25.5  & 35.4  & 44.5 \\\\\n      SSF-DAN~\\cite{Du_2019_ICCV}                  & 90.3 & 38.9  & 81.7  & 24.8 & 22.9  & 30.5 & 37.0  & 21.2 & 84.8  & 38.8 & 76.9 & 58.8   & 30.7  & 85.7 & 30.6  & 38.1 & 5.9   & 28.3  & 36.9  & 45.4 \\\\\n      ADVENT~\\cite{Vu_2019_CVPR}                   & 89.4 & 33.1  & 81.0  & 26.6 & 26.8  & 27.2 & 33.5  & 24.7 & 83.9  & 36.7 & 78.8 & 58.7   & 30.5  & 84.8 & 38.5  & 44.5 & 1.7   & 31.6  & 32.4  & 45.5 \\\\ \\hline\n      I2I Adapt~\\cite{murez2018image}              & 85.8 & 37.5  & 80.2  & 23.3 & 16.1  & 23.0 & 14.5  & 9.8  & 79.2  & 36.5 & 76.4 & 53.4   & 7.4   & 82.8 & 19.1  & 15.7 & 2.8   & 13.4  & 1.7   & 35.7 \\\\\n      Stylization~\\cite{dundar2018domain}          & 86.9 & 44.5  & 84.7  & 38.8 & 26.6  & 32.1 & 42.3  & 22.5 & 84.7  & 30.9 & 85.9 & 67.0   & 28.1  & 85.7 & 38.3  & 31.8 & 21.5  & 31.3  & 24.6  & 47.8 \\\\ \\hline\n      DCAN~\\cite{wu2018dcan}                       & 85.0 & 30.8  & 81.3  & 25.8 & 21.2  & 22.2 & 25.4  & 26.6 & 83.4  & 36.7 & 76.2 & 58.9   & 24.9  & 80.7 & 29.5  & 42.9 & 2.5   & 26.9  & 11.6  & 41.7 \\\\\n      DISE~\\cite{chang2019all}                     & 91.5 & 47.5  & 82.5  & 31.3 & 25.6  & 33.0 & 33.7  & 25.8 & 82.7  & 28.8 & 82.7 & 62.4   & 30.8  & 85.2 & 27.7  & 34.5 & 6.4   & 25.2  & 24.4  & 45.4 \\\\\n      FCAN~\\cite{Zhang_2018_CVPR}                  & 88.9 & 37.9  & 82.9  & 33.2 & 26.1  & \\textbf{42.8} & 43.2  & 28.4 & 86.5  & 35.2 & 78.0 & 65.9   & 22.8  & 86.7 & 23.7  & 34.9 & 2.7   & 24.0  & 41.9  & 46.6 \\\\ \\hline\n      FCTN~\\cite{zhang2018fully}                   & 72.2 & 28.4  & 74.9  & 18.3 & 10.8  & 24.0 & 25.3  & 17.9 & 80.1  & 36.7 & 61.1 & 44.7   & 0.0   & 74.5 & 8.9   & 1.5  & 0.0   & 0.0   & 0.0   & 30.5 \\\\\n      CBST~\\cite{zou2018unsupervised}              & 89.6 & \\textbf{58.9}  & 78.5  & 33.0 & 22.3  & 41.4 & 48.2  & 39.2 & 83.6  & 24.3 & 65.4 & 49.3   & 20.2  & 83.3 & 39.0  & 48.6 & 12.5  & 20.3  & 35.3  & 47.0 \\\\\n      PyCDA~\\cite{lian2019constructing}            & \\textbf{92.3} & 49.2  & 84.4  & 33.4 & 30.2  & 33.3 & 37.1  & 35.2 & 86.5  & 36.9 & 77.3 & 63.3   & 30.5  & 86.6 & 34.5  & 40.7 & 7.9   & 17.6  & 35.5  & 48.0 \\\\\n      MLSL~\\cite{iqbal2019mlsl}                    & 89.0 & 45.2  & 78.2  & 22.9 & 27.3  & 37.4 & 46.1  & 43.8 & 82.9  & 18.6 & 61.2 & 60.4   & 26.7  & 85.4 & 35.9  & 44.9 & \\textbf{36.4}  & 37.2  & 49.3  & 49.0 \\\\ \\hline\n      Curriculum~\\cite{zhang2017curriculum}        & 72.9 & 30    & 74.9  & 12.1 & 13.2  & 15.3 & 16.8  & 14.1 & 79.3  & 14.5 & 75.5 & 35.7   & 10    & 62.1 & 20.6  & 19   & 0     & 19.3  & 12    & 31.4 \\\\\n      Penalizing~\\cite{zhu2018penalizing}          & -    & -     & -     & -    & -     & -    & -     & -    & -     & -    & -    & -      & -     & -    & -     & -    & -     & -     & -     & 38.1 \\\\\n      Effective~\\cite{saleh2018effective}          & 79.8 & 29.3  & 77.8  & 24.2 & 21.6  & 6.9  & 23.5  & \\textbf{44.2} & 80.5  & 38.0 & 76.2 & 52.7   & 22.2  & 83.0 & 32.3  & 41.3 & 27.0  & 19.3  & 27.7  & 42.5 \\\\\n      MaxSquare~\\cite{Chen_2019_ICCV}              & 89.3 & 40.5  & 81.2  & 29.0 & 20.4  & 25.6 & 34.4  & 19.0 & 83.6  & 34.4 & 76.5 & 59.2   & 27.4  & 83.8 & 38.4  & 43.6 & 7.1   & 32.2  & 32.5  & 45.2 \\\\\n      Bidirectional~\\cite{li2019bidirectional}     & 91.0 & 44.7  & 84.2  & 34.6 & 27.6  & 30.2 & 36.0  & 36.0 & 85.0  & \\textbf{43.6} & 83.0 & 58.6   & 31.6  & 83.3 & 35.3  & \\textbf{49.7} & 3.3   & 28.8  & 35.6  & 48.5 \\\\ \\hline\\hline\n      \\textbf{FCN$_{adv}$+RPT$^{1}$}                                  & 88.7 & 37.0  & 85.2  & 36.6 & 27.7  & 42.6 & 49.1  & 30.0 & 86.9  & 37.6 & 80.7 & 66.8   & 27.5  & 88.1 & 30.3  & 39.5 & 22.5  & 28.0  & 53.0  & 50.4 \\\\\n      \\textbf{FCN$_{adv}$+RPT$^{3}$}                                  & 89.2 & 43.3  & 86.1  & 39.5 & 29.9  & 40.2 & 49.6  & 33.1 & 87.4  & 38.5 & 86.0 & 64.4   & 25.1  & 88.5 & 36.6  & 45.8 & 23.9  & 36.5  & 56.8  & 52.6 \\\\\n      \\textbf{FCN$_{adv}$+RPT$^{3}$}+MS                               & 89.7 & 44.8  & \\textbf{86.4}  & \\textbf{44.2} & \\textbf{30.6}  & 41.4 & \\textbf{51.7}  & 33.0 & \\textbf{87.8}  & 39.4 & \\textbf{86.3} & 65.6   & 24.5  & \\textbf{89.0} & 36.2  & 46.8 & 17.6 & \\textbf{39.1} & \\textbf{58.3} & \\textbf{53.2} \\\\ \\hline\n   \\end{tabular}\n   \\label{tab:GTA5}\n   \\vspace{-0.15in}\n\\end{table*}\n\n\\subsection{An Ablation Study}\nNext, we conduct an ablation study to assess the performance impacts of different design components. We separately assess the three regularizations in RPT: patch-based consistency regularization (\\textbf{PCR}), cluster-based consistency regularization (\\textbf{CCR}) and spatial logic regularization (\\textbf{SLR}). Table \\ref{tab:contribution} details the contribution of each component towards the overall performance. FCN$_{adv}$, by considering adaptive batch normalization and adversarial learning (ABN+ADV), successfully boosts mIoU from 32.3\\% to 47.2\\%. The result indicates the importance of narrowing the domain gap between synthetic data and real images. The three regularizations in target domain introduce 1.8\\%, 0.6\\% and 0.8\\% of improvement, respectively. Furthermore, by increasing the number of state updating during network optimization, additional 2.2\\% of improvement is observed from RPT$^{1}$ to RPT$^{3}$. Figure \\ref{fig:comparison} shows the gradual improvement on semantic segmentation of five images, when different design components are incrementally integrated.\n\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.48\\textwidth]{comparison.pdf}}\n   \\caption{\\small Examples of semantic segmentation results on GTA5-Cityscapes adaptation. The original images, their ground truth and comparative results at different stages of FCN$_{adv}$+RPT$^{3}$ are given.}\n   \\label{fig:comparison}\n   \\vspace{-0.15in}\n\\end{figure}\n\n\n\\subsection{Comparisons with State-of-the-Art}\nWe compare with several state-of-the-art techniques for unsupervised domain adaptation on GTA5~$\\to$~Cityscapes. Broadly, we can categorize the baseline methods into five categories: (1) representation-level domain adaptation by adversarial learning \\cite{chen2018road,Du_2019_ICCV,pmlr-v80-hoffman18a,hong2018conditional,luo2019taking,sankaranarayanan2018learning,Tsai_2018_CVPR}; (2) appearance-level domain adaptation by image translation~\\cite{dundar2018domain,murez2018image}; (3) appearance-level + representation-level adaptation \\cite{chang2019all,wu2018dcan,Zhang_2018_CVPR}; (4) self-learning \\cite{iqbal2019mlsl,lian2019constructing,zhang2018fully,zou2018unsupervised}; (5) others \\cite{Chen_2019_ICCV,li2019bidirectional,saleh2018effective,zhang2017curriculum,zhu2018penalizing}. The performance comparisons on GTA5~$\\to$~Cityscapes adaptation are summarized in Table~\\ref{tab:GTA5}.\nFCN$_{adv}$+RPT$^{3}$ achieves new state-of-the-art performance with mIoU of 52.6\\%. Benefiting from the proposed regularizations, FCN$_{adv}$+RPT$^{3}$ outperforms SSF-DAN~\\cite{Du_2019_ICCV} and ADVENT~\\cite{Vu_2019_CVPR}, which also adopt a similar adversarial mechanism, by additional improvement of 7.2\\% and 7.1\\%, respectively. The performance is also better than the most recently proposed FCAN~\\cite{Zhang_2018_CVPR} and Stylization~\\cite{dundar2018domain}, which exploit a novel appearance transferring module that is not considered in RPT. Comparing to the best reported result to-date by MLSL~\\cite{iqbal2019mlsl}, our proposed model still leads the performance by 3.6\\%. By further integrating with the multi-scale (MS) scheme, i.e, FCN$_{adv}$+RPT$^{3}$+MS, the mIoU boosts to 53.2\\% with 9 out of the 19 categories reach to-date the best reported performances.\n\n\nTo verify the generalization of RPT, we also test the performance on SYNTHIA~$\\to$~Cityscapes using the same settings. Following previous works~\\cite{iqbal2019mlsl,lian2019constructing,Vu_2019_CVPR,zou2018unsupervised}, the performances are reported in terms of mIoU@16 and mIoU@13 by not considering the different number of categories. The performance comparisons are summarized in Table \\ref{tab:SYNTHIA}. Similarly, FCN$_{adv}$+RPT$^{3}$+MS achieves the best performance with mIoU@16 = 51.7\\% and mIoU@13 = 59.5\\%. The performances are better than PyCDA, which reports the best known results, by 5\\% and 6.2\\% respectively.\n\\begin{table*}[]\n   \\centering\n   \\footnotesize\n   \\caption{\\small Comparisons with the state-of-the-art unsupervised domain adaptation methods on SYNTHIA~$\\to$~Cityscapes transfer.}\n   \\begin{tabular}{l@{~}|@{~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~}|@{~}c@{~~}c@{~}}\\hline\n                                                & road  & sdwlk & bldng & wall  & fence & pole  & light & sign  & vgttn & sky   & person & rider & car   & bus   & mcycl & bcycl & mIoU@16   & mIoU@13 \\\\ \\hline\n   Learning~\\cite{sankaranarayanan2018learning} & 80.1          & 29.1          & 77.5          & 2.8           & 0.4          & 26.8          & 11.1          & 18.0          & 78.1          & 76.7          & 48.2          & 15.2          & 70.5          & 17.4          & 8.7           & 16.7          & 36.1          & -             \\\\\n   ROAD~\\cite{chen2018road}                     & 77.7          & 30.0          & 77.5          & 9.6           & 0.3          & 25.8          & 10.3          & 15.6          & 77.6          & 79.8          & 44.5          & 16.6          & 67.8          & 14.5          & 7.0           & 23.8          & 36.2          & -             \\\\\n   AdaptSegNet~\\cite{Tsai_2018_CVPR}            & 84.3          & 42.7          & 77.5          & -             & -            & -             & 4.7           & 7.0           & 77.9          & 82.5          & 54.3          & 21.0          & 72.3          & 32.2          & 18.9          & 32.3          & -             & 46.7          \\\\\n   CLAN~\\cite{luo2019taking}                    & 81.3          & 37.0          & 80.1          & -             & -            & -             & 16.1          & 13.7          & 78.2          & 81.5          & 53.4          & 21.2          & 73.0          & 32.9          & 22.6          & 30.7          & -             & 47.8          \\\\\n   Conditional~\\cite{hong2018conditional}       & 85.0          & 25.8          & 73.5          & 3.4           & \\textbf{3.0} & 31.5          & 19.5          & 21.3          & 67.4          & 69.4          & 68.5          & 25.0          & 76.5          & 41.6          & 17.9          & 29.5          & 41.2          & -             \\\\\n   SSF-DAN~\\cite{Du_2019_ICCV}                  & 84.6          & 41.7          & 80.8          & -             & -            & -             & 11.5          & 14.7          & 80.8          & 85.3          & 57.5          & 21.6          & 82.0          & 36.0          & 19.3          & 34.5          & -             & 50.0          \\\\\n   ADVENT~\\cite{Vu_2019_CVPR}                   & 85.6          & 42.2          & 79.7          & 8.7           & 0.4          & 25.9          & 5.4           & 8.1           & 80.4          & 84.1          & 57.9          & 23.8          & 73.3          & 36.4          & 14.2          & 33.0          & 41.2          & 48.0          \\\\ \\hline\n   DCAN~\\cite{wu2018dcan}                       & 82.8          & 36.4          & 75.7          & 5.1           & 0.1          & 25.8          & 8.0           & 18.7          & 74.7          & 76.9          & 51.1          & 15.9          & 77.7          & 24.8          & 4.1           & 37.3          & 38.4          & -             \\\\\n   DISE~\\cite{chang2019all}                     & \\textbf{91.7} & \\textbf{53.5} & 77.1          & 2.5           & 0.2          & 27.1          & 6.2           & 7.6           & 78.4          & 81.2          & 55.8          & 19.2          & 82.3          & 30.3          & 17.1          & 34.3          & 41.5          & -             \\\\\\hline\n   CBST~\\cite{zou2018unsupervised}              & 53.6          & 23.7          & 75.0          & 12.5          & 0.3          & 36.4          & 23.5          & 26.3          & 84.8          & 74.7          & 67.2          & 17.5          & 84.5          & 28.4          & 15.2          & 55.8          & 42.5          & 48.4          \\\\\n   PyCDA~\\cite{lian2019constructing}            & 75.5          & 30.9          & 83.3          & 20.8          & 0.7          & 32.7          & 27.3          & \\textbf{33.5} & 84.7          & 85.0          & 64.1          & 25.4          & 85.0          & 45.2          & 21.2          & 32.0          & 46.7          & 53.3          \\\\\n   MLSL~\\cite{iqbal2019mlsl}                    & 59.2          & 30.2          & 68.5          & \\textbf{22.9} & 1.0          & 36.2          & 32.7          & 28.3          & \\textbf{86.2} & 75.4          & \\textbf{68.6} & 27.7          & 82.7          & 26.3          & 24.3          & 52.7          & 45.2          & 51.0          \\\\ \\hline\n   Curriculum~\\cite{zhang2017curriculum}        & 57.4          & 23.1          & 74.7          & 0.5           & 0.6          & 14.0          & 5.3           & 4.3           & 77.8          & 73.7          & 45.0          & 11.0          & 44.8          & 21.2          & 1.9           & 20.3          & 29.7          & -             \\\\\n   Penalizing~\\cite{zhu2018penalizing}          & -             & -             & -             & -             & -            & -             & -             & -             & -             & -             & -             & -             & -             & -             & -             & -             & 34.2          & 40.3          \\\\\n   MaxSquare~\\cite{Chen_2019_ICCV}              & 82.9          & 40.7          & 80.3          & 10.2          & 0.8          & 25.8          & 12.8          & 18.2          & 82.5          & 82.2          & 53.1          & 18.0          & 79.0          & 31.4          & 10.4          & 35.6          & 41.4          & 48.2          \\\\\n   Bidirectional~\\cite{li2019bidirectional}     & 86.0          & 46.7          & 80.3          & -             & -            & -             & 14.1          & 11.6          & 79.2          & 81.3          & 54.1          & \\textbf{27.9} & 73.7          & 42.2          & 25.7          & 45.3          & -             & 51.4          \\\\   \\hline\\hline\n   \\textbf{FCN$_{adv}$+RPT$^{1}$}                                  & 87.7          & 43.1          & 84.0          & 10.5          & 0.5          & \\textbf{42.2} & \\textbf{40.5} & 33.1          & 86.0          & 81.9          & 56.0          & 26.1          & 85.9          & 35.8          & 24.8          & 56.2          & 49.6          & 57.0        \\\\\n   \\textbf{FCN$_{adv}$+RPT$^{3}$}                                  & 88.9          & 46.5          & 84.5          & 15.1          & 0.5          & 38.5          & 39.5          & 30.1          & 85.9          & 85.8          & 59.8          & 26.1          & 88.1          & 46.8          & 27.7          & 56.1          & 51.2          & 58.9         \\\\\n   \\textbf{FCN$_{adv}$+RPT$^{3}$}+MS                               & 89.1          & 47.3          & \\textbf{84.6} & 14.5          & 0.4          & 39.4          & 39.9          & 30.3          & 86.1          & \\textbf{86.3} & 60.8          & 25.7          & \\textbf{88.7} & \\textbf{49.0} & \\textbf{28.4} & \\textbf{57.5} & \\textbf{51.7} & \\textbf{59.5}\\\\ \\hline\n   \\end{tabular}\n   \\vspace{-0.15in}\n   \\label{tab:SYNTHIA}\n\\end{table*}\n\\subsection{Examples of Regularization}\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.478\\textwidth]{case_loss.pdf}}\n   \\caption{\\small Examples showing the effectiveness of patch-based consistency and cluster-based consistency in RPT.}\n   \\label{fig:case_loss}\n   \\vspace{-0.15in}\n\\end{figure}\nFigure~\\ref{fig:case_loss} shows examples to demonstrate the effectiveness of patch-based and cluster-based consistency regularizations. Here, we crop some highlighted regions of input image, ground truth, prediction by FCN$_{adv}$ and prediction by FCN$_{adv}$+RPT$^{3}$, respectively.\nOn one hand, as shown in Figure~\\ref{fig:case_loss}(a), patch-based consistency encourages the pixels to be predicted as the dominative category of the superpixel. On the other hand, cluster-based consistency is able to correct the predictions with the cue of visual similarity across superpixels as illustrated in Figure~\\ref{fig:case_loss}(b).\nThese examples validate our motivation of enforcing label consistency within superpixel and cluster, where most semantic labels are correctly predicted in the target domain. Figure~\\ref{fig:case_loss_logic} further visualizes the merit of modeling spatial context by spatial logic regularization.\nGiven the segmentation results from FCN$_{adv}$, our proposed LSTM encoder-decoder outputs the logical probability of assigning current semantic labels to each region. The darkness indicates that the region is predicted with low logical probability.\nBetter results are achieved by penalizing the illogical predictions, such as \\textit{road} on the top of \\textit{vegetation} (1$st$ row) or \\textit{car} (2$nd$ row), \\textit{sky} below \\textit{building} (3$rd$ row), \\textit{fence} above \\textit{building} (4$th$ row).\n\n\\section{Conclusion}\n\\begin{figure}[!tb]\n   \\centering {\\includegraphics[width=0.45\\textwidth]{case_loss_logic.pdf}}\n   \\caption{\\small The examples of punished patches by spatial logic.}\n   \\label{fig:case_loss_logic}\n   \\vspace{-0.15in}\n\\end{figure}\nWe have presented Regularizer of Prediction Transfer (RPT) for unsupervised domain adaptation of semantic segmentation. RPT gives light to a novel research direction, by directly exploring the three intrinsic criteria of semantic segmentation to restrict the label prediction on the target domain. These criteria, when imposed as regularizers during training, are found to be effective in alleviating the problem of model overfitting.\nThe patch-based consistency attempts to unify the prediction inside each region by introducing its dominative category to the unconfident pixels. The cluster-based consistency further amends the prediction according to other visually similar regions which belong to the same cluster. In pursuit of suppressing illogical predictions, spatial logic is involved to regularize the spatial relation which is shared across domains.\nExperiments conducted on the transfer from GTA5 to Cityscapes show that the injection of RPT can consistently improve the domain adaptation across different network architectures. More remarkably, the setting of FCN$_{adv}$+RPT$^{3}$ achieves new state-of-the-art performance. A similar conclusion is also drawn from the adaptation from SYNTHIA to Cityscapes, which demonstrates the generalization ability of RPT.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nRecent work on parton distributions functions (PDF's)\nin the nucleon has focussed\non probing the sea and gluon distribution at small $x$. The valence \nquarks distribution has been thought to be relatively well understood.\nHowever, the precise knowledge of the $u$ and $d$ quark distribution\nat high $x$ is very important at collider energies in searches for signals\nfor new physics at high $Q^2$.\nIn addition, the value of $d\/u$ as $x \\rightarrow 1$ is of theoretical\ninterest.\nRecently, a proposed CTEQ toy model~\\cite{toy} included\n the possibility of an additional contribution to the $u$ quark distribution\n(beyond $x>0.75$) as an explanation for both the initial HERA high\n $Q^2$ anomaly~\\cite{highQ2},\nand for the jet excess at high-$P_t$  at CDF~\\cite{CDFjet}. In this\nLetter we conclude that a re-analysis of data from\nNMC and SLAC leads to a great improvement in our  knowledge\nof PDF's at high $x$, and rules out such toy models.\n\nInformation about valence quarks originates from the proton and neutron\nstructure function data. The $u$ valence quark distribution at high $x$\nis relatively well\nconstrained by the proton structure function $F_2^p$.\nHowever, the neutron structure function $F_2^n$, which is sensitive \nto the $d$ valence quark at high $x$,\nis actually extracted from deuteron data.\n Therefore, there is an uncertainty in the $d$ valence quark \ndistribution from the corrections for nuclear binding effects in the deuteron. \nIn past extractions of $F_2^n$ from deuteron data,\nonly Fermi motion corrections\nwere considered, and other \nbinding effects were assumed to be negligible.\nRecently, the corrections for nuclear binding effects in the deuteron,\n$F_2^d\/F_2^{n+p}$, have been extracted empirically from\nfits to the nuclear dependence\nof electron scattering data from SLAC experiments\nE139\/140~\\cite{GOMEZ}.\nThe empirical extraction uses a\nmodel proposed by Frankfurt and Strikman~\\cite{Frankfurt}, \nin which all binding effects \nin the deuteron and heavy nuclear targets\nare assumed to scale with the nuclear density.\nThe total correction for nuclear binding effects in the deuteron\n(shown in Fig. \\ref{fig:f2dp}(a)), \nis in a direction which is opposite\nto what is expected from the previous models which included only \nthe Fermi motion effects. \nThe suprisingly large correction extracted in this empirical way\nmaybe controversial, but is smaller than the recent theoretical prediction\n~\\cite{duSLAC} (dashed line in Fig. \\ref{fig:f2dp}(a))\n\nThe ratio $F_2^d\/F_2^p$ is directly related to $d\/u$. In leading order QCD,\n $2F_2^d\/F_2^p -1 \\simeq (1+4d\/u)\/(4+d\/u)$ at high $x$.\n  We perform a next-to-leading order (NLO) analysis on the precise\n  NMC $F_2^d\/F_2^p$ data~\\cite{NMCf2dp} to extract $d\/u$\n  as a function of $x$.\n We extract the ratio $F_2^{p+n}\/F_2^p$\n by applying the nuclear binding correction\n $F_2^d\/F_2^{n+p}$  to the $F_2^d\/F_2^p$ data.\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=f2dp_mor98_v2.ps,width=3.0in,height=3.0in}}\n\\caption{(a) The total correction for nuclear \neffects (binding and Fermi motion) in the deuteron,\n $F_2^d\/F_2^{n+p}$, as a function of $x$, extracted from fits to\nthe nuclear dependence of SLAC $F_2$ electron scattering\ndata (compared to theoretical model~[6]). ~(b) Comparison of NMC $F_2^{n+p}\/F_2^p$ (corrected for nuclear\neffects) and the prediction in NLO using the  MRS(R2) \nPDF\nwith and without our proposed modification to the $d\/u$ ratio.}\n\\label{fig:f2dp}\n\\end{figure}\n As shown in Fig. \\ref{fig:f2dp}(b), the\n standard PDF's~\\cite{MRSR2,CTEQ3M} do\n not describe the extracted $F_2^{p+n}\/F_2^p$.\n Since the $u$ distribution is relatively well constrained,\n we find a correction term to  $d\/u$ in the standard PDF's\n (as a function of $x$), by  varying only the $d$ distribution to fit the data.\n The correction term is  parametrized  \n as a simple quadratic form, $\\delta (d\/u) = (0.1\\pm0.01)(x+1)x$\n for the MRS(R2) PDF,\n where the corrected $d\/u$ ratio\n is $(d\/u)' = (d\/u) + \\delta (d\/u)$.\n Based on this correction,\n we obtain a  MRS(R2)-modified PDF as shown in Fig \\ref{fig:dou}(a).\n The correction to other PDF's such as CTEQ3M\/4M is similar.\n Note that since the $d$ quark level is small at large $x$,\n all the sum rules are easily satisfied with a very minute change at low $x$.\n The NMC data, when corrected for nuclear binding effects\nin the deuteron, clearly indicate that $d\/u$ in the\n standard PDF's is significantly underestimated \n at high $x$ as shown in Fig. \\ref{fig:dou}(a).\n It also shows that the modified $d\/u$ ratio \n approaches\n $0.2\\pm0.02$ as $x \\rightarrow 1$, in agreement with a QCD\n prediction~\\cite{Farrar}. In contract, if the\ndeuteron data is only corrected for Fermi motion effects (as\nwas mistakenly done in the past) both the $d\/u$ from data and the $d\/u$\nin the standard PDF's fits\napproach $0$ as $x \\rightarrow 1$.\n Figure~\\ref{fig:dou}(a) shows that  $d\/u$ values \nextracted from CDHSW~\\cite{du_cdhsw} $\\nu p$\/\n$\\overline{\\nu} p$ data (which are free from nuclear effects) \nalso favor the modified PDF's at high $x$.\n\nInformation (which is not affected by the corrections\nfor nuclear effects in the deuteron)\non $d\/u$ can be also extracted from $W$ production data in hadron\ncolliders.\nFigure~\\ref{fig:dou}(b) shows that the predicted $W$ asymmetry calculated\nwith the DYRAD NLO QCD program\nusing our modified PDF is\nin much better agreement with recent CDF data~\\cite{Wasym} at large\nrapidity than standard PDF's.\nWhen the modified PDF at $Q^2$=$16$ GeV$^2$ is evolved to $Q^2$=$10^4$ GeV$^2$\nusing the NLO QCD evolution, we find that\nthe modified $d$ distribution at $x=0.5$ is increased by about 40 \\% \nin comparison to the standard $d$ distribution.\nThe modified PDF's have a significant impact\non the \ncharged current cross sections~\\cite{zeushighq2}\nin the HERA high $Q^2$ region, shown in \nFig.~\\ref{fig:highq2}(a), because \nthe charged current scattering with positrons is on $d$ quark only.\nFigure~\\ref{fig:highq2}(b) shows that\nthe modified PDF's\nalso lead to an increase of 10\\% in the\nproduction rate\nof very high $P_T$ jets~\\cite{jet} in hadron colliders. \n\n\\begin{figure}[ht]\n\\centerline{\\psfig{figure=dou_mor98_v2.ps,width=3.0in,height=3.0in}}\n\\caption{(a) The $d\/u$ distributions at $Q^2$=$16$ GeV$^2$ \nas a function of $x$ for the standard and modified MRS(R2) PDF\ncompared to the CDHSW data. \n(b) Comparison of the CDF $W$ asymmetry data with NLO standard\nCTEQ3M, MRS(R2), and modified MRS(R2) as a function of the lepton rapidity.\nThe standard CTEQ3M with \na resummation calculation is also shown \nfor comparison.}\n\\label{fig:dou}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=zeus_highq2_diffx.ps,width=3.0in,height=1.5in}}\n\\centerline{\\psfig{figure=cdf-d0jet_cteq4mbase_v2.ps,width=3.0in,height=1.5in}}\n\n\\caption{ (a) The HERA charged current cross section data and \n(b) the CDF and D0 \ninclusive jet cross section data are compared \nwith both standard and modified PDF's.}\n\\label{fig:highq2}\n\\end{figure}\n\n\nSince all the standard PDF's, including our modified versions, are\nfit to data with $x$ less than 0.75, we now\ninvestigate the validity of the modified MRS(R2) at very high\n$x$  by comparing to  $F_2^p$ data at SLAC.\nAlthough the SLAC data at very high $x$ are at reasonable values\nof $Q^2$ $(7<Q^2<31$ GeV$^2)$, they are in a region in which\n non-perturbative effects such as target mass\n and higher twist are very large.\nWe use the Georgi-Politzer calculation~\\cite{GPtm}\nfor the target mass corrections (TM). These involve\nusing the scaling variable \n$\\xi=2x\/(1+\\sqrt{1+4M^2x^2\/Q^2})$ instead of $x$.\nSince a complete calculation of higher twist\neffects is not available, the very low $Q^2$ data are\nused to obtain information on the size of these terms.\n\nWe use two approaches in our study of the higher\ntwist effects:\nan empirical method, and the renormalon model. \nIn the empirical approach, the higher twist contribution is evaluated\nby adding a term $h(x)\/Q^2$ to the perturbative QCD (pQCD) prediction \nof the structure function (including target mass effects).\n The $x$ dependence of the higher\ntwist coefficients $h(x)$ is fitted to the global \ndeep-inelastic scattering (DIS) $F_2$ \n(SLAC, BCDMS, and NMC) data~\\cite{SLACF2,BCDMSF2,NMCF2} in the \nkinematic region \n($0.1<x<0.75$, $1.25<Q^2<260$ GeV$^2$)\nwith the following form; \n$F_2 = F_2^{pQCD+TM}(1+h(x)\/Q^2)f(x)$. \nHere $f(x)$ is a floating factor to investigate possible\n$x$ dependent corrections to our modified PDF.\nA functional form, $a(\\frac{x^b}{1-x}-c)$ \nfor $h(x)$ is used in the higher twist fit to\nestimate the size of the higher twist terms above $x=0.75$.\nThe SLAC and BCDMS data are normalized to the NMC data.\nIn the case of the BCDMS data, a systematic error shift $\\lambda$\n(in standard deviation units) is allowed\nto account for the correlated point-to-point  systematic errors.\nThe empirical higher twist fits with the modified NLO MRS(R2) pQCD\nprediction with TM\nhave been performed simultaneously\non the proton and deuteron $F_2$ data with 11 free parameters\n(two relative normalizations and three parameters for $h(x)$ per target\nand the BCDMS $\\lambda$).\nWe find that empirical higher twist fit describes the data well\n($\\chi^2\/d.o.f.=843\/805$), and the higher twist contributions\nin the  proton and deuteron are similar.\nThe magnitude is  almost half of the size from\na previous analysis of SLAC\/BCDMS\ndata~\\cite{Virchaux},\nbecause that analysis was based on \\mbox{$\\alpha_s$}(\\mbox{$M_Z^2$}) $=0.113$,\nwhile \\mbox{$\\alpha_s$}(\\mbox{$M_Z^2$}) $=0.120$ in the MRS(R2) PDF,\nwhich is close to the current world average.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=disht_f2_mor98.ps,width=3.1in,height=4.1in}}\n\\centerline{\\psfig{figure=disht_r_mor98.ps,width=3.1in,height=4.1in}}\n\\caption{The description of higher twist fit using the renormalon model\nwith the modified NLO MRS(R2) PDF. The CCFR neutrino data is also shown \nfor comparison.\n(a) Comparison of $F_2$ and NLO prediction\nwith and without higher twist contributions. \n(b) Comparison of $R$ and NLO prediction\nwith and without \nthe renormalon higher twist contributions.}\n\\label{fig:disht}\n\\end{figure}\nIn the renormalon model approach~\\cite{renormalon}, the \nmodel predicts the complete $x$ dependence of the higher twist\ncontributions \n to  $F_2$, $2xF_1$, and $xF_3$, with only two unknown parameters\n$A_2$ and $A_4$. \nWe extract the $A_2$ and $A_4$ parameters, which\ndetermine the overall level of the  $1\/Q^2$\nand $1\/Q^4$ terms by fitting to the global\ndata set for $F_2$ and $R [= F_2(1+4Mx^2\/Q^2)\/2xF_1 - 1]$.\nThe values of $A_2$ and $A_4$ for the proton and deuteron\nare same in this model.\nThe $x$ dependence of $2xF_1$ differs from that of $F_2$ but is the \nsame as that\nof $xF_3$ within a power correction of $1\/Q^2$. \nOur fits  can also be used to estimate the size of the higher twist effects\nin $xF_3$ [e.g. the Gross-Llewellyn Smith (GLS) sum rule].\nThe higher twist fit in this approach has employed the same procedure \nas the empirical method.  Figure~\\ref{fig:disht} shows that the model yields\ndescription of the $x$ dependence of higher twist terms in\n both $F_2$ and $R$\nwith just the two free parameters ($\\chi^2\/d.o.f.=1470\/928$).\nThe CCFR neutrino data~\\cite{CCFRF2}\nis shown for comparison though it is not used in the fit.\nThe extracted values of $A_2$ are $-0.093 \\pm 0.005$\n and $-0.101 \\pm 0.005$, for proton and deuteron,\n respectively. The contribution of $A_4$ is found  to be negligible. \nWe find that the floating factor $f(x)$ for the deuteron deviates from 1\nand is also bigger than\nthat for the proton, unless the modified MRS(R2) PDF is used.\nThis reflects our earlier conclusion\nthat the standard $d$ distribution is underestimated at high $x$ region.\nAs in the empirical fit, the extracted $A_2$ value \nis half of the previous estimated value~\\cite{renormalon} \nof $A_2$ based on SLAC\/BCDMS [\\mbox{$\\alpha_s$}(\\mbox{$M_Z^2$}) $=0.113$] analysis~\\cite{note}. \nSince both of these approaches yield a reasonable description for the higher\ntwist effects, we proceed to compare the predictions of the modified\nPDF's (including target mass and renormalon higher twist corrections) to the\nSLAC proton $F_2$ data at very high $x$ ($0.7<x<1$). \n\n\n\\begin{figure}[t]\n\\centerline{\\psfig{figure=f2_highx_highq2_comp_mrsr2base_v2.ps,width= 3.1in,height=3.8in}}\n\\caption{Comparison of SLAC $F_2^p$ data\nwith the predictions of the modified MRS(R2), CTEQ4M and the CTEQ toy model at high $x$\nand higher $Q^2$ ($20<Q^2<31$ GeV$^2$). ~~~(a) Ratio to pQCD, (b) ratio to pQCD \nwith TM effects,\nand (c) ratio to pQCD with TM and higher twist effects}\n\\label{fig:highx_highq2}\n\\end{figure}\n\nThere is a wealth of SLAC data~\\cite{SLACres} in the region up to $x=0.98$ \nand intermediate $Q^2$ $(7<Q^2<31$ GeV$^2)$. \nPrevious PDF fits have not used these data. \nWe use the estimate of the higher twist effects \nfrom the models, based on the data (below $x<0.75$) described above.\nNote that the data for $x>0.75$ is in the DIS region,\nand the data for $x>0.9$ is the resonance region.\nIt is worthwhile to investigate the resonance region also because\nfrom duality arguments~\\cite{Bloom} it is expected  \nthat the average behavior of the resonances and elastic peak\nshould follows the DIS scaling limit curve.\nFigure~\\ref{fig:highx_highq2} shows the ratio of the SLAC data to the predictions\nof the modified MRS(R2) at relatively large $Q^2$ ($21<Q^2<30$ GeV$^2$)\nwhere the elastic contribution is negligible.\nWith the inclusion of target mass and the renormalon higher twist\neffects, the very high $x$ data from SLAC is remarkably well described \nby the modified MRS(R2) up to $x=0.98$.  \nThe good description\nof the data by the modified MRS(R2) is also achieved \nusing the empirical estimate [$h(x)\/Q^2$] of higher twist effects\nas shown in Fig.~\\ref{fig:highx_highq2}(c).\nFigure~\\ref{fig:highx_highq2} also shows that the CTEQ Toy model\n(with an additional 0.5\\%\ncomponent of $u$ quarks beyond $x>0.75$) overestimates the SLAC data \nby a factor of three at $x = 0.9$ (DIS region).\nFrom these comparisons, we find that the SLAC $F_2$ data do not support\nthe CTEQ Toy model\nwhich proposed an additional $u$ quark contribution\nat high $x$ as an explanation of the initial HERA high $Q^2$ anomaly\nand the CDF high-$P_t$ jet excess. As indicated in\nFig.~\\ref{fig:highx_highq2}(c), the uncertainties in the PDF's at\nhigh $x$ are small. The difference between\nCTEQ4M and MRS(R2) (with our $d\/u$ modifications) is\nan estimate of the errors.\n\n\nIn conclusion, we find that nuclear binding effects in the deuteron\nplay a significant role in our understanding of $d\/u$\nat high $x$.\nWith the inclusion of target mass\nand  higher twist corrections, the modified PDF's\nalso describe all DIS data up to $x = 0.98$ and down to\n$Q^2 = 1$ GeV$^2$.\nThe modified PDF's with our $d\/u$ correction\nare in good agreement with the prediction of QCD at $x=1$, and with\nthe CDHSW $\\nu p$\nand $\\overline{\\nu} p$\ndata, the HERA CC cross section data,\nthe collider high-$P_t$ jet data, and with the\nCDF $W$ asymmetry data.\nA next-to-next leading order (NNLO) analysis~\\cite{note} of $R$ indicates  that\nthe higher twist effects extracted in the NLO fit \nat low $Q^2$ may originate from the missing NNLO terms.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIt is well known that the usual theory of $\\beta^{-}$ decay presumes that the decay of a neutron to proton is accompanied by the creation of an electron and an anti-neutrino in continuum states. However, in a stellar plasma where atoms get partially or fully ionized, this continuum decay is not the sole option. Nuclear $\\beta ^{-}$ decay to the bound states of the ionized atom is another probable channel. Also bare atoms have been produced terrestrially and $\\beta ^{-}$ decays have been studied in storage ring experiments. In 1947 Daudel \\!{\\em et al.\\ } \\cite{daudel} first proposed the concept of bound state $\\beta$ decay. This suggests that a nucleus has a possibility to undergo $\\beta ^{-}$ decay by creating an electron in a previously unoccupied atomic orbital instead of the continuum decay. It is important to understand that the bound state decay process does not occur subsequently from the $\\beta^{-}$ decay of an electron previously created in the continuum state, it is rather the direct creation of an electron in an atomic bound state accompanied by a mono-energetic anti-neutrino created in the free state carrying away the total decay energy. This process has been studied both theoretically as well as experimentally over the past seven decades. \n\n\nIn case of a neutral atom, available phase space for the creation of an electron in a vacant atomic orbital is very small and therefore the bound state decay is almost negligible compared to the contribution of the continuum decay. Contrarily, ionization of atoms may lead to drastic enhancement of bound state $\\beta$ decay probability due to the availability of more unoccupied atomic levels. In some previous theoretical works from 60's to 80's, various groups have studied the continuum and bound state  $\\beta$ decay for neutron, tritium \\cite{bahcall} and  fully ionized (bare) heavy atoms  \\cite{takahashi, takahashi1, takahashi2}. However, in most cases, previous theoretical works were based on very old data and\/or informatically incomplete. Simultaneously, the development of experimental techniques has served fruitfully to detect bound and continuum state $\\beta$ decay channels of fully ionized atoms. In 1992, Jung \\!{\\em et al.\\ } first observed the bound state $\\beta^{-}$ decay for the bare $^{163}$Dy atom \\cite{jung} by storing the fully ionized parent atom in a heavy-ion storage ring. In the same decade, Bosch \\!{\\em et al.\\ } studied the bound state  $\\beta ^{-} $ decay for fully ionized $^{187}$Re \\cite{bosch} which was helpful for the calibration of $^{187}$Re - $^{187}$Os   galactic chronometer \\cite{yokoi}. Further experiments with bare $^{207}$Tl \\cite{ohtsubo} showed the simultaneous measurement of bound and continuum state $\\beta^{-}$ decay. However, the authors have mentioned this decay as a single $\\beta^{-}$ transition process to a particular daughter level with 100 \\%  branching \\cite{ohtsubo} whereas, the present data \\cite{nndc} suggest three available levels among which the total $\\beta^{-}$ decay is distributed. \n \nIn earlier studies, Takahashi and Yokoi \\cite{takahashi, takahashi2} had investigated $\\beta$ transition (bound state $\\beta^{-}$ decay and orbital electron capture) processes of some selected heavy nuclei suitable for s-process studies. However, in their work, they had not given separately the bound state decay rate of bare atoms. Further, in another work, Takahashi  \\!{\\em et al.\\ } \\cite{takahashi1} had studied the $\\beta^{-}$ decay of some bare atoms for which bound state $\\beta^{-}$ decays produce significant enhancement in decay rates and proposed measurement in storage ring experiment.\nHowever, they did not take into account  the contribution of transitions to all possible energy levels of the daughter nucleus in total $\\beta^{-}$ decay rate enhancement. As an example, according to the present $\\beta^{-}$ decay data \\cite{nndc}, there are six possible  $\\beta^{-}$ transitions from the [117.59 keV, $6^{+}$] state of $^{110}$Ag to various states of $^{110}$Cd, but they had mentioned the contribution of only one transition.\n\n\n\nWith the availability of modern day experimental $\\beta$ decay half-lives in terrestrial condition for the neutral atom, experimental  Q-values  for $\\beta ^{-} $ decays and atomic physics inputs, it becomes inevitable to re-visit some of the earlier works. Moreover, in a previous work, Takahashi and Yokoi \\cite{takahashi} addressed a few  nuclei in their `case studies', undergoing $\\beta ^{-}$ transitions, as some of the essential turnabouts in $s$-process nucleosynthesis, where contributions from atoms with different states of ionization were considered. However, the explicit study of bound and continuum state $\\beta ^{-}$ transitions of bare atoms for most of these nuclei remained unevaluated till date both experimentally as well as theoretically.\n \n\nIn the present work, our aim is to study the $\\beta ^{-} $ decay of some elements, in the mass range (A $\\approx$ 60 - 240) which might be of interest for future experimental evaluations using storage ring. In particular, calculations of $\\beta^{-}$ decay rates to the continuum as well as bound state of these fully ionized atoms, where information for neutral atom experimental half-life and  $\\beta^{-}$ decay branchings are terrestrially available, have been performed. Most importantly the study of effective half-lives for bare atoms will be helpful to set a limit for the maximum enhancement in $\\beta^{-}$ decay rate due to the effect of bound state decay channels. \nMoreover, we have also discussed the effect of different nuclear structure and decay inputs (Q value, radius etc.) over the bound to continuum decay rate ratio. In addition, some interesting phenomena of changes in $\\beta^{-}$ decay branching for a number of bare atoms along with some notable change in branching (branching-flip) for a few of them, have been obtained. The branching-flip is obtained for the first time.\n\n\n\n \nThe paper is organized as follows: section \\ref{2} contains the methodology of our entire calculation for bound and continuum state $\\beta^{-}$ decay rates for bare atom,  as well as comparative half-life ($Log ft$) for neutral atom. In section \\ref{3A} we have discussed our results for the neutral atoms, whereas in section \\ref{3B}  results for the bare atoms have been discussed. The phenomenon of change in $\\beta^{-}$ decay branching for bare atom compared to that in neutral atom is also discussed explicitly in the section \\ref{3B}.  Conclusion of our work has been described in section \\ref{4}. Finally, we present a table for the calculated $\\beta^-$ decay rates in Appendix A followed by a discussion on the choice of spin-parity for unconfirmed states of neutral atom in Appendix B.\n \n\n\n\n\n\n\n\\section{{Methodology} \\label{2}}\n\n In this work, we have dealt with the allowed and first-forbidden $\\beta ^{-} $ transitions for neutral and fully ionized atoms. The contributions of higher-order forbidden transitions are negligible in the determination of the final $\\beta ^{-} $ decay rate and thus we have not tabulated the contributions for the same.    \n\nThe transition rates (in $sec^{-1}$) for allowed (a), non-unique first-forbidden(nu) and unique first-forbidden(u) transitions are given by \\cite{takahashi, takahashi1, takahashi2}\n\n\\begin{eqnarray}\n\\lambda = [(ln 2)\/f_0t](f^{*}_{m})     ~~~~~~ \\text {for m= a, nu}  \\\\ \\nonumber\n=[(ln 2)\/f_1t](f^{*}_{m})~~~~~~ \\text{for m= u ~~      }.\n\\end{eqnarray}\n\nHere $t$ is the partial half-life of the specific parent-daughter energy level combination for which transition rate has to be calculated and $f^{*}_{m}$ is the lepton phase volume part described in detail, below in this section. For allowed and non-unique first-forbidden $\\beta ^{-} $ decay, the expression for the decay rate function $f_0(Z,W_0)$ can be simplified to \\cite{gove, konopinski} \n\n\\begin{eqnarray}\nf_0(Z,W_0) =\\int^{W_0}_1\\sqrt{(W^2-1)} W (W_0-W)^2\\\\ \\nonumber\n\\times F_0(Z,W)L_0dW .\n\\end{eqnarray}\n\nThe certain combinations of electron radial wave functions evaluated at nuclear radius R ( in the unit of $\\hslash\/m_ec$) were first introduced by Konopinski and Uhlenbeck \\cite{konopinski} as $L_k$'s. The value for $k=0$ can be approximated  as\n \n\\begin{equation}\nL_0 = \\dfrac{1+\\sqrt{1-\\alpha^2 Z^2}}{2}.\n\\end{equation}\nHere, $\\alpha$ is the fine structure constant. In the work of Behrens and J\\\"anecke \\cite{behrens}, the authors had taken a different form of $L_0$, which includes a slight dependence on the momentum. However, we find that the $L_0$ approximation, adopted in our calculation, is in  good agreement with that from the Ref. \\cite{behrens} within the considered energy window.  \n\nIn Eq.(2), $W$ is the total energy of the $\\beta^{-}$ particle for a $Z-1\\rightarrow Z $ transition and $W_0 = Q_n\/m_ec^2+1 $ is the maximum energy available for the $\\beta^{-}$ particle. Here the mass difference between initial (parent) and final (daughter) states of neutral atoms are expressed as the decay $Q$ value ($Q_n$ in keV). The term $F_0(Z,W)$ is the Fermi function for allowed and non-unique transition, given by \\cite{konopinski}\n\n\\begin{eqnarray}\nF_0(Z,W)=\\dfrac{4}{ \\left|\\Gamma \\left( {1+2\\sqrt{1-\\alpha^2 Z^2 }}\\right)\\right|^2}\\\\ \\nonumber\n\\left(2R\\sqrt{W^2-1}\\right)^{2\\left(\\sqrt{1-\\alpha^2 Z^2} -1\\right)}exp\\left[\\dfrac{\\pi \\alpha Z W}{\\sqrt{W^2-1}}\\right] \\\\ \\nonumber\n\\times \\left|{\\Gamma{\\left(\\sqrt{1-\\alpha^2 Z^2} + i \\dfrac{\\alpha Z W}{\\sqrt{W^{2}-1}} \\right)}}\\right |^2.\n\\end{eqnarray}\n\nSimilarly, for the unique first-forbidden transition the decay rate function $f_1(Z,W_0)$ has the form reduced from Refs. \\cite{gove, konopinski} is given by,\n\n\\begin{eqnarray}\nf_1(Z,W_0) = \\int^{W_0}_1\\sqrt{(W^2-1)} W (W_0-W)^2 F_0(Z,W) \\\\ \\nonumber\n\\times\\left[(W_0-W)^2  L_0 + 9 L_1 \\right]dW ,\n\\end{eqnarray}\n\nwith $L_1$ given by,\n\n\\begin{equation}\nL_1 = \\dfrac{F_1(Z,W)}{F_0(Z,W)} \\left(\\dfrac{W^2-1}{9}\\right) \\dfrac{2+\\sqrt{4-\\alpha^2 Z^2}}{4}.\n\\end{equation}\n \nThe term $F_1(Z,W)$ for unique $\\beta ^{-} $ transition is given by \\cite{konopinski},\n\n\\begin{eqnarray}\nF_1(Z,W)=\\dfrac{(4!)^2}{ \\left|\\Gamma \\left( {1+2\\sqrt{4-\\alpha^2 Z^2 }} \\right) \\right|^2} \\\\ \\nonumber\n\\left(2R\\sqrt{W^2-1}\\right)^{2\\left(\\sqrt{4-\\alpha^2 Z^2} -2\\right)}exp\\left[\\dfrac{\\pi \\alpha Z W}{\\sqrt{W^2-1}}\\right] \\\\ \\nonumber\n\\times \\left|{\\Gamma{\\left(\\sqrt{4-\\alpha^2 Z^2} + i \\dfrac{\\alpha Z W}{\\sqrt{W^{2}-1}} \\right)}}\\right |^2.\n\\end{eqnarray}\n\n\n \nEqs. (2) and (5) are general forms of $f_0(Z,W_0)$ and $f_1(Z,W_0)$. For more precise calculation of f-factor, one should in principle, include various corrections in the integrand of Eqs. (2) and (5). Corrections due to atomic physics effects, radiative correction and finite nuclear size effects might be important for such studies.  For fully ionized atoms, corrections due to atomic physics effects, such as, imperfect overlap of initial and final atomic wave functions, exchange effects that comes from the anti-symmetrisation of the emitted electron with the atomic electrons \\cite{bahcall2}, screening of the nuclear charge due to the coulomb field of the atomic electronic cloud are not needed. For neutral atom, the decay to the atomic bound state should be negligible \\cite{bahcall2}. Also, the screening and exchange corrections together cancel a large part of the overlap correction \\cite{budick}. Further the non-orthogonality effect becomes rapidly smaller as Z increases \\cite{takahashi1}. Some of the corrections have positive sign and some of them have negative sign. So unless all the corrections are taken together, the treatment for corrections to f- factor will not be consistent. Therefore we have neglected these contributions both for bare and neutral atoms. We have included the correction due to the extended charge distribution of the nucleus on the $\\beta^{-}$ spectrum. This correction is $\\Lambda_k(Z,W) \\rightarrow \\Lambda_k(1+\\Delta\\Lambda_k(Z,W)$), where the term $\\Lambda_k$ can be written in terms of $L_k$ and $F_0(Z,W)$ as \\cite{gove, konopinski}\n\n\\begin{equation}\n\\Lambda_k(Z,W)= F_0(Z,W)L_{k-1}\\left[ \\dfrac {(2k-1)!!}{(\\sqrt{W^2-1})^{k-1}}\\right]^2 ,\n\\end{equation}\n  \nin such a way that it reduces to $\\left[ F_0(Z,W)L_0\\right]$ and $\\left[ 9F_0(Z,W)L_1\/(W^2-1)\\right]$ for $k=1$ and $2$, respectively. The correction term is given by \\cite{gove},\n\n\\begin{eqnarray}\n\\Delta\\Lambda_k (Z,W) =(Z-50) \\times \\\\ \\nonumber\n\\left[ -25\\times 10^{-4} - 4\\times10^{-6} W \\times (Z-50)\\right] \\\\ \\nonumber\n\\text {   for  } k = 1, Z > 50 , \\\\ \\nonumber\n= 0 ~~~~~~~~~~~~~~~~~~~~~    \\text {   for   } k = 1 , Z \\le 50, \\\\ \\nonumber\n= 0 ~~~~~~~~~~~~~~~~~~~~~~    \\text {for                      } k > 1~~~~~~~~~~~.\n\\end{eqnarray}\n\nThe screened energy of the emitted electron $(W')$ enters through $\\Delta\\Lambda_k(Z,W')$, where $W'=W-V_0(Z)$. We calculated $V_0(Z)$, following Gove and Martin \\cite{gove}, using expression from W. R. Garrett and C. P. Bhalla \\cite{bhalla}. This correction to the integrand in Eqs. (2) and (5) has effect in the fourth decimal place of the f-factor and this is consistent with Ref. \\cite{hayen} discussed for the allowed $\\beta^{-}$ decay. So we have dropped $W'$ and used $W$ in the integrand.\n\nIt is to be noted that in the present work we have used experimental quantities, such as Q - value, half-life, branching, which have uncertainties even up to the first decimal place \\cite{nndc, nist}. So, in our treatment we have neglected the screening effect too for neutral atom. Therefore, by using Eqs. (8) and (9)  in the integrand of Eq. (2) and Eq. (5) one can calculate the values for  $f_0(Z,W_0)$ and $f_1(Z,W_0)$  incorporating  only finite size correction.\n\n In the work of  A. Hayes \\!{\\em et al.\\ } \\cite{hayes}, the authors have taken a different form of the finite-size correction  involving the charge density, which has a complicated radial dependency. However, we find that the results from the present calculation are in agreement with the available experimental data \\cite{nndc}. \n\n Further, from the above expressions (Eqs.(4) and (7)), it is evident that the factors $F_0(Z,W)$ and $F_1(Z,W)$  depend on the radius, thereby making the terms $f_0$ and $f_1$ (Eqs.(2) and (5)), radius dependent. Thus, in our present study, we have used various radius values from different phenomenological models and experiments to study their effects on the final $ft$ values. In order to calculate $ft$ values for a nucleus, we have extracted the half-life $t$ for individual transition to daughter levels using the latest $\\beta$ decay\nbranching information available in the literature \\cite{nndc}. \n\n\nThe lepton phase volume $f^{*}_{m}$ \\cite{takahashi2} for the continuum state  $\\beta^{-}$ decay can thus be expressed as,\n\n\n\n\\begin{eqnarray}\nf^{*}_{m=a,nu}(Continuum) = \\int^{W_c}_1\\sqrt{(W^2-1)} \\\\ \\nonumber\n W (W_c-W)^2 F_0(Z,W) L_0  dW,\n\\end{eqnarray}\n \nand\n \n\\begin{eqnarray}\nf^{*}_{m=u}(Continuum) = \\int^{W_c}_1\\sqrt{(W^2-1)} \\\\ \\nonumber\nW (W_c-W)^2 F_0(Z,W) \\times \\\\ \\nonumber\n \\left[(W_c-W)^2 L_0 + 9L_1\\right]  dW,\n\\end{eqnarray}\n\nHere $W_c = Q_c\/m_ec^2 +1 $ is the maximum energy available to the emitted $\\beta^{-}$ particle, and $Q_c$ is given by,\n\n\\begin{eqnarray}\nQ_c = Q_n - \\left[ B_n(Z+1) - B_n(Z)\\right].\n\\end{eqnarray}\n     \n The term $\\left[ B_n(Z+1) - B_n(Z)\\right]$ denotes the difference of binding energies for bound electrons of the daughter and the parent atom. The experimental values for all the atomic data (binding energies\/ionization potential) are availed from Ref. \\cite{nist}. \n\nFurther, for the bound state  $\\beta^{-}$ decay of the bare atom  $f^{*}_{m}$ takes the form \\cite{takahashi2}\n\n\\begin{eqnarray}\nf^{*}_{m=a,nu}(Bound) = \\sum_x \\sigma_x \\left(\\pi\/2\\right) \\left[ f_x\\text{ or }g_x \\right]^2 b^2 \\\\ \\nonumber\n\\left(\\text {for  } x=ns_{1\/2},np_{1\/2}\\right),\n\\end{eqnarray}\n \nand\n \n\\begin{eqnarray}\nf^{*}_{m=u}(Bound) = \\sum_x \\sigma_x \\left(\\pi\/2\\right) \\left[ f_x\\text{ or }g_x \\right]^2 b^4  \\\\ \\nonumber\n\\left(\\text {for  } x=ns_{1\/2},np_{1\/2}\\right), \\\\ \\nonumber\n~~~~~~~~~~~~~~~~~~~~\\\\ \\nonumber\n= \\sum_x \\sigma_x \\left(\\pi\/2\\right) \\left[ f_x \\text{ or } g_x \\right]^2 b^2 \\left(9\/R^2\\right) \\\\ \\nonumber\n\\left(\\text{for  } x=np_{3\/2},nd_{3\/2}\\right).\n\\end{eqnarray}\n\nHere $\\left[ f_x\\text{ or }g_x \\right]$ is the larger component of electron radial wave function evaluated at the nuclear radius $R$ of the daughter for the orbit $x$. The $\\left[ f_x\\text{ or }g_x \\right]$ is obtained by solving Dirac radial wave equations using the Fortran subroutine RADIAL by Salvat \\!{\\em et al.\\ } \\cite{cpc}. In our case,  $\\sigma_x$ is the vacancy of the orbit, chosen as unity and $b$ is equal to $Q_b\/m_ec^2$ where,\n\n\\begin{eqnarray}\nQ_b = Q_n - \\left[ B_n(Z+1) - B_n(Z)\\right]-B_{shell}(Z+1).\n\\end{eqnarray}\n \nFor example, in case of a bare atom, if the emitted $\\beta^{-}$ particle gets absorbed in the atomic K  shell, then the last  term of Eq.(15) will be the ionization potential for the K electron denoted by $B_K(Z+1)$. \n\n\n\n\\section{Results and Discussion}\n\n\nIn this work, we have calculated  \n$\\beta^-$ decay transition rates to  bound and continuum states,  for a number of  fully ionized atoms in the mass range A $\\approx$ 60-240. One of the motivations is that there are some evidences where earlier works were not equipped enough to address the entire $\\beta^-$ decay scenario.\n  This might be due to the unavailability of information about all the energy levels participating in transition processes.\n\nAs an example, Takahashi \\!{\\em et al.\\ } \\cite{takahashi1} have considered transitions for allowed(a), first-forbidden non-unique(nu) and first-forbidden unique(u) decay of parent nuclei to a few energy levels of daughter nuclei. For instance, in the case of $^{228}$Ra nucleus, the authors have tabulated the decay from the ground state of the parent $[E$(keV), $J^\\pi] = [0.0, 0^+]$ nucleus to $[6.3, 1^-]$ and $[33.1, 1^+]$ states of the daughter nucleus  $^{228}$Ac. However, these two transitions cover only the 40\\% of the total $\\beta^{-}$ decay branching of neutral $^{228}$Ra atom from the ground state. With the latest experimental data \\cite{nndc}, we find that there are two more available states of  $^{228}$Ac where the rest amount of $\\beta^{-}$ decay from the ground state of $^{228}$Ra occur. In this section, it will be shown that the contributions of all these four states are extremely important in the determination of effective enhancement of $\\beta^{-}$ transition rates of bare $^{228}$Ra as well as to understand the phenomenon of  branching-flip, discussed in section \\ref{3B}.\n\n For simplicity, this section is subdivided into two parts. The first subsection involves the calculation of  $Log~ ft$ for the neutral atom, a necessary ingredient for the calculation of $\\beta^-$ decay rate of the bare atom. In the next subsection, the $\\beta^{-}$ decay transition rates of  bare atoms have been discussed with a detailed explanation of TABLE {\\red A.I}. The dependence of these decay rates on different parameters is also  examined in the same subsection. Finally, we have shown and discussed the change in individual level branchings in fully ionized atoms.\n\n\n\\subsection{{ $Log~ ft$ calculation for neutral atoms}\\label{3A}}\n\nIt is evident from Eqs.(1-9) that the calculation for $ft=f_0t (\/f_1t)$ is one of the essential components in the determination of the transition rate $\\lambda$, which in turn depends on radius R of the daughter nucleus. However, $Log~ ft$ data obtained from Ref. \\cite{nndc} can not provide the information of the $R$ dependence of $Log~ ft$.\nAs the present theoretical modelling  for bare atom depends on radius (see section \\ref{2}), we find it more accurate to calculate  $Log~ ft$ for neutral atom  for different choices of radii.\n\nIn Appendix \\ref{lognu}, we present a table for bound and continuum state $\\beta^-$ decay rates for bare atoms along with the values of $Log~ ft$ for corresponding neutral atoms at different radii and compare our calculations with existing theoretical as well as experimental results (see the supplemental material \\cite{supl} for details).\nAs explained in section \\ref{2}, we have tabulated $Log~ ft$ values only for allowed (a), first-forbidden non-unique (nu) and first-forbidden unique (u) transitions.\n\n  \nHere, in TABLE {\\red A.I}, $R_1$ is the phenomenological radius evaluated as $R_1=1.2 A^{1\/3}$ fm, whereas $R_2$ is the nuclear charge radius in fm \\cite{angeli} and $R_3$ is the half-density radius given by \\cite{gove} $R_3=(1.123A^{1\/3} - 0.941A^{-1\/3})$ fm. We have calculated  $Log~ ft$ values for $R_1$, $R_2$ and $R_3$ and compared them with the existing data \\cite{nndc}. Besides, we have tabulated the available values from previous calculations of Takahashi \\!{\\em et al.\\ } \\cite{takahashi1} in the same table. \n\nOne can see that the change in radius may cause a change in the $Log ~ft$ value mostly in the second decimal place. In the next subsection, we will show the effect of these variations on the transition rates for bare atoms.\n\nFurther, from TABLE {\\red A.I} and the supplemental material \\cite{supl}, it can be noted that our calculation matches with the experimental  $Log~ ft$ data \\cite{nndc} in most cases up to the first decimal place. The agreement of our result with experimental data \\cite{nndc} confirms the applicability of the methodology adopted in the present  study.\n\n\n\n\n\\subsection {{ Bound and Continuum decay rates of bare atoms}\\label{3B}}\n\n\n In  the ninth and the eleventh column of TABLE  {\\red A.I} of Appendix \\ref{lognu}, bound and continuum $\\beta^-$ decay rates of bare atoms are presented, respectively.\n\n It is observed that the dependence on radius affect the bound ($\\lambda_B$) and the continuum state ($\\lambda_C$)  decay rates  in first or second decimal places, and the ratio $\\lambda_B\/\\lambda_C$ remains almost unaffected up to the first decimal place for most of the examined cases. \n\nFurther, from TABLE  {\\red A.I} (also see the supplemental material \\cite{supl}), we find that the values for $\\lambda_B$ and $\\lambda_C$ from our calculation agree with those of the existing theoretical results \\cite{takahashi1} quite well. The possible reasons for the slight mismatch between our calculation and that from Takahashi \\!{\\em et al.\\ } \\cite{takahashi1} \nare mainly due to (i) the effect of the nuclear radius, (ii) the adoption of present day Q values (for all $Q_n ,  Q_c$ and $Q_b$), (iii) availability of present day  $\\beta^-$ decay branching of neutral atoms and (iv) the choice of significant digits. Despite that, the overall success of our calculation in reproducing available $\\lambda_B$ and $\\lambda_C$ for bare atoms once again justify the extension of the present calculation to previously unevaluated cases.  \n\n\\begin{figure*}\n\n{\\includegraphics[width=15cm,height=11cm]{lblc_4}}\n\\caption{(Color online) Ratio of $\\lambda_B\/\\lambda_C$ Vs the neutral atom Q-value $Q_n$ (in keV) for various $\\beta^{-}$ transitions (for the radius $R_{1}$). The dotted curves are obtained from fitting to Eq.(16).  See text for details.  \n\\label{lblc}}\n\\end{figure*}\n \n\n\nIt can again be shown from TABLE  {\\red A.I} that in a transition from the parent nucleus $^AX_{Z-1}$  to different energy levels of the daughter nucleus $^AX_{Z}$, the ratio $\\lambda_B\/\\lambda_C$ for all transitions are not same, rather it decreases with increasing $Q_n$ value. It can be understood from the expressions in Eqs.(10-15) where the factors $f^{*}_{Continuum}$ and $f^{*}_{Bound}$ depend on $Q_c$ and $Q_b$, respectively, which are again derived from the neutral atom Q value $Q_n$. Due to different $Q_n$ values for different transitions,  $\\lambda_B\/\\lambda_C$ can be identified as a function of $Q_n$. For the sake of understanding, in FIG. \\ref{lblc}, we have plotted the ratio $\\lambda_B\/\\lambda_C$ versus $ Q_n$ for the nuclei $^{115}$Cd, $^{123}$Sn, $^{136}$Cs and $^{152}$Eu. In each case, dependence on $Q_n$ is observed which can be fitted to the form\n \n\\begin{eqnarray}\n\\dfrac{\\lambda_{B}}{\\lambda_{C}}=a\\times({Q_n})^b\n\\end{eqnarray}\n\nwhere a and b are the nucleus dependent constants given in TABLE  \\ref{ab}.\n\n\\begin{table}[H]\n\n\\caption{Parameters a and b for Eq.(16) for the radius $R_1$.}\n\n\\vspace*{0.3 cm}\n\t\\centering\n\\resizebox{!}{!}{\n \\begin{tabular}{|c|c|c|}\n\t\t\\hline\n                 Parent $\\rightarrow$ Daughter        &              Parameter a           &       Parameter b \\\\ \\hline\n \\rule{0pt}{0.5 cm}\n                 $^{115}Cd$ $\\rightarrow$ $^{115}In$  &    3093.12 $\\pm$ 317.17            & -1.48 $\\pm$ 0.02   \\\\ \\hline\n\n \\rule{0pt}{0.5 cm}\n                 $^{123}Sn$ $\\rightarrow$ $^{123}Sb$  &    12657.22 $\\pm$ 1515.52            & -1.73 $\\pm$ 0.03   \\\\ \\hline\n\n \\rule{0pt}{0.5 cm}\n                 $^{136}Cs$ $\\rightarrow$ $^{136}Ba$  &    5178.76 $\\pm$ 654.04            & -1.52 $\\pm$ 0.02   \\\\ \\hline\n\n \\rule{0pt}{0.5 cm}\n                 $^{152}Eu$ $\\rightarrow$ $^{152}Gd$  &    18851.81 $\\pm$ 1065.69            & -1.68 $\\pm$ 0.01   \\\\ \\hline\n\n\\end{tabular}}\n\n\\label{ab}\n\\end{table}\n\n\n The TABLE \\ref{ab} confirms that Eq.(16) is a characteristic feature of $\\lambda_{B}\/ \\lambda_{C}$ ratio of the bare atom with particular Z and A values. If there is a mistake in the calculation of  $f^{*}$ for $\\lambda_{B}$ or $\/$ and $\\lambda_{C}$, then the ratio point will not fit to such a power law.\n\n \n\n\n In the fourteenth column of TABLE {\\red A.I}, the ratio of $\\lambda_{Bare}(=\\lambda_{B} + \\lambda_{C})\/\\lambda_{Neutral}$ (called here rate enhancement factor) has been tabulated. It is evident from these values that there must be an enhancement in the decay rate for each transitions (i.e. $\\lambda_{Bare}\/\\lambda_{Neutral} > 1$) because of the additional bound state decay channel. \n\nIn FIG. \\ref{enh}, the ratio of  $\\lambda_{Bare}\/\\lambda_{Neutral}$  for $^{110}$Ag, $^{155}$Eu and $^{227}$Ac have been shown. From the figure, it can be noted that rate enhancements  (a) are different for different transitions of a particular nucleus, (b) are dependent on $Q_n$ values : lower the $Q_n$, larger the enhancement. Moreover, this rate enhancement factor (c) also depends on Z and A of the atom; larger the value of Z and\/or A,  larger the enhancement.  \n\nFurther, in TABLE  {\\red A.I}, we have tabulated effective $\\beta^-$ decay half-lives for bare atoms and compared to those of neutral atoms. It should be noted that the neutral atom half-life given in the fifteenth column of the table is the total half-life corresponding to a, nu and u types of $\\beta^{-}$ transitions only.\n\n\\begin{figure}\n\n{\\includegraphics[width=85mm,height=65mm]{Figure_2b}}\n\\caption{(Color online) Ratio of $\\lambda_{Bare}\/\\lambda_{Neutral}$ Vs the neutral atom Q-value $Q_n$ (in keV) for various $\\beta^{-}$ transitions (for the radius $R_{1}$). See text for details.  \n\\label{enh}}\n\\end{figure}\n  \n\\vspace{0.2cm}\n\n\n\\underline {\\textbf{Transition details: case studies}}  \n\\vspace{0.2cm}\n\n\\begin{figure*}[t]\n\\centering\n{\\includegraphics[width=19.5cm,height=7.5cm]{Figure_3}}\n\\caption{(Color online) Comparison of the $\\beta^{-}$ decay branchings for neutral and bare $^{136}$Cs isotope (for the radius $R_{1}$). \n\\label{noflip}}\n\\end{figure*}\n\n\n\nThe dependence of the rate enhancement factor on $Q_n$ causes a change in $\\beta^-$ branching for the bare atom. In bare atom, branchings similar to the neutral atom can only be achieved if the factor $\\lambda_{Bare}\/\\lambda_{Neutral}$ remains constant with  $Q_n$, which is obviously not the case (FIG. \\ref{enh}). In other words, this change can be understood to be an outcome of the non-uniformity of the  $\\lambda_B \/ \\lambda_C $ ratio with $Q_n$. It is observed that the continuum decay rate for bare atom decreases with respect to that for the neutral atom (i.e. $\\lambda_{C} < \\lambda_{Neutral}$) due to the reduction of continuum Q value ($Q_c < Q_n$, Eq. (12)).  Further from FIG. \\ref{lblc}, it is clear that with the decrease in the $Q_n$ value, $\\lambda_B$ dominates more over $\\lambda_C$ and hence the effective decay rate of the bare atom $\\lambda_{Bare}=\\lambda_B + \\lambda_C$ does not follow the same branching as that of the neutral atom.\n\n\n\n Note: For the $\\beta^{-}$ transition having very low $Q_n$ value, bound state decay may be the only path of $\\beta^{-}$ decay. As an example, in the transition of $^{227}$Ac $[0.0,3\/2^-]$ to  $^{227}$Th $[37.9,3\/2^-]$ with $Q_n = 6.9$ keV, $Q_c$ for continuum decay of the bare atom becomes $-13.1$ keV. As evident from Eqs.(10-12), due to the negative value of  $Q_c$, the corresponding decay channel gets closed. On the other hand, as $(Q_b-Q_n) > 0$ for this transition, the total decay is governed by the bound state channel only.\n\n\n \nAs an example, in FIG. \\ref{noflip}, we have compared branchings for  neutral (left panel) and bare (right panel) $^{136}$Cs atom. It can be seen from FIG. \\ref{noflip} that the branchings for all $\\beta^-$ transitions of the bare atom have been changed from that of the neutral atom. However, the ordering of each branch remains unaltered in both cases, i.e. the $[0.0, 5^+] \\rightarrow [2207.1, 6^+]$ branch gets the maximum feeding followed by the $[0.0, 5^+] \\rightarrow [1866.6, 4^+]$ and $[0.0, 5^+] \\rightarrow [2140.2, 5^-]$ branches, whereas the minimum feed goes to  $[0.0, 5^+] \\rightarrow [2030.5, 7^-]$ channel for both the neutral and bare atoms.\n\nFurther, some notable observations and comments for some nuclei are given below.  \n\n $\\bullet$ In case of neutral $^{207}$Tl atom in terrestrial condition, the $[0.0, 1\/2^+]$ state of  $^{207}$Tl  decays to $[0.0, 1\/2^-]$ state of  $^{207}$Pb with 99.729\\% branching, whereas to $[569.6, 5\/2^-]$ state of the daughter has the branching  $>$0.00004\\% (in some places of Ref. \\cite{nndc} this value is given as  $<$0.00008\\%) and to $[897.8, 3\/2^-]$  state has 0.271\\% branching \\cite{nndc} (see supplemental material \\cite{supl} for details). For bare atom, Ohtsubo \\!{\\em et al.\\ } \\cite{ohtsubo} had observed bound state decay rate $\\lambda_B= 4.29(29) \\times 10^{-4}$ sec$^{-1}$ and continuum state decay rate $\\lambda_C=2.29 (012) \\times 10^{-3}$ sec$^{-1}$, by considering the transition to  $[0.0, 1\/2^-]$ state of $^{207}$Pb with 100\\% branching. In our calculation for bare atom, we have got bound state decay rate $\\lambda_B = 4.15 \\times 10^{-4}$ sec$^{-1}$ and continuum state decay rate $\\lambda_C= 2.54 \\times 10^{-3}$ sec$^{-1}$. The calculated branchings of bare $^{207}$Tl : $\\sim$ 99.6 \\% to $[0.0, 1\/2^-]$, $\\sim$ 0.00005\\% - 0.0001\\% to $[569.6, 5\/2^-]$ and $\\sim$ 0.4 \\% to $[897.8, 3\/2^-]$ states of the daughter $^{207}$Pb.\n\n In our study, we found some special cases where  effective branchings for the bare atom do not follow the same ordering as that of the neutral atom. This indicates a very interesting phenomenon of branching-flip, obtained for the first time in this work. Sometimes the additive contribution of $\\lambda_B$ and $\\lambda_C$ and the effect of these two competing channels can lead to this branching-flip. This  can be understood from FIG. \\ref{all}. In FIG. \\ref{all}, decay rates (sec$^{-1}$) for neutral ($\\lambda_{Neutral}$) and bare ($\\lambda_{Bare}$) atom along with all decay components ($\\lambda_B$ and $\\lambda_C$) of the bare atom versus $Q_n$ are shown for the ground state decay of $^{134}$Cs and $^{228}$Ra nuclei. One can see from FIG. \\ref{all} that the highest point corresponding to $\\lambda_{Neutral}$ (i.e. maximum $\\beta^{-}$ branching in neutral atom) and the highest point corresponding to $\\lambda_{Bare}$ (i.e. maximum $\\beta^{-}$ branching in bare atom) are coming from different transitions to the daughter nuclei (different $Q_n$ values), which clearly indicates the phenomenon of flip in the branching sequence.\n\n$\\bullet$ In the case of $^{134}$Cs,  $\\lambda_{Neutral}$ is maximum at $Q_{n} = 658.1$ keV, which is due to the maximum branching to the 1400.6 keV level (see supplemental material \\cite{supl} for details) of  $^{134}$Ba \\cite{nndc}. In contrary, for the same nucleus, $\\lambda_{Bare}$ is maximum at $Q_n = 88.8$ keV which therefore indicates the maximum branching to the 1969.9 keV level (see TABLE {\\red A.I}) of the daughter $^{134}$Ba for bare atom. \n\n$\\bullet$ Similarly for $^{228}$Ra, the maximum branching for the bare atom ($(\\lambda_{Bare})_{max}$ at $Q_n=12.7$ keV) shifts from that of the neutral atom ($(\\lambda_n)_{max}$ at $Q_n=39.1$ keV). In FIG. \\ref{flip}, we have shown the change and alteration of transition branchings for the $\\beta^-$ decay of $^{228}$Ra.  \nOne can see the branching-flips of the participating levels of the $^{228}$Ac atom in FIGS. \\ref{all} and \\ref{flip}. In case of the neutral $^{228}$Ra atom, maximum branching is 40\\% for the $[6.7, 1^+]$ level of the daughter \\cite{nndc}. After complete ionization, the major contribution of the total decay rate comes due to the bound state enhancement of $Q_n=$12.7 keV channel which has $\\sim$ 84.07\\% decay to the  $[33.1, 1^+]$  level (30\\% in neutral atom) of the daughter atom, whereas only $\\sim$ 5.81\\% of the total decay branching is observed for the level $[6.7, 1^+]$.   \n\nThere are a few more cases, where the branching-flips are observed. However, not necessarily, all the transition branches face the phenomenon of flip. It may also happen that only two or three branches change their sequence, whereas other branches remain in the same order as that of the neutral atom.\n\n$\\bullet$ In the $\\beta^-$ decay of $^{152}$Eu $[45.5998, 0^{-}]$ (see Table 1 of the Ref. \\cite{supl} for branching details), we find that in both cases (neutral and bare) the branching to  $[0.0, 0^+]$ branch of the daughter dominate over the rest, whereas a branching-flip is observed between $[344.3,2^+]$ and $[1314.6, 1^-]$ states.  \n\n$\\bullet$ Similarly for $^{227}$Ac, we find that there is a branching-flip between two transitions from   $[0.0, 3\/2^-]$ state of the parent to  $[0.0, 1\/2^+]$ and $[24.5, 3\/2^+]$ states of the daughter atom.\nThe ratio of branching for these two levels is 5.4:1 for neutral atom, which changes to 1:1.38 for bare atom. \n\n\nIt should be noted that the ultimate fate of individual branchings in the bare atom is decided by two factors: the initial branching (required to calculate $Log~t$ for each transition: a part of $Log~ft$ calculation) and Q value of the neutral atom. The competition between these two factors determines whether the branching-flip will occur or not.\n \n   \n\\textbf{Effect of uncertainties: } Furthermore, in order to get the complete picture of  $\\beta^-$ decay for bare atom, effects due to uncertainties in $\\beta^-$ decay half-life  and Q value need to be considered. The effect of uncertainty is appreciable depending on the numerical value of the half-life and Q value. In case of atoms with the  $\\beta^-$ decay half-life of the order of seconds\/minutes and having high Q value, no significant change is observed in the calculation of $Log~ ft$ due to the inclusion of experimental uncertainties. The contributions peek out for long lived nuclei  with large uncertainty or for transitions of high Q value having large uncertainty.   \n For example, in case of $^{93}$Zr atom, where the neutral atom half-life is equal to $1.61 \\times  10^6(5)$ years, $Log~ ft$ for the transition $[0.0,5\/2^+ \\rightarrow 30.8,1\/2^-]$ with the radius $R_1$ is given by ${10.234}^{+0.014}_{-0.013}$. Therefore, the final values for continuum and bare state $\\beta^-$ transitions including the uncertainties can be written as  $\\lambda_C={6.87}_{-0.21}^{+0.22} \\times 10^{-15}$ $sec^{-1}$ and $\\lambda_B={6.13}_{-0.19}^{+0.20} \\times 10^{-15}$ $sec^{-1}$, respectively. \n\n\n\n\n\\begin{figure*}[t]\n\\centering\n{\\includegraphics[width=17cm,height=6cm]{Figure_4}}\n\n\\caption{(Color online) Decay rates (in sec$^{-1}$) for neutral ($\\lambda_{Neutral}$) and bare($\\lambda_{Bare}$) atoms along with all the decay components ($\\lambda_B$ and $\\lambda_C$) of the bare atom (for the radius $R_{1}$) with the neutral atom Q-value $Q_n$ (in keV). See text for details.  \n\\label{all}}\n\\end{figure*}  \n\n\\begin{figure*}[t]\n\\centering\n{\\includegraphics[width=19cm,height=7cm]{Figure_5}}\n\\caption{(Color online) Comparison of level branchings on neutral and bare $^{228}$Ra isotope (for the radius $R_{1}$). Left Panel: neutral atom, Right Panel: bare atom.   \n\\label{flip}}\n\\end{figure*}\n\n\n\n\\section{{Conclusion}\\label{4}}\nTo summarize, in this work we have calculated individual contributions of bound and continuum state $\\beta^-$ decays to the effective $\\beta^-$ decay rate of the bare atom in the  A $\\approx$ 60 to 240 mass range where earlier information were partial and\/or old.\n\n Additionally, the dependence of transition rates over the nuclear radius and the Q value is illustrated clearly in the present study.  We found a power law dependence of $\\lambda_{B}\/ \\lambda_{C}$ of a bare atom on $Q_n$ for each value of Z and A. Along with the effective enhancement of transition rates, we have found that transition branchings for the bare atom differs from that of the neutral atom for all Z and A, which is an outcome of non-uniform enhancement amongst the participating branches. Most interestingly, we have found few nuclei, viz. $^{134}$Cs, $^{228}$Ra etc., where some flip in the branching pattern is found for their bare configuration. It will be interesting to see how these results help planning new experiments involving bare atoms.  The calculations will be extended to partially ionized atoms which will provide decay rate as function of density and temperature of the stellar plasma and will be useful for calculation of nucleosynthesis processes.     \n\n\n\n\n\\section*{ACKNOWLEDGEMENT}\nAG is grateful to DST-INSPIRE Fellowship (IF160297) for providing financial support. CL  acknowledges the grant from DST-NPDF (No. PDF\/2016\/001348) Fellowship.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\\label{sec:introduction}}\n\\IEEEPARstart{C}{reating} dynamic general clothes or garments on animated characters has been a long-standing problem in computer graphics (CG).\nIn the CG industry, physics-based simulations (PBS) are used to achieve realistic and detailed folding patterns for garment animations. \nHowever, it is time-consuming and requires expertise to synthesize fine geometric details since high-resolution meshes with tens of thousands or more vertices are often required.\nFor example, 10 seconds are required for physics-based simulation of a frame for detailed skirt animation shown in Fig.~\\ref{fig:lrhrsim1}.\nNot surprisingly, garment animation remains a bottleneck in many applications.\nRecently, data-driven methods provide alternative solutions to fast and effective wrinkling behaviors for garments.\nDepending on human body poses, some data-driven methods~\\cite{wang10example,Feng2010transfer,deAguiar10Stable,santesteban2019learning, wang2019learning} are capable of generating tight cloth animations successfully.\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{ccc}\n\t\\multicolumn{3}{c}{\n\t\\includegraphics[width=1.0\\linewidth]{pictures\/wireframe2_1.pdf}}  \\\\\n\t(a) coarse skirt & (b) tracked skirt & (c) fine skirt\n\t\\end{tabular}\n\t\\caption{\\small \\cl{One frame of \\YL{skirt in different representations.} (a) \\YL{coarse mesh} (207 triangles), (b) \\YL{tracked mesh} (13,248 triangles) and (c) \\YL{fine mesh} (13,248 triangles). \\YL{Both coarse and fine meshes are obtained by simulating the skirt using a physics-based method \\cl{\\cite{Narain2012AAR}}. The tracked mesh is obtained with physics-based simulation involving additional constraints to track the coarse mesh.} The tracked mesh  exhibits stiff folds while the wrinkles in the fine simulated mesh are more realistic.}%\n\t}\n\t\\label{fig:lrhrsim1} \n\\end{figure}\nUnfortunately, they are not suitable for loose garments, such as skirts, since the deformation of wrinkles cannot be defined by a static mapping from a character's pose.\nInstead of human poses, wrinkle augmentation on coarse simulations provides another alternative. \nIt utilizes coarse simulations with fast speed to cover a high-level deformation and leverages learning-based methods to add realistic wrinkles.\nPrevious methods~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} commonly require dense correspondences between coarse and fine meshes, so that local details can be added without affecting global deformation. \n\\YL{Such methods also require coarse meshes to be sufficiently close to fine meshes, as they only add details to coarse meshes.}\nTo maintain the correspondences for training data and ensure closeness between coarse and fine meshes, weak-form constraints such as various test functions~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} are applied to make fine meshes track the coarse meshes, \n\\YL{but as a result, the obtained high-resolution meshes do not fully follow physical behavior, leading to animations that lack realism. An example is shown in Fig.~\\ref{fig:lrhrsim1} where the tracked skirt (b) loses a large amount of wrinkles which should appear when simulating on fine meshes (c).}\n\n \nWithout requiring the constraints between coarse and fine meshes, we propose \n\\gl{the DeformTransformer network\nto synthesize detailed thin shell animations from coarse ones, based on deformation transfer.}\nThis is inspired by the similarity observed between pairs of coarse and fine meshes generated by PBS. %\nAlthough the positions of vertices from two meshes are not aligned, the overall deformation is similar, so it is possible to predict fine-scale deformation with coarse simulation results.\nMost previous works~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} use explicit vertex coordinates to represent 3D meshes, which are sensitive to translations and rotations,\nso they require good alignments between low- and high-resolution meshes. \nIn our work, we regard the cloth animations as non-rigid deformation and propose a novel representation for mesh sequences, called TS-ACAP (Temporal and Spatial As-Consistent-As-Possible) representation. \nTS-ACAP is a local deformation representation, capable of representing and solving large-scale deformation problems, while maintaining the details of meshes.\nCompared to the original ACAP representation~\\cite{gao2019sparse}, TS-ACAP is fundamentally designed to ensure the temporal consistency of the extracted feature sequences, \\YL{and meanwhile} it can maintain the original features of ACAP \\YL{to cope with large-scale deformations}.\nWith \\YL{TS-ACAP} representations for both coarse and fine meshes, we leverage a sequence transduction network to map the deformation from coarse to fine level to assure the temporal coherence of generated sequences.\nUnlike existing works using recurrent neural networks (RNN)~\\cite{santesteban2019learning}, we utilize the Transformer network~\\cite{vaswani2017attention}, an architecture consisting of frame-level attention mechanisms for our mesh sequence transduction task.\nIt is based entirely on attention without recursion modules so can be trained significantly faster than architectures based on recurrent %\nlayers.\nWith \\YL{temporally consistent features and the Transformer network, \\YL{our method achieves} stable general cloth synthesis with fine details in an efficient manner.}\n\nIn summary, the main contributions of our work are as follows:\n\\begin{itemize}\n\t\\item \\YL{We propose a novel framework for the synthesis of cloth dynamics, by learning temporally consistent deformation from low-resolution meshes to high-resolution meshes \\gl{with realistic dynamic}, which is $10 \\sim 35$ times faster than PBS \\cite{Narain2012AAR}.}\n\t\\item \\YL{To achieve this, we propose a \\cl{temporally and spatially as-consistent-as-possible deformation representation (TS-ACAP)} to represent the cloth mesh sequences. It is able to deal with large-scale deformation, essential for mapping between coarse and fine meshes, while ensuring temporal coherence.} \n    \\item \\gl{Based on the TS-ACAP, We further design an effective neural network architecture (named DeformTransformer) by improving Transformer network, which successfully enables high-quality synthesis of dynamic wrinkles with rich details on thin shells and maintains temporal consistency on the generated high-resolution mesh sequences.}\n    \n    \n\\end{itemize}\n\nWe qualitatively and quantitatively evaluate our method for various cloth types (T-shirts, pants, skirts, square and disk tablecloth) with different motion sequences. \nIn Sec.~\\ref{sec:related_work}, we review the work most related to ours. We then give the detailed description of our method in Sec.~\\ref{sec:approach}. \nImplementation details are presented in Sec.~\\ref{sec:implementation}. We present experimental results, including extensive\ncomparisons with state-of-the-art methods in Sec.~\\ref{sec:results}, and finally, we draw conclusions and \\YL{discuss future work} in Sec.~\\ref{sec:conclusion}.\n\n\n\\section{Related work} \\label{sec:related_work}\n\\subsection{Cloth Animation}\nPhysics-based techniques for realistic cloth simulation have been widely studied in computer graphics, \\YL{using methods such as} implicit Euler integrator \\cite{BW98,Harmon09asynchronous}, iterative optimization \\cite{terzopoulos87elastically,bridson03wrinkles,Grinspun03shell}, collision detection and response \\cite{provot97collision,volino95collision}, etc. \n\\YL{Although such techniques can generate realistic cloth dynamics, }they are time consuming for detailed cloth synthesis, and the robustness and efficiency of simulation systems are also of concern.\n\\YL{To address these, alternative methods have been developed to generate} the dynamic details of cloth animation via adaptive techniques \\cite{lee2010multi,muller2010wrinkle,Narain2012AAR}, data-driven approaches \\cite{deAguiar10Stable, Guan12DRAPE, wang10example, kavan11physics,zurdo2013wrinkles} and deep learning-based methods \\cite{chen2018synthesizing,gundogdu2018garnet,laehner2018deepwrinkles,zhang2020deep}, etc.\n\n Adaptive techniques \\cite{lee2010multi, muller2010wrinkle} usually simulate a coarse model by simplifying the smooth regions and \\YL{applying interpolation} to reconstruct the wrinkles, \\YL{taking normal or tangential degrees of freedom into consideration.}  \nDifferent from simulating a reduced model with postprocessing detail augmentation, Narain {\\itshape et al.} \\cite{Narain2012AAR} directly generate dynamic meshes in \\YL{the} simulation phase through adaptive remeshing, at the expense of increasing \\YL{computation time}.  \n\nData-driven methods have drawn much attention since they offer faster cloth animations than physical models.\nWith \\YL{a} constructed database of \\YL{high-resolution} meshes, researchers have proposed many techniques depending on the motions of human bodies with linear conditional models\\cite{deAguiar10Stable, Guan12DRAPE} or secondary motion graphs \\cite{Kim2013near, Kim2008drivenshape}.\nHowever, these methods are limited to tight garments and not suitable for skirts or cloth with more freedom.\nAn alternative line \\YL{of research} is to augment details on coarse simulations \\YL{by exploiting knowledge from a} database of paired meshes, to generalize the performance to complicated testing scenes.\nIn this line, in addition to wrinkle synthesis methods \\YL{based on} bone clusters \\cite{Feng2010transfer} or human poses \\cite{wang10example} for fitted clothes, there are some approaches \\YL{that investigate how to} learn a mapping from a coarse garment shape to a detailed one for general \\YL{cases} of free-flowing cloth simulation.\nKavan {\\itshape et al.} \\cite{kavan11physics} present linear upsampling operators to \\YL{efficiently} augment \\YL{medium-scale} details on coarse meshes.\nZurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} define wrinkles as local displacements and use \\YL{an} example-based algorithm to enhance low-resolution simulations.\n\\YL{Their approaches mean the} high-resolution cloth \\YL{is} required to track \\YL{the} low-resolution cloth, \\YL{and thus cannot} exhibit full high-resolution dynamics.\n\nRecently deep learning-based methods have been successfully applied for 3D animations of human \\YL{faces}~\\cite{cao2016real, jiang20183d}, hair \\cite{zhang2018modeling, yang2019dynamic} and garments \\cite{liu2019neuroskinning, wang2019learning}.\nAs for garment synthesis, some approaches \\cite{laehner2018deepwrinkles, santesteban2019learning, patel2020tailornet} are proposed to utilize a two-stream strategy consisting of global garment fit and local \\YL{wrinkle} enhancement.\nL{\\\" a}hner {\\itshape et al.} \\cite{laehner2018deepwrinkles} present DeepWrinkles, \\YL{which recovers} the global deformation from \\YL{a} 3D scan system and \\YL{uses a} conditional \\YL{generative adversarial network} to enhance a low-resolution normal map.\nZhang {\\itshape et al.} \\cite{zhang2020deep} further generalize the augmentation method with normal maps to complex garment types as well as various motion sequences.\n\\YL{These approaches add wrinkles on normal maps \\YL{rather than geometry}, and thus their effectiveness is restricted to adding fine-scale visual details, not large-scale dynamics.}\nBased on \\YL{the} skinning representation, some algorithms \\cite{gundogdu2018garnet,santesteban2019learning} use neural networks to generalize garment synthesis algorithms to multiple body shapes. \n\\YL{In addition, other works are} devoted to \\YL{generalizing neural networks} to various cloth styles \\cite{patel2020tailornet} or cloth materials \\cite{wang2019learning}.\nDespite tight garments dressed on characters, some deep learning-based methods \\cite{chen2018synthesizing, oh2018hierarchical} are %\n\\YL{demonstrated to work for cloth animation with higher degrees} of freedom.\nChen {\\itshape et al.} \\cite{laehner2018deepwrinkles} represent coarse and fine meshes via geometry images and use \\YL{a} super-resolution network to learn the mapping.\nOh {\\itshape et al.} \\cite{oh2018hierarchical} propose a multi-resolution cloth representation with \\YL{fully} connected networks to add details hierarchically.\nSince the \\YL{free-flowing cloth dynamics are harder for networks to learn} than tight garments, the results of these methods have not reached the realism of PBS. \\YL{Our method based on a novel deformation representation and network architecture has superior capabilities of learning the mapping from coarse and fine meshes, generating realistic cloth dynamics, while being much faster than PBS methods.}\n \n \\begin{figure*}[ht]\n \t\\centering\n \t\\includegraphics[width=1.0\\linewidth, trim=20 250 20 50,clip]{pictures\/mainpicture2.pdf} \n \t\\caption{\\small The overall architecture of our detail synthesis network. At data preparation stage, we generate low- and high-resolution \\gl{thin shell} animations via coarse and fine \\gl{meshes} and various motion sequences.\n \t Then we encode the coarse meshes and the detailed meshes to a deformation representation TS-ACAP, respectively.\n \t\\YL{Our algorithm then} learns to map the coarse features to fine features %\n \t\\YL{by designing a DeformTransformer network that consists of temporal-aware encoders and decoders, and finally reconstructs the detailed animations.}\n \t}\n \t\\label{fig:pipeline}\n \\end{figure*}\n\n\\subsection{Representation for 3D Meshes}\nUnlike 2D images with regular grid of pixels, \\YL{3D meshes have irregular connectivity which makes learning more difficult. To address this, existing deep learning based methods turn 3D meshes to a wide range of representations to facilitate processing~\\cite{xiao2020survey},} such as voxels, images \\YL{(such as depth images and multi-view images)}, point clouds, meshes, etc.\n\\YL{The volumetric representation has a regular structure, but it} often suffers from \\YL{the problem of extremely high space and time consumption.}\nThus Wang {\\itshape et al.} \\cite{wang2017cnn} propose an octree-based convolutional neural network and encode the voxels sparsely. \nImage-based representations including \\YL{depth images} \\cite{eigen2014depth,gupta2014learning} and multi-view images \\cite{Su2015mvcnn,li20193d} are proposed to encode 3D models in a 2D domain. \nIt is unavoidable that both volumetric and image-based representations lose some geometric details.\nAlternatively, geometry images are used in \\cite{sinha2016deep,Sinha2017surfnet,chen2018synthesizing} for mesh classification or generation\\YL{, which are obtained through cutting a 3D mesh to a topological disk, parameterizing it to a rectangular domain and regularly sampling the 3D coordinates in the 2D domain~\\cite{gu2002geometry}.}\n\\YL{However, this representation} may suffer from parameterization distortion and seam line problems.\n\nInstead of representing 3D meshes into other formats, recently there are methods \\cite{tan2017autoencoder, tan2017variational, hanocka2019meshcnn} applying neural networks directly to triangle meshes with various features.\nGao {\\itshape et al.} \\cite{gao2016efficient} propose a deformation-based representation, called the rotation-invariant mesh difference (RIMD) which is translation and rotation invariant.\nBased on the RIMD feature, Tan {\\itshape et al.} \\cite{tan2017variational}  propose a fully connected variational autoencoder network to analyze and generate meshes.\nWu {\\itshape et al.} \\cite{wu2018alive} use the RIMD to generate\na 3D caricature model from a 2D caricature image. \nHowever, it is expensive to reconstruct vertex coordinates from the RIMD feature due to the requirement of solving a very complicated optimization.\nThus it is not suitable for fast mesh generation tasks.\nA faster deformation representation based on an as-consistent-as-possible (ACAP) formulation \\cite{gao2019sparse} is further used to reconstruct meshes \\cite{tan2017autoencoder}, which is able to cope with large rotations and efficient for reconstruction.\nJiang {\\itshape et al.} \\cite{jiang2019disentangled} use ACAP to disentangle the identity and expression of 3D \\YL{faces}. \nThey further apply ACAP to learn and reconstruct 3D human body models using a coarse-to-fine pipeline \\cite{jiang2020disentangled}. \n\\YL{However, the ACAP feature is represented based on individual 3D meshes. When applied to a dynamic mesh sequence, it does not guarantee temporal consistency.}\nWe propose a \\cl{temporally and spatially as-consistent-as-possible (TS-ACAP)} representation, to ensure both spatial and temporal consistency of mesh deformation.\nCompared to ACAP, our TS-ACAP can also accelerate the computation of features thanks to the sequential constraints.  \n\n\\subsection{Sequence Generation with \\YL{DNNs (Deep Neural Networks)}}\nTemporal information is crucial for stable and \\gl{vivid} sequence generation. Previously, recurrent neural networks (RNN) have been successfully applied in many sequence generation tasks \\cite{mikolov2010recurrent, mikolov2011extensions}. However, it is difficult to train \\YL{RNNs} to capture long-term dependencies since \\YL{RNNs} suffer from the vanishing gradient problem \\cite{bengio1994learning}. To deal with this problem, previous works proposed some variations of RNN, including long short-term memory (LSTM) \\cite{hochreiter1997long} and gated recurrent unit (GRU) \\cite{cho2014properties}. These variations of RNN rely on the gating mechanisms to control the flow of information, thus performing well in the tasks that require capturing long-term dependencies, such as speech recognition \\cite{graves2013speech} and machine translation \\cite{bahdanau2014neural, sutskever2014sequence}. Recently, based on attention mechanisms, the Transformer network \\cite{vaswani2017attention} has been verified to outperform \\YL{many typical sequential models} for long sequences. This structure is able to inject the global context information into each input. Based on Transformer, impressive results have been achieved in tasks with regard to audio, video and text, \\textit{e.g. } speech synthesis \\cite{li2019neural, okamoto2020transformer}, action recognition \\cite{girdhar2019video} and machine translation \\cite{vaswani2017attention}.\nWe utilize the Transformer network to learn the frame-level attention which improves the temporal stability of the generated animation sequences.\n\n\\section{Approach} \\label{sec:approach}\nWith a simulated sequence of coarse meshes $\\mathcal{C} = \\{\\mathcal{C}_1, \\dots, \\mathcal{C}_n\\}$ as input, our goal is to produce a sequence of fine ones $\\mathcal{D} = \\{\\mathcal{D}_1, \\dots, \\mathcal{D}_n\\}$ which have similar non-rigid deformation as the PBS. Given two simulation sets of paired coarse and fine garments, we extract the TS-ACAP representations respectively, \\YL{and} then use our proposed DeformTransformer network to learn the \\YL{transform} \\YL{from the low-resolution space to the high-resolution space}. \\YL{As illustrated previously in Fig.~\\ref{fig:lrhrsim1}, such a mapping involves deformations beyond adding fine details.}\nOnce the network is trained by the paired examples, a consistent and detailed animation $\\mathcal{D}$ can be synthesized for each input sequence $\\mathcal{C}$. \n\n\\subsection{Overview}\nThe overall architecture of our detail synthesis network is illustrated in Fig. \\ref{fig:pipeline}.\nTo synthesize realistic \\gl{cloth animations}, we propose a method to simulate coarse meshes first and learn a \\YL{temporally-coherent} mapping to the fine meshes.    \nTo realize our goal, we construct datasets including low- and high-resolution cloth animations, \\textit{e.g. } coarse and fine garments dressed on a human body of various motion sequences. \nTo efficiently extract localized features with temporal consistency, we propose a new deformation representation, called TS-ACAP (temporal \\YL{and spatial} as-consistent-as-possible), which is able to cope with both large rotations and unstable sequences. It also has significant advantages: it is efficient to compute for \\YL{mesh} sequences and its derivatives have closed form solutions.\nSince the vertices of the fine models are typically more than ten thousand to simulate realistic wrinkles, it is hard to directly map the coarse features to the high-dimensional fine ones for the network.\nTherefore, \\YL{convolutional encoder networks are} \napplied to encode \\YL{coarse and fine meshes in the TS-ACAP representation} to \\YL{their latent spaces}, respectively.\nThe TS-ACAP generates local rotation and scaling\/shearing parts on vertices, so we perform convolution \\YL{operations} on vertices %\n\\YL{to learn to extract useful features using shared local convolutional kernels.}\nWith encoded feature sequences, a sequence transduction network is proposed to learn the mapping from coarse to fine TS-ACAP sequences.\nUnlike existing works using recurrent neural networks \\YL{(RNNs)}~\\cite{santesteban2019learning}, we use the Transformer \\cite{vaswani2017attention}, a sequence-to-sequence network architecture, based on frame-level attention mechanisms for our detail synthesis task, \\YL{which is more efficient to learn and leads to superior results.}\n\n\\subsection{Deformation Representation}\n\\YL{As discussed before, large-scale deformations are essential to represent \\gl{thin shell mode dynamics such as }cloth animations, because folding and wrinkle patterns during animation can often be complicated. Moreover, cloth animations are in the form of sequences, hence the temporal coherence is very important for the realistic. Using 3D coordinates directly cannot cope with large-scale deformations well, and existing deformation representations are generally designed for static meshes, and directly applying them to cloth animation sequences on a frame-by-frame basis does not take temporal consistency into account. }\nTo cope with this problem, we propose a mesh deformation feature with spatial-temporal consistency, called TS-ACAP, to represent the coarse and fine deformed shapes, which exploits the localized information effectively and reconstructs \\YL{meshes} accurately.\nTake \\YL{coarse meshes} $\\mathcal{C}$ for instance and \\YL{fine meshes $\\mathcal{D}$ are processed in the same way.} \\YL{Assume that a sequence} of coarse meshes contains $n$ models with the same topology, each denoted as $\\mathcal{C}_{t}$ \\YL{($1\\leq t \\leq n$)}. \n\\YL{A mesh with the same topology is chosen as the reference model, denoted as $\\mathcal{C}_{0}$. For example, for garment animation, this can be the garment mesh worn by  a character in the T pose.}\n$\\mathbf{p}_{t,i} \\in \\mathbb{R}^{3}$ is the $i^{\\rm th}$ vertex on\nthe $t^{\\rm th}$ mesh.\nTo represent the local shape deformation, the deformation gradient $\\mathbf{T}_{t,i} \\in \\mathbb{R}^{3 \\times 3}$ can be obtained by minimizing the following energy:\n\\begin{equation}\n\t\\mathop{\\arg\\min}_{\\mathbf{T}_{t,i}} \\ \\  \\mathop{\\sum}_{j \\in \\mathcal{N}_i} c_{ij} \\| (\\mathbf{p}_{t,i} - \\mathbf{p}_{t,j}) -  \\mathbf{T}_{t,i} (\\mathbf{p}_{0,i} - \\mathbf{p}_{0,j}) \\|_2^2  \\label{con:computeDG}\n\\end{equation}\nwhere $\\mathcal{N}_i$ is the one-ring neighbors of the $i^{\\rm th}$ vertex, and $c_{ij}$ is the cotangent weight $c_{ij} = \\cot \\alpha_{ij} + \\cot \\beta_{ij} $ \\cite{sorkine2007rigid,levi2014smooth}, where $\\alpha_{ij}$\nand $\\beta_{ij}$ are angles opposite to the edge connecting the $i^{\\rm th}$ and $j^{\\rm th}$ vertices.\n\nThe main drawback of the deformation gradient representation is that it cannot handle large-scale rotations, which often \\YL{happen} in cloth animation. \nUsing polar decomposition, the deformation gradient $\\mathbf{T}_{t,i} $ can be decomposed into a rotation part and a scaling\/shearing part $\\mathbf{T}_{t,i} = \\mathbf{R}_{t,i}\\mathbf{S}_{t,i}$.\nThe scaling\/shearing transformation $\\mathbf{S}_{t,i}$ is uniquely defined, while the rotation $\\mathbf{R}_{t,i}$ \\YL{corresponds to infinite possible rotation angles (differed by multiples of $2\\pi$, along with possible opposite orientation of the rotation axis)}. Typical formulation often constrain the rotation angle to be within $[0, \\pi]$ which is unsuitable for smooth large-scale animations. \n\nIn order to handle large-scale rotations, we first require the orientations of rotation axes and rotation angles of \\YL{spatially} adjacent vertices \\YL{on the same mesh} to be as consistent as possible. \nEspecially for our sequence data, we further add constraints for adjacent frames to ensure the temporal consistency of the orientations of rotation axes and rotation angles on each vertex.\n\nWe first consider consistent orientation for axes.\n\\begin{flalign}\\label{eqn:axis}\n\t\\arg\\max_{{o}_{t,i}} \\sum_{(i,j) \\in \\mathcal{E} }  {o}_{t,i}{o}_{t,j} \\cdot s(\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}, \\theta_{t,i}, \\theta_{t,j}) \\nonumber\\\\\n\t+ \\sum_{i \\in \\mathcal{V} }  {o}_{t,i} \\cdot s(\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t-1,i}, \\theta_{t,i}, \\theta_{t-1,i}) \\nonumber\\\\\n\t{\\rm s.t.} \\quad\n\t{o}_{t,1} = 1, {o}_{t,i} = \\pm 1 (i \\neq 1) \\quad  \n\\end{flalign}\nwhere $t$ is the \\YL{index} of \\YL{the} frame, $\\mathcal{E}$ is the edge set, and $\\mathcal{V}$ is the vertex set. \\YL{Denote by $(\\boldsymbol{\\omega}_{t,i}, \\theta_{t,i})$ one possible choice for the rotation axis and rotation angle that match $\\mathbf{R}_{t,i}$. $o_{t,i} \\in \\{+1, -1\\}$ specifies whether the rotation axis is flipped ($o_{t,i} = 1$ if the rotation axis is unchanged, and $-1$ if its opposite is used instead). }\\YL{The first term promotes spatial consistency while the second term promotes temporal consistency.} \n$s(\\cdot)$ is a function measuring orientation consistency, which is defined as follows:\n\\begin{equation}\n\ts(\\cdot)=\\left\\{\n\t\\begin{aligned}\n\t\t0 & , & |\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}|\\leq\\epsilon_1 \\; {\\rm or} \\;\n\t\t\\theta_{t,i}<\\varepsilon_2 \\; {\\rm or}  \\; \\theta_{t,j}<\\varepsilon_2 \\\\\n\t\t1 & , & {\\rm Otherwise~if}~\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}>\\epsilon_1 \\\\\n\t\t-1 & , & {\\rm Otherwise~if}~ \\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}<-\\epsilon_1 \\\\\n\t\\end{aligned}\n\t\\right.\n\\end{equation}\n\\YL{The first case here is to ignore cases where the rotation angle is near zero, as the rotation axis is not well defined in such cases.}\nAs for rotation angles, \\YL{we optimize the following}\n\\begin{flalign}\\label{eqn:angle}\n\\arg\\min_{r_{t,i}} &\\sum_{(i,j) \\in \\mathcal{E} } \\| (r_{t,i} \\cdot 2\\pi+{o}_{t,i}\\theta_{t,i}) - (r_{t,j}\\cdot 2\\pi+{o}_{t,j}\\theta_{t,j}) \\|_2^{2} &\\nonumber\\\\\n+ &\\sum_{i \\in \\mathcal{V} } \\| (r_{t,i} \\cdot 2\\pi+{o}_{t,i}\\theta_{t,i}) - (r_{t-1,i}\\cdot 2\\pi+{o}_{t,j}\\theta_{t-1,i}) \\|_2^{2} \\nonumber\\\\ \n{\\rm s.t.}& \\quad r_{t,i} \\in \\mathbb{Z},~~r_{t,1} = 0.\n\\end{flalign}\nwhere $r_{t,i} \\in \\mathbb{Z}$ specifies how many $2\\pi$ rotations should be added to the rotation angle.\n\\YL{The two terms here promote spatial and temporal consistencies of rotation angles, respectively. \nThese optimizations can be solved using integer programming, and we use the mixed integer solver CoMISo~\\cite{comiso2009} which provides an efficient \\gl{solver}. See~\\cite{gao2019sparse} for more details.}\nA similar process is used to compute the TS-ACAP representation of the fine meshes. \n\n\n\\cl{Compared to the ACAP representation, our TS-ACAP representation considers temporal constraints to represent nonlinear deformation for optimization of axes and angles, which is more suitable for consecutive large-scale deformation \\YL{sequences}.\nWe compare ACAP~\\cite{gao2019sparse} and our TS-ACAP using a simple example of a simulated disk-shaped cloth animation sequence. Once we obtain deformation representations of the meshes in the sequence, \nwe interpolate two meshes, the initial state mesh and a randomly selected frame, using linear interpolation of \\YL{shape representations}.\n\\YL{In Fig. \\ref{fig:interpolation}, we demonstrate the interpolation results with ACAP representation, which shows that it cannot handle such challenging cases with complex large-scale deformations. In contrast, with our temporally and spatially as-consistent-as-possible optimization, our TS-ACAP representation is able to produce consistent interpolation results.}\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{pictures\/acap_tacap1_1.pdf}%\n\t\\caption{\\small Comparison of shape interpolation results with different deformation representations, ACAP and TS-ACAP.  %\n\t(a) and (b) are the source (t = 0) and target (t = 1) models with large-scale deformation to be interpolated. \n\tThe first row shows the interpolation results by ACAP, and the second row show the results with our TS-ACAP. \n\t\\gl{The interpolated models with ACAP feature are plausible in each frame while they are not consistent in the temporal domain.}\n\t}\n\t\\label{fig:interpolation}\n\\end{figure}\n}\n\n\\subsection{DeformTransformer Networks}\nUnlike \\cite{tan2017variational, wang2019learning} which use fully connected layers for mesh encoder, we perform convolutions \\YL{on meshes to learn to extract useful features using compact shared convolutional kernels.} \nAs illustrated in Fig. \\ref{fig:pointconv}, we use a convolution operator on vertices  \\cite{duvenaud2015convolutional, tan2017autoencoder} where the output at a vertex is obtained as a linear combination of input in its one-ring neighbors along with a bias. \n\\YL{The input to our network is the TS-ACAP representation, which for the $i^{\\rm th}$ vertex of the $t^{\\rm th}$ mesh, we collect non-trivial coefficients from the rotation $\\mathbf{R}_{t, i}$ and scaling\/shearing $\\mathbf{S}_{t,i}$, which forms a 9-dimensional feature vector (see~\\cite{gao2019sparse} for more details). Denote by $\\mathbf{f}_i^{(k-1)}$ and $\\mathbf{f}_i^{k}$ the feature of the $i^{\\rm th}$ vertex at the $(k-1)^{\\rm th}$ and $k^{\\rm th}$ layers, respectively. The convolution operator is defined as follows:\n\\begin{equation}\n\t\\mathbf{f}_i^{(k)} =\n\t\\mathbf{W}_{point}^{(k)} \\cdot \\mathbf{f}_{i}^{(k-1)} + \n\t\\mathbf{W}_{neighbor}^{(k)} \\cdot \\frac{1}{D_i} \\mathop{\\sum}_{j=1}^{D_i} \\mathbf{f}_{n_{ij}}^{(k-1)}\n\t+ \\mathbf{b}^{(k)} \n\\end{equation}\nwhere $\\mathbf{W}_{point}^{(k)}$, $\\mathbf{W}_{neighbor}^{(k)}$ and $\\mathbf{b}^{(k)}$ are learnable parameters for the $k^{\\rm th}$ convoluational layer, $D_i$ is the degree of the $i^{\\rm th}$ vertex, $n_{ij}(1 \\leq j \\leq D_i )$ is the $j^{\\rm th}$ neighbor of the $i^{\\rm th}$ vertex.\n}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.48\\linewidth]{pictures\/pointconv.pdf} \n\t\\caption{\\small Illustration of the convolutional operator on meshes. \n\t\tThe result of convolution for each vertex is obtained by a linear combination from the input in the 1-ring neighbors of the vertex, along with a bias.\n\t}\n\t\\label{fig:pointconv}\n\\end{figure}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=0 50 0 150,clip]{pictures\/transformer.pdf} %\n\t\\caption{\\small The architecture of our DeformTransformer network.\n\t\tThe coarse and fine mesh sequences are embedded into feature vectors using the TS-ACAP representation which \\YL{is} defined \\YL{at} each vertex as a 9-dimensional vector. \n\t\tThen two convolutional \\YL{encoders} map coarse and fine features to \\YL{their latent spaces}, respectively.\n\t\tThese latent vectors are fed into the DeformTransformer network, \\cl{which consists of the encoder and decoder, each including a stack of $N=2$ identical blocks with 8-head attention,} to recover \\YL{temporally-coherent} deformations.\n\t\tNotice that in \\YL{the} training phase the input high-resolution TS-ACAP \\YL{features are those from the ground truth}, \n\t\t\\YL{but during testing, these features are initialized to zeros, and once a new high-resolution frame is generated, its TS-ACAP feature is added.}\n\t\tWith predicted feature vectors, realistic and stable cloth animations are generated.\n\t}\n\t\\label{fig:Transformer}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.4\\linewidth, trim=18 33 18 3,clip]{pictures\/tshirt06_08_poseswithhuman_collision\/temp0270keyshot_unsolve.png} \n\t\\includegraphics[width=0.4\\linewidth, trim=18 33 18 3,clip]{pictures\/tshirt06_08_poseswithhuman_collision\/temp0270keyshot_solve.png} \n\t\\caption{\\small For tight clothing, data-driven cloth deformations may suffer from apparent collisions with the body (left). We apply a simple postprocessing step to push \n\t\\YL{the collided} T-shirt vertices outside the body (right).\n\t}\n\t\\label{fig:collisionrefinement}\n\\end{figure}\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth, trim=50 150 100 150,clip]{pictures\/dataset.pdf} \n\t\\caption{\\small \n\t\tWe test our algorithm on 5 datasets including TSHIRT, PANTS, SKIRT, SHEET and DISK.\t\t \n\t\tThe former three are  garments (T-shirts, skirts, and pants) dressed on a template body and simulated with various motion sequences.\n\t\tThe SHEET dataset is a square sheet interacting with various obstacles.\n\t\tThe DISK dataset is a round tablecloth draping on a cylinder in the wind of various velocities. \n\t\tEach cloth shape has a coarse resolution (top) and a fine resolution (bottom). \n\t} \n\t\\label{fig:dataset}\n\\end{figure*}\nLet $\\mathcal{F}_\\mathcal{C} = \\{\\mathbf{f}_{\\mathcal{C}_1}, \\dots, \\mathbf{f}_{\\mathcal{C}_n}\\}$ be the sequence of coarse mesh features, and $\\mathcal{F}_\\mathcal{D} = \\{\\mathbf{f}_{\\mathcal{D}_1}, \\dots, \\mathbf{f}_{\\mathcal{D}_n}\\}$ be its counterpart, the sequence of detailed mesh features.\nTo synthesize $\\mathcal{F}_\\mathcal{D}$ from $\\mathcal{F}_\\mathcal{C}$, the DeformTransformer framework is proposed to solve this sequence-to-sequence problem.\nThe DeformTransformer network consists of several stacked encoder-decoder layers, \\YL{denoted} as $Enc(\\cdot)$ and $Dec(\\cdot)$. To take the order of the sequence into consideration, triangle positional embeddings \\cite{vaswani2017attention} are injected into frames of $\\mathcal{F}_\\mathcal{C}$ and $\\mathcal{F}_\\mathcal{D}$, respectively.\nThe encoder takes coarse mesh features as input and encodes it to a \\YL{temporally-dependent} hidden space.\nIt is composed of identical blocks \\YL{each} with two sub-modules, one is the multi-head self-attention mechanism, the other is the frame-wise fully connected feed-forward network. \nWe also employ a residual connection around these two sub-modules, followed \\YL{by} the layer normalization.\nThe multi-head attention is able to build the dependence between any frames, thus ensuring that each input can consider global context of the whole sequence. Meanwhile, compared with other sequence models, this mechanism splits \\YL{the} attention into several subspaces so that it can model the frame \\YL{relationships} in multiple aspects.\nWith the encoded latent vector $Enc(\\mathcal{F}_\\mathcal{C})$, the decoder network attempts to reconstruct a sequence of fine mesh features.\nThe decoder has two parts: \nThe first part takes fine mesh sequence $\\mathcal{F}_\\mathcal{D}$ as \\YL{input} and \nencodes it similar to the encoder. \n\\YL{Unlike the encoder, detailed meshes are generated sequentially, and when predicting frame $t$, it should not attend to subsequent frames (with the position after frame $t$). To achieve this, we utilize a masking process\nfor the self-attention module.} The second part performs multi-head attention over the output of the encoder, thus capturing the long-term dependence between coarse mesh features $\\mathcal{F}_\\mathcal{C}$ and fine mesh features $\\mathcal{F}_\\mathcal{D}$.\nWe train the Transformer network by minimizing the mean squared error between predicted detailed features and the ground-truth.\nWith predicted TS-ACAP feature vector, we reconstruct the vertex coordinates of \\YL{the} target mesh\\YL{, in the same way as reconstruction from ACAP features} (please refer to \\cite{gao2019sparse} for details).   \nOur training data is generated by PBS \\YL{and is collision-free}.\nSince human body \\YL{(or other obstacles)} information is unseen in our algorithm, it does not guarantee the predicted cloth \\YL{is free from any penetration}.\nEspecially for tight garment like T-shirts, it will be apparent if collision \\YL{between the garment and human body} happens.\nWe use a fast refinement method \\cite{wang2019learning} to push the cloth vertices colliding with the body outside \\YL{while} preserving the local wrinkle details (see Fig.~\\ref{fig:collisionrefinement}). \nFor each vertex detected inside the body, we find its closest point over the body surface with normal and position.\nThen the cloth mesh is deformed to update the vertices by minimizing the energy which penalizes the euclidean distance and Laplacian difference between the updated mesh and the initial one (please refer to \\cite{wang2019learning} for details).\nThe collision solving process usually takes less than 3 iterations to converge to a collision-free state.\n\n\\section{Implementation}\\label{sec:implementation}\nWe describe the details of the dataset construction and the network architecture in this section.\n\n\\textbf{\\YL{Datasets}.}\nTo test our method, we construct 5 datasets, called TSHIRT, PANTS, SKIRT, SHEET and DISK respectively.\nThe former three datasets are different types of garments, \\textit{i.e. }, T-shirts, skirts and pants worn on human bodies.\nEach type of garment \\YL{is represented by both low-resolution and  high-resolution meshes}, \\YL{containing} 246 and 14,190 vertices for the T-shirts, 219 and 12,336 vertices for the skirts, 200 and 11,967 vertices for the pants.\nGarments of the same type and resolution are simulated from a template mesh, which means \\YL{such meshes obtained  through cloth animations have the same number of vertices and the same connectivity}.\nThese garments are dressed on animated characters, which are obtained via driving a body \\YL{in the SMPL (Skinned Multi-Person Linear)  model} \\cite{loper2015smpl} with publicly available motion capture data from CMU \\cite{hodgins2015cmu}.\nSince the motion data is captured, there are some \\YL{self-collisions} or long repeated sequences.  \n\\YL{After removing poor quality data}, we select various motions, such as dancing, walking, running, jumping etc., including 20 sequences (\\YL{9031, 6134, 7680 frames in total} for TSHIRT, PANTS and SKIRT respectively).\nIn these motions, 18 sequences are randomly selected for training and the remaining 2 sequences for testing.\nThe SHEET dataset consists of a pole or a sphere of three different sizes crashing to a piece of \\YL{cloth sheet}.\nThe coarse mesh has 81 vertices and the fine mesh has 4,225 vertices.\nThere are \\YL{4,000} frames in the SHEET dataset, in which 3200 frames for training and \\YL{the remaining} 800 frames for testing.\nWe construct the DISK dataset by draping a round tablecloth to a cylinder in the wind, with 148 and 7,729 vertices for coarse and fine meshes respectively.\nWe adjust the velocity of the wind to get various animation sequences, in which 1600 frames for training and 400 frames for testing. \n\n\\begin{table*}[ht]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{ Statistics and timing (sec\/\\YL{frame}) of the testing examples including five types of \\YL{thin shell animations}.\n\t}\n\t\\label{table:runtime}\n\t\\centering\n\t\\begin{tabular}{cccccccccc}\n\t\t\\toprule[1.2pt] \n\t\tBenchmark & \\#verts & \\#verts & PBS & ours  & speedup & \\multicolumn{4}{c}{our components}              \\\\ \\cline{7-10} \n\t\t& LR      & HR      & HR    &       &         & coarse & TS-ACAP & synthesizing & refinement \\\\\n\t\t&         &         &         &       &         & sim.   & extraction & (GPU)        &            \\\\  \\hline \\hline\n\t\tTSHIRT    & 246     & 14,190   &  8.72    & 0.867 & \\textbf{10}     & 0.73   & 0.11      &  0.012 & 0.015\\\\\n\t\tPANTS     & 200     & 11,967   & 10.92  &0.904   & \\textbf{12}      & 0.80   & 0.078 & 0.013  & 0.013\\\\\n\t\tSKIRT     & 127     & 6,812 & 6.84 & 0.207  & \\textbf{33} & 0.081 & 0.10 & 0.014 & 0.012 \\\\  \n\t\tSHEET     & 81      & 4,225    & 2.48    & 0.157  & \\textbf{16} & 0.035  & 0.10 &   0.011      &  0.011 \\\\  \n\t\tDISK      & 148     & 7,729    & 4.93 & 0.139 & \\textbf{35}     & 0.078  & 0.041 &   0.012      &   0.008 \\\\  \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table*} \nTo prepare the above datasets, we generate both \\YL{low-resolution (LR)} and \\YL{high-resolution (HR)} cloth \\YL{animations} by PBS.\nThe initial state of the HR mesh is obtained by applying the Loop subdivision scheme \\cite{Thesis:Loop} to the coarse mesh and waiting for several seconds till stable.\nPrevious works \\cite{kavan11physics, zurdo2013wrinkles, chen2018synthesizing} usually constrain the high-resolution meshes by various tracking mechanisms to ensure that the coarse cloth \\YL{can be seen as} a low-resolution version of the fine cloth during the complete animation sequences.\nHowever, fine-scale wrinkle dynamics cannot be captured by this model, as wrinkles are defined quasistatically and limited to a \\YL{constrained} subspace.\nThus we \\YL{instead perform} PBS for the two resolution meshes \\emph{separately}, without any constraints between them.\nWe use a cloth simulation engine called ARCSim \\cite{Narain2012AAR} to produce all animation sequences of low- and high-resolution meshes with the same parameter setting. \nIn our experiment, we choose the Gray Interlock from a library of measured cloth materials \\cite{Wang2011DEM} as the material parameters for ARCSim simulation.\nSpecially for garments interacting with characters, to ensure collision-free, we manually put the coarse and fine garments on a template human body (in the T pose) and run the simulation to let the \\YL{clothing} relax. To this end, we define the initial state for all subsequent simulations.\nWe interpolate 15 frames between the T pose and the initial pose of each motion sequence, before applying the motion sequence, which is smoothed using a convolution operation.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfloat{\n\t\t\\includegraphics[width=0.5\\linewidth]{pictures\/hyper_inputframes-eps-converted-to.pdf} \n\t}\n\t\\subfloat{\n\t\t\\includegraphics[width=0.5\\linewidth]{pictures\/hyper_hiddensize-eps-converted-to.pdf} \n\t}\n\t\\caption{\\small Evaluation of hyperparameters in the Transformer network\\YL{, using the SKIRT dataset. }\n\t\t(Left) average error for the reconstructed results as a function of the number of input frames.\n\t\t(Right) error for the synthesized results under the condition of various dimensions of the latent layer.\n\t}\n\t\\label{fig:hyperpara}\n\\end{figure}\n\\textbf{Network architecture.}\nAs shown in Fig.~\\ref{fig:Transformer}, our transduction network consists of two components, namely convolutional \\YL{encoders} to map coarse and fine mesh sequences into latent spaces for improved generalization capability, and the Transformer network for \\YL{spatio-temporally} coherent deformation transduction.\nThe feature encoder module takes the 9-dimensional TS-ACAP features defined on vertices as input, followed by two convolutional layers with $tanh$ as the activation function. \nIn the last convolutional layer we abandon the activation function, similar to \\cite{tan2017autoencoder}.\nA fully connected layer is used to map the output of the convolutional layers into a 16-dimensional latent space.\nWe train one encoder for coarse \\YL{meshes} and another for fine \\YL{meshes} separately.\nFor the DeformTransformer network, its input includes the embedded latent vectors from both \\YL{the} coarse and fine domains.\nThe DeformTransformer network consists of sequential encoders and decoders, \neach \\YL{including} a stack of 2 identical blocks with 8-head attention.\nDifferent from variable length sequences used in natural language processing, we \\YL{fix} the number of input frames \\YL{(to 3 in our experiments)} since a motion sequence may include a thousand frames.\n\\YL{We perform experiments to evaluate the performance of our method with different settings.}\nAs shown in Fig.~\\ref{fig:hyperpara} \\YL{(left)}, using 3 input frames is found to perform well in our experiments.\nWe also evaluate the results generated with various dimensions of latent space shown in Fig. \\ref{fig:hyperpara} \\YL{(right)}.\nWhen the dimension of latent space is larger than 16, the network can \\YL{easily overfit}.\nThus we set the dimension of the latent space %\nto 16, which is sufficient for all the examples in the paper.\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{Quantitative comparison of reconstruction errors for unseen \\YL{cloth animations} in several datasets. We compare our results with Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} with LR meshes as a reference. \\YL{Three metrics, namely RMSE (Root Mean Squared Error), Hausdorff distance and STED (Spatio-Temporal Edge Difference)~\\cite{Vasa2011perception} are used. Since LR meshes have different number of vertices from the ground truth HR mesh, we only calculate its Hausdorff distance.}}\n \t\\label{table:compare_zurdo_chen2}\n\t\\centering \n\t\\begin{tabular}{ccccc} \n\t\t\\toprule[1.2pt]\n\t\t\\multirow{3}{*}{Dataset} & \\multirow{3}{*}{Methods} & \\multicolumn{3}{c}{Metrics} \\\\ \\cline{3-5}\n\t\t&                          & RMSE    & Hausdorff & STED  \\\\ \n\t\t&                          & $\\times 10^\\YL{-2}$ $\\downarrow$ & $\\times 10^\\YL{-2}$ $\\downarrow$ & $\\downarrow$ \\\\\n\t\t\\hline \\hline\n\t\t\\multirow{4}{*}{TSHIRT}   &  LR    & -          & 0.59     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 0.76   & 0.506    &  0.277     \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 1.04   & 0.480    &  0.281     \\\\ \\cline{2-5}\n\t\t&  Our                     & \\textbf{0.546}    &  \\textbf{0.416}   & \\textbf{0.0776}      \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{PANTS}   &  LR                   & -      &  0.761     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 1.82              & 1.09 & 0.176   \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 1.89                 & 0.983& 0.151    \\\\ \\cline{2-5}\n\t\t&  Our                    & \\textbf{0.663} & \\textbf{0.414}    & \\textbf{0.0420}      \\\\ \\hline \\hline\n\t\t\\multirow{4}{*}{SKIRT} &  LR                      & -      & 2.09     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 1.93 &  1.31    & 0.562      \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 2.19 & 1.52     & 0.178      \\\\ \\cline{2-5}\n\t\t&  Our                    & \\textbf{0.685} & \\textbf{0.681}    &  \\textbf{0.0241}      \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{SHEET} \n\t\t&  LR                      & -       & 2.61 &   -   \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 4.37  & 2.60 &   0.155 \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 3.02  & 2.34 &   0.0672    \\\\ \\cline{2-5}\n\t\t&  Our                     & \\textbf{0.585} & \\textbf{0.417}   &  \\textbf{0.0262}     \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{DISK} &  LR & -      & 3.12    & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}   & 7.03   & 2.27    &  0.244     \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.}  & 11.40  &  2.23           & 0.502           \\\\ \\cline{2-5}\n\t\t&  Our                      & \\textbf{2.16}   & \\textbf{1.30}   &  \\textbf{0.0557 }     \\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\n\\section{Results}\\label{sec:results}\n\\subsection{Runtime Performance}\nWe implement our method on a \\YL{computer with a} 2.50GHz \\YL{4-Core} Intel CPU for coarse simulation and TS-ACAP extraction,\nand \\YL{an} NVIDIA GeForce\\textsuperscript{\\textregistered}~GTX 1080Ti GPU for fine TS-ACAP generation by the network and mesh coordinate reconstruction.\nTable~\\ref{table:runtime} shows average per-frame execution time of our method for various cloth datasets.\nThe execution time contains four parts: coarse simulation, TS-ACAP extraction, high-resolution TS-ACAP synthesis, and collision refinement. \nFor reference, we also \\YL{measure} the time of a CPU-based implementation of  high-resolution PBS using ARCSim \\cite{Narain2012AAR}.\nOur algorithm is $10\\sim35$ times faster than the \\YL{PBS} HR simulation.\nThe low computational cost of our method makes it suitable for the interactive applications. \n\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/0crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/1crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/2crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/3crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/4crop0090down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/0crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/1crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/2crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/3crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/4crop0300down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/0crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/1crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/2crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/3crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/4crop0110down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/0crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/1crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/2crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/3crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/4crop0260down.png} \\\\ \n      \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data \\YL{on the TSHIRT} dataset.\n\t\t(a) coarse simulation,\n\t\t(b) results of \\cite{chen2018synthesizing},\n\t\t(c) results of \\cite{zurdo2013wrinkles},\n\t\t(d) our results,\n\t\t(e) ground truth generated by PBS.\n\t\tOur method produces the detailed shapes of higher quality than Chen {\\itshape et al.} and Zurdo {\\itshape et al.}, see the folds and wrinkles in the close-ups. Chen {\\itshape et al.} results suffer from seam line problems. The results of Zurdo {\\itshape et al.} exhibit clearly noticeable artifacts.}\n\t\\label{fig:comparetoothers_tshirt}\n\\end{figure}\n \\begin{figure}[!htb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0010down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0060down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0140down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0160down.png} \\\\  \n\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular} \n\t\\caption{Comparison of the reconstruction results for unseen data in the PANTS dataset.\n\t\t(a) coarse simulation results,\n\t\t(b) results of \\cite{chen2018synthesizing}, mainly smooth the coarse meshes and barely exhibit any wrinkles.\n\t\t(c) results of \\cite{zurdo2013wrinkles}, have clear artifacts on examples where LR and HR meshes are not aligned well, \\textit{e.g. } the trouser legs.\n\t\t(d) our results, ensures physically-reliable results.\n\t\t(e) ground truth generated by PBS.\n\t}\n\t\\label{fig:comparetoothers_pants}\n\\end{figure}  \n\\begin{figure*}[htb]\n\t\\centering\n\t\\subfloat[Input]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0080_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0110_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0140_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0160_00_skirtlrkeyshot.png} \n\t\\end{minipage}} \n\t\\subfloat[Chen {\\itshape et al.}]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0080keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0110keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0140keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0160keyshot.png} \n\t\\end{minipage}} \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0080_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0110_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0140_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0160_00_skirtlr_result.png} \n\t\\end{minipage}\n\t\\subfloat[Zurdo {\\itshape et al.}]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0080keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0110keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0140keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0160keyshot.png} \n\t\\end{minipage}} \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0080_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0110_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0140_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0160_00_skirthr.png} \n\t\\end{minipage}\n\t\\subfloat[Ours]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0080_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0110_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0140_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0160_00_skirthrkeyshot.png} \n\t\\end{minipage}}\n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0080_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0110_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0140_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0160_00_skirthr.png} \n\t\\end{minipage}  \n\t\\subfloat[GT]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0080_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0110_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0140_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0160_00_skirthrkeyshot.png} \n\t\\end{minipage}}\n\t\\begin{minipage}[b]{0.08\\linewidth} \n\t\t\\includegraphics[width=1.000000\\linewidth, trim=0 0 0 0,clip]{pictures\/bar.png}\n\t\\end{minipage}\n\t\n\t\\caption{Comparison of the reconstruction results for unseen data in the SKIRT dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing},\n\t\t(c) the results of \\cite{zurdo2013wrinkles},\n\t\t(d) our results,\n\t\t(e) the ground truth generated by PBS.\n\tThe reconstruction accuracy is qualitatively showed as a difference map. \n\tReconstruction errors are color-coded and warmer colors indicate larger errors. Our method leads to significantly lower reconstruction errors. }\n\t\\label{fig:comparetoothers_skirt}\n\\end{figure*}\n\n\\subsection{\\YL{Fine Detail} Synthesis Results and Comparisons}\nWe now demonstrate our method using various \\YL{detail enhancement}\nexamples \\YL{both} quantitatively and qualitatively, \\YL{including added wrinkles and rich dynamics.}\nUsing detailed meshes generated by PBS as ground truth, we compare our results with physics-based coarse simulations, our implementation of a deep learning-based method \\cite{chen2018synthesizing} and a conventional machine learning-based method \\cite{zurdo2013wrinkles}.\n\nFor quantitative comparison, we use \\YL{three} metrics: Root Mean Squared Error (RMSE), Hausdorff distance as well as spatio-temporal edge difference (STED) \\cite{Vasa2011perception} designed for motion sequences with a focus on `perceptual' error of models.\nThe results are shown in Table~\\ref{table:compare_zurdo_chen2}.\nNote that \\YL{for the datasets from the top to bottom in the table,} the Hausdorff \\YL{distances} between LR meshes and the ground truth are increasing. \\YL{This} tendency is in accordance with the deformation range from tighter T-shirts and pants to skirts and square\/disk tablecloth with higher degrees \\YL{of freedom}.\nSince the vertex position representation cannot handle rotations well, the larger scale the models deform, the more artifacts Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} would \\YL{bring in} in the reconstructed models, \\YL{leading  to increased} RMSE and Hausdorff distances.  \nThe results indicate that our method has better reconstruction results \\YL{quantitatively} than the compared methods \\YL{on} the 5 datasets with \\YL{all the three} metrics.\nEspecially \\YL{for} the SKIRT, SHEET and DISK \\YL{datasets} which \\YL{contain} loose cloth \\YL{and hence larger and richer deformation}, our \\YL{method} outperforms \\YL{existing methods significantly} since tracking between coarse and fine meshes \\YL{is} not required in our algorithm.\n\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0130down.png}\\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0180down.png} \\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0260down.png} \\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0320down.png} \\\\ \n\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data in the SHEET dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing}, with inaccurate and\nrough wrinkles different from the GT.\n\t\t(c) the results of \\cite{zurdo2013wrinkles}, show similar global shapes to coarse meshes with some wrinkles and unexpected sharp corner.\n\t\t(d) our results, show mid-scale wrinkles and similar global deformation as GT.\n\t\t(e) the ground truth generated by PBS.}\n\t\\label{fig:comparetoothers_crashball}\n\t\\vspace{-0.2cm}\n\\end{figure}   \n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.001} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/0crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/1crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/2crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/3crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/4crop0050down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0090down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0160down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0360down.png} \\\\ \n\t\t\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data in the DISK dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing}, cannot reconstruct credible shapes. \n\t\t(c) the results of \\cite{zurdo2013wrinkles}, show apparent artifacts near the flying tails since no tracking constraints applied.\n\t\t(d) our results, reproduce large-scale deformations, see the tail of the disk flies like a fan in the wind.\n\t\t(e) the ground truth generated by PBS.}\n\t\\label{fig:comparetoothers_disk}\n\\end{figure}    \n\n\\YL{We further make qualitative comparisons on the 5 datasets.}\nFig. \\ref{fig:comparetoothers_tshirt} shows \\YL{detail synthesis results} on the TSHIRT dataset.\nThe first and second \nrows \nare from \\YL{sequence} 06\\_08, a woman dribbling the basketball sideways and the \\YL{last two rows} are from \\YL{sequence} 08\\_11, a walking woman.\nIn this dataset of tight t-shirts on human bodies, Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} and our method are able to reconstruct the garment model completely with mid-scale wrinkles.\nHowever, Chen {\\itshape et al.} \\cite{chen2018synthesizing} suffer from the seam line problems due to \\YL{the use of geometry image representation}. \nA geometry image is a parametric sampling of the shape, which is \\YL{made a topological disk by cutting through some seams.} \nThe boundary of the disk needs to be fused so that the reconstructed mesh has the original topology.\n\\YL{The super-resolved geometry image corresponding to high-resolution cloth animations are not entirely accurate, and as a result the fused boundaries no longer match exactly, }\n\\textit{e.g. } clear seam lines on the shoulder and crooked boundaries on the left side of the waist \\YL{for the examples} in Fig.~\\ref{fig:comparetoothers_tshirt} (b)),\n\\YL{while} our method \\YL{produces} better results than \\cite{chen2018synthesizing} and \\cite{zurdo2013wrinkles} which have  \\YL{artifacts of unsmooth surfaces}.\n\nFig. \\ref{fig:comparetoothers_pants} shows comparative results of the animations of pants on a fixed body shape while changing the body pose over time. \nThe results of \\cite{chen2018synthesizing} \\YL{mainly} smooth the coarse meshes and barely exhibit \\YL{any} wrinkles.\nZurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} utilize tracking algorithms to ensure the %\n\\YL{close alignment}\nbetween coarse and fine meshes, and thus the fine meshes are constrained \\YL{and do not exhibit the behavior of full physics-based simulation.}\n\\YL{So on the PANTS dataset,} the results of \\cite{zurdo2013wrinkles} have clear artifacts on examples \\YL{where} LR and HR meshes are not aligned well, \\textit{e.g. } the trouser legs.\nDifferent from the two compared methods \\YL{that reconstruct displacements} or local coordinates, \nour method \\YL{uses} deformation-based features in both encoding and decoding \\YL{phases} which \\YL{does not suffer from such restrictions and ensures physically-reliable results.}\n\nFor looser garments like \\YL{skirts}, we show comparison results in Fig. \\ref{fig:comparetoothers_skirt}, with color coding to highlight the differences between synthesized results and the ground truth.\nOur method successfully reconstructs the swinging skirt \\YL{caused by} the body motion (see the small wrinkles on the waist and the \\YL{medium-level} folds on the skirt \\YL{hem}).\nChen {\\itshape et al.} are able to reconstruct the overall shape of the skirt, however there are many small unsmooth \\YL{triangles leading to noisy shapes}\ndue to the 3D coordinate representation with untracked fine meshes with abundant wrinkles.\nThis leads to unstable animation, please see the accompanying video.\nThe results of \\cite{zurdo2013wrinkles} have some problems of the global deformation, see the directions of the skirt hem and the large highlighted area in the color map.\nOur learned \\YL{detail} synthesis model provides better visual quality for shape generation \\YL{and the generated results look} closer to the ground truth.\n \nInstead of garments dressed on human bodies, we additionally show some results of free-flying tablecloth. \nThe comparison of the testing results \\YL{on} the SHEET dataset are shown in Fig.~\\ref{fig:comparetoothers_crashball}.\nThe results of \\cite{chen2018synthesizing} show inaccurate and rough wrinkles different from the ground truth. \nFor hanging sheets in the results of \\cite{zurdo2013wrinkles}, the global shapes are more like coarse \\YL{meshes} with some wrinkles and unexpected sharp corners, \\textit{e.g. } the left side in the last row of Fig. \\ref{fig:comparetoothers_crashball} (c),\nwhile ours show \\YL{mid-scale} wrinkles and similar global deformation \\YL{as} the high-resolution meshes. \n\nAs for the DISK dataset, from the visual results in Fig.~\\ref{fig:comparetoothers_disk}, we can see that Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} cannot handle large-scale rotations well and cannot reconstruct credible shapes in such cases. \n\\gl{Especially for Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles}, the impact of tracking is significant for their algorithm.}\nThey can reconstruct the top\nand part of tablecloth near the cylinder, but the flying tails have apparent artifacts.  \nOur algorithm does not have such drawbacks.\nNotice how our method successfully reproduces ground-truth deformations, including the overall drape (\\textit{i.e. }, how the tail of the disk flies like a fan in the wind) and mid-scale wrinkles.\n\n\\begin{table}[!htb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{User study results on cloth \\YL{detail} synthesis. We show the average ranking score of the three methods: Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles}, and ours. The\n\t\tranking ranges from 1 (the best) to 3 (the worst). The results are calculated\n\t\tbased on 320 trials. We see that our method achieves the best in terms of\n\t\twrinkles, temporal stability \\YL{and overall quality}.}\n\t\\label{table:userstudy}\n\t\\centering \n\t\\begin{tabular}{cccc}\n\t\t\\toprule[1.2pt]  \n\t\tMethod                    & Wrinkles   & Temporal stability & Overall   \\\\ \\hline \n\t\tChen {\\itshape et al.}    & 2.184 & 2.1258 &2.1319\\\\ \\hline  \n\t\tZurdo {\\itshape et al.}   & 2.3742 & 2.5215 & 2.4877\\\\ \\hline         \n\t\tOurs                      & \\textbf{1.4417} & \\textbf{1.3528} & \\textbf{1.3804}  \\\\\n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\\gl{We further conduct a user study to evaluate the stability and realistic of the synthesized dense mesh dynamics. 32 volunteers are involved for this user study.}\nFor every question, we give one sequence and 5 images of coarse meshes as references, \\YL{and} then let the user rank the corresponding outputs from  Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} and ours according to three different criteria (wrinkles, temporal stability and overall). \nWe shuffle the order of the algorithms each time we exhibit the question and show shapes from the three methods randomly \\YL{to avoid bias}. \nWe show the results of the user study in Table \\ref{table:userstudy}, where we observe that our generated \\YL{shapes} perform the best on all three criteria. \n\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{Per-vertex error (RMSE) on synthesized shapes with different feature representations: 3D coordinates, ACAP and TS-ACAP.}\n\t\\label{table:feature_compare}\n\t\\centering\n\t\\begin{tabular}{cccccc}\n\t\t\\toprule[1.2pt]\n\t\tDataset        & TSHIRT      & PANTS    &  \tSKIRT   & SHEET & DISK      \\\\ \\hline\n\t\t3D coordinates &  0.0101      & 0.0193   &    0.00941 &  0.00860   &  0.185 \\\\ \\hline\n\t\tACAP           &  0.00614     & 0.00785  &   0.00693  &  0.00606   &   0.0351     \\\\ \\hline\n\t\tTS-ACAP          &  \\textbf{0.00546}     & \\textbf{0.00663}  &    \\textbf{0.00685} &   \\textbf{0.00585} & \\textbf{0.0216}\\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.001}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}}\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0040.png} \\\\\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0075.png} \\\\\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0110.png} \\\\ \n \t\\vspace{0.3cm} \\small (a) Input & \\vspace{0.3cm}\\small (b) Coordinates & \\vspace{0.3cm}\\small (c) Ours & \\vspace{0.3cm}\\small (d) GT\n  \\end{tabular} \n\t\\caption{The evaluation of the TS-ACAP feature in our detail synthesis method. \n\t\t(a) input coarse \\YL{shapes},\n\t\t(b) the results using 3D coordinates, which can be clearly seen the rough appearance, unnatural deformation and some artifacts, especially in the highlighted regions with details shown in the close-ups.\n\t\t(c) our results, which show smooth looks and the details are more similar to the GT.\n\t\t(d)\tground truth.\n\t\t }\n\t\\label{fig:ablationstudy_coordiniates_skirt}\n\\end{figure}\n\\begin{figure}[htb]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.05cm}  \n    \\renewcommand\\arraystretch{0.001}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.02\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}}\n \t \\rotatebox{90}{\\small ACAP} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0103.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0104.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0105.png} \\\\\n \t\\rotatebox{90}{\\small TS-ACAP}  &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0103.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0104.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0105.png} \\\\  \n     \\vspace{0.3cm}    & \\vspace{0.3cm}  \\small $t = 103$ & \\vspace{0.3cm} \\small $t = 104$  & \\vspace{0.3cm} \\small $t = 105$ \n\t\\end{tabular} \n\t\\caption{\n\t\t Three consecutive frames from a testing sequence in the DISK dataset. First row: the results of ACAP. As shown in the second column, the enlarged wrinkles are different from the previous and the next frames.\n\t\t This causes jumping in the animation.\n\t\t Second row: the consistent results obtained via TS-ACAP feature, demonstrating that our TS-ACAP representation ensures the temporal coherence. \n\t}\n\t\\label{fig:jump_acap}\n\\end{figure}\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\fontsize{7.5}{9}\\selectfont\n\t\\caption{Comparison of RMSE between synthesized shapes and ground truth with different networks, \\textit{i.e. } without temporal modules, with RNN, with LSTM and ours with the Transformer network.}\n\t\\label{table:transformer_compare}\n\t\\centering\n\t\\begin{tabular}{cccccc}\n\t\t\\toprule[1.2pt]\n\t\tDataset        & TSHIRT      & PANTS    &  \tSKIRT   & SHEET       & DISK \\\\ \\hline\n\t\tWO Transformer &  0.00909     & 0.01142   &    0.00831 &   0.00739    &  0.0427    \\\\ \\hline\n\t\tWith RNN  &  0.0435      &  0.0357   & 0.0558     &   0.0273     & 0.157      \\\\ \\hline\n\t\tWith LSTM  &  0.0351      &  0.0218   &  0.0451    &  0.0114      &  0.102     \\\\ \\hline\n\t\tWith Transformer         &  \\textbf{0.00546}     & \\textbf{0.00663}  &    \\textbf{0.00685} &   \\textbf{0.00585} & \\textbf{0.0216} \\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table} \n\\begin{figure}[tb]\n \t\\centering\n    \\setlength{\\tabcolsep}{0.0cm}  \n    \\renewcommand\\arraystretch{-1.9}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.08\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}}\n \t\t\\rotatebox{90}{\\small (a) Input}& \n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0008.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0016.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0022.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0094.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0200.png}   \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (b) EncDec} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0008.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0016.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0022.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0094.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0200.png} \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (c) RNN} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0008.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0016.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0022.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0094.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0200.png} \n\t \t\\\\\n\t \t\\rotatebox{90}{\\small (d) LSTM}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0008.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0016.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0022.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0094.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0200.png} \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (e) Ours}& \n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0008.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0016.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0022.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0094.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0200.png} \n \t\t\\\\ \n \t\t  \\rotatebox{90}{\\small (f) GT}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0008.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0016.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0022.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0094.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0200.png} \n \t\\end{tabular} \n \t\\caption{The evaluation of the Transformer network in our model for wrinkle synthesis.\n \t\tFrom top to bottom we show (a) %\n \t\t\\gl{input coarse mesh with physical simulation}\n \t\t(b) the results with an encoder-decoder \\YL{dropping out temporal modules}, (c) the results with RNN \\cite{chung2014empirical}, (d) the results with LSTM \\cite{hochreiter1997long}, (e) ours, and (f) the ground truth generated by PBS.}\n \t\\label{fig:transformer_w_o_tshirt}\n \\end{figure}   \n\n\\subsection{\\YL{Evaluation of} Network Components}\nWe evaluate the effectiveness of our network components for two aspects: the \\YL{capability} of the TS-ACAP feature and the \\YL{capability} of the Transformer network. \nWe evaluate our method qualitatively and quantitatively on different datasets.\n\n\\textbf{Feature Representation Evaluation}.\nTo verify the effectiveness of our TS-ACAP feature, we compare per-vertex position errors to other features to evaluate the generated shapes in different datasets quantitatively. \nWe compare our method using TS-ACAP feature with our transduction methods using 3D vertex coordinates and ACAP, with network layers and parameters adjusted accordingly to optimize performance alternatively.\nThe details of numerical comparison are shown in Table \\ref{table:feature_compare}.\nACAP and TS-ACAP show quantitative improvements than 3D coordinates.  \nIn Fig. \\ref{fig:ablationstudy_coordiniates_skirt}, we exhibit several compared examples of animated skirts of coordinates and TS-ACAP. \n\\YL{The results using coordinates show rough appearance, unnatural deformation and some artifacts, \n I can't really see the two circles?\nespecially in the highlighted regions with details shown in the close-ups.} Our results with TS-ACAP are more similar to the ground truth than the ones with coordinates. \nACAP has the problem of temporal inconsistency, thus the results are shaking or jumping frequently. \n\\YL{Although the use of the Transformer network can somewhat mitigate this issue, such artifacts can appear even with the Transformer.}\n\\YL{Fig.~\\ref{fig:jump_acap} shows} three consecutive frames from a testing sequence in the DISK dataset.\nResults with TS-ACAP show more consistent wrinkles than the ones with ACAP thanks to the temporal constraints.\n\n\\textbf{Transformer Network Evaluation}.\nWe also evaluate the impact of the Transformer network in our pipeline. \nWe compare our method to an encoder-decoder network dropping out the temporal modules, our pipeline with the recurrent neural network (RNN) and with the long short-term memory (LSTM) \\YL{module}.\nAn example of T-shirts is given in Fig. \\ref{fig:transformer_w_o_tshirt}, \\YL{showing} 5 frames in order.\nThe results without any temporal modules show artifacts on the sleeves and neckline since these places have strenuous \\YL{forces}. %\nThe models using RNN and LSTM stabilize the sequence via eliminating dynamic and detailed deformation, but all the results keep wrinkles on the chest from the initial state\\YL{, lacking rich dynamics.}\nBesides, they are not able to generate stable and realistic garment animations \\YL{that look similar to} the ground truth,\n\\YL{while} \\YL{our} method with the Transformer network \\YL{apparently} improves the temporary stability, \\YL{producing results close to the ground truth.}\nWe also quantitatively evaluate the performance of the Transformer network \\YL{in our method} via per-vertex error. \nAs shown in Table \\ref{table:transformer_compare}, the RMSE of our model \\YL{is} smaller than the other models.\n\n\\section{Conclusion and Future Work}\\label{sec:conclusion}\nIn this paper, we introduce a novel algorithm for synthesizing robust and realistic cloth animations via deep learning.\nTo achieve this, we propose a geometric deformation representation named TS-ACAP which well embeds the details and ensures the temporal consistency.\n\\YL{Benefiting} from \\YL{the} deformation-based feature, there is no explicit requirement of tracking between coarse and fine meshes in our algorithm. \nWe also use the Transformer network based on attention mechanisms to map the coarse TS-ACAP to fine TS-ACAP, maintaining the stability of our generation.\nQuantitative and qualitative results reveal that our method can synthesize realistic-looking wrinkles in various datasets, such as draping tablecloth, tight or \\YL{loose} garments dressed on human bodies, etc. \n \nSince our algorithm synthesizes \\YL{details} based on the coarse meshes, the time for coarse simulation is unavoidable.\nEspecially for tight garments like T-shirts and pants, the collision solving phase is time-consuming.\nIn the future, we intend to generate coarse sequences for tight cloth via skinning-based methods in order to reduce the computation for our pipeline.\nAnother limitation is that our current network is not able to deal with all kinds of garments with different topology.\n\\newpage\n\\bibliographystyle{IEEEtran}\n\n\n\\section{Introduction}\\label{sec:introduction}}\n\\IEEEPARstart{C}{reating} dynamic general clothes or garments on animated characters has been a long-standing problem in computer graphics (CG).\nIn the CG industry, physics-based simulations (PBS) are used to achieve realistic and detailed folding patterns for garment animations. \nHowever, it is time-consuming and requires expertise to synthesize fine geometric details since high-resolution meshes with tens of thousands or more vertices are often required.\nFor example, 10 seconds are required for physics-based simulation of a frame for detailed skirt animation shown in Fig.~\\ref{fig:lrhrsim1}.\nNot surprisingly, garment animation remains a bottleneck in many applications.\nRecently, data-driven methods provide alternative solutions to fast and effective wrinkling behaviors for garments.\nDepending on human body poses, some data-driven methods~\\cite{wang10example,Feng2010transfer,deAguiar10Stable,santesteban2019learning, wang2019learning} are capable of generating tight cloth animations successfully.\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tabular}{ccc}\n\t\\multicolumn{3}{c}{\n\t\\includegraphics[width=1.0\\linewidth]{pictures\/wireframe2_1.pdf}}  \\\\\n\t(a) coarse skirt & (b) tracked skirt & (c) fine skirt\n\t\\end{tabular}\n\t\\caption{\\small \\cl{One frame of \\YL{skirt in different representations.} (a) \\YL{coarse mesh} (207 triangles), (b) \\YL{tracked mesh} (13,248 triangles) and (c) \\YL{fine mesh} (13,248 triangles). \\YL{Both coarse and fine meshes are obtained by simulating the skirt using a physics-based method \\cl{\\cite{Narain2012AAR}}. The tracked mesh is obtained with physics-based simulation involving additional constraints to track the coarse mesh.} The tracked mesh  exhibits stiff folds while the wrinkles in the fine simulated mesh are more realistic.}%\n\t}\n\t\\label{fig:lrhrsim1} \n\\end{figure}\nUnfortunately, they are not suitable for loose garments, such as skirts, since the deformation of wrinkles cannot be defined by a static mapping from a character's pose.\nInstead of human poses, wrinkle augmentation on coarse simulations provides another alternative. \nIt utilizes coarse simulations with fast speed to cover a high-level deformation and leverages learning-based methods to add realistic wrinkles.\nPrevious methods~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} commonly require dense correspondences between coarse and fine meshes, so that local details can be added without affecting global deformation. \n\\YL{Such methods also require coarse meshes to be sufficiently close to fine meshes, as they only add details to coarse meshes.}\nTo maintain the correspondences for training data and ensure closeness between coarse and fine meshes, weak-form constraints such as various test functions~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} are applied to make fine meshes track the coarse meshes, \n\\YL{but as a result, the obtained high-resolution meshes do not fully follow physical behavior, leading to animations that lack realism. An example is shown in Fig.~\\ref{fig:lrhrsim1} where the tracked skirt (b) loses a large amount of wrinkles which should appear when simulating on fine meshes (c).}\n\n \nWithout requiring the constraints between coarse and fine meshes, we propose \n\\gl{the DeformTransformer network\nto synthesize detailed thin shell animations from coarse ones, based on deformation transfer.}\nThis is inspired by the similarity observed between pairs of coarse and fine meshes generated by PBS. %\nAlthough the positions of vertices from two meshes are not aligned, the overall deformation is similar, so it is possible to predict fine-scale deformation with coarse simulation results.\nMost previous works~\\cite{kavan11physics,zurdo2013wrinkles,chen2018synthesizing} use explicit vertex coordinates to represent 3D meshes, which are sensitive to translations and rotations,\nso they require good alignments between low- and high-resolution meshes. \nIn our work, we regard the cloth animations as non-rigid deformation and propose a novel representation for mesh sequences, called TS-ACAP (Temporal and Spatial As-Consistent-As-Possible) representation. \nTS-ACAP is a local deformation representation, capable of representing and solving large-scale deformation problems, while maintaining the details of meshes.\nCompared to the original ACAP representation~\\cite{gao2019sparse}, TS-ACAP is fundamentally designed to ensure the temporal consistency of the extracted feature sequences, \\YL{and meanwhile} it can maintain the original features of ACAP \\YL{to cope with large-scale deformations}.\nWith \\YL{TS-ACAP} representations for both coarse and fine meshes, we leverage a sequence transduction network to map the deformation from coarse to fine level to assure the temporal coherence of generated sequences.\nUnlike existing works using recurrent neural networks (RNN)~\\cite{santesteban2019learning}, we utilize the Transformer network~\\cite{vaswani2017attention}, an architecture consisting of frame-level attention mechanisms for our mesh sequence transduction task.\nIt is based entirely on attention without recursion modules so can be trained significantly faster than architectures based on recurrent %\nlayers.\nWith \\YL{temporally consistent features and the Transformer network, \\YL{our method achieves} stable general cloth synthesis with fine details in an efficient manner.}\n\nIn summary, the main contributions of our work are as follows:\n\\begin{itemize}\n\t\\item \\YL{We propose a novel framework for the synthesis of cloth dynamics, by learning temporally consistent deformation from low-resolution meshes to high-resolution meshes \\gl{with realistic dynamic}, which is $10 \\sim 35$ times faster than PBS \\cite{Narain2012AAR}.}\n\t\\item \\YL{To achieve this, we propose a \\cl{temporally and spatially as-consistent-as-possible deformation representation (TS-ACAP)} to represent the cloth mesh sequences. It is able to deal with large-scale deformation, essential for mapping between coarse and fine meshes, while ensuring temporal coherence.} \n    \\item \\gl{Based on the TS-ACAP, We further design an effective neural network architecture (named DeformTransformer) by improving Transformer network, which successfully enables high-quality synthesis of dynamic wrinkles with rich details on thin shells and maintains temporal consistency on the generated high-resolution mesh sequences.}\n    \n    \n\\end{itemize}\n\nWe qualitatively and quantitatively evaluate our method for various cloth types (T-shirts, pants, skirts, square and disk tablecloth) with different motion sequences. \nIn Sec.~\\ref{sec:related_work}, we review the work most related to ours. We then give the detailed description of our method in Sec.~\\ref{sec:approach}. \nImplementation details are presented in Sec.~\\ref{sec:implementation}. We present experimental results, including extensive\ncomparisons with state-of-the-art methods in Sec.~\\ref{sec:results}, and finally, we draw conclusions and \\YL{discuss future work} in Sec.~\\ref{sec:conclusion}.\n\n\n\\section{Related work} \\label{sec:related_work}\n\\subsection{Cloth Animation}\nPhysics-based techniques for realistic cloth simulation have been widely studied in computer graphics, \\YL{using methods such as} implicit Euler integrator \\cite{BW98,Harmon09asynchronous}, iterative optimization \\cite{terzopoulos87elastically,bridson03wrinkles,Grinspun03shell}, collision detection and response \\cite{provot97collision,volino95collision}, etc. \n\\YL{Although such techniques can generate realistic cloth dynamics, }they are time consuming for detailed cloth synthesis, and the robustness and efficiency of simulation systems are also of concern.\n\\YL{To address these, alternative methods have been developed to generate} the dynamic details of cloth animation via adaptive techniques \\cite{lee2010multi,muller2010wrinkle,Narain2012AAR}, data-driven approaches \\cite{deAguiar10Stable, Guan12DRAPE, wang10example, kavan11physics,zurdo2013wrinkles} and deep learning-based methods \\cite{chen2018synthesizing,gundogdu2018garnet,laehner2018deepwrinkles,zhang2020deep}, etc.\n\n Adaptive techniques \\cite{lee2010multi, muller2010wrinkle} usually simulate a coarse model by simplifying the smooth regions and \\YL{applying interpolation} to reconstruct the wrinkles, \\YL{taking normal or tangential degrees of freedom into consideration.}  \nDifferent from simulating a reduced model with postprocessing detail augmentation, Narain {\\itshape et al.} \\cite{Narain2012AAR} directly generate dynamic meshes in \\YL{the} simulation phase through adaptive remeshing, at the expense of increasing \\YL{computation time}.  \n\nData-driven methods have drawn much attention since they offer faster cloth animations than physical models.\nWith \\YL{a} constructed database of \\YL{high-resolution} meshes, researchers have proposed many techniques depending on the motions of human bodies with linear conditional models\\cite{deAguiar10Stable, Guan12DRAPE} or secondary motion graphs \\cite{Kim2013near, Kim2008drivenshape}.\nHowever, these methods are limited to tight garments and not suitable for skirts or cloth with more freedom.\nAn alternative line \\YL{of research} is to augment details on coarse simulations \\YL{by exploiting knowledge from a} database of paired meshes, to generalize the performance to complicated testing scenes.\nIn this line, in addition to wrinkle synthesis methods \\YL{based on} bone clusters \\cite{Feng2010transfer} or human poses \\cite{wang10example} for fitted clothes, there are some approaches \\YL{that investigate how to} learn a mapping from a coarse garment shape to a detailed one for general \\YL{cases} of free-flowing cloth simulation.\nKavan {\\itshape et al.} \\cite{kavan11physics} present linear upsampling operators to \\YL{efficiently} augment \\YL{medium-scale} details on coarse meshes.\nZurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} define wrinkles as local displacements and use \\YL{an} example-based algorithm to enhance low-resolution simulations.\n\\YL{Their approaches mean the} high-resolution cloth \\YL{is} required to track \\YL{the} low-resolution cloth, \\YL{and thus cannot} exhibit full high-resolution dynamics.\n\nRecently deep learning-based methods have been successfully applied for 3D animations of human \\YL{faces}~\\cite{cao2016real, jiang20183d}, hair \\cite{zhang2018modeling, yang2019dynamic} and garments \\cite{liu2019neuroskinning, wang2019learning}.\nAs for garment synthesis, some approaches \\cite{laehner2018deepwrinkles, santesteban2019learning, patel2020tailornet} are proposed to utilize a two-stream strategy consisting of global garment fit and local \\YL{wrinkle} enhancement.\nL{\\\" a}hner {\\itshape et al.} \\cite{laehner2018deepwrinkles} present DeepWrinkles, \\YL{which recovers} the global deformation from \\YL{a} 3D scan system and \\YL{uses a} conditional \\YL{generative adversarial network} to enhance a low-resolution normal map.\nZhang {\\itshape et al.} \\cite{zhang2020deep} further generalize the augmentation method with normal maps to complex garment types as well as various motion sequences.\n\\YL{These approaches add wrinkles on normal maps \\YL{rather than geometry}, and thus their effectiveness is restricted to adding fine-scale visual details, not large-scale dynamics.}\nBased on \\YL{the} skinning representation, some algorithms \\cite{gundogdu2018garnet,santesteban2019learning} use neural networks to generalize garment synthesis algorithms to multiple body shapes. \n\\YL{In addition, other works are} devoted to \\YL{generalizing neural networks} to various cloth styles \\cite{patel2020tailornet} or cloth materials \\cite{wang2019learning}.\nDespite tight garments dressed on characters, some deep learning-based methods \\cite{chen2018synthesizing, oh2018hierarchical} are %\n\\YL{demonstrated to work for cloth animation with higher degrees} of freedom.\nChen {\\itshape et al.} \\cite{laehner2018deepwrinkles} represent coarse and fine meshes via geometry images and use \\YL{a} super-resolution network to learn the mapping.\nOh {\\itshape et al.} \\cite{oh2018hierarchical} propose a multi-resolution cloth representation with \\YL{fully} connected networks to add details hierarchically.\nSince the \\YL{free-flowing cloth dynamics are harder for networks to learn} than tight garments, the results of these methods have not reached the realism of PBS. \\YL{Our method based on a novel deformation representation and network architecture has superior capabilities of learning the mapping from coarse and fine meshes, generating realistic cloth dynamics, while being much faster than PBS methods.}\n \n \\begin{figure*}[ht]\n \t\\centering\n \t\\includegraphics[width=1.0\\linewidth, trim=20 250 20 50,clip]{pictures\/mainpicture2.pdf} \n \t\\caption{\\small The overall architecture of our detail synthesis network. At data preparation stage, we generate low- and high-resolution \\gl{thin shell} animations via coarse and fine \\gl{meshes} and various motion sequences.\n \t Then we encode the coarse meshes and the detailed meshes to a deformation representation TS-ACAP, respectively.\n \t\\YL{Our algorithm then} learns to map the coarse features to fine features %\n \t\\YL{by designing a DeformTransformer network that consists of temporal-aware encoders and decoders, and finally reconstructs the detailed animations.}\n \t}\n \t\\label{fig:pipeline}\n \\end{figure*}\n\n\\subsection{Representation for 3D Meshes}\nUnlike 2D images with regular grid of pixels, \\YL{3D meshes have irregular connectivity which makes learning more difficult. To address this, existing deep learning based methods turn 3D meshes to a wide range of representations to facilitate processing~\\cite{xiao2020survey},} such as voxels, images \\YL{(such as depth images and multi-view images)}, point clouds, meshes, etc.\n\\YL{The volumetric representation has a regular structure, but it} often suffers from \\YL{the problem of extremely high space and time consumption.}\nThus Wang {\\itshape et al.} \\cite{wang2017cnn} propose an octree-based convolutional neural network and encode the voxels sparsely. \nImage-based representations including \\YL{depth images} \\cite{eigen2014depth,gupta2014learning} and multi-view images \\cite{Su2015mvcnn,li20193d} are proposed to encode 3D models in a 2D domain. \nIt is unavoidable that both volumetric and image-based representations lose some geometric details.\nAlternatively, geometry images are used in \\cite{sinha2016deep,Sinha2017surfnet,chen2018synthesizing} for mesh classification or generation\\YL{, which are obtained through cutting a 3D mesh to a topological disk, parameterizing it to a rectangular domain and regularly sampling the 3D coordinates in the 2D domain~\\cite{gu2002geometry}.}\n\\YL{However, this representation} may suffer from parameterization distortion and seam line problems.\n\nInstead of representing 3D meshes into other formats, recently there are methods \\cite{tan2017autoencoder, tan2017variational, hanocka2019meshcnn} applying neural networks directly to triangle meshes with various features.\nGao {\\itshape et al.} \\cite{gao2016efficient} propose a deformation-based representation, called the rotation-invariant mesh difference (RIMD) which is translation and rotation invariant.\nBased on the RIMD feature, Tan {\\itshape et al.} \\cite{tan2017variational}  propose a fully connected variational autoencoder network to analyze and generate meshes.\nWu {\\itshape et al.} \\cite{wu2018alive} use the RIMD to generate\na 3D caricature model from a 2D caricature image. \nHowever, it is expensive to reconstruct vertex coordinates from the RIMD feature due to the requirement of solving a very complicated optimization.\nThus it is not suitable for fast mesh generation tasks.\nA faster deformation representation based on an as-consistent-as-possible (ACAP) formulation \\cite{gao2019sparse} is further used to reconstruct meshes \\cite{tan2017autoencoder}, which is able to cope with large rotations and efficient for reconstruction.\nJiang {\\itshape et al.} \\cite{jiang2019disentangled} use ACAP to disentangle the identity and expression of 3D \\YL{faces}. \nThey further apply ACAP to learn and reconstruct 3D human body models using a coarse-to-fine pipeline \\cite{jiang2020disentangled}. \n\\YL{However, the ACAP feature is represented based on individual 3D meshes. When applied to a dynamic mesh sequence, it does not guarantee temporal consistency.}\nWe propose a \\cl{temporally and spatially as-consistent-as-possible (TS-ACAP)} representation, to ensure both spatial and temporal consistency of mesh deformation.\nCompared to ACAP, our TS-ACAP can also accelerate the computation of features thanks to the sequential constraints.  \n\n\\subsection{Sequence Generation with \\YL{DNNs (Deep Neural Networks)}}\nTemporal information is crucial for stable and \\gl{vivid} sequence generation. Previously, recurrent neural networks (RNN) have been successfully applied in many sequence generation tasks \\cite{mikolov2010recurrent, mikolov2011extensions}. However, it is difficult to train \\YL{RNNs} to capture long-term dependencies since \\YL{RNNs} suffer from the vanishing gradient problem \\cite{bengio1994learning}. To deal with this problem, previous works proposed some variations of RNN, including long short-term memory (LSTM) \\cite{hochreiter1997long} and gated recurrent unit (GRU) \\cite{cho2014properties}. These variations of RNN rely on the gating mechanisms to control the flow of information, thus performing well in the tasks that require capturing long-term dependencies, such as speech recognition \\cite{graves2013speech} and machine translation \\cite{bahdanau2014neural, sutskever2014sequence}. Recently, based on attention mechanisms, the Transformer network \\cite{vaswani2017attention} has been verified to outperform \\YL{many typical sequential models} for long sequences. This structure is able to inject the global context information into each input. Based on Transformer, impressive results have been achieved in tasks with regard to audio, video and text, \\textit{e.g. } speech synthesis \\cite{li2019neural, okamoto2020transformer}, action recognition \\cite{girdhar2019video} and machine translation \\cite{vaswani2017attention}.\nWe utilize the Transformer network to learn the frame-level attention which improves the temporal stability of the generated animation sequences.\n\n\\section{Approach} \\label{sec:approach}\nWith a simulated sequence of coarse meshes $\\mathcal{C} = \\{\\mathcal{C}_1, \\dots, \\mathcal{C}_n\\}$ as input, our goal is to produce a sequence of fine ones $\\mathcal{D} = \\{\\mathcal{D}_1, \\dots, \\mathcal{D}_n\\}$ which have similar non-rigid deformation as the PBS. Given two simulation sets of paired coarse and fine garments, we extract the TS-ACAP representations respectively, \\YL{and} then use our proposed DeformTransformer network to learn the \\YL{transform} \\YL{from the low-resolution space to the high-resolution space}. \\YL{As illustrated previously in Fig.~\\ref{fig:lrhrsim1}, such a mapping involves deformations beyond adding fine details.}\nOnce the network is trained by the paired examples, a consistent and detailed animation $\\mathcal{D}$ can be synthesized for each input sequence $\\mathcal{C}$. \n\n\\subsection{Overview}\nThe overall architecture of our detail synthesis network is illustrated in Fig. \\ref{fig:pipeline}.\nTo synthesize realistic \\gl{cloth animations}, we propose a method to simulate coarse meshes first and learn a \\YL{temporally-coherent} mapping to the fine meshes.    \nTo realize our goal, we construct datasets including low- and high-resolution cloth animations, \\textit{e.g. } coarse and fine garments dressed on a human body of various motion sequences. \nTo efficiently extract localized features with temporal consistency, we propose a new deformation representation, called TS-ACAP (temporal \\YL{and spatial} as-consistent-as-possible), which is able to cope with both large rotations and unstable sequences. It also has significant advantages: it is efficient to compute for \\YL{mesh} sequences and its derivatives have closed form solutions.\nSince the vertices of the fine models are typically more than ten thousand to simulate realistic wrinkles, it is hard to directly map the coarse features to the high-dimensional fine ones for the network.\nTherefore, \\YL{convolutional encoder networks are} \napplied to encode \\YL{coarse and fine meshes in the TS-ACAP representation} to \\YL{their latent spaces}, respectively.\nThe TS-ACAP generates local rotation and scaling\/shearing parts on vertices, so we perform convolution \\YL{operations} on vertices %\n\\YL{to learn to extract useful features using shared local convolutional kernels.}\nWith encoded feature sequences, a sequence transduction network is proposed to learn the mapping from coarse to fine TS-ACAP sequences.\nUnlike existing works using recurrent neural networks \\YL{(RNNs)}~\\cite{santesteban2019learning}, we use the Transformer \\cite{vaswani2017attention}, a sequence-to-sequence network architecture, based on frame-level attention mechanisms for our detail synthesis task, \\YL{which is more efficient to learn and leads to superior results.}\n\n\\subsection{Deformation Representation}\n\\YL{As discussed before, large-scale deformations are essential to represent \\gl{thin shell mode dynamics such as }cloth animations, because folding and wrinkle patterns during animation can often be complicated. Moreover, cloth animations are in the form of sequences, hence the temporal coherence is very important for the realistic. Using 3D coordinates directly cannot cope with large-scale deformations well, and existing deformation representations are generally designed for static meshes, and directly applying them to cloth animation sequences on a frame-by-frame basis does not take temporal consistency into account. }\nTo cope with this problem, we propose a mesh deformation feature with spatial-temporal consistency, called TS-ACAP, to represent the coarse and fine deformed shapes, which exploits the localized information effectively and reconstructs \\YL{meshes} accurately.\nTake \\YL{coarse meshes} $\\mathcal{C}$ for instance and \\YL{fine meshes $\\mathcal{D}$ are processed in the same way.} \\YL{Assume that a sequence} of coarse meshes contains $n$ models with the same topology, each denoted as $\\mathcal{C}_{t}$ \\YL{($1\\leq t \\leq n$)}. \n\\YL{A mesh with the same topology is chosen as the reference model, denoted as $\\mathcal{C}_{0}$. For example, for garment animation, this can be the garment mesh worn by  a character in the T pose.}\n$\\mathbf{p}_{t,i} \\in \\mathbb{R}^{3}$ is the $i^{\\rm th}$ vertex on\nthe $t^{\\rm th}$ mesh.\nTo represent the local shape deformation, the deformation gradient $\\mathbf{T}_{t,i} \\in \\mathbb{R}^{3 \\times 3}$ can be obtained by minimizing the following energy:\n\\begin{equation}\n\t\\mathop{\\arg\\min}_{\\mathbf{T}_{t,i}} \\ \\  \\mathop{\\sum}_{j \\in \\mathcal{N}_i} c_{ij} \\| (\\mathbf{p}_{t,i} - \\mathbf{p}_{t,j}) -  \\mathbf{T}_{t,i} (\\mathbf{p}_{0,i} - \\mathbf{p}_{0,j}) \\|_2^2  \\label{con:computeDG}\n\\end{equation}\nwhere $\\mathcal{N}_i$ is the one-ring neighbors of the $i^{\\rm th}$ vertex, and $c_{ij}$ is the cotangent weight $c_{ij} = \\cot \\alpha_{ij} + \\cot \\beta_{ij} $ \\cite{sorkine2007rigid,levi2014smooth}, where $\\alpha_{ij}$\nand $\\beta_{ij}$ are angles opposite to the edge connecting the $i^{\\rm th}$ and $j^{\\rm th}$ vertices.\n\nThe main drawback of the deformation gradient representation is that it cannot handle large-scale rotations, which often \\YL{happen} in cloth animation. \nUsing polar decomposition, the deformation gradient $\\mathbf{T}_{t,i} $ can be decomposed into a rotation part and a scaling\/shearing part $\\mathbf{T}_{t,i} = \\mathbf{R}_{t,i}\\mathbf{S}_{t,i}$.\nThe scaling\/shearing transformation $\\mathbf{S}_{t,i}$ is uniquely defined, while the rotation $\\mathbf{R}_{t,i}$ \\YL{corresponds to infinite possible rotation angles (differed by multiples of $2\\pi$, along with possible opposite orientation of the rotation axis)}. Typical formulation often constrain the rotation angle to be within $[0, \\pi]$ which is unsuitable for smooth large-scale animations. \n\nIn order to handle large-scale rotations, we first require the orientations of rotation axes and rotation angles of \\YL{spatially} adjacent vertices \\YL{on the same mesh} to be as consistent as possible. \nEspecially for our sequence data, we further add constraints for adjacent frames to ensure the temporal consistency of the orientations of rotation axes and rotation angles on each vertex.\n\nWe first consider consistent orientation for axes.\n\\begin{flalign}\\label{eqn:axis}\n\t\\arg\\max_{{o}_{t,i}} \\sum_{(i,j) \\in \\mathcal{E} }  {o}_{t,i}{o}_{t,j} \\cdot s(\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}, \\theta_{t,i}, \\theta_{t,j}) \\nonumber\\\\\n\t+ \\sum_{i \\in \\mathcal{V} }  {o}_{t,i} \\cdot s(\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t-1,i}, \\theta_{t,i}, \\theta_{t-1,i}) \\nonumber\\\\\n\t{\\rm s.t.} \\quad\n\t{o}_{t,1} = 1, {o}_{t,i} = \\pm 1 (i \\neq 1) \\quad  \n\\end{flalign}\nwhere $t$ is the \\YL{index} of \\YL{the} frame, $\\mathcal{E}$ is the edge set, and $\\mathcal{V}$ is the vertex set. \\YL{Denote by $(\\boldsymbol{\\omega}_{t,i}, \\theta_{t,i})$ one possible choice for the rotation axis and rotation angle that match $\\mathbf{R}_{t,i}$. $o_{t,i} \\in \\{+1, -1\\}$ specifies whether the rotation axis is flipped ($o_{t,i} = 1$ if the rotation axis is unchanged, and $-1$ if its opposite is used instead). }\\YL{The first term promotes spatial consistency while the second term promotes temporal consistency.} \n$s(\\cdot)$ is a function measuring orientation consistency, which is defined as follows:\n\\begin{equation}\n\ts(\\cdot)=\\left\\{\n\t\\begin{aligned}\n\t\t0 & , & |\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}|\\leq\\epsilon_1 \\; {\\rm or} \\;\n\t\t\\theta_{t,i}<\\varepsilon_2 \\; {\\rm or}  \\; \\theta_{t,j}<\\varepsilon_2 \\\\\n\t\t1 & , & {\\rm Otherwise~if}~\\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}>\\epsilon_1 \\\\\n\t\t-1 & , & {\\rm Otherwise~if}~ \\boldsymbol{\\omega}_{t,i} \\cdot \\boldsymbol{\\omega}_{t,j}<-\\epsilon_1 \\\\\n\t\\end{aligned}\n\t\\right.\n\\end{equation}\n\\YL{The first case here is to ignore cases where the rotation angle is near zero, as the rotation axis is not well defined in such cases.}\nAs for rotation angles, \\YL{we optimize the following}\n\\begin{flalign}\\label{eqn:angle}\n\\arg\\min_{r_{t,i}} &\\sum_{(i,j) \\in \\mathcal{E} } \\| (r_{t,i} \\cdot 2\\pi+{o}_{t,i}\\theta_{t,i}) - (r_{t,j}\\cdot 2\\pi+{o}_{t,j}\\theta_{t,j}) \\|_2^{2} &\\nonumber\\\\\n+ &\\sum_{i \\in \\mathcal{V} } \\| (r_{t,i} \\cdot 2\\pi+{o}_{t,i}\\theta_{t,i}) - (r_{t-1,i}\\cdot 2\\pi+{o}_{t,j}\\theta_{t-1,i}) \\|_2^{2} \\nonumber\\\\ \n{\\rm s.t.}& \\quad r_{t,i} \\in \\mathbb{Z},~~r_{t,1} = 0.\n\\end{flalign}\nwhere $r_{t,i} \\in \\mathbb{Z}$ specifies how many $2\\pi$ rotations should be added to the rotation angle.\n\\YL{The two terms here promote spatial and temporal consistencies of rotation angles, respectively. \nThese optimizations can be solved using integer programming, and we use the mixed integer solver CoMISo~\\cite{comiso2009} which provides an efficient \\gl{solver}. See~\\cite{gao2019sparse} for more details.}\nA similar process is used to compute the TS-ACAP representation of the fine meshes. \n\n\n\\cl{Compared to the ACAP representation, our TS-ACAP representation considers temporal constraints to represent nonlinear deformation for optimization of axes and angles, which is more suitable for consecutive large-scale deformation \\YL{sequences}.\nWe compare ACAP~\\cite{gao2019sparse} and our TS-ACAP using a simple example of a simulated disk-shaped cloth animation sequence. Once we obtain deformation representations of the meshes in the sequence, \nwe interpolate two meshes, the initial state mesh and a randomly selected frame, using linear interpolation of \\YL{shape representations}.\n\\YL{In Fig. \\ref{fig:interpolation}, we demonstrate the interpolation results with ACAP representation, which shows that it cannot handle such challenging cases with complex large-scale deformations. In contrast, with our temporally and spatially as-consistent-as-possible optimization, our TS-ACAP representation is able to produce consistent interpolation results.}\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{pictures\/acap_tacap1_1.pdf}%\n\t\\caption{\\small Comparison of shape interpolation results with different deformation representations, ACAP and TS-ACAP.  %\n\t(a) and (b) are the source (t = 0) and target (t = 1) models with large-scale deformation to be interpolated. \n\tThe first row shows the interpolation results by ACAP, and the second row show the results with our TS-ACAP. \n\t\\gl{The interpolated models with ACAP feature are plausible in each frame while they are not consistent in the temporal domain.}\n\t}\n\t\\label{fig:interpolation}\n\\end{figure}\n}\n\n\\subsection{DeformTransformer Networks}\nUnlike \\cite{tan2017variational, wang2019learning} which use fully connected layers for mesh encoder, we perform convolutions \\YL{on meshes to learn to extract useful features using compact shared convolutional kernels.} \nAs illustrated in Fig. \\ref{fig:pointconv}, we use a convolution operator on vertices  \\cite{duvenaud2015convolutional, tan2017autoencoder} where the output at a vertex is obtained as a linear combination of input in its one-ring neighbors along with a bias. \n\\YL{The input to our network is the TS-ACAP representation, which for the $i^{\\rm th}$ vertex of the $t^{\\rm th}$ mesh, we collect non-trivial coefficients from the rotation $\\mathbf{R}_{t, i}$ and scaling\/shearing $\\mathbf{S}_{t,i}$, which forms a 9-dimensional feature vector (see~\\cite{gao2019sparse} for more details). Denote by $\\mathbf{f}_i^{(k-1)}$ and $\\mathbf{f}_i^{k}$ the feature of the $i^{\\rm th}$ vertex at the $(k-1)^{\\rm th}$ and $k^{\\rm th}$ layers, respectively. The convolution operator is defined as follows:\n\\begin{equation}\n\t\\mathbf{f}_i^{(k)} =\n\t\\mathbf{W}_{point}^{(k)} \\cdot \\mathbf{f}_{i}^{(k-1)} + \n\t\\mathbf{W}_{neighbor}^{(k)} \\cdot \\frac{1}{D_i} \\mathop{\\sum}_{j=1}^{D_i} \\mathbf{f}_{n_{ij}}^{(k-1)}\n\t+ \\mathbf{b}^{(k)} \n\\end{equation}\nwhere $\\mathbf{W}_{point}^{(k)}$, $\\mathbf{W}_{neighbor}^{(k)}$ and $\\mathbf{b}^{(k)}$ are learnable parameters for the $k^{\\rm th}$ convoluational layer, $D_i$ is the degree of the $i^{\\rm th}$ vertex, $n_{ij}(1 \\leq j \\leq D_i )$ is the $j^{\\rm th}$ neighbor of the $i^{\\rm th}$ vertex.\n}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.48\\linewidth]{pictures\/pointconv.pdf} \n\t\\caption{\\small Illustration of the convolutional operator on meshes. \n\t\tThe result of convolution for each vertex is obtained by a linear combination from the input in the 1-ring neighbors of the vertex, along with a bias.\n\t}\n\t\\label{fig:pointconv}\n\\end{figure}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth, trim=0 50 0 150,clip]{pictures\/transformer.pdf} %\n\t\\caption{\\small The architecture of our DeformTransformer network.\n\t\tThe coarse and fine mesh sequences are embedded into feature vectors using the TS-ACAP representation which \\YL{is} defined \\YL{at} each vertex as a 9-dimensional vector. \n\t\tThen two convolutional \\YL{encoders} map coarse and fine features to \\YL{their latent spaces}, respectively.\n\t\tThese latent vectors are fed into the DeformTransformer network, \\cl{which consists of the encoder and decoder, each including a stack of $N=2$ identical blocks with 8-head attention,} to recover \\YL{temporally-coherent} deformations.\n\t\tNotice that in \\YL{the} training phase the input high-resolution TS-ACAP \\YL{features are those from the ground truth}, \n\t\t\\YL{but during testing, these features are initialized to zeros, and once a new high-resolution frame is generated, its TS-ACAP feature is added.}\n\t\tWith predicted feature vectors, realistic and stable cloth animations are generated.\n\t}\n\t\\label{fig:Transformer}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.4\\linewidth, trim=18 33 18 3,clip]{pictures\/tshirt06_08_poseswithhuman_collision\/temp0270keyshot_unsolve.png} \n\t\\includegraphics[width=0.4\\linewidth, trim=18 33 18 3,clip]{pictures\/tshirt06_08_poseswithhuman_collision\/temp0270keyshot_solve.png} \n\t\\caption{\\small For tight clothing, data-driven cloth deformations may suffer from apparent collisions with the body (left). We apply a simple postprocessing step to push \n\t\\YL{the collided} T-shirt vertices outside the body (right).\n\t}\n\t\\label{fig:collisionrefinement}\n\\end{figure}\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth, trim=50 150 100 150,clip]{pictures\/dataset.pdf} \n\t\\caption{\\small \n\t\tWe test our algorithm on 5 datasets including TSHIRT, PANTS, SKIRT, SHEET and DISK.\t\t \n\t\tThe former three are  garments (T-shirts, skirts, and pants) dressed on a template body and simulated with various motion sequences.\n\t\tThe SHEET dataset is a square sheet interacting with various obstacles.\n\t\tThe DISK dataset is a round tablecloth draping on a cylinder in the wind of various velocities. \n\t\tEach cloth shape has a coarse resolution (top) and a fine resolution (bottom). \n\t} \n\t\\label{fig:dataset}\n\\end{figure*}\nLet $\\mathcal{F}_\\mathcal{C} = \\{\\mathbf{f}_{\\mathcal{C}_1}, \\dots, \\mathbf{f}_{\\mathcal{C}_n}\\}$ be the sequence of coarse mesh features, and $\\mathcal{F}_\\mathcal{D} = \\{\\mathbf{f}_{\\mathcal{D}_1}, \\dots, \\mathbf{f}_{\\mathcal{D}_n}\\}$ be its counterpart, the sequence of detailed mesh features.\nTo synthesize $\\mathcal{F}_\\mathcal{D}$ from $\\mathcal{F}_\\mathcal{C}$, the DeformTransformer framework is proposed to solve this sequence-to-sequence problem.\nThe DeformTransformer network consists of several stacked encoder-decoder layers, \\YL{denoted} as $Enc(\\cdot)$ and $Dec(\\cdot)$. To take the order of the sequence into consideration, triangle positional embeddings \\cite{vaswani2017attention} are injected into frames of $\\mathcal{F}_\\mathcal{C}$ and $\\mathcal{F}_\\mathcal{D}$, respectively.\nThe encoder takes coarse mesh features as input and encodes it to a \\YL{temporally-dependent} hidden space.\nIt is composed of identical blocks \\YL{each} with two sub-modules, one is the multi-head self-attention mechanism, the other is the frame-wise fully connected feed-forward network. \nWe also employ a residual connection around these two sub-modules, followed \\YL{by} the layer normalization.\nThe multi-head attention is able to build the dependence between any frames, thus ensuring that each input can consider global context of the whole sequence. Meanwhile, compared with other sequence models, this mechanism splits \\YL{the} attention into several subspaces so that it can model the frame \\YL{relationships} in multiple aspects.\nWith the encoded latent vector $Enc(\\mathcal{F}_\\mathcal{C})$, the decoder network attempts to reconstruct a sequence of fine mesh features.\nThe decoder has two parts: \nThe first part takes fine mesh sequence $\\mathcal{F}_\\mathcal{D}$ as \\YL{input} and \nencodes it similar to the encoder. \n\\YL{Unlike the encoder, detailed meshes are generated sequentially, and when predicting frame $t$, it should not attend to subsequent frames (with the position after frame $t$). To achieve this, we utilize a masking process\nfor the self-attention module.} The second part performs multi-head attention over the output of the encoder, thus capturing the long-term dependence between coarse mesh features $\\mathcal{F}_\\mathcal{C}$ and fine mesh features $\\mathcal{F}_\\mathcal{D}$.\nWe train the Transformer network by minimizing the mean squared error between predicted detailed features and the ground-truth.\nWith predicted TS-ACAP feature vector, we reconstruct the vertex coordinates of \\YL{the} target mesh\\YL{, in the same way as reconstruction from ACAP features} (please refer to \\cite{gao2019sparse} for details).   \nOur training data is generated by PBS \\YL{and is collision-free}.\nSince human body \\YL{(or other obstacles)} information is unseen in our algorithm, it does not guarantee the predicted cloth \\YL{is free from any penetration}.\nEspecially for tight garment like T-shirts, it will be apparent if collision \\YL{between the garment and human body} happens.\nWe use a fast refinement method \\cite{wang2019learning} to push the cloth vertices colliding with the body outside \\YL{while} preserving the local wrinkle details (see Fig.~\\ref{fig:collisionrefinement}). \nFor each vertex detected inside the body, we find its closest point over the body surface with normal and position.\nThen the cloth mesh is deformed to update the vertices by minimizing the energy which penalizes the euclidean distance and Laplacian difference between the updated mesh and the initial one (please refer to \\cite{wang2019learning} for details).\nThe collision solving process usually takes less than 3 iterations to converge to a collision-free state.\n\n\\section{Implementation}\\label{sec:implementation}\nWe describe the details of the dataset construction and the network architecture in this section.\n\n\\textbf{\\YL{Datasets}.}\nTo test our method, we construct 5 datasets, called TSHIRT, PANTS, SKIRT, SHEET and DISK respectively.\nThe former three datasets are different types of garments, \\textit{i.e. }, T-shirts, skirts and pants worn on human bodies.\nEach type of garment \\YL{is represented by both low-resolution and  high-resolution meshes}, \\YL{containing} 246 and 14,190 vertices for the T-shirts, 219 and 12,336 vertices for the skirts, 200 and 11,967 vertices for the pants.\nGarments of the same type and resolution are simulated from a template mesh, which means \\YL{such meshes obtained  through cloth animations have the same number of vertices and the same connectivity}.\nThese garments are dressed on animated characters, which are obtained via driving a body \\YL{in the SMPL (Skinned Multi-Person Linear)  model} \\cite{loper2015smpl} with publicly available motion capture data from CMU \\cite{hodgins2015cmu}.\nSince the motion data is captured, there are some \\YL{self-collisions} or long repeated sequences.  \n\\YL{After removing poor quality data}, we select various motions, such as dancing, walking, running, jumping etc., including 20 sequences (\\YL{9031, 6134, 7680 frames in total} for TSHIRT, PANTS and SKIRT respectively).\nIn these motions, 18 sequences are randomly selected for training and the remaining 2 sequences for testing.\nThe SHEET dataset consists of a pole or a sphere of three different sizes crashing to a piece of \\YL{cloth sheet}.\nThe coarse mesh has 81 vertices and the fine mesh has 4,225 vertices.\nThere are \\YL{4,000} frames in the SHEET dataset, in which 3200 frames for training and \\YL{the remaining} 800 frames for testing.\nWe construct the DISK dataset by draping a round tablecloth to a cylinder in the wind, with 148 and 7,729 vertices for coarse and fine meshes respectively.\nWe adjust the velocity of the wind to get various animation sequences, in which 1600 frames for training and 400 frames for testing. \n\n\\begin{table*}[ht]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{ Statistics and timing (sec\/\\YL{frame}) of the testing examples including five types of \\YL{thin shell animations}.\n\t}\n\t\\label{table:runtime}\n\t\\centering\n\t\\begin{tabular}{cccccccccc}\n\t\t\\toprule[1.2pt] \n\t\tBenchmark & \\#verts & \\#verts & PBS & ours  & speedup & \\multicolumn{4}{c}{our components}              \\\\ \\cline{7-10} \n\t\t& LR      & HR      & HR    &       &         & coarse & TS-ACAP & synthesizing & refinement \\\\\n\t\t&         &         &         &       &         & sim.   & extraction & (GPU)        &            \\\\  \\hline \\hline\n\t\tTSHIRT    & 246     & 14,190   &  8.72    & 0.867 & \\textbf{10}     & 0.73   & 0.11      &  0.012 & 0.015\\\\\n\t\tPANTS     & 200     & 11,967   & 10.92  &0.904   & \\textbf{12}      & 0.80   & 0.078 & 0.013  & 0.013\\\\\n\t\tSKIRT     & 127     & 6,812 & 6.84 & 0.207  & \\textbf{33} & 0.081 & 0.10 & 0.014 & 0.012 \\\\  \n\t\tSHEET     & 81      & 4,225    & 2.48    & 0.157  & \\textbf{16} & 0.035  & 0.10 &   0.011      &  0.011 \\\\  \n\t\tDISK      & 148     & 7,729    & 4.93 & 0.139 & \\textbf{35}     & 0.078  & 0.041 &   0.012      &   0.008 \\\\  \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table*} \nTo prepare the above datasets, we generate both \\YL{low-resolution (LR)} and \\YL{high-resolution (HR)} cloth \\YL{animations} by PBS.\nThe initial state of the HR mesh is obtained by applying the Loop subdivision scheme \\cite{Thesis:Loop} to the coarse mesh and waiting for several seconds till stable.\nPrevious works \\cite{kavan11physics, zurdo2013wrinkles, chen2018synthesizing} usually constrain the high-resolution meshes by various tracking mechanisms to ensure that the coarse cloth \\YL{can be seen as} a low-resolution version of the fine cloth during the complete animation sequences.\nHowever, fine-scale wrinkle dynamics cannot be captured by this model, as wrinkles are defined quasistatically and limited to a \\YL{constrained} subspace.\nThus we \\YL{instead perform} PBS for the two resolution meshes \\emph{separately}, without any constraints between them.\nWe use a cloth simulation engine called ARCSim \\cite{Narain2012AAR} to produce all animation sequences of low- and high-resolution meshes with the same parameter setting. \nIn our experiment, we choose the Gray Interlock from a library of measured cloth materials \\cite{Wang2011DEM} as the material parameters for ARCSim simulation.\nSpecially for garments interacting with characters, to ensure collision-free, we manually put the coarse and fine garments on a template human body (in the T pose) and run the simulation to let the \\YL{clothing} relax. To this end, we define the initial state for all subsequent simulations.\nWe interpolate 15 frames between the T pose and the initial pose of each motion sequence, before applying the motion sequence, which is smoothed using a convolution operation.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfloat{\n\t\t\\includegraphics[width=0.5\\linewidth]{pictures\/hyper_inputframes-eps-converted-to.pdf} \n\t}\n\t\\subfloat{\n\t\t\\includegraphics[width=0.5\\linewidth]{pictures\/hyper_hiddensize-eps-converted-to.pdf} \n\t}\n\t\\caption{\\small Evaluation of hyperparameters in the Transformer network\\YL{, using the SKIRT dataset. }\n\t\t(Left) average error for the reconstructed results as a function of the number of input frames.\n\t\t(Right) error for the synthesized results under the condition of various dimensions of the latent layer.\n\t}\n\t\\label{fig:hyperpara}\n\\end{figure}\n\\textbf{Network architecture.}\nAs shown in Fig.~\\ref{fig:Transformer}, our transduction network consists of two components, namely convolutional \\YL{encoders} to map coarse and fine mesh sequences into latent spaces for improved generalization capability, and the Transformer network for \\YL{spatio-temporally} coherent deformation transduction.\nThe feature encoder module takes the 9-dimensional TS-ACAP features defined on vertices as input, followed by two convolutional layers with $tanh$ as the activation function. \nIn the last convolutional layer we abandon the activation function, similar to \\cite{tan2017autoencoder}.\nA fully connected layer is used to map the output of the convolutional layers into a 16-dimensional latent space.\nWe train one encoder for coarse \\YL{meshes} and another for fine \\YL{meshes} separately.\nFor the DeformTransformer network, its input includes the embedded latent vectors from both \\YL{the} coarse and fine domains.\nThe DeformTransformer network consists of sequential encoders and decoders, \neach \\YL{including} a stack of 2 identical blocks with 8-head attention.\nDifferent from variable length sequences used in natural language processing, we \\YL{fix} the number of input frames \\YL{(to 3 in our experiments)} since a motion sequence may include a thousand frames.\n\\YL{We perform experiments to evaluate the performance of our method with different settings.}\nAs shown in Fig.~\\ref{fig:hyperpara} \\YL{(left)}, using 3 input frames is found to perform well in our experiments.\nWe also evaluate the results generated with various dimensions of latent space shown in Fig. \\ref{fig:hyperpara} \\YL{(right)}.\nWhen the dimension of latent space is larger than 16, the network can \\YL{easily overfit}.\nThus we set the dimension of the latent space %\nto 16, which is sufficient for all the examples in the paper.\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{Quantitative comparison of reconstruction errors for unseen \\YL{cloth animations} in several datasets. We compare our results with Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} with LR meshes as a reference. \\YL{Three metrics, namely RMSE (Root Mean Squared Error), Hausdorff distance and STED (Spatio-Temporal Edge Difference)~\\cite{Vasa2011perception} are used. Since LR meshes have different number of vertices from the ground truth HR mesh, we only calculate its Hausdorff distance.}}\n \t\\label{table:compare_zurdo_chen2}\n\t\\centering \n\t\\begin{tabular}{ccccc} \n\t\t\\toprule[1.2pt]\n\t\t\\multirow{3}{*}{Dataset} & \\multirow{3}{*}{Methods} & \\multicolumn{3}{c}{Metrics} \\\\ \\cline{3-5}\n\t\t&                          & RMSE    & Hausdorff & STED  \\\\ \n\t\t&                          & $\\times 10^\\YL{-2}$ $\\downarrow$ & $\\times 10^\\YL{-2}$ $\\downarrow$ & $\\downarrow$ \\\\\n\t\t\\hline \\hline\n\t\t\\multirow{4}{*}{TSHIRT}   &  LR    & -          & 0.59     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 0.76   & 0.506    &  0.277     \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 1.04   & 0.480    &  0.281     \\\\ \\cline{2-5}\n\t\t&  Our                     & \\textbf{0.546}    &  \\textbf{0.416}   & \\textbf{0.0776}      \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{PANTS}   &  LR                   & -      &  0.761     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 1.82              & 1.09 & 0.176   \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 1.89                 & 0.983& 0.151    \\\\ \\cline{2-5}\n\t\t&  Our                    & \\textbf{0.663} & \\textbf{0.414}    & \\textbf{0.0420}      \\\\ \\hline \\hline\n\t\t\\multirow{4}{*}{SKIRT} &  LR                      & -      & 2.09     & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 1.93 &  1.31    & 0.562      \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 2.19 & 1.52     & 0.178      \\\\ \\cline{2-5}\n\t\t&  Our                    & \\textbf{0.685} & \\textbf{0.681}    &  \\textbf{0.0241}      \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{SHEET} \n\t\t&  LR                      & -       & 2.61 &   -   \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}  & 4.37  & 2.60 &   0.155 \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.} & 3.02  & 2.34 &   0.0672    \\\\ \\cline{2-5}\n\t\t&  Our                     & \\textbf{0.585} & \\textbf{0.417}   &  \\textbf{0.0262}     \\\\ \\hline  \\hline\n\t\t\\multirow{4}{*}{DISK} &  LR & -      & 3.12    & -     \\\\ \\cline{2-5}\n\t\t&  Chen {\\itshape et al.}   & 7.03   & 2.27    &  0.244     \\\\ \\cline{2-5}\n\t\t&  Zurdo {\\itshape et al.}  & 11.40  &  2.23           & 0.502           \\\\ \\cline{2-5}\n\t\t&  Our                      & \\textbf{2.16}   & \\textbf{1.30}   &  \\textbf{0.0557 }     \\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\n\\section{Results}\\label{sec:results}\n\\subsection{Runtime Performance}\nWe implement our method on a \\YL{computer with a} 2.50GHz \\YL{4-Core} Intel CPU for coarse simulation and TS-ACAP extraction,\nand \\YL{an} NVIDIA GeForce\\textsuperscript{\\textregistered}~GTX 1080Ti GPU for fine TS-ACAP generation by the network and mesh coordinate reconstruction.\nTable~\\ref{table:runtime} shows average per-frame execution time of our method for various cloth datasets.\nThe execution time contains four parts: coarse simulation, TS-ACAP extraction, high-resolution TS-ACAP synthesis, and collision refinement. \nFor reference, we also \\YL{measure} the time of a CPU-based implementation of  high-resolution PBS using ARCSim \\cite{Narain2012AAR}.\nOur algorithm is $10\\sim35$ times faster than the \\YL{PBS} HR simulation.\nThe low computational cost of our method makes it suitable for the interactive applications. \n\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/0crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/1crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/2crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/3crop0090down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/4crop0090down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/0crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/1crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/2crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/3crop0300down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt06_08_poses\/4crop0300down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/0crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/1crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/2crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/3crop0110down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/4crop0110down.png} \\\\\n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/0crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/1crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/2crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/3crop0260down.png} &  \n     \\includegraphics[width=\\linewidth,  trim=17 0 37 0,clip]{pictures\/tshirt08_11_poses\/4crop0260down.png} \\\\ \n      \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data \\YL{on the TSHIRT} dataset.\n\t\t(a) coarse simulation,\n\t\t(b) results of \\cite{chen2018synthesizing},\n\t\t(c) results of \\cite{zurdo2013wrinkles},\n\t\t(d) our results,\n\t\t(e) ground truth generated by PBS.\n\t\tOur method produces the detailed shapes of higher quality than Chen {\\itshape et al.} and Zurdo {\\itshape et al.}, see the folds and wrinkles in the close-ups. Chen {\\itshape et al.} results suffer from seam line problems. The results of Zurdo {\\itshape et al.} exhibit clearly noticeable artifacts.}\n\t\\label{fig:comparetoothers_tshirt}\n\\end{figure}\n \\begin{figure}[!htb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0010down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0010down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0060down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0060down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0140down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0140down.png} \\\\  \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/0crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/1crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/2crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/3crop0160down.png} & \n \t\\includegraphics[width=\\linewidth, trim=28 0 28 5,clip]{pictures\/pants09_07_poses\/4crop0160down.png} \\\\  \n\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular} \n\t\\caption{Comparison of the reconstruction results for unseen data in the PANTS dataset.\n\t\t(a) coarse simulation results,\n\t\t(b) results of \\cite{chen2018synthesizing}, mainly smooth the coarse meshes and barely exhibit any wrinkles.\n\t\t(c) results of \\cite{zurdo2013wrinkles}, have clear artifacts on examples where LR and HR meshes are not aligned well, \\textit{e.g. } the trouser legs.\n\t\t(d) our results, ensures physically-reliable results.\n\t\t(e) ground truth generated by PBS.\n\t}\n\t\\label{fig:comparetoothers_pants}\n\\end{figure}  \n\\begin{figure*}[htb]\n\t\\centering\n\t\\subfloat[Input]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0080_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0110_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0140_00_skirtlrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/0\/frm0160_00_skirtlrkeyshot.png} \n\t\\end{minipage}} \n\t\\subfloat[Chen {\\itshape et al.}]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0080keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0110keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0140keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/1\/temp0160keyshot.png} \n\t\\end{minipage}} \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0080_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0110_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0140_00_skirtlr_result.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45 ,clip]{pictures\/skirt09_06_posescolormap\/1\/09_06_posesfrm0160_00_skirtlr_result.png} \n\t\\end{minipage}\n\t\\subfloat[Zurdo {\\itshape et al.}]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0080keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0110keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0140keyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/2\/temp0160keyshot.png} \n\t\\end{minipage}} \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0080_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0110_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0140_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/2\/frm0160_00_skirthr.png} \n\t\\end{minipage}\n\t\\subfloat[Ours]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0080_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0110_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0140_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/3\/frm0160_00_skirthrkeyshot.png} \n\t\\end{minipage}}\n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0080_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0110_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0140_00_skirthr.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_posescolormap\/3\/frm0160_00_skirthr.png} \n\t\\end{minipage}  \n\t\\subfloat[GT]{ \n\t\t\\begin{minipage}[b]{0.11\\linewidth} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0080_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0110_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0140_00_skirthrkeyshot.png} \n\t\t\t\\includegraphics[width=1.000000\\linewidth, trim=45 45 45 45,clip]{pictures\/skirt09_06_poses\/4\/frm0160_00_skirthrkeyshot.png} \n\t\\end{minipage}}\n\t\\begin{minipage}[b]{0.08\\linewidth} \n\t\t\\includegraphics[width=1.000000\\linewidth, trim=0 0 0 0,clip]{pictures\/bar.png}\n\t\\end{minipage}\n\t\n\t\\caption{Comparison of the reconstruction results for unseen data in the SKIRT dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing},\n\t\t(c) the results of \\cite{zurdo2013wrinkles},\n\t\t(d) our results,\n\t\t(e) the ground truth generated by PBS.\n\tThe reconstruction accuracy is qualitatively showed as a difference map. \n\tReconstruction errors are color-coded and warmer colors indicate larger errors. Our method leads to significantly lower reconstruction errors. }\n\t\\label{fig:comparetoothers_skirt}\n\\end{figure*}\n\n\\subsection{\\YL{Fine Detail} Synthesis Results and Comparisons}\nWe now demonstrate our method using various \\YL{detail enhancement}\nexamples \\YL{both} quantitatively and qualitatively, \\YL{including added wrinkles and rich dynamics.}\nUsing detailed meshes generated by PBS as ground truth, we compare our results with physics-based coarse simulations, our implementation of a deep learning-based method \\cite{chen2018synthesizing} and a conventional machine learning-based method \\cite{zurdo2013wrinkles}.\n\nFor quantitative comparison, we use \\YL{three} metrics: Root Mean Squared Error (RMSE), Hausdorff distance as well as spatio-temporal edge difference (STED) \\cite{Vasa2011perception} designed for motion sequences with a focus on `perceptual' error of models.\nThe results are shown in Table~\\ref{table:compare_zurdo_chen2}.\nNote that \\YL{for the datasets from the top to bottom in the table,} the Hausdorff \\YL{distances} between LR meshes and the ground truth are increasing. \\YL{This} tendency is in accordance with the deformation range from tighter T-shirts and pants to skirts and square\/disk tablecloth with higher degrees \\YL{of freedom}.\nSince the vertex position representation cannot handle rotations well, the larger scale the models deform, the more artifacts Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} would \\YL{bring in} in the reconstructed models, \\YL{leading  to increased} RMSE and Hausdorff distances.  \nThe results indicate that our method has better reconstruction results \\YL{quantitatively} than the compared methods \\YL{on} the 5 datasets with \\YL{all the three} metrics.\nEspecially \\YL{for} the SKIRT, SHEET and DISK \\YL{datasets} which \\YL{contain} loose cloth \\YL{and hence larger and richer deformation}, our \\YL{method} outperforms \\YL{existing methods significantly} since tracking between coarse and fine meshes \\YL{is} not required in our algorithm.\n\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.01} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n     \\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0130down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0130down.png}\\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0180down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0180down.png} \\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0260down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0260down.png} \\\\ \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/0crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/1crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/2crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/3crop0320down.png} & \n \t\\includegraphics[width=\\linewidth, trim=25 0 50 6,clip]{pictures\/crashballpole0.3withhuman\/4crop0320down.png} \\\\ \n\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data in the SHEET dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing}, with inaccurate and\nrough wrinkles different from the GT.\n\t\t(c) the results of \\cite{zurdo2013wrinkles}, show similar global shapes to coarse meshes with some wrinkles and unexpected sharp corner.\n\t\t(d) our results, show mid-scale wrinkles and similar global deformation as GT.\n\t\t(e) the ground truth generated by PBS.}\n\t\\label{fig:comparetoothers_crashball}\n\t\\vspace{-0.2cm}\n\\end{figure}   \n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.001} \n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}>{\\centering\\arraybackslash}m{0.2\\linewidth}} \n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/0crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/1crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/2crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/3crop0050down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=42 0 30 30,clip]{pictures\/disk4.300withhuman\/4crop0050down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0090down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0090down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0160down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0160down.png} \\\\\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/0crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/1crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/2crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/3crop0360down.png} &\n\t\t\t \\includegraphics[width=\\linewidth, trim=54 0 18 30,clip]{pictures\/disk4.300withhuman\/4crop0360down.png} \\\\ \n\t\t\t \\vspace{0.3cm} \\footnotesize (a) Input & \\vspace{0.3cm} \\hspace{-0.3cm} \\footnotesize (b) Chen {\\itshape et al.} & \\vspace{0.3cm} \\hspace{-0.2cm} \\footnotesize (c) Zurdo {\\itshape et al.} & \\vspace{0.3cm} \\footnotesize (d) Ours & \\vspace{0.3cm} \\footnotesize (e) GT  \n\t\\end{tabular}\n\t\\caption{Comparison of the reconstruction results for unseen data in the DISK dataset.\n\t\t(a) the coarse simulation,\n\t\t(b) the results of \\cite{chen2018synthesizing}, cannot reconstruct credible shapes. \n\t\t(c) the results of \\cite{zurdo2013wrinkles}, show apparent artifacts near the flying tails since no tracking constraints applied.\n\t\t(d) our results, reproduce large-scale deformations, see the tail of the disk flies like a fan in the wind.\n\t\t(e) the ground truth generated by PBS.}\n\t\\label{fig:comparetoothers_disk}\n\\end{figure}    \n\n\\YL{We further make qualitative comparisons on the 5 datasets.}\nFig. \\ref{fig:comparetoothers_tshirt} shows \\YL{detail synthesis results} on the TSHIRT dataset.\nThe first and second \nrows \nare from \\YL{sequence} 06\\_08, a woman dribbling the basketball sideways and the \\YL{last two rows} are from \\YL{sequence} 08\\_11, a walking woman.\nIn this dataset of tight t-shirts on human bodies, Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} and our method are able to reconstruct the garment model completely with mid-scale wrinkles.\nHowever, Chen {\\itshape et al.} \\cite{chen2018synthesizing} suffer from the seam line problems due to \\YL{the use of geometry image representation}. \nA geometry image is a parametric sampling of the shape, which is \\YL{made a topological disk by cutting through some seams.} \nThe boundary of the disk needs to be fused so that the reconstructed mesh has the original topology.\n\\YL{The super-resolved geometry image corresponding to high-resolution cloth animations are not entirely accurate, and as a result the fused boundaries no longer match exactly, }\n\\textit{e.g. } clear seam lines on the shoulder and crooked boundaries on the left side of the waist \\YL{for the examples} in Fig.~\\ref{fig:comparetoothers_tshirt} (b)),\n\\YL{while} our method \\YL{produces} better results than \\cite{chen2018synthesizing} and \\cite{zurdo2013wrinkles} which have  \\YL{artifacts of unsmooth surfaces}.\n\nFig. \\ref{fig:comparetoothers_pants} shows comparative results of the animations of pants on a fixed body shape while changing the body pose over time. \nThe results of \\cite{chen2018synthesizing} \\YL{mainly} smooth the coarse meshes and barely exhibit \\YL{any} wrinkles.\nZurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} utilize tracking algorithms to ensure the %\n\\YL{close alignment}\nbetween coarse and fine meshes, and thus the fine meshes are constrained \\YL{and do not exhibit the behavior of full physics-based simulation.}\n\\YL{So on the PANTS dataset,} the results of \\cite{zurdo2013wrinkles} have clear artifacts on examples \\YL{where} LR and HR meshes are not aligned well, \\textit{e.g. } the trouser legs.\nDifferent from the two compared methods \\YL{that reconstruct displacements} or local coordinates, \nour method \\YL{uses} deformation-based features in both encoding and decoding \\YL{phases} which \\YL{does not suffer from such restrictions and ensures physically-reliable results.}\n\nFor looser garments like \\YL{skirts}, we show comparison results in Fig. \\ref{fig:comparetoothers_skirt}, with color coding to highlight the differences between synthesized results and the ground truth.\nOur method successfully reconstructs the swinging skirt \\YL{caused by} the body motion (see the small wrinkles on the waist and the \\YL{medium-level} folds on the skirt \\YL{hem}).\nChen {\\itshape et al.} are able to reconstruct the overall shape of the skirt, however there are many small unsmooth \\YL{triangles leading to noisy shapes}\ndue to the 3D coordinate representation with untracked fine meshes with abundant wrinkles.\nThis leads to unstable animation, please see the accompanying video.\nThe results of \\cite{zurdo2013wrinkles} have some problems of the global deformation, see the directions of the skirt hem and the large highlighted area in the color map.\nOur learned \\YL{detail} synthesis model provides better visual quality for shape generation \\YL{and the generated results look} closer to the ground truth.\n \nInstead of garments dressed on human bodies, we additionally show some results of free-flying tablecloth. \nThe comparison of the testing results \\YL{on} the SHEET dataset are shown in Fig.~\\ref{fig:comparetoothers_crashball}.\nThe results of \\cite{chen2018synthesizing} show inaccurate and rough wrinkles different from the ground truth. \nFor hanging sheets in the results of \\cite{zurdo2013wrinkles}, the global shapes are more like coarse \\YL{meshes} with some wrinkles and unexpected sharp corners, \\textit{e.g. } the left side in the last row of Fig. \\ref{fig:comparetoothers_crashball} (c),\nwhile ours show \\YL{mid-scale} wrinkles and similar global deformation \\YL{as} the high-resolution meshes. \n\nAs for the DISK dataset, from the visual results in Fig.~\\ref{fig:comparetoothers_disk}, we can see that Chen {\\itshape et al.} \\cite{chen2018synthesizing} and Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} cannot handle large-scale rotations well and cannot reconstruct credible shapes in such cases. \n\\gl{Especially for Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles}, the impact of tracking is significant for their algorithm.}\nThey can reconstruct the top\nand part of tablecloth near the cylinder, but the flying tails have apparent artifacts.  \nOur algorithm does not have such drawbacks.\nNotice how our method successfully reproduces ground-truth deformations, including the overall drape (\\textit{i.e. }, how the tail of the disk flies like a fan in the wind) and mid-scale wrinkles.\n\n\\begin{table}[!htb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{User study results on cloth \\YL{detail} synthesis. We show the average ranking score of the three methods: Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles}, and ours. The\n\t\tranking ranges from 1 (the best) to 3 (the worst). The results are calculated\n\t\tbased on 320 trials. We see that our method achieves the best in terms of\n\t\twrinkles, temporal stability \\YL{and overall quality}.}\n\t\\label{table:userstudy}\n\t\\centering \n\t\\begin{tabular}{cccc}\n\t\t\\toprule[1.2pt]  \n\t\tMethod                    & Wrinkles   & Temporal stability & Overall   \\\\ \\hline \n\t\tChen {\\itshape et al.}    & 2.184 & 2.1258 &2.1319\\\\ \\hline  \n\t\tZurdo {\\itshape et al.}   & 2.3742 & 2.5215 & 2.4877\\\\ \\hline         \n\t\tOurs                      & \\textbf{1.4417} & \\textbf{1.3528} & \\textbf{1.3804}  \\\\\n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\\gl{We further conduct a user study to evaluate the stability and realistic of the synthesized dense mesh dynamics. 32 volunteers are involved for this user study.}\nFor every question, we give one sequence and 5 images of coarse meshes as references, \\YL{and} then let the user rank the corresponding outputs from  Chen {\\itshape et al.} \\cite{chen2018synthesizing}, Zurdo {\\itshape et al.} \\cite{zurdo2013wrinkles} and ours according to three different criteria (wrinkles, temporal stability and overall). \nWe shuffle the order of the algorithms each time we exhibit the question and show shapes from the three methods randomly \\YL{to avoid bias}. \nWe show the results of the user study in Table \\ref{table:userstudy}, where we observe that our generated \\YL{shapes} perform the best on all three criteria. \n\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{Per-vertex error (RMSE) on synthesized shapes with different feature representations: 3D coordinates, ACAP and TS-ACAP.}\n\t\\label{table:feature_compare}\n\t\\centering\n\t\\begin{tabular}{cccccc}\n\t\t\\toprule[1.2pt]\n\t\tDataset        & TSHIRT      & PANTS    &  \tSKIRT   & SHEET & DISK      \\\\ \\hline\n\t\t3D coordinates &  0.0101      & 0.0193   &    0.00941 &  0.00860   &  0.185 \\\\ \\hline\n\t\tACAP           &  0.00614     & 0.00785  &   0.00693  &  0.00606   &   0.0351     \\\\ \\hline\n\t\tTS-ACAP          &  \\textbf{0.00546}     & \\textbf{0.00663}  &    \\textbf{0.00685} &   \\textbf{0.00585} & \\textbf{0.0216}\\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table}\n\\begin{figure}[tb]\n\t\\centering\n\t\\setlength{\\fboxrule}{0.5pt}\n    \\setlength{\\fboxsep}{-0.01cm}\n\t\\setlength{\\tabcolsep}{0.00cm}  \n    \\renewcommand\\arraystretch{0.001}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}>{\\centering\\arraybackslash}m{0.25\\linewidth}}\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0040.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0040.png} \\\\\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0075.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0075.png} \\\\\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/0\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/1\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/2\/crop0110.png} &\n \t\\includegraphics[width=1.000000\\linewidth, trim=63 0 0 0,clip]{pictures\/skirt09_07_poseswithhuman\/3\/crop0110.png} \\\\ \n \t\\vspace{0.3cm} \\small (a) Input & \\vspace{0.3cm}\\small (b) Coordinates & \\vspace{0.3cm}\\small (c) Ours & \\vspace{0.3cm}\\small (d) GT\n  \\end{tabular} \n\t\\caption{The evaluation of the TS-ACAP feature in our detail synthesis method. \n\t\t(a) input coarse \\YL{shapes},\n\t\t(b) the results using 3D coordinates, which can be clearly seen the rough appearance, unnatural deformation and some artifacts, especially in the highlighted regions with details shown in the close-ups.\n\t\t(c) our results, which show smooth looks and the details are more similar to the GT.\n\t\t(d)\tground truth.\n\t\t }\n\t\\label{fig:ablationstudy_coordiniates_skirt}\n\\end{figure}\n\\begin{figure}[htb]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.05cm}  \n    \\renewcommand\\arraystretch{0.001}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.02\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}>{\\centering\\arraybackslash}m{0.31\\linewidth}}\n \t \\rotatebox{90}{\\small ACAP} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0103.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0104.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/0\/crop0105.png} \\\\\n \t\\rotatebox{90}{\\small TS-ACAP}  &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0103.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0104.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=90 0 0 60,clip]{pictures\/tacap_acap\/1\/crop0105.png} \\\\  \n     \\vspace{0.3cm}    & \\vspace{0.3cm}  \\small $t = 103$ & \\vspace{0.3cm} \\small $t = 104$  & \\vspace{0.3cm} \\small $t = 105$ \n\t\\end{tabular} \n\t\\caption{\n\t\t Three consecutive frames from a testing sequence in the DISK dataset. First row: the results of ACAP. As shown in the second column, the enlarged wrinkles are different from the previous and the next frames.\n\t\t This causes jumping in the animation.\n\t\t Second row: the consistent results obtained via TS-ACAP feature, demonstrating that our TS-ACAP representation ensures the temporal coherence. \n\t}\n\t\\label{fig:jump_acap}\n\\end{figure}\n\\begin{table}[tb]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\fontsize{7.5}{9}\\selectfont\n\t\\caption{Comparison of RMSE between synthesized shapes and ground truth with different networks, \\textit{i.e. } without temporal modules, with RNN, with LSTM and ours with the Transformer network.}\n\t\\label{table:transformer_compare}\n\t\\centering\n\t\\begin{tabular}{cccccc}\n\t\t\\toprule[1.2pt]\n\t\tDataset        & TSHIRT      & PANTS    &  \tSKIRT   & SHEET       & DISK \\\\ \\hline\n\t\tWO Transformer &  0.00909     & 0.01142   &    0.00831 &   0.00739    &  0.0427    \\\\ \\hline\n\t\tWith RNN  &  0.0435      &  0.0357   & 0.0558     &   0.0273     & 0.157      \\\\ \\hline\n\t\tWith LSTM  &  0.0351      &  0.0218   &  0.0451    &  0.0114      &  0.102     \\\\ \\hline\n\t\tWith Transformer         &  \\textbf{0.00546}     & \\textbf{0.00663}  &    \\textbf{0.00685} &   \\textbf{0.00585} & \\textbf{0.0216} \\\\ \n\t\t\\bottomrule[1.2pt]\n\t\\end{tabular}\n\\end{table} \n\\begin{figure}[tb]\n \t\\centering\n    \\setlength{\\tabcolsep}{0.0cm}  \n    \\renewcommand\\arraystretch{-1.9}\n \t\\begin{tabular}{>{\\centering\\arraybackslash}m{0.08\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}>{\\centering\\arraybackslash}m{0.18\\linewidth}}\n \t\t\\rotatebox{90}{\\small (a) Input}& \n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0008.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0016.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0022.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0094.png} &\n\t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/0\/0200.png}   \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (b) EncDec} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0008.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0016.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0022.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0094.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/5\/0200.png} \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (c) RNN} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0008.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0016.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0022.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0094.png} &\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/rnn\/0200.png} \n\t \t\\\\\n\t \t\\rotatebox{90}{\\small (d) LSTM}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0008.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0016.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0022.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0094.png}&\n\t \t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/lstm\/0200.png} \n \t\t\\\\\n \t\t \\rotatebox{90}{\\small (e) Ours}& \n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0008.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0016.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0022.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0094.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/3\/0200.png} \n \t\t\\\\ \n \t\t  \\rotatebox{90}{\\small (f) GT}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0008.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0016.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0022.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0094.png}&\n \t\t\\includegraphics[width=\\linewidth, trim=5 5 5 5,clip]{pictures\/tshirt06_08_poses\/4\/0200.png} \n \t\\end{tabular} \n \t\\caption{The evaluation of the Transformer network in our model for wrinkle synthesis.\n \t\tFrom top to bottom we show (a) %\n \t\t\\gl{input coarse mesh with physical simulation}\n \t\t(b) the results with an encoder-decoder \\YL{dropping out temporal modules}, (c) the results with RNN \\cite{chung2014empirical}, (d) the results with LSTM \\cite{hochreiter1997long}, (e) ours, and (f) the ground truth generated by PBS.}\n \t\\label{fig:transformer_w_o_tshirt}\n \\end{figure}   \n\n\\subsection{\\YL{Evaluation of} Network Components}\nWe evaluate the effectiveness of our network components for two aspects: the \\YL{capability} of the TS-ACAP feature and the \\YL{capability} of the Transformer network. \nWe evaluate our method qualitatively and quantitatively on different datasets.\n\n\\textbf{Feature Representation Evaluation}.\nTo verify the effectiveness of our TS-ACAP feature, we compare per-vertex position errors to other features to evaluate the generated shapes in different datasets quantitatively. \nWe compare our method using TS-ACAP feature with our transduction methods using 3D vertex coordinates and ACAP, with network layers and parameters adjusted accordingly to optimize performance alternatively.\nThe details of numerical comparison are shown in Table \\ref{table:feature_compare}.\nACAP and TS-ACAP show quantitative improvements than 3D coordinates.  \nIn Fig. \\ref{fig:ablationstudy_coordiniates_skirt}, we exhibit several compared examples of animated skirts of coordinates and TS-ACAP. \n\\YL{The results using coordinates show rough appearance, unnatural deformation and some artifacts, \n I can't really see the two circles?\nespecially in the highlighted regions with details shown in the close-ups.} Our results with TS-ACAP are more similar to the ground truth than the ones with coordinates. \nACAP has the problem of temporal inconsistency, thus the results are shaking or jumping frequently. \n\\YL{Although the use of the Transformer network can somewhat mitigate this issue, such artifacts can appear even with the Transformer.}\n\\YL{Fig.~\\ref{fig:jump_acap} shows} three consecutive frames from a testing sequence in the DISK dataset.\nResults with TS-ACAP show more consistent wrinkles than the ones with ACAP thanks to the temporal constraints.\n\n\\textbf{Transformer Network Evaluation}.\nWe also evaluate the impact of the Transformer network in our pipeline. \nWe compare our method to an encoder-decoder network dropping out the temporal modules, our pipeline with the recurrent neural network (RNN) and with the long short-term memory (LSTM) \\YL{module}.\nAn example of T-shirts is given in Fig. \\ref{fig:transformer_w_o_tshirt}, \\YL{showing} 5 frames in order.\nThe results without any temporal modules show artifacts on the sleeves and neckline since these places have strenuous \\YL{forces}. %\nThe models using RNN and LSTM stabilize the sequence via eliminating dynamic and detailed deformation, but all the results keep wrinkles on the chest from the initial state\\YL{, lacking rich dynamics.}\nBesides, they are not able to generate stable and realistic garment animations \\YL{that look similar to} the ground truth,\n\\YL{while} \\YL{our} method with the Transformer network \\YL{apparently} improves the temporary stability, \\YL{producing results close to the ground truth.}\nWe also quantitatively evaluate the performance of the Transformer network \\YL{in our method} via per-vertex error. \nAs shown in Table \\ref{table:transformer_compare}, the RMSE of our model \\YL{is} smaller than the other models.\n\n\\section{Conclusion and Future Work}\\label{sec:conclusion}\nIn this paper, we introduce a novel algorithm for synthesizing robust and realistic cloth animations via deep learning.\nTo achieve this, we propose a geometric deformation representation named TS-ACAP which well embeds the details and ensures the temporal consistency.\n\\YL{Benefiting} from \\YL{the} deformation-based feature, there is no explicit requirement of tracking between coarse and fine meshes in our algorithm. \nWe also use the Transformer network based on attention mechanisms to map the coarse TS-ACAP to fine TS-ACAP, maintaining the stability of our generation.\nQuantitative and qualitative results reveal that our method can synthesize realistic-looking wrinkles in various datasets, such as draping tablecloth, tight or \\YL{loose} garments dressed on human bodies, etc. \n \nSince our algorithm synthesizes \\YL{details} based on the coarse meshes, the time for coarse simulation is unavoidable.\nEspecially for tight garments like T-shirts and pants, the collision solving phase is time-consuming.\nIn the future, we intend to generate coarse sequences for tight cloth via skinning-based methods in order to reduce the computation for our pipeline.\nAnother limitation is that our current network is not able to deal with all kinds of garments with different topology.\n\\newpage\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHurricanes (also known as tropical cyclones or typhoons) are low-pressure weather systems with well-organized convection, which are among some of the most destructive disasters on Earth because they can induce strong winds, a rapid rise in local sea level, and heavy precipitation (Anthes \\cite{Anthes}, Emanuel \\cite{Emanuelb}). Hurricanes can also enhance the vertical mixing of heat and nutrients in the ocean, increase horizontal oceanic heat transport, and, subsequently, influence global climate (Emanuel \\cite{Emanuela}, Jansen \\&  Ferrari \\cite{Jansen}, Li \\& Sriver \\cite{LiH}). For instance, the curl of strong hurricane winds can cause divergence and convergence in the upper ocean, producing regions of up-welling and down-welling, enhancing the exchanges between surface and subsurface oceans. Therefore, it is an important and interesting issue to consider whether hurricanes can form on other potentially habitable planets beyond Earth.\n\nIn this study, we focus on tidally locked terrestrial planets around M dwarfs due to their relatively large planet-to-star ratios and frequent transits based on observations. These planets differ from Earth in three major aspects: the uneven distribution of stellar energy between the permanent day and night sides, the slow rotation rate due to strong tidal force, and the redder stellar spectrum than the Sun. Several atmospheric general circulation models (AGCMs) have been employed and modified to simulate and understand the atmospheric and climatic dynamics of the planets  (Joshi et al. \\cite{Joshi}, Merlis \\& Schneider \\cite{Merlis}, Edson et al. \\cite{Edson}, Pierrehumbert \\cite{Pierrehumberta}, Wordsworth et al. \\cite{Wordsworthetal}, Leconte et al. \\cite{Lecontea}, Menou \\cite{Menou}, Showman et al. \\cite{Showmanb}, Carone et al. \\cite{Carone}, Wordsworth \\cite{Wordsworth}, Shields et al. \\cite{Shields}, Turbet et al. \\cite{Turbetetal}, Boutle et al. \\cite{Boutle}, Checlair et al. \\cite{Checlair}, Haqq-Misra et al. \\cite{Haqq-Misra}, Kopparapu et al. \\cite{Kopparapu}, Noda et al. \\cite{Noda}, Wolf \\cite{Wolf2017}, Turbet et al. \\cite{Turbet}, Del Genio et al. \\cite{Del Genio}, Pierrehumbert \\& Hammond \\cite{Pierrehumbertb}, Yang et al. \\cite{Yang}). \n\nThese AGCMs have horizontal resolutions that are always equal to or larger than 300 km, so that most of their studies focus on planetary-scale phenomena, such as global-scale Walker circulation, equatorial superrotation, and forced Rossby and Kelvin waves, and their models are not able to properly simulate the characteristic features of hurricanes, such as the warm core, the eye-eyewall structure, and the spiral rain bands. No work has investigated synoptic phenomena apart from Bin et al. (\\cite{Bin}). Based on the output data of an AGCM, the authors estimated genesis potential index of hurricanes and showed that the probability of hurricane formation is low for planets in the middle range of the habitable zone of M dwarfs. However, the model resolution they used was not directly capable of simulating hurricanes and the question of whether the empirical index could be applied to exoplanets could not be answered. Moreover, they considered planets in the middle range of the habitable zone only. Here, we show that the possibility of hurricane formation increases with temperature and for planets with higher temperatures (closer to the inner edge of the habitable zone), the possibility is greater.\n\nIn this study, we explicitly simulate hurricane formation on tidally locked terrestrial planets with a high-resolution ($\\approx$50~km) AGCM. The structure of the paper is as follows. Section~2 describes our methods, Section~3 presents our results, and Section~4 gives our conclusions. \n\n\n\n\n\\section{Model description and experimental design}\nFor our model, we used the global Community Atmosphere Model version 4 (CAM4) with a dynamical core of finite volume (Neale et al. \\cite{Neale}). Deep convection was parameterized using the updated mass flux scheme of Zhang and McFarlane (\\cite{ZhangG}). Subgrid-scale momentum transport associated with convection was included (Richter \\& Rasch \\cite{Richter}). The parameterization of shallow moist convection is based on Hack (\\cite{Hack}). Condensation, evaporation, and precipitation parameterization is based on Zhang et al. (\\cite{ZhangM}) and Rasch and Kristjansson (\\cite{Rasch}). Cloud fraction is diagnosed from atmospheric stratification, convective mass flux, and relative humidity (Slingo \\cite{Slingo}, Hack et al. \\cite{Hack1993}, Kiehl et al. \\cite{Kiehl}, Rasch \\& Kristjansson \\cite{Rasch}). The realistic radiative transfer of water vapor, clouds, greenhouse gases, and aerosols are included as well (Ramanathan \\& Downey \\cite{Ramanathan}, Briegleb \\cite{Briegleb}, Collins et al. \\cite{Collins}, Neale et al. \\cite{Neale} ). \n\n\n\nThe horizontal resolution we employed is 0.47$^{\\circ}$\\,$\\times$\\,0.63$^{\\circ}$ in latitude and longitude, respectively. The number of vertical levels is 26. The planetary surface is covered by seawater throughout (namley, an aquaplanet). Because of the high resolution and limited computational power, we specify surface temperature (T$_S$) in the simulations. With a fixed T$_S$, the atmosphere reaches an equilibrium state within several years. If the model were coupled to a 50-m slab ocean, the surface and atmosphere would require tens of years to reach the equilibrium state, which is about one order of magnitude longer than that in the simulations with fixed T$_S$. The thermal inertia of the slab ocean is much larger than that of the atmosphere. If the model were coupled to a fully dynamical ocean with a depth of, for example, 3000 m, the model will require thousands of years to reach the equilibrium state due to the high thermal inertia and the slow motion of the deep ocean. For simulating hurricanes, a fixed surface temperature experiment is a good start and a useful method for understanding the formation and the properties of hurricanes, as found in hurricane simulations on Earth, carried out, for example, in the studies of Held and Zhao (\\cite{Held}) and Khairoutdinov and Emanuel (\\cite{Khairoutdinov}) and the recent review papers of Emanuel (\\cite{Emanuelb}) and Merlis and Held (\\cite{Merlis19}). The fixed-temperature surface acts as a boundary condition for the atmosphere system. Under a fixed T$_S$, the surface and the top of the atmosphere are not in energy balance, but the atmosphere itself is in energy balance; this is because the energy deficit or excess at the surface is approximately equal to that at the top of the atmosphere. In the coupled slab ocean experiment (shown in Sect. 4 below), the surface, the top of the atmosphere, and the atmosphere are all in energy balance.\n\n\nThe surface temperature is set according to previous simulations of lower resolution AGCMs coupled to a 50-m slab ocean (Yang et al. \\cite{Yang13}, Wolf et al. \\cite{Wolf}). On the day side, the surface temperature is a function of latitude and longitude: $(T_{max}-T_{min})cos(\\varphi)cos(\\lambda)+T_{min}$, or $(T_{max}-T_{min}){cos}^{1\/4}(\\varphi){cos}^{1\/4}(\\lambda)+T_{min}$, where $T_{max}$ is the maximum surface temperature, $T_{min}$ is the minimum surface temperature, $\\varphi$ is the latitude, and $\\lambda$ is the longitude. On the night side, the surface temperature is uniform with a value of $T_{min}$. Three groups of $T_{max}$ and $T_{min}$ are used. One is for planets near the inner edge of the habitable zone, 315 K \\& 310 K. The other two are for planets in the middle range of the habitable zone, 308 K \\& 275 K and 301 K \\& 268 K; the power of 1\/4 is used due to the very weak temperature gradients in the substellar region and strong temperature gradients near the terminators.\n\n\nThe planetary rotation period is set equal to the orbital period. Four rotation periods are examined: 6, 10, 20, and 40 Earth days. For other types of spin-orbit resonance, such as 3:2 as in the case of Mercury, the climate lies between the synchronous rotation and the rapid rotation of Earth (Yang et al. \\cite{Yang14}) and we have not carried out these kinds of experiments to date. Planetary radius and gravity are set to be the same as Earth, but both obliquity and eccentricity are set to zero. Stellar temperature is set to 2,600 or 3,700 K. The stellar radiation at the substellar point is set to 1,300 or 1,800 W m$^{-2}$. By default, the mean surface pressure is 1.0 bar with $\\approx$79\\% N$_2$ and $\\approx$21\\% O$_2$. For greenhouse gases, we set the CO$_2$ concentration to 367 parts per million by volume (ppmv), N$_2$O to 316 parts per billion by volume (ppbv), and CH$_4$ to 1760 ppbv. The ozone concentration is set to be the same as present-day Earth, which may influence the outflow temperature of hurricanes and the overshooting of extremely strong convection.\n\n\nIn order to briefly test the effect of atmospheric composition on hurricane formation, we did several ideal experiments in which the background gas is set to H$_2$, He, N$_2$, O$_2$, and CO$_2$, respectively. The corresponding mean molecular weights are 2.02, 4.00, 28.01, 31.99, and 44.00 g mole$^{-1}$ and the corresponding specific heats (Zhang \\& Showman \\cite{ZhangX}) are 28.9, 20.8, 29.1, 29.5, and 37.2 J mole$^{-1}$ K$^{-1}$. We modify these two constants only. The model we employed is incapable of calculating the radiative transfer of dense H$_2$, He, O$_2$, or CO$_2;$  meanwhile, surface temperatures under background gases that differ from Earth have not been seriously examined. Thus, we chose to use the globally uniform surface temperature (301 K) and stellar radiation (340 W m$^{-2}$) with neither a seasonal nor diurnal cycle. This idealized thermal boundary condition is unrealistic, but it is capable of avoiding the effects of any strong wind shear, the baroclinic zone, or other features that may inhibit hurricane formation or propagation (Merlis \\& Held \\cite{Merlis19}). We used two planetary rotation periods:\\ of one and of three Earth days.\n\n\n\nThe initial states of the experiments were based on long-term (of 40 Earth years) simulations using a lower resolution of 4$^{\\circ}$$\\times$5$^{\\circ}$ or 1.9$^{\\circ}$$\\times$2.5$^{\\circ}$ under the same experimental designs and parameterization schemes. Then each experiment was run for five Earth years under high resolution and the last four years were used to carry out the analysis, presented below.\n\n\nHurricane formation and tracking is based on six hourly model output variables using the Geophysical Fluid Dynamics Laboratory tracking algorithm (Zhao et al. \\cite{Zhao}). Candidate hurricanes are identified by finding regions that satisfy the following criteria: 1) the local 850-hPa relative vorticity maximum exceeds 3.5$\\times$10$^{-5}$ s$^{-1}$; 2) the 850-hPa warm-core temperature must be at least 0.5 K warmer than the surrounding local mean; 3) the distance between the local sea level pressure minimum and the vorticity maximum should be within a distance of 2$^{\\circ}$ latitude or longitude and so, this should also be the distance between the local sea level pressure minimum and the warm-core center; 4) the maximum 850-hPa wind speed exceeds $\\approx$33 m s$^{-1}$ at some point. These values for the thresholds impact the exact number of detected hurricanes but they do not affect the main conclusions of this study.\n\n\\begin{figure*\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.9\\textwidth]{fig1.PNG}\n\\caption{Snapshots of hurricanes on a tidally locked aqua-planet near the inner edge of the habitable zone in the control experiment. From left to right, the variables are: instantaneous wind speed at 850 hPa, surface air pressure, precipitation, and the vertical component of relative vorticity at 850 hPa, respectively. From upper to bottom, they are for three different moments. The four hurricanes are marked with black circles over the wind speed panels. The black cross is the substellar point in this figure and hereafter. See the supplementary video online for a visualisation of the evolution of the hurricanes.}\n\\label{fig1}\n\\end{figure*}\n\nFor Earth, a useful method for estimating the possibility of hurricane formation is the genesis potential index (GPI; Emanuel \\& Nolan \\cite{Emanuelc}), which is written as:\n\\begin{equation}\nGPI = |10^5 (\\zeta + f)|^{3\/2}(RH\/50)^3(V_{pot}\/70)^3(1.0+0.1V_{shear})^{-2},\n\\end{equation}\nwhere $\\zeta$ is the vertical component of relative vorticity, $f$ is the planetary vorticity, $RH$ is the relative humidity at the middle troposphere (600 hPa), $V_{shear}$ is the wind shear of horizontal winds between the upper and lower troposphere (300 minus 850 hPa; or called vertical wind shear), and $V_{pot}$ is potential intensity. The $V_{pot}$ is a measure of the maximum near-surface wind that can be maintained by hurricane under given environmental conditions. We note that these parameters are not entirely independent; for instance, vertical wind shear can influence relative humidity. The value of $V_{pot}$ is calculated based on a local balance between thermal energy import and mechanical energy dissipation (Emanuel \\cite{Emanuel}, Bister \\& Emanuel \\cite{Bister}), written as\n\n\\begin{equation}\n    V_{pot}^2 = \\frac{C_k}{C_D}\\frac{T_s}{T_o}\\,[CAPE^* - CAPE^b]|_m,\n\\end{equation}\nwhere $C_k$ is the exchange coefficient for enthalpy, $C_D$ is the drag coefficient, $T_s$ is the surface temperature, $T_o$ is the mean outflow temperature, ${CAPE}^\\ast$ is the convective available potential energy of air lifted from saturation at sea surface, and ${CAPE}^b$ is that of the boundary layer air. Both ${CAPE}^\\ast$ and ${CAPE}^b$ are computed at the radius of maximum surface wind.\n\nIn the discussion of the size of hurricanes, the Rossby deformation radius ($L_R$) is used, as described in Section 3.3 below. Here, $L_R$ is the length scale at which rotational effects become as important as the effects of gravity waves or buoyancy in the evolution of the flow in a disturbance. The $L_R$ is equal to $\\frac{NH\\ }{f}$, where $N$ is the Brunt-Vaisala frequency, and $f$ is the Coriolis parameter. Furthermore, $H$ is the scale height, equaling to $\\frac{R^\\ast\\bar{T}}{M_dg}$, where $R^\\ast$ is the universal gas constant, $\\bar{T}$ is the mean air temperature, $M_d$ is the molar weight of the atmosphere, and $g$ is the gravity (Wallace \\& Hobbs \\cite{Wallace}). The $L_R$ decreases as $M_d$ increases due to the reduction of $H$. In idealized experiments with uniform surface temperature or uniform rotation, it is one of the rough scales that can be used for understanding hurricane size (Held \\& Zhao \\cite{Held}); however, in more realistic conditions such as on Earth, $L_R$ is not a good scaling (Chavas et al. \\cite{Chavas16}, Chavas \\& Reed \\cite{Chavas19}). \n\n\n\n\n\n\\begin{figure*\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.8\\textwidth]{fig2.PNG}\n\\caption{Azimuthal-mean height-radius cross-section of a typical, mature hurricane on the day side of a tidally locked aqua-planet in the control experiment. (a) tangential wind speed ($v_\\theta$), (b) radial wind speed ($v_r$), (c) vertical velocity ($\\omega$), (d) relative humidity, (e) temperature anomaly from the environmental value ($\\Delta$T), and (f) equivalent potential temperature anomaly ($\\Delta \\theta_e$).}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure*\n   \\centering\n   \\includegraphics[width=0.8\\textwidth]{fig3.PNG}\n      \\caption{Mechanisms for hurricane formation in the control experiment. (a): Location of hurricane formation (dots) and surface air temperature (color shading). The number of hurricanes during the four-year integration is 154. (b): Long-term mean surface air pressure (shading) and winds at 850 hPa (vector). (c): Same as (b) but for an instantaneous. The life cycle of one hurricane on the day side: (d): Maximum surface wind speed (blue) and minimum surface pressure (red), (e): Relative vorticity (blue) and divergence (red) at 850 hPa, (f): Precipitation (blue) and vertical velocity at 850 hPa (red), and (g): Surface latent heat flux (blue). For (e)-(g), the variables are calculated for area mean of 500$\\times$500 km$^{2}$ around the low-pressure center.}\n         \\label{fig3}\n   \\end{figure*}\n\n\n\n\n\\section{Results}\n\\subsection{Hurricanes on tidally locked planets}\n\nFigure 1 and 2 show the results of the control experiment for an aquaplanet orbiting close to the inner edge of the habitable zone around a late M dwarf of 2600 K. The rotation period is set to six Earth days. Both the day and night sides are set to hot (310--315~K), and the day-to-night surface temperature contrast is small (see Fig. 3a). This experimental design is due to the fact that the day-to-night atmospheric latent heat transport is very efficient for planets near the inner edge (Haqq-Misra et al. \\cite{Haqq-Misra}, Yang et al. \\cite{Yang}); oceanic heat transport can act to further reduce the day-to-night temperature contrast (Yang et al. \\cite{Yang}). The high surface temperature, weak day-to-night contrast, and relatively fast rotation rate (comparing to planets in the middle range of the habitable zone or planets orbiting around hotter stars) benefit hurricane formation. The temperatures of 310-315 K are close to the conditions of a runaway greenhouse state. For tidally locked planets, the planets start to enter runaway greenhouse state when the maximum surface temperature is close to or higher than these values (Wolf et al. \\cite{Wolf}).\n\n\n\n\nClearly, there are hurricanes in the control experiment. In the mature stage of the hurricanes, maximum wind speed reaches $\\approx$30-50 m s$^{-1}$ (Fig.~1a), surface air pressure at the center is $\\approx$950-980 hPa (Fig.~1b), precipitation reaches as high as 200-500 mm per day due to strong convection especially near the eyewall (Fig.~1c), and relative vorticity near the surface is on the order of 10$^{-4}$ s$^{-1}$ (Fig.~1d), values that are close to those on Earth (Anthes \\cite{Anthes}, Emanuel \\cite{Emanuelb}). The surface wind speed increases as the eyewall is approached from outside, but inside the eyewall the winds as well as precipitation weaken rapidly. The winds rotate counter-clockwise in the northern hemisphere and clockwise in the southern hemisphere due to the Coriolis force, although in this experiment, it is much smaller than that on Earth. The precipitation exhibits well-defined spiral bands rather than uniformly distributed throughout the region of the hurricane. The clear patterns of eye-eyewall and spiral rain bands suggest that the model resolution of 50 km is good in resolving the hurricanes.\n\n\\begin{figure*\n   \\centering\n   \\includegraphics[width=0.8\\textwidth]{fig4.PNG}\n      \\caption{Environmental conditions for hurricane formation. (a): Genesis potential index (GPI); (b): Planetary vorticity; (c): Long-term mean relative vorticity at 850 hPa; (d): Relative humidity at 600 hPa; (e): Potential intensity; (f): Vertical shear of the horizontal winds between 300 and 850 hPa. We note the environmental vorticity in panel (c) is one order smaller than the vorticity of hurricanes shown in Fig.~1d.\n              }\n         \\label{fig4}\n   \\end{figure*}\n\n\n\n\n\nFor vertical cross structure (Fig.~2), the tangential wind component dominates the flow through the system, although the radial wind is also significant. In the boundary layer, the winds flow towards the low-pressure center. Within the hurricane, upward motion is robust and tilts radially outward. The maximum ascendance near the surface locates at a distance of $\\approx$250 km outward from the hurricane center rather than at the center itself. Indeed, weak downward motion takes up the center. Due to the ascendance and deep convection, relative humidity is high in the hurricane. Latent heat release from the convection and adiabatic warming by compression from the subsidence in the eye produces a warmer region of air with temperatures of $\\approx$4~K above the environmental value. This can be called a 'warm core,' which is one of the most characteristic features of a hurricane. The warm core can also be viewed from the equivalent potential temperature anomaly. Hurricanes on the night side (Fig.~S1) are smaller in horizontal size compared to those on the day side, $\\approx$500 versus 1500 km, but the vertical structures are similar.\n\n\nStatistical analysis shows that in the control experiment there are four preferred regions for hurricane genesis: the northern and southern tropics of the day side near the substellar point and the middle-to-high latitudes of the night side on each hemisphere (black dots in Fig.~3a). Hurricane formation is largely determined by small-scale convection, large-scale environmental conditions, and the interactions between them. Below, we explore the underlying mechanisms in two ways.\n\n\nOne way is the positive feedback between cumulus convection and larger-scale disturbance, known as the conditional instability of the second kind (CISK; Charney \\& Eliassen \\cite{Charney}, Smith \\cite{Smith}, Yamasaki \\cite{Yamasaki}, Wang \\cite{Wang}). On tidally locked planets, long-term mean atmospheric circulation is characterized by large-scale Rossby waves on the west and pole of the substellar point and Kelvin waves on the east of the substellar point (Fig.~3b), excited from the uneven stellar radiation distribution (Showman \\& Polvani \\cite{Showmana}, Showman et al. \\cite{Showmanb}). The wave pattern is similar to the tropical Matsuno-Gill pattern on Earth (Matsuno \\cite{Matsuno}, Gill \\cite{Gill}), but the meridional (south-north) scale is larger, $\\approx$10,000 versus 3,000 km. The Rossby waves have one low-pressure center on each hemisphere, whose corresponding environmental vertical motion is updrafts and relative vorticity is positive (negative) on the northern (southern) hemisphere. This low-pressure system favors the onset of the CISK feedback: surface winds spiral into the low-pressure center and create horizontal convergence; this low-level convergence enhances the relative vorticity through vortex stretching, increases upward motion following the conservation of mass, and, in the meantime, brings water vapor into the center, amplifying cumulus convection and release of latent heat. The latent heat release warms the air and lowers the air density through forcing more upper-level air to move outward away from the center, subsequently reducing the surface pressure; the lower surface pressure further enhances the low-level convergence and increases the growth rate of the relative vorticity through vortex stretching (Fig.~3d-f). This feedback is the key in promoting the growth of small-scale disturbances to hurricanes in the background low-pressure regions. It is similar to that on Earth: hurricane generally forms in the monsoon troughs and the confluence zones where the surface pressure is relatively low, collocated with high cyclonic vorticity, convergent surface winds, and divergent winds aloft (Anthes \\cite{Anthes}, Emanuel \\cite{Emanuelb}). Moreover, during the formation phase, the latent heat flux from the surface to the boundary layer increases strongly (Fig.~3g), which also contributes to intensifying the hurricane through the feedback of wind-induced surface energy exchange (WISHE; Emanuel \\cite{Emanuelb}, Wang \\cite{Wang}).\n\n\n The lifetime of the hurricanes on the day side is $\\approx$40-50 Earth days, longer than that on Earth. This is mainly due to the absence of continents and the warm surface everywhere in the experiment. On the night side, the lifetime is shorter, $\\approx$10-20 Earth days.\n\n \\begin{figure*\n   \\centering\n   \\includegraphics[width=0.8\\textwidth]{fig5.PNG}\n      \\caption{Effects of planetary rotation and surface temperature on hurricane formation (dots) and the GPI (color shading). Experiments are for varying rotation period (a-c), varying surface temperature (d-e), and varying values of both (f). Experimental designs are the same as the control experiment in Fig.~1 except that the rotation period is set to (a): 10, (b): 20, and (c): 40 Earth days; (d): Maximum surface temperature reduced from 315 to 308 K and the night-side surface temperature is reduced from 310 to 275 K (see Fig.~6a); (e): Same as (d) but for 301 K and 268 K, respectively (see Fig.~6b); and (f) same as (e) but for a rotation period of 40 earth days. The number of hurricanes is 88, 34, 21, 10, 2, and 0, respectively. See Fig.~7 for snapshots of typical hurricanes. The southern hemisphere always has more hurricanes than the northern hemisphere; this may be due to some asymmetry in initial state or some stochastic process in the model.\n              }\n         \\label{fig5}\n   \\end{figure*}\n\n\n\\begin{figure*\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.8\\textwidth]{fig6.PNG}\n\\caption{Surface air temperatures specified in the simulations of planets in the middle range of the habitable zone. (a): Maximum surface temperature is 308~K and the night-side surface temperature is uniform with a value of 275~K. (b): Same as (a) but the temperature is 7~K lower throughout.}\n\\label{fig6}\n\\end{figure*}\n\n\\begin{figure*\n\\centering\n\\setlength{\\abovecaptionskip}{0.1cm}\n\\includegraphics[width=0.8\\textwidth]{fig7.PNG}\n\\caption{Snapshots of instantaneous surface wind speed (m s$^{-1}$) of typical hurricanes in the experiments of varying rotation period (a-c), varying surface temperature (d-e), and variations of both (f). The six hurricanes are marked with black circles. There is no hurricane in panel (f). These experiments are the same as those shown in Fig.~5.}\n\\label{fig7}\n\\end{figure*}\n\n\nAnother more quantitative way is the empirical equation of the genesis potential index (GPI; Emanuel \\& Nolan \\cite{Emanuelc}, Camargo et al. \\cite{Camargo}, Bin et al. \\cite{Bin}). The index combines five environmental factors to predict the potential of hurricane formation, including planetary vorticity, relative vorticity, relative humidity, potential intensity, and wind shear. A comparison between Fig.~3a and~4a reveals a positive correlation between the location of hurricane genesis and large values of the GPI. In the four hurricane formation regions, GPI values are large because the relative vorticity is great, relative humidity is high over the substellar region, potential intensity is large, and vertical wind shear is weak (Fig.~4b-f). These properties favor hurricane formations in these scenarios. For example, when the shear is strong, an initial disturbance will be ventilated by cooler or drier air and thereby temperature and moisture anomalies are hard to maintain (Tang \\& Emanuel \\cite{Tang}). In this experiment, the vertical wind shear is strong, especially in the tropics of the night side associated with atmospheric superrotation (Showman et al. \\cite{Showmanb}, Pierrehumbert \\& Hammond \\cite{Pierrehumbertb}) and in the extratropics of the day side (Fig.~4f), so that there is nearly no hurricane formation there. The applicability of the empirical GPI index on Earth to the tidally locked planet is mainly due to the fact that we employed an Earth-like atmosphere here; when the atmospheric composition is quite different from that of Earth, GPI does not serve as a good index, as addressed below. \n\nThe formation of hurricanes on the night side is surprising because the night side has no stellar radiation and the long-term mean vertical motion is downwelling rather than upwelling. In this experiment, based on a planet close to the inner edge of the habitable zone, however, the night-side surface is warm and the surface temperature gradient is small (Fig.~3a). In addition, there are a few short-time, small, low-pressure regions (Fig.~3c), the planetary vorticity is relatively high (Fig.~4b), and the vertical wind shear is weak (Fig.~4f) at the middle-to-high latitudes. These factors promote hurricane genesis there. However, when the surface temperature is decreased or the rotation rate is slowed down, there are fewer or altogether no hurricanes on the night side (see below).\n\n\n\\begin{figure*\n   \\centering\n   \\includegraphics[width=0.8\\textwidth]{fig8.PNG}\n      \\caption{ Effects of background gases on hurricane formation. From (a) to (e), these are snapshots of instantaneous surface air pressure (hPa) under background gases of H$_2$, He, N$_2$, O$_2$, and CO$_2$, respectively. In all these experiments, the planetary rotation period is one Earth day and surface temperature is uniform (301 K). For experiments with a rotation period of three Earth days, the results are the same except that the hurricanes are larger in size but fewer in number in the latter three experiments.\n              }\n         \\label{fig8}\n   \\end{figure*}\n\n\\begin{figure*\n   \\centering\n   \\includegraphics[width=0.8\\textwidth]{fig9.PNG}\n      \\caption{Snapshots of (a) surface temperature, (b) surface air pressure, (c) near-surface wind strength, (d) precipitation, and (e) relative vorticity in one experiment coupled to a slab ocean. Panels c\u2013e: only the hurricane region is shown in order to more clearly exhibit the structure of the hurricane. In this experiment, rotation period (=orbital period) is ten Earth days, the CO$_2$ concentration is 300 ppmv, stellar flux is 1450 W m$^{-2}$, and the star temperature is 2600 K. No oceanic heat transport is involved in this run.      }\n         \\label{fig9}\n   \\end{figure*}\n\n\n\\subsection{Effects of planetary rotation rate and surface temperature} \nIn order to test the effect of the rotation rate, we performed three experiments in which rotation period is increased (i.e., the rotation rate is decreased) while other experimental design features are left to remain the same as those of the control experiment. In the case of ten days, the hurricane frequency does not change much on the day side but decreases significantly on the night side (Fig.~5a). In the case of 20 days, there are nearly no evidence of a hurricane on the night side and the number of hurricanes on the day side also decreases substantially (Fig.~5b). For the case of 40 days, the hurricane only forms at regions very close to the substellar point (Fig.~5c). This trend as a function of rotation period is mainly attributed to three factors: the direct weakness of planetary vorticity, the reduction of relative vorticity due to the fact that the atmosphere becomes so steady that waves and disturbances become less active, and the decrease of relative humidity on the night side (with an increase on the day side) due to the strengthening of the thermal-driven global Walker circulation (Fig.~S2).\n\n\nWhen the surface temperature is decreased, it is harder for a hurricane to form. As the maximum surface temperature is set to 308 K and the night-side surface temperature is set to 275 K (Fig.~6a), fewer hurricanes form in the vicinity of the substellar point and no hurricane on the night side (Fig.~5d). When the maximum surface temperature is set to 301 K and the night-side surface temperature is set to 268 K(Fig.~6b), there are only two hurricane events during the integration of four Earth years (Fig.~5e), which is consistent with the GPI prediction in Bin et al. (\\cite{Bin}). The temperature of 301 K is close to the tropical surface temperatures on Earth. This suggests that hurricane formation on tidally locked planets requires a warmer surface due to their slower rotation rates and stronger wind shears. For planets with both slow rotation and low temperature, no hurricane can form (Fig.~5f). The decreasing trend of hurricane formation as a function of reduced surface temperature is due to two main processes: the relative humidity and potential intensity decrease because of the cooler surface and weaker upwelling and convection, and the vertical wind shear becomes much stronger due to the enhanced temperature gradients between the day and night sides (Fig.~S3). Moreover, when the night-side surface temperature is low, air convergence from the night side to the day side brings cool air rather than warm air into the substellar region, suppressing hurricane formation there.\n\n\n\\subsection{Effect of bulk atmospheric composition}\nAtmospheric compositions on terrestrial exoplanets are as yet unknown. Here, we carry out a preliminary investigation of how atmospheric molecular weight influences hurricane formation under a uniform surface temperature of 301 K. When the background atmosphere is set to H$_2$ or He, there is no hurricane, in contrast to the experiments of N$_2$, O$_2$, and CO$_2$ (Fig.~8), although the GPI value is comparable to or even larger than that shown above. This is due to the fact that the condensate\u2013H$_2$O is heavier than H$_2$ and He, so that any disturbance that brings water vapor upward will cause the density of a moist parcel to be larger than its surrounding environment, similar to the conditions in Saturn's atmosphere (Guillot \\cite{Guillot}, Li \\& Ingersoll \\cite{LiC}, Leconte et al. \\cite{Leconteb}). This process induces a negative buoyancy and stabilizes the atmosphere against convection. It can simply be understood by using the ideal gas equation, $p=\\rho R_d T_v$ , and the virtual temperature ($T_v$),\n\\begin{center}\n\\begin{equation}\n  \\centering\n T_v=\\frac{p}{p+\\left(\\epsilon-1\\right)e}T\n,\\end{equation}  \n\\end{center}\n\n\\noindent\nwhere $p$ is total air pressure, $e$ is the partial pressure of the condensate, $\\rho$ is air density, $R_d$ is the gas constant of dry air, $\\epsilon$ is the molecular weight ratio of water vapor to the dry air, and $T$ is the air temperature. For a H$_2$-dominated (or He-dominated) atmosphere, $\\epsilon$ is equal to 9 (or 4.5), so that $T_v$ is smaller than $T$. Therefore, a moist parcel is heavier than a dry parcel under the same $p$ and $T$, and moist convection is inhibited, which is opposite to the conditions on Earth. Moreover, in the experiments with N$_2$, O$_2$, and CO$_2$, a clear trend is revealed, namely, that the size of the hurricane decreases as the mean molecular weight is increased. This is due to the fact that the atmospheric scale height is inversely proportional to the mean molecular weight and, subsequently, the Rossby deformation radius (see the last paragraph of Section 2 above) becomes smaller.\n\n\n\n\nIn the experiments of  Fig.~8, the value of Rossby deformation radius is $\\approx$500-1500 km, comparable to the hurricane size. However, the Rossby deformation radius is strongly latitude-dependent because $f$ is equal to $2\\Omega sin(\\varphi)$, where  $\\Omega$ is the rotation rate and $\\varphi$ is the latitude, but the hurricane size in the experiments does not exhibit the same dependency. A better scaling was not found because of the nonlinear dynamics of hurricane and the complex interactions between the hurricane and diabatic heating, environmental relative humidity, mesoscale convective system, and other features (Emanuel \\cite{Emanuelb}; Merlis \\& Held \\cite{Merlis19}). Under more  realistic conditions, such as on Earth and the simulations in Sections 3.1 and 3.2, the Coriolis effect is not constant between latitudes and there are strong interactions between hurricanes and mean circulation, so that the Rossby deformation radius is not a good scale for hurricane size (e.g., Chavas et al. \\cite{Chavas16}).\n\n\n\n\n\\section{Conclusions and discussions}\n\nWe find that hurricanes can form on tidally locked planets especially for those orbiting near the inner edge of the habitable zone of late M dwarfs. For planets in the middle range of the habitable zone, hurricanes are relatively fewer. Storm theories of Earth and Saturn can be used to understand the hurricane formation on tidally locked planets. Hurricanes can enhance the ocean mixing and oceanic heat transport from warmer to cooler regions in both horizontal and vertical directions. Hurricanes can also influence the transmission spectra of tidally locked planets. For instance, if a hurricane moves to the terminator, water vapor concentration would increase  (Fig.~S4), which can influence the transmission signals. Unfortunately, present-day telescopes are not capable of observing this feature (Morley et al. \\cite{Morley}, de Wit et al. \\cite{de Wit}) mainly due to the small-scale height of the atmosphere and the relatively small size of the hurricane compared to the planetary radius. Differentiating them requires the large space telescopes or ground-based extremely large telescopes of the future.\n\n\n\nFurthermore, future studies require the use of AGCMs coupled to a slab ocean or fully coupled atmosphere\u2013ocean models. The result of a test of the atmosphere coupled to a slab ocean is shown in Fig.~9. We can still find hurricanes in the experiment, whereas more results for different rotation periods, different stellar fluxes, and different CO$_2$ concentrations will be presented in a separate paper in the near future. Moreover, future works are required to examine how continents influence the results of such studies. Hurricanes always decay quickly when they move over land because of the dramatic reduction in evaporation and the increase in surface roughness. Global climate models with more realistic cloud schemes and regional cloud-resolving models with more accurate radiation transfer are required to simulate the hurricane genesis, especially for those who have quite different atmospheric compositions or air masses from Earth. One another weakness in this study is the convection scheme that was developed based on the knowledge of convection on Earth. Future studies using high-resolution models with explicit convection (e.g., Sergeev et al. \\cite{Sergeev}) are required. Moreover, convective self-aggregation (such as Bony et al. \\cite{Bony}, Pendergrass et al. \\cite{Pendergrass}, Wing et al. \\cite{Wing}) may have occurred in our simulations, particularly in the 310--315~K experiment.  A future work is required to analyze this feature.\n\nRecently, using Earth-based metrics for hurricane genesis, Komacek et al. (\\cite{Komacek}) found that\nhurricane genesis is most favorable on tidally locked terrestrial exoplanets with intermediate rotation periods of about 8\u201310 days in the habitable zones of late-type M dwarf stars, and that on slowly rotating planets hurricane generis is unfavorable. The latter is consistent with Bin et al. (\\cite{Bin}) and our results. Future simulations using hurricane-resolved models are required to verify the intermediate rotation periods conclusion shown in Komacek et al. (\\cite{Komacek}).\n\n\n\n\n\\begin{acknowledgements}\n     We thank the National Center for Atmospheric Research (NCAR) groups for developing the model CAM4 and making it available to public. We are grateful to the discussions with Hao Fu, Gan Zhang, Cheng Li, Weixin Xu, Yongyun Hu, Zhiyong Meng, Dorian S. Abbot, Thaddeus D. Komacek, and Fengyi Xie.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{ Introduction}\n\nThis paper continues our study of correlation functions\nin lattice integrable models \\cite{BJMST1,BJMST2,BJMST3,BJMST4,\nBJMST}.\nConsider the infinite XXZ spin chain  with the Hamiltonian\n\\begin{eqnarray}\nH_{\\rm XXZ}=\\textstyle{\\frac{1}{2}}\\sum\\limits_{k=-\\infty}^{\\infty}\n\\left( \n\\sigma_{k}^1\\sigma_{k+1}^1+\n\\sigma_{k}^2\\sigma_{k+1}^2+\n\\Delta\\sigma_{k}^3\\sigma_{k+1}^3\n\\right), \n\\label{eq:XXZ}\n\\end{eqnarray}\nwhere $\\sigma^a \\, (a=1,2,3)$ are \nthe Pauli matrices and\n\\begin{eqnarray*}\n\\Delta=\\cos\\pi\\nu\n\\end{eqnarray*}\nis a real parameter. \nWe use the usual notation\n$$\nq=e^{\\pi i \\nu}\n\\,.\n$$\nIn our previous work \\cite{BJMST}, \nwe obtained an algebraic representation for\ngeneral correlation functions of the XXZ model. \nHere we generalize this result to the situation when a disorder\noperator is present.\nIn the course we find a new interesting structure behind the model. \nWe consider only the massless regime $|\\Delta|<1$, $0<\\nu<1$, \nsince it is more important for physics because of its relation to\nconformal field theory (CFT) discussed below.\nExplanation about the massive regime $\\Delta>1$ will be given\nelsewhere.\n\nLet us introduce \n$$S(k)=\\textstyle{\\frac 1 2} \\sum\\limits_{j=-\\infty}^k\\sigma ^3_j\\,.$$\nDenote by $|\\text{vac}\\rangle$ the ground state of\nthe Hamiltonian, and let $\\alpha$ be a parameter. \nWe consider the normalized vacuum expectation values:\n\\begin{align}\n\\frac{\\langle\\text{vac}|q^{2\\alpha S(0)}\n\\mathcal{O}|\\text{vac}\\rangle}{\\langle\\text{vac}|q^{2\\alpha S(0)}\n|\\text{vac}\\rangle}\\label{exp}\n\\end{align}\nwhere $\\mathcal{O}$ is a local operator.\n\nLocality of $\\mathcal{O}$ implies that the operator\n$q^{2\\alpha S(0)}\\mathcal{O}$ stabilizes: there exist integers $k, l$ such that \nfor all $j>l$ (resp. $j< k$) this operator acts on the \n$j$-th lattice site as $1$ (resp. $q^{\\alpha\\sigma ^3}$).\nIf $k$ (resp. $l$) is the maximal (resp. minimal) integer with this property,\n$l-k+1$ will be called the length of the operator $q^{2\\alpha S(0)}\\mathcal{O}$. \nThe very formulation of the problem implies that we are\ninterested only in local operators $\\mathcal{O}$ of total spin $0$\n(otherwise the correlation function vanishes). \nNevertheless, for the sake of\nconvenience we introduce the spaces $\\mathcal{W}\n_{\\alpha,s}$ of operators $q^{2\\alpha S(0)}\\mathcal{O}$ of\nspin $s$:\n$$\n\\[S\\ ,\\ q^{2\\alpha S(0)}\\mathcal{O}\\]=s\\ q^{2\\alpha S(0)}\\mathcal{O},\n\\quad S=S(\\infty)\\,.\n$$\nAlso we set\n$$\n\\mathcal{W}_{\\alpha}=\\bigoplus\\limits _{s=-\\infty}^{\\infty}\n\\mathcal{W}_{\\alpha,s}\\,.\n$$\n\n\nThe leading long distance asymptotics of the XXZ spin chain is \ndescribed by CFT with $c=1$: \nthat of free bosons $\\phi,\\bar{\\phi}$ \nwith compactification radius $\\beta=\\sqrt{1-\\nu}$. \n{}For an extensive discussion about the XXZ model \nas an irrelevant perturbation of CFT, \nwe refer the reader to \\cite{luk}.  \nThe space $\\mathcal{W}_{\\alpha,s}$ corresponds to the space\nof descendants of the operator \n$$\ne^{\\frac i 2\\(\\alpha (\\beta ^{-1}-\\beta )(\\phi +\\bar{\\phi})+s\\beta (\\phi-\n\\bar{\\phi})\\)}\\,.\n$$\nSimilarly to the conformal case \\cite{BLZ1,BLZ2,BLZ3}, \nintroduction of the disorder parameter $\\alpha$ \nregularizes the problem, and allows to write\nmuch nicer formulae than in the case $\\alpha =0$ \n\\footnote{\nThe formulae are written initially for $|q^{\\alpha}|<1$\nand continued analytically in $\\alpha$, but $\\alpha =0$ is\none of singular points where l'H\\^opital's rule should be applied.}.\nAnother similarity is that it is very\nconvenient to consider, as an intermediate object which\ndoes not enter the final formulae, the following space:\n$$\n\\mathcal{W}_\n{[\\alpha]}=\\bigoplus\\limits _{k=-\\infty}^{\\infty}\\mathcal{W}_{\\alpha+k}\\ .\n$$\n\nIn this paper we \nshall introduce two \nanti-commuting families of operators $\\mathbf{b} (\\zeta )$ and\n$\\mathbf{c} (\\zeta)$ acting on $\\mathcal{W}_{[\\alpha]}$:\n\\begin{align}\n\\[\\mathbf{b} (\\zeta_1),\\mathbf{b} (\\zeta_2)\\]_+=\\[\\mathbf{b} (\\zeta_1),\\mathbf{c} (\\zeta_2)\\]_+=\\[\\mathbf{c} (\\zeta_1),\\mathbf{c} (\\zeta_2)\\]_+=0\\ .\\nonumber\n\\end{align}\nThe operators $\\mathbf{b} (\\zeta )$ and $\\mathbf{c} (\\zeta )$ have the following block\nstructure:\n$$\n\\mathbf{b} (\\zeta):\\ \\mathcal{W}_{\\alpha+k,s}\\to\n\\mathcal{W}_{\\alpha+k+1,s-1}, \n\\quad \\mathbf{c} (\\zeta ):\\ \\mathcal{W}_{\\alpha+k,s}\\to \n\\mathcal{W}_{\\alpha+k-1,s+1}\n\\ .\n$$\nHence the operator \n$\\mathbf{b} (\\zeta_1)\\mathbf{c} (\\zeta _2)$ acts from $\\mathcal{W}_{\\alpha,0}$ to itself.\n\n\nThe operators $\\mathbf{b} (\\zeta )$, $\\mathbf{c} (\\zeta )$ \nare formal series in $(\\zeta -1)^{-1}$.\nWhen applied to an operator $q^{2\\alpha S(0)}\\mathcal{O}$ of length $L$,   \nthe singularity is a pole of order $L$, in other words, the series terminates at\n$(\\zeta -1)^{-L}$.\nThe action of $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$ produces\noperators of the same or smaller length. \nThe coefficients of $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$ \ngive rise to an action of the Grassmann algebra with $2L$\ngenerators. In particular \n$$\n\\mathbf{b}(\\zeta_1)\\cdots\\mathbf{b}(\\zeta_{L+1})\n\\(q^{2\\alpha S(0)}\\mathcal{O}\\)=0,\n\\quad \n\\mathbf{c}(\\zeta_1)\\cdots\\mathbf{c}(\\zeta_{L+1})\n\\(q^{2\\alpha S(0)}\\mathcal{O}\\)=0\n\\,.\n$$\n\nWe introduce also the  linear functional\non $\\text{End}\\(\\mathbb{C}^2\\)$:\n\\begin{align}\n\\text{tr}^{\\alpha}(x)=\n\\frac 1 {q^{\\frac {\\alpha }2}+q^{-\\frac {\\alpha }2}}\n\\text{tr} \\(q^{-\\frac 1 2 \\alpha \\sigma ^3}x\\)\n\\label{tral}\n\\end{align}\nwith the obvious properties:\n$$\n\\text{tr}^{\\alpha}(1)=\\text{tr}^{\\alpha}(q^{\\alpha \\sigma ^3})=1\\,.\n$$\nThis gives rise to a\nlinear functional on $\\mathcal{W}_{\\alpha}$ \n$$\n\\mathbf{tr}^{\\alpha}(X)=\\cdots{\\rm tr} _1^{\\alpha}\\ {\\rm tr} _2^{\\alpha}\\ \n{\\rm tr} _3^{\\alpha}\\cdots (X)\\,.\n$$\n\nOur main result is:\n\\begin{align}\n\\frac{\\langle\\text{vac}|q^{2\\alpha S(0)}\n\\mathcal{O}|\\text{vac}\\rangle}{\\langle\\text{vac}|q^{2\\alpha S(0)}\n|\\text{vac}\\rangle}\\ =\n\\mathbf{tr}^{\\alpha}\\(e^{\\mbox{\\scriptsize\\boldmath{$\\Omega$}}}\\(q^{2\\alpha S(0)}\\mathcal{O}\\)\\)\\,,\n\\label{main}\n\\end{align}\nwhere\\footnote {In \\cite{BJMST} the operator $\\mbox{\\boldmath$\\Omega $}$ was denoted by $\\Omega ^*$.}\nthe operator $\\mbox{\\boldmath$\\Omega $}$ acts on $\\mathcal{W}_{[\\alpha ]}$:\n\\begin{align}\n\\mbox{\\boldmath$\\Omega $}=-\n{\\rm res} _{\\zeta_1=1}{\\rm res} _{\\zeta_2=1}\n\\(\\mbox{\\boldmath$\\omega $} \\(\\zeta_1\/\\zeta_2\\)\\mathbf{b} (\\zeta _1)\\mathbf{c} (\\zeta _2)\n\\frac{d\\zeta _1}{\\zeta _1}\\frac{d\\zeta _2}{\\zeta _2}\\)\\,,\\nonumber\n\\end{align}\nand $\\mbox{\\boldmath$\\omega $} (\\zeta)$ is a scalar operator on each $\\mathcal{W}_{\\alpha}$,  \n\\begin{align}\n&\n\\left. \\mbox{\\boldmath$\\omega $} (\\zeta)\\right|_{\\mathcal{W}_{\\alpha}}=\\omega (\\zeta ,\\alpha)1_{\\mathcal{W}_{\\alpha }},\\label{omega}\n\\end{align}\nthe scalar being\n\\begin{align}\n&\\omega (\\zeta ,\\alpha)\\nonumber\\\\\n&=\\frac {4(q\\zeta)^{\\alpha }} {\\(1+q^{\\alpha}\\)^2}\n\\(\n\\frac {q^{-\\alpha}}{1-q^{-2}\\zeta ^2}-\\frac {q^{\\alpha}}{1-q^2\\zeta ^2}\\)+\n\\int\\limits _{-i\\infty -0}^{i\\infty -0}\\zeta ^{u+\\alpha}\n\\frac {\\sin \\frac {\\pi} 2\\(u-\\nu(u+\\alpha)\\)}{\\sin \\frac {\\pi} 2 u\n\\cos \\frac {\\pi\\nu} 2\\(u+\\alpha\\)}du\\ \\nonumber.\n\\end{align}\n{}For any local operator of length $L$, \nthe trace is effectively taken over $\\(\\mathbb{C}^2\\)^{\\otimes L}$.\n\nComments are in order about the meaning of \\eqref{main}.\nIn \\cite{JM,JM2}, \nin the setting of inhomogeneous chains,  \nit was conjectured \nthat the thermodynamic limit of the ground state \naverages in the finite XXZ chain \nare certain specific solutions of the reduced qKZ \n(rqKZ) equation given by multiple integrals. \nSubsequently these integral formulas were also derived \nfrom the viewpoint of algebraic Bethe Ansatz \\cite{maillet}.   \nWe take these formulas as the definition of \nthe left hand side of \\eqref{main}.\nFollowing our previous works \\cite{BJMST2,BJMST}, \nwe present here another formula for solutions of rqKZ equations. \nThe right hand side of \\eqref{main}\nis its specialization to the homogeneous case.\nWe have no doubt that these two solutions coincide\n\\footnote{It is known to be the case in the massive regime, \nsee \\cite{BJMST2}. \nWe also confirm the coincidence at the free fermion point, \nsee section \\ref{XX}.   \n}. \nSince a mathematical proof is lacking at the moment,\nwe propose \\eqref{main} as conjecture. \nThe function $\\omega (\\zeta ,\\alpha)$ and $\\mathbf{tr}^{\\alpha}$ develop singularities\nat $\\alpha = \\pm 1\/\\nu$. \nIn view of this, we presume that the formula holds true throughout the range \n$|{\\rm Re}\\, \\alpha|<1\/\\nu$. \n\nIt will be shown that the operators $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$ commute\nwith the adjoint action of the shift operator $U$ and \nof local integrals of motion $I_p$ on $\\mathcal{W}_{[\\alpha]}$. \nSince $q^{-\\alpha S}$ commutes with $U,I_p$, \none immediately concludes that the vacuum expectation values of \n$U\\(q^{2\\alpha S(0)}\\mathcal{O}\\)U^{-1}\n -q^{2\\alpha S(0)}\\mathcal{O}$ and\n$\\[ I_p,q^{2\\alpha S(0)}\\mathcal{O}\\]$ \ngiven by (\\ref{main}) vanish, as it should be.\n\nIn our opinion the appearance of anti-commuting operators $\\mathbf{b} (\\zeta)$\nand $\\mathbf{c} (\\zeta)$ is quite remarkable. \nIn the next section we explain how these operators are\nconstructed using the $q$-oscillators. \nWe explain their relation to the \nJordan-Wigner fermions in the XX case in Section \\ref{XX}. \nIn Appendix we briefly discuss the generalization\nof our previous formulae \\cite{BJMST} to the case when\nthe disorder operator is present. \n\n{}For the sake of simplicity \nwe consider the homogeneous chain only. \nWe give brief explanations about the inhomogeneous case when needed.\nWe do not give complete  proofs, but just sketch the derivation\nof the main statements. \nWe tried to make this paper as brief as possible,\nleaving the details to a separate publication. \n\n\n\\section{Operators $\\mathbf{b} (\\zeta )$ and $\\mathbf{c} (\\zeta )$}\\label{bc}\n\n{}First we prepare our notation for the $L$-operators. \nConsider the quantum affine algebra $U_q(\\widehat{\\mathfrak{sl}}_2)$. \nThe universal $R$-matrix of this algebra belongs to the tensor product \n$\\mathfrak{b}_+\\otimes \\mathfrak{b}_-$ of its two Borel subalgebras. \nBy an $L$-operator we mean its image under an algebra map\n$\\mathfrak{b}_+\\otimes \\mathfrak{b}_-\\to N_1\\otimes N_2$, \nwhere $N_1,N_2$ are some algebras.  \nIn this paper we always take $N_2$ to be the \nalgebra $M=Mat(2,{\\mathbb C})$ of $2\\times 2$ matrices. \nAs for $N_1$ we make several choices:\n$U_q(\\mathfrak{sl}_2)$, $M$, \nthe $q$-oscillator algebra \n$Osc$ (see below) or $Osc\\otimes M^{\\pm}$, \nwhere $M^{\\pm}\\subset M$ are the subalgebras\nof upper and lower triangular matrices.\n{}For economy of symbols, we use the same letter $L$ \nto designate these various $L$-operators. \nWe always put indices, \nindicating to which tensor product of algebras they belong. \nWe use $j,k,\\cdots$ as labels for the lattice sites, \nand $a,b,\\cdots$ as those for the `auxiliary' two-dimensional space. \nAccordingly we write the matrix algebra as $M_j$ or $M_a$. \nCapital letters $A,B,\\cdots$ will indicate \nthe $q$-oscillator algebra $Osc$.  \n{}Finally, for $Osc\\otimes M^{\\pm}$ we use pairs of \nindices such as $\\{A,a\\}$.\n\nThe first case of $L$-operators is when $N_1=U_q(\\mathfrak{sl}_2)$:\n\\begin{align}\nL_j(\\zeta)=\\begin{pmatrix}\\zeta q^{\\frac{H+1}2}-\\zeta ^{-1}q^{-\\frac{H+1}2}\n&(q-q^{-1})Fq^{\\frac{H-1}2}\\cr (q-q^{-1})q^{-\\frac{H-1}2}E & \\zeta q^{-\\frac{H-1}2}-\\zeta ^{-1}q^{\\frac{H-1}2}\n\\end{pmatrix}_j\\in U_q(\\mathfrak{sl}_2)\\otimes\nM_j.\\label{Lop}\n\\end{align}\nHere $E,F,q^{\\pm H\/2}$ are the standard generators of $U_q(\\mathfrak{sl}_2)$. \nThe suffix $j$ in the right hand side \nmeans that it is considered as a $2\\times 2$ matrix in $M_j$. \nThis is an exceptional case when we do not put any index for\nthe first (`auxiliary') tensor factor; we shall never use several copies \nof $U_q(\\mathfrak{sl}_2)$.\nMapping further $U_q(\\mathfrak{sl}_2)$ to $M_a$, \nwe obtain the second $L$-operator \n$$\nL_{a,j}(\\zeta)\\in M_a\\otimes M_j\\,,\n$$  \nwhich actually \ncoincides with the standard $4\\times 4$ $R$-matrix. \n\nThe next case is due originally\nto Bazhanov, Lukyanov and Zamolodchikov \\cite{BLZ3}.\nLet us consider the $q$-oscillators $a$, $a^*$ satisfying\n$$ \naa^*-q^2a^*a=1-q^2.\n$$\nIt is convenient to introduce one more element $q^D$ such that \n\\begin{align}\n&q^{D}a^*=a^*\nq^{D+1},\\qquad q^{D}a=aq^{D-1}\\,,\\nonumber\\\\\n&a^*a=1-q^{2D}, \\quad aa^*=1-q^{2D+2}\\,.\\nonumber\n\\end{align}\nDenote by $Osc$ the algebra generated by $a,a^*,q^{\\pm D}$ \nwith the above relations.\nWe consider the following two representations of $Osc$, \n\\begin{align}\nW^+=&\\bigoplus\\limits _{k=0}^\\infty \\mathbb{C}|k\\rangle,\n\\quad a^*|k-1\\rangle=|k\\rangle,\n\\quad\nD|k\\rangle=k|k\\rangle ,\\quad a|0\\rangle=0;\\nonumber\\\\\nW^-=&\\bigoplus\\limits _{k=-\\infty}^{-1} \\mathbb{C}|k\\rangle,\n\\quad a|k+1\\rangle=|k\\rangle,\n\\quad\nD|k\\rangle=k|k\\rangle ,\\quad a^*|-1\\rangle=0\n\\,.\n\\nonumber\n\\end{align}\nIn the root of unity case, if $r$ is the smallest positive \ninteger such that $q^{2r}=1$, \nwe consider the $r$-dimensional quotient of $W^{\\pm}$ \ngenerated by $|0\\rangle$ or $|-1\\rangle$.\n\n\nThe $L$-operator associated with $Osc$ is given by\n\\begin{align}\n&L^+_{A,j}(\\zeta)=i\\zeta ^{-\\frac 1 2}q^{-\\frac 1 4}\n\\begin{pmatrix}1&  -\\zeta a_A^*\\\\[5pt]\n -\\zeta  a_A &  1- \\zeta ^2q^{2D_A+2} \\end{pmatrix}_j\n\\begin{pmatrix}q^{D_A} &0\\\\[5pt] 0 & q^{-D_A}  \\end{pmatrix}_j\n\\in \nOsc _A\\otimes M_j\\nonumber\\,.\n\\end{align}\nThis $L$-operator satisfies the crossing symmetry relation:\n$$\nL^+_{A,j}(\\zeta)^{-1}=\\frac 1 {\\zeta-\\zeta ^{-1}}\\overline{L}_{A,j}^+(\\zeta)\\,,\n$$\nwhere we have set\n$$\n\\overline{L}_{A,j}^+(\\zeta)=\\sigma ^2 _j L_{A,j}^+(\\zeta q^{-1}) ^{t _j}\\sigma^2_j\n\\,,\n$$\nand $t_j$ stands for the transposition in $M_j$.\nWe use also another $L$-operator\n$$\nL^-_{A,j}(\\zeta)=\\sigma ^1_jL^+_{A,j}(\\zeta)\\sigma ^1_j\\ .\n$$\n\nConsider the product $L ^+_{A,j}(\\zeta )L_{a,j}(\\zeta )$. \nIt is well known that this product can be \nbrought to a triangular form, giving rise \nin particular to Baxter's `$TQ$-equation' for transfer matrices. \nNamely, introducing\n$$G ^+_{A,a}=\n\\begin{pmatrix} q^{-D_A} &0\\\\ 0& q^{D_A} \\end{pmatrix}_a\n\\begin{pmatrix} 1& a^*_A\\\\ 0 &1  \\end{pmatrix}_a,\n\\quad G^-_{A,a}=\\sigma ^1_aG^+_{A,a}\\sigma ^1_a\\,,\n$$\none easily finds that \n\\begin{align}\n&L^+ _{\\{A,a\\},j}(\\zeta )\n=\\(G^+_{A,a}\\)^{-1}  L^+ _{A,j}(\\zeta)L_{a,j}(\\zeta )G^+_{A,a}\n\\label{fusionright+}\\\\ &=\n\\begin{pmatrix}\n(\\zeta q-\\zeta ^{-1}q^{-1})\nL^+_{A,j}(\\zeta q^{-1})q^{-\\frac{\\sigma^3_j} 2} &0\\\\\n(q-q^{-1})L^+_{A,j}(\\zeta q)\\sigma _j^+q^{-2D_A+\\frac 1 2}\n& (\\zeta -\\zeta ^{-1})\nL^+_{A,j}(\\zeta q )q^{\\frac{\\sigma^3_j} 2}\n\\end{pmatrix}_a\\in\\(Osc_A\\otimes M^-_a\\)\n\\otimes M_j \\,.\n\\nonumber\n\\end{align}\n\n{}For the inverse matrix one has:\n\\begin{align}\nL ^+_{\\{A,a\\},j}(\\zeta )^{-1}&=\\frac 1{(\\zeta -\\zeta ^{-1})\n(\\zeta q -\\zeta ^{-1}q^{-1})(\\zeta q^{-1}-\\zeta ^{-1}q)}\n\\label{fusioninv}\\\\ &\\times\n\\begin{pmatrix}\n(\\zeta -\\zeta ^{-1})\nq^{\\frac{\\sigma^3_j} 2}\\ \\overline{L }^+_{A,j}(\\zeta q^{-1})&   0  \\\\\n -(q-q^{-1}) \\sigma _j^+\\ \\overline{L }^+_{A,j}(\\zeta q)q^{-2D_A+\\frac 1 2}\n& (\\zeta q^{-1}-\\zeta ^{-1}q)q^{-\\frac{\\sigma^3_j} 2}\n\\ \\overline{ L }^+_{A,j}(\\zeta   q)\n\\end{pmatrix}_a\\,.\n\\nonumber\n\\end{align}\nAgain we shall use another $L$-operator:\n$$ L ^-_{\\{A,a\\},j}(\\zeta )=\\sigma ^1_a\\sigma ^1_jL ^+_{\\{A,a\\},j}(\\zeta )\n\\sigma ^1_a\\sigma ^1_j \\in\\(Osc_A\\otimes M^+_a\\)\n\\otimes M_j\n\\,.\n$$\nSome information will be needed about $R$-matrices which intertwine these \n$L$-operators.\n{}First, consider the Yang-Baxter equation:\n\\begin{align}\n&R_\n{A,B}(\\zeta_1\/\\zeta _2)L^{\\pm}_{A,j}(\\zeta _1)\nL^{\\pm}_{B,j}(\\zeta _2)=L_{B,j}^{\\pm}(\\zeta _2)L^{\\pm}_{A,j}(\\zeta _1)R_{A,B}(\\zeta_1\/\\zeta _2)\n\\,.\n\\label{YB+}\n\\end{align}\nThe $R$-matrix appearing in \\eqref{YB+} is given by \n$$\nR_{A,B}(\\zeta)=P_{A,B}h(\\zeta, u_{A,B})\\zeta ^{D_A+D_B}\\,,\n$$\nwhere $P_{A,B}$ is the permutation, \n$$\nu_{A,B}=a_A^*q^{-2D_A} a_B,\n$$\nand the function $h(\\zeta, u)$ is given by\n$$\nh(\\zeta,u)=\\sum\\limits _{n=0}^{\\infty}\\ \n\\(-uq^{-1}\\)^n\n\\prod_{j=1}^n\\frac{q^{j-1}\\zeta^{-1}-q^{-j+1}\\zeta}{q^j-q^{-j}}\n\\,.\n$$\nWhen $q$ is not a root of unity,   \nthe series for $R_{A,B}(\\zeta)$ \nis well defined because\nthe action of $u_{A,B}$ on $W^{\\pm}\\otimes W^{\\pm}$ \nis locally nilpotent.  \nOtherwise we replace the right hand side \nby the sum $\\sum_{n=0}^{r-1}$, \nif $r$ is the smallest positive \ninteger such that $q^{2r}=1$.\n\nSecond, consider the Yang-Baxter equation for the $L$-operators\n$L^+_{\\{A,a\\},j}$:\n\\begin{align}\n&R ^+_{\\{A,a\\},\\{B,b\\}}(\\zeta_1\/\\zeta _2)L^+_{\\{A,a\\},j}(\\zeta _1)L^+_{\\{B,b\\},j}(\\zeta _2)=\nL^+_{\\{B,b\\},j}(\\zeta _2)L^+_{\\{A,a\\},j}(\\zeta _1)\nR ^+_{\\{A,a\\},\\{B,b\\}}(\\zeta_1\/\\zeta _2)\\,.\n\\label{YB4}\n\\end{align}\nThe corresponding $R$-matrix has the form \n\\begin{align}\n&R ^+_{\\{A,a\\},\\{B,b\\}}(\\zeta )=\n\\begin{pmatrix}\n\\mathcal{R}_{1,1}(\\zeta)&0 & 0 &0\\\\\n\\mathcal{R} _{2,1}(\\zeta) &\\mathcal{R}_{2,2}(\\zeta)&0 & 0\\\\\n\\mathcal{R} _{3,1}(\\zeta) &0 &\\mathcal{R}_{3,3}(\\zeta)&0\\\\\n\\mathcal{R} _{4,1}(\\zeta) &\\mathcal{R} _{4,2}(\\zeta) &\\mathcal{R}_{4,3}(\\zeta) &\\mathcal{R} _{4,4}(\\zeta)\n\\end{pmatrix}_{a,b}\n\\,.\n\\label{R+ab}\n\\end{align}\nThe entries $\\mathcal{R} _{i,j}(\\zeta )$ can be found by a direct\ncalculation. In this paper we shall need only two of them:\n\\begin{align}\n&\n\\mathcal{R}_{1,1}(\\zeta)=q^{-D_A}R_{A,B}(\\zeta)q^{D_B},\n\\quad \\mathcal{R}_{4,4}(\\zeta)=-\\zeta ^2q^{D_A}R_{A,B}(\\zeta)q^{-D_B}\\,.\n\\nonumber\n\\end{align}\nUp to scalar coefficients depending on $\\zeta $,  \nthese operators\ncan be guessed immediately, but the coefficient, especially\nthe sign, in $\\mathcal{R} _{4,4}(\\zeta)$ is important for us. \nAs usual we define:\n$$\nR ^-_{\\{A,a\\},\\{B,b\\}}(\\zeta)\n=\\sigma ^1_a\\sigma ^1_bR^+_{\\{A,a\\}\\{B,b\\}}(\\zeta)\n\\sigma ^1_a\\sigma ^1_b\n\\,.\n$$\n\n\nNow we have everything necessary for the definition of\nthe operators $\\mathbf{b} (\\zeta )$ and $\\mathbf{c} (\\zeta )$. \n{}For two integers $k\\le l$ we set \n$$\nM_{[k,l]}=M_{k}\\otimes\\cdots \\otimes M_{l}\\,.\n$$\nThis is the algebra of linear operators on the `quantum space'\non the interval $[k,l]$.  \nOur main object is the monodromy matrix\n\\begin{align}\nT^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )=L^{\\pm}_{\\{A,a\\},l}(\\zeta )\\cdots L^{\\pm}_{\\{A,a\\},k}(\\zeta )\n\\in Osc_A\\otimes M^{\\mp}_a\\otimes M_{[k,l]}\\,.\n\\label{monodromy0}\n\\end{align}\nDefine further an element\n$\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta)\\in \nOsc_A\\otimes M^{\\mp}_a\\otimes \\text{End}(M_{[k,l]})$\nby setting\n\\begin{align}\n&\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta)(X_{[k,l]})= \nT^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )\\cdot \n(1_{A,a}\\otimes X_{[k,l]})\\cdot\nT^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )^{-1}\n\\, ,\n\\label{monodromy}\n\\end{align}\nwhere $1_{A,a}=1_A\\otimes 1_a$ and $X_{[k,l]}\\in M_{[k,l]}$. \nTo illustrate the definition, we have, for \n$x_{\\{A,a\\}}\\in Osc _A\\otimes M_a^{\\mp}$ and $X_{[k,l]}\\in\nM_{[k,l]}$, an equality in \n$Osc_A\\otimes M^{\\mp}_a\\otimes M_{[k,l]}$\n\\begin{align}\n&\\(\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )\\cdot x_{\\{A,a\\}}\\otimes id\\)\\(X_{[k,l]}\\)\\nonumber\\\\\n&=T^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )\\cdot\\(1_{\\{A,a\\}}\\otimes X_{[k,l]}\\)\\cdot\nT^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta )^{-1}\\cdot \n(x_{\\{A,a\\}}\\otimes 1_{[k,l]})\\,,\\nonumber\n\\end{align}\nwhere \n$id$ is the identity operator in $\\text{End}(M_{[k,l]})$.\n\nWe define $\\mathbb{T}^{\\pm}_{A,[k,l]}(\\zeta)\\in Osc_A\\otimes \\text{End}(M_{[k,l]})$ and $\\mathbb{T}_{a,[k,l]}(\\zeta)\\in M_a\\otimes \\text{End}(M_{[k,l]})$ in a similar manner.\n\n\nIn the following we shall use only $\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta)^{-1}$. \nWe understand certain inconvenience in using the inverse\noperators, but it has for us a \nhistorical reason:\nonce we define the transfer-matrix as in \\cite{JM}, the order\nof multipliers is fixed everywhere.\n\nWe have the Yang-Baxter equation \n\\begin{align}\n&\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta_1)^{-1}\\mathbb{T} ^{\\pm}_{\\{B,b\\},[k,l]}(\\zeta_2)^{-1}\nR^{\\pm}_{\\{A,a\\},\\{B,b\\}}(\\zeta_1\/\\zeta _2)\n\\label{rightYB}\n\\\\&\\quad=\nR^{\\pm}_{\\{A,a\\},\\{B,b\\}}(\\zeta_1\/\\zeta _2)\n\\mathbb{T} ^{\\pm}_{\\{B,b\\},[k,l]}(\\zeta_2)^{-1}\n\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta_1)^{-1}\\,, \n\\nonumber\n\\end{align}\nwhere the identity is in $Osc_A\\otimes M^{\\mp}_a\\otimes\nOsc_B\\otimes M^{\\mp}_b\\otimes \\text{End}(M_{[k,l]})$. \n\nOf particular importance are the $Q$-operators acting on local operators. \nThey are defined as \n\\begin{align}\n& \\mathbf{Q} ^+_{[k,l]}(\\zeta, \\alpha )=\n{\\rm tr} ^+_A\\(q^{2\\alpha D_A}\\ \\mathbb{T}^{+}_{A,[k,l]}(\\zeta)^{-1}\\)(1-q^{2(\\alpha-\\mathbf{S})}), \n\\label{Qop}\\\\\n & \n\\mathbf{Q}^-_{[k,l]}(\\zeta, \\alpha )={\\rm tr} ^-_A\\(q^{-2\\alpha (D_A+1)}\\ \\mathbb{T} ^{-}_{A,[k,l]}(\\zeta)\n ^{-1}\\)q^{2\\mathbf{S} }\n (1-q^{2(\\alpha-\\mathbf{S})})\\,,\n\\nonumber\n\\end{align}\nwhere $\\mathbf {S} $ stands for the adjoint action of the total spin\noperator \n$$ \n\\mathbf {S} (X_{[k,l]})=\\bigl[\nS(l)-S(k-1)\\ ,\\ X_{[k,l]}\\bigr]\\, ,\n\\qquad\nX_{[k,l]}\\in M_{[k,l]}\\,.\n$$\nThe trace functionals ${\\rm tr} ^{+}_A\\bigl(q^{2\\alpha D_A}Y_A\\bigr)$ and \n${\\rm tr} ^{-}_A\\bigl(q^{-2\\alpha (D_A+1)}Y_A\\bigr)$ for $Y_A\\in Osc_A$ are defined\nas analytic \ncontinuations with respect to $\\alpha$ of traces over\n$W^+$ and $W^-$ from the region $|q^{\\alpha}|<1$. \nThe $Q$-operators \\eqref{Qop} are mutually commuting families \nof operators. They are\nso normalized that $\\mathbf{Q}^{\\pm}_{[k,l]}(0,\\alpha)=1$.\n\n\nRegarding $\\mathbb{T} ^{\\pm}_{\\{A,a\\},[k,l]}(\\zeta)^{-1}$ \nas a matrix in $M^{\\mp }_a$, let us write its entries as\n\\begin{align}\n&\\mathbb{T} ^{+}_{\\{A,a\\},[k,l]}(\\zeta)^{-1}=\\begin{pmatrix}\n\\mathbb{A} ^+_{A,[k,l]}(\\zeta)&0\\\\\n\\mathbb{C} ^+_{A,[k,l]}(\\zeta)&\\mathbb{D} ^+_{A,[k,l]}(\\zeta)\n\\end{pmatrix}_a,\\nonumber\\\\\n&\\mathbb{T} ^{-}_{\\{A,a\\},[k,l]}(\\zeta)^{-1}=\\begin{pmatrix}\n\\mathbb{A} ^-_{A,[k,l]}(\\zeta)&\\mathbb{B} ^-_{A,[k,l]}(\\zeta)\\\\\n0&\\mathbb{D} ^-_{A,[k,l]}(\\zeta)\n\\end{pmatrix}_a\\, ,\n\\nonumber\n\\end{align}\nwhere $\\mathbb{A} ^+_{A,[k,l]}(\\zeta)$, etc., are elements of\n$Osc_A\\otimes \\text{End}(M_{[k,l]})$.\nIt follows from the definition that $\\mathbb{T} ^{\\pm\\ }_{\\{A,a\\},[k,l]}(\\zeta)^{-1}$ \nhave poles of order $l-k+1$ at the points \n$\\zeta^2=1, q^{\\pm 2}$. \nHowever, looking at the formulae (\\ref{fusionright+})--\n(\\ref{fusioninv}), one realizes that \nat the pole $\\zeta ^2=1$ only $\\mathbb{C} ^+_{A,[k,l]}(\\zeta)$ and\n$\\mathbb{B} ^-_{A,[k,l]}(\\zeta)$ are singular.  \nThis motivates, at least partly, the following definition:\n\\begin{align}\n&\\mathbf{c} _{[k,l]}(\\zeta,\\alpha)=q^{\\alpha -\\mathbf{S} }\\(1-q^{2(\\alpha-\\mathbf{S})}\\)\n\\text{sing}_{\\,\\zeta=1}\\left[\n\\zeta ^{\\alpha -\\mathbf{S}}{\\rm tr} ^+_A\\(q^{2\\alpha D_A}\\ \\mathbb{C} ^+_{A,[k,l]}(\\zeta)\n\\)\n\\right],\\label{defc1}\\\\\n&\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)={q^{2\\mathbf{S}}} \\text{\\,sing}_{\\,\\zeta=1}\\left[\n\\zeta ^{-\\alpha +\\mathbf{S}}{\\rm tr} ^-_A\\(q^{-2\\alpha(D_A+1)}\\ \\mathbb{B} ^-_{A,[k,l]}(\\zeta)\n\\)\n\\right]\\,.\n\\label{defb1}\n\\end{align}\nHere and after, $\\text{\\,sing}_{\\zeta=1}[f(\\zeta )]$ signifies the singular part of \n$f(\\zeta)$ at $\\zeta =1$:\n\\begin{align}\n\\text{\\,sing}_{\\zeta=1}\\[f(\\zeta)\\]=\n\\frac 1 {2\\pi i}\\int \\frac { f(\\xi)}  {\\zeta-\\xi}d\\xi\n\\,,\n\\label{int}\n\\end{align}\nwhere the integral is taken over a simple closed curve containing \n$\\xi=1$ inside, while \n$\\xi =\\zeta $ and other singular points of $f(\\xi)$ are outside. \nWe note that \n$$\n[\\mathbf{S},\\mathbf{c} _{[k,l]}(\\zeta,\\alpha)]=\\mathbf{c} _{[k,l]}(\\zeta,\\alpha),\n\\quad \n[\\mathbf{S},\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)]=-\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)\\,.\n$$\n\nThere are several important properties of operators $\\mathbf{c} _{[k,l]}(\\zeta,\\alpha)$ and\n$\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)$ which we formulate as Lemmas.\n\\begin{lem}\\label{lem1}\nThe operators $\\mathbf{c} _{[k,l]}(\\zeta,\\alpha)$ and\n$\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)$ satisfy the following anti-commutation relations:\n\\begin{align}\n&\\mathbf{c} _{[k,l]}(\\zeta_1,\\alpha-1)\\mathbf{c} _{[k,l]}(\\zeta _2,\\alpha)=-\\mathbf{c} _{[k,l]}(\\zeta _2,\\alpha-1)\\mathbf{c} _{[k,l]}(\\zeta_1,\\alpha)\n\\,,\n\\label{commcc}\\\\\n&\\mathbf{b} _{[k,l]}(\\zeta_1,\\alpha+1)\\mathbf{b} _{[k,l]}(\\zeta _2,\\alpha)=-\\mathbf{b} _{[k,l]}(\\zeta _2,\\alpha+1)\\mathbf{b} _{[k,l]}(\\zeta_1,\\alpha)\\,.\n\\label{commbb}\n\\end{align}\n\\end{lem}\n\n\n\\begin{proof}\nConsider the Yang-Baxter equations (\\ref{rightYB}) for $+$. Using the $R$-matrix (\\ref{R+ab}) one finds:\n\\begin{align}\n&\\zeta_1^{-2}\\mathbb{C}^+_{A,[k,l]}(\\zeta _1)\\mathbb{C}^+_{B,[k,l]}(\\zeta _2)\n\\label{CC}\\\\\n&+{\\zeta _2} ^{-2}q^{D_A}R_{A,B}(\\zeta_1\/\\zeta_2)q^{-D_B}\\cdot\n\\mathbb{C}^+_{B,[k,l]}(\\zeta _2)\\mathbb{C} ^+_{A,[k,l]}(\\zeta _1)\\cdot\nq^{-D_B}R_{A,B}(\\zeta_1\/\\zeta_2)^{-1}q^{D_A}=\\cdots\\nonumber\n\\end{align}\nwhere $\\cdots $ stands for a sum of \nterms which contain at least one\n$\\mathbb{A}^+_{[k,l]}(\\zeta _i)$ or $\\mathbb{D} ^+_{[k,l]}(\\zeta _i)$,  \nand hence have vanishing singular parts at $\\zeta _i=1$.\nMultiplying (\\ref{CC}) by $q^{2(\\alpha-1)D_A+2\\alpha D_B}$, \ntaking the trace and the singular part, \none immediately gets (\\ref{commcc}).\nSimilarly one proves (\\ref{commbb}) using (\\ref{rightYB}) for $-$.\n\\end{proof}\n\n\\begin{lem}\\label{lem2}\nWe have the following reduction relations:\n\\begin{align}\n&\\mathbf{c} _{[k,l]}(\\zeta ,\\alpha)\\(X_{[k,l-1]}\\cdot 1_l\\)=\\mathbf{c} _{[k,l-1]}(\\zeta ,\\alpha )\\(X_{[k,l-1]}\\)\\cdot 1_l\\,,\n\\label{redc+}\\\\\n&\\mathbf{b} _{[k,l]}(\\zeta ,\\alpha)\\(X_{[k,l-1]}\\cdot 1_l\\)=\\mathbf{b} _{[k,l-1]}(\\zeta ,\\alpha )\\(X_{[k,l-1]}\\)\\cdot 1_l\\,,\n\\label{redb+}\\\\\n&\\mathbf{c} _{[k,l]}(\\zeta ,\\alpha)\\(q^{\\alpha\\sigma ^3_k}\\cdot X_{[k+1,l]}\\)=\nq^{(\\alpha-1)\\sigma ^3_k}\\cdot\n\\mathbf{c} _{[k+1,l]}(\\zeta ,\\alpha)\\(X_{[k+1,l]}\\)\n\\,,\n\\label{redc-}\\\\\n&\\mathbf{b} _{[k,l]}(\\zeta ,\\alpha)\\(q^{\\alpha\\sigma ^3_k}\\cdot X_{[k+1,l]}\\)=\nq^{(\\alpha+1)\\sigma ^3_k}\\cdot\n\\mathbf{b} _{[k+1,l]}(\\zeta ,\\alpha)\\(X_{[k+1,l]}\\)\\label{redb-}\\,.\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nThe equations (\\ref{redc+}), (\\ref{redb+}) are\ntrivial consequences of the definition. \nIn contrast, eqs. (\\ref{redc-}), (\\ref{redb-})\nare far from being obvious. \n\nConsider the first of them. By definition we have:\n\\begin{align}\n&\\frac 1 {q^{\\alpha-\\mathbf{S} }\\(1-q^{2(\\alpha-\\mathbf{S})}\\)}\n\\mathbf{c} _{[k,l]}(\\zeta ,\\alpha)\\(q^{\\alpha\\sigma ^3_k}\\cdot X_{[k+1,l]}\\)\\nonumber\n\\\\\n&=\n\\text{\\,sing}_{\\zeta=1}\\left[\n{\\rm tr} ^+_A\\(q^{2\\alpha D_A}\\ \\mathbb{C}_{A,[k,l]}^+(\\zeta)\\(q^{\\alpha\\sigma ^3_k}\\cdot X_{[k+1,l]}\\)\n\\)\\zeta^{\\alpha -s-1}\n\\right]\\,,\n\\nonumber\n\\end{align}\nwhere $s$ is the spin of $X_{[k+1,l]}$.\n\nLet us simplify the trace.  \nWe will use the crossing symmetry\n\\begin{align}\n&\\mathcal{P} ^- _{j,\\bar j}\nL ^+_{A,j}(\\zeta q ^{-1})L ^+_{A, \\bar j}(\\zeta )=\n(\\zeta -\\zeta^{-1})\\mathcal{P} ^- _{j,\\bar j}\\,,\n\\label{cross1}\\\\\n&\\mathcal{P} ^- _{j,\\bar j}\nL ^+_{\\{A,a\\},j}(\\zeta q ^{-1})L ^+_{\\{A,a\\}, \\bar j}(\\zeta )=\n(\\zeta q -\\zeta^{-1}q^{-1})(\\zeta -\\zeta^{-1})(\\zeta q^{-1}-\\zeta^{-1}q)\\mathcal{P} ^- _{j,\\bar j}\n\\,,\n\\label{cross2}\n\\end{align}\nwhere $\\mathcal{P}^-_{j,\\bar j}$ is the anti-symmetrizer. \nIntroducing consecutively some additional two-dimensional spaces, we have \n\\begin{align}\n&{\\rm tr} ^+_A\\(q^{2\\alpha D_A}\\ \\mathbb{C}_{A,[k,l]}^+(\\zeta)\\(q^{\\alpha\\sigma ^3_k}\\cdot X_{[k+1,l]}\\)\\)\n\\label{trtrtr}\n\\\\\n&=\n{\\rm tr} _a{\\rm tr} ^+_A\\(\\sigma _a^+L_{\\{A,a\\},k}^+(\\zeta)^{-1}\\cdot\n\\mathbb{T}_{\\{A,a\\},[k+1,l]}^+(\\zeta)^{-1}\\(X_{[k+1,l]}\\)\\cdot\nq^{\\alpha\\sigma ^3_k} L_{\\{A,a\\},k}^+(\\zeta)q^{2\\alpha D_A}\\)\\nonumber\\\\&=\n\\frac {1}{(\\zeta -\\zeta ^{-1})\n(\\zeta q -\\zeta ^{-1}q^{-1})(\\zeta q^{-1}-\\zeta ^{-1}q)}\\nonumber\\\\&\\times\n{\\rm tr} _{\\bar k}{\\rm tr} _a{\\rm tr} ^+_A\\(\\sigma _a^+L_{\\{A,a\\},\\bar k}^+(\\zeta q^{-1})\n\\cdot\n2\\mathcal{P}^-_{k,\\bar k}\\cdot\n\\mathbb{T}_{\\{A,a\\},[k+1,l]}^+(\\zeta)^{-1}\n\\(X_{[k+1,l]}\\)\\cdot\nq^{\\alpha\\sigma ^3_k} \n\\cdot\nL_{\\{A,a\\},k}^+(\\zeta)q^{2\\alpha D_A}\\)\\nonumber\\,.\n\\end{align}\nNow use\n$$\nq^{\\alpha \\sigma ^3_k}L ^+_{A,k}(\\zeta)q^{2\\alpha D_A}\n=q^{2\\alpha D_A}L^+_{A,k}(\\zeta)q^{\\alpha \\sigma ^3_k}\n$$ \nand the cyclicity of the trace to simplify (\\ref{trtrtr}) further:\n\\begin{align}\n&{\\rm tr} ^+_A\\(q^{2\\alpha D_A}\\ \\mathbb{C}_{A,[k,l]}^+(\\zeta)\\(q^{\\alpha\\sigma ^3_k}\n\\cdot X_{[k+1,l]}\\)\\)\n\\label{exp1}\n\\\\\n&=\n\\frac {1}{(\\zeta -\\zeta ^{-1})\n(\\zeta q -\\zeta ^{-1}q^{-1})(\\zeta q^{-1}-\\zeta ^{-1}q)}\n\\nonumber\n\\\\\n&\\times\n{\\rm tr} _{\\bar k}{\\rm tr} _a{\\rm tr} ^+_A\\(\n\\mathbb{T}_{\\{A,a\\},[k+1,l]}^+(\\zeta)^{-1}\\(X_{[k+1,l]}\\)q^{2\\alpha D_A}\\mathcal{L}(\\zeta )\nq^{\\alpha\\sigma ^3_k} \\)\n\\,.\\nonumber\n\\end{align}\nIt is easy to see that\n\\begin{align}\n\\mathcal{L}(\\zeta )&=2\\mathcal{P}^-_{k,\\bar k}L_{\\{A,a\\},k}^+(\\zeta)\\sigma _a^+L_{\\{A,a\\},\\bar k}^+(\\zeta q^{-1})\n\\label{exp2}\\\\\n&=\n\\begin{pmatrix}\n\\zeta q-\\zeta^{-1}q^{-1} & 0\\\\\n0 &(q-q^{-1})q^{-2D_A-\\frac 1 2}\n\\end{pmatrix}_a\n2\\mathcal{P}^-_{k,\\bar k}L_{A,k}^+(\\zeta q^{-1})L_{A,\\bar k}^+(\\zeta )\n\\nonumber\\\\ \n&\\times\n\\begin{pmatrix}\nq^{-\\frac {\\sigma ^3_k} 2} \\sigma ^+_{\\bar k}  &    \nq^{\\frac {\\sigma ^3_{\\bar k}-\\sigma ^3_k} 2}\\\\[5pt]\n\\sigma ^+ _k \\sigma ^+_{\\bar k}        &      \\sigma ^+ _kq^{\\frac {\\sigma ^3_{\\bar k}} 2}\n\\end{pmatrix}_a\n\\begin{pmatrix}\n(q-q^{-1})q^{-2D_A+\\frac 1 2} & 0\\\\ 0 &\\zeta q^{-1}-\\zeta ^{-1}q\n\\end{pmatrix}_a\\,.\n\\nonumber\n\\end{align}\nwhere we used\n$$L^+_{A,j}(\\zeta q)\\sigma _j^+q^{-2D_A+\\frac 1 2}=\nq^{-2D_A-\\frac 1 2}L^+_{A,j}(\\zeta q^{-1})\\sigma _j^+\\,.$$\nIn view of \\eqref{cross1}, $\\mathcal{L}(\\zeta )$ is divisible by $\\zeta-\\zeta^{-1}$, and \nin (\\ref{exp1}) we can drop the diagonal elements of \n$\\mathbb{T}_{\\{A,a\\},[k+1,l]}^+(\\zeta)^{-1}$,  \narriving immediately at (\\ref{redc-}).\n\nThe proof of (\\ref{redb-}) is similar.\n\\end{proof}\n\n\\noindent\n{\\bf Remark.} \nThe above construction carries over to inhomogeneous chains \nwhere an independent spectral parameter $\\xi_j$ is attached to each site $j$. \nThe operators $\\mathbf{c} _{[k,l]}(\\zeta;\\xi _k,\\cdots ,\\xi _l)$, \n$\\mathbf{b} _{[k,l]}(\\zeta;\\xi _k,\\cdots ,\\xi _l)$ \nare defined via the above construction with two modifications: \n\\begin{enumerate}\n\\item In the definition (\\ref{monodromy0}), each \n$L ^{\\pm}_{\\{A,a \\},j}\\({\\zeta}\\)$ is replaced by \n$L ^{\\pm}_{\\{A,a \\},j}\\({\\zeta}\/{\\xi_j}\\)$.\n\\item The singular part is understood as an \nintegral (\\ref{int}) around the points $\\xi_k,\\cdots,\\xi_l$. \n\\end{enumerate}\nLemma \\ref{lem1} and Lemma \\ref{lem2} remain valid. \n\\hfill{\\qed}\n\\medskip\n\nLemma \\ref{lem2} allows us to define \nuniversal operators $\\mathbf{b} (\\zeta,\\alpha)$, $\\mathbf{c} (\\zeta ,\\alpha)$: \n\\begin{definition}\\label{defbc}\n{}For any operator\n$q^{2\\alpha S(0)}\\mathcal{O}\\in \\mathcal{W}_{\\alpha}$, \nlet \n$\\(q^{2\\alpha S(0)}\\mathcal{O} \\)_{[k,l]}$ be its restriction \nto the finite interval $[k,l]$ of the lattice.  \nWe define \n\\begin{align}\n&\\mathbf{b} (\\zeta,\\alpha):\\ \\mathcal{W}_{\\alpha,s}\\to \n\\mathcal{W}_{\\alpha+1,s-1}\n\\ ,\\label{actb}\\\\\n&\\mathbf{c} (\\zeta ,\\alpha ):\\ \\mathcal{W}_{\\alpha,s}\\to \n\\mathcal{W}_{\\alpha-1,s+1}\n\\label{actc}\\,,\n\\end{align}\nby setting \n\\begin{align}\n&\\mathbf{b} (\\zeta,\\alpha)\\(q^{2\\alpha S(0)}\\mathcal{O}\\)\n=\\lim _{k\\to-\\infty,l\\to \\infty}\n\\mathbf{b} _{[k,l]}(\\zeta,\\alpha)\\(\\(q^{2\\alpha S(0)}\\mathcal{O}\\)_{[k,l]}\\)\\,,\n\\label{defb}\\\\\n&\\mathbf{c} (\\zeta,\\alpha)\\(q^{2\\alpha S(0)}\\mathcal{O}\\)\n=\\lim _{k\\to-\\infty,l\\to \\infty}\n\\mathbf{c} _{[k,l]}(\\zeta,\\alpha)\\(\\(q^{2\\alpha S(0)}\\mathcal{O}\\)_{[k,l]}\\)\\,.\n\\label{defc}\n\\end{align}\n\\end{definition}\nIt follows from Lemma \\ref{lem2} that for any particular operator\n$q^{2\\alpha S(0)}\\mathcal{O}$ the expressions under the limit \nin (\\ref{defb}), (\\ref{defc}) stabilize for sufficiently \nlarge interval $[k,l]$.  \nHence the limit is well-defined. \nIn particular we have, for any $k$, \n\\begin{align}\n\\mathbf{b} (\\zeta,\\alpha)(q^{2\\alpha S(k)})=0, \\quad \\mathbf{c} (\\zeta,\\alpha)(q^{2\\alpha S(k)})=0\\,. \\label{zerobc}\n\\end{align}\n\nDenoting by $\\mathbf{b}(\\zeta)$ and $\\mathbf{c} (\\zeta)$ the operators acting on \nthe direct sum $\\mathcal{W}_{[\\alpha]}$ we have \nthe anti-commutativity\n\\begin{align}\n\\[\\mathbf{b} (\\zeta_1),\\mathbf{b} (\\zeta_2)\\]_+=\\[\\mathbf{c} (\\zeta_1),\\mathbf{c} (\\zeta_2)\\]_+=0\\ .\\nonumber\n\\end{align}\n\nIn Appendix, we give a brief summary of \nthe algebraic formula for the correlation functions in the \npresence of disorder. The result is expressed \nin terms of the operator\n\\begin{align}\n\\mbox{\\boldmath$\\Omega $}&= -\\text{res}_{\\zeta_1=1}\\text{res}_{\\zeta_2=1}\n \\(\\mathbf{X}(\\zeta_1,\\zeta _2)\n \\mbox{\\boldmath$\\omega $} (\\zeta_2\/\\zeta _1)\n \\frac {d\\zeta _1}{\\zeta _1} \\frac {d\\zeta _2}{\\zeta _2}\\)\\,,\n\\label{Omega2}\n\\end{align}\nwhere $\\left.\\mathbf{X} (\\zeta_1,\\zeta _2)\\right|_{\\mathcal{W}_{\\alpha}}=\\mathbf{X} (\\zeta_1,\\zeta _2,\\alpha)$, the operator $\\mathbf{X} (\\zeta_1,\\zeta _2,\\alpha)$ is  given in either of the two formulas \n(\\ref{App0}), (\\ref{App1}), $\\mbox{\\boldmath$\\omega $} (\\zeta)$ is given by (\\ref{omega}).  \nThe following result allows us to express $\\mbox{\\boldmath$\\Omega $}$ in terms of \n$\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta )$. \nAt the same time, \nthe existence of two equivalent representations  \nguarantees the anti-commutativity between the latter. \n\\begin{lem}\\label{lem3}\nThe operator $\\mathbf{X}(\\zeta _1,\\zeta _2)$ can be evaluated as follows:\n\\begin{align}\n\\left.\\mathbf{X} (\\zeta _1,\\zeta _2)\\right|_{\\mathcal{W}_{\\alpha}}\n=\\mathbf{b} (\\zeta _2,\\alpha-1)\\mathbf{c} (\\zeta _1,\\alpha)=-\\mathbf{c} (\\zeta_1,\\alpha +1)\\mathbf{b} (\\zeta _2,\\alpha)\\,.\n\\label{X=bc=cb}\n\\end{align}\n\\end{lem}\n\\begin{proof}\nConsider the  formula (\\ref{App0}). We have:\n\\begin{align}\n{\\rm tr} _{a,b}&\\(\n B^0_{b,a}(\\zeta_2\/\\zeta_1)\\mathbb{T}_a(\\zeta_1)^{-1}\\mathbb{T}_b(\\zeta _2)^{-1}\\)\\mathbf{Q}^+(\\zeta_1,\\alpha+1)\\mathbf{Q}^-(\\zeta_2,\\alpha +1)\n\\nonumber\\\\\n&=\n {\\rm tr} _{a,b}{\\rm tr} _A^+{\\rm tr} _B^-\\(B^0_{b,a}(\\zeta_2\/\\zeta_1)      \n \\mathbb{T}_a(\\zeta_1)^{-1}\\mathbb{T}_b(\\zeta _2)^{-1}\n \\mathbb{T}^{+}_A(\\zeta _1)^{-1}\\mathbb{T}^{-}_B(\\zeta _2)^{-1}\\right. \\nonumber\n\\\\&\\left.\\times q^{2(\\alpha+1)(D_A-D_B-1)}\\)(1-q^{2(\\alpha+1-\\mathbf{S})})^2\nq^{2\\mathbf{S}}\n\\,.\n\\nonumber\n \\end{align}\nWe move $\\mathbb{T}_b(\\zeta_2)^{-1}$ through $\\mathbb{T}^+_A(\\zeta _1)^{-1}$ using\nthe Yang-Baxter equation\n$$\nL ^+_{A,b}\\({\\zeta _1}\/{\\zeta _2}\\)\\mathbb{T}_b(\\zeta_2)^{-1}\\mathbb{T}^{+}_A(\\zeta _1)^{-1}\n=\\mathbb{T}^{+}_A(\\zeta _1)^{-1}\\mathbb{T}_b(\\zeta_2)^{-1}L^+_{A,b}\\({\\zeta _1}\/{\\zeta _2}\\)\\,.\n$$\nNow $\\mathbb{T}_a(\\zeta_1)^{-1} \\mathbb{T}^{+}_A(\\zeta _1)^{-1}$ and \n$\\mathbb{T}_b(\\zeta_2)^{-1} \\mathbb{T}^{-}_B(\\zeta _2)^{-1}$  come together. \nConjugating by $G ^+_{A,a}$, $G^-_{B,b}$,  \nwe can combine them into \nthe monodromy matrices \n$\\mathbb{T} ^{+}_{\\{A,a\\}}(\\zeta_1)^{-1}$, $\\mathbb{T}^{-}_{\\{B,b\\}}(\\zeta _2)^{-1}$.\nIn these monodromy matrices we drop\ndiagonal elements \nbecause they have no singularities at $\\zeta _i=1$. \nThen by a straightforward calculation we come to\n\\begin{align}\n&{\\rm tr} _{a,b}\\(\n B^0_{b,a}(\\zeta_2\/\\zeta_1)\n \\mathbb{T}_a(\\zeta_1)^{-1}\\mathbb{T}_b(\\zeta _2)^{-1}\n\\)\\mathbf{Q}^+(\\zeta_1,\\alpha+1)\\mathbf{Q}^-(\\zeta_2,\\alpha +1)\n\\label{x1}\\\\\n&\n\\simeq\n-\n {\\rm tr} _A^+{\\rm tr} _B^-\\(\\mathbb{C}^+_{A}(\\zeta _1)\\mathbb{B} ^-_B(\\zeta _2)q^{2(\\alpha+1) D_A-2\\alpha (D_B+1)-2}\\)(1-q^{2(\\alpha-\\mathbf{S}+1)})^2\nq^{2\\mathbf{S}}\n\\nonumber \n\\end{align}\nwhere $\\simeq$ means that the singular parts are identical. \nSimilarly we have:\n\\begin{align}\n&{\\rm tr} _{a,b}\\(\n B^1_{a,b}(\\zeta_1\/\\zeta_2)\\mathbb{T}_b(\\zeta _2)^{-1}\\mathbb{T}_a(\\zeta_1)^{-1}\\)\\mathbf{Q}^-(\\zeta_2,\\alpha -1)\\mathbf{Q}^+(\\zeta_1,\\alpha-1)\n\\label{x2} \\\\\n&\n\\simeq\n- {\\rm tr} _A^+{\\rm tr} _B^-\\(\\mathbb{B} ^-_B(\\zeta _2)\\mathbb{C}^+_{A}(\\zeta _1)\nq^{2\\alpha D_A-2(\\alpha-1)(D_B+1)}\\)\n(1-q^{2(\\alpha-\\mathbf{S}-1)})^2q^{2\\mathbf{S}}\n\\,.\n\\nonumber\n\\end{align}\nEq. (\\ref{X=bc=cb}) follows from (\\ref{x1}),\n(\\ref{x2}) and the definition of $\\mathbf{b} (\\zeta ,\\alpha)$, $\\mathbf{c} (\\zeta ,\\alpha)$.\n\\end{proof}\nThe main formula \\eqref{main} follows from \\eqref{Omega2},\n\\eqref{X=bc=cb} \nand \\eqref{sol}. \n\nLet $U$ be the shift operator by one lattice unit, which acts \non local operators by adjoint:\n$$\nU\\sigma ^a_jU^{-1}=\\sigma ^a_{j+1}.\n$$\nThere is also an infinite set of local integrals of motion\nwhich commute with $U$ and among themselves.\nThe last important property of $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$ \nis their invariance:\n\\begin{lem}\\label{lem4}\nThe operators $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$ commute with the\nadjoint action of the shift operator $U$ and of the local integrals of motion.\n\\end{lem}\n\\begin{proof}\n{}For $U$ the statement of this lemma follows immediately from\nthe definition, essentially it is a consequence of Lemma \\ref{lem2}.\n\nThe local integrals of motion are of the form\n\\begin{align}\nI_p=\\sum\\limits _{j=-\\infty}^{\\infty} d_{j,p}\\,,\n\\label{plocal}\n\\end{align}\nwhere $d_{j,p}$ is an operator acting non-trivially on the sites \n$j,\\cdots, j+p$. We shall call operators of the type (\\ref{plocal}) \n$p$-local operators.\n\nLet us write the $4\\times 4$ $R$-matrix as \n$\\check{R}_{j,k}(\\xi)=P_{j,k}L_{j,k}(\\xi)$. We set  \n$$ \nU_{[k,l]}(\\xi)=(q-q^{-1})^{k-l}\n\\check{R}_{l,l-1}(\\xi)\\cdots \\check{R}_{k+1,k}(\\xi)\\,.\n$$\n{}Following the remark after Lemma \\ref{lem2},  \nconsider $\\mathbf{c} _{[k,l]}$ with one inhomogeneity:\n$$\n\\mathbf{c} _{[k,l]}(\\zeta;\\xi, 1,\\cdots, 1)\\ \\text{and }\\ \\mathbf{c} _{[k,l]}(\\zeta;1,\\cdots, 1,\\xi)\\,.\n$$\nIt is clear from the definition that\n\\begin{align}\n&U_{[k,l]}(\\xi)\\cdot \n\\mathbf{c} _{[k,l]}(\\zeta;\\xi, 1,\\cdots, 1)\\(\\(q^{2\\alpha S(0)}\\mathcal{O}\\)_{[k,l]}\\)\\ \n\\cdot U_{[k,l]}(\\xi)^{-1}\n\\label{comm}\\\\\n&=\\mathbf{c} _{[k,l]}(\\zeta;1,\\cdots, 1,\\xi)\n\\(U_{[k,l]}(\\xi)\\cdot \\(q^{2\\alpha S(0)}\\mathcal{O}\\)_{[k,l]}\\cdot \nU_{[k,l]}(\\xi)^{-1}\\)\\,.\n\\nonumber \n\\end{align}\n \n \nLet $\\xi=1+\\epsilon$. Then \n$$\nU_{[k,l]}(\\xi)=\\exp\\(\\sum\\limits _{p=1}^{\\infty} \\epsilon^p I_{[k,l],p}\\)\\,.\n$$\nDue to the Campbell-Hausdorff formula,  \nthe operators $I_{[k,l],p}$ are $p$-local. \n{}For finite $k,l$ these operators do not \ncommute because of some boundary terms, \nbut in the limit $k\\to-\\infty$, \n$l\\to\\infty$ they coincide with the local integrals of motion \n$I_p$ which are combined into the generating function:\n$$\nU(\\xi)=\\exp\\bigl(\\sum\\limits _{p=1}^{\\infty}\\epsilon ^p I_{p}\\bigr)\\,.\n$$\n \nIn the right hand side of (\\ref{comm}) we have the expression\n$$\nU_{[k,l]}(\\xi)\\cdot \n\\(q^{2\\alpha S(0)}\\mathcal{O}\\)_{[k,l]}\\cdot \nU_{[k,l]}(\\xi)^{-1}=\\sum\\limits \\epsilon ^p \n\\(q^{2\\alpha S(0)}\\mathcal{O}\\)^{(p)}_{[k,l]}\\,.\n$$\nHere the $p$-local operators $I_{[k,l],p}$ act by multiple adjoint.\nIt is clear that for every given degree $p$ we can \nfind a large enough interval $[k,l]$ in order that \n$$ \n\\(q^{2\\alpha S(0)}\\mathcal{O}\\)^{(p)}_{[k,l]}\n=\\(\\bigl(q^{2\\alpha S(0)}\\mathcal{O}\\bigr)^{(p)}\\)_{[k,l]}\\,,\n$$\nwhere \n$$\nU(\\xi)\\cdot q^{2\\alpha S(0)}\\mathcal{O}\\cdot U(\\xi)^{-1}=\n\\sum\\limits \\epsilon ^p \\(q^{2\\alpha S(0)}\\mathcal{O}\\)^{(p)}\\,.\n$$\nObviously\n$$\n\\text{length}\\(\\bigl(q^{2\\alpha S(0)}\\mathcal{O}\\bigr)^{(p)}\\)\\le \n\\text{length}\\(q^{2\\alpha S(0)}\\mathcal{O}\\)+2p\\,. \n$$ \n        \nNow  considering (\\ref{comm}) order by\norder in $\\epsilon$, choosing for \nevery order sufficiently large interval\n$[k,l]$ and using the inhomogeneous version of\nLemma \\ref{lem2} and the definition of $\\mathbf{c} (\\zeta)$,  \nwe get:\n\\begin{align}\n&U(\\xi)\\cdot \n\\mathbf{c} (\\zeta)\\(q^{2\\alpha S(0)}\\mathcal{O}\\)\\cdot U(\\xi)^{-1}=\n\\mathbf{c} (\\zeta)\\(U(\\xi)\\cdot q^{2\\alpha S(0)}\\mathcal{O}\\cdot U(\\xi)^{-1}\\),\n\\label{comm1}\n\\end{align}\nwhich  is understood as an equality of power series in $\\epsilon$.\n\\end{proof}\n\n\\section{Free fermion point}\\label{XX}\n\nConsider the point $\\nu =1\/2$, $q=i$. \nFor this coupling constant\nthe Hamiltonian turns into\n$$\nH_{XX}=\\sum\\limits _{j=-\\infty}^{\\infty}\\(\\sigma _j^+\\sigma _{j+1}^-+\n\\sigma _j^-\\sigma _{j+1}^+\\)\\,,\n$$\nand can be diagonalized by the Jordan-Wigner transformation:\n$$\n\\psi _k ^{\\pm}=\\sigma ^{\\pm}_ke^{\\mp\\pi i S(k-1)\n}\\,.\n$$\nThe space $\\mathcal{W}_{[\\alpha]}$ becomes a\ndirect sum of two components:\n$$\\mathcal{W}_{[\\alpha]}=\\mathcal{W}_{\\alpha}\\oplus \\mathcal{W}_{\\alpha +1}\\,.$$\nWe set \n$$y= e^{\\frac{\\pi i \\alpha} 2}\\, ,$$\nso that the space $\\mathcal{W}_{\\alpha}$ consists of operators of the form \n$\ny^{ 2 S(0)}\\mathcal{O}\n$.\nThere are two fermion\noperators acting in the space of states, so, there are \nfour of them\nacting on the space of operators by left and right multiplication.\nIt is convenient to introduce the following four operators:\n\\begin{align}\n&\\Psi _{k}^{\\pm}(X)=\\psi ^{\\pm}_kX-(-1)^{F(X)} X\\psi ^{\\pm}_k,\n\\label{PsiPhi}\\\\\n&\\Phi_{\\alpha,k}^{\\pm}(X)=\n\\frac 1 {1-y^{\\mp 2}}\n\\(\\psi ^{\\pm}_kX-y^{\\mp 2}(-1)^{F(X)} X\\psi ^{\\pm}_k\\)\\, .\n\\nonumber\n\\end{align}\nwhere $F(X)$ is the fermionic number of the operator $X$.\n\nWe have $\\Phi_{\\alpha+2,k}^{\\pm}=\\Phi_{\\alpha,k}^{\\pm}$. \nThese operators are natural for us because $\\Psi_{k}^{\\pm}$\nannihilate $1$ while $\\Phi_{\\alpha,k}^{\\pm}$ annihilate $y^{2 S}$ \n(recall that at plus or minus infinity \n$y^{ 2 S(0)}\\mathcal{O}$ stabilizes to $1$ or  $y^{2 S}$). \nThe operators $\\Psi_{k}^{\\pm}$, $\\Phi_{\\alpha,k}^{\\pm}$ satisfy the \ncanonical anti-commutation relations:\n\\begin{align}\n&[\\Psi _{k}^{\\epsilon},\\Psi _{l}^{\\epsilon '}]_+=\n[\\Phi_{\\alpha,k}^{\\epsilon},\\Phi_{\\alpha,l}^{\\epsilon '}]_+=0,\\quad\n[\\Psi _{k}^{\\epsilon},\\Phi_{\\alpha,l}^{\\epsilon '}]_+\n=\\delta _{\\epsilon+\\epsilon ',0}\\delta _{k,l}\\,.\n\\label{comferm}\n\\end{align}\nIt is clear, however, that the operators\n$\\mathbf{b} (\\zeta,\\alpha)$, $\\mathbf{c} (\\zeta,\\alpha)$ cannot be constructed as linear \ncombinations of $\\Psi _{k}^{\\pm},\\ \\Phi_{\\alpha,k}^{\\pm}$. \nIndeed the operators\n$\\mathbf{b} (\\zeta,\\alpha)$, $\\mathbf{c} (\\zeta,\\alpha)$ are translationally invariant, in particular, they \nannihilate  $y^{2 S(k)}$ for any $k$, see (\\ref{zerobc}).\nClearly this is impossible for any linear combination of \n$\\Psi _{k}^{\\pm},\\ \\Phi_{\\alpha,k}^{\\pm}$. \nOur plan in this section is as\nfollows. First, we find a compact expression for $\\mathbf{b} (\\zeta,\\alpha)$ and\n$\\mathbf{c} (\\zeta,\\alpha)$ in terms of $\\Psi _{k}^{\\pm},\\ \\Phi_{\\alpha,k}^{\\pm}$. \nThen we show that our formula gives the same result for the correlators \nas the one \nobtained by a straightforward calculation based on normal ordering.\n\nThe calculation of $\\mathbf{b} (\\zeta,\\alpha)$, $\\mathbf{c} (\\zeta ,\\alpha)$ at the free fermon point\nis summarized by\n\\begin{lem}\\label{lem5}\nAt the free fermion point, the operators $\\mathbf{b} (\\zeta,\\alpha)$ \nand $\\mathbf{c}(\\zeta,\\alpha)$ are given by \n\\begin{align}\n&\\mathbf{b} (\\zeta ,\\alpha)=\\frac{2i^{-\\mathbf{S}}}{1-(-1)^{\\mathbf{S}}y^{2}}\\ \n{\\rm sing}_{\\zeta=1}\n\\[\\zeta ^{-\\alpha+\\mathbf{S}}\\Psi^-(\\zeta )E^{-}(\\zeta ,\\alpha-\\mathbf{S})\n\\frac{\\zeta}{1+\\zeta^2}\n\\]\\,,\n\\label{bcff}\\\\\n&\\mathbf{c} (\\zeta ,\\alpha)=2y\\ {\\rm sing}_{\\zeta=1}\n\\[\\zeta ^{\\alpha-\\mathbf{S}}\\Psi ^+(\\zeta )\nE^{+}(\\zeta ,\\alpha-\\mathbf{S})\n\\frac{\\zeta}{1+\\zeta^2}\n\\]\\,,\n\\nonumber\n\\end{align}\nwhere\n\\begin{align}\n\\Psi^{\\pm}(\\zeta )=\\sum\\limits _ {j=-\\infty}^{\\infty}\n\\Psi^{\\pm}_j\\(\\frac {1+\\zeta^2}\n{1-\\zeta ^2}\\)^{j}\\label{fermion}\n\\end{align}\nand\n\\begin{align}\nE^{\\pm}(\\zeta,\\alpha)=\\exp\\(\\mathcal{N}\\[\\Phi_{\\alpha}^{\\pm} \n\\log \\(I-\\zeta ^2M\\)\n\\Psi ^{\\mp}-\n\\Phi_{\\alpha}^{\\mp} \n\\log \\(I+\\zeta ^2M\\)\n\\Psi ^{\\pm}\\]\\)\\,.\\label{E}\n\\end{align}\nIn the last formula we consider $\\Phi_{\\alpha,j}^{\\pm}$ \n(resp. $\\Psi^{\\pm}_j$)\nas components of a row (resp. column) vector, \n$$ \nM=(1+u)(1-u)^{-1},\\qquad \n\\(u\\Psi ^{\\pm}\\)_j=\\Psi ^{\\pm}_{j+1}\\,,\n$$\nand \n$\\log \\(1\\pm\\zeta ^2M\\)$ are understood as Taylor series in $u$.\n$\\mathcal{N}[\\cdot]$ stands for the normal ordering \nwhich applies only to operators acting at the same site. For them we set\n\\begin{align}\n\\mathcal{N}[\\Phi_{\\alpha,j}^{\\epsilon}\\Psi_j^{\\epsilon'}]\n=\n\\begin{cases}\n\\Phi_{\\alpha,j}^{\\epsilon}\\Psi ^{\\epsilon'}_j&\\quad (j>0)\\,,\\\\\n-\\Psi ^{\\epsilon'}_j\\Phi_{\\alpha,j}^{\\epsilon}&\\quad (j\\le0 )\\,.\\\\\n\\end{cases}\n\\label{norm}\n\\end{align}\n\\end{lem} \nSince the $q$-oscillators become fermions at $q=i$,    \nLemma can be shown by \nmanipulations \nwith exponentials of quadratic forms in fermions. \nDetails will be given in another publication.\n\nWe remark that the exponent of (\\ref{E}) is well defined \nas an operator on  $\\mathcal{W}_{\\alpha}$. Indeed by definition\nit consists of $\\mathcal{N}\\(\\Phi_{\\alpha,k}^{\\pm}\\Psi ^{\\mp}_l\\)$ \nwith $l\\ge k$. \nOn a particular operator in $\\mathcal{W}_{\\alpha}$ \nonly a finite number of these operators do not vanish.\n\\medskip\n\nIt has been said that, unlike $\\mathbf{b} (\\zeta)$, $\\mathbf{c} (\\zeta)$,  \nformulae containing fermions necessarily\nbreak the translational invariance.  \nWe choose the point $k=1$ as the origin \nand consider only operators of the form \n\\begin{align}\ny^{2 S(0)}\\mathcal{O}_>\n\\label{O+}\n\\end{align}\nwhere $\\mathcal{O}_> $ acts only on the interval $[1,\\infty)$. \nAny operator in $\\mathcal{W}_{\\alpha}$ can be brought to the form (\\ref{O+}) by a shift, \nso we do not really lose generality.\nIn the sequel we need the operators on a half line:\n$$\n\\mathbf{b} _>(\\zeta,\\alpha)=\\mathbf{b}_{[1,\\infty)}(\\zeta,\\alpha),\\qquad\n\\mathbf{c} _>(\\zeta,\\alpha)=\\mathbf{c}_{[1,\\infty)}(\\zeta,\\alpha)\\,. \n$$\nThey are defined as in (\\ref{bcff}), replacing $E^{\\pm}(\\zeta,\\alpha)$, $\\Psi ^{\\pm}(\\zeta )$ and  \n$\\Phi_{\\alpha}^{\\pm}(\\zeta )$ by $E^{\\pm}_>(\\zeta,\\alpha)$, \n$\\Psi^{\\pm}_>(\\zeta )$ and  \n$\\Phi_{\\alpha,>}^{\\pm}(\\zeta )$, respectively. \nThe latter are  given by the same formulae (\\ref{fermion}), (\\ref{E}) \nwith  non-positive components of fermions removed. \n\nIn the free fermion case the function $\\omega (\\zeta ,\\alpha)$ can be\ncalculated explicitly.  \nPutting it together with \\eqref{bcff},  \nwe rewrite our main formula in the free fermion case as follows. \n\\begin{align}\n&\\frac{\\langle\\text{vac}|\\ y^{2 S(0)}\\mathcal{O}_>\n\\ |\\text{vac}\\rangle}{\\langle\\text{vac}|\\ y^{2 S(0)}\n\\ |\\text{vac}\\rangle}\n =\\mathbf{tr}_>^{\\alpha}\\(e^{\\mbox{\\scriptsize\\boldmath{$\\Omega$}}_>}\\(\\mathcal{O}_>\\)\\),\\label{mainferm}\\\\\n&\\mbox{\\boldmath$\\Omega $} _>=\n\\frac {i} {\\sin\\frac {\\pi  \\alpha} 2}{\\rm res} _{\\zeta _1=1}\n{\\rm res} _{\\zeta _2=1}\\(\n\\frac {\\zeta _1^{\\alpha}\\zeta_2^{-\\alpha}-1}{\\zeta _1^2+\\zeta _2^2}\nE^{-}_>(\\zeta _2,\\alpha)E^{+}_>(\\zeta _1,\\alpha)\n\\Psi_>^-(\\zeta _2)\\Psi_>^+(\\zeta _1)\n\\frac{d\\zeta_1^2}{1+\\zeta_1^2}\n\\frac{d\\zeta_2^2}{1+\\zeta_2^2}\n\\),\\nonumber\n\\end{align}\nwhere $\\mathbf{tr}_>^{\\alpha}$ means that the trace is calculated\nover the positive half of the chain only.\n\nNow\nnotice that \n\\begin{align}\n&\\Psi ^{\\pm}_>(\\zeta)(I)=0,\n\\quad \\mathbf{tr}_>^{2(\\alpha+1)}\n\\(\\Phi_{\\alpha,>}^{\\pm}(\\zeta)\\(\\mathcal{O}_>\\)\\)=0,\\label{cran}\\\\\n&\\psi ^{\\pm}_j\\mathcal{O}_>\n=\\(\\Phi_{\\alpha,j}^{\\pm}-\\frac {y^{\\mp 2}}{1-y^{\\mp 2}}\n\\Psi ^{\\pm}_j\\)(\\mathcal{O}_>)\\,.\n\\nonumber\n\\end{align}\nSo, by changing $ \\mathbf{tr}_>^{\\alpha}$ to $ \\mathbf{tr}_>^{2(\\alpha+1)}$, \nthe operators $\\Phi_{\\alpha,>}^{\\mp}$ and $\\Psi ^{\\pm}_>$ can be considered\nas creation-annihilation operators in the space of operators.\nFor efficient application of them we need \nthe following:\n\\begin{lem}\\label{lem6}\nThe following identity holds:\n\\begin{align}\n\\mathbf{tr}_>^{\\alpha}\\(e^{\\mbox{\\scriptsize\\boldmath{$\\Omega$}} _>}\\(\\mathcal{O}_>\\)\\)=\n\\mathbf{tr}_>^{2(\\alpha+1)}\\(e^{\\widetilde{\\mbox{\\scriptsize\\boldmath{$\\Omega$}}} _>}\\(\\mathcal{O}_>\\)\\)\n\\label{newtrace}\n\\end{align}\nwhere \n\\begin{align}\n\\widetilde{\\mbox{\\boldmath$\\Omega $}} _>= \n\\frac {i} {\\sin\\frac {\\pi  \\alpha} 2}{\\rm res} _{\\zeta _1=1}\n{\\rm res} _{\\zeta _2=1}\\(\n\\frac {\\zeta _1^{\\alpha}\\zeta_2^{-\\alpha}}{\\zeta _1^2+\\zeta _2^2}\n\\Psi_>^-(\\zeta _2)\\Psi_>^+(\\zeta _1)\n\\frac{d\\zeta_1^2}{1+\\zeta_1^2}\\frac{d\\zeta_2^2}{1+\\zeta_2^2}\n\\)\\,.\\nonumber\n\\end{align}\n\\end{lem}\n\nThe formulae (\\ref{cran}) and (\\ref{newtrace})\nallow an explicit calculation of correlators. One easily obtains:\n\\begin{align}\n\\frac{\\langle\\text{vac}|\\ y^{2 S(0)}\\psi ^+_{k_1}\\cdots \\psi ^+_{k_p}\n\\psi ^-_{l_p}\\cdots \\psi ^-_{l_1}\\ |\\text{vac}\\rangle }\n{\\langle\\text{vac}|y^{2 S(0)}\\ |\\text{vac}\\rangle }\n=\\text{det}\\(\\langle \\psi ^+_{k_i} \n\\psi ^-_{l_j}  \\rangle\\)_{i,j=1,\\cdots ,p}\\label{fermcorr}\n\\end{align}\nwhere \n\\begin{align}\n\\langle \\psi ^+_{k} \n&\\psi ^-_{l}  \\rangle =\\label{2point}\\\\\n&\\frac {i}{\\sin \\frac {\\pi\\alpha} 2}\n\\(-\\frac {y} 2\\delta _{k,l}\n+\\,{\\rm res} _{\\zeta_1=1}{\\rm res} _{\\zeta _2=1}\n\\frac {\\zeta _1^{\\alpha}\\zeta _2^{-\\alpha}}{\\zeta _1^2+\\zeta _2 ^2}\n\\(\\frac {1+\\zeta _1^2} {1-\\zeta _1^2} \\)^k\\(\\frac {1+\\zeta _2^2} {1-\\zeta _2^2} \\)^l\n\\frac {d\\zeta_1^2}{1+\\zeta _1 ^2}\\frac {d\\zeta_2^2}{1+\\zeta _2 ^2}\\)\\,.\n\\nonumber\n\\end{align}\n\nOn the other hand, \none can calculate the correlators (\\ref{fermcorr}) \ndirectly by normal ordering $y^{2 S(0)}$. \nThe result is the same: \n(\\ref{2point}) is the two-point function while (\\ref{fermcorr}) is\nobtained by the Wick theorem.\n\nThis calculation is unsatisfactory because\nwe had to pass through the fermions $\\Psi^{\\pm}_k$, \n$\\Phi_{\\alpha,k}^{\\pm}$.\nIt would be much better to find a basis in the space of local\noperators, on which the original\noperators $\\mathbf{b} (\\zeta ,\\alpha)$, $\\mathbf{c} (\\zeta ,\\alpha)$ act nicely. \nSuch a construction would have a chance to generalize to an arbitrary coupling constant.\n{}For the moment we cannot do that. \n\n\\section{Conclusion}\\label{sec:conclusion}\n\nThe main result of this paper can be formulated as follows.\nWe consider the space $\\mathcal{W}_{[\\alpha]}$ \nof local operators in the presence of a disorder field.\nWe have shown that the vacuum expectation values\nof operators in $\\mathcal{W}_{[\\alpha]}$ can be expressed \nin terms of two anti-commutative families of operators \n$\\mathbf{b}(\\zeta)$ and $\\mathbf{c}(\\zeta )$ acting on $\\mathcal{W}_{[\\alpha]}$ .\nAt present, we do not know \nhow to organize the space $\\mathcal{W}_{[\\alpha]}$ \nin order to describe efficiently the action of $\\mathbf{b}(\\zeta)$ and $\\mathbf{c}(\\zeta )$.\nThe operators $\\mathbf{b}(\\zeta)$ and $\\mathbf{c}(\\zeta )$ should be considered as \nannihilation operators, as both of them kill the `vacua', i.e., operators\n$q^{2\\alpha S(k)}$, for all $k$.  \nWhat is missing is a construction of creation operators.\nEven in the free fermion case, we were able rather to\nmake a detour than to actually solve the problem. \n\nIn fact, the problem of constructing creation operators\ncannot be solved literally, because \n$\\mathbf{b}(\\zeta )$ and $\\mathbf{c}(\\zeta )$ have a large common kernel. \nConsider the restricted operators\n$\\mathbf{b}_{[k,l]}(\\zeta ,\\alpha)$ and $\\mathbf{c}_{[k,l]}(\\zeta ,\\alpha )$ acting \non the space of dimension $4^{l-k+1}$.  \nIn the free fermion case, it can be shown that \nthe dimension of the kernel is $2^{l-k+1}$. \nNumerical experiments indicate that the dimension stays the same generically. \n\nBecause of this kernel, we cannot expect \noperators satisfying the canonical anti-commutation relations \nwith $\\mathbf{b}(\\zeta )$ and $\\mathbf{c}(\\zeta )$. \nSo the first problem is to understand the meaning of the kernel. \nObviously, the \ndifference of any two operators in the kernel has \nvanishing expectation value. \nThe origin of these operators with zero vacuum expectation \nvalues is a mystery to us. \nThe only operators for which this property \ncan be easily explained are the descendants generated\nby adjoint action of local integrals of motion, \nbut for them the vacuum expectation values vanish for a different \nreason: $\\mathbf{b}(\\zeta)$ and $\\mathbf{c}(\\zeta)$ commute with the adjoint action\nof local integrals of motion as is explained by Lemma \\ref{lem4}.\n\nUnderstanding the origin of the kernel of $\\mathbf{b}(\\zeta )$ and $\\mathbf{c}(\\zeta )$,  \nand the construction of creation operators, \nare the problem which we wish to solve.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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S}_{\\lambda}}\n\\def\\matsubsc#1#2#3{[\\relstack{#1}{#2}]_{#3}}\n\\def[1\\ x_i]_{1\\leq i\\leq n}{[1\\ x_i]_{1\\leq i\\leq n}}\n\\def[\\relstack{(x_i-\\kappa_k)_+}{1\\leq k\\leq K}]_{1\\leq i\\leq n}{[\\relstack{(x_i-\\kappa_k)_+}{1\\leq k\\leq K}]_{1\\leq i\\leq n}}\n\\def[\\relstack{(x_i-x_j)_+}{1\\leq i,j\\leq n}]{[\\relstack{(x_i-x_j)_+}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{C(|x_i-x_j|)}{1\\leq i,j\\leq n}]{[\\relstack{C(|x_i-x_j|)}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{C(\\Vert \\bx_i-\\bx_j\\Vert)}{1\\leq i,j\\leq n}]{[\\relstack{C(\\Vert {\\bf x}_i-{\\bf x}_j\\Vert)}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{C(\\Vert \\bx_i-\\bkappa_k\\Vert){[\\relstack{C(\\Vert {\\bf x}_i-{\\thick\\kappa}_k\\Vert)}\n{1\\leq k\\leq K}]_{1\\leq i\\leq n}}\n\\def[\\relstack{C(\\Vert\\bkappa_k-\\bkappa_{k'}\\Vert){[\\relstack{C(\\Vert{\\thick\\kappa}_k-{\\thick\\kappa}_{k'}\\Vert)}\n{1\\le k,k'\\le K}]}\n\\def\\bZradialfullhdlrknots^{-1\/2}{[\\relstack{C(\\Vert\\bkappa_k-\\bkappa_{k'}\\Vert)^{-1\/2}}\n\\def\\bZradialfull{[\\relstack{C(|x_i-x_j|)}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{|x_i-x_j|}{1\\leq i,j\\leq n}]{[\\relstack{|x_i-x_j|}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{-|x_i-x_j|+|x_i|+|x_j|}{1\\leq i,j\\leq n}]{[\\relstack{-|x_i-x_j|+|x_i|+|x_j|}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{-|x_i-x_j|}{1\\leq i,j\\leq n}]{[\\relstack{-|x_i-x_j|}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{|x_i-x_j|}{1\\leq i,j\\leq n}]^{1\/2}{[\\relstack{|x_i-x_j|}{1\\leq i,j\\leq n}]^{1\/2}}\n\\def[\\relstack{(-1)^m|x_i-x_j|^{2m-1}}{1\\leq i,j\\leq n}]{[\\relstack{(-1)^m|x_i-x_j|^{2m-1}}{1\\leq i,j\\leq n}]}\n\\def[\\relstack{e^{-|x_i-x_j|\/\\rho}}{1\\leq i,j\\leq n}]{[\\relstack{e^{-|x_i-x_j|\/\\rho}}{1\\leq i,j\\leq n}]}\n\\def\\bZtps{[\\relstack{\\Vert {\\bf x}_i-{\\thick\\kappa}_k\\Vert^2\\log\n\\Vert{\\bf x}_i-{\\thick\\kappa}_k\\Vert}{1\\le k\\le K}]_{1\\le i\\le n}}\n\\def\\bOmegatps{[\\relstack{\\Vert{\\thick\\kappa}_k-{\\thick\\kappa}_{k'}\\Vert^2\\log\n\\Vert{\\thick\\kappa}_k-{\\thick\\kappa}_{k'}\\Vert}{1\\le k,k'\\le K}]}\n\\def[\\relstack{(x-\\kappa_k)_+}{1\\leq k\\leq K}]{[\\relstack{(x-\\kappa_k)_+}{1\\leq k\\leq K}]}\n\\def[\\relstack{2(x-\\kappa_k)_+}{1\\leq k\\leq K}]{[\\relstack{2(x-\\kappa_k)_+}{1\\leq k\\leq K}]}\n\\def[\\relstack{3|x-\\kappa_k|(x-\\kappa_k)}{1\\leq k\\leq K}]{[\\relstack{3|x-\\kappa_k|(x-\\kappa_k)}{1\\leq k\\leq K}]}\n\\def\\bX\\bbeta+\\bZ\\bu{{\\bf X}{\\thick\\beta}+{\\bf Z}{\\bf u}}\n\\def\\bG_{\\smbtheta}{{\\bf G}_{{\\thick{\\scriptstyle{\\theta}}}}}\n\\def\\left[\\begin{array}{c{\\left[\\begin{array}{c}\n{\\widehat\\bbeta} \\\\\n{\\widehat \\bu}\\\\\n\\end{array}\\right]}\n\\def{\\widehat{\\smbu}}{{\\widehat{{\\thick{\\scriptstyle{\\rm u}}}}}}\n\\def\\bbeta^0{{\\thick\\beta}^0}\n\\def\\bu^0{{\\bf u}^0}\n\\def\\by^0{{\\bf y}^0}\n\\def\\bdf^0{{\\bf 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Sampling}}\n\\def\\index{ GLMM: to correct for nonconstant variance}{\\index{ GLMM: to correct for nonconstant variance}}\n\\def\\index{ Hat matrix}{\\index{ Hat matrix}}\n\\def\\index{ Heteroscedasticity}{\\index{ Heteroscedasticity}}\n\\def\\index{ Heteroscedasticity}{\\index{ Heteroscedasticity}}\n\\def\\index{ Iteratively reweighted least squares}{\\index{ Iteratively reweighted least squares}}\n\\def\\index{ Latent variables}{\\index{ Latent variables}}\n\\def\\index{ Lidar Data}{\\index{ Lidar Data}}\n\\def\\index{ Linear Mixed Model (LMM)}{\\index{ Linear Mixed Model (LMM)}}\n\\def\\index{ Logistic regression}{\\index{ Logistic regression}}\n\\def\\index{ Markov chain Monte--Carlo (MCMC)}{\\index{ Markov chain Monte--Carlo (MCMC)}}\n\\def\\index{ Markov chain Monte--Carlo (MCMC)}{\\index{ Markov chain Monte--Carlo (MCMC)}}\n\\def\\index{ Measurement error}{\\index{ Measurement error}}\n\\def\\index{ Metropolis--Hastings Algorithm}{\\index{ Metropolis--Hastings 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WinBUGS}}\n\\defna\\\"{\\i}ve{na\\\"{\\i}ve}\n\\def\\mbox{x}{\\mbox{x}}\n\\def\\mbox{z}{\\mbox{z}}\n\\def\\mbox{s}{\\mbox{s}}\n\\def\\mbox{t}{\\mbox{t}}\n\\def\\mbox{a}{\\mbox{a}}\n\\def\\mbox{b}{\\mbox{b}}\n\\def\\mbox{K}{\\mbox{K}}\n\\def\\mbox{L}{\\mbox{L}}\n\\def\\textcolor{red}{\\LARGE$\\bullet$}{\\textcolor{red}{\\LARGE$\\bullet$}}\n\\def\\textcolor{ruppertgreen}{\\LARGE$\\bullet$}{\\textcolor{ruppertgreen}{\\LARGE$\\bullet$}}\n\\def\\textcolor{gold}{\\LARGE$\\bullet$}{\\textcolor{gold}{\\LARGE$\\bullet$}}\n\\newcommand{\\plagiarism} {\n\\vfill\n\\begin{center}\nName: \\rule{8cm}{0.25pt}\\vspace{8pt}\nTutorial: \\rule{6cm}{0.25pt}\\vspace{16pt}\n\\fbox{\\parbox{\\textwidth}{\nI declare that this assessment item is my own work, except where\nacknowledged, and has not been submitted for academic credit elsewhere, and\nacknowledge that the assessor of this item may, for the purpose of assessing\nthis item:\n\\begin{itemize}\n\\item Reproduce this assessment item and provide a copy to another member of\nthe University; and\/or,\n\\item Communicate a copy of this assessment item to a plagiarism checking\nservice (which may then retain a copy of the assessment item on its database\nfor the purpose of future plagiarism checking).\n\\end{itemize}\nI certify that I have read and understood the University Rules in respect of\nStudent Academic Misconduct.\\vspace{24pt}\nSigned: \\rule{8cm}{0.25pt} \\hfill Date: \\rule{3cm}{0.25pt}\n}}\\end{center}\n\\vfill\n}\n\\def\\exerDBtags#1#2{\\null}\n\\newcommand{\\slide}[1]{\\foilhead{\\sf #1}}\n\\newcommand{\\fontsize{24pt}{24pt}\\sf}{\\fontsize{24pt}{24pt}\\sf}\n\\newenvironment{list1}%\n{\\begin{list}\n{\\bulletcolour$\\bullet$}\n{\\setlength{\\leftmargin}{30mm}\\setlength{\\itemsep}{0.5ex}}\\sf}\n{\\end{list}\\normalsize}\n\\newenvironment{list2}%\n{\\begin{list}{\\small\\bulletcolour$\\bullet$}\n{\\setlength{\\leftmargin}{0.75in}\\setlength{\\itemsep}{0.5ex}}\\small}\n{\\end{list}\\normalsize}\n\\def\\slidehead#1{\\slide{\\Large\\headingcolour #1}}\n\\def\\null{\\null}\n\n\\section{Introduction}\\label{sec:intro}\nIt is well-known that in the normal linear regression model,\n$$\nY_i | X_i \\sim X_i^T\\beta + \\epsilon_i\\ , \\quad \\epsilon_i \\stackrel{{\\tiny \\mbox{i.i.d.}}}{\\sim} N(0,\\sigma^2)\\ ,\n$$\nthe mean-model parameter $\\beta$ is orthogonal to the error variance $\\sigma^2$ \\citep[e.g.][Section 3.3]{CR1987}. \nThere are two important implications of this. \nFirst, the maximum likelihood estimator (MLE) of $\\beta$ is asymptotically efficient regardless of whether $\\sigma^2$ is known or estimated simultaneously from the data. Second, the MLE of $\\beta$ is independent of the MLE of $\\sigma^2$, which is central to deriving the usual $t$-tests for inferences on $\\beta$. (Note that orthogonal parameters are only asymptotically independent in general; finite-sample independence is special to the normal distribution.) When interest lies primarily in the mean-model, the error variance is often deemed a nuisance parameter.\n\nSimilar orthogonality results hold for other generalized linear models (GLMs). Specific examples include the gamma regression model, in which the mean-model parameter is orthogonal to a nuisance shape parameter \\citep[][Section 3.2]{CR1987}, and multinomial models for polytomous data, in which the mean-model parameter is orthogonal to a nuisance vector of baseline probability masses \\citep{RG2009}. Note that the orthogonality property holds for any link function. \n\nIn each of the above settings, the error distribution is characterized by a finite vector of nuisance parameters and orthogonality is established by showing that the Fisher information matrix is block-diagonal, with the blocks corresponding to the vector of mean-model parameters and the vector of nuisance parameters. It is usually straightforward to perform these calculations on a case-by-case basis, working with the specific family of distributions under consideration, but a general result for parametric GLMs can be found in \\citet{JK2004}.\n\nWhen we move away from specific parametric families to consider the class of all GLMs, we find that a general orthogonality property, although expected, may not be so easy to establish. This is because such a class constitutes a semiparametric model, and it is no longer feasible to compute and examine the Fisher information matrix in the presence of an infinite-dimensional parameter. \n\nIn this note, we show that the mean-model parameter is always orthogonal to the error distribution in GLMs, even when  \nthe error distribution is treated as an infinite-dimensional parameter, belonging to the space of all distributions having a Laplace transform in some neighborhood of zero. This class includes, as special cases, the classical normal, Poisson, gamma and multinomial distributions, as well as many interesting and non-standard distributions, such as the generalized Poisson distribution of \\citet{WF1997} for overdispersed counts and the class of all exponential dispersion models with constant dispersion \\citep{Jorg1987}. We note in Section \\ref{sec:2} that this class of distributions is as large as possible for GLMs, so the result here is indeed the most general possible. \n\nThat a general orthogonality property should hold is alluded to in \\citet[][Section 6.2]{JK2004}. In that paper, the notion of orthogonality between a finite-dimensional and infinite-dimensional parameter being considered is that orthogonality holds, in the usual Fisher information matrix sense, for every finite-dimensional submodel. In this note, we use a slightly stronger notion of orthogonality, namely, that the score function for the finite-dimensional parameter is orthogonal to the nuisance tangent space of the infinite-dimensional parameter. Recalling that the nuisance tangent space is the closure of all finite-dimensional submodel tangent spaces, we see that the notion of orthogonality here implies the notion considered in \\citet{JK2004}.\n\nOur proof proceeds along the following lines. We first use an exponential tilt representation of GLMs, introduced in \\citet{RG2009} and expanded upon in \\citet{Huang2013}, to index any GLM by just two parameters, namely, a finite-dimensional mean-model parameter $\\beta$ and an infinite-dimensional error distribution parameter $F$. The orthogonality of the two parameters is then characterized by the orthogonality of the score function for $\\beta$ to the nuisance tangent space for $F$. As it turns out, the nuisance tangent space for $F$ is rather difficult to work with directly, but by embedding the model into a larger class of models, we find that the required calculations become particularly simple. \n\nThe exponential tilt representation is derived in Section \\ref{sec:2} and the general orthogonality property is proven in Section \\ref{sec:3}. A connection with mean-variance models is outlined in Section \\ref{sec:4}. A brief discussion of the theoretical and practical implications of the our findings is given in Section \\ref{sec:5}, which concludes the note.\n\n\\section{An exponential tilt representation of generalized linear models}\n\\label{sec:2}\n\nRecall that a GLM \\citep{MN1989} for independent data pairs $(X_1,Y_1), \\ldots, (X_n, Y_n)$ is defined by two components. First, there is a conditional mean model for the responses,\n\\begin{equation}\nE(Y_i|X_i) = \\mu(X_i^T\\beta) \\ ,\n\\label{eq:meanmodel}\n\\end{equation}\nwhere $\\mu$ is a user-specified inverse-link function and $\\beta \\in \\mathbb{R}^q$ is a vector of unknown regression parameters. Second, the conditional distributions $F_i$ of each response $Y_i$ given covariate $X_i$ are assumed to come from some exponential family. Assuming the distributions $F_i$ have densities $dF_i$ with respect to some dominating measure, the second component can be written in the exponential tilt form\n\\begin{equation}\ndF_i(y) = \\exp\\{b(X_i; \\beta, F) + \\theta(X_i; \\beta, F) y\\} dF(y)\n\\end{equation}\nfor some reference distribution $F$, where \n\\begin{equation}\n\\label{eq:norm}\nb(X_i; \\beta, F) = -\\log \\int \\exp\\{\\theta(X_i; \\beta, F) y\\} dF(y)\n\\end{equation}\nis a normalizing function and, in order to satisfy (\\ref{eq:meanmodel}), the tilt $\\theta(X_i;\\beta,F)$ is implicitly defined as the solution to the mean constraint\n\\begin{equation}\n\\mu(X_i^T\\beta) = \\int y \\exp\\{b(X_i; \\beta, F) +\\theta(X_i;\\beta,F) y\\} dF(y) \\ .\n\\label{eq:mean}\n\\end{equation}\n\nIt is easy to see that the exponential tilt representation (\\ref{eq:meanmodel})--(\\ref{eq:mean}) encompasses all classical GLMs. For example, normal, Poisson and gamma regression models can be recovered by choosing $dF$ to be a Gaussian, Poisson or gamma kernel, respectively. The main advantage of this representation is that it naturally allows for the reference distribution $F$ to be considered as an infinite-dimensional nuisance parameter, along with the finite-dimensional parameter $\\beta$, in the model. It is this novel representation that allows us to conveniently \ncharacterize any GLM using just the two parameters $\\beta$ and $F$.\n\nAs with any GLM, the reference distribution $F$ is required to have a Laplace transform in some neighborhood of the origin so that the cumulant generating function (\\ref{eq:norm}) is well-defined. Thus, the parameter space for $F$ is the class of all distributions that have a Laplace transform in some neighborhood of the origin. Note that this class of distribution functions is as large as it can be for GLMs, because any distribution outside this class cannot be used to generate a valid model.\n\nThe exponential tilt representation (\\ref{eq:meanmodel})--(\\ref{eq:mean}) was first introduced in \\citet{RG2009}. In that paper, the representation is used to derive a useful alternative parametrization of the multinomial regression model for polytomous responses. The representation is also used in \\cite{Huang2013} to motivate a semiparametric extension of GLMs for arbitrary responses.\n\n\\section{The orthogonality of parameters}\n\\label{sec:3}\nIn parametric models, orthogonality of parameters can be characterized by the Fisher information matrix being block-diagonal. In semiparametric models however, orthogonality between a finite-dimensional parameter $\\beta$ and an infinite-dimensional parameter $F$ cannot be characterized in this way. Rather, it is characterized through the score function for $\\beta$ being orthogonal to the nuisance tangent space for $F$. Intuitively speaking, the projection of the score function for $\\beta$ on to the nuisance tangent space is a measure of the loss of information about $\\beta$ due to the presence of the nuisance parameter $F$ -- this is zero if and only if the score function is orthogonal to the nuisance tangent space. Note that this general notion of orthogonality reduces to the Fisher information matrix criterion when the nuisance parameter is finite-dimensional.\n\nNow, the loglikelihood function corresponding to model (\\ref{eq:meanmodel})--(\\ref{eq:mean}) is $l(\\beta, F|X,Y) = \\log dF(Y) + b(X; \\beta, F) + \\theta(X;\\beta, F) Y$. Thus, the\nscore function for $\\beta$ is given by\n$$S_{\\beta,F}(X,Y) := \\frac{\\partial}{\\partial \\beta} l(\\beta, F) = \\frac{\\partial}{\\partial \\beta} b(X;\\beta, F) + \\frac{\\partial}{\\partial \\beta} \\theta(X;\\beta, F) Y.$$\nImplicit differentiation of the defining equations for \n$b$ and $\\theta$ leads to the identities\n\\begin{eqnarray*}\n\\frac{\\partial }{\\partial \\beta} b(X;\\beta,F) = - \\mu (X^T\\beta) \\frac{\\partial }{\\partial \\beta} \\theta(X;\\beta,F)  \\quad \\mbox{ and } \\quad\n\\frac{\\partial }{\\partial \\beta} \\theta(X;\\beta,F) = \\frac{\\mu'(X^T\\beta) }{V(X;\\beta,F)} X\\ ,\n\\end{eqnarray*}\nwhere\n$V(X;\\beta,F) = E_{\\beta,F}[(Y-\\mu(X^T\\beta))^2|X]$\nis the conditional variance of $Y$ given $X$ under parameter value $(\\beta,F)$.  The score function for $\\beta$ therefore reduces to\n\\begin{equation}\nS_{\\beta,F}(X,Y) = X\\frac{\\mu'(X^T\\beta)}{V(X;\\beta,F)} \\left(Y-\\mu(X^T\\beta)\\right)  \\ ,\n\\label{eq:betascore}\n\\end{equation}\nwhich is of the same weighted least-squares form as for a parametric GLM. The difference here is that the variance function $V(X;\\beta,F)$ is not known because $F$ is not specified.\n\nThe orthogonality between $\\beta$ and $F$ now reduces to the score function (\\ref{eq:betascore}) being orthogonal to the nuisance tangent space for $F$. Although it is not hard to derive a score function for $F$ \\citep[e.g.][Section 3.3]{Huang2013}, it turns out to be rather difficult to compute the nuisance tangent space explicitly. This is also noted in \\citet[][Section 6.2]{JK2004}. We can work around this, however, by embedding model (\\ref{eq:meanmodel})--(\\ref{eq:mean}) into a more general class of ``semiparametric restricted moment models\" \\citep[e.g.][Section 4.5]{Tsia2006} for which the required calculations are much easier. This class is given by\n\\begin{equation}\n\\label{eq:srmm}\nY = \\mu(X,\\beta) + \\epsilon \\ ,\n\\end{equation}\nwhere the conditional distribution of $\\epsilon$ given $X$ is specified only up to the moment condition $E(\\epsilon|X) = 0$. It is clear that the semiparametric extension (\\ref{eq:meanmodel})--(\\ref{eq:mean}) is a subclass of the restricted moment model, with $\\mu(X,\\beta) = \\mu(X^T\\beta)$ and $E(\\epsilon|X) = E(Y-\\mu(X^T\\beta)|X) = 0$ by construction. The nuisance tangent space for $F$ in the semiparametric model (\\ref{eq:meanmodel})--(\\ref{eq:mean}) must therefore be a subspace of the nuisance tangent space for the restricted moment model.\n\nElementary calculations \\citep[see][pp.81--83]{Tsia2006} show that the nuisance tangent space for the restricted moment model is given by\n\\begin{equation*}\n\\Lambda = \\left\\{\n\\text{all $q \\times 1$ functions $a(X,Y)$ such that\n$E_{\\beta,F}[(Y-\\mu(X,\\beta))a(X,Y)|X] = 0$}\n\\right \\}\n\\end{equation*}\nand the projection operator $\\Pi_{\\beta,F}$ onto this nuisance tangent space is given by\n\\begin{equation*}\n\\Pi_{\\beta,F} s = s - \\frac{E_{\\beta,F} \\left[\\left(Y-\\mu(X,\\beta)\\right)s|X\\right]}{V(X;\\beta,F)} \\left(Y-\\mu(X,\\beta)\\right) \\ .\n\\end{equation*}\nApplying this operator to the score function (\\ref{eq:betascore}), with $\\mu(X,\\beta) = \\mu(X^T\\beta)$, gives\n\\begin{eqnarray*}\n\\Pi_{\\beta,F} S_{\\beta,F} &=& X\\frac{ \\mu'(X^T\\beta)}{V(X;\\beta,F)} \\left(Y-\\mu(X^T\\beta)\\right) \\\\\n& & - X\\frac{ \\mu'(X^T\\beta)}{V(X;\\beta,F)} E_{\\beta,F} \\left[\\frac{(Y-\\mu(X^T\\beta))^2}{V(X;\\beta,F)}\\Bigg|X \\right] (Y-\\mu(X^T\\beta)) \\\\\n&=& 0 \\ \\  \\mbox{ for all } (\\beta, F),\n\\end{eqnarray*}\nbecause $V(X;\\beta,F) = E_{\\beta,F}[(Y-\\mu(X^T\\beta))^2|X]$ by definition. \nThus, the score function (\\ref{eq:betascore}) is orthogonal to the nuisance tangent space in the restricted moment model (\\ref{eq:srmm}) and therefore necessarily orthogonal to the nuisance tangent space in the semiparametric model (\\ref{eq:meanmodel})--(\\ref{eq:mean}) also. \nWe summarize as follows:\n\n\\begin{Proposition}[Orthogonality]\n\\label{le:orth}\nThe mean-model parameter $\\beta$ and the error distribution $F$ in any generalized linear model are orthogonal.\n\\end{Proposition}\n\nNote that the nuisance tangent space for any parametric model (that is, a model in which $F$ is characterized by a finite number of parameters) is necessarily a subspace of the semiparametric nuisance tangent space. We therefore have the following corollary:\n\n\\begin{Corollary} [Orthogonality in parametric models]\nIf the error distribution is characterized by a finite vector of nuisance parameters $\\phi$, then the mean-model parameter $\\beta$ is orthogonal to $\\phi$.\n\\end{Corollary}\n\n\\section{A connection with quasilikelihood models}\n\\label{sec:4}\nA popular extension of GLMs is the class of quasilikelihood (QL) models, also known mean-variance models \\citep[e.g.][]{Wed1974}. These models make the assumption that $E(Y|X) = \\mu(X,\\beta)$ for some mean function $\\mu$ and $\\mbox{Var}(Y|X) = v(\\mu)$ for some positive variance function $v$. Such models can be characterized by their quasi-score functions for $\\beta$,\n\\begin{equation}\n\\label{eq:quasiscore}\n\\frac{\\partial \\mu}{\\partial \\beta}= \\frac{Y-\\mu}{v(\\mu)} \\ .\n\\end{equation}\nIn classical QL literature, the functional forms of both $\\mu$ and $v$ are usually specified, although there is growing literature on adaptive estimation in which the variance function is left unspecified and estimated nonparametrically from data \\citep[e.g.][]{DZ2002}. By comparing score equations (\\ref{eq:betascore}) and (\\ref{eq:quasiscore}), note that GLMs form a subset of QL models.\n\nFor models characterized by (\\ref{eq:quasiscore}), \\citet{JK2004} showed that the mean-model parameter $\\beta$ is orthogonal to the variance function $v$ whenever the latter can be characterized by a finite number of parameters. A general orthogonality result for arbitrary, infinite-dimensional $v$ remains elusive, however, perhaps because of the fact that not all QL score functions (\\ref{eq:quasiscore}) correspond to actual probability models. Indeed, such correspondences are atypical. Nevertheless, there is an interesting connection between QL models with unspecified variance functions and GLMs with unspecified error distributions in a certain asymptotic sense made more precise below.\n\nThe connection is based on a rather remarkable, but relatively obscure, result from \\citet{Hiejima1997}, who showed that GLMs can be considered ``dense\" in the class of QL models in the following asymptotic sense: for any mean-variance relationship, there exists an exponential family of distributions (i.e. a GLM with some error distribution $F$) whose score equations for $\\beta$ admit roots that are arbitrarily close to the roots of the corresponding QL score equation, as the sample size increases.  Thus, for large enough sample sizes, any adaptive QL model with unspecified variance function $v$ can be approximated arbitrarily well by a GLM with unspecified error distribution $F$, with the latter possessing the orthogonality property \\ref{le:orth}. We conjecture that this connection may be the best possible for adaptive QL models, mainly because of the aforementioned fact that QL score functions typically do not correspond to actual probability models.\n\n\\section{Practical implications}\n\\label{sec:5}\nThe orthogonality property \\ref{le:orth} naturally suggests the idea of estimating the error distribution nonparametrically and simultaneously with the mean-model, leading to a kind of ``adaptive GLM\". Indeed, if the joint estimation procedure is based on maximum semiparametric likelihood, then the estimator for $\\beta$ is guaranteed to be asymptotically efficient and asymptotically independent of the estimated error distribution. In other words, both estimation and inferences on $\\beta$ are asymptotically unaffected by having to also estimate the error distribution. In contrast, estimation and inferences in GLMs with misspecified error distributions, or QL models with misspecified variance functions, are generally not efficient. \n\nThe idea of jointly estimating the mean and error distribution in GLMs is considered in more detail in \\citet{Huang2013}. In that paper, it is demonstrated that inferences on $\\beta$ based on profiling out the error distribution $F$ in the likelihood can be more accurate than inferences based on QL methods. Here, we focus our attention on the accuracy of the  point estimates of $\\beta$. The results here complement those found in \\citet[][Section 6]{Huang2013}.\n\n\\begin{table}\n\\label{tab:1}\n\\centering\n\\caption{\\it Relative root mean-square errors of a semiparametric MLE ($\\hat \\beta_{SP}$) and the usual MLE ($\\hat \\beta_{MLE}$) in three simulation settings, based on 5000 replications each.\n}\n\\small{\n\\begin{tabular}{lllccccccc}\n\\hline\n & & & \\uline{\\hspace{3mm}Esti\\hspace{-2.5mm}} & \\uline{\\hspace{-2.5mm}mator\\hspace{3mm}} \\\\\nData &n\t& Parameter\t& $\\hat \\beta_{SP}$ & $\\hat \\beta_{MLE}$ \\\\\n\\hline\nExponential & 33 & Intercept \t& 0.199 & 0.194 \\\\\n& & Group effect \t\t\t \t  & 0.361  & 0.354 \\\\\n& & Common slope  \t\t & 0.464 & 0.455 \\\\\n& 66 & Intercept \t\t\t &  0.124 & 0.122  \\\\\n& & Group effect \t\t\t & 0.247 & 0.243 \\\\\n& & Common slope  \t \t& 0.300 & 0.297 \\\\\nPoisson & 44 & Intercept & 0.275 & 0.271 \\\\\n & & Coefficient of $X_1$ & 0.610 & 0.594 \\\\\n & & Coefficient of $X_2$ & 0.205 & 0.201\\\\\n & & Coefficient of $X_3$ & 0.546& 0.533\\\\\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\nIn Table 1, we compare the relative root mean-square error of a semiparametric MLE of $\\beta$ (with $F$ unknown) to that of the usual MLE (with $F$ set to the true distribution) from three sets of simulations. Recall that the relative root mean-square error of an estimator $\\hat \\beta$ is defined as the root mean-square error of $\\hat \\beta$ divided by the absolute value $|\\beta|$ of the parameter. The simulation settings are based on a leukemia survival dataset from \\citet{DH1997} and a mine injury dataset from \\citet{MMVR2010}, and are described in more detail in Sections 6.1 and 6.2 of \\citet{Huang2013}. The particular semiparametric estimation approach we use for estimating $\\beta$ \nis based on empirical likelihood and is described in more detail in Section 4 of \\citet{Huang2013}. \n\nWe see from Table 1 that the relative root mean-square errors of the two estimators are essentially the same, even for moderately small sample sizes. This supports the claim that maximum likelihood estimation of $\\beta$ is asymptotically efficient regardless of whether $F$ is known or completely unknown and estimated nonparametrically from data.\n\n\n\\section{Conclusion}\nIn this note, we have shown that orthogonality between the mean-model parameter and the error distribution holds for any GLM, parametric or nonparametric. This confirms, in greatest generality, what is well-known for the special cases of normal, gamma and multinomial regression. The result also has implications for applied statistical work, with our numerical results suggesting that little is lost by treating the error distribution nonparametrically, even in moderately sized problems. (It can also be said that little is gained by knowing the error distribution completely!) Nonparametric estimation of the error distribution can therefore safeguard against biases due to parametric model misspecification, without sacrificing much in terms of efficiency.\n\n\\section*{Acknowledgments}\nThe authors thank the Associate Editor and an anonymous referee for suggestions that improved the paper. Paul. J. Rathouz was funded by NIH grant R01 HL094786. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe resurgence of autoencoders (AE) \\cite{yann1987modeles,bourlard1988auto,hinton1994autoencoders} is an important component in the rapid development of modern deep learning \\cite{goodfellow2016deep}. Autoencoders have been widely adopted for modeling signals and images \\cite{poultney2007efficient, vincent2010stacked}.\nIts statistical counterpart, the variational autoencoder (VAE) \\cite{kingma2013auto}, has led to a recent wave of development in generative modeling due to its two-in-one capability, both representation and statistical learning in a single framework. Another exploding direction in generative modeling includes generative adversarial networks (GAN) \\cite{goodfellow2014generative}, but GANs focus on the generation process and are not aimed at representation learning (without an encoder at least in its vanilla version). \n\nCompared with classical dimensionality reduction methods like principal component analysis (PCA) \\cite{candes1933robust,Jolliffe2011principal} and Laplacian eigenmaps \\cite{belkin2003laplacian}, VAEs have demonstrated their unprecedented power in modeling high dimensional data of real-world complexity. However, there is still a large room to improve for VAEs to achieve a high quality reconstruction\/synthesis. Additionally, it is desirable to make the VAE representation learning more transparent, interpretable, and controllable.\n\nIn this paper, we attempt to learn a transparent representation by introducing guidance to the latent variables in a VAE. We design two strategies for our Guided-VAE, an unsupervised version (Fig.~\\ref{fig:model}.a) and a supervised version (Fig.~\\ref{fig:model}.b). The main motivation behind Guided-VAE is to encourage the latent representation to be semantically interpretable, while maintaining the integrity of the basic VAE architecture. Guided-VAE is learned in a multi-task learning fashion. The objective is achieved by taking advantage of the modeling flexibility and the large solution space of the VAE under a lightweight target. Thus the two tasks, learning a good VAE and making the latent variables controllable, become companions rather than conflicts.\n\nIn {\\bf unsupervised Guided-VAE}, in addition to the standard VAE backbone, we also explicitly force the latent variables to go through a lightweight encoder that learns a deformable PCA. As seen in Fig.~\\ref{fig:model}.a, two decoders exist, both trying to reconstruct the input data ${\\bf x}$:\nThe main decoder, denoted as $\\text{Dec}_{main}$, functions regularly as in the standard VAE \\cite{kingma2013auto}; the secondary decoder, denoted as $\\text{Dec}_{sub}$, explicitly learns a geometric deformation together with a linear subspace.\nIn {\\bf supervised Guided-VAE}, we introduce a subtask for the VAE by forcing one latent variable to be discriminative (minimizing the classification error) while making the rest of the latent variable to be adversarially discriminative (maximizing the minimal classification error). This subtask is achieved using an adversarial excitation and inhibition formulation. Similar to the unsupervised Guided-VAE, the training process is carried out in an end-to-end multi-task learning manner. The result is a regular generative model that keeps the original VAE properties intact, while having the specified latent variable semantically meaningful and capable of controlling\/synthesizing a specific attribute.\nWe apply Guided-VAE to the data modeling and few-shot learning problems and show favorable results on the MNIST, CelebA, CIFAR10 and Omniglot datasets.\n\nThe contributions of our work can be summarized as follows:\n{\n\\begin{itemize}\n \\setlength\\itemsep{0mm}\n \\setlength{\\itemindent}{0mm}\n\\item We propose a new generative model disentanglement learning method by introducing latent variable guidance to variational autoencoders (VAE). Both unsupervised and supervised versions of Guided-VAE have been developed. \n\\item In unsupervised Guided-VAE, we introduce deformable PCA as a subtask to guide the general VAE learning process, making the latent variables interpretable and controllable.\n\\item In supervised Guided-VAE, we use an adversarial excitation and inhibition mechanism to encourage the disentanglement, informativeness, and controllability of the latent variables.\n\\end{itemize}\n}\n\nGuided-VAE can be trained in an end-to-end fashion. It is able to keep the attractive properties of the VAE while significantly improving the controllability of the vanilla VAE.  It is applicable to a range of problems for generative modeling and representation learning.\n\n\\section{Related Work}\nRelated work can be discussed along several directions.\n\nGenerative model families such as generative adversarial networks (GAN) \\cite{goodfellow2014generative,WGAN} and variational autoencoder (VAE) \\cite{kingma2013auto} have received a tremendous amount of attention lately. Although GAN produces higher quality synthesis than VAE, GAN is missing the encoder part and hence is not directly suited for representation learning.\nHere, we focus on disentanglement learning by making VAE more controllable and transparent.\n\nDisentanglement learning \\cite{mathieu2016disentangling,szabo2017challenges,hu2018disentangling,achille2018emergence,gonzalez2018image,jha2018disentangling} recently becomes a popular topic in representation learning. Adversarial training has been adopted in approaches such as \\cite{mathieu2016disentangling,szabo2017challenges}.\nVarious methods \\cite{peng2017reconstruction,kim2018disentangling,lin2019exploring} have imposed constraints\/regularizations\/supervisions to the latent variables, but these existing approaches often involve an architectural change to the VAE backbone and the additional components in these approaches are not provided as secondary decoder for guiding the main encoder.\nA closely related work is the $\\beta$-VAE \\cite{higgins2017beta} approach in which a balancing term $\\beta$ is introduced to control the capacity and the independence prior. $\\beta$-TCVAE \\cite{chen2018isolating} further extends $\\beta$-VAE by introducing a total correlation term.\n\nFrom a different angle, principal component analysis (PCA) family \\cite{candes1933robust,Jolliffe2011principal,candes2011robust} can also be viewed as representation learning. Connections between robust PCA \\cite{candes2011robust} and VAE \\cite{kingma2013auto} have been observed \\cite{dai2018connections}. Although being a widely adopted method, PCA nevertheless has limited modeling capability due to its linear subspace assumption. To alleviate the strong requirement for the input data being pre-aligned, RASL \\cite{peng2012rasl} deals with unaligned data by estimating a hidden transformation to each input.\nHere, we take advantage of the transparency of PCA and the modeling power of VAE by developing a sub-encoder (see Fig. \\ref{fig:model}.a), deformable PCA, that guides the VAE training process in an integrated end-to-end manner. After training, the sub-encoder can be removed by keeping the main VAE backbone only.\n\nTo achieve disentanglement learning in supervised Guided-VAE, we encourage one latent variable to directly correspond to an attribute while making the rest of the variables uncorrelated. This is analogous to the excitation-inhibition mechanism \\cite{murphy2003multiplicative, yizhar2011neocortical}\nor the explaining-away \\cite{wellman1993explaining} phenomena. Existing approaches \\cite{liu2018detach,lin2019exploring} impose supervision as a conditional model for an image translation task, whereas our supervised Guided-VAE model targets the generic generative modeling task by using an adversarial excitation and inhibition formulation. This is achieved by minimizing the discriminative loss for the desired latent variable while maximizing the minimal classification error for the rest of the variables.\nOur formulation has a connection to the domain-adversarial neural networks (DANN) \\cite{ganin2016domain}, but the two methods differ in purpose and classification formulation. Supervised Guided-VAE is also related to the adversarial autoencoder approach \\cite{makhzani2016adversarial}, but the two methods differ in the objective, formulation, network structure, and task domain. In \\cite{ilse2019diva}, the domain invariant variational autoencoders method (DIVA) differs from ours by enforcing disjoint sectors to explain certain attributes.\n\nOur model also has connections to the deeply-supervised nets (DSN) \\cite{lee2015deeply}, where intermediate supervision is added to a standard CNN classifier. There are also approaches \\cite{engel2018latent,bojanowski2018optimizing} in which latent variables constraints are added, but they have different formulations and objectives than Guided-VAE. Recent efforts in fairness disentanglement learning \\cite{creager2019flexibly,song2018learning} also bear some similarity, but there is still a large difference in formulation.\n\n\n\\section{Guided-VAE Model}\n\\label{Guided-VAE}\nIn this section, we present the main formulations of our Guided-VAE models. The unsupervised Guided-VAE version is presented first, followed by introduction of the supervised version.\n\n\\begin{figure*}[!htp]\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tabular}{cc}\n\\hspace{-2mm}\n\\includegraphics[width=0.5\\textwidth]{.\/figures\/UnGuidedVAE.png}  &\n\\hspace{5mm}\n\\includegraphics[width=0.5\\textwidth]{.\/figures\/SuGuidedVAE.png}\\\\\n(a) Unsupervised Guided-VAE  &\n(b) Supervised Guided-VAE\\\\\n\\hspace{-2mm}\n\\end{tabular}\n}\n\\caption{Model architecture for the proposed Guided-VAE algorithms.}\n\\label{fig:model}\n\\vspace{-3mm}\n\\end{center}\n\\end{figure*}\n\n\\subsection{VAE}\n\nFollowing the standard definition in variational autoencoder (VAE) \\cite{kingma2013auto}, a set of input data is denoted as $\\text{X}=({\\bf x}_1,...,{\\bf x}_n)$ where $n$ denotes the number of total input samples. The latent variables are denoted by vector ${\\bf z}$. The encoder network includes network and variational parameters $\\bm{\\phi}$  that produces variational probability model $q_{\\bm{\\phi}}({\\bf z}|{\\bf x})$. The decoder network is parameterized by $\\bm{\\theta}$ to reconstruct sample $\\tilde{{\\bf x}}=f_{\\bm{\\theta}}({\\bf z})$. The log likelihood $\\log p({\\bf x})$ estimation is achieved by maximizing the Evidence Lower BOund (ELBO) \\cite{kingma2013auto}:\n\\begin{equation}\n\\begin{aligned}\n    ELBO(\\bm{\\theta}, \\bm{\\phi}; {\\bf x}) &= {\\mathbb{E}}_{q_{\\bm{\\phi}}({\\bf z}|{\\bf x})} [\\log(p_{\\bm{\\theta}}({\\bf x}|{\\bf z}))] \\\\\n    &- D_{\\mathrm{KL}}(q_{\\bm{\\phi}}({\\bf z}|{\\bf x}) || p({\\bf z})).\n\\label{eq:ELBO}\n\\end{aligned}\n\\end{equation}\n\nThe first term in Eq. (\\ref{eq:ELBO}) corresponds to a reconstruction loss $\\int q_{\\bm{\\phi}}({\\bf z}|{\\bf x}) \\times ||{\\bf x}-f_{\\bm{\\theta}}({\\bf z})||^2  d{\\bf z}$ (the first term is the \\emph{negative} of reconstruction loss between input ${\\bf x}$ and reconstruction $\\tilde{{\\bf x}}$) under Gaussian parameterization of the output.\nThe second term in Eq. (\\ref{eq:ELBO}) refers to the KL divergence between the variational distribution $q_{\\bm{\\phi}}({\\bf z}|{\\bf x})$ and the prior distribution $p({\\bf z})$.\nThe training process thus tries to optimize:\n\\begin{equation}\n    \\max_{\\bm{\\theta}, \\bm{\\phi}} \\left\\{\\sum_{i=1}^n ELBO(\\bm{\\theta}, \\bm{\\phi}; {\\bf x}_i)\\right\\}.\n\\label{eq:VAE}\n\\end{equation}\n\\vspace{-6.5mm}\n\\subsection{Unsupervised Guided-VAE }\nIn our unsupervised Guided-VAE, we introduce a deformable PCA as a secondary decoder to guide the VAE training. An illustration can be seen in Fig. \\ref{fig:model}.a. This secondary decoder is called $\\text{Dec}_{sub}$.\nWithout loss of generality, we let ${\\bf z}=({\\bf z}_{def}, {\\bf z}_{cont})$. ${\\bf z}_{def}$ decides a deformation\/transformation field, e.g. an affine transformation denoted as $\\tau({\\bf z}_{def})$. ${\\bf z}_{cont}$ determines the content of a sample image for transformation. The PCA model consists of $K$ basis $B=({\\bf b}_1,...,{\\bf b}_K)$. We define a deformable PCA loss as:\n\\begin{equation}\n\\begin{aligned}\n    & {\\mathcal{L}}_{DPCA}(\\bm{\\phi}, B)\\\\\n    &= \\sum_{i=1}^n  {\\mathbb{E}}_{q_{\\bm{\\phi}}({\\bf z}_{def},{\\bf z}_{cont}|{\\bf x}_i)}\\left[ ||{\\bf x}_i- \\tau({\\bf z}_{def}) \\circ ({\\bf z}_{cont} B^T) ||^2 \\right] \\\\\n    &+ \\sum_{k,j\\ne k} ({\\bf b}_{k}^T {\\bf b}_{j})^2,\n\\label{eq:DPCA}\n\\end{aligned}\n\\end{equation}\nwhere $\\circ$ defines a transformation (affine in our experiments) operator decided by $\\tau({\\bf z}_{def})$ and $\\sum_{k,j\\ne k} ({\\bf b}_{k}^T {\\bf b}_{j})^2$ is regarded as the orthogonal loss. A normalization term $\\sum_{k} ({\\bf b}_{k}^T {\\bf b}_{k}-1)^2$ can be optionally added to force the basis to be unit vectors. We follow the spirit of the PCA optimization and a general formulation for learning PCA can be found in \\cite{candes2011robust}.\n\nTo keep the simplicity of the method we learn a fixed basis $B$ and one can also adopt a probabilistic PCA model \\cite{tipping1999probabilistic}. Thus, learning unsupervised Guided-VAE becomes:\n\\begin{equation}\n\\begin{aligned}\n    \\max_{\\bm{\\theta}, \\bm{\\phi}, B} \\left \\{  \\sum_{i=1}^n  ELBO(\\bm{\\theta}, \\bm{\\phi}; {\\bf x}_i)   - {\\mathcal{L}}_{DPCA}(\\bm{\\phi}, B) \\right \\}.\n\\label{eq:ungvae}\n\\end{aligned}\n\\end{equation}\nThe affine matrix described in our transformation follows implementation in \\cite{jaderberg2015spatial}:\n\n\\begin{equation}\n A_\\theta=\n\\left[ {\n\\begin{array}{ccc}\n\\theta_{11} & \\theta_{12} & \\theta_{13}\\\\\n\\theta_{21} & \\theta_{22} & \\theta_{23}\n\\end{array}\n}\\right]\n\\label{eqn:attention}\n\\end{equation}\nThe affine transformation includes translation, scale, rotation and shear operation. We use different latent variables to calculate different parameters in the affine matrix according to the operations we need.\n\n\\begin{figure*}[!htp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\n\\includegraphics[width=1\\textwidth]{.\/figures\/MNIST.png}\\\\\n\\hspace{1.0em} (a)VAE \\hspace{6.0em} (b) $\\beta$-VAE \\hspace{4.0em} (c) CC$\\beta$-VAE \\hspace{4.0em} (d) JointVAE \\hspace{2.0em} (e) Guided-VAE (Ours) \n\\end{tabular}\n\n\\caption{\\small{\\textbf{Latent Variables Traversal on MNIST:} Comparison of traversal results from vanilla VAE  \\cite{kingma2013auto}, $\\beta$-VAE \\cite{higgins2017beta},  $\\beta$-VAE with\ncontrolled capacity increase (CC$\\beta$-VAE), JointVAE \\cite{dupont2018learning} and our Guided-VAE on the MNIST dataset. $z_{1}$ and $z_{2}$ in Guided-VAE are controlled.}}\n\\label{fig:mnist}\n\\vspace{-3mm}\n\\end{center}\n\\end{figure*}\n\n\\vspace{-1mm}\n\\subsection{Supervised Guided-VAE}\n\\vspace{-1mm}\nFor training data $\\text{X}=({\\bf x}_1,...,{\\bf x}_n)$, suppose there exists a total of $T$ attributes with ground-truth labels.\nLet ${\\bf z}=(z_t, {\\bf z}_t^{rst})$ where $z_t$ defines a scalar variable deciding the $t$-th attribute and ${\\bf z}_t^{rst}$ represents remaining latent variables. Let $y_t({\\bf x}_i)$ be the  ground-truth label for the $t$-th attribute of sample ${\\bf x}_i$; $y_t({\\bf x}_i) \\in \\{-1, +1\\}$. For each attribute, we use an adversarial excitation and inhibition method with term: \n\n\\begin{equation}\n\\begin{aligned}\n& {\\mathcal{L}}_{Excitation}(\\bm{\\phi}, t) \\\\\n&= \\max_{w_t}\\left\\{ \\sum_{i=1}^n {\\mathbb{E}}_{q_{\\bm{\\phi}}(z_t|{\\bf x}_i)} [\\log p_{w_t}(y=y_t({\\bf x}_i)|z_t)]\\right\\} ,\n\\end{aligned}\n\\end{equation}\nwhere $w_t$ refers to classifier making a prediction for the $t$-th attribute using the latent variable $z_t$.\n\nThis is an excitation process since we want latent variable $z_t$ to directly correspond to the attribute label. \n\n\nNext is an inhibition term.\n\\begin{equation}\n\\begin{aligned}\n& {\\mathcal{L}}_{Inhibition} (\\bm{\\phi}, t) \\\\\n&= \\max_{C_t} \\left\\{\\sum_{i=1}^n {\\mathbb{E}}_{q_{\\bm{\\phi}}({\\bf z}_t^{rst}|{\\bf x}_i)} [\\log p_{C_t}(y=y_t({\\bf x}_i)|{\\bf z}_t^{rst})] \\right\\},\n\\label{eq:inhibition}\n\\end{aligned}\n\\end{equation}\nwhere $C_t({\\bf z}_t^{rst})$ refers to classifier making a prediction for the $t$-th attribute using the remaining latent variables ${\\bf z}_t^{rst}$.\n$\\log p_{C_t}(y=y_t({\\bf x})|{\\bf z}_t^{rst})$ is a cross-entropy term for minimizing the classification error in Eq. (\\ref{eq:inhibition}).\nThis is an inhibition process since we want the remaining variables ${\\bf z}_t^{rst}$ as independent as possible to the attribute label in Eq. (\\ref{eq:sugvae}) below.\n\\begin{equation}\n\\begin{aligned}\n    &\\max_{\\bm{\\theta}, \\bm{\\phi}} {\\bigg\\{}  \\sum_{i=1}^n  ELBO(\\bm{\\theta}, \\bm{\\phi}; {\\bf x}_i)  \\\\\n    &+ \\sum_{t=1}^T \\left[{\\mathcal{L}}_{Excitation}(\\bm{\\phi}, t) - {\\mathcal{L}}_{Inhibition} (\\bm{\\phi},t) \\right] {\\bigg\\}}.\n\\label{eq:sugvae}\n\\end{aligned}\n\\end{equation}\n\nNotice in Eq. (\\ref{eq:sugvae}) the minus sign in front of the term ${\\mathcal{L}}_{Inhibition} (\\bm{\\phi}, t)$ for maximization which is an adversarial term to make ${\\bf z}_t^{rst}$ as uninformative to attribute $t$ as possible, by pushing the best possible classifier $C_t$ to be the least discriminative.\nThe formulation of Eq. (\\ref{eq:sugvae}) bears certain similarity to that in domain-adversarial neural networks \\cite{ganin2016domain} in which the label classification is minimized with the domain classifier being adversarially maximized. Here, however, we respectively encourage and discourage different parts of the features to make the same type of classification. \n\n\\section{Experiments}\n\\label{Experiments}\n\nIn this section, we first present qualitative and quantitative results demonstrating our proposed unsupervised Guided-VAE (Figure \\ref{fig:model}a) capable of disentangling latent embedding more favorably than previous disentangle methods \\cite{higgins2017beta, dupont2018learning, kim2018disentangling} on MNIST dataset \\cite{lecun2010mnist} and 2D shape dataset \\cite{dsprites17}. We also show that our learned latent representation improves classification performance in a representation learning setting. Next, we extend this idea to a supervised guidance approach in an adversarial excitation and inhibition fashion, where a discriminative objective for certain image properties is given (Figure \\ref{fig:model}b) on the CelebA dataset \\cite{liu2015faceattributes}. Further, we show that our method is architecture agnostic, applicable in a variety of scenarios such as image interpolation task on CIFAR 10 dataset \\cite{cifar10} and a few-shot classification task on Omniglot dataset \\cite{lake2015human}.\n\n\\subsection{Unsupervised Guided-VAE}\n\n\\subsubsection{Qualitative Evaluation}\n\nWe present qualitative results on the MNIST dataset first by traversing latent variables received affine transformation guiding signal in Figure \\ref{fig:mnist}. Here, we applied the Guided-VAE with the bottleneck size of 10 (i.e. the latent variables ${\\bf z} \\in \\mathbb{R}^{10}$). The first latent variable $z_{1}$ represents the rotation information, and the second latent variable $z_{2}$ represents the scaling information. The rest of the latent variables ${\\bf z}_{3:10}$ represent the content information. Thus, we present the latent variables as ${\\bf z} = ({\\bf z}_{def}, {\\bf z}_{cont}) = ({\\bf z}_{1:2}, {\\bf z}_{3:10})$.\n\nWe compare traversal results of all latent variables on MNIST dataset for vanilla VAE  \\cite{kingma2013auto}, $\\beta$-VAE \\cite{higgins2017beta}, JointVAE \\cite{dupont2018learning} and our Guided-VAE  ($\\beta$-VAE, JointVAE results are adopted from \\cite{dupont2018learning}). While $\\beta$-VAE cannot generate meaningful disentangled representations for this dataset, even with controlled capacity increased, JointVAE can disentangle class type from continuous factors. Our Guided-VAE disentangles geometry properties rotation angle at $z_{1}$ and stroke thickness at $z_{2}$ from the rest content information ${\\bf z}_{3:10}$.  \n\nTo assess the disentangling ability of Guided-VAE against various baselines, we create a synthetic 2D shape dataset following \\cite{dsprites17, higgins2017beta} as a common way to measure the disentanglement properties of unsupervised disentangling methods. The dataset consists 737,280 images of 2D shapes (heart, oval and square) generated from four ground truth independent latent factors: $x$-position information (32 values), $y$-position information (32 values), scale (6 values) and rotation (40 values). This gives us the ability to compare the disentangling performance of different methods with given ground truth factors. We present the latent space traversal results in Figure \\ref{fig:dsprites}, where the results of $\\beta$-VAE and FactorVAE are taken from \\cite{kim2018disentangling}. Our Guided-VAE learns the four geometry factors with the first four latent variables where the latent variables ${\\bf z} \\in \\mathbb{R}^{6} = ({\\bf z}_{def}, {\\bf z}_{cont}) = ({\\bf z}_{1:4}, {\\bf z}_{5:6})$. We observe that although all models are able to capture basic geometry factors, the traversal results from Guided-VAE are more obvious with fewer factors changing except the target one. \n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.85}{\n\\begin{tabular}{c}\n\\hspace{1.2em} $\\beta$-VAE \\hspace{8.6em} FactorVAE\\\\\n\\includegraphics[width=0.52\\textwidth]{.\/figures\/2dshape_factor.jpg}\\\\\n\\hspace{4.1em} VAE \\hspace{7.2em} Guided-VAE (Ours)\\\\\n\\includegraphics[width=0.52\\textwidth]{.\/figures\/2dshape_ours.jpg}\n\n\\end{tabular}\n}\n\\caption{\\small \\textbf{Comparison of qualitative results on 2D shape.} First row: originals. Second row: reconstructions. Remaining rows: reconstructions of latent traversals across each\nlatent dimension. In our results, $z_1$ represents the $x$-position information, $z_2$ represents the $y$-position information, $z_3$ represents the scale information and $z_4$ represents the rotation information. }\n\\label{fig:dsprites}\n\\vspace{-8mm}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htp]\n\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tabular}{c} \n\\includegraphics[width=0.5\\textwidth]{.\/figures\/Celeba_2.png}\\\\\n\\hspace{1.0em} Gender \\hspace{8.0em} Smile\n\\end{tabular}\n}\n\\caption{\\small\\textbf{Comparison of Traversal Result learned on CelebA:} Column 1 shows traversed images from male to female. Column 2 shows traversed images from smiling to no-smiling. The first row is from \\cite{higgins2017beta} and we follow its figure generation procedure.}\n\\label{fig:Comparison_CelebA}\n\\end{center}\n\\vspace{-5mm}\n\\end{figure}\n\\vspace{-3mm}\n\\subsubsection{Quantitative Evaluation}\n\nWe perform two quantitative experiments with strong baselines for disentanglement and representation learning in Table \\ref{tab:disentangle_results} and \\ref{tab:classification-mnist-methods}. We observe significant improvement over existing methods in terms of {\\em disentanglement} measured by Z-Diff score \\cite{higgins2017beta}, SAP score \\cite{kumar2017variational}, Factor score \\cite{kim2018disentangling} in Table \\ref{tab:disentangle_results}, and representation {\\em transferability} based on classification error in Table \\ref{tab:classification-mnist-methods}.\n\n\\begin{table}\n\\begin{center}\n\\scalebox{0.8}{\n\\begin{tabular}{l | cccc}\n\\textbf{Model ($d_{\\bf z}=6$)} & {Z-Diff $\\uparrow$} &  {SAP $\\uparrow$} & {Factor $\\uparrow$}  \\\\\n\\hline\n\\textbf{\\textsc{VAE}   \\cite{kingma2013auto}}\n            & 78.2 & 0.1696 & 0.4074   \\\\\n\\textbf{\\textsc{$\\beta$-VAE ($\\beta$=2)\\cite{higgins2017beta}}}\n            & 98.1 & 0.1772 & 0.5786  \\\\\n\\textbf{\\textsc{FactorVAE ($\\gamma$=5) \\cite{kim2018disentangling}}} \n            & 92.4 & 0.1770 & 0.6134  \\\\\n\\textbf{\\textsc{FactorVAE ($\\gamma$=35) \\cite{kim2018disentangling}}} \n            & 98.4 & 0.2717 & 0.7100  \\\\\n\\textbf{\\textsc{$\\beta$-TCVAE ($\\alpha$=1,$\\beta$=5,$\\gamma$=1) \\cite{chen2018isolating}}} \n            & 96.8 & 0.4287 & 0.6968  \\\\\n\\hline\n\\textbf{\\textsc{Guided-VAE (Ours)}}   \n            & \\textbf{99.2} & 0.4320 & 0.6660 \\\\ \n\\textbf{\\textsc{Guided-$\\beta$-TCVAE (Ours)}}   \n            & 96.3 & \\textbf{0.4477} & \\textbf{0.7294} \\\\ \n\\end{tabular}\n}\n\\caption{\\small \\textbf{Disentanglement:}  Z-Diff score, SAP score, and Factor score over unsupervised disentanglement methods on 2D Shapes dataset. [$\\uparrow$ means higher is better]}\n\\label{tab:disentangle_results}\n\\end{center}\n\\vspace{-2mm}\n\\end{table}\n\nAll models are trained in the same setting as the experiment shown in Figure \\ref{fig:dsprites}, and are assessed by three disentangle metrics shown in Table \\ref{tab:disentangle_results}. An improvement in the Z-Diff score and Factor score represents a lower variance of the inferred latent variable for fixed generative factors, whereas our increased SAP score corresponds with a tighter coupling between a single latent dimension and a generative factor. Compare to previous methods, our method is orthogonal (due to using a side objective) to most existing approaches. $\\beta$-TCVAE \\cite{chen2018isolating} improves $\\beta$-VAE \\cite{higgins2017beta} based on weighted mini-batches to stochastic training. Our Guided-$\\beta$-TCVAE further improves the results in all three disentangle metrics.\n\n\\begin{table}\n\\begin{center}\n\\scalebox{0.7}{\n\\begin{tabular}{l | ccc}\n\\textbf{Model } & {$d_{\\bf z} = 16$ $\\downarrow$} &  {$d_{\\bf z} = 32$ $\\downarrow$}  &  {$d_{\\bf z} = 64$ $\\downarrow$} \\\\\n\\hline\n\\textbf{\\textsc{VAE}   \\cite{kingma2013auto}}\n            & 2.92\\%$\\pm$0.12 & 3.05\\%$\\pm$0.42 & 2.98\\%$\\pm$0.14\\\\\n\\textbf{\\textsc{$\\beta$-VAE($\\beta$=2)}\\cite{higgins2017beta}} \n            & 4.69\\%$\\pm$0.18 & 5.26\\%$\\pm$0.22 & 5.40\\%$\\pm$0.33 \\\\\n\\textbf{\\textsc{FactorVAE($\\gamma$=5)} \\cite{kim2018disentangling}} \n            & 6.07\\%$\\pm$0.05 & 6.18\\%$\\pm$0.20 & 6.35\\%$\\pm$0.48 \\\\\n\\textbf{\\textsc{$\\beta$-TCVAE ($\\alpha$=1,$\\beta$=5,$\\gamma$=1)} \\cite{chen2018isolating}} \n            & 1.62\\%$\\pm$0.07 & 1.24\\%$\\pm$0.05 & 1.32\\%$\\pm$0.09 \\\\\n\\hline\n\\textbf{\\textsc{Guided-VAE (Ours)}}   \n            & 1.85\\%$\\pm$0.08 & 1.60\\%$\\pm$0.08  & 1.49\\%$\\pm$0.06 \\\\            \n\\textbf{\\textsc{Guided-$\\beta$-TCVAE (Ours)}}   \n            & \\textbf{1.47\\%$\\pm$0.12 } & \\textbf{1.10\\%$\\pm$0.03 } & \\textbf{1.31\\%$\\pm$0.06} \\\\\n\\end{tabular}\n}\n\\caption{\\small \\textbf{Representation Learning:} Classification error over unsupervised disentanglement methods on MNIST. [$\\downarrow$ means lower is better]{\\scriptsize \\textsuperscript{$\\dagger$} The 95 \\% confidence intervals from 5 trials are reported.}}\n\\label{tab:classification-mnist-methods}\n\\end{center}\n\\vspace{-5mm}\n\\end{table}\n\n\\begin{figure*}[!htp]\n\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tabular}{ccc}\n\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Bald.png}  &\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Bangs.png} &\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Black_Hair.png}\\\\\n(a) Bald  &\n(b) Bangs &\n(c) Black Hair\\\\\n\\vspace{+1mm}\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Mouth_Slightly_Open.png} &\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Receding_Hairlines.png}  &\n\\includegraphics[width=0.31\\textwidth]{.\/figures\/Young.png}\\\\\n(d) Mouth Slightly Open &\n(e) Receding Hairlines &\n(f) Young\\\\\n\n\\end{tabular}\n\n}\n\\caption{\\small\\textbf{Latent factors learned by Guided-VAE on CelebA:} Each image shows the traversal results of Guided-VAE on a single latent variable which is controlled by the lightweight decoder using the corresponding labels as signal.}\n\\label{fig:celeba_appendix}\n\\end{center}\n\\end{figure*}\n\nWe further study representation transferability by performing classification tasks on the latent embedding of different generative models. Specifically, for each data point ($\\mathbf{x}, y$), we use the pre-trained generative models to obtain the value of latent variable ${\\bf z}$ given input image ${\\bf x}$. Here ${\\bf z}$ is a $d_{{\\bf z}}$-dim vector. We then train a linear classifier $f(\\cdot)$ on the embedding-label pairs $\\{({\\bf z}, y)\\}$ to predict the class of digits. For the Guided-VAE, we disentangle the latent variables ${\\bf z}$ into deformation variables ${\\bf z}_{def}$ and content variables ${\\bf z}_{cont}$ with same dimensions (i.e. $d_{{\\bf z}_{def}}=d_{{\\bf z}_{cont}}$). We compare the classification errors of different models with multiple choices of dimensions of the latent variables in Table \\ref{tab:classification-mnist-methods}. In general, VAE \\cite{kingma2013auto}, $\\beta$-VAE \\cite{higgins2017beta}, and FactorVAE \\cite{kim2018disentangling} do not benefit from the increase of the latent dimensions, and $\\beta$-TCVAE \\cite{chen2018isolating} shows  evidence that its discovered representation is more useful for classification task than existing methods. Our Guide-VAE achieves competitive results compare to $\\beta$-TCVAE, and our Guided-$\\beta$-TCVAE can further reduce the classification error to $1.1\\%$ when $d_{\\bf z} = 32$, which is $1.95\\%$ lower than the baseline VAE.  \n\nMoreover, we study the effectiveness of ${\\bf z}_{def}$ and ${\\bf z}_{cont}$ in Guided-VAE separately to reveal the different properties of the latent subspace. We follow the same classification task procedures described above but use different subsets of latent variables as input features for the classifier $f(\\cdot)$. Specifically, we compare results based on the deformation variables ${\\bf z}_{def}$, the content variables ${\\bf z}_{cont}$, and the whole latent variables ${\\bf z}$ as the input feature vector. To conduct a fair comparison, we still keep the same dimensions for the deformation variables ${\\bf z}_{def}$ and the content variables ${\\bf z}_{cont}$. Table \\ref{tab:classification-mnist-disentanglement} shows that the classification errors on ${\\bf z}_{cont}$ are significantly lower than the ones on ${\\bf z}_{def}$, which indicates the success of disentanglement as the content variables should determine the class of digits. In contrast, the deformation variables should be invariant to the class. Besides, when the dimensions of latent variables ${\\bf z}$ are higher, the classification errors on ${\\bf z}_{def}$ increase while the ones on ${\\bf z}_{cont}$ decrease, indicating a better disentanglement between deformation and content with increased latent dimensions.\n\n\n\\begin{table}\n\\begin{center}\n\\scalebox{0.65}{\n\\begin{tabular}{l | cccccc}\n\\textbf{Model } & $d_{{\\bf z}_{def}}$  & {$d_{{\\bf z}_{cont}}$}  & {$d_{{\\bf z}}$} & {${\\bf z}_{def}\\,\\, Error$  $\\uparrow$} &  {${\\bf z}_{cont}   \\,\\, Error$  $\\downarrow$}  &  {${\\bf z} \\,\\,Error$ $\\downarrow$}\\\\\n\\hline\n\\textbf{\\textsc{Guided-VAE}} \n              &8 & 8 & 16   & 27.1\\% & 3.69\\% & 2.17\\% \\\\\n\\,\\,\\,\\,\\,\\,  &16 &16 & 32  & 42.07\\% & 1.79\\% & 1.51\\% \\\\\n\\,\\,\\,\\,\\,\\,  &32 & 32 & 64 & 62.94\\% & 1.55\\% & 1.42\\% \\\\\n\n\\end{tabular}\n}\n\\caption{\\small \\textbf{Classification on MNIST using different latent variables as features:} Classification error over Guided-VAE with different dimensions of latent variables [$\\uparrow$ means higher is better, $\\downarrow$ means lower is better]}\n\\label{tab:classification-mnist-disentanglement}\n\\end{center}\n\\vspace{-9mm}\n\\end{table}\n\n\\vspace{-2mm}\n\\subsection{Supervised Guided-VAE}\n\n\\subsubsection{Qualitative Evaluation}\nWe first present qualitative results on the CelebA dataset \\cite{liu2015faceattributes} by traversing latent variables of attributes shown in Figure \\ref{fig:Comparison_CelebA}  and Figure \\ref{fig:celeba_appendix}. In Figure \\ref{fig:Comparison_CelebA}, we compare the traversal results of Guided-VAE with $\\beta$-VAE for two labeled attributes (gender, smile) in the CelebA dataset. The bottleneck size is set to 16 ($d_{\\bf z} = 16$). We use the first two latent variables $z_1, z_2$ to represent the attribute information, and the rest ${\\bf z}_{3:16}$ to represent the content information. During evaluation, we choose $z_t \\in \\{z_1, z_2\\}$ while keeping the remaining latent variables ${\\bf z}_t^{rst}$ fixed. Then we obtain a set of images through traversing $t$-th attribute (e.g., smiling to non-smiling) and compare them over $\\beta$-VAE. In Figure \\ref{fig:celeba_appendix}, we present traversing results on another six attributes.\n\n$\\beta$-VAE performs decently for the controlled attribute change, but the individual ${\\bf z}$ in $\\beta$-VAE is not fully entangled or disentangled with the attribute. We observe the traversed images contain several attribute changes at the same time. Different from our Guided-VAE, $\\beta$-VAE cannot specify which latent variables to encode specific attribute information. Guided-VAE, however, is designed to allow defined latent variables to encode any specific attributes. Guided-VAE outperforms $\\beta$-VAE by only traversing the intended factors (smile, gender) without changing other factors (hair color, baldness).\n\\vspace{-5mm}\n\n\\subsubsection{Quantitative Evaluation}\n\nWe attempt to interpret whether the disentangled attribute variables can control the generated images from the supervised Guided-VAE. We pre-train an external binary classifier for $t$-th attribute on the CelebA training set and then use this classifier to test the generated images from Guided-VAE. Each test includes $10,000$ generated images randomly sampled on all latent variables except for the particular latent variable $z_t$ we decide to control. As Figure \\ref{fig:supervised-quantitative} shows, we can draw the confidence-$z$ curves of the $t$-th attribute where $z=z_t\\in [-3.0, 3.0]$  with $0.1$ as the stride length. For the gender and the smile attributes, it can be seen that the corresponding $z_t$ is able to enable ($z_t < -1$) and disable ($z_t > 1$) the attribute of the generated image, which shows the controlling ability of the $t$-th attribute by tuning the corresponding latent variable $z_t$.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\vspace{-3mm}\n\\scalebox{0.9}{\n\\begin{tabular}{c}\n\\includegraphics[width=0.35\\textwidth]{.\/figures\/conf.pdf}\n\n\\end{tabular}\n}\n\\vspace{-1mm}\n\\caption{\n\\small Experts (high-performance external classifiers for attribute classification) prediction for being negatives on the generated images. We traverse $z_1$ (gender) and $z_2$ (smile) separately to generate images for the classification test.\n}\n\\label{fig:supervised-quantitative}\n\\end{center}\n\\vspace{-7mm}\n\\end{figure}\n\n\\vspace{-5mm}\n\\subsubsection{Image Interpolation}\n\\vspace{-2mm}\n\nWe further show the disentanglement properties of using supervised Guided-VAE on the CIFAR10 dataset. ALI-VAE borrows the architecture that is defined in \\cite{ALI}, where we treat $G_z$ as the encoder and $G_x$ as the decoder. This enables us to optimize an additional reconstruction loss. Based on ALI-VAE, we implement Guided-ALI-VAE (Ours), which adds supervised guidance through excitation and inhibition shown in Figure \\ref{fig:model}. ALI-VAE and AC-GAN \\cite{acgan} serve as a baseline for this experiment. \n\nTo analyze the disentanglement of the latent space, we train each of these models on a subset of the CIFAR10 dataset \\cite{cifar10} (Automobile, Truck, Horses) where the class label corresponds to the attribute to be controlled. We use a bottleneck size of 10 for each of these models. We follow the training procedure mentioned in \\cite{acgan} for training the AC-GAN model and the optimization parameters reported in \\cite{ALI} for ALI-VAE and our model. For our Guided-ALI-VAE model, we add supervision through inhibition and excitation on $z_{1:3}$.  \n\n\\begin{table}\n\\begin{center}\n\\scalebox{0.75}{\n\\begin{tabular}{l | cc}\n\\textbf{Model }             & Automobile-Horse $\\downarrow$   &  Truck-Automobile $\\downarrow$   \\\\\n\\hline\n\\textbf{\\textsc{AC-GAN} \\cite{acgan}}    &   88.27             &   81.13 \\\\\n\\textbf{\\textsc{ALI-VAE}} \\textsuperscript{$\\dagger$}   &   91.96             &   78.92 \\\\\n\\textbf{\\textsc{Guided-ALI-VAE (Ours)}}      &   \\textbf{85.43}             &  \\textbf{72.31} \\\\\n\\end{tabular}\n}\n\\caption{\\small \\textbf{Image Interpolation: } FID score measured for a subset of CIFAR10 \\cite{cifar10} with two classes each. [$\\downarrow$ means lower is better] \\textsuperscript{$\\dagger$} ALI-VAE is a modification of the architecture defined in \\cite{ALI} }\n\n\\label{tab:cifar-fid}\n\\end{center}\n\\end{table}\nTo visualize the disentanglement in our model, we interpolate the corresponding $z$, $z_{t}$ and $z_{t}^{rst}$ of two images sampled from different classes. The interpolation here is computed as a uniformly spaced linear combination of the corresponding vectors. The results in Figure \\ref{fig:traverse-Interpolation} qualitatively show that our model is successfully able to capture complementary features in $z_{1:3}$ and $z_{1:3}^{rst}$. Interpolation in $z_{1:3}$ corresponds to changing the object type. Whereas, the interpolation in $z_{1:3}^{rst}$ corresponds to complementary features such as color and pose of the object.\n\nThe right column in Figure \\ref{fig:traverse-Interpolation} shows that our model can traverse in $z_{1:3}$ to change the object in the image from an automobile to a truck. Whereas a traversal in $z_{1:3}^{rst}$ changes other features such as background and the orientation of the automobile. We replicate the procedure on ALI-VAE and AC-GAN and show that these models are not able to consistently traverse in $z_{1:3}$ and $z_{1:3}^{rst}$ in a similar manner. Our model also produces interpolated images in higher quality as shown through the FID scores \\cite{fid} in Table \\ref{tab:cifar-fid}. \n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{c}\n\\vspace{-2mm}\n\\includegraphics[width=0.48\\textwidth]{.\/figures\/cifar10_new.png}\n\\end{tabular}\n\\caption{\\small Interpolation of images in $z$, $z_{1:3}$ and $z_{1:3}^{rst}$ for AC-GAN, ALI-VAE and Guided-ALI-VAE (Ours).\n}\n\\label{fig:traverse-Interpolation}\n\\end{center}\n\\vspace{-8mm}\n\\end{figure}\n\n\n\n\\subsection{Few-Shot Learning}\nPreviously, we have shown that Guided-VAE can perform images synthesis and interpolation and form better representation for the classification task. Similarly, we can apply our supervised method to VAE-like models in the few-shot classification. Specifically, we apply our adversarial excitation and inhibition formulation to the Neural Statistician \\cite{edwards2016towards} by adding a supervised guidance network after the statistic network. The supervised guidance signal is the label of each input. We also apply the Mixup method \\cite{zhang2017mixup} in the supervised guidance network. However, we could not reproduce exact reported results in the Neural Statistician, which is also indicated in \\cite{korshunova2018bruno}. For comparison, we mainly consider results from Matching Nets \\cite{vinyals2016matching} and Bruno \\cite{korshunova2018bruno} shown in Table \\ref{tab:Omniglot}. Yet it cannot outperform Matching Nets, our proposed Guided Neural Statistician reaches comparable performance as Bruno (discriminative), where a discriminative objective is fine-tuned to maximize the likelihood of correct labels.\n\n\\begin{table}[!htb]\n\\begin{center}\n\\scalebox{0.8}{\n\\begin{tabular}{l | cc|cc}\n\\textbf{Model} & \\multicolumn{2}{c|}{\\textbf{5-way}} &  \\multicolumn{2}{|c}{\\textbf{20-way}} \\\\\n\\textbf{Omniglot } & \\textbf{ 1-shot} & \\textbf{5-shot} & \\textbf{1-shot} & \\textbf{ 5-shot}\\\\\n\\hline\n\\textbf{\\textsc{Pixels} \\cite{vinyals2016matching}} & 41.7\\%  &63.2\\% &26.7\\% & 42.6\\%\\\\\n\\textbf{\\textsc{Baseline Classifier} \\cite{vinyals2016matching}} & 80.0\\%  &95.0\\% &69.5\\% & 89.1\\%\\\\\n\\textbf{\\textsc{Matching Nets} \\cite{vinyals2016matching}} & 98.1\\%  &98.9\\% &93.8\\% & 98.5\\%\\\\\n\\textbf{\\textsc{Bruno} \\cite{korshunova2018bruno}} & 86.3\\%  &95.6\\% &69.2\\% & 87.7\\%\\\\\n\\textbf{\\textsc{Bruno (discriminative)} \\cite{korshunova2018bruno}} & 97.1\\%  &99.4\\% &91.3\\% & 97.8\\%\\\\\n\n\\hline\n\\textbf{\\textsc{Baseline} } & 97.7\\%  &99.4\\% &91.4\\% & 96.4\\%\\\\\n\\textbf{\\textsc{Ours (discriminative)}} &97.8\\% &99.4\\%  &92.1\\% &96.6\\%\\\\\n\\end{tabular}\n}\n\\caption{\\small \\textbf{Few-shot classification:} Classification accuracy for a few-shot learning task on the Omniglot dataset.}\n\\label{tab:Omniglot}\n\\vspace{-5mm}\n\\end{center}\n\\end{table}\n\n\n\n\n\\vspace{-4mm}\n\\section{Ablation Study}\n\\label{others}\n\n\\subsection{Deformable PCA}\nIn Figure \\ref{fig:pca}, we visualize the sampling results from PCA and $Dec_{sub}$. By applying a deformation layer into the PCA-like layer, we show deformable PCA has a more crispy sampling result than vanilla PCA. \n\n\\begin{figure}[!htb\n\\begin{center}\n\\scalebox{0.7}{\n\\begin{tabular}{c}\n\\includegraphics[width=0.52\\textwidth]{.\/figures\/pca_image.pdf}\\\\\n \n\\end{tabular}\n\\vspace{-2mm}\n}\n\n\\caption{\\small (Top) Sampling Result Obtained from PCA (Bottom) Sampling Result obtained from learned deformable PCA (Ours)}\n\\label{fig:pca}\n\\end{center}\n\\end{figure}\n\n\\vspace{-6mm}\n\\subsection{Guided Autoencoder}\n\\vspace{-2mm}\n\nTo further validate our concept of ``guidance'', we introduce our lightweight decoder to the standard autoencoder (AE) framework. We conduct MNIST classification tasks using the same setting in Figure \\ref{tab:classification-mnist-methods}. As Table \\ref{tab:classification-mnist-ae} shows, our lightweight decoder improves the representation learned in autoencoder framework. Yet a VAE-like structure is indeed not needed if the purpose is just reconstruction and representation learning. However, VAE is of great importance in building generative models. The modeling of the latent space of ${\\bf z}$ with e.g., Gaussian distributions is again important if a probabilistic model is needed to perform novel data synthesis (e.g., the images shown in Figure \\ref{fig:Comparison_CelebA}  and Figure \\ref{fig:celeba_appendix}).\n\n\\begin{table}[h!]\n\\begin{center}\n\\scalebox{0.8}{\n\\begin{tabular}{l | ccc}\n\\textbf{Model } & {$d_{\\bf z} = 16$ $\\downarrow$} &  {$d_{\\bf z} = 32$ $\\downarrow$}  &  {$d_{\\bf z} = 64$ $\\downarrow$} \\\\\n\\hline\n\\textbf{\\textsc{Auto-Encoder (AE)}}\n              & \\textbf{1.37}\\%$\\pm$0.05 & 1.06\\%$\\pm$0.04 & 1.34\\%$\\pm$0.04 \\\\\n\\textbf{\\textsc{Guided-AE (Ours)}}\n              & 1.46\\%$\\pm$0.06 & \\textbf{1.00}\\%$\\pm$0.06 & \\textbf{1.10}\\%$\\pm$0.08 \\\\\n\\end{tabular}\n}\n\\caption{\\footnotesize Classification error over AE and Guided-AE on MNIST.}\n\\label{tab:classification-mnist-ae}\n\\end{center}\n\\vspace{-7mm}\n\\end{table}\n\n\\subsection{Geometric Transformations}\n\\vspace{-2mm}\n\nWe conduct an experiment by excluding the geometry-guided part from the unsupervised Guided-VAE. In this way, the lightweight decoder is just a PCA-like decoder but not a deformable PCA. The setting of this experiment is exactly the same as described in  Figure \\ref{fig:mnist}. The bottleneck size of our model is set to 10 of which the first two latent variables $z_1, z_2$ represent the rotation and scaling information separately. As a comparison, we drop off the geometric guidance so that all 10 latent variables are controlled by the PCA-like light decoder. As shown in Figure \\ref{fig:ablation} (a) (b), it can be easily seen that geometry information is hardly encoded into the first two latent variables without a geometry-guided part.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tabular}{cc}\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/DeformablePCA_ablation.png}&\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/sample_9.png}\n\\\\ \n\\includegraphics[width=0.23\\textwidth]{.\/figures\/mnist_ablation_4.png}&\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/mnist_4.png}\n\\\\\n\\small{(a) Unsupervised Guided-VAE} & \\small{(b) Unsupervised Guided-VAE}\\\\\n\\small{without Geometric Guidance} & \\small{with Geometric Guidance}\\\\\n\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/ablation_DEI_2.png}\n& \\hspace{-2mm}\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/ablation_DEI_1.png}\n\\\\ \n\\includegraphics[width=0.23\\textwidth]{.\/figures\/celeba_ablation_smile.png}\n& \\hspace{-2mm}\n\\includegraphics[width=0.23\\textwidth]{.\/figures\/celeba_smile.png}\n\\\\ \n\\small{(c) Supervised Guided-VAE} & \\small{(d) Supervised Guided-VAE}\\\\\n\\small{without Inhibition} & \\small{with Inhibition}\\\\\n\\end{tabular}\n}\n\\vspace{1mm}\n\\caption{\\small{Ablation study on Unsupervised Guided-VAE and Supervised Guided-VAE}}\n\\label{fig:ablation}\n\\end{center}\n\\vspace{-5mm}\n\\end{figure}\n\n\\vspace{-2mm}\n\\subsection{Adversarial Excitation and Inhibition}\n\\vspace{-2mm}\nWe study the effectiveness of adversarial inhibition using the exact same setting described in the supervised Guided-VAE part. As shown in Figure \\ref{fig:ablation} (c) and (d), Guided-VAE without inhibition changes the smiling and sunglasses while traversing the latent variable controlling the gender information.\nThis problem is alleviated by introducing the excitation-inhibition mechanism into Guided-VAE.\n\n\\vspace{-2mm}\n\\section{Conclusion}\n\\vspace{-2mm}\nIn this paper, we have presented a new representation learning method, guided variational autoencoder (Guided-VAE), for disentanglement learning. Both unsupervised and supervised versions of Guided-VAE utilize lightweight guidance to the latent variables to achieve better controllability and transparency. Improvements in disentanglement, image traversal, and meta-learning over the competing methods are observed. Guided-VAE maintains the backbone of VAE and it can be applied to other generative modeling applications. \\\\\n\\hspace{-1mm}{\\bf Acknowledgment}. \\small{This work is funded by NSF IIS-1618477, NSF IIS-1717431, and Qualcomm Inc. ZD is supported by the Tsinghua Academic Fund for Undergraduate Overseas Studies. We thank Kwonjoon Lee, Justin Lazarow, and Jilei Hou for valuable feedbacks.}\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSystems consisting of diffusing particles or random walks interacting by means of a long-range  potential  are non-equilibrium systems, which describe different phenomena in physics, chemistry and biology. From a physical perspective they are used to study   metastable supercooled liquids \\cite{Supercool, Dean}, \nmelting in type-II high-temperature superconductors \\cite{Nelson},  electron transport in quasi-one-dimensional conductors \\cite{Quasi1d} and carbon nanotubes \\cite{Nanotube}.  From a chemical viewpoint the interest in these systems lies in the fact that some diffusion-controlled reactions processes rely on the diffusion of long-range interacting particles which react after they are closer than an effective capture distance. Some examples include radiolysis in liquids \\cite{ParkDeem}, \n electronic energy transfer reactions \\cite{Klafter} and a large variety of chemical reactions in amorphous media \\cite{RDreview}. From a biological viewpoint, the investigation of these systems is helpful in  understanding \nthe dynamics of interacting populations in terms of predator-prey models \\cite{Krap-Redner, Bray} and  \n membrane inclusions with curvature-mediated interactions \\cite{Reynwar1, Reynwar2}.  \n\nVicious walks (VW) are a class of non-intersecting random walks, where the process is terminated upon the first encounter between walkers \\cite{Fisher}. \nThe fundamental physical quantity describing VW is the survival probability which is defined  as the  probability  that no pair of particles has   collided up to time $t$. Diffusing particles or walks that are not allowed to meet each other but otherwise remain free, we call pure VW.  The behavior  of pure VW is generally well-known. The survival probability for such a system has been computed in the framework of renormalization group theory in arbitrary spatial dimensions up to  two-loop order \\cite{Cardy, Bhat1, Bhat2}. These approximations have been confirmed by exact results available in one dimension from the solution of the boundary problem of the Fokker-Plank equation \\cite{Krap-Redner, Bray}, using matrix model formalism \\cite{Katori} and Bethe ansatz technique \\cite{Derrida}. On the other hand the effect of long range interactions has been extensively investigated in many-body problems.\nIt has been shown that the existence of  long-range disorder leads to a rich phase diagram with interesting crossover effects \\cite{Halp, Bla, Prud}. If the   potential is Coulomb-like ($\\sim r^{-1-\\sigma}$) then systems in one dimension \n behave similar to a one-dimensional version of a Wigner crystal \\cite{Wigncrist}\nfor $\\sigma<0$ and similar to  a Luttinger liquid for $\\sigma\\ge 0$ \\cite{Mor-Zab}. If the potential is logarithmic then in the long-time limit the dynamics of  particles \nare described by non-intersecting paths \\cite{Hinrichsen,Katori}.\nThe generalization of VW that includes the effect of long range interactions has not attracted much attention in the literature. Up to our knowledge there was one attempt to study long-range VW \\cite{Bhat3}. Here the authors  considered the case of a  long-range potential decaying as $gr^{-\\sigma-d}$, where $g$  is a coupling constant. It was shown by applying the Wilson momentum shell renormalization group that only one of the critical exponents characterize long-range VW.\nFor a specific value of $\\sigma$ ($\\sigma=2-d$) they show that the exponent $\\gamma$, which determines the decay of the asymptotic survival probability with time, is given by the expression:\n\\beq\\label{sp_bhat}\n\\gamma= \\frac{p(p-1)}{4}u_1,\n\\end{equation}\n where $p$ the number of VW in the system, $u_1=(\\varepsilon\/2+[(\\varepsilon\/2)^2 +g]^{1\/2})$ and $\\varepsilon=2-d$. There are limitations to the above approach. First, it is restricted to a single form of the potential ($\\sim r^{-2}$) and systems such as membrane inclusions and chemical reactions have different power-law potentials. Second, it considers  identical walkers but one would like to have results if the diffusion constant of all walkers are different. Finally it is not convenient to compute higher-loop corrections using the Wilson formalism.\n\nIn this paper we reconsider the problem of long-range VW using methods of Callan-Symanzyk renormalized field theory in\nconjunction with an expansion in $\\varepsilon=2-d$ and $\\delta=2-d-\\sigma$. We note that it is more convenient to compute logarithmic and higher loop corrections by using this method. We derive the asymptotics of the survival and reunion probability for all values of the parameters $(\\sigma,d)$ for the first time.  \n\n\\begin{table}\n\\vspace{0.3cm}\n\\caption{\\label{tab:table1}%\nOne-loop survival probability of $p$ sets of particles with $n_j$ particles in each set large-time asymptotic at different regions of the $\\sigma-d$ plane. We refer to Figure 3 for specific value of $\\sigma$ and $d$ in each region.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nRegion & Survival probability & \\\\\n\\hline\n I & $t^{-(d-2)\/2}+t^{-(d+\\sigma-2)\/2}$ & \\\\\n II & $t^{-\\frac{1}{2}\\sum_{ij}n_in_j\\varepsilon}$ &   \\\\\n III & $t^{-\\frac{u_1}{2}\\sum_{ij}n_in_j (1+\\delta\/2\\log t)} $& \\\\\n IV & $t^{-(d-2)\/2}$ &  \\\\\n V, $d=2$ & $t^{-\\frac{\\sqrt{g_0}}{2}\\sum_{ij}n_in_j (1+\\delta\/2\\log t)} $ & \\\\\n VI,  $\\sigma=2-d$ & $t^{-\\frac{u_1}{2}\\sum_{ij}n_in_j}$ \\footnote{$u_1$ is defined by the formula (\\ref{sp_bhat}).} & \\\\\n \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nIn this paper we will show that there are several regions in $\\sigma-d$  plane in which we have different behavior of the critical exponent. Our results are summarized in  Table I. We note that results on the line $\\sigma+d=2$ have been obtained before \\cite{Bhat3}. Regions I and IV correspond to Gaussian or mean-field behavior (see Figure 3). In region II we found that the system reproduces pure VW. Logarithmic corrections in region III and at the short-range upper critical dimension $d=2$  have been obtained as series expansion in $\\delta=2-\\sigma-d$.\n\nThe remainder of this paper is organized as follows: \nSection~\\ref{sec:model} reviews the field theoretic formulation of long range VW and describes Feynman rules and dimensionalities of various quantities.  In section~\\ref{sec:LR} we derive the value of all fixed points and study their stability. Section~\\ref{sec:results} presents results for the critical exponents and logarithmic corrections of various dynamical observables. Section~\\ref{sec:concl} contains our concluding remarks. In Appendix A we give the details of the computation of some integrals that appear in Section~\\ref{sec:LR}.\n\n\n\n\\section{Modelling VW with long-range interations}\n\\label{sec:model}\n\n\nAs the  starting point of the description of our model we consider $p$ sets of diffusing particles or random walks with $n_i$ particles in each set $i=1\\dots p$, with a pairwise intraset interaction which includes a local or short-range part and a non-local or long-range tail. The local part determines the vicious nature of walks: if two walks belonging to the different sets are brought close to each other, both are  annihilated. Walks belonging to the same set are supposed to be independent. At $t=0$ all particles start in the vicinity of the origin. We are interested in the survival and reunion probabilities of walks at time $t>0$.\n\n\nA continuum description of a system of $N$ Brownian particles $X_i$ with two-body interactions is simplified by the coarse-graining procedure in which a large number of microscopic degrees of freedom are averaged out. Their influence is simply modelled as a Gaussian noise-term in the Langevin equations. A convenient starting point for the description of the stochastic dynamics is the path-integral formalism. Then the system under consideration is modeled by the classical action \n\\beq\\label{langLR} S= \\int\\limits_0^{+\\infty}dt  \\left(\\sum\\limits_{i=1}^N \\dot X_i^2\/(2D_i) + \\sum\\limits_{i<j}V(X_i-X_j) \\right)\\end{equation}\nwhere $t$ is (imaginary-)time, $X_i (t)$ is the $d$-dimensional vector denoting the position of $i$th particle at time $t$. \n$D_i$ is an $i$th particle diffusion coefficient. The path-integral representation of the probability density function for the particle displacements from their original positions is given by the functional ${\\cal Z}=\\int{\\cal D}X\\exp[-S]$. \n The survival probability is defined as the expectation value \n \\beq\\label{sp}P(t)=\\br\\prod_{i,j}[1-\\delta(X_i(t)-X_j(t))]\\ke \\end{equation} \n with respect to the functional ${\\cal Z}$. It is computed in the framework of usual perturbation theory and will be a sum of integrals over internal degrees of freedom. It is more convenient to perform these integrations in Fourier space. To do this we would need the Fourier transform of the interaction potential $V(r)$. We note that it is comprised of a short-range part of the form $V_0(r) = \\lambda\\delta(r)$ and a long-range part which decays with the distance $r$ as a power law, $V_l(r)= g r^{-d-\\sigma}$. The Fourier transform of the latter is divergent if $\\sigma\\geq0$.\nWe introduce the cut-off parameter $a$ to regularize the singularity \n$V_l(r)=g(r^{2}+a^{2})^{-(d+\\sigma)\/2}$.\nFourier transformation of this function is given by the expression \\beq\\label{Vft}V_l(q)= g\\frac{\\pi^{d\/2}2^{\\sigma}}{\\Gamma(\\frac{d+\\sigma}{2})}(q\/a)^{\\sigma\/2}K_{\\sigma\/2}(aq),\\end{equation} \nwhere $K_{\\sigma}$ is the modified Bessel-function with index $\\sigma$. Small $a$ expansion of (\\ref{Vft}) at leading order yields  \n\n\\beq V_l(q)  \\sim g\\begin{cases} \nq^{\\sigma}, & \\mbox{if} \\quad\\sigma\\ne 0\\\\\n\\log (aq) , & \\mbox{if } \\quad\\sigma=0 \\\\\n\\end{cases} \\end{equation}\nwhere we used the property $K_{-\\sigma}(x)=K_{\\sigma}(x)$ of the Bessel function. The non-universal coefficient coming from the Taylor expansion can be absorbed by the appropriate renormalization of the constant $g$. Special cases when $\\sigma$ is even gives  logarithmic behavior. Effectively it does not change  our results. So we focus on the typical term $ q^{\\sigma}$.\n\n\nThe second quantized version of the action (\\ref{langLR}) can be constructed using  standard methods \n\\cite{Doi, Peliti}. The generalization of the action to the long-range interacting case is also known \\cite{Mahan, Fetter}. The result is\n\n\\begin{widetext} \\beq\\label{hamil}\nS (\\phi_i,\\phi^{\\dagger}_i) = \\int dtd^dx \\{\\sum_i [ \\phi^{\\dagger}_i\\pd_{t} \\phi_i + D_i \\nabla\\phi^{\\dagger}_i\\nabla \\phi_i]\\}  +\\int dtd^dxd^dy \\sum\\limits_{i<j}  \\phi^{\\dagger}_i(t,x)\\phi_i(t,x)V_{ij}(x-y)\\phi^{\\dagger}_j(t,y)\\phi_j(t,y) \\;. \\end{equation}\n\\end{widetext}\n\nThe first term describes the evolution of free random walks with diffusion constants $D_i$. \nThe potential is \n\\beq V_{ij}(x-y) = \\lambda_{ij}\\delta(x-y) + g_{ij} V(x-y),\\end{equation} \nand we refer to $\\lambda_{ij}, g_{ij} $ as short-range and long-range coupling constants respectively. \n\nA dynamic response functional  associated with the action  (\\ref{hamil}) is\n\\beq\\label{Z}\n\\mathcal{Z} = \\int \\mathcal{D}\\phi\\mathcal{D}\\phi^{\\dagger} e^{-S(\\phi_i,\\phi^{\\dagger}_i) }\n\\end{equation}\nwhere $\\phi_i\\left(x,t\\right)$ is the complex scalar field.  After the quantization we may treat $\\phi^{\\dagger}_i\\left( x,t\\right) $ as the creation operator which creates a particle of sort $i$ at point $x$ at time $t$. \nHaving the dynamic response functional, correlation functions can be computed as functional averages (path integrals) of monomials of $\\phi$ and $\\phi^{\\dagger}$ with the weight $\\exp\\left\\{  -S(\\phi,\\phi^{\\dagger})\\right\\}$. \n\n\\begin{figure}\n\\includegraphics[scale=0.35]{fr.eps}\n\\caption{\\label{fig:frules} Feynman rules for the theory (\\ref{hamil}). Notice that both $\\lambda$ and $g$ vertices appear with differenet $i$ and $j$ indices and that $g$ has momentum dependence.}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[scale=0.6]{fd.eps}\n\\caption{\\label{fig:fdiag} One-loop Feynman diagrams contributing to $\\lambda_{Rij}$}\n\\end{figure*}\n\nAs a first step towards the renormalization group  analysis of this model, we discuss the dimensions of various quantities in (\\ref{hamil}) expressed in terms of momentum: \\beq[t]=p^{-2}\\quad [\\phi]=p^d \\quad [\\lambda]=p^{2-d} \\quad [g]=p^{2-d-\\sigma}.\\end{equation} \nThe naive dimension of the coupling constant $g$ allows us to identify the upper critical dimension $d_{c}(\\sigma)=2-d-\\sigma$.\nFor $\\sigma>0$, the short-range term naively dominates the long-range term and we expect to have the behavior of the system similar to the  case of pure VW. \nWe will reserve the symbol $\\varepsilon$ ($\\varepsilon=2-d$) to denote deviations from the short-range critical dimension $d_c=2$,   and  $\\delta$ ($\\delta=2-d-\\sigma$) for the deviations from the long-range critical dimension $d_{c}(\\sigma)$.\nIf $\\sigma=0$ then the critical dimension of the long-range part coincides with the short-range part and we have the non-trivial correction to the asymptotic behavior due to long-range interactions.\n This boundary separates mean-field or Gaussian behavior  from  long-range behavior. \nFor $\\sigma<0$ the long-range term  dominates the short-range term and we expect to have non-trivial corrections to the behavior of the system. \n\nNow we consider diagrammatic representation elements of model (\\ref{hamil}).\nIn zero-loop approximation the vertex 4-point function takes a simpler form after  Laplace-Fourier transformation: \n\\beq \\Gamma^{(2,2)}_{ij}(s,p) = V_{ij}(p_1+p_2) \\delta(\\sum\\limits_k p_k).\\end{equation}\nThe same transformation applied to the bare propagator yields:\n\\beq\\label{prop} \\Gamma^{(1,1)}_{j}(s,p) = (s+D_ip^2)^{-1} \n\\end{equation}\nWe note that there are no vertices in (\\ref{hamil}) that produce diagrams which dress the propagator, implying there is no field renormalization. As a consequence the bare propagator (\\ref{prop}) is the full propagator for the theory. Feynman rules are summarized in Figure 1. There are two vertices in the theory: one is a short-range $\\lambda$-vertex and another is a long-range momentum dependent $g$-vertex. Each external line of the vertex corresponds to a functionally independent field. The propagator is formed by contracting appropriate lines from different vertices. We recall the propagator is the correlation function of $\\phi_i$ and $\\phi^{\\dagger}_i$ fields only. \n\nPhysical observables are  computed with the help of correlation functions. The probability that $p$ sets of particles with $n_i$ particles in each set  start at the proximity of the origin and finish at $x_{i,\\alpha_i}$ ($i$ index  enumerates different sets and $\\alpha_i$ index enumerates particles in set $i$) without intersecting each other can be obtained by generalizing eqn (\\ref{sp}). \nIn the field theoretical formulation, this probability becomes the following correlation function:\n\\begin{equation}\n\\label{sp-G}\n G(t)\n=\\int\\prod_{i=1}^p\\prod_{\\alpha_i=1}^{n_i}d^dx_{i,\\alpha_i}\n\\langle\\phi_i(t,x_{i,\\alpha_i})(\\phi^{\\dagger}_i(0,0))^{n_i}\\rangle,\n\\end{equation}\nIn the Feynman representation it is the vertex with $2N$ ($N=\\sum_j n_j$) external lines. In the first order of the perturbation theory one needs to contract these lines with corresponding lines of the vertices in Figure 1. Since there are many independent fields in the correlation function (\\ref{sp-G}) this operation can be done in many ways. It yields a combinatorial factor, $n_in_j$, in  front of each diagram, which is the number of ways of constructing a loop from the $n_i$ lines of type $i$ and $n_j$ lines of type $j$ on the one hand and one line of type $i$ and one line of type $j$ on the other hand. From the next section we will see that  the survival probability scales as $G(t)\\sim t^{-\\gamma}$, where $\\gamma$ is the critical exponent. If all walks are free, $\\gamma=0$. In the presence of interactions we expect $\\gamma$ to be a universal quantity that does not depend on the intensity of the short-range interaction $\\lambda_{ij}$. It is convenient to introduce the so called truncated correlation function which is obtained from (\\ref{sp-G}) by factoring out external lines:\n\\beq\\label{tcf} \\Gamma(t) = G(t)\/(\\Gamma^{(1,1)})^{2N}\\end{equation}\n\n Another physical observable, the reunion probability, is defined as the probability that $p$ sets of particles with $n_i$ particles in each set  start at the proximity of the origin and without colliding into each other finish at the proximity of some point at time $t$: \n\\beq\\label{rp}\n R(t)\n=\\int d^dx \\prod_{i=1}^p \\langle\\phi_i(t,x)^{n_i}(\\phi^{\\dagger}_i(0,0))^{n_i}\\rangle,\n\\end{equation}\nIn the Feynman representation  it is depicted as the watermelon diagram with  $2N$ stripes.\nWe note that if the theory is free this expression is the product of free propagators and at the large-time limit the return probability  scales  as $R_{\\cal O}(t) \\sim t^{-(N-1)d\/2}$.\nIf interactions are taken into account it becomes $R(t) \\sim t^{-(N-1)d\/2 -2\\gamma}$, where $\\gamma$ is survival probability exponent. The reason that it enters with the factor 2 is the following. If we cut a watermelon diagram of the reunion probability correlation function in the middle then it produces  two vertex diagrams with $2N$ external lines of the survival probability correlation function. As a result the reunion probability is the product of two survival probabilities. It remains true in all orders of perturbation theory. For a rigorous proof we refer to \\cite{Bhat2}.\n\n\\section{The Renormalization of observables}\n\\label{sec:LR}\n\n\n  \\begin{figure}[b]\n\\includegraphics[scale=0.25]{fs.eps}\n\\caption{\\label{fig:sigmad} The critical behavior of vicious walks with long-range interactions in the different regions of the $(\\sigma,d)$ plane. Region I and IV correspond to the mean field short-range behavior, in region II will be critical short-range behavior, region III is the long-range behavior. The lines $d=2$ and $\\sigma+d=2$ represent regions V and VI respectively.}\n\\end{figure}\n\nWhile computing correlation functions like (\\ref{sp-G})  perturbatively one faces divergent integrals when $d=d_c$.\nThe convenient scheme developed for dealing with these divergences follows Callan-Symanzik renormalization-group analysis \\cite{Zinn, Amit}. Within this scheme we start with the bare correlation function \n$G(t;\\lambda,g)$, where $\\lambda =\\{\\lambda_{ij}\\}$, and $g=\\{g_{ij}\\}$ denote the set of bare short-range and long-range coupling constants. \nIn the renormalized theory it becomes $G_{R}(t;\\lambda_R,g_R,\\mu)$. From  dimensional analysis it follows that \n\\beq\\label{diman} G_{R}(t;\\lambda_R,g_R,\\mu) = G_{R}(t\\mu;\\lambda_R,g_R),\\end{equation}\nwhere $\\mu$ is the renormalization scale. The scale invariance leads to the expression \n\\beq\\label{scaleinv} G_{R}(t;\\lambda_R,g_R,\\mu) = Z(\\lambda_R, g_R, \\mu)G(t;\\lambda,g).\\end{equation}\nHere functions $Z$ are chosen in such a way that   $G_{R}(t,\\lambda_R,g_R,l)$ remains finite when the cut-off is removed at each order  in a  series expansion of $\\lambda_R$, $g_R$, $\\varepsilon$ and $\\delta$. From the fact that $G(t,\\lambda,g)$ does not depend on the renormalization scale $\\mu$ we get the Callan-Symanzik equation \n\\beq\\label{cseq} \\left(\\mu\\frac{\\pd}{\\pd \\mu} + \\beta_g \\frac{\\pd}{\\pd g} + \\beta_u \\frac{\\pd}{\\pd u} - \\gamma \\right) G_{R}=0, \\end{equation} \nwhere the $\\beta$-functions are defined by \n\\beq\\label{beta} \\beta_{\\lambda} (\\lambda_R, g_R)= \\mu  \\frac{\\pd}{\\pd \\mu} \\lambda_R\\qquad  \\beta_{g} (\\lambda_R, g_R)=\\mu \\frac{\\pd}{\\pd \\mu} g_R \\end{equation} and the function  $\\gamma$ by \\beq\\label{gamma}\\gamma(\\lambda_R, g_R) = \\mu \\frac{\\pd}{\\pd \\mu} \\ln Z.\\end{equation}\nThe renormalization group functions are understood as the expansion in double series of coupling constants $\\lambda$ and $g$ and deviations from the critical dimension $\\varepsilon$ and $\\delta$. We take $\\delta = O(\\varepsilon)$.\nThe coefficient $Z(\\lambda_R, g_R, \\mu)$ is fixed by the normalization conditions. It is more convenient to impose these conditions on the Laplace transform of the truncated correlation function (\\ref{tcf}). One sets the following condition then \n\\beq\\label{norm}  \\Gamma_{R}(\\mu) =1,\\end{equation}\nwhen $s=\\mu$. \nWe note that the same  multiplicative renormalization factor $Z$ yields $\\Gamma$ finite. From this fact one can infer that \n\\beq\\label{scalGamma}\\Gamma(\\mu;\\lambda,g) = Z(\\mu;\\lambda,g)^{-1}.\\end{equation}\nIf we express unrenormalized couplings in terms of renormalized ones (\\ref{scalGamma}) we will obtain the  equation for finding $Z$ explicitly. \n\n\nThe equation (\\ref{cseq}) can be solved by the method of characteristics. Within this method we let couplings depend on the scale which is parametrized by $\\mu(x)=x\\mu $. Here $x$ is introduced as a parametrization variable of the RG flow and is not to be confused with position. Henceforth $x$ will refer to this parametrization variable. We introduce running couplings $\\bar \\lambda(x)$ and $\\bar g(x)$. They satisfy the equations \\beq x\\frac{d}{dx}\\bar g(x)=\\beta_g(\\bar \\lambda(x),\\bar g(x))\\quad x\\frac{d}{dx}\\bar \\lambda(x)=\\beta_{\\lambda}(\\bar \\lambda(x),\\bar g(x)).\\end{equation} The renormalized value should be defined by the initial conditions $\\bar \\lambda(1)=\\lambda_R$ and $\\bar g(1)=g_R$. the solution of the equation  is then\n\\beq\\label{solRG} G_{R}(t)\n= e^{\\int\\limits_1^{\\mu t} \\gamma(\\bar \\lambda(x), \\bar g(x))dx\/x}G_{R}(\\mu^{-1};\\bar\\lambda(\\mu t),\\bar g(\\mu t),\\mu) \\end{equation}\n\nNext we calculate the first-order contribution to the   renormalized vertices.\nThe $\\lambda$-vertex  is   renormalized by the set of diagrams that are shown in Figure 2. We notice that there are no diagrams producing the momentum dependent $g$-vertex in the theory (\\ref{hamil}). \nThis statement is the corollary of the fact that only independent fields of power one enter into the expression of the vertex and there are no higher powers of fields. Also we keep in mind that the renormalized couplings are defined by the value of the vertex function  taken at zero external momenta. It produces the following expression:\n\\beq\\label{Run}\\begin{cases} \n\\lambda_{Rij} &= \\lambda_{ij} -\\frac{1}{2} (\\lambda_{ij}^2 I_1 + 2\\lambda_{ij} g_{ij}I_2 + g_{ij}^2I_3)\\\\\ng_{Rij} &= g_{ij} \\\\\n\\end{cases} \\end{equation}\n where $I_k=I_k(\\sigma;D_i,D_j)$ are one-loop integrals corresponding to the diagrams $a$, $b$, $c$ in the Figure 2 respectively. Using the Feynman rules we can explicitly write them down:\n \\beq\\label{int} I_k = \\int \\frac{d^dq}{(2\\pi)^d} \\frac{q^{(k-1)\\sigma}}{2s+(D_i+D_j)q^2}, \\quad k=1,2,3. \\end{equation}\n  We will use dimensional regularization procedure to compute these integrals. The details of the computation are summarized in  Appendix A.  We note that integrals will diverge logarithmically at different values of the spatial dimension $d$. For this reason it leads to different critical behavior in different regions of the $\\sigma-d$ plane (see Figure 3). These regions correspond to four possibilities for $\\varepsilon=2-d$ and $\\delta=2-d-\\sigma$ to be positive or negative. Only if $\\delta=O(\\varepsilon)$ or, in other words, if both $\\varepsilon$ and $\\delta$ are infinitesimally small but the ratio $\\varepsilon\/\\delta$ is finite we expect non-zero fixed points of the renormalization group flow. Similar approximation have been used before \\cite{Halp} but for different models with long-range disorder. It allows us to follow the standard procedure of deriving the $\\beta$-functions which consists of two steps.  \n  \n  First, we express  unrenormalized couplings in terms of the renormalized.   For the short-range  coupling constant $\\lambda$ it can be done by solving the quadratic equation in (\\ref{Run}). Expanding the square root and keeping terms up to the second order we infer that \n\\beq\\label{unR}\\begin{cases} \n\\lambda_{ij} &= \\lambda_{Rij} +\\frac{1}{2} (\\lambda_{Rij}^2 \\frac{a_d}{\\varepsilon} + 2\\lambda_{Rij} g_{Rij}\\frac{b_d}{\\delta} + g_{Rij}^2\\frac{c_d}{2\\delta-\\varepsilon})\\\\\ng_{ij} &= g_{Rij} \\\\\n\\end{cases} \\end{equation}\nwhere $a_d$, $b_d$ and $c_d$coefficients have been found explicitly in  Appendix A. Now we introduce dimensionless renormalized couplings \n\\beq\\label{dless} \\bar{g}_{Rij} = a_d(2s)^{-\\delta\/2} \\quad \\bar{\\lambda}_{Rij} = b_d(2s)^{-\\varepsilon\/2}.\\end{equation} An important observation is that $c_da_d=b_d^2$ which can be verified by  explicit substitution (see Appendix A). Multiplying the first and second equation in (\\ref{unR}) by the factors $a_d$ and $b_d$ respectively, and using redefinitions (\\ref{dless}) we can condense all pre-factors in the right hand side of the equations into the dimensionless constants. \n\nSecond, we differentiate equations (\\ref{unR}) with respect to the  scaling parameter $\\mu$. Using definitions (\\ref{beta}) and the fact that bare couplings do not depend on the scale,   we derive \n\\beq\\label{betalmg}\\begin{cases} \n\\beta_{\\lambda,ij} &= -\\varepsilon \\bar\\lambda_{Rij} + (\\bar\\lambda_{Rij}+ \\bar g_{Rij})^2\\\\\n\\beta_{g,ij} &= -\\delta \\bar g_{Rij} \\\\\n\\end{cases} \\end{equation}\nwhere the right hand side is understood as the leading contribution to the $\\beta$-functions from the double expansions in \n$\\lambda,g $ and $\\varepsilon,\\delta$. \n From (\\ref{betalmg}) we see that it is convenient to introduce new coupling constants $u_{Rij}= \\bar \\lambda_{Rij} +\\bar g_{Rij}$. After this step the  renormalization group equations read\n\n \\beq\\label{flow}\\begin{cases} \n\\beta_{u,ij} &= -\\varepsilon u_{Rij} + u_{Rij}^2 -g_{Rij} \\\\\n\\beta_{g,ij} &= -\\delta g_{Rij} \\\\\n\\end{cases} \\end{equation}\nWe note that in the last equations $g$ coupling constant  has been redefined $\\sigma \\bar g_{Rij}\\to g_{Rij}$.\n\nFixed points are zeros of the $\\beta$-functions. If $\\delta \\neq 0$ then the last equation in (\\ref{flow}) is zero only when $g_*=0$. Then the first equation has two solutions $u=0$ and $u=\\varepsilon$. If $\\delta=0$ then $g$ plays the role of a parameter and the fixed points are determined by the roots of the quadratic equation\n\\beq 0=-\\varepsilon u + u^2 -g\\end{equation}\nwhich are real if $g\\ge-(\\varepsilon\/2)^2$ and we find\n\\beq u_{1,2} = \\varepsilon\/2 \\pm \\sqrt{(\\varepsilon\/2)^2 + g}.\\end{equation}\nAll fixed points are listed in the Table II. The stability of these fixed points is determined by the matrix of partial derivatives \n\\beq\\label{stab} \\beta_* = - \n \\left( \\begin{array}{cc}\n\\pd\\beta_u\/\\pd u & \\pd\\beta_u\/\\pd g \\\\\n\\pd\\beta_g\/\\pd u & \\pd\\beta_g\/\\pd g  \n \\end{array} \\right)_{u=u_*, g=g_*} \\end{equation}\nEigenvalues are listed in the Table \\ref{tab:fp}. The Gaussian fixed point is stable in all directions for $\\varepsilon<0$ and $\\delta<0$ which corresponds to region I in  Figure 3. In this region we find both  short-range(pure VW) and  long-range  mean-field behavior depending on the sign of $\\sigma$. \nOn the contrary, for $\\varepsilon>0$ and $\\delta>0$ we find that the Gaussian fixed point is unstable(irrelevant) in all directions and the  short-range (pure VW) fixed point is stable(relevant) only in $u$-direction. It means that  long-range interactions will play a leading  role. This region corresponds to region  III in  Figure 3. Next for $\\varepsilon>0$ and $\\delta<0$ we find that the short-range (pure VW) fixed point is stable in all directions. It means that the system is insensitive to the long-range tail.  This region corresponds to region  II in  Figure 3. Finally for $\\varepsilon<0$ and $\\delta>0$ we find that the short-range (pure VW) fixed point is unstable in all directions and the system will be described by  mean-field at long time. \n\n\n\n\n\\begin{table}[b]\n\\caption{\\label{tab:fp}\nFixed points for flow equations (\\ref{flow}) and the corresponding eigenvalues $(\\lambda_1,\\lambda_2)$ of the stability matrix (\\ref{stab}). We note that $u_1$ and $u_2$ are values of the }\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nFixed point & $(u_*, g_*)$ & $(\\lambda_1, \\lambda_2)$ \\\\\n\\hline\n Gaussian & $(0,0)$ & $(\\varepsilon,\\delta)$ \\\\\nPure VW  & $(\\varepsilon,0)$ & $(-\\varepsilon,\\delta)$ \\\\\n LR stable  & $(u_{1},0)$ & $(-\\sqrt{\\varepsilon^2-4g},0)$ \\\\\n LR unstable & $(u_{2},0)$ & $(\\sqrt{\\varepsilon^2-4g},0)$\n \\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\section{Calculation of  critical exponents and discussion}\n\\label{sec:results}\n\nHere we describe our  method of computing critical exponents. It is based on the formula (\\ref{scalGamma}) from the previous section. First, we obtain the leading divergent part of the correlation function. \nThe renormalized correlation function depends on the scale $\\mu$ but it appears in all formulas in combination with time: $\\mu t$. \nSecond, since we have found the  bare coupling constant as a function of renormalized (dressed) couplings we express correlation function in terms of dressed couplings. \nFinally using the normalization condition (\\ref{norm}) and the definition (\\ref{gamma}) we differentiate $Z$ with respect to $\\mu \\pd\/\\pd\\mu$ to obtain the exponent $\\gamma$. The poles should cancel after this operation. \n\n In section 2 it was explained  that the truncated correlation function in the one-loop approximation is given by the formula\n\\beq\\label{oneloop} \\Gamma(t;\\lambda,g)= 1- \\sum_{i,j} n_i n_j\\left(\\lambda_{ij} I_1 +g_{ij}I_2\\right).\\end{equation}\nHere integrals are the same as in (\\ref{int}). \n\nWe start our analysis with the region I. Notice that truncated correlation function $\\Gamma(t)$ and survival probability $G(t)$ have similar large time behavior. We use large momentum cut-off to compute  integrals $I_1$ and $I_2$ as in formula (\\ref{mfint}) in Appendix A. The renormalization of coupling constants is trivial in this case.\nTherefore the leading contribution to the survival probability is given by \n\\beq G(t)\\sim t^{(2-d)\/2}+g_0t^{(2-d-\\sigma)\/2},\\end{equation} \nwhere $g_0$ is non-universal coefficient and we will not need its exact value. We notice that if $\\sigma>0$ the second term will decay faster than the first term and in the long-time limit it will produce the same behavior as mean-field pure VW. On the other hand if $\\sigma<0$\nthe first term will decay faster and long-range interactions will play a leading role.  Many authors observed similar behavior in various systems with long-range defects \\cite{Halp, Bla, Prud}. Intuitively if potential falls fast with distance than the system effectively represent system with short-range potential where particle interact when they are close to each other. \n\nRegion IV exhibits similar behavior. Now the integral $I_2$ is computed with the help of the dimensional regularization (\\ref{intres}) and the integral $I_1$ remains the same. From the fact (\\ref{diman}) one can infer that\nthe survival probability scales as \n\\beq G(t)\\sim t^{(2-d)\/2}.\\end{equation}\nShort-range behavior dominates because  the running coupling constant will flow towards the  Gaussian fixed point  at long time limit which is the only stable fixed in this region. This result is exact regardless the number of loops one takes into account.\n\n\nIn Region II the computation is as follows. \n\\beq\\label{rtwo} \\ln Z = \\sum n_in_j\\left( \\lambda_{ij}\\frac{a_d}{\\varepsilon} + g_{ij}t^{(2-d-\\sigma)\/2}\\right),\\end{equation}\nso plugging the result from (\\ref{ad}) to (\\ref{rtwo}) we obtain at the fixed point $(\\lambda_*=\\varepsilon, g=0)$\n\\beq\\label{gtwo} \\gamma = -\\frac{1}{2}\\sum n_in_j\\varepsilon \\end{equation}\nAnd we reproduce the pure VW behavior.  This result is the reflection of the fact that the renormalization-group trajectories run away to stable pure VW fixed point. It is with agreement with the results obtained by Katori in \\cite{Katori} for $d=1$, and the logarithmic intraset particle interactions. The irrelevance of the long-range interaction in lower dimensions is a typical phenomenon observed in a various out of equilibrium interacting\nparticle systems. \n \n\nWe now consider regions III, V and VI.  Integrals in (\\ref{oneloop}) are computed via dimensional regularization. Taking the inverse of (\\ref{oneloop}) and then logarithm one can obtain at the leading order:  \n\\beq\\label{lnZ} \\log Z = \\sum n_in_j\\left(\\lambda_{ij}\\frac{a_d}{\\varepsilon} + g_{ij}\\frac{b_d}{\\delta}\\right)\\end{equation}\nwhere  $a_d$ and $b_d$ are defined in Appendix A in (\\ref{ad}) and (\\ref{bd}).\nWe note that after taking the derivative the poles in (\\ref{lnZ}) will cancel in the limit of $\\delta=O(\\varepsilon)$. Also one recalls the expansion (\\ref{unR}) and the redefinitions in (\\ref{dless}). Using (\\ref{gamma}) we show that the expression for the function $\\gamma$ which determines critical exponent takes the form\n\\beq \\gamma =- \\frac{1}{2}\\sum_{ij} n_in_j u_R \\end{equation}\n Evaluated at the stable fixed point  $(u_1=\\varepsilon\/2+ \\sqrt{(\\varepsilon\/2)^2 +g}$ it gives the following result:\n\\beq\\label{gthree} \\gamma = -\\frac{1}{2}\\sum_{ij}  n_in_j u_1,  \\end{equation}\nand the survival probability scales as  $G(t)\\sim t^{\\gamma}$. \n\n\nWe will now find the logarithmic corrections to this scaling law. The running coupling constant can be found from the flow equation (\\ref{flow}):  $\\bar{g}(x) = e^{-\\delta x} g$. In the  case $\\delta,\\varepsilon=0$ (the intersection of regions V and VI) the flow equation for  $\\bar u(x)$ is \n\\beq\\label{flowu} x\\frac{d\\bar u(x)}{dx} = -\\bar{u}^2(x) +g \\end{equation} and the solution is\n\\beq\\label{tanh} \\bar{u}(x) = \\sqrt{g} \\tanh (\\sqrt{g} \\log x +\\phi_0)\\sim \\sqrt{g} \\tanh (\\sqrt{g} \\log x),\\end{equation}\nwhere $\\phi_0$ is the initial condition and we do not need its exact form. After plugging this expression into the (\\ref{solRG}) we infer \n\\beq\\label{gammaint} \\int\\limits_1^{\\mu t} \\gamma(\\bar u, \\bar g)\\frac{dx}{x}  \\sim \\log (\\cosh (\\sqrt{g} \\log \\mu t))\\end{equation}\n Thus the survival probability is\n\\beq\\label{dnulld2} G(t) \\sim \\cosh (\\sqrt{g} \\log t)^{-\\frac{1}{2}\\sum n_in_j } \\end{equation}\nIn the limit of large time $\\cosh (\\sqrt{g} \\log t)\\sim t^{\\sqrt{g}} $ implying $gamma =- \\frac{1}{2}\\sum_{ij} n_in_j \\sqrt{g}$ which is consistent with equation (\\ref{gthree}). \nFor negative coupling constant $g<0$ the solution in (\\ref{tanh}) becomes\n\\beq\\label{tan} \\bar{u}(x) \\sim -\\sqrt{|g|} \\tan (\\sqrt{|g|} \\log x)\\end{equation}\nThe integral (\\ref{gammaint}) is divergent if $t>\\exp(\\pi\/2\\sqrt{|g|})$ which leads to the result that the survival probability is zero beyond this time. For smaller times one has  $G(t) \\sim \\cos (\\sqrt{|g|} \\log t)^{-\\frac{1}{2}\\sum n_in_j }$. Thus, upto one-loop order approximation, It implies that if walks are attracted to each other then all of them will annihilate at some finite time. This might be a signature of faster than power law decay and we expect to have corrections to this behavior at higher loop approximation.\n\nNext we consider the case when $\\varepsilon=0$ and $\\delta\\ne 0$ but $\\delta$ remains small i.e. region V. The flow equation for the $\\bar u(x)$ is \n\\beq\\label{flowug} x\\frac{d\\bar u(x)}{dx} = -\\bar{u}^2(x) +g x^{-\\delta}\\end{equation} and the solution can be found by the method of perturbation. \nUp to the first order\n\\beq \\bar u(x) = \\sqrt{g}\\tanh (\\sqrt{g} \\log x)+\\delta\\sqrt{g}\\log(x)\\tanh (\\sqrt{g} \\log x)\\end{equation}\nAfter plugging this expression into eqn (\\ref{solRG}) we infer \n\\beq \\int\\limits_1^{\\mu t} \\gamma(\\bar u, \\bar g)\\frac{dx}{x}  \\sim -\\frac{1}{2}\\sum n_in_j\\left(\\log(t^{\\sqrt{g}}) +\\frac{1}{2}\\delta\\sqrt{g} \\log^2 (t)\\right) \\end{equation}\nTherefore we have the correction to the survival probability in the form\n  \\beq G\\sim  t^{-\\frac{1}{2}\\sum n_in_j \\sqrt{g} (1 + \\delta\/2(\\log t))}\\end{equation}\n\n\n\nNow we extend our analysis to the case when $\\varepsilon>0$, corresponding to regions III and VI. The evolution of the coupling constant is \\beq x\\frac{d}{dx}\\bar u(x) = \\varepsilon\\bar u-\\bar u^2 +g x^{\\delta}\\end{equation}\nWe choose the ansatz in the form $\\bar u(x) = u_0(x)+\\delta v(x)$. For $\\delta=0$ (i.e. region VI) the equation for  $u_0(x)$ reads \n\\beq \\label{u0ex} x\\frac{d}{dx} u_0(x) = \\varepsilon u_0- u_0^2 +g  \\end{equation}\nand we reproduce the result (\\ref{gthree}). We now extend to the case where $\\varepsilon,\\delta>0$ (region III). Here we will need the exact solution to (\\ref{u0ex}) to find the corrections:\n\\beq  u_0(x) = \\frac{Cx^{u_1-u_2} u_1 +u_2}{1+Cx^{u_1-u_2}}, \\end{equation} where $C =(u_R-u_2)\/(u_1-u_R)$. The logarithmic correction follows from the form of the perturbation. The equation for $v(x)$ is \n\\beq x\\frac{d}{dx}v(x) = \\varepsilon v-2u_0v -g \\log x \\end{equation}\n The solution can be found explicitly as a combination of hypergeometric functions. \n  In the most interesting case, $\\varepsilon=1$ ($d=1$) the hypergeometric functions are degenerate and become linear functions. Corrections to the integral then read\n\\beq \\int\\limits_1^{\\mu t} \\gamma dx\/x\\sim \\frac{1}{2}\\delta u_1 \\log^2( t)\n+\\log( t)  ( t)^{u_1-u_2}) \\end{equation}\nIn the limit of large time only the first term contributes to the exponent and the survival probability scales as \n\n\\beq G\\sim  t^{-\\frac{1}{2}\\sum n_in_j u_1 (1 + \\delta\/2\\log t)}\\end{equation}\n\n\n\n\\section{Conclusion}\n\\label{sec:concl}\nIn summary, we studied long-range vicious walks using the methods of Callan-Symanzik renormalized field theory.\n Our work confirms the previously known RG fixed point structure including their stability regions.\n We calculated \nthe critical exponents for all values of $\\sigma$ and $d$ to first order in $\\varepsilon$ expansion and to all orders in $\\delta$ expansion, which have hitherto been known only for $d+\\sigma=2$. Our results indicate that, depending on the exact values of $d$ and $\\sigma$, the system can be dominated by either short range (pure VW) or long range behaviors.\nIn addition, we calculated the leading logarithmic corrections for several dynamical observables that are typically measured in simulations.\n\nWe hope that our work stimulates further interest in long-range vicious walks. It would be interesting to see further simulation results for the critical exponents for $d>1$ and for logarithmic corrections. Also, it would be interesting to have analytical and numerical results for other universal quantities such as scaling functions and amplitudes. \n\n\n\\section{Acknowledgments}\n  AG would like to acknowledge UC Merced start-up funds and a James S. McDonnell Foundation Award for Studying Complex Systems.\n\n\n\n\n\n\n\\section*{Appendix A}\n\nEffective four-point function (one-particle irreducible, 1PI) that appeared in (\\ref{Run}) is composed of usual short-range  and new momentum dependent vertices. This gives rise to integrals (\\ref{int}). The first integral $\\mu=1$ has been evaluated in \\cite{Cardy} by using  alpha representation $1\/(q^2+s) = \\int_0^{+\\infty} d\\alpha e^{i(q^2+s)\\alpha}$ and the result is \\beq I_1=K_d (2s)^{-\\varepsilon\/2}\\Gamma(\\varepsilon\/2).\\end{equation} We notice that since there is no angular dependence one can perform $d-1$ integrations and one will be left with one dimensional integral. To compute this integral we use the formula \\cite{GR}:\n\n\\beq \\int\\limits_{0}^{+\\infty} dx \\frac{x^{\\nu-1}}{P+Qx^2} = \\frac{1}{2P} \\left(\\frac{P}{Q}\\right)^{\\nu\/2} \\Gamma\\left(\\frac{\\nu}{2}\\right)\\Gamma\\left(1-\\frac{\\nu}{2}\\right)\\end{equation}\nWe see that in our case $P=s$, $Q=(D_i+D_j)$ and $\\nu =d+(\\mu-1)\\sigma$. This immediately gives the result:\n\n\\begin{align}  I_{\\mu} =&  \\frac{K_d}{2} \\left(\\frac{1}{(D_i+D_j)}\\right)^{\\frac{d+(\\mu-1)\\sigma}{2}} s^{\\frac{d+(\\mu-1)\\sigma}{2}-1} \\times\n\\nonumber\\\\\n&\\times\\Gamma\\left(\\frac{d+(\\mu-1)\\sigma}{2}\\right)\\Gamma\\left(1-\\frac{d+(\\mu-1)\\sigma}{2}\\right)\n\\,, \\label{intres}%\n\\end{align}\nwhere $K_d = 2^{d-1}\\pi^{-d\/2}\\Gamma^{-1}(d\/2)$ is the surface area of $d$-dimensional unit sphere. \n\nIt is convenient to define \\beq\\label{ad} a_d = \\frac{K_d}{2} \\left(\\frac{2}{(D_i+D_j)}\\right)^{d\/2} (2s)^{-\\varepsilon\/2}\\end{equation}\n\\beq\\label{bd} b_d = \\frac{K_d}{2} \\left(\\frac{2}{(D_i+D_j)}\\right)^{(d+\\sigma)\/2} (2s)^{-\\delta\/2}\\end{equation}\n\\beq\\label{cd} c_d = \\frac{K_d}{2} \\left(\\frac{2}{(D_i+D_j)}\\right)^{(d+2\\sigma)\/2} (2s)^{-(2\\delta-\\varepsilon)\/2}\\end{equation}\nSo integral $I_{\\mu}$ in the limit of $\\delta=O(\\varepsilon)$ can be written as:\n\\beq I_1=\\frac{a_d}{\\varepsilon},\\quad I_2=\\frac{b_d}{\\delta},\\quad I_3=\\frac{c_d}{2\\delta-\\varepsilon}.\\end{equation}\nWe used an expansion $\\Gamma(\\varepsilon\/2) \\sim 2\/\\varepsilon$ for small $\\varepsilon$.\nAn important property of coefficients (\\ref{ad}) - (\\ref{cd}) is that \\beq c_da_d=b^2_d,\\end{equation}\nwhich can be verified by direct substitution. \n\n\n\nNow we compute mean field integrals:\n\n\\beq\\label{mfint} I_{\\mu} = \\int d^dq dt q^{d+\\sigma}\\exp(-t(D_i+D_j)q^2)\\sim t^{-(d+\\sigma-2)\/2}, \\end{equation}\nwhere we assumed that the large momentum cut-off is imposed and corresponding coupling constants have been renormalized. The non-universal coefficient  is not important. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nDistributional reinforcement learning  \\cite{jaquette73markov,sobel82variance,white88mean,morimura2010nonparametric,c51} focuses on the intrinsic randomness of returns within the reinforcement learning (RL) framework. As the agent interacts with the environment, irreducible randomness seeps in through the stochasticity of these interactions, the approximations in the agent's representation, and even the inherently chaotic nature of physical interaction \\cite{yu2016more}. Distributional RL aims to model the distribution over returns, whose mean is the traditional value function, and to use these distributions to evaluate and optimize a policy.\n\nAny distributional RL algorithm is characterized by two aspects: the parameterization of the return distribution, and the distance metric or loss function being optimized. Together, these choices control assumptions about the random returns and how approximations will be traded off. Categorical DQN \\citep[C51]{c51} combines a categorical distribution and the cross-entropy loss with the Cram\\'er-minimizing projection \\cite{rowland2018analysis}. For this, it assumes returns are bounded in a known range and trades off mean-preservation at the cost of overestimating variance.\n\nC51 outperformed all previous improvements to DQN on a set of 57 Atari 2600 games in the Arcade Learning Environment \\cite{bellemare13arcade}, which we refer to as the Atari-57 benchmark. Subsequently, several papers have built upon this successful combination to achieve significant improvements to the state-of-the-art in Atari-57 \\cite{hessel2018rainbow,gruslys2018reactor}, and challenging continuous control tasks \\cite{barthmaron2018d4pg}.\n\nThese algorithms are restricted to assigning probabilities to an a priori fixed, discrete set of possible returns. \\citet{dabney2017qr} propose an alternate pair of choices, parameterizing the distribution by a uniform mixture of Diracs whose locations are adjusted using quantile regression. Their algorithm, QR-DQN, while restricted to a discrete set of quantiles, automatically adapts return quantiles to minimize the Wasserstein distance between the Bellman updated and current return distributions. This flexibility allows QR-DQN to significantly improve on C51's Atari-57 performance.\n\nIn this paper, we extend the approach of \\citet{dabney2017qr}, from learning a discrete set of quantiles to learning the full quantile function, a continuous map from probabilities to returns. When combined with a base distribution, such as $U([0,1])$, this forms an implicit distribution capable of approximating any distribution over returns given sufficient network capacity. Our approach, \\textit{implicit quantile networks} (IQN), is best viewed as a simple distributional generalization of the DQN algorithm \\cite{mnih15nature}, and provides several benefits over QR-DQN. \n\nFirst, the approximation error for the distribution is no longer controlled by the number of quantiles output by the network, but by the size of the network itself, and the amount of training. Second, IQN can be used with as few, or as many, samples per update as desired, providing improved data efficiency with increasing number of samples per training update. Third, the implicit representation of the return distribution allows us to expand the class of policies to more fully take advantage of the learned distribution. Specifically, by taking the base distribution to be non-uniform, we expand the class of policies to $\\epsilon$-greedy policies on arbitrary distortion risk measures \\cite{yaari1987dual,wang1996premium}.\n\nWe begin by reviewing distributional reinforcement learning, related work, and introducing the concepts surrounding risk-sensitive RL. In subsequent sections, we introduce our proposed algorithm, IQN, and present a series of experiments using the Atari-57 benchmark, investigating the robustness and performance of IQN. Despite being a simple distributional extension to DQN, and forgoing any other improvements, IQN significantly outperforms QR-DQN and nearly matches the performance of Rainbow, which combines many orthogonal advances. \nIn fact, in human-starts as well as in the hardest Atari games (where current RL agents still underperform human players) IQN improves over Rainbow.\n\n\\section{Background \/ Related Work}\n\\label{sec:background}\n\nWe consider the standard RL setting, in which the interaction of an agent and\nan environment is modeled as a Markov Decision Process \n$(\\mathcal{X}, \\mathcal{A}, R, P, \\gamma)$ \\cite{puterman94markov}, \nwhere $\\mathcal{X}$ and $\\mathcal{A}$ denote the state and action spaces, \n$R$ the (state- and action-dependent) reward function,\n$P(\\cdot | x, a)$ the transition kernel, \nand $\\gamma \\in (0, 1)$ a discount factor. A policy $\\pi(\\cdot | x)$ maps a state to a distribution over actions.\n\nFor an agent following policy $\\pi$, the discounted sum of future \nrewards is denoted by the random variable $Z^\\pi(x, a) = \\sum_{t=0}^\\infty \\gamma^t R(x_t, a_t)$, where $x_0 = x$, $a_0 = a$, $x_t \\sim P(\\cdot | x_{t-1}, a_{t-1})$, and $a_t \\sim \\pi(\\cdot | x_{t})$. The action-value function is defined as\n$Q^\\pi(x,a) = \\mathbb{E}\\left[ Z^\\pi(x,a)\\right]$, and can be characterized by the Bellman equation \n\\begin{equation*}\nQ^\\pi(x,a) = \\mathbb{E} \\left[ R(x,a) \\right] + \n\\gamma \\mathbb{E}_{P,\\pi} \\left[ Q^\\pi(x',a')\\right]. \n\\end{equation*}\nThe objective in RL is to find an optimal policy $\\pi^*$, which maximizes $\\mathbb{E}[Z^\\pi]$,\ni.e.~$Q^{\\pi^*}(x,a) \\geq Q^{\\pi}(x,a)$ for all $\\pi$ and all $x, a$. One approach is to find the unique fixed point $Q^* = Q^{\\pi^*}$ of the Bellman optimality operator \\cite{bellman57dynamic}: \n\\begin{equation*}\nQ(x,a) = \\mathcal{T} Q(x,a) := \\mathbb{E}\\left[ R(x,a) \\right] + \\gamma \\mathbb{E}_{P}  \\max_{a'} Q(x',a').\n\\end{equation*}\nTo this end, Q-learning \\cite{watkins1989learning} iteratively improves an estimate, $Q_\\theta$, of the optimal action-value function, $Q^*$, by repeatedly applying the Bellman update:\n\\begin{equation*}\nQ_\\theta(x,a) \\leftarrow \\mathbb{E} \\left[ R(x,a) \\right] + \n\\gamma \\mathbb{E}_{P} \\left[\\max_{a'} Q_\\theta(x',a')\\right]. \n\\end{equation*}\nThe action-value function can be approximated by a parameterized function $Q_\\theta$ (e.g.~a neural network), and trained by minimizing the squared temporal difference (TD) error,\n\\begin{equation*}\n    \\delta_t^2 = \\left[ r_t + \\gamma \\max_{a' \\in \\mathcal{A}} Q_\\theta(x_{t+1}, a') - Q_\\theta(x_t, a_t) \\right]^2,\n\\end{equation*}\nover samples $(x_t, a_t, r_t, x_{t+1})$ observed while following an $\\epsilon$-greedy policy over $Q_\\theta$. This policy acts greedily with respect to $Q_\\theta$ with probability $1 - \\epsilon$ and uniformly at random otherwise.\nDQN \\cite{mnih15nature} uses a convolutional neural network to parameterize $Q_\\theta$ and the Q-learning algorithm to achieve human-level play on the Atari-57 benchmark.\n\n\n\\subsection{Distributional RL}\n\nIn distributional RL, the distribution over returns (the law of $Z^\\pi$) is considered instead of the scalar value function $Q^\\pi$ that is its expectation. This change in perspective has yielded new insights into the dynamics of RL \\cite{azar2012sample}, and been a useful tool for analysis \\cite{lattimore2012pac}. Empirically, distributional RL algorithms show improved sample complexity and final performance, as well as increased robustness to hyperparameter variation \\cite{barthmaron2018d4pg}.\n\nAn analogous distributional Bellman equation of the form\n\\begin{equation*}\nZ^\\pi(x,a) \\stackrel{D}{=} R(x,a) + \\gamma  Z^\\pi(X',A')\n\\end{equation*}\ncan be derived, where $A \\stackrel{D}{=} B$ denotes that \ntwo random variables $A$ and $B$ have equal probability laws, and the \nrandom variables $X'$ and $A'$ are distributed according to $P(\\cdot | x, a)$ and $\\pi(\\cdot | x')$, respectively.\n\n\\citet{morimura10parametric} defined the distributional Bellman operator explicitly in terms of conditional probabilities, parameterized by the mean and scale of a Gaussian or Laplace distribution, and minimized the Kullback-Leibler (KL) divergence between the Bellman target and the current estimated return distribution. However, the distributional Bellman operator is not a contraction in the KL.\n\nAs with the scalar setting, a distributional Bellman optimality operator can be defined by\n\\begin{equation*}\n\\mathcal{T} Z(x,a) \\stackrel{D}{:=} R(x,a) + \\gamma Z(X',\\argmax_{a' \\in \\mathcal{A}} \\expect Z(X', a')),\n\\end{equation*}\nwith $X'$ distributed according to $P(\\cdot | x, a)$. While the distributional Bellman operator for policy evaluation is a contraction in the $p$-Wasserstein distance \\cite{c51}, this no longer holds for the control case. Convergence to the optimal policy can still be established, but requires a more involved argument.\n\n\\citet{c51} parameterize the return distribution as a categorical distribution over a fixed set of equidistant points and minimize the KL divergence to the projected distributional Bellman target. Their algorithm, C51, outperformed previous DQN variants on the Atari-57 benchmark. Subsequently, \\citet{hessel2018rainbow} combined C51 with enhancements such as prioritized experience replay \\cite{schaul16prioritized}, $n$-step updates \\cite{sutton1988learning}, and the dueling architecture \\cite{wang2016dueling}, leading to the Rainbow agent, current state-of-the-art in Atari-57.\n\nThe categorical parameterization, using the projected KL loss, has also been used in recent work to improve the critic of a policy gradient algorithm, D4PG, achieving significantly improved robustness and state-of-the-art performance across a variety of continuous control tasks \\cite{barthmaron2018d4pg}.\n\n\\subsection{$p$-Wasserstein Metric}\n\nThe $p$-Wasserstein metric, for $p \\in [1, \\infty]$, plays a key role in recent results in distributional RL \\cite{c51,dabney2017qr}. It has also been a topic of increasing interest in generative modeling \\cite{wgan,bousquet2017optimal,tolstikhin2017wasserstein}, because unlike the KL divergence, the Wasserstein metric inherently trades off approximate solutions with likelihoods.\n\nThe $p$-Wasserstein distance is the $L_p$ metric on inverse cumulative distribution functions (c.d.f.), also known as quantile functions \\cite{muller1997integral}. For random variables $U$ and $V$ with quantile functions $F_U^{-1}$ and $F_V^{-1}$, respectively, the $p$-Wasserstein distance is given by\n\\begin{equation*}\n    W_p(U, V) = \\left( \\int_0^1 |F_U^{-1}(\\omega) - F_V^{-1}(\\omega)|^p d\\omega \\right)^{1\/p}.\n\\end{equation*}\n\nThe class of optimal transport metrics express distances between distributions in terms of the minimal cost for transporting mass to make the two distributions identical. This cost is given in terms of some metric, $c\\colon \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R}^{\\geq0}$, on the underlying space $\\mathcal{X}$. The $p$-Wasserstein metric corresponds to $c = L_p$. We are particularly interested in the Wasserstein metrics due to the predominant use of $L_p$ spaces in mean-value reinforcement learning.\n\n\\subsection{Quantile Regression for Distributional RL}\n\\citet{c51} showed that the distributional Bellman operator is a contraction in the $p$-Wasserstein metric, but as the proposed algorithm did not itself minimize the Wasserstein metric, this left a theory-practice gap for distributional RL. Recently, this gap was closed, in both directions. First and most relevant to this work, \\citet{dabney2017qr} proposed the use of \\textit{quantile regression} for distributional RL and showed that by choosing the quantile targets suitably the resulting projected distributional Bellman operator is a contraction in the $\\infty$-Wasserstein metric. Concurrently, \\citet{rowland2018analysis} showed the original class of categorical algorithms are a contraction in the Cram\\'er distance, the $L_2$ metric on cumulative distribution functions.\n\nBy estimating the quantile function at precisely chosen points, QR-DQN minimizes the Wasserstein distance to the distributional Bellman target \\cite{dabney2017qr}. This estimation uses \\textit{quantile regression}, which has been shown to converge to the true quantile function value when minimized using stochastic approximation \\cite{qrbook}. \n\nIn QR-DQN, the random return is approximated by a uniform mixture of $N$ Diracs,\n\\begin{equation*}\n    Z_\\theta(x,a) := \\tfrac{1}{N} \\sum_{i=1}^N \\delta_{\\theta_i(x,a)},\n\\end{equation*}\nwith each $\\theta_i$ assigned a fixed quantile target, $\\hat{\\tau}_i = \\frac{\\tau_{i-1} + \\tau_{i}}{2}$ for $1 \\le i \\le N$, where $\\tau_i = i\/N$. These quantile estimates are trained using the \\citet{huber1964robust} quantile regression loss, with threshold $\\kappa$,\n\\begin{align*}\n    \\rho^\\kappa_\\tau(\\delta_{ij}) &= |\\tau - \\mathbb{I}{\\{ \\delta_{ij} < 0 \\}}| \\frac{\\mathcal{L}_\\kappa(\\delta_{ij})}{\\kappa},\\ \\quad \\text{with}\\\\\n    \\mathcal{L}_\\kappa(\\delta_{ij}) &= \\begin{cases}\n        \\frac{1}{2} \\delta_{ij}^2,\\quad \\ &\\text{if } |\\delta_{ij}| \\le \\kappa\\\\\n        \\kappa (|\\delta_{ij}| - \\frac{1}{2}\\kappa),\\quad \\ &\\text{otherwise}\n    \\end{cases},\n   \n\\end{align*}\non the pairwise TD-errors\n$$\\delta_{ij} = r + \\gamma \\theta_j(x', \\pi(x')) - \\theta_i(x, a).$$\n\n\nAt the time of this writing, QR-DQN achieves the best performance on Atari-57, human-normalized mean and median, of all agents that do not combine distributional RL, prioritized replay, and $n$-step updates \\cite{dabney2017qr,hessel2018rainbow,gruslys2018reactor}.\n\n\\subsection{Risk in Reinforcement Learning}\n\nDistributional RL algorithms have been theoretically justified for the Wasserstein and Cram\\'er metrics \\cite{c51,rowland2018analysis}, and learning the distribution over returns, in and of itself, empirically results in significant improvements to data efficiency, final performance, and stability \\cite{c51,dabney2017qr,gruslys2018reactor,barthmaron2018d4pg}. However, in each of these recent works the policy used was based entirely on the mean of the return distribution, just as in standard reinforcement learning. A natural question arises: can we expand the class of policies using information provided by the distribution over returns (i.e.~to the class of risk-sensitive policies)? Furthermore, when would this larger policy class be beneficial?\n\nHere, `risk' refers to the uncertainty over possible outcomes, and \\emph{risk-sensitive} policies are those which depend upon more than the mean of the outcomes. At this point, it is important to highlight the difference between \\emph{intrinsic uncertainty}, captured by the distribution over returns, and \\emph{parametric uncertainty}, the uncertainty over the value estimate typically associated with Bayesian approaches such as PSRL \\cite{osband2013more} and Kalman TD \\cite{geist2010kalman}. Distributional RL seeks to capture the former, which classic approaches to risk are built upon\\footnote{One exception is the recent work \\cite{moerland2017efficient} towards combining both forms of uncertainty to improve exploration.}.\n\nExpected utility theory states that if a decision policy is consistent with a particular set of four axioms regarding its choices then the decision policy behaves as though it is maximizing the expected value of some utility function $U$ \\cite{von1947theory},\n$$\\pi(x) = \\argmax_a \\expect_{Z(x, a)} [U(z)].$$\nThis is perhaps the most pervasive notion of risk-sensitivity. A policy maximizing a linear utility function is called \\emph{risk-neutral}, whereas concave or convex utility functions give rise to \\emph{risk-averse} or \\emph{risk-seeking} policies, respectively. Many previous studies on risk-sensitive RL adopt the utility function approach \\cite{howard1972risk,marcus1997risk,maddison2017particle}.\n\n\nA crucial axiom of expected utility is \\textit{independence}: given random variables $X$, $Y$ and $Z$, such that $X \\succ Y$ ($X$ preferred over $Y$), any mixture between $X$ and $Z$ is preferred to the same mixture between $Y$ and $Z$ \\cite{von1947theory}. Stated in terms of the cumulative probability functions, $\\alpha F_X + (1 - \\alpha) F_Z \\ge \\alpha F_Y + (1 - \\alpha) F_Z,\\ \\forall \\alpha \\in [0, 1]$. This axiom in particular has troubled many researchers because it is consistently violated by human behavior \\cite{tversky1992advances}. The Allais paradox is a frequently used example of a decision problem where people violate the independence axiom of expected utility theory \\cite{allais1990allais}.\n\nHowever, as \\citet{yaari1987dual} showed, this axiom can be replaced by one in terms of convex combinations of outcome values, instead of mixtures of distributions. Specifically, if as before $X \\succ Y$, then for any $\\alpha \\in [0, 1]$ and random variable $Z$, $\\alpha F_X^{-1} + (1 - \\alpha) F_Z^{-1} \\ge \\alpha F_Y^{-1} + (1 - \\alpha)F_Z^{-1}$. This leads to an alternate, dual, theory of choice than that of expected utility. Under these axioms the decision policy behaves as though it is maximizing a distorted expectation, for some continuous monotonic function $h$:\n$$\\pi(x) = \\argmax_a \\int_{-\\infty}^{\\infty} z \\frac{\\partial}{\\partial z} (h \\circ F_{Z(x, a)})(z) \\,dz.$$\n\nSuch a function $h$ is known as a \\textit{distortion risk measure}, as it distorts the cumulative probabilities of the random variable \\cite{wang1996premium}. That is,\nwe have two fundamentally equivalent approaches to risk-sensitivity.\nEither, we choose a utility function and follow the expectation of this utility. Or, we choose a reweighting of the distribution and compute expectation under this distortion measure. Indeed, \\citet{yaari1987dual} further showed that these two functions are inverses of each other. The choice between them amounts to a choice over whether the behavior should be invariant to mixing with random events or to convex combinations of outcomes.\n\nDistortion risk measures include, as special cases, cumulative probability weighting used in cumulative prospect theory \\cite{tversky1992advances}, conditional value at risk \\cite{chow2014algorithms}, and many other methods \\cite{morimura2010nonparametric}. Recently \\citet{majumdar2017should} argued for the use of distortion risk measures in robotics.\n\n\n\\section{Implicit Quantile Networks}\n\\label{sec:analysis}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.48\\textwidth]{figures\/network_arch.pdf}\n\\end{center}\n\\caption{Network architectures for DQN and recent distributional RL algorithms.}\\label{fig:network_arch}\n\\end{figure}\n\n\nWe now introduce the \\textit{implicit quantile network} (IQN), a deterministic parametric function trained to reparameterize samples from a base distribution, e.g.~$\\tau \\sim U([0,1])$, to the respective quantile values of a target distribution.\nIQN provides an effective way to learn an implicit representation of the return distribution, yielding a powerful function approximator for a new DQN-like agent.\n\nLet $F^{-1}_Z(\\tau)$ be the quantile function at $\\tau \\in [0, 1]$ for the random variable $Z$. For notational simplicity we write $Z_\\tau := F^{-1}_Z(\\tau)$, thus for $\\tau \\sim U([0,1])$ the resulting state-action return distribution sample is $Z_\\tau(x, a) \\sim Z(x, a)$.\n\nWe propose to model the state-action quantile function as a mapping from state-actions and samples from some base distribution, typically $\\tau \\sim U([0,1])$, to $Z_\\tau(x, a)$, viewed as samples from the implicitly defined return distribution.\n\nLet $\\beta\\colon [0, 1] \\to [0, 1]$ be a distortion risk measure, with identity corresponding to risk-neutrality. Then, the \\textit{distorted expectation} of $Z(x, a)$ under $\\beta$ is given by\n\\begin{equation*}\n    Q_\\beta(x, a) := \\expect_{\\tau \\sim U([0,1])} \\left[ Z_{\\beta(\\tau)}(x, a) \\right].\n\\end{equation*}\nNotice that the distorted expectation is equal to the expected value of $F^{-1}_{Z(x,a)}$ weighted by $\\beta$, that is, $Q_\\beta = \\int_0^1 F^{-1}_Z(\\tau) d\\beta(\\tau)$. The immediate implication of this is that for any $\\beta$, there exists a sampling distribution for $\\tau$ such that the mean of $Z_\\tau$ is equal to the distorted expectation of $Z$ under $\\beta$, that is, any distorted expectation can be represented as a weighted sum over the quantiles \\cite{dhaene2012remarks}. Denote by $\\pi_\\beta$ the risk-sensitive greedy policy\n\\begin{equation}\\label{eqn:rs_policy}\n    \\pi_\\beta(x) = \\argmax_{a \\in \\mathcal{A}} Q_\\beta(x, a).\n\\end{equation}\n\nFor two samples $\\tau, \\tau' \\sim U([0,1])$, and policy $\\pi_\\beta$, the sampled temporal difference (TD) error at step $t$ is\n\\begin{equation}\\label{eqn:sampledTD}\n    \\delta^{\\tau,\\tau'}_t = r_t + \\gamma Z_{\\tau'}(x_{t+1}, \\pi_\\beta(x_{t+1})) - Z_{\\tau}(x_t, a_t).\n\\end{equation}\nThen, the IQN loss function is given by\n\\begin{equation}\\label{eqn:iqn_loss}\n    \\mathcal{L}(x_t, a_t, r_t, x_{t+1}) = \\frac{1}{N'} \\sum_{i=1}^{N} \\sum_{j=1}^{N'} \\rho_{\\tau_i}^\\kappa \\left( \\delta_t^{\\tau_i, \\tau_j'} \\right),\n   \n\\end{equation}\nwhere $N$ and $N'$ denote the respective number of iid samples $\\tau_i, \\tau_j' \\sim U([0,1])$ used to estimate the loss.\nA corresponding sample-based risk-sensitive policy is obtained by approximating $Q_\\beta$ in Equation~\\ref{eqn:rs_policy} by $K$ samples of $\\tilde \\tau \\sim U([0,1])$:\n\\begin{equation*}\n    \\tilde \\pi_\\beta(x) = \\argmax_{a \\in \\mathcal{A}} \\frac{1}{K}\\sum_{k=1}^K  Z_{\\beta(\\tilde\\tau_k)}(x, a).\n\\end{equation*}\n\nImplicit quantile networks differ from the approach of \\citet{dabney2017qr} in two ways. First, instead of approximating the quantile function at $n$ fixed values of $\\tau$ we approximate it with $Z_\\tau(x, a) \\approx f( \\psi(x), \\phi(\\tau))_a$ for some differentiable functions $f$, $\\psi$, and $\\phi$. If we ignore the distributional interpretation for a moment and view each $Z_\\tau(x, a)$ as a separate action-value function, this highlights that implicit quantile networks are a type of \\textit{universal value function approximator} (UVFA) \\cite{schaul2015universal}. There may be additional benefits to implicit quantile networks beyond the obvious increase in representational fidelity. As with UVFAs, we might hope that training over many different $\\tau$'s (goals in the case of the UVFA) leads to better generalization between values and improved sample complexity than attempting to train each separately.\n\nSecond, $\\tau$, $\\tau'$, and $\\tilde \\tau$ are sampled from continuous, independent, distributions. Besides $U([0,1])$, we also explore risk-sentive policies $\\pi_\\beta$, with non-linear $\\beta$. The independent sampling of each $\\tau$, $\\tau'$ results in the sample TD errors being decorrelated, and the estimated action-values go from being the true mean of a mixture of $n$ Diracs to a sample mean of the implicit distribution defined by reparameterizing the sampling distribution via the learned quantile function.\n\n\n\\subsection{Implementation}\n\\label{sec:algorithm}\nConsider the neural network structure used by the DQN agent \\cite{mnih15nature}. Let $\\psi\\colon \\mathcal{X} \\to \\mathbb{R}^d$ be the function computed by the convolutional layers and $f\\colon \\mathbb{R}^d \\to \\mathbb{R}^{|\\mathcal{A}|}$ the subsequent fully-connected layers mapping $\\psi(x)$ to the estimated action-values, such that $Q(x, a) \\approx f(\\psi(x))_a$. For our network we use the same functions $\\psi$ and $f$ as in DQN, but include an additional function $\\phi\\colon [0, 1] \\to \\mathbb{R}^d$ computing an embedding for the sample point $\\tau$. We combine these to form the approximation $Z_\\tau(x, a) \\approx f(\\psi(x) \\odot \\phi(\\tau))_a$, where $\\odot$ denotes the element-wise (Hadamard) product.\n\nAs the network for $f$ is not particularly deep, we use the multiplicative form, $\\psi \\odot \\phi$, to force interaction between the convolutional features and the sample embedding. Alternative functional forms, e.g.~concatenation or a `residual' function $\\psi \\odot (1 + \\phi)$, are conceivable, and $\\phi(\\tau)$ can be parameterized in different ways. To investigate these, we compared performance across a number of architectural variants on six Atari 2600 games (\\textsc{Asterix, Assault, Breakout, Ms.Pacman, QBert, Space Invaders}).\nFull results are given in the Appendix. Despite minor variation in performance, we found the general approach to be robust to the various choices. Based upon the results we used the following function in our later experiments, for embedding dimension $n = 64$:\n\\begin{equation}\\label{eqn:iqn_architecture}\n    \\phi_j(\\tau) := \\operatorname{ReLU}(\\sum_{i=0}^{n-1} \\cos(\\pi i \\tau)w_{ij} + b_j).\n\\end{equation}\n\nAfter settling on a network architecture, we study the effect of \nthe number of samples, $N$ and $N'$, used in the estimate terms of Equation~\\ref{eqn:iqn_loss}.\n\nWe hypothesized that $N$, the number of samples of $\\tau \\sim U([0,1])$, would affect the sample complexity of IQN, with larger values leading to faster learning, and that with $N = 1$ one would potentially approach the performance of DQN. This would support the hypothesis that the improved performance of many distributional RL algorithms rests on their effect as auxiliary loss functions, which would vanish in the case of $N = 1$.\nFurthermore, we believed that $N'$, the number of samples of $\\tau' \\sim U([0,1])$, would affect the variance of the gradient estimates much like a mini-batch size hyperparameter. Our prediction was that $N'$ would have the greatest effect on variance of the long-term performance of the agent.\n\nWe used the same set of six games as before, with our chosen architecture, and varied $N, N' \\in \\{1, 8, 32, 64\\}$. In Figure~\\ref{fig:num_atoms} we report the average human-normalized scores on the six games for each configuration. Figure~\\ref{fig:num_atoms} (left) shows the average performance over the first ten million frames, while (right) shows the average performance over the last ten million (from 190M to 200M). \n\nAs expected, we found that $N$ has a dramatic effect on early performance, shown by the continual improvement in score as the value increases.\nAdditionally, we observed that $N'$ affected performance very differently than expected: it had a strong effect on early performance, but minimal impact on long-term performance past $N' = 8$. \n\nOverall, while using more samples for both distributions is generally favorable, $N = N' = 8$ appears to be sufficient to achieve the majority of improvements offered by IQN for long-term performance, with variation past this point largely insignificant. To our surprise we found that even for $N = N' = 1$, which is comparable to DQN in the number of loss components, the longer term performance is still quite strong ($\\approx3\\times$ DQN).\n\nIn an informal evaluation, we did not find IQN to be sensitive to $K$, the number of samples used for the policy, and have fixed it at $K = 32$ for all experiments.\n\n\n\\section{Risk-Sensitive Reinforcement Learning}\n\\label{sec:risky_rl}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{figures\/n_atoms.pdf}\n\\end{center}\n\\caption{Effect of varying $N$ and $N'$, the number of samples used in the loss function in Equation~\\ref{eqn:iqn_loss}. Figures show human-normalized agent performance, averaged over six Atari games, averaged over first 10M frames of training (left) and last 10M frames of training (right). Corresponding values for baselines: DQN ($32, 253$) and QR-DQN ($144, 1243$).}\\label{fig:num_atoms}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figures\/atari_risky_iqn.pdf}\n\\end{center}\n\\caption{Effects of various changes to the sampling distribution, that is various cumulative probability weightings.}\\label{fig:risk_atari}\n\\end{figure*}\n\nIn this section, we explore the effects of varying the distortion risk measure, $\\beta$, away from identity. This only affects the policy, $\\pi_\\beta$, used both in Equation~\\ref{eqn:sampledTD} and for acting in the environment. As we have argued, evaluating under different distortion risk measures is equivalent to changing the sampling distribution for $\\tau$, allowing us to achieve various forms of risk-sensitive policies. We focus on a handful of sampling distributions and their corresponding distortion measures. The first one is the cumulative probability weighting parameterization proposed in cumulative prospect theory \\cite{tversky1992advances,gonzalez1999shape}:\n\\begin{equation*}\n    \\operatorname{CPW}(\\eta, \\tau) = \\frac{\\tau^{\\eta}}{(\\tau^{\\eta} + (1 - \\tau)^\\eta)^{\\frac{1}{\\eta}}}.\n\\end{equation*}\nIn particular, we use the parameter value $\\eta = 0.71$ found by \\citet{wu1996curvature} to most closely match human subjects. This choice is interesting as, unlike the others we consider, it is neither globally convex nor concave. For small values of $\\tau$ it is locally concave and for larger values of $\\tau$ it becomes locally convex. Recall that concavity corresponds to risk-averse and convexity to risk-seeking policies.\n\nSecond, we consider the distortion risk measure proposed by \\citet{wang2000class}, where $\\Phi$ and $\\Phi^{-1}$ are taken to be the standard Normal cumulative distribution function and its inverse:\n\\begin{equation*}\n    \\operatorname{Wang}(\\eta, \\tau) = \\Phi(\\Phi^{-1}(\\tau) + \\eta).\n\\end{equation*}\nFor $\\eta < 0$, this produces risk-averse policies and we include it due to its simple interpretation and ability to switch between risk-averse and risk-seeking distortions.\n\nThird, we consider a simple power formula for risk-averse ($\\eta < 0$) or risk-seeking ($\\eta > 0$) policies:\n\\begin{equation*}\n    \\operatorname{Pow}(\\eta, \\tau) =  \\begin{cases}\n        \\tau^{\\frac{1}{1 + |\\eta|}},\\quad \\ &\\text{if } \\eta \\ge 0\\\\\n        1 - (1 - \\tau)^{\\frac{1}{1 + |\\eta|}},\\quad \\ &\\text{otherwise}\n    \\end{cases}.\n\\end{equation*}\n\nFinally, we consider conditional value-at-risk (CVaR):\n\\begin{equation*}\n    \\operatorname{CVaR}(\\eta, \\tau) = \\eta \\tau.\n\\end{equation*}\nCVaR has been widely studied in and out of reinforcement learning \\cite{chow2014algorithms}. Its implementation as a modification to the sampling distribution of $\\tau$ is particularly simple, as it changes $\\tau \\sim U([0,1])$ to $\\tau \\sim U([0,\\eta])$. Another interesting sampling distribution, not included in our experiments, is denoted $\\operatorname{Norm}(\\eta)$ and corresponds to $\\tau$ sampled by averaging $\\eta$ samples from $U([0,1])$.\n\nIn Figure~\\ref{fig:risk_atari} (right) we give an example of a distribution (Neutral) and how each of these distortion measures affects the implied distribution due to changing the sampling distribution of $\\tau$. $\\operatorname{Norm}(3)$ and $\\operatorname{CPW}(.71)$ reduce the impact of the tails of the distribution, while $\\operatorname{Wang}$ and $\\operatorname{CVaR}$ heavily shift the distribution mass towards the tails, creating a risk-averse or risk-seeking preference. Additionally, while CVaR entirely ignores all values corresponding to $\\tau > \\eta$, $\\operatorname{Wang}$ gives these non-zero, but vanishingly small, probability.\n\nBy using these sampling distributions we can induce various risk-sensitive policies in IQN. We evaluate these on the same set of six Atari 2600 games previously used. Our algorithm simply changes the policy to maximize the distorted expectations instead of the usual sample mean. Figure~\\ref{fig:risk_atari} (left) shows our results in this experiment, with average scores reported under the usual, risk-neutral, evaluation criterion.\n\nIntuitively, we expected to see a qualitative effect from risk-sensitive training, e.g.~strengthened exploration from a risk-seeking objective. Although we did see qualitative differences, these did not always match our expectations.\nFor two of the games, \\textsc{Asterix} and \\textsc{Assault}, there is a very significant advantage to the risk-averse policies. Although $\\operatorname{CPW}$ tends to perform almost identically to the standard risk-neutral policy, and the risk-seeking $\\operatorname{Wang}(1.5)$ performs as well or worse than risk-neutral, we find that both risk-averse policies improve performance over standard IQN. However, we also observe that the more risk-averse of the two, $\\operatorname{CVaR}(0.1)$, suffers some loss in performance on two other games (\\textsc{QBert} and \\textsc{Space Invaders}). \n\nAdditionally, we note that the risk-seeking policy significantly underperforms the risk-neutral policy on three of the six games. It remains an open question as to exactly why we see improved performance for risk-averse policies. There are many possible explanations for this phenomenon, e.g.~that risk-aversion encodes a heuristic to stay alive longer, which in many games is correlated with increased rewards.\n\n\\section{Full Atari-57 Results}\n\\label{sec:atari57}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figures\/atari57_full.pdf}\n\\end{center}\n\\caption{Human-normalized mean (left) and median (right) scores on Atari-57 for IQN and various other algorithms. Random seeds shown as traces, with IQN averaged over 5, QR-DQN over 3, and Rainbow over 2 random seeds.}\\label{fig:atari57}\n\\end{figure*}\n\nFinally, we evaluate IQN on the full Atari-57 benchmark, comparing with the state-of-the-art performance of Rainbow, a distributional RL agent that combines several advances in deep RL \\cite{hessel2018rainbow}, the closely related algorithm QR-DQN \\cite{dabney2017qr}, prioritized experience replay DQN \\cite{schaul16prioritized}, and the original DQN agent \\cite{mnih15nature}. Note that in this section we use the risk-neutral variant of the IQN, that is, the policy of the IQN agent is the regular $\\epsilon$-greedy policy with respect to the mean of the state-action return distribution.\n\nIt is important to remember that Rainbow builds upon the distributional RL algorithm C51 \\cite{c51}, but also includes prioritized experience replay \\cite{schaul16prioritized}, Double DQN \\cite{vanhasselt16deep}, Dueling Network architecture \\cite{wang2016dueling}, Noisy Networks \\cite{fortunato2017noisy}, and multi-step updates \\cite{sutton1988learning}. In particular, besides the distributional update, $n$-step updates and prioritized experience replay were found to have significant impact on the performance of Rainbow. Our other competitive baseline is QR-DQN, which is currently state-of-the-art for agents that do not combine distributional updates, $n$-step updates, and prioritized replay.\n\nThus, between QR-DQN and the much more complex Rainbow we compare to the two most closely related, and best performing, agents in published work. In particular, we would expect that IQN would benefit from the additional enhancements in Rainbow, just as Rainbow improved significantly over C51.\n\nFigure~\\ref{fig:atari57} shows the mean (left) and median (right) human-normalized scores during training over the Atari-57 benchmark. IQN dramatically improves over QR-DQN, which itself improves on many previously published results. At 100 million frames IQN has reached the same level of performance as QR-DQN at 200 million frames. Table~\\ref{fig:perc_scores} gives a comparison between the same methods in terms of their best, human-normalized, scores per game under the 30 random no-op start condition. These are averages over the given number of seeds. Additionally, using human-starts, IQN achieves $162\\%$ median human-normalized score, whereas Rainbow reaches $153\\%$ \\cite{hessel2018rainbow}, see Table~\\ref{fig:perc_scores_human}.\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{ l | r | r | r | c }\n\\multicolumn{1}{c}{} & \\mbox{\\textbf{Mean}} & \\mbox{\\textbf{Median}} & \\mbox{\\textbf{Human Gap}} & \\mbox{\\textbf{Seeds}} \\\\\n\\hline\n\\textsc{DQN}  &   228\\% & 79\\% & 0.334 & 1 \\\\\n\\textsc{Prior.}   &   434\\% & 124\\% & 0.178  & 1 \\\\\n\\textsc{C51}   &   701\\% & 178\\% & 0.152  & 1 \\\\\n\\textsc{Rainbow}   &   \\textbf{\\textcolor{blue}{1189\\%}} & \\textbf{\\textcolor{blue}{230\\%}} & 0.144  & 2 \\\\\n\\textsc{QR-DQN}   &   864\\% & 193\\% & 0.165  & 3  \\\\\n\\hline\n\\textsc{IQN}   &   1019\\% & 218\\% & \\textbf{\\textcolor{blue}{0.141}} & 5  \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Mean and median of scores across 57 Atari 2600 games, measured as percentages of human baseline \\cite{nair15massively}. Scores are averages over number of seeds.}\n\\label{fig:perc_scores}\n\\end{table}\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\multicolumn{6}{c}{\\textbf{Human-starts (median)}}  \\\\\n\\hline\n\\textsc{DQN}  & \\textsc{Prior.} & \\textsc{A3C}   & \\textsc{C51}   & \\textsc{Rainbow} & \\textsc{IQN}   \\\\\n\\hline\n68\\% & 128\\%  & 116\\% & 125\\% & 153\\%   & \\textbf{\\textcolor{blue}{162\\%}}\n\\end{tabular}\n\\end{center}\n\\caption{Median human-normalized scores for human-starts.}\n\\label{fig:perc_scores_human}\n\\end{table}\n\nFinally, we took a closer look at the games in which each algorithm continues to underperform humans, and computed, on average, how far below human-level they perform\\footnote{Details of how this is computed can be found in the Appendix.}. We refer to this value as the \\textit{human-gap}\\footnote{Thanks to Joseph Modayil for proposing this metric.} metric and give results in Table~\\ref{fig:perc_scores}. Interestingly, C51 outperforms QR-DQN in this metric, and IQN outperforms all others. This shows that the remaining gap between Rainbow and IQN is entirely from games on which both algorithms are already super-human. The games where the most progress in RL is needed happen to be the games where IQN shows the greatest improvement over QR-DQN and Rainbow.\n\n\\section{Discussion and Conclusions}\n\\label{sec:discussion}\n\nWe have proposed a generalization of recent work based around using quantile regression to learn the distribution over returns of the current policy. Our generalization leads to a simple change to the DQN agent to enable distributional RL, the natural integration of risk-sensitive policies, and significantly improved performance over existing methods. The IQN algorithm provides, for the first time, a fully integrated distributional RL agent without prior assumptions on the parameterization of the return distribution.\n\nIQN can be trained with as little as a single sample from each state-action value distribution, or as many as computational limits allow to improve the algorithm's data efficiency. Furthermore, IQN allows us to expand the class of control policies to a large class of risk-sensitive policies connected to distortion risk measures. Finally, we show substantial gains on the Atari-57 benchmark over QR-DQN, and even halving the distance between QR-DQN and Rainbow.\n\nDespite the significant empirical successes in this paper there are many areas in need of additional theoretical analysis. We highlight a few particularly relevant open questions we were unable to address in the present work. First, sample-based convergence results have been recently shown for a class of categorical distributional RL algorithms \\cite{rowland2018analysis}. Could existing sample-based RL convergence results be extended to the QR-based algorithms?\n\nSecond, can the contraction mapping results for a fixed grid of quantiles given by \\citet{dabney2017qr} be extended to the more general class of approximate quantile functions studied in this work? Finally, and particularly salient to our experiments with distortion risk measures, theoretical guarantees for risk-sensitive RL have been building over recent years, but have been largely limited to special cases and restricted classes of risk-sensitive policies. Can the convergence of the distribution of returns under the Bellman operator be leveraged to show convergence to a fixed-point in distorted expectations? In particular, can the control results of \\citet{c51} be expanded to cover some class of risk-sensitive policies?\n\nThere remain many intriguing directions for future research into distributional RL, even on purely empirical fronts. \\citet{hessel2018rainbow} recently showed that distributional RL agents can be significantly improved, when combined with other techniques. Creating a Rainbow-IQN agent could yield even greater improvements on Atari-57. We also recall the surprisingly rich return distributions found by \\citet{barthmaron2018d4pg}, and hypothesize that the continuous control setting may be a particularly fruitful area for the application of distributional RL in general, and IQN in particular.\n\n\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Main divisibility theorem}\\label{intro}\nThe divisibility by primes of the coefficients in the integer powers $\\ell(x)^t$ of the power series for $\\log(1+x)\/x$,\ngiven by\n$$\\ell(x):=\\sum_{i=0}^\\infty(-1)^i\\frac{x^i}{i+1},$$\nhas been applied in several ways in algebraic topology.  See, for example, \\cite{AT} and \\cite{SS}.\nOur main divisibility result, \\ref{mainthm}, says that, in an appropriate range, this divisibility is the same as that\nof the coefficients of $(1\\pm \\frac{x^{p-1}}p)^t$. Here $p$ is any prime and $t$ is any integer. We denote by $\\nu_p(-)$ the exponent\nof $p$ in an integer, and by $[x^n]f(x)$ the coefficient of $x^n$ in a power series $f(x)$.\n\\begin{thm} \\label{mainthm} If $t$ is any integer and $m\\le p^{\\nu_p(t)}$, then $$\\nu_p\\left([x^{(p-1)m}]\\ell(x)^t\\right)=\\nu_p(t)-\\nu_p(m)-m.$$\n\\end{thm}\nThus, for example, if $\\nu_3(t)=2$, then, for $m=1,\\ldots,9$, the exponent of 3 in $[x^{2m}]\\ell(x)^t$ is, respectively,\n$1$, $0$, $-2$, $-2$, $-3$, $-5$, $-5$, $-6$, and $-9$, which is the same as in $(1\\pm \\frac{x^2}3)^t$.\nIn Section \\ref{sec2}, we will discuss what we can say about $\\nu_p([x^n]\\ell(x)^t)$ when $n$ is not divisible by $(p-1)$ and $n<(p-1)p^{\\nu_p(t)}$ .\n\nThe motivation for Theorem \\ref{mainthm} was provided by ongoing thesis work of Karen McCready at Lehigh University, which\nseeks to apply the result when $p=2$ to make more explicit some nonimmersion results for complex projective\nspaces described in \\cite{SS}. Proving Theorem \\ref{mainthm} led the author to discover an interesting modification\nof multinomial coefficients.\n\n\\begin{defin} For an ordered $r$-tuple of nonnegative integers $(i_1,\\ldots,i_r)$, we define\n$$c(i_1,\\ldots,i_r):=\\frac{(\\sum i_jj)(\\sum i_j -1)!}{i_1!\\cdots i_r!}.$$\n\\end{defin}\nNote that $c(i_1,\\ldots,i_r)$ equals $(\\sum i_jj)\/\\sum i_j$ times a multinomial coefficient.\nSurprisingly, these numbers satisfy the same recursive formula as  multinomial coefficients.\n\\begin{defin} For positive integers $k\\le r$, let $E_k$ denote the\nordered $r$-tuple whose only nonzero entry is a 1 in position $k$.\\end{defin}\n\\begin{prop} \\label{recurs} If $I=(i_1,\\ldots,i_r)$ is an ordered $r$-tuple of nonnegative integers, then\n\\begin{equation}\\label{receq}c(I)=\\sum_{i_k>0}c(I-E_k).\\end{equation}\n\\end{prop}\nIf we think of a multinomial coefficient $\\binom{\\sum i_j}{i_1,\\cdots,i_r}:=(i_1+\\cdots+i_r)!\/((i_1)!\\cdots(i_r)!)$ as being determined\nby the unordered $r$-tuple $(i_1,\\ldots,i_r)$ of nonnegative integers, then it satisfies the recursive\nformula analogous to that of  (\\ref{receq}). For a multinomial coefficient, entries which are 0\ncan be omitted, but that is not the case for $c(i_1,\\ldots,i_r)$.\n\n\\begin{proof}[Proof of Proposition \\ref{recurs}] The right hand side of (\\ref{receq}) equals\n\\begin{eqnarray*}&&\\sum_k i_k\\frac{(\\sum i_j -2)!}{(i_1)!\\cdots(i_r)!}\\left(\\sum_j i_jj -k\\right)\\\\\n&=&\\frac{(\\sum i_j -2)!}{(i_1)!\\cdots(i_r)!}\\left(\\left(\\sum i_k\\right)\\left(\\sum i_jj\\right)-\\sum i_kk\\right)\\\\\n&=&\\frac{(\\sum i_j -2)!}{(i_1)!\\cdots(i_r)!}\\left(\\sum i_jj\\right)\\left(\\sum i_j-1\\right),\n\\end{eqnarray*}\nwhich equals the left hand side of (\\ref{receq}).\n\\end{proof}\n\\begin{cor} If $\\sum i_j>0$, then $c(i_1,\\ldots,i_r)$ is a positive integer.\\label{cor1}\\end{cor}\n\\begin{proof} Use (\\ref{receq}) recursively to express $c(i_1,\\ldots,i_r)$ as a sum of various $c(E_k)=k$.\n\\end{proof}\n\\begin{cor}\\label{corr2} For any ordered $r$-tuple $(i_1,\\ldots,i_r)$ of nonnegative integers and any prime $p$,\n\\begin{equation}\\nu_p\\left(\\sum i_j\\right)\\le\\nu_p\\left(\\sum i_jj\\right)+\\nu_p\\binom{\\sum i_j}{i_1,\\cdots,i_r}.\\label{cor2}\\end{equation}\n\\end{cor}\n\\begin{proof} Multiply numerator and denominator of the definition of $c(i_1,\\ldots,i_r)$ by $\\sum i_j$\nand apply Corollary \\ref{cor1}.\n\\end{proof}\n\nThe proof of Theorem \\ref{mainthm}  utilizes Corollary \\ref{corr2} and also the following lemma.\n\\begin{lem} \\label{lem} If $t$ is any integer and $\\sum i_j\\le p^{\\nu_p(t)}$, then\n\\begin{equation}\\label{nueq}\\nu_p\\binom{t}{t-\\sum i_j,i_1,\\ldots,i_r}=\\nu_p(t)+\\nu_p\\binom{\\sum i_j}{i_1,\\ldots,i_r}-\\nu_p\\left(\\sum i_j\\right).\\end{equation}\n\\end{lem}\n\\begin{proof} For any integer $t$, the multinomial coefficient on the left hand side of (\\ref{nueq}) equals\n$t(t-1)\\cdots(t+1-\\sum i_j)\/\\prod i_j!$, and so\nthe left hand side of (\\ref{nueq}) equals $\\nu_p(t(t-1)\\cdots(t+1-\\sum i_j))-\\sum\\nu_p(i_j!)$.\nSince $\\nu_p(t-s)=\\nu_p(s)$ provided $0<s<p^{\\nu_p(t)}$, this becomes $\\nu_p(t)+\\nu_p((\\sum i_j -1)!)-\\sum\\nu_p(i_j!)$,\nand this equals the right hand side of (\\ref{nueq}).\\end{proof}\n\n \n\n\\begin{proof}[Proof of Theorem \\ref{mainthm}] By the multinomial theorem,\n$$[x^{(p-1)m}]\\ell(x)^t=(-1)^{(p-1)m}\\sum_IT_I,$$ where \\begin{equation}T_I=\\binom{t}{t-\\sum i_j,i_1,\\ldots,i_r}\\frac1{\\prod(j+1)^{i_j}},\\label{Teq}\\end{equation}\nwith the sum taken over all $I=(i_1,\\ldots,i_r)$ satisfying $\\sum i_jj=(p-1)m$.\nUsing Lemma \\ref{lem}, we have\n$$\\nu_p(T_I)=\\nu_p(t)+\\nu_p\\binom{\\sum i_j}{i_1,\\ldots,i_r}-\\nu_p\\left(\\sum i_j\\right)-\\sum i_j\\nu_p(j+1).$$\nIf $I=mE_{p-1}$, then $\\nu_p(T_I)=\\nu_p(t)+0-\\nu_p(m)-m$. The theorem will follow once we show that\nall other $I$ with $\\sum i_jj=(p-1)m$ satisfy $\\nu_p(T_I)>\\nu_p(t)-\\nu_p(m)-m$. Such $I$ must have\n$i_j>0$ for some $j\\ne p-1$.  This is relevant because $\\frac1{p-1}j\\ge \\nu_p(j+1)$ with equality if and only if\n$j=p-1$. For  $I$ such as we are considering, we have\n\\begin{eqnarray}&&\\nu_p(T_I)-(\\nu_p(t)-\\nu_p(m)-m)\\label{eqs}\\\\\n&=&\\nu_p\\binom{\\sum i_j}{i_1,\\ldots,i_r}-\\nu_p(\\sum i_j)-\\sum i_j\\nu_p(j+1)+\\nu_p(\\sum i_jj)+\\tfrac1{p-1}\\sum i_jj\\nonumber\\\\\n&\\ge&\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))\\nonumber\\\\\n&>&0.\\nonumber\\end{eqnarray}\nWe have used (\\ref{cor2}) in the middle step.\n\\end{proof}\n\n\\section{Zero coefficients}\\label{zsec}\nWhile studying coefficients related to Theorem \\ref{mainthm}, we noticed the following result about occurrences\nof coefficients of powers of the reciprocal log series which equal 0.\n\\begin{thm}\\label{cor1p} If $m$ is odd and $m>1$, then $[x^m]\\bigl(\\frac x{\\log(1+x)}\\bigr)^m=0$, while if $m$ is even and $m>0$, then $[x^{m+1}]\\bigl(\\frac x{\\log(1+x)}\\bigr)^m=0$.\\end{thm}\n\nMoreover, this property  characterizes the reciprocal log series.\n\\begin{cor}\\label{cor2p} A power series $f(x)=1+\\sum\\limits_{i\\ge1}c_ix^i$ with $c_1\\ne0$ has  $[x^m](f(x)^m)=0$\nfor all odd $m>1$, and $[x^{m+1}](f(x)^m)=0$ for all even $m>0$ if and only if $f(x)=\\frac{2c_1x}{\\log(1+2c_1x)}$.\\end{cor}\n\\begin{proof} By Theorem \\ref{cor1p}, the reciprocal log series satisfies the stated property. Now assume that $f$ satisfies this property and let $n$ be a positive integer and $\\epsilon=0$ or 1. Since $$[x^{2n+1}]f(x)^{2n+\\epsilon}=(2n+\\epsilon)(2n+\\epsilon-1)c_1c_{2n}+(2n+\\epsilon)c_{2n+1}+P,$$\nwhere $P$ is a polynomial in $c_1,\\ldots,c_{2n-1}$, we see that\n$c_{2n}$ and $c_{2n+1}$ can be determined from the $c_i$ with $i<2n$.\n\\end{proof}\n\nOur proof of Theorem \\ref{cor1p} is an extension of arguments of \\cite{AT} and \\cite{Dlog}.\nIt benefited from ideas of Francis Clarke. It can be derived from results in \\cite[ch.6]{GKP}, but\nwe have not seen it explicitly stated anywhere.\n\n\\begin{proof}[Proof of Theorem \\ref{cor1p}]\n Let $m>1$ and\n$$\\left(\\frac x{\\log(1+x)}\\right)^m=\\sum_{i\\ge0}a_i x^i.$$\nLetting $x=e^y-1$, we obtain\n\\begin{equation}\\label{eq}\\left(\\frac{e^y-1}y\\right)^m=\\sum_{i\\ge0}a_i(e^y-1)^i.\\end{equation}\nLet $j$ be a positive integer, and multiply both sides of (\\ref{eq}) by $y^me^y\/(e^y-1)^{j+1}$, obtaining\n\\begin{eqnarray}\\label{eq1}(e^y-1)^{m-j-1}e^y&=&y^m\\sum_{i\\ge0}a_i(e^y-1)^{i-j-1}e^y\\\\\n&=&y^m\\left(a_j\\frac{e^y}{e^y-1}+\\sum_{i\\ne j}\\textstyle\\frac{a_i}{i-j}\\textstyle\\frac d{dy}(e^y-1)^{i-j}\\right).\\nonumber\\end{eqnarray}\n  Since the derivative of a Laurent series has no $y^{-1}$-term, we conclude\nthat the coefficient of $y^{m-1}$ on the RHS of (\\ref{eq1}) is $a_j[y^{-1}](1+\\frac 1y\\frac y{e^y-1})=a_j$.\n\nThe Bernoulli numbers $B_n$ are defined by $\\frac y{e^y-1}=\\sum \\frac{B_n}{n!}y^n$.\nSince $\\frac y{e^y-1}+\\frac12y$ is an even function of $y$, we have the well-known result that\n$B_n=0$ if $n$ is odd and $n>1$.\n\nLet\n$$j=\\begin{cases}m&m\\text{ odd}\\\\\nm+1&m\\text{ even.}\\end{cases}$$\nFor this $j$, the LHS of (\\ref{eq1}) equals\n$$\\begin{cases}1+\\sum\\frac{B_i}{i!}y^{i-1}&m\\text{ odd}\\\\\n-\\frac d{dy}(e^y-1)^{-1}=-\\sum\\frac{(i-1)B_i}{i!}y^{i-2}&m\\text{ even,}\\end{cases}$$\nand comparison of coefficient of $y^{m-1}$ in (\\ref{eq1}) implies\n$$\\begin{cases}a_m=\\frac{B_m}{m!}=0&m\\text{ odd}\\\\\na_{m+1}=-\\frac{mB_{m+1}}{(m+1)!}=0&m\\text{ even,}\\end{cases}$$\nyielding the theorem.\\end{proof}\n\n\\section{Other coefficients}\\label{sec2}\nIn this section, a sequel to Theorem \\ref{mainthm}, we describe what can be easily said about\n$\\nu_p([x^{(p-1)m+\\Delta}]\\ell(x)^t)$ when $0<\\Delta<p-1$ and $m< p^{\\nu_p(t)}$.\nThis is not relevant in the motivating case, $p=2$. Our first result says that these exponents are at least as large\nas those of\n$[x^{(p-1)m}]\\ell(x)^t$.\nHere $t$ continues to denote any integer, positive or negative.\n\\begin{prop} If $0<\\Delta<p-1$ and $m< p^{\\nu_p(t)}$, then\n$$\\nu_p\\left([x^{(p-1)m+\\Delta}]\\ell(x)^t\\right)\\ge\\nu_p(t)-\\nu_p(m)-m.$$\\label{delprop}\n\\end{prop}\n\\begin{proof} We consider terms $T_I$ as in (\\ref{Teq}) with $\\sum i_jj=(p-1)m+\\Delta$.\nSimilarly to (\\ref{eqs}), we obtain\n\\begin{eqnarray} &&\\nu_p(T_I)-(\\nu_p(t)-\\nu_p(m)-m)\\nonumber\\\\\n&=&\\nu_p\\binom{\\sum i_j}{i_1,\\ldots,i_r}-\\nu_p\\left(\\sum i_j\\right)-\\sum i_j\\nu_p(j+1)\\nonumber\\\\\n&&+\\nu_p(m)+m\\label{A}.\\end{eqnarray}\n\nFor $I=(i_1,\\ldots,i_r)$, let\n\\begin{eqnarray*}{\\widetilde\\nu}_p(I)&:=&\\nu_p\\binom{\\sum i_j}{i_1,\\ldots,i_r}-\\nu_p\\left(\\sum i_j\\right)\\\\\n&=&\\nu_p\\left(\\frac 1{i_j}\\binom{\\sum i_j-1}{i_1,\\ldots,i_j-1,\\ldots,i_r}\\right),\\end{eqnarray*}\nfor any  $j$. Thus\n\\begin{equation}\\label{vt}{\\widetilde\\nu}_p(I)\\ge-\\min\\limits_j\\nu_p(i_j).\\end{equation}\n\nIgnoring the term $\\nu_p(m)$, the expression (\\ref{A}) is\n\\begin{equation}\\label{C}\\ge{\\widetilde\\nu}_p(I)+\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))-\\textstyle\\frac{\\Delta}{p-1}.\\end{equation}\nNote that\n$$\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))-\\textstyle\\frac{\\Delta}{p-1}=m-\\sum i_j\\nu_p(j+1)$$\nis an integer and is greater than $-1$, and hence is $\\ge0$.\n\nBy (\\ref{vt}), if ${\\widetilde\\nu}_p(I)=-e$ with $e\\ge0$, then all $i_j$ are divisible by $p^e$. Thus $\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))$\nis positive and divisible by $p^e$. Hence it is $\\ge p^e$. Therefore, (\\ref{C}) is $\\ge-e+p^e-1\\ge0$.\nWe obtain the desired conclusion, that, for each $I$, (\\ref{C}), and hence (\\ref{A}), is $\\ge0$.\n\\end{proof}\n\n\n\nFinally, we address the question of when does equality occur in Proposition \\ref{delprop}.\nWe give a three-part result, but by the third it becomes clear that obtaining additional results is probably more\ntrouble than it is worth.\n\\begin{prop} In Proposition \\ref{delprop},\n\\begin{itemize}\n\\item[a.] the inequality is strict ($\\ne$) if $m\\equiv0\\ (p)$;\n\\item[b.] equality holds if $\\Delta=1$ and $m\\not\\equiv0,1 \\ (p)$;\n\\item[c.] if $\\Delta=2$ and $m\\not\\equiv0,2\\ (p)$, then equality holds\nif and only if $3m\\not\\equiv 5\\ (p)$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof} We begin as in the proof of \\ref{delprop}, and note that, using (\\ref{cor2}), (\\ref{A}) is\n\\begin{equation}\\ge\\nu_p(m)-\\textstyle\\frac{\\Delta}{p-1}+\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))-\\nu_p((p-1)m+\\Delta)\\label{B}.\n\\end{equation}\n\n(a) If $\\nu_p(m)>0$, then $\\nu_p((p-1)m+\\Delta)=0$ and so (\\ref{B}) is greater than 0.\n\nIn (b) and (c), we exclude consideration of the case where $m\\equiv\\Delta\\ (p)$ because then\n$\\nu_p((p-1)m+\\Delta)>0$ causes complications.\n\n(b) If $\\Delta=1$ and $m\\not\\equiv0,1\\ (p)$, then for $I=E_1+mE_{p-1}$, (\\ref{A}) equals\n$$\\nu_p(m+1)-\\nu_p(m+1)-m+\\nu_p(m)+m=0,$$\nwhile for other $I$, (\\ref{B}) is\n$$0-\\tfrac1{p-1}+\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))>0.$$\n\n\n(c) Assume $\\Delta=2$ and $m\\not\\equiv0,2\\ (p)$. Then\n\\begin{eqnarray}&&T_{2E_1+mE_{p-1}}+T_{E_2+mE_{p-1}}\\nonumber\\\\\n&=&\\frac{t(t-1)\\cdots(t-m-1)}{2!m!}\\frac1{4p^m}+\\frac{t(t-1)\\cdots(t-m)}{m!}\\frac1{3p^m}\\nonumber\\\\\n&=&(-1)^m\\textstyle\\frac t{p^m}\\bigl(\\tfrac18(-m-1+A)+\\tfrac13(1+B)\\bigr)\\nonumber\\\\\n&=&(-1)^m\\textstyle\\frac t{24p^m}(-3m+5+(3A+8B)).\\label{D}\\end{eqnarray}\nHere $A$ and $B$ are rational numbers which are divisible by $p$. This is true because $\\nu_p(t)>\\nu_p(i)$\nfor all $i\\le m$.  Since $p>3$, (\\ref{D}) has $p$-exponent $\\ge\\nu_p(t)-m$\nwith equality if and only if $3m-5\\not\\equiv0\\ (p)$. Using (\\ref{B}), the other terms $T_I$ satisfy\n$$\\nu_p(T_I)-(\\nu_p(t)-m)\\ge\\sum i_j(\\tfrac1{p-1}j-\\nu_p(j+1))-\\tfrac2{p-1}>0.$$\n\\end{proof}\n\n\\def\\rule{.6in}{.6pt}{\\rule{.6in}{.6pt}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Related Work}\n\n\\begin{figure*}[h]\n\\includegraphics[width=\\textwidth]{module.pdf}\n\\caption{Parameter-free Average Attention Module (PfAAM).}\n\\label{fig:pfaam}\n\\end{figure*}\n\n\nIn this section, we briefly review related model architectures and attention mechanisms.\n\n\\paragraph{Network Architecture.}\n\nWith the release of AlexNet \\cite{krizhevsky2012imagenet} in 2012 for ImageNet LSVRC-2012 competition \\cite{russakovsky2015imagenet}, outperforming all other submissions by a large margin, convolutional neural networks became state-of-the-art for computer vision tasks, having only recently been challenged by vision transformer models \\cite{dosovitskiy2020image}, which, however, require expensive pre-training on huge amounts of data. Several improvements to the neural network architecture have been proposed to improve their discriminative abilities. The application of deeper architectures \\cite{simonyan2014very, szegedy2015going, wang2017residual}, wider networks \\cite{zagoruyko2016wide}, increased connectivity \\cite{huang2017densely}, grouped convolutions \\cite{xie2017aggregated}, depthwise convolution \\cite{xie2017aggregated} or reduced computational requirements \\cite{howard2017mobilenets, sandler2018mobilenetv2} have lead to a plethora of potential architectures. Attempts have been made to automatically search for the best network architecture reducing the manual design of neural network architectures \\cite{tan2019efficientnet, real2017large}. However, building blocks that boost network performance at low computational costs and can be incorporated in any network architecture without the need for manual adjustment or hyperparameter search remain of high value and motivated the design of PfAAM. \\par\n\n\\paragraph{Attention Modules.} Human perception is highly selective and filters information based on its relevance for decision making. According to this, so called attention mechnisms have been proposed for computer vision models. See \\cite{guo2022attention} for a comprehensive overview of attention mechanisms. Generally, attention mechanisms can be split into methods related to channel attention \\cite{hu2018squeeze, wang2020eca} focusing on 'what is important' in the image and spatial attention \\cite{oktay2018attention} highlighting 'where is the important information' in the image or a combination of both \\cite{woo2018cbam,yang2021simam}. However, most of these attention modules are implemented by adding learnable model parameters \\cite{hu2018squeeze, wang2020eca, oktay2018attention, woo2018cbam} during training increasing computational cost and model size, relate only on spatial or channel attention, or depend on tunable hyperparameters \\cite{yang2021simam}. As an extension to existing modules, PfAAM captures channel and spatial attention without adding parameters or hyperparameters and is simple by design, promoting a self-reinforcing effect by averaging activations.\n\n\\section{Conclusion}\n\nIn this work, we present a novel attention mechanism PfAAM based on highlighting areas of high activation. When PfAAM is used in different network architectures for classification and semantic segmentation, the performance increases for all tested architectures, while the network size remained unchanged and the computational cost is low. Even though PfAAM does not add additional trainable parameters to the network and does not rely on other theoretical considerations, its positive effect is surprisingly robust, suggesting that it leads to a self-focusing effect on relevant features. In summary, PfAAM provides a simple novel building block that might be considered for future neural network design in computer vision tasks.\n\\section{Experiments}\n\nIn this section we tested the optimal PfAAM setup and network integration in an ablation study and performed experiments for classification and semantic segmentation with different network architectures.\n\n\n\\subsection{Ablation Study}\n\nTo maximize the effect of the PfAAM block we tested different implementation options. First, we tested averaging versus maximizing as channel and spatial pooling operations within PfAAM to analyze their effect on the overall performance. Max pooling enhance the effect of individual strong activation whereas averaging increases areas with overall strong activation. The performance of a baseline ResNet-164 \\cite{he2016deep} with PfAAM blocks added to each residual block of the network were compared based on classification error. Furthermore, we tested the influence of an additional Batch Normalization \\cite{ioffe2015batch} before each PfAAM. The accuracy of the network was tested using the CIFAR-10 dataset \\cite{krizhevsky2009learning}, that consists of 50k training and 10k test images with a size of 32 x 32 RGB pixels belonging to 10 different classes. See table \\ref{tab:ablation} for classification errors of the PfAAM implementations. In total, there is no large difference in the resulting classification error and all PfAAM implementations improve the performance compared to the baseline model, from which we concluded that each of the implementations performs reasonably well. Averaging without Batch Normalization showed the best performance overall, which is why we continued further experiments with this configuration, unless stated otherwise.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Comparison of different PfAAM implementations CIFAR-10 using averaging or maximizing and an additional Batch Normalization (BN). The lowest classification error is shown in bold.}\n\\begin{tabular}{ l | c }\n\\hline\n &   error (\\%) \\\\\n\\hline\nResNet-164 \\cite{he2016deep} & 5.46 \\\\\nResNet-164+PfAAM(max)  & 4.79  \\\\\nResNet-164+PfAAM(avg) & \\textbf{4.76} \\\\\nResNet-164+BN+PfAAM(max)  & 4.94  \\\\\nResNet-164+BN+PfAAM(avg)  & 4.86 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:ablation}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Experiments}\n\nTo analyze the effect of PfAAM on neural network performance we used baseline architectures for classification and semantic segmentation and compared the performance of the regular architecture to the same architecture but with additional PfAAM blocks incorporated.\n\n\\subsubsection{Image Classification}\n\nTo investigate the effect of PfAAM in a classification task, we conducted experiments using CIFAR-10 and CIFAR-100 as benchmark. Both data sets have the same size, but are divided into 10 and 100 classes, respectively. As model architectures Residual Networks \\cite{he2016deep} and Wide Residual Networks \\cite{zagoruyko2016wide} with varying depth and width were used to cover basic architectures from shallow to deep and thin to wide. Results in table \\ref{tab:exp} show an reduction of the classification error for all tested architectures with integrated PfAAM. For deeper architectures the effect of PfAAM is larger showing a reduction of the error rate of  over 12\\% for ResNet-110 and ResNet-164 on CIFAR-10, thus ResNet-110+PfAAM almost matches the performance of the regular ResNet-164 which has 40\\% more trainable parameters. For wider but shallower architectures with fewer residual blocks, the effect of PfAAM is smaller (1.4\\% reduction for WRN-16-8 on CIFAR-10), suggesting that the effect scales with the number of PfAAM units per network. Since PfAAM does not introduce additional learnable parameters, it generally increases the performance of the network in image classification by improving the utilization of the existing parameters.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Classification error (\\%) on CIFAR-10 and CIFAR-100. The lowest error per model architecture and data set is shown in bold.}\n\\begin{tabular}{ l | c | c | c }\n\n\\hline\n\\multicolumn{4}{c}{CIFAR-10}\\\\\n\\hline\n &  \\# params & original  & +PfAAM\\\\\n\\hline\nResNet-110 \\cite{he2016deep} & 1.2M & 6.37 & \\textbf{5.57} \\\\\nResNet-164 \\cite{he2016deep} & 1.7M & 5.46 &\\textbf{4.76} \\\\\nWRN-28-2 \\cite{zagoruyko2016wide} & 1.5M & 5.73 & \\textbf{5.29} \\\\\nWRN-16-8 \\cite{zagoruyko2016wide} & 11M & 4.27 & \\textbf{4.21} \\\\\n\\hline\n\\multicolumn{4}{c}{CIFAR-100}\\\\\n\\hline\n &  \\# params & original  & +PfAAM\\\\\n\\hline\nResNet-110 \\cite{he2016deep} & 1.2M & 26.88 & \\textbf{24.22} \\\\\nResNet-164 \\cite{he2016deep} & 1.7M & 24.33 & \\textbf{23.05} \\\\\nWRN-28-2 \\cite{zagoruyko2016wide} & 1.5M & 26.69 & \\textbf{25.38} \\\\\nWRN-16-8 \\cite{zagoruyko2016wide} & 11M & 20.43 & \\textbf{20.33} \\\\\n\\hline\n\\end{tabular}\n\\label{tab:exp}\n\\end{center}\n\\end{table}\n\n\n\n\\subsubsection{Semantic Segmentation}\n\nTo test PfAAM for semantic segmentation, we used the PASCAL VOC 2012 segmentation dataset \\cite{everingham2010pascal} consisting of 1464 training and 1449 validation images of 20 categories and an additional background class. Following previous work \\cite{zhao2017pyramid,chen2014semantic, long2015fully}, we used the extended dataset with annotations from \\cite{hariharan2011semantic} resulting in 10582 training images. We trained a U-Net \\cite{ronneberger2015u} and a Feature Pyramid Network (FPN) \\cite{lin2017feature} on the training images and compared the results to the same architectures with added PfAAM. Each model used a ResNet-50 \\cite{he2016deep} as encoder-backbone, which was pre-trained on the ImageNet dataset \\cite{russakovsky2015imagenet}. The results in table \\ref{tab:seg} show the mean intersection over union (mIoU) on the validation images. Both models show increased performance when trained with PfAAM increasing the mIoU by 7.7\\% for U-Net and 5.3\\% for FPN, respectively. The averaged validation mIoU for U-Net with and without PfAAM during training are depicted in figure \\ref{tab:seg} showing a clear improvement for the PfAAM-model. Similar to classification, the introduction of the PfAAM in the model architecture improves the performance, underlining its general applicability as neural network building block enhancing model performance. \n\n\\begin{table}\n\\begin{center}\n\\caption{Segmentation results (mIoU, \\%) on PASCAL VOC 2012 validation set. Best results per model architecture are shown in bold.}\n\\begin{tabular}{ l | c | c }\n\\hline\n &   original & +PfAAM\\\\\n\\hline\n U-Net  & 55.7 & \\textbf{60.3}\\\\\n FPN   & 56.5 & \\textbf{59.7}\\\\\n\\hline\n\\end{tabular}\n\\label{tab:seg}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{seg.pdf}\n\\caption{Validation mIoU during training for a regular U-Net and the same architecture extended by PfAAM, shown is $mean \\pm std$.}\n\\end{center}\n\\label{fig:seg}\n\\end{figure}\n\n\n\\subsubsection{Implementation Details}\nFor CIFAR training we followed the established standard training procedure used by the original publications \\cite{he2016deep,zagoruyko2016wide}. Each 32x32 image or its its horizontally mirrored version was padded by 4 pixels and randomly cropped back to 32x32 pixel. The neural networks were trained for 200 epochs by optimizing the cross-entropy loss using SGD (stochastic gradient decent) with a momentum of 0.9, a weight decay of 0.0005 a mini-batch size of 128 and an initial learning rate of 0.1. The learning rate was step-wise decreased after 60 epochs, 120 epochs and 160 by a factor of 0.2. \\par\nFor semantic segmentation using the PASCAL VOC dataset, the training images were randomly horizontally flipped and scaled by a factor of 0.5 to 2 for each axis, from which random 224x224 patches were cut and fed into the neural network. Optimization was performed using SGD with a momentum of 0.9 and a constant learning rate of 0.0001 for 200 epochs, optimizing the cross-entropy loss function excluding pixels labeled as \\textit{void}.\\par\nUnless stated otherwise, all results are reported as the median over 5 runs.\n\n\\section{Introduction}\n\nConvolutional neural networks have demonstrated an impressive ability to solve a broad range of computer vision tasks \\cite{krizhevsky2012imagenet, long2015fully, redmon2016you, he2017mask}. Typically, a convolutional neural network is built modular and the local receptive field is increasing step-wise with network depth. According to that architecture, the network captures hierarchical patterns based on input image representations within the network. Increasing the representational power of neural networks is of ongoing research interest to emphasize the most important features for a given task. Previous work has shown improvements based on adaptations regarding the inner connectivity \\cite{huang2017densely, chollet2017xception} or in utilizing an attention mechanism that globally highlight relevant features \\cite{hu2018squeeze, woo2018cbam, wang2020eca}. \\par\nHowever, existing attention mechanisms rely on trainable parameters, only regard spatial or channel-wise attention, or introduce additional tunable hyperparameters. Here, we introduce Parameter-free Average Attention Module (PfAAM) which improves performance solely by basic mathematical operations and is based on averaging input feature maps. PfAAM can be introduced into network architectures of arbitrary form and do not add trainable parameters or non-trainable hyperparameters and therefore do not change the overall size or complexity of the network. Furthermore, we show that the network performance of different architectures is enhanced by using PfAAM for both classification and semantic segmentation tasks. While most previous work has focused on hand-crafted modules with additional parameters, we present PfAAM as a lightweight plug-and-play module that is compatible with most neural network architectures, enhancing their performance and can be used for various computer vision tasks.\n\\section{Parameter-free Average Attention Module}\n\n\nThe general structure and computation of PfAAM is shown in figure \\ref{fig:pfaam}.\nConsider a feature map $F \\in  \\mathbb{R} ^{H \\times W \\times C} $ as intermediate input, the PfAAM separates the input in a spatial attention part $A_{sp} \\in  \\mathbb{R} ^{H \\times W \\times 1}$ by averaging the input along its channels and a channel attention part $A_{ch} \\in  \\mathbb{R} ^{1 \\times 1 \\times C}$ by calculating the average of each channel. The resulting attention maps are then expanded along their reduced dimensions and recombined to depict the attention to the most important parts of the feature input map. The final recombined attention map uses a sigmoidal gating mechanism to enhance the representational power of the input.\n\nThe overall process can be summarized as follows:\n\n\\begin{equation}\nF' = \\sigma ( A_{sp} \\otimes A_{ch} ) \\otimes F,\n\\end{equation}\n\nwith $\\otimes$ denoting the element-wise multiplication, $\\sigma$ the sigmoid function $\\sigma(x) = \\frac{1}{1+e^{-x}}$ and $F'$ the output of PfAAM. By element-wise multiplication of $A_{sp}$ and $A_{ch}$ the values are broadcasted (copied) along the axes by which they were reduced during averaging to regain their input sizes.\n\nIn contrast to attention modules that emphasize features by learned parameters PfAAM is parameter-free and is solely highlighting features via averaging spatially and along channels.\n\n\\subsection{Spatial Attention Component}\n\nTo emphasize the spatial attention in a feature map we produce a spatial attention map. This is performed by averaging the spatial features along their channels. As a result, the attention is focused on parts in the feature map where a feature is detected.\nThe average of each spatial element $x_{H \\times W} \\in \\mathbb{R} ^{C}$ can be calculated as follows:\n\\begin{equation}\nA_{sp}(x_{H \\times W}) = \\frac{1}{C}\\sum\\limits_{i=1}^C x_{H \\times W}(i).\n\\end{equation}\n\nBy averaging along the channels, the dimension is reduced and produces a spatial map where each element represents the average across channels. As a consequence the spatial areas with high activations are emphasized while areas with low activations are suppressed, thus highlighting positions with detected features.\n\n\\subsection{Channel Attention Component}\nIn accordance to spatial attention the channel attention is calculated by averaging along the spatial dimensions of the feature map. Formally, for each channel $y_C \\in \\mathbb{R} ^{H \\times W}$ the average along its spatial dimensions can be calculated as:\n\\begin{equation}\nA_{ch}(y_C) = \\frac{1}{H \\times W}\\sum\\limits_{i=1}^H \\sum\\limits_{j=1}^W y_C(i,j).\n\\end{equation}\n\nBy averaging along the spatial dimensions, channels are emphasized in which a feature is detected and reducing the influence of channels with low activations for their corresponding features.\n\n\n\\subsection{Model Integration}\nThe three-dimensional input to PfAAM is processed into a matrix with the same dimensions, which can be used as an element-wise multiplier to amplify the activations within the input.  Because of its simplicity, the PfAAM block can be easily integrated into different network architectures and positions, allowing it to be used as a general building block for convolutional neural networks. In the following section, the position of PfAAM within the residual blocks and different pooling operations were analyzed, showing that averaging is slightly preferred over maximization. Finally, PfAAM was successfully tested in different network architectures for classification and segmentation, showing an increase in performance.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMulti-instance learning (MIL) \\cite{Dietterich1997} was originally proposed for drug activity prediction where the task is to predict whether a molecule is suitable for binding to a receptor. \nSince each molecule may take many distinct low-energy conformations and scientists only know its suitability for binding at the molecule level, MIL is proposed to model the molecules as bags and the conformations of the molecules as instances, where only the bag labels are provided but the instance labels are unknown to the learner.\n\nSince its inception, MIL has been studied extensively in various applications where the tasks have inherently structured representations or the fine-grained instance labels are expensive to obtain.\nFor example, MIL has been studied in text categorization \\cite{Andrews:2002:SVM:2968618.2968690,Zhou2009,Ji2020} where articles are represented by bags of sentences with only article-level labels, and in image classifications \\cite{Chen2006,Ilse2018,Skrede2020,Yao2020} where the images are divided into bags of patches with only image-level labels. \n\nAs many real-world objects are inherently structured, an important advantage of multi-instance learning is that by representing objects as bags of instances it can convey more information than using a flat single-instance representation. Since instances within a bag correspond to parts of an object, they share structural and contextual information inherited from the bag and are unlikely to be independent.\nTo see this, let us consider the example from drug activity prediction where instances within a bag represent low-energy conformations of the same molecule. \nThe instances are evidently not independently and identically distributed (i.i.d.) since all conformations share the same structure of single bonds and only differ in how the single bonds rotate.\n\nUnfortunately, despite the fact that neither the instances within a bag are  independently and identically distributed, nor should they be treated as such \\cite{Zhou2009,Zhang2014}, most of the existing MIL algorithms approach the problem by either explicitly assume all instances to be i.i.d. and directly predict the instance labels,\nor focus solely on predicting the bag labels by transforming the bags into a single-vector embedded space which buries instance-level information and prohibits instance label prediction.\n\n\nIn this paper, we propose the Multi-Instance Variational Auto-Encoder (MIVAE) algorithm which explicitly models the instances within a bag as non-i.i.d. using a generative model consisted of a shared bag-level latent factor and instance-level latent factors specific to each of the instances (Figure 1). \nBy using the bag-level factor to capture the structural and contextual dependencies among the instances and using the instance-level factors to capture the instance-specific variations, MIVAE excels at the tasks of both instance label and bag label prediction.\nOn the one hand, integrating the shared bag-level factor and the individualized instance-level factors encapsulate sufficient information for predicting the bag labels. \nOn the other hand, since the shared structural and contextual dependencies are captured by the bag-level factor and the instance-level factors only capture the variations of the individual instances, the instance-level factors promotes better prediction of the instance labels .\n\nThe contributions of this work are three-fold:\n(i) We propose the Multi-instance Variational Auto-Encoder (MIVAE) model, to the best of our knowledge, the first neural network-based MIL algorithm that simultaneously models the instances as non-i.i.d.\n(ii) We extend the powerful variational auto-encoder framework to MIL and instantiate an implementation of MIVAE for simultaneously inferencing the bag-level and instance-level latent factors and predicting both instance labels and bag labels.\n(iii) We empirically demonstrate the effectiveness of MIVAE over classic and neural network-based MIL algorithms using a suite of MIL benchmark datasets and end-to-end medical image diagnosis datasets for both instance label and bag label prediction.\n\n\nThe rest of this paper is organized as follows. We discuss related work in Section 2 and present the proposed MIVAE framework in Section 3. In Section 4 we report the experimental results. Finally, we conclude the paper in Section 5.\n\n\\section{Related Work}\nExisting MIL algorithms can be roughly divided into two categories depending whether the algorithm operates at the level of instances or the level of bags \\cite{Amores2013}.\nThe first group of algorithms directly tackle the MIL problem at the instance-level by training a discriminative instance classifier which separates positive instances in the positive bags from the negative ones \\cite{Andrews:2002:SVM:2968618.2968690,Li2009,Kim2010,Kandemir2014,Haussmann2017}.\nThe second group of methods operates at the bag-level, either by embedding the bags into a single-vector representation and solve a single-instance learning problem at the embedded space \\cite{Chen2006,Wei2017}, or designing multi-instance kernels for measuring similarities between the bags \\cite{Gaertner2002,Zhou2009,Xu2019}.\n\n\nRecently, several neural network-based approaches have been investigated for MIL. \n\\cite{Wang2018} extends feed-forward neural networks to MIL by using permutation-invariant pooling operations on the learned instance embeddings. \n\\cite{Ilse2018,Shi2020} propose to use the attention mechanism in the pooling layer such that the attention weights can be interpreted as how much the instances contribute to the bag label.\nTo the best of our knowledge, all of the existing neural network-based MIL algorithms assume the instances to be i.i.d. and the proposed MIVAE is the first deep generative model for MIL.\n\nSeveral algorithms has approached MIL with generative models which directly predicting the instance labels by either assuming the Gaussian Process \\cite{Kim2010,Haussmann2017} or the Dirichlet Process \\cite{Kandemir2014}. \nHowever, all of them explicitly assume the instances within a bag to be i.i.d.\nIn this work we show that such assumption not only does not agree with the real-world scenarios, but also hurts empirical performance for both bag and instance level label prediction.\n\nPerhaps the work most related to ours is the proposal \\cite{Doran2016}, where they model the multi-instance bags as distributions over instances. \nHowever, their contribution are mainly theoretical while in this work we propose the MIVAE algorithm which is not only applicable to classical MIL tasks with pre-computed features, but also excels at end-to-end learning from weakly-labeled data with neural networks.\n\n\\section{Method}\n\\begin{figure}[t!]\n\t\\centering\n\t\t\\includegraphics[height=1.5in]{generative.pdf}  \n\t\\caption{The generative model for the proposed Multi-Instance Variational Autoencoder (MIVAE). $Y_i$ is the bag label for bag $B_i$, $x_{ij} \\in B$ are the instances of $B_i$. $\\mathbf{z}^B$ is the bag-level factor shared by all of the instances within $B_i$, and $\\mathbf{z}^I_{ij}$ are the instance-level factors specific to each instance.}\n\t\\label{model}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\begin{subfigure}[b]{0.95\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=1.3in]{archi_generative.pdf}\n\t\t\\caption{Generative Model.}\n\t\\end{subfigure}\n\t~\n\t\\begin{subfigure}[b]{0.95\\columnwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=1.5in]{archi_inference.pdf}\n\t\t\\caption{Inference Model.}\n\t\\end{subfigure}\n\t\\caption{Architecture of the model network and the inference network for the Multi-Instance Variational AutoEncoder (MIVAE). White nodes correspond to parametrized deterministic neural network transformations, gray nodes correspond to drawing samples from the respective distribution. Dashed arrows in the inference model represent the auxiliary classifiers $q_{\\omega_y} (y|\\mathbf{z}^B,\\mathbf{z}^I)$.}\n\t\\label{architecture}\n\\end{figure*}\n\nLet $\\mathcal{X} = \\mathbb{R}^d$ denote the space of the instances and $\\mathcal{Y} = \\{0,1\\}$ denote the space of the label. \nThe learner is given a dataset of $m$ bags multi-instance bags $\\mathcal{B} = \\{\\mathbf{B}_1, \\cdots, \\mathbf{B}_i,\\cdots, \\mathbf{B}_m \\}$, where each bag $\\mathbf{B}_i = \\{\\pmb{x}_{i1}, \\cdots, \\pmb{x}_{ij}, \\cdots, \\pmb{x}_{x_{in_i}} \\}$ is a set containing an arbitrary number of  $n_i$ instances with $\\pmb{x}_{ij} \\in \\mathcal{X}$. \nDuring training, each bag $\\mathbf{B}_i$ is provided with a bag label $y_i \\in \\mathcal{Y}$; however, the label of the instances are unknown. The goal of multi-instance learning as two-fold, to predict both the bag label and the instance labels of unseen bags.\nWhen the context is clear, we drop the subscript $i$ from $\\pmb{x}_{ij}$, $\\mathbf{B}_i$ and $y_i$ for the conciseness of the notations.\n\nAt the heart of this work lies the claim that the instances within the same bag are not independently and identically distributed and the i.i.d. assumption is only applicable at the bag-level.\nOn the contrary, instances within the same bag share structural and contextual dependencies inherited from the bag and are only independent to each other when the dependencies have been considered.\nApart from drug activity prediction, the dependencies of instances exists widely in MIL applications. \nFor example, in text classification instances are paragraphs\/sentences of an article and obviously share the styles of the author and contexts of the topic \\cite{Ji2020}. \nIn medical image diagnosis instances are patches of an organ from a patient which naturally share the pathology features and prognoses of the patient \\cite{Skrede2020}.\n\nIn order to capture both the shared dependencies and the individual variations of the instances, we propose the MIVAE model as depicted in Figure \\ref{model}.\nSpecifically, it assumes that for each bag the observed instances are generated from two type of latent factors: a bag-level latent factor $\\mathbf{z}^B$ shared by all instances, and $n_i$ instance-level latent factors $\\mathbf{z}^I_j, j\\in\\{1,\\cdots,n_i\\}$ specific to each of the instances.\nWhile the shared bag-level factor $\\mathbf{z}^B$ controls the structural and contextual dependencies among the instances, the instance-level factors $\\mathbf{z}^I_j$ are responsible for the variations of each instances.\nWhen conditioning on the shared bag-level factor $\\mathbf{z}^B$, the instances becomes independent to each other.\n\nAt the level of bags, MIL is supervised since each bag is labeled during training. Therefore, we assume the bag-level factor $\\mathbf{z}^B$ to be dependent on $y$. \nHowever, since both the instance labels and how they relate to the bag labels are unknown, assuming fixed generative process would restrict the expressiveness of the model. Therefore, we address the prediction of instance labels at inference time.\n\n\n\\subsection{Multi-Instance Variational Auto-Encoder}\n\nIn practice we only observe the bags and bag labels,\nour goal is thus to infer the unobservable posterior distributions for both the bag-level and the instance-level latent factors $p_\\theta(\\mathbf{z}^B|\\mathbf{B})$, $p_\\theta(\\mathbf{z}^I_j|\\pmb{x}_j)$, and the posterior distribution of the observable \n$p_\\theta(\\mathbf{B}|\\mathbf{z}^B,\\mathbf{z}^I_1,\\cdots,\\mathbf{z}^I_j)$\nwhere $\\theta$ denotes the generative parameters.\nSince exact inference of the posterior is intractable due to the fact that both the marginal likelihood and the posterior lack analytical solution, we employ variational autoencoder \\cite{Kingma2014} parameterized by neural networks for efficient approximate inference.  \n\nAs depicted in Figure \\ref{architecture}(a),\nwe utilize two separately-organized encoding networks $q_{\\phi_B}(\\mathbf{z}^B|\\mathbf{B})$ and $q_{\\phi_I}(\\mathbf{z}^I_j|\\pmb{x}_j)$, where $\\phi_B$ and $\\phi_I$ denote the parameters, to serve as variational posteriors of the bag-level and instance-level latent factors. \nThe encoded latent factors are then feed into a single decoder $p_\\theta(\\pmb{x}_j|\\mathbf{z}^B,\\mathbf{z}^I_j)$ for reconstructing the bag of instances. \nFollowing standard VAE design, the prior distributions of the latent factors $p(\\mathbf{z}^B)$ and $p(\\mathbf{z}^I_j)$ are chosen as multivariate Gaussians. \nSpecifically, the prior distributions of the factors and the generative model are factorized as:\n\\begin{align*}\n\t&p(\\mathbf{z}^B|y) = \\prod_{k=1}^{D_{z_B}} \\mathcal{N}(z^{B}_k|f_{y}(y),1);\\\\\n\t&p(\\mathbf{z}^I_j) = \\prod\\limits_{k=1}^{D_{z_I}} \\mathcal{N}(z^{I}_{jk}|0,1); \\nonumber\\\\\n\t& p(\\pmb{x}_j|\\mathbf{z}^B,\\mathbf{z}^I_j) = \\prod\\limits_{k=1}^{d} p(x_{jk}|\\mathbf{z}^B,\\mathbf{z}^I_j), \n\\end{align*}\nwhere $p(x_{jk}|\\mathbf{z}^B,\\mathbf{z}^I_j)$ is the suitable distribution for the $k$-th feature of the instance, i.e., Gaussians for continuous features or Bernoulli for binary features;\n$f_y$ is a function parameterized by neural network using the bag label as input;\n$D_{z_B}$ and $D_{z_I}$ are the parameters that determine the dimensionality of the bag-level and instance-level latent factors. \n\n\nComparing to a standard VAE, the inference model of MIVAE needs to overcome two challenges specific to MIL as depicted in Figure \\ref{architecture}(b). \nFirstly, apart from inferring the instance-level factors $\\mathbf{z}^I$ specific to each instance, MIVAE needs to infer a common bag-level factor $\\mathbf{z}^B$ shared by $n_i$ individual instances $\\pmb{x}_{j}, j = 1,\\cdots,n_i$; however, in standard VAE the latent variables are specific to each of the inputs. \nSecondly, without any instance labels, MIVAE needs to infer the instance labels from the instance-level factors $\\mathbf{z}^I_{j}$.\n\n\nGiven instances $\\pmb{x}_j$, the instance-level factors $\\mathbf{z}^I_j$ can be inferred directly with a standard encoder $q_{\\phi_I}(\\mathbf{z}^{I}_{j}|\\pmb{x}_{j})$ parameterized by neural networks. \nFor the shared bag-level factor $\\mathbf{z}^B$, a straightforward approach would be directly encoding from the concatenation of all instances within the bag; however, this approach unfortunately violates the permutation-invariant property of the instances, i.e., the encoded bag-level factor may dependent on the ordering of the instances.\n\nTo avoid this problem, we design MIVAE to first encode the intermediate bag-level factors $\\mathbf{\\hat{z}}^{B}_j$ using $q_{\\phi_B}(\\mathbf{\\hat{z}}^{B}_{j}|\\pmb{x}_{j})$ for each instance, \nand then utilize a permutation-invariant function for encouraging the intermediate instance-specific factors into a shared bag-level factor. \nSpecifically, the variational approximations for the instance-level factors and the intermediate bag-level factors are defined as\n\\begin{align}\n\t&q_{\\phi_{I}}(\\mathbf{z}_j^I|\\pmb{x}_j) = \\prod_{k=1}^{D_{z_I}} \\mathcal{N} (\\mu = f_{\\phi_{I}}^\\psi(\\pmb{x}_j), \\sigma^2 = f_{\\phi_{I}}^\\pi(\\pmb{x}_j)),\\\\\n\t&q_{\\phi_{B}}(\\mathbf{\\hat{z}}^{B}_j|\\pmb{x}_j) = \\prod_{k=1}^{D_{z_B}} \\mathcal{N}(\\mu = f_{\\phi_{B}}^\\psi(\\pmb{x}_j), \\sigma^2 = f_{\\phi_{B}}^\\pi(\\pmb{x}_j) ),\n\\end{align}\nfor $\\pmb{x}_j$ with $j=1,\\cdots,n_i$, where $f_{\\phi_{I}}^\\psi, f_{\\phi_{B}}^\\psi$ and $f_{\\phi_{I}}^\\pi,  f_{\\phi_{B}}^\\pi$ are the means and variances for the Gaussian distributions parameterized by neural networks.\nThen, the shared bag-level factor is calculated as the mean of the intermediate factors as\n\\begin{align}\n\t\t\\resizebox{0.91\\linewidth}{!}{\n\t\t\t$\n\tq_{\\phi_B}(\\mathbf{z}_B | \\mathbf{B}) = \\prod\\limits_{k=1}^{D_{z_B}} \n\t\\mathcal{N} ( \\frac{1}{n_i}\\sum\\limits_{j=1}^{n_i} f_{\\phi_{B}}^\\psi (\\pmb{x}_{j}),  \\frac{1}{n_i}\\sum\\limits_{j=1}^{n_i} f_{\\phi_{B}}^\\pi (\\pmb{x}_{j})).\n\t$\n}\n\\end{align} \nIn other words, the encoding neural networks $f_{\\phi_{B}}^\\psi$ and $f_{\\phi_{B}}^\\pi$ first infer the means and variances of the posterior distributions for the intermediate factors from the instances $\\pmb{x}_{j}$, then the shared bag-level factor is calculated as the average of all the intermediate bag-level factors. \nIt is worth noting that the variance of $\\mathbf{z}^B$ then become the average of all the intermediate factor variances. \nIn this way, as the decoder reconstructs each of the instances using the shared bag-level factor $\\mathbf{z}^B$ and individual instance-level factor $\\mathbf{z}^I_j$, all of the intermediate bag-level factors are encouraged to converge to the same value during optimization. \nTherefore, $\\mathbf{z}^B$ will become a shared bag-level latent factor common to all $\\pmb{x}_{j}$, while each $\\mathbf{z}^I_j$ captures as the instance-specific variations.  \n\nThe generative and inference models of MIVAE can be learned by maximizing the marginal likelihood over the bags\n\\begin{align}\n\t\\log p_\\theta(\\mathbf{B},y) &= \\mathbb{E}_{q_{\\phi_B} q_{\\phi_I}} [\\log \\frac{p_\\theta(\\mathbf{B},y, \\mathbf{z}^B, \\mathbf{z}^I)} {q_\\phi(\\mathbf{z}^B, \\mathbf{z}^I |\\mathbf{B},y)}] \\nonumber\\\\\n\t& + \\mathbb{E}_{q_{\\phi_B} q_{\\phi_I}} [\\log \\frac{q_\\phi(\\mathbf{z}^B, \\mathbf{z}^I |\\mathbf{B},y)} {p_\\theta(\\mathbf{z}^B, \\mathbf{z}^I |\\mathbf{B},y)}] ,\n\t\\label{likelihood}\n\\end{align}\nwhere the second term is the Kullback-Leibler (KL) divergence from the variational approximation to the true posterior. \nBecause the KL-divergence is non-negative, the first term serves as a lower bound of the marginal likelihood which is also called the evidence lower bound (ELBO).\nMaximizing the marginal likelihood is equivalent to maximizing the ELBO, which can be re-written as\n\\begin{align}\n\t\\mathcal{L}_{\\textrm{ELBO}}(\\mathbf{B},y) = & \\mathbb{E}_{q_{\\phi_B}{q_{\\phi_I}}} [\\log p_\\theta (\\mathbf{B},y|\\mathbf{z}^I, \\mathbf{z}^B)] \\nonumber\\\\\n\t& -  D_{KL} [q_{\\phi_B}(\\mathbf{z}^B|\\mathbf{B})|| p_{\\theta_B}(\\mathbf{z}^B|y)] \\nonumber\\\\\n\t& -  \\sum\\limits_{j=1}^{n_i} D_{KL} [q_{\\phi_I}(\\mathbf{z}^I_j|\\pmb{x}_j)|| p_{\\theta_I}(\\mathbf{z}^I_j)].\n\\end{align}\n\nA missing piece of the puzzle is how to predict the instance labels from the inferred instance-level factors $\\mathbf{z}^I$ and predict the bag labels from both bag-level and instance-level factors. \nInstead of assuming fixed generative distributions which may restrict the applicability of the model, we employ an auxiliary classifier $q_{\\omega_I}(y|\\mathbf{z}^B, \\mathbf{z}^I_1, \\cdots,  \\mathbf{z}^I_{n_i} )$ attached to the evidence lower bound for end-to-end prediction of both instance labels and bag labels. \nUsing the auxiliary classifier allows MIVAE to have the flexibility of utilizing any permutation-invariant multi-instance pooling techniques, i.e., max-pooling, LSE-pooling, and attention pooling \\cite{Ilse2018}. \nIn order to demonstrate that disentangling the dependencies among instances into $\\mathbf{z}^B$ and using only the instance-level factors  $\\mathbf{z}^I_j$ promotes better instance label prediction performance, we use the straightforward max-pooling operation which is defined as\n\\begin{align}\n\tf_{I} = \\max\\limits_{i=1,\\cdots,n_i}\\{f_{\\omega_I}(\\mathbf{z}^I_1),\\cdots,f_{I}(\\mathbf{z}^I_{n_i})\\},\n\\end{align}\nwhere $f_{\\omega_I}$ is a neural network with a sigmoid activation to map the instance-level factors to probabilities. \nThen, the auxiliary for predicting the bag label can be expressed as\n\\begin{align}\nq_{\\omega} (y|\\mathbf{z}^B,\\mathbf{z}^I_1,\\cdots, \\mathbf{z}^I_{n_i}) = q_{\\omega} (y|f_{\\omega_B}(\\mathbf{z}^B), f_{I}),\n\\end{align}\nwhere $f_{\\omega_B}$ is also a parameterized neural network that outputs probabilities.\nFinally, the MIVAE objective can be expressed as the sum of the ELBO and the auxiliary classifier,\n\\begin{align}\n\t\\resizebox{0.91\\hsize}{!}{%\n\t$\\mathcal{L}_{\\text{MIVAE}} =  \\mathcal{L}_{\\textrm{ELBO}}(\\mathbf{B},y)\n\t + \\alpha \\mathbb{E}_{q_{\\phi_{B}} q_{\\phi_{I}}} [\\log q_{\\omega}(y|\\mathbf{z}^B,\\mathbf{z}^I_1,\\cdots, \\mathbf{z}^I_{n_i})],\n\t\\label{loss_function}\n\t$%\n}\n\\end{align}\nwhere $\\alpha$ is a weighting parameter of the auxiliary objective. \nUsing the re-parameterization trick \\cite{Kingma2014}, the objective of MIVAE can be efficiently optimized using stochastic gradient descent. \n\nFor predicting the instance labels of unseen bags, we use the trained instance-level encoder $q_{\\phi_I}(\\mathbf{z}^I_j|\\pmb{x}_j)$ to infer the mean of the posteriors for each $\\pmb{x}_j \\in B$, and use the intermediate output $f_{\\omega_I}(\\mathbf{z}^I_{nj})$ from the auxiliary classifier. \nFor predicting the bag labels, we use the bag-level encoder $q_{\\phi_B}(\\mathbf{z}^B|\\mathbf{B})$ to infer the mean of the bag-level latent factor $\\mathbf{z}^B$, and directly use the output of the auxiliary classifier $q_{\\omega} (y|\\mathbf{z}^B,\\mathbf{z}^I_1,\\cdots, \\mathbf{z}^I_{n_i})$ for prediction.\n\n\\section{Experiments}\nWe empirically evaluate MIVAE against state-of-the-art MIL algorithms for both bag label prediction and instance label prediction tasks.\nFirstly, we compare against SVM-based algorithms including mi-SVM \\cite{Andrews:2002:SVM:2968618.2968690}, miFV \\cite{Wei2017}, mi-Graph \\cite{Zhou2009}, and KI-SVM \\cite{Li2009}. \nSecondly, we compare against neural network-based algorithms including mi-Net and MI-Net \\cite{Wang2018}, and AttentionMIL \\cite{Ilse2018}. \nThirdly, we compare against generative MIL algorithms for predicting instance labels, including Gaussian Process MIL (GPMIL) \\cite{Kim2010}, Dirichlet Process MIL (DPMIL) \\cite{Kandemir2014}, and Variational Gaussian Process MIL (VGPMIL) \\cite{Haussmann2017}.\n\nWe implement MIVAE using PyTorch and provide the source code, datasets, network structures, and parameter grids in the supplementary. Unless otherwise stated, the reported results are obtained from averaging the test results of 10 times repeated 10-fold cross validation as in previous MIL literature. Training for MIVAE are conducted for 100 epochs and parameters are tuned using validation loss evaluate with 10\\% of the training bags. \nAll experiments are conducted using a PC with a AMD Ryzen 3700x CPU with 32GB RAM and a single Nvidia GTX 1080Ti GPU with 11GB memory.\n\n\\subsection{Bag Classification: MIL Benchmarks}\n\\begin{table}[!t]\n\t\\tiny\n\t\\centering\n\t\\caption{Bag label prediction results on five benchmark MIL benchmarks. The highest average accuracy is marked in bold.}\t\n\t\\begin{tabular}{ l | c c c c c} \n\t\t\\hline\n\t\tMethod & Musk1 & Musk2 & Fox & Tiger & Elephant \\\\\n\t\t\\hline\n\t\tmi-SVM & .874$\\pm$.120 & .836$\\pm$.088 & .582$\\pm$.102 & .789$\\pm$.089 & .820$\\pm$.073 \\\\\n\t\n\t\tMILES & .842$\\pm$.081 & .838$\\pm$.095 & \\textbf{.760$\\pm$.045} & .840$\\pm$.081 & \\textbf{.891$\\pm$.053}\\\\\n\t\tmi-Graph &  .889$\\pm$.073 & \\textbf{.903$\\pm$.086} & .616$\\pm$.079 & .801$\\pm$.083 & .869$\\pm$.078 \\\\\n\t\tmiFV & .878$\\pm$.013 & .868\t$\\pm$.094 & .621$\\pm$.109 & .813$\\pm$.083 & .852$\\pm$.081 \\\\\n\t\n\t\t\\hline\n\t\tmi-Net & .892$\\pm$.040 & .858$\\pm$.048 & .615$\\pm$.043 & .839$\\pm$.064& .868$\\pm$.052\\\\\t\t\n\t\tAttentionMIL & .900$\\pm$.050 & .863$\\pm$.042 & .603$\\pm$.059 & .845$\\pm$.048 & .857$\\pm$.057\\\\\n\t\t\\hline\n\t\t$\\text{MIVAE}$ & \\textbf{.904$\\pm$.050} & .890$\\pm$0.62 & .626$\\pm$.055 & \\textbf{.850$\\pm$.051} & .870$\\pm$.064\\\\\n\t\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{benchmark}\n\\end{table}\n\n\n\\begin{table*}[!hbt]\n\t\\centering\n\n\t\\caption{Average test Area Under the Precision-Recall Curve (AUC-PR) of the 20 Newsgroup datasets. The results for the compared methods are obtained from the literature where standard deviations are not reported. The highest average AUC-PR is marked in bold. The last row summarizes the average of the 20 datasets.}\n\t\\begin{tabular}{ l| H c c c c c  H | H c H }\n\t\t\\hline\n\t\n\t\n\t\t& MI-SVM & mi-SVM & KI-SVM  & GPMIL  & DPMIL  & VGPMIL & AttenMIL & MIVAE-att & MIVAE & MIVAE-max-emb \\\\\n\t\t\\hline\n\t\talt.atheism & 0.38 & 0.53 & 0.68 & 0.44 & 0.67 & 0.70 &  & 0.72 & \\textbf{0.745$\\pm$.030} & \\textbf{0.745$\\pm$.029} \\\\\n\t\tcomp.graphics & 0.07& 0.65 & 0.47 & 0.49 & 0.79 & 0.79 &  0& 0.828$\\pm$0.038 & \\textbf{0.800$\\pm$.042} &79.0$\\pm$3.6\\\\\n\t\tcomp.os.ms-windows.misc & 0.03 & 0.42 & 0.38 & 0.36 & 0.51 & 0.52 & & 0.487$\\pm$0.033 & \\textbf{0.548$\\pm$.038} & 53.7$\\pm$4.0 \\\\\n\t\tcomp.sys.ibm.pc.hardware & 0.10 & 0.57 & 0.31 & 0.35 & 0.67 & 0.70 & & 0.668$\\pm$0.039 & \\textbf{0.711$\\pm$.034}\\\\\n\t\tcomp.sys.mac.hardware & 0.27 & 0.56 & 0.39 & 0.54 & 0.76 & \\textbf{0.79} & & &0.783$\\pm$.035 \\\\\n\t\tcomp.windows.x & 0.04 & 0.56 & 0.37 & 0.36 & 0.73 & 0.69 & & & \\textbf{0.754$\\pm$.032}\\\\\n\t\tmisc.forsale & 0.10 & 0.31 & 0.29 & 0.33 & 0.45 & 0.54 & & &\\textbf{0.553$\\pm$.334} \\\\\n\t\trec.autos & 0.34 & 0.51 &  0.45 & 0.38 & 0.76 & 0.71 & & & \\textbf{0.720$\\pm$.024}\\\\\n\t\trec.motorcycles & 0.27 & 0.09 & 0.52 & 0.46 & 0.69 & 0.76 & & &\\textbf{0.766$\\pm$.029} \\\\\n\t\trec.sport.baseball & 0.22 & 0.18 & 0.52 & 0.38 & 0.74 & 0.76 & & & \\textbf{0.764$\\pm$.036}\t\\\\\n\t\trec.sport.hockey & 0.75 & 0.27 & 0.66 & 0.43 & 0.91 & \\textbf{0.94} & & & 0.925$\\pm$.020\\\\\n\t\tsci.crypt & 0.32 & 0.57 & 0.47 & 0.31 & 0.68 & \\textbf{0.82} & & & 0.773$\\pm$.036\\\\\n\t\tsci.electronics & 0.34 & 0.83 & 0.42 & 0.71 & 0.90 & 0.92 & & & \\textbf{0.928$\\pm$.020} \\\\\n\t\tsci.med & 0.44 & 0.37 & 0.55 & 0.32 & 0.73 & 0.73 & & & \\textbf{0.745$\\pm$.025}\\\\\n\t\tsci.space & 0.20 & 0.46 & 0.51 & 0.32 & 0.70 & 0.74 &  &0.738$\\pm$0.031 & \\textbf{0.748$\\pm$.027}\\\\\n\t\tsoc.religion.christian & 0.40 & 0.05 & 0.53 & 0.45 & 0.72 & 0.73 & & & \\textbf{0.753$\\pm$.035}\\\\\n\t\ttalk.politics.guns & 0.01 & 0.57 & 0.43 & 0.38 & 0.64 & \\textbf{0.72} & & & 0.714$\\pm$.038 \\\\\n\t\ttalk.politics.mideast & 0.60 & 0.77 & 0.60 & 0.46 & 0.80 & \\textbf{0.87} & & 0.752$\\pm$0.05\\ & 0.840$\\pm$.020\\\\\n\t\ttalk.politics.misc  & 0.30 & 0.61 & 0.50 & 0.29 & 0.60 & 0.64 & & 0.665$\\pm$0.037 & \\textbf{0.650$\\pm$.044}\\\\\n\t\ttalk.religion.misc & 0.04 & 0.08 & 0.32 & 0.32 & 0.51 & 0.49 & & & \\textbf{0.525$\\pm$.035} \\\\\n\t\t\\hline\n\t\tW\/T\/L & 20\/0\/0 & 20\/0\/0 & 20\/0\/0 & 20\/0\/0 & 20\/0\/0 & 15\/0\/5 & -\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{text}\n\\end{table*}\nWe first study the performance of MIVAE for the bag label predictions on five MIL benchmark datasets including the drug activity prediction datasets Musk1 and Musk2, and three content-based image retrieval tasks Fox, Tiger, and Elephant. Following previous works, we report the average accuracy and the standard deviation obtained from repeating 10-fold cross validation for 10 times.\n\nSince these benchmark datasets only contains a small number of bags (approximately 100 bags for the Musk datasets and 200 bags for the image retrieval datasets) and the pre-computed feature vectors, we do not expect MIVAE to significantly outperform existing methods since its deep generative model may not be well suited for these type of tasks. \nHowever, from Table \\ref{benchmark} we can see that the performance of MIVAE is competitive to the compared baselines. Specifically, MIVAE achieves the highest average accuracy on Musk1 and Tiger datasets, and performs competitively on the others.\nThe results demonstrate that the model of MIVAE is sucessful for handling the uncertainties of multi-instance learning.\n\n\\subsection{Instance Classification: 20 NewsGroups}\n\nTo study the performance for predicting instance labels, we evaluate MIVAE against baselines using the 20 Newsgroup dataset \\cite{Zhou2009}. \nIt contains a collection of 20 datasets where each of them is consisted of 100 bags.\nEach bag contains approximately 40 instances of articles from 20 different topics where each instance represents one article described by the top 200 TF-IDF features. A bag is considered to be positive if at least one of its instances belongs to a specific topic.\nThe labels of the bags are distributed evenly; however, in each positive bag only approximately 3\\% of the instances are positive. Because of the proportions of positive instances are small, the dataset has been widely used as a standard benchmark for instance label prediction performances.\n\nFollowing previous work, we report results obtained from repeating 10-fold cross validation for 10 times using the train\/test splits provided in \\cite{Zhou2009}.  \nBecause the ground truth instance labels are highly imbalanced, we use Area Under the Precision-Recall Curve (AUC-PR) instead of the area under the receiver operating characteristic curve. Furthermore, in order to demonstrate that the advantage of MIVAE lies in its model instead of parameter tuning, the parameters of MIVAE are selected using only the first of the 20 Newsgroup datasets (\\textit{alt.atheism}), and the same parameters are applied across all the rest of the  datasets.\n\nAs shown by the Table \\ref{text}, without tuning parameters for each specific dataset, MIVAE outperforms the compared baselines on 15 out of 20 datasets for predicting instance labels despite the small number of training bags.\nSince all compared algorithms assume the instances to be i.i.d., \nthese results confirm the advantage of MIVAE for modeling the dependencies among the instances.\n\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Bag label and instance label prediction results on the test data of the Colon Cancer dataset. Experiments were repeated for 5 times and the average accuracy $\\pm$ standard deviation of 10-fold cross validations are reported.}\n\t\\begin{tabular}{l| c H H H c}\n\t\t\\hline\n\t\tMethod & Accuracy & Precision & Recall & F-Score & AUC-PR \\\\\n\t\t\\hline\n\t\tmi-Net & 0.824$\\pm$0.021 & 0.866$\\pm$0.017 & 0.816$\\pm$0.031 & 0.813$\\pm$0.023 & 0.491$\\pm$0.028\\\\\n\t\tMI-Net & 0.845$\\pm$0.015 & 0.884$\\pm$0.014 & 0.753$\\pm$0.020 & 0.839$\\pm$0.017 & 0.466$\\pm$0.031\\\\\n\t\t\\hline\n\t\tAttentionMIL & 0.900$\\pm$0.021 & 0.953$\\pm$0.014 &0.855$\\pm$0.017 & 0.901$\\pm$0.011 & 0.500$\\pm$0.034\\\\\n\t\tGated-AttentionMIL & 0.895$\\pm$0.026 & 0.944$\\pm$0.016& 0.851$\\pm$0.035 & 0.893$\\pm$0.022 & 0.513$\\pm$0.023\\\\\n\t\t\\hline\n\t\tMIVAE&  \\textbf{0.925$\\pm$0.014} & & & & \\textbf{0.747$\\pm$0.032}\\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{histopathology}\n\\end{table}\n\\begin{figure*}[!t]\n\t\\centering\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\\includegraphics[height=1.2in]{img80_original.jpg}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\\includegraphics[height=1.2in]{img80_truth.jpg}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img80_max.jpg}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img80_attention.jpg}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img80_MIVAE.jpg}\n\t\\end{subfigure}\n\t\\\\\n\t\\vspace{1mm}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\\includegraphics[height=1.2in]{img9_original.jpg}\n\t\\caption{Original image}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img9_truth.jpg}\n\t\t\\caption{Ground truth}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img9_max.jpg}\n\t\t\\caption{mi-Net}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img9_attention.jpg}\n\t\t\\caption{AttentionMIL}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.18\\textwidth}\n\t\t\\includegraphics[height=1.2in]{img9_MIVAE.jpg}\n\t\t\\caption{MIVAE}\n\t\\end{subfigure}\n\t\\caption{(a) Original H\\&E stained images of the Colon Cancer dataset. (b) Ground truths. (c) Prediction heatmaps of mi-Net. (d) Prediction heatmaps of AttentionMIL. (e) Prediction heatmaps of MIVAE. This figure is best viewed in color.}\n\t\\label{colon_cancer}\n\\end{figure*}\n\\subsection{Histopathology Images: Colon Cancer}\nComputer-aided histopathology using hematoxylin and eosin (H\\&E) stained whole-slide images is an important task for MIL since traditional supervised approaches require pixel-level annotations, which is extremely labor-intensive for pathologists \\cite{Skrede2020}. \nTherefore, a label efficient multi-instance approach using only image-level labels is essential. \nFor this purpose, we evaluate MIVAE against state-of-the-art end-to-end neural network-based MIL algorithms on both bag and instance label prediction using H\\&E histopathology images from the colon cancer dataset provided in \\cite{Sirinukunwattana2016}.\n\nThe dataset contains 100 H\\&E images where each image is from a patient's tissue of either normal or malignant regions. \nFor each image bag, the instances are generated as patches $27\\times 27$ pixels using markings of major nuclei for each cell. A total amount of 22,444 nuclei instances are provided with ground truth instance class labels, i.e. epithelial, inflammatory, fibroblast, and miscellaneous.\nThe bag is considered to be positive if it contains at least one epithelial patch.\n\nFor fairness of comparison, the training images are color normalized and augmented with the same random horizontal flipping\/vertical flipping, and random rotation procedures.\nFurthermore, they also use the same convolutional structures for feature extraction (please refer to the supplementary for details).\nWhen evaluating bag label prediction, we use accuracy since the bag labels are evenly distributed.\nBecause the instances label are imbalanced, we use AUC-PR for quantitative evaluating instance label prediction performance as in the last section. The reported results are the average performances from 5 times repeated 10-fold cross-validation.\n\nAs shown in the results of Table \\ref{histopathology}, MIVAE achieves both the highest classification accuracy for predicting bag labels and the best AUC-PR scores for predicting instance labels, indicating that MIVAE outperforms state-of-the-art neural network-based MIL algorithms at both tasks. \n\nIn Figure \\ref{colon_cancer}, we use heatmaps to visually compare the instance label prediction performances among MIVAE against mi-Net and AttentionMIL. \nFor each image, the heatmaps are obtained by multiplying the instances prediction scores of each algorithm (normalized to the range of $[0,1]$) to the pixel values of the image patches. \nFrom Figure \\ref{colon_cancer} (b)-(e), it is evident that the heatmaps generated by MIVAE are the closest to the ground truths, while mi-Net exhibits lower recalls and the predictions of AttentionMIL are not as confident as the predictions of MIVAE.\n\nIt is worth noting that both mi-Net and AttentionMIL assume the instances to be independently distributed.\nMoreover, mi-Net also uses max-pooling when linking instance prediction scores to bag label. \nTherefore, contrasting the heatmaps of MIVAE and mi-Net paints a clear picture for the benefits of modelling the instance dependencies: \nmi-Net only selects a small portion of positive patches, \nwhile MIVAE is able to identify most of them because the shared dependencies are excluded from instance label predictions.\nFurthermore, comparing MIVAE with AttentionMIL shows that by using the instance-level factors, max-pooling provides more confident predictions than the attention weights.\nFor more examples, please kindly refer to the supplementary materials.\n\nTo sum up, the experiments results demonstrates that the proposed MIVAE algorithm is effective at both bag label and instance label prediction tasks. \nMoreover, the instance prediction scores of MIVAE can be used for explaining the region of interests\nwhich is valuable for weakly-labeled tasks where fine-grained labels are costly to obtain.\n\n\n\n\\section{Conclusion}\nIn this paper, we propose MIVAE, a generative multi-instance learning algorithm which models the instances within a bag as non-i.i.d.. \nExperiments on several MIL benchmarks and medical imaging datasets demonstrate that MIVAE is effective for both instance-level and bag-level label prediction. \nAn interesting future direction would be to investigate how other MIL pooling techniques affect the performance of MIVAE. Furthermore, adapting MIVAE to address the distribution change problem is also worth exploring.\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S:intro}\n\nThe heavy computational demands of high-fidelity computational fluid dynamics (CFD) simulations, caused by fine grid resolutions and time step sizes in addition to complex physical models, are the primary bottleneck preventing most industrial and academic researchers from performing and using such studies. Reactive-flow simulations considering detailed chemistry in particular pose prohibitive computational demands due to (1) chemical stiffness, caused by rapidly depleting species and\\slash or fast reversible reactions, and (2) the large and ever-increasing size of detailed reaction mechanisms. While reaction mechanisms for fuels relevant to hypersonic engines, such as hydrogen or ethylene, may contain 10--70 species~\\cite{Burke:2011fh,Qin:2000ki}, a recent surrogate mechanism for gasoline consists of about \\num{1550} species and \\num{6000} reactions~\\cite{Mehl:2011}; a surrogate mechanism for biodiesel contains almost \\num{3300} species and over \\num{10000} reactions~\\cite{Herbinet:2010}. Strategies for incorporating such large, realistic reaction mechanisms in reactive-flow simulations are beyond the scope of this paper; for example, Lu and Law~\\cite{Lu:2009gh} recently reviewed strategies for mechanism reduction.\n\nEven compact mechanisms pose challenges due to stiffness. In the presence of stiffness, explicit integration algorithms generally require time step sizes on the same order as the fastest chemical time scales, which can be many orders of magnitude smaller than the flow time scale~\\cite{Lu:2009gh}. Due to the resulting computational inefficiency, most reactive-flow simulations rely on specialized integration algorithms such as high-order implicit solvers based on backward differentiation formulas (BDFs)~\\cite{Byrne:1987wp,Hairer:2010gq}. However, these implicit solvers involve expensive linear algebra operations, so techniques for removing stiffness via reduced chemistry have also been developed~\\cite{Lu:2009gh}.\n\nExploiting graphics processing unit (GPU) acceleration offers another avenue for enabling the use of accurate, detailed reaction mechanisms in high-fidelity reactive-flow simulations. Most reactive-flow codes rely on the operator-splitting or fractional-step method~\\cite{Strang:1968wh,Knio:1999vd,Day:2000ek,Sportisse:2000gc,Oran:2001ui,Bourlioux:2003ip,Najm:2005hi,Ren:2008kd}, where the large system of governing partial differential equations (PDEs) is separated such that different physical processes are evaluated separately. For the chemistry---typically the most time-consuming portion of the simulation, accounting for 90\\% or more of the total simulation time in some cases---this results in a system of independent ordinary differential equations (ODEs) for the conservation of species mass in each spatial location (i.e., at each grid point or volume).\n\nDue to the independent nature of the integration for the systems of ODEs governing chemistry in all locations, the entire set can be integrated simultaneously. One option is to parallelize the chemistry integration on multiple central processing unit (CPU) cores or processors using the Message Passing Interface (MPI)~\\cite{MPI-Forum:2009} or OpenMP~\\cite{Dagum:1998hb,Chandra:2001ts,OpenMP:2008}, but the massive parallelism and increasing performance of GPUs---as well as the potential to reduce capital costs through improved energy efficiency---make them an attractive option for accelerating reactive-flow codes. General CFD applications also benefit from GPU acceleration due to the inherent data parallelism of most calculations for both finite difference and finite volume methods. Vanka et al.~\\cite{Vanka:2011vc} surveyed some of the literature on using GPUs to accelerate general CFD simulations; more recently, Niemeyer and Sung~\\cite{Niemeyer:2013} comprehensively reviewed advances in this area for both nonreactive and reactive flows. In the following, we will summarize important results related to GPU-based reactive-flow simulations.\n\nThe first effort in this area came from Spafford et al.~\\cite{Spafford:2010ky}, who accelerated the species rate evaluations in the direct numerical simulation (DNS) code S3D~\\cite{Hawkes:2005eh,Chen:2009gs} on the GPU. In their approach, the CPU handles the time integration of the chemical source terms using an explicit fourth-order Runge--Kutta method. Each integration step requires four species rate evaluations, and for each evaluation the CPU invokes the GPU to evaluate the species rates of change for all grid points simultaneously. Using an ethylene reaction mechanism with 22 species, Spafford et al.~\\cite{Spafford:2010ky} achieved performance speedups of around 15$\\times$ and 9$\\times$ for single- and double-precision calculations, respectively.\n\nMost recent efforts follow the spatially-independent acceleration paradigm introduced by Spafford et al.~\\cite{Spafford:2010ky}, beginning with Niemeyer et al.~\\cite{Niemeyer:2011uw}, who developed a GPU-based explicit integration algorithm for nonstiff chemistry. Using a compact hydrogen mechanism with 9 species and 38 irreversible reactions~\\cite{Yetter:1991}, Niemeyer et al.~\\cite{Niemeyer:2011uw} demonstrated a computational speedup of up to 75$\\times$ compared to a single-core CPU over a wide range of independent ODE systems. Shi et al.~\\cite{Shi:2012cl} presented a hybrid CPU\\slash GPU chemistry integration strategy where the GPU simultaneously integrates nonstiff chemistry in grid cells using an explicit algorithm and the CPU handles spatial locations with stiff chemistry using a standard implicit integrator. This combined approach, paired with a reactive-flow code, achieved an overall performance speedup of 11--46$\\times$ over the algorithms executed on a single CPU core.\n\nLe et al.~\\cite{Le:2013kt} developed the first reactive-flow solver where the GPU evaluates both the fluid transport and chemical kinetics terms. As with most other approaches, they used operator splitting to decouple and independently solve the fluid transport and chemistry terms. They handled the stiff chemical kinetics terms in parallel on the GPU using a first-order implicit method (the backward Euler method), employing a direct Gaussian elimination to solve the resulting linear system of equations. Compared against an equivalent CPU version executed on a single processor core, their combined GPU solver performed up to 40 times faster using a reaction mechanism for methane with 36 species and reversible 308 reactions, on a grid with greater than \\num{e4} cells. However, the low order of the chemistry solver---first order---should be noted.\n\nStone et al.~\\cite{Stone:2013jf} implemented two chemistry integrators on the GPU: (1) a fourth-order adaptive Runge--Kutta--Fehlberg (RKF45) method and (2) the standard fifth-order accurate implicit CVODE method. Applied to a reduced mechanism for ethylene with 19 species and 15 global reaction steps~\\cite{Zambon:2007go} and compared against equivalent single-core CPU versions over a range of ODE numbers, the RKF45 and CVODE methods achieved up to 28.6$\\times$ and 7.7$\\times$ speedup, respectively. The GPU-based RKF45 method performed $20.2\\times$ faster than the CPU-based DVODE solver operating on a single core. It should be noted that the reduced mechanism used by Stone et al.~\\cite{Stone:2013jf} may not exhibit much stiffness, since it was developed by applying the quasi-steady-state approximation to certain radical species and eliminating fast elementary reactions~\\cite{Zambon:2007go}.\n\nAlternative approaches for GPU acceleration of chemical kinetics have also been presented that exploit other areas of data independence. Shi et al.~\\cite{Shi:2011hv} used the GPU to (1) simultaneously calculate all the reaction rates for a single kinetic system (i.e., a single computational volume\\slash grid point) and (2) accelerate the matrix inversion step of the implicit time integration using a commercial GPU linear algebra library, CULA~\\cite{Humphrey:2010ga}. They found this approach beneficial for large reaction mechanisms (e.g., more than \\num{1000} species), accelerating homogenous autoignition simulations up to 22$\\times$, but for moderate-size mechanisms (e.g., less than 100 species) their GPU-based implementation performed slower than the original CPU version. More recently, Sankaran~\\cite{Sankaran:2013he} presented a new approach for accelerating the chemistry in turbulent combustion simulations where the GPU solves the unsteady laminar flamelet equations; the controlling CPU handles the main flow solver. This method involves three levels of concurrency on the GPU: (1) the solution of species reaction rates, thermochemical properties, and molecular transport rates; (2) the solution of the discretized flamelet equations in an regular grid in the mixture fraction space; and (3) the solution of multiple flamelets.\n\nHere, we demonstrate new strategies for accelerating chemical kinetics with moderate levels of stiffness using GPU-based explicit integration algorithms. Building upon our earlier work using the standard fourth-order Runge--Kutta algorithm~\\cite{Niemeyer:2011uw}, we demonstrate the potential performance improvement using a related explicit fifth-order adaptive method for nonstiff chemical kinetics. In addition, we introduce a stabilized explicit Runge--Kutta method that can handle moderate stiffness, and show that it can be used on GPUs to achieve significant computational speedup.\n\nThe rest of the paper is structured as follows. First, we discuss some topics related to GPU computing in Section~\\ref{S:gpu}. Next, in Section~\\ref{S:gov-eq} we provide the governing equations for chemical kinetics in reactive-flow simulations, then in Sections~\\ref{S:rkck} and \\ref{S:rkc} we describe the explicit integration algorithms used in this study. In Section~\\ref{S:results} we demonstrate the performance of the GPU-accelerated integration algorithms using four reaction mechanisms with increasing levels of stiffness and discuss these results. Finally, we summarize our work in Section~\\ref{S:conclusions} and outline future research directions.\n\n\\section{Methodology}\n\\label{S:method}\n\n\\subsection{GPU computing}\n\\label{S:gpu}\n\nWhile an in-depth discussion about GPU computing is beyond the scope of this work, we will briefly introduce important concepts. Interested readers should see the textbooks, e.g., by Kirk and Hwu~\\cite{Kirk:2010we} and Sanders and Kandrot~\\cite{Sanders:2010tq}. The current generation of application programming interfaces, such as CUDA~\\cite{NVIDIA:2011wf} and OpenCL~\\cite{Munshi:2011wk}, enables a C-like programming experience while exposing the massively parallel architecture of graphics processors. This avoids programming in the graphics pipeline directly. Our efforts are based in CUDA, a programming platform created and supported by NVIDIA, but the programming model of OpenCL, an open-source framework, is similar.\n\nIn addition, recently a new avenue for GPU parallelization has been introduced: OpenACC~\\cite{OpenACC:2011vn,Reyes:2012}, which uses compiler directives (e.g., \\texttt{\\#pragma} statements)  placed in Fortran, C, and {C\\nolinebreak[4]\\hspace{-.05em}\\raisebox{.4ex}{\\tiny\\bf ++}}{} codes to identify sections of code to be run in parallel on GPUs. This approach is similar to OpenMP~\\cite{Dagum:1998hb,Chandra:2001ts,OpenMP:2008} for parallelizing work across multiple CPUs or CPU cores that share memory.\n\nGPUs operate on the ``single instruction, multiple thread'' (SIMT) parallelization paradigm, similar to vector computing, where a large number of processing units independently and simultaneously execute the same instructions on different data elements. A parallel GPU function is a kernel, which---in the CUDA programming model---consists of a grid of thread blocks. Each block is made up by threads, the fundamental CUDA processing unit. Physically, GPUs consist of a number of streaming multiprocessors (e.g., 14), which can each execute 32 operations simultaneously. Thread blocks are subdivided into warps consisting of 32 threads; the streaming multiprocessors execute instructions for threads in a particular warp simultaneously. For optimal performance, all 32 threads within a warp should follow the same instruction pathway. If threads in a warp encounter different instructions (e.g., through a conditional branch), the warp diverges and significant loss in performance may result---in the worst case, by a factor of 32, if each thread follows a different instruction pathway.\n\n\\subsection{Governing equations}\n\\label{S:gov-eq}\n\nGiven a vector of state variables $\\mathbf{\\Phi} = \\lbrace \\phi_1, \\dotsc, \\phi_n \\rbrace$, the governing equations for scalars in a general reactive-flow simulation are\n\\begin{equation}\n\\frac{\\partial \\phi_i}{\\partial t} = \\nabla \\cdot \\left( \\mathbf{A}_i + \\mathbf{D}_i \\right) + R_i , \\quad i = 1, 2, \\dotsc, n,\n\\end{equation}\nwhere \\textbf{A} and \\textbf{D} represent the advective and diffusive fluxes, respectively, and \\emph{R} represents the change due to chemical reactions. Solving this stiff, coupled system for a large number of grid points\\slash volumes is challenging, so many reactive-flow modeling approaches rely on operator splitting (also known as the fractional step method)~\\cite{Strang:1968wh,Knio:1999vd,Day:2000ek,Sportisse:2000gc,Oran:2001ui,Bourlioux:2003ip,Najm:2005hi,Ren:2008kd}. This technique separates the integration of the stiff reaction terms from the spatially discretized transport terms, resulting in a large number of independent systems of ODEs---one for each spatial location---to solve.\n\nWhen the reaction terms are separated from physical transport, the species equations are\n\\begin{align}\n\\frac{d Y_i }{dt} &= \\frac{W_i \\omega_i}{\\rho} , \\quad i = 1, 2, \\dotsc, n_S , \\label{E:mass} \\\\\n\\omega_i &= \\sum_{j = 1}^{n_R} \\left( \\nu_{i j}^{\\prime \\prime} - \\nu_{i j}^{\\prime} \\right) \\Omega_j , \\label{E:spec-rate}\n\\end{align}\nwhere $Y_i$ denotes the mass fraction of the \\emph{i}th chemical species, $n_S$ and $n_R$ are the numbers of species and reactions, respectively, $\\rho$ is the mixture density, $W_i$ is the molecular weight of the \\emph{i}th species, $\\Omega_j$ is the rate of progress of reaction \\emph{j}, and $\\nu_{i j}^{\\prime \\prime}$ and $\\nu_{i j}^{\\prime}$ are the reverse and forward stoichiometric coefficients for the \\emph{i}th species in reaction \\emph{j}. The rate of progress of an irreversible reaction without pressure dependence is given by\n\\begin{equation}\n\\Omega_j = k_j \\prod_{k = 1}^{n_S} C_k^{\\nu_{k j}^{\\prime}} , \\label{E:rxn-rate}\n\\end{equation}\nwhere $C_k$ is the concentration of species \\emph{k}. Third-body and pressure-dependent reactions were also considered, depending on the formulation given for the particular reaction; see, for example, Law~\\cite{Law:2006}, or the Chemkin manual~\\cite{Kee:1996vd}, for details. The reaction rate coefficient $k_j$ follows the Arrhenius formulation\n\\begin{equation}\nk_j = A_j T^{\\beta_j} \\exp \\left( \\frac{-E_j}{\\mathcal{R} T} \\right) , \\label{E:rxn-const}\n\\end{equation}\nwhere $\\mathcal{R}$ is the universal gas constant, $A_j$ is the pre-exponential factor, $\\beta_j$ is the temperature exponent, and $E_j$ is the activation energy for reaction \\emph{j}. In general, reactions may be reversible; those without explicitly defined Arrhenius reverse rate parameters (i.e., \\emph{A}, $\\beta$, and $E$) require evaluation of the equilibrium constant to obtain their reverse rate coefficients. To avoid the conditional statements that may cause thread divergence on GPUs (as will be discussed in Section~\\ref{S:results}) required by this evaluation, we converted all such reversible reactions into two irreversible reactions for each following the procedure given in \\ref{A:irrev}.\n\nIn addition to the species equations, we consider a constant-pressure energy equation\n\\begin{equation}\n\\frac{dT}{dt} = -\\frac{1}{\\rho c_p} \\sum_{i = 1}^{n_S} h_i \\omega_i W_i , \\label{E:energy}\n\\end{equation}\nwhere $c_p$ is the mass-averaged constant-pressure specific heat and $h_i$ is the specific enthalpy of the \\emph{i}th species. Together, the coupled mass and energy equations model the time-dependent behavior of an adiabatic, homogenous gas mixture in a closed system. The number of unknowns is equal to the number of species plus one (temperature), $N = n_S + 1$, and the vector of dependent variables consists of temperature and the species mass fractions, $\\mathbf{y} (t) = \\lbrace T, Y_1, Y_2, \\dotsc, Y_{n_S} \\rbrace$.\n\nTypically, reactive-flow simulation codes use BDF-based implicit algorithms to solve Eqs.~\\eqref{E:mass} and \\eqref{E:energy}. While explicit algorithms tend to offer greater general efficiency and lower startup costs---important in operator-split formulations where the transport terms modify the thermochemical conditions and invalidate any saved information, such as the Jacobian matrix, between reaction integration steps---stiffness-induced instabilities force the use of extremely small time step sizes. Implicit algorithms offer greater stability and therefore allow larger time step sizes in the presence of stiffness, resulting in better performance overall. However, implicit methods involve complex control algorithms and linear algebra subroutines, with logical tests for convergence and controlling error. As such, these implicit methods may not be suitable for operating on GPUs, where the complex control flow in such operations could cause threads in a warp to diverge due to slightly different conditions. Stone et al.~\\cite{Stone:2013jf} ported the implicit CVODE solver to GPU operation, and found that it performed only slightly better than a multi-core CPU version would. Explicit algorithms, on the other hand, involve simpler logical flow, and may be better-suited for GPU operation, especially with little-to-moderate stiffness in the chemical kinetics.\n\n\\subsection{Runge--Kutta--Cash--Karp method}\n\\label{S:rkck}\n\nWhen the chemical kinetics exhibits little to no stiffness, we can solve the system of equations given by Eqs.~\\eqref{E:mass} and \\eqref{E:energy} using an explicit integration method such as the fifth-order Runge--Kutta method developed by Cash and Karp~\\cite{Cash:1990}, namely, the RKCK method. This approach uses an embedded fourth-order method to determine the truncation error and adaptively select the step size; our methodology is taken from Press et al.~\\cite{Press:1992}.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{@{}c c c c c c c c c@{}}\n\\toprule\n\\emph{i} & $a_i$ & \\multicolumn{5}{c}{$b_{i j}$} & $c_i$ & $c^*_i$ \\\\ \\midrule\n1 & &  & & & & & $\\frac{37}{378}$ & $\\frac{2825}{27648}$ \\\\\n2 & $\\frac{1}{5}$ & $\\frac{1}{5}$ & & & & & 0 & 0 \\\\\n3 & $\\frac{3}{10}$ & $\\frac{3}{40}$ & $\\frac{9}{40}$ & & & & $\\frac{250}{621}$ & $\\frac{18575}{48384}$ \\\\\n4 & $\\frac{3}{5}$ & $\\frac{3}{10}$ & $-\\frac{9}{10}$ & $\\frac{6}{5}$ & & & $\\frac{125}{594}$ & $\\frac{13525}{55296}$ \\\\\n5 & 1 & $-\\frac{11}{54}$ & $\\frac{5}{2}$ & $-\\frac{70}{27}$ & $\\frac{35}{27}$ & & 0 & $\\frac{277}{14336}$ \\\\\n6 & $\\frac{7}{8}$ & $\\frac{1631}{55296}$ & $\\frac{175}{512}$ & $\\frac{575}{13824}$ & $\\frac{44275}{110592}$ & $\\frac{253}{4096}$ & $\\frac{512}{1771}$ & $\\frac{1}{4}$ \\\\ \\midrule\n\\multicolumn{2}{c}{\\emph{j}} & 1 & 2 & 3 & 4 & 5 & & \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Coefficients for the fifth-order Runge--Kutta--Cash--Karp method, adopted from Press et al.~\\cite{Press:1992}.}\n\\label{T:rkck}\n\\end{center}\n\\end{table}\n\nIf $\\mathbf{y}_n$ is the approximation to the exact solution $\\mathbf{y}(t)$ at $t = t_n$, and $\\delta t_n = t_{n+1} - t_n$ is the current step size, then the RKCK formulas, which also apply to any general fifth-order Runge--Kutta method, are\n\\begin{align}\n\\mathbf{k_1} &= \\delta t \\, \\mathbf{f} \\left( t_n, \\mathbf{y}_n \\right) \\\\\n\\mathbf{k}_2 &= \\delta t \\, \\mathbf{f} \\left( t_n + a_2 \\, \\delta t, \\mathbf{y}_n + b_{2 1} \\mathbf{k}_1 \\right) , \\\\\n\\mathbf{k}_3 &= \\delta t \\, \\mathbf{f} \\left( t_n + a_3 \\, \\delta t, \\mathbf{y}_n + b_{3 1} \\mathbf{k}_1 + b_{3 2} \\mathbf{k}_2 \\right) , \\\\\n\\mathbf{k}_4 &= \\delta t \\, \\mathbf{f} \\left( t_n + a_4 \\, \\delta t, \\mathbf{y}_n + b_{4 1} \\mathbf{k}_1 + b_{4 2} \\mathbf{k}_2 + b_{4 3} \\mathbf{k}_3 \\right) , \\\\\n\\mathbf{k}_5 &= \\delta t \\, \\mathbf{f} \\left( t_n + a_5 \\, \\delta t, \\mathbf{y}_n + b_{5 1} \\mathbf{k}_1 + b_{5 2} \\mathbf{k}_2 + b_{5 3} \\mathbf{k}_3 + b_{5 4} \\mathbf{k}_4 \\right) , \\\\\n\\mathbf{k}_6 &= \\delta t \\, \\mathbf{f} \\left( t_n + a_6 \\, \\delta t, \\mathbf{y}_n + b_{6 1} \\mathbf{k}_1 + b_{6 2} \\mathbf{k}_2 + b_{6 3} \\mathbf{k}_3 + b_{6 4} \\mathbf{k}_4 + b_{6 5} \\mathbf{k}_5 \\right) , \\\\\n\\mathbf{y}_{n+1} &= \\mathbf{y}_n + c_1 \\mathbf{k}_1 + c_2 \\mathbf{k}_2 + c_3 \\mathbf{k}_3 + c_4 \\mathbf{k}_4 + c_5 \\mathbf{k}_5 + c_6 \\mathbf{k}_6 , \\\\\n\\mathbf{y}^*_{n+1} &= \\mathbf{y}_n + c^*_1 \\mathbf{k}_1 + c^*_2 \\mathbf{k}_2 + c^*_3 \\mathbf{k}_3 + c^*_4 \\mathbf{k}_4 + c^*_5 \\mathbf{k}_5 + c^*_6 \\mathbf{k}_6 ,\n\\end{align}\nwhere $\\mathbf{y}_{n+1}$ is the fifth-order solution and $\\mathbf{y}^*_{n+1}$ is the solution of the embedded fourth-order method. The vector $\\mathbf{f} (t, \\mathbf{y} ) = d \\mathbf{y} (t, \\mathbf{y}) \/ dt $ represents the evaluation of the right-hand side of Eqs.~\\eqref{E:mass} and \\eqref{E:energy}. The RKCK coefficients are given in Table~\\ref{T:rkck}. The fourth- and fifth-order solutions are used to estimate the error of the step $\\mathbf{\\Delta}_{n+1}$,\n\\begin{equation}\n\\mathbf{\\Delta}_{n+1} = \\mathbf{y}_{n+1} - \\mathbf{y}^*_{n+1} = \\sum_{i = 1}^6 \\left( c_i - c^*_i \\right) \\mathbf{k}_i .\n\\end{equation}\nThis error is then compared against a desired accuracy, $\\mathbf{\\Delta_0}$, defined by\n\\begin{equation}\n\\mathbf{\\Delta}_0 = \\epsilon \\left( | \\mathbf{y}_n | + \\left| \\delta t \\, \\mathbf{f} \\left(t_n, \\mathbf{y}_n \\right) \\right| + \\delta \\right) ,\n\\end{equation}\nwhere $\\epsilon$ is a tolerance level and $\\delta$ represents a small value (e.g., \\num{e-30}). If the estimated error of the current step is larger than the desired accuracy ($ \\mathbf{\\Delta}_{n+1} > \\mathbf{\\Delta}_0 $), the step is rejected and a smaller step size is calculated; if the error is smaller than the desired accuracy ($ \\mathbf{\\Delta}_{n+1} \\leq \\mathbf{\\Delta}_0 $), the step is accepted and the step size for the next step is calculated. The following is used to calculate a new step size based on error and the current step size:\n\\begin{equation}\n\\delta t_{\\text{new}} =\n\\begin{dcases}\nS \\, \\delta t_n \\, \\max_i \\left( \\left| \\frac{\\Delta_{0,i}}{\\Delta_{n+1, i}} \\right| \\right)^{1\/5} \\quad \\text{if } \\mathbf{\\Delta_{n+1}} \\leq \\mathbf{\\Delta}_0 , \\\\\nS \\, \\delta t_n \\, \\max_i \\left( \\left| \\frac{\\Delta_{0,i}}{\\Delta_{n+1, i}} \\right| \\right)^{1\/4} \\quad \\text{if } \\mathbf{\\Delta_{n+1}} > \\mathbf{\\Delta}_0 .\n\\end{dcases}\n\\label{E:hnew}\n\\end{equation}\nHere, \\emph{i} represents the \\emph{i}th element of the related vector and \\emph{S} denotes a safety factor slightly smaller than unity (e.g., 0.9). Eq.~\\eqref{E:hnew} is used to calculate the next time step size for an accepted step and a new, smaller step size when the error is too large (and therefore the step is rejected). In practice, step size decreases and increases are limited to factors of ten and five, respectively.\n\n\n\\subsection{Runge--Kutta--Chebyshev method}\n\\label{S:rkc}\n\nFor stiff problems, standard explicit integration methods become unsuitable due to stability issues, requiring unreasonably small time step sizes~\\cite{Hairer:2010gq}. Traditionally, implicit integration algorithms such as those based on BDFs have been used to handle stiff problems, but these require expensive linear algebra operations on the Jacobian matrix. In addition, the complex logical flow would result in highly divergent instructions across different initial conditions, making implicit algorithms unsuitable for operation on GPUs. One alternative to implicit algorithms for problems with moderate levels of stiffness is a stabilized explicit scheme such as the Runge--Kutta--Chebyshev (RKC) method~\\cite{Houwen:1980,Verwer:1990tg,vanderHouwen:1996ti,Verwer:1996vo,Sommeijer:1997uv,Verwer:2004gf}. While the RKC method is explicit, it is capable of handling stiffness through additional stages---past the first two required for second-order accuracy---that extend its stability domain along the negative real axis.\n\nOur RKC implementation is taken from Sommeijer et al.~\\cite{Sommeijer:1997uv} and Verwer et al.~\\cite{Verwer:2004gf}. Following the same terminology as in the description of the RKCK method in Section~\\ref{S:rkck}, where $\\mathbf{y}_n$ is the approximation to the exact solution $\\mathbf{y}(t)$ at $t = t_n$ and $\\delta t_n = t_{n+1} - t_n$ is the current step size, the formulas for the second-order RKC are\n{\\allowdisplaybreaks \\begin{IEEEeqnarray}{rCl}\n\\mathbf{w}_0 & = & \\mathbf{y}_n , \\label{E:rkc0} \\\\\n\\mathbf{w}_1 & = & \\mathbf{w}_0 + \\tilde{\\mu}_1 \\, \\delta t \\, \\mathbf{f}_0 , \\label{E:rkc1} \\\\\n\\mathbf{w}_j & = & (1 - \\mu_j - \\nu_j ) \\mathbf{w}_0 + \\mu_j \\mathbf{w}_{j - 1} \\nonumber \\\\\n& & +\\: \\nu_j \\mathbf{w}_{j - 2} + \\tilde{\\mu}_j \\, \\delta t \\, \\mathbf{f}_{j - 1} + \\tilde{\\gamma}_j \\, \\delta t \\, \\mathbf{f}_0, \\quad j = 2, \\dotsc, s ,  \\label{E:rkcj} \\\\\n\\mathbf{y}_{n + 1} & = & \\mathbf{w}_s , \\label{E:rkcs}\n\\end{IEEEeqnarray}}%\nwhere \\emph{s} is the total number of stages. The $\\mathbf{w}_j$ are internal vectors for the stages, and $\\mathbf{f}_j$ are evaluations of the right-hand-side function of the governing equations at each stage, where $\\mathbf{f}_j = \\mathbf{f} (t_n + c_j \\, \\delta t , \\mathbf{w}_j )$. Note the recursive nature of $\\mathbf{w}_j$, which requires only five arrays for storage. The coefficients used in Eqs.~\\eqref{E:rkc1} and \\eqref{E:rkcj} are available analytically for any $s \\geq 2$:\n\\begin{align}\n\\tilde{\\mu}_1 &= b_1 \\omega_1 , \\\\\n\\mu_j = \\frac{2 b_j \\omega_0}{b_{j - 1}} , \\quad \\nu_j &= \\frac{-b_j}{b_{j-2}}, \\quad \\tilde{\\mu}_j = \\frac{2 b_j \\omega_1}{b_{j-1}}, \\quad \\tilde{\\gamma}_j = -a_{j-1} \\tilde{\\mu_j} \\\\\nb_0 = b_2, \\quad b_1 &= \\frac{1}{\\omega_0}, \\quad b_j = \\frac{T_j^{\\prime \\prime} (\\omega_0) }{ \\left( T_j^{\\prime} (\\omega_0) \\right)^2 }, \\\\\nw_0 = 1 + \\frac{\\kappa}{s^2} , \\quad \\omega_1 &= \\frac{ T_s^{\\prime} (\\omega_0) }{ T_s^{\\prime \\prime} (\\omega_0) } ,\n\\end{align}\nwhere $\\kappa \\geq 0$ is the damping parameter (we used $\\kappa = 2 \/ 13$~\\cite{Sommeijer:1997uv,Verwer:2004gf}). $T_j(x)$ are the Chebyshev polynomials of the first kind, defined recursively as\n\\begin{equation}\nT_j (x) = 2 x T_{j - 1} (x) - T_{j - 2} (x), \\quad j = 2, \\dotsc, s,\n\\end{equation}\nwhere $T_0 (x) = 1$, $T_1 (x) = x$, and $T_j^{\\prime}(x)$ and $T_j^{\\prime \\prime}(x)$ are the first and second derivatives of $T_j (x)$, respectively. The $c_j$ used in the function evaluations are\n\\begin{align}\nc_1 &= \\frac{c_2}{T_2^{\\prime}(\\omega_0)} \\approx \\frac{c_2}{4}, \\\\\nc_j &= \\frac{T_s^{\\prime} (\\omega_0)}{T_s^{\\prime \\prime} (\\omega_0)} \\frac{T_j^{\\prime \\prime} (\\omega_0)}{T_j^{\\prime} (\\omega_0)} \\approx \\frac{j^2 - 1}{s^2 - 1}, \\quad 2 \\leq j \\leq s - 1, \\\\\nc_s &= 1.\n\\end{align}\n\nThe RKC method can also be used with an adaptive time stepping method for error control, as given by Sommeijer et al.~\\cite{Sommeijer:1997uv}. After taking the step $t_{n+1} = t_n + \\delta t_n$ and calculating $\\mathbf{y}_{n+1}$, the error in the calculation at the current step is estimated using\n\\begin{equation}\n\\mathbf{\\Delta}_{n+1} = \\frac{4}{5} (\\mathbf{y}_n - \\mathbf{y}_{n+1}) + \\frac{2}{5} \\delta t_n (\\mathbf{f}_n + \\mathbf{f}_{n+1}) .\n\\end{equation}\nThese error estimates are used with absolute and relative tolerances to define the weighted RMS norm of error:\n\\begin{align}\n\\| \\mathbf{\\Delta}_{n+1} \\|_{\\text{rms}} &= \\left \\| \\frac{\\mathbf{\\Delta}_{n+1}}{\\mathbf{T} \\sqrt{N}} \\right \\|_2 , \\label{E:rmsE} \\\\\n\\mathbf{T} &= \\mathbf{A} + R \\cdot \\max \\left( | \\mathbf{y}_n |, | \\mathbf{y}_{n+1} | \\right) ,\n\\end{align}\nwhere \\emph{N} represents the number of unknown variables (here, $N = n_S + 1$ as defined previously), $\\mathbf{A}$ is the vector of absolute tolerances, and \\emph{R} is the relative tolerance. The norm $\\| \\cdot \\|_2$ indicates the Euclidean or $L_2$ norm. The step is accepted if $ \\| \\mathbf{\\Delta}_{n+1} \\|_{\\text{rms}} \\leq 1 $; otherwise, it is rejected and redone using a smaller step size. The weighted RMS norm of error for the current and prior steps, and the associated step sizes, are then used to predict the new step size, using\n\\begin{align}\n\\delta t_{n+1} &= \\min \\left( 10, \\max( 0.1, f ) \\right) \\delta t_n , \\\\\nf &= 0.8 \\left( \\frac{ \\| \\mathbf{\\Delta}_n \\|_{\\text{rms}}^{1 \/ (p + 1)} }{ \\| \\mathbf{\\Delta}_{n+1} \\|_{\\text{rms}}^{1 \/ (p + 1)} } \\frac{\\delta t_n}{\\delta t_{n - 1}} \\right) \\frac{1}{ \\| \\mathbf{\\Delta}_n \\|_{\\text{rms}}^{1 \/ (p + 1)} } ,\n\\end{align}\nwhere \\emph{p} is the order of the algorithm---two, in this case. When a step is rejected, we use a similar equation to calculate a new step size:\n\\begin{equation}\nf = \\frac{0.8}{ \\| \\mathbf{\\Delta}_n \\|_{\\text{rms}}^{1 \/ (p + 1)} } .\n\\end{equation}\n\nIn order to determine the initial time step size, we first use a tentative step size calculated as the inverse of the spectral radius $\\sigma$---the magnitude of the largest eigenvalue---of the Jacobian. After predicting the error associated with this tentative step, we then set the initial step size as one-tenth of the step size that would satisfy error control based on the tentative step:\n\\begin{align}\n\\delta t_0 &= \\frac{1}{ \\sigma } , \\\\\n\\mathbf{\\Delta}_0 &= \\delta t_0 \\left( \\mathbf{f}(t_0 + \\delta t_0, \\mathbf{y}_0 + \\delta t_0 \\, \\mathbf{f}(t_0, \\mathbf{y}_0)) - \\mathbf{f} (t_0, \\mathbf{y}_0) \\right), \\\\\n\\delta t_1 &= 0.1 \\frac{\\delta t_0}{ \\| \\mathbf{\\Delta_0} \\|_{\\text{rms}}^{1\/2} } ,\n\\end{align}\nwhere $\\|\\mathbf{\\Delta_0}\\|_{\\text{rms}}$ is evaluated in the same manner as $\\| \\mathbf{\\Delta}_{n+1} \\|_{\\text{rms}}$ using Eq.~\\eqref{E:rmsE}.\n\nAfter selecting the optimal time step size to control local error, the algorithm then determines the optimal number of RKC stages in order to remain stable. Due to stiffness, too few stages would lead to instability. The local stiffness is determined using the spectral radius and time step size. The number of stages are determined by\n\\begin{equation}\ns = 1 + \\sqrt{1 + 1.54 \\, \\delta t_n \\, \\sigma} ,\n\\end{equation}\nas suggested by Sommeijer et al.~\\cite{Sommeijer:1997uv}, where the value 1.54 is related to the stability boundary of the algorithm. Note that \\emph{s} may vary between time steps due to a changing spectral radius and time step size. In our RKC implementation, we used a nonlinear power method~\\cite{Sommeijer:1997uv} to calculate the spectral radius; this choice costs an additional vector to store the computed eigenvector, but avoids storing or calculating the Jacobian matrix. Depending on the problem type, alternative methods such as the Gershgorin circle theorem~\\cite{Gersgorin:1931,Horn:1990} could be used to obtain an upper-bound estimate for the spectral radius. In our experience, however, the circle theorem tended to overestimate the spectral radius, resulting in unnecessarily large numbers of stages---this induced greater computational expense compared to using the power method. Following Sommeijer et al.~\\cite{Sommeijer:1997uv}, in our RKC implementation the spectral radius is estimated every 25 (internal) steps or after a step rejection. In addition, the computed eigenvector is saved to be used as the initial guess in the next evaluation.\n\n\\section{Results and discussion}\n\\label{S:results}\n\nIn order to study the performance of the GPU-based RKCK and RKC solvers (termed RKCK-GPU and RKC-GPU, respectively), we tested their performance with four reaction mechanisms, representing different levels of stiffness. We varied the problem size, meaning number of chemical kinetics ODEs, over a wide range from \\num{e2} to \\num{e6}, representing a wide range of grid resolutions in an operator-split reactive-flow code.\n\nFirst, we studied the performance of RKCK-GPU using a nonstiff hydrogen mechanism. Next, we considered (separately) mechanisms for hydrogen\\slash carbon monoxide and methane with moderate levels of stiffness and use these to study the performance of RKC-GPU. Finally, we examined the performance of RKC-GPU in a case where stiffness is more severe, using an ethylene mechanism. In all four cases, we compared the performance of the GPU algorithm against an equivalent CPU version. In the presence of stiffness, we also compared the performance of RKC-GPU against an implicit CPU-based code, VODE\\_F90~\\cite{Byrne:2006}, a Fortran 90 version of the well-known VODE solver.\n\nIn both the CPU and GPU algorithms used here, we generated the subroutines needed for chemical kinetics source terms (e.g., species rates, reaction rates, thermodynamic properties) using an open-source Python tool that we created~\\cite{Niemeyer:2013cs}, which takes Chemkin-format reaction mechanisms as input. Further, we converted all reversible reactions in the reaction mechanisms used here into two irreversible reactions for each in order to avoid the computation of equilibrium constants, as described in \\ref{A:irrev}. We developed an additional Python tool implementing this procedure that is also available online~\\cite{Niemeyer:2013im}. We paired VODE with CHEMKIN-III~\\cite{Kee:1996vd} to evaluate the chemical kinetics and species thermodynamic properties. All calculations were performed in double precision and at constant pressure---although the generated subroutines are also capable of constant volume conditions.\n\nAll calculations reported here were performed using a single GPU and single CPU; we measured the serial CPU performance using a single core as well as parallelized CPU performance---via OpenMP~\\cite{OpenMP:2008}---on six cores. The GPU calculations were performed using an NVIDIA Tesla c2075 GPU with \\SI{6}{\\gigaB} of global memory. An Intel Xeon X5650 CPU, running at \\SI{2.67}{\\giga\\hertz} with \\SI{256}{\\kiloB} of L2 cache memory per core and \\SI{12}{\\megaB} of L3 cache memory, served as the host processor for the GPU calculations and ran the CPU single- and six-core OpenMP calculations. We used the GNU Compiler Collection (gcc) version 4.6.2 (with the compiler options ``\\texttt{-O3 -ffast-math -std=c99 -m64}'') to compile the CPU programs and the CUDA 5.0 compiler nvcc version 0.2.1221 (``\\texttt{-O3 -arch=sm\\_20 -m64}'') to compile the GPU versions. The function \\texttt{cudaSetDevice()} was used to hide any device initialization delay in the CUDA implementations prior to the timing.\n\nImposing identical initial conditions for all ODEs would not represent the situation in a reactive-flow simulation where conditions vary across space, so we generated initial conditions for the ODEs by sampling the solutions obtained from constant pressure homogeneous ignition simulations. For all four fuels studied, we used starting conditions of \\SI{1600}{\\kelvin}, \\SI{1}{atm}, and an equivalence ratio of one. This resulted in a set of initial conditions covering a wide range of temperatures and species mass fractions. For example, some data points came from the pre-ignition induction period, some from the transient regime when temperature increases rapidly, and some from the post-ignition stage where conditions approach equilibrium. We distributed the resulting initial conditions in two ways. First, we assigned initial conditions sequentially to ODEs, where consecutive data points---taken from consecutive time steps---contain similar conditions. This emulated adjacent spatial locations with similar but not identical conditions. Further, this procedure represents a more realistic performance measure compared to the previous work of Niemeyer et al.~\\cite{Niemeyer:2011uw}, where identical initial conditions and a constant time step size were used. For the GPU-based algorithms, similar---but not identical---initial conditions will result in threads within warps that may follow divergent pathways, due to varying time step sizes, for example. In order to further explore the impact of divergence on performance, we also assigned initial conditions to threads in a second manner: randomly shuffling the order. Compared to using similar conditions, randomly selected initial conditions represent a worst-case potential for divergence.\n\nOther potential sources of thread divergence could be conditional statements in the source terms, because, e.g., thermodynamic properties are typically fitted as polynomials across different temperature ranges, certain reaction pressure-dependence formulations are described in different pressure ranges. We attempted to minimize the occurrence of such conditional statements by converting each reversible reaction in the reaction mechanisms into a pair of irreversible reactions (as described above). This avoided the temperature conditional statement required for evaluating the Gibbs function polynomial, in turn needed for the equilibrium constants. Regarding the conditional statements required to evaluate the species thermodynamic properties for the energy equation or reaction rates for particular pressure-dependence formulations, in the current work, neither of these contributed to thread divergence because (1) all temperatures experienced by threads fell within the same polynomial fitting range and (2) none of the pressure-dependent reactions considered in the reaction mechanisms were formulated using multiple pressure ranges. However, in general cases, conditional statements on temperature or pressure could cause additional thread divergence.\n\nThe integration algorithms take as input initial conditions and a global time step, performing internal sub-stepping as necessary. The computational times, or wall-clock times, reported represent the average over ten global time steps. For the GPU implementations, the reported computational time per global time step included the overhead required for transmitting data between the CPU and GPU before and after each integration step. The integrator restarts at each global time step,  not storing any data from the previous step---although any sub-stepping performed by the algorithm within these larger steps does benefit from retained information from prior sub-steps. This is done to emulate a true operator-split code, where the transport integration step would update the thermochemical conditions independently from the chemistry and therefore invalidate any retained information between global steps. This reduces the efficiency somewhat, by forcing the integrator to take initially large test steps, but the startup costs of the explicit integration algorithms considered here pale in comparison to those of implicit integrators such as VODE, where the Jacobian matrix must be re-evaluated.\n\nIn the GPU-based algorithms, threads independently integrated each chemical kinetics ODE. The total number of threads then equaled the number of ODEs; blocks consisted of 64 threads each. For problem sizes of \\num{4194304} or larger, where a block size of 64 threads would exceed the maximum limit on number of blocks per grid (\\num{65535}) in one dimension, we used a block size of $N_t \/ \\num{32768}$, where $N_t$ is the total number of threads. We kept the block size as a multiple of 32 to ensure blocks contained whole thread warps.\n\n\\subsection{Hydrogen kinetics}\n\\label{S:h2}\n\nFirst, we considered a case where stiffness in the chemical kinetics does not pose a challenge, using the hydrogen oxidation mechanism of Yetter et al.~\\cite{Yetter:1991} with 9 species and 38 irreversible reactions. We employed the explicit RKCK method, with a tolerance level $\\epsilon$ of \\num{1e-10}, and performed 10 global integration steps of \\SI{1e-8}{\\second} (or \\SI{10}{\\nano\\second}) each. The average time needed per step is reported. This application is relevant particularly for DNS and studies of high-speed flows, which use extremely short time step sizes in order to resolve the Kolmogorov scales and capture the short time scales due to high flow velocity, respectively. Adjacent ODEs used similar initial conditions as described in the previous section.\n\nThe lack of stiffness in this case was due to both the particular chemistry considered and the short global time step sizes used. Quantifying stiffness is somewhat difficult~\\cite{Hairer:2010gq}, but in general explicit methods are more efficient for nonstiff problems than implicit or other stiff integrators (e.g., stabilized explicit methods like RKC). In terms of computational time, RKCK-GPU performed nearly 2.3$\\times$ and 2.6$\\times$ faster than RKC-GPU for problem sizes of \\num{65536} and \\num{262144} independent ODEs, respectively, so we consider this case nonstiff.\n\nFigure~\\ref{F:h2-rkck} shows the performance results of the CPU- and GPU-based RKCK algorithms for problem sizes ranging from 64 to \\num{8388608}. RKCK-GPU performed faster than the single-core RKCK-CPU for problem sizes of 128 ODEs and larger, and faster than the six-core CPU version when the number of ODEs is 512 or larger. Note that the speedup of the GPU implementation increased with growing problem size. For the largest problem sizes, RKCK-GPU ran up to 126$\\times$ and 25$\\times$ faster than the single- and six-core RKCK-CPU versions. On six cores RKCK-CPU ran between five and six times faster than on a single core, due to the data independent nature of the problem.\n\n\\begin{figure}[tbp] \\begin{center}\n\\includegraphics[width=0.8\\linewidth]{h2-rkck.pdf}\n\\caption{Performance comparison of (single- and six-core) CPU and GPU integration of the nonstiff hydrogen mechanism using the explicit RKCK method. Note that both axes are displayed in logarithmic scale.}\n\\label{F:h2-rkck}\n\\end{center} \\end{figure}\n\nWe also studied the effect of different initial conditions on the performance of RKCK-GPU, by randomly shuffling the data points used for this purpose such that neighboring threads no longer contained similar data. This resulted in thread divergence, since different threads in each warp will require different inner time step sizes---therefore some threads will require a greater number of steps, while others will finish sooner. Figure~\\ref{F:h2-rkck-random} shows the comparison of performance for RKCK-GPU between threads with similar initial conditions and threads where initial conditions were randomly selected (and are therefore different). The divergence caused by randomized initial conditions reduced performance by up to a factor of 2.3, with a greater reduction at larger problem sizes. We note that some divergence was also present for threads with similar---but not identical---initial conditions.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{h2-rkck-random.pdf}\n\\caption{Performance comparison of RKCK-GPU integration of the nonstiff hydrogen mechanism where neighboring threads have similar and randomized initial conditions.}\n\\label{F:h2-rkck-random}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{h2-rkck-divergence.pdf}\n\\caption{Warp thread divergence comparison of RKCK-GPU for nonstiff hydrogen kinetics for similar and random initial conditions, where the number of occurrences of the divergence measure \\emph{D} is plotted for \\num{2048} thread warps.}\n\\label{F:h2-rkck-diverge}\n\\end{center}\n\\end{figure}\n\nIn order to quantify this divergence, we introduce a measure for the divergence in a thread warp, \\emph{D}, proposed by Stone~\\cite{Stone:2013} and Sankaran~\\cite{Sankaran:2013}, defined by\n\\begin{equation}\nD = \\frac{ \\sum_{i = 1}^{32} d_i }{32 \\, \\max_{i} d_i } ,\n\\end{equation}\nwhere $d_i$ denotes the number of right-hand function (i.e., derivative) evaluations in the \\emph{i}th thread over a certain number of global time steps. We used this to represent the cost of integration per global step for each thread within a warp. Values of \\emph{D} approaching one represent a warp with completely converged threads, while values approaching zero represent a situation where a small number of threads perform significantly more work than other threads. However, it should be noted that \\emph{D} is not a perfect measure of divergence in general applications, where threads may follow different instructions but perform similar amounts of work. Figure~\\ref{F:h2-rkck-diverge} shows the distribution of \\emph{D} for \\num{65536} ODEs, corresponding to \\num{2048} thread warps, where the sum of the derivative evaluations over ten global time steps was used to evaluate \\emph{D}. For similar initial conditions, the divergence remained low as measured by \\emph{D}, while with randomized initial conditions the divergence was greater, with \\emph{D} ranging between 0.3--0.8 and showing peaks around 0.4 and 0.65. This divergence likely caused the reduced performance of RKCK-GPU with randomly chosen initial conditions compared to similar initial conditions.\n\n\\subsection{Hydrogen\\slash carbon monoxide kinetics}\n\\label{S:h2-co}\n\nNext, we studied a kinetic system with moderate stiffness, using the hydrogen\\slash carbon monoxide reaction mechanism of Burke et al.~\\cite{Burke:2011fh}, which consists of 13 species and 27 reversible (converted to 54 irreversible) reactions. Here, we chose a global time step size of \\SI{1e-6}{\\second} and reported the average computational time for ten steps. This value represents step sizes used in large-eddy simulations of reactive flows~\\cite{Wang:2011kq,Bulat:2013ds}. We consider this problem to be ``moderately'' stiff because RKC-GPU performed more than three times faster than RKCK-GPU. For RKC, we used a relative tolerance of \\num{1e-6} and an absolute tolerance of \\num{1e-10}. Adjacent ODEs used similar initial conditions.\n\nFigure~\\ref{F:h2-co-rkc} shows the performance comparison between the single- and six-core RKC-CPU and RKC-GPU for problem sizes ranging from 64 to \\num{4194304}. As with the RKCK algorithm, at smaller problem sizes RKC-GPU compared less favorably against RKC-CPU, but the speedup increased with increasing problem size. RKC-GPU outperformed RKC-CPU on a single CPU core for the entire range of ODE numbers considered here, while it performed faster than the six-core version for problem sizes of 512 ODEs and larger. While the exact speedup varied, for ODE numbers of \\num{262144} and higher RKC-GPU demonstrated performance speedups of 59$\\times$ and 10$\\times$ compared to RKC-CPU on one and six CPU cores, respectively.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{h2-co_rkc.pdf}\n\\caption{Performance comparison of (single- and six-core) CPU and GPU integration of the moderately stiff hydrogen\\slash carbon monoxide mechanism using the stabilized explicit RKC method.}\n\\label{F:h2-co-rkc}\n\\end{center}\n\\end{figure}\n\nSimilar to our analysis of divergence for RKCK-GPU, we also studied the effect of randomized initial conditions on the performance of RKC-GPU. In this case, there are now three potential sources of thread divergence: (1) varying numbers of iterations for the nonlinear power method used to estimate the spectral radius, (2) varying numbers of stages due to different spectral radii, and (3) varying numbers of steps due to different time step sizes. Figure~\\ref{F:h2-rkc-random} shows the performance comparison for RKC-GPU between threads with similar and randomized initial conditions. Thread divergence caused by random initial conditions reduced the performance of RKC-GPU by up to a factor of 3.3. As expected, RKC-GPU exhibited a greater performance loss than RKCK-GPU, where the major source of thread divergence was varying numbers of time steps.\n\n\\begin{figure}[tbp] \\begin{center}\n\\includegraphics[width=0.8\\linewidth]{h2-co_rkc-random.pdf}\n\\caption{Performance comparison of RKC-GPU integration of the hydrogen\\slash carbon monoxide mechanism where neighboring threads have similar and randomized initial conditions.}\n\\label{F:h2-rkc-random}\n\\end{center} \\end{figure}\n\nThe greater divergence of RKC-GPU is also demonstrated in Fig.~\\ref{F:h2-rkc-diverge}, where the number of occurrences of \\emph{D} is counted for \\num{65536} ODEs (\\num{2048} warps). In this case, even similar initial conditions caused some divergence. This was likely the reason for the reduced performance speedup of RKC-GPU compared to that of RKCK-GPU relative to their respective CPU versions. With randomly distributed initial conditions, \\emph{D} is distributed normally around $\\sim$0.55. Compared to the distribution of \\emph{D} for RKCK, RKC shows a higher incidence of low values, likely the cause behind the greater reduction in performance for the randomized initial condition case with RKC.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{h2-co-rkc-divergence.pdf}\n\\caption{Warp thread divergence comparison of RKC-GPU for hydrogen\\slash carbon monoxide kinetics for similar and random initial conditions, where the number of occurrences of the divergence measure \\emph{D} is plotted for \\num{2048} thread warps.}\n\\label{F:h2-rkc-diverge}\n\\end{center}\n\\end{figure}\n\n\\subsection{Methane kinetics}\n\\label{S:methane}\n\nNext, we analyzed the performance of the CPU and GPU versions of RKC in another case with moderate stiffness, using the GRI-Mech 3.0~\\cite{Smith:2010} mechanism for methane oxidation, which consists of 53 species and 325 reaction steps (converted to 634 irreversible reactions). As in the previous section, we chose a global time step size of \\SI{1e-6}{\\second} and reported the average computational time for ten steps. Adjacent ODEs used similar initial conditions. In this case, RKC-GPU performed nearly eight times faster than RKCK-GPU in terms of computational time, suggesting more significant stiffness compared to Section~\\ref{S:h2-co}. Consequently, we also compared the performance of RKC-GPU with the CPU-based implicit solver VODE. In both RKC and VODE, we selected a relative tolerance of \\num{1e-6} and an absolute tolerance of \\num{1e-10}.\n\nFigure~\\ref{F:ch4-rkc} shows the performance comparison between the single- and six-core RKC-CPU and RKC-GPU for problem sizes ranging from 64 to \\num{2097152}. As before, RKC-GPU performed better at larger problem sizes. Similar to the  hydrogen\\slash carbon monoxide mechanism results, RKC-GPU outperformed RKC-CPU using a single CPU core for all ODE numbers considered here and faster than the six-core version for problem sizes of 512 and larger. At larger problem sizes, RKC-GPU compared slightly more favorably than in the previous section, performing up to 69$\\times$ and 13$\\times$ faster than RKC-CPU on one and six CPU cores, respectively. The jump in computational time between 512 and \\num{1024} ODEs corresponded to the addition of initial conditions with greater stiffness.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{ch4-rkc.pdf}\n\\caption{Performance comparison of (single- and six-core) CPU and GPU integration of the moderately stiff methane mechanism using the stabilized explicit RKC method.}\n\\label{F:ch4-rkc}\n\\end{center}\n\\end{figure}\n\nSince this problem exhibited greater stiffness compared to the previous case, we also studied the performance of VODE on the CPU compared against RKC-GPU. Figure~\\ref{F:ch4-rkc-vode} shows the computational time for VODE on six CPU cores and RKC-GPU for problem sizes ranging from 64 to \\num{1048576}. At all numbers of ODEs considered, RKC-GPU performed faster than VODE, demonstrating a speedup of up to 57$\\times$. Though it is not shown here, we also note that RKC-CPU outperformed VODE---both on six CPU cores---by a factor of three to six.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{ch4-rkc-vode.pdf}\n\\caption{Performance comparison between VODE running on six CPU cores and RKC-GPU with the moderately stiff methane mechanism over a wide range of problem sizes (i.e., number of ODEs).}\n\\label{F:ch4-rkc-vode}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{F:ch4-rkc-random} shows the performance of RKC-GPU for methane kinetics when the initial conditions in neighboring threads were similar and randomized. The behavior demonstrated here was similar to that in the previous section, with increasing disparity in performance for larger numbers of ODEs. In this case, with randomly selected initial conditions RKC-GPU performed up to nearly four times slower than when threads contained similar initial conditions. This drop in performance can also be seen in the distribution of \\emph{D} for \\num{65536} ODEs (\\num{2048} warps) in Fig.~\\ref{F:ch4-diverge}. The divergence, as measured by \\emph{D}, showed similar behavior to that of hydrogen\\slash carbon monoxide in Fig.~\\ref{F:h2-rkc-diverge}. Here, for similar conditions, \\emph{D} is clustered near one, and for randomized initial conditions normally distributed around 0.45---slightly lower than with the hydrogen\\slash carbon monoxide mechanism. This likely explains the slightly greater drop in performance for randomly ordered initial conditions, compared to the previous section.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{ch4-rkc-random.pdf}\n\\caption{Performance comparison of RKC-GPU integration of the methane mechanism where neighboring threads have similar and randomized initial conditions.}\n\\label{F:ch4-rkc-random}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{ch4-divergence.pdf}\n\\caption{Warp thread divergence comparison of RKC-GPU for methane kinetics for similar and random initial conditions, where the number of occurrences of the divergence measure \\emph{D} is plotted for \\num{2048} thread warps.}\n\\label{F:ch4-diverge}\n\\end{center}\n\\end{figure}\n\n\\subsection{Ethylene kinetics}\n\\label{S:ethylene}\n\nFinally, we studied the performance of RKC-GPU in a case where stiffness is more severe: ethylene oxidation using the USC Mech version II mechanism~\\cite{Wang:2007}, which consists of 111 species and 784 reactions (converted to \\num{1566} irreversible reactions). In both RKC and VODE, we selected a relative tolerance of \\num{1e-6} and an absolute tolerance of \\num{1e-10}. Adjacent ODEs used similar initial conditions.\n\nFigure~\\ref{F:c2h4-rkc-vode} shows the computational time for RKC-CPU, RKC-GPU, and VODE for numbers of ODEs ranging from 64 to \\num{131072}. As before, we chose a global time step size of \\SI{1e-6}{\\second} and reported the average computational time for ten steps. Both CPU-based algorithms were executed on six CPU cores. Here, we omit the single-core RKC-CPU results; the performance ratio between the single- and six-core version showed similar scaling (4--6$\\times$) to that shown in the previous sections. At problem sizes smaller than 256 ODEs, both RKC-CPU and VODE performed faster than RKC-GPU. RKC-GPU and VODE showed nearly indistinguishable performance for \\num{1024} and \\num{2048} ODEs. For numbers of ODEs greater than \\num{8192}, RKC-GPU performed 12--18$\\times$ faster than RKC-CPU and 2.5--4.5$\\times$ faster than VODE.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{c2h4-rkc.pdf}\n\\caption{Average computational time required for a \\SI{1e-6}{\\second} global time step using RKC-CPU, RKC-GPU, and VODE with the ethylene oxidation mechanism, for a wide range of ODE numbers. Both RKC-CPU and VODE were performed on six CPU cores.}\n\\label{F:c2h4-rkc-vode}\n\\end{center}\n\\end{figure}\n\nNext, we increased the global time step size to \\SI{1e-4}{\\second} to further increase the severity of stiffness. Figure~\\ref{F:c2h4-1e-4} shows the performance of RKC-GPU and six-core VODE for numbers of ODEs ranging from 64 to \\num{65536}. For all problem sizes here, RKC-GPU is slower than VODE. At best, RKC-GPU ran 2.5$\\times$ slower than VODE for \\num{16384} ODEs.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{c2h4-rkc-vode-1e-4.pdf}\n\\caption{Performance comparison between VODE running on six CPU cores and RKC-GPU with the ethylene mechanism over a wide range of problem sizes for a global time step size of \\SI{1e-4}{\\second}.}\n\\label{F:c2h4-1e-4}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Discussion}\n\\label{S:discussion}\n\nThe results shown above demonstrate that GPUs may be used to significantly reduce the cost of incorporating detailed chemistry in reactive-flow simulations. When stiffness is low due to the chemistry or small time step sizes used---such as those used in high-speed flow or DNS studies---explicit algorithms such as RKCK offer significantly higher performance on CPUs than implicit methods. Implementing RKCK on the GPU compounds this performance benefit over an order of magnitude, performing up to factor of 25 faster than the equivalent six-core CPU version in this study. As such, GPU-accelerated explicit methods are an attractive choice for nonstiff problems.\n\nHowever, many chemical kinetics problems exhibit stiffness and therefore implicit algorithms are typically chosen to integrate the chemistry terms. As shown here, though, stabilized explicit methods such as RKC offer another option when stiffness is moderate. Demonstrated with methane kinetics, the RKC-GPU solver performed up to nearly 60$\\times$ faster than the implicit VODE solver on six CPU cores. In fact, even the CPU implementation of RKC outperformed VODE. Based on these results, we suggest that a GPU-accelerated stabilized explicit method like RKC should be used in place of the standard implicit solvers in reactive-flow simulations---when stiffness is moderate. Typically, high-fidelity simulations use mechanisms with less than around 100 species---the size of those used in this study---so applying the GPU-based RKC integrator could significantly reduce the cost of chemistry in such studies. In addition, it could allow the use of larger, more complex mechanisms.\n\nIn the presence of more severe stiffness, as with the ethylene oxidation mechanism here, the GPU-accelerated RKC still showed significant speedup over the CPU version. Unfortunately, the comparison between VODE (on six CPU cores) and RKC-GPU became less favorable, with the speedup dropping to a factor of 4.5 for a global time step size of \\num{1e-6}. The performance of RKC-GPU compared to VODE dropped further when the global time step size was increased---due to the greater stiffness this induced. For example, a time step size of \\SI{1e-4}{\\second} may be used for engine simulations; in this case, RKC-GPU performed at best 2.5$\\times$ slower than VODE (on six CPU cores). As Stone et al.~\\cite{Stone:2013jf} demonstrated, porting VODE to the GPU may not yield much benefit over a multi-core CPU implementation. Therefore, for problems with severe stiffness, an integration algorithm appropriate for GPU acceleration needs to be developed.\n\nIn all cases shown here, the speedup of the GPU-based algorithm compared to the equivalent CPU-based algorithm improved with increasing numbers of ODEs; at the smallest numbers, the six-core CPU-based algorithms performed better. This trend agrees with that observed in previous efforts using various integration algorithms~\\cite{Niemeyer:2011uw,Shi:2012cl,Stone:2013jf}. At smaller problem sizes, the overhead due to memory transfer between the GPU and controlling CPU dominates, while at larger problem sizes the time required for actual computation comprises most of the total wall-clock time.\n\nFurther, we observed a general trend of increasing RKC-GPU to RKC-CPU speedup with increasing mechanism size. For mechanisms with 13, 53, and 111 species, RKC-GPU performed up to 10$\\times$, 13$\\times$, and 18$\\times$ faster, respectively, than the six-core RKC-CPU for a global time step size of \\SI{1e-6}{\\second}.\n\nWe also found that the the performance of both RKCK-GPU and RKC-GPU dropped by factors of up to 2.5 and 4.0, respectively, when adjacent threads (corresponding to spatial locations) used randomly shuffled---rather than similar---initial conditions. This was due to divergence from threads following different instruction pathways, since different conditions will result in varying inner time step sizes. RKC-GPU exhibited greater thread divergence due to additional sources from the spectral radius estimation and varying number of stages, and correspondingly with randomized initial conditions this method displayed a larger reduction in performance compared to RKCK-GPU relative to the respective CPU versions. In general, we consider the case of threads with similar initial conditions more realistic, since in reactive-flow simulations---particularly with structured grids---neighboring volumes\\slash grid points will contain similar thermochemical states. However, a reduction in performance due to divergence could result in some cases, such as with unstructured grids, where neighboring locations may not be stored consecutively in memory.\n\n\\section{Conclusions}\n\\label{S:conclusions}\n\nIn the present work we demonstrated new strategies for accelerating reactive-flow simulations using graphics processing units (GPUs). Most approaches for such simulations rely on the operator-splitting technique, which separates the chemistry and transport terms in each time step for separate evaluation. This results in a large number of ordinary differential equations (ODEs) governing the evolution of the species mass fractions for each discretized spatial location (i.e., grid point or volume) that need to be solved each time step. Here, we demonstrated that explicit algorithms used to integrate the chemistry ODEs in parallel on GPUs can perform significantly faster than equivalent CPU versions. We employed the explicit fifth-order Runge--Kutta--Cash--Karp (RKCK) and second-order Runge--Kutta--Chebyshev (RKC) methods for nonstiff and moderately stiff kinetics, respectively.\n\nWe studied the performance of the RKCK algorithm using a nonstiff hydrogen mechanism with with 9 species and 38 irreversible reactions~\\cite{Yetter:1991}, and the performance of the RKC algorithm using three mechanisms with increasing sizes and levels of stiffness: (1) hydrogen\\slash carbon monoxide with 13 species and 54 irreversible reactions~\\cite{Burke:2011fh}, (2) methane with 53 species and 634 irreversible reactions~\\cite{Smith:2010}, and (3) ethylene with 111 species and \\num{1566} irreversible reactions~\\cite{Wang:2007}. By comparing the performance of the CPU and GPU versions of RKCK and RKC, as well as the CPU-based implicit VODE solver, over a wide range of problem sizes (i.e., number of chemistry ODEs), we drew the following conclusions:\n\\begin{itemize}\n\\item For cases without stiffness, the GPU-based RKCK outperformed the six-core CPU version by a factor of 25 at best.\n\\item For cases with moderate levels of stiffness, the GPU-based RKC performed faster than the six-core RKC-CPU by, at best, factors of 10 with a hydrogen\\slash carbon monoxide mechanism, 13 with a methane mechanism, and 18 with an ethylene mechanism.\n\\item In the presence of moderate stiffness in the methane mechanism, RKC-GPU outperformed the implicit VODE solver---on six CPU cores---by a maximum factor of 57.\n\\item For cases with moderate stiffness, even the CPU-based RKC outperformed VODE.\n\\item With increased stiffness in the case of the ethylene mechanism, RKC-GPU performed only 4.5$\\times$ faster at best than VODE on six CPU cores.\n\\item When stiffness became more severe due to a larger time step size used with the ethylene mechanism, RKC-GPU became less efficient than six-core VODE, performing at best 2.5$\\times$ slower.\n\\item At small problem sizes (less than 512 ODEs), the six-core RKC-CPU was more efficient, but RKC-GPU outperformed the serial (single-core) CPU version in all cases considered here.\n\\item Due to thread divergence, the performance of the GPU solvers degraded with randomized (and therefore different) initial conditions in adjacent memory locations, by up to a factor of four slower compared to using similar initial conditions.\n\\end{itemize}\n\nFinally, we note that while we used a second-order accurate RKC algorithm here, higher order RKC methods exist. For example, Abdulle~\\cite{Abdulle:2002ws} developed a fourth-order RKC with similar traits to the current method. Our future work will involve implementing these higher order algorithms where such accuracy is needed, as well as developing a GPU-based stiff integrator that can handle severe stiffness.\n\n\\section*{Acknowledgements}\n\nThis work was supported by the National Science Foundation under grant number 0932559, the US Department of Defense through the National Defense Science and Engineering Graduate Fellowship program, the National Science Foundation Graduate Research Fellowship under grant number DGE-0951783, and the Combustion Energy Frontier Research Center---an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-SC0001198.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-0.05in}\nIn recent years, we have made significant advances in standard recognition tasks such as image classification~\\cite{he2016deep}, detection~\\cite{ren2015faster} or segmentation~\\cite{chen2016attention}. Most of these gains are a result of using feed-forward end-to-end learned ConvNet models. Unlike humans where visual reasoning about the space and semantics is crucial~\\cite{biederman1982scene}, our current visual systems lack any context reasoning beyond convolutions with large receptive fields. Therefore, a critical question is how do we incorporate both \\emph{spatial} and \\emph{semantic} reasoning as we build next-generation vision systems.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{teaser}\n\\caption{{\\small Current recognition systems lack the reasoning power beyond convolutions with large receptive fields, whereas humans can explore the rich space of spatial and semantic relationships for reasoning: \\eg inferring the fourth ``window'' even with occlusion, or the ``person'' who drives the ``car''. To close this gap, we present a generic framework that also uses relationships to iteratively reason and build up estimates.}\\label{fig:teaser}}\n\\vspace{-0.2in}\n\\end{figure}\n\nOur goal is to build a system that can not only extract and utilize hierarchy of convolutional features, but also improve its estimates via spatial and semantic relationships. But what are spatial and semantic relationships and how can they be used to improve recognition? Take a look at Fig.~\\ref{fig:teaser}. An example of spatial reasoning (top-left) would be: if three regions out of four in a line are ``window'', then the fourth is also likely to be ``window''. An example of semantic reasoning (bottom-right) would be to recognize ``school bus'' even if we have seen few or no examples of it -- just given examples of ``bus'' and knowing their connections. Finally, an example of spatial-semantic reasoning could be: recognition of a ``car'' on road should help in recognizing the ``person'' inside ``driving'' the ``car''.\n\nA key recipe to reasoning with relationships is to \\emph{iteratively} build up estimates. \nRecently, there have been efforts to incorporate such reasoning via top-down modules~\\cite{ronneberger2015u,wei2016convolutional} or using explicit memories~\\cite{xiong2016dynamic,marino2016more}. In the case of top-down modules, high-level features which have class-based information can be used in conjunction with low-level features to improve recognition performance. An alternative architecture is to use explicit memory. For example, Chen \\& Gupta~\\cite{chen2017spatial} performs sequential object detection, where a \\emph{spatial memory} is used to store previously detected objects, leveraging the power of ConvNets for extracting dense context patterns beneficial for follow-up detections. \n\nHowever, there are two problems with these approaches: a) both approaches use stack of convolutions to perform \\emph{local} pixel-level reasoning~\\cite{divvala2009empirical}, which can lack a \\emph{global} reasoning power that also allows regions farther away to directly communicate information; b) more importantly, both approaches assume enough examples of relationships in the training data -- so that the model can learn them from scratch, but as the relationships grow exponentially with increasing number of classes, there is not always enough data. A lot of semantic reasoning requires learning from few or no examples~\\cite{fei2006one}. Therefore, we need ways to exploit additional structured information for visual reasoning.\n\nIn this paper, we put forward a generic framework for both spatial and semantic reasoning. Different from current approaches that are just relying on convolutions, our framework can also learn from structured information in the form of knowledge bases~\\cite{chen2013neil,zhu2015building} for visual recognition. The core of our algorithm consists of two modules: the local module, based on spatial memory~\\cite{chen2017spatial}, performs pixel-level reasoning using ConvNets. We make major improvements on efficiency by parallel memory updates. Additionally, we introduce a global module for reasoning beyond local regions. In the global module, reasoning is based on a \\emph{graph} structure. It has three components: a) a knowledge graph where we represent classes as nodes and build edges to encode different types of semantic relationships; b) a region graph of the current image where regions in the image are nodes and spatial relationships between these regions are edges; c) an assignment graph that assigns regions to classes. Taking advantage of such a structure, we develop a reasoning module specifically designed to pass information on this graph. Both the local module and the global module roll-out iteratively and cross-feed predictions to each other in order to refine estimates. Note that, local and global reasoning are not isolated: a good image understanding is usually a compromise between background knowledge learned \\emph{a priori} and image-specific observations. Therefore, our full pipeline joins force of the two modules by an attention~\\cite{chen2016attention} mechanism allowing the model to rely on the most relevant features when making the final predictions.\n\nWe show strong performance over plain ConvNets using our framework. For example, we can achieve $8.4\\%$ absolute improvements on ADE~\\cite{zhou2016semantic} measured by per-class average precision, where by simply making the network deeper can only help ${\\sim}1\\%$. \n\n\\vspace{-0.05in}\n\\section{Related Work}\n\\vspace{-0.05in}\n\\noindent{\\bf Visual Knowledge Base.} Whereas past five years in computer vision will probably be remembered as the successful resurgence of neural networks, acquiring visual knowledge at a large scale -- the simplest form being labeled instances of objects~\\cite{russakovsky2015imagenet,lin2014microsoft}, scenes~\\cite{zhou2016semantic}, relationships~\\cite{krishna2016visual} \\etc -- deserves at least half the credit, since ConvNets hinge on large datasets~\\cite{chensun2017}. Apart from providing labels using crowd-sourcing, attempts have also been made to accumulate structured knowledge (\\eg relationships~\\cite{chen2013neil}, $n$-grams~\\cite{divvala2014learning}) automatically from the web. However, these works fixate on building knowledge bases rather than using knowledge for reasoning. Our framework, while being more general, is along the line of research that applies visual knowledge base to end tasks, such as affordances~\\cite{zhu2015building}, image classification~\\cite{marino2016more}, or question answering~\\cite{wu2016ask}.\n\n\\noindent{\\bf Context Modeling.} Modeling context, or the interplay between scenes, objects and parts is one of the central problems in computer vision. While various previous work (\\eg scene-level reasoning~\\cite{torralba2003context}, attributes~\\cite{farhadi2009describing,parikh2011relative}, structured prediction~\\cite{krahenbuhl2011efficient,desai2011discriminative,tu2010auto}, relationship graph~\\cite{johnson2015image,lu2016visual,xu2017scene}) has approached this problem from different angles, the breakthrough comes from the idea of feature learning with ConvNets~\\cite{he2016deep}. On the surface, such models hardly use any explicit context module for reasoning, but it is generally accepted that ConvNets are extremely effective in aggregating local pixel-to-level context through its ever-growing receptive fields~\\cite{zeiler2014visualizing}. Even the most recent developments such as top-down module~\\cite{xie2016top,lin2016feature,tdm_cvpr17}, pairwise module~\\cite{santoro2017simple}, iterative feedback~\\cite{wei2016convolutional,newell2016stacked,carreira2016human}, attention~\\cite{yang2016stacked}, and memory~\\cite{xiong2016dynamic,chen2017spatial} are motivated to leverage such power and depend on variants of convolutions for reasoning. Our work takes an important next step beyond those approaches in that it also incorporates learning from structured visual knowledge bases directly to reason with spatial and semantic relationships.\n\n\\noindent{\\bf Relational Reasoning.} The earliest form of reasoning in artificial intelligence dates back to symbolic approaches~\\cite{newell1980physical}, where relations between abstract symbols are defined by the language of mathematics and logic, and reasoning takes place by deduction, abduction~\\cite{hobbs1988interpretation}, \\etc. However, symbols need to be grounded~\\cite{harnad1990symbol} before such systems are practically useful. Modern approaches, such as path ranking algorithm~\\cite{lao2011random}, rely on statistical learning to extract useful patterns to perform relational reasoning on structured knowledge bases. As an active research area, there are recent works also applying neural networks to the graph structured data~\\cite{scarselli2009graph,henaff2015deep,li2015gated,kipf2016semi,niepert2016learning,das2016chains,marino2016more}, or attempting to regularize the output of networks with relationships~\\cite{deng2014large} and knowledge bases~\\cite{hu2016deep}. However, we believe for visual data, reasoning should be both local and global: discarding the two-dimensional image structure is neither efficient nor effective for tasks that involve regions.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{local-global}\n\\caption{{\\small Overview of our reasoning framework. Besides a plain ConvNet that gives predictions, the framework has two modules to perform reasoning: a local one (Sec.~\\ref{conv}) that uses spatial memory $\\mathcal{S}_i$, and reasons with another ConvNet $\\mathcal{C}$; and a global one (Sec.~\\ref{beyond}) that treats regions and classes as nodes in a graph and reasons by passing information among them. Both modules receive combined high-level and mid-level features, and roll-out iteratively (Sec.~\\ref{iter}) while cross-feeding beliefs. The final prediction $f$ is produced by combining all the predictions $f_i$ with attentions $a_i$ (Sec.~\\ref{attend}).}\\label{fig:overview}}\n\\vspace{-0.2in}\n\\end{figure*}\n\n\\vspace{-0.05in}\n\\section{Reasoning Framework}\n\\vspace{-0.05in}\nIn this section we build up our reasoning framework. Besides plain predictions $p_0$ from a ConvNet, it consists of two core modules that reason to predict. The first one, local module, uses a spatial memory to store previous beliefs with parallel updates, and still falls within the regime of convolution based reasoning (Sec.~\\ref{conv}). Beyond convolutions, we present our key contribution -- a global module that reasons directly between regions and classes represented as nodes in a graph (Sec.~\\ref{beyond}). Both modules build up estimation iteratively (Sec.~\\ref{iter}), with beliefs cross-fed to each other. Finally taking advantage of both local and global, we combine predictions from all iterations with an attention mechanism (Sec.~\\ref{attend}) and train the model with sample re-weighting (Sec.~\\ref{train}) that focuses on hard examples (See Fig.~\\ref{fig:overview}).\n\n\\subsection{Reasoning with Convolutions\\label{conv}}\nOur first building block, the local module, is inspired from~\\cite{chen2017spatial}. At a high level, the idea is to use a spatial memory $\\mathcal{S}$ to store previously detected objects at the very location they have been found. $\\mathcal{S}$ is a tensor with three dimensions. The first two, height $H$ and width $W$, correspond to the reduced size ($1\/16$) of the image. The third one, depth $D$ (${=}512$), makes each cell of the memory $c$ a vector that stores potentially useful information at that location.\n\n$\\mathcal{S}$ is updated with both high-level and mid-level features. For high-level, information regarding the estimated class label is stored. However, just knowing the class may not be ideal -- more details about the shape, pose \\etc can also be useful for other objects. For example, it would be nice to know the pose of a ``person'' playing tennis to recognize the ``racket''. In this paper, we use the logits $f$ before soft-max activation, in conjunction with feature maps from a bottom convolutional layer $h$ to feed-in the memory. \n\nGiven an image region $r$ to update, we first crop the corresponding features from the bottom layer, and resize it to a predefined square ($7{\\times}7$) with bi-linear interpolation as $h$. Since high-level feature $f$ is a vector covering the entire region, we append it to all the $49$ locations. Two $1{\\times}1$ convolutions are used to fuse the information~\\cite{chen2017spatial} and form our input features $f_r$ for $r$. The same region in the memory $\\mathcal{S}$ is also cropped and resized to $7{\\times}7$, denoted as $s_r$. After this alignment, we use a convolutional gated recurrent unit (GRU)~\\cite{chung2014empirical} to write the memory:\n\\begin{equation}\\label{gru}\n    s'_r = u \\circ s_r + (1 - u) \\circ \\sigma(W_f f_r + W_s(z \\circ s_r) + b),\n\\end{equation}\nwhere $s'_r$ is the updated memory for $r$, $u$ is update gate, $z$ is reset gate, $W_f$, $W_s$ and $b$ are convolutional weights and bias, and $\\circ$ is entry-wise product. $\\sigma(\\cdot)$ is an activation function. After the update, $s'_r$ is placed back to $\\mathcal{S}$ with another crop and resize operation\\footnote{Different from previous work~\\cite{chen2017spatial} that introduces an inverse operation to put the region back, we note that crop and resize \\emph{itself} with proper extrapolation can simply meet this requirement.}.\n\n\\noindent {\\bf Parallel Updates.} Previous work~\\cite{chen2017spatial} made sequential updates to memory. However, sequential inference is inefficient and GPU-intensive -- limiting it to only give ten outputs per image~\\cite{chen2017spatial}. In this paper we propose to update the regions in parallel as an approximation. In overlapping cases, a cell can be covered multiple times from different regions. When placing the regions back to $\\mathcal{S}$, we also calculate a weight matrix $\\Gamma$ where each entry $\\gamma_{r,c}{\\in}[0,1]$ keeps track of how much a region $r$ has contributed to a memory cell $c$: $1$ meaning the cell is fully covered by the region, $0$ meaning not covered. The final values of the updated cell is the weighted average of all regions. \n\nThe actual reasoning module, a ConvNet $\\mathcal{C}$ of three $3{\\times}3$ convolutions and two $4096$-D fully-connected layers, takes $\\mathcal{S}$ as the input, and builds connections within the local window of its receptive fields to perform prediction. Since the two-dimensional image structure and the location information is preserved in $\\mathcal{S}$, such an architecture is particularly useful for relationships with spatial reasoning.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\linewidth]{paths}\n\\caption{{\\small Illustration of directly passing information on a graph with multiple edge types. Here four nodes are linked with two edge types. Each node is represented as an input feature vector $m_i$ (aggregated as $M$). Weight matrix $W_j$ is learned for edge type $j$ to transform inputs. Then adjacency matrix $A_j$ is applied to pass information to linked nodes. Finally, output $G$ is generated by accumulating all edge types and apply activation function.}\\label{fig:paths}}\n\\vspace{-0.2in}\n\\end{figure}\n\n\\subsection{Beyond Convolutions\\label{beyond}}\nOur second module goes beyond local regions and convolutions for global reasoning. Here the meaning of \\emph{global} is two-fold. First is \\emph{spatial}, that is, we want to let the regions farther away to directly communicate information with each other, not confined by the receptive fields of the reasoning module $\\mathcal{C}$. Second is \\emph{semantic}, meaning we want to take advantage of visual knowledge bases, which can provide relationships between classes that are globally true (\\ie commonsense) across images. To achieve both types of reasoning, we build a graph $\\mathcal{G}=(\\mathcal{N}, \\mathcal{E})$, where $\\mathcal{N}$ and $\\mathcal{E}$ denote node sets and edge sets, respectively. Two types of nodes are defined in $\\mathcal{N}$: region nodes $\\mathcal{N}_r$ for $R$ regions, and class nodes $\\mathcal{N}_c$ for $C$ classes.  \n\nAs for $\\mathcal{E}$, three groups of edges are defined between nodes. First for $\\mathcal{N}_r$, a spatial graph is used to encode spatial relationships between regions ($\\mathcal{E}_{r{\\rightarrow}r}$). Multiple types of edges are designed to characterize the relative locations. We begin with basic relationships such as ``left\/right'', ``top\/bottom'' and we define edge weights by measuring the pixel-level distances between the two. Note that we do not use the raw distance $x$ directly, but instead normalizing it to $[0,1]$ with a kernel $\\kappa(x){=}\\exp(-x\/\\Delta)$ (where $\\Delta{=}50$ is the bandwidth), with the intuition that closer regions are more correlated. The edge weights are then used directly in the adjacency matrix of the graph. Additionally, we include edges to encode the coverage patterns (\\eg intersection over union, IoU~\\cite{everingham2010pascal}), which can be especially helpful when two regions overlap. \n\nA second group of edges lie between regions and classes, where the assignment for a region to a class takes place. Such edges shoulder the responsibility of propagating beliefs from region to class ($e_{r{\\rightarrow}c}$) or backwards from class to region ($e_{c{\\rightarrow}r}$). Rather than only linking to the most confident class, we choose full soft-max score $p$ to define the edge weights of connections to all classes. The hope that it can deliver more information and thus is more robust to false assignments. \n\nSemantic relationships from knowledge bases are used to construct the third group of edges between classes ($\\mathcal{E}_{c{\\rightarrow}c}$). Again, multiple types of edges can be included here. Classical examples are ``is-kind-of'' (\\eg between ``cake'' and ``food''), ``is-part-of'' (\\eg between ``wheel'' and ``car''), ``similarity'' (\\eg between ``leopard'' and ``cheetah''), many of which are universally true and are thus regarded as commonsense knowledge for humans. Such commonsense can be either manually listed~\\cite{russakovsky2015imagenet} or automatically collected~\\cite{chen2013neil}. Interestingly, even relationships beyond these (\\eg actions, prepositions) can help recognition~\\cite{marino2016more}. Take ``person ride bike'' as an example, which is apparantly more of an image-specific relationship. However, given less confident predictions of ``person'' and ``bike'', knowing the relationship ``ride'' along with the spatial configurations of the two can also help prune other spurious explanations. To study both cases, we experimented with two knowledge graphs in this paper: one created in-house with mostly commonsense edges, and the other also includes more types of relationships accumulated at a large-scale. For the actual graphs used in our experiments, please see Sec.~\\ref{data} for more details.\n\nNow we are ready to describe the graph-based reasoning module $\\mathcal{R}$. As the input to our graph, we use $M_r{\\in}\\mathbb{R}^{R\\times D}$ to denote the features from all the region nodes $\\mathcal{N}_r$ combined, where $D$ (${=}512$) is the number of feature channels. For each class node $n_c$, we choose off-the-shelf word vectors~\\cite{joulin2016fasttext} as a convenient representation, denoted as $M_c{\\in}\\mathbb{R}^{C\\times D}$. We then extend previous works~\\cite{scarselli2009graph,niepert2016learning} and pass messages directly on $\\mathcal{G}$ (See Fig.~\\ref{fig:paths}). Note that, because our end-goal is to recognize regions better, all the class nodes should only be used as intermediate ``hops'' for better region representations. With this insight, we design two reasoning paths to learn the output features $G_r$: a \\emph{spatial} path on which only region nodes are involved:\n\\begin{equation}\\label{spatial}\n    G^{spatial}_r = \\sum_{e{\\in} \\mathcal{E}_{r{\\rightarrow}r}}{A_e M_r W_e},\n\\end{equation}\nwhere $A_e{\\in}\\mathbb{R}^{r\\times r}$ is the adjacency matrix of edge type $e$, $W_e{\\in}\\mathbb{R}^{d\\times d}$ is weight (bias is ignored for simplicity). The second reasoning path is a \\emph{semantic} one through class nodes:\n\\begin{equation}\\label{semantic}\n    G^{semantic}_c = \\sum_{e{\\in} \\mathcal{E}_{c{\\rightarrow}c}}{A_e \\sigma(A_{e_{r{\\rightarrow}c}} M_r W_{e_{r{\\rightarrow}c}} + M_c W_c) W_e},\n\\end{equation}\nwhere we first map regions to classes through $A_{e_{r{\\rightarrow}c}}$ and $W_{e_{r{\\rightarrow}c}}$, combine the intermediate features with class features $M_c$, and again aggregate features from multiple types of edges between classes.\nFinally, the output for regions $G_r$ are computed by merging these two paths:\n\\begin{equation}\\label{output}\n    G_r = \\sigma(G^{spatial}_r + \\sigma(A_{e_{c{\\rightarrow}r}} G^{semantic}_c W_{e_{c{\\rightarrow}r}})),\n\\end{equation}\nwhich first propagates semantic information back to regions, and then applies non-linear activation (See Fig.~\\ref{fig:ss}).\n\nJust like convolution filters, the above-described paths can also be stacked, where the output $G_r$ can go through another set of graph operations -- allowing the framework to perform joint spatial-semantic reasoning with deeper features. We use three stacks of operations with residual connections~\\cite{he2016deep} in $\\mathcal{R}$, before the output is fed to predict.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.9\\linewidth]{spatial-semantic}\n\\caption{{\\small Two reasoning paths used in our global reasoning module $\\mathcal{R}$. Taking the region and class inputs $M_r$ and $M_c$, the spatial path directly passes information in the region graph with region-to-region edges $\\mathcal{E}_{r{\\rightarrow}r}$, whereas the semantic path first assigns regions to classes with $e_{r{\\rightarrow}c}$, passes the information on to other classes with class-to-class edges $\\mathcal{E}_{c{\\rightarrow}c}$, and then propagates back. Final outputs are combined to generate output region features $G_r$.}\\label{fig:ss}}\n\\vspace{-0.2in}\n\\end{figure}\n\n\\subsection{Iterative Reasoning\\label{iter}}\nA key ingredient of reasoning is to iteratively build up estimates. But how does information pass from one iteration to another? Our answer is \\emph{explicit} memory, which stores all the history from previous iterations. The local module uses spatial memory $\\mathcal{S}$, and the global module uses another memory $\\mathcal{M}$ but without spatial structures. At iteration $i$, $\\mathcal{S}_i$ is followed by convolutional reasoning module $\\mathcal{C}$ to generate new predictions $f_i^l$ for each region. Similarly, global module also gives new predictions $f_i^g$ from $\\mathcal{R}$. These new predictions as high-level features can then be used to get the updated memories $\\mathcal{S}_{i+1}$ and $\\mathcal{M}_{i+1}$. The new memories will lead to another round of updated $f_{i+1}$s and the iteration goes on. \n\nWhile one can do local and global reasoning in isolation, both the modules work best in conjunction. Therefore, for our full pipeline we want to join force of both modules when generating the predictions. To this end, we introduce \\emph{cross-feed} connections. After reasoning, both the local and global features are then concatenated together to update the memories $\\mathcal{S}_{i+1}$ and $\\mathcal{M}_{i+1}$ using GRU. In this way, spatial memory can benefit from global knowledge of spatial and semantic relationships, and graph can get a better sense of the local region layouts. \n\n\\subsection{Attention\\label{attend}}\nInspired from the recent work on attention~\\cite{chen2016attention}, we make another modification at the model output. Specifically, instead of only generating scores $f$, the model also has to produce an ``attention'' value $a$ that denotes the relative confidence of the current prediction compared to the ones from other iterations or modules. Then the fused output is a weighted version of all predictions using attentions. Mathematically, if the model roll-outs $I$ times, and outputs $N{=}2I{+}1$ (including $I$ local, $I$ global and $1$ from plain ConvNet) predictions $f_n$, using attentions $a_n$, the final output $f$ is calculated as:\n\\begin{equation}\\label{att}\n    f = \\sum_{n}{w_n f_n}, \\quad\\mathrm{where}\\quad w_n=\\frac{\\exp(-a_n)}{\\sum_{n'}{\\exp(-a_{n'})}}.\n\\end{equation}\nNote again that here $f_n$ is the logits before soft-max, which is then activated to produce $p_n$. The introduction of attention allows the model to intelligently choose feasible predictions from different modules and iterations.\n\n\\subsection{Training\\label{train}}\nFinally, the overall framework is trained end-to-end, with a total loss function consists of: a) plain ConvNet loss $\\mathcal{L}_{0}$; b) local module loss $\\mathcal{L}^l_{i}$; c) global module loss $\\mathcal{L}^g_{i}$; and d) the final prediction loss with attentions $\\mathcal{L}_f$.\n\nSince we want our reasoning modules to focus more on the harder examples, we propose to simply \\emph{re-weight} the examples in the loss, based on predictions from previous iterations. Formally, for region $r$ at iteration $i{\\ge}1$, the cross-entropy loss for both modules is computed as: \n\\begin{equation}\\label{reweight}\n    \\mathcal{L}_{i}(r) = \\frac{\\max(1. - p_{i-1}(r), \\beta)}{\\sum_{r'}\\max(1. - p_{i-1}(r'), \\beta)}\\log(p_{i}(r)),\n\\end{equation}\nwhere $p_{i}(r)$ is the soft-max output of the ground-truth class, and $\\beta{\\in}[0,1]$ controls the entropy of the weight distribution: when $\\beta{=}1$, it is uniform distribution; and when $\\beta{=}0$, entropy is minimized. In our experiments, $\\beta$ is set to $0.5$. $p_{i-1}(r)$ is used as features without back-propagation. For both local and global, $p_{0}(r)$ is the output from the plain ConvNet. \n\n\\vspace{-0.05in}\n\\section{Experiments}\n\\vspace{-0.05in}\nIn this section we evaluate the effectiveness of our framework. We begin with our experimental setups, which includes the datasets to work with (Sec.~\\ref{data}), the task to evaluate on (Sec.~\\ref{task}) and details of our implementation (Sec.~\\ref{details}). We discuss our results and analyze them in Sec.~\\ref{results} and Sec.~\\ref{ablative} respectively. \n\n\\subsection{Datasets and Graphs\\label{data}}\nDatasets are biased~\\cite{torralba2011unbiased}. For context reasoning we would naturally like to have scene-focused datasets~\\cite{zhou2016semantic} as opposed to object-focused ones~\\cite{russakovsky2015imagenet}. To showcase the capabilities of our system, we need densely labeled dataset with a large number of classes. Finally, one benefit of using knowledge graph is to transfer across classes, therefore a dataset with \\emph{long-tail} distribution is an ideal test-bed. Satisfying all these constraints, ADE~\\cite{zhou2016semantic} and Visual Genome (VG)~\\cite{krishna2016visual} where regions are densely labeled in open vocabulary are the main picks of our study. \n\nFor ADE, we use the publicly released training set ($20,210$) images for training, and split the validation set ($2,000$ images) into {\\tt val-1k} and {\\tt test-1k} with $1,000$ images each. The original raw names are used due to a more detailed categorization~\\cite{zhou2016semantic}. We filter out classes with less than five instances, which leaves us with $1,484$ classes. With the help of parts annotations in the dataset, a commonsense knowledge graph is created with five types of edges between classes: a) ``is-part-of'' (\\eg ``leg'' and ``chair''); b) ``is-kind-of'' (\\eg ``jacket'' and ``clothes''); c) ``plural-form'' (\\eg ``tree'' and ``trees''); d) ``horizontal-symmetry'' (\\eg ``left-arm'' and ``right-arm''); e) ``similarity'' (\\eg ``handle'' and ``knob''). Notice that the first four types are directed edges, hence we also include their inverted versions. \n\nFor VG, the latest release (v$1.4$) is used. We split the entire set of $108,077$ images into $100$K, $4,077$ and $4$K as {\\tt train}, {\\tt val} and {\\tt test} set. Similar pre-processing is done on VG, except that we use synsets~\\cite{russakovsky2015imagenet} instead of raw names due to less consistent labels from multiple annotators. $3,993$ classes are used. For knowledge graph between classes, we take advantage of the relationship annotations in the set, and select the top $10$ most frequent relationships to automatically construct edges beyond commonsense relationships constructed for ADE. For each type of relationships, the edge weights are normalized so that each row of the adjacency matrix is summed-up to one. While this approach results in a noisier graph, it also allows us to demonstrate that our approach is scalable and robust to noise.\n\nFinally, we also show experiments on COCO~\\cite{lin2014microsoft}. However, since it is detection oriented -- has only $80$ classes picked to be mutually-exclusive, and covers less percentage of labeled pixels, we only report results a) without the knowledge graph and b) without a test split ({\\tt trainval35k}~\\cite{chen2017spatial} for training and {\\tt minival} for evaluation). This setup is for analysis purposes only.\n\n\\begin{table}[t]\n\\centering\n\\renewcommand{\\arraystretch}{1.1}\n\\renewcommand{\\tabcolsep}{1.2mm}\n\\caption{\\label{tab:final}{Main results on ADE {\\tt test-1k} and VG {\\tt test}. AP is average precision, AC is classification accuracy. Superscripts show the improvement $\\nabla$ over the baseline.}}\n\\resizebox{1.0\\linewidth}{!}{\n\\begin{tabular}{@{} C{0.5cm} !{\\vrule} L{2.5cm} !{\\vrule} x{1.2cm} x{1.2cm} !{\\vrule} x{1.2cm} x{1.2cm} @{}}\n\\Xhline{1pt}\n\\multirow{2}{*}{$\\%$} & \\multirow{2}{*}{\\textbf{Method}} & \\multicolumn{2}{c!{\\vrule}}{per-instance} & \\multicolumn{2}{c}{per-class} \\\\\n\\Xcline{3-6}{0.5pt}\n& & AP\\textsuperscript{$\\nabla$} & AC\\textsuperscript{$\\nabla$} & AP\\textsuperscript{$\\nabla$} & AC\\textsuperscript{$\\nabla$} \\\\\n\\Xhline{1pt}\n\\parbox[t]{2.5mm}{\\multirow{7}{*}{\\rotatebox[origin=c]{90}{\\small ADE}}} & Baseline & 67.0 & 67.0 & 40.1 & 33.2 \\\\\n& ~~~~{\\small w\/ ResNet-101} & 68.2 & 68.3 & 40.8 & 34.4 \\\\\n& ~~~~{\\small w\/ $800$-input} & 68.2 & 68.2 & 41.0 & 34.3 \\\\\n& ~~~~{\\small Ensemble} & 68.7 & 68.8 & 42.9 & 35.3 \\\\\n\\Xcline{2-6}{0.5pt}\n& Ours\\textsubscript{-Local} & 71.6\\textsuperscript{+4.6} & 71.7\\textsuperscript{+4.7} & 47.9\\textsuperscript{+7.8} & 38.7\\textsuperscript{+5.7} \\\\\n& Ours\\textsubscript{-Global} & 69.8\\textsuperscript{+2.8} & 69.8\\textsuperscript{+2.8} & 44.5\\textsuperscript{+4.4} & 36.8\\textsuperscript{+3.6} \\\\\n& Ours\\textsubscript{-Final} & {\\bf 72.6}\\textsuperscript{+5.6} & {\\bf 72.6}\\textsuperscript{+5.6} & {\\bf 48.5}\\textsuperscript{+8.4} & {\\bf 39.5}\\textsuperscript{+6.3} \\\\\n\n\\Xhline{0.5pt}\n\\parbox[t]{2.5mm}{\\multirow{7}{*}{\\rotatebox[origin=c]{90}{\\small VG}}} & Baseline & 49.1 & 49.6 & 16.9 & 12.1 \\\\\n& ~~~~{\\small w\/ ResNet-101} & 50.3 & 50.8 & 18.0 & {\\bf 13.0} \\\\\n& ~~~~{\\small w\/ $800$-input} & 49.5 & 50.0 & 17.0 & 12.2 \\\\\n& ~~~~{\\small w\/ Ensemble} & 50.2 & 50.7 & 17.7 & 12.3 \\\\ \n\\Xcline{2-6}{0.5pt}\n& Ours\\textsubscript{-Local} & 51.4\\textsuperscript{+2.3} & 51.9\\textsuperscript{+2.3} & 18.8\\textsuperscript{+1.9} & 12.8\\textsuperscript{+0.7} \\\\\n& Ours\\textsubscript{-Global} & 50.9\\textsuperscript{+1.8} & 51.5\\textsuperscript{+1.9} & 18.3\\textsuperscript{+1.4} & 12.6\\textsuperscript{+0.5} \\\\\n& Ours\\textsubscript{-Final} & {\\bf 51.7}\\textsuperscript{+2.6} & {\\bf 52.2}\\textsuperscript{+2.6} & {\\bf 19.1}\\textsuperscript{+2.2} & 12.9\\textsuperscript{+0.8} \\\\\n\n\\Xhline{1pt}\n\\end{tabular}\n}\n\\vspace{-0.2in}\n\\end{table}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{examples}\n\\caption{{\\small Qualitative examples from ADE {\\tt test-1k} (best if zoomed-in). For regions highlighted in blue, the predictions from baseline and our model are compared. Other regions are also listed to provide the context. For example, the ``right-leg'' is less confused with ``left-leg'' after reasoning (top-left); the ``mouse'' on the ``desk'' is predicted despite low resolution (top-third); and ``detergent-dispenser'' is recognized given the context of ``washing-machine'' (top-right). At bottom-right we show a failure case where context does not help ``remote-control'', probably because it has never appeared on the ``night-table'' before and no semantic relationship is there to help.}\\label{fig:examples}}\n\\vspace{-0.2in}\n\\end{figure*}\n\n\\subsection{Task and Evaluation\\label{task}}\nWe evaluate our system on the task of region classification, where the goal is to assign labels to designated regions denoted by rectangular bounding boxes. For both training and testing, we use provided ground-truth locations. We picked this task for three reasons. The {\\bf first} one is on evaluation. As the number of classes increases in the vocabulary, \\emph{missing} labels are inevitable, which is especially severe for object parts (\\eg ``rim'', ``arm'') and related classes (\\eg ``shoes'' \\vs ``sneakers'') where external knowledge is valuable. If there are missing labels, fair evaluation becomes much more difficult since accuracy becomes impossible to evaluate -- cannot tell if a prediction is wrong, or the label itself is missing. Interestingly, such an issue also happens to other research areas (\\eg recommendation systems~\\cite{sarwar2001item} and link prediction~\\cite{liben2007link}). Borrowing ideas from them, a practical solution is to evaluate \\emph{only} on what we already know -- in our case ground-truth regions. {\\bf Second}, although region classification is a simplified version of object detection and semantic segmentation, it maintains a richer set of labels, especially including ``stuff'' classes like ``road'', ``sky'', and object instances. Modeling ``stuff-object'' and instance-level relationships is a crucial capability which would be missed in a pure detection\/segmentation setting. {\\bf Finally} as our experiment will show (Sec.~\\ref{ablative}), while object detectors can be used off-the-shelf, the additional manually defined parameters and components (\\eg overlapping threshold for a region to be positive\/negative, predefined scale\/aspect ratio sets of anchors~\\cite{ren2015faster}) in its pipeline pose limitations on how much context can benefit. For example, after non-maximal suppression (NMS), highly overlapping objects (\\eg ``window'' and ``shutter'') will be suppressed, and ironically this is exactly where context reasoning could have helped. On the other hand, by feeding fixed regions directly for end-to-end learning, we can at least factorize the \\emph{recognition} error from the \\emph{localization} one~\\cite{hoiem2012diagnosing}, and get a clean focus on how context can help discriminating confusing classes.\n\nSince ADE is a segmentation dataset, we convert segmentation masks to bounding boxes. For object classes (\\eg ``person''), each instance is created a separate box. Part (\\eg ``head'') and part-of-part (\\eg ``nose'') are also included. For VG and COCO, boxes are directly used.\n\nFor evaluation, we use classification accuracy (AC) and average precision (AP)~\\cite{everingham2010pascal}. Note that since all the regions are fixed with known labels, there is no need to set a region overlap threshold for AP. Results can be aggregated in two ways: the first way (``per-class'') computes metrics separately for each class in the set, and take the mean; since the final scores are all taken from a calibrated soft-max output, a second way (``per-instance'') that computes metrics simultaneously for all classes. Intuitively, ``per-class'' assigns more weights to instances from rare classes.\n\n\\subsection{Implementation Details\\label{details}}\nA simplified version of \\emph{tf-faster-rcnn}\\footnote{\\url{https:\/\/github.com\/endernewton\/tf-faster-rcnn}} is used to implement our baseline for region classification, with region proposal branch and bounding box regression components removed. Unless otherwise noted, ResNet-50~\\cite{he2016deep} pre-trained on ImageNet~\\cite{russakovsky2015imagenet} is used as our backbone image classifier, and images are enlarged to shorter size $600$ pixels during both training and testing. Specifically, full-image shared convolutional feature maps are computed till the last \\emph{conv4} layer. Then the ground-truth boxes are used as regions-of-interest to compute region-specific features (crop and resize to $7{\\times}7$ without max-pool). All layers of \\emph{conv5} and up are then adopted to obtain the final feature for the baseline prediction $p_0$. Batch normalization parameters are fixed.\n\nFor the local module, we use the last \\emph{conv4} layer as our mid-level features to feed the spatial memory $\\mathcal{S}$. For the global module, mid-level features are the final \\emph{conv5} ($2048$-D) layer after avg-pool. Both features are fused with the logits before soft-max $f$, and then fed into the memory cells. Word vectors from fastText~\\cite{joulin2016fasttext} are used to represent each class, which extracts sub-word information and generalizes well to out-of-vocabulary words. ReLU is selected as the activation function. We roll-out the reasoning modules $3$ times and concurrently update all regions at each iteration, as more iterations do not offer more help.\n\nWe apply stochastic gradient descent with momentum to optimize all the models, and use the validation set to tune hyper-parameters. Our final setups are: $5e^{-4}$ as the initial learning rate, reduced once ($0.1{\\times}$) during fine-tuning; $1e^{-4}$ as weight decay; $0.9$ as momentum. For ADE, we train $320$K iterations and reduce learning rate at $280$K. For VG and COCO the numbers are $640$K\/$500$K and $560$K\/$320$K, respectively\\footnote{Training longer still reduces cross-entropy, but drops both AP and AC.}. We use a single image per step, and the only data augmentation technique used during training is left-right flipping\\footnote{The labels for class pairs like ``left-hand'' and ``right-hand'' are swapped for flipped images.}. No augmentation is used in testing. \n\n\\subsection{Main Results\\label{results}}\nQuantitative results on ADE {\\tt test-1k} and VG {\\tt test} are shown in Tab.~\\ref{tab:final}. Besides plain ConvNet $p_0$, we also add three more baselines. First, we use ResNet-101 as the backbone to see the performance can benefit from deeper networks. Second, we increase the input image size with a shorter side $800$ pixels, which is shown helpful especially for small objects in context~\\cite{lin2016feature}. Finally, to check whether our performance gain is a result of more parameters, we include model ensemble as the third baseline where the prediction of two separate baseline models are averaged.\n\nAs can be seen, our reasoning modules are performing much better than all the baselines on ADE. The local module alone can increase per-class AP by $7.8$ absolute points. Although the global module alone is not as effective ($4.4\\%$ improvement), the performance gain it offers is \\emph{complementary} to the local module, and combining both modules we arrive at an AP of $48.5\\%$ compared to the baseline AP $40.1\\%$. On the other hand, deeper network and larger input size can only help ${\\sim}1\\%$, less than model ensembles. Additionally, our models achieve higher per-class metric gains than per-instance ones, indicating that \\emph{rare} classes get helped more -- a nice property for learning from few examples. Some qualitative results are listed in Fig.~\\ref{fig:examples}. \n\nWe also report the speed for future reference. On Titan Xp, the final model on ADE trains at 0.344s per iteration, compared to the baseline ResNet-50 at $0.163$s and ResNet-101 at $0.209$s. For testing, our model takes $0.165$s, whereas ResNet-50 $0.136$s, ResNet-101 $0.156$s. We believe the additional\ncost is minimal with regard to the extra accuracy.\n\nWe see a similar but less significant trend on VG. This can potentially be a result of \\emph{noisier} labels -- for ADE (and COCO shown later), the per-instance AP and AC values are within $0.1\\%$, intuitively suggesting that \\emph{higher} scores usually correspond to correct classifications. However, on VG the difference is at ${\\sim}0.5\\%$, meaning more of the highly confident predictions are not classified right, which are likely caused by missing ground-truths. \n\n\\subsection{Analysis\\label{ablative}}\nOur analysis is divided into two major parts. In the first part, we conduct thorough ablative analysis on the framework we have built. Due to space limitation, we only report results on ADE here at Tab.~\\ref{tab:contribute}, for more analysis on VG, please check our supplementary material. \n\nAs can be seen, re-weighting hard examples with Eq.~\\ref{reweight} helps around $0.5\\%$ regardless of reasoning modules. Spatial memory $\\mathcal{S}$ is critical in the local module -- if replaced by feeding last \\emph{conv4} layer directly the performance drops almost to baseline. Local context aggregator $\\mathcal{C}$ is less influential for ADE since the regions including background are densely labeled. A different story takes place at the global module: removing the reasoning module $\\mathcal{R}$ steeply drops performance, whereas further removing memory $\\mathcal{M}$ does not hurt much. Finally, for our full pipeline, removing cross-feeding and dropping the number of iterations both result in worse performance.\n\n\\begin{table}[t]\n\\centering\n\\renewcommand{\\arraystretch}{1.1}\n\\renewcommand{\\tabcolsep}{1.2mm}\n\\definecolor{LightGreen}{rgb}{0.75,1,0.75}\n\\definecolor{LightRed}{rgb}{1,0.75,0.75}\n\\definecolor{LightBlue}{rgb}{0.75,0.75,1}\n\\caption{\\label{tab:contribute}{Ablative analysis on ADE {\\tt test-1k}. In the first row of each block we repeat Local, Global and Final results from Tab.~\\ref{tab:final}. Others see Sec.~\\ref{ablative} for details.}}\n\\resizebox{1.0\\linewidth}{!}{\n\\begin{tabular}{@{} C{0.5cm} !{\\vrule} L{2.5cm} !{\\vrule} x{1.2cm} x{1.2cm} !{\\vrule} x{1.2cm} x{1.2cm} @{}}\n\\Xhline{1pt}\n\\multirow{2}{*}{$\\%$} & \\multirow{2}{*}{\\textbf{Analysis}} & \\multicolumn{2}{c!{\\vrule}}{per-instance} & \\multicolumn{2}{c}{per-class} \\\\\n\\Xcline{3-6}{0.5pt}\n& & AP & AC & AP & AC \\\\\n\\Xhline{1pt}\n\\parbox[t]{2.5mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\small Local}}} & Ours\\textsubscript{-Local} & 71.6 & 71.7 & 47.9 & 38.7 \\\\\n& ~~~~{\\small w\/o re-weight} & 71.3 & 71.3 & 46.7 & 37.9 \\\\\n& ~~~~{\\small w\/o $\\mathcal{C}$} & 70.9 & 71.0 & 46.1 & 37.5 \\\\\n& ~~~~{\\small w\/o $\\mathcal{S}$} & 67.6 & 67.6 & 42.1 & 34.4 \\\\\n\n\\Xhline{0.5pt}\n\\parbox[t]{2.5mm}{\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\small Global}}} & Ours\\textsubscript{-Global} & 69.8 & 69.8 & 44.5 & 36.8 \\\\\n& ~~~~{\\small w\/o re-weight} & 69.2 & 69.2 & 43.8 & 36.7 \\\\\n& ~~~~{\\small w\/o spatial} & 67.8 & 67.8 & 41.5 & 35.0 \\\\\n& ~~~~{\\small w\/o semantic} & 69.1 & 69.2 & 43.9 & 35.9 \\\\\n& ~~~~{\\small w\/o $\\mathcal{R}$} & 67.1 & 67.2 & 41.5 & 34.5 \\\\\n& ~~~~{\\small w\/o $\\mathcal{M}$ \\& $\\mathcal{R}$} & 67.1 & 67.1 & 41.0 & 34.0 \\\\\n\n\\Xhline{0.5pt}\n\\parbox[t]{2.5mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\small Final}}} & Ours\\textsubscript{-Final} & 72.6 & 72.6 & 48.5 & 39.5 \\\\\n& ~~~~{\\small w\/o re-weight} & 72.1 & 72.2 & 47.3 & 38.6 \\\\\n& ~~~~{\\small w\/o cross-feed} & 72.2 & 72.2 & 47.6 & 39.0 \\\\\n& ~~~~{\\small $2$ iterations} & 71.9 & 72.0 & 48.1 & 39.0 \\\\\n\n\\Xhline{1pt}\n\\end{tabular}\n}\n\\vspace{-0.1in}\n\\end{table}\n\n\\noindent{\\bf Missing Regions.} So far we have shown results when all the regions are present. Next, we want to analyze if our framework is robust to missing regions: if some percentage of regions are not used for reasoning. This will be a common scenario if we use our framework in the detection setting -- the underlying region proposal network~\\cite{ren2015faster} may itself miss some regions. We perform this set of experiments on COCO, since its regions are object-focused.\n\nWe test three variations. In the first variation, the same region classification pipeline is applied as-is. In the other two, we drop regions. While we could have done it randomly, we simulate the real-world scenario by using region proposals from faster R-CNN~\\cite{ren2015faster} ($1190$K\/$900$K, {\\tt minival} detection mAP $32.4\\%$) for testing, where $300$ region proposals after NMS are applied to filter the ground-truth regions (max IoU${>}\\delta$). Evaluation is only done on the remaining regions. Here we choose not to use region proposals directly, since the model has seen ground truth regions only. We test two variations: a) ``pre'', where the regions are filtered before inference, \\ie only the remaining ground-truths are fed for reasoning; ``post'', where regions are filtered after inference. Note that for the baseline, ``pre'' and ``post'' makes no difference performance-wise.\n\n\\begin{table}[t]\n\\centering\n\\renewcommand{\\arraystretch}{1.1}\n\\renewcommand{\\tabcolsep}{1.2mm}\n\\caption{\\label{tab:coco}{Results with missing regions when region proposals are used. COCO {\\tt minival} is used since it is more detection oriented. {\\bf pre} filters regions before inference, and {\\bf post} filters after inference.}}\n\\resizebox{1.0\\linewidth}{!}{\n\\begin{tabular}{@{} L{1.8cm} !{\\vrule} C{0.6cm} C{0.6cm}  !{\\vrule} x{1.2cm} x{1.2cm} !{\\vrule} x{1.2cm} x{1.2cm} @{}}\n\\Xhline{1pt}\n\\multirow{2}{*}{\\textbf{Method}} & \\multirow{2}{*}{\\bf \\small pre} & \\multirow{2}{*}{\\bf \\small post} & \\multicolumn{2}{c!{\\vrule}}{per-instance} & \\multicolumn{2}{c}{per-class} \\\\\n\\Xcline{4-7}{0.5pt}\n& & & AP\\textsuperscript{$\\nabla$} & AC\\textsuperscript{$\\nabla$} & AP\\textsuperscript{$\\nabla$} & AC\\textsuperscript{$\\nabla$} \\\\\n\\Xhline{1pt}\nBaseline & & & 83.2 & 83.2 & 83.7 & 75.9 \\\\\nOurs\\textsubscript{-Local} & & & 84.9\\textsuperscript{+1.7} & 84.9\\textsuperscript{+1.7} & 85.8\\textsuperscript{+2.1} & 77.6\\textsuperscript{+1.7} \\\\\nOurs\\textsubscript{-Global} & & & 85.6\\textsuperscript{+2.4} & 85.7\\textsuperscript{+2.5} & 86.9\\textsuperscript{+3.2} & 78.2\\textsuperscript{+2.3} \\\\\nOurs\\textsubscript{-Final} & & & {\\bf 86.0}\\textsuperscript{+2.8} & {\\bf 86.0}\\textsuperscript{+2.8} & {\\bf 87.4}\\textsuperscript{+3.7} & {\\bf 79.0}\\textsuperscript{+3.1} \\\\\n\\Xhline{0.5pt}\nBaseline & - & - & 87.0 & 87.0 & 87.7 & 80.2 \\\\\nOurs\\textsubscript{-Final} & \\cmark & & 88.6\\textsuperscript{+1.6} & 88.6\\textsuperscript{+1.6} & 89.9\\textsuperscript{+2.2} & {\\bf 82.6}\\textsuperscript{+2.4} \\\\\nOurs\\textsubscript{-Final} & & \\cmark & {\\bf 88.8}\\textsuperscript{+1.8} & {\\bf 88.8}\\textsuperscript{+1.8} & {\\bf 90.1}\\textsuperscript{+2.4} & 82.5\\textsuperscript{+2.3} \\\\\n\\Xhline{1pt}\n\\end{tabular}\n}\n\\vspace{-0.1in}\n\\end{table}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.\\linewidth]{trend-new.pdf}\n\\caption{{\\small Trends of recall and per-class AP when varying IoU threshold $\\delta$ from $0$ to $.9$ to drop regions. See text for details.}\\label{fig:trend}}\n\\vspace{-0.2in}\n\\end{figure}\n\nThe results are summarized in Tab.~\\ref{tab:coco}. Interestingly, despite lacking a knowledge graph, our global module works better than the local module with the region graph alone, likely due to its power that allows direct region-to-region communication even for farther-away pairs. Combining the two, we report $3.7\\%$ absolute advantage on per-class AP over the baseline even with all classes being objects -- no ``stuff'' classes involved.\n\nIn Fig.~\\ref{fig:trend}, we vary $\\delta$ from $0$ to $.9$: with $0$ keeping all regions and $0.9$ dropping the most. As the trend shows, while the reasoning module suffers when regions are dropped, it is quiet resilient and the performance degradation is smooth. For example (listed in Tab.~\\ref{tab:coco}), with an IoU threshold $\\delta$ of $0.5$ that recalls $78.1\\%$ of the ground truth boxes, we still outperform the baseline by $2.4\\%$ in the ``post'' setting, and $2.2\\%$ in ``pre'' where not all regions can be fed for reasoning. The lower gap implies a) region proposals are usually corresponding to easy examples where less context is needed, and b) context reasoning frameworks like ours benefit from more known regions. At $\\delta{=}.8$ the recall ($30.5\\%$) is so small that it cannot afford much reasoning, and at $\\delta{=}.9$ (recall $3.9\\%$), reasoning even hurts the performance.\n\n\\vspace{-0.05in}\n\\section{Conclusion}\n\\vspace{-0.05in}\nWe presented a novel framework for iterative visual reasoning. Beyond convolutions, it uses a graph to encode spatial and semantic relationships between regions and classes and passes message on the graph. We show strong performance over plain ConvNets, \\eg achieving an $8.4\\%$ absolute gain on ADE and $3.7\\%$ on COCO. Analysis also shows that our reasoning framework is resilient to missing regions caused by current region proposal approaches. \n\n\\noindent {\\bf Acknowledgements}: This work was supported in part by ONR MURI N000141612007. XC would also like to thank Shengyang Dai and Google Cloud AI team for support during the internship. \n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{The $U(3)\\times U(3)$ NJL model}\n\nThe $U(3)\\times U(3)$ NJL model with scalar-pseudoscalar and vector-axial-vector sectors is\nused in the present work. To solve the $U_A(1)$ problem, the six-quark t`Hooft interaction is\nadded to the Lagrangian of the model \\cite{Klimt:1989pm,Klevansky:1992qe}\n\\begin{eqnarray}\n \\mathcal{L}& =& {\\bar q}(i{\\hat \\partial} - m^0)q\n  + \\frac{G}{2}\\sum_{i=0}^8 [({\\bar q} {\\lambda}_i q)^2 +({\\bar q}i{\\gamma}_5{\\lambda}_i q)^2]\n  + \\frac{G_V}{2}\\sum_{i=0}^8 [({\\bar q} {\\gamma}_\\mu {\\lambda}_i q)^2 +({\\bar q}{\\gamma}_5{\\gamma}_\\mu{\\lambda}_i  q)^2]\n   \\nonumber \\\\\n &&- K \\left( {\\det}[{\\bar q}(1+\\gamma_5)q]+{\\det}[{\\bar q}(1-\\gamma_5)q] \\right),\n\\label{Ldet}\n\\end{eqnarray}\nwhere $\\lambda_i$ (i=1,...,8) are the Gell-Mann matrices and $\\lambda^0 =\n{\\sqrt{\\frac{2}{3}}}${\\bf 1}, with {\\bf 1} being the unit matrix; $m^0$ is the current quark\nmass matrix with diagonal elements $m^0_u$, $m^0_d$, $m^0_s$ $(m^0_u \\approx m^0_d)$, $G$ and\n$G_V$ are the scalar--pseudoscalar and vector--axial-vector four-quark coupling constants; $K$\nis the six-quark coupling constants. The six-quark interaction can be reduced to an effective\nfour-fermion vertex after the contraction of one of the quark pairs. The details are given in\nappendix A.\n\nLight current quarks transform to massive constituent quarks as a result of spontaneous chiral\nsymmetry breaking. Constituent quark masses can be found from the Dyson-Schwinger equation for\nthe quark propagators (gap equations)\n\\begin{eqnarray}\nm_u&=&m_u^0 + 8 m_u G I_1(m_u)+32 m_u m_s K I_1(m_u) I_1(m_s)\\nonumber\\\\\nm_s&=&m_s^0 + 8 m_s G I_1(m_s)+32 K \\left(m_uI_1(m_u)\\right)^2, \\label{gapNJL}\n\\end{eqnarray}\nwhere $I_1(m)$ is the quadratically divergent integral. The modified Pauli-Villars (PV)\nregularization with two substractions with same $\\Lambda$ is used for the regularization of\ndivergent integrals\\footnote{Any function $f(m^2)$ of mass $m^2$ is regularized by using the\nrule\n\\begin{eqnarray}\nf(m^2)\\to f(m^2)-f(m^2+\\Lambda^2)+\\Lambda^2 f^\\prime(m^2+\\Lambda^2).\\nonumber\n\\end{eqnarray}} (see\n\\cite{Bernard:1992mp,Bernard:1995hm,Bajc:1996gt,Schuren:1991sc}). In this case the\nquadratically and logarithmically divergent integrals $I_1(m)$ and $I_2(m)$ have the same form\nas in the four-momentum cut-off scheme\n\\begin{eqnarray}\nI_1(m) &=& \\frac{N_c}{4 \\pi^2}\n\\left[\\Lambda^2-m^2\\ln\\left(\\frac{\\Lambda^2}{m^2}+1\\right)\\right], \\quad\nI_2 (m) =\\frac{N_c}{4 \\pi^2}\n\\left[\\ln\\left(\\frac{\\Lambda^2}{m^2}+1\\right)-\\left(1+\\frac{m^2}{\\Lambda^2}\\right)^{-1}\\right]\n\\nonumber.\n\\end{eqnarray}\n\nMoreover, the Pauli-Villars regularization is suitable for the description of the vector\nsector because it preserves gauge invariance.\n\nMasses and vertex functions of the mesons can be found from the Bethe-Salpeter equation. The\nexpression for the quark-antiquark scattering matrix is\n\\begin{eqnarray}\n\\hat{T}=\\mathbf{G}+\\mathbf{G}\\mathbf{\\Pi}(p^2)\\hat{T}=\\frac{1}{\\mathbf{G}^{-1}-\\mathbf{\\Pi}(p^2)},\n\\end{eqnarray}\nwhere $\\mathbf{G}$ and $\\mathbf{\\Pi}(p^2)$ are the corresponding matrices of the four-quark\ncoupling constant and polarization loops. The particle mass can be found from the equation\n$\\mathrm{det}(\\mathbf{G}^{-1}-\\mathbf{\\Pi}(M^2))=0$ and near the poles the corresponding part\nof the $\\hat{T}$ matrix can be expressed in the form\n\\begin{eqnarray}\n\\hat{T}=\\frac{\\bar{V} \\otimes V}{p^2-M^2},\n\\end{eqnarray}\nwhere $V$ and $M$ are the vertex function and mass of the meson, and $\\bar{V} = \\gamma^0\nV^\\dag \\gamma^0$. Details of calculations for different channels are presented in appendices\nB, C. Here we discuss only general properties.\n\nThe most simple situation takes place for the vector and the isovector scalar meson with equal\nquark masses (say $\\rho$ and $a_0$). In this case, the coupling constant and polarization\noperator are just numbers (not matrices). For pseudoscalar mesons, additional axial-vector\ncomponents appear in the vertex function due to the pseudoscalar--axial-vector mixing (in the\nscalar case this transition loop is proportional to the difference of quark masses). An\nadditional complication takes place for $\\eta$ and $\\eta^\\prime$ due to the singlet-octet\nmixing (or mixing of strange and non-strange quarks due to the t`Hooft interaction).\nTherefore, the vertex function of this meson has four components: strange and non-strange\npseudoscalar and axial-vector.\n\n\\section{Fixing model parameters}\n\nThe model has six parameters: the coupling constants $G$, $G_V$, $K$, PV cut-off $\\Lambda$,\nand constituent quark masses $m_u$ and $m_s$. We use two parametrization schemes. In the first\none, the model parameters are defined using masses of the pion, kaon, $\\rho$ and $\\eta$ mesons\nand the weak pion decay constant $f_\\pi$. Note that the number of input parameters is greater\nthan the number of physical observables by one. This allows us, following\n\\cite{Bernard:1995hm}, to take the mass of the $u$ quark slightly larger than the half of the\n$\\rho$-meson mass.\nAs a result, we have the following set (set I) of model parameters\n\\begin{eqnarray}\n m_u = 390\\,\\mathrm{MeV},\\,\n m_s=496\\,\\mathrm{GeV},\\,\n G=6.62\\,\\mathrm{GeV}^{-2},\\,\n G_V=-11.29\\,\\mathrm{GeV}^{-2},\\,\n K = 123\\,\\mathrm{GeV}^{-5},\\,\n\\Lambda=1\\,\\mathrm{GeV}.\n\\end{eqnarray}\nThe values of the current quark masses $m^0_u,m^0_s$ are defined from the gap\nequations (\\ref{gapNJL}) $m^0_u=3.9$ MeV and $m^0_s=70$ ($m^0_u\/m^0_s=18$).\n\nFor this set of model parameters, the two-photon decay width of the $\\eta$ meson\n$\\Gamma_{\\eta\\to \\gamma\\gamma}=0.37$ KeV, is smaller than the experimental one:\n$\\Gamma^{\\mathrm{exp}}_{\\eta\\to\\gamma\\gamma}=0.510\\pm0.026$ \\cite{PDBook}.\n\nIn the set II the model parameters are fixed in order to reproduce the two-photon decay width\nof the $\\eta$ meson instead of its mass (the $\\eta$ meson mass in this case $M_\\eta=530$ MeV)\n\\begin{eqnarray}\n m_u = 390\\,\\mathrm{MeV},\\,\n m_s=506\\,\\mathrm{GeV},\\,\n G=8.04\\,\\mathrm{GeV}^{-2},\\,\n G_V=-11.29\\,\\mathrm{GeV}^{-2},\\,\n K = 77\\,\\mathrm{GeV}^{-5},\\,\n\\Lambda=1\\,\\mathrm{GeV}.\n\\end{eqnarray}\nThe current quark masses are $m^0_u=3.9$ MeV and $m^0_s=78$ MeV ($m^0_u\/m^0_s=20$).\n\n\\section{Decay $\\eta \\to \\pi^0 \\gamma \\gamma$}\n\\begin{figure}\n\\resizebox{0.8\\textwidth}{!}{\\includegraphics{EtPiGaGa}} \\caption{\\label{fig:etpigaga}\nDiagrams contributing to the amplitude of the process $\\eta \\to \\pi^0 \\gamma \\gamma$.}\n\\end{figure}\n\nThe general form of the $\\eta \\to \\pi^0 \\gamma \\gamma$ decay amplitude contains two\nindependent tensor structures \\cite{Ecker:1987hd}\n\\begin{eqnarray}\nT= T^{\\mu\\nu}\\epsilon^{1}_\\mu\\epsilon^{2}_\\nu,\\quad T^{\\mu\\nu}=A(x_{1},x_{2})(q_{1}^{\\nu}\nq_{2}^{\\mu} - q_{1} \\cdot q_{2} g^{\\mu\\nu}) + B(x_{1},x_{2}) \\left[ -M_{\\eta}^{2} x_{1}x_{2}\ng^{\\mu\\nu} - \\frac{q_{1} \\cdot q_{2}}{M_{\\eta}^{2}} p^{\\mu}p^{\\nu} + x_{1} q_{2}^{\\mu} p^{\\nu}\n+ x_{2} p^{\\mu} q_{1}^{\\nu} \\right], \\label{ampform}\n\\end{eqnarray}\nwhere $p$, $q_1$, $q_2$ are the momentum of the $\\eta$ meson and photons, $\\epsilon^{1}_\\mu$\nand $\\epsilon^{2}_\\nu$ are the polarization vectors of the photons, and $x_i= p\\cdot\nq_i\/M_\\eta^2$.\n\nThe $\\eta \\to \\pi^0 \\gamma \\gamma$ decay width has the form\n\\begin{eqnarray}\n \\Gamma & = & \\frac{M_{\\eta}^{5}}{256 \\pi^{2}}\n  \\int\\limits_{0}^{(1-y)\/2} dx_{1} \\int\\limits_{x_2^{\\mathrm{min}} }^{x_2^{\\mathrm{max}}} dx_{2}\n  \\left\\{ \\left| A(x_{1},x_{2}) + \\frac{1}{2} B(x_{1},x_{2}) \\right|^{2} \\left[ 2(x_{1}+x_{2})\n+y -1 \\right]^{2} \\right. \\nonumber \\\\\n  &  + &  \\left. \\frac{1}{4} \\left| B(x_{1},x_{2}) \\right| ^{2} \\left[ 4 x_{1} x_{2}\n- \\left[ 2(x_{1}+x_{2})+ y-1 \\right] \\right] ^{2} \\right\\}  ,\\\\\n&&x_2^{\\mathrm{min}}={(1-2x_1-y)\/2 },\\quad x_2^{\\mathrm{max}}={(1-2x_1-y)\/2(1-2x_1)},\n\\quad y =M_\\pi^2\/M_\\eta^2.\\nonumber\n\\end{eqnarray}\n\nIn the NJL model the amplitude for the $\\eta \\to \\pi^0 \\gamma \\gamma$ decay process is\ndescribed by three types of diagrams (see Fig. \\ref{fig:etpigaga}): the quark box and exchange\nof scalar($a_0$) and vector ($\\rho,\\omega$) resonances. Let us consider theses contributions\nin detail.\n\nThe scalar meson exchange has the simplest form. It gives a contribution only to $A(x_1,x_2)$.\nThis contribution consists of three parts and can be written in the form (see appendices B and\nC for the definition of polarization loops and vertex functions):\n\\begin{eqnarray}\n A(x_1,x_2) &=& \\frac{g_{a_0 \\eta \\pi}(2q_1\\cdot q_2)g_{a_0 \\gamma \\gamma}(2q_1\\cdot q_2)}{G_{a_0}^{-1}-\\Pi_{SS}^{uu}(2 q_1\\cdot q_2 )}\n, \\quad q_1\\cdot q_2 = M_{\\eta}^{2}\\left(x_1+x_2-\\frac{1}{2}\\right)+\\frac{M_\\pi^2}{2}\\nonumber\\\\\ng_{a_0\\gamma\\gamma}(p^2)&=& \\frac{1}{2 \\pi^2}  \\int \\limits_0^1 dx_1 \\int \\limits_0^{1-x_1}\ndx_2 \\frac{m_u(1-4x_1x_2)}{(p^2 x_1 x_2-m_u^2-\\Lambda^2)^2 (p^2 x_1 x_2-m_u^2)} \\\\\ng_{a_0 \\eta \\pi}(p^2)&=& -i 2 N_c N_f\\int \\frac{d^4_\\Lambda k}{(2\\pi)^4}\n  \\mathrm {Tr}_D\\left\\{V_{a_0}S_u(k+q_1)V_{\\pi}S_u(k)V_{\\eta}S_u(k-q_2)\\right\\}.\\nonumber\n\\end{eqnarray}\nhere $\\mathrm {Tr}_D$ is the trace over Dirac indices, index $\\Lambda$ in the measure of\nintegration means PV regularization of the integral and $S_j(p)=(\\hat{p}-m_j)^{-1}$.\n\nThe amplitude with the vector meson ($\\rho,\\omega$) exchanges consists of two quark triangles\nof anomalous type (see appendix D) and the vector meson propagator. It gives the following\ncontributions\n\\begin{eqnarray}\n B(x_{1},x_{2})&=&\\sum\\limits_{j=\\rho,\\omega}\\,\\sum\\limits_{i=1,2}\n \\frac{g_{\\eta j \\gamma}(M_\\eta^2,M_\\eta^2(1-2 x_i),0)g_{\\pi j \\gamma}(M_\\pi^2,M_\\eta^2(1-2 x_i),0)}\n{G_{2}^{-1}-\\Pi_{VV}^{uu}(M_\\eta^2(1-2 x_i))},\\\\\n A(x_{1},x_{2})&=&\\sum\\limits_{j=\\rho,\\omega}\\, \\sum\\limits_{i=1,2}\n \\frac{g_{\\eta j \\gamma}(M_\\eta^2,M_\\eta^2(1-2 x_i),0)g_{\\pi j \\gamma}(M_\\pi^2,M_\\eta^2(1-2 x_i),0)M_\\eta^2(1-x_i)}\n{G_{2}^{-1}-\\Pi_{VV}^{uu}(M_\\eta^2(1-2 x_i))}.\\nonumber\n\\end{eqnarray}\n\nThe box diagram is of a more complicated structure. It consists of three types of boxes (plus\nthree crossed) and contains the diagrams with pseudoscalar and axial-vector components of the\n$\\pi$ and $\\eta$ mesons\n\\begin{eqnarray}\n  T_{\\mu\\nu}=-i e^{2} \\int \\frac{d^4_\\Lambda k}{(2\\pi)^4}\n  \\mathrm {Tr}_D \\biggl(\\biggr. &&V_{\\pi} S(k) V_{\\eta}     S(k+p-q_1-q_{2}) \\gamma_{\\nu} S(k+p-q_1)   \\gamma_{\\mu} S(k+p)+\\nonumber\\\\\n&&+V_{\\pi} S(k) V_{\\eta}     S(k+q_2)         \\gamma_{\\nu} S(k+p-q_1)   \\gamma_{\\mu} S(k+p)\\\\\n&&+V_{\\pi} S(k) \\gamma_{\\nu} S(k+q_2)         \\gamma_{\\mu} S(k+q_1+q_2) V_{\\eta}     S(k+p)\n  +\\left\\{q_1 \\leftrightarrow q_2 ,\\mu \\leftrightarrow \\nu \\right\\}\\biggl.\\biggr)\\nonumber\n\\end{eqnarray}\nWe calculate these diagrams numerically. In order to check the integration procedure, we\ncalculate all coefficients of different tensor structures and verify if they have gauge\ninvariant form (\\ref{ampform}).\n\nThe obtained results for the decay width are given in the Table 1 for two sets of model\nparameters. The main contribution comes from the box diagram. The contribution from vector\nmesons has a constructive interference while the scalar $a_0$ contribution has a destructive\none. The results are in satisfactory agreement with Crystal Ball data $0.45\\pm0.12$\n\\cite{Prakhov:2005vx} and the present value $0.57\\pm0.21$ given in PDG\\cite{PDBook}.\n\n\\begin{table}\n\\caption{$\\eta\\to\\pi^0\\gamma\\gamma$ decay width.}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n Contribution        &model 1& model 2  \\\\\n \\hline\n vector mesons       & 0.17  & 0.20     \\\\\n scalar meson        & 0.03  & 0.12    \\\\\n vector+scalar mesons& 0.10  & 0.12    \\\\\n box                 & 0.28  & 0.35      \\\\\n box+vector          & 0.78  & 0.95    \\\\\n total               & 0.53  & 0.45 \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\nIt is also very instructive to consider the invariant mass distribution. In Figures\n\\ref{fig:DifEtaPiGaGaS1} and \\ref{fig:DifEtaPiGaGaS2} the invariant mass distribution of the\ntwo-photons is shown for the scalar meson contribution, vector mesons contribution, scalar +\nvector mesons and total. In Figure \\ref{fig:DifEtaPiGaGaS1S2Os}, the results of our\ncalculations of the invariant mass distribution are compared with the calculation in the\nchiral unitary approach \\cite{Oset:2002sh}.\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG1v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS1} Invariant mass distribution of the two-photons of the\nscalar meson contribution(dots), vector meson contributions(short dash), scalar + vector\nmesons(dash-dot), quark box(long dash) and total(continuous line) for the set I.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG2v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS2} Invariant mass distribution of the two-photons of the\nscalar meson contribution(dots), vector meson contributions(short dash), scalar + vector\nmesons(dash-dot) , quark box(long dash) and total(continuous line) for the set II.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG123v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS1S2Os} Invariant mass distribution of the two-photons of the\ntotal contributions for the set I(dashes), set II(dots) together with the results of the\nchiral unitary approach \\cite{Oset:2002sh}.}\n\\end{figure}\n\\section{Conclusions}\n\nEarlier calculations of the process $\\eta \\to \\pi^0 \\gamma\\gamma$ in the NJL model do not\ninclude the momentum dependence of quark loops and pseudoscalar--axial-vector transitions and\nare in satisfactory agreement with the GAMS experiment.\n\nRecently, the new experimental data on this decay have been obtained and the value of the\ndecay width is almost two times smaller. A number of theoretical estimates is also obtained,\nand it seems that the momentum dependence of amplitudes is important for a correct description\nof this process ( ``all-order'' estimate in ChPT).\n\nIn the present work, the contributions from quark box, scalar and vector pole diagrams are\nconsidered with the full momentum dependence. The pseudoscalar--axial-vector transitions are\nalso taken into account.\n\nThe obtained result is consistent with recent experiments, theoretical estimates of ChPT\n\\cite{Ametller:1991dp,Bellucci:1995ay} and the chiral unitary approach\\cite{Oset:2002sh}.\n\n\nIn future, we plan to consider the polarizability of pions and also decays of vector mesons\n$\\rho(\\omega)\\to\\eta(\\pi)\\pi\\gamma$.\n\\begin{acknowledgments}\nThe authors thank I. V. Anikin, A. E. Dorokhov, A. A. Osipov and V. L. Yudichev for useful\ndiscussions. The authors acknowledge the support of the Russian Foundation for Basic Research,\nunder contract 05-02-16699.\n\\end{acknowledgments}\n\n\n\\section*{Appendixes}\n\\subsection{Lagrangian}\n\nLagrangian (\\ref{Ldet}) can be rewritten in the form (see\n\\cite{Klimt:1989pm,Klevansky:1992qe})\n\\begin{eqnarray}\n&&\\mathcal{L} =\n {\\bar q}(i{\\hat \\partial} - m^0)q +\n  {\\frac{1}{2}} \\sum_{i=1}^9\n   [G_i^{(-)} ({\\bar q}{\\lambda^\\prime}_i q)^2 +G_i^{(+)}({\\bar q}i{\\gamma}_5{\\lambda^\\prime}_i q)^2] +\n     \\nonumber \\\\\n  &&\\qquad+ G^{(-)}_{us}({\\bar q}           {\\lambda}_u q)({\\bar q}            {\\lambda}_s q)\n          + G^{(+)}_{us}({\\bar q}i{\\gamma}_5{\\lambda}_u q)({\\bar q}i {\\gamma}_5{\\lambda}_s q)\n  + \\frac{G_V}{2}\\sum_{i=0}^8 [({\\bar q}{\\gamma}_\\mu {         \\lambda}_i q)^2\n                              +({\\bar q}{\\gamma}_5{\\gamma}_\\mu{\\lambda}_i q)^2],\n\\label{LGus}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&{\\lambda^\\prime}_i={\\lambda}_i ~~~ (i=1,...,7),~~~\\lambda^\\prime_8 = \\lambda_u =\n({\\sqrt 2}\n\\lambda_0 + \\lambda_8)\/{\\sqrt 3},\\nonumber\\\\\n&&\\lambda^\\prime_9 = \\lambda_s = (-\\lambda_0 + {\\sqrt 2}\\lambda_8)\/{\\sqrt 3}, \\label{DefG}\\\\\n&&G_1^{(\\pm)}=G_2^{(\\pm)}=G_3^{(\\pm)}= G \\pm 4Km_sI_1 (m_s), \\nonumber \\\\\n&&G_4^{(\\pm)}=G_5^{(\\pm)}=G_6^{(\\pm)}=G_7^{(\\pm)}= G \\pm 4Km_uI_1 (m_u),\n\\nonumber \\\\\n&&G_u^{(\\pm)}= G \\mp 4Km_sI_1(m_s), ~~~ G_s^{(\\pm)}= G, ~~~ G_{us}^{(\\pm)}= \\pm 4{\\sqrt\n2}Km_uI_1 (m_u).\\nonumber\n\\end{eqnarray}\n\n\\subsection{Polarization loops}\nPolarization loops in different channels after the PV regularization\n\\begin{eqnarray}\ne^{-izm_im_j} \\to R_{ij}(z)=e^{-izm_im_j}\\left[1-(1+iz\\Lambda^2)e^{-iz\\Lambda^2}\\right]\n\\end{eqnarray}\ntake the form (see \\cite{Bernard:1995hm} for the expressions for the polarization loops with\nequal indices)\n\\begin{eqnarray}\n \\Pi_{PP}^{ij}(p^2) &=&\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\left[-\\frac{i}{z}+\\frac{1}{2}p^2(1-y^2)-\\frac{1}{2}\\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\right] ,\\nonumber\\\\\n \\Pi_{SS}^{ij}(p^2) &=&\\Pi_{PP}^{ij}(p^2)-2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA} ,\\nonumber\\\\\n \\Pi_{VV}^{ij,{\\mu\\nu}}(p^2) &=&\\left(g^{\\mu\\nu}-\\frac{p^{\\mu}p^{\\nu}}{p^2}\\right) \\Pi_{VV}^{ij}(p^2) + \\frac{p^{\\mu}p^{\\nu}}{p^2} \\Pi_{VV}^{ij,L}(p^2),\\nonumber\\\\\n \\Pi_{VV}^{ij,L}(p^2) &=&\\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\nonumber,\\\\\n \\Pi_{VV}^{ij}(p^2) &=& \\Pi_{VV}^{ij,L}(p^2)- p^2 \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy(1-y^2)\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\nonumber,\\\\\n \\Pi_{AA}^{ij,{\\mu\\nu}}(p^2) &=&\\left(g^{\\mu\\nu}-\\frac{p^{\\mu}p^{\\nu}}{p^2}\\right) \\Pi_{AA}^{ij,T}(p^2) + \\frac{p^{\\mu}p^{\\nu}}{p^2} \\Pi_{AA}^{ij}(p^2),\\nonumber\\\\\n \\Pi_{AA}^{ij,T}(p^2) &=&\\Pi_{VV}^{ij}(p^2)+2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA},\\\\\n \\Pi_{AA}^{ij}(p^2) &=&\\Pi_{VV}^{ij,L}(p^2)+2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n \\Pi_{PA}^{ij,\\mu}(p^2) &=& \\frac{p^{\\mu}}{\\sqrt{p^2}} \\Pi_{PA}^{ij}(p^2) = p^{\\mu} i(m_i+m_j) \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n \\Pi_{AP}^{ij,\\mu}(p^2) &=& \\frac{p^{\\mu}}{\\sqrt{p^2}} \\Pi_{AP}^{ij}(p^2) =-p^{\\mu} i(m_i+m_j) \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n                &&A=\\frac{p^2}{4}(1-y^2)-\\frac{1}{2}\\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\nonumber.\n\\end{eqnarray}\n\n\n\\subsection{Vertex functions}\n\nThe most simple form have the vertex functions for the vector $\\rho$ and the isovector scalar\nmeson $a_0$, namely \\footnote{We suppress flavor indices.}:\n\\begin{eqnarray}\nV_{a_0}=g_{a_0} \\mathbf{I} a_0, \\quad V_{\\rho}= g_{\\rho} \\gamma_\\mu \\rho^\\mu.\n\\end{eqnarray}\nThe matrices $\\mathbf{G}$ and $\\mathbf{\\Pi}$ for $a_0$ and $\\rho$ mesons have the form\n\\begin{eqnarray}\n\\mathbf{G}_{a_0} &=& G_1^{(-)},\\, \\mathbf{\\Pi}_{a_0}(p^2)  = \\Pi_{SS}^{uu}(p^2), \\\\\n\\mathbf{G}_{\\rho}&=&G_2  \\quad     , \\mathbf{\\Pi}_{\\rho}(p^2) = \\Pi_{VV}^{uu}(p^2)\\nonumber\n\\end{eqnarray}\n\nFor the pion and kaon,  additional axial-vector components appear in the vertex function due\nto pseudoscalar--axial-vector mixing\n\\begin{eqnarray}\n V_{\\pi}=g_{\\pi} i \\gamma_5 (1+\\Delta_\\pi \\hat{p}) \\pi,\\,\\quad\n V_{K}  =g_{K} i\\gamma_5(1+\\Delta_K \\hat{p}) K\n\\end{eqnarray}\nHere $\\mathbf{G}$ and $\\mathbf{\\Pi}$ are\n\\begin{eqnarray}\n\\mathbf{G}_{\\pi} =\n\\begin{pmatrix} G_1^{(+)}&0\\\\\n                0        &G_2\\end{pmatrix}\n, \\mathbf{\\Pi}_{\\pi}(p^2) =\n\\begin{pmatrix}\n\\Pi^{uu}_{PP}(p^2) & \\Pi^{uu}_{PA}(p^2)\\\\\n\\Pi^{uu}_{AP}(p^2) & \\Pi^{uu}_{AA}(p^2)\n\\end{pmatrix},\\\\\n\\mathbf{G}_{K} =\n\\begin{pmatrix} G_4^{(+)}&0\\\\\n                0        &G_2\\end{pmatrix}\n, \\mathbf{\\Pi}_{K}(p^2) =\n\\begin{pmatrix}\n\\Pi^{us}_{PP}(p^2) & \\Pi^{us}_{PA}(p^2)\\\\\n\\Pi^{us}_{AP}(p^2) & \\Pi^{us}_{AA}(p^2)\n\\end{pmatrix}.\\nonumber\n\\end{eqnarray}\n\nTherefore, the vertex function of the $\\eta$ meson have four components: strange and\nnon-strange pseudoscalar and axial-vector\n\\begin{eqnarray}\nV_{\\eta}&=&\n  g_{\\eta_u} i \\gamma_5 (1+\\Delta_{\\eta_u} \\hat{p}) \\eta_u\n +g_{\\eta_s} i \\gamma_5 (1+\\Delta_{\\eta_s} \\hat{p}) \\eta_s =\\\\\n &=& g_{\\eta} i \\gamma_5( \\cos\\Theta_\\eta\\eta_u-\\sin\\Theta_\\eta\\eta_s +\\Delta_{\\eta} \\hat{p}( \\cos\\widetilde{\\Theta}_\\eta \\eta_u\n -\\sin\\widetilde{\\Theta}_\\eta\\eta_s)),\\nonumber\n\\end{eqnarray}\nwhere $\\Theta_\\eta$ and $\\widetilde{\\Theta}_\\eta$ are the mixing angles for pseudoscalar and\naxial-vector components. The matrices $\\mathbf{G}$ and $\\mathbf{\\Pi}(p^2)$ are four-by-four\nmatrices\n\\begin{eqnarray}\n\\mathbf{G} =\n\\begin{pmatrix} \\mathbf{G}^{(+)}&0\\\\\n                0        &\\mathbf{G_2}\\end{pmatrix}\n, \\mathbf{G}^{(+)} =\n\\begin{pmatrix} G_u^{(+)}&G_{us}^{(+)}\\\\\n                G_{us}^{(+)}&G_s^{(+)}\\end{pmatrix},\\mathbf{G_2}=\\mathrm{diag}\\{G_2,G_2\\}\\\\\n\\mathbf{\\Pi}(p^2) =\n\\begin{pmatrix}\n\\mathbf{\\Pi}_{PP}(p^2) & \\mathbf{\\Pi}_{PA}(p^2)\\\\\n\\mathbf{\\Pi}_{AP}(p^2) & \\mathbf{\\Pi}_{AA}(p^2)\n\\end{pmatrix}\n, \\mathbf{\\Pi}_{ij}(p^2) = \\mathrm{diag}\\{\\Pi^{uu}_{ij(p^2)},\\Pi^{ss}_{ij}(p^2)\\},\ni,j=P,A\\nonumber\n\\end{eqnarray}\n\n\n\\subsection{Amplitudes $\\eta\\to\\gamma\\gamma$, $\\rho\\to\\eta(\\pi)\\gamma$}\n\nThe amplitude for the two-photon decay width of the pseudoscalar meson has the form\n\\begin{eqnarray}\nA(P\\to\\gamma\\gamma) \\ = \\ e^2\\; g_{P\\gamma\\gamma}(M_P^2,q_1^2,q_2^2)\\\n\\epsilon_{\\mu\\nu\\alpha\\beta} \\ \\epsilon_1^{\\mu} \\epsilon_2^{\\nu} \\;q_1^\\alpha\nq_2^\\beta\\ ,\n\\end{eqnarray}\nwhere $q_1$, $q_2$ are the momentum of photons and $\\epsilon^{1}_\\mu$, $\\epsilon^{2}_\\nu$ are\nthe polarization vectors of the photons,\n\\begin{eqnarray}\ng_{\\pi\\gamma\\gamma}(M_\\pi^2,q_1^2,q_2^2) & = & I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi},  \\\\\ng_{\\eta\\gamma\\gamma}(M_\\eta^2,q_1^2,q_2^2) & = &\\frac{5}{3}\nI_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}-\\frac{\\sqrt 2}{3} I_s(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_s}\n.\\nonumber\n\\end{eqnarray}\nThe loop integrals $I_j(M_P^2)$ are given by\n\\begin{eqnarray}\nI_j(M_P^2,q_1^2,q_2^2) & = &\n \\frac{1}{2 \\pi^2}  \\int \\limits_0^1 dx_1 \\int \\limits_0^{1-x_1} dx_2\n \\frac{m_j}{m_j^2-x_1(1-x_1-x_2)q_1^2-x_2(1-x_1-x_2) q_2^2-x_1x_2 M_P^2}.\n\\end{eqnarray}\nThe amplitudes for the processes $\\rho(\\omega)\\to\\eta(\\pi)\\gamma$ have the form\n\\begin{eqnarray}\nA(P V \\gamma) \\ = \\ g_\\rho e\\; g_{P\\rho\\gamma}(M_P^2,q_1^2,q_2^2)\\\n\\epsilon_{\\mu\\nu\\alpha\\beta} \\ \\epsilon_1^{\\mu} \\epsilon_2^{\\nu} \\;q_1^\\alpha q_2^\\beta\\ ,\n\\end{eqnarray}\nhere $q_1$ and $\\epsilon^{1}_\\mu$ are the momentum and the polarization vector of\n$\\rho(\\omega)$ meson.\n\\begin{eqnarray}\ng_{\\pi \\rho\\gamma}(M_\\pi^2,q_1^2,q_2^2)  & = & I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi},\\quad\ng_{\\eta\\rho\\gamma}(M_\\eta^2,q_1^2,q_2^2)  = 3I_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}, \\nonumber\\\\\ng_{\\pi \\omega\\gamma}(M_\\pi^2,q_1^2,q_2^2)  & = &3I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi} ,\\quad\ng_{\\eta\\omega\\gamma}(M_\\eta^2,q_1^2,q_2^2)  =  I_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}.\n\\end{eqnarray}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Calculation Details}\\label{sec:calculation_details}\n\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Lattice Volume & $M_{\\pi}$ (MeV) & $N_{\\rm cfgs}$ & $N_{\\rm tsrcs}$ for $c\\bar{c}$, $c\\bar{s}$, $c\\bar{l}$ & $N_{\\rm vecs}$ \\\\\n\\hline\n $24^{3}\\times 128$ & 391 & 553 & 32, 16, 16 & 162 \\\\\n\\hline\n $32^3\\times 256$ & 236 &  484 & 1, 1, 2 & 384  \\\\\n\\hline\n\\end{tabular}\n\\caption{The lattice gauge field ensembles used.  The volume is given as\n$(L\/a_s)^3 \\times (T\/a_t)$ where $L$ and $T$ are respectively the spatial and temporal extents of the lattice.  The number of gauge field configurations used, $N_{\\rm cfgs}$, and the number\nof perambulator time-sources used per configuration, $N_{\\rm tsrcs}$, are shown along with the number of eigenvectors used in the distillation framework~\\cite{Peardon:2009}, $N_{\\rm vecs}$.}\n\\label{tab:lattice_details}\n\\end{center}\n\\end{table}\n\nIn this study we use an anisotropic lattice formulation where the temporal lattice spacing, $a_t$, is smaller than the spatial lattice spacing, $a_s \\approx 0.12$~fm, with an anisotropy $\\xi \\equiv a_s \/ a_t \\approx 3.5$.  The gauge sector is described by a tree-level Symanzik-improved anisotropic action, while the fermionic sector uses a tadpole-improved anisotropic Sheikholeslami-Wohlert (clover) action with stout-smeared gauge fields~\\cite{Morningstar:2003} and $N_{f}=2+1$ flavours of dynamical quarks.  For both ensembles the heavier dynamical quark is tuned to approximate the physical strange quark, but the ensembles differ in light quark mass giving the two different pion masses.  Table~\\ref{tab:lattice_details} summarises these lattice ensembles -- full details are given in Refs.~\\cite{Edwards:2008,Lin:2009}.\n\nWe use the same relativistic action for the charm quark as for the light and strange quarks (with tadpole-improved tree-level clover coefficients). The charm-quark mass and anisotropy parameters are tuned to reproduce the physical $\\eta_c$ mass and a relativistic dispersion relation -- this process was described for the $M_{\\pi} \\sim 400$ MeV ensemble in Ref.\\ \\cite{Liu:2012}. Throughout this work we do not correct experimental data for electromagnetic effects. For the $M_{\\pi} \\sim 240$ MeV ensemble, the momentum dependence of the $\\eta_c$ energy after tuning is shown in Figure~\\ref{fig:dispersion}.  The momentum is quantised by the periodic boundary conditions on the cubic spatial volume, $\\vec{p} = \\frac{2\\pi}{L} \\vec{n}$, where $\\vec{n}=(n_x,n_y,n_z)$ and $n_i \\in \\{0, 1, 2, \\dots, L\/a_s - 1 \\}$.  A reasonable fit to the dispersion relation,\n\\begin{equation}\n\\label{equ:dispersion}\n(a_t E)^2 = (a_t M)^2 + \\left( \\frac{2\\pi}{\\xi L\/a_s} \\right)^2 n^2 \\, ,\n\\end{equation}\nis obtained giving $\\xi_{\\eta_c} = 3.456(4)$, in agreement with the anisotropy measured from the pion dispersion relation on this ensemble, $\\xi_{\\pi} = 3.453(6)$~\\cite{Wilson:2015dqa}.\nThe fit gives $M_{\\eta_c} = 2945(17)$ MeV compared to the experimental value $2983.6(6)$~MeV~\\cite{PDG2015}, and so we estimate that the systematic uncertainty from tuning the charm-quark mass is of order 1\\%.\nFigure~\\ref{fig:dispersion} also shows the momentum dependence of the $D$ meson energy; a fit to Eq.~\\ref{equ:dispersion} gives $\\xi_{D} = 3.443(7)$, in reasonable agreement with $\\xi_{\\pi}$ and $\\xi_{\\eta_c}$.\n\n\\begin{figure}[tb]\n\\begin{minipage}{.5\\linewidth}\n\\includegraphics[width=0.95\\textwidth]{Plots\/dispersion_etac.pdf}\n\\end{minipage}\n\\begin{minipage}{.5\\linewidth}\n\\includegraphics[width=0.95\\textwidth]{Plots\/dispersion_D.pdf}\n\\end{minipage}\n\\caption{Points show the dependence of the $\\eta_c$ (left panel) and $D$ (right panel) energy on momentum; error bars show the one sigma statistical uncertainty on either side of the mean.  Lines are fits to the relativistic dispersion relation, Eq.~\\ref{equ:dispersion}, giving $\\xi_{\\eta_c} = 3.456(4)$ ($\\chi^{2}\/N_\\mathrm{d.o.f} = 1.08$) and $\\xi_D = 3.443(7)$ ($\\chi^{2}\/N_\\mathrm{d.o.f} = 0.38)$.}\n\\label{fig:dispersion}\n\\end{figure}\n\nTo give results in physical units, we set the scale via $a_t^{-1} = M_\\Omega^{\\mathrm{phys}} \/ (a_t M_\\Omega)$ using the $\\Omega$ baryon mass measured on this ensemble, $a_t M_\\Omega = 0.2789(16)$~\\cite{Wilson:2015dqa}, leading to $a_t^{-1} = 5997$ MeV. When quoting masses we reduce the already small systematic uncertainty from tuning the charm-quark mass by subtracting $M_{\\eta_c}$ ($\\tfrac{1}{2} M_{\\eta_c}$) from the mass of charmonia (open-charm mesons), rendering it negligible compared to other systematic uncertainties.\n\nThe aim of this work is to study how the spectra change as we vary the light-quark mass and only statistical uncertainties are given in the spectra we present in the following sections. While a full error budget is beyond the scope of this work, the uncertainties arising from working at a finite lattice spacing and in a finite volume were discussed in Ref.~\\cite{Liu:2012}, where they were estimated to be small and have no overall qualitative effect on the spectrum. The uncertainty arising from the ambiguity in how to set the scale can be estimated by choosing a different reference observable. For example, setting the scale on the $M_\\pi \\sim 240$ MeV ensemble using the $h_c$ -- $\\eta_c$ mass splitting gives $a_t^{-1} = 5960$ MeV, $0.6\\%$ lower than from using $M_\\Omega$. On the other hand, using the $\\eta_c(2S)$ -- $\\eta_c(1S)$ mass splitting gives $a_t^{-1} = 5787$ MeV, $4\\%$ lower than when using the $\\Omega$ baryon mass. \n\nAnother source of systematic uncertainty comes from ignoring the unstable nature of states above threshold (see Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub}). Although this is difficult to estimate, for a narrow resonance a conservative approach is to consider the uncertainty to be of the order of the width~\\cite{Dudek:2010}.\n\n\n\n\\subsection{Calculation of spectra}\n\nWe follow the methodology presented in Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} to compute the spectra.  In brief, meson masses and other spectral information are obtained from the analysis of the time dependence of two-point Euclidean correlation functions,\n\\begin{equation}\nC_{ij}(t) = \\langle 0 | \\mathcal{O}_i(t) \\mathcal{O}^\\dagger_j(0) | 0 \\rangle~,\n\\end{equation}\nwhere $\\mathcal{O}^\\dagger(0)$ [$\\mathcal{O}(t)$] is the creation operator [annihilation operator] and $t$ is the time separation.  When computing charmonium correlators, disconnected Wick diagrams, where the charm quark and antiquark annihilate, are not included -- these are OZI suppressed and so are expected to only give a small contribution in charmonium.  There are no such disconnected contributions to the open-charm meson correlators considered here.\n\nThe hypercubic lattice has a reduced symmetry compared to an infinite volume continuum so states at rest are labelled by the irreducible representations (\\emph{irreps}), $\\Lambda$, of the octahedral group, $O_h$, rather than spin~\\cite{Johnson:1982yq}.  \nA method to ameliorate this issue and determine the continuum spin, $J$, of extracted states is given in Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} which also contain demonstrations of its efficacy.  Parity, $P$, and any relevant flavour quantum numbers, e.g.\\ charge-conjugation, $C$, are still good quantum numbers in our lattice formulation.\n\nIn each quantum-number channel, the distillation technique~\\cite{Peardon:2009} is used to compute correlation functions involving a large basis of derivative-based fermion-bilinear interpolating operators~\\cite{Dudek:2010}.\\footnote{To investigate more completely the resonant nature of states above threshold we would need to supplement the basis with operators of additional structures, e.g.~multi-meson operators, as in Ref.~\\cite{Moir:2016srx}.}  The resulting matrices of correlation functions, $C_{ij}(t)$, are analysed using a variational procedure~\\cite{Michael:1985,Luscher:1990,Blossier:2009kd} as described in Ref.~\\cite{Dudek:2010}. This amounts to solving a generalised eigenvalue problem,\n$C_{ij}(t) v^{\\mathfrak{n}}_j = \\lambda^{\\mathfrak{n}}(t,t_0) C_{ij}(t_0) v^{\\mathfrak{n}}_j $, where $t_{0}$ is a carefully chosen reference time-slice.  For sufficiently large times, the eigenvalues, $\\lambda^{\\mathfrak{n}}(t,t_0)$, known as principal correlators, are proportional to $e^{-M_{\\mathfrak{n}}(t-t_{0})}$ where $M_{\\mathfrak{n}}$ is the energy of the $\\mathfrak{n}^{th}$ state.  \nEnergies are extracted from a fit to the form, $(1 - A_{\\mathfrak{n}}) e^{-M_{\\mathfrak{n}}(t-t_0)} + A_\\mathfrak{n} e^{-M'_{\\mathfrak{n}} (t-t_0)}$, \nwhere the fit parameters are $M_{\\mathfrak{n}}$, $A_{\\mathfrak{n}}$ and $M'_{\\mathfrak{n}}$.  The second exponential proves useful in stabilising the fit because it `mops up' excited state contamination.\nThe eigenvectors, $v^{\\mathfrak{n}}_{j}$, are related to the operator-state overlaps (or matrix elements), $Z_i^{(\\mathfrak{n})} \\equiv \\langle \\mathfrak{n} | \\mathcal{O}_i^\\dagger | 0 \\rangle$, and contain information on the structure of a state -- they are used in our method for determining the continuum spin.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmprincorr.pdf}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmoverlaps.pdf}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmoverlaps24.pdf}\n\\end{center}\n\\caption{Top row: principal correlators for a selection of low-lying charmonium states in the $T_1^{--}$ irrep on the $M_\\pi \\sim 240$ MeV ensemble. The data (points) and fits (curves) for $t_0 = 11$ are plotted as $\\lambda^{(\\mathfrak{n})} e^{M_\\mathfrak{n}(t-t_0)}$ showing the central values and one sigma statistical uncertainties.  In each case the fit is reasonable with $\\chi^2\/N_\\mathrm{d.o.f} \\sim 1$.  Red parts of the curves show the time regions used in the fits; blue points were not included in the fits.  \nMiddle row: the operator-state overlaps, $Z$, for the state above, normalised so that the largest value for an operator across all states is equal to unity.  Colour coding is described in the text and the error bars indicate the one sigma statistical uncertainty.\nBottom row: overlaps for the corresponding state on the $M_\\pi \\sim 400$ MeV ensemble.}\n\\label{fig:prin_corrs}\n\\end{figure}\n\nFigure~\\ref{fig:prin_corrs} shows a selection of principal correlators from charmonium correlation functions in the $\\Lambda^{PC} = T_{1}^{--}$ irrep on the $M_\\pi \\sim 240$ MeV ensemble.  The leading time dependence, $e^{-M_{\\mathfrak{n}}(t-t_0)}$, has been divided out yielding a plateau when a single exponential dominates.\nBeneath each principal correlator we show the overlap, $Z$, of each operator onto that state and below that, for comparison, the overlaps for the corresponding state on the $M_\\pi \\sim 400$ MeV ensemble.  The operators were constructed to have definite $J^{PC}$ in the continuum: red bars correspond to $J=1$, blue to $J=3$ and yellow to $J=4$.  It is clear that each state is dominated by operators from a given $J$, demonstrating that the spin-identification methodology~\\cite{Dudek:2010} can be used -- this pattern is repeated for each of the spectra we determine. The darker shade of red and lighter shade of blue represent operators that are proportional to the spatial part of the field strength tensor, $F_{ij}$. We identify a state as hybrid, i.e. a meson with excited gluonic degrees of freedom~\\cite{Dudek:2010}, when overlaps from these operators onto a given state are large compared to their overlaps onto other states\\footnote{In Figure~\\ref{fig:prin_corrs} the apparently considerable overlap of the $J=3$ state with hybrid operators is an artefact of the normalisation; in absolute terms these overlaps are small and we do not identify that state as a hybrid.}.\n\n\n\\section{Comparison of the spectra at two light quark masses}\\label{sec:comparison}\n\nThe principal difference between the spectra presented in Refs.~\\cite{Liu:2012,Moir:2013ub} and this work is the light quark mass, corresponding to $M_\\pi\\sim 400$ MeV in those references and $M_\\pi\\sim 240$ MeV here.  Figures~\\ref{fig:charmonium_comparison}, \\ref{fig:Ds_comparison} and \\ref{fig:D_comparison} show comparisons of the charmonia, $D_s$ and $D$ spectra at the two light quark masses -- it can be seen that, in general, we observe only a mild light quark mass dependence throughout the entire spectra, with no change in the overall pattern of states. The systematic uncertainties were discussed in Section \\ref{sec:calculation_details}.\n\nWe note in passing that we achieve a greater statistical precision on the $M_{\\pi} \\sim 400$ MeV ensemble due to the larger number of time-sources used (see Table \\ref{tab:lattice_details}).  In the discussion that follows some notable features in each spectrum are highlighted and in Section \\ref{sec:mixing} we investigate the mixing between spin-triplet and spin-singlet open-charm mesons.\n\n\n\\subsection{Charmonium}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/pionmasscomparison_2.pdf}\n\\caption{Charmonium spectrum, labelled by $J^{PC}$, with $M_\\pi\\sim 240$ MeV (left column for each $J^{PC}$) compared to the spectrum with $M_\\pi\\sim 400$ MeV from Ref.~\\cite{Liu:2012} (right column for each $J^{PC}$). As in earlier figures, red and blue boxes highlight states identified as constituents of, respectively, the lightest and first-excited supermultiplet of hybrid mesons. Dashed lines show some of the lower thresholds using computed masses for $M_\\pi\\sim 240$ MeV (coarse dashing) and $M_\\pi\\sim 400$ MeV (fine dashing): green is $\\eta_c \\pi \\pi$, red is $D \\bar{D}$ and blue is $D \\bar{D}^*$.}\n\\label{fig:charmonium_comparison}\n\\end{figure}\n\nIn charmonium the light quark dependence enters through the sea quark content in the dynamical gauge field ensembles. As shown in Figure~\\ref{fig:charmonium_comparison}, for the low-lying states the masses are generally consistent between the two ensembles within statistical uncertainties.  An exception is the hyperfine splitting, $M_{J\/\\psi} - M_{\\eta_c}$, where we find a small but statistically significant increase when the light quark mass is decreased.\n\nA second notable feature is that the masses of states higher up in the spectrum are generally larger on the $M_\\pi \\sim 240$ MeV ensemble.  This is particularly the case for the hybrids, implying a small increase in their mass as $M_\\pi$ is reduced; as a consequence the splitting between the hybrids and low-lying conventional mesons increases, albeit in a rather mild fashion. However, it is important to note that at higher energies the statistical uncertainties are larger and neglecting the unstable nature of states may be more important. We emphasise that the overall pattern of hybrid mesons is unaffected by decreasing the light quark mass. \n\n\n\\subsection{$D_s$ mesons}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/Ds_840_860_comp_2.pdf}\n\\caption{As Figure~\\ref{fig:charmonium_comparison} but for the $D_{s}$ meson spectrum labelled by $J^P$.}\n\\label{fig:Ds_comparison}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/D_840_860_comp_2.pdf}\n\\caption{As Figure~\\ref{fig:charmonium_comparison} but for the $D$ meson spectrum labelled by $J^P$.}\n\\label{fig:D_comparison}\n\\end{figure}\n\nAs for charmonium, and shown in Fig.\\ \\ref{fig:Ds_comparison}, only mild dependence on the light quark mass is observed throughout the $D_s$ meson spectrum. The largest change in the low-lying states is for the lightest $0^{+}$ (our candidate for the $D^{*}_{s0}(2317)$). However, this state is expected to be heavily influenced by the nearby $DK$ threshold to which it can couple in $S$ wave, and interestingly, it has decreased just enough to remain below the threshold, in agreement with the experimental situation.\n\nOnce again we observe a tendency for the hybrid states, coloured red in Fig.\\ \\ref{fig:Ds_comparison}, to increase in mass, and hence the splitting between the hybrids and the lowest conventional $D_{s}$ mesons to increase, as $M_\\pi$ is reduced.  However, there is no change to their overall pattern.\n\n\n\\subsection{$D$ mesons}\n\nFigure~\\ref{fig:D_comparison} shows that, in the $D$ meson spectrum, the light quark mass dependence is also relatively mild and there is no change to the pattern of states.  As expected, the masses generally decrease with decreasing pion mass -- a $D$ meson contains a valence light quark unlike a charmonium or $D_s$ meson.\nThe most significant differences are observed for the lightest $0^+$ and $1^+$ states and this may be because they are strongly influenced by nearby thresholds that can couple in $S$ wave, namely $D \\pi$ and $D^* \\pi$ respectively.  However, the mass of the second-lightest $1^+$, which is also in the vicinity of the $D^* \\pi$ threshold, does not change significantly.  This may be because the mass difference between the charm quark and the light quark is large enough for the expectations of the heavy-quark limit to be a reasonable guide.  In this limit, one of the $1^+$ states can decay to $D^{*}\\pi$ only in $S$ wave, whereas the other can decay to $D^{*}\\pi$ only in $D$ wave~\\cite{Isgur:1991wq}; the latter would be expected to be influenced less by the position of the $D^* \\pi$ threshold.\n\nAt higher energies in the spectrum, there are generally only small or statistically insignificant mass shifts while, as for charmonia and $D_{s}$ mesons, there is a general trend for the hybrid mesons to become heavier as the light-quark mass decreases. This change is somewhat less clear in the $D$ meson spectrum because of the opposing trend for mesons to become lighter as the light-quark mass decreases.\n\n\n\n\n\\subsection{Mixing of spin-triplet and spin-singlet open-charm mesons}\\label{sec:mixing}\n\nAs discussed in Section~\\ref{sec:spectra:opencharm}, charge conjugation is not a good quantum number for open-charm mesons and, consequently, quark model spin-singlet $(^{1}L_{J=L})$ and spin-triplet $(^{3}L_{J=L})$ states with the same $J=L$ can mix. Quantifying this mixing at different light quark masses can provide an insight into the flavour symmetry breaking. Using a two-state hypothesis and assuming energy-independent mixing we can determine the mixing angle defined in Eq.~(6.1) of~\\cite{Moir:2013yfa} from ratios of operator overlaps (interpreted non-relativistically) as described in that reference.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{c c c|ccc|c}\n&&&&$|\\theta|\/^{\\circ}$&&\\\\\n & $J^P$ & $M_\\pi \\, \/ \\, \\mathrm{MeV}$ & $\\sim(\\rho - \\rho_2)$ &  $\\sim \\pi$ & $ \\sim \\pi_2$ & Heavy-quark limit\\\\\n\\hline \\hline\nc-s &$1^+$         &240 &60.2(0.4) &63.1(0.7) &65.4(0.7) &\\multirow{2}{*}{54.7 or 35.3} \\\\\n    &              &400 &60.9(0.6) &64.9(0.2) &66.4(0.4) & \\\\ \\cline{2-7}\n    &$2^-$         &240 &56.3(0.9) &60.7(0.8) &63.5(0.9) &\\multirow{2}{*}{50.8 or 39.2} \\\\\n    &              &400 &64.9(1.9) &68.7(2.0) &70.9(1.8) & \\\\ \\cline{2-7}\n   &$1^-$ (hybrid) &240 &58.9(1.0) &66.2(1.9) &65(2.0) & \\\\\n    &              &400 &59.9(1.7) &67.9(0.9) &67.3(0.9) & \\\\\n\\hline \\hline\nc-l &$1^+$         &240 &52.7(0.9) &61.4(0.4) &67.1(1.0)   &\\multirow{2}{*}{54.7 or 35.3} \\\\\n    &              &400 &60.1(0.4) &62.6(0.2) &65.4(0.2) & \\\\ \\cline{2-7}\n    &$2^-$         &240 &50.4(0.7) &57.5(0.8) &61.4(0.9) &\\multirow{2}{*}{50.8 or 39.2} \\\\\n    &              &400$^{\\star}$  &63.3(2.2) &67.8(3.7) &71.1(3.9) & \\\\ \\cline{2-7} \n   &$1^-$ (hybrid) &240 &57.8(1.1) &71.4(2.2) &69.9(2.5) & \\\\\n    &              &400 &59.7(1.1) &68.4(0.8) &67.4(0.9) & \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{Absolute value of the mixing angles for the lightest pairs of $1^+$, $2^-$ and hybrid $1^-$ states in the charm-strange (c-s) and charm-light (c-l) sectors on the two ensembles.  The mixing angles expected in the heavy-quark limit are also shown.  In the $M_\\pi \\sim 400$ MeV case highlighted by the $^{\\star}$, we have subtracted the angle given in Ref.~\\cite{Moir:2013yfa} from $90^{\\circ}$ so that the mass ordering of the states is consistent between the two ensembles.}\n\\label{tab:mixing}\n\\end{center}\n\\end{table}\n\nIn Table \\ref{tab:mixing}, we show the mixing angles for the lightest pairs of $P$-wave ($J^P=1^+$), $D$-wave ($J^{P} = 2^-$) and $ J^{P} = 1^-$ hybrid states extracted using three different operators for the two different ensembles.  The variation between mixing angles determined using the three different operators gives an estimate of the size of the systematic uncertainties as discussed in Ref.~\\cite{Moir:2013yfa}.  The $1^+$ mixing angle from the $\\rho - \\rho_2$ operator in the charm-light sector is closer to the heavy-quark limit value on the $M_\\pi \\sim 240$ MeV ensemble, but the analogous angle in the charm-strange sector does not differ significantly between the ensembles.  For both charm-light and charm-strange mesons, the $2^{-}$ mixing angle is closer to the heavy-quark limit value for the lighter pion mass whereas the $1^-$ hybrid mixing angle shows no significant difference between the two ensembles.\n\nIn all cases on both ensembles, the determined mixing angles lie between the flavour-symmetry limit ($0^{\\circ}$ or $90^{\\circ}$) and the heavy-quark limit values.  This is expected since the charm quark, although much heavier than the light and strange quarks, is not heavy enough for the heavy-quark limit to apply strictly.\n\n\\section{Summary and Outlook}\\label{sec:conclusions}\n\nWe have presented spectra of excited hidden and open-charm mesons obtained from dynamical lattice QCD calculations with a pion mass of approximately $240$ MeV.  The use of distillation in combination with large variational bases of interpolating operators allows us to extract highly excited mesons, while the spin identification scheme has allowed a robust identification of the $J^{P(C)}$ of states as high as spin four, including states with exotic quantum numbers.  \nThe majority of mesons we extract can be interpreted in terms of the $n^{2S+1}L_{J}$ pattern expected from quark potential models.  However, excess states, with both exotic and non-exotic quantum numbers, that do not fit this pattern are also determined.  By examining the operator overlaps we identify these as hybrid mesons, i.e. having excited gluonic degrees of freedom.  The supermultiplets of hybrid mesons follow a pattern consistent with a quark-antiquark combination in $S$ or $P$-wave coupled to a $1^{+-}$ gluonic excitation.\nThe pattern and energy scale of hybrids are the same as that found in the light meson and baryon sectors~\\cite{Dudek:2010,Dudek:2011b,Dudek:2012,Edwards:2012fx,Dudek:2013yja}, studies of charmed baryons~\\cite{Padmanath:2013zfa,Padmanath:2015jea} and in our earlier work on charmonia and open-charm mesons~\\cite{Liu:2012,Moir:2013ub}.\n\nComparing the spectra to those from a similar lattice calculation with a pion mass of approximately $400$ MeV, we find that the overall qualitative features \nare the same and, even in the case of charm mesons with a valence light quark, we find only small quantitative differences. \nThe hybrid mesons appear to show a mild increase in mass as the pion mass is decreased but the pattern of states and supermultiplet structure is unchanged.\n\nWe also compared the spin-singlet -- spin-triplet mixing angles for the lightest pairs of charm-strange and charm-light $P$-wave $(J^{P} = 1^{+})$, $D$-wave $(J^{P} = 2^{-})$ and hybrid $(J^{P} = 1^{-})$ states between the two lattice ensembles.  Using a non-relativistic interpretation of operator overlaps, our results suggest that the mixing angles for the charm-light $1^{+}$ and the charm-light and charm-strange $2^{-}$ states become closer to those expected in the heavy-quark limit as the pion mass is reduced.  Conversely, we find no significant difference in the hybrid $1^{-}$ mixing angles between the two ensembles.\n\nAs discussed earlier, a limitation of these calculations is that we have not accounted for the unstable nature of states above threshold.  \nThis issue has already been addressed for a variety of mesons appearing as bound states and resonances in coupled-channel $D\\pi$, $D\\eta$ and $D_{s}\\bar{K}$ scattering~\\cite{Moir:2016srx}.  The work presented here lays the foundation for extending these scattering calculations to pion masses \ncloser to the physical value, as well as to other scattering channels involving hidden and open-charm mesons. \n\n\\section{Introduction}\\label{sec:introduction}\n\nThe experimental status of the charm sector of Quantum Chromodynamics (QCD) has changed dramatically over the last decade \\cite{PDG2015}.  The discovery of a plethora of unexpected charmonium-like states, commonly known as ``$X, Y, Z$'s'', has highlighted the need for a more complete theoretical understanding of the spectrum.  Many different interpretations have been put forward: some are suggested to be hybrid mesons (a quark-antiquark pair with excited gluonic degrees of freedom) and others two quarks and two antiquarks in a tightly-bound configuration (tetra-quark), a molecular-like combination of two mesons, or a charmonium-like core surrounded by light degrees of freedom (hadro-quarkonium).  There are similar puzzles in the open-charm sector ($D$ and $D_s$ mesons) where the measured masses and widths of the low-lying $D^{*}_{s0}(2317)^{\\pm}$ and $D_{s1}(2460)^{\\pm}$ states are significantly smaller and narrower than expected from quark models.  For some recent reviews see Refs.~\\cite{Brambilla:2010cs,Brambilla:2014jmp,Olsen:2015zcy,Swanson:2015wgq,Prencipe:2015kva}.\n\nIn principle these states can be understood within Quantum Chromodynamics (QCD) using lattice QCD, a non-perturbative, \\textit{ab initio} formulation of the theory.  Spurred on by the experimental situation, there have been many lattice QCD calculations of hidden and open-charm mesons. The majority have focused on lowest-lying states below threshold, achieving unprecedented precision with the various systematic effects under control (some recent examples can be found in Refs.~\\cite{Namekawa:2011wt, McNeile:2012qf, Dowdall:2012ab, Donald:2012ga, Galloway:2014tta}). On the other hand, there have been a number of investigations of excited charmonia and open-charm mesons~\\cite{Dudek:2007,Bali:2011rd, Bali:2011dc, Mohler:2011ke, Bali:2015lka, Kalinowski:2015bwa, Cichy:2016bci}, all of which have some systematic uncertainties not fully accounted for and extract a more limited set of states than we consider here.\n\nIn a previous lattice QCD study, the Hadron Spectrum Collaboration used large bases of interpolating operators with various structures to robustly extract many excited and high-spin states and, crucially, to identify their continuum quantum numbers.  Highlights included the presence of states with exotic quantum numbers (i.e.\\ those forbidden with solely a quark-antiquark pair) and the identification of ``supermultiplets'' of hybrid mesons.  However, these calculations were performed with unphysically-heavy light quarks corresponding to $M_{\\pi} \\sim 400$ MeV.  The results provided useful benchmarks for other approaches such as nonrelativistic effective field theories, for example see Ref.~\\cite{Berwein:2015vca}.\n\nThe current work extends these earlier investigations by performing similar calculations with light-quark masses significantly closer to their physical values, corresponding to $M_{\\pi} \\sim 240$ MeV.  The spectra at the two light quark masses are compared, focusing on the overall qualitative picture and, in particular, whether changes in the pattern of states with exotic quantum numbers or other hybrid mesons are observed.  This allows us to explore the light-quark mass dependence of excited heavy quarkonia which has been suggested to be significant~\\cite{Guo:2012tg}.\n\nIn this study the unstable nature of states above threshold is not considered -- a point discussed in~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} -- and so the spectra should only be considered a guide to the pattern of resonances.  In the charm sector, we have already addressed this limitation for a variety of states appearing as bound-states and resonances in coupled-channel $D\\pi$, $D\\eta$ and $D_{s}\\bar{K}$ scattering~\\cite{Moir:2016srx} for $M_{\\pi} \\sim 400$ MeV and investigations of various other channels involving charm quarks are underway.  This paper lays the foundation for extending those studies to $M_{\\pi} \\sim 240$ MeV, where the additional light-quark mass, closer to the physical value, will enable us to study the evolution with light-quark mass of hidden and open-charm bound-states and resonances.\n\nA number of other investigations of near-threshold bound states, scattering and resonances in the charm sector have appeared over the last few years~\\cite{Ozaki:2012ce, Mohler:2012na, Liu:2012zya, Prelovsek:2013cra, Prelovsek:2013xba, Mohler:2013rwa, Chen:2014afa, Lang:2014yfa, Prelovsek:2014swa, Padmanath:2015era, Lang:2015sba, Chen:2015jwa, Chen:2016lkl,Ikeda:2016zwx}.  There have also been studies addressing the existence of four-quark configurations (mostly considering static heavy quarks)~\\cite{Bicudo:2012qt, Brown:2012tm, Ikeda:2013vwa, Bicudo:2015vta, Bicudo:2015kna, Francis:2016hui, Alberti:2016dru, Peters:2016isf,Bicudo:2016jwl}. However, these are mainly exploratory and more comprehensive calculations as described in Ref.~\\cite{Moir:2016srx} are called for.\n\nThe remainder of the manuscript is organised as follows.  In Section~\\ref{sec:calculation_details} we describe the lattice ensembles used in this study, provide some details on the tuning of the anisotropy and charm-quark mass, and give a brief overview of the analysis of two-point correlation functions.  In Section~\\ref{sec:spectra} we present and interpret the charmonium, $D_s$ and $D$ meson spectra from the calculations with $M_{\\pi} \\sim 240$ MeV.  In Section~\\ref{sec:comparison} we compare these spectra to those from earlier computations with $M_{\\pi} \\sim 400$ MeV and we present a summary in Section~\\ref{sec:conclusions}.\n\n\\section{Charmonium and Open-Charm Spectra}\n\\label{sec:spectra}\n\nIn this section we present the spectra, labelled by $J^{P(C)}$, computed on the $M_{\\pi} \\sim 240$ MeV ensemble.  \nResults for charmonium are described first followed by those for $D_s$ and $D$ mesons.\n\n\n\\subsection{Charmonium}\n\\label{sec:spectra:charmonium}\n\n\n\\begin{figure}[tb]\n\\begin{center}\n     \\includegraphics[width=0.99\\textwidth]{Plots\/charmonium_spinplot.pdf}\n\\end{center}\n    \\caption{Charmonium spectrum up to around $4.5$ GeV labelled by $J^{PC}$; the left (right) panel shows the negative (positive) parity states.  Green, red and blue boxes are the masses computed on our $M_\\pi\\sim 240$ MeV ensemble while black boxes are experimental values from the PDG summary tables~\\cite{PDG2015}. As discussed in the text, we show the calculated (experimental) masses with the calculated (experimental) $\\eta_c$ mass subtracted.  The vertical size of the boxes represents the one-sigma statistical (or experimental) uncertainty on either side of the mean.  Red and blue boxes correspond to states identified as hybrid mesons grouped into, respectively, the lightest and first-excited supermultiplet, as described in the text.  Dashed lines show the location of some of the lower thresholds for strong decay using computed (coarse green dashing) and experimental (fine grey dashing) masses.}   \n  \\label{fig:charmonium_spectrum}\n\\end{figure}\n\nThe charmonium spectrum computed on the $M_\\pi\\sim 240$ MeV ensemble is shown in Figure~\\ref{fig:charmonium_spectrum} and the results are tabulated in \nAppendix~\\ref{app:tables}.\nFor flavour singlets such as charmonium, charge-conjugation, $C$, and parity, $P$, are both good quantum numbers and so states are labelled by $J^{PC}$.  As discussed above, masses are presented after subtracting the $\\eta_c$ mass to reduce the systematic uncertainty arising from tuning the charm quark mass. \nDashed lines indicate the location of some thresholds for strong decay: $\\eta_c \\pi \\pi$ (the lowest threshold if the charm quark and antiquark do not annihilate), $D\\bar{D}$ and $D\\bar{D}^*$.  Since the resonant nature of states above threshold is not investigated in this work, a conservative approach is to only consider the mass values accurate up to the order of the hadronic width~\\cite{Dudek:2010}.\n\nAs found in Ref.~\\cite{Liu:2012}, many of the states with non-exotic $J^{PC}$ follow the $n^{2S+1}L_{J}$ pattern predicted by quark potential models, where $J$ is the total spin of the meson with relative orbital angular momentum $L$, quark-antiquark spin $S$ and radial quantum number $n$.  \nWe find all states up to $J=4$ expected by such models.\n\nFigure~\\ref{fig:charmonium_spectrum} also shows the states (coloured red and blue) that do not fit the $n^{2S+1}L_{J}$ pattern.  Four of these have exotic $J^{PC}$ quantum numbers, $0^{+-}, 1^{-+}, 2^{+-}$, and we find that they, as well as the excess states with non-exotic quantum numbers, have relatively large overlaps onto operators that are proportional to the spatial components of the field strength tensor, $F_{ij}$ (i.e. operators that have a non-trivial gluonic structure), something not seen for the other states in the spectrum. Furthermore, on removing operators proportional to $F_{ij}$ from the variational basis we generally observe a reduction in the quality of the signal for these states. We therefore follow Refs.~\\cite{Dudek:2010,Liu:2012} and interpret these excess states as hybrid mesons.\n\nAs discussed in detail in Ref.~\\cite{Liu:2012}, the hybrid states can be grouped into supermultiplets.  We find that the set $\\left[(0^{-+}, 1^{-+}, 2^{-+}), 1^{--}\\right]$, highlighted in red in Figure~\\ref{fig:charmonium_spectrum}, forms the lightest charmonium hybrid supermultiplet, while the\nstates highlighted in blue, $(0^{++}, 1^{++}, 2^{++})$, $(0^{+-}, 1^{+-}, 1^{+-}, 1^{+-}, 2^{+-}, 2^{+-}, 3^{+-})$, form the first excited hybrid supermultiplet. These patterns are consistent with a quark-antiquark pair coupled to a $1^{+-}$ gluonic excitation; the lightest hybrid supermutiplet has the quark-antiquark pair in $S$-wave and the first excited hybrid supermultiplet has it in $P$-wave.\nThe lightest hybrids appear $\\sim 1.2$ - $1.3$ GeV above the lightest $S$-wave meson multiplet.\nThis pattern of hybrids and their energy scale are consistent with what was found in the light meson and baryon sectors~\\cite{Dudek:2010,Dudek:2011b,Dudek:2012,Edwards:2012fx,Dudek:2013yja}, studies of charmed baryons~\\cite{Padmanath:2013zfa,Padmanath:2015jea} and in our previous work on charmonia and open-charm mesons~\\cite{Liu:2012,Moir:2013ub}.\n\nAs noted in Section~\\ref{sec:calculation_details}, these calculations are performed at a single spatial lattice spacing. On the $400$ MeV ensemble we estimated a scale of $40$ MeV for the discretisation uncertainty arising from $\\mathcal{O}(a_{s})$ corrections to charmonia~\\cite{Liu:2012}. Since the $240$ MeV ensemble has the same spatial lattice spacing, we expect the 40 MeV scale to also be a reasonable estimate for the discretisation uncertainty here.\n\n\n\\subsection{$D_s$ and $D$ mesons}\n\\label{sec:spectra:opencharm}\n\nFor flavoured mesons, such as $D_{s}$ and $D$, charge conjugation is no longer a good quantum number and states are labelled only by $J^{P}$. Figures~\\ref{fig:Ds_spectrum} and \\ref{fig:D_spectrum} show the $D_s$ and $D$ meson spectra respectively; these results are tabulated in Appendix~\\ref{app:tables}.\nMasses are presented with half the mass of the $\\eta_c$ subtracted in order to reduce the systematic uncertainty arising from tuning the charm quark mass.\nDashed lines indicate some of the lower strong-decay thresholds ($DK$ for the $D_s$ spectrum and $D\\pi$ and $D^{*}\\pi$ for the $D$ meson spectrum).\n\nAs for charmonium, the $D_s$ and $D$ spectra can be interpreted in terms of a $n^{2S+1}L_{J}$ pattern and we identify complete $S,P,D$ and $F$-wave multiplets.\nWithin the negative parity sector of both spectra, there are four states, highlighted in red, that do not appear to fit this pattern.  Due to their relatively large overlap with operators featuring a non-trivial gluonic structure, these are identified as the members of the lightest hybrid meson supermultiplet in each flavour sector.  The pattern is again consistent with a $1^{+-}$ gluonic excitation coupled to an $S$-wave quark-antiquark pair and they appear at an energy $\\sim 1.2$ - $1.3$ GeV above the lightest conventional multiplet.  However, unlike in charmonium, the first excited hybrid supermultiplet is not \nrobustly determined for open-charm mesons and is not shown here. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Plots\/Ds_spin_spectrum.pdf}\n\\caption{$D_s$ meson spectrum labelled by $J^P$; the left (right) panel shows the negative (positive) parity states.  Green and red boxes are the masses computed on the $M_\\pi\\sim 240$ MeV ensemble while black boxes are experimental masses of the neutral $D$ mesons from the PDG summary tables~\\cite{PDG2015}.  As discussed in the text, the calculated (experimental) masses are shown with with half the calculated (experimental) $\\eta_c$ mass subtracted.  The vertical size of the boxes indicates the one-sigma statistical (or experimental) uncertainty on either side of the mean.  Red boxes show states identified as constituting the lightest hybrid supermultiplet, as described in the text.  \nDashed lines indicate the $DK$ threshold using computed (coarse green dashing) and experimental (fine grey dashing) masses.}\n\\label{fig:Ds_spectrum}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Plots\/D_spin_spectrum.pdf}\n\\caption{As Figure~\\ref{fig:Ds_spectrum} but for the $D$ meson spectrum.\nDashed lines show the $D\\pi$ and $D^\\ast\\pi$ thresholds using computed (coarse green dashing) and experimental (fine grey dashing) masses.}\n\\label{fig:D_spectrum}\n\\end{center}\n\\end{figure}\n\n\n\\section{Tables of Results}\\label{app:tables}\n\nIn Tables \\ref{tab:charmonium_summary}, \\ref{tab:Ds_summary} and \\ref{tab:D_summary} we present numerical values for, respectively, the charmonium, $D_{s}$ and $D$ meson masses obtained for $M_\\pi \\sim 240$~MeV. \nMasses are given in MeV with either the mass of the $\\eta_{c}$ subtracted (charmonium) or half the mass of the $\\eta_{c}$ subtracted (open-charm mesons)\nin order to minimise the systematic uncertainty in tuning the charm quark mass. \nIn all cases the quoted error corresponds to the (one-sigma) statistical uncertainty.  As discussed earlier, above the lowest multi-hadron threshold in each channel states can decay strongly into lighter hadrons and, aside from any other systematic uncertainties, we only expect the masses to be correct up to around the width of the state~\\cite{Dudek:2010}.\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^{PC}$ &  &  & $M-M_{\\eta_c}$ &\\hspace{-0.7cm} $(MeV)$  &  & \\\\\n\\hline\n$0^{-+}$ & 0 & 679(6) & 1197(7) & 1295(18)&  &  \\\\\n$1^{--}$ & 88(1) & 728(7) & 865(7) & 1316(17)& 1345(27) & 1427(17) \\\\\n$2^{--}$ & 879(7) & 1352(21) &  & &  &  \\\\\n$2^{-+}$ & 888(7) & 1414(24) & 1472(21) &  &  &  \\\\\n$3^{--}$ & 902(6) & 1442(18) & 1484(40) & &  &  \\\\\n$4^{-+}$ & 1474(19)  &  &  & &  &  \\\\\n$4^{--}$ & 1450(18) &  &  & &  &  \\\\\n\\hline\n$0^{++}$ & 466(3)  & 989(10) & 1485(25) & 1607(46)&  &  \\\\\n$1^{++}$ & 531(4) & 1038(12) & 1486(25) & 1534(35)&  &  \\\\\n$1^{+-}$ & 545(4) & 1041(12) & 1454(23) & 1587(27) & 1643(47) & 1681(53)   \\\\\n$2^{++}$ & 571(4) & 1065(13) & 1154(11) & 1173(11) & 1639(32) &  \\\\\n$3^{++}$ & 1166(11) &  &  & &  &  \\\\\n$3^{+-}$ & 1173(11) & 1660(34) &  & &  &  \\\\\n$4^{++}$ & 1181(12) &  &  & &  &  \\\\\n\\hline\n$1^{-+}$ & 1326(23) &  &  & &  &  \\\\\n$0^{+-}$ & 1453(27) &  &  & &  &  \\\\\n$2^{+-}$ & 1518(18) & 1647(26) &  & &  &  \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the charmonium spectrum presented in Figure \\ref{fig:charmonium_spectrum}.  Masses are shown with $M_{\\eta_{c}}$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:charmonium_summary}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^P$ &  &  & $M-M_{\\eta_c}\/2$& \\hspace{-0.7cm}$(MeV)$  &  &   \\\\\n\\hline\n$0^-$ & 467(11) & 1225(17) & 1679(27) & 1873(31) & & \\\\\n$1^-$ & 593(12) & 1286(12) & 1399(21) & 1740(30) & 1891(33) & 1898(38)\\\\\n$2^-$ & 1424(19) & 1440(20) & 1952(35) & 1993(36) & 2002(32) & \\\\\n$3^-$ & 1481(19) & 2029(28) &  &  &  & \\\\\n$4^-$ & 2075(29) & 2109(31) &  &  &  & \\\\\n\\hline\n$0^+$ & 886(14)  & 1567(35) & 1934(51) & &  & \\\\\n$1^+$ & 1022(15) & 1064(16) & 1612(25) & 1670(26) & 1929(44) & 2030(35) \\\\\n$2^+$ & 1100(15) & 1675(24) & 1773(23) & 2000(37) & & \\\\\n$3^+$ & 1766(22) & 1779(22) &  &  &  & \\\\\n$4^+$ & 1811(24) &  &  &  &  & \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the $D_{s}$ meson spectrum presented in Figure \\ref{fig:Ds_spectrum}.  Masses are shown with $M_{\\eta_{c}} \/ 2$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:Ds_summary}\n\\end{center}\n\\end{table}\n\n\n\\clearpage\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^P$ &  &  & $M-M_{\\eta_c}\/2$ &\\hspace{-0.7cm}$(MeV)$  &  &   \\\\\n\\hline\n$0^-$ & 382(10) & 1138(17) & 1569(26) & 1783(29) & 2176(37)  &\\\\\n$1^-$ & 509(11) & 1233(22) & 1315(21) & 1610(33) & 1801(34) & 1838(36)\\\\\n$2^-$ & 1352(19) & 1429(20) & 1912(34) & 1935(34) & & \\\\\n$3^-$ & 1441(19) & 2032(26) &  &  &  & \\\\\n$4^-$ & 2037(29) & &  &  &  & \\\\\n\\hline\n$0^+$ & 770(15) & 1494(25) & 2201(45) & 1874(26) &  & \\\\\n$1^+$ & 881(17) & 984(14) & 1559(27) & 1603(26) & &\\\\\n$2^+$ & 1020(16) & 1623(26) & 1665(29) & 1925(36) & & \\\\\n$3^+$ & 1724(21) & 1743(21) &  &  &  & \\\\\n$4^+$ & 1804(22) &  &  &  &  & \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the $D$ meson spectrum presented in Figure \\ref{fig:D_spectrum}.  Masses are shown with $M_{\\eta_{c}} \/ 2$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:D_summary}\n\\end{center}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThis paper studies the dynamics of fluid drops or bubbles immersed in another fluid filling a porous media under the action of gravity. This process is governed by the classical Darcy's law \n\\begin{equation}\\label{Darcy}\n\\frac{\\mu(x,t)}{\\kappa(x,t)} u(x,t)=-\\nabla p(x,t)-g(0,\\rho(x,t)), \n\\end{equation} \nwhere  $u$ is the velocity of the fluid, $p$ is the pressure, $\\rho$ is the density and $\\mu$ is the viscosity of the fluid. Above $x\\in\\mathbb{R}^2$ and $t\\geq 0$.  Here the medium is assumed to be homogeneous so  that the permeability $\\kappa(x,t)=\\kappa \\ge0$ is constant, as is the gravitational acceleration $g>0$. While Darcy's law was first derived experimentally \\cite{Darcy56}, it can be rigorously obtained through homogenization \\cite{Tartar89,Hornung97}. This physical scenario is mathematically analogous to the evolution of an incompressible flow in a Hele-Shaw cell \\cite{Gancedo2017} where the fluid is set inside two parallel plates that are close enough together so that the resulting dynamics are two dimensional. In particular, the results in this paper can be applied to the Hele-Shaw problem. \n\n\n\n\nThe presence of two immiscible fluids is modeled by taking the viscosity $\\mu$ and the density $\\rho$ as piece-wise constant functions:\n\\begin{equation}\\label{patchsolution}\n\\mu(x,t)=\\left\\{\\begin{array}{rl}\n\\mu^1,& x\\in D(t),\\\\\n\\mu^2,& x\\in \\mathbb{R}^2\\smallsetminus D(t),\n\\end{array}\\right.\n\\quad \n\\rho(x,t)=\\left\\{\\begin{array}{rl}\n\\rho^1,& x\\in D(t),\\\\\n\\rho^2,& x\\in \\mathbb{R}^2\\smallsetminus D(t),\n\\end{array}\\right.\n\\end{equation}\nwhere $D(t)$ is a simply connected bounded domain, namely, the \\textit{bubble}. Thus, there is a sharp interface between the fluids, moving with the flow, which we assume to be incompressible:\n\\begin{equation}\\label{incom}\n\\nabla \\cdot u(x,t)=0.\n\\end{equation}\nWe consider the physically relevant case where surface tension at the free boundary is taken into consideration. The Laplace-Young's formula then states that \\cite{Hou94}:\n\\begin{equation}\\label{surfacetension}\np^1(x,t)-p^2(x,t)=\\sigma K(x,t),\\hspace{1cm}x\\in\\partial D(t),\n\\end{equation}\nwhere $K(x,t)$ denotes the curvature of the curve $\\partial D(t)$, $\\sigma>0$ is the constant surface tension coefficient and $p^1(x,t)$, $p^2(x,t)$ are the limits of the pressure at $x$ from inside and outside, respectively.\nWe are then dealing with the Muskat problem, where the main mathematical interest is to study the dynamics of the free boundary $\\partial D(t)$, especially between water and oil \\cite{Muskat34}. \nIt is remarkable that the evolution equation for the free boundary is well-defined even though the velocity is not continuous. The discontinuity in the velocity is due to the density, viscosity and pressure jumps. But the interface evolution is dictated only by the normal velocity, which is continuous by the incompressibility condition. \n\nIn this sense it is indeed possible to obtain a self-evolution equation for the interface $\\partial D(t)$ which is called the contour evolution system.   This system is equivalent to the Eulerian-Lagrangian formulation \\eqref{Darcy}, \\eqref{patchsolution}, \\eqref{incom}, \\eqref{surfacetension} understood in a weak sense.\nDue to the irrotationality of the velocity in each domain $D(t)$, the vorticity is concentrated on the interface $\\partial D(t)$. That is, the vorticity is given by a delta distribution as follows\n$$\n\\nabla^\\perp\\cdot u(x,t)=\\omega(\\alpha,t)\\delta(x=z(\\alpha,t)),\n$$\nwhere $\\omega(\\alpha,t)$ is the amplitude of the vorticity and $z(\\alpha,t)$ is a parameterization of $\\partial D(t)$ with\n$$\n\\partial D(t)=\\{z(\\alpha,t)=(z_1(\\alpha,t), z_2(\\alpha,t)):\\, \\alpha\\in[-\\pi,\\pi]\\}.\n$$\nThe Biot-Savart law then yields that\n\\begin{equation*}\nu(x,t)=\\frac1{2\\pi} \\text{pv}\\int_{-\\pi}^\\pi \\frac{(x-z(\\beta,t))^\\perp}{|x-z(\\beta,t)|^2}\\omega(\\beta,t)d\\beta,\\hspace{1cm}x\\neq z(\\beta,t),\n\\end{equation*}\nand taking limits in the normal direction to $z(\\alpha,t)$ one finds\n\\begin{equation*}\n\\begin{aligned}\nu_1(z(\\alpha,t),t)&=BR(z,\\omega)(\\alpha,t)-\\frac12\\frac{\\omega(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^2}\\partial_{\\alpha}z(\\alpha,t),\\\\\nu_2(z(\\alpha,t),t)&=BR(z,\\omega)(\\alpha,t)+\\frac12\\frac{\\omega(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^2}\\partial_{\\alpha}z(\\alpha,t),\n\\end{aligned}\n\\end{equation*}\nwhere $BR$ is the Birkhoff-Rott integral that is given by\n\\begin{equation}\\label{eqBR}\nBR(z,\\omega)(\\alpha,t)=\\frac1{2\\pi} \\text{pv}\\int_{-\\pi}^\\pi \\frac{(z(\\alpha,t)-z(\\beta,t))^\\perp}{|z(\\alpha,t)-z(\\beta,t)|^2}\\omega(\\beta,t)d\\beta.\n\\end{equation}\nTaking the dot product with $\\partial_\\alpha z$ in the above equations for $u_1$ and $u_2$ and subtracting one from the other, one then finds that the vorticity strength is given by the jump in the tangential velocity\n\\begin{equation*}\n\\omega(\\alpha,t)=\\left(u_2(z(\\alpha,t),t)-u_1(z(\\alpha,t),t)\\right)\\cdot\\partial_{\\alpha}z(\\alpha,t).\n\\end{equation*}\nThen using Darcy's law \\eqref{Darcy} yields the non-local implicit identity\n\\begin{multline}\\label{eqOmega}\n\\omega(\\alpha,t)=2 A_\\mu |\\partial_{\\alpha}z(\\alpha,t)| \\mathcal{D}(z,\\omega)(\\alpha,t) +2A_\\sigma\\partial_{\\alpha}K(z(\\alpha,t))-2A_\\rho\\partial_{\\alpha}z_2(\\alpha,t),\n\\end{multline}\nwhere\n\\begin{equation}\\label{Dz}\n\\mathcal{D}(z,\\omega)(\\alpha,t)=-BR(z,\\omega)(\\alpha,t)\\cdot\\frac{\\partial_{\\alpha}z(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|},\n\\end{equation}\nand\n\\begin{equation}\\label{AmuAsigmaArho}\nA_\\mu=\\frac{\\mu_2-\\mu_1}{\\mu_2+\\mu_1}, \\quad \nA_{\\sigma}=\\frac{\\kappa \\sigma}{\\mu_2+\\mu_1}, \\quad \nA_\\rho = g\\kappa \\frac{\\rho_2-\\rho_1}{\\mu_2+\\mu_1}.\n\\end{equation}\nFurther, in \\eqref{eqOmega} the curvature is given by\n\\begin{equation}\\label{curvature}\n\tK(\\alpha,t)=\\frac{\\partial_{\\alpha}z(\\alpha,t)^\\perp\\cdot\\partial_{\\alpha}^2z(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^3}.\n\\end{equation}\nSince the fluids are immiscible, the interface is just advected by the normal velocity of the fluid flow:\n\\begin{equation*}\nz_t(\\alpha,t)\\cdot \\partial_\\alpha z(\\alpha,t)^\\perp=BR(z,\\omega)(\\alpha,t)\\cdot \\partial_\\alpha z(\\alpha,t)^\\perp.\n\\end{equation*}\nTherefore a tangential velocity $T(z(\\alpha,t))$ can be introduced to change the parametrization of the interface, without altering its shape.  Let $\\partial_\\alpha z(\\alpha,t)=z_\\alpha(\\alpha,t)$. Then we denote the unit tangent and normal vectors by\n\t\\begin{equation}\\label{vectors}\n\t\\tau(\\alpha,t)=\\frac{z_\\alpha(\\alpha,t)}{|z_\\alpha(\\alpha,t)|},\\quad n(\\alpha,t)=\\frac{z_\\alpha(\\alpha,t)^\\perp}{|z_\\alpha(\\alpha,t)|}.\n\t\\end{equation}\nWithout changing the shape of the interface we can replace the above equation by \t\n\\begin{equation}\\label{eqcurve}\nz_t(\\alpha,t)=\\big(BR(z,\\omega)(\\alpha,t)\\cdot n(\\alpha,t)\\big)n(\\alpha,t)+T(z(\\alpha,t))\\tau(\\alpha,t).\n\\end{equation}\nTherefore we have a closed system of equations for the the contour evolution system with (\\ref{eqcurve}), (\\ref{eqBR}), (\\ref{eqOmega}), and  (\\ref{Dz}).\n\nGiven its origins in petrochemical engineering and its mathematical equivalence with Hele-Shaw flows \\cite{SaffmanTaylor58}, the Muskat problem has long attracted a lot of attention from physics \\cite{Bear72,Saffman86}. Mathematically, the Muskat problem poses many challenges, since even the well-posedness of the problem is not always guaranteed. Indeed, when one neglects surface tension, the well-posedness depends on the  Rayleigh-Taylor condition (which is also called the Saffman-Taylor condition for the Muskat problem). If the fluids have different densities, this condition requires the denser fluid to be below the less dense fluid. When this condition is satisfied, i.e., in the stable setting \\cite{Ambrose04}, local-in-time existence for large initial data is known for both density and viscosity jump cases in 2d and 3d \\cite{CG07,CCG11, CCG13,CGS16}\nfor subcritical spaces\n\\cite{CGSV17,Mat16,AlazardOneFluid2019,AlazardLazar2019,NguyenPausader2019}.\nHowever, finite time singularities can arise even from these stable configurations. As a matter of fact, the Muskat problem was the first incompressible model where blow-up was proved starting with well-posed initial data \\cite{CCFGL12,CCFG13,GG14,CCFG16}.\n\nFrom the previous considerations, it is an important question to determine under which conditions the solution exists and remains regular globally in time.  For the non-surface tension case, the global existence in the stable setting was first obtained  for small enough initial data in subcritical norms, allowing both density and viscosity jumps \\cite{SCH04,CG07,EM11, CGS16} and later for some critical norms \\cite{BSW14,CGSV17}. Very recently, global well-posedness results appeared that allow initial data of \\textit{medium} size in critical spaces, meaning initial data explicitly bounded independent of any parameter: first only for the density jump case \\cite{CCGS13,CCGRS16,Cam17,Cam20}, and later extended to the density-viscosity jump case \\cite{GGPS19}.  In particular in \\cite{GGPS19} there is a medium-size bound for the initial data that is independent of any parameter of the system when $|A_\\mu|=1$, and that value of the bound for the initial data is improved when $|A_\\mu|<1$. In all these results, the magnitude of the slope of the first derivative appears as a crucial quantity. However, this restriction is removed in \\cite{CordobaLazar2018,GancedoLazar2020} by assuming smallness in the critical $L^2$ based Sobolev norm.\n\n\n\n On the other hand, in the unstable scenario, the problem is ill-posed in all Sobolev spaces $H^s$, $s>0$ \\cite{GGPS19}, unless surface tension is taken into account. In that case, surface tension controls the instabilities at large scales, giving well-posedness. Classical results for this scenario can be found in \\cite{DuchonRobert1984,Chen93,ES97}. See the recent work \\cite{NguyenST2019} for low regularity initial data. Unstable scenarios are known \\cite{Ott97} which exhibit exponential growth locally in time of low order norms \\cite{GHS07}, and finger shaped unstable stationary solutions were also studied \\cite{EM11}. In particular, Rayleigh-Taylor stable solutions with surface tension converge to the solution without surface tension \\cite{Ambrose2014} with optimal decay rate or low regularity \\cite{FlynnNguyen2020}. Here, this is not the case as the scenario we deal with is Rayleigh-Taylor unstable. \nRecently, while writing this paper, unstable fluid layers have been proved to exist globally in time for initial near flat configurations \\cite{GG-BS2019}.\n \nIn this paper, we aim to improve the understanding of the effects produced by the surface tension for bubble-shaped interfaces. In particular, we consider the movement of fluid bubbles under the effect of gravity in another fluid with both different densities and viscosities. This is a highly unstable situation, as the Rayleigh-Taylor condition cannot hold for a closed curve. The function that provides the Rayleigh-Taylor condition has mean zero in this scenario. Moreover, as one expects, we will show that a less dense bubble moves upwards. But this means that on the top part of the interface, the less viscous fluid may push the more viscous one and the denser one is on top of the lighter one: both classic conditions in the linear Rayleigh-Taylor analysis are violated here. So that in our scenario, gravity effects make hard to find global-in-time control.\nPrevious results dealing with this setting \\cite{CP93,YT11,YT12} assumed no gravity force (i.e., $g=0$ or no density jump) and required small initial data in high regularity spaces (such as $H^r$ for $r\\geq 4$).\n\n We show here that even without surface tension, circle shaped curves are steady state solutions evolving vertically due to gravity. Furthermore, this surprising state in this unstable configuration allows to find global-in-time existence for capillarity bubbles. We will show that if the initial interface of a bubble is close to a circle with respect to a  constant depending on the dimensionless constants\n$$|A_\\mu|\\quad\\mbox{and}\\quad  \\frac{R^2|A_\\rho|}{A_\\sigma},\\quad\\mbox{with}\\quad \\pi R^2=|D(0)|,$$\n then the solution exists globally in time and, moreover, it becomes instantly analytic. In particular, in our proof it is possible to compute the explicit numerical condition that the initial data must satisfy. It is interesting to notice that only two quantities are involved, where the second represents the ratio between gravity force per length and surface tension, \n$$\\frac{|A_{\\rho}|R^2}{A_\\sigma}=\\frac{gR^2|\\rho_2-\\rho_1|}{\\sigma}.$$\nWe will also show that these bubbles converge exponentially fast in time to a circle that moves vertically with constant velocity equal to $A_\\rho$ (upwards if $A_\\rho>0$). Due to the incompressibility condition, the area of the bubble is preserved during the process.\nWe give precise statements of these results in Section \\ref{MainResults}. In next section, we provide the contour equations we use throughout the paper.\n\nNote that the parameterization that is used in for instance \\cite{CG07,CCGS13,CCGRS16,Cam17,GGPS19} is difficult to use in our scenario (those results are close to a horizontal line while in contrast the results in this paper are close to a circle) because our system in general sends the solution to a nearby circular steady state, and not to the one that we linearize around. The steady state that the solution converges to is determined by the dynamics and, without another conservation law that may not exist, then the limit can not be predicted by the initial data alone. In particular, the standard parametrization for star-shape bubbles given by\n\\begin{equation*}\n    z(\\alpha,t)=R(1+f(\\alpha,t))(\\cos{(\\alpha)},\\sin{(\\alpha)})\n\\end{equation*}\ndoes not do a good job of describing the nearby circular steady states, and therefore it is hard to use in this context. In particular, unless the center is the origin, which one cannot know \\textit{a priori}, circles parametrized in this form do not have a simple expression.\nFrom the analytical point of view, \nlooking at the decay on the Fourier side, it is easy to find that there is no dissipation at the linear level for the $\\pm 1$ Fourier coefficients, which corresponds to the fact that the center of a circle parametrized in this way is given by the $\\pm 1$ Fourier coefficients of $f$. At the nonlinear level, these Fourier coefficients are present and mixed in the evolution together with the rest of them, making it difficult to control globally in time. In order to handle this issue we reparametrize the interface getting a tangent vector to the curve with length independent of the parameter $\\alpha$  so that $\\partial_\\alpha |z_\\alpha|=0$  \\cite{Hou94}. Therefore, the system can be reformulated in terms of the angle formed between the tangent and the horizontal, $\\alpha+\\vartheta(\\alpha,t)$, and the length of the curve as follows\n$$\nz_\\alpha(\\alpha,t)=\\frac{L(t)}{2\\pi}(\\cos(\\alpha+\\vartheta(\\alpha,t)),\\sin(\\alpha+\\vartheta(\\alpha,t))).\n$$\nThe main unknown to control in this setting is $\\vartheta$. \n\nIn this parametrization circles correspond to a constant value of $\\vartheta$. \nThe evolution  of the zero frequency of $\\vartheta$ is decoupled from the rest.\nWhile the $\\pm 1$ are also neutral in this formulation, the simple compatibility condition\n$$\n\\int_{-\\pi}^\\pi z_\\alpha(\\alpha,t)d\\alpha=0\n$$\nused in \\cite{YT11,YT12}, allows us to control the $\\pm 1$ Fourier coefficients of $\\vartheta$ in terms of the higher modes.   For the higher Fourier modes we can use the dissipation due to surface tension. All the frequencies, together with the initial condition, determine the evolution of the center of the bubble.\n\nOn the other hand, in the analysis done around circles, it is possible to check that the Fourier coefficients of different frequencies interact together in the evolution, even at the linear level. If the ratio $|A_\\rho|R^2\/A_\\sigma$  between gravity and surface tension forces is large, it is not straightforward how to take advantage of the dissipation. Thus it is not clear how to obtain the global-in-time result in terms only of the size of the initial data and not upon the size of the parameters. In order to obtain a global result that does not rely on the size of the physical parameters of the problem, we preform a transformation in Fourier space of the infinite-dimensional nonlinear system and we prove that this transformation \\textit{diagonalizes} the linear system so that our result holds for any size of the physical parameters.  In particular, we show that it is possible to obtain explicitly the size of the smallness constant. Finally, the analysis we perform is for low regularity initial data, ($z_0(\\alpha)\\in C^{1,\\frac12}(\\mathbb T)$), allowing unbounded  initial curvature and providing instant (analytic) smoothing. \n\n\n\n\\subsection{Outline} The rest of the paper is structured as follows.  In Section \\ref{formulation}, we explain the contour dynamics formulation of the system of equations for the interface and we derive the full linearization.  Section \\ref{MainResults} records the notation that will be used in the rest of the paper and explains the main theorem proving global existence, uniqueness and exponential large time decay.  Then Section \\ref{IFTSection} gives the proof of the implicit function theorem to obtain the implicit relation between $\\hat{\\theta}(\\pm 1)$ and the higher Fourier modes.\nIn Section \\ref{sec:FourierRandS} we prove the Fourier multiplier estimates for the operators $\\mathcal{R}$ and $\\mathcal{S}$.\nSection \\ref{secw} proves the a priori estimates on the vorticity strength $\\omega$.\nIn Section \\ref{secanalytic} we use all the previous estimates to prove the global existence and instant analyticity of solutions.\nLastly in Section \\ref{sec:uniqueness} we explain the proof of uniqueness.\n\n\n\n\n\n\\section{Contour dynamics formulation}\\label{formulation}\n\n\nIn this section we introduce the contour evolution equations that will be used throughout the paper. We suppress the dependence in $t$ for clarity of notation. We note that in the introduction it was convenient to introduce the system using vector notation.  However in the rest of the paper, we will study the equation using complex notation.    In Section \\ref{sec:ComplexVector} we explain some complex notation used in the rest of this paper.  In Section \\ref{subsec:parametrization}, we rewrite the equations (\\ref{eqcurve}), (\\ref{eqBR}), (\\ref{eqOmega}), (\\ref{Dz}) in terms of the length of the curve and the angle of the tangent vector \\cite{Hou94}.  Then in Section \\ref{subsec:evolutionSystem} we derive an equivalent expression for the evolution of the length of the curve.  \nIn Section \\ref{sec:FourierTransCalc}, we explain the calculations that we will use involving the Fourier transform in our further decompositions.\nLastly in Section \\ref{sec:linearization}  we decompose the equations into linear and nonlinear parts.  In particular, the calculation of the expression for the linearized operator is given in Proposition \\ref{linearfourier}.\n\n\n\\subsection{Complex notation and vector notation}\\label{sec:ComplexVector}\nIn particular given $x=(x_1, x_2)$ and $y=(y_1, y_2)$ in vector notation and given $z=x_1+ix_2$ and $w=y_1+iy_2$ in complex notation, then the inner product is expressed as\n\\begin{equation}\\notag\n    x\\cdot y = x_1 y_1 + x_2 y_2 = \\text{Re}\\hspace{0.05cm}(\\overline{z} w).    \n\\end{equation}\nHere $\\overline{z} = x_1 - i x_2$ is the complex conjugate.  Similarly in two dimensions in vector notation we can write  $x \\wedge y \\overset{\\mbox{\\tiny{def}}}{=} x_1 y_2 - x_2 y_1$ in vector notation and this is equal to $\\text{Im}\\hspace{0.05cm}{(\\overline{x}y)}$ in complex notation.  Then for a vector the perpendicular is $x^\\perp = (-x_2, x_1)$, and in complex notation the perpendicular is $iz= -x_2 + i x_1$.  We will use the complex notation in most of the rest of the paper.\n\n\n\n\n\\subsection{Parametrization}\\label{subsec:parametrization}\n\nNow we define $\\vartheta(\\alpha)$ so\nthat $\\alpha+\\vartheta(\\alpha)$ is the angle formed between the tangent to the curve and the horizontal. In complex notation, this means that\n\\begin{equation}\\label{param}\n    z_\\alpha(\\alpha)=|z_\\alpha(\\alpha)|e^{i(\\alpha+\\vartheta(\\alpha))}.\n\\end{equation}\nIn this formulation the normal and tangential vectors from \\eqref{vectors} are \n\\begin{equation*}\n    n(\\alpha)=ie^{i(\\alpha+\\vartheta(\\alpha))}, \\quad \\tau(\\alpha) = e^{i(\\alpha+\\vartheta(\\alpha))}.\n\\end{equation*}\nWe will then denote the normal velocity by $U(\\alpha)$ with\n\\begin{equation}\\label{U}\n    U(\\alpha)=\\text{Re}\\hspace{0.05cm}(\\overline{BR}(z,\\omega)(\\alpha) n(\\alpha))=\\text{Re}\\hspace{0.05cm}(\\overline{BR}(\\omega)(\\alpha) ie^{i(\\alpha+\\vartheta(\\alpha))}),  \n\\end{equation}\nwith the Birkhoff-Rott integral \\eqref{eqBR} given by\n\\begin{equation*}\n    \\overline{BR}(z,\\omega)(\\alpha)=\\frac{1}{2\\pi i}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\beta)}{z(\\alpha)-z(\\beta)}d\\beta.\n\\end{equation*}\nNote that $\\overline{BR}(z,\\omega)$ is the complex conjugate of \\eqref{eqBR} written in complex notation, this holds in particular since $\\omega(\\beta)$ is seen to be real as in \\eqref{omega} below.\nThese expressions can be written in terms of $z_\\alpha(\\alpha)$ by noticing that\n\\begin{equation*}\n    z(\\alpha)-z(\\alpha-\\beta)=\\int_0^\\alpha z_\\alpha(\\eta)d\\eta-\\int_0^{\\alpha-\\beta} z_\\alpha(\\eta)d\\eta =\\beta\\int_0^1 z_\\alpha(\\alpha+(s-1)\\beta)ds.\n\\end{equation*}\nThe equation \\eqref{eqcurve} then reads as follows\n\\begin{equation}\\label{eqcurve.theta}\n    z_t(\\alpha)=U(\\alpha)n(\\alpha)+T(\\alpha)\\tau(\\alpha).\n\\end{equation}\nTaking a derivative in $\\alpha$ and projecting into normal and tangential components, we obtain the evolution equations for $\\vartheta(\\alpha)$ and $|z_\\alpha(\\alpha)|$:\n\\begin{equation}\\label{thetaeqaux}\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{1}{|z_\\alpha(\\alpha)|}\\Big(U_\\alpha(\\alpha)+T(\\alpha)(1+\\vartheta_\\alpha(\\alpha))\\Big),\\\\\n    |z_\\alpha(\\alpha)|_t&=T_\\alpha(\\alpha)-(1+\\vartheta_\\alpha(\\alpha))U(\\alpha).\n\\end{aligned}\n\\end{equation}\nNow, we can choose a tangential velocity $T(\\alpha)$ so that the parametrization of $z(\\alpha)$ has a tangent vector whose modulus does not depend on $\\alpha$. Indeed, we impose\n\\begin{equation}\\label{modul}\n    |z_\\alpha(\\alpha)|=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}|z_\\alpha(\\eta)|d\\eta=\\frac{L(t)}{2\\pi},\n\\end{equation}\nwhere $L(t)$ is the length of the curve at time $t$. We then differentiate in time the equation above and use equation \\eqref{thetaeqaux}$_2$ twice to obtain that\n\\begin{equation}\\label{T}\n    T(\\alpha)=\\int_0^{\\alpha}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta+T(0),\n\\end{equation}\nwhere $T(0)$ simply provides a change of frame in the parametrization. \nTherefore, after substitution of this expression of $T(\\alpha)$ into \\eqref{thetaeqaux} and using the relation $|z_\\alpha(\\alpha)|=\\frac{L(t)}{2\\pi}$, the evolution system in terms of $\\vartheta(\\alpha)$ and $L(t)$ is the following\n\\begin{equation}\\label{eqcurve3}\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{2\\pi}{L(t)} U_\\alpha(\\alpha)+\\frac{2\\pi}{L(t)}T(\\alpha)(1+\\vartheta_\\alpha(\\alpha)),\\\\\n    L_t(t)&=-\\int_{-\\pi}^{\\pi} (1+\\vartheta_\\alpha(\\alpha))U(\\alpha)d\\alpha,\n\\end{aligned}\n\\end{equation}\nwhere $T(\\alpha)$ is defined in \\eqref{T}, with $T(0)$ free to choose, \nand $U(\\alpha)$ is given by \\eqref{U} with\n\\begin{equation}\\label{BR}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{1}{ i L(t) }\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\vartheta(\\alpha+(s-1)\\beta)}ds}\\frac{d\\beta}{\\beta}.\n\\end{equation}\nRecalling the expression of the curvature in terms of the angle using \\eqref{curvature}, \n\\begin{equation*}\n    K(z)(\\alpha)=\\frac{2\\pi}{L(t)}(1+\\vartheta_\\alpha(\\alpha)),\n\\end{equation*}\nthe equation for the vorticity strength $\\omega(\\alpha)$ in \\eqref{eqOmega} reads as follows\n \\begin{equation}\\label{omega}\n    \\omega(\\alpha)=2A_\\mu \\frac{L(t)}{2\\pi} \\mathcal{D}(\\omega)(\\alpha)+2A_\\sigma\\frac{2\\pi}{L(t)} \\vartheta_{\\alpha\\alpha}(\\alpha)-2A_\\rho\\frac{L(t)}{2\\pi}\\sin{(\\alpha+\\vartheta(\\alpha))},\n\\end{equation}\n with $\\mathcal{D}(z,\\omega)(\\alpha)$ in \\eqref{Dz} given by\n\\begin{equation}\\label{md}\n    \\mathcal{D}(\\omega)(\\alpha)=-\\text{Re}\\hspace{0.05cm}(\\overline{BR}(\\omega)(\\alpha) e^{i(\\alpha+\\vartheta(\\alpha))}).\n\\end{equation}\nIn addition, \nwe notice that because $|z_\\alpha(\\alpha)|$ is constant in $\\alpha$ and $z(\\alpha)$ is a closed curve, then the following constraint must hold\n\\begin{equation}\\label{constraint}\n    0=\\int_{-\\pi}^{\\pi} \\frac{z_\\alpha(\\alpha)}{|z_\\alpha(\\alpha)|} d\\alpha=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\vartheta(\\alpha))}d\\alpha.\n\\end{equation}\nFinally, once the system \\eqref{T}-\\eqref{md} is solved, one can track the evolution of a single point, say that with $\\alpha=0$, by integrating in time \\eqref{eqcurve} (notice that the right hand side of \\eqref{eqcurve} has been shown to depend only on $z_\\alpha$, given by \\eqref{param}, \\eqref{modul}, and \\eqref{eqcurve3}).\n\n\\subsection{Evolution System}\\label{subsec:evolutionSystem}\nFor our purposes, equation \\eqref{eqcurve3}$_2$ is not  convenient  to study $L(t)$. Instead, we will make use of the fact that the fluid is incompressible, and thus the volume is preserved.\nThe volume is given in terms of the curve $z(\\alpha)$ by\n\\begin{equation*}\n    V=\\frac12\\int_{-\\pi}^\\pi z(\\alpha)\\wedge  z_\\alpha(\\alpha)d\\alpha,\n\\end{equation*}\nwhich in complex notation reads as\n\\begin{equation*}\n    V=\\frac12\\text{Im}\\hspace{0.05cm}\\int_{-\\pi}^\\pi  \\overline{z(\\alpha)} z_\\alpha(\\alpha) d\\alpha.\n\\end{equation*}\nSince $U(\\alpha)$ in \\eqref{U} is a total derivative in $\\alpha$, the conservation of volume is obtained by simply taking a time derivative in the equation above.\nNow, from \\eqref{param} and \\eqref{modul} we have that\n$$\nz_\\alpha(\\alpha)=\\frac{L(t)}{2\\pi}e^{i(\\alpha+\\vartheta(\\alpha))},\n$$\nand we can write\n\\begin{equation*}\n    z(\\alpha)=z(0)+\\int_0^\\alpha z_\\alpha(\\eta)d\\eta.\n\\end{equation*}\nThen the conservation of volume  writes as follows\n\\begin{equation}\\label{volumelength}\n\\begin{aligned}\n    V_0&=\\pi R^2\n    \\\\\n    &=\\frac{1}{2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\text{Im}\\hspace{0.05cm} \\Big(\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)}e^{i(\\vartheta(\\alpha)-\\vartheta(\\eta))}d\\eta d\\alpha\\Big)\\\\\n    &=\\frac{1}{2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\text{Im}\\hspace{0.05cm} \\Big(2\\pi i\n    +\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\vartheta(\\alpha)-\\vartheta(\\eta))^n d\\eta d\\alpha\\Big).\n\\end{aligned}\n\\end{equation}\nThis yields the following equation for $L(t)$:\n\\begin{equation}\\label{Lequation}\n\\Big(\\frac{L(t)}{2\\pi}\\Big)^2=R^2\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\vartheta(\\alpha)-\\vartheta(\\eta))^n d\\eta d\\alpha\\Big)^{-1}.\n\\end{equation}\nThis is the equation for $L(t)$ that will be use later in the paper,\n\nConversely, it is not hard to check that \\eqref{Lequation} implies \\eqref{eqcurve3}$_2$. In fact, taking a time derivative of \\eqref{volumelength}, and assuming $L(t)\\neq0$, gives that\n\\begin{equation*}\n    \\begin{aligned}\n    L'(t)&=\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha i e^{i(\\alpha-\\eta)} e^{i(\\vartheta(\\alpha)\\!-\\!\\vartheta(\\eta))}(\\vartheta_t(\\alpha)\\!-\\!\\vartheta_t(\\eta)) d\\eta d\\alpha\\\\\n    &=\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\vartheta_t(\\alpha)\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}d\\eta d\\alpha\\\\\n    &\\quad-\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta d\\alpha.\n    \\end{aligned}\n\\end{equation*}\nThus writing in the first term $e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\vartheta_t(\\alpha)=\\partial_\\alpha\\int_0^\\alpha e^{i\\eta} e^{i\\vartheta(\\eta)}\\vartheta_t(\\eta)$ and integrating by parts we obtain that\n\\begin{equation}\\label{auxL}\n    \\begin{aligned}\n    L'(t)=\\frac{-1}{2\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta d\\alpha.\n    \\end{aligned}\n\\end{equation}\nUsing the equation for $\\vartheta$ in \\eqref{eqcurve3}$_1$, we have\n\\begin{equation*}\n    \\begin{aligned}\n     \\frac{L(t)}{2\\pi}&\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta=\n    \\int_0^\\alpha  e^{-i\\eta-i\\vartheta(\\eta)}\\big(U_\\alpha(\\eta)+ T(\\eta)(1+\\vartheta_\\alpha(\\eta))\\big)\n     d\\eta\\\\\n     &=e^{-i\\alpha-i\\vartheta(\\alpha)}U(\\alpha)-e^{-i\\vartheta(0)}U(0)+i\\int_0^\\alpha  e^{-i\\eta-i\\vartheta(\\eta)}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta\\\\\n     &\\quad+i\\int_0^\\alpha  \\partial_\\eta\\big(e^{-i\\eta-i\\vartheta(\\eta)}\\big)T(\\eta) d\\eta.\n    \\end{aligned}\n\\end{equation*}\nWe then integrate by parts once more in the last term, also using \\eqref{T}, to obtain\n\\begin{equation*}\n    \\begin{aligned}\n    i&\\int_0^\\alpha  \\partial_\\eta\\big(e^{-i\\eta-i\\vartheta(\\eta)}\\big)T(\\eta) d\\eta=i e^{-i\\alpha-i\\vartheta(\\alpha)}T(\\alpha)-i e^{-i\\vartheta(0)}T(0)\\\\\n    &-i\\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta +\\frac{i}{2\\pi}\\int_{-\\pi}^\\pi(1+\\theta_\\alpha(\\eta))U(\\eta)d\\eta \\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}d\\eta.\n    \\end{aligned}\n\\end{equation*}\nSubstituting into the previous equation we find that\n\\begin{equation*}\n    \\begin{aligned}\n     \\frac{L(t)}{2\\pi}&\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta=\n    e^{-i\\alpha-i\\vartheta(\\alpha)}U(\\alpha)-e^{-i\\vartheta(0)}U(0)+i e^{-i\\alpha-i\\vartheta(\\alpha)}T(\\alpha)\\\\\n    &-i e^{-i\\vartheta(0)}T(0)\n     +\\frac{i}{2\\pi}\\int_{-\\pi}^\\pi(1+\\theta_\\alpha(\\eta))U(\\eta)d\\eta \\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}d\\eta.\n    \\end{aligned}\n\\end{equation*}\nThus, plugging this back into \\eqref{auxL} and using relation \\eqref{constraint} gives\n\\begin{equation*}\n    \\begin{aligned}\n    L'(t)=\\frac{-1}{2\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2 \\Big(\\text{Im}\\hspace{0.05cm}\\! \\int_{-\\pi}^\\pi \\! \\!\\!e^{i\\alpha+i\\vartheta(\\alpha)}\\! \\int_0^\\alpha\\! \\! e^{-i\\eta-i\\vartheta(\\eta)} d\\eta d\\alpha\\Big)\\!\\int_{-\\pi}^\\pi\\!\\! (1\\!+\\!\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta,\n    \\end{aligned}\n\\end{equation*}\nwhich recalling \\eqref{volumelength} implies \\eqref{eqcurve3}$_2$. \n\nWe will later show (see Subsection \\ref{sec:Lestimate}) that the condition on the initial data will guarantee that $L(t)>0$ for all time, and thus the formulations using \\eqref{eqcurve3}$_2$ and \\eqref{Lequation} are equivalent.\n\nIn summary, the closed system of equations that define the evolution of the Muskat bubble can be expressed by \\eqref{eqcurve3}$_1$ and \\eqref{Lequation}, together with \\eqref{constraint}.  We will study the evolution of this system to prove our main results.\n\n\n\n\n\n\\subsection{Calculations Involving the Fourier Transform}\\label{sec:FourierTransCalc}\n\nIn this section we recall basic calculations for the periodic Fourier transform that will be used in the next section.  In particular we define the Fourier transform of a periodic function $g$ with domain $\\mathbb{T}=[-\\pi, \\pi]$ as: \n\\begin{equation}\\notag\n    \\mathcal{F}(g)(k)\\overset{\\mbox{\\tiny{def}}}{=}\\widehat{g}(k)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi g(\\alpha)e^{-i k\\alpha}d\\alpha,\n\\end{equation}\nand the corresponding Fourier series\n\\begin{equation}\\notag\n    g(\\alpha)=\\sum_{k\\in\\mathbb{Z}}\\widehat{g}(k)e^{i k\\alpha}. \n\\end{equation}\nFor later use, we also define the periodic Hilbert transform as \n\\begin{equation}\\label{defHilbertTransform}\n    \\mathcal{H} (g)(\\alpha)\\overset{\\mbox{\\tiny{def}}}{=}\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{g(\\alpha-\\beta)}{\\tan{(\\beta\/2)}}.\n\\end{equation}\nWe notice that $\\mathcal{H} (g)(\\alpha)=-\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{g(\\alpha+\\beta)}{\\tan{(\\beta\/2)}}$.  Adding half of these together one can calculate that $\\mathcal{H} (c)=0$ if $c\\in \\mathbb{C}$ is a constant and \n$\n\\mathcal{H} (g)(\\alpha)=-i\\sum_{k \\ne 0} \\text{sgn}(k) \\widehat{g}(k)e^{i k\\alpha}.\n$\nAnd further $\\mathcal{F}(\\mathcal{H} (g))(k) = -i \\text{sgn}(k) \\widehat{g}(k)$.  Then we define the operator $\\Lambda$ using the Fourier transform as \n$\n\\mathcal{F}(\\Lambda g)(k)\\overset{\\mbox{\\tiny{def}}}{=} |k| \\widehat{g}(k).\n$\nAnd we observe that $\\mathcal{H} (g_\\alpha )(\\alpha)=\\sum_{k\\in \\mathbb{Z}} |k| \\widehat{g}(k)e^{i k\\alpha} = \\Lambda g$.    And furthermore $$\\partial_\\alpha \\mathcal{H} (g_{\\alpha\\alpha} )(\\alpha)=-\\sum_{k\\in \\mathbb{Z}} |k|^3 \\widehat{g}(k)e^{i k\\alpha} = -\\Lambda^3 g.$$  \nAlso one can compute by plugging in the Fourier series that\n\\begin{equation}\\label{fourierCalcAlpha}\n        \\mathcal{F}\\Big(\\int_0^\\alpha g(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^\\pi g(\\eta)d\\eta \\Big)(k)=\\left\\{\n    \\begin{aligned}-\\frac{i}{k} \\widehat{g}(k),\\quad k\\neq0,\\\\\n        \\sum_{j\\neq0}\\frac{i}{j}\\widehat{g}(j),\\quad k=0,\n    \\end{aligned}\\right.\n\\end{equation}\nThese calculations will be used in the next section when we take the Fourier transform of the linearization.\n\n\n\\subsection{Linearization and Nonlinear Expansion}\\label{sec:linearization}\n\nWe proceed next to decompose the equation for $\\vartheta$ in the system \\eqref{T}-\\eqref{md} into linear and nonlinear parts. We will Taylor expand the nonlinear terms around the zero frequency of $\\vartheta(\\alpha)$. Define\n\\begin{equation}\\label{theta}\n    \\theta(\\alpha)=\\vartheta(\\alpha)-\\hat{\\vartheta}(0).\n\\end{equation}\nTaking into account that\n\\begin{equation*}\n    \\int_0^1 e^{i(\\alpha+(s-1)\\beta)}ds=e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta},\n\\end{equation*}\nwe write the denominator of \\eqref{BR} as follows\n\\begin{multline*}\n    \\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\hat{\\vartheta}(0)}e^{i\\theta(\\alpha+(s-1)\\beta)}ds=\\\\\n    e^{i\\hat{\\vartheta}(0)}e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta}\\left(e^{-i\\alpha}\\frac{i\\beta}{1-e^{-i\\beta}}\\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\theta(\\alpha+(s-1)\\beta)}ds-1+1\\right).\n\\end{multline*}\nThen after performing a Taylor expansion, \\eqref{BR} is given by \n\\begin{multline*}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{e^{-i\\hat{\\vartheta}(0)}}{iL(t)} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\beta e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta}}\n    \\\\\\cdot\\sum_{n\\geq 0}\\left(1-\\frac{i\\beta}{1-e^{-i\\beta}}\\int_0^1 e^{i(s-1)\\beta}e^{i\\theta(\\alpha+(s-1)\\beta)}ds\\right)^nd\\beta.\n\\end{multline*}\nBy further Taylor expanding the exponential term, we find that\n\\begin{multline*}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{e^{-i\\hat{\\vartheta}(0)}e^{-i\\alpha}}{iL(t)}\\sum_{n\\geq 0}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\beta }\\frac{(-1)^n(i\\beta)^{n+1}}{(1-e^{-i\\beta})^{n+1}}\\\\\n    \\cdot\\Big(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\Big)^nd\\beta.\n\\end{multline*}\nWe further Taylor expand $e^{i\\theta(\\alpha)}$.  Then plugging these expansions into \\eqref{U} provides the series for $U(\\alpha)$\n\\begin{multline*}\n    U(\\alpha)=\\text{Re}\\hspace{0.05cm}\\left(\\frac{1}{L(t)}\\sum_{n,l\\geq0}\\frac{(i\\theta(\\alpha))^l}{l!}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)(-1)^n(i\\beta)^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n    \\right.\\\\\n    \\left.\\cdot\\left(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\right)^nd\\beta\\right).\n\\end{multline*}\nFor convenience, we introduce the following notation for the operators $\\mathcal{R}$ and $\\mathcal{S}$.  We first define $\\mathcal{R}$:  \n\\begin{equation}\\label{R}\n    \\mathcal{R}(f)(\\alpha)\\!=\\!\\frac{i}{\\pi}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\!\\!\\frac{f(\\alpha\\!-\\!\\beta)}{\\beta}\\frac{\\beta^2}{(1\\!-\\!e^{-i\\beta})^2}\\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta}\\theta(\\alpha\\!+\\!(s-1)\\beta) ds d\\beta.\n\\end{equation}\nAbove $\\mathcal{R}$ is chosen to be a linear function in $\\theta$, it corresponds to $l=0$, $n=1$ and $m=1$ in $U(\\alpha)$ above. Then, we further define the operator $\\mathcal{S}$ to be the nonlinear in $\\theta$ terms inside $U(\\alpha)$ above:\n\\begin{multline}\\label{S}\n    \\mathcal{S}(f)(\\alpha)=\n    \\frac{1}{\\pi}\\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\theta(\\alpha))^l}{l!}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\\\\\n    \\cdot\n    \\left(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\right)^n d\\beta\\\\\n    +\\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{2}}{\\beta(1-e^{-i\\beta})^{2}}\\sum_{m\\geq 2}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds d\\beta.\n\\end{multline}\nThe $\\mathcal{S}$ operator corresponds to the terms in $U(\\alpha)$ above where $n,l \\ge 0$ and $n+l\\ge 2$ plus the case where $l=0$, $n=1$ and $m\\ge 2$. \n\nFor the cases in $U(\\alpha)$ where $n=l=0$ and $n=0$, $l=1$ we further notice that\n\\begin{equation}\\label{hilbert}\n    \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)}{1-e^{-i\\beta}}d\\beta=-i\\mathcal{H}  f(\\alpha)+\\hat{f}(0),\n\\end{equation}\nwhere $\\mathcal{H}  f$ denotes the periodic Hilbert transform of $f$ as given in \\eqref{defHilbertTransform}.  The previous identity \\eqref{hilbert} is obtained multiplying above and below by $1-e^{i\\beta}$ and using the trigonometric identities\n$1-\\cos{(\\beta)}=2\\sin^2{(\\beta\/2)}$ and  $\\sin{(\\beta)}=2\\sin{(\\beta\/2)}\\cos{(\\beta\/2)}.\n$\nFurther $\\omega(\\alpha)$ in \\eqref{omega} can be written as an exact derivative, its mean value is zero and therefore $\\hat{\\omega}(0)=0$. \nThese calculations show that we can write the expression inside \\eqref{U} as\n\\begin{equation}\\notag\n    i\\overline{BR}(\\omega)(\\alpha) e^{i(\\alpha+\\vartheta(\\alpha))}\n    =\n    \\frac{\\pi}{L(t)} \\left(i \\theta(\\alpha) \\mathcal{H}(\\omega) + \\mathcal{H}(\\omega) +\\mathcal{R}(\\omega)(\\alpha)+\\mathcal{S}(\\omega)(\\alpha)\\right).\n\\end{equation}\nThus, noticing that the term with $n=0$, $l=1$ vanishes in $U(\\alpha)$ in \\eqref{U} because it is purely imaginary, using the notation above we can write $U(\\alpha)$ in the following manner\n\\begin{equation}\\label{Udecomp}\n    U(\\alpha)=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega(\\alpha)+ \\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega)(\\alpha)+ \\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{equation}\nProceeding similarly, in \\eqref{md}, one finds that\n\\begin{equation}\\label{Ddecomp}\n    \\mathcal{D}(\\omega)(\\alpha)=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega(\\alpha)+ \\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega)(\\alpha)+ \\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{equation}\nWe shall now split all the terms into zero, first or higher order polynomials of $\\theta(\\alpha)$. First, the vorticity strength \\eqref{omega} is split as follows\n\\begin{equation*}\n    \\omega(\\alpha)=\\omega_0(\\alpha)+\\omega_1(\\alpha)+\\omega_{\\geq2}(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{omegasplit}\n\\left\\{\\begin{aligned}\n    \\omega_0(\\alpha)&\\!=\\!-A_\\rho\\frac{L(t)}{\\pi}\\sin{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n    \\omega_1(\\alpha)&\\!=\\!A_\\mu\\frac{L(t)}{\\pi}\\mathcal{D}_1(\\omega_0)(\\alpha)\\!+\\!2A_\\sigma\\frac{2\\pi}{L(t)}\\theta_{\\alpha\\alpha}\\!\n    \\\\\n    &\\hspace{1.6cm}-\\!A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha),\\\\\n    \\omega_{\\geq2}(\\alpha)&\\!=\\!A_\\mu\\frac{L(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega)(\\alpha)\\!\n    \\\\ &\\hspace{1.6cm} -\\!A_\\rho\\frac{L(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta(\\alpha))^{2j}}{(2j)!}\\\\\n    &\\hspace{1.6cm}-A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta(\\alpha))^{1+2j}}{(1+2j)!},\\\\\n    \\omega_{\\geq1} &\\!=\\! \\omega_{1}+\\omega_{\\geq2}.\n\\end{aligned}\\right.\n\\end{equation}\nAbove we used the trigonometric identity\n$\\sin(a+b) = \\sin(a)\\cos(b)+\\cos(a)\\sin(b)$,\nas well as the Taylor expansions for sine and cosine. \nThen $\\mathcal{D}_1(\\omega_0)(\\alpha)$ and $\\mathcal{D}_{\\geq2}(\\omega)(\\alpha)$ are obtained, in turn, by introducing \\eqref{omegasplit} into \\eqref{Ddecomp} as follows\n\\begin{equation*}\n    \\mathcal{D}(\\omega)(\\alpha)=\\mathcal{D}_1(\\omega_0)(\\alpha)+\\mathcal{D}_{\\geq2}(\\omega)(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{mdsplit}\n\\left\\{\\begin{aligned}\n    \\mathcal{D}_1(\\omega_0)(\\alpha)&=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega_0(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega_0)(\\alpha)\\Big),\\\\\n    \\mathcal{D}_{\\geq2}(\\omega)(\\alpha)&=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega_{\\geq1}(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{aligned}\\right.\n\\end{equation}\nAnalogously, the splitting for $U(\\alpha)$ from \\eqref{Udecomp} is\n\\begin{equation*}\n    U(\\alpha)=U_0(\\alpha)+U_1(\\alpha)+U_{\\geq2}(\\alpha),\n\\end{equation*}\\vspace{-0.5cm}\nwith\n\\begin{equation}\\label{Usplit}\n\\left\\{\\begin{aligned}\n    U_0(\\alpha)&=\\frac{\\pi}{L(t)}\\mathcal{H}  \\omega_0(\\alpha),\\\\\n    U_1(\\alpha)&=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega_1(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega_0)(\\alpha)\\Big),\\\\\n    U_{\\geq2}(\\alpha)&=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega_{\\geq2}(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_{\\geq1})(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{aligned}\\right.\n\\end{equation}\nRecalling the expression for $T(\\alpha)$ in \\eqref{T}, we find that\n\\begin{equation*}\n    T(\\alpha)=T_0(\\alpha)+T_1(\\alpha)+T_{\\geq2}(\\alpha),\n\\end{equation*}\nwhere, using $\\int_{-\\pi}^{\\pi} U_0(\\eta) d\\eta =0$, we have\n\\begin{equation}\\label{Tsplit}\n\\left\\{\\begin{aligned}\n    T_0(\\alpha)=& T(0)+\\int_0^\\alpha U_0(\\eta)d\\eta,\\\\\n    T_1(\\alpha)=& \\int_0^\\alpha  U_1(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}U_1(\\eta)d\\eta \\\\\n    &+\\int_0^\\alpha\\theta_\\alpha(\\eta)U_0(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)U_0(\\eta)d\\eta,\\\\\n    T_{\\geq2}(\\alpha)=&\\int_0^\\alpha (1+\\theta_\\alpha(\\eta))U_{\\geq2}(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}(1+\\theta_\\alpha(\\eta))U_{\\geq2}(\\eta)d\\eta\\\\\n    &+\\int_0^\\alpha\\theta_\\alpha(\\eta)U_1(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)U_1(\\eta)d\\eta.\n\\end{aligned}\\right.\n\\end{equation}\nThese are all the splittings that we will use in the following.\n\nWe first examine the zero order terms from \\eqref{thetaeqaux}$_1$.   The zero order terms on the right side of the equality \\eqref{thetaeqaux}$_1$ would be\n\\begin{equation}\\notag\n        \\Theta(\\alpha)=(U_0)_\\alpha(\\alpha)+T_0(\\alpha).\n\\end{equation}\nNow a direct calculation from \\eqref{omegasplit} shows that \n\\begin{equation}\\label{hilbertTcalc}\n    \\mathcal{H}(\\omega_0)(\\alpha) \n    = A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha+\\hat{\\vartheta}(0))}.\n\\end{equation}\nThen we plug this into \\eqref{Usplit}$_1$ and \\eqref{Tsplit}$_1$ to obtain\n\\begin{equation}\\label{u0t0}\n\\left\\{\n\\begin{aligned}\n    U_0(\\alpha)&=A_\\rho \\cos{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n     T_0(\\alpha)&=A_\\rho\\sin{(\\alpha+\\hat{\\vartheta}(0))}-A_\\rho\\sin{\\hat{\\vartheta}(0)}+T(0).\n\\end{aligned}\\right.\n\\end{equation}\nIn particular then the zero order term $\\Theta(\\alpha)$ does not depend on $\\alpha$,\n\\begin{equation*}\n    \\Theta(\\alpha)=-A_\\rho \\sin{\\hat{\\vartheta}(0)}+T(0).\n\\end{equation*}\nNow we choose \n\\begin{equation}\\label{T0}\n    T(0)=A_\\rho \\sin{\\hat{\\vartheta}(0)}.\n\\end{equation}\nThus the parametrization of the circle solution is independent of time (see Proposition \\ref{circles} below).  Further $\\Theta(\\alpha)=0.$\n\n\nNow we introducing the splittings \\eqref{Usplit} and \\eqref{Tsplit} into the equation for $\\vartheta$ in \\eqref{eqcurve3}, we find that\n\\begin{equation}\\label{system}\n\\left\\{\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{2\\pi}{L(t)}\\Big(\\mathcal{L}(\\alpha)+N(\\alpha)\\Big),\\\\\n    \\mathcal{L}(\\alpha)&=(U_1)_\\alpha(\\alpha)+T_1(\\alpha)+T_0(\\alpha)\\theta_\\alpha(\\alpha),\\\\\n    N(\\alpha)&=(U_{\\geq 2})_\\alpha(\\alpha)+T_{\\geq 2}(\\alpha)(1+\\theta_\\alpha(\\alpha))+T_1(\\alpha)\\theta_\\alpha(\\alpha).\n\\end{aligned}\\right.\n\\end{equation}\nNow we will expand the linear terms in $\\mathcal{L}(\\alpha)$ in \\eqref{system}.  To do this we first split $U_1(\\alpha)$ in \\eqref{Usplit} into parts corresponding to the parameters $A_\\rho$, $A_\\sigma$ and $A_\\mu$ respectively as  \n\\begin{equation*}\n    U_1(\\alpha)=A_\\rho U_{1\\rho}(\\alpha)+4A_\\sigma\\frac{\\pi^2}{(L(t))^2} U_{1\\sigma}(\\alpha)+A_\\mu A_\\rho U_{1\\mu}(\\alpha),\n\\end{equation*}\nTo calculate these terms we plug $\\omega_0(\\alpha)$ and $\\omega_1(\\alpha)$ from \\eqref{omegasplit} into $U_1(\\alpha)$ in \\eqref{Usplit} using also $\\mathcal{D}_1(\\omega_0)(\\alpha)$ from \\eqref{mdsplit} and \\eqref{hilbertTcalc}.  \nWe obtain\n\\begin{equation}\\label{uterms}\n\\left\\{\n\\begin{aligned}\n    U_{1\\rho}(\\alpha)&=-\\mathcal{H} \\big(\\theta(\\alpha)\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\big)-\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))}),\\\\\n    U_{1\\sigma}(\\alpha)&=\\mathcal{H}  \\theta_{\\alpha\\alpha}(\\alpha),\\\\\n    U_{1\\mu}(\\alpha)&= - \\mathcal{H}  \\big(\\theta(\\alpha)\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\big)+\\mathcal{H} \\hspace{0.03cm}\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))}).\n\\end{aligned}\\right.\n\\end{equation}\nWe analogously write the linear part, $\\mathcal{L}(\\alpha)$ in \\eqref{system}, \nas follows\n\\begin{equation*}\n    \\mathcal{L}(\\alpha)=A_\\rho \\mathcal{L}_\\rho(\\alpha)+4A_\\sigma\\frac{\\pi^2}{(L(t))^2}\\mathcal{L}_\\sigma(\\alpha)+A_\\mu A_\\rho\\mathcal{L}_\\mu(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{linearterms}\n\\left\\{\n\\begin{aligned}\n    \\mathcal{L}_{\\rho}(\\alpha)&=(U_{1\\rho})_\\alpha(\\alpha)+\\int_0^\\alpha U_{1\\rho}(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}U_{1\\rho}(\\eta)d\\eta\n    \\\\\n    &\\quad+\\int_0^{\\alpha}\\theta_\\alpha(\\eta)\\cos{(\\eta+\\hat{\\vartheta}(0))}d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)\\cos{(\\eta+\\hat{\\vartheta}(0))}d\\eta\\\\\n    &\\quad+\\theta_\\alpha(\\alpha)\\sin{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n    \\mathcal{L}_\\sigma(\\alpha)&=-\\Lambda^3 \\theta(\\alpha)+\\int_0^\\alpha \\mathcal{H}  \\theta_{\\alpha\\alpha}(\\eta)d\\eta,\\\\\n    \\mathcal{L}_\\mu(\\alpha)&=(U_{1\\mu})_\\alpha(\\alpha)+\\int_0^\\alpha U_{1\\mu}(\\eta)d\\eta.\n\\end{aligned}\\right.\n\\end{equation}\nHere we used that $\\partial_\\alpha \\mathcal{H}\\theta_{\\alpha\\alpha} = -\\Lambda^3 \\theta(\\alpha)$.  We also used that \n$$\\int_{-\\pi}^{\\pi}\\mathcal{H}  \\theta_{\\alpha\\alpha}(\\eta)d\\eta = \\int_{-\\pi}^{\\pi}U_{1\\mu}(\\eta)d\\eta =0$$ since both integrals are of a Hilbert transform and thus have zero value for the zero Fourier frequency.\nThis completes our decomposition of the equation \\eqref{eqcurve} into \\eqref{system}.  \n\nIn the following, we explain the steady states circles for equation \\eqref{eqcurve} using the reformulation of the equations given above.  \n\n\\begin{prop}\\label{circles}\n\tA circle of radius $R$, defined by \\eqref{circles} and \\eqref{modul} with $\\vartheta(\\alpha)=\\widehat{\\vartheta}(0)$ constant in time and $L(t)=2\\pi R$, is a time-independent solution of \\eqref{T}-\\eqref{md} with $T(0)$ given by \\eqref{T0}.\n\tIt corresponds to the solution of \\eqref{eqcurve} given by a circle of radius $R$ moving vertically with velocity $A_\\rho$.\t\n\\end{prop}\n\n\\begin{proof}\nFor $\\vartheta(\\alpha)=\\widehat{\\vartheta}(0)$, all the linear and nonlinear terms in the decompositions \\eqref{omegasplit}-\\eqref{Tsplit} are zero. Thus, with $L(t)=2\\pi R$,  as in \\eqref{u0t0} with \\eqref{T0} we have\n\\begin{align*}\n    U(\\alpha)&=U_0(\\alpha)=A_\\rho\\cos{(\\alpha+\\widehat{\\vartheta}(0))},\\\\\n    T(\\alpha)&=T_0(\\alpha) = A_\\rho\\sin{(\\alpha+\\widehat{\\vartheta}(0))}.\n\\end{align*} \nBoth equations in \\eqref{eqcurve3} are then trivially satisfied; equation \\eqref{eqcurve3}$_1$ is decomposed as  \\eqref{system} with $\\mathcal{L}(\\alpha)=N(\\alpha)=0$.\t\n\t\nThen we integrate \\eqref{param} to obtain\n\\begin{equation*}\n\tz(\\alpha,t)=z(0,t)+R\\int_0^\\alpha e^{i(\\eta+\\widehat{\\vartheta}(0))} d\\eta.\n\\end{equation*}\nWe differentiate the above in time, and then use \\eqref{eqcurve.theta} to obtain\n\\begin{equation*}\n\\begin{aligned}\n\tz_t(\\alpha,t)&=z_t(0,t)=U(0,t)n(0,t)+T(0,t)\\tau(0,t)\\\\\n\t&=A_\\rho\\cos{(\\widehat{\\vartheta}(0))}ie^{i\\widehat{\\vartheta}(0)}+A_\\rho\\sin{(\\widehat{\\vartheta}(0))}e^{i\\widehat{\\vartheta}(0)}=0+i A_\\rho.\n\\end{aligned}\t\n\\end{equation*}\t\nThis completes the proof.\n\\end{proof}\n\nNext, we compute the Fourier transform of the linearized system. Because the function $\\theta(\\alpha)$ is real and has zero average, we only need to compute the positive frequencies.\n\n\n\n\\begin{prop}\\label{linearfourier}\n\t(Linear system in Fourier variables.)\n\tFor $k\\geq 1$, $k\\neq2$, the Fourier transform of the linear terms \\eqref{linearterms} are given by  \n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    \\widehat{\\mathcal{L}}(k)&=-A_\\sigma \\frac{4\\pi^2}{L(t)^2}k(k^2\\!-\\!1)\\hat{\\theta}(k)\\!-\\!(1\\!+\\!A_\\mu)A_\\rho\\frac{(k^2\\!-\\!1)(k\\!+\\!1)}{k(k\\!+\\!2)}e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k+1),\t\n\t\\end{aligned}\n\t\\end{equation*}\n\tand for $k=2$,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    \\widehat{\\mathcal{L}}(2)&=-A_\\sigma \\frac{4\\pi^2}{L(t)^2}6\\hat{\\theta}(2)+(1-A_\\mu)A_\\rho\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(1)\\\\\n\t    &\\quad-(1+A_\\mu)A_\\rho\\frac{9}{8}e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3).\n\t\\end{aligned}\n\t\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nFirst, we note that, for a general function $f(\\alpha)$ and $k\\neq0$ we have \\eqref{fourierCalcAlpha}.\nTherefore, for $k\\geq 1$, the Fourier coefficients of\n\\eqref{linearterms} are given by\n\\begin{equation}\\label{Lfourier}\n\\begin{aligned}\n    \\widehat{\\mathcal{L}}_\\sigma(k)&=-k(k^2-1)\\hat{\\theta}(k),\\\\\n    \\widehat{\\mathcal{L}}_\\mu(k)&=i\\left(k-\\frac1{k}\\right)\\widehat{U}_{1\\mu}(k),\\\\\n    \\widehat{\\mathcal{L}}_\\rho(k)\n    &=\n    \\left(k-\\frac1{k}\\right)\n    \\left(i\\widehat{U}_{1\\rho}(k)+\\frac{e^{i\\hat{\\vartheta}(0)}}{2}\\hat{\\theta}(k-1)-\\frac{e^{-i\\hat{\\vartheta}(0)}}{2}\\hat{\\theta}(k+1)\\right),\n\\end{aligned}\n\\end{equation}\nso it remains to compute $\\widehat{U}_{1\\mu}(k)$ and $\\widehat{U}_{1\\rho}(k)$.\nFrom \\eqref{uterms}, we can write \n\\begin{align*}\n    \\widehat{U}_{1\\mu}(k)\n    =&\n    \\frac{i}{2}\\Big(e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!\n    +\\!e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!+\\!1)\\Big)\\!\n    \\\\\n    &-\\!i\\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\Big)(k),\n\\end{align*}\nand\n\\begin{align*}\n    \\widehat{U}_{1\\rho}(k)\n    =&\\frac{i}2 \\Big(e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!+\\!e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!+\\!1)\\Big)\\!\n    \\\\\n    &-\\!\\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\Big)(k).\n\\end{align*}\nRecalling the expression of $\\mathcal{R}$ in \\eqref{R}, we have that\n\\begin{multline*}\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))})\\Big)(k)=\\\\\n    \\frac1{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\int_0^1\\!\\! \\text{Im}\\hspace{0.05cm}\\left(\\!\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\!\\right)\\!\\mathcal{F}\\Big(\\theta(\\alpha+(s-1)\\beta)\\sin{(\\alpha\\!-\\!\\beta\\!+\\!\\hat{\\vartheta}(0))}\\Big)(k)ds d\\beta.\n\\end{multline*}\nUsing that\n\\begin{equation*}\n    \\text{Im}\\hspace{0.05cm}\\left(\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\right)=-\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}},\n\\end{equation*}\nand computing the Fourier transform inside the integral, \nwe obtain that\n\\begin{multline}\\nota\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\\\\\n    -\\frac{ e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}ds d\\beta\\\\\n    +\\frac{ e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}ds d\\beta.\t\n\\end{multline}\nTaking into account that\n\\begin{equation*}\n    \\text{Re}\\hspace{0.05cm}\\left(\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\right)=\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}},\n\\end{equation*}\nand proceeding analogously, the following expression is found for the real part: \n\\begin{multline}\\notag\n    \\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\\\\\n    -\\frac{i e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\n    \\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}ds d\\beta\n    \\\\\n    +\\frac{i e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}ds d\\beta.\t\n\\end{multline}\nThe above integrals are calculated in Lemma \\ref{lemmaI1I2} below. Plugging in their values, we have \n\\begin{multline}\\label{fourierimag}\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\n    \\frac{ e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\frac{-k\\pi}{2+k} 1_{k \\ge 1}\\\\\n    -\\frac{ e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\n        \\left(-\\pi 1_{k \\ge 1, k\\ne 2}+ \\pi \\left(\\frac{1}{2}-\\log 4 \\right)1_{k =2} \\right),\t\n\\end{multline}\nand\n\\begin{multline}\\label{fourierreal}\n    \\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\n    -\\frac{i e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\\pi \\left(\\log 4-\\frac{3}{2} \\right)1_{k =2} \n    \\\\\n    +\\frac{i e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\n    \\frac{2\\pi}{2+k} 1_{k \\ge 1}.\t\n\\end{multline}\nWe conclude the Fourier transform of $U_{1\\mu}$ and $U_{1\\rho}$ are given by\n\\begin{equation*}\n    \\widehat{U}_{1\\mu}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &\\frac{i}2e^{-i\\hat{\\vartheta}(0)}\\Big(1+\\frac{k}{2+k}\\Big)\\hat{\\theta}(k+1),\\qquad\\hspace{1.8cm} k=1,3,4,\\dots,\\\\\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\Big(\\frac32-\\log{(4)}\\Big)\\hat{\\theta}(1)+\\frac{3i}4e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3),\\qquad k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\t\nand \n\\begin{equation*}\n    \\widehat{U}_{1\\rho}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!+\\!\\frac{i}2e^{-i\\hat{\\vartheta}(0)}\\Big(1\\!-\\!\\frac{2}{2\\!+\\!k}\\Big)\\hat{\\theta}(k\\!+\\!1),\\hspace{0.3cm} k=1,3,4,\\dots,\\\\\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\Big(-\\frac12+\\log{4}\\Big)\\hat{\\theta}(1)+\\frac{i}4e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2.\n    \\end{aligned}\\right.\n\\end{equation*}\t\nSubstituting these expressions into \\eqref{Lfourier} gives that\n\\begin{equation*}\n    \\widehat{\\mathcal{L}}_{\\mu}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &-e^{-i\\widehat{\\vartheta}(0)}\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k+1),\\hspace{0.4cm} k=1,3,4,\\dots,\\\\\n        &-\\frac34\\Big(\\frac32-\\log{4}\\Big)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1)-\\frac98e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\nand\n\\begin{equation*}\n    \\widehat{\\mathcal{L}}_{\\rho}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &-e^{i\\widehat{\\vartheta}(0)}\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k-1),\\hspace{0.4cm} k=1,3,4,\\dots,\\\\\n        &\\frac34\\Big(\\frac32-\\log{4}\\Big)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1)-\\frac98e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\nFinally, adding them according to \\eqref{linearterms}, the result follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lemmaI1I2}\nFor $k\\in\\mathbb{Z}\\setminus\\{0\\}$, define the integrals \n\\begin{equation*} \n\tI_1(k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}dsd\\beta,\n\\end{equation*}\n\\begin{equation*}\n\tI_2(k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}dsd\\beta.\n\\end{equation*}\nFor $k\\geq 1$ and $k\\neq 2$,\n\\begin{equation*}\n\tI_1(k)=-\\pi,\\qquad I_2(k)=0,\n\\end{equation*}\nwhile for $k\\leq -1$. \n\\begin{equation*}\n\tI_1(k)=\\frac{-k\\pi}{2-k},\\qquad I_2(k)=\\frac{2\\pi}{2-k}.\n\\end{equation*}\t\nThe value $k=2$ is given by\n\\begin{equation*}\n\tI_1(2)=\\pi\\Big(\\frac12-\\log{4}\\Big),\\qquad I_2(2)=\\pi\\Big(\\log{4}-\\frac32\\Big).\n\\end{equation*}\n\\end{lemma}\n\n\n\\begin{remark}\nNotice that\n\\begin{equation*} \n\tI_1(-k)=-\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}dsd\\beta,\n\\end{equation*}\nand \n\\begin{equation*}\n\tI_2(-k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}dsd\\beta.\n\\end{equation*}\nThus Lemma \\ref{lemmaI1I2} covers all the integrals in \\eqref{fourierimag} and \\eqref{fourierreal}.\n\\end{remark}\n\n\n\n\\begin{proof}\n\tBoth integrals are computed similarly. We only show the details for $I_1(k)$. \n\tFirst, consider the case $k\\neq2$.\n    Using complex exponentials, we write the numerator as follows\n    \\begin{equation*}\n        \\begin{aligned}\n            \\cos{(\\beta s)} \\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}&=   \\frac{1}{4i}\\Big(e^{ik(s-1)\\beta}-e^{-i(k(s-1)\\beta-2s\\beta)}\\\\\n            &\\quad+e^{i(k(s-1)\\beta-2s\\beta)}-e^{-i(k(s-1)\\beta)}\\Big).   \n        \\end{aligned}\n    \\end{equation*}\n    Thus integration in $s$ gives that\n    \\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!=\\!\\frac{-1}{16}\\!\\int_{-\\pi}^{\\pi}\\!\\Big(\\frac{1\\!-\\!e^{-i k\\beta}}{k}\\!+\\!\\frac{e^{2i\\beta}\\!-\\!e^{i k\\beta}}{k-2}\\!+\\!\\frac{e^{-2i\\beta}\\!-\\!e^{-i k\\beta}}{k-2}\\!+\\!\\frac{1\\!-\\!e^{i k\\beta}}{k}\\Big)\\frac{d\\beta}{\\sin^2{(\\beta\/2)}}\\\\\n            &\\!= \\!\\frac{-1}{16}\\!\\int_{-\\pi}^{\\pi}\\!\\Big(\\frac{2\\!-\\!e^{i k\\beta}\\!-\\!e^{-i k\\beta}}{k}\\!+\\!\\frac{e^{2i\\beta}\\!+\\!e^{-2i \\beta}}{k-2}\\!-\\!\\frac{e^{i k\\beta}\\!+\\!e^{-i k\\beta}}{k-2}\\Big)\\frac{d\\beta}{\\sin^2{(\\beta\/2)}}.\n        \\end{aligned}\n    \\end{equation*}\nWe then write the denominator in complex form too\n\\begin{equation*}\n    \\begin{aligned}\n        \\sin^2{(\\beta\/2)}=\\frac{-1}4\\big(e^{i\\beta\/2}-e^{-i\\beta\/2}\\big)^2,\n    \\end{aligned}\n\\end{equation*}\nand formally expand it\n\\begin{equation*}\n    \\begin{aligned}\n        \\big(\\sin{\\beta\/2)}\\big)^{-2}\\!=-4e^{-i\\beta}\\big(1\\!-\\!e^{-i\\beta}\\big)^{-2}\\!=-4e^{-i\\beta}\\sum_{l\\geq1} l e^{-i(l-1)\\beta}\\!=\\!-4\\sum_{l\\geq1}l e^{-i l\\beta},\n    \\end{aligned}\n\\end{equation*}\nwhere we have used that\\begin{equation*}\n\\frac{1}{(1-x)^2}=\\sum_{l\\geq1} l x^{l-1}.    \n\\end{equation*}\nTherefore, for $k\\neq0,2$, we have\n \\begin{equation*}\n        \\begin{aligned}\n            I_1(k)\\!= \\!\\frac{1}{4}\\sum_{l\\geq1}l\\!\\int_{-\\pi}^{\\pi}\\!\\Big(&\\frac{2e^{-i l\\beta}\\!-\\!e^{i (k-l)\\beta}\\!-\\!e^{-i (k+l)\\beta}}{k}\\!+\\!\\frac{e^{i(2-l)\\beta}\\!+\\!e^{-i(2+l) \\beta}}{k-2}\\\\\n            &\\quad-\\frac{e^{i (k-l)\\beta}\\!+\\!e^{-i( k+l)\\beta}}{k-2}\\Big)d\\beta.\n        \\end{aligned}\n    \\end{equation*}\nThus performing the integral in $\\beta$ we obtain that\n\\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!= \\!\\frac{1}{4}\\sum_{l\\geq1}l\\Big(\\frac{2\\pi}{k}\\big(-\\delta(k-l)-\\delta(k+l)\\big)\\\\\n            &\\hspace{1.7cm}+\\!\\frac{2\\pi}{k\\!-\\!2}\\big(\\delta(2\\!-\\!l)\\!+\\!\\delta(2\\!+\\!l)\\!-\\!\\delta(k\\!-\\!l)\\!-\\!\\delta(k\\!+\\!l)\\big)\\Big)\\\\\n            &=\\frac{1}{4}\\Big(-2\\pi \\text{sgn}(k)+\\frac{2\\pi}{k-2}2-\\frac{2\\pi}{k-2}|k|\\Big),\n        \\end{aligned}\n    \\end{equation*}\n\twhich gives\n\t\\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!=\\frac{\\pi}{2}\\Big(\\!- \\text{sgn}(k)\\!+\\!\\frac{2\\!-\\!|k|}{k\\!-\\!2}\\Big)=\\frac{\\pi}{2}\\frac{-2|k|\\!+\\!2(\\text{sgn}(k)\\!+\\!1)}{k\\!-\\!2}.\n        \\end{aligned}\n    \\end{equation*}\n\tIt follows then that for $k\\geq1$ ($k\\neq2$),\n\t\\begin{equation*}\n\t    I_1(k)=-\\pi,\n\t\\end{equation*}\n    and for $k\\leq-1$,\n    \\begin{equation*}\n        I_1(k)=\\frac{\\pi}2\\frac{2k}{k-2}=\\pi\\frac{|k|}{2+|k|}.\n    \\end{equation*}\n    The above computations can be justified by writing $x=\\lambda e^{-i\\beta}$ with $0<\\lambda<1$. Then, $\\big(1-\\lambda e^{-i\\beta}\\big)^{-2}=\\sum_{l\\geq1} l\\lambda^{l-1}e^{-i(l-1)\\beta}$ converges uniformly and one can repeat the steps above and take the limit $\\lambda\\to1$.\n    \n    Lastly, we deal with the case $k=2$. We first rewrite it as follows\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)=\\int_{-\\pi}^{\\pi}\\int_0^1\\frac{\\beta}{8\\sin^2{(\\beta\/2)}}\\left(\\sin{(2\\beta s-2\\beta)}-\\sin{(2\\beta)}\\right)ds d\\beta,\n\t\\end{aligned}\n\t\\end{equation*}\n\tso after integration in $s$ we obtain\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)=\\int_{-\\pi}^{\\pi}\\frac{\\beta}{8\\sin^2{(\\beta\/2)}}\\left(\\frac{-1+\\cos{(2\\beta)}}{2\\beta}-\\sin{(2\\beta)}\\right)ds d\\beta.\n\t\\end{aligned}\n\t\\end{equation*}\n\tUsing repeatedly the double angle formula, we find that\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)&=-\\frac12\\int_{-\\pi}^{\\pi}\\cos^2{(\\beta\/2)}d\\beta-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta)}\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta\t\\\\\n\t    &=-\\frac{\\pi}2-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta)}\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta,\n\t\\end{aligned}\n\t\\end{equation*}\n\twhich can be further simplified\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)&=-\\frac{\\pi}2+\\int_{-\\pi}^{\\pi}\\beta\\sin{(\\beta\/2)}\\cos{(\\beta\/2)}d\\beta -\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta\\\\\n\t    &=\\frac{\\pi}2-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta.\n\t\\end{aligned}\n\t\\end{equation*}\n\tIntegration by parts gives that\n\t\\begin{equation*}\n\t    I_1(2)=\\frac{\\pi}2+2\\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta.\n\t\\end{equation*}\n\tThis integral above is related to the known \\text{Clausen function} \\cite{Cl32} of order two (whose value at $\\pi$ is zero):\n\t\t\\begin{equation*}\n\t\t0=Cl_2(\\pi) =\n\t    \\int_0^\\pi \\log{|2\\sin{(\\beta\/2)}|}d\\beta=\n\t    \\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta+\\pi\\log{2},\n\t\\end{equation*}\n\tWe thus conclude \n\t\t\\begin{equation*}\n\t    I_1(2)=\\frac{\\pi}2+2\\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta=\\frac{\\pi}2-2\\pi\\log{2}=\\pi\\Big(\\frac12-\\log{4}\\Big).\n\t\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\nWe notice that in Proposition \\ref{linearfourier} the first frequency mode is neutral at the linear level. However, the restriction \\eqref{constraint} is an equation that relates this frequency with all the higher ones. Thus, the rough idea is to proceed as follows: apply an implicit function theorem to \\eqref{constraint} to solve $\\widehat{\\theta}(-1)$ and $\\widehat{\\theta}(1)$ in terms of $\\widehat{\\theta}(k)$ for $|k|\\geq2$; use \\eqref{system} to compute $\\widehat{\\theta}(k)$ for $|k|\\geq2$, for which the linear operator provides dissipation when we are able to control the nonlinear terms.  Then we will use \\eqref{Lequation} to control $L(t)$ in terms of $\\theta(t)$.  Finally we can compute the evolution of the zero frequency $\\widehat{\\vartheta}(0)$ from \\eqref{system}. \n\n\n\n\n\n\n\\section{Notation and main results}\\label{MainResults}\n\nWe introduce the notation that will be used in the rest of the paper in Section \\ref{sec:Notation}.   We will then state the main results in Section \\ref{sec:globalTHM}.\n\n\\subsection{Notation}\\label{sec:Notation}\nWe recall the complex vector notation introduced in Section \\ref{sec:ComplexVector} and the Fourier transform notation introduced in Section \\ref{sec:FourierTransCalc}.  We use $\\mathbb{T}=[-\\pi,\\pi]$ as our domain with periodic boundary conditions. \n\nWe introduce the space $\\mathcal{F}^{0,1}$ to denote the Wiener algebra, i.e., the space of absolutely convergent Fourier series. The norm in this space is\n\\begin{equation}\\notag \\label{fzerone.def}\n\\|f\\|_{\\mathcal{F}^{0,1}}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\in\\mathbb{Z}} |\\hat{f}(k)|.\n\\end{equation}\nWe analogously introduce the homogeneous spaces $\\dot{\\mathcal{F}}^{s,1}$ with norm\n\\begin{equation}\\notag \\label{fsone.def}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\ne 0} |k|^s|\\hat{f}(k)|, \\quad s\\ge 0.\n\\end{equation}\nFurther in Section \\ref{sec:uniqueness} we will use the notation\n\\begin{equation}\\label{max.fcns}\n    \\|f_{1}, f_{2}, \\ldots , f_{k}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\overset{\\mbox{\\tiny{def}}}{=} \n    \\sum_{j=1}^{k}\\|f_{j}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nMoreover, we will use the spaces of analytic functions $\\dot{\\mathcal{F}}^{s,1}_\\nu$ where these norms include exponential weights as follows:\n\\begin{equation}\\label{fzeroonenunorm}\n\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\in\\mathbb{Z}}e^{\\nu(t)|k|}|\\hat{f}(k)|,\n\\end{equation}\nand\n\\begin{equation}\\label{fsonenorm}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\ne 0}e^{\\nu(t)|k|}|k|^s|\\hat{f}(k)|, \\quad s\\ge 0,\n\\end{equation}\nwith  \n\\begin{equation}\\label{nu}\n    \\nu(t)\\overset{\\mbox{\\tiny{def}}}{=}\\nu_0 \\frac{t}{1+t}>0,\n\\end{equation}\nwhere $0<\\nu'(t)\\leq \\nu_0$ is bounded and small enough for all time when $\\nu_0>0$ is chosen small enough. \n\n\nWe recall the embeddings $\\dot{\\mathcal{F}}^{s_2,1}_\\nu\\hookrightarrow \\dot{\\mathcal{F}}^{s_1,1}_\\nu$ for $0<s_1\\leq s_2$, with norm inequality\n\\begin{equation}\\label{embed}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s_1,1}_\\nu}\\leq \\|f\\|_{\\dot{\\mathcal{F}}^{s_2,1}_\\nu}, \\qquad  0<s_1\\leq s_2.\n\\end{equation}\nThis inequality also holds if $s_1=0$ with $\\mathcal{F}^{0,1}$ in the lower bound provided that $\\hat{f}(0)=0$.\nWe also recall the interpolation inequality \n\\begin{equation}\\label{interpolation}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\|f\\|_{\\dot{\\mathcal{F}}^{s_1,1}_{\\nu}}^{1-\\sigma}\\|f\\|_{\\dot{\\mathcal{F}}^{s_2,1}_{\\nu}}^\\sigma,\\qquad s=(1-\\sigma)s_1+\\sigma s_2,\n\\end{equation}\nfor $0 \\le \\sigma \\le 1$, $s, s_1, s_2 \\ge 0$.\n\nWe further will use the discrete delta function $\\delta(k)$ for $k\\in \\mathbb{Z}$ which is defined  as $\\delta(0)=1$ and $\\delta(k)=0$ for $k\\ne 0$.  We also define $1_A$ to be the standard indicator function of the set $A$ so that $1_A(x)=1$ if $x\\in A$ and $1_A(x)=0$ if $x\\notin A$.  \n\nWe define the $\\ell^p=\\ell^p(\\mathbb{Z})$ norms for $1\\le p < \\infty$ of a sequence $a=\\{a_k\\}_{k\\in \\mathbb{Z}}$ as\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^p} \\overset{\\mbox{\\tiny{def}}}{=} \\left( \\sum_{k\\in\\mathbb{Z}} |a_k|^p \\right)^{1\/p},\n\\end{equation}\nand for $p=\\infty$ we use\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^\\infty} \\overset{\\mbox{\\tiny{def}}}{=} \\sup_{k\\in\\mathbb{Z}} |a_k|.\n\\end{equation}\nWe also define the following notation for $s\\ge 0$:\n\\begin{equation}\\label{bfcn.def}\n    b(n,s) \n    \\overset{\\mbox{\\tiny{def}}}{=} \\left\\{\n    \\begin{aligned} \n    1,\\quad 0 \\le s \\le 1,\\\\\n        n^{s-1}, \\quad s >1,\n    \\end{aligned}\\right.\n\\end{equation}\nFurther define the high frequency cut-off operator $\\mathcal{J}_N$ for $N\\ge 0$ by\n\\begin{equation}\\label{CutOffHigh}\n    \\widehat{\\mathcal{J}_N f}(k) \\overset{\\mbox{\\tiny{def}}}{=} 1_{|k|\\leq N}\\widehat{f}(k).\n\\end{equation}\nWe will use these norms and notations in the rest of the paper.\n\n\nThroughout the paper, we will denote \n\\begin{equation*}\n    C_i=C_i(x)=C_i\\left(x; A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}\\right)>0\n\\end{equation*}\nas functions that are increasing in $x\\geq0$ and might depend on the physical parameters such as $A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}$,  with the property that $C_i(x)\\approx 1$  for all $-1\\leq A_\\mu\\leq1$, $\\frac{|A_\\rho|R^2}{A_\\sigma}\\geq0$. Typically, $x$ will be the norm $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$ with $s=0$ or $s=1\/2$.\n\n\nWe denote $f*g$ as the standard convolution of $f$ and $g$.\nWe use the iterated convolution notation\n\\begin{equation}\\label{iteratedConvlution}\n*^kf = \\underbrace{f *  \\cdots *f}_\\text{$k-1$ convolutions of $k$ copies of $f$}\n\\end{equation}\nThus for instance $*^2f = f*f$.  We then sometimes also use the notation $g*^kf = g * \\underbrace{f *  \\cdots *f}_\\text{$k-1$ convolutions}$ to avoid an additional $*$ in the notation.\n\n\n\n\n\n\\subsection{Main Results}\\label{sec:globalTHM} \n\nThe main result of this paper states that, for any value of the physical parameters $A_\\rho$, $A_\\mu$, $A_\\sigma$, a bubble in a porous medium, with arbitrary volume $\\pi R^2$ and shape that is not too far from a circle, converges exponentially fast to a circle that rises (or falls) with constant velocity proportional to the difference between the inner and outer density. The initial interface needs to have barely more than a  continuous tangent vector, allowing in particular for unbounded curvature.  In particular since we suppose $\\theta_0\\in \\dot{\\mathcal{F}}^{\\frac12,1}$ then the initial interface has regularity $W^{\\frac32,\\infty}$,  in terms of the tangent vector the initial regularity is $W^{\\frac12,\\infty}$.  In particular the initial curvature doesn't need to be bounded.\nMoreover, the interface becomes instantaneously analytic.\n\nWe summarize here the system that models our problem. For clarity, we write the zero frequency of $\\vartheta$ apart because it is decoupled from the rest, and the equation for $\\theta$ with the linear and nonlinear terms separated:\n\\begin{equation}\\label{finalsystem}\n\\left\\{\\begin{aligned}\n \\widehat{\\vartheta}_t(0)&=\\frac{2\\pi}{L(t)}\\widehat{T}*\\widehat{(1+\\theta_\\alpha)}(0),\\\\\n     \\theta_t(\\alpha)&=\\frac{2\\pi}{L(t)}\\Big(\\mathcal{L}(\\alpha)+N(\\alpha)\\Big),\\\\\n    L(t)&=2\\pi R\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta(\\alpha)-\\theta(\\eta))^n d\\eta d\\alpha\\Big)^{-\\frac12},\\\\\n    0&=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\theta(\\alpha))}d\\alpha,\n    \\end{aligned}\\right.\n\\end{equation}\nwhere the linear and nonlinear terms $\\mathcal{L}(\\alpha)$, $N(\\alpha)$ are given by \\eqref{system}\nwith $T(\\alpha)$ defined in \\eqref{T}, \\eqref{Tsplit}, and \\eqref{T0}, $U(\\alpha)$ in \\eqref{U} and \\eqref{Usplit}.\n\n\n\\begin{thm}\\label{thm:global}\n\tFix $A_\\mu\\in[-1,1]$, $A_\\rho\\in\\mathbb{R}$, $A_\\sigma>0$, and $R>0$.\n\tAssume that the initial data  $\\vartheta_0(\\alpha)=\\widehat{\\vartheta}_0(0)+\\theta_0(\\alpha)\\in \\dot{\\mathcal{F}}^{\\frac12,1}$ satisfies the medium-size condition \n    \\begin{equation}\\label{condition}\n     \\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}<K\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n    \\end{equation}\n    where $K$ is defined in \\eqref{Kdef}.\n\tThen, for any $T>0$, there exists a unique global strong solution   $\\vartheta(\\alpha, t)=\\widehat{\\vartheta}(0,t)+\\theta(\\alpha,t)$ to the system  \\eqref{finalsystem}, which lies in the space\n\t\\begin{equation*}\t    \\vartheta\\in C([0,T];\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu)\\cap L^1([0,T];\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu),\\quad 0<T<\\infty,\n\t\\end{equation*}\n\twith $\\nu(t)$ given by \\eqref{nu}.\n\tIn particular the solution becomes instantaneously analytic. Moreover, the following energy inequality is satisfied for $0\\leq t\\leq T$:\n\t\\begin{equation}\\label{estimatef12}\n    \t\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)+ \\frac{A_\\sigma}{R^3}  \\mathcal{D} \\int_0^t \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}(\\tau) d\\tau \\leq C_S^2\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n\t\\end{equation}\n\twith $$\\mathcal{D}=\\mathcal{D}\\Big(C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma},\n       A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)>0$$\n       defined in \\eqref{dissipation}, and $C_S=C_S\\big(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma}\\big)$ defined in \\eqref{CSbounds}. \n       In addition,  we have the uniform in time estimate\n\t\\begin{equation}\\label{decay}\n\t    \\begin{aligned}\n\t        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)\\leq C_S^2 \\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}e^{-\\frac{A_\\sigma}{R^3}\\mathcal{D} t}.\n\t    \\end{aligned}\n\t\\end{equation}\n\tFurthermore, the zero frequency $\\widehat{\\vartheta}(0)$ remains bounded for all times\n\t\\begin{equation}\\label{zeroBoundUniform}\n\t     |\\widehat{\\vartheta}(0,t)|\\leq |\\widehat{\\vartheta}_0(0)|+C_{42}\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n\t\\end{equation}\n\twhere $C_{42}$ is defined in \\eqref{C42}.\n\\end{thm}\n\nWe remark that none of the uniform constants in Theorem \\ref{thm:global} depend upon our choice of $T>0$, and $T$ can be taken arbitrarily large.\n\n\\begin{remark}\nFrom Proposition \\ref{circles}, the large time decay in \\eqref{decay} implies the exponential convergence to rising or falling circles. Moreover, as part of the proof, it will be proven in \\eqref{Lboundaux} that the length satisfies for all times $t\\geq0$ that\n\\begin{equation}\\label{Lfinalbound}\n     \\begin{aligned}\n        \\frac{R}{\\sqrt{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}(t)}-1\\big)}}\\leq\\frac{L(t)}{2\\pi}\\leq \\frac{R}{\\sqrt{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}(t)}-1\\big)}},\n    \\end{aligned}\n    \\end{equation}\n    which also shows that $L(t)\\to 2\\pi R$ as $t\\to \\infty$.\n\\end{remark}\n\n\nThe size condition \\eqref{condition} of the theorem above is explicit: for any value of the physical parameters, it gives a bound for the norm of the initial data that can be computed. We also notice that, thanks to the diagonalization performed in Section \\ref{secanalytic}, the dissipative character of the equation is shown in \\eqref{estimatef12} for any $A_\\sigma>0$, no matter how large the gravity effects are.\n\n\n\n\n\\section{Implicit function theorem}\\label{IFTSection}\n\n\nIn this section, we will prove an explicit uniform upper bound for the $\\pm 1$ frequencies of $\\theta$ in terms of the higher Fourier frequencies of $\\theta$.  The main result of this section is the implicit function theorem in Proposition \\ref{IFTprop}.\n\n\n\n\n\nFor ease of notation, only in this section we will use the following space.  For $s \\ge 0$, we define the normed space\n\\begin{equation}\\label{tildefsone}\n    \\tilde{\\mathcal{F}}^{s,1} \\overset{\\mbox{\\tiny{def}}}{=} \\left\\{ u: \\mathbb{T} \\rightarrow \\mathbb{T} \\ | \\ \\hat{u}(0) = \\hat{u}(\\pm 1) = 0 \\text{ and } \\|u\\|_{\\tilde{\\mathcal{F}}^{s,1}}  < \\infty \\right\\}.\n\\end{equation}\nHere we use the norm\n\\begin{equation*}\n    \\|u\\|_{\\tilde{\\mathcal{F}}^{s,1}} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{|k|\\geq 2} |k|^{s}|\\hat{u}(k)|.\n\\end{equation*}\nIn view of \\eqref{tildefsone}, recalling \\eqref{CutOffHigh}, we consider \n$\n\\tilde{\\theta} \\overset{\\mbox{\\tiny{def}}}{=} \\left(I-\\mathcal{J}_2 \\right)\\theta.\n$\nWe then remark that \n$$\n\\tilde{\\theta}(\\alpha) = \n\\left(I-\\mathcal{J}_2 \\right)\\theta(\\alpha)=\\sum_{|k|\\ge 2} \\hat{\\theta}(k) e^{ik\\alpha}.\n$$\nAn implicit relation between the frequencies $\\hat{\\theta}(\\pm 1)$ and the function $\\tilde{\\theta}$ can be derived from \\eqref{constraint} and is given by\n\\begin{equation}\\label{implicitrelation}\n\t\\int_{-\\pi}^{\\pi} e^{i\\alpha + i\\hat{\\theta}(-1)e^{-i\\alpha} + i\\hat{\\theta}(1)e^{i\\alpha} + i\\tilde{\\theta}(\\alpha)} d\\alpha = 0.\n\\end{equation}\nNote that $\\hat{\\theta}(-1) = \\overline{\\hat{\\theta}(1)}=\\text{Re}\\hspace{0.05cm} \\hat{\\theta}(1)- i \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1)$ since $\\theta$ is real.  Further\n\\begin{equation}\\notag\n\\hat{\\theta}(-1)e^{-i\\alpha} + \\hat{\\theta}(1)e^{i\\alpha}\n=\n2\\text{Re}\\hspace{0.05cm} \\hat{\\theta}(1) \\cos(\\alpha) - \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1)\\sin(\\alpha).\n\\end{equation}\nWe will also use the following notation\n\\begin{equation}\\label{psi}\n    \\psi(\\alpha, x, u) = \\alpha + 2(x_{1}\\cos(\\alpha) - x_{2}\\sin(\\alpha)) + u(\\alpha),\n    \\quad x\\in\\mathbb{R}^{2}.\n\\end{equation}\nThen we express the integral in \\eqref{implicitrelation} as a vector as follows\n\\begin{equation}\\label{g}\n\tg(u, x) = \n\t\\begin{bmatrix}\n\t\t\\int_{-\\pi}^{\\pi} \\cos(\\psi(\\alpha, x, u))d\\alpha\n\t\t\\\\\n\t\t\\int_{-\\pi}^{\\pi} \\sin(\\psi(\\alpha, x, u))d\\alpha\n\t\\end{bmatrix}\n\t=\t\\begin{bmatrix}\n\t\tg_1(u, x)\n\t\t\\\\\n\t\tg_2(u, x)\n\t\\end{bmatrix}.\n\\end{equation}\nHere $g: \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2} \\rightarrow \\mathbb{R}^{2}$.\nNow we can rewrite the relation \\eqref{implicitrelation} as\n$$\ng(\\tilde{\\theta}, (\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))) = 0.\n$$\nThe main result in this section is the following proposition.\n\n\n\n\\begin{prop}\\label{IFTprop}\nFix $0<r<\\frac{1}{2}\\log(\\frac{5}{4})$.\nConsider $\\|\\theta\\|_{\\mathcal{F}^{0,1}} < r$.  Let\n$$\nB_{\\tilde{\\mathcal{F}}^{0,1}}(0,r) \\overset{\\mbox{\\tiny{def}}}{=} \\{ u \\in \\tilde{\\mathcal{F}}^{0,1}: \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < r\\}.\n$$\nThen there exists a unique function $F:B_{\\tilde{\\mathcal{F}}^{0,1}}(0,r)\\to \\mathbb R^2$ such that \n$F(\\tilde{\\theta})=(\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))$ and \\eqref{implicitrelation} is satisfied.\nWe further have\n\\begin{equation}\\label{frequencyrelation}\n\t\t\t|\\hat{\\theta}(+1)|+|\\hat{\\theta}(-1)|  \\leq   C_{I}(r)r \\sum_{|k|\\ge 2} |\\hat{\\theta}(k)|.\n\\end{equation}\nHere the constant $C_I(r)>0$ is given by \n\\begin{equation}\\label{c1.implicit}\nC_I(r) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{r}\\frac{2\\exp(r)(\\exp(r)-1)}{1- 4(\\exp(2r)-1)}.\n\\end{equation}\nWe note that $C_I(r)$ is an increasing function of $r$ and that $C_I(r)\\to 2$ as $r\\to 0$ and $C_I(r)\\to \\infty$ as $r\\to \\frac{1}{2}\\log(\\frac{5}{4})$. \n\\end{prop}\n\nThis proposition is shown by an implicit function theorem argument on the function described in \\eqref{g} around the value $g(0,0) = 0$. The remainder of this section is dedicated to proving Proposition \\ref{IFTprop}.\n\nFirst, we compute the Fr{\\'e}chet derivatives with respect to $u \\in \\tilde{\\mathcal{F}}^{0,1}$ and $x\\in \\mathbb{R}^{2}$.  Below we use the notation $D_u$ to denote the one component Fr{\\'e}chet derivative of $g(u,x)$ so that $D_u g(u,x)$ is a two dimensional vector, and we denote $D_x$ to denote the two component Fr{\\'e}chet derivative of $g(u,x)$ so that $D_xg(u,x)$ is a $2\\times 2$ matrix.  We will also use the notation $D=(D_u, D_x)$ to denote the derivative in the variables $(u,x)$ and then $Dg(u,x)$ can be represented as a $3\\times 3$ matrix.\n\n\n\\begin{lemma}\\label{Frechet:Lemma}\n\tThe Fr\\'echet derivatives of $g$, recalling \\eqref{psi}, are given by\n\t\\begin{equation}\\label{Frechetu}\n\t\tD_{u}g(u,x)h\n\t\t=\n\t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\sin(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\cos(\\psi(\\alpha, x, u))\n\t\\end{bmatrix},\n\t\\end{equation}\n\tfor $h\\in \\tilde{\\mathcal{F}}^{0,1}$ and\n\t\\begin{equation}\\label{Frechetx}\n\t\tD_{x}g(u,x)y \n\t\t\t\t=\n\t\t\\begin{bmatrix}\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\cos\\alpha & 2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\sin\\alpha\n\t\t\\\\\n\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\cos\\alpha \n\t\t & -2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\sin\\alpha\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\t y_{1}\n\t\t\\\\\n\t\ty_{2}\n\t\\end{bmatrix},\n\t\\end{equation}\n\tfor $y = \\begin{bmatrix}\n\t\t y_{1}\n\t\t\\\\\n\t\ty_{2}\n\t\\end{bmatrix} \\in \\mathbb{R}^{2}$.\n\\end{lemma}\n\nFor simplicity, we write \\eqref{g} in complex notation as\n\\begin{equation}\\notag\n\tg(u, x) = \\int_{-\\pi}^{\\pi} e^{i\\alpha + 2i(x_{1}\\cos(\\alpha) -x_{2}\\sin(\\alpha)) + iu(\\alpha)}d\\alpha.\n\\end{equation}\nIn particular then in complex notation \\eqref{Frechetu} takes the form\n\t\\begin{equation}\\notag\n\t\tD_{u}g(u,x)h = \\int_{-\\pi}^{\\pi} i h(\\alpha) e^{i\\psi(\\alpha, x, u)} d\\alpha.\n\t\\end{equation}\nWe will prove Lemma \\ref{Frechet:Lemma} using this expression for $g(u,x)$.\n\n\n\\begin{proof}[Proof of Lemma \\ref{Frechet:Lemma}]\n\tComputing the derivative with respect to $u$, we have\n\t\\begin{multline*}\n\t\t|g(u+h, x) - g(u,x) - D_{u}g(u,x)h| \\\\= \\Big|\\int_{-\\pi}^{\\pi}  e^{i\\psi(\\alpha, x, u)}(e^{i h(\\alpha)} - 1 - ih(\\alpha)) d\\alpha \\Big|\n\t\t\\\\\n\t\t\\leq 2\\pi \\|e^{ih(\\alpha)} - 1 - ih(\\alpha)\\|_{L^{\\infty}} \\leq 2\\pi \\sum_{n=2}^{\\infty} \\frac{\\|h\\|_{L^{\\infty}}^{n}}{n!} \n\t\t\\\\\n\t\t\\leq   \\Big(2\\pi \\sum_{n=1}^{\\infty} \\frac{\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}^{n}}{(n+1)!} \\Big) \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\n\t\t\t\t\\leq   \\Big(2\\pi e^{\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}} \\Big) \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}},\n\t\\end{multline*}\n\tsince for $h\\in\\tilde{\\mathcal{F}}^{0,1}$ we have that $\\hat{h}(0) = \\hat{h}(\\pm 1) = 0$ and so\n\t$$ \\|h\\|_{L^{\\infty}} \\leq \\sum_{k\\in\\mathbb{Z}} |\\hat{h}(k)| = \\sum_{|k| \\geq 2}  |\\hat{h}(k)|.$$\n\tThus, as $h \\rightarrow 0$, we obtain validation of \\eqref{Frechetu}. Then \\eqref{Frechetx} is proven in a similar way.  \\end{proof}\n\nWe now prove Proposition \\ref{IFTprop}.\n\n\\begin{proof}[Proof of Proposition \\ref{IFTprop}]\nFirst notice that from \\eqref{Frechetx} we have \n\\begin{equation}\\label{dxg00}\n    \\begin{aligned}\n\tD_{x} g(0,0)y\n\t& = \t\t\\begin{bmatrix}\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\alpha) \\cos\\alpha & 2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\alpha) \\sin\\alpha\n\t\t\\\\\n\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\alpha) \\cos\\alpha \n\t\t & -2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\alpha) \\sin\\alpha\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\ty_{1}\\\\\n\t\ty_{2}\n\t\\end{bmatrix} \n\t\\\\\n\n\t&= 2\\pi \\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix} \\begin{bmatrix}\n\t\ty_{1}\\\\\n\t\ty_{2}\n\t\\end{bmatrix}\n\t=2\\pi\\begin{bmatrix}\n\t\ty_{2}\\\\\n\t\ty_{1}\n\t\\end{bmatrix},\n\t\\end{aligned}\n\\end{equation}\nTherefore $D_{x} g(0,0)^{-1} = \\frac{1}{2\\pi}\\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix}$.  \nFor simplicity, we normalize the function $g$ around $(0,0)$ by defining\n\\begin{equation}\\label{tildeg}\n\t\\tilde{g}(u,x) = \\begin{bmatrix}\n\t\t\\tilde{g}_{1}(u,x)\\\\\n\t\t\\tilde{g}_{2}(u,x)\n\t\\end{bmatrix} \n\t=  D_{x}g(0,0)^{-1}\\begin{bmatrix}\n\t\tg_{1}(u,x)\\\\\n\t\tg_{2}(u,x)\n\t\\end{bmatrix}\n\t=\\frac{1}{2\\pi}\\begin{bmatrix}\n\t\tg_{2}(u,x)\\\\\n\t\tg_{1}(u,x)\n\t\\end{bmatrix},\n\\end{equation}\nand thus, $D_{x}\\tilde{g}(0,0) = \\mathbb{I}_{\\mathbb{R}^{2}}$ which is the identity matrix on $\\mathbb{R}^{2}$. Next, define the function $\\phi: \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2} \\rightarrow \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$ by\n\\begin{equation}\\label{phi.def}\n\\phi(u,x) = [D\\tilde{\\phi}(0,0)]^{-1}\\tilde{\\phi}(u,x), \\quad\\text{  where  } \\quad \\tilde{\\phi}(u,x) \n=\n\\begin{bmatrix}\nu   \\\\\n\t\\tilde{g}_{1}(u,x)\\\\\n\t\t\\tilde{g}_{2}(u,x)\n\\end{bmatrix}.\n\\end{equation}\nThen also $D\\tilde{\\phi}(u,x)$ is a $3 \\times 3$ matrix.  We will obtain the implicit function $F$ given in Proposition \\ref{IFTprop} by inverting $\\phi$ in a neighborhood of the point $(0, 0) \\in \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$. We will also calculate $[D\\tilde{\\phi}(0,0)]^{-1}$,  notice that \n\\begin{equation}\\notag\nD\\tilde{\\phi}(0,0) \n=\n    \\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(0,0) & D_{x}\\tilde{g}(0,0)\n\t\\end{bmatrix}\n\t=\n\t\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix}.\n\\end{equation}\nHere $\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}}$ is the identity map on $\\tilde{\\mathcal{F}}^{0,1}$.\nThe last equality holds since\n\\begin{equation*}\n\\begin{aligned}\nD_{u}\\tilde{g}(0,0)h \n&= D_{x}g(0,0)^{-1}D_{u}g(0,0)h \n\\\\\n&=\\frac{1}{2\\pi}\n\t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix}\t\\begin{bmatrix}\n\t\t -\\sin(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\cos(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}=  0,\n\\end{aligned}\n\\end{equation*}\nsince $h\\in\\tilde{\\mathcal{F}}^{0,1}$ so that $\\hat{h}(\\pm 1) =0$.  Therefore $[D\\tilde{\\phi}(0,0)]^{-1} = \t\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix}$.\n\n\nFor two norms $\\| \\cdot \\|$ and $| \\cdot |$, we will use the notation that \n\\begin{equation}\\label{littleo}\n   f=o(|h|), \\quad \\text{if} \\quad \n   \\|f\\|\\to 0 \\quad \\text{and} \\quad \\frac{\\|f\\|}{|h|}\\to 0 \\quad \\text{as}\\quad  |h|\\to 0.\n\\end{equation}\nNow, to invert $\\phi$, we first define the function \n\\begin{equation}\\label{tau.def}\n    \\tau(u,x) \\overset{\\mbox{\\tiny{def}}}{=} (u,x) - \\phi(u,x).\n\\end{equation}\nWe will show that $\\tau(u,x)$ is a contraction map by computing $D\\tau$. We will calculate that\n$$\nD\\tau(u,x) = \\mathbb{I} - [D\\tilde{\\phi}(0,0)]^{-1} D\\tilde{\\phi}(u,x),\n$$\nwhere $\\mathbb{I}$ is the identity map on $\\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$. To this end, we compute \n\\begin{multline}\\notag\n\t\\tilde{\\phi}(u+h,x+y) - \\tilde{\\phi}(u,x)\n\t\\\\ = \\tilde{\\phi}(u+h,x+y) - \\tilde{\\phi}(u,x+y) + \\tilde{\\phi}(u,x+y) - \\tilde{\\phi}(u,x)\n\t\\\\\n\t= \\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(u,x) & D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix} + o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + |y|).\n\\end{multline}\nThen using, \n\\eqref{littleo},  the above holds as $\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + |y| \\to 0$.\n Hence,\n\\begin{equation*}\n\\begin{aligned}\n\tD\\tau(u,x) &=  \\mathbb{I} - \n\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix} \\!\\!\n\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(u,x) & D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix} \\\\\n\t&=\n\t\\begin{bmatrix}\n\t\t0  & 0\\\\\n\t\t-D_{u}\\tilde{g}(u,x) & \\mathbb{I}_{\\mathbb{R}^{2}}- D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix}.\n\t\\end{aligned}\n\\end{equation*}\nTo obtain $\\tau$ as a contraction on some ball $B_{R}(0) \\subset \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$, it is sufficient to show that the following holds\n\\begin{equation}\\label{contraction}\n\t\\Big\\|D\\tau(u,x) \\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix}\\Big\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} < r \\|(h,y)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}},\n\\end{equation}\nfor some $0< r< 1$ and\nfor any $(h,y)\\in B_{R}(0) \\subset \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$  where \n\\begin{equation}\\notag\n    B_{R}(0) \\overset{\\mbox{\\tiny{def}}}{=} \\{(u,x) \\ | \\ |x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < R \\}, \\quad R>0.\n\\end{equation}\nHere we also denote $\\|(h,y)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} = \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}+ |y|$.\n\n\n\n\n\n\nSince we have \\eqref{dxg00} then, using \\eqref{Frechetu} and \\eqref{Frechetx}, condition \\eqref{contraction} becomes the condition that the inequality\n\\begin{multline}\\label{contractioncondition}\n\\Big\\|D\\tau(u,x) \\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix}\\!\\Big\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}\\!\\!\n\t=\n\t\\\\\n\t\\Bigg|\\frac{1}{2\\pi} \t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\cos(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\sin(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}\n\t\\\\\n\t+\\! \\frac{1}{2\\pi} \n\t\t\t\\begin{bmatrix}\n\t\t\t\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right) - 2\\pi y_1\n\t\t \\\\\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right)- 2\\pi y_2\n\t\\end{bmatrix}\n\t\\Bigg|\n\t\\\\\n\t< r\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + r|y|,\n\\end{multline}\nholds for a fixed $0< r< 1$, for all $(u,x)\\in B_{R}(0)$, and for any $(h,y)\\in \\tilde{\\mathcal{F}}^{0,1}\\times \\mathbb{R}^{2}$.  We also use the definition \\eqref{psi} in the integral above.\n\n\nWe further {\\it claim} that condition \\eqref{contractioncondition} is satisfied for any ball \n$B_{R}(0)$ such that $2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(2)$ holds for all $(u,x)\\in B_{R}(0)$. \n\n\n\n {\\it Proof of the claim:}\tLet\n  \t\\begin{equation}\\label{A1}\n \tA_{1} = \n \t\\begin{bmatrix}\n\t\tA_{11}\n\t\t\\\\\n\t\tA_{12}\n\t\\end{bmatrix}\n \t=\n \t \\frac{1}{2\\pi} \t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\cos(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\sin(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}.\\end{equation}\nWe will use the complex exponential representation of $\\cos(\\psi(\\alpha, x, u))$ and $\\sin(\\psi(\\alpha, x, u))$ with \\eqref{psi}.  We will further Taylor expand $\\exp\\left(i 2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+i u(\\alpha)) \\right)$ \n\tand use that\n\t$\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} h(\\alpha) e^{\\pm i\\alpha} d\\alpha = \\hat{h}(\\mp 1) = 0$ since $h\\in \\tilde{\\mathcal{F}}^{0,1}$.   We obtain\n\t\t\\begin{multline*}\n\t\t|A_{11}| \n\t\t\\le \\frac{1}{4\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\\\\\n+ \\frac{1}{4\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{-i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|.\n\t\\end{multline*}\n\tFor simplicity we estimate the term with $e^{i\\alpha}$ below:\n\\begin{multline}\\notag\n\\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\n\\\\\n\t\t= \\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k} \\frac{2^{k}(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))^{k}u(\\alpha)^{n-k}}{n!} d\\alpha   \\Big|\n\\\\\n\t\t\\leq \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!}\\Big|\\Big(\\hat{h} \\ast^{k} \\mathcal{F}\\{2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))\\}\\ast^{n-k}\\hat{u}\\Big)(-1)\\Big|\n\\\\\n\\leq \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!} \\|\\hat{h}\\|_{\\ell^{1}}\\|\\hat{u}\\|_{\\ell^{1}}^{n-k}2^{k-1}|x|^{k}.\n\\end{multline}\nThe last line is obtained using Young's inequality for convolutions and \n\t$$\\widehat{\\cos}(k) = \\frac{1}{2}(\\delta(1-k) + \\delta(-1-k)), \\quad   \\widehat{\\sin}(k) = \\frac{-i}{2}(\\delta(1-k) - \\delta(-1-k)),$$\n and thus the $\\ell^{1}$ norm of the Fourier transform is\n \\begin{equation}\\label{ell1ft.def}\n    \\|(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha)))^{\\wedge}\\|_{\\ell^{1}} = 2|x|,\n\\end{equation}\n\tand the $\\ell^{\\infty}$ norm of the Fourier transform is\n\t$$\\|(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha)))^{\\wedge}\\|_{\\ell^{\\infty}} = |x|.$$\nThen, rewriting the series in function form using Taylor's theorem, we get\n\\begin{equation}\\notag\n    \t|A_{11}| \\leq \\frac{1}{2}\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\nDoing the same for the second term in \\eqref{A1} we obtain\n\\begin{equation}\\label{A1estimate}\n    \t|A_{1}| \\leq \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\n\tSince the latter term in \\eqref{contractioncondition} does not involve $h$, for this term we need \n$$\n e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1 < r < 1.\n$$\nThese estimates give the upper bound in terms of $\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}$ in \\eqref{contractioncondition}.\n\t\n\t\n\t\n\n\n\n\t\n\tThe other term in \\eqref{contractioncondition} will give us the upper bound in terms of $|y|$. Now let\n\t\\begin{equation*}\n\t    A_{2} = \\begin{bmatrix} A_{21} \\\\ A_{22} \\end{bmatrix}=  \\frac{1}{2\\pi} \n\t\t\t\\begin{bmatrix}\n\t\t\t\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right) - 2\\pi y_1\n\t\t \\\\\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right)- 2\\pi y_2\n\t\\end{bmatrix}.\n\t\\end{equation*}\n\tThen using \\eqref{psi}, again we use the complex exponential form of $\\cos(\\psi)$ and by Taylor expansion we have\n\t\\begin{multline}\\notag\n\t\tA_{21} = \n\t\t\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos(\\alpha)-y_{2}\\sin(\\alpha))e^{i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!} \n\t\t\\\\\n+\t\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos(\\alpha)-y_{2}\\sin(\\alpha))e^{-i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!}.\n\t\\end{multline}\nWe estimate the term with $e^{i\\alpha}$ as \n\t\\begin{multline}\\notag\n\t\t\\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)e^{i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!} \\Big|\n\t\t\\\\\n =\\Big| \\sum_{n=1}^{\\infty}\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} (y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)e^{i\\alpha}\\frac{(2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\n\t\t\\\\\n\t\t\\leq \\frac{1}{2}\\sum_{n=1}^{\\infty} \\frac{1}{n!}\\|2(y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)^{\\wedge}\\|_{\\ell^{\\infty}}\\|(2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)+u(\\alpha))^{\\wedge}\\|_{\\ell^{1}}^{n}\n\t\t\\\\\n\t\t\\leq \\frac{1}{2}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) |y|.\n\t\\end{multline}\n\tAll the other terms are estimated in the same way.\n\tAdding all the estimates together we obtain the condition \n\\begin{equation}\\notag\n    |A_2| \\le 2\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) |y|.\n\\end{equation}\nThis is a bigger coefficient in front of $|y|$ than the coefficient in the bound for $A_{1}$. Thus, the following condition\n$$\n2\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) < 1,\n$$ \nis sufficient. This yields the claim that condition \\eqref{contractioncondition} is satisfied on any ball contained in the set of $(x,u)$ such that $$2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(3\/2).$$  This completes the proof of the {\\it claim}.\n\t\n\n\nNow  $\\tau$ satisfies the contraction \\eqref{contraction} for $(u,x)\\in B_{R}(0)$ for $R$ chosen to satisfy \\eqref{contractioncondition}.  Recalling \\eqref{phi.def} and \\eqref{tau.def}, for $v,w\\in\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}$ and for $0<r<1$ fixed in \\eqref{contractioncondition}, we obtain\n\\begin{align*}\n\t\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} &\\leq \\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} + \\|\\tau(v)-\\tau(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}\\\\ &\\leq \\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} + r\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}.\n\\end{align*} \nWe conclude that\n\\begin{equation*}\n\t\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} \n\t\\leq (1-r)^{-1}\\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}.\n\\end{equation*}\nHence, $\\phi$ is an injection on $B=B_R(0)$ and so there exists an inverse $\\phi^{-1} : \\phi(B)\\rightarrow B$. Now we define the map $F(u)$ by\n$$\nF(u) \\overset{\\mbox{\\tiny{def}}}{=} \\pi_{2}\\circ \\phi^{-1}(u,(0,0)) \\text{ \\ \\ where \\ \\ } \\pi_{2}(a,(b_{1},b_{2})) = (b_{1},b_{2}).\n$$\nFurther define the map $\\pi_{1}$ by $\\pi_{1}(a,(b_{1},b_{2})) = a$.\n\nGiven $u\\in \\pi_{1}(B)$, by the definition of $F(u)$ there exists $u'\\in \\pi_{1}(B)$ such that $(u,(0,0)) = \\phi(u',F(u))$.  Then $u=u'$ by \\eqref{phi.def}. Further $\\phi(u,F(u)) = (u,\\tilde{g}(u,F(u)))$ and therefore $\\tilde{g}(u,F(u)) = 0$ using \\eqref{phi.def} and \\eqref{tildeg}.  We have thus shown that $g(u,F(u)) = 0$ from \\eqref{g}.  We conclude that $(u,(0,0)) \\in \\phi(B)$  for all $u\\in \\pi_{1}(B)$ and so $(u,F(u)) = \\phi^{-1}(u,(0,0)) \\in B$.  Therefore $g(u,F(u)) = 0$ for any $u\\in \\pi_{1}(B)$. This concludes the proof of the existence of the function $F$ described in Proposition \\ref{IFTprop}. \n\n\n\n\n\n\n\n\nTo obtain the estimate \\eqref{frequencyrelation} on $B$, first note that for $x = (x_{1},x_{2})$ we have\n\\begin{equation*}\n\\begin{aligned}\n|x| &= |\\pi_{2}\\circ \\phi^{-1}(u,(0,0))| = |\\pi_{2}\\circ \\phi^{-1}(u,(0,0)) - \\pi_{2}\\circ \\phi^{-1}(0,(0,0))|\\\\\n&\\leq \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\|D_{u}(\\pi_{2}\\circ\\phi^{-1})\\|.\n\\end{aligned}\n\\end{equation*}\nAbove note that $g(0,F(0))=0$ and so $F(0)=0$.  \nAlso the norm on $D_{u}(\\pi_{2}\\circ\\phi^{-1})$ on is the operator norm.   We calculate $D_{u}(\\pi_{2}\\circ\\phi^{-1})$  as follows:\n$$\\pi_{2}\\circ \\phi^{-1}(u+h,y)-\\pi_{2}\\circ \\phi^{-1}(u,y) = \\pi_{2}(u+h,x_{h}) - \\pi_{2}(u,x) = x_{h}-x,$$\nwhere we suppose $(u,y) \\in \\phi(B)$ and $(u+h,y) \\in \\phi(B)$ for small $h$.  Therefore\n\\begin{equation*}\n\\tilde{g}(u+h,x_{h}) = y \\text{ \\ \\ and \\ \\ }  \\tilde{g}(u,x) = y.    \n\\end{equation*}\nWe conclude that $[\\tilde{g}(u+h,x_{h})- \\tilde{g}(u,x)] = 0$.\nThus\n$$0 = \\tilde{g}(u+h,x_{h}) - \\tilde{g}(u,x_{h}) +\\tilde{g}(u,x_{h})- \\tilde{g}(u,x).$$\nThen, using \\eqref{littleo}, we have\n$$\n0 = D_{u}\\tilde{g}(u,x_{h})h +o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}) + D_{x}\\tilde{g}(u,x)(x_{h}-x) +o(|x_{h}-x|).\n$$\nWe can now conclude that\n\\begin{multline}\\notag\n    x_{h}-x + D_{x}\\tilde{g}(u,x)^{-1}o(|x_{h}-x|) \\\\ = -D_{x}\\tilde{g}(u,x)^{-1}D_{u}\\tilde{g}(u,x)h -D_{x}\\tilde{g}(u,x)^{-1}o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}).\n\\end{multline}\nWe will show that $D_{x}\\tilde{g}(u,x)^{-1}$ is a bounded operator with bound \\eqref{detandTbound} for $(x,u)$ satisfying \\eqref{xucriteria}. Thus, for $(x,u)$ satisfying \\eqref{xucriteria}, we have\n$$D_{u}(\\pi_{2}\\circ \\phi^{-1}) = -D_{x}\\tilde{g}(u,x)^{-1}D_{u}\\tilde{g}(u,x).$$\nThen we conclude that\n\\begin{equation}\\label{xinitialbound}\n|x| \\leq \\|D_{x}\\tilde{g}(u,x)^{-1}\\| \\|D_{u}\\tilde{g}(u,x)\\| \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}.\n\\end{equation}\nThe term $D_{u}\\tilde{g}(u,x)$, using \\eqref{Frechetu} with \\eqref{dxg00}, is given by\n$$D_{u}\\tilde{g}(u,x) = D_{x}g(0,0)^{-1}D_{u}g(u,x) = -A_{1},$$\nwhere  $A_{1}$ is given by \\eqref{A1} with \\eqref{psi}. Hence, using \\eqref{A1estimate}, we have \n\\begin{equation}\\label{Dugbound}\n\\|D_{u}\\tilde{g}(u,x)\\| \\leq \t\\Big(e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\nConsider the operator $D_{x}\\tilde{g}(u,x)^{-1}$.  \nUsing \\eqref{Frechetx} and \\eqref{dxg00} with \\eqref{psi}, we calculate that\n\\begin{equation*}\n    D_{x}\\tilde{g}(u,x) =\\! \\frac{1}{\\pi}\\! \\begin{bmatrix}\n    \\!\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha & -\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\\\\\\\\\n    -\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha &  \\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha\n    \\end{bmatrix}\\!.    \n\\end{equation*}\nInverting this matrix, we obtain\n\\begin{equation}\\label{aux1}\n    \\begin{aligned}\n    D_{x}\\tilde{g}(u,x)^{-1} = \\frac{1}{\\det(D_{x}\\tilde{g}(u,x))} T(\\psi),\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{T.psi.def}\n    T(\\psi)=\\frac{1}{\\pi} \\begin{bmatrix}\n\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha\\ \\ \\ \\ \\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\n\\\\\n\\\\\n\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha \\ \\ \\ \\  \\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\n\\end{bmatrix}.\n\\end{equation}\nWe will calculate a lower bound for $d\\overset{\\mbox{\\tiny{def}}}{=}\\det(D_{x}\\tilde{g}(u,x))$:  \n\\begin{align}\nd\n&\\!=\\! \\frac1{\\pi^{2}}\\!\\int_{-\\pi}^{\\pi}\\!\\!\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\\! \\int_{-\\pi}^{\\pi}\\!\\!\\sin(\\beta)\\sin(\\psi(\\beta)) d\\beta \\nonumber\\\\ \\label{det}\n&\\quad- \\frac1{\\pi^{2}}\\! \\int_{-\\pi}^{\\pi}\\!\\!\\!\\!\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha \\!\\int_{-\\pi}^{\\pi}\\!\\!\\!\\sin(\\beta)\\cos(\\psi(\\beta)) d\\beta\\\\\n&=\\frac1{\\pi^{2}}\\int_{-\\pi}^{\\pi}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\!\\cos(\\alpha)\\sin(\\beta)\\sin(\\psi(\\beta)-\\psi(\\alpha))d\\alpha d\\beta.\\nonumber\n\\end{align}\nNext, recalling \\eqref{psi}, define \n$$\nR(\\alpha,\\beta) \\overset{\\mbox{\\tiny{def}}}{=} 2x_{1}(\\cos\\beta-\\cos\\alpha)-2x_{2}(\\sin\\beta-\\sin\\alpha) + u(\\beta)-u(\\alpha).\n$$\nThen we have\n\\begin{align}\n|\\sin(\\psi(\\beta)-\\psi(\\alpha))& -\\sin(\\beta - \\alpha)|  = |\\sin(\\beta-\\alpha + R(\\alpha,\\beta))-\\sin(\\beta - \\alpha)| \\nonumber\\\\\\label{sinAalphbeta}\n&=  |\\sin(\\beta-\\alpha)(\\cos(R(\\alpha,\\beta)) - 1)+\\cos(\\beta-\\alpha)\\sin(R(\\alpha,\\beta))| \\\\\\nonumber\n&\\leq |\\cos(R(\\alpha,\\beta)) - 1| + |\\sin(R(\\alpha,\\beta))| .\n\\end{align}\nSince $|R(\\alpha,\\beta)| \\leq 4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}$, we have that\n\\begin{multline}\\label{sinAalphbetabound}\n|\\cos(R(\\alpha,\\beta)) - 1| + |\\sin(R(\\alpha,\\beta))| \n\\\\\n\\leq \\sum_{n=1}^{\\infty} \\frac{(4|x|\\! +\\! 2\\|u\\|_{\\mathcal{F}^{0,1}})^{2n}}{(2n)!}\\! +\\! \\sum_{n=0}^{\\infty}\\frac{(4|x| \\!+\\! 2\\|u\\|_{\\mathcal{F}^{0,1}})^{2n+1}}{(2n+1)!} = e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} \\!-\\! 1,\n\\end{multline}\nwhich tends to zero as $(u,x)$ go to zero. Hence, by \\eqref{sinAalphbeta} and \\eqref{sinAalphbetabound} we have\n\\begin{multline*}\n\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)(\\sin(\\psi(\\beta)-\\psi(\\alpha)) -\\sin(\\beta - \\alpha))d\\alpha d\\beta \\\\ \\geq -4\\pi^{2}\\|\\sin(\\psi(\\beta)-\\psi(\\alpha)) -\\sin(\\beta - \\alpha)\\|_{L^{\\infty}} \\\\\n\\geq -4\\pi^{2} (e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} - 1)\n\\end{multline*}\nand\n$$\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta) \\sin(\\beta - \\alpha)d\\alpha d\\beta = \\pi^{2}.$$\nWe thus conclude \n\\begin{multline*}\n    \\Big|\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)\\sin(\\psi(\\alpha)-\\psi(\\beta))d\\alpha d\\beta\\Big| \\\\\n    = \\Big|\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)(\\sin(\\psi(\\alpha)-\\psi(\\beta)) -\\sin(\\alpha - \\beta))d\\alpha d\\beta\\\\ \n    + \\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta) \\sin(\\alpha - \\beta)d\\alpha d\\beta\\Big|\\\\\n    \\geq -4\\pi^{2} (e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} - 1) + \\pi^{2}.\n\\end{multline*}\nThus, the determinant given by \\eqref{det} is bounded from below by\n\\begin{equation}\\label{detbound}\n|\\det(D_{x}\\tilde{g}(u,x))| \\geq 1+4\\big(1- e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}}\\big).\n\\end{equation}\nTherefore, it only remains to estimate the norm $\\|T(\\psi)\\|$ to obtain a bound for the norm of $D_{x}\\tilde{g}(u,x)^{-1}$ in \\eqref{aux1}.\n\n\n\n\n\nRecalling \\eqref{T.psi.def}, we calculate that \n\\begin{equation}\\notag\n    T(\\psi)\\begin{bmatrix}\ny_{1}\\\\\ny_{2}\n\\end{bmatrix}\n=\n    \\frac{1}{\\pi} \\begin{bmatrix}\n\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)d\\alpha\n\\\\\n\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha))\\right) d\\alpha\n\\end{bmatrix}.\n\\end{equation}\nWe have for $\\phi(\\alpha) = \\psi(\\alpha) - \\alpha = 2(x_{1}\\cos(\\alpha) - x_{2}\\sin(\\alpha)) + u(\\alpha)$ that\n\\begin{multline}\\notag\n\\sin\\alpha\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)\n\\\\\n    =\n    -\\frac{1}{4}y_1\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}-e^{-i\\phi}-e^{i\\phi}\n    +e^{-i\\phi}e^{-2i\\alpha}\n    \\right)\n    \\\\\n        +\\frac{1}{4i}y_2\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}+e^{-i\\phi}-e^{i\\phi}\n    -e^{-i\\phi}e^{-2i\\alpha}\n    \\right).\n\\end{multline}\nAnd similarly\n\\begin{multline}\\notag\n\\cos\\alpha\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)\n\\\\\n    =\n    \\frac{1}{4i}y_1\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}-e^{-i\\phi}+e^{i\\phi}\n    -e^{-i\\phi}e^{-2i\\alpha}\n    \\right)\n    \\\\\n        +\\frac{1}{4}y_2\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}+e^{-i\\phi}+e^{i\\phi}\n    +e^{-i\\phi}e^{-2i\\alpha}\n    \\right).\n\\end{multline}\nWe conclude that\n\\begin{multline}\\notag\n    T(\\psi)\n=\n\\\\\n\\frac{1}{2}\n\\begin{bmatrix}\n\\left(\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)}-\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)} \\right)\n&\n\\left(\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}-\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n\\\\\n\\left(\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}-\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n& \\left(\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}+\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n\\end{bmatrix}.\n\\end{multline}\nHence \n\\begin{equation*}\n\\Big|T(\\psi) \\begin{bmatrix}\ny_{1}\\\\\ny_{2}\n\\end{bmatrix} \\Big| \\leq 2 |y|\\|\\widehat{e^{i\\phi(\\alpha)}}\\|_{\\ell^{1}}.\n\\end{equation*}\nHence, using \\eqref{ell1ft.def}, we obtain the bound\n\\begin{align*}\n\\|T(\\psi)\\| &\\leq \\sum_{n=0}^{\\infty} \\frac{\\|\\widehat{\\phi^{n}}\\|_{\\ell^{1}}}{n!}\\\\ \n&\\leq  \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!}\\Big\\| \\ast^{k} |\\mathcal{F}\\{2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)\\}|\\ast^{n-k}|\\hat{u}|\\Big\\|_{\\ell^{1}}\\\\\n&\\leq  \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!} 2^{k}|x|^{k}\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}^{n-k}.\n\\end{align*}\nThis finally yields the result \n\\begin{equation}\\label{Tbound}\n\\|T(\\psi)\\|\\leq 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}.\n\\end{equation}\nFrom \\eqref{aux1}, using \\eqref{detbound} and \\eqref{Tbound}, we obtain \n\\begin{equation}\\label{detandTbound}\n\\|D_{x}\\tilde{g}(u,x)^{-1}\\| \\leq \\frac{ 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}}{1+ 4(1- e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}})}.\n\\end{equation}\nIn conclusion, we have from \\eqref{xinitialbound}, \\eqref{Dugbound} and \\eqref{detandTbound} that\n\\begin{equation}\\label{xbound}\n2|x| \\leq  \\frac{ 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}(e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1)}{1- 4( e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}-1)} \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}.\n\\end{equation}\nFrom the denominator term, we need $$ e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1 < \\frac{1}{4}.$$\nThis yields the condition\n\\begin{equation}\\label{xucriteria}\n2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\frac{1}{2}\\log\\big(5\/4\\big),\n\\end{equation}\nwhich ensures that the condition \n$2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(3\/2)$ is also satisfied.   Using $|x| = |\\hat{\\theta}(\\pm 1)|$ and $u = \\tilde{\\theta}$, we obtain \n$$2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} = |\\hat{\\theta}(1)| +  |\\hat{\\theta}(-1)| + \\|\\tilde{\\theta}\\|_{\\tilde{\\mathcal{F}}^{0,1}} =  \\|\\theta\\|_{\\mathcal{F}^{0,1}}.$$\nThus, for $\\|\\theta\\|_{\\mathcal{F}^{0,1}} <\\frac{1}{2}\\log\\big(\\frac{5}{4}\\big)$ and using \\eqref{xbound}, we obtain Proposition \\ref{IFTprop}.\n\\end{proof}\n\n\nThe result in Proposition \\ref{IFTprop} will be needed to close the \\textit{a priori} estimates in Subsection \\ref{subsecGlobal} . In particular, Proposition \\ref{linearfourier} show that the modes $\\pm 1$ are neutral at the linear level (see also \\eqref{auxx2}). \n\n\n\n\\section{Fourier multiplier estimates}\\label{sec:FourierRandS}\n\n\n\n\nIn this section, we will prove the crucial estimates for the operators $\\mathcal{R}$ from \\eqref{R}  (in Proposition \\ref{Restimates.prop}) and $\\mathcal{S}$ from \\eqref{S} (in Proposition \\ref{Smultiplierprop}) in the norms ${\\mathcal{F}^{0,1}_\\nu}$ and  ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ for $s> 0$ from \\eqref{fzeroonenunorm} and \\eqref{fsonenorm} respectively.  \n\nWe will use the following facts throughout the section:\n\n\\begin{lemma}\\label{triangleprop}\nWe have the estimate\n\t\\begin{equation}\\label{zeroproduct}\n\t\t\\|g_{1}g_{2}\\cdots g_{n}\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \\prod_{k=1}^{n} \\|g_{k}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\tFor $s>0$, recalling \\eqref{bfcn.def}, we have\n\t\\begin{equation}\\label{sproduct}\n\t\t\\|g_{1}g_{2}\\cdots g_{n}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq b(n,s)\\sum_{j=1}^{n} \\|g_{j}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t\t\\prod_{\\substack{k=1 \\\\ k\\ne j}}^{n}  \\|g_{k}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\\end{lemma}\n\n\n\\begin{remark}\\label{nuremark}\nWe note that these results and all the apriori estimates in this paper further hold with ${\\mathcal{F}^{0,1}_\\nu}$ and ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ replaced by ${\\mathcal{F}^{0,1}}$ and ${\\dot{\\mathcal{F}}^{s,1}}$ respectively since we can simply take $\\nu=0$.\n\\end{remark}\n\n\n\n\n\\begin{proof}\nFirst consider the case $0< s \\leq 1$ and $n=2$ in \\eqref{sproduct}. Then we use the inequalities\n$$|k|^{s} \\leq |k-j|^{s} + |j|^{s}, $$\nand\n$$e^{\\nu(t)|k|} \\leq e^{\\nu(t)|k-j|}e^{\\nu(t)|j|}, $$\nto see that\n\\begin{align*}\n\t\\|g_1 g_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &= \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|} e^{\\nu(t)|k|} |k|^{s} \n\t\\left| \\widehat{g_1 g_2}(k) \\right|\n\t\\\\\n \t&\\leq \\sum_{j,k\\in\\mathbb{Z}} e^{\\nu(t)|k-j|}e^{\\nu(t)|j|}(|k-j|^{s} + |j|^{s})  \\left| \\hat{g_1}(k-j)\\hat{g_2}(j) \\right| \\\\\n\t&\\leq \\|g_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|g_2\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|g_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\|g_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{align*}\nFor general $n$ and $s>0$, recalling \\eqref{bfcn.def}, we use the following inequality\n\\begin{equation}\\label{s.inequality}\n    |k|^{s} \\leq b(n,s) (|k-k_{1}|^{s} + |k_{1}-k_{2}|^{s} + \\ldots + |k_{n-2} - k_{n-1}|^{s} + |k_{n-1}|^{s}).\n\\end{equation}\nThen the proofs of all the other cases in \\eqref{zeroproduct} and \\eqref{sproduct} follow similarly.  \n\\end{proof}\n\n\n\nWe will also use the following repeatedly in this section:\n\\begin{prop}\nWe have the useful bound for $\\beta \\in \\mathbb{T}=[-\\pi, \\pi]$ and $l \\ge 1$:\n\\begin{equation}\\label{usefuloperatorbound}\n\\Big|\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1 \\Big| \\leq |\\beta|\\frac{l}{2}\\Big(\\frac{\\pi}{2}\\big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nBy the mean value theorem we have\n\\begin{equation*}\n\t\t\\Big|\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1 \\Big| \\leq |\\beta| \\Big\\|\\frac{d}{d\\beta}\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}  \\Big\\|_{L^{\\infty}({\\mathbb{T}})}.\n\t\\end{equation*}\n\tWe compute the above derivative as:\n\t\\begin{equation*}\n\t\t\\frac{d}{d\\beta}\\Big( \\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l}  = l\\Big(\\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l-1}\\frac{d}{d\\beta}\\Big(\\frac{i\\beta}{1-e^{-i\\beta}}\\Big).\n\t\\end{equation*}\n\tFirst, for $\\beta\\in\\mathbb{T}$, we have \n\t\\begin{align*}\n\t\t\\Big|\\frac{\\beta}{(1-e^{-i\\beta})}\\Big|&= \\frac{|\\beta|}{\\sqrt{(1-\\cos(\\beta))^{2}+\\sin^{2}(\\beta)}}\n\t\t= \\frac{|\\beta|}{\\sqrt{2-2\\cos(\\beta)}}\\\\\n\t\t&= \\frac{|\\beta|}{2\\sin(\\beta\/2)}\n\t\t\\leq \\frac{\\pi}{2}.\n\t\\end{align*}\n\t\tNext, we have\n\t\\begin{align*}\n\t\t\\Big|\\frac{d}{d\\beta}\\Big(\\frac{\\beta}{1-e^{-i\\beta}}\\Big)\\Big| & = \\frac{|1-e^{-i\\beta}-i\\beta e^{-i\\beta}|}{|(1-e^{-i\\beta})^{2}|}\\\\\n\t\t&=\\frac{|(1-\\cos(\\beta)-\\beta\\sin(\\beta)) + i (\\sin(\\beta)-\\beta\\cos(\\beta))|}{4\\sin^{2}(\\beta\/2)}.\n\t\\end{align*}\n\tIn absolute value, the real term in the numerator is\n\t$$|1-\\cos(\\beta)-\\beta\\sin(\\beta)| = |2\\sin^{2}(\\beta\/2) -2\\beta\\sin(\\beta\/2)\\cos(\\beta\/2)|.$$\n\tHence for $\\beta\\in\\mathbb{T}$ we have  \n\t$$\\frac{|1-\\cos(\\beta)-\\beta\\sin(\\beta)| }{4\\sin^{2}(\\beta\/2)} \\leq \\Big|\\frac{1}{2} - \\frac{\\beta}{2\\tan(\\beta\/2)}\\Big|\\leq \\frac{1}{2}.$$\n\tNext, the imaginary part can be checked to have a maximum of \n\t$$ \\frac{|\\sin(\\beta)-\\beta\\cos(\\beta)|}{4\\sin^{2}(\\beta\/2)} \\leq \\frac{\\pi}{4}.$$\n\tThus we have \n\t\\begin{equation}\\notag\n\t\\Big|\\frac{d}{d\\beta}\\Big(\\frac{\\beta}{1-e^{-i\\beta}}\\Big)\\Big|\\leq \\frac{1}{2}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\t\\end{equation}\n\tWe conclude that\n\t$$\\Big|\\frac{d}{d\\beta}\\Big[\\Big( \\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l}-1\\Big] \\Big| \\leq \\frac{1}{2}l\\Big(\\frac{\\pi}{2}\\Big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.$$\n\tThis yields \\eqref{usefuloperatorbound}.\n\\end{proof}\n\nWe will now estimate the operator $\\mathcal{R}$ from \\eqref{R} as follows. \n\n\\begin{prop}\\label{Restimates.prop}\n\tThe operator $\\mathcal{R}$ from \\eqref{R} satisfies the estimates\n\t\\begin{equation}\\label{Restimates}\n\\begin{aligned}\n\t\t\\|\\mathcal{R}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &\\leq {C_{\\mR}}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\t\n\t\t\\\\\n\t\t\\|\\mathcal{R}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &\\leq {b(2,s)} {C_{\\mR}}(\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}),\\qquad s>0.\n\\end{aligned}\n\\end{equation}\nwhere $b(2,s)$ is from \\eqref{bfcn.def} and the constant\n\\begin{equation}\\label{CR}\n{C_{\\mR}}\t\\overset{\\mbox{\\tiny{def}}}{=} 1+\\tilde{\\mathcal{C}}>0,    \n\\end{equation}\nand $\\tilde{\\mathcal{C}}>0$ is defined in \\eqref{catalanconstant}.\n\\end{prop}\n\nIn the rest of this section we will adopt the convention that \n\t\\begin{equation}\\label{convention}\n\t\\frac{1-e^{-i\\beta(1+k_1)}}{1+k_1}  = i\\beta \\quad \\text{for} ~k_1=-1. \n\t\\end{equation}\nThis convention will allow us to write many formula's succinctly.\n\n\\begin{proof}\nTaking the Fourier transform of the operator $\\mathcal{R}$ from \\eqref{R} and using the convention \\eqref{convention} we obtain \n\t\\begin{equation*}\n\t   \\begin{aligned}\n\t   \t\\widehat{\\mathcal{R}(f)}(k) &=\\!\\frac{i}{\\pi}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k)e^{-ik\\beta}\\beta}{(1\\!-\\!e^{-i\\beta})^2}\\ast \\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta(1+k)}\\hat{\\theta}(k) dsd\\beta\\\\\n\t\t&=\\!\\frac{i}{\\pi}\\sum_{k_1\\in\\mathbb{Z}}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k-k_1)e^{-i(k-k_{1})\\beta}\\beta}{(1\\!-\\!e^{-i\\beta})^2} \\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta(1+k_1)}\\hat{\\theta}(k_1) dsd\\beta\\\\\n\t\t&= \\!\\frac{1}{\\pi}\\sum_{k_1\\in\\mathbb{Z}}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k-k_1)e^{-i(k-k_{1})\\beta}}{(1\\!-\\!e^{-i\\beta})^2} \\frac{1-e^{-i\\beta(1+k_1)}\\hat{\\theta}(k_1)}{1+k_1} d\\beta.\n\t    \\end{aligned}\n\t\\end{equation*}\n\t\tUsing the convention \\eqref{convention}, we have\n\t\t\\begin{equation*}\n\t\\widehat{\\mathcal{R}(f)}(k)=  \\sum_{k_1\\in\\mathbb{Z}} \\hat{f}(k-k_1)\\hat{\\theta}(k_1)I(k,k_1),\n\t\\end{equation*}\n   where\n\t\\begin{equation}\\label{I.function.def}\n\t I(k,k_1)\\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1})\\beta}}{(1\\!-\\!e^{-i\\beta})^2} \\frac{1-e^{-i\\beta(1+k_1)}}{1+k_1} d\\beta.\n\t\\end{equation}\n\tFor $k_{1} = -1$, using \\eqref{convention}, we split \\eqref{I.function.def} as \n\t\\begin{equation*}\n\t\tI(k,-1) = \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} d\\beta \n\t\t+ \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} \\Big(\\frac{i\\beta}{1-e^{-i\\beta}} - 1 \\Big).\n\t\\end{equation*}\nWe will first calculate each of these two integrals separately below.\t\n\t\n\n\t\nWe can calculate the general integral formula:\n\\begin{equation}\\label{j1}\n \\frac{1}{\\pi} \\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i \\ell \\beta}}{1-e^{-i\\beta}} d\\beta\n =\n 1_{\\ell \\le 0}(\\ell) -  1_{\\ell \\ge 1}(\\ell).\n\\end{equation}\nThis will be used several times below.  It is proven using  \\eqref{hilbert} and noticing that $\\mathcal{F}(e^{-i \\ell \\beta})(0)=\\delta(\\ell)$.  Further from \\eqref{defHilbertTransform} we have\n$$\n-i\\mathcal{H} (e^{-i \\ell \\beta})(0) = \n-\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\sin(\\ell\\beta)}{\\tan{(\\beta\/2)}} d\\beta\n= 1_{\\ell \\le -1}(\\ell) -  1_{\\ell \\ge 1}(\\ell).\n$$  \nCombining these calculations gives \\eqref{j1}.\n\t\n\tFrom \\eqref{j1}, the first term in $I(k,-1)$ is $\\pm 1$\n\tdepending on the sign of $k+1$. For the second integral, we use \\eqref{usefuloperatorbound} for $l=1$ to obtain\n\t\\begin{align*}\n\t\\Big|\\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} \\Big(\\frac{i\\beta}{1-e^{-i\\beta}} - 1 \\Big)\\Big| &\\leq \\frac{1}{\\pi}\\sqrt{\\frac{1}{4} + \\frac{\\pi^{2}}{16}} \\int_{-\\pi}^{\\pi} \\frac{|\\beta|}{2|\\sin(\\beta\/2)|} d\\beta\\\\\n\t&\\leq \\tilde{\\mathcal{C}},\n\t\\end{align*}\n\twhere \n\t\\begin{equation}\\label{catalanconstant}\n\t\t\\tilde{\\mathcal{C}} \\overset{\\mbox{\\tiny{def}}}{=} \\frac{4}{\\pi}\\mathcal{V}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\t\\end{equation}\nIn this calculation we used that $|1-e^{-i\\beta}| = 2|\\sin(\\beta\/2)|$.  Here $\\mathcal{V} \\approx 0.916$ is Catalan's constant:\n\t\\begin{equation}\\notag\n\t    \\mathcal{V} = \\frac{1}{4}\\int_{-\\pi\/2}^{\\pi\/2} \\frac{\\beta}{\\sin\\beta} d\\beta\n\t    =\\frac{1}{16}\\int_{-\\pi}^{\\pi} \\frac{\\beta}{\\sin(\\beta\/2)} d\\beta.\n\t\\end{equation}\n\tHence, for $k_{1}= -1$, using \\eqref{CR}, we have the bound\n\t\\begin{equation}\\label{k1negativeone}\n\t|I(k,-1)| \\leq 1+ \\tilde{\\mathcal{C}} = {C_{\\mR}}.\n\t\\end{equation}\nThis will be our main estimate for the case for $k_{1}= -1$.\t\n\t\n\n\t\nNow generally in \\eqref{I.function.def} for $k_1>-1$ we have\n\t\\begin{equation}\\label{big.one.def}\n    \t\\frac{1-e^{-i\\beta (1+k_{1})}}{1-e^{-i\\beta}} = \\sum_{r=0}^{k_{1}} e^{-ir\\beta}\n\\end{equation}\n\tand\n    if $k_1 \\leq -2$ then we have\n    \\begin{equation}\\label{small.one.def}\n\\frac{1-e^{-i\\beta (1+k_{1})}}{1-e^{-i\\beta}} = -\\sum_{r=1}^{-1-k_{1}} e^{i\\beta r}.\n\\end{equation}\nThus for $k_1>-1$, using \\eqref{I.function.def},  \\eqref{j1} and \\eqref{big.one.def} we have\n\\begin{multline*}\n\tI(k,k_1) \n\t= \\frac{1}{1+k_1}\\sum_{r=0}^{k_1} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1}+r)\\beta}}{1\\!-\\!e^{-i\\beta}}  d\\beta\n\t\\\\\n\t=\\frac{1}{1+k_1}\\sum_{r=0}^{k_1} \\left( 1_{k-k_{1}+r \\le 0} -  1_{k-k_{1}+r \\ge 1} \\right),\n\t\\end{multline*}\n\twhile for $k_1 \\leq -2$ we similarly, using \\eqref{small.one.def}, have\n\t\\begin{multline*}\n\tI(k,k_1) = - \\frac{1}{1+k_1}\\sum_{r=1}^{-1-k_1} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1}-r)\\beta}}{1\\!-\\!e^{-i\\beta}}  d\\beta\n\t\\\\\n\t=\\frac{-1}{1+k_1}\\sum_{r=1}^{-1-k_1} \\left( 1_{k-k_{1}-r \\le 0} -  1_{k-k_{1}-r \\ge 1} \\right),\n\t\\end{multline*}\n\tIn both cases, $k_1>-1$ and $k_1 \\leq -2$, we conclude that \n\t    \\begin{equation}\\label{I.est.rest}\n    |I(k,k_1)| \\leq 1.\n    \\end{equation}\nHence we conclude that\n\t\\begin{equation*}\n\t|\\widehat{\\mathcal{R}(f)}(k)| \\leq  \t{C_{\\mR}}|(\\hat{f}\\ast \\hat{\\theta})(k)|.   \n\t\\end{equation*}\n\tApplying ${\\mathcal{F}^{0,1}_\\nu}$ and ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ norms to both sides, using Lemma \\ref{triangleprop}, gives the result.\n\\end{proof}\n\n\n\n\t\t\nNext, we proceed with the estimates for $\\mathcal{S}$ from \\eqref{S}.\n\\begin{prop}\\label{Smultiplierprop}\n\tThe operator $\\mathcal{S}$ in \\eqref{S} satisfies the estimates\n\t\\begin{equation}\\label{Sestimates}\n    \\begin{aligned}\n    \t\\|\\mathcal{S}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &\\leq C_1 \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n\t\t\t\\|\\mathcal{S}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}& \\leq C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},     \\quad s>0,\n    \\end{aligned}\n\t\\end{equation}\n\twhere the positive constant $C_1$ is given by \\eqref{C1}.  Further the positive constants $C_{3}$ and $C_{4}$ for $s>0$ are given in \\eqref{C3}.\n\\end{prop}\n\n\\begin{proof}\nFrom \\eqref{S}, we split  the operator $\\mathcal{S}$ as: \n\t\\begin{equation}\\label{Ssum}\n\t\\mathcal{S}(f)(\\alpha) = \n\t{\\mathcal{A}}(f)(\\alpha) + {\\mathcal{B}}(f)(\\alpha),\n\t\\end{equation}\n\twhere\n\t\\begin{multline}\\label{breveS}\n\t{\\mathcal{A}}(f)(\\alpha) \\overset{\\mbox{\\tiny{def}}}{=} \\\\ \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{2}}{\\beta(1-e^{-i\\beta})^{2}}\n\t\\sum_{m\\geq 2}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds d\\beta,\n\t\\end{multline}\n\tand\n\t\\begin{equation}\\label{tildeS}\n\t\t{\\mathcal{B}}(f)(\\alpha)\\overset{\\mbox{\\tiny{def}}}{=} \\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\theta(\\alpha))^l}{l!} \\mathcal{S}_{n}(f)(\\alpha).\n\t\\end{equation}\nHere for $n\\ge0$ we define\n\t\\begin{equation}\\label{Snl}\n\t\t\\mathcal{S}_{n}(f)(\\alpha) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n\t\tM(\\alpha,\\beta)^{n} d\\beta,\n\t\\end{equation}\n\twith\n\t\\begin{equation}\\label{M}\n\t\tM(\\alpha,\\beta) \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds.\n\t\\end{equation}\nWe take the Fourier transform to obtain\n\t\\begin{equation}\\label{fourierS}\n\t\t\\widehat{{\\mathcal{B}}}(f)(k) = \\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\hat{\\theta}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k),\n\t\\end{equation}\n\twhere \n\t\\begin{equation}\\label{fourierSnl}\n\t\t\\widehat{\\mathcal{S}_{n}(f)}(k) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\hat{f}(k)e^{-ik\\beta}\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n\t\t\\ast^{n}\\widehat{M}(k,\\beta) d\\beta.\n\t\\end{equation}\nFor $k_{1} \\neq -1$, from \\eqref{M} we have\n\t\\begin{align*}\n\t\t\\widehat{M}(k_{1},\\beta) &= \\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}\\Big(\\ast^{m}(\\hat{\\theta}(k_{1}) e^{ik_{1}(s-1)\\beta})\\Big) ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!} \\int_0^1 e^{i(s-1)\\beta}\\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) e^{i(k_{j}-k_{j+1})(s-1)\\beta}\\right)\\\\\n\t\t&\\hspace{3in} \\cdot\\hat{\\theta}(k_{m}) e^{ik_{m}(s-1)\\beta}  ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta(1+k_{1})}\\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) \\right) \\hat{\\theta}(k_{m}) ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!} \\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) \\right) \\hat{\\theta}(k_{m}) \\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}\\\\\n\t\t&=  \\sum_{m\\geq 1}\\frac{i^m}{m!} \\left(\\ast^{m}\\widehat{\\theta}(k_{1})\\right) \\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}.\n\t\\end{align*}\nAnd when $k_{1} = -1$,  analogously we have\n\t$$\\widehat{M}(k_{1},\\beta) =  \\sum_{m\\geq 1}\\frac{i^m}{m!} \\left(\\ast^{m}\\widehat{\\theta}(k_{1})\\right).$$\nThus we define\n\t\\begin{equation}\\label{P}\n\t\tP(k) \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 1}\\frac{i^m}{m!} (\\ast^{m}\\widehat{\\theta}(k)).\n\t\\end{equation}\nNow we use the convention \\eqref{convention} and plug the above into \\eqref{fourierSnl} to obtain\n\t\\begin{align}\n\t\t\\widehat{\\mathcal{S}_{n}}(k_{1}) &= \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\hat{f}(k_{1})e^{-ik_{1}\\beta}\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\\ast\n\t\t\\left(\\ast^{n}P(k_{1})\\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}\\right) d\\beta \\nonumber\\\\\n\t\t&= \\frac{1}{i^{n}}\\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}I(k_{1},\\ldots,k_{n+1})\\hat{f}(k_{n+1})\\prod_{j=1}^{n}P(k_{j}-k_{j+1}) \\hspace{0.05cm}, \\label{Snfourier}\n\t\\end{align}\n\twhere $I = I(k_{1},\\ldots,k_{n+1})$ is defined by \n\t\\begin{equation}\\label{I}\n\t\tI \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{n+1}\\beta}}{1-e^{-i\\beta}}\\prod_{j=1}^{n}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{(1+k_{j}-k_{j+1})(1-e^{-i\\beta})}d\\beta.\n\t\\end{equation}\n\tWe suppose that $l$ elements of $\\{k_{j}-k_{j+1}\\}_{j=1}^n$ satisfy $k_{j} - k_{j+1} =  -1$ for $0\\le l \\le n$.  Then ordering the subscripts such that $k_{j} - k_{j+1} \\neq  -1$ for $j= 1,\\ldots, n-l$, we obtain with the convention \\eqref{convention} that the integral $I = I(k_{1},\\ldots, k_{n+1})$ given by \\eqref{I} becomes \n\t\\begin{equation*}\n\t\tI = \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{n+1}\\beta}}{1-e^{-i\\beta}}\\frac{(i\\beta)^{l}}{(1-e^{-i\\beta})^l}\n\t\t\\prod_{j=1}^{n-l}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{(1+k_{j}-k_{j+1})(1-e^{-i\\beta})}d\\beta.\n\t\\end{equation*}\n\tNow, if $k_{j}-k_{j+1} > -1$ then similar to \\eqref{big.one.def} we have\n\\begin{equation}\\notag \n    \t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}} = \\sum_{r_{j}=0}^{k_{j}-k_{j+1}} e^{-ir_{j}\\beta}\n\\end{equation}\n\tand\n    if $k_{j}-k_{j+1} \\leq -2$ then similar to \\eqref{small.one.def} we have\n    \\begin{equation}\\notag \n\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}} = \\sum_{r_{j}=1}^{-1-(k_{j}-k_{j+1})} -e^{i\\beta r_j}.\n\\end{equation}\n    Hence, if $k_{j}-k_{j+1} \\leq -2$ only for $j=m,\\ldots,n-l$ then\n    \\begin{multline*}\n    \\prod_{j=1}^{n-l}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}}\\\\ = \\sum_{r_{1}=0}^{k_{1}-k_{2}} e^{-ir_{1}\\beta} \\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}} e^{-ir_{m-1}\\beta} \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} -e^{ir_{m}\\beta}\\\\ \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}} -e^{ir_{n-l}\\beta}\\\\\n\t\t=\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}(-1)^{n-l-m+1} e^{-i\\tilde{A}\\beta}\n\t\t\\end{multline*}\n\t\twhere\n\t\t$$\\tilde{A} = r_{1} + \\ldots + r_{m-1} -r_{m} -\\ldots -r_{n-l}.$$\n\t\tHence, in this case\n\t\\begin{multline}\\label{IJ}\n\t    I =  \\prod_{j=1}^{n-l}\n\t\t\\frac{1}{1+k_{j}-k_{j+1}}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\\\ \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}(-1)^{n-l-m+1}\n\t\t\\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}(i\\beta)^{l}}{(1-e^{-i\\beta})^{l+1}} d\\beta\n\t\\end{multline}\n\twhere\n\t$$A = k_{n+1}+\\tilde{A}.$$\n\tFor the inner integral, which we define as $J$, we have\n\t\\begin{equation}\\label{j1j2}\n\t\tJ \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{i^{l}\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}(i\\beta)^{l}}{(1-e^{-i\\beta})^{l+1}} d\\beta = \\frac{1}{i^{l}}( J_{1} + J_{2}), \n\t\\end{equation}\n\twhere we calculate $J_1$ as in \\eqref{j1}\n\tand\n\t\\begin{equation}\\label{j2}\n\t\tJ_{2} \\overset{\\mbox{\\tiny{def}}}{=}  \\frac{1}{\\pi} \\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}}{1-e^{-i\\beta}}\\Big[\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1\\Big] d\\beta.\n\t\\end{equation}\n\tWe can bound $J_{2}$ as well. Using the bound \\eqref{usefuloperatorbound} in \\eqref{j2}, we have \n\t\\begin{align*}\n\t\t|J_{2}| &\\leq \\frac{1}{2\\pi}l\\Big(\\frac{\\pi}{2}\\Big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}} \\int_{-\\pi}^{\\pi} \\frac{|\\beta|}{2|\\sin(\\beta\/2)|} d\\beta\\\\\n\t\t&=\\tilde{\\mathcal{C}} l\\Big(\\frac{\\pi}{2}\\Big)^{l-1},\n\t\\end{align*}\n\twhere the calculation above is similar to the calculation above \\eqref{catalanconstant} and the constant $\\tilde{\\mathcal{C}}$ is given by \\eqref{catalanconstant}.  \n\n\t\n\t\nThus, from \\eqref{IJ} and \\eqref{j1j2}, we have\n\t\\begin{align*}\n\t\t|I|&\\leq\\prod_{j=1}^{n-l}\n\t\t\\frac{1}{|1+k_{j}-k_{j+1}|}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\\\ \n\t\t&\\hspace{2in}\\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}  \\left(1 + \\tilde{\\mathcal{C}} l\\big(\\frac{\\pi}{2}\\big)^{l-1}\\right)\\\\\n\t\t&\\leq\\prod_{j=1}^{n-l}\n\t\t\\frac{1}{|1+k_{j}-k_{j+1}|}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \n\t\t\\\\ \n\t\t&\\hspace{2in}\\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}} \\left( 1 + \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\right).\n\t\\end{align*}\nThen we denote\n\\begin{equation}\\label{an.def}\n\t    a_n\\overset{\\mbox{\\tiny{def}}}{=} 1 + \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}.\n\\end{equation}\nTherefore we have\n\t\\begin{equation}\\label{Ibound}\n\t\t|I(k_{1},\\ldots,k_{n+1})| \\leq a_n \\ \\ \\forall  (k_{1},\\ldots, k_{n+1}) \\in \\mathbb{Z}^{n+1}.\n\t\\end{equation}\t\n\tPlugging \\eqref{Ibound} into the estimate for \\eqref{Snfourier}, we obtain\n\t\\begin{align}\n\t\t|\\widehat{\\mathcal{S}_{n}}(k_{1})|&\\leq  \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}} |I(k_{1},\\ldots,k_{n+1})||\\hat{f}(k_{n+1})|\n\t\t\\prod_{j=1}^n |P(k_{j}-k_{j+1})| \\nonumber\\\\\n\t\t&\\leq  \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}  a_n |\\hat{f}(k_{n+1})|\\prod_{j=1}^n |P(k_{j}-k_{j+1})|\\nonumber\\\\\n\t\t& = a_n(|\\hat{f}|\\ast^{n} |P|)(k_{1}). \\label{Snfbound}\n\t\\end{align}\t\n\tWe will use this estimate below to obtain the appropriate upper bound for $\\mathcal{S}(f)$ given by \\eqref{fourierS}. \n\t\n\n\n\t\nRecalling \\eqref{nu}, in the rest of this proof for convenience of notation we define the $\\ell^1_\\nu$ norm of a sequence $a=\\{a_k\\}_{k\\in \\mathbb{Z}}$ as\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^{1}_\\nu} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{k\\in\\mathbb{Z}} e^{\\nu |k|} |a_k|^p.\n\\end{equation}\nThen for $s=0$ using \\eqref{tildeS} and \\eqref{fourierS} we have\n\t\\begin{align}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &= \\Big\\|\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\hat{\\theta}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k) \\Big\\|_{\\ell^{1}_{\\nu}} \\nonumber \\\\\n\t\t&\\leq \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\n+l \\geq 2}}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}\\|\\widehat{\\mathcal{S}_{n}(f)}(k)\\|_{\\ell^{1}_{\\nu}} \\label{SSn0}.\n\t\\end{align}\n\tLet us examine the bound on the $\\widehat{\\mathcal{S}_{n}(f)}$ term in \\eqref{SSn0} using \\eqref{Snfbound}. For the quantity $P$ given by \\eqref{P}, \twe have\n\t\\begin{equation}\\label{P0}\n\t\t\\|P\\|_{\\ell^{1}_{\\nu}} \\leq \\sum_{m\\geq 1}\\frac{1}{m!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m} = \\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1.\n\t\\end{equation}\n\tFrom \\eqref{Snfbound} and \\eqref{an.def}, we then have\n\t\\begin{equation}\\label{Snfzero}\n\t\t\\|\\mathcal{S}_{n}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq   a_n\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{equation}\n\tHence, from \\eqref{SSn0}, we have\n\t\\begin{equation}\\label{SSn00}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \n\t\t\\! \\! \\! \\! \\! \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\n+l \\geq 2}}\\frac{a_n}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{equation}\nWe will separately look at the above sum when $n=0$, $n=1$ and $n\\geq 2$. \n\n\n\t\t\t\nFirst, when $n=0$ then $a_0=1$ and we have\n\t\\begin{equation}\\label{SSn00n0}\n\t\t\\sum_{l\\geq 2}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\tWhen $n=1$ then $a_1=1+\\tilde{\\mathcal{C}}$ and we have\n\t\\begin{multline}\\label{SSn0n1}\n\t\t\\sum_{l\\geq 1}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}\\Big(1 + \\tilde{\\mathcal{C}} \\Big)(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} \\\\ =\\Big(1 + \\tilde{\\mathcal{C}}\\Big)\\frac{(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{multline}\n\tFinally, for $n\\geq 2$ using also \\eqref{an.def} we have\n\t\\begin{multline}\\label{SSn0n2}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{a_n}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \\\\= \\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}\\\\ \n\t\t+ \\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\n\t\t\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\n\t\t\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \n\t\t\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{multline}\nNow considering the first sum on the right hand side of \\eqref{SSn0n2}, we have\n\t\\begin{multline}\\label{SSn0n21}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}\\\\ = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\sum_{n\\geq 2} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n-2}\\\\\n\t\t=\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(2-\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}))} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}.\n\t\\end{multline}\nThen the second sum on the right hand side of \\eqref{SSn0n2} is\n\\begin{multline*}\n\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \n\\\\\n=\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\\\ \\cdot \\sum_{n\\geq 2}\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{n-1} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\\end{multline*}\nNow using that $$\\sum_{ n\\geq 2} nx^{n-1} = -1 + \\sum_{n\\geq 1} nx^{n-1} = -1 + \\frac{1}{(1-x)^{2}} = \\frac{x(2-x)}{(1-x)^{2}}, $$\n\twe obtain\n\\begin{multline}\\label{SSn00n22}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \\\\\n=\\frac{\\pi\\tilde{\\mathcal{C}}}{2}\n\t\t\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\\\\\cdot\\Big( \\frac{2-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\big(1-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{2}}\\Big).\n\\end{multline}\t\n\tCombining \\eqref{SSn00n0}, \\eqref{SSn0n1}, \\eqref{SSn0n2}, \\eqref{SSn0n21} and \\eqref{SSn00n22} into \\eqref{SSn00}, we obtain in the space $\\mathcal{F}^{0,1}_\\nu$ with $\\nu=0$ that\n\t\\begin{equation*}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \\tilde{C}_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\n\t\\end{equation*} \n\twhere $\\tilde{C}_1\\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_1(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ is an increasing function of $\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}$ given by\n\t\\begin{multline}\\label{tildeC1}\n\t\t\\tilde{C}_1=\n\t\t\\\\\n\t\t\\frac{\\pi\\tilde{\\mathcal{C}}}{2}\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\frac{2-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\big(1-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{2}} \\\\+ \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} + \\Big(1 + \\tilde{\\mathcal{C}}\\Big)\\frac{(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\\\+\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(2-\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}))}.\n\t\\end{multline}\n This completes the estimate \\eqref{Sestimates} for the operator \\eqref{tildeS} and for $s=0$.\t\n\t\n\n\n\n\t\t\t\t\nFor the operator \\eqref{tildeS} and $s>0$, from \\eqref{fourierS}, \\eqref{fourierSnl}, \\eqref{P}, \\eqref{Snfourier} with \\eqref{I} we have\n\t\\begin{multline}\\label{S1S2S3}\n\t\\|{\\mathcal{B}}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t= \\Big\\||k_{1}|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\ast^{l}\\hat{\\theta}(k_{1}))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k_{1}) \\Big\\|_{\\ell^{1}_{\\nu}} \n\t\\\\\n\t= \\Big\\||k_{1}|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(\\ast^{l}\\hat{\\theta}(k_{1}))}{l!} \\ast \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}I(k_{1},\\ldots,k_{n+1})\\hat{f}(k_{n+1})\\prod_{j=1}^{n}P(k_{j}-k_{j+1})  \\Big\\|_{\\ell^{1}_{\\nu}}\n\t\\\\\n\t\\leq S_{1} + S_{2} + S_{3},\n\t\\end{multline}\n\twhere we use $a_n$ from \\eqref{an.def} and \\eqref{bfcn.def} so that,\n\tusing \\eqref{s.inequality} and \\eqref{Ibound} with \\eqref{P}, the terms $S_i$ are given by\n\t\\begin{equation*}\n\t    \\begin{aligned}\n\tS_{1}  &=  \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n l\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l-1}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|P\\|_{\\ell^{1}_{\\nu}}^{n},\\\\\n\tS_{2} &= \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|P\\|_{\\ell^{1}_{\\nu}}^{n},\\\\\n\tS_{3} &= \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n n \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|P\\|_{_{\\ell^{1}_{\\nu}}}^{n-1}\\|\\mathcal{F}^{-1}(P)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} .\n\t    \\end{aligned}\n\t\\end{equation*}\n\tWe recall from \\eqref{P} and \\eqref{P0} that\n\t$\\|P\\|_{\\ell^{1}_{\\nu}}\\le \\exp{(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})}-1$  and notice with \\eqref{s.inequality} that for $s>0$ we have\n\t\\begin{equation}\\label{Ps.estimate}\n\t\\|\\mathcal{F}^{-1}(P)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq \\sum_{m\\geq 1}\\frac{{b}(m,s)}{(m-1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\t\t\\end{equation}\n\tThen, we have that\n\t\\begin{equation*}\n\tS_{2}\\le \\tilde{C}_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\t\\end{equation*}\n\twhere\n\t\\begin{equation}\\label{tildeC3}\n\t\\begin{aligned}\n\t \\tilde{C}_3&\\overset{\\mbox{\\tiny{def}}}{=} \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!} a_n\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{n+l-2}\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^n,\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\\begin{equation*}\nS_{1}  +S_{3}\\le  \\tilde{C}_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{equation*}\nwhere \n\\begin{equation}\\label{tildeC4}\n\\begin{aligned}\n\\tilde{C}_4&\\overset{\\mbox{\\tiny{def}}}{=} \\!\\!\\!\\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\ n+l\\geq 2}}\\!\\!\\!\\! \\frac{{b}(l\\!+\\!n\\!+\\!1,s)}{l!} a_n\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{n+l-2}\\Big(l\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\!-\\!1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^n+\\! \\tilde{C}_5\\Big),\n\\end{aligned}\n\\end{equation}\nwith \n\\begin{equation}\\label{C5}\n\\tilde{C}_5\\overset{\\mbox{\\tiny{def}}}{=} n\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\!-\\!1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^{n-1}\\sum_{m\\geq 1}\\frac{{b}(m,{s})}{(m-1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}.    \n\\end{equation}\nFinally, going back to \\eqref{S1S2S3}, we obtain the result for $s>0$ that\n\\begin{equation*}\n    \\|{\\mathcal{B}}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\tilde{C}_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\tilde{C}_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\\end{equation*}\nThis completes the desired estimate for ${\\mathcal{B}}$ in \\eqref{Sestimates} for $s>0$.\n\n\n\n\n\t\n\t\t\t\t\nNow it only remains to bound ${\\mathcal{A}}(f)(\\alpha)$ as defined by \\eqref{breveS} in \\eqref{Sestimates}.  Analogously to \\eqref{Snfourier} and \\eqref{I}, one can obtain that\n\\begin{equation*}\n\\widehat{{\\mathcal{A}}}(k_{1}) = \\sum_{k_{2}\\in\\mathbb{Z}} I(k_{1},k_{1}-k_{2})\\hat{f}(k_{2})P_2(k_{1}-k_{2})\n\\end{equation*}\nwhere\n$$ P_2(k) = \\sum_{m\\geq 2} \\frac{i^{m}}{m!} (\\ast^{m}\\hat{\\theta}(k))$$\nand $I(k_{1},k_{1}-k_{2})$ is given by \\eqref{I.function.def} with $k=k_1$ and $k_1$ in \\eqref{I.function.def} replaced by $k_{1}-k_{2}$ using also \\eqref{convention}, so that in particular\n$$ \nI(k_{1},k_{1}-k_{2}) =  \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{2}\\beta}}{1-e^{-i\\beta}}\n\t\t\\frac{1-e^{-i\\beta (1+k_{1}-k_{2})}}{i(1+k_{1}-k_{2})(1-e^{-i\\beta})}d\\beta.\n$$\nThen analogously to \\eqref{k1negativeone} and \\eqref{I.est.rest} we have $$| I(k_{1},k_{1}-k_{2})| \\le C_\\mathcal{R},$$ with $C_\\mathcal{R}$ from \\eqref{CR}.  Note that $\\widehat{{\\mathcal{A}}}$ is the operator $\\widehat{\\mathcal{S}_{1}}$ in \\eqref{Snfourier} with $n=1$ if you replace $P$ from \\eqref{P} with $P_2$ above.  Then we have the same estimate as \\eqref{Snfbound} with $P_2$ replacing $P$ and $n=1$ recalling also \\eqref{an.def}.   We estimate $P_2$ similarly to \\eqref{P0} (except that $m\\ge 2$).  We conclude that\n\\begin{equation}\\label{breveS0bound}\n\\|{\\mathcal{A}}\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq C_\\mathcal{R} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} \\|P_2\\|_{\\mathcal{F}^{0,1}_\\nu} \n\\leq C_\\mathcal{R} \\breve{C}_{1}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\end{equation}\nwhere\n\\begin{equation}\\label{breveS1constant1}\n\\breve{C}_{1} = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}.\n\\end{equation}\nThus, recalling \\eqref{CR}, \\eqref{catalanconstant}, \\eqref{tildeC1} and \\eqref{breveS1constant1}, then we have\n\\begin{equation}\\label{C1}\nC_{1} = \\tilde{C}_{1} + C_\\mathcal{R} \\breve{C}_{1}.\n\\end{equation}\nWe obtain \\eqref{Sestimates} for $s=0$ by combining \\eqref{breveS0bound} with the bound above \\eqref{tildeC1}. \n\n\n\n\n\n\nWe turn to the estimate for ${\\mathcal{A}}$ in \\eqref{breveS} for $s>0$.    We will use \\eqref{s.inequality} since $s>0$ and \\eqref{bfcn.def}.  We will in this case estimate $P_2$ similarly to \\eqref{Ps.estimate} (except with $m\\ge 2$).\nWe then have the estimate\n\\begin{equation*}\n \\|{\\mathcal{A}}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq C_\\mathcal{R} \\sum_{m\\geq 2} \\frac{{b}(m+1,s)}{m!} \\big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} + m \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\big).\n\\end{equation*}\nNow define\n\\begin{equation}\\label{breveCs}\n\\breve{C}_{3} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 0} \\frac{{b}(m+3,s)}{(m+2)!}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m}, \\quad\n\\breve{C}_{4} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 0} \\frac{{b}(m+3,s)}{(m+1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m},\n\\end{equation}\nthen we have\n\\begin{equation}\\label{breveSsbound}\n\\|{\\mathcal{A}}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq C_\\mathcal{R} \\breve{C}_{3}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} + C_\\mathcal{R} \\breve{C}_{4} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{equation}\nHence further define\n\\begin{equation}\\label{C3}\nC_{3} \\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_{3} + C_\\mathcal{R} \\breve{C}_{3}, \\quad C_{4} \\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_{4} + C_\\mathcal{R} \\breve{C}_{4}.\n\\end{equation}\nThen using \\eqref{tildeC3}, \\eqref{tildeC4} (and the estimate below them) and \\eqref{breveCs} (and the estimate above it) we obtain \\eqref{Sestimates} for $s>0$.\n\\end{proof}\n\nIn the next section we prove the \\textit{a priori} estimates on the vorticity strength $\\omega$.\n\n\\section{\\textit{A priori} estimates on the vorticity strength}\\label{secw}\n\nThis section includes the \\textit{a priori} estimates for the vorticity strength $\\omega$ from \\eqref{omega} that will be used in particular in Subsection \\ref{subsecGlobal}.  The main result of the section is Proposition \\ref{vorticityestimatess}.\n\n\n\\begin{prop}\\label{vorticityestimatess}\n\tThe linear part of the vorticity, $\\omega_1$  in \\eqref{omegasplit}, satisfies the following estimates:\n\t\\begin{equation}\\label{omega1f0}\n    \\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n    \\end{equation}\n    for $s>0$, recalling \\eqref{bfcn.def}, we have\n    \\begin{equation}\\label{omega1fss}\n    \\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq A_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}2b(2,s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n    \\end{equation}\nThe nonlinear part  $\\omega_{\\geq2}$ from \\eqref{omegasplit} satisfies\n\\begin{equation}\\label{omega2f0}\n\t\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_{\\nu}}\n\t\\leq |A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_9\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n\t\n\t+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{10}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n\\end{equation}\nwhile $s>0$ we have\n\\begin{multline}\\label{omega2fss}\n\t\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t\\!\\leq\\!\n\t|A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\\!\n\t\\\\\n+\n\t\\!|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{multline}\nwhere $C_9$ and $C_{10}$ are defined in \\eqref{C9C10} and $C_{13}$ and $C_{14}$ are in \\eqref{C13C14}.\n\\end{prop}\n\n\\begin{proof}\nFirst, recalling \\eqref{bfcn.def}, we note that \n\\begin{equation}\\label{cosine.calc.FT}\n\\begin{aligned}\n    \\|\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n    \\|\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\\qquad s>0,\n\\end{aligned}    \n\\end{equation}\nas similar to the calculations in the proof of Lemma \\ref{triangleprop}.  We point out that the same calculations also hold with $\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}$ replaced by $\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}$.  \nFor example, to see the above in the case $0 < s \\leq 1$, \nwe have \n\\begin{multline}\\label{proof.angle.nu.ineq}\n\\|\\cos{(\\alpha\\!+\\!\\widehat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \n= \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|}|k|^{s}|((\\cos{(\\alpha\\!+\\!\\widehat{\\vartheta}(0))})^{\\wedge}\\ast \\widehat{\\theta}) (k)|\n\\\\\n= \\sum_{k, k_{1}\\in\\mathbb{Z}} e^{\\nu(t)|k|} |k|^{s} |\\frac{1}{2}(\\delta(1-k_{1})\ne^{i\\widehat{\\vartheta}(0) }\n+ \\delta(1+k_{1})\ne^{-i\\widehat{\\vartheta}(0)}\n) \\widehat{\\theta} (k-k_{1})|\\\\\n\\leq  \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|}|k|^{s}  \\frac{1}{2} (|\\widehat{\\theta} (k-1)|+|\\widehat{\\theta} (k+1)|)\\\\\n\\leq  \\frac{1}{2}\\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k-1|}e^{\\nu(t)}(|k-1|^{s}+1)   |\\widehat{\\theta} (k-1)|\\\\ \\hspace{0.5in}+e^{\\nu(t)|k+1|}e^{\\nu(t)}(|k+1|^{s}+1) |\\widehat{\\theta} (k+1)|\\\\\n= e^{\\nu(t)}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}) \\leq 2e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{multline}\nThis explains the difference between \\eqref{omega1f0} and \\eqref{omega1fss}.\nWe will explain the proof of \\eqref{omega1fss} for $s>0$.  The proof of the other case, \\eqref{omega1f0} when $s=0$, is analogous.  It follows from \\eqref{omegasplit} and \\eqref{cosine.calc.FT} that\n\\begin{equation*}\n\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\leq \n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\!+\\!\nA_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\n\\!+\\!\n|A_\\rho| \\frac{L(t)}{\\pi}2b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{equation*}\nand from \\eqref{mdsplit} with \\eqref{cosine.calc.FT} and \\eqref{hilbertTcalc} we have\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq\\frac{\\pi}{L(t)}\\Big(|A_\\rho|\\frac{L(t)}{\\pi}2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\|\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big). \n\\end{aligned}\n\\end{equation*}\nRecalling  \\eqref{fourierimag} together with Lemma  \\eqref{cosine.calc.FT} gives the following estimate \n\\begin{equation*}\n    \\|\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq |A_\\rho|\\frac{L(t)}{\\pi}2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{equation*}\nTherefore,  \n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq 4|A_\\rho|e^{\\nu(t)}b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}, \n\\end{aligned}\n\\end{equation*}\nand thus the estimate \\eqref{omega1fss} for $\\omega_1$ is complete.  The estimate for \\eqref{omega1f0} is proven in the same way.\n\n\n\nNow we proceed to bound $\\omega_{\\geq 2}$ from \\eqref{omegasplit} in $\\mathcal{F}^{0,1}_\\nu$.  In this case we have\n\\begin{equation}\\label{omega2aux}\n\\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n&\\leq \n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\n+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq2}\\frac{\\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^j}{j!}\n\\\\\n&=\n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_6\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C6}\nC_6=\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}-1-\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2}.\n\\end{equation}\nFrom \\eqref{mdsplit}, one has that\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big).\n\\end{aligned}\n\\end{equation*}\nUsing the estimates \\eqref{Restimates} and \\eqref{Sestimates} and splitting the vorticity terms as $\\omega_{\\geq1} = \\omega_{1}+\\omega_{\\geq2}$, we obtain the estimate\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}&\\|_{\\mathcal{F}^{0,1}_\\nu}\\leq \\frac{\\pi}{L(t)}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+\n{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\Big),\n\\end{aligned}\n\\end{equation*}\nso, substituting back in \\eqref{omega2aux} and solving for $\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq C_8\\Big( |A_\\mu|C_7\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+|A_\\mu||A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\\\\n&\\quad\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_6\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big),\n    \\end{aligned}\n\\end{equation*}\nwhere we defined\n\\begin{equation}\\label{C7C8}\n    C_7=1+{C_{\\mR}}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\qquad C_8=\\frac{1}{1-|A_\\mu|C_7\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}},\n\\end{equation}\nwith $C_1$ given in \\eqref{C1}.\nSubstituting in \\eqref{omega1f0}  we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq 2|A_\\mu|A_\\sigma \\frac{2\\pi}{L(t)}C_7 C_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n    &+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_8\\Big(|A_\\mu|(1+2|A_\\mu|)C_7+|A_\\mu|C_1+C_6\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n    \\end{aligned}\n\\end{equation*}\nwhich gives the  estimate  \\eqref{omega2f0} by defining\n\\begin{equation}\\label{C9C10}\n\\begin{aligned}\nC_9&=C_7C_8,\\\\\nC_{10}&=|A_\\mu|(1+2|A_\\mu|)C_7C_8+|A_\\mu|C_1C_8+C_6C_8,\n\\end{aligned}\n\\end{equation}\nwhere $C_1$, $C_6$, $C_7$, and $C_8$ were previously defined in \\eqref{C1}, \\eqref{C6}, and \\eqref{C7C8}.\n\n\n\n\nWe consider now the case $s>0$ in \\eqref{omega2fss}. From \\eqref{omegasplit}, also using \\eqref{proof.angle.nu.ineq}, we have\n\\begin{multline}\\label{omega2aux2}\n\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\leq |A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq2}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j}}{j!}\n\\\\\n\\quad\n+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq1}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j}}{j!}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n\\leq\n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{multline}\nwhere \n\\begin{equation}\\label{C11}\nC_{11}=\\sum_{j\\geq 2}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j-2}}{j!}+\\sum_{j\\geq1}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j-1}}{j!}.\n\\end{equation}\nRecalling \\eqref{mdsplit}, and \\eqref{s.inequality}, one has that\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\mathcal{F}^{0,1}_\\nu}+ b(2,s)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n&\\qquad\\qquad+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big).\n\\end{aligned}\n\\end{equation*}\nUsing the estimates \\eqref{Restimates} and \\eqref{Sestimates} with $s>0$ and splitting the vorticity terms again, we obtain\n\\begin{multline*}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\frac{\\pi}{L(t)}\\bigg((1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\\\\n+\n(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n+\n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n+\\!C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!\\!\\bigg),\n\\end{multline*}\nwhich becomes \n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3\\!+\\!C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+\\!C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big),\n\\end{aligned}\n\\end{equation*}\nwhere \n\\begin{equation}\\label{C12}\n\\begin{aligned}\nC_2&=1+b(2,s){C_{\\mR}}+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\nC_{12}&=1+b(2,s){C_{\\mR}}+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nwith $C_\\mathcal{R}$ and $C_4$ given in \\eqref{CR} and \\eqref{C3}, respectively.\nSubstituting back in \\eqref{omega2aux2}, and solving for $\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$, we have\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq |A_\\mu|\\tilde{C}_8 \\Big( \n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&+\nC_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\\\\n&+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\tilde{C}_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n    \\end{aligned}\n\\end{equation*}\nwhere we defined\n\\begin{equation}\\label{tildeC8}\n\\tilde{C}_8=\\frac{1}{1-|A_\\mu|C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}.\n\\end{equation}\nWe introduce the estimates \\eqref{omega2f0}, \\eqref{omega1f0}, and \\eqref{omega1fss} to obtain that\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}&\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq |A_\\mu|\\tilde{C}_8\\bigg( \n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+\nA_\\sigma \\frac{4\\pi}{L(t)}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+A_\\sigma\\frac{4\\pi}{L(t)} C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}}+(1\\!+\\!2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}2b(2,s) e^{\\nu(t)}C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+|A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+|A_\\rho|\\frac{L(t)}{4\\pi}e^{\\nu(t)}C_{10}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\!\n\\bigg)\\!\\!+\\!|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\tilde{C}_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nNext, we use the interpolation inequality \\eqref{interpolation} to find that\n\\begin{equation}\\label{interp}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\leq \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu},\n\\end{equation}\nand therefore\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq |A_\\mu|A_\\sigma \\frac{4\\pi}{L(t)}\\tilde{C}_8\\Big(C_2+C_{12}+|A_\\mu|C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\\\\\n    &+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\tilde{C}_8\\Big(|A_\\mu|(C_3+C_4)+|A_\\mu|(1+2|A_\\mu|)(2C_2+C_{12})\\\\\n    &+|A_\\mu|C_{10}C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_{11}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{aligned}\n\\end{equation*}\nwhich gives the inequality \\eqref{omega2fss} by defining\n\\begin{equation}\\label{C13C14}\n\\begin{aligned}\nC_{13}&=\\tilde{C}_8\\Big(C_2+C_{12}+|A_\\mu|C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big),\\\\\nC_{14}&=|A_\\mu|\\tilde{C}_8\\Big(C_3\\!+\\!C_4\\!+\\!(1\\!+\\!2|A_\\mu|)(2C_2\\!+\\!C_{12})\\!+\\!C_{10}C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\!\n\\\\\n& \\hspace{1cm}+\\!C_{11}\\tilde{C}_8,\n\\end{aligned}\n\\end{equation}\nwhere $C_3$ and $C_4$ in \\eqref{C3}, $C_2$ in \\eqref{C12}, $\\tilde{C}_8$ in \\eqref{tildeC8}, $C_9$ and $C_{10}$ in\\eqref{C9C10}, $C_{11}$ in \\eqref{C11}, and $C_{12}$ in \\eqref{C12} were previously defined.\n\\end{proof}\n\n\nIn the next section we put together all the previous results in the paper to prove the global existence and instant analyticity as in Theorem \\ref{thm:global}\n\n\n\n\\section{Global existence and instant analyticity}\\label{secanalytic}\n\nThis section is dedicated to the proof of Theorem \\ref{thm:global}. In Subsection \\ref{sec:Lestimate} we prove the claimed bound for the length $L(t)$ from \\eqref{Lfinalbound}.  Then Subsection \\ref{subsecGlobal} proves the main \\textit{a priori}  estimates for a solution $\\theta$.  We prove the global energy inequality from \\eqref{estimatef12}.  In particular we diagonalize the linearized operator in Proposition \\ref{diagonalization}.  We prove operator bounds for the corresponding linear transformations in Lemma \\ref{lemmaS}.   In Subsection \\ref{sec:NonLinearEst} we prove the corresponding non-linear estimates that were used in the a priori estimates from the previous subsection.  In particular we prove Theorem \\ref{thm:Nbound}.  Lastly, in Subsection \\ref{subregular} we describe the scheme of the proof of our main theorem via a regularization argument.\n\n\n\n\\subsection{Estimate for $L(t)$}\\label{sec:Lestimate}\n\nIn this section, we prove \nthe bound for $L(t)$ from \\eqref{Lfinalbound}.  Equation \\eqref{Lequation}, together with the bound\n\\begin{equation}\\label{lengthauxbound}\n\\begin{aligned}\n    \\text{Im}\\hspace{0.05cm}\\Big(\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta(\\alpha)-\\theta(\\eta))^n d\\eta d\\alpha\\Big)&\\leq \\pi^2\\sum_{n\\geq1}\\frac{2^n\\|\\theta\\|_{\\mathcal{F}^{0,1}}^n}{n!}\\\\\n    &\\leq \\pi^2\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big),\n\\end{aligned}\n\\end{equation}\nimplies that\n\\begin{equation}\\label{Lboundaux}\n    \\begin{aligned}\n        \\frac{R^2}{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}\\leq\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\leq \\frac{R^2}{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)},\n    \\end{aligned}\n\\end{equation}\nand therefore\n\\begin{equation}\\label{Lbound}\n    \\begin{aligned}\n        2\\pi R C_{37}\\leq L(t)\\leq 2\\pi RC_{38},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C37C38}\n    C_{37}=\\frac{1}{\\sqrt{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}},\\quad C_{38}=\\frac{1}{\\sqrt{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}}.\n\\end{equation}\nWe have thus shown that the length of the curve is controlled  for all times. In particular, we also have the following estimates\n\\begin{equation}\\label{Lestimates}\n\\begin{aligned}\n    \\Big|R^2\\Big(\\frac{2\\pi}{L(t)}\\Big)^2-1\\Big|&\\leq  \\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big),\n    \\\\\n    \\Big|R\\Big(\\frac{2\\pi}{L(t)}\\Big)-1\\Big|&\\leq  \\sqrt{1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}-1=C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}},\n    \\\\\n    \\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|&\\leq  \\Big(1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)^{3\/2}-1=C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}},\n    \\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\\label{C39C40}\n    C_{39}=\\frac{\\sqrt{1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}}},\\quad C_{40}=\\frac{\\Big(1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)^{3\/2}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}}}.\n\\end{equation}\nThese are the main estimates that we will use for $L(t)$.  We remark that all the estimates in this section hold with the same proof in the norms $\\mathcal{F}^{0,1}_\\nu$ and $\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu$.   In the next section we prove the main a priori estimates for $\\theta$.\n\n\n\n\\subsection{\\textit{A priori} Estimates for $\\theta$}\\label{subsecGlobal}\n\nIn this section we obtain the \\textit{a priori} estimates that guarantee the global existence of the solutions, the instant in time analyticity, and the exponentially fast convergence to a circle. In particular, the main goal is to show the energy inequality \\eqref{estimatef12}.\n\n\nThe results of this section are ordered as follows: First, we write the system \\eqref{system} using the linearization \\eqref{linearfourier} of Subsection \\ref{sec:linearization}; then, we diagonalize the system according to Proposition \\ref{diagonalization} to show its dissipative character; the bounds for this change of frame are proven in Lemma \\ref{lemmaS}; the estimates for the nonlinear terms are proven in Subsection \\ref{sec:NonLinearEst}, together with the control of the length in Subsection \\ref{sec:Lestimate}.\n\n\n\\begin{proof}[Proof of the global energy inequality from  \\eqref{estimatef12}]\nEquation \\eqref{system} and  Proposition \\ref{linearfourier} show that the equation for $\\widehat{\\theta}(k)$, $1\\leq k\\neq2$, is given by\n\\begin{equation*}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(k)&=-A_\\sigma \\Big(\\frac{2\\pi}{L(t)}\\Big)^3k(k^2\\!-\\!1)\\widehat{\\theta}(k)\\!-\\!\\big(1\\!+\\!A_\\mu\\big)A_\\rho\\frac{2\\pi}{L(t)}\\frac{(k^2\\!-\\!1)(k\\!+\\!1)}{k(k\\!+\\!2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k\\!+\\!1)\\\\\n   &\\quad+\\frac{2\\pi}{L(t)}\\widehat{N}(k),\n    \\end{aligned}\n\\end{equation*}\nand for $k=2$, we have\n\\begin{equation*}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(2)&=-A_\\sigma \\Big(\\frac{2\\pi}{L(t)}\\Big)^36\\widehat{\\theta}(2)\\!-(1+A_\\mu)A_\\rho\\frac{2\\pi}{L(t)}\\frac{9}{8}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3)\\\\\n   &\\quad +(1-A_\\mu)A_\\rho\\frac{2\\pi}{L(t)}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1) +\\frac{2\\pi}{L(t)}\\widehat{N}(k).\n    \\end{aligned}\n\\end{equation*}\nWe notice in the equation above that the terms that are linear in $\\widehat{\\theta}(k)$ have time-dependent coefficients. However, this dependency happens only through $L(t)$. We will show that $L(t)$ is bounded from below and above (see Subsection \\ref{sec:Lestimate}). In fact, it is not hard to see from \\eqref{Lequation} that, to leading order, $L(t)$ equals $2\\pi R$. Thus we rewrite the equation above as follows\n\\begin{equation}\\label{thetaequation}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(k)&=-\\frac{A_\\sigma}{R^3} k(k^2-1)\\widehat{\\theta}(k)-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k+1)\\\\\n   &\\quad+\\frac{2\\pi}{L(t)}\\widehat{N}(k)-\\frac{A_\\sigma}{R^3} k(k^2-1)\\widehat{\\theta}(k)\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big)\\\\\n   &\\quad-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k+1)\\Big(R\\frac{2\\pi}{L(t)}-1\\Big),\n    \\end{aligned}\n\\end{equation}\nfor $1\\leq k\\neq2$, and we decompose the equation analogously for $k=2$.\n\nNext, we write the corresponding linear system as follows\n\\begin{equation}\\label{linearZ}\n    \\begin{aligned}\n    z_t(k)=\\sum_{j\\geq1}M_{k,j}z(j),\\quad k\\geq1,\n    \\end{aligned}\n\\end{equation}\nwhere we denote\n\\begin{equation*}\n    M_{k,j}=\\left\\{\n    \\begin{aligned}\n    &-a(k),\\quad j=k,\\\\\n    &b(k),\\hspace{0.8cm} j=k+1,\\\\\n    &c(1),\\hspace{0.8cm}j=1,k=2,\\\\\n    &0,\\hspace{1.2cm} \\text{otherwise},\n    \\end{aligned}\\right.\n\\end{equation*}\nwith \n\\begin{equation}\\label{coeffs}\n    a(k)=\\frac{A_\\sigma}{R^3} k(k^2-1),\\quad b(k)=-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)},\n\\end{equation}\n\\begin{equation*}\n    c(1)=(1-A_\\mu)\\frac{A_\\rho}{R}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}.\n\\end{equation*}\nNotice that this is an upper triangular system except for the entry $k=2$,$j=1$.  This entry, $k=2$ with $j=1$, will require special attention.  The eigenvalues of this system are $-a(k)$.  This is given in the following proposition.\n\n\n\n\n\\begin{prop}[Diagonalization]\\label{diagonalization}\nConsider $z=(z(k))_{k\\geq1},y=(y(k))_{k\\geq1}\\in \\ell^1$ and the linear operator \\begin{equation*}\n    \\begin{aligned}\n    S^{-1}:\\ell^1&\\rightarrow \\ell^1\\\\\n    z&\\mapsto y=S^{-1}z,\n    \\end{aligned}\n\\end{equation*}\ndefined by\n    \\begin{equation*}\n        y(k)=\\sum_{j\\geq1}S^{-1}_{k,j}z(j),\n    \\end{equation*}\n    with\n    \\begin{equation}\\label{Sm1}\n        S^{-1}_{k,j}=\\left\\{\n        \\begin{aligned}\n        (&-1)^{j-k}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k)-a(k+l)},\\hspace{0.4cm}j\\geq k\\geq2,\\\\\n        &1,\\hspace{1.2cm}j=k=1,\\\\\n        -&\\frac{c(1)}{a(2)},\\hspace{1.2cm}k=2,j=1,\\\\\n        &0,\\hspace{1.72cm} \\text{otherwise}.\n        \\end{aligned}\\right.\n    \\end{equation}\nThen, the inverse operator $S$ is given by\n    \\begin{equation*}\n        S_{k,j}=\\left\\{\n        \\begin{aligned}\n        &\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k-1+l)-a(j)},\\hspace{0.4cm}j\\geq k\\geq2,\\\\\n        &1,\\hspace{1.2cm}j=k=1,\\\\\n        &\\frac{c(1)}{a(2)},\\hspace{1.72cm}k=2,j=1,\\\\\n        &0,\\hspace{1.72cm} \\text{otherwise}.\n        \\end{aligned}\\right.\n    \\end{equation*}\nMoreover, the linear operator $S^{-1}$ diagonalizes the system \\eqref{linearZ} as follows\n\\begin{equation*}\n    y_t(k)=-a(k)y(k),\\quad k\\geq1.\n\\end{equation*}\n\\end{prop}\n\nWe remark that since $\\prod_{l=1}^{0}=1$ by definition then $S_{k,k} = S^{-1}_{k,k} =1$ when $j=k$ for all $k\\ge 1$.   We also have the following lemma which gives uniform bounds for the operators $S$ and $S^{-1}$.\n\n\\begin{lemma}\\label{lemmaS}\n    The operator norms in $\\ell^1$ of the linear operators $S$ and $S^{-1}$ satisfy the following bounds\n    \\begin{equation}\\label{CSbounds}\n        \\begin{aligned}\n        \\|S^{-1}\\|_{\\ell^1\\rightarrow \\ell^1}\\leq C_{S},\\qquad  \\|S\\|_{\\ell^1\\rightarrow \\ell^1}\\leq C_{S},\n        \\end{aligned}\n    \\end{equation}\nwith $C_S = C_S(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})$ where\n\\begin{equation*}\n    C_S\n    \\overset{\\mbox{\\tiny{def}}}{=} \\max\\Big\\{1+\\frac{1}{4}\\big(1\\!-\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big),6\\frac{I_3\\Big(2\\sqrt{\\big(1\\!+\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big(1\\!+\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\Big\\}.\n\\end{equation*}\n\\end{lemma}\n\nIn the constant $C_S$ above we used the modified Bessel function of the first kind of order three.  In general, for an integer $n\\ge 0$, we define\n\\begin{equation}\\label{bessel.In.def}\nI_n(z) \\overset{\\mbox{\\tiny{def}}}{=} \\left(\\frac{z}{2} \\right)^n\n \\sum_{j=0}^{\\infty}\n \\frac{(z\/2)^{2j}}{j!(j+n)!}.\n\\end{equation}\nWe will prove Proposition \\ref{diagonalization} and Lemma \\ref{lemmaS} after we finish the proof of the main global energy inequality. \n\n\n\\begin{remark}\\label{analytic.norm.remark.S}\nThe results in Proposition \\ref{diagonalization} and Lemma \\ref{lemmaS} also hold in the space $\\ell^1$ with weight $e^{\\nu(t)|k|}|k|^s$ for any $s\\ge 0$ without any change to the proof; i.e.\n\\begin{equation*}\n    \\|z\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq \n     C_S\\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n     \\quad \n         \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq \n     C_S\\|z\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{equation*}\nwhere $y(k)$ and $z(k)$ are defined in Proposition \\ref{diagonalization}.\n\\end{remark}\n\nAccording to Proposition \\ref{diagonalization}, we apply the linear transformation $S^{-1}$ to \\eqref{thetaequation} to rewrite the system with the linear part in diagonal form\n\\begin{equation}\\label{thetaeqdiag}\n    \\begin{aligned}\n   (S^{-1}\\widehat{\\theta})_t(k)&=-\\frac{A_\\sigma}{R^3} k(k^2-1)(S^{-1}\\widehat{\\theta})(k)+\\frac{2\\pi}{L(t)}(S^{-1}\\widehat{N})(k)\\\\\n   &\\quad-\\frac{A_\\sigma}{R^3} (S^{-1}u)(k)\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big)\\\\\n   &\\quad-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}e^{-i\\widehat{\\vartheta}(0)}(S^{-1}v)(k)\\Big(R\\frac{2\\pi}{L(t)}-1\\Big)\\\\\n   &\\quad+(1-A_\\mu)\\frac{A_\\rho}{R}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}\\Big(R\\frac{2\\pi}{L(t)}-1\\Big)(S^{-1}w)(k),\n    \\end{aligned}\n\\end{equation}\nwhere we introduced the notation\n\\begin{equation*}\n    \\begin{aligned}\n    u(k)=k(k^2-1)\\widehat{\\theta}(k),\\quad v(k)=\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k+1),\\quad w(k)=1_{k=2}\\widehat{\\theta}(1).\n    \\end{aligned}\n\\end{equation*}\nTaking into account that $\\theta(\\alpha)$ is a real-valued function, we can write the norm \\eqref{fsonenorm} in terms of the positive frequencies alone\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}=\\sum_{k\\in\\mathbb{Z}}e^{\\nu(t)|k|}|k|^{1\/2}|\\widehat{\\theta}(k)|=2\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}|\\widehat{\\theta}(k)|.\n\\end{aligned}\n\\end{equation*}\nDefine\n\\begin{equation*}\n    \\begin{aligned}\n    \\widehat{y}(k)=(S^{-1}\\widehat{\\theta})(k),\\quad \\widehat{y}(-k)=\\overline{\\widehat{y}(k)},\\quad k\\geq1, \\quad \\widehat{y}(0)=0,\\quad y=\\mathcal{F}^{-1}\\widehat{y},\n    \\end{aligned}\n\\end{equation*}\nand consider the evolution of the quantity\n\\begin{equation*}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}=2\\sum_{k\\geq1} e^{\\nu(t)k}k^{1\/2}|\\widehat{y}(k)|.\n\\end{equation*}\nTaking the derivative in time we obtain that\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\!=\\!2\\!\\sum_{k\\geq1}\\!\\nu'(t) k^{3\/2}e^{\\nu(t)k}|\\widehat{y}(k)|\\!+\\!2\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{\\widehat{y}_t(k)\\overline{\\widehat{y}(k)}\\!+\\!\\widehat{y}(k)\\overline{\\widehat{y}_t(k)}}{|\\widehat{y}(k)|}.\n\\end{aligned}\n\\end{equation*}\nTherefore, substituting \\eqref{thetaeqdiag}, one finds the following equation\n\\begin{equation}\\label{auxx}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\!&=\\!2\\!\\sum_{k\\geq1}\\!\\nu'(t) k^{3\/2}e^{\\nu(t)k}|\\widehat{y}(k)|-2\\frac{A_\\sigma}{R^3}\\!\\!\\sum_{k\\geq1}\\!e^{\\nu(t)k}k^{3\/2}(k^2-1)|\\widehat{y}(k)|\n    \\\\\n    &\\quad+2\\frac{2\\pi}{L(t)}\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{(S^{-1}\\widehat{N})(k)\\overline{\\widehat{y}(k)}+\\overline{(S^{-1}\\widehat{N})(k)}\\widehat{y}(k)}{|\\widehat{y}(k)|}\\\\\n    &\\quad+Y_u+Y_v+Y_w,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    Y_u=-2\\frac{A_\\sigma}{R^3}\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3\\!-\\!1\\Big)\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}\\frac{(S^{-1}u)(k)\\overline{\\widehat{y}(k)}+\\overline{(S^{-1}u)(k)}\\widehat{y}(k)}{2|\\widehat{y}(k)|},\n\\end{equation*}\n\\begin{equation*}\n\\begin{aligned}\n    Y_v&=-2(1\\!+\\!A_\\mu)\\frac{A_\\rho}{R}\\Big(R\\frac{2\\pi}{L(t)}\\!-\\!1\\Big)\n    \\\\\n    &\\quad\\times\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}\\frac{e^{-i\\widehat{\\vartheta}(0)}(S^{-1}v)(k)\\overline{\\widehat{y}(k)}\\!+\\!e^{i\\widehat{\\vartheta}(0)}\\overline{(S^{-1}v)(k)}\\widehat{y}(k)}{2|\\widehat{y}(k)|},\n\\end{aligned}\n\\end{equation*}\n\\begin{equation*}\n\\begin{aligned}\n    Y_w&=2(1\\!-\\!A_\\mu)\\frac{A_\\rho}{R}\\Big(R\\frac{2\\pi}{L(t)}\\!-\\!1\\Big) \\frac32\\Big(\\frac34\\!-\\!\\log{2}\\Big)\n    \\\\\n    &\\quad\\times\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{e^{+i\\widehat{\\vartheta}(0)}(S^{-1}w)(k)\\overline{\\widehat{y}(k)}\\!+\\!e^{-i\\widehat{\\vartheta}(0)}\\overline{(S^{-1}w)(k)}\\widehat{y}(k)}{|\\widehat{y}(k)|}.\n    \\end{aligned}\n\\end{equation*}\nWe have the bounds\n\\begin{equation*}\n    \\begin{aligned}\n    |Y_u|&\\leq 2\\frac{A_\\sigma}{R^3}\\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}|(S^{-1}u)(k)|,\n    \\\\\n    |Y_v|&\\leq 2(1+A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big|\\!\\!\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}|(S^{-1}v)(k)|,\n    \\\\\n    |Y_w|&\\leq 2(1-A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big| \\frac32\\Big(\\frac34-\\log{2}\\Big)\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}|(S^{-1}w)(k)|.\n    \\end{aligned}\n\\end{equation*}\nNoticing that $(S^{-1}w)(k)=w(k)=1_{k=2}\\widehat{\\theta}(1)$ and using the inequalities given by Lemma \\ref{lemmaS}, we have\n\\begin{equation*}\n    \\begin{aligned}\n    |Y_u|&\\leq2\\frac{A_\\sigma}{R^3}\\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\sum_{k\\geq2}e^{\\nu(t)k}k^{1\/2}|u(k)|,\n    \\\\\n    |Y_v|&\\leq2(1+A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big|C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\sum_{k\\geq2}e^{\\nu(t)k}k^{1\/2}|v(k)|,\n    \\\\\n    |Y_w|&\\leq 2^{1\/2}3(1-A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big| \\Big(\\frac34-\\log{2}\\Big)e^{2\\nu(t)}|\\widehat{\\theta}(1)|.\n    \\end{aligned}\n\\end{equation*}\nSince \n\\begin{equation*}\n    k(k^2-1)\\leq k^3,\\quad \\text{for }k\\geq2,\\qquad \\frac{(k^2-1)(k+1)}{k^{1\/2}(k+2)}\\leq (k+1)^{3\/2}, \\quad \\text{for }k\\geq 1,\n\\end{equation*}\nit holds that\n\\begin{equation*}\n    |u(k)|\\leq  k^3 \\left| \\widehat{\\theta}(k)\\right|, \\qquad |v(k)|\\leq (k+1)\\widehat{\\theta}(k+1).\n\\end{equation*}\nNow we use the inequalities for $L(t)$ from  \\eqref{Lestimates} (see Subsection \\ref{sec:Lestimate}) \nto obtain that\n\\begin{equation}\\label{Ybounds}\n    \\begin{aligned}\n    |Y_u|&\\leq 2\\frac{A_\\sigma}{R^3}C_{S}C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\sum_{k\\geq2}e^{\\nu(t)k}k^{\\frac72}|\\widehat{\\theta}(k)|,\\\\\n    |Y_v|&\\leq2(1+A_\\mu)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq2}e^{\\nu(t)k}(k+1)^{3\/2}|\\widehat{\\theta}(k+1)|,\\\\\n    |Y_w|&\\leq 2^{1\/2}3(1-A_\\mu)\\frac{|A_\\rho|}{R}C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\Big(\\frac34-\\log{2}\\Big)e^{2\\nu(t)}|\\widehat{\\theta}(1)|.\n    \\end{aligned}\n\\end{equation}\nAbove we wrote $C_{S}=C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})$.\nGoing back to \\eqref{auxx} with the bound \\eqref{Lbound} and the inequality\n\\begin{equation*}\n    k^{3\/2}(k^2-1)\\geq \\frac34 k^{7\/2},\\quad \\text{for }k\\geq 2,\n\\end{equation*}\nwe find using also \\eqref{CSbounds} that\n\\begin{equation}\\label{auxx2}\n    \\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\nu'(t)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}_\\nu} -\\frac32\\frac{A_\\sigma}{R^3}\\sum_{k\\geq2}e^{\\nu(t)k}k^{7\/2}|\\widehat{y}(k)|\n    \\\\\n    &\\quad+C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}+|Y_u|+|Y_v|+|Y_w|\n    .\n    \\end{aligned}\n\\end{equation}\nWe will next control the $k=1$ frequency in the dissipation part above.\nTo that end recall from \\eqref{frequencyrelation} that, if $\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\leq \\frac12\\log{\\big(\\frac54\\big)}$, then we have \n\\begin{equation*}\n    2|\\widehat{\\theta}(1)|\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\sum_{k\\geq2}|\\widehat{\\theta}(k)|,\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &=2  e^{\\nu(t)} |\\widehat{\\theta}(1)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|\\\\\n    &\\leq 2\\left(C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1\\right)\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|,\\qquad s\\geq0.\n\\end{aligned}\n\\end{equation*}  \nWe will find an analogous inequality in terms of $y$.\nSince $\\widehat{y}(1)=\\widehat{\\theta}(1)$, for $s\\geq0$ we have that\n\\begin{equation*}\n\\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&=2  e^{\\nu(t)} |\\widehat{\\theta}(1)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|\\\\\n    &\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|,\n\\end{aligned}\n\\end{equation*}\nthus substituting $\\widehat{\\theta}(k)=(S\\widehat{y})(k)$ and using \\eqref{equivaux2} with \\eqref{sigmabound} and \\eqref{bessel.In.def}, we obtain\n\\begin{equation*}\n    \\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} e^{2\\nu(t)}2^s\\Big|\\frac{c(1)}{a(2)}\\Big||\\widehat{y}(1)|\\\\\n    &\\quad+2\\bigg(C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1\\bigg)\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|.\n    \\end{aligned}\n\\end{equation*}\nSubtracting the $\\widehat{y}(1)$ term and using that  $e^{\\nu(t)}\\leq e^{\\nu_0}$ for all $t\\geq0$, we find\n\\begin{equation}\\label{ybound.imlicit}\n    \\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2C_y\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|, \\qquad s\\geq0,\n    \\end{aligned}\n\\end{equation}\nwhere using also \\eqref{coeffs} we define\n\\begin{equation}\\label{Cy}\n    \\begin{aligned}\n    C_y&=\\frac{C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1}{1-C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) 2^s e^{\\nu_0}\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}.\n    \\end{aligned}\n\\end{equation}\nThis is the bound for $\\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$ that we will use in \\eqref{auxx2}.\n\n\n\nNotice that in \\eqref{Ybounds}, regarding $Y_w$, that $3 \\sqrt{2} \\left(\\frac{3}{4}-\\log(2)\\right) \\le 1.91\\le 2$.  Also in Lemma \\ref{lemmaS} we have $C_S \\ge 1$.  Thus, from \\eqref{Ybounds}, we have \n$$\n|Y_v|+|Y_w|\n\\leq\n    2\\left(1+|A_\\mu|\\right)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\n    e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq1}e^{\\nu(t)k}k^{3\/2}|\\widehat{\\theta}(k)|,\n$$\nNow we go back to \\eqref{auxx2}, use \\eqref{ybound.imlicit}, and substitute in the bounds for $Y_u, Y_v, Y_w$ \\eqref{Ybounds} to obtain\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\nu'(t)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}_\\nu} -\\frac34\\frac{A_\\sigma}{R^3}C_y^{-1}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n    \\\\\n    &\\quad+C_{S}\n    C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\\\\n    &\\quad\n    +2\\frac{A_\\sigma}{R^3}C_{S}\n    C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\sum_{k\\geq2}e^{\\nu(t)k}k^{\\frac72}|\\widehat{\\theta}(k)|\n    \\\\\n    &\\quad \n    +\n    2\\left(1+|A_\\mu|\\right)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\n    e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq1}e^{\\nu(t)k}k^{3\/2}|\\widehat{\\theta}(k)|,\n\\end{aligned}\n\\end{equation*}    \nwhere we have combined the bounds for $Y_v$ and $Y_w$ as above.\nThe reverse inequalities \\eqref{CSbounds} and the embeddings \\eqref{embed} give that\n\\begin{equation}\\label{balance}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq  \\frac{A_\\sigma}{R^3}\\bigg(-\\frac34C_y^{-1}+\\nu'(t)\\frac{R^3}{A_\\sigma}+2\\big(C_{S}\\big)^2C_{40}\\|y\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu} \\\\\n    &\\quad+2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma} \\big(C_{S}\\big)^2C_{39}e^{\\nu(t)}\\|y\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu}\\bigg)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\\\\\n    &+ \n    C_{S} C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}.\n\\end{aligned}\n\\end{equation}    \nNext, we will use \\eqref{Nbound} to control the nonlinear term $\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$. Together with \\eqref{Lbound}, we obtain\n\\begin{equation*}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\frac{A_\\sigma}{R^2}C_{35}C_{37}^{-2}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n        +|A_\\rho|e^{\\nu(t)}C_{36}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThus using also \\eqref{CSbounds} we have\n\\begin{equation}\\label{nonlinearaux}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\frac{A_\\sigma}{R^2}\\big(C_{S}\\big)^2\\Big(C_{35}C_{37}^{-2}\n        +\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nSubstitution of \\eqref{nonlinearaux}  into \\eqref{balance}, and using once more \\eqref{embed}, provides that\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq - \\frac{A_\\sigma}{R^3}\\bigg(\\frac34C_y^{-1}-\\nu'(t)\\frac{R^3}{A_\\sigma}-2\\big(C_{S}\\big)^2C_{40}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu} \\\\\n    &-2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma} \\big(C_{S}\\big)^2C_{39}e^{\\nu(t)}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu} \\\\   &-\\big(C_{S}\\big)^3C_{37}^{-1}\\Big(C_{35}C_{37}^{-2}\\!+\\!\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\n    \\bigg)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n\\end{aligned}\n\\end{equation*}  \nSince $C_y(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{Cy}, $C_{35}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ and $C_{36}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{C35C36}, $\\big(C_{37}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\big)^{-1}$ in \\eqref{C37C38}, $C_{39}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ and $C_{40}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{C39C40}, are increasing functions of the argument, we can bound all of them by evaluating at the bigger quantity $C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ since \n$\\|\\theta\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu} \\le \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\le C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$.\nHere we suppress the dependency on $A_\\mu$, $|A_\\rho|R^2\/A_\\sigma$ from $C_S$. \n\nFor clarity of notation, in formulas \\eqref{dissipation} and \\eqref{C41} below, we understand that all of these functions are evaluated at $C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$.\nWe conclude that\n\\begin{equation}\\label{balance2}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq -\\frac{A_\\sigma}{R^3} \\mathcal{D} \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\\label{dissipation}\n    \\begin{aligned}\n       \\mathcal{D}\\Big(\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma},\n       A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)&=\\frac34C_y^{-1}-\\nu'(t)\\frac{R^3}{A_\\sigma}-C_{41}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{multline}\\label{C41}\n    C_{41}=\n    \\\\\n    C_S^2\\Big(\n       2C_{40}+2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma}C_{39}e^{\\nu(t)}+C_{S}C_{37}^{-1}\\Big(C_{35}C_{37}^{-2}\\!+\\!\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\Big).\n\\end{multline}\nNote that we can initially choose $\\nu_0>0$ in \\eqref{nu} to be arbitrarily small, and that $\\nu'(t) = \\frac{\\nu_0}{(1+t)^2}\\le \\nu_0$.  \n\nFinally, suppose that the following condition holds initially\n\\begin{equation}\\label{dissipationcond}\n     \\frac34C_y^{-1}-C_{41}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}>0,\n\\end{equation}\nwhere $C_y$ was defined in \\eqref{Cy}.\nFor $\\nu_0$ small enough, using \\eqref{balance2} and the fact that $C_{41}$ decreases as $\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ decreases, then this condition will be propagated in time.   Thus it holds that\n\\begin{equation}\\label{auxbal}\n    \t\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)+ \\frac{A_\\sigma}{R^3}  \\mathcal{D}\n    \n      \n       \\!\\int_0^t\\! \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}(\\tau) d\\tau \\leq \\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}.\n\\end{equation}\nAbove $\\mathcal{D}=\\mathcal{D}\\Big(\\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)$.\nNow since $C_y$ and $C_{41}$ are monotonically increasing in $\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$, the positivity condition \\eqref{dissipationcond} is equivalent to a medium-size condition on the initial data\n\\begin{equation*}\n    \\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}<h\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n\\end{equation*}\nwhere the function $h$ is implicitly defined via \\eqref{dissipationcond}. \nRecalling the bound for the inverse relation $\\widehat{\\theta}(k)=(S\\widehat{y})(k)$ in \\eqref{CSbounds}, and denoting \\begin{equation}\\label{Kdef}\n    K\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big)=C_S\\big(A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}\\big) h\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n\\end{equation}\nwe obtain the medium-size condition for $\\theta_0$ given in \\eqref{condition}.  This completes the proof of \\eqref{condition} and \\eqref{estimatef12}.  The proof of \\eqref{decay} follows from applying \\eqref{embed} to \\eqref{balance2}.\n\n\n\nTo perform the estimate for $\\widehat{\\vartheta}(0)$ from \\eqref{zeroBoundUniform}, we use \\eqref{eqcurve3}$_1$ to obtain\n\\begin{equation*}\n    \\widehat{\\vartheta}_t(0)=\\frac{2\\pi}{L(t)}\\widehat{T}*\\widehat{(1+\\theta_\\alpha)}(0).\n\\end{equation*}\nNote that $\\widehat{U_\\alpha}(0)=0$.  We use the splitting for $T$ from \\eqref{Tsplit}, and notice from \\eqref{u0t0} with \\eqref{T0} that \n\\begin{equation*}\n    \\widehat{T_0}(k)=0 \\text{ for }|k|\\neq 1, \\qquad |\\widehat{T_0}(k)|\\leq \\frac{|A_\\rho|}2 \\text{ for }|k|= 1,\n\\end{equation*}\nthus $\\|T_0\\|_{\\mathcal{F}^{0,1}}\\leq A_\\rho$. Then we have\n\\begin{equation*}\n    |\\widehat{\\vartheta}(0)|\n    \\leq |\\widehat{\\vartheta}_0(0)|+\n    \\int_0^t \\frac{2\\pi}{L(\\tau)}\\Big(A_\\rho\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}+\\big(\\|T_1\\|_{\\mathcal{F}^{0,1}}+\\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}}\\big)\\big(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\big)\\Big) d \\tau.\n\\end{equation*}\nThe estimates \\eqref{T1aux} and \\eqref{T2aux} with $s=0$ in Subsection \\ref{sec:NonLinearEst}, together with the bounds for $L(t)$ in \\eqref{Lbound}, give \n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_1\\|_{\\mathcal{F}^{0,1}}&\\leq \\frac{A_\\sigma}{R^2}\\Big(2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\!+\\!C_{37}^{-2}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}}\\!+\\!2\\frac{|A_\\rho|R^2}{A_\\sigma}\\big((1\\!+\\! 2|A_\\mu|)\\!+\\!2{C_{\\mR}}\\big)e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\Big),\n    \\\\\n    \\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}}&\\leq \\frac{A_\\sigma}{R^2}\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}C_{34}e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\!+\\!C_{37}^{-2}C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}}\\Big).\n    \\end{aligned}\n\\end{equation*}\nTherefore, reordering and using \\eqref{auxbal}, \\eqref{interp} and \\eqref{embed}, we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    |\\widehat{\\vartheta}(0)|&\\leq |\\widehat{\\vartheta}_0(0)|+C_{42}\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n    \\end{aligned}\n\\end{equation*}\nwhere \n\\begin{multline}    \\label{C42}\n    C_{42}=\n    \\\\ \n    \\frac{1}{\\mathcal{D}}C_S^2C_{37}^{-1}\\Bigg(\\frac{|A_\\rho|R^2}{A_\\sigma}+C_{37}^{-2}+2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\\Big(1+\\big((1+2|A_\\mu|)+2{C_{\\mR}}\\big)\\Big)\n    \\\\+C_S^2\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\bigg(C_{37}^{-2}+C_{37}^{-2}C_{33}(1+C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})\n    \\\\\n    +\n    2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\\Big(1+\\big((1+2|A_\\mu|)+2{C_{\\mR}}\\big)+\\frac{1}{2}C_{34}(1+C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})\\Big)\\bigg)\\Bigg).\n\\end{multline} \nIn $C_{42}$ above  the quantities ($\\mathcal{D}$ in \\eqref{dissipation}, $C_{37}$ in \\eqref{C37C38}, $C_{33}$ and $C_{34}$ in \\eqref{C33C34}) are evaluated at\n$C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$, with $C_S$ defined in \\eqref{CSbounds}.\nThis completes the proof of \\eqref{zeroBoundUniform}.\n\\end{proof}\n\n\nWe will now prove Proposition \\ref{diagonalization}.\n\n\\begin{proof}[Proof of Proposition \\ref{diagonalization}]\nBecause $S^{-1}$ has trivial kernel, it suffices to prove that $S$ is a right inverse of $S^{-1}$, i.e., that\n\\begin{equation}\\label{rightinverse.proof}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\delta_{k,m},\n\\end{equation}\nwhere $\\delta_{k,m}$ is the Kronecker delta. \nThe cases $k=1$ and $k=2, m=1$ are straightforward. For the rest of cases, we have that\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}&=\\sum_{j\\geq1}(-1)^{j-k} \\delta_{j\\geq k}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k)-a(k+l)}\\delta_{m\\geq j}\\prod_{n=1}^{m-j}\\frac{b(j-1+n)}{a(j-1+n)-a(m)}\\\\\n    &=\\sum_{j=k}^m (-1)^{j-k}\\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)b(j-1+n)}{(a(k)-a(k+l))(a(j-1+n)-a(m))}.\n    \\end{aligned}\n\\end{equation*}\nIf $m=k$, then clearly $\\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=1$. Now, we notice that\n\\begin{equation*}\n    \\begin{aligned}\n    \\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} b(k\\!-\\!1\\!+\\!l)b(j\\!-\\!1\\!+\\!n)&=b(k)b(k\\!+\\!1)\\dots b(j\\!-\\!2)b(j\\!-\\!1)b(j)b(j\\!+\\!1)\\dots b(m\\!-\\!1)\\\\\n    &=\\prod_{n=0}^{m-k-1}b(k+n)\n    \\end{aligned}\n\\end{equation*}\nis independent of $j$, and thus comes out of the sum as\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\prod_{n=0}^{m-k-1}b(k+n) \\sum_{j=k}^m (-1)^{j-k}A(j),\n    \\end{aligned}\n\\end{equation*}\nwhere we define\n\\begin{equation*}\n    A(j)=\\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} \\frac{1}{(a(k)\\!-\\!a(k\\!+\\!l))(a(j\\!-\\!1\\!+\\!n)\\!-\\!a(m))}.\n\\end{equation*}\nChanging $n\\leftarrow j-k-1+n$, it becomes\n\\begin{equation*}\n    A(j)=\\prod_{n=j-k}^{m-k-1}\\frac{1}{a(k+n)\\!-\\!a(m)}\\prod_{l=1}^{j-k} \\frac{1}{a(k)\\!-\\!a(k\\!+\\!l)}.\n\\end{equation*}\nThus, multiplying and dividing by the product of $a(k+n)-a(m)$ from $n=0$ to $n=j-k-1$, we rewrite it as follows\n\\begin{equation*}\n    \\begin{aligned}\n    A(j)=\\prod_{n=0}^{m-k-1}\\frac{1}{a(k+n)-a(m)}\\prod_{l=1}^{j-k}\\frac{a(k+l-1)-a(m)}{a(k)-a(k+l)}.\n    \\end{aligned}\n\\end{equation*}\nSince the first product does not depend on $j$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\prod_{n=0}^{m-k-1}\\frac{b(k+n)}{a(k+n)-a(m)}\\sum_{j=k}^m (-1)^{j-k}\\prod_{l=1}^{j-k}\\frac{a(k+l-1)-a(m)}{a(k)-a(k+l)}.\n    \\end{aligned}\n\\end{equation*}\nChanging the index $j\\leftarrow j-k$ and denoting $\\tilde{m}=m-k$, we are left to show that, for all $\\tilde{m}\\geq1$,\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j=0}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\frac{a(k+l-1)-a(\\tilde{m}+k)}{a(k)-a(k+l)}=0.\n    \\end{aligned}\n\\end{equation*}\nWriting the sum with a common denominator, the identity that we need to prove is\n\\begin{equation*}\n    \\begin{aligned}\n    W=\\sum_{j=0}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big)=0.\n    \\end{aligned}\n\\end{equation*}\nWe add the first two terms in the sum by noticing that they only differ in one term, which partially cancels as\n\\begin{equation*}\n    \\begin{aligned}\n    W&=-\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\prod_{l=2}^{\\tilde{m}}\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &+\n    \\sum_{j=2}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nWe now realize that we can repeat the same step with the new first two terms to obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    W&=\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k\\!+\\!2)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\prod_{l=3}^{\\tilde{m}}\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &+\n    \\sum_{j=3}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nWe can continue the process to find that\n\\begin{equation*}\n    \\begin{aligned}\n    &W=\\!(-1)^r\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k\\!+\\!2)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\\\ &\\hspace{3cm}\\dots\\big(a(k\\!+\\!r)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\!\\!\\!\\!\n    \\prod_{l=r+1}^{\\tilde{m}}\\!\\!\\!\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &\\hspace{1cm}+\n    \\sum_{j=r+1}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nThus for $r=\\tilde{m}-1$, we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    W\n    &=\\!(-1)^{\\tilde{m}-1}\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\dots \\big(a(k\\!+\\!\\tilde{m}\\!-\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\\\\n    &+ (-1)^{\\tilde{m}}\\prod_{l=1}^{\\tilde{m}}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)=0.\n    \\end{aligned}\n\\end{equation*}\nThis proves \\eqref{rightinverse.proof}.\n\nWe next prove that $S^{-1}$ in \\eqref{Sm1} diagonalizes \\eqref{linearZ}. Notice that $S^{-1}_{k,k}=1$ and $S^{-1}_{k,j}=0$ for $j<k$, $k\\geq2$. Then, for $k\\geq3$, we can write\n\\begin{equation*}\n    \\begin{aligned}\n    y_t(k)=\\sum_{j\\geq k}S^{-1}_{k,j}z_t(j)=-a(k)S^{-1}_{k,k}z(k)+\\sum_{j\\geq k+1}\\Big(-a(j)S^{-1}_{k,j}+b(j-1)S^{-1}_{k,j-1}\\Big)z(j).\n    \\end{aligned}\n\\end{equation*}\nFinally, notice that \n\\begin{equation*}\n    -a(j)S^{-1}_{k,j}+b(j-1)S^{-1}_{k,j-1}=-a(k)S^{-1}_{k,j},\\qquad j-1\\geq k\\geq 2,\n\\end{equation*}\nprecisely when $S^{-1}_{k,j}$ is given by \\eqref{Sm1}. The case $k=1$ is trivial, while for $k=2$ it holds that\n\\begin{equation*}\n    \\begin{aligned}\n    y_t(2)&=\\sum_{j\\geq 2}S^{-1}_{2,j}z_t(j)=c(1)z(1)\\!-\\!a(2)z(2)\\!+\\!\\sum_{j\\geq 3}\\!\\Big(\\!-a(j)S^{-1}_{2,j}\\!+\\!b(j\\!-\\!1)S^{-1}_{2,j-1}\\Big)z(j)\\\\\n    &=-a(2)\\frac{c(1)}{a(2)}z(1)-a(2)\\sum_{j\\geq2}S^{-1}_{2,j}z(j)=-a(2)y(2).\n    \\end{aligned}\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\nWe are now ready to prove Lemma \\ref{lemmaS}.\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemmaS}]\nDenote for $j \\ge k \\ge 2$\n\\begin{equation*}\n\\begin{aligned}\n    \\beta(k,j)=(-1)^{j-k}\\prod_{m=1}^{j-k} \\frac{b(k-1+m)}{a(k)-a(k+m)}.\n\\end{aligned}\n\\end{equation*}\nPlugging in the expressions for $b(k)$ and $a(k)$  in \\eqref{coeffs}, we find that\n\\begin{equation*}\n    \\begin{aligned}\n    \\beta(k,j)\\!=\\!\\Big(\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big)^{j-k}\\!\\!\\!\\!e^{-i(j-k)\\widehat{\\vartheta}(0)}\\!\\!\\prod_{m=1}^{j-k}\\!\\!\\frac{(k\\!+\\!m)^2(k\\!-\\!2\\!+\\!m)}{(k\\!-\\!1\\!+\\!m)(k\\!+\\!1\\!+\\!m)}\\frac{1}{k^3\\!-\\!(k\\!+\\!m)^3\\!+\\!m}.\n    \\end{aligned}\n\\end{equation*}\nThen, we can write \n\\begin{equation}\\label{equivaux}\n    \\begin{aligned}\n    \\|y\\|_{\\ell^1}&=\\sum_{k\\geq1}|y(k)|\\leq \\Big(1+\\Big|\\frac{c(1)}{a(2)}\\Big|\\Big)|z(1)|+\\sum_{k\\geq2}\\sum_{j\\geq k}|\\beta(k,j)||z(j)|\\\\\n    &=\\Big(1+\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+\\sum_{l\\geq2}\\sum_{j=0}^{l-2}|\\beta(l-j,l)||z(l)|\\\\\n    &\\leq \\Big(1+\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+\\sum_{l\\geq2}\\lambda(l)|z(l)|,\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    \\lambda(l)=\\sum_{j=0}^{l-2}|\\beta(l-j,l)|.\n\\end{equation*}\nIn the following we will estimate $\\lambda(l)$ to control $\\beta$.\n\nThe identity $k^3-(k+m)^3+m=-m\\big(m^2+3m k+3k^2-1\\big)$ gives the bound\n\\begin{equation}\\label{index.lower}\n    |(l-j)^3-(l-j+m)^3+m|\\geq m\\big(m^2+2m(l-j)+(l-j)^2\\big)=m(m+l-j)^2.\n\\end{equation}\nThus\n\\begin{equation}\\label{betabound}\n    \\begin{aligned}\n    |\\beta(l-j,l)|&\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{l-j\\!-\\!2\\!+\\!m}{m(l-j\\!-\\!1\\!+\\!m)(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{1}{m(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big(\\big|\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j}\\prod_{m=1}^{j}\\frac{1}{m(m+3)},\n    \\end{aligned}\n\\end{equation}\nwhere the last step is due to the fact that $j\\leq l-2$. Therefore, we have that\n\\begin{equation*}\n    \\begin{aligned}\n    \\lambda(l)&\\leq \\sum_{j=0}^{l-2}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^j\\frac{6}{j!(j+3)!} \\leq \\sum_{j=0}^{\\infty}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^j\\frac{6}{j!(j+3)!}\\\\\n    &=6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}},\n    \\end{aligned}\n\\end{equation*}\nwhere $I_3$ is the modified Bessel function of the first kind of order three as in \\eqref{bessel.In.def}.\nWe conclude thereby from \\eqref{equivaux} that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|y\\|_{\\ell^1}&\\leq\n    \\Big(1\\!+\\!\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\sum_{l\\geq2}|z(l)|\\\\\n    &\\leq C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\|z\\|_{\\ell^1},\n    \\end{aligned}\n\\end{equation*}\nwith $C_{S}$ given in \\eqref{CSbounds}. \n\nWe proceed to prove the reverse direction, $z=S y$. For $j \\ge k \\ge 2$, denote\n\\begin{equation*}\n\\begin{aligned}\n    \\alpha(k,j)=\\prod_{m=1}^{j-k} \\frac{b(k-1+m)}{a(k-1+m)-a(j)}\n.\n\\end{aligned}\n\\end{equation*}\nRecalling \\eqref{coeffs}, it becomes\n\\begin{equation*}\n    \\begin{aligned}\n    \\alpha(k,j)&=(-1)^{j-k}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j-k}e^{-i(j-k)\\widehat{\\vartheta}(0)}\\\\\n    &\\quad\\cdot\\prod_{m=1}^{j-k}\\frac{(k\\!+\\!m)^2(k\\!-\\!2\\!+\\!m)}{(k\\!-\\!1\\!+\\!m)(k\\!+\\!1\\!+\\!m)}\\frac{1}{(k\\!-\\!1\\!+\\!m)^3\\!-\\!j^3\\!+\\!j\\!-\\!k\\!+\\!1\\!-\\!m}.\n    \\end{aligned}\n\\end{equation*}\nNext, proceeding as in \\eqref{equivaux}, we have\n\\begin{equation}\\label{equivaux2}\n    \\begin{aligned}\n    \\|z\\|_{\\ell^1}&=\\sum_{k\\geq1}|z(k)|\\leq \\big(1+\\Big|\\frac{c(1)}{a(2)}\\Big|\\big)|y(1)|+\\sum_{k\\geq2}\\sum_{j\\geq k}|\\alpha(k,j)||y(j)|\\\\\n    &\\leq \\Big(1+\\frac14\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|y(1)|+\\sum_{l\\geq2}\\sigma(l)|y(l)|,\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    \\sigma(l)=\\sum_{j=0}^{l-2}|\\alpha(l-j,l)|.\n\\end{equation*}\nThe inequality below follows from \\eqref{index.lower} with $M+L-J = j$ and $L-J=m+k-1$\n\\begin{equation*}\n    \\begin{aligned}\n    |(k-1+m)^3-j^3+j-k+1-m|\\geq (j-k+1-m)j^2.\n    \\end{aligned}\n\\end{equation*}\nUsing the inequality above,  and $m \\le j \\le l -2$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    |\\alpha(l-j,l)|&\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{(l-j\\!+\\!m)^2(l-j-2+m)}{(l-j\\!-\\!1\\!+\\!m)(l-j\\!+\\!1\\!+\\!m)}\\frac{1}{l^2(j\\!+\\!1\\!-\\!m)}\\\\\n    &\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{1}{m(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j}\\prod_{m=1}^{j}\\frac{1}{m(m+3)},\n    \\end{aligned}\n\\end{equation*}\nwhich coincides with the bound found for $\\beta$ in \\eqref{betabound}. Therefore, \n\\begin{equation}\\label{sigmabound}\n    \\begin{aligned}\n    \\sigma(l)&\\leq6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}.\n    \\end{aligned}\n\\end{equation}\nAbove we recall $I_3$ from \\eqref{bessel.In.def}.  Then going back to \\eqref{equivaux2} we have\n\\begin{equation*}\n    \\begin{aligned}\n    \\|z\\|_{\\ell^1}&\\leq \n     C_S\\left(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma}\\right)\\|y\\|_{\\ell^1},\n    \\end{aligned}\n\\end{equation*}\nwith $C_S$ defined in \\eqref{CSbounds}.   This completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Nonlinear Estimates in $\\dot{\\mathcal{F}}^{s,1}_\\nu$} \\label{sec:NonLinearEst}\nWe recall that the nonlinear terms \\eqref{system} are given by\n\\begin{equation*}\n    N(\\alpha)=N_1+N_2+N_3,\n\\end{equation*}\nwith\n\\begin{equation}\\label{N1N2N3}\n\\begin{aligned}\n    N_1=(U_{\\geq 2})_\\alpha(\\alpha),\\quad N_2=T_{\\geq 2}(\\alpha)(1+\\theta_\\alpha(\\alpha)),\\quad\n    N_3=T_1(\\alpha)\\theta_\\alpha(\\alpha),\n\\end{aligned}\n\\end{equation}\nwhere $U$ and $T$ are described in \\eqref{Usplit} and \\eqref{Tsplit}.\n\nWe will first perform the estimates for $N_1$ for $s\\ge 0$.  From \\eqref{Usplit}, we have\n\\begin{multline}\\label{N1f0}\n    \\|N_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}=\\|U_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n    \\\\\n    \\!\\leq\\! \\frac{\\pi}{L(t)}\\Big(\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\!+\\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\!+\\| \\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\Big).\n\\end{multline}\nUsing the bound \\eqref{Restimates}$_2$ and splitting $\\omega_{\\geq1}=\\omega_{1}+\\omega_{\\geq2}$ gives that\n\\begin{equation*}\n    \\begin{aligned}\n        \\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq   b(2,s){C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+b(2,s){C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+b(2,s){C_{\\mR}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\n        +b(2,s){C_{\\mR}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThen introducing the vorticity estimates \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, and \\eqref{omega2fss}, and ordering the terms in $A_\\rho$ and $A_\\sigma$, it follows that\n\\begin{equation*}\n    \\begin{aligned}\n        \\| \\mathcal{R}&(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}b(2,s){C_{\\mR}}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        +|A_\\mu|C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\\\\\n        &\\quad+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\mu|C_{9}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\Big)\\\\\n        &\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}b(2,s){C_{\\mR}}\\Big(2(1+2|A_\\mu|)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad +(1+2|A_\\mu|)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_{10}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big).\n    \\end{aligned}\n\\end{equation*}\nInterpolation as in \\eqref{interpolation} yields that\n\\begin{equation}\\label{est1}\n    \\begin{aligned}\n        \\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq A_\\sigma\\frac{4\\pi}{L(t)}C_{21}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        \\\\\n        &\\quad +|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{22}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C21C22}\n\\begin{aligned}\n    C_{21}&=2b(2,s){C_{\\mR}}+|A_\\mu|b(2,s){C_{\\mR}}(C_{9}+C_{13})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n    C_{22}&=3b(2,s){C_{\\mR}}(1+2|A_\\mu|)+ b(2,s){C_{\\mR}}(C_{10}+C_{14})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nand $C_\\mathcal{R}$ in \\eqref{CR}, $C_9$ and $C_{10}$ in \\eqref{C9C10}, $C_{13}$ and $C_{14}$ in \\eqref{C13C14} were defined previously.\n\n\n\n\nNext, we split $\\omega=\\omega_0+\\omega_1+\\omega_{\\geq2}$ as in \\eqref{omegasplit} and use the bound \\eqref{Sestimates}$_2$ to obtain \n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq |A_\\rho|\\frac{L(t)}{\\pi}(C_3+C_4)e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n        +\n        C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n        +C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n        &\\quad+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nSo again we introduce the vorticity estimates \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, and \\eqref{omega2fss}, and order the terms in $A_\\rho$ and $A_\\sigma$, to obtain \n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\mathcal{S}&(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}\\Big( C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}+|A_\\mu|C_3C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu} \\\\\n        &\\quad+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\mu|C_9C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\Big)\\\\\n        &\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\Big( (C_3\\!+\\!C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+2b(2,s) (1\\!+\\!2|A_\\mu|)C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+C_3C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+C_4(1+2|A_\\mu|)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+C_4C_{10}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\Big).\n    \\end{aligned}\n\\end{equation*}\nApplying interpolation \\eqref{interpolation} and the  embedding \\eqref{embed}, \nwe obtain that\n\\begin{multline}\\label{est2}\n        \\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}C_{23}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        \\\\\n          +|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{24}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n\\end{multline}\nwhere\n\\begin{equation}\\label{C23C24}\n    \\begin{aligned}\n        C_{23}&=C_3+C_4+|A_\\mu|(C_3C_{13}+C_4C_9)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n        C_{24}&=C_3+C_4+(1+2|A_\\mu|)(2b(2,s)C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\n        \\\\\n        & \\quad +(C_3C_{14} +C_4C_{10})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n    \\end{aligned}\n\\end{equation}\nwith $C_3$ and $C_4$ in \\eqref{C3}, $C_9$ and $C_{10}$ in \\eqref{C9C10}, $C_{13}$ and $C_{14}$ in \\eqref{C13C14} previously defined.     \n\nFinally, we combine estimates \\eqref{est1} and \\eqref{est2} together with vorticity estimate \\eqref{omega2fss} to close the bound \\eqref{N1f0}.  We obtain\n\\begin{equation}\\label{N1final}\n        \\|N_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2\n        C_{25}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}+|A_\\rho|e^{\\nu(t)}\n        C_{26}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n\\end{equation}\nwhere\n\\begin{equation}\\label{C25C26}\n    \\begin{aligned}\n        C_{25}=|A_\\mu| C_{13}+C_{21}+C_{23}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\qquad       C_{26}=C_{14}+C_{22}+C_{24}.\n    \\end{aligned}\n\\end{equation}\nAbove  $C_{13}$ and $C_{14}$ are defined in \\eqref{C13C14}, $C_{21}$ and $C_{22}$ in \\eqref{C21C22}, and $C_{23}$ and $C_{24}$ in \\eqref{C23C24}.  This completes the estimate for $N_1$.\n\n\n\n\nWe proceed now to estimate the term $N_2$ from \\eqref{N1N2N3}.  For simplicity in this case we perform the estimate for $0\\le s \\le 1$.  The case $s>1$ would add several complications to the notation, but can also be proven similarly, as we will see in the following.  For $s\\ge 0$ we use \\eqref{bfcn.def} and \\eqref{s.inequality} to obtain\n\\begin{equation}\\label{N2aux}\n\\begin{aligned}\n    \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}(1+b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})+b(2,s)\\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}.\n\\end{aligned}\n\\end{equation}\nWe recall that for a function $f(\\alpha)$ one has \\eqref{fourierCalcAlpha} and then for $s\\ge 0$ we have\n\\begin{equation}\\label{negative.norm.bound}\n    \\begin{aligned}\n        \\left|\\left|\\int_0^\\alpha f(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^\\pi f(\\eta)d\\eta\\right|\\right|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\left( 1 + 1_{s=0}\\right)\\|f-\\widehat{f}(0)\\|_{\\dot{\\mathcal{F}}^{-1+s,1}_\\nu} \\\\\n        &\\leq C(s)\\|f\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere $(s-1)^+ = s-1$ if $s\\ge 1$ and $(s-1)^+ = 0$ if $s \\le 1$ as usual.  We also define\n\\begin{equation}\\label{def.CS}\n    C(s) = \\left( 1 + 1_{s=0}\\right).\n\\end{equation}\nIf we performed the estimate below for $s>1$ we would need work with the norm of $\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu$ instead of the simpler $\\mathcal{F}^{0,1}_\\nu$.  Thus we only do $0 \\le s \\le 1$ for the $N_2$ term.\n\n\n\nTherefore, for $0 \\le s \\le 1$ recalling the expression for $T_{\\geq2}$ in \\eqref{Tsplit}, we have\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq C(s)\\|(1+\\theta_\\alpha)U_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+C(s)\\|\\theta_\\alpha U_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n    &\\leq C(s)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|U_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+C(s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|U_1\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThen introducing $U_{\\geq2}$ and $U_1$  from\\eqref{Usplit} we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  C(s)\\frac{\\pi}{L(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\big(\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\\big)\n    \\\\\n    &\\quad+C(s)\\frac{\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\big(\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{0})\\|_{\\mathcal{F}^{0,1}_\\nu}\\big).\n    \\end{aligned}\n\\end{equation*}\nWe split the vorticity terms, $\\omega_{\\geq1} = \\omega_{1}+ \\omega_{\\geq2}$, and use $\\omega_0$ in \\eqref{omegasplit}.  We further use  the estimates for $\\mathcal{R}$ and $\\mathcal{S}$ in \\eqref{Restimates} and \\eqref{Sestimates} and use \\eqref{cosine.calc.FT} to obtain\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  C(s)\\frac{\\pi}{L(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\Big((1+{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n    &\\quad+({C_{\\mR}}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+A_\\rho\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big)\n    \\\\\n    &\\quad+C(s)\\frac{\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\Big(\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+A_\\rho\\frac{L(t)}{\\pi}e^{\\nu(t)}{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big).\n    \\end{aligned}\n\\end{equation*}\nThen we introduce the vorticity estimates \\eqref{vorticityestimatess} and \\eqref{omega2f0} to obtain \n\\begin{equation*}\n    \\begin{aligned}    \\|T_{\\geq2}&\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq  \n    A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C(s)\n\\Big(|A_\\mu|C_9\\big(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\big)\\\\\n&\\quad+(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\Big)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n\\\\\n&\\quad+A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 C(s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu} \\!+\\!|A_\\rho|C(s)\\Big(C_{10}(1\\!+\\!C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\\\\n&\\quad+(1+2|A_\\mu|)(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_1\\Big)e^{\\nu(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\\\\n&\\quad\n+|A_\\rho|C_\\mathcal{R}C(s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\n+(1+2|A_\\mu|)|A_\\rho|C(s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThe above expression reduces to\n\\begin{equation}\\label{T2aux}\n    \\begin{aligned}    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  \n    A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C33C34}\n    \\begin{aligned}    \n    C_{33}&=C(s)+C(s)\\Big(|A_\\mu|C_9\\big(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\big)\\\\\n    &\\qquad+(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) \\Big)(1+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}),\\\\\n    C_{34}&=C(s)\\Big(C_{10}(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\\\\n    &\\quad+(1+2|A_\\mu|)(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_1\\Big)(1+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\n    \\\\\n    &\\quad+C(s)C_\\mathcal{R}+C(s)(1+2|A_\\mu|),\n    \\end{aligned}\n\\end{equation}\nand $C_\\mathcal{R}$ is defined in \\eqref{CR}, $C(s)$ in \\eqref{def.CS}, $C_1$ in \\eqref{C1}, $C_9$ and  $C_{10}$ \\eqref{C9C10}.  \n\n\n\nThen, going back to \\eqref{N2aux}, we find that \n\\begin{equation}\\notag\n    \\begin{aligned}\n        \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\Big(A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\Big)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu})\\\\\n        &\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu} \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n        \\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ 1\\right)\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ 1\\right).\n    \\end{aligned}\n\\end{equation}\nFinally, interpolation \\eqref{interpolation} gives that\n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\leq \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}},\n    \\end{aligned}\n\\end{equation*}\nThus we have\n\\begin{equation}\\label{N2bound}\n    \\begin{aligned}\n        \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n        &\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}}+ \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu} \\right)\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\ \\left( 2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\right).\n    \\end{aligned}\n\\end{equation}\nThis completes our estimates for $N_2$.  \n\n\n\nLastly, the estimate for $N_3$ in \\eqref{N1N2N3} for $s\\ge 0$ is obtained with \\eqref{bfcn.def} as follows\n\\begin{equation*}\n    \\begin{aligned}\n        \\|N_3\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq b(2,s) \\left(\\|T_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}+\\|T_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\right).\n    \\end{aligned}\n\\end{equation*}\nThe bound for $T_1$ in \\eqref{Tsplit} for $s\\ge 0$ from \\eqref{negative.norm.bound} using \\eqref{u0t0} and \\eqref{cosine.calc.FT} is \n\\begin{equation}\\label{T1aux}\n    \\begin{aligned}\n        \\|T_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\|U_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}+\\|\\theta_\\alpha U_0\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        \\\\\n        &\\leq \\|U_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}+2|A_{\\rho}| b(2,s) e^{\\nu(t)} \\|\\theta_\\alpha\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        \\\\\n        &\\leq \\frac{\\pi}{L(t)}\\big(\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        +\\|\\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\big)\n        \\\\\n        &\\quad +2|A_{\\rho}| b(2,s) e^{\\nu(t)} \\|\\theta_\\alpha\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nWe use \\eqref{omega1f0}, \\eqref{omega1fss} and \\eqref{Restimates} with \\eqref{omegasplit} and  \\eqref{cosine.calc.FT} to obtain\n\\begin{multline*}\n\\frac{\\pi}{L(t)}\\big(\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        +\\|\\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\big)\n        \\\\\n        \\leq \n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+(s-1)^+,1}_\\nu}\n        +2b(2,s)|A_\\rho|\\Big((1+2|A_\\mu|)+2 C_\\mathcal{R}\\Big)e^{\\nu(t)}\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}.\n\\end{multline*}\nThus\n\\begin{equation}\\label{N3boud}\n    \\begin{aligned}\n        \\|N_3\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 \n        b(2,s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+(s-1)^+,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_3^{\\prime}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nHere\n\\begin{equation}\\label{C3prime.prime}\n        C_3^{\\prime}= 2b(2,s)^2\\left(1+(1+2|A_\\mu|)+2 C_\\mathcal{R} \\right).\n\\end{equation}\nThis completes our estimates for $N_3$.  \n\nIn summary, we can combine the bounds \\eqref{N1final}, \\eqref{N2bound}, and \\eqref{N3boud} to prove the following theorem.\n\n\\begin{thm}\\label{thm:nonLinear}\nFor $0\\le s \\le 1$ we have the estimate for $N$ from \\eqref{system} as\n\\begin{equation}\\label{thm:Nbound}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_1(N)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\\\\\n        &\\quad+\n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_2(N)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation}\\label{C1NC2N}\n    \\begin{aligned}\n        C_1(N)&=C_{25}+C_{33}+b(2,s),\\\\\n        C_2(N)&=C_{26}+C_{34}(1+2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_{3}^\\prime.\n    \\end{aligned}\n\\end{equation}\nFurther $C_{25}$ and $C_{26}$ are defined in \\eqref{C25C26}, $C_{33}$ and $C_{34}$ in \\eqref{C33C34}, and $C_{3}^\\prime$ is previously defined in \\eqref{C3prime.prime}.\n\\end{thm}\n\n\n\n\nPlugging in $s=1\/2$ in the bound for the nonlinear term in Theorem \\ref{thm:nonLinear} in  \\eqref{thm:Nbound},\nwe find that\n\\begin{equation}\\label{Nbound}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{35}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n        +|A_\\rho|e^{\\nu(t)}C_{36}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C35C36}\n    \\begin{aligned}\n        C_{35}&=C_{35}(\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu})=C_1(N)+2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\\\\n        C_{36}&=C_{36}(\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu})=C_2(N),\n    \\end{aligned}\n\\end{equation}\nand $C_1(N)$ and $C_2(N)$ are defined in \\eqref{C1NC2N}, and  $C_{33}$ is defined in \\eqref{C33C34}. Notice that in the definition of $C_{35}$ and $C_{36}$ we can evaluated all the previous functions $C_i$ in the norm $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ instead of $\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}$, which could be done due to \\eqref{embed} and the fact that these $C_i$ are increasing functions of the norm.\n\n\n\n\\subsection{Regularization scheme and completion of the proof of Theorem \\ref{thm:global}}\\label{subregular}\n\nWe will now put all the pieces together to complete the proof of Theorem \\ref{thm:global}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:global}]\nThe proof then follows a standard regularization argument. Recall the high-frequency cut-off operator $\\mathcal{J}_N$ by \\eqref{CutOffHigh}.  Denote $f_N=\\mathcal{J}_N f$ and consider the regularized version of system \\eqref{finalsystem}\n:\n\\begin{equation*}\n    \\begin{aligned}\n     (\\vartheta_N)_t&=\\mathcal{J}_N\\big(\\frac{2\\pi}{L_N(t)} (U_N)_\\alpha+\\frac{2\\pi}{L_N(t)}T_N(1+(\\vartheta_N)_\\alpha)\\Big),\\\\\n     \\frac{L_N(t)}{2\\pi}&=R\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta_N(\\alpha)-\\theta_N(\\eta))^n d\\eta d\\alpha\\Big)^{-\\frac12},\\\\\n     0&=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\theta_N(\\alpha))}d\\alpha.\n    \\end{aligned}\n\\end{equation*}\nWe abused notation in the definition of $L_N(t)$ above since we are not using $\\mathcal{J}_N$ from \\eqref{CutOffHigh}. Solving the last constraint by the implicit function theorem (see Proposition \\ref{IFTprop} in Section \\ref{IFTSection}) gives $F(\\theta_N)=(\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))$ which can be solved for $\\widehat{\\theta}(\\pm 1)$.   \nThus substituting in as well the expression for $L_N(t)$, we obtain one equation for $\\varphi_N=\\widehat{\\vartheta}(0)+\\mathcal{F}^{-1}(1_{|k|\\neq1}\\widehat{\\theta_N}(k))$. We thereby have the system written as an ODE of the form \n\\begin{equation*}\n    \\dot{\\varphi}_N=\\mathcal{J}_N \\mathcal{G}\\big(\\varphi_N\\big),\\qquad \\varphi_N(0)=\\widehat{\\vartheta}_0(0)+\\mathcal{F}^{-1}(1_{|k|\\neq1}\\widehat{\\theta_{N,0}}(k)),\n\\end{equation*}\nfor a certain nonlinear function $\\mathcal{G}$.  Here $\\theta_{N,0}=\\mathcal{J}_N\\theta_{0}$ is the initial condition.  Therefore, Picard's theorem on Banach spaces yields the local existence of regularized solutions $\\varphi_N\\in C^1([0,T_N);H_N^m)$, where the space $H_N^m$ is defined by $H_N^m=\\{f\\in H^m(\\mathbb{T}): \\text{supp}(\\widehat{f})\\subset [-N,N]\\}.$ Furthermore, is clear that the \\textit{a priori} estimates in Subsection \\ref{subsecGlobal}, in particular the energy balance \\eqref{estimatef12}, hold for the regularized system, which provides uniform bounds for $\\varphi_N$ in the space $L^\\infty(\\mathbb{R}_+; \\dot{\\mathcal{F}}^{\\frac12,1}_\\nu)\\cap L^1(\\mathbb{R}_+; \\dot{\\mathcal{F}}^{\\frac72,1}_\\nu)$. One can then apply a version of the Aubin-Lions lemma to get the strong convergence to the full system, up to a subsequence, of the approximated problems and thereby conclude the existence result. We refer Section 5 of \\cite{GG-BS2019} for further details of such an approximation argument.   \n\\end{proof}\n\n\n\n\n\\section{Uniqueness}\\label{sec:uniqueness}\n\n\nIn this last section we will prove uniqueness in $\\dot{\\mathcal{F}}^{\\frac12,1}$ of solutions to \\eqref{finalsystem} with initial data of the size given by the constraint in \\eqref{condition}.  In particular the main result of this section is  Theorem \\ref{uniquenessproposition} just below.  To prove this theorem, in Subsection \\ref{sec:unique.length} we prove the required estimates for the differences of the lengths.  Then in Subsection \\ref{subsec:uniq.vort} we prove the estimates on the differences of the vorticity strength.  After that in Subsection \\ref{subsec:uniq.nonlinear} we prove the main estimates on the differences of the non-linear terms.  Lastly in Subsection \\ref{subsec:uniq.proof} we collect all the previous estimates to prove the uniqueness of the solutions to \\eqref{system} as in Theorem \\ref{uniquenessproposition}.\n\n\n\n\n\nThroughout the proof of Theorem \\ref{uniquenessproposition}, we define coefficients that will be used in the rest of this section.\n\n\\begin{defn}\\label{uniquenessimplicitdef}\nWe use the symbol $\\mathcal{E}>0$ to denote any coefficient that is integrable in time and may depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$, recalling \\eqref{max.fcns}, which is bounded and  $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ which is time integrable.\n\nThe symbol $\\mathcal{C}>0$ will denote any coefficient that is bounded and can depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$.  \n\nThe symbol $\\mathcal{C}_{s}>0$ will denote any coefficient that is bounded and can depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$.  \n\nThe symbol $\\mathcal{E}_{s}>0$ will denote any coefficient that may depend upon $\\mathcal{C}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}$.\n\\end{defn}\n\n\n\n\\begin{thm}\n\\label{uniquenessproposition}\nConsider two solutions $\\vartheta_{1}$ and $\\vartheta_{2}$ of \\eqref{system} with the same initial data satisfying the medium size condition, as in Theorem \\ref{thm:global}.   Then these solutions satisfy the following differential inequality\n\\begin{multline}\\label{uniqineq}\n\\frac{d}{dt}\\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big) \n\\\\\n\\leq (\\mathcal{C}+\\mathcal{E})(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}).\n\\end{multline}\n\\end{thm}\n\n\nWith Theorem \\ref{uniquenessproposition}, we can conclude by Gronwall's inequality that for any $T>0$, we have\n\\begin{multline*}\n \\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big) \\Big|_{t=T} \n \\\\\n \\leq \\exp\\left(\\int_{0}^{T} (\\mathcal{C}+\\mathcal{E}) dt\\right)\n \\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big)\\Big|_{t=0}= 0.\n \\end{multline*}\nThis holds since $\\mathcal{E}=\\mathcal{C}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ with \\eqref{max.fcns}, is time integrable by Theorem \\ref{thm:global}.\n\n\nThe rest of this section is devoted to proving Theorem \\ref{uniquenessproposition}.  Given two solutions $\\vartheta_{1}(\\alpha,t)$ and $\\vartheta_{2}(\\alpha,t)$ with initial data $\\vartheta_{1}(\\alpha,0) = \\vartheta_{2}(\\alpha,0)$ that satisfies \\eqref{condition}; their respective evolution equations are given by \\eqref{system} as follows\n$$(\\vartheta_{i})_t(\\alpha)=\\frac{2\\pi}{L_{i}(t)}\\Big(\\mathcal{L}_{i}(\\alpha)+N_{i}(\\alpha)\\Big),$$\nwhere the vorticity terms are denoted by $\\omega_{1}$ and $\\omega_{2}$ respectively. The evolution of $\\vartheta_{1} - \\vartheta_{2}$ is then given by\n\\begin{align}\\notag\n (\\vartheta_{1}-\\vartheta_{2})_t(\\alpha)\n &= \\Big(\\frac{2\\pi}{L_{1}(t)} -\\frac{2\\pi}{L_{2}(t)}\\Big) \\Big(\\mathcal{L}_{1}(\\alpha)+N_{1}(\\alpha)\\Big) + \\frac{2\\pi}{L_{2}(t)}(\\mathcal{L}_{1}(\\alpha) - \\mathcal{L}_{2}(\\alpha))  \\\\&\\hspace{2in}+ \\frac{2\\pi}{L_{2}(t)}(N_{1}(\\alpha) - N_{2}(\\alpha)). \\label{theta1theta2}\n\\end{align}\nUsing the evolution equation \\eqref{theta1theta2}, \nwe will prove \\eqref{uniqineq}. In Proposition \\ref{lengthdifferenceprop}, we give an estimate to control the length difference in the first term on the right hand side of \\eqref{theta1theta2}. By the estimates from Section \\ref{sec:NonLinearEst}, the coefficient, $\\mathcal{L}_{1}(\\alpha)+N_{1}(\\alpha)$, of the length difference in the first term is bounded by $\\mathcal{C}+\\mathcal{E}$.  The bound on the length difference is shown in in Proposition \\ref{lengthdifferenceprop}. The second term on the RHS of \\eqref{theta1theta2} gives a linear coercive estimate for the time evolution. Lastly, Section \\ref{subsec:uniq.nonlinear} is dedicated to controlling the third terms on the RHS of \\eqref{theta1theta2} using the idea of Proposition \\ref{uniquenessprop} and the non-linear estimates as in Section \\ref{sec:NonLinearEst}.\n\n\n\nAdditionally, we will use the following idea repeatedly:\n\\begin{prop}\\label{uniquenessprop}\nConsider two functions $f$ and $g$ in $\\dot{\\mathcal{F}}^{s,1}$ for some $s \\geq 0$.  We also consider some operator $T$. Then, for any $n \\in \\mathbb{N}$ we have\n\\begin{multline}\\label{uniqueoperator}\n\\|   f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} + \\Big(\\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\Big)\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline}\nFor $s>0$, we have\n\\begin{multline}\\label{uniqueoperatorsnorm}\n\\|   f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\ \\leq b(n+1,s)\\Big(\\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\dot{\\mathcal{F}}^{s,1}}  + n\\|f\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} \\\\+  n\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f-g\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}} + n\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\+ n(n-1)\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-2}\\|f,g\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}\\Big),\n\\end{multline}\nwhere we recall the definition \\eqref{max.fcns}.  In the special case where $T = \\frac{d}{d\\alpha^{j}}$ for some $j\\in \\mathbb{N}$, we obtain\n\\begin{multline}\\label{uniquenessestimate}\n\\|   f(\\alpha)^{n}\\frac{d^{j}}{d\\alpha^{j}}f(\\alpha)  - g(\\alpha)^{n} \\frac{d^{j}}{d\\alpha^{j}}g(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|f-g\\|_{\\mathcal{F}^{j,1}} + \\Big(\\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\Big)\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|g\\|_{\\mathcal{F}^{j,1}}.\n\\end{multline}\n\\end{prop}\n\n\\begin{proof}\nSince\n\\begin{align*}\n f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha) &=   f(\\alpha)^{n}Tf(\\alpha)  - f(\\alpha)^{n} Tg(\\alpha) \\\\&\\hspace{0.5in}+  f(\\alpha)^{n}Tg(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\\\ &= f(\\alpha)^{n}(Tf(\\alpha)  -  Tg(\\alpha)) +  (f(\\alpha)^{n}-g(\\alpha)^{n})Tg(\\alpha).\n\\end{align*}\nWe obtain\n\\begin{multline*}\n\\| f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} + \\|f(\\alpha)^{n} - g(\\alpha)^{n}\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline*}\nNext, we have\n\\begin{align*}\n\\|f(\\alpha)^{n} - g(\\alpha)^{n}\\|_{\\mathcal{F}^{0,1}} &= \\Big\\|\\sum_{k=0}^{n-1}f(\\alpha)^{n-k} g(\\alpha)^{k} -f(\\alpha)^{n-k-1} g(\\alpha)^{k+1} \\Big\\|_{\\mathcal{F}^{0,1}}\\\\ &\\leq \\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\|f-g\\|_{\\mathcal{F}^{0,1}}.\n\\end{align*}\nThis yields \\eqref{uniqueoperator}.  Then \\eqref{uniqueoperatorsnorm} is proven similarly.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Estimates for the differences of the lengths}\\label{sec:unique.length}\n\nWe need to control the difference in the lengths $L_{1}(t)$ and $L_{2}(t)$, for example, to control the first term on the right hand side of \\eqref{theta1theta2}. In this section, we prove the following proposition on the differences of the lengths.\n\n\\begin{prop}\\label{lengthdifferenceprop}\nConsider the lengths, $L_{1}(t)$ and $L_{2}(t)$, of two solutions, $\\vartheta_{1}$ and $\\vartheta_{2}$ respectively, to \\eqref{system} as defined by \\eqref{modul}. Then, we have\n\\begin{equation}\\label{lengthdifference}\n\\left|\\frac{L_{1}(t)}{2\\pi}-\\frac{L_{2}(t)}{2\\pi}\\right| \\leq  C_L \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\end{equation}\nwith $C_{L}$ defined by \\eqref{CL}.  \n\\end{prop}\n\n\n\\begin{proof}\nWe recall equation \\eqref{Lequation}.  Thus, denoting $\\Delta f=f(\\alpha)-f(\\eta)$, we have that \n\\begin{multline}\\label{lenghtaux}\n    \\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\n    \\\\\n    =\\frac{\\frac{R^2}{2\\pi}\\Big(\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\\Big)}{\\Big(1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)\\!\\Big(1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)}.\n\\end{multline}\nRecalling the estimates in \\eqref{lengthauxbound} - \\eqref{C37C38}, the denominator is bounded by\n\\begin{multline*}\n\\bigg(\\prod_{m=1}^2\\Big(\\!1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_m)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)\\!\\bigg)^{-1}\n\\\\\n\\leq \n\\bigg(\\prod_{m=1}^2\n\\Big(1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta_m\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)\n\\bigg)^{-1}.\n\\end{multline*}\nFurther the numerator has the upper bound\n\\begin{equation*}\n\\begin{aligned}\n&\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\\\\\n&=\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2-\\Delta\\theta_1)\\sum\\limits_{m=0}\\limits^{n-1}\\big(\\Delta\\theta_1\\big)^m\\big(\\Delta\\theta_2\\big)^{n-m-1} d\\eta d\\alpha\\\\\n&\\leq  \\pi^2  \\sum\\limits_{n\\geq1}\\frac{2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}}{n!}\\sum\\limits_{m=0}\\limits^{n-1}\\big(2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\big)^m\\big(2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}\\big)^{n-m-1}.\n\\end{aligned}\n\\end{equation*}\nThus we further obtain the estimate\n\\begin{equation*}\n\\begin{aligned}\n&\n\\left| \\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\n\\right|\n\\\\\n&\\leq \\frac{2\\pi^2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}-2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}\\sum\\limits_{n\\geq1}\\frac{1}{n!}\\Big(\\big(2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\big)^n-\\big(2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}\\big)^n\\Big)\\\\\n&=2\\pi^2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}\\frac{e^{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}}-e^{2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}-2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}.\n\\end{aligned}\n\\end{equation*}\nWe substitute this back into \\eqref{lenghtaux} to obtain that\n\\begin{equation}\\label{diflength}\n\\begin{aligned}\n\\Big|\\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\\Big|\\leq  R^2 C_{L,2} \\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}},\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\notag\n    \\begin{aligned}\n    C_{L,2}=\n    \\pi\\frac{e^{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}}\\!-\\!e^{2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\!-\\!2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}\n    \\bigg(\\prod_{m=1}^2\n\\Big(1-\\frac\\pi2\\big(e^{2\\|\\theta_m\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)\n\\bigg)^{-1}.\n    \\end{aligned}\n\\end{equation}\nThe estimate \\eqref{diflength} allows to easily bound terms like $L_1(t)-L_2(t)$ or $L_1(t)^{-1}-L_2(t)^{-1}$. In fact, using \\eqref{Lbound}, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\Big|\\frac{L_1(t)}{2\\pi}-\\frac{L_2(t)}{2\\pi}\\Big|=\\frac{1}{\\frac{L_1(t)}{2\\pi}+\\frac{L_2(t)}{2\\pi}}\\Big|\\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\\Big|\\leq  C_L\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}},\n    \\end{aligned}\n\\end{equation*}\nwhere, with $C_{37}$ is defined in \\eqref{C37C38}, we have\n\\begin{equation}\\label{CL}\n    \\begin{aligned}\n    C_{L}=R\\frac{C_{L,2}}{2C_{37}}.\n    \\end{aligned}\n\\end{equation}\nThis completes the proof of \\eqref{lengthdifference}.\n\\end{proof}\n\n\n\n\n\\subsection{Estimates for the differences of the vorticity strength}\\label{subsec:uniq.vort}\nIn this subsection we will estimate the differences of the vorticity strength terms.\nWe use the splitting in \\eqref{omegasplit} as\n\\begin{multline*}\n\\omega_{1}(\\alpha)-\\omega_{2}(\\alpha) \n\\\\\n=\n(\\omega_{1})_{0}(\\alpha)-(\\omega_{2})_{0}(\\alpha) + (\\omega_{1})_{1}(\\alpha)-(\\omega_{2})_{1}(\\alpha) + (\\omega_{1})_{\\geq 2}(\\alpha)-(\\omega_{2})_{\\geq 2}(\\alpha).\n\\end{multline*}\nAbove $(\\omega_{i})_{0}(\\alpha)$ is the zero component, as defined in \\eqref{omegasplit}, of the vorticity term $\\omega_{i}$ for $i=1,2$ etc.  In this section, we prove the following estimates on each difference in the vorticity decomposition.\n\n\\begin{prop}\\label{prop:diff.vorticity.ests}\nFor $s \\ge 0$, we have the estimates\n\\begin{equation}\\label{omega0diffest}\n\\|(\\omega_{1})_{0}-(\\omega_{2})_{0}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq   2 \\left|A_\\rho \\right| C_{L}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\frac{A_\\rho}{\\pi}L_{2}(t)|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|,\n\\end{equation}\nand\n\\begin{multline}\\label{linearomegadiffnorm}\n\\|(\\omega_{1})_{1}-(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq \n\\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)\\\\ + \\mathcal{E} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}  + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{multline}\nFor $s>0$ \nwe further have\n\\begin{multline}\\label{omeganonlinearest}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\tilde{\\Gamma}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\n\\end{multline}\nwhere $\\tilde{\\Gamma}$ is given by \\eqref{tildeGamma}.\n\\end{prop}\n\nWe further give the estimate of the form \\eqref{omeganonlinearest} when $s=0$ in \\eqref{omega.diff.zero.est}.  It is important to notice that $\\tilde{\\Gamma}$ given by \\eqref{tildeGamma} is smaller than the corresponding coefficient of $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s+2,1}}$ in the estimate of \\eqref{omega2fss}.\n\n\\begin{proof}\nFor the zero-th order term in the splitting we have\n\\begin{align*}\n(\\omega_{1})_{0}(\\alpha)-(\\omega_{2})_{0}(\\alpha) &= -2A_\\rho\\Big(\\frac{L_{1}(t)}{2\\pi}-\\frac{L_{2}(t)}{2\\pi} \\Big)\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\\\ &\\hspace{0.5in}- 2A_\\rho\\frac{L_{2}(t)}{2\\pi}(\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}).\n\\end{align*}\nHence, for $s \\ge 0$, \nwe have the estimate\n\\begin{align*}\n\\|(\\omega_{1})_{0}-(\\omega_{2})_{0}\\|_{\\dot{\\mathcal{F}}^{s,1}} &\\leq \\frac{\\left| A_\\rho \\right| }{\\pi} |L_{1}(t)-L_{2}(t)|\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\&\\hspace{0.25in}+  \\frac{\\left| A_\\rho \\right|}{\\pi}L_{2}(t)\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{align*}\nWe have\n$\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 1$\nand\n\\begin{multline*}\n\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 2\\Big|\\sin\\Big(\\frac{\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big|\\\\ \\cdot \\Big\\|\\cos\\Big(\\alpha +\n\\frac{\\widehat{\\vartheta}_{1}(0)+\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{multline*}\nWe have \n$\\Big\\|\\cos\\Big(\\alpha +\\frac{\\widehat{\\vartheta}_{1}(0)+\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 1$\nand since we assume the difference is small we have\n\\begin{equation}\\label{sinezero}\n\\Big|\\sin\\Big(\\frac{\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big| \\leq \\frac{1}{2} |\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|.\n\\end{equation}\nHence, we obtain \\eqref{omega0diffest}, which completes the difference estimates for the zero order vorticity strength terms.\n\nNext, the linear difference in the vorticity strength terms from \\eqref{omegasplit} is\n\\begin{equation}\\label{linearomegadiff}\n(\\omega_{1})_{1}(\\alpha)-(\\omega_{2})_{1}(\\alpha) = \\frac{A_\\mu}{\\pi}W_{1} + 4A_\\sigma\\pi W_{2} -\\frac{A_\\rho}{\\pi}  W_{3}\n\\end{equation}\nwhere\n\\begin{equation}\\notag\nW_{1} =L_{1}(t)\\mathcal{D}_1((\\omega_{1})_0)(\\alpha) - L_{2}(t)\\mathcal{D}_1((\\omega_{2})_0)(\\alpha)\n\\end{equation}\nand\n\\begin{equation}\\notag\nW_{2} =\\Big(\\frac{1}{L_{1}(t)} -\\frac{1}{L_{2}(t)}\\Big) (\\theta_{1})_{\\alpha\\alpha} + \\frac{1}{L_{2}(t)}((\\theta_{1})_{\\alpha\\alpha} - (\\theta_{2})_{\\alpha\\alpha})\n\\end{equation}\nand\n\\begin{multline}\\notag\nW_{3} =(L_{1}(t)-L_{2}(t))\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\theta_{1}(\\alpha) \\\\+ L_{2}(t)[\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}]\\theta_{1}(\\alpha)\\\\ + L_{2}(t)\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}[\\theta_{1}(\\alpha)-\\theta_{2}(\\alpha)].\n\\end{multline}\nWe will estimate each of the terms  $W_{1}$,  $W_{2}$ and  $W_{3}$ in the following.\n\nFor $W_{1}$ we have using \\eqref{mdsplit} that\n\\begin{multline*}\nW_{1} = \\theta_{1}(\\alpha)\\mathcal{H}  ( (\\omega_{1})_0)(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}((\\omega_{1})_0)(\\alpha)- \\theta_{2}(\\alpha)\\mathcal{H}  ((\\omega_{2})_0)(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}((\\omega_{2})_0)(\\alpha).\n\\end{multline*}\nIt can be shown by the estimates in $\\mathcal{R}$ and $L_{1}-L_{2}$ that for $s \\ge 0$ we have\n\\begin{equation}\\label{D1est}\n\\|W_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + |\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+ \\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nFor $W_{2}$, we have\n\\begin{equation}\\label{W2est}\n\\|W_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\frac{1}{L_{1}(t)L_{2}(t)}C_{L}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{s+2,1}}+ \\frac{1}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{equation}\nThe important term in $W_{2}$ for the purposes of the uniqueness argument is the second term that has the difference $\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}$.  For $W_{3}$, using \\eqref{lengthdifference}, we obtain that\n\\begin{equation}\\label{W3est2}\n    \\|W_{3}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+ \\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nHence, from \\eqref{D1est}, \\eqref{W2est} and \\eqref{W3est2}, we obtain from \\eqref{linearomegadiff} that\n\\begin{multline*}\n\\|(\\omega_{1})_{1}-(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq \n\\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)\\\\ + \\mathcal{E} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}  + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{multline*}\nNote that the coefficient in front of $\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}$ is the same as that in \\eqref{omega1fss} in front of $\\|\\theta\\|_{\\mathcal{F}^{s+2,1}_{\\nu}}$.\nThis completes the difference estimates for the linear terms in the vorticity strength given by \\eqref{linearomegadiffnorm}.\n\nNext we estimate differences of the nonlinear terms in the vorticity strength from \\eqref{omegasplit}.  We decompose the terms as\n\\begin{equation}\\label{omegadiff2}\n(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2} = W_{21} + W_{22} + W_{23}\n\\end{equation}\nwhere the terms $W_{21}$, $W_{22}$, and $W_{23}$ are given by\n\\begin{equation}\\notag\nW_{21} = A_\\mu\\frac{L_{1}(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega_{1})(\\alpha) - A_\\mu\\frac{L_{2}(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega_{2})(\\alpha),\n\\end{equation}\nand\n\\begin{multline}\\notag\nW_{22} = A_\\rho\\frac{L_{2}(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\widehat{\\vartheta}_{2}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta_{2}(\\alpha))^{2j}}{(2j)!}\\\\ - A_\\rho\\frac{L_{1}(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\widehat{\\vartheta}_{1}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta_{1}(\\alpha))^{2j}}{(2j)!},\n\\end{multline}\nand\n\\begin{multline}\\notag\nW_{23} = A_\\rho\\frac{L_{2}(t)}{\\pi}\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta_{2}(\\alpha))^{1+2j}}{(1+2j)!}\\\\-A_\\rho\\frac{L_{1}(t)}{\\pi}\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta_{1}(\\alpha))^{1+2j}}{(1+2j)!}.\n\\end{multline}\nFirst, we use \\eqref{mdsplit} to observe that\n\\begin{multline}\\label{W21expansion}\nW_{21} = -A_{\\mu}\\Big(\\theta_{1}(\\alpha)\\mathcal{H}  ( (\\omega_{1})_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}((\\omega_{1})_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega_{1})(\\alpha)\\\\ -\\theta_{2}(\\alpha)\\mathcal{H}  ( (\\omega_{2})_{\\geq1})(\\alpha)-\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}((\\omega_{2})_{\\geq1})(\\alpha)-\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega_{2})(\\alpha) \\Big)\\\\\n= -A_{\\mu}(W_{211}+W_{212}+W_{213}),\n\\end{multline}\nwhere the differences of like terms with either subscript $1$ or $2$ are combined in each $W_{21j}$. Then for $s \\ge 0$ we have\n\\begin{multline}\\notag\n\\|W_{211}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq b(2,s) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} + b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{1} -(\\omega_{2})_{1}\\|_{\\mathcal{F}^{0,1}} +  b(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{1} -(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\ + b(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} +b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline}\nThus, using \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, \\eqref{omega2fss}  and \\eqref{linearomegadiffnorm} we have\n\\begin{multline}\\label{W211est}\n\\|W_{211}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq\n\\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|) +\\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\\\\n+\\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}} \n+\n\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2+s,1}} \n\\\\\n+ \nb(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n+b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}},\n\\end{multline}\nwhere $\\mathcal{E}_{s} = \\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\\mathcal{E}$  and some bounded constant $\\mathcal{E}$ and $C$ is a bounded constant depending on $\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}$.\n\nWe now consider the term $W_{212}$.  We have from \\eqref{Restimates} that\n\\begin{multline}\\label{Rdiffgeq1}\n\\|\\mathcal{R}((\\omega_{1})_{\\geq 1}) - \\mathcal{R}((\\omega_{2})_{\\geq 1})\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ \\leq b(2,s){C_{\\mR}}(\\|(\\omega_{1})_{\\geq 1}-(\\omega_{2})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}} + \\|(\\omega_{1})_{\\geq 1}-(\\omega_{2})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\+\\|(\\omega_{2})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+\\|(\\omega_{2})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}).\n\\end{multline}\nHence, using \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, \\eqref{omega2fss} and \\eqref{linearomegadiffnorm}, we obtain\n\\begin{multline}\\label{W212est}\n\\|W_{212}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq  \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|) +\\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\\n+ \\mathcal{E}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\n+\\frac{4A_{\\sigma}{C_{\\mR}} \\pi}{L_{2}(t)}b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2+s,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\\\ \n+ \nb(2,s){C_{\\mR}} (\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}}).\n\\end{multline}\nThis is the estimate for $W_{212}$.\n\n\n\nNext for $W_{213}$ containing the difference in $\\mathcal{S}$, we actually have to consider two differences.\nWe recall the splitting of $\\mathcal{S}$ from \\eqref{Ssum} and we will use $f_i = \\omega_i$ below. First, it can be shown from  \\eqref{tildeS} that\n\\begin{align}\\label{Boperatordiff}\n\\|{\\mathcal{B}}(f_{1}) - {\\mathcal{B}}(f_{2})\\|_{\\dot{\\mathcal{F}}^{s,1}}\n&\\leq  B_{1} + B_{2}\n\\end{align}\nwhere\n\\begin{equation}\\notag\nB_{1} = \\Big\\||k|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\widehat{\\theta}_{1}(k))-\\ast^{l}\\widehat{\\theta_{2}}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f_{1})}(k) \\Big\\|_{\\ell^{1}}\n\\end{equation}\nand\n\\begin{equation}\\notag\nB_{2} = \\Big\\||k|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^{n} i^{l+n+1}\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast (\\widehat{\\mathcal{S}_{n}(f_{1})}(k)-\\widehat{\\mathcal{S}_{n}(f_{2})}(k)) \\Big\\|_{\\ell^{1}}.\n\\end{equation}\nFor $B_{1}$, we obtain using similar arguments to Proposition \\ref{uniquenessprop} that\n\\begin{multline}\\notag\nB_{1} \\leq \n\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{b(l+1,s)}{(l-1)!} \\Big( \n \\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-1}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n \\\\\n+ \n(l-1)\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-2}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\mathcal{F}^{0,1}}\n\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\\\ \n+ \\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-1}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big).\n\\end{multline}\nFor $B_{2}$, we first consider the difference in the operator $\\mathcal{S}_{n}$. By \\eqref{Snfourier}, we have for two functions $f_{1}$ and $f_{2}$, with $a_{n}$ given by \\eqref{an.def}, that\n\\begin{align*}\n|i^{n}(\\widehat{\\mathcal{S}_{n}(f_{1})}-\\widehat{\\mathcal{S}_{n}(f_{2})})(k)| &= \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}|I(k_{1},\\ldots,k_{n+1})|\\\\\n&\\cdot\\Big(|\\hat{f_{1}}(k_{n+1})\\prod_{j=1}^{n}P_{1}(k_{j}-k_{j+1}) - \\hat{f_{2}}(k_{n+1})\\prod_{j=1}^{n}P_{2}(k_{j}-k_{j+1})|\\Big)\\\\\n&\\leq a_{n} |\\hat{f}_{1}-\\hat{f}_{2}|\\ast^{n} |P_{1}| + a_{n} |\\hat{f}_{2}|\\ast|\\ast^{n} P_{1}-\\ast^{n} P_{2}|.\n\\end{align*}\nHence using the estimates as in \\eqref{S1S2S3} we have\n\\begin{align}\\label{B2est}\n\\begin{split}\nB_{2} &\\leq \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} a_{n} \\Big\\||k|^{s}\\frac{\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast(|\\hat{f}_{1}-\\hat{f}_{2}|\\ast^{n} |P_{1}|) \\Big\\|_{\\ell^{1}} + \\tilde{B}_{2}\\\\\n&\\leq \\tilde{C}_3\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^2\\|f_{1}-f_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}+\\tilde{C}_4\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|f_{1}-f_{2}\\|_{\\mathcal{F}^{0,1}} + \\tilde{B}_{2}\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\\notag\n\\tilde{B}_{2} = \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} a_{n} \\Big\\||k|^{s}\\frac{\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast(|\\hat{f}_{2}|\\ast|\\ast^{n} P_{1}-\\ast^{n} P_{2}|) \\Big\\|_{\\ell^{1}}.\n\\end{align}\nWe have for $s > 0$ that\n\\begin{multline}\\notag\n\\||k|^{s}(P_{1}-P_{2})\\|_{\\ell^{1}} \\leq \\sum_{m\\geq 1} \\frac{b(m,s)}{(m-1)!}(\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{m-1}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + (m-1)\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{m-2}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}})\n\\end{multline}\nwith an analogous estimate holding in the case $s=0$.\nHence we have\n\\begin{multline}\\notag\n\\tilde{B}_{2} \\leq \\mathcal{E}\\Big[(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}+\\|f_{i}\\|_{\\dot{\\mathcal{F}}^{s,1}})\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + (\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+\\|f_{i}\\|_{\\mathcal{F}^{0,1}})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big].\n\\end{multline}\nWe also similarly estimate ${\\mathcal{A}}$ as in \\eqref{breveSsbound} to obtain\n\\begin{multline}\\notag\n\\|{\\mathcal{A}}(f_{1})(\\alpha) -\t{\\mathcal{A}}(f_{2})(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}\\Big[\\|f_{i}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\|f_{i}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big]\\\\ + \\mathcal{C}_{\\mathcal{R}} C_{4}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|f_{1}-f_{2}\\|_{\\mathcal{F}^{0,1}} + \\mathcal{C}_{\\mathcal{R}} C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\|f_{1}-f_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{multline}\nIn summary, for $s>0$, using \\eqref{omega0diffest} and \\eqref{linearomegadiffnorm}, we obtain\n\\begin{multline}\\label{W213est}\n    \\|W_{213}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|)+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\\\\ + C_{4}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{1})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}} + C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}},\n\\end{multline}\nwhere $C_{3}$ and $C_{4}$ are given by \\eqref{C3} and ${C_{\\mR}}$ is given by \\eqref{CR}.\n\n\n\nFurther using Proposition \\ref{uniquenessprop}, then $W_{22}$ and $W_{23}$ can be estimated by a bound of the type:\n\\begin{equation}\\label{W22W23}\n\\|W_{2j}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+\\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{equation}\nfor $j=2,3$.\nHence, using \\eqref{W211est}, \\eqref{W212est}, \\eqref{W213est}, we obtain from \\eqref{omegadiff2} that\n\\begin{multline}\\label{omeganonlinearestinitial}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\tilde{\\Gamma}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}} \\\\\n+|A_{\\mu}|C_{12}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}} +|A_{\\mu}|C_{2}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{multline}\nwhere\n\\begin{equation}\\label{tildeGamma}\n\\tilde{\\Gamma} =|A_{\\mu}| \\frac{4A_{\\sigma}\\pi}{L_{2}(t)} C_{2}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\end{equation}\nwith $C_{2}$ given by \\eqref{C12}.\n\nComputing an estimate analogous to \\eqref{omeganonlinearestinitial} for $s=0$ yields the estimate\n\\begin{multline}\\label{omega.diff.zero.est}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\Gamma\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\n\\end{multline}\nwhere\n\\begin{equation}\\label{Gamma}\n\\Gamma = |A_{\\mu}|\\tilde{C}_{8}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\n\\end{equation}\nfor $\\tilde{C}_{8}$ given by \\eqref{tildeC8} and the other constants given by Definition \\ref{uniquenessimplicitdef}. \n\\end{proof}\n\n\n\n\\subsection{Estimates for the differences of the main nonlinear term}\\label{subsec:uniq.nonlinear}\nIn this section, we show the following bound on the nonlinear difference term of \\eqref{theta1theta2}.\n\n\\begin{prop}\\label{nonlineardiffprop}\nWe have the following estimate for $\\delta>0$ given by \\eqref{delta} and for $\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ that can be chosen to be arbitrarily small:\n\\begin{equation*}\n\\|N_{1}-N_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|) + (\\delta+\\epsilon) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\n\\end{equation*}\nwhere ${\\mathcal{E}}>0$ is a time integrable coefficient depending on $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ and $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  Further $\\epsilon = \\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})>0$ can be chosen arbitrarily small.\n\\end{prop}\n\nWe remark that all the terms involving the upper bound of ${\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|)$ follow similarly to the estimates from Section \\ref{secanalytic} also using the idea of Proposition \\ref{uniquenessprop} and the vorticity estimates in Proposition \\ref{prop:diff.vorticity.ests}.  Our proof below is focused on the estimates of the term $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ and the constant $\\delta>0$. In the proof of Proposition \\ref{nonlineardiffprop}, when we compute the difference of nonlinear terms in $\\dot{\\mathcal{F}}^{\\frac12,1}$, any terms where the difference occurs as $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $s<7\/2$ can be absorbed by interpolation and the Young's inequality into the term ${\\mathcal{E}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ and contributes an arbitrarily small term $\\epsilon \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ to be absorbed into the linear decay, e.g. see the estimate to obtain \\eqref{samplediff1}. This term is then taken care of by the Gronwall argument described in the comment below Theorem \\ref{uniquenessproposition}. In this way, most of the nonlinear terms can be easily estimated and only the few terms of order $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ need be computed.\n\n\\begin{proof}\nFor the nonlinear terms, we will denote the decomposition given in \\eqref{N1N2N3} of $N_1$ for $\\theta_{1}$ and $N_2$ for $\\theta_{2}$ respectively by \n$$ N_{1} = N_{11}+N_{12}+N_{13},\n\\quad\nN_{2} = N_{21}+N_{22}+N_{23}.$$\nWe now consider the differences of $N_{1j}- N_{2j}$ for $j=1,2,3$. We will only explicitly compute the constant in front of terms with difference $\\| \\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  We make this idea clear in the following. Denoting $(U_{i})_{\\geq 2}$ for the term containing $\\theta_{i}$, we have\n\\begin{multline}\\label{N1diff}\n\\|N_{11}-N_{21}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq \\Big|\\frac{\\pi}{L_{1}(t)}-\\frac{\\pi}{L_{2}(t)}\\Big|\\|\\frac{L_{1}(t)}{\\pi}(U_{1})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\\\+ \\frac{1}{L_{2}(t)}\\|\\frac{L_{1}(t)}{\\pi}(U_{1})_{\\geq 2}-\\frac{L_{2}(t)}{\\pi}(U_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}.\n\\end{multline}\nIn the latter term of \\eqref{N1diff}, we have a term of the form\n\\begin{multline}\\label{Romegadiff}\n    \\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 1})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 1})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \n    \\\\\n    \\leq \\ldots + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\n    + \\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 2})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}.\n\\end{multline}\nAbove the dots ``$\\ldots$'' represent that other terms are present which turn out to be lower order.  We similarly denote $\\mathcal{R}_{i}$ for the term \\eqref{R} which contains $\\theta_{i}$.   Then for the first term present in the upper bound above we have the estimate\n\\begin{align}\\notag\n\\begin{split}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &\\leq  \\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha}-(\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\ &\\hspace{0.5in}+ \\|\\mathcal{R}_{1}((\\theta_{2})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\\n\\leq \\sqrt{2}{C_{\\mR}}&(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}} + \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\ &+\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\|\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}).\n\\end{split}\n\\end{align}\nWe therefore have\n\\begin{multline}\\label{r1r2diffaa}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\n\\\\\n\\leq \\sqrt{2}{C_{\\mR}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}+\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2})\n\\\\\n+\\sqrt{2}{C_{\\mR}}(\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2} \\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3}).\n\\end{multline}\nHence, if we apply Young's inequality, e.g.\n\\begin{multline*}\n\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2}\\\\ \\leq \\frac{1}{4\\epsilon}\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{2\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + \\epsilon \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{4\/3} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\n\\end{multline*}\nwe obtain\n\\begin{multline}\\label{samplediff1}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\\\ \\leq \\mathcal{E}(\\|\\theta_{1},\\theta_{2}\\|_{_{\\dot{\\mathcal{F}}^{\\frac72,1}}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +c_{\\epsilon}(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{2\/3}+\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{3\/4})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}) \\\\+\\epsilon (\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{4\/3}+\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{3\/2})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}+ \\sqrt{2}{C_{\\mR}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}\n\\end{multline}\nfor a constant $c_{\\epsilon}$. The first two terms in \\eqref{samplediff1} are linear in $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $0 \\leq s \\leq 1\/2$ and the constants are time integrable on $[0,T]$ for any $T>0$ since $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ is integrable in time. The final two terms with the difference $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ needs to be absorbed in the linear decay coming from $\\mathcal{L}_{1}-\\mathcal{L}_{2}$. For the other term in \\eqref{Romegadiff}, we have similarly,\n\\begin{multline}\n\\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 2})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\leq \\ldots + \\sqrt{2}{C_{\\mR}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\Gamma\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\\\ + \\epsilon\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}},\n\\end{multline}\nwhere we use \\eqref{Gamma} and $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small.\n\nThe only other terms containing a term like $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ are the other two terms that also come from $N_{11}-N_{12}$, as can be observed from the terms which contain $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ in the estimates of Section \\ref{secanalytic}. The first term is\n\\begin{align}\n\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{H}  ( (\\omega_{1})_{\\geq2} )-\\mathcal{H}  ((\\omega_{2})_{\\geq2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &= \\frac{\\pi}{L_{2}(t)}\\| (\\omega_{1})_{\\geq2}-(\\omega_{1})_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber \\\\\n&\\leq \\ldots + (\\frac{\\pi\\Gamma}{L_{2}(t)} + \\epsilon)\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}, \\label{Hdiff}\n\\end{align}\nwhere the unwritten terms are lower order due to being linear in $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $0 \\leq s \\leq 1\/2$ with time integrable coefficients as done in \\eqref{samplediff1}. Again, $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small. Similarly, the final term that we need to compute is\n\\begin{align}\n\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}(\\omega_{1})-\\mathcal{S}(\\omega_{2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &\\leq \\ldots +\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}((\\omega_{1})_{1})-\\mathcal{S}((\\omega_{2})_{1})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber\\\\\n&\\hspace{0.5in}+\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}((\\omega_{1})_{\\geq 2})-\\mathcal{S}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber \\\\\n\\leq \\ldots& + (\\frac{\\pi\\Gamma C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}}{L_{2}(t)}+\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}))\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}, \\label{Sneededdiff}\n\\end{align}\nwhere we use the estimate on $\\mathcal{S}(f_{1})-\\mathcal{S}(f_{2})$ computed to give \\eqref{W213est} and so $C_{3}$ is given by \\eqref{C3} and $\\Gamma$ is given by \\eqref{Gamma}.\n\n\nAll remaining nonlinear terms in the estimates of the evolution of $\\theta_{1}-\\theta_{2}$ are lower order due to having no futher upper bounds in terms of the highest order difference $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  Hence, in total the difference of nonlinear terms yields\n\\begin{equation}\\notag\n\\|N_{1}-N_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|) + (\\tilde{\\delta}+\\epsilon) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}},\n\\end{equation}\nwhere $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small; and for $\\Gamma= |A_{\\mu}|\\tilde{C}_{8}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}$ given by \\eqref{Gamma}, we have $\\tilde{\\delta}$ given by\n\\begin{equation*}\n\\tilde{\\delta} = \\frac{\\pi}{L_{2}(t)}(\\sqrt{2}{C_{\\mR}}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)} + b(2,3\/2){C_{\\mR}}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\Gamma + \\Gamma + \\Gamma C_{3}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2})\n\\end{equation*}\nand so by \\eqref{Lbound}, setting\n\\begin{multline}\\label{delta}\n\\delta =\n\\\\\n\\frac{A_{\\sigma}}{C_{37}^{2}R^{2}}(\\sqrt{2}{C_{\\mR}}+ b(2,3\/2){C_{\\mR}}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}|A_{\\mu}|\\tilde{C}_{8}+ |A_{\\mu}|\\tilde{C}_{8} +|A_{\\mu}|\\tilde{C}_{8} C_{3}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}),\n\\end{multline}\nwe obtain the result of Proposition \\ref{nonlineardiffprop}.\n\\end{proof}\n\n\\subsection{Proof of uniqueness}\\label{subsec:uniq.proof}\nWe are now ready to prove Theorem \\ref{uniquenessproposition}. \n\n \n\n\\begin{proof}[Proof of Theorem \\ref{uniquenessproposition}]\nFrom Proposition \\ref{linearfourier}, the difference of the linear terms is\n\\begin{multline}\\label{lineardifference}\n\\widehat{\\mathcal{L}}_{1}(k) - \\widehat{\\mathcal{L}}_{2}(k) \n=-A_\\sigma \\frac{4\\pi^2}{L_{2}(t)^2}k(k^2-1)(\\widehat{\\theta}_{1}(k)- \\widehat{\\theta}_{2}(k))\n\\\\\n-(1+A_\\mu)A_\\rho\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}_{2}(0)} (\\widehat{\\theta}_{1}(k+1)- \\widehat{\\theta}_{2}(k+1)) \n\\\\\n+ \\tilde{L}_{1}(k) \n+ \\tilde{L}_{2}(k),\n\\end{multline}\nwhere\n\\begin{equation*}\n\\tilde{L}_{1}(k) = -4\\pi^2A_\\sigma k(k^2-1)\\widehat{\\theta}_{1}(k)\\Big(\\frac{1}{L_{1}(t)^2}-\\frac{1}{L_{2}(t)^2}\\Big)\n\\end{equation*}\nand \n\\begin{equation*}\n\\tilde{L}_{2}(k) = -(1+A_\\mu)A_\\rho\\frac{(k^2-1)(k+1)}{k(k+2)}(e^{-i\\widehat{\\vartheta}_{1}(0)}-e^{-i\\widehat{\\vartheta}_{2}(0)})\\widehat{\\theta}_{1}(k+1).\n\\end{equation*}\nAnd similarly for $k=2$.\nFor $\\tilde{L}_{1}$, \nwe have\n\\begin{equation}\\label{L1unique}\n\\|\\tilde{L}_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\leq 4\\pi^{2}A_\\sigma \\Big|\\frac{1}{L_{1}(t)^{2}} - \\frac{1}{L_{2}(t)^{2}}\\Big|\\||k|^{3\/2}(k^{2}-1)\\widehat{\\theta}_{1}(k)\\|_{\\ell^{1}}.\n\\end{equation}\nFor $\\tilde{L}_{2}$, we have\n\\begin{multline}\\label{L2unique}\n\\|\\tilde{L}_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\n\\\\\n\\leq |(1+A_\\mu)A_\\rho| |e^{-i\\widehat{\\vartheta}_{1}(0)}-e^{-i\\widehat{\\vartheta}_{2}(0)}|\\Big\\|\\frac{|k|^{1\/2}|k^2-1||k+1|}{2|k||k+2|}|\\widehat{\\theta}_{1}(k+1)|  \\Big\\|_{\\ell^{1}}.\n\\end{multline}\nUsing similar arguments as earlier, we obtain from \\eqref{L1unique} and \\eqref{L2unique} that\n\\begin{equation}\\label{linearnonlinearparts}\n\\|\\tilde{L}_{1},\\tilde{L}_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|)\n\\end{equation}\nwhere ${\\mathcal{E}}$ is a time integrable coefficient. Hence, the new quantities from \\eqref{linearnonlinearparts} do not need to be absorbed by the linear decay. The coefficient $\\delta$ of the norm $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ is less than the coefficients in \\eqref{Nbound}, and hence, $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ satisfying \\eqref{condition} and taking $\\epsilon$ arbitrarily small is sufficient for $(\\delta+\\epsilon)\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ from Proposition \\ref{nonlineardiffprop} to be absorbed into the linear decay terms of \\eqref{lineardifference} by following the similar procedure to Section \\ref{subsecGlobal}.\n\\end{proof}\n\n\n\n\n\\subsection*{Acknowledgements}\nFG and EGJ were partially supported by the grant MTM2017-89976-P (Spain). FG, EGJ and NP were partially supported by the ERC through the Starting Grant project H2020-EU.1.1.-639227. RMS was partially supported by the NSF grant DMS-1764177 (USA).\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nApplications of random matrix theory cover many\nbranches of physics and cross-disciplinary\nfields~\\cite{RMTGENERAL} involving multivariate analysis of large\nand noisy data sets~\\cite{MULTIVARIATE}. The standard random matrix\nformulation belongs to the Gaussian basin, with a measure that is\nGaussian or polynomial with finite second moment. The ensuing\nmacroscopic spectral distribution is localized with finite supports\non the real axis. The canonical distribution for a Gaussian measure\nis Wigner's semi-circle.\n\nThe class of stable (L\\'{e}vy) distributions~\\cite{LEVY} is however much\nlarger (the Gaussian class represents only one fixed point in the\nstability basin of the L\\'{e}vy class), and one is tempted to ask, why\nthe theory of random L\\'{e}vy matrices is not so well established.\nThe case of L\\'{e}vy randomness is far from being academic, and many\ndistributions in physics and outside (finance, networks) exhibit\npower-like behavior referred to as {\\it fat} or\n{\\it heavy tails}~\\cite{FATTAIL}.\n\nOne of the chief reasons for why the theory of\nrandom L\\'{e}vy matrices is not yet well understood is the\ntechnical difficulty inherent to these distributions. First,\neven for one-dimensional stable distributions,\nthe explicit form of the probability distribution\nfunctions (pdf) is known analytically only in\nfew cases~\\cite{FELLER,ZOLOTARIEV}.\nSecond, L\\'{e}vy distributions have divergent moments, and the\nfiniteness of the second moment (condition for the Gaussian stability\nclass) is usually a key for many of the techniques established in\nRandom Matrix Theory. Third, numerical studies involving\npower-like behavior on infinite supports require enormous statistics\nand are very sensitive to systematic errors.\n\nIn 1994, Bouchaud and Cizeau~\\cite{BC} (hereafter BC)\nconsidered large $N\\times N$  random symmetric\nmatrices with entries sampled from one-dimensional, stable\ndistributions. In the large $N$ limit they obtained analytical\nequations for the entries of the resolvent, and then\nchecked their predictions for the spectra using numerically\ngenerated spectra by random sampling. The agreement was fair, although not\nperfect. Contrary to the standard Gaussian-like ensembles, the\nmeasure in the BC approach was not rotationally invariant.\n\nIn 2002, following the work in~\\cite{BERVOIC}, we have suggested\nanother L\\'{e}vy-type ensemble~\\cite{FREELEVY} (hereafter FRL).\nBy construction its measure is rotationally invariant.\nThe average spectral distribution in this ensemble is stable\nunder matrix the convolution of two independent but identical\nensembles. It is similar to the stability property of \none-dimensional L\\'{e}vy distributions. \nThe measure is non-analytic in the matrix $H$ and\nuniversal at large $H$ with a potential $V(H)\\approx {\\rm ln}\\,H^2$. This\nweak logarithmic rise in the asymptotic potential is at the origin\nof the long tail in the eigenvalue spectra. \n\nThe present work will compare the WL and FRL results as advertised\nin~\\cite{KRACOW}. In section~2 we reanalyze and correct the original\narguments for the resolvent presented in~\\cite{BC}. Our integral\nequations for the resolvent and spectral density are different from\nthe ones in~\\cite{BC}. We carry explicit analytical\ntransformations and expansions to provide insights to the spectrum.\nWe show that there is a perfect agreement between the analytical and\nnumerical results obtained by sampling large L\\'{e}vy matrices. \nWe also discuss the relation of ours and BC's results. \nIn section~3 we recall the key concepts behind FRL ensembles. In large\n$N$ the resolvent obeys a simple analytic equation. The resulting\nspectra are compared to the spectra following from the corrected BC\nanalysis. \nThe WL and FRL matrices represent two types of stability under\nmatrix convolution. In both cases we have a power behavior in the\ntails of the spectrum. By a pertinent rescaling we may in fact\nenforce the same tail behavior and compare the spectra. The observed\ndifferences disappear in the Gauss limit and become more pronounced\nin the Cauchy limit. In sectin 4 we explain the relation between \nthe two types of stability on a simple example: we construct sums \nof WL matrices rotated by random $O(N)$ matrices and show that \nthe spectrum of these sums converges by a matrix central \nlimit theorem to the pertinent symmetric FRL spectrum. Our \nconclusions are in section 5.\n\n\\section{Wigner-L\\'{e}vy Matrices}\n\n\\subsection*{Definition of the ensemble}\nIn a pioneering study on random L\\'{e}vy matrices, Bouchaud and\nCizeau~\\cite{BC} discussed a Wigner ensemble of $N\\times N$\nreal symmetric random matrices with elements being iid random variables:\nwith probability density function following a L\\'{e}vy\ndistribution $P(x)\\equiv N^{1\/\\mu} L_\\mu^{C,\\beta}\\left(N^{1\/\\mu} x\\right)$,\nwith $\\mu$ being the stability index, $\\beta$ -- the asymmetry parameter,\nand $C$ -- the range of the distribution (see below). We shall\ncall these matrices Wigner-L\\'evy (WL) or Bouchaud-Cizeau (BC) matrices.\nThe probability measure for the ensemble of such matrices \nis given by:\n\\begin{eqnarray}\nd\\mu_{WL}(H) =\\prod_{i\\le j}\\,P(H_{ij})\\,dH_{ij}\n\\label{measure}\n\\end{eqnarray}\nThe scaling factor $N^{1\/\\mu}$ in pdf makes the\nlimiting eigenvalue density independent of the matrix \nsize $N$ when $N\\rightarrow \\infty$. \nAlternatively one can think of the matrix elements\n$H_{ij}$ as if they were calculated as \n$H_{ij} = h_{ij}\/N^{1\/\\mu}$ with $h_{ij}$ being \niid random numbers independent of $N$:\n$p(x) = L_\\mu^{C,\\beta}(x)$.\n\nL\\'{e}vy distributions are notoriously hard to write explicitly\n(except in few cases), but their characteristic functions are more\nuser friendly~\\cite{FELLER}\n\\begin{eqnarray}\nL_{\\mu}^{C,\\,\n\\beta}(x)=\\frac{1}{2\\pi} \\int dk \\hat{L}(k)e^{ikx}\n\\label{levydef}\n \\end{eqnarray}\nwhere  the characteristic function is given by\n\\begin{eqnarray}\n\\log \\hat{L}(k)= -C|k|^{\\mu} (1+i \\beta ~{\\rm sign}(k) ~\\tan\n(\\pi \\mu \/2)).\n\\label{logchar}\n\\end{eqnarray}\nThe parameters $\\mu$, $\\beta$ and $C$ are related\nto the asymptotic behavior of $L_{\\mu}^{C,\\beta}(x)$\n\\begin{eqnarray}\n \\lim_{x \\rightarrow \\pm \\infty}  L_{\\mu}^{C ,\\beta}(x)=\n\\gamma(\\mu) \\frac{C (1\\pm \\beta)}\n{|x|^{\\mu+1}}\n\\label{levy1}\n\\end{eqnarray}\nwith  the $\\mu$-dependent parameter $\\gamma(\\mu)$ given by\n\\begin{eqnarray}\n\\gamma(\\mu)=\\Gamma(1+\\mu)\\sin(\\frac{\\pi\\mu}{2})\n\\end{eqnarray}\nHere $\\mu$ is the stability index defined in the interval $(0,2]$,\n$-1 \\le\\beta\\le 1$ measures the asymmetry of the distribution\nand the range $C > 0$ is the analogue of the variance, in a sense\nthat a typical value of $x$ is $C^{1\/\\mu}$. A standard choice\ncorresponds to $C=1$.\n\nWe shall consider here only the stability index in the range $(1,2)$,\nalthough as will be shown later, results obtained in this range seem to be\nvalid also for $\\mu=1$. We also assume that all random variables have zero\nmean.\n\n\\subsection*{Determination of the eigenvalue density}\n\nA method of calculating the eigenvalue density of the Wigner-L\\'evy\nmatrices was invented by Bouchaud and Cizeau \\cite{BC}. Let us \nin this section briefly recall the main steps of the method. \nIt is convenient to introduce the resolvent, called also Green's \nfunction:\n\\begin{eqnarray}\ng(z) =\\frac{1}{N}\\langle {\\rm Tr} \\; G(z) \\rangle\n\\end{eqnarray}\nwhere elements of the matrix $G(z)$ are\n\\begin{eqnarray} G_{ij}(z)=\n(z-H)_{ij}^{-1}\n\\label{resolvent}\n\\end{eqnarray}\nand the averaging is carried out using the measure (\\ref{measure}).\nThe resolvent contains the same information as the eigenvalue density \n$\\rho(\\lambda)$. Indeed if one approaches the real axis one finds\nthat $\\rho(\\lambda) = -1\/\\pi \\lim_{\\epsilon \\rightarrow 0^+}\n{\\rm Im} \\; g(\\lambda + i\\epsilon)$. This is\nhow one usually calculates $\\rho(\\lambda)$ from $g(z)$. \nFor the Wigner-L\\'evy ensemble, where individual matrix\nelements have large scale-free statistical fluctuations\na slightly different method turns out to be more practical --\na method which allows one to avoid problems in\ntaking the double limit (first $N\\rightarrow \\infty$ \nand $\\epsilon \\rightarrow 0^+$) in which the\nfluctuations are suppressed in an uncontrollable way in\nthe presence of an imaginary part of $z$.\nIn the BC method $z$ is kept on the real axis and\nfluctuations are not suppressed, so one can safely take\nthe large $N$ limit.\n\nThe method goes as follows \\cite{BC}. \nOne first generates a symmetric $N\\times N$ random matrix $H$\nusing the measure (\\ref{measure}) \nand then by inverting $H\\!-\\!z$ one\ncalculates the resolvent $G(z)$ (\\ref{resolvent}).\nNext one adds a new row (and a symmetric column) \nof independent numbers identically distributed as those in the old matrix $H$.\nOne obtains a new $(N\\!+\\!1)\\times (N\\!+\\!1)$ matrix $H^{+1}$, \nwhere $+1$ emphasizes that it has one more row and \none more column than $H$. \nIt is convenient to number elements of the original\nmatrix $H_{ij}$ by indices running over the range\n$i,j=1,\\dots, N$, and assign the index $0$ to the new row and the new column \nso that now indices of $H^{+1}$ run over the range $0,\\dots, N$.\nIf one inverts the matrix $H^{+1}\\! - \\!z$ one obtains a new \n$(N\\!+\\!1)\\times (N\\!+\\!1)$ resolvent $G^{+1}(z)$.\nOne can show that it obeys a recursive relation~\\cite{BC}\n\\begin{eqnarray}\nz-\\frac{1}{G_{00}^{+1}(z)}= H_{00} + \\sum_{i,j}^N H_{0i}\nH_{0j} G_{ij}(z) = \\frac{h_{00}}{N^{1\/\\mu}} + \\sum_{i,j}^N\n\\frac{h_{0i}h_{0j} G_{ij}(z)}{N^{2\/\\mu}}.\n\\label{diagoo}\n\\end{eqnarray}\nwhich relates the element $G_{00}^{+1}(z)$ \nof the $(N\\!+\\!1)\\times (N\\!+\\!1)$ resolvent to \nthe elements of the old $N\\times N$ resolvent $G(z)$.\nOne can also derive similar equations for off-diagonal elements\nof $G^{+1}(z)$:\n\\begin{eqnarray}\n\\frac{G_{0j}^{+1}(z)}{G_{00}^{+1}(z)}=\n\\sum_i^N \\frac{h_{0i}G_{ij}(z)}{N^{2\/\\mu}}\n\\label{off-diag}\n\\end{eqnarray}\nThe difference between the probability distribution of the \nelements of $G^{+1}(z)$ and of $G(z)$ dissapears \nin the limit $N\\rightarrow \\infty$. \nThe diagonal elements\nof the matrix $G^{+1}(z)$ are identically distributed as the diagonal \nelements of the matrix $G(z)$. The same holds\nfor off-diagonal ones. Moreover in this limit\nall elements of the $G$ matrix\nbecome independent of each other. In particular\nall diagonal elements of $G(z)$ \nbecome independent identically distributed (iid)\nrandom variables as $N\\rightarrow \\infty$.\nOne can use equations (\\ref{diagoo}) and\n(\\ref{off-diag}) to derive self-consistency equations \nfor the probability density function (pdf)\nfor diagonal and the pdf for off-diagonal\nelements. We are here primarily interested in \nthe distribution of the diagonal elements, as we shall\nsee below. The self-consistency\nequation for the probability distribution of diagonal\nelements follows from the equation (\\ref{diagoo}) \nand is independent of the distribution of the off-diagonal elements.\nThis can be seen as follows.\nLet us first define after Bouchaud and Cizeau~\\cite{BC} \na quantity:\n\\begin{eqnarray}\nS_0(z) = z - \\frac{1}{G^{+1}_{00}(z)}\n\\label{SG}\n\\end{eqnarray}\nIt is merely a convenient change of variables  suited\nto the left hand side of equation (\\ref{diagoo}).\nIt is clear that if one determines the pdf for $S=S_0(z)$,\none will also be able to determine\nthe pdf for $G=G^{+1}_{00}(z)$ since the two pdfs can\nbe obtained from each other by the \nchange of variables (\\ref{SG}):\n\\begin{eqnarray}\nP_G(G) = \\frac{1}{G^2} P_S\\left(z-\\frac{1}{G}\\right)\n\\label{vchange}\n\\end{eqnarray}\nwhere $P_G$ and $P_S$ are pdfs for $G^{+1}_{00}(z)$\nand $S_0(z)$ respectively. Since the probability distribution\n$P_G$ is identical for all diagonal elements of $G$,\nit remains to determine \nthe probability distribution $P_S$ for the quantity $S_0(z)$.\n\nLet us sketch how to do that.\nFirst observe that the first term in the equation (\\ref{diagoo})\ncan be neglected at large $N$, so the equation assumes the\nform:\n\\begin{eqnarray}\nS_0(z)=  \\sum_{i}^N\n\\frac{h^2_{0i} G_{ii}(z)}{N^{2\/\\mu}}\n+\\sum_{i\\neq j}\n\\frac{h_{0i}h_{0j} G_{ij}(z)}{N^{2\/\\mu}}\\,\\,.\n\\label{SELF1}\n\\end{eqnarray}\nNow observe that by construction $h_{0i}$ are independent\nof $G_{ij}(z)$. We shall now show that for large $N$\nthe first term in (\\ref{SELF1}) dominates over the\nsecond one. We shall modify here the  \nargument used in \\cite{BC}, where it was assumed that the off-diagonal \nelements $G_{ij}N(z),~i\\ne j$ are suppressed by a factor $1\/N^{1\/\\mu}$\nwith respect to the diagonal\nelements. Instead we note that by construction the quantities\n$G_{ij}(z)$ are statistically independent of $h_{0i},~i=0,\\dots,N$.\nSince we shall only be interested by the\ndiagonal elements of the resolvent matrix, we may replace the contribution\nof the off-diagonal elements $h_{0i}h_{0j}G_{ij}(z),~i\\ne j$ by their\naveraged values. As a result, the contribution of the off-diagonal terms\naverages out. Note that this is also true in the Gaussian limit $\\mu=2$\nwhere following \\cite{BC} we may replace all elements of (\\ref{SELF1}) by\ntheir respective averages. Taking this into account and omitting the\nsubleading contribution $H_{00}$, we get\n\\begin{equation}\nS_0(z) = \\sum_i^N \\frac{h_{0i}^2 G_{ii}(z)}{N^{2\/\\mu}}.\n\\label{SELF2}\n\\end{equation}\nThus the problem was simplified to \nan equation where the left hand side ($S_0=z-1\/G^{+1}_{00}$)\nand the right hand side depend only of the diagonal elements of \nthe $G$-matrix, which as we mentioned before, are \nidentically distributed in the limit $N \\rightarrow \\infty$.\nUsing (\\ref{SELF2}) one can derive a self-consistency equation for the \nprobability density function (pdf) $P_G$ for diagonal elements\nof the matrix $G$.\n\n\\subsection*{Generalized central limit theorem}\n\nTo proceed further we apply with Bouchaud and Cizeau~\\cite{BC}\nthe {\\it generalized central limit theorem} to derive\nthe universal behavior of the sum on the right hand side of\n(\\ref{SELF2}) in the limit $N\\rightarrow \\infty$:\n\\begin{itemize}\n\\item {\\bf i.} If the $h_{0 i}$'s are sampled from\nthe L\\'{e}vy distribution $L_{\\mu}^{C,\\beta}$, the squares $t_i=h_{0i}^2$\nfor large $t_i$ are distributed solely along the positive real axis,\nwith a heavy tail distribution:\n\\begin{equation}\n\\propto \\gamma(\\mu) \\frac{C dt_i}{t_i^{1+\\mu\/2}}\n\\end{equation}\nirrespective of $\\beta$. The sum\n\\begin{equation}\n\\sum_i^N \\frac{t_i}{N^{2\/\\mu}}\n\\end{equation} is distributed following\n$L_{\\mu\/2}^{C',1}(t_i)$. The range parameters $C$ and\n$C'$ are related by~\\rf{logchar}\n\\begin{equation}\n2 C' \\gamma(\\mu\/2) = C \\gamma(\\mu)\n\\label{scaling}\n\\end{equation}\nThe factor 2 on the left hand side corresponds to the sum $(1+\\beta)+(1-\\beta)$\nappearing as a contribution from positive and negative values of the original\ndistribution of $h_{0i}$.\nThis relation  is important when\ncomparing to the numerical results below where $C=1$ is used.\nFrom now on (and to simplify the equations) we assume instead that $C'=1$.\n\\item {\\bf ii.} By virtue of the central limit theorem \nthe following sum \n\\begin{equation}\n\\sum_i^N \\frac{G_{ii}(z) t_i}{N^{2\/\\mu}}\n\\end{equation}\nof iid heavy tailed numbers $t_i$ is \nfor $G_{ii}(z)={\\cal O}(N^0)$ and $N\\rightarrow \\infty$\nL\\'evy distributed with the pdf:\n$L_{\\mu\/2}^{C(z),\\beta(z)}$, which has the stability index $\\mu\/2$\nand the effective range $C(z)$ and the asymmetry parameter $\\beta(z)$\ncalculated from the equations:\n\\begin{equation}\nC(z)=\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}\n\\label{ceff}\n\\end{equation}\nand\n\\begin{equation}\n\\beta(z)=\\frac{\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}{\\rm sign}(G_{ii}(z))}\n{\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}}\\label{beta}\\,\\,.\n\\end{equation}\nwhich follow from the composition rules for the tail amplitudes\nof iid heavy tailed numbers $t_i$ defined above.\n\\end{itemize}\n\n\\subsection*{Integral Equations}\n\nWe saw in the previous section that \nthe generalized central limit theorem implies for large\n$N$ that the ``self-energy'' $S=S_0(z)$ is distributed according to the\nL\\'evy law $P_S(S) = L_{\\mu\/2}^{C(z),\\beta(z)}(S)$ with the\nstability index $\\mu\/2$ being a half of the stability index\nof the L\\'evy law\ngoverning the distribution of individual elements\nof the matrix $H$, and with the\neffective range parameter $C(z)$ and the\nasymmetry parameter $\\beta(z)$ which can be calculated\nfrom the equations (\\ref{ceff}) and (\\ref{beta}), respectively.\nOne should note that the effective parameters $C(z),\\beta(z)$\nof the distribution $P_S(S)$ are calculated\nfor $C=1$ and that they are independent of $\\beta$ of\nthe probability distribution: $L_\\mu^{C,\\beta}(h_{ij})$ of \nthe $H$-matrix elements.\n\nThe sums on the right hand side of the two equations \n(\\ref{ceff}), (\\ref{beta}) for $C(z)$ and $\\beta(z)$ \nhave a common form $\\frac{1}{N} \\sum_i f(G_{ii}(z))$.\nSince in the limit $N\\rightarrow \\infty$, the diagonal\nelements become iid, the sums on can be substituted\nby integrals over the probability density for $G_{ii}$:\n\\begin{eqnarray}\n\\frac{1}{N}\\sum_i^N \\langle f(G_{ii}(z))\\rangle = \n\\int dG \\; P_G(G) \\; f(G)  = \n\\int \\frac{dG}{G^2} P_S\\left (z-\\frac{1}{G}\\right) \\; f(G)\n\\end{eqnarray}\nwhere in the second step we used the equation (\\ref{vchange}).\nSince the distribution\n$P_S(S) = L_{\\mu\/2}^{C(z),\\beta(z)}(S)$ is known\nup to the values of two effective parameters\n$C(z)$ and $\\beta(z)$\nthe equations (\\ref{ceff}) and (\\ref{beta}) can be written\nas self-consistency relations for $\\beta(z)$ and $C(z)$:\n\\begin{eqnarray}\nC(z) &=& -\\hspace{-11pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2}\n|G|^{\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}\n(z-1\/G)\\nonumber \\\\\n\\beta(z)&=&\\frac{-\\hspace{-10pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2}\n|G|^{\\mu\/2}{\\rm sign}(G) L_{\\mu\/2}^{C(z),\\beta(z)}(z-1\/G)}\n{-\\hspace{-10pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2} |G|^{\\mu\/2}\nL_{\\mu\/2}^{C(z),\\beta(z)}(z-1\/G)}.\\label{correct}\n\\end{eqnarray}\nThe symbol $-\\hspace{-11pt}\\int$ stands for principal value of the\nintegral.  Notice here the difference between our second equation\nand that in~\\cite{BC}. In addition to~\\cite{BC} we also note\nthat the resolvent takes the form:\n\\begin{equation}\ng(z) = \\frac{1}{N} \\sum_i G_{ii}(z) \\ \\longrightarrow \\ \ng(z) = -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dG}{G^2}~G~ L_{\\mu\/2}^{C(z),\\beta(z)} (z-1\/G)\n\\label{G}\n\\end{equation}\nThe integrals \\rf{correct} and \\rf{G} can be rewritten using the\nnew integration variable $x=1\/G$ as\n\\begin{eqnarray}\nC(z)&=&\\int^{+\\infty}_{-\\infty} \ndx |x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)\\nonumber\\\\\n\\beta(z)&=&\n\\frac{\\int^{+\\infty}_{-\\infty} \ndx ~{\\rm sign}(x)|x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)}\n{\\int^{+\\infty}_{-\\infty} dx |x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)}\n\\label{correct1}\n\\end{eqnarray}\nand\n\\begin{equation}\ng(z)=-\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dx}{x}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)\n\\end{equation}\nAll the steps above require both $z$ and $G_{ii}(z)$\nto be strictly real. The argument cannot be extended to the complex $z$ plane.\nSo all integrals above should be interpreted as principal value integrals, \nwherever it is necessary.\nSince $\\mu < 2$ all integrals in \\rf{correct1} are convergent also\nin the usual sense.\n\nThe equation for $g(z)$ can be rewritten as\n\\begin{equation} g(z)= -\\hspace{-11pt}\\int_{-\\infty}^\\infty \\frac{dx}{z-x}L_{\\mu\/2}^{C(z),\\beta(z)}(x).\n\\label{gdef1} \\end{equation} \nNotice the nontrivial dependence on $z$ in the\nparameters of the L\\'{e}vy distribution. Above equation can be a\nsource of  confusion, since its structure  resembles another\nrepresentation of $g(z)$\n\\begin{equation}\ng(z) =  -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{d\\lambda}{z-\\lambda}\\rho(\\lambda)\n\\label{hilbert}\n\\end{equation}\nwhich superficially looks as if one could identify in \\rf{gdef1} $x$ and\n$\\lambda$ and $L_{\\mu\/2}^{C(z),\\beta(z)}(x)$ with $\\rho(\\lambda)$. This is\nnot the case, and one should instead invert \n(\\ref{gdef1}) using the inverse Hilbert transform:\n\\begin{equation}\n\\rho(\\lambda) = \\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dz}{z-\\lambda} g(z).\n\\label{disp}\n\\end{equation}\nIn other words, one first has to reconstruct numerically  the real\npart of the resolvent, and only then  compute numerically the\nspectral function  $\\rho(\\lambda)$, using the \"dispersive relation\"\n(\\ref{disp}). This is a difficult and rather subtle procedure.\nIn the next section we give some analytical insights to the\nintegral equations that would help solve them and extract the\nspectral function.\n\n\\subsection*{Analytical properties useful for numerics}\n\nOne cannot do the integrals from previous section analytically.\nAs mentioned, one cannot even write down an explicit form\nof the L\\'evy distribution. It is a great numerical challenge\nto solve the problem numerically even if all the expressions\nare given. One realizes that already\nwhen one tries to compute the Fourier integral (\\ref{levydef})\nof the characteristic function since one immediately\nsees that the integrand in the form (\\ref{levydef})\nis a strongly oscillating function making the numerics\nunstable. Fortunately using the power of the complex\nanalysis one can change this integral to a form which\nis numerically stable. So in this section we present \nsome analytic tricks which allow one to reduce the problem \nof computing the eigenvalue density \nas formulated in the previous section \nto a form which is well suited to \nthe numerical computation.\n\nTo simplify \\rf{correct1} we proceed in steps. First, we make\nuse of the L\\'{e}vy distribution through its characteristic\n\\begin{equation}\nL_{\\mu\/2}^{C,\\beta}(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}dk\ne^{ikx} e^{-C|k|^{\\mu\/2}(1+i\\zeta ~{\\rm sign}(k))}. \\label{Levyf}\n\\end{equation}\nwith $\\zeta=\\beta~\\tan(\\pi\\mu\/4)$. By rescaling through\n\n\\begin{eqnarray} k &=& C^{-2\/\\mu}k',\\\\\n\\nonumber x &=& C^{2\/\\mu}x', \\\\ \\label{scale1} z &=& C^{2\/\\mu}z',\n\\nonumber\n\\end{eqnarray}\nwe can factor out the range\n$L_{\\mu\/2}^{C,\\beta}(x)=C^{-2\/\\mu}L_{\\mu\/2}^{1,\\beta}(x')$.\nSecond, we make use of the following integrals,\n\n\\begin{eqnarray}\n-\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dx}{z-x}e^{ikx}&=&-2ie^{ikz}{\\rm sign}k,\\nonumber \\\\\n\\int_0^{\\infty}dx'~\\frac{\\cos(k'x')}{x'^{\\mu\/2}}&=&|k'|^{\\mu\/2-1}\\Gamma(1-\\mu\/2)\n\\sin(\\pi\\mu\/4),\\\\ \\nonumber\n\\int_0^{\\infty}dx'~\\frac{\\sin(k'x')}{x'^{\\mu\/2}}&=&|k'|^{\\mu\/2-1}{\\rm\nsign}(k') \\Gamma(1-\\mu\/2)\\cos(\\pi\\mu\/4).\n\\end{eqnarray}\nLast, we make use of the change of variables $p=k'^{\\mu\/2}$.\nWith this in mind, we obtain\n\n\\begin{eqnarray}\nC^2(z')&=&\\frac{4}{\\pi\\mu}\\Gamma\\left(1-\\frac{\\mu}{2}\\right)\n\\sin\\left(\\frac{\\pi\\mu}{4}\\right) \\int_0^{\\infty} dp\n\\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p},\\\\ \\label{C}\n\\zeta(z')&=&\n\\frac{\\int_0^{\\infty} dp \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}}\n{\\int_0^{\\infty} dp \\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}}, \\label{bet}\n\\end{eqnarray}\nwith $\\zeta(z')=\\tan(\\pi\\mu\/4)~\\beta(z')$.\nFor every $z'$ we can iteratively solve the equation for $\\zeta(z')$,\nthen we determine $C(z')$ and use $z = C^{2\/\\mu}(z') z'$ to express\neverything in terms of $z$. These transformations solve~\\rf{correct1}.\nUsing the same method we rewrite the equation for $g(z)$ as\n\n\\begin{equation}\n\\bar{g}(z')=C(z')^{2\/\\mu}g(z) =\\frac{2}{\\mu}\\int_0^{\\infty}dp~p^{(2-\\mu)\/\\mu}\n\\sin(p^{2\/\\mu}z' - p~\\zeta(z'))~e^{-p}.\n\\end{equation}\n\nThese integral forms are useful to study the small-$z'$ limit. In this\ncase $\\zeta(z')$ is an antisymmetric function of $z'$ and has an expansion\nin powers of $z'$. Using\n$\\zeta(z')=k_1 z'+ {\\cal O}(z'^3) $ we can recursively obtain\nthe coefficients of this expansion. The first term is\n$k_1=\\Gamma(1+2\/\\mu)\/2$. Similarly $C(z')$ is a symmetric function in $z'$\n\n\\begin{equation}\nC^2(z')=\\frac{4}{\\pi\\mu}\\Gamma\\left(1-\\frac{\\mu}{2}\\right)\n\\sin\\left(\\frac{\\pi\\mu}{4}\\right)+{\\cal O}(z'^2)\n\\end{equation}\n\nFor $|z'|$ large ($z' \\to \\pm \\infty$) a different approach\nis needed. In this case we  follow\nNolan~\\cite{NOLAN} and treat the two integrals (numerator and\ndenominator) of \\rf{bet} together, i.e. we  consider the integral\n\n\\begin{equation} \\int_0^{\\infty} dp e^{-h(p)}, \\end{equation} where \\begin{equation} h(p) =\n(1-i\\zeta)p + iz' p^{2\/\\mu}.\n\\end{equation}\nNolan's idea is to close the contour of integration in the\ncomplex $p$ plane in the\nfollowing way: at $p \\to \\infty$ we add an arc and afterwards\ncontinue until $p=0$ along the line where $\\Im h(p)$=0. Using the\nparameterization $p=re^{i\\theta}$ we  get the parametric equation\nfor $r(\\theta)$ along this line, valid for $z'>0$\n\n\\begin{equation} r(\\theta)=\\left(\\frac{\\sin(\\theta_0-\\theta)}{z' \\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}\\theta)}\\right)^{\\mu\/(2-\\mu)}.\n\\end{equation}\nThe angle\n$\\theta$ for the curve we need ($\\mu \\ge 1$) is bounded between\n$\\theta_0 = {\\rm arctan}(\\zeta(z'))$, where $r=0$ and\n$-\\theta_1=-\\pi\\mu\/4$, where $\\cos(\\frac{2}{\\mu}\\theta)$ is zero.\nLet us now  introduce a new variable $\\psi$ through\n$\\theta=\\theta_0-\\psi$ and $0 \\le \\psi \\le \\theta_0+\\theta_1$. In\nthis range we have \\begin{eqnarray}\nV(\\psi)&=&\\left(\\frac{\\sin(\\psi)}{\\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))} \\right)^{\\mu\/(2-\\mu)}, \\\\\n\\nonumber\n\\Re h(\\psi)&=&\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)\n\\frac{\\cos(\\frac{2-\\mu}{\\mu}\\psi\n-\\frac{2}{\\mu}\\theta_0)}{\\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}. \\end{eqnarray}\nAfter some manipulations we obtain\n\n\\begin{eqnarray} \\int \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&=&\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)e^{-\n\\Re h(\\psi)}\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{2-\\mu}\\frac{\\cos(\\frac{2-\\mu}{\\mu}(\\psi-\\theta_0))}\n{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0)}-\\frac{\\mu}{2-\\mu}\\frac{\\sin\\theta_0}\n{\\sin\\psi}\\right)\\\\ \\nonumber \\int \\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&=&\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)e^{-\n\\Re h(\\psi)}\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{2-\\mu}\\frac{\\sin(\\frac{2-\\mu}{\\mu}(\\psi-\\theta_0))}\n{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}+\\frac{\\mu}{2-\\mu}\\frac{\\cos\\theta_0}\n{\\sin\\psi}\\right). \\label{integrals} \\end{eqnarray}\n\nThe resulting integrals look complicated, however they contain\nboth the small-$z'$ and the large-$z'$ asymptotics. For $z' \\to 0$\nwe have $\\psi = z' p^{2\/\\mu-1}$, which reproduces the small-$z'$\nexpansion presented above. For $z' \\to \\infty$ we have $\\psi =\n\\theta_0+\\theta_1-u\/z'^{\\mu\/2}$. Note that in this limit \\begin{equation}\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))=\\sin(\\frac{2}{\\mu}\\frac{u}{z'^{\\mu\/2}})\n\\end{equation} and the $z'$ dependence in ${\\rm Re}h(\\psi)$ vanishes in\nleading order.\n\nThe large-$z'$ asymptotics requires some work. For the\nleading orders we have\n\n\\begin{eqnarray}\n\\int \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&\\approx&\\Gamma(1+\\mu\/2)\\sin\\theta_1~(z')^{-\\mu\/2}\\\\ \\nonumber \\int \\cos\n(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p} &\\approx&\\Gamma(1+\\mu\/2)\n\\cos\\theta_1~(z')^{-\\mu\/2}.\n\\end{eqnarray}\nThus for $z' \\to \\infty$ we have\n\n\\begin{equation}\n\\zeta(z')=\\tan(\\theta_1)+{\\cal O}(1\/z'^{\\mu\/2}) \\end{equation} and \\begin{eqnarray}\nC(z')&=& z'^{-\\mu\/4}(1+{\\cal O}(1\/z'^{\\mu\/2}))\\\\ \\nonumber z\n&=&\\sqrt{z'}(1+{\\cal O}(1\/z'^{\\mu\/2})).\n\\end{eqnarray}\nAll the formulas above apply to the case $z'>0$. One can also derive similar\nformulas for $z'<0$, it is however more practical to use the symmetry\nproperties of the functions $C(z')$ and $\\zeta(z')$.\n\nAs a final check let us compute $\\bar{g}(z')$. A rerun of the\nabove transformations on the integrals give\n\n\\begin{eqnarray}\n\\bar{g}(z')&=&\\frac{2}{2-\\mu}\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{2\/(2-\\mu)}V(\\psi)^{2\/\\mu}e^{-{\\rm\nRe}h(\\psi)}\n\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{\\mu}\\frac{1}{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}\n+\\frac{\\sin(\\frac{2-\\mu}{\\mu}\\psi-\\frac{2}{\\mu}\\theta_0)}\n{\\sin\\psi}\\right). \\end{eqnarray}\nNotice that both asymptotics follow from this representation.\nFor $z' \\to \\infty$ we have $\\bar{g}(z')=1\/z' + \\cdots$.\nwhich implies $g(z) = 1\/z +\\cdots$. This can be viewed as a check\nof the correct normalization of the eigenvalue density distribution.\n\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.95.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distributions for $\\mu=1.95$}}\n\\label{fig1}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.75.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red)  eigenvalue distribution for $\\mu=1.75$}} \\label{fig2}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.50.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.50$}} \\label{fig3}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.25.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.25$}} \\label{fig4}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.00.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.00$}} \\label{fig5}\n\\end{figure}\n\n\n\n\\subsection*{Numerical Comparison}\n\nIn this section we show perfect agreement between the theoretical\nanalysis for the eigenvalue distribution~\\rf{disp}\nand the numerically generated eigenvalue distribution. The latter\nis obtained by diagonalizing $N\\times N$ random L\\'{e}vy matrices\nsampled using the measure~\\rf{measure}. The former was generated\nby calculating numerically $g(z)$ as detailed above.\nWe performed numerically the inverse Cauchy transform~\\rf{disp}.\nIt is important to note that the integral transforms entering into\nthe definition of the eigenvalue density $\\rho(\\lambda)$ in\n\\rf{disp} converge slowly for $z\\to \\infty$. For that, we have used\nthe asymptotic expansion of $g(z)$ to perform the large-$z$ part of\nthe integrals.\n\nAs noted above, all the above analytical results were obtained\nusing a specific choice of the scale factor~\\rf{scaling}. The\ncomparison with the numerically generated eigenvalue distribution\ngenerated using $L_{\\mu}^{1,0}$ distribution requires rescaling\nthrough $\\lambda \\to \\phi\\lambda$ and\n$\\rho(\\lambda)\\to\\rho(\\lambda)\/\\phi$ with\n\\begin{equation} \\phi=\n\\left(\\frac{\\Gamma(1+\\mu)\n\\cos(\\frac{\\pi\\,\\mu}{4})}{\\Gamma(1+\\frac{\\mu}{2})}\\right)^{1\/\\mu}\n\\label{phi}\n\\end{equation}\nBelow we  show a sequence of results for\n$\\mu=1.95,~1.75,~1.50,~1.25$ and $1.00$ with this rescaling.\nThe comparison is for high statistics $400\\times 400$ samples (red).\nWe have checked that the convergence is good already for $100\\times 100$\nsamples, with no significant difference between $N=100, 200, 400$.\nThe numerical results are also not sensitive to the choice $\\beta\\neq 0$.\nThe agreement between the results following from the integral equations\nand the numerically generated spectra is perfect.\nThis is true even for $\\mu=1$, where in principle the arguments used in\nthe derivation may not be valid.\n\n\n\\subsection*{Numerical observation}\n\nIn the mean-field approximation \\cite{BC} one can assume that\nthere are no correlations between large eigenvalues of the \nWigner-L\\'evy random matrix. In this case the eigenvalue \ndensity takes the form:\n\\begin{equation} \n\\widehat{\\rho}_\\mu(\\lambda) = L_{\\mu\/2}^{C(\\lambda),\\beta(\\lambda)}(\\lambda).\n\\label{gdef1hat} \n\\end{equation} \nIt is natural to ask how good this mean-field approximation is.\nThis can be done by comparing the mean-field eigenvalue \ndistribution $\\widehat{\\rho}(\\lambda)$ (\\ref{gdef1hat})\nto the eigenvalue distribution $\\rho(\\lambda)$ calculated \nby the inverse Hilbert transform (\\ref{disp}) of \nthe resolvent $g(z)$ (\\ref{gdef1}) as we did\nin previous section. We made this comparison numerically. \nThe result of this numerical experiment was that\nwithin the numerical accuracy which we achieved \nthe two curves representing $\\widehat{\\rho}(\\lambda)$\nand $\\rho(\\lambda)$ lied on top of each other \nin the whole studied range of $\\lambda$. Since our numerical\ncodes are written in {\\em Mathematica} we could push the \nnumerical accuracy very far, being only limited by the\nexecution time of the code. We have not seen any sign \nof deviation between the shapes of the two curves. \nThis provides us with a strong numerical evidence that\nthe mean-field argument \\cite{BC} gives \nan exact result but so far we have not managed to prove it.\nThe value of the eigenvalue density $\\widehat{\\rho}_\\mu(0)$ \nfor $\\lambda=0$ can be calculated analytically for the mean-field\ndensity (\\ref{gdef1hat}). Rescaling the density \n$\\widehat{\\rho}_\\mu(\\lambda) \\rightarrow \\widehat{\\rho}_\\mu(\\phi \\lambda)\/\\phi$ \nby the factor $\\phi$ (\\ref{phi}) we eventually obtain\n\\begin{equation}\n\\widehat{\\rho}_\\mu(0) = \n\\frac{\\Gamma(1+2\/\\mu)}{\\pi}\\left(\\frac{\\Gamma(1+\\mu\/2)^2}{\\Gamma(1+\\mu)}\\right)^\n{1\/\\mu}\n\\label{r0bc}\n\\end{equation}\nWe draw this function in Fig.\\ref{fig6}. In the same figure \nwe also show points representing numerically evaluated values of \nthe corresponding density $\\rho_\\mu(0)$ (\\ref{disp}) at some values\nof $\\mu$. Within the numerical accuracy $\\rho_\\mu(0)$ and \n$\\widehat{\\rho}_\\mu(0)$ assume the same values.\n\\begin{figure\n\\psfrag{M}{\\bf{\\Large $\\mu$}}\n\\psfrag{RH}{\\bf{\\Large $\\rho(0)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{rh000.ps}}}}\n\\caption[phased]{{\\small The line represents the function\n$\\widehat{\\rho}_\\mu(0)$ (\\ref{r0bc}) while the circles the values of  \n$\\rho_\\mu(0)$ computed numerically for $\\mu=1.0,1.25,1.5,1.75,1.95,2.0$ \nfrom the equation (\\ref{disp}).}} \n\\label{fig6}\n\\end{figure} \n\nThe physical meaning of the mean-field argument\nis that indeed one can think of the large eigenvalues as \nindependent of each other. A similar observation has\nbeen made recently \\cite{SF}. The mathematical meaning \nof the mean-field argument \nis more complex as we shall discuss below.\n\nLet us for brevity denote the function \n$L_{\\mu\/2}^{C(z),\\beta(z)}(x)$, which is a function \nof two real arguments, by \n$f(z,x) \\equiv L_{\\mu\/2}^{C(z),\\beta(z)}(x)$.\nThe equation (\\ref{gdef1}) can be now written as:\n\\begin{equation} \ng(z)= -\\hspace{-11pt}\\int_{-\\infty}^\\infty  d x \\frac{f(z,x)}{z-x}\n\\label{gdef2} \n\\end{equation} \nand (\\ref{disp}) as\n\\begin{equation}\n\\rho(\\lambda) = \\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz \\frac{g(z)}{z-\\lambda} .\n\\end{equation} \nWhen we insert (\\ref{gdef2}) into the last equation we have:\n\\begin{equation}\n\\rho(\\lambda) = \n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx \\frac{f(z,x)}{(z-\\lambda)(z-x)}.\n\\end{equation}\nA question is when this exact expression for $\\rho(\\lambda)$ is\nequal to the mean-field solution: \n$\\widehat{\\rho}(\\lambda) = f(\\lambda,\\lambda)$.\nRecall the Poincar\\'e-Bertrand theorem.\nIt tells us that the following equation holds\n\\begin{equation}\nf(\\lambda,\\lambda) = \n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx \\frac{f(z,x)}{(z-\\lambda)(z-x)} -\n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz \\frac{f(z,x)}{(z-\\lambda)(z-x)}.\n\\label{pb}\n\\end{equation} \nWe see that the density $\\rho(\\lambda)$ is given by the mean-field\nresult: $\\rho(\\lambda)=\\widehat{\\rho}(\\lambda) \\equiv f(\\lambda,\\lambda)$\nif the second term on the right-hand side of (\\ref{pb}) vanishes.\nUnfortunately we have not managed to show that this is really\nthe case for $f(z,x) = L_{\\mu\/2}^{C(z),\\beta(z)}(x)$.\nOne can however trivially observe that it would be the case if\n$f(z,x)$ had the following form \n$f(z,x)=f(x,x)=L_{\\mu\/2}^{C(x),\\beta(x)}(x)$,\nand probably also if\n$f(z,x)$ were a slowly varying function of $z$ for $z$ \nclose to $x$, in which case the integral (\\ref{gdef2}) would\npick up only the contribution from $f(x,x)$ leading to\n(\\ref{gdef1hat}).\n\n\\section{Free Random L\\'{e}vy Matrices}\n\n\\subsection*{Rotationally invariant measure}\n\nClearly Wigner L\\'evy matrices are not rotationally invariant.\nIn this section we shall discuss orthogonally (or unitary) invariant\nensembles of L\\'evy matrices. It can be shown that maximally\nrandom measures for such matrices have the form \\cite{FREELEVY,BAL}:\n\\begin{eqnarray}\nd\\mu_{FR}(H)=\\prod_{i\\le j}\\,dH_{ij}\\,e^{-\\,N{\\rm Tr}V(H)}\n\\label{frmeasure}\n\\end{eqnarray}\nWe shall be interested only in potentials with have tails\nwhich lead to eigenvalue distributions (spectral densities)\nwith heavy tails $\\rho(\\lambda) \\sim \\lambda^{-1-\\mu}$ \nbelonging to the L\\'{e}vy domain of attraction.\nA generic form of $V(\\lambda)$ at asymptotic eigenvalues $\\lambda$\nis in this case\n\\begin{eqnarray}\nV(\\lambda)={\\rm ln}\\lambda^2+{\\cal O}(1\/\\lambda^\\mu)\n\\end{eqnarray}\nIn general the potential does not have to be an analytic function.\nWe shall be interested here only in stable ensembles in\nthe sense that the spectral measure~\\rf{frmeasure} \nfor the convolution of two independent and identical ensembles \nhas the same form as the measure of the individual\nensembles. In other words, the spectral measure for\na matrix constructed as a sum of two independent matrices\ntaken from the ensemble has exactly the same spectral measure\n(eigenvalue density) modulo linear transformations.\n\nIt turns out that one can classify all the stable spectral\nmeasures thanks to the relation of the problem to free probability\ncalculus. The matrix ensemble (\\ref{frmeasure}) is in the large\n$N$ limit a realization of free random variables\n\\cite{FREELEVY}, so one can use theorems developed in free random\nprobability~\\cite{BERVOIC}. In particular we can use the fact that\nin free probability theory stable laws are classified. They actually\nparallel stable laws (\\ref{logchar})\nof classical probability theory.\nIn free probability the analogue of the {\\em logarithm} of the\ncharacteristic function (\\ref{logchar}) is the\nR-transform, introduced  by Voiculescu~\\cite{VOICULESCU}.\nThe R-transform linearizes the matrix convolution, generating \nspectral cumulants, which are additive under convolution.\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\langle{\\langle}\n\\def\\rangle{\\rangle}\n\\def{\\textstyle \\frac{1}{2}}{{\\textstyle \\frac{1}{2}}}\n\\def{\\cal O}{{\\cal O}}\n\\def{\\cal B}{{\\cal B}}\n\\newcommand{{\\mbox{e}}}{{\\mbox{e}}}\n\\def\\partial{\\partial}\n\\def{\\vec r}{{\\vec r}}\n\\def{\\vec k}{{\\vec k}}\n\\def{\\vec q}{{\\vec q}}\n\\def{\\vec p}{{\\vec p}}\n\\def{\\vec P}{{\\vec P}}\n\\def{\\vec \\tau}{{\\vec \\tau}}\n\\def{\\vec \\sigma}{{\\vec \\sigma}}\n\\def{\\vec J}{{\\vec J}}\n\\def{\\vec B}{{\\vec B}}\n\\def{\\hat r}{{\\hat r}}\n\\def{\\hat k}{{\\hat k}}\n\\def\\roughly#1{\\mathrel{\\raise.3ex\\hbox{$#1$\\kern-.75em%\n\\lower1ex\\hbox{$\\sim$}}}}\n\\def\\roughly<{\\roughly<}\n\\def\\roughly>{\\roughly>}\n\\def{\\mbox{fm}}{{\\mbox{fm}}}\n\\def{\\vec x}{{\\vec x}}\n\\def{\\rm EM}{{\\rm EM}}\n\\def{\\mbox{\\boldmath $\\pi$}}{{\\mbox{\\boldmath $\\pi$}}}\n\\def{\\mbox{\\boldmath $\\rho$}}{{\\mbox{\\boldmath $\\rho$}}}\n\\def{\\mbox{\\boldmath $\\alpha$}}{{\\mbox{\\boldmath $\\alpha$}}}\n\\def{\\mbox{\\boldmath $\\Sigma$}}{{\\mbox{\\boldmath $\\Sigma$}}}\n\\def{\\mbox{\\boldmath $\\Pi$}}{{\\mbox{\\boldmath $\\Pi$}}}\n\\def{\\mbox{\\boldmath $\\Gamma$}}{{\\mbox{\\boldmath $\\Gamma$}}}\n\\def{\\rm Tr}\\,{{\\rm Tr}\\,}\n\\def{\\bar p}{{\\bar p}}\n\\def{z \\bar z}{{z \\bar z}}\n\\def{\\cal M}_s{{\\cal M}_s}\n\\def\\abs#1{{\\left| #1 \\right|}}\n\\def{\\vec \\epsilon}{{\\vec \\epsilon}}\n\\def\\nlo#1{{\\mbox{N$^{#1}$LO}}}\n\\def{\\mbox{M1V}}{{\\mbox{M1V}}}\n\\def{\\mbox{M1S}}{{\\mbox{M1S}}}\n\\def{\\mbox{E2S}}{{\\mbox{E2S}}}\n\\def{\\cal R}_{\\rm M1}}\\def\\rE{{\\cal R}_{\\rm E2}{{\\cal R}_{\\rm M1}}\\def\\rE{{\\cal R}_{\\rm E2}}\n\\def\\J#1#2#3#4{ {#1} {\\bf #2} (#4) {#3}. }\n\\defPhys. Rev. Lett.{Phys. Rev. Lett.}\n\\defPhys. Lett.{Phys. Lett.}\n\\defPhys. Lett. B{Phys. Lett. B}\n\\defNucl. Phys.{Nucl. Phys.}\n\\defNucl. Phys. A{Nucl. Phys. A}\n\\defNucl. Phys. B{Nucl. Phys. B}\n\\defPhys. Rev.{Phys. Rev.}\n\\defPhys. Rev. C{Phys. Rev. C}\n\\def{\\cal M}{{\\cal M}}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newcommand{\\f}[2]{\\frac{#1}{#2}}\n\\newcommand{\\lambda}{\\lambda}\n\\newcommand{\\Lambda}{\\Lambda}\n\\newcommand{\\longrightarrow}{\\longrightarrow}\n\\newcommand{\\mbox{\\rm tr}}{\\mbox{\\rm tr}}\n\\newcommand{\\alpha}{\\alpha}\n\\newcommand{\\beta}{\\beta}\n\\newcommand{\\varepsilon}{\\varepsilon}\n\\newcommand{\\delta}{\\delta}\n\n\\subsection*{Stable laws in free probability}\nThe remarkable achievement by Bercovici and Voi\\-cu\\-les\\-cu~\\cite{BERVOIC}\nis an explicit derivation of all R-transforms defined by the equation\n$R(G(z))=z-1\/G(z)$ where $G(z)$ is the resolvent  for all free\nstable distributions. \nWe just note that $R(G(z))$ is a sort of self-energy for rotationally\nsymmetric FRL ensembles which is the analogue of (\\ref{SELF2}) which\nwe previously defined for Wigner-L\\'evy ensembles. \nIt is self-averaging and additive.\n\nFor stable laws $R(z)$ is known. It \ncan has either the trivial form $R(z)=a$ or\n\\begin{eqnarray}\nR(z) = b z^{\\mu-1}\n\\end{eqnarray}\nwhere $0<\\mu<2$, $b$ is a parameter which can be related to the\nstability index $\\mu$, the asymmetry parameter $\\beta$, and the range $C$ \nknown from the corresponding stable laws (\\ref{logchar}) of\nclassical probability \\cite{BERVOIC,PATA}\n\\indent{\\begin{eqnarray}\nb = \\left\\{ \\begin{array}{cl}\nC\\ e^{i(\\frac{\\mu}2-\\!1)(1\\!+\\!\\beta)\\pi}\n     {\\rm ~~for~~} 1 <\\mu<2 \\nonumber \\\\\n C \\ e^{i[\\pi+\\frac{\\mu}2(1\\!+\\!\\beta)\\pi]}\n      {\\rm ~~for~~} 0 <\\mu <1 \\nonumber\n    \\end{array} \\right. \\, .\n\\end{eqnarray}}\nIn the marginal case: $\\mu=1$, $R(z)$ reads:\n\\begin{eqnarray}\nR(z) = - i C(1+\\beta) -\n\\frac{2\\beta C}{\\pi} \\ln C z\n\\end{eqnarray}\nThe branch cut structure of $R(z)$ is chosen in such a way that the\nupper complex half plane is mapped to itself.\nRecalling that $R=z-1\/G$ in the large $N$ limit,\none finds that for the trivial case $R(z)=0$,\nthe resolvent: $G(z)=z^{-1}$\nand the spectral distribution\nis a Dirac delta, $\\rho(\\lambda)=\\delta(\\lambda)$.\nOtherwise, on the upper half-plane, the resolvent fulfills\nan algebraic equation\n\\begin{eqnarray}\nbG^{\\mu}(z)-zG(z)+1=0\\,\\,,\n\\label{Levygreen}\n\\end{eqnarray}\nor in the marginal case ($\\mu=1$):\n\\begin{eqnarray}\n\\bigg(\\!z\\!+i C(1\\!+\\!\\beta)\\!\\bigg) G(z) +\\frac{2\\beta C}{\\pi}\nG(z)\\ln C G(z) - 1\\! = \\!0.\n\\end{eqnarray}\nOn the lower half-plane $G(\\bar{z})=\\bar{G}(z)$ \\cite{BERVOIC}.\n\nThe equation for the resolvent (\\ref{Levygreen})\nhas explicit solutions only for the following values:\n$\\mu=1\/4,1\/3,1\/2, 2\/3,3\/4,4\/3,3\/2$ and $2$.\nIn all other cases the equation is transcendental and one\nhas to apply numerical procedures to unravel the spectral distribution.\nAgain the form of the potential generating\nstable free L\\'{e}vy ensembles is highly non-trivial and is\nonly known in few cases~\\cite{FREELEVY}. We refer to~\\cite{FREELEVY}\nfor further references and discussions.\n\n\\subsection*{Comparison of free L\\'evy and Wigner-L\\'evy spectra}\n\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.95new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.95$}} \\label{fig7}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.75new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.75$}} \\label{fig8}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.50new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.50$}} \\label{fig9}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.25new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.25$}} \\label{fig10}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.00new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.00$}} \\label{fig11}\n\\end{figure}\nWe present in Figures 7-11 several comparisons between\nthe free random L\\'{e}vy spectra (FRL) following from the\nsolution to the transcendental equation (red) and the\nrandom L\\'{e}vy spectra (BC) obtained by solving the coupled\nintegral equations (black), for zero asymmetry ($\\beta=0$)\nand different tail indices $\\mu$. The FRL spectra are normalized\nto agree with the BC spectra in the tails of the distributions. \nWe recall that the FRL spectra asymptote\n$\\rho(\\lambda)\\approx {\\rm sin}(\\pi \\mu\/2)\/\\pi$. \nThe comparison in bulk shows that the spectra are similar, in particular \nclose to the Gaussian limit $\\mu=2$, where both approaches become equivalent.\nFor smaller $\\mu$ there are differences.\n\nWL and FRL matrices represent two types of random matrices \nspectrally stable under matrix\naddition. For the WL matrices it follows from the measure, since each\nmatrix elements is generated from a stable L\\'{e}vy distribution and therefore\nthe sum of $\\cal N$ WL matrices, scaled by $1\/{\\cal N}^{1\/\\mu}$ is equivalent \nto the original WL ensemble. The important point is that the WL measure is\nnot symmetric while the FRL one is. \n\n\\section{Spectral stability and maximal entropy principle}\n\nThe matrix ensembles discussed in this paper are\nstable with respect to matrix addition in the sense\nthat the eigenvalue distribution for the \nmatrix constructed as a sum of \ntwo independent matrices from the original ensemble \n$H=H_1+H_2$ is identical as the the original one up to \na trivial rescaling.\nWigner-L\\'evy matrices are obviously stable, since\nthe probability distribution for individual matrix elements\nof the sum $H_{ij} = H_{1,ij}+H_{2,ij}$ is stable. \nA sum of two Wigner-L\\'evy matrices is again a Wigner-L\\'evy\nmatrix. The Wigner-L\\'evy matrices are not rotatationally \ninvariant. This means in particular \nthat the eigenvalue distribution itself\ndoes not provide the whole information about the underlying\nmatrix ensemble. Indeed, if $O$ is a fixed orthogonal matrix,\nand $H$ is a Wigner-L\\'evy matrix, then the matrix $OHO^T$\nis not anymore a Wigner-L\\'evy matrix but it has\nexactly the same eigenvalue distribution as $H$.\nIn other words, an ensemble of Wigner matrices is not \nmaximally random among ensembles with the same\neigenvalue distribution.\nOne expects that maximally random ensemble with the given\nspectral properties should be rotationally invariant. In this\ncase one also expects that the stability holds\nnot only for the sum $H=H_1+H_2$ but also for the sum \nof relatively rotated matrices: \n$H= H_1 + O H_2 O^T$,\nwhere $O$ is an arbitrary orthogonal matrix.\nIt can be shown \\cite{BAL} that the ensemble of random matrices\nwhich maximizes randomness (Shannon's entropy) for \na given spectral density has the probability measure\nexactly of the form (\\ref{frmeasure}) as discussed here.\n\nStable laws are important because they define domains\nof attractions. For example, if one thinks of a matrix addition \none expects that a sum of many independent\nidentically distributed random matrices \n$H = H_1 + \\dots + H_n$ should for $n \\rightarrow \\infty$ \nbecome a random matrix from a stable ensemble.\n\nMaximally random spectrally stable ensembles which we discussed\nin the section on free random matrices play a special role \nsince they can serve as an attraction point for the sums of\niid rotationally invariant matrices. Moreover one expects \nthat even for not rotationally invariant random matrices \n$H_i$, the sums of the form \n$B = O_1 H_1 O_1^T  + \\dots + O_n H_n O_n^T$ where\n$O_i$ are random orthogonal matrices, will for large $n$\ngenerate a maximally random matrix $B$ from a spectrally\nstable ensemble. In this spirit one can expect that\nif ones adds many randomly rotated Wigner-L\\'evy matrices:\n\\begin{eqnarray}\nB = \\frac{1}{{\\cal N}^{1\/\\mu}}\\sum_i^{\\cal N} O_i A_i O_i^T\n\\end{eqnarray}\nthat for ${\\cal N}\\to \\infty$ the matrices $B$ should \nbecome rotationally invariant, maximally random with\na distribution governed by the FRL symmetric distribution. \nIn Figures 12-14 we show that this is indeed the case.\nThe plots illustrate the two types of stability discussed above. \nIn each case we generate $N=100$ WL matrices \nand combine ${\\cal N}=100$ of them either as a simple sum (black) or a \nrotated sum (red) with the appropriate scale factor. The plots\nrepresent the numerically measured spectra for the two cases.\nWe present results for $\\mu=1.5,~1.25$ and 1, which all show that a simple sum\nreproduces the BC result, while the rotated sum reproduces the symmetric\nFRL distribution.\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.50.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1.5$}} \\label{fig12}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.25.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1.25$}} \\label{fig13}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.00.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1$}} \\label{fig14}\n\\end{figure}\n\\section{Conclusions}\n\nWe have given a detailed analysis of the macroscopic limit of\ntwo distinct random matrix theories based on L\\'{e}vy type\nensembles. The first one was put forward by Bouchaud and Cizeau\n\\cite{BC} and uses a non-symmetric measure under the orthogonal\ngroup, and the second one was suggested by us~\\cite{FREELEVY} and\nuses a symmetric measure.\n\nAfter correcting the original analysis\nin~\\cite{BC}, in particular our formulae\n(\\ref{correct1}), (\\ref{correct}) replace (10b) and (12b) in~\\cite{BC},\nand their eq. (15) is replaced by the pair of ``dispersion relations''\n(\\ref{G}) {\\em and} (\\ref{disp}), we  found perfect agreement between\nthe analytical and numerical spectra. The WL measure is easy to\nimplement numerically for arbitrary asymmetry parameter $\\beta$\nin the L\\'{e}vy distributions. The spectrum of\nWL matrices does not depend on $\\beta$ and remains symmetric and universal,\ndepending only on $\\mu$.\n\nWe have also shown that the spectra generated analytically for symmetric\nFRL matrices are similar to the ones generated from WL matrices.\nUnlike the WL ensemble, the FRL ensemble allows for\nboth symmetric and asymmetric L\\'{e}vy distributions. Both ensembles\nare equally useful for addressing issues of recent\ninterest~\\cite{LEVYFIN,BPBOOK2}.\n\nLet us finish the paper with two remarks. \n\\begin{itemize}\n\\item {\\bf i.} The application of free probability calculus to asymptotically free\nmatrix realizations allows one to derive spectral density of the\nmatrices from the underlying matrix ensemble\nbut it does not tell one how to calculate eigenvalue correlation\nfunctions, or joint probabilities for many eigenvalues. Actually\ndifferent matrix realizations of free random variables may have\ncompletely different structure of eigenvalue correlations even\nif they are realizations of the same free random variables.\nTo fix correlations or joint probabilities for \ntwo or more eigenvalues, one has to introduce the concept\nof higher order freeness \\cite{2F}. We think however that \nif one imposes on a matrix\nrealization of free random variables an additional requirement\nthat it has to be maximally random in the sense of maximizing\nShannon's entropy \\cite{BAL} then this aditional\nrequirement automatically fixes the probability\nmeasure (\\ref{frmeasure}) for the ensemble\nand thus also all multi-eigenvalues correlations.\n\n\\item {\\bf ii.} Large eigenvalues behave differently for Wigner-L\\'evy and\nmaximally random free random L\\'evy matrices\ndiscussed in this paper. As pointed out recently \\cite{SF},\nthe largest eigenvalues fluctuate independently \nfor Wigner-L\\'evy ensemble, a little bit like in the mean-field\nargument \\cite{BC} mentioned before, while for the maximally \nrandom matrix ensemble (\\ref{frmeasure}) even large eigenvalues\nare correlated \\cite{FREELEVY}. \n\n\\end{itemize}\n\n\n\\begin{acknowledgments}\nWe thank Jean-Philippe Bouchaud for discussions\nwhich prompted us to revisit and compare the two approaches\npresented in this paper. This work was partially supported\nby the Polish State Committee for Scientific Research (KBN) grant\n2P03B 08225 22 (MAN, ZB), Marie Curie TOK programme \"COCOS -\nCorrelations in Complex Systems\" (Contract MTKD-CT-2004-517186),\n(MAN, JJ, ZB, GP), the National Office for Research and Technology\ngrant RET14\/2005 (GP) and by US-DOE grants DE-FG02-88ER40388 and\nDE-FG03-97ER4014 (IZ).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction:}\nBiomolecules ($DNA$ \\& $proteins$) live in the crowded, constrained regime. Such molecules are soft object and therefore\ncan be easily squeezed into spaces that are much smaller than the natural size of the molecule in the bulk. For instance,\n$actin$ filaments in $eukaryotic$ cell \\cite{1} or $protein$ encapsulated in $Ecoli$ \\cite{2} is found in nature are\nexamples of confined biomolecules serves the basis for understanding numerous phenomenon observed in polymer technology,\nbio-technology and many other molecular processes occurring in the living cells. The conformational properties of single\nbiomolecules have attracted considerable attention in recent years due to developments in single molecule based observations\nand experiments \\cite{3,4,5,6,7,8}. Under confined geometrical condition, excluded volume effects and effect of geometrical constraint\ncompete with entropy of the molecule. Therefore, constraints can modify conformational properties and\nadsorption desorption transition behaviours of macromolecules.\n\nBehaviour of the flexible polymer molecule in a good solvent, confined to different geometries have been studied \nfor past few years \\cite{9,10}. For example, Brak and his coworkers \\cite{11} used directed self avoiding\nwalk model to study behaviour of flexible polymer chain confined between two parallel walls on square lattice. However,\nin present investigation we have considered a linear semiflexible copolymer chain on square lattice constrained by a one dimensional\nstair shaped surface. For two dimensional space, surface is a line and copolymer chain is constrained by the such surface. The chain\nis made of two type of monomers $A$ and $B$ distributed along the back bone of chain in a sequence $A-B-A-B\\dots$. \nTherefore, it is a sequential or alternating copolymer chain.\nWe have considered sequential copolymer chain because it serve as a paradigmatic \nmodel of actually disordered macromolecule {\\it e. g.} $proteins$.\n\nTo analyze the effect of constraint, we have used\nfully directed self avoiding walk model introduced by Privmann {\\it et. al} \\cite{12} and used generating function technique \nto solve the model analytically.  \nThe results so obtained is used to discuss the behaviour of homopolymer chain in constrained geometry and also to compare results obtained\nfor conformational properties of the polymer chain for the case, when chain is in the bulk, in absence of one \ndimensional stair to the case when there is stair to constrain the chain. \nIf constraint is an attractive surface, it contributes an energy $\\epsilon_s$\n($<0$) for each step of walk making on the surface. This leads\nto an increased probability defined by a Boltzmann weight\n$\\omega=\\exp(-\\epsilon_s\/k_BT)$ of making a step on the surface\n($\\epsilon_s < 0$ or $\\omega > 1$, $T$ is temperature and\n$k_B$ is the Boltzmann constant). \nThe chain gets adsorbed on the surface at an appropriate  value of $\\omega$ or $\\epsilon_s$. \nTherefore, transition\nbetween adsorbed and desorbed phases is marked by a critical value of adsorption\nenergy $\\epsilon_s$ or $\\omega_c$. \nThe crossover\nexponent $\\phi$ at the adsorption transition point is defined as, $N_s \\sim N^{\\phi}$, where $N$ is the total\nnumber of monomers in the chain while $N_s$ is the number of monomers adsorbed onto the surface. \nA comparison is also made to discuss effect of constraint on adsorption transition point. \n\n\nThe paper is organized as follows: In Sec. 2 lattice model of\ndirected self avoiding walk is described for a linear semiflexible sequential copolymer chain \nin good solvent. In sub-section 2.1, we have discussed behaviour \nof the copolymer chain in the bulk in presence of one dimensional stair shaped surface. The results\nobtained in constrained geometry case is compared with those obtained in absence of constraint \nfor semiflexible homopolymer chain.\nSub-section 2.2, is devoted to discuss adsorption of copolymer chain on a one dimensional flat surface \nin presence of constraint and results obtained is compared for the case when there is no constraint near the \nhomopolymer and copolymer chains. In Sub-section 2.3, adsorption of copolymer chain on constraint is discussed. While\nin sub-section 2.4, general expression of the partition function of copolymer chain which\nis interacting with constraint and flat surface is discussed.\nFinally, in Sec. 3 we summarize the results obtained.\n\n\\section{Model and method}\nA lattice model of fully directed self-avoiding walk $\\cite{12}$ is used to\nmodel a semiflexible alternating copolymer chain in two dimensions and generating function technique has\nbeen used to calculate partition function of the copolymer chain in presence of an impenetrable surface (line) having\nshape like a stair (as shown schematically in figure 1). Since copolymer chain is fully directed \ntherefore, walker is allowed to take steps along $+x$, $+y$ directions on the square lattice.\nThe stiffness of the chain is accounted by introducing an energy barrier for each bend in the walk of the chain. \nThe stiffness weight $k=exp(-\\beta\\epsilon_{b})$ where $\\beta=(k_BT)^{-1}$ is \ninverse of the temperature and $\\epsilon_b(>0)$ is the energy associated with \neach bend of the walk of copolymer chain. For $k=1$ or $\\epsilon_{b}=0$ the chain is said to be flexible and for \n$0<k<1$ or $0<\\epsilon_{b} <\\infty$ the chain is  \nsemiflexible. However, when $\\epsilon_{b}\\to\\infty$ or $k\\to0$, \nthe copolymer chain is like a rigid rod.\n\nDirected self avoiding walk model can be solved \nanalytically and therefore gives exact value of conformational properties and \nadsorption transition point of the chain. However,  \ndirected walk model is restrictive in the sense that bending angle has same value {\\it i. e.}\n$90^0$ (for present study) for each bend and directedness of the walk introduces certain extent of stiffness in the \npolymer chain because all directions of the space are not treated equally by the walker. \n\n\\begin{figure}[htbp] \n\\centering \n\\epsfxsize=14cm\\epsfbox{fig01.eps} \n\\caption{In this figure polymer chain is shown (I) in absence of constraint (one dimensional stair shaped surface) while in (II) there\nis constraint. Walk of the chain starts from point O and directed along both the directions of space.}\n\\label{Figure1}\n\\end{figure} \n\nThe partition function of a semiflexible sequential copolymer chain made of two type of monomers (A \\& B) can be written as, \n\n\\begin{equation}\nZ(k,x_1,x_2)={\\sum}^{N=\\infty}_{N=0}\\sum_{ all\\hspace{0.07cm}walks\\hspace{0.07cm}of\\hspace{0.05cm}N\\hspace{0.05cm}steps} {x_1}^{N\/2}{x_2}^{N\/2}k^{N_b}\n\\end{equation}\nwhere, $N_b$ is the total number of bends in a self avoiding walk of $N$ steps (monomers),\n$x_1$ and $x_2$ is the step fugacity of each of the A and B type monomers, respectively.\nFor the sake of mathematical simplicity we assume here and onwards $x_1=x_2=x$. Method\nof analysis discussed in this paper can also be extended to the case when $x_1\\ne x_2$. \n\nWe consider an impenetrable line, located in the $x-y$ plane (as shown schematically in the figure 1) has shape like a stair, one end\nof the copolymer chain is grafted to origin $O$. Since stair is impenetrable, therefore, it constrains the\npolymer chain. We are interested in analyzing effect of this constrain on the conformation properties\nof the copolymer chain in the bulk and also in calculating adsorption desorption phase transition point of the\ncopolymer chain in presence of constraint.\n\n\\subsection{\\bf \\it Copolymer chain in the bulk}\n\nThe partition function of a sequential semiflexible copolymer chain in the bulk is calculated using the\nmethod of analysis discussed by Mishra {\\it et al.}, $\\cite{13}$ and \ncomponents of the partition function $Z(k,x)$ of the copolymer chain in the bulk is written as follows:\n\\begin{equation}\nX_0(k,x)= x+x^2k(1+Y+kX_0)\n\\end{equation}\nwhere, $X_n(k,x)$ (here and onwards $n$=0,1,2,3,4...) is the sum of Boltzmann weight of all walks with \nfirst step along $+x$-direction and $n$ step distant from the stair\nwhile $Y(k,x)$ is sum of Boltzmann weight of all the walks of the copolymer chain having first step along $+y$ direction.\nAbove equation can be re-written as,\n\\begin{equation}\nX_0(k,x)= \\frac{x(1+kx)}{1-x^2k^2}+\\frac{x^2kY}{1-x^2k^2}\n\\end{equation}\nSimilarly, we have\n\\begin{equation}\n\\hspace{-2.5cm}X_n(k,x)=x+x^2+..+x^n+x^n\\frac{x(1+kx)}{1-x^2k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{x^2k}{1-x^2k^2}\\large)Y\n\\end{equation}\nand substituting $n=\\infty$, we recover $X_\\infty(k,x)=X(k,x)$ \\cite{13},\n\\begin{equation}\nX_\\infty(k,x)= \\frac{x}{1-x}+\\frac{xkY'}{1-x}\n\\end{equation}\nWhere, $Y'$ is the other component of the partition function in absence of constraint. \nBut, another component of the partition function of the semiflexible copolymer chain in presence of constraint is,\n\\begin{equation}\nY(k,x)=x+x(x+kX_1)+x^2(x+kX_2)+x^3(x+kX_3)+.......\n\\end{equation}\nso that,\n\\begin{equation}\n\\hspace{-2.5cm}Y(k,x)=\\frac{x(1-x^2)(1-x^2k^2)+kx^2(1-k^2x^2)+kx^2(1-x)(x+kx^2)}{(1-x)(1-x^2)(1-k^2x^2)-k^2x^2(1-x^2k^2)-k^2x^4(1-x)}\n\\end{equation}\nThus, the partition function of the copolymer chain \nin the bulk is written as,\n\\begin{equation}\nZ(k,x)=X_0(k,x)+Y(k,x)\n\\end{equation}\nsuch that\n\\begin{equation}\n\\hspace{-2.5cm}Z(k,x)=\\frac{x(2+(-1+2k)x+(-2+k-2k^2)x^2+(1-2k+2k^2-2k^3)x^3)}{1-x-(1+2k^2)x^2+(1+k^2)x^3+k^4x^4}\n\\end{equation}\nFrom singularity of the partition function $Z(k,x)$, we obtain critical value of step fugacity, $x_c$, needed for\npolymerization of an infinitely long linear semiflexible copolymer chain. Since, we have considered $x_1=x_2=x$, we can \ntreat this result for semiflexible homopolymer chain {\\it i. e.} both the monomers of copolymer chain are identical. \nFor, $k=1$, we have $x_c$(with constraint)=0.532089 while $x_c$(without constraint)=0.5 \\cite{13} for flexible\nhomopolymer chain for fully directed self avoiding walk model on square lattice. Variation of $x_c$ with $k$ in\npresence as well as absence of stair (constraint) is shown in figure 2.\n\nPersistent length, $l_p$, of the chain can be calculated using definition of Mishra {\\it et. al} \\cite{13}, \n$l_p${\\large $(=\\frac{x\\frac {\\partial Log Z(k,x)}{\\partial x}}{k\\frac{\\partial Log Z(k,x)}{\\partial k}})$}, as average distance \nof the polymer chain between its two successive bends. The variation of the persistent length \nwith $k$ is shown in figure 2. A comparison of $l_p$ is also made for the case when there no constraint \\cite{13}.\n\n\\subsection{Adsorption on one dimensional flat line}\nIn order to analyze the effect of presence of constraint on adsorption transition point, we consider an impenetrable line\nlocated at $x$=0 and calculated components of the partition function of the sequentional copolymer chain, which is interacting with\nimpenetrable flat line (while there is no attraction between monomer and constraint) \nusing method of analysis used by Mishra {\\it et. al}, we have component of the partition function along the surface as,\n \n\\begin{equation}\n\\hspace{-2.5cm}S^f(k,\\omega_1,\\omega_2,x)=s_1(1+s_2+kX_1)+s_1s_2(1+s_2+kX_2)+s_1^2s_2(1+s_2+kX_3)+\\dots\n\\end{equation}\nhere $A$ type monomer of the copolymer chain is grafted at $O$, we have $s_1(=\\omega_1x)$, $s_2=(\\omega_2x)$ is the \nBoltzmann weight of $A$ and $B$ type monomers of the copolymer chain\nwith flat surface, respectively. While $X_n(k,x)$ is the component of the partition function\nperpendicular to the flat surface {\\it i. e.} along $x$-axis. The value of $X_n(k,x)$ we have used from Eq. (4) to calculate partition function,\n$Z^f(k,\\omega_1,\\omega_2,x)$, of the copolymer chain (provided $s_1s_2<1$) as follows,\n\\begin{equation}\n\\hspace{-2.5cm}S^f(k,\\omega_1,\\omega_2,x)=\\frac{s_1+s_1s_2}{1-s_1s_2}+\\frac{s_1kx(1+s_2)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1s_2kx^2(1+s_1)}{(1-s_1s_2)(1-s_1s_2x^2)}+u_1+u_2\n\\end{equation}\n\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig02.eps}}\n\\caption{This figure compares the value of conformational properties of semiflexible\nhomopolymer chain when there is constraint (an impenetrable stair shaped surface) to that of the case if there \nis no constraint. We have also compared the value of adsorption desorption transition point of the semiflexible \nhomopolymer chain on a flat surface in presence and absence of the constraint.}\n\\label{Figure2}\n\\end{figure}\nwhere,\n\n$u_1$={\\Large$\\frac{s_1kx(1+s_2x)(x+kx^2)}{(1-s_1s_2x^2)(1-x^2k^2)}$}\n\nand\n\n$u_2$={\\Large$(\\frac{s_1k^2x(1+s_2)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1s_2k^2x^2(1+s_1)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1(1+s_2x)k^2x^3}{(1-s_1s_2x^2)(1-k^2x^2)})$}Y\nTherefore, partition function of the copolymer chain with $A$ type monomer grafted at $O$ can be calculated as,\n\n\\begin{equation}\nZ^f(k,\\omega_1,\\omega_2,x)=S^f(k,\\omega_1,\\omega_2,x)+X_0(k,x)\\hspace{2cm} (s_1s_2<1)\n\\end{equation}\n\\begin{equation}\n\\hspace{-2.5cm}Z^f(k,\\omega_1,\\omega_2,x)=\\frac{f(k,\\omega_1,\\omega_2,x)}{(u_3(1-x-(1+2k^2)x^2+(1+k^2)x^3+k^4x^4)))}\\hspace{1cm} (s_1s_2<1)\n\\end{equation}\n\nwhere,\n\n$u_3=(-1+s_1s_2)(-1+s_1s_2x^2)$\n\nWe are interested in the singularity of the partition function. Therefore, comparing the denominator of the partition\nfunction $Z^f(k,\\omega_1,\\omega_2,x)$ equal to zero, we obtain critical value of monomer surface attraction\nrequired for adsorption of copolymer chain, $\\omega_{c1}=\\frac{1}{\\omega_{c2}x^2}$ on the flat surface. Effect of\nconstraint on adsorption transition of copolymer chain on flat surface is analyzed through figure (3). \nReplacing $s_1$ with $s_2$ in Eq. (13) we\ncould obtain partition function for copolymer having first monomer of $B$ type. The value of adsorption transition\npoint remain unchanged when grafted monomer of the copolymer chain is of $A$ or $B$ type. \nSubstituting $\\omega_{c1}=\\omega_{c2}=\\omega_c$, we get value of $\\omega_c$ required for adsorption of homopolymer\nsemiflexible chain on an impenetrable line in presence of constraint. This result is shown in figure 2 and comparison\nof this result with those obtained in absence of constraint \\cite{13} is shown for different values of $k$.\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig03.eps}}\n\\caption{This figure compares the value of $\\omega_{c1}$ required for adsorption of copolymer chain\non the flat surface for $\\omega_{c2}=0.5,0.9$ in presence and absence of the constraint.}\n\\label{Figure3}\n\\end{figure}\n\n\\subsection{Adsorption of copolymer chain on constraint (one dimensional stair)}\nThe stair shaped one dimensional line (surface) is located in the $x-y$ plane as shown schematically in figure (1). The copolymer chain is interacting\nwith this surface. While, flat surface (line) located at $x=0$ is neutral for copolymer chain. The components of the partition function of\nsurface interacting copolymer chain can be calculated using method discussed above, as,\n\n\\begin{equation}\n\\hspace{-2.5cm}S_{0A}(k,\\omega_3,\\omega_4,x)= s_3+s_3s_4k(1+Y_{A}+s_3k)+s_3^2s_4k^2(1+Y_{A}+s_4k)+s_3^2s_4^2k^3(1+Y_{A}+s_3k)+\\dots\n\\end{equation}\n\nwhere, $s_3(=\\omega_3x)$, $s_4(=\\omega_4x)$ is the Boltzmann weight of interaction energy of $A$ and $B$ type monomers with stair shaped surface\nrespectively. $S_{nA}(k,\\omega_3,\\omega_4,x)$ [$A$ indicates that first monomer grafted at $O$ of the copolymer chain is of $A$ type] \nis the sum of Boltzmann weight of all walks having first step along $+x$-direction and $n$ step distant from the stair\nwhile $Y_A(k,\\omega_3,\\omega_4,x)$ is sum of Boltzmann weight of all the walks of the surface interacting copolymer chain having first step \nalong $+y$ direction. In the same fashion one can define $S_{nB}(k,\\omega_3,\\omega_4,x)$ and $Y_B(k,\\omega_3,\\omega_4,x)$. \nWe assume that first monomer of copolymer chain is of $A$ type therefore for $s_3s_4<1$, we have,\n\\begin{equation}\nS_{0A}(k,\\omega_3,\\omega_4,x)= \\frac{s_3(1+s_4k)}{1-s_3s_4k^2}+\\frac{s_3s_4kY_A}{1-s_3s_4k^2}\n\\end{equation}\nalso, for odd $n$=$p$,\n\\begin{equation}\n\\hspace{-2.5cm}S_{pA}(k,\\omega_3,\\omega_4,x)=x+x^2+..+x^n+x^n\\frac{s_3(1+ks_4)}{1-s_3s_4k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{s_3s_4k}{1-s_3s_4k^2}\\large)Y_A\n\\end{equation}\nand for even $n$=$q$,\n\\begin{equation}\n\\hspace{-2.5cm}S_{qA}(k,\\omega_3,\\omega_4,x)=x+x^2+..+x^n+x^n\\frac{s_4(1+ks_3)}{1-s_3s_4k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{s_3s_4k}{1-s_3s_4k^2}\\large)Y_A\n\\end{equation}\nThe component of the partition function of surface interacting semiflexible copolymer chain along $y$-axis is,\n\\begin{equation}\nY_A(k,\\omega_3,\\omega_4,x)=x+x(x+kS_{1B})+x^2(x+kS_{2A})+x^3(x+kS_{3B})+\\dots\n\\end{equation}\nsubstituting the value of $S_{qA}(k,\\omega_3,\\omega_4,x)$ and $S_{pB}(k,\\omega_3,\\omega_4,x)$ in Eq. (18) we get,\n\\begin{equation}\n\\hspace{-2.5cm}Y_A(k,\\omega_3,\\omega_4,x)=\\frac{x(1-x^2)(1-s_3s_4k^2)+kx^2(1-k^2s_3s_4)+kx^2(1-x)(s_3+ks_3s_4)}{(1-x)(1-x^2)(1-k^2s_3s_4)-k^2x^2(1-s_3s_4k^2)-k^2s_3s_4x^2(1-x)}\n\\end{equation}\nThus, we calculated partition function of copolymer chain interacting with a stair shaped impenetrable surface as,\n\\begin{equation}\nZ^s(k,\\omega_3,\\omega_4,x)=S_{0A}(k,\\omega_3,\\omega_4,x)+Y_A(k,\\omega_3,\\omega_4,x)\n\\end{equation}\ntherefore, partition function of the copolymer chain which is interacting with an impenetrable stair shaped line is,\n\\begin{equation}\n\\hspace{-2.5cm}Z^s(k,\\omega_3,\\omega_4,x)=\\frac{N^r(k,\\omega_3,\\omega_4,x)}{D^r(k,\\omega_3,\\omega_4,x)}\n\\end{equation}\n\nwhere,\n\n$N^r$=$x+kx^2-x^3-s_3(2k^3s_4x^2-(-1+x)^2(1+x)+k^2x(s_4+x-2s_4x)+k(-1+x)(s_4+s_4x+x^2))$\n\nand,\n\n$D^r=k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)$\n\nFrom singularity of the partition function $Z^s(k,\\omega_3,\\omega_4,x)$, we \nfound that $\\omega_3=\\frac{-1+x+x^2+k^2x^2-x^3}{k^2\\omega_4x^2(-1+x+k^2x^2)}$ \n({\\it i. e.} on comparing $D^r(k,\\omega_3,\\omega_4,x)=0$) required for adsorption of semiflexible copolymer chain and it is independent of bending \nenergy or stiffness $k$ of the polymer chain. \nThis we have discussed through numerical results shown in figure (4) for $\\omega_{c2}=0.5, 0.9 \\& 1.0$. \nThis is because, the polymer chain is directed towards the surface (constraint). \nReplacing $\\omega_3$ and $\\omega_4$ with $\\omega_c$, we have adsorption \ntransition point for homopolymer chain. The value of $\\omega_c$ for homopolymer chain is close to unity (1$\\pm$0.0001) \nfor the values of $k$, $0.1\\le k<1$.\n\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig04.eps}}\n\\caption{This figure is used to show that copolymer chain adsorption on constraint appears independent\nof stiffness of the copolymer chain. However, in the very stiff chain limit adsorption of the chain on \ncontsraint may not be possible.}\n\\label{Figure4}\n\\end{figure}\n\n\\subsection{General expression of the partition function}\n\nWe now derive expression of the partition function of the  sequential semiflexible linear \ncopolymer chain for the case when copolymer chain is interacting with flat surface and constraint. \nThe flat impenetrable surface (line) located at $x=0$ and another\nsurface is constraint (an impenetrable stair shaped line located along the line $x=y$; schematically shown in the figure (1)).\nWe assume that $A$ type monomer of copolymer chain is first monomer of the chain and one end of the chain is grafted at (0,0) or $O$. \nLet $S_A$ is the component of the partition function having first step on to the flat surface, then,\n\\begin{equation}\n\\hspace{-2.5cm}S_{A}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)= s_1+s_1s_2k(1+S_{1B}+s_2k)+s_1^2s_2k^2(1+S_{2A}+s_2k)+\\dots\n\\end{equation}\n\nSubstituting value of $S_{qA}$, $S_{pB}$ from Eqs. (16\\& 17) and using following relation we have general expression of the partition function,\n\\begin{equation}\n\\hspace{-2.5cm}Z^{s\\&f}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)=S_{A}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)+S_{0A}(k,\\omega_3,\\omega_4,x)\n\\end{equation}\n\\begin{equation}\n\\hspace{-2.5cm}Z^{s\\&f}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)=\\frac{s_1(1+s_2)}{1-s_1s_2}+u_4+u_5-\\frac{u_6}{u_7}+\\frac{u_8}{u_{9}}\\hspace{.5cm}(s_1s_2<1 ; s_3s_4<1)\n\\end{equation}\n\nwhere,\n\n\\vspace{.2cm}\n\n$u_4$={\\Large$\\frac{ks_3(1+ks_4)s_1x(1+s_2x)}{(-1+k^2s_3s_4)(-1+s_1s_2x^2)}$}\n\n\\vspace{.2cm}\n\n\n$u_5$={\\Large$\\frac{ks_1(1+s_2)x(1+s_2x)}{(-1+s_1s_2)(-1+s_1s_2x^2)}$}\n\n\\vspace{.2cm}\n\n\n$u_6=s_3(-(k^2(-1+s_4)x^2)+k^3s_4x^2-(-1+x)^2(1+x)+ks_4(-1+x^2))$\n\n\\vspace{.2cm}\n\n\n$u_7=k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)$\n\n\\vspace{.2cm}\n\n\n$u_8=(k^2s_1x^2(-1-k^2s_3s_4(-1+x)+k^3s_3s_4x+k(-1+s_3(-1+x))x+x^2)(-1-s_2(1+x+s_1x)+s_3s_4((-1+s_1s_2)(1+s_2x)+k^2(1+s_2+s_2x+s_1s_2x))))$\n\n\\vspace{.2cm}\n\n\nand\n\n\\vspace{.2cm}\n\n\n$u_{9}=((-1+k^2s_3s_4)(-1+s_1s_2)(-1+s_1s_2x^4)(k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)))$\n\n\\vspace{.2cm}\n\n\nWe call this relation (Eq. 24) as general expression of the partition function because from this expression we can recover \nexpression of partition function discussed in above sub-sections by suitable substitutions. For example, replacing $s_1$, $s_2$,\n$s_3$ and $s_4$ by x we have partition function of the copolymer chain in the bulk {\\it i. e.} $Z(k,x)$. \n\n\\section{Result and discussion}\n\nWe have considered a linear semiflexible sequential copolymer chain made of two type of monomers and fully directed\nself avoiding walk model is used to model the copolymer chain on square lattice. \nTechnique of generating function is used to calculate partition function of the chain in presence of constraint (an impenetrable one\ndimensional line) analytically. \nThe variation of\ncritical value of step fugacity required for polymerization of an infinitely long linear copolymer chain and \npersistent length under good solvent have been discussed. \nThe results obtained with constraint is compared to case when there is no constraint. \nWe have also compared the value of monomer surface attraction required for\nadsorption of semiflexible copolymer chain on a flat surface and constraint. The surface is a one dimensional line\nand walks of the chain is directed along both the directions of the space.\n\nWe have found that adsorption of the homopolymer chain on constraint occurs at a very small value of monomer and surface attraction. \nThis is because, walks of the polymer\nchain is directed along the stair shaped surface, walks of the chain follows curvature of the constraint and major \ncontribution in the partition function of copolymer chain is due to walks lying on the constraint. \nIn the case of stair shaped impenetrable surface, critical value of\nmonomer surface attraction required for the copolymer\nchain adsorption appears independent of bending energy\nof the chain (excluding the case of very stiff chains). While adsorption of copolymer and homopolymer chain on the flat surface \nis function of semiflexibility of the chain. The stiffer chain adsorption occurs at a smaller\nvalue of monomer surface attraction than the flexible polymer chain.\nThese features of adsorption\non an impenetrable flat and curved surfaces are similar to the semiflexible polymer adsorption $\\cite{13,14,15,16,17}$.\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nIn mathematical imaging the problem of segmentation refers to the process of automatically detecting regions, objects or patterns of interest. This is of particular importance in biomedical imaging for cell or organ quantification as well as in materials science and engineering. The main goal of this work is to present a new multiscale segmentation method combining variational inverse scale-space methods with nonlinear spectral analysis.\n\n\n\\textit{Image segmentation.} The task of image segmentation can sometimes be addressed by simple methods which are solely based on intensity or histogram thresholds. However those methods quickly fail in more complex experimental scenarios, where challenges in terms of different sizes, intensities, contrasts or uncertainties like noise occur. Thus, a class of commonly used mathematical techniques to overcome some of the difficulties are nonlinear variational methods. In general there are two different ways of describing a region, either by its edge or by the interior of the region. Therefore two main concepts of addressing segmentation by an energy minimization problem evolved in the previous decades: edge-based and region-based segmentation. While edge-based segmentation separates regions based on discontinuity information \\cite{Kass1988,Kichenassamy1995,Caselles1997}, region-based segmentation separates by similarity measures within the regions. In this paper we focus on a region-based approach.\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Sizes]{\\includegraphics[height=0.15\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_1.png}}\\qquad\n  \\subfigure[Intensities]{\\includegraphics[height=0.15\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_1.png}}\\qquad\n  \\subfigure[Shapes]{\\includegraphics[height=0.15\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_1.png}}\\\\\n  \\caption{Minimal examples where multiscale image segmentation could offer added value}\n\\label{fig:overview_goal_segmentation}\n\\end{figure}\n\nThe proposed method in this work is mainly built upon the Chan-Vese (CV) model \\cite{Chan2001}. For a given image function $f:\\Omega \\rightarrow \\mathbb{R}$ the domain $\\Omega \\subset \\mathbb{R}^d$ should be separated into two regions $\\Omega_1$ and $\\Omega_2 := \\Omega \\setminus \\Omega_1$ via the following variational energy minimization method\n\\vspace{-2pt}\n\\begin{equation}\\label{eq:CVorig}\n\\int_{\\Omega_1} (f(x) - c_1)^2 \\mathrm{d}x + \\int_{\\Omega_2} (f(x) - c_2)^2 \\mathrm{d}x + \\alpha \\cdot Per(C)\\longrightarrow \\min_{C,c_1,c_2}\n\\end{equation}\n\\vspace{-2pt}\nwhere $C$ is the desired contour separating $\\Omega_1$ and $\\Omega_2$ and $c_i$ stands for a desired average intensity value within region $\\Omega_i$. $Per(C)$ denotes the length of the contour $C$ separating $\\Omega_1$ and $\\Omega_2$ and can be controlled via a regularization parameter $\\alpha > 0$. An easy example is the image $f$ given in Figure \\ref{fig:overview_goal_segmentation}(a) where the domain $\\Omega$ can be segmented with respect to foreground and background. In contrast to simple histogram thresholding methods this holds true even in the case of strong noise if the parameter $\\alpha$ is chosen adequately. A generalization of this model was introduced in \\cite{Vese2002} and in \\cite{Chan2000} the model has also been extended to vector-valued images like color images. Recently in \\cite{zosso2015} an extension of the CV model that can handle artifacts and illumination bias in images has been proposed.\n\n\n\\textit{Challenges\/Questions.} The CV model is very useful in segmenting regions of interest which have very similar intensity values, e.g. Figure \\ref{fig:overview_goal_segmentation}(a). However, automatically detecting single objects based on their size is more challenging. Even with a varying parameter $\\alpha$ controlling the contour length (forward scale-space), it is for example not possible to detect the smallest object as a singleton. A similar challenge occurs when segmenting separate objects due to their intensity values, e.g. Figure \\ref{fig:overview_goal_segmentation}(b). Increasing the number of constants $c_i$ to four is suboptimal because we usually a-priori don't know the number of objects. Varying a threshold or parameter $\\alpha$ could lead to a correct segmentation but the estimated intensity constants $c_1$, $c_2$ will likely be incorrect. \\textit{Hence, is it possible to automatically detect multiple scales in a nonlinear variational image segmentation model, for instance with respect to different object sizes or object intensities? Can the segmentation of an image automatically be decomposed with respect to those scales?}\n\nMany region-based segmentation methods only use constraints on the contour length or curvature as regularization. However, in view of shape optimization and dictionary learning an approach that could also automatically separate objects with respect to their shape (cf. Figure \\ref{fig:overview_goal_segmentation}(c)) would be very interesting. \\textit{Hence, what is the role of geometric shapes in a multiscale segmentation approach?}\n\n\n\\textit{Scale-space methods.}\nIn the previous decades there has been a continuous interest in the analysis of different scales and the construction of scale spaces in imaging. In general it is desired to automatically detect all scales present in an image and simultaneously determine which scales are informative and contribute most to the image. For segmentation this problem is addressed in the fundamental works by Witkin and Koenderink \\cite{Witkin1983,Koenderink1984}. In relation to those, several methods to detect and analyze interesting scales have been proposed, see for example \\cite{Lifshitz1990,Vincken1997,Tabb1997,Lindeberg1998,Florack2000,Letteboer2004}. The underlying scale-space that is examined is defined by a linear diffusion process. A drawback of those approaches is that linear diffusion smoothes edge information and is therefore in general not suitable for applications where one is interested in retaining sharp edge information. Especially in biomedical image applications this is often the case. Therefore those theories were extended to non-linear diffusion processes, see \\cite{Niessen1997,Niessen1999,Dam2000}. A drawback of these approaches is that their analysis of scales is not fully automatic and can only be used in a forward approach, thus going from fine to coarse scales and then trying to find a backward relation. In this work we concentrate on nonlinear diffusion processes for segmentation where scale automatically relates to intensity and size.\n\nA prominent example of a variational method for nonlinear diffusion is the ROF model \\cite{Osher2005}. With increasing regularization parameter $\\alpha$ a sequence of functionals generates a nonlinear forward scale space flow that filters signals from fine to coarse. However in this process the total variation regularization functional is known to lead to a systematic contrast loss in the filtered image $u$ \\cite{Meyer2001}, whereas the main discontinuities in the signal remain at their position in the domain. To tackle the problem of intensity loss Osher et al. proposed in \\cite{Osher2005} an iterative contrast enhancement procedure based on Bregman distances. This approach is known to generate a nonlinear inverse scale space flow generating filtered signals from coarse to fine and with improved quality. This idea was successfully applied to more general inverse problems. \\cite{Burger2007,Brune2011,Benning2013}\n\n\\textit{Spectral methods.} Recently Gilboa \\cite{Gilboa2013,Gilboa2014} developed a framework to detect scales based on the nonlinear total variation diffusion process. The total variation is known to retain edge information while smoothing the signal apart from the edges. In this framework scales are detected based on a spectral decomposition of the given image into TV eigenfunctions \\cite{Meyer2001}. This concept does not only hold true for higher-order regularization functionals \\cite{Papafitsoros2015511,Poeschl2015,Benning2013} but more generally for convex, one-homogenous functionals $J$ with corresponding nonlinear eigenvalue equations \\cite{Burger2015,Burger2016} of the following form\n\\vspace{-5pt}\n\\begin{equation*}\n\t\\lambda u \\in \\partial J(u) \\ \n\\end{equation*}\nwhere $\\partial J(\\cdot)$ denotes the subdifferential of $J$. When minimizing for instance the total variation $J$, those eigenfunctions $u$ simply loose contrast whereas the overall structure of the function remains the same. The magnitude of contrast loss is related to the eigenvalue $\\lambda$. The eigenfunctions shape is determined by the chosen norm in the TV functional which can be adapted to the application of interest. In this way signals are not linearly smoothed to overcome scales but are step-by-step transformed to a composition of nonlinear eigenfunctions at coarser scales. A spectral response function can be used to examine which scales have a strong contribution to the original signal and to design filters of certain scales. Moreover, such spectral method can be combined with forward and inverse scale space approaches \\cite{Burger2016}.\\newpage\n\n\n\\textit{Main contribution.} In this paper we will extend the idea of (inverse) scale space methods known for nonlinear diffusion processes to segmentation and shape detection problems.\n\\begin{itemize}\n\\setlength{\\itemsep}{3pt}\n\t\\setlength{\\itemindent}{-.2in}\n\t\\item First goal: A novel inverse multiscale segmentation method based on Bregman iterations\n\t\\item Second goal: An adaptive regularization parameter strategy for $\\alpha$ (independent of $c_1$, $c_2$)\n\t\\item Third goal: A spectral analysis for segmentation regarding shapes of eigenfunctions\n\\end{itemize}\nA TV-based forward scale space for segmentation can easily be derived from the CV model via an increasing regularization parameter. We extend this framework by an inverse scale space for segmentation, still based on the CV model and therefore on a nonlinear diffusion process. For this purpose, we make use of Bregman iterations, among others well-known for improving total variation denoising results. The relation between the forward scale space and the inverse scale space is examined. Both iterative strategies are accomplished by a spectral transform and response function, which are used to easily examine scales and to filter certain scales. Since our method uses the total variation as a nonlinear diffusion process, we can make use of relatively easy and fast numerical and parallel implementation schemes developed in recents years.\n\n\\textit{Organization.} This work is organized as follows.\nIn Section \\ref{sec:CVmodel} we start with a revision of the segmentation model by Chan and Vese including its convexification. Together with a revision of the iterative denoising strategy using Bregman iterations in Section \\ref{ref:bregmandnoising}, the combination of those two concepts forms the first ingredient of our new inverse scale space method for multiscale segmentation introduced in Section \\ref{sec:bregman-cv}. An interesting interpretation of the method as an adaptive regularization method is presented in Section \\ref{sec:reg-param}. \nSection \\ref{sec:spec-analysis} deals with nonlinear spectral methods and contains the second main ingredient of our new approach. In \\ref{sec:genTV} we start with a brief summary of generalized nonlinear total variation functionals addressing different eigenshapes and continue in \\ref{sec:spec-analysisdenoising} with recent works by Gilboa et al. solving related nonlinear eigenvalue problems in imaging. In Section \\ref{sec:spec-analysissegm} we extend those ideas from nonlinear image denoising to image segmentation.\nIn Section \\ref{sec:numrealization} we describe the numerical realization of our approach using primal-dual convex optimization methods. \nIn Section \\ref{sec:results} we first illustrate the strengths and limitations of our multiscale segmentation method by studies on synthetic datasets with a certain focus on eigenfunctions and shapes. Moreover, we underline the potential and wide applicability by three different biomedical imaging applications. Its reliable performance is demonstrated on real fluorescence microscopy images that contain Circulation Tumor Cells, with various shapes and sizes, among white blood cells and debris in \\ref{sec:resultsCTC}. Besides, we present results on electron microscopy images suffering from inhomogeneous backgrounds in \\ref{sec:resultsEM} and interesting results on network-like shapes representing vascular systems in \\ref{sec:resultsnetwork}.\nWe end with a conclusion and an outlook to future possible perspectives in Section \\ref{sec:conl}.\n\\section{Modeling Segmentation with Inverse Scale Spaces}%\nIn the following section we will shortly describe the model by Chan and Vese \\cite{Chan2001} for segmentation and the adaption of the ROF model for denoising using Bregman distances \\cite{Osher2005}. Afterwards we will introduce our novel Bregman-CV model for segmentation and show some advantages of our model. \n\\subsection{Globally Convex Segmentation}\\label{sec:CVmodel}\nThe idea of the CV model has originally been derived from the more general model for image segmentation introduced by Mumford and Shah (\\cite{Mumford1989}). Here, one seeks for a solution of the variational energy\n\\begin{equation*}\nJ^{\\text{MS}}(u,C) = \\int_{\\Omega} |f(x) - u(x)|^2 \\mathrm{d}x +  \\alpha \\cdot Per(C) + \\beta \\cdot \\int_{\\Omega\\setminus C} |\\nabla u(x)| ^2 \\mathrm{d}x \\longrightarrow \\min_{u,C}\n\\end{equation*}\nwhere $u$ is a differentiable function that is allowed to be discontinuous on $C$. $C$ describes the union of the boundaries and thereby represents the contour defining the segmentation. Thus, $u$ is a smooth approximation of the original image $f$ and is composed of several regions $\\Omega_i$. Within each region $\\Omega_i$, $u$ is smooth. If we restrict this model so that $u$ is composed of only two regions $\\Omega_1$ and $\\Omega_2$ we derive with $\\beta \\rightarrow \\infty$ the CV model for segmentation\n\\begin{equation*}\nJ^{\\text{CV}}(c_1,c_2,C) = \\int_{\\Omega_1} (f(x) - c_1)^2 \\mathrm{d}x + \\int_{\\Omega_2} (f(x) - c_2)^2 \\mathrm{d}x + \\alpha \\cdot Per(C)\\longrightarrow \\min_{C,c_1,c_2}.\n\\end{equation*}\nHere, $c_i$ is the intensity value of $u$ within the corresponding region $\\Omega_i$. These values need to be determined together with contour the $C$. Note that in comparison to the original model we omit the area regularization term (cf. \\eqref{eq:CVorig}).\n\nThe contour $C$ can be indirectly represented via a level-set function $\\Phi$ (\\cite{Osher1988}) with\n\\begin{equation*} \nC = \\{x \\in \\Omega: \\Phi(x) = 0\\} \n\\end{equation*} \nand $\\Phi(x)$ being positive if and only if $x \\in \\Omega_1$. Together with the Heaviside function\n\\begin{equation*}\n\tH(\\Phi(x)) = \\begin{cases}1 &\\text{ if }\\Phi(x)\\geq 0 \\\\ 0 &\\text{ if }\\Phi(x)< 0\\end{cases}\n\\end{equation*}\nand its regularized version $H_{\\epsilon}$ this results in\n\\begin{align*}\nJ^{\\text{CV2}}(c_1,c_2,\\Phi) = \\int_{\\Omega} (f(x) - c_1)^2 H_{\\epsilon}(\\Phi(x)) \\mathrm{d}x & + \\int_{\\Omega} (f(x) - c_2)^2 (1 - H_{\\epsilon}(\\Phi(x)))\\mathrm{d}x \\\\&+ \\alpha \\cdot \\int_{\\Omega}|\\nabla H_{\\epsilon}(\\Phi(x))|\\mathrm{d}x~\\longrightarrow~ \\min_{\\Phi,c_1,c_2}.\n\\end{align*}\nThe contour $C$ evolves during minimization until it reaches a minimum which, in the ideal case, describes the object boundaries. Besides the original minimization strategy by gradient descent, several minimization methods to solve the CV model have been developed, see for example \\cite{He2007,Zehiry2007,Badshah2008,Bae2009}. One disadvantage of the model is its non-convexity which makes the solution depending on the used initialization. With a badly chosen initialization the minimization might get stuck in a local minimum that corresponds to a bad or meaningless segmentation.\n\nFor a better understanding of the relation between the nonlinear denoising model by Rudin, Osher and Fatemi (\\cite{Rudin1992}) and the CV segmentation model we use the total variation defined as\n\\begin{equation}\\label{eq:TV}\nTV(u) := \\sup\\limits_{\\substack{\\varphi \\in C_0^{\\infty}(\\Omega; \\mathbb{R}^2)\\\\ ||\\varphi||_{\\infty} <1}} \\int_{\\Omega} u \\nabla\\cdot \\varphi \\;\\mathrm{d}\\mu \\text {\\ \\  with \\ \\ } BV(\\Omega) := \\{ u \\in L^{1}(\\Omega)|TV(u) < \\infty\\}.\n\\end{equation}\nThe total variation of a characteristic $u(x) = \\begin{cases} 1 &\\text{\\  if  \\ } x \\in \\Omega_1 \\cup C\\\\ 0  &\\text{ if } x \\in \\Omega_2\\end{cases}$ corresponds to the contour length $|C|$ which can be shown by the co-area-formula.\nTherefore we can formulate the segmentation problem as\n\\begin{equation} \\label{eq:CV3} J^{CV3}(c_1,c_2,u) = \\int_{\\Omega} u\\left((f(x)-c_1)^2 - (f(x) - c_2)^2\\right)\\mathrm{d}x + \\alpha \\; TV(u)\n\\longrightarrow \\min_{\\substack{u\\in BV(\\Omega),c_1,c_2\\\\u(x)\\in \\{0,1\\}}}.\n\\end{equation}\nFor fixed $c_1, c_2$ the solution of \\eqref{eq:CV3} corresponds to the solution of an ROF problem with binary constraint (\\cite{Burger2012}):\n\\begin{equation}\\label{eq:CV-ROF}\n \\min_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in \\{0,1\\}}}\\frac{1}{2}|| u(x) - r(x)||_2^2 + \\alpha  TV(u)\n \\end{equation}\nwith $r(x) = (f(x)-c_2)^2 \\!-\\! (f(x) - c_1)^2 \\!-\\!\\frac{1}{2}$.\n\nThe regularization parameter $\\alpha$ in the segmentation model \\eqref{eq:CV3} has the role of a scale parameter, meaning that $\\alpha$ determines the scale of the objects that are segmented. The CV model describes a forward scale approach, thus a small parameter $\\alpha$ corresponds to small scales that are segmented. An increased regularization parameter results in a solution where the smaller scales are not segmented but only larger ones. The meaning of scale is determined by the regularization functional, in this case the total variation. The total variation encodes a measure of the contour length as well as the height of piecewise constant areas. One disadvantage is that due to the 1-homogeneity of $TV$ our method cannot distinguish between height and contour length. Thus, a small object with a bright intensity can have the same scale as a large object with a less bright intensity. For more details see section \\ref{sec:results}.\n\\\n\\paragraph{Convexification}\nThe CV segmentation model \\eqref{eq:CV3} as well as the binary ROF model \\eqref{eq:CV-ROF} are both not convex. Even for fixed values of $c_1$ and $c_2$ both models are non-convex due to the binary constraint on $u$. As mentioned before this might result in local instead of global minimum solutions. Approaches to overcome this difficulty and find global minima of the CV model are presented for example in\\cite{Chan2006,Bresson2007,Goldstein2010,Brown2012}. In \\cite{Chan2006} the authors showed that global minimizers of \\eqref{eq:CV3} for any given fixed $c_1,c_2 \\in \\mathbb{R}$ can be found by solving\n \\begin{equation}\\label{eq:convCV}\nJ^{CV3}(c_1,c_2,v) = \\int_\\Omega v ((f(x)-c_1)^2 - (f(x)-c_2)^2) dx + \\alpha~TV(v) \\longrightarrow \\min_{v \\in BV(\\Omega),~v(x) \\in [ 0,1 ]}\n\\end{equation}\nand defining $ u(x) := v^{\\ast}(x) \\geq \\mu$ for a.e. $\\mu \\in [0,1]$.\n\nThus, the binary constraint can be relaxed and combined with a thresholding. Here, the variational model to solve is convex, though not strictly convex. One should bear in mind that the found solution is therefore not unique. Yet solutions of \\eqref{eq:convCV} are close to binary even if the constraint is relaxed. Thus for most choices of $\\mu$ we derive the same solution which means that the choice of $\\mu$ has only a very limited impact on our method. Therefore we don't see any disadvantages when choosing the global but not unique minimum $u(x) := v^{\\ast}(x) \\geq 0.5$. A fully convex formulation (including the constants $c_1$ and $c_2$) of problem \\eqref{eq:CV3} can be found in \\cite{Brown2012}. This method is computationally less efficient and currently we don't think that, in our method, the advantages of the full convexity outweigh the increased computational time. \n\\subsection{Inverse Scale Space for TV-Denoising}\\label{ref:bregmandnoising}\nBefore introducing our new segmentation model in the following subsection we will first recall some properties of the well-known ROF model \\cite{Rudin1992} and its extension by Bregman distances introduced in \\cite{Osher2005}. To denoise an image corrupted by additive Gaussian noise, \\cite{Rudin1992} proposed to solve the nonlinear variational problem  \n\\begin{equation}\\label{eq:ROF}\n\\frac{1}{2}||u - f||_2^2 + \\alpha \\ TV(u) \\longrightarrow \\min_{u \\in BV(\\Omega)}\n\\end{equation}\nreferred to as the ROF model. \nSimilar to the CV model this generates a forward scale space flow regarding the scale parameter $\\alpha$. An increased parameter $\\alpha$ leads again to a solution $u$ where fine scales are removed and vice versa. The total variation regularization functional is known to lead to a systematic contrast loss in the denoised image $u$ \\cite{Meyer2001}. To tackle this problem Osher et al. proposed in \\cite{Osher2005} an iterative contrast enhancement procedure based on Bregman distances. Instead of using the total variation regularization functional as before, information about the solution $u$ that we gained from a prior solution of problem \\eqref{eq:ROF} is included. Therefore, problem \\eqref{eq:ROF} is replaced by a sequence of variational problems \n\\begin{equation}\\label{eq:Bregman-ROF}\nu_{k+1} = \\argmin_{u\\in BV(\\Omega)} \\frac{1}{2}||u - f||_2^2 + \\alpha\\ D^{p_k}_{TV}(u,u_k).\n \\end{equation}\nThe regularization $D^{p_k}_{TV}(u,u_k):=TV(u)-TV(u_k)-\\langle p_k,u-u_k\\rangle$ is the Bregman distance of $u$ to the previous iterate $u_k$ with respect to the total variation. $p_k \\in \\partial TV(u_{k})$ is an element in the subdifferential of the total variation of the prior solution $u_{k}$. Although this subdifferential might be multivalued, the iterative regularization algorithm automatically selects a unique subgradient based on the optimality condition. For $k = 0$ we set $u_0 = p_0 = 0$.\nThe iterative strategy of this model is as follows: We start with a large parameter $\\alpha$ that results in an oversmoothed solution $u_1$ that consists of only large scales. In every iteration step finer scales are added back to the solution. Thus, the scale parameter that determines the range of the scales present in $u$ is the iteration parameter $k$. In contrast to the forward approaches presented before, a small $k$ corresponds to coarse scales and a large $k$ to very fine scales. Therefore, the Bregman-ROF denoising approach is an inverse scale space approach. The authors showed that this strategy leads to enhanced contrast of the final solution $u_{k_{\\text{max}}}$ compared to the solution of \\eqref{eq:ROF}. Hence, solving \\eqref{eq:Bregman-ROF} instead of \\eqref{eq:ROF} with increasing $\\alpha$ is not only an inverted way of detecting scales. This method rather allows for a detection of solutions which cannot be obtained by an adequate choice of $\\alpha$ in the original ROF model.\\\\\n\n\\subsection{Bregman-CV Segmentation Model}\\label{sec:bregman-cv}\nIn the following section we will introduce our new inverse scale space approach for segmentation. It is based on the similarity of the ROF functional and the CV functional shown in \\eqref{eq:CV-ROF}. Similar to the Bregman-ROF denoising problem we replace the total variation regularization by an iterative regularization based on Bregman distances. Thus, the resulting novel segmentation model is given by\n\\begin{equation*}\nu_{k+1} = \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right)\n+ \\alpha \\ D^{p_k}_{TV}(u,u_k)\n\\end{equation*}\nBy inserting the definition of the Bregman distance $D^{p_k}_{TV}(u,u_k):=TV(u)-TV(u_k)-\\langle p_k,u-u_k\\rangle$ and ignoring the parts independent of $u$ we derive the following model.\n\\begin{equation}\\label{eq:Bregman-CV}\nu_{k+1}  = \\argmin_{u\\in BV(\\Omega)} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\chi_{[0,1]}(u)\n+ \\alpha \\ \\left(TV(u)-<u,p_k>\\right)\n\\end{equation}\nwith $p_k \\in \\partial TV(u_k)$, $p_0 = 0$ and $ \\chi_{[0,1]}(u) = 0$ if $u(x) \\in [0,1]$ and equal to infinity elsewhere. The range of the scales present in $u_{k+1}$ is again determined by the iteration index $k$. This model is an inverse scale space approach, thus a small $k$  corresponds to a large scale segmentation and vice versa. \n\nBy definition of the Bregman distance it is $p_k \\in \\partial TV(u_k)$ where the subdifferential is multivalued. Therefore we need to determine a rule to choose $p_k$. One way is to derive an update strategy based on the optimality condition of \\eqref{eq:Bregman-CV} (cf. \\cite{Osher2005}):\n\\begin{equation*}\n0 = \\left((f-c_1)^2 - (f-c_2)^2\\right) + q_{k+1} + \\alpha p_{k+1} - \\alpha p_k\\text{\\ \\ \\ (opt.cond.)}.\n\\end{equation*}\nHere, $q_{k+1}$ is an element in the subdifferential of the characteristic function $\\chi_{[0,1]}(u_{k+1})$. The subdifferential of a characteristic function is a normal cone and is in our case given by \n\\begin{equation*}\nq_k(x) \\in \\begin{cases} (-\\infty,0] &\\mbox{if} \\quad u_k(x) = 0 \\\\ \\{ 0 \\} &\\mbox{if} \\quad 0 < u_k(x) < 1 \\\\ [0,\\infty) &\\mbox{if}\\quad u_k(x) = 1\\end{cases}.\n\\end{equation*}\nThus, we can choose $q_k = 0$ and neglect it from hereon. The update strategy for $p_{k+1}$ is then given by\n\\begin{equation}\np_{k+1} = p_k - \\frac{1}{\\alpha}\\left((f-c_1)^2 - (f-c_2)^2\\right) = -\\frac{k+1}{\\alpha}\\left((f-c_1)^2 - (f-c_2)^2\\right)\\label{eq:update_p}.\n\\end{equation}\nThis update is independent of $u_k$, thus it is a pointwise constant update in every iteration.\n\n\\subsection{Interpretation as an Adaptive Regularization Approach}\\label{sec:reg-param}\nOne important question is whether solutions of this model are in some sense improved compared to the solutions of the original CV model. We mentioned before that Bregman iterations lead to a contrast enhancement when applied to the ROF functional and that solving the CV model corresponds to solving a binary ROF. Yet, one should bear in mind that a contrast enhancement is meaningless in the case of a binary image since the contrast is already determined by the binary constraint. This is supported by the following observation: when inserting \\eqref{eq:update_p} into \\eqref{eq:Bregman-CV} we get\n\\begin{align*}\nu_{k+1} & = \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}}\\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\alpha \\left(TV(u)-<u,p_k>\\right)\\\\\n& =  \\argmin_{\\substack{u\\in BV(\\Omega)\\\\u(x)\\in [0,1]}} \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right) + \\frac{\\alpha}{(k+1)} \\ TV(u)\n\\end{align*}\nWith this, it is straightforward to see that all solutions $u \\in BV(\\Omega)$ derived by the Bregman-CV model can also be found by the original CV model. Nevertheless, there are advantages of using the iterative update strategy.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Linearly spaced $\\alpha$'s from 50 to 1.]{\\includegraphics[width=0.35\\textwidth]{img\/normal_alphas.png}}\\label{fig:reg_params_linear }\\quad \n  \\subfigure[\"$\\alpha$'s\" resulting from 50 Bregman steps with $\\alpha = 50$.]{\\includegraphics[width=0.35\\textwidth]{img\/bregman_alphas.png}}\\label{fig:reg_params_bregman}\\quad \n  \\caption{Comparison between linearly spaced regularization parameters and the automatically chosen parameters in the Bregman-CV model.}\n\\label{fig:reg_params} \n\\end{figure}\nIn Figure \\ref{fig:reg_params} b) the resulting $\\tilde\\alpha = \\frac{\\alpha}{k+1}$ for $\\alpha = 50$ and 50 Bregman iterations are shown. It is obvious that in the first iterations the decrease of the regularization parameter is much larger compared to later iterations. This is reasonable since first the large scales are reconstructed and in later Bregman iterations the finer scales are incorporated in the result. Making large steps with $\\alpha$ in large scales and becoming finer is therefore a reasonable strategy. A large decrease in the later iterations would probably miss some scales in between while in the first iterations too small steps are not reasonable. With this strategy the problem of automatically choosing the regularization parameter is less severe. By choosing a large $\\alpha$ and performing multiple Bregman iterations, a broad spectrum of scales is detected. Yet one should bear in mind that a strategy to automatically detect important scales is needed for a fully automated framework. \n\\section{A Spectral Method for Multiscale Segmentation}\\label{sec:spec-analysis} \nThe analysis of eigenvalues and spectral decomposition is a well-known theory in the field of linear signal and image processing (see e.g. \\cite{MarpleJr1987} or \\cite{Stoica2005} for a more recent overview). Since nonlinear regularization became popular in the last years, there is a growing interest in generalizing this theory to nonlinear operators. In \\cite{Benning2012} Benning et al. examined singular values for nonlinear, convex regularization functionals and in  \\cite{Gilboa2013,Gilboa2014} Gilboa transferred the idea of spectral decompositions to the nonlinear total variation functional and related operators. The general idea is to examine solutions of the nonlinear eigenvalue problem \n\\begin{equation}\\label{eq:eigenval}\n\\lambda u \\in \\partial J(u)\n\\end{equation}\nwhere $\\partial J$ denotes the subdifferential of a (one-homogeneous) convex functional usually representing regularization in inverse imaging problems. \nBy transferring solutions of \\eqref{eq:eigenval} to sparse peaks in a spectral domain, advanced filters enhancing or suppressing certain image components can easily be designed. This concept was first introduced for the total variation functional and later generalized to one-homogenous functions and analyzed in \\cite{Burger2015}, \\cite{Gilboa2015} and \\cite{Burger2016}. \n\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Solution of the ROF model for different $\\alpha$.]{\\includegraphics[height=0.22\\textheight]{img\/tv_block_1d.png}}\\label{fig:TVblock }\\quad \n  \\subfigure[Solution of the inverse Bregman-ROF model for different Bregman iterations.]{\\includegraphics[height=0.22\\textheight]{img\/bregmantv_block_1d.png}}\\label{fig:BregmanTVblock}\\quad \n  \\caption{Solutions of the ROF and Bregman-ROF model in case the signal is a TV eigenfunction. Different regularization parameters $\\alpha$ in the forward scale (left) and Bregman steps in the inverse scale space (right).}\n\\label{fig:1dEF} \n\\end{figure}\nIn the case of a one dimensional signal the eigenfunction of the total variation corresponds to a single block with constant height, see Figure \\ref{fig:1dEF}. When increasing the regularization parameter in the ROF model \\eqref{eq:ROF}, i.e. when minimizing the total variation of this block signal, the block looses it height but the edges remain at the same position (cf. Figure \\ref{fig:1dEF} (a)). Therefore all signals can be seen as the original signal multiplied by a scalar and are eigenfunction of the total variation. The same holds true for solutions of the Bregman-ROF model \\eqref{eq:Bregman-ROF}. The only difference is that now eigenfunctions of the TV functional (blocks) are reshaped with increasing number of Bregman iterations instead of removed (see Figure \\ref{fig:1dEF}(b)). This reflects the inverse scale approach of the Bregman-ROF model. Different to our novel approach, the Bregman updates in the Bregman-ROF model cannot be reformulated as an adaptive regularization parameter choice. As far as we know it is not entirely clear how the forward and inverse TV flow for denoising relate to each other.\n\n\n\\subsection{Generalized Definition of the Total Variation}\\label{sec:genTV}\nIn the previous paragraph we mentioned that in the one dimensional case the eigenfunction of the total variation functional is a block with constant height. If we want to proceed to the two dimensional case  the eigenfunctions are not as clearly defined as in the 1D case. Different than before we now have the freedom to choose the norm body that is used within the infinity norm in the TV definition \\eqref{eq:TV}. This choice determines the shape of the eigenfunctions. To reflect this dependence on the chosen norm in the TV definition we introduce a generalized version of the TV definition:\n\\begin{equation}\\label{eq:TVgen}\nTV_\\gamma(u) :=  \\sup\\limits_{\\substack{\\varphi \\in C_C^{1}(\\Omega; \\mathbb{R}^d)\\\\ \\varphi(x) \\cdot n <\\gamma(n) \\forall n\\in\\mathbb{R}^{d}}}\\ -\\int_{\\Omega} u \\nabla\\cdot \\varphi  \\mathrm{d}x.\n\\end{equation}\nHere, $\\gamma : \\mathbb{R}^d \\rightarrow \\mathbb{R} $ is a convex, positively 1-homogeneous function such that  $\\gamma (x) > 0$ for $x \\neq 0$.\nIf $u$ is a function in $\\mathcal{W}^{1,1}(\\Omega)$ the primal definition of equation \\eqref{eq:TVgen} is given by\n\\begin{equation*}\nTV_\\gamma(u) :=  \\int_{\\Omega} \\gamma(\\nabla u) \\mathrm{d}x.\n\\end{equation*}\nIn both definitions the choice of $\\gamma$ determines the shape of the eigenfunctions. We refer to \n \\begin{equation*}\n \\mathcal{F}_{\\gamma} := \\large\\{ z \\in \\mathbb{R}^{d} : \\gamma(z) \\leq 1\\large\\}\n \\end{equation*}\nas the Frank diagram and the corresponding Wulff shape is defined as\n\\begin{align*}\\mathcal{W}_{\\gamma} :=& \\large\\{ z \\in \\mathbb{R}^{d} : z \\cdot x \\leq \\gamma(x) \\text{ for all } x\\in \\mathbb{R}^{d} \\large\\}\\\\\n= &\\large\\{ z \\in \\mathbb{R}^{d} : \\gamma^{\\ast}(x) := \\sup\\limits_{x\\in \\mathbb{R}^{d}} \\ \\frac{z \\cdot x}{ \\gamma(x)} \\leq 1 \\large\\}.\\end{align*}\nNote that from definition \\eqref{eq:TVgen} together with the definition of the Wulff shape we can conclude that $\\varphi(x) \\in \\mathcal{W}_{\\gamma}$. Therefore, for every choice of $\\gamma$, functions with the same shape as the Wulff shape are eigenfunctions of $TV_\\gamma$ (see \\cite{esedoglu2004} Theorem 4.1). The Frank diagram $\\mathcal{F}_{\\gamma}$ and the Wulff shape $\\mathcal{W}_{\\gamma}$ for the three most common choices of $\\gamma$ are presented in Table \\ref{tab:shapes}. For simplicity we will omit the subscript $\\gamma$ in $TV_\\gamma$ from hereon, but we will come back to it in the numerical results in section \\ref{sec:results}.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n$\\gamma(x)$ & $\\mathcal{F}_{\\gamma}$ & $\\mathcal{W}_{\\gamma} $\\\\[7pt]\n\\hline\n$\\gamma = \\| \\cdot \\|_{1}$ & \\includegraphics[height = 0.05\\textheight]{img\/square_rot.png} & \\includegraphics[height = 0.05\\textheight]{img\/square.png}\\\\[6pt]\n\\hline\n$\\gamma = \\| \\cdot \\|_{2}$& \\includegraphics[height = 0.05\\textheight]{img\/circle.png} & \\includegraphics[height = 0.05\\textheight]{img\/circle.png}\\\\[6pt]\n\\hline\n $\\gamma= \\| \\cdot \\|_{\\infty}$ &\\includegraphics[height = 0.05\\textheight]{img\/square.png} & \\includegraphics[height = 0.05\\textheight]{img\/square_rot.png}\\\\\n\\hline\n\\end{tabular}\n\\caption{Examples for $\\gamma$ and the corresponding Frank diagrams $\\mathcal{F}_{\\gamma}$ and Wulff shapes $\\mathcal{W}_{\\gamma}$. The Wulff shape corresponds to the shape of the eigenfunctions of the $TV_{\\gamma}$ functional.}\n\\label{tab:shapes}\n\\end{table}\n\n\\subsection{Spectral Analysis for Nonlinear Functionals}\\label{sec:spec-analysisdenoising}\nIn the following section we will first review the basic ideas of spectral TV analysis before transferring those ideas to segmentation in the next section. The idea to decompose an image based on the basic TV elements was presented by Gilboa in \\cite{Gilboa2013,Gilboa2014}. These basic TV elements, called eigenfunctions of the total variation functional, are all functions $u \\in BV(\\Omega)$ that solve the nonlinear eigenvalue problem\n\\begin{equation}\\label{eq:eigenvalProb}\n\\lambda u \\in \\partial TV(u).\n\\end{equation}\nIn the previous section we already presented some examples of TV eigenfunctions (see e.g. Figure \\ref{fig:1dEF} and Table \\ref{tab:shapes}). A  more general description of these eigenfunctions in case $\\gamma$ is isotropic is given in \\cite{Bellettini2002}. Bellettini et al. showed that all indicator functions $ \\mathbbm{1}_C(x)$ of a convex and connected set $C$ with finite perimeter $Per(C)$ which admit\n\\begin{equation}\\label{eq:l2eig}\n\\esssup_{p \\in \\partial C} \\kappa (p) \\leq \\frac{Per(C)}{Area(C)}\n\\end{equation}\nwhere $Area(C)$ denotes the area of $C$ and $\\kappa$ the curvature of $\\partial C \\in C^{1,1}$ are solutions of \\eqref{eq:eigenvalProb} and therefore eigenfunctions. Obtaining an analog condition for anisotropic, smooth and strictly convex choices of $\\gamma$ is straightforward but more challenging in the case of non-smooth or non-strictly convex choices of $\\gamma$. Candidates of these shapes are presented in \\cite{bellettini2001}. In \\cite{esedoglu2004} Esedoglu and Osher gave an example of a TV eigenfunction for $\\gamma = \\| \\cdot \\|_{1}$ that is not a Wulff shape (see section \\ref{sec:resultsshapes} for more details).\n\nIn order to detect eigenshapes at different scales in a given signal $f$ a scale space approach is needed. One way to define a scale space based on the total variation is given by the total variation flow \\cite{Andreu2001,Andreu2002,Bellettini2002,Burger2007,Steidl2004}. The TV flow arises when minimizing the total variation with steepest descent method and is defined as \n\\begin{equation}\\label{eq:tvflow}\n\\begin{aligned}\nu_t(t,x) &= -p(t,x) \\hspace{15pt} \\text{for } p(t,x) \\in \\partial TV(u(t,x))\\\\\nu(0,x) &= f(x)\n\\end{aligned}\n\\end{equation}\nwith Neumann boundary conditions. For $f(x) = \\mathbbm{1}_C(x)$, with $C$ defined as above, the unique solution of \\eqref{eq:tvflow} is $u(t,x) = (1 - \\lambda_C t) ^{+}  \\mathbbm{1}_C(x)$ with $\\lambda_C = \\frac{Per(C)}{Area(C)}$ (cf. \\cite{Bellettini2002}). Hence, the time derivative $u_t(t,x)$ is given by the original signal multiplied with a scalar and $u$ is an eigenfunction. To obtain a suitable framework to decompose or filter images based on these eigenfunction Gilboa proposed to define a spectral framework that transforms eigenfunctions to peaks in the spectral domain. If $u(t,x)$ is a solution of \\eqref{eq:tvflow} the TV spectral transform and spectral response can be defined as\n\\begin{equation}\\label{eq:tvtransform}\n\\phi (t,x) = u_{tt}(t,x)\\cdot t \\quad\\text{and}\\quad S(t) = || \\phi(t,x) ||_{L^{1}(\\Omega)}.\n\\end{equation}\nNote, that there are alternative definitions of the spectral response presented in \\cite{Burger2015}. \n\nAnother approach to construct an forward scale space is based on the variational ROF problem. Instead of solving \\eqref{eq:tvflow} the ROF model\n\\begin{equation}\\label{eq:ROFscalespace}\n \\min_{u\\in BV(\\Omega)}\\frac{1}{2}|| u(x) - f(x)||_2^2 + t \\cdot  TV(u).\n\\end{equation}\nis solved for different regularization parameters. Hence, $t$ is the (artificial) time variable that determines the scale comparable to $t$ in the TV-flow approach \\eqref{eq:tvflow}. One drawback of this approach is that there is no clear rule for the choice of different $t$'s. The spectral transform and response function can be equivalently defined as in \\eqref{eq:tvtransform}. \n\nA third, but significantly different, scale space approach is an inverse scale approach. The inverse scale space flow is defined as\n\\begin{equation}\\label{eq:inverseflow}\n\\begin{aligned}\np_s(s,x) &= f(x) - u(s,x) \\hspace{15pt} \\text{for } p(s,x) \\in \\partial TV(u(s,x))\\\\\nu(0,x) &= p(0,x) = 0.\n\\end{aligned}\n\\end{equation}\nThus, the flow is now defined on the dual variable $p \\in \\partial TV(u)$ and $s$ is the time variable determining the scale. Note that in \\cite{Burger2005} it was shown that the iterative Bregman-L2-TV model \\eqref{eq:Bregman-ROF} can be associated with a discretization of \\eqref{eq:inverseflow} for $\\frac{1}{\\alpha} \\rightarrow 0$.\nAs \\eqref{eq:inverseflow} is an inverse approach the time variable $t$ can be associated with $\\frac{1}{s}$. In this case the spectral transform and response functions are defined as\n\\begin{equation}\\label{eq:tvtransform2}\n\\phi (s,x) = u_{s}(s,x) \\quad\\text{and}\\quad S(s) = || \\phi(s,x) ||_{L^{1}(\\Omega)},\n\\end{equation}\nwhere $u(s,x)$ is the solution of \\eqref{eq:inverseflow}. Note, that for small $s$ $\\phi(s,x)$ now measures changes in the coarse scales. See \\cite{Burger2015} for more details.\n\nWith all three approaches we are able to transform a signal to the spectral domain and detect different scales based on TV eigenfunctions. If we assume that $\\phi(t,x)$ is integrable over time, the original signal $f$ can be reconstruct via\n\\begin{equation*}\nf(x) = \\int_0^{\\infty} \\phi(t,x) dt + \\bar{f}\n\\end{equation*}\nwhere $\\bar{f}$ is the average of $f$. Filters can be defined with $\\phi_{H}(t,x) = H(t)\\phi(t,x)$ via \n\\begin{equation*}\nf_{H}(x) = \\int_0^{\\infty} \\phi_{H}(t,x) dt + H(\\infty)\\bar{f}.\n\\end{equation*}\n\n\n\\subsection{Spectral Response of Multiscale Segmentation}\\label{sec:spec-analysissegm} \nIn section \\ref{sec:bregman-cv} we presented two variational models to detect segmentations of a given image $f$ at different scales. To decompose the segmentation into different scales and detect important scales or clusters of scales in the segmentation we want to transfer the idea of spectral analysis based on the total variation (cf. sec. \\ref{sec:spec-analysisdenoising}) to segmentation. Therefore we need to find a suitable transformation of the segmentation $u$ to the spectral domain and vice versa. Note that our goal is not to reconstruct the original signal $f$ or filtered versions of $f$ but the reconstructed function should be a segmentation itself. To do so we make use of the idea that eigenfunctions of the TV functional should be transformed to peaks in the spectral domain. In the following we will derive this spectral transform function for a forward scale space and an inverse scale space approach. To represent the forward scale space we associate the regularization parameter $\\alpha$ in the convex version of the original model by Chan and Vese \\eqref{eq:convCV} with the artificial time variable $t$. That means, we solve\n \\begin{equation}\\label{eq:convCV-scale}\n\\int_\\Omega u ((f(x)-c_1)^2 - (f(x)-c_2)^2) dx + t \\cdot TV(u) \\longrightarrow \\min_{u \\in BV(\\Omega),~u(x) \\in [ 0,1 ]}\n\\end{equation}\nfor different $t$.This is comparable to the variational approach in \\eqref{eq:ROFscalespace}. So far, we could not find a forward scale space representation that can be associated with a flow on $u$ comparable to the TV-flow. One difficulty is that the optimality condition of this model, i.e. $0 = (f-c_1)^2 - (f-c_2)^2 + \\alpha p$ with $p \\in TV(u)$, has no direct dependence on $u$. The inverse scale space representation is based on the Bregman-CV model we introduced in \\eqref{eq:Bregman-CV}. The optimality condition in each step of the iterative Bregman-CV strategy is given as\n \\begin{equation*}\n 0 = \\left((f-c_1)^2 - (f-c_2)^2\\right) + \\alpha \\left( p_k - p_{k-1}\\right) \\text{ with } p_k \\in TV(u_k) \\ \\forall \\ k \\\\.\n \\end{equation*}\nThe resulting equation\n\\begin{equation*}\n\\frac{p_k - p_{k-1}}{\\frac{1}{\\alpha}} = (f-c_2)^2 - (f-c_1)^2\n \\end{equation*}\ncan be interpreted as a discretization with stepsize $\\frac{1}{\\alpha}$ of\n\\begin{equation}\\label{eq:invsegslow}\n\\begin{aligned}\np_s(s,x) &=  (f(x)-c_2)^2 - (f(x)-c_1)^2\\hspace{15pt} \\text{with } p(s,x) \\in \\partial TV(u(s,x))\\\\\np(0,x) &= 0.\n\\end{aligned}\n\\end{equation}\nAgain, $s$ is the time variable that is inverse to the time variable $t$ in \\eqref{eq:convCV-scale}. We refer to this flow as the inverse scale space segmentation flow. Note that within this flow description there is no direct dependence on $u$ but $u$ is only indirectly given by $p \\in \\partial TV(u)$. Yet, when looking for solutions of this flow at different times $s$, we solve the Bregman-CV model for multiple Bregman iterations and there the corresponding $u$ is available. \n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Evolution of $u$ in \\eqref{eq:convCV-scale}.]{\\includegraphics[width=0.4\\textwidth]{img\/peak_segm.png}}\\quad \\quad\n  \\subfigure[Evolution of $u$ in \\eqref{eq:invsegslow}.]{\\includegraphics[width=0.4\\textwidth]{img\/peak_segm_bregman.png}}\n  \\caption{Illustration of the evolution of $u\\in\\{0,1\\}$ over time at a fixed point when $f$ is a TV eigenfunction shown for the forward (a) and inverse scale space approach (b). Note, due to the binary constraint on u the evolution of $u$ is already a step function, i.e. $u(x)$ is either part of the segmentation (u(x)=1) or not (u(x)=0).}\n\\label{fig:peaks} \n\\end{figure}\n\nTo transform the eigenfunctions to peaks in the spectral domain we define the spectral transform function as follows:\n\\begin{equation}\\label{eq:spectransseg}\n\\phi (t,x) =\\begin{cases} -u_{t}(t,x)& \\text{ (forward case)}\\\\\n\\ \\ \\ u_t(t,x)& \\text{ (inverse case)}\\end{cases}\\\\[3pt]\n\\end{equation}\nThis definition is motivated by Figure \\ref{fig:peaks} where the evolution of TV eigenfunctions over time in \\eqref{eq:convCV-scale} and \\eqref{eq:invsegslow} is illustrated. We can see, that the first derivative (in a distributional sense) is a peak at that time point where the eigenfunction vanishes or appears respectively. For the spectral response function we use the definition\n\\begin{equation}\n\tS(t) = || \\phi(t,x) ||_{L^{1}(\\Omega)}\n\t\\label{eq:spectralResponse}\n\\end{equation}\nintroduced by Gilboa (\\cite{Gilboa2013,Gilboa2014}). Here, the influence of certain scales to the segmentation is encoded. Using this function $\\phi$ and assuming integrability over time we can get the following backtransformation:\n\\begin{equation*}\nf^{\\text{seg}}(x) = \\int_0^{\\infty} \\phi(t,x) dt,\n\\end{equation*}\nwhere $f^{\\text{seg}}(x):= (f - c_2)^2 - (f-c_1)^2 + \\frac{1}{2} < \\frac{1}{2}$ is the results of the simple clustering problem\n$\\min_{u(x)\\in \\{0,1\\}} \\int_{\\Omega} u\\left((f(x)-c_1)^2 - (f(x) - c_2)^2\\right)\\mathrm{d}x$. By choosing a function $H$ with $H(t) \\in \\{ 0,1 \\}$ certain scales of interest can be filtered via\n\\begin{equation}\\label{eq:filterseg}\nf^{\\text{seg}}_{H}(x) = \\int_0^{\\infty} \\phi_{H}(t,x) dt \\ \\ \\text{ with } \\ \\ \\phi_{H}(t,x) = H(t)\\phi(t,x).\n\\end{equation}\nWith this framework we can easily segment an image and simultaneously detect important scales in this segmentation instead of using the spectral TV analysis as in Section \\ref{sec:spec-analysisdenoising} and segmenting separately. Moreover, using the filtering approach we can easily construct segmentations of only certain scales by filtering (cf. \\eqref{eq:filterseg}). Examples are shown in section \\ref{sec:results}.%\n\\section{Numerical Methods}\\label{sec:numrealization}\nOur novel approach introduced in the two previous sections consists of two parts. First we have to solve the Bregman-CV model  \\eqref{eq:Bregman-CV} introduced in section \\ref{sec:bregman-cv}. Afterwards we can analyze the detected scales using the spectral response function \\eqref{eq:spectralResponse} introduced in section \\ref{sec:spec-analysissegm}. Therefore an efficient solution of the Bregman-CV model is required. In the following section we will give a very brief introduction into primal-dual optimization schemes and then show how this can be used to solve \\eqref{eq:Bregman-CV}. We close the section by a speed comparison between a Matlab and a parallelized C\/Mex implementation of our code. \n\n\\subsection{Primal-Dual Optimization Methods}\nTo solve nonlinear problems of the form\n\\begin{equation}\\label{eq:primalprob}\n \\min_{x \\in X} \\ F(Kx) + G(x)\n \\end{equation}\nwith $F$ and $G$ being proper, convex, lower-semicontinuous functions, primal-dual optimization methods became very popular in the last years. Instead of minimizing the primal problem  \\eqref{eq:primalprob} they make use of the primal-dual formulation of this problem given by\n\\begin{equation}\\label{eq:primaldualprob}\n\\min_{x\\in X} \\max_{y\\in Y}\\  \\langle Kx,y\\rangle + G(x) - F^{\\ast}(y)\n\\end{equation}\nwith $G: X \\rightarrow [0, +\\infty]$ and $F^{\\ast}:Y \\rightarrow [0, +\\infty]$ being the convex-conjugate of $F$. By updating in every iteration step a primal and a dual variable these methods are able to avoid some difficulties that arise when working on the purely primal or dual problem. One example is the minimization of variational methods with TV regularization. If the gradient is zero, the TV functional is not differentiable which leads to problems in purely primal minimization schemes like gradient descent. Some examples for primal-dual minimization algorithms are the PDHG algorithm \\cite{Zhu2008}, a generalization of PDHG by Esser et al. \\cite{Esser2010}, the Split Bregman algorithm by Goldstein and Osher \\cite{Goldstein2009}, Bregman iterative algorithms \\cite{Yin2008}, the second-order CGM algorithm \\cite{Chan1999} and inexact Uzawa methods \\cite{Zhang2011}.\n\nIn \\cite{Chambolle2011} Chambolle and Pock proposed an algorithm which can be seen as a generalization of PDHG as well. This algorithm was originally proposed by Pock et al. in \\cite{Pock2009} to minimize a convex relaxation of the Mumford-Shah functional. The efficient first-order primal-dual algorithm to minimize general problems of the form \\eqref{eq:primaldualprob} is presented in Algorithm \\ref{alg:cp}. \n$ (I + \\sigma \\partial F^{\\ast})^{-1}$ and $(I + \\tau \\partial G)^{-1}$ are the resolvent operators of $F^{\\ast}$ and $G$ respectively which are defined through \n\\eq{y = (I + \\tau \\partial G)^{-1}(x) = \\argmin_{y} \\left\\{ \\frac{\\| y-x\\|^2}{2\\tau} + G(y) \\right\\}.}\n\\begin{algorithm}\n\t\\caption{First-order primal-dual algorithm by Chambolle and Pock (\\cite{Chambolle2011})}\n\t\\label{alg:cp}\n\t{\\fontsize{10}{10}\\selectfont\n\t\\begin{algorithmic}\n\t\t\\State \\textbf{Parameters:} $\\tau, \\sigma > 0$, $\\theta \\in [0,1]$\n\t\t\\State \\textbf{Initialization:} $n=0,\\ x^0 \\in X,\\ y^0 \\in Y, \\bar{x}^0 = x^0$\\\\\n\t\t\\State \\textbf{Iteration:}\\\\\n                \\State  \\textbf{for } $(n\\geq 0)$\\ \\textbf{ do }\n\t\t\\begin{enumerate}\\itemsep5pt\n\t\t\t\\item $y^{n+1} = (I + \\sigma \\partial F^{\\ast})^{-1}(y^n + \\sigma K \\bar{x}^n)$.\n\t\t\t\\item $x^{n+1} = (I + \\tau \\partial G)^{-1}(x^n - \\tau K^{\\ast} y^{n+1})$.\n\t\t\t\\item $\\bar{x}^{n+1} = x^{n+1} + \\theta (x^{n+1} - x^{n})$.\n\t\t\t\\item Set $n=n+1$.\n\t\t\\end{enumerate}\\\\\n               \n\t\t\\State \\textbf{end for}\\\\\n\t\t\\State \\Return $x^n$\n\t\\end{algorithmic}\n\t}\n\\end{algorithm}\n\n\n\\subsection{Numerical Realization Bregman-CV}\nIn order to use Algorithm \\ref{alg:cp} to solve the constraint problem \\eqref{eq:Bregman-CV} we reformulate the problem into\n\\begin{equation}\\label{eq:primBregCV}\n \\int_{\\Omega} u\\left((f-c_1)^2 - (f-c_2)^2\\right)\n+ \\text{id}_{[0,1]}(u) + \\alpha \\ \\left(TV(u)-<u,p_k>\\right) \\longrightarrow \\min_{u\\in BV(\\Omega)}\n\\end{equation}\nwith $p_k \\in \\partial TV(u_k)$ and $p_0 = 0$. $\\text{id}_{[0,1]}(u)$ is the indicator function of the interval $[0,1]$ defined as 0 if $u \\in [0,1]$ and $\\infty$ otherwise. To derive an minimization strategy based on Algorithm \\ref{alg:cp} we set $x = u$, $K(u) = \\grad u$ and\n\\begin{equation*}\nF(u) = \\| u\\|_{1} \\text{ and } G(u) = \\text{id}_{[0,1]}(u) +  \\int_{\\Omega} u\\left( (f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right).\n\\end{equation*}\nThe convex-conjugate of $F(u) = \\| u\\|_{1}$ is given by $F^{\\ast}(p) = \\delta_{P}(p)$ with $P = \\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}$ and\n\\begin{equation}\n\t\\delta_{P}(p) = \\begin{cases} 0 &\\mbox{if } p \\in P\\\\ \\infty &\\mbox{if } p\\notin P\\end{cases} .\n\\end{equation}\nTogether with \\eqref{eq:primaldualprob} we derive the primal-dual variant of \\eqref{eq:primBregCV}:\n\\eqn{\\label{eq:dualBregCV}\\langle \\grad u,p\\rangle + \\text{id}_{[0,1]}(u) +  \\int_{\\Omega} u\\left[ (f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right] - \\alpha \\delta_{P}(p) \\longrightarrow \\min_{u}\\max_{p}.}\nThe resolvent operators for $G$ and $F^{\\ast}$ are defined through\n\\begin{align}\nu = (I + \\tau \\partial G)^{-1}(\\tilde{u}) &= \\text{Proj}_{[0,1]}\\left[\\tilde{u} - \\tau\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right]\\notag\\\\\n& = \\max\\left(0,\\min\\left(1, \\tilde{u} - \\tau\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right)\\right)\\label{eq:proxprimal}\n\\end{align}\nand\n\\begin{equation}\np = (I + \\sigma \\partial F^{\\ast})^{-1}(\\tilde{p}) = \\text{Proj}_{\\left\\{\\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}\\right\\}}\\left(\\tilde{p_{ij}}\\right).\\label{eq:proxdual}\n\\end{equation}\nNote that the $L^{\\infty}$ norm $\\| p\\|_{\\infty}$ is in the discrete setting defined as $\\| p\\|_{\\infty} = \\max_{i,j} \\large\\{\\gamma^{\\ast}(p_{i,j})\\large\\}$ where the choice of $\\gamma$ determines the shape of the eigenfunctions of the TV functional. For $\\gamma = \\| \\cdot \\|_{\\ell^{p}}$ with $p = 1$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | \\max\\{|x|,|y|\\} \\leq 1\\right\\}$$ is an TV eigenfunction, for $p = 2$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | \\sqrt{|x|^2+|y|^2} \\leq 1\\right\\}$$ and for $p=\\infty$ the unit ball defined by $$\\left\\{ (x,y) \\in \\Omega | (|x|+|y|) \\leq 1\\right\\}.$$ However these are not the only eigenfunctions.\nWith \\eqref{eq:proxprimal} and \\eqref{eq:proxdual}, we derive the primal-dual algorithm presented in Algorithm \\ref{alg:cpforcv} to minimize  \\eqref{eq:Bregman-CV}. Note that we are not updating the constants $c_1$ and $c_2$, but start with a good estimate and leave it fixed. To a certain extend the varying regularization parameter can compensate for an error in those constants. Some examples are presented in Section \\ref{sec:results}.\n\\begin{algorithm}\n\t\\caption{First-order primal-dual algorithm to solve \\eqref{eq:Bregman-CV}.}\n\t\\label{alg:cpforcv}\n\t{\\fontsize{10}{10}\\selectfont\n\t\\begin{algorithmic}%\n\t\t\\State \\textbf{Parameters:} data $f$, reg. param. $\\alpha \\geq 0$, $\\tau, \\sigma > 0$, $\\theta \\in [0,1]$, \t\t\t   $maxIts \\in \\mathbb{N}, \\ maxBregIts \\in \\mathbb{N}$\n\t\t\\State \\textbf{Initialization:} $l=1,\\ u^0_1=0, \\ p_0:=0, \\bar{u}^0 = u^0$\\\\\n\t\t\\State \\textbf{Iteration:}\\\\\n\t\t\\State \\textbf{while } \\big($k<maxBregIts$\\big) \\textbf{ do }\n                \\begin{enumerate}\\itemsep5pt\n\t\t\t\\item Set n=0.\n\t\t\\end{enumerate}\n\t\t\\begin{quote}\n\t\t \\State  \\textbf{while } \\big($n<maxIts$\\big) \\textbf{ do }\\\\\n\t\t\\begin{enumerate}\\renewcommand{\\labelenumi}{\\alph{enumi})}\\itemsep5pt\n\t\t\t\\item $p^{n+1} = \\text{Proj}_{\\left\\{\\left\\{p: \\| p\\|_{\\infty} \\leq 1\\right\\}\\right\\}}\\left( p^n + \\sigma \\grad \\bar{u}^n \\right)$.\n\t\t\t\\item $u^{n+1} = \\max\\left(0,\\min\\left(1, u^n + \\tau \\div p^{n+1} - \\frac{\\tau}{\\alpha}\\left((f - c_1)^2 - (f - c_2)^2 - \\alpha p_k\\right)\\right)\\right)$.\n\t\t\t\\item $\\bar{u}^{n+1} = u^{n+1} + \\theta (u^{n+1} - u^{n})$.\n\t\t\t\\item Set $n=n+1$.\n\t\t\\end{enumerate}\\\\\n\t\t\\State \\textbf{end while}\\\\\n\t\t\\end{quote}\n\t\t\\begin{enumerate}\\itemsep5pt\n\t\t\\setcounter{enumi}{1}\n\t\t\\item Update $p_{k+1} = p_k + \\frac{1}{\\alpha}\\left((f - c_2)^2 - (f - c_1)^2\\right)$.\n\t\t\\item Set $u^0_{k+1}=u^n_{k}$.\n\t\t\\item Set $k=k+1$.\n\t\t\\end{enumerate}\\\\\n\t\t\\State \\textbf{end while}\\\\\n\t\t\\State \\Return $u^0_{k}$\n\t\\end{algorithmic}\n\t}\n\\end{algorithm}\n\\begin{figure}[t]\n  \\centering \n  \\includegraphics[width=0.6\\textwidth]{img\/speed_test.png}\n  \\caption{\\textit{Comparison of runtimes for Matlab and C\/Mex code.} The code was tested on 2 different computers: a Mac with an Intel Core i5 processor with two cores of 2.6 GHz and 8GB of memory (solid graphs) and a Linux computer with an Intel Xeon processor with 16 cores of  2.6GHz and 32 GB of memory (dashed graphs). On both computers we have tested a full Matlab implementation (blue), a C\/MEX implementation without parallelization (yellow) and a C\/MEX implementation with parallelization using openMP (green).}\n\\label{fig:speed} \n\\end{figure}\n\nWe implemented two versions of the code, the first one is written in MATLAB and the second one is written in C and called via MATLAB using the MEX interface. For speed comparisons we've tested both code versions (always with 1 Bregman iteration) on two different machines. First, a MacBook Pro with an Intel Core i5 processor with two cores of 2.6 GHz and 8GB of memory (solid graphs in Figure \\ref{fig:speed}) and the second machine was a Dell Linux workstation with in Intel Xeon processor with 16 cores of 2.6GHz and 32GB of memory (dashed graphs in Figure \\ref{fig:speed}). Moreover, we ran the C\/Mex code with and without parallelization using openMP. The results of our speed test can be seen in Figure \\ref{fig:speed}. We see that especially for large images the computational time is significantly decreased when using the C implementation (yellow and green graphs) compared to the pure MATLAB code (blue graphs). All C versions behave very similar for small images but deviate for larger images. Without parallelization the C code is slightly faster on the Mac (solid yellow graph). When activating openMP and parallelizing parts of the code the computational time could be further improved for both computers. The difference is much more significant when using the computer with more cores. For the Mac we see a speed-up of about 30 percent (solid green graph) for large images while on the Linux computer (dashed green graph) we could achieve a speed-up of about 65 percent compared to the C implementation without parallelization.\n\\section{Experimental Results}\\label{sec:results}\nIn this section we illustrate the main properties, advantages and limitations of our novel multiscale segmentation framework via several synthetic experiments as well as real experiments from biological cell imaging and retina imaging. The main framework, we apply and study throughout this section, consists of Algorithm \\ref{alg:cpforcv}, performing the inverse scale space method Bregman-CV in \\eqref{eq:Bregman-CV}, together with an evaluation of the spectral response $S$ in \\eqref{eq:spectralResponse}.\n\nIn the first set of segmentation experiments we focus on simple objects (e.g. discs) and investigate how underlying object intensities and sizes can automatically be detected as multiple scales. In the second part we concentrate on the stability of our method under uncertainty like noise. In the third part of this section we generalize our study to other simple shapes (e.g. blocks or diamonds), explore the connection to underlying eigenfunctions of the nonlinear $TV_{\\gamma}$ functional and the potential for shape detection respectively shape optimization. Finally, we apply our framework to real images obtained from the CellSearch system and other automatic scanning microscopes used for the enumeration of Circulating Tumor Cells \\cite{allard2004} offering new automatic procedures for tumor cell identification, quantification and classification. Moreover, we show potential applications of our algorithm in the field of electron microscopy of biological samples and retina imaging.\n\\subsection{Multiscale Segmentation: Size versus Intensity}\\label{sec:resultsscale}\nIn this first subsection we start with the investigation of basic segmentation tasks. We try to find round objects (discs) with varying scale (e.g. differences in size or intensity) in front of homogeneous backgrounds to verify the basic properties of our model. Since discs are the unit balls of the $\\ell^2$ norm we chose $\\gamma = \\gamma^{\\ast} = \\| \\cdot \\|_2$ in the TV norm \\eqref{eq:TVgen}.\n\\begin{figure}[htb]\n  \\centering \n    \\subfigure[1st Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[9th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_9.png}}\\quad \n  \\subfigure[12th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_12.png}}\\quad \n    \\subfigure[15th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_16.png}}\\quad \n      \\subfigure[22nd Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_24.png}}\\\\\n        \\subfigure[Spectral response $S$]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_size_nonoise\/S_gesamt.png}}\\quad\n         \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_size_nonoise_new\/back_gray.png}}\n  \\caption{\\textit{Detection of size scales}. Automatic segmentation of discs of different sizes under fixed intensity (height). Segmentation is visualized via red contour. (a)-(e) Segmentation after different number of Bregman Iterations. (f) Associated spectral response function $S$. The number of Bregman iterations is plotted on the x-axis. Every peak in $S$ corresponds to one object that appears in the segmentation (compare to (a) - (e)). The decreasing size of the peaks reflects the decreasing disc size. }\n\\label{fig:ballssize} \n\\end{figure}\n\nIn the first example in Figure \\ref{fig:ballssize}(a) we can automatically identify four discs with different diameters (sizes) via the Bregman-CV inverse scale space algorithm \\ref{alg:cpforcv}. As visualized in Figure \\ref{fig:ballssize}(b)-(e) the method automatically picks out the discs one by one while iterating (w.r.t $t \\approx \\alpha_k$) through the scale space. At the end, the full foreground-background segmentation is available. In addition to this, the spectral response \\eqref{eq:spectralResponse} of our iterates regarding $t$, nicely reflects the sparse occurrence of new contours, see Figure \\ref{fig:ballssize}(f). Note that in this example the intensity of peaks decreases due to the different size of each disc. The height of $S$ at each time point reflects the number of pixels that occurred (or vanished) in the segmentation in this step. In Figure \\ref{fig:ballssize}(g) we plotted the function $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$ in color coding. To obtain an easily interpretable visualization we use a gray background. Since the function $\\Phi(x,t)$ for given $x$ is equal to 1 exactly at the time point $t_x$ where $x$ occurs in the segmentation and 0 elsewhere, $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$ corresponds for every point $x$ to $t_x$. We can see that apart from the four time steps where the four discs appear nearly nothing is appearing in between, reflecting the sparse behavior of our methods. Only very few pixels at the object boundaries occur in later iterations which is caused by discretization inaccuracies. In this way the multiscale approach of our method can be seen since each color reflects one segmented scale.\\\\\n\\begin{figure}[htb]\n  \\centering \n\\subfigure[1st Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[3rd Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_3.png}}\\quad \n  \\subfigure[5th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_5.png}}\\quad \n    \\subfigure[10th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_10.png}}\\quad \n      \\subfigure[29th Bregman iteration.]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_int_nonoise\/bregman_cv_29.png}}\\\\\n        \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_int_nonoise\/S_gesamt.png}}\\quad\n         \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.16\\textheight]{results_new\/balls_int_nonoise_new\/back_gray.png}}\n\n  \\caption{\\textit{Detection of intensity scales}. Automatic segmentation of discs of different intensities (heights) under fixed size. Segmentation is visualized via red contour. (a)-(e) Segmentation after different number of Bregman Iterations. (f) Associated spectral response function $S$. The number of Bregman iterations is plotted on the x-axis. Every peak in $S$ corresponds to one object that appears in the segmentation (compare to (a) - (e)).}\n\\label{fig:ballsint} \n\\end{figure}\n\nThe results of the same experiment but now with intensity scales are shown in the next Figure.  The four discs in Figure \\ref{fig:ballsint}(a) all have\nthe same size but vary in their intensity from $0.55$ to $1$. Note that in some sense this contradicts the assumption of the CV model that the data roughly consists of two different intensity levels $c_1$ and $c_2$. Nevertheless our multiscale approach is able to detect all scales reliably. Like in all our examples we estimated the constants beforehand as $c_1 = \\text{mean}{\\{f(x)| f(x) < \\frac{1}{2}\\cdot \\max{f(x)}\\}}$ and $c_2 = \\text{mean}{\\{f(x)| f(x) \\geq \\frac{1}{2}\\cdot \\max{f(x)}\\}} $ and leave them fixed throughout the whole iteration process. Figure \\ref{fig:ballsint}(b)-(e) visualizes the result of the four Bregman iterations in which the different discs appear. The color-coded result of multiples scales ($\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$) is shown in Figure \\ref{fig:ballsint}(g). Again, only a few pixels occur at time steps apart from the four significant ones. It is remarkable that the first three scales appear shortly after each other while the last scale appears strikingly later. This can be explained by the way the regularization parameter is adapted automatically through the Bregman distance (cf. Figure \\ref{fig:reg_params} ). In later iterations the change in $\\alpha_k$ becomes smaller so that the time until a certain object appears is larger compared to the previous iterate. Although this might lead to late appearance of smaller scales we can thereby guarantee to not miss scale differences in the finer scales. The late appearance of the last disc is also reflected in the spectral response function $S$ in Figure \\ref{fig:ballsint}(f). Again, the response function has a sparse representation and the last peak occurs delayed. Apart from small variations caused by discretization artifacts the peaks now have all the same height reflecting their similar size.\n\n\\paragraph{Limitations:} In the previous two examples we showed how we can reliably detect different size scales and different intensity scales with our method. One limitation of our approach is the simultaneous detection of size and intensity scales in an image. \n\\begin{figure}[htb]\n  \\centering \n  \\subfigure[Data $f$ with intensity of 0.68 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_068_bw.png}}\\quad \\quad\n  \\subfigure[Data $f$ with intensity of 0.69 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_069_bw.png}}\\quad \\quad\n  \\subfigure[Data $f$ with intensity of 0.70 (large disc).]{\\includegraphics[width=0.18\\textwidth]{results_new\/balls_varying_size_and_int\/f_070_bw.png}}\\\\\n   \\subfigure[Spectral response $S$ for (a).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_068.png}}\\quad \n      \\subfigure[Spectral response $S$ for (b).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_069.png}}\\quad \n        \\subfigure[Spectral response $S$ for (c).]{\\includegraphics[width=0.2\\textwidth]{results_new\/balls_varying_size_and_int\/S_gesamt_070.png}}\n  \\caption{\\textit{Scale ambiguity w.r.t. size and intensity.} (a) - (c) Three different datasets that differ only in the intensity of the larger circle (0.68\/0.69\/0.7). The associated spectral responses $S$ are shown below. (a) together with (d) shows that out method is not able to reliably distinguish size and intensity scales but only a slight decrease of the intensity leads to two separate peaks in the spectral response function ((b) together with (e) and (c) together with (f)). }\n\\label{fig:criticalcase} \n\\end{figure}\nFigure \\ref{fig:criticalcase}(a)-(c) shows three different data sets, each with a small disc with intensity 1 and a larger discs with intensity $0.68-0.7$ respectively. For all three examples we computed the spectral response with fixed parameters $\\alpha = 200$ and 40 Bregman iterations shown in Figure \\ref{fig:criticalcase}(d)-(f). We can see that in (d) the spectral method was not able to detect the difference between a large object with a lower intensity and a smaller object with a high intensity. Both objects appear at exactly the same point in time. Figure \\ref{fig:criticalcase}(e) and (f) shows that already a small increase of the lower intensity circumvents the problem and both objects are represented as individual peaks in the spectral response. The height of each peaks reveals which object occurs. Yet we hope to solve this issue in future versions; solution ideas are presented in the summary and conclusion section \\ref{sec:conl}.\n\n\\subsection{Robustness against Noise}\nIn this second subsection we investigate the robustness of our methods with respect to uncertainties like noise. \n\\begin{figure}[htb]\n  \\centering \n\n  \\subfigure[Noise level $\\sigma = 0.25$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 0.5$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 0.75$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_strongnoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Noise level $\\sigma = 1$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_verystrongnoise\/bregman_cv_1.png}}\\\\\n   \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_strongnoise\/S_gesamt.png}}\\quad \n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_verystrongnoise\/S_gesamt.png}}\\\\\n   \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_littlenoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_mediumnoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_highnoise_new\/back_gray.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_veryhighnoise_new\/back_gray.png}}\n      \n\\caption{\\textit{Robustness against noise}. (a) - (d) shows the same underlying binary signal but with different levels of noise added. Exactly the same model parameters are used to process all 4 datasets. The associated spectral response functions are shown in (e) - (h) and the color-coded segmentation results are shown in (i) - (l). We can see that our method is very robust against noise, we only see a slight shift to the right in the spectral response function for higher noise levels. For high noise levels some artifacts at the objects boundary occur but we reliably find all 4 objects independent of the noise level. }\n\\label{fig:robustness} \n\\end{figure}\nFigure \\ref{fig:robustness}(a)-(d) shows the same underlying image but with different noise levels. The underlying signal is a binary signal showing four squares with different size. We added Gaussian noise with mean $0$ and standard deviation $\\sigma = 0.25, 0.5, 0.75$ and $1$ respectively. Since squares are the unit balls of the $\\ell^{\\infty}$ norm we chose $\\gamma = \\| \\cdot \\|_{1}$ and therefore $\\gamma^{\\ast} = \\| \\cdot \\|_{\\infty}$. The results of our inverse Bregman-CV model are plotted in terms of the spectral response function $S$ and the color-coded segmentation of all scales $\\int_{t}\\Phi(x,t) \\cdot t \\mathrm{d} t$. Note that we chose the same parameter setting for all four examples ($\\alpha = 100$ and 30 Bregman iterations). In Figure \\ref{fig:robustness}(e)-(h) we can see that the spectral response function has for all noise levels a sparse representation indicating very clearly the four scales in the underlying signal. It is also remarkable that the peaks are roughly at the same points in time, shifting only slightly to the right for higher noise levels. This is even more obvious in the color-coded representation Figure \\ref{fig:robustness}(i)-(l). The color bar next to each image shows that the color coding is consistent over all four examples. Thus, the same colors (see for example Figure \\ref{fig:robustness}(i) and (j)) reflect that the objects occur at the exact same iteration unaffected of the changed noise level. Even in case of high noise ($\\sigma$ = 0.75) only the second and the fourth largest object appear one iteration later although the smaller objects are now affected by an incomplete segmentation. In case of very high noise ($\\sigma = 1$) we cannot perfectly segment the objects, especially the smaller ones \"get lost\" in the noise. Nevertheless the spectral response function is still close to sparse and for all four objects we reliably segment most of the object. These results show that our method is very robust to noise and circumvents to some extend the problem of choosing a good regularization parameter $\\alpha$. The parameter setting that we used yielded a reliable extraction of all scales in the underlying signal independent of the noise present in our data. \n\\subsection{Segmentation regarding Shapes of Eigenfunctions}\\label{sec:resultsshapes}\nThis subsection is about the choice of the $\\gamma$-function and how this choice is reflected in our spectral analysis framework. As mentioned in Section \\ref{sec:genTV} by varying $\\gamma$ in the generalized definition of the total variation \\eqref{eq:TVgen} the shape of the eigenfunctions is changed. In the following two examples we will first show different examples of eigenfunctions that lead to a sparse representation of the spectral response function. Here we differentiate between eigenfunctions that are Wulff shapes of the chosen $\\gamma$ function and those that are not. In addition, we investigate what happens if the choice of $\\gamma$ does not match the shapes present in the image.\n\n\\begin{figure}[htb]\n  \\centering \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise\/bregman_cv_10.png}}\\quad \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise\/bregman_cv_24.png}}\\quad \n  \\subfigure[Data $f$ with segmentation shown in red.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise\/bregman_cv_24.png}}\\\\\n    \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise\/S_gesamt.png}}\\quad \n      \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise\/S_gesamt.png}}\\quad \n        \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise\/S_gesamt.png}}\\\\\n    \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/l1_size_nonoise_new\/back_gray.png}}\\quad \n      \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/balls_size_nonoise_new\/back_gray.png}}\\quad \n        \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.22\\textwidth]{results_new\/linf_size_nonoise_new\/back_gray.png}}\n  \\caption{\\textit{Varying norm bodies and eigenshapes.} Automatic detection of different signals composed of eigenshapes for different norm bodies. Column one shows the results for $\\gamma^{\\ast} = \\| \\cdot \\|_1$, column two for $\\gamma^{\\ast} = \\| \\cdot \\|_2$ and column three for $\\gamma^{\\ast} = \\| \\cdot \\|_{\\infty}$. (a) - (c) Segmentation results visualized via red contour after 30 Bregman Iterations. Associated spectral response function $S$ is shown in (d) - (f) and the color-coded segmentation in (g) - (i). With the correct choice of $\\gamma$ all eigenshapes can be detected in a single step. }\n\\label{fig:differentnorms} \n\\end{figure}\nIn Figure \\ref{fig:differentnorms}(a)-(c) each images shows a composite of eigenfunction for three different $\\ell^p$-norm choices for $\\gamma$ (namely $p = \\infty$, $2$ and $1$). In each of the images, the input data is shown together with the final segmentation result in red. The associated spectral responses are shown below (Figure \\ref{fig:differentnorms}(d)-(f)). We can see that as long as the choice of $\\gamma$ and the given dataset match (in the sense that the components of $f$ have the same shape as the unit-ball of $\\gamma^{\\ast}$ (cf. Table \\ref{tab:shapes})) we get a sparse response function. In Figure \\ref{fig:differentnorms}(g)-(i) the color-coded segmentation of all scales is shown. We can see that for rectangular shapes ((g) and (i)) no discretization artifacts at the boundary occur. Moreover, we can again use the same parameter setting for all three data sets. The similar color coding in (h) and (i) indicates that the objects occur roughly at the same time point independent of their shape and $\\gamma$. The data set in the first column is larger than the other ones thus the objects occur earlier and we need less iterations. \n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[TV eigenfunction for $\\gamma = \\| \\cdot \\|_2$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/nonWulff.png}}\\quad \n   \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/S_gesamt.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_l2_new\/back_gray.png}}\\\\\n    \\subfigure[TV eigenfunction for $\\gamma = \\| \\cdot \\|_1$.]{\\includegraphics[width=0.235\\textwidth]{results_new\/nonWulf_linf_new\/nonWulff2.png}}\\quad \n \\subfigure[Spectral response $S$.]{\\includegraphics[height=0.14\\textheight]{results_new\/nonWulf_linf_new\/S_gesamt.png}} \\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.091\\textheight]{results_new\/nonWulf_linf_new\/back_gray.png}}\n  \\caption{\\textit{Reconstruction of eigenfunctions that are not Wulff shapes.} (a) and (d) show two TV eigenfunctions that are not Wulff shapes for $\\gamma = \\| \\cdot \\|_2$ and $\\gamma = \\| \\cdot \\|_1$ respectively. The associated spectral responses are shown in (b) and (e) and the color-coded segmentation results in (c) and (f). It is obvious that eigenfunctions that are not Wulff shapes do have a sparse spectral response representation and appear in one step in the corresponding segmentation.}\n\\label{fig:nonWulff} \n\\end{figure}\n In Figure \\ref{fig:nonWulff} (a) and (d) we present two eigenfunctions which are not Wulff shapes. The round object in (a) fulfills condition \\eqref{eq:l2eig} and is therefore a TV eigenfunction for $\\gamma = \\| \\cdot \\|_2$. The corresponding spectral response function in (b) and the color-coded segmentation in (c) show that not only eigenfunctions that are Wulff shapes appear in one step but also more general eigenfunctions. Another example (now for $\\gamma = \\| \\cdot \\|_1$) is shown in Figure \\ref{fig:nonWulff} (d). Esedoglu and Osher showed in \\cite{esedoglu2004} that a rectangle whose contours are parallel to the x- and y-axis respectively is an eigenfunction of the anisotropic TV functional. This is reflected in the spectral response function Figure \\ref{fig:nonWulff} (e) and the color-coded segmentation in \\ref{fig:nonWulff} (f) which show that the rectangle appears in one step although it is not a Wulff shape of the 1-norm. For natural images, where objects are often not Wulff shapes, this allows objects with various shapes to appear in one step. Some examples can be seen in the subsections \\ref{sec:resultsCTC} - \\ref{sec:resultsnetwork}.\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Given Data.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_1.png}}\\quad \n  \\subfigure[Bregman iteration 10.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_10.png}}\\quad \n  \\subfigure[Bregman iteration 16.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_16.png}}\\\\\n  \\subfigure[Bregman iteration 18.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_18.png}}\\quad \n  \\subfigure[Bregman iteration 31.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_31.png}}\\quad \n  \\subfigure[Bregman iteration 41.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/bregman_cv_41.png}}\\\\\n  \\subfigure[Spectral response $S$.]{\\includegraphics[width=0.23\\textwidth]{results_new\/varying_shapes_nonoise\/S_gesamt.png}}\\quad\n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.31\\textwidth]{results_new\/varying_shapes_nonoise_new\/back_gray.png}}  \n  \\caption{\\textit{Reconstruction of mixed shapes.} Automatic detection of a signal consisting of different size scales and shapes. The binary signal is shown in (a). In (b) - (f) the segmentation results of the Bregman iterations where a significant peak in the spectral response $S$ in (g) appeared are shown. The color-coded segmentation is shown in (h). Here the behavior of non-eigenfunction over time can be seen. They do not appear in one step but reshape and evolve into the shape of the original signal.  }\n\\label{fig:scalesmix} \n\\end{figure}\nIn Figure \\ref{fig:scalesmix} we investigated how our multiscale approach behaves in case of shape mixtures and shapes that do not match the prior shape assumption made in $\\gamma$. The given data (Figure \\ref{fig:scalesmix}) consists of different size scales and different sizes with only two objects matching the prior assumption of round objects ($\\gamma = \\| \\cdot \\|_2$). Figure \\ref{fig:scalesmix}(b)-(f) shows all time steps where a new scale occurs in the segmentation (shown in red). We can see that the round objects appear in one step while all other ones appear with rounded edges and reshape over time. This can be seen more clearly in the color-coded segmentation of all steps shown in Figure \\ref{fig:scalesmix}(h). Especially for the large triangle in the middle we can nicely see how non-eigenfunctions are reshaped over time and how the segmentation is very round at the beginning and then propagates into the edges. Nevertheless we cannot get clear edges, neither for the triangle nor for the squares. The spectral transform function is nevertheless close to sparse but due to this reshaping behavior there is a small signal also between the clear peaks. It might be suitable to use the $\\ell_0$-norm of $S$ as a measure in order to evaluate if a certain signal matches the prior shape assumption made. However for this idea the influence of discretization artifacts as shown in some examples before, would have to be clarified in further detail.\n\\subsection{Fluorescence microscopy images containing Circulating Tumor Cells}\\label{sec:resultsCTC}\nIn this subsection we demonstrate how our multiscale segmentation approach can be applied to the analysis of Circulating Tumor Cells (CTCs). Before presenting some experimental data sets we will first give a short introduction on CTCs, associated research questions and challenges for diagnosis and treatment of cancer patients.\n\nCirculating Tumor Cells are cells that dissociate from a primary tumor and invade the blood stream. In recent literature it was shown that the number of CTCs present in the blood is associated with the survival chance of a patient and can be used to guide therapy of cancer patients. The elimination of CTCs indicates effective therapy whereas increase or failure to eliminate indicates a futile therapy. A challenge in the identification of CTCs is that they are very rare and therefore difficult to detect among other cells in the pool of blood cells. The gold standard for CTC enumeration is the CellSearch system and in prospective multicenter studies a threshold of 5 CTCs \/ 7.5 ml of blood was used to separate patients with metastatic disease into those with favorable and unfavorable prognosis \\cite{cristofanilli2004,debono2008}. Currently all known CTC analysis tools are based on subjective morphological criteria and objects can be classified differently by different operators which results in different interpretation of the status of the patients. The low cell counts and subsequent statistical analyses further increases the chances of mistakes. The problem of subjectivity in the CTC analysis is currently addressed by the development of an open-source toolbox to automatically detect and classify CTCs in datasets from various machines and institutes. This project is part of a European consortium called CANCER-ID that aims at validating blood-based biomarkers and is funded by the European Union. One key component of this toolbox is the development of a (nearly) parameter-free segmentation algorithm which performs reliably and efficiently on multiple tumor cell datasets (see Figure \\ref{fig:cells}).\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{img\/CTC_classes.png}\n   \\caption{\\textit{Schematic overview of different CTC classes.} For each subclass a cartoon image of a typical cell is shown. Next to it on the right different examples for every class are shown. All images consist of two overlaid color channels where green is the tumor cell marker and magenta is the nucleus marker. The relative frequencies of each subclass are shown on the left. By looking only at the green signal it is obvious that the size scale varies between different subclasses.}\n\\label{fig:cellclasses} \n\\end{figure}\nMoreover, in recent studies \\cite{coumans2010} it was hypothesized that not only the definition used to classify objects as CTC predicted clinical outcome but also those objects that did not match the criteria of an intact cell. A schematic overview of different cell classes and their relative frequencies is shown in Figure \\ref{fig:cellclasses}. Here, green is a fluorescence marker indicating the presence of cytokeratin expressed in cancer cells but not on white blood cells and magenta is a fluorescence marker indicating the presence of DNA thus showing the nucleus of the cell. When comparing the green marker among the different CTC classes we observe that the signal clusters strongly vary in size (area) and therefore motivate the usage of our multiscale segmentation approach.\n\n\\begin{figure}[ht]\n  \\centering \n  \\subfigure[Fluorescence image from the CellSearch system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/cell_comp1.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp1\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from a filter system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/cell_comp2.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp2\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from the CellSearch system with more clustered cells.]{\\includegraphics[width=0.27\\textwidth]{results_new\/cell_comp3\/cell_comp3.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp3\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp3\/back_gray.png}}\\\\\n  \n    \\subfigure[Fluorescence image from the ARIOL system.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/cell_comp4.png}}\\quad \n  \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/S_gesamt_jet.png}}\\quad \n  \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[width=0.25\\textwidth]{results_new\/cell_comp4\/back_gray.png}}\\\\\n\n   \\caption{\\textit{Experimental cell data.} Automatic detection of (a) different sparse cell sizes (scales) with nearly constant background, (d) cells in front of an inhomogeneous background, (g) clustered cells with nearly constant background and (j) different intensity and size scales. All results are obtained with the exact same parameters. In column one single channel images obtained from different fluorescence microscopes are shown. The corresponding spectral responses are shown in the second column and the color-coded segmentation in column three. It can be seen that more complex images (inhomogeneous background or mixtures of scales) lead to more complexity in the spectral response function (response becomes less sparse).}\n\\label{fig:cells} \n\\end{figure}\nIn Figure \\ref{fig:cells} (a), (d), (g) and (j) experimental data sets are shown. From the original image sets, consisting of three different fluorescent color channels, we extracted the tumor cell marker (green) and used those images as input for our multiscale segmentation approach. Although the difficulties vary between all images (inhomogeneous background, noise, cell clusters, mixture of size and intensity scales), we can process all images with our multiscale segmentation approach with the \\textit{exact same parameters}. This is essential  for the development of a user-friendly (parameter-free) toolbox for CTC analysis. Note that the dim spots in image (d) are not cells but only pores of a filter used to collect cells (bright spots) and therefore it is not desired to segment them. The resulting spectral response functions for all four images are shown in Figure \\ref{fig:cells} (b), (e), (h) and (k) with a color coding corresponding to the coding used in the segmentation results in (c), (f), (i) and (l). The color coding of the response function shows that all objects which appear later in our segmentation and therefore belong to finer scales have a yellow to reddish color in the color-coded segmentation. The very large and intact cells are blue (with some small artifacts at the boundary) and smaller cells (or large fragments) are shown in light blue to green. We can nicely observe that the object colors cover the whole color scale range. For images that are more complex (e.g. (d) and (j)) also the spectral response function is more complex but the color-coded segmentation shows that nearly every object appears in one step and thereby has a clearly defined scale that we use as a feature in our classification approach. Here, we profit from the fact that not only Wulff shapes (perfectly circular objects) but also other eigenshapes appear in one step in our segmentation.\nHence, this segmentation approach not only provides all contours automatically without any parameter adaptations but simultaneously also a simple classification of cells based on their size (scale resp. color\/appearance time). This analysis can be applied to all color channels separately and be used together with more features in a subsequent automatic classification approach. The constants $c_1$ and $c_2$ can again simply be estimated a-priori from the data by a simple thresholding and averaging approach and are fixed throughout the iterative process.\n\n\\subsection{Electron microscopy cell images with nonuniform background}\\label{sec:resultsEM}\nIn this subsection we show that our multiscale segmentation approach can also be used for real-world images where the background is less homogenous as in the cell images before. In Figure \\ref{fig:emcells} (a) we show an electron microcopy dataset of acinar cells provided by \\cite{riedel}. For a more obvious visual interpretation we reverted the contrast from the original EM dataset (bright background with dark cells) to a dark, but not uniform, background with brighter cells. The intensity of the cells is also inhomogeneous and varies from very bright cells (left) to cells with an intensity closer to the background (right). Figure \\ref{fig:emcells} (b) shows the color-coded spectral response function and (c) the color-coded segmentation of the cells. We see that the inhomogeneous intensity of the cells is reflected in the scales found by our method. The inhomogeneity of the background is not influencing the segmentation result and the constants $c_1$ and $c_2$ can still be estimated beforehand from the data without further adaption. One problem that occurs is the ambiguity of size and intensity scales which can be observed when comparing small but bright cells on the right (visualized in light green) with larger cell clusters on the right (also visualized in light green). Currently our method is not able to differentiate their scales. A useful extension of our method would be a combination of our multiscale approach with a watershed type algorithm that is able to split cell clusters into individual cells of different scales.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Electron microscopy data of acinar cells in the parotid gland provided by \\cite{riedel}.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/em1.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.15\\textheight]{results_new\/em1\/back_gray.png}}\n  \\caption{\\textit{Segmentation of cells in front of an inhomogeneous background.} (a) shows the input image obtained by electron microscopy, (b) the corresponding spectral response function and (c) the color-coded segmentation. This result shows that our method can detect inhomogeneities in the object we want to segment while inhomogeneities in the background (up to a certain degree) do not affect the result of our segmentation method. }\n\\label{fig:emcells} \n\\end{figure}\n\n\\subsection{Segmentation and clustering of biological network structures}\\label{sec:resultsnetwork}\nIn many biological and medical applications network-like structures occur as part of vascular systems. Examples are blood-vessels in retinal images, anisotropic structures in brain or myocardium imaging or at the microscopic level even cells consisting of a round core with several meandering arms of different diameters. Although these networks have a much more complex shape than the examples before, our multiscale method is very useful for a segmentation or clustering of these networks. An example of a round object with \"arms\" of different diameters is shown in Figure \\ref{fig:cellarms} (a). In (b) the corresponding spectral response function is shown and the color-coded segmentation in (c). We can see that the spectral response function has sparse clusters where each cluster corresponds to branches of a certain diameter. Although these \"arms\" are not eigenfunctions, they do appear in one step and therefore have a distinct scale if they have constant diameter. A similar behavior can be seen in Figure \\ref{fig:retina}. As input for our multiscale method we used a manual blood vessel segmentation (b) of a retinal image (a) from the STARE dataset \\cite{hoover2000}. The resulting spectral response is shown in Figure \\ref{fig:retina} (c) and the color-coded segmentation in (d). We see that the different times of appearance of the vessels (indicated by the different colors) lead to a clustering of the underlying network based on the diameter of each branch. With this approach we can get segmentations of the network that contain only branches of certain diameter scales or directly get an estimate of how many different vessel sizes are present and how often they occur. In order to apply our segmentation approach directly on raw retinal images, instead of using it as a postprocessing step, a decomposition method for retinal images similar to Zosso \\cite{zosso2015} could be combined with our multiscale approach.\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Given Data.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/network.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded segmentation of multiple scales.]{\\includegraphics[height=0.2\\textheight]{results_new\/network\/back_gray.png}}\n  \\caption{\\textit{Reconstruction of different scales in network-like structures.} The input data is shown in (a) with the corresponding spectral response function (b) and the color-coded segmentation (c). In this example the usability of our method in case of more complex, network-like structures is demonstrated. Interestingly, the spectral response function shows distinct sparse clusters where every cluster corresponds to \"arms\" or \"branches\" of a specific diameter. Although they are not shaped as an eigenfunction they appear (nearly) in one step and are represented by one specific scale.}\n\\label{fig:cellarms} \n\\end{figure}\n\n\n\\begin{figure}[t]\n  \\centering \n  \\subfigure[Retinal image of the STARE dataset \\cite{hoover2000}.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2\/network_retina_orig.png}}\\quad \n     \\subfigure[Manually segmented blood vessel network \\cite{hoover2000}.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/network_retina2-2.png}}\\quad \n   \\subfigure[Color-coded spectral response $S$.]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/S_gesamt_jet.png}}\\quad \n \\subfigure[Color-coded clustering of network in (b).]{\\includegraphics[height=0.2\\textheight]{results_new\/network_retina2-2\/back_gray.png}}\n  \\caption{\\textit{Clustering of networks.}  (a) A retinal image of a human eye taken from the STARE dataset \\cite{hoover2000} with a manual segmentation of the blood vessel network in (b). We applied our multiscale segmentation method to the binary image in (b) and thereby obtain the spectral response function (c) and a clustering of the network based on the diameter of the network branches (d). We see that large blood vessels are represented by blue colors, medium vessels by light blue to green colors and the very fine branches by yellow to red colors. Although the representation of (c) is not as sparse as in the previous example we can get a meaningful clustering of the graph based on the resulting colors in (d).}\n\\label{fig:retina} \n\\end{figure}\n\n\n\\section{Summary and Conclusion}\\label{sec:conl}\n\nIn this paper, we have studied a novel multiscale segmentation method which is a combination of an inverse scale space variational approach and a nonlinear spectral analysis. Using our approach we can automatically detect objects of different scales where scale can refer to intensity or size. These scales can be evaluated using a spectral response function and we can easily decompose the segmentation with respect to those scales using the spectral transform function. Thus we have extended the useful spectral analysis that was introduced for $TV$ denoising applications before to a segmentation approach. This novel approach can to some extent circumvent the problem of choosing a suitable regularization parameter. The iterative procedure using Bregman distances can be interpreted as an automatic regularization parameter choice method and is very robust with respect to noise or errors in the estimated $c_1$ and $c_2$ values in the CV model. We have shown that our method works very reliable even for applications with more than two intensity scales although this actually contradicts the models binary assumption. The framework can easily be adapted to different prior shape assumptions (Wulff shapes). With regard to real experimental datasets such an assumption is not limiting the usefulness of the method, due to its ability to even detect compositions of eigenfunctions not belonging to the restricted class of Wulff shapes. For the numerical implementation we used a primal-dual optimization scheme that, especially with a C\/MEX version, lead to a very efficient framework, even for large datasets. We illustrated the strengths and limitations of our method on synthetic datasets with a certain focus on eigenfunctions, shapes as well as robustness against noise and background artifacts. Three different biomedical imaging applications with different shapes and challenges emphasize the potential and wide applicability of our approach. The code is also part of an open-source toolbox called ACCEPT. With this software users are able to automatically detect and classify CTCs from fluorescence images, but can also extract important information related to the level of expression of certain therapy targets on the CTC (see the application and results described in section \\ref{sec:resultsCTC}).\n\nOne limitation that we currently have in our model is an ambiguity in scale. It is known for variational methods with an $l_2$ type data fidelity term combined with total variation regularization that these methods are not able to distinguish between intensity and size scales. In practice it seems not very likely that two objects with different size and different intensity fall onto exactly the same peak in the spectral response. Nevertheless to reliably interpret the spectral response function it is advantageous to have only one dominating scale in the data that is analyzed. Another limitation is the flexibility with regard to different shapes. Currently only one shape assumption at the moment is possible. \n\n\n\\textit{Outlook.} There are four main questions that we want to address in our future research. To allow a clear interpretation of the spectral response function even in the case of size and intensity scales we want to adapt the dataterm in the way that it can differentiate both scales. For denoising this can be done using an $l_1$ type data fidelity and we want to generalize this to our segmentation model (which is not obvious in terms of gradient flows). For the experimental cell datasets we currently analyze every fluorescent channel separately. In a future work we would like to investigate if a combined segmentation in all channels can lead to an improved segmentation and a richer response function. Moreover, we are interested in how far we can use the spectral response function to investigate if a shape assumption matches the data. One idea is to use the $l_0$ norm of the spectral response function as a measure for shape optimization. A further development of this idea could lead to an approach where we could learn an optimal $\\gamma$ function and thereby an optimal shape of eigenfunctions for a specific dataset. Another very interesting consideration is to transfer our method from the currently local setup to a nonlocal setup. A first foundation for this was recently laid with a work on non-local TV spectral theory by Aujol et al. in \\cite{Aujol2015}. More mathematical spectral theory for nonlocal TV, its usefulness for solving and improving complex segmentation tasks and a direct comparison with spectral clustering and graph cuts will be of high interest for the community and our future research.\n\\section*{Acknowledgements}%\nLZ, LT, CB acknowledge support by the EUFP7  program \\#305341 CTCTrap and the IMI EU program \\#115749 CANCER-ID. CB acknowledges support by the NWO via Veni grant 613.009.032 within the NDNS+ cluster.\n\n\\bibliographystyle{alpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supranuclear density matter}\n\n\\subsection{Introduction}  \n\nNeutron stars are the densest observable objects in the Universe, attaining physical conditions of matter that cannot be replicated on Earth. Inside neutron stars, the state of matter ranges from ions (nuclei) embedded in a sea of electrons at low densities in the outer crust, through increasingly neutron-rich ions in the inner crust and outer core, to the supranuclear densities reached in the center, where particles are squeezed together more tightly than in atomic nuclei, and theory predicts a host of possible exotic states of matter (Figure \\ref{nscut}). The nature of matter at such densities is one of the great unsolved problems in modern science, and this makes neutron stars unparalleled laboratories for nuclear physics and quantum chromodynamics (QCD) under extreme conditions.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{nsV3less.jpeg}\n\\caption{Schematic structure of a neutron star. The outer layer is a solid ionic crust supported by electron degeneracy pressure. Neutrons begin to leak out of ions (nuclei) at densities $\\sim 4 \\times 10^{11} $ g\/cm$^3$ (the neutron drip density, which separates inner from outer crust), where neutron degeneracy also starts to play a role.  At densities $\\sim 2\\times 10^{14}$ g\/cm$^3$, the nuclei dissolve completely.  This marks the crust-core boundary. In the core, densities reach several times the nuclear saturation density $\\rho_\\mathrm{sat} = 2.8\\times 10^{14}$ g\/cm$^3$ (see text).}\n\\label{nscut}\n\\end{figure}\n\nThe most fundamental macroscopic diagnostic of dense matter is the pressure-density-temperature relation of bulk matter, the equation of state (EOS). The EOS can be used to infer key aspects of the microphysics, such as the role of many-body interactions at nuclear densities or the presence of deconfined quarks at high densities (Section \\ref{nmat}).  Measuring the EOS of supranuclear density matter is therefore of major importance to nuclear physics.  However it is also critical to astrophysics.  The dense matter EOS is clearly central to understanding the powerful, violent, and enigmatic objects that are neutron stars. However, neutron star\/neutron star and neutron star\/black hole binary inspiral and merger, prime sources of gravitational waves and the likely engines of short gamma-ray bursts \\citep{Nakar07}, also depend sensitively on the EOS \\citep{Shibata11,Faber12,Bauswein12,Lackey12,Takami14}.  The EOS affects merger dynamics, black hole formation timescales, the precise gravitational wave and neutrino signals, any associated mass loss and r-process nucleosynthesis, and the attendant gamma-ray bursts and optical flashes \\citep{Metzger10,Hotokezaka11,Rosswog15,Kumar15}.  The EOS of dense matter is also vital to understanding core collapse supernova explosions and their associated gravitational wave and neutrino emission \\citep{Janka07}\\footnote{Note that whilst most neutron stars, even during the binary inspiral phase, can be described by the cold EOS that is the focus of this Colloquium (see Section \\ref{mrtoeos}), temperature corrections must be applied when describing either newborn neutron stars in the immediate aftermath of a supernova, or the hot differentially rotating remnants that may survive for a short period of time following a compact object merger.  The cold and hot EOS must of course connect and be consistent with one another.}.\n\n\\subsection{The nature of matter:  major open questions}\n\\label{nmat}\n\nThe properties of neutron stars, like those of atomic nuclei, depend\ncrucially on the interactions between protons and neutrons (nucleons)\ngoverned by the strong force. This is evident from the seminal work of\nOppenheimer and Volkoff \\citep{Oppenheimer39}, which showed\nthat the maximal mass of neutron stars consisting of non-interacting\nneutrons is 0.7 M$_\\odot$. To stabilize heavier neutron stars, as\nrealized in nature, requires repulsive interactions between nucleons,\nwhich set in with increasing density. At low energies, and thus low\ndensities, the interactions between nucleons are attractive, as they\nhave to be to bind neutrons and protons into nuclei. However, to\nprevent nuclei from collapsing, repulsive two-nucleon and\nthree-nucleon interactions set in at higher momenta and\ndensities.   Because neutron stars reach densities exceeding those in atomic nuclei, this makes them particularly sensitive to many-body forces \\citep[see, for example,][]{Akmal98}, and recently it was shown that the dominant uncertainty at nuclear densities is due to three-nucleon forces \\citep{Hebeler10b,Gandolfi12}\n\nAt low energies, effective field theories based on QCD provide a systematic basis for nuclear forces \\citep{Epelbaum09}, which make unique predictions for many-body forces \\citep{Hammer13} and\nneutron-rich matter \\citep{Tolos08,Hebeler14,Hebeler15}. While two-nucleon interactions are well constrained,\nthree-nucleon forces are a frontier in nuclear physics, especially for\nneutron-rich nuclei \\citep[see, for example,][]{Wienholtz13}. Such exotic nuclei are the focus of present and upcoming\nlaboratory experiments. Neutron star observations probe the same\nnuclear forces at extremes of density and neutron richness. In\naddition, to effective field theories, there are nuclear potential\nmodels, such as the Argonne two-nucleon and Urbana\/Illinois\nthree-nucleon potentials, which are fit to two-body scattering data and\nlight nuclei \\citep{Gandolfi14,Carlson14}.\n\nAt high densities, neutron stars may be affected by exotic states of\nmatter. This regime is not accessible to first principle QCD\ncalculations due to the fermion sign problem \\citep[see, for example, the discussions in][]{Hands07,Miller13a}. Therefore, at present, one has to resort to models, and experiment and observation are vital\nto test theories and drive progress. In addition, recently,\nperturbative QCD calculations have been performed at very high\ndensities (above 10 GeV\/fm$^3$,  $\\sim 64 \\rho_\\mathrm{sat}$), and used to interpolate to the EOS at low densities\n\\citep{Kurkela14}.\n\nFor symmetric matter (with an equal number of neutrons and protons) at the\nnuclear saturation density $\\rho_\\mathrm{sat} = 2.8 \\times 10^{14}$ g\/cm$^3$ (the central density in very large nuclei when the Coulomb interaction is neglected)\nthere is a range of experimental constraints. This includes nuclear\nmasses and charge\nradii~\\citep[see, for example,][]{Klupfel09,Kortelainen10,Kortelainen14,Niksic15} as well\nas giant dipole resonances and dipole\npolarizabilities~\\citep{Trippa08,Tamii11,Piekarewicz12}. Neutron-rich\nmatter can be probed by measuring the neutron skin thickness of heavy\nnuclei \\citep{Horowitz01,RocaMaza11}. However, all of these laboratory\nexperiments probe only matter at nuclear densities and\nbelow. Low-energy heavy-ion collisions probe hot and dense matter, but\nhave uncontrolled extrapolations to zero temperature and to extreme\nneutron-richness \\citep{Tsang09}.  Neutron stars therefore provide a\nunique environment for testing our understanding of the physics of the\nstrong interaction and dense matter.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{Trhom_RMP2.pdf}\n\\caption{Hypothetical states of matter accessed by neutron stars and \ncurrent or planned laboratory experiments (Large Hadron Collider and other heavy ion collision experiments, shown by black arrows), in the parameter space of\ntemperature against baryon chemical potential (1-2 GeV corresponds to\n$\\sim$ 1-6 times the density of normal atomic nuclei). Quarkyonic\nmatter: a hypothesised phase where cold dense quarks experience\nconfining forces \\citep{McLerran07,Fukushima11}. The stabilizing\neffect of gravitational confinement in neutron stars permits\nlong-timescale weak interactions (such as electron captures) to reach\nequilibrium, generating matter that is neutron rich (see Figure 2 of\n\\citealt{Watts14}) and may involve matter with strange quarks. This\nmeans that neutron stars access unique states of matter that can only\nbe created with extreme difficulty in the laboratory: nuclear\nsuperfluids, strange matter states with hyperons, deconfined quarks,\nand color superconducting phases.}\n\\label{paramspace}\n\\end{figure}\n\nAt very high densities, possibly reached in neutron star cores,\ntransitions to non-nucleonic states of matter may occur. Some of\nthe possibilities involve strange quarks: unlike heavy-ion collision\nexperiments, which always produce very short-lived and hot dense\nstates, the stable gravitationally confined environment of a neutron\nstar permits slow-acting weak interactions that can form states of\nmatter with a high net strangeness. Strange matter possibilities\ninclude the formation of hyperons \\citep[strange baryons,][]{Ambartsumyan60,Glendenning82,Balberg99,Vidana15}, deconfined quarks \\citep[forming a hybrid star,][]{Collins75}, or color superconducting\nphases \\citep{Alford08}. It is even possible that the entire star\nmight convert into a lower energy self-bound state consisting of up,\ndown and strange quarks, known as a strange quark star\n\\citep{Bodmer71,Witten84,Haensel86}. Other states that have been hypothesized\ninclude Bose-Einstein condensates of mesons \\citep[pions or kaons, the\nlatter containing a strange quark, see for example][]\n{Kaplan86,Kunihiro93}. The densities at which such phases may\nappear are highly uncertain.\n\nFigure \\ref{paramspace} compares the parameter space that can be\naccessed within the laboratory to that which can be explored with\nneutron stars. The physical ground state of dense matter is neutron\nrich, which develops via weak interactions, and it is unbound so \ngravitational confinement is necessary to realize the ground state\nof dense matter in nature. Only neutron stars sample this low\ntemperature regime of the dense matter EOS. The exotic non-nucleonic\nstates of matter described in the previous paragraph can be reached\nonly with extreme difficulty in the laboratory.\n\n\\subsection{Methodology: how neutron star mass and radius specify the EOS}\n\\label{mrtoeos}\n\nThe relativistic stellar structure equations relate the EOS to macroscopic observables including the mass $M$ and radius $R$ of the neutron star.  The dependence of the EOS on temperature can be neglected in computing bulk structure for neutron stars older than $\\sim 100$ s: by this point the neutron star has cooled far below the Fermi temperature of the particles involved, the matter is degenerate, and hence temperature effects are negligible \\citep[see for example][]{Haensel07}.  For non-rotating and non-magnetic stars, the classic Tolman-Oppenheimer Volkoff stellar structure equations would apply \\citep{Tolman39,Oppenheimer39}.  However rotation is important, and the equations must be modified accordingly. For neutron stars spinning at a few hundred Hz the slow rotation (to second order) Hartle-Thorne metric is appropriate for most applications \\citep{Hartle68}.  One can also compute full GR models for stars spinning at up to break-up speeds using a variety of methods implemented in well-tested codes \\citep[for a review see][]{Stergioulas03}.  Codes such as \\texttt{rotstar} \\citep{Bonazzola98} and \\texttt{rns} \\citep{Stergioulas95} generate masses and radii for rapidly rotating neutron stars that are accurate to better than one part in $10^{-4}-10^{-5}$. \n\nThere is a one to one map from the EOS to the $M$-$R$ relation \\citep[see, for example,][]{Lindblom92}. Some examples are shown in Figure \\ref{eos2mr}.  A few general features are worthy of note.  For each EOS there is a maximum mass that is a direct consequence of General Relativity \\citep[for a review of this topic see][]{Chamel13}, and there are plausible astrophysical mechanisms (formation or accretion) that might lead to this being reached in real neutron stars.  The minimum observable mass, by contrast, is more likely to be set by evolution than by stability.   Radius tends to reduce as mass increases (although for some EOS models, radius increases slightly with increasing mass in the mid-range of masses), and current models suggest radii in the range 8-15 km for masses above 1 M$_\\odot$.   \n\nIn terms of dependence on the nuclear physics, the maximum mass is determined primarily by the behavior of the cold EOS at the very highest densities \\citep[$\\sim 5-8 \\rho_\\mathrm{sat}$,][]{Lattimer05,Ozel09,Read09,Hebeler13}. The presence of non-nucleonic phases (such as hyperons or condensates) softens the EOS, reducing pressure support and leading to a smaller maximum mass. Radius on the other hand, depends more strongly on the behavior of the EOS at $\\sim (1-2) \\rho_\\mathrm{sat}$ \\citep{Lattimer01}. The nucleonic EOS at these densities is highly sensitive to three-nucleon forces \\citep{Hebeler10,Gandolfi12}, whilst the presence of non-nucleonic phases tends to reduce $R$. The slope of the $M$-$R$ relation (i.e. whether $R$ increases or decreases with $M$), for masses $\\gtrsim 1.2$ M$_\\odot$ (the observed minimum, consistent with expectations from formation models), depends on the pressure at $\\sim 4 \\rho_\\mathrm{sat}$ \\citep{Ozel09}.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{eossmall.png}\n\\caption{The pressure density relation (EOS, left) and the corresponding $M$-$R$ relation (right) based on models\nwith different microphysics. Red: nucleonic EOS from \\citet{Lattimer01}. Black solid: Hybrid models (strange quark core; \\citealt{Zdunik13}). Black dashed: Hyperon core models \\citep{Bednarek12}. Magenta: A self-bound strange quark star model \\citep{Lattimer01}. Grey band: range of a parameterized family of nucleonic EOS based on chiral effective field theory at low densities, which provides\na systematic expansion for nuclear forces that allows one to estimate the theoretical uncertainties involved, combined with using a general extrapolations to high densities \\citep[see Figure 12 of][for examples of specific representative EOS lying within this band]{Hebeler13}.}\n\\label{eos2mr}\n\\end{figure*}\n\nIn testing EOS models, there are two potential approaches.  One is simply to compute, for a given EOS model, the resulting $M$-$R$ relation, and then determine the likelihood of obtaining the measured values of $M$, $R$ (with uncertainties) if this model is correct.   The other option is to perform the inverse process, and to map from the measured values of $M$-$R$ (with their uncertainties) to the EOS.  The first attempt to address this problem, which made no assumptions about the form of the EOS, was made by \\citet{Lindblom92}.  Since then the approach has been refined by several authors.  Newer analyses rely on parameterized representations of the EOS that are a good characterization of many specific EOS models, but contain the lowest possible number of adjustable parameters (e.g. piecewise polytropic fits as employed by \\citet{Ozel09,Read09,Steiner10}, or spectral representations as employed by \\citet{Lindblom12,Lindblom14}).  \\citet{Ozel09} showed that given three measurements of $M$, $R$ with accuracies of $\\sim 5$\\%, current EOS models could be distinguished at the 3$\\sigma$ level using a piecewise-polytropic representation.    Using spectral representations, \\citet{Lindblom12} showed that the inversion process itself (reliance on discrete measurements, and the use of a generalized model) introduces errors that are typically less than $\\sim 1$ \\%\\footnote{This was the case for all models tested apart from those with a very strong phase transition, which are not well described by spectral representations with only a few parameters, and are better tested using the alternative method.}.  At this level the anticipated measurement errors on $M$, $R$ (which are at the few percent level) would be the primary determinant of the uncertainty on the inferred EOS.  \n\n\n\\subsection{Current observational constraints on the cold dense EOS}\n\\label{currentobs}\n\nThe cleanest constraints on the EOS to date have come from radio pulsar timing, where the mass of neutron stars in compact binaries can be measured very precisely using relativistic effects \\citep{Lorimer08}. Since any given EOS has a maximum stable mass (Figure \\ref{eos2mr}), high mass stars can rule out particular EOS.   The most massive pulsars have masses $\\approx 2$ M$_\\odot$ \\citep{Demorest10,Antoniadis13}, and that these results would have an impact on EOS studies was immediately clear \\citep{Demorest10,Ozel10}.    The requirement to generate neutron stars with masses of at least  2 M$_\\odot$ is now an integral part of EOS model development.  \\citet{Hebeler13} and \\citet{Lattimer14}, for example, have combined insights from  nuclear physics with the new maximum mass requirement in their models. Meanwhile \\citet{Kurkela14} have generated models that are constrained to approach the EOS of quark matter at high density computed from state of the art perturbative QCD calculations (which are robust in the very highest density regime).   \n\nOne of the largest impacts of the maximum mass measurements has been on studies of hyperons. The presence of hyperons in neutron stars is energetically probable, but should induce a strong softening of the EOS that would lead to maximum masses below 2 M$_\\odot$. The solution of this {\\it hyperon puzzle} \\citep[see for example][]{Chen11,Weissenborn12} requires some additional repulsion to make the EOS stiffer. Possible mechanisms include stiffer hyperon-nucleon and\/or hyperon-hyperon interactions, the inclusion of three-body forces with one or more hyperons \\citep{Takatsuka08,Logoteta13}, or the appearance of a phase transition to quark matter (see discussion of hybrid stars in Section \\ref{nmat}). \n\nWhilst measuring masses using radio pulsars yields very precise results, measuring radius (or mass and radius simultaneously) is more challenging and, with the methods currently in use, more model-dependent. \nCurrent efforts to constrain the full $M$-$R$ relation focus primarily on spectroscopic measurements of the surface emission from accreting neutron stars in quiescence \\citep{Rutledge99,Heinke03,Heinke06,Webb07,Servillat12,Catuneanu13,Guillot13,Guillot14,Heinke14}, and when they exhibit thermonuclear (Type I) X-ray bursts due to unstable nuclear burning in accreted surface layers \\citep{vanParadijs79,vanParadijs87,Ozel06,Ozel09a,Guver10a,Guver10b,Suleimanov11,Ozel12,Guver13}. \n\nThe essence of the method is the fact that the surface spectrum is\nclose to a diluted blackbody with a color temperature that is larger\nthan the effective temperature of the star $T_{\\rm c}=f_c T_{\\rm eff}$\nby the color-correction factor $f_{\\rm c}\\approx 1.3-2$ \\citep{London86,Zavlin96,Madej04,Heinke06,Suleimanov11b,Suleimanov12}.  This factor depends on the\natmospheric composition and on the effective surface gravity. For a slowly\nspinning neutron star at a known distance $D$, measuring the flux $F$\nof surface emission and the color temperature $T_{\\rm c}$ gives a relation between $M$ and $R$:\n\\begin{equation}\nR^2\\left(1-\\frac{2GM}{Rc^2}\\right)^{-1}=\\frac{F D^2 f_{\\rm c}^4}\n{\\sigma T_{\\rm c}^4}\\;.\n\\end{equation}\nwhere $\\sigma$ is the Stefan-Boltzmann constant.  Spectroscopic measurements using quiescent neutron stars, for which no additional information is available, result only in broad\nconstraints between the neutron star masses and radii \\citep[see][]{Heinke14,Guillot14}, although ensemble constraints may be tighter \\citep{Ozel15}.   \n\n\nStudies of the spectra during thermonuclear X-ray bursts allow \nadditional constraints to be obtained that can break the degeneracy\nbetween $M$ and $R$ \\citep[see reviews by][]{Lewin93,Ozel06}.  \nThe radius expansion bursts (during which radiation forces lift\nthe neutron star photospheres) serve as a good laboratory\nbecause, from the burst flux $F_{\\rm Edd}$ at the touchdown point,\none can get a measurement of the Eddington luminosity, which is a\ndifferent function of $M$ and $R$\n\\begin{equation}\nL_{\\rm Edd} = 4\\pi D^2 F_{\\rm Edd} = \\frac{4\\pi GMc}{\\kappa_e} \\left(\n1-\\frac{2GM}{Rc^2}\\right)^{1\/2}\\;.\n\\end{equation}\nHere $\\kappa_e=0.2(1+X)$~cm$^2$~g$^{-1}$ is the electron scattering\nopacity and $X$ is the hydrogen mass function.  Combining this with\nthe measurement of the surface area based on the thermal flux leads to\na weakly correlated inference of both $M$ and $R$ \\citep{Ozel09,Guver10a,Guver10b,Suleimanov11,Ozel12,Guver13,Poutanen14}.\n\nDifferent spectroscopic methods and burst selection criteria have been used to obtain constraints on the EOS. \\citet{Ozel10,Steiner10,Steiner13} find that large ($\\ge 13$ km) radii are ruled out, while others \\citep{Suleimanov11,Poutanen14} get radii in the range 11-15 km. Clearly, the $\\approx 5\\%$ accuracy in radius required to pinpoint the value of the pressure beyond the nuclear saturation density has not yet been achieved. \n\nBoth source types are affected by uncertainties in the composition of the atmosphere and (in many cases) lack of prior knowledge of the distance to each source  \\citep[see for example][]{Heinke14}\\footnote{Note that in principle, spectroscopic measurements rely on the absolute flux calibration of X-ray telescopes.  However, a limited number of studies currently seem to indicate that there is no significant flux calibration bias \\citep{Suleimanov11,Guver15}.}.  In the case of quiescent neutron stars, there are additional sources of uncertainty related to the composition of the intervening interstellar medium \nas well as the effects of residual accretion \\citep{Heinke14,Guillot14}, but see also \\citet{Bahramian15}. In the case of the X-ray\nbursters, there are several uncertainties: a systematic spread in the angular size of the source and the Eddington flux \\citep{Guver12a,Guver12b} which may reflect some level of non-uniform emission over the stellar surface;  identification of the touchdown point \\citep{Galloway08b,Guver12a,Miller13a}; and the role of accretion \\citep{Kajava14,Poutanen14}.  \n\n \nPerhaps more importantly, both types of spectroscopic measurements of\nneutron star masses and radii rely on the particular astrophysical\ninterpretations of two types of observations: that the quiescent\nemission is powered by deep crustal heating and that the Eddington critical flux can be estimated accurately from the data.\nThe timing techniques discussed later in this article, on the other\nhand, do not rely on these interpretations and, therefore, provide an\nindependent measurement of masses and radii, with orthogonal\nsystematic uncertainties and biases. Obtaining multiple measurements\nof masses and radii with different techniques, and in many cases for\nthe same sources, will allow us to address and correct for the\nsystematic uncertainties of each method.\n\nTiming-based techniques for constraining $M$ and $R$ rely on the presence of surface inhomogeneities, leading to emission that varies periodically as the star rotates. Three types of neutron star systems are suitable for a timing analysis, and several attempts have already been made:  for the accretion-powered pulsars \\citep{Poutanen03,Leahy04,Leahy09,Leahy11,Morsink11}; for accreting neutron stars that show oscillations during their thermonuclear X-ray bursts \\citep{Bhattacharyya05c}; and for rotation-powered X-ray pulsars \\citep{Bogdanov07,Bogdanov08,Bogdanov09,Bogdanov13}.  All of the $M$-$R$ constraints coming from these observations to date have very large error bars. However the method has great promise (see Section \\ref{pulse}). \n\nLaboratory experiments have also provided some constraints on the EOS.  Neutron star radii, for example, have been shown to depend strongly on the density dependence of the nuclear symmetry energy close to $\\rho_\\mathrm{sat}$ \\citep[see, for example,][]{Lattimer14}, and the behaviour of this quantity for densities up to $\\rho_\\mathrm{sat}$ is now being probed by laboratory experiments \\citep{Tsang12}.  Other experimental observables, such as K$^+$ meson production in nuclear collisions at subthreshold energies \\citep{Sturm01} and the nuclear elliptic flow in heavy ion collisions \\citep{Danielewicz02}, have been used as a sensitive probe for the stiffness of nuclear matter for high densities.  K$^+$ meson production seems to suggest a soft EOS for two to three times saturation density.  Indeed, these results on heavy-ion collisions have been used\nto constrain the features of neutron stars \\citep{Sagert12}.  Meanwhile results on elliptic flow rule out strongly repulsive nuclear EOS from relativistic mean field theory and weakly repulsive EOS with phase transitions at densities less than three times that of stable nuclei, but not EOS softened at higher densities due to a transformation to quark matter.  However, heavy-ion observables are obtained in highly-symmetric systems at high temperatures, thus the analysis of the EOS for asymmetric zero-temperature systems must be treated with caution. A reliable analysis of the EOS for zero temperature in asymmetric matter can be only achieved by neutron star measurements.\n\n\n\\subsection{Future observational constraints on the cold dense EOS}\n\nThe dense matter EOS will be a target science area for a number of different telescopes over the next decade, operating in very different wavebands.  To set the techniques that can be exploited in the hard X-ray band in context, we first review the advances that are expected in other wavebands. \n     \nThe next decade will see a major expansion in our ability to detect galactic radio pulsars, with the advent of the Square Kilometer Array \\citep[SKA,][]{Bourke15} and its pathfinders LOFAR \\citep{vanHaarlem13}, ASKAP \\citep{Schinckel12} and MeerKAT \\citep{Booth12}. The SKA is both more sensitive than current telescopes and will have increased timing precision.  By finding more radio pulsars in binary systems, and being able to determine post-Keplerian orbital parameters more precisely, the SKA should increase the number of measured neutron star masses by a factor of at least $\\sim 10$ \\citep{Watts14}.  New EOS constraints will result if the maximum mass record is broken, as discussed in Section \\ref{currentobs}.   Radio observations can also deliver radius via measurements of the neutron star moment of inertia (determined from spin-orbit coupling).  The moment of inertia of the only known double pulsar system PSR J0737-3039 \\citep{Burgay03}, will be determined to within 10\\% within the next 20 years \\citep{Lattimer05b,Kramer09}, resulting in a constraint on $R$ $\\sim 5$\\% \\citep[see also the discussion in Section 4.2 of][]{Watts14}.  The SKA may discover more systems for which a measurement of moment of inertia is possible, although it will be challenging since the requirements on system geometry are quite restrictive \\citep{Watts14}.  \n\nThe upgraded gravitational wave telescopes Advanced LIGO \\citep{LIGO15} and Advanced VIRGO \\citep{Acernese15} begin to enter service from 2015 and will operate well into the next decade. Gravitational waves from the late inspirals of binary neutron stars are sensitive to the EOS, with departures from the point particle waveform constraining $M$ and $R$. Global seismic oscillations excited by coalescence also depend on the EOS \\citep{Bauswein12}.  Estimates are that Advanced LIGO and VIRGO could achieve uncertainties of ~10\\% (1$\\sigma$) in $R$, for the closest detected binaries \\citep{Read13}, although event rates are highly uncertain. \n\nNICER-SEXTANT is a NASA Explorer Mission of Opportunity experiment that is due to be mounted on the International Space Station in late 2016 \\citep{Arzoumanian14}.  NICER has an effective area that is 0.2~m$^2$ at 2~keV dropping to 0.06~m$^2$ at 6~keV.  The primary focus of NICER will therefore be on soft X-ray sources, in particular rotation-powered pulsars that emit in both the X-ray and radio bands.   The X-ray emission from rotation-powered pulsars is expected to be steady, which means that long accumulation of data is possible, even though they are dim. Radii will then be inferred using soft X-ray waveform modeling \\citep[][and see Section \\ref{pulse}]{Bogdanov08}.  If the mass of a neutron star and the pattern of radiation from its surface are known accurately a priori, NICER observations will\nachieve an accuracy of $\\simeq 2$\\% in the measurement of  radius \\citep{Gendreau12,Bogdanov13}. In practice, the measurement will limited by uncertainties in these two\nrequirements.   The uncertainty in the mass measurement of\nNICER's primary target, the bright pulsar PSR~J0437$-$4715, is $\\sim$5\\% \\citep{Reardon15}. Other main targets of NICER will be the nearby isolated radio millisecond pulsars PSR J0030+0451 and J2124-3358, however these will produce less stringent constraints since there is no prospect for measuring their masses. The temperature profile on the\nneutron star surface, which is believed to be determined by the flux\nof return currents circulating in the magnetosphere \\cite[see e.g.][]{Bai10,Philippov14} is also highly uncertain, and development of ab initio numerical models that might fully address these questions is still a work in progress.  Waveform modelling for accretion-powered millisecond pulsars may also be possible \\citep[see the discussion in][and Section \\ref{pulse}]{Gendreau12}.\n\nAthena, the soft X-ray observatory selected for the European Space Agency's L2 launch slot in the late 2020s, has an effective area requirement of 2 m$^2$ at 1 keV, dropping to 0.25 m$^2$ at 6 keV \\citep{Nandra13}.  Dense matter is not a primary science goal for Athena.  However it could in principle be used for waveform modelling for the same isolated X-ray pulsars as NICER, and spectral modeling of neutron stars in quiescence or the cooling tails of X-ray bursts \\citep{Motch13}.  These techniques have already been discussed in detail both above and in Section \\ref{currentobs}.    \n\nASTROSAT \\citep{Singh14}, a multiwavelength Indian astronomy\nsatellite was launched in September 2015. LAXPC, a 3--80 keV X-ray\ntiming instrument, has an effective area of 0.8 m$^2$ in the 5--20 keV\nrange.  SXT, a co-aligned 0.3-8 keV X-ray imager, has an effective\narea of 0.01 m$^2$ in the 1--2 keV range.  This combination is\nwell-suited for spectral modeling of the cooling tails in X-ray\nbursts.  LAXPC can also in principle be used for waveform modeling of\naccretion-powered millisecond pulsars or X-ray burst oscillations.\n\n\\section{Hard X-ray timing techniques that deliver $M$ and $R$}\n\nHard X-ray timing enables three primary techniques that have the capability of delivering $M$ and $R$:  waveform (or {\\it pulse profile}) modeling, spin measurements, and asteroseismology.  These techniques involve different classes of neutron star: accreting neutron stars with thermonuclear bursts, accretion-powered X-ray pulsars, and isolated highly magnetic neutron stars known as magnetars.  The use of multiple techniques and different source types allows cross-calibration of techniques, and independent cross-checks on the EOS.  In this Section we will explore each technique in turn, and discuss the contraints on the EOS that would be achievable with a large area  ($\\sim 10$ m$^2$) hard X-ray (2-30 keV) timing telescope.  Much of the work presented here was developed as part of the science case for the proposed Large Observatory for X-ray Timing \\citep[LOFT:][and the LOFT ESA M3 Yellow Book, \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}]{Feroci12,Feroci14}.  \n\n\\subsection{Waveform modelling}\n\nMillisecond X-ray oscillations are observed from accretion-powered pulsars \\citep{Patruno12}, from some thermally-emitting rotation-powered (non-accreting) pulsars \\citep{Becker01} and during some thermonuclear bursts on accreting neutron stars \\citep[burst oscillations, see][]{Watts12}.  These oscillations are thought to be produced by X-ray emission from a region on the surface of the star that is hotter than the rest of the stellar surface and is offset from the rotational pole of the star.  As the hotter region rotates around the star, the X-ray flux seen by a distant observer is modulated at or near the rotation frequency of the star (Figure \\ref{pulse}).   Accretion-powered pulsations are formed as material is channeled onto the magnetic poles of accreting X-ray pulsars.  X-ray emission comes from both a hotspot that forms at the magnetic poles, and from the shock that forms just above the star's surface as the channeled material decelerates abruptly.  Burst oscillations, by contrast, are due to hotspots that form during thermonuclear X-ray bursts on accreting neutron stars \\citep{Lewin93,Galloway08}.  Ultimately one would hope to be able to use both types of pulsation when modelling the resulting pulse profile, or waveform, to recover $M$ and $R$: several sources show both phenomena, providing an important cross-check on the results.   \n\n\\subsubsection{Factors affecting the waveform}\n\\label{factorswaveform}\n\nAs the photons propagate through the curved space-time of the star, information about $M$ and $R$ is encoded into the shape and energy-dependence of the waveform.  General relativistic light-bending, which depends on compactness $M$\/$R$, affects the amplitude of the pulsations.  Special relativistic Doppler boosting, aberration, and the magnitude of time delays depend on the relative orientation of the hot spot and\nthe line of sight \\citep{Braje00}. This introduces a number of effects that lead to a direct measurement of the neutron star\nradius. \n\nIn the absence of special relativity, the peak flux occurs when the hot spot is facing the observer and the time for the\nflux to rise from minimum to maximum is the same as the time to fall from maximum to minimum. The Doppler boosting effect makes the\nblueshifted side of the star appear brighter than the redshifted side, so that the flux maximum occurs earlier in phase than if the special\nrelativistic effects did not occur. This asymmetry between the rise and fall time is approximately proportional to the projected line-of-sight velocity. Since the angular velocity is known from the pulse frequency, this provides a constraint on $R$ if the inclinations of the\nspot and observer are known. The great advantage of the very rapidly rotating neutron stars (with spin frequencies of 400 Hz and larger) is that the Doppler boosting effect is more pronounced in the data which reduces degeneracies. The asymmetry in the rise\/fall times is independent of the normalization, so errors in flux calibration are not important.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{newfig.pdf}\n\\caption{Left: As the neutron star rotates, emission from a surface hotspot generates a pulsation. The figure shows observer inclination $i$ and hotspot inclination $\\alpha$.  The invisible surface is smaller than a hemisphere due to relativistic light-bending.  Right: A schematic from \\citet{Psaltis14b} to illustrate the effects on the waveform arising from relativistic effects: the waveform is modified from a pure sinusoid, and the temperature (indicated by the color, which is the ratio of the number of photons with energies above to those below the blackbody temperature) also varies even though the underlying hotspot has a uniform temperature. The shape of the waveform and its energy dependence can be used to recover $M$ and $R$. See also \\citet{Viironen04} for examples of waveforms that include the expected variation in X-ray polarization.}\n\\label{pulse}\n\\end{figure*}\n\n\nThere are, of course, other factors that affect the waveforms and these must be taken into account when fitting for $M$ and $R$. The\ngeometric parameters $\\alpha$, the angle between the spin axis and the spot center, and $i$, the observer's inclination angle (see Figure \\ref{pulse}) introduce\ndegeneracies with $M$ and $R$. As the compactness ratio $M\/R$ increases, the pulse fraction decreases. However a decrease in the quantity\n$\\sin(\\alpha)\\sin(i)$ also decreases the pulse fraction. Similarly an increase in $\\sin(\\alpha)\\sin(i)$ increases the projected line-of-sight\nvelocity, leading to a degeneracy with the star's equatorial radius. Other geometric parameters such as the hotspot's shape and\nsize as well as emission from the rest of the star and the disk affect the shape of the light curve. Fortunately, the resulting parameter\ndependencies can be resolved when the detailed structure of the waveforms and their dependence on photon energy is taken into account (see below), allowing us to recover $M$ and $R$.\n\nA very important contribution to the parameter degeneracy is the beaming pattern of the radiation. A highly beamed emission pattern could, in some ways mimic the effects of decreased gravitational light bending and special relativistic Doppler boosting.  In the case of the thermal emission from X-ray bursts, this pattern is very well understood from theoretical modelling: it is very close to the limb-darkened pattern of a scattering dominated atmosphere \\citep{Madej91,Suleimanov12}, and is not a significant source of uncertainty \\citep{Miller13b}.  In contrast, in the case of the accretion-powered pulsations, the theoretical beaming pattern due to Compton scattering introduces a free parameter that is poorly constrained by the observations, leading\nto the weak constraints on $M$ and $R$ that result for these stars \\citep{Poutanen03, Leahy09, Leahy11, Morsink11}. The beaming pattern for the hydrogen atmosphere models used to compute the light curves of the rotation-powered pulsars \\citep{Bogdanov08}  do not formally have free parameters, although there are still some open questions regarding the beaming from such atmospheres that are thought to be heated by relativistic particles from the magnetosphere.  \n\nAdditional complications to the waveform modeling from accreting sources come from the fact that the neutron star is surrounded by the accretion disk,  which may block radiation coming from the `southern hemisphere'.  For accreting pulsars, the eclipses by the disk may lead to appearance of strong harmonic structure to the waveform that completely dominates all other sources producing harmonics \\citep{Poutanen08}. The evolution of the waveform from the best-studied pulsar SAX J1808.4$-$3658 during the outburst can be explained by varying just one parameter - the disk inner radius \\citep{Poutanen09,Ibragimov09,Kajava11}. The way out is to either use data at high accretion rate, when the southern pole  is completely blocked, or to model the eclipse by the disk, but this requires knowledge of the inclination.  In addition the accretion steam may block hotspot emission at some phases,  introducing a dip \\citep{Ibragimov09}. It is therefore important to use constraints from different techniques where possible to cross-check.\n\n\n\\subsubsection{Space-time of spinning neutron stars}\n\nAs mentioned in Section \\ref{factorswaveform}, rapid rotation is desirable for waveform modelling since it can break degeneracies\\footnote{The known accretion-powered pulsars and rapidly-rotating burst oscillation sources, the prime targets for this technique, have spin rates in the range 180-620 Hz \\citep{Patruno12,Watts12}.  More rapid rotation rates are seen in the radio pulsar population, and rotation rates exceeding 1000 Hz are theoretically possible (see Section \\ref{spinm})}.  Extensive work on gravitational lensing in spinning neutron star spacetimes has fully quantified the various levels of approximation  and their effects on the generation of pulse profiles.  The qualitative character of the general-relativistic effects depends on the ratio\n\n \\begin{equation}\n\\frac{f_{\\rm s}}{f_0}=0.24\n\\left(\\frac{f_{\\rm s}}{600~{\\rm Hz}}\\right)\n\\left(\\frac{M}{1.8 ~\\mathrm{M}_\\odot}\\right)^{-1\/2}\n\\left(\\frac{R}{10~{\\rm km}}\\right)^{3\/2}\\;,\n\\end{equation}\nof the spin frequency of the neutron star, $f_{\\rm s}$, and the characteristic frequency $f_0=\\sqrt{GM\/R^3}\/(2\\pi)$. To zeroth order in $f_{\\rm s}\/f_0$, the\nneutron star is spherically symmetric and its external spacetime is described by the Schwarzschild metric, which depends only on the mass\nof the star. Waveform calculations in this limit were performed in the pioneering work of \\citet{Pechenick83}. A simple approximation for\nthe light-bending integral, introduced by \\citet{Beloborodov02}, is also useful in many cases.\n\nTo first order in $f_{\\rm s}\/f_0$, the external spacetime of a slowly rotating neutron star and the exterior of the Kerr metric are the same \\citep{Hartle67}.  In this case, the amount of gravitational lensing depends also on the spin angular momentum of the star (because of the effects of\nframe dragging), which in turn depends on the density profile inside the star and hence on the equation of state. The effects of frame\ndragging on gravitational lensing are negligible \\citep{Braje00} and, therefore, the external spacetime (for the purpose of calculating\nwaveforms) is still well-described by the Schwarzschild metric. However, at the same order, special relativistic effects (Doppler boosts and aberration) as well as time delays become significant and can be calculated in the Schwarzschild + Doppler approximation \\citep{Miller98,Poutanen03,Poutanen06}.  This Schwarzschild + Doppler approach has been shown to be an excellent approximation for spin frequencies less than about 300 Hz \\citep{Cadeau07} through comparisons with waveforms calculated in exact, numerically generated neutron star spacetimes.\n\nTo second order in $f_{\\rm s}\/f_0$, the neutron star becomes oblate and its external spacetime is described by the Hartle--Thorne metric \\citep{Hartle68},\nwhich depends on the mass of the neutron star, on its spin angular momentum, and on its quadrupole mass moment. For spin frequencies\n$\\lesssim 600$~Hz, the effect of the stellar oblateness on the waveforms can be as large as 10-30\\% \\citep{Morsink07,Psaltis14a,Miller15} whereas the effect of the spacetime quadrupole is of the order of $1-5$\\% \\citep{Psaltis14a}. The\neffects of the stellar oblateness alone can be incorporated into an approximation that makes use of the light-bending formula arising from\nthe Schwarzschild metric, and a shape function that only depends on $M\/R$ and the spin frequency \\citep{Morsink07,Baubock13,AlGendy14}.\n\nFinally, to even higher orders in $f_{\\rm s}\/f_0$, the external spacetime of the neutron star depends on multipole moments that are of\nincreasing order. In principle the external spacetime of a rapidly spinning neutron star can be accurately described by the analytic\nsolution of \\citet{Manko00a,Manko00b}, see \\citep{Berti04,Berti05}.  However, the form of this metric is impractical for use in ray-tracing applications. On the other hand, spacetimes of rapidly spinning neutron stars can be calculated numerically \\citep[see for example][]{Cook94a,Stergioulas95,Bonazzola98}.  Simulations of waveforms for such rapidly rotating neutron stars with particular choices of the equation of state have been performed by \\citet{Cadeau07}.\n\n\\subsubsection{Inversion: from waveform to $M$ and $R$}\n\n The key question, when considering how to apply the waveform modelling technique in practice, is how many photons must be accumulated for a given source.  Some insight is given by the study of \\citet{Psaltis14b}, which generated simulated waveforms under various assumptions (including that of a small hotspot - angular radius less than 20$^\\circ$ - and isotropic emission from the hotspot), and studied the dependence of some key waveform properties on $M$, $R$ and other relevant parameters.  By using information about the shape and energy-dependence of the waveform, the different dependence on $M$ and $R$ of the various observables can in principle be used to break the degeneracies with the geometric parameters to recover $M$ and $R$.   This study also resulted in an order of magnitude estimate for the number of photons from the hotspot that would need to be accumulated in order to reach precisions of a few \\% in $R$, $\\sim 10^6$ counts.  \n\nThe inversion problem has also been examined by \\citet{Lo13} and \\citet{Miller15}.  These studies employed a Bayesian approach and Markov Chain Monte Carlo sampling methods in an extensive parameter estimation study that estimated $M$ and $R$ by fitting waveform models to synthetic waveform data, and determined confidence regions in the $M$-$R$ plane. Some example results are shown in Figure \\ref{contours}.  The uncertainties in $M$ and $R$ estimates are most sensitive to the stellar rotation rate (rapid rotation leading to smaller uncertainties), the spot inclination, and the observer inclination. They also depend on the background (be that from the accretion disk, astrophysical background, or instrumental background), but much more weakly \\citep[see also][]{Psaltis14b}. This technique does not require knowledge of the distance because one fits the fractional amplitude of the pulsations, not the absolute value. \\citet{Lo13} showed that for fixed values of the other system properties that affect the waveform, uncertainties in $M$ and $R$ estimates scale as ${\\cal R}^{-1}$, where ${\\cal R} \\equiv N_{\\rm osc}\/\\sqrt{N_{\\rm tot}} = 1.4f_{\\rm rms}\\sqrt{N_{\\rm tot}}$. Here $N_{\\rm osc}$ is the total number of counts in the oscillating component of the waveform, $N_{\\rm tot}$ is the total number of counts collected, and $f_{\\rm rms}$ is the fractional rms amplitude of the oscillation. If the stellar rotation rate is $\\gtrsim\\,$300~Hz and the spot center and the observer's sightline are both within 30$^\\circ$ of the star's rotational equator, burst oscillation waveform data with ${\\cal R} \\gtrsim\\,$400 allow $M$ and $R$ to be determined with uncertainties $\\lesssim\\,$10\\%.\n\n\\begin{figure*}\n\\centering\n\\includegraphics{LOFT3panel.pdf}\n\\caption{Constraints on $M$ and $R$ obtained by fitting a waveform model to synthetic observed waveform data \\citep[from][]{Miller15}. The black square indicates the mass and radius used to construct the synthetic data, the solid red lines show the 1$\\sigma$ constraints, and the blue dashed lines show the 2$\\sigma$ constraints. Here the spin rate $f_s$=600 Hz, hotspot inclination $\\alpha$ and observer inclination $i$ are 90$^\\circ$, the fractional rms amplitude of the oscillations is 10.8\\%, and the total number of counts is $10^7$. $M$ and $R$ are tightly constrained: $\\Delta M\/M$=2.8 \\% and $\\Delta R\/R$=2.9\\%. The parameter values used to generate the waveform data used in Panel (b) are the same as in Panel (a), except the rotation rate, which is much lower (300 Hz), causing the constraints on $M$ and $R$ to be weaker: $\\Delta M\/M$=5.6\\% and $\\Delta R\/R$=7.6 \\%. Panel (c) shows that when the spot is at an intermediate colatitude (here 60$^\\circ$), the constraints on $M$ and $R$ are weaker ($\\Delta M\/M$=6.5\\% and $\\Delta R\/R$=6.7 \\%), even if the star has a large radius (here 15 km) and is rapidly rotating (here, at 600 Hz).}\n\\label{contours}\n\\end{figure*}\n\nA key result from the \\citet{Lo13} and \\citet{Miller15} studies is their conclusions on the effects of systematic errors. They considered the consequences of differences in the actual spot shape, beaming pattern and energy spectrum from what was assumed in the model.  None of these cases yielded simultaneously (1) a statistically good fit, (2) apparently tight constraints (at the desired few percent level) on $M$ and $R$, and (3) significantly biased masses and radii.  Thus if an analysis yields a good fit with tight constraints, the inferred mass and radius are reliable.  This statement is currently unique among proposed methods to measure neutron star radii.\n\n\\subsubsection{Instrument requirements and observing strategy}\n\nIt is the necessity of obtaining ${\\cal R} \\approx 400$ for systems with favorable geometry that drives the requirement for large-area instruments to properly exploit this technique.  The observing times necessary to achieve this have been studied in detail as part of the LOFT assessment process (see LOFT ESA M3 Yellow Book, \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}), with observing strategy focusing on the burst oscillation sources given their well understood spectrum \\citep{Suleimanov12} and very low ($\\sim$ 1\\%) atmospheric model uncertainties \\citep{Miller13b}.  Using burst and burst oscillation properties observed with RXTE \\citep[burst brightness, burst oscillation fractional amplitude, as summarized in][]{Galloway08} for the known burst oscillation sources (such as 4U 1636-536), it was shown that with a $\\sim 10$ m$^2$ instrument one would need to combine data from $\\sim 10-30$ bursts with oscillations to meet the required target\\footnote{For a more detailed discussion of how one would combine data from different bursts, and how one can account for potential changes in the size and position of the hotspot, see \\citet{Lo13}.}.  Given the percentage of bursts that show oscillations (which is not 100\\%), and the mean burst rate \\citep[again using properties observed with RXTE,][]{Galloway08} this would require observing sources for a few hundred ks, easily feasible within anticipated mission lifetimes.  Burst oscillations do seem to occur preferentially in certain accretion states \\citep{Muno04}, and targeting these states with a suitable all-sky monitor would reduce the necessary observing time.  There are at present 27 known sources with burst oscillations and\/or accretion-powered pulsations spinning at 100 Hz or faster.  Given that some are transient sources with long periods of quiescence, new discoveries are to be expected.   Having such a large number of sources to choose from will help to select a sample with optimal observational characteristics (such as flux, pulse amplitude and harmonic content), the latter being easily achievable. \n\nAs indicated above, for favorable geometry that will be reflected in higher harmonic content of the pulsations, ${\\cal R} \\approx 400$ is sufficient to measure $M$ and $R$ to a few percent precision (Figure \\ref{ellipses}).  Less favorable geometries would require more observing time, since errors on $M$ and $R$ scale roughly as the inverse square root of the total number of counts  \\citep{Lo13,Psaltis14b}. Since a mix of geometries among sources is a reasonable expectation, observing strategy must be flexible enough to ensure that the goals can be met no matter what system geometries we encounter. Flexibility can also allow responsiveness to preliminary findings:  ideally one would schedule longer observations of the sources that are most constraining in the $M$-$R$ plane, in order to further reduce the size of their error ellipses. By thus tailoring observations one can confirm key findings at a much higher confidence. Figure \\ref{ellipses} illustrates the type of constraints on the $M$-$R$ relation that could be delivered by such a strategy.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{m4ellipses.png}\n\\caption{$1\\sigma$ confidence regions illustrating the constraints on the EOS that would be expected from a $\\sim 10$ m$^2$ hard X-ray timing telescope. For the sake of illustration, the regions are assumed to trace the $M$--$R$ curve of a proposed EOS that produces a neutron star with a strange-matter core. The larger regions assume observing time has been dedicated to reach ${\\cal R}\\approx 400$ for the 50\\% of known burst oscillation systems (here 10) that have optimal spot and observer inclinations ($i=60^\\circ$--$90^\\circ$) if their orientations are randomly distributed. The size and shape of these regions are from the Bayesian analyses of waveform data by \\citet{Lo13,Miller15} that assume the hot spots and observers are in the equatorial plane and that there is no independent knowledge of any of the model waveform parameters. The smaller $1\\sigma$ confidence regions show the constraints that could be achieved by a deep follow-up to halve the fractional uncertainties in $M$ and $R$ for the 3--4 stars that are most constraining, in terms of their locations in the $M$--$R$ plane, the potential for reducing their uncertainties, and the availability of complementary constraints from other observations. Constraints like those shown will exclude many of the EOS models shown in this figure. In this illustration, the models shown as grey lines are excluded. }\n\\label{ellipses}\n\\end{figure}\n\nIndependent knowledge of any of the relevant parameters improves the uncertainties, with the biggest improvement coming from knowledge of the observer inclination.  There are very good prospects in the next $\\sim 10$ years for determining the angle of our line of sight to the axis of the binary orbit using Fe line modeling \\citep{Cackett10,Egron11}, Doppler shifting of burst oscillation frequencies \\citep{Strohmayer02,Casares06}, and burst echo mapping \\citep[using the time delay between X-ray burst emission from the neutron star and the optical echo of that emission as it is reprocessed by the surface of the companion star,][note that this requires simultaneous optical observations]{Casares10}.  Since mass transfer is expected to cause the stellar rotation axis to align with the orbital axis in our target systems \\citep{Hills83,Bhattacharya91,Guillemot14}, this will yield the observer inclination. \n\nUsing accreting sources has the advantage that it enables independent cross-checks. Several potential targets show both accretion-powered pulsations and burst oscillations, allowing checks using two independent waveform models.   One can also use the continuum spectral modelling constraints outlined in Section \\ref{currentobs}, and there are potential new continuum fitting techniques that would be enabled by high quality spectra enabled from a large area detector \\citep{Lo13}.  Observation of an identifiable surface atomic line in the hot-spot emission would also provide a tight contraint \\citep[see for example][]{Rauch08}.   The rotational broadening of an atomic line depends on $R$, so combining this with its centroid (which depends on $M\/R$) yields a measurement of $M$ and $R$ independently, modulo the unknown inclination \\citep{Ozel03}.  To avoid the degeneracy with the inclination, one can also use the equivalent width of the line, which depends on the effective gravitational acceleration $g_\\mathrm{eff} = GM (1-2GM\/Rc^2)^{-1\/2}\/R^2$ \\citep{Chang05}.  Combining these two pieces of information yields separate measurements of $M$ and $R$, that are complementary to and independent of the constraints obtained from waveform fitting.   Note that whilst the effects of frame dragging can be neglected \\citep{Bhattacharyya05b}, the effect of stellar oblateness and the space-time quadrupole on the line profile must be taken into account for rotation rates $\\gtrsim 300$ Hz \\citep{Baubock13}.  \\citet{Lo13} explore how the constraints that would result from the detection of an atomic line could be combined with the results of waveform fitting.  \n\n\\subsection{Spin measurements}\n\\label{spinm}\n\nThe spin distribution of neutron stars offers another way of constraining the EOS, and a large-area instrument, with a correspondingly high sensitivity to pulsations, offers a unique opportunity to fully characterise this function.  \n\n\\subsubsection{Rapid rotation}\n\nAt the very simplest level, one can obtain constraints from the most rapidly rotating neutron stars.  The limiting spin rate $f_\\mathrm{max}$, at which the equatorial surface velocity is comparable to the local orbital velocity and mass-shedding occurs, is a function of $M$ and $R$ and hence fast spins constrain the EOS.  The mass-shedding frequency is given to good approximation \\citep{Haensel09}  by the empirical formula \n\n\\begin{equation}\nf_\\mathrm{max} \\approx {\\cal C} \\left[\\frac{M}{M_\\odot}\\right]^\\frac{1}{2} \\left[\\frac{R}{10 ~\\mathrm{km}} \\right]^{-\\frac{3}{2}} ~\\mathrm{kHz}\n\\end{equation}\nwhere $R$ is the radius of the non-rotating star of mass $M$.  Softer EOS have smaller $R$ for a given $M$ and hence have higher limiting spin rates.  More rapidly spinning neutron stars place increasingly stringent constraints on the EOS.  The deviation of ${\\cal C}$ from its Newtonian value of 1.838 depends, in GR, \\citep[as computed by][]{Haensel09} on the neutron star interior mass distribution.  For a hadronic EOS (one that consists of baryons or mesons), ${\\cal C}$ = 1.08, whilst for a strange star with a crust, ${\\cal C} \\approx$ 1.15.   This can be recast as a limit on $R$\n\n\\begin{equation}\nR<10 {\\cal C}^\\frac{2}{3} \\left[\\frac{M}{M_\\odot}\\right]^\\frac{1}{3} \\left[\\frac{f_s}{1 ~\\mathrm{kHz}}\\right]^{-\\frac{2}{3}} ~\\mathrm{km} \n\\end{equation}\nwhich places a constraint on the EOS (Figure \\ref{spinlim}). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{spind.png}\n\\caption{Spin limits on the EOS, reproduced from \\citet{Watts14}.  Neutron stars of a given spin rate must lie to the left of the relevant limiting line in the $M$-$R$ plane (shown in blue for various spins). The current record holder, which spins at 716 Hz \\citep{Hessels06} is not constraining.  However given a high enough spin, individual EOS can be ruled out.  Between 1 kHz and 1.25 kHz, for example, some individual EOS in the grey band \\citep[which represents a parameterized family of EOS models from][]{Hebeler13} would be excluded.}\n\\label{spinlim}\n\\end{figure}\n\nThe most rapidly spinning neutron star known, a 716 Hz radio pulsar \\citep{Hessels06}, is not spinning rapidly enough to be constraining.  However the discovery of a neutron star with sub-millisecond spin would pose a strong and clean constraint on the EOS.  What then are the prospects therefore for finding more rapidly spinning stars?  Since the standard formation route for the millisecond radio pulsars (MSPs) is via spin-up due to accretion \\citep{Alpar82,Radhakrishnan82,Bhattacharya91}, it is clear that we should look in the X-ray as well as the radio, and theory has long suggested that accretion could spin stars up close to the break-up limit \\citep{Cook94b}.  \n\nFigure \\ref{spindist} shows the current spin distributions for the MSPs and their proposed progenitors, the rapidly spinning accreting neutron stars. For the far more numerous radio pulsars, there is a clear fall off in the distribution at high spin.  Until a few years ago, the sometimes prohibitive computational costs of large surveys sensitive to very rapidly rotating objects limited radio pulsar searches, and this is likely still reflected in the measured distributions \\citep[for a more in-depth discussion, see][]{Watts14}.  The improvement in computational capabilities in recent years, and targeted searches for sources from the Fermi catalogues have led to the discovery of several new binary millisecond radio pulsars with spin frequencies above 500 Hz.  Interestingly the most rapidly rotating stars, including the 716 Hz spin record holder, appear to be in eclipsing systems, where matter from the companion obscures the radio pulsations for large fractions of the orbit.  This suggests that there may be an observational bias against finding these systems in the radio \\citep[see][for further discussion]{Watts14}.  \n\nFor the accreting systems, it is clear that we are still in the regime of small number statistics: however the drop-off at high spin rates seen in the radio is not apparent.  There are moreover physical reasons why it may have been difficult to find the most rapidly-spinning accreting sources.  Rapid spin is most likely in sources that accrete at high rates, for example, and when accretion rate is high episodes of channeled accretion are expected to be intermittent \\citep{Romanova08}.   Strong accretion may also drive a star towards alignment, making pulsations weak \\citep{Ruderman91,Lamb09b}.   For bursters, the most rapid spins may suppress flame spread \\citep{Spitkovsky02,Cavecchi13}, weakening and shortening bursts.  This means that the existence of rapidly spinning stars, that have been out of reach of current detectors, is certainly plausible both on observational and theoretical grounds.  \n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{mergedcorr.pdf}\n\\caption{Spin distributions of neutron stars with rotation rates above 100 Hz.  Top panel:  radio pulsars.  Lower panel:  accreting neutron stars (accretion powered millisecond pulsars and burst oscillation sources).}\n\\label{spindist}\n\\end{figure}\n\nAccretion-powered pulsations and burst oscillations are strongest in the hard X-ray (2-30 keV) band, and sensitivity scales directly with signal to noise.  As such, a large area hard X-ray timing instrument is necessary to discover more neutron star spins.  Intermittent accretion-powered pulsations have already been detected from a number of rapidly spinning sources \\citep{Galloway07,Casella08,Altamirano08}:  to detect more of these events, sensitivity to brief pulsation trains is key.  A 10 m$^2$ instrument with response similar to that of RXTE would be able to detect 100-s duration pulse trains down to an amplitude of $\\sim$ 0.4\\% rms for a 100 mCrab source (5$\\sigma$), $\\sim 100$ s being the duration of intermittent pulsations observed from Aql X-1 by \\citet{Casella08}.   RXTE would have needed 15 times as long to reach the same sensitivity, so that 100 s pulse trains were severely diluted; longer pulse trains suffered from Doppler smearing.   Searches for weak (rather than intermittent) accretion-powered pulsations using the sophisticated techniques being used for the Fermi pulsar surveys \\citep{Atwood06,Abdo09,Messenger11,Pletsch12}, which compensate for orbital Doppler smearing, would yield 5 $\\sigma$ multi-trial pulsation sensitivities of $\\sim 0.03-0.003$ \\% (rms) in bright neutron stars ($>$ 100 mCrab) and $\\sim 0.2-0.04$ \\% in faint neutron stars (10-100 mCrab), taking into account expected observing times and prior knowledge of the orbits.   A 10 m$^2$ instrument would also be able to detect oscillations in individual Type I X-ray bursts down to amplitudes of 0.6\\% rms (for a 1s exposure and a typical burst of brightness of 4 Crab); by stacking bursts sensitivity would improve.  \n\n\\subsubsection{Spin distribution and evolution}\n\nMapping the spin distribution more fully, so that the accreting neutron star sample is no longer limited by small number statistics, is also extremely valuable.  One of the big open questions in stellar evolution is how precisely the recycling scenario progresses - and whether it does indeed account for the formation of the entire MSP population.  The discovery of the first accreting millisecond X-ray pulsar by \\citet{Wijnands98}, and the recent detection of transitional pulsars, that switch from radio pulsars to accreting X-ray sources \\citep{Archibald09,Papitto13,Bassa14,Patruno14,Stappers14,Bogdanov14,Bogdanov15}, seems to confirm the basic picture.  However key details of the evolutionary process, in particular the specifics of mass transfer and magnetic field decay, remain to be resolved \\citep[see for example the discussion in][]{Tauris12}. Comparison of the spin distributions of the MSPs and the accreting neutron stars is a vital part of that effort.  \n\nThe torques that operate on rapidly spinning accreting neutron stars also remain an important topic of investigation.  Accretion torques, mediated by the interaction between the star's magnetic field and the accretion flow \\citep[first explored in detail by][]{Ghosh78,Ghosh79a,Ghosh79b}, clearly play a very large role \\cite[see][for a review of more recent work]{Patruno12}.  There are also several mechanisms, such as core r-modes \\citep[a global oscillation of the fluid, restored by the Coriolis force, see][for a recent review]{Haskell15} and crustal mountains \\citep[see][for a recent review]{Chamel13}, that may generate gravitational waves and hence a spin-down torque.  These mechanisms are expected to depend in part on the EOS \\citep[see, for example,][]{Ho11,Moustakidis15}.  In addition there are potential interactions between internal magnetic fields and an unstable r-mode that may be important \\citep[see for example][]{Mendell01}, and the physics of the weak interaction at high densities also becomes relevant, since weak interactions control the viscous processes that are an integral part of the gravitational wave torque mechanisms \\citep{Alford12}.  \n\nTorque mechanisms can be probed in two ways:  firstly, by examining the maximum spin reached, which may be below theoretical break-up rates, since both magnetic torques and gravitational wave torques may act to halt spin-up \\citep{Bildsten98,Lamb05,Andersson05}; and secondly by high precision tracking of spin evolution, enabled by increased sensitivity to pulsations.  Whilst extracting EOS information from the spin distribution and spin evolution will clearly be more challenging than the clean constraint that would come from the detection of a single rapid spin, it is nonetheless an important part of the models and one that can be tested.  Ultimately, more and better quality timing data are needed to confirm if it is, indeed, the magnetic field that regulates the spin of the fastest observed accreting neutron stars or if additional torques are needed. On the one hand, it has been argued that the spin-evolution during and following an accretion outburst of IGR~J00291+5934 is consistent with the `standard' magnetic accretion model \\cite{Falanga05,Patruno10,Hartman11}. On the other hand, the results are not quite consistent and there is still room for refinements and\/or additional torques \\cite{Andersson14,Ho14}. Whether this means that there is scope for a gravitational-wave element or not remains unclear \\cite{Ho11,Haskell12}, but a large area X-ray instrument should take us much closer to the answer.\n\nPrecision ephemerides from X-ray timing are very important enablers for simultaneous gravitational wave searches, since one has to fold long periods of data to detect the weak signals, and the gravitational wave frequency depends on spin rate in both mountain and r-mode mechanisms. Without such ephemerides, the number of templates that must be searched makes detection of continuous wave emission from sources like Sco X-1 very difficult \\citep{Watts08b}. This is very clear when one compares the limits currently obtained for continuous wave gravitational wave searches where ephemerides are known \\citep[the radio pulsars,][]{Abbott07,Abbott08} to those obtained for systems where the spin is not known (non-pulsing systems like Cas A, \\citealt{Abadie10} and Sco X-1, \\citealt{Aasi14}). A direct detection of gravitational waves from such a system would of course have immediate consequences for potential gravitational wave emission mechanisms, and any EOS dependence.  \n\n\n\\subsection{Asteroseismology}\n\nAsteroseismology is now firmly established as a precision technique for the study of the interiors of normal stars.  As such the detection of seismic vibrations in neutron stars was one of RXTE's most exciting discoveries.  They were found in magnetars, young, highly magnetized neutron stars that emit bursts of hard X-ray\/gamma-rays powered by decay of the strong magnetic field \\citep[see][for a review]{Woods06}.  What triggers the flares remains unknown, but most likely involves either starquakes or magnetospheric instabilities.  Rapid reconfiguration\/reconnection powers the electromagnetic burst:  however the events are so powerful that it had already been suggested that they might set the star vibrating \\citep{Duncan98}.   These vibrations, which manifest as Quasi-Periodic Oscillations (QPOs) in hard X-ray emission, were first detected in the several hundred second long tails of the most energetic giant flares from two magnetars \\citep{Israel05,Strohmayer05,Strohmayer06a,Watts06}.  Similar QPOs have since been discovered during storms of short, low fluence bursts from several magnetars \\citep{Huppenkothen13,Huppenkothen14a,Huppenkothen14b}.   The QPOs have frequencies that range from 18 to 1800 Hz.  \n\nSeismic vibrations offer us a unique way to explore the interiors of neutron stars.  The QPOs were initially tentatively identified with torsional shear modes of the neutron star crust, and torsional Alfv\\'en modes of the highly magnetized fluid core. These identifications were based on the expected mode frequencies, which are set by both the size of the resonant volume (determined by the star's radius) and the relevant wave speed. The fact that the oscillations must be computed in a relativistic framework introduces additional dependences, and for this reason they can be used to diagnose $M$ and $R$ (see for example \\citealt{Samuelsson07} for relativistic crust modes, and \\citealt{Sotani08} for relativistic core Alfv\\'en modes). Seismic vibrations also take us beyond the simple $M$-$R$ relation, constraining the non-isotropic components of the stress tensor of supranuclear density material.\n\nIn fact, for a star with a magnetar strength magnetic field, crustal vibrations and core vibrations should couple together on very short timescales \\citep{Levin07}. The current viewpoint is that the QPOs are associated with global magneto-elastic axial (torsional) oscillations of the star \\citep{Glampedakis06,Lee08,Andersson09,Steiner09,vanHoven11,vanHoven12,Colaiuda11,Colaiuda12,Gabler12,Gabler13a, Passamonti13,Passamonti14,Asai14,Glampedakis14}. Since coupled oscillations depend on the same physics, they have frequencies in the same range as the natural frequencies of the isolated elements. \n\nCurrent magneto-torsional oscillation models can in principle easily explain the presence of oscillations at 155 Hz and below. Until recently there was a significant problem with the higher frequency QPOs, which appeared to persist much longer than the models predicted, but this has now been resolved \\citep{Huppenkothen14c}. Issues currently being addressed include questions of emission \\citep{Timokhin08,DAngelo12,Gabler14}, excitation \\citep{Link14}, coupling to polar Alfv\\'en modes \\citep{Lander10,Lander11,Colaiuda12}, and resonances between the crust and core that might develop as a result of superfluid effects \\citep{Gabler13b,Passamonti14}. The latter in particular can have a large effect on the characteristics of the mode spectrum, and since superfluidity is certainly present in neutron stars, mode models must start to take this into account properly before we can make firm mode identifications. What is now clear is that mode frequencies depend not only on $M$ and $R$, but also on magnetic field strength\/geometry, superfluidity, and crust composition.\n\nSeveral papers have specifically explored EOS dependencies in neutron star asteroseismology \\citep{Strohmayer05,Strohmayer06a,Watts07,Samuelsson07,Sotani08,Steiner09,Gabler12}.  Figure \\ref{seis} illustrates the constraints that result when one models the QPOs detected in the SGR 1806--20 hyperflare as torsional shear oscillations of the neutron star crust, \\citep{Samuelsson07}.   This model is simple, in that it does not include magnetic coupling between crust and core.  However it gives some idea of the types of constraints on $M$ and $R$ that can result from the detection of several frequencies in a single event, where having multiple simultaneous frequencies assists mode identification (the burst storm identifications discussed above involve combining data from multiple bursts, so are less useful in this regard).  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{seis.png}\n\\caption{$M$-$R$ diagram showing the seismological constraints for the soft gamma-ray repeater SGR 1806--20 using the relativistic torsional crust oscillation model of \\citet{Samuelsson07}, in which the 29 Hz QPO is identified as the fundamental and the 625 Hz QPO as the first radial overtone. The neutron star lies in the box where the constraints from the two frequency bands overlap. Once QPOs are detected, frequency measurement errors are negligible for this purpose.  This model is very simple (it does not include) crust-core coupling, but it gives some idea of the type of constraints that might result from the detection of a harmonic sequence of seismic vibrations.  More sophisticated models that take into account coupling and the other relevant physical effects are under development.}\n\\label{seis}\n\\end{figure}\n\nSadly giant flares are rare, occurring only every $\\sim$ 10 years.   Ideally therefore we would like the ability to make similar detections in the more frequent but less bright events.    Intermediate flares, which are detected roughly once per year, have similar peak fluxes and spectra to the tails of the giant flares, but are too brief ($\\sim$ 1 s) to permit detection of similar QPOs with current instrumentation. A $\\sim 10$ m$^2$ hard X-ray timing instrument would be sensitive to QPOs in intermediate flares with similar fractional amplitudes as those observed in the tails of giant flares, provided that the collimator permitted the transmission of higher energy (above 30 keV) photons.  The latter is important since intermediate flares are unpredictable and likely to be observed off-axis, although one can increase the odds of capturing them by scheduling pointed observations during periods of high burst activity \\citep{Israel08}.  Theoretically the expectation of similar fractional amplitudes is justified: mode excitation at substantial amplitude even by events releasing energies typical of intermediate flares is feasible \\citep{Duncan98}.  Empirically, QPOs in giant flares tend to appear rather late in the tails, when luminosities are similar to those in intermediate flares, and given that they appear and disappear multiple time in these tails, may be triggered by magnetic starquakes at these 'low' fluxes \\citep{Strohmayer06a}.  The development of similar fractional amplitude QPOs in intermediate flares is thus considered plausible.  This idea has also been given a boost by the discovery of QPOs in short burst storms from two different magnetars \\citep{Huppenkothen13,Huppenkothen14a,Huppenkothen14b} including one that had also shown QPOs in a giant flare, since individually these bursts are much less energetic than the intermediate flares. The amplitudes at which the oscillations were detected in the burst storms are comparable to those of the detections in the giant flares. Upper limits on the presence of QPOs in the intermediate flares observed by current instruments, however, are above this level.  \n\n\n\\section{Summary}\n\nNeutron stars are unique testing grounds for fundamental nuclear physics, the only place where one can study the  equation of state of cold matter in equilibrium, at up to ten times normal nuclear densities.  The stable gravitational confinement permits the formation of matter which is extremely neutron rich, and which may involve matter with strange quarks.  The relativistic stellar structure equations show that there is a one to one mapping between the bulk properties of neutron stars, in particular their mass and radius, and the dense matter EOS.   Efforts to measure neutron star properties for this purpose are being made by both electromagnetic and gravitational wave astronomers.  In this Colloquium, we explored the techniques available using hard X-ray timing instruments:  waveform fitting, spin measurements, and asteroseismology. Hard X-ray timing offers unique advantages in terms of the numbers of known sources, and the potential for cross-checks using independent techniques and source classes.   \n\nThe previous generation of hard X-ray timing telescopes, in particular RXTE (a 0.6 m$^2$ telescope which operated from 1995 to 2012), uncovered many of the phenomena described in this Colloquium.  To exploit them to measure the EOS, however, requires larger area instruments, and various mission concepts are now being proposed.  These have included the 3 m$^2$ Advanced X-ray Timing Array \\citep[AXTAR:][]{Ray10}, and the 8.5-10 m$^2$ Large Observatory for X-ray Timing, LOFT \\citep[][see also the LOFT ESA M3 Yellow Book \\texttt{http:\/\/sci.esa.int\/loft\/53447-loft-yellow-book}]{Feroci12,Feroci14}.   None have yet been successful in securing a launch slot.   However the advantages that such a telescope would offer in terms of measuring the dense matter equation of state are sufficiently highly compelling that mission concept development continues apace.  \n\n\\begin{acknowledgments}\nThe authors would like to thank all of the members of the LOFT Consortium, in particular the members of the LOFT Dense Matter Working Group, for useful discussions.  ALW acknowledges support from NWO Vidi Grant 639.042.916, NWO Vrije Competitie Grant 614.001.201, and ERC Starting Grant 639217 CSINEUTRONSTAR.  The work of KH and AS is supported by ERC Grant No. 307986 STRONGINT and the DFG through Grant SFB 634.  MF, GI, and LS acknowledge support from the Italian Space Agency (ASI) under contract I\/021\/12\/0.  LT acknowledges support from the Ramon y Cajal Research Programme and from Contracts No. FPA2010-16963 and No. FPA2013-43425-P of Ministerio de Economia y Competitividad, from FP7-PEOPLE-2011-CIG under Contract No. PCIG09-GA-2011-291679 as well as NewCompStar (COST Action MP1304).  SMM acknowledges support from NSERC. JP acknowledges the Academy of Finland grant 268740. AP acknowledges support from NWO Vidi Grant 639.042.319.  AWS was supported by the U.S. Department of Energy Office of Nuclear Physics.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe field of gravitational lensing has seen exponential growth\nin its physical \\cite{SEF,KSW} and mathematical \\cite{PLW,PWreview,Petters} infrastructure,\nyielding diverse applications in astronomy and cosmology.\nIn this paper we address gravitational lensing in the setting of one of the\nmost important non-spherically symmetric and non-static solutions\nof the Einstein equations, namely, Kerr black holes.  This has already been the focus of many studies.  Indeed, several authors have explored the Kerr's caustic structure, as well as Kerr black hole lensing in the\nstrong deflection limit, focusing on leading order effects in light\npassing close to the region of photon capture\n(e.g., Rauch \\& Blandford \\cite{RB}, Bozza \\cite{Bozza,Bozza2,Bozza3}, V\\'asquez \\& Esteban \\cite{VqzE}, \nBozza, De Luca, Scarpetta, and Sereno \\cite{Betal}, Bozza, De Luca, and Scarpetta \\cite{Betal2}, and Bozza \\& Scarpetta \\cite{BS2}).\n\nStudies of Kerr lensing have also been undertaken in the weak deflection limit.  In particular, Sereno \\& De Luca \\cite{SerenoDeLuca,SerenoDeLuca2} gave an analytic treatment of caustics and two lensing observables for Kerr lensing in the weak deflection limit, while Iyer \\& Hansen \\cite{Iyer1,Iyer2} found an asymptotic expression for the equatorial bending angle.  Werner \\& Petters \\cite{WernerPetters} used magnification relations for weak-deflection Kerr lensing to address the issue of Cosmic Censorship (for lensing and Cosmic Censorship in the spherically symmetric case, see Virbhadra \\& Ellis \\cite{VE2}).  \n\nIn Papers I and II of our series, we are developing a comprehensive analytic framework for Kerr black\nhole lensing, with a focus on regimes\nbeyond the weak deflection limit (but not restricted to the strong\ndeflection limit).  In three earlier papers \\cite{KP1,KP2,KP3}, Keeton \\& Petters\nstudied lensing by static, spherically symmetric compact objects in\ngeneral geometric theories of gravity.  In \\cite{KP1,KP2}, the authors found\nuniversal relations among lensing observables for Post-Post-Newtonian (PPN) models that\nallowed them to probe the PPN spacetime geometry beyond the\nquasi-Newtonian limit.  In \\cite{KP3} they considered braneworld gravity,\nwhich is an example of a model outside the standard PPN family as it\nhas an additional independent parameter arising from an extra\ndimension of space.  They developed a wave optics theory (attolensing)\nto test braneworld gravity through its signature in interference\npatterns that are accessible with the Fermi Gamma-ray Space\nTelescope.  Papers I and II present a similar analysis of Kerr black hole lensing beyond the weak deflection limit.\n\nThe outline of this paper is as follows.  In Section~\\ref{sec:gen-lenseq} we present\na new, general lens equation and magnification formula governing lensing by a thin deflector.\nBoth equations are applicable for non-equatorial observers and \nassume that the source and observer are in the asymptotically\nflat region.\nIn addition, our lens equation incorporates \nthe displacement for a general setting that Bozza \\& Sereno \\cite{BS} \nintroduced for the case of a spherically symmetric deflector.\nThis occurs when the light ray's tangent lines\nat the source and observer do not intersect on the lens plane.\nSection~\\ref{sec:lenseq-Kerr} gives an explicit expression for this\ndisplacement when the observer is in the equatorial plane\nof a Kerr black hole as well as\nfor the case of spherical symmetry.   The lens equation itself is derived in Appendix~\\ref{app:Kerr-null-geodesics}.\n\nIn Paper II we solve our lens equation perturbatively to obtain analytic\nexpressions for five lensing observables (image positions, magnifications,\ntotal unsigned magnification, centroid, and time delay) for the regime of\nquasi-equatorial lensing.\n\n\n\n\n\\section{General Lens Equation with Displacement}\n\\label{sec:gen-lenseq}\n\n\n\\subsection{Angular Coordinates on the Observer's Sky}\n\\label{sec:obs-coords}\n\nLet us define Cartesian coordinates $(x,y,z)$ centered on the\ncompact object and oriented such that the observer lies on the\npositive $x$-axis.  We assume that gravitational lensing will take place outside the photon region, which is outside the ergosphere, so that $(x,y,z)$ are always spatial coordinates.  We also point out that we will not be considering distances more than a Hubble time, so that our formalism ignores the expansion of the universe.\n\nAssume that the observer in the asymptotically flat region is at\nrest relative to the $(x,y,z)$ coordinates.  {\\em All equations\nderived in this section are relative to the asymptotically flat\ngeometry of such an observer.}\nThe natural coordinates for the observer to use in gravitational\nlensing are angles on the sky.  To describe these angles, we\nintroduce ``spherical polar\" coordinates\ndefined with respect to the observer and the optical axis (from the observer to\nthe lens), and the $yz$-planes at the deflector and the light source.  The vector to the image position is then described by the\nangle $\\vth$ it makes with the optical axis, and an azimuthal\nangle $\\vphi$.  Similarly, the vector to the source position is\ndescribed by the angle $\\cb$ it makes with the optical axis and\nby an azimuthal angle $\\chi$.  These angles are shown in\n\\reffig{ObsCoords}.  Note that the optical axis is the $x$-axis.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.64]{ObsCoords.eps}\n\\end{center}\n\\caption{\nAngles on the observer's sky.\nAn image's position is determined by $(\\vth, \\vphi)$.  The source's position is given by $(\\cb,\\chi)$.}\n\\label{fig:ObsCoords}\n\\end{figure}\nWe adopt the usual convention for spherical polar coordinates:\nthe image position has $\\vth > 0$ and $0 \\le \\vphi < 2\\pi$,\nwhile the source position has $\\cb \\ge 0$ and $0 \\le \\chi < 2\\pi$.\nIn fact, since we only need to consider the ``forward'' hemisphere from the observer\nwe can limit $\\vth$ to the interval $(0,\\pi\/2)$ and $\\cb$ to the interval $[0,\\pi\/2)$.\n\nThe\n``lens plane'' is the plane perpendicular to the optical axis\ncontaining the lens, and the ``source plane'' is the plane\nperpendicular to the optical axis containing the source; these are also shown in \\reffig{ObsCoords}.  Define\nthe distances $d_L$ and $d_S$ to be the perpendicular distances\nfrom the observer to the lens plane and source plane, respectively,\nwhile $d_{LS}$ is the perpendicular distance from the lens plane\nto the source plane.  Some investigators define $d_S$ to be the\ndistance from the observer to the source itself, as opposed to\nthe shortest distance to the source plane.  We shall comment on\nthis distinction in Section~\\ref{sec:spherical}.\n\n\n\n\n\\subsection{General Lens Equation via Asymptotically Flat Geometry}\n\\label{sec:gen-lens-eqn}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.67]{LensGeom.eps}\n\\end{center}\n\\caption{\nA lensing scenario demonstrating that the tangent line to the segment of the ray arriving at the observer and the tangent line of the ray at the source need not intersect on the lens plane; i.e., $A' \\neq B'$ in general.  The angles $\\cb$ and $\\vartheta$ are as in \\reffig{ObsCoords} (or rather, they are their projections onto the $xz$-plane), $\\hat{\\alpha}$ is the ``bending angle,\" and $d_L$, $d_{S}$, and $d_{LS}$ are the perpendicular distances between the lens plane and observer, the source plane and observer, and the lens and source planes, respectively.}\n\\label{fig:LensGeom}\n\\end{figure}\n\nConsider the lensing geometry shown in \\reffig{LensGeom}.  With respect to the light ray being lensed, there are two tangent lines of interest: the tangent line to the segment of the ray arriving at the observer and the tangent line to the ray emanating from the source.  \nAs first emphasized in \\cite{BS}, {\\it it is important to realize that\nthese two tangent lines need not intersect.}  If they do intersect (as\nfor a spherical lens, since in that case the tangent lines are\ncoplanar), the intersection point need not lie in the lens plane.\nThis effect has often been neglected, and while it may be small in the\nweak deflection limit (see Section~\\ref{sec:spherical} below) we should include it for greater generality.\nA simple way to capture this displacement is to\nconsider the points where the two tangent lines cross the lens\nplane, namely, the points $A'$ and $B'$ in \\reffig{LensGeom}.\nIf the tangent lines do intersect in the lens\nplane, then $A' = B'$.  Otherwise, as can be seen in greater detail in \\reffig{Lens-Geom3}, there is a displacement on the lens plane that\nwe quantify by defining\n\\beq \\label{eq:disp-def}\n\\cd_y = B'_y - A'_y\\,, \\qquad \\cd_z = B'_z - A'_z\\,.\n\\eeq\nNote from \\reffig{Lens-Geom3} that the\ntangent line to the segment of the ray arriving at the observer \nis determined by $(\\vth,\\vphi)$.  The tangent line to the ray\nemanating from the source can likewise be described by the angles\n$(\\vth_S,\\vphi_S)$, where $-\\pi\/2 < \\vth_S < \\pi\/2$ and $0 \\leq \\vphi_S < 2\\pi$.  As shown in \\reffig{Lens-Geom3}, $\\vth_S$ has vertex $B'$ and is measured from the line joining the points $B'$ and $B''$, which runs parallel to the optical axis.  We adopt the following sign convention for $\\vth_S$: if $\\vth_S$ goes {\\it toward} the optical axis, then it will be positive; otherwise it is negative (e.g., the $\\vth_S$ shown in \\reffig{Lens-Geom3} is positive).\nWe will obtain the general lens equation by considering the coordinates\nof the points $A'$ and $B'$ in \\reffig{Lens-Geom3}.  Using the\nasymptotically flat geometry of the observer, we have\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.62]{LensGeom-without-nu.eps}\n\\end{center}\n\\caption{\nA detailed diagram of lensing with displacement.  The tangent line to the segment of the ray arriving at the observer is determined by $(\\vth,\\vphi)$ and intersects the lens plane at $A'$, while the tangent line to the ray emanating from the source is determined by $(\\vth_S,\\vphi_S)$ and intersects the lens plane at $B'$.  The distance between these two points is quantified by the displacement amplitude $\\cd$, whose horizontal and vertical components we denote by $\\cd_y$ and $\\cd_z$, respectively.  The deflector could be a Kerr black hole and the light ray may dip below the $xy$-plane.}\n\\label{fig:Lens-Geom3}\n\\end{figure}\n\\beqa\nA'_x &=& 0\\,, \\nonumber\\\\\nA'_y &=& d_L \\tan\\vth \\, \\cos\\vphi\\,, \\label{eq:Acoordsy}\\\\\nA'_z &=& d_L \\tan\\vth \\, \\sin\\vphi\\,,\n  \\label{eq:Acoords}\\\\\nB'_x &=& 0\\,, \\nonumber\\\\\nB'_y &=& d_S \\tan\\cb \\, \\cos\\chi + d_{LS} \\tan\\vth_S \\, \\cos(\\pi-\\vphi_S)\\,,\n  \\\\\nB'_z &=& d_S \\tan\\cb \\, \\sin\\chi - d_{LS} \\, \\tan \\vth_S \\, \\sin (\\pi-\\vphi_S)\\,.\n  \\label{eq:Bcoords}\n\\eeqa\nSubstituting eqns.~(\\ref{eq:Acoordsy})--(\\ref{eq:Bcoords}) into eqn.~(\\ref{eq:disp-def}) yields\n\\beqa\nd_S \\tan\\cb \\, \\cos\\chi & = & d_L \\tan\\vth \\, \\cos\\vphi\n  \\ + \\ d_{LS} \\tan \\vth_S \\, \\cos \\vphi_S\n  \\ + \\ \\cd_y\\,,\n  \\label{eq:le-h} \\\\\nd_S \\tan\\cb \\, \\sin\\chi & = & d_L  \\tan\\vth \\, \\sin\\vphi\n  \\ + \\ d_{LS} \\tan \\vth_S \\, \\sin \\vphi_S\n  \\ + \\ \\cd_z\\,.\n  \\label{eq:le-v}\n\\eeqa\nThe left-hand sides involve only the source position, while the\nright-hand sides involve only the image position.\nIn other words, {\\em this pair of equations\nis the general form of the gravitational lens equation for\nsource and observer in the asymptotically flat region, for a general isolated compact object.}  Note that apart from the asymptotic flatness assumption, these equations use no properties specific to Kerr black holes; and if the deflector was a Kerr black hole, then neither the observer nor the source has been assumed to be equatorial.\nWe shall refer to eqns.~(\\ref{eq:le-h}) and (\\ref{eq:le-v}), respectively, as the\n``horizonal'' and ``vertical'' components of the lens equation\ndue to the cosine\/sine dependence on $\\chi$.\n\nConsider now the case when the light ray and\nits tangent lines lie in a plane which contains the optical axis.  This forces $\\chi = \\vphi$ or $\\chi = \\vphi+\\pi$ depending on whether\nthe source is on the same or opposite side of the lens as the image.  To distinguish these two cases, it is useful to define\n$\\snq = \\cos(\\chi-\\vphi)$ to be a sign that indicates whether the\nsource is on the same side of the lens as the image ($\\snq=+1$)\nor on the opposite side ($\\snq=-1$).  The condition $A' \\neq B'$ will still hold in general, but the line in the lens plane from the origin to the point $B'$ will now make the same angle with respect to the $y$-axis as the point $A'$, namely, the angle $\\vphi$ (see Fig.~\\ref{fig:Lens-Geom3}).  As a result, the line in the source plane from the origin to the point $B''$ will also make the angle $\\vphi$ with respect to the $y$-axis.  Thus we will have $\\vphi_S = \\vphi+\\pi$.  Given these conditions,\neqns.~(\\ref{eq:le-h}) and (\\ref{eq:le-v}) reduce to the single lens\nequation\n\\beq \\label{eq:le-sph}\n  d_S\\,\\snq\\,\\tan\\cb = d_L \\tan\\vth - d_{LS} \\tan(\\hat{\\alpha}-\\vth) + \\cd \\,,\n\\eeq\nwhere the displacement amplitude is $\\cd = \\cd_y\/\\!\\cos\\vphi = \\cd_z\/\\!\\sin\\vphi$ (in the case of planar rays),\nand to connect with traditional descriptions of gravitational\nlensing we have introduced the light bending angle\n$\\hat{\\alpha} \\equiv \\vth + \\vth_S$.  (If desired, the sign $\\snq$ can be\nincorporated into the tangent so that the left-hand side is\nwritten as $\\tan(\\snq\\cb)$, where we think of $\\snq\\cb$ as the\nsigned source position.)  {\\it Eqn.~(\\ref{eq:le-sph}) is the general form of the lens equation in the case of planar rays.}  If the displacement $\\cd$ is ignored,\nthen eqn.~(\\ref{eq:le-sph}) matches the spherical lens equation given\nby \\cite{virbetal2}.  We consider the displacement term in\n\\refsec{spherical}.\n\n\n\n\\subsection{General Magnification Formula}\n\\label{sec:gen-mag}\n\nThe magnification of a small source is given by the ratio of the\nsolid angle subtended by the image to the solid angle subtended\nby the source\n(e.g.,\n\\cite[p.~82]{PLW}).  As measured by the observer, if $\\ell$ is the\ndistance to the image (as opposed to the perpendicular distance),\nthen the small solid angle subtended by the image is\n\\beqa\n  d\\Omega_I = \\frac{|(\\ell \\, d \\vth)\\, (\\ell \\sin\\vth \\, d\\vphi)|}{\\ell^2}\n=|\\sin\\vth\\ d\\vth\\ d\\vphi| = |d(\\cos\\vth)\\ d\\vphi| .\\nonumber\n\\eeqa\nSimilarly, the small solid angle subtended by the source is\n\\beqa\n  d\\Omega_S = |\\sin\\cb\\ d\\cb\\ d\\chi| = |d(\\cos\\cb)\\ d\\chi| .\\nonumber\n\\eeqa\nWe then have the absolute magnification\n\\beqa\n  |\\mu| = \\frac{d\\Omega_I}{d\\Omega_S} = |\\det J|^{-1} ,\\nonumber\n\\eeqa\nwhere $J$ is the Jacobian matrix\n\\beqa\n  J = \\frac{\\partial (\\cos \\cb, \\chi)}{\\partial(\\cos\\vth, \\vphi)}\n  = \\left[\\matrix{\n    \\frac{\\partial\\cos\\cb}{\\partial\\cos\\vth}\n  & \\frac{\\partial\\cos\\cb}{\\partial\\vphi   } \\cr\n    \\frac{\\partial\\chi   }{\\partial\\cos\\vth}\n  & \\frac{\\partial\\chi   }{\\partial\\vphi   }\n  }\\right].\\nonumber\n\\eeqa\nWriting out the determinant and dropping the absolute value in\norder to obtain the signed magnification, we get\n\\beq \\label{eq:mu-general}\n  \\mu = \\left[ \\frac{\\sin\\cb}{\\sin\\vth} \\left(\n      \\frac{\\partial\\cb}{\\partial\\vth }\\ \\frac{\\partial\\chi}{\\partial\\vphi}\n    - \\frac{\\partial\\cb}{\\partial\\vphi}\\ \\frac{\\partial\\chi}{\\partial\\vth }\n  \\right) \\right]^{-1} .\n\\eeq\nIn the case of spherical symmetry, the image and source lie in\nthe same plane, so $\\partial\\cb\/\\partial\\vphi=0$ and\n$\\partial\\chi\/\\partial\\vphi=1$, recovering  the familiar result\n\\beqa\n  \\mu = \\left( \\frac{\\sin\\cb}{\\sin\\vth}\\ \\frac{\\partial\\cb}{\\partial\\vth}\n    \\right)^{-1} .\\nonumber\n\\eeqa\n\n\n\n\n\n\\section{Lens Equation for Kerr Black Holes}\n\\label{sec:lenseq-Kerr}\n\n\n\\subsection{Kerr Metric}\n\\label{sec:Kerr-metric}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.62]{BHCoords2.eps}\n\\end{center}\n\\caption{\nCartesian $(X,Y,Z)$ and spherical polar $(r,\\kpolar,\\kazym)$\ncoordinates centered on the black hole, where $\\kpolar = \\pi\/2 - \\wp$ with $\\wp$ the polar angle; note that $-\\pi\/2 \\leq \\kpolar \\leq \\pi\/2$.  The black hole spins about the $Z$-axis, which corresponds to\n$\\kpolar=\\pi\/2$, in the direction of\nincreasing $\\kazym$.  The equatorial plane\nof the black hole corresponds to $\\kpolar = 0$ or the\n$(X,Y)$-plane.}\n\\label{fig:BHCoords}\n\\end{figure}\n\nNow let the deflector in Fig.~\\ref{fig:Lens-Geom3} be a Kerr black hole.  The Kerr metric is the unique axisymmetric,\nstationary, asymptotically flat, vacuum solution of the Einstein\nequations describing a stationary black hole with mass $\\bhpm$\nand spin angular momentum $\\bhpJ$ (see, e.g., \\cite[pp.~322-324]{wald}).\nConsider the Kerr metric in Boyer-Lindquist coordinates\n$(t,r,\\wp,\\kazym)$, where $\\wp$ is the polar angle\nand $\\kazym$ the azimuthal angle.  For our purposes, it is\nconvenient to transform $\\wp$ to $\\kpolar = \\pi\/2-\\wp$\nand work with the slightly  modified Boyer-Lindquist coordinates\n$(t,r,\\kpolar,\\kazym)$; note that $-\\pi\/2 \\leq \\kpolar \\leq \\pi\/2$.  The spatial coordinates are shown in\n\\reffig{BHCoords}.\n\nThe metric takes the form\n\\beqa\n\\label{eq:app:kerr-metric}\n  ds^2  = g_{tt}\\, dt^2  + g_{rr}\\,dr^2 \n    + g_{\\kpolar \\kpolar}\\,d\\kpolar^2\n    + g_{\\kazym \\kazym}\\,d\\kazym^2  \n    + 2\\,g_{t \\kazym}\\,dt\\,d\\kazym\\,,\\nonumber\n\\eeqa\nwhere $t = c {\\tt t}$ with ${\\tt t}$ being physical time.  The\nmetric coefficients are\n\\beqa \\label{eq:app:kerr-components}\n  g_{tt} &=& - \\frac{r(r-2\\gravr) + \\bha^2 \\sin^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ , \\label{gtt}\\\\\n  g_{rr} &=& \\frac{r^2 + \\bha^2 \\sin^2\\kpolar}{r(r-2\\gravr) + \\bha^2}\\ , \\label{grr}\\\\\n  g_{\\kpolar \\kpolar} &=& r^2 + \\bha^2 \\sin^2\\kpolar\\,, \\label{gss}\\\\\n  g_{\\kazym \\kazym} &=& \\frac{(r^2+\\bha^2)^2\n    - \\bha^2(\\bha^2+r(r-2\\gravr))\\cos^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ \\cos^2\\kpolar\\,, \\label{gphiphi}\\\\\n  g_{t \\kazym} &=& - \\frac{2 \\gravr \\bha \\, r \\cos^2\\kpolar}\n    {r^2 + \\bha^2 \\sin^2\\kpolar}\\ .\\label{gtphi}\n\\eeqa\nThe parameter $\\gravr$ is the gravitational radius, and $\\bha$ \nis the angular momentum per unit mass:\n\\beqa\n  \\gravr = \\frac{G \\bhpm}{c^2}\\ , \\qquad\n  \\bha = \\frac{\\bhpJ}{c \\bhpm}\\ .\\nonumber\n\\eeqa\nNote that both $\\gravr$ and $\\bha$ have dimensions of length.\nIt is convenient to define a dimensionless spin parameter:\n\\beqa\n\\label{eq:ahat}\n  \\ahat = \\frac{\\bha}{\\gravr}\\ .\\nonumber\n\\eeqa\nUnless stated to the contrary, the black hole's spin is\nsubcritical; i.e., $\\ahat^2 < 1$.\n\n\n\n\\subsection{Lens Equation for an Equatorial Observer}\n\\label{sec:Kerr-lens-eqn}\n\n{\\em We now specialize to the case when the observer lies\nin the equatorial plane of the Kerr black hole,} so the coordinates\n$(x,y,z)$ in \\reffig{Lens-Geom3} coincide with the coordinates\n$(X,Y,Z)$ in \\reffig{BHCoords}.  Note that we still consider\ngeneral source positions.\n\nIn \\refapp{Kerr-null-geodesics} we carefully analyze null\ngeodesics seen by an observer in the equatorial plane.  By\nconsidering constants of the motion, we derive the following\nlens equation:\n\\beqa\n  d_S \\tan\\cb \\cos\\chi &=& d_{LS} \\tan\\vth_S \\cos\\vphi_S\n    \\ + \\ d_L\\ \\frac{\\sin\\vth \\cos\\vphi}{\\cos\\vth_S}\\ ,\n    \\label{eq:le-h-Kerr} \\\\\n  d_S \\tan\\cb \\sin\\chi &=& d_{LS} \\tan\\vth_S \\sin\\vphi_S\n    \\ + \\ \\frac{d_L \\sin\\vth}{1 - \\sin^2\\vth_S \\sin^2\\vphi_S} \\times\n    \\label{eq:le-v-Kerr} \\\\\n  &&\\qquad \\left[ \\cos\\vphi \\sin\\vth_S \\tan\\vth_S \\sin\\vphi_S \\cos\\vphi_S\n    + \\left( \\sin^2\\vphi - \\sin^2\\vth_S \\sin^2\\vphi_S \\right)^{1\/2} \\right] .\n    \\nonumber\n\\eeqa\n{\\it This is the general form of the lens equation for an equatorial\nobserver in the Kerr metric for observer and source in the\nasymptotically flat region.}  \nIt is valid for all light rays, whether they loop\naround the black hole or not, as long as they lie outside the\nregion of photon capture.  No small-angle approximation is required.\n\n\n\nNote that eqns. (\\ref{eq:le-h}) and (\\ref{eq:le-v}) represent the general\nform of the lens equation, with the displacement terms explicitly\nwritten, while eqns. (\\ref{eq:le-h-Kerr}) and (\\ref{eq:le-v-Kerr}) give\nthe exact lens equation for an equatorial Kerr observer, with\nthe displacement terms implicitly included.  Demanding that these\ntwo pairs of equations be equivalent allows us to identify the\ndisplacement terms for an equatorial Kerr observer:\n\\beqa\n  \\cd_y &=& d_L \\sin\\vth \\cos\\vphi \\left( \\frac{1}{\\cos\\vth_S}\n    - \\frac{1}{\\cos\\vth} \\right) , \\label{eq:dispy}\\\\\n  \\cd_z &=& - d_L \\tan\\vth \\sin\\vphi \\ + \\ \n    \\frac{d_L \\sin\\vth}{1 - \\sin^2\\vth_S \\sin^2\\vphi_S} \\times \\label{eq:dispz}\\nonumber\\\\\n  &&\\qquad \\left[ \\cos\\vphi \\sin\\vth_S \\tan\\vth_S \\sin\\vphi_S \\cos\\vphi_S\n    + \\left( \\sin^2\\vphi - \\sin^2\\vth_S \\sin^2\\vphi_S \\right)^{1\/2} \\right].\\label{eq:dispz}\n\\eeqa\n\n\n\n\\subsection{Schwarzschild Case}\n\\label{sec:spherical}\n\nIn the case of a spherically symmetric lens we have $\\vphi_S = \\vphi + \\pi$, and either $\\chi = \\vphi$\nor $\\chi = \\vphi+\\pi$, depending on whether the source lies on the\nsame or opposite side of the lens as the image.  Once again, we define $\\snq = \\cos(\\chi-\\vphi)$ to be\na sign that distinguishes these two cases.  With these conditions\neqns.~(\\ref{eq:le-h-Kerr}) and (\\ref{eq:le-v-Kerr}) combine to form the\nsingle lens equation with displacement for a Schwarzschild black hole:\n\\beqa \\label{eq:le-sph-Kerr}\n  d_S\\,\\snq\\,\\tan\\cb = d_L\\ \\frac{\\sin\\vth}{\\cos\\vth_S} \\ - \\ \n    d_{LS}\\,\\tan\\vth_S\\,.\n\\eeqa\nTwo comments are in order.  First, our spherical lens equation\n(\\ref{eq:le-sph-Kerr}) is equivalent to the spherical lens\nequation recently derived by Bozza \\& Sereno \\cite{BS,Bozza2} (up to the sign $\\snq$,\nwhich was not discussed explicitly in \\cite{BS,Bozza2}).\nThe second comment refers to the amplitude of the displacement.\nBy comparing our general planar-ray lens equation (\\ref{eq:le-sph}) with eqn.~(\\ref{eq:le-sph-Kerr}),\nwe can identify the displacement\n\\beqa\n\\label{eq:disp-sph}\n  \\cd = d_L \\sin\\vth \\left[ \\frac{1}{\\cos(\\alpha-\\vth)} - \\frac{1}{\\cos\\vth}\n    \\right]\\,,\n\\eeqa\nwhere we have switched from $\\vth_S$ to the bending angle\n$\\alpha = \\vth+\\vth_S$.  Now let\n$\\delta\\alpha = \\alpha \\, {\\rm mod} \\, 2 \\pi$, and assume that\n$\\vth$ and $\\delta\\alpha$ are small.  (Note that we need not\nassume $\\alpha$ itself is small, only that $\\delta\\alpha$ is small.\nThis means that our analysis applies to all light rays, including those that loop\naround the lens.)  Taylor expanding the displacement in $\\vth$\nand $\\alpha$ yields\n\\beqa\n  \\cd = \\frac{d_L}{2} \\, (\\vth \\ \\delta \\alpha) \n         (\\delta \\alpha - 2 \\vth)  \\ + \\ {\\cal O}(4).\\nonumber\n\\eeqa\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nRecently a lens equation was developed for Schwarzschild lensing with displacement (see Section~\\ref{sec:spherical} above), when the light ray's tangent lines\nat the source and observer do not meet on the lens plane.  In this paper we found a new generalization of the lens equation with displacement for axisymmetric lenses that extends the previous work to a fully three-dimensional setting.  Our formalism assumes that the source and observer are in the asymptotically flat region, and does not require a small angle approximation.  Furthermore, we found a new magnification formula applicable to this more general context.  Our lens equation is thus applicable to non-spherically symmetric compact bodies, such as Kerr black holes.  We gave explicit formulas for the\ndisplacement when the observer is in the equatorial plane\nof a Kerr black hole and \nfor the situation of spherical symmetry.  \n\n\n\n\n\\begin{acknowledgments}\n\nABA and AOP would especially like to thank Marcus C. Werner for helpful discussions.  AOP acknowledges the support of NSF Grant DMS-0707003.\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLanguage and facial expression are strong indicators for behavior analysis. There is numerous research trying to solve the emotion recognition problem based on these data. However, the non-linguistic vocalizations are understudied even though they are very useful information. Analyzing and applying these signals are interesting topics and require more attention from researchers. The A-VB competition was conducted for that reason and is expected to explore advanced improvements in emotion science. The competition \\cite{b2} consists of 4 individual tasks as below:\n\n\\begin{itemize}\n\\item High-dimension task (A-VB-HIGH): a multi-output regression task generating 10 values in the range of [0,1] corresponding to levels of Awe, Excitement, Amusement, Awkwardness, Fear, Horror, Distress, Triumph, Sadness, and Surprise.\n\\item Two-dimension task (A-VB-TWO): a multi-output regression task generating 2 values in the range of [0,1] corresponding to levels of Valence and Arousal.\n\\item Culture task (A-VB-CULTURE): a multi-output regression task generating 40 values in the range of [0,1] corresponding to levels of culture including China, US, South Africa, and Venezuela combined with  levels of emotion including Awe, Excitement, Amusement, Awkwardness, Fear, Horror, Distress, Triumph, Sadness, Surprise.\n\\item Type task (A-VB-TYPE): a multi-class classification task predicting the type of expressive vocal including Gasp, Laugh, Cry, Scream, Grunt, Groan, Pant, Other.\n\\end{itemize}\n\nThe evaluation metric for three regression tasks is the Concordance Correlation Coefficient (CCC) and the metric for the categorical task is the Unweighted Average Recall (UAR). The detail is listed below:\n\\begin{itemize}\n\\item A-VB-HIGH: The metric is the average CCC score of 10 emotions.\n\\item A-VB-TWO: The metric is the average CCC score of valence-arousal.\n\\item A-VB-CULTURE: The metric is the average CCC score of 40 culture-emotion levels.\n\\item A-VB-TYPE: The metric is the UAR score of 8 classes of vocalizations.\n\\end{itemize}\n\nAll of the above metrics are in the range of [0,1]. The greater the score is, the better the model performs. We should also note that all the results in this paper are in percentage. \n\nIn this paper, we propose a straightforward approach using a pre-trained Wav2vec network to resolve the problem. The model accomplishes a noticeable improvement compared to the baseline provided by the organizers. Because of its simplicity, our model can be considered a new baseline for all tasks in the competition.\n\n\\section{Related work}\nIn the baseline paper \\cite{b2}, the authors introduce two different approaches, which are feature-based and end-to-end approaches. In the feature-based option, the OpenSMILE toolkit \\cite{b11} is leveraged to extract the COMPARE (COMputational PARalinguistics ChallengE \\cite{b12}) feature and EGEMAPS (The extended Geneva Minimalistic Acoustic Parameter Set \\cite{b13}) feature from an input sample. The features are then fed to a 3-layer fully-connected neural network. Mean Squared Error (MSE) loss is used for regression tasks while the classification task applies the Cross-entropy (CE) loss function.\n\nIn the end-to-end manner, the baseline model uses End2You \\cite{b14}, a multimodal profiling toolkit that is capable of training and evaluating models from raw input. Particularly, Emo-18 architecture \\cite{b15} is chosen for the competition. The model includes 1-D Convolutional Neural Network (CNN) layers to derive the features from audio frames and a Recurrent Neural Network to learn the temporal information.\n\nFor the speaker recognition task, Nik Vaessen and David A. van Leeuwen \\cite{b4} conducted fine-tuning the Wav2vec2 model by using a shared fully-connected layer. Their model and ours have one thing in common: exploiting the pre-trained Wav2vec. However, there are considerable differences between the two methods. Basically, speaker recognition is a classification problem so the authors optimize the model with CE or Binary Cross-entropy (BCE) loss. In our method, we consider two loss options for the regression tasks, which are MSE and CCC loss. Additionally, besides using a shared fully-connected layer, we also take advantage of the RNN to explore the temporal behavior.\n\\section{Method}\nThe sequence embeddings are obtained from the waveform signal by the audio extractor. They are then fed into an RNN to enrich the sequence information. Afterward, a fully connected layer changes the embeddings' dimension to the output sizes depending on the particular task. Finally, a pooling layer is used to reduce variable-length sequence embeddings to fix-size speaker embedding. The dimension of the final prediction would be 2, 10, 40, or 8 corresponding to A-VB-TWO, A-VB-HIGH, A-VB-CULTURE or A-VB-TYPE task, respectively. Figure~\\ref{fig:model} describes the architecture of our method.\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[height=8cm]{model.PNG}}\n\\caption{Block diagram of our proposed model.}\n\\label{fig:model}\n\\end{figure}\n\n\\subsection{Audio extractor}\n\nWe take advantage of the Pre-trained Wav2vec 2.0 models \\cite{b3} provided by Pytorch. They are trained with large unlabeled audio corpora so they can effectively capture the audio features. We conducted experiments with 4 versions of the Wav2vec2 model described below:\n\\begin{itemize}\n\\item BASE: use the base configuration of transformer trained with 960 hours of unlabeled audio from LibriSpeech \\cite{b5}.\n\\item LARGE: use the large configuration of transformer trained with 960 hours of unlabeled audio from LibriSpeech \\cite{b5}.\n\\item LARGE-LV60K: use the large configuration of transformer trained with 60,000 hours of unlabeled audio from Libri-Light \\cite{b6}.\n\\item XLSR53: use base configuration of transformer \\cite{b10} trained with 56,000 hours of unlabeled audio from multiple datasets (Multilingual LibriSpeech  \\cite{b7}, CommonVoice  \\cite{b8} and BABEL  \\cite{b9}).\n\\end{itemize}\n\n\\subsection{Pooling Method}\nInspired by the model of \\cite{b4}, we use 4 options of pooling to fix the length of embedding: first (take the first sequence embedding to be the speaker embedding), last (take the last sequence embedding to be the speaker embedding), max, and average pooling. The performance of models with various types of pooling layers is recorded to study their impact on the result. The operation of the pooling can be described as:\n\\begin{equation}\ns_{i} = Pooling(e_1, e_2, ..., e_{m_i})\n\\end{equation}\nwhere $s_{i}$ is the speaker embedding of $i^{th}$ sample; $e_1, e_2, ..., e_{m_i}$ are the temporal embeddings and $m_i$ is the sequence length of corresponding sample.\n\n\n\\subsection{Loss function}\nWe use the CE loss for the classification task. In the remaining tasks, we want to test the effect of the loss function on the performance of the model so we did the experiments and compared the result of the model using Mean Square Error (MSE) and Concordance Correlation Coefficient (CCC) loss. The CCC loss is formulated as below:\n\\begin{equation}\n\\mathcal{L} = 1 - CCC = 1 - \\frac{2s_{xy}}{s^{2}_x + s^{2}_y  + \\left(\\overline{x}-\\overline{y}\\right)^2}\n\\end{equation}\nwhere $\\overline{x}$ and $\\overline{y}$ are the mean values of ground truth and predicted values, respectively, $s_x$ and $s_x$ are corresponding variances and $s_{xy}$ is the covariance value.\n\n\\section{Dataset and Experiments}\n\\subsection{Dataset}\nThe database used for the competition is the HUME-VB dataset  \\cite{b1} which consists of 59201 audio files and is split into 3 sets (training, validation, and test) of similar size. Each file has 53 labels corresponding to 4 tasks, one label is a categorical label that is used in the classification task and the remains are values in the range [0,1] representing the emotional level. The organizers provide 2 versions of the HUME-VB dataset: the raw version sampled at 48kHz and the processed version where audio files are converted to 16kHz and normalized to -3 decibels. In our experiments, we take the processed version as the input of our model.\n\n\n\\subsection{Experiments}\nOur model was implemented using the Pytorch framework. The experiments were conducted on a machine with NVIDIA RTX 2080 Ti GPU. All scenarios were run in 20 epochs, and the model with the best performance on the validation set was recorded. The batch size is 16 and the initial learning rate is $1e-4$. We used Adam optimizer with the weight decay coefficient of $0.0625$.\n\nIn our setting, we take the output from the $12^{th}$ layer of the Wav2vec network to be the sequence embeddings. The input size of the RNN network depends on the configuration of the pre-trained audio extractor, which is 768 for base configuration and 1024 for large configuration. It includes 2 LSTM layers and the hidden size is fixed to 512.\n\n\\section{Discussion}\nWe tested the performance of the model with various audio extractors to explore their effect. Table~\\ref{tab:result} shows the performance on four tasks of the competition with 4 pre-trained Wav2vec extractors. As a result, the XLSR53 pre-trained model achieves the best performance in A-VB-TWO and A-VB-TYPE when LARGE-LV60K attains the highest scores in A-VB-HIGH and A-VB-CULTURE. In the meanwhile, the BASE model produces the lowest score in A-VB-TWO and A-VB-TYPE due to its simple architecture.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different extractors. Experimented with CCC loss and Last Pooling}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|l|l|l|}\n    \\hline\n    Model & TWO & HIGH & CULTURE & TYPE \\\\\n    \\hline\n    \\textit{End2You Baseline} & \\textit{49.88}  & \\textit{56.38} & \\textit{43.59} & \\textit{41.66} \\\\\n    \\hline\n    BASE & 54.65  & 58.69 & 47.18 & 41.65     \\\\\n    \\hline\n    LARGE & 55.42 & 58.00 & 47.12 & 43.96      \\\\\n    \\hline\n    LARGE-LV60K & 61.59 & \\textbf{65.41} & \\textbf{53.39} & 47.22  \\\\\n    \\hline\n    XLSR53 & \\textbf{61.94} & 65.32 & 52.50 & \\textbf{49.89}  \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:result}\n\\end{center}\n\\end{table}\n\nRegarding the pooling method, we examined their influence on the results in A-VB-HIGH. As shown in Table~\\ref{tab:pool}, in both LARGE-LV60K and XLSR53 model, the Last pooling outperforms the other options while the First pooling gets the lowest CCC score among the four methods. The result of Avg pooling is slightly better than Max pooling in both LARGE-LV60K and XLSR53 scenarios. It can be inferred that the last embedding of the sequence contains the most useful information for the prediction when using other embeddings or combining them may degrade the accuracy.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different pooling methods. Experimented on A-VB-HIGH with CCC loss}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|}\n    \\hline\n    Pooling & LARGE-LV60K & XLSR53 \\\\\n    \\hline\n     First & 53.56 & 58.20 \\\\\n    \\hline\n     Max & 60.08 & 61.41 \\\\\n    \\hline\n     Avg & 61.49 & 62.40 \\\\\n    \\hline\n     Last & \\textbf{65.41} & \\textbf{65.32} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:pool}\n\\end{center}\n\\end{table}\nNext, we conducted the training processes with MSE and CCC to explore their advantage. As a consequence, the model trained with CCC loss gives better performance on the validation set compared to the one trained with MSE loss. The detail is shown in Table~\\ref{tab:loss}.\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with different loss functions. Experimented on A-VB-HIGH with Last pooling}\n\\begin{center}\n  \\centering\n  \\begin{tabular}{|l|l|l|}\n  \\hline\n    Loss function & LARGE-LV60K & XLSR53 \\\\\n    \\hline\n    MSE & 63.77 & 63.94 \\\\\n    \\hline\n    CCC & \\textbf{65.41} & \\textbf{65.32} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:loss}\n\\end{center}\n\\end{table}\n\nIn addition, we carried out the ablation study to analyze the role of the RNN. According to Table~\\ref{tab:rnn}, using the RNN can significantly boost the accuracy of the model in all four tasks. It can be explained by the capability of learning temporal information of the LSTM, which can enhance the overall operation of the model.\n\n\\begin{table}\n \\caption{Evaluation score on the HUME-VB validation set with and without RNN. Experimented on A-VB-HIGH with LARGE-LV60K model, CCC loss, and Last pooling}\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|}\n    \\hline\n    Model & TWO & HIGH & CULTURE & TYPE \\\\\n    \\hline\n    Without RNN & 47.38 & 50.70 & 40.20 & 39.10 \\\\\n    \\hline\n    With RNN & \\textbf{61.59} & \\textbf{65.41} & \\textbf{53.39} & \\textbf{47.22} \\\\\n    \\hline\n  \\end{tabular}\n  \\label{tab:rnn}\n\\end{center}\n\\end{table}\n\nAfter conducting the above experiments, we concluded that the best configuration of our model is combining either LARGE-LV60K or XLSR53 pre-trained model with last pooling method and utilizing CCC loss. This setting was used to train separated model for each task in order to obtain unbiased evaluation on test set. We decided to choose LARGE-LV60K for A-VB-HIGH and A-VB-CULTURE, XLSR53 for A-VB-TWO and A-VB-TYPE. This time we trained each model for 50 epochs and applied early stopping by monitoring the validation result. Our best models and their evaluations on test set and validation set are listed in  Table~\\ref{tab:test}.\n\n\\begin{table}\n\\caption{Evaluation score on the HUME-VB validation and test sets. Experimented with CCC loss and Last pooling for 50 epochs}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n Task & Pre-trained &\\multicolumn{2}{|c|}{Performance} \\\\\n\\cline{3-4} \n name & Audio Extractor & Val set & Test set \\\\\n\\hline\n TWO & XLSR53 & 61.94 & 62.02 \\\\\n\\hline\n HIGH & LARGE-LV60K & 66.76 & 66.77 \\\\\n\\hline\n CULTURE & LARGE-LV60K & 54.93 & 54.95 \\\\\n\\hline\n TYPE & XLSR53 & 49.89 & 49.70 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:test}\n\\end{center}\n\\end{table}\n\n\\section{Conclusion}\nThis paper presents our proposed method for all sub-challenges of the A-VB competition. Particularly, we fine-tuned the pre-trained Wav2vec and combined it with basic neural networks and a proper pooling method. The CCC loss and Last pooling show the best performance on four tasks among the other options. Our model outperforms the baseline of the organizer on the test set, with the CCC score of 62.02 for A-VB-TWO, 66.77 for A-VB-HIGH, 54.95 for A-VB-CULTURE and UAR metric of 49.70 for A-VB-TYPE.\n\n\\section*{Acknowledgment}\n\nThis work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT). (NRF-2020R1A4A1019191).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMassive multiple-input multiple-output (MIMO), also known as\nlarge-scale or very-large MIMO, is a promising technology to meet\nthe ever growing demands for higher throughput and better\nquality-of-service of next-generation wireless communication\nsystems \\cite{RusekPersson13,LarssonEdfors14,ChenSun16}. Massive\nMIMO systems are those that are equipped with a large number of\nantennas at the base station (BS) simultaneously serving a much\nsmaller number of single-antenna users sharing the same\ntime-frequency slot. By exploiting the asymptotic orthogonality\namong channel vectors associated with different users, massive\nMIMO systems can achieve almost perfect inter-user interference\ncancelation with a simple linear precoder and receive combiner\n\\cite{Marzetta10}, and thus have the potential to enhance the\nspectrum efficiency by orders of magnitude.\n\n\n\nDespite all these benefits, massive MIMO systems pose new\nchallenges for system design and hardware implementation. Due to\nthe large number of antennas at the BS, the hardware cost and\npower consumption could become prohibitively high if we still\nemploy expensive and power-hungry high-resolution\nanalog-to-digital convertors (ADCs) \\cite{ChenZhao14}. To address\nthis obstacle, recent studies (e.g.\n\\cite{RisiPersson14,FanJin15,ZhangDai16,JacobssonDurisi15,WangLi15,WenWang16,ChoiMo16})\nconsidered the use of low-resolution ADCs (e.g. 1-3 bits) for\nmassive MIMO systems. It is known that the hardware complexity and\npower consumption grow exponentially with the resolution (i.e. the\nnumber of bits per sample) of the ADC. Therefore lowering the\nresolution of the ADC can effectively reduce the hardware cost and\npower consumption. In particular, for the extreme one-bit case,\nthe ADC becomes a simple analog comparator. Also, automatic gain\ncontrol (AGC) is no longer needed when one-bit ADCs are used,\nwhich further simplifies the hardware complexity.\n\n\n\nMassive MIMO with low-resolution ADCs has attracted much attention\nover the past few years. Great efforts have been made to\nunderstand the effects of low-resolution ADCs on the performance\nof MIMO and massive MIMO systems. Specifically, by assuming full\nknowledge of channel state information (CSI), the capacity at both\nfinite and infinite signal-to-noise ratio (SNR) was derived in\n\\cite{MoHeath15} for one-bit MIMO systems. For massive MIMO\nsystems with low-resolution ADCs, the spectral efficiency and the\nuplink achievable rate were investigated in\n\\cite{RisiPersson14,FanJin15,ZhangDai16,LiangZhang16} under\ndifferent assumptions. The theoretical analyses suggest that the\nuse of the low cost and low-resolution ADCs can still provide\nsatisfactory achievable rates and spectral efficiency.\n\n\n\n\n\n\n\n\n\nIn this paper, we consider the problem of channel estimation for\nuplink multiuser massive MIMO systems, where one-bit ADCs are used\nat the BS in order to reduce the cost and power consumption.\nChannel estimation is crucial to support multi-user MIMO operation\nin massive MIMO systems\n\\cite{AdhikaryNam13,ChoiLove14,SunGao15,GaoDai15,FangLi17}. To\nreach the full potential of massive MIMO, accurate downlink CSI is\nrequired at the BS for precoding and other operations. Most\nliterature on massive MIMO systems, e.g.\n\\cite{Marzetta10,RusekPersson13,YinGesbert13,MullerCottatellucci14},\nassumes a time division duplex (TDD) mode in which the downlink\nCSI can be immediately obtained from the uplink CSI by exploiting\nchannel reciprocity. Nevertheless, channel estimation for massive\nMIMO systems with one-bit ADCs is challenging since the magnitude\nand phase information about the received signal are lost or\nseverely distorted due to the coarse quantization. It was shown in\n\\cite{RisiPersson14} that one-bit massive MIMO systems require an\nexcessively long training sequence (e.g. approximately 50 times\nthe number of users) to achieve an acceptable performance. The\nwork \\cite{JacobssonDurisi15} showed that for one-bit massive MIMO\nsystems, a least-squares channel estimation scheme and a\nmaximum-ratio combining scheme are sufficient to support both\nmultiuser operation and the use of high-order constellations.\nNevertheless, a long training sequence is still a requirement. To\nalleviate this issue, a Bayes-optimal joint channel and data\nestimation scheme was proposed in \\cite{WenWang16}, in which the\nestimated payload data are utilized to aid channel estimation. In\n\\cite{ChoiMo16}, a maximum likelihood channel estimator, along\nwith a near maximum likelihood detector, were proposed for uplink\nmassive MIMO systems with one-bit ADCs.\n\n\n\nDespite these efforts, channel estimation using one-bit quantized\ndata still incur much larger estimation errors as compared with\nusing the original unquantized data, and require considerably\nhigher training overhead to attain an acceptable estimation\naccuracy. To address this issue, in this paper, we study one-bit\nquantizer design and examine the impact of the choice of\nquantization thresholds on the estimation performance.\nSpecifically, the optimal design of quantization thresholds as\nwell as the training sequences is investigated. Note that one-bit\nquantization design is an interesting and important issue but\nlargely neglected by existing massive MIMO channel estimation\nstudies. In fact, most channel estimation schemes, e.g.\n\\cite{RisiPersson14,JacobssonDurisi15,WenWang16,ChoiMo16}, assume\na fixed, typically zero, quantization threshold. The optimal\nchoice of the quantization threshold was considered in\n\\cite{KochLapidoth13,Verdu02}, but addressed from an\ninformation-theoretic perspective. Our theoretical results reveal\nthat, given that the quantization thresholds are optimally\ndevised, using one-bit ADCs can achieve an estimation error close\nto (with an increase only by a factor of $\\pi\/2$) the minimum\nachievable estimation error attained by using infinite-precision\nADCs. The optimal quantization thresholds, however, are dependent\non the unknown channel parameters. To cope with this difficulty,\nwe propose an adaptive quantization (AQ) scheme by which the\nthresholds are dynamically adjusted in a way such that the\nthresholds converge to the optimal thresholds, and a random\nquantization (RQ) scheme which randomly generates a set of\nnon-identical thresholds based on some statistical prior knowledge\nof the channel. Simulation results show that our proposed schemes,\nbecause of their wisely devised quantization thresholds, present a\nsignificant performance improvement over the fixed quantization\nscheme that use a fixed (say, zero) quantization threshold. In\nparticular, the AQ scheme, even with a moderate number of pilot\nsymbols (about 5 times the number of users), can provide an\nachievable rate close to that of the perfect CSI case.\n\n\nThe rest of the paper is organized as follows. The system model\nand the problem of channel estimation using one-bit ADCs are\ndiscussed in Section \\ref{sec:system-model}. In Section\n\\ref{sec:MLE-CRB}, we develop a maximum likelihood estimator and\ncarry out a Cram\\'{e}r-Rao bound analysis of the one-bit channel\nestimation problem. The optimal design of quantization thresholds\nand the pilot sequences is studied in Section\n\\ref{sec:optimal-design}. In Section \\ref{sec:AQ-RQ}, we develop\nan adaptive quantization scheme and a random quantization scheme\nfor practical threshold design. Simulation results are provided in\nSection \\ref{sec:experiments}, followed by concluding remarks in\nSection \\ref{sec:conclusion}.\n\n\n\n\n\n\\section{System Model and Problem Formulation} \\label{sec:system-model}\nConsider a single-cell uplink multiuser massive MIMO system, where\nthe BS equipped with $M$ antennas serves $K$ ($M\\gg K$)\nsingle-antenna users simultaneously. The channel is assumed to be\nflat block fading, i.e. the channel remains constant over a\ncertain amount of coherence time. The received signal at the BS\ncan be expressed as\n\\begin{align}\n\\boldsymbol{Y}=\\boldsymbol{H}\\boldsymbol{X}+\\boldsymbol{W}\n\\label{data-model}\n\\end{align}\nwhere $\\boldsymbol{X}\\in\\mathbb{C}^{K\\times L}$ is a training\nmatrix and its row corresponds to each user's training sequence\nwith $L$ pilot symbols, $\\boldsymbol{H}\\in\\mathbb{C}^{M\\times K}$\ndenotes the channel matrix to be estimated, and\n$\\boldsymbol{W}\\in\\mathbb{C}^{M\\times L}$ represents the additive\nwhite Gaussian noise with its entries following a circularly\nsymmetric complex Gaussian distribution with zero mean and\nvariance $2\\sigma^2$.\n\nTo reduce the hardware cost and power consumption, we consider a\nmassive MIMO system which uses one-bit ADCs at the BS to quantize\nthe received signal. Specifically, at each antenna, the real and\nimaginary components of the received signal are quantized\nseparately using a pair of one-bit ADCs. Thus in total $2M$\none-bit ADCs are needed. The quantized output of the received\nsignal, $\\boldsymbol{B}\\triangleq [b_{m,l}]$, can be written as\n\\begin{align}\n\\boldsymbol{B}=\\mathcal{Q}(\\boldsymbol{Y})\n\\label{conventional-quantizer}\n\\end{align}\nwhere $\\mathcal{Q}(\\boldsymbol{Y})$ is an element-wise operation\nperformed on $\\boldsymbol{Y}$, and for each element of\n$\\boldsymbol{Y}$, $y_{m,l}$, we have\n\\begin{align}\n\\mathcal{Q}(y_{m,l})=\\text{sgn}(\\Re(y_{m,l}))+j\\text{sgn}(\\Im(y_{m,l}))\n\\end{align}\nin which $\\Re(y)$ and $\\Im(y)$ denote the real and imaginary\ncomponents of $y$, respectively, and the sign function\n$\\text{sgn}(\\cdot)$ is defined as\n\\begin{align}\n\\text{sgn}(y) \\triangleq \\left\\{ \\begin{array}{ll}\n1 & \\textrm{if $y\\ge 0$}\\\\\n-1 & \\textrm{otherwise}\n\\end{array} \\right.\n\\end{align}\nTherefore the quantized output belongs to the set\n\\begin{align}\nb_{m,l}\\in \\{1+j,-1+j,1-j,-1-j\\}\\quad \\forall m,l\n\\end{align}\nNote that in (\\ref{conventional-quantizer}), we implicitly assume\na zero threshold for one-bit quantization. Nevertheless, using\nidentically a zero threshold for all measurements is not\nnecessarily optimal, and it is interesting to analyze the impact\nof the quantization thresholds on the channel estimation\nperformance. Such an issue (i.e. choice of quantization\nthresholds), albeit important, was to some extent neglected by\nmost existing studies. To examine this problem, let\n$\\boldsymbol{T}\\triangleq [\\tau_{m,l}]$ denote the thresholds used\nfor one-bit quantization. The quantized output of the received\nsignal, $\\boldsymbol{B}$, is now given as\n\\begin{align}\n\\boldsymbol{B}=\\mathcal{Q}(\\boldsymbol{Y}-\\boldsymbol{T})\n\\label{quantizer}\n\\end{align}\n\nTo facilitate our analysis, we first convert (\\ref{data-model})\ninto a real-valued form as follows\n\\begin{align}\n\\boldsymbol{\\widetilde{Y}}=\\boldsymbol{\\widetilde{A}}\\boldsymbol{\\widetilde{H}}+\\boldsymbol{\\widetilde{W}}\n\\end{align}\nwhere\n\\begin{align}\n\\boldsymbol{\\widetilde{Y}}\\triangleq & [ \\Re(\\boldsymbol{Y}) \\\n\\Im(\\boldsymbol{Y})]^T \\nonumber\\\\\n\\boldsymbol{\\widetilde{H}} \\triangleq & [ \\Re(\\boldsymbol{H}) \\\n\\Im(\\boldsymbol{H})]^T \\nonumber\\\\\n\\boldsymbol{\\widetilde{W}} \\triangleq & [ \\Re(\\boldsymbol{W}) \\\n\\Im(\\boldsymbol{W})]^T \\nonumber\n\\end{align}\nand\n\\begin{align}\n\\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\label{A-X-relationship}\n\\end{align}\nVectorizing the real-valued matrix $\\boldsymbol{\\widetilde{Y}}$,\nthe received signal can be expressed as a real-valued vector form\nas\n\\begin{align}\n\\boldsymbol{y}=\\boldsymbol{A}\\boldsymbol{h}+\\boldsymbol{w}\n\\label{data-model-vector}\n\\end{align}\nwhere\n$\\boldsymbol{y}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{Y}})$,\n$\\boldsymbol{A}\\triangleq\\boldsymbol{I}_{M}\\otimes\\boldsymbol{\\widetilde{A}}$,\n$\\boldsymbol{h}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{H}})$,\nand\n$\\boldsymbol{w}\\triangleq\\text{vec}(\\boldsymbol{\\widetilde{W}})$.\nIt can be easily verified $\\boldsymbol{y}\\in\\mathbb{R}^{2ML}$,\n$\\boldsymbol{A}\\in\\mathbb{R}^{2ML\\times 2MK}$, and\n$\\boldsymbol{h}\\in\\mathbb{R}^{2MK}$. Accordingly, the one-bit\nquantized data can be written as\n\\begin{align}\n\\boldsymbol{b}=\\text{sgn}(\\boldsymbol{y}-\\boldsymbol{\\tau})\n\\label{quantized-data-model-vector}\n\\end{align}\nwhere $\\boldsymbol{\\tau}\\triangleq \\text{vec}([\n\\Re(\\widetilde{\\boldsymbol{T}}) \\\n\\Im(\\widetilde{\\boldsymbol{T}})]^T)$ and\n$\\boldsymbol{\\tau}\\in\\mathbb{R}^{2ML}$. For simplicity, we define\n$N\\triangleq 2ML$. \n\nOur objective in this paper is to estimate the channel\n$\\boldsymbol{h}$ based on the one-bit quantized data\n$\\boldsymbol{b}$, examine the best achievable estimation\nperformance and investigate the optimal thresholds\n$\\boldsymbol{\\tau}$ as well as the optimal training sequences\n$\\boldsymbol{X}$. To this objective, in the following, we first\ndevelop a maximum likelihood (ML) estimator and carry out a\nCram\\'{e}r-Rao bound (CRB) analysis. The optimal choice of the\nquantization thresholds as well as the training sequences is then\nstudied based on the CRB matrix of the unknown channel parameter\nvector $\\boldsymbol{h}$.\n\n\n\n\n\\section{ML Estimator and CRB Analysis} \\label{sec:MLE-CRB}\n\\subsection{ML Estimator}\nBy combining (\\ref{data-model-vector}) and\n(\\ref{quantized-data-model-vector}), we have\n\\begin{align}\nb_n=\\text{sgn}(y_n-\\tau_n)\n=\\text{sgn}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}+w_n-\\tau_n), \\quad\n\\forall n\n\\end{align}\nwhere, by allowing a slight abuse of notation, we let $b_n$,\n$y_n$, $\\tau_n$, and $w_n$ denote the $n$th entry of\n$\\boldsymbol{b}$, $\\boldsymbol{y}$, $\\boldsymbol{\\tau}$, and\n$\\boldsymbol{w}$, respectively; and $\\boldsymbol{a}_n^T$ denotes\nthe $n$th row of $\\boldsymbol{A}$. It is easy to derive that\n\\begin{align}\nP(b_n=1;\\boldsymbol{h}) & =P(w_n\\geq-(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n);\\boldsymbol{h}) \\nonumber \\\\\n& =F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)\n\\end{align}\nand\n\\begin{align}\nP(b_n=-1;\\boldsymbol{h}) & =P(w_n < -(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n);\\boldsymbol{h}) \\nonumber \\\\\n& =1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)\n\\end{align}\nwhere $F_{w}(\\cdot)$ denotes the cumulative density function (CDF)\nof $w_n$, and $w_n$ is a real-valued Gaussian random variable with\nzero-mean and variance $\\sigma^2$. Therefore the probability mass\nfunction (PMF) of $b_n$ is given by\n\\begin{align}\np(b_n;\\boldsymbol{h})=& [1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]^{(1-b_n)\/2} \\nonumber \\\\\n&\\cdot[F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]^{(1+b_n)\/2}\n\\end{align}\nSince $\\{b_n\\}$ are independent, the log-PMF or log-likelihood\nfunction can be written as\n\\begin{align}\nL(\\boldsymbol{h}) & \\triangleq  \\log p(b_1,\\dots,b_N;\\boldsymbol{h}) \\nonumber \\\\\n& = \\sum_{n=1}^{N} \\bigg\\{ \\frac{1-b_n}{2}\\log [1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)]   \\nonumber \\\\\n& \\qquad \\quad + \\frac{1+b_n}{2} \\log\n[F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)] \\bigg\\}\n\\label{log-PMF}\n\\end{align}\nThe ML estimate of $\\boldsymbol{h}$, therefore, is given as\n\\begin{align}\n\\hat{\\boldsymbol{h}}=\\arg\\max_{\\boldsymbol{h}} \\ L(\\boldsymbol{h})\n\\label{MLE}\n\\end{align}\nIt can be proved that the log-likelihood function\n$L(\\boldsymbol{h})$ is a concave function. Hence computationally\nefficient search algorithms can be used to find the global\nmaximum. The proof of the concavity of $L(\\boldsymbol{h})$ is\ngiven in Appendix \\ref{appA}.\n\n\n\n\n\n\n\\subsection{CRB}\nWe now carry out a CRB analysis of the one-bit channel estimation\nproblem (\\ref{quantized-data-model-vector}). The CRB results help\nunderstand the effect of different system parameters, including\nquantization thresholds as well as training sequences, on the\nestimation performance. We first summarize our derived CRB results\nin the following theorem.\n\n\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem} \\label{theorem1}\nThe Fisher information matrix (FIM) for the estimation problem\n(\\ref{quantized-data-model-vector}) is given as\n\\begin{align}\n\\boldsymbol{J}(\\boldsymbol{h})=\\sum_{n=1}^{N}\ng(\\tau_n,\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T\n\\end{align}\nwhere $g(\\tau_n,\\boldsymbol{a}_n)$ is defined as\n\\begin{align}\ng(\\tau_n,\\boldsymbol{a}_n) \\triangleq \\frac {f_{w}^2\n(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)}\n{F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)(1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n))}\n\\label{g-function}\n\\end{align}\nin which $f_{w}(\\cdot)$ denotes the probability density function\n(PDF) of $w_n$. Accordingly, the CRB matrix for the estimation\nproblem (\\ref{quantized-data-model-vector}) is given by\n\\begin{align}\n\\text{CRB}(\\boldsymbol{h})=\\boldsymbol{J}^{-1}(\\boldsymbol{h}) =\n\\left( \\sum_{n=1}^{N} g(\\tau_n,\\boldsymbol{a}_n) \\boldsymbol{a}_n\n\\boldsymbol{a}_n^T \\right)^{-1} \\label{CRB}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{appB}.\n\\end{proof}\n\nAs is well known, the CRB places a lower bound on the estimation\nerror of any unbiased estimator \\cite{Kay93} and is asymptotically\nattained by the ML estimator. Specifically, the covariance matrix\nof any unbiased estimate satisfies:\n$\\text{cov}(\\hat{\\boldsymbol{h}})-\\text{CRB}(\\boldsymbol{h})\n\\succeq \\boldsymbol{0}$. Also, the variance of each component is\nbounded by the corresponding diagonal element of\n$\\text{CRB}(\\boldsymbol{h})$, i.e., $\\text{var}(\\hat{h}_i) \\ge\n[\\text{CRB}(\\boldsymbol{h})]_{ii}$.\n\nWe observe from (\\ref{CRB}) that the CRB matrix of\n$\\boldsymbol{h}$ depends on the quantization thresholds\n$\\boldsymbol{\\tau}$ as well as the matrix $\\boldsymbol{A}$ which\nis constructed from training sequences $\\boldsymbol{X}$ (cf.\n(\\ref{A-X-relationship})). Naturally, we wish to optimize\n$\\boldsymbol{\\tau}$ and $\\boldsymbol{A}$ (i.e. $\\boldsymbol{X}$)\nby minimizing the trace of the CRB matrix, i.e. the overall\nestimation error asymptotically achieved by the ML estimator. The\noptimization therefore can be formulated as follows\n\\begin{align}\n\\min_{\\boldsymbol{X},\\boldsymbol{\\tau}}\\quad &\n\\text{tr}\\left\\{\\text{CRB}(\\boldsymbol{h})\\right\\} = \\mathrm{tr}\n\\left\\{ \\left( \\sum_{n=1}^{N} g(\\tau_n,\\boldsymbol{a}_n)\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T \\right)^{-1} \\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad &\n\\boldsymbol{A}=\\boldsymbol{I}_M\\otimes\\boldsymbol{\\widetilde{A}}\n\\nonumber\\\\\n& \\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\nonumber\\\\\n& \\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P\n \\label{opt1}\n\\end{align}\nwhere $\\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P$ is a\ntransmit power constraint imposed on the pilot signals. Such an\noptimization is examined in the following section, where it is\nshown that the optimization of $\\boldsymbol{X}$ can be decoupled\nfrom the optimization of the threshold $\\boldsymbol{\\tau}$.\n\n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{g-function.eps}\n\\caption{The function value of $g(\\tau_n,\\boldsymbol{a}_n)$ vs.\n$(\\boldsymbol{a}_n^T\\boldsymbol{h}-\\tau_n)$, where $\\sigma^2=1$.}\n\\label{fig1}\n\\end{figure}\n\n\n\n\n\\section{Optimal Design and Performance Analysis} \\label{sec:optimal-design}\n\\subsection{Optimal Quantization Thresholds and Pilot\nSequences} Before proceeding, we first introduce the following\nresult.\n\n\\newtheorem{proposition}{Proposition}\n\\begin{proposition} \\label{proposition1}\nFor the Gaussian random variable $w_n$,\n$g(\\tau_n,\\boldsymbol{a}_n)$ defined in (\\ref{g-function}) is a\npositive and symmetric function attaining its maximum when\n$\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{h}$ (see Fig. \\ref{fig1}).\n\\end{proposition}\n\\begin{proof}\nPlease see Appendix \\ref{appE}.\n\\end{proof}\n\nHence, given a fixed $\\boldsymbol{A}$ (i.e. $\\boldsymbol{X}$), the\noptimal quantization thresholds conditional on $\\boldsymbol{A}$\nare given by\n\\begin{align}\n\\tau_n^{\\star}=\\boldsymbol{a}_n^{T}\\boldsymbol{h}, \\quad \\forall\nn\\in\\{1,\\ldots,N\\} \\label{optimumthreshold}\n\\end{align}\nThe result (\\ref{optimumthreshold}) comes directly by noting that\n\\begin{align}\n\\sum_{n=1}^N\ng_n(\\tau_n^{\\star},\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T-\\sum_{n=1}^N\ng_n(\\tau_n,\\boldsymbol{a}_n)\\boldsymbol{a}_n\\boldsymbol{a}_n^T\\succeq\\mathbf{0}\n\\end{align}\nand resorting to the convexity of $\\text{tr}(\\boldsymbol{P}^{-1})$\nover the set of positive definite matrix, i.e. for any\n$\\boldsymbol{P}\\succ\\mathbf{0}$, $\\boldsymbol{Q}\\succ\\mathbf{0}$,\nand $\\boldsymbol{P}-\\boldsymbol{Q}\\succeq \\mathbf{0}$, the\nfollowing inequality\n$\\text{tr}(\\boldsymbol{P}^{-1})\\leq\\text{tr}(\\boldsymbol{Q}^{-1})$\nholds (see \\cite{BoydVandenberghe03}).\n\nWe see that the optimal choice of the quantization threshold\n$\\tau_n$ is dependent on the unknown channel $\\boldsymbol{h}$. To\nfacilitate our analysis, we, for the time being, suppose\n$\\boldsymbol{h}$ is known. Substituting (\\ref{optimumthreshold})\ninto (\\ref{opt1}) and noting that\n\\begin{align}\ng(\\tau_n^{\\star},\\boldsymbol{a}_n)=\\frac {f_{w}^2 (0)}\n{F_{w}(0)(1-F_{w}(0))}  = \\frac{2}{\\pi\\sigma^2} \\quad \\forall n\n\\end{align}\nthe optimization (\\ref{opt1}) reduces to\n\\begin{align}\n\\min_{\\boldsymbol{X}} \\quad & \\frac{\\pi\\sigma^2}{2} \\text{tr}\n\\left\\{ \\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad &\n\\boldsymbol{A}=\\boldsymbol{I}_M\\otimes\\boldsymbol{\\widetilde{A}}\n\\nonumber\\\\\n& \\boldsymbol{\\widetilde{A}} \\triangleq \\left[\\begin{array}{ccc}\n\\Re(\\boldsymbol{X}) & \\Im(\\boldsymbol{X}) \\\\\n-\\Im(\\boldsymbol{X}) & \\Re(\\boldsymbol{X})\n\\end{array}\\right]^T \\nonumber\\\\\n& \\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P \\label{opt2}\n\\end{align}\nwhich is now independent of $\\boldsymbol{h}$. We have the\nfollowing theorem regarding the solution to the optimization\n(\\ref{opt2}).\n\n\n\n\\begin{theorem} \\label{theorem2}\nThe minimum achievable objective function value of (\\ref{opt2}) is\ngiven by $(\\pi\\sigma^2 MK^2)\/P$ and can be attained if the pilot\nmatrix $\\boldsymbol{X}$ satisfies\n\\begin{align}\n\\boldsymbol{X}\\boldsymbol{X}^H = (P\/K) \\boldsymbol{I}\n\\label{theorem2:eqn1}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{appC}.\n\\end{proof}\n\nTheorem \\ref{theorem2} reveals that, for one-bit massive MIMO\nsystems, users should employ orthogonal pilot sequences in order\nto minimize channel estimation errors. Although it is a convention\nto use orthogonal pilots to facilitate channel estimation for\nconventional massive MIMO systems, to our best knowledge, its\noptimality in one-bit massive MIMO systems has not been\nestablished before.\n\n\n\\subsection{Performance Analysis}\nWe now investigate the estimation performance when the optimal\nthresholds are employed, and its comparison with the performance\nattained by a conventional massive MIMO system which assumes\ninfinite-precision ADCs. Substituting the optimal thresholds\n(\\ref{optimumthreshold}) into the CRB matrix (\\ref{CRB}), we have\n\\begin{align}\n\\text{CRB}_{\\text{OQ}}(\\boldsymbol{h})=\\frac{\\pi\\sigma^2}{2}\n\\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1} \\label{CRB-Q}\n\\end{align}\nwhere for clarity, we use the subscript OQ to represent the\nestimation scheme using optimal quantization thresholds. On the\nother hand, when the unquantized observations $\\boldsymbol{y}$ are\navailable, it can be readily verified that the CRB matrix is given\nas\n\\begin{align}\n\\text{CRB}_{\\text{NQ}}(\\boldsymbol{h})=\\sigma^2 \\left(\n\\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1} \\label{CRB-NQ}\n\\end{align}\nwhere we use the subscript NQ to represent the scheme which has\naccess to the unquantized observations. Comparing (\\ref{CRB-Q})\nwith (\\ref{CRB-NQ}), we can see that if optimal thresholds are\nemployed, then using one-bit ADCs for channel estimation incurs\nonly a mild performance loss relative to using infinite-precision\nADCs, with the CRB increasing by only a factor of $\\pi\/2$, i.e.\n\\begin{align}\n\\text{CRB}_{\\text{OQ}}(\\boldsymbol{h})=\\frac{\\pi}{2}\n\\text{CRB}_{\\text{NQ}}(\\boldsymbol{h})\n\\end{align}\nWe also take a glimpse of the estimation performance as the\nthresholds deviate from their optimal values. For simplicity, let\n$\\tau_n=\\tau_n^{\\star}+\\delta=\\boldsymbol{a}_n^{T}\\boldsymbol{h}+\\delta,\n\\forall n$, in which case the CRB matrix is given by\n\\begin{align}\n\\text{CRB}_{\\text{Q}}(\\boldsymbol{h})=\\frac\n{F_{w}(\\delta)(1-F_{w}(\\delta))}{f_{w}^2 (\\delta)}\\left(\n\\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\end{align}\nSince $(F_{w}(\\delta)(1-F_{w}(\\delta)))\/f_{w}^2(\\delta)$ is the\nreciprocal of $g(\\tau_n,\\boldsymbol{a}_n)$, from Fig. \\ref{fig1},\nwe know that the function value\n$(F_{w}(\\delta)(1-F_{w}(\\delta)))\/f_{w}^2(\\delta)$ grows\nexponentially as $|\\delta|$ increases. This indicates that a\ndeviation of the thresholds from their optimal values results in a\nsubstantial performance loss.\n\n\nIn summary, the above results have important implications for the\ndesign of one-bit massive MIMO systems. It points out that a\ncareful choice of quantization thresholds can help improve the\nestimation performance significantly, and help achieve an\nestimation accuracy close to an ideal estimator which has access\nto the raw observations $\\boldsymbol{y}$.\n\n\n\nThe problem lies in that the optimal thresholds\n$\\boldsymbol{\\tau}$ are functions of $\\boldsymbol{h}$, as\ndescribed in (\\ref{optimumthreshold}). Since $\\boldsymbol{h}$ is\nunknown and to be estimated, the optimal thresholds\n$\\boldsymbol{\\tau}$ are also unknown. To address this difficulty,\nwe, in the following, propose an adaptive quantization (AQ) scheme\nby which the thresholds are dynamically adjusted from one\niteration to another, and a random quantization (RQ) schme which\nrandomly generates a set of nonidentical thresholds based on some\nstatistical prior knowledge of the channel.\n\n\n\n\n\n\n\n\n\\section{Practical Threshold Design Strategies} \\label{sec:AQ-RQ}\n\\subsection{Adaptive Quantization}\nOne strategy to overcome the above difficulty is to use an\niterative algorithm in which the thresholds are iteratively\nrefined based on the previous estimate of $\\boldsymbol{h}$.\nSpecifically, at iteration $i$, we use the current quantization\nthresholds $\\boldsymbol{\\tau}^{(i)}$ to generate the one-bit\nobservation data $\\boldsymbol{b}^{(i)}$. Then a new estimate\n$\\hat{\\boldsymbol{h}}^{(i)}$ is obtained from the ML estimator\n(\\ref{MLE}). This estimate is then plugged in\n(\\ref{optimumthreshold}) to obtain updated quantization\nthresholds, i.e. $\\boldsymbol{\\tau}^{(i+1)}=\\boldsymbol{A}\n\\hat{\\boldsymbol{h}}^{(i)}$, for subsequent iteration. When\ncomputing the ML estimate $\\hat{\\boldsymbol{h}}^{(i)}$, not only\nthe quantized data from the current iteration but also from all\nprevious iterations can be used. The ML estimator (\\ref{MLE}) can\nbe easily adapted to accommodate these quantized data since the\ndata are independent across different iterations. Due to the\nconsistency of the ML estimator for large data records, this\niterative process will asymptotically lead to optimal quantization\nthresholds, i.e. $\\boldsymbol{\\tau}^{(i)} \\stackrel{i \\to\n\\infty}{\\longrightarrow} \\boldsymbol{A} \\boldsymbol{h}$. In fact,\nour simulation results show that the adaptive quantization scheme\nyields quantization thresholds close to the optimal values within\nonly a few iterations.\n\nFor clarity, we summarize the adaptive quantization (AQ) scheme as\nfollows.\n\n\\begin{center}\n\\textbf{Adaptive Quantization Scheme}\n\\end{center}\n\\vspace{0cm} \\noindent\n\\begin{tabular}{lp{7.7cm}}\n\\hline 1.& Select an initial quantization threshold\n$\\boldsymbol{\\tau}^{(0)}$ and the maximum number of iterations $i_{\\text{max}}$. \\\\\n2.& At iteration $i=1,2,\\ldots$: Based on $\\boldsymbol{y}$ and\n$\\boldsymbol{\\tau}^{(i)}$, calculate the new binary data\n$\\boldsymbol{b}^{(i)}=\\text{sgn}(\\boldsymbol{y}-\\boldsymbol{\\tau}^{(i)})$. \\\\\n3.& Compute a new estimate of $\\boldsymbol{h}$,\n$\\hat{\\boldsymbol{h}}^{(i)}$,\nvia (\\ref{MLE}). \\\\\n4.& Calculate new thresholds according to $\\boldsymbol{\\tau}^{(i+1)}=\\boldsymbol{A}\\hat{\\boldsymbol{h}}^{(i)}$. \\\\\n5.& Go to Step 2 if $i < i_{\\text{max}}$. \\\\\n\\hline\n\\end{tabular}\n\n\nNote that during the iterative process, the channel\n$\\boldsymbol{h}$ is assumed constant over time. Thus the AQ scheme\ncan be used to estimate channels that are unchanged or slowly\ntime-varying across a number of consecutive frames. For example,\nfor the scenario where the relative speeds between the mobile\nterminals and the base station are slow, say, 2 meters per second,\nthe channel coherence time could be up to tens of milliseconds,\nmore precisely, about 60 milliseconds if the carrier frequency is\nset to 1GHz, according to the Clarke's model\n\\cite{TseViswanath05}. Suppose the time duration of each frame is\n10 milliseconds which is a typical value for practical LTE\nsystems. In this case, the channel remains unchanged across 6\nconsecutive frames. We can use the AQ scheme to update the\nquantization thresholds at each frame based on the channel\nestimate obtained from the previous frame. In this way, we can\nexpect that the quantization thresholds will come closer and\ncloser to the optimal values from one frame to the next, and as a\nresult, a more and more accurate channel estimate can be obtained.\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{AQ.eps}\n\\caption{An off-line implementation of the AQ scheme.}\n\\label{fig2}\n\\end{figure}\n\nThe above scheme assumes a static or slowly time-varying channel\nacross multiple frames. Another way of implementing the AQ scheme\nrequires no such an assumption, but at the expense of increased\nhardware complexity. The idea is to use a number of\nsample-and-hold (S\/H) circuits to sample the analog received\nsignals and to store their values for subsequent offline\nprocessing. Specifically, each antenna\/RF chain is followed by\n$2L$ S\/H circuits which are equally divided into two groups to\nsample and store the real and imaginary components, respectively\n(see Fig. \\ref{fig2}). Through a precise timing control, we ensure\nthat at each antenna, say, the $m$th antenna, the $l$th S\/H\ncircuit pair in the two groups are controlled to store the real\nand imaginary components of the $l$th received pilot symbol, i.e.\n$\\Re(y_{m,l})$ and $\\Im(y_{m,l})$, respectively. Also, to avoid\nusing a one-bit ADC for each S\/H circuit, a switch can be used to\nconnect a single one-bit ADC with multiple S\/H circuits. Once the\nanalog signals $\\boldsymbol{y}$ have been stored, the AQ scheme\ncan be implemented in an offline manner. Clearly, this offline\napproach can be implemented on a single frame basis, and thus no\nlonger requires a static channel assumption. Nevertheless, such an\nimplementation requires a number of S\/H circuits as well as\nprecise timing control for sampling and quantization. Also, this\noffline processing may cause a latency issue which should be taken\ncare of in practical systems.\n\n\n\n\n\n\n\n\n\\subsection{Random Quantization}\nThe AQ scheme requires the channel to be (approximately)\nstationary, or needs to be implemented with additional hardware\ncircuits. Here we propose a random quantization (RQ) scheme that\ndoes not involve any iterative procedure and is simple to\nimplement. The idea is to randomly generate a set of non-identical\nthresholds based on some statistical prior knowledge of\n$\\boldsymbol{h}$, with the hope that some of the thresholds are\nclose to the unknown optimal thresholds. For example, suppose each\nentry of $\\boldsymbol{h}$ follows a Gaussian distribution with\nzero mean and variance $\\sigma_h^2$. Note that different entries\nof $\\boldsymbol{h}$ may have different variances due to the reason\nthat they may correspond to different users. Nevertheless, we\nassume the same variance for all entries for simplicity. We\nrandomly generate $N$ different realizations of $\\boldsymbol{h}$,\ndenoted as $\\{\\boldsymbol{\\tilde{h}}_n\\}$, following this known\ndistribution. The $N$ quantization thresholds are then devised\naccording to\n\\begin{align}\n\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{\\tilde{h}}_n, \\quad \\forall\nn\\in\\{1,\\ldots,N\\} \\label{multi-thresholding}\n\\end{align}\nOur simulation results suggest that this RQ scheme can achieve a\nconsiderable performance improvement over the conventional fixed\nquantization scheme which uses a fixed (typically zero) threshold.\nThe reason is that the thresholds produced by\n(\\ref{multi-thresholding}) are more likely to be close to their\noptimal values.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Simulation Results} \\label{sec:experiments}\nWe now carry out experiments to corroborate our theoretical\nanalysis and to illustrate the performance of our proposed one-bit\nquantization schemes, i.e. the AQ and the RQ schemes. We compare\nour schemes with the conventional fixed quantization (FQ) scheme\nwhich employs a fixed zero threshold for one-bit quantization, and\na no-quantization scheme (referred to as NQ) which uses the\noriginal unquantized data for channel estimation. For the NQ\nscheme, it can be easily verified that its ML estimate is given by\n\\begin{align}\n\\boldsymbol{\\hat{h}}=(\\boldsymbol{A}^T\\boldsymbol{A})^{-1}\\boldsymbol{A}^T\\boldsymbol{y}\n\\end{align}\nand its associated CRB is given by (\\ref{CRB-NQ}). For other\nschemes such as the RQ and the FQ, although a close-form\nexpression is not available, the ML estimate can be obtained by\nsolving the convex optimization (\\ref{MLE}). In our simulations,\nwe assume independent and identically distributed (i.i.d.)\nrayleigh fading channels, i.e. all elements of $\\boldsymbol{H}$\nfollow a circularly symmetric complex Gaussian distribution with\nzero mean and unit variance. Training sequences $\\boldsymbol{X}$\nwhich satisfy (\\ref{theorem2:eqn1}) are randomly generated. The\nsignal-to-noise ratio (SNR) is defined as\n\\begin{align}\n\\text{SNR}=\\frac{P}{KL\\sigma^2}\n\\end{align}\n\n\n\\begin{figure}[!t]\n \\centering\n\\begin{tabular}{c}\n\\includegraphics[width=3.5in]{msevsitr1}\\\\\n(a). $K=8$, $L=32$ and $\\text{SNR}=15$ dB. \\\\\n\\includegraphics[width=3.5in]{msevsitr2}\\\\\n(b). $K=16$, $L=40$ and $\\text{SNR}=15$ dB.\n\\end{tabular}\n  \\caption{MSEs of the AQ scheme as a function of the number of iterations.}\n   \\label{fig3}\n\\end{figure}\n\n\nWe first examine the estimation performance of our proposed AQ\nscheme which adaptively adjusts the thresholds based on the\nprevious estimate of the channel. Fig. \\ref{fig3} plots the\nmean-squared errors (MSEs) vs. the number of iterations for the AQ\nscheme, where we set $K=8$, $L=32$ for Fig. (a) and $K=16$, $L=40$\nfor Fig. (b). The SNR is set to 15dB. The MSE is calculated as\n\\begin{align}\n\\text{MSE}=\\frac{1}{K M}\n\\|\\boldsymbol{H}-\\boldsymbol{\\hat{H}}\\|_F^2\n\\end{align}\nTo better illustrate the effectiveness of the AQ scheme, we also\ninclude the CRB results in Fig. \\ref{fig3}. in which the CRB-OQ,\ngiven by (\\ref{CRB-Q}), represents the theoretical lower bound on\nthe estimation errors of any unbiased estimator using optimal\nthresholds for one-bit quantization, and the CRB-NQ, given by\n(\\ref{CRB-NQ}), represents the lower bound on the estimation\nerrors of any unbiased estimator which has access to the original\nobservations. From Fig. \\ref{fig3}, we see that our proposed AQ\nscheme approaches the theoretical lower bound CRB-OQ within only a\nfew (say, 5) iterations, and achieves performance close to the CRB\nassociated with the NQ scheme. This result demonstrates the\neffectiveness of the AQ scheme in searching for the optimal\nthresholds. In the rest of our simulations, we set the maximum\nnumber of iterations, $i_{\\text{max}}$, equal to 5 for the AQ\nscheme.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{msevsL}\n\\caption{MSEs vs. number of pilot symbols, where $K=8$ and\n$\\text{SNR}=15$ dB.} \\label{fig4}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{msevsSNR}\n\\caption{MSEs vs. SNR(dB), where $K=8$ and $L=96$.} \\label{fig5}\n\\end{figure}\n\n\n\nWe now compare the estimation performance of different schemes.\nFig. \\ref{fig4} plots the MSEs of respective schemes as a function\nof the number of pilot symbols, $L$, where we set $K=8$ and\n$\\text{SNR}=15\\text{dB}$. The corresponding CRBs of these schemes\nare also included. Note that the CRBs for the FQ and the RQ\nschemes can be obtained by substituting the thresholds into\n(\\ref{CRB}). Results are averaged over $10^3$ independent runs,\nwith the channel and the pilot sequences randomly generated for\neach run. From Fig. \\ref{fig4}, we can see that the proposed AQ\nscheme outperforms the FQ and RQ schemes by a big margin. This\nresult corroborates our analysis that an optimal choice of the\nquantization thresholds helps achieve a substantial performance\nimprovement. In particular, the AQ scheme needs less than 30 pilot\nsymbols to achieve a decent estimation accuracy with a MSE of\n0.01, while the FQ and RQ schemes require a much larger number of\npilot symbols to attain a same estimation accuracy. On the other\nhand, we should note that although the AQ scheme has the potential\nto achieve performance close to the NQ scheme, the implementation\nof the AQ is more complicated since it involves an iterative\nprocess to learn the optimal thresholds. In contrast, our proposed\nRQ scheme is as simple as the FQ scheme to implement, meanwhile it\npresents a clear performance advantage over the FQ scheme. We can\nsee from Fig. \\ref{fig4} that the RQ requires about 100 symbols to\nachieve a MSE of 0.1, whereas the FQ needs about 250 pilot symbols\nto reach a same estimation accuracy. The reason why the RQ\nperforms better than the FQ is that some of the thresholds\nproduced according to (\\ref{multi-thresholding}) are likely to be\nclose to the optimal thresholds. In Fig. \\ref{fig5}, we plot the\nMSEs of respective schemes under different SNRs, where we set\n$K=8$ and $L=96$. Similar conclusions can be made from Fig.\n\\ref{fig5}.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{servsL}\n\\caption{SERs vs. number of pilot symbols, where $K=8$, $M=64$ and\n$\\text{SNR}=5\\text{dB}$.} \\label{fig6}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{ratevsL}\n\\caption{Achievable rates vs. number of pilot symbols, where\n$K=8$, $M=64$ and $\\text{SNR}=5\\text{dB}$.} \\label{fig7}\n\\end{figure}\n\n\n\n\nNext, we examine the effect of channel estimation accuracy on the\nsymbol error rate (SER) performance. For each scheme, after the\nchannel is estimated, a near maximum likelihood detector\n\\cite{ChoiMo16} developed for one-bit massive MIMO is adopted for\nsymbol detection. For a fair comparison, in the symbol detection\nstage, the quantization thresholds are all set equal to zero, as\nassumed in \\cite{ChoiMo16}. In our experiments, QPSK symbols are\ntransmitted by all users. Fig. \\ref{fig6} plots the SERs of\nrespective schemes vs. the number of pilot symbols, where we set\n$K=8$, $M=64$, and $\\text{SNR}=5\\text{dB}$. Results are averaged\nover all $K$ users. The SER performance obtained by assuming\nperfect channel knowledge is also included. It can be seen that\nthe SER performance improves as the number of pilot symbols\nincreases, which is expected since a more accurate channel\nestimate can be obtained when more pilot symbols are available for\nchannel estimation. We also observe that the AQ scheme, using a\nmoderate number (about 120 symbols that is only 15 times the\nnumber of users) of pilot symbols, can achieve SER performance\nclose to that attained by assuming perfect channel knowledge.\nMoreover, the SER results, again, demonstrate the superiority of\nthe RQ over the FQ scheme. In order to attain a same SER, say,\n$10^{-3}$, the RQ requires about 60 pilot symbols, whereas the FQ\nrequires about 100 pilot symbols.\n\nIn Fig. \\ref{fig7}, the achievable rates of respective schemes vs.\nthe number of pilot symbols are depicted, where we set $K=8$,\n$M=64$, and $\\text{SNR}=5\\text{dB}$. The achievable rate for the\n$k$th user is calculated as \\cite{MollenChoi17}\n\\begin{align}\nR_k \\triangleq \\log_2 \\left( 1+ \\frac{\n|\\mathbb{E}\\left[s_k^*(t)\\hat{s}_k(t)\\right]|^2 }\n{\\mathbb{E}\\left[|\\hat{s}_k(t)|^2\\right] -\n|\\mathbb{E}\\left[s_k^*(t)\\hat{s}_k(t)\\right]|^2} \\right)\n\\end{align}\nwhere $s_k(t)$ is the transmit symbol of the $k$th user at time\n$t$, $()^{*}$ denotes the conjugate, and $\\hat{s}_k(t)$ is the\nestimated symbol of $s_k(t)$, which is obtained via the near\nmaximum likelihood detector by using the channel estimated by\nrespective schemes. The achievable rate we plotted is averaged\nover all $K$ users. It can be seen that, even with a moderate\nnumber of pilot symbols (about 5 times the number of users), the\nAQ scheme can provide an achievable rate close to that of the\nperfect CSI case, whereas the achievable rates attained by the\nother two schemes are far below the level of the AQ scheme.\nCompared to the FQ, the RQ scheme achieves an increase of about 30\npercent in the achievable rate.\n\n\n\n\n\n\\section{Conclusions} \\label{sec:conclusion}\nAssuming one-bit ADCs at the BS, we studied the problem of one-bit\nquantization design and channel estimation for uplink multiuser\nmassive MIMO systems. Specifically, based on the derived CRB\nmatrix, we examined the impact of quantization thresholds on the\nchannel estimation performance. Our theoretical analysis revealed\nthat using one-bit ADCs can achieve an estimation error close to\nthat attained by using infinite-precision ADCs, given that the\nquantization thresholds are optimally set. Our analysis also\nsuggested that the optimal quantization thresholds are dependent\non the unknown channel parameters. We developed two practical\nquantization design schemes, namely, an adaptive quantization\nscheme which adaptively adjusts the thresholds such that the\nthresholds converge to the optimal thresholds, and a random\nquantization scheme which randomly generates a set of\nnon-identical thresholds based on some statistical prior knowledge\nof the channel. Simulation results showed that the proposed\nquantization schemes achieved a significant performance\nimprovement over the fixed quantization scheme that uses a fixed\n(typically zero) quantization threshold, and thus can help\nsubstantially reduce the training overhead in order to attain a\nsame estimation accuracy target.\n\n\n\n\n\n\n\n\n\\useRomanappendicesfalse\n\\appendices\n\n\\section{Proof of Concavity of The Log-Likelihood Function (\\ref{log-PMF})} \\label{appA}\nIt can be easily verified that\n$f_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$ is log-concave\nin $\\boldsymbol{h}$ since the Hessian matrix of $\\log\nf_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$, which is given\nby\n\\begin{align}\n\\frac {\\partial^2 \\log\nf_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)} {{\\partial\n\\boldsymbol{h} \\partial \\boldsymbol{h}^T}} = -\n\\frac{\\boldsymbol{a}_n \\boldsymbol{a}_n^{T}} {\\sigma^2}\n\\end{align}\nis negative semidefinite. Consequently the corresponding\ncumulative density function (CDF) and complementary CDF (CCDF),\nwhich are integrals of the log-concave function\n$f_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)$ over convex\nsets $(-\\infty,\\tau_n)$ and $(\\tau_n,\\infty)$ respectively, are\nalso log-concave, and their logarithms are concave. Since\nsummation preserves concavity, $L(\\boldsymbol{h})$ is a concave\nfunction of $\\boldsymbol{h}$.\n\n\n\\section{Proof of Theorem \\ref{theorem1}} \\label{appB}\nDefine a new variable $z_n\\triangleq\n\\boldsymbol{a}_n^{T}\\boldsymbol{h}$ and define\n\\begin{align}\nl(z_n) & \\triangleq  \\frac{1-b_n}{2}  \\mathrm{log} [1-F_{w}(z_n-\\tau_n)]   \\nonumber \\\\\n& \\quad + \\frac{1+b_n}{2} \\mathrm{log} [F_{w}(z_n-\\tau_n)].\n\\end{align}\nThe first and second-order derivative of $L(\\boldsymbol{h})$ are\ngiven by\n\\begin{align}\n\\frac {\\partial L(\\boldsymbol{h})} {\\partial \\boldsymbol{h}} =\n\\sum_{n=1}^{N} \\frac {\\partial l(z_n)} {\\partial z_n} \\frac\n{\\partial z_n} {\\partial \\boldsymbol{h}} = \\sum_{n=1}^{N} \\frac\n{\\partial l(z_n)} {\\partial z_n} \\boldsymbol{a}_n\n\\end{align}\nand\n\\begin{align}\n\\frac {\\partial^2 L(\\boldsymbol{h})} {\\partial \\boldsymbol{h}\n\\partial \\boldsymbol{h}^T}\n&= \\sum_{n=1}^{N} \\boldsymbol{a}_n \\frac {\\partial^2 l(z_n)}\n{\\partial z_n^2}\n\\frac {\\partial z_n} {\\partial \\boldsymbol{h}^T} \\nonumber \\\\\n&= \\sum_{n=1}^{N} \\frac {\\partial^2 l(z_n)} {\\partial z_n^2}\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T .\n\\end{align}\nwhere\n\\begin{align}\n\\frac {\\partial l(z_n)} {\\partial z_n} &= \\frac{1-b_n}{2}  \\frac{f_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)-1}   \\nonumber \\\\\n& \\quad + \\frac{1+b_n}{2}\n\\frac{f_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)} \\label{deriv1}\n\\end{align}\nand\n\\begin{align}\n\\frac {\\partial^2 l(z_n)} {\\partial z_n^2} =& \\frac{1-b_n}{2}\n\\bigg[\n\\frac{f'_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)-1} \\nonumber \\\\\n& -\\frac{f_{w}^2 (z_n-\\tau_n)}{(F_{w}(z_n-\\tau_n)-1)^2} \\bigg] + \\frac{1+b_n}{2}   \\nonumber \\\\\n&  \\cdot \\bigg[ \\frac{f'_{w}(z_n-\\tau_n)}{F_{w}(z_n-\\tau_n)} -\n\\frac{f_{w}^2 (z_n-\\tau_n)}{F_{w}^2 (z_n-\\tau_n)} \\bigg]\n\\label{deriv2}\n\\end{align}\nwhere $f_{w}(x)$ denotes the probability density function (PDF) of\n$w_n$, and $f'_{w}(x)\\triangleq\\frac{\\partial f_{w}(x)}{\\partial\nx}$.\n\nTherefore, the Fisher information matrix (FIM) of the estimation\nproblem is given as\n\\begin{align}\nJ(\\boldsymbol{h}) & = -E \\left[\\frac {\\partial^2\nL(\\boldsymbol{h})} {\\partial \\boldsymbol{h} \\partial\n\\boldsymbol{h}^T} \\right]\n= - \\sum_{n=1}^{N} E_{b_n} \\left[ \\frac {\\partial^2 l(z_n)} {\\partial z_n^2} \\right]\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T  \\nonumber \\\\\n& \\stackrel {(a)}{=} \\sum_{n=1}^{N} \\frac {f_{w}^2\n(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)}\n{F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n)(1-F_{w}(\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n))}\n\\boldsymbol{a}_n \\boldsymbol{a}_n^T\n\\end{align}\nwhere $E_{b_n}[\\cdot]$ denotes the expectation with respect to the\ndistribution of $b_n$, and $(a)$ follows from the fact that\n$b_{n}$ is a binary random variable with\n$P(b_{n}=1|\\tau_n,z_n)=F_{w}(z_n-\\tau_n)$ and\n$P(b_{n}=-1|\\tau_n,z_n)=1-F_{w}(z_n-\\tau_n)$. This completes the\nproof.\n\n\n\n\n\n\n\n\\section{Proof of Proposition \\ref{proposition1}} \\label{appE}\nBefore proceeding, we first introduce the following lemma.\n\\newtheorem{lemma}{Lemma}\n\\begin{lemma} \\label{lemma2}\nFor $x\\ge 0$, define\n\\begin{align}\n\\bar{F}(x) \\triangleq \\int_{0}^{x} f(u)\\mathrm{d}u\n\\end{align}\nwhere $f(\\cdot)$ denotes the PDF of a real-valued Gaussian random\nvariable with zero-mean and unit variance. We have $\\bar{F}(x)$\nupper bounded by\n\\begin{align}\n\\bar{F}(x) \\le \\frac{1}{2} \\sqrt{ 1-e^{-\\frac{2x^2}{\\pi}} } .\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{appF}.\n\\end{proof}\n\n\nDefine the function\n\\begin{align}\n\\bar{g}(x) \\triangleq \\frac {f^2 (x)} {F(x)(1-F(x))}\n\\end{align}\nwhere $f(\\cdot)$ and $F(\\cdot)$ denotes the PDF and CDF of a\nreal-valued Gaussian random variable with zero-mean and unit\nvariance respectively. Invoking Lemma \\ref{lemma2}, we have\n\\begin{align}\n\\bar{g}(x) =\\frac {f^2 (x)} {\\frac{1}{4}-\\bar{F}^2(x)} \\le\n\\frac{2}{\\pi} e^{-(1-\\frac{2}{\\pi})x^2} \\le \\frac{2}{\\pi}.\n\\end{align}\nand $\\bar{g}(x) =\\frac{2}{\\pi}$ if and only if $x=0$. Noting that\n\\begin{align}\n\\frac{1}{\\sigma^2} \\bar{g} \\left(\n\\frac{\\boldsymbol{a}_n^{T}\\boldsymbol{h}-\\tau_n}{\\sigma} \\right) =\ng(\\tau_n,\\boldsymbol{a}_n) .\n\\end{align}\nTherefore $g(\\tau_n,\\boldsymbol{a}_n)$ attains its maximum when\n$\\tau_n=\\boldsymbol{a}_n^{T}\\boldsymbol{h}$. The proof is\ncompleted here.\n\n\n\\section{Proof of Lemma \\ref{lemma2}} \\label{appF}\nDefine two i.i.d. Gaussian random variables with zero-mean and\nunit variance, namely, $X$ and $Y$. The joint distribution\nfunction of $X$ and $Y$ is $f_{XY}(x,y)=f(x)f(y)$. Define two\nregions $D_1 \\triangleq \\{(u,v)\\mid 0 \\le u \\le x , 0 \\le v \\le x\n\\}$ and $D_2 \\triangleq \\{(u,v)\\mid u \\ge 0 , v \\ge 0, u^2+v^2\\le\n\\frac{4x^2}{\\pi} \\}$. Obviously, the areas of $D_1$ and $D_2$ are\nthe same, i.e., $\\mu(D_1)=\\mu(D_2)$, where $\\mu(\\cdot)$ denote the\narea of a region. The probabilities of $(X,Y)$ belonging in these\ntwo regions can be computed as\n\\begin{align}\nP((X,Y)\\in D_1) &= \\iint_{D_1} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\nonumber \\\\\n&= \\bar{F}^2(x) \\\\\nP((X,Y)\\in D_2) &= \\iint_{D_2} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\nonumber \\\\\n&= \\frac{1}{4} \\left(1-e^{-\\frac{2x^2}{\\pi}}\\right)\n\\end{align}\nLet $S_1\\setminus S_2$ denote the set obtained by excluding\n$S_2\\cap S_1$ from $S_1$. Clearly, we have\n\\begin{align}\n\\mu(D_1 \\setminus D_2)=\\mu(D_2 \\setminus D_1) \\label{appF:eqn1}\n\\end{align}\nAlso, according to the definition of $D_1$ and $D_2$, we have\n\\begin{align}\nf_{XY}(u,v)&\\le \\frac{1}{2\\pi} e^{-\\frac{2x^2}{\\pi}}, \\quad (u,v)\\in D_1 \\setminus D_2 \\\\\nf_{XY}(u,v)&\\ge \\frac{1}{2\\pi} e^{-\\frac{2x^2}{\\pi}}, \\quad\n(u,v)\\in D_2 \\setminus D_1 \\label{appF:eqn2}\n\\end{align}\nCombining (\\ref{appF:eqn1})--(\\ref{appF:eqn2}), we arrive at\n\\begin{align}\n\\iint_{D_1 \\setminus D_2} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v \\le\n\\iint_{D_2 \\setminus D_1} f_{XY}(u,v) \\mathrm{d}u\\mathrm{d}v\n\\end{align}\nFrom the above inequality, we have $P((X,Y)\\in D_1)\\le P((X,Y)\\in\nD_2)$, i.e.\n\\begin{align}\n\\bar{F}^2(x)\\leq\\frac{1}{4}\n\\left(1-e^{-\\frac{2x^2}{\\pi}}\\right)\\Rightarrow \\bar{F}(x) \\le\n\\frac{1}{2} \\sqrt{ 1-e^{-\\frac{2x^2}{\\pi}} }\n\\end{align}\nThis completes the proof.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{theorem2}} \\label{appC}\nNote that from the constraint\n$\\text{tr}(\\boldsymbol{X}\\boldsymbol{X}^H)\\leq P$, we can easily\nderive that\n\\begin{align}\n\\text{tr}(\\boldsymbol{A}^T \\boldsymbol{A})\\leq 2M P\n\\end{align}\nTo prove Theorem \\ref{theorem2}, let us first consider a new\noptimization that has the same objective function as (\\ref{opt2})\nwhile with a relaxed constraint:\n\\begin{align}\n\\min_{\\boldsymbol{A}} \\quad & \\frac{\\pi\\sigma^2}{2}\\text{tr}\n\\left\\{ \\left( \\boldsymbol{A}^T \\boldsymbol{A} \\right)^{-1}\n\\right\\}\n\\nonumber\\\\\n\\text{s.t.} \\quad & \\text{tr}(\\boldsymbol{A}^T\\boldsymbol{A})\\leq\n2M P  \\label{appC:opt1}\n\\end{align}\nClearly, the feasible region defined by the constraints in\n(\\ref{opt2}) is a subset of that defined by (\\ref{appC:opt1}).\nSince $\\text{tr}(\\boldsymbol{Z}^{-1})$ is convex over the set of\npositive definite matrix, the optimization (\\ref{appC:opt1}) is\nconvex. Its optimum solution is given as follows.\n\\begin{lemma} \\label{lemma1}\nConsider the following optimization problem\n\\begin{align}\n\\min_{\\boldsymbol{Z}}\\quad &\\text{tr}(\\boldsymbol{Z}^{-1})\n\\nonumber\\\\\n\\text{s.t.}\\quad & \\text{tr}(\\boldsymbol{Z})\\leq P_0\n\\label{appC:opt2}\n\\end{align}\nwhere $\\boldsymbol{Z}\\in\\mathbb{R}^{p\\times p}$ is positive\ndefinite. The optimum solution to (\\ref{appC:opt2}) is given by\n$\\boldsymbol{Z}=(P_0\/p)\\boldsymbol{I}$ and the minimum objective\nfunction value is $p^2\/P_0$.\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{appD}.\n\\end{proof}\n\nFrom Lemma \\ref{lemma1}, we know that any $\\boldsymbol{A}$\nsatisfying\n\\begin{align}\n\\boldsymbol{A}^T \\boldsymbol{A} = (P\/K) \\boldsymbol{I}\n\\label{appC:eqn1}\n\\end{align}\nis an optimal solution to (\\ref{appC:opt1}). Note that the set of\nfeasible solutions (\\ref{appC:opt1}) subsumes the feasible\nsolution set of (\\ref{opt2}). Hence, if the optimal solution to\n(\\ref{appC:opt1}) is meanwhile a feasible solution of\n(\\ref{opt2}), then this solution is also an optimal solution to\n(\\ref{opt2}). It is easy to verify that if (\\ref{theorem2:eqn1})\nholds valid, the resulting $\\boldsymbol{A}$ satisfies\n(\\ref{appC:eqn1}) and is thus an optimal solution to\n(\\ref{appC:opt1}). As a consequence, it is also an optimal\nsolution to (\\ref{opt2}). This completes the proof.\n\n\n\n\n\n\n\n\n\\section{Proof of Lemma \\ref{lemma1}} \\label{appD}\nLet $\\boldsymbol{Z}=\\boldsymbol{U}\\boldsymbol{D}\\boldsymbol{U}^T$\ndenote the eigenvalue decomposition of $\\boldsymbol{Z}$, where\n$\\boldsymbol{U}\\in\\mathbb{R}^{p\\times p}$ and\n$\\boldsymbol{D}\\in\\mathbb{R}^{p\\times p}$. By replacing\n$\\boldsymbol{Z}$ with\n$\\boldsymbol{U}\\boldsymbol{D}\\boldsymbol{U}^T$, the optimization\n(\\ref{appC:opt2}) is reduced to determining the diagonal matrix\n$\\boldsymbol{D}\\triangleq \\text{diag}(d_1,\\dots,d_{p})$\n\\begin{align}\n\\min_{\\{d_i\\}} \\ & \\sum_{i=1}^{p} \\frac{1}{d_i}\n\\nonumber\\\\\n\\text{s.t.} \\ \\,\n& \\sum_{i=1}^{p} {d_i} \\leq P_0 \\nonumber\\\\\n& d_i > 0, \\qquad \\forall i\\in\\{1,\\dots,p\\}\\label{opt5}\n\\end{align}\nThe Lagrangian function associated with (\\ref{opt5}) is given by\n\\begin{align}\nL(d_i;\\lambda;\\nu_i)=\\sum_{i=1}^{p} \\frac{1}{d_i} + \\lambda \\left(\n\\sum_{i=1}^{p} {d_i} - P_0 \\right) - \\sum_{i=1}^{p} {\\nu_i d_i}\n\\end{align}\nwith KKT conditions \\cite{BoydVandenberghe03} given as\n\\begin{align}\n-\\frac{1}{d_i^2}+\\lambda-\\nu_i=0 & , \\quad \\forall i \\nonumber\\\\\n\\lambda\\left(\\sum_{i=1}^{p} {d_i} - P_0\\right)  =0 & \\nonumber\\\\\n\\lambda\\geq 0 & \\nonumber\\\\\n\\nu_i d_i=0 & ,\\quad \\forall i \\nonumber\\\\\nd_i >0 & , \\quad \\forall i \\nonumber\\\\\n\\nu_i\\ge 0 & , \\quad \\forall i \\nonumber\n\\end{align}\nFrom the last three equations, we have $\\nu_i=0$, $\\forall i$.\nThen from the first equation we have\n\\begin{align}\n\\lambda=\\frac{1}{d_i^2}>0 \\label{lambda}\n\\end{align}\nand\n\\begin{align}\nd_1=d_2=\\dots=d_{p}.\n\\end{align}\nFrom (\\ref{lambda}) and the second equation, we have\n$\\sum_{i=1}^{p} {d_i} - P_0 =0$, from which $d_i$ can be readily\nsolved as $d_i=P_0\/p$, $\\forall i$, i.e., the optimal\n$\\boldsymbol{D}$ is given by $\\boldsymbol{D}^{\\star}= (P_0\/p)\n\\boldsymbol{I}$. Consequently we have $\\boldsymbol{Z}^{\\star}=\n(P_0\/p) \\boldsymbol{I}$. This completed the proof.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Conclusions}\n\\label{sec:conclusions}\nIn the last two decades, a huge amount of results were obtained on operational state complexity of regular languages.\nResults are roughly  split between: individual and combined operations; regular and different classes of subregular languages; deterministic and nondeterministic complexity; different alphabet sizes; and worst case versus average case. \nIn general, all this work also suggest new directions of research and open problems.\n \nAs it is evident by this survey, \nmany results on this area are functions parametrized by some measures, mostly the state complexities of the operation arguments. Given the amount and diversity of these functions, it is useful to have a software tool that helps to structurally organize, visualize and manipulate this information. Towards this goal,  a first step was taken by the development of \\textsf{DesCo},  a Web-based information system  for descriptional complexity results~\\cite{desco,nabais13a:_desco}. \\textsf{DesCo} keeps information about language classes, languages operations, models of computation, measures of complexity and complexity functions (both operational and transformational). \nFor instance, given an operation, it is possible to obtain \nthe complexity functions for all language classes and all complexity measures (that are registered in the database).\n\n\nTo obtain a witness for a tight upper bound, many authors performed experiments using computer software. The reason why some witnesses would work for several  (or almost all) complexity bounds only recently has been addressed.\nUniversal witnesses (and their variants) for operational state complexity of regular languages can be considered a major breakthrough. Conditions for a family of languages  to be \\emph{universal} include also other measures as the syntactic complexity and the number of atoms.\nThe study of necessary and\/or sufficient conditions for the maximality of all these measures is a new direction of research. Other open problems are how and whether this approach  extends to other classes of subregular languages and to other complexity measures, in particular to  nondeterministic state complexity and transition complexity.\n\nBesides the worst-case complexity  of an operation, researchers also studied the range of possible values that can be achieved, as a function of the complexities of the arguments and the alphabet size. A magic value  is a value that cannot occur (for that kind of complexity, operation and alphabet size). In general, if growing alphabet sizes are allowed no magic numbers exists (and even for binary alphabets they are rare). The distribution of possible complexity values and the density of languages (or tuples of languages) that achieve that values can also be valuable for average-case analysis. \n\nWitnesses with alphabets of increasing size were used in the quest of magic numbers, for the state complexity of certain operations over subregular languages, and almost for all results on combined operations with an arbitrary number of operands.\nThis suggest the question of whether the alphabet size should be a parameter of the  complexity under study. In particular, it should be investigated which situations cannot be characterized without increasing alphabets, and the ones for which languages with fixed alphabets can exists but are not  yet known.\n\n\nFor many automata applications, a major direction of research is average-case state complexity. An essential question for average results is the probability distribution  that is chosen for the models. The few results that exist use a uniform distribution, and even in this case the problem is very difficult. Recently, using the framework of analytic combinatorics, some average-case results were obtained for the size of NFAs\\xspace equivalent to a given regular expression~\\cite{nicaud09a:_averag_size_of_glush_autom,broda11:_averag_state_compl_of_partial_deriv_autom,broda12:_averag_size_of_glush_and,broda13:_hitch_guide_descr_compl_analy_combin}. It is also worthwhile to mention the average-case computational complexity analysis of the Brzozowski minimization algorithm carried on by Felice and Nicaud~\\cite{felice13:_brzoz_algor_is_gener_super,felice14:_averag_compl_of_brzoz_algor}. This work can be specially relevant for the operational state complexity because the authors give some  characterizations of the state complexity of reversal. Another approach for average-case analysis is to consider  experimental results based on samples of uniformly random generated automata. There are some random generators for non-isomorphic \n DFAs\\xspace\\cite{almeida07_c:_enumer_gener_strin_autom_repres,bassino07:_theor_comput_scien,felice13:_random_gener_of_deter_acycl}, but for NFAs\\xspace, the fact that  there is no generic polynomial algorithm for graph isomorphism, the problem seems unfeasible in general.\n\n\n\n\\section{Acknowledgements}\n\\label{sec:ack}\nThis work was partially supported by the\n    European Regional Development Fund through the programme COMPETE\n    and by the Portuguese Government through the FCT under \n    projects PEst-C\/MAT\/UI0144\/ 2013 and PTDC\/EIA-CCO\/101904\/2008.\nThe writing of this paper started in the fall of 2010 when Nelma Moreira\nand Rog\u00e9rio Reis were visiting Sheng Yu at University of Western\nOntario. When Sheng passed away in January 2012, the paper was already\nalmost in the current form.  We deeply thank Kai Salomaa for his\nencouragement, suggestions, and corrections. We also thank Davide\nNabais and Eva Maia for proofreading.\nComments and criticisms of the anonymous referee were most valuable for improving the \nfinal version of the paper.\n\n\n\\section{References}\n\n\\section{State Complexity of Combined Operations}\n\\label{sec:scco}\nThe number of standard individual operations on regular languages is clearly\nlimited and almost all of their state complexities have been already\nobtained. However, in many practical cases, not only these\nindividual operations but also their combinations are used,\nfor example, the operations expressed by the regular expressions in\nthe programming language Perl. These combinations are called\ncombined operations.\n\nIn 2011, Salomaa \\emph{et al.} \\cite{SaSaYu11} proved that it cannot exist an algorithm such that, for a given composition of basic regularity preserving operations, computes the state complexity of the corresponding composed operation. The undecidability result holds already for arbitrary compositions of intersection and marked concatenation and the proof relies on a reduction from Hilbert's Tenth Problem. Although the composition of\nstate complexities of individual component operations of a combined operation\nwould give an upper bound for the state complexity of the\ncombined operation, the upper bound is usually too high to be\nmeaningful~\\cite{LiMaSaYu08,salomaa07a:_state_compl_of_combin_operat,Yu06}. For example, for two regular\nlanguages $L_1$ and $L_2$ accepted by $m$-state and an $n$-state\nDFA\\xspace, respectively, the exact state\ncomplexity of $(L_1\\cup L_2)^*$ is actually\n$2^{m+n-1}-2^{m-1}-2^{n-1}+1$, while the composition of their\nindividual state complexities is $2^{mn-1}+2^{mn-2}$. Clearly,\n$O(2^{m+n})$ and $O(2^{mn})$ are totally different.\n\nSince the number of combined operations is unlimited and the state\ncomplexities of many of them are very difficult to\ncompute, it would be good if we have a general estimation\nmethod that generates close upper bounds of the state complexities\nof combined operations which are good enough\nto use in practice. Such an estimation method has been proposed by \u00c9sik \\textit{et al.}~\\cite{EsGaLiYu09}, and Salomaa and Yu~\\cite{SaYu07}. A further concept in this direction,\napproximation of state complexity has been introduced Gao and Yu~\\cite{GaYu12}.\n\nIn the following, we will survey both the results of state\ncomplexities of combined operations and the results of estimations\nand approximations of state complexities of combined operations.\n\n\n\\subsection{State Complexity of Combined Operations on Regular Languages}\n\nThe state complexities of a number of basic combined operations on regular languages have been\nstudied. Most of these combined operations are composed of two basic individual operations. The results are shown in Table~\\ref{tab:sc-some-combined-regular}.\n\nIn 1996, Birget~\\cite{birget96:_state_compl_of_oever_sigma} obtained the the state complexity of $\\overline{\\Sigma^\\star \\overline{L}}$, where $L$ is a regular language. This combination of complementation, catenation and star is the first combined operation composed of different individual operations whose state complexity was established. In 2007, Salomaa \\emph{et al.}~\\cite{salomaa07a:_state_compl_of_combin_operat} pointed out that the mathematical composition of state complexities of individual component operations of a combined operation is usually much higher than the state complexity of the combined operation. This is because the result of a component operation of the combined operation may not be among the worst-cases of\nthe succeeding component operation. They established the state complexity of $(L_1 \\cup L_2)^*$ and indicated that the state complexity of $(L_1 \\cap L_2)^*$ should be at least reasonably close to the mathematical composition of state complexities of intersection and star. Later, Jir\\'askov\\'a and Okhotin~\\cite{jiraskova11:_state_compl_of_star_of} proved that the state complexity of $(L_1 \\cap L_2)^*$ is exactly the same as the mathematical composition of state complexities of intersection and star.\n\nGao \\emph{et al.}~\\cite{GaSaYu08},  in 2008, established the state complexities of $(L_1 L_2)^*$ and $(L_1^R)^*$, where $L_1$ and $L_2$ are regular languages. The state complexity of $(L_1 L_2)^*$ is $2^{m +n -1}-2^{m -1}-2^{n -1}+1$ which is lower than the mathematical composition of the state complexity of catenation and star. Interestingly, the state complexity of $(L_1^R)^*$ is the same as that of $L_1^R$ which is $2^m$. The worst-case example over a three-letter alphabet for $L_1^R$ \\cite{yu94:_state_compl_of_some_basic} also works for $(L_1^R)^*$.\n\n\n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Regular}\\\\\n\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{$|\\Sigma|$}\\\\\n\\hline\n\\hline\n$\\overline{\\Sigma^\\star \\overline{L_1}}$ & $2^{m-1}$ (\\cite{birget96:_state_compl_of_oever_sigma}) & 2\\\\\n\\hline\n$\\overline{L_1^*}^*$ & $2^{\\theta(m\\log m)}$ (\\cite{jiraskova12:_state_compl_of_star_compl_star}) & 7\\\\\n\\hline\n$(L_1 \\cup L_2)^*$ & $2^{m +n -1}-2^{m -1}-2^{n -1}+1$ (\\cite{jiraskova11:_state_compl_of_star_of,salomaa07a:_state_compl_of_combin_operat}) & 2\\\\\n\\hline\n$(L_1 \\cap L_2)^*$ & $2^{mn -1}+2^{mn -2}$ (\\cite{jiraskova11:_state_compl_of_star_of}) & 6\\\\\n\\hline\n$(L_1 L_2)^*$   &  $2^{m+n-1}+2^{m+n-4}-2^{m-1}-2^{n-1}+m+1$ (\\cite{GaSaYu08}) & 4\\\\\\hline\n$(L_1^R)^*=(L_1^*)^R$   &  $2^m$  (\\cite{GaSaYu08}) & 3\\\\\n\\hline\n$(L_1\\cup L_2)^R$ & $2^{m+n}-2^m-2^n+2$ (\\cite{LiMaSaYu08}) & 3\\\\\n    \\hline\n$(L_1\\cap L_2)^R$ & $2^{m+n}-2^m-2^n+2$ (\\cite{LiMaSaYu08})  & 3\\\\\n    \\hline\n$(L_1L_2)^R$ & $3\\cdot 2^{m+n-2}-2^n+1$ (\\cite{CuGaKaYu12,LiMaSaYu08}) & 4\\\\\n    \\hline\n    $L_1^*L_2$   &  $5 \\cdot 2^{m+n-3} - 2^{m-1} - 2^{n} +1$ (\\cite{CuGaKaYu12})   & 4\\\\\n\\hline\n$L_1L_2^*$   &  $(3m-1)2^{n-2}$ (\\cite{CGKY12-cat-sr}) & 3\\\\\n\\hline\n$L_1^RL_2$   &  $3\\cdot 2^{m+n-2}$ (\\cite{CuGaKaYu12}) & 4\\\\\n\\hline\n$L_1L_2^R$   &  $m 2^{n}-2^{n-1}-m+1$ (\\cite{CGKY12-cat-sr}) & 3\\\\\n\\hline\n$L_1(L_2\\cup L_3)$   &  $(m-1)(2^{n+p}-2^{n}-2^{p}+2)+2^{n+p-2}$ (\\cite{CGKY11-cat-ui}) & 4\\\\\n\\hline\n$L_1(L_2\\cap L_3)$   &  $m 2^{np}-2^{np-1}$ (\\cite{CGKY11-cat-ui}) & 4\\\\\n\\hline\n$L_1^*\\cup L_2$   &  $3\\cdot 2^{m-2}\\cdot n-n+1$ (\\cite{GaYu10}) & 3\\\\\n\\hline\n$L_1^*\\cap L_2$   &  $3\\cdot 2^{m-2}\\cdot n-n+1$ (\\cite{GaYu10})  & 3\\\\\n\\hline\n$L_1^R\\cup L_2$   &  $2^{m}\\cdot n-n+1$ (\\cite{GaYu10}) & 4\\\\\n\\hline\n$L_1^R\\cap L_2$   &  $2^{m}\\cdot n-n+1$ (\\cite{GaYu10}) & 4\\\\\n\\hline\n$(L_1 \\cup L_2)L_3$ & $mn2^p - (m+n-1)2^{p-1}$ (\\cite{CuGaKaYu12}) & 4\\\\\n\\hline\n$(L_1 \\cap L_2) L_3$ & $mn2^{p}-2^{p-1}$ (\\cite{CuGaKaYu12})\n & 4\\\\\n\\hline\n$L_1L_2 \\cup L_3$ & $(m2^{n}-2^{n-1})p$ (\\cite{CuGaKaYu12}) & 4\\\\\n\\hline\n$L_1L_2 \\cap L_3$ & $(m2^n-2^{n-1})p$ (\\cite{CuGaKaYu12}) & 3\\\\\n\\hline\n$ L_1 L_2 L_3$ & $m2^{n+p}-2^{n+p-1}-(m-1)2^{n+p-2}$ & 5\\\\\n    & $-2^{n+p-3}-(m-1)(2^p-1)$ (\\cite{EsGaLiYu09}) & \\\\\n\\hline\n\\end{tabular}\n  \\centering\n  \\caption{\\small{State complexities of some basic combined operations on regular languages}}\\label{tab:sc-some-combined-regular}\n\\end{table}\n\nIn 2008, Liu \\emph{et al.}~\\cite{LiMaSaYu08} studied the state complexities of $(L_1\\cup L_2)^R$, $(L_1\\cap L_2)^R$, and $(L_1L_2)^R$, where $L_1$ and $L_2$ are regular languages. The tight bounds for $(L_1\\cup L_2)^R$ was proved and the state complexity of $(L_1\\cap L_2)^R$ is the same as that of $(L_1\\cup L_2)^R$ because of De Morgan's laws and $\\overline{L^R}=\\overline{L}^R$. They also gave an upper bound for the last combined operation which was proved to be tight, in 2012, by Cui \\emph{et al.}~\\cite{CuGaKaYu12}.\n\nCui \\emph{et al.}~\\cite{CGKY11-cat-ui} established the state complexities of $L_1(L_2\\cup L_3)$ and $L_1(L_2\\cap L_3)$ in 2011. The state complexity of $L_1(L_2\\cup L_3)$ is lower than the mathematical composition of the state complexities of union and catenation, whereas the state complexity of $L_1(L_2\\cap L_3)$ is the same as the corresponding composition.\n\nIn 2012, Jir{\\'a}skov{\\'a} and Shallit~\\cite{jiraskova12:_state_compl_of_star_compl_star} proved the state complexity of the combined operation $\\overline{L_1^*}^*$ to be $2^{\\theta(m\\log m)}$, where $L_1$ is a regular language accepted by an $m$-state DFA\\xspace. A seven-letter alphabet was used in the proof for the lower bound.\n\nGao \\emph{et al.} presented the state complexities of four combined operations: $L_1^*\\cup L_2$, $L_1^*\\cap L_2$, $L_1^R\\cup L_2$, and $L_1^R\\cap L_2$, where $L_1$ and $L_2$ are regular languages accepted by $m$ and $n$-state DFAs\\xspace, respectively. The state complexities of the four combined operations are all $n-1$ less than the mathematical composition of the state complexities of their component operations. Although gaps are the same, the reasons causing them are different. For $L_1^*\\cup L_2$ and $L_1^*\\cap L_2$, the gap $n-1$ exists because there are $n-1$ unreachable states in the constructions of resulting DFAs\\xspace. For $L_1^R\\cup L_2$ and $L_1^R\\cap L_2$, it is because $n$ states are equivalent and can be merged into one in the constructions.\n\nCui \\emph{et al.}~\\cite{CGKY12-cat-sr,CuGaKaYu12} gave the state complexities of a number of combined operations including: $L_1^*L_2$, $L_1L_2^*$, $L_1^RL_2$, $L_1L_2^R$, $(L_1 \\cup L_2)L_3$, $(L_1 \\cap L_2)L_3$, $L_1L_2 \\cup L_3$, and $L_1L_2 \\cap L_3$. The state complexities of the first five combined operations are less than the corresponding mathematical compositions and the state complexities of the others are the same as the compositions. The state complexity of $L_1L_2^R$ is equal to that of catenation combined with antimorphic involution $(L_1 \\theta(L_2))$ in biology \\cite{CGKY12-cat-sr}. Up to now, the state complexities of all the combined operations composed of two basic individual operations have been obtained. These results will serve as the basis of the research on the state complexities of combined operations with more complex structures in the future.\n\n\n\n\nBesides these basic combined operations, a few combined operations on $k$ operand regular languages have also been investigated, e.g. $(\\bigcup\\limits_{i=1}^{k} L_i)^*$, $k\\ge 2$. These results are summarized in Table~\\ref{tab:sc-classes-of-combined-regular}. The state complexity of $L_1\\cap L_2\\cap \\ldots \\cap L_k$, $k\\ge 2$ was shown to be $n_1n_2\\cdots n_k$ by Birget \\cite{birget91:_inter_of_regul_languag_and_state_compl}, and Yu  and Zhuang~\\cite{yu91:_state_compl_of_inter_of_regul_languag} in 1991, where $L_i$ is a regular language accepted by an $n_i$-state DFA\\xspace, $1\\le i\\le k$. \\'Esik \\emph{et al.}~\\cite{EsGaLiYu09} later extended the result to combined Boolean operations. A combined Boolean operation $f(L_1,L_2,\\ldots,L_k)$ is a function which can be constructed from the projection functions and the binary union, intersection and the complementation operations by function composition, e.g. $\\overline{L_1}\\cup L_2\\cap L_2\\cap \\ldots \\cap L_k$. Its state complexity was proved to be also $n_1n_2\\cdots n_k$.\\ \\'Esik \\emph{et al.}~\\cite{EsGaLiYu09} presented the state complexities of $L_1L_2L_3$ and $L_1L_2L_3L_4$ in the same paper. The worst-case examples for the two combined operations are modifications of the worst-case examples proposed by Yu \\emph{et al.}~\\cite{yu94:_state_compl_of_some_basic} for catenation. On the basis of these results, Gao~\\cite{Ga10} established the state complexity of $L_1 L_2 \\ldots L_k$, which formula is too complex to figure here.\n\nIn 2012, Gao \\emph{et al.}~\\cite{GaKaYu12-union-and-intersection-of-square-and-reversal} gave the state complexities of a series of combined operations composed of arbitrarily many individual operations, including: $(\\bigcup\\limits_{i=1}^{k} L_i)^*$, $(\\bigcup\\limits_{i=1}^{k} L_i)^2$, $\\bigcup\\limits_{i=1}^{k} L_i^*$, $\\bigcap\\limits_{i=1}^{k} L_i^*$, $\\bigcup\\limits_{i=1}^{k} L_i^2$, $\\bigcap\\limits_{i=1}^{k} L_i^2$, $\\bigcup\\limits_{i=1}^{k} L_i^R$, and $\\bigcap\\limits_{i=1}^{k} L_i^R$. Tight bounds were established for all these combined operations.\n\nIn Table~\\ref{tab:sc-classes-of-combined-regular}, we can see that all the results on the state complexities of combined operations on $k$ operand languages were proved with increasing alphabets. Clearly, it is comparatively easier to design worst-case examples with increasing alphabets than fixed ones. However, the most crucial reason is that it is impossible to design a worst-case example for a combined operation on\narbitrary $k$ operand languages which are over a fixed alphabet and accepted by arbitrary $n_1$, $n_2$, $\\ldots$, $n_k$-state DFAs\\xspace, respectively. This is because there exist only a limited number of different DFAs\\xspace\nwith a fixed number of states if the alphabet is fixed. Therefore, when $k$ is large enough and $n_i$ is an arbitrary positive integer, $1\\le i\\le k$,\nsome of the DFAs\\xspace may have the same number of states and some of them may be indeed\nthe same according to pigeonhole principle \\cite{GaKaYu12-union-and-intersection-of-square-and-reversal}. Thus, the research on the state\ncomplexities of combined operations on $k$ operand languages uses increasing alphabets\nin general.\n\n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Regular}\\\\\n\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{$|\\Sigma|$}\\\\\n\\hline\n\\hline\n$(\\bigcup\\limits_{i=1}^{k} L_i)^*$ & $\\prod\\limits_{i=1}^{k}(2^{n_i-1}-1)+2^{\\sum\\limits_{j=1}^{k}n_j-k}$ (\\cite{GaKa12}) & $2k+1$\\\\\n\\hline\n$(\\bigcup\\limits_{i=1}^{k} L_i)^2$ & $\\prod\\limits_{h=1}^{k}(n_h-1)[\\prod\\limits_{i=1}^{k}(2^{n_i}-1)+1]$ & $2k+1$\\\\\n  & $+[\\prod\\limits_{j=1}^{k}n_j -\\prod\\limits_{l=1}^{k}(n_l-1)]2^{\\sum\\limits_{m=1}^{k}n_m-k}$ (\\cite{GaKa12}) & \\\\\n\\hline\n$\\bigcup\\limits_{i=1}^{k} L_i^*$   &  $(\\frac{3}{4}) ^k 2^{g}-\\sum\\limits_{i=1}^{k}[\\prod\\limits_{j=1}^{i-1}(\\frac{3}{4}2^{n_j}-1)\\prod\\limits_{t=i+1}^{k}(\\frac{3}{4}2^{n_t})]\n+1$ (\\cite{GaKaYu12-union-and-intersection-of-star}) & $2k$\\\\\n\\hline\n$\\bigcap\\limits_{i=1}^{k} L_i^*$   &  $(\\frac{3}{4}) ^k 2^{g}-\\sum\\limits_{i=1}^{k}[\\prod\\limits_{j=1}^{i-1}(\\frac{3}{4}2^{n_j}-1)\\prod\\limits_{t=i+1}^{k}(\\frac{3}{4}2^{n_t})]\n+1$ (\\cite{GaKaYu12-union-and-intersection-of-star}) & $2k$\\\\\n\\hline\n$\\bigcup\\limits_{i=1}^{k} L_i^2$   &  $\\prod\\limits_{i=1}^{k}(n_i 2^{n_i}-2^{n_i-1})$ (\\cite{GaKaYu12-union-and-intersection-of-square-and-reversal}) & $2k$\\\\\n\\hline\n$\\bigcap\\limits_{i=1}^{k} L_i^2$   &  $\\prod\\limits_{i=1}^{k}(n_i 2^{n_i}-2^{n_i-1})$ (\\cite{GaKaYu12-union-and-intersection-of-square-and-reversal}) & $2k$\\\\\n\\hline\n$\\bigcup\\limits_{i=1}^{k} L_i^R$   &  $\\prod\\limits_{i=1}^{k} (2^{n_i} - 1) + 1$ (\\cite{GaKaYu12-union-and-intersection-of-square-and-reversal}) & $3k$\\\\\n\\hline\n$\\bigcap\\limits_{i=1}^{k} L_i^R$   &  $\\prod\\limits_{i=1}^{k} (2^{n_i} - 1) + 1$ (\\cite{GaKaYu12-union-and-intersection-of-square-and-reversal}) & $3k$\\\\\n    \\hline\nA Boolean\n & $n_1n_2\\cdots n_k$ (\\cite{birget91:_inter_of_regul_languag_and_state_compl,EsGaLiYu09,yu91:_state_compl_of_inter_of_regul_languag}) & $2k$\\\\\noperation & & \\\\\n$f(L_1,\\ldots,L_k)$ & & \\\\\n    \\hline\n    $L_1L_2\\cdots L_k$   &  see details in \\cite{EsGaLiYu09,Ga10,GaYu09}    & $2k-1$\\\\\n\\hline\n\\end{tabular}\n  \\centering\n  \\caption{\\small{State complexities of some combined operations on $k$ regular languages, $k\\ge 2$}}\\label{tab:sc-classes-of-combined-regular}\n\\end{table}\n\n\\subsection{State Complexity of Combined Operations on Prefix-free Regular Languages}\nSince the research history of combined operations is much shorter than that of individual operations, there remains a lot of work to be done on state complexity of combined operations for subregular language classes. The state complexities of several combined operations on prefix-free regular languages were obtained by Han \\emph{et al.}~\\cite{HaSaYu10}, in 2010. These results are shown in Table~\\ref{tab:sc-combined-prefix-free}.\n\n\n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Prefix-Free Regular}\\\\\n\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{$|\\Sigma|$}\\\\\n\\hline\n\\hline\n$(L_1 \\cup L_2)^*$ & $5\\cdot 2^{m +n -6}$ (\\cite{HaSaYu10}) & 4\\\\\n\\hline\n$(L_1 \\cap L_2)^*$ & $mn-2(m+n)+6$ (\\cite{HaSaYu10}) & 4\\\\\n\\hline\n$(L_1 L_2)^*$   &  $m+n-2$ (\\cite{HaSaYu10}) & 2\\\\\n\\hline\n$(L_1^R)^*=(L_1^*)^R$   &  $2^{m-2}+1$ (\\cite{HaSaYu10}) & 3\\\\\n\\hline\n\\end{tabular}\n  \\centering\n  \\caption{\\small{State complexities of some combined operations on prefix-free regular languages}}\\label{tab:sc-combined-prefix-free}\n\\end{table}\n\n\n\\subsection{Estimation and Approximation of State Complexity of\nCombined Operations}\n\\label{sec:estapp}\nWe can summarize at least two problems concerning the state complexities for combined operations.\n First, the state complexities of combined operations composed of large numbers of individual operations are extremely difficult to compute. Second, a large proportion of results that have been obtained are pretty complex and impossible to comprehend~\\cite{GaYu09}.\n For example, \u00c9sik et al.~\\cite{EsGaLiYu09} shown that the state complexity of the catenation for four\nregular languages with state complexities $m, n, p, q$, respectively, is\n$$9(2m-1)2^{n+p+q-5}-3(m-1)2^{p+q-2}-(2m-1)2^{n+q-2}+(m-1)2^q+(2m-1)2^{n-2}\n.$$\n\nClearly, in these situations, close estimations and approximations of state complexities are usually good enough to use.\n\n\\subsubsection{Estimation of State Complexity of Combined Operations}\n\nAn estimation method through nondeterministic state complexity to\nobtain the upper bound was first introduced by Salomaa and Yu~\\cite{SaYu07}.\n Assume we are considering the combination of a language operation\n$g_1$ with $k$ arguments together with operations  $g_2^i$, $i = 1,\n\\ldots, k$. The {\\em nondeterministic estimation upper bound,} or\n{\\em NEU-bound\\\/} for the deterministic state complexity of the\ncombined operation $g_1(g_2^1, \\ldots, g_2^k)$ is calculated as\nfollows:\n\n\\begin{itemize}\n\\item[{\\rm (i)}]\nLet the arguments of the operation $g^i_2$ be DFAs\\xspace $A^i_j$ with\n$m^i_j$ states,  $i = 1, \\ldots, k$, $j = 1, \\ldots, r_i$, $r_i \\geq\n1$.\n\\item[{\\rm (ii)}] The nondeterministic\nstate complexity of the combined operation is at most the\ncomposition of the individual state complexities, and hence the\nlanguage\n$$\ng_1( g_2^1(L(A^1_1), \\ldots, L(A^1_{r_1})), \\ldots, g_2^k(L(A^k_1),\n\\ldots, L(A^k_{r_k})))\n$$\nhas an NFA\\xspace with at most\n$$\n{\\rm nsc}(g_1)( {\\rm nsc}(g^1_2)(m^1_1, \\ldots, m^1_{r_1}), \\ldots,\n{\\rm nsc}(g^k_2)(m^k_1, \\ldots, m^k_{r_k}))\n$$\nstates, where {\\rm nsc}$(g)$ is the nondeterministic state\ncomplexity (as a function) of the language operation $g$.\n\\item[{\\rm (iii)}] Consequently,\nthe deterministic state complexity of the combined operation\n$g_1(g_2^1, \\ldots, g_2^k)$ is upper bounded by\n\\begin{equation}\n\\label{NEU} 2^{ {\\rm nsc}(g_1)( {\\rm nsc}(g^1_2)(m^1_1, \\ldots,\nm^1_{r_1}), \\ldots, {\\rm nsc}(g^k_2)(m^k_1, \\ldots, m^k_{r_k})) }\n\\end{equation}\n\\end{itemize}\n\n\nTable~\\ref{tab:sc-neu} shows the state complexities and their\ncorresponding NEU-bounds of the four combined operations~\\cite{SaYu07}: (1) star of union, (2) star of intersection, (3)\nstar of catenation, and (4) star of reversal.\n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Regular}\\\\\n\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{NEU-bound}\\\\\n\\hline\n\\hline\n$(L_1\\cup L_2)^*$ & $2^{m+n-1}-2^{m-1}-2^{n-1}+1$\n& $2^{m+n+2}$\\\\\n\\hline $(L_1\\cap L_2)^*$ & $ 3\/4\\; 2^{mn}$\n& $2^{mn+1}$ \\\\\n\\hline $(L_1L_2)^*$ & $2^{m+n-1}+2^{m+n-4}-2^{m-1}-2^{n-1}+m+1$ &\n$2^{m+n+1}$ \\\\\n\\hline\n$(L_1^R)^*$ & $2^m$ & $2^{m+2}$\\\\\n\\hline\n\\end{tabular}\n\\centering\n  \\caption{\\small{State complexities of four combined operations and their\ncorresponding NEU-bounds on regular languages \\cite{GaYu12}}}\\label{tab:sc-neu}\n\\end{table}\nThis method works well when a\ncombined operation ends with the star operation. However, it\ndoes not work well in general for combined operations that are ended\nwith reversal \\cite{EsGaLiYu09,SaYu07}. For example, the state complexity of $(L(A)\\cap L(B))^*$ is $2^{m + n} - 2^{m} - 2^{n} + 2$, where $A$ and $B$ are $m$-state and $n$-state DFAs\\xspace, respectively. But using the above method, we would\nobtain an estimate $2^{mn + 1}$. We note that in this particular case if reversal is distributed over intersection we can again recover a good estimate. Thus, it may be possible to have a general estimation method that takes in account algebraic properties of the considered model.\\footnote{This observation was made to us by an annonymous referee.}\n\n\\subsubsection{Approximation of State Complexity of Combined Operations}\n\nAlthough an estimation of the state complexity of a combined\noperation is simpler and more convenient to use, it does not show\nhow close it is to the state complexity. To solve this\nproblem, the concept of approximation of state complexity was\nproposed by Gao and Yu~\\cite{GaYu09}.\nThe idea of approximation of state complexity comes from the notion of\napproximation algorithms \\cite{GaGrUl72,Jo72,Jo73}. A large number of\npolynomial-time approximation algorithms have been proposed for many NP-complete problems, e.g. the\ntraveling-salesman problem, the set-covering problem, and\nthe subset-sum problem, etc. Since it is considered intractable to obtain an optimal solution for an\nNP-complete problem, near optimal\nsolutions obtained by approximation algorithms are often good enough to use in practice. Assume there is a maximization or a minimization problem. An\napproximation algorithm is said to have a ratio bound of $\\rho(n)$\nif for any input of size $n$, the cost $C$ of the solution produced\nby the algorithm is within a factor of $\\rho(n)$ of the cost $C^*$\nof an optimal solution \\cite{CoLeRi90:_Intro_to_algorithms}:\n$$\\max\\left(\\frac{C}{C^*}, \\frac{C^*}{C}\\right) \\leq \\rho(n).$$\nThe concept of approximation of state complexity is\nsimilar to that of approximation algorithms. An approximation of state complexity of an operation\nis a close estimation of the state complexity of the operation with a ratio bound showing the error range of the\napproximation \\cite{GaYu09}. In spite of similarities, there are some fundamental differences\nbetween an approximation\nalgorithm and approximation of state complexity. The efforts in the area of approximation algorithms are\nin designing polynomial algorithms for NP-complete problems such that\nthe results of the algorithms approximate the optimal results whereas the efforts in approximation of state complexity are in\nsearching directly for the estimations of state complexities such\nthat they are within some certain ratio bounds \\cite{GaYu09}. The aim of designing an\napproximation algorithm is to transform an intractable problem into\none that is easier to compute and the result is not optimal but still acceptable. In comparison, an approximation of state complexity may\nhave two different effects: \n\\begin{enumerate}[(1)]\n\\item it gives a reasonable estimation of a\ncertain state complexity, with some bound, the exact value of which\nis difficult or impossible to compute; or \n\\item it gives a simpler and\nmore comprehensible formula that approximates a known state\ncomplexity \\cite{GaYu12}.\n\\end{enumerate}\nGao \\emph{et al.} gave a formal definition of approximation of state complexity in \\cite{GaYu12}. Let $\\xi$ be a combined operation on $k$ regular languages. Assume that the state complexity of $\\xi$ is $\\theta$. We say that $\\alpha$\nis an approximation of the state complexity of the operation $\\xi$\nwith the ratio bound $\\rho$ if, for any large enough positive\nintegers $n_1, \\ldots, n_k$, which are the numbers of states of the\nDFAs\\xspace that accept the argument languages of the operation,\nrespectively,\n$$\\max\\left(\\frac{\\alpha(n_1, \\ldots, n_k)}{\\theta(n_1, \\ldots,\nn_k)}, \\frac{\\theta(n_1, \\ldots, n_k)}{\\alpha(n_1, \\ldots,\nn_k)}\\right) \\leq \\rho(n_1, \\ldots, n_k).$$ Note that in many cases,\n$\\rho$ is a constant.\n Some examples of approximation of state complexity of combined operations are shown in Table~\\ref{tab:approximation}.\n\n\n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Regular}\\\\\n\\hline\n&\\multicolumn{1}{c}{Approximation}&\\multicolumn{1}{c|}{Ratio bound}\\\\\n\\hline\n\\hline\n$(L_1\\cup L_2)^*$ & $2^{m+n+2}$  & $\\approx 8$  \\cite{GaYu12}\\\\\n\\hline $(L_1\\cap L_2)^*$ & $2^{mn+1}$ & $8\/3$  \\cite{GaYu12}\\\\\n\\hline $(L_1L_2)^*$  &$2^{m+n+1}$ & $\\approx 4$   \\cite{GaYu12}\\\\\n\\hline\n$(L_1^R)^*$ & $2^{m+2}$ & $4$  \\cite{GaYu12}\\\\\n\\hline\n$(L_1\\backslash R)^*$ & $2^{m-1}+2^{m-2}$ & $\\frac{4}{3}$ \\cite{GaYu09} \\\\\n\\hline\n$L_1\\backslash R^*$ & $2^{m+1}$ & $\\frac{8}{3}$ \\cite{GaYu09} \\\\\n\\hline\n\\end{tabular}\n\\centering\n  \\caption{\\small{Approximations of state complexities of six combined operations and their\ncorresponding ratio bounds on regular languages}}\\label{tab:approximation}\n\\end{table}\n\n\n\n\n\\section{State Complexity and Nondeterministic State Complexity}\n\\label{sec:scnsc}\nThe \\emph{state complexity} of a regular language $L$, $sc(L)$, is the\nnumber of states of its minimal DFA\\xspace.  The \\emph{nondeterministic state\ncomplexity} of a regular language $L$, $nsc(L)$, is the number of\nstates of a minimal NFA\\xspace that accepts $L$. Since a DFA\\xspace is in particular an NFA\\xspace, for any regular language $L$\none has $sc(L)\\leq nsc(L)$. It is well known that any $m$-state NFA\\xspace\ncan be converted, via the \\emph{subset construction}, into an equivalent\nDFA\\xspace with at most\n$2^m$ states~\\cite{rabin59:_finit_autom_and_their_decis_probl}\n(we call this conversion \\emph{determination}). Thus, $sc(L)\\leq 2^{nsc(L)}$.\n\\begin{figure}[htb]\n\\centering\n  \\begin{tabular}[h]{lc}\n{\\small(i)}&\\raisebox{-.7\\height}{\\includegraphics[width=9.5cm]{diag1}}\n\\ignore{\\SmallPicture{\\VCDraw{\n      \\begin{VCPicture}{(-2,-3)(17,1)}\n       \\State[0]{(0,0)}{0}\\Initial{0}\n       \\State[1]{(3,0)}{1}\n       \\State[2]{(6,0)}{2}\n       \\SetStateLineStyle{none}\n       \\State[\\cdots]{(9,0)}{3}\n       \\SetStateLineStyle{solid}\n       \\StateVar[m-2]{(12,0)}{m2}\n       \\FinalStateVar[m-1]{(16,0)}{m1}\n       \\EdgeL{0}{1}{a}\\LoopS{0}{b}\n       \\EdgeL{1}{2}{a,b}\n       \\EdgeL{2}{3}{a,b}\n       \\EdgeL{3}{m2}{a,b}\n       \\EdgeL{m2}{m1}{a,b}\n       \\ArcL[.5]{m1}{1}{a}\n       \\VArcL[.5]{arcangle=20}{m1}{0}{a}\n     \\end{VCPicture}\n   }\n }}\\\\\n{\\small(ii)}&\n\\raisebox{-.5\\height}{\\includegraphics[width=8.5cm]{diag2}}\n\\ignore{\\SmallPicture{\\VCDraw{\n    \\begin{VCPicture}{(-2,-2)(15,2)}\n      \\FinalState[0]{(0,0)}{0}\\Initial{0}\n      \\State[1]{(3,0)}{1}\n      \\State[2]{(6,0)}{2}\n      \\SetStateLineStyle{none}\n      \\State[\\cdots]{(9,0)}{3}\n      \\SetStateLineStyle{solid}\n      \\StateVar[m-1]{(12,0)}{m1}\n      \\ArcL{0}{1}{a,b}\n      \\EdgeL[.2]{1}{0}{b,c}\n      \\LoopN{1}{c}\n      \\EdgeL{1}{2}{a}\n      \\LoopN{2}{b,c}\n      \\EdgeL{2}{3}{a}\n      \\EdgeL{3}{m1}{a}\n      \\LoopN{m1}{b,c}\n      \\VArcL{arcangle=20}{m1}{0}{a}\n    \\end{VCPicture}\n  }\n}}\n\\\\{\\small(iii)}&\n\\raisebox{-.5\\height}{\\includegraphics[width=10cm]{diag3}}\n\\ignore{\\SmallPicture{\\VCDraw{\n    \\begin{VCPicture}{(-2,-3)(18,2)}\n      \\FinalState[0]{(0,0)}{0}\\Initial{0}\n      \\State[1]{(3,0)}{1}\n      \\State[2]{(6,0)}{2}\n      \\SetStateLineStyle{none}\n      \\State[\\cdots]{(9,0)}{3}\n      \\SetStateLineStyle{solid}\n      \\StateVar[m-2]{(12,0)}{m2}\n      \\StateVar[m-1]{(16,0)}{m1}\n      \\ArcL{0}{1}{a}\n      \\EdgeL[.1]{1}{0}{b}\\LoopN{1}{b}\\EdgeL{1}{2}{a}\n      \\EdgeL{2}{3}{a}\\VArcL[.1]{arcangle=20}{2}{0}{b}\\LoopN{2}{b}\n      \\EdgeL{3}{m2}{a}\n      \\LoopN{m2}{b}\\VArcL[.1]{arcangle=20}{m2}{0}{b}\\EdgeL{m2}{m1}{a}\n      \\LoopN{m1}{b}\\VArcL[.2]{arcangle=20}{m1}{0}{a,b}\n    \\end{VCPicture}\n  }\n }}\n\n\\end{tabular}    \n  \\caption{\\small{Moore {\\small(i)}, Lupanov {\\small(ii)}, and Meyer \\&\n    Fischer {\\small(iii)} minimal $m$-state NFAs\\xspace\n    with equivalent minimal $2^m$-state DFAs\\xspace}}\n \\label{fig:nfadfaMooreLupanovMeyerFisher}\n\\end{figure}\nTo show that this upper bound is tight one must exhibit a family of languages\n$(L_m)_{m\\geq 1}$ such that $nsc(L_m)=m$ and $sc(L_m)=2^m$, for every\n$m\\geq 1$.\nIn 1963, Lupanov~\\cite{lupanovn):_compar_of_two_types_of_finit_sourc}\nshowed that this upper bound is tight using a family of ternary\nlanguages.  In 1971, Moore~\\cite{moore71:_bound_for_state_set_size} and\nMeyer and Fischer~\\cite{meyer71:_econom_of_descr_by_autom} presented\ndifferent families of binary languages. All three families of NFAs\\xspace are\nrepresented in Figure~\\ref{fig:nfadfaMooreLupanovMeyerFisher}.\nHowever, for unary languages that upper bound is not\nachievable~\\cite{lyubich64:_estim_for_optim_deter_of,chrobak86:_finit_autom_and_unary_languag,chrobak03:_errat_to}.\nChrobak~\\cite{chrobak86:_finit_autom_and_unary_languag,chrobak03:_errat_to}\nproved that if $L$ is a unary language with $nsc(L)=m$, then\n$sc(L)=O(F(m))$ where\n\\begin{equation}\n  \\label{eq:fm}\nF(m)=\\max\\{\\operatorname{lcm}(x_1,\\ldots,x_l)\\mid x_1,\\ldots, x_l\\geq 1 \\text{ and }\nx_1+\\cdots+x_l=m\\}  \n\\end{equation}\n\\noindent is the Landau's function and $\\operatorname{lcm}$ denotes the least common\nmultiple. It\nis known that $F(m)=e^{\\Theta{(\\sqrt{m\\ln m})}}$, so\n$sc(L)=e^{\\Theta{(\\sqrt{m\\ln m})}}$. This asymptotic bound is tight,\ni.e., for every $m$ there exists a unary language $L_m$ such that\n$nsc(L_m)\\leq m$ and $sc(L_m)= F(m-1)$.  Other related bounds were\nstudied by Meregethi and\nPighizzini~\\cite{mereghetti00:_optim_simul_between_unary_autom}. \n\nFor a general finite language $L$,  if $nsc(L)=m$ then\n$sc(L)=\\Theta(k^{\\frac{m}{1+\\log k}})$, $k=|\\Sigma| > 1$, and this bound is\ntight~\\cite{salomaa97:_nfa_to_dfa_trans_for}. In the case of finite\nbinary languages, $\\Theta(2^{\\frac{m}{2}})$ is a tight\nbound. In 1973, Mandl~\\cite{mandl73:_precis_bound_assoc_with_subset} had\nalready proved that, for any finite binary language $L$, if $nsc(L)=m$\nthen $sc(L)\\leq 2 \\cdot 2^{m\/2}- 1$ if $m$ is even, and $sc(L)\\leq\n3\\cdot 2^{\\lfloor m\/2\\rfloor}-1$ if $m$ is odd, and that these bounds are\ntight. Finally, for finite unary  languages, nondeterminism does not lead to\nsignificant improvements. If $L$ is a finite unary language with\n$nsc(L)=m$, then $sc(L)\\leq\nm+1$~\\cite{mandl73:_precis_bound_assoc_with_subset,salomaa97:_nfa_to_dfa_trans_for}.\n\nIn Section~\\ref{sec:subregularlanguages} the state complexity of determination of other subregular languages is reviewed. As it will be evident from the results in the following sections, the complexity of determination plays a fundamental role in the operational complexity and thus the importance of its study \\textit{per se}. \n\nThe possible gap between state complexity and nondeterministic state\ncomplexity for general regular languages lead to the notion of\n\\emph{magic number} introduced in 2000 by Iwama \\emph{et\n  al.}~\\cite{iwama00:_tight_bound_number_of_states,iwama00:_famil_of_nfas_which_need}.\nA number $\\alpha$, such that $\\alpha\\in [m,2^m]$, is \\emph{magic} for\n$m$ with respect to a given alphabet of size $k$, if there is no minimal\n$m$-state NFA\\xspace whose equivalent minimal DFA\\xspace has $\\alpha$ states. This\nnotion has been extensively researched in the last decade and has been\nextended to other gaps between two state complexity\nvalues~\\cite{lyubich64:_estim_for_optim_deter_of,chrobak86:_finit_autom_and_unary_languag,jiraskova01:_note_minim_finit_autom,\ngeffert05:_non_deter_and_size_of,geffert07:_magic_number_in_state_hierar,\ngeffert07:_state_hierar_for_one_way_finit_autom,jirasek08:_deter_blow_ups_of_minim,jiraskova09:_magic_number_and_ternar_alpha,jiraskova11:_magic_number_and_ternar_alphab,holzer12:_magic_number_probl_for_subreg_languag_famil}.\nWe summarize here some of the obtained results.  The general observation is\nthat, apart from unary languages, magic numbers are hard to find.  For\nbinary languages, it was shown that if $\\alpha=2^m-2^n$ or\n$\\alpha=2^m-2^n-1$, for $n\\in\n[0,m\/2-2]$~\\cite{iwama00:_tight_bound_number_of_states}, and\n$\\alpha=2^m-n$ for $n\\in [5, 2m-2]$ and some coprimality condition\nholds for $n$~\\cite{iwama00:_famil_of_nfas_which_need}, then $\\alpha$\nis not magic. Also, for a binary alphabet, all numbers $\\alpha \\in\n[m,m+2^{\\lfloor m\/3\\rfloor}]$ have been shown to be\nnon-magic~\\cite{jiraskova08:_state_compl_of_compl_stars}, which\nimproves previous results,\n$[m,m^2\/2]$~\\cite{jiraskova01:_note_minim_finit_autom} and\n$[m,2^{\\sqrt[3]{m}}]$~\\cite{geffert05:_non_deter_and_size_of}.\nFor ternary or quaternary regular languages, and for languages over an\nalphabet of exponential growing size there are no magic\nnumbers~\\cite{jiraskova01:_note_minim_finit_autom,jirasek08:_deter_blow_ups_of_minim,jiraskova09:_magic_number_and_ternar_alpha,jiraskova11:_magic_number_and_ternar_alphab}.\nFor the unary case, however, trivially all numbers between\n$e^{(1+o(1))\\sqrt{m\\ln m}}$ and $2^m$ are\nmagic~\\cite{lyubich64:_estim_for_optim_deter_of,chrobak86:_finit_autom_and_unary_languag,geffert07:_magic_number_in_state_hierar}.\nMoreover, it has been shown that there are much more magic than\nnon-magic numbers in the range from $m$ to $e^{(1+o(1))\\sqrt{m\\ln\n    m}}$~\\cite{geffert07:_magic_number_in_state_hierar}.  In the case\nof finite languages, partial results were obtained by Holzer\n\\emph{et\n  al.}~\\cite{holzer12:_magic_number_probl_for_subreg_languag_famil}.\nAll numbers $\\alpha\\in[m+1, (\\frac{m}{2})^2 +\\frac{m}{2} + 1]$,\n if $m$ even, and $\\alpha\\in[m+1, (\\frac{m-1}{2})^2  + m + 1]$, if $m$\n is odd, are non-magic. Moreover, all numbers of the form\n $3\\cdot2^{\\frac{m}{2}-1} + 2^i -1$, with $m$ even, and\n $2^{\\frac{m+1}{2}} + 2^i - 1$, with\n $m$ odd, for some integer $i\\in [1,\\lceil\\frac{m-1}{2}\\rceil]$\nare non-magic. In the same paper, the magic number problem is also\nstudied for other subregular language classes.\n\n\n\\subsection{State Complexity versus Quotient Complexity}\n\\label{sec:scqc}\n\nQuotient complexity, introduced in 2009 by\nBrzozowski~\\cite{brzozowski09:_quotien_compl_of_regul_languag,brzozowskiar:_quotien_compl_of_regul_languag},\ncoincides, for regular languages, with the notion of state complexity\nbut it is defined in terms of languages and their (left) quotients.  The\n\\emph{left quotient} of a language $L$ by a word $w$ is defined as the\nlanguage $w^{-1}L = \\{x \\in \\Sigma^\\star \\mid wx \\in L\\}$. The\n\\emph{quotient complexity} of $L$, denoted by $\\kappa(L)$, is the number of distinct languages\nthat are left quotients of $L$ by some word. It is well known that, for a regular language $L$, the\nnumber of left quotients is finite and is exactly the number of\nstates of the minimal DFA\\xspace accepting $L$. So, in the case of regular\nlanguages, state complexity and quotient complexity\ncoincide. Considering that quotient complexity is given in terms of\nlanguages, and their left quotients, some language algebraic\nproperties can be used in order to obtain upper bounds for the\ncomplexity of operations over languages. Actually, the proof that the\nset of\n(left) quotients of a regular language is\nfinite~\\cite{brzozowski64:_deriv_of_regul_expres} was one of the\nearliest studies of state complexity.  Quotient complexity can also be\nuseful to show that an upper bound is tight.  If a given operation\ncan be expressed as a function of other operations (for example,\n$L_1-L_2 = L_1\\cap \\comp{L_2}$), then, witnesses for the worst-case complexity of those\noperations can be used to provide a witness for the complexity of the\nfirst operation.\n\n\n\\section{State Complexity of Individual Operations}\n\\label{sec:sciop}\n\nThe \\emph{state complexity of an operation} (or \\emph{operational\n  state complexity}) on regular languages is the worst-case state\ncomplexity of a language resulting from the operation, considered as a\nfunction of the state complexities of the operands. Adapting a formulation from Holzer and\nKutrib~\\cite{holzer09:_nondet_finit_autom_recen_resul}, given a binary\noperation $\\circ$, the \\emph{$\\circ$-language operation state\n  complexity problem} can be stated as follows\n \\begin{itemize}\n \\item Given an $m$-state DFA\\xspace $A_1$ and an $n$-state DFA\\xspace  $A_2$.\n \\item How many states are sufficient and necessary, in the worst case,\n   to accept the language $L(A_1) \\circ L(A_2)$ by a DFA\\xspace?\n \\end{itemize}\n\n This formulation can be generalized for operations with other arities, other kinds of \n automata and classes of languages.\n  An upper bound can be obtained by providing an algorithm that, given\n DFAs\\xspace for the operands, constructs  a DFA\\xspace that accepts the resulting\n language. The number of states of the resulting DFA\\xspace is an upper bound for the \n state complexity of the referred operation.\n\n\n\n\n To show that an upper\n bound is tight, for each operand a family of languages (one language,\n for each possible value of the state complexity) must be given such\n that the resulting automata achieve that bound.  We can call those\n families \\emph{witnesses}.  The same approach is used to obtain the\n nondeterministic state complexity of an operation on regular\n languages.  No proofs are here presented for the stated results,\n although several examples of families of languages, for which the\n operations achieve a certain upper bound, are given.\nThere are very few results of the study of state complexity on the\naverage case. However, whenever some results are known they are\nmentioned together with the corresponding worst-case analysis.\n\n\n In this section, the following notation is used.  When considering\nunary operations, let $L$ be regular language with $sc(L)=m$\n($nsc(L)=m$) and let $A=(Q,\\Sigma,\\delta,q_0,F)$ be the complete\nminimal DFA\\xspace (a minimal NFA\\xspace) such that $L=L(A)$. Furthermore,\n$|\\Sigma|=k$ or $|\\Sigma|=f(m)$ if a growing alphabet is taken into\naccount, $|F|=f$, and $|F-\\{q_0\\}|=l$.  In the same way, for binary\noperations let $L_1$ and $L_2$ be regular languages over the same\nalphabet with $sc(L)=m$ ($nsc(L)=m$) and $sc(L_2)=n$ ($nsc(L_2)=n$),\nand let $A_i=(Q_i,\\Sigma,\\delta_i,q_i,F_i)$ be complete minimal DFAs\\xspace\n(minimal NFAs\\xspace) such that $L_i=L(A_i)$, for $i\\in[1,2]$. Furthermore,\n$|\\Sigma|=k$ or $|\\Sigma|=f(m,n)$ if a growing alphabet is taken into\naccount, $|F_i|=f_i$, and $|F_i-\\{q_i\\}|=l_i$, for $i\\in [1,2]$.\n\n\n\n\\subsection{Basic Operations}\n\nIn this section we review the main results related with state\ncomplexity (and nondeterministic state complexity) of some basic\noperations on regular languages: Boolean operations (mainly union,\nintersection, and complement), catenation, star (and plus), and\nreversal. For some  classes of languages, left and right quotients \nare also\nconsidered. Because their particular characteristics, that were\nalready pointed out in Section~\\ref{sec:scnsc}, for each operation the\nlanguages are divided into \\emph{regular} ($k\\geq 2$\nand infinite), \\emph{finite} ($k\\geq 2$), \\emph{unary } (infinite) and\n\\emph{finite unary}. Some other subregular languages are considered in\nSection~\\ref{sec:subregularlanguages}. Whenever known, results on the range of\ncomplexities that can be reached for each operation are also\npresented. This extension of the notion of magic number to operational state complexity is now an active topic of research.\n\nThere are some other survey papers that partially review the results here\npresented and that were a reference to our\npresentation~\\cite{yu97:_handb_formal_languag,yu01:_state_compl_of_regul_languag,yu02:_state_compl_of_finit_and,hromkovic02:_descr_compl_of_finit_autom,yu05:_state_compl,salomaa07:_descr_compl_of_nondet_finit_autom,holzer09:_nondet_finit_autom_recen_resul,brzozowski09:_quotien_compl_of_regul_languag,holzer09:_descr_and_comput_compl_of_finit_autom,holzer11:_descr_and_comput_compl_of}.\n\n\\subsubsection{General Regular Languages}\n\\label{sec:generalregularbop}\nTable~\\ref{tab:scnscregular} summarizes the results for general\nregular languages. The (fifth) third column contains the smallest alphabet\nsize of the witness languages for the (nondeterministic) state\ncomplexity given in the (fourth) second column, respectively.  Columns\nwith this kind of information also appear in several tables to\nfollow.\n \n\\begin{table}[htbp]\n\\begin{tabular}{|l||c|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Regular}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{$|\\Sigma|$}&\\multicolumn{1}{c}{nsc}&\\multicolumn{1}{c|}{$|\\Sigma|$}\\\\\\hline\\hline\n$L_1\\cup L_2$&$mn$&2&$m+n+1$&2\\\\\\hline\n$L_1\\cap L_2$&$mn$&2&$mn$&2\\\\\\hline\n\\mbox{$\\comp{L}$}&$m$&1&$2^{m}$&2\\\\\\hline\n  $(L_1- L_2)$&$mn$ &2&&\\\\\\hline\n $(L_1\\oplus L_2)$&$mn$&2&&\\\\\\hline\n\\multirow{2}{*}{$L_1L_2$}&$m2^n-f_12^{n-1}$, if $\\ n>1$&2&\\multirow{2}{*}{$m+n$}&\\multirow{2}{*}{2}\\\\\\cline{2-3}\n&$m$, if $n=1$& 1&&\\\\\\hline\n\\multirow{3}{*}{$L^\\star$}&\n$2^{m-1}+2^{m-l-1}$, if $m>1,\\, l>0$&2&\\multirow{3}{*}{$m+1$}&\\multirow{3}{*}{2}\\\\\\cline{2-3}\n&$m$, if $m>1,\\, l=0$&1&&\\\\\\cline{2-3}\n\n&$m+1$, if $m=1$&1&&\\\\\\hline\n\n$L^{+}$&$2^{m-1}+2^{m-l-1}-1$&2&$m$&2\\\\\\hline\n$L^{R}$&$2^{m}$&2&$m+1$&2\\\\\\hline\n$L_2\\setminus L_1$&$2^m-1$&2&&\\\\\\hline\n$L_1\\,\/\\,L_2$&$m$&1&&\\\\\\hline\n$w^{-1} L$&$m$&$1$&$O(m+1)$&\\\\\\hline\n$Lw^{-1}$&$m$&1&$m$&1\\\\\\hline\n\\end{tabular}\n  \\centering\n  \\caption{\\small{State complexity and nondeterministic state complexity for basic operations on regular languages}}\\label{tab:scnscregular}\n\\end{table}\n\nIn 1994, Yu \\emph{et al.}~\\cite{yu94:_state_compl_of_some_basic}\nstudied the state complexity of catenation, star, reversal, union,\nintersection, and  left and right quotients.  They also studied the\nstate complexity of some operations for unary languages.  More than two decades before, in\n1970, Maslov~\\cite{maslov70:_estim_of_number_of_states} had presented\nsome estimates for union, catenation, and star. Although Maslov\nconsidered possible incomplete DFAs\\xspace, and the paper has some\nincorrections, the binary languages presented are tight witnesses for\nthe upper bounds for that three\noperations~\\cite{brzozowski09:_quotien_compl_of_regul_languag}. Rabin and\n Scott~\\cite{rabin59:_finit_autom_and_their_decis_probl}\nindicated the upper bound $mn$ for the intersection (that also applies\nto union). Maslov and Yu \\emph{et al.} gave similar witnesses of\ntightness, both for union and intersection. The families of languages\ngiven by Yu \\emph{et al.} for intersection are $\\{x\\in\\{a,b\\} \\mid\n\\#_a(x) = 0\\pmod{m}\\}$ and $\\{x\\in\\{a,b\\} \\mid \\#_b(x)= 0\n\\pmod{n}\\}$. Their complements are witnesses for\nunion. Hricko~\\emph{et al.}~\\cite{hricko05:_union_and_inter_of_regul}\nshowed that for any integers $m\\geq 2$, $n\\geq 2$, and $\\alpha\\in\n[1,mn]$, there exist binary languages $L_1$ and $L_2$ such that\n$sc(L_1) = m$, $sc(L_2) = n$, and $sc(L_1\\cup L_2) = \\alpha$. Thus, there are no magic numbers for the union. The same\nholds for intersection.\n\nComplementation for DFAs\\xspace is trivial (one has only to exchange the final\nstates) and thus, the state complexity of the complement is the same one of the \noriginal language, i.e., $sc(\\comp{L})=sc(L)$. For other Boolean\noperations (set difference, symmetric difference, exclusive\ndisjunction, etc.) the state complexity can be obtained by expressing\nthem as a function of union, intersection and\ncomplement~\\cite{brzozowski09:_quotien_compl_of_regul_languag}.\n\n\nFor catenation, Yu \\emph{et al.} gave the upper bounds\n$m2^n-f_12^{n-1}$, if $m\\geq 1, n\\geq 2$; and $m$, if $m\\geq 1,\nn=1$. They presented binary languages tight bound witnesses for $m\\geq\n1,\\; n=1$ and $m=1,\\; n\\geq2$, but ternary languages tight bound\nwitnesses for $m> 1,\\; n\\geq 2$. However, the bound is tight for the\nfollowing binary language families presented by Maslov:\n$\\{w\\in\\{a,b\\}^\\star\\mid \\#_a(w)=(m-1)\\pmod{m}\\}$ and $L((a^\\star\nb)^{n-2}(a+b)(b+a(a+b))^\\star)$, for all $m,n\\geq 2$ and\n$f_1=1$. Other families of binary languages for which the catenation\nachieves the upper bound were presented by\nJir\\'askov\\'a~\\cite{jiraskova05:_state_compl_of_some_operat}.  Concerning the possible existence of magic numbers, the\nsame\nauthor~\\cite{jiraskova09:_concat_of_regul_languag_and_descr_compl,jiraskova11:_concat_of_regul_languag_and_descr_compl}\nproved that, for all $m$, $n$ and $\\alpha$ such that either $n=1$ and\n$\\alpha\\in [1,m]$, or $n\\geq 2 $ and $\\alpha\\in[1,m2^n-2^{n-1}]$,\nthere exist languages $L_1$ and $L_2$ with $sc(L_1)=m$ and\n$sc(L_2)=n$, defined over a growing alphabet, such that\n$sc(L_1L_2)=\\alpha$. Moreover, Jir\\'asek \\emph{et\n  al.}~\\cite{jirasek05:_state_compl_of_concat_and} showed that the\nupper bound $m2^n - f_12^{n-1}$ on the catenation of two languages\n$L_1$ and $L_2$, with $sc(L_1)=m\\geq 2$ and $sc(L_2)=n\\geq 2$\nrespectively, are tight for any integer $f_1$ with $f_1\\in\n[1,m-1]$. The witness language families are binary and accepted by the\nDFAs\\xspace presented in Figure~\\ref{fig:witnesscatenationstar}.\n\n \\begin{figure}[htb]\n   \\centering\n   \\begin{tabular}[t]{lc}\n    {\\small(i)}& \\raisebox{-.5\\height}{\\includegraphics[width=10cm]{diag4}}\n\t\t\\ignore{\\SmallPicture{\\VCDraw{\n             \\begin{VCPicture}{(-2,-2)(20,2)}\n               \\State[0]{(0,0)}{0}\\Initial{0}\n               \\State[1]{(2.5,0)}{1}\n               \\SetStateLineStyle{none}\n               \\State[\\cdots]{(5.5,0)}{2}\n               \\State[\\cdots]{(11.5,0)}{mf1}\n               \\SetStateLineStyle{solid}\n               \\FinalStateVar[m-f_1]{(8.5,0)}{mf}\n               \\FinalStateVar[m-2]{(14.5,0)}{m2}\n               \\FinalStateVar[m-1]{(18,0)}{m1}\n               \\LoopN{0}{b}\n               \\LoopN{1}{b}\n               \\LoopN{mf}{b}\n               \\LoopN{m2}{b}\n               \\LoopN{m1}{b}\n               \\EdgeL{0}{1}{a}\n               \\EdgeL{1}{2}{a}\n               \\EdgeL{2}{mf}{a}\n               \\EdgeL{mf}{mf1}{a}\n               \\EdgeL{mf1}{m2}{a}\n               \\EdgeL{m2}{m1}{a}\n               \\VArcL{arcangle=10}{m1}{0}{a}\n             \\end{VCPicture}\n           }\n         }}\n    \\\\{\\small(ii)}& \\raisebox{-.5\\height}{\\includegraphics[width=7.5cm]{diag5}}\n\t\t\\ignore{\\SmallPicture{\\VCDraw{\n             \\begin{VCPicture}{(-1,-2)(14,3)}\n               \\State[0]{(0,0)}{0}\\Initial{0}\n               \\State[1]{(3,0)}{1}\n               \\SetStateLineStyle{none}\n               \\State[\\cdots]{(6,0)}{2}\n               \\SetStateLineStyle{solid}\n               \\StateVar[m-2]{(9,0)}{m2}\n               \\FinalStateVar[m-1]{(13,0)}{m1}\n               \\LoopN{0}{b}\\EdgeL{0}{1}{a}\n               \\EdgeL{1}{2}{a,b}\n               \\EdgeL{2}{m2}{a,b}\n               \\EdgeL{m2}{m1}{a,b}\n               \\ArcL{m1}{0}{a,b}\n             \\end{VCPicture}\n           }\n         }}\n\n   \\end{tabular}\n   \\caption{\\small{Witness DFAs\\xspace for all range of state complexities of the catenation}}\n   \\label{fig:witnesscatenationstar}\n \\end{figure}\n\n\n The state complexity for the star on a regular language $L$ was\n studied by Yu \\emph{et al.}. A lower bound of $2^{m-1}$ was presented\n before, by Ravikumar and\n Ibarra~\\cite{ravikumar89:_relat_type_of_ambig_of,ravikumar90:_some_applic_of_techn_of}.\n If $sc(L)=1$ then either $L=\\Sigma^\\star$, and $sc(L^\\star)=1$, or\n $L=\\emptyset$, and $sc(L^\\star)=2$. If $sc(L)=m>1$, but $l=0$,\n i.e., the minimal DFA\\xspace accepting $L$ has the initial state as the\n only final state, then $sc(L^\\star)=m$, as $L=L^\\star$. Finally, if\n $sc(L)=m>1$, and $l>0$, then $sc(L^\\star)\\leq\n 2^{m-1}+2^{m-l-1}$. The upper bound $2^{m-1}+2^{m-2}$ is achieved\n for the language $\\{w\\in \\{a,b\\}^\\star\\mid \\#_a(w) \\text{ is odd}\\}$, if\n $m=2$; if $m> 2$, for the family of binary languages accepted by\n the DFAs\\xspace\\label{starwitness} presented in\n Figure~\\ref{fig:witnesscatenationstar}:(ii).\nWe note that although the upper bound given by Maslov is not correct\n($\\frac{3}{4}2^{m}-1$ instead of $\\frac{3}{4}2^{m}$), the family of\nlanguages he presented are witnesses for the above bound (for $m>2$). Those\nlanguages are accepted by the  DFAs\\xspace presented in Figure~\\ref{fig:maslovstar}.\n\n\n\\begin{figure}[htb]\n  \\centering\n\n\\includegraphics[width=8cm]{diag7}\\ignore{\\SmallPicture{\\VCDraw{\n      \\begin{VCPicture}{(-1,-2.5)(14,3)}\n        \\State[0]{(0,0)}{0}\\Initial{0}\n        \\State[1]{(3,0)}{1}\n        \\SetStateLineStyle{none}\n        \\State[\\cdots]{(6,0)}{2}\n        \\SetStateLineStyle{solid}\n        \\StateVar[m-2]{(9,0)}{m2}\n        \\FinalStateVar[m-1]{(13,0)}{m1}\n        \\LoopN{0}{b}\\ArcL{0}{1}{a}\n        \\ArcL{1}{2}{a}\\ArcL{1}{0}{b}\n        \\ArcL{2}{1}{b}\\ArcL{2}{m2}{a}\n        \\ArcL{m2}{m1}{a}\\ArcL{m2}{2}{b}\n        \\VArcL{arcangle=30}{m1}{0}{a}\\ArcL[.5]{m1}{m2}{b}\n      \\end{VCPicture}\n     }\n   }}\n\n  \\caption{\\small{Maslov's witness DFAs\\xspace for the state complexity of the\n    star}}\\label{fig:maslovstar}\n\\end{figure}\nJir\\'askov\\'a~\\cite{jiraskova08:_state_compl_of_compl_stars} proved that for all integers $m$ and $\\alpha$ with either $m=1$ and $\\alpha \\in\n[1,2]$, or $m \\geq 2$ and $\\alpha \\in [1, 2^{m-1} +2^{m-2}]$, there\nexists a language $L$ over an alphabet of size $2^m$ such that\n$sc(L) = m$ and $sc(L^\\star) = \\alpha$. This result was improved by~Jir\\'askov\\'a \\emph{et al.}~\\cite{jiraskova14:_kleen_closur_regul_and_prefix_free_languag} by using an alphabet of size atmost $2m$. Again, no gaps or magic numbers exist for the Kleene star operation. \n\nThe state complexity for the plus on a regular language $L$ ($L^{+} = LL^\\star$) coincides with the one for star in the first two cases,\nbut for $m>1$ and $l>0$ one state is saved (as a new initial state is\nnot needed).\n\n\nIn 1966, Mirkin~\\cite{mirkin66:_dual_autom} pointed out that the\nreversal of the NFAs\\xspace given by Lupanov as an example of a tight bound\nfor determination (see\nFigure~\\ref{fig:nfadfaMooreLupanovMeyerFisher}:(ii)), were\ndeterministic. This yields that $2^m$ is a tight upper bound for the\nstate complexity of reversal of a (at least ternary) language $L$\nsuch that\n$sc(L)=m$. Leiss~\\cite{leiss85:_succin_repres_of_regul_languag}\nstudied also this problem and proved the tightness of the bound for\nanother family of ternary languages. Yu \\emph{et al.} presented also\n(independently) Lupanov example. Salomaa \\emph{et\n  al.}~\\cite{salomaa04:_state_compl_of_rever_of_regul_languag} studied\nseveral classes of languages where the upper bound is\nachieved. Nevertheless, a family of binary languages therein presented\nas meeting the upper bound for $m\\geq 5$ was later shown not to be\nso~\\cite{jiraskova10:_compl_in_prefix_free_regul_languag}. A family of\nbinary languages for which the upper bound for reversal is tight was\ngiven by Jir\\'askov\\'a and S\\v ebej~\\cite{regular10:_juraj_s_ebej,jiraskova11:_note_rever_of_binar_regul_languag} and their minimal\nDFAs\\xspace are represented in Figure~\\ref{fig:binarywitnessreversal}.\n\\begin{figure}[htb]\n  \\centering\n\n\n \\includegraphics[width=9cm]{diag6}\n \\ignore{\\SmallPicture{\\VCDraw{\n        \\begin{VCPicture}{(-2,-1)(18,2)}\n          \\State[0]{(0,0)}{0}\\Initial{0}\n          \\State[1]{(2.5,0)}{1}\n          \\State[2]{(5,0)}{2}\n          \\State[3]{(7.5,0)}{3}\n          \\State[3]{(10,0)}{4}\n          \\SetStateLineStyle{none}\n          \\State[\\cdots]{(12.5,0)}{5}\n          \\SetStateLineStyle{solid}\n          \\StateVar[m-2]{(15,0)}{m2}\n          \\FinalStateVar[m-1]{(18.5,0)}{m1}\n          \\LoopN{0}{b}\\EdgeL{0}{1}{a}\\ArcL{2}{0}{a} \\VArcR{arcangle=-30}{1}{0}{b}\n          \\EdgeL{1}{2}{a}\n          \\ArcL{2}{3}{b}         \\ArcL{3}{2}{b}\n          \\EdgeL{3}{4}{a}\\ArcL{m1}{3}{a}\n          \\LoopN{4}{b}\n          \\EdgeL{4}{5}{a}\n          \\EdgeL{5}{m2}{a}         \\LoopN{m2}{b}\n          \\EdgeL{m2}{m1}{a}\n          \\LoopN{m1}{b}\n        \\end{VCPicture}\n      }\n }}\n\n\n  \\caption{\\small{Witness DFAs\\xspace for the state complexity of the reversal}}\n  \\label{fig:binarywitnessreversal}\n\\end{figure}\n\nIn the paper cited\nabove~\\cite{jiraskova08:_state_compl_of_compl_stars}, Jir\\'askov\\'a shown that for all $m$ and $\\alpha$ with $2\\leq m\\leq \\alpha\\leq 2m$, there exists a binary languague $L$ such that $sc(L)=m$ and $sc(L^R)=\\alpha$. Allowing alphabets of size $2^m$ and $m\\geq 3$, the reversal operation has no magic numbers in the range~$[\\log m, 2^m]$. This result was improved by S\\v ebej~\\cite{sebej13:_rever_regul_languag_and_descr_compl} considering an alphabet of size $2m-2$.  S\\v ebej gives also some enhanced partial results for the binary case.\n\n\n\nYu \\emph{et al.} showed that the state complexity for the left\nquotient of a regular language $L_1$ by an arbitrary language $L_2$,\n$L_2\\setminus L_1$, is less or equal to $2^m-1$, with $sc(L_1)=m$, and\nthat this bound is tight for the family of binary languages given in\nFigure~\\ref{fig:witnesscatenationstar}:(ii)\nand considering $L_2=\\Sigma^\\star$. In\n1971, Conway~\\cite{conway71:_regul_algeb_and_finit_machin}\nhad already stated that if $L_2$ is a regular language then\n$sc(L_2\\setminus L_1)\\leq 2^m$. For the right quotient of a regular\nlanguage $L_1$ by an arbitrary language $L_2$ one has $sc(L_1\/L_2)\\leq\nm$. The minimal DFA\\xspace accepting $L_1\/L_2$ coincides with the one for\n$L_1$, except that the set of final states is the set of states $q\\in\nQ_1$ such that there exists a word of $w\\in L_2$ such that\n$\\delta_1(q,w)\\in F_1$. The bound is tight for $L_2=\\{\\varepsilon\\}$.\nFor the left and the right quotients of a regular language $L$ by a\nword $w\\in \\Sigma^\\star$ it is then easy to see that\n$sc(w^{-1}L)=sc(Lw^{-1})\\leq m$. As a family of languages for which\nthe upper bound is tight consider\n$(a^m)^\\star$\nand\n$w\\in\n\\{a\\}^\\star$~\\cite{ellul03:_descr_compl_measur_of_regul_languag}.\n\n\nThe state complexity of basic operations on NFAs\\xspace was first studied\nby Holzer and Kutrib~\\cite{holzer03:_state_compl_of_basic_operat}, and\nalso by Ellul~\\cite{ellul03:_descr_compl_measur_of_regul_languag}.  We\nnote that for state complexity purposes it is tantamount to consider\nNFAs\\xspace with or without $\\varepsilon$-transitions. NFAs\\xspace are considered\nwith only one initial state and trimmed, i.e., all states are\naccessible from the initial state and from all states a final state is\nreached.\n\nFor union, only a new initial state with $\\varepsilon$ transitions for\neach of the operands initial states is needed, thus $sc(L_1\\cup\nL_2)\\leq m+n+1$. To see that the upper bound is tight, consider the\nfamilies $(a^m)^\\star$ and $(b^n)^\\star$ over a binary\nalphabet. For intersection, a product construction is needed.\n\nThe nondeterministic state complexity of the complementation is,\ntrivially, at most $2^m$. That this upper bound is tight even for\nbinary languages was proved by\nJir\\'askov\\'a~\\cite{jiraskova05:_state_compl_of_some_operat}, using a\n\\emph{fooling-set lower-bound\n  technique}~\\cite{birget91:_inter_of_regul_languag_and_state_compl,glaister96:_lower_bound_techn_for_size,hromkovic97:_commun_compl_and_paral_comput}.\nThose languages are accepted by the NFAs\\xspace presented in\nFigure~\\ref{fig:witnessnfas} (for $m>2$).\n\\begin{figure}[htb]\n  \\centering\n\\includegraphics[width=7cm]{diag8}\n\\ignore{\\SmallPicture{\\VCDraw{\n        \\begin{VCPicture}{(-2,-2)(14,2)}\n          \\State[0]{(0,0)}{0}\\Initial{0}\n          \\State[1]{(3,0)}{1}\n          \\SetStateLineStyle{none}\n          \\State[\\cdots]{(6,0)}{2}\n          \\SetStateLineStyle{solid}\n          \\StateVar[m-2]{(9,0)}{m2}\n          \\FinalStateVar[m-1]{(13,0)}{m1}\n          \\LoopN{0}{a}\n          \\EdgeL[.6]{0}{1}{a,b}\n          \\EdgeL[.6]{1}{2}{a,b}\n          \\EdgeL{2}{m2}{a,b}\n          \\EdgeL{m2}{m1}{a,b}\n          \\ArcL[.1]{1}{0}{a}\n          \\VArcL{arcangle=20}{m2}{0}{a}\n          \\VArcR[.8]{arcangle=-25}{m1}{m2}{a}\n          \\VArcR[.8]{arcangle=-25}{m1}{1}{a}\n          \\VArcR[.7]{arcangle=-25}{m1}{0}{a}\n        \\end{VCPicture}\n      }\n    }}\n\n  \\caption{\\small{Witness NFAs\\xspace for the nondeterministic state\n      complexity of complementation}}\n  \\label{fig:witnessnfas}\n\\end{figure}\n\nSee Holzer and Kutrib~\\cite{holzer09:_nondet_finit_autom_recen_resul}\nfor other witness languages.  Using the same techniques, Jir\\'askov\\'a\nand Szabari~\\cite{jirasek05:_state_compl_of_concat_and} proved that  for all integers $m\\geq 1$\n  and $\\alpha\\in [\\log m, 2^m]$, there exists a language $L$  over an alphabet of exponential\ngrowing size, such\n that $nsc(L) = m$ and $nsc(\\comp{L}) = \\alpha$.  This result  was improved to a five-symbol alphabet by\nJir\\'askov\\'a~\\cite{jiraskova08:_state_compl_of_compl_stars}. \n\n\nMera and Pighizzini~\\cite{mera05:_compl_unary_nondet_autom} proved a\nrelated \\emph{best case} result, i.e., for every $m\\geq 2$ there\nexists a language $L$ such that $nsc(L)=m$, $nsc(\\comp{L})\\leq\nm+1$ and $sc(L)=sc(\\comp{L})=2^m$. However, as we will see below,\nthis result does not hold if unary languages are considered.\n \nThe upper bound for the nondeterministic state complexity of\ncatenation is $m+n$ and this bound can be reached considering the\nwitness binary languages given for\nunion\\label{witnessnsccatenation}. All the values $\\alpha\\in [1,m+n]$\ncan be obtained as nondeterministic state complexity of catenation of\nunary\nlanguages~\\cite{jiraskova09:_concat_of_regul_languag_and_descr_compl}.\n\nFor the plus of a regular language $L$, we have $nsc(L^+)\\leq\nnsc(L)=m$: an NFA\\xspace accepting $L^+$ coincides with one accepting\n$L$ except that each final state has also the transitions to the\ninitial state. In the case of the star, one more state can be needed\n(if $L$ does not accept the empty word), i.e., $sc(L^\\star)\\leq\nm+1$.  Witness languages of the tightness of these bounds are $\\{w\\in\n\\{a,b\\}^\\star\\mid \\#_a(w)=(m-1) \\pmod{m}\\}$. All range of values $\\alpha\\in\n[1,m+1]$ can be reached for the nondeterministic state complexity of\nthe star of binary\nlanguages~\\cite{jiraskova08:_state_compl_of_compl_stars}.\n\nFor the reversal, at most one more state will be\nneeded, so $nsc(L^R)\\leq m+1$. Witness ternary languages were\npresented by Holzer and Kutrib, but the bound is tight even for the family\nof binary languages $(m>1)$ which minimal NFAs\\xspace are presented in\nFigure~\\ref{fig:jirasreveralnfa}~\\cite{jiraskova05:_state_compl_of_some_operat}.\nIf $nsc(L)=m\\geq 3$ the possible values for  $nsc(L^R)$ are $m-1$, $m$ or\n$m+1$~\\cite{jiraskova08:_state_compl_of_compl_stars}. The first value\nis reached by the reversals of the above binary languages and the\nsecond considering the languages $\\{w \\in \\{a,b\\}^\\star \\mid \\; |w|=0 \\pmod{m}\\}$.\n\n\\begin{figure}[htb]\n  \\begin{center}\n\n\\includegraphics[width=7cm]{diag9}\n\\ignore{\\SmallPicture{\\VCDraw{\n        \\begin{VCPicture}{(-2,-2)(12,1)}\n          \\FinalState[0]{(0,0)}{0}\\Initial{0}\n          \\FinalState[1]{(3,0)}{1}\n          \\FinalStateVar[m-2]{(8,0)}{m2}\n          \\SetStateLineStyle{none}\n          \\State[\\cdots]{(5,0)}{2}\n          \\SetStateLineStyle{solid}\n          \\FinalStateVar[m-1]{(12,0)}{m1}\n          \\EdgeL{0}{1}{a}\n          \\EdgeL{1}{2}{a}\n          \\EdgeL{2}{m2}{a}\n          \\EdgeL{m2}{m1}{a}\n          \\VArcL{arcangle=16}{m1}{0}{b}\n        \\end{VCPicture}\n      }\n     }}\n\n    \\caption{\\small Witness NFAs\\xspace for the nondeterministic state complexity\n      of reversal}\n  \\label{fig:jirasreveralnfa}\n  \\end{center}\n\\end{figure}\n\nThe nondeterministic state complexity of left and right quotients by a\nword were studied by\nEllul~\\cite{ellul03:_descr_compl_measur_of_regul_languag}. Given a\nminimal NFA\\xspace $A=(Q,\\Sigma,\\delta,q_0,F)$ accepting $L$, an NFA\\xspace $C$\naccepting $Lw^{-1}$, for $w\\in \\Sigma$, coincides with $A$ except\nthat the set of final states is $\\{q\\in Q\\mid \\delta(q,w)\\cap\nF\\not=\\emptyset\\}$. Thus $nsc(Lw^{-1})\\leq nsc(L)$.  The witness\nlanguages used for the state complexity of right quotient show that\nthe bound is tight. An upper bound for $nsc(w^{-1}L)$ can be obtained\nby considering an NFA\\xspace $C$ with one new initial state $q_0'$ and\n$\\varepsilon$-transitions from $q_0'$ to each state of $A$ reached\nwhen inputing $w$.\n\n\\paragraph{Universal Witnesses}\n\\label{sec:universalwitnesses}\nBrzozowski~\\cite{brzozowski12:_in_searc_of_most_compl_regul_languag,brzozowski13:_in_searc_of_most_compl_regul_languag}\nidentified a ternary family of languages $U_m(a,b,c)$ which provides witnesses for the state\ncomplexity of all operations considered in the previous section.  The\nfamily, presented in Figure~\\ref{fig:universalwitness}, fulfills also\nother conditions that, according to the same author, should be verified by\n\\emph{the most difficult (regular) languages}. For a language $L_m$\nthe suggested conditions are:\n\\begin{enumerate}[(1)]\n\\item \\label{u:1} The state complexity should be $m$.\n\\item  \\label{u:2} The state complexity of each quotient of $L_m$ should be $m$.\n\\item  \\label{u:3} The number of atoms of $L_m$ should be $2^m$. An atom of a\n  regular language with quotients $K_0,\\ldots,K_{m-1}$ is a non-empty\n  intersection of the form $\\widetilde{K}_0\\cap \\cdots \\cap \\widetilde{K}_{m-1}$, \nwhere $\\widetilde{K}_i$ is either $K_i$ or $\\overline{K}_i$. Thus the\nnumber of atoms is bounded from above by $2^m$, and it was proved by Brzozowski \\textit{et al.}~\\cite{brzozowski11:_theor_of_atomat,brzozowski14:_theor_of_atomat} that this bound is\ntight\\footnote{We also notice that the number of atoms of a language $L$ is equal to the state complexity of $L^{R}$.}. Every quotient of $L_m$  is a union of atoms.\n\\item  \\label{u:4}The state complexity of each atom of $L_m$ should be maximal. \nIt was shown~\\cite{brzozowski12:_quotien_compl_of_atoms_of_regul_languag} that the  complexity of the atoms with 0 or $m$ complemented quotients is bounded from above by $2^m-1$, and the complexity of any atom with $r$ complemented quotients, where $1\\le r\\le m-1$, by \n\\begin{equation*}\nf(m,r)=1 + \\sum_{k=1}^{r} \\sum_{h=k+1}^{m-r+k} \\binom{m}{h}\\binom{h}{k}.\n\\end{equation*}\n\\item  \\label{u:5} The syntactic semigroup  of $L_m$ should have cardinality \n  $m^m$, which is well known to be a tight upper\n  bound~\\cite{maslov70:_estim_of_number_of_states}. This measure,\n  which is called the \\emph{syntactic complexity} of a language, has been\n  recently studied for many classes of subregular languages~\\cite{\n  holzer04:_deter_finit_autom_and_syntac_monoid_size,\n  krawetz05:_state_compl_and_monoid_of,\n  brzozowski12:_syntac_compl_of_prefix_suffix,\n  brzozowski11:_syntac_compl_of_ideal_and_closed_languag,\n  brzozowski12:_syntac_compl_of_some_class,\n  brzozowski14:_syntactic_comp_trivial,  brzozowski14:_bounds_syntactic_compl}.\n\\end{enumerate}\nThe following result~\\cite{brzozowski12:_in_searc_of_most_compl_regul_languag, brzozowski13:_in_searc_of_most_compl_regul_languag} can be considered a milestone in the operational state complexity for regular languages,  where $U_m$ is depicted in Figure \\ref{fig:universalwitness}:\n\n\\begin{quotation}\n  \\emph{$(U_m(a,b,c)\\mid m\\geq 3)$ meets conditions \\ref{u:1}--\\ref{u:5} and\n  is a witness for the reversal and the star. The families   $(U_m(a,b,c)\\mid m\\geq 3)$ and\n    $(U_n(b,a,c)\\mid n\\geq 3)$ are witnesses for the Boolean\n    operations, whereas $(U_m(a,b,c)\\mid m\\geq 3)$ and\n    $(U_n(a,b,c)\\mid n\\geq 3)$ are witnesses for catenation.}\n\\end{quotation}\n\n\nVariants of the universal witness were also given for several combined\noperations. The question of whether there are universal witnesses for other\noperations,  classes of subregular languages or other complexity\nmeasures is an open problem (see~\\cite{brzozowski14:_most_complex_right}). However, when searching for witnesses for\na given upper bound, to ensure that the above conditions (or some of\nthem) are verified, can be a good starting point. Moreover, the study of properties that may enforce  (some of) the conditions (\\ref{u:1}) -- \n (\\ref{u:5}) is fundamental for a better understanding of the operational state complexity \\cite{brzozowski14:_maxim_atomic_languag}.\n\n\\begin{figure}[htb]\n  \\begin{center}\n\\includegraphics[width=8.5cm]{diag10}\\ignore{\\SmallPicture{\\VCDraw{\n      \\begin{VCPicture}{(-2,-4)(15,2)}\n      \\State[0]{(0,0)}{0}\\Initial{0}\n      \\State[1]{(3,0)}{1}\n      \\State[2]{(6,0)}{2}\n      \\SetStateLineStyle{none}\n      \\State[\\cdots]{(9,0)}{3}\n      \\SetStateLineStyle{solid}\n      \\StateVar[m-2]{(12,0)}{m2}\n      \\FinalStateVar[m-1]{(16,0)}{m1}\n      \\EdgeR{0}{1}{a,b}\n      \\ArcR[.40]{1}{0}{b}\n      \\LoopN{0}{c}\n      \\LoopN{1}{c}\n      \\EdgeL{1}{2}{a}\n      \\LoopN{2}{b,c}\n      \\EdgeL{2}{3}{a}\n      \\EdgeL{m2}{m1}{a}\n      \\EdgeL{3}{m2}{a}\n      \\LoopN{m2}{b,c}\n      \\LoopN{m1}{b}\n      \\VArcR{arcangle=35}{m1}{0}{a,c}\n    \\end{VCPicture}\n      }\n     }}\n\n    \\caption{\\small Universal witness DFAs\\xspace,  $U_m(a,b,c)$.}\n\n  \\label{fig:universalwitness}\n  \\end{center}\n\\end{figure}\n\n\n\\subsubsection{Unary Regular Languages}\n\\label{sec:unaryregularbop}\nTable~\\ref{tab:scnscbounary} presents the main state complexity\nresults of the basic operations on unary languages. Given the\nconstraints on both DFAs\\xspace and NFAs\\xspace over a one symbol alphabet, and\nthe results presented in Section~\\ref{sec:scnsc}, the state complexity\nfor several operations on unary languages is much lower than what\nis predicted by the general results of state complexity. Some results on the average-case state complexity of operations on\nunary languages were presented by\nNicaud~\\cite{nicaud99:_averag_state_compl_of_operat_unary_autom,nicaud00:_du_compor_en_moyen_des}.\n\n\\begin{table}[htbp]\n  \\centering\n  \\begin{tabular}{|l||c|c|c|}\\hline\n    \\multicolumn{4}{|c|}{Unary Regular}\\\\\\hline\n    &sc&nsc&asc\\\\\\hline\\hline\n    $L_1\\cup L_2$&$\\sim mn$ &\n      $m+n+1$, if $m\\not=\\dot{n}$\n&$\\sim \\displaystyle{\\frac{3\\zeta(3)}{2\\pi^2}}mn$\n    \\\\\\hline\n    $L_1\\cap L_2$&$\\sim mn$ &$mn$, if $(m,n)=1$&\n\n$\\sim \\displaystyle{\\frac{3\\zeta(3)}{2\\pi^2}}mn$\n\\\\\\hline \n    $\\comp{L}$&$m$&$e^{\\Theta(\\sqrt{m\\log m})}$&\\\\   \\hline\n    $L_1L_2$&$\\sim mn$&\\begin{tabular}{c}$[m+n-1,m+n]$,\\\\ if $m,n>1$\n    \\end{tabular}\n    &$O(1)$, $n<P(m)$ \n\\\\\\hline\n    $L^\\star$&\n\\begin{tabular}{c}$(m-1)^2+1$, \\\\if $m>1$, $l>1$\n\\end{tabular}\n&$m+1$, if $m>2$\n&$O(1)$\n\\\\\\hline \n    $L^{+}$&$(m-1)^2$&$m$, if $m>2$\n&\\\\\\hline\n    $L^{R}$&$m$&$m$&\\\\\\hline\n    $w^{-1}L$&$m$&$m$&\\\\\\hline\n    $Lw^{-1}$&$m$&$m$&\\\\\\hline\n  \\end{tabular}\n  \\caption{\\small{State complexity ($sc$), nondeterministic state complexity ($nsc$) and\n      average state complexity ($asc$) of\n      basic operations on unary languages. The $\\sim$ symbol means that the complexities are asymptotically equal to the given values. The upper bounds of state\n      complexity for union, intersection and catenation  are exact if the greatest common divisor of $m$ and $n$,      $(m,n)$ is $1$. For the average state complexity of intersection and\n      union, $\\zeta(n)$ is the function $\\zeta$ of Riemman. \nFor the average state complexity of catenation, $n$ must\n      be bounded by a polynomial $P$ in $m$. \n    }\n  }\n  \\label{tab:scnscbounary}\n\\end{table}\nA DFA\\xspace that accepts a unary language is characterized by a noncyclic\npart (the tail) and a cyclic part (the loop).  A characterization and\nthe enumeration of minimal unary DFAs\\xspace was given by\nNicaud~\\cite{nicaud99:_averag_state_compl_of_operat_unary_autom}.\n\nThe state complexity of the reversal of a unary language $L$ is\ntrivially equal to the state complexity of $L$.  The state complexities of Boolean operations on unary languages coincide asymptotically with the ones on\n general regular\nlanguages. Yu~\\cite{yu01:_state_compl_of_regul_languag} shown that the\nbound was tight for union (and thus, for intersection) if $m$ and $n$\nare coprimes and the witness languages are $(a^m)^\\star$ and\n$(a^n)^\\star$. The state complexity of catenation and star was\nproved by Yu \\emph{et al.}~\\cite{yu94:_state_compl_of_some_basic} and\nthe tightness for the first was also shown for $m$ and $n$\ncoprimes. The witnesses for the catenation are $(a^m)^\\star a^{m-1}$ and $(a^n)^\\star a^{n-1}$. For the star, if $m=2$ a\nwitness is $(aa)^\\star$, and for each $m> 2$ a witness is $(a^m)^\\star\na^{m-1}$.  The state complexity when $m$ and $n$ are not necessarily\ncoprimes was studied by Pighizzini and\nShallit~\\cite{pighizzini01:_unary_languag_concat_and_its_state_compl,pighizzini02:_unary_languag_operat_state_compl}. In\nthis case, the tight bounds are given by the number of states in the\ntail and in the loop of the resulting automata.  The state complexity\nfor left and right quotient by a word on unary languages coincide\nwith the general case.\n\n Nicaud~\\cite{nicaud99:_averag_state_compl_of_operat_unary_autom,nicaud00:_du_compor_en_moyen_des}\n proved that the state complexity of union, intersection and\n catenation on two languages $L_1$ and $L_2$ is asymptotically\n equivalent to $mn$, where $m=sc(L_1)$ and $n=sc(L_2)$.  Let $D_n$  be  the set  of\n unary (complete and initially connected) DFAs\\xspace\n with $n$ states. The \\emph{average state complexity}\n (asc) of a\n binary operation $\\circ$ on regular languages is given by \n$$\\frac{\\displaystyle{\\sum_{A_1\\times A_2\\in D_m\\times D_n}}sc(L(A_1)\\circ\n  L(A_2))}{|D_m\\times D_n|}$$\n\n\\noindent This definition can be generalized to  operations with other \narities, other kinds of automata and classes of languages.\n\nAs shown in Table~\\ref{tab:scnscbounary}, the average state complexities \nof catenation and star  on unary languages are bounded by a\nconstant, and for intersection (and union)  note that \n$\\frac{3\\zeta(3)}{2\\pi^2}\\approx 0.1826907423$.\n Magical numbers for the star operation on unary languages was studied by \\v Cevorov\u00e1~\\cite{cevorova13}. Considering the gap between the worst-case upper bound, $n^2-2n+2$, and the average case (less than a constant), it is not a surprise that for every $n$ no more than $4$ complexities are attainable between $n^2-4n+6$ and the upper bound. In the same paper, the author also establishes a relation between this problem and the Frobenius problem.\n\nThe nondeterministic state complexity of basic operations on unary\nlanguages was studied by Holzer and\nKutrib~\\cite{holzer03:_unary_languag_operat_and_their}, and also by\nEllul~\\cite{ellul03:_descr_compl_measur_of_regul_languag}. For union\nand intersection, the upper bound coincides with the general\ncase. However, it was proved to be achievable for union if $m$ is not\na divisor or multiple of $n$. As in the deterministic case, the\nwitnesses for intersection are $(a^m)^\\star$ and $(a^n)^\\star$, if\n$m$ and $n$ are coprimes. The nondeterministic state complexity of the\ncomplementation is $O(F(m))$ (where $F$ is the Landau's function\nof equation (\\ref{eq:fm})),\nwhich is directly related with the state complexity of\ndetermination. Holzer and\nKutrib~\\cite{holzer03:_unary_languag_operat_and_their} proved that\nthis upper bound is tight in order of magnitude, i.e.,  for any integer $m > 1$ there exists a unary language $L$ such that\n $nsc(L)=m$ and $nsc(\\comp{L})=\\Omega(F(m))$.\nMoreover, Mera and Pighizzini~\\cite{mera05:_compl_unary_nondet_autom}\nhave shown that for each $m\\geq 1$ and unary language $L$, such that\n$nsc(L)=m$ and $sc(L)=sc(\\comp{L})=e^{O(\\sqrt{m\\log m})}$, then\n$nsc(\\comp{L})\\geq m$. The upper bound $m+n$ for the catenation of two unary languages is not\nknow to be tight. The known lower bound is $m+n-1$ achieved by the\ncatenation of  $\\{a^l\\mid l= (m-1) \\pmod{m}\\}$ and $\\{a^l\\mid l= (n-1)\n\\pmod{n}\\}$~\\cite{holzer03:_unary_languag_operat_and_their}.\nThe same languages can be used to show the tightness of the bound\n$m+1$ for the star (and the plus) operation. For the left and right quotients, notice that in the unary case\n$w^{-1}L=Lw^{-1}$, and the results for the general case apply.\n\n\n\\subsubsection{Finite Languages}\n\\label{sec:finitebop}\nFinite languages are an important subset of regular languages. They\nare accepted by complete DFAs\\xspace that are acyclic apart from a loop on\nthe \\emph{sink} (or dead) state, for all alphabetic symbols. Minimal\nDFAs\\xspace have also special graph properties that lead to a linear time\nminimization algorithm~\\cite{revuz92:_minim_of_acycl_deter_autom}, and where\nthe length of the longest word accepted by the language plays an important role. Table~\\ref{tab:scncsfiniteregular} shows that the (nondeterministic)\nstate complexity of operations on finite languages are, in general,\nlower than in the general case.\n\n\\begin{table} [htbp]\n  \\centering\n\\begin{tabular}{|l||c|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Finite}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&\\multicolumn{1}{c|}{$|\\Sigma|$}&\\multicolumn{1}{c}{nsc}&\\multicolumn{1}{c|}{$|\\Sigma|$}\\\\\\hline\\hline\n$L_1\\cup L_2$&$mn-(m+n)$&$f(m,n)$&$m+n-2$&2\\\\\\hline\n$L_1\\cap L_2$&$mn - 3(m + n) + 12$&$f(m,n)$&$mn$&2\\\\\\hline\n$\\comp{L}$&$m$&1&$\\Theta(k^{\\frac{m}{1+\\log k}})$&2\\\\\\hline\n\\multirow{2}{*}{$L_1L_2$}&$(m-n+3)2^{n-2}-1$, $m+1\\geq n$&$2$&\\multirow{2}{*}{$m+n-1$}&\\multirow{2}{*}{$2$}\\\\\\cline{2-3}\n&$m+n-2$, if $l_1=1$&$1$&&\\\\\\hline\n\\multirow{2}{*}{$L^\\star$}&\n$2^{m-3}+2^{m-l-2}$, $l\\geq 2$, $m\\geq 4$&3&\\multirow{2}{*}{$m-1$, $m>1$}&\\multirow{2}{*}{$1$}\\\\\\cline{2-3}\n&$m-1$, if $f=1$&$1$&&\\\\\\hline\n$L^{+}$&$m$&$1$&$m$, $m>1$&1\\\\\\hline\n$L^{R}$&$O(k^{\\frac{m}{1+\\log k}})$&2&$m$&2\\\\\\hline\n\\end{tabular}\n\\caption{\\small{State complexity and nondeterministic state complexity of\n  basic operations on finite languages}}\n\\label{tab:scncsfiniteregular}\n\\end{table}\n  C\\^ampeanu \\emph{et\n  al.}~\\cite{campeanu01:_state_compl_of_basic_operat_finit_languag}\npresented the first formal study of state complexity of operations on\nfinite languages. Yu~\\cite{yu01:_state_compl_of_regul_languag}\npresented upper bounds of $O(mn)$ for the union and the intersection.\nThe tight upper bounds were given by Han and\nSalomaa~\\cite{han08:_state_compl_of_union_and} using growing size\nalphabets. The upper bound for union and intersection cannot be\nreached with a fixed alphabet when $m$ and $n$ are arbitrarily large.\nC\\^ampeanu \\emph{et al.} gave tight upper bounds for catenation, star\nand reversal. For catenation the bound $(m-n+3)2^{n-2}-1$ is tight for\nbinary languages, if $m+1\\geq n> 2$. The DFAs\\xspace of the witness\nlanguages are presented in Figure~\\ref{fig:finiteintersection}.\n\n\\begin{figure}[htb]\n  \\centering\n\\begin{tabular}[c]{c}\n\n  \\raisebox{-.5\\height}{\\includegraphics[width=6cm]{diag11}}\n\t\\ignore{\\SmallPicture{\\VCDraw{\n        \\begin{VCPicture}{(-1,-1)(11,3)}\n          \\FinalState[0]{(0,0)}{0}\\Initial{0}\n          \\FinalState[1]{(2,0)}{1}\n          \\SetStateLineStyle{none}\n          \\State[\\cdots]{(4,0)}{2}\n          \\SetStateLineStyle{solid}\n          \\FinalStateVar[m-2]{(7,0)}{m2}\n          \\StateVar[m-1]{(11,0)}{m1}\n          \\EdgeL{0}{1}{a,b}\n          \\EdgeL{1}{2}{a,b}\n          \\EdgeL{2}{m2}{a,b}\n          \\EdgeL{m2}{m1}{a,b}\n          \\LoopN{m1}{a,b}\n         \\end{VCPicture}\n       }\n     }}\n  \\\\\\raisebox{-.5\\height}{\\includegraphics[width=6cm]{diag12}}\n\t\\ignore{\\SmallPicture{\\VCDraw{\n        \\begin{VCPicture}{(-1,-1)(11,3)}\n          \\State[0]{(0,0)}{0}\\Initial{0}\n          \\State[1]{(2,0)}{1}\n          \\SetStateLineStyle{none}\n          \\State[\\cdots]{(4,0)}{2}\n          \\SetStateLineStyle{solid}\n          \\FinalStateVar[m-2]{(7,0)}{m2}\n          \\StateVar[m-1]{(11,0)}{m1}\n          \\EdgeL{0}{1}{b}\\ArcR{0}{m1}{a}\n          \\EdgeL{1}{2}{a,b}\n          \\EdgeL{2}{m2}{a,b}\n          \\EdgeL{m2}{m1}{a,b}\n          \\LoopN{m1}{a,b}\n        \\end{VCPicture}\n      }\n    }}\n  \\end{tabular}    \n\\caption{\\small{Witness DFAs\\xspace for the state complexity  of catenation on\n  finite languages}}\n\\label{fig:finiteintersection}\n\\end{figure}\n\nFor star, C\\^ampeanu \\emph{et al.}  shown that the bound\n$2^{m-3}+2^{m-4}$ is tight for ternary languages.  The tight upper\nbound for the reversal of a finite binary language is\n$3\\cdot2^{p-1}-1$, if $m=2p$, and $2^{p-1}-1$ if $m=2p-1$.\n\nNondeterministic state complexity of basic operations on finite\nlanguages were studied by Holzer and\nKutrib~\\cite{holzer03:_state_compl_of_basic_operat}.  Minimal NFAs\\xspace\naccepting finite languages without the empty word can be assumed to\nhave only a final state (with no transitions); and if the empty word is in\nthe language, the initial state is also final.  Because there\nare no cycles, for the union of two finite languages three states can\nbe avoided: no new initial state is needed, and the initial states and\nthe final states can be merged. The upper bound $m+n-2$ is tight for\nthe languages ${a^{m-1}}$ and ${b^{n-1}}$, for $m,n\\geq 2$.  In the case\nof the intersection, the upper bound coincides with the general case,\nand it is tight for the binary languages $\\{w\\in\\{a,b\\}^\\star \\mid\n\\#_a(w)=0 \\pmod{m}\\}$ and $\\{w\\in\\{a,b\\}^\\star \\mid \\#_b(w) = 0\n\\pmod{n}\\}$. Considering the upper bound of determination for finite\nlanguages, the nondeterministic state complexity for complement is\nbounded by $O(k^{\\frac{m}{1+\\log k}})$. The lower bound\n$\\Omega(k^{\\frac{m}{2\\log k}})$ is reached for alphabets\n$\\Sigma=\\{a_1,\\ldots,a_k\\}$ of size $k\\geq 2$, and the languages\n$\\Sigma^ja_1\\Sigma^iy$, where $i\\geq 0$, $0 \\leq j \\leq i$, $y \\in\n\\Sigma\\setminus\\{a_1\\}$, and $m>2$. However, a tighter lower bound \ncan be achieved by the determination lower bound of $\\Omega(k^{\\frac{m}{1+\\log k}})$.\n For catenation of finite languages represented by NFAs\\xspace, one state\ncan be saved. Witness languages for the tightness of the bound $m+n-1$\ncan be the ones used for union. Two states are also saved for the\nstar, and for plus the nondeterministic state complexity coincides with\nthe one for the general case. Witness languages are $a^{m}$\nand $a^{m-1}$, respectively. NFAs\\xspace for the reversal are exponentially more succinct then DFAs\\xspace. In\nthe case of finite languages, and like other operations, one \nstate can be spared. Witness languages are $\\{a,b\\}^{m-1}$.\n\n\n\\subsubsection{Finite Unary Languages}\n\\label{sec:scnscfiniteunary}\nTable~\\ref{tab:scncsfiniteunary} summarizes the state complexity and\nnondeterministic state complexity results of basic operations on\nfinite unary\nlanguages~\\cite{campeanu01:_state_compl_of_basic_operat_finit_languag,yu01:_state_compl_of_regul_languag,holzer03:_unary_languag_operat_and_their}.\nState complexity of union, intersection and catenation on finite unary\nlanguages are linear, while they are quadratic for general unary\nlanguages. In this setting, nondeterminism is only relevant for the\nstar (and plus), as unary regular languages are obtained.  As \nalready stated, for a finite unary language $L$, one has $sc(L)\\leq\nnsc(L)+1$, and $sc(L)-2$ is the length of the longest word in the\nlanguage. If a operation preserves finiteness, for state complexity\nonly the longest words must be considered.\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|l||c|c|}\\hline\n\\multicolumn{3}{|c|}{Finite Unary}\\\\\\hline\n&sc&nsc\\\\\\hline\\hline\n$L_1\\cup L_2$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$\\\\\\hline\n\n$L_1\\cap L_2$&$\\min\\{m,n\\}$\n&$\\min\\{m,n\\}$\n\\\\\\hline\n$\\comp{L}$&$m$\n  &$m+1$\\\\\\hline\n  $(L_1- L_2)$&$m$ & \\\\\\hline \n  $(L_1\\oplus L_2)$&$\\max\\{m,n\\}$&\\\\\\hline\n$L_1L_2$&$m+n-2$&$m+n-1$\\\\\\hline\n$L^\\star$&\n\\begin{tabular}{c}\n$2$, if $m=3$\\\\\n$m-1$, if $f=1$\\\\\n$m^2-7m+13$, if $m>4$, $f\\geq 3$\n\\end{tabular}&\n$m-1$\\\\\\hline\n$L^{+}$&$m$&$m$\\\\\\hline\n$L^{R}$&$m$&$m$\\\\\\hline\n\\end{tabular}\n\\caption{\\small{State complexity and nondeterministic state complexity of\n  basic operations on finite unary languages}}\n\\label{tab:scncsfiniteunary}\n\\end{table}\n\n\n\n\n\\subsection{Other Regularity Preserving Operations}\n\\label{sec:otherpreservingregularityoperations}\n\nTable~\\ref{tab:scnscregularitypreservingoperations} presents the\nresults for the state complexity of some regularity preserving\noperations, that are detailed in the next paragraphs.\n\n\\paragraph{Proportional removals}\n\\label{sec:scnscpropotionalremovals}\nThe \\emph{proportional removals} preserving\nregularity were studied by\nHartmamis~\\cite{stearns63:_regul_preser_modif_of_regul_expres} and\nwere full characterized by\nSeiferas and McNaughton~\\cite{seiferas76:_regul_preser_relat}.\n For any binary relation $r \\subseteq \\mathbb{N} \\times \\mathbb{N} $ \nand any language $L \\subseteq \\Sigma^\\star$, let the language $P (r, L)$ be defined as\n$$P(r,L)=\\{x\\in\\Sigma^\\star\\mid \\exists y\\in\\Sigma^\\star \\text{ such\n  that } xy\\in L \\,\\wedge\\, r(|x|,|y|)  \\}.$$  \n\n\\noindent A relation $r$ is\n\\emph{regularity-preserving} if $P(r,L)$ is regular for every regular\nlanguage $L$. Seiferas and\nMcNaughton~\\cite{seiferas76:_regul_preser_relat} gave sufficient and\nnecessary conditions of regularity preservation in this context.  For the special case where $r$ is the identity, the correspondent\nlanguage is denoted by $\\frac{1}{2}(L)$.\nDomaratzki~\\cite{domaratzki02:_state_compl_of_propor_remov} proved\nthat for a regular language $L$,\n$sc(\\frac{1}{2}(L))=O(sc(L)F(sc(L)))$ (where $F$ is the Landau's\nfunction of equation (\\ref{eq:fm})) and this bound is tight for ternary languages. In the\ncase of $L$ be a unary language, one gets $sc(\\frac{1}{2}(L))=sc(L)$.\nFollowing Nicaud's work on average-case complexity, mentioned above,\nDomaratzki showed that the average state complexity of the\n$\\frac{1}{2}(\\cdot)$ operation on a $m$-state unary automaton is\nasymptotically equivalent to $\\frac{5}{8} m + c$, for some constant\n$c$. Domaratzki also studied the state complexity of polynomial removals.  Let $f \\in \\mathbb{Z}[x]$ be a strictly monotonic polynomial such\n   that $f(\\mathbb{N}) \\subset \\mathbb{N}$. Then, the relation $r_f =\n   \\{(n, f (n))\\mid n \\geq 0\\}$ preserves regularity, and $sc(P\n   (r_f,L)) \\leq O(sc(L)F(sc(L))).$\n\n\n\\noindent In 1970, Maslov~\\cite{maslov70:_estim_of_number_of_states} had already\nstudied the language $\\frac{p}{q}(L)$, i.e., a language $P(r,L)$ such\nthat $r$ is defined by $\\{(m,n)\\mid mq=pn \\}$ with $p, q\\in\n\\mathbb{N}$.  An open problem is to obtain the state complexity of\n$P(r,L)$ where $r$ belongs to the broader class of regularity preserving\nrelations studied by Seiferas and McNaughton. \n\nNondeterministic state complexity of polynomial removals was studied by Go\\v c \\emph{et al.}~\\cite{GocPS13}. The authors showed an $O(n^2)$ upper bound and a matching lower bound in the case where the polynomial is a sum of monomials and a constant, or when the polynomial has rational roots.\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||c|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Regular}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&$|\\Sigma|$&\\multicolumn{1}{c}{nsc}&$|\\Sigma|$\\\\\\hline\\hline\n\\multirow{2}{*}{$\\frac{1}{2}(L)$}& $me^{\\Theta(\\sqrt{m\\log m})}$&3&\\multirow{2}{*}{$O(m^2)$}&\\\\\\cline{2-3}\n&$m$&1&&\\\\\\hline\n\n\\multirow{2}{*}{$L^i$}&$\\Theta(m2^{(i-1)m})$&$6$&\\multirow{3}{*}{$im$}&\\multirow{3}{*}{$2$}\\\\\\cline{2-3}\n&$im-i+1$&$1$&&\\\\\\cline{1-3}\n$L^3$&$\\frac{6m-3}{8}\\scriptstyle{4^m-(m-1)2^m-m}$&$4$&&\\\\\\hline\n\\multirow{3}{*}{$\\shift{L}$}&$2^{m^2+m\\log m-O(m)}$&$4$&$1$, if $m=1$&\\multirow{2}{*}{$2$}\\\\\\cline{2-4}\n&$2^{\\Theta(m^2)}$& $2$,$3$&$2m^2+1$, if $m\\geq 2$&\\\\\\cline{2-5}\n&$m$&$1$&$m$&$1$\\\\\\hline\n$L_1\\shuffle L_2$&$O(2^{mn}-1)\n&$5$&$O(mn)\n&$5$\\\\\\hline\n\\multirow{2}{*}{$L_1\\odot_{\\bot}L_2$}&$m2^{n-1}-2^{n-2}$,&\\multirow{2}{*}{$4$}&\n\\multirow{2}{*}{$m+n$}&\\multirow{2}{*}{$2$}\\\\\n&if $m\\geq 3$, $n\\geq  4$&&&\\\\\\hline\n\\multicolumn{5}{|c|}{Unique Regular Operations}\\\\\\hline\n$L_1\\stackrel{\\circ}{\\cup}L_2$&$mn$&$2$&&\\\\\\hline\n$L_1\\circ L_2$&$O(m3^n-f_13^{n-1})$&&$\\geq 2^{O(h)}$&\\\\\\hline\n${L}^{\\circ 2}$&$m3^m-3^{m-1}$&$2$&&\\\\\\hline\n${L}^{\\circ}$&\n$\\begin{array}{c}\n  O(3^{m-1} + (f+2)3^{m-f-1}\\\\\n  - (2^{m-1} + 2^{m-f-1}-2))\n\\end{array}$\n&&&\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity and nondeterministic state complexity of\n  some regularity preserving operations: proportional removals for\n  the identity relation ($\\frac{1}{2}(L))$; power $L^i$ where\n  $i\\geq 2$; cyclic shift $\\shift{L}$; shuffle $L_1\\shuffle L_2$;\n orthogonal catenation\n  $L_1\\odot_{\\bot}L_2$; unique operations: for unique star $L^{\\circ}$,\n  $\\varepsilon\\notin L$; for the nondeterministic state complexity of $L_1\\circ L_2$,\n  the combined state complexity of $L_1$ and $L_2$ is $O(h)$, for $h\\geq 0$.}\n \\label{tab:scnscregularitypreservingoperations}\n\\end{table}\n\n\\paragraph{Power}\n\\label{sec:scnscpower}\n\nGiven a regular language $L$ and $i\\geq 2$, an upper bound of the\nstate complexity of the language $L^i$ is given by considering the state\ncomplexity of catenation. However, a tight upper bound is obtained if\nthis operation is studied individually.  Domaratzki and\nOkhotin~\\cite{domaratzki09:_state_compl_of_power} proved that\n$sc(L^i)=\\Theta(m2^{(i-1)m})$, for $i\\geq 2$.  \nThe bound is tight\nfor a family of languages over a six-symbol alphabet. In the case $i =\n3$, $sc(L^3)=\\frac{6m - 3}{8} 4^m - (m -1)2^m - m$, for $m\\geq 3$, and the\ntightness is witnessed by a family of languages over a four-symbol\nalphabet. For the square, i.e. if $i=2$, the upper bound is the one given by the state complexity of catenation, $sc(L^2)=m2^m -2^{m-1}$ and it is met by a language accepted by a $m$-state DFA with only one final state. In the case of multiple $l$ final states, the upper bound is $(m-l)2^m+l2^{m-1}$. \\v Cevorov\u00e1 \\emph{et al.}~\\cite{CevorovaJK14} proved that those upper bounds are tight in the ternary case for every $l\\in[1,m-2]$. The nondeterministic state complexity of $L^i$ is\nproved to be $im$. This bound is shown to be tight over a binary\nalphabet, for $m\\geq 2$.\nThe power of unary languages was studied by\nRampersad~\\cite{rampersad06:_state_compl_of_l_and_l_k}. If $L$ is a\nunary language with $sc(L)=m\\geq 2$, then $sc(L^i)=im-i+1$. For the square,  \\v Cevorov\u00e1 \\emph{et al.} showed that all the complexities in the range $[1,2m-1]$ can be\nattained for $m\\geq 5$.\n\n\\paragraph{Cyclic Shift}\n\\label{sec:scnsccyclicshift}\n\n\nThe \\emph{cyclic shift} of a language $L$ is defined as $\\shift{L} = \\{\nvu \\mid uv \\in L \\}$.\nMaslov~\\cite{maslov70:_estim_of_number_of_states} gave an upper bound\nof $(m2^m-2^{m-1})^m$ for the state complexity of cyclic shift and an\nasymptotic lower bound of $(m-3)^{m-3} \\cdot 2^{(m-3)^2}$, by\nconsidering languages over a growing alphabet (if complete DFAs\\xspace\nare considered). Jir\\'askov\\'a and\nOkhotin~\\cite{jiraskova08:_state_compl_of_cyclic_shift} reviewed and\nimproved Maslov results. Using a fixed four-symbol alphabet, they\nobtained a lower bound of $(m-1)! \\cdot 2^{(m-1)(m-2)}$, $m\\geq 3$,  which shows\nthat $sc(\\shift{L})=2^{m^2 +m\\log m - O(m)}$\nfor alphabets of size greater than $3$. For binary and ternary languages, they\nproved that the state complexity is $2^{\\Theta(m^2)}$. \nAs this function grows faster than the number\nof DFAs\\xspace for a given $m$, there must exist some \\emph{magic numbers}\nfor the  state complexity of the cyclic shift over languages of a fixed alphabet. \n\nThe nondeterministic state complexity of this operation was shown to\nbe $2^{m^2}+1$, for $m\\geq 2$, and the upper bound is tight for binary\nlanguages. Although the hardness of this operation on the\ndeterministic case, its  nondeterministic state complexity is\nrelatively low.  For a unary language $L$, as $\\shift{L}=L$, one\ngets $sc(\\shift{L})=nsc(\\shift{L})=sc(L)$.\n\n\n\n\\paragraph{Shuffle}\\label{sec:scnscshuffle}\n\nThe \\emph{shuffle} operation  of two words $w_1,w_2\\in \\Sigma^\\star$ is\ndefined by\n\\begin{eqnarray*}\n  w_1\\shuffle w_2&=&\\{u_1 v_1\\ldots u_mv_m\\mid\\\\\n&& u_i,v_i\\in\\Sigma^\\star,\\  i\n\\in[1,m],\\, w_1=u_1\\ldots u_m \\text{ and } w_2=v_1\\ldots v_m \\}.\n\\end{eqnarray*}\nThis operation is extended trivially to languages. If two regular\nlanguages are regular, their shuffle is also a regular language.\nC\\^ampeanu \\emph{et al.}~\\cite{campeanu02:_tight_lower_bound_for_state}\nshowed that the state complexity of the shuffle of two regular languages\n$L_1$ and $L_2$ is less or equal to $2^{mn}-1$.\nThey proved that this bound is tight for witness\nlanguages over a five symbols alphabet and if minimal incomplete\nDFAs\\xspace are considered (see Figure~\\ref{fig:shufflewitnesses}). Thus,\n$sc(L_1\\shuffle L_2)$ is at least $2^{(sc(L_1)-1)(sc(L_2)-1)}$.\n\\begin{figure}[ht]\n  \\centering\n\n\n  \\begin{tabular}[t]{c}\n    \\raisebox{-.5\\height}{\\includegraphics[width=6.5cm]{diag13}}\n\t\t\\ignore{\\SmallPicture{\\VCDraw{\n            \\begin{VCPicture}{(-2,-2)(11,2)}\n              \\FinalState[0]{(0,0)}{0}\\Initial{0}\n              \\State[1]{(3,0)}{1}\n              \\SetStateLineStyle{none}\n              \\State[\\cdots]{(6,0)}{4}\n              \\SetStateLineStyle{solid}\n              \\StateVar[m-1]{(10,0)}{m1}\n              \\LoopN{0}{d}\n              \\LoopN{1}{d,f}\n              \\LoopN{m1}{d,f}\n              \\EdgeL{0}{1}{a,c}\n              \\EdgeL{1}{4}{a,c}\n              \\EdgeL{4}{m1}{a,c}\n              \\ArcL{m1}{0}{a}\n            \\end{VCPicture}\n          }\n        }}\n    \\\\\\raisebox{-.5\\height}{\\includegraphics[width=6.5cm]{diag14}}\n\t\t\\ignore{\\SmallPicture{\\VCDraw{\n            \\begin{VCPicture}{(-2,-2)(11,2)}\n              \\FinalState[0]{(0,0)}{0}\\Initial{0}\n              \\State[1]{(3,0)}{1}\n              \\SetStateLineStyle{none}\n              \\State[\\cdots]{(6,0)}{4}\n              \\SetStateLineStyle{solid}\n              \\StateVar[m-1]{(10,0)}{m1}\n              \\LoopN{0}{c}\n              \\LoopN{1}{c,f}\n              \\LoopN{m1}{c,f}\n              \\EdgeL{0}{1}{b,d}\n              \\EdgeL{1}{4}{b,d}\n              \\EdgeL{4}{m1}{b,d}\n              \\ArcL{m1}{0}{b}\n             \\end{VCPicture}\n           }\n         }}\n\n  \\end{tabular}\n  \\caption{\\small Incomplete DFAs\\xspace for the tight upper bound of state\n    complexity of shuffle.}\n\\label{fig:shufflewitnesses}\n\\end{figure}\n\nVarious restrictions and generalizations of the shuffle operation have\nbeen studied. Mateescu \\emph{et al.}~\\cite{mateescu98:_shuff_trajec}\nintroduced the shuffle operation of two languages $L_1$ and $L_2$ on a\nset of trajectories $T\\subseteq \\{0,1\\}^\\star$, $L_1\\shuffle_T L_2$.\nWhen $L_1$, $L_2$, and $T$ are regular languages $L_1\\shuffle_T L_2$\nis a regular language. In particular, if $T=\\{0,1\\}^\\star$, then\n$L_1\\shuffle_T L_2=L_1\\shuffle L_2$; and if\n$T=\\{0\\}^\\star\\{1\\}^\\star$, then $L_1\\shuffle_T L_2=L_1L_2$.\nDomaratzki and\nSalomaa~\\cite{domaratzki04:_state_compl_of_shuff_trajeccd} studied the\nstate complexity of the shuffle on regular trajectories. In general,\n$sc(L_1\\shuffle_T L_2)\\leq 2^{nsc(L_1)nsc(L_2)nsc(T)}$. If $T$ belongs\nto special families of regular languages, tight bounds were also\npresented.\n\n\n\n\\paragraph{Orthogonal Catenation}\n\\label{sec:scnscorthogonalcatenation}\n\nA language $L$ is the \\emph{orthogonal catenation} of $L_1$ and $L_2$,\nand denoted by $L = L_1 \\odot_\\bot L_2$, if every word $w$ of $L$ can\nbe obtained in just one way as a catenation of a word of $L_1$ and a\nword of $L_2$. If catenation uniqueness is not verified for every word\nof $L$, orthogonal catenation of $L_1$ and $L_2$ is undefined,\notherwise $L_1$ and $L_2$ are \\emph{orthogonal}. Daley \\emph{et\n  al.}~\\cite{daley10:_orthog_concat}\nstudied the state complexity of orthogonal catenation and generalized\northogonality to other operations.  Although it is a restricted\noperation, its state complexity is only half of the one for the\ngeneral catenation, i.e., $m2^{n-1}-2^{n-2}$ for $m\\geq 3$ and $n\\geq\n4$. The tight bound was obtained for languages over a four-symbol\nalphabet.  Concerning nondeterministic state complexity, one has\n$nsc(L_1\\odot_\\bot L_2)=nsc(L_1)+nsc(L_2)$, which coincides with the\none for (general) catenation. Witness languages presented for the\ncatenation are orthogonal (see page \\pageref{witnessnsccatenation}),\nthus apply to orthogonal catenation.\n\n\n\n\\paragraph{Unique Regular Operations}\n\\label{sec:scnscuniqueoperations}\n\nSimilar to orthogonality is the concept of \\emph{unique operation}\nintroduced by Rampersad \\emph{et\n  al.}~\\cite{rampersad09:_state_compl_of_unique_ration_operat}. However,\ninstead of demanding that every pair of words of the operand languages\nlead to a distinct word on the resulting language, the language\nresulting from a \\emph{unique operation} only contains the words that\nare uniquely obtained through the given operation.  Rampersad \\emph{et\n  al.} studied\nseveral properties of unique operations and of their \\emph{poly}\ncounterpart (i.e. where each resulting word must be obtained in more\nthan one way), such as closure, ambiguity, and  membership and\nnon-emptiness decision problems. Results on\nstate complexity and nondeterministic state complexity were obtained\nfor \\emph{unique union} ($L_1\\stackrel{\\circ}{\\cup}L_2$),\n \\emph{unique catenation} ($L_1\\circ L_2$), \\emph{unique\n  square} ($L\\circ L={L}^{\\circ 2}$), and \\emph{unique star} (${L}^\\circ$).\nThe state complexity of $L_1\\stackrel{\\circ}{\\cup}L_2$ is\n$mn$, and witness binary languages are $\\{x\\in\\{a,b\\} \\mid\n\\#_a(x) = (m-1)\\pmod{m}\\}$ and $\\{x\\in\\{a,b\\} \\mid \\#_b(x)= (n-1)\n\\pmod{n}\\}$, for $m,n\\geq 3$ (that were also used by Maslov~\\cite{maslov70:_estim_of_number_of_states} for general union).\nFor unique catenation, $sc(L_1\\circ L_2)\\leq m3^n-f_13^{n-1}$ which is\nmuch higher than the one for general catenation. It is an open problem\nto know if this bound is tight, although  several examples, for specific values of\n$m$ and $n$, were presented.  However, for the unique square\n$sc({L}^{\\circ 2})=m3^m-3^{m-1}$, and the bound is\ntight for binary languages and $m\\geq 3$.  For the nondeterministic state\ncomplexity of unique catenation, a  exponential lower\nbound was provided.\nAn upper bound for the state complexity of the unique star is\n$3^{m-1} + (f+2)3^{m-f-1} - (2^{m-1} + 2^{m-f-1}-2)$. But,\nagain,  it is an open problem to know if this upper bound is tight.\n\n\\subsection{Other Subregular Languages}\n\\label{sec:subregularlanguages}\n\nBesides finite and unary languages, several other subregular languages\nare used in many applications and are now theoretically well studied.\nPrefix-free or suffix-free languages are examples of codes that are\nfundamental in coding\ntheory~\\cite{jurgensen97:_codes,berstel10:_codes_and_autom}. Prefix-closed,\nfactor-closed, or subword-closed languages were studied by several\nauthors~\\cite{haines69:_free_monoid_partial_order_by_embed,thierrin72:_convex_languag,luca90:_some_combin_proper_of_factor_languag,gill74:_multip_entry_finit_autom}. These\nlanguages belong to a boarder set of languages, the \\emph{convex\n  languages}, for which a general framework have\nbeen recently\naddressed by Ang and\nBrzozowski~\\cite{ang09:_languag_convex_with_respec_to} and Brzozowski\n\\emph{et al.}~\\cite{brzozowski09:_decis_probl_for_convex_languag}. A\ndetailed survey on complexity topics was presented\nby Brzozowski~\\cite{brzozowski10:_compl_in_convex_languag}.  Partially\nbased on that survey, here we summarize some of the results concerning\nthe state complexity of preserving regularity operations over some of\nthe convex subregular languages. Star-free languages are other family of \nsubregular languages well studied~\\cite{schutzenberger65:_finit_monoid_havin_only_trivial_subgr,mcnaughton71:_count_free_autom}. We\n briefly address recent results on the (nondeterministic) state complexity of basic\nregular operations on these languages.\n \n\n\\subsubsection{Convex Subregular Languages}\n\\label{sec:convexlanguages}\nWe begin by some definitions and results\non determination for these languages. Let $\\unlhd$ be a partial order on $\\Sigma^\\star$, and let $\\unrhd$ be\nits converse. A language $L$ is $\\unlhd$-convex if $u \\unlhd v$ and $v\n\\unlhd w$ with $u,w \\in L$ implies $v \\in L$. It is $\\unlhd$-free if\n$v \\unlhd w$ and $w \\in L$ implies $v \\notin L$. It is $\\unlhd$-closed if\n$v \\unlhd w$ and $w \\in L$ implies $v \\in L$. \nIt is $\\unrhd$-closed if $v \\unrhd w$ and $w \\in L$ implies $v \\in L$.  \nThe closure and the\nconverse closure operations are:\n$$_\\unlhd L=\\{v\\mid v \\unlhd w \\text{ for some } w\\in L\\},$$\n$$L_\\unlhd=\\{v\\mid w \\unlhd v \\text{ for some }\\ w\\in L\\}.$$\nThe \\emph{freeness} operation, $L^\\unlhd$ can defined for a language $L$, by\n$$L^\\unlhd \\subseteq L \\text{ and } \\forall w\\in L^\\unlhd,\n\\forall v\\in \\Sigma^\\star,\\; v\\lhd w\\text{ implies }v\\notin L^\\unlhd.$$\nThe following proposition is\nfrom~\\cite{ang09:_languag_convex_with_respec_to}, except for the last\nitem.\n\\begin{propos}\nLet $\\unlhd$ be an arbitrary relation on $\\Sigma^\\star$. Then \n\\begin{enumerate}\n\\item  A language is $\\unlhd$-convex if and only if it is\n  $\\unrhd$-convex.\n\\item  A language is $\\unlhd$-free if and only if it is $\\unrhd$-free.\n\\item  Every $\\unlhd$-closed language and every $\\unrhd$-closed\n  language is $\\unlhd$-convex. \n\\item  A language is $\\unlhd$-closed if and only if its complement is\n  $\\unrhd$-closed.\n \\item A language L is $\\unlhd$-closed ($\\unrhd$-closed) if and only\n   if $L = _\\unlhd L$ ($L = L_\\unlhd$).\n \\item A language L is $\\unlhd$-free if and only   if $L = L^\\unlhd$.\n\\end{enumerate}\n\\end{propos}\n\n\\noindent We consider $\\unlhd$ to be:\n\n\\begin{itemize}\n\\item $\\leq$: if $u,v,w\\in \\Sigma^\\star$ and $w=uv$, then $u$ is\n  \\emph{prefix} of $w$, and we write $u\\leq w$.\n\\item $\\preceq$:  if $u,v,w\\in \\Sigma^\\star$ and $w=uv$, then $v$ is\n  \\emph{suffix} of $w$, and we write $v\\preceq w$ \n\\item $\\sqsubseteq$: if $u,v,w\\in \\Sigma^\\star$ and $w=uxv$, then $x$ is\n  \\emph{factor} of $w$, and we write $x\\sqsubseteq w$.  Note that a prefix or\n  suffix of $w$ is also a factor of $w$. This relation is also  called \\emph{infix}.\n\\item $\\Subset$: if $w=w_0a_1w_1\\cdots a_nw_n$, where $a_1,\\ldots,a_n \\in \\Sigma$, and\n$w_0,\\ldots,w_n\\in \\Sigma^\\star$, then $v = a_1\\cdots a_n$ is a\n\\emph{subword} of $w$; and we write $v\\Subset w$. Note that every\nfactor of $w$ is a subword  of $w$.\n\\end{itemize}\n\nIf a language is both prefix- and suffix-convex it is\n\\emph{bifix-convex}. In the same way are defined \\emph{bifix-free} and\n\\emph{bifix-closed} languages. Ideals are languages directly related with closed languages. A\nnon-empty language $L\\subseteq \\Sigma^\\star$ is a\n\\begin{itemize}\n\\item \\emph{right ideal} if $L=L\\Sigma^\\star$ (also called \\emph{ultimate\n  definite} \\cite{paz65:_ultim_defin_and_symmet_defin}); the\n  complement is prefix converse-closed.\n\\item \\emph{left ideal} if $L=\\Sigma^\\star L$ (also called \\emph{reverse\n  ultimate definite} \\cite{paz65:_ultim_defin_and_symmet_defin}); the\n  complement is suffix converse-closed.\n\\item \\emph{two-sided ideal} if $L=\\Sigma^\\star L\\Sigma^\\star$ (also\n  called \\emph{central definite}); the complement is bifix converse-closed.\n\\item \\emph{all-sided ideal} if $L=\\Sigma^\\star \\shuffle L$; the complement\n  is subword converse-closed; also studied by Haines ~\\cite{haines69:_free_monoid_partial_order_by_embed} and\n  Thierrin~\\cite{thierrin72:_convex_languag}.\n\\end{itemize}\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c|c||c|c||c|c|}\\hline\n  \\multicolumn{6}{|c|}{Free}\\\\\\hline\n  \\multicolumn{1}{|c}{$\\leq$}&$|\\Sigma|$&\\multicolumn{1}{c}{$\\preceq$}&$|\\Sigma|$&\\multicolumn{1}{c}{$\\sqsubseteq$}&$|\\Sigma|$\n\\\\\\hline \n  $2^{m-1}+1$&3&$2^{m-1}+1$&$3$&$2^{m-2}+2$&$3$\\\\\\hline\n\\multicolumn{2}{|c||}{$]m,2^{m-1}+1]$}&\\multicolumn{2}{|c||}{$]m,2^{m-1}+1]$}&\\multicolumn{2}{|c|}{$]m,2^{m-2}+2]$}\\\\\\hline\n  \\multicolumn{6}{|c|}{Closed}\\\\\\hline\n  \\multicolumn{1}{|c}{$\\leq$}&$|\\Sigma|$&\\multicolumn{1}{c}{$\\preceq$}&$|\\Sigma|$\n  &\\multicolumn{1}{c}{$\\sqsubseteq$}&$|\\Sigma|$\n\\\\\\hline\n  $2^m$&$3$&$2^{m-1}+1$&$4$&$2^{m-1}+1$&$4$\\\\\\hline\n\\multicolumn{2}{|c||}{$]m,2^{m}]$}&\\multicolumn{2}{|c||}{$[m,2^{m-1}+1]$}&\\multicolumn{2}{|c|}{$]m,2^{m-1}+1]$}\\\\\\hline\n  \\multicolumn{6}{|c|}{Ideal}\\\\\\hline\n  \\multicolumn{1}{|c}{right}&$|\\Sigma|$&\\multicolumn{1}{c}{left}&$|\\Sigma|$\n  &\\multicolumn{1}{c}{two-sided}&$|\\Sigma|$\n\\\\\\hline\n  $2^{m-1}$&$2$&$2^{m-1}+1$&$3$&$2^{m-2}+1$&$3$\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity of determination of free, closed and ideal\n  languages considering prefix, suffix and factor partial orders,\n  respectively. For each free and closed of languages, the range of correspondent non-magic numbers appears on the second row.}\n  \\label{tab:scdeterminationconvex}\n\\end{table}\n\nSome of the languages defined above are also characterized in terms of\nproperties of the finite automata that accept them. In particular:\nprefix-closed languages are accepted by NFAs\\xspace where all states are\nfinal; suffix-closed languages are accepted by NFAs\\xspace where all states\nare initial; factor-closed languages are accepted by NFAs\\xspace where all\nstates are initial and final; prefix-free languages are accepted by\nnon-exiting NFAs\\xspace (i.e. there are no transitions from the final\nstates); suffix-free languages are accepted by non-returning NFAs\\xspace\n(i.e. there are no transitions to the initial state); and factor-free\nlanguages are accepted by non-returning and non-exiting NFAs\\xspace.\n\n The state complexity of the determination on some subregular\n languages (or for the kind of NFAs\\xspace they are defined by) was\n recently studied by Bordihn \\emph{et\n   al.}~\\cite{bordihn09:_deter_of_finit_autom_accep_subreg_languag},\n Jui-Yi Kao \\emph{et al.}~\\cite{kao09:_nfas_where_all_states_are}, and\n Jir\\'askov\\'a \\emph{et\n   al.}~\\cite{jiraskova10:_compl_in_prefix_free_regul_languag}.\n Table~\\ref{tab:scdeterminationconvex} presents some of the values for\n the languages considered above.  The existence of magic numbers for\n some subregular languages was studied by Holzer \\emph{et\n   al.}~\\cite{holzer12:_magic_number_probl_for_subreg_languag_famil}. As\n can be seen in Table~\\ref{tab:scdeterminationconvex}, $m$ is the only\n magic number for all free languages and for both prefix- and\n factor-closed languages (except if $m=1$, where $m$ is\n non-magic). Suffix-closed languages have no magic numbers.\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Prefix-free}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&$|\\Sigma|$&\\multicolumn{1}{c}{nsc}&$|\\Sigma|$\\\\\\hline\n$L_1\\cup L_2$&$mn-2$&$2$&$m+n$&$2$\\\\\\hline\n$L_1\\cap L_2$&$mn-2(m+n-3)$& $2$  &$mn-(m+n)+2$&$2$\\\\\\hline\n$\\comp{L}$&$m$&$1$&$2^{m-1}$&$3$\\\\\\hline\n $(L_1- L_2)$&$mn-m-2n+4$ &$3$&$(m-1)2^{n-1}+1$&$4$\n  \\\\\\hline \n  $(L_1\\oplus L_2)$&$mn-2$&$2$&&\\\\\\hline\n$L_1L_2$&$m+n-2$&$1$&$m+n-1$&$1$\\\\\\hline\n$L_1\/L_2$&\n\\begin{tabular}{c}\n$n-1$\\\\\n$n-m+2$\n\\end{tabular}&\n\\begin{tabular}{c}\n$2$ \\\\\n$1$\n\\end{tabular}\n&&\\\\\\hline\n$L^\\star$&\n\\begin{tabular}{c}\n$m$\\\\\n$m-2$\n\\end{tabular}\n&\\begin{tabular}{c}\n$2$ \\\\\n$1$\n\\end{tabular}&\n$m$&$1$\\\\\\hline\n$L^{R}$&$2^{m-2}+1$&$3$&$m$&$1$\\\\\\hline\n$\\shift{L}$&$(2m-3)^{m-2}$&$6$&$2m^2-4m+3$&$2$\\\\\\hline\n\\end{tabular}\n  \\caption{\\small State complexity and nondeterministic state complexity of some operations on prefix-free languages}\n  \\label{tab:scnscprefixfree}\n\\end{table}\n\n\\paragraph{Free languages}\n\\label{sec:freelanguages}\n\nTable~\\ref{tab:scnscprefixfree} summarizes state complexity results of\nindividual operations on prefix-free\nlanguages~\\cite{han09:_nondet_state_compl_of_basic,\nhan09:_operat_state_compl_of_prefix,\njiraskova10:_compl_in_prefix_free_regul_languag,\nbrzozowski11:_quotien_compl_of_bifix_factor,\njiraskova14:_kleen_closur_regul_and_prefix_free_languag,\njiraskova14:_complement_pref_free,\njirasek14:_prefix_free_languag,\neom13:_state_compl_of_k_union}.\nIn the case of state complexity, the results are valid for Boolean operations  if\n$m,n\\geq 3$; for catenation if $m,n\\geq 2$; for star if  $k=1$, then\n$m\\geq 3$, if $k=2$ then $m\\not=3$, and else $m\\geq 2$; and for reversal if\n$m\\geq 4$ and the tight bound cannot be reached if\n$k=2$~\\cite{jiraskova10:_compl_in_prefix_free_regul_languag}. The state complexty of right quotient is $1$, if  $k=1$ and $m=1$ or $m>n$, and if $k=2$ and $m=1$ or  $n=1$; furthermore, if $m=2$ then $sc(L_1\/L_2)=n$~\\cite{jirasek14:_prefix_free_languag}.\n\nNote\nthat here the state complexity of the catenation and the star are much lower\nthan on general regular languages. Moreover, for the star, the only complexities attained are $m-2$, $m-1$, and $m$~\\cite{jiraskova14:_kleen_closur_regul_and_prefix_free_languag}.\n\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{|c||c|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Suffix-free}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&$|\\Sigma|$&\\multicolumn{1}{c}{nsc}&$|\\Sigma|$\\\\\\hline\n$L_1\\cup L_2$&$mn-(m+n-2)$&$2$&$m+n-1$&$2$\\\\\\hline\n$L_1\\cap L_2$&$mn-2(m+n -3)$&$2$&$mn-(m+n-2)$&$2$\\\\\\hline\n$\\comp{L}$&$m$&$1$&\n\\begin{tabular}{c}\n$2^{m-1}$\\\\\n$\\leq 2^{m-1}+2^{m-3}+1$\\\\\n$\\Theta(\\sqrt{m})$\n\\end{tabular}\n&\n\\begin{tabular}{c}\n$3$\\\\\n$2$\\\\\n$1$\n\\end{tabular}\n\\\\\\hline\n$L_1- L_2$&$mn-(m+2n-4)$&$4$&&\\\\\\hline\n$L_1\\oplus L_2$&$mn-(m+n-2)$&$5$&&\\\\\\hline\n$L_1L_2$&$(m-1)2^{n-2}+1$&$4$&&\\\\\\hline\n$L^\\star$&$2^{m-2}+1$&$4$&&\\\\\\hline\n$L^{R}$&$2^{m-2}+1$&$3$&&\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity and nondeterministic state complexity of some operations on suffix-free languages}\n  \\label{tab:scncssuffixlanguages}\n\\end{table}\nTable~\\ref{tab:scncssuffixlanguages} summarizes the state complexity\nof some regular operations on suffix-free languages.  Han and Salomaa\nshowed that all bounds, except for complementation, difference, and symmetric\ndifference, are tight\n\\cite{han09:_state_compl_of_basic_operat,han10:_nondet_state_compl_for_suffix}. Jir\\'askov\\'a\nand Olej\\'ar~\\cite{jiraskova09:_state_compl_of_union_and} provided\nbinary witnesses for intersection and union. They also proved that for\nall integer $\\alpha$ between $1$ and the respective bound there are\nlanguages $L_1$ and $L_2$ such that $(n)sc(L_1\\circ L_2)=\\alpha$, for\n$\\circ\\in\\{\\cap,\\cup\\}$ (and witnesses ternary, except for $nsc(L_1\\cap\nL_2)$ for which the witnesses are over a four-symbol alphabet). The bounds for\ndifference and symmetric difference are from Brzozowski \\emph{et al.}~\\cite{brzozowski11:_quotien_compl_of_bifix_factor}. Jir\u00e1skov\u00e1 \\emph{et al.}~\\cite{jiraskova14:_complement_pref_free} proved the results for complementation.\n\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||p{4.5cm}|c|c|c|}\\hline\n\\multicolumn{5}{|c|}{Free}\\\\\\hline\n&&\\multicolumn{1}{|c}{$\\leq\\cup\\preceq$}&\\multicolumn{1}{|c}{$\\sqsubseteq$}\n&\\multicolumn{1}{|c|}{$\\Subset$}\\\\\\hline\n&\\multicolumn{1}{|c|}{sc}&\\multicolumn{3}{|c|}{$|\\Sigma|$}\\\\\\hline\n$L_1\\cup  L_2$&$mn-m-n$&$5$&$5$&$< m+n-3$\\\\\\hline\n$L_1\\cap L_2$\n&$mn-3m-3n+12$, $m,n\\geq 4$&$3$&$3$&$m+n-7$\\\\\\hline\n$L_1-  L_2$&$mn-2m-3n+9$&$4$&$4$&$<  m+n-6$\\\\\\hline\n$L_1\\oplus  L_2$&$mn-m-n$&$5$&$5$&$m+n-3$\\\\\\hline\n$L_1L_2$&$m+n-2$, $m,n>1$&1&1&1\\\\\\hline\n$L^\\star$&$m-1$, $m>2$&2&2&2\\\\\\hline\n$L^{R}$&$2^{m-3}+2$, $m\\geq 3$&2&2&$2^{m-3}-1$\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity of basic operations on bifix-, factor-, and subword-free languages}\n  \\label{tab:scfixlanguages}\n\\end{table}\n\nIf a language is subword-free then it is factor-free, and if it is\nfactor-free then it is bifix-free.  Table~\\ref{tab:scfixlanguages}\nsummarizes the state complexity of some regular operations on bifix-,\nfactor-, and subword-free\nlanguages~\\cite{brzozowski11:_quotien_compl_of_bifix_factor}.  The\ntight upper bounds for the state complexity of these operations on the\nthree classes of languages coincide.\n\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||c|c||c|c||c|c|c|}\\hline\n  \\multicolumn{8}{|c|}{Closed}\\\\\\hline\n  &\\multicolumn{1}{c}{$\\leq$}&$|\\Sigma|$&\\multicolumn{1}{c}{$\\preceq$}&\n$|\\Sigma|$&\\multicolumn{1}{c}{$\\sqsubseteq$,$\\Subset$}&\\multicolumn{1}{c}{$|\\Sigma|_{\\sqsubseteq}$}&\\multicolumn{1}{c|}{$|\\Sigma|_{\\Subset}$}\\\\\\hline\n  $L_1\\cup L_2$&$mn$&$2$&$mn$&$4$&$mn$&$2$&$2$\\\\\\hline\n  $L_1\\cap  L_2$&$\\scriptstyle{mn-m-n+2}$&$2$&$mn$&$2$&$\\scriptstyle{mn-m-n+2}$&$2$&$2$\\\\\\hline\n  $L_1-L_2$&$mn-n+1$&$2$&$mn$&$4$&$mn-n+1$&$2$&$2$\\\\\\hline\n  $L_1\\oplus\n  L_2$&$mn$&$2$&$mn$&$2$&$mn$&$2$&$2$\\\\\\hline\n  $L_1L_2$&$\\scriptstyle{m2^{n-2}+2^{n-2}}$&$3$&$\\scriptstyle{mn-fn+f}$&$3$&$m+n-1$&$2$&$2$\\\\\\hline\n  $L^\\star$&$2^{m-2}+1$&$3$&$m$&$2$&$2$&$2$&$2$\\\\\\hline\n  $L^R$&$2^{m-1}$&$2$&$2^{m-1}+1$&$3$&$2^{m-2}+1$&$3$&$2m$\\\\\\hline\\hline\n  $_\\unlhd  L$&$m$&$1$&$2^{m-1}$&$2$&$2^m-1$&$2$&\\\\\\hline\n$_\\Subset  L$&&&&&\\begin{tabular}{c}\n$2^{m-2}+1$\\\\\n$2^{\\Omega(\\frac{m}{3})}$\n\\end{tabular}&&\n\\begin{tabular}{c}\n$m-2$\\\\\n$2$\n\\end{tabular}\n\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity of some operations on prefix-, suffix-,\n  factor-, and subword-closed languages. \n  The last two columns correspond to factor and subword, respectively.\n  The last but one row contains the state complexity of the  closure of\n  prefix, suffix, and factor respectively. The last row contains the state\n complexity of the subword closure, considering unbounded and binary alphabets, respectively.}\n  \\label{tab:scclosedlanguages}\n\\end{table}\n\n\\paragraph{Closed Languages and Ideals}\n\\label{sec:closedlanguagesideals}\n\nTable~\\ref{tab:scclosedlanguages} shows the state complexity\nof some basic operations on prefix-, suffix-, factor-, and\nsubword-closed\nlanguages.  A\nlanguage is factor-closed if and only if it is subword-closed. So the\nstate-complexity results of operations are the same for those classes.\nThe state complexity of the closure on the respective partial orders\nis also considered. Subword and converse subword closures were first\nstudied by \nGruber \\emph{et al.}~\\cite{gruber07:_size_of_higman_haines_sets,gruber09:_more_size_of_higman_haines_sets}\nand Okhotin~\\cite{okhotin10:_state_compl_of_scatt_subst_and_super}.\nBrzozowski \\emph{et al.}~\\cite{brzozowski10:_quotien_compl_of_closed_languag,brzozowski14:_quotien_compl_of_closed_languag} presented the tight upper bound, but using a growing alphabet. Karandikar and   Schoebelen~\\cite{karandikar14:_state_compl_of_closur_and} shown that the exponential blown up is also required in the binary case. Given a regular\nlanguage $L$ with $sc(L)=m$, $nsc(_\\Subset L)=nsc({L}_\\Subset)=m$ and\nthese upper bounds are tight for witness binary languages.\nPrefix, suffix, and factor closures (respectively, $_\\leq L$, $_\\preceq L$, and $_\\sqsubseteq L$) were studied by Kao \\emph{et\n  al.}~\\cite{kao09:_nfas_where_all_states_are}. \nIf $L$ does not have $\\emptyset$ as a quotient, Brzozowski \\emph{et al.} shown that the state complexity of the suffix closure is $2^m -1$ (instead of $2^{m-1}$).\n  \n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||c|c||c|c||c|c|c|}\\hline\n  \\multicolumn{8}{|c|}{Ideal}\\\\\\hline\n  &\\multicolumn{1}{c}{right}&$|\\Sigma|$&\n\\multicolumn{1}{c}{left}&$|\\Sigma|$&\n\\multicolumn{1}{c}{-sided}&\\multicolumn{1}{c}{$|\\Sigma|_{\\text{two}}$}&$|\\Sigma|_{\\text{all}}$\\\\\\hline\n  $L_1\\cup\n  L_2$&$\\scriptstyle{mn-m-n+2}$&$2$&$mn$&$4$&$\\scriptstyle{mn-m-n+2}$&$2$&$2$\\\\\\hline\n  $L_1\\cap\n  L_2$&$mn$&$2$&$mn$&$2$&$mn$&$2$&$2$\\\\\\hline\n  $L_1-L_2$&$\\scriptstyle{mn-m+1}$&$2$&$mn$&$4$&$\\scriptstyle{mn-m+1}$&$2$&$2$\\\\\\hline\n  $L_1\\oplus\n  L_2$&$mn$&$2$&$mn$&$2$&$mn$&$2$&$2$\\\\\\hline\n  $L_1L_2$&$\\scriptstyle{m+2^{n-2}}$&$1$&$\\scriptstyle{m+n-1}$&$1$&$\\scriptstyle{m+n-1}$&$1$&$3$\\\\\\hline\n  $L^\\star$&$m+1$&$2$&$m+1$&$2$&$m+1$&$2$&$2$\\\\\\cline{2-8}\n&\\multicolumn{7}{|l|}{If $\\varepsilon\\in L$, then $L=\\Sigma^\\star$ and\n  $sc(L^\\star)=1$.}\\\\\\hline\n  $L^R$&$2^{m-1}$&$2$&$2^{m-1}+1$&$3$&$2^{m-2}+1$&$3$&$\\scriptstyle{2m-4}$\\\\\\hline\n \\end{tabular}\n  \\caption{\\small State complexity of basic operations on ideals. The last\n    two columns correspond to two-sided  and all-sided ideals, respectively.}\n  \\label{tab:scideals}\n\\end{table}\n\n\n\n  \nIf $L$ is a right (respectively, left, two-sided, all-sided) ideal, any\nlanguage $G \\subseteq \\Sigma^\\star$ such that $L=G\\Sigma^\\star$\n(respectively, $L=\\Sigma^\\star G$, $L=\\Sigma^\\star\nG\\Sigma^\\star$,$L=\\Sigma^\\star\\shuffle\tG$) is a \\emph{generator} of $L$.  Brzozowski and\nJir\\'askov\\'a~\\cite{brzozowski10:_quotien_compl_of_ideal_languag} studied\nstate complexity on ideals. Table~\\ref{tab:scideals} presents the\nstate complexity of basic operations on ideals.  As stated before\nclosed languages and ideals are related. In particular, the state\ncomplexity of basic operations on two-sided and all-sided ideals\ncoincide. Brzozowski~\\cite{brzozowski10:_compl_in_convex_languag}\nobserved that for the four types of convex languages (prefix, suffix,\nfactor and subword) the state complexity of the Boolean operations is\n$mn$.\n\n\\paragraph{Unary convex languages}\nIn the case of unary languages, prefix, suffix, factor, and subword partial orders\ncoincide. Table~\\ref{tab:scunaryfreeclosedideals} summarizes the state\ncomplexity of basic operations on unary free, unary closed, unary\nideals and unary convex languages.\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||c|c|c|c|}\\hline\n  \\multicolumn{5}{|c|}{Unary}\\\\\\hline\n  &\\multicolumn{1}{c|}{Free}&\\multicolumn{1}{c|}{Closed}&\\multicolumn{1}{c|}{Ideal}&\\multicolumn{1}{c|}{Convex}\\\\\\hline\n  $L_1\\cup L_2$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$&$\\min\\{m,n\\}$&$\\max\\{m,n\\}$\\\\\\hline\n  $L_1\\cap L_2$&$m=n$&$\\min\\{m,n\\}$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$\\\\\\hline\n  $L_1-L_2$&$m$&$m$&$n$&$\\max\\{m,n\\}$\\\\\\hline\n  $L_1\\oplus  L_2$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$&$\\max\\{m,n\\}$\\\\\\hline\n  $L_1L_2$&$m+n-2$&$m+n-2$&$m+n-1$&$m+n-1$\\\\\\hline\n  $L^\\star$&$m-2$&$2$&$m-1$&$n^2-7n+13$\\\\\\hline\n  $L^R$&$m$&$m$&$m$&$m$\\\\\\hline\n\\end{tabular}\n  \\caption{\\small State complexity of basic operations on unary convex languages}\n  \\label{tab:scunaryfreeclosedideals}\n\\end{table}\n\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|c||c|c||c|c|}\\hline\n&\\multicolumn{2}{|c||}{Regular}&\\multicolumn{2}{c|}{Ideal}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&$|\\Sigma|$&\\multicolumn{1}{c}{sc}&$|\\Sigma|$\\\\\\hline\n$L^\\leq$&$m+1$&$2$&$m+1$&$2$\\\\\\hline\n$L^\\preceq$&$(m-1)2^{m-2}+2$, $m\\geq 4$&$4$&$\\frac{n(n-1)}{2}+2$&$1$\\\\\\hline\n$L^\\sqsubseteq$&$(m-2)2^{m-3}+3$, $m\\geq 4$&$3$&$n+1$&$1$\\\\\\hline\n$L^b$& $(m-2)2^{m-2}+3$, $m\\geq4$&$4$&&\\\\\\hline\n\\end{tabular}\n  \\caption{\\small State complexity  of prefix, suffix, factor and bifix\n    operations on  regular languages and on ideals (right, left and\n    two sided, respectively).}\n  \\label{tab:scfreeness}\n\\end{table}\n\n\\paragraph{Freeness Operations} \nHere we analyse the state complexity of freeness\noperations for prefix, suffix, bifix and factor orders that were\nstudied by Pribavkina and\nRodaro~\\cite{pribavkina10:_state_compl_of_prefix_suffix}.\nGiven a regular language $L$, the $\\unlhd$-free language\n$L^\\unlhd$ for $\\unlhd\\in\\{\\leq,\\preceq,\\sqsubseteq\\}$, is  respectively\\footnote{In~\\cite{pribavkina10:_state_compl_of_prefix_suffix}\nthe superscripts for prefix, suffix and factor operations were respectively $p$,\n$s$ and~$\\iota$.}:\n\\begin{itemize}\n\\item prefix: $L^\\leq=L-L\\Sigma^+$\n\\item suffix: $L^\\preceq=L-\\Sigma^+L$\n\\item factor: $L^\\sqsubseteq=L-(\\Sigma^+ L \\Sigma^\\star\n  \\cup \\Sigma^\\star L \\Sigma^+ )$\n\\end{itemize}\n\\noindent The bifix operation is defined by $L^b=L^\\leq \\cap\nL^\\preceq$.  If $L$ is an ideal, prefix, suffix and factor operations\nwere studied by Brzozowski and\nJir\\'askov\\'a~\\cite{brzozowski10:_quotien_compl_of_ideal_languag}. In this\ncase, the resulting languages are minimal generators for left, right\nand two sided ideals, respectively. Table~\\ref{tab:scfreeness}\npresents the state complexity of prefix, suffix, factor and bifix\noperations on regular languages (and correspondent ideals). The state\ncomplexity of this operations is much lower in the case of right and\ntwo-sided ideals than for general regular languages.\n\n\\subsubsection{Star-free Languages}\n\\label{sec:starfree}\nStar-free languages are the smallest class containing the finite\nlanguages and closed under Boolean operations and catenation. This\nclass of languages correspond exactly to the regular languages of star\nheight $0$. The minimal DFAs\\xspace of star-free languages are\n\\emph{permutation-free} (i.e. no word performs a non-trivial\npermutation of a subset of its states).  Bordhin \\emph{et\n  al.}~\\cite{bordihn09:_deter_of_finit_autom_accep_subreg_languag}\nshowed that the state complexity of the determination of a star-free\nlanguage $L$ is $2^{nsc(L)}$. Figure~\\ref{fig:starfreedet} presents a\nfamily of ternary NFAs\\xspace for which the bound is tight. Holzer \\emph{et\n  al.}~\\cite{holzer12:_magic_number_probl_for_subreg_languag_famil}\nshowed that star-free languages have no magic numbers.\n   \\begin{figure}[hbt]\n     \\centering\n\n    \\includegraphics[width=8cm]{diag15}\n\t\t\\ignore{\\SmallPicture{\\VCDraw{\n              \\begin{VCPicture}{(-2,-1)(14,2)}\n                \\State[0]{(0,0)}{0}\\Initial{0}\n                \\State[1]{(3,0)}{1}\n                \\State[2]{(6,0)}{2}\n                \\SetStateLineStyle{none}\n                \\State[\\cdots]{(9,0)}{3}\n                \\SetStateLineStyle{solid}\n                \\FinalStateVar[m-1]{(12,0)}{m1}\n                \\LoopN{0}{b,c}\\EdgeL{0}{1}{a,b}\n                \\LoopN{1}{b}\\EdgeL{1}{2}{a,c}\n                \\LoopN{2}{b}\\EdgeL{2}{3}{a,c}\n                \\EdgeL{3}{m1}{a,c}\n                \\LoopN{m1}{b}\n              \\end{VCPicture}\n             }\n           }  }   \n\n  \\caption{\\small{}Minimal $m$-state NFAs\\xspace with equivalent minimal $2^m$-state DFA\\xspace for star-free languages}\n     \\label{fig:starfreedet}\n   \\end{figure}\n\n   Brzozowski and\n   Liu~\\cite{brzozowski11:_quotien_compl_of_star_free_languag} studied\n   the state complexity of the basic regular operations on star-free\n   languages, and their results are summarized in\n   Table~\\ref{tab:starfree}. The bounds obtained for general regular\n   languages are reached except in the catenation for $n=2$, the\n   reversal, and operations on unary languages. Holzer \\emph{et\n     al.}~\\cite{holzer11:_nondet_state_compl_of_star_free_languag,holzer12:_nondet_state_compl_of_star_free_languag}\n   studied the same languages for the operational nondeterministic\n   state complexity. The bounds  coincide with the ones for general\n   regular languages and are tight for binary languages. The witness languages\n   for union and catenation are $a^{m-1}(ba^{m-1})^\\star$ and\n   $b^{n-1}(ab^{n-1})^\\star$. For intersection, witnesses are  $b^\\star(ab^\\star)^{m-1}$ and\n   $a^\\star(ba^\\star)^{n-1}$. The first witness for union is also a\n   witness for the star operation. The language family  presented in\n   Figure~\\ref{fig:jirasreveralnfa} is star-free and thus  a witness for the reversal operation.\n  On unary star-free languages, the upper bounds for  operational nondeterministic\n   state-complexity coincide with general case, except for the\n   complementation. Holzer \\emph{et\n     al.}~\\cite{holzer12:_nondet_state_compl_of_star_free_languag}\n   showed that for reversal and star the bounds are tight.  For union, the presented lower bound\nmisses the upper bound by one state. For intersection, the presented\nbound is tight in the order of magnitude ($\\Theta(mn)$) and the bound for\ncomplementation is $\\Theta(n^2)$. The  lower bound  for catenation  misses the upper\nbound for unary general languages by one state.\n\n\\begin{table}[htbp]\n  \\centering\n\\begin{tabular}{|l||c|c||c|}\\hline\n\\multicolumn{4}{|c|}{Star-free}\\\\\\hline\n&\\multicolumn{1}{c}{sc}&$|\\Sigma|$&\\multicolumn{1}{c|}{Unary}\\\\\\hline\\hline\n$L_1\\circ L_2$&$mn$&2&$\\max\\{m,n\\}$\\\\\\hline\n\\multirow{2}{*}{$L_1L_2$}&$(m-1)2^n+2^{n-1}$, if $n\\geq 3$&$4$&\\multirow{2}{*}{$m+n-1$}\\\\\\cline{2-3}\n&$[3m-2,3m-1]$, if $n=2$\n&$3$&\\\\\\hline\n\\multirow{3}{*}{$L^\\star$}& $2$, if $m=1$&$1$& $2$, if $m=1$\\\\\\cline{2-4}\n&$2^{m-1}+2^{m-2}$, if $m\\geq 2$&$4$&$m$, if $m\\in[2,5]$\\\\\\cline{2-4}\n&&&$m^2-7m+13$, if $m>5$\n\\\\\\hline\n$L^R$&$2^m-1$&$m-1$&$m$\\\\\\hline\n\\end{tabular}\n\\caption{\\small State complexity  of basic regular operations on star-free\n  regular and unary languages, where $\\circ\\in\n  \\{\\cup,\\cap,\\setminus,\\oplus\\}$. For non-unary star-free languages\n  and $n=2$, $m\\geq 2$.\n  For non-unary star-free languages if $m\\in [1,2]$, the bound for\n  reversal is tight  for $|\\Sigma|\\geq m$, and if $m\\geq 3$, for $|\\Sigma|\\geq m-1$.}\n  \\label{tab:starfree}\n\\end{table}\n\n\n\n\\subsection{Some More Results}\nWe briefly cite some more work on operational state complexity.\nC\\^ampeanu and Ho~\\cite{campeanu04:_maxim_state_compl_for_finit_languag}\nand Brzozowski and\nKonstantinidis~\\cite{brzozowski09:_state_compl_hierar_of_unifor}\nconsidered uniform finite languages.  Krieger \\emph{et al.} studied\ndecimations of\nlanguages~\\cite{krieger09:_decim_of_languag_and_state_compl}.\nC\\^ampeanu and\nKonstantinidis~\\cite{campeanu08:_state_compl_of_subwor_closur}\nanalysed a subword closure operation.  Union-free languages were\nconsidered by Jir\\'askov\\'a and\nMasopust~\\cite{jiraskova10:_compl_in_union_free_regul_languag,jiraskova11:_compl_in_union_free_regul_languag}.\nThe same authors studied the state complexity of projected languages~\\cite{jiraskova11:_state_compl_of_projec_languag}.\nThe chop (or \\emph{fusion}) of two words is their catenation where the\ntouching symbols are merged if equal, or is undefined otherwise. The\nchop operation and its iterated variants (star and plus) where studied\nby Holzer \\emph{et al.}~\\cite{holzer11:_chop_of_languag,\nholzer11:_chop_operat_and_expres,holzer12:_state_compl_of_chop_operat}. The\n(nondeterministic) state complexity results are similar to  the ones\nfor catenation, star and plus, with the exception of chop-star where\nthe complexities also depend on the alphabet size. This  comes as a\nsurprise as chop based regular expressions are known to be\nexponentially more succinct than classical catenation based ones. Bassino \\emph{et al.}~\\cite{bassino10:_compl_of_operat_cofin_languag}\nprovided upper bounds of the state complexity of basic operations on\ncofinite languages as a function of the size the of complementary finite\nlanguage (taken as the summation of the lengths of all its words).  The\naverage state complexity on finite languages is addressed in two\nworks.  Gruber and\nHolzer~\\cite{gruber07:_averag_state_and_trans_compl} analysed the\naverage state complexity of DFAs\\xspace and NFAs\\xspace based on a uniform\ndistribution over finite languages whose longest word is of length at\nmost $n$.  Based on the size of finite languages as the summation of\nthe lengths of all its words and a correspondent uniform distribution,\nBassino \\emph{et al.}~\\cite{bassino10:_averag_state_compl_of_ration}\nestablish that the average state complexities of the basic regular\noperations are asymptotically linear.\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\nAutomata theory is one of the oldest research areas in computer\nscience. Much research has been done on automata theory since 1950's. Work in many subareas of automata theory is still ongoing these days due to its new applications in areas such as software engineering, programming languages, parallel programming, network security, formal verification and natural language and speech processing \\cite{Mo96,PeRi96,MoAlRoRoCoDe00,Sc06,LuYu09,wang13:_handb_of_finit_state_based}. \n\nDescriptional complexity and, in particular, state complexity is one of such active subareas. Generally speaking, the study of complexity mainly focuses on the following two kinds of issues: time and space complexity issues, i.e.\\ time and space needed for the execution of the processes; or descriptional complexity issues, i.e.\\ the succinctness of the model representations \\cite{yu01:_state_compl_of_regul_languag}. In general, having succinct objects will improve our control on software, which may become smaller, more efficient and easier to certify.\n\nState complexity is a type of descriptional complexity based on the finite machine model, and, in the domain of regular languages, it is related to the basic question of how to measure the size of a finite automaton. For the deterministic finite automaton (DFA\\xspace) case, the three usual answers are: the number of states, the number of transitions, or a combination of the two~\\cite{yu01:_state_compl_of_regul_languag}. \nFor a complete DFA\\xspace, whose transition function is defined for every state and every possible input symbol, the number of transitions is linear with the number of states, for each fixed alphabet.\nThus, the number of states becomes the key measure for the size of a complete DFA\\xspace. \nWhen considering the descriptional complexity of nondeterministic finite automata (NFA\\xspace), because this notion of completeness is not present, the measures based on the number of states and on the number transitions, are much more loosely related.\n\nSince a regular language can be accepted by many DFAs\\xspace with different number of states but only by one unique minimal, complete DFA\\xspace, the deterministic state complexity of a regular language is defined as the number of states of the minimal, complete DFA\\xspace accepting it. If we replace the minimal, complete DFA\\xspace with minimal NFA\\xspace, we have the definition of nondeterministic state complexity. Since state complexity is used as a natural abbreviation of deterministic state complexity by most researchers working in the area, we also follow the convention in this paper.\n\nComplexity can be studied in two different flavours: in the worst case~\\cite{yu01:_state_compl_of_regul_languag} and in the average case~\\cite{nicaud99:_averag_state_compl_of_operat_unary_autom}. The worst-case complexity of a  class of regular languages is the supremum of the complexities of all the languages in the\nclass~\\cite{yu01:_state_compl_of_regul_languag} whereas the average-case complexity, it is the average value of the complexities of those languages. Although its evident practical importance, there is still very few research on average-case state complexity. For that reason, in this paper, we mainly review worst-case results.\n\nResults on descriptional complexity can be, roughly, divided into representational (or transformational) and operational. Representational complexity studies the complexity of transformations between models, by comparing the sizes of different representations of formal languages~\\cite{Sa09-UWO-talk}. For example, given an $n$-state NFA\\xspace for a regular language, the DFA\\xspace which is equivalent to it has at most $2^n$ states, and this result, established in 1957, is considered the first state complexity result~\\cite{rabin59:_finit_autom_and_their_decis_probl}. Operational state complexity studies the state complexity of operations on languages. When we speak about the state complexity of an operation on regular languages, we mean the state complexity of the class of resulting languages from the operation~\\cite{yu01:_state_compl_of_regul_languag}. For example, when we say the state complexity of the intersection operation on two regular languages, accepted by $m$-state and $n$-state DFAs\\xspace, respectively, is $mn$, we mean that $mn$ is the worst-case state complexity of the class of regular languages that can be represented as the intersection of an $m$-state DFA\\xspace language and an $n$-state DFA\\xspace language. Note that this implies that the intersection of any $m$-state DFA\\xspace language $L_1$ and $n$-state DFA\\xspace $L_2$ language has a DFA\\xspace with at most $mn$ states (upper bound) and that there exist languages $L_1$ and $L_2$ such that the minimal DFA for $L_1 \\cap L_2$ has exactly $mn$ states (lower bound).\n\nIn this survey, we mostly concentrate in operational state complexity results. \nAlthough first studies go back to the 1960's and 1970's, research in\nthe area has been most active in the last two decades.  This can be\npartially explained by the fact that back then, descriptional\ncomplexity issues were not a priority for applications, as they are\ntoday. But, also, due to its combinatorial nature many of the current\nresearch is only possible with the help  of new high-performance symbolic manipulation software and powerful computers~\\cite{GaYu12}.\n\nThe paper is organized as follows. After some preliminares in the next section, the notions of deterministic and nondeterministic state complexity  are considered in Section~\\ref{sec:scnsc}. To better understand the possible gap between both measures is a main topic of research.\nIn Section~\\ref{sec:sciop}, we review the state complexities of individual regularity preserving language operations, like,  Boolean operations, catenation, star, reversal, shuffle, orthogonal catenation, proportional removal, and cyclic shift, etc. These individual operations are fundamental and important in formal languages and automata theory research and applications. Results in these two sections are given for different classes of (sub)regular languages, e.g. general infinite, finite, unary, star-free, etc. In Section~\\ref{sec:scco},  we revisit the state complexities of combined operations which are combinations of individual operations, e.g., star of union, star of intersection, star of catenation, star of reversal, union of star, intersection of star, etc. The state complexities of most of these combined operations are much lower than the mathematical composition of the state complexities of their component individual operations. We also review the methods of estimation and approximation of state complexity of combined operations which can be used for very complex combined operations.\nSection~\\ref{sec:conclusions} concludes this survey with some discussion on the results presented, highlighting some open problems and directions of future research.\n\n\\section{Preliminaries}\n\\label{sec:preliminares}\nHere we recall some basic definitions related to finite\nautomata and regular languages. For a more complete presentation\nthe reader is referred to~\\cite{yu97:_handb_formal_languag}.\n\nThe set of natural numbers is denoted by $\\mathbb{N}$ and for $i, j \\in \\mathbb{N}$,\n$[i, j] = \\{ x \\in \\mathbb{N} \\mid i \\leq x \\leq j \\}$. The power set of a set $S$ is denoted by $2^S$ and the cardinality of\na finite set $S$ is $|S|$. \nIn the following, $\\Sigma$ stands always for a finite alphabet,\nthe empty word is represented by $\\varepsilon$ and the set of all words\nover $\\Sigma$ by $\\Sigma^\\star$. A language is a subset\nof $\\Sigma^\\star$. We say that $L \\subseteq \\Sigma^\\star$ is a unary\n(respectively, binary, ternary) language if\n$|\\Sigma| = 1$ (respectively, $|\\Sigma| = 2$, $|\\Sigma| = 3$).\nNote this definition does not require that all symbols of\n$\\Sigma$ actually appear in words of $L$ and hence every unary\nlanguage is also a binary language and a binary language is always\na ternary language. A language $L$ is said to be finite if  $L$\nis a finite subset of $\\Sigma^\\star$.\n\nA {\\em nondeterministic finite automaton\\\/} (NFA\\xspace) is a tuple\n$A = (Q, \\Sigma, \\delta, q_0, F)$ where $Q$ is a finite set of states,\n$\\Sigma$ is a finite alphabet, $\\delta : Q \\times \\Sigma \\rightarrow 2^Q$\nis the (multi-valued) transition function, $q_0 \\in Q$ is the initial\nstate and $F \\subseteq Q$ is the set of final (accepting) states.\nThe transition function is extended as a function\n$\\widehat{\\delta} : Q \\times \\Sigma^\\star \\rightarrow Q$ by setting\n$\\widehat{\\delta}(q,\\varepsilon) = q$ for $q\\in Q$ and for $w \\in \\Sigma^\\star$,\n$x \\in \\Sigma$, $\\widehat{\\delta}(q,wx) = \\delta( \\widehat{\\delta}(q, w),x)$. To simplify notation, we  denote $\\widehat{\\delta}$ \nby $\\delta$. The language recognized by the NFA\\xspace $A$ is \n$\nL(A) = \\{ w \\in \\Sigma^\\star \\mid \\delta(q_0, w) \\cap F \\neq \\emptyset \\}.\n$\n\nAn NFA\\xspace $A = (Q, \\Sigma, \\delta, q_0, F)$ is a\n{\\em complete deterministic finite automaton\\\/} (DFA\\xspace) if the transition function\n$\\delta$ is one-valued, that is, $\\delta$ is a function\n$Q \\times \\Sigma  \\rightarrow Q$. An {\\em incomplete} DFA\\xspace allows\nthe possibility that some transitions may be undefined, that is,\n$\\delta$ is a partial function $Q \\times \\Sigma \\rightarrow Q$.\n\nBoth the DFAs\\xspace and the NFAs\\xspace define the class of regular languages~\\cite{yu97:_handb_formal_languag}.\nIt is well known that any regular language has a unique\nminimal (complete or incomplete) DFA\\xspace, that is, a unique\nDFA\\xspace with the smallest number of states. For a given regular\nlanguage the sizes of the minimal, complete DFA\\xspace and minimal, incomplete\nDFA\\xspace differ by at most one state. Furthermore, for a given DFA\\xspace\nthere exists an $n \\log n$ time algorithm to compute\nthe minimal DFA\\xspace~\\cite{yu97:_handb_formal_languag}. On the other hand, for a given regular language  there may\nbe more than one minimal NFA\\xspace and NFA\\xspace minimization is\nPSPACE-hard~\\cite{holzer11:_descr_and_comput_compl_of,yu97:_handb_formal_languag}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Excursion Set Theory initially introduced by Bond et al. \\cite{Bond1991} \nprovides a self-consistent mathematical framework to infer the properties of the halo\nmass distribution from the statistics of the initial density field.\nThe formalism generalizes the original Press-Schechter idea \\cite{PressSchechter1974} \nby formulating the halo mass counting problem as one of stochastic calculus. The \nstarting point is the realization that at any location in space the linear matter density\nfluctuation field performs a random walk as function of a filtering scale $R$.\nIn average, this scale naturally defines a mass scale $M=\\bar{\\rho}\\,V(R)$,\n where $\\bar{\\rho}$ is the mean matter \ndensity and $V(R)$ the enclosed spatial volume. By counting the number of trajectories which first-cross a collapse threshold it is then possible to \ncompute the fraction of mass elements in halos $F(M)$ and consequently derive the halo mass function $dn\/dM = (1\/V)\\,dF\/dM$.\nThe requirement of first-crossing is key to solving the so called ``cloud-in-cloud'' \nproblem affecting the original Press-Schechter result. In fact, the first-crossing condition guarantees that in the small scale limit \n($R\\rightarrow 0$) and independently of the properties of the random walks\nthe fraction of mass into collapsed objects always tends to unity. \n\nThe Excursion Set is analytically solvable in the case of uncorrelated (Markov) random walks for which the evaluation\nof the first-crossing distribution is reduced to solving a standard diffusion problem. However, uncorrelated random walks are generated by\na special filtering of the linear density field which corresponds to a non-physical halo mass definition. In contrast any filtering \nwhich specifies a physically meaningful mass generates correlated random walks for which $F(M)$ can be inferred only through a numerical computation. \nThis has represented a major limitation \nsince Monte Carlo simulations are computationally expensive and moreover do not provide the same level of physical insight \nof analytic solutions. The seminal work by Maggiore \\& Riotto \\cite{MaggioreRiotto2010a} has made a major step forward in this direction. \nUsing the path-integral formulation of the Excursion Set Theory the authors have shown that the first-crossing distribution\nof correlated random walks can be computed as a perturbative expansion about the Markovian solution.\nUsing this methodology it has been possible to derive analytical formulae for the halo mass function under different halo collapse model \nassumptions as well as Gaussian and non-Gaussian (NG) initial conditions \\cite{MaggioreRiotto2010b,MaggioreRiotto2010c,DeSimone2011}. \n\nIn a series of papers \\cite{CorasanitiAchitouv2011a,CorasanitiAchitouv2011b,AchitouvCorasaniti2012}\nwe have used this formalism to evaluate the imprint of the non-spherical collapse of halos\non the mass function. To this end we have introduced an effective stochastic Diffusive Drifting Barrier (DDB) model which \nparametrizes the main features of the ellipsoidal collapse of halos. \nAccounting for such effects can reproduce the halo mass function from N-body simulations with remarkable \naccuracy both for Gaussian and non-Gaussian initial conditions. \n\nHere, we extend the work presented \\cite{AchitouvCorasaniti2012} to derive a more accurate expression of the\ncontribution to the halo mass function of the primordial bispectrum expanded in the large scale \nlimit to next-to-leading order and compute the leading order contribution of the primordial trispectrum.\n\nThe paper is organized as follows. In Section~\\ref{secI} we review the path-integral formulation of the Excursion Set and its application to\nGaussian and non-Gaussian initial conditions in the case of the DDB model. In Section~\\ref{secII} we evaluate a lower limit on the interval validity\nof the bispectrum expansion at next-to-leading order. In Section~\\ref{secIII} and~\\ref{secIV} compute the bispectrum and trispectrum\ncontribution to the mass function respectively. Finally, we present our conclusion in Section~\\ref{secV}.\n\n\\section{Excursion Set Mass Function and Diffusive Drifting Barrier}\\label{secI}\nHere, we briefly review the main features of the path-integral formulation of the Excursion Set Theory. \nFirst, let us introduce the variance of the linear density\nfield $\\delta$ filtered on a scale $R$:\n\\begin{equation}\n\\sigma^2(R)\\equiv S(R)=\\frac{1}{2\\pi^2}\\int dk\\,k^2P(k)\\tilde{W}^2(k,R),\n\\end{equation}\nwhere $P(k)$ is the linear matter power spectrum and $\\tilde{W}^2(k,R)$ is the Fourier transform of the filter\nfunction in real space, $W(x,R)$. As already mentioned, by selecting a volume $V(R)=\\int d^3x\\,W(x,R)$ the filter naturally associates a mass to the enclosed region,\n$M=\\bar{\\rho}\\,V(R)$. Thus we have a one-to-one relation between $R$ (or $M$) and $S$.\n\nIn the Excursion Set Theory the filtered density field $\\delta(x,R)$ at any random point in space \nperforms a random walk as function of $R$, $M$ or equivalently $S$ which plays\nthe role of a pseudo-time variable. The random walks start at $S=0$ with $\\delta=0$, since in the large scale limit ($R\\rightarrow \\infty$) we have\n$S\\rightarrow 0$ and the matter density distribution tends toward homogeneity, i.e. $\\delta\\rightarrow 0$. We are interested in counting trajectories that at a given value of $S$ \ncross for the first time a collapse density threshold $B$ such that $\\delta=B$. This threshold encodes all informations on the gravitational\ncollapse of halos. In order to model features of the non-spherical collapse, the absorbing barrier $B$ is promoted to a stochastic variable \n(see e.g. \\cite{Audit1997,Sheth2001,MaggioreRiotto2010b})\nalso performing a random walk as function of $S$ (i.e.~$R$ or $M$). In such a case it is convenient to introduce $Y=B-\\delta$, which performs\na random walk starting at $Y(0)=Y_0$ with barrier crossing at $Y_c=0$. \n\nOur goal is to compute the probability distribution $\\Pi(Y_0,Y,S)$ \nof trajectories starting at $Y_0$ which reach the value $Y$ at $S$ without ever touching the barrier $Y_c=0$. \nThis can be computed as a path-integral over the ensemble of the random trajectories (see \\cite{MaggioreRiotto2010a} for a detailed\nderivation). Let us discretize the pseudo-time variable $S$ in equally spaced steps, \n$\\Delta S=\\epsilon$, such that $S_k=k \\epsilon$ with $k=1,..,n$. The probability distribution of trajectories starting at $Y_0$ and ending in $Y_n$\nat $S_n$ and that have never crossed the barrier before is given by\n\\begin{equation}\n\\Pi_\\epsilon(Y_0,Y_n,S_n)=\\int_{Y_c}^{\\infty} dY_1..\\int_{Y_c}^{\\infty} dY_{n-1} W(Y_0,..,Y_n,S_n),\\label{piepsilon}\n\\end{equation}\nwhere \n\\begin{equation}\nW(Y_0,..,Y_n,S_n)=\\int\\mathcal{D}\\lambda\\, \n  e^{i\\sum_{i=1}^n\\lambda_i Y_i}\\langle e^{-i\\sum\\limits_{i=1}^n\\lambda_i Y(S_i)}\\rangle,\\label{densitymark}\n\\end{equation}\nwhere the brackets $\\langle...\\rangle$ refer to an ensemble average of the random walks\nand the averaged quantity is the explonential of\n\\begin{equation}\\label{partition}\nZ=\\sum_{p=1}^{\\infty}\\frac{(-i)^p}{p!}\\sum_{i_1=1}^{n}...\\sum\\limits_{i_p=1}^{n}\\lambda_{i_1}...\\lambda_{i_p}\\langle Y_{i_1}..Y_{i_p}\\rangle_c\n\\end{equation}\nwhich is the partition function of the system written in terms of the $p$-point connected correlation functions\n$\\langle Y_{i_1}..Y_{i_p}\\rangle_c$ of the random walks. Thus, the \nthe properties of the stochastic system are entirely determined by the connected correlators. Once these are specified\nthen Eq.~(\\ref{piepsilon}) can be integrated in the continuous limit to finally obtain \nthe first-crossing distribution\n\\begin{equation}\\label{firstcross}\n\\frac{dF}{dS}\\equiv\\mathcal{F}(S)=-\\frac{\\partial}{\\partial S}\\left[\\int_{Y_c}^{\\infty} dY\\,\\Pi(Y_0,Y,S)\\right],\n\\end{equation}\nand the halo mass function is given by\n\\begin{equation}\n\\frac{dn}{dM}=f(\\sigma)\\dfrac{\\bar{\\rho}}{M^2}\\frac{d \\log\\sigma^{-1}}{d\\log M},\n\\end{equation}\nwhere $f(\\sigma)=2S\\mathcal{F}(S)$ is the so called ``multiplicity'' function.\n\n\\subsection{Gaussian Initial Conditions}\nLet us consider a Gaussian density field smoothed with a top-hat filter in real space. On average the density field is homogeneous,\nthus implying that $\\langle\\delta(S)\\rangle_c=0$. Therefore, due to the Gaussian nature of the field the only non-vanishing \nconnected correlator is the 2-point function $\\langle\\delta(S)\\delta(S')\\rangle_c$, while\nall higher-order connected correlators identically vanish. Maggiore \\& Riotto \\cite{MaggioreRiotto2010a} have shown that for \nstandard cosmological scenarios with Cold Dark Matter power spectra, the 2-point function smoothed with a sharp-x\nis well approximated by $\\langle\\delta(S)\\delta(S')\\rangle_c={\\rm min}(S,S')+\\Delta(S,S')$, where the first term corresponds to Markov random walks\ngenerated by a sharp-k filter and the second term is well approximated by $\\Delta(S,S')=\\kappa S\/S' (S'-S)$ with a nearly constant amplitude $\\kappa<1$. \nThus, the pseudo-time correlations induced by the filter function can be treated as small correction about the Markovian \ncase and the mass function obtained using a perturbative expansion of the partition function in the path-integral\nin powers of $\\kappa$.\n \nConcerning the barrier random walks, in \\cite{CorasanitiAchitouv2011a,CorasanitiAchitouv2011b} we have introduced\na stochastic model with linear drift and Gaussian diffusion characterized by $\\langle B(S)\\rangle=\\delta_c+\\beta S$ and \n$\\langle B(S)B(S')\\rangle_c=D_B\\,{\\rm min}(S,S')$, where $\\delta_c$ is the linearly extrapolated critical spherical collapse density, $\\beta$ is \nthe average linear rate of deviation from the spherical collapse prediction and $D_B$ is the amplitude of the scatter about the average \n\\footnote{In the Excursion Set the barrier diffusion coefficient parametrizes the stochasticity inherent to the ellipsoidal collapse of halos. However, \nit is important to keep in mind that in the Excursion Set halos can form out of any random position. On the other hand, numerical simulations show\nthat halos form preferentially out of peaks of the linear density field as suggested by the hierarchical model of structure formation. \nThus when comparing with N-body results the value of $D_B$ can be biased by the underlying assumption of the Excursion Set approach \n(see \\cite{AseemRavi2012} for an extension of the formalism to random walks around density peaks).}. \nIn such a case the non-vanishing connected correlators of the $Y$ variable are \n\\begin{eqnarray}\n\\langle Y(S)\\rangle_c & = &\\delta_c+\\beta S \\label{yave}\\\\\n\\langle Y(S)Y(S')\\rangle_c &=& (1+D_B){\\rm min}(S,S')+\\Delta(S,S').\\label{yvar}\n\\end{eqnarray}\nSubstituting these expressions in Eq.~(\\ref{partition}) and performing a double expansion in $\\kappa$ and $\\beta$ \nwe have derived the Gaussian multiplicity function\n\\begin{equation}\\label{fddb}\nf_G(\\sigma)=f_{0}(\\sigma)+f_{\\kappa=1}(\\sigma)\n\\end{equation}\nwhere $f_0(\\sigma)$ is the Markovian contribution and $f_{\\kappa=1}(\\sigma)$ is the filter correction to \nfirst order in $\\kappa$ and up to second order in $\\beta$ which read as\n\\begin{equation}\nf_0(\\sigma)=\\frac{\\delta_c}{\\sigma}\\sqrt{\\frac{2a}{\\pi}}\\,e^{-\\frac{a}{2\\sigma^2}(\\delta_c+\\beta\\sigma^2)^2}\\label{fsigma0}\n\\end{equation}\nand\n\\begin{equation}\\label{fkappa1}\nf_{\\kappa=1}(\\sigma)=f_{1,\\beta=0}^{m-m}(\\sigma)+f_{1,\\beta^{(1)}}^{m-m}(\\sigma)+f_{1,\\beta^{(2)}}^{m-m}(\\sigma)\n\\end{equation}\nwith\n\\begin{equation}\nf_{1,\\beta=0}^{m-m}(\\sigma)=-\\tilde{\\kappa}\\dfrac{\\delta_c}{\\sigma}\\sqrt{\\frac{2a}{\\pi}}\\left[e^{-\\frac{a \\delta_c^2}{2\\sigma^2}}-\\frac{1}{2} \\Gamma\\left(0,\\frac{a\\delta_C^2}{2\\sigma^2}\\right)\\right],\n\\end{equation}\n\\begin{equation}\nf_{1,\\beta^{(1)}}^{m-m}(\\sigma)=- a\\,\\delta_c\\,\\beta\\left[\\tilde{\\kappa}\\,\\text{Erfc}\\left( \\delta_c\\sqrt{\\frac{a}{2\\sigma^2}}\\right)+ f_{1,\\beta=0}^{m-m}(\\sigma)\\right],\n\\end{equation}\n\\begin{equation}\\label{beta2}\nf_{1,\\beta^{(2)}}^{m-m}(\\sigma)=-a\\,\\beta\\left[\\frac{\\beta}{2} \\sigma^2 f_{1,\\beta=0}^{m-m}(\\sigma)+\\delta_c \\,f_{1,\\beta^{(1)}}^{m-m}(\\sigma)\\right],\n\\end{equation}\nwhere $\\tilde{\\kappa}=a\\,\\kappa$ and $a=1\/(1+D_B)$. Equation~(\\ref{beta2}) includes a term $\\mathcal{O}(\\beta^2)$ which was missing in the original derivation presented in \\cite{CorasanitiAchitouv2011a,CorasanitiAchitouv2011b}. As we explain in Appendix~\\ref{app} this is due to having neglected a factor $\\exp(-\\beta^2S\/2)$ in the computation of the probability $\\Pi_\\epsilon(Y_c,Y_c,S)$\\footnote{We thank Ruben van Drongelen for pointing this to us.} and which enters the calculation of the memory-of-memory term to first order in $\\kappa$.\n\n\\subsection{Non-Gaussian Initial Conditions}\nIn the case of non-Gaussian initial conditions the higher-order connect correlators of the linear density field are non-vanishing. \nLet us consider the case of primordial non-Gaussianity sourced by a bispectrum term, hence in addition to Eq.~(\\ref{yave}) \nand (\\ref{yvar}), the partition function contains the contribution of a non-vanishing 3-point connected correlation function \n$\\langle Y(S_i)Y(S_j)Y(S_k)\\rangle_c=-\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c$ with\n\\begin{equation}\\label{3pts}\n\\begin{split}\n&\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c=\\int\\dfrac{d^3k_i}{(2\\pi)^3}\\dfrac{d^3k_j}{(2\\pi)^3}\\dfrac{d^3k_k}{(2\\pi)^3}\\tilde{W}(k_i,R_i[S_i])\\\\\n&\\times\\tilde{W}(k_j,R_j[S_j])\\tilde{W}(k_k,R_k[S_k])\\mathcal{M}(k_i)\\mathcal{M}(k_j)\\mathcal{M}(k_j)\\times\\\\\n&\\times\\langle\\zeta(\\textbf{k}_i)\\zeta(\\textbf{k}_j)\\zeta(\\textbf{k}_k)\\rangle_c,\n\\end{split}\n\\end{equation} \nwhere $\\tilde{W}$ is the Fourier transform of the sharp-x filter, $\\mathcal{M}(k)=2\/(5 H_{0}^{2}\\Omega_m)T(k)k^2$, \n$H_0$ is the Hubble constant, $\\Omega_m$ the matter density, $T(k)$ the transfer function and $\\zeta(k)$ \nis the curvature perturbation with\n\\begin{equation}\n\\langle\\zeta(\\textbf{k}_i)\\zeta(\\textbf{k}_j)\\zeta(\\textbf{k}_k)\\rangle_c=(2\\pi)^3\\delta_D(\\textbf{k}_i+\\textbf{k}_j+\\textbf{k}_k) B(k_i,k_j,k_k),\n\\end{equation}\nwhere $B(k_i,k_j,k_k)$ is the so called ``reduced'' bispectrum. \n\nBy expanding Eq.~(\\ref{densitymark}) in powers of the amplitude of the reduced bispectrum (usually parametrized by the coefficient $f_{NL}$),\nwe obtain to first-order in $f_{NL}$ the non-Gaussian part of the first-crossing distribution\n\\begin{equation}\n\\mathcal{F}_{NG}(S)=-\\frac{\\partial}{\\partial S} F_{NG}(S)\n\\end{equation}\nwhere $F_{NG}(S)$ is the continuous limit of\n\\begin{equation}\n\\begin{split}\\label{Fng}\n&F_{NG}(S)=\\frac{1}{6}\\sum_{i,j,k=0}^{n}\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c\\times\\\\\n&\\times\\int_{Y_c}^{\\infty}dY\\int_{Y_c}^{\\infty}dY_1...dY_{n-1}\\partial_i\\partial_j\\partial_k W_{0}(Y_0,...,Y,S),\n\\end{split}\n\\end{equation}\nwhere $W_0(...)$ is the Gaussian Markovian probability density distribution.\nEq.~(\\ref{Fng}) can be evaluated provided we have an analytical expression for the primordial bispectrum. \nIn \\cite{AchitouvCorasaniti2012} we have used the standard approach of considering a triple Taylor series \nof the primordial bispectrum in the large scale limit. \nIn the next Section we will study in detail the range of validity of such an expansion and infer the\nrelevant contribution to the non-Gaussian halo mass function.\n\n\\section{Interval Validity of Primordial Bispectrum Expansion}\\label{secII}\nLet us expand the bispectrum Eq.~(\\ref{3pts}) in a triple Taylor series around $S_i\\sim S_j\\sim S_k\\sim S$ (for convenience we set $S=S_n$):\n\\begin{equation}\\label{Som}\n\\begin{split}\n&\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c =\\sum_{p,q,r=0}^{\\infty}\\dfrac{(-1)^{p+q+r}}{p!q!r!}\n(S-S_i)^p\\times\\\\&\\times(S-S_i)^q(S-S_k)^r G_{3}^{(p,q,r)}(S)\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\nG_{3}^{(p,q,r)}(S)\\equiv\\frac{d^p}{dS_{i}^{p}}\\frac{d^q}{dS_{j}^{q}}\\frac{d^r}{dS_{k}^{r}} \\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c\\bigg|_{i,j,k=n}.\n\\end{equation}\n\nWe expect the signature of primordial non-Gaussianity to be stronger at large scales ($S\\rightarrow 0$) where the evolution\nof the density field remains linear. Hence, the leading order contribution to the halo mass function is given by \nthe lowest order term of the bispectrum expansion. This corresponds to having $p+q+r=0$ which gives the leading order \nterm $\\langle\\delta^3(S)\\rangle$ that can be computed numerically using Eq.~(\\ref{3pts}) for a given\ntype of primordial NG. It is convenient to introduce the normalized skewness $\\mathcal{S}_3(S)=\\langle\\delta^3(S)\\rangle\/S^2$. \nIn \\cite{AchitouvCorasaniti2012} we have provided fitting formula\nfor $\\mathcal{S}_3(S)$ and its derivatives accurate to a few percent for local and equilateral non-Gaussianity.\n \nThe next-to-leading order contribution is given by three terms corresponding\nto the case $p+q+r=1$. Hence, up to next-to-leading order the primordial bispectrum reads as\n\\begin{equation}\\label{expg3}\n\\begin{split}\n&\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c =\\langle\\delta^3(S)\\rangle-(S-S_i)\\,G_{3}^{(1,0,0)}(S)+\\\\&-(S-S_j)\\,G_{3}^{(0,1,0)}(S)-(S-S_k)\\,G_{3}^{(0,0,1)}(S),\n\\end{split}\n\\end{equation}\nsince $S_i\\sim S_j\\sim S_k$ we can collect the terms in $(S-S_i)$, moreover by computing $G_3^{(1,0,0)}$, $G_3^{(0,1,0)}$ and $G_3^{(0,0,1)}$ as\nderivatives of Eq.~(\\ref{3pts}) one can notice that \n\\begin{equation}\\label{sumg3}\nG_{3}^{1,0,0}(S)+G_{3}^{0,1,0}(S)+G_{3}^{0,0,1}(S)=\\dfrac{dR}{dS}\\dfrac{d}{dR}\\langle\\delta^3(R)\\rangle\n\\end{equation}\nand introducing \n\\begin{equation}\n\\mathcal{U}_3(S)=\\dfrac{1}{S}\\dfrac{dR}{dS}\\dfrac{d}{dR}\\langle\\delta^3(R(S))\\rangle,\n\\end{equation}\nwe can rewrite Eq.~(\\ref{expg3}) as\n\\begin{equation}\\label{bispdec}\n\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\rangle_c\\approx S^2\\mathcal{S}_3(S) - (S-S_i) S\\,\\mathcal{U}_3(S).\n\\end{equation}\nWe can now derive a lower limit, $S_{\\rm min}$, on the value of $S_i,S_j,S_k$ for which such an expansion remains valid.\nThis is obtained by imposing the next-to-leading order term to be smaller than the leading one.\nWe find \n\\begin{equation}\nS_{\\rm min}=S\\left[1-\\frac{\\mathcal{S}_3(S)}{\\mathcal{U}_3(S)}\\right],\n\\end{equation}\nusing the fitting formulae for $\\mathcal{S}_3(S)$ and $\\mathcal{U}_3(S)$ derived in \\cite{AchitouvCorasaniti2012} we find that\nto good approximation $S_{\\rm min}=S\/\\alpha$ with $\\alpha^{-1}_{\\rm loc}= 0.373$ and $\\alpha^{-1}_{\\rm equi}= 0.382$ \nfor local and equilateral NG respectively.\n\n\\section{Non-Gaussian Halo Mass Function and Bispectrum Expansion Accuracy}\\label{secIII}\nHaving inferred a lower limit on the interval validity of the bispectrum expansion up to next-to-leading\norder we can infer a more accurate estimate of its contribution to the multiplicity function.\nGiven the bispectrum expansion Eq.~(\\ref{bispdec}) we can split Eq.~(\\ref{Fng}) as\n\\begin{equation}\nF_{NG}(S)=F_{NG}^L(S)+F_{NG}^{NL}(S).\n\\end{equation}\nAs shown in \\cite{AchitouvCorasaniti2012} the leading order term is given by \n\\begin{equation}\nF_{NG}^L(S)=\\frac{1}{6}S^2 S_3 (S)\\int_{Y_c}^{\\infty}dY\\,\\frac{\\partial^3}{\\partial Y_{c}^3} \\Pi_0(Y_0,Y,S)\n\\end{equation}\nwhere $\\Pi_0(Y_0,Y,S)$ is the probability distribution of the Gaussian random walks in the case of the Diffusing Drifting Barrier model given by Eq.~(8) in \\cite{CorasanitiAchitouv2011a}. The integral can be computed analytically to finally obtain the \nleading order contribution to the multiplicity function \\cite{AchitouvCorasaniti2012}:\n\\begin{widetext}\n\\begin{equation}\\label{fngl}\n\\begin{split}\nf_{NG}^{L}(\\sigma)=\\frac{a}{6}\\sqrt{\\frac{2a}{\\pi}}\\sigma e^{-\\dfrac{a(\\delta_c+\\beta\\sigma^2)^2}{2\\sigma^2}}\\biggl\\{S_3(\\sigma)\\biggl[\\dfrac{a^2}{\\sigma^4}\\delta_c^4\n-2\\dfrac{a}{\\sigma^2}\\delta_c^2-1+3\\dfrac{a^2}{\\sigma^2}\\beta\\delta_c^3+3 a\\beta \\delta_c+a^2\\beta^3\\sigma^2 \\delta_c+3 a^2\\beta^2\\delta_c^2+13 a \\beta^2\\sigma^2\\biggr]+\\\\\n+\\dfrac{dS_3(\\sigma)}{d\\log\\sigma}\\biggl[\\dfrac{a}{\\sigma^2}\\delta_c^2-1+3 a\\beta \\delta_c+4 a\\beta^2\\sigma^2\\biggr] \\biggr\\}\n+\\dfrac{2}{3}a^3\\beta^3\\sigma^4 e^{-2a\\beta \\delta_c}\\text{Erfc}\\biggl[\\sqrt{\\dfrac{a}{2\\sigma^2}}(\\delta_c-\\beta\\sigma^2)\\biggr]\\biggl\\{ 4 S_3(\\sigma)+\\dfrac{dS_3(\\sigma)}{d\\log\\sigma}\\biggr\\} \n\\end{split}\n\\end{equation}\n\\end{widetext}\n\nOn the other hand let us detail more the derivation of the next-to-leading order term which differs from\nthat of \\cite{AchitouvCorasaniti2012}. In such a case we have from Eq.~(\\ref{Fng})\nthat\n\\begin{equation}\n\\begin{split}\nF_{NG}^{NL}(S)&=-\\frac{1}{6}S\\,U_3 (S)\\sum_{i_{\\rm{min}}}^{i_{\\rm{max}}}(S-S_i)\\times\\\\\n&\\times\\int_{Y_c}^{\\infty} dY\\sum_{j,k=1}^{n}\\int_{Y_c}^{\\infty}dY_1...\\int_{Y_c}^{\\infty} dY_{n-1} \\,\\partial_i\\partial_j\\partial_k W_0,\n\\end{split}\n\\end{equation}\nwhere we have decomposed the sum in Eq.~(\\ref{Fng}) and kept only the terms up to $n-1$. In fact, as shown in \n\\cite{AchitouvCorasaniti2012} the integral in $dY_n$ of the terms with $i,j,k=n$ \nvanishes since the integrands are total derivatives. Furthermore it is easy to show that \n$\\sum_{j,k} \\rightarrow {\\partial^2}\/{\\partial Y_c^2}$, thus\n\\begin{equation}\n\\begin{split}\nF_{NG}^{NL}(S)&=-\\frac{1}{6}S\\,U_3 (S)\\sum_{i=i_{\\rm{min}}}^{n-1}(S-S_i)\\times\\\\\n&\\times\\int_{Y_c}^{\\infty} dY\\frac{\\partial^2}{\\partial Y_c^2} \\biggl[ \\int_{Y_c}^{\\infty}dY_1...\\int_{Y_c}^{\\infty} dY_{n-1} \\,\\partial_i W_0\\biggr],\n\\end{split}\n\\end{equation}\nwhere the sum is bounded from below due to the fact that $S_{min}<S_i<S$.\nThe multiple integral in the above expression can be computed by part and using the fact that $W_0$ obeys the Chapman-Kolmogorov\nequation \n\\begin{equation}\n\\begin{split}\n&W_0(Y_0,..,\\hat{Y}_i,..,Y,S)=W_0(Y_0,..,Y_c,S_i-S_{i_{\\rm{min}}})\\times\\\\\n&\\times W_0(Y_c,..,Y,S-S_i+S_{i_{\\rm{min}}})\n\\end{split}\n\\end{equation}\nwe have\n\\begin{equation}\n\\begin{split}\n&F_{NG}^{NL}=-\\frac{1}{6}S\\,U_3 (S)\\sum_{i=i_{\\rm{min}}}^{n-1}(S-S_i)\\times\\\\\n&\\times\\int_{Y_c}^{\\infty}dY\\frac{\\partial^2}{\\partial Y_c^2}\\biggl[\\Pi\\biggl(Y_0,Y_c,S_i-\\frac{S}{\\alpha}\\biggr)\\Pi\\biggl(Y_c,Y,S-S_i+\\frac{S}{\\alpha}\\biggr)\\biggr].\n \\end{split}\n\\end{equation}\nwhere $\\Pi(Y_0,Y_c,S_i-S\/\\alpha)$ and $\\Pi(Y_c,Y,S-S_i+S\/\\alpha)$ are given by Eq.~(3.20) and (3.21) in \\cite{AchitouvCorasaniti2012}\nrespectively. In the continuous limit and taking into account that the bispectrum expansion up to next-to-leading order is valid in the range\n $S_{min}\\approx S\/\\alpha<S_i<S$ we have $\\sum_{i=i_{\\rm{min}}}^{n-1}\\rightarrow \\int_{S\/\\alpha}^{S}dS_i$. \n\nFinally, we \nobtain the first-crossing distribution to next-to-leading order which reads as\n\\begin{equation}\n\\mathcal{F}_{NG}^{NL}(S)=-\\frac{1}{6}\\left\\{\\frac{\\partial^2 I_{NL}^{\\alpha}}{dY_{c}^2}\\left[U_3+S\\frac{dU_3}{dS}\\right]+S U_3\\frac{\\partial}{\\partial S}\\frac{\\partial^2 I_{NL}^{\\alpha}}{dY_{c}^2}\\right\\}\n\\end{equation}\nwith \n\\begin{equation}\\label{d2idyc2}\n\\begin{split}\n&\\frac{\\partial^2 I_{NL}^{\\alpha}}{\\partial Y_{c}^2}=-\\frac{a}{\\pi}\\int_{\\frac{S}{\\alpha}}^{S}\\frac{dS_i}{S_{i}^{3\/2}}e^{-\\frac{a}{2S_i}(\\delta_c+\\beta S_i)^2}\\left[S\\left(1-\\frac{1}{\\alpha}\\right)-S_i\\right]\\times\\\\\n&\\times\\left\\{-2 a \\beta+a^2\\beta^2 \\delta_c+a\\frac{\\delta_c}{S_i}\\left[-3+2 a \\beta \\delta_c+a\\frac{\\delta_c^2}{S_i}\\right]\\right\\}\\times\\\\\n&\\times\\left\\{e^{-\\frac{a}{2}\\beta^2(S-S_i)}+\\frac{\\beta}{2}\\sqrt{\\pi\\,a(S-S_i)}\\text{Erfc}\\left[-\\beta\\sqrt{\\frac{a}{2}(S-S_i)}\\right]\\right\\}\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\\label{ddsd2idyc2}\n\\begin{split}\n&\\frac{\\partial}{\\partial S}\\frac{\\partial^2 I_{NL}^{\\alpha}}{\\partial Y_{c}^2}=-\\frac{a}{\\pi}\\int_{\\frac{S}{\\alpha}}^{S}\\frac{dS_i}{S_{i}^{3\/2}} e^{-\\frac{a}{2S_i}(Y_0+\\beta S_i)^2}\\times\\\\\n&\\times\\left\\{-2 a \\beta+a^2\\beta^2 \\delta_c+a\\frac{\\delta_c}{S_i}\\left[-3+2 a \\beta \\delta_c+a\\frac{\\delta_c^2}{S_i}\\right]\\right\\}\\times\\\\\n&\\times\\left\\{\\frac{e^{-\\frac{a}{2}\\beta^2(S-S_i)}}{2\\sqrt{S-S_i}}\\left[1-\\frac{1}{\\alpha}+\\frac{1}{\\alpha}\\frac{S_i}{S-S_i}+\\frac{1}{\\alpha}a\\beta^2\\frac{S+S_i}{\\sqrt{S-S_i}}\\right]+\\right.\\\\\n&\\left.+\\sqrt{\\frac{a\\pi}{2}}\\beta\\left(1-\\frac{1}{\\alpha}\\right)\\text{Erfc}\\left[-\\beta\\sqrt{\\frac{a}{2}(S-S_i)}\\right]\\right\\}.\n\\end{split}\n\\end{equation}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.35]{fig1a.eps}\\\\\n\\includegraphics[scale=0.35]{fig1b.eps}\n\\end{tabular}\n\\caption{Leading order (red dash line), next-to-leading order (blue dot line) contribution to the non-Gaussian multiplicity function (black solid line) \nfor local (top panel) and equilateral (bottom panel) non-Gaussianities respectively.}\\label{fig1}\n\\end{figure}\n\nThe integrals in Eq.~(\\ref{d2idyc2}) and (\\ref{ddsd2idyc2}) can be computed numerically to finally evaluate the next-to-leading order contribution to the \nmultiplicity function, \n$f_{NG}^{NL}(\\sigma)=2\\sigma^2 \\mathcal{F}_{NG}^{NL}(\\sigma^2)$.\n\nIn Fig.~\\ref{fig1} we plot $f_{NG}^{L}(\\sigma)$, $f_{NG}^{NL}(\\sigma)$ and their sum in units of $f_{NL}$\nfor local (top panel) and equilateral (bottom panel) non-Gaussianity. We have set the\nDDB model parameters to the values best fitting the Gaussian halo mass function inferred from Gaussian N-body simulations in \\cite{PPH8}. \nThe mass interval shown here corresponds to that probed by these numerical simulations. We may notice that in the local and equilateral cases the leading\norder term is larger than the next-to-leading order one at high-masses ($\\log_{10}\\sigma^{-1}>0.1$), while at lower masses the next-to-leading order is larger. \nThis is expected since as already mentioned the signature of primordial non-Gaussianity at large masses results of\nthe lowest order in the bispectrum expansion. \n\nWe can now evaluate the overall contribution to the halo multiplicity function, $f(\\sigma)=f_G(\\sigma)+f_{NG}(\\sigma)$.\nIn Fig.~\\ref{fig3} we plot the relative difference of the NG halo mass function  with and without next-to-leading order term for local (top panel)\nand (bottom panel) equilateral non-Gaussianity respectively in the case of $f_{NL}=150$. As we can see the differences is no larger than $2\\%$\nin the low mass range, hence the next-to-leading term remains negligible even for large non-Gaussianities in the mass range\ncorresponding to halos with $M>10^{13} M_\\odot$ and can be neglected for practical purposes.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.35]{fig3modi.eps}\n\\end{tabular}\n\\caption{Relative difference bewteen of the non-Gaussian halo mass function with and without next-leading order contribution\nin the case of local (panel a) and equilateral non-Gaussianity (panel b).}\\label{fig3}\n\\end{figure}\n\n\n\\section{Trispectrum contribution to the Non-Gaussian Halo Mass Function}\\label{secIV}\nA number of scenarios of primordial inflation predict deviation from Gaussianity of the form (see e.g. \\cite{Sasaki2006,Enqvist2008})\n\\begin{equation}\\label{zeta4}\n\\zeta=\\zeta_G+\\frac{3}{5}f_{NL}(\\zeta_{G}^2-\\langle\\zeta_{G}^2\\rangle)+\\frac{9}{25}g_{NL}\\zeta_{G}^3+\\mathcal{O}(\\zeta_{G}^{4}),\n\\end{equation}\nwhere $\\zeta_{G}$ is the primordial curvature perturbation and $g_{NL}$ is the amplitude of the cubic term which give rises to\na non-vanishing 4-point connected correlation function of the linear density field given by\n\\begin{equation}\\label{4pts}\n\\begin{split}\n&\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\delta(S_l)\\rangle_c=\\int\\dfrac{d^3k_i}{(2\\pi)^3}\\dfrac{d^3k_j}{(2\\pi)^3}\\dfrac{d^3k_k}{(2\\pi)^3} \\dfrac{d^3k_l}{(2\\pi)^3}\\\\\n&\\tilde{W}(k_i,R_i)\\tilde{W}(k_j,R_j)\\tilde{W}(k_k,R_k)\\tilde{W}(k_l,R_l)\\times\\\\\n&\\times\\mathcal{M}(k_i)\\mathcal{M}(k_j)\\mathcal{M}(k_k)\\mathcal{M}(k_l)\\langle\\zeta(\\textbf{k}_i)\\zeta(\\textbf{k}_j)\\zeta(\\textbf{k}_k)\\zeta(\\textbf{k}_l)\\rangle_c\n\\end{split}\n\\end{equation}\nwhere $\\langle\\zeta(\\textbf{k}_i)\\zeta(\\textbf{k}_j)\\zeta(\\textbf{k}_k)\\zeta(\\textbf{k}_l)\\rangle_c=(2\\pi)^3\\delta_D(\\textbf{k}_i+\\textbf{k}_j+\\textbf{k}_k+\\textbf{k}_l) T(k_i,k_j,k_k,k_l)$\nand $T(k_i,k_j,k_k,k_l)$ is the trispectrum. \n\nWe can compute the trispectrum contribution to the multiplicity function by including the 4-point connect correlator\nin the partition function and expand the path-integral for small values of the trispectrum amplitude. As in the case of\nthe bispectrum, to leading order in a large scale expansion the trispectrum can be approximated as\n\\begin{equation}\n\\langle\\delta(S_i)\\delta(S_j)\\delta(S_k)\\delta(S_l)\\rangle_c\\simeq\\langle\\delta^4(S)\\rangle_c\n\\end{equation}\nTo first-order in the trispectrum amplitude we have\n\\begin{equation}\\label{fcrstri}\n\\mathcal{F}_{NG}^{\\rm Tri,L}(S)=-\\dfrac{\\partial}{\\partial S}F_{NG}^{\\rm Tri,L}(S),\n\\end{equation}\nwhere $F_{NG}^{\\rm Tri,L}(S)$ is the continuous limit of \n\\begin{equation}\n\\begin{split}\n&F_{NG}^{\\rm Tri,L}(S)=-\\dfrac{1}{4!}\\sum_{i,j,k,l}\\langle\\delta^4(S)\\rangle_c\\times\\\\\n&\\times\\int_{Y_c}^{\\infty}dY \\int_{Y_c}^{\\infty}dY_1...dY_{n-1}\\partial_i\\partial_j\\partial_k\\partial_l W_0.\n\\end{split}\n\\end{equation}\nUsing the fact that $\\sum_{i,j,k,l} \\rightarrow {\\partial^4}\/{\\partial Y_c^4}$ we obtain\n\\begin{equation}\\label{fngtri}\nF_{NG}^{\\rm Tri,L}(S)=-\\dfrac{1}{4!}\\langle\\delta^4(S)\\rangle_c\\frac{\\partial^4}{\\partial Y_{c}^{4}}\\int_{Y_c}^{\\infty}\\Pi_0(Y_0,Y,S)\\,dY,\n\\end{equation}\nthe integral can be computed analytically, \n\\begin{equation}\n\\begin{split}\n&\\frac{\\partial^4}{\\partial Y_{c}^{4}}\\int_{Y_c}^{\\infty} \\Pi(Y_0,Y,S)\\,dY=\\sqrt{\\frac{a}{2\\pi S}}e^{\\frac{a}{2S}(\\delta_c+\\beta S)^2}\\times\\\\\n&\\times\\biggl[-16(a\\beta)^3+8\\frac{a^2}{S}\\beta+6\\frac{a^2}{S^2}Y_0-14\\frac{a^3}{S}\\beta^2 \\delta_c-8\\frac{a^3}{S^2}\\beta \\delta_c^{2}+\\\\\n&-2\\frac{a^3}{S^3}\\delta_c^3\\biggr]-8(a\\beta)^4e^{-2a\\beta Y_0}\\text{Erfc}\\left[\\sqrt{\\frac{a}{2S}}(Y_0-\\beta S)\\right].\n\\end{split}\n\\end{equation}\n\nAs in the case of the bispectrum, it is convenient to introduce the 4th-order reduced cumulant,\n$\\mathcal{S}_{4}(R)\\equiv\\langle\\delta^4(R)\\rangle\/S^3$.\nSubstituting in Eq.~(\\ref{fngtri}) and evaluating the first-crossing distribution Eq.~(\\ref{fcrstri})\nwe finally obtain the trispectrum contribution to the multiplicity function:\n\\begin{equation}\n\\begin{split}\\label{fL4}\n&f_{NG}^{\\rm Tri,L}(\\sigma)=2\\,\\mathcal{S}_4(\\sigma)\\sigma^6 (a\\beta)^4 e^{-2a\\beta \\delta_c}\\times\\\\\n&\\times\\text{Erfc}\\left[\\sqrt{\\frac{a}{2\\sigma^2}}(\\delta_c-\\beta\\sigma^2)\\right]+\\mathcal{S}_4(\\sigma)e^{-\\frac{a}{2\\sigma^2}(\\delta_c+\\beta\\sigma^2)^2}\\times\\\\\n&\\times(a\\sigma)^2\\sqrt{\\dfrac{2a}{\\pi S}}\\biggl[-\\frac{1}{2}\\beta\\sigma^2+\\frac{11}{6}a\\beta^3\\sigma^4-\\frac{1}{8}\\delta_c+a(\\beta\\sigma)^2\\delta_c+\\\\\n&+\\frac{1}{24}a^2(\\beta\\sigma)^4\\delta_c+\\frac{1}{6}(a\\sigma \\delta_c)^2\\beta^3+\\frac{1}{4}(a\\beta)^2\\delta_c^3-\\frac{1}{6}a\\frac{\\delta_c^3}{\\sigma^2}+\\\\\n&+\\frac{1}{6}\\frac{a^2}{\\sigma^2}\\beta\\delta_c^4+\\frac{1}{24}a^2\\frac{\\delta_c^5}{\\sigma^4}\\biggr]+\\frac{1}{3}\\frac{d\\mathcal{S}_4(\\sigma)}{d\\log\\sigma}\\sigma^6 (a\\beta)^4 e^{-2a\\beta \\delta_c}\\times\\\\\n&\\times\\text{Erfc}\\left[\\sqrt{\\frac{a}{2\\sigma^2}}(\\delta_c-\\beta\\sigma^2)\\right]+\\frac{d\\mathcal{S}_4(\\sigma)}{d\\log\\sigma}e^{-\\frac{a}{2\\sigma^2}(\\delta_c+\\beta\\sigma^2)^2}\\times\\\\\n&\\times(a\\sigma)^2\\sqrt{\\dfrac{2a}{\\pi S}}\\biggl[-\\frac{1}{6}\\beta\\sigma^2+\\frac{1}{3}a\\beta^3\\sigma^4-\\frac{1}{8}\\delta_c+\\\\\n&+\\frac{7}{24}a(\\beta\\sigma)^2 \\delta_c+\\frac{1}{6}a\\beta \\delta_c^2+\\frac{1}{24}a\\frac{\\delta_c^3}{\\sigma^2}\\biggr]\n\\end{split}\n\\end{equation}\n\nIt can be noticed that by setting the barrier model parameters to the spherical collapse values $a=1$ and $\\beta=0$ we recover the formula\nderived in \\cite{MaggioreRiotto2010d}. It is also worth noticing that in the spherical collapse limit and \nneglecting the filter correction to first order in $\\kappa$, the NG multiplicity function given by the sum of the Markovian term, the bispectrum and trispectrum leading order contributions has the same functional form as that derived in \\cite{LV} using the Edgeworth expansion to describe the non-Gaussian probability distribution of the initial density perturbations.\n\nThe above formula has been derived without making any assumption on the mechanism that generates the non-vanishing 4-point correlation function\nof the primordial density field, namely the specific form of the trispectrum, $T(k_i,k_j,k_k,k_l)$. Furthermore, the  \namplitude of the trispectrum is affected not only by the cubic term in \nEq.~(\\ref{zeta4}) and parametrized in terms of $g_{NL}$, but also by the skewness which is parametrized by $f_{NL}$. \nIn models where curvature perturbations are sourced by a single scalar field, as in the case of the curvaton model \n(see e.g. \\cite{LythWands2002}) the skewness contribution to the kurtois (parametrized in terms of $\\tau_{NL}$) \nis given by $\\tau_{NL}=36\/25 f_{NL}^2$. In \\cite{SuyamaYamaguci2008} it has been shown that for a variety\nof inflationary scenarios holds the disequality $\\tau_{NL}\\ge 36\/25 f_{NL}^2$. Recently, \nthe authors of \\cite{Biagettietal2012} have argued that violating such an inequality would imply\nsome non-trivial new physics, since the inequality results on the one hand from the fact that NG is generated on super-horizon scales \nand on the other hand on the positivity of the 2-point correlation function. Thus, testing such inequality may provide \nhints of fundamental physics at the epoch of inflation.\n\nIt is beyond the scope of this paper to compute the trispectrum\nfor specific primordial non-Gaussian scenarios for which the forth-order reduced cumulant needs\nto be numerically computed for a given trispectrum template. Hence, for simplicity we limit to local type of non-Gaussianity for\nfitting functions of $\\mathcal{S}_4(\\sigma)$ has been computed in \\cite{Yokoyama2011}. Even in such a restricted case\nwe can still infer some relevant information on the imprint of the trispectrum on the halo mass function and the implication\nfor testing the Suyama-Yamaguchi inequality.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.35]{fig4a.eps}\\\\\n\\includegraphics[scale=0.35]{fig4b.eps}\n\\end{tabular}\n\\caption{Leading order contribution of the bispectrum (red dash line), the trispectrum (blue dot line) \nand their sum (black solid line) for local type of primordial non-Gaussianity for $f_{NL}=100$, $\\tau_{NL}=10^4$ in the case of \n$g_{NL}=0$ (bottom panel) and $g_{NL}=10^6$ (top panel) respectively.}\\label{fig4}\n\\end{figure}\nTo this purpose we set the DDB model parameters to their Gaussian value\\footnote{As shown in \\cite{AchitouvCorasaniti2012} \nthis is a good approxiamtion for $f_{NL}<150$.} and plot in Fig.~\\ref{fig4}\nthe contribution of the bispectrum (red dash line), trispectrum (blue dot line) and their sum (black solid line) to the multiplicity function\nfor values of primordial non-Gaussian amplitude which are consistent with Cosmic Microwave Background (CMB) limits \\cite{Smidt2010}. \nIn particular, we set $f_{NL}=100$, $\\tau_{NL}=10^4$ and consider $g_{NL}=0$ (bottom panel) and $g_{NL}=10^6$ (top panel) respectively. \nWe can see that even for $g_{NL}=10^6$ and $\\tau_{NL}=10^4$ the trispectrum signal exceeds that of the bispectrum \nonly in a very limited low-mass range. In contrast, for $g_{NL}=0$ and $\\tau_{NL}=10^4$ the trispectrum  \ncontribution to the non-Gaussian multiplicity function remains very small compared to that of the bispectrum.\nThis suggests that within current CMB limits, a violation of the Suyama-Yamaguci inequality\nwill be hardly detectable solely using the halo mass function. It is possible that measurements of the galaxy bias\nmay be more informative as investigated in \\cite{Biagettietal2012}. On the other hand constraints on the halo mass function\nfrom cluster counts may still be a useful probe when used in combination with estimates of the halo bias \nfrom the clustering of massive clusters. As shown in \\cite{Pillepich2012} for the case of primordial bispectrum, these tests can provide \nimproved constraints on primordial non-Gaussianity from the upcoming generation of cosmic structure surveys.\n\n\\section{Conclusion}\\label{secV}\nThe halo mass function carries an imprint of the statistics of the primordial density field as well as the properties of the halo\ncollapse process. The path-integral formulation of the Excursion Set theory provides a powerful and self-consistent mathematical\nframework to account for these effects on the halo mass function. Here, we have extended a previous analysis \\cite{AchitouvCorasaniti2012} and \nperformed a more accurate derivation of the contribution of the primordial bispectrum expanded in the large\nscale limit to next-to-leading order for the Diffusive Drifting Barrier model introduced \nin \\cite{CorasanitiAchitouv2011a,CorasanitiAchitouv2011b}. We have shown that \nthe next-to-leading order term of the primordial bispectrum decomposition contributes to no more than $\\sim2\\%$ of the \nnon-Gaussian mass function. Thus, for all practical purposes it can be neglected. We have also derived an analytic formula\nfor the trispectrum contribution. As in the case of the bispectrum, the multiplicity function depends on terms which couple the \nparameters encoding the ellipsoidal collapse of halos with the primordial four-point correlation function.\nAlso in this case we find that in the spherical collapse limit the trispectrum contribution reduces to the functional \nform derived in the Press-Schechter formalism using the Edgeworth expansion. However, in order to reproduce N-body simulation results the\nlatter requires two ad-hoc prescriptions. First, the non-Gaussian prediction is rescaled by a Gaussian simulation \ncalibrated multiplicity function, such as to account for the imprints of the ellipsoidal collapse. Thus, implicitly assuming that the effect of the non-spherical collapse of halos on the mass function is independent of the amplitude of primordial non-Gaussianity. A good modelling of the collapse parameters is even more relevant for primordial trispectrum terms which act at low masses where the simple spherical collapse is not valid since qualitatively the trispectrum signature on the halo mass function can be mimic by a higher value of $\\beta$. In principle, the later can probe multi-field inflation by testing the validity of the Suyama-Yamaguchi inequality. However, assuming values of $f_{NL}$ and $\\tau_{NL}$ consistent with current CMB limits, we find that bispectrum contribution is always the dominant NG signal. It is possible that tests of the scale dependent halo bias can be more informative regarding the trispectrum signature. In such a case, it will be interesting to investigate, in the context of the peak background split,  how the mass filtering corrections and the coupling between non-spherical collapse parameters and primordial non-Gaussian amplitudes alter the linear halo bias prediction.\n\n\\begin{acknowledgments}\nWe thank James G. Bartlett for his support and his advices. We also thank Ruben van Drongelen and Koenraad Schalm for useful discussions. I. Achitouv is supported by a scholarship of the `Minist\\`ere de l'Education Nationale, de la Recherche et de la Technologie' (MENRT). The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7\/2007-2013 Grant Agreement no. 279954).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTranscription profiling by deep sequencing (RNA-seq) has become the technology\nof choice for quantifying gene expression, because of its high throughput, its\nlarge dynamic range, and the possibility to identify novel transcripts.\nHowever, noise in RNA-seq experiments is not yet well understood. Hence,\nappropriate statistical models to quantify gene expression and assess\nstatistical significance of differential expression (DE) are needed and are the\nobject of intense research \\citep[e.g.,][]{Garber2011}.\n\nDue to biological variation, gene-level RNA-seq count data are overdispersed\nrelative to the Poisson distribution. Most methods assume that the\nread counts come from negative binomial distributions and differ in their\ntreatment of the (over-)dispersion parameters.\nThe edgeR method \\citep{Robinson2008,robinson2010edger} estimates \ndispersion parameters of individual genes robustly by shrinking them toward a\ngenome-wide parametric function relating dispersion to mean, whereas\nDEseq \\citep{Anders2010} models this relation nonparametrically.\n\\citet{Lund2012} noticed the importance of taking uncertainty\nin the estimation of the dispersion parameter into account and were able to\nintegrate it into their statistical test. More recently, \\citet{Wu2013}\nobserved systematic over- and under-shrinkage of the gene-specific\ndispersion parameters in edgeR and DEseq, respectively. To circumvent\nthis issue, \\citet{Wu2013} stayed within a frequentist paradigm but placed a prior\non dispersion parameters and thus were able to obtain better DE detection in their analyses.\n\nMoreover, calling DE genes is typically the starting\npoint for further analyses, most prominently gene set enrichment analyses.\nHowever, the power for calling a gene DE typically increases with\nits average read counts. Frequentist approaches that do not take this variation in power into account are bound to mischaracterize uncertainty regarding whether sets are enriched.\nTo cope with the difference in testing power among genes, dedicated gene set\nenrichment methods have been investigated \\citep{Young2010,Mi2012}.\n\n\nAltogether, the limitations met by frequentist approaches --- for example, properly quantifying uncertainty for identifying\nDE genes and for downstream analyses\n--- call for a fully Bayesian treatment. \\citet{Hardcastle2010} followed an\nempirical Bayesian approach called baySeq. However, the prior parameters are\nestimated using quasi-likelihood methods, which ignores uncertainty in the parameter estimation. Two\nBayesian approaches, MMseq \\citep{Turro2011}, which focuses on Poisson noise, and\nthe more recent Bitseq \\citep{Glaus2012}, which accounts for overdispersed\ndata, were developed to tackle the more complicated problem of estimating expression levels\nof transcript isoforms originating from the same locus, leading to highly coupled\nestimates. In order to address this difficult task, BitSeq required some\nshortcuts in the inference procedure. Moreover, the approach does not return explicit\nposterior probabilities for genes to be differentially expressed. Instead, the authors advise to use the ranking of the most probable up- and\ndown-regulated genes.\n\n\n\nHere, we propose a fully Bayesian analysis of differential expression in RNA\nsequencing data (BADER). Because we perform full posterior inference, our\napproach takes account of all uncertainties in a natural way and is theoretically\njustified.\n\nBADER allows the dispersion parameters to be gene-specific, yet it is robust even\nfor datasets with very small sample sizes due to borrowing of information across\ngenes through a common hyperprior.\nWe model the log fold change of a gene (\\emph{i.e.}, the log ratio of the mean\nexpression levels between two groups) by a continuous random variable\nwith a point mass at zero. This provides a probabilistic indicator whether a gene\nexhibits true DE, or if the difference in expression is\ncaused by random variability. It also yields a density estimate for the magnitude\nof the log fold change in the case of DE. Using simulated data, we show that\nBADER exhibits excellent performance in terms of DE detection.\n\nFinally, we demonstrate how posterior samples of the DE indicators can be integrated into downstream\nanalyses in the case of gene set enrichment. Our results show that this approach\nhas more power to detect enriched gene sets than a frequentist counterpart.\n\n\n\n\\section{Methods}\n\\subsection{Model Description \\label{sec:model}}\n\nAssume that from an RNA-seq experiment we have a set of read counts of the form:\n\\begin{equation}\n\\label{data}\n\\{k_{ij}^T \\!: \\; i = 1,\\dots,n_T ; \\, j=1,\\dots,m; \\, T=A,B \\},\n\\end{equation}\nwhere $i$ denotes the library or sample, $j$ denotes the genomic region, and $T$\ndenotes the experimental condition (treatment). ``Genomic region'' here could be a gene, an exon, or a set of exon, but henceforth we will refer to it as a gene for simplicity.\n\nIn many recent publications \\citep[e.g.,][]{Robinson2008,Anders2010}, the\nnegative-binomial distribution is the overdispersed-Poisson data model of\nchoice for such data. This distribution can be written as the marginal\ndistribution of a Poisson random variable whose rate parameter follows a gamma\ndistribution. Here we assume that the Poisson rate parameter follows a\nlognormal distribution, leading to the following two-stage data model (see Appendix \\ref{app:analyticcomparison} for more details):\n\\begin{equation}\n\\label{lognormalmodel}\n\\begin{split}\n   k_{ij}^T | \\lambda_{ij}^T & \\stackrel{ind.}{\\sim} Poi( s_i^T e^{ \\lambda_{ij}^T}); \\quad \\text{for all} \\; i, j, T, \\\\\n   \\lambda_{ij}^T | \\mu_j^T, \\alpha_j^T & \\stackrel{ind.}{\\sim} N( \\mu_j^T, e^{\\alpha_j^T}  ); \\quad \\text{for all} \\; i, j, T,\n\\end{split}\n\\end{equation}\nwhere $s_i^T$ denotes the \\textit{sampling depth} of sample or library $i$ in\ngroup $T$, which we estimate using the median ratio estimator of\n\\citet{Anders2010}. The data model \\eqref{lognormalmodel} was chosen for\ncomputational reasons. While inference from the gamma and lognormal\ndistribution typically leads to the same conclusions (\\citealp{Atkinson1982};\n\\citealp{McCullagh1989}, pp.\\ 286\/293), the\nlatter allows us to obtain many MCMC parameter updates in closed form (see\nSection \\ref{sec:inference} below), which results in more efficient MCMC\nsampling (see Appendix \\ref{app:numericalcomparison}).\n\n\nThe log mean rates for the two groups $A$ and $B$ are noted by $\\mu_j^A$\nand $\\mu_j^B$, respectively. For $\\mu_j^A$, we\nassume a noninformative prior with density $[\\mu_j^A] \\propto 1$, independently\nfor $j = 1,\\dots,m$. A crucial component of our model is the log-fold-change\nparameter, $ \\gamma_j \\colonequals \\mu_j^B - \\mu_j^A$, whose prior distribution\nis assumed to be a mixture distribution of a point mass at zero --- indicating\nno differential expression (DE) for gene j --- and a normal distribution: \n\\begin{equation}\n\\label{foldchange}\n\t\\gamma_j | I_j, \\sigma_\\gamma^2 \\stackrel{ind.}{\\sim}\n           \\begin{cases} 0, & I_j = 0 \\\\ N(0,\\sigma_\\gamma^2), & I_j = 1 \\end{cases} ; \\quad\n            j=1,\\dots,m,\n\\end{equation}\nwhere $I_j | \\pi \\stackrel{iid}{\\sim} B(\\pi) $, $j=1,\\dots,m$, are Bernoulli DE indicators.\n\nThe parameters $\\alpha_j^T$ in \\eqref{lognormalmodel} determine the degree of\noverdispersion (for $\\alpha_j^T \\rightarrow - \\infty$, the data model for\n$k_{ij}^T$ in \\eqref{lognormalmodel} is exactly Poisson). We assume\n\\begin{equation}\n\\label{dispersionprior}\n\\alpha_j^T \\stackrel{iid}{\\sim} N(\\psi_0,\\tau^2); \\quad j=1,\\ldots,m; \\, T = A,B.\n\\end{equation}\n\nFor all hyperparameters in \\eqref{foldchange} and \\eqref{dispersionprior}, we\nassume independent noninformative priors: $\\pi \\sim U(0,1)$, $[\\psi_0] \\propto\n1$, $[\\sigma_\\gamma^2 ] \\propto 1\/\\sigma_\\gamma^2$, and $[\\tau^2] \\propto\n1\/\\tau^2$. The latter two priors are the Jeffreys prior for the variance\nparameter of a normal distribution.\n\n\n\n\\subsection{Posterior Inference \\label{sec:inference}}\n\nGiven a set of RNA-seq counts, $\\{k_{ij}^T\\}$, as in \\eqref{data}, the primary interest is in the\nposterior distribution of the log-fold-change parameters, $\\{ \\gamma_j \\}$. Once\nwe have obtained this (joint) distribution, it is trivial to derive quantities of\ninterest, like the individual posterior DE probabilities, $ p_j \\colonequals\nP(I_j = 1 | \\{k_{ij}^T\\})$; $j=1,\\dots,m$. A great advantage of Bayesian\ninference is that we obtain \\emph{joint} posterior distributions, and so we do\nnot have to account for multiple testing explicitely (see,\n\\citep[e.g.,][]{Scott2006}. We can also use the results for further downstream analyses\n(e.g., for gene set enrichment; see Section \\ref{sec:enrichment} below).\n\n\nBecause there is no closed-form expression for the joint posterior of all\nunknown variables in the model, we resort to sampling-based inference via a Markov chain\nMonte Carlo (MCMC) algorithm. We use a Gibbs sampler \\citep{Geman1984}, which\ncycles through all unknowns and updates each parameter based on its so-called\nfull conditional distribution (FCD). For a generic parameter (or set of\nparameters) $\\theta$, the FCD is defined as the conditional distribution of\n$\\theta$ given the data and all other variables in the model, and we denote the\nFCD as $[\\theta | \\,\\cdot\\,]$.\n\n\nThe FCDs of the rate and dispersion parameters are not available in closed form,\n\\begin{align*}\n\\textstyle[\\lambda_{ij}^T | \\, \\cdot \\, ]\\propto\\, & \\textstyle Poi( k_{ij}^T | s_i^T e^{\\lambda_{ij}^T} ) \\,N(\\lambda_{ij}^T | \\mu_j^A + \\gamma_j I(T\\!=\\!B), \\exp\\{\\alpha_j^T\\}) , \\displaybreak[0]\\\\\n\\textstyle[\\alpha_j^T | \\,\\cdot \\, ]\\propto\\, &\\textstyle\\big( \\prod_{i} N(\\lambda_{ij}^T | \\mu_j^A +  \\gamma_j I(T\\!=\\!B), \\exp\\{\\alpha_j^T\\}) \\big) N(\\alpha_j^T | \\psi_0 ,\\tau^2),\n\\end{align*}\nand so we use adaptive Metropolis-Hastings steps \\citep{Haario2001} to update these parameters for all $i,j,T$.\n\nThe updates of the hyperparameters are given by:\n\\begin{align*}\n\t\\pi |\\,\\cdot\\,\\sim\\,& \\textstyle Beta(1+ \\sum_{j} I_j, 1+ \\sum_{j} (1-I_j) ), \\displaybreak[0]\\\\\n\t\\sigma_\\gamma^2 |\\, \\cdot\\, \\sim \\,&\\textstyle InvGamma(\\sum_{j} I_j\/2,\\sum_{j } \\gamma_j^2 I_j \/2), \\displaybreak[0]\\\\\n\t\\psi_0 |\\, \\cdot \\,\\sim\\,& \\textstyle N (\\sum_{j,T}\\alpha_j^T\/2m,\\tau^2\/2m), \\\\\n\t\\tau^2 |\\, \\cdot\\, \\sim\\,&\\textstyle InvGamma(m, \\sum_{j,T}(\\alpha_j^T-\\psi_0)^2\/2).\n\\end{align*}\n\nInitial tests showed heavy posterior dependence between $I_j$, $\\mu_j^A$ and\n$\\gamma_j$. Fortunately, our choice of the lognormal distribution in\n\\eqref{lognormalmodel} allows updating these parameters jointly from $[\\gamma_j,\n\\mu_j^A, I_j | \\cdot ]$ by integrating both $\\mu_j^A$ and $\\gamma_j$ out when\nupdating $I_j$ and by integrating $\\gamma_j$ out when updating $\\mu_j^A$. Let\n$\\Omega$ denote the set of all parameters, and define $\\overline{\\lambda_j^T}\n\\colonequals  \\sum_i \\lambda_{ij}^T\/n_T$, $v_0 \\colonequals \\exp\\{\\alpha_j^B\\}$\nand $v_1\\colonequals \\exp\\{\\alpha_j^B\\} + n_T \\sigma_\\gamma^2$. The updates are\nas follows:\n\\begin{align*}\n\\textstyle I_j | \\Omega \\backslash \\{\\mu_j^A, \\gamma_j, I_j\\} & \\textstyle\\stackrel{ind.}{\\sim} B\\big(\\pi N_1\/( \\pi N_1 + (1-\\pi) N_0) \\big), \\; \\text{where} \\\\\nN_l & \\textstyle\\colonequals N\\big(\\overline{\\lambda_j^A} | \\overline{\\lambda_j^B}, (\\exp\\{\\alpha_j^A\\} + v_l)\/n_T\\big); \\quad l \\in \\{0,1\\}, \\displaybreak[0]\\\\\n\\textstyle\\mu_j^A |  \\Omega \\backslash \\{\\mu_j^A, \\gamma_j \\} &\\textstyle\\stackrel{ind.}{\\sim} \nN\\Big(\\frac{\\overline{\\lambda^A_j} v_{I_j} + \\overline{\\lambda_j^B} \\exp\\{\\alpha_j^A\\}}{\\exp\\{\\alpha_j^A\\} + v_{I_j}},\\frac{\\exp\\{\\alpha_j^A\\} v_{I_j}}{n_T(\\exp\\{\\alpha_j^A\\} + v_{I_j})}\\Big),\\displaybreak[0]\\\\\n\\textstyle \\gamma_j | \\Omega \\backslash \\{\\gamma_j\\} & \\textstyle \\stackrel{ind.}{\\sim} \n\\begin{cases} 0, & I_j = 0, \\\\ N \\big( \\frac{\\sigma_\\gamma^2 \\sum_i{(\\lambda_{ij} - \\mu_j^A)}}{v_1}, \\frac{\\sigma_\\gamma^2 \\exp\\{\\alpha_j^B\\}}{v_1}\\big), &I_j = 1, \n\\end{cases} \n\\end{align*}\nfor $j=1,\\ldots,m$.\nFurther details and proofs can be found in Appendix \\ref{postinferencedetails}.\n\nTypically, we run the MCMC algorithm for 20,000 iterations, discarding the first 10,000 as burn-in steps and saving every 10th remaining step. In a setup with 10,000 genes and 2 samples in each group, one iteration takes around 0.13 seconds on a 2 GHz 64-bit processor with the BADER software. Computation times should change approximately linearly for other numbers of genes or samples, as the required number of computations for posterior inference is $\\mathcal{O}(mn)$.\n\n\n\\section{Results}\n\n\\subsection{Inference on the Dispersion Parameter \\label{sec:dispersion}}\n\nTo assess inference on dispersion parameters in a realistic case,\nwe used a large dataset published by \\citet{Pickrell2010},\navailable from\n\\url{http:\/\/bowtie-bio.sourceforge.net\/recount\/} \\citep{Frazee2011}.\nOf the $69$ samples in the dataset, we took subsamples of $n \\in \\{2,5,10,20\\}$,\ncorresponding to typical sample sizes in RNA-seq datasets. Genes\nwhere the sum of the remaining counts for all samples were less or equal to 5\nwere dropped from the analysis. We assumed that the posterior medians of\n$\\mu_j$ and $\\alpha_j$ from the full dataset were the ``true'' values, and\ncalculated the empirical coverage of central $80\\%$ posterior credible\nintervals for these parameters derived from the subsamples. For both parameters\nand for all (sub-)sample sizes, the empirical coverage of the ``true'' values\nwas between about $70\\%$ and $90\\%$, indicating accurate\ncalibrations of the posterior distributions of the gene-specific dispersion\nparameters.\n\n\n\\subsection{Determining Differential Expression \\label{sec:DE}}\n\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.65\\textwidth]{scatterLog2.pdf}\n\\caption{Scatterplot of the posterior mean of the log fold change parameter from BADER against the normalized mean count for genes in the dataset from \\citet{Katz2010}.\nThe red stars indicate genes with a posterior probability of DE higher than $0.9$. The plot inserts show the posterior distribution of the log fold change parameter of a gene found to be DE with a low mean count, and of a gene with a high absolute log fold change parameter but posterior probability of DE smaller than $0.9$.}\n\\label{scatterMouse}\n\\end{figure}\n\nWe ran BADER on a dataset from \\citet{Katz2010}, where we dropped all genes\nfor which the sum of all counts were less or equal to 5. Figure \\ref{scatterMouse} \nshows a scatterplot of the posterior means of the log fold change parameters against the normalized mean counts, with red stars indicating the genes with a posterior probability of DE higher than $0.9$. Interestingly, we find some\ngenes with high absolute value of the log fold change parameter but smaller\nposterior probability of DE than $0.9$, due to the heterogeneous gene-specific posterior uncertainty allowed by our flexible model.\n\n\nBecause we do not know the ground truth, these results do not tell us whether BADER improves upon existing methods in terms of DE detection. Hence, we then compared methods on simulated datasets where we do know the underlying truth. We generated 100 datasets, each consisting of $n_A = n_B =2$ samples for $m=5000$ ``genes,'' divided into 250 sets of 20 genes each. Of these groups, $90\\%$ were assumed to be not enriched and we chose a small probability of DE ($0.1$) for a gene in these groups. The remaining $10\\%$ of groups were assumed to be enriched and consequently, genes in these sets were DE with a high probability of $0.75$.\nThe individual gene expression counts were simulated according to model \\eqref{lognormalmodel} with hyperparameters $\\psi_0 = -3$, $\\tau = 0.8$, and $\\sigma_\\gamma = \\log_2(1.5)$, which were chosen to be as realistic as possible. The mean log expression level for gene set $l$ was chosen as $\\bar{\\mu}^A_l \\sim N(5,1)$ for $l=1,\\ldots,250$, with individual gene log expression levels in set $l$ simulated as $\\mu_j^A | \\bar{\\mu}^A_l \\sim N(\\bar{\\mu}^A_l, 0.5^2)$. \nIn this subsection, we will focus on determining DE for individual genes, and we will consider inference on enriched gene sets in Section \\ref{sec:enrichment}.\n\nUsing these 100 simulated datasets, we compared BADER (version 0.99.2) to edgeR \\citep{robinson2010edger} (version 2.6.10), DESeq \\citep{Anders2010} (version 1.12.1), baySeq\n\\citep{Hardcastle2010} (version 1.14.1), and DSS \\citep{Wu2013} (version 1.4.0), applying the methods from the packages' vignettes. We compared results averaged over all 100 datasets using a receiver operating characteristic (ROC) curve, which plots the true positive rate versus the false positive rate when varying over a threshold parameter $q$\n(Figure~\\ref{ROCfull}). In the case of edgeR, DESeq, and DSS, we take all genes to be DE for which the p-value is lower than $q$. In the case of baySeq and BADER, we take genes to be DE if the corresponding posterior probability is larger than $1-q$. This analysis showed that BADER outperformed all other approaches in calling genes differentially expressed.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\textwidth]{ROCbig.pdf}\n\\caption{DE-detection ROC curves for simulated data: Comparing BADER to four competitors}\n\\label{ROCfull}\n\\end{figure}\n\nIt is important to note again that, while BADER performs well in terms of DE detection, the output from our algorithm is much richer than just posterior DE\nprobabilities for individual genes. Among other things, we obtain the joint posterior distribution of the log-fold-change parameters $\\gamma_j$ (see\nEquation \\eqref{foldchange}), which, for example, also contains the posterior distribution of the magnitude of the fold change for DE genes. In addition,\nthis output allows for further downstream analyses, an example of which is given in the next section.\n\n\n\n\\subsection{Gene Set Enrichment \\label{sec:enrichment}}\n\nOnce we have obtained samples from the posterior distribution as described in Section \\ref{sec:inference}, inference on sets or groups of genes (``gene set enrichment'') can proceed without much difficulty. For example, one could use the posterior distributions of DE indicators as an input to the algorithm of \\citet{Bauer2010}, for which groups of genes are defined to be enriched if all genes in that group are DE. \n\nHere, for each set of genes $\\mathcal{S} \\subset \\{1,\\ldots,m\\}$, we test the ``competitive null hypothesis'' \\citep{Goeman2007} that the genes in $\\mathcal{S}$ are at most as often differentially expressed as the remaining genes by considering the posterior probability of the event\n\\begin{equation}\n\\label{geneSetDE}\n  \\textstyle\\sum_{j \\in \\mathcal{S}} I_j \\, \/ \\, | \\mathcal{S} | \\,> \\, \\sum_{j \\in \\mathcal{S}^c} I_j \\, \/ \\, | \\mathcal{S}^c |,\n\\end{equation}\nwhere the $I_j$ are the DE indicators introduced in \\eqref{foldchange}.\n\nA problem with gene set enrichment has been that for genes with low counts, the\npower for DE detection is low. Hence, in testing for category enrichment,\ncategories or groups of genes with low counts are rarely determined to be\nenriched. \\citet{Young2010} and \\citet{Mi2012} attempt to remedy this problem by accounting for differences in gene length. However, we believe that different gene lengths are not the only cause of bias in enrichment analysis, and it is necessary to account for differences in gene-specific DE uncertainty directly.\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=0.6\\textwidth]{geneSet_roc.pdf}\n\\caption{ROC curves for detection of enriched gene sets: As determined by BADER in red, as determined by a combination of DESeq and Fisher' exact test (FF) in blue}\n\\label{geneSet_roc}\n\\end{figure}\n\nWe considered the same 100 simulated datasets as described in Section \\ref{sec:DE}. For BADER, we obtained the posterior probability for a gene set to be enriched by estimating the posterior probability of \\eqref{geneSetDE}. We compared our results to a frequentist approach (henceforth referred to as FF) that consisted of testing for DE for each gene using DESeq \\citep{Anders2010} with a significance level of 0.05, followed by calculating the p-value of Fisher's exact test for over-representation of DE genes in the set under\nconsideration. Figure \\ref{geneSet_roc} shows the ROC curves for the two methods. BADER considerably outperforms the FF approach. The difference in performance is especially large when focusing on gene sets with low mean expression level (e.g., $\\bar{\\mu}^A_l < 3$; figure not shown).\n\n\n\n\\section{Discussion}\n\nWe developed a fully Bayesian model for the analysis of RNA-seq\ncount data (BADER). We described proper MCMC-based posterior inference, taking into account all uncertainty without shortcuts.\nBADER is distributed open-source on Bioconductor as an \\texttt{R} package with an efficient \\texttt{C++}\nback-end.\n\nWe demonstrated the value of posterior samples of our log-fold-change parameters with a point mass at zero. Their use in\nDE detection is highly competitive and natural, avoiding explicit\nadjustments for multiple testing. Moreover, their integration with\ndownstream analyses is not only conceptually easier, but also suffers\nless from biases than many frequentist approaches. We demonstrated this point\nby using BADER posterior samples for gene set\nenrichment analysis, showing more power to detect enriched gene\nsets.\n\n\n\n\\section*{Acknowledgments}\n\nWe thank Daniel Bader, Jens Preussner, and several anonymous reviewers for helpful feedback on the\nmanuscript. J.G.\\ is supported by the Bavarian Research Center for Molecular Biosystems.\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nThe $C^*$-algebra of a directed graph $E=(E^0,E^1,s,r)$ is a\nuniversal object generated by Hilbert space operators satisfying\ncertain relations, where the relations reflect the path structure\nof the graph \\cite{BPRS,CK,FLR,KPR,KPRR,RS}. The vertices $v$ of\n$E$ correspond to projections $P_v$ onto mutually orthogonal\nsubspaces, and the edges $e$ correspond to partial isometries\n$S_e$ which map between these subspaces. Given any subset $X$ of\nvertices, the sum $\\Sigma_{v\\in X}P_v$ converges strictly to a\nprojection, which we denote by $P_X$, in the multiplier algebra\n$M(C^*(E))$ \\cite{BPRS}. Corners of the type $P_X C^*(E)P_X$\nassociated to certain sets $X$ of vertices arise often in the\nstudy of graph algebras: see \\cite{BP,CG,D,DT,HSQLS,SzK}, and\nSection \\ref{coactions}.\n\nIt is in general very useful to be able to identify an abstract\n$C^*$-algebra with a graph algebra. This is because the graphical\npresentation encodes a great deal of structural information about\nits associated $C^*$-algebra, and allows one to compute the\nalgebra's invariants via straightforward calculations\n\\cite{BHRS,DT2,HSPIS,RS}. So, given a directed graph $E$ and a\nsubset $X$ of vertices, it might be useful to realize the corner\n$P_XC^*(E)P_X$ itself as the $C^*$-algebra of a directed graph. It\nis to this goal that the first part of this paper is devoted. That\nis, we shall give a construction which, in certain cases, produces\na graph for this corner from the graph $E$.\n\nProbably the best known example of such a construction is the\nprocedure described in the literature as ``adding a tail to a\nsink'', which is used to approximate the $C^*$-algebra of a graph\ncontaining sinks as a full corner of the algebra of a graph\nwithout sinks \\cite[Lemma 1.2]{BPRS}. In a similar vein, if the\ngraph $E$ contains infinite-emitters, one may realize $C^*(E)$ as\na full corner of the $C^*$-algebra of a row-finite graph, via a\nconstruction due to Drinen and Tomforde \\cite{DT}. This\nconstruction was generalized in \\cite[Section 4]{BP}, and further\nin \\cite{CG}.\n\nThe conditions of \\cite[Theorem 3.1]{CG} are in practice quite\nlimiting: for example, the theorem is not applicable if the\nhereditary complement of the set $X$ contains loops, sinks or\ninfinite-emitters. In order to overcome these restrictions, the\napproach taken here is substantially different to that of\n\\cite{BP,CG,DT}. The prototype for our present construction is\n\\cite[Section 2]{SzK}, where the graph $E$ was assumed to be\nfinite, and the set $X$ to consist of a single vertex. We\ngeneralize and simplify this construction, and fix up a slight\nerror. Unfortunately, this new approach is not applicable to our\nbasic (nonunital) examples, adding a tail at a sink and the\ndesingularization of \\cite{DT}. However, our construction applies\nin particular to all unital graph algebras (Lemma \\ref{extree}),\nand it has been shown that any graph algebra can be approximated\nas a direct limit of unital graph algebras \\cite{RS}.\n\nOne context in which corners of graph algebras arise naturally is\nas fixed point algebras of certain discrete coactions on graph\nalgebras. Indeed, the motivating example for this research was the\nconstruction in \\cite{HSQLS} of quantum lens spaces as the fixed\npoint algebras of certain actions of finite cyclic groups on\nquantum spheres, by analogy with construction of the classical\nlens spaces. The actions in question arise from labellings, in the\nsense of Kumjian and Pask \\cite{KP}, who showed that the crossed\nproduct of a graph algebra $C^*(E)$ by such a group action is\nitself isomorphic to the $C^*$-algebra of a directed graph, called\nthe skew product graph. The work of Kumjian and Pask was\ngeneralized in \\cite{DPR,KQR} to cover labellings of directed\ngraphs by discrete (not necessarily abelian) groups. Labellings of\nthis sort give rise to discrete group coactions, rather than the\ncompact group actions of \\cite{KP}, but the realization of the\ncrossed product as the graph algebra of a skew product still\nworks. In Theorem \\ref{fixedpoint} we show that, just as in the\ncase of the quantum lens spaces, the fixed point algebras of these\ndiscrete coactions may be recovered as corners of the skew product\ngraph algebras, and then give a condition on labellings which\nensures that we may use the construction of Section \\ref{corners}\nto realize these corners as graph algebras.\n\n\\section{Preliminaries}\\label{preliminaries}\n\nWe adopt the standard nomenclature of directed graphs and graph\nalgebras, as found in \\cite{BPRS}, for example, with the following\nadditions:\n\n\\subsection*{Directed graphs} Let $E$ be a directed graph, and let $m,n\\in{\\mathbb N}\\cup\\{\\infty\\}$ be\nsuch that $n\\geq m$. If $\\mu\\in E^m$ and $\\nu\\in E^n$ are paths of\nlength $m$ and $n$ respectively, such that $\\nu_i=\\mu_i$ for all\n$i=1,\\ldots,m$, then we say that $\\mu$ is an \\emph{initial\nsubpath} of $\\nu$, and write $\\mu\\prec\\nu$.\n\nEach finite path $\\mu\\in E^*$ gives a finite sequence\n$s(\\mu_1),r(\\mu_1),r(\\mu_2),\\ldots,r(\\mu)$ of vertices. The path\n$\\mu$ is called \\emph{vertex-simple} if this sequence contains no\nrepeated vertices (i.e. if $\\mu$ contains no loops). Similarly, an\ninfinite path $\\nu\\in E^\\infty$ is called vertex-simple if its\ncorresponding right-, left- or bi-infinite sequence of vertices\ncontains no repetition. Each path of length zero (i.e. each\nvertex) is also defined to be vertex-simple. A graph $E$ for which\n$E^*\\cup E^\\infty$ contains only vertex-simple paths is called an\n\\emph{acyclic} graph. A graph $E$ for which $E^\\infty$ contains no\nvertex-simple paths is called a \\emph{path-finite} graph.\n\nIf $E$ is a directed graph and $F$ a subgraph of $E$, then for\nvertices $u,v\\in E^0$ we write $u\\geq_F v$ to mean that there is a\npath $\\mu\\in F^*$ with $s(\\mu)=u$ and $r(\\mu)=v$. A subset\n$X\\subseteq E^0$ is said to be \\emph{hereditary} if it has the\nproperty that for all $v\\in X$ and $u\\in E^0$, $v\\geq_E u$ implies\n$u\\in X$. For any subset $Y\\subseteq E^0$ we shall denote by\n$H_E(Y)$ the smallest hereditary subset of $E^0$ containing $Y$.\nThe set $H_E(Y)\\setminus Y$ is referred to as the \\emph{hereditary\ncomplement} of $Y$ in $E$.\n\nA subgraph $T$ of a directed graph $E$ is called a \\emph{directed\nsubtree} of $E$ if it is acyclic and if $|T^1\\cap r^{-1}(v)|\\leq\n1$ for each vertex $v\\in T^0$ (that is, if each vertex in $T^0$\nreceives at most one edge in $T^1$). If $T$ is a directed subtree\nof $E$, let $T^r$ denote the subset of $T^0$ consisting of those\nvertices $v$ with $|T^1\\cap r^{-1}(v)|=0$ (these vertices are\ncalled the \\emph{roots} of $T$). Let $T^l$ denote the subset of\n$T^0$ consisting of those vertices $v$ with $|T^1\\cap\ns^{-1}(v)|=0$ (these vertices are called the \\emph{leaves} of\n$T$).\n\nThe concept of a directed subtree (in particular, a row- and\npath-finite one) is central to our construction in Section\n\\ref{corners}, and the following lemma points out several basic\nand useful facts about such graphs.\n\n\\begin{lemma}\\label{T} Let $T$ be a row-finite, path-finite directed\nsubtree of a directed graph $E$. Then the following hold:\n\\begin{enumerate} \\item For each $v\\in T^0$ there exists a\nunique path $\\tau(v)$ in $T^*$ with source in $T^r$ and range $v$.\nThen for $u,v\\in T^0$, $v\\geq_T u$ if and only if\n$\\tau(v)\\prec\\tau(u)$. \\item For each $v\\in T^0$ there exist at\nmost finitely many vertices $u\\in T^0$ with $v\\geq_T u$. \\item For\neach $v\\in T^0$ there exists at least one $u\\in T^l$ such that\n$v\\geq_T u$. \\item Suppose $u,v\\in T^0$ have $\\tau(v)\\prec\\tau(u)$\nand $u\\neq v$. Then there exists a unique edge $e\\in s^{-1}(v)\\cap\nT^1$ such that $\\tau(v)e\\prec\\tau(u)$. If $f\\in s^{-1}(v)\\cap T^1$\nsatisfies $\\tau(u)\\prec\\tau(v)f$ then $f=e$ and $\\tau(v)e=\\tau(u)$.\n\\end{enumerate}\\end{lemma}\n\n\\begin{proof} (1) Fix $v\\in T^0$. If $v\\in T^r$ then $\\tau(v)=v$.\nIf not, then $v$ receives exactly one edge $e_1\\in T^1$. If\n$s(e_1)$ is in $T^r$ then $\\tau(v)=e_1$. If not, then $s(e_1)$\nreceives exactly one edge $e_2\\in T^1$. As $T$ is path-finite we\nget a path $\\tau(v)=e_n e_{n-1}\\ldots e_1$ with source in $T^r$\nafter finitely many iterations of this construction. Uniqueness of\n$\\tau(v)$ follows from the fact that each $u\\in T^0$ receives at\nmost one edge, and this same fact gives the equivalence $v\\geq_T\nu\\iff \\tau(v)\\prec\\tau(u)$.\n\n(2) Suppose $v\\in T^0$ is such that infinitely many such $u$\nexist. As $T$ is row-finite, there is an edge $e_1\\in\ns^{-1}(v)\\cap T^1$ such that $e_1$ is the first edge in infinitely\nmany distinct paths in $T^*$ (by the pigeonhole principle). We may\napply this same argument to the vertex $r(e_1)$, giving an edge\n$e_2\\in s^{-1}(r(e_1))\\cap T^1$ such that $e_1 e_2$ is an initial\nsubpath of infinitely many distinct paths in $T^*$. Continuing\nthis construction gives a path $e_1 e_2\\ldots\\in T^\\infty$, which\nmust be vertex-simple because $T$ is acyclic. This contradicts the\nassumption that $T$ is path-finite, proving the claim.\n\n(3) Fix $v\\in T^0$. If $v\\in T^l$ then we are done. Otherwise\nchoose $e_1\\in s^{-1}(v)\\cap T^1$. If $r(e)\\in T^l$ then we are\ndone; otherwise find $e_2\\in s^{-1}(r(e_1))\\cap T^1$. This\nconstruction must terminate after finitely many iterations,\nbecause $T$ is path-finite.\n\n(4) Fix $u,v\\in T^0$ with $\\tau(v)\\prec\\tau(u)$ and $u\\neq v$.\nClearly there exists an edge $e\\in s^{-1}(v)\\cap T^1$ such that\n$\\tau(v)e\\prec\\tau(u)$. Suppose $e'$ is another such edge. Then\n$r(e)=r(e')\\in T^0$, contradicting that each edge in $T^0$\nreceives at most one edge. Hence $e$ is unique. Now suppose $f\\in\ns^{-1}(v)\\cap T^1$ has $\\tau(u)\\prec\\tau(v)f$. Since\n$\\tau(v)\\prec\\tau(u)$ we must then have $\\tau(u)=\\tau(v)f$, so\n$\\tau(v)f\\prec\\tau(u)$ and $f=e$ by uniqueness of $e$.\n\\end{proof}\n\n\\subsection*{Graph $C^*$-algebras} A Cuntz-Krieger $E$-family is a set\n$\\{P_v,S_e:v\\in E^0,e\\in E^1\\}$ of operators on a Hilbert space\nsuch that the elements $P_v$ are mutually orthogonal projections\nand the elements $S_e$ are partial isometries with mutually\northogonal ranges, satisfying the following relations:\n\n\\begin{enumerate}\\item[(CK1)] $S_e^* S_e=P_{r(e)}$ for all $e\\in E^1$;\n\\item[(CK2)] $S_e S_e^*\\leq P_{s(e)}$ for each $e\\in E^1$;\n\\item[(CK3)] $P_v=\\sum_{e\\in s^{-1}(v)} S_e S_e^*$ for each $v\\in\nE^0$ with $0<|s^{-1}(v)|<\\infty$.\\end{enumerate}\n\nThe $C^*$-algebra of $E$, denoted $C^*(E)$, is defined to be the\nuniversal $C^*$-algebra generated by a Cuntz-Krieger $E$-family.\n\nFor any subset $X\\subseteq E^0$, the sum $\\sum_{v\\in X} P_v$\nconverges strictly to a projection $P_X$ in $M(C^*(E))$\n\\cite[Lemma 1.1]{BPRS}.\n\n\\section{Corners of directed graphs} \\label{corners}\n\nIn this section we describe our procedure for constructing a graph\nfor the corner $P_XC^*(E)P_X$ of a graph algebra $C^*(E)$\nassociated to a vertex set $X$. This construction is given in the\nfollowing definition, and its relation to $P_XC^*(E)P_X$ is shown\nin Theorem \\ref{main}.\n\n\\begin{definition}\\label{corner} Let $E$ be a directed graph, $X\\subseteq E^0$, and let\n$T$ be a row-finite, path-finite directed subtree of $E$ with\n$T^r=X$ and $T^0=H_E(X)$. Define a new directed graph, denoted\n$E(T)$ and called the \\emph{$T$-corner of $E$}, as follows:\n\\begin{align*}& E(T)^0:=T^0\\setminus\\{v\\in T^0:\\emptyset\\neq\ns^{-1}(v)\\subseteq T^1\\}\\\\ &E(T)^1:=\\{e_u:e\\in\ns^{-1}(T^0)\\setminus T^1,u\\in E(T)^0,r(e)\\geq_T u\\}\\\\\n&s(e_u)=s(e),\\ r(e_u)=u.\\end{align*}\n\\end{definition}\n\n\\begin{example} Suppose $X$ is a hereditary subset of $E^0$. We then have $T^0=X=T^r$, and\nsince no root may receive an edge in $T$ we infer that $T^1$ is\nempty. Thus each $v\\in T^0$ is either a sink, or emits an edge\nwhich does not belong to $T^1$; this implies that $E(T)^0=T^0=X$.\nFurthermore, for each edge $e$ with source in $T^0$, and each\nvertex $u\\in T^0$, $r(e)\\geq_T u$ if and only if $r(e)=u$, because\n$T^*=T^0$. Hence $E(T)^1=\\{e_{r(e)}:s(e)\\in X\\}$, where each\n$e_{r(e)}$ has the same range and source as $e$. Thus $E(T)$ is\nnothing but the graph $(X,s^{-1}(X),s,r)$.\n\\end{example}\n\n\\begin{example}\\label{O_2} Let $E$\nbe the graph \\[\\xygraph{{v_0}=\"v\":_*{e_1}\n@\/_1pc\/[dl(2)]{v_1}=\"dl\":_*{e_2}@\/_1pc\/[r(4)]{v_2}=\"dr\":_*{e_0}@\/_1pc\/\"v\":_-*{f_2}\n@\/_\/\"dr\":_-*{f_1}@\/_\/\"dl\":_-*{f_0}@\/_\/\"v\"}\\]\n\n\\noindent and let $T^0=E^0$, $T^1=\\{e_1,f_2\\}$. $T$ is a row- and\npath-finite directed subtree of $E$ with root set $X=\\{v_0\\}$,\nsuch that $T^0=H_E(X)$. The vertex $v_0$ is not a sink, and each\nedge with source $v_0$ belongs to $T^1$, so $v_0\\not\\in E(T)^0$.\nOn the other hand, both $v_1$ and $v_2$ emit edges in $E$ which\nare not part of $T$ (for example, $f_0$ and $e_0$ respectively),\nso both belong to $E(T)^0$.\n\nNow constructing the edge set $E(T)^1$, we consider in turn each\nedge in $s^{-1}(T^0)\\setminus T^1=\\{e_0,e_2,f_0,f_1\\}$; let us\nstart with $e_0$. The range $v_0$ of $e_0$ satisfies $v_0\\geq_T\nv_1$, because $e_1$ is a path in $T^*$ with source $v_0$ and range\n$v_1$. Since $v_1$ belongs to $E(T)^0$ there will be an edge\n$(e_0)_{v_1}$ in $E(T)^1$, with $s((e_0)_{v_1})=s(e_0)=v_2$ and\n$r((e_0)_{v_1})=v_1$. Similarly, there will be an edge\n$(e_0)_{v_2}$ with source $v_2$ and range $v_2$. Notice that,\nalthough $v_0=r(e_0)\\geq_T v_0$, there is no edge $(e_0)_{v_0}$\nbecause $v_0\\not\\in E(T)^0$. Considering the remaining edges\n$e_2,f_0,f_1\\in s^{-1}(T^0)\\setminus T^1$ in a similar way, we\nobtain that\n$E(T)^1=\\{(e_0)_{v_1},(e_0)_{v_2},(e_2)_{v_2},(f_0)_{v_1},(f_0)_{v_2},(f_1)_{v_1}\\}$,\nso $E(T)$ is the following graph: \\[\\xygraph{\n{v_1}=\"v_1\":_{(f_0)_{v_1}}@(ul,dl)\"v_1\":^{(f_0)_{v_2}}@\/^3pc\/[r(2)]{v_2}=\"v_2\"\n:_{(e_0)_{v_2}}@(dr,ur)\"v_2\":^{(e_0)_{v_1}}@\/^3pc\/\"v_1\":^{(e_2)_{v_2}}@\/^1pc\/\"v_2\":^{(f_1)_{v_1}}@\/^1pc\/\"v_1\"}\n\\]\n\\end{example}\n\nThere will often be more than one choice of subtree with the\ndesired properties, giving nonisomorphic graphs $E(T)$:\n\n\\begin{example}\\label{pqr} Let $p,q,r$ be positive integers, and let $E$ be as shown:\n\\begin{equation*}E=\\xygraph{{u}:@(ul,ur)\"u\":^{\\#p}_e[r(2)]{v}:\n@(ul,ur)\"v\":^{\\#q}_f[r(2)]{w}:@(ul,ur)\"w\",\n\"u\":@\/_3pc\/^{\\#r}_g\"w\"}\\end{equation*}\n\n\\noindent Here the label ``$\\#n$'' above an arrow indicates that\nthat arrow represents $n$ edges, and a label of ``$x$'' below an\narrow means that we will distinguish one of those edges and call\nit $x$. Then the subgraph $T_1$ with $T_1^0=E^0$ and\n$T_1^1=\\{e,g\\}$ is a finite directed subtree of $E$ with root\n$X_1=\\{u\\}$, satisfying $T_1^0=H_E(X_1)$. For this $T_1$, the\nconstruction of Definition \\ref{corner} gives $E(T_1)\\cong E$. On\nthe other hand, let $T_2$ be the finite subtree $T_2^0=E^0$,\n$T_2^1=\\{e,f\\}$. This tree also has root set $X_2=X_1=\\{u\\}$ and\n$T_2^0=H_E(X_2)$, but now the construction gives the following\ngraph:\n\\begin{equation*}E(T_2)=\\xygraph{{u}:@(ul,ur)\"u\":^{\\#p}[r(2)]{v}:\n@(ul,ur)\"v\":^{\\#q}[r(2)]{w}:@(ul,ur)\"w\",\n\"u\":@\/_3pc\/^{\\#(r+p)}\"w\"}\\end{equation*}\n\\end{example}\n\n\\begin{theorem}\\label{main} Let $E$, $X$, $T$ and $ E(T)$ be as\nin Definition \\ref{corner}. Then $C^*(E(T))\\cong\nP_{X}C^*(E)P_{X}$.\n\\end{theorem}\n\nNotice that the important properties of the tree are row- and\npath-finiteness, as one can find a subtree with the other\nproperty, for any root set  $X$, using an inductive construction\nas in the proof of Lemma \\ref{extree}. As an aside, the following\nlemma indicates the scope of Theorem \\ref{main}:\n\n\\begin{lemma}\\label{extree} If $H_E(X)\\setminus X$ is finite, then there is a subtree\nwith the desired properties. If $X$ is finite and $H_E(X)$\ninfinite, then there is no such subtree. \\end{lemma}\n\n\\begin{proof} First suppose $H_E(X)\\setminus X$ is finite. For\neach $v\\in H_E(X)$ let $d(v)$ be the length of a shortest path in\n$E^*$ with source in $X$ and range $v$ (such a path exists because\n$H_E(X)$ is hereditary). Let $T^0=H_E(X)$ and construct the edge\nset $T^1$ recursively as follows. For each $n\\in{\\mathbb N}$ and each $v\\in\nH_E(X)\\setminus X$ with $d(v)=n$ choose one edge $e_v\\in E^1$ such\nthat $r(e_v)=v$ and $d(s(e_v))=n-1$. Let $T^1=\\{e_v:v\\in\nH_E(X)\\setminus X\\}$. Then the subgraph $T$ of $E$ is row- and\npath-finite by finiteness of $H_E(X)\\setminus X$. On the other\nhand, suppose $X$ is finite and $H_E(X)$ infinite, and suppose $T$\nis a directed subtree of $E$ with roots $X$ and vertex set\n$H_E(X)$. By Lemma \\ref{T}(1) and the pigeonhole principle, there\nmust be a vertex $v\\in X$ such that $v\\geq_T u$ for infinitely\nmany vertices $u\\in H_E(X)$. Hence, by part (2) of Lemma \\ref{T},\n$T$ cannot be row- and path-finite.\n\\end{proof}\n\nThe proof of Theorem \\ref{main} will proceed in three main steps:\nfirst we find a Cuntz-Krieger family for $E(T)$ inside $C^*(E)$,\nso that the universal property of $C^*(E(T))$ gives a homomorphism\n$\\phi:C^*(E(T))\\to C^*(E)$. Next we show that this $\\phi$ is\ninjective, using the gauge-invariant uniqueness theorem\n\\cite[Corollary 1.4]{SzCKU}. Finally we show that the range of\n$\\phi$ is equal to $P_{X}C^*(E)P_{X}$ using an inductive argument.\n\nFor the first step, let $\\{P_v,S_e\\}$ be the canonical\nCuntz-Krieger generators of $C^*(E)$. For each $v\\in T^0$ let\n$\\tau(v)\\in T^*$ be the path given by part (1) of Lemma \\ref{T}\n(in particular, for $v\\in X$, $\\tau(v)=v$). Now for each $v\\in\nT^0$, define\n\\[Q_v:=S_{\\tau(v)}S_{\\tau(v)}^*-\\sum_{e\\in T^1\\cap\ns^{-1}(v)}S_{\\tau(v)e}S_{\\tau(v)e}^*.\\] Since $T$ is row-finite,\nthis sum is finite and each $Q_v$ is an element of $C^*(E)$. The\nrelations (CK1)--(CK3) in $C^*(E)$ imply that each $Q_v$ is a\nprojection. These projections will correspond to the vertex\nprojections of $E(T)$, and we shall need to know that they are\nnonzero:\n\n\\begin{lemma}\\label{Q_v} For each $v\\in T^0$, $Q_v=0$ if and only if $\\emptyset\\neq s^{-1}(v)\\subseteq\nT^1$. Also,\n\\begin{equation}\\label{Q_v1}S_{\\tau(v)}S_{\\tau(v)}^*=\\sum_{u\\in\nT^0,\\,v\\geq_T u}Q_u.\\end{equation}\\end{lemma}\n\n\\begin{proof} For the first claim, first suppose $\\emptyset\\neq s^{-1}(v)\\subseteq\nT^1$. The subgraph $T$ is row-finite, and each edge with source\n$v$ belongs to $T^1$, so $0<|s^{-1}(v)|<\\infty$. The Cuntz-Krieger\nrelation (CK3) in $C^*(E)$ then gives $P_v=\\sum_{e\\in\ns^{-1}(v)}S_e S_e^*$, so we have \\begin{align*}\nQ_v&=S_{\\tau(v)}S_{\\tau(v)}^*-\\sum_{e\\in\ns^{-1}(v)}S_{\\tau(v)e}S_{\\tau(v)e}^*\\qquad\\text{(since $T^1\\cap s^{-1}(v)=s^{-1}(v)$)}\\\\\n&=S_{\\tau(v)}P_vS_{\\tau(v)}^*-S_{\\tau(v)}\\left(\\sum_{e\\in\ns^{-1}(v)}S_e S_e^*\\right)S_{\\tau(v)}^*=0.\\end{align*} Conversely,\nif $v$ is a sink in $E$ then $Q_v=S_{\\tau(v)}S_{\\tau(v)}^*\\neq 0$.\nIf $v$ emits an edge $f\\in E^1\\setminus T^1$ then the relations\n(CK1)--(CK3) in $C^*(E)$ imply that $S_{\\tau(v)f}S_{\\tau(v)f}^*$\nis a subprojection of $S_{\\tau(v)}S_{\\tau(v)}^*$ orthogonal to\n$\\sum_{e\\in T^1\\cap s^{-1}(v)}S_{\\tau(v)e}S_{\\tau(v)e}^*$, so\n$Q_v\\geq S_{\\tau(v)f}S_{\\tau(v)f}^*\\neq 0$.\n\nFor the second claim, first notice that the sum is finite by part\n(2) of Lemma \\ref{T}. For each vertex $v\\in T^0$, let $c(v)$ be\nthe number of elements in the set $\\{u\\in T^0:v\\geq_T u\\}$. The\nformula \\eqref{Q_v1} will be derived by induction on $c(v)$. For\nthe basis step, note that if $c(v)=1$ then $T^1\\cap\ns^{-1}(v)=\\emptyset$, so $Q_v=S_{\\tau(u)}S_{\\tau(u)}^*$ as\ndesired. For $n\\in{\\mathbb N}$, suppose the formula \\eqref{Q_v1} holds for\nall $w\\in T^0$ with $c(w)\\leq n-1$, and let $v\\in T^0$ have\n$c(v)=n$. Now\n\\begin{equation}\\label{Q_v2}S_{\\tau(v)}S_{\\tau(v)}^*=Q_v+\\sum_{e\\in T^1\\cap\ns^{-1}(v)}S_{\\tau(v)e}S_{\\tau(v)e}^*,\\end{equation} and for each\n$e\\in T^1\\cap s^{-1}(v)$ we have $\\tau(v)e=\\tau(r(e))$ and\n$c(r(e))<c(v)$, so $S_{\\tau(v)e}S_{\\tau(v)e}^*=\\sum_{u\\in\nT^0,\\,r(e)\\geq_T u}Q_u$ by the inductive hypothesis. Substituting\nthis into \\eqref{Q_v2} gives the formula \\eqref{Q_v1} for the\nvertex $v$.\n\\end{proof}\n\nNow for $e_u\\in E(T)^1$ define\n$T_{e_u}:=S_{\\tau(s(e))e}S_{\\tau(r(e))}^*Q_u$.\n\n\\begin{prop}\\label{CK} The family\n$\\{Q_v,T_{e_u}:v\\in E(T)^0,\\,e_u\\in E(T)^1\\}$ is a Cuntz-Krieger\nfamily for the graph $ E(T)$.\n\\end{prop}\n\n\\begin{proof} The proof of this proposition requires some technical manipulations\nof the relations (CK1)--(CK3), but is theoretically\nstraightforward. Lemma \\ref{Q_v} implies that for each $v\\in\nE(T)^0$, $Q_v$ is a nonzero projection. To see that they are\nmutually orthogonal, first notice that for each $v$, $Q_v$ is a\nsubprojection of $S_{\\tau(v)}S_{\\tau(v)}^*$. Suppose $v$ and $w$\nare distinct elements of $ E(T)^0$ such that $Q_vQ_w\\neq 0$. We\nmust have $S_{\\tau(v)}^*S_{\\tau(w)}\\neq 0$, and hence one of\n$\\tau(v)$ and $\\tau(w)$ is an initial subpath of the other (this\nimplication is a consequence of the fact that the $S_e$ have\nmutually orthogonal ranges). Assume, without loss of generality,\nthat $\\tau(w)\\prec\\tau(v)$, and let $f\\in T^1\\cap s^{-1}(w)$ be\nthe edge given by Lemma \\ref{T}(4). Then\n\\[\\sum_{e\\in T^1\\cap\ns^{-1}(w)}S_{\\tau(v)}^*S_{\\tau(w)e}S_{\\tau(w)e}^*=S_{\\tau(v)}^*S_{\\tau(w)f}S_{\\tau(w)f}^*,\\]\nbecause $f$ is the unique edge in $T^1\\cap s^{-1}(w)$ with the\nproperty that $\\tau(w)f\\prec\\tau(v)$. Now\n$S_{\\tau(v)}^*S_{\\tau(w)f}S_{\\tau(w)f}^*=S_{\\tau(v)}^*$, and thus\n\\begin{align*}Q_vQ_w&=Q_vS_{\\tau(v)}S_{\\tau(v)}^*\\left(S_{\\tau(w)}S_{\\tau(w)}^*-\\sum_{e\\in T^1\\cap\ns^{-1}(w)}S_{\\tau(w)e}S_{\\tau(w)e}^*\\right)\\\\\n&=Q_v\\left(S_{\\tau(v)}S_{\\tau(v)}^*-S_{\\tau(v)}S_{\\tau(v)}^*\\right)\\\\&=0.\\end{align*}\nHence $Q_vQ_w\\neq 0$ if and only if $v=w$.\n\nTurning our attention to the $T_{e_u}$, fix $e_u\\in E(T)^1$. By\ndefinition of $ E(T)^1$ we must have $\\tau(r(e))\\prec\\tau(u)$, so\n$Q_u\\leq S_{\\tau(u)}S_{\\tau(u)}^*\\leq\nS_{\\tau(r(e))}S_{\\tau(r(e))}^*$. Therefore\n\\begin{align*}\nT_{e_u}^*T_{e_u}&=Q_uS_{\\tau(r(e))}(S_{\\tau(s(e))e}^*S_{\\tau(s(e))e})S_{\\tau(r(e))}^*Q_u\n=Q_uS_{\\tau(r(e))}P_{r(e)}S_{\\tau(r(e))}^*Q_u\\\\&=Q_u(S_{\\tau(r(e))}S_{\\tau(r(e))}^*)Q_u\n=Q_u.\\end{align*} Thus the $T_{e_u}$ are nonzero partial\nisometries with $T_{e_u}^*T_{e_u}=Q_{r(e_u)}$. To see that they\nhave mutually orthogonal ranges, take $e_u$ and $f_v$ in $ E(T)^1$\nand suppose $T_{e_u}^*T_{f_v}\\neq 0$. Now\n\\begin{equation}\\label{mutorth}T_{e_u}^*T_{f_v}=Q_u\nS_{\\tau(r(e))}S_{\\tau(s(e))e}^*S_{\\tau(s(f))f}S_{\\tau(r(f))}^*Q_v,\\end{equation}\nand in order for this product to be nonzero we must have either\n$\\tau(s(f))f\\prec\\tau(s(e))e$ or $\\tau(s(e))e\\prec\\tau(s(f))f$.\nSince neither $e$ nor $f$ belongs to $T^1$ (so that neither may be\npart of any $\\tau(w)$), this implies that\n$\\tau(s(e))e=\\tau(s(f))f$, and so $e=f$. Putting $e=f$ in\n\\eqref{mutorth} gives\n\\[T_{e_u}^*T_{f_v}=Q_uS_{\\tau(r(e))}S_{\\tau(r(e))}^*Q_v=Q_uQ_v,\\]\nand in order for this product to be nonzero we must have $u=v$.\nThus $e_u=f_v$.\n\nFor the inequality $T_{e_u}T_{e_u}^*\\leq Q_{s(e_u)}$, we calculate\n\\begin{align*}S_{\\tau(s(e))e}^*Q_{s(e)}&=S_{\\tau(s(e))e}^*\\left(S_{\\tau(s(e))}S_{\\tau(s(e))}^*-\\sum_{f\\in\nT^1\\cap s^{-1}(s(e))}S_{\\tau(s(e))f}S_{\\tau(s(e))f}^*\\right)\\\\\n&=S_{\\tau(s(e))e}^*S_{\\tau(s(e))e}S_{\\tau(s(e))e}^*-0=S_{\\tau(s(e))e}^*,\\end{align*}\nsince $e\\not\\in T^1$ implies that $\\tau(s(e))e$ is not an initial\nsubpath of any $\\tau(s(e))f$ for $f\\in T^1$. Thus\n\\[T_{e_u}T_{e_u}^*Q_{s(e_u)}=T_{e_u}Q_vS_{\\tau(r(e))}(S_{\\tau(s(e))e}^*Q_{s(e)})\n=T_{e_u}Q_vS_{\\tau(r(e))}S_{\\tau(s(e))e}^*=T_{e_u}T_{e_u}^*.\\]\n\nTo prove the remaining identity (CK3), we need to know the\nfollowing fact about singular vertices in $E(T)$:\n\n\\begin{lemma}\\label{sing} Each edge $e$ in $E^1\\setminus T^1$ with\n$s(e)\\in T^0$ gives at least one edge in $ E(T)$ with source\n$s(e)$. In particular, if $v\\in T^0$ is a singular vertex of $E$\nthen $v$ is a singular vertex of $ E(T)$.\\end{lemma}\n\n\\begin{proof} Let $e$ be an edge in $E^1\\setminus T^1$ with\n$s(e)\\in T^0$, and let $s(e)=v$, $r(e)=u$. Since $T^0$ is a\nhereditary subset of $E^0$ we must have $u\\in T^0$. By part (3) of\nLemma \\ref{T}, there exists at least one vertex $u'\\in T^0$ with\n$u\\geq_T u'$ and $s^{-1}(u')\\cap T^1=\\emptyset$. Then by\ndefinition of $ E(T)$ we have $u'\\in E(T)^0$, and there is an edge\n$e_{u'}$ in $ E(T)^1$ with source $v$.\n\nFor the second claim, suppose $v\\in T^0$ is a sink in $E$. Then\n$\\emptyset=s^{-1}(v)$ gives $v\\in E(T)$, and since there is no\nedge in $E^1$ with source $v$ there is no edge $e_u$ in $ E(T)^1$\nwith source $v$. Hence $v$ is a sink in $E(T)$. On the other hand,\nsuppose $v\\in T^0$ emits infinitely many edges. Since $T$ is\nrow-finite, infinitely many of these edges must belong to\n$E^1\\setminus T^1$. Each of these edges gives at least one edge in\n$ E(T)$ with source $v$ by the preceding paragraph, so $v$ is an\ninfinite-emitter in $E(T)$.\\end{proof}\n\nNow suppose $v\\in E(T)^0$ is nonsingular in $ E(T)$. Then $v$ is\nnonsingular in $E$ by the preceding lemma, so the Cuntz-Krieger\nrelation (CK3) in $C^*(E)$ gives $P_v=\\sum_{e\\in\ns^{-1}(v)}S_eS_e^*$. Now\n\\begin{align} Q_v&=S_{\\tau(v)}S_{\\tau(v)}^*-\\sum_{e\\in T^1\\cap\ns^{-1}(v)}S_{\\tau(v)e}S_{\\tau(v)e}^*=S_{\\tau(v)}P_vS_{\\tau(v)}^*-S_{\\tau(v)}\\left(\\sum_{e\\in\nT^1\\cap s^{-1}(v)}S_eS_e^*\\right)S_{\\tau(v)}^*\\notag\\\\\n&=S_{\\tau(v)}\\left(P_v-\\sum_{e\\in T^1\\cap\ns^{-1}(v)}S_eS_e^*\\right)S_{\\tau(v)}^*=\\sum_{e\\in\ns^{-1}(v)\\setminus\nT^1}S_{\\tau(v)e}S_{\\tau(v)e}^*.\\label{Q_vsum}\\end{align}\n\nFix an edge $e\\in s^{-1}(v)\\setminus T^1$. This edge gives one\nedge $e_u$ in $E(T)$ with source $v$ for each vertex $u\\in E(T)^0$\nwith $r(e)\\geq_T u$. The formula \\eqref{Q_v1} of Lemma \\ref{Q_v}\ngives \\begin{align*}\nS_{\\tau(v)e}S_{\\tau(v)e}^*&=S_{\\tau(v)e}P_{r(e)}^3S_{\\tau(v)e}^*=S_{\\tau(v)e}S_{\\tau(r(e))}^*(S_{\\tau(r(e))}S_{\\tau(r(e))}^*)^2S_{\\tau(r(e))}S_{\\tau(v)e}^*\\\\\n&=S_{\\tau(v)e}S_{\\tau(r(e))}^*(S_{\\tau(r(e))}S_{\\tau(r(e))}^*)\\left(S_{\\tau(v)e}S_{\\tau(r(e))}^*(S_{\\tau(r(e))}S_{\\tau(r(e))}^*)\\right)^*\\\\\n&=\\left(S_{\\tau(v)e}S_{\\tau(r(e))}^*\\left(\\sum_{u\\in\nT^0,\\,r(e)\\geq_T\nu}Q_u\\right)\\right)\\left(S_{\\tau(v)e}S_{\\tau(r(e))}^*\\left(\\sum_{u\\in\nT^0,\\,r(e)\\geq_T u}Q_u\\right)\\right)^*\\\\\n&=\\left(\\sum_{u\\in  E(T)^0,\\,r(e)\\geq_T\nu}T_{e_u}\\right)\\left(\\sum_{u\\in  E(T)^0,\\,r(e)\\geq_T u}\nT_{e_u}^*\\right).\\end{align*} Since for $u\\neq u'$ we have\n$T_{e_u}T_{e_{u'}}^*=0$, this product expands as\n\\[S_{\\tau(v)e}S_{\\tau(v)e}^*=\\sum_{u\\in  E(T)^0,\\,r(e)\\geq_T\nu}T_{e_u}T_{e_u}^*.\\] Substituting into \\eqref{Q_vsum} now gives\nthe Cuntz-Krieger identity $Q_v=\\sum_{s(e_u)=v}T_{e_u}T_{e_u}^*$,\nand this final identity completes the proof of the proposition.\n\\end{proof}\n\nNow the universal property of $C^*(E(T))$ gives a $*$-homomorphism\n$\\phi:C^*( E(T))\\to C^*(E)$ which maps each canonical generator of\n$C^*( E(T))$ to its corresponding element of the family\n$\\{Q_v,T_{e_u}\\}$. The following two propositions show that $\\phi$\nis injective and has range $P_{X}C^*(E)P_{X}$.\n\n\\begin{prop}\\label{inj} The map $\\phi$ defined above is\ninjective.\\end{prop}\n\n\\begin{proof} Arguing as in \\cite[Section 1]{BPRS}, the universal property of $C^*(E)$ implies that there exists\nan action $\\sigma:{\\mathbb T}\\to\\operatorname{Aut}(C^*(E))$ given on generators by\n$\\sigma_t(P_v)=P_v$ for all $v\\in E^0$,\nand\\begin{equation*}\\sigma_t(S_e)=\\begin{cases}S_e&\\text{for }e\\in\nT^1\\\\tS_e &\\text{otherwise.}\\end{cases}\\end{equation*} This action\ndoes not move any $S_{\\tau(v)}$ for $v\\in E(T)^0$, and hence does\nnot move any $Q_v$ either. For $e_u\\in E(T)^1$ we have $e\\in\nE^1\\setminus T^1$, and so for $t\\in{\\mathbb T}$,\n\\[\\sigma_t(T_{e_u})=\\sigma_t(S_{\\tau(s(e))}S_eS_{\\tau(r(e))}^*Q_v)=S_{\\tau(s(e))}tS_eS_{\\tau(r(e))}^*Q_v=tT_{e_u}.\\]\nThus if $\\gamma$ denotes the gauge action on $C^*(E(T))$ we have\n$\\phi\\circ\\gamma=\\sigma\\circ\\phi$, and all $Q_v$ are nonzero, so\nthe gauge-invariant uniqueness theorem \\cite[Corollary 1.4]{SzCKU}\nimplies that $\\phi$ is injective.\n\\end{proof}\n\n\\begin{prop}\\label{surj} $\\phi(C^*( E(T)))=P_{X}C^*(E)P_{X}$.\n\\end{prop}\n\n\\begin{proof} For $v\\in E(T)^0$ we have $P_X Q_v P_X=P_{s(\\tau(v))}Q_vP_{s(\\tau(v))}=Q_v$, and for\n$e_u\\in E(T)^1$ we have $P_X T_{e_u}P_X\n=P_{s(\\tau(s(e)))}T_{e_u}P_{s(\\tau(u))}=T_{e_u}$. Hence\n$\\phi(C^*(E(T)))\\subseteq P_X C^*(E)P_X$, and it remains to show\nthe opposite inclusion. To do this, we must show that the range of\n$\\phi$ contains all products $S_\\mu S_\\nu^*$ such that $\\mu,\\nu\\in\nE^*$, $s(\\mu),s(\\nu)\\in X$ and $r(\\mu)=r(\\nu)$. Since for such\n$\\mu$ and $\\nu$ we have $S_\\mu S_\\nu^*=S_\\mu\nS_{\\tau(r(\\mu))}^*S_{\\tau(r(\\mu))}S_\\nu^*=(S_\\mu\nS_{\\tau(r(\\mu))}^*)(S_\\nu S_{\\tau(r(\\nu))}^*)^*$, we may assume\nthat $\\nu=\\tau(r(\\mu))$. The proof is by induction on the length\nof $\\mu$.\n\nIf $|\\mu|=0$ then $\\mu=s(\\mu)\\in X$ and so $\\mu=\\tau(r(\\mu))$.\nThen $S_\\mu\nS_{\\tau(r(\\mu))}^*=S_{\\tau(r(\\mu))}S_{\\tau(r(\\mu))}^*$, which is\nin the range of $\\phi$ by Lemma \\ref{Q_v}. Now for $n\\in{\\mathbb N}$,\nsuppose $|\\mu|=n$ and suppose that $S_\\nu S_{\\tau(r(\\nu))}^*$\nbelongs to the range of $\\phi$ for all paths $\\nu$ of length\n$n-1$. Let $e$ be the final edge of $\\mu$, and write $\\mu=\\mu'e$.\nThen \\begin{align*}S_\\mu\nS_{\\tau(r(\\mu))}^*&=S_{\\mu'}S_eS_{\\tau(r(e))}^*\n=S_{\\mu'}P_{r(\\mu')}S_eS_{\\tau(r(e))}^*=S_{\\mu'}(S_{\\tau(r(\\mu'))}^*S_{\\tau(r(\\mu'))})S_eS_{\\tau(r(\\mu))}^*\n\\\\&=(S_{\\mu'}S_{\\tau(r(\\mu'))}^*)(S_{\\tau(r(\\mu'))e}S_{\\tau(r(e))}^*),\\end{align*}\nwhere $S_{\\mu'}S_{\\tau(r(\\mu'))}^*$ belongs to the range of $\\phi$\nby the inductive hypothesis. If $e\\in T^1$ then\n$\\tau(r(\\mu'))e=\\tau(r(e))$, and so\n$S_{\\tau(r(\\mu'))e}S_{\\tau(r(e))}^*$ belongs to the range of\n$\\phi$ by Lemma \\ref{Q_v}. If $e$ does not belong to $T^1$, then\nonce again we use Lemma \\ref{Q_v} to give\n\\begin{align*}S_{\\tau(r(\\mu'))e}S_{\\tau(r(e))}^*&=S_{\\tau(r(\\mu'))e}S_{\\tau(r(e))}^*(S_{\\tau(r(e))}S_{\\tau(r(e))}^*)\\\\\n&=S_{\\tau(s(e))e}S_{\\tau(r(e))}^*\\left(\\sum_{u\\in\nE(T)^0,\\,r(e)\\geq_T u}Q_u\\right)\\\\&=\\sum_{u\\in E(T)^0,\\,r(e)\\geq_T\nu}T_{e_u}\\end{align*} which belongs to the range of $\\phi$. This\ncompletes the proof by induction.\n\\end{proof}\n\nPropositions \\ref{CK}, \\ref{inj} and \\ref{surj} prove Theorem\n\\ref{main}.\n\n\\begin{example} Theorem \\ref{main} is already known in the case\nwhere $X$ is hereditary: in this case we have seen that $E(T)$ is\njust the subgraph of $E$ consisting of the vertices $X$ and each\nedge whose source belongs to this set. Now the isomorphism of\n$C^*(E(T))$ with the corner $P_X C^*(E)P_X$ is implied by\n\\cite[Theorem 4.1(c)]{BPRS} and its proof.\n\\end{example}\n\n\\begin{example} Let $p,q,r$ be positive integers, and for each nonnegative integer $n$ let\n\\begin{equation*}E_n=\\xygraph{{u}:@(ul,ur)\"u\":^{\\#p}[r(2)]{v}:\n@(ul,ur)\"v\":^{\\#q}[r(2)]{w}:@(ul,ur)\"w\",\n\"u\":@\/_3pc\/^{\\#(r+np)}\"w\"}\\end{equation*}\n\n\\noindent Example \\ref{pqr} and Theorem \\ref{main} show that for\neach $n$ we have $C^*(E_n)\\cong P_uC^*(E_0)P_u$. Graphs of this\ntype arise, for example, in the study of quantum lens spaces\n\\cite{HSQLS}.\n\\end{example}\n\n\\section{Labellings of directed graphs and discrete\ncoactions}\\label{coactions}\n\nCorners of graph algebras arise as the fixed point algebras of\ncertain discrete coactions on graph algebras. In this section we\nmake this precise, and then link up with Theorem \\ref{main}.\n\nThroughout this section, $G$ will be a group with identity element\n$1_G$, or $0_G$ if $G$ is abelian (we'll sometimes omit the\nsubscript $G$ to avoid clutter). We will assume throughout that\n$G$ is discrete, with the exception of Corollary\n\\ref{abelianfixedpoint} and the paragraph immediately following\nthe statement of Theorem \\ref{fixedpoint}. We denote by $C^*(G)$\nthe full group $C^*$-algebra, and for $s\\in G$ we simply write $s$\nto denote the image of the group element under the canonical\nmapping $G\\to C^*(G)$. We denote by $\\lambda$ the left-regular\nrepresentation of $G$ (and $C^*(G)$) on $l^2(G)$, and by $M$ the\nrepresentation of $C_0(G)$ on $l^2(G)$ by multiplication. For\n$s\\in G$, $\\chi_s$ denotes the characteristic function of $\\{s\\}$.\nAll $C^*$-algebra tensor products considered here will involve at\nleast one nuclear $C^*$-algebra, and we write $A\\otimes B$ to mean\nthe completion of the algebraic tensor product in its unique\n$C^*$-norm.\n\n\\subsection*{Coactions of discrete groups on $C^*$-algebras} In line with \\cite{DPR} we adopt\nthe following notations and conventions. Let $\\delta_G:C^*(G)\\to\nC^*(G)\\otimes C^*(G)$ be the \\emph{comultiplication} $s\\mapsto\ns\\otimes s$ for $s\\in G$. A \\emph{coaction} of $G$ on $A$ is an\ninjective nondegenerate homomorphism $\\delta:A\\to A\\otimes C^*(G)$\n(where ``nondegenerate'' means that $\\operatorname{span}\\delta(A)A\\otimes C^*(G)$\nis dense in $A\\otimes C^*(G)$) such that $(\\delta\\otimes\\operatorname{id})\\circ\n\\delta=(\\operatorname{id}\\otimes\\delta_G)\\circ\\delta$. For each $s\\in G$ let\n$A_s=\\{a\\in A:\\delta(a)=a\\otimes s\\}$, and write $a_s$ to denote a\ngeneric element of $A_s$. The span of the subalgebras $A_s:s\\in G$\nis dense in $A$ (here it is important that $G$ be discrete). The\n\\emph{fixed point algebra} $A^\\delta$ of $\\delta$ is defined as\n$A^\\delta=A_{1_G}$.\n\nA \\emph{covariant representation} of the triple $(A,G,\\delta)$ is\na pair $(\\pi,\\mu)$, where $\\pi:A\\to B(\\mathcal H)$ and\n$\\mu:C_0(G)\\to B(\\mathcal H)$ are nondegenerate representations on\na Hilbert space $\\mathcal H$ satisfying\n$\\pi(a_s)\\mu(\\chi_t)=\\mu(\\chi_{st})\\pi(a_s)$ for all $s,t\\in G$\nand $a_s\\in A_s$. Given a nondegenerate representation $\\pi$ of\n$A$ on $\\mathcal H$, and letting $\\lambda$ denote the left-regular\nrepresentation of $C^*(G)$ on $l^2(G)$ and $M$ the representation\nof $C_0(G)$ on $l^2(G)$ by multiplication, there is a covariant\nrepresentation $((\\pi\\otimes\\lambda)\\circ\\delta,1\\otimes M)$ of\n$(A,G,\\delta)$ on $\\mathcal H\\otimes l^2(G)$, called the\n\\emph{regular covariant representation induced by} $\\pi$\n\\cite[Proposition 2.6]{R}. The coaction $\\delta$ is \\emph{normal}\nif there is a covariant representation $(\\pi,\\mu)$ with $\\pi$\nfaithful. The \\emph{crossed product} $A\\times_\\delta G$ is the\nuniversal $C^*$-algebra generated by a covariant representation\n$(j_A,j_G)$, and is densely spanned by elements of the form\n$j_A(a_s)j_G(\\chi_t)$. The nondegenerate representations of\n$A\\times_\\delta G$ are in one-to-one correspondence with the\ncovariant representations of $(A,G,\\delta)$ \\cite[Definition\n2.8]{R}. In particular, any nondegenerate representation $\\pi$ of\n$A$ on a Hilbert space $\\mathcal H$ induces a nondegenerate\nrepresentation of the crossed product on $\\mathcal H\\otimes\nl^2(G)$, corresponding to the regular covariant representation\ninduced by $\\pi$. We denote this representation of $A\\times_\\delta\nG$ by $\\operatorname{Ind}\\pi$.\n\n\\subsection*{Labellings of directed graphs} Once again, conventions are\nadopted from \\cite{DPR}; see also \\cite{GT,KP}. A \\emph{labelling}\nof a directed graph $E$ by a discrete group $G$ is a function\n$c:E^1\\to G$. Given a labelling $c$, the \\emph{skew product}\n$E\\times_c G$ (called the \\emph{voltage graph} in \\cite{GT}) is\nthe graph with vertex set $E^0\\times G$, edge set $E^1\\times G$\nand whose source and range maps are given by $s(e,s)=(s(e),c(e)s)$\nand $r(e,s)=(r(e),s)$ (there are several slightly different\ndefinitions of the skew product, all yielding isomorphic graphs\n\\cite{DPR,GT,KQR,KP}). As further notation, if $c$ is a labelling\nof a directed graph $E$ by a group $G$ and if $\\mu$ is an element\nof $E^*$, we shall denote by $c(\\mu)$ the element\n$c(\\mu_1)c(\\mu_2)\\dots c(\\mu_{|\\mu|})$ of $G$.\n\n\\begin{example}\\label{O_22} Let $E$ be the graph with one vertex $v$ and two\nedges $e$ and $f$. Define a labelling $c$ of $E$ by ${\\mathbb Z}_3$ as\n$c(e)=2$, $c(f)=1$. Then the skew product graph $E\\times_c{\\mathbb Z}_3$ is\nas follows:\n\n\\[E=\\xygraph{ v=\"v\":@(ul,dl)_e\"v\":@(dr,ur)_f\"v\"}\\qquad\\qquad E\\times_c{\\mathbb Z}_3=\\xygraph{[u]{(v,0)}=\"v\":_{(e,1)}\n@\/_1pc\/[dl(2)]{(v,1)}=\"dl\":_{(e,2)}@\/_1pc\/[r(4)]{(v,2)}=\"dr\":_{(e,0)}@\/_1pc\/\"v\":_-{(f,2)}\n@\/_\/\"dr\":_-{(f,1)}@\/_\/\"dl\":_-{(f,0)}@\/_\/\"v\"   } \\]\n\\end{example}\n\nAny labelling of a graph $E$ by a discrete group $G$ induces a\nnormal coaction of $G$ on $C^*(E)$, and the graph $C^*$-algebra of\nthe skew product is naturally isomorphic to the crossed product of\n$C^*(E)$ by this coaction; this is the content of the following\nresults from \\cite{KQR} and \\cite{DPR}, which we recall here for\nconvenience:\n\n\\begin{lemma}\\label{KQR2.3}\\cite[Lemma 2.3]{KQR}\\cite[Lemma 3.3]{DPR} Let $c$ be a\nlabelling of a directed graph $E$ by a discrete group $G$. Then\nthere is a normal coaction $\\delta$ of $G$ on $C^*(E)$ such that\n\\[\\delta(S_e)=S_e\\otimes c(e)\\quad\\text{and}\\quad\n\\delta(P_v)=P_v\\otimes 1_G\\quad\\text{for $e\\in E^1$, $v\\in\nE^0$.}\\]\n\\end{lemma}\n\n\\begin{theorem}\\label{DPR3.4}\\cite[Theorem 3.4]{DPR} Let $c$ be a\nlabelling of a directed graph $E$ by a discrete group $G$, with\ncorresponding coaction $\\delta$. Then\n\\[C^*(E\\times_c G)\\cong C^*(E)\\times_{\\delta} G\\] under the\nisomorphism $\\phi$ given on generators by\n\\[\\phi(P_{(v,s)})=j_{C^*(E)}(P_v)j_G(\\chi_s)\\quad\\text{and}\\quad\n\\phi(S_{(e,s)})=j_{C^*(E)}(S_e)j_G(\\chi_s)\\] for $v\\in E^0$, $e\\in\nE^1$ and $s\\in G$.\n\\end{theorem}\n\n\\begin{example}\\label{Cayley} Let $G$ be a discrete group and $S\\subseteq G$ a\ngenerating subset of cardinality $n\\in{\\mathbb N}\\cup\\{\\infty\\}$.\n\\cite[Theorem 2.2.3]{GT} implies that any Cayley graph $\\Gamma$\nfor the pair $(G,S)$ is isomorphic to a skew product, by $G$, of\nthe graph $B_n$ with one vertex and $n$ edges. Hence by\n\\cite[Theorem 3.4]{DPR}, there is a normal coaction $\\delta$ of\n$G$ on $C^*(B_n)$ (=$\\mathcal O_n$, the Cuntz algebra generated by\n$n$ nonunitary isometries with mutually orthogonal ranges\n\\cite{C}), such that \\[C^*(\\Gamma)\\cong \\mathcal O_n\\times_\\delta\nG.\\]\n\\end{example}\n\n\\begin{remark} \\cite[Theorem 3.4]{DPR} is more general than the\nversion used here. \\cite{DPR} considered ``coactions of a\nhomogeneous space'' $G\/H$, and defined an analogue of the crossed\nproduct $A\\times_\\delta(G\/H)$. Similarly, one may define skew\nproducts of graphs by discrete homogeneous spaces rather than\ndiscrete groups. \\cite[Theorem 3.4]{DPR} then says that\n$C^*(E\\times_c(G\/H))\\cong C^*(E)\\times_\\delta(G\/H)$. At this stage\nit is not clear how to extend our Theorem \\ref{fixedpoint} to this\nmore general setting, as we don't have an obvious analogue of the\nfixed point algebra.\\end{remark}\n\n\\subsection*{Fixed point algebras associated to labellings} We shall use the isomorphism of \\cite[Theorem 3.4]{DPR}\nto prove the following:\n\n\\begin{theorem}\\label{fixedpoint} Let $c$ be a labelling of a directed graph $E$ by a discrete group\n$G$, with corresponding coaction $\\delta$. Then\n\\[C^*(E)^{\\delta}\\cong P_{E^0\\times\\{1_G\\}}C^*(E\\times_c\nG)P_{E^0\\times\\{1_G\\}}.\\]\\end{theorem}\n\nThis will follow immediately from the next lemma, which is the\nanalogue for coactions of a well-known fact about compact group\nactions on $C^*$-algebras: if $\\alpha:G\\to\\operatorname{Aut}(A)$ is such an\naction and $i_G:G\\to M(A\\times_\\alpha G)$ the canonical embedding,\nthen $i_G(1)$ is a projection which compresses the crossed product\nto a copy of the fixed point algebra $A^\\alpha$. Our lemma is a\nweaker version of a result proved by Quigg \\cite[Corollary\n2.5]{Q}; we give a proof here for the sake of completeness.\n\n\\begin{lemma}\\label{fixedpointlemma} Let $\\delta$ be a coaction of\na discrete group $G$ on a $C^*$-algebra $A$, and let $(j_A,j_G)$\nbe the universal covariant representation of $(A,G,\\delta)$. Then\nthe fixed point algebra $A^\\delta$ is isomorphic to the corner\n$j_G(\\chi_1)(A\\times_\\delta G)j_G(\\chi_1)$.\n\\end{lemma}\n\n\\begin{proof} Consider the linear map $\\psi:A^\\delta\\to\nA\\times_\\delta G$, $a_1\\mapsto j_{A}(a_1)j_G(\\chi_1)$. For\n$a_1,b_1\\in A^\\delta$ the covariance property gives\n\\[j_A(a_1b_1)j_G(\\chi_1)=j_A(a_1)j_A(b_1)j_G(\\chi_1)^2=j_A(a_1)j_G(\\chi_1)j_A(b_1)j_G(\\chi_1)\\]\nand\n\\[j_A(a_1^*)j_G(\\chi_1)=\\left(j_G(\\chi_1)j_A(a_1)\\right)^*=\\left(j_A(a_1)j_G(\\chi_1)\\right)^*,\\]\nso $\\psi$ is a homomorphism. Furthermore, it is injective: let\n$\\pi$ be a faithful representation of $A$ on a Hilbert space\n$\\mathcal H$, and consider the induced representation $\\operatorname{Ind}\\pi$ of\n$A\\times_\\delta G$ on $\\mathcal H\\otimes l^2(G)$. Then for each\nnonzero $a_1\\in A^\\delta$ we have\n\\[\\operatorname{Ind}\\pi(\\psi(a_1))=(\\pi(a_1)\\otimes\\lambda_1)(1\\otimes\nM(\\chi_1))=\\pi(a_1)\\otimes M(\\chi_1)\\neq 0.\\] Hence we have shown\nthat $A^\\delta\\cong j_A(A^\\delta)j_G(\\chi_1)$.\n\nFor $s,t\\in G$ and $a_s\\in A_s$ the covariance of $(j_A,j_G)$\nimplies\n\\[\nj_G(\\chi_1)\\left(j_A(a_s)j_G(\\chi_t)\\right)j_G(\\chi_1)=\\begin{cases}j_A(a_s)j_G(\\chi_1)&\\text{if\n}s=t=1\\\\ 0&\\text{otherwise.}\\end{cases}\\] The span of the elements\n$j_A(a_s)j_G(\\chi_t)$ is dense in $A\\times_\\delta G$, and\neverything in sight is continuous, so this shows that\n\\[j_G(\\chi_1)\\left(A\\times_\\delta\nG\\right)j_G(\\chi_1)=\\overline{j_A(A^\\delta)j_G(\\chi_1)}\\cong\nA^\\delta.\\]\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{fixedpoint}] To simplify the notation, write $P$ for the element\n$P_{E^0\\times \\{1_G\\}}$ of $ M(C^*(E\\times_c G))$. The isomorphism\n$\\phi$ of Theorem \\ref{DPR3.4} extends to a strictly continuous\nisomorphism $\\bar{\\phi}$ of $ M(C^*(E\\times_c G))$ onto $\nM(C^*(E)\\times_{\\delta} G)$, such that\n\\[\\bar{\\phi}(P)=\\sum_{v\\in E^0}\\phi(P_{(v,1)})=\\sum_{v\\in\nE^0}j_{C^*(E)}(P_v)j_G(\\chi_1)=j_{C^*(E)}\\left(\\sum_{v\\in\nE^0}P_v\\right)j_G(\\chi_{1})=j_G(\\chi_{1}).\\] (Note that in writing\n``$j_{C^*(E)}\\left(\\sum_{v\\in E^0}P_v\\right)$'' we are actually\nusing the strongly continuous extension of $j_{C^*(E)}$ to the\nmultiplier algebra $M(C^*(E))$.) Thus the corner in which we are\ninterested is isomorphic to $j_G(\\chi_{1})(C^*(E)\\times_\\delta G)\nj_G(\\chi_{1})$, and so Lemma \\ref{fixedpointlemma} proves the\ntheorem.\n\\end{proof}\n\n\\begin{example}\\label{O_23} Continuing on from Examples \\ref{O_2} and \\ref{O_22}, the coaction\n$\\delta:C^*(E)\\to C^*(E)\\otimes C^*({\\mathbb Z}_3)$ induced by the\nlabelling $c$ is defined on generators by $\\delta(P_v)=P_v\\otimes\n0_{{\\mathbb Z}_3}$, $\\delta(S_e)=S_e\\otimes 2$ and $\\delta(S_f)=S_f\\otimes\n1$. Proposition \\ref{fixedpoint} says that the fixed point algebra\n$C^*(E)^\\delta$ is isomorphic to the corner\n$P_{(v,0)}C^*(E\\times_c{\\mathbb Z}_3)P_{(v,0)}$. As in Example \\ref{O_2},\nlet $T$ be the row- and path-finite directed subtree of\n$E\\times_c{\\mathbb Z}_3$ with vertices $E^0\\times{\\mathbb Z}_3$ and edges $(e,1)$\nand $(f,2)$. Then Theorem \\ref{main} implies that $C^*(E)^\\delta$\nis isomorphic to $C^*(E(T))$, where $E(T)$ is the graph\n\\[\\xygraph{\n{\\cdot}=\"v_1\":@(ul,dl)\"v_1\":@\/^2pc\/[r(2)]{\\cdot}=\"v_2\"\n:@(dr,ur)\"v_2\":@\/^2pc\/\"v_1\":@\/^1pc\/\"v_2\":@\/^1pc\/\"v_1\"}\n\\]\\end{example}\n\nWhen the group $G$ is abelian, coactions of $G$ correspond (via\nthe Fourier transform) to actions of the dual group $\\widehat G$\nof group homomorphisms $\\chi:G\\to{\\mathbb T}$ \\cite[Remark 2.7]{EQ}. The\nfollowing corollary applies this fact to Theorem \\ref{fixedpoint}.\n\n\\begin{cor} \\label{abelianfixedpoint} Let $\\alpha$ be an action of a compact abelian group $G$\non a graph algebra $C^*(E)$, such that for each $t\\in G$, each\n$v\\in E^0$ and each $e\\in E^1$, $\\alpha_t(P_v)=P_v$ and\n$\\alpha_t(S_e)=\\chi(e,t)S_e$ for some $\\chi(e,t)\\in{\\mathbb C}$. Then there\nis a labelling $c$ of $E$ by $\\widehat G$ such that\n\\[C^*(E)^\\alpha\\cong P_{E^0\\times\\{1_G\\}}C^*(E\\times_c \\widehat\nG)P_{E^0\\times\\{1_G\\}}.\\]\n\\end{cor}\n\n\\begin{proof} Each $\\alpha_t$ is an automorphism of $C^*(E)$, so each\n$\\chi(e,t)S_e$ is a partial isometry. This implies that for all\n$e\\in E^1$ and $t\\in G$, $\\chi(e,t)\\in{\\mathbb T}$. Now as\n$\\alpha:G\\to\\operatorname{Aut}(C^*(E))$ is a group homomorphism, we must have\neach $\\chi(e,\\cdot):G\\to{\\mathbb T}$ a group homomorphism, so\n$\\chi(e,\\cdot)\\in\\widehat G$ for each $e\\in E^1$. Let $c$ be the\nlabelling $e\\mapsto\\chi(e,\\cdot)$ of $E$ by the discrete group\n$\\widehat G$, and let $\\delta$ be the induced coaction of\n$\\widehat G$ on $C^*(E)$. The Fourier transform then gives\n$C^*(E)^\\alpha=C^*(E)^\\delta$, so the corollary follows from\nTheorem \\ref{fixedpoint}.\n\\end{proof}\n\nThe following key example was first brought to our attention by\nDavid Pask:\n\n\\begin{example} Let $E$ be a directed graph, and let $c:E^1\\to{\\mathbb Z}$\nbe the labelling $c(e)=1$ for all $e\\in E^1$. This labelling\ncorresponds to the canonical gauge action $\\gamma$ of ${\\mathbb T}$ on\n$C^*(E)$, so Corollary \\ref{abelianfixedpoint} implies that the\n$AF$-core $C^*(E)^\\gamma$ is isomorphic to the corner $\nP_{E^0\\times\\{0_{\\mathbb Z}\\}}C^*(E\\times_c{\\mathbb Z})P_{E^0\\times\\{0_{\\mathbb Z}\\}}$.\n\\end{example}\n\nTheorem \\ref{fixedpoint} and its corollary lead us to seek\nconditions on the labelling $c$ which allow us to apply Theorem\n\\ref{main} to find a graph for $C^*(E)^{\\delta}$. That is, we seek\nconditions on $c$ which ensure that $E\\times_c G$ has a row- and\npath-finite directed subtree with roots $E^0\\times\\{1_G\\}$ and\nvertex set $H_{E\\times_c G}(E^0\\times\\{1_G\\})$. If $E^0$ and $G$\nare both finite, the graph $E\\times_c G$ has finitely many\nvertices and so Lemma \\ref{extree} tells us that we can always\nfind such a tree. More generally, when $E$ is row-finite we have\nthe following:\n\n\\begin{prop} \\label{Kirchhoff} Let $c$ be a labelling of a row-finite directed\ngraph $E$ by a discrete group $G$ such that for each $\\mu\\in\nE^\\infty$, there exists $i\\in{\\mathbb N}$ such that\n$c(\\mu_1\\ldots\\mu_i)=1_G$. Then $E\\times_c G$ has a row- and\npath-finite directed subtree with roots $E^0\\times\\{1_G\\}$ and\nvertex set $H_{E\\times_c G}(E^0\\times\\{1_G\\})$.\\end{prop}\n\n(Note that when $E$ is finite and $G={\\mathbb Z}$, the above condition is\nequivalent to the condition that $c(\\lambda)=0$ for each loop\n$\\lambda\\in E^*$, in analogy with Kirchhoff's voltage law.)\n\n\\begin{proof} We may apply the iterative method used in the proof\nof Lemma \\ref{extree} to find a directed subtree $T$ of $E\\times_c\nG$ with the desired roots and vertex set. This tree is row-finite\nbecause $E$ is, so it remains to show that it is path-finite.\nSuppose it is not, and let $\\mu=(\\mu_1,t_1)(\\mu_2,t_2)\\ldots$ be\nan infinite path in $T$. Lemma \\ref{T}(1) implies that we may\nassume that the source of $\\mu$ is in $E^0\\times\\{1_G\\}$, so that\n$t_1=c(\\mu_1)^{-1}$, and then by definition of the source and\nrange maps in $E\\times_c G$ we must have $r(\\mu_i)=s(\\mu_{i+1})$\nand $t_i =c(\\mu_1\\ldots\\mu_i)^{-1}$ for each $i\\in{\\mathbb N}$. Now\n$\\mu_1\\mu_2\\ldots$ is an infinite path in $E^\\infty$, so by\nassumption there exists an index $i\\in{\\mathbb N}$ such that\n$c(\\mu_1\\ldots\\mu_i)=1_G$. For this $i$ we have $r\\left((\\mu_i,t_i\n)\\right)=(r(\\mu_i),c(\\mu_1\\ldots\\mu_i)^{-1})=(r(\\mu_i),1_G)$, so\nthe vertex $(r(\\mu_i),1_G)$ receives an edge in the subtree $T$.\nThis is a contradiction, since each vertex in $E^0\\times\\{1_G\\}$\nis a root of $T$, and so we conclude that $T$ is path-finite.\n\\end{proof}\n\n\\begin{remark} We can also prove the following analogue of Proposition\n\\ref{Kirchhoff} for coactions of homogeneous spaces: when $E$ is\nrow-finite and $H$ is a subgroup of $G$ such that for each $\\mu\\in\nE^\\infty$ there exists $i\\in{\\mathbb N}$ with $c(\\mu_1\\ldots\\mu_i)\\in H$,\nthen there is a directed subtree in $E\\times_c (G\/H)$ with the\ndesired properties. The proof is virtually identical to the proof\nof the preceding proposition, and is therefore\nomitted.\\end{remark}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA monthly\/quarterly\/annually updated sizeable remote sensing image containing the entire region of interest is crucial for downstream change detection tasks, geographic information system, and many other remote sensing applications, e.g., disaster monitoring, environmental monitoring, resource monitoring and geographical mapping. However, limited by the imaging width, mechanism and cloud coverage, a single remote sensing image often cannot cover the entire region of interest. However, suppose several remote sensing images obtained from different satellites, sensors or imaging times are directly mosaicked. In that case, a significant flower-cloth effect will appear (i.e., the mosaic image is not visually continuous.). This phenomenon is since different remote sensing images may be affected by imaging time, illumination effects, atmospheric attenuation, and other factors (\\cite{teillet1986image},\\cite{kim1990normalization}), resulting in radiometric distortion on the same object. Therefore, eliminating these distortions is important.\n\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[the detailed direct merge result at red point]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/direct_0255_merge_red.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area at red point]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/histogram_matching_0255_merge_red.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area with Poisson editing at red point \\textbf{(no seam lines)}]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/poisson_tiling_merge_0255_red.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area with GIMP manual seam line elimination at red point ]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/GIMP_merge_02553-changed_red.jpg}}}\n\\caption{The comparison figure of the four methods at red point}\n\\label{fig:Comparison figure of the four methods at red point}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[the detailed direct merge result at yellow point]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/direct_0255_merge_yellow.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area at yellow point]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/histogram_matching_0255_merge_yellow.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area with Poisson editing at yellow point \\textbf{(no seam lines)}]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/poisson_tiling_merge_0255_yellow.jpg}}}\n\\subfigure[the detailed merge result using histogram matching on overlap area with GIMP manual seam lines elimination at yellow point]{%\n\\resizebox*{3.3cm}{!}{\\includegraphics{figure\/GIMP_merge_02553-changed_yellow.jpg}}}\n\\caption{The comparison figure of the four methods at yellow point}\n\\label{fig:Comparison figure of the four methods at yellow point}\n\\end{figure}\n\n\n\nResearchers have proposed many methods to alleviate these distortions, but these distortions are not solved effectively because of the seam lines. Relative radiometric normalization(RRN) aims to reduce the global numerical colour\/radiometric differences among multiple remote sensing images by modifying the original image's colour\/radiometric properties to match the reference image. Wallis filtering (\\cite{zhang1999image}, \\cite{Li2019Parallel}) is one kind of methods, forcing different regions to have the same mean and variance. Some methods bases on histogram matching (\\cite{chavez1994automatic},\\cite{horn1979destriping},\\cite{weinreb1989destriping}) and some further bases on matching on the overlap area (\\cite{helmer2005cloud}). There are also other approaches including pseudo-invariant feature (\\cite{schott1988radiometric}), least square regression (\\cite{zhang2008automatic}), establishing reference database (\\cite{zhang2017mixed}) and etc. However, the methods mentioned above still need post-processing on seam lines to achieve a smooth transition. Besides, in the complex scenarios when there are lots of remote sensing images with different intersecting modes or extreme topological relationships, it becomes hard to use automatic feathering and blending methods to remove seam line. Therefore,  people cost a lot of staffing and time to use Photoshop and other image editing software to manually edit remote sensing images.\n\n\nCurrently, seam lines eliminating methods, including mosaic line optimization, feathering and blending, utilizes overlapped area or vertical\/horizontal seam lines shape, failing when the overlapped area is small. For instance, \\cite{santellani2018seamless} utilizes Voronoi decomposition and colour contrast cost minimum distance mosaic line optimization to minimize the colour contrast on both sides of the mosaic line. Instead of eliminating the specified seam lines, this kind of methods generates a better mosaic line. However, this kind of methods usually cannot be perfectly seamless and may become ineffective when the overlapped area is small or the overlapped area has inner ring. \\cite{lu2014fusion} assumes the rectangular shape of the overlapped area and utilizes the 2-D linear weighted blending on the low frequency overlapped parts, and merges the images based on the mosaic line optimized by the high frequency overlapped parts. However, when the overlapped shape is not rectangular, or the overlapped area is small, this method becomes intractable or unhelpful. Laplacian pyramid (\\cite{burt1987Laplacian}) blending utilizes the mask on the overlapped area to construct different levels of blending Gaussian and Laplacian pyramids to reconstruct the blending image to eliminate seam lines. This kind of methods may hopefully overcome arbitrary seam lines on the rectangular overlapped area. However, when the overlapped area is small or complicated, the directly smoothing and downsampling procedures of Laplacian pyramid blending become intractable. Moreover, Laplacian pyramid blending has to change the colour\/radiometric of all the images on both sides of seam lines, and this can be a kind of distortion.\n\nThis paper presents a novel automatic approach for RRN mosaicking that uses the histogram matching on the overlap area to achieve relative radiometric normalization, Poisson editing to remove the seam line and sequence merging to determine the normalization transfer order. Compared with the traditional seam lines elimination methods, advantages of our method are as follows: \\begin{enumerate}\n\\item \\textbf{fully automatic};\n\\item \\textbf{widely applicable} (it can overcome arbitrary seam lines shapes, complex overlapped shapes and small overlapped area.);\n\\item \\textbf{truly seamless with low distortion} (instead of generating a new mosaic line, it achieves seamless transition by eliminating the colour contrast on one side of seam lines, reducing the distortion.).\n\\end{enumerate}\n\n\n\n\n\\section{Methodology}\nSuppose that we have a reference image and a set of other images. We can regard the reference image as the starting image of image mosaicking. The proposed method iteratively selects an original image intersected with the reference image and sequentially implements histogram matching, Poisson editing, and merging procedures to generate a new reference image. Repeating the procedures, we will finally obtain the complete mosaicking reference image.\n\nIn the following sections, we will first introduce the histogram matching method based on overlap area, and then introduce the Poisson editing on the complex boundary, and finally introduce the merging procedure.\n\n\\subsection{Histogram matching based on overlap area}\nFirst, suppose the range of a remote sensing image is $[j_{min},j_{max}]$, we can map it to $[i_{min},i_{max}]$ where $i_{min},i_{max}$ are integers. The density histogram of an image is \\begin{equation}\\label{density histogram}\n  P(i)=\\frac{N_i }{N },\n\\end{equation}\nwhere $N_i$ is the number of pixels that their grey scales equal $i$ and $N$ is the total number of pixels. The cumulative probability of the image is\n\\begin{equation}\\label{cumulative probability}\nCP(i) = P(t\\le i) = \\frac{\\sum_{t=1}^{i}N_t}{N}.\n\\end{equation}\nSuppose the overlapped area points set is $S^{overlapped}$, and ${N_t^{overlapped}}_O$ denotes the number of pixels in overlapped original image that grey scales equal $t$ and ${N_t^{overlapped}}_R$ denotes the number of pixels in overlapped reference image that grey scales equal $t$, it yields the cumulative probability of the overlap of the original image and the reference image are $CP_O^{overlapped}$ and $CP_R^{overlapped}$ respectively:\n\\begin{eqnarray}\\label{cumulative probability overlapped}\n  CP_O^{overlapped}(i) &=& \\frac{\\sum_{t=1}^{i}{N_t^{overlapped}}_O}{\\vert S^{overlapped}\\vert }, \\\\\n  CP_R^{overlapped}(i) &=& \\frac{\\sum_{t=1}^{i}{N_t^{overlapped}}_R}{\\vert S^{overlapped}\\vert }.\n\\end{eqnarray}\n\nFor pixel value $i_O$ with cumulative probability $ CP_O^{overlapped}(i_O)$, we can find the corresponding $i_R$, which share the similar probability that\n\\begin{equation}\\label{find cumulative}\ni_R = \\arg\\min_{i}\\vert CP_R^{overlapped}(i)- CP_O^{overlapped}(i_O)\\vert.\n\\end{equation}\nIt yields the mapping $T^{overlap}:i_O\\rightarrow i_R$.\nWe apply the $T^{overlap}$ to the original image then we achieve the histogram matching.\n\nSince objects in the overlap area of the images are supposed consistent, when using the mapping $T^{overlap}$ obtained from the histogram matching on the overlap area of the original image and the reference image, the radiometric value of the processed origin image incline to be consistent to the reference image's radiometric value. Notice that if the reference image and original image only intersect on a small area and the overlapped area's colour\/radiometric diversity is insufficient, we skip this procedure and move to the Poisson editing procedure.\n\nHowever, although the radiometric value can be consistent, there might be noticeable seam lines, especially when the intersection of the original image and reference image is complex or a slight difference occurs in the overlap area. Some feathering and blending methods are only suitable for simple intersection modes such as large rectangle intersection. However, they lose efficacy when the intersection is complex, or the intersection area is small, or the seam lines shapes are complicated, or the radiometric value contrast is too large. An efficient seam lines elimination is needed.\n\n\\subsection{Poisson editing on complex boundary}\nPoisson editing introduces a guidance vector field from the source image to transplant the source image's gradient vector field into a target region to achieve seamless filling (\\cite{perez2003Poisson}).\nThe central guiding ideology of Poisson editing assumes that seamless image function to be solved in $\\Omega\\in R^2$ space is $f$, the source image's gradient vector field is $\\boldsymbol v$ and the boundary of the seamless image defined on $\\partial \\Omega$ is $f^*$ where $\\partial \\Omega$ is the boundary points of $\\Omega$. The task is to find the optimal $f^{optimal}$ such that\n\\begin{equation}\\label{Poisson function}\n  f^{optimal}=\\arg\\min\\limits_{f}\\int\\int_{\\Omega}\\Vert \\nabla f -\\boldsymbol v\\Vert^2 \\quad \\mathrm{with} \\ f\\vert_{ \\partial\\Omega} = f^*\\vert_{ \\partial \\Omega},\n\\end{equation}\nwhere $\\Vert \\cdot \\Vert$ is the $L_2$ norm and $\\nabla$ is the gradient operator. The minimizing problem is equivalent to solve:\n\\begin{equation}\\label{Poisson function}\n  f^{optimal}=\\arg\\min\\limits_{f}\\triangle f =\\mathrm{div} \\ \\boldsymbol v \\ \\mathrm{over} \\ \\Omega \\quad \\mathrm{with} \\ f\\vert_{ \\partial\\Omega} = f^*\\vert_{ \\partial \\Omega},\n\\end{equation}\nwhere $\\triangle$ is the laplace operator and $\\mathrm{div}$ is divergence operator.\n\nThe above equations are generalized to the discrete case (linear equations) to make the Poisson editing algorithm trackable, and we can further solve it by iteration methods. The detailed can refer to \\cite{perez2003Poisson}.\n\nThe important thing left is to determine the source point set $\\Omega$ and the boundary function $f^*$ on $\\partial \\Omega$. The following shows how to obtain the $\\Omega$ and $f^*$ on $\\partial \\Omega$.\n\nFirst, we have the processed original image $I_O$ with point sets $S_O=\\{\\boldsymbol p|\\boldsymbol p \\in origin\\ image\\}$ and the reference image $I_R$ with point set $S_R=\\{\\boldsymbol p|\\boldsymbol p \\in reference\\ image\\}$. The example is illustrated in the following figure~\\ref{fig:The illustration of the reference and origin point set}.\n\\begin{figure}[ht]\n\\centering\n\\subfigure[reference point set $S_R$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/reference.png}}}\\hspace{5pt}\n\\subfigure[original point set $S_O$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/origin_set.png}}}\\hspace{5pt}\n\\subfigure[whole point set $S_O\\cup S_R$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/whole_set.png}}}\n\\caption{The illustration of the reference and original point set. The dark blue points are the intersection points.} \\label{fig:The illustration of the reference and origin point set}\n\\end{figure}\n\nSecond, the target image $I_T$ is with the region $S_T = S_O-S_R^\\circ$ where $-$ is the set difference and $S_R^\\circ$ is the interior point of $S_R$. By the interior point of $S_R$, we mean that \\begin{equation}\\label{interior point}\nS_R^\\circ=\\{\\boldsymbol q|\\boldsymbol q\\in S_R ,\\ Neighbor(\\boldsymbol q)\\in S_R \\ \\mathrm{and} \\ \\vert Neighbor(\\boldsymbol q)\\vert =4\\}\n     \\end{equation}\nwhere $Neighbor(\\boldsymbol q)$ is the set of $\\boldsymbol q$'s 4-connected neighbors. Besides,\n\\begin{equation}\\label{target image}\n  I_T(p)=\\left\\{\n            \\begin{aligned}\nI_O(p) \\ \\boldsymbol p \\in  (S_O-S_R^\\circ)-\\partial S_R\\\\\nI_R(p) \\ \\boldsymbol p \\in   (S_O-S_R^\\circ)\\cap \\partial S_R\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\partial S_R = S_R - S_R^\\circ$.\n\\begin{figure}[ht]\n\\centering\n\\subfigure[the reference interior point set $S_R^\\circ$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/reference_interior.png}}}\\hspace{5pt}\n\\subfigure[the target point set $S_T= S_O-S_R^\\circ$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/target_set.png}}}\\hspace{5pt}\n\\subfigure[the source point set $S_S= S_T^\\circ$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/source_set.png}}}\n\\caption{The illustration of the reference interior, target and source point set. The dark blue points are the intersection points. The white points which don't belong to the point set demonstrate original set's boundary.}\n\\label{fig:The illustration of the reference interior, target and source point set}\n\\end{figure}\n\n\nThe source image $I_S$ is with the region $S_S=S_T^\\circ$ and $I_S={I_O}_{\\vert S_S}$, that is $I_S(\\boldsymbol p)=I_O(\\boldsymbol p)\\ \\forall \\boldsymbol p\\in S_S$.\nIt is obvious that the source image is contained in the target image, and the target image is modified slightly on the boundary point of the $S_O-S_R^\\circ$. The point value of $I_R$ on $(S_O-S_R^\\circ)\\cap \\partial S_R$ replaces of $I_O$ on $(S_O-S_R^\\circ)\\cap \\partial S_R$. Notice that replacement introduce the seam line between the boundary and the interior points. The example of the aforementioned point set is illustrated in the figure~\\ref{fig:The illustration of the reference interior, target and source point set}.\n\nFinally, let $\\Omega = S_T$ and $f^*\\vert_{\\partial \\Omega} = I_T\\vert_{\\partial S_T}$, we can implement the Poisson editing to transplant source image to the target image and obtained the seamless image where the color contrast of the boundary and interior points are eliminated due to the guided interpolation.\n\nNotice that the size of source image $I_S$ might be pretty large, and it will cost many resources to solve the Poisson equation. Empirically,  only the pixel value of the region in $S_S$ that near  $(S_O-S_R^\\circ)\\cap \\partial S_R$ will be changed drastically after the Poisson editing. Therefore, we consider reducing the size of $I_S$ in the actual implementation. Let \\begin{equation}\\label{Sphere of a point}\n  B_d(\\boldsymbol p)=\\{\\boldsymbol q|\\vert q_1 -p_1\\vert +\\vert q_2-p_2\\vert \\le d\\},\n\\end{equation}\nand\n\\begin{equation}\\label{Sphere of a set}\n   B_d(S)=\\cup_{\\boldsymbol p \\in S}   B_d(\\boldsymbol p) .\n\\end{equation}\nWe build a new source image $I_S^{new}$ with $S_S^{new} = S_S \\cap B_d(S_T\\cap \\partial S_R)$  and $S_T^{new} = S_T \\cap B_{d+1}(S_T\\cap \\partial S_R)$. Let $\\Omega = S_T^{new}$ and $f^*\\vert_{\\partial \\Omega} = I_T\\vert_{\\partial S_T^{new}}$ implement the Poisson editing of the new source image to the target image. The choice of $d$ depends on the size of $I_O$ and depends on whether we implement the histogram matching procedure. If we skip the histogram matching procedure, $d$ should be set to the maximum value to also eliminate global color contrast. The $S_S^{new}$ is illustrated in the figure~\\ref{fig:new source point set}.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[the new source point set $S_S^{new}$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/source_new_set.png}}}\n\\subfigure[the new target point set $S_T^{new}$]{%\n\\resizebox*{4cm}{!}{\\includegraphics{figure\/target_new_set.png}}}\n\\caption{The illustration of the new source point set with $d=3$. The white points which don't belong to the point set demonstrate original set's boundary. The dark blue points are the intersection points.}\n\\label{fig:new source point set}\n\\end{figure}\n\\subsection{Merging}\nIn order to let this procedure continue, we merge the seamless target image and the reference image. Notice that since the seam lines are removed, the merged image tends to be seamless and can be the new reference image shown in the figure~\\ref{new reference image} to participate in the latter repeated processing.\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=4cm]{figure\/new_reference.png}\n  \\caption{the merged new reference image}\\label{new reference image}\n\\end{figure}\n\n\\FloatBarrier\n\\section{Experiments}\n\\begin{figure}[ht]\n\\centering\n\\subfigure[the direct merge result]{%\n\\resizebox*{7cm}{!}{\\includegraphics{figure\/direct_0255_merge.jpg}}}\n\\subfigure[the merge result using histogram matching on overlap area]{%\n\\resizebox*{7cm}{!}{\\includegraphics{figure\/histogram_matching_0255_merge.jpg}}}\\hspace{5pt}\n\\subfigure[the merge result using histogram matching on overlap area with Poisson editing]{%\n\\resizebox*{7cm}{!}{\\includegraphics{figure\/poisson_tiling_merge_0255.jpg}}}\n\\subfigure[the merge result using histogram matching on overlap area with GIMP manual seam line elimination]{%\n\\resizebox*{7cm}{!}{\\includegraphics{figure\/GIMP_merge_02553-changed.jpg}}}\n\\caption{The comparison figure of the four methods  }\n\\label{fig:Comparison figure of the four methods}\n\\end{figure}\n\\subsection{Material and environment}\nWe implement the algorithm and apply 34 resampled remote sensing multispectral images of GF-1 and GF-2 satellites in image mosaicking task.\n\nThese images partially cover the range of Qingdao, Shandong Province of China. These images were acquired in May 2019.\n\nWe performed the preprocessing steps of RPC orthorectication, radiometric calibration, atmospheric correction and  Gram-Schmidt PanSharpening (\\cite{laben2000process}) on these images, and finally resampled the spatial resolution to  16m. The first three spectral are displayed.\n\nWe implement the algorithm on the computer with Inter(R) Xeon(R) W-2123 3.60GHz CPU  and 32G RAM.\n\n\\subsection{Image mosaicking}\n\n\nWe use three automatic methods and one manual method with the help of GIMP image editing software(\\cite{gimp}). We use python to accomplish the automatic algorithm. Precisely, we first calculate the intersection of the square envelope of the reference image and the square envelope of the original image to speed up the calculation of the overlapped area and the determination of the blue boundary points($S_T\\cap\\partial S_R$). The parameter $d$ defined in equation~\\ref{Sphere of a point} is set to  $150$ in this experiment. Then we use the sparse matrix of scipy library(\\cite{2020SciPy}) to construct the Poisson matrix and coefficient and use the algebraic multi-grid method with pyamg(\\cite{OlSc2018}) library to solve the Poisson equations. We use gdal library to merge the images.\n\n\nThe time-consuming algebraic multi-grid solving step of  Poisson editing on about $4000-750000$ points costs about $1-21$ second. The average processing time on single remote sensing image ranging from about $(2000\\times1500)$ to $(2900\\times2400)$ is about 1.5 minute.\n\n\n\nFigure~\\ref{fig:Comparison figure of the four methods} demonstrates the direct mosaic result, histogram matching result, histogram matching with Poisson editing result and histogram matching with GIMP manual seam line elimination. From figure~\\ref{fig:Comparison figure of the four methods}, we can tell from the figure that histogram matching eliminates the colour contrast.\n\nWe provide the detailed image of the red point and yellow point in the figure~\\ref{fig:Comparison figure of the four methods at red point} and figure~\\ref{fig:Comparison figure of the four methods at yellow point}.\n\nAccording to figure~\\ref{fig:Comparison figure of the four methods at red point} and \\ref{fig:Comparison figure of the four methods at yellow point}, histogram matching on overlap area cannot deal with the boundary seam lines in the complex situation, including the boundary lake and multiple intersection areas. The colour difference on the different parts of the intersection area can cause colour inconsistency after histogram matching. The first case happened in the area shown in the figure~\\ref{fig:Comparison figure of the four methods at yellow point}, and these two cases happened together in the area shown in the figure~\\ref{fig:Comparison figure of the four methods at red point}. The upper-right remote sensing image in the figure~\\ref{fig:Comparison figure of the four methods at red point} has more intersection with the image above rather than the intersection of image on the left, while the colour is different on these two images. As a result, it causes the colour difference on the lake. The manual edition blurs and makes the feathering effect on the image and alleviate the seam lines. In contrast, the Poisson editing can eliminate the seam lines on the complex situation and spread the colour to the expansion region and make the transition smooth. Compare to the four methods,  the histogram matching on overlap area with Poisson editing method has the best effect.\n\\FloatBarrier\n\n\\section{Conclusion}\n\nThis paper presents a novel automatic approach for eliminating seam lines in RRN mosaicking tasks. It utilizes the histogram matching on the overlap area to achieve colour\/radiometric normalization, Poisson editing to remove the complicated seam lines in diverging intersecting modes and merging to achieve seamless mosaicking finally. It is fully automatic, overcoming arbitrary seam lines shapes, complex intersecting modes and small overlapped area. Our method visually surpasses the automatic methods without Poisson editing in the experiment and the manual blurring and feathering method using GIMP software.\n\nCurrently, our method is independent of the mosaic line optimization algorithm, and it can easily incorporate with the optimized mosaic line. Besides, our method also naturally applies to cloud inpainting task. In the future, we will investigate how to accelerate the Poisson editing procedure to let this algorithm be more efficient than the current version.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Shengbing Zou's guidance and leadership. We would like to thank Jian Liao for discussion the Poisson editing algorithm. We would like to thank Fan Wang for discussing the computation of the overlapped area.\n\n\\section*{Disclosure statement}\n\nNone.\n\n\\bibliographystyle{tfcad}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe production of $\\phi$ mesons is considered to be one of the key observables to \nprobe the state of matter produced in relativistic heavy-ion collisions. \nStrangeness enhancement in relativistic nucleus-nucleus\ncollisions compared to nucleon-nucleon collisions\nhas been originally suggested as a possible signal for the formation of\na deconfined plasma of quarks and gluons during the initial \nstate of the reaction \n\\cite{Rafelski:1982pu,Koch:1986ud,Shor:1984ui}. The dominant production\nof $s\\bar{s}$ pairs via gluon-gluon interaction in the plasma\nmay result in an enhanced number of strange and multistrange\nparticles produced after hadronization; in particular, free\n$s\\bar{s}$ pairs would coalesce to form $\\phi$ mesons \n\\cite{Shor:1984ui}, whereas their\nproduction in $pp$ collisions is suppressed according to the\nOkubo-Zweig-Iizuka rule \\cite{Okubo:1963fa,Zweig:1964,Iizuka:1966fk}. \nIt is moreover expected that $\\phi$ mesons decouple from the rest of the \nsystem earlier than other non-strange hadrons. \nAt RHIC energies, an early decoupling of the $\\phi$ \nfrom the hadronic rescattering dynamics was found in Refs. \\cite{Cheng:2003as} and \\cite{Hirano:2007ei}. \nSimilar conclusions were obtained with the RQMD cascade in \\cite{vanHecke:1998yu} for the \n$\\Omega$ baryons at SPS energies.\n\nThe dominant hadronic decay of the $\\phi$ meson is \n$\\phi\\rightarrow K \\bar{K}$; \nadditionally, being a neutral vector meson, the $\\phi$ \ncontributes to dilepton production via the direct decays  \n$\\phi\\rightarrow e^+e^-$ and $\\phi\\rightarrow \\mu^+\\mu^-$.\nPhi meson production was investigated extensively at \nthe SPS by several experiments in \nthe kaon (NA49 \\cite{Afanasev:2000uu,Friese:2002re,Alt:2004wc,Alt:2008iv} \nand CERES \\cite{Adamova:2005jr}), \ndielectron (CERES \\cite{Adamova:2005jr}), and dimuon \n(NA50 \\cite{Alessandro:2003gy}and NA60 \\cite{Arnaldi:2009wr}) channels. \nThe reconstruction of in-matter $\\phi\\rightarrow K \\bar{K}$ decays, \nhowever, might be partially prevented by kaon absorption and \nrescattering \\cite{Johnson:1999fv,Santini:2006cm,Filip:2001st} and a priori a careful investigation \nof kaon final state interactions cannot be avoided in a\nquantitative comparison to experimental data.\nOn the contrary, the dilepton channel is \nnot  affected by such a shortcoming: dileptons \nleave their production site essentially undistorted, and \na comparison of model calculations to measurements in the dilepton channel \nis surely more straightforward. Recently, \nhigh statistics measurements of \n$\\phi\\rightarrow \\mu^+\\mu^-$ meson production in In-In collisions \nat  $E_{\\rm lab}$=158$A$ GeV  have been  presented by the NA60 Collaboration \n\\cite{Arnaldi:2009wr}. This reaction will be addressed in the present work.\n\nTo investigate  $\\phi\\rightarrow \\mu^+\\mu^-$ production \nin  In-In collisions at  $E_{\\rm lab}$=158$A$ GeV we employ an integrated \nBoltzmann+hydrodynamics hybrid approach based on the Ultrarelativistic \nQuantum Molecular Dynamics (UrQMD) transport model with an intermediate \nhydrodynamic stage. In this approach, initial conditions and \ncontinuous decoupling up to freeze-out are treated within a \nfull non-equilibrium transport approach, whereas \nhydrodynamics is used to describe the intermediate equilibrated phase.\nThis allows to reduce the parameters for the initial conditions and provides \na consistent freeze-out description. \nMoreover, by comparing the hybrid approach to pure transport calculations, \nwe are able to directly investigate the consequences that \na dynamical approach involving local thermal and chemical equilibrium \nand one based on full non-equilibrium dynamics have on $\\phi$ \nmeson production.\n\n\nThe paper is structured in the following way: \nIn Sec. \\ref{The model}, we briefly discuss the hybrid model and \npresent the procedure used to evaluate the  $\\phi\\rightarrow \\mu^+\\mu^-$ \nemission and, consequently, \nreconstruct the $\\phi$ meson. \nIn Sec. \\ref{Results} we present calculations for $\\phi$-meson \ntransverse mass spectra as a function \nof centrality. The various contributions to the spectra associated \nwith different stages of the reaction dynamics are shown and a \ncomparison to the NA60 data is presented.\nFinally, a summary and conclusions are given in Sec. \\ref{Conclusions}.\n\n\n\\section{The model \\label{The model}}\n\n\\subsection{The hybrid approach}\n\nTo simulate the dynamics\nof the In+In collisions we employ a transport approach with an\nembedded three-dimensional ideal relativistic one fluid\nevolution for the hot and dense stage of the reaction\nbased on the UrQMD\n model \\cite{Petersen:2008dd}. The present hybrid approach has \nbeen extensively described \nin  Ref. \\cite{Petersen:2008dd}. Here, we limit ourselves \nto briefly describe  its main features and refer the reader to Ref. \n\\cite{Petersen:2008dd} for details.\n\nUrQMD \\cite{Bass:1998ca,Bleicher:1999xi,Petersen:2008kb} is a hadronic transport approach which simulates multiple \ninteractions of ingoing\nand newly produced particles, the excitation and fragmentation\nof color strings and the formation and decay\nof hadronic resonances.  \nThe coupling between the\nUrQMD initial state and the hydrodynamical evolution\nproceeds when the two Lorentz-contracted nuclei have\npassed through each other. Here, the spectators continue to propagate \nin the cascade and all other hadrons\nare mapped to the hydrodynamic grid. This treatment\nis especially important for non-central collisions which\nare also studied in the present work. Event-by-event fluctuations are \ndirectly taken into account via initial conditions\ngenerated by the primary collisions and string fragmentations in the \nmicroscopic UrQMD model. This leads to\nnon-trivial velocity and energy density distributions for\nthe hydrodynamical initial conditions \\cite{Steinheimer:2007iy,Petersen:2009vx}. \nSubsequently, a full (3+1) dimensional\nideal hydrodynamic evolution is performed using\nthe SHASTA algorithm \\cite{Rischke:1995ir,Rischke:1995mt}. The hydrodynamic\nevolution is gradually merged into the hadronic cascade: \nto mimic an iso-eigentime hypersurface, full\ntransverse slices, of thickness $\\Delta z$ = 0.2fm, are transformed to particles whenever in all\ncells of each individual slice the energy density  drops below\nfive times the ground state energy density. \nThe employment of such gradual transition allows to \nobtain a rapidity independent transition temperature \nwithout artificial time dilatation effects \\cite{Petersen:2009gu} and has \nbeen explored in detail in various recent works \n\\cite{Li:2008qm,Petersen:2009mz,Petersen:1900zz,Petersen:2009gu} \ndevoted to SPS conditions. \nWhen merging, the hydrodynamic fields are transformed to particle\ndegrees of freedom via the Cooper-Frye equation in the computational\nframe. The created particles proceed in their evolution in\nthe hadronic cascade where rescatterings and\nfinal decays occur until all interactions cease and\nthe system decouples.\n\nAn input for the hydrodynamical calculation is\nthe equation of state (EoS). In this work we employ\na hadron gas equation of state, describing a\nnon-interacting gas of free hadrons \\cite{Zschiesche:2002zr}. Included here\nare all reliably known hadrons with masses up to $\\approx$ 2\nGeV, which is equivalent to the active degrees of freedom\nof the UrQMD model.\n\n\n\\subsection{Reconstruction of the phi meson in the dimuon channel }\nThe reconstruction of the $\\phi$ meson from the dimuon channel requires the \nevaluation of the $\\phi\\rightarrow\\mu^+\\mu^-$ emission. This is calculated \nperturbatively in the evolution stage that precedes or follows the \nhydrodynamical phase, and from thermal rates in the latter. In the following, \nthe terms \npre-equilibrium\/pre-hydro (post-equilibrium\/post-hydro), will be used \nto indicate the stage preceding (following) the mapping \nUrQMD$\\rightarrow$hydrodynamical\n (hydrodynamical$\\rightarrow$UrQMD) evolution description. \nNote that in the pre- and post-hydro stages the particles are the explicit \ndegree of freedom and their interactions are explicitly treated within the cascade \ntransport approach.  \n\n\\subsubsection{Reconstruction from pre-equilibrium and post-equilibrium emission}\n\nGiven the number $N^\\phi_{\\mu^+\\mu^-}$ of $\\mu^+\\mu^-$ pairs emitted in  \n$\\phi\\rightarrow \\mu^+\\mu^-$ decays, the number $N^{\\rm rec}_{\\phi}$ of \n$\\phi$ meson reconstructed \nin the dimuon channel is \n\\begin{equation}\nN^{\\rm rec}_{\\phi}=N^\\phi_{\\mu^+\\mu^-}\/BR^\\phi_{\\mu^+\\mu^-},\n\\label{eq1}\n\\end{equation}\nwhere $BR^\\phi_{\\mu^+\\mu^-}$ is the $\\phi\\rightarrow \\mu^+\\mu^-$ branching \nratio. The latter is given by the ratio between the dimuon and the total width \nof the $\\phi$ meson, i.e. $BR^\\phi_{\\mu^+\\mu^-}=\\Gamma^\\phi_{\\mu^+\\mu^-}\/\n\\Gamma^\\phi_{\\rm tot}$.\nThus,\n\\begin{equation}\nN^{\\rm rec}_{\\phi}=N^\\phi_{\\mu^+\\mu^-}~\\frac{\\Gamma^\\phi_{\\rm tot}}\n{\\Gamma^\\phi_{\\mu^+\\mu^-}}.\n\\label{eq2}\n\\end{equation}\n \nIn the pre-equilibrium and post-equilibrium phase dimuon emission from the \n$\\phi$-meson can be calculated perturbatively as\n\\begin{equation}\nN^\\phi_{\\mu^+\\mu^-}=\\sum_n^{N_{\\phi}}\\int_{\\tau_i(n)}^{\\tau_f(n)}\n\\Gamma^\\phi_{\\mu^+\\mu^-}~d\\tau\n\\label{eq3}\n\\end{equation}\nwhere now $N_{\\phi}$ indicates the number of $\\phi$ mesons present, \nin some stage of the evolution, in the system and  $\\tau_i(n)$ ($\\tau_f(n)$) \nare the times at which the \n$n$-th $\\phi$ meson appeared in (disappeared from) the system and are \nevaluated in the meson rest-frame.\nThis perturbative method is known as time integration method or \n``shining method'' and has long been \napplied in the transport description of dilepton emission \n(see e.g. \\cite{Li:1994cj,Vogel:2007yu,Schmidt:2008hm}).\nCombining Eqs.(\\ref{eq1})--(\\ref{eq3}) one has:\n\\begin{equation}\nN^{\\rm rec}_{\\phi}=\\sum_n^{N_{\\phi}}\\int_{\\tau_i(n)}^{\\tau_f(n)}\n\\Gamma^\\phi_{\\rm tot}~d\\tau\n\\label{cascmaster}\n\\end{equation}\nwhich expresses the fact that the number of reconstructed mesons is \nproportional to the typical life-time of the meson in the system.\n\n\nSome considerations are now in order. During the pre-equilibrium phase, \n$\\tau_i(n)$ coincides with the time at which a $\\phi$ meson is typically \nproduced from a nucleon-nucleon scattering. With exception of some \nvery rare almost instantaneous interaction, $\\tau_f(n)$ typically coincides \nwith the time at which the hydrodynamical phase starts. In other words, \nthe probability of dimuon emission in the pre-hydro stage is evaluated up to \nthe moment $\\phi$ mesons are merged into \nthe hydrodynamical phase.\nFrom then on, $\\phi$ emission will be treated as thermal. \n\nIn the post-equilibrium phase, $\\tau_i(n)$  coincides either with the time at \nwhich the \ntransition  from the hydrodynamical to the transport description is \nperformed (for those $\\phi$ mesons produced \nvia the Cooper-Frye equation) or with the time at which a $\\phi$ meson is \nproduced from hadronic interactions still occurring in the cascade phase, \nas result of the fact that the whole system is not yet completely decoupled; \n$\\tau_f(n)$ is, in this case, the time at which the \n$\\phi$ meson decays or eventually rescatters. \n\n\n\\subsubsection{Reconstruction from equilibrated thermal emission}\nThermal dimuon emission from the $\\phi$ meson can be expected to be \nsignificantly smaller than the post-equilibrium emission, \ndue to the fact that the lifetime of the fireball (7-10 fm) is much smaller \nthan the  $\\phi$ (vacuum) lifetime of $\\sim$44 fm. \nTo determine the thermal dimuon production rate from  $\\phi$-meson decays \nwe observe that the mass of the $\\phi$ meson ($m_\\phi$=1.019 GeV) is larger \nthan the typical \nlocal temperature of the thermalized fireball, thus  \nthe particle number distribution function can be reasonably evaluated in \nBoltzmann approximation.\nMoreover, the $\\phi$ meson being a very narrow resonance \n($\\Gamma^\\phi_{\\rm tot}=0.00426$ GeV and $\\Gamma^\\phi_{\\rm tot}\/m_\\phi\\approx 0.4\\%$ ), for simplicity we \nneglect its small width and approximate the mass distribution of \nthe meson with a $\\delta(m^2-m_0^2)$ function (pole approximation).\n\nIn the Boltzmann and pole approximation the $\\phi$ particle phase space \ndistribution function is given by \n\\begin{equation}\n\\frac{dN_\\phi}{d^3x~d^3q} \\left(q_0;T(\\mathbf{x},t)\\right)=\\frac{3}{(2\\pi)^3}~\ne^{-q_0\/T(\\mathbf{x},t)},\n\\end{equation}\nwhere $T(\\mathbf{x},t)$ is the  local temperature, $(q_0,\\mathbf{q})$ the meson 4-momentum in the thermal frame and the dependence of the \ntemperature from the \n(discretized) space-time point of the (3+1) grid has \nbeen explicitly indicated. \nThe number of dimuons produced per unit phase space volume and unit time from $\\phi\\rightarrow \\mu^+\\mu^-$ decays is\n\\begin{equation}\n\\frac{dN_{\\mu^+\\mu^-}}{d^4x~d^3q} \\left(q_0;T(\\mathbf{x},t)\\right)=\n\\frac{dN_\\phi}{d^3x~d^3q}~ \\frac{\\Gamma^\\phi_{\\mu^+\\mu^-}}{\\gamma_\\phi(q_0)}.\n\\label{dimuonrate}\n\\end{equation}\nHere $\\Gamma^\\phi_{\\mu^+\\mu^-}$ is the dimuon width as defined in the meson \nrest-frame and $\\gamma_\\phi=q_0\/m_\\phi$ is the Lorentz factor \ntransforming from the meson to the thermal rest frame.\nThe number of dimuons emitted per unit space-time volume can be obtained by integrating Eq. (\\ref{dimuonrate}) over momentum. The integration can be performed \nanalytically and one finds: \n\\begin{equation}\n\\frac{dN_{\\mu^+\\mu^-}}{d^4x} \\left(T(\\mathbf{x},t)\\right)= \\frac{3}{2\\pi^2}~m_\\phi^2~T(\\mathbf{x},t)~\nK_1\\left( \\frac{m_\\phi}{T(\\mathbf{x},t)}\\right)\\Gamma^\\phi_{\\mu^+\\mu^-},\n\\label{momintrate}\n\\end{equation} \nwhere $K_1$ is the modified Bessel function of first order. \nThe contribution of a single cell to the dimuon emission in the \ntime step $\\Delta t$ is therefore:\n\\begin{equation}\n\\left( N_{\\mu^+\\mu^-} \\right)_{\\rm cell}=\\frac{dN_{\\mu^+\\mu^-}}{d^4x}\\left(T_{\\rm cell}\\right)  ~V_{\\rm cell} ~\\Delta t.\n\\end{equation}\nThe cell contribution to the dimuon emission \nis calculated according to the above expression. \nTo compare with experimental data, dimuon momenta in the c.m. frame are \nthen generated with a Monte Carlo procedure according to the distribution function \n\\begin{equation}\n\\frac{dN_{\\mu^+\\mu^-}}{d^3x~d^3p}\\sim\\frac{m_\\phi}{p_0}~\nf_B\\left(\\frac{p_\\nu u^\\nu}{T_{\\rm cell}}\\right)\n\\end{equation}\nwith  $u^\\nu$ the fluid cell 4-velocity and $f_B$ the boson distribution function.\n\nThe total dimuon yield from thermal $\\phi$ mesons is  \nobtained summing the above expression over all fluid cells and all time \nsteps of the (3+1) grid that are spanned by the system during the \nhydrodynamical evolution until the transition criterium is reached. \nLet us express this summation symbolically as:\n\\begin{equation}\nN_{\\mu^+\\mu^-}^{\\phi}=\\int_{V\\subset \\mathbb{R}^4} d^4x~~\n\\left\\{ \\frac{3}{2\\pi^2}~m_\\phi^2~T(\\mathbf{x},t)~\nK_1\\left( \\frac{m_\\phi}{T(\\mathbf{x},t)}\\right)\\Gamma^\\phi_{\\mu^+\\mu^-}\n\\right\\}.\n\\end{equation}\nThe corresponding number of $\\phi$ mesons reconstructed in the dimuon channel \nis then: \n\\begin{equation}\nN^{\\rm rec}_{\\phi}=\\int_{V\\subset \\mathbb{R}^4}d^4x~~\n\\left\\{ \\frac{3}{2\\pi^2}~m_\\phi^2~T(\\mathbf{x},t)~\nK_1\\left( \\frac{m_\\phi}{T(\\mathbf{x},t)}\\right)\\Gamma^\\phi_{\\rm tot}\n\\right\\}.\n\\label{thermalmaster}\n\\end{equation}\n\nBefore proceeding, we would like to make two considerations. \nThe first one concerns an implication \nof the EoS used  for the hydrodynamical evolution.\nAs previously stated, a hadronic EoS has been used in the present work. \nThis implies that dimuon emission from the $\\phi$ meson is here \nevaluated during \nthe whole hydrodynamical phase. In general, however, if a phase transition \nfrom quark-gluon to hadronic matter occurs, \nthe hadronic thermal rate would be only a fraction of the total thermal rate.\nIn this sense, the results presented in the next section on thermal dimuon \nemission from the $\\phi$ meson should be regarded as an upper limit \nof the possible amount of dimuons indeed produced by $\\phi$ mesons during the \nhigh temperature\/high density stage of the heavy-ion collision.\n\nThe second consideration regards \nthe validity of the pole approximation used.\nThis approximation is justified as long as the $\\phi$ meson maintains \nits properties of narrow resonance.\nIn medium, the $\\phi$ meson is expected  to broaden, \nas suggested by hadronic many body calculations. \nIf the amount of broadening is significant or\/and the spectral \nfunction of the meson develops a complex ``structured'' shape in the medium, \nthe pole approximation may loose validity. \nOn a quantitative base, there is no general consensus \non the specific amount of  broadening of the  $\\phi$\nmeson spectral function to be expected in a high temperature\/high \ndensity environment.  \nEarly evaluations of medium effects based on \nhadronic rescattering at finite temperature have indicated quite moderate \nchanges of both the $\\phi$ meson  mass and width \\cite{Ko:1993id,Haglin:1994ap,\nHaglin:1994xu,Smith:1997xu}. In a subsequent calculation, \ncollision rates in a meson gas have been estimated to amount to a \nbroadening by $\\sim$  20 MeV at $T$=150 MeV \\cite{AlvarezRuso:2002ib}. \nThe dressing of the kaon cloud is presumably\nthe main effect for $\\phi$ modifications in nuclear\nmatter, increasing its width by $\\sim$ 25 MeV at normal nuclear\ndensity \\cite{Cabrera:2002hc}. In hot and baryon poor hadronic matter the \nin-medium  properties of the $\\phi$ have been schematically explored in \nRef. \\cite{Rapp:2000pe}: the meson was found to retain its resonance structure \nand an in-medium width of $\\sim$ 32 MeV at $(T,\\mu_N)$=(180,27) MeV was estimated.\nIn a recent work \\cite{Vujanovic:2009wr}, the spectral density of the \n$\\phi$ meson in a hot bath of nucleons and pions has been \nmicroscopically calculated \nfrom the forward scattering amplitude in a two component approach. \nThe authors found a considerable broadening of the meson width, e.g., \n$\\Gamma^\\phi_{\\rm med}\\sim$ 100 MeV at saturation density and temperature \n$T=150$ MeV.\n\nIn fact, there is no visible evidence for a strong in-medium scenario for \nthe $\\phi$ meson from the NA60 data. \nFor all the analyzed centrality bins, the  measured  invariant \nmass distribution can be described in terms of a vacuum-like spectral function and the \nextracted values for the mass and the width are compatible with the PDG values and independent of centrality. Of course, one should remind that \nthe extraction of the in-medium modified component of the total dimuon emission \nfrom the experimental data is a non-trivial task, since this component \nwould lie under the large unmodified peak produced from the decays occurring at\nthe freeze-out.\nCertainly, the study of in-medium modifications of the $\\phi$ meson \nproperties is in itself an interesting research topic. However,  \nin the present work will not address this issue and assume that \nalso in-medium the resonance maintains its narrow width, so that \nthe pole approximation is still valid. \nFor the meson pole mass $m_\\phi$, moreover, the vacuum value will be used. \nIn any case, as we will show below, from our analysis it emerges \nthat the amount of $\\phi$'s from thermal emission is by far smaller than the abundance from the cascade \npart, \nso that an eventual in-medium modification of the thermal rate \nwill most likely not alter the results on the total transverse mass spectra \ndiscussed in the next section.\n\n\n\\section{Results \\label{Results}}\nIn this section we investigate the relative abundances of \n$\\phi\\rightarrow \\mu^+\\mu^-$ production in the various stages \nof the system evolution and present results \nfor $\\phi$ meson invariant transverse mass spectra as a function of the collision centrality. Calculations for $\\phi\\rightarrow \\mu^+\\mu^-$ production in \nIn-In collisions at  $E_{\\rm lab}$=158$A$ GeV have been performed for 5 centrality classes.\n In agreement with the treatment of the experimental data, \neach class  was identified by the range of charged particle multiplicity \n$dN_{ch}\/d\\eta$ in the pseudorapidity interval 3$\\leq\\eta\\leq$4.\nThe relation between a specific range of $dN_{ch}\/d\\eta$ and the corresponding centrality bin was specified by the NA60 collaboration and can be found in \nTable 1 of Ref. \\cite{Arnaldi:2009wr}. The correspondence between the \n5 ranges of $dN_{ch}\/d\\eta$ and 5 ranges of impact parameters was obtained \nby the analysis of the charged particles obtained within the \nUrQMD hybrid model as a function of the impact parameter selected in the \nMonte Carlo simulation.\n\n \\subsection{Relative abundances in the various evolution stages and $m_T$ spectra}\n\nFirst, let us investigate the relation between the amount of $\\phi$ mesons \nreconstructed from $\\phi\\rightarrow \\mu^+\\mu^-$ and the stage of the evolution \nprobed by the dimuon emission. We start by discussing separately and in detail results for the most central bin. Later, an analogous analysis is presented as function of centrality.\n\n\nThe contribution to the $\\phi$ \ntransverse mass spectra of \nthose $\\phi$ mesons emitting during the hydrodynamical and the \ncascade stage are separately shown in Fig.\\ref{fig1}. \nThe emission from the hydrodynamic stage is found almost one order of magnitude \nsmaller than the emission from the cascade. As already mentioned, this is due \nto the relative smallness of the duration of the hydrodynamical phase when \ncompared to the cascade phase. Thus, the dominant contribution comes from the \ncascade stage. \nThis stage contains pre- and post-equilibrium emission. However, \nas shown in Fig.\\ref{fig1-b}, the pre-equilibrium \nemission is negligible \n(two orders of magnitude smaller than \nthe post-equilibrium emission) and will not be discussed further. \nThe post-equilibrium emission can be divided in two categories: (i) the \nemission from $\\phi$ particles produced via Cooper-Frye at the transition \npoint and (ii) the emission from $\\phi$ particles produced \n\\emph{during} the cascade (Fig. \\ref{fig1-b}). \nIn the first case, the particles have a momentum distribution \nthat reflects the thermal properties of the transition point, although \ntheir dimuon decay occurs later in a non-thermal environment. \nIn this sense, this copious emission, though not specifically thermal \n(i.e. not described by thermal rate equations) still carries information about the preceding thermal phase. \nIt represents the most direct ``remain'' of the thermally equilibrated \nphase previously experienced by the system.\nThe second contribution, on the contrary, can be labeled as a \n``purely cascade'' one. This is the contribution of $\\phi$ particles produced in the \nnon-equilibrium environment on the way to final decoupling. This second contribution is characterized by steeper transverse mass spectra. the shape of the total spectra is found to be composed by the interplay of both \nemissions.\n\nIt is instructive to compare the hybrid model calculations to the pure \ncascade calculations, in which no assumption of an intermediate equilibrium \nphase is made. The comparison is presented in Fig. \\ref{compare} (top) together with the experimental data. As one can see, \nthe absence of an intermediate thermal phase results in a steepening of the \ntransverse mass spectra not supported by the data. \nMoreover, the pure cascade calculation strongly underestimates the $\\phi$ meson yield, a feature already emerged in recent independent investigations \\cite{Alt:2008iv} performed in relation to measurements obtained by the NA49 experiment. \nThere, it was found that a  statistical hadron gas model with undersaturation of strangeness \\cite{Becattini:2005xt} could account for the measured yields \\cite{Alt:2008iv}.\n\nFrom this first analysis, we can conclude that, \ndespite the smallness of the \nspecifically thermal $\\phi\\rightarrow \\mu^+\\mu^-$ emission, the presence of the thermal phase is essential in order to obtain an appropriate yield \nand slope of the $m_T$ distributions from the cascade emission. In this sense, \nwe speak about the presence of ``thermal relics''. \n\n\\begin{figure}\n\\includegraphics[width=.46\\textwidth]{figures\/totyield_c1.eps}\n\\caption{Transverse mass distributions of the $\\phi$ meson in central \nindium-indium collisions. The $\\phi$ is reconstructed \nin the dimuon channel (see text).\nDouble-dotted-dashed line: contribution to the $\\phi\\rightarrow \\mu^+\\mu^- $ production of the hydrodynamical stage. Dashed line: contribution of the cascade stage. Full line: total   $\\phi\\rightarrow \\mu^+\\mu^- $ production. \\label{fig1}}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=.46\\textwidth]{figures\/cascade_yield_c1.eps}\n\\caption{ Decomposition of the post-equilibrium \n$\\phi\\rightarrow \\mu^+\\mu^- $ production in: (i) \nemission from  $\\phi$ mesons which are merged into the cascade at the \ntransition point via the Cooper-Frye equation and \nemit in the cascade stage of the evolution (dashed line); (ii) emission from \n$\\phi$ mesons which are produced and emit in the cascade stage \n(dotted-dashed line). The pre-equilibrium emission is denoted by the dotted line.\nThe full line represents the total cascade emission. Due to the smallness of the \npre-equilibrium emission the latter practically coincides with the \ntotal post-equilibrium emission. \\label{fig1-b}}\n\\end{figure}\n \\begin{figure}\n \\includegraphics[width=.46\\textwidth]{figures\/comp_urqmd-hyd.eps}%\n \\caption{Transverse mass distributions of the $\\phi$ meson in central (top) and peripheral (bottom) indium-indium collisions. The hybrid model calculation (full line) is compared to the pure UrQMD transport calculation (dashed line) \nand to experimental data \\cite{Arnaldi:2009wr}. Bin-widths which coincide with the ones of the experimental data have been here used. \\label{compare}}\n \\end{figure}\n\nAn analogous  analysis has been performed for the further 4 centrality bins \nand results are presented in Fig.\\ref{fig3} and Fig.\\ref{fig4}. \nFor all centrality classes the thermal rate from the hydrodynamic evolution is found to be much smaller than the \ncascade emission. With decreasing centrality, the pure cascade \nemission (dash-dotted line) becomes less and less important and the spectra \nis more and more determined by the emission from those mesons emerging \nfrom the thermal stage into the cascade at the transition point.\n\n\n \\begin{figure}\n \\includegraphics[width=.46\\textwidth]{figures\/totyield_c2-c5.eps}%\n \\caption{Same as in Fig. \\ref{fig1}, \nbut for different centrality classes. Double-dotted-dashed line: contribution to the $\\phi\\rightarrow \\mu^+\\mu^- $ production of the hydrodynamical stage. Dashed line: contribution of the cascade stage. \\label{fig3}}\n \\end{figure}\n \\begin{figure}\n \\includegraphics[width=.46\\textwidth]{figures\/cascade_yield_c2-c5.eps}%\n \\caption{Same as in Fig. \\ref{fig1-b}, \nbut for different centrality classes. Dashed line: emission from  $\\phi$ mesons which are merged into the cascade at the \ntransition point via the Cooper-Frye equation and \nemit in the cascade stage of the evolution; Dotted-dashed line: emission from \n$\\phi$ mesons which are produced and emit in the cascade stage. \\label{fig4}}\n \\end{figure}\n\n \\begin{figure}\n \\includegraphics[width=.46\\textwidth]{figures\/mtspec.eps}%\n \\caption{Transverse mass distributions of the $\\phi$ meson in indium-indium collisions as a function \nof centrality; from top to bottom: central to peripheral spectra. The hybrid model calculations are compared \nwith NA60 data \\cite{Arnaldi:2009wr}. Bin-widths which coincide with the ones of the experimental data have been here used. \\label{mtspectra}}\n \\end{figure}\n\n\nFinally, the $\\phi$ meson transverse mass spectra calculated within the \nhybrid model for the 5 centrality classes considered are compared to NA60 data in Fig. \\ref{mtspectra}.  The hybrid model can account pretty well for both slope and yield in \nthe  first 4 centrality classes. Small deviations  are observed for the utmost peripheral bin where, most likely, this kind of hybrid models are already at the limit of \napplicability. They rely on the assumption that an equilibrium phase is indeed reached, which for very peripheral reactions is at least questionable \\cite{Petersen:2009zi}.\n\nPure transport calculations fail however too in describing this very peripheral \nreaction (see Fig. \\ref{compare}, bottom), though the comparison with experimental data suggests that \ndeviations of the resulting slope from the measured one are much smaller than \nthe in the case of central collisions. \n\n\n \n \n\\section{\\label{Conclusions}Summary and conclusions}\nIn this work we employed an integrated Boltzmann+hydrodynamics \nhybrid approach based on the Ultrarelativistic \nQuantum Molecular Dynamics  \ntransport model with an intermediate hydrodynamic stage \nto analyze $\\phi$-meson production in In+In collisions at  $E_{\\rm lab}$=158$A$ GeV from its \nreconstruction in the $\\phi\\rightarrow\\mu^+\\mu^-$ channel.\nWe find that the hybrid model fairly describes $\\phi$ yields and \ntransversal mass spectra at various \ncollision centralities. \nIn particular, the analysis points out that \nthe underlying assumption of the existence of an \nintermediate equilibrated phase seems to play an essential role \nin order to  catch the main aspects of the physics emerging from \nthe $\\phi$ dimuon emission at top SPS energies. \n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nThe authors acknowledge G. Torrieri for fruitful discussions and the \nNA60 Collaboration, M. Floris in particular, for providing the experimental data.\nThis work was supported by BMBF, GSI, DFG and the Hessen Initiative \nfor Excellence (LOEWE) through the Helmholtz International Center for FAIR (HIC for FAIR). We thank the Center for Scientific Computing for providing computational resources.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGalaxy clusters are  the largest gravitationally bound\nstructures in the Universe, and hence are  a powerful probe of the evolution and structure formation in  the Universe at redshifts $z < 2$ (see the \\cite{Mantz:2014xba,Vikhlininrev} for  detailed reviews).  They are also wonderful laboratories for fundamental Physics measurements~\\cite{Carbone,Desai18}\nOne can obtain constraints on the cosmological parameters,  if we assume that the X-ray gas mass\nfraction, $f_{gas}$, of hot, massive and relaxed galaxy clusters does not evolve with redshift \\cite{1996PASJ...48L.119S,Allen2007,Ettori2009,Mantz:2014xba,pen97,white93}.  In order to quantify the baryon content  and its possible time evolution, crucial to cosmological tests with  gas mass fraction measurements, hydrodynamic simulations have been used to calibrate the gas depletion factor, $g(z)$, viz. the ratio by which the gas mass fraction of galaxy clusters is depleted with respect to the  universal  mean of baryon fraction. \\citet{2013MNRAS.431.1487P} and \\citet{2013ApJ...777..123B},  have shown using simulations  of hot, massive and dynamically relaxed galaxy clusters ($M_{500}>10^{14} M_{\\odot}$) involving   different physical processes, that by modelling  a possible time evolution for $g(z)$, given by $g(z) \\equiv g_0(1+g_1 z)$ \\footnote{Note that elsewhere in literature (eg~\\cite{Bora2021EPJC}),the parametric form for the  gas depletion factor is usually denoted by  $ \\gamma(z) = \\gamma_0(1+\\gamma_1 z)$},    one  obtains: $0.55 \\leq g_0 \\leq 0.79$ and $-0.04 \\leq g_1 \\leq 0.07$, depending on the physical processes that are included in the simulations (see Table 3 in \\citep{2013MNRAS.431.1487P} for values found at $r_{2500}$). Therefore, no significant trend of $g(z)$ as a function of redshift was found in their simulations. For their analysis, ~\\citeauthor{2013MNRAS.431.1487P} considered $f_{gas}$ as a cumulative quantity into $r_{2500}$. Recently, by using the cosmo-OverWhelmingly Large Simulation,   \\citeauthor{Brun2017MNRAS.466.4442L} showed that in most realistic models of intra-cluster medium, which include active galactic nuclei feedback, the slopes of the various mass-observable relations deviate substantially from the self-similar ones, particularly at late times and for low-mass clusters. \n\n \n \nOn the other hand, there have been a number of works, which   have estimated the depletion factor using only cosmological observations. For instance,  \\citeauthor{Holanda2017JCAP}  carried out an analysis using 40 $f_{gas}$ measurements  observed by the Chandra X-ray telescope~\\cite{Mantz:2014xba}  along with Type Ia supernovae observations,  with  priors  on $\\Omega_b$ and $\\Omega_M$ from from the Planck  results \\cite{Planck2015}, and assuming the validity of the cosmic distance duality relation (CDDR) \\cite{2007GReGr..39.1047E}. No evolution of  the gas depletion factor with redshift was found from this aforementioned analysis.\nIn a follow-up work, ~\\citeauthor{Holanda2018} used 38 Chandra X-ray  $f_{gas}$ measurements   in the redshift range $0.14 \\leqslant z \\leqslant 0.89$~\\cite{Laroque2006ApJ}, along with angular diameter distances from X-ray\/SZ measurements and priors from Planck results \\cite{Planck2015}. Unlike~\\cite{Holanda2017JCAP}, they did not use CDDR to derive the angular diameter distance. The aforementioned work reported $g_0$ =  $0.76\\pm0.14$ and $g_1$ = $-0.42_{-0.40}^{+0.42}$, which also implies no redshift evolution for the  gas depletion factor. In these analyses the gas mass fraction was calculated at $r_{2500}$.\n\nConstraints on the evolution of $g(z)$ using gas mass fraction measurements at $r_{500}$ have also been obtained.\n\\citeauthor{Zheng2019EPJC} used 182 galaxy clusters detected by the Atacama Cosmology Telescope and  cosmic chronometers, and found a mild decrease of $g(z)$ as a function of redshift.\nMost recently, \\citet{Bora2021EPJC} also looked for a time evolution of $g(z)$ using gas mass fraction measurements at $r_{500}$,  from  both the SPT-SZ \\cite{2018MNRAS.478.3072C} and Planck Early Sunyaev-Zeldovich effect cluster data \\cite{Planck2011A&A...536A...8P} along with cosmic chronometers and priors from Planck results \\cite{Planck2020A&A}. Conflicting  results between both the samples were found: for SPT-SZ, $g(z)$ was found to be decreasing as a function of redshift (at more than 5$\\sigma$), whereas a positive trend with redshift was found for Planck ESZ data (at more than 4$\\sigma$). Their results implied that  one cannot use  $f_{gas}$ values  at $r_{500}$ as a stand-alone probe  for any model-independent cosmological tests. \n\nIn this letter, we propose a new observational test for   the  time evolution of the gas depletion factor by using  multiple large scale structure probes, namely, galaxy cluster gas mass fraction measurements~\\cite{Mantz:2014xba} along with  strong gravitational lenses (SGL) observations obtained from SLOAN Lens ACS + BOSS Emission-line Lens Survey (BELLS) + Strong Legacy Survey SL2S + SLACS \\cite{Leaf2018}. For our analysis, we assume the validity of the CDDR \\cite{2007GReGr..39.1047E}. By considering a possible time evolution for $g(z)$, such as $g(z) = g_0(1+g_1 z)$, and analyses by using three SGL sub-samples  separately (divided by their mass interval), we obtain an error-weighted average given by:  $g_1 = -0.15 \\pm 0.055$ at $1\\sigma$ c.l.\nTherefore, we infer  a mild time evolution at about $2.7\\sigma$ from our analysis.\nWe note that unlike previous works in this area, our results are independent  of the  Planck  cosmological parameters.\n\n\nThis letter is organized as follows. Section~\\ref{methodology} explains the methodology adopted in this work. In section~\\ref{data}, we present the data sample used for our analysis. Section~\\ref{sec:analysis} describes our analysis and results. Our conclusions are presented in Section~\\ref{sec:conclusions}. \n\n\\section{Methodology}\n\\label{methodology}\n\nIn this section, we discuss some aspects of SGL systems and gas mass fractions, and show how one can combine these observations in order to put constraints on the gas depletion factor.\n\n\n\n\\subsection{Strong Gravitational Lensing Systems}\n\n Strong gravitational lensing has proved to be a poweful diagnostic  of modified gravity theories  and cosmological models, as well as  fundamental physics~\\cite{1992grle.book.....S,2010CQGra..27w3001B}. Usually,  a strong lens system  consists of  a foreground galaxy or  a cluster of galaxies positioned between a source (quasar) and an observer, where the multiple-image separation from the source depends only on the lens and the source angular diameter distance (see, for instance,  Refs.~\\cite{Cao2015ApJ,Holanda:2016msr,Cao2018,Amante:2019xao,Lizardo:2020wxw,boraDMevolution}, where SGL systems were used recently as a cosmological tool). By using  the simplest model assumption (the  singular isothermal sphere) to describe the SGL systems,  the Einstein radius ($\\theta_E$), a fundamental quantity,  can be defined as \\cite{2010CQGra..27w3001B,Cao2015ApJ}:\n \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=10cm, height=8cm]{low.png} \n    \\caption{\\textbf{For Low mass range SGL sample:} The 1D marginalized likelihood distributions along with 2D marginalized constraints showing the 68\\%, 95\\%, and 99\\% credible regions for the parameters $\\gamma$ (from the power law model describing the SGL systems) and $g_1$ (from the gas depletion factor $g(z)$), obtained using the {\\tt Corner} python module~\\cite{corner}.}\n   \\label{fig:low}\n    \n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=10cm, height=8cm]{inter.png} \n    \\caption{\\textbf{For Intermediate mass range SGL sample:} The 1D marginalized likelihood distributions along with 2D marginalized constraints showing the 68\\%, 95\\%, and 99\\% credible regions for the parameters $\\gamma$ (from the power law model describing the SGL systems)  and $g_1$ (from the gas depletion factor $g(z)$), obtained using the {\\tt Corner} python module~\\cite{corner}.}\n    \\label{fig:intermediate}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=10cm, height=8cm]{high.png} \n    \\caption{\\textbf{For High mass range SGL sample:} The 1D marginalized likelihood distributions along with 2D marginalized constraints showing the 68\\%, 95\\%, and 99\\% credible regions for the parameters $\\gamma$  (from the power law model describing  the  SGL  systems) and $g_1$ (from the gas depletion factor $g(z)$), obtained using the {\\tt Corner} python module~\\cite{corner}.}\n    \\label{fig:high}\n\\end{figure}\n\n\n\\begin{equation}\n\\theta_E = 4\\pi \\frac{D_{A_{ls}}}{D_{A_{s}}} \\frac{\\sigma_{SIS}^{2}}{c^2}\n\\label{eq:thetaE_SIS}\n\\end{equation}\nIn this equation,  $D_{A_{ls}}$ is the angular diameter distance from the lens to the source; $D_{A_{s}}$ denotes the angular diameter distance from the observer to the source; $\\sigma_{SIS}$ is the velocity dispersion caused by the lens mass distribution; and $c$ is the speed of light.\n\nFor  our analyses, a  flat universe is considered and the following  observational quantity derived  for SGL systems is used~\\cite{Liao19}: \n\\begin{equation}\n\\label{eq:D_SIS}\nD={\\frac{D_{A_{ls}}} {D_{A_s}}}=\\frac{{\\theta}_E c^2}{4{\\pi} \\sigma^2_{SIS}}\n\\end{equation}\nFor a flat universe, the comoving distance $r_{ls}$ is given  by\n$r_{ls}=r_s-r_l$~\\cite{2010CQGra..27w3001B}, and using $r_s=(1+z_s)D_{A_s}$, $r_l=(1+z_l)D_{A_l}$  and $r_{ls}=(1+z_s)D_{A_{ls}}$, we obtain the following:\n\\begin{equation}\n\\label{eq:D}\nD= 1 - \\frac{(1+z_l)D_{A_{l}}}{(1+z_s)D_{A_{s}}}\n\\end{equation}\nFinally, by the validity of the CDDR relation\n $D_L=(1+z)^2D_A$~\\cite{2007GReGr..39.1047E}, Eq.~\\ref{eq:D} can be re-cast as follows:\n\\begin{equation}\n\\label{eq:final_D}\n\\frac{(1+z_s)}{(1+z_l)}= (1-D)\\frac{D_{L_s}}{D_{L_l}}\n\\end{equation}\n\n\n\\subsection{Gas mass fraction}\n\n\nThe cosmic gas mass fraction can be  defined as $f_{gas} \\equiv \\Omega_b\/\\Omega_M$ (where $\\Omega_b$ and $\\Omega_M$ are the baryonic and total matter density parameters, respectively), and the constancy  of this quantity within massive, relaxed clusters at $r_{2500}$ has been used to constrain cosmological parameters by using the following equation (see, for instance, \\cite{Mantz:2014xba}) \n\\begin{equation}\n\\label{fgas}\nf_{gas}(z) = A(z) g(z) K \\left[\\frac{\\Omega_b}{\\Omega_{M}}\\right] \\left(\\frac{D_L^*}{D_L}\\right)^{3\/2}\\\n\\end{equation}\nHere,  the observations are done in the X-ray band, the asterisk denotes the corresponding quantities for the fiducial model used in the observations to obtain  $f_{gas}$ (usually a flat $\\Lambda$CDM model with Hubble constant $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ and the present-day total matter density parameter $\\Omega_M=0.3$), $A(z)$ represents the angular correction factor, which is very close to unity in almost all cases, and hence can be neglected. The $K$ factor  is the instrument calibration constant, which also accounts for any bias in the  mass due to non-thermal pressure and bulk motions in the  baryonic gas~\\cite{Allen2007,Mantz:2014xba,2013MNRAS.431.1487P,2013ApJ...777..123B}.  Twelve galaxy clusters used in this present work are also part of the ``Weighing the Giants'' sample \\cite{2016MNRAS.457.1522A}, and this work found the $K$ factor to be constant, i.e.,  no significant trends with\nmass, redshift, or any  morphological indicators were found. Therefore, we also posit the same in  this work. The quantity $g(z)$, which is of interest here is again modelled in the same way as previous works, i.e. $g(z)=g_0(1+g_1z)$. The ratio in the parenthesis of Eq.~\\ref{fgas} encapsulates the expected variation in $f_{gas}$ when the underlying cosmology is varied, which makes the analyses with gas mass fraction measurements model-independent. Finally, it is important to stress that the Eq.~\\ref{fgas} is obtained only after assuming the  validity of CDDR (see Ref.\\cite{Gon2012} for details). \nIn the last decade, a plethora of analyses using myriad observational data have been undertaken in order to establish whether or\nnot the CDDR holds in practice.   A succinct summary of the latest observational constraints on the CDDR\ncan be found in Ref.\\cite{2021arXiv210401614H,BoraCDDR}, which demonstrates the validity of CDDR  to within 2$\\sigma$.\n\nThe key equation to our method can be  obtained from combining Eq.~\\ref{eq:final_D} and \\ref{fgas}, and  by incorporating a possible departure from the gas depletion factor, such as $g(z)=g_0(1+g_1z)$. In this way, we now obtain:\n\n\\begin{equation}\n\\label{main_equation}\n\\left[\\frac{(1+g_1 z_s)}{(1+g_1 z_l)}\\right] = \\left[\\frac{f_{gas}(z_s)}{f_{gas}(z_l)}\\right] \\left[\\frac{(1+z_s)D^*_{L_l}}{(1+z_l)D^*_{L_s}}\\right]^{3\/2}(1-D)^{-3\/2}\n\\end{equation}\n As we can see, unlike Ref.\\cite{Holanda:2017cmc,Bora2021EPJC}, our results are independent of the $K$ factor, $g_0$, and   $\\Omega_b\/\\Omega_M$.    \n \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=9.8cm, height=6.8cm]{evolution.png} \n    \\caption{The evolution of the normalized gas depletion factor along with $1\\sigma$ error band as a function of redshift using 40 X-ray $f_{gas}$ measurements near $r_{2500}$ from ~\\cite{Mantz:2014xba}.  This figure shows a mild evolution of $g(z)$ with respect to  redshift.}\n    \\label{fig:evolution}\n\\end{figure}\n\n\\section{Cosmological data}\n\\label{data}\nThe data used in our analysis were as follows:\n\\begin{itemize}\n\\item 40 X-ray gas mass fraction measurements of hot ($kT \\geq  5$ keV), massive and morphologically relaxed  galaxy clusters from  Chandra archive. The galaxy cluster redshift range is $0.078 \\leq z \\leq 1.063$~\\cite{Mantz:2014xba}.  The bias in the mass measurements from X-ray data arising by assuming hydrostatic equilibrium was calibrated using robust mass estimates for the target clusters from weak gravitational lensing \\cite{2016MNRAS.457.1522A}, reducing systematic uncertainties. Furthermore, these authors measured the gas mass fraction in spherical shells at radii near $r_{2500}$, instead of using  the cumulative fraction integrated over all radii ($< r_{2500}$) as in previous works.\n\\item We also consider sub-samples  from a specific catalog containing 158 confirmed sources of strong gravitational lensing systems \\cite{Leaf2018}. This  compilation includes 118 SGL systems identical to the compilation of \\cite{Cao2015ApJ}, which were obtained from a combination of SLOAN Lens ACS, BOSS Emission-line Lens Survey (BELLS), and Strong Legacy Survey SL2S, along with 40 additional systems recently discovered by SLACS and pre-selected by \\cite{2017ApJ...851...48S} (see Table I in \\cite{Leaf2018}).  Several studies have shown that the slopes of  the density profiles for individual galaxies show a non-negligible deviation from the Singular isothermal sphere profile~\\cite{Koopmans2009,Auger2010,Barnab2011,Sonnenfeld2013,Cao2016,HolandaSaulo,Chen2019}. Therefore, for the mass distribution of lensing systems, the  power-law model is assumed. This model considers  a spherically symmetric mass distribution with a more general power-law index $\\gamma $, namely $\\rho \\propto r^{-\\gamma}$. In this approach $\\theta_E$ is given by:\n\\begin{equation}\n\\theta_E =   4 \\pi\n\\frac{\\sigma_{ap}^2}{c^2} \\frac{D_{ls}}{D_s} \\left[\n\\frac{\\theta_E}{\\theta_{ap}} \\right]^{2-\\gamma} f(\\gamma),\n\\label{thetaE}\n\\end{equation}\nwhere $\\sigma_{ap}$ is the  stellar velocity dispersion inside an aperture of size $\\theta_{ap}$ and\n\\begin{eqnarray} \\label{f factor}\nf(\\gamma) &=& - \\frac{1}{\\sqrt{\\pi}} \\frac{(5-2 \\gamma)(1-\\gamma)}{(3-\\gamma)} \\frac{\\Gamma(\\gamma - 1)}{\\Gamma(\\gamma - 3\/2)}\\nonumber\\\\\n          &\\times & \\left[ \\frac{\\Gamma(\\gamma\/2 - 1\/2)}{\\Gamma(\\gamma \/ 2)} \\right]^2\n\\end{eqnarray}\nThus, we obtain: \n\\begin{equation} \n\\label{NewObservable}\n D=\\frac{D_{A_{ls}}}{D_{A_{s}}} = \\frac{c^2 \\theta_E }{4 \\pi \\sigma_{ap}^2} \\left[ \\frac{\\theta_{ap}}{\\theta_E} \\right]^{2-\\gamma} f^{-1}(\\gamma)\n\\end{equation}\nFor $\\gamma = 2$,  the singular isothermal spherical distribution is recovered. All the  terms necessary  to evaluate $D$ can be found in Table 1 of \\cite{Leaf2018}. However, the complete dataset (158 points) is culled to  98 points, after the following cuts: $z < 1.061$ and  $D \\pm \\sigma_D <1 $ ($D > 1$ represents a non physical region). The compilation considered here contains only those SGL systems with early type galaxies acting as lenses. Each data point contains estimated apparent and Einstein radius, with spectroscopically measured stellar apparent velocity dispersion, as well as both the lens and the source redshifts.\n\n\nWe should point out that  some works have recently explored a possible time evolution of the mass density power-law index \\cite{Cao2016,HolandaSaulo,Amante:2019xao,Chen2019}. No significant evolution has been found in these works. On the other hand, these results indicate that it is prudent to  use low, intermediate, and high-mass galaxies separately in any cosmological analyses. As commented in \\cite{Cao2016}, elliptical galaxies with velocity dispersions smaller than $200$ km\/s may be classified roughly as relatively low-mass galaxies, while those with velocity dispersion larger than $300$ km\/s may be treated as relatively high-mass galaxies. Naturally, elliptical galaxies with velocity dispersion between $200-300$ km\/s may be classified as intermediate-mass galaxies. Therefore, in our analyses, we work with three distinct sub-samples consisting of 26, 63, and 9 data points with low, intermediate, and high velocity dispersions, respectively.\n\n\nIn order to constrain $g_1$ by using  Eq.~\\ref{main_equation},   gas mass fraction measurements  at the  lens and source redshifts (for each SGL system) are required. These quantities are obtained by applying Gaussian Process Regression~\\cite{seikel12} using the 40 gas mass fraction measurements compiled by  Ref.\\cite{Mantz:2014xba} in order to estimate the gas mass fraction at any arbitrary redshift.\n\\end{itemize}\n\\vspace{0.2cm}\n\n\n\\begin{table*}[]\n\\caption{\\label{tab:table1}. Constraints on the parameters $\\gamma$ and $g_1$ for different $\\sigma_{ap}$ range used in this analysis as discussed in Sect.~\\ref{sec:analysis}.}\n    \\centering\n    \\begin{tabular}{|l|c|c|c|r|} \\hline\n    \\textbf{Sample} & \\boldmath$\\sigma_{ap}$\\textbf{(km\/sec)} & \\boldmath$\\gamma$ & \\boldmath$g_1$\\\\ \\hline \n      Low\\ & $\\sigma_{ap}< 200$ & $1.913^{+0.009}_{-0.010}$  &  $0.172^{+0.173}_{-0.167}$ \\\\\n      Intermediate & $200 < \\sigma_{ap}< 300$ & $2.041\\pm0.012$ & $-0.188^{+0.060}_{-0.059}$ \\\\\n      High &  $\\sigma_{ap}> 300$  & $1.983^{+0.082}_{-0.081}$ & $-0.137^{+0.319}_{-0.254}$\\\\\n     \n      \n      \\hline \n      \n    \\end{tabular}\n\n\\end{table*}\n\\section{Analysis and Results} \n\\label{sec:analysis}\n\nThe joint constraints on the power law index, $\\gamma$ and redshift dependent part of the gas depletion factor, $g_1$  can be obtained by maximizing the likelihood distribution function, ${\\cal{L}}$  given by\n\n\\begin{widetext}\n\\begin{equation}\n    \\label{eq:logL}\n   -2\\ln\\mathcal{L} = \\sum_{i=1}^{n} \\ln 2\\pi{\\sigma_{i}^2}+ \\sum_{i=1}^{n}\\frac{\\left(\\Phi(g_1,z_i)- \\left[\\frac{f_{gas}(z_s)}{f_{gas}(z_l)}\\right] \\left[\\frac{(1+z_s)D^*_{L_l}}{(1+z_l)D^*_{L_s}}\\right]^{3\/2}(1-D)^{-3\/2}\\right)^2}{\\sigma_{i}^2} , \n\\end{equation}          \\end{widetext} \n\n\nwhere \n\\begin{equation}\n  \\Phi(g_1,z_i) =  \\left[\\frac{(1+g_1 z_s)}{(1+g_1 z_l)}\\right]\n\\end{equation}\n\n\n\nHere, $\\sigma_i$  denotes the statistical errors associated with the gravitational lensing observations and gas mass fraction measurements, and are obtained by using standard propagation errors techniques. We maximize our likelihood function using the  $\\tt{emcee}$ MCMC sampler~\\cite{emcee} in order to estimate the free parameters used in Eq.~\\ref{eq:logL}, viz. $\\gamma$ and $g_1$. \n\nThe one-dimensional marginalized posteriors for each parameter along with the 68\\%, 95\\%, and 99\\% 2-D marginalized credible intervals, are shown in Fig.~\\ref{fig:low}, Fig.~\\ref{fig:intermediate}, and Fig.~\\ref{fig:high} for the Low, Intermediate, and High samples, respectively. As we can see, the analyses with the low and high mass SGL sub-samples are in full agreement with no evolution for the depletion factor, while  the intermediate sub-sample shows a non-negligible evolution of $g(z)$ (see table~\\ref{tab:table1}). \nWe calculate an error-weighted average using the three estimates  and obtained: $g_1 = -0.15 \\pm 0.055$ at  1$\\sigma$, showing a mild evolution for  $g(z)$ at 2.7$\\sigma$. (see Fig.~\\ref{fig:evolution}). \n\n We should point out that although a non-constant gas depletion factor has previously been obtained using measurements at $r_{500}$~\\cite{Zheng2019EPJC,Bora2021EPJC}, this is the first result  which  finds  a time-varying depletion factor, using gas mass fraction measurements at $r_{2500}$. Previous works using cosmological simulations (see  \\cite{2013MNRAS.431.1487P} and \\cite{2013ApJ...777..123B})  did not obtain a significant trend of  $g(z)$ as a function of redshift  in their cosmological  hydrodynamic simulations for gas mass fraction measurements at $r_{2500}$. However, it is important to comment that models describing the physics of the intra-cluster medium used in hydrodynamic simulations may not span the entire range of physical process allowed by our current understanding.\n\nFinally, as we can see, the high SGL sub-sample is in full agreement with the SIS model ($\\gamma=2$) (see table~\\ref{tab:table1}) while the low and intermediate mass SGL sub-samples are not compatible with this model  even at 3$\\sigma$ c.l.. Therefore, our results also reinforce the need for  segregating the  lenses into low, intermediate, and high velocity dispersions, and analyzing them separately.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\n \nIn this letter, we have explored a possible  time evolution of gas depletion factor, using a sample of 40 galaxy clusters with their X-ray gas mass fraction obtained in spherical shells at radii near $r_{2500}$. The analyses were performed by using the  gas mass fraction measurements in conjunction with  sub-samples of strong gravitational lens systems and the cosmic distance duality relation validity. The depletion factor was modelled assuming $g(z)=g_0 (1 + g_1 z)$, and we found a mild evolution at 2.7$\\sigma$, i.e. $g_1 = -0.15 \\pm 0.055$  (see Fig.~\\ref{fig:evolution}). This estimate is obtained by  calculating an error-weighted average by combining  the different   values in Table ~\\ref{tab:table1}, which contain the results for  each of the sub-samples.  \n\nOur method did not use the Planck cosmological results on $\\Omega_b\/\\Omega_M$,  as in previous works in this area.  This is the first work in literature which finds a non-negligible evolution of the gas depletion factor using gas mass fraction measurements at $r_{2500}$. This result is in tension with results from hydrodynamical simulations.\nMoreover,  our results also reinforce the need for  segregating the  lenses into low, intermediate and high velocity dispersions, and analyzing them separately.  The  mass-sheet degeneracy in the gravitational lens system (see \\cite{2016JCAP...08..020B} and references therein) and its effect on our results also could be explored as an interesting extension of this work. \n\nIt is important to comment that the method discussed here can be used in a near future with data set   from the X-ray survey eROSITA \\cite{eRosita}, that is expected to detect $\\approx$ 100,000 galaxy clusters, along with followup optical and infrared data from EUCLID mission, Vera Rubin LSST, and Nancy Grace Rowan space telescope, that will discover thousands of strong lensing systems.\n\n\n\n\n\n\n\\section*{ACKNOWLEDGEMENT}\nKB would like to express his gratitude towards the Department of Science and Technology, Government of India for providing the financial support under DST-INSPIRE Fellowship program. RFLH\nthanks CNPq No.428755\/2018-6 and 305930\/2017-6.\n\n\n\n    \n\n\n \n  \n    \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\n\\par Structured prediction is a field of machine learning where\noutputs are expected to obey some predefined discrete\nstructure. Instances of structured prediction with various output\nstructures occur in many applications, including computer vision\n(e.g., scene graph generation \\citep{johnson2015image} with\ngraph-structured output), natural language processing (e.g.,\nlinguistic parsing \\citep{niculae2018sparsemap} with tree-structured\noutput, relation extraction \\citep{roth2004linear} with\ntuple-structured output) or modeling the spatial structure of physical\nentities and processes \\citep{9006803}.\n\nA key difficulty faced by all models is to tractably model\ninterdependencies between different parts of the output.  As output\nspaces tend to be combinatorially large, special techniques,\napproximations, and independence assumptions must be used to work with\nthe associated probability distributions. Considerable research has\ninvestigated specific structures for which such approaches make\nmachine learning tractable.  For instance, when outputs are trees over\na fixed set of nodes, maximal arborescence algorithms allow for exact\ninference \\citep{chu1965shortest, branchings}; when outputs are\ngraph-structured, loopy belief propagation can be used for approximate\ninference \\citep{loopy}.\n\nIn the specific case where outputs are structured as a linear\nsequence, a commonly made independence assumption is the Markov\nassumption. This requires that model outputs depend only on their\nimmediate neighbors, and do not depend directly on more distant ones.\nA common model that makes use of this assumption is the linear-chain\nconditional random field (CRF) \\citep{lafferty2001conditional}, which\nhas found ubiquitous use across common sequence labeling tasks,\nincluding part-of-speech tagging \\citep{pos-twitter} and named entity\nrecognition \\citep{lample-etal-2016-neural}. This model became popular\nwith the use of hand-crafted feature vectors in the 2000s, and is\nnowadays commonly used as an output layer in neural networks to\nencourage the learning of structural properties of the output\nsequence.\n\nThe Markov assumption makes training tractable, but also limits the\nCRF's expressive power, which can hamper performance, especially for\nlong sequences \\citep{scheible2016model}.  Semi-Markov CRFs\n\\citep{sarawagi2004semi} and skip-chain CRFs\n\\citep{sutton2004collective} are techniques for relaxing the Markov\nassumption, but both come with drawbacks in performance and\nexpressiveness.\n \nIn this work, we propose a new method to tractably relax the Markov\nassumption in CRFs.\nSpecifically, we show how to constrain the output of CRFs to \\textit{a\n  regular language}, such that the resulting\n\\textit{regular-constrained CRF (RegCCRF)} is guaranteed to output\nlabel sequences from that language. Since regular languages can encode\nlong-distance dependencies between the symbols in their strings,\nRegCCRFs provide a simple model for structured prediction that\nprovides a guarantee to respect these non-local constraints. The\ndomain knowledge specifying these constraints is defined via regular\nexpressions, a straightforward, well understood formalism.\nWe show that our method is distinct from the common family of\napproaches that enforce constraints at decoding time, and that our\nconstrained training approach better approximates the true data\ndistribution. We also test our method empirically as the output layer\nof a neural network, and attain state-of-the-art performance for\nsemantic role labeling on the OntoNotes corpus \\citep{ontonotes,\npradhan-etal-2012-conll}. To encourage the use of RegCCRFs, we release our code\nas a Python library under the Apache 2.0 license,\nto be used as a drop-in replacement for traditional CRFs in PyTorch\nprojects.\n\\footnote{Available at \\url{https:\/\/www.ims.uni-stuttgart.de\/data\/regccrf}}\n\n\\section{Related work}\n\\label{sec:related-work}\n\nWe identify three areas of structured prediction that are particularly\nrelevant to our current work: constrained decoding, which can enforce\noutput constraints at decoding time, techniques for weakening the Markov\nassumption of CRFs to learn long-distance dependencies,\nand structure learning for learning constraints from data in graphical models.\n  \n\\paragraph{Constrained decoding.}\nA common approach to enforcing constraints in model output is\n\\textit{constrained decoding}: Models are trained agnostic to\nconstraints, but during decoding it is ensured that the model output\nsatisfies the constraints.  This almost always corresponds to finding\nor approximating a conditionalized version of the model's\ndistribution, conditionalized on the output obeying the specified\nconstraints.  We discuss constrained coding as it relates to RegCCRFs\nin Section~\\ref{sec:constrained_decoding}.\n\nFor CRFs, \\citet{kristjansson2004interactive} present \\textit{constrained\nconditional random fields}, which can enforce that\nparticular tokens either are or are not assigned particular labels\n(positive and negative constraints, respectively). They propose a\nconstrained Viterbi algorithm for MAP inference and a constrained\nforward-backward algorithm for marginal inference. In their\ntask, interactive information extraction, constraints are not\navailable at training time, and so they focus on constrained\ndecoding.  Formally, our work is a strict generalization of\nthis approach, as position-wise constraints can be formulated as a\nregular language, but regular languages go beyond position-wise\nconstraints.\n  \nOther existing works treat decoding with constraints as a search\nproblem, searching for the most probable decoding path which satisfies\nall constraints.  In one example of this approach, \\cite{he2017deep}\ntrain a neural network to predict token-wise output probabilities for\nsemantic role labeling following the BIO label-alphabet \\citep{ramshaw1999text},\nand then use\nexact A* search to ensure that the output forms a valid BIO sequence\nand that particular task-specific constraints are satisfied. For\nautoregressive models, constrained beam search\n\\citep{hokamp2017lexically, anderson-etal-2017-guided,\n  hasler-etal-2018-neural} can enforce regular-language constraints\nduring search.\n  \n\\paragraph{Markov relaxations.}\nWhile our approach can relax the Markov assumption of CRFs \nthrough nonlocal hard constraints,\nanother strand of work has developed models which can directly\n\\textit{learn} nonlocal dependencies in CRFs:\n\nSemi-Markov CRFs \\citep{sarawagi2004semi} relax the\nMarkov property to the semi-Markov property.\nIn this setting, CRFs are tasked with\nsegmentation, where individual segments may depend only on their\nimmediate neighbors, but model behavior within a particular segment\nneed not be Markovian. As such, semi-Markov CRFs are capable of\ncapturing nonlocal dependencies between output variables, but only to\na range of one segment and inside of a segment.\n   \nSkip-chain CRFs \\citep{sutton2004collective} change the\nexpressiveness of CRFs by relaxing the assumption that only the linear\nstructure of the input matters: they add explicit dependencies between\ndistant nodes in an otherwise linear-chain CRF. These dependencies are\npicked based on particular properties, e.g., input variables of the\nsame value or which share other properties. In doing so, they add\nloops to the model's factor graph, which makes exact training and\ninference intractable, and leads to the use of approximation\ntechniques such as loopy belief propagation and Gibbs sampling.\n\n\\paragraph{Structure learning for graphical models.}\nWhile our work focuses on incorporating known constraints into machine\nlearning methods, a related task is learning in the face of unknown\nconstraints, which must be learned from the data.  In the field of\nprobabilistic graphical models, this is most naturally framed as\nstructure learning. As the graphical model's graph structure encodes\nconditional independence information between variables, learning that\ngraph structure can be seen as learning potential constraints.  Graph\nstructures can be learned globally, as in \\citet{NIPS2006_a4380923}\nand \\citet{Schmidt2008StructureLI}, or independence relationships can\nbe estimated locally, as in \\citet{independence}.\n\n\n\\section{Preliminaries and notation}\nAs our construction involves finite-state automata and conditional random\nfields, we define these here and specify the notation we will use in\nthe remainder of this work.\n\n\\subsection{Finite-state automata}\n\\label{sec:NFA}\nAll automata are taken to be nondeterministic finite-state automata (NFAs) without epsilon transitions.\nLet such an NFA be defined as a 4-tuple $(\\Sigma, Q, F, E)$,\nwhere\n\\begin{itemize}\n\\item $\\Sigma = \\{a_1, a_2, ..., a_{|\\Sigma|}\\}$ is a finite alphabet\n  of symbols,\n\\item $Q = \\{q_1, q_2, ..., q_{|Q|}\\}$ is a finite set of states. By convention, $q_1$ is the sole starting state.\n  \\item $F\\subseteq Q$ is a set of accepting states.\n  \\item $E \\subseteq Q \\times \\Sigma \\times Q$ is a set of directed,\n    symbol-labeled edges between states.  These edges define the NFA's\n    transition function $\\Delta: Q \\times \\Sigma \\rightarrow 2^Q$,\n    with $r \\in \\Delta(q, a) \\leftrightarrow (q, a, r) \\in E$.\n\\end{itemize}\nAn automaton is said to accept a string $\\bm{s} \\in \\Sigma^*$ if there exists a contiguous path of edges from $q_1$ to some\naccepting state whose edge labels are exactly the symbols of $\\bm{s}$.\nThe \\textit{language} defined by an automaton is the set of all such accepted strings.\nA language is \\textit{regular} if and only if it is the language of some NFA.\n\n\\subsection{Linear-chain conditional random fields}\n\\label{sec:CRF}\nLinear-chain conditional random fields (CRFs)\n\\cite{lafferty2001conditional} are a sequence labeling model.\nParameterized by $\\theta$, they use a global exponential model to\nrepresent the conditional distribution over label sequences\n$\\bm{y} = \\langle y_1, y_2, ..., y_t \\rangle$ conditioned on input\nsequences $\\bm{x} = \\langle x_1, x_2, ..., x_t \\rangle$:\n\\begin{equation}\n  \\label{eqn:expmodel}\n  \\Pu \\propto \\exp \\sum_j f_{\\theta}^j(\\bm{x}, \\bm{y}),\n\\end{equation}\nwith individual observations $x_i$ coming from some observation space\n$X$, and outputs $y_i$ coming from\nsome finite alphabet $Y$. In this\nwork, we use CRFs for sequence labeling problems, but the dataset\nlabels do not correspond directly to the CRF's outputs $y_i$.  In\norder to avoid ambiguity, and since the term ``state'' already has a\nmeaning for NFAs, we call $\\bm{y}$ the CRF's \\textit{tag sequence}, and\neach $y_i$ a \\textit{tag}. The terms \\textit{label sequence} and\n\\textit{label} will thus unambiguously refer to the original dataset\nlabels.\n\n\nEach $f_\\theta^j$ is a potential function of $\\bm{x}$ and $\\bm{y}$,\nparameterized by $\\theta$.  Importantly, in a linear-chain CRF, these\npotential functions are limited to two kinds: The \\textit{transition\n  function} $g_{\\theta}(y_i, y_{i+1})$ assigns a potential to each\npair $(y_i, y_{i+1})$ of adjacent tags in $\\bm{y}$, and the\n\\textit{emission function} $h_{\\theta}(y_i\\mid\\bm{x}, i)$ assigns a\npotential to each possible output tag $y_i$ given the observation\nsequence $\\bm{x}$ and its position $i$.\n\nWith these, the distribution defined by a CRF is\n\\begin{equation}\n  \\Pu \\propto \\exp \\left(\\sum_{i=1}^{t-1} g_{\\theta}(y_i, y_{i+1}) + \\sum_{i=1}^{t} h_{\\theta}(\\bm{x},y_i,i)\\right).\n\\end{equation}\n\nLimiting our potential functions in this way imposes a Markov\nassumption on our model, as potential functions can only depend on a\nsingle tag or a single pair of adjacent tags.  This makes learning and\ninference tractable: the forward algorithm \\citep{jurafskymartin} can\ncalculate negative log-likelihood loss during training, and the\nViterbi algorithm \\citep{viterbi, jurafskymartin} can be used for\ninference.  Both are linear in $t$, and quadratic in $|Y|$ in both\ntime and space.\n\n\\iffalse\nIn practice, the transition function $g_{\\theta}$ is represented\nexplicitly as a $|Y|\\times|Y|$ parameter matrix, and the emission\nfunction $h_{\\theta}$ can be an arbitrary learnable function --\nincreasingly commonly represented as a deep neural network.\n\\fi\n\n\\section{Regular-constrained CRFs}\n\nGiven a regular language $\\mathcal{L}$, we would like to\nconstrain a\nCRF to $\\mathcal{L}$.  We formalize this notion of constraint with\nconditional probabilities -- a CRF constrained to $\\mathcal{L}$ is described\nby a (further) conditionalized version of that CRF's distribution $\\Pu$, conditioned\non the event that the tag  sequence $\\bm{y}$ is in $\\mathcal{L}$. \nWe write this distribution as\n\\begin{equation}\n\\Pc = \\begin{cases}\n\\alpha \\cdot \\Pu & \\text{if } \\bm{y} \\in \\mathcal{L} \\\\\n0 & \\text{otherwise},\\\\\n\\end{cases}\n\\end{equation}\nwith $\\alpha \\geq 1$ defined as $\\alpha = \\frac{1}{\\sum_{\\bm{y} \\in \\mathcal{L}} \\Pu}$.\n\nIn order to utilize this distribution for machine learning, we\nneed to be able to compute log-likelihood losses and perform MAP\ninference.  As discussed in Section~\\ref{sec:CRF}, both of these are\nefficiently computable for CRFs.  Thus, if we can construct a separate\nCRF whose output distribution can be interpreted as\n$P(\\bm{y}\\mid\\bm{x}, \\mathcal{L})$, both of these operations will be available.  We\ndo this in the next section.\n\n\\subsection{Construction}\n\n\\begin{figure}\n{\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,\n                    semithick]\n  \n  \\tikzstyle{inpath}=[draw=ForestGreen, ultra thick, fill=ForestGreen]\n \n\n  \\node[initial,accepting,state] at (0, 0) (q1) {$q_1$};\n  \\node[state] at (3, 0) (q2)  {$q_2$};\n  \\node[accepting,state] at (0, -2) (q3)  {$q_3$};\n  \\node[state] at (3, -2) (q4)  {$q_4$};\n  \n  \\path (q1)  edge [loop above] node {O} (q1);\n  \\path[inpath, preaction={draw,line width=1mm}] (q1)  edge  node {\\textcolor{ForestGreen}{\\textbf{B}}} (q2);\n  \\path (q2)  edge [loop above] node {I} (q2);\n  \\path[inpath] (q2) edge node {\\textcolor{ForestGreen}{\\textbf{O}}} (q4);\n  \\path (q4)  edge [loop below] node {O} (q4);\n  \\path[inpath] (q4)  edge node {\\textcolor{ForestGreen}{\\textbf{B}}} (q3);\n  \\path (q2)  edge [bend left=15] node {B} (q3);\n  \\path (q3)  edge [bend left=15] node {B} (q2);\n  \\path[inpath] (q3)  edge [loop below] node {\\textcolor{ForestGreen}{\\textbf{I}}} (q3);\n  \\path (q3)  edge node {O} (q1);\n\n  \\draw [-] (5,.5) -- (5,-2.5);\n\n\\end{tikzpicture}\n\\hspace{1cm}\n\\begin{tikzpicture}\n\\tikzstyle{var}=[draw, circle, minimum size=1cm, inner sep=0cm]\n\\tikzstyle{given}=[fill=lightgray]\n\\tikzstyle{factor}=[draw, rectangle, fill=white]\n\n\\node[var] at (0, 0) (y1) {$\\edge{q_1}{B}{q_2}$};\n\\node[var] at (4, 0) (y2) {$\\edge{q_2}{O}{q_4}$};\n\\node[var] at (8, 0) (y3) {$\\edge{q_4}{B}{q_3}$};\n\\node[var] at (12, 0) (y4) {$\\edge{q_3}{I}{q_3}$};\n\\node[var, given] at (6, 3) (x) {$\\bm{x}$};\n\\draw (x) -- (y1) node [midway, factor] {$h_\\theta(\\bm{x},\\text{B}, 1)$};\n\\draw (x) -- (y2) node [midway, factor] {$h_\\theta(\\bm{x},\\text{O}, 2)$};\n\\draw (x) -- (y3) node [midway, factor] {$h_\\theta(\\bm{x},\\text{B}, 3)$};\n\\draw (x) -- (y4) node [midway, factor] {$h_\\theta(\\bm{x},\\text{I}, 4)$};\n\\draw (y1) -- (y2) node [midway, factor] {$g_\\theta(\\text{B}, \\text{O})$};\n\\draw (y2) -- (y3) node [midway, factor] {$g_\\theta(\\text{O}, \\text{B})$};\n\\draw (y3) -- (y4) node [midway, factor] {$g_\\theta(\\text{B}, \\text{I})$};\n\\end{tikzpicture}\n}}\n\\vspace{.15cm}\\\\\n\\resizebox{\\textwidth}{!}{\n$Y' = \\left\\{\\edge{q_1}{O}{q_1}, \\edge{q_1}{B}{q_2}, \\edge{q_2}{I}{q_2}, \\edge{q_2}{B}{q_3}, \\edge{q_2}{O}{q_4}, \\edge{q_3}{O}{q_1}, \\edge{q_3}{B}{q_2}, \\edge{q_3}{I}{q_3}, \\edge{q_4}{B}{q_3}, \\edge{q_4}{O}{q_4}\\right\\}$\n}\n\\caption{Example for a RegCCRF, showing NFA and unrolled factor\n  graph. $\\mathcal{L}$ describes the language\n  $(\\text{O}\\mid\\text{BI}^*\\text{O}^*\\text{BI}^*)^*$, the language of\n  valid BIO sequences for an even number of spans.  We would like to\n  calculate $\\Pc$ for\n  $\\bm{y} = \\langle \\text{B}, \\text{O}, \\text{B}, \\text{I} \\rangle$.  We\n  show an unambiguous automaton $M$ for $\\mathcal{L}$ (left), and a\n  factor graph (right) for the auxiliary CRF computing\n  $P_\\theta(\\bm{y'}\\mid\\bm{x})$, where $\\bm{y'} \\in Y'^*$ corresponds to\n  the sole accepting path of $\\bm{y}$ through $M$ (marked).}\n\\label{fig:construction}\n\\end{figure}\n\nLet $M := (\\Sigma, Q, F, E)$ be an NFA that describes $\\mathcal{L}$.  We\nassume that $M$ is \\textit{unambiguous} -- i.e., every\nstring in $\\mathcal{L}$ is accepted by exactly one path through $M$.  As\nevery NFA can be transformed into an equivalent unambiguous NFA\n\\citep{mohri2012disambiguation}, we can make this assumption with no\nloss of generality.\n\nOur plan is to represent $\\Pc$ by constructing a separate CRF with a\ndistinct tag set, whose output sequences can be interpreted directly\nas paths through $M$. As $M$ is unambiguous, each label sequence in\n$\\mathcal{L}$ corresponds to exactly one such path.  We parameterize this\nauxiliary CRF identically to our original CRF -- that is, with a\nlabel-wise (not tag-wise) transition and emission functions. Thus, for\nall parameterizations $\\theta$, both distributions $\\Pu$ and $\\Pc$ are\nwell defined.\n\n\n\n\nThere are many ways to construct such a CRF.  As CRF training and\ninference are quadratic in the size of the tag set $Y$, we would\nprefer a construction which minimizes $|Y|$.  However, for clarity, we\nwill first present a conceptually simple construction, and then\ndiscuss approaches to reduce $|Y|$.\n\nWe start with our original CRF, parameterized by $\\theta$, with tag\nset $Y = \\Sigma$, transition function $g_\\theta$, and emission\nfunction $h_\\theta$, describing the probability distribution\n$\\Pu$, $\\bm{y} \\in \\Sigma^*$.  From this, we construct a new CRF, also\nparameterized by the same $\\theta$, but with tag set $Y'$, transition\nfunction $g'_\\theta$, and emission function $h'_\\theta$. This\nauxiliary CRF describes the distribution $P'_\\theta(\\bm{y'}\\mid\\bm{x})$\n(with $\\bm{y'} \\in Y'^*$), which we will be able to interpret as\n$\\Pc$.  The construction is as follows:\n\\begin{gather}\nY' = E \\\\\ng'_{\\theta}((q, a, r),(q', a', r')) =\n\\begin{cases}\n    g_{\\theta}(a, a') & \\text{if } r = q'\\\\\n    -\\infty & \\text{otherwise} \\\\\n\\end{cases} \\\\\nh'_{\\theta}(\\bm{x}, (q, a, r), i) = \\begin{cases}\n    -\\infty & \\text{if } i = 1, q \\neq q_1\\\\\n    -\\infty & \\text{if } i = t, r \\not\\in F\\\\\n    h_{\\theta}(\\bm{x}, a, i) & \\text{otherwise.}\\\\\n    \\end{cases} \n\\end{gather}\nThis means that the tags of our new CRF are the edges of $M$, the\ntransition function assigns zero probability to transitions between\nedges which do not pass through a shared NFA state, and the emission\nfunction assigns zero probability to tag sequences which do not begin\nat the starting state or end at an accepting state.  Apart from these\nconstraints, the transition and emission functions depend only on edge\nlabels, and not on the edges themselves, and agree with the standard\nCRF's $g_\\theta$ and $h_\\theta$ when\nno constraints are violated.\n\nAs $M$ is unambiguous, every tag sequence $\\bm{y}$ corresponds\nto a single path through $M$, representable as an edge sequence\n$\\bm{\\pi} = (\\pi_1, \\pi_2, ..., \\pi_t), \\pi_i \\in E$.  Since this path is a\ntag sequence for our auxiliary CRF, we can directly calculate the\nauxiliary CRF's $P'_\\theta(\\bm{\\pi}\\mid\\bm{x})$.  From the construction of\n$g'_\\theta$ and $h'_\\theta$, this must be equal to $\\Pc$, as it is\nproportional to $\\Pu$ for $\\bm{y} \\in \\mathcal{L}$, and zero (or, more\ncorrectly, undefined) otherwise.  Figure~\\ref{fig:construction}\nillustrates this construction with a concrete example.\n\n\\subsection{Tag-set minimization}\n\\label{sec:tagmin}\nAs the Viterbi and forward algorithms are quadratic in $|Y|$,\nvery large tag sets can lead to performance issues during training and inference.\nTo alleviate this, we would like $|Y|$ to be as\nsmall as possible. Without changing $\\mathcal{L}$, there are two approaches we can\ntake here:\n\nFirst, we can select $M$ to have as few edges as possible. This\nproblem for unambiguous NFAs is NP-complete in the general case\n\\citep{jiang1991minimal}, and, assuming $\\text{P} \\neq \\text{NP}$, is\nnot even efficiently approximable\n\\citep{gruber2007inapproximability}. Nonetheless, in practice many\nlanguages can be minimized manually, and heuristic\napproaches can reduce, if not minimize, the size of NFAs.\n\nSecond, we can reduce the size of $|Y|$ by adjusting our construction.\nInstead of making each edge of $M$ a tag, we can adopt equivalence\nclasses of edges.  Reminiscent of standard NFA minimization, we define\n$(q, a, r) \\sim (q', a', r') \\leftrightarrow (q, a) = (q', a')$. When\nconstructing our CRF, whenever a transition would have been allowed\nbetween two edges, we allow a transition between their corresponding\nequivalence classes.  We do the same to determine which classes are\nallowed as initial or final tags.  As each equivalence class\ncorresponds (non-uniquely) to a single symbol $a$, we can translate\nbetween tag sequences and strings of $\\mathcal{L}$ just as before.\n\n\n\\section{Constrained training vs. constrained decoding}\n\\label{sec:constrained_decoding}\nOur construction suggests two possible use cases for a RegCCRF:\n\\textit{constrained decoding}, where a CRF is trained unconstrained, and\nthe learned weights are then used in a RegCCRF at decoding time, and\n\\textit{constrained training}, where a RegCCRF is both trained and decoded\nwith constraints.\nIn this section, we compare between these two approaches and a standard,\n\\textit{unconstrained CRF}.\nWe assume a machine learning\nsetting where we have access to samples from some data distribution\n$\\widetilde{P}(\\bm{x}, \\bm{y})$, with each $\\bm{x} \\in X^*$, and each $\\bm{y}$ of\nmatching length in some regular language $\\mathcal{L} \\subseteq{\\Sigma^*}$.\nWe wish to model the conditional distribution\n$\\widetilde{P}(\\bm{y}\\mid\\bm{x})$ with either a CRF or a RegCCRF, by way of minimizing the\ncross-entropy between model and data distributions.\n\nThe unconstrained CRF corresponds to a CRF that has been trained, without constraints,\non data points from $\\widetilde{P}(\\bm{x}, \\bm{y})$, and is subsequently used\ndirectly. As such, it makes no use of the language $\\mathcal{L}$.\nThe model's output distribution is $\\Pu[\\theta_u]$, with parameter\nvector\n\\begin{equation}\n\\theta_u = \\argmin_\\theta \\E{- \\ln \\Pu}\n\\label{eq:thetau}\n\\end{equation}\nminimizing CRF cross-entropy with the data distribution.\n\n\nIn constrained decoding, a CRF is trained unconstrained, \nbut its weights are used in a RegCCRF at decoding time. The output distribution\nof such a model is\n$\\Pc[\\theta_u]$.\nIt is parameterized by the same\nparameter vector $\\theta_u$ as the unconstrained CRF, as the training procedure is\nidentical, but the output distribution is conditioned on $\\bm{y} \\in \\mathcal{L}$.\n\n\nConstrained training involves directly optimizing a\nRegCCRF's output distribution, avoiding any asymmetry between training and decoding time.\nIn this case, the output distribution of the model is $\\Pc[\\theta_c]$,\nwhere\n\\begin{equation}\n\\theta_c = \\argmin_\\theta \\E{-\\ln \\Pc}\n\\end{equation}\nis the parameter vector which minimizes the RegCCRF's cross-entropy\nwith the data distribution.\n\nIn the remainder of this section, we will demonstrate that these three approaches\nform a hierarchy of sorts, when compared by their ability to match the data distribution.\nIn particular, when compared by cross-entropy, we show that\n\\begin{equation}\n  H_\\text{unconstrained} \\geq H_\\text{constrained decoding} \\geq H_\\text{constrained training},\n\\end{equation}\nwith each $H$ corresponding to that model's cross-entropy with the data distribution.\nThis suggests we should prefer the constrained training regimen.\n\n\n\\paragraph{An unconstrained CRF cannot outperform constrained decoding.}\nHere we compare the distributions $\\Pu[\\theta_u]$ and $\\Pc[\\theta_u]$.\nWe wish to demonstrate that $\\Pu[\\theta_u]$ can never achieve lower cross-entropy\nwith the data distribution than $\\Pc[\\theta_u]$, and that the two distributions\nachieve identical cross-entropy only when $\\Pu[\\theta_u] = \\Pc[\\theta_u]$ i.e. when constraints have no effect.\n\\begin{proof} We demonstrate the more general case for arbitrary $\\theta$.\nSince every $\\bm{y}$ in $\\widetilde{P}$ is in $\\mathcal{L}$,\n\\begin{equation}\n\\Pc = \\alpha \\cdot \\Pu,\n\\end{equation}\nwith $\\alpha \\geq 1$.\nThus, the cross-entropy of the regular-constrained CRF is\n\\begin{equation}\n\\E{- \\ln \\Pc} = \\E{- \\ln \\Pu} - \\ln \\alpha.\n\\end{equation}\nThis differs from the cross-entropy of the unconstrained CRF only by the term $- \\ln \\alpha$.\nAs $\\alpha \\geq 1$, the regular-constrained CRF's cross-entropy is less than or equal to that of the unconstrained CRF,\nwith equality only when $\\alpha = 1$ and therefore $\\Pu = \\Pc$. \n\\end{proof}\n\n\\paragraph{Constrained decoding cannot outperform constrained training.}\nIn this case, we compare the distributions $\\Pc[\\theta_u]$ and $\\Pc[\\theta_c]$.\nWe will demonstrate that the former\ncan never outperform the latter, again as measured by cross-entropy with the data distribution.\n\\begin{proof}\nThis follows directly from our definitions, as we\ndefine $\\theta_c$ to minimize cross-entropy between $\\Pc$ and the data\ndistribution.  Thus, $\\Pc[\\theta_u]$ could never yield a lower cross-entropy\nthan $\\Pc[\\theta_c]$, as that would contradict our definition of $\\theta_c$.\n\\end{proof}\n\n\n\\section{Synthetic Data Experiments}\nWhile we have shown that constrained training cannot underperform\nconstrained decoding, the conditions where it is strictly better\ndepend on exactly how the partition and emission functions are\nparameterized, and are not easily stated in general terms.  In this\nsection, we compare the two regimens empirically to demonstrate\nsimple cases where the two are not equivalent.\n\n\nThe procedure is similar for both synthetic data experiments.  For\neach experiment, we specify a regular language $\\mathcal{L}$, an observation\nalphabet $X$, and a joint data distribution $\\widetilde{P}(\\bm{x}, \\bm{y})$\nover observation sequences in $X^*$ and label sequences in $\\mathcal{L}$.\nWe then train two models, one with a RegCCRF, parameterized by\n$\\theta_{c}$, and one with an unconstrained CRF, parameterized by\n$\\theta_{u}$.  For each model, we initialize parameters randomly,\nthen use stochastic gradient descent to minimize cross-entropy with $\\widetilde{P}$.\nWe directly generate samples from\n$\\widetilde{P}$ to use as training data.  After optimizing\n$\\theta_{c}$ and $\\theta_{u}$, construct a RegCCRF with $\\theta_u$\nfor use as a constrained-decoding model, and we compare the constrained-training\ndistribution $\\Pc[\\theta_c]$ with the constrained-decoding\ndistribution $\\Pc[\\theta_u]$.\n\nWe use a simple architecture for our models, with both the transition functions\n$g_\\theta$ and emission functions $h_\\theta$ represented as parameter matrices.\nWe list training hyperparameters in the Appendix~\\ref{sec:appendix}.\n\n\\subsection{Arbitrary differences in cross-entropy}\n\n\\begin{figure}\n\\resizebox{\\textwidth}{!}{\n\\input{arbitrary_p.tex}\n\\input{arbitrary_h.tex}\n}\n\\caption{Model output probabilities, and cross-entropy with the data\n  distribution, plotted against sequence length $k$.  As $k$\n  increases, constrained decoding becomes a progressively worse\n  approximation for the data distribution, while constrained training\n  is consistently able to match the data distribution.}\n\\label{fig:arbitrary}\n\\end{figure}\nWe would like to demonstrate that, when comparing constrained training\nto constrained decoding in terms of cross-entropy with the data\ndistribution, constrained training can outperform constrained decoding\nby an arbitrary margin.  We choose\n$\\mathcal{L} = (\\texttt{ac})^*\\mid(\\texttt{bc})^*$ to make conditional\nindependence particularly relevant -- as every even-indexed label is\n\\texttt{c}, an unconstrained CRF must model odd-indexed labels\nindependently, while a constrained CRF can use its constraints to\naccount for nonlocal dependencies.  For simplicity, we hold the input\nsequence constant, with $X = \\{\\texttt{o}\\}$.\n\nOur approach is to construct sequences of various lengths. For $k \\in \\mathbb{N}$, we let our data distribution be\n\\begin{equation}\n\\widetilde{P}(\\texttt{o}^{2k}, (\\texttt{ac})^k) = \\frac{3}{4} \\text{\n  and }\n\\widetilde{P}(\\texttt{o}^{2k}, (\\texttt{bc})^k) = \\frac{1}{4}.\n\\end{equation}\nWe train and evaluate individual models for each sequence lengths\n$k$.\nFigure~\\ref{fig:arbitrary} plots model probabilities and\ncross-entropies for various $k$.  We see that, regardless of $k$,\n$\\Pc[\\theta_c]$ is able to match $\\widetilde{P}(\\bm{y}\\mid\\bm{x})$ almost\nexactly, with only small deviations due to sampling noise in SGD.  On\nthe other hand, as sequence length increases, $\\Pc[\\theta_u]$ becomes\nprogressively ``lop-sided'', assigning almost all of its probability\nmass to the string $(\\texttt{ac})^k$.  This behavior is reflected the\nmodels' cross-entropies with the data distribution -- constrained\ntraining stays at near-constant cross-entropy for all $k$, while the\ncross-entropy of constrained decoding grows linearly with $k$.\n\n\\subsection{Differences in MAP inference}\nWe show here that constrained training and constrained decoding can disagree\nabout which label sequence they deem most likely.\nFurthermore, in this case, MAP inference agrees with the data distribution's\nmode for constrained training, but not for constrained decoding.\nTo do this, we construct a fixed-length output language\n$\\mathcal{L} = \\texttt{acd}\\mid\\texttt{bcd}\\mid\\texttt{bce}$, where an\nunconstrained CRF is be limited by the Markov property to predict $\\bm{y}$'s prefix\nand suffix independently.\nWe select our data distribution,\n\\begin{equation}\n\\widetilde{P}(\\texttt{ooo}, \\texttt{acd}) = 0.4 \\text{ and }\n\\widetilde{P}(\\texttt{ooo}, \\texttt{bcd}) = 0.3 \\text{ and }\n\\widetilde{P}(\\texttt{ooo}, \\texttt{bce}) = 0.3,\n\\end{equation}\nto be close to uniform, but with one label sequence holding\nthe slight majority, and we ensure that the majority label sequence is \\textit{not}\nthe label sequence with both the majority prefix and the majority suffix (i.e. \\texttt{bcd}).\nAs before, we hold the observation sequence as a constant ($\\texttt{ooo}$).\nWe train a constrained and an unconstrained CRF to convergence, and compare $\\Pc[\\theta_u]$\nto $\\Pc[\\theta_c]$.\n\n\\begin{table}\n\\centering\n\\caption{Output distributions for constrained decoding ($\\Pc[\\theta_u]$)\nand constrained training ($\\Pc[\\theta_c]$), compared to the data distribution $\\widetilde{P}(\\bm{y}\\mid\\bm{x})$.\nConstrained decoding cannot learn the data distribution exactly, and yields a\nmode which disagrees with that of the\ndata distribution.}\n\\label{tab:map}\n\\begin{tabular}{lccc}\n\\toprule\n$\\bm{y}$ & $\\widetilde{P}(\\bm{y}\\mid\\bm{x})$ & $\\Pc[\\theta_u]$ & $\\Pc[\\theta_c]$ \\\\\n\\cmidrule(r){1-1}\\cmidrule(rl){2-2}\\cmidrule(rl){3-3}\\cmidrule(l){4-4}\n\\texttt{acd} & \\textbf{0.4} & 0.32 & \\textbf{0.40}\\\\\n\\texttt{bcd} & 0.3 & \\textbf{0.48} & 0.30\\\\\n\\texttt{bce} & 0.3 & 0.20 & 0.40\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{tab:map} shows $\\Pc[\\theta_u]$ and $\\Pc[\\theta_c]$ as they compare to\n$\\widetilde{P}(\\bm{y}\\mid\\bm{x})$.\nWe find that, while the mode of $\\widetilde{P}(\\bm{y}\\mid\\bm{x})$ is \\texttt{acd},\nthe mode of constrained decoding distribution $\\Pc[\\theta_u]$ is \\texttt{bcd},\nthe string with the majority prefix and the majority suffix.  Conversely, the constrained\ntraining distribution $\\Pc[\\theta_c]$ matches the data distribution almost\nexactly, and predicts the correct mode.\n\n\\section{Real-world data experiment: semantic role labeling}\n\n\\paragraph{Task.} As a final experiment, we apply our RegCCRF to the\nNLP task of semantic role labeling (SRL) in the popular PropBank\nframework \\citep{palmer-etal-2005-proposition}. In line with previous\nwork, we adopt the \\textit{known-predicate setting}, where events are\ngiven and the task is to mark token spans as (types of) event\nparticipants. PropBank assumes 7 semantic \\textit{core roles} (ARG0\nthrough ARG5 plus ARGA) plus 21 \\textit{non-core roles} for modifiers\nsuch as times or locations. For example, in [$_{\\text{ARG0}}$\n\\textit{Peter}] \\textbf{saw} [$_{\\text{ARG1}}$ \\textit{Paul}]\n[$_{\\text{ARGM-TMP}}\\, \\textit{yesterday}]$, the argument labels\ninform us who does the seeing (ARG0), who is seen (ARG1), and when the\nevent took place (ARGM-TMP). In addition, role spans may be labeled as\n\\textit{continuations} of previous role spans, or as\n\\textit{references} to another role span in the sentence.\n\nSRL can be framed rather naturally as a sequence labeling task\n\\citep{he2017deep}. However, the task comes with a number of hard\nconstraints that are not automatically satisfied by standard CRFs,\nnamely: (1) Each core role may occur at most once per event; (2)\ncontinuations require that the same span type occurs previously in the\nsentence; (3) references require that the same span type occurs\nelsewhere in the sentence (before or after).\n\n\\paragraph{Data.} In line with previous work\n\\citep{ouchi-etal-2018-span}, we work with the OntoNotes corpus as\nused in the CoNLL 2012 shared task\\footnote{\nAs downloaded from \\url{https:\/\/catalog.ldc.upenn.edu\/LDC2013T19}, and\npreprocessed according to \\url{https:\/\/cemantix.org\/data\/ontonotes.html}\n}\n \\citep{ontonotes, pradhan-etal-2012-conll},\nwhose training set comprises 66 roles (7 core roles, 21 non-core roles,\n19 continuation types, and 19 reference types).\n\n\\paragraph{Model.} To encode the three constraints listed above in a\nRegCCRF, we define a regular language describing valid BIO\nsequences \\citep{ramshaw1999text} over the 66 roles. A minimal\nunambiguous NFA for this language has more than\n$2^2\\cdot3^{19}$ states, which is too large to run the Viterbi\nalgorithm on our hardware.  However, as many labels are very rare,\nwe can shrink our automaton by discarding them at the cost of\nimperfect recall. We achieve further reductions in size by ignoring\nconstraints on reference roles, treating them identically to non-core\nroles. Our final automaton recognizes 5 core role types (ARG0 through\nARG4), 17 non-core \/ reference roles, and one continuation role type\n(for ARG1). This automaton has 672 states, yielding a RegCCRF with\n2592 tags.\n\nOur model is then given by this RegCCRF, trained constrained, with emission scores provided\nby a linear projection of the output of a pretrained RoBERTa network\n\\cite{liu2019roberta}.  In order to provide the model with event information,\nthe given predicates are prefixed by a special reserved token in the input\nsequence. RoBERTa parameters are fine-tuned during the\nlearning of transition scores and projection weights -- a full\ndescription of the training procedure is provided in the Appendix~\\ref{sec:appendix}.\n\nAs RegCCRF loss is only finite for label sequences in $\\mathcal{L}$, we must\nensure that our training data do not violate our constraints.  For the\nroles which we discard, we simply remove the offending labels from the\ntraining data. In six cases, training instances directly conflict with\nthe constraints specified --- all of these cases involve continuation\nroles missing a valid preceding role.  We discard these instances.\n\n\\paragraph{Baseline.} As a baseline model, we use the same\narchitecture with a standard CRF replacing the RegCCRF. Since for this\nbaseline, we are not limited by GPU memory, we include all role\ntypes present in the training set, and use the complete training set.\n\n\n\\paragraph{Results and analysis.}\nWe evaluate both regular-constrained and unconstrained CRF models against the\nevaluation partition, and measure performance using $F_1$ score for exact span\nmatches.\nFor comparability with prior work, we use the evaluation script\\footnote{As \navailable from \\url{https:\/\/www.cs.upc.edu\/~srlconll\/soft.html}.}\nfor the CoNLL-2005 shared task \\citep{carreras-marquez-2005-introduction}.\nThese results, averaged over five trials, are presented in Table~\\ref{tab:srl_results}.\n\nBoth our RegCCRF and the unconstrained CRF baseline outperform the\nexisting state-of-the-art ensemble model\n\\cite{ouchi-etal-2018-span}. We ascribe this improvement over the\nexisting literature to our use of RoBERTa -- prior work in SRL tends\nto rely on ELMo \\citep{peters-etal-2018-deep}, which has been show to\ndo worse than pretrained transformer-based models in downstream tasks\n\\citep{devlin-etal-2019-bert}.\n\nBetween our RegCCRF and baseline models, we find that the former\nsignificantly\\footnote{All significance results are at the $p<0.05$\n  level, as measured by a two-tailed permutation test.}  outperforms\nthe latter in terms of $F_1$ score. This is despite the fact that our\nRegCCRF is unable to predict rare role types, while the CRF has no\nsuch limitation. In fact, just under 1\\% of all role spans in the test\ndata are of a type not included in the RegCCRF, leaving it with a\ntheoretical maximum of 99\\% recall. Nonetheless, our RegCCRF actually\nachieves a higher average recall than the CRF, though this difference\nis not significant. In contrast, unsurprisingly, the RegCCRF show\nsignificantly better precision than the CRF.\nNo single model from the literature beats the RegCCRF in precision, only\nthe ensemble of \\cite{ouchi-etal-2018-span}.\n\n\n\n\\begin{table}\n\\robustify\\bfseries\n\\centering\n  \\caption{\n  \\label{tab:srl_results}\n  Results from our experiments (averaged over 5 trials), along with selected\n  reported results from recent literature.\n  }\n  \\begin{tabular}{llSSS}\n  \\toprule\n&   Model& {Precision} & {Recall} & {$F_1$}\\\\\n  \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(l){5-5}\n\\multirow{2}{*}{Our experiments} &   RoBERTa + CRF & 86.82 & 87.73 & 87.27 \\\\\n&   RoBERTa + RegCCRF & 87.22 & \\bfseries 87.79 & \\bfseries 87.51 \\\\\n  \\cmidrule(r){1-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(l){5-5}\n\\multirow{4}{*}{\\parbox{25mm}{Results from\\\\ literature}}&  \\cite{he2017deep} & {---} & {---} & 85.5\\\\\n&    \\cite{ouchi-etal-2018-span} & 87.1 & 85.3 &  86.2\\\\\n&   \\cite{ouchi-etal-2018-span} (ensemble) & \\bfseries 88.5 & 85.5 & 87.0\\\\\n& \\cite{Li_He_Zhao_Zhang_Zhang_Zhou_Zhou_2019} &  85.7 & 86.3 & 86.0 \\\\\n  \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\section{Conclusion and Future Work}\nIn this work, we have presented a method to constrain the output of\nCRFs to a regular language. Our construction allows constraints to be\nused at training or prediction time, and we demonstrate that training\nwith constraints better captures the data distribution. To test our\nmethod empirically, we carry out experiments on synthetic data, and\nincorporate a RegCCRF into a deep neural model for semantic role labeling,\ndemonstrating improvements over the state-of-the-art.\n\nFor enhanced expressibility, future work could investigate\nnon-binary constraints, i.e., regular\nlanguage-based constraints with learnable weights. \nAdditionally, regular language\ninduction (e.g. \\cite{regex_induction1, regex_induction2}) could be\nused to learn languages automatically, reducing manual specification\nand identifying non-obvious constraints.\n\nAnother avenue for continuing research lies in identifying further\napplications for RegCCRFs. We believe that the NLP task of relation extraction\ncould be a fruitful target -- RegCCRFs offer a mechanism to make the\nproposal of a relation conditional on the presence of the right\nnumber and type of arguments.\n\nWhile the construction we have proposed cannot be lifted directly to\ncontext-free languages due to the unbounded state space of the\ncorresponding pushdown automata, there is plentiful work on regular\napproximations of context-free languages \\citep{mohri2001regular}. On\nthis basis, for example, a RegCCRF backed by a regular language describing trees of\na limited depth could also be applied to tasks with context-free\nconstraints.\n\nOur implementation of RegCCRFs, as well as scripts for replicating our\nexperiments, are available at\n\\url{https:\/\/www.ims.uni-stuttgart.de\/data\/regccrf}.\n\n\\section*{Acknowledgements}\nThis work is supported by IBM Research AI through the IBM AI Horizons\nNetwork.\n\n\\bibliographystyle{plainnat-nourl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe formation of galaxies is one of the biggest challenges in\nastronomy and astrophysics. It is the golden era to solve this\nproblem, because we have had good background of the formation and\nevolution of stars and the universe. In fact, many works have been\ndone and some progress has been presented. However, it is far from\nwell understanding galaxies and further researches are needed.\nEvolutionary stellar population synthesis is a widely used technique\nfor studying galaxies. It can give reliable studies to galaxies via\ntheir stellar contents, because all galaxies contain a great deal of\nstars and stars contribute mainly to the light of galaxies. A lot of\nworks studied galaxies using their stellar populations\n\\cite{Li:2006}. However, most previous works studied the stellar\npopulations of nearby galaxies, via spectra-like methods, while\noptical colours are thought to be unusable for determining two\nstellar-population parameters (age and metallicity). This is very\nthe well-known stellar age--metallicity degeneracy. However, some\nstudies also showed that colours in different bands are sensitive to\ndifferent stellar-population parameters and can possibly be used to\ngive estimates to stellar populations of distant galaxies, and then\ninvestigations to galaxy formation and evolution. I will review our\nworks on trying to use colours to study the stellar-population\nparameters of galaxies. This will be useful for future studies,\nbecause colours can be obtained much more easily than the spectra of\ngalaxies.\n\nThe structure of the paper is as follows. In section 2, I introduce\nthe sensitivities of colours to stellar-population parameters and\nsome potential colours for stellar population studies. In section 3,\nI summary a few points that should be considered when using colours\nto study stellar populations. In section 4, I introduce a new\nstellar population model for using colours to study stellar\npopulations of galaxies. Finally, in section 5, I give a short\nconclusion.\n\n\n\\section{Sensitivities of colours}\\label{sec:sens}\nSpectra line strength indices can be used to disentangle the stellar\nage--metallicity degeneracy because they have different\nsensitivities to the age and metallicity of stellar populations.\nHowever, optical colours are thought to be useless for disentangling\nstellar age--degeneracy because they have similar sensitivities to\nstellar-population parameters. In order to investigate the\npossibility of using colours in different bands to study the ages\nand metallicities of stellar populations, we investigated the\nsensitivities of colours to stellar-population parameters\n\\cite{Li:2007}. In that work, a simple stellar population model\n\\cite{Bruzual:2003} and a relative sensitivity method were used. The\nresults showed that colours in different bands have various\nsensitivities to the inputs of stellar populations. In detail, some\ncolours related to optical bands, e.g., $(B-V)$, $(U-R)$, $(R-I)$\nand $(V-I)$, are more sensitive to stellar age, while some other\nones related to near-infrared bands, e.g., $(B-K)$, $(R-K)$, $(V-K)$\nand $(I-K)$, to stellar metallicity. However, it showed that every\ncolour is affected by both stellar age and metallicity. This\nsuggests that it is impossible to determine stellar age or\nmetallicity using one colour index. However, using a pair of colours\nthat consist of an age-sensitive colour and a metallicity-sensitive\ncolour, the stellar ages and metallicities of galaxies can be\ndetermined. One can refer to our paper \\cite{Li:2007} for more\ndetails. When we tried to study the sensitivities of some colours\n(hereafter composite colours) including magnitudes on different\nphotometry systems, we found that some composite colours [e.g.,\n$(r-K)$, $(u-K)$, $(u-R)$ and $(i-I)$] have good sensitivities to\nstellar-population parameters. Thus they can be used for stellar\npopulation studies. If taken the usual observational errors for\nmagnitudes, the abilities of different pairs of colours for\ndisentangling the well-known stellar age--metallicity degeneracy can\nbe valued \\cite{Li:2008a}. It is shown that pairs such as [$(r-K)$,\n$(u-R)$] and [$(r-K)$, $(u-r)$] are better for usual stellar\npopulation studies. However, the uncertainties in final stellar ages\nand metallicities are somewhat large (near 100\\%). This results from\nthe large observational uncertainties in colours. Because different\nsurveys have different observational uncertainties, the errors in\nthe results of various surveys can be different a lot. In future\nsurveys, because colour uncertainties will be possibly reduced, it\nwill be possible to give more accurate constraints on stellar ages\nand metallicities of galaxies via colours.\n\n\\section{Points should be noted}\\label{sec:points}\nColours related to different bands have various sensitivities to\nstellar-population parameters and can help us to determine the\nstellar ages and metallicities of galaxies. However, some points\nshould be noted, because simple stellar population models are\nusually used, but colours can be affected by, e.g., dust, young\nstars, and binary stars in galaxies, and observed colours are\nrelated to the distances of galaxies. In the following part, I will\nmention a few points.\n\n\\subsection{Effects of young stars}\nBecause early-type galaxies were thought to have some homogeneous\nand old stellar populations, some simple stellar population models\nwere widely used in the studies of the stellar populations of\nearly-type galaxies. However, more and more observations showed that\nthere are recent star formations in those galaxies. It suggests that\nyoung stars are common in all type of galaxies and there should be\nmore than one stellar populations in a galaxy. In this case, it is\nnecessary to consider the effects of young stars on the\ndetermination of stellar ages and metallicities of galaxies, as\nyoung stars are usually bright and can contribute much to the light.\nOne \\cite{Li:2007b} of our works studied how young stars can affect\nthe stellar-population parameters determined by colours. That work\nshows that if there were two stellar populations (an old and a young\none with the same metallicity) in a galaxy, the younger the age or\nthe larger the mass fraction of the young component, the bluer the\ncolours of the galaxy. When one gives estimates to stellar ages and\nmetallicities via comparing a pair of colours of galaxies to those\nof theoretical simple stellar populations, younger ages and richer\nmetallicities are usually obtained. Therefore, one should take the\neffects of young stars into account when studying stellar-population\nparameters using colours.\n\n\\subsection{Effects of binary interactions}\nBecause it is easier to model stellar populations via single stars,\nmost widely used stellar population models are single-star stellar\npopulation models (ssSSPs), which do not take the effects of binary\ninteractions into account. However, binary stars are common and they\nevolve differently from single stars. Two of our works\n\\cite{Li:2008b,Li:2008c} show that binary interactions make stellar\npopulations less luminous, while making colours bluer, age-sensitive\nline strength indices larger and metallicity-sensitive indices less\ncompared to ssSSPs. When using colours to determine the\nstellar-population parameters of galaxies, different stellar ages\nand metallicities will be obtained via ssSSPs and bsSSPs. Usually,\npoorer metallicities and similar ages will be given by ssSSP models,\ncompared to the results obtained via bsSSP models. Therefore, when\ninvestigating the stellar metallicities of galaxies, the effects of\nbinary interactions should be taken into account. In addition,\nalthough ssSSP and bsSSP models can give similar ages for galaxies,\nit is actually much more complicated in practical works, because the\neffects of binary interactions are degenerate with\nstellar-population mixing. This needs further investigations.\n\n\\subsection{Effects of dust and corrections}\nMost stellar population models do not take the effects of dust into\naccount, but dust exists in galaxies and can change the colours of\ngalaxies. Therefore, the effects of dust should be considered. Many\nworks about the dust of galaxies have been done, but there is a long\nway to go. Although it is difficult to give accurate corrections for\nthe dust in galaxies, we can reduce the effects of dust in final\nresults via defining our galaxy samples carefully. If we study\nluminous and relatively blue early-type galaxies, dust will affect\nour results much more slightly, because there is less dust in such\ngalaxies. Moreover, there are larger uncertainties in the results of\nof galaxies with large (e.g., $>$ 1) red shifts, because the\ncorrections of the colours of such galaxies have much uncertainties\nand they can lead to large uncertainties in final results. However,\nthis will possibly be improved in the future, following the\ndevelopment of telescopes and the process of observational data.\n\n\\section{A new model for stellar population studies}\\label{sec:model}\n\nAlthough there are many available stellar population models and some\nof them are widely used, there are some limitations in those models.\nIt is necessary to build some new and more advanced models for\nstellar population studies and galaxy studies. I introduce a new\nmodel for studying stellar populations of galaxies via colours or\nlow resolution spectral energy distributions (SEDs). This is a model\nthat takes both the effects of binary interactions and population\nmixing into account. Because binary insteraction and population\nmixing are common in galaxies, the theoretical populations of the\nnew model are closer to those of galaxies.\n\\begin{figure*}\n  \\includegraphics[angle=-90,width=0.8\\textwidth]{aprim.ps}\n  \\caption{The spectral energy distributions of three composite stellar populations with different metallicities. A fraction of 50\\% is taken for binaries.}\n\\end{figure*}\n\nThe model is built using an isochrone database \\cite{Li:2008d} of\nboth single-star and binary-star stellar populations, in which the\nevolution of stars was calculated via the rapid stellar evolution\ncode of Hurley et al. (2002). When building the new model, we used\nBaSeL 3.1 spectra library \\cite{Westera:2002} to transform the\nresults of stellar evolution into the SEDs of stellar populations.\nAlthough a galaxy may contains many populations, we build our\npopulation via two stellar populations (an old and a young one).\nThis makes it easier to calculate the model and possible to estimate\nthe characteristics of the main components of galaxies. Following\nthe results of Thomas et al. (2005), a exponentially declining law\nwith age is taken for the mass fractions of young components in our\nwork. In such a case, when the model is used for studies, some\nmass-weighted stellar ages and metallicities will be obtained for\nthe components of galaxies. As the main results, the SEDs and\ncolours of composite stellar populations are calculated. In Fig. 1,\nthe SEDs of a few populations are shown as an example. We see that\nthe UV-upturns observed in elliptical galaxies are reproduced by our\nnew model. In fact, UV SEDs are sensitive to the recent star\nformations of galaxies. When we try to study the sensitivities of\ncolours to the inputs of stellar populations, photometries in\ndifferent bands are found to be sensitive to different model inputs.\nThis will be useful for studying the metallicities and the ages of\nthe components of the stellar populations of galaxies, and then the\nstar formation histories of galaxies.\n\n\n\n\\section{Conclusion and discussion}\\label{sec:discuss}\n\nColours can help us to explore the stellar populations of galaxies,\nbecause different colours have various sensitivities to the inputs\nof stellar populations. If the galaxy sample is well selected and\nsuitable stellar population model is used, some credible results can\nbe obtained via colours of galaxies. It is will be useful for\nstudying the formation and evolution of galaxies. However, some\npoints should be noted in studies.\n\n\\section*{Acknowledgment}\nWe are grateful to Profs. Zhanwen Han, Xu Kong, Licai Deng, and Gang\nZhao for useful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:0}\nOur purpose is to establish factorization of matrices $M_{N}(\\mathcal{A})$ \nover certain rings $\\mathcal{A}$ of functions, among them the ring of \npolynomials, and the  $L^{\\infty}$ functions on the circle group $\\mathbb{T}$. \nAn equivalent \nformulation is the study of functions on $\\mathbb{T}$ which take values in the \n$N \\times N$ scalar matrices. The general setting is as follows: Fix $N$, and \nconsider the group $SL_{N}(\\mathcal{A})$ where the ``$S$\" is for determinant \n$= 1$. The object is then to factor arbitrary elements in \n$SL_{N}(\\mathcal{A})$ as alternating products of upper and lower triangular \nmatrix functions; equivalently, upper and lower triangular elements in \n$M_{N}(\\mathcal{A})$ with the constant $1$ in the diagonal.\n\nIn digital signal or image-processing one makes use of subdivisions of various \nfamilies of signals into frequency bands. This is of relevance in modern-day \nwireless signal and image processing, and the choice of a number $N$ of \nfrequency bands may vary from one application to the next.\n\nThere is a certain representation theoretic framework which has proved \nsuccessful: one builds a representation of the basic operations on signals, \nfiltering, down-sampling (in the complex frequency variable), up-sampling, and \ndual filter. These operations get represented by a system of operators in \nHilbert spaces of states, say $\\mathcal{H}$.\n\nA multiresolution (see Fig. 1) then takes the form of a family of closed \nsubspaces in $\\mathcal{H}$.  In this construction, ``non-overlapping frequency \nbands\" correspond to orthogonal subspaces in $\\mathcal{H}$; or equivalently to \nsystems of orthogonal projections. Since the different frequency bands must \nexhaust the signals for the entire system, one looks for orthogonal \nprojections which add to the identity operator in $\\mathcal{H}$. This leads \nto the study of certain representations of the Cuntz algebra $\\mathcal{O}_N$, \ndetails below. Since time\/frequency-analysis is non-commutative, one is \nfurther faced with a selection of special families of commuting orthogonal \nprojections. When these iteration schemes (repeated subdivision sequences) are \napplied to the initial generators, one arrives at new bases and frames; and, \nin other applications, to wavelet families as recursive scheme. \n\nOur study of iterated matrix-factorizations are motivated by such questions \nfrom signal processing, and arising in multi-resolution analyses. In this \ncase, elements in the group $SL_{N}(\\mathcal{A})$ of matrix-functions act on \nvector-functions $f$ in a complex frequency variable, where the components in \n$f$ correspond to a specified system of $N$ frequency-bands. When a \nmatrix-factorization is established, then the action of the respective upper \nand lower triangular elements in $M_{N}(\\mathcal{A})$ are especially simple, \nin that a lower triangular filter filters a low band, and then adds it to one \nof the higher bands; and similarly for the action of upper triangular matrix \nfunctions.\n\nOur analysis depend on a certain representation of the Cuntz algebra \n$\\mathcal{O}_N$,  where $\\mathcal{O}_N$ is an algebra generated by the basic \noperations on signal representations, filtering, down-sampling (in the complex \nfrequency variable), up-sampling, and dual filter; see Fig 1. \n\n\n\n\n\\[\n  \\sum_{n}b_{n}z^{nN}=b_{0}+b_{1}z^{N}+b_{2}z^{2N}+\\cdots \\text{ ;}\n\\]\nso\n\\[\n  c_{n}=\n  \\begin{cases}\n    b_{n\/N}  \\quad \\text{if } N | n \\\\\n    0   \\quad \\text{if } N \\nmid n\n  \\end{cases}\n\\]\n\n\n\\[\n  \\frac{1}{N}\\sum_{w \\in \\mathbb{T}, w^{N}=z}b_{n}w^{n}=b_{0}+b_{N}z\n  +b_{2N}z^{2}+\\cdots\n\\]\n\n\n\n\\section{Factorization Algorithm}\n\\label{sec:2}\n\nIn order to illustrate our use of representations of the Cuntz algebra \n$O_{N}$ in algorithms for factorization, we begin with the case of $N=2$. The \nskeleton of these algorithms has three basic steps which we now outline.\n\n\\subsection*{The Algorithm}\nGiven\n\\[\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  \\in SL_{2}(\\mathcal{F})\n\\]\nwhere $\\mathcal{F}$ is some fixed ring of functions defined on a subset\n$\\Omega \\subset \\mathbb{C}$ such that $\\mathbb{T} \\subset \\Omega$.\n\n\\subsubsection*{Step 1:}\nGiven\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix},\n  \\quad AD-BC\\equiv 1 \\quad \\text{on $\\mathbb{T}$,}\n\\]\nand set\n\\begin{equation}\n\\label{eq:2.1}\n  \\mathcal{A}(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    A(z^{2})+zB(z^{2}) \\\\\n    C(z^{2})+zD(z^{2})\n  \\end{pmatrix}.\n\\end{equation}\nLet $S_{i}$, $i=0,1$ be  \n\n\\begin{equation}\n\\label{eq:2.2}\n  \\begin{cases}\n    S_{0}f(z)=f(z^{2}) \\\\\n    S_{1}f(z)=zf(z^{2})\n  \\end{cases}\n\\end{equation}\nFor the corresponding adjoint operators we therefore get:\n\\begin{equation}\n\\label{eq:2.3}\n  \\begin{cases}\n    S_{0}^{*}f(z)=\\frac{1}{2}\\sum_{\\omega^{2}=z}f(\\omega) \\\\\n    S_{1}^{*}f(z)=\\frac{1}{2}\\sum_{\\omega^{2}=z}\\overline{\\omega}f(\\omega)\n \\end{cases}\n\\end{equation}\nwhere the summation in (\\ref{eq:2.2}), (\\ref{eq:2.3}) are over points\n$z, \\omega \\in \\mathbb{T}$.\n\nThen $(S_{i})_{i=0,1}$ are isometries in $L^{2}(\\mathbb{T})$, and\n$S_{i}^{*}S_{j}=\\delta_{i,j}I$, $\\sum_{i=0}^{1}S_{i}S_{j}^{*}=I$ where $I$\ndenotes the identity operator in the Hilbert space $L^{2}(\\mathbb{T})$.\nWe will want $\\mathcal{F}$ to be a ring of meromorphic functions, such that\nthey are determined by their values on\n$\\mathbb{T}=\\{z \\in \\mathbb{C}, |z|=1\\}$;\nor we are simply working with functions on $\\mathbb{T}$.\n\n\\subsubsection*{Step 2:}\nFind functions $L$ such that\n\\begin{equation}\n\\label{eq:2.4}\n  \\begin{pmatrix}\n    l & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\mathcal{A}_{new}\n  =\\mathcal{A}.\n\\end{equation}\nSolution: Apply (\\ref{eq:2.4}) to\n\\[\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix},\n\\]\nand set\n\\[\n  \\mathcal{A}_{new}(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    f_{0}(z) \\\\\n    f_{1}(z)\n  \\end{pmatrix};\n\\]\nthen\n\\begin{equation}\n\\label{eq:2.5}\n  \\begin{cases}\n    f_{0}=A(z^{2})+zB(z^{2}) \\\\\n    L(z^{2})f_{0}(z)+f_{1}(z)=C(z^{2})+zD(z^{2}).\n  \\end{cases}\n\\end{equation}\nApply $S_{i}^{*}$, $i=0,1$, to (\\ref{eq:2.5})\n\\begin{equation}\n\\label{eq:2.6}\n  \\begin{cases}\n    S_{0}^{*}f_{0}=A, \\quad S_{1}^{*}f_{0}=B \\\\\n    LS_{0}^{*}f_{0}+S_{0}^{*}f_{1}=C \\\\\n    LS_{1}^{*}f_{0}+S_{1}^{*}f_{1}=D. \\\\\n    \\Rightarrow L=\\frac{C-S_{0}^{*}f_{1}}{A}; \\quad\n    L=\\frac{D-S_{1}^{*}f_{1}}{B}.\n  \\end{cases}\n\\end{equation}\n\n\\begin{corollary}\n  $A(S_{1}^{*}f_{1})-B(S_{0}^{*}f_{1})=1$.\n\\end{corollary}\n\\begin{proof}\n  Consider (\\ref{eq:2.6}) with $det \\mathcal{A}=1$.\n  \\[\n    \\mathcal{A}=\n    \\begin{pmatrix}\n      A & B \\\\\n      C & D\n    \\end{pmatrix}\n  \\]\n  with $AD-BC=1$.\n  So\n  \\[\n    \\begin{pmatrix}\n      A & B \\\\\n      S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n    \\end{pmatrix}\n    \\in SL_{2}(\\mathcal{F}).\n  \\]\n  We now assume $S_{i}\\mathcal{F} \\subset \\mathcal{F}$, and\n  $S_{i}^{*}\\mathcal{F} \\subset \\mathcal{F}$, for all $i=0,1$.\n\\end{proof}\n\n\\subsubsection*{Step 3:}\nHaving form $L$, from (\\ref{eq:2.4}) we get\n\\[\n  \\mathcal{A}_{new} =\n  \\begin{pmatrix}\n    l & 0 \\\\\n    -L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n      A & B \\\\\n      C & D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n      A & B \\\\\n      -LA+C & -LB+D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n      A & B \\\\\n      S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\]\n\n\\subsubsection*{Step 4:}\n\\[\n  \\begin{pmatrix}\n    l & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\mathcal{A}_{up}\n  =\\mathcal{A}_{new}.\n\\]\nSet\n\\[\n  \\mathcal{A}_{up}(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    g_{0}(z) \\\\\n    g_{1}(z)\n  \\end{pmatrix};\n\\]\nand we get\n\\[\n  \\begin{cases}\n    g_{0}(z)+U(z^{2})g_{1}(z)=A(z^{2})+B(z^{2})z \\\\\n    g_{1}(z)=(S_{0}^{*}f_{1})(z^{2})+(S_{1}^{*}f_{1})(z^{2})z\n  \\end{cases}.\n\\]\nApply $S_{i}^{*}$, $i=0,1$\n\\[\n  \\Rightarrow\n  \\begin{cases}\n  S_{0}^{*}g_{0}+US_{0}^{*}g_{1}=A, \\quad S_{1}^{*}g_{0}+US_{1}^{*}g_{1}=B \\\\\n  S_{0}^{*}g_{1}=S_{0}^{*}f_{1}, \\quad S_{1}^{*}g_{1}=S_{1}^{*}f_{1} \\\\\n  \\Rightarrow U=\\frac{A-S_{0}^{*}g_{0}}{S_{0}^{*}f_{1}} \\quad \\text{and} \\quad\n  U=\\frac{B-S_{1}^{*}g_{0}}{S_{1}^{*}f_{1}}\n  \\end{cases}\n\\]\nand continue.\n\n\\[\n  S_{0}^{*}f_{0}=A, \\quad S_{1}^{*}f_{0}=B\n\\]\n\\[\n  LS_{0}^{*}f_{0}+S_{0}^{*}f_{0}=C\n\\]\n\\[\n  LS_{1}^{*}f_{0}+S_{1}^{*}f_{0}=D\n\\]\n\\[\n  L=\\frac{C-S_{0}^{*}f_{1}}{A}=\\frac{D-S_{1}^{*}f_{1}}{B}\n\\]\n\\[\n  A(D-S_{1}^{*}f_{1})=B(C-S_{0}^{*}f_{1})\n\\]\n\\[\n  1=AS_{1}^{*}f_{1}-BS_{0}^{*}f_{1}\n\\]\n\\[\n  A_{new}^{(1)} =\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    -L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    A & B \\\\\n    -LA+C & -LB+D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\]\nso\n\\[\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}.\n\\]\n\n\n\\section{Factorization Cases}\n\\label{sec:2}\nIn the infinite-dimensional group \n$SL_{2}(L^{\\infty}(\\mathbb{T}))$, consider elements \n$\\mathcal{A}$ with factorization as in (\\ref{eq:3.1}):\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n\\]\n\n\\begin{equation}\n\\label{eq:3.1}\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(1)},\n  \\quad L \\in L^{\\infty}(\\mathbb{T}), \\quad \\mathcal{A}^{(1)} \\in\n  SL_{2}(L^{\\infty}(\\mathbb{T}))\n\\end{equation}\nOptimal\n\\[\n  \\mathcal{A}^{(1)}(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    f_{0} \\\\\n    f_{1}\n  \\end{pmatrix}\n\\]\n\\begin{equation}\n\\label{eq:3.2}\n  \\begin{cases}\n    A(z^{2})+zB(z^{2})=f_{0}(z) \\\\\n    C(z^{2})+zD(z^{2})=L(z^{2})f_{0}(z)+f_{1}(z)\n  \\end{cases}\n  \\quad \\{S_{i}\\}_{i=0} \\in REP(\\mathcal{O}_{2}, L^{2}(\\mathbb{T}))\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:3.3}\n  \\begin{cases}\n    S_{0}^{*}f_{0}=A, S_{1}^{*}f_{0}=B \\\\\n    LS_{0}^{*}f_{0}+S_{0}^{*}f_{1}=C \\\\\n    LS_{1}^{*}f_{0}+S_{1}^{*}f_{1}=D\n  \\end{cases}\n\\end{equation}\n\\begin{equation}\n\\label{eq:3.4}\n  \\iff\n  \\begin{cases}\n    S_{0}^{*}f_{1}=C-LA  \\\\\n    S_{1}^{*}f_{1}=D-LB\n  \\end{cases}\n\\end{equation}\n\\begin{equation}\n\\label{eq:3.5}\n  \\Rightarrow\n  f_{1}=(S_{0}S_{0}^{*}+S_{1}S_{1}^{*})f_{1}=S_{0}(C-LA)+S_{1}(D-LB)\n\\end{equation}\n\\[\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  \\longrightarrow\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} &  S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\]\n\nSince $S_{i}$ is isometric for $i=1,2$.\n\\begin{equation}\n\\label{eq:3.6}\n  \\|f_{1}\\|^{2}=\\|C-LA\\|^{2}+\\|D-LB\\|^{2} \\quad \\text{where $\\|\\cdot\\|$ is\n  the $L^{2}(\\mathbb{T})-$norm.}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:3.7}\n  \\langle u, v \\rangle=\\int_{\\mathbb{T}}\\overline{u}v \\quad \\text{with\n  respect to Haar measure on $\\mathbb{T}$.}\n\\end{equation}\nSo any functions\n\\begin{equation}\n\\label{eq:3.8}\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(1)}\n\\end{equation}\nwe pick the one with $f_{1}$ attaching its minimum in (\\ref{eq:3.6})\n\\begin{equation}\n\\label{eq:3.7-1}\n  \\inf\\{ (\\ref{eq:3.6})| \\text{factorization } (\\ref{eq:3.8}) \\text{ holds}\\}\n\\end{equation}\nCalculating $L$ on $\\mathcal{A}$\n\\[\n  L_{M}(\\epsilon)=L+\\epsilon M, \\quad M \\in L^{\\infty}(\\mathbb{T}).\n\\]\n\n\\begin{equation}\n\\label{eq:3.8-1}\n  \\frac{d}{d\\epsilon}\\bigg\\vert_{\\epsilon=0} (\\ref{eq:3.6})=0 \\quad\n  \\text{ at a minimum.}\n\\end{equation}\n\n\\begin{align*}\n  &\\iff \\\\\n  &\\langle MA, C-LA \\rangle + \\langle C-LA, MA \\rangle +\n  \\langle MB, D-LB \\rangle + \\langle D-LB, MB \\rangle \\\\\n  &=Re(\\langle MA, C-LA \\rangle+\\langle MB, D-LB \\rangle)=0 \\quad\n  \\forall M \\in L^{\\infty}(\\mathbb{T}).\n\\end{align*}\n\n\\begin{equation}\n\\label{eq:3.9-1}\n  \\overline{A}(C-LA)+\\overline{B}(D-LB)=0 \\quad \\text{pointwise a. e. on\n  $\\mathbb{T}$.}\n\\end{equation}\nSet $det\\mathcal{A}=1$,\n\\[\n  \\|A\\|^{2}+\\|B\\|^{2}>0 \\quad \\text{a. e. on $\\mathbb{T}$.}\n\\]\nSo\n\\begin{equation}\n\\label{eq:3.10}\n  L= \\frac{\\overline{A}C+\\overline{B}D}{|A|^{2}+|B|^{2}} \\quad\n  \\text{pointwise a. e. $\\mathbb{T}$.}\n\\end{equation}\nSolving for matrices $\\mathcal{A}^{(1)}$ in (\\ref{eq:3.8-1}), we get\n\\[\n  \\mathcal{A}^{(1)}=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    -L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    A & B \\\\\n    C-LA & D-LB\n  \\end{pmatrix}.\n\\]\nSo\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(1)}\n\\]\nWith the above $L$ in (\\ref{eq:3.10}) we see that\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(1)}\n\\]\nis the \\underline{optimal} factorization with a lower matrix as a\nleft-factor.\n\n\\begin{corollary}\n\\label{C:3.2}\nGiven\n  \\[\n    \\begin{pmatrix}\n      A & B \\\\\n      C & D\n    \\end{pmatrix}\n    \\in GL_{2}(L^{\\infty}(\\mathbb{T}));\n  \\]\nthen the optimal solution (\\ref{eq:3.10}) to the factorization problem\n\\begin{equation}\n\\label{eq:3.17}\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\end{equation}\nhas the matrix\n\\[\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\]\non the right hand side in (\\ref{eq:3.17}) orthogonal, i.e.,\n\\begin{equation}\n\\label{eq:3.18}\n  \\overline{A}(S_{0}^{*}f_{1})+\\overline{B}(S_{1}^{*}f_{1}) \\equiv 0\n  \\quad \\text{on $\\mathbb{T}$.}\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nWhen the function $L$ in (\\ref{eq:3.10}) is used in the computation of\n\\[\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix},\n\\]\nwe see that for any $z \\in \\mathbb{T}$,\n$((S_{0}^{*}f_{1})(z), (S_{1}^{*}f_{1})(z))$ in $\\mathbb{C}$ is in the\northogonal complement of $(A(z), B(z))$; indeed with (\\ref{eq:3.10}) we get\n\\begin{align*}\n  &\\overline{A}(S_{0}^{*}f_{1})+\\overline{B}(S_{1}^{*}f_{1}) \\\\\n  &=\\overline{A}\\left(C-\\frac{\\overline{A}C+\\overline{B}D}{|A|^{2}+|B|^{2}}A\\right)+\n  \\overline{B}\\left(D-\\frac{\\overline{A}C+\\overline{B}D}{|A|^{2}+|B|^{2}}B\\right) \\\\\n  &=\\overline{A}C+\\overline{B}D-(\\overline{A}C+\\overline{B}D)\\equiv 0;\n\\end{align*}\ni.e., a pointwise identity for functions on $\\mathbb{T}$.\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{C:3.1}\n  If $\\mathcal{A} \\in SU(L^{\\infty}(\\mathbb{T}))$ (i.e., unitary) then $L$ in\n(\\ref{eq:3.10}) is $0$ and so $\\mathcal{A}=\\mathcal{A}^{(1)}$ so the\nfactorization steps.\n\\end{corollary}\n\\begin{proof}\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix},\n\\]\nso unitary makes that the rows are orthogonal $\\overline{A}C+\\overline{B}D=0$\nin the inner product on $\\mathbb{C}^{2}$\n\\[\n  \\langle z, w \\rangle = \\overline{z_{1}}w_{1}+\\overline{z_{2}}w_{2}\n\\]\nand $|A|^{2}+|B|^{2}=1$.\n\\end{proof}\n\\begin{equation}\n\\label{eq:3.10-1}\n  \\mathcal{A}^{(1)}=\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\end{equation}\nusing $\\longrightarrow$\n\\[\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}.\n\\]\nNote this using the repeated on any\n$\\mathcal{A}^{(1)} \\in SL_{2}(L^{\\infty}(\\mathbb{T}))$ each time pick $L$ such\nthat the infimum in (\\ref{eq:3.6}) is attained.\n\nWith the same argument, we factor matrix\n\\[\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\quad U \\in L^{\\infty}(\\mathbb{T})\n\\]\n\\begin{equation}\n\\label{eq:3.11}\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(2)}, \\quad \\mathcal{A}^{(2)} \\in SL_{2}(L^{\\infty}(\\mathbb{T})).\n\\end{equation}\nSet\n\\begin{equation}\n\\label{eq:3.12}\n  \\begin{pmatrix}\n    g_{0} \\\\\n    g_{1}\n  \\end{pmatrix}\n  =\\mathcal{A}^{(2)}(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n\\end{equation}\n\\[\n  \\begin{cases}\n    A(z^{2})+zB(z^{2})=g_{0}+U(z^{2})g_{1} \\\\\n    C(z^{2})+zD(z^{2})=g_{1}\n  \\end{cases}\n\\]\n\\begin{equation}\n\\label{eq:3.13}\n  \\begin{cases}\n    A=S_{0}^{*}g_{0}+US_{0}^{*}g_{1} \\\\\n    B=S_{1}^{*}g_{0}+US_{1}^{*}g_{1} \\\\\n    C=S_{0}^{*}g_{1}, D=S_{1}^{*}g_{1}\n  \\end{cases}\n  \\quad S_{0}^{*}g_{0} = A-UC, \\quad S_{1}^{*}g_{0}=B-UD\n\\end{equation}\n\\[\n  g_{0}=S_{0}S_{0}^{*}g_{0}+S_{1}S_{1}^{*}g_{1}\n  =S_{0}(A-UC)+S_{1}(B-UD)\n\\]\n\\begin{equation}\n\\label{eq:3.14}\n  \\|g_{0}\\|^{2}=\\|A-UC\\|^{2}+\\|B-UD\\|^{2}\n\\end{equation}\nsuch that (\\ref{eq:3.11}) holds.\n\\[\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  \\longrightarrow\n  \\begin{pmatrix}\n    A-UC & B-UD \\\\\n    C & D\n  \\end{pmatrix},\n  \\quad S_{0}^{*}g_{0} = A-UC, \\quad S_{1}^{*}g_{0}=B-UD.\n\\]\nPick $U$ such that\n\\[\n  \\overline{C}(A-UC)+\\overline{D}(B-UD)=0\n\\]\n\\begin{equation}\n\\label{eq:3.15}\n  U=\\frac{\\overline{C}A+\\overline{D}B}{|C|^{2}+|D|^{2}}\n\\end{equation}\n\\begin{equation}\n\\label{eq:3.16}\n  \\mathcal{A}^{(2)}=\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\\n    C & D\n  \\end{pmatrix}\n\\end{equation}\nin (\\ref{eq:3.11}).\n\nIf\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  \\in SL_{2}(L^{\\infty}(\\mathbb{T}))\n\\]\nthen\n\\[\n  U=\\frac{\\overline{C}A+\\overline{D}B}{|C|^{2}+|D|^{2}} =0.\n\\]\nSee (\\ref{eq:3.15}) so the factorization\n\\[\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\mathcal{A}^{(2)}\n\\]\nin (\\ref{eq:3.11}) is then, $U=0 \\Rightarrow \\mathcal{A}=\\mathcal{A}^{(2)}$.\nThen following factorization results:\n\\[\n  \\mathcal{A}=(\\prod(lower)(upper))SL_{2}(L^{\\infty}(\\mathbb{T}))\n\\]\n\\begin{equation}\n\\label{eq:3.16-1}\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  \\underset{\\text{factor out lower matrix on the left}}{\\longrightarrow}\n  \\begin{pmatrix}\n    A & B \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\end{equation}\n\\[\n  \\underset{\\text{factor out upper matrix on the left}}{\\longrightarrow}\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}.\n\\]\nOr equivalently,\n\\begin{equation}\n\\label{eq:3.17}\n  \\begin{pmatrix}\n    A & B \\\\\n    C & D\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\\n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}.\n\\end{equation}\n\n\\begin{corollary}\n\\label{C:3.2}\nConsider $\\mathcal{A}\\in SL_{2}(L^{\\infty}(\\mathbb{T}))$, and the \nfactorization\n\\begin{equation}\n\\label{eq:3.18}\n  \\mathcal{A}=\n  \\begin{pmatrix}\n    1 & 0 \\\\  \n    L_{1} & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U_{1} \\\\  \n    0 & 1\n  \\end{pmatrix}\n  \\cdots \n  \\begin{pmatrix}\n    1 & 0 \\\\  \n    L_{p} & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U_{p} \\\\  \n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\  \n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n\\end{equation}\nresulting from an iteration of the algorithm from (\\ref{eq:3.17}).  Then \nthe last factor in (\\ref{eq:3.18}) is of diagonal form if and only if the \nfollowing hold:\nThere are functions $\\varphi, \\psi \\in L^{2}(\\mathbb{T})$ such that \n\\begin{equation}\n\\label{eq:3.19}\n  g_{0}(z)=\\varphi(z^{2}), \\quad \\text{and} \\quad f_{1}(z)=z\\psi(z^{2});\n\\end{equation}\nand, in this case, the last factor in (\\ref{eq:3.18}) is as follows:\n\\begin{equation}\n\\label{eq:3.20}\n  \\begin{pmatrix}\n    S_{0}^{*}g_{0} & S_{1}^{*}g_{0} \\\\  \n    S_{0}^{*}f_{1} & S_{1}^{*}f_{1}\n  \\end{pmatrix}\n=\n  \\begin{pmatrix}\n    \\varphi & 0 \\\\  \n    0 & \\psi\n  \\end{pmatrix}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nThis follows from (\\ref{eq:3.17}), and the Cuntz-relations:\n\\begin{equation}\n\\label{eq:3.21}\n  S_{i}^{*}S_{j}=\\delta_{i,j}, \\quad \\sum_{i}S_{i}S_{i}^{*}=I.\n\\end{equation}\n\n\n\n\\end{proof}\n\n\n\n\\subsection{Factorizations}\n\\label{sec:4.1.1}\n\nWe fix a value of $N > 1$ (i.e., the given number of frequency bands), and we \nbegin with the formula for a canonical system of $N$ isometries  $S_{i}$ which \ndefine an associated representation of the Cuntz algebra $O_{N}$.  Said \ndifferently: The system of isometries $\\{S_{i} \\}$ satisfies the Cuntz \nrelations with reference to the Hilbert space $L^{2}(\\mathbb{T})$ where \n$\\mathbb{T}$ is the circle group (one-torus) with its normalized invariant \nHaar measure. When the value of $N$ is fixed, then the multi-resolution \nfilters will then take the form of $N \\times N$ matrix functions; the matrix \nentries might be polynomials, or, more generally, functions from \n$L^{\\infty}(\\mathbb{T})$. Hence the questions about matrix factorization \ndepends on the context. In the case of polynomial entries we will make use of \ndegree, but this is not available for the more general case of entries from the \nalgebra $L^{\\infty}(\\mathbb{T})$. In every one of the settings, we develop \nfactorization algorithms, and the particular representation of the Cuntz \nalgebra will play an important role. \n\nThe standard representation of $O_{N}$, which we will use below, is given by \nthe system of isometries $\\{S_{j}\\}$ as follows:\n\n\\begin{equation}\n\\label{E:3.10}\n  (S_{j}\\varphi)(z)=f_{j}(z)\\varphi(z^{N}).\n\\end{equation}\n\\begin{lemma}\n\\label{L:3.10}\n\\cite{JoSo10} Let $N\\in \\mathbb{Z}_{+}$ be given and let $F=(f_{j})_{j\\in \\mathbb{Z}_{+}}$ be a function\nsystem.  Then $F \\in \\mathcal{O}\\mathcal{F}_{N}$ if and only if the operators\n$S_{j}$ \\ref{E:3.10}) satisfy\n\\begin{equation}\n\\label{eq:3.11}\n  S_{j}^{*}S_{k}=\\delta_{j,k}I\n\\end{equation}\n\\begin{equation}\n\\label{eq:3.12}\n  \\sum_{j \\in \\mathbb{Z}_{N}}S_{j}S_{j}^{*}=I,\n\\end{equation}\nwhere $I$ denotes the identity operator in $\\mathcal{H}=L^{2}(\\mathbb{T})$.\n\\end{lemma}\n\nWe say that the isometries $\\{ S_{j} \\}_{j \\in \\mathbb{Z}_{N}}$ define a\nrepresentation of the Cuntz-algebra $\\mathcal{O}_{N}$,\n$(S_{j}) \\in Rep(\\mathcal{O}_{N}, L^{2}(\\mathbb{T}))$.\n\n\\begin{lemma}\n\\label{L:4.1}\n\\cite{JoSo10} Let $N\\in\\mathbb{Z}_{+}$ be fixed, $N>1$, and let $A=(A_{j,k})$ be an $N \\times N$\nmatrix-function with $A_{j,k} \\in L^{2}(\\mathbb{T})$.  Then the following two\nconditions are equivalent:\n\\begin{enumerate} [(i)]\n\\item For $F=(f_{j})\\in \\mathcal{F}_{2}(N)$, we have $F(z)=A(z^{N})b(z)$.\n\\item $A_{i,j}=S_{j}^{*}f_{i}$ where the operators $S_{i}$ are from the\n  Cuntz-relations (\\ref{eq:3.11}, \\ref{eq:3.12}).\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(i) $\\Rightarrow$ (ii). Writing out the matrix-operation in (i), we get\n\\begin{equation}\n\\label{E:4.2}\n  f_{i}(z)=\\sum_{j}A_{i,j}(z^{N})z^{j}=\\sum_{j}(S_{j}A_{i,j})(z).\n\\end{equation}\nUsing $S_{j}^{*}S_{k}=\\delta_{j,k}I$, we get $A_{i,j}=S_{j}^{*}f_{i}$ which\nis (ii).\n\nConversely, assuming (ii) and using $\\sum_{j}S_{i}S_{j}^{*}=I$, we get\n$\\sum_{j}S_{j}A_{i,j}=f_{i}$ which is equivalent to (i) by the computation in\n(\\ref{E:4.2}) above.\n\\end{proof}\n\n\\begin{theorem}\n\\label{T:4.2.5}\n(Sweldens \\cite{SwRo91}, \\cite{JoSo10})\nLet $A \\in SL_{2}\\text{(pol)}$, then there are $l, p \\in \\mathbb{Z}_{+}$,\n$K \\in \\mathbb{C} \\setminus \\{0\\}$ and polynomial functions $U_{1}, \\ldots, U_{p}$,\n$L_{1}, \\ldots, L_{p}$ such that\n\\begin{equation}\n\\label{E:4.2.12}\n  A(z)=z^{l}\n  \\begin{pmatrix}\n    K & 0 \\\\\n    0 & K^{-1}\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U_{1}(z) \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L_{1}(z) & 1\n  \\end{pmatrix}\n  \\cdots\n  \\begin{pmatrix}\n    1 & U_{p}(z) \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L_{p}(z) & 1\n  \\end{pmatrix}.\n\\end{equation}\n\\end{theorem}\n\nThe filter algorithm corresponding to the matrix-factorization in \n(\\ref{E:4.2.12}) is as follows:\nAnd in steps:\n\n\\begin{remark}\n\\label{R:4.2.6}\n\\cite{JoSo10} Note that if\n\\[\n  \\begin{pmatrix}\n    \\alpha & \\beta \\\\\n    \\gamma & \\delta\n  \\end{pmatrix}\n  \\in SL_{2}\\text{(pol)},\n\\]\nthen one of the two functions $\\alpha(z)$ or $\\delta(z)$ must be a monomial.\n\\end{remark}\n\n\n\\subsection{The $2 \\times 2$ case: Polynomials}\n\\label{sec:5.1}\n\\cite{JoSo10} To highlight the general ideas, we begin with some details worked out in the\n$2 \\times 2$ case; see equation (\\ref{eq:3.16-1}).\n\nTo get finite algorithms, we should assume in the present subsection that the\nmatrix-entries are polynomials.\n\nFirst note that from the setting in Theorem \\ref{T:4.2.5}, we may assume\nthat matrix entries have the form $f_{H}(z)$ as in section \\ref{sec:2} but\nwith $H \\subset \\{0,1,2, \\cdots\\}$, i.e., $f_{H}(z)=a_{0}+a_{1}z+ \\cdots$.\nThis facilitates our use of the Euclidean algorithm.\n\nSpecifically, if $f$ and $g$ are polynomials (i.e.,\n$H \\subset \\{0,1,2, \\cdots\\})$ and if deg$(g)\\leq$ deg$(f)$, the Euclidean\nalgorithm yields\n\\begin{equation}\n\\label{E:5.1.1}\n  f(z)=g(z)q(z)+r(z)\n\\end{equation}\nwith deg$(r) < $ deg$(g)$.  We shall write\n\\begin{equation}\n\\label{E:5.1.2}\n  q=quot(g,f), \\quad \\text{and} \\quad r=rem(g,f).\n\\end{equation}\n\nSince\n\\begin{equation}\n\\label{E:5.1.2}\n  \\begin{pmatrix}\n    K & 0 \\\\\n    0 & K^{-1}\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & K^{2}U \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    K & 0 \\\\\n    0 & K^{-1}\n  \\end{pmatrix},\n\\end{equation}\nwe may assume that the factor\n\\[\n  \\begin{pmatrix}\n    K & 0 \\\\\n    0 & K^{-1}\n  \\end{pmatrix}\n\\]\nfrom the equation (\\ref{E:5.1.2}) factorization occurs on the rightmost place.\n\n\\begin{equation}\n\\label{E:4.2.9}\n  F=U_{N}[b],\n\\end{equation}\nwhere $U$ is a unitary matrix-function, where\n\\[\n  b=\n  \\begin{pmatrix}\n    1 \\\\\n    z \\\\\n    z^{2} \\\\\n    \\vdots \\\\\n    z^{N-1}\n  \\end{pmatrix}\n\\]\nand where $U_{N}[b](z)=U(z^{N})b(z)$.\n\nLet $U$ represent scalar valued matrix entry in a matrix function.\nWe now proceed to determine the polynomials $U_{1}(z), L_{1}(z), \\cdots$,\netc. inductively starting with\n\\[\n  A=\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  B,\n\\]\nwhere $U$ and $B$ are to be determined.  Introducing \\ref{E:4.2.9}), this\nreads\n\\begin{equation}\n\\label{E:5.1.3}\n  A(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & U(z^{2}) \\\\\n    0 & 1\n  \\end{pmatrix}\n  B(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & U(z^{2}) \\\\\n    0 & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    h(z) \\\\\n    k(z)\n  \\end{pmatrix}.\n\\end{equation}\nBut the matrix function\n\\[\n  A=\n  \\begin{pmatrix}\n    \\alpha & \\beta \\\\\n    \\gamma & \\delta\n  \\end{pmatrix}\n\\]\nis given and fixed see Remark \\ref{R:4.2.6}.  Hence\n\\begin{equation}\n\\label{E:5.1.4}\n  \\gamma(z^{2})+\\delta(z^{2})z=k(z)\n\\end{equation}\nis also fixed.  The two polynomials to be determined are $u$ and $h$ in\n(\\ref{E:5.1.3}).  Carrying out the matrix product in (\\ref{E:5.1.3}) yields:\n\\[\n  \\alpha(z^{2})+\\beta(z^{2})z=h(z)+u(z^{2})k(z)\n  = h_{0}(z)+h_{1}(z^{2})z+u(z^{2})\\{\\gamma(z^{2})+\\delta(z^{2})z\\}\n\\]\nwhere we used the orthogonal splitting\n\\begin{equation}\n\\label{E:5.1.5}\n  L^{2}(\\mathbb{T})=S_{0}S_{0}^{*}L^{2}(\\mathbb{T})\\oplus\n  S_{1}S_{1}^{*}L^{2}(\\mathbb{T})\n\\end{equation}\nfrom Lemma \\ref{L:3.10}.  Similarly, from (\\ref{E:5.1.4}), we get\n\\[\n  \\gamma(z^{2})+\\delta(z^{2})z=k_{0}(z^{2})+k_{1}(z^{2})z;\n\\]\nand therefore $\\gamma = k_{0}$ and $\\delta=k_{1}$, by Lemma \\ref{L:4.1}.\n\nCollecting terms and using the orthogonal splitting (\\ref{E:5.1.5}) we arrive\nat the following system of polynomial equations:\n\\begin{equation}\n\\label{E:5.1.6}\n\\begin{cases}\n  \\alpha = h_{0} + u\\gamma \\\\\n  \\beta = h_{1} + u\\delta ;\n\\end{cases}\n\\end{equation}\nor more precisely,\n\\[\n\\begin{cases}\n  \\alpha(z) = h_{0}(z) + u(z)\\gamma(z) \\\\\n  \\beta(z) = h_{1}(z) + u(z)\\delta(z).\n\\end{cases}\n\\]\nIt follows that the two functions $u$ and $h$ may be determined from the\nEuclidean algorithm.  With (\\ref{E:5.1.2}), we get\n\\begin{equation}\n\\label{E:5.1.7}\n\\begin{cases}\n  u=quot(\\gamma, \\alpha) \\\\\n  h_{0}=rem(\\gamma, \\alpha) \\\\\n  h_{1}=rem(\\delta, \\beta).\n\\end{cases}\n\\end{equation}\n\n\\begin{remark}\n\\label{R:5.1.1}\n\\cite{JoSo10}\nThe relevance of the determinant condition we have from Theorem \\ref{T:4.2.5}\nis as follows:\n\\[\n  detA=\\alpha\\delta-\\beta\\gamma \\equiv 1.\n\\]\nSubstitution of (\\ref{E:5.1.6}) into this yields:\n\\[\n  h_{0}\\delta - h_{1}\\gamma \\equiv 1.\n\\]\n\nSolutions to (\\ref{E:5.1.6}) are possible because the two polynomials\n$\\delta(z)$ and $\\gamma(z)$ are mutually prime.  The derived matrix\n\\[\n  \\begin{pmatrix}\n    h_{0} & h_{1} \\\\\n    \\gamma & \\delta\n  \\end{pmatrix}\n\\]\nis obtained from $A$ via a row-operation in the ring of polynomials.\n\nFor the inductive step, it is important to note:\n\\begin{equation}\n\\label{E:5.1.8}\n  deg(h_{0}) < deg(\\gamma), \\quad \\text{and} \\quad\n  deg(h_{1}) < deg(\\delta).\n\\end{equation}\nThe next step, continuing from (\\ref{E:5.1.3}) is the determination of a\nmatrix-function $C$ and three polynomials $p, q,$ and $L$ such that\n\\begin{equation}\n\\label{E:5.1.9}\n  \\begin{pmatrix}\n    1 & -U \\\\\n    0 & 1\n  \\end{pmatrix}\n  A=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  C\n\\end{equation}\nand\n\\begin{equation}\n\\label{E:5.1.10}\n  \\begin{pmatrix}\n    1 & -U(z^{2}) \\\\\n    0 & 1\n  \\end{pmatrix}\n  A(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L(z^{2}) & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    p(z) \\\\\n    q(z)\n  \\end{pmatrix}.\n\\end{equation}\nHere\n\\[\n  \\begin{pmatrix}\n    p \\\\\n    q\n  \\end{pmatrix}\n  =C(z^{2})\n  \\begin{pmatrix}\n    1 \\\\\n    z\n  \\end{pmatrix}.\n\\]\nThe reader will notice that in this step, everything is as before with the\nonly difference that now\n\\[\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n\\]\nis lower diagonal in contrast with\n\\[\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n\\]\nin the previous step.\n\nThis time, the determination of the polynomial $p$ in (\\ref{E:5.1.10}) is\nautomatic.  With\n\\[\n  p(z)=p_{0}(z^{2})+zp_{1}(z^{2})\n\\]\n(see (\\ref{E:5.1.5})) and we get the following system:\n\\[\n  \\begin{cases}\n    p_{0}=\\alpha - u\\gamma = h_{0} \\\\\n    p_{1}=\\beta - u\\delta = h_{1} ; \\quad \\text{and}\n  \\end{cases}\n\\]\n\\[\n  \\begin{cases}\n    \\gamma=L(\\alpha-u\\gamma)+q_{0}=Lh_{0}+q_{0} \\\\\n    \\delta=L(\\beta-u\\delta)+q_{1}=Lh_{1}+q_{1} \\\\\n  \\end{cases}.\n\\]\nSo the determination of $L(z)$ and $q(z)=q_{0}(z^{2})+zq_{1}(z^{2})$ may be\ndone with Euclid:\n\\begin{equation}\n\\label{E:5.1.11}\n  \\begin{cases}\n    L= quot(\\alpha-u\\gamma, \\gamma)=quot(h_{0}, \\gamma) \\\\\n    q_{0}= rem(\\alpha-u\\gamma, \\gamma)=rem(h_{0}, \\gamma) \\\\\n    q_{1}= rem(\\beta-u\\delta, \\delta)=rem(h_{1}, \\delta).\n  \\end{cases}\n\\end{equation}\n\nCombining the two steps, the comparison of degrees is as follows:\n\\begin{equation}\n\\label{E:5.1.12}\n  \\begin{cases}\n    deg(q_{0}) < deg(h_{0}) < deg(\\gamma) \\\\\n    deg(q_{1}) < deg(h_{1}) < deg(\\delta)\n  \\end{cases}.\n\\end{equation}\nTwo conclusions now follow:\n\\begin{enumerate} [(i)]\n  \\item the procedure may continure by recursion;\n  \\item the procedure must terminate.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{remark}\n\\label{R:5.1.2}\nIn order to start the algorithm in (\\ref{E:5.1.7}) with direct reference to\nEuclid, we must have\n\\begin{equation}\n\\label{E:5.1.13}\n  deg(\\gamma) \\leq deg(\\alpha)\n\\end{equation}\nwhere\n\\[\n  A=\n  \\begin{pmatrix}\n    \\alpha & \\beta \\\\\n    \\gamma & \\delta\n  \\end{pmatrix}\n\\]\nis the initial $2 \\times 2$ matrix-function.\n\nNow, suppose (\\ref{E:5.1.13}), i.e., that\n\\[\n  deg(\\gamma) > deg(\\alpha).\n\\]\nThen determine a polynomial $L$ such that\n\\begin{equation}\n\\label{E:5.1.14}\n  deg(\\gamma-L\\alpha) \\leq deg(\\alpha).\n\\end{equation}\nWe may then start the procedure (\\ref{E:5.1.7}) on the matrix function\n\\[\n  \\begin{pmatrix}\n    \\alpha & \\beta \\\\\n    \\gamma -L\\alpha & \\delta\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    -L & 1\n  \\end{pmatrix}\n  A.\n\\]\nIf a polynomial $U$ and a matrix function $B$ is then found for\n\\[\n  \\begin{pmatrix}\n    \\alpha & \\beta \\\\\n    \\gamma -L\\alpha & \\delta\n  \\end{pmatrix}\n\\]\nthen the factorization\n\\[\n  A=\n  \\begin{pmatrix}\n    1 & 0 \\\\\n    L & 1\n  \\end{pmatrix}\n  \\begin{pmatrix}\n    1 & U \\\\\n    0 & 1\n  \\end{pmatrix}\n  B\n\\]\nholds; and the recursion will then work as outlined.\n\nIn the following, starting with a matrix-function $A$, we will always assume\nthat the degrees of the polynomials $(A_{i,j})_{i,j\\in \\mathbb{Z}_{N}}$ have\nbeen adjusted this way, so the direct Euclidean algorithm can be applied.\n\\end{remark}\n\n\\subsection{The $3 \\times 3$ case}\n\\label{sec:5.2}\nThe thrust of this section is the assertion that Theorem \\ref{T:4.2.5} holds\nwith small modifications in the $3 \\times 3$ case.\n\n\\subsubsection{Comments:}\nIn the definition of $A \\in SL_{3}(\\text{pol})$, it is understood that $A(z)$\nhas $detA(z)\\equiv 1$ and that the entries of the inverse matrix $A(z)^{-1}$\nare again polynomials.\n\nNote that if $L, M, U$ and $V$ are polynomials, then the four matrices\n\\begin{equation}\n\\label{E:5.2.1}\n  \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    L & 1 & 0 \\\\\n    0 & M & 1\n  \\end{pmatrix},\n  \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    0 & 1 & 0 \\\\\n    L & 0 & 1\n  \\end{pmatrix},\n  \\begin{pmatrix}\n    1 & U & 0 \\\\\n    0 & 1 & V \\\\\n    0 & 0 & 1\n  \\end{pmatrix}\n  \\quad \\text{and} \\quad\n  \\begin{pmatrix}\n    1 & 0 & U \\\\\n    0 & 1 & 0 \\\\\n    0 & 0 & 1\n  \\end{pmatrix}\n\\end{equation}\nare in $SL_{3}(\\text{pol})$ since\n\n\\begin{equation}\n\\label{E:5.2.2}\n  \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    L & 1 & 0 \\\\\n    0 & M & 1\n  \\end{pmatrix}^{-1}\n  =\n  \\begin{pmatrix}\n    1 & 0 & 0 \\\\\n    -L & 1 & 0 \\\\\n    LM & -M & 1\n  \\end{pmatrix} \\quad \\text{and}\n\\end{equation}\n\n\\begin{equation}\n\\label{E:5.2.3}\n  \\begin{pmatrix}\n    1 & U & 0 \\\\\n    0 & 1 & V \\\\\n    0 & 0 & 1\n  \\end{pmatrix}^{-1}\n  =\n  \\begin{pmatrix}\n    1 & -U & UV \\\\\n    0 & 1 & -V \\\\\n    0 & 0 & 1\n  \\end{pmatrix}.\n\\end{equation}\n\n\\begin{theorem}\n\\label{T:5.2.1}\n\\cite{JoSo10}\nLet $A \\in SL_{3}(\\text{pol})$; then the conclusion in Theorem \\ref{T:4.2.5}\ncarries over with the modification that the alternating upper and lower\ntriangular matrix-functions now have the form (\\ref{E:5.2.1}) or\n(\\ref{E:5.2.2})-(\\ref{E:5.2.3}) where the functions $L_{j}, M_{j}, U_{j}$ and\n$V_{j}$, $j=1,2, \\cdots$ are polynomials.\n\\end{theorem}\n\n\\subsection{The $N \\times N$ case}\n\\label{sec:5.3}\n\nBelow we outline the modifications to our algorithms from the \n$2 \\times 2$ case needed in order to deal with filters with $N (> 2)$ bands, \nhence factorization of $N \\times N$ matrix functions. The main difference when \nthe number of frequency bands $N$ is more than $2$ is that in our \nfactorizations, both the lower and the upper triangular factors, must take \ninto account operations which cross between any pair of the total system of \n$N$ frequency bands.\n\n\\begin{theorem}\n\\label{T:5.3.1}\n\\cite{JoSo10}\nLet $N \\in \\mathbb{Z}_{+}$, $N>1$, be given and fixed.  Let\n$A \\in SL_{N}(\\text{pol})$; then the conclusions in Theorem \\ref{T:4.2.5} carry\nover with the modification that the alternative factors in the product are\nupper and lower triangular matrix-functions in $SL_{N}(\\text{pol})$.  We may\ntake the lower triangular matrix-factors\n$\\mathcal{L}=(L_{i,j})_{i,j \\in \\mathbb{Z}_{N}}$ of the form\n\\[\n  \\begin{pmatrix}\n  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n  0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n  L_{p} & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n  0     & L_{p+1} & 0  & 1 & 0 & 0 & 0 & 0 \\\\\n  0 & 0 & . & 0  &1 & 0 & 0 & 0 \\\\\n  0 & 0 & 0 & . & 0 & 1 & 0 & 0 \\\\\n  0 & 0 & 0 & 0 & . & 0 & 1 & 0  \\\\\n  0 & 0 & 0 & 0 & 0 & L_{N-1} & 0 & 1\n  \\end{pmatrix}\n\\]\npolynomial entries\n\\begin{equation}\n\\label{E:5.3.1}\n  \\begin{cases}\n  L_{i,i} \\equiv 1, \\\\\n  L_{i,j}(z)=\\delta_{i-j, p}L_{i}(z);\n  \\end{cases}\n\\end{equation}\nand the upper triangular factors of the form\n$\\mathcal{U}=(U_{i,j})_{i,j\\in \\mathbb{Z}_{N}}$ with\n\\begin{equation}\n\\label{E:5.3.2}\n  \\begin{cases}\n  U_{i,i} \\equiv 1, \\\\\n  L_{i,j}(z)=\\delta_{i-j, p}U_{i}(z).\n  \\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\\textbf{Notation.}  Let $U_{1}, \\cdots, U_{N}$, $L_{1}, \\cdots, L_{N}$ be\npolynomials and set\n\\begin{equation}\n\\label{E:5.3.3}\n  \\mathcal{U}_{N}(U)=\n  \\begin{pmatrix}\n  1 & U_{1} & 0 & 0 & 0 & 0 & 0  \\\\\n  0 & 1 & U_{2} & 0 & 0 & 0 & 0  \\\\\n  0 & 0 & 1 & . & 0 & 0 & 0  \\\\\n  0 & 0 & 0 & 1 & . & 0 & 0  \\\\\n  0 & 0 & 0 & 0 & 1 & . & 0  \\\\\n  0 & 0 & 0 & 0 & 0 & 1 & U_{N-1}  \\\\\n  0 & 0 & 0 & 0 & 0 & 0 & 1\n  \\end{pmatrix}\n\\end{equation}\n\n\\begin{equation}\n\\label{E:5.3.4}\n  \\mathcal{L}_{N}(L)=\n  \\begin{pmatrix}\n  1 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n  L_{1} & 1 & 0 & 0 & 0 & 0 & 0  \\\\\n  0 & L_{2} & 1 & 0 & 0 & 0 & 0  \\\\\n  0 & 0 & . & 1 & 0 & 0 & 0  \\\\\n  0 & 0 & 0 & . & 1 & 0 & 0  \\\\\n  0 & 0 & 0 & 0 & . & 1 & 0  \\\\\n  0 & 0 & 0 & 0 & 0 & L_{N-1} & 1\n  \\end{pmatrix}\n\\end{equation}\n\nNote that both are in $SL_{N}(\\text{pol})$; and we have\n\\[\n  \\mathcal{U}_{N}(U)^{-1}=\\mathcal{U}_{N}(-U) \\quad \\text{and}\n\\]\n\\[\n  \\mathcal{L}_{N}(L)^{-1}=\\mathcal{L}_{N}(-L).\n\\]\n\n\\textbf{Step 1:} Starting with $A=(A_{i,j})\\in SL_{N}(\\text{pol})$.  Then\nleft-multiply with a suitably chosen $\\mathcal{U}_{N}(-U)$ such that the\ndegrees in the first column of $\\mathcal{U}_{N}(-U)A$ decrease, i.e.,\n\\begin{equation}\n\\label{E:5.3.5}\n  deg(A_{0,0}) \\leq deg(A_{1,0}-u_{2}A_{1,0}) \\leq \\cdots deg(A_{N-1,0}).\n\\end{equation}\nIn the following, we shall use the same letter $A$ for the modified\nmatrix-function.\n\n\\textbf{Step 2:} Determine a system of polynomials $L_{1}, \\cdots, L_{N-1}$\nand a polynomial vector-function\n\\[\n  \\begin{bmatrix}\n    f_{0} \\\\\n    f_{1} \\\\\n    \\ldots \\\\\n    f_{N-1}\n  \\end{bmatrix}\n\\]\nsuch that\n\\begin{equation}\n\\label{E:5.3.6}\n  A_{N}\n  \\begin{bmatrix}\n    1 \\\\\n    z \\\\\n    z^{2} \\\\\n    \\ldots \\\\\n    z^{N-1}\n  \\end{bmatrix}\n  = \\mathcal{L}_{N}(L)_{N}\n  \\begin{bmatrix}\n    f_{0} \\\\\n    f_{1} \\\\\n    \\ldots \\\\\n    f_{N-1}\n  \\end{bmatrix},\n\\end{equation}\nor equivalently\n\\[\n  \\sum_{j=0}^{N-1}A_{i,j}(z^{N})z^{j}=\n  \\begin{cases}\n    f_{0}(z) &\\text{ if $i=0$} \\\\\n    L_{i}(z^{N})f_{i-1}(z)+f_{i}(z) &\\text{ if $i>0$}\n  \\end{cases}.\n\\]\n\n\\textbf{Step 3:} Apply the operators $S_{j}$ and $S_{j}^{*}$ from section\n\\ref{sec:2} to both sides in (\\ref{E:5.3.6}).  First (\\ref{E:5.3.6}) takes\nthe form:\n\\[\n  \\sum_{j=0}^{N-1}S_{j}A_{i,j}=\n  \\begin{cases}\n    f_{0} &\\text{ if $i=0$} \\\\\n    S_{f_{i-1}}L_{i}+f_{i} &\\text{ if $i>0$}\n  \\end{cases}.\n\\]\nFor $i=1$, we get\n\\begin{equation}\n\\label{E:5.3.7}\n  A_{1,j}=L_{1}A_{0,j}+k_{j} \\quad \\text{where} \\quad k_{j}=S_{j}^{*}f_{1}.\n\\end{equation}\n\nBy (\\ref{E:5.3.5}) and the assumptions on the matrix-functions, we note that\nthe system (\\ref{E:5.3.7}) may now be solved with the Euclidean algorithm:\n\\begin{equation}\n\\label{E:5.3.8}\n  \\begin{cases}\n    L_{1}=quot(A_{0,j}, A_{1,j}) \\\\\n    k_{j}=rem(A_{0,j}, A_{1,j})\n  \\end{cases}\n\\end{equation}\nwith the same polynomial $L_{1}$ for $j=0,1, \\cdots, N-1$.\n\nFor the polynomial function $f_{1}$ we then have\n\\begin{equation}\n\\label{E:5.3.9}\n  f_{1}=\\sum_{j=0}^{N-1}S_{j}k_{j};\n\\end{equation}\ni.e.\n\\[\n  f_{1}(z)=k_{0}(z^{N})+k_{1}(z^{N})z+\\cdots+k_{N-1}(z^{N-1})z^{N-1}.\n\\]\n\nThe process now continues recursively until all the functions\n$L_{1}, L_{2}, \\cdots, f_{1}, f_{2}, \\cdots$ have been determined.\n\n\\textbf{Step 4:} The formula (\\ref{E:5.3.6}) translates into a\nmatrix-factorizations as follows: With $L$ and $F$ determined in\n(\\ref{E:5.3.6}), we get\n\\begin{equation}\n\\label{E:5.3.10}\n  A=\\mathcal{L}_{N}(L)B\n\\end{equation}\nas a simple matrix-product taking $B=(B_{i,j})$ and\n\\begin{equation}\n\\label{E:5.3.11}\n  B_{i,j}=S_{j}^{*}f_{i},\n\\end{equation}\nwhere we used Lemmas \\ref{L:3.10} and \\ref{L:4.1}.\n\n\\textbf{Step 5:} The process now continues with the polynomial matrix-function\nfrom (\\ref{E:5.3.10}) and (\\ref{E:5.3.11}).  We determine polynomials\n$U_{1}, \\cdots, U_{N-1}$ and a third matrix function\n\\[\n  C=(C(z))=(C_{i,j}(z)) \\quad \\text{such that} \\quad B=\\mathcal{U}_{N}(U)C.\n\\]\n\n\\textbf{Step 6:} As each step of the process we alternate $L$ and $U$; and\nat each step, the degrees of the matrix-functions is decreased.  Hence the\nrecursion must terminate as stated in Theorem \\ref{T:5.3.1}.\n\\end{proof}\n\n\n\\subsection{$L^{\\infty}(\\mathbb{T})$-matrix entries.}\n\\label{sec:3.1}\nWhile the case $N=2$ is motivated by application to the high-pass v.s.\nlow-pass filters, may result for the $N=2$ case carry over.  To see this, we\nfirst define the Cuntz-algebra $\\mathcal{O}_{N}$ in general the relations are\n\\begin{equation}\n\\label{eq:3.1.1}\n  S_{i}^{*}S_{j}=\\delta_{i, j}I, \\quad \\sum_{i}S_{i}S_{i}^{*}=I,\n\\end{equation}\nwhen the elements $(S_{i})_{i=0}^{N-1}$ are given symmetrically.\n\nEach case (\\ref{eq:3.1.1}) has many representations; for example if\n$(m_{i}(z))_{i=0}^{N-1}$, $z \\in \\mathbb{T}$, is a system of filters\ncorresponding to $N$ frequency bands, we may obtain a representation of\n$\\mathcal{O}_{N}$ acting on the Hilbert space $L^{2}(\\mathbb{T})$ as\nfollows\n\\begin{equation}\n\\label{eq:3.1.2}\n  (S_{i}\\psi)(z)=m_{i}(z)\\psi(z^{N}), \\quad \\forall z \\in \\mathbb{T},\n  \\quad \\psi \\in L^{2}(\\mathbb{T}).\n\\end{equation}\n\nFor $i \\in \\{0, 1, \\cdots, N-1 \\}$, the adjoint operator of $S_{i}$ in\n(\\ref{eq:3.1.2}) is\n\\begin{equation}\n\\label{eq:3.1.3}\n  (S_{i}^{*}\\psi)(z)=\\frac{1}{N}\\sum_{w^{N}=z}\\overline{m_{i}(z)}\\psi(z^{N}),\n  \\quad z \\in \\mathbb{T}.\n\\end{equation}\nA direct verification shows that the Cuntz-relation (\\ref{eq:3.1.1}) are\nsatisfied for the operators $(S_{i})_{i=0}^{N-1}$ in (\\ref{eq:3.1.2}) if\nand only if the system $(m_{i})_{i=0}^{N-1}$ is a multi-band filter\ncovering the $N$ frequency bands.\n\nThe simplest example of the representation in (\\ref{eq:3.1.2}) is the case\nwhere $m_{i}(z)=z^{i}$, $i=0,1, \\cdots, N-1$; and so\n\\begin{equation}\n\\label{eq:3.1.4}\n  (S_{i}\\psi)(z)=z^{i}\\psi(z^{N}), \\quad i=0,1, \\cdots, N-1,\n  \\quad z \\in \\mathbb{T}, \\quad \\psi \\in L^{2}(\\mathbb{T}).\n\\end{equation}\n\n\\begin{theorem}\n\\label{T:3.1.1}\nLet $g=(g_{ij})_{i,j=0}^{N-1}\\in SL_{N}(L^{\\infty}(\\mathbb{T})$, i.e.,\n$g_{ij}(\\cdot) \\in L^{\\infty}(\\mathbb{T}))$, and\n\\begin{equation}\n\\label{eq:3.1.5}\n  det g(\\cdot)\\equiv 1 \\quad \\text{ on $\\mathbb{T}$}\n\\end{equation}\nthen for every factorization\n\\begin{equation}\n\\label{eq:3.1.6}\n  g(z)=\n  \\begin{pmatrix}\n    1 & 0 & 0 & \\cdots & 0 \\\\\n    L_{1}(z) & 1 & 0 & \\cdots & 0 \\\\\n    L_{2}(z) & 0 & 1 & \\ddots & 0 \\\\\n    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n    \\vdots & \\vdots & \\vdots & \\ddots &\\vdots \\\\\n    L_{N-1}(z) & 0 & \\cdots & 0 & 1\n  \\end{pmatrix}\n  g^{(new)}(z) \\quad \\text{(matrix product)}\n\\end{equation}\nthere is a unique $f_{i} \\in L^{\\infty}(\\mathbb{T})$ such that\n\\begin{equation}\n\\label{eq:3.1.7}\n  g_{0,j}^{(new)}(z)=g_{0,j}(z), \\quad \\text{and}\n\\end{equation}\n\\begin{equation}\n\\label{eq:3.1.8}\n  g_{i,j}^{(new)}(z)=S_{j}^{*}f_{i}, \\quad \\text{for $i=1,2, \\cdots, N-1$}\n\\end{equation}\nwhere $\\{S_{j}\\}_{j=0}^{N-1}$ is the system of Cuntz-isometries from\n(\\ref{eq:3.1.4}).\n\\end{theorem}\n\\begin{proof}\nWith the arguments above, in the space $\\mathcal{O}_{N}$ of $N=2$, we now\nget matrix, the system:\n\\begin{equation}\n\\label{eq:3.1.9}\n  g^{(new)}(z^{N})\n  \\begin{pmatrix}\n  1 \\\\\n  z \\\\\n  z^{2} \\\\\n  \\vdots \\\\\n  z^{N-1}\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n  f_{0}(z) \\\\\n  f_{1}(z) \\\\\n  f_{2}(z) \\\\\n  \\vdots \\\\\n  f_{N-1}(z)\n  \\end{pmatrix},\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:3.1.10}\n  \\begin{cases}\n    S_{j}^{*}f_{0}=g_{0,j}, \\\\\n    L_{i}g_{0,j}+S_{j}^{*}f_{i}=g_{i,j},\n  \\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:3.1.11}\n  f_{i}=\\sum_{j=0}^{N-1}S_{j}S_{j}^{*}f_{i}\n  =\\sum_{j=0}^{N-1}S_{j}(g_{i,j}-L_{i}g_{0,j})\n\\end{equation}\nfor $i=1,2, \\cdots, N-1$, which is desired conclusion.\n\\end{proof}\n\n\\subsection{Optimal factorization in the case of\n$SL_{N}(L^{\\infty}(\\mathbb{T}))$}\n\\label{sec:3.2}\nFix $N>2$, and consider the usual inner product in $\\mathbb{C}^{N}$,\n\\begin{equation}\n\\label{eq:3.2.1}\n  \\langle z, w \\rangle:=\\sum_{j=0}^{N-1}\\overline{z_{j}}w_{j},\n\\end{equation}\ndefined for all $z=(z_{0}, \\cdots, z_{N-1})$, and $w=(w_{0}, \\cdots, w_{N-1})$.\n\nFor $g=(g_{ij}(z))_{i,j=0}^{N-1} \\in SL_{N}(L^{\\infty}(\\mathbb{T}))$, set\n\\[\n  \\tilde{g_{0}}(z)=(g_{0j}(z))_{j=0}^{N-1}, \\quad a.e.,\n\\]\nthe first row in the matrix-function\n$\\mathbb{T} \\ni Z \\mapsto (g(z)) \\in SL_{N}(L^{\\infty}(\\mathbb{T}))$.  Let\n$P(z)=P^{(g)}(z)$ denote the projection of $\\mathbb{C}^{N}$ onto the\none-dimensional subspace generated by $\\tilde{g_{0}}(z) \\in \\mathbb{C}^{N}$.\n\nNote that $(P(z))_{z \\in \\mathbb{T}}$ is a field of orthogonal rank-$2$\nprojection in $\\mathbb{C}^{N}$.  Setting\n\\begin{equation}\n\\label{eq:3.2.2}\n  \\|\\tilde{g_{0}}(z)\\|_{2}^{2}=\\sum_{j=0}^{N-1}|g_{0,j}(z)|^{2},\n\\end{equation}\nwe have:\n\\begin{equation}\n\\label{eq:3.2.3}\n  P(z)\\xi=\\sum_{j=0}^{N-1}\\frac{\\overline{g_{0,j}(z)}\\xi_{j}}\n  {\\|\\tilde{g_{0}}(z)\\|_{2}^{2}}\\text{ }g_{0,j}(z)\n  \\quad \\text{for all } \\xi=(\\xi{0}, \\cdots, \\xi_{N-1}) \\in \\mathbb{C}^{N};\n\\end{equation}\nand set\n\\begin{equation}\n\\label{eq:3.2.4}\n  \\tilde{g_{j}}^{(new)}(z)=\\tilde{g_{0}}(z)-P(z)\\tilde{g_{j}}(z).\n\\end{equation}\n\n\\begin{corollary}\n\\label{C:3.2.1}\n\\begin{enumerate} [(i)]\n  \\item For the factorization (\\ref{eq:3.1.6}) in Theorem \\ref{T:3.1.1}, the\noptimal choice is that given by the matrix-factor $f^{(new)}$ having as rows\nthe vector fields $\\tilde{g_{i}}^{(new)}(z)$ specified in (\\ref{eq:3.2.4}).\n  \\item With the resolution of row-vector fields,\n\\begin{equation}\n\\label{eq:3.2.5}\n  \\tilde{g_{j}}^{(new)}(z)=(S_{0}^{*}f_{i}, S_{1}^{*}f_{i}, \\cdots,\n  S_{N-1}^{*}f_{i})\n\\end{equation}\nfrom (\\ref{eq:3.1.8}), the optimal solution is attained; and it is the unique\nminimizer for the following system of optimization problems:\n\\begin{equation}\n\\label{eq:3.2.6}\n  min_{f_{i}\\in L^{2}(\\mathbb{T})}\\|f_{i}\\|_{L^{2}(\\mathbb{T}}^{2}, \\quad\n  1 \\leq i < N,\n\\end{equation}\nwhere each choice $(f_{i})_{i=1}^{N-1}$ yields a matrix-factor\n$\\mathcal{A}^{(new)}$ via (\\ref{eq:3.2.5}).\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n  The proof of the conclusions in (i)-(ii) in the corollary follows from the\narguments in the proof of Theorem \\ref{T:3.1.1} above.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsubsection*{Acknowledgments}\nWe thank Mitsuru Kusumoto and Kohei Hayashi for their helpful comments.\nWe also thank Prof. Wayne Luk at Imperial College London for his overall support for Ruizhe Zhao's PhD study and hardware resources.\n\n\n\n\\section{Adaptive Loss Scaling}\n\n\\input{tex\/method\/overview.tex}\n\\input{tex\/method\/loss_scaled_backprop.tex}\n\\input{tex\/method\/loss_scale_calc.tex}\n\\subsection{Preliminary}\\label{sec:background:preliminary}\n\nAs mentioned above, MPT~\\citep{Micikevicius2017} uses FP16 for storing the activations and gradients and for the most compute-intensive tasks, while FP32 is used only where increased precision is required.\nFP16 has three fewer exponent bits than FP32, limiting its dynamic range to magnitudes between $u_{min}=2^{-24}$ and $u_{max}=65505$. In practice, the gradients often have a larger range than this, resulting in numerical issues when using MPT.\nIn FP16, if the absolute actual value of a gradient $|g|$ is smaller than $u_{min}$, it will become 0; and if it is larger than $u_{max}$, it will be infinite.\nAlso, even if a value is in $[u_{min}, u_{max})$, the closer it comes to either bound, the less accurate its FP16 form is regarding absolute rounding error, e.g., 1024.1 is rounded to 1024.\nUnderflow motivates loss scaling, while the overflow and rounding error are what loss scaling should be careful with.\n\n\\input{tex\/background\/backprop_alg.tex}\n\n\\Figref{fig:alg:two_backprop} shows the basic loss scaling algorithm (\\Algref{alg:loss_scaled}) compared to standard backpropagation without loss scaling (\\Algref{alg:std_backprop}). Note that they differ only in that in \\Algref{alg:loss_scaled}, the initial error gradients $\\delta_{N+1}$ from the output layer are scaled by $\\alpha$ before the start of the backward pass, and that the weight gradient update \\texttt{GradW} is then unscaled by the same $\\alpha$ just before the weight update. Recall that $\\alpha$ should be chosen large enough to prevent underflow issues from affecting training, while also being small enough to prevent overflow.\nEven when kept within this range, using a larger value than necessary could introduce more absolute rounding error as aforementioned.\nAlso, since loss scaling amplifies the ratio between the largest and the smallest elements within each gradient, \\textbf{swamping}~\\citep{Higham1993}, the phenomenon that summing small values with larger ones is inaccurate in floating-point, becomes more likely and may hinder training~\\citep{Wang2018b}.\n\n\\subsection{Related Work}\\label{sec:background:related_work}\n\nMany recent works focus on reducing rounding error to improve the training performance.\n\\citet{Wang2018b} devise a chunk-based accumulation mechanism to mitigate the swamping issue.\n\\citet{Sakr2019} improve the solution to the same problem by finding lower precision for accumulation by variance analysis.\nAlternatively, \\citet{Hoffer2018} identify the numerical issues caused by batch normalization and propose to replace it by a more numerically stable and efficient alternative.\nThese methods are orthogonal to loss scaling, and we plan to study the effect of applying them together with adaptive loss scaling as future work.\n\nLoss scaling that aims to improve mixed precision training by reducing the underflow rate in the computed gradients can be traced back to \\citet{Micikevicius2017}.\nThey originally suggest to choose a constant loss scale either empirically, or using a factor that cannot scale the maximal absolute gradient value to overflow.\n\\citet{Kuchaiev2018} propose two improved versions: one is called \\emph{backoff}, which simply makes the loss scale smaller if a numerical error is encountered during training; the other is \\emph{logmax}, which models the maximal absolute gradient value across training iterations by log-normal distribution, in order to estimate the proper loss scale value for the \\emph{next} iteration.\nWe argue that these new solutions are still not ideal, since backoff is simply trial-and-error and can waste training workload; and logmax is risky to use when we do not have much gradient values to model that log normal distribution, or this assumption cannot apply.\n\\citet{Mellempudi2019} further study the effect of the backoff method for 8-bit floating point.\n\n\\section{Introduction}\n\nTraining deep neural networks (DNNs) is well-known to be time and energy consuming, motivating the development of new methods and hardware to make training more efficient. One way to improve training efficiency is to use numerical representations that are more hardware-friendly. This is the reason that the IEEE 754 32-bit single-precision floating point format (FP32) is more widely used for training DNNs than the more precise double precision format (FP64), which is commonly used in other areas of high-performance computing. In an effort to further improve hardware efficiency, there has been increasing interest in using data types with even lower precision than FP32 for training~\\citep{Micikevicius2017, Kuchaiev2018, Wang2018b, Kalamkar2019, Mellempudi2019, Sakr2019}.\nOf these, the IEEE half-precision floating-point (FP16) format is already well supported by modern GPU vendors~\\citep{nvidia-volta}.\nUsing FP16 for training can reduce the memory footprint by half compared to FP32 and significantly improve the runtime performance and power efficiency.\nNevertheless, numerical issues like overflow, underflow, and rounding errors frequently occur when training in low precision only.\n\nRecent works propose various improvements, of which \\textbf{mixed precision training (MPT)}~\\citep{Micikevicius2017} is the state-of-the-art.\nIts core idea is to use FP16 for the compute-intensive yet precision-insensitive operations, such as matrix multiplication, for computational efficiency, while using FP32 for the operations that require high precision, such as batch normalization~\\citep{Ioffe2015} and gradient update accumulation.\nActivations and gradients, which largely contribute to memory consumption, are stored in FP16, while the weights are stored in FP32 for more accurate accumulation of gradient updates.\nEven though MPT seems promising and has wide support from both hardware and software frameworks, it still suffers from reliability issues, mainly due to the more limited dynamic range of FP16 being unable to adequately cover possible gradient values during training.\nThe most common issue is for small gradients to fall into the underflow gap and become zero, which makes training less effective.\n\n\\textbf{Loss scaling}~\\citep{Micikevicius2017,Kuchaiev2018,Mellempudi2019} addresses the range limitation in FP16 by introducing a hyperparameter $\\alpha$ to scale the loss value before the start of the backward pass so that the computed (scaled) gradients can then be properly represented in FP16 without significant underflow. For an appropriate choice of $\\alpha$, loss scaling can achieve state of the art results that are competitive with regular FP32 training. Unfortunately, there is no single value of $\\alpha$ that will work in arbitrary models, and so it often needs to be tuned per model. Its value must be chosen large enough to prevent underflow issues from affecting training accuracy. However, if $\\alpha$ is chosen too large, it could amplify rounding errors caused by \\textbf{swamping}~\\citep{Higham1993} or even result in overflow.\nThis observed sensitivity to the particular choice of loss scale is also reported by \\citet{Mellempudi2019}, who find that different values can lead to very different ResNet-50 MPT convergence behavior.\nFurthermore, the data distribution of gradients can vary both between layers and between iterations (\\Figref{fig:overview}), which implies that a single scale is insufficient.\nFor instance, gradients closer to the input require a higher loss scale that may cause overflow or severe rounding errors if the same value were used in layers closer to the output. Including the time spent tuning $\\alpha$, the total training time of MPT can even exceed regular FP32 training.\n\n\\begin{figure}[!t]\n    \\begin{center}\n        \\begin{subfigure}[t]{0.45\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/SSD512_underflow_rate}\n            \\caption{Underflow rate is calculated by counting the absolute gradients below $2^{-24}$, the smallest positive FP16 number.}\n            \\label{fig:overview:underflow}\n        \\end{subfigure}\n        ~ \n        \\begin{subfigure}[t]{0.45\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/SSD512_expected_loss_scale}\n            \\caption{Expected loss scale of each layer is calculated by 1 over the $(0.01N)$-th smallest absolute gradient, where $N$ is the size of each gradient and $0.01$ is the largest underflow rate permitted.}\n            \\label{fig:overview:loss_scale}\n        \\end{subfigure}\n    \\end{center}\n    \\caption{Statistics of activation gradients collected from training SSD~\\citep{Liu2015a} by FP32. Data are collected from different training iterations (120k in total). Layer ID are assigned in the topological order of backpropagation computation. Layers with higher ID are closer to the input.}\n    \\label{fig:overview}\n\\end{figure}\n\nWe introduce a loss scaling-based training method called \\textbf{adaptive loss scaling} that makes MPT easier and more practical to use. We hope that this will help to utilize better existing hardware with support for fast FP16 operations. Our method improves the usability of MPT compared to existing methods by removing the need to tune a model-specific loss scale hyperparameter, while retaining (and in some cases surpassing) the accuracy of regular FP32 training.\nWe achieve this by introducing layer-wise loss scale values which are automatically computed and dynamically updated during training to deal with underflow more effectively than existing methods.\nExperimental results on several examples show that MPT with adaptive loss scaling can achieve the best model accuracy and the shortest overall training time, especially when training deep models on large datasets.\n\n\n\n\n\\section{Background}\\label{sec:background}\n\n\\input{tex\/background\/preliminary.tex}\n\\input{tex\/background\/related_work.tex}\n\n\\section{Introduction}\n\nTraining deep neural networks in data types with lower precision than single-precision floating-point (FP32) is gradually supported by recent hardware platforms.\nFor instance, IEEE half-precision floating-point (FP16), which takes 16 bits per number and uses 5 bits for exponent, 11 bits for the significand, is being supported by newly released GPU platforms \\citep{nvidia-volta}.\nThe benefits of leveraging them are appealing: they require less memory footprint, consume less energy, and run faster compared with FP32 on the same workload.\nNevertheless, due to the relatively narrow dynamic range and low resolution of numeric values representable by them, the numerical instability of training DNN can be amplified and models cannot even be trained.\n\nDiving into a training procedure based on SGD with decaying learning rate, we notice that problems caused by low-precision floating-point (LFP) may appear in these phases:\n\n\\begin{enumerate}[label=\\textbf{(FM\\arabic*)}]\n    \\item During the first few epochs after initialization, since the model loss is large and the learning rate is normally large, gradients are more likely to explode and step out of the dynamic range of LFP, i.e., overflow, without proper initialization.\\label{fm:init}\n    \\item In the middle phase, some numerically unstable units, e.g., variance calculation in Batch Normalization, may produce abnormal values (NaN or Inf) and destroy training. Rounding errors introduced by LFP may also tweak the optimization direction.\\label{fm:round}\n    \\item Around the last epochs when the learning rate is small, multiplying the learning rate by small gradients can possibly vanish to 0 due to underflow, which hinders further training.\\label{fm:under}\n\\end{enumerate}\n\nThese three \\emph{failure modes} are what we have empirically noticed while training DNN in LFP. Examples for them are shown in Figure~\\ref{fig:fm}.\n\n\\begin{figure}[h]\n\\centering\n\\caption{Failure modes from training in LFP.}\n\\label{fig:fm}\n\\end{figure}\n\nRecent literature proposes various solutions to address these failure cases.\nMixed precision training~\\citep{Micikevicius2017} provides a comprehensive, state-of-the-art approach: for~\\ref{fm:round}, numerically unstable units are computed in FP32 and model weights are stored in FP32 rather than LFP to avoid inaccurate accumulation by small gradient updates; loss scaling to scale up gradient updates to prevent underflow for~\\ref{fm:under}.\nStill, a proper solution for \\ref{fm:init} is missing here and the training can still go wrong as in \\figref{fig:mpt-fail}.\nAlso, tuning new hyperparameters from the loss scaling mechanism, as well as the necessity of supporting both FP32 and LFP are troublesome.\nThere are some other papers that can make further improvement.\n\\cite{Hoffer2018} promote the usage of $L_1$ or $L_{\\infty}$ based BN for better numerical stability.\nFocusing on reducing the rounding error caused by adding large and small floating-point values, i.e., swamping~\\citep{Higham1993}, is an alternative approach to improve LFP training performance~\\citep{Wang2018b, Sakr2019}.\n\\cite{Aberger2019} improve both the arithmetic precision by bit-centering and intend to address the exploding\/vanishing gradient problem by reducing gradient variance through SVRG~\\citep{Johnson2013}.\nFloating-point itself can be improved as well, e.g., \\cite{Kalamkar2019} mention the \\texttt{bfloat16} that increases the ratio of the exponent to enlarge the dynamic range can empirically improve the training performance, and \\cite{Johnson2018} digs into core arithmetic floating-point arithmetic for better accuracy.\nIn a word, almost none of these prior works provides a universal approach that addresses all the three failure modes without introducing much overhead.\n\n\\begin{figure}[h]\n\\centering\n\\caption{How mixed precision training fails when the initialization is wrong and gradient explodes.}\n\\label{fig:mpt-fail}\n\\end{figure}\n\nOur paper aims to resolve all these failure modes in an efficient manner.\nWe stick to FP16 due to its universality on recent hardware platforms, and we prefer a hyperparameter-free and FP32-free approach to reduce the overhead for usage.\nFor \\ref{fm:init}, we first dive into the initialization problem to ensure that at least in the first few epochs, gradients and activations will stay in the dynamic range.\nWe extend the \\emph{fixup} initialization method~\\citep{Zhang2019}, which produces the state-of-the-art performance on over 10k layers DNN, to take into account the dynamic range of FP16.\nThe numerical instability mentioned in~\\ref{fm:round} caused by BN can be addressed by fixup in the sense that they replace BN by a trainable affine transformation.\nSurprisingly, those rounding errors that are deliberately addressed by some prior works, are not significantly harmful to the validation accuracy based on our empirical study, especially when the model is overparameterized.\nRegarding the vanishing gradient problem incurred by underflow as in~\\ref{fm:under}, instead of treating the small learning rate as it is, we choose to improve the learning rate schedule such that the total amount of underflow will be constrained by a threshold.\nMoreover, our method is parameterised regarding the floating-point configuration, the number of exponent and significand bits $(N_E, N_S)$ is populated to the solutions we give. \n\nTo be specific, our method is a combination of applying the following solutions:\n\n\\begin{enumerate}[label=\\textbf{(FS\\arabic*)}, itemsep=0em]\n    \\item Improve the fixup initializer to take into account the dynamic range of FP16 and avoid exploding gradient.\n    \\item Empirically study the effect of removing BN and using BN alternatives, as well as the rounding errors.\n    \\item Change the learning rate schedule of SGD-based and adaptive optimizers to reduce the amount of underflow gradient updates.\n\\end{enumerate}\n\nOur method can achieve ...\n\n\\section{Rounding Error}\n\\section{Optimization}\n\\section{Initialization}\n\n\\section{Conclusion}\n\nThis paper presents adaptive loss scaling, a method that calculates layer-wise loss scale during runtime, to improve the performance and usability of MPT.\nEmpirically we find it works better than plain MPT, existing loss scaling methods, and even FP32 in some cases, regarding model accuracy and the time taken to converge.\nFuture work includes evaluating adaptive loss scaling on other tasks and models, especially those for Natural Language Processing; and trying to find a tighter upper bound of loss scale for each layer, e.g., based on the variance analysis in~\\citep{Sakr2019}, such that each layer can be scaled more effectively; extending it to FP8 is also intriguing to try.\n\n\n\n\\section{Experiment}\\label{sec:experiment}\n\nThis section presents various experiments to show the benefit of using adaptive loss scaling over other loss scaling approaches.\nThe models we focus on are basically for computer vision tasks, including image classification and object detection, and their topologies vary, ranging from sequential architecture to skip connection based ones.\nWe implemented our approach using Chainer v6.1.0~\\citep{tokui2019chainer}, a deep learning framework that supports efficient automatic differentiation.\nFor comparison, we choose FP32 training, loss scaling by a fixed value~\\citep{Micikevicius2017}, and dynamic loss scaling by backoff~\\citep{Kuchaiev2018} as baselines.\n\n\\input{tex\/experiment\/cifar.tex}\n\\input{tex\/experiment\/imagenet.tex}\n\\input{tex\/experiment\/object_detection.tex}\n\n\\subsection{Object Detection}\\label{sec:exp:object_detection}\n\nWe select the Single-Shot Detector model~\\citep{Liu2015a} SSD512 (VGG-16 backbone, 512 input resolution) as our baseline for the object detection task.\nThe basic training schedule stays the same as the original paper.\nSSD is a rather challenging model for MPT: its VGG-16 backbone is not interleaved by batch normalization layers, which implies that gradients are not normalized, and their distribution can vary a lot across layers at different depth.\nIt also has a multi-branched topology, in which each branch detects objects at a different scale and passes different values.\nAs seen in~\\Figref{fig:overview}, SSD512 cannot be properly scaled by a fixed loss scale.\n\n\\begin{table}[h]\n  \\centering\n  \\caption{Test performance of SSD512 models trained by different loss scaling methods, measured in mAP (\\%). FP32 denotes the baseline results trained in FP32, and we select the golden value from~\\citep{Fu2017a} for the case using 32 as the batch size. None means no loss scaling is used, ``Fixed (best)'' stands for the best performance we can find after trying out several fixed loss scale values (including \\{8, 128, 1024, 2048\\}), and ``Dynamic'' shows the results of dynamic loss scaling. }\n  \\label{table:exp:object_detection}\n\n  \\begin{tabular}{l|lllll}\n    \\hline\n    \\textbf{Batch} & \\textbf{FP32} & \\textbf{None} & \\textbf{Fixed (best)} & \\textbf{Dynamic} & \\textbf{Adaptive} \\\\\\hline\n    8  & 78.94 & diverged & 79.11 & 75.04 & \\textbf{79.24} \\\\ \n    32 & 79.50 & diverged & 80.01 & 80.17 & \\textbf{80.31} \\\\ \n    \\hline\n  \\end{tabular}\n\\end{table}\n\nTable~\\ref{table:exp:object_detection} shows the comparison result.\nWe examine two scenarios with different batch sizes yet the same total number of iterations.\nWithout loss scaling training diverges at a very early stage.\nLoss scaling can make mixed precision training stabler, e.g., the best fixed loss scaling results perform similarly to the FP32 baseline.\nWhen the batch size is small, dynamic loss scaling performs poorly since NaN is more frequent and the current dynamic algorithm finds it hard to choose a good scale. \n\nOur adaptive method gives the best result even compared to the FP32 baseline.\n\\Figref{fig:exp:diff_in_underflow_rate} shows the large amount of underflow rate reduced by using adaptive loss scaling, which can give a hint about why it performs better.\nRegarding the speed overhead of calculating loss scale, computing the statistics takes around 27\\% of the overall training time if we update loss scale per iteration.\nThe update frequency of adaptive loss scaling results per 100 iterations, which reduces the overhead to 0.27\\%.\nCompared to fixed loss scaling, which performs worse and requires 3 rounds of training to find the best scale, our approach seems appealing for the reduction in total training time it provides.\n\n\\subsection{Ablation Study}\\label{sec:exp:ablation}\n\n\n\n\\subsection{Image Classification}\\label{sec:exp:image_cls}\n\nWe also compare loss scaling methods on ILSVRC2012~\\citep{JiaDeng2009}.\nResNet-18 and 50~\\citep{He2015} are baseline.\nBased on the training method proposed in the original paper, we set the data type of each layer by the default mixed precision training setting, and change only the loss scaling method.\nDue to the limitation of resources, we can only select 128 as the fixed loss scale.\nWe also compare with dynamic loss scaling using the backoff strategy~\\citep{Kuchaiev2018}.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Image classification evaluation for different loss scaling methods. Numbers showed here are top-1 test accuracy (\\%). None means no loss scaling applied.}\n    \\label{table:exp:image_cls:imagenet}\n    \\begin{tabular}{l|lllll}\n        \\hline\n        \\textbf{Model} & \\textbf{FP32} & \\textbf{None} & \\textbf{Fixed (128)} & \\textbf{Dynamic} & \\textbf{Adaptive} \\\\\\hline\n        ResNet-18 & 69.76 & 71.24 & 71.39 & 71.39 & \\textbf{71.44} \\\\\n        ResNet-50 & 76.15 & 76.07 & 76.02 & 76.12 & \\textbf{76.22} \\\\ \n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{figure}[!t]\n    \\centering\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/ResNet-18_loss_scales.pdf}\n        \\caption{Calculated loss scale across different layers and iterations during ResNet-18 training.}\n        \\label{fig:exp:imagenet_loss_scale}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.47\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/SSD512_diff_underflow_rate.pdf}\n        \\caption{The difference in underflow rate of each layer's activation gradient comparing fixed and adaptive loss scaling.}\n        \\label{fig:exp:diff_in_underflow_rate}\n    \\end{subfigure}\n    \\caption{Examples that show the benefit from using adaptive loss scaling. Note that the underflow rate in \\Figref{fig:exp:diff_in_underflow_rate} is collected by subtracting the percentage of zeros of the FP16 result and the cast-to-FP32 result. The higher $\\Delta$ is, the more effective that adaptive loss scaling can mitigate underflow.}\n\\end{figure}\n\nResults are listed in Table~\\ref{table:exp:image_cls:imagenet}.\nIn general, adaptive loss scaling performs the best among all MPT and FP32 training results.\nNote that loss scaling with a fixed arbitrary scale (128) even reduces model test accuracy for ResNet-50 compared with no loss scaling, and on the contrary, there is no hassle in hyperparameter using adaptive loss scaling.\nLoss scale of each layer in ResNet-18 is listed in~\\Figref{fig:exp:imagenet_loss_scale}.\nIt shows that our calculated loss scale is much smaller than 128, which implies 128 is too large and may cause rounding error.\nWe further compare the \\textbf{maximum standard deviation} $\\sigma_{max}$ of all gradients at different iterations between adaptive and fixed, and we observe that the ratio of $\\sigma_{max}$ of fixed over adaptive is around \\textbf{20 times}, both for ResNet-18\/50, which can be the major cause for accuracy drop since that high variance can increase accumulation error~\\citep{Sakr2019}. \n\\subsection{Deep MLP}\n\nDeep MLP is frequently used to study the exploding and vanishing gradient problem in training deep learning models.\nTherefore, testing different loss scaling methods under the mixed precision framework for deep MLP helps us measure the effectiveness of adaptive loss scaling for addressing such problems.\nWe train MLP with the same number of hidden units per layer, but different depth, with various loss scaling methods on the MNIST dataset.\nEach layer in this MLP is followed by ReLU only, without batch normalization.\nEach MLP instance is trained by the same schedule for all loss scaling methods, specifically, taking 10 epochs and using $1e^{-3}$ learning rate.\nWeights among all layers are initialized through the approach in~\\citep{He2015a}.\n\n\n\\subsection{CIFAR}\\label{sec:exp:cifar}\n\nWe first evaluate our method on CIFAR-10\/100~\\citep{Krizhevsky2009} image classification using ResNet models~\\citep{He2015} of depth 20, 56, and 110.\nWe explore three different loss scaling options: without loss scaling, fixed loss scaling~\\citep{Micikevicius2017} with scales selected from $\\{16, 128, 1024, 4096, 8192, 16384\\}$, and adaptive loss scaling ($T_{uf}$ is set to $1e^{-3}$).\nThe other training settings are the same as specified in the original paper~\\citep{He2015}.\n\n\\begin{table}[h]\n    \\caption{Test accuracy results for ResNet models trained on CIFAR-10\/100, each \\textbf{averaged} from 4 runs with different random seeds. The best result for each combination of dataset and model is bolded. Fixed loss scaling is tied with ours for the best result on ResNet-20 (C100).}\n    \\label{table:exp:image_cls:cifar}\n    \\begin{center}\n        \\begin{tabular}{l|l|lllll}\n            \\hline\n            \\textbf{CIFAR}\n            & \\textbf{Depth}\n            & \\textbf{FP32}\n            & \\textbf{None}\n            & \\textbf{Fixed (best)}\n            & \\textbf{Fixed (worst)}\n            & \\textbf{Adaptive} \\\\ \\hline\n            \\multirow{3}{*}{C10}\n            & 20 &\n            91.25\\% &\n            92.16\\% &\n            \\textbf{92.24\\%} &\n            92.16\\% &\n            \\textbf{92.26\\%} \\\\\n            & 56 &\n            93.03\\% &\n            92.79\\% &\n            \\textbf{93.29\\%} &\n            92.78\\% &\n            93.22\\% \\\\\n            & 110 &\n            93.57\\% &\n            93.54\\% &\n            93.85\\% &\n            93.73\\%&\n            \\textbf{93.90\\%} \\\\ \\hline\n            \\multirow{3}{*}{C100}\n            & 20 &\n            67.94\\% &\n            68.31\\% &\n            \\textbf{68.48\\%} &\n            68.18\\%&\n            \\textbf{68.47\\%} \\\\\n            & 56 &\n            71.15\\% &\n            71.17\\% &\n            \\textbf{71.56\\%}&\n            71.26\\%&\n            71.26\\% \\\\\n            & 110 &\n            71.14\\% &\n            72.05\\% &\n            72.46\\% &\n            72.34\\% &\n            \\textbf{72.66\\%} \\\\\n            \\hline\n        \\end{tabular}\n    \\end{center}\n\\end{table}\n\nResults are in Table~\\ref{table:exp:image_cls:cifar}.\nFirst of all, ResNet models overfit on CIFAR (train accuracy reaches 100\\%), such that reducing arithmetic error may decrease its regularization effect and then result in worse test accuracy.\nThat is why adaptive loss scaling is not very beneficial for ResNet-20 and 56.\nBut regarding ResNet-110, since it is much deeper than the others, gradients are harder to propagate and underflow is more harmful, and arithmetic errors hinder training rather than regularizing it. \nMore importantly, in terms of the total training time, to find the best fixed loss scale we should train \\textbf{6 times to cover all candidates}, while adaptive loss scaling only needs one round.\n\\subsubsection{Backpropagation Algorithm}\n\n\\begin{algorithm}\n    \\DontPrintSemicolon\n    \\caption{Backpropagation algorithm with adaptive loss scaling.\n    Assuming each layer has a single output.}\n    \\label{alg:backprop}\n\n    \\SetKwFunction{FGetLayerOutput}{GetLayerOutput}\n    \\SetKwFunction{FGetLossScale}{GetLossScale}\n    \\SetKwFunction{FBackprop}{Backprop}\n    \\SetKwFunction{FOp}{OP}\n\n    $\\sgrad{\\alpha_{N+1}}{{\\bm{\\delta}}_{N+1}} \\gets $ initial loss scale, and error gradient \\emph{scaled} by $\\alpha_0$\\;\n    \\For{$i \\gets $ layer ID in a reversed topological order}{\n        $j$         $\\gets$ \\FGetLayerOutput{$i$}\\;\n        $\\beta_i$   $\\gets$ \\FGetLossScale{\\FOp{$i$}, $\\sgrad{\\alpha_j}{{\\bm{\\delta}}_j}$}\\;\n        ${\\bm{\\delta}}_i$ $\\gets$ \\FBackprop{\\FOp{$i$}, $\\beta_i{\\bm{\\delta}}_j$}\\;\n        $\\alpha_i$  $\\gets$ $\\alpha_j\\beta_i$\\;\n    }\n\\end{algorithm}\n\nIn general, the adaptive loss scaled backpropagation works as shown in the pseudocode of \\Algref{alg:backprop}.\nAt the beginning, we have an initial error gradient, directly calculated from the loss value and the final output.\nWe may optionally scale this gradient by a manually set value $\\alpha_{N+1}$.\nNext, we traverse the computation graph of the neural network in a reversed topological order, i.e., move to a new layer until all its output layers have been visited.\nThis stays the same as the canonical way of backpropagation.\nWe cover single-output cases for simplicity at this stage, and we will dive into detail about how to handle branch output in the following sections.\nWhen visiting each layer $i$, we retrieve the tuple $\\sgrad{\\alpha_j}{{\\bm{\\delta}}_j}$ propagated to it from output layers and calculate a local loss scale value $\\beta_i$.\nThis local value will be multiplied to ${\\bm{\\delta}}_j$ \\emph{before} we calculate the gradient for previous layers, as well as the gradient update for weights of layer $i$ (omitted in the algorithm).\nSince $\\beta_i$ contributes to the magnitude of the gradient ${\\bm{\\delta}}_i$, we can calculate the loss scale value corresponds to ${\\bm{\\delta}}_i$ by $\\alpha_j\\beta_i$.\nLooking back to the $N$-layer MLP example, from this algorithm, we can realize that the loss scale value from layer $i$ is an accumulation of $\\beta_i$ all its following layers' scales, i.e., $\\alpha_i=\\alpha_{N+1}\\prod_{j=i}^{N} \\beta_j$.\n\n\n\\subsubsection{Loss Scaled Gradient}\n\n\n\\subsubsection{Post-processing}\\label{sec:impl}\n\nA raw loss scale value calculated from \\Eqref{eq:loss_scale:gemm} should be post-processed by the following rules.\nMost importantly, the raw loss scale should be rounded down to the nearest \\textbf{power-of-two} number.\nOtherwise, due to the nature of floating-point, a scaled gradient cannot always be unscaled by the same scale, i.e., $(\\alpha x)\/\\alpha \\neq x$ if $\\alpha$ is not a power of two~\\citep{Muller}.\n\nThe upper bound of layer-wise loss scale is determined by avoiding overflow.\nFor the GEMM case, it can be simply calculated by choosing the maximal numbers from both operands, multiplying them together, and then taking $u_{max}$ over that multiplication result as the largest possible loss scale, i.e., $u_{max} \/(\\max({\\bm{W}}) \\times \\max({\\bm{\\delta}}))$.\nThis is a loose bound since these selected maximal numbers may not be multiplied together when calculating the output activation gradient.\nIn practice, this upper bound is much larger than the lower bound, and we simply choose the lower bound as the loss scale value, and only switch to the upper bound only when the upper bound is smaller than the lower bound.\nOther operators will not update loss scale (except branching, which has been discussed in \\Secref{sec:loss_scale_calc:elemwise}), we assume they will not cause overflow.\n\n\n\n\\subsubsection{GEMM}\\label{sec:loss_scale_calc:gemm}\n\nWe start by considering a linear layer with input activations ${\\bm{X}}$, weights ${\\bm{W}}$, and output activations ${\\bm{Y}}$. The GEMM computation for the forward-pass is then given by ${\\bm{Y}} = {\\bm{X}} {\\bm{W}}^T$ (ignoring the bias term without loss of generality). Now consider the backward pass for this same layer, in which the ``input'' to the layer consists of the tuple $\\sgrad{\\alpha}{{\\bm{\\delta}}}$ received from the next downstream layer, where ${\\bm{\\delta}} = \\alpha\\partgrad{L}{{\\bm{Y}}}$. Note that $\\alpha$ represents the total loss scale (i.e., product of all downstream layer-wise scales). As shown earlier, the weight gradients for ${\\bm{W}}$ are computed by $({\\bm{X}}^T{\\bm{\\delta}})\/\\alpha$.\n\n\nWe assume that both ${\\bm{W}}$ and ${\\bm{\\delta}}$ can be characterized by two i.i.d. normal random variables ${\\textnormal{w}}$ and ${\\textnormal{g}}$, respectively, so that  ${\\textnormal{w}} \\sim \\mathcal{N}(\\mu_w, \\sigma_w^2)$ and ${\\textnormal{g}} \\sim \\mathcal{N}(\\mu_g, \\sigma_g^2)$.\nWe also assume that their product ${\\textnormal{p}} = {\\textnormal{w}} {\\textnormal{g}}$ is another random variable with a zero-mean normal distribution, i.e., ${\\textnormal{p}} \\sim \\mathcal{N}(0, \\sigma_p^2)$.\nThese assumptions are standard in the literature on weight initialization~\\citep{He2015a,Glorot2010} and low-precision floating-point training~\\citep{Sakr2019}.\nSince ${\\textnormal{w}}$ and ${\\textnormal{g}}$ are uncorrelated, $\\sigma_p^2$ is given by $(\\sigma_w^2 + \\mu_w^2)(\\sigma_g^2 + \\mu_g^2)$ based on product distribution rules, and can be computed from the corresponding empirical statistics.\n\n${\\textnormal{p}}$ characterizes the distribution of the intermediate results before the final GEMM reduction happens, so that the output is the sum of $N$ values sampled from ${\\textnormal{p}}$, where $N$ is the number of columns.\nIntuitively, if the probability that ${\\textnormal{p}}$ experiences underflow is reduced, the final result will have less underflow rate as well.\nLet the upper bound for an underflow positive value be $u$, which can take the minimal subnormal value of the given low-precision data type, e.g., $u = u_{min} = 2^{-24}$ for FP16, then our objective corresponds to reducing $P(|{\\textnormal{p}}| \\leq u)$ by scaling ${\\textnormal{w}}$ or ${\\textnormal{g}}$.\n\n\\begin{equation}\\label{eq:loss_scale:gemm}\n    P(\\alpha|{\\textnormal{p}}| \\leq u) \\leq T_{uf}\n    \\Leftrightarrow\n    \\mathrm{erf}\\left(\\frac{u}{\\alpha\\sigma_p\\sqrt{2}}\\right) \\leq T_{uf}\n    \\Leftrightarrow\n    \\alpha \\geq \\frac{u}{\\sigma_p\\sqrt{2}\\times \\mathrm{erf}^{-1}(T_{uf})}\n\\end{equation}\n\nWe introduce a new term, $T_{uf}$ that specifies the threshold for the probability of underflow for ${\\textnormal{p}}$ in each layer, which can also be interpreted as the upper bound of underflow rate.\nSuppose either ${\\textnormal{w}}$ or ${\\textnormal{g}}$ is scaled by $\\alpha$ before the multiplication, then $P(\\alpha|{\\textnormal{p}}|\\leq u) \\leq T_{uf}$ becomes the expected outcome of loss scaling.\nSince ${\\textnormal{p}}$ is assumed to be $\\mathcal{N}(0, \\sigma_p^2)$, it implies that $|{\\textnormal{p}}|$ is a random variable with half-normal distribution, and $P(\\alpha|{\\textnormal{p}}|\\leq u) = \\mathrm{erf}(u \/ (\\alpha\\sigma_p\\sqrt{2}))$.\nTherefore, we can deduce the \\textbf{lower bound} of loss scale for each GEMM-based layer by~\\eqref{eq:loss_scale:gemm} with $T_{uf}$ and $u$.\n\n\\input{tex\/method\/loss_scale_calc\/calc_alg.tex}\n\nIn practice, we take this lower bound term as the loss scale value (\\Secref{sec:impl} presents the details for the corresponding upper bound to prevent overflow).\n\\Algref{alg:loss_scale:gemm} illustrates the steps of the loss scale calculation for a GEMM-based layer.\nNote that the computation of $\\sigma_p$ requires the statistics of ${\\textnormal{w}}$ and ${\\textnormal{g}}$. These in turn require computing the sample mean and variance of ${\\bm{W}}$ and ${\\bm{\\delta}}$, which is the main source of computational overhead, with around the same computation budget as batch normalization. In our current implementation, these statistics are calculated on GPU and then transferred to CPU to finish the loss scale calculation. There is potential to optimize the data transfer mechanism in the future, and for now we simply reduce the frequency of loss scale update if the overhead is large.\nFor FP16-based mixed precision training, we set $u$ to $2^{-24}$ as required by the FP16 representation range.\n$T_{uf}$ is set to $1.0 \\times 10^{-3}$ in all of our experiments, which corresponds to allowing an underflow rate of 0.1\\%.\n\n\\subsubsection{Element-wise and Branching Operations}\\label{sec:loss_scale_calc:elemwise}\n\nElement-wise operations can take one input argument (unary), such as activation functions; or two operands (binary), e.g., element-wise multiplication and addition.\nBatch normalization also falls into this category.\nHeuristically, we do not update the loss scale for these operations, because normally they will not significantly change the amount of underflow values, and there are no statistical properties that we can directly make use of without introducing much computational overhead.\n\nOne particular element-wise operation that requires special treatment is \\textbf{branching}.\nIt is used mainly in networks that employ skip connections, such as ResNet~\\citep{He2015}, DenseNet~\\citep{Huang2016a}, etc.; it also appears in object detection models that have multiple outputs, such as SSD~\\citep{Liu2015a}. \nIt copies its single input to multiple branches in the forward pass, and sums all received gradients during backpropagation.\nThe special case that we should treat deliberately is when the gradients to sum have different loss scales.\nIf we were to directly sum these gradients, we would no longer be able to compute the output gradient's loss scale, preventing subsequent layers from restoring the correct gradient magnitude in their weight updates.\n\nOur solution to this issue is to \\textbf{rescale} input gradients before the summation happens, as shown in \\Algref{alg:loss_scale_calc:branch}.\nSuppose we have $N$ input tuples $\\{\\sgrad{\\alpha_i}{{\\bm{\\delta}}_i}\\}_{i=1}^N$, and $\\alpha_j$ is the maximum loss scale among them, then the rescaling works by multiplying any gradient other than ${\\bm{\\delta}}_j$ by a factor to produce the gradient that has the same loss scale as $j$, which is calculated by $\\alpha_j\/\\alpha_i$.\nAnd if $\\alpha_j$ is too large and may cause overflow in other gradients,which is decided by checking the scaled maximum absolute gradient, we will search in a descending order of all scales among inputs until we find one.\n\\subsection{Loss Scaled Backpropagation}\\label{sec:loss_scaled_backprop}\n\nWe use a 2-tuple notation to denote the propagated entity from layer $i$: $\\sgrad{\\alpha_i}{{\\bm{\\delta}}_i}$, in which $\\alpha_i$ is the loss scale value for layer $i$ and ${\\bm{\\delta}}_i$ is the gradient that \\emph{has been scaled} by $\\alpha_i$. \nTo be more specific about this notation, we can take a $N$-layer MLP as an example (see \\Figref{fig:ada_loss} for the notation).\nIn this case, layer $i$ takes in $\\sgrad{\\alpha_{i+1}}{{\\bm{\\delta}}_{i+1}}$, updates its weight by $({\\bm{y}}_{i-1}^T{\\bm{\\delta}}_{i+1})\/\\alpha_{i+1}$, and produces $\\sgrad{\\alpha_i}{{\\bm{\\delta}}_i}$ for the previous layer $i-1$.\nWe will elaborate more on how $\\alpha_i$ is calculated in the following sections.\n\n\\RestyleAlgo{ruled}\n\\begin{algorithm}\n    \\DontPrintSemicolon\n    \\caption{Backpropagation algorithm with adaptive loss scaling,\n    assuming each layer has a single output. \\Secref{sec:loss_scale_calc:elemwise} shows how multiple outputs work.}\n    \\label{alg:backprop}\n\n    \\SetKwFunction{FGetLayerOutput}{GetLayerOutput}\n    \\SetKwFunction{FGetLossScale}{GetLossScale}\n    \\SetKwFunction{FGetWeightGrad}{GetWeightGradient}\n    \\SetKwFunction{FBackprop}{Backprop}\n    \\SetKwFunction{FOp}{OP}\n\n    $\\sgrad{\\alpha_{N+1}}{{\\bm{\\delta}}_{N+1}} \\gets $ initial loss scale, and error gradient \\emph{scaled} by $\\alpha_0$\\;\n    \\For{$i \\gets $ layer ID in a reversed topological order}{\n        $j$         $\\gets$ \\FGetLayerOutput{$i$}\\;\n        ${\\bm{W}}_i$     $\\gets$ ${\\bm{W}}_i + $ \\FGetWeightGrad{${\\bm{\\delta}}_j$} $\/\\alpha_j$\\;\n        $\\beta_i$   $\\gets$ \\FGetLossScale{\\FOp{$i$}, $\\sgrad{\\alpha_j}{{\\bm{\\delta}}_j}$}\\;\n        $\\sgrad{\\alpha_i}{{\\bm{\\delta}}_i} \\gets \\sgrad{\\alpha_j\\beta_i}{ \\texttt{Backprop}(\\texttt{OP}(i),\\beta_i{\\bm{\\delta}}_j)}$\\;\n    }\n\\end{algorithm}\n\n\\Algref{alg:backprop} shows the pseudocode for adaptive loss scaled backpropagation for the case where each layer has a single output (we describe how to handle the multiple-output case in \\Secref{sec:loss_scale_calc:elemwise}):\n\n\\begin{enumerate}[itemsep=0em]\n\\item\nWe start with the error gradients $\\delta_{N+1}$ computed from the output loss value for the last layer $N+1$. We may optionally scale this gradient by $\\alpha_{N+1}$. Normally we keep it as 1.\n\n\\item\nAs visiting each previous layer $i$ in a reversed topological order of the computational graph, we retrieve the tuple $\\sgrad{\\alpha_j}{{\\bm{\\delta}}_j}$ propagated to it from the next downstream layer that represents the scaled loss for layer $i$'s output. We calculate a local loss scale value $\\beta_i$, which will be used to scale ${\\bm{\\delta}}_j$ \\emph{before} we calculate the activation gradients for the previous layer.\n\n\\item\nWe use ${\\bm{\\delta}}_j$ and other cached inputs (omitted) to compute the gradients for ${\\bm{W}}_i$. However, since these gradients have been scaled, we must unscale them using $\\alpha_j$ before performing the weight update.\n\n\\item \nSince $\\beta_i$ contributes to the magnitude of the gradient ${\\bm{\\delta}}_i$, we calculate the loss scale value ${\\bm{\\delta}}_i$ to be passed to the next previous layer as $\\alpha_j\\beta_i$.\n\\end{enumerate}\n\n\n\n\n\\subsection{Problem Formulation}\n\nHere we consider a multi-layer perceptron (MLP) model that consists of matrix multiplication.\nSuppose ${\\bm{X}}_i$ is the input to layer $i$, ${\\bm{W}}_i$ indicates its weights, the output ${\\bm{X}}_{i+1}$ equals to ${\\bm{W}}_i {\\bm{X}}_i$.\nGiven the empirical loss as $\\mathcal{L}$, the gradient backpropagated to layer $i$ as $\\nicefrac{\\partial\\mathcal{L}}{\\partial{\\bm{X}}_{i+1}}$, and ,\nThe gradient backpropagated to ${\\bm{X}}_i$ is scaled by $\\prod_{j=i+1}^{L+1} \\alpha_{j}$, i.e., $\\prod_{j=i+1}^{L+1} \\alpha_{j} \\partial \\mathcal{L}\/\\partial {\\bm{Y}}_i$.\nIn total there are $L$ sequentially connected layers, and our objective is that, given an initial $\\alpha_{L+1}$, we need to decide all the $\\alpha_i$, $1 \\leq i \\leq L$ such that the amount of elements in activation gradients $\\partial \\mathcal{L}\/\\partial {\\bm{Y}}_i$ is largely reduced.\n\n\\begin{equation}\n    \\begin{aligned}\n    \\alpha_i^t = \\argmin_{\\alpha}\\ \n        \\mathcal{C}\n        \\left(\n            \\alpha\n            \\left( \\prod_{j=i+1}^{L+1} \\alpha_{j}^t \\right)\n            \\Delta {\\bm{X}}_{i+1}^t\n            {\\bm{W}}_i^{t-1}\n        \\right)\n    \\end{aligned}\n\\end{equation}\n\nWe assume that the product $p$ between an input gradient and a weight is drawn from a normal distribution $\\mathcal{N}(\\mu_p, \\sigma_p^2)$~\\citep{He2015a}, weights are generated i.i.d from a distribution with mean and variance $\\sigma_w^2$, and elements in $\\Delta {\\bm{X}}_{i+1}^t$ are also i.i.d from a distribution with mean $\\mu_g$ and variance $\\sigma_g^w$.\nWe can further deduce $\\mu_p$ and $\\sigma_p^2$ by~\\eqref{eq:pprob}.\n\n\\begin{equation}\\label{eq:pprob}\n   \\mu_p = \\mu_w \\mu_g\\quad\n   \\sigma_p^2 = (\\sigma_w^2 + \\mu_w^2) (\\sigma_g^2 + \\mu_g^2) - \\mu_g^2 \\times \\mu_w^2\n\\end{equation}\n\nFurthermore, we assume $\\mu_w$ to be 0. $p$ then has zero mean, and the absolute value of $p$ forms a half-normal distribution.\nWe intend to constrain the amount of $|p|$ values that are below the minimal denormal number of FP16 $u_{\\min}$, denoted by $N_{uf}$, within a threshold $t$.\nWe can derive $N_{uf}$ by the CDF of $|p|$, which is $\\mathrm{erf}(\\nicefrac{u_{\\min}}{\\sqrt{2}\\sigma_p})$.\nIf we scale the input gradient by $\\alpha$ in order to constrain $N_{uf}$ under $t$, we can deduce the minimal value of $\\alpha$ that can be taken:\n\n\\begin{equation}\n   t \\geq P(\\alpha \\times |p| \\leq u_{\\min}) = \\mathrm{erf}(\\frac{u_{\\min}}{\\sqrt{2}\\alpha\\sigma_p}) \n   \\Longleftrightarrow\n   \\alpha \\geq \\frac{u_{\\min}}{\\mathrm{erf}^{-1}(t) \\sqrt{2} \\sigma_p}\n\\end{equation}\n\n\\begin{equation}\n    p = g \\times w\n\\end{equation}\n\n\\begin{equation}\n    p \\sim \\mathcal{N}(0, \\sigma_p^2)\n\\end{equation}\n\n\\subsubsection{Post-processing}\\label{sec:impl}\n\nA raw loss scale value calculated from \\Eqref{eq:loss_scale:gemm} should be post-processed by the following rules:\n\n\\begin{enumerate}\n\\item\nMost importantly, it should be rounded down to the nearest \\textbf{power-of-two} number.\nOtherwise, due to the nature of floating-point, a scaled gradient cannot always be unscaled by the same scale, i.e., $(\\alpha x)\/\\alpha \\neq x$ if $\\alpha$ is not a power of two~\\citep{Muller}.\n\n\\item\nBesides this rule, we also give an upper bound of the total accumulated loss scale value to prevent it going too large and amplifying low-precision arithmetic errors, although it is pretty rare based on our empirical evaluation and we fix this bound to 2048 as a constant.\n\n\\item\nThe upper bound of loss scale is determined by avoiding overflow.\nFor the GEMM case, it can be simply calculated by choosing the maximal numbers from both operands, multiplying them together, and then taking 1 over that multiplication result as the largest possible loss scale, i.e., $1\/(\\max({\\bm{W}}) \\times \\max({\\bm{\\delta}}))$.\nThis is a loose bound since these selected maximal numbers may not be multiplied together when calculating the output activation gradient.\nIn practice, this upper bound is much larger than the lower bound, and we simply choose the lower bound as the loss scale value, and only switch to the upper bound only when the upper bound is smaller than the lower bound.\nOther operators will not update loss scale (except branching, which has been discussed in \\Secref{sec:loss_scale_calc:elemwise}), we assume they will not cause overflow.\n\\end{enumerate}\n\n\n\\subsection{Loss Scale Calculation}\\label{sec:loss_scale_calc}\n\n\nThis section describes how to compute the layer-wise loss scales for the various operation types that are sufficient to support its implementation in general DNNs, which broadly consists of general matrix multiplication (\\textbf{GEMM}) and \\textbf{element-wise} operations. GEMM is the basis of linear layers, while the element-wise category covers batch normalization~\\citep{Ioffe2015}, activation functions, and math operations such as the element-wise addition commonly used in skip connections~\\citep{He2015}.\n\n\\input{tex\/method\/loss_scale_calc\/gemm.tex}\n\\input{tex\/method\/loss_scale_calc\/elemwise.tex}\n\\input{tex\/method\/loss_scale_calc\/post_proc.tex}\n\\section{Detailed Analysis on CIFAR Results}\n\nTable~\\ref{table:exp:image_cls:cifar} shows that adaptive loss scaling is beneficial for training ResNet-110, while less advantageous for ResNet-20 and ResNet-56.\nWe hypothesize the reason behind is that underflow causes more numerical problems when the model is deeper.\nFor shallower models, the difference between the oracle gradient values and the underflowing ones is moderate and can even be viewed as a form of regularization.\nThis argument is supported by the fact that the training accuracy of ResNet models on CIFAR can always reach 100\\%.\nIn this way, even though adaptive loss scaling can improve the accuracy of the computed gradients, this does not necessarily always translate to improved test accuracy.\n\n\\begin{table}[h]\n\\centering\n\\caption{The effect of different fixed loss scales on the test accuracy, which is measured for ResNet-20 and ResNet-56 on CIFAR-10. Numbers on the first row give the fixed loss scales.}\n\\label{table:resnet_on_cifar}\n\\begin{tabular}{l|lllllll}\n\\hline\n\\textbf{Model} & 1 & 16 & 128 & 1024 & 4096 & 8192 & 16384 \\\\\n\\hline\nResNet-20 &\n92.16\\% &\n92.19\\% &\n92.16\\% &\n92.20\\% &\n92.24\\% &\n92.24\\% &\n92.24\\% \\\\\n\\hline\nResNet-56 &\n92.80\\% &\n93.28\\% &\n92.79\\% &\n93.08\\% &\n93.19\\% &\n93.19\\% &\n93.19\\%\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe dive deeper into this argument by reviewing Table~\\ref{table:resnet_on_cifar}, which shows the test accuracy of the two shallower ResNet models on CIFAR-10.\nFor both models, the test accuracy first increases to a maxima at 16, then there is a sudden drop at 128, and finally it climbs up to a plateau.\nOur hypothetical interpretation is as follows:\n\n\\begin{enumerate}[itemsep=0em]\n    \\item Initially the test accuracy is low. Here the underflow rate is expected to be at its highest, and it is the major cause for the low test accuracy.\n    \\item The test accuracy then increases with loss scale, mainly due to the mitigation of underflow by loss scaling. However, as the gradients become more accurate, the regularizing effect from underflow is also reduced and the test accuracy will drop, until the loss scale reaches around 128.\n    \\item If the loss scale continues to increase, the high rounding error and swamping problem caused by large scales will arise. It adds another kind of regularization, which is relatively more harmful than what underflow may cause, and the test accuracy cannot improve much.\n\\end{enumerate}\n\nEven though this interpretation is hypothetical, this empirical evaluation in Table~\\ref{table:resnet_on_cifar} shows that the relationship between the goodness of a loss scaling scheme and test accuracy is complicated when the model tends to overfit.\n\n\\section{Effect from different loss scales on SSD}\n\nHere we present how changing fixed loss scales will affect the SSD training result, in order to understand the benefits of both training time and model accuracy from using adaptive loss scaling.\n\n\\begin{table}[h]\n\\centering\n\\caption{Changes in mAP after changing the fixed loss scales. Batch size = 8.}\n\\label{table:ssd_all}\n\\begin{tabular}{l|llllll}\n\\hline\n\\textbf{Loss scale} &\n1 &\n8 &\n16 &\n128 &\n1024 &\n2048 \\\\\\hline\nmAP (\\%) &\ndiverged &\n24.62  &\ndiverged &\n79.04 &\n79.04 &\n79.11 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{table:ssd_all} gives all the current empirical results.\nNo loss scaling and fixed loss scaling with scale as 16 are both diverged.\n8 almost does not improve during training.\n\\{128, 1024, 2048\\} all give good results compared to other scale candidates.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGenerative Adversarial Nets (GAN) \\citep{GO:14} are one of the state-of-the-art solutions in the field of generative models in computer vision. Recent studies on GANs also show their great capabilities in feature extraction \\citep{RA:15,DO:16} and classification tasks \\citep{SA:16}. These are mainly achieved by incorporating feature matching technique in training the generator and multitask training of the discriminator, that plays an additional role as a classifier. \n\nInspired by the semi-supervised framework described by \\cite{SA:16}, we propose a novel model for triplet network learning. The feature layer that is involved in feature matching technique during generative part of the training is further used as a triplet output in supervised part of training the discriminator. As a consequence, the feature representation of the triplet output is enriched by the consequences of GAN's training. This benefit is especially observed when the access to the labeled triplets is limited.   \n\nIn the paper, we make the following contributions: 1) we propose the novel method for training triplet network using GAN framework; 2) we show, how to obtain stronger representation from GAN model if we have access to some portion of labeled triplets; 3) we show in the experiment, that with only $16$ features our model has been able to produce competitive classification performance using the simple $k$-nn method.     \n\n\\section{Triplet Networks with GAN }\n\\label{architecture}\n\n\\subsection{Triplet Networks}\n\\label{triplet}\n\nTriplet networks \\citep{HO:15} are one of the most commonly used techniques in deep learning metric \\citep{YO:16,ZH:16}. The main idea that stays behind them is to take the set of triplets (training data), where each triplet is composed of  query $\\mathbf{x}^{q}$ (assumed to be positive), positive $\\mathbf{x}^{+}$ and negative $\\mathbf{x}^{-}$ examples, and train the network $T(\\mathbf{x})$ to construct the effective feature extractor. The model makes use of the probability ($p_{T(\\mathbf{x}^{q},\\mathbf{x}^{+},\\mathbf{x}^{-})}:=p_T$) that the distance of the query example to the negative example is greater than its distance to the positive one: $p_T = \\frac{\\exp\\{{d_T(\\mathbf{x}^{q},\\mathbf{x^{-}})\\}}}{\\exp\\{{d_T(\\mathbf{x}^{q},\\mathbf{x^{-}})\\}} + \\exp\\{{d_T(\\mathbf{x}^{q},\\mathbf{x^{+}})\\}}}$, where $d_T(\\mathbf{x}_1,\\mathbf{x}_2)$ is defined as Euclidean distance of the outputs of $T(\\cdot)$: $d_T(\\mathbf{x}_1,\\mathbf{x}_2) = || T(\\mathbf{x}_1) - T(\\mathbf{x}_2 )||_2$.\n\n\nThe loss function for a single triplet $(\\mathbf{x}^{q}_n,\\mathbf{x}^{+}_n,\\mathbf{x}^{-}_n)$ can be defined as $L_T = -\\log{(p_{T(\\mathbf{x}^{q}_n,\\mathbf{x}^{+}_n,\\mathbf{x}^{-}_n)})}$. We propose to use slightly different lost function from that was defined by \\cite{HO:15} because we want to be consistent with \\emph{log-prob} discriminative part of learning for GAN model. \n\n\n\\subsection{Generative Adversarial Nets (GAN)}\n\\label{gan}\n\nThe main idea of GANs is based on game theory and assumes training of two competing network structures, generator $G(\\mathbf{z})$ and discriminator $D(\\mathbf{x})$. The goal of GANs is to train generator $G$ to sample from the data distribution $p_{data}(\\mathbf{x})$ by transforming the vector of noise $\\mathbf{z}$. The discriminator $D$ is trained to distinguish the samples generated by $G$ from the samples from $p_{data}(\\mathbf{x})$. The training problem formulation is as follows:  $\\min_{G} \\max_{D} V(D,G) = \\mathrm{E}_{\\mathbf{x} \\sim p_{data}(\\mathbf{x})}{[\\log{(D(\\mathbf{x}}))]} +\\mathrm{E}_{z \\sim p_{\\mathbf{z}}(\\mathbf{z})}{[\\log{(1 - D(G(\\mathbf{z})}))]}$, where $p_{\\mathbf{z}}(\\mathbf{z})$ is prior over $\\mathbf{z}$.  \n\nThe model is usually trained with the SGD approach by sampling minibatch of fakes from $p_{\\mathbf{z}}(\\mathbf{z})$  and minibatch of data samples from $p_{data}(\\mathbf{x})$. They are used to maximize $V(D,G)$ with respect to parameters of $D$ by assuming a constant $G$, and then minimize $V(D,G)$  with respect to parameters of $G$ by assuming a constant $D$. The procedure is repeated for each of the epochs. \n\n\\subsection{Semi-supervised training with GAN}\n\\label{ssgan}\n\nThe most recent studies on GAN show the great benefit of using them in semi-supervised (and supervised) classification. The main idea of this approach is to incorporate the discriminator $D$ into an additional classification task. As a consequence, $D$ is trained both to distinguish fake and true samples and to classify the examples to one of the predefined classes in the classification. \n\nThe loss function for the discriminator can be defined as a sum of supervised and unsupervised parts, $L_D = L_s + L_{u}$. The supervised part is defined as $L_{s}= -\\mathrm{E}_{\\mathbf{x},y \\sim p_{data}(\\mathbf{x},y)}{[\\log{(p(y|\\mathbf{x})})]}$, where $y$ denotes class label. The unsupervised part of the criterion is defined as $L_u = - V(D,G)$.\n\nTo improve the quality of prediction and to obtain better feature representation, \\cite{SA:16} recommend, that generator $G$ is trained using so-called \\emph{feature matching} procedure. The objective to train the generator $G$ is $L_G = ||\\mathrm{E}_{\\mathbf{x} \\sim p_{data}(\\mathbf{x})}\\mathbf{f}(\\mathbf{x}) - \\mathrm{E}_{z \\sim p_{\\mathbf{z}}(\\mathbf{z})}{\\mathbf{f}(G(\\mathbf{z}))}||_2^2$, where $\\mathbf{f}(\\mathbf{x})$ denotes the intermediate activation layer of the discriminator.\n \n\\subsection{Triplet training with GAN}\n\\label{sstgan}\n\nIn our approach we make use of benefits of using triplet networks for metric learning and the effectiveness of semi-supervised GAN in classification tasks. The main idea behind our approach is to incorporate discriminator in a metric learning task instead of involving it in classification. As a consequence, we aim at obtaining a good feature representation in generative part of training, but also in supervised part of training the discriminator. \n\nWe assume the output of the proposed triplet network is characterized by $M$ features, $T(\\mathbf{x}) = [t_1(\\mathbf{x}),\\dots,t_M(\\mathbf{x})]^T$. Inspired by \\cite{SA:16}, we define the discriminator $D_T(\\mathbf{x}) $ in the following manner: $D_T(\\mathbf{x}) = \\frac{\\sum_{m=1}^{M}\\exp(t_m(\\mathbf{x})) }{\\sum_{m=1}^{M}\\exp(t_m(\\mathbf{x}))+1}$. It indicates the posteriori probability of being real examples, while the posteriori probability for fake examples is just $1 -D_T(\\mathbf{x})$. Certainly, we can train an additional layer on top of this feature layer as in common practice. However, we find that this does not give clear advantage over the $D_T(\\mathbf{x})$ defined as above, since we essentially only need a mapping from the features to the probability. So this $D_T(\\mathbf{x})$ is employed in our model for the sake of efficiency.\n\nThe loss function used for training triplet discriminator is composed of triplet-based and unsupervised components $L_{TD} = L_{Ts} + L_{Tu}$. We define the triplet based component as follows $L_{Ts}= -\\mathrm{E}_{\\mathbf{x}^{q},\\mathbf{x}^{+},\\mathbf{x}^{-} \\sim p_{data}(\\mathbf{x}^{q},\\mathbf{x}^{+},\\mathbf{x}^{-} )}{[\\log{(p_T)}]}$ and the unsupervised part remains unchanged $L_{Tu} = - V(D_T,G)$. \n\nFor supervised loss component ($L_{Ts}$) we sample labeled triplets from data distribution $p_{data}(\\mathbf{x}^{q},\\mathbf{x}^{+},\\mathbf{x}^{-} )$. The unsupervised loss component is trained in classical manner using triplet discriminator $D_T$.\n\nThe generative part of the model utilizes the feature matching, where the vector of matched outputs represents the output of triplet network, $T(\\mathbf{x}) = \\mathbf{f}(\\mathbf{x})$. The triplet output is further used in $k$-nn-based search with a Euclidean distance.  \n  \n\\section{Experiments}\n\nThe goal of the experiment is to evaluate the quality of the triplet model using benchmark datasets. We make use of the structures proposed by \\cite{SA:16} and modify the output layer to triplet manner. The number of output features $M$ is set as low as $16$. We use standard data split for the benchmark datasets, $50000$ training and $10000$ test examples (Cifar10), $60000$ training and $10000$ test cases (MNIST). We take only $100$ labeled examples for MNIST, and $5000$ for Cifar10 from their training splits. The weights of our model are initialized using the GAN model pretrained in unsupervised manner. $500$ epochs are used during using MNIST dataset and $700$ epochs for Cifar10.  \n\n\\begin{table}[t]\n\\caption{Classification accuracy on benchmark datasets. $9$-nn is used for evaluation on $16$ features. }\n\\label{tab}\n\\begin{center}\n \\begin{tabular}{|l|c|c|}\n\\emph{Method} & \\textbf{MNIST} & \\textbf{Cifar10} \\\\\n    \\hline\n  Only Triplet & 81.27 & 70.76 \\\\\n  Only GAN & 96.48 & 55.39 \\\\\n  Our method & 97.50 & 80.97 \\\\\n  \\end{tabular}\n\\end{center}\n\\end{table}\n\nThe number of possible triplets for Cifar10 was large, so we applied simple hard-mining technique before each of the training epochs. For each of $K$ labeled query examples we determine the $N$ positive examples that are, according to model $T$ on current training stage, the most distant to the considered example. We select also $N$ negatives that are the closest to the considered example. As a consequence, the query example, the most distant positive and the closest negative form one triplet. We continue creating the triplets by taking closer positive and more distant negative to obtain $K$ triplets for one query example. Using selected positives and negatives we form the $N\\cdot K$ triplets to balance the number of unsupervised data. For MNIST the total number of possible triplets is $90000$, which is not very large. So we simply randomly select $60000$ for each epoch. \n          \n\nThe results of initial experiments are presented in Table \\ref{tab}. We compared our approach with the triplet network trained only on labeled examples and the GAN model that is trained only on unlabeled data.  The proposed approach outperforms the two methods and the improvement on Cifar10 is significant. In addition, we trained our model using all $50000$ labeled examples for Cifar10 data ($16$ features, $9$-nn classifier) and we obtained the accuracy of classification equal $88.04$. This result is promising comparing to the classification accuracy of triplet network reported by \\cite{HO:15} ($87.10$), taking into account that we use only 16 features, $9$-nn classifier, no data augmentation and no models pretrained on external data are performed. Currently, we aim at better model selection and better balance during training procedure to improve the performance in further. We also investigated the case of more features and obtained $98.68$ classification accuracy on MNIST dataset ($256$ features, $9$-nn classifier).\n       \n\n\\section{Conclusion}\n\nIn this work we present a novel framework for learning triplet models that makes use of GAN model to obtain better feature representation. The presented model can be easily applied for image retrieval tasks, where only a small portion of labeled data could be accessed. This model shows promising results even when the number of features is low, which is computationally desirable for the methods like $k$-nn search. Very recently, we noticed that the paper \\citep{AR:17} on improving GAN with Wasserstein distance could be beneficial to our research. We plan to incorporate this approach into our model to improve its performance in further.   \n\n\\subsubsection*{Acknowledgments}\nThis work was undertaken with financial support of a Thelxinoe grant in the context of the EMA2\/S2 THELXINOE: Erasmus Euro-Oceanian Smart City Network project, grant reference number: 545783-EM-1-2013-1-ES-ERA MUNDUS-EMA22.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn non-central heavy-ion collision(HIC), large vorticity and strong magnetic field are expected to be generated in extremely hot quark gluon plasma(QGP). Straightforward electromagnetical(EM) computation shows the magnetic field would reach about $O(10^{14})T$\\cite{Kharzeev:2007jp}\nin the early stage of HIC, while kinetic and hydrodynamic simulations\\cite{Becattini:2007sr,Jiang:2016woz} indicate the  local vorticity would exceed $0.5 fm^{-1}$ with the total angular momentum of QGP at a range of $O(10^{4})-O(10^{5})\\hbar$. Known as the Barnett and magnetization effects, spin particles are polarized by these pseudo vector field and thus distribute differently from the normal thermal distributions. Besides chiral effects induced by such pseudo vector fields\\cite{Kharzeev:2007tn, Son:2009tf, Kharzeev:2010gr}, studies on these distribution modifications would be helpful to understand the hadronization mechanism of the strong interaction as well. Inspired by the large amplitude and retention by the angular momentum conservation, vorticity has attracted more and more interests recently.\n\nComparing with magnetic field effects, the rotation-related effects are electric charge blind, and only involve kinetic properties of the QGP and  strong interaction which we are mostly interested in. Experimentally, in order to screen out the EM effects neutral particles with finite spin numbers are chosen as carriers of the vorticity polarization effects. As it is difficult to detect the chargeless particle directly the distribution of its charged daughter particle serves as an alternative observable for the global polarization effect. With the help of the $\\Lambda$ hyperon the average magnitude of the vorticity of QGP has been extracted by the STAR collaboration\\cite{STAR:2017ckg}. In these measurements the expectation of $\\Lambda$ polarization as well as the vorticity behavior of collision energy have been confirmed as well. All the results seem to be understandable by considering the energy shift induced by the voriticity polarization to spins. However the theory became a little vague when the $K^{*0}$ and $\\phi$ mesons' measurements were presented in \\cite{Acharya:2019vpe}. The mismatch between these measurement indicates the fine structure of hadrons may play an non-negligible role in polarization processes.\n\nThe mass is one of the most fundamental attributes of a hadron. For a composite particle it will be modified by the single-particle dispersion relation of the fundamental degree of freedom as well as the interaction among them. Studies of hadron masses would help us to discover many clues of the environment where hadrons are born. As a well-known example, $\\sigma$ meson and pion masses would change with the growing temperature and chemical potential because of chiral restoration~\\cite{Klevansky:1992qe}. And recently people have studied the vector meson $\\rho$ mass in external magnetic field as well by taking the polarization effect on quarks into account. A lattice calculation demonstrates that charged $\\rho$ meson mass decreases firstly and increases finally, leaving a minimum around $eB\\simeq 1GeV^{2}$~\\cite{Hidaka:2012mz}. And by using effective models, such as the Nambu--Jona-Lasinio(NJL) model with vector channel, the $\\rho$ meson with different spin components have been studied~\\cite{Liu:2014uwa,Liu:2018zag}. Therefore a natural question is what about the mass behavior under a background vorticity field which is a little like the magnetic case at the first sight. For the rotating effect, the co-rotating frame~\\cite{Yamamoto:2013zwa} is usually adopted and a nontrivial spin connection term will be introduced~\\cite{Matsuo:2015}, which serves as a polarization term for angular momentums. With this extended NJL model it is suggested that chiral phase transition would take place as angular velocity increasing~\\cite{Jiang:2016wvv}. Furthermore, people have established more complicated phase diagrams which combine rotation and other physical conditions such as chemical potential, isospin and magnetic field~\\cite{Wang:2018sur,Zhang:2018ome,Chen:2015hfc}. In those NJL models, at the quark level, the rotation always behaves as an effective chemical potential. This analogy has been understood with a  Hamiltonian shifting $\\hat{H}\\rightarrow\\hat{H}-\\vec{\\omega}\\cdot\\hat{J}$ and the latter term may be corresponding to an effective chemical potential~\\cite{Chen:2015hfc,Matsuo:2012wv}. At the same quark level, holographic models also contribute to elaborate the property of rotating quark matter by setting up a four-dimensional AdS-Kerr-Newman black hole to construct a rotation-magnetism analogy~\\cite{McInnes:2016dwk}. While for the composite hadrons, such as vector mesons, there have been few works on the mass behaviors.\n\n\n\nIn this paper, we focus on the scalar and the vector meson and investigate their masses under the rotation at finite chemical potential. In Sec. ~\\ref{NJL model under rotating frame}, in order to deal with both the finite temperature and density cases, we introduce the two- flavor NJL model with vector channel in the co-rotating frame. In this framework, we generate the dynamical quark mass with chiral symmetry spontaneous breaking and construct scalar and vector masons with the dressed quark propagator and extract the corresponding masses with the well-known random phase approximation(RPA) in Sec. ~\\ref{Scalar and vector meson mass under rotating frame} and show their numerical results in Sec. ~\\ref{Numerical results and discussion}. Because of the rich phase structure at large chemical potential we only study the  range of $\\mu_q<200$MeV in this work and leave the discussion of the rotating color superconductivity in our following works.  We have found that masses of scalar mesons are controlled by the chiral phase transition which could be driven by temperature, density and rotation. While the vector meson, which carries net angular momentum, is governed by the polarization effect on the total angular momentum before chiral symmetry restoration. At large angular velocity the  mass of spin component $s_z=1$ for the vector meson vanishes. This indicates the macroscopic condensate of spin component $s_z=1$ of vector meson $\\langle \\rho^{s_z=1}\\rangle$  thus spontaneous spin polarization would be induced in the ultra-fast rotating system. In Sec.~\\ref{Conclusion},  we summarize our main results and give an outlook.\n\n\\section{NJL model in co-rotating frame}\n\\label{NJL model under rotating frame}\nNJL model is an effective model with 4-fermion interaction which is widely used to study quark-quark and quark-antiquark pairing which corresponding to chiral phase transition, superfluidity and superconductivity and so on. Besides the usual scalar channels we take account the vector channels in order to construct the vector $\\rho$ mesons. The Lagrangian of the two-flavor NJL model in the co-rotating frame is given by~\\cite{Wang:2018sur,Bernard:1988db}:\n\\begin{equation}\n\\mathcal{L} = \\bar{\\psi}[i\\bar{\\gamma}^{\\mu}(\\partial_{\\mu}+\\Gamma_{\\mu})-m]\\psi+G_S[(\\bar{\\psi}\\psi)^2+(\\bar{\\psi}i\\gamma_5\\vec{\\tau}\\psi)^2]-G_V[(\\bar{\\psi}\\gamma_{\\mu}\\psi)^2+(\\bar{\\psi}\\gamma_{\\mu}\\gamma_5\\psi)^2],\n\\end{equation}\nwhere $m$ is the current quark mass. $G_{S}$ and $G_{V}$ are the coupling constants in the scalar and vector channels, respectively. In the curved co-rotating frame the gamma matrices $\\bar{\\gamma}^{\\mu}$ should be defined according to the corresponding Clifford algebra. The curved gamma matrices are connected with the flat ones with the vierbein as $\\bar{\\gamma}^{\\mu}=e_{a}^{\\ \\mu}\\gamma^{a}$ and where $e_{a}^{\\ \\mu}$ should be chosen to satisfy $g_{\\mu\\nu}=\\eta_{ab}e^{a}_{\\ \\mu}e^{b}_{\\ \\nu}$, where $\\eta_{ab}$ is the metric of flat space-time and $\\gamma^{a}$ is flat gamma matrices. In our case a simple enough choice is $e^{a}_{\\ \\mu}=\\delta^a_{\\ \\mu}+  \\delta^a_{\\ i}\\delta^0_{\\ \\mu} \\, v_i$ and $e_{a}^{\\ \\mu}=\\delta_a^{\\ \\mu} -  \\delta_a^{\\ 0}\\delta_i^{\\ \\mu} \\, v_i$, where $v_i$ is the linear velocity $\\vec{v} =\\vec{\\omega}\\times\\vec{x}$ under the presence of  a constant angular velocity $\\vec{\\omega}$. The so-called spinor connection is given by $\\Gamma_\\mu=\\frac{1}{4}\\times\\frac{1}{2}[\\gamma^a,\\gamma ^b] \\, \\Gamma_{ab\\mu}$, where $\\Gamma_{ab\\mu}=\\eta_{ac}(e^c_{\\ \\sigma} G^\\sigma_{\\ \\mu\\nu}e_b^{\\ \\nu}-e_b^{\\ \\nu}\\partial_\\mu e^c_{\\ \\nu})$ and $G^\\sigma_{\\ \\mu\\nu}$ is the usual Christoffel connection determined by $g_{\\mu\\nu}$~\\cite{Yamamoto:2013zwa,Jiang:2016wvv,Matsuo:2015}. In the slow velocity limit $|\\vec{\\omega}\\times\\vec{x}|\\ll c$ we could only keep the $O(\\vec{v})$ terms which can be reduced to the ordinary polarization form as $\\vec{\\omega}\\cdot\\vec{J}$, where $\\vec{J}=\\vec{x}\\times\\vec{p}+\\vec{S}$ is the total angular momentum~\\cite{Jiang:2016wvv,Matsuo:2015} and $\\vec{S}=\\frac{1}{2}\\left(\n                      \\begin{array}{cc}\n                        \\vec{\\sigma} & 0 \\\\\n                        0 & \\vec{\\sigma} \\\\\n                      \\end{array}\n                    \\right)\n$ is the spin operator.\n\nApplying the mean field approximation and choosing the direction of rotation as the $z$-axis, the bilinear part of the Lagrangian at finite chemical potential is given by~\\cite{Wang:2018sur}\n\\begin{equation}\n\\mathcal{L}=\\bar{\\psi}[i\\gamma^{\\mu}\\partial_{\\mu}+\\gamma^{0}(\\omega \\hat{J_{z}}+\\mu)-M]\\psi-\\frac{(M-m)^{2}}{4G_{S}},\n\\end{equation}\nwhere $J_{z}$ is the third component of total angular momentum $\\vec{J}$, and $\\mu$ is the quark chemical potential, it is seen that the angular velocity plays similar role as the chemical potential, and $M$ is the constituent quark mass which is given by the chiral condensate as $M = m - 2G_S\\left<\\bar\\psi\\psi\\right>$.\nThe general grand potential is given by~\\cite{Jiang:2016wvv,Wang:2018sur}:\n\\begin{eqnarray}\n\\label{omg}\n\\Omega(T,\\mu,M,\\omega) &=&\\int d^3 \\mathbf{r}~ \\bigg\\{\\frac{(M-m)^2}{4G_S} \\nonumber \\\\\n     & & -\\frac{N_c N_f}{16\\pi^2} T \\sum_n\\int dk_t^2 \\int dk_z[J_n(k_t r)^2+J_{n+1}(k_t r)^2]\\left[ \\ln(1+e^{(E_k-(n+\\frac{1}{2})\\omega-\\mu)\/T})\\right.\\nonumber\\\\\n& & + \\left. \\ln(1+e^{-(E_k-(n+\\frac{1}{2})\\omega-\\mu)\/T})+\\ln(1+e^{-(E_k+(n+\\frac{1}{2})\\omega+\\mu)\/T})+ \\ln(1+ e^{(E_k+(n+\\frac{1}{2})\\omega+\\mu)\/T})\\right]\\bigg\\}.\\nonumber\\\\\n\\end{eqnarray}\nwhere $E_k=\\sqrt{k_t^2+k_z^2+M^2}$ and $k_{t, z}$ are the transverse and longitudinal momentum respectively. Obviously the local potential approximation $\\partial_r M(r)\\simeq 0$ has been adopted during solving the eigen modes. In the following computation we choose $N_c=3$ and $N_f=2$. In this work we will neglect the four-fermion contributions to the ground state, which means the chiral condensate is completely computed by the gap equation as\n$\\frac{\\partial \\Omega}{\\partial M} = 0$\nwith the constraint $\\frac{\\partial^2 \\Omega}{\\partial M^2} > 0$. In Sec. \\ref{Numerical results and discussion}, we will show the numerical result of the constituent quark mass $M$. It serves as the environment where mesons are given birth, and thus modifies their masses. In the mean field approximation the gap equation is just the one-loop diagram of the quark propagator which reads as\n\\begin{equation}\n\\label{propagator}\n\\begin{aligned}\nS(\\tilde{r};\\tilde{r'})&=\\frac{1}{(2 \\pi )^2}\\sum _n\\int\\frac{d k_0}{2\\pi } \\int k_t d k_t \\int d k_z \\frac{e^{i n \\left(\\phi -\\phi '\\right)}e^{-i k_0 \\left(t-t'\\right)+i k_z \\left(z-z'\\right)}}{[k_{0}+(n+\\frac{1}{2})\\omega]^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\\epsilon} \\\\\n&\n\\times\\{[[k_{0}+(n+\\frac{1}{2})\\omega]\\gamma^{0}-k_{z}\\gamma^{3}+M][J_{n}(k_{t}r)J_{n}(k_{t}r')\\mathcal{P}_{+}+e^{i (\\phi-\\phi)'}J_{n+1}(k_{t}r)J_{n+1}(k_{t}r')\\mathcal{P}_{-}]\\\\\n&-i\\gamma^{1}k_{t}e^{i \\phi}J_{n+1}(k_{t}r)J_{n}(k_{t}r')\\mathcal{P}_{+}-\\gamma^{2}k_{t}e^{-i\\phi '}J_{n}(k_{t}r)J_{n+1}(k_{t}r')\\mathcal{P}_{-}\n\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{P}_{\\pm}=\\frac{1}{2}(1\\pm i\\gamma^{1}\\gamma^{2})$ are projection operators and $\\tilde{r}=(t,r,\\theta,\\phi)$ are the coordinates in the cylindrical frame.\n\n\\section{Scalar and vector meson mass under rotation}\n\\label{Scalar and vector meson mass under rotating frame}\n\\subsection{The scalar meson}\nIn the NJL model, meson is regarded as  $q \\bar q $ bound states or resonances, which can be obtained from the quark-antiquark scattering amplitude\n\\cite{Buballa:2003qv,He:1997gn,Rehberg:1995nr}. In the random phase approximation (RPA), the full propagator of $\\sigma$ meson $D_{\\sigma}(q^2)$ can be expressed to leading order in $1\/N_c$ as an infinite sum of quark-loop chains:\n\\begin{equation}\\label{meson propagator}\n\tD_{\\sigma}(q^2)=\\frac{2G_{S}}{1-2G_{S}\\Pi_{s}(q^2)},\n\\end{equation}\nwhere $\\Pi_{s}(q^2)$ is the quark one-loop polarization function and takes the form of\n\\begin{eqnarray}\n\\label{pola}\n\\Pi_{s}(q)=-i \\int d^{4}\\tilde{r}Tr_{sfc}[i S(0;\\tilde{r})i S(\\tilde{r};0)]e^{i q\\cdot \\tilde{r}},\n\\end{eqnarray}\n where $Tr_{sfc}$ means trace in spin, flavor and color space.  After a tedious calculation in Appendix.~\\ref{appendix:a}, the polarization function could be simplified as this form\n\\begin{equation}\n\\begin{aligned}\n\\Pi_{s}(q^2)&=\n-2 i N_{f}N_{c}\\int \\frac{d^{4} p}{(2\\pi)^{4}} \\\\\n&\\times\n\\left\\{ \\frac{\\left(p_{0}+q_{0}+\\frac{1}{2}\\omega \\right) \\left(p_{0}+\\frac{1}{2}\\omega \\right)+M^2-(\\vec{p}+\\vec{q})\\cdot \\vec{p}}\n{\\left[\\left(p_{0}+q_{0}+\\frac{1}{2}\\omega\\right)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}\\right]\n\\left[\\left(p_{0}+\\frac{1}{2}\\omega\\right)^{2}-\\vec{p}^{2}-M^{2}\\right]}\\right.\\\\\n&+ \\left.\\frac{\\left(p_{0}+q_{0}-\\frac{1}{2}\\omega \\right) \\left(p_{0}-\\frac{1}{2}\\omega \\right)+M^2-(\\vec{p}+\\vec{q})\\cdot \\vec{p}}\n{\\left[\\left(p_{0}+q_{0}-\\frac{1}{2}\\omega\\right)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}\\right]\n\\left[\\left(p_{0}-\\frac{1}{2}\\omega\\right)^{2}-\\vec{p}^{2}-M^{2}\\right]}\n\\right\\}.\n\\end{aligned}\n\\end{equation}\nIf we use finite temperature theory with chemical potential \\cite{Kapusta}, the polarization function will be:\n\\begin{equation}\n\t\\Pi_{s}(\\vec{q},i\\nu_{n})=2N_{f} N_{c}T \\sum_{s=\\pm}\\sum_N\\int \\frac{d^{3}\\vec{p}}{(2\\pi)^{3}}\n\t\\frac{[(i \\tilde{\\omega}_{N}+i \\nu_{n})+\\frac{1}{2}s\\omega+\\mu][i \\tilde{\\omega}_{N}+\\frac{1}{2}s\\omega+\\mu]+M^{2}-(\\vec{p}+\\vec{q})\\cdot\\vec{p}}{[(i \\tilde{\\omega}_{N}+i \\nu_{n}+\\frac{1}{2}s\\omega+\\mu)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}][(i \\tilde{\\omega}_{N}+\\frac{1}{2}s\\omega+\\mu)^{2}-\\vec{p}^{2}-M^{2}]},\n\\end{equation}\nwhere $\\tilde{\\omega}_{N}=(2N+1)\\pi T$ is  Matsubara frequency.\nConsidered analytic continuation $\\Pi_{s}(\\vec{q},\\tilde{\\nu})=\\Pi_{s}(\\vec{q},i\\nu_{n})|_{\\tilde{\\nu}+i \\eta}$ and set $\\vec{q}=0$, an explicit form of $\\Pi_{s}(0,\\tilde{\\nu})$ is shown in Appendix~\\ref{appendix:a}\n\nFrom the pole of above propagator in Eq.(\\ref{meson propagator}), the $\\sigma$ mass can be obtained by solving:\n\\begin{equation}\n\t1-2 G_{S} \\Pi_{s}(0,\\tilde{\\nu})=0,\n\\end{equation}\nWe have similar operation for pseudoscalar meson $\\pi$. The operators in polarization functions are defined as $\\tau^{\\pm}=\\frac{1}{\\sqrt{2}}(\\tau_{1}\\pm i \\tau_{2})$ where $\\tau_{i}$ are Pauli Matrice. In polarization functions, we choose $\\tau^a=\\tau^3,\\tau^b=\\tau^3$ for neutral pion and $\\tau^a=\\tau^+,\\tau^b=\\tau^-$ for charged pion. However, polarization functions have the same form for different charged mesones. \n\\begin{equation}\n\\begin{aligned}\n\\Pi_{ps}(q^2)&=-i \\int d^{4}\\tilde{r}Tr_{sfc}[i \\gamma^{5}\\tau^{a}i S(0;\\tilde{r})i \\gamma^{5}\\tau^{b}i S(\\tilde{r};0)]e^{i q\\cdot \\tilde{r}}\\\\\n&=4 i N_{f}N_{c}\\int \\frac{d^{4} p}{(2\\pi)^{4}} \\\\\n&\\times\n\\left\\{ \\frac{\\left(p_{0}+q_{0}+\\frac{1}{2}\\omega \\right) \\left(p_{0}+\\frac{1}{2}\\omega \\right)-M^2-(\\vec{p}+\\vec{q})\\cdot \\vec{p}}\n{\\left[\\left(p_{0}+q_{0}+\\frac{1}{2}\\omega\\right)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}\\right]\n\\left[\\left(p_{0}+\\frac{1}{2}\\omega\\right)^{2}-\\vec{p}^{2}-M^{2}\\right]}\\right.\\\\\n&+ \\left.\\frac{\\left(p_{0}+q_{0}-\\frac{1}{2}\\omega \\right) \\left(p_{0}-\\frac{1}{2}\\omega \\right)-M^2-(\\vec{p}+\\vec{q})\\cdot \\vec{p}}\n{\\left[\\left(p_{0}+q_{0}-\\frac{1}{2}\\omega\\right)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}\\right]\n\\left[\\left(p_{0}-\\frac{1}{2}\\omega\\right)^{2}-\\vec{p}^{2}-M^{2}\\right]}.\n\\right\\}\n\\end{aligned}\n\\end{equation}\nFor finite temperature formalism with chemical potential, the polarization function will be:\n\\begin{equation}\n\t\\Pi_{ps}(\\vec{q},i\\nu_{n})=-4N_{f} N_{c}T \\sum_{s=\\pm}\\sum_N\\int \\frac{d^{3}\\vec{p}}{(2\\pi)^{3}}\n\t\\frac{[(i \\tilde{\\omega}_{N}+i \\nu_{n})+\\frac{1}{2}s\\omega+\\mu][i \\tilde{\\omega}_{N}+\\frac{1}{2}s\\omega+\\mu]-M^{2}-(\\vec{p}+\\vec{q})\\cdot\\vec{p}}{[(i \\tilde{\\omega}_{N}+i \\nu_{n}+\\frac{1}{2}s\\omega+\\mu)^{2}-(\\vec{p}+\\vec{q})^{2}-M^{2}][(i \\tilde{\\omega}_{N}+\\frac{1}{2}s\\omega+\\mu)^{2}-\\vec{p}^{2}-M^{2}]}.\n\\end{equation}\nConsidered analytic continuation $\\Pi_{ps}(\\vec{q},\\tilde{\\nu})=\\Pi_{ps}(\\vec{q},i\\nu_{n})|_{\\tilde{\\nu}+i \\eta}$ and set $\\vec{q}=0$, an explicit form of $\\Pi_{ps}(0,\\tilde{\\nu})$ is shown in Appendix~\\ref{appendix:a}\n\nFrom the pole of above propagator, the pion mass can be obtained by solving:\n\\begin{equation}\n\t1-2 G_{S} \\Pi_{ps}(0,\\tilde{\\nu})=0.\n\\end{equation}\n\n\\subsection{The $\\rho$ meson}\nFollowing the Ref.~\\cite{Liu:2014uwa}, we construct the vector meson in a similar way with the rotation-modified quark propagators. For the 2-flavor model we take the vector $\\rho$ meson for example, its 1-loop polarization function reads as\n\\begin{equation}\n\\Pi^{\\mu\\nu,ab}(q)=-i \\int d^{4}\\tilde{r}Tr_{sfc}[i \\gamma^{\\mu}\\tau^{a}S(0;\\tilde{r})i \\gamma^{\\nu}\\tau^{b}S(\\tilde{r};0)]e^{i q\\cdot \\tilde{r}}.\n\\end{equation}\nAs there is no isospin breaking in the quark propagators $S(0;\\tilde{r})$, the polarization functions of charged and neutral $\\rho$ mesons are supposed to be the same under rotation. Nonzero elements of the matrix reads as\n\\begin{equation}\n\\Pi^{\\mu\\nu}_{\\rho}=\\left(\n                      \\begin{array}{cccc}\n                        0 & 0 & 0 & 0 \\\\\n                        0 & \\Pi^{11} & \\Pi^{12} & 0 \\\\\n                        0 & \\Pi^{21} & \\Pi^{22} & 0 \\\\\n                        0 & 0 & 0 & \\Pi^{33} \\\\\n                      \\end{array}\n                    \\right).\n\\end{equation}\nThe explicit expressions of matrix elements are shown in Appendix~\\ref{appendix:b}. The analysis of the Lorentz structure suggests the tensor can be decomposed according to its polarization directions as follows\n\\begin{equation}\n\\Pi^{\\mu\\nu}_{\\rho}=A_{1}^{2}P^{\\mu\\nu}_{1}+A_{2}^{2}P^{\\mu\\nu}_{2}+A_{3}^{2}L^{\\mu\\nu}+A_{4}^{2}u^{\\mu}u^{\\nu},\n\\end{equation}\nwhere $u^{\\mu}$ is the four momentum in the rest frame. $u^\\mu=(1,0,0,0)$ is a unit vector. And the projection operators are given as:\n\\begin{equation}\n\\begin{aligned}\nP^{\\mu\\nu}_{1}&=-\\epsilon^{\\mu}_{1}\\epsilon^{\\nu}_{1},(s_{z}=-1 \\text{ for } \\rho \\text{ meson }),\\\\\nP^{\\mu\\nu}_{2}&=-\\epsilon^{\\mu}_{2}\\epsilon^{\\nu}_{2},(s_{z}=+1\\text{ for } \\rho \\text{ meson }),\\\\\nL^{\\mu\\nu}&=-b^{\\mu}b^{\\nu},(s_{z}=0 \\text{ for } \\rho \\text{ meson }).\n\\end{aligned}\n\\end{equation}\nwhere in flat frame $\\epsilon^{\\mu}_{1}=\\frac{1}{\\sqrt{2}}(0,1,i,0)$ and $\\epsilon^{\\mu}_{2}=\\frac{1}{\\sqrt{2}}(0,1,-i,0)$ are the right and left-hand polarization vectors respectively. And $b^{\\mu}=(0,0,0,1)$ is the direction of rotation. As a result the $\\rho$ meson propagator can be decomposed in the similar way as:\n\\begin{equation}\nD^{\\mu\\nu}_{\\rho}(q^{2})=D_{1}(q^{2})P^{\\mu\\nu}_{1}+D_{2}(q^{2})P^{\\mu\\nu}_{2}+D_{3}(q^{2})L^{\\mu\\nu}+D_{4}(q^{2})u^{\\mu}u^{\\nu},\n\\end{equation}\nwhere coefficients $D_{i}$ have the RPA summation forms as:\n\\begin{equation}\nD_{i}(q^2)=\\frac{2G_{V}}{1+2G_{V}A_{i}^2}.\n\\end{equation}\nAgain the momentum poles here are corresponding to masses of vector $\\rho$ mesons which are solutions to equations:\n\\begin{equation}\n\\label{vectorpole}\n1+2G_{V}A_{i}^{2}=0,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{coeff}\n\\begin{aligned}\nA_{1}^{2}&=-(\\Pi_{11} - i \\Pi_{12}),(s_{z}=-1 \\text{ for } \\rho \\text{ meson }),\\\\\nA_{2}^{2}&=-\\Pi_{11} - i \\Pi_{12},(s_{z}=+1\\text{ for } \\rho \\text{ meson }),\\\\\nA_{3}^{2}&=-\\Pi_{33},(s_{z}=0 \\text{ for } \\rho \\text{ meson }).\n\\end{aligned}\n\\end{equation}\n\n\\section{Numerical results and discussion}\n\\label{Numerical results and discussion}\nIn order to evaluate the mass of $\\rho$ meson at finite chemical potential and relatively large vorticity, we choose the soft cut-off scheme to avoid the leakage of the energy scale. The cut-off function is:\n\n\\begin{equation}\nf_{\\Lambda}(\\bm{p})=\\frac{\\Lambda^{10}}{\\Lambda^{10}+\\bm{p}^{10}},\n\\end{equation}\nwhere $\\Lambda=582$MeV. In numerical calculation, Momentum integrals are understood as follows~\\cite{Frasca:2011zn}\n\\begin{equation}\n\\int \\frac{d \\bm{p}}{2 \\pi}\\rightarrow\\int \\frac{d \\bm{p}}{2 \\pi}f_{\\Lambda}(\\bm{p}).\n\\end{equation}\nThe other parameters are chosen as those in Ref~\\cite{Liu:2014uwa} , i.e. $G_{S}\\Lambda^{2}=2.388$ and $G_{V}\\Lambda^{2}=1.73$ and the current quark mass $m_{0}=5$MeV.\n\nBy neglecting mesons' fluctuations it is easy to solve the gap equation of chiral condensate at finite temperature as well as chemical potential under rotation. As the phase diagram shown in  Ref.~\\cite{Jiang:2016wvv,Wang:2018sur,Chen:2015hfc} the vorticity serves as another kind of chemical potential which would weaken the chiral condensate at finite temperature case and complement the chemical potential at finite density case. As shown in Fig.(\\ref{subfig:mesonmassT3}) there is a crossover at medium temperature along the angular velocity. While at low temperature the increase of chemical potential will change the 1st order chiral restoration to a crossover in Fig.(\\ref{subfig:mesonmass1}), (\\ref{subfig:mesonmass2}) and (\\ref{subfig:mesonmass3}). As the phase structure determines the macroscopic properties of the system it is reasonable to expect that the dependence of meson masses on the angular velocity would be smooth at medium temperature and density systems, while kinked at the 1st order point for the low density systems.\n\n\n\\subsection{The scalar meson}\n Because of carrying no net angular momentum, the profile of scalar meson mass is completely determined by the chiral symmetry in our model. For the zero chemical potential case shown in Fig.(\\ref{subfig:mesonmass1}) and (\\ref{subfig:mesonmassT3}), as angular velocity increases the chiral condensate behaves the same as that in the \\cite{Jiang:2016wvv}. At extremely low temperature the chiral restoration is 1st order and thus the masses keep invariant and then jump together at the critical angular velocity. While in hot matter the condensate keeps melting slowly until the crossover range $\\omega\\sim 0.6$GeV. As the consequence, $\\sigma$ meson mass stays almost static and pions serve as Goldstone particles in the chiral breaking phase. When the $\\omega$ close to the crossover range they approach each other and eventually become almost degenerate because of the chiral symmetry restoration. The behavior at finite density could be understood with chiral symmetry as well by noticing the order of phase transition. As Fig.(\\ref{subfig:mesonmass1}), (\\ref{subfig:mesonmass2}) and (\\ref{subfig:mesonmass3}) shown, at low density, i.e. $\\mu<100$MeV and zero temperature, there is a 1st order gap at $\\omega\\simeq 0.8$GeV for the dependence of the chiral condensate on angular velocity.  After that the pion would break the constraint of Goldstone theorem, that is the mass increases to meet that of $\\sigma$ meson which driven by the chiral symmetry. As the chemical potential increase further, the phase transition would be weaken into the crossover, and the mass dependence on the angular velocity would become more and more smooth as shown in Fig.(\\ref{subfig:mesonmass2}) and (\\ref{subfig:mesonmass3}).\n\n\n\\begin{figure}[!thb]\n\t\\centering\n\t\\includegraphics[width=0.65\\textwidth]{quarkmassV1.pdf}\\\\\n\t\\caption{The constituent quark mass as a function of angular velocity for different chemical potentials. }\n\t\\label{fig:quarkmass}\t\n\\end{figure}\n\nThe constituent quark mass calculated from $M = m - 2G_S\\left<\\bar\\psi\\psi\\right>$ as a function of angular velocity is shown in Fig.\\ref{fig:quarkmass} for different chemical potentials. It is seen that the chiral condensate shows 1st order phase transition\nat large angular velocity for small chemical potentials and at small angular velocity for large chemical potentials, this is in agreement\nwith the results in \\cite{Wang:2018sur}, where it has been observed that the 1st order phase transition shows up in two corners of the 3D $T-\\mu-\\omega$ phase diagram.\n\t\n\\begin{figure}[t!]\n\\subfloat[scalar meson mass as a function of angular velocity at $\\mu=0 MeV$]{\\includegraphics[width=200pt]{scalarmesonmassatmu0.pdf}\\label{subfig:mesonmass1}\\hspace{1pt}}\\hspace{30pt}\n\\subfloat[scalar meson mass as a function of angular velocity at $T=150 MeV$]{\\includegraphics[width=200pt]{scalarmesonmassatT150MeV.pdf}\\label{subfig:mesonmassT3}\\hspace{1pt}}\\\\\n\\subfloat[scalar meson mass as a function of angular velocity at $\\mu=100 MeV$]{\\includegraphics[width=200pt]{scalarmesonmassatmu100MeV.pdf}\\label{subfig:mesonmass2}\\hspace{1pt}}\\hspace{30pt}\n\\subfloat[scalar meson mass as a function of angular velocity at $\\mu=200 MeV$]{\\includegraphics[width=200pt]{scalarmesonmassatmu200MeV.pdf}\\label{subfig:mesonmass3}\\hspace{1pt}}\n\\caption{scalar meson mass as a function of angular velocity at different chemical potential and temperature.}\\label{fig:mesonmass}\n \\end{figure}\n\nFrom the numerical result it is clear that the angular velocity and chemical potential are complementary to each other when driven the chiral restoration. At low chemical potential the critical\/crossover angular velocity is larger and become smaller when the chemical potential is larger.\nHowever it is obvious that the chemical potential and angular velocity are not exactly equivalent to each other. Because physically the chemical potential is the energy shift from the difference between particle and anti-particle, while the shift induced by the rotation polarization is from the spin up and down difference. From this aspect the $\\pm \\frac{1}{2}\\omega$ could be treated as the {\\it spin chemical potential}. Analytically the difference could explicitly observed in the gap equation and the polarization functions as follows\n\n\\begin{equation}\n\\begin{aligned}\n\\Pi_{s}(0,i \\nu_{n})=&N_{f}N_{c}\\sum_{s=\\pm}\\int\\frac{d^{3}\\vec{p}}{(2\\pi)^{3}}\\left[\nRes1(\\vec{p},\\nu_{n}) \\theta \\left(-\\mu -\\frac{s\\omega }{2}+E_{p}\\right) n_{f}\\left(E_{p}-\\mu -\\frac{s\\omega }{2},T\\right)\\right.\\\\\n&+Res3(\\vec{p},\\nu_{n}) \\theta \\left(-\\mu -\\frac{s\\omega }{2}+E_{p}\\right) n_{f}\\left(E_{p}-\\mu -\\frac{s\\omega }{2},T\\right)\\\\\n&-Res1(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right) n_{f}\\left(-E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\n-Res2(\\vec{p},\\nu_{n}) n_{f}\\left(E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\\\\\n&-Res3(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right) n_{f}\\left(-E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\n-Res4(\\vec{p},\\nu_{n}) n_{f}\\left(E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\\\\\n&+Res1(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right)\n+Res3(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right)\\\\\n&-Res1\\left(\\vec{p},\\nu_{n}\\right)-Res3\\left(\\vec{p},\\nu_{n}\\right)\n\\Big],\n\\end{aligned}\n\\end{equation}\n\nwhere $Res1(\\vec{p},\\nu_{n}),Res2(\\vec{p},\\nu_{n}),Res3(\\vec{p},\\nu_{n})$ and $Res4(\\vec{p},\\nu_{n})$ are residues in Eq.(\\ref{residues1}).\n\nAnd the polarization function $\\Pi_{ps}$ is given as: \n\\begin{equation}\n\\begin{aligned}\n\\Pi_{ps}(0,i \\nu_{n})=&N_{f}N_{c}\\sum_{s=\\pm}\\int\\frac{d^{3}\\vec{p}}{(2\\pi)^{3}}\\left[\nRes1'(\\vec{p},\\nu_{n}) \\theta \\left(-\\mu -\\frac{s\\omega }{2}+E_{p}\\right) n_{f}\\left(E_{p}-\\mu -\\frac{s\\omega }{2},T\\right)\\right.\\\\\n&+Res3'(\\vec{p},\\nu_{n}) \\theta \\left(-\\mu -\\frac{s\\omega }{2}+E_{p}\\right) n_{f}\\left(E_{p}-\\mu -\\frac{s\\omega }{2},T\\right)\\\\\n&-Res1'(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right) n_{f}\\left(-E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\n-Res2'(\\vec{p},\\nu_{n}) n_{f}\\left(E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\\\\\n&-Res3'(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right) n_{f}\\left(-E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\n-Res4'(\\vec{p},\\nu_{n}) n_{f}\\left(E_{p}+\\mu +\\frac{s\\omega }{2},T\\right)\\\\\n&+Res1'(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right)\n+Res3'(\\vec{p},\\nu_{n}) \\theta \\left(\\mu +\\frac{s\\omega }{2}-E_{p}\\right)\\\\\n&-Res1'(\\vec{p},\\nu_{n})-Res3'(\\vec{p},\\nu_{n})\\Big].\n\\end{aligned}\n\\end{equation}\nwhere $Res1'(\\vec{p},\\nu_{n}),Res2'(\\vec{p},\\nu_{n}),Res3'(\\vec{p},\\nu_{n})$ and $Res4'(\\vec{p},\\nu_{n})$ are residues in Eq.(\\ref{residues2}).\n\nIt is clear that the functions depend on both the $\\mu\\pm\\omega\/2$ combinations. However it is also reasonable that the critical\nbehavior would take place at one of the angular velocitys which satisfy $E_p-\\mu\\pm\\frac{\\omega}{2}=0$. If we choose both the chemical\npotential and angular velocity positive, the $\\omega=2(m-\\mu)$ part would dominate the critical behavior. Hence the chemical potential\nand angular velocity appear to be complementary to each other on the determination of the critical point.\n\n\n\n\n\\subsection{The $\\rho$ meson}\nTaking the direction of rotation as the $z$-axies, and the three components of a massive vector meson can be represented as $s_{z}=\\pm 1$ and $s_{z}=0$. And the nonzero spin ones would be polarized by the so-call Barnett effect which introduces the shift as $-{\\vec\\omega}\\cdot{\\vec S}$ to the energy levels under rotation. In our 2-flavor model we take the $\\rho$ meson for example to explore the rotation-induced energy shift with the self-consistent numerical calculations at the quark level. Fig.\\ref{fig:rhomesonmass} shows the numerical results for $\\rho$ masses with $s_{z}=\\pm 1$ and $s_{z}=0$ as functions of angular velocity at temperature $T=10$MeV. It is obvious that the splitting mass curves have shown the different influence of rotation. For the $s_z=0$ case there is no net angular momentum for the particle polarization by the rotation. This makes the mass dependence on the rotation is almost the same as the scalar case which stay invariant as the chiral condensate below the critical angular velocity.\nWhile for the $s_z=\\pm 1$ cases the rotation polarization would generate the energy shift $\\mp \\omega$ to the corresponding masses. This is confirmed by the numerical results in Fig.\\ref{fig:rhomesonmass}. The mass dependence on the angular velocity is two straight lines for the $s_z=\\pm 1$ components. The behavior could be analytically proven with explicit form of the polarization functions. In the pole approximation the masses are determined by the pole of the meson propagators as Eq.(\\ref{vectorpole}). With straightforward computation in the Appendix the polarization functions of vector meson satisfy\n\\begin{equation}\n\\frac{1}{2}A_{1}^{2}(m_{\\rho}+\\omega)+\\frac{1}{2}A_{2}^{2}(m_{\\rho}-\\omega)\n=A_{3}^{2}(m_{\\rho}).\n\\end{equation}\nThis means the $m_\\rho(\\omega=0)-s_z\\omega$ are exactly the masses of $s_z=0,\\pm 1$ components. The mass of $s_z=1$ spin component decreases linearly with the angular velocity, and reaches zero at the critical angular velocity $\\omega_c=m_\\rho(\\omega=0)$. Beyond the critical angular velocity $\\omega_c$, the $s_z=1$ spin component of vector meson will develop condensation in the vacuum and this indicates that the system will be spontaneously spin polarized under strong rotation.\n\n\n\\begin{figure}[t]\n \t\\includegraphics[width=0.65\\textwidth]{rhomesonmass.pdf}\\\\\n \t\\caption{$\\rho$ meson masses as a function of angular velocity at temperature $T=10 MeV$.}\\label{fig:rhomesonmass}\n \\end{figure}\n\n\n\\section{Conclusion}\n\\label{Conclusion}\nUsing the NJL model with vector channel interaction we have calculated the scalar, pseudoscalar and vector mesons' masses at finite temperature, chemical potential and angular velocity. In the RPA and pole approximation the mesons are treated as the effective degree of freedoms which transmit the interaction between quarks. And the masses are determined by the polarization functions. This approximation could preserve the Goldstone theorem explicitly although\nthe back reaction of meson to the phase transition is neglected. Because of the four-fermion point interaction and pole approximation the microscopic details of mesons have been lost. And all of them behave as fundamental particles which are polarized by rotation according to their net spin angular momentum. For the scalar and pseudoscalar cases the mass spectra are controlled by the chiral condensate which is the main mechanism generating the hadron mass in NJL model.\nAt low temperature and chemical potential the chiral restoration is 1st order  which make the meson masses a sudden jump at the critical angular velocity. While as the temperature or chemical potential increasing the phase transition would degenerate to crossovers which also smoothen the mass curves of mesons along the angular velocity.  It is easy to expect that at large enough angular velocity the vector condensate vacuum would be preferred and the corresponding effective mass should be zero. That is why we have only studied the vector meson's mass behavior below the $\\omega=m_\\rho$. It is found that although the polarization function computation is complicated masses of the three components $s_z=0, \\pm 1$ are the same as the result by treating them as the fundamental particles, that is $m_\\rho$ and $m_\\rho\\pm \\omega$.  Once the chiral restored the vector condensate would emerge simultaneously which will be studied in our next work.\n\nIn non-central heavy-ion collisions, the created system carries large angular momentum. The properties of particles will be changed under rotating medium. In this paper, we investigated the behavior of scalar and vector meson mass under the rotation.\nIt is found that the behavior of scalar and pseudoscalar meson masses under the angular velocity $\\omega$ is similar to that at finite chemical potential, both rely on the behavior of constituent quark mass and reflect the property related to the chiral symmetry. However, masses of vector meson have more profound relation with rotation. After tedious calculation, it turns out that at low temperature and small chemical potenial, the mass for spin component $s_z=0,\\pm 1$ of vector meson under rotation shows very simple mass splitting relation $m_{\\rho}^{s_z}(\\omega)=m_\\rho(\\omega=0)-\\omega s_z$, similar to the Zeeman splitting of charged meson under magnetic fields. Especially it is noticed that the mass of spin component $s_z=1$ vector meson $\\rho$ decreases linearly with $\\omega$ and reaches zero at $\\omega_c=m_\\rho(\\omega=0)$, this indicates the system will develop $s_z=1$ vector meson condensation and the system will be spontaneously spin polarized under rotation. It deserves further study to compare the spin polarization with $s_z=1$ vector meson condensation and the spin polarization defined by the\ncondensation of $<{\\bar\\psi}i\\sigma^{\\mu\\nu}\\psi>$ proposed in \\cite{Tatsumi-spinpolarization}.\n\n\n\n\\begin{acknowledgements}\n\tWe thank Kun Xu for useful discussion. M.H.is supported by the NSFC under Grant Nos. 11725523 and 11735007,  Chinese Academy of Sciences under Grant No. XDPB09, the start-up funding from University of Chinese Academy of Sciences(UCAS), and the Fundamental Research Funds for the Central Universities .\tY.J. is  supported by NSFC under Grant No.11875002 and the Zhuobai Program of Beihang University.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe beginning of cosmic dawn is the most exciting target of 21-cm observations since it is the earliest period with a strong signal that is feasible to probe with upcoming 21-cm experiments \\citep{madau97}. During this period, high-redshift galaxies drove the 21-cm signal as the non-ionizing ultraviolet photons emitted from stars were redshifted by cosmic expansion to the nearest Lyman-band frequency. Many photons reached the Ly-${\\alpha}${} frequency (directly or by cascade), near which the photons were absorbed and re-emitted hundreds of thousands of times by intergalactic atomic hydrogen before being redshifted out of the line \\citep{barkana16,Mesingerbook}. During this absorption and re-emission, through the subtle Wouthuysen - Field effect \\citep{wouthuysen52, field58} these photons drove the spin temperature (defined as the effective temperature describing the occupation ratio of hyperfine levels in the ground state of hydrogen) very close to the kinetic temperature of the gas. This is in contrast with the earlier approximate equality between the spin temperature and the cosmic microwave background (CMB) temperature, an equality that was broken as a result of the formation of the first stars. This so-called Ly-${\\alpha}${} coupling transition is expected to be observable as a prominent 21-cm absorption feature since the kinetic temperature was much lower than the CMB temperature at these redshifts. The two essential ingredients in simulating the fluctuations of the 21-cm signal during the coupling transition are the clustering properties of galaxies, which were the sources of the Ly-${\\alpha}${} photons, and the distribution around these sources of the photons that were absorbed and thus produced the coupling effect. X-ray heating and other astrophysical effects were likely insignificant at this early time (see below).\n \nAccurate predictions of the 21-cm signal at high redshift require us to follow the evolution of large volumes ($> 100$~Mpc on a side), for several reasons: the radiation (including Ly-${\\alpha}$) that drove the 21-cm signal reached out to these distances from each source; upcoming observations will be limited by resolution and other constraints to imaging scales from $\\sim 10$~Mpc to a few hundred; and most importantly, the first galaxies represented rare peaks in the cosmic density field, leading to surprisingly large fluctuations in their number density on large scales \\citep{barkana04}, which drove observable 21-cm fluctuations during the Ly-${\\alpha}${} coupling era \\citep{barkana05}. Full numerical simulations that capture these large scales cannot resolve the small halos expected to dominate star formation at cosmic dawn. Indeed, large-scale simulations run at cosmic dawn \\citep{Ghara17,semelin17} can resolve halos down to a mass of $4 \\times 10^9~M_\\odot$ at best. Instead, most simulations focus on the later era of cosmic reionization \\citep{CODA} or on achieving sufficient resolution within much smaller simulated volumes \\citep{ahn15,xu16} \\citep[generally too small for 21-cm cosmology, ][and more specifically, smaller than a {\\it single} bubble of the type we present below]{barkana04}. At the other extreme, fully analytical approaches have often been used to introduce novel ideas including Ly-${\\alpha}${} fluctuations \\citep{barkana05}, but such calculations require crude approximations (usually including the assumption of small, linear fluctuations in many quantities), and so are too inaccurate. Thus, the most realistic predictions of the 21-cm signal from cosmic dawn have come from various intermediate methods termed semi-numerical simulations \\citep{mesinger11,visbal12,kaurov18,munoz19}, which combine analytical models normalized to the results of full simulations on small scales, with a detailed numerical integration of the relevant radiation fields on large scales. \n\nSemi-numerical simulations usually generate galaxy distributions based on models \\citep{press74, sheth99, barkana04} for the mean number of dark matter halos per volume, given the density field. This approach produces a good fit to the average number of halos found in full numerical simulations, but it fails to capture the Poisson fluctuations (shot noise), which are necessarily prominent at sufficiently high redshifts, when the number density of star-forming halos was low. The possible role of Poisson fluctuations in 21-cm observations at cosmic dawn has been previously investigated with approximate analytical calculations \\citep{barkana05} or with approximate simulation-based methods \\citep{kaurov18} that resolved halos of mass $2 \\times 10^9\\ M_\\odot$ and higher. Sub-grid methods have been used to approximately insert low-mass halos into full radiative-transfer simulations that investigated cosmic heating and reionization \\citep{ross17}, but not the Ly-${\\alpha}${} era (the saturated coupling limit was assumed). Publications that use the semi-numerical code \\texttt{21cmFast} \\citep{mesinger11} apparently do not include Poisson fluctuations. The \\texttt{SimFast21} code has been run with Poisson-generated halos but only for a case dominated by low-mass halos \\citep{santos11}, in which the signatures we highlight here are too weak to be observable. \n\nWe have modified an existing semi-numerical simulation \\citep{visbal12, fialkov14a, cohen16} to fully incorporate the shot-noise contribution to the clustering of galaxies, for all halo masses including those expected to dominate star formation at cosmic dawn. To do this we generated all star-forming halos individually from a Poisson model centered on the expected mean distribution of halo masses. It is important to keep an open mind and consider a wide range of possible galactic halo masses, as we do below, since on the one hand, low-mass halos are the most abundant at early times, but on the other hand, efficient star formation may occur only in massive halos, as suggested by both extrapolations of low-redshift observations \\citep{mirocha} and the results of numerical simulations that achieve high resolution in small volumes \\citep{xu16}.\n \nThe second ingredient necessary for realistic predictions of the 21-cm signal from the Ly-${\\alpha}${} coupling era is the radiative transfer of the photons. The path that Ly-${\\alpha}${} photons travel through the intergalactic hydrogen, from emission through cosmological redshift and until absorption near the Ly-${\\alpha}${} line center, is commonly approximated by a straight line. In reality, photons emitted in the range of frequencies between Ly-${\\alpha}${} and Ly-${\\beta}${} usually scatter from the blue wing of the Ly-${\\alpha}${} line, long before reaching the line center. Such multiple scattering results in photons traveling shorter effective distances from their sources before absorption, compared to the no-scattering approximation. While the Ly-${\\alpha}${} photons can travel up to hundreds of Mpc, multiple scattering creates an over-concentrated halo of Ly-${\\alpha}${} photons at a characteristic comoving distance (from where the line center is reached) of \\citep{loeb99} \n\\begin{equation}\n    R_* = 21 \\times \\left(\\frac{\\Omega_b \/ \\Omega_m}{0.157}\\right) \\left(\\frac{1+z}{20}\\right)\\  {\\rm Mpc}\\ ,\n\\end{equation}at redshift $z$. Analytical and numerical calculations \\citep{chuzhoy07b,semelin07,naoz08,vonlanthen11,higgins12} have suggested that this should boost 21-cm fluctuations, but large-scale simulations \\citep{vonlanthen11,semelin17} that incorporate radiative transfer of Ly-${\\alpha}${} are severely limited, only resolving halos above a mass of $9 \\times 10^{10}\\ M_\\odot$. In order to make realistic predictions for the halo masses expected to host galaxies at cosmic dawn, we have added this effect to our semi-numerical simulations, using a Monte-Carlo calculation of the effective distance distribution of Ly-${\\alpha}${} photons as a function of the emission and absorption redshifts. \n\nThis paper is organized as follows: In \\cref{sec:methods} we present our semi-numerical 21-cm simulation and discuss the modifications introduced in this work. In \\cref{sec:results} we show the results of the upgraded simulation, focusing on cosmic dawn. We summarize in \\cref{sec:summary}.\n\n\n\\section{Methods}\n\\label{sec:methods}\n\nIn this work we have extended an independent 21-cm semi-numerical simulation code that we previously developed \\citep{visbal12, fialkov14a, cohen17}. The approach used in our code  was originally inspired by \\texttt{21cmFast} \\citep{mesinger11}. Our code simulates the evolution of the 21-cm signal in a 3-dimensional volume composed of 128 voxels on a side, each with a  size of 3 comoving Mpc. The simulation produces a realization of the 21-cm signal from cosmic dawn, arising from the coupling transition due to Ly-${\\alpha}${} photons from the first stars ($z \\sim 20-30$), through the heating of the intergalactic medium by the first X-ray sources, until cosmic reionization ($z \\sim 6-10$). In this work we have focused on the high-redshift coupling transition, the earliest era of galaxy formation that is feasible to detect with upcoming observations. While our simulation includes  heating of the IGM by X-ray photons, and reionization by UV photons, these do not play an important role in the models we consider here.\n\n\n\\subsection{Simulating the high-redshift galaxy population}\n\nThe first step of the simulation is obtaining a sample  of dark matter halos in the simulation volume. We start by creating a random realization of the large-scale, linear, density field, given its statistical properties (specifically, the power spectrum of the initial Gaussian random density field) as measured by the Planck satellite \\citep{planckcollaboration18}. Note that fluctuations on the scale of the voxel size (3 comoving Mpc) are still rather linear at the high redshifts considered. Given the large-scale, linear density field, we obtain the population of collapsed dark matter halos inside each voxel, using a modified Press-Schechter model \\citep{press74, sheth99, barkana04} that was fitted to match the results of full cosmological simulations. \n\nA major modification to this procedure, introduced in this work, is adding Poisson fluctuations to the number of halos predicted by the modified Press-Schechter model. In each time-step of the simulation, we calculate the predicted number of {\\it new halos formed in the time step}, in different mass bins, and draw the created halos from a Poisson distribution with the predicted number acting as the mean. Adding Poisson fluctuations to the number of halos created in each time step allows us to create a complete realization of the time evolution of the 21-cm signal. While this simplified calculation of the halo population neglects correlations between different mass bins, it is sufficient for the era we focus on here, where almost every pixel in the box has either a single galactic halo or none. As noted in the introduction, publications that use the semi-numerical code \\texttt{21cmFast} do not seem to include Poisson fluctuations; e.g., Fig.~3 of a paper \\citep{Kern17} that used \\texttt{21cmFast} shows an example that corresponds to $V_c \\sim 37$~km\/s and $f_* = 0.1$, yet at the high-redshift end the 21-cm power spectrum is flat and does not show the break that we find due to Poisson-enhanced individual halo bubbles. More clearly, in a recent paper \\citep{park19} that added into \\texttt{21cmFast} a duty cycle for star-forming halos, the duty cycle was inserted as a simple multiplicative factor into the various radiative emissivities, with no mention of the additional effect of an increase in Poisson fluctuations that would be found in a code that {\\it did}\\\/ include individual halos.\n\nGiven a dark matter halo of mass $M$, the baryon fraction contained in the halo is assumed to be the cosmic mean, except for a reduction due the streaming velocity \\citep{tseliakhovich11, fialkov12}. A halo forms stars if $M > M_{\\rm min}$ where $M_{\\rm min}$ is the minimum mass for star formation, determined by gas cooling and\/or feedback.  In this paper this minimum mass for star formation is parameterized by the circular velocity (a more direct measure of the depth of the potential well, and also the virial temperature), defined as the velocity of a circular orbit at the halo virial radius. For a halo of mass $M$,\n\\begin{equation}\n    V_c = 16.9 \\left(\\frac{M}{10^8 M_{\\odot}}\\right)^{1\/3}\\left(\\frac{1+z}{10}\\right)^{1\/2}  \\left(\\frac{\\Omega_m h^2}{0.141}\\right)^{1\/6} \\left(\\frac{\\Delta_c}{18 \\pi^2}\\right)^{1\/6}\\   {\\rm km}\\;{\\rm s}^{-1}\\ ,\n\\end{equation}\nwhere $\\Delta_c$ is the ratio between the collapsed  density and the critical density at the time of collapse, which equals $18 \\pi^2$ for spherical collapse.\n\nThe stellar mass M$_{\\star}$ in each star-forming halo is the gas mass times the star formation efficiency f$_{\\star}$. The luminosity of the galaxy is assumed to be proportional to the star formation rate (SFR). We apply two commonly-used approaches to obtain the SFR from M$_{\\star}$ \\citep{mesinger11,park19}: \n\\begin{equation}\n    {\\rm SFR} = \\frac{d{\\rm M}_{\\star}}{dt}\\ ,\n\\end{equation}\ncorresponding to a bursting mode in newly-accreted gas, and\n\\begin{equation}\n    {\\rm SFR} = \\frac{{\\rm M}_{\\star}}{t_{\\star} H(z)^{-1}}\\ ,\n\\end{equation}\ncorresponding to a quiescent mode in previously-accreted gas. Here $H(z)^{-1}$ is the Hubble time, and $t_{\\star}$ is an additional parameter that we set to $0.2$ so that $t_{\\star} H(z)^{-1}$ corresponds approximately to the characteristic dynamical time of a halo (at its virial density). We have performed tests using each of the two star-formation modes separately, and found that while these two SFR prescriptions result in a somewhat different time evolution of the SFR, there is no significant difference to the coupled bubbles picture. Since in reality both modes are likely to contribute, in our examples here we have assumed that the total SFR is given by the sum of the two modes (and then the bursting mode usually dominates at the redshifts considered here). \n\nIn the examples shown in this work we have assumed $f_* = 0.1$ as our standard value, and used $f_* = 0.3$ to illustrate a case with higher $f_*$. The increased abundance of star-forming halos in models with very low $V_c$ allows such models to reach 21-cm milestones at reasonable redshifts with lower $f_*$, so we illustrated these models with $f_* = 0.03$ for $V_c = 16.5$~km\/s, and $f_* = 0.007$ for $V_c = 4.2$~km\/s. For $V_c$ we have used characteristic values for molecular hydrogen cooling ($4.2$~km\/s), atomic cooling ($16.5$~km\/s), ten times the halo mass of atomic cooling ($35.5$~km\/s), 100 times the atomic cooling mass ($76.5$~km\/s), and additional intermediate values ($25$ and $50$~km\/s). For the excess-radio model \\citep{fialkov19} we set $f_* = 0.1$ and the radio background assumed a Galactic-like synchrotron spectrum that has an amplitude three times the CMB brightness temperature at 78~MHz. \n\n\\subsection{Full numerical simulations of cosmic dawn}\n\nAccurate predictions of the 21-cm signal at high redshift require us to follow the evolution of large volumes for several important reasons. The required volumes begin from a minimum of $100$~Mpc on a side, but a more advisable number is a few hundred Mpc. N-body simulations in which the dark matter halo distribution is realistically generated can achieve a minimum resolved halo of $4 \\times 10^9~M_\\odot$ in a volume that is 430~Mpc on a side \\citep{Ghara17}. These are gravity-only numerical simulations on top of which approximate methods are later used to add star formation and the various astrophysical radiation fields. Approximate simulation-based methods \\citep{kaurov18} that were used to explore Poisson fluctuations have been able to generate halos of mass down to $2 \\times 10^9\\ M_\\odot$ in a volume 910~Mpc on a side. These works all assumed optically thin Ly-${\\alpha}${} evolution (with no scattering except at the center of the Ly-${\\alpha}${} line). Numerical simulations run at cosmic dawn with numerical radiative transfer of the Ly-${\\alpha}${} photons \\citep{semelin17} resolved halos of mass $8 \\times 10^8~M_\\odot$ in a volume 30~Mpc on a side (barely able to capture the Ly-${\\alpha}${} halo around a single galaxy), or $9 \\times 10^{10}~M_\\odot$ in a box of side 150~Mpc.\n\nIn listing these various numbers of minimum resolved halos, we have adopted the common assumption of 20 simulation particles needed in order to resolve a halo. This number, however, is quite optimistic. Numerical resolution studies \\citep{springel03} suggest that $>100$ particles are necessary in order to determine even the overall mass of an individual halo to within a factor of two, and even more particles are needed for quantities such as the overall star formation rate in the halo (which is sensitive to the merger history and thus to the small precursor halos that are less well-resolved). \n\n\\subsection{Ly-${\\alpha}${} photon distance distribution}\n\nGiven the population of galaxies obtained as described above, we calculated the spatial distribution of the Ly-${\\alpha}${} photons that they produce. As explained in the introduction, Ly-${\\alpha}${} photons were the driver of the early 21-cm signal at cosmic dawn. These Ly-${\\alpha}${} photons originated as continuum photons emitted at frequencies between Ly-${\\alpha}${} and the Lyman limit. The emitted photons produced Ly-${\\alpha}${} photons by two different mechanisms: (i) Photons emitted at frequencies between Ly-${\\alpha}${} and  Ly-${\\beta}${} were redshifted directly to  the Ly-${\\alpha}${} frequency by cosmic expansion,  and (ii) photons emitted at higher frequencies were absorbed in higher Lyman series frequencies and created atomic cascades;  $\\sim 30 \\%$ of  cascades originating from Ly-${\\gamma}${} and above produced Ly-${\\alpha}${} photons, while no Ly-${\\alpha}${} photons were produced by cascades originating from Ly-${\\beta}${} \\citep{hirata06, pritchard06}.  In this work, the distribution of emitted photons is calculated assuming Population II stars, while Population III stars would lower the Ly-${\\alpha}${} output by about a factor of two \\citep{barkana05,bromm01} (while substantially increasing the ionizing photon output, which is unimportant at the redshifts that we consider here); such a change is nearly degenerate with a change in $f_*$ (only nearly because of the effect of the stellar spectrum which, however, is small due to the narrow relevant frequency range).\n\nIn previous semi-numerical simulations, the intensity of Ly-${\\alpha}${} photons was calculated with the assumption that photons travel in a straight line between emission and absorption at the line center. This assumption made it easy to find the Ly-${\\alpha}${} intensity at a point by integrating over previous redshifts, where at each redshift sources contribute only at a single distance from the final arrival point. Thus, the contribution of sources at redshift $z_{\\rm emission}$ to the distribution of Ly-${\\alpha}${} photons at a lower redshift $z_{\\rm absorption}$ was found by convolving the distribution of sources at $z_{\\rm emission}$ with a spherical shell window function, with a radius corresponding to the distance that photons travel between $z_{\\rm emission}$ and $z_{\\rm absorption}$.\n\nInstead, photons actually scatter elastically off hydrogen atoms in the blue wing of the Ly-${\\alpha}${} line before reaching the line center, and thus reach $z_{\\rm absorption}$ in a distribution of distances from their source, for any given $z_{\\rm emission}$. The straight-line distance is the upper limit of this more realistic distribution that is found when multiple scattering is accounted for. This effect is important for photons emitted between Ly-${\\alpha}${} and  Ly-${\\beta}${}, but not for Ly-${\\alpha}${} photons injected from the higher Lyman lines, since the effective distance corresponding to the wing of the line is very small for those transitions \\citep{naoz08}. In this work we include multiple scattering by first calculating the effective distance distribution for photons emitted between Ly-${\\alpha}${} and  Ly-${\\beta}${}, using a Monte Carlo code inspired by previous work \\citep{loeb99,naoz08}. We then construct a window function that gives a good fit to this distance distribution, and use it instead of the previously used simple, spherical shell window function. We run the Monte Carlo code and construct the window function separately for each combination of emission and absorption redshifts. Two examples are shown in Fig. \\ref{fig:example_fit_res}. \n\n   \\begin{figure*}\n    \\centering\n {\\includegraphics[width=0.49\\textwidth]{fitting_res_za_25_shell_10.pdf}} \n  {\\includegraphics[width=0.49\\textwidth]{fitting_res_za_25_shell_20.pdf}} \\\\[4pt]\n   {\\includegraphics[width=0.49\\textwidth]{wf_25_10.png}}  \n {\\includegraphics[width=0.49\\textwidth]{wf_25_20.png}}\n    \\caption{{\\bf Calculation of the multiple scattering of Ly-${\\alpha}${} photons.} {\\bf Top panels:} Example distributions of the distance from the source at which photons are absorbed at the Ly-${\\alpha}${} frequency given an emitted and absorbed redshift, generated using a photon scattering Monte Carlo code. The black dots show the number of photons per bin of log distance, as obtained by the Monte Carlo code, for a total of 250,000 photons per panel. The light-green line shows our fit to the distance distribution, while the dark-green vertical line shows the straight-line distance that all these photons would have traveled without the effect of multiple scattering. {\\bf Bottom panels:} The corresponding window functions that are used in our semi-numerical simulation of the 21-cm signal. The window function represents the distribution of photons per volume emitted from a point source at the center (normalized for display purposes to a volume integral of $10^6$). Multiple scattering substantially changes the window functions from the previously-used spherical-shell window functions, which are shown for comparison. All panels show photons emitted and absorbed at specific redshifts: For all panels $z_{\\rm absorption} = 25$, with $z_{\\rm emission} = 25.82$ for the panels on the left and $z_{\\rm emission} = 26.64$ on the right. Fully incorporating these window functions into our code and exploring a wide range of possible astrophysical parameters allows us to go well past previous investigations of the effect of Ly-${\\alpha}${} scattering \\citep{chuzhoy07b,semelin07,naoz08,vonlanthen11,higgins12}.} \n    \\label{fig:example_fit_res}\n\\end{figure*} \n\nWe note that only in our group the semi-numerical simulations since early on  \\citep{Complete} have included a rough approximation to the effect of multiple scattering on the 21-cm power spectrum based on an analytical study \\citep{naoz08}; while this did boost the power spectrum we have now found that it underestimated the boost by a typical factor of 1.5 and did not capture the correct dependence on wave number or on the astrophysical parameters. We also note that while X-ray heating and UV ionizing radiation do not play a significant role in the 21-cm signal at the early times that we have focused on, these effects are included in our semi-numerical simulations. Ly-${\\alpha}${} photons themselves can contribute to the heating of the IGM, but this effect is small during the cosmic dawn, as shown previously  \\citep{furlanetto06b} and as we have also verified with our simulation (but note that Ly-${\\alpha}${} heating can become important at later redshifts, when the Ly-${\\alpha}${} intensity field is significantly larger than required to produce the WF effect). \n\n\\section{Results}\n\\label{sec:results}\n\nIn this work we present the combined effect of Poisson fluctuations and multiple scattering of Ly-${\\alpha}${} photons on the 21-cm signal, over a wide range of astrophysical parameters that have never been probed this realistically before. Including these effects in our simulation, we obtain a cosmic dawn 21-cm signal that is substantially different from previous predictions without these effects (Fig.~\\ref{fig:vc50_slices}; see Fig.~\\ref{fig:vc25_slices} in Appendix B for another example that shows that the effect remains striking even with halos of significantly lower mass). We clarify that we refer henceforth as \"previous work\" to models run with the same parameters as our full case but including neither Poisson fluctuations nor multiple scattering (above we cited previous publications related to these effects and laid out in detail their limitations). \n\nIn the results corresponding to previous work, all simulation voxels produced a non-zero contribution to the Ly-${\\alpha}${} intensity field, but with Poisson fluctuations taken into account, at these high redshifts only a small fraction of voxels contain star-forming halos (initially only one per voxel). The stellar Ly-${\\alpha}${} photons produce coupling between the spin temperature and the kinetic gas temperature and produce a 21-cm absorption halo around each star-forming halo. If we increase the Ly-${\\alpha}${} intensity (which corresponds to increasing the galaxy brightness), the 21-cm signal approaches saturation as the spin temperature approaches the kinetic temperature of the gas. Once this limit is reached near a halo, further increasing the Ly-${\\alpha}${} intensity cannot make the nearby absorption even deeper, but it does increase the size of the coupled bubble around the halo, which makes the bubble easier to observe. Meanwhile, the disappearance of halos from many pixels (where the previous fractional numbers of halos became zero after the implementation of Poisson fluctuations) clears out the regions between the bubbles, further increasing their relative contrast.\n\n\\begin{figure*}\n    \\centering\n \\includegraphics[width=0.7\\textwidth]{vc50_slices.png}  \n    \\caption{{\\bf Simulated images of the cosmic dawn 21-cm signal.} Since early galaxies in this model were rare, we find it useful to show a kind of projected image, defined as showing the minimum value of the signal in the direction perpendicular to the image (obtained from a simulation box that is 384 comoving Mpc on a side; each image is made of square pixels of side $3$~Mpc). All panels correspond to the same simulated volume which illustrates a model with a star-formation efficiency f$_{\\star}$ = 0.1 and a minimum circular velocity V$_{\\rm c}$ = 50 km\/s, corresponding to a minimum star-forming halo mass of $M_{\\rm min} \\sim 8 \\times 10^8 \\; {\\rm M}_{\\odot}$ at the redshift shown, $z = 21$ (Fig. \\ref{fig:vc25_slices} shows similarly striking effects for V$_{\\rm c}$ = 25 km\/s). {\\bf Left panels:} Results from previous work, that is, without the effects of Poisson fluctuations and multiple scattering, shown on the scale set by the right-hand panels, for easy comparison. {\\bf Right panels:} Results from this work. {\\bf Top panels:} Ideal images (i.e., showing the direct simulation outputs). {\\bf Bottom panels:} Projections of the same simulated volumes as in the top panel but as mock SKA images (see text); such a smoothed projection can be similarly obtained from real images. In this work, the signal is composed of large \"coupled bubbles\" around individual galaxies. The large size and depth of the bubbles helps them retain sufficient contrast in the mock SKA projected image to enable their detection. The locations of the $>3 \\sigma$ peaks as found in the smoothed SKA box are marked in both the ideal and SKA boxes, for easy comparison. The peaks correspond to individual coupled bubbles in the ideal image, while in the SKA box there is a minor contribution smoothed in from nearby smaller bubbles. Note that some additional peaks with a lower significance can be seen in the SKA box, corresponding to smaller coupled bubbles in the ideal image. Also note that the SKA boxes are shown with respect to the cosmic mean brightness temperature, but the plotted values are negative due to our choice of showing projected minimum values.}\n    \\label{fig:vc50_slices}\n\\end{figure*}\n\nIn order to study the observational consequences of a cosmic dawn signal dominated by individual coupled bubbles as described above, we used the Square Kilometer Array \\citep{koopmans15} (hereafter SKA) as an example target instrument, and created mock SKA images that account for the expected angular resolution, thermal noise, and foreground effects, all as a function of redshift. There are two common approaches to dealing with the bright foreground expected in 21-cm images. {\\it Foreground removal}\\\/ involves modeling the foreground in order to subtract it accurately from the images (often with the help of additional observations obtained at higher resolution than needed for the cosmological 21-cm signal itself), while {\\it foreground avoidance}\\\/ involves removing regions in $k$-space that are expected to be contaminated by the foreground. Recent work \\citep{datta10, dillon14, pober14, pober15} has shown that the foreground is expected to contaminate a wedge-like region in the $k_{||}$ vs. $k_{\\perp}$ plane (where we separate the wavevector to components parallel and perpendicular to the line of sight), with more foreground-free $k_{||}$ modes available at lower $k_{\\perp}$ values. Since the SKA is designed to produce high-resolution deep sky images, we assume that  foreground subtraction will allow the remaining wedge of foreground avoidance to be relatively small. We refer to this reduced foreground avoidance, assumed to result from combining it with reasonably accurate foreground subtraction, as {\\it mild foreground avoidance}. To the SKA images we then add a first analysis step of three-dimensional spherical smoothing, which we find to be helpful for reducing the noise \\citep{quantiles} and bringing out the Ly-${\\alpha}${} bubbles.\n\nA cosmic dawn signal dominated by coupled bubbles is predicted to feature prominently in the 21-cm power spectrum (Fig.~\\ref{fig:power_spec}), producing a distinct power spectrum shape that is strongly correlated with the typical size of the bubbles (and thus the typical brightness of early galaxies). Coupled bubbles of size $R_{\\rm bubble}$  suppress fluctuations on scales smaller than the typical bubble size, and thus result in a break in the power spectrum at $k_{\\rm break} \\sim 2 \\pi \/ R_{\\rm bubble}$. Meanwhile, on large scales the power spectrum is boosted compared to previous predictions by a factor that is between 2 and 7 depending on the astrophysical parameters. Thus, the signature of discrete galaxies is also a promising goal for radio arrays targeting the 21-cm power spectrum at cosmic dawn, such as the Hydrogen Epoch of Reionization Array \\citep{HERA} (HERA) and the New Extension in Nan\\c{c}ay Upgrading LOFAR  \\citep{zarka12} (NenuFAR).\n\n\\begin{figure*}\n    \\centering\n \\includegraphics[width=0.7\\textwidth]{power_spectrum.png}  \n \n    \\caption{{\\bf The 21-cm power spectrum at cosmic dawn.} We show the power spectrum at the Ly-${\\alpha}${} peak (defined as the redshift at cosmic dawn where the power spectrum at $k = 0.1 \\; {\\rm Mpc}^{-1}$ peaks), for various cases with or without the effects of Poisson fluctuations and multiple scattering of Ly-${\\alpha}${} photons (solid lines and dashed lines, respectively). Unlike the gradual decline with $k$ previously expected (dashed lines, shown in three cases: $V_{\\rm c} = 25$, $35.5$, and $50$~km\/s), we find a strong enhancement of the fluctuations on large scales (small $k$), and a clear break in the power spectrum. The break location corresponds to the typical size of the coupled bubbles, which in turn correlates strongly with the typical brightness of individual galaxies. If the power spectrum can be measured with a noise level that approaches the expected thermal noise of the SKA (black line at $z=20$), then the break may be detectable even if star-formation occurred in halos as small as $V_{\\rm c} \\sim 25$~km\/s (corresponding to $M_{\\rm min} = 1.1 \\times 10^8 \\; {\\rm M}_{\\odot}$ at $z=20$). We also show examples of including Poisson fluctuations but not the multiple scattering of Ly-${\\alpha}${} photons (dotted lines, corresponding to the same models as the dashed lines); both effects play a substantial role in the results shown in this work, with multiple scattering having a larger relative role in models with the lowest $M_{\\rm min}$. For example, for the intermediate case of $V_{\\rm c} = 35.5\\ $km\/s (which corresponds to ten times the minimum mass for atomic cooling, or a minimum halo mass of $M_{\\rm min} \\sim 3 \\times 10^8 \\; {\\rm M}_{\\odot}$ at $z=20$), the power spectrum at $k=0.1$ is enhanced by a factor of 3.8 compared to previous work, while Poisson fluctuations alone would produce an enhancement by a factor of 2.2. We also show one excess radio model motivated by the EDGES measurement (see text). For all cases shown, the power spectrum was averaged over 18 different runs of the simulation with different initial conditions and (for relevant cases) Poisson realizations.}\n    \\label{fig:power_spec}\n\\end{figure*}\n\nWhile it will be intriguing to detect individual coupled bubbles in 21-cm images (as illustrated in Fig.~\\ref{fig:vc50_slices}), it is important to also construct an effective statistic, to be applied to 21-cm images of cosmic dawn, that aggregates together the individual bubbles and takes advantage of this feature in order to distinguish among models. We propose the {\\it total peak profile}\\\/ statistical probe, which measures a combination of the abundance, spatial extent, and brightness-temperature depth of the coupled 21-cm bubbles. We first detect both minima and maxima in the smoothed SKA box (see \\cref{sec:ska_box}); an example of such detected peaks is shown in Fig.~\\ref{fig:vc50_slices} (only peak minima are shown in the figure, for comparison with the image which shows projected minimum values of the signal). We restrict ourselves to strong peaks, defined as having a value higher (in absolute value) than $3 \\sigma$, where $\\sigma$ is the standard deviation of the SKA box voxel values. We calculate the radial profile around each peak that passes this threshold, and sum the profiles. The summing is done with the signed (that is, not absolute) value, in order to explicitly capture the asymmetry between maxima and minima. Indeed, any symmetric field (about its mean) would give a total result approaching zero; thus, this statistic is inherently non-Gaussian, and naturally brings out the effect of individual galaxies over thermal noise, and over any Gaussian component of the 21-cm fluctuations. Also, we sum (rather than average) these peak profiles, in order to maintain the sensitivity to the number density of peaks. In order to avoid a direct dependence on the size of the observed volume, we normalize the result by scaling it to a volume of 1~Gpc$^3$ (which corresponds almost exactly to the volume of an SKA field of view at $z=20$ with a depth of 10~MHz, or to 18 of our simulation volumes). The resulting total peak profile per volume, $\\mathcal{T}_{21}(r)$, is expected to be negative during the coupling era of cosmic dawn, and measuring it as such would imply stronger minima than maxima, thus confirming the detection of coupled bubbles of 21-cm absorption above the noise level. Fig.~\\ref{fig:radial_profiles} shows $\\mathcal{T}_{21}(r)$ as calculated from the SKA boxes for a variety of possible parameters of the early galaxies. The expected scatter in $\\mathcal{T}_{21}(r)$ due to cosmic variance and noise, for an SKA field of view, is fairly small and is shown in Appendix A (Fig.~\\ref{fig:radial_profiles_scatter}).\n\n\n\n\\begin{figure}\n    \\centering\n\n \\includegraphics[width=0.49\\textwidth]{radial_profile_cases.png}  \n \n    \\caption{{\\bf The total peak profile at cosmic dawn.} We show the total radial profile around peaks, $\\mathcal{T}_{21}(r)$, calculated from our simulated SKA data as a sum over all peaks that is then normalized per Gpc$^3$. Negative $\\mathcal{T}_{21}(r)$ corresponds to the detection of coupled bubbles in 21-cm absorption on top of the SKA noise and foreground. A noise-dominated image (or any Gaussian signal) would instead give a result near zero. We show $\\mathcal{T}_{21}(r)$ for the same astrophysical cases as in Fig.~\\ref{fig:power_spec}. For each case, the redshift with the highest value of $\\mathcal{T}_{21}(r)$ at $r=0$ is shown. We obtain significant values of $\\mathcal{T}_{21}(r)$ for a wide variety of astrophysical cases. The most prominent  $\\mathcal{T}_{21}(r)$ is seen for cases with higher star formation efficiencies and minimum masses for star formation. Such models produce larger and rarer (and thus higher contrast) coupled bubbles. In results corresponding to previous work, $\\mathcal{T}_{21}(r)$ is weaker (in absolute value) by a factor of 2--4, with Poisson fluctuations playing the dominant role in producing the large $\\mathcal{T}_{21}(r)$ obtained in this work. With Poisson fluctuations included, the prominent peaks have such a high Ly-${\\alpha}${} intensity that their surroundings are already strongly coupled even without accounting for multiple scattering, so that the additional effect of multiple scattering on their 21-cm profile is small; however, multiple scattering consistently has a strong effect on less prominent, more typical fluctuations, as measured by the 21-cm power spectrum (Fig.~\\ref{fig:power_spec}; see also Fig. \\ref{fig:vc25_slices}). For each case shown, we averaged results obtained from 18 different runs of the simulation, with independent realizations of the initial conditions, Poisson fluctuations, and SKA noise. For further discussion see \\cref{sec:ska_box}.}\n    \\label{fig:radial_profiles}\n\\end{figure}\n\nSince 21-cm coupling requires a quite low Ly-${\\alpha}${} intensity \\citep{madau97}, it is expected to occur early enough that the observational probes considered here should be nearly unaffected \\citep{cohen18} by other astrophysical radiation such as Ly-${\\alpha}${} heating, X-ray heating or UV ionizing radiation; indeed, we have focused on signatures that occur early on, well before Ly-${\\alpha}${} coupling approaches saturation. This means that observations at this early time depend only on the mass distribution and star-formation efficiency of halos. Higher masses of star-forming halos and higher star-formation efficiencies increase the sizes of individual Ly-${\\alpha}${} bubbles, making it easier to detect them as well as the corresponding power spectrum break (which is moved to lower $k$). Now, higher halo masses also delay star formation and push the Ly-${\\alpha}${} peak to a lower redshift (where observations are easier), while high efficiencies go the other way. Overall, the masses and star formation efficiencies of star-forming halos can be deduced separately given the multiple measures available, namely the amplitude and shape of the power spectrum and of the total peak profile, plus the redshifts at which these statistics peak (or, more generally, their redshift evolution). \n\nIn both Figs.~\\ref{fig:power_spec}  and \\ref{fig:radial_profiles} we have included a case motivated by the recent, intriguing but not yet independently verified, EDGES measurement of the sky averaged 21-cm signal from cosmic dawn \\citep{bowman18}. The EDGES measurement implies a larger than expected ratio between the background radiation and gas temperatures at cosmic dawn. This can be explained either with a lower than expected kinetic gas temperature due to a baryon - dark matter interaction \\citep{barkana18,munoz18,Liu20}, or an excess radio background that raises the effective radiation temperature \\citep{bowman18, feng18, fialkov19}. We include here an example of the latter model \\citep{fialkov19} with parameters that are consistent with the amplitude of the absorption detected by EDGES. If such an excess radio background exists, it should give the Ly-${\\alpha}${} bubbles a much higher contrast ($> 1000$~mK), making them even more prominent in the SKA boxes, and more easily detectable through the total peak profile $\\mathcal{T}_{21}(r)$ as well as the break in the 21-cm power spectrum (which is boosted tremendously in this case). We note that here we have assumed a uniform radio background \\citep{fialkov19}, but if the excess radio radiation was emitted by the same galaxies that emitted the Ly-${\\alpha}${} radiation and created the bubbles, then the radio intensity should be higher near the galaxies thus creating an even stronger contrast for the coupled bubbles \\citep{reis20c}.\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this work we have presented results of an upgraded simulation of the 21-cm signal from cosmic dawn, and discussed implications for planned experiments such as the SKA or the HERA. We introduced two new effects to our simulation: Poisson fluctuations in the number of galaxies, and multiple scattering of Ly-${\\alpha}${} photons. Compared to results neglecting these effects, we found a 21-cm signal with enhanced contrast, and showing the signature of individual galaxies. In particular, the 21-cm power spectrum is enhanced by a factor of 2--7 on large scales, with a significantly different shape. Simulating SKA images we found that it should be possible to detect individual galaxies at cosmic dawn, depending on the astrophysical scenario and advancements in data analysis techniques. We also discussed the {\\it total peak profile} - an effective statistic that could be applied to future observations to distinguish between models.\n\n\nFor simplicity, in this work we have assumed a constant value of the star formation efficiency f$_{\\star}$ for all star-forming halos. While this approach is common, in reality we expect significant scatter in the f$_{\\star}$ value among galaxies, due to different merger and accretion histories, and as found in simulations \\citep{xu16}. We have tested the effect of a galaxy to galaxy variance in the star formation efficiency and found that a significant variance can strongly enhance the coupled bubble signature in the cosmic dawn signal. Even for a lower {\\it average}\\\/ star formation efficiency than we have assumed, a variance in this parameter should still result in a few galaxies bright enough to produce large coupled bubbles that can be detected by the SKA. This highlights again the fact that the Ly-${\\alpha}${} bubble cosmic dawn signal predicted here is produced by individual galaxies and affected by small number statistics at the tail of the brightness distribution. This is in contrast to previous work predicting a signal dominated by large scale structure and determined by the average properties of the galaxy population. Our results are a boon to planned 21-cm observations of cosmic dawn, as they predict favorable observational targets in the form of an enhanced 21-cm power spectrum and a strongly non-Gaussian 21-cm signal, even if most star-forming halos were small as is generally expected. Finally, we note that the novel effects investigated here also affect the global 21-cm signal (due to the non-linearity of the 21-cm fluctuations) but only marginally, at the few to ten percent level at the redshifts investigated here.\n\n\\section*{Acknowledgments}\n\nWe acknowledge the usage of the DiRAC HPC. AF was supported by the Royal Society University Research Fellowship. This project was made possible for I.R.\\ and R.B.\\ through the support of the ISF-NSFC joint research program (grant No.\\ 2580\/17). \n\nThis research made use of:\n {\\fontfamily{cmtt}\\selectfont  SciPy}  \\citep[including {\\fontfamily{cmtt}\\selectfont  pandas} and {\\fontfamily{cmtt}\\selectfont NumPy, }][]{2020SciPy, numpy},  {\\fontfamily{cmtt}\\selectfont IPython}  \\citep[][]{perez07}, {\\fontfamily{cmtt}\\selectfont matplotlib}  \\citep[][]{hunter07}, {\\fontfamily{cmtt}\\selectfont Astropy}  \\citep[][]{astropy-collaboration13}, {\\fontfamily{cmtt}\\selectfont Numba}  \\citep{lam15},  the SIMBAD database  \\citep[][]{wenger00}, and the NASA Astrophysics Data System Bibliographic Services.\n\n\n\\section*{Data availability}\nNo new data were generated or analysed in support of this research.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the first experimental realization of the Bose-Hubbard model\n\\citep{Fisher_PRB_1989} with ultracold atoms in optical lattices\n\\citep{Greiner_Nature_2002}, ultracold gases in optical lattices\nhave become one of the most important experimental platforms \\citep{Bloch_RMP_2008}\nto investigate rich physics that are relevant for a wide range of\ndifferent fields, ranging from solid-state physics, over condensed\nmatter physics, to high energy physics. This is attributed to the\ntremendous experimental development in ultracold gases in optical\nlattices in the last two decades, including realizing, for instance,\noptical lattices with different dimensionality and geometries \\citep{Stoferle_PRL_2004,Spielman_PRL_2007,Becker_NJP_2010,Jo_PRL_2012},\ndifferent types of interactions \\citep{Moses_Science_2015,Baier_Science_2016,Landig_Nature_2016},\nartificial gauge fields \\citep{Aidelsburger_PRL_2011,Duca_Science_2015,Wu_Science_2016},\netc, which at the same time gives rise to considerable types of systems\ntheat support rich unconventional quantum phases with nontrivial spatial\nstructures. \n\nA case in point is ultracold Bose gases in optical lattices, where\nfor this type of systems with, for instance, long-range interactions\n\\citep{Moses_Science_2015,Baier_Science_2016,Landig_Nature_2016},\nartificial gauge fields \\citep{Cooper_PRL_2011,Aidelsburger_PRL_2011,Duca_Science_2015,Wu_Science_2016},\netc, experimental and theoretical investigations \\citep{Aidelsburger_PRL_2011,Duca_Science_2015,Landig_Nature_2016,Yi_PRL_2007,Li_PRA_2013,He_PRA_2021,Cole_PRL_2012,Radic_PRL_2012,Cai_PRA_2012,Hickey_PRL_2014,He_PRA_2015}\nhave shown that they can support exotic phases with nontrivial spatial\nstructures, such as change-density waves, spin-density waves, unconventional\nsuperfluid, etc. In investigations of these systems, straightforward\nmean-field decoupling approach \\citep{Oosten_PRA_2001,Sachdev_QPT_2011}\nand bosonic Gutzwiller mean-field variational approach \\citep{Jaksch_PRL_1998,Krauth_PRB_1992}\nare two of the most widely employed theoretical tools that have efficiently\nrevealed considerable nontrivial physics of these systems at zero\ntemperature. However, concerning finite temperature properties of\nthese systems, which are naturally indispensable for relevant experiments\nand the thorough understanding of their physical behavior, these two\napproaches seem not as efficient to be employed as in the zero temperature\ninvestigations \\citep{Hickey_PRL_2014}, not even to mention that\nthe legitimacy of the straightforward mean-field decoupling is not\nself-evident in different application scenarios. This thus poses the\nnatural demand for a reliable mean-field theory that could efficiently\ninvestigate the finite temperature properties of these systems.\n\nWe address this demand for the type of mean-field theory where the\nmean-field treatment is performed on the hopping term of the system\nto investigate the superfluid transition. Using single-component Bose\ngases in optical lattices as the concrete type of systems, we show\nthat the widely used straightforward mean-field decoupling of the\nhopping term \\citep{Oosten_PRA_2001,Sachdev_QPT_2011} generally results\nin problematic mean-field Hamiltonians with no lower energy bounds\nonce the spatial dependence of the mean-field superfluid order parameter\nis taken into account (cf.~Eq.~(\\ref{eq:H_SMFD_two_sub_lat}) and\nthe discussion below). We solve this problem by establishing the proper\nfinite temperature mean-field theory {[}cf.~Eq.~(\\ref{eq:Z_MF}){]}\nvia a systematic functional integral approach. In particular, we find\nthe proper mean-field Hamiltonian generally assumes an intrinsic \\emph{non-hermitian}\nstructure {[}cf.~Eqs.~(\\ref{eq:HMF_general_form}, \\ref{eq:HSS_general_form}){]}\nthat originates from the indefiniteness of the hopping matrix of the\nsystem. Based on this non-hermitian mean-field Hamiltonian, we develop\nan efficient and versatile approach for investigating the physics\nof the system at finite temperatures, where properties of the system\ncan be calculated via straightforward investigations on the saddle\npoints of an effective potential function for the order parameter\n{[}cf.~Eqs.~(\\ref{eq:Z_MF_Omega}, \\ref{eq:Omega_general}) and\nFig.~\\ref{Fig_1_Finite_T_Phase_diagram}{]}. We illustrated our approach\nby using both a homogeneous and an inhomogeneous two-sublattice finite\ntemperature mean-field theory to investigate the finite temperature\nsuperfluid transition of Bose gases in optical lattices. We directly\nmap out the finite temperature phase diagram of the system (cf.~Fig.~\\ref{Fig_1_Finite_T_Phase_diagram})\nand show how Mott lobes ``melt'' in the presence of thermal fluctuations\n(cf.~Fig.~\\ref{Fig_2_Mott_lobes_at_different_T_Hom_MFT} and Fig.~\\ref{Fig_3_Mott_lobes_at_different_T_two_sublattice_MFT}).\nSince the underlying finite temperature mean-field theory is quite\ngeneral, this approach can be straightforwardly applied to efficiently\ninvestigate the finite temperature properties of related systems with\nphases possessing nontrivial spatial structures. \n\n\\section{Model Hamiltonian and limitations of the straightforward mean-field\ndecoupling approach}\n\n\\subsection{Model and its straightforward mean-field decoupled Hamiltonian }\n\nTo proceed with the discussion on a concrete basis, we consider the\nsimplest single-band Bose-Hubbard model \\citep{Fisher_PRB_1989} on\na two-dimensional (2D) square lattice which is widely used in describing\nthe physics of ultracold gases in optical lattice \\citep{Bloch_RMP_2008}.\nIts explicit form reads\n\\begin{equation}\n\\hat{H}=-t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(\\hat{b}_{\\mathbf{i}}^{\\dagger}\\hat{b}_{\\mathbf{j}}+\\mathrm{h.c.}\\right)+\\sum_{\\mathbf{i}}\\left(\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\right),\\label{eq:BHM}\n\\end{equation}\nwhere $\\hat{b}_{\\mathbf{i}}^{\\dagger}$ ($\\hat{b}_{\\mathbf{i}}$)\nis the bosonic creation (annihilation) operator for the Wannier state\non the site $\\mathbf{i}$ in the lowest band, $\\hat{n}_{\\mathbf{i}}\\equiv\\hat{b}_{\\mathbf{i}}^{\\dagger}\\hat{b}_{\\mathbf{i}}$\nis the particle number operator. Here, $t$ is the positive hopping\namplitude of bosons between nearest neighboring lattice sites denoted\nby $\\langle\\mathbf{i},\\mathbf{j}\\rangle$, $U$ is the strength of\nthe on-site interaction and $\\mu$ is the chemical potential. \n\nTo investigate the properties of the system, straightforward mean-field\ndecoupling for the hopping term is frequently employed \\citep{Oosten_PRA_2001,Sachdev_QPT_2011}.\nIt is done by plugging the decomposition $\\hat{b}_{\\mathbf{i}}=\\psi_{\\mathbf{i}}+\\left(\\hat{b}_{\\mathbf{i}}-\\psi_{\\mathbf{i}}\\right)$\ninto the hopping term of $\\hat{H}$ and keeping only up to the first-order\nterms in $(\\hat{b}_{\\mathbf{i}}-\\psi_{\\mathbf{i}})$. This gives rise\nto the straightforward mean-field decoupled (SMFD) Hamiltonian $\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\})$,\nthe explicit form of which reads\n\\begin{align}\n & \\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\})=t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(\\psi_{\\mathbf{i}}^{*}\\psi_{\\mathbf{j}}+\\psi_{\\mathbf{i}}\\psi_{\\mathbf{j}}^{*}\\right)\\label{eq:H_SMFD_general}\\\\\n & -t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(\\psi_{\\mathbf{i}}^{*}\\hat{b}_{\\mathbf{j}}+\\hat{b}_{\\mathbf{i}}^{\\dagger}\\psi_{\\mathbf{j}}+\\mathrm{h.c.}\\right)+\\sum_{\\mathbf{i}}\\left[\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\right],\\nonumber \n\\end{align}\nHere, $\\psi_{\\mathbf{i}}$ is a complex variable and assumes the physical\ninterpretation of the local superfluid (SF) order parameter. At zero\ntemperature, the value of $\\psi_{\\mathbf{i}}$ is expected to be determined\nby minimizing the ground state energy of $\\hat{H}_{\\mathrm{SMFD}}(\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\})$.\n\n\\subsection{Limitations of the straightforward mean-field decoupling approach}\n\nWhen utilizing the straightforward mean-field decoupled Hamiltonian\n$\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\})$,\na homogeneous ansatz for $\\psi_{\\mathbf{i}}$, i.e., $\\psi_{\\mathbf{i}}=\\psi$\nfor $\\forall\\mathbf{i}$ is usually further assumed (see for instance\nRefs.\\citep{Oosten_PRA_2001,Sachdev_QPT_2011}). Under this homogeneous\nansatz, $\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}$ assumes the form \n\n\\begin{align}\n\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\psi^{*},\\psi)= & 2N_{s}tz\\psi^{*}\\psi-\\sum_{\\mathbf{i}}2zt(\\hat{b}_{\\mathbf{i}}^{\\dagger}\\psi+\\mathrm{h}.\\mathrm{c}.)\\nonumber \\\\\n & +\\sum_{\\mathbf{i}}\\left[\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\right],\\label{eq:H_SMFD_homogeneous}\n\\end{align}\nwith $z\\equiv\\sum_{\\mathbf{j}=\\langle\\mathbf{i}\\rangle}=4$ being\nthe coordination number of the 2D square lattice and $N_{s}$ being\nthe total number of the lattice sites. Direct calculations based on\n$\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\psi^{*},\\psi)$ can give reasonable\nground state properties of the system \\citep{Oosten_PRA_2001,Sachdev_QPT_2011}\nand are consistent with related results from other mean-field approaches,\nfor instance, bosonic Gutzwiller variational wave function approach\n\\citep{Krauth_PRB_1992,Jaksch_PRL_1998}. In this regard, one would\nnaturally expect this straightforward mean-field decoupling approach\nwith a generic ansatz for the mean-field $\\psi_{\\mathbf{i}}$ that\ntakes into account its possible spatial dependence, should also be\nable to give reasonable predictions. \n\nSomewhat unexpected, one actually finds this natural expectation is\n\\emph{not true} in general. To see this point, let us first take a\nsimple inhomogeneous ansatz for $\\psi_{\\mathbf{i}}$ as an example,\nwhere $\\psi_{\\mathbf{i}}$ assumes the same value on each of the two\nsub-lattices of the 2D square lattice and can assume different values\non different sub-lattice, i.e., $\\psi_{\\mathbf{i}}=\\psi_{\\sigma}$\nif $\\mathbf{i}\\in\\mathring{\\sigma}$ with $\\sigma=e,o$. Here, $\\mathring{\\sigma}$\ndenotes the set of all lattice sites of the sub-lattice with the index\n$\\sigma$, and we denote two sub-lattice indices as $e$ and $o$.\nThe straightforward mean-field decoupled Hamiltonian based on this\nansatz can be directly obtained. Its explicit form reads \n\\begin{align}\n & \\hat{H}_{\\mathrm{SMFD}}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})=N_{s}zt(\\psi_{e}^{*}\\psi_{o}+\\psi_{o}^{*}\\psi_{e})\\label{eq:H_SMFD_two_sub_lat}\\\\\n & +\\sum_{\\sigma}\\sum_{\\mathbf{i}\\in\\mathring{\\sigma}}\\left\\{ \\frac{1}{2}U_{s}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}-2zt\\left(\\hat{b}_{\\mathbf{i}}^{\\dagger}\\psi_{\\bar{\\sigma}}+\\mathrm{h}.\\mathrm{c}.\\right)\\right\\} ,\\nonumber \n\\end{align}\nwhere $\\bar{\\sigma}$ is defined by $\\bar{\\sigma}=o$ if $\\sigma=e$\nand $\\bar{\\sigma}=e$ if $\\sigma=o$. We notice that the first term\nof $\\hat{H}_{\\mathrm{SMFD}}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})$\nis a quadrature with respect to $\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\}$\nthat is \\emph{not} positive definite, i.e., $N_{s}zt(\\psi_{e}^{*}\\psi_{o}+\\psi_{o}^{*}\\psi_{e})=(\\psi_{e}^{*},\\psi_{o}^{*})\\left(\\begin{array}{cc}\n0 & N_{s}zt\\\\\nN_{s}zt & 0\n\\end{array}\\right)(\\psi_{e},\\psi_{o})^{T}$ with $\\left(\\begin{array}{cc}\n0 & N_{s}zt\\\\\nN_{s}zt & 0\n\\end{array}\\right)$ clearly\\emph{ not} being a positive definite matrix. This directly\nindicates that $\\hat{H}_{\\mathrm{SMFD}}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})$\ngenerally assumes no lower energy bound with respect to $\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\}$,\nhence making the mean-field theory Eq.~(\\ref{eq:H_SMFD_two_sub_lat})\nnot reliable anymore. \n\nMore generally, we notice from Eq.~(\\ref{eq:H_SMFD_general}) that\nthe straightforward mean-field decoupled Hamiltonian $\\hat{H}_{\\mathrm{\\mathrm{SMFD}}}(\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\})$\nis \\emph{not} guaranteed to assume a lower energy bound with respect\nto $\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\}$, since its first\nterm is a quadrature of $\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\}$\nthat is \\emph{not }guaranteed to be positive definite. This thus indicates\nthe reliability of the mean-field theory Eq.~(\\ref{eq:H_SMFD_general})\nis limited to the applications combined with the ansatzes for $\\psi_{\\mathbf{i}}$\nthat give positive definite quadratures of $\\{\\psi_{\\mathbf{i}}^{*},\\psi_{\\mathbf{i}}\\}$.\nIn this regard, it is desirable that a reliable construction for the\nmean-field theory beyond this limitation can be established. Indeed,\nas we shall present in the following, such a finite temperature mean-field\ntheory that can reliably work together with generic ansatzes for $\\psi_{\\mathbf{i}}$\ncan be established via formulating the exact partition function of\nthe system in the functional integral form and taking the classical\nlimit of the quantum order parameter field introduced by the standard\nHubbard-Stratonovich transformation \\citep{Hubbard_PRL_1959,Stratonovich_HST_1957,Altland_CFT_2010}.\n\n\\section{Finite temperature mean-field theory with intrinsic non-hermitian\nstructure }\n\nTo solve the issue of the mean-field theory constructed by the straightforward\nmean-field decoupling approach, we first write down the exact partition\nfunction of the system $Z=\\mathrm{tr}[e^{-\\beta\\hat{H}}]$ in the\nstandard coherent state functional integral formulation, the explicit\nform of which reads \n\\begin{equation}\nZ=\\int\\mathcal{D}(\\{b_{\\mathbf{i}}^{*}(\\tau),b_{\\mathbf{i}}(\\tau)\\})\\,e^{-S[\\{b_{\\mathbf{i}}^{*}(\\tau),b_{\\mathbf{i}}(\\tau)\\}]},\n\\end{equation}\nwith the action $S[\\{b_{\\mathbf{i}}^{*}(\\tau),b_{\\mathbf{i}}(\\tau)\\}]$\nassuming the explicit form\n\\begin{align}\n & S[\\{b_{\\mathbf{i}}^{*}(\\tau),b_{\\mathbf{i}}(\\tau)\\}]=\\int_{0}^{\\beta}d\\tau\\left\\{ -t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(b_{\\mathbf{i}}^{*}b_{\\mathbf{j}}+\\mathrm{c.c.}\\right)\\right.\\nonumber \\\\\n & \\left.+\\sum_{\\mathbf{i}}\\left(b_{\\mathbf{i}}^{*}\\partial_{\\tau}b_{\\mathbf{i}}+\\frac{U}{2}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}b_{\\mathbf{i}}-\\mu b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}\\right)\\right\\} ,\\label{eq:Exact_action}\n\\end{align}\nwhere $b_{\\mathbf{i}}^{*}(\\tau)$ ($b_{\\mathbf{i}}(\\tau)$) is the\ncomplex field that corresponds to the bosonic operator $\\hat{b}_{\\mathrm{i}}^{\\dagger}$\n($\\hat{b}_{\\mathrm{i}}$). Via the standard Hubbard-Stratonovich transformation\n(HST) \\citep{Hubbard_PRL_1959,Stratonovich_HST_1957,Altland_CFT_2010},\nthe fluctuating quantum superfluid order parameter field $\\psi_{\\mathbf{i}}(\\tau)$\ncan be directly introduced into the partition function to decouple\nthe hopping term in the action as what has been done routinely in\nliterature (see, for instance, Refs.~\\citep{Fisher_PRB_1989,Sachdev_QPT_2011}).\nHowever, we remark here that one should pay special attention to the\nconvergence of the complex Gaussian integral involved in the HST,\nwhich is frequently overlooked in literature, since the ``hopping\nmatrix'' (to be defined explicitly below) is generically an \\emph{indefinite}\nmatrix.\n\n\\subsection{Hubbard-Stratonovich transformation in the presence of the indefinite\nhopping matrix}\n\nTo perform the HST in the presence of the indefinite hopping matrix,\nlet us first reformulate the hopping term in the action, i.e., $-t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(b_{\\mathbf{i}}^{*}b_{\\mathbf{j}}+\\mathrm{c.c.}\\right)$,\ninto a matrix form, i.e., \n\\begin{equation}\n-t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(b_{\\mathbf{i}}^{*}b_{\\mathbf{j}}+\\mathrm{c.c.}\\right)=\\sum_{\\mathbf{i},\\mathbf{j}}b_{\\mathbf{i}}^{*}\\mathbf{T}_{\\mathbf{i}\\mathbf{j}}b_{\\mathbf{j}}=B^{\\dagger}\\mathbf{T}B,\\label{eq:hopping_term_before_HST}\n\\end{equation}\nwhere $\\mathbf{T}$ is the ``hopping matrix'' with its matrix elements\ndenoted by $\\mathbf{T}_{\\mathbf{i}\\mathbf{j}}$, $B$ is an $N_{s}$\ndimensional column vector that collects all $b_{\\mathbf{i}}$, i.e.,\n$B\\equiv(\\cdots,b_{\\mathbf{i}},\\cdots)^{T}$. Since $\\mathbf{T}$\nis indefinite in general, the quadrature in Eq.~(\\ref{eq:hopping_term_before_HST})\nis indefinite too. We separate the positive definite part of the quadrature\nfrom its negative definite part by first diagonalizing $\\mathbf{T}$\nwith a unitary matrix $U$, i.e., $U\\mathbf{T}U^{\\dagger}=E_{(+)}\\bigoplus E_{(-)}$,\nwhere $E^{(+)}$ ($E^{(-)}$) is an $N^{+}$ ($N^{-}$) dimensional\ndiagonal matrix that contains all the $N^{+}$ ($N^{-}$) positive\n(negative) eigenvalues of $\\mathbf{T}$. Then, the hopping term $B^{\\dagger}\\mathbf{T}B=\\tilde{B}^{\\dagger}(E_{(+)}\\bigoplus E_{(-)})\\tilde{B}$\nwith $\\tilde{B}\\equiv UB$ and can be further written as a sum of\nthe positive and the negative definite part, i.e., \n\n\\begin{align}\nB^{\\dagger}\\mathbf{T}B & =\\tilde{B}_{(+)}^{\\dagger}E_{(+)}\\tilde{B}_{(+)}+\\tilde{B}_{(-)}^{\\dagger}E_{(-)}\\tilde{B}_{(-)}\\label{eq:qudrature_of_hopping_positive_negative_separation}\n\\end{align}\nwhere $\\tilde{B}_{(+)}$ ($\\tilde{B}_{(-)}$) is an $N_{+}$ ($N_{-}$)\ndimensional column vector that contains the first $N_{+}$ (last $N_{-}$)\nelements of $\\tilde{B}$, i.e., $\\tilde{B}_{(+)}=\\left(\\begin{array}{cc}\n\\mathbf{I}_{(+)} & \\mathbf{0}_{(+)}\\end{array}\\right)\\tilde{B}$ and $\\tilde{B}_{(-)}=\\left(\\begin{array}{cc}\n\\mathbf{0}_{(-)} & \\mathbf{I}_{(-)}\\end{array}\\right)\\tilde{B}$, with $\\mathbf{I}_{(\\pm)}$ being an $N_{\\pm}$ dimensional identity\nmatrix and $\\mathbf{0}_{(\\pm)}$ being an $N_{\\pm}\\times N_{\\mp}$\nzero matrix. \n\nThe HST of the whole hopping term is facilitated by performing two\nstandard HST for the positive definite part and the negative definite\npart, separately. For the positive definite part, the HST reads\n\\begin{align}\n & e^{-\\tilde{B}_{(+)}^{\\dagger}E_{(+)}\\tilde{B}_{(+)}}\\label{eq:HST_positive_part}\\\\\n= & \\int d(\\tilde{\\Psi}_{(+)}^{\\dagger},\\tilde{\\Psi}_{(+)})e^{-\\tilde{\\Psi}_{(+)}^{\\dagger}E_{(+)}\\tilde{\\Psi}_{(+)}+i\\left(\\tilde{\\Psi}_{(+)}^{\\dagger}E_{(+)}\\tilde{B}_{(+)}+\\mathrm{c.c.}\\right)},\\nonumber \n\\end{align}\nwhile for the negative definite part, the corresponding HST reads\n\\begin{align}\n & e^{-\\tilde{B}_{(-)}^{\\dagger}E_{(-)}\\tilde{B}_{(-)}}\\label{eq:HST_negative_part}\\\\\n & =\\int d(\\tilde{\\Psi}_{(-)}^{\\dagger},\\tilde{\\Psi}_{(-)})e^{\\tilde{\\Psi}_{(-)}^{\\dagger}E_{(-)}\\tilde{\\Psi}_{(-)}+\\left(\\tilde{\\Psi}_{(-)}^{\\dagger}E_{(-)}\\tilde{B}_{(-)}+\\mathrm{c.c.}\\right)},\\nonumber \n\\end{align}\nwith $\\tilde{\\Psi}_{(\\pm)}$ being an $N_{\\pm}$ dimensional complex\nvector variable. In particular, we notice from Eq.~(\\ref{eq:HST_positive_part})\nthat the imaginary unit $i$ appears as an overall prefactor of the\ncoupling term between $\\tilde{\\Psi}_{(+)}$ and $\\tilde{B}_{(+)}$\nin the HST for the positive definite part, while this is not the case\nfor the negative definite one. \n\nWith the two separate HST in Eqs.~(\\ref{eq:HST_positive_part},~\\ref{eq:HST_negative_part}),\nwe can straightforwardly express the hopping term in the partition\nfunction as an integral with respect to the $N_{s}$ dimensional complex\nvector variable $\\Psi\\equiv U^{\\dagger}(\\tilde{\\Psi}_{(+)}^{T}\\tilde{\\Psi}_{(-)}^{T})^{T}$,\ni.e., \n\\begin{equation}\ne^{-B^{\\dagger}\\mathbf{T}B}=\\int d(\\Psi^{\\dagger},\\Psi)e^{-\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{H}}\\Psi+\\left(\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{NH}}B+B^{\\dagger}\\mathbf{T}_{\\mathrm{NH}}\\Psi\\right)},\\label{eq:hopping_term_complete_HST}\n\\end{equation}\nwhere \n\\begin{align}\n\\mathbf{T}_{\\mathrm{H}} & \\equiv U^{\\dagger}\\left(E_{(+)}\\bigoplus(-E_{(-)})\\right)U,\\label{eq:T_H}\\\\\n\\mathbf{T}_{\\mathrm{NH}} & \\equiv U^{\\dagger}\\left((iE_{(+)})\\bigoplus(E_{(-)})\\right)U.\\label{eq:T_NH}\n\\end{align}\nIn particular, we notice that $T_{\\mathrm{H}}$ is a positive definite\nhermitian matrix, while $\\mathbf{T}_{\\mathrm{NH}}$ is a \\emph{non-hermitian}\nmatrix in general, and gives rise to the \\emph{non-hermitian} structure\nof the mean-field Hamiltonian as we shall see in the following.\n\n\\subsection{Non-hermitian structure of the mean-field Hamiltonian}\n\nWith the HST in Eq.~(\\ref{eq:hopping_term_complete_HST}) for the\nhopping term, we can straightforwardly reformulate the partition function\n$Z$ as a functional integral with respect to both $B(\\tau)$ and\n$\\Psi(\\tau)$, the explicit form of which reads \n\\begin{align}\nZ= & \\int\\mathcal{D}(\\Psi^{\\dagger}(\\tau),\\Psi(\\tau),B^{\\dagger}(\\tau),B(\\tau))\\label{eq:Z_after_HST}\\\\\n & \\times e^{-S[\\Psi^{\\dagger}(\\tau),\\Psi(\\tau),B^{\\dagger}(\\tau),B(\\tau)]},\\nonumber \n\\end{align}\nwhere \n\\begin{align}\n & S[\\Psi^{\\dagger}(\\tau),\\Psi(\\tau),B^{\\dagger}(\\tau),B(\\tau)]=\\int_{0}^{\\beta}d\\tau\\Psi^{\\dagger}T_{\\mathrm{H}}\\Psi\\nonumber \\\\\n & +\\int_{0}^{\\beta}d\\tau\\sum_{\\mathbf{i}}\\left\\{ b_{\\mathbf{i}}^{*}\\partial_{\\tau}b_{\\mathbf{i}}+\\frac{U}{2}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}b_{\\mathbf{i}}-\\mu b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}\\right.\\nonumber \\\\\n & \\left.-\\left[\\left(\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{NH}}\\right)_{\\mathbf{i}}b_{\\mathbf{i}}+b_{\\mathbf{i}}^{*}\\left(\\mathbf{T}_{\\mathrm{NH}}\\Psi\\right)_{\\mathbf{i}}\\right]\\right\\} .\\label{eq:complete_action_aftere_HST}\n\\end{align}\nTo proceed further, we make the mean-field (classical) approximation\nthat neglect the quantum fluctuations of the order parameter field\n$\\Psi(\\tau)$, i.e., assume the superfluid order parameter field $\\Psi(\\tau)$\ndoes not depend on $\\tau$, i.e., $\\Psi(\\tau)=\\Psi$. This gives rise\nto the mean-filed partition function $Z_{\\mathrm{MF}}$ with the explicit\nform\n\\begin{equation}\nZ_{\\mathrm{MF}}=\\int d(\\Psi^{\\dagger},\\Psi)\\mathcal{D}(B^{\\dagger}(\\tau),B(\\tau))e^{-S_{\\mathrm{MF}}[\\Psi^{\\dagger},\\Psi,B^{\\dagger}(\\tau),B(\\tau)]},\\label{eq:mean-field_partition_function}\n\\end{equation}\nwhere the mean-field action $S_{\\mathrm{MF}}$ reads\n\\begin{align}\n & S_{\\mathrm{MF}}[\\Psi^{\\dagger},\\Psi,B^{\\dagger}(\\tau),B(\\tau)]=\\beta\\Psi^{\\dagger}T_{\\mathrm{H}}\\Psi\\nonumber \\\\\n & +\\int_{0}^{\\beta}d\\tau\\sum_{\\mathbf{i}}\\left\\{ b_{\\mathbf{i}}^{*}\\partial_{\\tau}b_{\\mathbf{i}}+\\frac{U}{2}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}b_{\\mathbf{i}}-\\mu b_{\\mathbf{i}}^{*}b_{\\mathbf{i}}\\right.\\nonumber \\\\\n & \\left.-\\left[\\left(\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{NH}}\\right)_{\\mathbf{i}}b_{\\mathbf{i}}+b_{\\mathbf{i}}^{*}\\left(\\mathbf{T}_{\\mathrm{NH}}\\Psi\\right)_{\\mathbf{i}}\\right]\\right\\} .\\label{eq:mean_field_complete_action_aftere_HST}\n\\end{align}\nNoticing that the $\\tau$ dependence of the second term of the mean-field\naction $S_{\\mathrm{MF}}$ only comes from $\\{b_{\\mathbf{i}}^{*}(\\tau),b_{\\mathbf{i}}(\\tau)\\}$,\nwe can reformulate $Z_{\\mathrm{MF}}$ as (see Appendix~\\ref{sec:Transformation-from-functional-back-to-operator}\nfor more details) \n\\begin{equation}\nZ_{\\mathrm{MF}}=\\int d(\\Psi^{\\dagger},\\Psi)\\mathrm{tr}\\left[e^{-\\beta\\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)}\\right],\\label{eq:Z_MF}\n\\end{equation}\nwhere the mean-field Hamiltonian $\\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)$\nassumes the explicit form\n\\begin{equation}\n\\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)=\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{H}}\\Psi+\\sum_{\\mathbf{i}}\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\Psi^{\\dagger},\\Psi),\\label{eq:HMF_general_form}\n\\end{equation}\nwith $\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\Psi^{\\dagger},\\Psi)$\nbeing a single site Hamiltonian that only involves operators on site\n$\\mathbf{i}$, the explicit of which reads\n\\begin{align}\n & \\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\Psi^{\\dagger},\\Psi)\\equiv\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\nonumber \\\\\n & -\\left[\\left(\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{NH}}\\right)_{\\mathbf{i}}\\hat{b}_{\\mathbf{i}}+\\hat{b}_{\\mathbf{i}}^{\\dagger}\\left(\\mathbf{T}_{\\mathrm{NH}}\\Psi\\right)_{\\mathbf{i}}\\right].\\label{eq:HSS_general_form}\n\\end{align}\n\nWe notice $\\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)$ is generally\na \\emph{non-hermitian} Hamiltonian due to the general non-hermiticity\nof $\\mathbf{T}_{\\mathrm{NH}}$. Such a non-hermitian structure is\nreminiscent of non-hermitian Hamiltonians widely used to describe\nphysics of open quantum systems (see Ref.~\\citep{Ashida_Adv_Phys_2020}\nfor a recent review), where the non-hermiticity originates from dissipations.\nIn contrast, here, it originates from the hermitian hopping term with\nan indefinite hopping matrix. We remark here that the above derivation\nof the mean-field Hamiltonian is completely general and can be directly\napplied to systems with complex hopping terms, for instance, systems\nwith spin-orbit coupling \\citep{Cole_PRL_2012,Radic_PRL_2012,Cai_PRA_2012,Hickey_PRL_2014,He_PRA_2015,Wu_Science_2016},\neffective magnetic flux \\citep{Cooper_PRL_2011,Aidelsburger_PRL_2011,Duca_Science_2015},\netc.\n\n\\subsection{Finite temperature mean-field theory in terms of the potential function\nfor the order parameter field}\n\nTo make the mean-field theory Eq.~(\\ref{eq:Z_MF}) a useful and efficient\ntool for investigating both zero temperature and finite temperature\nproperties of the system, we further reformulate the integrand in\nthe expression for $Z_{\\mathrm{MF}}$ in Eq.~(\\ref{eq:Z_MF}) as\nan exponential of a potential function of the superfluid order parameter\nfield alone, i.e., \n\\begin{equation}\nZ_{\\mathrm{MF}}=\\int d(\\Psi^{\\dagger},\\Psi)e^{-\\beta N_{s}\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)},\\label{eq:Z_MF_Omega}\n\\end{equation}\nwhere the potential function $\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)$\nreads\n\\begin{align}\n & \\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)\\nonumber \\\\\n= & \\frac{1}{N_{s}}\\left(\\Psi^{\\dagger}\\mathbf{T}_{\\mathrm{H}}\\Psi-\\frac{1}{\\beta}\\ln\\prod_{\\mathbf{i}}\\mathrm{tr}\\left[e^{-\\beta\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\Psi^{\\dagger},\\Psi)}\\right]\\right).\\label{eq:Omega_general}\n\\end{align}\n\nThe most appealing feature of this reformulation is that we can analyze\nthe properties of the system by simply investigating the saddle points\nof the potential function $\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)$.\nThis is due to the fact that in the thermodynamic limit ($N_{s}\\rightarrow+\\infty$),\nthe partition function $Z_{\\mathrm{MF}}$ is exactly determined by\nthe saddle point value of $\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)$,\nhence the value of the SF order parameter $\\bar{\\Psi}$ at temperature\n$T$ is determined by the value of $\\Psi$ that minimize $\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)$.\nAlthough the explicit form of $\\Omega_{t,U,\\mu,\\beta}(\\Psi^{\\dagger},\\Psi)$\ncan not be obtained analytically due to the trace in its expression,\nhowever its value at given $(\\Psi^{\\dagger},\\Psi)$ can be calculated\nvery efficiently at sufficiently high accuracy by employing a large\nenough cut-off $n_{\\mathrm{max}}$ in the dimension of the local Hilbert\nspace for the site $\\mathbf{i}$. For the temperature regime of most\ninterests, where thermal fluctuations compete strongly with the two\nother energy scales in the system (the ones associated with on-site\ninteraction and hopping), i.e., $k_{B}T\\sim U\\sim zt$, a cut-off\n$n_{\\mathrm{max}}$ of $\\mathcal{O}(10)$ is already large enough. \n\n\\subsection{Finite temperature mean-field theory combined with the homogeneous\nand the inhomogeneous two-sublattice ansatz for $\\psi_{\\mathbf{i}}$}\n\nTo illustrate concretely how the finite temperature mean-field theory\nis applied, we used it together with the homogeneous and an inhomogeneous\ntwo-sublattice ansatz to investigate the superfluid transition of\nthe system at finite temperature. To this end, let us first diagonalize\nthe hopping term that appears in the action in Eq.~(\\ref{eq:Exact_action})\nby a straightforward lattice Fourier transformation, i.e., \n\\begin{align}\n-t\\sum_{\\langle\\mathbf{i},\\mathbf{j}\\rangle}\\left(b_{\\mathbf{i}}^{*}b_{\\mathbf{j}}+\\mathrm{c}.\\mathrm{c}.\\right) & =\\sum_{\\mathbf{k}}\\mathcal{E}(\\mathbf{k})\\tilde{b}_{\\mathbf{k}}^{*}\\tilde{b}_{\\mathbf{k}},\\label{eq:hopping_term_diagonal_form}\n\\end{align}\nwhere $\\tilde{b}_{\\mathbf{k}}\\equiv\\left(L_{x}L_{y}\\right)^{-1\/2}\\sum_{\\mathbf{i}}e^{-i\\mathbf{k}\\cdot\\mathbf{i}}b_{\\mathbf{i}},$\n$\\mathcal{E}(\\mathbf{k})\\equiv-4t\\left(\\cos k_{x}+\\cos k_{y}\\right)$,\nand $k_{\\alpha}=0\\cdot2\\pi\/L_{\\alpha},1\\cdot2\\pi\/L_{\\alpha},2\\cdot2\\pi\/L_{\\alpha},\\ldots,(L_{\\alpha}-1)\\cdot2\\pi\/L_{\\alpha}$,\nwith $\\alpha=x,y$ and $L_{\\alpha}$ being the number of lattice sites\nalong the direction $\\alpha$. This enables us to write down the explicit\nform of the mean-field hamiltonian $\\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)$\n(see Appendix \\ref{App:Explicit_form_HMF} for derivation details),\nwhich reads\n\\begin{align}\n & \\hat{H}_{\\mathrm{MF}}(\\Psi^{\\dagger},\\Psi)=\\sum_{\\mathbf{i}}\\left(\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\right)\\label{eq:HMF_tight_binding_general_form}\\\\\n & +\\frac{1}{N_{s}}\\sum_{\\mathbf{i},\\mathbf{i}',\\mathbf{k}}\\psi_{\\mathbf{i}}^{*}\\left([\\theta(-\\mathcal{E}(\\mathbf{k}))-\\theta(\\mathcal{E}(\\mathbf{k}))]\\mathcal{E}(\\mathbf{k})e^{i\\mathbf{k}\\cdot(\\mathbf{i}-\\mathbf{i}')}\\right)\\psi_{\\mathbf{i}'}\\nonumber \\\\\n & +\\frac{1}{N_{s}}\\sum_{\\mathbf{i},\\mathbf{i}',\\mathbf{k}}\\psi_{\\mathbf{i}}^{*}\\left([\\theta(-\\mathcal{E}(\\mathbf{k}))+i\\theta(\\mathcal{E}(\\mathbf{k}))]\\mathcal{E}(\\mathbf{k})e^{i\\mathbf{k}\\cdot(\\mathbf{i}-\\mathbf{i}')}\\right)\\hat{b}_{\\mathbf{i}'}\\nonumber \\\\\n & +\\frac{1}{N_{s}}\\sum_{\\mathbf{i},\\mathbf{i}',\\mathbf{k}}\\hat{b}_{\\mathbf{i}}^{\\dagger}\\left([\\theta(-\\mathcal{E}(\\mathbf{k}))+i\\theta(\\mathcal{E}(\\mathbf{k}))]\\mathcal{E}(\\mathbf{k})e^{i\\mathbf{k}\\cdot(\\mathbf{i}-\\mathbf{i}')}\\right)\\psi_{\\mathbf{i}'},\\nonumber \n\\end{align}\nwhere $\\theta(x)$ is the Heaviside step function, i.e., $\\theta(x)=1$,\nif $x>0$, and $\\theta(x)=0$, if $x<0$.\n\n\\subsubsection{Homogeneous finite temperature mean-filed theory}\n\nNow we can easily construct the simplest mean filed theory by assuming\nthat the superfluid order parameter field is homogenous, i.e., $\\psi_{\\mathbf{i}}=\\psi$.\nPlugging this homogeneous ansatz for $\\psi_{\\mathbf{i}}$ into Eq.~(\\ref{eq:HMF_tight_binding_general_form}),\none can directly obtain (see Appendix \\ref{App:Explicit_form_HMF}\nfor derivation details)\n\\begin{align}\n & \\hat{H}_{\\mathrm{MF}}(\\psi^{*},\\psi)=\\sum_{\\mathbf{i}}\\left[2tz\\psi^{*}\\psi+\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\psi^{*},\\psi)\\right],\\label{eq:HMF_homogeneous_ansatz}\n\\end{align}\nwith \n\\begin{equation}\n\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\psi^{*},\\psi)=-2zt(\\hat{b}_{\\mathbf{i}}^{\\dagger}\\psi+\\mathrm{h}.\\mathrm{c}.)+\\frac{U}{2}\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}},\n\\end{equation}\nThe corresponding mean-field partition function $Z_{\\mathrm{MF}}$\nin terms of the potential function $\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)$\nreads \n\\begin{align}\nZ_{\\mathrm{MF}} & =\\int d(\\psi^{*},\\psi)e^{-\\beta N_{s}\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)},\\label{eq:ZMF_homogeneous_ansatz}\n\\end{align}\nwith\n\\begin{equation}\n\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)=2tz\\psi^{*}\\psi-\\frac{1}{\\beta}\\ln\\mathrm{tr}\\left[e^{-\\beta\\hat{H}_{\\mathrm{SS}}(\\psi^{*},\\psi)}\\right].\\label{eq:Omega_homogeneous_ansatz}\n\\end{equation}\nThe trace in Eq.~(\\ref{eq:Omega_homogeneous_ansatz}) can be calculated\nby first diagonalizing the single-site Hamiltonian $\\hat{H}_{\\mathrm{SS}}(\\psi^{*},\\psi)$\nin the local occupation number basis with a cut-off $n_{\\mathrm{max}}$\nfor the occupation number, and then performing the summation $\\sum_{n=0}^{n_{\\max}}e^{-\\beta\\varepsilon_{n}(\\psi^{*},\\psi)}$,\nwith $\\varepsilon_{n}(\\psi^{*},\\psi)$ being the eigenenergy of $\\hat{H}_{\\mathrm{SS}}(\\psi^{*},\\psi)$.\nThis enables us to extract the dependence of $\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)$\non $\\psi$ at generic fixed system parameters $\\{t,U,\\mu,\\beta\\}$,\nand to obtain the saddle point. \n\nIn the upper panels of Fig.~\\ref{Fig_1_Finite_T_Phase_diagram},\nwe show two typical landscapes of the potential function $\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)$\nat different temperatures (other system parameters are kept the same\nwith $2zt\/U=0.24$ and the filling factor $\\rho\\equiv N_{s}^{-1}\\sum_{\\mathbf{i}}\\langle\\hat{n}_{\\mathbf{i}}\\rangle=1$).\nWe see that at the low temperature ($k_{B}T\/U=0.1$ in this case)\nthe potential function assumes a typical Mexican hat structure with\nits minimums located at a ring where $\\psi$ assumes non-zero modulus,\nwhile at the high temperature ($k_{B}T\/U=0.3$ in this case) the potential\nfunction assumes a bowl structure with its unique minimum located\nat $\\psi=0$. This kind of change in the structure of the potential\nfunction directly indicates the temperature-driven phase transition\nfrom the superfluid to the normal phase is a second-order transition.\nIndeed, from the lower-left panel of Fig.~\\ref{Fig_1_Finite_T_Phase_diagram},\nwe see that the modulus of the superfluid order parameter $|\\bar{\\psi}|$\nof the system continuously decreases to zero as $T$ increases. The\ncomplete finite temperature phase diagram of the system at unit filling\n($\\rho=1$) is shown in the lower-right panel of Fig.~\\ref{Fig_1_Finite_T_Phase_diagram},\nwhere we notice that the critical temperature $T_{C}$ for the superfluid\nto normal phase transition only show obvious increasing behavior with\nrespect to the hopping amplitude once the two competing energy scales,\ni.e., $2zt$ and $k_{B}T$, are comparable with each other. \n\n\\begin{figure}\n\\includegraphics[width=1.7in]{Omega_Landscape_SSB_Phase}\\includegraphics[width=1.7in]{Omega_Landscape_Symmetric_Phase}\n\n\\includegraphics[width=1.6in]{psi_vs_T_at_fixed_tHop}\\includegraphics[width=1.8in]{Fin_Tem_Phase_Diagram_unit_filling}\n\n\\caption{Upper panels: Two typical landscapes of the potential function $\\Omega_{t,U,\\mu,\\beta}(\\psi^{*},\\psi)$\nin the homogeneous finite temperature mean-field theory at different\ntemperatures. Other system parameters are kept the same in these two\nplots, with $2zt\/U=0.24$ and the filling factor $\\rho\\equiv N_{s}^{-1}\\sum_{\\mathbf{i}}\\langle\\hat{n}_{\\mathbf{i}}\\rangle=1$.\nLower-left panel: Superfluid order parameter $|\\bar{\\psi}|$ as a\nfunction of the temperature at unit filling ($\\rho=1$) with $2zt\/U=0.24$.\nLower-right panel: Finite temperature phase diagram of the system\nat unit filling ($\\rho=1$). See text for more details.}\n\\label{Fig_1_Finite_T_Phase_diagram}\n\\end{figure}\n\nMoreover, away from unit filling, we calculated the superfluid order\nparameter $|\\bar{\\psi}|$ as a function of the chemical potential\n$\\mu$ and hopping amplitude $t$ at different temperatures as shown\nin Fig.~\\ref{Fig_2_Mott_lobes_at_different_T_Hom_MFT}. We see that\nat nearly zero temperature ($k_{B}T\/U=10^{-2}$), the distribution\nof $|\\bar{\\psi}|$ on the $\\mu$-$t$ plane still manifests a clear\nMott-lobe structure, which is consistent with zero temperature results\nfrom other mean-field type theories \\citep{Fisher_PRB_1989,Jaksch_PRL_1998,Oosten_PRA_2001,Sachdev_QPT_2011}.\nAs $T$ increases the superfluid region with relatively small hopping\namplitude $t$ vanishes first, making the Mott-lobe structure less\nand less apparent.\n\n\\begin{figure}\n\\includegraphics[width=1.7in]{Fin_Tem_MFT_Homgeneous_psi_T_0d01}\\includegraphics[width=1.7in]{Fin_Tem_MFT_Homgeneous_psi_T_0d03}\n\n\\includegraphics[width=1.7in]{Fin_Tem_MFT_Homgeneous_psi_T_0d07}\\includegraphics[width=1.7in]{Fin_Tem_MFT_Homgeneous_psi_T_0d10}\n\n\\caption{Superfluid order parameter $|\\bar{\\psi}|$ as a function of the chemical\npotential $\\mu$ and hopping amplitude $t$ at different temperatures.\nThe calculations are performed within the homogeneous finite temperature\nmean-filed theory. At nearly zero temperature ($k_{B}T\/U=10^{-2}$),\nthe distribution of $|\\bar{\\psi}|$ on the $\\mu$-$t$ plane still\nmanifests a clear Mott-lobe structure. As the temperature $T$ increases\nthe superfluid region with relatively small hopping amplitude $t$\nvanishes first, making the Mott-lobe structure less and less apparent.\nSee text for more details.}\n\\label{Fig_2_Mott_lobes_at_different_T_Hom_MFT}\n\\end{figure}\n\nFinally, we remark that by comparing Eq.~(\\ref{eq:HMF_homogeneous_ansatz})\nwith Eq.~(\\ref{eq:H_SMFD_homogeneous}), we notice that under the\nhomogenous ansatz for $\\psi_{\\mathbf{i}}$, the mean-field Hamiltonian\nconstructed by the straightforward mean-field decoupling assumes exactly\nthe same form as the one obtained via the systematic functional integral\nconstruction. On the one hand, this provides a solid theoretical ground\nfor the straightforward mean-field decoupling approach combined with\nthe homogenous ansatz. On the other hand, we should emphasize that\nsuch a formal consistency is purely due to the coincidence caused\nby the form of the homogeneous ansatz, under which only the negative\ndefinite part of the quadrature associated with the hopping term {[}cf.~Eq.~(\\ref{eq:qudrature_of_hopping_positive_negative_separation}){]}\neventually appears in the mean-field Hamiltonian. Indeed, as we shall\nsee explicitly in the following concrete example with an inhomogeneous\ntwo-sublattice ansatz, the functional integral construction gives\nrise to a mean-field Hamiltonian with a lower energy bound with respect\nto the superfluid order parameter field, which is in sharp contrast\nto the one constructed by the straightforward mean-field decoupling\nshown in Eq.~(\\ref{eq:H_SMFD_two_sub_lat}). \n\n\\subsubsection{Two-sublattice finite temperature mean-field theory}\n\nNow, let us proceed to construct the mean-field theory by using a\nmore general ansatz for $\\psi_{\\mathbf{i}}$, which assumes a two-sublattice\nstructure, i.e., $\\psi_{\\mathbf{i}}=\\psi_{\\sigma}$ if $\\mathbf{i}\\in\\mathring{\\sigma}$\nwith $\\sigma=e,o$, and $\\mathring{\\sigma}$ denoting the set of all\nlattice sites of the sub-lattice with the index $\\sigma$. Without\nlosing any generality, the explicit form of $\\psi_{\\mathbf{i}}$ under\nthis ansatz reads\n\\begin{equation}\n\\psi_{\\mathbf{i}}=\\frac{1+\\cos(\\mathbf{K}\\cdot\\mathbf{i})}{2}\\psi_{e}+\\frac{1-\\cos(\\mathbf{K}\\cdot\\mathbf{i})}{2}\\psi_{o},\\label{eq:two-sublattice_psi_explicit_form}\n\\end{equation}\nwith $\\mathbf{K}\\equiv(\\pi,\\pi)^{T}$. Directly plugging this ansatz\ninto Eq.~(\\ref{eq:HMF_tight_binding_general_form}), we can directly\nobtain (see Appendix \\ref{App:Explicit_form_HMF} for derivation details)\n\n\\begin{align}\n & \\hat{H}_{\\mathrm{MF}}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})\\nonumber \\\\\n= & N_{s}zt(\\psi_{e}^{*}\\psi_{e}+\\psi_{o}^{*}\\psi_{o})+\\sum_{\\mathbf{i}}\\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\}),\\label{eq:HMF_two-sublattice_ansatz}\n\\end{align}\nwhere\n\\begin{align}\n & \\hat{H}_{\\mathrm{SS}}^{(\\mathbf{i})}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})=\\frac{1}{2}U\\hat{n}_{\\mathbf{i}}(\\hat{n}_{\\mathbf{i}}-1)-\\mu\\hat{n}_{\\mathbf{i}}\\label{eq:HSS_two-sublattice_ansatz}\\\\\n & +zt\\left\\{ \\left[\\hat{b}_{\\mathbf{i}}^{\\dagger}(\\psi_{e}+\\psi_{o})+\\mathrm{h.c.}\\right]+ie^{i\\mathbf{K}\\cdot\\mathbf{i}}\\left[\\hat{b}_{\\mathbf{i}}^{\\dagger}(\\psi_{e}-\\psi_{o})+\\mathrm{h.c.}\\right]\\right\\} .\\nonumber \n\\end{align}\nThe corresponding mean-field partition function $Z_{\\mathrm{MF}}$\nreads \n\\begin{align}\nZ_{\\mathrm{MF}} & =\\int d(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})e^{-\\beta N_{s}\\Omega_{t,U,\\mu,\\beta}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})},\\label{eq:ZMF_two-sublattice_ansatz}\n\\end{align}\nwhere \n\\begin{align}\n & \\Omega_{t,U,\\mu,\\beta}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})\\equiv zt\\left(\\psi_{e}^{*}\\psi_{e}+\\psi_{o}^{*}\\psi_{o}\\right)\\\\\n & -\\frac{1}{2\\beta}\\ln\\left(\\mathrm{tr}\\left[e^{-\\beta\\hat{H}_{\\mathrm{SS}}^{(e)}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})}\\right]\\cdot\\mathrm{tr}\\left[e^{-\\beta\\hat{H}_{\\mathrm{SS}}^{(o)}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})}\\right]\\right),\\nonumber \n\\end{align}\nwith $\\hat{H}_{\\mathrm{SS}}^{(\\sigma)}(\\psi_{\\sigma}^{*},\\psi_{\\sigma})$\nassuming the explicit form\n\\begin{align}\n & \\hat{H}_{\\mathrm{SS}}^{(\\sigma)}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})=\\frac{1}{2}U\\hat{n}(\\hat{n}-1)-\\mu\\hat{n}\\\\\n & +zt\\left\\{ \\left[\\hat{b}^{\\dagger}(\\psi_{e}+\\psi_{o})+\\mathrm{h.c.}\\right]+i\\eta_{\\sigma}\\left[\\hat{b}^{\\dagger}(\\psi_{e}-\\psi_{o})+\\mathrm{h.c.}\\right]\\right\\} ,\\nonumber \n\\end{align}\nwhere $\\eta_{\\sigma}\\equiv+1,-1$ for $\\sigma=e,o$, respectively.\nFrom the explicit form of $\\hat{H}_{\\mathrm{SS}}^{(\\sigma)}(\\psi_{\\sigma}^{*},\\psi_{\\sigma})$,\nwe notice that $\\hat{H}_{\\mathrm{SS}}^{(e)}$ is the hermitian conjugate\nof $\\hat{H}_{\\mathrm{SS}}^{(o)}$, i.e., $(\\hat{H}_{SS}^{(e)})^{\\dagger}=\\hat{H}_{SS}^{(o)}$.\nThis indicates the product $\\mathrm{tr}[e^{-\\beta\\hat{H}_{SS}^{(e)}}]\\cdot\\mathrm{tr}[e^{-\\beta\\hat{H}_{SS}^{(o)}}]$\nis guaranteed to assume positive real values. Therefore, the potential\nfunction $\\Omega_{t,U,\\mu,\\beta}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})$\nis still a real-valued function despite $\\hat{H}_{\\mathrm{MF}}(\\{\\psi_{\\sigma}^{*},\\psi_{\\sigma}\\})$\nbeing a non-hermitian Hamiltonian.\n\nThe concrete calculations based on the two-sublattice finite temperature\nmean-field theory Eq.~(\\ref{eq:ZMF_two-sublattice_ansatz}) can proceed\nin a similar way as the one for the homogeneous mean-field theory.\nAt different temperatures, the superfluid order parameter on the even\nand odd sublattice, i.e., $|\\bar{\\psi_{e}}|$ and $|\\bar{\\psi_{o}}|$,\nas a function of the chemical potential $\\mu$ and hopping amplitude\n$t$ are shown in Fig.~\\ref{Fig_3_Mott_lobes_at_different_T_two_sublattice_MFT}.\nWe directly observe that $|\\bar{\\psi_{e}}|$ and $|\\bar{\\psi_{o}}|$\nshow exactly the same behavior, indicating the superfluid order parameter\nbeing homogeneous over the whole system as expected. Moreover, compared\nwith results from the homogeneous finite temperature mean-field theory\nshown in Fig.~\\ref{Fig_2_Mott_lobes_at_different_T_Hom_MFT}, we\ncan easily see that $|\\bar{\\psi_{e}}|$ and $|\\bar{\\psi_{o}}|$ manifest\nthe same behavior as $|\\bar{\\psi}|$, indicating the two-sublattice\nfinite temperature mean-field theory is constant with the homogeneous\none. This is also consistent with the natural physical expectation\nthat equilibrium phases of the system should be homogeneous. \n\n\\begin{figure}\n\\includegraphics[width=1.7in]{Fin_Tem_MFT_Two_SubL_psi_e_T_0d01}\\includegraphics[width=1.7in]{Fin_Tem_MFT_Two_SubL_psi_o_T_0d01}\n\n\\includegraphics[width=1.7in]{Fin_Tem_MFT_Two_SubL_psi_e_T_0d07}\\includegraphics[width=1.7in]{Fin_Tem_MFT_Two_SubL_psi_o_T_0d07}\n\n\\caption{Superfluid order parameter on the even and odd sublattice, i.e., $|\\bar{\\psi_{e}}|$\nand $|\\bar{\\psi_{o}}|$, as a function of the chemical potential $\\mu$\nand hopping amplitude $t$ at different temperatures. The calculations\nare performed within the two-sublattice finite temperature mean-field\ntheory. The distribution of $|\\bar{\\psi}_{e}|$ on the $\\mu$-$t$\nplane is exactly the same as the one of $|\\bar{\\psi}_{o}|$, indicating\nthe superfluid order parameter is always homogeneous over the whole\nsystem as expected. Moreover, the $|\\bar{\\psi}_{e}|$ and the $|\\bar{\\psi}_{o}|$\ndistribution on the $\\mu$-$t$ plane are exactly the same as the\n$|\\bar{\\psi}|$ distribution obtained within the homogeneous mean-field\ntheory shown in Fig.~\\ref{Fig_2_Mott_lobes_at_different_T_Hom_MFT}.\nSee text for more details.}\n\\label{Fig_3_Mott_lobes_at_different_T_two_sublattice_MFT}\n\\end{figure}\n\n\n\\section{Conclusions }\n\nThe proper mean-field Hamiltonian can generally manifest distinct\nintrinsic structure from its original one, as the finite temperature\nmean-field theory for the Bose gases in optical lattices shows: due\nto the general indefiniteness of the hopping matrix, the proper mean-field\ntreatment of the hopping term directly gives rise to a mean-field\nHamiltonian that is \\emph{non-hermitian}. This non-hermitian structure\nposes no hindrance to the subsequent calculations. On the contrary,\nit facilitates the application of the mean-field theory in combination\nwith generic space-dependent ansatzes for the order parameter field,\nbased on which an efficient and versatile approach for calculating\nfinite temperature properties of the system can be developed, as illustrated\nin the investigation of the finite temperature superfluid transition\nof Bose gases in optical lattices. We believe our work will stimulate\nfurther effort to investigate finite temperature properties of the\nunconventional superfluids in ultracold atom systems with spin-orbit\ncoupling, effective magnetic flux, etc, and also other distinct intrinsic\nstructures of the mean-field Hamiltonians that could appear in various\nmean-field treatments.\n\\begin{acknowledgments}\nThis work was supported by NSFC (Grant No.~11874017, No.~11674334,\nand No.~11947302), GDSTC under Grant No.~2018A030313853, Science\nand Technology Program of Guangzhou (Grant No.~2019050001), and START\ngrant of South China Normal University.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{The Physics and Status of Core Collapse Supernova Simulations}\\label{sec:foundation}\n\nCore collapse supernovae (CCSNe) are initiated by the collapse of the iron cores of massive stars at the ends of their lives. The collapse proceeds to ultrahigh densities, in excess of the densities of nucleons in the nucleus of an atom (``super-nuclear'' densities). The inner core becomes incompressible under these extremes, bounces, and, acting like a piston, launches a shock wave into the outer stellar core. This shock wave will ultimately propagate through the stellar layers beyond the core and disrupt the star in a CCSN explosion. However, the shock stalls in the outer core, losing energy as it plows through it, and exactly how the shock wave is revived remains an open question, although, important progress is being made, particularly with two-dimensional (2D) models. The means to this revival is the central question in CCSN theory. (For a more complete review, the reader is referred to \\cite{Mezz05}, \\cite{Jank12}, and \\cite{KoTaSu12}.) \n\nAfter core bounce, $\\sim10^{53}$~ergs of energy in neutrinos and antineutrinos of all three flavors  are released from the newly formed proto-neutron star (PNS) at the center of the explosion. \nThe typical observationally estimated CCSN explosion energy  is $\\sim 10^{51}$~ergs ($\\equiv1$~Bethe), with estimates for individual supernovae ranging from 0.3--5~Bethe \\cite{Hamu03,NoToUm06,Smar09}.\nPast simulations \\cite{Wils85,BeWi85} demonstrated that energy in the form of neutrinos emerging from the PNS can be deposited behind the shock and may revive it. \nThis neutrino reheating is central to CCSN models today. \nHowever, while a prodigious amount of neutrino energy emerges from the PNS, the neutrinos are weakly coupled to the material below the shock. \nThe neutrino heating is very sensitive to the distribution of neutrinos in energy (or frequency) and direction of propagation, at any given spatial point behind the shock \n\\cite{BuGo93,JaMu96,MeMeBr98,MeCaBr98b,MeLiMe01,Jank01}.\nRealistic CCSN simulations require a neutrino transport method that can reproduce the angular and energy distributions of the neutrinos in the critical heating region.\n\nNormal iron core stars do not explode when modeled in spherical symmetry  \\cite[cf.,][]{LiMeTh01a,RaJa02,ThBuPi03}, thus multidimensional effects are required.  \nFluid instabilities ({\\it e.g.}, convection) in the PNS may boost the luminosity of this central neutrino source and consequent neutrino heating \\cite{SmWiBa81,WiMa93,MiPoUr02,BrRaMe04,BuRaJa06}.  \nNeutrino-driven convection between the PNS and the shock fundamentally alters the nature of energy flow and shock revival \\cite{HeBeHi94,BuHaFr95,JaMu96,FrWa04,BuRaJa06,BrDiMe06} relative to the spherically symmetric case, allowing simultaneous down-flows that fuel the neutrino luminosities and neutrino-heated up-flows that bring energy to the shock. \nThe standing accretion shock instability (SASI), a computationally discovered instability of the shock wave itself \\cite{BlMeDe03}, dramatically alters the shock and explosion dynamics \n\\cite{BlMeDe03,JaBuKi05,BuLiDe06,OhKoYa06,HaMuWo13}. Recent axisymmetric (2D) models \\cite{MuJaHe12,BrMeHi13} demonstrate that neutrino heating in conjunction with neutrino-driven convection and the SASI are able to generate explosions, although the quantitative predictions --- in particular, the explosion energies --- differ between these two groups. However, it is important to note that our predictions are consistent with observations \\cite{BrLeHi14} across a range of observables: explosion energy, $^{56}$Ni mass, neutron star mass, and neutron star kicks.\nDespite these differences, these advances suggest that the SASI  may be the ``missing link'' that will enable the Wilson delayed-shock, neutrino-heating mechanism to operate successfully in multiple spatial dimensions, especially for more massive progenitors. \n\nThere are many other inputs to the physics of the core collapse supernova (CCSN) mechanism that must also be included in simulations. The strength of these effects have been tested in many one-dimensional (1D) simulations and some multidimensional simulations.\nThe PNS in a developing CCSN is sufficiently compact to require the inclusion of general relativistic effects to gravity and neutrino propagation \\cite{BaCoKa85,LiMeTh01a,LiMeTh01b,BrDeMe01,MaDiJa06,OtDiMa07,MuJaDi10,LeMeMe12a,MuJaMa12}.\nGetting the correct radiative coupling requires inclusion of all neutrino--matter interactions (opacities) that affect the neutrino transport, heating, and cooling. Several recent studies have considered the effects of neutrino opacities, including inelastic scattering of neutrinos on electrons, nucleons, and nuclei, detailed nuclear electron capture, and nuclear medium effects on the neutrino interactions \\cite{HiMeBr03,BuJaKe03,KeRaJa03,ThBuPi03,MaJaBu05,MaLiFr06,LaMaMu08,JuLaHi10,RoReSh12,LeMeMe12b}.\nA nuclear equation of state for both nuclear matter in the PNS and the nuclei and nucleons in the surrounding matter is required. Several equations of state have been proposed \\cite{BeBrAp79,ElHi80,Coop85,LaSw91,WiMa93,ShToOy98b,HeSc10,ShHoTe11,StHeFi13} and their impact in CCSNe has been examined \\cite{SwLaMy94,RaBuJa02,SuYaSu05,MaJaMu09,LeHiBa10,Couc13a}.\nFinally, the nuclear composition must be evolved in the outer regions where nuclear statistical equilibrium (NSE) does not apply.\n\nThe centrifugal effects of stellar core rotation, especially for rapid rotation, can also change supernova dynamics qualitatively and quantitatively \\cite{FrWa04,BuRaJa06}. \nAn additional level of complexity is added by models with dynamically important magnetic fields, amplified by rapid rotation and the magnetorotational instability, that may  play a significant role in driving, and perhaps collimating, some CCSNe \\cite{Symb84,AkWhMe03,BuDeLi07} and \\emph{collapsars} (jets generated by accretion disks about newborn black holes producing combined CCSNe\/$\\gamma$-ray bursts). \nRecent  observations of shock breakout \\cite{ScJuWo08} disfavor a strongly collimated jet as the driver for explosions for ordinary supernovae \\cite{CoWhMi09} --- i.e., cases where rotation likely does not play a major role. \nMagnetic fields are expected to become important in the context of rapidly rotating progenitors, where significant rotational energy can be tapped to develop strong and organized magnetic fields (e.g., see \\cite{BuDeLi07}). State-of-the-art stellar evolution models for massive stars \\cite{wohe07} do not predict large core rotation rates. For non-rapidly rotating progenitors, magnetic fields are expected to serve more of a supporting role, for neutrino shock reheating (e.g., see \\cite{ObJaAl14}).\n \nWhile the list of major macroscopic components in any CCSN clearly indicates this is a 3D phenomenon, 3D studies have been relatively rare and, until recently, generally have skimped, largely for practical reasons, on key physics to which prior studies (noted above) have indicated careful attention must be paid.  \n3D simulations have examined aspects of the CCSN problem using a progression of approximations.\n3D, hydrodynamics-only simulations of the SASI, which isolate the accretion flow from the feedbacks of neutrino heating and convection, have identified the spiral ($m=1$) mode, with self-generated counter-rotating flows that can spin the PNS to match the $\\sim$50~ms periods of young pulsars \\cite{Blon05a,Blon05b,BlMe07} and examined the generation of magnetic fields \\cite{EnCaBu10} and turbulence \\cite{EnCaBu12} by the SASI.\nAnother often-used formulation for approximate 3D simulations is the neutrino ``lightbulb'' approximation, where a proscribed neutrino luminosity determines the heating rate, with the neutrino heating and cooling parameterized independently. \nNeutrino lightbulb simulations have been used successfully to study the development of NS kicks \\cite{NoBrBu12,WoJaMu12,WoJaMu13}, mixing in the ejecta \\cite{HaJaMu10}, and, in 2D simulations, the growth of the SASI with neutrino feedbacks \\cite{ScJaFo08}. Lightbulb simulations have also been used to examine the role of dimensionality (1D-2D-3D) in CCSNe \\cite{MuBu08,NoBuAl10,HaMaMu12,Couc13b}.\nA more sophisticated approximate neutrino transport method is the ``leakage'' scheme. Leakage schemes use the local neutrino emission rate and the opaqueness of the overlying material to estimate the cooling rate and from that the neutrino luminosity and heating rate. \nLeakage models have been used by Ott et al. \\cite{OtAbMo13}, including the full 3D effects of GR.\nFryer and Warren \\cite{FrWa02,FrWa04} employed a \\emph{gray} neutrino transport scheme in three dimensions. In such schemes, one evolves the spatial neutrino energy and momentum densities with a \nparameterization of the neutrino spectra. As a neutrino angle- and energy-integrated scheme, the dimensionality of the models is greatly reduced, which is ideal for performing a larger number of exploratory studies.\nThese 3D studies, and other recent studies \\cite[cf.][]{TaKoSu12,BuDoMu12,HaMuWo13,CoOc13}, confirm the conclusion that CCSN simulations must ultimately be performed in three spatial dimensions.  \n\nThe modeling of CCSNe in three dimensions took an important step forward recently. The Max Planck (MPA) group launched the first 3D CCSN simulation with multifrequency neutrino transport with relativistic corrections and state-of-the-art neutrino opacities, and general relativistic gravity. Results from the first 400 ms after stellar core bounce were reported in \\cite{HaMuWo13} for a 27 \\ensuremath{M_{\\odot}}\\ progenitor. At present, the ``Oak Ridge'' group is performing a comparable simulation beginning with the 15~\\ensuremath{M_{\\odot}}\\ progenitor used in our 2D studies. We have evolved approximately the first half second after bounce (for further discussion, see Section~\\ref{sec:current3D}). \n\n\\section{Lessons from Spherical Symmetry}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{fig2_color.pdf}\n\\caption{Shock trajectories in km, versus time after bounce, for models with decreasing physics \\cite{LeMeMe12}.}\n\\label{fig:shockvphysics}\n\\end{figure}\n\nRecent studies carried out in the context of general relativistic, spherically symmetric CCSN models with Boltzmann neutrino transport demonstrate that (i) a general relativistic treatment of gravity, (ii) special and general relativistic corrections to the neutrino transport, such as the gravitational redshift of neutrinos, and (iii) the use of a complete set of weak interactions and a realistic treatment of those interactions are indispensable \\cite{LeMeMe12a}. As shown in Figure \\ref{fig:shockvphysics}, the impact of moving to a Newtonian description of gravity from a fully general relativistic treatment has a significant impact on the shock trajectory. The Newtonian simulation neglects general relativity in the description of gravity {\\it per se}, as well as general relativistic transport effects such as gravitational redshift. Thus, the switch from a general relativistic description to a Newtonian description impacts more than just the treatment of gravity. In turn, if we continue to simplify the model, this time reducing the set of weak interactions included and the realism with which these weak interactions are included, we see a further significant change in the shock trajectory, with fundamentally different behavior early on after bounce. In this instance, we have neglected the impact of nucleon correlations in the computation of electron capture on nuclei (see \\cite{HiMeMe03}), energy exchange in the scattering of neutrinos on electrons, corrections due to degeneracy and nucleon recoil in the scattering of neutrinos on nucleons, and nucleon--nucleon bremsstrahlung. Finally, if we continue to simplify the neutrino transport by neglecting special relativistic corrections to the transport, such as the Doppler shift, we obtain yet another significant change. The spread in the shock radii at $t>$120 ms after bounce is approximately 60 km. Its relative fraction of the average of the shock radii across the four cases at $t>$ 120 ms is $>$33\\%. Moreover, the largest variation in the shock radii in our 2D models is obtained at $\\sim$ 120 ms after bounce, which is around the time when the shock radii in our one- and two-dimensional models begin to diverge (see Figure \\ref{fig:label1Dv2D}). In all four of our 2D models, the postbounce evolution is quasi-spherical until $\\sim$110 ms after bounce. Thus, the use of the \\textsc{Agile-BOLTZTRAN}\\ code, which solves the general relativistic Boltzmann equation with a complete set of neutrino weak interactions for the neutrino transport in the context of spherically symmetric models, to determine the physics requirements of more realistic two- and three-dimensional modeling is possible. Indeed, the conclusions of our studies are corroborated by similar studies carried out in the context of 2D multi-physics models \\cite{MuJaMa12}. Taken together, these studies establish the {\\it necessary} physics that must be included in CCSN models in the future. Whether or not the current treatments of this physics in the context of two- and three-dimensional models is {\\it sufficient}, as we will discuss, remains to be determined.\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{1Dv2D.pdf}\n\\caption{Shock trajectories in km, versus time after bounce, for our 1D and 2D models \\cite{BrMeHi13}. The 1D and 2D evolution begins to diverge between 100 and 125 ms after bounce.}\n\\label{fig:label1Dv2D}\n\\end{figure}\n\n\\section{Our Code}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{rbr.pdf}\n\\caption{A depiction of the ``ray-by-ray'' (RbR) approach. Each ray corresponds to a separate spherically symmetric problem. In the limit of spherical symmetry, the RbR approach is exact. Each ray solve gives what would be obtained in a spherically symmetric solve for conditions at the base of the ray, on the proto-neutron star surface. For a {\\it persistent} hot spot, such as the one depicted here at the base of ray 1, the RbR approximation would overestimate the angular variations in the neutrino heating at the points 1 and 2 above the surface. In spherical symmetry, the condition at the base of each ray is assumed to be the same over the entire portion of the surface subtended by the backward causal cone for that ray. Thus, for ray 1, the entire subtended surface would be considered hotter than it is, whereas for ray 2 the contribution from the hot spot at the base of ray 1 to the heating at point 2 above the surface would be ignored.\n\\label{fig:rbr}\n}\n\\end{figure}\n\n\\textsc{Chimera}\\ is a parallel, multi-physics code built specifically for multidimensional simulation of CCSNe.\nIt is the chimeric combination of separate codes for hydrodynamics and gravity; neutrino transport and opacities; and a nuclear EoS and reaction network, coupled by a layer that oversees data management, parallelism, I\/O, and control.\n\nThe hydrodynamics are modeled using a dimensionally-split, Lagrangian-Remap (PPMLR) scheme  \\cite{CoWo84} as implemented in VH1 \\cite{HaBlLi12}.\nSelf-gravity is computed by multipole expansion \\cite{MuSt95}.\nWe include the most important effects of GR by replacing  the Newtonian monopole term  with a GR monopole computed from the TOV equations \\cite[][Case~A]{MaDiJa06}.\n\nNeutrino transport is computed in the ``ray-by-ray-plus'' (RbR+) approximation \\cite{BuRaJa03}, where an independent, spherically symmetric transport solve is computed for each ``ray'' (radial array of zones with the same $\\theta$, $\\phi$). (It is very important to note that the RbR+ approximation does not restrict the neutrinos to strict radial propagation only. In spherical symmetry, neutrinos propagate along arbitrary rays, not just radial rays, but the {\\em net} angular flux is zero, leaving only radial flux. Each RbR+ solve is a {\\em full} spherically symmetric solve (see Figure \\ref{fig:rbr}). The 3D problem is broken up into $N_{\\theta}\\times N_{\\phi}$ spherically symmetric problems, where $N_{\\theta,\\phi}$ are the number of latitudinal and longitudinal zones, respectively. RbR+ is exact (physically speaking, modulo numerical error) if the neutrino source is spherically symmetric. Thus, if accreted material raining down on the PNS surface via the non-spherical accretion funnels, obvious in Figures~\\ref{fig:entropy} and \\ref{fig:entropy3D}, and creating hot spots, spreads rapidly over the surface relative to the neutrino-heating and shock-revival time scales, which we find it does, and in the absence of significant rotation, the RbR+ approximation is a reasonable approximation, at least initially. There are practical benefits to the approximation, as well, which we will discuss later.)\n\nThe transport solver for each ray is an improved and updated version of the multi-group flux-limited diffusion transport solver of Bruenn \\cite{Brue85} enhanced for GR \\cite{BrDeMe01}, with an additional geometric flux limiter to prevent an overly-rapid transition to free streaming of the standard flux-limiter.  All $O(v\/c)$ observer correction terms have been included.\n\n\\textsc{Chimera}\\ solves for all three flavors of neutrinos and antineutrinos with four coupled species: \\ensuremath{\\nu_{e}}, \\ensuremath{\\bar \\nu_e}, $\\ensuremath{\\nu_{\\mu\\tau}}=\\{\\ensuremath{\\nu_{\\mu}},\\ensuremath{\\nu_{\\tau}}\\}$, $\\ensuremath{\\bar \\nu_{\\mu\\tau}}=\\{\\ensuremath{\\bar \\nu_{\\mu}},\\ensuremath{\\bar \\nu_{\\tau}}\\}$, with typically 20 energy groups covering two decades in neutrino energy.\nOur standard, modernized, neutrino--matter interactions include emission, absorption, and non-isoenergetic scattering on free nucleons \\cite{RePrLa98}, with weak magnetism corrections \\cite{Horo02}; emission\/absorption (electron capture) on nuclei \\cite{LaMaSa03}; isoenergetic scattering on nuclei, including ion-ion correlations; non-isoenergetic scattering on electrons and positrons; and pair emission from $e^+e^-$-annihilation \\cite{Brue85} and nucleon-nucleon bremsstrahlung \\cite{HaRa98}.\n\\textsc{Chimera}\\ generally utilizes the $K = 220$~\\mbox{MeV}\\ incompressibility version of the Lattimer--Swesty \\cite{LaSw91} EoS for  $\\rho>10^{11}\\,\\ensuremath{{\\mbox{g~cm}}^{-3}}$ and a modified version of the Cooperstein \\cite{Coop85} EoS for  $\\rho<10^{11}\\,\\ensuremath{{\\mbox{g~cm}}^{-3}}$, where nuclear statistical equilibrium (NSE) applies.\nMost \\textsc{Chimera}\\ simulations have used a 14-species $\\alpha$-network ($\\alpha$, \\isotope{C}{12}-\\isotope{Zn}{60}) for the non-NSE regions \\cite{HiTh99a}. In addition,\n\\textsc{Chimera}\\ utilizes a 17-nuclear-species NSE calculation for the nuclear component of the EOS for $Y_{\\rm e}>26\/56$ to provide a smooth join with the non-NSE regime\n\nDuring evolution, the radial zones are gradually and automatically repositioned to track changes in the mean radial structure.\nTo minimize restrictions on the time step from the Courant limit, the lateral hydrodynamics for a few inner zones are ``frozen'' during collapse, and after prompt convection fades, the laterally frozen region expands to the inner 6--8~km.\nIn the ``frozen'' region the  radial hydrodynamics and neutrino transport are  computed in spherical symmetry.\n\nThe supernova code most closely resembling \\textsc{Chimera}\\ \nis the \\textsc{PROMETHEUS-VERTEX}\\ code developed by the Max Planck group \\cite{BuRaJa03,BuRaJa06,BuJaRa06,MuJaDi10}. This code utilizes a RbR+ approach to neutrino transport, solving the first two multifrequency angular moments of the transport equations with a variable Eddington closure that is solved at intervals using a 1D approximate Boltzmann equation.\n\n\\textsc{Chimera}\\ does not yet include magnetic fields. Studies with \\textsc{Chimera}\\ that include magnetic fields will be part of future efforts. \n\n\\section{Our Approach in Context}\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{2DApproaches.pdf}\n\\caption{An overview of the approaches used in the context of 2D CCSN modeling by various groups around the world \\cite{SuKoTa10,TaKoSu14,NaTaKu14,DoBuZh14,maja09,MuJaMa12,BrMeHi13}.  \n\\label{fig:label2DApproaches}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.0in]{3DApproaches.pdf}\n\\caption{An overview of the approaches used in the context of 3D CCSN modeling by several groups around the world \\cite{TaKoSu12,HaMuWo13,LeBrHi15}.\n\\label{fig:label3DApproaches}}\n\\end{figure}\n\nA number of 2D simulations have been performed to date with multi-frequency neutrino transport. These break down into two classes, those that have implemented the RbR neutrino transport approximation and those that have not --- i.e., those that have implemented 2D transport. Figure \\ref{fig:label2DApproaches} provides an overview of the approaches used by various supernova groups in producing these 2D models. It is clear the RbR approximation has enabled the inclusion of general relativity and state-of-the-art neutrino interactions, at the expense of the added spatial dimensionality of the transport, whereas the non-RbR approach includes the second spatial dimension in the neutrino transport, but does so at the expense of realism in the treatment of gravity and the neutrino interactions with stellar matter. The reason for this is simple: In the RbR approach, transport codes that have been used in spherically symmetric studies, such as \\textsc{Agile-BOLTZTRAN}\\ , can be deployed. These codes already, or at least can more easily, include all relativistic transport corrections and full weak interaction physics. To achieve the same level of sophistication in two and three spatial dimensions is more difficult and far more computationally intensive. For example, a 3D multi-frequency approach (e.g., flux-limited diffusion or a variable Eddington tensor method) will require the sustained-petaflop performance of present-day leadership-class computing facilities. In light of the practical difficulties associated with including more physics in fully 3D simulations, the RbR approximation provides an alternative approach that can be used in the interim. The use of both approaches by the community as it moves forward will be essential, as simulations with RbR neutrino transport with approximate general relativity and full weak interaction physics must be gauged by non-RbR approaches that can test the efficacy of the RbR approach. Ultimately, the two approaches must merge, with 3D simulations performed with 3D (i.e., not RbR) general relativistic neutrino transport, general relativistic hydrodynamics and gravity, and a full weak interaction set. Figure \\ref{fig:label3DApproaches} gives an overview of the 3D simulations performed to date, using multi-frequency neutrino transport. It is obvious that fewer groups have attempted this, and far fewer simulations have been performed. It is also evident they have all been performed with RbR and not 3D neutrino transport.\n\n\\section{Results from our 2D Core Collapse Supernova Models}\\label{sec:current2D}\n\nWe \\cite{BrMeHi13,BrLeHi14} have performed four 2D simulations with \\textsc{Chimera}\\ beginning with the 12, 15, 20, and \n25~\\ensuremath{M_{\\odot}}\\ progenitors of Woosley and Heger \\cite{wohe07}.\nOne result of these simulations is the realization that a fully developed (and therefore final) explosion energy will require much more lengthy simulations than anticipated in the past.\nIn the explosion energy plot, Figure~\\ref{fig:energy}, the dashed lines show the growth of the ``diagnostic energy'' (the sum of the gravitational potential energy, the kinetic energy, and the internal energy in each zone --- i.e., the total energy in each zone --- for all zones having a total energy greater than zero) along with more refined estimates of the final explosion energy that account for the work required to lift the as-yet-unshocked envelope ``overburden'' (dash-dotted lines) and, in addition, the estimated energy released from recombination of free nucleons and alpha particles into heavier nuclei (solid lines). We expect these latter two measures to bracket the final kinetic energy of the fully developed explosion. Using the definition of the explosion energy that includes both the energy cost to lift the overlying material and the energy gain associated with nuclear recombination, we can define $t_{\\rm explosion}$, the explosion time, which is the time at which the explosion energy becomes positive and, therefore, the explosion can be said to have been initiated. For the 12, 15, 20, and 25 M$_\\odot$ models, $t_{\\rm explosion}$ is approximately 320, 320, 500, and 620 ms after bounce, respectively. \n\nMoving now to a comparison with observations: All four models have achieved explosion energies that are in the $\\approx $0.4--1.4 Bethe range of observed Type~II supernovae (see Figure \\ref{fig:energycomparison}). Figures \\ref{fig:nickelmass} and \\ref{fig:pnsmass} compare our predictions for the mass of $^{56}$Ni produced and the final proto-neutron star (baryonic) masses produced, respectively, with observations. Note, the large systematic errors in observed progenitor masses preclude any detailed comparison between our results and observations {\\em as a function of progenitor mass}. Nonetheless, comparisons of our predicted {\\em ranges} of explosion energies, $^{56}$Ni masses, etc. with observed ranges is meaningful and demonstrates we are making progress toward developing predictive models.\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{movie.jpg}\n\\caption{Evolution of the entropy (upper half) and radial velocity (lower half) at 150, 300, and 600~ms after bounce  for the 12~\\ensuremath{M_{\\odot}}\\ model of Bruenn et al. \\cite{BrMeHi13}.  \n\\label{fig:entropy}}\n\\end{figure}\n\nThree snapshots of hydrodynamic motion are visible in \nFigure~\\ref{fig:entropy}, \nwhich shows the entropy (upper half) and radial velocity (lower half) for the 12 \\ensuremath{M_{\\odot}}\\ model at 150~ms, 300~ms, and 600~ms after bounce.  \nAt 150~ms, roughly 100~ms before rapid shock expansion heralds the onset of a developing explosion, asphericity is developing as a result of vigorous neutrino-driven convection and the SASI.  \nBy 300~ms large-scale, high-entropy, buoyant plumes are evident, as the explosion continues to develop.  \nHowever, low-entropy down-flows still connect the unshocked regions with the PNS surface, continuing to supply accretion energy to power the neutrino luminosities driving the development of the explosion. By 600~ms, these down-flows have been cut off by the expanding ejecta, but their remnants continue to accrete onto the PNS, allowing the explosion to continue to gain in strength.\n\nThough these simulations have run further into explosion than previous simulations, the final explosion energies --- in particular, for the 20 and 25 M$_\\odot$ models --- are clearly still developing. \nThese simulations will therefore continue. Additional 2D simulations --- e.g., using different progenitor masses --- are planned.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Expl_E_vs_t_12M_25M_Comp.pdf}\n\\caption{Diagnostic energy (\\ensuremath{E^{+}}; dashed lines) versus post-bounce time for all of our published 2D models \\cite{BrMeHi13,BrLeHi14}. Dash-dotted lines (\\ensuremath{E^{+}_{\\rm ov}}) include binding energy of overburden and dashed lines (\\ensuremath{E^{+}_{\\rm ov, rec}}) also include estimated energy gain from nuclear recombination.}\n\\label{fig:energy}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{Explosion_Energy_Comparisons.pdf}\n\\caption{\nObserved explosion energies for a number of CCSNe, along with predicted explosion energies from our 12, 15, 20, and 25 M$_\\odot$ progenitor models (red dots) \\cite{BrLeHi14}. The arrows indicate that our explosion energies are still increasing at the end of each run. The length of each arrow is a measure of the rate of change of the explosion energy at the end of the corresponding run.\n\\label{fig:energycomparison}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{Nickel56_Comparisons.pdf}\n\\caption{\nObserved production of $^{56}$Ni for a number of CCSNe, along with our predictions from our 12, 15, 20, and 25 M$_\\odot$ progenitor models (red dots) \\cite{BrLeHi14}.\n\\label{fig:nickelmass}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{N_Star_Mass.pdf}\n\\caption{\nTime evolution of the proto-neutron star (baryonic) mass in each of our 4 2D models, beginning with 12, 15, 20, and 25 M$_\\odot$ progenitors \\cite{BrLeHi14}.\n\\label{fig:pnsmass}\n}\n\\end{figure}\n\n\\section{Preliminary Results from our 3D Core Collapse Supernova Model}\\label{sec:current3D}\n\n\\begin{figure}\n\\includegraphics[width=3.1in]{1D2D3DShockTrajectories.pdf}\n\\caption{Evolution of the shock trajectory from our 1D model and the angle-averaged shock trajectories from our 2D and 3D models, all for the 15~\\ensuremath{M_{\\odot}}\\ case \\cite{LeBrHi15}. The 1D model does not develop an explosion, whereas an explosion is obtained in both our 2D and our 3D models.\n\\label{fig:1D2D3DShockTrajectories}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.15in]{3D441msYZ.pdf}\n\\caption{Snapshot of the equatorial cross section of the entropy in our ongoing 3D simulation for the 15~\\ensuremath{M_{\\odot}}\\ case at $\\sim$441 ms after bounce \\cite{LeBrHi15}. Red indicates high-entropy, expanding, rising material. Green\/blue indicates cooler, denser material. Evident are significant (green) down flows fueling the neutrino luminosities.\n\\label{fig:entropy3D}\n}\n\\end{figure}\n\nFew 3D multiphysics models with necessary realism (as defined above) have been performed. Notable among these is the recently published model of Hanke et al. \\cite{HaMuWo13}. Preliminary results from the Oak Ridge group \\cite{LeBrHi15} in the context of a model similar to the Garching group's model -- i.e., with essentially the same physics and treatment of this physics -- are presented here, although we begin with the same 15 M$_\\odot$ Woosley--Heger progenitor used in our 2D models, whereas they began with the 27 M$_\\odot$ Woosley--Heger progenitor. \n\nFigure \\ref{fig:1D2D3DShockTrajectories} shows the angle-averaged shock trajectories from our one-, two-, and three-dimensional models, all run with the \\textsc{Chimera}\\ code beginning with the same 15 M$_\\odot$ Woosley--Heger progenitor and including the same (full) physics. Explosion is evident in both the 2D and the 3D cases. Explosion is not obtained in 1D. Comparing the two- and three-dimensional trajectories, we see that the development of the explosion in the 3D case is slower. In the 2D case, the shock radius changes rapidly beginning at about 200 ms after bounce. In the 3D case, the shock radius does not begin to climb dramatically until approximately 100 ms later, at $\\sim$300 ms after bounce. The 1D and 2D\/3D angle-averaged shock radii diverge at approximately 125 ms after bounce, and the 2D and 3D angle-averaged shock radii diverge later, at about 200 ms after bounce.\n\nFigure \\ref{fig:entropy3D} is a snapshot of a 2D slice of our ongoing 3D model at approximately 441 ms after bounce. Shown is the stellar core entropy. The shock wave is clearly outlined by the jump in entropy across it. Neutrino-driven convection is evident in the slice. Hotter (red) rising plumes bring neutrino-heated material up to the shock, while cooler (green) down flows replace the fluid below. Distortion of the shock away from axisymmetry and the nonaxisymmetric patterns of convection beneath the shock are also evident. Conclusive evidence for $l=1$, ``sloshing'' and $m=1$, ``spiral'' modes of the SASI will require a modal analysis, although the 2D slice clearly does not rule out either mode. \n\nThis simulation utilizes 32,400 rays (solid angle elements) with 2\\ensuremath{^\\circ}\\ resolution in longitude and a resolution in latitude that varies from 8\\ensuremath{^\\circ}\\ at the pole to better than 0.7\\ensuremath{^\\circ}\\ at the equator, but is uniform in the cosine of the colatitude. \nDue to the Courant limit, the coordinate pole in standard spherical-polar coordinates creates a strong restriction on the time step size and therefore lengthens the total run time compared to a similar resolution 2D simulation.  \nOur constant cosine-of-colatitude grid seeks to minimize this impact without resorting to a grid that is coarse at all latitudes or implementing unevolved (frozen) regions near the pole. The simulation will consume approximately 100 M core--hours to complete. {\\em (This gives a strong indication of how the physics included in the models, even in the RbR+ approximation, significantly drives upward their computational cost.)}\nAs this 3D simulation for a 15~\\ensuremath{M_{\\odot}}\\ progenitor evolves, we will be able to examine the nature of the CCSN explosion mechanism without the assumption of axisymmetry that is inherent in the 2D models. {\\em The} key question: Will this model yield a robust explosion? And will other predictions agree with observations? As indicated by all of our 2D models, our current 3D model will need to be run significantly longer, and detailed computations of the explosion energy and other observables will need to be completed before we can begin to answer these questions.\n\n\\section{Conclusions and Outlook}\n\nThe most sophisticated spherically symmetric models developed to date do not exhibit core collapse supernova explosions. Despite the prodigious amount of gravitational binding energy tapped during stellar core collapse and radiated via neutrinos, neutrino heating of the stellar core material beneath the supernova shock wave, unaided by other physics, is unable to power such explosions. On the other hand, with the aid of neutrino-driven convection beneath the shock, and the SASI, robust explosions have been obtained in both two- and three-dimensional models, with model predictions consistent with observations of multiple quantities (explosion energy, $^{56}$Ni mass, neutron star mass, neutron star kick velocity).\n\nOne- and two-dimensional studies have identified a list of key physics needed in CCSN models. The addition of new physics (e.g., magnetic fields) will likely add to this list as the new physics is added to today's most advanced models (e.g., see \\cite{ObJaAl14}). It is also possible that the addition of new physics will render some of the physics currently included less important. However, it is unlikely that the impact of general relativity and of important neutrino physics (e.g., relativistic transport corrections such as gravitational redshift and the full physics of electron capture and neutrino scattering) will be significantly lessened by adding new physics. The quantum leap in CCSN modeling that occurred two decades ago, where axisymmetry replaced spherical symmetry, did not reduce the importance of this physics --- case in point, both Lentz et al. \\cite{LeMeMe12} and Mueller et al. \\cite{MuJaMa12} reached the same conclusions. Moreover, the development of magnetic fields will depend on the environment established by accretion and neutrino heating.\nFuture modeling --- in particular, the direction we choose to take --- should rely on the predictions of the best {\\em available} models, more so than on speculation of what physics may or may not be important. With this in mind, the task at hand is, therefore, to build 3D models with the minimum physics set identified in the studies mentioned above. \n\nIn this brief review, we outlined the approaches used by the various supernova modeling groups around the world, focusing on two- and three-dimensional, multi-frequency models. While a comparative analysis of the results of these studies can shed light on the impact of (a) Newtonian versus general relativistic gravity, hydrodynamics, and neutrino transport, and\/or (b) including a reduced versus a complete set of neutrino weak interactions, the latter of which would include detailed nuclear electron capture and neutrino energy scattering, results from simulations cutting across these various levels of sophistication should not be compared with the expectation that the outcomes --- in particular, whether or not robust explosions are obtained --- should be the same. For example, comparing a Newtonian and a general relativistic model, with all other physics in the models kept the same, allows us to understand the role of general relativity, but we should not expect the Newtonian and general relativistic models to agree quantitatively, or even qualitatively.\n\nHaving said this, a comparison between, for example, the results obtained by the Oak Ridge and Garching groups can be made given the similarity of their approaches and the physics included in each of their model sets. In this context, it is important to note that the results of the Garching group differ between simulations performed with their \\textsc{PROMETHEUS-VERTEX}\\ code \\cite{maja09}, which uses a general relativistic monopole correction to the Newtonian self-gravitational potential, derived from the Tolman-Oppenheimer-Volkov equation of the spherically-averaged fluid and thermodynamic quantities in the stellar core, and with their \\textsc{COCONUT-VERTEX}\\ code \\cite{MuJaMa12}, which instead uses the conformal flatness approximation to the general relativistic gravitational field. \\textsc{PROMETHEUS-VERTEX}\\ is the code most similar to \\textsc{Chimera}\\ . Unfortunately, to date, results from the \\textsc{PROMETHEUS-VERTEX}\\ code using the more modern Woosley--Heger progenitor set \\cite{wohe07} have not been published, so a direct comparison is not yet possible.\n\nFocusing once again on the ongoing 3D simulations cited here: Will robust neutrino-driven explosions be obtained? If the answer is no, three explanations are possible: (1) Removing current approximations in the models (e.g., the use of RbR neutrino transport) and\/or making other improvements (e.g., increasing the spatial resolution) may fundamentally alter the outcomes. (2) We are missing essential physics. (3) A combination of additional physics and improved modeling may be needed to alter the outcomes. \nWith regard to (1)-(3):\n\n(A) All of the simulations documented here were initiated from state-of-the-art (e.g., the \\citet{wohe07} series) spherically-symmetric progenitor models. \nCouch and Ott \\cite{CoOt13} point out that multidimensional simulations of the advanced stages of stellar evolution of massive stars yield large deviations from \nspherical symmetry in the Si\/O layer (see \\cite{Arnett14} and the references cited therein).\nThey demonstrate that such (expected) deviations from spherical symmetry can qualitatively alter the \npost-stellar-core-bounce evolution, triggering an explosion in a model that otherwise fails to explode. Such a qualitative change in outcome \ndemands better initial conditions, which can be obtained when spherically symmetric models, currently able to complete stellar evolution through \nsilicon burning and the formation of the iron core (multidimensional models are not yet capable of this), are informed by 3D stellar\nevolution models of earlier burning stages.\n\n(B) Given the importance of the SASI in the explosion models developed thus far, and given that the SASI is a long-wavelength instability, how will the SASI and the turbulence it induces, or neutrino-driven convection and the turbulence it induces, interact? There is evidence, for example, that the energy in long-wavelength modes of the SASI are sapped by the very turbulence the SASI seeds, as a result of the significant shear between counterrotating flows induced by its $m=1$ spiral mode in three dimensions \\citep{EnCaBu12}. On the other hand, Couch and Ott \\cite{CoOt14} recently showed that turbulent ram pressure may be important in driving the shock outward, relieving some of the work from the thermal pressure associated with neutrino heating. Moreover, significant deviations from spherical symmetry in the progenitor, as would be expected based on the current 3D stellar evolution models discussed above, would seed turbulence and, thus, potentially enhance the contribution of turbulence to the outward pressure driving the shock.\n\n(C) If we maintain that CCSNe are neutrino-driven, it may be logical to assume that we are missing something essential in the neutrino sector. Motivated by the experimental and observational measurement of neutrino mass, recent efforts to explore its impact on neutrino transport in stellar cores have uncovered new and increasingly complex physical scenarios \\citep{dufuqi10,chcafr12,chcafr13,VlFuCi14}. Now that the quantum kinetic equations for neutrinos in stellar cores have been derived (e.g., see \\cite{VlFuCi14}), efforts can begin in earnest to extend Boltzmann models to include the quantum mechanical coherent effects associated with neutrino mass. This is, of course, a long-term goal. It is not clear that physics associated with neutrino mass will have an impact on the explosion mechanism, but it has been demonstrated that such physics may impact terrestrial CCSN neutrino signatures significantly (e.g., see \\cite{DuFuCa07}).\n\nSince Colgate and White first proposed that CCSNe are neutrino-driven \\cite{CoWh66}, nearly five decades have passed. Ascertaining the CCSN explosion mechanism has certainly been a challenge. Each new piece of physics, each new dimension, has brought both breakthroughs and additional challenges. Nonetheless, the last decade of CCSN modeling has led to rapid progress. This progress --- in particular, the recent progress outlined here --- and the growing capability of available supercomputing platforms, encourage us that a solution to this long-standing astrophysics problem is achievable with a continued, systematic effort in perhaps the not-too-distant future.\n\n\n\n\\section{The Physics and Status of Core Collapse Supernova Simulations}\\label{sec:foundation}\n\nCore collapse supernovae (CCSNe) are initiated by the collapse of the iron cores of massive stars at the ends of their lives. The collapse proceeds to ultrahigh densities, in excess of the densities of nucleons in the nucleus of an atom (``super-nuclear'' densities). The inner core becomes incompressible under these extremes, bounces, and, acting like a piston, launches a shock wave into the outer stellar core. This shock wave will ultimately propagate through the stellar layers beyond the core and disrupt the star in a CCSN explosion. However, the shock stalls in the outer core, losing energy as it plows through it, and exactly how the shock wave is revived remains an open question, although, important progress is being made, particularly with two-dimensional (2D) models. The means to this revival is the central question in CCSN theory. (For a more complete review, the reader is referred to \\cite{Mezz05}, \\cite{Jank12}, and \\cite{KoTaSu12}.) \n\nAfter core bounce, $\\sim10^{53}$~ergs of energy in neutrinos and antineutrinos of all three flavors  are released from the newly formed proto-neutron star (PNS) at the center of the explosion. \nThe typical observationally estimated CCSN explosion energy  is $\\sim 10^{51}$~ergs ($\\equiv1$~Bethe), with estimates for individual supernovae ranging from 0.3--5~Bethe \\cite{Hamu03,NoToUm06,Smar09}.\nPast simulations \\cite{Wils85,BeWi85} demonstrated that energy in the form of neutrinos emerging from the PNS can be deposited behind the shock and may revive it. \nThis neutrino reheating is central to CCSN models today. \nHowever, while a prodigious amount of neutrino energy emerges from the PNS, the neutrinos are weakly coupled to the material below the shock. \nThe neutrino heating is very sensitive to the distribution of neutrinos in energy (or frequency) and direction of propagation, at any given spatial point behind the shock \n\\cite{BuGo93,JaMu96,MeMeBr98,MeCaBr98b,MeLiMe01,Jank01}.\nRealistic CCSN simulations require a neutrino transport method that can reproduce the angular and energy distributions of the neutrinos in the critical heating region.\n\nNormal iron core stars do not explode when modeled in spherical symmetry  \\cite[cf.,][]{LiMeTh01a,RaJa02,ThBuPi03}, thus multidimensional effects are required.  \nFluid instabilities ({\\it e.g.}, convection) in the PNS may boost the luminosity of this central neutrino source and consequent neutrino heating \\cite{SmWiBa81,WiMa93,MiPoUr02,BrRaMe04,BuRaJa06}.  \nNeutrino-driven convection between the PNS and the shock fundamentally alters the nature of energy flow and shock revival \\cite{HeBeHi94,BuHaFr95,JaMu96,FrWa04,BuRaJa06,BrDiMe06} relative to the spherically symmetric case, allowing simultaneous down-flows that fuel the neutrino luminosities and neutrino-heated up-flows that bring energy to the shock. \nThe standing accretion shock instability (SASI), a computationally discovered instability of the shock wave itself \\cite{BlMeDe03}, dramatically alters the shock and explosion dynamics \n\\cite{BlMeDe03,JaBuKi05,BuLiDe06,OhKoYa06,HaMuWo13}. Recent axisymmetric (2D) models \\cite{MuJaHe12,BrMeHi13} demonstrate that neutrino heating in conjunction with neutrino-driven convection and the SASI are able to generate explosions, although the quantitative predictions --- in particular, the explosion energies --- differ between these two groups. However, it is important to note that our predictions are consistent with observations \\cite{BrLeHi14} across a range of observables: explosion energy, $^{56}$Ni mass, neutron star mass, and neutron star kicks.\nDespite these differences, these advances suggest that the SASI  may be the ``missing link'' that will enable the Wilson delayed-shock, neutrino-heating mechanism to operate successfully in multiple spatial dimensions, especially for more massive progenitors. \n\nThere are many other inputs to the physics of the core collapse supernova (CCSN) mechanism that must also be included in simulations. The strength of these effects have been tested in many one-dimensional (1D) simulations and some multidimensional simulations.\nThe PNS in a developing CCSN is sufficiently compact to require the inclusion of general relativistic effects to gravity and neutrino propagation \\cite{BaCoKa85,LiMeTh01a,LiMeTh01b,BrDeMe01,MaDiJa06,OtDiMa07,MuJaDi10,LeMeMe12a,MuJaMa12}.\nGetting the correct radiative coupling requires inclusion of all neutrino--matter interactions (opacities) that affect the neutrino transport, heating, and cooling. Several recent studies have considered the effects of neutrino opacities, including inelastic scattering of neutrinos on electrons, nucleons, and nuclei, detailed nuclear electron capture, and nuclear medium effects on the neutrino interactions \\cite{HiMeBr03,BuJaKe03,KeRaJa03,ThBuPi03,MaJaBu05,MaLiFr06,LaMaMu08,JuLaHi10,RoReSh12,LeMeMe12b}.\nA nuclear equation of state for both nuclear matter in the PNS and the nuclei and nucleons in the surrounding matter is required. Several equations of state have been proposed \\cite{BeBrAp79,ElHi80,Coop85,LaSw91,WiMa93,ShToOy98b,HeSc10,ShHoTe11,StHeFi13} and their impact in CCSNe has been examined \\cite{SwLaMy94,RaBuJa02,SuYaSu05,MaJaMu09,LeHiBa10,Couc13a}.\nFinally, the nuclear composition must be evolved in the outer regions where nuclear statistical equilibrium (NSE) does not apply.\n\nThe centrifugal effects of stellar core rotation, especially for rapid rotation, can also change supernova dynamics qualitatively and quantitatively \\cite{FrWa04,BuRaJa06}. \nAn additional level of complexity is added by models with dynamically important magnetic fields, amplified by rapid rotation and the magnetorotational instability, that may  play a significant role in driving, and perhaps collimating, some CCSNe \\cite{Symb84,AkWhMe03,BuDeLi07} and \\emph{collapsars} (jets generated by accretion disks about newborn black holes producing combined CCSNe\/$\\gamma$-ray bursts). \nRecent  observations of shock breakout \\cite{ScJuWo08} disfavor a strongly collimated jet as the driver for explosions for ordinary supernovae \\cite{CoWhMi09} --- i.e., cases where rotation likely does not play a major role. \nMagnetic fields are expected to become important in the context of rapidly rotating progenitors, where significant rotational energy can be tapped to develop strong and organized magnetic fields (e.g., see \\cite{BuDeLi07}). State-of-the-art stellar evolution models for massive stars \\cite{wohe07} do not predict large core rotation rates. For non-rapidly rotating progenitors, magnetic fields are expected to serve more of a supporting role, for neutrino shock reheating (e.g., see \\cite{ObJaAl14}).\n \nWhile the list of major macroscopic components in any CCSN clearly indicates this is a 3D phenomenon, 3D studies have been relatively rare and, until recently, generally have skimped, largely for practical reasons, on key physics to which prior studies (noted above) have indicated careful attention must be paid.  \n3D simulations have examined aspects of the CCSN problem using a progression of approximations.\n3D, hydrodynamics-only simulations of the SASI, which isolate the accretion flow from the feedbacks of neutrino heating and convection, have identified the spiral ($m=1$) mode, with self-generated counter-rotating flows that can spin the PNS to match the $\\sim$50~ms periods of young pulsars \\cite{Blon05a,Blon05b,BlMe07} and examined the generation of magnetic fields \\cite{EnCaBu10} and turbulence \\cite{EnCaBu12} by the SASI.\nAnother often-used formulation for approximate 3D simulations is the neutrino ``lightbulb'' approximation, where a proscribed neutrino luminosity determines the heating rate, with the neutrino heating and cooling parameterized independently. \nNeutrino lightbulb simulations have been used successfully to study the development of NS kicks \\cite{NoBrBu12,WoJaMu12,WoJaMu13}, mixing in the ejecta \\cite{HaJaMu10}, and, in 2D simulations, the growth of the SASI with neutrino feedbacks \\cite{ScJaFo08}. Lightbulb simulations have also been used to examine the role of dimensionality (1D-2D-3D) in CCSNe \\cite{MuBu08,NoBuAl10,HaMaMu12,Couc13b}.\nA more sophisticated approximate neutrino transport method is the ``leakage'' scheme. Leakage schemes use the local neutrino emission rate and the opaqueness of the overlying material to estimate the cooling rate and from that the neutrino luminosity and heating rate. \nLeakage models have been used by Ott et al. \\cite{OtAbMo13}, including the full 3D effects of GR.\nFryer and Warren \\cite{FrWa02,FrWa04} employed a \\emph{gray} neutrino transport scheme in three dimensions. In such schemes, one evolves the spatial neutrino energy and momentum densities with a \nparameterization of the neutrino spectra. As a neutrino angle- and energy-integrated scheme, the dimensionality of the models is greatly reduced, which is ideal for performing a larger number of exploratory studies.\nThese 3D studies, and other recent studies \\cite[cf.][]{TaKoSu12,BuDoMu12,HaMuWo13,CoOc13}, confirm the conclusion that CCSN simulations must ultimately be performed in three spatial dimensions.  \n\nThe modeling of CCSNe in three dimensions took an important step forward recently. The Max Planck (MPA) group launched the first 3D CCSN simulation with multifrequency neutrino transport with relativistic corrections and state-of-the-art neutrino opacities, and general relativistic gravity. Results from the first 400 ms after stellar core bounce were reported in \\cite{HaMuWo13} for a 27 \\ensuremath{M_{\\odot}}\\ progenitor. At present, the ``Oak Ridge'' group is performing a comparable simulation beginning with the 15~\\ensuremath{M_{\\odot}}\\ progenitor used in our 2D studies. We have evolved approximately the first half second after bounce (for further discussion, see Section~\\ref{sec:current3D}). \n\n\\section{Lessons from Spherical Symmetry}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{fig2_color.pdf}\n\\caption{Shock trajectories in km, versus time after bounce, for models with decreasing physics \\cite{LeMeMe12}.}\n\\label{fig:shockvphysics}\n\\end{figure}\n\nRecent studies carried out in the context of general relativistic, spherically symmetric CCSN models with Boltzmann neutrino transport demonstrate that (i) a general relativistic treatment of gravity, (ii) special and general relativistic corrections to the neutrino transport, such as the gravitational redshift of neutrinos, and (iii) the use of a complete set of weak interactions and a realistic treatment of those interactions are indispensable \\cite{LeMeMe12a}. As shown in Figure \\ref{fig:shockvphysics}, the impact of moving to a Newtonian description of gravity from a fully general relativistic treatment has a significant impact on the shock trajectory. The Newtonian simulation neglects general relativity in the description of gravity {\\it per se}, as well as general relativistic transport effects such as gravitational redshift. Thus, the switch from a general relativistic description to a Newtonian description impacts more than just the treatment of gravity. In turn, if we continue to simplify the model, this time reducing the set of weak interactions included and the realism with which these weak interactions are included, we see a further significant change in the shock trajectory, with fundamentally different behavior early on after bounce. In this instance, we have neglected the impact of nucleon correlations in the computation of electron capture on nuclei (see \\cite{HiMeMe03}), energy exchange in the scattering of neutrinos on electrons, corrections due to degeneracy and nucleon recoil in the scattering of neutrinos on nucleons, and nucleon--nucleon bremsstrahlung. Finally, if we continue to simplify the neutrino transport by neglecting special relativistic corrections to the transport, such as the Doppler shift, we obtain yet another significant change. The spread in the shock radii at $t>$120 ms after bounce is approximately 60 km. Its relative fraction of the average of the shock radii across the four cases at $t>$ 120 ms is $>$33\\%. Moreover, the largest variation in the shock radii in our 2D models is obtained at $\\sim$ 120 ms after bounce, which is around the time when the shock radii in our one- and two-dimensional models begin to diverge (see Figure \\ref{fig:label1Dv2D}). In all four of our 2D models, the postbounce evolution is quasi-spherical until $\\sim$110 ms after bounce. Thus, the use of the \\textsc{Agile-BOLTZTRAN}\\ code, which solves the general relativistic Boltzmann equation with a complete set of neutrino weak interactions for the neutrino transport in the context of spherically symmetric models, to determine the physics requirements of more realistic two- and three-dimensional modeling is possible. Indeed, the conclusions of our studies are corroborated by similar studies carried out in the context of 2D multi-physics models \\cite{MuJaMa12}. Taken together, these studies establish the {\\it necessary} physics that must be included in CCSN models in the future. Whether or not the current treatments of this physics in the context of two- and three-dimensional models is {\\it sufficient}, as we will discuss, remains to be determined.\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{1Dv2D.pdf}\n\\caption{Shock trajectories in km, versus time after bounce, for our 1D and 2D models \\cite{BrMeHi13}. The 1D and 2D evolution begins to diverge between 100 and 125 ms after bounce.}\n\\label{fig:label1Dv2D}\n\\end{figure}\n\n\\section{Our Code}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{rbr.pdf}\n\\caption{A depiction of the ``ray-by-ray'' (RbR) approach. Each ray corresponds to a separate spherically symmetric problem. In the limit of spherical symmetry, the RbR approach is exact. Each ray solve gives what would be obtained in a spherically symmetric solve for conditions at the base of the ray, on the proto-neutron star surface. For a {\\it persistent} hot spot, such as the one depicted here at the base of ray 1, the RbR approximation would overestimate the angular variations in the neutrino heating at the points 1 and 2 above the surface. In spherical symmetry, the condition at the base of each ray is assumed to be the same over the entire portion of the surface subtended by the backward causal cone for that ray. Thus, for ray 1, the entire subtended surface would be considered hotter than it is, whereas for ray 2 the contribution from the hot spot at the base of ray 1 to the heating at point 2 above the surface would be ignored.\n\\label{fig:rbr}\n}\n\\end{figure}\n\n\\textsc{Chimera}\\ is a parallel, multi-physics code built specifically for multidimensional simulation of CCSNe.\nIt is the chimeric combination of separate codes for hydrodynamics and gravity; neutrino transport and opacities; and a nuclear EoS and reaction network, coupled by a layer that oversees data management, parallelism, I\/O, and control.\n\nThe hydrodynamics are modeled using a dimensionally-split, Lagrangian-Remap (PPMLR) scheme  \\cite{CoWo84} as implemented in VH1 \\cite{HaBlLi12}.\nSelf-gravity is computed by multipole expansion \\cite{MuSt95}.\nWe include the most important effects of GR by replacing  the Newtonian monopole term  with a GR monopole computed from the TOV equations \\cite[][Case~A]{MaDiJa06}.\n\nNeutrino transport is computed in the ``ray-by-ray-plus'' (RbR+) approximation \\cite{BuRaJa03}, where an independent, spherically symmetric transport solve is computed for each ``ray'' (radial array of zones with the same $\\theta$, $\\phi$). (It is very important to note that the RbR+ approximation does not restrict the neutrinos to strict radial propagation only. In spherical symmetry, neutrinos propagate along arbitrary rays, not just radial rays, but the {\\em net} angular flux is zero, leaving only radial flux. Each RbR+ solve is a {\\em full} spherically symmetric solve (see Figure \\ref{fig:rbr}). The 3D problem is broken up into $N_{\\theta}\\times N_{\\phi}$ spherically symmetric problems, where $N_{\\theta,\\phi}$ are the number of latitudinal and longitudinal zones, respectively. RbR+ is exact (physically speaking, modulo numerical error) if the neutrino source is spherically symmetric. Thus, if accreted material raining down on the PNS surface via the non-spherical accretion funnels, obvious in Figures~\\ref{fig:entropy} and \\ref{fig:entropy3D}, and creating hot spots, spreads rapidly over the surface relative to the neutrino-heating and shock-revival time scales, which we find it does, and in the absence of significant rotation, the RbR+ approximation is a reasonable approximation, at least initially. There are practical benefits to the approximation, as well, which we will discuss later.)\n\nThe transport solver for each ray is an improved and updated version of the multi-group flux-limited diffusion transport solver of Bruenn \\cite{Brue85} enhanced for GR \\cite{BrDeMe01}, with an additional geometric flux limiter to prevent an overly-rapid transition to free streaming of the standard flux-limiter.  All $O(v\/c)$ observer correction terms have been included.\n\n\\textsc{Chimera}\\ solves for all three flavors of neutrinos and antineutrinos with four coupled species: \\ensuremath{\\nu_{e}}, \\ensuremath{\\bar \\nu_e}, $\\ensuremath{\\nu_{\\mu\\tau}}=\\{\\ensuremath{\\nu_{\\mu}},\\ensuremath{\\nu_{\\tau}}\\}$, $\\ensuremath{\\bar \\nu_{\\mu\\tau}}=\\{\\ensuremath{\\bar \\nu_{\\mu}},\\ensuremath{\\bar \\nu_{\\tau}}\\}$, with typically 20 energy groups covering two decades in neutrino energy.\nOur standard, modernized, neutrino--matter interactions include emission, absorption, and non-isoenergetic scattering on free nucleons \\cite{RePrLa98}, with weak magnetism corrections \\cite{Horo02}; emission\/absorption (electron capture) on nuclei \\cite{LaMaSa03}; isoenergetic scattering on nuclei, including ion-ion correlations; non-isoenergetic scattering on electrons and positrons; and pair emission from $e^+e^-$-annihilation \\cite{Brue85} and nucleon-nucleon bremsstrahlung \\cite{HaRa98}.\n\\textsc{Chimera}\\ generally utilizes the $K = 220$~\\mbox{MeV}\\ incompressibility version of the Lattimer--Swesty \\cite{LaSw91} EoS for  $\\rho>10^{11}\\,\\ensuremath{{\\mbox{g~cm}}^{-3}}$ and a modified version of the Cooperstein \\cite{Coop85} EoS for  $\\rho<10^{11}\\,\\ensuremath{{\\mbox{g~cm}}^{-3}}$, where nuclear statistical equilibrium (NSE) applies.\nMost \\textsc{Chimera}\\ simulations have used a 14-species $\\alpha$-network ($\\alpha$, \\isotope{C}{12}-\\isotope{Zn}{60}) for the non-NSE regions \\cite{HiTh99a}. In addition,\n\\textsc{Chimera}\\ utilizes a 17-nuclear-species NSE calculation for the nuclear component of the EOS for $Y_{\\rm e}>26\/56$ to provide a smooth join with the non-NSE regime\n\nDuring evolution, the radial zones are gradually and automatically repositioned to track changes in the mean radial structure.\nTo minimize restrictions on the time step from the Courant limit, the lateral hydrodynamics for a few inner zones are ``frozen'' during collapse, and after prompt convection fades, the laterally frozen region expands to the inner 6--8~km.\nIn the ``frozen'' region the  radial hydrodynamics and neutrino transport are  computed in spherical symmetry.\n\nThe supernova code most closely resembling \\textsc{Chimera}\\ \nis the \\textsc{PROMETHEUS-VERTEX}\\ code developed by the Max Planck group \\cite{BuRaJa03,BuRaJa06,BuJaRa06,MuJaDi10}. This code utilizes a RbR+ approach to neutrino transport, solving the first two multifrequency angular moments of the transport equations with a variable Eddington closure that is solved at intervals using a 1D approximate Boltzmann equation.\n\n\\textsc{Chimera}\\ does not yet include magnetic fields. Studies with \\textsc{Chimera}\\ that include magnetic fields will be part of future efforts. \n\n\\section{Our Approach in Context}\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{2DApproaches.pdf}\n\\caption{An overview of the approaches used in the context of 2D CCSN modeling by various groups around the world \\cite{SuKoTa10,TaKoSu14,NaTaKu14,DoBuZh14,maja09,MuJaMa12,BrMeHi13}.  \n\\label{fig:label2DApproaches}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.0in]{3DApproaches.pdf}\n\\caption{An overview of the approaches used in the context of 3D CCSN modeling by several groups around the world \\cite{TaKoSu12,HaMuWo13,LeBrHi15}.\n\\label{fig:label3DApproaches}}\n\\end{figure}\n\nA number of 2D simulations have been performed to date with multi-frequency neutrino transport. These break down into two classes, those that have implemented the RbR neutrino transport approximation and those that have not --- i.e., those that have implemented 2D transport. Figure \\ref{fig:label2DApproaches} provides an overview of the approaches used by various supernova groups in producing these 2D models. It is clear the RbR approximation has enabled the inclusion of general relativity and state-of-the-art neutrino interactions, at the expense of the added spatial dimensionality of the transport, whereas the non-RbR approach includes the second spatial dimension in the neutrino transport, but does so at the expense of realism in the treatment of gravity and the neutrino interactions with stellar matter. The reason for this is simple: In the RbR approach, transport codes that have been used in spherically symmetric studies, such as \\textsc{Agile-BOLTZTRAN}\\ , can be deployed. These codes already, or at least can more easily, include all relativistic transport corrections and full weak interaction physics. To achieve the same level of sophistication in two and three spatial dimensions is more difficult and far more computationally intensive. For example, a 3D multi-frequency approach (e.g., flux-limited diffusion or a variable Eddington tensor method) will require the sustained-petaflop performance of present-day leadership-class computing facilities. In light of the practical difficulties associated with including more physics in fully 3D simulations, the RbR approximation provides an alternative approach that can be used in the interim. The use of both approaches by the community as it moves forward will be essential, as simulations with RbR neutrino transport with approximate general relativity and full weak interaction physics must be gauged by non-RbR approaches that can test the efficacy of the RbR approach. Ultimately, the two approaches must merge, with 3D simulations performed with 3D (i.e., not RbR) general relativistic neutrino transport, general relativistic hydrodynamics and gravity, and a full weak interaction set. Figure \\ref{fig:label3DApproaches} gives an overview of the 3D simulations performed to date, using multi-frequency neutrino transport. It is obvious that fewer groups have attempted this, and far fewer simulations have been performed. It is also evident they have all been performed with RbR and not 3D neutrino transport.\n\n\\section{Results from our 2D Core Collapse Supernova Models}\\label{sec:current2D}\n\nWe \\cite{BrMeHi13,BrLeHi14} have performed four 2D simulations with \\textsc{Chimera}\\ beginning with the 12, 15, 20, and \n25~\\ensuremath{M_{\\odot}}\\ progenitors of Woosley and Heger \\cite{wohe07}.\nOne result of these simulations is the realization that a fully developed (and therefore final) explosion energy will require much more lengthy simulations than anticipated in the past.\nIn the explosion energy plot, Figure~\\ref{fig:energy}, the dashed lines show the growth of the ``diagnostic energy'' (the sum of the gravitational potential energy, the kinetic energy, and the internal energy in each zone --- i.e., the total energy in each zone --- for all zones having a total energy greater than zero) along with more refined estimates of the final explosion energy that account for the work required to lift the as-yet-unshocked envelope ``overburden'' (dash-dotted lines) and, in addition, the estimated energy released from recombination of free nucleons and alpha particles into heavier nuclei (solid lines). We expect these latter two measures to bracket the final kinetic energy of the fully developed explosion. Using the definition of the explosion energy that includes both the energy cost to lift the overlying material and the energy gain associated with nuclear recombination, we can define $t_{\\rm explosion}$, the explosion time, which is the time at which the explosion energy becomes positive and, therefore, the explosion can be said to have been initiated. For the 12, 15, 20, and 25 M$_\\odot$ models, $t_{\\rm explosion}$ is approximately 320, 320, 500, and 620 ms after bounce, respectively. \n\nMoving now to a comparison with observations: All four models have achieved explosion energies that are in the $\\approx $0.4--1.4 Bethe range of observed Type~II supernovae (see Figure \\ref{fig:energycomparison}). Figures \\ref{fig:nickelmass} and \\ref{fig:pnsmass} compare our predictions for the mass of $^{56}$Ni produced and the final proto-neutron star (baryonic) masses produced, respectively, with observations. Note, the large systematic errors in observed progenitor masses preclude any detailed comparison between our results and observations {\\em as a function of progenitor mass}. Nonetheless, comparisons of our predicted {\\em ranges} of explosion energies, $^{56}$Ni masses, etc. with observed ranges is meaningful and demonstrates we are making progress toward developing predictive models.\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{movie.jpg}\n\\caption{Evolution of the entropy (upper half) and radial velocity (lower half) at 150, 300, and 600~ms after bounce  for the 12~\\ensuremath{M_{\\odot}}\\ model of Bruenn et al. \\cite{BrMeHi13}.  \n\\label{fig:entropy}}\n\\end{figure}\n\nThree snapshots of hydrodynamic motion are visible in \nFigure~\\ref{fig:entropy}, \nwhich shows the entropy (upper half) and radial velocity (lower half) for the 12 \\ensuremath{M_{\\odot}}\\ model at 150~ms, 300~ms, and 600~ms after bounce.  \nAt 150~ms, roughly 100~ms before rapid shock expansion heralds the onset of a developing explosion, asphericity is developing as a result of vigorous neutrino-driven convection and the SASI.  \nBy 300~ms large-scale, high-entropy, buoyant plumes are evident, as the explosion continues to develop.  \nHowever, low-entropy down-flows still connect the unshocked regions with the PNS surface, continuing to supply accretion energy to power the neutrino luminosities driving the development of the explosion. By 600~ms, these down-flows have been cut off by the expanding ejecta, but their remnants continue to accrete onto the PNS, allowing the explosion to continue to gain in strength.\n\nThough these simulations have run further into explosion than previous simulations, the final explosion energies --- in particular, for the 20 and 25 M$_\\odot$ models --- are clearly still developing. \nThese simulations will therefore continue. Additional 2D simulations --- e.g., using different progenitor masses --- are planned.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Expl_E_vs_t_12M_25M_Comp.pdf}\n\\caption{Diagnostic energy (\\ensuremath{E^{+}}; dashed lines) versus post-bounce time for all of our published 2D models \\cite{BrMeHi13,BrLeHi14}. Dash-dotted lines (\\ensuremath{E^{+}_{\\rm ov}}) include binding energy of overburden and dashed lines (\\ensuremath{E^{+}_{\\rm ov, rec}}) also include estimated energy gain from nuclear recombination.}\n\\label{fig:energy}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{Explosion_Energy_Comparisons.pdf}\n\\caption{\nObserved explosion energies for a number of CCSNe, along with predicted explosion energies from our 12, 15, 20, and 25 M$_\\odot$ progenitor models (red dots) \\cite{BrLeHi14}. The arrows indicate that our explosion energies are still increasing at the end of each run. The length of each arrow is a measure of the rate of change of the explosion energy at the end of the corresponding run.\n\\label{fig:energycomparison}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{Nickel56_Comparisons.pdf}\n\\caption{\nObserved production of $^{56}$Ni for a number of CCSNe, along with our predictions from our 12, 15, 20, and 25 M$_\\odot$ progenitor models (red dots) \\cite{BrLeHi14}.\n\\label{fig:nickelmass}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.00in]{N_Star_Mass.pdf}\n\\caption{\nTime evolution of the proto-neutron star (baryonic) mass in each of our 4 2D models, beginning with 12, 15, 20, and 25 M$_\\odot$ progenitors \\cite{BrLeHi14}.\n\\label{fig:pnsmass}\n}\n\\end{figure}\n\n\\section{Preliminary Results from our 3D Core Collapse Supernova Model}\\label{sec:current3D}\n\n\\begin{figure}\n\\includegraphics[width=3.1in]{1D2D3DShockTrajectories.pdf}\n\\caption{Evolution of the shock trajectory from our 1D model and the angle-averaged shock trajectories from our 2D and 3D models, all for the 15~\\ensuremath{M_{\\odot}}\\ case \\cite{LeBrHi15}. The 1D model does not develop an explosion, whereas an explosion is obtained in both our 2D and our 3D models.\n\\label{fig:1D2D3DShockTrajectories}\n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.15in]{3D441msYZ.pdf}\n\\caption{Snapshot of the equatorial cross section of the entropy in our ongoing 3D simulation for the 15~\\ensuremath{M_{\\odot}}\\ case at $\\sim$441 ms after bounce \\cite{LeBrHi15}. Red indicates high-entropy, expanding, rising material. Green\/blue indicates cooler, denser material. Evident are significant (green) down flows fueling the neutrino luminosities.\n\\label{fig:entropy3D}\n}\n\\end{figure}\n\nFew 3D multiphysics models with necessary realism (as defined above) have been performed. Notable among these is the recently published model of Hanke et al. \\cite{HaMuWo13}. Preliminary results from the Oak Ridge group \\cite{LeBrHi15} in the context of a model similar to the Garching group's model -- i.e., with essentially the same physics and treatment of this physics -- are presented here, although we begin with the same 15 M$_\\odot$ Woosley--Heger progenitor used in our 2D models, whereas they began with the 27 M$_\\odot$ Woosley--Heger progenitor. \n\nFigure \\ref{fig:1D2D3DShockTrajectories} shows the angle-averaged shock trajectories from our one-, two-, and three-dimensional models, all run with the \\textsc{Chimera}\\ code beginning with the same 15 M$_\\odot$ Woosley--Heger progenitor and including the same (full) physics. Explosion is evident in both the 2D and the 3D cases. Explosion is not obtained in 1D. Comparing the two- and three-dimensional trajectories, we see that the development of the explosion in the 3D case is slower. In the 2D case, the shock radius changes rapidly beginning at about 200 ms after bounce. In the 3D case, the shock radius does not begin to climb dramatically until approximately 100 ms later, at $\\sim$300 ms after bounce. The 1D and 2D\/3D angle-averaged shock radii diverge at approximately 125 ms after bounce, and the 2D and 3D angle-averaged shock radii diverge later, at about 200 ms after bounce.\n\nFigure \\ref{fig:entropy3D} is a snapshot of a 2D slice of our ongoing 3D model at approximately 441 ms after bounce. Shown is the stellar core entropy. The shock wave is clearly outlined by the jump in entropy across it. Neutrino-driven convection is evident in the slice. Hotter (red) rising plumes bring neutrino-heated material up to the shock, while cooler (green) down flows replace the fluid below. Distortion of the shock away from axisymmetry and the nonaxisymmetric patterns of convection beneath the shock are also evident. Conclusive evidence for $l=1$, ``sloshing'' and $m=1$, ``spiral'' modes of the SASI will require a modal analysis, although the 2D slice clearly does not rule out either mode. \n\nThis simulation utilizes 32,400 rays (solid angle elements) with 2\\ensuremath{^\\circ}\\ resolution in longitude and a resolution in latitude that varies from 8\\ensuremath{^\\circ}\\ at the pole to better than 0.7\\ensuremath{^\\circ}\\ at the equator, but is uniform in the cosine of the colatitude. \nDue to the Courant limit, the coordinate pole in standard spherical-polar coordinates creates a strong restriction on the time step size and therefore lengthens the total run time compared to a similar resolution 2D simulation.  \nOur constant cosine-of-colatitude grid seeks to minimize this impact without resorting to a grid that is coarse at all latitudes or implementing unevolved (frozen) regions near the pole. The simulation will consume approximately 100 M core--hours to complete. {\\em (This gives a strong indication of how the physics included in the models, even in the RbR+ approximation, significantly drives upward their computational cost.)}\nAs this 3D simulation for a 15~\\ensuremath{M_{\\odot}}\\ progenitor evolves, we will be able to examine the nature of the CCSN explosion mechanism without the assumption of axisymmetry that is inherent in the 2D models. {\\em The} key question: Will this model yield a robust explosion? And will other predictions agree with observations? As indicated by all of our 2D models, our current 3D model will need to be run significantly longer, and detailed computations of the explosion energy and other observables will need to be completed before we can begin to answer these questions.\n\n\\section{Conclusions and Outlook}\n\nThe most sophisticated spherically symmetric models developed to date do not exhibit core collapse supernova explosions. Despite the prodigious amount of gravitational binding energy tapped during stellar core collapse and radiated via neutrinos, neutrino heating of the stellar core material beneath the supernova shock wave, unaided by other physics, is unable to power such explosions. On the other hand, with the aid of neutrino-driven convection beneath the shock, and the SASI, robust explosions have been obtained in both two- and three-dimensional models, with model predictions consistent with observations of multiple quantities (explosion energy, $^{56}$Ni mass, neutron star mass, neutron star kick velocity).\n\nOne- and two-dimensional studies have identified a list of key physics needed in CCSN models. The addition of new physics (e.g., magnetic fields) will likely add to this list as the new physics is added to today's most advanced models (e.g., see \\cite{ObJaAl14}). It is also possible that the addition of new physics will render some of the physics currently included less important. However, it is unlikely that the impact of general relativity and of important neutrino physics (e.g., relativistic transport corrections such as gravitational redshift and the full physics of electron capture and neutrino scattering) will be significantly lessened by adding new physics. The quantum leap in CCSN modeling that occurred two decades ago, where axisymmetry replaced spherical symmetry, did not reduce the importance of this physics --- case in point, both Lentz et al. \\cite{LeMeMe12} and Mueller et al. \\cite{MuJaMa12} reached the same conclusions. Moreover, the development of magnetic fields will depend on the environment established by accretion and neutrino heating.\nFuture modeling --- in particular, the direction we choose to take --- should rely on the predictions of the best {\\em available} models, more so than on speculation of what physics may or may not be important. With this in mind, the task at hand is, therefore, to build 3D models with the minimum physics set identified in the studies mentioned above. \n\nIn this brief review, we outlined the approaches used by the various supernova modeling groups around the world, focusing on two- and three-dimensional, multi-frequency models. While a comparative analysis of the results of these studies can shed light on the impact of (a) Newtonian versus general relativistic gravity, hydrodynamics, and neutrino transport, and\/or (b) including a reduced versus a complete set of neutrino weak interactions, the latter of which would include detailed nuclear electron capture and neutrino energy scattering, results from simulations cutting across these various levels of sophistication should not be compared with the expectation that the outcomes --- in particular, whether or not robust explosions are obtained --- should be the same. For example, comparing a Newtonian and a general relativistic model, with all other physics in the models kept the same, allows us to understand the role of general relativity, but we should not expect the Newtonian and general relativistic models to agree quantitatively, or even qualitatively.\n\nHaving said this, a comparison between, for example, the results obtained by the Oak Ridge and Garching groups can be made given the similarity of their approaches and the physics included in each of their model sets. In this context, it is important to note that the results of the Garching group differ between simulations performed with their \\textsc{PROMETHEUS-VERTEX}\\ code \\cite{maja09}, which uses a general relativistic monopole correction to the Newtonian self-gravitational potential, derived from the Tolman-Oppenheimer-Volkov equation of the spherically-averaged fluid and thermodynamic quantities in the stellar core, and with their \\textsc{COCONUT-VERTEX}\\ code \\cite{MuJaMa12}, which instead uses the conformal flatness approximation to the general relativistic gravitational field. \\textsc{PROMETHEUS-VERTEX}\\ is the code most similar to \\textsc{Chimera}\\ . Unfortunately, to date, results from the \\textsc{PROMETHEUS-VERTEX}\\ code using the more modern Woosley--Heger progenitor set \\cite{wohe07} have not been published, so a direct comparison is not yet possible.\n\nFocusing once again on the ongoing 3D simulations cited here: Will robust neutrino-driven explosions be obtained? If the answer is no, three explanations are possible: (1) Removing current approximations in the models (e.g., the use of RbR neutrino transport) and\/or making other improvements (e.g., increasing the spatial resolution) may fundamentally alter the outcomes. (2) We are missing essential physics. (3) A combination of additional physics and improved modeling may be needed to alter the outcomes. \nWith regard to (1)-(3):\n\n(A) All of the simulations documented here were initiated from state-of-the-art (e.g., the \\citet{wohe07} series) spherically-symmetric progenitor models. \nCouch and Ott \\cite{CoOt13} point out that multidimensional simulations of the advanced stages of stellar evolution of massive stars yield large deviations from \nspherical symmetry in the Si\/O layer (see \\cite{Arnett14} and the references cited therein).\nThey demonstrate that such (expected) deviations from spherical symmetry can qualitatively alter the \npost-stellar-core-bounce evolution, triggering an explosion in a model that otherwise fails to explode. Such a qualitative change in outcome \ndemands better initial conditions, which can be obtained when spherically symmetric models, currently able to complete stellar evolution through \nsilicon burning and the formation of the iron core (multidimensional models are not yet capable of this), are informed by 3D stellar\nevolution models of earlier burning stages.\n\n(B) Given the importance of the SASI in the explosion models developed thus far, and given that the SASI is a long-wavelength instability, how will the SASI and the turbulence it induces, or neutrino-driven convection and the turbulence it induces, interact? There is evidence, for example, that the energy in long-wavelength modes of the SASI are sapped by the very turbulence the SASI seeds, as a result of the significant shear between counterrotating flows induced by its $m=1$ spiral mode in three dimensions \\citep{EnCaBu12}. On the other hand, Couch and Ott \\cite{CoOt14} recently showed that turbulent ram pressure may be important in driving the shock outward, relieving some of the work from the thermal pressure associated with neutrino heating. Moreover, significant deviations from spherical symmetry in the progenitor, as would be expected based on the current 3D stellar evolution models discussed above, would seed turbulence and, thus, potentially enhance the contribution of turbulence to the outward pressure driving the shock.\n\n(C) If we maintain that CCSNe are neutrino-driven, it may be logical to assume that we are missing something essential in the neutrino sector. Motivated by the experimental and observational measurement of neutrino mass, recent efforts to explore its impact on neutrino transport in stellar cores have uncovered new and increasingly complex physical scenarios \\citep{dufuqi10,chcafr12,chcafr13,VlFuCi14}. Now that the quantum kinetic equations for neutrinos in stellar cores have been derived (e.g., see \\cite{VlFuCi14}), efforts can begin in earnest to extend Boltzmann models to include the quantum mechanical coherent effects associated with neutrino mass. This is, of course, a long-term goal. It is not clear that physics associated with neutrino mass will have an impact on the explosion mechanism, but it has been demonstrated that such physics may impact terrestrial CCSN neutrino signatures significantly (e.g., see \\cite{DuFuCa07}).\n\nSince Colgate and White first proposed that CCSNe are neutrino-driven \\cite{CoWh66}, nearly five decades have passed. Ascertaining the CCSN explosion mechanism has certainly been a challenge. Each new piece of physics, each new dimension, has brought both breakthroughs and additional challenges. Nonetheless, the last decade of CCSN modeling has led to rapid progress. This progress --- in particular, the recent progress outlined here --- and the growing capability of available supercomputing platforms, encourage us that a solution to this long-standing astrophysics problem is achievable with a continued, systematic effort in perhaps the not-too-distant future.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and results}\n      Recently, Bauke and Mertens have proposed in \\cite{BaMe} a\n        new and original  look at disordered spin systems.\nThis point of view consists of studying the micro-canonical\nscenario,\n  contrary to the canonical formalism, that has become\n  the favorite tool to treat  models of statistical mechanics.\n  More precisely, they analyze the statistics of spin\n  configurations whose energy is very close to a given value.\n  In discrete spin systems, for a given system size,\n  the Hamiltonian will take on a finite number\n  of random values, and generally\n  (at least, if the disorder is continuous)\n        a given value $E$ is attained\n     with probability $0$.\n   One may, however, ask :\n   How close to $E$ the best approximant is\n      when the system size grows and, more generally,\n  what the distribution of the energies that come closest to\n  $E$ is~? Finally, how the values of the corresponding\n  configurations are distributed in configuration space~?\n\n   The original motivation for this viewpoint came from\n   a reformulation of a problem in combinatorial optimization,\n   the number partitioning problem\n   (this is the problem of\n    partitioning $N$ (random) numbers into two subsets such that\n    their sums in these subsets are as close as possible)\n          in terms of a spin system\n    Hamiltonian \\cite{BFM, M1,M2}. Mertens conjecture stated in these\n      papers has been proven  to be correct in \\cite{BCP}\n        (see also \\cite{BCMP}),\n    and generalized in \\cite{BK1}\n        for the partitioning into $k>2$ subsets.\n\n     Some time later, Bauke and Mertens generalized this conjecture\n       in the following sense : let $(H_N(\\sigma))_{\\s \\in \\Sigma_N}$\n         be the Hamiltonian\n   of any disordered spin system with discrete spins\n      ($\\Sigma_N$ being the configuration space)  and\n   continuously  distributed couplings, let $E$ be any given number,\n   then the distribution of the close to optimal approximants of the\n     level $\\sqrt{N}E$ is asymptotically\n       (when the volume of the system $N$ grows to infinity)\n             the same as if the energies $H_N(\\s)$\n       are replaced by independent Gaussian random\n        variables with the same mean and variance as $H_N(\\s)$\n     (that is the same as for Derrida's Random Energy spin glass Model \\cite{D1},\n        that is why it is called the REM conjecture).\n\n    What this distribution for independent Gaussian random variables\n    is ? Let $X$ be a standard Gaussian random variable,\n      let $\\delta_N \\to 0$ as $N \\to \\infty$, $E \\in {\\bf R}$,\n          $b>0$.\n       Then it is easy to compute that\n       $$ \\mathop{\\hbox{\\sf P}}\\nolimits(   X  \\in [E -\\detla_N b, E+ \\delta_N b])= (2 \\delta_N b)\n       \\sqrt{1\/(2\\pi)}e^{-E^2\/2}(1+o(1))\\ \\ \\ N \\to \\infty.$$\n   Let now $(X_\\s)_{s \\in \\Sigma_N}$ be  $|\\Sigma_N|$\n   independent standard Gaussian random variables.\n    Since they are independent, the number of them\n    that are in the interval $[E -\\detla_N b, E+ \\delta_N\n     b]$ has a Binomial distribution with parameters\n   $(2 \\delta_N b)\n       \\sqrt{1\/(2\\pi)}e^{-E^2\/2}(1+o(1))$ and $ |\\Sigma_N|$. If we put\n          $$\\delta_N =|\\Sigma_N|^{-1} \\sqrt{2\\pi}(1\/2)e^{E^2\/2},$$\n   by a well known theorem of the course of elementary Probability,\n   this random number converges in law  to the Poisson\n     distribution with  parameter $b$ as $N \\to \\infty$. More generally,\n  the point process\n   $$ \\sum_{\\sigma \\in \\Sigma_N} \\delta_{ \\{\\delta_N^{-1} N^{-1\/2}\n        |\\sqrt{N} X_\\s-\n     \\sqrt{N}E|\\}} $$\n converges, as $N \\to \\infty$, to the Poisson point process\n      in ${\\bf R}_+$ whose intensity measure is the Lebesgue measure.\n\n  So,  Bauke and Mertens conjecture states that\n    for the Hamiltonian $(H_N(\\s))_{\\s \\in \\Sigma_N}$\n    of any disordered spin system and\n       for a suitable normalization $C(N,E)$\n  the sequence of point processes\n  $$ \\sum_{\\sigma \\in \\Sigma_N} \\delta_{ \\{C(N,E)|H_N(\\s)-\n     \\sqrt{N}E|\\}} $$\n      converges, as $N \\to \\infty$, to the Poisson point process\n      in ${\\bf R}_+$ whose intensity measure is the Lebesgue measure.\n           In other words, the best approximant\n     to $\\sqrt{N} E$ is at distance $C^{-1}(N,E)W$,\n      where $W$ is an exponential random variable of mean $1$.\n    More generally, the  $k$th best approximant\n     to $\\sqrt{N} E$ is at distance $C^{-1}(N,E)(W_1+\\cdots +W_k)$,\n       where $W_1,\\ldots, W_k$ are  independent\n       exponential random variables of mean $1$, $k=1,2\\ldots$\n            It appears rather surprising that such a result\n              holds in great generality.\n      Indeed, it is well known that the correlations of the random\n        variables are strong enough to modify e.g.\\ the maxima of the\n        Hamiltonian.\n    This conjecture\n      has been proven in \\cite{BK2}\n   for a rather large class of disordered spin systems\n     including\n     short range lattice spin systems as well as\n    mean-field  spin glasses,\n        like $p$-spin Sherringthon-Kirkpatrick (SK) models with\n          Hamiltonian $H_N(\\s)=N^{1\/2-p\/2} \\sum_{i_1,\\ldots, i_p}\n             \\s_{i_1}\\cdots \\s_{i_p}J_{1\\leq i_1,\\ldots, i_p\\leq N}$\n               where $J_{i_1,\\ldots, i_p}$ are\n        independent standard Gaussian random variables, $p\\geq 1$.\n              See also \\cite{BCMN1}\n     for the detailed study of the case $p=1$.\n\n   Two questions naturally pose themselves.\n    (i) Consider instead of $E$, $N$-dependent\n      energy levels, say, $E_N={\\rm const} N^\\alpha$.\n        How fast can we allow $E_N$ to grow with $N \\to \\infty$\n          for the same behaviour\n (i.e.\\ convergence to the standard Poisson point process under  a\n  suitable  normalization) to hold ?\n     (ii) What type of behaviour can we expect\n        once $E_N$ grows faster than this value ?\n\n    The first question (i) has been investigated\n    for Gaussian disordered spin systems in \\cite{BK2}.\n           It turned out that for short range\n     lattice spin systems on ${\\bf Z}^d$  this\n     convergence is still true up to $\\alpha<1\/4$.\n              For mean-field  spin glasses,\n        like $p$-spin SK models with\n          Hamiltonian $H_N(\\s)=N^{1\/2-p\/2} \\sum_{i_1,\\ldots, i_p}\n             \\s_{i_1}\\cdots \\s_{i_p}J_{i_1,\\ldots, i_p}$\n               mentioned above,\n     this conjecture holds true up to $\\alpha<1\/4$\n       for $p=1$ and up to $\\alpha<1\/2$ for $p\\geq 2$.\n             It has been proven in \\cite{BCMN2}\n               that the conjecture fails at\n                 $\\alpha=1\/4$ for $p=1$ and $\\alpha=1\/2$\n                   for $p=2$.\n              The paper \\cite{BCMN2}\n              extends also these results for non-Gaussian\n   mean-field $1$-spin SK models with $\\alpha>0$.\n\n  The second question (ii), that is the local behaviour\n      beyond  the critical value of $\\alpha$,\n          where Bauke and Mertens conjecture fails,\n               has been investigated\n      for Derrida's Generalized Random\n   Energy Models  (\\cite{D2}) in \\cite{BK3}.\n\n    Finally,\n    the paper \\cite{BGK} introduces a new REM conjecture,\n    where the range of energies involved is not reduced to a small\n    window. The authors prove that for  large class of random Hamiltonians\n     the point process of properly normalized energies\n       restricted to a sparse enough random subset of spin\n     configuration space converges to the same point process\n       as for the Random Energy Model, i.e. Poisson point process\n      with intensity measure $\\pi^{-1\/2}e^{-t\\sqrt{2\\ln 2}}dt$.\n\n      In this paper we study Bauke and Merten's conjecture\n          on the local behaviour of energies not\n      for  disordered spin systems but for directed\n      polymers in random environment.\n       These models have received enough of attention\n       of mathematical community over past fifteen years,\n        see e.g.\\ \\cite{CSY} for a survey of the main results\n        and references therein.\n           Let $(\\{w_n\\}_{n\\geq 0}, P)$ is  a simple\n             random walk on the $d$-dimensional lattice\n               ${\\bf Z}^d$. More precisely,\n               we let $\\Omega$ be the path space\n$\\Omega=\\{\\omega=(\\omega_n)_{n\\geq 0};\n  \\omega_n\\in {\\bf Z}^d, n\\geq 0\\}$,\n    ${\\cal F}$ be the cylindrical $\\sigma$-field on $\\Omega$\n     and for all $n\\geq 0$, $\\omega_n: \\omega \\to \\omega_n$\n       be the projection map.\n         We consider the unique probability measure $P$\n         on $(\\Omega, {\\cal F})$ such that $\\omega_1-\\omega_0,\n         \\ldots, \\omega_n-\\o_{n-1}$ are independent and\n         $$ P(\\o_0=0)=1,\\ \\\n           P(\\o_n-\\o_{n-1}=\\pm \\delta_j)=(2d)^{-1}, \\ \\\n             j=1,\\ldots, d,$$\n where $\\delta_j=(\\delta_{kj})_{k=1}^d$ is the $j$th\n    vector of the canonical basis of ${\\bf Z}^d$.\n       We will denote by $S_N=\\{\\omega^N=(i,\\omega_i)_{i=0}^N\\}$\n          ($(i,\\omega_i)\\in {\\bf N}\\times {\\bf Z}^d$)\n         the space of paths of length $N$.\n   We define the energy of the path $\\omega^N=(i,\\omega_i)_{i=0}^N$\n   as\n\\begin{equation}\n\\label{enn}\n     \\eta(\\omega^N)=N^{-1\/2}\\sum_{i=1}^N \\eta(i,\\o_i)\n     \\end{equation}\nwhere $\\{\\eta(n,x) : n \\in {\\bf N}, x\\in {\\bf Z}^d\\}$\n   is a sequence of independent identically distributed\n     random variables  on a probability space $(H, {\\cal G}, \\mathop{\\hbox{\\sf P}}\\nolimits)$.\n  We assume that they have  mean zero  and variance $1$.\n\nOur first theorem  extends Bauke and Merens conjecture\n  for directed polymers.\n\n\\begin{theo}\n\\label{th0}\n Let $\\eta(n,x)$, $\\{\\eta(n,x) : n \\in {\\bf N}, x\\in {\\bf\nZ}^d\\}$,\n   be the i.i.d. random variables of the  third moment finite and\n     with the Fourier transform\n     $\\phi(t)$ such that $|\\phi(t)|=O(|t|^{-1})$, $|t|\\to \\infty$.\n        Let $E_N=c \\in {\\bf R}$ and let\n \\begin{equation}\n     \\label{delta0}\n      \\delta_N = \\sqrt{\\pi\/2} e^{c^2\/2}\n           ((2d)^N)^{-1}.\n      \\end{equation}\n    Then the point process\n\\begin{equation}\n\\label{th1e0}\n  \\sum_{\\o^N\\in S_N} \\delta_{\\{\\delta_N^{-1}\n  |\\eta(\\o^N)-E_N|\\}}\n\\end{equation}\n   converges weakly as $N \\uparrow \\infty$ to the Poisson\n      point process ${\\cal P}$ on ${\\bf R}_+$\n        whose intensity measure is the Lebesgue measure.\n    Moreover, for any $\\epsilon>0$ and any $b \\in {\\bf R}_+$\n\\begin{equation}\n\\label{th1b0} \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall N_0\\ \\exists N \\geq N_0,\\\n \\exists \\o^{N,1}, \\o^{N,2}\\ : \\\n   {\\rm cov}\\,(\\eta(\\o^{N,1}), \\eta(\\o^{N,2}))>\\epsilon\\ :$$\n    $$  |\\eta(\\o^{N,1})-E_N|\\leq  |\\eta(\\o^{N,2})-E_N|\\leq \\delta_N\n      b)=0.\n\\end{equation}\n\\end{theo}\n\n  The decay assumption on the Fourier transform is not optimal,\n  we believe that it can be weaken but we did not try to\n    optimize it. Nevertheless, some condition\n  of this type is needed, the result can not be extended\n  for discrete distributions where the number of possible\n  values the Hamiltonian takes on would be finite.\n\nThe next two theorems\n     prove Bauke and Mertens conjecture\n  for directed polymers in Gaussian environment for growing levels\n  $E_N=cN^{\\alpha}$.\n    We are able to prove that this conjecture\n      holds true for $\\alpha<1\/4$ for polymers\n        in dimension $d=1$ et\n      and $\\alpha<1\/2$ in dimension\n        $d\\geq 2$.\n          We leave this investigation\n         open for non-Gaussian environments.\n\n     The values $\\alpha=1\/4$ for $d=1$ and\n     $\\alpha=1\/2$ for $d\\geq 2$ are likely to be the true\n     critical values. Note that these are the same\n    as for Gaussian SK-spin glass models\n    for $p=1$ and $p=2$ respectively according to\n     \\cite{BCMN2}, and likely for $p\\geq 3$ as well.\n\n\\begin{theo}\n\\label{th1}\n Let $\\eta(n,x)$, $\\{\\eta(n,x) : n \\in {\\bf N}, x\\in {\\bf\nZ}^d\\}$, be independent standard Gaussian random variables.\n Let $d=1$. Let $E_N=c N^{\\alpha}$ with\n  $c \\in {\\bf R}$, $\\alpha \\in [0, 1\/4[$ and\n     \\begin{equation}\n     \\label{delta}\n      \\delta_N = \\sqrt{\\pi\/2} e^{E_N^2\/2}\n           (2^N)^{-1}.\n           \\end{equation}\n   Then the point process\n\\begin{equation}\n\\label{th1e}\n  \\sum_{\\o^N\\in S_N} \\delta_{\\{\\delta_N^{-1}\n  |\\eta(\\o^N)-E_N|\\}}\n\\end{equation}\n   converges weakly as $N \\uparrow \\infty$ to the Poisson\n      point process ${\\cal P}$ on ${\\bf R}_+$\n        whose intensity measure is the Lebesgue measure.\n    Moreover, for any $\\epsilon>0$ and any $b \\in {\\bf R}_+$\n\\begin{equation}\n\\label{th1b} \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall N_0\\ \\exists N \\geq N_0,\\\n \\exists \\o^{N,1}, \\o^{N,2}\\ : \\\n   {\\rm cov}\\,(\\eta(\\o^{N,1}), \\eta(\\o^{N,2}))>\\epsilon\\ :$$\n    $$  |\\eta(\\o^{N,1})-E_N|\\leq  |\\eta(\\o^{N,2})-E_N|\\leq \\delta_N\n      b)=0.\n\\end{equation}\n\\end{theo}\n\n\\begin{theo}\n\\label{th2}\nLet $\\eta(n,x)$, $\\{\\eta(n,x) : n \\in {\\bf N}, x\\in {\\bf\nZ}^d\\}$ be independent standard Gaussian random variables.\n Let $d \\geq 2$. Let $E_N=c N^{\\alpha}$ with\n  $c \\in {\\bf R}$, $\\alpha \\in [0, 1\/2[$  and\n\\begin{equation}\n\\label{delta1}\n      \\delta_N = \\sqrt{\\pi\/2} e^{E_N^2\/2}\n           ((2d)^N)^{-1}.\n           \\end{equation}\n   Then the point process\n\\begin{equation}\n\\label{th2e}\n  \\sum_{\\o^N\\in S_N} \\delta_{\\{\\delta_N^{-1}\n  |\\eta(\\o^N)-E_N|\\}}\n\\end{equation}\n   converges weakly as $N \\uparrow \\infty$ to the Poisson\n      point process ${\\cal P}$ on ${\\bf R}_+$\n        whose intensity measure is the Lebesgue measure.\n Moreover, for any $\\epsilon>0$ and any $b \\in {\\bf R}_+$\n\\begin{equation}\n\\label{th2b} \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall N_0\\ \\exists N \\geq N_0,\\\n \\exists \\o^{N,1}, \\o^{N,2}\\ : \\\n   {\\rm cov}\\,(\\eta(\\o^{N,1}), \\eta(\\o^{N,2}))>\\epsilon\\ :$$\n    $$  |\\eta(\\o^{N,1})-E_N|\\leq  |\\eta(\\o^{N,2})-E_N|\\leq \\delta_N\n      b)=0.\n\\end{equation}\n\\end{theo}\n\n\\noindent{\\bf Acknowledgements.}\n  The author thanks Francis Comets for introducing him\n to the area of directed polymers. He also thanks\n  Stephan Mertens and Anton Bovier for attracting\n  his attention to the local behavior of disordered spin systems\n and interesting discussions.\n\n\\section{Proofs of the theorems.}\n\n\nOur approach  is based on the following  sufficient condition\n   of convergence to the Poisson point process.\n   It has been proven in a somewhat more general form\n   in \\cite{BK1}.\n\n\\begin{theo}\n\\label{tc}\n Let $V_{i,M}\\geq 0$, $i\\in {\\bf N}$, be a family of\nnon-negative random\n  variables satisfying the following assumptions : for any\n   $l \\in {\\bf N}$ and all sets of constants $b_j>0$,\n   $j=1,\\ldots,l$\n   $$ \\lim_{ M \\to \\infty} \\sum_{(i_1,\\ldots, i_l) \\in \\{1,\\ldots,\n   M\\} }\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{j=1}^{l} V_{i_j, M}<b_j) \\to \\prod_{j=1}^l b_j $$\n   where the sum is taken over all possible sequences of\n   \\underline{different}\n    indices $(i_1,\\ldots, i_l)$.\n  Then the point process\n  $$ \\sum_{i=1}^M \\delta_{\\{V_{i,M}\\}}$$\n  on ${\\bf R}_+$ converges weakly in distribution as\n  $M \\to \\infty$\nto the Poisson\n      point process ${\\cal P}$ on ${\\bf R}_+$\n        whose intensity measure is the Lebesgue measure.\n\\end{theo}\n\n  Hence, in  all our proofs,\n   we just have to verify the hypothesis of Theorem \\ref{tc}\n        for $V_{i,M}$ given by\n  $\\delta_N^{-1}\n  |\\eta(\\o^{N,i})-E_N|$, i.e.\\ we must show that\n\\begin{equation}\n\\label{zet}\n   \\sum_{(\\o^{N,1}, \\ldots, \\o^{N,l}) \\in S_N^{\\otimes l}}\n       \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} :  |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)\n             \\to b_1\\cdots b_l\n \\end{equation}\n   where the sum is taken over all sets of different paths\n$(\\o^{N,1}, \\ldots, \\o^{N,l})$.\n\n\\bigskip\n\n\\noindent{\\bf Informal proof of Theorem \\ref{th0}.}\n  Before proceeding with rigorous proofs let us give some\n  informal arguments supporting  Theorem~\\ref{th0}.\n\nThe random variables $\\eta(\\o^{N,i})$, $i=1,\\ldots,l$,\n  are the sums of independent identically distributed\n  random variables with mean $0$ and the covariance matrix\n   $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ with $1$ on the diagonal\n    and the covariances\n       ${\\rm cov}\\, (\\eta(\\o^{N,i}),\n         \\eta(\\o^{N,j}))= N^{-1}\\#\\{m : \\o^{N,i}_m=\\o^{N,j}_m\\}\n          \\equiv b_{i,j}(N)$.\n\n   The number of sets $(\\o^{N,1},\\ldots, \\o^{N,l})$\n   with $b_{i,j}(N)=o(1)$ ($o(1)$ should be chosen of an appropriate order)\n        for all pairs $i\\ne j$,\n    $i,j=1,\\ldots,l$, as $N\\to \\infty$, is $(2d)^{Nl}(1-\\gamma(N))$\n      as $N \\to \\infty$ where $\\gamma(N)$ is exponentially small\n        in $N$.\n      For all such sets\n        $(\\o^{N,1},\\ldots, \\o^{N,i})$,\n          by the local Central Limit Theorem,\n    the random variables $\\eta(\\o^{N,i})$, $i=1,\\ldots,l$,\n    should behave asymptotically as\n    Gaussian random variables with covariances\n      $b_{i,j}(N)=o(1)$ and the determinant of the covariance matrix\n   $1+o(1)$. Therefore, the probability\n         that these random variables belong to $[-\\delta_N b_i+c, \\delta_N\n    b_i+c]$ respectively for $i=1,\\ldots,l$,\n       equals $$(2\\delta_N b_1)\\cdots (2\\delta_N b_l)\n       (\\sqrt{2\\pi})^{-l} e^{-c^2l\/2}=\n          b_1\\cdots b_l 2^{-Nl}(1+o(1)).$$\n          Since the number of such sets\n         $(\\o^{N,1},\\ldots, \\o^{N,l})$\n            is $(2d)^{Nl}(1+o(1))$,\n   the sum (\\ref{zet}) over them converges to $b_1\\cdots b_l$.\n\n   Let us turn to the remaining tiny part of $S_N^{\\otimes l}$\n     where $(\\o^{N,1},\\ldots, \\o^{N,l})$\n  are such that the covariances\n     $b_{i,j}(N)\\ne o(1)$ with $o(1)$ of an appropriate order\n         for some $i\\ne j$,\n   $i,j=1,\\ldots, l$, $N\\to \\infty$.\n   The number of such sets is exponentially smaller than\n     $(2d)^{Nl}$.\n       Here two possibilities should be\nconsidered differently.\n\n\n The first one is when the covariance matrix\n  is non-degenerate. Then, invoking again the Central Limit Theorem,\n    the probabilities $\\mathop{\\hbox{\\sf P}}\\nolimits(\\cdot)$\n      in this case are not greater than\n      $$ ({\\rm det}B_N(\\o^{N,1},\\ldots, \\o^{N,l}))^{-1\/2}\n           (2\\delta_N b_1)\\cdots(2\\delta_N b_l)\n       (\\sqrt{2\\pi})^{-l}.$$\n    From the definition of the covariances\n    of $\\eta(\\o^{N,i})$,\n    ${\\rm det}\\,B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ is a finite polynomial in\n    the variables $1\/N$. Therefore the probabilities $\\mathop{\\hbox{\\sf P}}\\nolimits(\\cdot)$ are bounded\n      by $(2d)^{-Nl}$ up to a polynomial term, while\n   the number of sets $(\\o^{N,1},\\ldots, \\o^{N,l})$\n    such that $b_{i,j}(N)\\ne o(1)$ some $i\\ne j$,\n   $i,j=1,\\ldots, l$, is exponentially smaller than\n     $(2d)^{Nl}$. Therefore the sum (\\ref{zet})\n   over such sets $(\\o^{N,1},\\ldots, \\o^{N,l})$\n     converges to zero exponentially fast.\n\n        Let now $(\\o^{N,1},\\ldots, \\o^{N,l})$\n          be such that $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n   is degenerate of the rank $r<l$.\n     Then, without loss of generality, we may assume\n         that $\\eta(\\o^{N,1}),\\ldots, \\eta(\\o^{N,r})$\n     are linearly independent, while\n   $\\eta(\\o^{N,r+1}),\\ldots, \\eta(\\o^{N,l})$\n   are their linear combinations.\n        Then the  probabilities $\\mathop{\\hbox{\\sf P}}\\nolimits(\\cdot)$\n  are bounded by the probabilities\n  that only $\\eta(\\o^{N,1}),\\ldots, \\eta(\\o^{N,r})$\n   belong to the corresponding  intervals,\n  which are at most $2^{-Nr}$ up to a polynomial term as\n   previously.\n      Moreover, we will show that for no one $m=0,1,\\ldots, N$,\n  $\\o^{N,1}_m,\\ldots, \\o^{N,r}_m$ can not be all different.\n     Otherwise, each of $\\o^{N,r+1},\\ldots, \\o^{N,l}$\n       would coincide with one of\n    $\\o^{N,1},\\ldots, \\o^{N,r}$, which is impossible since the\n      sum (\\ref{zet}) is taken over sets of different(!) paths.\n     This implies that the number of such sets\n       $(\\o^{N,1},\\ldots, \\o^{N,r})$\n   is exponentially smaller than $2^{Nr}$. Furthermore,\n     the number of possibilities to complete each of these sets\n       by  $\\o^{N,r+1},\\ldots, \\o^{N,l}$\n   such that  $\\eta(\\o^{N,r+1}),\\ldots, \\eta(\\o^{N,l})$\n   are linear combinations\n of     $\\eta(\\o^{N,1}),\\ldots, \\eta(\\o^{N,r})$\n         is $N$-independent. Thus the number of sets\n   $(\\o^{N,1},\\ldots, \\o^{N,l})$ in this case being exponentially\n   smaller than $2^{Nr}$, and the probabilities\n      being $2^{-Nr}$ up to a polynomial term,\n          the corresponding sum (\\ref{zet})\n   converges to zero. This completes the informal proof\n   of (\\ref{th1e0}) in Theorem \\ref{th0}.\n\n\n We now give rigorous proofs. We start with proofs of Theorems\n  \\ref{th1} and \\ref{th2} in Gaussian environment\n  and give the proof of Theorem \\ref{th0} after that.\n\n\\bigskip\n\n\\noindent{\\bf Proof of Theorem~\\ref{th1}.}\n  For  $\\eta \\in ]0, 1\/2[$  let us denote by\n\\begin{equation}\n\\label{rr}\n  {\\cal R}_{N,l}^{\\eta}=\\{ (\\o^{N,1},\\ldots, \\o^{N,l}) :\n    \\ {\\rm cov}(\\eta(\\o^{N,i}),\\eta( \\o^{N,j}))\\leq N^{\\eta-1\/2},\\\n    \\forall i,j=1,\\ldots,l,\\\n      i\\ne j\\}.\n      \\end{equation}\n\n \\noindent{\\it Step 1.} As a first preparatory step, we need to estimate the capacity\n     of ${\\cal R}_{N,l}^{\\eta}$ in (\\ref{zgu}).\n    Let us first note that for any two paths\n     $\\o^{N,1}, \\o^{N,2} \\in S_N$\n$${\\rm cov}(\\eta(\\o^{N,1}), \\eta(\\o^{N,2}))=s\/N$$ if and only if\n  $$ \\#\\{m : (\\omega_m^1, m)=(\\omega_m^2,m) \\}=s,$$\n    i.e. the number of moments of time within the period\n    $[0,N]$ when the trajectories $\\o^{N,1}$ and $\\o^{N,2}$\n       are at the same point of the space ${\\bf Z}$ equals $s$.\n      But due to the symmetry of the simple random walk\n     \\begin{equation}\n       \\label{rr}\n       \\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                  \\#\\{m \\in[0,\\ldots,N] :\n                \\omega_m^1 - \\o_m^2=0\\} =s \\Big\\}$$\n       $$ = \\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                   \\#\\{m \\in[0,\\ldots,N] :\n             \\omega_m^1 + \\o_m^2=0\\} =s \\Big\\}.\n             \\end{equation}\n  Taking into account the fact that the random walk starting from $0$\n    can not visit $0$ at odd moments of time, we obtain that\n      (\\ref{rr})\n      equals\n        $$ \\#\\Big\\{\\omega^{2N} :\n                    \\#\\{m \\in[0,\\ldots,2N] :\n                \\omega_m = 0\\}=s \\Big\\}.$$\n  This last number is well-known for the simple random walk\n      on ${\\bf Z}$ : it equals\n     $ 2^{2N} 2^{s-2(2N)}{2N \\choose 2(2N)-s}$\n       (see e.g. \\cite{Fe})\n         which is, by Stirling's formula,\n                  when $s=[N^{1\/2+\\eta}]$,\n          $\\eta \\in ]0, 1\/2[$, equivalent to\n     $2^{2N} (2\\pi N)^{-1\/2} e^{-s^2\/(2(2N))}=\n        2^{2N} (2\\pi N)^{-1\/2} e^{-N^{2\\eta}\/4}$ as $N \\to\\infty$.\n        Finally, we obtain that for all $N \\geq 0$\n            the number  (\\ref{rr})\n          it is not greater than $2^{2N}e^{-h N^{2\\eta}}$\n            with some constant $h>0$.\n It follows that for all $N>0$\n\\begin{eqnarray}\n \\lefteqn{ |S_N^{\\otimes,l}\\setminus {\\cal R}_{N,l}^{\\eta}|\n }\\nonumber \\\\\n      &\\leq & (l(l-1)\/2) 2^{N(l-2)}\n        \\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                  \\#\\{m \\in[0,\\ldots,N] :\n                \\omega_m^1 - \\o_m^2=0\\} \\geq N^{1\/2+\\eta}\\Big\\}\n                \\nonumber \\\\\n    &\\leq & 2^{Nl} C N \\exp(-h N^{2\\eta}) \\label{zgu}\n\\end{eqnarray}\n   where $C>0$, $h>0$ are some constants.\n\n\\medskip\n\n\\noindent{\\it Step 2.} The second preparatory step\n     is the estimation (\\ref{es1}) and (\\ref{es2})\n        of the probabilities in the sum (\\ref{zet}).\n Let $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n    be the covariance matrix of the random variables\n        $\\eta(\\o^{N,i})$  for\n    $i=1,\\ldots, l$.\n    Then, if $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ is non-degenerate,\n    \\begin{equation}\n    \\label{mia}\n \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} : |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)\n = \\int_{C(E_N)} \\frac{e^{-(\\vec z B_N^{-1}(\\o^{N,1},\\ldots,\n \\o^{N,l})\\vec z)\/2}}{(2\\pi)^{l\/2}\n   \\sqrt{{\\rm det} B_N(\\o^{N,1},\\ldots,\n \\o^{N,l})} }\\, d\\vec z\n \\end{equation}\n   where\n   $$ C(E_N)=\\{\\vec z=(z_1,\\ldots, z_l) : |z_i- E_N|\\leq\n   \\delta_N b_i, \\forall i=1,\\ldots,l \\}.$$\n\n     Let $\\eta \\in ]0, 1\/2[$.\n      Since $\\delta_N$ is exponentially small in $N$, we see that\n      uniformly for\n        $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       R}_{N,l}^{\\eta}$,\n  the probability (\\ref{mia}) equals\n   \\begin{equation}\n   \\label{es1}\n   (2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\ne^{-(\\vec E_N B_N^{-1}(\\o^{N,1},\\ldots,\n \\o^{N,l})\\vec E_N)\/2}(1+o(1))$$\n$$ =(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\n   e^{-\\|E_N\\|^2(1+O(N^{\\eta-1\/2}))\/2}(1+o(1))\n \\end{equation}\n   where we denoted by $\\vec E_N$ the vector\n$(E_N, \\ldots, E_N)$.\n\n\n         We will also need a more rough estimate of the probability\n             (\\ref{mia})  out of\n   the set ${\\cal\n       R}_{N,l}^{\\eta}$.\n       Let now the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n   be of the rank $r\\leq l$.\n      Then, if $r<l$, there are $r$ paths among\n$\\o^{N,1},\\ldots, \\o^{N,l}$ such that\n   corresponding $r$ random variables  $\\eta(\\o^{N,i})$ form the\n   basis.\n        Without loss of generality we may assume that these\n   are $\\o^{N,1},\\ldots, \\o^{N,r}$.\n      Then the matrix\n$B_N(\\o^{N,1},\\ldots, \\o^{N,r})$\n  is non-degenerate\n  and $\\eta(\\o^{N,r+1}),\\ldots,\\eta(\\o^{N,l})$\n  are linear combinations of\n$\\eta(\\o^{N,1}),\\ldots,\\eta(\\o^{N,r})$.\n  We may now estimate from above the probabilities (\\ref{zet})\n    by the probabilities\n    $\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{r} :  |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)\n$ that can be expressed in terms of the $r$-dimmensional\n   integrals\nlike (\\ref{mia}). Consequently, in this case\n\\begin{equation}\n\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} : |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)\n  \\leq  \\frac{ (2\\delta_N\/\\sqrt{2 \\pi})^r b_1\\cdots b_r }\n      {\\sqrt{ {\\rm det }B_N(\\o^{N,1},\\ldots, \\o^{N,r}) }}.\n       \\end{equation}\n   From the definition of the matrix elements,\n      one sees that ${\\rm det }B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n        is a finite polynomial in the variables $1\/N$.\n          Hence, if the rank of $B(\\o^{N,1},\\ldots, \\o^{N,r})$\n          equals $r$, we have for all $N>0$\n\\begin{equation}\n\\label{es2}\n \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} : |\\eta(\\o^{N,i})-E_N|<b_i\n\\delta_N) \\leq\n     2^{-Nr}e^{c^2 r N^{2\\alpha}\/2}N^{k(r)}\n \\end{equation}\n for some $k(r)>0$.\n\n\\medskip\n\n \\noindent{\\it Step 3.}\n      Armed with (\\ref{zgu}), (\\ref{es1}) and (\\ref{es2}),\n    we now proceed with the proof of the theorem.\n\n  For given $\\alpha \\in ]0, 1\/4[$, let us choose\n   first\n    $\\eta_0 \\in ]0, 1\/4[$ such that\n\\begin{equation}\n\\label{eta_0}\n  2\\alpha-1\/2+\\eta_0<0.\n  \\end{equation}\n     Next, let us choose $\\eta_1>\\eta_0$\n       such that\n\\begin{equation}\n\\label{keta_0}\n  2\\alpha-1\/2+\\eta_1<2\\eta_0,\n  \\end{equation}\n  then $\\eta_2>\\eta_1$ such that\n\\begin{equation}\n\\label{eta_1}\n  2\\alpha-1\/2+\\eta_2<2\\eta_1,\n  \\end{equation}\netc. After $i-1$ steps we choose  $\\eta_i >\\eta_{i-1}$ such that\n\\begin{equation}\n\\label{eta_i}\n  2\\alpha-1\/2+\\eta_i<2\\eta_{i-1}.\n  \\end{equation}\n   Let us take e.g.\\ $\\eta_i=(i+1)\\eta_0$.\n    We stop the procedure at\n  $n = [\\alpha\/\\eta_0]$th step, that is\n\\begin{equation}\n \\label{eta_n}\n  n=\\min\\{i\\geq 0 : \\alpha <\\eta_i\\}.\n\\end{equation}\n  Note that $\\eta_{n-1}\\leq \\alpha<1\/4$, and then\n   $\\eta_n=\\eta_{n-1}+\\eta_0<1\/2$.\n\n    We will prove that the sum\n(\\ref{zet}) over ${\\cal R}_{N,l}^{\\eta_0}$\n  converges to $b_1\\cdots b_l$, while those over\n   ${\\cal R}_{N,l}^{\\eta_i}\\setminus {\\cal R}_{N,l}^{\\eta_{i-1}}$\n     for $i=1,2,\\ldots,n$ and the one over\n$S_N^{\\otimes l} \\setminus {\\cal R}_{N,l}^{\\eta_{n}}$\n  converge o zero.\n\nBy (\\ref{es1}), each term of the sum (\\ref{zet})\n           over ${\\cal R}^{\\eta_0}_{N,l}\n $ equals\n$$(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\ne^{- \\|\\vec E_N\\|^2 (1+O(N^{\\eta_0-1\/2}))\/2 }(1+o(1)).\n$$\n   Here $e^{\\|\\vec E_N\\|^2 \\times O(N^{\\eta_0-1\/2})}\n      =1+o(1)$ by the choice  (\\ref{eta_0}) of $\\eta_0$.\n  Then, by the definition of $\\delta_N$\n    (\\ref{delta}),  each term of the sum (\\ref{zet})\n           over ${\\cal R}^{\\eta_0}_{N,l}$ is\n$$ (b_1\\cdots b_l) 2^{-Nl}(1+o(1))$$\n  uniformly for $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       R}_{N,l}^{\\eta_0}$.\n         The number of terms in this\n           sum is $|{\\cal\n       R}_{N,l}^{\\eta_0}|$, that is\n         $2^{Nl}(1+o(1))$ by (\\ref{zgu}).\n     Hence, the sum (\\ref{zet}) over\n ${\\cal R}^{\\eta_0}_{N,l}\n $ converges to $b_1\\cdots b_l$.\n\n\n    Let us  consider the sum over\n${\\cal R}_{N,l}^{\\eta_i}\\setminus {\\cal R}_{N,l}^{\\eta_{i-1}}$\n     for $i=1,2,\\ldots,n$.\n        Each term in this sum equals\n$$(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\ne^{- \\|\\vec E_N\\|^2 (1+O(N^{\\eta_i-1\/2})\/2 }(1+o(1))\n$$\nuniformly for $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       R}_{N,l}^{\\eta_i}$.  Then, by the definition\n         of  $\\delta_N$ (\\ref{delta}),  it is bounded by\n        $2^{-Nl} C_i e^{h_i N^{2\\alpha -1\/2+\\eta_i}}$\n   with some constants $C_i, h_i>0$.\n    The number of terms in this sum\nis not greater than $|S_{N}^{\\otimes l} \\setminus {\\cal\nR}_{N,l}^{\\eta_{i-1}}|$\n   which is bounded due to  (\\ref{zgu})\n     by $C N 2^{Nl}\\exp(-h N^{2\\eta_{i-1}})$.\n   Then by the choice of $\\eta_i$\n        (\\ref{eta_i}) this sum converges to zero\n     exponentially fast.\n\n Let us now treat the sum over\n$S_N^{\\otimes l} \\setminus {\\cal R}_{N,l}^{\\eta_{n}}$.\n     Let us first study the sum\nover $(\\o^{N,1},\\ldots, \\o^{N,l})$ such that\n  the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ is non-degenerate.\n       By (\\ref{es2})  each term in this sum\n   is bounded by\n $ 2^{-Nl}e^{c^2 l N^{2\\alpha}\/2}N^{k(l)}$\n   for some $k(l)>0$.\n         The number of terms in this sum is bounded by\n  $ C N 2^{Nl}\\exp(-h N^{2\\eta_{n}})$ by (\\ref{zgu}). Since\n     $\\alpha<\\eta_n$ by (\\ref{eta_n}),\n     this sum converges to zero exponentially fast.\n\n     Let us finally turn to\nthe sum over $(\\o^{N,1},\\ldots, \\o^{N,l})$ such that\n  the matrix $B(\\o^{N,1},\\ldots, \\o^{N,l})$\n   is degenerate of the rank $r<l$.\nBy (\\ref{es2})  each term in this sum\n   is bounded by\n   \\begin{equation}\n   \\label{pp}\n  2^{-Nr}e^{c^2 r N^{2\\alpha}\/2}N^{k(r)}\n  \\end{equation}\n   for some $k(r)>0$.\n\n      There are $r$ paths among\n$\\o^{N,1},\\ldots, \\o^{N,l}$ such that\n   corresponding $\\eta(\\o^{N,i})$ form the basis.\n        Without loss of generality we may assume that these\n   are $\\o^{N,1},\\ldots, \\o^{N,r}$.\n     Note that $\\o^{N,1},\\ldots, \\o^{N,r}$\n  are such that it can not be for no one\n     $m \\in [0,\\ldots, N]$ that\n         $\\o^1_m,\\ldots, \\o^r_m$ are all different.\n       In fact, assume that\n $\\o^1_m,\\ldots, \\o^r_m$ are all different. Then\n     $\\eta(m, \\o^{1}_m),\\ldots, \\eta(m, \\o^{r}_m)$\n     are independent identically distributed random variables\n         and $\\eta(m, \\o^{r+1}_m)=\n           \\mu_1 \\eta(m, \\o^{1}_m)+\\cdots + \\mu_r \\eta(m,\n           \\o^{r}_m)$.\n    If $\\o^{r+1}_m$ is different from all $\\o^1_m,\\ldots, \\o^r_m$,\n         then $\\eta(m, \\o^{r+1}_m)$ is independent from\n           all of $\\eta(m, \\o^{1}_m),\\ldots,\\eta(m,\n           \\o^{r}_m)$, then  the linear coefficients, being the\n             covariances of $\\eta(m, \\o^{r+1}_m)$\n              with  $\\eta(m, \\o^{1}_m),\\ldots, \\eta(m, \\o^{r}_m)$,\n    are $\\mu_1=\\cdots=\\mu_r=0$.\n     So, $\\eta(\\o^{N,r+1})$\n         can not be a non-trivial linear combination\n    of $\\eta(\\o^{N,1}),\\ldots, \\eta(\\o^{N,r})$.\n If $\\o^{r+1}_m$ equals one of $\\o^1_m,\\ldots, \\o^r_m$,\n   say $\\o^{i}_m$, then again by computing the\n  covariances of  $\\eta(m, \\o^{r+1}_m)$\n              with  $\\eta(m, \\o^{1}_m),\\ldots, \\eta(m, \\o^{r}_m)$,\n    we get $\\mu_i=1$, $\\mu_j=0$\n     for $j=1,\\ldots, i-1,i+1,\\ldots,r$.\n            Consequently,\n   $\\eta(\\o^{i}_k)=\\eta(\\o^{r+1}_k)$\n     for all $k=1,\\ldots, N$, so that\n        $\\o^{N,i}=\\o^{N,r+1}$. But this is impossible\n          since the sum  (\\ref{zet})\n               is taken over \\underline{different\\\/}  paths\n          $\\o^{N,1},\\ldots, \\o^{N,l}$.\n  Thus the sum is taken only over paths\n$\\o^{N,1},\\ldots, \\o^{N,r}$ where at each moment of time\n  at least two of them are at the same place.\n\n      The number of such sets of $r$ different\n            paths is exponentially smaller than\n      $2^{Nr}$ : there exists $p>0$ such that\n  is does not exceed $2^{Nr}e^{-pN}$.\n(In fact, consider $r$  independent simple\n   random walks on ${\\bf Z}$ that at a given moment of time\n     occupy any $k<r$ different points of ${\\bf Z}$.\n   Then with probability not less than\n   $(1\/2)^r$, at the next moment of time, they\n      occupy at least $k+1$ different points.\n   Then with probability not less than $((1\/2)^r)^r$\n at least once during $r$ next moments of time\n   they will occupy $r$ different points.\n  So, the number of sets of different $r$ paths that\n         at each moment of time during $[0,N]$\n   occupy at most $r-1$ different points\n    is not greater than $2^{Nr}(1-(1\/2^{r})^r)^{[N\/r]}$.)\n\n    Given any set of $r$ paths with\n      $\\eta(\\o^{N,1}),\\ldots, \\eta(\\o^{N,r})$ linearly independent,\n            there is an $N$-independent\n    number of possibilities to complete it by linear\n    combinations $\\eta(\\o^{N,r+1}),\\ldots \\eta(\\o^{N,l})$.\n      To see this, first consider the equation\n $\\lambda_1\\eta(\\o^{N,1})+\\cdots+\\lambda_r\\eta(\\o^{N,r})=0$\n   with unknown $\\lambda_1,\\ldots, \\lambda_r$.\n     For any moment of time $m \\in [0,N]$\n       this means $\\lambda_1 \\eta(m, \\o_m^{1})+\\cdots +\\lambda_r\n       \\eta(m, \\o_m^r)=0$.\n             If $\\o_m^{i_1}=\\o_m^{i_2}=\\cdots \\o_m^{i_k}$\n               but $\\o_m^j\\ne \\o_m^{i_1}$ for all $\n     j\\in \\{1,\\ldots,r\\}\\setminus\\{i_1,\\ldots, i_k\\}$,\n     then $\\lambda_{i_1}+\\cdots + \\lambda_{i_k}=0$.\n         Then for any\n           $m \\in [0, N]$\n   the equation $\\lambda_1 \\eta(m, \\o_m^{1})+\\cdots +\\lambda_r\n       \\eta(m, \\o_m^r)=0$ splits into a certain number\n         $n(m)$ ($1\\leq n(m) \\leq r$)\n   equations of  type $\\lambda_{i_1}+\\cdots + \\lambda_{i_k}=0$.\n       Let us construct a matrix $A$ with $r$ columns\n   and at least $N$ and at most $rN$ rows in the following way.\n      For any $m>0$, according to given $\\o_m^1,\\ldots, \\o_m^r$,\n      let us add to A  $n(m)$ rows : each equation\n          $\\lambda_{i_1}+\\cdots + \\lambda_{i_k}=0$ gives\n       a row with $1$ at places $i_1,\\ldots, i_k$ and\n          $0$ at all other places.\n          Then the equation\n           $\\lambda_1\\eta(\\o^{N,1})+\\cdots+\\lambda_r\\eta(\\o^{N,i})=0$\n             is equivalent $A \\vec \\lambda =\\vec 0$\n               with $\\vec \\lambda=(\\lambda_1,\\ldots, \\lambda_r)$.\n      Since this equation has only a trivial solution $\\vec \\lambda=0$,\n    then the rank of $A$ equals $r$.\n         The matrix $A$ contains at most $2^r$ different rows.\n              There is less than  $(2^r)^r$ possibilities\n    to choose $r$ linearly independent of them.\n    Let $A^{r \\times r}$ be an $r \\times r$\n        matrix consisting of  $r$ linearly independent rows of $A$.\n        The fact that $\\eta(\\omega^{N,r+1})$ is\n     a linear combination\n      $\\mu_1\\eta(\\o^{N,1})+\\cdots+\\mu_r\\eta(\\o^{N,r})=\\eta(\\o^{N,r+1})$\n      can be written as  $A^{r \\times r} \\vec \\mu =\\vec b$\n   where the vector $\\vec b$ contains only  $1$\n   and $0$ : if a given row $t$ of the matrix\n     $A^{r \\times r}$ corresponds to the $m$th step\n       of the random walks and has $1$ at places\n   $i_1,\\ldots,i_k$ and $0$ elsewhere, then\n     we put $b_t=1$ if $\\o_m^{i_1}=\\o_m^{r+1}$\n      and $b_t=0$ if $\\o_m^{i_1}\\ne \\o_m^{r+1}$.\n   Thus, given\n     $\\o^{N,1},\\ldots, \\o^{N,r}$,\n           there is an $N$ independent number\n     of possibilities to write the system $A^{r \\times r} \\vec \\mu =\\vec b$\n        with non degenerate matrix $A^{r \\times r}$\n      which determines uniquely linear coefficients\n         $\\mu_1,\\ldots, \\mu_r$ and consequently\n   $\\eta(\\o^{N,r+1})$ and the path $\\o^{N,r+1}$\n     itself through these linear coefficients.\n   Hence, there is not more possibilities to\n     choose $\\o^{N,r+1}$\n   than the number of  non-degenerate matrices\n    $A^{r \\times r}$ multiplied by the number of vectors $\\vec\n    b$, that is roughly not more than $2^{r^2+r}$.\n\n  These observations lead to the fact that the sum (\\ref{zet})\n     with the  covariance matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ of the\n       rank  $r$ contains at most $(2^{r^2+r})^{l-r} 2^{Nr}e^{-p N}$\n  different terms with some constant $p>0$.\n    Then, taking into account the estimate (\\ref{pp})\n      of each term with $2\\alpha<1$, we deduce that it converges to zero\n     exponentially fast.\n This finishes the proof\n     of  (\\ref{th1e}).\n\n     To show (\\ref{th1b}), we have been already noticed\n       that the sum of terms\n       $\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{2} : |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)$\n         over all pairs  of different paths $\\o^{N,1}, \\o^{N,2}$\n           in $S_N^{\\otimes l} \\setminus\n                {\\cal R}_{N,l}^{\\eta_0}$\n          converges to zero exponentially fast.\n   Then (\\ref{th1b}) follows from the Borel-Cantelli lemma.\n\n\\medskip\n\n\\noindent{\\bf Proof of Theorem \\ref{th2}.}\n    We  have again to verify the hypothesis of Theorem \\ref{tc}\n         for $V_{i,M}$ given by\n  $\\delta_N^{-1}\n  |\\eta(\\o^{N,i})-E_N|$, i.e.\\ we must show (\\ref{zet}).\n\n  For  $\\beta \\in ]0, 1[$  let us denote by\n $$ {\\cal K}_{N,l}^{\\beta}=\\{ (\\o^{N,1},\\ldots, \\o^{N,l}) :\n    \\ {\\rm cov}(\\eta(\\o^{N,i}),\\eta( \\o^{N,j}))\\leq N^{\\beta-1},\\\n    \\forall i,j=1,\\ldots,l,\\\n      i\\ne j\\}.$$\n\n\\noindent{\\it Step 1.}  In this step we estimate the capacity of the\ncomplementary\n     set to ${\\cal K}_{N,l}^{\\beta}$ in (\\ref{kk}) and (\\ref{kkk}).\n\n   We have :  \\begin{eqnarray}\n   \\lefteqn{ |S_N^{\\otimes,l}\\setminus {\\cal K}_{N,l}^{\\beta}|\n }\\label{zgu1} \\\\\n      &\\leq & (l(l-1)\/2) (2d)^{N(l-2)}\n        \\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                  \\#\\{m \\in[0,\\ldots,N] :\n                \\omega_m^1 - \\o_m^2=0\\} > N^{\\beta}\\Big\\}.\n         \\nonumber\n\\end{eqnarray}\n  It has been shown in the proof of Theorem \\ref{th1} that\n   the number\n$$\\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                  \\#\\{m \\in[0,\\ldots,N] :\n                \\omega_m^1 - \\o_m^2=0\\} > N^{\\beta}\\Big\\}$$\n equals the number of paths of a simple random walk\n within the period $[0,2N]$ that visit the origin\n  at least $[N^\\beta]+1$ times.\n\n      Let $W_r$ be the time of the $r$th return to the origin\n  of a simple random walk\n    ($W_1=0$), $R_N$ be the number of returns\n   to the origin in the first $N$ steps.\n          Then for any integer $q$\n      $$P(R_N \\leq q)=P(W_1+(W_2-W_1)+\\cdots +(W_q-W_{q-1}) \\geq N)\n   \\geq \\sum_{k=1}^{q-1} P(E_k)$$\n     where $E_k$ is the event that exactly $k$ of the variables\n        $W_s-W_{s-1}$ are greater or equal than $N$,\n          and $q-1-k$ are less than $N$. Then\n$$\\sum_{k=1}^{q-1} P(E_k)=\\sum_{k=1}^{q-1} {q-1 \\choose k}\n     P(W_2-W_1 \\geq N)^k (1-  P(W_2-W_1 \\geq N))^{q-1-k}$$\n     $$=\n       1-(1-  P(W_2-W_1 \\geq N))^{q-1}.$$\n    It is shown in \\cite{ET}\n    that in the case $d=2$\n    $$P(W_2-W_1 \\geq N)\n       =\\pi (\\log N)^{-1}(1+ O((\\log N)^{-1})), \\ \\ \\ N \\to \\infty.$$\n  Then\n  $$ P(R_N >q) \\leq \\Big(1-\\pi (\\log N)^{-1}(1+o(1))\\Big)^{q-1}.$$\n  Consequently,\n  $$ \\#\\Big\\{\\omega^{N,1},\\omega^{N,2} :\n                  \\#\\{m \\in[0,\\ldots,N] :\n                \\omega_m^1 - \\o_m^2=0\\} > N^{\\beta}\\Big\\}\n                $$\n                $$=(2d)^{2N} P(R_{2N}>[N^\\beta])$$\n                  $$\\leq\n                 (2d)^{2N} \\Big(1-\\pi (\\log 2N)^{-1}(1+o(1)) \\Big)^{[N^\\beta]-1}\n                   \\leq (2d)^{2N} \\exp(- h (\\log 2N)^{-1} N^{\\beta}) $$\n      with some constant $h>0$.\n         Finally for $d=2$ and all $N>0$\n           by (\\ref{zgu1})\n \\begin{eqnarray}\n    |S_N^{\\otimes l}\\setminus {\\cal K}_{N,l}^{\\eta}|\n  \\leq  (2d)^{lN} \\exp(- h_2 (\\log 2N)^{-1} N^{\\beta}) \\label{kk}\n  \\end{eqnarray}\n   with some constant $h_2>0$.\n\n   In the case $d\\geq 3$ the random walk is transient and\n $$P(W_2-W_1 \\geq N)\\geq P(W_2-W_1 =\\infty)=\\gamma_d>0.$$\n   It follows that $\\mathop{\\hbox{\\sf P}}\\nolimits(R_N>q)\\leq (1-\\gamma_d)^{q-1}$ and\n   consequently\n\\begin{eqnarray}\n    |S_N^{\\otimes,l}\\setminus {\\cal K}_{N,l}^{\\beta}|\n  \\leq  (2d)^{lN} \\exp(- h_d N^{\\beta}) \\label{kkk}\n  \\end{eqnarray}\n  with some constant $h_d>0$.\n\n\\medskip\n\n \\noindent{\\it Step 2.}    Proceeding exactly as in the proof of Theorem\n     \\ref{th1}, we obtain that uniformly for\n        $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       K}_{N,l}^{\\beta}$,\n     \\begin{equation}\n     \\label{est1}\n     \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} : |\\eta(\\o^{N,i})-E_N|<b_i \\delta_N)\n        $$\n        $$ =(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\n   e^{-\\|E_N\\|^2(1+O(N^{\\beta-1}))\/2}(1+o(1))\n \\end{equation}\n   where we denoted by $\\vec E_N$ the vector\n$(E_N, \\ldots, E_N)$.\n   Moreover, if\nthe covariance the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n   is of the rank $r\\leq l$\n     (using the fact that its determinant is a finite polynomial\n        in the variables $1\/N$)\n            we get as in the proof of Theorem \\ref{th1} that\n\\begin{equation}\n\\label{est2}\n \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} : |\\eta(\\o^{N,i})-E_N|<b_i\n\\delta_N) \\leq\n     (2d)^{-Nr}e^{c^2 r N^{2\\alpha}\/2}N^{k(r)}\n \\end{equation}\n for some $k(r)>0$.\n\n\\medskip\n\n \\noindent{\\it Step 3.}\n   Having (\\ref{kk}), (\\ref{kkk}), (\\ref{est1}) and (\\ref{est2}),\n      we are able to carry out the proof of the theorem.\n    For given $\\alpha \\in ]0, 1\/2[$, let us choose\n   first $\\beta_0>0$ such that\n\\begin{equation}\n\\label{beta_0}\n  2\\alpha-1+\\beta_0<0.\n  \\end{equation}\n     Next, let us choose $\\beta_1>\\beta_0$\n       such that\n\\begin{equation}\n\\label{bketa_0}\n  2\\alpha-1+\\beta_1<\\beta_0,\n  \\end{equation}\n  then $\\beta_2>\\beta_1$ such that\n\\begin{equation}\n\\label{beta_1}\n  2\\alpha-1+\\beta_2<\\beta_1,\n  \\end{equation}\netc. After $i-1$ steps we choose  $\\beta_i >\\beta_{i-1}$ such that\n\\begin{equation}\n\\label{beta_i}\n  2\\alpha-1+\\beta_i<\\beta_{i-1}.\n  \\end{equation}\n   Let us take e.g.\\ $\\beta_i=(i+1)\\beta_0$.\n    We stop the procedure at\n  $n = [2\\alpha\/\\beta_0]$th step, that is\n\\begin{equation}\n \\label{beta_n}\n  n=\\min\\{i\\geq 0 : 2\\alpha <\\beta_i\\}.\n\\end{equation}\n  Note that $\\beta_{n-1} \\leq 2\\alpha$, and then\n   $\\beta_n=\\beta_{n-1}+\\beta_0<2\\alpha+ 1-2\\alpha=1$.\n\n    We will prove that the sum\n(\\ref{zet}) over ${\\cal K}_{N,l}^{\\beta_0}$\n  converges to $b_1\\cdots b_l$, while those over\n   ${\\cal K}_{N,l}^{\\beta_i}\\setminus {\\cal K}_{N,l}^{\\beta_{i-1}}$\n     for $i=1,2,\\ldots,n$ and the one over\n$S_N^{\\otimes l} \\setminus {\\cal K}_{N,l}^{\\beta_{n}}$\n  converge o zero.\n\nBy (\\ref{est1}), each term of the sum (\\ref{zet})\n           over ${\\cal K}^{\\beta_0}_{N,l}\n $ equals\n$$(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\ne^{- \\|\\vec E_N\\|^2 (1+O(N^{\\beta_0-1}))\/2 }(1+o(1)).\n$$\n   Here $e^{\\|\\vec E_N\\|^2 \\times O(N^{\\beta_0-1})}\n      =1+o(1)$ by the choice  (\\ref{beta_0}) of $\\beta_0$.\n  Then, by the definition of $\\delta_N$\n    (\\ref{delta1}), each term of the sum (\\ref{zet})\n           over ${\\cal K}^{\\beta_0}_{N,l}$ is\n$$ (b_1\\cdots b_l) (2d)^{-Nl}(1+o(1))$$\n  uniformly for $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       K}_{N,l}^{\\eta_0}$.\n         The number of terms in this\n           sum is $|{\\cal\n       K}_{N,l}^{\\beta_0}|$, that is\n         $(2d)^{Nl}(1+o(1))$ by (\\ref{kk}) and (\\ref{kkk}).\n     Hence, the sum (\\ref{zet}) over\n ${\\cal K}^{\\beta_0}_{N,l}\n $ converges to $b_1\\cdots b_l$.\n\n    Let us  consider the sum over\n${\\cal K}_{N,l}^{\\beta_i}\\setminus {\\cal K}_{N,l}^{\\beta_{i-1}}$\n     for $i=1,2,\\ldots,n$. By (\\ref{est1})\n        each term in this sum equals\n$$(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\ne^{- \\|\\vec E_N\\|^2 (1+O(N^{\\beta_i-1})\/2 }(1+o(1))\n$$\nuniformly for $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       K}_{N,l}^{\\beta_i}$.  Then, by the definition\n         of  $\\delta_N$ (\\ref{delta1}),  it is bounded by\n       the quantity $(2d)^{-Nl} C_i e^{h_i N^{2\\alpha -1+\\beta_i}}$\n   with some constants $C_i, h_i>0$.\n    The number of terms in this sum\nis not greater than $|S_{N}^{\\otimes l} \\setminus {\\cal\nK}_{N,l}^{\\beta_{i-1}}|$\n   which is bounded\n     by $(2d)^{Nl}\\exp(-h_2 N^{\\beta_{i-1}} (\\log 2 N)^{-1})$\n       in the case $d=2$ due to (\\ref{kk})\n        and\n  by the quantity $(2d)^{Nl}\\exp(-h_d N^{\\beta_{i-1}} )$\n  in the case $d\\geq 3$ due to (\\ref{kkk}).\n   Then by the choice of $\\beta_i$\n        (\\ref{beta_i}) this sum converges to zero\n     exponentially fast.\n\n Let us now treat the sum over\n$S_N^{\\otimes l} \\setminus {\\cal K}_{N,l}^{\\beta_{n}}$.\n     Let us first analyze the sum\nover $(\\o^{N,1},\\ldots, \\o^{N,l})$ such that\n  the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ is non-degenerate.\n       By (\\ref{est2})  each term in this sum\n   is bounded by\n $ (2d)^{-Nl}e^{c^2 l N^{2\\alpha}\/2}N^{k(l)}$\n   for some $k(l)>0$.\n         The number of terms in this sum is bounded by\n the quantity  $ (2d)^{Nl}\\exp(-h_2 N^{\\beta_{n}} (\\log 2N)^{-1})$\n  in the case $d=2$ and by\n $ (2d)^{Nl}\\exp(-h_d N^{\\beta_{n}})$\n  in the case $d\\geq 3$ respectively by (\\ref{kk}) and (\\ref{kkk}) . Since\n     $2\\alpha<\\beta_n$ by (\\ref{beta_n}),\n     this sum converges to zero exponentially fast.\n\nLet us finally turn to the sum over $(\\o^{N,1},\\ldots, \\o^{N,l})$\nsuch that\n  the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n   is degenerate of the rank $r<l$.\nBy (\\ref{est2})  each term in this sum\n   is bounded by $\n  (2d)^{-Nr}e^{c^2 r N^{2\\alpha}\/2}N^{k(r)}$\n   for some $k(r)>0$, while exactly by\n    the same arguments as in the proof of Theorem \\ref{th1},\n     (they are, indeed, valid in all dimensions)\n    the number of terms in this sum is\n    less than $O((2d)^{Nr})e^{-p N}$\n    with some constant $p>0$.\n       Hence, this last sum converges to zero\n         exponentially fast as $2\\alpha <1$.\n             This finishes the proof  of (\\ref{th2e}).\n          The proof of (\\ref{th2b}) is completely\n           analogous to the one of\n(\\ref{th1b}).\n\n\\medskip\n\n\n\\noindent{\\bf Proof of Theorem \\ref{th0}.}\n     We again concentrate on  the proof in the sum (\\ref{zet})\n       with $E_N=c$.\n\n \\noindent{\\it Step 1.}  First of all,  we  need a rather rough estimate\n    of the probabilities of (\\ref{zet}).\n  Let $(\\o^{N,1},\\ldots, \\o^{N,r})$\n    be such that the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,r})\n     $ is  non-degenerate.\n       We  prove in this step that there exists  a constant $k(r)>0$\n       such that for any $N>0$ and any $(\\o^{N,1},\\ldots, \\o^{N,r})$\n    with non-degenerate $B_N(\\o^{N,1},\\ldots, \\o^{N,r})$, we have:\n\\begin{equation}\n\\label{gg2} \\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{r} :  |\\eta(\\o^{N,i})-c|<b_i \\delta_N)\n   \\leq (2d)^{-Nr}N^{k(r)}.\n\\end{equation}\n   Let\n   $$f^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(t_1,\\ldots, t_r)\n      = \\mathop{\\hbox{\\sf E}}\\nolimits \\exp\\Big( i\\sum_{k=1}^r t_k \\eta(\\o^{N,k})\n      \\Big)$$\n be the Fourier transform  of  $(\\eta(\\o^{N,1}),\\ldots,\n \\eta(\\o^{N,r}))$. Then\n\\begin{eqnarray}\n \\lefteqn{\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{r} :  |\\eta(\\o^{N,i})-c|<b_i \\delta_N)}\n \\nonumber\\\\\n&= &\\frac{1}{(2\\pi)^r}\n  \\int\\limits_{{\\bf R}^r} f^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(\\vec t)\n      \\prod_{k=1}^r \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} dt_1\\cdots d t_r \\label{fou}\n\\end{eqnarray}\n   provided that the integrand is in $L^1({\\bf R}^d)$.\n  We will show that this is the case  due to the assumption\n   made on $\\phi$  and deduce the bound (\\ref{gg2}).\n\n     We know that the function $f_N^{\\o^{N,1},\\ldots, \\o^{N,r}}(\\vec\n  t)$ is the product of $N$ generating functions :\n   \\begin{equation}\n   \\label{zey}\n      f^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(\\vec t)=\n      \\prod_{n=1}^N \\mathop{\\hbox{\\sf E}}\\nolimits \\exp\\Big( i N^{-1\/2}\n         \\sum_{k=1}^r t_k  \\eta(n, \\o^{N,k}_n)\n      \\Big).\n      \\end{equation}\n   Moreover, each of these functions\n   is itself a product of (at minimum $1$ and at maximum $r$)\n     generating functions of type $\\phi((t_{i_1}+\\cdots\n     +t_{i_k})N^{-1\/2})$.\n   More precisely, let us construct the matrix $A$ with\n      $r$ columns and at least $N$ and at most $rN$\n   rows as in the proof of Theorem~\\ref{th1}. Namely,\n          for each step $n=0,1,2,\\ldots,N$,\n    we add to the matrix $A$ at least $1$ and at most $r$ rows\n  according to the following rule:\n      if $\\o_n^{N,i_1}=\\o_n^{N,i_2}=\\cdots=\\o_n^{N,i_k}$\n          and $\\o_n^{N,j}\\ne \\o_n^{N,i_1}$\n            for any $j\\in \\{1,\\ldots,r\\}\n            \\setminus \\{i_1,\\ldots,i_k\\}$, we add to $A$\n              a row with $1$ at places $i_1,\\ldots, i_k$ and\n       $0$ at other $r-k$ places.\n  Then\n   \\begin{equation}\n   \\label{aa}\nf^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(\\vec t)=\n  \\prod_{j} \\phi (N^{-1\/2}(A \\vec t)_j).\n   \\end{equation}\n  Since $B_N(\\o^{N,1},\\ldots, \\o^{N,r})\n     $ is non-degenerate, the rank of the matrix $A$\n       equals $r$.  Let us choose in $A$\n         any $r$ linearly independent rows,\n           and let us denote by $A^r$\n      the $r \\times r$ matrix\n        constructed by them.\n  Then by the assumption made on $\\phi$\n\\begin{equation}\n   \\label{aaa}\n|f^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(\\vec t)| \\leq\n  \\prod_{j=1}^r |\\phi (N^{-1\/2}(A^r \\vec t)_j)|\n    \\leq \\prod_{j=1}^r  \\min \\Big(1, \\frac{ C N^{1\/2} }{ |(A^r \\vec\n    t)_j|} \\Big) \\leq C^r N^{r\/2}\n       \\prod_{j=1}^r  \\min \\Big(1, \\frac{1 }{ |(A^r \\vec\n    t)_j|} \\Big)\n   \\end{equation}\n with some constant $C>0$. Furthermore\n\\begin{equation}\n\\label{bb}\n \\Big|\\prod_{k=1}^r \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} \\Big| \\leq\n         \\prod_{k=1}^r \\min \\Big( (2\\delta_N)b_k, \\\n           \\frac{2}{|t_k|}\\Big) \\leq C'\\prod_{k=1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n\\end{equation}\n  with some $C'>0$. Hence,\n  \\begin{eqnarray}\n \\lefteqn{\\frac{1}{(2\\pi)^r}\n  \\int\\limits_{{\\bf R}^r} \\Big|f^{\\o^{N,1},\\ldots, \\o^{N,r}}_N(\\vec t)\n      \\prod_{k=1}^r \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k}\\Big| dt_1\\cdots d t_r }\\nonumber\\\\\n & \\leq & C_0 N^{r\/2}\n       \\int\n\\prod_{k=1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(A^r \\vec\n    t)_k|} \\Big) d \\vec t \\label{ss}\n    \\end{eqnarray}\n  with some constant $C_0>0$  depending on the function $\\phi$ and on\n    $b_1,\\ldots, b_r$ only.\n       Since the matrix $A^r$ is non-degenerate,\n     using easy arguments\n    of linear algebra, one can show that for some\n      constant $C_1>0$ depending on the matrix $A^{r}$ only,\n  we have\n\\begin{equation}\n \\int \\prod_{k=1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(A^r \\vec\n    t)_k|} \\Big)d\\vec t\n \\leq   C_1 \\int \\prod_{k=1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\Big(1, \\frac{1 }{ |\n    t_k|} \\Big) d\\vec t. \\label{sdg}\n\\end{equation}\n   The proof of (\\ref{sdg}) is given in Appendix.\nBut the right-hand of (\\ref{sdg}) is finite.\n  This shows that the integrand in (\\ref{fou})\n     is in $L^1({\\bf R}^d)$ and\n   the inversion formula (\\ref{fou}) is valid.\n  Moreover, the right-hand side of (\\ref{sdg})  equals $C_1 (2((2d)^{-N}\n+  (2d)^{-N} N \\ln 2d + (2d)^{-N}))^r$.\n Hence, the probabilities above are bounded by the quantity\n   $C_0 N^{r\/2} C_1 2^r(2+N \\ln (2d))^r (2d)^{-Nr}$\n     with $C_0$ depending on $\\phi$ and $b_1,\\ldots, b_r$ and\n          $C_1$ depending on the choice of $A^r$.\nTo conclude the proof of (\\ref{gg2}), it remains to remark\n that there is an $N$-independent number of possibilities\n   to construct a matrix $A^{r}$ (at most $2^{r^2}$),\nsince it contains only $0$ or $1$.\n\n\n\\medskip\n\n \\noindent{\\it Step 2.}  We keep the notation ${\\cal R}_{N,l}^{\\eta}$\n   from (\\ref{rr}) for $\\eta \\in ]0,1\/2[$.\n  The capacity of this set for $d=1$ is estimated in (\\ref{zgu}).\n    Moreover by (\\ref{kk}) for $d=2$\n $$ |S_{N}^{\\otimes l}\\setminus {\\cal R}_{N,l}^{\\eta}|=\n  |S_{N}^{\\otimes l}\\setminus {\\cal K}_{N,l}^{\\eta+1\/2}|\n     \\leq (2d)^{Nl} \\exp(-h_2 (\\log 2N)^{-1} N^{1\/2+\\eta})$$\n     and by (\\ref{kkk}) for $d\\geq 3$\n  $$ |S_{N}^{\\otimes l}\\setminus {\\cal R}_{N,l}^{\\eta}|\n     =    |S_{N}^{\\otimes l}\\setminus {\\cal K}_{N,l}^{\\eta+1\/2}|\n           \\leq (2d)^{Nl}\\exp(-h_d N^{1\/2+\\eta}),$$\n  so that, for all $d\\geq 1$ there are $h_d, C_d>0$\n  such that for all $N> 0$\n\\begin{equation}\n\\label{rrs}\n |S_{N}^{\\otimes l}\\setminus {\\cal R}_{N,l}^{\\eta}|\n              \\leq (2d)^{Nl}C_d N \\exp(-h_d N^{2\\eta}).\n              \\end{equation}\n\n\\medskip\n\n\n\\noindent{\\it Sep 3.}   In this step we show that uniformly for\n$(\\o^{N,1},\n     \\ldots, \\o^{N,l}) \\in\n{\\cal R}_{N,l}^{\\eta}$\n  \\begin{equation}\n \\label{gg}\n\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} :  |\\eta(\\o^{N,i})-c|<b_i \\delta_N)=\n(2d)^{-Nl}b_1\\cdots b_l(1+o(1)).\n\\end{equation}\n\n  For any $(\\o^{N,1},\\ldots \\o^{N,l}) \\in\n{\\cal R}_{N,l}^{\\eta}$, we can represent the probabilities in the\n sum (\\ref{zet}) as\nsums of four terms :\n\\begin{eqnarray}\n \\lefteqn{\\mathop{\\hbox{\\sf P}}\\nolimits(\\forall_{i=1}^{l} :  |\\eta(\\o^{N,i})-c|<b_i \\delta_N)}\n \\nonumber\\\\\n&= &\\frac{1}{(2\\pi)^l}\n  \\int_{{\\bf R}^l} f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)\n      \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} dt_1\\cdots d t_l \\nonumber\\\\\n      & = &\\sum_{m=1}^4 I_N^m(\\o^{N,1},\\ldots, \\o^{N,l})\\label{zuzu}\n\\end{eqnarray}\n where\n \\begin{eqnarray}\n  I_N^1& =& \\frac{1}{(2\\pi)^l} \\int\\limits_{{\\bf R}^l}\n       \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} e^{-\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l})\n      \\vec t\/2}d\\vec t \\label{dd} \\\\\n       &&{}-\\frac{1}{(2\\pi)^l}  \\int\\limits_{\\|t\\|>\\epsilon N^{1\/6}}\n       \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} e^{-\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l})\n      \\vec t\/2}d\\vec t.\\nonumber\\\\\n\n  I_N^2& =& \\frac{1}{(2\\pi)^l} \\int\\limits_{\\|t\\|<\\epsilon N^{1\/6}}\n   \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} \\nonumber \\\\\n       && \\ \\ \\ \\ \\ \\ \\ \\  \\  \\  \\  \\  \\  \\  \\  \\  \\ \\ \\  \\ \\ {} \\times  \\Big(\n    f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)-\n       e^{-\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l})\n      \\vec t\/2}\\Big) d\\vec t\\label{ff}\\\\\n  I_N^3& =&\\frac{1}{(2\\pi)^l} \\int\\limits_{\\epsilon N^{1\/6}<\\|t\\|<\\delta N^{1\/2}}\n   \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k}\n    f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)\n     d\\vec t\\nonumber\\\\\n  I_N^4& =& \\frac{1}{(2\\pi)^l}\n  \\int\\limits_{ \\|t\\|>\\delta N^{1\/2}}\n   \\prod_{k=1}^l \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k}\n    f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)\n     d\\vec t\\nonumber\n\\end{eqnarray}\n with $\\epsilon, \\delta>0$ chosen according to the following\n   Proposition \\ref{pr1}.\n\\begin{pro}\n\\label{pr1}\n There exist constants $N_0,C,\\epsilon, \\delta, \\zeta>0$ such\nthat\n   for all $(\\o^{N,1},\\ldots \\o^{N,l}) \\in\n{\\cal R}_{N,l}^{\\eta}$  and all $N \\geq N_0$\n  the following estimates hold:\n\\begin{equation}\n\\label{base1} \\Big|f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t) -\ne^{-\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l}) \\vec t\/2} \\Big|\n  \\leq \\frac{C \\|t\\|^3}{\\sqrt{N}}\n  e^{ -\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l}) \\vec t\/2},\\ \\ \\ \\\n  \\hbox{for all  } \\|t\\|\\leq \\epsilon N^{1\/6}.\n\\end{equation}\n\n\\begin{equation}\n\\label{base2} \\Big| f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)\\Big|\n\\leq e^{-\\zeta \\|t\\|^2} \\ \\ \\\n \\hbox{for all  } \\|t\\|<\\delta \\sqrt{N}.\n\\end{equation}\n\\end{pro}\n\n     The proof of this proposition mimics the one of the Berry-Essen\n  inequality and is given in Appendix.\n\n    The first part of\n$I_N^1$ is just the probability that\n    $l$ Gaussian random variables with\n        zero mean and covariance matrix\n    $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ belong\n          to the intervals\n $[-\\delta_N b_k+c, \\delta_N b_k+c]$ for $k=1,\\ldots, l$ respectively.\n     This is\n      \\begin{equation}\n    \\label{mia0}\n   \\int\\limits_{|z_j- c|\\leq\n   \\delta_N b_j, \\forall_{j=1}^l }\n   \\frac{e^{-(\\vec z B^{-1}(\\o^{N,1},\\ldots,\n \\o^{N,l})\\vec z)\/2}}{(2\\pi)^{l\/2}\n   \\sqrt{{\\rm det} B(o^{N,1},\\ldots,\n \\o^{N,l})} }\\, d\\vec z\n $$\n$$ =  (2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l) e^{-(\\vec c\nB^{-1}(\\o^{N,1},\\ldots,\n \\o^{N,l})\\vec c)\/2}(1+o(1))$$\n$$ =(2\\delta_N\/\\sqrt{2\\pi})^l (b_1\\cdots b_l)\n   e^{-lc^2(1+O(N^{\\eta-1\/2}))\/2}(1+o(1))\n   = (2d)^{-Nl}b_1\\cdots b_l(1+o(1))\n \\end{equation}\n     uniformly for\n        $(\\omega^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal\n       R}_{N,l}^{\\eta}$, where we denoted by $\\vec c$ the vector\n$(c, \\ldots, c)$.\n     Since\n     \\begin{equation}\n     \\label{pr}\n   \\prod_{k=1}^l \\Big| \\frac{ e^{-i t_k (-b_k \\delta_N+c)} -  e^{-i t_k (b_k\n      \\delta_N +c)}}{it_k} \\Big|\n        \\leq  (2 \\delta_N b_1)\\cdots (2\\delta_N b_l)= O((2d)^{-Nl})\n        \\end{equation}\nand the elements of the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\nout\nof the\n  diagonal are $O(N^{\\eta-1\/2})=o(1)$ as  $N \\to \\infty$,\n     the second part of $I_N^1$\n       is   smaller than\n  $(2d)^{-Nl}$ exponentially (with exponential\n    term $\\exp(-h N^{1\/3})$ for some $h>0$).\n\n     There is a constant $C>0$ such that\n    the term $I_N^2$ is bounded by\n     $C (2d)^{-Nl} N^{-1\/2}$\n      for any $(\\o^{N,1},\\ldots \\o^{N,l}) \\in\n{\\cal R}_{N,l}^{\\eta}$ and all $N$ large enough.\n     This follows from  (\\ref{pr}),\n       the estimate (\\ref{base1})\n     and again the fact that\n   the elements of the matrix $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ out of the\n  diagonal are $O(N^{\\eta-1\/2})=o(1)$ as  $N \\to \\infty$.\n\n     The third term $I_N^3$ is exponentially smaller than\n     $(2d)^{-Nl}$\n     by (\\ref{pr}) and  the estimate (\\ref{base2}).\n\n    Finally, by (\\ref{pr})\n$$ |I_N^4|\\leq (2 \\delta_N b_1)\\cdots (2\\delta_N b_l)\n        \\int\\limits_{\\|t\\|>\\delta \\sqrt{N}}\n            |f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)| d\\vec t=\n              O((2d)^{-Nl})\n             \\int\\limits_{\\|t\\|>\\delta \\sqrt{N}}\n            |f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)| d\\vec t.$$\n  The function\n   $ f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)$\n   is the product of $N$ generating  functions (\\ref{zey}).\n         Note that for any pair $\\o^{N,i}, \\o^{N,j}$\n  of  $(\\o^{N,1},\\ldots, \\o^{N,l})\\in {\\cal R}_{N,l}^{\\eta}$,\n              there are at most $N^{\\eta+1\/2}$\n steps $n$ where\n            $\\o^{N,i}_n= \\o^{N,j}_n$.\n         Then there are at least $N -[l(l-1)\/2]N^{\\eta+1\/2}=a(N)$\n          steps where all  $l$ coordinates  $\\o^{N,i}$, $i=1,\\ldots,\n          l$,\n               of the vector\n$(\\o^{N,1},\\ldots, \\o^{N,l}) \\in {\\cal R}_{N,l}^{\\eta}$\n               are\n           different.\n           In this case\n$$\\mathop{\\hbox{\\sf E}}\\nolimits \\exp\\Big( i N^{-1\/2}\\sum_{k=1}^l t_k  \\eta(n, \\o^{N,k}_n)\n\\Big)=\\phi(t_1 N^{-1\/2})\\cdots \\phi(t_k N^{-1\/2}).$$\n    By the assumption made on $\\phi$,\n      this function is aperiodic and thus $|\\phi(t)|<1$\n        for $t\\ne 0$.\n    Moreover, for any $\\delta>0$ there exists\n      $h(\\delta)>0$ such that $|\\phi(t)|\\leq 1-h(\\delta)$\n         for $|t|>\\delta\/l$.\n  Then\n  $$\\int\\limits_{\\|t\\|>\\delta \\sqrt{N}}\n            |f^{\\o^{N,1},\\ldots, \\o^{N,l}}_N(\\vec t)| d\\vec t\n  \\leq \\int\\limits_{\\|t\\| >\\delta \\sqrt{N}} |\n    \\phi(t_1 N^{-1\/2})\\cdots \\phi(t_k N^{-1\/2})|^{a(N)}d\\vec t\n$$\n$$ = N^{l\/2} \\int\\limits_{\\|s\\| >\\delta  }\n    |\\phi(s_1 )\\cdots \\phi(s_k )|^{a(N)}d\\vec s\n      \\leq N^{l\/2}(1-h(\\delta))^{a(N)-2}\n     \\int\\limits_{\\|s\\| >\\delta  }\n    |\\phi(s_1 )\\cdots \\phi(s_k )|^2d\\vec s $$\n      where $a(N)=N(1+o(1))$ and\n         the last integral converges due to the assumption\n       made on $\\phi(s)$.\n    Hence $I_N^4$ is exponentially smaller than $(2d)^{-Nl}$.\n    This finishes the proof of (\\ref{gg}).\n\n\\medskip\n\n \\noindent{\\it Step 4.} We are now able to prove the theorem using\n  the estimates (\\ref{gg2}),(\\ref{rrs}) and  (\\ref{gg}).\n       By (\\ref{gg}),\n   the sum (\\ref{zet}) over ${\\cal R}_{N,l}^{\\eta}$\n   (with fixed $\\eta \\in ]0,1\/2[$) that contains\n   by (\\ref{rrs})$(2d)^{Nl}(1+o(1))$ terms,\n    converges to  $b_1 \\cdots b_l$.\n     The sum (\\ref{zet}) over $(\\o^{N,1},\\ldots, \\o^{N,l}) \\not \\in\n   {\\cal R}_{N,l}^{\\eta}$ but with $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$\n     non-degenerate, by (\\ref{rrs}) has only at most\n         $(2d)^{Nl} C N \\exp(-h N^{2\\eta})$ terms,\n      while each of its terms by (\\ref{gg2}) with $r=l$\n   is of the order $(2d)^{-Nl}$ up to a polynomial term. Hence,\n      this sum converges to zero.\n   Finally,\ndue to the fact that in any\n         set $(\\o^{N,1},\\ldots, \\o^{N,l})$\n    taken into account in (\\ref{zet}) the paths\n         are all different,\n    the sum over $(\\o^{N,1},\\ldots, \\o^{N,l}) \\not \\in\n   {\\cal R}_{N,l}^{\\eta}$ with\n     $B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ of the rank $r<l$\n      has an\n     exponentially smaller number of terms than $(2d)^{Nr}$.\n        This has been shown  in detail\n        in the proof of Theorem \\ref{th1} where\n          the arguments did not depend on the dimension\n  of the random walk.\n        Since  by (\\ref{gg2}) each of these terms\n   is of the order $(2d)^{-Nr}$ up to a polynomial term,\n     this sum converges to zero.\n    This concludes the proof of (\\ref{th1e0}).\n      The proof of (\\ref{th1b0}) is completely analogous\n        to the one of (\\ref{th1b}).\n\n\n\\section{Appendix}\n\n\\noindent{\\it Proof of (\\ref{sdg}).}\n   It is carried out via trivial arguments of linear\n   algebra.\n\n   Let  $m=1,2,\\ldots, r+1$,\n       $D_{m-1}$ be a non-degenerate $r\\times r$\n   matrix  with the first $m-1$ rows having $1$ on the\n  diagonal and $0$ outside of the diagonal.\n    (Clearly, $D_0$ is just a non-degenerate matrix\n      and  $D_r$ is the diagonal matrix with $1$ everywhere\n     on the diagonal.)\n  Let us introduce the integral\n\\begin{eqnarray}\n \\lefteqn{ J^{m-1}(D_{m-1})}\\nonumber\\\\\n & =& \\int \\prod_{k=1}^r\n   \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(D_{m-1} \\vec\n    t)_k|} \\Big)  d\\vec t \\nonumber \\\\\n&=&\n \\int \\prod_{k=1}^{m-1}\\min\\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n          \\min\\Big(1, \\frac{1}{|t_k|} \\Big)\n     \\prod_{k=m}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(D_{m-1} \\vec\n     t)_k|} \\Big)  d\\vec t.\\nonumber\n \\end{eqnarray}\n  Sice $D_{m-1}$ is non-degenerate, there exists $i \\in \\{m,\\ldots,r\\}$\n     such that $d_{m,i} \\ne 0$\n             and the matrix $D_m$\n which is obtained from the matrix $D_{m-1}$ by replacing its $m$th\n row\n   by the one with $1$ at the place $(m,i)$ and $0$\n     at all places $(m,j)$ for $j \\ne i$\n       is non-degenerate.\n  Without loss of generality we may assume that $i=m$\n    (otherwise juste permute the $m$th with the $i$th column\n      in $D_{m-1}$ and $t_i$ with $t_m$ in the integral $J^{m-1}(D_{m-1})$ above).\n Since either\n  $|t_{m-1}| < |(D_{m-1} \\vec t)_{m-1}|$\n   or $|t_{m-1}| \\geq |(D_{m-1} \\vec t)_{m-1}|$,\n  we can estimate $J^{m-1}(D_{m-1})$ roughly by the sum of the following\n   two terms :\n\\begin{eqnarray}\n  \\lefteqn{J^{m-1}(D_{m-1})}\\nonumber \\\\\n& \\leq &\n  \\int \\prod_{k=1}^{m}\\min\\Big((2d)^{-N}, \\frac{1}{|t_k|}\n\\Big)\n          \\min\\Big(1, \\frac{1}{|t_k|} \\Big)\n     \\prod_{k=m+1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(D_{m-1} \\vec\n    t)_k|} \\Big)  d\\vec t \\nonumber\\\\\n &&{} +\n  \\int \\prod_{k=1}^{m-1}\\min\\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n          \\min\\Big(1, \\frac{1}{|t_k|} \\Big)\n\\min\\Big((2d)^{-N}, \\frac{1}{|(D_{m-1} \\vec t)_m|} \\Big)\n          \\min\\Big(1, \\frac{1}{| (D_{m-1} \\vec t)_m|} \\Big)\\nonumber \\\\\n   &&\\ \\ \\ \\ \\ \\ \\ \\ {}\\times  \\prod_{k=m+1}^r\n  \\min \\Big((2d)^{-N}, \\frac{1}{|t_k|} \\Big)\n     \\min \\Big(1, \\frac{1 }{ |(D_{m-1} \\vec\n    t)_k|} \\Big)  d\\vec t. \\label{dd}\n\\end{eqnarray}\n      The first term here is just $J^{m}( D_m)$.\n  Let us make a change of variables\n   in the second one : let $\\vec z =B_{D_{m-1}} \\vec t$,\n    where the matrix $B_{D_{m-1}}$ is chosen such that\n              $z_1=t_1,\\ldots, z_{m-1}=t_{m-1},\n          z_m =(D_{m-1} \\vec t)_m, z_{m+1}=t_{m+1},\n       \\ldots, z_{r}=t_r$. (Clearly, its $m$th row\n         is the same as in the matrix $D_{m-1}$,\n           and it has $1$ on the diagonal in all other\n               $r-1$ rows and $0$ outside of it.)\n  Since $d_{m,m}\\ne 0$, the matrix $B$ is non-degenerate.\n      Then $ D_{m-1} \\vec t=D_{m-1} B^{-1}_{D_{m-1}} \\vec z$,\n  where the matrix $D_{m-1} B^{-1}_{D_{m-1}} $ is non-degenerate,\n   and, moreover, it has the first $m$ rows with $1$ on the\n   diagonal and $0$ outside of it, as we have\n      $(D_{m-1} \\vec t)_1=t_1=z_1, \\ldots, (D_{m-1} \\vec t)_{m-1}=t_{m-1}\n        =z_{m-1}, (D_{m-1} \\vec t)_m=z_m$.\n  Then (\\ref{dd}) can be written as\n\\begin{equation}\n\\label{ddd}\n J^{m-1}(D_{m-1}) \\leq J^m(D_{m})+ d_{m,m}^{-1}\n    J^m( D_{m-1} B^{-1}_{D_{m-1}}).\n\\end{equation}\n\n   Now, observe that the left-hand side  of (\\ref{sdg}) is\n        $J^0(A^r)$.\n  By (\\ref{ddd}) it is bounded by\n   $J^1(A^r_1)+ a_{1,1}^{-1} J^1(A^r B^{-1}_{A^r})$.\n    Again by (\\ref{ddd}) each of these two terms can be estimated\n        by a sum of two terms\n  of type $J^2(\\cdot)$ etc. After $2^r$ applications\n    of (\\ref{ddd}) $J^{0}(A^r)$\n    is bounded by a sum of $2^r$ terms\n         of type $J^r(D_r)$ multiplied by some constants\n    depending only on the initial matrix $A_r$.\n        But all these $2^r$ terms $J^r(D_r)$ are the same\n   as in the right-hand side of (\\ref{sdg}).\n\n\\medskip\n\n\\noindent{\\it Proof of Proposition \\ref{pr1}.}\n   We use the representation (\\ref{aa}) of\n     $f_N^{\\o^{N,1},\\ldots, \\o^{N,l}}(\\vec t)\n   $ as the product of a certain number\n      $K(N,\\o^{N,1},\\ldots, \\o^{N,l})$\n         (denote it shortly by $K(N,\\o)$, clearly\n          $ N \\leq K(N, \\o) \\leq lN$)\n             of generating functions $ \\phi(N^{-1\/2}(A \\vec t)_j)$\n    where at most $2^l$ are different.\n      Each of them is\n        of the form $\\mathop{\\hbox{\\sf E}}\\nolimits \\exp(i N^{-1\/2}(t_{i_1}+\\cdots +t_{i,k})X)$\n     with $X$ a standard Gaussian random variable.\n  Applying the fact that $|e^{iz}-1-iz-(iz)^2\/2!|\\leq |z|^3\/3!$\n    for any $z\\in {\\bf R}$, we can write\n\\begin{equation}\n\\label{jj} \\phi(N^{-1\/2}(A \\vec t)_j)\n  = 1- \\frac{ ((A \\vec t)_j)^2}{2! N}-\\theta_j \\frac{((A \\vec t)_j)^3\n      \\mathop{\\hbox{\\sf E}}\\nolimits|X|^3}{ 3!\n  N^{3\/ 2}} \\equiv 1-\\alpha_j\n  \\end{equation}\n     with some complex $\\theta_j$ with $|\\theta_j|<1$.\n      It follows that there are some constants $C_1,C_2>0$ such that\n         for any $( \\o^{N,1},\\ldots, \\o^{N,l}  ) \\in\n         {\\cal R}_{N,l}^{\\eta}$ and any $j$\n we have:       $|\\alpha_j|\\leq C_1 \\|\\vec t\\|^2 N^{-1}+ C_2\\|\\vec t\\|^3\n       N^{-3\/2}$. Then $|\\alpha_j|<1\/2$ and $|\\alpha_j|^2 \\leq\n         C_3 \\|\\vec t\\|^3\n       N^{-3\/2}$  with some $C_3>0$\n                 for all $\\vec t$ of the absolute\n              value $\\|\\vec t\\|\\leq \\delta \\sqrt{N}$\n           with $\\delta>0$ small enough.\n            Thus $ \\ln \\phi(N^{-1\/2}(A \\vec t)_j)\n              =-\\alpha_j+\\tilde \\theta_j \\alpha_j^2\/2$\n  (using the expansion $\\ln(1+z)=z +\\tilde \\theta z^2\/2$\n   with some $\\tilde \\theta$\n     of the absolute value $|\\tilde \\theta|<1$\n        which is true for all $z$ with $|z|<1\/2$)\n        for all $( \\o^{N,1},\\ldots, \\o^{N,l}  ) \\in\n         {\\cal R}_{N,l}^{\\eta}$ and for all $\\vec t$ with\n          $\\|\\vec t\\|\\leq \\delta \\sqrt{N}$\n        with some $\\tilde \\theta_j$ such that $|\\tilde \\theta_j|<1$.\n          It follows that\n\\begin{equation}\n\\label{zgh}\n      f_N^{\\o^{N,1},\\ldots, \\o^{N,l}}(\\vec t)\n   = \\exp\\Big( -\\sum_{j=1}^{K(N,\\o)}\n        \\alpha_j+ \\sum_{j=1}^{K(N,\\o)}\n            \\tilde \\theta_j \\alpha_j^2\/2\\Big).\n            \\end{equation}\n          Since $A^*A=B_N(\\o^{N,1},\\ldots,\n       \\o^{N,l})$,   here  $-\\sum_{j=1}^{K(N,\\o)}\n        \\alpha_j =-\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l}) \\vec t\/2\n           +\\sum_{j=1}^{K(N,\\o)} p_j$ where\n             $|p_j| \\leq C_2\\|\\vec t\\|^3\n       N^{-3\/2}$.  Then\n\\begin{equation}\n\\label{zgh1}\n    f_N^{\\o^{N,1},\\ldots, \\o^{N,l}}(\\vec t)=\n       \\exp \\Big( -\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l}) \\vec t\/2 \\Big)\n          \\exp\\Big(\\sum_{j=1}^{K(N,\\o)}p_j+\n            \\tilde \\theta_j \\alpha_j^2\/2\\Big)\n\\end{equation}\n              where\n           $|p_j|+  |\\tilde \\theta_j \\alpha_j^2\/2| \\leq\n      (C_2+C_3\/2)\\|\\vec t\\|^3 N^{-3\/2}$ for all $j$.\n   Since $K(N,\\o) \\leq l N$, we have\n   \\begin{equation}\n   \\label{qk}\n     \\Big| \\sum_{j=1}^{K(N,\\o)}\n         p_j+  \\tilde \\theta_j \\alpha_j^2\/2 \\Big|\\leq (C_2+C_3\/2)l\\|t\\|^3\n      N^{-1\/2}.\n      \\end{equation}\n          It follows that for $\\epsilon>0$ small enough\n         $| \\exp (\\sum_{j=1}^{K(N,\\o)}\n         p_j+  \\tilde \\theta_j \\alpha_j^2\/2)-1|\n           \\leq C_4 \\|\\vec t\\|^3 N^{-1\/2}$\n             for all $\\vec t$ with $\\|\\vec t\\| \\leq \\epsilon N^{1\/6}$.\n          This proves (\\ref{base1}).\n   Finally\n\\begin{equation}\n\\label{zlt}\n |f_N^{\\o^{N,1},\\ldots, \\o^{N,l}}(\\vec t)| \\leq\n       \\exp \\Big( -\\vec t B_N(\\o^{N,1},\\ldots, \\o^{N,l}) \\vec t\/2 \\Big)\n    \\exp \\Big((C_2+C_3\/2)l\\|t\\|^3\n      N^{-1\/2}\\Big).\n   \\end{equation}\nTaking into account the fact that the elements of\n$B_N(\\o^{N,1},\\ldots, \\o^{N,l})$ out of the diagonal are at most\n   $N^{-1\/2+\\eta}=o(1)$ as $N \\to \\infty$,\n       one deduces from (\\ref{zlt})\n           that for $\\delta>0$ small enough\n(\\ref{base2})  holds true with some $\\zeta>0$\n   for all $N$ large enough and all $\\vec t$ with\n   $\\|\\vec t\\|\\leq \\delta \\sqrt{N}$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\nIn quantum world there are different kinds of correlations between multiparticle quantum states. Understanding the nature of correlations is one of the\nchallenges in the development of quantum information\nscience. Given a bipartite or multipartite state one usually tries to characterize the amount of classical correlation, quantum correlation and quantum entanglement \ncontained in the composite system. Different correlations can arise depending on the state preparation procedure and\nmeasurements performed on the system. These correlations can account for many counter-intuitive\nfeatures in the quantum world.\nIn particular, entanglement is a physical property that has been successfully employed to interpret several phenomena which cannot be understood\nusing the laws of classical physics \\cite{Horodecki1}. It has also been identified as the basic ingredient for different quantum communication protocols like super-dense coding\n\\cite{Wiesner}, quantum teleportation \\cite{Brassard}, quantum cryptography \\cite{Bennett}, remote-state preparation \\cite{Pati, Bennett2}\nand quantum computational tasks such as the one-way quantum computer \\cite{bri}.\n\n\nOne fundamental property of quantum correlations in multiparty quantum states is that it can be monogamous \\cite{Coffman}. To state this in a qualitative way, if a correlation \nmeasure is monogamous, then\nthis says that in a composite quantum state, if two subsystems are more correlated with each other, then they will share a less amount of correlation with the other subsystems \nwith respect to that measure of correlation. In other words, it puts a restriction on the shareability of correlation between the different parties of a composite quantum state. \nSpecifically, \nif the two subsystems are maximally quantum correlated with each other, then they cannot get correlated to any other subsystem at the same time.\nThe measures of classical correlation are never monogamous and  therefore are considered to be freely shareable. But, not all measures of quantum correlation\nsatisfy monogamy \\cite{Osborne,Adesso,Hiroshima,Seevinck,Lee}. For example, the square of concurrence and the squashed entanglement satisfy the  monogamy inequality \\cite{Matthias1},\nwhereas the relative entropy of entanglement, the entanglement of formation and other measures do not satisfy monogamy in general. Recently, it has been shown that the monogamous \ncharacter is not an intrinsic property of other quantum correlation measures. In particular, the quantum discord \\cite{Zurek} for tripartite states does not obey monogamy in general \n\\cite{Prabhu,gio,alex}. However, interestingly, though a quantum correlation measure may not satisfy monogamy, yet the quantum correlation measure raised to a power will certainly \nobey monogamy \\cite{Salini}. It has been shown that the square of the concurrence, which is a monotonic function of entanglement of formation, is monogamous. Similarly, it has been shown that \nthe square of the quantum discord also satisfies monogamy.\nThe concept of monogamy, not only is important from fundamental point of view, it finds practical importance too.\nFor example, the monogamy of quantum correlations plays a crucial role in the security of quantum cryptography \\cite{Gisin1}.\n\nWhile the monogamy is an important property to study for various correlation measures, still there remains other desirable properties that the \ncorrelation measures are expected to obey from the perspective of being physically meaningful. \nOne such property is the additivity on tensor product of density matrices \\cite{Shor1}. \nThe property of additivity on tensor product states dictates that a correlation measure is an additive measure if the value of that measure on the tensor product of density matrices is \nsimply equal to the addition of the values of that correlation measure on the individual density matrices forming the tensor product state. \nThe quantum mutual information is an additive measure of total correlation and the squashed entanglement is another additive measure of quantum\ncorrelation \\cite{Matthias}.\nHowever, all correlation measures are not yet proved to be additive \\cite{Werner1}. \nThere are measures of entanglement and \ncapacity of channels that have been proved to be non-additive \\cite{Werner,Hastings,Hayden1,Smith}. For example the relative entropy of entanglement is proved to be non-additive \n\\cite{Shor2} and there is\nstrong indication that the bipartite distillable entanglement is also non-additive \\cite{Werner1}. Also, the additivity of entanglement of formation still remains an open question,\n and it is conjectured to obey a strong super-additivity condition \\cite{Pomeransky}. Thus, the question of additivity of the different correlation measures is\n one of the intriguing and yet to be solved question in the realm of quantum information theory.\n\n\nA measure of total correlation was proposed by\nTerhal {\\it et al}. \\cite{Terhal} called the entanglement of purification. It should be emphasized that entanglement of purification\nis not a measure of entanglement, but a measure of total correlation defined in units of pure state entanglement.  This definition was motivated operationally, \ntrying to see if quantum states could be\nconstructed from EPR pairs, i.e. the Einstein-Podolsky-Rosen\npairs, with vanishing amount of communication asymptotically. It\nis based on the entanglement-separability paradigm, \ntrying to capture the classical and quantum correlations in an unified way. \nIt was shown to be satisfying the properties of a genuine measure of total correlation. \nAlso, a monogamy relation involving the entanglement of purification and the quantum advantage of dense coding was given by Horodecki {\\it et al}. \\cite{Horodecki}.\nHowever, the conditions for the monogamy or polygamy nature of entanglement of purification have not been found yet. The present paper is motivated from the fact that the mutual \ninformation, \na measure of total correlation is monogamous for any tripartite pure states \\cite{gio}. Therefore, if the entanglement of purification is a measure of total correlation can it be \nstrictly monogamous for all tripartite pure states? We find that the entanglement of purification of a tripartite pure state $\\rho_{ABC}$ across $A:BC$ partition is never less than\nits sum for the reduced density matrices $\\rho_{AB}$ and $\\rho_{AC}$, and is mostly polygamous. This observation calls for further investigation in understanding\nthe nature of correlation captured by the entanglement of purification. At first, we prove that similar to the mutual information, the entanglement of purification does not increase \nupon discarding ancilla. Thereafter, we explore the monogamy, polygamy and the additivity properties of the entanglement of purification for pure as well as mixed tripartite\nstates. Furthermore, we find analytically the lower bound and actual value of the entanglement of purification for different classes of mixed states.\nWe also present some conditions for the monogamy of entanglement of purification in terms of monogamy of entanglement of formation and other entropic inequalities.\nWe use these properties of entanglement of purification to explore the monogamy and additivity properties of the quantum advantage of dense coding. \nThe above definition as a theory of 'all correlation' may have important applications in quantum information theory.\n\nThe paper is organized as follows. In section II, we provide the definition of the monogamy of correlations.\nIn section III, we discuss the measures of total correlation namely the quantum mutual information \nand the entanglement of purification, mentioning specifically the monogamy \nproperties of the quantum mutual information. Here we also state the definition of interaction information and discuss some of its properties briefly.\nThen, we move on to find the relation between the entanglement of purification and the quantum mutual information of the purified state in section IV.\nHere, we discuss what happens to the entanglement of purification upon discarding of subsystem of a composite quantum system. Thereafter, we\nobtain the lower bounds and exact values of entanglement of purification for some mixed quantum states, specifically for a class of tripartite states and\nhigher dimensional bipartite states. In section V,\nwe derive the results for the monogamy and polygamy nature  of the entanglement of purification for pure as well as mixed states, extending to the case of $n$ parties. \nHere, we discuss a relation between the monogamy of entanglement of purification with that of the entanglement of formation and quantum discord in\nthe case of tripartite pure states. Also, in this section, the monogamy conditions for the mixed states are explored  with specific examples\nand cases where the the states are polygamous. \nIn section VI, we find out that if entanglement of purification is not additive on tensor product of density matrices, then \nit has to be a sub-additive quantity. Next, using the results we derive in the previous sections,\nwe find out the monogamy, super-additivity (on tensor products) properties for the quantum advantage of dense coding in section VII, \nwhere we also obtain the upper bounds for some states and identify the states with no quantum advantage of dense coding. We end with conclusions and outlook in section VIII.\n\n\n\\section{Monogamy and Polygamy of correlations}\nMonogamy is a property of a multiparticle quantum state that can be studied with respect to a particular correlation measure. \nIt is an important property that tells us about the nature of the correlation at our disposal, in particular, whether it is freely shareable or not. \nClassical correlations \\cite{Henderson} are always polygamous, whereas certain quantum correlation\nmeasures satisfy this property and some others do not \\cite{Adesso,Hiroshima,Prabhu,Matthias}. For example, the quantum discord is not in general a monogamous quantity for even \nsome cases of the pure tripartite states, whereas the total correlation given by the quantum mutual information is strictly monogamous for all tripartite pure states.\nTherefore, the monogamy or polygamy nature of the total correlation measure that supposedly contains some amount of quantum and classical\ncorrelation is an important question to consider. \nNow, according to the definition of monogamy, it is a property which does not allow the free sharing of correlation between the \nsubparts of a composite system. Mathematically, if a correlation measure $Q(\\rho)$ satisfies \n\\begin{equation}\n Q(A:BC)\\geq Q(A:B)+Q(A:C)  \n\\end{equation}\nfor any tripartite state $\\rho_{ABC}$, then the correlation measure is called monogamous, otherwise it is called polygamous. This definition can be extended to the case of \n$n$ parties as well. A correlation measure $Q$ is said to be $n$ partite monogamous if the following inequality is satisfied\n\\begin{equation}\n  Q(A_1:A_2..A_n)\\geq  Q(A_1:A_2)+Q(A_1:A_3)+..Q(A_1:A_n)\\nonumber\n\\end{equation}\nand otherwise it is called $n$ partite polygamous.\n\n\n\\section{ Measures of total correlation}\n\nWe consider multiparticle quantum system with each subsystem defined on a finite dimensional Hilbert space ${\\cal H}$. Let ${\\cal L}(\\cal H)$ be the \nset of all linear operators acting on ${\\cal H}$ and $D({\\cal H})$ be the set of all density operators $\\rho$ with $\\rho \\ge 0$ and $\\t Tr(\\rho) =1$. \nThe composite state $\\rho_{ABC} \\in D({\\cal H}_{ABC} )$ is a general state that may contain classical and quantum correlations including entanglement. \nThe von-Neumann entropy for a density operator $\\rho_A$ is defined as \n$S(A) = -{\\t Tr}(\\rho_A \\log_2 \\rho_A)$, where $ \\rho_{A}=Tr_{BC}(\\rho_{ABC})$.\nIn this section we discuss two important measures of total correlation in the bipartite scenario, namely, \nthe quantum mutual information and the entanglement of purification. The measures of total correlation try to capture quantitatively the total correlations comprising of the \nclassical as well as the quantum correlations in a bipartite state $\\rho_{AB} = {\\t Tr}_C (\\rho_{ABC})$.\n\n\n\n\\subsection{ Quantum Mutual Information}\n\nThe quantum mutual information is a measure of total correlation in a quantum system.\nIt is a straightforward generalization of the classical mutual information. The quantum mutual information is obtained by just replacing the Shannon entropy \nby the von-Neumann entropy for the respective terms in the expression for the classical mutual information. \nThus, for a bipartite quantum state, the quantum mutual information of the state $\\rho_{AB}$ is defined as $ I(A:B)= S(A)-S(A|B)$, where $\nS(A|B) = S(AB)- S(B)$ is the quantum conditional entropy \\cite{Cerf}. Quantum mutual information satisfies some natural properties, all of which,  \na total correlation measure is expected to satisfy. \nFirst, it never increases \nupon discarding of quantum systems, i.e., $I(A:BC)\\geq I(A:B)$. Secondly, the quantum mutual information is additive on tensor product of density matrices, which is \n$I(AC:BD)=I(A:B)+I(C:D)$ for $\\rho_{AB}\\otimes\\sigma_{CD}$. \nApart from these, the monogamy properties of the mutual information have been studied in Ref.\\cite{gio}. There, it was shown that a necessary and sufficient condition \nfor the monogamy of quantum mutual information can be stated in terms of the interaction information \\cite{Prabhu,gio}. Specifically, it can be shown that for any pure tripartite state \n$\\vert\\Psi\\rangle_{ABC}$, we have \n\\begin{align}\nI(A:B) + I(A:C) = I(A:BC), \\nonumber\n\\end{align}\nwhich implies that the quantum mutual information is strictly monogamous for a pure tripartite state. The necessary and sufficient criteria for quantum mutual information to be \nmonogamous for mixed tripartite state is that the interaction information should be positive \\cite{gio}. \nIn classical information theory, interaction information of state $\\rho_{ABC}$ is defined as $\\tilde{I}(\\rho_{ABC})= H(AB)+H(BC)+H(AC)-H(A)-H(B)-H(C)-H(ABC)$, where $H(AB)$ denote the Shannon entropies \n\\cite{Thomas}.\nReplacing the Shannon entropies by the von-Neumann entropies we obtain the quantum generalization of the interaction information. The quantum interaction information is therefore\nnothing but $\\tilde{I}(\\rho_{ABC})= S(AB)+S(BC)+S(AC)-S(A)-S(B)-S(C)-S(ABC)$, where $S(AB)$ denote the von-Neumann entropy of the density matrix $\\rho_{AB}$. \nInteraction information is a measure of the effect of \nthe presence of a third party $C$ on the amount of correlations shared by the other two parties as it is given by the difference between the information shared between the parties \n$A$ and $B$ when $C$ is present and when $C$ is not present. Quantum interaction information can be positive as well as negative. It is invariant under the action of local unitaries \nand non-increasing under the action of unilocal measurements \\cite{Prabhu}. It has been used to provide necessary and sufficient conditions for the monogamy of quantum discord in\nRef.\\cite{Prabhu}.\nQuantum mutual information is an important measure of correlation and finds application in\na large number of settings primarily in studying the channel capacities \\cite{Bennett1,Benjamin}. Also, an operational interpretation has been given of the quantum mutual \ninformation in Ref.\\cite{Groissman}. There, it was interpreted as the total amount of randomness or noise needed to erase the correlations in a bipartite quantum state \ncompletely.\n\n\\subsection{ Entanglement of purification}\n\n\nThe entanglement of purification is a measure of total correlation along a bipartition in a quantum state, \\cite{Terhal} defined \nusing the notion of the entanglement separability paradigm. Interestingly, in this approach the authors in \\cite{Terhal} have treated both the quantum entanglement and the classical correlation in a unified framework, \nby defining a measure of total correlation namely the entanglement of purification in units of pure state entanglement.\nBy their definition, the entanglement of purification is expressed as the entanglement of the purified version of the mixed state as follows. Suppose we have a mixed state $\\rho_{AB}$, and we purify it \nto a pure state $\\vert\\Psi\\rangle_{ABA'B'}$. Then, the entanglement of purification is defined as\n\\begin{equation}\nE_p(A:B)=\\min_{A'B'} E_f(AA':BB'),\n\\end{equation}\nwhere $E_p(A:B)$ denotes the entanglement of purification of the state $\\rho_{AB}$ across $A:B$ partition, and $E_f(AA':BB')$ is the entanglement of \nformation across the bipartition $ AA':BB'$ of the pure state $\\vert\\Psi\\rangle_{ABA'B'}$, \nobtained from $\\rho_{AB}$ by any standard\npurification procedure such as $\\vert\\Psi_s\\rangle_{AA':BB'}= \\sum_{i}\\sqrt{\\lambda_i}\\vert \\Psi_i\\rangle_{AB}\\otimes\\vert 0\\rangle_{A'}\\vert i\\rangle_{B'}$. Here, the $\\lambda_i$  \nare the Schmidt coefficients and $ \\vert \\Psi_i\\rangle $ are the corresponding Schmidt vectors in $\\cal H_{AB}$. \nThe above expression can be reformulated in terms of the trace preserving completely positive (TCP) maps, \nsince every quantum operation can be written in terms of the TPCP maps. Following Ref.\\cite{Terhal}, from Eq(4), we \nget $E_p(A:B)$ of $\\rho_{AB}$ as the following minimum over unitary matrices as \n\\begin{align}\nE_p(A:B)=\\min_{U_{A'B'}}E_f(AA':BB'),\n\\end{align}\nwhere $E_f(AA':BB')$ is the entanglement of formation across the $AA':BB'$ partition of the pure state \n${(I_{AB}\\otimes U_{A'B'})(\\vert\\Psi_s\\rangle\\langle\\Psi_s\\vert)(I_{AB}\\otimes U_{A'B'})^{\\dagger}}$ obtained from $\\rho_{AB}$ by a standard purification procedure and then acting \nunitary matrices over the ancilla part. This is nothing but the entropy $\\min_{U_{A'B'}}S(Tr_{AA'}((I_{AB}\\otimes U_{A'B'})(\\vert\\Psi_s\\rangle\\langle\\Psi_s\\vert)(I_{AB}\\otimes U_{A'B'})^{\\dagger}))$. Now by tracing out the\n$ AA'$ part from the pure state as well as the unitary operator, one obtains the following equivalent form of entanglement of purification in terms of the TCP map\n\\begin{eqnarray}\nE_p(A:B)=\\min_{\\Lambda_{B'}}S ((I_{B}\\otimes\\Lambda_{B'})(\\mu_{BB'}(\\rho_{AB})));\\nonumber\\\\\n\\Lambda_{B'}(\\nu) = {\\t Tr}_{A'}(U_{A'B'}(\\nu_{B'}\\otimes\\vert0\\rangle\\langle 0\\vert_{A'})U^\\dagger_{A'B'});\\nonumber\\\\\n\\mu_{BB'}(\\rho_{AB})= {\\t Tr}_{AA'}(\\vert\\Psi\\rangle\\langle\\Psi\\vert),\n\\end{eqnarray}\nwhere $ \\Lambda_{B'}$ is a TCP map. The above form is derived in \\cite{Terhal}. Therefore, the \nminimization over unitary matrices in Eq(3) is now represented as a minimization over all TPCP maps $\\Lambda_{B'}$, since a TCP map is equivalently represented as an unitary transformation on the larger system followed by \ntracing over the ancilla.\nIt was shown that the above optimization can be successfully performed in a Hilbert space of a limited dimension $ d_{A'}=d_{AB}$ and $d_{B'}=d_{AB}^2 $, due to the result by \nTerhal {\\it et al}. \\cite{Terhal}.\nFor pure states, the entanglement of purification is equal to the entanglement of formation and for a mixed state $\\rho_{AB}$, one has $E_p(A:B)\\geq E_f(A:B)$.\nAlongside, the authors have introduced the regularised entanglement of purification $E_p^\\infty(A:B)$. It was shown that the asymptotic cost of preparing $n$ copies of \n$\\rho_{AB}$ from singlets using only local operations and an asymptotically vanishing\namount of quantum or classical communication is equal to the regularised entanglement of purification.\nThis implies that the regularised entanglement of purification is actually the \nentanglement cost (with $LO_q$) of the quantum states $\\rho$ on $\\cal H_d\\otimes\\cal H_d$ \\cite{Terhal}, i.e., $E_{LO_q}(A:B)=E_p^\\infty(A:B)$.\nLater, from an operational point of view it was shown that if it is additive on tensor product states then $ E^\\infty_P(A:B)$ is actually the optimal visible compression rate for mixed \nstates \\cite{Hayashi}. Other operational interpretations have been explored for this quantity. In particular, the regularized entanglement of purification was shown to be equal to the\nentanglement assisted noisy channel capacity \\cite{Nilanjana}. On another note it was shown that the regularized entanglement of purification $E_{LO_q}(A:B)$ gives the communication cost \nof simulating a channel without the presence of prior entanglement \\cite{Shor}. However, the entanglement of purification is mostly an unexplored quantity since \nit is a difficult quantity to calculate analytically owing to the optimization needed to be done in a larger Hilbert space. But, using the monogamy property of entanglement,\nthe authors in Ref.\\cite{Matthias1} have found the entanglement of purification for a class of bipartite states supported in symmetric or antisymmetric subspaces analytically to be \n$S(A)$. However, one of the unanswered question regarding the entanglement of purification is the property of additivity. \nIt is still not known whether the entanglement of purification is additive on tensor product states or not. But, some progress has been made in this direction by, where\nentanglement of purification has been proved to be non-additive within a certain numerical tolerance \\cite{Chen}. The entanglement of purification has been related to some other \ninformation theoretic quantities as well. It has also been shown that the entanglement of purification is related to the partial quantum information, through its monogamy relation \nwith the quantum advantage of dense coding \\cite{Horodecki}.\n\n\n\n\\section{ Entanglement of purification in terms of quantum mutual information\n: Lower bound and exact values}\n\nThe entanglement of purification can be rewritten in terms of the quantum mutual information.\nFor the pure state $\\vert\\Psi\\rangle_{ABA'B'}$, which is the\noptimally purified state for the mixed state $\\rho_{AB}$ for evaluating the entanglement of purification, the quantum mutual information between parties $AA'$ and $BB'$ is given by \n$ I(AA':BB')= S(AA')+S(BB')-S(AA'BB')$. Since $\\vert\\Psi\\rangle_{ABA'B'}$ is a pure state, we have\n\\begin{equation}\n E_p(A:B) = \\frac{I(AA':BB')}{2}.\\nonumber\n\\end{equation}\nTherefore, the entanglement of purification is actually half of the optimised quantum mutual information of the purified version of the mixed density\nmatrix. The above equations are then used to prove a better lower bound for the entanglement of purification. Before that, we prove an important property of entanglement of \npurification, an attribute of a measure of total correlation.\n\\vskip 10pt\n{\\textbf{ Proposition 1}}: The entanglement of purification never increases upon discarding of quantum system, i.e.,\n\\begin{equation}\n E_p(A:BC)\\geq E_p(A:B).\n\\end{equation}\n\n\\textit{Proof}: \nIf $\\rho_{ABC}$ is pure, then $E_p(A:BC)=S(A)$.  Also, we know that $E_p(A:B)\\leq S(A)$. This leads to $E_p(A:BC)\\geq E_p(A:B)$.\nIn case of mixed states $\\rho_{ABC}$, we note that the set of all the pure states for calculating $E_p(A:BC)$ is a subset of the set of all pure states taken for calculating\n$E_p(A:B)$.\nThis clearly implies that $ min[I(AA':BB')]\\leq min[I(AA':BC(BC)')]$. From here we thus conclude that $E_p(A:BC)\\geq E_p(A:B)$. Thus, like\nthe quantum mutual information, the entanglement of purification also never increases upon discarding of quantum systems. This is a desired property that the total\ncorrelation should not increase upon discarding of quantum system. It is easily seen that the equality condition holds when $\\rho_{AB}$ is supported in the symmetric\nor antisymmetric subspace.\n\nWe now state some simple inequalities for entanglement of purification which will be later used for deriving the monogamy and polygamy conditions for it. Let \n$\\vert\\Psi\\rangle_{ABA'B'}$ be the optimal pure state for evaluating the entanglement of purification of $\\rho_{AB}$. \nUsing the sub-additivity of conditional entropy \\cite{Chuang} for a composite quantum system of four parties, i.e., \n$S(AB\\vert A'B')\\leq S(A\\vert A')+S(B\\vert B')$, we get $S(ABA'B')-S(AB)\\leq S(AA')-S(A)+S(BB')-S(B)$. But we know $E_p(A:B)=S(AA')=S(BB')$ and $S(ABA'B')=0$, \nsince according to the definition of entanglement of purification $\\rho_{ABA'B'}$ is a pure state. Using this in the above inequality,\nwe get $2E_p(A:B)\\geq I(A:B)$. Therefore, we have the following lower bound on $E_p(A:B)$\n\\begin{equation}\n E_p(A:B)\\geq \\frac{I(A:B)}{2}.\n\\end{equation}\nExtending this to the asymptotic limit, one easily obtains $E_p^{\\infty}(A:B)=E_{LO_q}(A:B)\\geq\\frac{I(A:B)}{2}$, by using the fact that the quantum mutual information is additive on \ntensor product of quantum states. The above lower bound was known for the entanglement of purification, but only in the asymptotic limit,\nand it was obtained from an operational point of view in \\cite{Terhal}. Here we obtain this bound for a single copy of $\\rho_{AB}$, \nand easily extend this to the asymptotic limit as $E_{LO_q}(A:B)\\geq \\frac{I(A:B)}{2}$ and get back the result given in Ref.\\cite{Terhal}. \nAlso, the lower bound given in \\cite{Terhal} for a single copy of $\\rho_{AB}$ is $E_f(A:B)$. However, we know that for some states one has \n$E_f(A:B)\\leq \\frac{I({A:B})}{2}$. Therefore, for these states we get a better lower bound\nfor a single copy of $\\rho_{AB}$. Now, we use the equation for entanglement of purification in terms of \nquantum mutual information to derive a lower bound for tripartite mixed states which is different from half of its quantum mutual information.\n\\vskip 10pt\n{\\textbf {Proposition 2}}:\nFor any pure or mixed tripartite quantum state:\n\\begin{equation}\n E_p(A:BC)\\geq S(A)-\\frac{1}{2}[S(A\\vert B)+S(A\\vert C)].\n\\end{equation}\n\n{\\textit {Proof}}:\nLet $\\vert\\Psi\\rangle_{ABCA'D'}$ be the optimal pure state for evaluating the entanglement of purification of $\\rho_{ABC}$. \nTherefore, we have $E_p(A:BC)= \\frac{I(AA':BCD')}{2}$.\nNote that the quantum mutual information of pure states satisfy the monogamy equality condition. Therefore, $E_p(A:BC)= \\frac{I(AA':B)}{2}+\\frac{I(AA':CD')}{2}$. \nAgain, the mutual information is non-increasing upon discarding of quantum systems, hence we have \n\\begin{align}\nE_p(A:BC)\\geq \\frac{I(A:B)}{2}+\\frac{I(A:C)}{2}.\n\\end{align} \nThis implies\n$E_p(A:BC)\\geq S(A)- (\\frac{S(A\\vert B)}{2}+\\frac{S(A\\vert C)}{2})$. In general, from the previous literature we know that $E_p(A:BC)\\geq \\frac{I(A:BC)}{2} $.\nHowever, for the states with $I(A:BC)\\leq I(A:B)+I(A:C)$, i.e, with the negative interaction information, we then have\n$E_p(A:BC)\\geq \\frac{I(A:B)}{2}+\\frac{I(A:C)}{2}\\geq\\frac{I(A:BC)}{2}$. Therefore, for these class of states, the entanglement of purification is upper and lower bounded as\n$S(A)\\geq E_p(A:BC)\\geq S(A)- (\\frac{S(A\\vert B)}{2}+\\frac{S(A\\vert C)}{2})$. \nExtending this to the asymptotic limit we obtain $E_{LO_q}(A:BC)\\geq \\frac{I(A:B)}{2}+\\frac{I(A:C)}{2}$, using the fact that quantum mutual information is \nadditive on tensor product of density matrices.\nWe note that the tripartite quantum states with negative interaction information are always \npolygamous for the quantum mutual information. Therefore, for these states, the above bound is always greater than the previous bound $\\frac{I(A:BC)}{2}$. This may give\na better lower bound than $\\frac{I(A:B)}{2}$ or the regularised classical mutual information \\cite{Terhal} for states consisting of quantum as well as classical correlations, \ndepending on the negativity of interaction information. One may extend this to the case of $n$ parties as well, such that for a $n$ partite density matrices $\\rho_{A_1A_2...A_n}$,\nwe get $E_p(A_1:A_2A_3..A_n)\\geq \\max[\\frac{(I(A_1:A_iA_j..)}{2}+\\frac{(I(A_1:A_kA_l..)}{2}]$ etc. where one takes all possible combinations of bipartitions between $A_1A_2...A_n$\n(keeping the node $A_1$ same for the reduced density matrices)\nto achieve the maximum value of the lower bound. Therefore, the quantum states with negative interaction information across any bipartition will have either the regularised \nclassical mutual information or this as the better lower bound than half of its quantum mutual information.\n\\vskip 10pt\n{\\textbf {Corollary}}:\nThe entanglement of purification for the class of tripartite mixed states satisfying the sub-additivity equality condition is given by $S(A)$.\n\\vskip 10pt\n{\\textit {Proof}}: From the previous paragraph we see that when $S(A\\vert B)+S(A\\vert C)=0$, we get $E_p(A:BC)\\geq S(A)$. But again, from the upper bound of\nentanglement of purification we have $E_p(A:BC)\\leq S(A) $. Therefore combining the above two equations, one obtains $E_p(A:B)=S(A)$ for the states which\nsatisfy the strong sub-additivity equality condition. Also, we know that mixtures\nof the tripartite mixed states each satisfying the strong sub-additivity equality condition and satisfying an additional constraint of biorthogonality if the third party is\ntraced out, satisfy the strong sub-additivity equality, and hence their entanglement of purification is also $S(A)$. Hence the proof. \nThe structure of the states obeying the sub-additivity equality condition has been precisely given in Ref.\\cite{Hayden}. There it was shown that every separable state can \nbe extended to a state that obeys the sub-additivity equality condition. Therefore, from these observations\nwe can comment that all separable states can be extended to a tripartite mixed state which has the maximum amount of total correlation as $S(A)$.\nFrom the viewpoint of the structure of the states \\cite{Hayden},\nthe structure states satisfying the SSA equality has been given as $ \\rho_{ABC} = \\bigoplus_j q_j\\rho_{Ab_j^L}\\otimes\\rho_{b_j^RC}$,\nwith states $\\rho_{Ab_j^L} $ on Hilbert space $ H_A\\otimes H_{b^L_j}$ and $\\rho_{b^R_jC}$ on $ H_{b^R_j}\\otimes H_C$ with probability distribution $q_j$. \nThus, all states of this form and all extensions of this class of states\nhave the maximal amount of total correlation given by the entanglement of purification as $S(A)$.\nNow we discuss the lower bound and exact values with some specific examples as given below.\n\n\\begin{figure}\n\\includegraphics[scale=0.63]{W_clscl_delta.pdf}\n\\caption{Difference between lower bounds for state $ p\\vert W\\rangle\\langle W\\vert +(1-p)[a\\vert 000\\rangle\\langle 000\\vert +(1-a)\\vert 111\\rangle \\langle 111\\vert]$.\nThe difference between the new lower bound and the previous one is always positive in this case.}\n\\end{figure}\n\n\\textit {Examples of exact values}:\n\nFirst we state the value of entanglement of purification for the following class of bipartite mixed states. \nThe entanglement of purification of the states satisfying the Araki-Lieb equality condition is $S(A)$.\nWe know $S(A)\\geq E_p(A:B)\\geq \\frac{1}{2}I(A:B)$. But $\\frac{1}{2}I(A:B)=S(A)+\\frac{1}{2}[S(B)-S(A)-S(AB)]$. The states satisfying the Araki-Lieb equality\ncondition have $S(B)-S(A)=S(AB)$. Then, we have $S(A)\\geq E_p(A:B)\\geq S(A)$. Therefore, $ E_p(A:B)= S(A)$ for these states.\nThe structure of states satisfying the Araki-Lieb equality condition is given in Ref.\\cite{Zhang}. There, it was shown that the states satisfy the Araki-Lieb equality condition\nif and only if the following conditions are satisfied. First, $\\cal{H_A}$ can be factorized as $\\cal {H_L}\\otimes\\cal{H_R}$ and secondly \n$\\rho_{AB}= \\rho_L\\otimes\\vert\\Psi_{RB}\\rangle\\langle\\Psi_{RB}\\vert$, where $\\vert\\Psi_{RB}\\rangle\\in\\cal{H_R}\\otimes\\cal{H_B}$. \nThe structure of such states that satisfy the Araki-Lieb equality condition is therefore of the form $\\rho_{AB}= \\rho_L\\otimes\\vert\\Psi_{RB}\\rangle\\langle\\Psi_{RB}\\vert$.\nTherefore, the value of entanglement of purification for these states is $S(A)$.\n\nFor the case of tripartite states, the entanglement of purification of states of the form \n$\\rho_{ABC}= p\\vert GHZ\\rangle\\langle GHZ\\vert^{\\underline{+}}+(1-p)[b\\vert 000\\rangle\\langle 000\\vert+(1-b)\\vert 111\\rangle \\langle 111\\vert]$\nis $S(A)$ for all values of $\\{p,a,b\\} \\in [0,1]$, where $\\vert GHZ\\rangle^{\\underline{+}} = \\sqrt{a}\\vert 000\\rangle \\underline{+} \\sqrt{(1-a)}\\vert111\\rangle$ is the \ngeneralized GHZ state \\cite{GHZ}. This holds for $n$ party as well, i.e., for the following state\n$\\rho_{ABC}= p\\vert GHZ_n\\rangle\\langle GHZ_n\\vert^{\\underline{+}} +(1-p)[b\\vert 0\\rangle\\langle 0\\vert^{\\otimes n} +(1-b)\\vert 1\\rangle \\langle 1\\vert^{\\otimes n}]$\nwhere $ \\vert GHZ_n\\rangle^{\\underline{+}} =\\sqrt{a}\\vert 0\\rangle^{\\otimes n} \\underline{+} \\sqrt{(1-a)}\\vert1\\rangle^{\\otimes n}$.\nThe proof is as follows. We know that for tripartite states $E_p(A:BC)\\geq \\frac{1}{2}[I(A:B)+I(A:C)]$. For the state given above, $I(A:B)+I(A:C)=2I(A:B)=2[S(A)+S(B)-S(AB)]=2S(A)$.\nThe first equality follows owing to the symmetry of the state between parties $B$ and $C$. The third equality follows from the fact that the nonzero eigenvalues of the density \nmatrices $\\rho_{AB}$ and $\\rho_{B}$ are exactly equal. Therefore, for the given state $ S(A)\\geq E_p(A:BC)\\geq S(A)$. Thus, $E_p(A:BC)=S(A)$. Let us consider another \nexample. The tripartite mixed state as a mixture of the  $\\vert GHZ\\rangle^+$ and $\\vert GHZ\\rangle^-$, i.e., if \n$\\rho_{ABC}= p\\vert GHZ\\rangle\\langle GHZ\\vert^+ +(1-p)\\vert GHZ\\rangle\\langle GHZ\\vert^-$ then it also has $E_p(A:B)= S(A)$ according to\nour previous argument. Here, the states are generalized $\\vert GHZ\\rangle$ states. And similar to the above,\nthis is also true for the arbitrary mixture of $n$ partite generalized $\\vert GHZ\\rangle$ states.\n\n\\textit {Examples of lower bounds}:\nAmong other examples, for the tripartite states of the form\n$ \\rho_{ABC}=p\\vert W\\rangle\\langle W\\vert +(1-p)[a\\vert 000\\rangle\\langle 000\\vert +(1-a)\\vert 111\\rangle \\langle 111\\vert]$,\nwhere $\\vert W\\rangle=\\frac{1}{\\sqrt{3}}[\\vert 100\\rangle+\\vert 010\\rangle+\\vert 001\\rangle]$ is the $\\vert W\\rangle$ state, a\nbetter lower bound\nis provided by $\\frac{1}{2}[I(A:B)+I(A:C)]\\geq \\frac{1}{2}I(A:BC)$, since the quantum mutual information is polygamous for these classes of states. This holds even for the regularised\nversion of the entanglement of purification, i.e., $E_{LO_q}(A:BC)\\geq \\frac{1}{2}[I(A:B)+I(A:C)]$, owing to the additivity of the \nquantum mutual information on tensor product of density matrices. The difference $\\Delta_{LB}$ between the two lower bounds equal to\n$\\frac{1}{2}[I(A:B)+I(A:C)-I(A:BC)]$ is plotted in Fig 1, which shows that it is always positive. Again we may consider the state\n$ \\rho_{ABC}= p\\vert W\\rangle\\langle W\\vert +\\frac{(1-p)}{8}I_3$ and the difference between the lower bounds are plotted in Fig 2.\n\n\\begin{figure}\n\\includegraphics[scale=0.83]{W_Id_delta1.pdf}\n\\caption{Difference between lower bounds for state $ \\rho= p\\vert W\\rangle\\langle W\\vert +\\frac{(1-p)}{8}I_3$. The difference between the new and the old lower bound is always positive\nhere. The difference in lower bounds is given by the amount of polygamy of quantum mutual information.}\n\\end{figure}\n\nOne can use the polygamy of the quantum mutual information to lower bound the entanglement of purification in higher dimensional bipartite states. If a sub-party is of higher \ndimension, and if the quantum mutual information is polygamous for the lower dimensional subparts obtained by breaking the higher dimensional subparty, then it gives  a better\nlower bound for the entanglement of purification than just half of the quantum mutual information of the state $\\rho_{AB}$. \n\nSuppose for a $ 2^n$ dimensional party $B$ in $\\rho_{AB}$, we break it down into two lower dimensional subparties $B_1$ and $B_2$ \\cite{Cornello}. Then, from Eq(8) we have\n$E_p(A:B)\\geq \\frac{1}{2}[I(A:B_1)+I(A:B_2)]$.\nFor negative interaction information between $B_1$ and $B_2$, i.e., $S(AB_1)+S(AB_2)+S(B_1B_2)-S(A)-S(B_1)-S(B_2)-S(AB_1B_2)< 0$, \nthe R.H.S is greater than $\\frac{I(A:B)}{2}$ \\cite{gio}. Thus it gives a better lower bound. We may say \nthat this better lower bound arises as a result of a second order polygamy relation of quantum mutual information. \nOne can easily extend to the asymptotic limit as well, thus we obtain the lower bound  \n$E_{LO_q}(A:B)\\geq \\frac{1}{2}[I(A:B_1)+I(A:B_2)]> \\frac{I(A:B)}{2}$. For these states $ E_{LO_q}(A:B)$ quantifies more correlation than $\\frac{I(A:B)}{2}$ as given in the original paper.\nFor these states, one now has to compare the quantity $\\frac{1}{2}[I(A:B_1)+I(A:B_2)]$ with the classical mutual information for obtaining a better lower bound.\nThe above equation can also be written as \n\\begin{equation}\nE_p(A:B)\\geq S(A)-\\frac{1}{2}[S(A\\vert B_1)+S(A\\vert B_2)]. \\nonumber\n\\end{equation} \nFrom this equation we can say that for the $2^n$ dimensional party $B$ \nin the bipartite state $\\rho_{AB}$, if the internal structure of $B$ is such that across any subpartition inside it, the sub-additivity equality condition is satisfied then \nthe entanglement of purification of that state is $S(A)$. Therefore, with the aid of the new lower bound as half of the summation of the quantum mutual information of the subparties, \nwe are able to conclude about the new exact values of entanglement of purification for these classes of the higher dimensional bipartite states.\n\n\n\\section{Monogamy and polygamy of entanglement of purification}\n\nHere we explore various conditions under which the entanglement of purification will be polygamous or monogamous for pure and mixed states.\n\n\\subsection{ Monogamy and polygamy of entanglement of purification for pure tripartite states}\n\\vskip 10pt\n{\\textbf {Theorem 1}}:\nThe entanglement of purification is polygamous for a tripartite pure state $\\rho_{ABC}$:\n\\begin{equation}\nE_p(A:B)+E_p(A:C)\\geq E_p(A: BC).\n\\end{equation}\n\\textit {Proof}:\nFrom Eq(6) we know that $E_p(A:B)\\geq \\frac{I(A:B)}{2}$. Therefore, we have\n$E_p(A:B)+E_p(A:C)\\geq \\frac{I(A:B)}{2}+\\frac{I(A:C)}{2}$. In\ncase of the tripartite pure state $\\rho_{ABC}$ the right hand side of the inequality just gives $S(A)$. This implies that\n\\begin{equation}\n E_p(A:B)+E_p(A:C)\\geq S(A).\\nonumber\n\\end{equation}\nSince for pure tripartite state $\\rho_{ABC}$, $E_p(A: BC)=S(A)$, we obtain:\n\\begin{equation}\n E_p(A:B)+E_p(A:C)\\geq E_p(A:BC).\\nonumber\n\\end{equation}\nThis shows the polygamous nature of the entanglement of purification for pure tripartite state $\\rho_{ABC}$.\nOne can directly see that the same relation holds for the regularised entanglement of purification:\n$E_{LO_{q}}(A:B)+E_{LO_{q}}(A:C)\\geq E_{LO_{q}}(A:BC)$, i.e., the regularised entanglement of purification is also a polygamous quantity.\nThis proves that entanglement of purification for any tripartite pure state is in general a polygamous quantity.\nAn implication of this is that the sum of the asymptotic entanglement cost of preparing $\\rho_{AB}$ and $\\rho_{AC}$ will \nnot be restricted by the asymptotic cost of preparing $\\rho_{A:BC}$.\n\nThe polygamy inequality above shows that there can be states satisfying the equality condition in the inequality. To analyse the states that may satisfy the equality condition we \nfind a following relation to the monogamy of entanglement of formation for those states.\nGiven a pure state $\\rho_{ABC}$, if entanglement of formation violates monogamy, then entanglement of purification will\nviolate monogamy equality for the same. However the converse is not true.\nThe proof is as follows. If entanglement of formation $E_f(A:BC)$ violates monogamy for some pure state $\\rho_{ABC}$, then we have\n$E_f(A:BC)< E_f(A:B)+E_f(A:C)$.\nBut for a pure state $\\rho_{ABC}$, we know that $E_f(A:BC)=E_p(A:BC)$. Therefore, replacing this\nin the above equation we get $ E_p(A:BC)< E_f(A:B)+E_f(A:C)$. Also, it is known that for \nany state $\\rho_{AB}$, we have $E_f(A:B)\\leq E_p(A:B) $.\nThis implies $E_p(A:BC)< E_p(A:B)+E_p(A:C)$\nwhich shows that the entanglement of purification also violates monogamy. Hence the proof. However the vice versa may not be true.\nWe know that for pure states the monogamy of entanglement of formation is equivalent to\nthe monogamy of quantum discord \\cite{gio}. Therefore, we conclude that the polygamy of quantum discord will also imply the polygamy of entanglement of purification likewise.\nIn other words, monogamy of entanglement of formation or quantum discord is a necessary condition for the tripartite state $\\rho_{ABC}$ to satisfy the \nmonogamy equality condition for entanglement of purification.\nNow let us try to compare the monogamy inequality of the entanglement of formation with the entanglement of purification for\nmixed tripartite state $\\rho_{ABC}$. Before that, we define a quantity called correlation of classical and quantum origin $E_{cq}(A:B)$ of the state $\\rho_{AB}$ as\n\\begin{align}\nE_{cq}(A:B) = E_{p}(A:B) - E_{f}(A:B).\\nonumber\n\\end{align}\nThis quantity is positive for mixed states and vanishes for pure bipartite states. Intuitively, this may contain some classical \ncorrelation and some amount of quantum correlation beyond entanglement that is captured by the entanglement of formation.\nFrom the definition it is clear that for a \ngiven mixed state $\\rho_{ABC}$, if $E_{cq}(A:B)$ and $E_{f}(A:B)$ are monogamous (polygamous), then the entanglement of purification will be (monogamous) polygamous.\nOne can also show that for three-qubit states if the the correlation of classical and quantum origin obeys monogamy and entanglement of formation satisfies \\cite{fanch} \n \\begin{align}\nE_{f}(A:B) + E_{f}(A:C) \\le 1.18\\nonumber\n\\end{align}\nthen the entanglement of purification will obey a weak monogamy relation as given by\n\\begin{align}\nE_{p}(A:B) + E_{p}(A:C) \\le   E_{p}(A:BC) + 1.18.\n\\end{align}\n\n\\subsection{Mixed states}\n\nThe entanglement of purification $E_p(A:BC)$ of a mixed tripartite state $\\rho_{ABC}$ is $\\frac{I(AA':BC(BC)')}{2}$, where the optimal pure state of $\\rho_{ABC}$ is\n$\\vert\\Psi_{ABCA'(BC)'}\\rangle$. Similarly, the entanglement of purification $E_p(A:B)$ of $\\rho_{AB}$ is $\\frac{I(AA'':BB'')}{2}$, where the optimal pure state for $\\rho_{AB}$ is\n$\\vert\\Phi_{ABA''B''}\\rangle$,  and the entanglement of purification $E_p(A:C)$ of $\\rho_{AC}$ is $\\frac{I(AA''':CC''')}{2}$, where the optimal pure state for $\\rho_{AC}$ is\n$\\vert\\xi_{ACA'''C'''}\\rangle$. Therefore, the monogamy inequality for a mixed tripartite state $\\rho_{ABC}$ is\n$I(AA':BC(BC)')\\geq I(AA'':BB'')+I(AA''':CC''')$. But owing to the largely difficult optimization needed, we may not be able to check this equation directly. Instead, we analyze \nsome specific cases of mixed states that are polygamous for entanglement of purification as follows.\n\nAt first we note that the tripartite mixed states satisfying the strong sub-additivity equality condition are polygamous for entanglement of purification. To see this,\nlet $\\vert\\Psi_{ABA'B'}\\rangle$ and $\\vert\\Psi_{ACA''C''}\\rangle$ be the optimal pure states for $\\rho_{AB}$ and $\\rho_{AC} $ respectively.\nThen $E_p(A:B)+E_p(A:C)\\geq \\frac{1}{2}[I(A:B)+I(A:C) + I(AA':B')+I(AA'':C'')]$. But $ I(A:B)+I(A:C) = 2S(A)-(S(A\\vert B)+S(A\\vert C))$,\nand if the strong sub-additivity equality condition is satisfied then we have  $S(A\\vert B)+S(A\\vert C) = 0$. \nPutting these in the equation, we get $ E_p(A:B)+E_p(A:C)\\geq S(A) + \\frac{1}{2}[I(AA':B')+I(AA'':C'')] $.\nBut the last two terms on the R.H.S are positive in general, as the quantum mutual information is always positive and vanishes only for the maximally mixed state. \nAlso, we know $E_p(A:BC)= S(A)$. Thus, combining these inequalities together we obtain \n$ E_p(A:B)+E_p(A:C)\\geq E_p(A:BC)$. Thus, the entanglement of purification is polygamous for the class of states that satisfy the strong sub-additivity equality.\nAmong other classes of states, if anyone of the reduced density matrices $\\rho_{AB}$, $\\rho_{AC}$  of a mixed state $\\rho_{ABC}$ are entirely supported on\nthe symmetric or antisymmetric subspaces, then the state will violate monogamy of entanglement of purification.\nThis follows from the result by Winter \\textit{et al}.\\cite{Matthias1}. The entanglement of purification of such bipartite density matrices (with the same dimension for both parties) is \n$S(A)$. But the entanglement of purification of the tripartite mixed state is also $S(A)$ and in general $E_p(A:C)\\geq 0$. Therefore, the polygamy inequality follows directly by \ncombining the above observations. Also, any tripartite extension of bipartite mixed states that satisfy the Araki-Lieb equality condition for their von-Neumann entropy\nis polygamous for entanglement of purification. We know that the states that satisfy the Araki Lieb equality condition have $E_p(A:B)= S(A)[S(B)]$. \nHowever, the other reduced density matrix has some non zero correlation and therefore non-zero entanglement of purification. Thus, in this case we have \n$E_p(A:B)+E_p(A:C)\\geq S(A)=E_p(A:BC)$, making entanglement of purification a polygamous measure of total correlation. Though for pure tripartite states we could prove the general \npolygamy inequality, for mixed states it is not clear whether such general inequality exists or not.\n\nNext, we discuss the relation to polygamy of quantum mutual information. \nSuppose $E_p(A:B)+E_p(A:C)= S(B\\vert A)+S(C\\vert A)+I(A:B)+I(A:C)$. This is greater than $ S(BC\\vert A)+I(A:B)+I(A:C)$ which is again\ngreater than $ E_p(A:BC)+I(A:B)+I(A:C)-I(A:BC)$. From the above equations one can see that if the mutual information is polygamous,\nthen here the entanglement of purification becomes polygamous. Again, a sufficient condition for monogamy of $E_p$\nis  $\\frac{I(A:BC)}{2}\\geq E_p(A:B)+E_p(A:C)$. This implies $\\frac{I(A:BC)}{2}\\geq \\frac{I(A:B)}{2}+\\frac{I(A:C)}{2}$, which is nothing but\n$I(A:BC)\\geq I(A:B)+I(A:C)$, i.e., the monogamy inequality for the quantum mutual information. This says that the states satisfying this particular sufficient\ncondition for monogamy of $E_p$ will also satisfy the monogamy inequality of quantum mutual information. \n\n\\subsection{Polygamy of entanglement of purification for multiparty}\n\nNow we investigate the polygamy of entanglement of purification in case of $n$ partite density matrices. The conditions for the polygamy for mixed states\nalso get translated here as sufficient conditions for polygamy. To put it in other words, the $n$-partite density matrices, pure or mixed, are polygamous if\nany one of the reduced density matrices of the subsystem satisfy the Araki-Lieb equality condition, strong sub-additivity equality condition or is supported on the symmetric\nor antisymmetric subspace. Now we state a simple sufficient condition for the polygamy of entanglement of purification and construct some examples.\n\\vskip 10pt\n{\\textbf{Proposition 3}}: All the $n$-partite states, pure or mixed with $\\sum_{i=1}^n I(A:A_i)\\geq 2S(A)$ are polygamous for entanglement of purification.\n\\vskip 10pt\n\n\\textit{Proof}: We have  $\\sum_{i=1}^n E_p(A:A_i)\\geq\\frac{1}{2}[\\sum_{i=1}^n I(A:A_i)]$. From this we get\n$\\sum_{i=1}^n E_p(A:A_i)\\geq S(A)+\\frac{1}{2}[\\sum_{i=1}^n I(A:A_i)]-2S(A)$. Thus, we get the condition in the proposition as the sufficient condition for polygamy of\nentanglement of purification. A large number of states will satisfy this condition, and thus will be polygamous. However, some states will violate this condition,\nand it will be inconclusive about the polygamous nature in case of those states. \n\nUsing the above relation, we easily see that\nthe $n$-party generalized $\\vert GHZ\\rangle $ and the $n$-party $ \\vert W\\rangle$ states are polygamous with respect to the entanglement of purification. We can explicitly see the\nproofs as follows. We have the generalized $GHZ$ state as $\\vert GHZ\\rangle = \\sqrt{p}\\vert 0\\rangle^{\\otimes n} + \\sqrt{1-p}\\vert 1\\rangle^{\\otimes n}$, where $0\\leq p\\leq1$ \n\\cite{GHZ}.\nBut we have obtained before that for tripartite pure states, $ E_p(A:A_1)+E_p(A:A_2)\\geq S(A)$. Thus, it holds true for the tripartite generalized $\\vert GHZ\\rangle$ states as well.\nNow for $n\\geq 3$, we see that all the reduced density matrices are exactly the same and L.H.S becomes $\\sum_{i=1}^n E_p(A:A_i)$. \nThis is nothing but $ E_p(A:A_1)+E_p(A:A_2)+\\sum_{i=3}^n E_p(A:A_i)$. Since each of the two party reduced density matrices are exactly the same as the two party reduced density matrices \nin the case of tripartite pure state, therefore using the above two equations we obtain $\\sum_{i=1}^n E_p(A:A_i)\\geq S(A)+\\sum_{i=3}^n E_p(A:A_i)$. The last term on R.H.S is \nalways positive. Therefore we obtain $ \\sum_{i=1}^n E_p(A:A_i)\\geq S(A)$, rendering the entanglement of purification polygamous for all $n$ in the case of generalized \n$\\vert GHZ\\rangle$ state. This is expected since every reduced density matrices share only classical correlation with the other reduced density matrices.\nWe now consider $\\vert W\\rangle = \\frac{1}{\\sqrt{n}}[\\vert 10..0\\rangle + \\vert 01..0\\rangle + .. ]$, where there are $n$ terms within the parenthesis \\cite{Dur}.\nWe show that this state is also polygamous for all values of $n$. To see this, first we note that all the two party reduced density matrices $\\rho_{AA_i}$ of this state \nare exactly same due to the symmetry of the state. Specifically each $\\rho_{AA_i} = \\frac{1}{n}[(n-2)\\vert 00\\rangle\\langle 00\\vert]+2\\vert\\Phi^+\\rangle\\langle\\Phi^+\\vert]$, where\n$\\vert\\Phi^+\\rangle=\\frac{1}{\\sqrt{2}}[\\vert 10\\rangle+\\vert 01\\rangle]$ is the Bell state. Now we calculate $\\frac{1}{2}[\\sum_{i=1}^n I(A:A_i)]=\\frac{n}{2}I(A:A_1)$, since \nall the two party reduced density matrices are same. Evaluating the eigenvalues in terms of $n$, we find that $S(A)=S(A_1)=2\\log_2 n-\\log_2(n-1)$ and $S(AA_1)=2\\log_2 n-1-\\log_2(n-2)$.\nPutting these values in the equation above, we get\n$\\frac{n}{2}I(A:A_1)-S(A)=\\frac{n}{2}+\\frac{n}{2}\\log_2 (n-2) +(n-1)\\log_2 \\frac{n}{n-1}$. This value is always positive for all values of $n>2$. \nThus combining the earlier result of tripartite pure state with the above finding, we conclude that the entanglement of purification is polygamous \nfor $n$ party $\\vert W\\rangle$ state. \n\nLikewise the case for mixed states, where we state some conditions relating monogamy of entanglement of purification with that of quantum mutual information,\nwe now state a proposition connecting the polygamy of quantum mutual information to the polygamy of entanglement of purification for a pure state of $n$ parties.\n\\vskip 10pt\n{\\textbf{Proposition 4}}: All the $n$ party pure states for which the quantum mutual information is $(n-1)$ partite polygamous for at least any one of the $(n-1)$ party reduced density \nmatrices of the pure state, is $n$ partite polygamous for both the entanglement of purification as well as the quantum mutual information.\n\\vskip 10pt\n{\\textit{Proof}}: Note that for $n$ partite pure state, we have $\\sum_{i=2}^n E_p(A_1:A_i)\\geq \\frac{1}{2}\\sum_{i=2}^n I(A_1:A_i)$. Now, let us take a reduced density\nmatrix $\\rho_{A_1A_2...A_{n-1}}$ to be polygamous for quantum mutual information, i.e., $\\sum_{i=2}^{n-1} I(A_1:A_i)\\geq I(A_1:A_2...A_{n-1})$. Then, we have \n$I(A_1:A_n)+\\sum_{i=2}^{n-1} I(A_1:A_i)\\geq I(A_1:A_n)+I(A_1:A_2...A_{n-1})$. Since the $n$ partite quantum state we are considering is a pure state,\ntherefore by virtue of monogamy of quantum mutual information, the R.H.S. of this equation is nothing but $ I(A_1:A_2A_3...A_n)$. But, we know for a pure state \n$ I(A_1:A_2A_3...A_n)=2S(A_1)$. From here it then follows that $\\sum_{i=2}^n E_p(A_1:A_i)\\geq S(A_1)$ and also $\\sum_{i=2}^n I(A_1:A_i)\\geq 2S(A_1)$. \nThese two equations are just the equations of polygamy for the entanglement of purification and the quantum mutual respectively for a $n$ partite pure state. It is easy to see that\none could take any one of the possible $(n-1)$ different reduced density matrices possible of the $n$ partite pure state (keeping the node $A_1$ intact for each reduced density matrix)\nas the one polygamous for the quantum mutual information and eventually get back the polygamy equation for both the entanglement of purification and quantum mutual information.\nAs a specific example of this proposition, we easily see that all the four party pure states with negative interaction information across any two pair of its bipartite reduced density \nmatrices, are polygamous for entanglement of purification.\n\n\n\\section{ Sub-additivity on tensor products}\nAdditivity is a desirable property to hold for a given measure of total correlation. Quantum mutual information is an additive measure of correlation, however entanglement of \npurification may not be an additive measure. Using strong numerical support this has been shown in Ref.\\cite{Chen}. Here we prove that if it is non-additive then it has to be a\nsub-additive quantity. We have the following theorem.\n\\vskip 5pt\n{\\textbf {Theorem 2}}:\nThe entanglement of purification is sub-additive in the tensor product of density matrices, i.e., for a tensor product density matrix ${\\rho_{AB}\\otimes\\sigma_{CD}}$, the following equation\nholds\n\\begin{equation}\n E_p(AC: BD)\\leq E_p(A: B)+E_p(C: D).\\nonumber\n\\end{equation}\nwith equality if and only if the optimal pure state for the tensor product of density matrices is\nthe tensor product of optimal pure states of the corresponding density matrices upto a local unitary equivalence.\n\\vskip 10pt\n{\\textit {Proof}}:\nLet us suppose $\\vert\\Psi_{ABA'B'}\\rangle$ and $\\vert\\Phi_{CDC'D'}\\rangle$ are the optimal purification for $\\rho_{AB}$ and $\\sigma_{CD}$ \ncorresponding to the value of entanglement of \npurification. Then $ \\vert\\Psi_{ABA'B'}\\rangle\\otimes\\vert\\Phi_{CDC'D'}\\rangle $ is a valid purification for $\\rho_{AB}\\otimes\\sigma_{CD} $, \nhowever not generally the optimal one. Now, we know \nthat $ E_p(A: B)=\\frac{I(AA': BB')}{2}$ and $ E_p(C: D)=\\frac{I(CC': DD')}{2}$ . \nAdding these two quantities we get $E_p(A: B)+E_p(C: D)=\\frac{I(AA': BB')}{2}+\\frac{I(CC': DD')}{2}$.\nBut the quantum mutual information is additive on tensor product of quantum states. Therefore, $\\frac{I(AA': BB')}{2}+\\frac{I(CC': DD')}{2}=\\frac{I(AA'CC': BB'DD')}{2}$ \nwhere $I(AA'CC': BB'DD')$ is the quantum mutual information of the state $\\vert\\Psi_{ABA'B'}\\rangle\\otimes\\vert\\Phi_{CDC'D'}\\rangle$. Thus, we have\n\\begin{align} \\nonumber E_p(A: B)+E_p(C: D) =\\frac{I(AA'CC': BB'DD')}{2}.\\nonumber\n\\end{align}\nSince $\\vert\\Psi_{ABA'B'}\\rangle\\otimes\\vert\\Phi_{CDC'D'}\\rangle$ is only one such purification of $\\rho_{AB}\\otimes\\sigma_{CD}$ \nand the optimization for $E_p(AC:BD)$ is over all possible purifications of $\\rho_{AB}\\otimes\\sigma_{CD}$ denoted by the set of pure states \n$\\{\\vert\\xi_{ABCDA''B''}\\rangle\\}$, therefore we have\n\\begin{align}\\nonumber\n \\min_{A''B''} \\frac{I(ACA'': BDB'')}{2}\n \\leq \\frac{I(ACA'C': BDB'D')}{2}, \\nonumber\n\\end{align}\nwhere $I(ACA'': BDB'')$ is the quantum mutual information of any such purification $ \\vert\\xi_{ABCDA''B''}\\rangle\\ $ and the minimum is over all such purification of \n$\\rho_{AB}\\otimes\\sigma_{CD}$ by the addition of ancilla part $A''B''$ to it. Hence we easily see that the above equation is nothing but the following inequality,\n\\begin{align}\\nonumber\nE_p(AC: BD) \\nonumber\n\\leq \\frac{I(ACA'C':BDB'D')}{2},\\nonumber\n\\end{align}\nwhich directly implies that, $E_p(AC: BD)\\leq E_p(A:B)+E_p(C:D)$ for the four partite tensor product density matrix $\\rho_{AB}\\otimes\\sigma_{CD}$.\nNow, in the following paragraph we check the equality condition. \n\nWhile checking the equality condition, we now omit the subscripts and write $\\vert\\Psi_{ABA'B'}\\rangle $ \nas $\\vert\\Psi\\rangle$, $\\vert\\Phi_{CDC'D'}\\rangle$ as $ \\vert\\Phi\\rangle$ and $\\vert\\xi_{ABCDA''B''}\\rangle$ as $\\vert\\xi\\rangle$ for simplicity.\nFirst, we check that if $ \\vert\\xi\\rangle=\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle$, then whether the \ndimensionality of the optimal purifying state agrees with the dimension  of the Hilbert space of the ancilla part, as given in Ref.\\cite {Terhal}. \nWe note that if $ \\vert\\xi\\rangle=\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle$, then $d_{A''}(\\vert\\xi\\rangle)=d_{A'}(\\vert\\Psi\\rangle)d_{C'}(\\vert\\Phi\\rangle),\nd_{B''}(\\vert\\xi\\rangle)=d_{B'}(\\vert\\Psi\\rangle)d_{D'}(\\vert\\Phi\\rangle)$.\nAccording to the theorem given in Ref.\\cite {Terhal}, $d_{A'}(\\vert\\Psi\\rangle)=d_{AB}(\\rho_{AB})$, $d_{C'}(\\vert\\Phi\\rangle)= d_{CD}(\\sigma_{CD})$ and \n$ d_{A''}(\\vert\\xi\\rangle)=d_{ABCD}(\\rho_{AB}\\otimes\\sigma_{CD})$. Similarly by the same theorem, we have \n$d_{B'}(\\vert\\Psi\\rangle)=d_{AB}^2(\\rho_{AB})$, $d_{D'}(\\vert\\Phi\\rangle)= d_{CD}^2(\\sigma_{CD})$ and \n$ d_{B''}(\\vert\\xi\\rangle)=d_{ABCD}^2(\\rho_{AB}\\otimes\\sigma_{CD})$. Now, we verify if the above two equations are consistent with dimensions proposed in Ref.\\cite{Terhal}\nfor $\\vert\\xi\\rangle$. Putting the values of $d_{A'}$ and $d_{B'}$ in terms of $d_{AB}$, we get \n$d_{A''}(\\vert\\xi\\rangle)=d_{AB}(\\rho_{AB})d_{CD}(\\sigma_{CD})$ and $d_{B''}(\\vert\\xi\\rangle)=d_{AB}^2(\\rho_{AB})d_{CD}^2(\\sigma_{CD})$. These values can be reframed as\nthe dimensions of the tensor product of the corresponding density matrices, i.e., $d_{AB}(\\rho_{AB})d_{CD}(\\sigma_{CD})= d_{ABCD}(\\rho_{AB}\\otimes\\sigma_{CD})$. Similarly \n$d_{AB}^2(\\rho_{AB})d_{CD}^2(\\sigma_{CD})= d_{ABCD}^2(\\rho_{AB}\\otimes\\sigma_{CD})$. This holds true even when \n$\\vert\\xi\\rangle=U_{A'C'}\\otimes U_{B'D'}\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle$, since the unitary matrices do not map density matrices from Hilbert space of a given dimension\nto that of a different dimension. \nThis shows that the dimensions are in agreement with those given by the theorem in \nRef.\\cite{Terhal}.\n\nWe now move on to the equality condition for the mutual information. \nFor this purpose, let us note that if $\\vert\\xi\\rangle=U_{A'C'}\\otimes U_{B'D'}\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle$, \nthen owing to the additivity of quantum mutual information and its invariance under the action of local unitaries, \none has $I(ACA'':BDB'')=I(AA':BB')+I(CC'':DD')$, where the mutual information terms are that of $\\vert\\xi\\rangle$, $\\vert\\Psi\\rangle$, \nand $\\vert\\Phi\\rangle$ respectively.\nThis implies that $E_p(AC:BD)=E_p(A:B)+E_p(C:D)$ for $\\rho_{AB}\\otimes\\sigma_{CD}$. This proves the if part of \ntheorem above. \n\nFor the only if condition we see that if $ \\vert\\xi\\rangle\\neq U_{A'C'}\\otimes U_{B'D'}\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle$, \nthen $I(\\vert\\xi\\rangle)\\neq I(\\vert\\Psi\\rangle\\otimes\\vert\\Phi\\rangle)$. This is because, the action of a non-local unitary will \nchange the probability distributions of the reduced density matrices and thus will change the value of quantum mutual information across $ACA'C':BDB'D'$ partition. As a result, \nthe equality holds only if the optimal pure state for the tensor product of the density matrices is the tensor \nproduct of corresponding optimal pure states, upto the local unitary equivalence $U_{A'C'}\\otimes U_{B'D'}$. \nTherefore, we see that if the entanglement of purification is non-additive, it is actually sub-additive. Thus, the above theorem rules out the \nsuper-additivity of entanglement of purification. The sub-additivity has been shown numerically in Ref.\\cite{Chen} for the Werner states.\nIt is important to note that, according to the result by authors in Ref.\\cite{Terhal}, one is guaranteed to find the optimal pure state in the Hilbert space of the aforementioned \ndimensionality. In that case our equality condition holds for the tensor product of the optimal pure states. However it does not rule out the existence of optimal pure states in \nHilbert space of other dimensions. Thus, in addition to the optimal pure state in Hilbert space of the dimensions given by the theorem, one may find other optimal pure states in \nHilbert space of higher or lower dimension. In particular, one might be able to find optimal pure state in Hilbert space of lower dimension.\nAs an example we have the Werner state and its optimal pure state for entanglement of purification can be found in Hilbert space of dimensions $4\\times 4$ as proved numerically\nin Ref.\\cite{Terhal}.\n\nUsing the results we have obtained on entanglement of purification, \nwe identify the classes of states that are additive on tensor products for the entanglement of purification as follows. We see that\nthe bipartite states satisfying the equality condition in Araki-Lieb inequality, the higher dimensional bipartite states satisfying the equality condition in strong sub-additivity\nwhen any party of it can be broken down into two lower dimensional subparties, the tripartite states satisfying the strong sub-additivity equality condition\nare additive on tensor products for entanglement of purification.\nThus, for the above class of states, the regularised entanglement of purification and their optimal visible\ncompression rate is given by the entanglement of purification. Apart from this, we are able to also draw the conclusion that\nthe entanglement of purification is additive on tensor products if and only if it is also super-additive on tensor products for all quantum states.\nHowever, whether there can be states $\\rho_{AB}\\otimes\\sigma_{CD}$ for which $ E_p(AC:BD) < E_p(A:B)_{\\rho_{AB}}+E_p(C:D)_{\\sigma_{CD}}$ is still an open question. We note that the \nquestion of non-additivity is now reduced to only the sub-additivity condition, ruling out the possibility of\n$E_p(AC:BD) > E_p(A:B)_{\\rho_{AB}}+E_p(C:D)_{\\sigma_{CD}}$ for $\\rho_{AB}\\otimes\\sigma_{CD}$.\n\n\n\\section{ Implications on the quantum advantage of dense coding}\n\nQuantum dense coding is a quantum communication protocol where one sends classical information beyond the classical capacity of the\nquantum channel with the help of a quantum state shared between two distant observers, and a noiseless quantum channel. The quantum advantage of dense coding is the\nincrease in the rate of classical information transmission due to shared entanglement. Mathematically, the quantum advantage of dense coding of a quantum state $\\rho_{AB}$is \ndefined in terms of the\ncoherent information as $\\Delta(A\\rangle B) = S(B)-inf_{\\Lambda_A}S[(\\Lambda_A\\otimes I_B)\\rho_{AB}]=sup_{\\Lambda_A}I'(A\\rangle B)$, \nwhere the infimum or supremum is performed over all the maps $\\Lambda_A$ acting on the state $\\rho_{AB}$ and $ I'(A\\rangle B)=S(B)-S(AB)$ is the coherent information of $\\rho_{AB}$.\nThere, it was proved that the quantum advantage of dense coding is a non-negative quantity. Again, \na quantum state is said to be dense codeable if the above quantity $ \\Delta(A\\rangle B)$ is strictly positive.\n It was shown in the paper by Horodecki \\cite{Horodecki} that it suffices to consider\nonly the extremal TPCP maps in evaluating the infimum or supremum for the above quantity, owing to the concavity of the von-Neumann entropy. It was also shown that the\nquantum advantage of dense coding may be non-additive, though not proved definitely. Apart from the aforementioned properties, the quantum advantage of dense coding\nwas shown to obey a monogamy relation with the entanglement of purification as $ S(B)\\geq \\Delta(A\\rangle B)+E_p(B:C)$ \\cite{Horodecki}, for any tripartite state $\\rho_{ABC}$, with equality\nfor pure tripartite states.\n\nTherefore, from the monogamy inequality and the polygamy of\nentanglement of purification for pure tripartite states as well as some of the mixed tripartite states mentioned here previously, it follows that\n\\begin{equation}\n\\triangle(B\\rangle A)+\\triangle(C\\rangle A)\\leq \\triangle(BC\\rangle A),\\nonumber\n\\end{equation}\nimplying that the quantum advantage of dense coding is strictly monogamous for the tripartite pure states as well as the other tripartite mixed states mentioned previously. \nThis property is straight forwardly carried over to the asymptotic limit as well.\nThus, we have $\\triangle^{\\infty}(B\\rangle A)+\\triangle^{\\infty}(C\\rangle A)\\leq \\triangle^{\\infty}(BC\\rangle A)$ for those same set of states. Also, it is easy to see that for the \nmixed states satisfying SSA equality \ncondition, the symmetric (antisymmetric) subspace condition and the states satisfying the Araki-Lieb equality condition and the cases for the $n$ partite pure states, monogamy is \nfollowed.\n\nIn the same way as that of the entanglement of purification, we conclude that the quantum advantage of dense coding is super-additive on tensor product of density matrices, i.e., \nfor a four partite tensor product state $\\rho_{AB}\\otimes\\sigma_{CD}$, we have the following equation\n\\begin{equation}\n \\Delta(AC\\rangle BD)\\geq \\Delta(A\\rangle B) +\\Delta(C\\rangle D).\\nonumber\n\\end{equation}\nThe proof is as follows. By definition, we have $\\Delta(A>B)=sup_{\\Lambda_A}I'(A>B)$. Thus, for the density matrix $\\rho_{AB}\\otimes\\sigma_{CD}$ we have \n$\\Delta(A\\rangle B)+\\Delta(C\\rangle D)=sup_{\\Lambda_A}I'(A\\rangle B)+sup_{\\Lambda_C}I'(C\\rangle D)= \nsup_{\\Lambda_{A}\\otimes\\Lambda_{C}}I'(AC\\rangle BD)$.\nThe second equation follows from the fact that the von-Neumann entropies are additive on tensor products of density matrices. Again for $\\rho_{AB}\\otimes\\sigma_{CD}$, by definition we have \n$\\Delta(AC\\rangle BD)=sup_{\\Lambda_{AC}}I'(AC\\rangle BD)$.\nHowever, the optimization for $\\rho_{AB}\\otimes\\rho_{CD}$ is over all $\\Lambda_{AC}$, and $\\{\\Lambda_A\\otimes\\Lambda_C\\}$ is only a subset of $\\{\\Lambda_{AC}\\}$. Thus, \n$sup_{\\Lambda_{AC}}I'(AC\\rangle BD\\geq sup_{\\Lambda_A\\otimes\\Lambda_C}I'(AC\\rangle BD)$ for the same four partite product state $\\rho_{AB}\\otimes\\sigma_{CD}$.\nWith the last equation we arrive at the super-additivity equation\nfor the quantum advantage of dense coding for tensor product states of the form $\\rho_{AB}\\otimes\\sigma_{CD}$, \ni.e., $\\Delta(AC\\rangle BD)\\geq \\Delta(A\\rangle B) +\\Delta(C\\rangle D)$ for $\\rho_{AB}\\otimes\\sigma_{CD}$.\n\n\nNot only super-additivity, but the monogamy inequality with entanglement of purification has other implications on the quantum advantage of dense coding as well. \nFrom the lower bound and some of the actual value of entanglement of purification, \nusing the property of monogamy with it and non-negativity of the quantum advantage of dense coding, we can identify some of the quantum states that have no \nquantum advantage of dense and also put an upper bound on it for some specific cases.\n\nLet $\\rho_{ABCD}$ be a quantum state, such that the sub-additivity equality condition is satisfied for the reduced density matrix $\\rho_{ABC}$, i.e., $S(B\\vert A)+S(B\\vert C)=0$.\nThen, from the monogamy inequality with entanglement of purification, we get $ S(B)\\geq \\Delta(D\\rangle B)+E_p(B:AC)$. But, in this case $E_p(B:AC)=S(B)$. Thus, putting this\nvalue, we have $ \\Delta(D\\rangle B)\\leq 0$. But, since $ \\Delta(D\\rangle B)\\geq 0$, thus we have $ \\Delta(D\\rangle B)=0$ for the states $\\rho_{BD}$, i.e., the quantum advantage of\ndense coding vanishes precisely for these states. Similarly, for any tripartite state, pure or mixed $\\rho_{ABC}$, if the state $\\rho_{BC}$ satisfies the Araki-Lieb equality condition,\nthen the quantum advantage of dense coding $\\Delta(A\\rangle B)$ of $\\rho_{AB}$ also becomes zero. Apart from the above exact values,\nthe lower bound on entanglement of purification puts an upper bound on the quantum advantage of dense coding via its monogamy relation with the quantum advantage of dense coding.\n\n\n\\section{ Conclusions and Outlook}\nIn this paper, we find that the monogamous nature of correlations is not unique to quantum correlations, but can also be the case for the total correlations for certain quantum \nstates. Thus, monogamy is not a property of the quantum correlation alone. Contrary to the monogamy nature of the mutual \ninformation for tripartite pure states, we have proved that the entanglement of purification can be polygamous for such states.\nThis shows that even though the mutual information and the entanglement of purification are supposed to capture total correlation, the nature of these correlations\ncan be completely opposite at least for tripartite systems.\nIn case of pure and mixed states, the monogamy of entanglement of purification is related to the monogamy of entanglement of formation. Also, we have found a\nnecessary condition for monogamy of entanglement of purification for a special class of mixed states, in terms of the interaction information or the polygamy of the\nquantum mutual information. A new lower bound of the entanglement of purification has been given for the tripartite mixed states and higher dimensional bipartite systems. \nUsing the formula for the lower bound we have been able to find the exact values\nof entanglement of purification for some classes of states. Furthermore,\nin this paper we have also shown that if entanglement of purification is not additive, it has to be a sub-additive quantity. \nUsing these results we have also shown that the quantum advantage of dense coding\nis strictly monogamous for all tripartite pure states and it is super-additive on tensor products. We have also identified some of the quantum states with no \nquantum advantage of dense coding.\nWe have brought forward these important aspects of the measure of total correlation as well as that of the quantum advantage of dense coding to the forefront. \nThese will help us understand better the nature of total and quantum correlations of composite quantum states. \nThis calls for more explorations and a deeper understanding of the total correlation present in a composite\nmixed state. The total correlation quantified by the mutual information can be split into quantum correlation and classical correlation. However,\nwe still do not know whether we can express the entanglement of purification as the sum of quantum and classical correlations. In view of the\npolygamy nature of entanglement of purification, can it be the case that the entanglement of purification contains more classical like\ncorrelation than the quantum correlation. This will be a topic of future investigation.\n\n\\section{ Acknowldgements}\nSB acknowledges discussions with Ujjwal Sen, Aditi Sen de, Andreas Winter, Debbie Leung and Atri Bhattacharya. SB acknowledges cluster computing facility at HRI.\nSB and AKP acknowledge financial support from DAE, Govt. of India.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Introduction}\nEvidence-based medicine intends to optimize healthcare decision-making by using evidence from well-designed and conducted research \\citep{guyatt2002users, moher2006systematic, egger2008systematic}. It classifies evidence by its epistemological strength and recommends using evidence from randomized controlled trials (RCTs), systematic reviews, and meta-analyses when available, to inform guidelines and policies. When conducted properly, systematic reviews and meta-analyses provide the most reliable evidence for synthesizing the benefits and harms associated with various treatment options, and can provide patients, caregivers, and doctors with integrated information for healthcare decision-making \\citep{moher1999improving,bossuyt2003towards, moher2009preferred,stewart2015preferred}.\n\nAlmost all RCTs measure and report more than one outcome, and often these outcomes are correlated with each other. For example, in a cardiovascular trial, the reduction in lipids level may be correlated with risk of clinical events such as stroke and myocardial infarction. In many RCTs, there is a balance between safety and efficacy; an experimental treatment may have greater efficacy than the placebo or standard therapy, but it may also have higher risk of adverse side effects such as transient toxicity or death. In practice, clinical decision-making relies on both efficacy and safety, so these outcomes must be considered simultaneously. Multivariate meta-analysis (MMA) is one technique proposed to jointly analyze multiple outcomes. MMA can borrow information from the potential correlation among the outcomes to improve the estimation of the pooled effect sizes \\citep{riley2007bivariate, riley2007evaluation,jackson2011multivariate}.\n\nOn the other hand, since multiple outcomes are simultaneously considered in medical decision-making, biases in some of the outcomes can affect the overall decision of treatment. Recently, empirical studies have provided convincing evidence of the existence of selective reporting. \\citet{chan2004empirical} compared the protocols of 102 trials with 122 published reports. Their investigation showed that, on average, 50\\% of efficacy outcomes and 65\\% of safety outcomes in each trial were incompletely reported; 62\\% of the 82 trials had major inconsistencies between outcomes stated in the trial protocols and those reported in publications. They also found that, compared with nonsignificant outcomes, statistically significant outcomes are more likely to have higher odds of being reported for both efficacy outcomes (odds ratio = 2.4) and safety outcomes (odds ratio = 4.7).  Other studies have found similar results, such as in the reporting of toxicity in seven different medical areas \\citep{hemminki1980study, chan2004outcome, hazell2006under, al2008selective, chowers2009reporting, mathieu2009comparison}, and in the reporting of safety outcomes for breast cancer treatments \\citep{vera2013bias}. In the present article's motivating study for the effects of interventions on hospital readmission and quality of life for heart failure patients, 11 studies out of 45 do not report readmission, while 30 studies do not report quality of life. \n\nAs such, outcome reporting bias (ORB), defined as ``{\\it{the selective reporting of some outcomes but not others, depending on the nature and direction of the results}}'' \\citep{sterne2016chapter}, may lead to biased inference in the pooled estimates of the outcomes and negatively affect patient outcomes. In addition to biased inference, ORB can also invalidate results from meta-analyses. For example, in this article's case study, the significant decrease in relative risk (RR) of hospital readmission for heart failure patients in the intervention group (95\\% confidence interval [CI] 0.862--0.993) is no longer present \\textit{after} we adjust for ORB (95\\% CI 0.876--1.051). In our meta-evaluation of 748 bivariate meta-analyses from the Cochrane Database of Systematic Reviews in Section \\ref{meta-meta}, we also found that 157 reviews experienced a change in statistical significance for at least one outcome \\textit{after} correcting for ORB.\n\nUntil recently, ORB has been understudied, especially compared to the well-studied publication bias (PB) problem, defined as ``{\\it{the publication or nonpublication of research findings, depending on the nature and direction of the results}}'' \\citep{sterne2016chapter}. In the presence of PB, the published studies form a biased selection of the research in certain areas, which then leads to biased estimates \\citep{jackson2007assessing}. The ORB problem is different from the PB problem in that, although outcomes of a study have been selectively reported under ORB, the remaining outcomes are still available. For PB, on the other hand, studies are completely missing, and we do not even know the number of studies that have been conducted but not published. Thus, the strategy for addressing ORB differs from that for PB, especially when leveraging the partially observed outcomes to infer the unreported outcomes.\n\nSince part of the outcomes are available, the current strategy for MMA with missing outcomes has focused on joint modeling of multiple outcomes that ``borrow strength'' across correlated outcomes \\citep{riley2009multivariate,kirkham2012multivariate,frosi2015multivariate}. The idea is that the set of studies with outcomes reported can inform the correlations among multiple outcomes, which can be used to ``impute'' the missing outcomes from the reported outcomes. Unfortunately, this joint modeling strategy alone is insufficient as an approach to account for ORB because it relies on the {\\emph{missing at random}} (MAR) assumption. This assumption is often not true in RCTs, since evidence suggests that the majority of missing outcomes are \\textit{selectively} unreported \\citep{chan2004empirical, vera2013bias}. It is also unclear if joint modeling alone can lead to less biased estimates in the presence of ORB. \n\nThe evaluation of ORB has been included as a key component by the Cochrane risk of bias tool  \\citep{higgins2011cochrane}, which is becoming a standard procedure in conducting a systematic review. However, the \\textit{Cochrane Handbook for Systematic Reviews of Interventions} (Chapter 8.14.2, version 5.1.0, 2011), has acknowledged that ``statistical methods to detect within-study selective reporting (i.e., outcome-reporting bias) are, as yet, not well developed'' \\citep{higgins2011chapter}. The \\textit{Journal of Clinical Epidemiology} has also stressed that ``guidance is needed for using multiple outcomes and results in systematic reviews'' \\citep{mayo2017multiple}. \n\nMotivated by the critical need for statistical models that can adjust for and evaluate the impact of ORB, we develop {\\bf{A}} {\\bf{B}}aysian {\\bf{S}}election model for correcting {\\bf{ORB}} (abbreviated as ABSORB henceforth) in this article. Specifically, we rely on selection models where multivariate latent variables are used to model the process of selective reporting of multiple outcomes in a flexible way. We then use a Bayesian approach to conduct estimation by placing appropriate priors on the unknown parameters. {\\emph{From a modeling point of view}}, the distributions of the latent variables that govern the reporting processes are allowed to be correlated with not only the significance of the outcomes but also the characteristics of the study. \n{\\emph{From a statistical inference point of view}}, the Bayesian approach allows the implementation of the model straightforwardly using Markov chain Monte Carlo (MCMC) and naturally provides uncertainty quantification for the model parameters through their posterior distributions. While there have been several approaches proposed for quantifying PB in \\textit{univariate} meta-analysis \\citep{lin2018quantifying, BaiLinBolandChen2020}, we are not aware of any existing approaches to quantify the impact of \\textit{ORB} in \\textit{multivariate} meta-analyses. By taking the Hellinger distance between the bias-corrected and non-bias corrected posterior densities for model parameters, we propose a measure to quantify the impact of outcome reporting bias.\n\nThe rest of the article is structured as follows. Section~\\ref{MotivatingData} describes the motivating case study of the effects of interventions on quality of life and hospital readmission for heart failure patients. Section~\\ref{ABSORB} introduces our proposed ABSORB model and our measure for quantifying the impact of ORB using our model. Section~\\ref{meta-meta} empirically evaluates these approaches through a meta-evaluation of bivariate meta-analyses from the Cochrane Database of Systematic Reviews. Section~\\ref{Application} applies our approaches to the case study of heart failure patients. Section~\\ref{Discussion} concludes the article with a discussion of our findings and potential extensions for future work.\n\n\\section{A Motivating Meta-Analysis on Interventions for Heart Failure Patients} \\label{MotivatingData}\n\nFor heart failure (HF) patients, readmission (ReAd) after discharging from the hospital is not rare, which places substantial burdens on both the patients and the health system. According to Medicare, the median risk-standardized 30-day readmission rate for HF was 23.0\\% \\citep{ZiaeianFonarow2016}. Due to the high cost of HF, preventing ReAd for HF patients has received particular attention from clinicians, researchers, and policymakers. For example, the Affordable Care Act has instituted a financial penalty for excessive readmissions for hospitals that is capped at 3\\% of a hospital's total Medicare payments for 2015 and beyond \\citep{ZiaeianFonarow2016}. On the other hand, quality of life (QoL) is an outcome that attracts more attention from patients, and the factors that affect the QoL of HF patients include anxiety, depression, and physical disability. A literature review by \\citet{Celano2018} found that these adverse QoL outcomes were associated with poor function, reduced adherence to treatment, and elevated mortality in HF patients.  \n\nTelemonitoring (TM) and structured telephone support (STS) are two common interventions and are demonstrated to be effective in reducing HF-specific readmission \\citep{inglis2015structured}. A series of RCTs measuring both the all-cause ReAd and QoL provides a good opportunity to systematically evaluate the effects of interventions (TM or STS) on these two outcomes for patients with heart failure. Moreover, \\citet{Celano2018} found that QoL was significantly associated with rehospitalization rates for HF patients. Therefore, it is of practical interest to \\textit{jointly} model the effects of interventions on both ReAd and QoL in order to capture the inherent correlations between these two outcomes.\n\nAfter a systematic search of scientific literature, 45 intervention studies were included in our analysis. For ReAd, we calculated the RR in order to quantify the change in risk of readmission due to the interventions compared to the usual care. Since the quantitative measure of QoL differed across studies, we calculated the standardized mean difference (SMD) in order to quantify the change of QoL between the intervention group and the group with usual care. \n\nFor multiple studies in our meta-analysis, either ReAd or QoL was missing. For each of the studies, there were three possible scenarios: 1) the study reported both ReAd \\textit{and} QoL, 2) the study reported \\textit{only} ReAd, and 3) the study reported \\textit{only} QoL. Among the 45 studies, only 8 studies published the results for both ReAd and QoL, 33 studies published only one of the two outcomes, and four studies did not publish either ReAd or QoL. Among the 41 studies with at least one outcome reported, 34 studies published the effect size of interventions on ReAd, and 15 studies published the effect size of interventions on QoL.\n\n\n\\begin{table}[!htbp]\n\t\\centering\n\t\\caption{Number of studies in the meta-analysis of interventions for HF patients, summarized by outcomes (columns) and by missingness scenarios (rows).  \\checkmark : reported, \\text{\\sffamily X}: missing.}\n\t\\medskip \n\t\n\t\\begin{tabular}{ccccccc}\n\t\t\\hline\n\t\t& \\multicolumn{3}{c}{Published studies} & \\multicolumn{3}{c}{Updated studies}\\\\\n\t\t\\cline{2-7}\n\t\t\\multirow{2}{*}{\\parbox{2cm}{\\centering Scenario}} & \\multicolumn{2}{c}{Outcome} & \\multirow{2}{*}{\\parbox{1.5cm}{\\centering No.\\ of studies}} &\\multicolumn{2}{c}{Outcome}&\\multirow{2}{*}{\\parbox{1.5cm}{\\centering No.\\ of studies}} \\\\\n\t\t\\cline{2-3}\\cline{5-6} \n\t\t& ReAd & QoL &&  ReAd & QoL \\\\\n\t\t\\hline\n\t\t1 & \\checkmark & \\checkmark & 8   &\\checkmark &\\checkmark & 11\\\\\n\t\t2 & \\checkmark & \\text{\\sffamily X} & 26 &\\checkmark &\\text{\\sffamily X} & 23 \\\\\n\t\t3 & \\text{\\sffamily X} & \\checkmark & 7 &\\text{\\sffamily X} &\\checkmark & 10 \\\\\n\t\tNo.\\ of studies & 34 & 15 & 41 & 34 & 21 & 44 \\\\\n\t\t\\hline\n\t\\end{tabular} \\label{ReAdQoLTable}\n\\end{table}\n\nWe queried the corresponding authors for the studies that did not report either QoL or ReAd, and only half of the authors we contacted replied. Specifically, we obtained QoL results for six of the 30 studies that did not publish results on QoL.\nThese new results gave us an updated sample with 11 studies that reported both ReAd and QoL and three \\textit{new} studies that \\textit{only} reported QoL. Table~\\ref{ReAdQoLTable} summarizes the outcome reporting in our initial dataset (i.e., published studies) and in our new dataset \\textit{after} obtaining six unpublished results on QoL from corresponding authors (i.e., updated studies).\n\nAs a preliminary investigation, we conducted Begg's test \\citep{Begg1994} and Egger's test \\citep{Egger1997} for PB on ReAd and QoL separately. These tests suggested strong evidence of publication bias for ReAd (Begg's test: p-value = 0.02, Egger's test: p-value = 0.01). For QoL, there was moderate evidence of publication bias (Egger's test: p-value=0.10). However, by plotting the funnel plot for QoL, the evidence of selective reporting became more pronounced. As depicted in the left panel of Figure~\\ref{funnelplots}, there was a moderate degree of asymmetry in the funnel plot for the published studies, as evidenced by a missing chunk out of the funnel on the left hand side. In the updated studies (right panel of Figure~\\ref{funnelplots}), we found that \\textit{all} six missing studies' QoL (represented by diamonds) were statistically \\textit{nonsignificant}. This strongly suggested the existence of selective reporting of QoL. Even though the six missing studies were updated, there was still evidence of outcome reporting bias, as shown by the Egger's regression \\citep{Egger1997}, i.e., the intercept was found to deviate from zero. \n\nWhile our initial investigation analyzed the ReAd and QoL outcomes separately, ReAd and QoL are likely to be correlated \\citep{Celano2018} in practice, and biased estimation in one outcome can affect estimation in the other. In addition, given the evidence that many missing outcomes in RCTs are selectively unreported rather than missing at random \\citep{chan2004empirical} (including in our case study), we were motivated to: 1) \\textit{jointly} model ReAd and QoL in such a way that \\textit{adjusts} for potential ORB, and 2) \\textit{quantify} the impact of ORB on our MMA. We detail our novel modeling approaches in Section~\\ref{ABSORB}.\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[width=.9\\linewidth]{funnel_plot_QoL_updated.png}\n\t\\caption{Contour-enhanced funnel plots for QoL in the published studies (left panel) and the updated studies (right panel). There is slightly less asymmetry in the funnel plot for the updated studies, suggesting the existence of selective reporting for QoL.}  \\label{funnelplots}\n\\end{figure}\n\n\n\n\\section{Statistical Methods} \\label{ABSORB}\n\nBased on our motivating case study, we focus on meta-analyses where two outcomes are of interest (or \\textit{bivariate} meta-analysis). In practice, a bivariate meta-analysis of studies of diagnostic test accuracy is the most common medical application of MMA \\citep{jackson2011multivariate, reitsma2005bivariate, chu2006bivariate}. In studies for drugs and other medical treatments, clinical efficacy and safety are also typically the two outcomes of greatest interest \\citep{chan2004empirical}. However, the extension of ABSORB to meta-analyses with more than two outcomes is relatively straightforward and is discussed in Section~\\ref{Discussion}. \n\nIn a bivariate meta-analysis, our main parameter of interest is an unknown vector of two population treatment effects $\\bm{\\mu} = (\\mu_1, \\mu_2)'$. For example, the first endpoint $\\mu_1$ could be a quantitative measure of the efficacy of a treatment, while the second endpoint $\\mu_2$ is a quantitative measure for the treatment's safety. In our case study, $\\mu_1$ is the RR of readmission, and $\\mu_2$ is the SMD of the quality of life for heart failure patients. We let $\\bm{y} = ( y_{1}, y_{2})'$ denote the reported effects for $\\bm{\\mu}$.\n\n\\subsection{The ABSORB Model} \\label{ABSORBModel}\n\nAs discussed in Section~\\ref{Introduction}, a common difficulty with conducting MMA is that in practice, outcomes are frequently unreported \\citep{jackson2011multivariate}. Selective reporting of $y_1$ or $y_2$ might lead to biased estimation and misleading inference about $\\bm{\\mu}$. With the ABSORB model, we aim to adjust for this ORB.\n\n\\subsubsection{Model Specification and Assumptions} \\label{ModelSpecification}\n\nBuilding upon the selection model literature for correcting PB in meta-analysis \\citep{copas1999works, copas2000meta, copas2001sensitivity, BaiLinBolandChen2020}, our goal is to explicitly model the selective reporting mechanism for partially reported outcomes. We assume that for each outcome $y_j, j = 1, 2$, there is a latent variable $z_j$ which determines the likelihood of $y_j$ being reported. \n\nLet $n$ denote the number of studies in our MMA. We assume that\n\\begin{align} \\label{YgivenZ}\ny_{ij} \\mid ( z_{ij} > 0 ) = \\mu_j + \\tau_j u_{ij} + s_{ij} \\epsilon_{ij},  \\hspace{.5cm} i = 1, \\ldots, n, \\hspace{.2cm} j = 1, 2,\n\\end{align}\nwhere $y_{ij}$ is the reported outcome for the $j$th endpoint for the $i$th study, $\\mu_j$ is the mean effect for the $j$th endpoint, and $s_{ij}$ is the reported standard error for $y_{ij}$. We assume that $u_{ij}$ and $\\epsilon_{ij}$ are marginally distributed as $\\mathcal{N}(0,1)$ and that $\\textrm{corr}(u_{ij}, \\epsilon_{ij}) = 0$. The $u_{ij}$'s are random effects that capture the between-study heterogeneity for the $j$th endpoint, while $\\tau_j > 0$ quantifies the amount of between-study heterogeneity. Meanwhile, the within-study random error is captured by $\\epsilon_{ij}$. Under \\eqref{YgivenZ}, we assume that $y_{ij}$ is only reported if the associated latent variable $z_{ij}$ is greater than zero. We further assume that the $z_{ij}$'s are generated according to\n\\begin{align} \\label{latentZ}\n\tz_{ij} = \\gamma_{0j} + \\gamma_{1j} \/ s_{ij} + \\delta_{ij}\n\\end{align}\nwhere $\\delta_{ij} \\sim \\mathcal{N}(0, 1)$. In \\eqref{latentZ}, the parameter $\\gamma_{0j}$ determines the overall probability of reporting $y_{ij}$, while $\\gamma_{1j}$ determines how the likelihood of reporting depends on sample size. In general, $\\gamma_{1j} \\geq 0$, so that studies with larger sample sizes are\nmore likely to report their outcomes. We assume that\n\\begin{align} \\label{EpsilonDeltaCorrelation}\n\\textrm{corr} (\\epsilon_{ij}, \\delta_{ij}) = \\rho_j\n\\end{align}\nthat is, the reported outcome $y_{ij}$ and the latent variables $z_{ij}$ are correlated through $\\rho_j$. The correlation parameters $\\rho_1$ and $\\rho_2$ in \\eqref{EpsilonDeltaCorrelation} control how the probability of reporting for the first and second endpoint respectively is influenced by the effect size of the study. When ORB for both endpoints is present, then $\\rho_1 \\neq 0$ and $\\rho_2 \\neq 0$. In this case, standard meta-analyses may lead to \\textit{biased} estimation of $\\bm{\\mu}$. \n\nIn line with standard bivariate meta-analysis \\citep{jackson2011multivariate}, we further assume that there is both within-study correlation between the $\\epsilon_{ij}$'s in \\eqref{YgivenZ}, as well as between-study correlation for the two endpoints. To model the within-study correlation, we assume that\n\\begin{align} \\label{WithinStudyCorrelation}\n\t\\textrm{corr} ( \\epsilon_{i1}, \\epsilon_{i2} ) = \\rho_\\text{W}\n\\end{align}\nAlthough the assumption that the within-study correlation is a constant $\\rho_\\text{W}$ across all the studies may be strong, this approach is commonly adopted in practice for MMA \\citep{RileyThompsonAbrams2007, LinChu2018} in order to keep the model parsimonious.\n\nTo model the between-study correlation, we assume that the random effects $(u_{i1}, u_{i2})'$ for the two endpoints in \\eqref{YgivenZ} are also correlated. That is, we assume that\n\\begin{align} \\label{BetweenStudyCorrelation}\n\t\\textrm{corr} ( u_{i1}, u_{i2} ) = \\rho_\\text{B}\n\\end{align}\nFinally, we assume that\n\\begin{align} \\label{ZeroCorrelations}\n\t\\textrm{corr} ( \\epsilon_{i1}, \\delta_{i2}) = \\textrm{corr} ( \\epsilon_{i2}, \\delta_{i1} ) = \\textrm{corr}(\\delta_{i1}, \\delta_{i2}) = 0\n\\end{align} \nAssumption \\eqref{ZeroCorrelations} implies that $y_{i1} \\mid (z_{i1} > 0, z_{i2}) = y_{i1} \\mid (z_{i1} > 0)$ and $y_{i2} \\mid ( z_{i1}, z_{i2} > 0) = y_{i2} \\mid (z_{i2} > 0)$. In other words, $y_{i1}$ is reported only if $z_{i1} > 0$ and does not depend on the value of $z_{i2}$. Similarly, $y_{i2}$ does not depend on $z_{i1}$, and $z_{i1}$ does not depend on $z_{i2}$. We stress that the outcomes $y_{1}$ and $y_{2}$ themselves are likely to be correlated, and this is captured in our model through the within-study correlation $\\rho_\\text{W}$ \\eqref{WithinStudyCorrelation} and the between-study correlation $\\rho_\\text{B}$ \\eqref{BetweenStudyCorrelation}. However, the probability of \\textit{reporting} each individual outcome should depend only on the associated latent variable.\n\n\\subsection{Estimation in the ABSORB Model} \\label{PriorSpecification}\nThe basic ABSORB model is given in \\eqref{YgivenZ}--\\eqref{latentZ}, while additional assumptions about the correlation structure of different parameters are encoded in \\eqref{EpsilonDeltaCorrelation}--\\eqref{ZeroCorrelations}. In summary, we have a total of 12 unknown parameters $(\\mu_1, \\mu_2, \\tau_1, \\tau_2, \\gamma_{01}, \\gamma_{02}, \\gamma_{11}, \\gamma_{12}, \\rho_1, \\rho_2, \\rho_\\text{W}, \\rho_\\text{B})'$ under the ABSORB model. We propose a Bayesian approach to estimating all these parameters by placing appropriate priors on them. \n\nFor the mean treatment effects $\\bm{\\mu}$, we place the vague priors,\n\\begin{align} \\label{muprior}\n\\mu_j \\sim \\mathcal{N}(0, 10^4)\n\\end{align}\nand for the heterogeneity parameters $(\\tau_1, \\tau_2)'$, we place vague half-Cauchy priors,\n\\begin{align} \\label{tauprior}\n\\tau_j \\sim \\mathcal{C}^{+} (0, 1)\n\\end{align}\nNext, we consider priors for $(\\gamma_{01}, \\gamma_{02}, \\gamma_{11}, \\gamma_{12})'$, the parameters that control the overall likelihood of reporting for the first and second endpoint respectively. To induce weakly informative priors on these parameters, we follow \\cite{BaiLinBolandChen2020} and specify the priors as,\n\\begin{align} \\label{gamma0prior}\n\\gamma_{0j} \\sim \\mathcal{U} (-2, 2)\n\\end{align}\n\\begin{align} \\label{gamma1prior}\n\\gamma_{1j} \\sim \\mathcal{U} (0, \\max_i s_{ij} )\n\\end{align}\nThe priors \\eqref{gamma0prior}--\\eqref{gamma1prior} ensure that most of the mass for each of the latent variables $z_{ij}$ lies in the interval $(-2, 3)$, leading to selection probabilities between 2.5\\% and 99.7\\%. Finally, in order to complete the prior specification, we place noninformative uniform priors on each of the correlation parameters,\n\\begin{align} \\label{rhoprior}\n\\rho_1, \\rho_2, \\rho_\\text{W}, \\rho_\\text{B} \\sim \\mathcal{U}(-1, 1).\n\\end{align}\nThe Bayesian approach is especially appealing for several reasons. First, we can implement the model straightforwardly using MCMC, thus avoiding the difficulties of maximum likelihood estimation (MLE). The main issue with the MLE in selection models is that it can face non-convergence \\citep{copas2001sensitivity}. This can arise from poor initializations, a flat plateau in the likelihood, or instability in the computation of a $12 \\times 12$ Hessian matrix during the optimization procedure \\citep{copas2001sensitivity, ning2017maximum}. MCMC sampling does not encounter such difficulties (provided that we run the MCMC for enough iterations), and we can monitor convergence for the posteriors $p(\\mu_1 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ and $p(\\mu_2 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ using trace plots or the effective sample size (ESS). Besides the computational advantages of the Bayesian approach, we can also obtain natural uncertainty quantification for the model parameters through their posterior distributions. These posterior densities will ultimately allow us to quantify the \\textit{impact} of outcome reporting bias, as we discuss in Section~\\ref{QuantifyingORB}.\n\n\\subsection{The ABSORB Likelihood and Implementation} \\label{LikelihoodImplementation}\n\nIn order to perform Bayesian inference under the ABSORB model \\eqref{YgivenZ}--\\eqref{ZeroCorrelations}, we need to obtain the likelihood function and then place the priors \\eqref{muprior}--\\eqref{rhoprior} on the model parameters. In this section, we describe how to derive the ABSORB likelihood for the $n$ studies in our MMA and perform posterior inference under this likelihood. \n\nBecause not all studies report both $y_{1}$ and $y_{2}$, we may not have an equal number of observations for $y_1$ and $y_2$. Consequently, we need to consider three separate cases for the reported outcomes in our meta-analysis: 1) both endpoints are reported, 2) only the first endpoint is reported, or 3) only the second endpoint is reported. Without loss of generality, suppose that the first $m_1$ studies report both endpoints, the next $m_2$ studies report only the first endpoint $y_1$, and the remaining $m_3 = n-(m_1+m_2)$ studies report only the second endpoint $y_2$. \n\nFirst note that we can reparameterize the ABSORB model \\eqref{YgivenZ}--\\eqref{ZeroCorrelations} as a hierarchical model by introducing further latent parameters $(\\theta_{i1}, \\theta_{i2})'$ for the $m_1$ studies that report both endpoints, $\\widetilde{\\theta}_{i1}$ for the $m_2$ studies that report only $y_1$, and $\\check{\\theta}_{i2}$ for the $m_3$ studies that report only $y_2$. The main reason for introducing these additional latent parameters is to ensure that the joint densities in our likelihood can be written explicitly. We denote $\\bm{\\Xi}$ as the collection of all unknown parameters, including these latent parameters.\n\nFor the $m_1$ studies that report both outcomes,  we rewrite \\eqref{YgivenZ} as\n\\begin{equation} \\label{ABSORBreparam}\n\\begin{array}{rl}\ny_{i1} \\mid ( z_{i1} > 0) \\sim & \\mathcal{N} ( \\theta_{i1}, s_{i1}^2 ); \\\\\ny_{i2} \\mid (z _{i2} > 0) \\sim & \\mathcal{N} ( \\theta_{i2}, s_{i2}^2),\n\\end{array} \\hspace{.5cm} i = 1, \\ldots, m_1,\n\\end{equation}\nwhere $(\\theta_{i1}, \\theta_{i2})$ is jointly distributed as\n\\begin{align} \\label{latentTheta}\n\\begin{pmatrix} \\theta_{i1}\\\\ \\theta_{i2} \\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix} \\mu_1 \\\\ \\mu_2 \\end{pmatrix}, \\begin{pmatrix} \\tau_1^2 & \\rho_\\text{B} \\tau_1 \\tau_2 \\\\ \\rho_\\text{B} \\tau_1 \\tau_2 & \\tau_2^2 \\end{pmatrix} \\right), \\hspace{.5cm} i = 1, \\ldots, m_1.\n\\end{align}\n For studies $i = 1, \\ldots, m_1$ that report both outcomes, the ABSORB model can be represented as\n\\begin{align} \\label{JointYZbothoutcomes}\n\\begin{pmatrix} y_{i1} \\\\ y_{i2} \\\\ z_{i1} \\\\ z_{i2} \\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix} \\theta_{i1} \\\\ \\theta_{i2} \\\\ \\gamma_{01} + \\gamma_{11} \/ s_{i1} \\\\ \\gamma_{02} + \\gamma_{12}\/s_{i2} \\end{pmatrix}, \\begin{pmatrix} s_{i1}^2 & \\rho_\\text{W} s_{i1} s_{i2} & \\rho_1 s_{i1} & 0 \\\\ \\rho_\\text{W} s_{i1} s_{i2} & s_{i2}^2 & 0 & \\rho_2 s_{i2} \\\\ \\rho_1 s_{i1} & 0 & 1 & 0 \\\\ 0 & \\rho_2 s_{i2} & 0 & 1 \\end{pmatrix} \\right) \\mathbb{I}_{ [ z_{i1} > 0 \\cap z_{i2} > 0]}.\n\\end{align}\nNamely, for each of these $m_1$ studies, the joint density of $(y_{i1}, y_{i2}, z_{i1}, z_{i2})$ is a truncated normal density that contains both endpoints $(y_{i1}, y_{i2})'$ only because \\textit{both} associated latent variables $z_{1}$ and $z_{2}$ are greater than zero. The off-diagonal entries in the covariance matrix in \\eqref{JointYZbothoutcomes} capture the correlations between $y_{i1}$, $y_{i2}$, $z_{i1}$, and $z_{i2}$. The likelihood function for $\\bm{\\Xi}$ in these $m_1$ studies is easily seen to be\n\\begin{equation} \\label{LikelihoodBothYs}\n\tL_1 (\\bm{\\Xi} ) = \\prod_{i=1}^{m_1} f( y_{i1}, y_{i2}, z_{i1}, z_{i2} \\mid \\theta_{i1}, \\theta_{i2}, \\gamma_{01}, \\gamma_{11}, \\gamma_{02}, \\gamma_{12}, \\rho_1, \\rho_2, \\rho_\\text{W} ),\n\\end{equation}\nwhere $f(y_{i1}, y_{i2}, z_{i1}, z_{i2} \\mid \\cdot )$ is the probability density function (pdf) for the truncated normal density in \\eqref{JointYZbothoutcomes}. \n\nFor the $m_2$ studies that only report the first endpoint $y_{1}$ but not $y_{2}$, we can also represent the model with a truncated normal density. However, since we do not observe $y_{2}$ for these studies, we can only write the joint density of $(y_{i1}, z_{i1}, z_{i2})'$ for studies $i = m_1+1, \\ldots, m_1+m_2$ as follows: \n\\begin{align} \\label{jointYZfirstoutcomeonly}\n\\begin{pmatrix} y_{i1} \\\\ z_{i1} \\\\ z_{i2} \\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix} \\widetilde{\\theta}_{i1} \\\\ \\gamma_{01} + \\gamma_{11} \/ s_{i1} \\\\ \\gamma_{02} + \\gamma_{12} \/ s_{i2} \\end{pmatrix}, \\begin{pmatrix} s_{i1}^2 & \\rho_1 s_{i1} & 0 \\\\ \\rho_1 s_{i1} & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\right) \\mathbb{I}_{[ z_{i1} > 0 \\cap z_{i2} < 0 ]},\n\\end{align} \nwhere $\\widetilde{\\theta}_{i1}$ is marginally distributed as\n\\begin{equation} \\label{latenttheta1}\n    \\widetilde{\\theta}_{i1} \\sim \\mathcal{N} (\\mu_1, \\tau_1^2), \\hspace{.5cm} i = m_1+1, \\ldots, m_1+m_2.\n\\end{equation}\nThe representation in \\eqref{jointYZfirstoutcomeonly} ensures that the first endpoint $y_1$ is only reported because the corresponding latent variable $z_1$ is greater than zero, while the second endpoint $y_2$ is \\textit{not} reported because the corresponding latent variable $z_2$ is \\textit{less} than zero. The main issue that we encounter with \\eqref{jointYZfirstoutcomeonly} is that the standard errors $s_{i2}$'s are not available for these $m_2$ studies (since $y_2$ was not reported for any of them), and our model requires these standard errors in order to parameterize the mean of $z_{i2}$. Nevertheless, we can estimate the missing $s_{i2}$'s using the approach given in Section~3.5 of \\cite{copas2014model}. Specifically, we use the relationship that $1 \/ s_{i2}^2 = k_2 n_i$, where $k_2$ is a constant and $n_i$ is the sample size of the $i$th study. Based on the other $n-m_2$ studies that reported $s_{i2}$ (i.e., the $m_1$ studies where both endpoints were reported and the $m_3$ studies that only reported the second endpoint $y_2$) and the corresponding sample sizes for these studies, we then estimate $k_2$ as\n\\begin{align*}\n\t\\widehat{k}_2 = \\frac{\\sum_{i \\in R_2} 1 \/ s_{i2}^2}{\\sum_{i \\in R_2} n_i },\n\t\\end{align*}\nwhere $R_2$ is the index set of the $n-m_2$ studies that have reported $s_{i2}$. The missing $s_{i2}$'s for the $m_2$ studies in \\eqref{jointYZfirstoutcomeonly} can then be estimated as $\\widehat{s}_{i2} = \\sqrt{1 \/ ( \\widehat{k}_2 n_i ) }$ \\citep{copas2014model}. Substituting the $s_{i2}$'s with their estimates $\\widehat{s}_{i2}$'s, the likelihood function for the $m_2$ studies that only report $y_1$ but not $y_2$ can be written as\n\\begin{equation} \\label{LikelihoodY1Only}\n\tL_2 (\\bm{\\Xi}) = \\prod_{i=m_1+1}^{m_1+m_2} f( y_{i1}, z_{i1}, z_{i2} \\mid \\widetilde{\\theta}_{i1}, \\gamma_{01}, \\gamma_{11}, \\gamma_{02}, \\gamma_{12}, \\rho_1) ,\n\\end{equation}\nwhere $f(y_{i1}, z_{i1}, z_{i2} \\mid \\cdot )$ is the pdf of the truncated normal density in \\eqref{jointYZfirstoutcomeonly}.\n\nFinally, for the remaining $m_3$ studies that only report the second endpoint $y_{2}$ but not $y_1$, we can similarly represent the model as follows. For $i = m_1 + m_2 + 1, \\ldots, n$, we have \n\\begin{align} \\label{jointYZsecondoutcomeonly}\n\\begin{pmatrix} y_{i2} \\\\ z_{i1} \\\\ z_{i2} \\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix} \\check{\\theta}_{i2} \\\\ \\gamma_{01}+ \\gamma_{11} \/ s_{i1} \\\\ \\gamma_{02} + \\gamma_{12} \/ s_{i2} \\end{pmatrix}, \\begin{pmatrix} s_{i2}^2 & 0 & \\rho_2 s_{i2} \\\\ 0 & 1 & 0 \\\\ \\rho_2 s_{i2} & 0 & 1 \\end{pmatrix} \\right) \\mathbb{I}_{[ z_{i1} < 0 \\cap z_{i2} > 0]},\n\\end{align} \nwhere $\\check{\\theta}_{i2}$ is marginally distributed as\n\\begin{equation}\\label{latenttheta2}\n\\check{\\theta}_{i2} \\sim \\mathcal{N}(\\mu_2, \\tau_2^2), \\hspace{.5cm} i = m_1+m_2+1, \\ldots, n.\n\\end{equation}\nThe truncated normal density in \\eqref{jointYZsecondoutcomeonly} ensures that the second endpoint $y_2$ is only reported because the corresponding latent variable $z_2$ is greater than zero, while $y_1$ is \\textit{not} reported because $z_1$ is \\textit{less} than zero. For these $m_3$ studies, we do not observe the standard errors $s_{i1}$'s since none of these studies reported $y_1$. As we require these $s_{i1}$'s in order to parameterize the mean of $z_{i1}$, we again follow the approach of \\cite{copas2014model} and first estimate\n\\begin{align*}\n\t\\widehat{k}_1 = \\frac{ \\sum_{i \\in R_1} 1 \/ s_{i1}^2}{\\sum_{i \\in R_1} n_i},\n\\end{align*}\nwhere $R_1$ is the index set for the $m_1+m_2$ studies that have reported $s_{i1}$. The missing $s_{i1}$'s for the $n-(m_1+m_2)$ studies in (3.7) are then estimated as $\\widehat{s}_{i1} = \\sqrt{1\/(\\widehat{k}_1 n_i)}$. Similarly as in \\eqref{LikelihoodY1Only}, the likelihood for these $n-(m_1+m_2)$ studies after substituting the $s_{i1}$'s with the $\\widehat{s}_{i1}$'s is\n\\begin{equation} \\label{LikelihoodY2Only}\n\tL_3 (\\bm{\\Xi} ) = \\prod_{i=m_1+m_2+1}^{n} f( y_{i2}, z_{i1}, z_{i2} \\mid \\check{\\theta}_{i2}, \\gamma_{01}, \\gamma_{11}, \\gamma_{02}, \\gamma_{12}, \\rho_2 ),\n\\end{equation}\nwhere $f(y_{i2}, z_{i1}, z_{i2} \\mid \\cdot )$ is the pdf of the truncated normal density in \\eqref{jointYZsecondoutcomeonly}. Combining \\eqref{LikelihoodBothYs}, \\eqref{LikelihoodY1Only}, and \\eqref{LikelihoodY2Only}, we see that the complete likelihood function for all $n$ studies is\n\\begin{align} \\label{ABSORBLikelihood}\nL (  \\bm{\\Xi} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n ) = L_1 (\\bm{\\Xi} )  L_2 (\\bm{\\Xi}) L_3 (\\bm{\\Xi}).\n\t\\end{align}\n Under \\eqref{ABSORBLikelihood}, the joint posterior distribution for $\\bm{\\Xi}$ is then\n\\begin{align} \\label{ABSORBposterior}\np ( \\bm{\\Xi} \\mid  \\bm{y}_1, \\ldots, \\bm{y}_n ) \\propto L ( \\bm{\\Xi} \\mid  \\bm{y}_1, \\ldots, \\bm{y}_n ) p ( \\bm{\\Xi} ),\n\\end{align}\nwhere $p (\\bm{\\Xi})$ is the product of the priors \\eqref{muprior}--\\eqref{rhoprior}, \\eqref{latentTheta}, \\eqref{latenttheta1}, and \\eqref{latenttheta2} on the model parameters. The main challenge with the ABSORB model is sampling from the truncated densities \\eqref{JointYZbothoutcomes}, \\eqref{jointYZfirstoutcomeonly}, and \\eqref{jointYZsecondoutcomeonly} in the full likelihood \\eqref{ABSORBLikelihood}. In Appendix~\\ref{Sampling}, we describe how to approximately sample from these truncated densities. With the prior for $\\bm{\\Xi}$ specified, the complete ABSORB model can then be implemented in any standard MCMC software to approximate the posterior distributions $p( \\mu_1 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n )$ and $p ( \\mu_2 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n )$.  For our implementation, we use the \\texttt{JAGS} software. \n\nNote that it may be the case that only one of the endpoints $y_1$ or $y_2$ in our MMA contains missing values. When there are no missing outcomes for $y_2$, the number of studies that only report $y_1$ but not $y_2$ is $m_2 = 0$, and we replace \\eqref{ABSORBLikelihood} with $L(\\bm{\\Xi} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n) = L_1 (\\bm{\\Xi}) L_3(\\bm{\\Xi})$. Similarly, if there are no missing outcomes for $y_1$, then $m_3 = 0$ and we replace \\eqref{ABSORBLikelihood} with $L(\\bm{\\Xi} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n ) = L_1 (\\bm{\\Xi}) L_2 (\\bm{\\Xi})$. In Appendix~\\ref{ABSORBISM}, we describe how to further extend the ABSORB model to incorporate studies that are \\textit{completely} missing due to publication bias (i.e., studies that do not report \\textit{either} $y_1$ or $y_2$). Such an extension of ABSORB to account for PB, in addition to ORB, is possible if we know the \\textit{number} of missing studies.\n\n\\subsection{Quantifying the Impact of Outcome Reporting Bias} \\label{QuantifyingORB}\n\nIn addition to correcting the bias in estimation of $\\bm{\\mu}$, it is also of practical interest to evaluate the \\textit{impact} of ORB on MMA. To the best of our knowledge, there are no existing approaches to quantify the impact of ORB, either frequentist or Bayesian. The Bayesian approach has a natural way of doing this through comparing the bias-corrected posteriors for $\\mu_1$ and\/or $\\mu_2$ under the ABSORB model against their \\textit{non}-bias corrected posteriors.\n\n\\subsubsection{Estimation for the Non-Bias Corrected Model} \\label{NonBiasCorrectedModel}\n\nWe first describe how to estimate the parameters in MMA with missing outcomes \\textit{without} accounting for ORB. The ABSORB model \\eqref{YgivenZ}--\\eqref{ZeroCorrelations} explicitly models the selective reporting mechanism through the latent variables $z_{1}$ and $z_{2}$. These variables control whether or not the corresponding outcomes $y_{1}$ or $y_{2}$ are reported, and thus, we obtain bias-corrected estimates of $\\bm{\\mu}$ under ABSORB. The likelihood of reporting $y_1$ and $y_2$ ultimately depends on the correlation parameters $\\rho_1$ and $\\rho_2$ in \\eqref{EpsilonDeltaCorrelation}. However, if $\\rho_1 = \\rho_2 = 0$, then $\\textrm{corr}(y_{ij}, z_{ij}) = 0$ for all $i = 1, \\ldots, n, j = 1, 2$, and the model \\eqref{YgivenZ}--\\eqref{ZeroCorrelations} reduces to\n\\begin{equation} \\label{ABSORBNoCorrelations}\n\t\\begin{array}{lll}\n\ty_{i1} & = \\mu_1 + \\tau_1 u_{i1} + s_{i1} \\epsilon_{i1}, & \\textrm{corr}(\\epsilon_{i1}, \\epsilon_{i2}) = \\rho_\\text{W}; \\\\\n\ty_{i2} & = \\mu_2 + \\tau_2 u_{i2} + s_{i2} \\epsilon_{i2}, & \\textrm{corr}(u_{i1}, u_{i2}) = \\rho_\\text{B}.\n\\end{array}\n\\end{equation}\nIn other words, when $\\rho_1 = \\rho_2 = 0$, the dependence of $y_{i1}$ and $y_{i2}$ on $z_{i1}$ and $z_{i2}$ respectively is removed in \\eqref{ABSORBNoCorrelations}, and we \\textit{only} have the unknown parameters $(\\mu_1, \\mu_2, \\tau_1, \\tau_2, \\rho_\\text{W}, \\rho_\\text{B})'$. In this case, the ABSORB model reduces to a joint model with a bivariate random effects model for the $m_1$ studies that report both $(y_1, y_2)'$ and univariate random effects models for the $m_2$ studies that report only $y_1$ and the $m_3$ studies that report only $y_2$. We call model \\eqref{ABSORBNoCorrelations} the \\textit{non}-bias corrected model because we ignore the selection process that was induced through the latent variables $z_1$ and $z_2$.\n\nSimilar to the bias-corrected ABSORB model, we introduce the latent parameters $(\\theta_{i1}, \\theta_{i2})'$ for $i = 1, \\ldots, m_1$, $\\widetilde{\\theta}_{i1}$ for $i = m_1+1, \\ldots, m_1+m_2$, and $\\check{\\theta}_{i2}$ for $i = m_1+m_2+1, \\ldots, n$, as in \\eqref{latentTheta}, \\eqref{latenttheta1}, and \\eqref{latenttheta2}. Let $\\bm{\\Omega}$ denote all the unknown parameters in the non-bias corrected model, including these latent parameters. Note that $\\bm{\\Omega}$ does not include the parameters $(\\rho_1, \\rho_2, \\gamma_{01}, \\gamma_{11}, \\gamma_{02}, \\gamma_{12})'$, because $\\rho_1$ and $\\rho_2$ are fixed at zero and we no longer need to condition on the latent variables $(z_1, z_2)'$ in our analysis. In the non-bias corrected model, we model the $m_1$ studies that report both outcomes as\n\\begin{align} \\label{StandardBivariateMetaAnalysis}\n\t\\begin{pmatrix} y_{i1} \\\\ y_{i2} \\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix} \\theta_{i1} \\\\ \\theta_{i2} \\end{pmatrix}, \\begin{pmatrix} s_{i1}^2 & \\rho_\\text{W} s_{i1} s_{i2} \\\\ \\rho_\\text{W} s_{i1} s_{i2} & s_{i2}^2 \\end{pmatrix} \\right), \\hspace{.5cm} i = 1, \\ldots, m_1,\n\\end{align} \nwhere the joint distribution of $(\\theta_{i1}, \\theta_{i2})'$ is given in \\eqref{latentTheta}. The likelihood function for these $m_1$ studies in the non-bias corrected model is\n\\begin{align} \\label{NonBiasCorrectedBothOutcomes}\n\tL_1 ( \\bm{\\Omega} ) = \\prod_{i=1}^{m_1}  f(y_{i1}, y_{i2} \\mid \\theta_{i1}, \\theta_{i2}, \\rho_\\text{W} ),\n\\end{align}\nwhere $f( y_{i1}, y_{i2} \\mid \\cdot)$ is the pdf of the bivariate normal density in \\eqref{StandardBivariateMetaAnalysis}. For the $m_2$ studies that only report $y_{1}$ but not $y_{2}$, the non-bias corrected model reduces to $y_{i1} \\sim \\mathcal{N} ( \\widetilde{\\theta}_{i1}, s_{i1}^2)$, where $\\widetilde{\\theta}_{i1} \\sim \\mathcal{N}(\\mu_1, \\tau_1^2)$. The corresponding likelihood function for these $m_2$ studies is\n\\begin{align} \\label{NonBiasCorrectedFirstOutcome}\n\tL_2 ( \\bm{\\Omega} )  = \\prod_{i=m_1+1}^{m_1+m_2} f(y_{i1} \\mid \\widetilde{\\theta}_{i1} ) ,\n\\end{align}\nwhere $f(y_{i1} \\mid \\widetilde{\\theta}_{i1})$ is the pdf for $\\mathcal{N}( \\widetilde{\\theta}_{i1}, s_{i1}^2)$. Similarly, for the $m_3$ studies that only report $y_{2}$ but not $y_{1}$, the non-bias corrected model reduces to $y_{i2} \\sim \\mathcal{N} (\\check{\\theta}_{i2}, s_{i2}^2)$, where $\\check{\\theta}_{i2} \\sim \\mathcal{N}( \\mu_2, \\tau_2^2)$. The corresponding likelihood for these $m_3$ studies is\n\\begin{align} \\label{NonBiasCorrectedSecondOutcome}\n\tL_3 (\\bm{\\Omega})  = \\prod_{i=m_1+m_2+1}^{n} f(y_{i2} \\mid \\check{\\theta}_{i2} ),\n\\end{align}\nwhere $f(y_{i2} \\mid \\check{\\theta}_{i2})$ is the pdf for $\\mathcal{N}(\\check{\\theta}_{i2}, s_{i2}^2)$. Altogether, the joint likelihood for all $n$ studies in the \\textit{non}-bias corrected model is the product of the likelihoods in \\eqref{NonBiasCorrectedBothOutcomes}--\\eqref{NonBiasCorrectedSecondOutcome}:\n\\begin{equation} \\label{NonBiasCorrectedLikelihood}\n\tL (\\bm{\\Omega} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n ) = L_1(\\bm{\\Omega})  L_2 ( \\bm{\\Omega} )  L_3 (\\bm{\\Omega}).\n\\end{equation}\nFrom \\eqref{NonBiasCorrectedLikelihood}, we conduct posterior inference for $\\bm{\\Omega}$ by placing the priors \\eqref{latentTheta}, \\eqref{latenttheta1}, and \\eqref{latenttheta2} on the latent variables $(\\theta_{i1}, \\theta_{i2})'$, $\\widetilde{\\theta}_{i1}$, and $\\check{\\theta}_{i2}$ respectively, and the priors \\eqref{muprior} on $\\bm{\\mu}$, \\eqref{tauprior} on $(\\tau_1, \\tau_2)'$, and \\eqref{rhoprior} on $(\\rho_\\text{W}, \\rho_\\text{B})'$. We thus obtain the posterior for $\\bm{\\Omega}$ as\n\\begin{align} \\label{NonBiasCorrectedPosterior}\n\tp( \\bm{\\Omega} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n) \\propto L ( \\bm{\\Omega} \\mid \\bm{y}_1, \\dots, \\bm{y}_n) p(\\bm{\\Omega}).\n\\end{align}\nWith the model fully specified, we can approximate the marginal posteriors $p(\\mu_1 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ and $p(\\mu_2 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ using MCMC. As before, if $m_2=0$, then we replace \\eqref{NonBiasCorrectedLikelihood} with $L(\\bm{\\Omega} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n) = L_1 (\\bm{\\Omega})  L_3 (\\bm{\\Omega})$, and if $m_3 = 0$, then we replace \\eqref{NonBiasCorrectedLikelihood} with $L(\\bm{\\Omega} \\mid \\bm{y}_1, \\ldots, \\bm{y}_n) = L_1(\\bm{\\Omega})  L_2 (\\bm{\\Omega})$.\n\n\\subsubsection{The $D$ Measure for Quantifying the Impact of ORB} \\label{DMeasure}\n\nTo quantify the impact of publication bias in \\textit{univariate} meta-analysis, \\cite{BaiLinBolandChen2020} proposed the $D$ measure as a way of measuring the difference between a publication bias-corrected and \\textit{non}-bias-corrected posterior for a mean treatment effect. Here, we extend the $D$ measure to quantify the impact of ORB bias in MMA. \n\nLet $p_\\text{ABS} ( \\mu_j \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ denote the posterior for $\\mu_j, j=1,2$ under the ABSORB model as described in Section~\\ref{LikelihoodImplementation}, and let $p_{NBC} ( \\mu_j \\mid \\bm{y}_1, \\ldots, \\bm{y}_n )$ denote the posterior for $\\mu_j$ under the non-bias corrected model described in Section~\\ref{NonBiasCorrectedModel}. To quantify the impact of ORB for each individual endpoint $\\mu_j$, we propose taking the Hellinger distance between $p_\\text{ABS}(\\mu_j \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ and $p_{NBC} (\\mu_j \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$. If we are instead interested in quantifying the joint impact from ORB on both endpoints, we can take the Hellinger distance between the joint posteriors $p_\\text{ABS}(\\mu_1, \\mu_2 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$ and $p_{NBC} ( \\mu_1, \\mu_2 \\mid \\bm{y}_1, \\ldots, \\bm{y}_n)$. \n\nLet $\\bm{x}$ be either a random scalar or a random vector. The Hellinger distance between densities $f$ and $g$ is defined as\n\\begin{equation}  \\label{Hellinger}\n\tH(f,g) = \\left[ 1 - \\displaystyle \\int \\sqrt{f (\\bm{x}) g (\\bm{x}) } d \\bm{x} \\right]^{1\/2},\n\\end{equation}\nThe Hellinger distance is an appealing way to quantify the dissimilarity between two probability densities. Unlike other probability distance measures such as the Kullback-Leibler distance, the Hellinger distance is symmetric \\textit{and} always bounded between zero and one. This gives the Hellinger distance a clear interpretation. Values close to zero indicate that $f$ and $g$ are nearly identical distributions, while values close to one indicate that the majority of the probability mass in $f$ does \\textit{not} overlap with that of $g$. \n\nFor shorthand, let $p_\\text{ABS}$ and $p_{NBC}$ be the posteriors for either $\\mu_1$, $\\mu_2$, or $\\bm{\\mu}$. Unfortunately, these posterior distributions are intractable and therefore need to be approximated. In the present context, we approximate the posteriors $p_\\text{ABS}$ and $p_{NBC}$ using MCMC samples to obtain kernel density estimates, $\\widehat{p}_\\text{ABS}$ and $\\widehat{p}_{NBC}$. We then use numerical integration to estimate the Hellinger distance \\eqref{Hellinger} between $\\widehat{p}_\\text{ABS}$ and $\\widehat{p}_{NBC}$. In short, our measure for the impact of ORB is\n\\begin{equation} \\label{Dmeasure}\n\tD = H \\left( \\widehat{p}_\\text{ABS}, \\widehat{p}_{NBC}  \\right),\n\\end{equation}\nThe $D$ measure \\eqref{Dmeasure} quantifies the degree to which the ABSORB posterior changes from the non-bias corrected posterior. Smaller values of $D$ ($D \\approx 0$) indicate that $p_\\text{ABS}$ and $p_{NBC}$ are almost identical. Thus, we conclude that there is negligible impact from ORB on the MMA. Meanwhile larger values of $D$ ($D \\approx 1$) indicate that ORB has a strong impact on the estimation of $\\bm{\\mu}$. In this case, the ABSORB posterior differs quite drastically from the non-bias corrected posterior. In the next section, we provide several illustrations of the $D$ measure on real systematic reviews from the Cochrane Database of Systematic Reviews.\n\n\\section{Meta-Evaluation with the Cochrane Database of Systematic Reviews}\n\\label{meta-meta}\n\nTo evaluate the performance of our model, we conducted a meta-evaluation of 748 systematic reviews from the Cochrane Database of Systematic Reviews (hereinafter refer to as the ``Cochrane Database''). We describe how we arrived at these 748 eligible reviews in Section A of the Supplementary Material. For dichotomous outcomes, we performed a log transformation to risk ratio and odds ratio outcomes. For each of the reviews in our meta-evaluation, we fit the ABSORB and non-bias corrected models of Section~\\ref{ABSORB}.  For both models, we ran three separate chains of the MCMC algorithm for 50,000 iterations, discarding the first 10,000 samples as burn-in. This left us with a total of 120,000 samples from three chains with which to approximate the posteriors and calculate the $D$ measures \\eqref{Dmeasure}. We monitored the convergence of the MCMC using ESS; if the ESS was below 100 for $\\mu_1$ or $\\mu_2$, then we increased the number of iterations to 100,000, 200,000, etc.\\ as needed.\n\nWe present three representative meta-analyses from our meta-evaluation, which we denote as MMA1, MMA2, and MMA3. Table~\\ref{ThreeMetaAnalyses} provides the details of these meta-analyses, including the review topic, the effect measure, and descriptions of the bivariate treatment effects of interest. The results from these three meta-analyses are depicted in Figure~\\ref{meta:CochraneExamples}. In Figure~\\ref{meta:CochraneExamples}, we plot the bias-corrected posteriors under the ABSORB model (solid line) against their non-biased corrected posteriors (dashed line) for $\\mu_1$ and $\\mu_2$, as well as the contour plots for the bias-corrected and non-biased corrected joint posteriors of $\\bm{\\mu}$. We also report the $D$ measures for $\\mu_1$, $\\mu_2$, and $\\bm{\\mu}$. For MMA1 (panels (a)-(c)), we see that there is a negligible impact from ORB for both endpoints, and thus the $D$ measures are all close to zero. In MMA2 (panels (d)-(f)), there is a fairly strong impact from ORB for the first endpoint ($D=0.41$) and a negligible impact ($D=0.12)$ for the second endpoint. In MMA3 (panels (g)-(i)), there is a very strong impact from ORB for the first endpoint ($D=0.98$) and a fairly strong impact ($D=0.49$) for the second endpoint. The bottom left graph in Figure~\\ref{meta:CochraneExamples} shows very little overlap between the bias-corrected and non-bias corrected posteriors for $\\mu_1$ in MMA3, and hence, we obtained a $D$ measure close to one. \n\n\n\\begin{table}[t!]\n    \\centering\n    \\caption{Three representative meta-analyses from the Cochrane Database.}\n    \\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lllll}\n    \\hline \n    & Topic & Outcome & Effect measure & Analysis  \\\\\n    \\hline\n    \\multirow{2}{*}{MMA1\\tablefootnote{Cochrane Database ID:CD000990, DOI: 10.1002\/14651858.CD000990.pub4.}}& Exercise for intermittent  & $\\mu_1$ & Mean difference & Change in maximal walking distance or time  \\\\\n    & claudication & $\\mu_2$ & Mean difference & Ankle brachial index  \\\\\n    \\hline\n     \\multirow{2}{*}{MMA2\\tablefootnote{Cochrane Database ID:CD000335, DOI: 10.1002\/14651858.CD000335.pub2.}} & Exercise therapy for treatment & $\\mu_1$ & Mean difference & Function measure  \\\\ \n    & of non-specific low back pain & $\\mu_2$ & Mean difference & Pain measure  \\\\\n    \\hline\n    \\multirow{2}{*}{MMA3\\tablefootnote{Cochrane Database ID:CD001886, DOI: 10.1002\/14651858.CD001886.pub4.}}& Anti-fibrinolytic use for minimizing & $\\mu_1$ & Risk ratio & Number of patients exposed to allogeneic blood \\\\ \n    & perioperative allogeneic blood transfusion & $\\mu_2$ & Mean difference & Units of allogeneic blood transfused \\\\\n    \\hline\n\\end{tabular}}\n\\end{center} \\label{ThreeMetaAnalyses}\n\\end{table}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.52\\linewidth]{Figure1_legend.png} \\\\\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu1_plots_case2.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu2_plots_case2.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_contour_plots_case2.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu1_plots_case1.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu2_plots_case1.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_contour_plots_case1.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu1_plots_case3.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_mu2_plots_case3.pdf}\n\\includegraphics[width=.25\\textwidth]{meta_ABSORB_contour_plots_case3.pdf}\n\t\\caption{Illustrations of three meta-analyses from the Cochrane Database. Panels~(a)--(c) show the results for MMA1, panels (d)--(f) show the results for MMA2, and panels (g)--(i) show the results for MMA3. In panels (g) and (i), $\\mu_1$ is plotted on the log-RR scale. }\\label{meta:CochraneExamples}\n\t\\end{figure}\n\nIn particular, for MMA2, the 95\\% posterior credible interval for $\\mu_1$ (i.e., the mean change in function measure after exercise therapy for lower back pain) shifted from $(-3.52, -0.38)$ under the non-bias corrected posterior to $(-2.53, 0.69)$ under the ABSORB bias-corrected posterior. This indicates that after adjusting for ORB, the 95\\% bias-corrected posterior interval contained zero, and the mean change in function measure after exercise therapy was \\textit{no longer} statistically significant. As a consequence of non-negligible ORB, 12.03\\% of all 748 meta-analyses in our meta-evaluation (90 reviews) had a change in statistical significance for the first outcome, and 10.56\\% (79 reviews) had a change in statistical significance for the second outcome. For 12 reviews, the statistical significance changed for \\textit{both} $\\mu_1$ and $\\mu_2$. These results demonstrate that non-negligible ORB can have a profound effect on the conclusions from MMA.  \n\n In Appendix~\\ref{AdditionalResults}, we provide the specific quantiles of the $D$ measure from our analysis. Based on these quantiles, we determined the following guidelines for interpreting the $D$ measure:\n\\begin{itemize}\n    \\item 0.00 to 0.20: probably no impact from ORB;\n    \\item 0.10 to 0.40: may represent moderate impact from ORB;\n    \\item 0.30 to 0.60: may represent substantial impact from ORB;\n    \\item 0.50 to 1.00: may represent severe impact from ORB.\n\\end{itemize}\nOur intervals were inspired by the guidelines given for the $I^2$ statistic  \\citep{higgins2002quantifying} in the Cochrane Handbook for Systematic Reviews of Interventions.\\footnote{\\url{https:\/\/handbook-5-1.cochrane.org\/chapter_9\/9_5_2_identifying_and_measuring_heterogeneity.htm}.} The $I^2$ statistic (for measuring heterogeneity in univariate meta-analyses) also lies between 0 and 1, and the Cochrane Handbook provides overlapping intervals for ``unimportant,'' ``moderate,'' ``substantial,'' and ``considerable'' heterogeneity based on $I^2$, so as to avoid setting hard cutoffs for its interpretation. \n\nIn our experience, a $D$ measure of 0.20 or higher usually suggested non-negligible ORB or the potential to qualitatively change the conclusions from meta-analyses. Meanwhile, a $D$ measure below 0.10 normally ruled out any impact from ORB (as illustrated in panels~(a)--(c) of Figure~\\ref{meta:CochraneExamples}). However, there were a few reviews where the statistical significance changed for an outcome even when $D<0.10$. This occurred when one of the CI endpoints was extremely close to zero -- in this case, the 95\\% CIs before and after bias correction were very similar to each other, but even a tiny discrepancy near zero changed the conclusion. Thus, the systematic reviewer should also investigate the CIs, not just the $D$ measure.\n\nOur meta-evaluation of the Cochrane Database found that 50.00\\% of MMAs had $D < 0.10$ for the first endpoint, 48.80\\% had $D < 0.10$ for the second endpoint, and 52.94\\% had $D < 0.10$ for both endpoints. However, there were also a few reviews where ORB had a very high impact. Namely, 26 reviews had $D$ measures greater than 0.50 for the first endpoint, and 11 reviews had $D$ measures greater than 0.50 for the second endpoint. Figure~\\ref{CochraneHistograms} plots the empirical histograms for the $D$ measure from our meta-evaluation. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=.32\\textwidth]{meta_D1.pdf}\n\t\\includegraphics[width=.32\\textwidth]{meta_D2.pdf}\n\t\\includegraphics[width=.32\\textwidth]{meta_D12.pdf}\n\t\\caption{Empirical histograms of the $D$ measure for $\\mu_1$ (left panel), $\\mu_2$ (middle panel) and $\\bm{\\mu}$ (right panel) from all 748 reviews in our meta-evaluation.}\\label{CochraneHistograms}\n\\end{figure}\n\nInstead of using the provided guidelines, an alternative is to simply use the quantiles from our meta-evaluation for interpretation. The quantiles for the $D$ measures are provided in Table \\ref{Dtable1} of Appendix \\ref{AdditionalResults}. Using this table, the systematic reviewer can locate the percentile of the $D$ measure obtained from his or her dataset among the $D$ measures from the Cochrane database and conclude that the evidence for ORB in his or her study is in the top, e.g., 20\\% of all analyzed datasets. \n\nOur meta-evaluation of real systematic reviews from the Cochrane Database illustrates the potential of the ABSORB model for adjusting the estimates of effect sizes in the presence of ORB and the $D$ measure for quantifying the impact of ORB. In Appendix~\\ref{Simulations}, we further validate our model through simulation studies under a variety of degrees of between-study heterogeneity and missingness. To summarize briefly, our simulation studies demonstrate that when ORB is present, the ABSORB model has lower bias and better empirical coverage than alternative approaches that remove studies with missing data, that impute the missing outcomes, or that ignore the correlations between the two endpoints.\n\n\n\n\n\n\n\n\\section{Case Study Results} \\label{Application}\n\nWe now apply the ABSORB model to the meta-analysis introduced in Section~\\ref{MotivatingData} on the effects of intervention on ReAd and QoL for HF patients. As in Section~\\ref{ABSORB}, $m_1$ denotes the number of studies that reported both ReAd and QoL, $m_2$ is the number of studies that reported only ReAd, and $m_3$ is the number of studies that reported only QoL. As discussed in Section~\\ref{MotivatingData}, our sample contained 45 studies on the effects of interventions on HF patients. We initially had $n=41$ published studies that reported at least one of ReAd or QoL ($m_1 = 8$, $m_2 = 26$, $m_3 = 7$). After querying the corresponding authors, we were able to obtain six additional results for QoL and a total of $n = 44$ studies with results for at least one of ReAd or QoL ($m_1 = 11$, $m_2 = 23$, $m_3 = 10$). \n\nWith our \\textit{a priori} knowledge about which studies had QoL results after querying the corresponding authors, we proceeded to conduct a two-stage analysis. In this first stage, we applied the ABSORB model to only the $n=41$ published studies that reported at least one of the ReAd or QoL outcomes (i.e., \\textit{before} we had queried the authors). In the second stage, we performed our analysis with the $n=44$ updated studies (i.e., \\textit{after} querying the authors). Our purpose for conducting this two-stage analysis was to see how our results changed after we were able to partially mitigate some of the ORB for QoL by querying corresponding authors. For our analysis, we did not include the studies that failed to report either ReAd or QoL. In Appendix~\\ref{AdditionalHFResults}, we apply an augmented model (introduced in Appendix~\\ref{ABSORBISM}) to \\textit{all} 45 intervention studies in both the published and the updated data. \n\nTo quantify the impact of outcome reporting bias in our MMA, we fit the ABSORB model of Section~\\ref{ABSORBModel} and the non-bias corrected model of Section~\\ref{NonBiasCorrectedModel} and used their posterior samples to compute the $D$ measure \\eqref{Dmeasure}. For both models, we ran three MCMC chains of 100,000 iterations, discarding the first 10,000 iterations as burn-in. In Appendix~\\ref{AdditionalHFResults}, we provide trace plots for these models, which show that the three chains mixed well and that the number of iterations we used was sufficient to achieve convergence.\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\\begin{figure}[!htbp]\n\\centering\n\\hspace{.5cm} \\includegraphics[width=.5\\linewidth]{Table3_legend.png} \\\\\n\\includegraphics[width=.4\\textwidth]{Table3_ReAd.pdf}\n\\includegraphics[width=.4\\textwidth]{Table3_QoL.pdf}\n\\caption{Plots of the posterior means and 95\\% posterior credible intervals for our case study on interventions for HF patients under the non-bias corrected and ABOSRB models. Panel~(a) plots the results for ReAd and panel~(b) plots the results for QoL.} \\label{HeartFailureResults}\n\\end{figure}\n \n \nFigure \\ref{HeartFailureResults} shows the posterior mean effect sizes and 95\\% posterior credible intervals for ReAd and QoL. For ReAd (panel (a) of Figure \\ref{HeartFailureResults}), there was little difference between the MMA results obtained from the published and updated datasets. The ABSORB model estimated a mean RR of 0.955 with a 95\\% CI of (0.876, 1.051) for the published data, and a mean RR of 0.956 with a 95\\% CI of (0.877, 1.054) for the updated data, which indicated \\textit{no} significant reduction of risk for hospital readmission for the intervention group. However, there \\textit{was} a qualitative difference in the clinical conclusions for ReAd from the \\textit{non}-bias corrected models. In addition to slightly lower mean estimates of RR for hospital readmission (0.931 for both the published and the updated data), the non-bias corrected models estimated 95\\% CIs of (0.862, 0.993) for the published data and (0.862, 0.994) for the updated data. This indicates that \\textit{without} correcting for ORB with the ABSORB model, our meta-analysis would have concluded that there was a \\textit{significant} reduction in risk of hospital readmission. \n\nAs for QoL (panel (b) of Figure \\ref{HeartFailureResults}), the ABSORB model estimated an SMD of 0.15 for QoL between intervention and control groups in the published data, which was slightly larger than the result obtained from the updated data (0.138). The 95\\% CI for the updated data (0.031, 0.232) was also narrower than the interval for the published data (0.009, 0.278). Based on these 95\\% CIs, there was a significant improvement in QoL for heart failure patients in the intervention group. Meanwhile, in the \\textit{non}-bias corrected model, the point estimates obtained from the published data and the updated data were both higher than their corresponding estimates under the ABSORB model. However, there was no change in the clinical conclusion from our ORB correction, since the non-bias corrected model also showed a significant improvement in QoL for the intervention group. \n\n \n\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\hspace{.4cm} \\includegraphics[width=.55\\linewidth]{Figure1_legend.png} \\\\\n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_mu1_plots_before.pdf}\n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_mu2_plots_before.pdf}\n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_contour_plots_before.pdf} \\\\\n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_mu1_plots_after.pdf} \n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_mu2_plots_after.pdf} \n\t\\includegraphics[width=.28\\textwidth]{app_ABSORB_contour_plots_after.pdf}\n\t\\caption{Panels~(a)--(c) show the results for the meta-analysis of interventions on HF patients using the \\textit{published} data. Panels~(d)--(f) show the results using the \\textit{updated} data.  ReAd is plotted on the log-RR scale in panels (a), (c), (d), and (f). }\\label{HeartFailurePlots}\n\\end{figure}\n\nIn Figure~\\ref{HeartFailurePlots}, we plot the posterior distributions of ReAd and QoL for the ABSORB model (solid line) and the non-bias corrected model (dashed line). In panels (a)-(c), we plot the results based on the published data, and in panels (d)-(f), we plot the results based on the updated data. For ReAd, we obtained $D = 0.25$ on the published data and $D=0.26$ on the updated data. These $D$ measures reflect the non-negligible impact from outcome reporting bias. In this case, the shift in the ReAd posterior towards the null side was enough to qualitatively change the conclusions from our meta-analysis.\n\nFor QoL, we obtained $D=0.25$ on the published data and $D=0.17$ on the updated data. By procuring more QoL outcomes from some missing studies, the updated data was less subject to ORB. This was consistent with the lower $D$ measure for QoL in our second stage analysis. Panel (b) of Figure~\\ref{HeartFailureResults} and the middle two panels of Figure~\\ref{HeartFailurePlots}  illustrate that the unadjusted and adjusted results for QoL were more similar to each other in the updated data than in the published data. In particular, the Jaccard index for QoL was 0.62 in the published data and 0.72 in the updated data. The Jaccard index (length of the intersection of two intervals divided by the length their union) gives a measure of consistency between the non-bias corrected and bias-corrected 95\\% CIs, with a larger value indicating greater similarity. In many practical situations, it may not be possible for systematic reviewers to obtain an updated dataset. However, this case study validates that our method produces bias-corrected results that are more consistent with the unadjusted analyses when researchers \\textit{are} able to mitigate some of the ORB.\n\nOur findings have important implications for clinicians, policymakers, and HF patients. Reducing hospital readmission for HF patients has been the primary objective of these stakeholders \\citep{ZiaeianFonarow2016}, and this has been the rationale for employing interventions like TM and STS. However, our results suggest that these interventions may not significantly reduce the risk of readmission. On the other hand, there seems to be a significant improvement in quality of life for HF patients who receive these interventions, compared to the patients who receive usual care. Therefore, we may conclude that TM and STS are still beneficial for the quality of life of patients, but that other approaches may be needed to significantly reduce the risk of hospital readmission.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{Discussion}\n\nIn this article, we have introduced a Bayesian selection model for correcting and quantifying the impact of outcome reporting bias (ABSORB) in multivariate meta-analysis. Our model enables us to not only correct the estimates of treatment effects, but also quantifies their uncertainty due to the presence of ORB. We employed the $D$ measure \\eqref{Dmeasure} to quantify the \\textit{impact} of ORB on the results of MMA by measuring the dissimilarity between the bias-corrected and non-bias corrected posterior densities. Our approaches were empirically evaluated through a meta-evaluation of 748 real systematic reviews from the Cochrane Database. In addition, we applied the ABSORB model to a meta-analysis on the effects of interventions on quality of life and hospital readmission for heart failure patients. Our results show that the presence of ORB can lead to qualitative differences in the conclusions from MMA. In particular, the relative risk of hospital readmission for HF patients in the intervention group shifted from a significant decrease (RR: 0.931, 95\\% CI 0.862--0.993) to a statistically \\textit{nonsignificant} effect (RR: 0.955, 95\\% CI 0.876--1.051) once we adjusted for ORB. Furthermore, we found in our meta-evaluation that after correcting for ORB, 157 out of 748 bivariate meta-analyses from the Cochrane Database \\textit{also} had a change in statistical significance for at least one outcome. Our study demonstrates the importance of accounting for ORB when conducting MMA.\n\nIn this paper, we focused on bivariate meta-analysis. However, the ABSORB model can also be extended to models with more than two outcomes. Suppose that we have $p$ outcomes of interest. When $p>2$, we can model each of the outcomes $y_{ij}$, $j = 1, \\ldots, p$, exactly as we did in the bivariate case through \\eqref{YgivenZ}--\\eqref{latentZ}. We also model each of the correlation parameters that controls the likelihood of reporting, $\\rho_j := \\textrm{corr}(\\epsilon_{ij}, \\delta_{ij})$, and the correlations between $\\epsilon_{ij}$, $\\epsilon_{ij'}$, $u_{ij}$ and $u_{ij'}$ for $j \\neq j'$ similarly as in \\eqref{EpsilonDeltaCorrelation}--\\eqref{BetweenStudyCorrelation}. While this extension of ABSORB to $p>2$ endpoints is straightforward, the potential downside is that the number of correlation parameters to estimate can be very large if $p$ is even moderately large. Thus, it may be desirable to simplify the correlation structure when $p$ is large so that the model remains parsimonious.\n\nAnother limitation when $p > 2$ is that the ABSORB model requires consideration of $2^p-1$ scenarios to completely specify its likelihood (e.g., studies with no missing endpoints, studies with only the first endpoint missing, studies with only the first two endpoints missing, etc.). While this is feasible for small $p$, it may become cumbersome if $p$ is moderately large. In the future, we plan to explore computationally efficient ways to extend the ABSORB model to handle a larger number of endpoints. This will make our model more appealing not just for MMA, but also for network meta-analysis (NMA). NMA expands the scope of a pairwise meta-analysis by simultaneously making comparisons across trials based on a common comparator (e.g., a standard treatment) \\citep{Lumley2002}. NMA combines direct evidence and indirect evidence under the assumption of evidence consistency. Ignoring the impact of ORB in NMA can lead to bias in both direct evidence and indirect evidence. Thus, it is critical to develop new approaches to account for ORB in the NMA framework.\n\nWhile the $D$ measure \\eqref{Dmeasure} that we introduced in Section~\\ref{QuantifyingORB} is a useful statistic for summarizing the \\textit{sensitivity} of the results from MMA to ORB, there are several limitations to it. First, the $D$ measure does not take into account the \\textit{direction} of the bias. Second, the $D$ measure does not have a variance estimate associated with it. Thus, unlike the $I^2$ statistic \\citep{higgins2002quantifying} or other measures for quantifying PB \\citep{LinChu2018}, there is no natural way of forming $100(1-\\alpha)\\%, \\alpha \\in (0,1)$, uncertainty intervals for the $D$ measure. One possibility is to calculate the $D$ measure on many independent, slightly perturbed datasets and to use the quantiles of the subsequent empirical distribution to obtain an interval estimate for $D$. However, this approach is also limited, and it is desirable to find more straightforward ways of obtaining \\textit{interval} estimates for $D$. In the future, we hope to develop measures that not only quantify the impact of ORB, but that can also take into account both the direction of the bias and the inherent uncertainty of the measure itself.\n\n\\section*{Code}\n\nAn \\textsf{R} package for implementing the model in this paper is available at \\url{https:\/\/github.com\/raybai07\/ABSORB}.\n\n\\section*{Acknowledgments}\nWe acknowledge Dr. Brian Finkelman for his help in collecting the intervention studies for the case study in this paper. This work was initiated when the first listed author was a postdoctoral researcher at the University of Pennsylvania under the supervision of the last listed author.\n\n\\section*{Funding}\nThis research was supported in part by generous funding from the College of Arts and Sciences at the University of South Carolina (RB), National Science Foundation grant OIA-1655740 (RB), National Institutes of Health (NIH) grants 1R01LM012607, 1R01AI130460, 1R01AG073435, 1R56AG074604, 1R01LM013519, 1R56AG069880 (XL and YC), and NIH grants R01LM012982 and UL1TR002494 (HC). This work was supported partially through Patient-Centered Outcomes Research Institute (PCORI) Project Program Awards (ME-2019C3-18315 and ME-2018C3-14899). All statements in this report, including its findings and conclusions, are solely those of the authors and do not necessarily represent the views of the Patient-Centered Outcomes Research Institute (PCORI), its Board of Governors or Methodology Committee.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{S1. Electronic band structures}\n\\setlength{\\parindent}{0pt}\n\\begin{figure}[h]\n    \\centering\n    {\\includegraphics[width=0.8\\textwidth]{SI\/Fig_S1_band.pdf}} \\\\\n    \\caption{Electronic band structures of (a) \\ce{Sb2S3} and (b) \\ce{Sb2Se3}.}\n    \\label{fig_band}\n\\end{figure}\n\n\\section*{S2. Fr{\\\"o}hlich polaron coupling constants}\n\\begin{table}[h]\n \\label{tab_alpha}\n \\caption{Parameters used to calculate Fr{\\\"o}hlich polaron coupling constant $\\alpha$ in this paper. The effective phonon frequency ($\\omega$) is in THz}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}ccccccc}\n\\hline\n\\multirow{2}{*}{Material} & \\multirow{2}{*}{} & \\multirow{2}{*}{\\textit{\\textepsilon}$_{\\infty}$} & \\multirow{2}{*}{\\textit{\\textepsilon}$_0$} & \\multirow{2}{*}{\\textit{$\\omega$}} & \\multicolumn{2}{c}{\\textit{m}$^*$} \\\\ \\cline{6-7} \n &  &  &  &  & e & h \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2S3}} & avg & \\multirow{4}{*}{10.26} & \\multirow{4}{*}{68.76} & \\multirow{4}{*}{3.49} & 0.40 & 0.64 \\\\\n & \\textit{x} &  &  &  & 0.16 & 0.47 \\\\\n & \\textit{y} &  &  &  & 0.92 & 0.65 \\\\\n & \\textit{z} &  &  &  & 5 & 0.97 \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2Se3}} & avg & \\multirow{4}{*}{13.52} & \\multirow{4}{*}{76.27} & \\multirow{4}{*}{2.57} & 0.35 & 0.9 \\\\\n & \\textit{x} &  &  &  & 0.14 & 0.85 \\\\\n & \\textit{y} &  &  &  & 0.81 & 0.55 \\\\\n & \\textit{z} &  &  &  & 7 & 3 \\\\ \\hline\n\\end{tabular*}\n\\end{table}\n~\\\\\nThe long-range electron-longitudinal optical phonon coupling can be expressed by the dimensionless Fr{\\\"o}hlich polaron coupling constant $\\alpha$\\cite{frohlich1952interaction}\n\\begin{equation}\n\\alpha=\\frac{e^2}{\\hbar}(\\frac{1}{\\varepsilon_\\infty}-\\frac{1}{\\varepsilon_0})\\sqrt{\\frac{m^*}{2\\hbar\\omega}},\n\\end{equation}\nwhere \\textit{\\textepsilon}$_{\\infty}$ and \\textit{\\textepsilon}$_0$ are the high-frequency and static dielectric constants, respectively, \\textit{m}$^*$ is the effective mass and $\\omega$ is the effective phonon frequency. The effective mass and effective frequency were calculated using the AMSET package\\cite{ganose2021efficient}.\nThe isotropic $\\alpha$ was obtained using the harmonic mean of the effective masses and the arithmetic average of the dielectric constants. The anisotropic $\\alpha$ was calculated using the anisotropic (direction-dependent) effective masses, consistent with previous work\\cite{guster2021frohlich}.\n~\\\\\n\\section*{S3. Effect of grain boundary scattering}\n\n\\setlength{\\parindent}{0pt}\n\\begin{figure}[ht]\n    \\centering\n    {\\includegraphics[width=1.0\\textwidth]{SI\/Fig_S2_mobility_mfp.pdf}} \\\\\n    \\caption{Calculated component and total mobilities with mean free path of (a) \\SI{100}{\\nanometre} and (b) \\SI{10}{\\nm} as a function of temperature.}\n    \\label{fig_mfp}\n\\end{figure}\n\nThe effect of grain boundary scattering on the mobility in \\ce{Sb2X3} was evaluated by incorporating an average grain size using the AMSET package\\cite{ganose2021efficient}. The grain boundary scattering lifetime is set to $v_g\/L$, where $v_g$ is the group velocity and \\textit{L} is the mean free path. In this work, the mean free path of \\SI{10} and \\SI{100}{\\nm} were tested. The carrier concentration and defect concentration were assumed to be \\SI{e13}{\\conc} and \\SI{e17}{\\conc}, respectively. According to our results (Fig. S2), at temperatures between 100 and \\SI{500}{\\kelvin}, the total mobility is not limited by the grain boundary scattering.\n\n\\section*{S4. Workflow of localising a polaron in \\ce{Sb2X3}}\n\nWe attempted to localise an electron or a hole in \\ce{Sb2S3} and \\ce{Sb2Se3} by the bond distortion method and electron attractor method. A 3$\\times$1$\\times$1 supercell (with the dimension of 11.40$\\times$11.20$\\times$\\SI{11.39}{\\cubic\\angstrom} and 11.85$\\times$11.55$\\times$\\SI{11.93}{\\cubic\\angstrom} for \\ce{Sb2S3}  and \\ce{Sb2Se3}, respectively) was constructed. In each system, one electron per supercell was added or removed to introduce an electron or a hole.\n~\\\\\n~\\\\\nWe first applied the bond distortion method to introduce distortions around one designated atom (Sb for adding an electron and S\/Se for adding a hole) and add small random displacements to all atoms. These are implemented by the ShakeNBreak package\\cite{shakenbreak_github,mosquera2021search}. Different distortions between 20\\% and 40\\% with both compression and stretching were considered and a standard deviation of 0.15 was used for the random displacements. However, after structural optimisation, all structures relaxed to perfect configurations.\n~\\\\\n~\\\\\nWe further combined the bond distortion method with the electron attractor method to confirm the formation of hole polarons in \\ce{Sb2S3}. The electron attractor method refers to attracting electrons or holes to a particular atomic site by replacing one certain atom. Phosphorous has stronger attraction to holes than sulfur as it contains fewer protons and has a lower electronegativity. Here, we used one P to replace one S in a supercell, and also introduced some local perturbations around the P atom. The number of electrons were kept the same as the neutral replaced system, suggesting one extra hole in \\ce{Sb2S3}. The structure with the substituted atom and local distortions was fully relaxed. Finally, we replaced back the S atom and relaxed the configuration again. Nevertheless, all structures went back to perfect configurations, indicating that the localised polarons are unlikely to form.\n\n\\section*{S5. Parameters used to calculate mobilities in \\ce{Sb2X3}}\nThe \\textit{k}-point meshes used to calculate transport properties were tested (shown in Fig. \\ref{fig_con}) and a \\textit{k}-point mesh of 169$\\times$57$\\times$57 is used for all calculations. The carrier concentration was set to \\SI{e13}{\\conc} according to previous experimental results in \\ce{Sb2X3} \\cite{chen2017characterization,liu2016green,zhou2014solution,yuan2016rapid,li2021defect,chalapathi2020influence,black1957electrical}. The calculated effective phonon frequency is 3.49 for \\ce{Sb2S3} and 2.57 for \\ce{Sb2Se3}. The calculated deformation potentials, elastic constants and dielectric constants are shown in Table \\ref{tab_def}, \\ref{tab_ela} and \\ref{tab_die}, respectively.\n\n\\begin{figure}[ht]\n    \\centering\n    {\\includegraphics[width=0.8\\textwidth]{SI\/Fig_S3_mobility_convergence.pdf}} \\\\\n    \\caption{The convergence of mobility in \\ce{Sb2X3} under different \\textit{k}-point meshes. The defect concentration is set to be \\SI{e14}{\\conc} and the temperature is set to be \\SI{300}{\\kelvin}.}\n    \\label{fig_con}\n\\end{figure}\n\n\\begin{table}[ht]\n\\caption{Calculated deformation potentials (D, eV) for the upper valence and lower conduction bands of \\ce{Sb2S3} and \\ce{Sb2Se3}}\n  \\label{tab_def}\n  \\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}cccccc}\n\\hline\nMaterial & \\multicolumn{2}{l}{} & D$_{XX}$ & D$_{YY}$ & D$_{ZZ}$ \\\\ \\hline\n\\multirow{6}{*}{\\ce{Sb2S3}} & \\multirow{3}{*}{VBM} & D$_{XX}$ & 5.41 & 0.26 & 0.07 \\\\\n &  & D$_{YY}$ & 0.26 & 0.10 & 0.02 \\\\\n &  & D$_{ZZ}$ & 0.07 & 0.02 & 1.27 \\\\ \\cline{2-6}\n & \\multirow{3}{*}{CBM} & D$_{XX}$ & 5.26 & 0.42 & 0.17 \\\\\n &  & D$_{YY}$ & 0.42 & 2.43 & 3.35 \\\\\n &  & D$_{ZZ}$ & 0.17 & 3.35 & 2.62 \\\\ \\hline\n\\multirow{6}{*}{\\ce{Sb2Se3}} & \\multirow{3}{*}{VBM} & D$_{XX}$ & 0.53 & 0.16 & 0.05 \\\\\n &  & D$_{YY}$ & 0.16 & 2.86 & 0.03 \\\\\n &  & D$_{ZZ}$ & 0.05 & 0.03 & 2.47 \\\\ \\cline{2-6}\n & \\multirow{3}{*}{CBM} & D$_{XX}$ & 3.31 & 0.36 & 0.09 \\\\\n &  & D$_{YY}$ & 0.36 & 0.39 & 0.29 \\\\\n &  & D$_{ZZ}$ & 0.09 & 0.29 & 1.38 \\\\ \\hline\n\\end{tabular*}\n\\end{table}\n\n\\begin{table}[ht]\n \\caption{Calculated elastic constants (in GPa) of \\ce{Sb2S3} and \\ce{Sb2Se3}}\n  \\label{tab_ela}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c}\n\\hline\nMaterial &  & C$_{XX}$ & C$_{YY}$ & C$_{ZZ}$ & C$_{XY}$  & C$_{YZ}$ & C$_{ZX}$ \\\\ \\hline\n\\multirow{6}{*}{\\ce{Sb2S3}} & C$_{XX}$ & 93.75 & 28.00 & 18.50 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{YY}$ & 28.00 & 57.25 & 15.39 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{ZZ}$ & 18.50 & 15.39 & 37.69 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{XY}$ & 0.00 & 0.00 & 0.00 & 31.68 & 0.00 & 0.00 \\\\\n & C$_{YZ}$ & 0.00 & 0.00 & 0.00 & 0.00 & 17.11 & 0.00 \\\\\n & C$_{ZX}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 8.77 \\\\ \\hline\n\\multirow{6}{*}{\\ce{Sb2Se3}} & C$_{XX}$ & 77.15 & 25.63 & 17.11 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{YY}$ & 25.63 & 54.15 & 17.03 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{ZZ}$ & 17.11 & 17.03 & 31.75 & 0.00 & 0.00 & 0.00 \\\\\n & C$_{XY}$ & 0.00 & 0.00 & 0.00 & 23.42 & 0.00 & 0.00 \\\\\n & C$_{YZ}$ & 0.00 & 0.00 & 0.00 & 0.00 & 18.41 & 0.00 \\\\\n & C$_{ZX}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 5.08 \\\\ \\hline\n\\end{tabular*}\n\\end{table}\n\n\\begin{table}[ht]\n\\caption{Calculated static (\\textit{\\textepsilon}$_0$) and high-frequency (\\textit{\\textepsilon}$_{\\infty}$) dielectric constants of \\ce{Sb2S3} and \\ce{Sb2Se3}}\n \\label{tab_die}\n\\begin{tabular*}{1\\textwidth}{@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c@{\\extracolsep{\\fill}}c}\n \\hline\n \\multirow{2}{*}{Material} & \\multicolumn{3}{c}{\\textit{\\textepsilon}$_0$} & \\multicolumn{3}{c}{\\textit{\\textepsilon}$_{\\infty}$} \\\\\n  & \\textit{x}     & \\textit{y}      & \\textit{z}    & \\textit{x}      & \\textit{y}     & \\textit{z}    \\\\ \\hline\n  Sb$_2$S$_3$                               & 98.94 & 94.21  & 13.14 & 11.55      & 10.97     & 8.25    \\\\ \n  Sb$_2$Se$_3$                              & 85.64 & 128.18 & 15.00  & 15.11  & 14.92 & 10.53 \\\\ \\hline\n\\end{tabular*}\n\\end{table}\n\n\\clearpage\n\n\\section{Introduction}\nAntimony chalcogenides (\\ce{Sb2X3}; X=S, Se) have emerged as promising light absorbing materials due to their attractive electronic and optical properties, including ideal band gaps (\\SIrange{1.1}{1.8}{\\electronvolt}) and high optical absorption coefficients (\\textgreater \\SI{e5}{\\per\\cm})\\cite{versavel2007structural,liu2016green,messina2009antimony,lai2012preparation,chen2015optical,vadapoo2011self,vadapoo2011electronic,nasr2011electronic,savory2019complex,lei2019review,wang2022lone}. \nThey are binary compounds with earth-abundant, low-cost and non-toxic constituents.\nThe \\ac{PCEs} in \\ce{Sb2X3} solar cells have improved rapidly over the past decade, with record efficiencies reaching \\SI{7.50}{\\percent} and \\SI{10.12}{\\percent} for \\ce{Sb2S3} and \\ce{Sb2Se3}, respectively\\cite{choi2014highly,duan2022effi}. Nevertheless, efficiencies are still well below those seen in state-of-the-art CdTe or hybrid halide perovskite devices, which have reached above \\SI{25}{\\percent} under laboratory conditions\\cite{green2021solar}.\n\nThe underlying efficiency bottleneck is unclear. While the structural, electronic and optical properties of \\ce{Sb2X3} have been widely investigated,\nthe charge carrier dynamics, which critically affect conversion efficiencies, remain controversial. Charge carrier transport in \\ce{Sb2X3} has been reported by several studies\\cite{yang2019ultrafast,grad2021charge,zhang2021suppressing,grad2020photoexcited,chen2017characterization}, but there are several fundamental questions that remain unanswered. The first is whether the nature of carrier transport is band-like or thermally-activated hopping. \\citet{yang2019ultrafast} studied the charge carrier dynamics in \\ce{Sb2S3} and ascribed the observed \\SI{0.6}{\\electronvolt} Stokes shift to self-trapped excitons, suggesting hopping transport. \nIn contrast, \\citet{liu2022ultrafast} and \\citet{zhang2021suppressing} argued against self-trapping in \\ce{Sb2Se3} due to the saturation of fast signal decay with increasing carrier density. \nConsidering it is challenging for direct measurements to distinguish whether the photoexcited carriers are intrinsically self-trapped or trapped at defect sites\\cite{ramo2007theoretical}, a systematic theoretical study on the carrier transport in \\ce{Sb2X3} is necessary. \nThe second issue is about the resulting charge carrier mobility. \nMeasured mobilities in \\ce{Sb2X3} show a large variation\\cite{chen2017characterization,liu2016green,zhou2014solution,yuan2016rapid,li2021defect,chalapathi2020influence,black1957electrical}, in part due to different synthesis and characterisation methods.\nAs such, the intrinsic limits to mobility in \\ce{Sb2X3} are unclear and the scattering physics underlying transport are not yet understood.\n\nIn this work, we studied the tendency for polaron trapping and its effect on charge carrier transport in \\ce{Sb2X3} by first-principles \\ac{DFT} and Boltzmann transport calculations. The electron-lattice interaction in \\ce{Sb2X3} was explored through the Fr{\\\"o}hlich polaron coupling constant and Schultz polaron radius. Modelling of electron and hole polarons in \\ce{Sb2X3} indicates the intrinsic formation of large polarons and contrast to recent suggestions of small polarons (i.e.~self-trapped carriers)\\cite{yang2019ultrafast,grad2021charge}. The prediction of large polaron formation is further reinforced by the results of carrier transport calculations. \nThe isotropically averaged mobilities are larger than \\SI{10}{\\mob} at room temperature and decrease with increasing temperature for both electrons and holes, further confirming the band-like transport in \\ce{Sb2X3}. We find the intrinsic mobility is limited by scattering from polar optical phonons at low and moderate defect concentrations, while at high charged defect concentrations (\\textgreater \\SI{e18}{\\conc}) impurity scattering dominates. We expect our results will enable the design of \\ce{Sb2X3} devices with improved efficiencies.\n\n\\ce{Sb2X3} crystallise in the orthorhombic \\textit{Pnma} space group and are comprised of strongly bonded quasi-\\ac{1D} [Sb$_4$X$_6$]$_n$ ribbons oriented along the [100] direction (Fig.~\\ref{fig_structure}). Ribbon formation is driven by the Sb lone pair with ribbons stacked together by weak interactions\\cite{wang2022lone}. According to our previous optimization using the HSE06 hybrid functional and D3 dispersion correction,\\cite{wang2022lone} the calculated lattice parameters are 3.80\/\\SI{3.95}{\\angstrom}, 11.20\/\\SI{11.55}{\\angstrom} and 11.39\/\\SI{11.93}{\\angstrom} for \\ce{Sb2S3}\/\\ce{Sb2Se3} along the \\textit{a}, \\textit{b} and \\textit{c} axes, respectively.\n\\ce{Sb2X3} are indirect band gap semiconductors with calculated indirect\/direct band gaps of 1.79\/\\SI{1.95}{\\electronvolt} and 1.42\/\\SI{1.48}{\\electronvolt}, respectively, which are in reasonable agreement with previous experimental \\cite{yesugade1995structural,el1998substrate,versavel2007structural,liu2016green,torane1999preparation,messina2009antimony,lai2012preparation,chen2015optical} and theoretical studies\\cite{vadapoo2011self,vadapoo2011electronic,caracas2005first,nasr2011electronic,savory2019complex}. The electronic band structures are shown in Fig.~S1 of the Supplementary Information.\nIt has been widely suggested that efficient transport can only happen along the ribbons, based on the understanding that \\ce{Sb2X3} are \\ac{1D} semiconductors\\cite{caruso2015excitons,song2017highly,guo2018tunable,yang2018adjusting,gusmao2019antimony}. However, neither the structural dimensionality nor the electronic dimensionality of \\ce{Sb2X3} is \\ac{1D}.\\cite{deringer2015vibrational,wang2022lone} \n\n\\begin{figure}[ht]\n    \\centering\n    {\\includegraphics[width=0.5\\textwidth]{Fig_1_structure}} \\\\\n    \\caption{Ground-state crystal structure (\\textit{Pnma} space group) of \\ce{Sb2X3}. The conventional unit cell is represented by a rectangle.}\n    \\label{fig_structure}\n\\end{figure}\n\n\n\n\\begin{table}[ht]\n \\caption{Calculated Fr{\\\"o}hlich parameter ($\\alpha$) and Schultz polaron radius (r$_f$) for electrons (e$^-$) and holes (h$^+$) in \\ce{Sb2S3} and \\ce{Sb2Se3} at T = \\SI{300}{\\kelvin}}\n \\label{tab_alpha}\n \\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}cccccc}\n\\hline\n\\multicolumn{1}{c}{\\multirow{2}{*}{Material}} & \\multicolumn{1}{c}{\\multirow{2}{*}{}} & \\multicolumn{2}{c}{$\\alpha$} & \\multicolumn{2}{c}{r$_f$ (\\AA)} \\\\ \\cline{3-6} \n\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{e$^-$} & \\multicolumn{1}{c}{h$^+$} & \\multicolumn{1}{c}{e$^-$} & \\multicolumn{1}{c}{h$^+$} \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2S3}} & avg & 1.6 & 2.0 & 45.5 & 40.4 \\\\\n & \\textit{x} & 1.0 & 1.8 & 57.3 & 43.7 \\\\\n & \\textit{y} & 2.4 & 2.1 & 36.9 & 40.3 \\\\\n & \\textit{z} & 5.7 & 2.5 & 23.7 & 36.4 \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2Se3}} & avg & 1.3 & 2.1 & 40.5 & 31.9 \\\\\n & \\textit{x} & 0.8 & 2.0 & 50.9 & 32.4 \\\\\n & \\textit{y} & 2.0 & 1.6 & 32.8 & 36.1 \\\\\n & \\textit{z} & 5.8 & 3.8 & 18.8 & 23.5 \\\\ \\hline\n \\end{tabular*}\n\\end{table}\n\nCharge carriers in crystals are formally described as quasi-particles due to their interaction with the extended structure. In polar semiconductors, the charge carriers and the surrounding lattice deformation form a so-called polaron,\\cite{emin2013polarons} which determines the nature of carrier transport.\nPolarons can be classified into two types based on the strength of electron-phonon coupling. Stronger coupling leads to larger local lattice distortion which provides the driving force for small polarons to form. Thus, for a small polaron, the lattice deformation is usually confined to one unit cell, and a carrier's motion is typically incoherent with thermally activated hops which lead to low mobility ($\\ll$ \\SI{1}{\\mob}). By contrast, the lattice deformation in a large polaron is usually moderate and spreads over multiple unit cells, resulting in a larger mobility (\\textgreater {} \\SI{1}{\\mob}). \nIn polar crystals, the electron-phonon interaction is usually dominated by the coupling of charge carriers to the \\ac{LO} phonons, which can be described within the Fr{\\\"o}hlich model\\cite{frohlich1952interaction}. \n\nWe first evaluate the Fr{\\\"o}hlich interaction by the coupling constant $\\alpha$. The calculated $\\alpha$ (shown in Table \\ref{tab_alpha}) shows an isotropically averaged value of $\\sim$2 for both \\ce{Sb2S3} and \\ce{Sb2Se3}, which falls in the intermediate electron-phonon coupling regime (defined as 0.5 $\\lesssim \\alpha \\lesssim$ 6).\\cite{stoneham2001theory} The magnitude of $\\alpha$ along the [100] and [010] directions is quite close ($\\Delta \\alpha$ = 1.2--1.4 and 0.3--0.4 for electrons and holes, respectively), suggesting similar electron-phonon interaction strengths along these two directions. We further estimate the size of polarons in \\ce{Sb2X3} by the Schultz polaron radius (r$_f$)\\cite{schultz1959slow}. The large values of electron and hole polaron radii (which extend over multiple structural units) indicate the polarons are delocalised in both \\ce{Sb2S3} and \\ce{Sb2Se3}. The details of parameters used and the procedure for averaging $\\alpha$ can be found in Section S2 of the Supplementary Information.\n\nFor an alternative assessment, we performed direct first-principles \\ac{DFT} calculations to model charge carriers in \\ce{Sb2X3}.\nThere are two challenges for reliable polaron modelling.\nThe first is the self-interaction error\\cite{parr1989w} arising from the approximate form of the exchange-correlation functional which causes electrons to spuriously delocalise\\cite{pacchioni2008modeling,pham2020efficient}. This is typically resolved by employing a hybrid functional\\cite{finazzi2008excess,deak2011polaronic,di2006electronic} which incorporates a certain amount of exact Fock exchange or by a Hubbard correction (DFT+U)\\cite{dudarev1998electron,anisimov1997first}.  \nSecondly, the formation of localised polarons is dependent on the initial geometries and wavefunctions. Different methods have been proposed to break the crystal symmetry and promote the formation of localised states. Among them, the bond distortion method and electron attractor method have proved reliable across a range of structures and chemistries\\cite{ramo2007theoretical,pham2020efficient,deskins2011distribution,deskins2009localized,shibuya2012systematic,hao2015coexistence,liu2019photocatalytic}. The former involves introducing local perturbations in a supercell in a region where the polaron is expected to localise, while the latter uses a temporarily-substituted atom to attract an electron, which is then removed and the structure re-relaxed.\nIn this work, all polaron calculations were performed using the HSE06 hybrid functional. We attempted to localise electron and hole polarons by adding or removing an electron from a \\ce{Sb2X3} supercell using both these distortion methods. The full computational details are provided in Section S4. No energy lowering distortions were found in any case. The electrons and holes always preferred to delocalise rather than localise in both \\ce{Sb2S3} and \\ce{Sb2Se3}, indicating again that small polarons are unlikely to form intrinsically by self-trapping. This is also supported by recent experimental evidence that the trap states in \\ce{Sb2Se3} are saturated by moderate density photocarriers and the free carrier lifetime is sensitive to the impurity density, which together exclude the possibility of self-trapping in \\ce{Sb2Se3}\\cite{liu2022ultrafast}.\n\nWe next consider the possibility of forming self-trapped excitons. Firstly, the large dielectric constants ($\\sim$ 100) and small effective masses ($\\sim$ 0.1) in \\ce{Sb2X3}\\cite{wang2022lone} suggest that the Coulomb interaction is strongly screened and a large exciton radius is favoured. The small experimental exciton binding energies (\\SIrange{0.01}{0.05}{\\electronvolt} for \\ce{Sb2S3} and \\SI{0.04}{\\electronvolt} for \\ce{Sb2Se3})\\cite{caruso2015excitons,lawal2018investigation} further indicate weak electron-hole interactions in \\ce{Sb2X3}. Additionally, experimental measurements of the imaginary part of the frequency-dependent complex photoconductivity in \\ce{Sb2Se3} do not reveal any negative components\\cite{wang2019both} that can be a signal of exciton formation. \nConsequently, we conclude that self-trapped excitons in \\ce{Sb2X3} are unlikely.\n\n\\begin{figure*}[t]\n    \\centering\n    {\\includegraphics[width=1.0\\textwidth]{Fig_2_mobility}} \\\\\n    \\caption{(a) Calculated average mobilities of electrons and holes in \\ce{Sb2S3} and \\ce{Sb2Se3} as a function of temperature with different defect concentrations. (b) Calculated total and component mobilities as a function of bulk defect concentration at \\SI{300}{\\kelvin}. ADP, acoustic deformation potential; POP, polar optical phonon; IMP, ionized impurity. $N_D$, defect concentration.}\n    \\label{fig_mobility_avg}\n\\end{figure*}\n\n\\begin{figure}[t]\n    \\centering\n    {\\includegraphics[width=0.5\\textwidth]{Fig_3_ani_mobility}} \\\\\n    \\caption{The anisotropic net carrier mobilities including all scattering mechanisms in \\ce{Sb2S3} and \\ce{Sb2Se3} as a function of temperature with a bulk defect concentration of \\SI{e14}{\\conc}.}\n    \\label{fig_mobility_ani}\n\\end{figure}\n\n\\begin{table}[ht]\n \\caption{Calculated mobilities of electrons ($\\mu_e$) and holes ($\\mu_h$) in \\ce{Sb2X3} at \\SI{300}{\\kelvin} under different defect concentrations ($N_D$) and experimental values for comparison. The anisotropy ratio (\\textit{a}$_r$) is defined as the ratio of\nmaximum to minimum mobility}\n \\label{tab_mobility}\n  \\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}ccccccc}\n \\hline\n Material & \\multicolumn{2}{c}{} & \\multicolumn{3}{c}{Calculated (\\si{\\mob})} & Experiment (\\si{\\mob}) \\\\\n\\hline\n &  &  & \\multicolumn{3}{c}{$N_D$ (cm$^{-3}$)}  &  \\\\ \n &  &  & 10$^{14}$ & 10$^{17}$ & 10$^{20}$ &  \\\\ \\cline{4-6} \n\\multirow{8}{*}{\\ce{Sb2S3}} & \\multirow{4}{*}{$\\mu_e$} & \\textit{x} & 53.90 & 44.72 & 0.96 &  \\\\\n &  & \\textit{y} & 9.60 & 7.13 & 0.07 &  \\\\\n &  & \\textit{z} & 1.88 & 1.35 & 0.01 &  \\\\\n &  & avg & 21.79 & 17.73 & 0.35 &  \\\\ \n &  & \\textit{a}$_r$ & 28.67 & 33.13 & 96.00 & \\\\ \\cline{2-7} \n & \\multirow{4}{*}{$\\mu_h$} & \\textit{x} & 18.58 & 15.90 & 0.38 &  \\\\\n &  & \\textit{y} & 13.53 & 11.33 & 0.19 &  \\\\\n &  & \\textit{z} & 9.34 & 8.35 & 0.22 &  \\\\\n &  & avg & 13.82 & 11.86 & 0.26 & 6.4-12.8\\cite{liu2016green}, 32.2-54.0\\cite{chalapathi2020influence} \\\\ \n  &  & \\textit{a}$_r$ & 1.99 & 1.90 & 2.00 & \\\\ \\hline\n\\multirow{8}{*}{\\ce{Sb2Se3}} & \\multirow{4}{*}{$\\mu_e$} & \\textit{x} & 89.97 & 76.38 & 1.96 &  \\\\\n &  & \\textit{y} & 16.74 & 11.65 & 0.11 &  \\\\\n &  & \\textit{z} & 1.94 & 1.41 & 0.01 &  \\\\\n &  & avg & 36.22 & 29.81 & 0.70 & 15\\cite{black1957electrical} \\\\ \n  &  & \\textit{a}$_r$ & 46.38 & 54.17 & 196.00 & \\\\ \\cline{2-7}\n & \\multirow{4}{*}{$\\mu_h$} & \\textit{x} & 9.50 & 8.38 & 0.17 & 2.59\\cite{chen_characterization_2017} \\\\\n &  & \\textit{y} & 16.95 & 14.63 & 0.25 & 1.17\\cite{chen_characterization_2017} \\\\\n &  & \\textit{z} & 2.22 & 1.95 & 0.06 & 0.69\\cite{chen_characterization_2017} \\\\\n &  & avg & 9.55 & 8.32 & 0.16 & 5.1\\cite{zhou2014solution}, 3.7-21.88\\cite{yuan2016rapid}, 45\\cite{black1957electrical} \\\\ \n  &  & \\textit{a}$_r$ & 7.64 & 7.50 & 4.17 & \\\\ \\hline\n\\end{tabular*}\n\\end{table}\n\n\nTo further understand the nature of transport in \\ce{Sb2X3} the first-principles carrier mobility\\cite{ganose2021efficient} was calculated. Both \\textit{n}-type and \\textit{p}-type doping were investigated, with calculations including scattering from \\ac{IMP}, \\ac{ADP} and \\ac{POP}.\nPiezoelectric scattering was not considered due to the centrosymmetric crystal structure.\nThe isotropically averaged mobilities are reasonably high at room temperature (T = \\SI{300}{\\kelvin}) for both electrons ($\\sim$\\SI{40}{\\mob}) and holes ($\\sim$\\SI{15}{\\mob}), at low and moderate defect concentrations ($<$\\SI{1e18}{\\conc}), indicating band-like transport (Fig.~\\ref{fig_mobility_avg}a).\nThe hole mobilities are a little lower than the electron mobilities in both \\ce{Sb2S3} and \\ce{Sb2Se3}, suggesting that \\textit{n}-type doping could be beneficial for carrier collection in photovoltaic devices.\nThis is in contrast to experimental measurements that have indicated higher mobility for \\textit{p}-type \\ce{Sb2Se3},\\cite{black1957electrical} however, this may be related to the doping asymmetry in these materials.\nThe intrinsic mobility is limited by Fr{\\\"o}hlich-type polar optical phonon scattering suggesting that large polarons are responsible for the transport behaviour (Fig.~\\ref{fig_mobility_avg}b).\nWe note that large deformation potentials have been suggested as the origin of self-trapping in the bismuth double perovskites\\cite{wu2021strong}.\nHowever, in \\ce{Sb2X3}, acoustic deformation potential scattering is weak (due to small deformation potentials $<$ \\SI{6}{\\electronvolt}), similar to that seen in the hybrid halide perovskites\\cite{wright2016electron,lu2017piezoelectric}, indicating self-trapping is unlikely to occur via coupling with acoustic vibrations.\n\nThe scattering from ionized impurities increases with the defect concentration. \nAt concentrations around \\SI{e18}{\\conc}, \\ac{IMP} and \\ac{POP} scattering are roughly the same strength and cause the mobility to reduce by a factor of a half (Fig.~\\ref{fig_mobility_avg}b).\nAt higher defect concentrations transport is entirely dominated by ionized impurities.\nOur results indicate that careful control of defect concentrations are essential for preventing degradation of device efficiencies.\nThis agrees well with previous experimental reports that the defect density is crucial to the carrier transport in \\ce{Sb2X3}, whereby bulk defect densities above \\SI{e15}{\\conc} led to significant degradation in conversion efficiency \\cite{islam2020two,li2020simulation,khadir2022performance}. \nFurthermore, considering that most experimental mobility measurements in \\ce{Sb2X3} were obtained from thin films where grain boundary scattering will further lower the mobility, we also tested the inclusion of mean free path scattering. According to our results (Fig. S2), the mobilities in \\ce{Sb2X3} are not significantly affected by grain boundary scattering even with grain sizes down to \\SI{10}{\\nm}, much smaller than the domain sizes typically seen in experiments\\cite{rijal2021influence,maghraoui2010structural,perales2008optical,lokhande2001novel}.\nAccordingly, our results suggest that grain boundary scattering is unlikely to be a dominant source of scattering in \\ce{Sb2X3} thin films, in agreement with previous studies\\cite{gonzalez2022deciphering}.\n\nThe anisotropy of mobility was also considered. As shown in Table \\ref{tab_mobility} and Fig.~\\ref{fig_mobility_ani}, our calculated mobilities are in reasonable agreement with the range of measured values.\nFor electron transport, there is considerable anisotropy with the [100] direction showing roughly 5 times the mobility of the [010] direction and over 25 times the mobility of the [001] direction in both \\ce{Sb2S3} and \\ce{Sb2Se3}.\nFor holes in \\ce{Sb2S3}, there is a high mobility in the (001) plane where the transport is roughly isotropic and approximately twice that of the [001] direction.\nFor holes in \\ce{Sb2Se3}, the picture is slightly altered with the highest mobility seen along [010], roughly 2 times the mobility along [100] and 8 times the mobility along [001].\nThe anisotropy in mobility follows the anisotropy in the calculated effective masses and the Fermi-surface dimensionality\\cite{wang2022lone}.\nDespite the anisotropic behaviour, even at moderate defect concentrations the electron and hole mobilities are still reasonably large ($>$\\SI{10}{\\mob}) in at least two directions.\nThe common description of \\ce{Sb2X3} as a \\ac{1D} semiconductors\\cite{zhou2015thin,liang2020crystallographic} oversimplifies the nature of transport.\nAccordingly, it may be possible to obtain high mobility thin films, even when the grains are not fully aligned along the direction of the quasi-1D ribbons.\n\nIn summary, we investigated the nature of charge carriers in \\ce{Sb2X3} semiconductors. \nOur results strongly suggest that self-trapping (i.e. the formation of small polarons) is unlikely to occur and that instead charge transport involves large polarons. \nIn particular, we found:\ni) moderate Fr{\\\"o}hlich coupling constants ($\\sim$2); ii) high Schultz polaron radii ($\\sim$\\SI{40}{\\angstrom}); \niii) the absence of electron or hole polaron formation in density functional theory calculations using the bond distortion and electron attractor methods; and iv) large carrier mobilities \\textgreater 10 cm$^2$\/Vs at room temperature for both electrons and holes (in agreement with experiments).\nWe conclude that there is no theoretical evidence for small polaron formation in pristine \\ce{Sb2X3} and self-trapping is unlikely to be the origin of the low open-circuit voltages in \\ce{Sb2X3} devices as reported in previous studies\\cite{yang2019ultrafast,grad2021charge}.\nAccordingly, the low photovoltages may not be a bulk property of these materials and could be surmountable with improved fabrication and processing conditions to engineer the defect and interfacial properties of devices.\n\n\\section{Methods}\nThe Fr{\\\"o}hlich polaron properties were solved using the open-source package \\textsc{PolaronMobility}\\cite{Frost2017}. \nThe first-principles carrier scattering rates and resulting mobilities were calculated using \\textsc{AMSET}\\cite{ganose2021efficient}. \nThe set of materials parameters used for these predictions are provided in Table S1--S4.\nThe crystal structure was plotted using \\textsc{Blender}\\cite{blender} and \\textsc{Beautiful Atoms}\\cite{Beautiful_Atoms2022}.\n\n\nAll of the underlying electronic structure calculations were performed based on Kohn-Sham density-functional theory\\cite{kohn1965self,dreizler1990density} as implemented in \\ac{VASP}\\cite{kresse1996efficient}. The projector augmented-wave (PAW) method\\cite{kresse1999ultrasoft} was employed with a plane-wave energy cutoff of \\SI{400}{\\electronvolt}. All calculations were carried out using the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06)\\cite{heyd2003hybrid,krukau2006influence} with the D3 dispersion correction\\cite{grimme2004accurate}. The atomic positions were optimised until the Hellman-Feynman forces on each atom were below \\SI{0.0005}{\\electronvolt\\per\\angstrom} for unit cells and \\SI{0.01}{\\electronvolt\\per\\angstrom} for 3$\\times$1$\\times$1 supercells. \nThe energy convergence criterion was set to \\SI{e-6}{\\electronvolt}. $\\varGamma$-centered \\textit{k}-point meshes were set to 7$\\times$2$\\times$2 and 2$\\times$2$\\times$2 for geometry optimisation with primitive unit cells and supercells, respectively. For uniform band structure calculations which were used as inputs for AMSET, a denser \\textit{k}-point mesh of 19$\\times$10$\\times$10 was used which is consistent with our previous calculations of carrier effective masses\\cite{wang2022lone}. Detailed settings and convergence data are presented in Section S5.\n\n\\section*{Acknowledgements}\nX.W. thanks Jarvist M. Frost and Yuchen Fu for valuable discussions.\nWe are grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP\/P020194\/1 and EP\/T022213\/1). X.W. acknowledges Imperial College London for a President's PhD Scholarship. A.M.G. was supported by EPSRC Fellowship EP\/T033231\/1. S.R.K. acknowledges the EPSRC Centre for Doctoral Training in the Advanced Characterisation of Materials (CDT-ACM)(EP\/S023259\/1) for a PhD studentship. \n\n\\section*{Author Contributions}\nThe author contributions have been defined following the CRediT system.\nX.W.: Conceptualization, Investigation, Formal analysis, Methodology, Visualization, Writing \u2013 original draft. \nA.M.G.: Methodology, Supervision, Writing \u2013 review \\& editing. \nS.R.K.: Methodology, Writing \u2013 review \\& editing. \nA.W.: Conceptualization, Methodology, Supervision, Writing \u2013 review \\& editing.\n\n\\section*{Data Access Statement}\n\nThe data supporting the findings reported in this study are openly available from \\url{https:\/\/nomad-lab.eu} at [DOI:xxx].\n\n\n\\section{Introduction}\nAntimony chalcogenides (\\ce{Sb2X3}; X=S, Se) have emerged as promising light absorbing materials due to their attractive electronic and optical properties, such as ideal band gaps and high optical absorption coefficients (\\textgreater 10$^{5}$ cm$^{-1}$)\\cite{wang2022lone}. They are simple binary compounds with earth-abundant, low-cost and non-toxic constituents\\cite{zeng2016antimony,lei2019review,dong2021boosting}. The \\ac{PCEs} in \\ce{Sb2X3} solar cells have improved rapidly over the past decade, with the record efficiencies reaching 7.5\\% and 9.2\\% for \\ce{Sb2S3} and \\ce{Sb2Se3}, respectively\\cite{choi2014highly,li20199}. Nevertheless, they are still far from those competing systems such as CdTe or perovskites solar cells (above 21\\% under laboratory conditions\\cite{green2021solar}) and have stagnated for a few years. \n\nThe underlying bottleneck of such unsatisfied efficiencies is elusive. While the structural, electronic and optical properties of \\ce{Sb2X3} have been widely investigated \\cite{wang2022lone,tideswell1957crystal,kyono2002low,messina2009antimony,chen2015optical,kocc2012first}, the charge carrier dynamics, which critically affect the conversion efficiencies, is still controversial in \\ce{Sb2X3}. Charge carrier transport in \\ce{Sb2X3} has been reported by several studies\\cite{yang2019ultrafast,grad2021charge,zhang2021suppressing,grad2020photoexcited,chen2017characterization}, but there are several fundamental questions remaining unclear. The first one is whether the nature of carrier transport in \\ce{Sb2X3} is band-like or thermally activated hopping. Yang et al. \\cite{yang2019ultrafast} studied the charge carrier dynamics in \\ce{Sb2S3} and ascribed the observed 0.6 eV Stokes shift to self-trapped excitons, which correspond to hopping transport. Other papers have not reached an agreement if self-trapping process occurs in \\ce{Sb2Se3}\\cite{zhang2021suppressing,grad2020photoexcited}. Considering it is challenging for experimentalists to distinguish whether the photoexcited carriers in \\ce{Sb2X3} are intrinsically self-trapped or trapped at extrinsic defect sites due to the inevitable imperfection in samples and small energies of self-trapping \\cite{ramo2007theoretical}, a systematically theoretical study on the carrier transport in \\ce{Sb2X3} is necessary, but still lacking. The second issue is about the resulting charge mobilities. Measured mobilities in \\ce{Sb2X3} show a large variation\\cite{chen2017characterization,liu2016green,zhou2014solution,yuan2016rapid,li2021defect,chalapathi2020influence,black1957electrical} due to different synthesis conditions of samples and different methods of measurement, and the scattering mechanism that limits the mobility in \\ce{Sb2X3} remains unknown. Besides, the direction of the carrier transport in \\ce{Sb2X3}, which is considered to be 1D (i.e. only efficient along the direction of ribbons)by previous papers\\cite{zhou2015thin,liang2020crystallographic}, has not been investigated theoretically. \n\nIn this work, we studied the polaron trapping and its effect on the charge carrier transport property in \\ce{Sb2X3} by first-principles \\ac{DFT} calculations. The electron-lattice interaction in \\ce{Sb2X3} was investigated by Fr{\\\"o}hlich polaron coupling constant and Schultz polaron radius. The results, together with our modelling of electron and hole polarons in \\ce{Sb2X3}, indicate the intrinsic formation of large polarons instead of small polarons (i.e. self-trapped carriers). The interpretation of large polaron formation is further reinforced by the results of carrier mobility. Our calculated mobilities are larger than 10 cm$^2$\/Vs at room temperature and decrease with increasing temperature for both electrons and holes, further confirming the band-like transport in \\ce{Sb2X3}. What's more, the limiting scattering mechanism was identified. We demonstrate that the theoretical achievable mobilities in \\ce{Sb2X3} are limited by the polar optical phonon scattering at low and moderate defect concentrations, while at high defect concentration ($\\textgreater 10^{18}$ cm$^{-3}$) they are limited by \\ac{IMP} scattering.\n\nAs shown in the Fig. \\ref{fig_structure}, \\ce{Sb2X3} have orthorhombic crystallographic phase (\\textit{Pnma} space group) and are comprised of strongly bonded quasi-1D [Sb$_4$X$_6$]$_n$ ribbons along the [100] direction. Those ribbons are separated by the Sb lone pairs and stacked together by weak interactions \\cite{wang2022lone}. According to our previous calculations\\cite{wang2022lone} with the HSE06 hybrid functional and D3 dispersion correction, the calculated lattice parameters are 3.80 (3.95), 11.20 (11.55) and 11.39 (11.93) {\\AA } for \\ce{Sb2S3} (\\ce{Sb2Se3}) along the a, b and c direction, respectively.\n\\ce{Sb2X3} are indirect band gap semiconductors with our calculated indirect (direct) band gaps of 1.79 (1.95) and 1.42 (1.48) eV, respectively, which are in reasonable agreement with previous experimental \\cite{yesugade1995structural,el1998substrate,versavel2007structural,liu2016green,torane1999preparation,messina2009antimony,lai2012preparation,chen2015optical} and theoretical studies \\cite{vadapoo2011self,vadapoo2011electronic,caracas2005first,nasr2011electronic,kocc2012first,savory2019complex}. The electronic band structures are shown in the Fig. S1.\nIt has been widely reported experimentally that in \\ce{Sb2X3}, efficient transport can only happen along the ribbons, based on the understanding that \\ce{Sb2X3} are 1D semiconductors\\cite{caruso2015excitons,song2017highly,guo2018tunable,yang2018adjusting,gusmao2019antimony}. However, our previous paper demonstrates that neither the structural dimensionality nor the electronic dimensionality of \\ce{Sb2X3} is 1D. Our results of Fermi surfaces suggest a combination of 3D (holes in \\ce{Sb2S3}) and quasi-2D transport. What's more, the small effective masses in \\ce{Sb2X3} also potentially favour the transport properties.\n\n\\begin{figure}[h]\n    \\centering\n    {\\includegraphics[width=0.5\\textwidth]{Fig_1_structure}} \\\\\n    \\caption{Ground-state crystal structures (\\textit{Pnma} space group) of \\ce{Sb2X3}. The unit cells are represented by rectangles.}\n    \\label{fig_structure}\n\\end{figure}\n\n\n\\begin{table}[h]\n \\caption{Calculated Fr{\\\"o}hlich parameter and Schultz polaron radius (\\AA) in \\ce{Sb2S3} and \\ce{Sb2Se3} at 300 K}\n \\label{tab_alpha}\n \\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}cccccc}\n\\hline\n\\multicolumn{1}{c}{\\multirow{2}{*}{Material}} & \\multicolumn{1}{c}{\\multirow{2}{*}{}} & \\multicolumn{2}{c}{$\\alpha$} & \\multicolumn{2}{c}{r$_f$} \\\\ \\cline{3-6} \n\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{e} & \\multicolumn{1}{c}{h} & \\multicolumn{1}{c}{e} & \\multicolumn{1}{c}{h} \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2S3}} & avg & 1.61 & 2.04 & 45.50 & 40.40 \\\\\n & \\textit{x} & 1.02 & 1.75 & 57.30 & 43.69 \\\\\n & \\textit{y} & 2.44 & 2.05 & 36.85 & 40.25 \\\\\n & \\textit{z} & 5.69 & 2.51 & 23.70 & 36.36 \\\\ \\hline\n\\multirow{4}{*}{\\ce{Sb2Se3}} & avg & 1.29 & 2.07 & 40.46 & 31.90 \\\\\n & \\textit{x} & 0.81 & 2.01 & 50.92 & 32.36 \\\\\n & \\textit{y} & 1.96 & 1.61 & 32.76 & 36.11 \\\\\n & \\textit{z} & 5.76 & 3.77 & 18.82 & 23.48 \\\\ \\hline\n \\end{tabular*}\n\\end{table}\n\nNevertheless, the precise carrier transport property is largely determined by the polaron's property instead of the bare carrier\\cite{hayes2012defects}. When photogenerated carriers interact with a deformable lattice, they can result in displacement of atoms from their equilibrium positions. The charge carriers and the surrounding lattice deformation form a so-called polaron\\cite{emin2013polarons}. Polarons can be classified into two types based on the spatial extent of lattice deformation which depends on the strength of electron-phonon coupling. Stronger electron-phonon coupling leads to larger local lattice distortion which provides the driving force for small polarons. Thus, in a small polaron, the lattice deformation is usually confined to one unit cell, and a carrier's motion is typically incoherently which leads to a low mobility ($\\ll$ 1 cm$^2$\/Vs). By contrast, the lattice deformation in a large polaron is usually moderate spreading over several unit cells, and the mobility is larger ($\\textgreater$ 1 cm$^2$\/Vs). \nIn polar crystals, the electron-phonon interaction is usually dominant by the coupling of charge carriers to the longitudinal \\ac{LO} phonons, which can be described within the Fr{\\\"o}hlich model\\cite{frohlich1952interaction}. We first evaluate the Fr{\\\"o}hlich interaction by the Fr{\\\"o}hlich coupling constant $\\alpha$. Our calculated $\\alpha$ (shown in the Table. \\ref{tab_alpha}) shows an isotropic value of $\\sim$ 2 for both \\ce{Sb2S3} and \\ce{Sb2Se3}, which falls in intermediate electron-phonon coupling regime (defined as 1\/2 $\\lesssim \\alpha \\lesssim$ 6 \\cite{stoneham2001theory}). The magnitude of $\\alpha$ along [100] and [010] directions are quite close, suggesting similar electron-phonon interaction along these two directions. We further estimated the size of polarons in \\ce{Sb2X3} by the Schultz polaron radius (r$_f$)\\cite{schultz1959slow}. The large values of electron and hole polaron radii (extend over several structural units) indicate the polarons tend to delocalise in both \\ce{Sb2S3} and \\ce{Sb2Se3}. The details of parameters used can be found in the Table. S1.\n\nFor a more accurate assessment, we further performed first-principles \\ac{DFT} calculations to model polarons in \\ce{Sb2X3}. Although polarons have been successfully simulated by DFT in many systems\\cite{deskins2007electron,hao2015coexistence,castleton2019benchmarking,ramo2007theoretical}, two main issues should be addressed in order to get reasonable results. The first one is the \\ac{SIE} in any approximate form of the exchange-correlation functional. Approaches to this problem include employing a hybrid functional which incorporates a certain amount of exact exchange from Hartree-Fock theory, or employing a Hubbard correction (DFT+U) which accounts for strong on-site Coulomb interaction of localised electrons.  \nAnother challenge is how to accurately model localised states. Simply applying corrections to \\ac{SIE} cannot guarantee the formation of stable polarons. The formation of localised polarons is particularly dependent on initial geometries and wavefunctions. Different methods have been proposed to break the crystal symmetry and identify localised states. Among them, bond distortion method\\cite{pham2020efficient} and electron attractor method are especially efficient and popular. The former mainly involves introducing some local perturbations in a supercell, while the latter uses a substituted atom to attract an electron. \nTherefore, in this paper, all polaron calculations were conducted using the HSE06 hybrid functional, which has been proved to be able to well describe the structural and electronic properties in perfect \\ce{Sb2X3} crystals (i. e. in the absence of an excess electron\/hole) according to our previous study\\cite{wang2022lone}. We attempted to localise an electron or a hole by adding or removing an electron from the \\ce{Sb2X3} supercell using both the bond distortion method and electron attractor method. The details of modelling are shown in the SI. However, according to our results, electron and hole polarons prefer to delocalise rather than localise in both \\ce{Sb2S3} and \\ce{Sb2Se3}, indicating small polarons are unlikely to form intrinsically.\n\nThe formation of self-trapped carriers has been excluded. In the following, we then consider the possibility of forming self-trapped excitons in \\ce{Sb2X3}. First, large dielectric constants and small effective masses in \\ce{Sb2X3}\\cite{wang2022lone} suggest that the Coulomb interaction is strongly screened and large exciton radius is more favoured. The small binding energies (0.01-0.05 eV and 0.04 eV for \\ce{Sb2S3} and \\ce{Sb2Se3}, respectively\\cite{caruso2015excitons,lawal2018investigation}) further indicate weak electron-hole attraction in \\ce{Sb2X3}. Besides, it is supported by experimental evidence that a negative imaginary part of frequency-dependent complex photoconductivity was not observed in \\ce{Sb2Se3}\\cite{wang2019both}. Consequently, exciton formation in \\ce{Sb2X3} is unlikely.\n\n\\begin{figure*}[t]\n    \\centering\n    {\\includegraphics[width=1.0\\textwidth]{Fig_2_mobility}} \\\\\n    \\caption{Calculated component and total mobilities of electrons and holes in (a) \\ce{Sb2S3} and (b) \\ce{Sb2Se3} as a function of bulk defect concentration at different temperatures. ADP: acoustic deformation potential; POP: polar optical phonon; IMP: ionized impurity}\n    \\label{fig_mobility_ref}\n\\end{figure*}\n\nIn order to further verify our argument and quantify the effect of polarons on the carrier transport property in \\ce{Sb2X3}, the anisotropic mobilities were calculated by the AMSET package (shown in Fig. \\ref{fig_mobility_ref}). Different scattering mechanisms including \\ac{ADP}, \\ac{IMP} and \\ac{POP} scattering were considered. Piezoelectric scattering was not considered due to the centrosymmetric crystal structure of \\ce{Sb2Se3}. It can be seen that the mobility arising from IMP scattering shows a linear decrease as the defect concentration increases, and thus the limiting scattering mechanism is sensitive to the defect concentration. If the bulk defect concentration is low, the mobility in \\ce{Sb2X3} is limited by POP scattering, while when the defect concentration is high enough (on the order of 10$^{18}$ cm$^{-3}$), IMP scattering becomes the limiting one. At intermediate defect concentration, the mobility due to POP and IMP are comparable and control the mobility together. This critical defect concentration depends on the temperature: the higher the temperature, the larger the critical value. It agrees with the reported papers that the defect density is crucial to the carrier transport and in \\ce{Sb2X3}, bulk defect densities above 10$^{15}$ cm$^{-3}$ lead to significant degradation in the conversion efficiency \\cite{islam2020two,li2020simulation,khadir2022performance}. \n\nThe anisotropy of mobility was also considered. Considering that the measurements of carrier mobilities vary widely from different laboratories, it is hard to directly compare our results with experiments. However, our calculated mobilities are basically close to the experimental values. From Fig. \\ref{fig_mobility_ref}, the electron mobilities along [100] and [010] directions are larger than those along [001] direction. For example, assuming the defect density is 10$^{17}$ cm$^{-3}$, the electron mobilities at room temperature in \\ce{Sb2S3} (\\ce{Sb2Se3}) are 49.25 (83.63), 8.10 (13.52) and 1.56 (1.61) cm$^2$\/Vs along \\textit{x}, \\textit{y} and \\textit{z} directions, respectively. However, the hole mobility shows less anisotropy than the electron mobility, especially in \\ce{Sb2S3}. These are in good agreement with the quasi-2D electron Fermi surfaces in both \\ce{Sb2S3} and \\ce{Sb2Se3}, and 3D hole Fermi surface in \\ce{Sb2S3}\\cite{wang2022lone}. Despite anisotropic behaviour, at a reasonable defect concentration, both electron and hole mobilities along each different direction show a large value ($\\textgreater$ 1 cm$^2$\/Vs) which indicates large polaron formation in \\ce{Sb2X3}. Moreover, the mobilities decrease as the temperature increases, which provides further evidence that large polarons instead of small polarons are more likely to form in \\ce{Sb2X3}. The results above demonstrate that the common understanding of \\ac{1D} transport in \\ce{Sb2X3} \\cite{zhou2015thin,liang2020crystallographic} is not comprehensive. The isotropic charge carrier dynamics has also been proved experimentally\\cite{grad2020photoexcited}.   \nFurthermore, considering that most of the experimental values of mobility in \\ce{Sb2X3} were obtained in thin films and grain boundary scattering could also lower the mobility, we also included the mean free path to study the effect of grain boundary scattering in these systems. Acccording to our results (Fig. S2), the mobilities in \\ce{Sb2X3} are not limited by the grain boundary scattering, which agrees with the previous study\\cite{gonzalez2022deciphering}.\n\nConclusively, all our results above give no evidence for self-trapping in \\ce{Sb2X3}. Nevertheless, we note that our results are not contradictory with the experimental observations in \\ce{Sb2S3} which were explained in the framework of self-trapping by Yang et al.\\cite{yang2019ultrafast}. The experimental observations include: \\textrm{i}) a Stokes shift of 0.6 eV;  \\textrm{ii}) picosecond carrier trapping dynamics and broad \\ac{PL} peak; \\textrm{iii}) large photoexcited carrier density and \\textrm{iv}) polarized light emission in \\ce{Sb2S3} single crystal. We clarify that those observations could but not necessarily be the signals of self-trapping process by the following points: \n1) It is widely acknowledged that a large Stokes shift is attributed to emission from trap states instead of band-edge states, but the origin of the trap states can either be self-trapped carriers\/excitons or defect states\\cite{baimuratov2019giant}.\n2) Ultrafast decay and broad \\ac{PL} emission are complex phenomena which are still under intense debate in the literature\\cite{baimuratov2019giant}. The timescale for the decay in \\ac{TA} measurement in self-trapping is typical subpicoseconds or a few picoseconds\\cite{buizza2021charge,kastl2022picoseconds,dexheimer2000femtosecond}, while Yang et al.\\cite{yang2019ultrafast} showed a timescale of $\\sim$20 ps for \\ce{Sb2S3} polycrystalline film and $\\sim$40 ps for \\ce{Sb2S3} single crystal.\n3) The large photoexcited carrier density could also be originated from the \\ac{PIA}. This can also be supported by the large trap density of 2.1 $\\times$ 10$^{20}$ cm$^{-3}$ reported in \\ce{Sb2Se3} and the authors demonstrate the absence of self-trapping process\\cite{zhang2021suppressing}.\n4) Besides experimental observations, Yang et al.\\cite{yang2019ultrafast} also mentioned the hopping transport mechanism in \\ce{Sb2S3} by citing another paper\\cite{roy1978electrical}. However, that was just simply deduced by narrow bands in \\ce{Sb2S3} and more convincing evidence is needed.\nTherefore, we conclude that it is possible for electrons or\/and holes to be trapped in \\ce{Sb2X3} with the assistance of extrinsic defects. \n\nIn summary, by systematic first-principles DFT calculations, we demonstate the intrinsic formation of large polarons and their effects on the charge carrier transport in \\ce{Sb2X3}. We studied the electron-lattice interaction including the Fr\u00f6hlich coupling constant and polaron radii, and modelled the electron and hole polarons in \\ce{Sb2X3} via bond distortion method and electron attractor method. All our results support that electron and hole polarons in \\ce{Sb2X3} tend to delocalise instead of localise. The large polarons further result in large carrier mobilities in \\ce{Sb2X3} (an isotropic value of \\textgreater 10 cm$^2$\/Vs at room temperature for both electrons and holes). Therefore, we conclude that there is no theoretical evidence for self-trapping of carriers in \\ce{Sb2X3}. Besides, we revealed that the maximum achievable mobilities in \\ce{Sb2X3} are limited by the polar optical phonon scattering at low and moderate defect concentrations, while at high defect concentration ($\\textgreater 10^{18}$ cm$^{-3}$), they are limited by ionized impurity scattering. Our study provides guidance for designing \\ce{Sb2X3} based solar cells with high efficiencies.\n\n\\section{Methods}\nAll calculations were performed based on Kohn-Sham density-functional theory (DFT)\\cite{kohn1965self,dreizler1990density} as implemented in the \\ac{VASP}\\cite{kresse1996efficient}. The projector augmented-wave (PAW) method\\cite{kresse1999ultrasoft} was employed with a plane-wave energy cutoff of 400 eV. All calculations were carried out using the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06)\\cite{heyd2003hybrid,krukau2006influence} and the D3 dispersion correction\\cite{grimme2004accurate}. The atomic positions were optimised until the Hellman-Feynman forces on each atom were below 0.0005 eV \\AA$^{-1}$ for unit cells and 0.01 eV \\AA$^{-1}$ for 3$\\times$1$\\times$1 supercells. The energy convergence criterion was set to 10$^{-6}$ eV. $\\varGamma$-centered \\textit{k}-point meshes were set to 7$\\times$2$\\times$2 and 2$\\times$2$\\times$2 for geometry optimisation with primitive unit cells and supercells, respectively. \nThe Fr{\\\"o}hlich parameters were calculated by the PolaronMobility package\\cite{Frost2017}. The mobility was calculated using the AMSET package \\cite{ganose2021efficient}. The carrier concentration was set to 10$^{13}$ cm$^{-3}$ according to previous experimental results\\cite{chen2017characterization,liu2016green,zhou2014solution,yuan2016rapid,li2021defect,chalapathi2020influence,black1957electrical}.\nThe crystal structures were plotted using CrystalMaker$^{\\circledR}$\\cite{crystalmaker}. \n\n\\section*{Acknowledgements}\nWe are grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP\/P020194\/1 and EP\/T022213\/1). Xinwei Wang acknowledges Imperial College London for the funding of a President's PhD Scholarship. Alex M. Ganose was supported by EPSRC Fellowship EP\/T033231\/1. Se\u00e1n R. Kavanagh acknowledges the EPSRC Centre for Doctoral Training in the Advanced Characterisation of Materials (CDT-ACM)(EP\/S023259\/1) for funding a PhD studentship. Xinwei Wang thanks Jarvist Moore Frost, Ye Yang and Yuchen Fu for valuable discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nGalaxies are not distributed randomly in the cosmic web \\citep{Joeveer78,Zeldovich82,Shandarin83,Einasto84,Bond96,Aragon10}, but are arranged in filaments and sheets surrounding cosmic voids and connecting clusters of galaxies \\citep{Pimbblet04,Aragon10,Jasche10,Tempel14b,Cautun14}. The properties of filaments affect the abundance, shape, and evolution of galaxies \\citep{Aragon07b,Hahn07a,Hahn07,Libeskind12,Libeskind13,Cautun13,Tempel13,Tempel13b}, and depend on the properties of the initial density fluctuations generated in the very early Universe. Therefore probes of the large-scale filaments enable us to test current physical and cosmological theories.\n\nProbes of the filaments require efficient algorithms for finding filaments. Several sophisticated algorithms for identifying filaments, both in the three dimensions and two dimensions, have been developed. Generally, there are three types of algorithms to identify a filament based on: (1) the distribution of galaxies or clusters of galaxies \\citep{Peebles80,Peacock98,Novikov06,Aragon07a,Sousbie11,Shandarin12,Cautun13}; (2) the gravitational tidal tensor -- the Hessian of the gravitational potential \\citep{Lee08,Forero09,Bond10a,Bond10b,Wang12}; and (3) the velocity field induced from the dynamics of the underlying density field \\citep{Hahn07,Shandarin11,Hoffman12,Libeskind12,Libeskind15,Falco14}. Recently, \\cite{Falco14} found that the line-of-sight velocities $v_{\\rm{los}}$ of the galaxies around a cluster of galaxies with the projected distances to the cluster center satisfying $2.5r_{\\rm{vir}}\\lesssim r\\lesssim 8r_{\\rm{vir}}$, where $r_{\\rm{vir}}$ is the virial radius, are as a function of $r$, if the galaxies are arranged in one filament or sheet, since these galaxies are gravitationally affected by the cluster. Therefore they plotted the $(r,v_{\\rm{los}})$ map for the galaxies, and identified the filamentary structures on the map as filaments or sheets (see the paper of Falco et al. 2014 for details). They proved that the line-of-sight velocities can be probes of filaments and sheets.\n\nIn this paper, we develop a new method of identifying filaments using the orientations of galaxies. \\cite{Tempel13b,Tempel15} found that the spin axes of bright spiral galaxies have a weak tendency to be aligned parallel to filaments, while the major axes of elliptical\/S0 galaxies are significantly aligned with their host filaments. Therefore the galaxies alignment may be an additional probe of filaments. We describe the method, and apply it to the galaxies assembled around the Coma cluster in section 2. The results are compared with the detected standard filaments using the method of \\cite{Falco14} in this section, and discussed in section 3. In section 4, we summarize the work. We adopt the WMAP7 cosmological parameters: $\\Omega_{\\rm{M}}=0.27$, $\\Omega_{\\rm{\\Lambda}}=0.73$, and $H_0=71\\ \\rm km\\ s^{-1}\\ Mpc^{-1}$.\n\n\\section{Identification of Filaments by Galaxies Alignment}\n\\subsection{Galaxies in a filament}\n\nFor the galaxies arranged in a filament, the major axes of red galaxies and spin axes of blue galaxies are related to their host filaments, e.g., the major axes of the elliptical\/S0 (red) galaxies are significantly aligned with the orientations of the filament \\citep{Tempel15}. Therefore, according to the distribution of orientations of the projected red galaxies, we may identify the filaments in two-dimensional (2D) images.\n\nIn order to quantitatively indicate the orientations of the projected ellipses, we define the east direction in the celestial coordinate system as $x$-axis, and the north direction as $y$-axis, and define the position angle of a red galaxy, $\\xi$, as the angle between the major axis of the projected ellipse and the $x$-axis, as shown in Fig.~\\ref{ske_pa}. The range of $\\xi$ is $0\\--90^{\\circ}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{ske.pdf}\n\\caption{Illustrations of the coordinate system, position angle $\\xi$, and projected distance $r$. O denotes the center of a cluster of galaxies.}\n\\label{ske_pa}\n\\end{figure}\n\nAnalogous to the work of \\cite{Falco14} using the $(r,v_{\\rm{los}})$ map, we can plot the $(r, \\xi)$ map for the projected red galaxies in an image, where the horizontal coordinate $r$ denotes the distance between a projected ellipse to the origin of the coordinates, and the longitudinal coordinate is the position angle $\\xi$. Since we aim to identify the large-scale filaments around a cluster, the origin of the coordinates is preferentially set as the center of a cluster. The $(r, \\xi)$ map sufficiently uses the 2D information of the orientations and positions of the red galaxies. If the red galaxies are not arranged in some filament structures, $\\xi$ should be uniformly random within $0\\--90^{\\circ}$. Conversely, if the galaxies are located in a filament, we should expect the non-random distribution of $\\xi$ and inhomogeneous distribution of $r$. This manner of detecting filaments is called location-alignment-method (LAM).\n\n\\subsection{Data Analysis}\n\nWe will apply LAM to the data of the galaxies around the Coma cluster, and then compare the results with the filaments obtained by the method of \\cite{Falco14}. We take the galaxy NGC~4874 as the center of the Coma cluster \\citep{Kent82} and origin of the coordinates, which is located at RA = $12^{\\rm{h}}59^{\\rm{m}}35^{\\rm{s}}.7$, and DEC = $+27^{\\circ}57'33''$. The galaxies used here are selected from the Sloan Digital Sky Survey data release 12 (SDSS DR12; Alam et al. 2009) with $r$-band model magnitudes (modelMag), $m_r$, satisfying $12\\ {\\rm{mag}}<m_r\\leq 18\\ \\rm{mag}$, classified as galaxies photometrically by the SDSS data reduction pipelines (i.e.,photoObj.type=3), and distributed within $9^{\\circ}$ from the position of the Coma center and with redshifts between 0.01 and 0.037 (i.e., velocities along the line of sight are between 3000 and 11000~km~$\\rm{s}^{-1}$; Falco et al. 2014). 2590 galaxies around the Coma cluster are obtained.\n\nIn order to obtain the position angles of the red galaxies, first we plot the $u-r$ versus $r$ color-magnitude diagram (CMD) and distribution of colors, $u-r$, of the 2590 numbers of galaxies, which are shown in Fig.~\\ref{cmd}. Second, we select the red galaxies. The red sequence \\citep{Bower92} is formed by old elliptical galaxies whose spectra show similar $4000\\ \\AA$ breaks (a characteristic of old stellar populations ubiquitous in elliptical galaxies; Pereira \\& Kuhn 2005) resulting from photospheric absorptions of heavy elements, and the $u$ and $r$ filters nicely probe the spectral region across the $4000\\AA$ break. A linear fitting to the red sequence,\n\\begin{equation}\n(u-r)=k\\cdot r+d,\n\\label{lf}\n\\end{equation}\nwhere $k\\simeq -0.106$, $d\\simeq 4.154$ are the slope and intercept respectively, is applied to obtain the slope of the ridgeline \\citep{Bower92} in the CMD. As shown in the distribution of colors of the galaxies, the colors of the red and blue galaxies approximately follow two Gaussian distributions, and the standard error $\\sigma\\simeq 0.2\\ \\rm{mag}$ for the colors of the red galaxies is obtained, therefore we select the galaxies confined within $2\\sigma\\simeq 0.4\\ \\rm{mag}$ around the ridgeline as the red galaxies, which are known as the collections of elliptical\/S0 galaxies and dwarf ``ellipticals'' \\citep{Hung10}. The rest bluer galaxies are mostly spiral galaxies. The selection criterion satisfies the Butcher-Oemler condition, i.e., blue galaxies are those at least 0.2 mag bluer than the cluster ridgeline \\citep{Butcher84}. Subsequently, $\\xi$ of the red galaxies are calculated by the parameters ``photoObj.deVPhi\\_r'' (parameter of the de Vaucouleurs surface brightness fitting model) from the SDSS data reduction pipelines, since the de Vaucouleurs $R^{1\/4}$ surface brightness profile fits well to many elliptical galaxies \\citep{Vaucouleurs48}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.49]{cmd.pdf}\n\\caption{The distribution of color and CMD of the galaxies around the Coma cluster. $u-r$ color is plotted against the $r$ magnitude of the galaxies. The elliptical\/S0 galaxies and spiral galaxies are denoted by red and blue points, respectively. The black solid line denotes the ridgeline of red galaxies, and the two dashed lines confine the galaxies within $2\\sigma$ color range around the ridgeline.}\n\\label{cmd}\n\\end{figure}\n\nOur goal is to find structures confined in a relatively small region in the $x-y$ plane. Fig.~\\ref{pa_filament_detect} shows the Coma cluster and its environment up to $9^{\\circ}$ from the cluster center. The 536 galaxies distributed within about $1.3^{\\circ}$ (corresponding to the virial radius $r_{\\rm{vir}}\\simeq2.3\\ \\rm{Mpc}$; Keshet et al. 2014) are mainly the members of Coma cluster and should be removed. Additionally, since the method of \\cite{Falco14} can only detect the galaxies in filaments with $r\\gtrsim 2.5r_{\\rm{vir}}$, therefore the galaxies distributed within $r\\lesssim 3^{\\circ}$ are also removed in order to compare our results with those obtained by the method of \\cite{Falco14}. All of the removed galaxies are denoted by the points in the central yellow round region in Fig.~\\ref{pa_filament_detect}. The image of the galaxies except the central removed galaxies is split into eight wedges, and our goal is to detect the filaments in two wedges (W1 and W2), which are denoted by the points of the orange regions in Fig.~\\ref{pa_filament_detect}. For each selected wedge, the $(r, \\xi)$ map of the red galaxies in this wedge is obtained; the $(r, \\xi)$ map is divided into $9\\times9$ cells with $\\Delta r\\times \\Delta \\xi=1^{\\circ}\\times 10^{\\circ}$, where $\\Delta r$ and $\\Delta \\xi$ are the bins of $r$ and $\\xi$, respectively. We need to compare the galaxy number density $n_i$ in each cell $i$, with the expected number of background galaxies in this cell $n^{\\rm{bg}}_i$, to determine whether there is an excess in the $(r, \\xi)$ map. The five background wedges are represented by the green regions shown in Fig.~\\ref{pa_filament_detect}, and we artificially set $\\xi$ of the background red galaxies uniformly random in $0\\--90^{\\circ}$ instead of their real orientations, since the background galaxies may be also arranged in filaments and thus grouped into some specific $\\xi$ ranges. In other words, we only use the information of the background density, but not their orientations. We exclude the two wedges (yellow wedges) adjacent to each selected wedge, since any structure in the selected wedge might stretch to the closest two wedges.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.6]{w_f.pdf}\n\\caption{The Coma cluster and its environment up to $9^{\\circ}$ from the cluster center. The left and right panels show all the galaxies (red and blue) and only the red galaxies in this region, respectively. The region within $3^{\\circ}$ from the cluster center is denoted by the central yellow round. The two selected wedges (W1 and W2), corresponding background wedges, and two wedges adjacent to each selected wedge, are colored orange, green, and yellow, respectively.}\n\\label{pa_filament_detect}\n\\end{figure*}\n\nThe excess in cell $i$ is defined as\n\\begin{equation}\nm_i=\\frac{n_i-n_i^{\\rm{bg}}\/5}{n_i^{\\rm{bg}}\/5}.\n\\label{ov_eq}\n\\end{equation}\nIf $m_i>0$, it means that there is an excess in the cell $i$. In order to select statistically significant excesses, we repeatedly set $\\xi$ of the background galaxies $10^5$ times. Each time, the cells with $m_i>0$ are selected. During the $10^5$ times of tests, some cells are always selected, but some other cells are selected only a few times. Finally, we only choose the cells with the accumulated selected times larger than a given criteria, for instance $1\\sigma$ (about 68264 times), $2\\sigma$ (about 95456 times), and $3\\sigma$ (about 99725 times); if a cell is chosen, all of the red galaxies in the cell are selected. Representatively, we plot the $(r,\\xi)$ maps and the final chosen-cells with the confidence level of $3\\sigma$ for the two selected wedges, as shown in Fig.~\\ref{pas}. For each $\\xi$ bin (bin of $10^{\\circ}$), we also plot the histogram of $r$ of the red galaxies in this $\\xi$ bin, i.e., the number of red galaxies as a function of $r$ (as denoted by the orange histogram in Fig.~\\ref{pas}), and the histogram of the simulated background galaxies in this $\\xi$ bin, i.e., $\\bar{n}^{\\rm{bg}}\/5$ as a function of $r$ (as denoted by the green histogram in Fig.~\\ref{pas}), where $\\bar{n}^{\\rm{bg}}_i$ is the mean value of $n^{\\rm{bg}}_i$ during the $10^5$ times of tests. The final chosen-cells are denoted by the blue bins.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.6]{cell_pas.pdf}\n\\caption{The left and right panels show the $(r,\\xi)$ map and distribution of $r$ of the red galaxies for W1 and W2, respectively. The top panel shows the $(r,\\xi)$ maps for the red galaxies in the two selected wedges. The red galaxies inside and outside the central $3.0^{\\circ}$ region are denoted by the black and orange points, respectively. The selected cells with the confidence level of $3\\sigma$ are colored blue. The lower nine panels show the distributions of $r$ of the red galaxies within the nine $\\xi$ bins. The $3.0^{\\circ}$ region from the cluster center is colored gray. The orange and green histograms denote the distributions of $r$ of the red galaxies in the selected and background wedges, respectively. The final chosen cells are also denoted by the blue bins.}\n\\label{pas}\n\\end{figure*}\n\n\n\\subsection{Results and Comparison}\n\nAs described in introduction, \\cite{Falco14} developed a method of identifying large-scale filaments and sheets around a cluster using the $v_{\\rm{los}}$ and 2D positions of galaxies $r$, which has been proved to be robust for both the simulations and observations \\citep{Falco14,Lee15}. In this paper, we use this method to identify the filaments in the same two wedges around the Coma cluster, and treat the detected filaments as the ``standard'' ones to be compared with the results by LAM. The detailed process of detecting the standard filaments can be found in \\cite{Falco14}.\n\nThe red galaxies in W1 and W2 detected by LAM are shown in Figs.~\\ref{w1} and \\ref{w2}, and denoted by the blue diamonds. The upper-left, upper-right, lower-left panels of the two figures show the results with the criteria $1\\sigma$, $2\\sigma$ and $3\\sigma$, respectively; and the lower-right panels show the red galaxies of the standard filaments (denoted by the blue triangles) in the corresponding wedges using the method of \\cite{Falco14}. It is worth noting that the standard filaments can only be detected with the $1\\sigma$ confidence level.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.6]{w1_f.pdf}\n\\caption{The selected red galaxies by LAM with the $1\\sigma$, $2\\sigma$, and $3\\sigma$ confidence levels for W1 are denoted by the blue diamonds in the upper-left, upper-right, and lower-left panels, respectively. The detected standard filaments by the method of Falco et al. (2014) are denoted by the blue triangles in the lower-right panel.}\n\\label{w1}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.6]{w2_f.pdf}\n\\caption{Analogous to Fig.~\\ref{w1}, this figure shows the selected red galaxies by LAM and method of Falco et al. (2014) for W2.}\n\\label{w2}\n\\end{figure*}\n\nIn Table~\\ref{N_comp}, the filaments obtained by LAM are compared with the standard filaments in the two wedges. For each wedge, we count the numbers of galaxies in the detected filaments ($N_{\\rm{det}}$) by LAM and standard filaments ($N_{\\rm{std}}$), respectively, and the number of galaxies in both the detected and standard filaments ($N_{\\rm{c}}$). Thus the fraction of the duplicated galaxies in the detected filaments, $N_{\\rm{c}}\/N_{\\rm{det}}$, and standard filaments, $N_{\\rm{c}}\/N_{\\rm{std}}$, are obtained. The latter suggests the detection efficiency of LAM: a higher $N_{\\rm{c}}\/N_{\\rm{std}}$ indicates more effective LAM.\n\n\\begin{table} \\scriptsize\n\\begin{tabular}{@{}cccc|c|c@{}}\n\\hline\n\\hline\n\\multicolumn{6}{c}{W1}\\\\\n\\hline\n  & \\multicolumn{3}{c}{LAM} & Alignment Only & No Redshifts\\\\\n  & $1\\sigma$ & $2\\sigma$ & $3\\sigma$ & $1\\sigma$ & $1\\sigma$\\\\\n \\hline\n $N_{\\rm{std}}$ & 59 & 59 & 59 & 59 & 59\\\\\n $N_{\\rm{det}}$ & 96 & 81 & 79 & 97 & 87\\\\\n $N_{\\rm{c}}$   & 56 & 45 & 44 & 41 & 49\\\\\n $N_{\\rm{c}}\/N_{\\rm{std}}$ & 95.0\\% & 76.3\\% & 74.6\\% & 69.5\\% & 83.0\\%\\\\\n $N_{\\rm{c}}\/N_{\\rm{det}}$ & 58.3\\% & 55.6\\% & 55.7\\% & 42.3\\% & 56.3\\%\\\\\n\\hline\n\\hline\n\\multicolumn{6}{c}{W2}\\\\\n\\hline\n & \\multicolumn{3}{c}{LAM} & Alignment Only & No Redshifts\\\\\n & $1\\sigma$ & $2\\sigma$ & $3\\sigma$ & $1\\sigma$ & $1\\sigma$\\\\\n\\hline\n$N_{\\rm{std}}$ & 66 & 66 & 66 & 66 & 66\\\\\n$N_{\\rm{det}}$ & 79 & 66 & 60 & 61 & 69\\\\\n$N_{\\rm{c}}$   & 63 & 55 & 54 & 42 & 58\\\\\n$N_{\\rm{c}}\/N_{\\rm{std}}$ & 95.4\\% & 83.3\\% & 81.8\\% & 63.6\\% & 87.9\\%\\\\\n$N_{\\rm{c}}\/N_{\\rm{det}}$ & 79.7\\% & 83.3\\% & 90.0\\% & 68.8\\% & 84.0\\%\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{The results of the detected galaxies for W1 and W2. The second to fourth columns list the results by LAM with the $1\\sigma$, $3\\sigma$, and $5\\sigma$ confidence levels, respectively. The fifth column lists the results with the $1\\sigma$ confidence level, when we only use the information of galaxies alignments. The last column lists the results without restricting the redshifts of galaxies.}\n\\label{N_comp}\n\\end{table}\n\nAccording to Table~\\ref{N_comp}, two points are concluded. First, with $1\\sigma$ confidence level, the detection efficiency of LAM (denoted by $N_{\\rm{c}}\/N_{\\rm{std}}$) is better than $95\\%$, indicating that LAM effectively find out most galaxies found by the method of \\cite{Falco14} in the filaments also with $1\\sigma$ confidence level. The detection efficiency of LAM decreases with the increasing confidence level, but is always better than $75\\%$, suggesting that LAM is still valid with such high confidence levels. Second, $N_{\\rm{c}}\/N_{\\rm{det}}$ is relatively low, suggesting that LAM detects some other galaxies in filaments which cannot be found by the method of \\cite{Falco14}.\n\nFinally, we plot the orientations (denoted by the black bars) of the selected red galaxies by LAM with the $3\\sigma$ confidence level for each wedge in the top panels of Fig.~\\ref{f_ori}. In order to clearly show how many filaments have been detected and the overall orientations of the detected filaments, we divide the two wedges into $1^{\\circ}\\times1^{\\circ}$ cells, as shown in Fig.~\\ref{f_ori}, and calculate the average orientation of the selected red galaxies in each cell, which is denoted by a red bar in the middle panels of Fig.~\\ref{f_ori}. The overall orientations of filaments are clearly revealed by the red bars. According to the average orientations, there are two main filaments in W1 (F1 and F2), and also two filaments, or more precisely, two sheets (S1 and S2; the filaments are located in the planes of the sheets; Falco et al. 2014; Tempel \\& Libeskind 2013) in W2. These filaments are circled by the blue lines in Fig.~\\ref{f_ori}. For each filament, we also plot the distribution of orientations of the red bars in the filament region, as shown in the bottom panels of Fig.~\\ref{f_ori}. Kolmogorov-Smirnov test (K-S test) is performed to detect the deviation of each orientation distribution from a uniform distribution, and the $p$ values of all of the four distributions are small ($p=0.13$ for F1, $p=0.21$ for F2, $p=0.20$ for S1, and $p=0.05$ for S2), suggesting that the orientations of the red galaxies in each filament are indeed anisotropic.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.7]{f_detect.pdf}\n\\caption{The left and right panels show the orientations of the red galaxies and distribution of each filament in W1 and W2, respectively. The top panels show the orientations of the selected red galaxies with the $3\\sigma$, which are denoted by the black bars. The dashed grids divide W1 and W2 into $1^{\\circ}\\times 1^{\\circ}$ cells. The middle panels show the average orientations (denoted by the red bars) of the black bars in each cell. The four filaments F1, F2, S1, and S2 are circled by the blue lines. The bottom panels show the distribution histograms of the orientations of red bars in each filament region, and the results of K-S test are illustrated by the $p$ values.}\n\\label{f_ori}\n\\end{figure*}\n\n\\section{Discussion}\n\nThe method of \\cite{Falco14} can only identify the filaments gravitationally bound to the clusters of galaxies, i.e., only the galaxies whose radial velocities are influenced by the gravity from the cluster matter can be detected; whereas LAM can theoretically find all of the filaments either influenced or not by the gravity of the clusters, since the orientations of red galaxies are not related directly to the gravity of the clusters of galaxies. Therefore the low value of $N_{\\rm{c}}\/N_{\\rm{det}}$ in W1 might indicate that there are some galaxies located in filaments but not or only weakly influenced by the gravity of the Coma cluster in W1.\n\n\\subsection{Effect of alignment of red galaxies}\n\nWe are interested in whether the selected galaxies are due to both the alignments and inhomogeneous distributions of the galaxies in the 2D image, or just due to the latter. For example, perhaps the detected filament (sheet) S1 in W2 is identified because of the high number density of galaxies in the $360\\--540\\ \\rm{arcmin}$ radius range rather than the alignments of red galaxies. In order to test the effect of alignment, the galaxies in the selected wedges themselves are treated as the background galaxies; meanwhile we artificially set $\\xi$ of the background galaxies uniformly random in $0\\--90^{\\circ}$ and test the excess $m_i'=(n_i-n_i^{\\rm{bg}})\/n_i^{\\rm{bg}}$ in each cell for $10^5$ times. The results are also listed in Table~\\ref{N_comp}. We find that the filaments can also be identified with the $1\\sigma$ confidence level; however the detection efficiency ($N_{\\rm{c}}\/N_{\\rm{std}}$) is much worse than that by LAM. The effect of alignments can be characterized by the fraction $f=N_{\\rm{c}}'\/N_{\\rm{c}}$, where $N_{\\rm{c}}$ and $N_{\\rm{c}}'$ are the numbers of detected galaxies by LAM and alignments only; $f=73.2\\%$ for W1, and $f=66.7\\%$ for W2. Therefore the galaxies selected by alignments account for substantial parts of the galaxies selected by LAM, suggesting that the alignments of red galaxies play an important role in LAM.\n\n\\subsection{LAM without redshift information}\n\nThe galaxies alignments and 2D distributions of galaxies are independent of the redshifts, therefore we expect that LAM should be still effective without the information of redshifts of the galaxies. However in this case, some interlopers, i.e., the foreground and background galaxies, will be included. The orientations of the foreground and background galaxies should be isotropic, if there is no strong gravitational lens in front of these galaxies, and the galaxies are not arranged in other filaments. Therefore the interlopers may be contaminations for LAM. With these in mind, we retrieve the catalog of galaxies around the Coma cluster (within $9^{\\circ}$) from SDSS DR12 again, without restricting the range of redshifts. In order to reduce the contamination of the interlopers in the image, the range of the $r$ band magnitudes of the selected red galaxies need to be carefully chosen. Here we only use the bright red galaxies within $12\\ {\\rm{mag}}<m_r<16\\ \\rm{mag}$ to detect the filaments, where the faint end of $16\\ \\rm{mag}$ is brighter than the previous value of $18\\ \\rm{mag}$, since the alignments of the brighter red galaxies are more significant \\citep{Tempel15}, and a brighter faint end induces fewer background interlopers; the other selecting criteria of the red galaxies remain unchanged. We apply LAM to the new sample of red galaxies, and the result for each wedge (W1 and W2) is also compared with the standard filaments, which is listed in the last column of Table~\\ref{N_comp}. We find that the results for both W1 and W2 are slightly worse than the results of redshifts restricted; however, the detection efficiency (83.0\\% for W1, 87.9\\% for W2) is also acceptable. Additionally, $N_{\\rm{c}}\/N_{\\rm{det}}$ (56.3\\% for W1, 84.0\\% for W2) for each wedge is also close to that with the redshifts restricted (58.3\\% for W1, 79.7\\% for W2), implying that the selected filaments without the redshifts information of galaxies are almost the same as those with the redshifts restricted.\n\nWe find that, in W1 and W2, 24 and 15 galaxies of the selected red galaxies by LAM (with $1\\sigma$ confidence level) are not found without the redshifts information, respectively. Among these red galaxies which are not found without using the redshifts information, 18 (for W1) and 11 (for W2) galaxies are fainter than $m_r=16\\ \\rm{mag}$, suggesting that these galaxies are not found is mainly due to the fact that their magnitudes are out of range. Therefore the results without the redshifts information are slightly worse than the results of LAM, because some faint ($m_r>16\\ \\rm{mag}$) galaxies are not included in the sample ($12\\ {\\rm{mag}}<m_r<16\\ \\rm{mag}$) we used here.\n\n\\subsection{Alignment of ``background\" galaxies?}\n\nWithout restricting the redshifts of galaxies, we select $N_{\\rm{det}}=87$ red galaxies in W1, and $N_{\\rm{det}}=69$ galaxies in W2. Among these selected red galaxies, 72 (for W1) and 64 (for W2) red galaxies were found by LAM before, and only 15 (for W1) and 5 (for W2) red galaxies were not found by LAM before; meanwhile among these red galaxies which were not found by LAM, 11 (for W1) and 4 (for W2) galaxies are located at the redshifts $z>0.037$ (i.e., the given lower redshift limit of background galaxies), suggesting that almost all of the galaxies which were not found by LAM are possible background interlopers. We directly plot the orientations of the possible background interlopers for W1 and W2 in Fig.~\\ref{ori_inter}, and find that the orientations of the possible background interlopers tend to be parallel to the overall orientations of filaments. In order to test whether the tendency is caused by physical reasons or just by accident, we need to study the orientations of the all possible background red galaxies ($z>0.037$, $12\\ {\\rm{mag}}<m_r<16\\ \\rm{mag}$) behind the four filaments\/sheets.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.7]{ori_inter.pdf}\n\\caption{The upper and lower panels show the orientations of ``background\" interlopers in W1 and W2, respectively. The red bars denoting the overall orientations of filaments are the same as those in the middle panels of Fig.~\\ref{f_ori}, and the green bars denote the orientations of the ``background\" interlopers in the sample of selected red galaxies without restricting redshifts.}\n\\label{ori_inter}\n\\end{figure*}\n\nFirst, we choose all possible background red galaxies ($z>0.037$, $12\\ {\\rm{mag}}<m_r<16\\ \\rm{mag}$) in the four filament regions in the 2D image as circled by the blue lines in Fig.~\\ref{f_ori}, and plot their orientations in the upper panels of Fig.~\\ref{ori_bg}. Second, we calculate the average orientation of the possible background galaxies in each $1^{\\circ}\\times1^{\\circ}$ spatial cell, with the same manner as used for the filament galaxies (see section 2.3 for detail), and plot the average orientations in the middle panels of Fig.~\\ref{ori_bg}. Finally, analogous to the lower panels in Fig.~\\ref{f_ori}, we plot the distribution histogram of the average orientations in each filament region, and use K-S test to test the deviation from a uniform distribution; $p$ values are shown in the lower panels of Fig.~\\ref{ori_bg}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.7]{ori_bg.pdf}\n\\caption{The orientations of the red ``background\" galaxies in each filament region and corresponding distribution of average orientations of the ``background\" galaxies. Analogous to Fig.~\\ref{f_ori}, the left and right panels correspond to W1 and W2, respectively. The top panels show the orientations of the red ``background\" ($z>0.037$) galaxies in each filament region, which are denoted by the green bars. The dashed grids divide W1 and W2 into $1^{\\circ}\\times 1^{\\circ}$ cells. The middle panels show the average orientations (denoted by the cyan bars) of the green bars in each cell. The four filament regions F1, F2, S1, and S2 are circled by the blue lines. The bottom panels show the distribution histograms of the orientations of cyan bars in each filament region, and the results of K-S test are illustrated by the $p$ values.}\n\\label{ori_bg}\n\\end{figure*}\n\nWe find that the orientation distribution of the possible background galaxies in F1 region significantly deviates from a uniform distribution ($p=0.09$), whereas the distributions of possible background galaxies in F2, S1, and S2 regions are more likely to be uniform; meanwhile, the most significant excess in the average orientation distribution of the possible background galaxies behind F1 is located at about $40^{\\circ}\\--50^{\\circ}$, which is consistent with the excess location of the red galaxies in F1 filament (see the lower left panel of Fig.~\\ref{f_ori}). Therefore the selection of the possible background interlopers in F1 region by LAM is more likely due to a physical reason rather than accident. For F2 region, the uniform distribution ($p=0.40$) may be due to the small number of statistics. Therefore for detecting F2, the effect of the possible background galaxies is not clear. For S1 and S2 regions, the possible background interlopers are more likely to be selected by accident, because of the high values of $p$ ($p=0.97$ for S1 region, $p=0.85$ for S2 region). Therefore for detection of S1 and S2, the possible background interlopers are contaminations.\n\nThere are some possible mechanisms that may result in the relatively significant alignment signal of the possible background galaxies behind F1. For example, according to many previous papers about the effect of gravitational lensing, theoretically the images of background galaxies might be stretched along the orientation of the foreground filament by the shear of the filament (e.g., Higuchi et al. 2014). However, the gravitational lensing by filaments is extremely weak \\citep{Dolag06,Mead10}, and changes the observed ellipticity of a background galaxy by only $2|\\gamma|\\simeq 0.01$ \\citep{Waerbeke01}, where $\\gamma$ is the shear of gravitational lensing. Therefore it is very unlikely that gravitational lensing contributes to the detected significant alignment signal shown in the lower left panel of Fig.~\\ref{ori_bg}.\n\nAnother possibility is that if some member galaxies in F1 are classified as background galaxies even if $z>0.037$, these galaxies may result in the detected alignment signal. In order to test this possibility, we plot the distribution of relative redshifts $z-z_0$ of all red galaxies in F1 region, as shown in Fig.~\\ref{dis_z}; $z$ and $z_0=0.02393$ are the observed redshifts of a red galaxy and NGC~4874 (center of Coma cluster), respectively. The dot-dashed and dashed lines denote $z=z_0$ (i.e., the redshift of Coma center) and $z=0.037$ (i.e., the redshift lower limit of the possible background galaxies), respectively. If there is no filament in F1 region, we should expect a decreasing number of galaxies with increasing $z-z_0$, in the range of $z>z_0$. However, a significant excess is located at about $0.0342<z<0.0377$ (denoted by the grey bin in Fig.~\\ref{dis_z}), indicating that there may be a filament (F1). Therefore a significant fraction of the member galaxies in F1 are indeed classified as ``background'' galaxies which may explain the significant alignment signal of ``background'' galaxies.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.49]{dis_z.pdf}\n\\caption{Distribution histogram of relative redshifts of the all galaxies in F1 region in the 2D image. The dot-dashed and dashed lines denote $z=z_0=0.02393$ and $z=0.037$, respectively. The excess with the range of $0.0342<z<0.0377$ is denoted by the grey bin.}\n\\label{dis_z}\n\\end{figure}\n\nTo further test whether the excess in Fig.~\\ref{dis_z} really reveals a filament (F1), we use LAM again with the red galaxies in F1 region with $0.0342<z<0.0377$ (the redshift range of the excess in Fig.~\\ref{dis_z}), and plot the orientations of the final selected red galaxies (with $3\\sigma$ confidence) and distribution of orientations in Fig.~\\ref{ori_nf}. We find that F1 is indeed detected with these galaxies, suggesting that the excess in Fig.~\\ref{dis_z} is resulted from the member galaxies in F1. The $p$ value ($p=0.09$) is smaller than that with $0.01<z<0.037$ ($p=0.13$ as shown in Fig.~\\ref{f_ori}), also indicating that the red galaxies with $0.037<z<0.0377$ in F1 region indeed are member galaxies of F1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.49]{ori_nf.pdf}\n\\caption{Analogous to Fig.~\\ref{f_ori}, from the upper to lower panels, we plot the orientations of the selected (with $3\\sigma$ confidence) red galaxies with $0.0342<z<0.0377$ in F1 region, average orientation in each spatial cell, and distribution of the average orientations. The result of K-S test is illustrated by the $p$ value.}\n\\label{ori_nf}\n\\end{figure}\n\nTherefore for W1, we need to subtract the galaxies with $0.037<z<0.0377$ from the eleven selected background galaxies shown in Fig.~\\ref{ori_inter}. After removing these galaxies, we find that the contaminations by the left background interlopers are not strong when LAM is used without the information of galaxy redshifts, since there are only seven ($8.0\\%$ of the total selected galaxies) and four ($5.8\\%$ of the total selected galaxies) background interlopers in W1 and W2, respectively. Therefore, an advantage of LAM is that the method is independent of the redshifts of galaxies, and thus can be applied to detecting filaments at relatively high redshifts, where there are photometric images but lack of spectroscopic data and thus the previous algorithms of detecting filaments (e.g., using the velocities field) fail.\n\n\\section{Summary and Conclusion}\n\nSince the red galaxies are preferentially aligned with their host filaments, we developed a new method, called location-alignment-method (LAM), of detecting filaments around clusters of galaxies, which uses both the alignments of the red galaxies and their distributions in 2D images. For the first time, the orientations of red galaxies are used to identify filaments. We applied LAM to the environment of Coma cluster, and compared our results with the ``standard'' filaments detected by the line-of-sight velocities and 2D positions of the galaxies \\citep{Falco14}. We summarize our results as follows.\n\n1) LAM can effectively find out the filaments around a cluster with $1\\sigma$ confidence level, and even relatively higher confidence levels.\n\n2) Four filaments (two filaments are located in sheets) are found in the two selected regions (W1 and W2),\n\n3) The alignment of red galaxies is important in LAM, since a substantial part ($73.2\\%$ for W1, and $66.7\\%$ for W2) of the selected red galaxies is due to the alignment.\n\n4) Applying LAM to the samples of bright (brighter than $16\\ \\rm{mag}$) red galaxies without the information of redshifts, we find that the filaments can still be detected (for W1 and W2, 83.0\\% and 87.9\\% red galaxies in the filaments are detected, respectively). The contaminatons by background interlopers are not strong.\n\nIn conclusion, there are two main advantages of LAM: (1) LAM can clearly reveal the number and overall orientations of the detected filaments; (2) LAM is independent of the redshifts of galaxies; therefore, we can use LAM to select the red galaxies in filaments among the sample of galaxies without the information of redshifts. Thus LAM can be applied at relatively high redshifts where there are photometric images but lack of spectroscopic data.\n\n\\section*{Acknowledgments}\n\nWe much thank the referee for his\/her helpful discussion. SNZ acknowledges partial funding support by 973 Program of China under grant 2014CB845802, by the National Natural Science Foundation of China under grant Nos. 11133002 and 11373036, by the Qianren start-up grant 292012312D1117210, and by the Strategic Priority Research Program ``The Emergence of Cosmological Structures'' of the Chinese Academy of Sciences under grant No. XDB09000000.\n\n\\bibliographystyle{mn2e}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\\subsection{Overview}\n\n\n\n\nLet $E$ and $E'$ be elliptic curves defined over a finite field $k$.  If $E$ and $E'$ are isogenous, then the  groups $E(k)$ and $E'(k)$ have the same order, but might not be isomorphic. For example, if $k$ is the field of 5 elements and $E$ and $E'$ are defined by\n\\begin{align*}\nE: y^2 &= x^3+x \\\\\nE': y^2 &= x^3 + x +2,\n\\end{align*}\nthen $E(k) \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$ and $E'(k) \\simeq \\mathbf{Z}\/4\\mathbf{Z}$. However, if $E(k) \\simeq E'(k)$, does that imply that $E(K) \\simeq E'(K)$, as $K$ ranges over finite extensions of $k$?  It is similarly easy to see that the answer is no:\n\n\\begin{exm} \\label{exm1}\nLet $k$ be the field of 17 elements and $K$ the field of $17^2$ elements.  Let \n\\begin{align*}\nE: y^2 &= x(x+6)(x+12) \\\\\nE': y^2 &= (x+1)(x+4)(x-4).\n\\end{align*} \nObserve that $E' = E\/\\langle (0,0) \\rangle$, so $E$ and $E'$ are 2-isogenous.  One can check that\n\\[\nE(k) \\simeq E'(k) \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/10\\mathbf{Z},\n\\]\nbut $E(K) = \\mathbf{Z}\/8\\mathbf{Z} \\times \\mathbf{Z}\/40\\mathbf{Z}$ and $E'(K)= \\mathbf{Z}\/4\\mathbf{Z} \\times \\mathbf{Z}\/80\\mathbf{Z}$.\n\\end{exm}\n\nIt is even possible to come up with examples where $E(K) \\simeq E'(K)$ for all finite extensions $K\/k$, but $E$ and $E'$ are not isomorphic as curves.  However, extreme examples such as these, and routine ones like \\ref{exm1},  only happen under very specific circumstances.  In fact, Example \\ref{exm1} can be viewed as a ``worst case scenario'' in the sense that, under mild rationality conditions, it is only in the context of rational 2-isogenies and quadratic extensions that we can have $E(k) \\simeq E'(k)$ and $E(K) \\not \\simeq E'(K)$.  (We will explain these rationality assumptions in detail below.)  In this paper we aim to understand  \\emph{how often} examples such as \\ref{exm1} occur.\n\n\\subsection{Reduction to 2-isogenies}  Throughout this paper we consider the simplest case where $E$ and $E'$ are related by an isogeny of prime degree $\\ell$, coprime to the characteristic of $k$.  If we additionally assume that $E(k)$ and $E'(k)$ have order divisible by $\\ell$ and the kernel of the isogeny $E \\to E'$ is generated by a rational point of order $\\ell$, then we recall the main theorem of \\cite{cullinan1}:\n\n\\begin{thm}[Theorem 1 of \\cite{cullinan1}] \\label{cull1thm}\nLet $k$ be a finite field of odd characteristic $p$, $\\ell \\ne p$ an odd prime, and $E$ and $E'$ ordinary elliptic curves over $k$.  Suppose $E$ and $E'$\n\\begin{itemize}\n\\item each have a point of order $\\ell$ defined over $k$, and \n\\item are $\\ell$-isogenous with kernel generated by a point defined over $k$.\n\\end{itemize}\nThen $E(k) \\simeq E'(k)$ if and only if $E(K) \\simeq E'(K)$ for all finite extensions $K\/k$.\n\\end{thm}\n\nConcretely, \\emph{when $\\ell$ is odd}, if $E$ and $E'$ are $\\ell$-isogenous and each has a non-trivial point of order $\\ell$ defined over $k$, then $E(k) \\simeq E'(k)$ implies $E(K) \\simeq E'(K)$ in all finite extensions $K\/k$.   \n\n\\bigskip\n\nBecause it is known how the $\\ell$-part of the groups of rational points grow in towers \\cite{lauter}, we can take from this theorem that the group structure of $E(k)$ completely determines the group structure of all $\\ell$-isogenous curves to $E$, in all finite extensions $K\/k$, under the hypothesis that the isogeny is generated by a $k$-rational point of order $\\ell$.  (In case the isogeny has degree $\\ell$, but the groups $E(k)$ and $E'(k)$ have order coprime to $\\ell$, then one must perform an initial base-field extension to (say) $L$ to obtain a point of order $\\ell$.  Then we can apply Theorem \\ref{cull1thm} taking $L$ as the base field.)\n\nExample \\ref{exm1} shows that Theorem \\ref{cull1thm} cannot be true when $\\ell=2$.  But it also exemplifies the only way for Theorem \\ref{cull1thm} to fail.  More precisely, in \\cite{cullinan2} the first author proved the following theorem, showing exactly under which circumstances the groups of rational points of 2-isogenous curves fail to be isomorphic in towers:\n\n\\begin{thm}[Theorem 2 of \\cite{cullinan2}] \\label{cull2thm}\nLet $E$ and $E'$ be ordinary, 2-isogenous elliptic curves defined over a finite field $k$ such that the isogeny is also defined over $k$. Suppose $E(k) \\simeq E'(k)$.  Let the endomorphism ring of each curve be an order in the quadratic imaginary ring $\\mathbf{Z}[\\omega]$ and write $\\pi = a+b\\omega \\in \\mathbf{Z}[\\omega]$, where $a$ is odd and $b$ is even, for the Frobenius endomorphism.  \nThen $E(K) \\simeq E'(K)$ for all finite extensions $K\/k$ unless the following holds:\n\\[\nv_2(a-1) = 1 \\text{  and  } v_2(a+1) > v_2(b) - s_2.\n\\]\nIn that case, $E(K) \\simeq E'(K)$ for odd-degree extensions $K\/k$ only.\n\\end{thm}\n\n\\begin{rmk}\nIn the published version of \\cite[Theorem 2]{cullinan2} there is a typographical error in the statement of the theorem.  There is an extra ``+1'' in the inequality for $v_2(a+1)$.  The corrected statement is listed above.\n\\end{rmk}\n\nIn the theorem, $s_2$ is a positive integer related to the conductors of the endomorphism rings of $E$ and $E'$. The upshot of this result is that there are precisely two possibilities. Either \n\\begin{enumerate}\n\\item $E(K) \\simeq E'(K)$ for all finite extensions $K\/k$, or \n\\item we can detect that $E(K) \\not \\simeq E'(K)$ in the unique quadratic extension $K\/k$. Moreover, we can detect this failure by performing computations \\emph{exclusively over $k$}.   \n\\end{enumerate}\nWe will review all of this background in detail in later sections of the paper. \n\n\\subsection{Setup and Statement of the Main Results} Granting this background, we now set our notation and aims for the paper.  Let $E$ and $E'$ be 2-isogenous elliptic curves defined over a field $k$ such that the isogeny is also defined over $k$.  We call such a pair $(E,E')$ \\textbf{rationally 2-isogenous}.  In this paper we  focus exclusively on the cases $k= \\mathbf{F}_p$ and $k = \\mathbf{Q}$. \n\nFix an odd prime $p$. If $E$ and $E'$ are rationally 2-isogenous over $\\mathbf{F}_p$, then $E(\\mathbf{F}_p)$ has a point $P$ of order 2 and \n\\[\nE' = E\/\\langle P \\rangle.\n\\]\nWe say that the pair $(E,E')$ is an  \\textbf{anomalous pair} if $E$ and $E'$ are rationally 2-isogenous over $\\mathbf{F}_p$, $E(\\mathbf{F}_p) \\simeq E'(\\mathbf{F}_p)$, and  $E(\\mathbf{F}_{p^2}) \\not \\simeq E'(\\mathbf{F}_{p^2})$. As explained above, this is precisely the obstruction for rationally 2-isogenous curves having isomorphic group structures in towers over $\\mathbf{F}_p$.\n\n\nHere is the point of view we take for the paper.  Fix a pair of rationally 2-isogenous curves $(E,E')$ over $\\mathbf{Q}$. We assume henceforth that $E$ and $E'$ do not have CM.  However, we will address the CM case in a forthcoming paper \\cite{rnt}; see Section \\ref{preview_2} for further details. To streamline notation, we will also use $E$ and $E'$ to denote the reductions modulo $p$ of the curves over $\\mathbf{Q}$.   We call a prime $p$ of good reduction \\textbf{anomalous} for $(E,E')$ if $E(\\mathbf{F}_p) \\simeq E'(\\mathbf{F}_p)$ and $E(\\mathbf{F}_{p^2}) \\not \\simeq E'(\\mathbf{F}_{p^2})$.  Therefore, at an anomalous prime for the pair $(E,E')$ defined over $\\mathbf{Q}$, we have that $(E,E')$ is an anomalous pair.  Depending on whether $p$ or $E$ is fixed, the two usages of ``anomalous'' should not be in conflict.   \n\nGiven this setup, we seek to understand the ratio\n\\begin{align} \\label{anomlim}\n\\mathcal{P}(X)  = \\frac{\\#\\lbrace \\text{anomalous}~p \\leq X \\rbrace}{\\pi(X)},\n\\end{align}\nwhere $\\pi(X)$ is the prime counting function, and also the limit $\\mathcal{P} = \\lim_{X \\to \\infty} \\mathcal{P}(X)$, if it exists. We note that $\\mathcal{P}(X)$ and $\\mathcal{P}$ depend on both $E$ and $E'$ (more specifically, they depend on the images of the 2-adic representations over $\\mathbf{Q}$ for each curve).  In this paper we only make one computation explicit: the case where the 2-adic images are isomorphic and as large as possible given the constraints of the setup.  The following examples show that there exist pairs $(E,E')$ for which anomalous primes exist, and there exist pairs for which they do not.  Throughout this paper when we refer to a proportion of primes with some property, or the probability that a prime has some property, we mean it is in this sense of counting primes up to $X$ and taking a limit.\n\n\\begin{exm}\nLet $E$ be the elliptic curve \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/210\/e\/5}{{\\tt 210e5}} and $E'$ the curve \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/210\/e\/4}{{\\tt 210e4}} of the \\texttt{LMFDB} \\cite{lmfdb}.  Then $E$ and $E'$ are 2-isogenous, with $\\mathbf{Q}$-torsion subgroups $\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/8\\mathbf{Z}$ and $\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/4\\mathbf{Z}$, respectively.  There are no anomalous primes for these curves, a consequence (as we will see below) of the sizes of $E(\\mathbf{Q})_\\mathsf{tors}$ and $E'(\\mathbf{Q})_\\mathsf{tors}$.   \n\\end{exm}\n\n\\begin{exm}\nThe isogeny class \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/10608\/y\/}{{\\tt 10608y}} consists of two elliptic curves, $E$ and $E'$, such that \n\\[\nE(\\mathbf{Q})  \\simeq E'(\\mathbf{Q}) \\simeq \\mathbf{Z}\/2\\mathbf{Z};\n\\]\nthese are the smallest Mordell-Weil groups possible given that $E$ and $E'$ are rationally 2-isogenous.  Moreover, the mod 2 representation of each curve has order 2, and the 2-adic representation of each has index 3 in $\\operatorname{GL}_2(\\mathbf{Z}_2)$, \\emph{i.e.},~is as large as possible given the hypotheses on each curve.     We consider all primes up to some bound and count those that are anomalous: \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\ntotal $\\#p$& $\\# p \\mid E(\\mathbf{F}_p) \\not \\simeq E'(\\mathbf{F}_p)$ & $\\# p \\mid E(\\mathbf{F}_p) \\simeq E'(\\mathbf{F}_p)$ & \\# anomalous \\\\\n\\hline\n1000 &  539 & 457 & 30 \\\\\n\\hline\n10000  & 5324 & 4672 & 335 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{exm}\nConverting the number of good primes to a value of $X$, we see that \n\\begin{align*}\n\\mathcal{P}(7919) &= \\frac{30}{1000} \\sim 0.03,\\text{ and} \\\\\n\\mathcal{P}(104741) &= \\frac{335}{10000} \\sim 0.0335,\n\\end{align*}\nsuggesting that the limit might exist.  \n\n\nThe main result of this paper is that the limit does exist and can be computed using the image of the 2-adic representations attached to $E$ and $E'$.  In general, a pair of rationally 2-isogenous elliptic curves define adjacent vertices on an {isogeny-torsion} graph over $\\mathbf{Q}$.  In \\cite{chil-alvaro} and \\cite{chiloyan}, the authors give a classification of all isogeny-torsion graphs over $\\mathbf{Q}$.  Moreover, the classification of Rouse and Zureick-Brown \\cite{rzb} of the possible images of the 2-adic representation of elliptic curves over $\\mathbf{Q}$ presents us with a finite list of graphs and images to consider for $E$ and $E'$.\n\nIn a forthcoming paper \\cite{rnt} we work out, among other things,  the possible values that can occur for elliptic curves over $\\mathbf{Q}$, including the CM case.  In this paper, we consider only one case and prove the following theorem.\n\n\\begin{thm} \\label{mainthm2}\nLet $E$ and $E'$ be rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ such that $[\\operatorname{GL}_2(\\mathbf{Z}_2): \\operatorname{im} \\rho_{E,2}] = [\\operatorname{GL}_2(\\mathbf{Z}_2):\\operatorname{im} \\rho_{E',2}] = 3$, \\emph{i.e.},~both curves have maximal 2-adic image given that each has a rational 2-torsion point.  Then $\\mathcal{P} = 1\/30$.\n\\end{thm}\n\n\\begin{rmk}\nThe elliptic curves of Theorem \\ref{mainthm2} are parameterized by the curve \\href{http:\/\/users.wfu.edu\/rouseja\/2adic\/X6.html}{$\\mathtt{X_6}$} of the \\href{http:\/\/users.wfu.edu\/rouseja\/2adic\/}{\\texttt{RZB}} database.\n\\end{rmk}\n\n\\begin{rmk}\nSee Section \\ref{preview_2} for a discussion of the non-maximal cases and setup to be addressed in \\cite{rnt}.\n\\end{rmk}\n\nIn order to get the result that $\\mathcal{P} = 1\/30$, we make full use of the structure of the \\textbf{2-isogeny volcano} $V_p$ of $E$ at $p$.  The 2-isogeny volcano is a graph, the connected components of which consist of vertices (elliptic curves over $\\mathbf{F}_p$) and edges ($\\mathbf{F}_p$-rational 2-isogenies), that organizes the curves into levels (we reserve the term \\emph{height} for the entire volcano and review our conventions in later sections).  All curves at the same level have isomorphic endomorphism rings, which implies that all curves at the same level have isomorphic group structures over $\\mathbf{F}_p$.  \n\nA 2-isogeny $E \\to E'$ defined over $\\mathbf{F}_p$ can be \\textbf{vertical} ($[\\operatorname{End}(E):\\operatorname{End}(E')] = 2$ or $1\/2$) or \\textbf{horizontal} ($\\operatorname{End}(E) \\simeq \\operatorname{End}(E')$); horizontal isogenies necessarily preserve the group structure over $\\mathbf{F}_p$, while vertical isogenies may or may not.  At an anomalous prime $p$, we have the following confluence of events:\n\\begin{itemize}\n\\item the $\\mathbf{Q}$-isogeny $E \\to E'$ reduces to a vertical isogeny over $\\mathbf{F}_p$, and\n\\item $E(\\mathbf{F}_p)[2^\\infty] \\simeq E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$ so that the volcano $V_p$ has the rough structure\n\\begin{center}\n\\begin{tikzpicture}[scale=1.0,sizefont\/.style={scale = 2}]\n\\draw[ultra thick] (1,5) node {${\\bullet}$};\n\\draw[ultra thick, dotted] (1,5) -- (1,4);\n\\draw[ultra thick] (1,4) node {${\\bullet}$};\n\\draw[thick] (1,3) -- (1,4);\n\\draw[ultra thick] (1,3) node {${\\bullet}$};\n\\draw[ultra thick] (0.5,2) node {${\\bullet}$};\n\\draw[ultra thick] (1.5,2) node {${\\bullet}$};\n\\draw[ultra thick, dotted] (0.5,2) -- (0.5,1);\n\\draw[ultra thick, dotted] (1.5,2) -- (1.5,1);\n\\draw[ultra thick] (1.5,1) node {${\\bullet}$};\n\\draw[ultra thick] (0.5,1) node {${\\bullet}$};\n\\draw[thick] (1,3) -- (0.5,2);\n\\draw[thick] (1,3) -- (1.5,2);\n\n\\draw(4,5) node {$\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$};\n\\draw(4,4) node {$\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$};\n\\draw(4,3) node {$\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$};\n\\draw(4,2) node {$\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$};\n\\draw[thick, dotted] (4,1.75) -- (4,1.25);\n\\draw(4,1) node {$\\mathbf{Z}\/4\\mathbf{Z}$};\n\n\\draw(-1,3) node {$E$};\n\\draw(-1,2) node {$E'$};\n\n\\end{tikzpicture}\n\\end{center}\nwhere either $E$ or $E'$ lies at least two levels above the floor, and\n\\item $E(\\mathbf{F}_{p^2})[2^\\infty] \\not \\simeq E'(\\mathbf{F}_{p^2})[2^\\infty]$ and $E$ and $E'$ are situated on $V_{p^2}$ as follows\n\n\\begin{center}\n\\begin{tikzpicture}[scale=1.0,sizefont\/.style={scale = 2}]\n\\draw[ultra thick] (1,5) node {${\\bullet}$};\n\\draw[ultra thick, dotted] (1,5) -- (1,4);\n\\draw[ultra thick] (1,4) node {${\\bullet}$};\n\\draw[thick] (1,3) -- (1,4);\n\\draw[ultra thick] (1,3) node {${\\bullet}$};\n\\draw[ultra thick] (0.5,2) node {${\\bullet}$};\n\\draw[ultra thick] (1.5,2) node {${\\bullet}$};\n\\draw[ultra thick, dotted] (0.5,2) -- (0.5,1);\n\\draw[ultra thick, dotted] (1.5,2) -- (1.5,1);\n\\draw[ultra thick] (1.5,1) node {${\\bullet}$};\n\\draw[ultra thick] (0.5,1) node {${\\bullet}$};\n\\draw[thick] (1,3) -- (0.5,2);\n\\draw[thick] (1,3) -- (1.5,2);\n\n\\draw(4,5) node {$\\vdots$};\n\\draw(4,4) node {$\\vdots$};\n\\draw(4,3) node {$\\mathbf{Z}\/2^{m+1}\\mathbf{Z} \\times \\mathbf{Z}\/2^{u-1}\\mathbf{Z}$};\n\\draw(4,2) node {$\\mathbf{Z}\/2^m\\mathbf{Z} \\times \\mathbf{Z}\/2^{u}\\mathbf{Z}$};\n\\draw(4,1) node {$\\mathbf{Z}\/2^{m+u}\\mathbf{Z}$};\n\\draw[thick, dotted] (4,1.75) -- (4,1.25);\n\\draw(-1,3) node {$E$};\n\\draw(-1,2) node {$E'$};\n\n\\end{tikzpicture}\n\\end{center}\n\\end{itemize}\n\nWe interpret the value $\\mathcal{P} = 1\/30$ as the sum of a geometric series, where the summands reflect the group theory of $\\operatorname{im} \\rho_{E,2}$ and $\\operatorname{im} \\rho_{E',2}$.  In particular, we filter the anomalous primes by \\textbf{defect} (which we explain in detail in the sections below).  Briefly, an anomalous prime has defect $(a,b)$ if $E(\\mathbf{F}_{p^2})$ has full $2^a$-torsion but not full $2^{a+1}$-torsion and $E(\\mathbf{F}_{p^2})$ has full $2^b$-torsion, but not full $2^{b+1}$-torsion.  It is a fact about adjacent vertices on an isogeny volcano that a prime can only have defect $(m+1,m)$ or $(m,m+1)$.  (This is exemplified in the figure above.)  Filtering by defect, and weighting each defect by the size of the kernels of the homomorphisms $\\operatorname{im} \\overline{\\rho}_{E,2^{m+1}} \\to \\operatorname{im} \\overline{\\rho}_{E,2^{m}}$ and $\\operatorname{im} \\overline{\\rho}_{E',2^{m+1}} \\to \\operatorname{im} \\overline{\\rho}_{E',2^{m}}$, we obtain the summands in the geometric series.\n\nTo ease the cumbersome notation, we let $G = \\operatorname{im} \\rho_{E,2}$ and $G' = \\operatorname{im} \\rho_{E',2}$.  If $p$ is a good prime, let $F \\in G$ and $F' \\in G'$ denotes representatives of the class of Frobenius.  Note that even though as a quadratic irrational number $\\pi = a+b\\omega$ (the Frobenius endomorphism) is represented in $\\operatorname{End}(E)$ and $\\operatorname{End}(E')$ by the same integral expression, the interpretation in each ring is different when those rings are not isomorphic.  Given all of this, we prove the following finer version of Theorem \\ref{mainthm2}.\n\n\\begin{thm} \\label{equithm}      \nLet $E$ and $E'$ be rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ such that $[\\operatorname{GL}_2(\\mathbf{Z}_2): G] = [\\operatorname{GL}_2(\\mathbf{Z}_2):G'] = 3$. Let $p$ be a prime such that $F \\equiv -I \\pmod{2^{m}}$ but $F\\not\\equiv-I \\pmod{2^{m+1}}$.  Then with probability 1\/2, $F' \\equiv -I \\pmod{2^{m}}$ and $p$ is not anomalous, and with probability 1\/2, $F' \\equiv -I \\pmod{2^{m-1}}$ and $F'\\not\\equiv -I \\pmod{2^m}$ and $p$ is anomalous of defect $(m+1,m)$.  Furthermore, this characterizes all anomalous primes of defect $(m+1,m)$.\n\\end{thm}\n\n\\begin{rmk}\nA similar result holds for primes of defect $(m,m+1)$.\n\\end{rmk}\n\nThis brings us to the final portion of the paper where we re-interpret our results on anomalous primes and their defects in terms of a probabilistic model of the distribution of heights of volcanoes and the discriminants of the endomorphism rings at each level.  \n\n\n\n\n\n\n\\subsection{Organization of the Paper}\nIn the next section we review background on elliptic curves over finite fields, in particular the relationship between the endomorphism ring and rational points.  We also recall the relevant history of this problem as well as the results in \\cite{cullinan1} and \\cite{cullinan2} that are applicable in this context.\n\n\nAs a rough guide to the results, the main point of Section \\ref{general} is to determine the structure of the 2-Sylow subgroup of $E(\\mathbf{F}_p)$ and $E'(\\mathbf{F}_p)$ at anomalous primes.  This leads to the notion of the defect of an anomalous prime.\n\nIn Section \\ref{Q} we prove Theorem \\ref{mainthm2} by filtering the anomalous primes by defect, determining the exact proportion of each defect, and then summing over all defects.  We determine the exact proportion of each defect by re-interpreting the criteria of Section \\ref{general} for a prime to be anomalous in terms of matrix conditions in the 2-adic representations attached to $E$ and $E'$.  Following this,  we interpret the defect of an anomalous prime as determining where on the isogeny volcano of the pair $(E,E')$ lies and give numerical data suggesting a finer relationship between anomalous primes and endomorphism rings.\n\nSection \\ref{preview_2} is dedicated to future work.  In particular, we contextualize the results of the present paper within the goals of a follow-up paper in which we explore the range of  values of $\\mathcal{P}$ that can occur for elliptic curves over $\\mathbf{Q}$, including CM curves.  \n\n\\subsection{Databases} We use two online databases in this work: the $L$-Functions and Modular Forms Database (\\href{https:\/\/www.lmfdb.org\/}{\\texttt{LMFDB}}), and the classification of 2-adic images of Galois representations attached to elliptic curves over $\\mathbf{Q}$, due to Rouse and Zureick-Brown (\\href{http:\/\/users.wfu.edu\/rouseja\/2adic\/}{\\texttt{RZB}}), based on the paper \\cite{rzb}.  Whenever we use an entry in the database, such as an isogeny class or elliptic curve in the \\texttt{LMFDB}, or a modular curve in the \\texttt{RZB} database, we link to that entry in the database.\n\n\\subsection{Notation} We will explain any specialized notation in main body of the paper, but we remind the reader of some standard conventions.  If $k$ is a field and $k^s$ a separable closure of $k$, then we write $\\operatorname{Gal}_k$ for the Galois group of $k^s\/k$.  If $E$ is an elliptic curve over $k$ and $\\ell$ is a prime number, then we write $T_\\ell E$ for the $\\ell$-adic Tate module of $E$ and \n\\begin{align*}\n&\\rho_{E,\\ell}: \\operatorname{Gal}_k \\to \\operatorname{Aut} (T_\\ell E), \\text{ and} \\\\\n&\\overline{\\rho}_{E,\\ell^n}: \\operatorname{Gal}_k \\to \\operatorname{Aut} (T_\\ell E \\otimes \\mathbf{Z}\/\\ell^n\\mathbf{Z}) \n\\end{align*}\nfor the $\\ell$-adic and mod $\\ell^n$ representations of $E$, respectively.  If $G \\subseteq \\operatorname{GL}_2(\\mathbf{Z}_{\\ell})$ is the image of the $\\ell$-adic representation, then we write $G(\\ell^n) \\subseteq \\operatorname{GL}_2(\\mathbf{Z}\/\\ell^n\\mathbf{Z})$ for its reduction modulo $\\ell^n$. \n\nIf $R$ is a ring, then we write $M_n(R)$ for the ring of $n\\times n$ matrices with entries in $R$.  Finally, if $p$ is a prime number, then we write $v_p: \\mathbf{Q}^\\times \\to \\mathbf{Z}$ for the $p$-adic valuation. \n\n\\subsection{Acknowledgments} We would like to thank Andrew Sutherland for supplying us with the initial computations that suggested the correct value of $\\mathcal{P}$.  The second author was supported by NSF Grants DMS 1802281 and DMS 2154223.\n\n\\section{Elliptic Curves over Finite Fields} \\label{background}\n\n\\subsection{Endomorphism Rings and Rational Points} \\label{first_background} Let $q$ be a power of an odd prime $p$ and let $E$ be an ordinary elliptic curve defined over $\\mathbf{F}_{q}$; we will address supersingular curves in Section \\ref{supsect}. Since $E$ is ordinary, its endomorphism ring $\\operatorname{End}(E)$ is isomorphic to an order $\\mathcal{O}$ in an imaginary quadratic field $K = \\mathbf{Q}(\\sqrt{D})$ for a squarefree negative integer $D$, and all endomorphisms of $E$ are defined over $\\mathbf{F}_q$.  \n\nLet $\\mathcal{O}_{K}$ denote the ring of integers of $K$.  Write $d_K$ for the discriminant of $\\mathcal{O}_{K}$, the maximal order of $K$.  Then \n\\[\nd_K = \\begin{cases}\n4D & \\text{ if } D \\equiv 2,3\\pmod{4} \\\\\nD & \\text{ if } D \\equiv 1 \\pmod{4}.\n\\end{cases}\n\\]\nRecall that if $g$ is a positive integer, then we denote by $\\mathcal{O}_g:=\\mathbf{Z} \\oplus \\mathbf{Z} g\\omega$  the order of conductor $g$ in $\\mathcal{O}_{K}$, where \n\\[\n\\omega = \\begin{cases}\n (1+\\sqrt{D})\/2 & \\text{if } D \\equiv 2,3\\pmod{4}\\\\\n \\sqrt{D} & \\text{if } D \\equiv 1 \\pmod{4}.\n \\end{cases}\n\\] \nWe may therefore write $\\mathbf{Z}[\\pi] = \\mathcal{O}_f$ and $\\mathcal{O}_{K} = \\mathcal{O}_1$.  Since $\\operatorname{End}(E) = \\mathcal{O}$ contains $\\mathbf{Z}[\\pi]$, we may write $\\mathcal{O} = \\mathcal{O}_g$ for some $g \\mid f$ with \n\\[\n\\mathcal{O}_f \\subseteq \\mathcal{O}_g \\subseteq \\mathcal{O}_1.\n\\]\nIf $\\Delta_g$ denotes the discriminant of $\\mathcal{O}_g$, then $\\Delta_g = g^2d_K$. \n\nIdentifying $\\operatorname{End}(E)$ with an order in $\\mathcal{O}_{K}$, we may write the Frobenius endomorphism $\\pi \\in \\operatorname{End}(E)$ explicitly as an element of $\\mathcal{O}_{K}$.  We now review how to do this.  Recall the well-known formulas relating the cardinality of $E(\\mathbf{F}_{q})$, the fundamental discriminant of $K$, and the trace $t$ of $\\pi$:\n\\begin{align}\n\\#E(\\mathbf{F}_{q}) &= 1 + q - t \\\\\n4q &= t^2 - \\beta^2\\Delta_g,\n\\end{align}\nwhere $t$ is the trace of Frobenius, $\\beta$ is a positive integer, and $\\Delta_g = g^2d_K$, as above.\n\nThen $\\pi$ has a unique integral representation $\\pi = a+b\\omega \\in \\mathbf{Z}[\\omega]$ given by \n\\begin{align*}\na &= \\begin{cases} t\/2 & \\text{ if } D\\equiv 2,3\\pmod{4} \\\\ (t-\\beta g)\/2 & \\text{ if }D \\equiv 1\\pmod{4}, \\end{cases} \\\\\nb&= \\beta g.\n\\end{align*}\nWe also recall a fundamental result of Lenstra \\cite{lenstra}, which gives the structure of $E(\\mathbf{F}_{q^m})$ for all positive integers $m$:\n\\begin{equation}\\label{eq_lenstra}\nE(\\mathbf{F}_{q^m}) \\simeq \\frac{\\mathcal{O}}{(\\pi^m - 1)}.\n\\end{equation}\n\n\\subsection{Isogenies} Keeping with the notation above, suppose that $E$ and $E'$ are isogenous (ordinary) elliptic curves defined over $\\mathbf{F}_{q}$.  Then the groups $E(\\mathbf{F}_q)$ and $E'(\\mathbf{F}_q)$ have the same cardinality, as do the groups $E(\\mathbf{F}_{q^m})$ and $E'(\\mathbf{F}_{q^m})$, for all positive integers $m$. \n\nLet $\\ell \\ne p$ be a prime number.  If $E$ and $E'$ have endomorphism rings $\\mathcal{O}$ and $\\mathcal{O}'$, respectively, and are $\\ell$-isogenous, then by a result of Kohel \\cite[Prop.~21]{kohel} we have \n\\[\n[\\mathcal{O}:\\mathcal{O}'] = \\ell, \\ell^{-1}, \\text{ or } 1.\n\\]\nIn the first two cases, the isogeny is called \\emph{vertical} (ascending\/descending, depending on the inclusion) and in the latter it is \\emph{horizontal}.\n\nIsogenous elliptic curves have the same trace of Frobenius.  In the case of a vertical isogeny, $\\mathcal{O}$ and $\\mathcal{O}'$ are orders in $\\mathcal{O}_{K}$ of relative index $\\ell$.  We explain what happens when $\\mathcal{O}' \\subseteq \\mathcal{O}$.  (There is a completely analogous setup when $\\mathcal{O} \\subseteq \\mathcal{O}'$.)  There exist divisors $g$ and $g'$ of $f$ such that $g'\/g = \\ell$ and $\\mathcal{O} = \\mathcal{O}_g$, $\\mathcal{O}' = \\mathcal{O}_{g'}$ with  \n$$\n\\mathbf{Z}[\\pi] = \\mathcal{O}_f \\subseteq \\mathcal{O}_{g'} \\subseteq \\mathcal{O}_{g} \\subseteq \\mathcal{O}_1 = \\mathcal{O}_{K}.\n$$\n\nTurning to the group structures of isogenous curves, we recall that the main results of \\cite{heuberger} and \\cite{wittmann}  give criteria for any pair of  isogenous elliptic curves to have isomorphic groups of $\\mathbf{F}_{q^{m}}$-rational points in terms of the prime divisors of the integral components of $\\pi^m$.  We now recall some of the special notation introduced in \\cite{heuberger} that we will adopt throughout the rest of this paper. \n\nDefine a finite set of prime numbers $\\mathbf{P}$ as follows, incorporating the notation above:\n\\[\n\\mathbf{P} = \\lbrace p\\text{ prime} \\mid v_p(g) \\ne v_p(g') \\rbrace.\n\\]\nFor each $p \\in \\mathbf{P}$ we set \n\\[\ns_p  = \\max \\lbrace v_p(g), v_p(g') \\rbrace,\n\\]\nwhence $s_p \\geq 1$.   With this notation in place, write \n\\[\n\\pi^m = a_m + b_m\\omega,\n\\] \nfor integers $a_m,b_m$.  Finally, we recall the criterion of \\cite[Thm.~2.4]{heuberger} for $E(\\mathbf{F}_{q^m})$ and $E'(\\mathbf{F}_{q^m})$ to be isomorphic:\n\\begin{align} \\label{heuberger_criterion}\nE(\\mathbf{F}_{q^m}) \\simeq E'(\\mathbf{F}_{q^m}) \\Longleftrightarrow v_p(a_m-1) \\leq v_p(b_m) - s_p,\n\\end{align}\nfor all $p \\in \\mathbf{P}$. \n\nNow we specialize to the situation that is the primary focus of this paper.  When the degree of the vertical isogeny $E \\to  E'$ is a prime number $\\ell$, then $g'\/g = \\ell^{\\pm 1}$ and so $\\mathbf{P} = \\lbrace \\ell \\rbrace$.  For descending isogenies we have $v_\\ell(g') = 1 + v_\\ell(g)$ and for ascending isogenies we have $v_\\ell(g) = 1+ v_\\ell(g')$.   Specializing further, we set $\\ell=2$ for the remainder of the paper.  In \\cite[Thm.~2]{cullinan2} the first author proved that if $E(\\mathbf{F}_{q}) \\simeq E'(\\mathbf{F}_{q})$ and $E(\\mathbf{F}_{q^2}) \\simeq E'(\\mathbf{F}_{q^2})$, then $E(\\mathbf{F}_{q^m}) \\simeq E(\\mathbf{F}_{q^m})$ for all positive integers $m$.  Theorem \\ref{cull2thm} gives the precise conditions under which the second isomorphism fails, given the first.  \n\n\\subsection{Supersingular Curves} \\label{supsect}\n\nIn the case where $E$ and $E'$ are supersingular curves over $\\mathbf{F}_p$ the situation is (perhaps surprisingly)  much simpler.  We recall the following result of Wittmann.\n\n\\begin{thm}[Theorem 4.1 of \\cite{wittmann}] \\label{wittmann1}\nLet $E\/\\mathbf{F}_p$ be a supersingular elliptic curve.  Then \n$$\nE(\\mathbf{F}_{p^{2k}}) \\simeq \\mathbf{Z}\/((-p)^k-1)\\mathbf{Z} \\times \\mathbf{Z}\/((-p)^k-1)\\mathbf{Z}. \n$$\nFurther:\n\\begin{itemize}\n\\item If $p \\not \\equiv 3 \\pmod{4}$ or $p \\equiv 3 \\pmod{4}$ and $E[2] \\not \\subseteq E(\\mathbf{F}_p)$ we have \n$$\nE(\\mathbf{F}_{p^{2k+1}}) \\simeq \\mathbf{Z}\/(p^{2k+1} + 1)\\mathbf{Z} \\text{ and } \\operatorname{End}_{\\mathbf{F}_p}(E) \\simeq \\mathbf{Z}[\\sqrt{-p}].\n$$\n\\item If $p \\equiv 3 \\pmod{4}$ and $E[2] \\subseteq E(\\mathbf{F}_p)$ we have\n$$\nE(\\mathbf{F}_{p^{2k+1}}) \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/\\left( \\frac{p^{2k+1}+1}{2} \\right)\\mathbf{Z} \\text{ and } \\operatorname{End}_{\\mathbf{F}_p}(E) \\simeq \\mathbf{Z}[(1+\\sqrt{-p})\/2].\n$$\n\\end{itemize}\n\\end{thm}\n\nIn \\cite{cullinan2} we observed that this immediately implies that when when $E$ and $E'$ are supersingular, then the group structure over $\\mathbf{F}_p$ completely determines the group structure over any finite extension:\n\n\\begin{cor}[Corollary 1 of \\cite{cullinan2}] \\label{cor1}\nLet $p$ be a prime.  Let $E_1$ and $E_2$ be supersingular, isogenous elliptic curves defined over $\\mathbf{F}_p$.  Suppose $E_1(\\mathbf{F}_p) \\simeq E_2(\\mathbf{F}_p)$.  Then $E_1(K) \\simeq E_2(K)$ for every finite extension $K\/\\mathbf{F}_p$.\n\\end{cor}\n\n\\section{General Properties of Anomalous Primes and Curves} \\label{general}\n\nWe retain the notation and setup of the previous sections, in particular we assume $E$ and $E'$ are ordinary.  We start with a general property of anomalous pairs.\n\n\\begin{prop}\nLet $(E,E')$ be an anomalous pair of elliptic curves defined over the finite field $\\mathbf{F}_p$.  Then $p \\equiv 1 \\pmod{4}$.\n\\end{prop}\n\n\\begin{proof} \nSuppose $p \\equiv 3 \\pmod{4}$.  We distinguish between the cases where $|E(\\mathbf{F}_p)| \\equiv 2\\pmod{4}$ versus $|E(\\mathbf{F}_p)| \\equiv 0 \\pmod{4}$. Recall that if $(E,E')$ is an anomalous pair then in the representation $\\pi = a+b\\omega$ of Frobenius as an element of $\\mathcal{O}_{K}$, we have that $b$ is even; write $b=2b'$.\n\nIf $|E(\\mathbf{F}_{p})| \\equiv 0 \\pmod{4}$, then $t \\equiv 0 \\pmod{4}$; write $t = 4t'$.  Since \n\\[\n4p = t^2 - b^2d_K = 16(t')^2 - 4(b')^2d_K,\n\\]\nwe must have $p = 4(t')^2 - (b')^2d_K$.  Thus $b'$ and $d_K$ are odd.  In particular, $v_2(b) =1$.  But since $(E,E')$ is an anomalous pair, we have \n\\[\nv_2(a-1) = 1 \\leq v_2(b) - s_2 = 1 - s_2,\n\\]\nwhence $s_2 =0$.  But this means $\\operatorname{End}(E) \\simeq \\operatorname{End}(E')$, contradicting the fact that $(E,E')$ are an anomalous pair.\n\nIf $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$, then $t \\equiv 2 \\pmod{4}$, so write $t = 2t'$ with $t'$ odd.  But then \n\\[\np  = (t')^2 - (b')^2d_K.\n\\]\nSince $p \\equiv 3 \\pmod{4}$ and $(t')^2 \\equiv 1 \\pmod{4}$, we must have $(b')^2d_K \\equiv 2 \\pmod{4}$.  But since $b'$ is odd and $d_K \\equiv 0$ or $1 \\pmod{4}$, this is impossible.  We conclude that if $p \\equiv 3 \\pmod{4}$ then $p$ cannot be anomalous. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lem} \\label{2torspt}\nIf $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$ then $E(\\mathbf{F}_p) \\simeq E'(\\mathbf{F}_p)$.\n\\end{lem}\n\n\\begin{proof}\nSince $E$ and $E'$ are 2-isogenous, the prime-to-2 parts of $E(\\mathbf{F}_p)$ and $E'(\\mathbf{F}_p)$ are isomorphic \\cite[Cor.~3]{cullinan1}.  Since each has a single point of order 2, the result follows by the structure theorem for finite abelian groups.\n\\end{proof}\n\n\\begin{thm}\nIf $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$ then $E(\\mathbf{F}_{p^2}) \\simeq E'(\\mathbf{F}_{p^2})$.\n\\end{thm}\n\n\\begin{proof}\nIf $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$ then by  Lemma \\ref{2torspt} we have $E(\\mathbf{F}_p) \\simeq E'(\\mathbf{F}_p)$. If, in addition, \n$E(\\mathbf{F}_{p^2}) \\not \\simeq E'(\\mathbf{F}_{p^2})$, then $(E,E')$ is anomalous whence $p \\equiv 1 \\pmod{4}$.  Writing $\\pi = a + b\\omega$ in the notation of Section 2, we have \n\\begin{enumerate}\n\\item $v_{2}(a-1) = 1 \\leq v_2(b) - s_2$, \\text{ and} \n\\item $v_2(a+1) >  v_2(b) - s_2$.\n\\end{enumerate}\nSince $v_2(a-1) = 1$, we have $a \\equiv 3 \\pmod{4}$.  We also have \n\\begin{align} \\label{n1_is_2_mod_4}\n|E(\\mathbf{F}_p)| = 1 + p - t \\equiv 2\\pmod{4},\n\\end{align}\nhence $t \\equiv 0 \\pmod{4}$.  Now we divide the argument into two cases based on $D \\pmod{4}$, where $D$ is the squarefree integer for which $K = \\mathbf{Q}(\\sqrt{D})$ is the endomorphism algebra of $E$ (and $E'$).\n\nIf $D \\equiv 2,3\\pmod{4}$, then $a=t\/2$ and so $t \\equiv 6\\pmod{8}$, a contradiction.  If $D \\equiv 1 \\pmod{4}$, then we first recall the inequality (1).  Since $(E,E')$ is an anomalous pair, we must have $s_2 \\geq1$ (otherwise, $\\operatorname{End}(E) \\simeq \\operatorname{End}(E')$), and so we conclude that $v_2(b) \\geq 2$.  But when $D \\equiv 1 \\pmod{4}$, we have $a = (t-b)\/2$. Since both $t$ and $b$ must be divisible by 4, we get that $a$ is even.  This contradicts $a \\equiv 3 \\pmod{4}$, established above. \n\\end{proof}\n\n\\begin{cor} \\label{2mod4cor}\nIf $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$ then $E(\\mathbf{F}_{p^m}) \\simeq E'(\\mathbf{F}_{p^m})$ for all positive integers $m$.\n\\end{cor}\n\n\\begin{proof}\nThis follows from \\cite[Thm.~2]{cullinan2}: if $E(\\mathbf{F}_{p^m}) \\simeq E'(\\mathbf{F}_{p^m})$ for $m \\in \\lbrace 1,2 \\rbrace$, then $E(\\mathbf{F}_{p^m}) \\simeq E'(\\mathbf{F}_{p^m})$ for all positive integers $m$. \n\\end{proof}\n\nTherefore, \\emph{every} pair of curves $E,E'$ over $\\mathbf{F}_p$ with $|E(\\mathbf{F}_p)| \\equiv 2 \\pmod{4}$  and that are rationally 2-isogenous have isomorphic Mordell-Weil groups in all finite extensions.  Therefore, any anomalous pair must have $|E(\\mathbf{F}_p)| \\equiv 0 \\pmod{4}$ and $p \\equiv 1 \\pmod{4}$.  \n\nNext, we define a finer notion of $(E,E')$ being an anomalous pair.  This will carry over to a refined notion of $p$ being an anomalous prime, which will be an important topic in the following sections. Because the 2-Sylow subgroups of $E(\\mathbf{F}_{p^2})$ and $E'(\\mathbf{F}_{p^2})$ have the same size, but are not isomorphic, we can ask how they differ.  We describe this difference using the notion of defect.\n\n\\begin{defn}\nLet $E \\to E'$ be rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ and let $p$ be an anomalous prime.  If \n\\begin{align*}\na  &= \\max \\lbrace i \\in \\mathbf{N} ~|~ E(\\mathbf{F}_{p^2})[2^\\infty] \\supseteq \\mathbf{Z}\/2^{i}\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}^{i} \\rbrace,\\text{ and} \\\\\na'  &= \\max \\lbrace i \\in \\mathbf{N} ~|~ E'(\\mathbf{F}_{p^2})[2^\\infty] \\supseteq \\mathbf{Z}\/2^{i}\\mathbf{Z} \\times 2^{i}\\mathbf{Z} \\rbrace,\n\\end{align*}\nthen we say that $p$ has \\textbf{defect} $(a,a')$.\n\\end{defn}\n\n\\begin{rmk}\nIt is a well-known property of the \\textbf{$\\ell$-isogeny volcano} (which we will recall in Section \\ref{volcanoes}) that if $E$ and $E'$ are $\\ell$-isogenous elliptic curves over a finite field $k$ and the $\\ell$-Sylow subgroups of $E(k)$ and $E'(k)$ are not isomorphic, then $E(k)[\\ell^\\infty] \\simeq \\mathbf{Z}\/\\ell^u\\mathbf{Z} \\times \\mathbf{Z}\/\\ell^v\\mathbf{Z}$ and $E'(k)[\\ell^\\infty] \\simeq \\mathbf{Z}\/\\ell^{u-1}\\mathbf{Z} \\times \\mathbf{Z}\/\\ell^{v+1}\\mathbf{Z}$ or $E'(k)[\\ell^\\infty] \\simeq \\mathbf{Z}\/\\ell^{u+1}\\mathbf{Z} \\times \\mathbf{Z}\/\\ell^{v-1}\\mathbf{Z}$ for some positive integer $u$ and nonnegative integer $v$.  Theorem \\ref{establish_defect} establishes a similar result and relates the defect of an anomalous prime to the 2-valuation of the Frobenius endomorphism. \n\\end{rmk}\n\nWe now make an observation concerning the 2-Sylow subgroups of anomalous pairs.\n\n\\begin{lem} \\label{22lem}\nSuppose $(E,E')$ is an anomalous pair.  Then $E(\\mathbf{F}_p)[2^\\infty] \\simeq  E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$.  \n\\end{lem}\n\n\\begin{proof}\nIf $(E,E')$ is an anomalous pair, then we must have $p \\equiv 1 \\pmod{4}$ and $|E(\\mathbf{F}_p)| \\equiv 0 \\pmod{4}$, as previously established.  If neither curve has full 2-torsion defined over $\\mathbf{F}_p$, then the 2-Sylow subgroups of $E(\\mathbf{F}_p)$ and $E'(\\mathbf{F}_p)$ are cyclic and the curves are rationally 2-isogenous.  By \\cite[Thm.~1.2]{aw}, this is not possible.  This establishes that $E(\\mathbf{F}_p)[2] \\simeq E'(\\mathbf{F}_p)[2] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$.\n\nTo see that $E(\\mathbf{F}_p)[2^\\infty] \\simeq  E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$ as well, recall from \\cite[p.~742]{miret2} that if $E$ and $E'$ are 2-isogenous and have isomorphic group structures over $\\mathbf{F}_p$, then it must be the case that $E(\\mathbf{F}_p)[2^\\infty] \\simeq  E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2^k\\mathbf{Z} \\times \\mathbf{Z}\/2^k\\mathbf{Z}$ for some $k$, hence $|E(\\mathbf{F}_p)| = p+1-t \\equiv 0 \\pmod{2^{2k}}$.  Suppose $k>1$.  Then both curves will have at least full $2^{k+1}$-torsion over $\\mathbf{F}_{p^2}$, and at least one will have full $2^{k+2}$-torsion (since $E(\\mathbf{F}_{p^2}) \\not \\simeq E'(\\mathbf{F}_{p^2})$).  Therefore ,\n\\[\n|E(\\mathbf{F}_{p^2})| = (p+1-t)(p+1+t) \\equiv 0 \\pmod{2^{2k+4}},\n\\]\nand so $p+1+t \\equiv 0 \\pmod{16}$.  Since $k>1$, we have $p+1-t \\equiv 0 \\pmod{16}$ as well, which implies that $t \\equiv 0 \\pmod{8}$.  But this contradicts the fact that for an anomalous pair we must have $t \\equiv 2 \\pmod{4}$.  This completes the proof.\n\\end{proof}\n\n\nWe will apply the following result in the proof of Theorem \\ref{establish_defect}.\n\\begin{lem} \\label{2_torsion_frob}\nLet $E$ be an ordinary elliptic curve defined over a finite field $\\mathbf{F}_{q}$ of odd characteristic.  Let $\\pi \\in \\operatorname{End}(E)$ be the Frobenius endomorphism.  If $v$ is the largest integer such that $\\pi^m -1$ factors as $2^v\\alpha$ in $\\operatorname{End}(E)$, then $E(\\mathbf{F}_{q^m})$ has full $2^v$-torsion but not full $2^{v+1}$-torsion. \n\\end{lem}\n\n\\begin{proof}\nBy Lenstra's  theorem \\eqref{eq_lenstra} \\cite[Thm.~1(a)]{lenstra}, we have $E(\\mathbf{F}_{q^m}) \\simeq \\operatorname{End}(E)\/(\\pi^m-1)$.   If $\\pi^m-1$ factors as $2^v\\alpha$, then clearly $E(\\mathbf{F}_{q^m})$ has full $2^v$-torsion. By factoring isogenies via \\cite[Thm.~25.1.2]{galbraith}, any $\\mathbf{F}_q$-rational endomorphism of $E$ whose kernel contains the $2^{v+1}$-torsion points would have to factor as $2^{v+1}\\beta$ in $\\operatorname{End}(E)$. Thus $E(\\mathbf{F}_{q^m})$ has full $2^v$-torsion but not full $2^{v+1}$-torsion.  \n\\end{proof}\n\n\n\\begin{thm} \\label{establish_defect}\nLet $E \\to E'$ be 2-isogenous elliptic curves over $\\mathbf{Q}$ and let $p$ be an anomalous prime.  Suppose $\\operatorname{End}(E) = \\mathcal{O}_g$ and $\\operatorname{End}(E) = \\mathcal{O}_{g'}$ are orders of conductor $g$ and $g'$, respectively, in the the imaginary number ring $\\mathcal{O}_{K} = \\mathbf{Z} + \\mathbf{Z}\\omega$; write $\\pi = a + b\\omega$ with $b = \\beta g = \\beta'g'$.  Then $p$ has defect $(m+1,m)$ or $(m,m+1)$ for some integer $m \\geq 2$, where $m = v_2(\\beta)$.\n\\end{thm}\n\n\n\\begin{proof}\nThe isogeny $E \\to E'$, initially defined over $\\mathbf{Q}$, reduces modulo $p$ to a vertical isogeny (if the reduction were horizontal then $\\mathcal{O}_g = \\mathcal{O}_{g'}$ and $p$ would not be anomalous).  For the remainder of the proof we assume the isogeny is descending and will conclude that $p$ has defect $(m+1,m)$; an identical argument for ascending isogenies would show that $p$ has defect $(m,m+1)$.  \n\nWrite $\\operatorname{End}(E) = \\mathcal{O}_g = \\mathbf{Z} + g\\mathbf{Z}\\omega$ and $\\operatorname{End}(E') = \\mathcal{O}_{g'} = \\mathbf{Z} + g'\\mathbf{Z}\\omega$.  We have $g' = 2g$ and also write $\\pi = a+ b\\omega$ with $b = \\beta g$ as established in Section \\ref{first_background}. Since $p$ is anomalous and since $v_2(g') = v_{2}(g) + 1$, we have \n\\[\nv_2(a-1) = 1 \\leq v_2(b)-s_2 = v_{2}(\\beta) - 1 < v_2(a+1).\n\\]\nObserve that $v_2(\\beta) \\geq 2$.\n\nNow we compute \n\\[\n\\pi^2 -1 = \\begin{cases} (a^2 - 1 + b^2D) + 2ab\\omega & \\text{if $d_K = 4D$ and $D \\equiv 2,3\\pmod{4}$, and} \\\\\n(a^2-1 + b^2 \\left(\\frac{D-1}{4} \\right)) + (2ab+b^2)\\omega & \\text{if $d_K = D$ with $D \\equiv 1\\pmod{4}$}.\\end{cases}\n\\]\n\nIn $\\mathcal{O}_g$ we can factor, \n\\[\n\\pi^2 - 1 =\n\\begin{cases}\n&(a^2-1+\\beta^2g^2D) + (2\\beta) a g\\omega,\\text{ or}\\\\\n&(a^2-1 + \\beta^2g^2 \\left(\\frac{D-1}{4} \\right)) + 2\\beta (a + (\\beta\/2))g\\omega,\n\\end{cases}\n\\]\ndepending on $d_K \\pmod{4}$. In the first case (since $a$ is odd) and in the second case (since $a$ is odd and $\\beta\/2$ is even), $\\pi^2 - 1$ is divisible in $\\mathcal{O}_g$ by $2^{v_2(\\beta)+1}$ and no higher power of 2.\n\nSimilarly, in $\\mathcal{O}_{g'} = \\mathcal{O}_{2g}$, $\\pi^2-1$ is divisible by $2^{v_2(\\beta)}$ and no higher power of 2.  By Lemma \\ref{2_torsion_frob}, $E(\\mathbf{F}_{p^2})$ has full $2^{v_{2}(\\beta)+1}$-torsion (and no higher) and $E'(\\mathbf{F}_{p^2})$ has full $2^{v_2(\\beta)}$-torsion (and no higher).  Thus $p$ has defect $(m+1,m)$ for some integer $m = v_2(\\beta) \\geq 2$.  \n\\end{proof}\n\nIn the next section we interpret anomalous primes and their defects in relation to isogeny volcanoes.\n\n\\section{Isogeny Volcanoes of Elliptic Curves} \\label{volcanoes}\n\nFollowing a brief recap of the theory of isogeny volcanoes of ordinary elliptic curves, our purpose in this section is to prove a key proposition in service of Theorems \\ref{mainthm2} and \\ref{equithm}. We do not intend for this to be a complete treatment of the background material; we refer the reader to \\cite{sutherland_volcano} for further details and proofs.\n\nLet $q$ be a power of a prime $p$ and $E$ an ordinary elliptic curve over $\\mathbf{F}_{q}$.  Let $V_q$ be the connected component of the 2-isogeny graph (volcano) containing $E$.  Then $V_q$ is a graph whose vertices correspond to elliptic curves defined over $\\mathbf{F}_{q}$ that are 2-power $\\mathbf{F}_{q}$-rationally isogenous to $E$ and edges are $\\mathbf{F}_{q}$-rational 2-isogenies.  Thus, in our setup, $E$ and $E'$ represent adjacent vertices on the graph $V_p$; note that $V_p$ is a subgraph of $V_{p^2}$. \n\nLet $q$ be a power of $p$ and $T$ the trace of Frobenius over $\\mathbf{F}_{q}$.  Let $\\mathsf{sqf}(m)$ denote the squarefee part of an integer $m$.  where $\\mathcal{O}_0$ is the endomorphism ring of an elliptic curve lying on the crater of $V_q$. Let $K = \\mathbf{Q}(\\sqrt{T^2-4q}) = \\mathbf{Q}(\\sqrt{D})$ where $D = \\mathsf{sqf}(T^2-4q)$.  Then \n\\[\n\\operatorname{disc} \\mathcal{O}_K = \\begin{cases} D & \\text{ if } D \\equiv 1 \\pmod{4}, \\text{ and} \\\\ 4D & \\text{ if } D \\equiv 2,3\\pmod{4}. \\end{cases}\n\\]\nA theorem of Kohel \\cite[Theorem 7(5)]{sutherland_volcano} shows that for a 2-isogeny volcano $2 \\nmid [\\mathcal{O}_K\\colon \\mathcal{O}_0]$.  \n\nThe \\emph{height} of the volcano $V_q$ is given by \\cite[Thm.~7]{sutherland_volcano}\n\\begin{equation}\\label{eqn:volcano_height}\nh(V_q) = \\frac{1}{2} v_2 \\left( \\frac{T^2 - 4q}{\\operatorname{disc} \\mathcal{O}_0} \\right) = \\frac{1}{2} v_2 \\left( \\frac{T^2 - 4q}{\\operatorname{disc} \\mathcal{O}_K} \\right).\n\\end{equation}\nWe choose the opposite labeling of the height as defined in \\cite{sutherland_volcano} (there it is called the \\textbf{depth}) and declare the floor of the volcano to have height 0.  In the case that $V_q$ consists of an isolated vertex, we set $h(V_q)=0$.  The subgraph of vertices at level $h(V_q)$ is called the \\textbf{crater} of the volcano.    This labeling is more convenient for interpreting the defect of an anomalous prime in terms of the location of $E$ and $E'$.\n\nThe endomorphism rings of the elliptic curves at the same level of the volcano are isomorphic, hence the 2-Sylow subgroups at the same level are isomorphic.  Elliptic curves on the floor of a volcano have cyclic 2-Sylow subgroups \\cite[\\S3]{sutherland_volcano}, say of order $2^\\nu$.  Then, for each $0 < m \\leq  h_{\\rm stab}$, we have the 2-Sylow subgroup at height $m$ is $\\mathbf{Z}\/2^{m}\\mathbf{Z} \\times \\mathbf{Z}\/2^{\\nu - m}\\mathbf{Z}$.  If $h_{\\rm stab} < h(V_p)$ then the volcano is called \\textbf{irregular} and $h_{\\rm stab}$ is called the \\textbf{stability level} \\cite[p.~742]{miret2}. By \\cite[\\S4]{miret2}, all curves between the stability level and the crater have isomorphic 2-Sylow subgroups. We refer to the levels of the volcano between the stability level and the crater as the \\textbf{stability zone}. \n\n\\begin{lem} \\label{heightlem}\nLet $E$ and $E'$ be 2-isogenous elliptic curves defined over $\\mathbf{F}_p$.  Let $V_p$ be the isogeny volcano which contains $E$ and $E'$ as adjacent vertices; let $V_{p^2}$ be the isogeny volcano over $\\mathbf{F}_{p^2}$.  Suppose $t\\equiv 2\\pmod{4}$.  Then $h(V_{p^2}) = h(V_p)+1$.\n\\end{lem}\n\n\\begin{proof}\nLet $t$ be the trace of $\\pi_E$ and $T$ the trace of $\\pi_E^2$.  By assumption $v_2(t)=1$. We have $T = t^2 - 2p$ since $|E(\\mathbf{F}_{p^2})| = (p+1-t)(p+1+t)$.  Then\n\\begin{align*}\nh(V_{p^2}) &=  \\frac{1}{2} v_2 \\left( \\frac{T^2 - 4p^2}{\\operatorname{disc} \\mathcal{O}_0} \\right) \n= \\frac{1}{2} v_2 \\left( \\frac{(T-2p)(T+2p)}{\\operatorname{disc} \\mathcal{O}_0} \\right)\n= \\frac{1}{2} v_2 \\left( \\frac{t^2 - 4p}{\\operatorname{disc} \\mathcal{O}_0} \\, t^2 \\right) \n= h(V_p) + 1.\n\\end{align*}\n\\end{proof}\n\n\\begin{rmk} The hypothesis that $t \\equiv 2 \\pmod{4}$ means that this lemma will be applicable to the case of anomalous pairs of elliptic curves.\n\\end{rmk}\n\n\\begin{prop}\nLet $E$ and $E'$ be 2-isogenous elliptic curves defined over a finite field $\\mathbf{F}_p$ and suppose $(E,E')$ is an anomalous pair.  Then:\n\\begin{itemize}\n\\item $V_p$ is irregular, and\n\\item $E$ and $E'$ represent adjacent edges on $V_p$ in the stability zone, and\n\\item $E$ and $E'$ do not both lie in the stability zone on $V_{p^2}$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nThis is just a matter of terminology.  Since $(E,E')$ is an anomalous pair, they are vertically isogenous.  By Lemma \\ref{22lem}, we have $E(\\mathbf{F}_p)[2^\\infty] \\simeq E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$, hence neither curve lies on the floor of the volcano $V_p$.  Since the 2-Sylow subgroups are isomorphic, $V_p$ is an irregular volcano and the curves must lie in the stability zone.  However, over $\\mathbf{F}_{p^2}$ the 2-Sylow subgroups are not isomorphic, hence at least one curve lies outside the stability zone.\n\\end{proof}\n\nNote that since $\\operatorname{disc} \\mathcal{O}_0 = \\operatorname{disc} \\mathcal{O}_K [\\mathcal{O}_K \\colon \\mathcal{O}_0]^2$ and $[\\mathcal{O}_K \\colon \\mathcal{O}_0]$ is odd, we have that $\\operatorname{disc} \\mathcal{O}_0 \\equiv \\operatorname{disc} \\mathcal{O}_K \\pmod{8}$.  Turning now to the endomorphism rings, we distinguish between the congruence classes $\\operatorname{disc} \\mathcal{O}_0 \\equiv 0,1,4,5 \\pmod{8}$.  In these cases, the shape of the crater corresponds to the discriminant in the following way, as established by \\cite[Thm.~7]{sutherland_volcano}.  When $\\operatorname{disc} \\mathcal{O}_0 \\equiv 0 \\pmod{4}$ or $\\operatorname{disc} \\mathcal{O}_0 \\equiv 5 \\pmod{8}$ then the volcanoes have shapes\n\\begin{center}\n\\begin{tikzpicture}[scale=1.5,sizefont\/.style={scale = 2}]\n\n\\draw[ultra thick] (1,3) node {${\\color{red}\\bullet}$};\n\\draw[ultra thick] (3,3) node {${\\color{red}\\bullet}$};\n\\draw[thick,red] (1,3) -- (3,3);\n\n\\draw[ultra thick] (0.5,2) node {$\\bullet$};\n\\draw[ultra thick] (1.5,2) node {$\\bullet$};\n\\draw[ultra thick] (2.5,2) node {$\\bullet$};\n\\draw[ultra thick] (3.5,2) node {$\\bullet$};\n\\draw[thick] (1,3) -- (0.5,2);\n\\draw[thick] (1,3) -- (1.5,2);\n\\draw[thick] (3,3) -- (2.5,2);\n\\draw[thick] (3,3) -- (3.5,2);\n\n\\draw[dashed] (.25,1.5) -- (0.5,2);\n\\draw[dashed] (.75,1.5) -- (0.5,2);\n\\draw[dashed] (2.25,1.5) -- (2.5,2);\n\\draw[dashed] (2.75,1.5) -- (2.5,2);\n\\draw[dashed] (1.25,1.5) -- (1.5,2);\n\\draw[dashed] (1.75,1.5) -- (1.5,2);\n\\draw[dashed] (3.25,1.5) -- (3.5,2);\n\\draw[dashed] (3.75,1.5) -- (3.5,2);\n\n\\draw[ultra thick] (6,3) node {${\\color{red}\\bullet}$};\n\\draw[ultra thick] (5,2) node {$\\bullet$};\n\\draw[ultra thick] (6,2) node {$\\bullet$};\n\\draw[ultra thick] (7,2) node {$\\bullet$};\n\n\\draw[dashed] (5.25,1.5) -- (5,2);\n\\draw[dashed] (4.75,1.5) -- (5,2);\n\\draw[dashed] (6.25,1.5) -- (6,2);\n\\draw[dashed] (5.75,1.5) -- (6,2);\n\\draw[dashed] (7.25,1.5) -- (7,2);\n\\draw[dashed] (6.75,1.5) -- (7,2);\n\n\\draw[thick] (6,3) -- (6,2);\n\\draw[thick] (6,3) -- (7,2);\n\\draw[thick] (6,3) -- (5,2);\n\n\\draw[thick] (4.4,2.5) node {or};\n\\end{tikzpicture}\n\\end{center}\nrespectively (with the crater highlighted in red).  If $\\operatorname{disc} \\mathcal{O}_0 \\equiv 1 \\pmod{8}$ then the crater forms a cycle whose length is the order of a certain element in the class group of $\\mathcal{O}_0$, as depicted in the following figure.\n\n\n\\begin{center}\n\\begin{tikzpicture}[scale=1.5,sizefont\/.style={scale = 2}]\n\\draw[ultra thick] (0,1) node {${\\color{red}\\bullet}$};\n\\draw[ultra thick] (-1,0) node {${\\color{red}\\bullet}$};\n\\draw[ultra thick] (-2\/3,-1) node {${\\color{red}\\bullet}$};\n\\draw[ultra thick] (2\/3,-1) node {${\\color{red}\\bullet}$};\n\n\\draw[ultra thick,red] (0,1) -- (-1,0);\n\\draw[ultra thick,red, dotted] (0,1) -- (1,0);\n\\draw[ultra thick,red,dotted] (1,0) -- (2\/3,-1);\n\\draw[ultra thick,red]  (-2\/3,-1) -- (2\/3,-1);\n\\draw[ultra thick,red]  (-2\/3,-1) -- (-1,0);\n\n\n\n\n\n\n\n\n\n\n\\draw[ultra thick] (-1,-4\/3) node {${\\bullet}$};\n\\draw[ultra thick] (-4\/3,-4\/3) node {${\\bullet}$};\n\\draw[ultra thick] (-1,-5\/3) node {${\\bullet}$};\n\n\\draw[ultra thick] (-1,-4\/3) -- (-4\/3,-4\/3);\n\\draw[ultra thick] (-1,-4\/3) -- (-1,-5\/3);\n\\draw[ultra thick] (-1,-4\/3) -- (-2\/3,-1);\n\n\\draw[ultra thick, dotted] (-4\/3,-4\/3) -- (-5\/3,-5\/3);\n\\draw[ultra thick, dotted] (4\/3,-4\/3) -- (5\/3,-5\/3);\n\n\\draw[ultra thick, dotted] (-4\/3,-4\/3) -- (-5\/3,-3\/3);\n\\draw[ultra thick, dotted] (4\/3,-4\/3) -- (5\/3,-3\/3);\n\n\\draw[ultra thick, dotted] (-3\/3,-5\/3) -- (-4\/3,-6\/3);\n\\draw[ultra thick, dotted] (3\/3,-5\/3) -- (4\/3,-6\/3);\n\n\\draw[ultra thick, dotted] (-3\/3,-5\/3) -- (-2\/3,-6\/3);\n\\draw[ultra thick, dotted] (3\/3,-5\/3) -- (2\/3,-6\/3);\n\n\\draw[ultra thick, dotted] (-6\/3,1\/3) -- (-8\/3,1\/3);\n\\draw[ultra thick, dotted] (6\/3,1\/3) -- (8\/3,1\/3);\n\n\\draw[ultra thick, dotted] (-6\/3,-1\/3) -- (-8\/3,-1\/3);\n\\draw[ultra thick, dotted] (6\/3,-1\/3) -- (8\/3,-1\/3);\n\n\\draw[ultra thick, dotted] (-6\/3,1\/3) -- (-6\/3,3\/3);\n\\draw[ultra thick, dotted] (6\/3,1\/3) -- (6\/3,3\/3);\n\n\\draw[ultra thick, dotted] (-6\/3,-1\/3) -- (-6\/3,-3\/3);\n\\draw[ultra thick, dotted] (6\/3,-1\/3) -- (6\/3,-3\/3);\n\n\\draw[ultra thick, dotted] (1\/3,6\/3) -- (1\/3,8\/3);\n\\draw[ultra thick, dotted] (1\/3,6\/3) -- (3\/3,6\/3);\n\n\\draw[ultra thick, dotted] (-1\/3,6\/3) -- (-1\/3,8\/3);\n\\draw[ultra thick, dotted] (-1\/3,6\/3) -- (-3\/3,6\/3);\n\n\n\n\n\\draw[ultra thick] (1,-4\/3) node {${\\bullet}$};\n\\draw[ultra thick] (4\/3,-4\/3) node {${\\bullet}$};\n\\draw[ultra thick] (1,-5\/3) node {${\\bullet}$};\n\n\\draw[ultra thick] (1,-4\/3) -- (4\/3,-4\/3);\n\\draw[ultra thick] (1,-4\/3) -- (1,-5\/3);\n\\draw[ultra thick] (1,-4\/3) -- (2\/3,-1);\n\n\\draw[ultra thick] (-5\/3,0) node {${\\bullet}$};\n\\draw[ultra thick] (-6\/3,1\/3) node {${\\bullet}$};\n\\draw[ultra thick] (-6\/3,-1\/3) node {${\\bullet}$};\n\n\\draw[ultra thick] (-1,0) -- (-5\/3,0);\n\\draw[ultra thick] (-5\/3,0) -- (-6\/3,1\/3);\n\\draw[ultra thick] (-5\/3,0) -- (-6\/3,-1\/3);\n\n\n\\draw[ultra thick,dotted] (1,0) -- (5\/3,0);\n\\draw[ultra thick,dotted] (5\/3,0) -- (6\/3,1\/3);\n\\draw[ultra thick,dotted] (5\/3,0) -- (6\/3,-1\/3);\n\n\\draw[ultra thick] (0,5\/3) node {${\\bullet}$};\n\\draw[ultra thick] (1\/3,6\/3) node {${\\bullet}$};\n\\draw[ultra thick] (-1\/3,6\/3) node {${\\bullet}$};\n\n\\draw[ultra thick] (0,1) -- (0,5\/3);\n\\draw[ultra thick] (0,5\/3) -- (1\/3,6\/3);\n\\draw[ultra thick] (0,5\/3) -- (-1\/3,6\/3);\n\\end{tikzpicture}\n\\end{center}\n\n\\begin{rmk} \\label{descending_remark}\nObserve that when $\\operatorname{disc} \\mathcal{O}_0 \\equiv 5 \\pmod{8}$ and $E$ is on the crater, then all 2-isogenies from $E$ are descending. \n\\end{rmk}\n\nWe now discuss some aspects of the volcano $V_q$ in terms of a matrix representation of Frobenius.  We will use this material in the proof of Theorem \\ref{nathan_thm}.  We continue with the notation from earlier in this section.  If $p$ is a prime number, then the Frobenius endomorphism at $p$ has a representative conjugacy class in $\\operatorname{GL}_2(\\mathbf{Z}_2)$ via the 2-adic representation.  Let $F \\in \\operatorname{GL}_2(\\mathbf{Z}_2)$ be a matrix in this conjugacy class. For any positive integer $k$, we have $\\det(F) \\equiv q \\pmod{2^k}$.  We note that a unit in $\\mathbf{Z}_2$ is a square in $\\mathbf{Z}_2$ if and only if it is $1$ modulo $8$.  Therefore, it still makes sense to take $\\mathsf{sqf}(\\alpha) \\pmod{8}$ for an $\\alpha\\in \\mathbf{Z}_2$.\n\nSuppose $F = -I + 2^m M$ where $m \\ge 2$ and $M =\\left( \\begin{smallmatrix} x & y \\\\ z & w\\end{smallmatrix} \\right)\\in \\operatorname{M}_2(\\mathbf{Z}_2)$. This implies \n\\[\nq \\equiv (-1+2^m x)(-1+2^m w)- 2^{2m} yz \\equiv 1 -2^m(x+w) - 2^{2m}(yz-xw) \\pmod{2^k},\n\\]\nand\n\\begin{eqnarray*}\nt^2 - 4q & \\equiv & (-2+2^m (x+ w))^2 - 4 \\left((-1+2^m x)(-1+2^m w)- 2^{2m} (yz-xw)\\right) \\pmod{2^k}\\\\\n& \\equiv & 2^{2m} \\left((x-w)^2+4yz \\right) \\pmod{2^k}.\n\\end{eqnarray*}\nMoreover, \n\\[\n\\mathsf{sqf}(t^2-4q) \\equiv \\mathsf{sqf}\\left((x-w)^2 + 4yz\\right) \\pmod{8}.\n\\]\nTherefore, we have the following:\n\\begin{enumerate}\n\\item $v_2(\\operatorname{disc} \\mathcal{O}_0)$ is determined by $\\mathsf{sqf}\\left((x-w)^2 + 4yz\\right) \\pmod{8}$, and \n\\item $h(V_q)$ is determined by \n\\begin{itemize}\n\\item $v_2\\left((x-w)^2+4yz\\right)$, and \n\\item $\\mathsf{sqf}\\left((x-w)^2 + 4yz\\right) \\pmod{8}$.\n\\end{itemize}\n\\end{enumerate}\n\n\\section{Elliptic Curves over $\\mathbf{Q}$} \\label{Q}\n\nWe now turn to the proof of Theorem \\ref{mainthm2}.  Let $E, E'$ be rationally 2-isogenous elliptic curves defined over $\\mathbf{Q}$.  Because the 2-isogeny is defined over $\\mathbf{Q}$, each curve has at least a rational 2-torsion point.  The exact proportion of anomalous primes is determined by the images of the 2-adic representations of $E$ and $E'$, as we will see below.  For the remainder of this section we will assume that both $G \\colonequals \\operatorname{im} \\rho_{E,2}$ and $G' \\colonequals \\operatorname{im} \\rho_{E',2}$ have index 3 in $\\operatorname{GL}_2(\\mathbf{Z}_2)$.  Up to isomorphism, $\\operatorname{GL}_2(\\mathbf{Z}_2)$ has a unique subgroup of index 3. \n\n\\subsection{Frobenius at Anomalous Primes} \\label{rep_setup} \nIn this  section we will describe the conjugacy class in $\\operatorname{GL}_2(\\mathbf{Z}_2)$ associated to Frobenius at an anomalous prime $p$.\n\nIf $p$ is anomalous then both $E$ and $E'$ have $E(\\mathbf{F}_p)[2^\\infty] \\simeq E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2 \\mathbf{Z}$ by Lemma \\ref{22lem}.  Write $F$ and $F'$ for matrix representatives of the Frobenius classes of $E$ and $E'$, respectively, as elements of $\\operatorname{GL}_2(\\mathbf{Z}_2)$.  It follows that \n\\[\nF \\equiv F' \\equiv I \\pmod{2}\n\\]\nand that neither $F \\pmod{4}$ nor $F' \\pmod{4}$ fixes a cyclic subgroup of $\\mathbf{Z}\/4\\mathbf{Z} \\times \\mathbf{Z}\/4\\mathbf{Z}$ of order $4$. \n\nSince anomalous primes can be partitioned by defect as in Theorem \\ref{establish_defect}, let us fix $m \\geq 2$ and suppose that $p$ has defect $(m+1,m)$.  In particular, we assume that the isogeny $E \\to E'$ is descending. Then we have\n\\[\nE(\\mathbf{F}_{p^2})[2^{\\infty}] \\simeq \\mathbf{Z}\/2^{a}\\mathbf{Z} \\times \\mathbf{Z}\/2^{m+1}\\mathbf{Z} \\text{ \\qquad and \\qquad  } E'(\\mathbf{F}_{p^2})[2^{\\infty}] \\simeq \\mathbf{Z}\/2^{a+1}\\mathbf{Z} \\times \\mathbf{Z}\/2^{m}\\mathbf{Z},\n\\]\nwhere $a \\geq m+1$.  Therefore\n\\begin{align*}\nF^2 &\\equiv I \\pmod{2^{m+1}} \\text{ but } F^2\\not\\equiv I \\pmod{2^{m+2}},\\text{ and} \\\\\n(F')^2 &\\equiv I \\pmod{2^{m}} \\text{ but } (F')^2\\not\\equiv I \\pmod{2^{m+1}}. \n\\end{align*}\n\nWe are thus led to the problem of determining, for fixed $m \\geq 2$, matrices $A \\in \\operatorname{GL}_2(\\mathbf{Z}_2)$ such that the following are simultaneously satisfied \n\\begin{itemize}\n\\item $A \\equiv I \\pmod{2}$, and\n\\item $A \\pmod{4}$ does not fix any cyclic subgroup of $\\mathbf{Z}\/4\\mathbf{Z} \\times \\mathbf{Z}\/4\\mathbf{Z}$ of order 4, and\n\\item $A^2 \\equiv I \\pmod{2^{m+1}}$ \\text{ but } $A^2 \\not\\equiv I \\pmod{2^{m+2}}$.\n\\end{itemize}\nIt is now an exercise in squaring matrices (which we omit) to conclude that there exist matrices $M,M' \\in \\operatorname{M}_2(\\mathbf{Z}_2)$ such that neither $M$ nor $M'$ is $\\equiv 0 \\pmod{2}$ and that $F$ and $F'$ are, up to conjugation, given by\n\\begin{align*}\nF &= -I + 2^mM \\\\\nF' &= -I + 2^{m-1}M'.\n\\end{align*}\n\nWe finish this subsection by collecting some known results on the Galois theory of torsion point fields and their consequences for anomalous primes.  The important point is that if $k$ is a number field and $E\/k$ is an elliptic curve for which $k(E[\\ell^n])\/k$ has Galois group $\\operatorname{GL}_2(\\mathbf{Z}\/\\ell^n\\mathbf{Z})$, then the normal subgroup $\\lbrace \\pm I \\rbrace$ of $\\operatorname{GL}_2(\\mathbf{Z}\/\\ell^n\\mathbf{Z})$ is the Galois group of $k(E[\\ell^n])\/k(x(E[\\ell^n]))$, with clear implications for the Frobenius at anomalous primes.\n\n\\begin{prop} \\label{adelman_prop}\nLet $k$ be a number field and $E\/k$ an elliptic curve.  Let $\\ell$ be a prime number and $n \\geq 1$ an integer.  Let $k(E[\\ell^n])$ be the $\\ell^n$-torsion field of $E$ and $k(x(E[\\ell^n]))$ the subfield generated by the $x$-coordinates of the points of $E[\\ell^n]$.  Let $G(\\ell^n) = \\operatorname{im} \\overline{\\rho}_{{E,\\ell^n}} \\subseteq \\operatorname{GL}_2(\\mathbf{Z}\/\\ell^n\\mathbf{Z})$ be the image of the mod $\\ell^n$ representation.  Then  $[k(E[\\ell^n]):k(x(E[\\ell^n]))] \\leq 2$ with $\\operatorname{Gal}(k(E[\\ell^n])\/k(x(E[\\ell^n]))) \\simeq G(\\ell^n) \\cap \\lbrace \\pm I \\rbrace$.\n\\end{prop} \n\n\\begin{proof}\nThis is contained in \\cite[Ch.~5]{adelman}; see especially Figs.~5.4, 5.5, 5.7.\n\\end{proof}\n\n\\begin{lem} \\label{adelman_lem}\nLet $E$ be an elliptic curve over $\\mathbf{Q}$ and suppose $p\\ne 2$ is a good prime for $E$.  Let $K_{2^n} = \\mathbf{Q}(E[2^n])$ with Galois group $\\operatorname{Gal}(K_{2^n}\/\\mathbf{Q}) \\simeq G(2^n) \\subseteq \\operatorname{GL}_2(\\mathbf{Z}\/2^n\\mathbf{Z})$.  Suppose $\\operatorname{Frob}_p \\in \\operatorname{Gal}(K_{2^n}\/\\mathbf{Q})$ is a lift of the Frobenius automorphism at $p$ (so that the decomposition group of $K_{2^n}$ is generated by $\\operatorname{Frob}_p$) and suppose that $\\overline{\\rho}_{E,2^n}(\\operatorname{Frob}_p) = F = - I \\in G(2^n)$. Then $\\mathbf{F}_p(x(E[2^n])) = \\mathbf{F}_p$ and $\\mathbf{F}_p$ contains no $y$-coordinate of any $2^n$-torsion point of $E$.\n\\end{lem}\n\n\\begin{proof}\nThis is a matter of translating the arithmetic of elliptic curves into the Galois theory of torsion point fields and the  behavior of Frobenius at unramified primes. In particular, it is the ``reduction modulo $p$'' of Proposition \\ref{adelman_prop}.  \n\nSince $p$ is an odd prime of good reduction for $E$ it is unramified in $K_{2^n}$, hence we can appeal to the explicit polynomial descriptions in \\cite[Table 5.1]{adelman}.  Let $K_{2^n}$ be the splitting field of the polynomial $T_{2^n}(x)$ and $\\mathbf{Q}(x(E[2^n]))$ the splitting field of $\\Lambda_{2^n}(x)$. \n\nIn general, the field extension $K_{2^n}\/\\mathbf{Q}(x(E[2^n]))$ has degree 1 or 2, depending on whether $\\mathbf{Q}(x(E[2^n]))$ contains any $y$-coordinates of any $2^n$-torsion points (note that if $G(2^n) = \\operatorname{GL}_2(\\mathbf{Z}\/2^n\\mathbf{Z})$ then the extension has degree 2).  We have $\\operatorname{Gal}(K_{2^n}\/\\mathbf{Q}(x(E[2^n]))) \\simeq \\lbrace \\pm I \\rbrace \\cap G(2^n)$ by Proposition \\ref{adelman_prop}, $\\Lambda_{2^n}(x)$ splits completely in $\\mathbf{Q}(x(E[2^n]))$, and that $K_{2^n}$ is generated over $\\mathbf{Q}(x(E[2^n]))$ by a single $y$-coordinate of a single $2^n$-torsion point (see \\cite[p.~74]{adelman}).  \n\nThe Galois theory of number fields then says that either $T_{2^n}(x)$ splits completely over $\\mathbf{Q}(x(E[2^n]))$ or factors as a product of irreducible quadratic polynomials, each of them Galois-conjugate.  In either case, the Galois group $\\operatorname{Gal}(K_{2^n}\/\\mathbf{Q}(x(E[2^n])))$ is the decomposition group at $p$, which is isomorphic to $\\langle \\operatorname{Frob}_p \\rangle$.  The hypothesis that $F \\equiv -I \\pmod{2^n}$ means that the polynomial $\\Lambda_{2^n}(x)$ splits completely modulo $p$, hence $\\mathbf{F}_p(x(E[2^n])) = \\mathbf{F}_p$.  The fact that Frobenius is non-trivial implies that $\\mathbf{F}_{p}(E[2^n])$ is a quadratic extension of $\\mathbf{F}_p$, hence contains no $y$-coordinate of any $2^n$-torsion point of $E$.\n\\end{proof}\n\nNext, we recall a basic fact about towers of torsion fields.  \n\n\\begin{thm} \\label{lattes_tower}\nLet $k$ be a field, $\\ell$ a prime number, and $E\/k$ an elliptic curve. Then we have the following inclusions of fields for all $n \\geq 1$:\n\\[\n\\xymatrix{\n& \\\\\n  & k(E[\\ell^n]) \\ar@{-}[d] \\ar@{--}[u] \\\\\n\\ar@{-}[ur] k(x(E[\\ell^n])) \\ar@{-}[d] \\ar@{--}[u]& k(E[\\ell^{n-1}]) \\ar@{-}[d] \\\\\n\\ar@{-}[ur] k(x(E[\\ell^{n-1}])) \\ar@{--}[d] & k(E[\\ell^{n-2}]) \\ar@{--}[d] \\\\\n\\ar@{-}[ur] & }\n\\]\n\\end{thm}\n\n\n\\begin{cor} \nWith all notation as above, suppose $E\/\\mathbf{F}_p$ is an elliptic curve such that $\\mathbf{F}_p(x(E[2^n])) = \\mathbf{F}_p$.  Then $\\mathbf{F}_p(x(E[2^k])) = \\mathbf{F}_p$ for all $k \\leq n$. \n\\end{cor}\n\n\\begin{proof}\nThis follows immediately.\n\\end{proof}\n\n\\begin{rmk}\nOne can see this from a representation theory point of view too: if $F \\equiv -I \\pmod{2^n}$, then $F \\equiv -I \\pmod{2^k}$ for all $k\\leq n$ as well.\n\\end{rmk}\n\nThe next proposition shows that over a finite field $\\mathbf{F}_p$, if $E \\to E'$ is descending and $F \\equiv -I \\pmod{2^m}$ then we automatically get that $F' \\equiv -I \\pmod{2^{m-1}}$.  This does not immediately imply that that $p$ is anomalous because it could further be the case that $F' \\equiv -I \\pmod{2^{m}}$.  This will be used in the proof of Theorem \\ref{m_proportion} below where we argue that $F' \\equiv -I \\pmod{2^m}$ for half of the primes for which $F \\equiv -I \\pmod{2^m}$  and $F' \\equiv -I \\pmod{2^{m-1}}$ for the other half.\n\n\n\n\n\\begin{prop} \\label{F'form}\nLet $E$ and $E'$ be ordinary 2-isogenous elliptic curves defined over $\\mathbf{F}_p$ and suppose that the isogeny $E \\to E'$ is descending.  Suppose $E(\\mathbf{F}_p)[2^\\infty] \\simeq E'(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z}  \\times \\mathbf{Z}\/2\\mathbf{Z}$ and that $F \\equiv -I \\pmod{2^m}$.  Then $F' \\equiv -I \\pmod{2^{m-1}}$.\n\\end{prop}\n\n\\begin{proof}\nSince $F \\equiv -I \\pmod{2^m}$, we have $E(\\mathbf{F}_p)[2^\\infty] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$ and $E(\\mathbf{F}_{p^2})[2^{m+1}] \\simeq \\mathbf{Z}\/2^{m+1}\\mathbf{Z} \\times \\mathbf{Z}\/2^{m+1}\\mathbf{Z}$. Since $E$ and $E'$ are isogenous, the groups $E(\\mathbf{F}_{p^2})$ and $E'(\\mathbf{F}_{p^2})$ have the same size, hence their 2-Sylow subgroups have the same size.  \n\nIf the 2-Sylow subgroups over $\\mathbf{F}_{p^2}$ are isomorphic, then $(F')^2 \\equiv I \\pmod{2^{m+1}}$. It is also the case that $F' \\equiv I \\pmod{2}$ and $F$ does not fix a cyclic subgroup of $\\mathbf{Z}\/4\\mathbf{Z} \\times \\mathbf{Z}\/4\\mathbf{Z}$ of order 4.  A calculation with matrices shows that $F' = -I \\in \\operatorname{GL}_2(\\mathbf{Z}\/2^m\\mathbf{Z})$ is the unique matrix satisfying these conditions simultaneously.  Thus, $F' \\equiv -I \\pmod{2^m}$. Hence it is also true that $F' \\equiv -I \\pmod{2^{m-1}}$.  \n\nIf the 2-Sylow subgroups of $\\mathbf{F}_{p^2}$ are not isomorphic, then because the isogeny is descending we have $E(\\mathbf{F}_{p^2})[2^{m}] \\simeq \\mathbf{Z}\/2^{m}\\mathbf{Z} \\times \\mathbf{Z}\/2^{m}\\mathbf{Z}$. Hence $F'$ is a matrix such that $F' \\equiv I \\pmod{2}$, does not stabilize a cyclic subgroup of $\\mathbf{Z}\/4 \\mathbf{Z} \\times \\mathbf{Z}\/4\\mathbf{Z}$ of order 4, and satisfies $(F')^2 \\equiv I \\pmod{2^m}$.  Therefore $F' \\equiv -I \\pmod{2^{m-1}}$ by the same reasoning.\n\\end{proof}\n\n\\begin{rmk}\nThis proposition tells us that if $\\mathbf{F}_p(x(E[2^n])) = \\mathbf{F}_p$ then $\\mathbf{F}_p(x(E'[2^{n-1}])) = \\mathbf{F}_p$.\n\\end{rmk}\n\nTo finish off this section we will record a technical lemma that we will need in the proof of Theorem \\ref{m_proportion} below.\n\n\\begin{lem} \\label{splitting_lem}\nLet $E$ be an elliptic curve over a field $k$ of characteristic $p>3$ and write \n\\[\nE:~y^2 = x^3 + ax+b.\n\\]  \nSuppose that $k$ contains $x(E[2^n])$.  Let $P = (\\xi,\\eta)$ be a point of order $2^{n+1}$ and let $\\langle P \\rangle$ denote the cyclic subgroup of $E[2^{n+1}]$ generated by $P$.  Then the set of $x$-coordinates of the points in $\\langle P \\rangle$ are contained in $k$ if and only if the the polynomial\n\\[\nx^4 - 4\\xi x^3 - 2ax^2 + (-4\\xi a - 8b)x + (a^2 - 4\\xi b)\n\\]\nsplits in $k$.\n\\end{lem}\n\n\\begin{proof}\nThe difference between any two points in $\\langle P \\rangle$ is a point of order dividing $2^{n}$.  By Theorem \\ref{lattes_tower}, since $k$ contains $x(E[2^n])$, it contains the $x$-coordinates of all points of order dividing $2^n$.  Thus, one point of $\\langle P \\rangle$ of exact order $2^{n+1}$ will have rational $x$-coordinate if and only if they all do.  Therefore, all the points of $\\langle P \\rangle$ will have rational $x$-coordinates if and only if the points of exact order $2^{n+1}$ do.  Such a point $P$ is the preimage under the duplication map of a point of order $2^n$ in $\\langle P \\rangle$, hence by \\cite[III.2.3(d)]{silverman} the $x$-coordinate of $P$ is $k$-rational if and only if the quartic polynomial (whose roots are the $x$-coordinates of these points of order $2^{m+1}$)\n\\[\nx^4 - 4\\xi x^3 - 2ax^2 + (-4\\xi a - 8b)x + (a^2 - 4\\xi b)\n\\]\nhas all its roots defined over $k$.  \n\\end{proof}\n\n\\subsection{The Proportion of Anomalous Primes} \n\nFix $m \\ge 2$.  The key step in proving Theorem \\ref{mainthm2} is the following.\n\n\\begin{thm} \\label{m_proportion}\nSuppose $E$ and $E'$ are rationally 2-isogenous elliptic curves defined over $\\mathbf{Q}$ such that $G$ and $G'$ each have index 3 in $\\operatorname{GL}_2(\\mathbf{Z}_2)$.  Let $m \\geq 2$.  Then the proportion of anomalous primes of defect $(m+1,m)$ is \n\\[\n\\frac{1}{2} \\cdot \\frac{1}{|G(2^m)|} = \\frac{1}{2^{4m+2}}.\n\\]\n\\end{thm}\n\n\\noindent We break this proof into two steps, starting with a Lemma.\n\n\\begin{lem} \\label{lemA}\nSuppose $p$ is a prime for which $F \\equiv  -I + 2^mM \\pmod{2^{m+1}}$ with $M \\not \\equiv 0 \\pmod{2}$.  Then $p$ is anomalous of defect $(m+1,m)$ if and only if $\\mathbf{F}_p(x(E'[2^m])) \\ne  \\mathbf{F}_p$.\n\\end{lem}\n\n\\begin{proof}\nThis follows from  the results of the previous section. We have that $p$ is anomalous of defect $(m+1,m)$ if and only if $E(\\mathbf{F}_p)[2^\\infty] \\simeq E'(\\mathbf{F}_p)[2^{\\infty}] \\simeq \\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$, $E(\\mathbf{F}_{p^2})[2^\\infty] \\simeq \\mathbf{Z}\/2^a\\mathbf{Z} \\times \\mathbf{Z}\/2^{m+1}\\mathbf{Z}$, and $E'(\\mathbf{F}_{p^2}) \\simeq \\mathbf{Z}\/2^{a+1}\\mathbf{Z} \\times \\mathbf{Z}\/2^{m}\\mathbf{Z}$.  By our matrix calculations, this is true if and only if $F \\equiv -I + 2^mM \\pmod{2^{m+1}}$ and $F' \\equiv -I + 2^{m-1}M'\\pmod{2^{m}}$ with neither $M$ nor $M' \\equiv 0 \\pmod{2}$.  By Proposition \\ref{F'form}, $F \\equiv -I + 2^mM\\pmod{2^{m+1}}$ implies $F' \\equiv -I + 2^{m-1}M'\\pmod{2^{m}}$.  By the Galois theory of torsion point fields from Lemma \\ref{adelman_lem}, $M' \\not \\equiv 0 \\pmod{2}$ if and only if $\\mathbf{F}_p(x(E'[2^m])) \\ne  \\mathbf{F}_p$.\n\\end{proof}\n\nWe now make some global choices for $E$ that we use in the next proof.  Fix a basis $P,Q$ for $T_2E$ and write $P_{2^k}$, $Q_{2^k}$ for the reductions modulo $2^k$ of $P$ and $Q$, respectively.  Since $E$ and $E'$ are rationally 2-isogenous, there exists a 2-isogeny $\\varphi: E \\to E'$ defined over $\\mathbf{Q}$.  Since $\\varphi$ is defined over $\\mathbf{Q}$, we may write $E' = E\/\\langle S \\rangle$ for some $\\mathbf{Q}$-rational 2-torsion point $S$ of $E$.  We choose our basis so that $S = P_2$. \n\nLet $P' = \\varphi(P)$ and $Q' = \\varphi(Q)$ with $P'_{2^k}$ and $Q'_{2^k}$ defined similarly.  We have that $Q'_{2^k}  = Q_{2^k} + \\langle P_2 \\rangle$ is a $2^k$-torsion point of $E'$, but $P'_{2^k}$ is not necessarily independent of $Q'_{2^k}$. We fix a basis $Q',R'$ for $T_2E'$ so that for all $m,\\ Q'_{2^m}$ and $R'_{2^m}$ form a basis for $E'[2^m]$. \n\nBy applying V\\'elu's explicit formulas, we see that there exists a change of coordinates such that $E$ and $E'$ are given by the explicit Weierstrass equations\n\\begin{align*}\nE:~y^2 &= (x + a_2)(x^2 - 4a_4) \\\\\nE':~y^2 &= x(x^2+a_2x + a_4),\n\\end{align*}\nwhere $P_2 = (-a_2,0)$ and $P_2' = (0,0)$.  Write $R'_{2^{m-1}} = (\\xi_{m-1},\\eta_{m-1})$ with $\\xi_{m-1},\\eta_{m-1} \\in \\overline{\\mathbf{Q}}$.  Then the $x$-coordinate $\\xi_m$ of $R'_{2^m}$ is given by one of the roots the quartic \n\\begin{align} \\label{quartic}\nx^4 - 4\\xi_{m-1} x^3 + (-4\\xi_{m-1} a_2 + 6a_4)x^2 + (4a_4a_2 - 8a_4)x + (-4a_4a_2 + (a_4^2 - 4\\xi_{m-1} a_4)).\n\\end{align}\n\nNext we show the existence of anomalous primes of defect $(m+1,m)$ for all $m \\geq 2$.\n\n\\begin{thm} \\label{nathan_thm}\nLet $E$ and $E'$ be rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ and suppose that $G$ and $G'$ each have index 3 in $\\operatorname{GL}_{2}(\\mathbf{Z}_2)$.  Then for all $m \\geq 2$ there exist anomalous primes of defect $(m+1,m)$. \n\\end{thm}\n\n\\begin{proof}\nFix $m \\geq 2$. By the assumption on the size of $G$ and $G'$ and the Chebatorev Density Theorem, there exist infinitely many primes p for which $F \\pmod{2^{m+1}}$  is in the conjugacy class of \n\\[\n-I + 2^m \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}.\n\\]\nLet $V_p$ be the isogeny volcano which contains $E$ and $E'$ as adjacent vertices and let $V_{p^2}$ be the corresponding volcano over $\\mathbf{F}_{p^2}$.  By our work in Section \\ref{volcanoes} we know that $E$ is on level at least $m$ of $V_p$.\n\n\\medskip\n\n\\noindent \\textbf{Claim.} \nThe height of $V_p$ is $m$ and $\\operatorname{disc} \\mathcal{O}_0 \\equiv 5 \\pmod{8}$.\n\n\\medskip\n\nWe now prove the claim.\nWrite $F = -I + 2^m \\left(\\begin{smallmatrix} x&y \\\\ z & w \\end{smallmatrix} \\right) \\in \\operatorname{GL}_2(\\mathbf{Z}_2)$.  \nAs in the end of Section~\\ref{volcanoes}, \n\\begin{align*}\nt^2 - 4p &\\equiv  2^{2m}((x-w)^2+4yz) \\pmod{2^{2m+1}}.\n\\end{align*}\nTherefore $v_2(t^2-4p) = 2m + v_2((x-w)^2+4yz)$.\n\nWe have $\\mathsf{sqf}(t^2-4p) \\equiv \\mathsf{sqf}( (x-w)^2+4yz)) \\pmod{8}$.  If $x-w$ is odd, then $v_2((x-w)^2+4yz) = 0$ and $v_2(t^2-4p) = 2m$.  In this case it is also true that $\\mathsf{sqf}( (x-w)^2+4yz)) \\equiv 5 \\pmod{8}$ if and only if $yz$ is odd.  If this holds, $\\operatorname{disc} \\mathcal{O}_0 \\equiv 5 \\pmod{8}$.  Consequently, since $\\operatorname{disc} \\mathcal{O}_0 \\equiv 1 \\pmod{4}$, equation \\eqref{eqn:volcano_height} shows that the height of $V_p$ is $v_2(t^2-4p)\/2 = m$.\n\nNow set $\\left(\\begin{smallmatrix} x&y \\\\z & w \\end{smallmatrix} \\right) = \\left(\\begin{smallmatrix} 1&1 \\\\ 1 & 0 \\end{smallmatrix} \\right)$ as above.  We saw in Section \\ref{volcanoes} that for a volcano $V_p$ in which $\\operatorname{disc} \\mathcal{O}_0 \\equiv 5 \\pmod{8}$, there is a unique vertex on the crater.  Since $E$ lies on level at least $m$ of $V_p$ and $h(V_p)= m$, we see that $E$ is the unique vertex on the crater of $V_p$.  \n\n\\medskip\n\nReturning to the proof of the theorem, we conclude from the Claim that $V_{p^2}$ has height $m+1$ and the group structure on the crater is $\\mathbf{Z}\/2^a\\mathbf{Z} \\times \\mathbf{Z}\/2^{m+1}\\mathbf{Z}$.  Since a vertex on the floor of $V_{p^2}$ has a cyclic group of rational points, it must be the case that the curves on each level of $V_{p^2}$ have different group structures.\nSo in particular, $E'(\\mathbf{F}_{p^2})[2^\\infty] = \\mathbf{Z}\/2^{a-1}\\mathbf{Z} \\times \\mathbf{Z}\/2^m\\mathbf{Z}$.\nThis means that $p$ must have defect $(m+1,m)$.\n\\end{proof}\n\n\nWe now finish the proof of Theorem \\ref{m_proportion}.\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{m_proportion}]\nBy Lemma \\ref{lemA} a prime $p$ is anomalous of defect $(m+1,m)$ if and only if $F = -I + 2^mM$ and $F' = -I + 2^{m-1}M'$, with neither $M$ nor $M' \\equiv 0 \\pmod{2}$.   We will interpret our proportion $1\/{2^{4m+2}}$ as a conditional probability.  Suppose $p$ is a prime such that $F = -I + 2^mM$.  By the Chebotarev Density Theorem, the proportion of such primes is $1\/|G(2^m)| = 1\/2^{4m+1}$.  By Proposition \\ref{F'form}, we have that $F' \\equiv -I \\pmod{2^{m-1}}$ at these primes as well.  We will show that for a proportion of 1\/2 of these primes we have $F' \\not\\equiv -I \\pmod{2^m}$ and so $p$ is anomalous with defect $(m+1,m)$.  Now we compute in the basis we have set above:\n\\begin{align*}\nF(P_{2^m}) &= -P_{2^m} \\\\\nF(Q_{2^m}) &= -Q_{2^m}\n\\end{align*}\nso that\n\\[\nF'(Q'_{2^m}) = F'(Q_{2^m} + \\langle P_2 \\rangle) = F(Q_{2^m}) + F(\\langle P_2 \\rangle) =  -Q_{2^m} + \\langle P_2 \\rangle = -Q'_{2^m} \n\\]\nbecause $P_2$ is defined over $\\mathbf{Q}$.   \n\nTherefore, we have determined that $F'$ acts on $E'[2^m]$ via \n\\[\nF' \\equiv \\begin{pmatrix}\n-1 & * \\\\ 0 & *\n\\end{pmatrix} \\pmod{2^m}.\n\\]\nBut since $p \\equiv \\det(F) \\pmod{2^m}$ and $\\det(F) \\equiv 1 \\pmod{2^m}$, we must additionally have $F' \\equiv \\left(\\begin{smallmatrix} -1 & * \\\\ 0 & -1 \\end{smallmatrix} \\right) \\pmod{2^m}$.  Therefore, Proposition \\ref{adelman_prop} shows that in this setting $p$ is not anomalous of defect $(m+1,m)$ if and only all the $x$-coordinates of the $2^m$-torsion points of $E'$ are defined over $\\mathbf{F}_p$.  To determine when this happens, we examine the quartic (\\ref{quartic}).\n\nBy Propositions \\ref{adelman_prop} and \\ref{F'form}, we know that the the $x$-coordinates of the $2^{m-1}$-torsion points on $E'$ are $\\mathbf{F}_p$-rational.  Any two choices of $R'_{2^m}$ differ by a $2^{m-1}$-torsion point. Therefore, the $x$-coordinate of any choice of $R'_{2^m}$ is defined over $\\mathbf{F}_p$ if and only if  the $x$-coordinate of one choice of of $R'_{2^m}$ is defined over $\\mathbf{F}_p$.\n\nWith notation as above, consider the quartic polynomial given in \\eqref{quartic}.  The roots of this polynomial give the $x$-coordinates of the $2^m$-torsion points of all preimages of $R_{2^{m-1}}$.  \n\n\nSince the $x$-coordinates of all of the $2^m$-torsion points of $E$ are defined over $\\mathbf{F}_{p^2}$, the quartic polynomial in \\eqref{quartic} must factor over $\\mathbf{F}_p$ as a product of irreducible polynomials each of degree at most $2$.  In particular, it is reducible over $\\mathbf{F}_p$.\n\nWe have shown that if the quartic (\\ref{quartic}) has one root defined over $\\mathbf{F}_p$, then it splits completely into linear factors over $\\mathbf{F}_p$.  Therefore, since (\\ref{quartic}) is reducible over $\\mathbf{F}_p$, it factors as a product of two conjugate quadratic polynomials over $\\mathbf{F}_p$.  If it were the case that these polynomials split into linear factors over $\\mathbf{F}_p$ for every $p$, there would not exist any primes of defect $(m+1,m)$, contradicting Theorem \\ref{nathan_thm}. Thus they must be irreducible for 1\/2 of the primes considered in this proof and and split for the complementary primes, and so the proportion of primes of defect $(m+1,m)$ is $(1\/2) \\cdot (1\/2^{4m+1}) = 1\/2^{4m+2}$, as claimed.\n\\end{proof}\n\nWe now complete the proof of Theorem \\ref{mainthm2} as a corollary.\n\n\\begin{cor}\nWith all notation as above, we have $\\mathcal{P} = 1\/30$.\n\\end{cor}\n\n\\begin{proof}\nFor all $m \\geq 2$, Theorem \\ref{m_proportion} shows that the proportion of anomalous primes of defect $(m+1,m)$ is $2^{-4m-2}$.  By symmetry via the dual isogeny, the proportion of anomalous primes of defect $(m,m+1)$ is $2^{-4m-2}$ as well.  Therefore, the proportion of anomalous primes $\\mathcal{P}$ is given by the geometric series\n\\[\n\\mathcal{P} = 2\\sum_{m =2}^{\\infty} \\frac{1}{2^{4m + 2}} = \\frac{1}{32} \\sum_{k=0}^{\\infty} \\frac{1}{16^k} = \\frac{1}{30}.\n\\]\n\\end{proof}\n\n\n\n\\section{The Distribution of Anomalous Primes by Volcano Height}\n\nIn this section we take a different point of view and explore how the defect of an anomalous prime corresponds to the height and shape of the associated volcano.  These results are motivated by experiments with the pair $(E,E'$) of rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ where where $E$ has \\texttt{LMFDB} label \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/69\/a\/2}{{\\tt 69a2}}  and $E'$ has label \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/69\/a\/1}{{\\tt 69a1}}.  We computed the anomalous primes $p$ up to $2\\cdot 10^7$ and divided them up by defect, the height of the associated volcano $h(V_p)$, and $\\operatorname{disc} \\mathcal{O}_0 \\pmod{8}$, which determines the shape of the crater of $V_p$.  We include the data for anomalous primes of defect $(3,2)$ and for anomalous primes of defect $(4,3)$ in Appendix \\ref{calculations}.  \n\nLet $S_m$ be the set of anomalous primes of defect $(m+1,m)$.  For $i \\in \\{0,1,4,5\\}$ and a positive integer $H \\ge m$, let $S_m(i,H)$ be the subset of $p\\in S_m$ for which $\\operatorname{disc} \\mathcal{O}_0 \\equiv i \\pmod{8}$ and $h(V_p) = H$.  Let $S'_m(i,H)$ denote the proportion of primes in $S_m$ that lie in $S_m(i,H)$.  The data we have collected strongly suggest the following results.  \n\\begin{conj}\\label{volcano_conj}\nLet $E$ and $E'$ be rationally 2-isogenous elliptic curves over $\\mathbf{Q}$ such that $[\\operatorname{GL}_2(\\mathbf{Z}_2): \\operatorname{im} \\rho_{E,2}] = [\\operatorname{GL}_2(\\mathbf{Z}_2):\\operatorname{im} \\rho_{E',2}] = 3$.  For any $H \\ge m$, we have\n\\[\nS_m'(1,H) = S_m'(5,H) = 4^{-(H-(m-1))}\n\\]\nand \n\\[\nS_m'(0,H) = S_m'(4,H) =  \\frac{1}{2}\\cdot 4^{-(H-(m-1))}.\n\\]\n\\end{conj}\nWe give one quick check that this conjecture is reasonable.  Since every $p \\in S_m$ lies in exactly one of the sets $S_m(i,H)$, it must be the case that \n\\[\n\\sum_{i\\in \\{0,1,4,5\\}} \\sum_{H \\ge m} S_m(i,H) = 1.\n\\]  \nFor $i \\in \\{1,5\\}$ we have \n\\[\n\\sum_{H \\ge m} S_m(i,H) = \\sum_{H \\ge m} 4^{-(H-(m-1))} = \\frac{1}{4} \\cdot \\frac{1}{1-\\frac{1}{4}} = \\frac{1}{3}.\n\\]\nFor $i \\in \\{0,4\\}$ we have \n\\[\n\\sum_{H \\ge m} S_m(i,H) = \\sum_{H \\ge m} \\frac{1}{2} \\cdot 4^{-(H-(m-1))} = \\frac{1}{8} \\cdot \\frac{1}{1-\\frac{1}{4}} = \\frac{1}{6}.\n\\]\n\nThere are three possibilities for the shape of the crater of the volcano $V_p$, depending on whether $\\operatorname{disc} \\mathcal{O}_0$ is congruent to $1$ modulo $8,\\ 5$ modulo $8$, or $0$ modulo $4$.  This calculation suggests that among the set of all anomalous primes, these three shapes are equally likely, and further that if we divide up the volcanoes of a fixed height $H \\ge m$, all three crater shapes are equally likely.  Another nice consequence of this conjecture is that for any fixed $i \\in \\{0,1,4,5\\}$ it is clear how $S_m(i,H)$ changes with $H$, as it predicts that \n\\[\n\\frac{S_m(i,H+1)}{S_m(i,H)} = \\frac{1}{4}.\n\\]\n\n\nIn Section \\ref{Q} we saw that if $p$ is anomalous of defect $(m+1,m)$ and $F \\in \\operatorname{GL}_2(\\mathbf{Z}_2)$ is in the conjugacy class of Frobenius, then \n\\[\nF = -I + 2^m \\left(\\begin{array}{cc} x & y \\\\ z & w\n\\end{array}\\right)\n\\]\nwhere $x,y,z,w$ are not all $0\\pmod{2}$.  At the end of Section \\ref{volcanoes} we saw that $\\operatorname{disc} \\mathcal{O}_0 \\pmod{8}$ is determined by $\\mathsf{sqf}((x-w)^2 + 4yz) \\pmod{8}$ and that $h(V_p)$ is determined by both $\\operatorname{disc} \\mathcal{O}_0 \\pmod{8}$ and $v_2\\left((x-w)^2+4yz\\right)$.\n\nThe goal of this section is to show that if the matrix $\\left(\\begin{smallmatrix} x & y \\\\ z & w \\end{smallmatrix}\\right)$ were distributed like a Haar random matrix in $\\operatorname{M}_2(\\mathbf{Z}_2)$ subject to the additional constraint that $v_2(y) = 0$, we would see the behavior predicted in Conjecture \\ref{volcano_conj}.  We do not currently have a satisfactory explanation of why Frobenius at anomalous primes of defect $(m+1,m)$ should correspond to these `random matrices with $y$ odd'.  \n\nFix a positive integer $m \\ge 2$.  We now explain our model for anomalous primes of defect $(m+1,m)$.  Let $E$ be an elliptic curve over $\\mathbf{F}_p$ with trace of Frobenius $t$ and $V_p$ be the associated 2-isogeny volcano over $\\mathbf{F}_p$. Let $K = \\mathbf{Q}(\\sqrt{t^2-4p}) = \\mathbf{Q}(\\sqrt{D})$ where $D = \\mathsf{sqf}(t^2-4p)$.  Recall from Section \\ref{volcanoes} that $h(V_p) = H$ if and only if\n\\[\nv_2(t^2 -4p) = 2m + v_2((x-w)^2 + 4yz) =  \\begin{cases} 2H & \\text{if }  D\\equiv 1 \\pmod{4} \\\\\n2H+2 & \\text{if } D\\equiv 3 \\pmod{4} \\\\\n2H+3 & \\text{if } D\\equiv 2 \\pmod{4}. \n\\end{cases}\n\\]\nAlso recall that $\\operatorname{disc} \\mathcal{O}_0\\equiv \\operatorname{disc} \\mathcal{O}_K\\pmod{8}$.\n\nInstead of starting from an elliptic curve over $\\mathbf{F}_p$ we consider a Haar random matrix $M = \\left(\\begin{smallmatrix} x & y \\\\ z & w \\end{smallmatrix}\\right)$ with entries in $\\mathbf{Z}_2$ subject to the additional constraint that $v_2(y) = 0$.  We use $\\det(-I + 2^m M)$ in place of $p$ and $\\operatorname{trace}\\left(-I+2^m M\\right) = -2+2^m(x+w)$ in place of $t$.  Note that for any fixed $x$, the map taking $w$ to $\\alpha = x-w$ is a bijection on $\\mathbf{Z}_2$.  For the rest of the proof we usually do not refer to $x$ and $w$, but only to $\\alpha$.  Let $\\alpha$ and $z$ be random elements of $\\mathbf{Z}_2$ distributed with respect to Haar measure, and $y \\in \\mathbf{Z}_2^*$ be a random unit in $\\mathbf{Z}_2$.  We write $\\Prob(\\cdot)$ to denote the proportion of $\\alpha,y,z$ for which some property holds.\n\nWe define a kind of height associated to the matrix $M$.  Let \n\\[\nH_M = \\begin{cases}\nm + \\frac{v_2((x-w)^2 + 4yz)}{2} & \\text{if } \\mathsf{sqf}((x-w)^2 + 4yz) \\equiv 1 \\pmod{4} \\\\\nm -1 + \\frac{v_2((x-w)^2 + 4yz)}{2} & \\text{if } \\mathsf{sqf}((x-w)^2 + 4yz) \\equiv 3 \\pmod{4} \\\\\nm -1 + \\frac{v_2((x-w)^2 + 4yz)-1}{2} & \\text{if } \\mathsf{sqf}((x-w)^2 + 4yz) \\equiv 2 \\pmod{4}\n\\end{cases}.\n\\]\n\n\\begin{thm} \\label{height_dist}\nLet $m \\ge 2$ and $H \\ge m$ be positive integers.  Let $M = \\left(\\begin{smallmatrix} x & y \\\\ z & w \\end{smallmatrix}\\right) \\in \\operatorname{M}_2(\\mathbf{Z}_2)$ be a Haar random matrix subject to the additional constraint that $v_2(y) = 0$.  \n\\begin{enumerate}\n\\item For $i \\in \\{1,5\\}$, the probability that $\\mathsf{sqf}((x-w)^2 + 4yz) \\equiv i \\pmod{8}$ and $H_M = H$ is $4^{-(H-(m-1))}$.  \n\\item For $i \\in \\{2,3\\}$, the probability that $\\mathsf{sqf}((x-w)^2 + 4yz) \\equiv i \\pmod{4}$ and $H_M = H$ is $\\frac{1}{2}\\cdot 4^{-(H-(m-1))}$.\n\\end{enumerate}\n\\end{thm}\n\nTheorem \\ref{height_dist} follows from the following stronger result.\n\\begin{thm}\\label{valuation_thm}\n\\begin{enumerate}\n\\item \n\\begin{eqnarray*}\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right) & = &  \n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 5\\pmod{8} \\end{array}\\right) \\\\\n& = & \n\\begin{cases} \n0 & \\text{if } k \\text{ is odd},\\\\\n2^{-(k+2)} & \\text{if } k \\text{ is even}.\n\\end{cases}\n\\end{eqnarray*}\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4} \\end{array}\\right) = \n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k=0,\\\\\n2^{-(k+1)} & \\text{if } k\\ge 2 \\text{ is even}.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k  \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\end{array}\\right) = \n\\begin{cases} \n0 & \\text{if } k \\text{ is even or } k=1,\\\\\n2^{-k} &  \\text{if } k\\ge 3 \\text{ is odd}.\n\\end{cases}\n\\]\n\\end{enumerate}\n\\end{thm}\nIt is straightforward to check that this result implies Theorem \\ref{height_dist} by checking the base case $H = m$ and then thinking about what happens to these probabilities as we increase $H$, dividing things into cases based on $\\mathsf{sqf}(\\alpha^2+4yz)\\pmod{8}$ and using the definition of $H_M$.\n\n\n\n\n\n\n\n\nWe prove this result by dividing the set of all $\\alpha, z \\in \\mathbf{Z}_2$ and $y \\in \\mathbf{Z}_2^*$ based on the relative sizes of $v_2(\\alpha^2)$ and $v_2(4yz) = 2 + v_2(z)$.  More precisely, we prove Theorem \\ref{valuation_thm} in three parts, where each part is divided into cases based on $\\mathsf{sqf}(\\alpha^2+4yz)\\pmod{8}$.\n\n\\begin{lem}\\label{alpha small}\n\\begin{enumerate}\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\\\  v_2(\\alpha^2)< v_2(4yz)  & & \\end{array}  \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd},\\\\\n2^{-(3k\/2+2)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 5\\pmod{8} \\\\  v_2(\\alpha^2)< v_2(4yz) & & \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd},\\\\\n2^{-(3k\/2+2)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4}  \\\\  v_2(\\alpha^2)< v_2(4yz) & & \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k = 0,\\\\\n2^{-(3k\/2+1)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\\item \n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\\\ v_2(\\alpha^2)< v_2(4yz) & &  \\end{array} \\right)\n=\n0.\n\\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{alpha big}\n\\begin{enumerate}\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\\\ v_2(\\alpha^2) > v_2(4yz) & &\\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd},\\\\\n2^{-(3k\/2+2)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 5\\pmod{8} \\\\ v_2(\\alpha^2) > v_2(4yz) & &\\end{array}\\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd},\\\\\n2^{-(3k\/2+2)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4}\\\\ v_2(\\alpha^2) > v_2(4yz) & & \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k=0,\\\\\n2^{-(3k\/2+1)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\\\ v_2(\\alpha^2) > v_2(4yz) & & \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is even or } k=1,\\\\\n2^{-(3k\/2-1\/2)} &  \\text{otherwise}.\n\\end{cases}\n\\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{alpha equal}\n\\begin{enumerate}\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\\\ v_2(\\alpha^2) = v_2(4yz) = 2\\beta \\end{array}\\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k \\in \\{0,2\\},\\\\\n2^{-(k+\\beta+2)} &  \\text{if } k \\text{ is even and } 2 \\le 2\\beta \\le k-1.\n\\end{cases}\n\\]\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 5\\pmod{8} \\\\ v_2(\\alpha^2) = v_2(4yz) = 2\\beta \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k \\in \\{0,2\\},\\\\\n2^{-(k+\\beta+2)} &  \\text{if } k \\text{ is even and } 2 \\le 2\\beta \\le k-1.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4} \\\\ v_2(\\alpha^2) = v_2(4yz) = 2\\beta \\end{array}  \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is odd or } k \\in \\{0,2\\},\\\\\n2^{-(k+\\beta+1)} &  \\text{if } k \\text{ is even and } 2 \\le 2\\beta \\le k-1.\n\\end{cases}\n\\]\n\n\\item\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = & k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\\\ v_2(\\alpha^2) = v_2(4yz) = 2\\beta  \\end{array} \\right)\n=\n\\begin{cases} \n0 & \\text{if } k \\text{ is even or } k=1,\\\\\n2^{-(k+\\beta)} & \\text{if } k \\text{ is odd and } 2 \\le 2\\beta \\le k-1.\n\\end{cases}\n\\]\n\\end{enumerate}\n\\end{lem}\n\nBefore proving these individual results, we see how they imply Theorem \\ref{valuation_thm}.  We divide this argument into cases.  Combining these three lemmas, it is clear that \n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right)  =   \n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\  \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 5\\pmod{8} \\end{array}\\right),\n\\]\nand that these probabilities are $0$ when $k$ is odd.  When $k = 0$ we have \n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &0 \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right) \n= \n2^{-2} + 0 + 0,\n\\]\nand when $k =2$ we have \n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &2 \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right) \n= \n2^{-5} + 2^{-5} + 0 = 2^{-4}.\n\\]\nNow suppose $k \\ge 4$ is even.  Note that $\\lfloor \\frac{k-1}{2}\\rfloor = k\/2-1$. We have \n\\begin{eqnarray*}\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right)  & = & \n2^{-(3k\/2+2)} + 2^{-(3k\/2+2)} + \\sum_{\\beta = 1}^{\\lfloor\\frac{k-1}{2}\\rfloor} 2^{-(k+\\beta+2)} \\\\ \n& = & 2^{-(3k\/2+1)} + 2^{-(k+2)} \\sum_{\\beta=1}^{k\/2- 1} 2^{-\\beta}.\n\\end{eqnarray*}\nWe write \n\\[\n\\sum_{\\beta=1}^{k\/2- 1} 2^{-\\beta} = 2^{-1} \\sum_{\\beta=0}^{k\/2- 2} 2^{-\\beta} = 2^{-1}\\left(2^{-(k\/2-2)} + 2^{-(k\/2-3)} + \\cdots + 2^{-1} + 2^0\\right).\n\\]\nWe see that \n\\begin{eqnarray*}\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 1\\pmod{8} \\end{array}\\right) \n& = & 2^{-(3k\/2+1)} + 2^{-(k+3)}(2^{-(k\/2-2)}+ 2^{-(k\/2-3)} + \\cdots +2^{-1} +2^{0}) \\\\\n& = &  2^{-(3k\/2+1)} + 2^{-(3k\/2+1)} +2^{-(3k\/2+2)} + \\cdots  + 2^{-(k+3)} \\\\\n& = & 2^{-(k+2)}.\n\\end{eqnarray*}\nWe next consider the analogous computation for the case where $\\mathsf{sqf}(\\alpha^2+4yz)  \\equiv  3\\pmod{4}$.  Combining the lemmas above, we see that\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4} \\end{array}\\right) = 0\n\\]\nif $k$ is odd or $k = 0$.  Suppose that $k \\ge 2$ is even.  We have\n\\begin{eqnarray*}\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 3\\pmod{4} \\end{array}\\right)  & = & \n2^{-(3k\/2+1)} + 2^{-(3k\/2+1)} + 2^{-(k+2)} \\sum_{\\beta=1}^{k\/2- 1} 2^{-\\beta},\n\\end{eqnarray*}\nwhere for $k=2$ the empty sum in the final term is $0$.  Arguing as above, it is now clear that this sum is $2$ times the analogous one for $\\mathsf{sqf}(\\alpha^2+4yz)  \\equiv  1\\pmod{8}$.  \n\nFinally, we consider the computation for the case where $\\mathsf{sqf}(\\alpha^2+4yz)  \\equiv  2\\pmod{4}$.  Combining the lemmas above, we see that\n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\end{array}\\right) = 0\n\\]\nif $k$ is even or $k =1$.  Suppose $k \\ge 3$ is odd.  We have \n\\[\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\end{array}\\right) = \n2^{-(3k\/2-1\/2)} + \\sum_{\\beta=1}^{\\frac{k-1}{2}} 2^{-(k+\\beta)}.\n\\]\nNote that \n\\[\n\\sum_{\\beta=1}^{\\frac{k-1}{2}} 2^{-(k+\\beta)} = 2^{-(k+1)} \\sum_{\\beta = 0}^{\\frac{k-3}{2}} 2^{-\\beta} = 2^{-(k+1)} \n\\left(2^{-(k\/2-3\/2)} + 2^{-(k\/2-4\/2)} + \\cdots + 2^{-1} + 2^0\\right).\n\\]\nThis gives\n\\begin{eqnarray*}\n\\Prob\\left(\\begin{array}{ccc} v_2(\\alpha^2+4yz) & = &k \\\\ \\mathsf{sqf}(\\alpha^2+4yz) & \\equiv & 2\\pmod{4} \\end{array}\\right) & = &\n2^{-(3k\/2-1\/2)} + \\left(2^{-(3k\/2-1\/2)} +2^{-(3k\/2-3\/2)} +\\cdots + 2^{-(k+1)}\\right) \\\\\n& = &\n2^{-k}. \n\\end{eqnarray*}\n\nWe now prove the three lemmas.\n\\begin{proof}[Proof of Lemma \\ref{alpha small}]\nSuppose $v_2(\\alpha^2) = k < v_2(4yz)$.  Therefore $k$ is even and $v_2(\\alpha^2+4yz) = k$.  The probability that $v_2(\\alpha) = k\/2$ is $2^{-(k\/2+1)}$.  The probability that $v_2(4yz) = 2 + v_2(z) > k$ is the probability that $v_2(z) \\ge k-1$, which is $1$ if $k=0$ and is $2^{-(k-1)}$ if $k \\ge 2$ is even.\n\nWrite $\\alpha = 2^{k\/2} u$ where $u \\in \\mathbf{Z}_2^*$ and $4yz = 2^{k+1} \\nu$ where $\\nu \\in \\mathbf{Z}_2$.  If $k = 0$ we must have $v_2(\\nu) \\ge 1$.  By varying $z$, we see that $\\nu$ is a Haar random element of $2\\mathbf{Z}_2$ when $k = 0$ and is a Haar random element of $\\mathbf{Z}_2$ otherwise.\n\nWe have\n\\[\n\\mathsf{sqf}(\\alpha^2 + 4yz) = \\mathsf{sqf}(u^2 + 2\\nu) \\equiv u^2+2\\nu \\pmod{8}.\n\\]\nSince $u^2 \\equiv 1 \\pmod{8}$ we see that\n\\[\nu^2+2\\nu  \\equiv \\begin{cases}\n1 \\pmod{8} & \\text{ if } v_2(\\nu) \\ge 2,\\\\\n5 \\pmod{8} & \\text{ if } v_2(\\nu) = 1,\\\\\n3 \\pmod{4} & \\text{ if } v_2(\\nu) = 0.\n\\end{cases}\n\\]\nWe see that $\\mathsf{sqf}(\\alpha^2 + 4yz) \\equiv 1 \\pmod{8}$ if and only if $v_2(z) \\ge 2k+1$, which happens with probability $2^{-(2k+1)}$, that  $\\mathsf{sqf}(\\alpha^2 + 4yz) \\equiv 5 \\pmod{8}$ if and only if $v_2(z) = 2k$, which happens with probability $2^{-(2k+1)}$, and that $\\mathsf{sqf}(\\alpha^2 + 4yz) \\equiv 3 \\pmod{4}$ if and only if $v_2(z) = 2k-1$, which happens with probability $2^{-2k}$ if $k \\ge 2$ and probability $0$ if $k =0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{alpha big}]\nSuppose $v_2(4yz) = 2+v_2(z) = k < v_2(\\alpha^2)$.  So $v_2(\\alpha^2+4yz) = k$.  The probability that $v_2(z) = k-2$ is $2^{-(k-1)}$ if $k \\ge 2$ and is $0$ otherwise.  If $k\\ge 2$ is even,\n\\[\n\\Prob(v_2(\\alpha^2) > k) = \\Prob(v_2(\\alpha) \\ge k\/2 + 1) = 2^{-(k\/2+1)},\n\\]\nand if $k$ is odd, \n\\[\n\\Prob(v_2(\\alpha^2) > k) = \\Prob(v_2(\\alpha) \\ge k\/2 + 1\/2) = 2^{-(k\/2+1\/2)}.\n\\]\nSuppose $v_2(z) = k-2$ where $k \\ge 2$.  We write $z = 2^{k-2} u$ where $u \\in \\mathbf{Z}_2^*$, so $4yz = uy 2^k$. Suppose $v_2(\\alpha^2) > k$.  If $k$ is even, then $\\alpha = \\gamma 2^{k\/2+1}$ where $\\gamma \\in \\mathbf{Z}_2$ is not necessarily a unit.  In this case, $\\mathsf{sqf}(\\alpha^2 + 4yz) = \\mathsf{sqf}(4 \\gamma + yu)$.  For a fixed value of $z$, by varying $y$ we see that $yu$ is a Haar random element of $\\mathbf{Z}_2^*$.  Therefore, the probability that $\\mathsf{sqf}(\\alpha^2 + 4yz) \\equiv 3\\pmod{4}$ is $1\/2$, and the probability that $\\mathsf{sqf}(\\alpha^2 + 4yz) \\equiv i \\pmod{8}$ is $1\/4$ for $i \\in \\{1,5\\}$.  This completes the proof in the case that $k$ is even. If $k$ is odd, then $\\alpha = \\gamma 2^{k\/2+1\/2}$ where $\\gamma \\in \\mathbf{Z}_2$ is not necessarily a unit.  In this case, $\\mathsf{sqf}(\\alpha^2 + 4yz) = \\mathsf{sqf}(4 \\gamma + 2yu) \\equiv 2\\mod{4}$.  This completes the proof when $k \\ge 3$ is odd.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{alpha equal}]\nSuppose that $v_2(\\alpha^2) = v_2(4yz) = 2 + v_2(z)$.  Since $v_2(\\alpha^2) = 2 v_2(\\alpha)$, we must have $v_2(\\alpha^2) = v_2(4yz) = 2 + v_2(z) = 2\\beta$ with $\\beta \\ge 1$.\n\nSuppose $v_2(\\alpha) = \\beta$ and write $\\alpha = 2^\\beta u$ where $u \\in \\mathbf{Z}_2^*$.  Suppose that $v_2(z) = 2\\beta -2$ and write $z = 2^{2\\beta-2} \\nu$ where $\\nu \\in \\mathbf{Z}_2^*$.  So $4yz = 2^{2\\beta} y \\nu$.  For a fixed value of $z$, varying $y$ shows that $y\\nu$ is a Haar random element of $\\mathbf{Z}_2^*$.\n\nWe have $v_2(\\alpha^2+4yz) = 2\\beta + v_2(u^2+y\\nu)$.  Since $u^2$ and $y\\nu$ are both units, $v_2(u^2+y\\nu) \\ge 1$ and we can write $u^2+y\\nu = 2\\delta$ where $\\delta \\in \\mathbf{Z}_2$.  Since $y\\nu$ is a Haar random element of $\\mathbf{Z}_2^*$, we see that $\\delta$ is a Haar random element of $\\mathbf{Z}_2$.  Suppose that $k - 2\\beta \\ge 0$.  Therefore,\n\\[\n\\Prob(v_2(u^2+y\\nu) = k-2\\beta) = \\begin{cases} \n0 & \\text{if } k - 2\\beta = 0\\\\ 2^{-(k-2\\beta)} & \\text{otherwise}.\n\\end{cases}\n\\]\nWe have \n\\[\n\\mathsf{sqf}(\\alpha^2+4yz) = \\mathsf{sqf}(u^2+y\\nu) = \\mathsf{sqf}(2\\delta).\n\\]\nIf $v_2(\\delta)$ is even, then $\\mathsf{sqf}(2\\delta) \\equiv 2 \\pmod{4}$.  If $v_2(\\delta)$ is odd, then for some nonnegative integer $r$ we have $2\\delta = 2^{2 r} \\delta'$, where $\\delta' \\in \\mathbf{Z}_2^*$, and $\\mathsf{sqf}(2\\delta) = \\mathsf{sqf}(\\delta')$.  If we restrict to any particular value of $r$, since $\\delta$ is a Haar random element of $\\mathbf{Z}_2$, we see that $\\delta'$ is a Haar random element of $\\mathbf{Z}_2^*$.  In particular, the probability that $\\mathsf{sqf}(\\alpha^2+4yz) \\equiv 3 \\pmod{4}$ is $1\/2$ and the probability that $\\mathsf{sqf}(\\alpha^2+4yz) \\equiv i \\pmod{8}$ is $1\/4$ for $i \\in \\{1,5\\}$.\n\nThe probability that $v_2(\\alpha) = \\beta$ is $2^{-(\\beta+1)}$.  The probability that $v_2(z) = 2\\beta -2$ is $2^{-(2\\beta-1)}$ if $\\beta \\ge 1$ and is $0$ if $\\beta = 0$.  We note that \n\\[\n2^{-(\\beta+1)} 2^{-(2\\beta-1)}2^{-(k-2\\beta)} = 2^{-(k+\\beta)}.\n\\]\nConsidering the different cases for $\\mathsf{sqf}(\\alpha^2+4yz)$ modulo $4$ and $8$ completes the proof.\n\\end{proof}\n\n\n\\section{Future Work} \\label{preview_2}\n\nIf the groups $G$ and $G'$ are not as large as possible (\\emph{i.e.},~do not have index 3 in $\\operatorname{GL}_2(\\mathbf{Z}_2)$), or if $G \\not \\simeq G'$, then the proportion $\\mathcal{P}$ of anomalous primes might be quite different than $1\/30$, as the following example shows.\n\n\\begin{exm}\nLet $E$ be the elliptic curve  \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/1200\/e\/5}{{\\tt 1200e5}} and $E'$ the curve \\href{https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/1200\/e\/2}{{\\tt 1200e2}}.  Both mod 4 representations have order 4 and neither mod 8 representation contains $-I$.  By inspecting the 2-adic representations, one can check that the only possible defects of anomalous primes are $(3,2)$ and $(2,3)$. In fact, more is true.\n\nIf we look explicitly at the images of the mod 4 representations, we see\n\\begin{align*}\nG(4) &= \\left\\{ \\begin{pmatrix} \\pm 1 & 0 \\\\ 0 & \\pm 1 \\end{pmatrix} \\right\\} \\\\\nG'(4) &= \\left\\{ \\begin{pmatrix}  1 & 0 \\\\ 0 & \\pm 1 \\end{pmatrix} ,  \\begin{pmatrix}  -1 & 2 \\\\ 0 & \\pm 1 \\end{pmatrix} \\right\\}.\n\\end{align*}\nIf $p$ is anomalous, then using the fact that $p \\equiv 1 \\pmod{4}$ and that the 2-Sylow subgroups of $E(\\mathbf{F}_p)$ and $E'(\\mathbf{F}_p)$ are both $\\mathbf{Z}\/2\\mathbf{Z} \\times \\mathbf{Z}\/2\\mathbf{Z}$, we must have $F \\equiv  -I \\pmod{4}$ and $F' \\equiv \\left(\\begin{smallmatrix}  -1 & 2 \\\\ 0 & -1 \\end{smallmatrix} \\right) \\pmod{4}$. Therefore, every anomalous prime has defect $(3,2)$ and by the Chebotarev density theorem this is exactly 1\/4 of all primes.\n\\end{exm}\n\nIn a forthcoming paper \\cite{rnt}, we take up the problem of determining all possible values of $\\mathcal{P}$, for all pairs of rationally 2-isogenous elliptic curves over $\\mathbf{Q}$, including the case where $E$ and $E'$ have CM.  What makes this a finite task is that \n\\begin{enumerate}\n\\item all images of 2-adic representations have been classified (\\cite{rzb} for the non-CM case and \\cite{alvaro} for the CM case), and\n\\item all isogeny-torsion graphs over $\\mathbf{Q}$ have been classified in \\cite{chiloyan} and \\cite{chil-alvaro}. \n\\end{enumerate}\nThere are additional consequences for the isogeny volcanoes attached to these curves that we explore as well, including how the torsion point fields $\\mathbf{Q}(E[2^m])$ and $\\mathbf{Q}(E'[2^m])$ are ``entangled''.   For example, we are able to show the following two results.\n\\begin{itemize}\n\\item If $\\mathbf{Q}(E[2]) = \\mathbf{Q}(E'[2])$ then $G$ and $G'$ must each have index greater than $3$ in $\\operatorname{GL}_2(\\mathbf{Z}_2)$.\n\\item If there are no primes of defect $(m+1,m)$ then we must have $\\mathbf{Q}(E[2^m]) = \\mathbf{Q}(E'[2^m])$ and $\\mathbf{Q}(x(E[2^m])) = \\mathbf{Q}(x(E'[2^m]))$.\n\\end{itemize}\nWe explore the consequences of these and similar results for anomalous primes.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTime-Dependent Density Functional Theory (TDDFT)~\\cite{Runge1984,Marques2012a,Ullrich2012} which represents a generalization of density functional theory (DFT)~\\cite{Hohenberg1964,Kohn1965} to time-dependent systems is  well established as a computationally efficient first-principles methodology for accessing excited state properties in both molecular and solid state materials.  While linear-response TDDFT~\\cite{CASIDA1995} (LR-TDDFT) has been a standard feature in first-principles quantum chemistry codes for over two decades~\\cite{Casida2012,Burke2005}, real-time TDDFT (RT-TDDFT)~\\cite{Yabana1996,Bertsch2000} has grown significantly in prominence within the last ten years driven primarily by the need for theoretical developments to complement experimental efforts employing time-domain laser spectroscopies. To date, a number of implementations of RT-TDDFT have been reported  ~\\cite{Yabana1996,Bertsch2000,Tsolakidis2002,Takimoto2007,Meng2008,Lopata2011,Andrade2012,Wang2013,Krieger2015,Goings2016,Provorse2016,Yost2017,salmon,Pemmaraju2018,Jia2018,Lian2018} with applications to both molecular and solid-state systems. In particular, for the  non-perturbative treatment of light-matter interactions in periodic solids especially outside the linear-response regime, the velocity gauge (VG) formulation of RT-TDDFT originally proposed by Yabana and Bertsch~\\cite{Yabana1996,Bertsch2000} has proven useful and numerical implementations of this scheme employing real-space-grids\\cite{Yabana2012,salmon,Andrade2012}, full-potential linearized augmented planewaves (FP-LAPW) \\cite{Dewhurst2004,Krieger2015} and atomic-orbital basis sets~\\cite{Pemmaraju2018,Lian2018} have been demonstrated in connection with a wide range of applications.\n\nAnalogously to the case of DFT for ground state properties, the predictive accuracy of TDDFT for excited states depends on the functional approximation utilized to describe exchange-correlation (XC) effects characterizing the many-electron system~\\cite{Ullrich2012,Marques2012a}.  A large body of literature investigating exact properties of the time-dependent XC potential in TDDFT as well as a hierarchy of practical XC kernel approximations for routine TDDFT simulations has been developed over the years~\\cite{Burke2005,Marques2012a,Ullrich2012,Casida2012,Baer2010,Baer2018}.  The Adiabatic Local Density Approximation (ALDA) wherein the XC potential is both a space and time-local multiplicative quantity represents the simplest XC approximation within  Kohn-Sham TDDFT~\\cite{Marques2012a,Ullrich2012}.  The ALDA was widely utilized in the early days of TDDFT but is characterized by a number of short-comings. Among the most well-known failures of the ALDA are its inability to describe charge transfer excitations in molecules and the lack of exciton binding in solids~\\cite{Marques2012a,Ullrich2012, Baer2010, Casida2012,Kronik2016,Turkowski2017,Kummel2017}. Since the advent of Generalized Kohn-Sham (GKS)~\\cite{Seidl1996,Baer2010,Perdew2017} theory over two decades ago, the ALDA has been largely superseded within molecular quantum chemistry by a wide array of hybrid-DFT functionals and the corresponding non-local XC kernels that provide much improved accuracy in both linear-response and real-time TDDFT applications~\\cite{Marques2012a,Ullrich2012, Baer2010, Casida2012,Kronik2016,Turkowski2017,Kummel2017}. These developments have recently been placed on a formally rigorous footing by Baer and Kronik within \\textit{Generalized Kohn-Sham} TDDFT (GKS-TDDFT)~\\cite{Baer2018}. \n\nIn solid-state systems, the importance of long-range nonlocal exchange for capturing excitonic effects within TDDFT has been widely discussed~\\cite{Botti2004,Marques2012a,Ullrich2012, Yang2015, Refaely-Abramson2015a, Kronik2016, Turkowski2017}, but practical implementations of nonlocal kernels grounded in GKS theory have taken longer to emerge primarily due to the computational cost. In recent years, the development of efficient algorithms for nonlocal exchange within planewave basis set implementations has led to wide spread utilization of GKS XC approximations within ground state DFT simulations of periodic solids~\\cite{Marsman2008,Betzinger2010,Bylaska2011,Lin2016,Barnes2017}.  Concomitantly, a number of atom-centered Gaussian-type orbital (GTO)~\\cite{Causa1988,Heyd2003,Tymczak2005a,Hutter2014} or Numerical atomic orbital (NAO)~\\cite{Fernandez2003,Shang2011,Levchenko2015,Qin2015} based condensed matter DFT platforms have also been developed to take advantage of GKS XC functionals in periodic systems. Several notable studies have extended these developments to TDDFT calculations of excited states in solids at the GKS level of theory~\\cite{Paier2008,Hutter2014,Yang2015,Sato2015,Refaely-Abramson2015a,Turkowski2017,Jia2018a}. In particular recent studies employing tuned range-separated hybrid functionals within planewave basis set implementations demonstrated good predictive accuracy for linear-response TDDFT with regards to excitonic effects in condensed matter systems~\\cite{Refaely-Abramson2015a}. \n\nTaking advantage of the aforementioned developments,  in this article a linear combination of atomic orbitals (LCAO) approach is described for performing velocity gauge real-time TDDFT (VG-RT-TDDFT)~\\cite{Yabana1996,Yabana2012,Pemmaraju2018} simulations of light-matter interaction and electron dynamics in periodic solids employing range-separated hybrid DFT functionals.  LCAO basis sets offer distinct computational efficiencies with regards to the treatment of localized core-electrons and the linear scaling with system size of the nonlocal Fock matrix relevant to GKS functionals~\\cite{Tymczak2005a,Levchenko2015,Qin2015}. The present approach aims to utilize these efficiencies to enable GKS VG-RT-TDDFT simulations for light-matter interactions spanning near infrared (NIR) to soft X-ray energies while treating valence and core electronic excitations on the same footing. Such a development is useful in two respects: Firstly, whereas the vast majority of RT-TDDFT simulations in solids to date, with some notable exceptions~\\cite{Sato2015,Yost2017,Jia2018a}, have been limited to ALDA or equivalent approximations, a framework such as the one discussed here could enable routine  VG-RT-TDDFT simulations at the GKS level of theory. Secondly, the unified treatment of core and valence excitations represents a useful first-principles theoretical complement to emerging ultrafast spectroscopic studies in solids utilizing high-harmonic generation (HHG) and X-ray free electron laser (XFEL) light sources to probe  electron dynamics from the point of view of core excitations~\\cite{Schultze2014,Zhang2015,Moulet2017}.     \n\\section{Numerical implementation details}\nThe developments reported in this article extend an LCAO VG-RT-TDDFT framework at the ALDA level of theory that has been previously described in detail~\\cite{Pemmaraju2018}. Therefore aspects related to generalizing the methodology beyond ALDA towards employing nonlocal XC functionals form the main focus of the following discussion.  The primary equations of interest are the time-dependent Generalized Kohn-Sham (TDGKS) equations for electron dynamics in the velocity gauge (VG)~\\cite{Baer2018,Sato2015}:\n\\begin{align}\\label{vgtdks}\n&\\imath\\hbar\\frac{\\partial}{\\partial t}\\tilde{\\psi}_i(\\overrightarrow{r},t)=\\hat{\\tilde{H}}_{GKS}\\tilde{\\psi}_i(\\overrightarrow{r},t)\n\\end{align} \nwherein $\\tilde{\\psi}_i(\\overrightarrow{r},t)$ represent the VG time-dependent GKS orbitals and the VG Hamiltonian $\\hat{\\tilde{H}}_{GKS}$ given by\n\\begin{align}\\label{HGKS}\n\\hat{\\tilde{H}}_{GKS}=\\frac{1}{2m}\\left[  \\overrightarrow{p} +\\frac{e}{c}\\overrightarrow{A}(t)\\right]^2  + \\hat{\\tilde{V}}_{ion}+\\int d\\overrightarrow{r}^\\prime \\frac{e^2}{|\\overrightarrow{r}-\\overrightarrow{r}^\\prime|}n(\\overrightarrow{r}^\\prime,t) + \\hat{V}_{XC}[\\rho(\\overrightarrow{r},\\overrightarrow{r}',t)]\n\\end{align}\nincludes the kinetic term incorporating coupling to time-dependent external fields via the vector potential $\\overrightarrow{A}(t)$, the VG electron-nuclear interaction $\\hat{\\tilde{V}}_{ion}$, the Hartree potential and in general a non-multiplicative XC operator $\\hat{V}_{XC}$ which is a functional of the instantaneous single-particle density matrix $\\rho(\\overrightarrow{r},\\overrightarrow{r}',t)$~\\cite{Baer2018}.  Propagating the TDGKS equation~\\ref{vgtdks} in time yields the time-dependent density matrix,\n\\begin{equation*}\n\\rho(\\overrightarrow{r},\\overrightarrow{r}',t)=\\sum_{i}^{N_{occ}} \\tilde{\\psi}_i(\\overrightarrow{r},t)\\tilde{\\psi}_i^*(\\overrightarrow{r}',t)\n\\end{equation*}\ndensity,\n\\begin{equation*}\nn(\\overrightarrow{r},t)=\\rho(\\overrightarrow{r},\\overrightarrow{r},t)=\\sum_{i}^{N_{occ}}|\\tilde{\\psi}_i(\\overrightarrow{r},t)|^2\n\\end{equation*}\nand macroscopic current\n\\begin{equation}\\label{macI}\n\\overrightarrow{I}(t)=-\\frac{e}{\\Omega}\\int_{\\Omega}d\\overrightarrow{r}\\overrightarrow{j}(\\overrightarrow{r},t).\n\\end{equation}\nThe time-dependent current density $\\overrightarrow{j}(\\overrightarrow{r},t)$ in (\\ref{macI}) is defined as \n\\begin{equation}\n\\overrightarrow{j}(\\overrightarrow{r},t)=\\sum_{i}\\frac{e}{2m} \\left\\lbrace \\tilde{\\psi}^*_i(\\overrightarrow{r},t) \\overrightarrow{\\pi}\\tilde{\\psi}_i(\\overrightarrow{r},t) + c.c \\right\\rbrace\n\\end{equation}\nand includes the generalized momentum\n\\begin{equation}\n\\overrightarrow{\\pi} = \\frac{m}{\\imath\\hbar} [\\overrightarrow{r}, \\hat{\\tilde{H}}_{GKS}].\n\\end{equation}\nFrom the above, observables related to the time-dependent density or current are therefore readily available and frequency domain quantities can be calculated through Fourier transforms\\cite{Marques2012a,Bertsch2000,Yabana2012,Pemmaraju2018}.\n\nIn the present implementation, the XC potential $\\hat{V}_{XC}$ in $\\hat{\\tilde{H}}_{GKS}$ is obtained from employing the range-separated hybrid (RSH) functional~\\cite{Baer2010,Refaely-Abramson2015a} form for the exchange-correlation energy $E_{XC}$:\n\\begin{align}\\label{ERSH}\nE_{XC} = \\alpha E^\\mathrm{SR}_{HFX} + (1-\\alpha) E^\\mathrm{SR}_{LDAX} +  (\\alpha+\\beta)E^\\mathrm{LR}_{HFX} + [1-(\\alpha+\\beta)]E^\\mathrm{LR}_{LDAX}\n+E_{LDAC}\n\\end{align}\nwhere $E^\\mathrm{SR}_{HFX}, E^\\mathrm{LR}_{HFX}$ represent short-range (SR) and long-range (LR) nonlocal Hartree-Fock exchange (HFX) respectively, $E^\\mathrm{SR}_{LDAX},E^\\mathrm{LR}_{LDAX}$ represent  short-range and long-range LDA exchange (LDAX)~\\cite{Toulouse2004} respectively and $E_{LDAC}$ represents the LDA correlation energy~\\cite{Perdew1981}. The coefficients $\\alpha, \\beta$ determine the fractions of each of the above SR and LR quantities within the total XC energy.  The SR and LR forms of the HF and LDA exchange energies are obtained by partitioning the Coulomb operator as:\n\\begin{align}\\label{RSHC}\n\\frac{1}{|\\overrightarrow{r}-\\overrightarrow{r}^\\prime|} = \\frac{ \\alpha + \\beta~\\mathrm{erf}(\\omega |\\overrightarrow{r}-\\overrightarrow{r}^\\prime| )}{|\\overrightarrow{r}-\\overrightarrow{r}^\\prime|} + \\frac{1-\\lbrace \\alpha + \\beta~\\mathrm{erf}(\\omega |\\overrightarrow{r}-\\overrightarrow{r}^\\prime| )\\rbrace}{|\\overrightarrow{r}-\\overrightarrow{r}^\\prime|}\n\\end{align}\nThe parameters $\\alpha, \\beta$ and the range-separation parameter $\\omega$ together determine the overall mix of LDAX and HFX at different length-scales~\\cite{Baer2010,Refaely-Abramson2015a}.  In this work, for $E^\\mathrm{SR}_{LDAX}$, the Coulomb attenuated form of the LDA exchange due to Toulouse \\textit{et al}~\\cite{Toulouse2004} adapted from the implementation within the libxc library~\\cite{Lehtola2018} is employed. For the evaluation of HFX, a standard real space approach for periodic systems involving the calculation of four-center electron repulsion integrals (ERIs) between atomic orbital basis functions distributed over an extended auxiliary supercell is used~\\cite{Causa1988,Levchenko2015,Qin2015}. Accordingly, matrix elements  of the HFX  operator $\\hat{X}^{\\sigma}$ are first constructed in real-space as:\n\\begin{align}\\label{HFXM}\nX^{\\sigma}_{ij}(\\mathbf{R}) = \\sum_{\\mathbf{R}_1\\mathbf{R}_2}\\sum_{p,q} D^{\\sigma}_{pq}(\\mathbf{R}_2-\\mathbf{R}_1)[\\phi_i^\\mathbf{0} \\phi_p^{\\mathbf{R}_1}||\\phi_q^{\\mathbf{R}_2}\\phi_j^{\\mathbf{R}}]\n\\end{align}\n where $\\sigma$ is the spin index, $\\mathbf{0}$ represents the reference unitcell and $\\mathbf{R},\\mathbf{R}_1,\\mathbf{R}_2$ are lattice vectors spanning the auxiliary supercell about the reference cell, $\\phi_n^{\\mathbf{L}}$ represents a basis function with index $n$ within the lattice cell at $\\mathbf{L}$, $D^{\\sigma}_{nm}(\\mathbf{L})$ is the real-space density matrix (DM) element connecting orbitals $\\phi_n^{\\mathbf{0}},\\phi_m^{\\mathbf{L}}$ and the quantity within square brackets is an ERI expressed in the general case of a range-separated Coulomb operator as\n \\begin{align}\\label{ERI}\n [\\phi_i^\\mathbf{0} \\phi_p^{\\mathbf{R}_1}||\\phi_q^{\\mathbf{R}_2}\\phi_j^{\\mathbf{R}}] = \\int \\int  d\\overrightarrow{r} d\\overrightarrow{r}^\\prime \\frac{\\phi_i^\\mathbf{0*}(\\overrightarrow{r})\\phi_p^{\\mathbf{R}_1*}(\\overrightarrow{r})\\mathrm{erf}(\\omega |\\overrightarrow{r}-\\overrightarrow{r}^\\prime| )\\phi_q^{\\mathbf{R}_2}(\\overrightarrow{r}^\\prime)\\phi_j^{\\mathbf{R}}(\\overrightarrow{r}^\\prime)}{|\\overrightarrow{r}-\\overrightarrow{r}^\\prime|}\n \\end{align}\nThe momentum space HFX matrix is subsequently obtained at points within the first Brillouin zone (BZ) through Fourier transformation as\n \\begin{align}\\label{HFXK}\n \\mathcal{X}^{\\sigma}_{ij}(\\mathbf{k}) = \\sum_\\mathbf{R} e^{\\imath \\mathbf{k} \\mathbf{R}} X^{\\sigma}_{ij}(\\mathbf{R})\n \\end{align}\nIn the case of range-separated hybrid functionals exchange contributions from both the standard $1\/r$ and range-separated $\\mathrm{erf}(\\omega r)\/r$  Coulomb operators are calculated separately and weighted according to the $\\alpha$ and $\\beta$ parameters in equation \\ref{RSHC} to form the total XC contribution within the GKS Hamiltonian~\\cite{Baer2010,Refaely-Abramson2015a}. The HFX like contribution to the total XC energy  is easily evaluated in the real-space approach as \n\\begin{align}\nE_{HFX}=-\\frac{1}{2} \\sum_{\\sigma} \\sum_{ij \\mathbf{R}} D^{\\sigma}_{ij}(\\mathbf{R}) X^{\\sigma}_{ij}(\\mathbf{R})\n\\end{align}\n\nWithin the above scheme, the primary computationally demanding tasks are the evaluation of the ERIs in equation~\\ref{ERI} and the summations in equation~\\ref{HFXM}. The ease of computing the ERIs strongly depends on the choice of orbital basis functions employed~\\cite{Causa1988,Qin2015,Levchenko2015}. Gaussian type orbitals (GTOs) which are standard in molecular quantum chemistry feature analytical expressions for the ERIs that facilitate rapid computation~\\cite{Helgaker1995,}. ERI schemes for other choices such as Slater type orbitals~\\cite{TeVelde2001} and fully numerical atomic orbitals (NAO)~\\cite{Levchenko2015} have also been reported in the literature. The present VG-RT-TDDFT implementation is based on a development version of the SIESTA~\\cite{Soler2002} code which features NAO basis functions in its standard operating mode. While the direct evaluation of ERIs between NAO basis functions is possible as has been demonstrated by Levchenko et al~\\cite{Levchenko2015}, the numerical integration schemes necessary for NAOs are typically slower compared to those for GTO basis functions and furthermore require significant algorithmic infrastructure to be built from the ground up.  On the other hand, ERI techniques for GTOs have been under development for over five decades and are readily accessible through modern open source libraries such as libint~\\cite{Libint2} and libcint~\\cite{Sun2015}. Therefore, for the evaluation of ERIs a hybrid approach based on expanding individual NAOs over a small number of GTOs is adopted in this instance. The method is closely related to the one already reported by Qin et al~\\cite{Qin2015} and consists of the following steps:\\\\ \n(i) NAO basis functions for the system of interest are  generated using standard procedures within SIESTA~\\cite{Soler2002} at the DFT LDA level.\\\\  \n(ii) Each NAO basis function is fit to a small number (typically 3-4) of primitive GTOs using a combination of Levenberg-Marquardt~\\cite{Press2007} and simplex~\\cite{Press2007} optimization. Thus each NAO is approximated as a contracted GTO. \\\\\n(iii) The GTO exponents and contraction coefficients determined in (ii) are used in conjunction with the integral evaluation scheme for real-spherical GTOs provided by the libcint~\\cite{Sun2015} library to calculate effective ERIs between NAOs.\\\\\nAs shown by Qin et al~\\cite{Qin2015} and also later on in this article, the above scheme leads to hybrid DFT simulations in good agreement with planewave basis set approaches where the HFX matrix is evaluated through momentum space algorithms~\\cite{Marsman2008}. Once an ERI calculation procedure is selected the summations over direct lattice sites implied in equation~\\ref{HFXM} need to be carried out. For a detailed discussion of the real-space convergence properties of the HFX matrix and related integral screening algorithms the reader is referred to the extensive existing literature on the subject~\\cite{Causa1988, Heyd2003,Levchenko2015}. Briefly, for a short-ranged $\\mathrm{erfc}(\\omega r)\/r$ form of the Coulomb interaction, real-space convergence of the HFX matrix $X^{\\sigma}_{ij}(\\mathbf{R})$ is straightforward and practically achieved by extending the lattice sums in (\\ref{HFXM}) over an auxiliary supercell whose dimensions are consistent with the length-scale of the interaction set by the range-separation parameter $\\omega$. For an unscreened $1\/r$ form of the Coulomb interaction the HFX matrix is in general long-ranged and the real-space decay of the density matrix primarily determines the range of the auxiliary lattice summations necessary for convergence~\\cite{Causa1988,Heyd2003,Levchenko2015}. In practice therefore, when long-range HFX is included, convergence has to be tested on a case by case basis. In the current implementation, the size of the real-space auxiliary supercell is included as a tunable parameter to allow for convergence to be verified to within a chosen tolerance level or to the extent permitted by computational feasibility. Because of the correspondence between real-space and momentum-space, similar considerations apply in reciprocal-space algorithms where the size of the $\\mathbf{q}$-point grid used for evaluating HFX is analogous to the size of the auxiliary supercell in real-space methods~\\cite{Marsman2008,Paier2008,Levchenko2015}.  \nOnce the HFX matrix is constructed as above it is combined with the LDAX and correlation contributions according to the recipe in equation~\\ref{ERSH}  to form the full range-separated hybrid XC matrix that directly enters the VG-GKS Hamiltonian in equation~\\ref{HGKS}. Because of the unitary invariance of the HFX matrix no additional complications are introduced by transformation to the velocity gauge. The VG-TD-GKS equations are then evolved using a standard Crank-Nicholson~\\cite{Crank1947} scheme with a predictor-corrector step~\\cite{Takimoto2007,Pemmaraju2018}. \n\\begin{table}[htbp]\n\\begin{center}\n\t\\begin{tabular}{ |c|c|c| }\n\t\t\\hline\n\t\t\\multicolumn{3}{|c|}{Computational parameters for bulk solids common to LCAO and PAW} \\\\\n\t\t\\hline\n\t\t& bulk Si &  bulk LiF \\\\\n\t\t\\hline\n\t\tlattice constant   & 5.429 \\AA   & 4.017 \\AA \\\\\n\t\t\\hline\n\t\tLDA pseudopotential & Si: [Ne]3$s^2$,3$p^2$ & Li: [He]2$s^1$\\\\\n\t\tvalence configuration &  & F:[He]2$s^2$,2$p^5$\\\\\n\t\t\\hline\n\t\tDFT SCF $\\mathbf{k}$-point grid & $\\Gamma-6\\times6\\times6$ & $\\Gamma-8\\times8\\times8$\\\\\n\t\t\\hline\n\t\tIPA optics $\\mathbf{k}$-point grid &$\\Gamma-30\\times30\\times30$&$\\Gamma-24\\times24\\times24$\\\\\n\t\t\\hline\n\t\t\\multicolumn{3}{|c|}{LCAO specific parameters} \\\\\n\t\t\\hline\n\t\t\\multirow{2}{*}{basis set ($nl$-$\\zeta$)} &  Si: $3s$-2,$4s$-1,$3p$-2,$3d$-2  & Li: $2s$-2, $2p$-2  \\\\\n\t\t&    & F: $2s$-2, $2p$-2, $3d$-1  \\\\\n\t\t\\hline\n\t\tReal-space mesh-cutoff & 364 Ry & 526 Ry  \\\\\n\t\t\\hline\n\t\tHFX auxiliary supercell & $6\\times6\\times6$ & $8\\times8\\times8$\\\\\n\t\t\\hline\n\t\tVG-RT-TDDFT $\\mathbf{k}$-point grid & $\\Gamma-30\\times30\\times30$ & $\\Gamma-24\\times24\\times24$\\\\\n\t\t\\hline\n\t\tVG-RT-TDDFT time step & 0.08 a.u & 0.08 a.u\\\\\n\t\t\\hline\n\t\t\\multicolumn{3}{|c|}{PAW specific parameters} \\\\\n\t\t\\hline\n\t\tplanewave cutoff  & 400 eV & 400 eV \\\\\n\t\t\\hline\n\t\tHFX $\\mathbf{q}$-point grid & $6\\times6\\times6$ & $8\\times8\\times8$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\caption{Computational parameters used for bulk Si and LiF simulations. To model the interaction with ionic cores, norm-conserving LDA pseudopotentials generated using the Troullier-Martins~\\cite{Troullier1991} scheme are employed within LCAO runs while PAW simulations use standard LDA-PAW datasets provided with the VASP~\\cite{Hafner2008} distribution. The same reference valence electronic configurations are used in both cases.  $\\Gamma$-centered $\\mathbf{k}$-point grids of the same dimensions are used in both codes to converge the DFT self-consistent field (SCF) and finer $\\mathbf{k}$-point grids are employed to subsequently calculate linear optical response at the level of the independent particle approximation (IPA). The same $\\mathbf{k}$-point grid dimensions used for IPA optical response are also used within LCAO based VG-RT-TDDFT simulations. Auxiliary supercell dimensions used to calculate HFX in the real-space LCAO approach are chosen to be consistent with $\\mathbf{q}$-point grids used to calculate HFX in momentum space within the PAW method. The LCAO basis set is indicated using $nl$-$\\zeta$ notation where $n,l$ are principal and azimuthal quantum numbers respectively and $\\zeta$ is the number of functions of each $nl$ type. }\\label{tab1}\n\\end{table}\n\\section{Results}\nIn the following, results obtained using the LCAO GKS implementation described above are presented. A number of prototypical materials such as the intermediate gap covalent semiconductor bulk Si, the wide band gap ionic insulator bulk LiF and the low-dimensional insulator monolayer hexagonal-BN (h-BN) are considered. Each one of the above systems requires a significantly different fraction of long-range asymptotic HFX and the results therefore span a broad parameter range in the context of RSH DFT and TDDFT~\\cite{Baer2010,Refaely-Abramson2015a}. The results are presented in two distinct steps: Firstly, the validity of the LCAO GKS implementation is verified at the level of GKS-DFT band structure calculations by comparison with corresponding planewave projector augmented wave (PAW)~\\cite{Blochl1994,Kresse1996,Marsman2008} simulations also carried out as a part of this work. Subsequently, GKS VG-RT-TDDFT results obtained with the LCAO scheme are compared against pre-existing planewave results in the literature and\/or experiment.  The LCAO simulations are performed with an in-house development version~\\cite{Pemmaraju2018} of the SIESTA~\\cite{Soler2002} code. The PAW calculations are carried out using the VASP~\\cite{Kresse1996,Hafner2008} package (version 5.4.1). Computational parameters employed for the bulk Si and LiF systems within the linear-combination of atomic orbital (LCAO) and planewave projector augmented wave (PAW) simulations carried out in this work are summarized in table~\\ref{tab1}.  \n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.44]{Fig1.pdf}\n\t\\caption{(a,b)~LCAO and PAW band dispersions for bulk Si (a) and LiF (b) obtained at the LDA and short-range-corrected hybrid (SRCH) DFT levels of theory are compared. Parameters characterizing the SRCH functional are listed in table~\\ref{prsh}. (c,d) Imaginary parts of the frequency dependent dielectric function for Si (c) and LiF (d) obtained at the LDA and SRCH levels from LCAO and PAW simulations are shown.}\n\t\\label{silif}\n\\end{figure}\n\\subsection{Groundstate bandstructures}\nIn figure~\\ref{silif}, band dispersions for bulk Si (Fig.~\\ref{silif}(a)) and LiF (Fig.~\\ref{silif}(b)) obtained from LCAO and PAW simulations are compared at the LDA and short-range-corrected hybrid (SRCH) DFT levels of theory.  The SRCH functional used here employs the same fraction of short-range HFX and range-separation parameter as the popular HSE06~\\cite{Krukau2006} functional. However the semilocal DFT XC components in the SRCH are based on the LDA where as the corresponding quantities within HSE06 are based on the generalized gradient approximation (GGA)~\\cite{Krukau2006}. The HFX parameters characterizing the SRCH are reported in table~\\ref{prsh}. The equivalent settings within VASP are AEXX=0.25, HFSCREEN=0.2. Several observations can be made with regards to figures~\\ref{silif}(a,b): LCAO and PAW band dispersions at the LDA level exhibit close agreement within the valence band (VB) and upto a few eV into the conduction band (CB). At higher energies into the CB, noticeable differences appear as the LCAO basis sets have less variational freedom compared to planewaves. Importantly, band-structures obtained using the SRCH functional also show a similar level of agreement as those at the LDA level especially near the Fermi energy. Numerical values of the band gaps in Si and LiF obtained at the LDA and SRCH DFT levels are also reported in table~\\ref{bgap} and show reasonably good agreement. At the SRCH DFT level, the bottom of the VB in bulk-Si comes out slightly more dispersive in the LCAO result, being 0.24 eV lower in energy than the PAW result of -13.28 eV at $\\Gamma$. Elsewhere in the VB and within the first 5 eV into the CB, in both Si and LiF, LCAO and PAW band energy differences are small enough to be barely noticeable. This suggests firstly that the real-space GKS DFT implementation in the LCAO code is consistent with a standard PAW implementation for SRCH functionals and furthermore that the LCAO basis sets which are constructed at the LDA level have sufficient flexibility to capture electronic structure changes induced by adopting SRCH functionals. A related perspective is provided by looking at the imaginary part of the frequency-dependent linear dielectric function $\\epsilon_2(\\omega)$ calculated within the independent particle approximation (IPA). As shown in figure~\\ref{silif}(c,d) for the LCAO basis sets employed here, a consistent level of agreement with PAW results is obtained for calculated IPA-$\\epsilon_2(\\omega)$ line-shapes at both the LDA and SRCH DFT levels.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.46]{Fig2.pdf}\n\t\\caption{(a) Band dispersions of LiF calculated using global hybrid functionals featuring 0\\%,50\\% and 100\\% Hartree-Fock exchange (HFX). Results from LCAO (red) and PAW (blue) simulations are plotted together.  (b) Variation of the calculated band gap in LiF as a function of \\%HFX also comparing LCAO and PAW results.}\n\t\\label{ghlif}\n\\end{figure}\n\\begin{table}[htbp]\n\t\\begin{center}\n\t\t\\begin{tabular}{ |c|c|c|c| }\n\t\t\t\\hline\n\t\t\t&Si &LiF&2D h-BN\\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{SRCH} & $\\alpha$=0.25&  $\\alpha$=0.25&  \\\\\n\t\t\t& $\\beta$=$-$0.25&  $\\beta$=$-$0.25 & - \\\\\n\t\t\t& $\\omega$=0.2~$\\mathrm{\\AA}^{-1}$&  $\\omega$=0.2~$\\mathrm{\\AA}^{-1}$ &  \\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{LRCH} & $\\alpha$=0.2&  $\\alpha$=0.2&  $\\alpha$=0\\\\\n\t\t\t& $\\beta$=$-$0.1167&  $\\beta$=0.326 &  $\\beta$=1\\\\\n\t\t\t& $\\omega$=0.208~$\\mathrm{\\AA}^{-1}$&  $\\omega$=0.695~$\\mathrm{\\AA}^{-1}$ &  $\\omega$=0.238~$\\mathrm{\\AA}^{-1}$\\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{GH} & &  &  $\\alpha$=0.5\\\\\n\t\t\t&- & - &  $\\beta$=0\\\\\n\t\t\t& & &  $\\omega$=0~$\\mathrm{\\AA}^{-1}$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Parameters characterizing the range-separated hybrid DFT functionals employed in this study. The fraction of HFX in the short-range is indicated by $\\alpha$ while the fraction of long-range HFX  in each case is given by ($\\alpha+\\beta$). The short-range-corrected hybrid (SRCH) functionals therefore only include HFX in the short-range. The long-range-corrected hybrid (LRCH) functionals (LRC) include a non-vanishing fraction of long-range HFX and the global hybrid (GH) functional includes the same fraction of HFX at all length-scales.}\\label{prsh}\n\\end{table}\n\\begin{table}[htbp]\n\t\\begin{center}\n\t\t \\begin{adjustbox}{width=1\\textwidth}\n\t\t\\begin{tabular}{ |c|c|c|c|c|c|c|c| }\n\t\t\t\\hline\n\t\t\t&\\multicolumn{2}{|c|}{LDA} &\\multicolumn{2}{|c|}{SRCH}&LRCH&GH&Reference\\\\\n\t\t\t\\hline\n\t\t\t&LCAO&PAW&LCAO&PAW&LCAO&LCAO& \\\\\n\t\t\t\\hline\n\t\t\tSi&0.538&0.463&1.151&1.154&1.191&-&1.17~\\cite{Lautenschlager1987} [Expt.]\\\\\n\t\t\tLiF&8.870&8.961&11.345&11.325&14.229&-&14.2$\\pm$0.02~\\cite{Piacentini1976} [Expt.]\\\\\n\t\t\t{2D h-BN}&4.588&-&-&-&7.819&9.046&7.77~\\cite{Ferreira2018}[GW]\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\end{center}\n\t\\caption{Band gaps in eV obtained for different choices of GKS functionals are shown for the solid-state materials considered in this study. For Si and LiF band gaps calculated at the LDA and SRCH DFT levels using LCAO basis sets are compared against planewave PAW results. For 2D h-BN the direct band gap at the K point in the Brillouin zone is shown as obtained from the LCAO method using LDA, LRCH and GH functionals in comparison with a literature GW value~\\cite{Ferreira2018} in the last column. Parameters characterizing the different functionals are shown in table~\\ref{prsh}. Reference values for the band gaps taken from the literature are shown in the last column.}\\label{bgap}\n\\end{table}\nNext the inclusion of non-zero long-range HFX exchange in the LCAO implementation is investigated in comparison with the PAW approach by adopting a global-hybrid (GH) functional form for the XC energy where by the same non-zero fraction of HFX operates at all length-scales. This is achieved within the LCAO scheme by setting $\\beta=0, \\omega=0$ in equations~\\ref{ERSH},\\ref{RSHC} and choosing $\\alpha$ to be between 0 and 1.  The equivalent choice in VASP would be to choose AEXX=$\\alpha$ and HFSCREEN=0~\\cite{Marsman2008,Hafner2008}. It is also worth noting in this instance that as shown in table~\\ref{tab1}, the size of the real-space auxiliary supercell and the reciprocal space $\\mathbf{q}$-point grid used for calculating HFX are consistent between the LCAO and PAW approaches respectively. In figure~\\ref{ghlif} band structures for LiF obtained from LCAO and PAW simulations employing 0\\%, 50\\% and 100\\% global HFX are compared. Once again, satisfactory agreement is demonstrated in that any discrepancies between LCAO and PAW results are a small compared to the magnitude of the changes induced in the band structures by varying the fraction HFX. Band gaps of LiF obtained from the two approaches are plotted as function of \\%HFX in figure~\\ref{ghlif}(b). The largest difference between LCAO and PAW band gaps is 0.252 eV observed at 100\\% HFX, when, the PAW predicted band gap is 21.807 eV. Thus the LCAO band gap is within 1.2\\% of the PAW value across the \\%HFX parameter range and given that significantly lower fractions of HFX are needed in practically useful simulations, the discrepancies in practice are expected to be smaller. To summarize the above discussion, for both SRCH and GH DFT functionals the present LCAO implementation is able to reproduce band dispersions that are in agreement with a standard PAW framework to within a few percent.  The validity of the LCAO scheme for general long-range-corrected hybrid (LRCH) DFT functionals featuring a non-zero fraction of long-range HFX follows from the fact that LRCH functionals of the form implied by equation~\\ref{ERSH}, can be constructed as linear combinations of SRCH and GH functionals. \n\n\\subsection{Time-domain simulations}\nIn this subsection generalized Kohn-Sham (GKS) VG-RT-TDDFT~\\cite{Baer2018,Sato2015} results obtained using the LCAO implementation are presented. It is noted in this context that a precursor to this implementation has been benchmarked against established real-space grid based VG-RT-TDDFT showing good agreement at the ALDA level of theory both for linear-response and strong-field induced dynamics~\\cite{Pemmaraju2018}. Therefore in the following, the emphasis is primarily on the results generated by extending the framework to the GKS level of theory.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.56]{Fig3.pdf}\n\t\\caption{Band structures of Si (left) and LiF (right) obtained using the long-range-corrected hybrid (LRCH) functional employed in subsequent VG-RT-TDDFT simulations are shown in comparison with those at the LDA and SRCH DFT levels. Note that in Si, the SRCH and LRCH functionals predict very similar band-structures where as in LiF the LRCH functional leads to a significant band gap increase. See table~\\ref{prsh} for parameters characterizing the different functionals in each system.}\n\t\\label{bslrch}\n\\end{figure}\n\\subsubsection{Long-range-corrected hybrid functionals}\nIn the subsequent discussion, LRCH functionals are investigated within the VG-RT-TDDFT formulation described in the previous section. The main focus is their performance with regards to describing exciton binding in solids. The $\\alpha$, $\\beta$ and $\\omega$ parameters characterizing  LRCH functionals in this study are chosen according to the general arguments put forth in previous works on excitons in solids based on frequency-domain linear-response TDDFT~\\cite{Yang2015,Refaely-Abramson2015a}.  Firstly, since the long-range decay of the Coulomb interaction in a dielectric medium is of the form $\\frac{1}{\\epsilon_{\\infty} r}$, the fraction of long-range asymptotic HFX determined by the sum $(\\alpha + \\beta)$, is set to match $\\frac{1}{\\epsilon_{\\infty}}$. Here $\\epsilon_{\\infty}$, which represents the macroscopic dielectric constant characterizing the ion-clamped electronic screening response in the static limit, can be chosen either from experiment or first-principles results.  Once $(\\alpha + \\beta)$ is constrained, as far as valence electronic structure is concerned, $\\alpha$ which determines the short-range HFX fraction can be chosen on a heuristic basis to be around 0.2-0.25. Finally, having chosen $\\alpha$ and $\\beta$ the range-separation parameter $\\omega$ is determined by requiring the single-particle GKS band-gap predicted by the LRCH functional to match the fundamental gap from first-principles GW calculations or experiment. The above prescription for LRCH functionals therefore ensures that both the fundamental single-particle band gap and qualitative long-range screening behavior are correctly accounted for and on this basis, TDDFT is able to describe electron-hole excitations satisfactorily in solids~\\cite{Refaely-Abramson2015a}. The $\\alpha$, $\\beta$, $\\omega$ parameters chosen for the LRCH functionals in this work are shown in table~\\ref{prsh} and the band-structures of Si and LiF obtained using these functionals within the LCAO framework are shown in figure~\\ref{bslrch}. Corresponding values for fundamental band gaps are reported in table~\\ref{bgap} . From figure~\\ref{bslrch} it is apparent that in semiconductors like bulk Si, where functionals such as HSE06~\\cite{Krukau2006,Paier2006,Paier2008} and the related SRCH functional employed here already predict accurate single-particle band-dispersions, the LRCH functional does not lead to further significant changes in band structure. This is achieved by compensating for the additional non-zero long-range HFX in the LRCH functional through a smaller fraction of short-range HFX relative to the SRCH case (see table ~\\ref{prsh}). On the other hand, in wide gap insulators like LiF, SRCH functionals are known to underestimate the band gap~\\cite{Paier2006,Paier2008} where as by construction, the LRCH functional induces further corrections to reproduce the fundamental gap. \n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.56]{Fig4.pdf}\n\t\\caption{(a) Time dependence of the current induced in bulk Si by a 0.001 a.u delta-function electric field pulse applied at time $t$=0. Results obtained from time-evolution employing different functionals are shown. (b) Imaginary part of the frequency dependent linear dielectric function $\\epsilon_2(\\omega)$ corresponding to the time-dependent currents shown in (a). (lower-left) $\\epsilon_2(\\omega)$ from LCAO RT-LDA and the LDA independent particle approximation (IPA)  are compared. (lower-middle) The LCAO RT-SRCH result from this work is compared with the TD-HSE06 result from~reference~\\citenum{Paier2008}. (lower-right)  The LCAO RT-LRCH result from this work is compared with experimental data from reference~\\cite{Lautenschlager1987}. The LRCH IPA result is also shown.}\n\t\\label{rt-si}\n\\end{figure}\n\\subsubsection{Linear optical response in bulk Si}\n In figure~\\ref{rt-si}, the time-dependent current in bulk-Si in response to a 0.001 a.u delta function electric-field pulse applied at time $t$=0 is shown as obtained from VG-RT-TDDFT simulations employing LDA, SRCH and LRCH functionals. At very early times the induced current is similar in all cases but differences start to emerge roughly around 1~fs. Primarily, RT-SRCH and RT-LRCH exhibit a shorter time-period for the current oscillations relative to RT-LDA derived from the higher energy onset for optical excitations with the GKS functionals. Furthermore, as shown in the inset within figure~\\ref{rt-si}(a), the currents from RT-SRCH and RT-LRCH also exhibit some differences as time progresses. The imaginary part of the frequency dependent dielectric function $\\epsilon_2(\\omega)$ obtained from these simulations is shown in figure~\\ref{rt-si}(b). As expected, the RT-LDA response is very similar to that obtained at the independent-particle approximation (IPA) level of theory. RT-SRCH meanwhile accounts for some fraction of excitonic effects in bulk-Si producing intensity enhancement near the onset of optical absorption around $\\sim$3.5 eV. Importantly,  in figure~\\ref{rt-si}(b) the RT-SRCH response obtained from the present LCAO scheme is compared against previously published linear-response TD-HSE06~\\cite{Paier2008} results in the literature and shows good agreement for the overall line-shape. In the rightmost panel of  figure~\\ref{rt-si}(b), the optical response from RT-LRCH is compared against experimental data~\\cite{Lautenschlager1987} also showing satisfactory agreement. In particular, the peaks at $\\sim$3.5 eV and $\\sim$4.3 eV observed in the experimental $\\epsilon_2(\\omega)$ are reproduced to within 0.2 eV largely consistent with previous LR-TDDFT results~\\cite{Refaely-Abramson2015a}. For completeness, the independent-particle  $\\epsilon_2(\\omega)$ obtained with the LRCH functional is also shown, clearly demonstrating the significant role of excitonic binding in going from IPA to RT-LRCH in bulk Si. \n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.6]{Fig5.pdf}\n\t\\caption{(a) Time dependence of the current induced in bulk LiF by a 0.001 a.u delta-function electric field pulse applied at time $t$=0. Results obtained from time-evolution employing different functionals are shown. (b) Imaginary part of the frequency dependent linear dielectric function $\\epsilon_2(\\omega)$ corresponding to the time-dependent currents shown in (a). (lower-left) $\\epsilon_2(\\omega)$ from LCAO RT-LDA and the LDA independent particle approximation (IPA)  are compared. (lower-middle) The LCAO RT-SRCH result from this work is compared with the TD-HSE06 result from~reference~\\citenum{Paier2008}. (lower-right)  The LCAO RT-LRCH result from this work is compared with experimental data from reference~\\citenum{Piacentini1976}. The LRCH IPA result is also shown }\n\t\\label{rt-lif}\n\\end{figure}\n\\subsubsection{Linear optical response in bulk LiF}\nIn figure~\\ref{rt-lif}, results for the more extreme case of the wide gap insulating ionic solid LiF are shown. Figure~\\ref{rt-lif}(a) plots the time-dependent current in GKS-VG-RT-TDDFT induced by a small 0.001 a.u delta-function electric-field pulse applied at t=0 to a bulk unitcell of LiF.  Results for RT-LDA, RT-SRCH and RT-LRCH time-propagation are compared. As in the case of bulk Si, at very short times under $\\sim$ 1 fs, the induced current looks similar with all three functionals. Subsequently, the RT-SRCH and RT-LRCH currents deviate from the RT-LDA one. Nevertheless, the RT-SRCH and RT-LDA currents look qualitatively similar in that the current oscillations exhibit a significant amplitude reduction at later times due to dephasing. In contrast, the RT-LRCH current differs markedly starting from around $\\sim$1.5 fs and exhibits the clear onset of a strong quasi-monochromatic oscillation that undergoes very little amplitude reduction over the time-frame shown. The imaginary part of the frequency dependent dielectric function $\\epsilon_2(\\omega)$ derived from the above time-propagation is plotted in figure~\\ref{rt-lif}(b). Once again, the RT-LDA result closely resembles that of the IPA showing no exciton binding. The RT-SRCH $\\epsilon_2(\\omega)$  shown in the middle panel of ~\\ref{rt-lif}(b) predicts a higher onset of absorption but is otherwise qualitatively similar to RT-LDA.  As is already well known from previous studies~\\cite{Paier2008,Yang2015,Refaely-Abramson2015a}, SRCH functionals do not have sufficient long-range nonlocal exchange to reproduce the strongly-bound Frenkel exciton in LiF. The LCAO result obtained for the RT-SRCH functional is in reasonable agreement with the planewave TD-HSE06 reported previously~\\cite{Paier2008}. Note that the TD-HSE06 results from reference~\\citenum{Paier2008} are averaged over several different $\\mathbf{q}$-point grids and therefore exhibit a smoother line shape than the LCAO results which correspond to a single 8$\\times$8$\\times$8 real-space auxiliary cell. Nevertheless this auxiliary supercell size is consistent with the fine $\\mathbf{q}$-point grid used for LiF in recent optimally-tuned RSH LR-TDDFT simulations~\\cite{Refaely-Abramson2015a}. In contrast to RT-LDA and RT-SRCH, the RT-LRCH result shown on the right of~\\ref{rt-lif}(b) reproduces the qualitatively different optical response measured experimentally~\\cite{Piacentini1976}. The pronounced quasi-monochromatic oscillations characterizing the time-dependent current in the RT-LRCH simulation translate in the frequency domain to an intense excitonic peak at 12.45 eV which is $\\sim$0.15 eV from the experimental peak at $\\sim$12.6 eV~\\cite{Piacentini1976}. Besides the main excitonic peak, two smaller peaks at 13.78 eV and 14.47 eV are also observed in the RT-LRCH $\\epsilon_2(\\omega)$. Although features at these energies are significantly broadened out in experiment, two peaks at very similar energies are clearly observed also in previous planewave LR-TDDFT results~\\cite{Refaely-Abramson2015a,Yang2015}. Nevertheless, within the LCAO basis set, the intensity of the second peak near $\\sim$13.78 eV seems to be overestimated relative to planewave LR-TDDFT~\\cite{Refaely-Abramson2015a,Yang2015}. To summarize the discussion on bulk solids, Si and LiF, the LCAO VG-RT-TDDFT results for both short-range-corrected and long-range-corrected hybrid functionals are generally consistent with equivalent planewave results or experiment and successfully describe the qualitative improvements afforded by the GKS framework in comparison to conventional KS ALDA. Quantitative agreement with respect to excitonic peak positions is within 0.2 eV of reference values. \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.6]{Fig6.pdf}\n\t\\caption{(a) Time dependence of the current induced in monolayer h-BN (2D h-BN) by a 0.001 a.u in-plane-polarized delta-function electric field pulse applied at time $t$=0.  Results obtained at the RT-LDA and RT-LRCH levels are shown. (b) Unitcell (dotted line) employed for VG-RT-TDDFT simulations of 2D h-BN. The light-blue dots represent centers of $ghost$ basis functions included in the vacuum region near the monolayer. (c) Imaginary part of the frequency dependent linear dielectric function $\\epsilon_2(\\omega)$ corresponding to the time-dependent currents shown in (a) is compared against that from other relevant approximations. The GW-RPA and GW-BSE results are from reference~\\citenum{Ferreira2018} }\n\t\\label{rt-hbn}\n\\end{figure}\n\\begin{table}[htbp]\n\t\\begin{center}\n\t \\begin{adjustbox}{width=1\\textwidth}\n\t\t\\begin{tabular}{ |c|c|c|c| }\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{|c|}{LCAO computational parameters for 2D h-BN simulations} \\\\\n\t\t\t\\hline\n\t\t\t& no core-states &  B 1$s$ in valence  & N 1$s$ in valence \\\\\n\t\t\t\\hline\n\t\t\tlattice constant (\\AA)  &  \\multicolumn{3}{|c|}{2.504} \\\\\n\t\t\t\\hline\n\t\t\tLDA pseudopotential & B: [He]2$s^2$,2$p^1$ & B: 1$s^2$,2$s^2$,2$p^1$&B: [He]2$s^2$,2$p^1$\\\\\n\t\t\tvalence configuration & N: [He]2$s^2$,2$p^3$ & N:[He]2$s^2$,2$p^3$&N:1$s^2$,2$s^2$,2$p^3$\\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}{*}{basis set ($nl$-$\\zeta$)} &  B: $2s$-2,$2p$-2,$3d$-1  & B: $1s$-2,$2s$-2,$2p$-2,$3d$-1  & B: $2s$-2,$2p$-2,$3d$-1   \\\\\n\t\t\t&N: $2s$-2,$2p$-2,$3d$-1  & N: $2s$-2,$2p$-2,$3d$-1  & N:$1s$-2,$2s$-2,$2p$-2,$3d$-1   \\\\\n\t\t\t& 8$\\times ghost:2s$-1&  8$\\times ghost:2s$-1&  8$\\times ghost:2s$-1   \\\\\n\t\t\t\\hline\n\t\t\tReal-space mesh-cutoff& 408 Ry  & \\multicolumn{2}{|c|}{940 Ry} \\\\\n\t\t\t\\hline\n\t\t\tHFX auxiliary supercell & \\multicolumn{3}{|c|}{$12\\times12\\times1$}  \\\\\n\t\t\t\\hline\t\t\t\n\t\t\tDFT SCF $\\mathbf{k}$-point grid &\\multicolumn{3}{|c|}{$\\Gamma-24\\times24\\times1$}\\\\\n\t\t\t\\hline\n\t\t\tIPA optics $\\mathbf{k}$-point grid &$\\Gamma-48\\times48\\times1$&-&-\\\\\n\t\t\t\\hline\n\t\t\tVG-RT-TDDFT $\\mathbf{k}$-point grid & \\multicolumn{3}{|c|}{$\\Gamma-24\\times24\\times1$}\\\\\n\t\t\t\\hline\n\t\t\tVG-RT-TDDFT time step & 0.08 a.u & \\multicolumn{2}{|c|}{0.0075 a.u} \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{|c|}{Computational parameters for planewave core-hole (CH) DFT approach} \\\\\n\t\t\t\\hline\n\t\t\t& - & B K-edge & N K-edge \\\\\n\t\t\t\\hline\n\t\t\tGGA ultrasoft  & - & B: 1$s^2$,2$s^2$,2$p^1$&B: [He]2$s^2$,2$p^1$\\\\\n\t\t\tpseudopotential   & - & N:[He]2$s^2$,2$p^3$&N:1$s^2$,2$s^2$,2$p^3$\\\\\n\t\t    configuration  & - & B$_\\mathrm{CH}$:1$s^1$,2$s^2$,2$p^4$&N$_\\mathrm{CH}$:1$s^1$,2$s^2$,2$p^4$\\\\\n\t\t\t\\hline\n\t\t\tplanewave cutoff  &  & \\multicolumn{2}{|c|}{25 Ry} \\\\\n\t\t\t\\hline\n\t\t\tsupercell dimensions & & \\multicolumn{2}{|c|}{15.024$\\times$15.024$\\times$16 $\\AA^3$}  \\\\\n\t\t\t\\hline\n\t\t\tDFT SCF $\\mathbf{k}$-point grid & &  \\multicolumn{2}{|c|}{$\\Gamma$-2$\\times$2$\\times$1}  \\\\\n\t\t\t\\hline\n\t\t\tFine $\\mathbf{k}$-point grid for XAS& &  \\multicolumn{2}{|c|}{$\\Gamma-$5$\\times$5$\\times$5}  \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\\end{center}\n\t\\caption{Computational parameters used for valence and core excitation simulations of 2D h-BN. To model the interaction with ionic cores, norm-conserving LDA pseudopotentials generated using the Troullier-Martins~\\cite{Troullier1991} scheme are employed within the LCAO code while planewave CH-DFT simulations use ultrasoft~\\cite{Vanderbilt1990} GGA pseudopotentials previously benchmarked for h-BN XAS computations~\\cite{Huber2015} using the DFT-XCH~\\cite{Prendergast2006,Prendergast2009a} approach.  $\\Gamma$-centered $\\mathbf{k}$-point grids  of the dimensions listed are used to converge different aspects of the simulations. The LCAO basis set is indicated using $nl$-$\\zeta$ notation where $n,l$ are principal and azimuthal quantum numbers respectively and $\\zeta$ is the number of functions of each $nl$ type. Additional $ghost$ basis functions in the vacuum region surrounding the 2D h-BN layer within the LCAO approach are also indicated.}\\label{hbncomp}\n\\end{table}\n\\subsubsection{Valence optical response in monolayer hexagonal BN}\nThe final material system considered in this article is monolayer hexagonal-BN (h-BN) which is a paradigmatic two-dimensional (2D) material with characteristically different screening properties compared to bulk solids~\\cite{Thygesen2017a}. Both low energy valence excitations in the UV and high energy core excitations in the soft X-ray range are considered in this context. In 2D materials the underscreening of the Coulomb interaction leads to strong exciton binding on the order of  1 eV even when the fundamental band gap is not very large~\\cite{Thygesen2017a}. According to recent first-principles many-body perturbation theory (MBPT) results based on the GW+BSE\\cite{Hybertsen1986,Rohlfing2000} approach,  2D h-BN features a $\\sim$7.77 eV gap at the K point in the Brillouin zone (BZ) along with a large exciton binding energy of $2.19$ eV~\\cite{Ferreira2018}. Because of its simple structure, chemical composition and interesting excitonic response, 2D h-BN represents a good test platform to investigate the performance of GKS-VG-RT-TDDFT with regards to describing the optical properties of 2D materials. In figure~\\ref{rt-hbn}(a), the structure of the unitcell used in the LCAO VG-RT-TDDFT simulations is shown. Because 2D systems are surrounded on either side by vacuum, within LCAO schemes, additional basis functions that are not centered on any specific atom but cover the region near the material are often necessary to adequately describe the exponential decay of the electron density into the vacuum region~\\cite{Paier2009}. In this work four layers of such \\textit{ghost} orbitals of B-$2s$ character are introduced on either side of the h-BN layer. A vacuum region of 40\\AA~is used to separate neighboring h-BN periodic images along the direction normal to the layer. Since HFX is constructed in real-space, any nonlocal exchange interaction between periodic images along the normal is trivially excluded within the current scheme. Other computational parameters employed to model 2D h-BN are listed in table~\\ref{hbncomp}. \nThe LRCH functional used in this instance is constructed according to the following rationale: Firstly, as discussed previously by Huser et al~\\cite{Huser2013}, the long-wavelength (q$_{||}$$\\rightarrow$0) dielectric constant in 2D materials tends to unity. This limit is adopted here setting $\\epsilon_{\\infty}$=1 which therefore implies ($\\alpha + \\beta$)=1 or 100\\% HFX in the long-range. Second, short-range HFX fraction is set to zero, i.e., $\\alpha$=0. This choice, determined through trial and error to optimize intermediate-range exchange, deviates from the heuristic value of $\\alpha$=0.2 employed above in bulk solids. The range-separation parameter $\\omega$ is then tuned to approximate the fundamental band gap in 2D h-BN predicted by GW calculations as listed in table~\\ref{bgap}.  In figure~\\ref{rt-hbn}(d) the time-dependent current induced in 2D h-BN for a small delta-function electric-field at $t$=0, polarized in the plane of the monolayer is shown at the RT-LDA and RT-LRCH levels of theory. RT-LRCH exhibits clear differences from RT-LDA in the form of stronger current oscillations that do not show significant dephasing. In the frequency-domain this manifests as intense excitonic peaks in the corresponding $\\epsilon_2(\\omega)$ predicted by RT-LRCH that are in reasonably close agreement with recent GW-BSE results (see figure~\\ref{rt-hbn}(c))~\\cite{Ferreira2018}. In contrast $\\epsilon_2(\\omega)$ from RT-LDA closely resembles the lineshape obtained at the level of the IPA completely lacking any exciton binding. Note that in figure~\\ref{rt-hbn}(c), the IPA $\\epsilon_2(\\omega)$ from employing the LRCH functional is compared against the GW-RPA result from reference~\\citenum{Ferreira2018}, also showing very good agreement.  Therefore, the parameters chosen for the LRCH functional employed in this instance simultaneously approximate MBPT results for the band gap, the IPA response and excitonic effects though VG-RT-TDDFT.  \n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.5]{Fig7.pdf}\n\t\\caption{(a,b) Time dependence of the current induced in monolayer h-BN (2D h-BN) by a 0.001 a.u in-plane-polarized delta-function electric field pulse applied at time $t$=0.  Results obtained at the RT-LDA, RT-LRCH and RT-GH levels are shown for the case of B-$1s$ core-states (a) and N-$1s$ core-states (b) included in the valence. (c) Imaginary part of the frequency dependent linear dielectric function $\\epsilon_2(\\omega)$ corresponding to the time-dependent currents shown in (a,b) is plotted at the B (left) and N (right) K-edge energies.}\n\t\\label{rt-bknk}\n\\end{figure}\n\\subsubsection{Core-excitations in monolayer hexagonal-BN}\nHaving analyzed the performance of GKS VG-RT-TDDFT for valence excitations in both bulk and 2D solids the extension to high energy core-excitations in the soft X-ray energy range is considered next. A comprehensive exploration of core-level spectra in a range of solid-state materials is not undertaken here. Instead, 2D h-BN is employed as a paradigmatic test system to highlight some differences between valence- and core-excitations in the context of VG-RT-TDDFT. Note that the performance of LCAO based VG-RT-TDDFT with respect to describing core-excitations in solids at the ALDA level has been previously investigated in comparison with established FP-LAPW~\\cite{Dewhurst2004} methods showing satisfactory agreement~\\cite{Pemmaraju2018}. The following discussion therefore proceeds from that baseline. Computational parameters relevant to core-excitation calculations are also listed in table~\\ref{hbncomp}. In figures~\\ref{rt-bknk}(a,b) the time-dependent current obtained for an in-plane-polarized delta-function electric field perturbation at time $t$=0 is shown for two separate simulations one including B 1$s$ core-states (figure~\\ref{rt-bknk}(a))  and the  other including N 1$s$ core-states (figure~\\ref{rt-bknk}(b)) in the description.  Three classes of functionals namely RT-LDA, RT-LRCH and RT-GH are considered, the parameters for the latter two being listed in table~\\ref{prsh}. Note that the 1$s$ core electrons are explicitly included in the simulation on the same footing as the 2$s$ and 2$p$ electrons mimicking an all-electron description~\\cite{Pemmaraju2018}.  In general, excitations from deep-lying core-levels introduce high-frequency modulations of the current superimposed on a slowly varying valence background the latter roughly resembling the current profile obtained if no core-states are included. Insets within figures~\\ref{rt-bknk}(a,b) show a magnified view of the current where the core-excitation induced fast oscillations are clearly apparent. Furthermore, the frequency of the fast oscillations is higher for the N K-edge case than the B K-edge case as the former is much higher in energy. \n\\begin{table}[htbp]\n\t\\begin{center}\n\t\t\\begin{tabular}{ |c|c|c|c|c| }\n\t\t\t\\hline\n\t\t\t& LDA & LRCH & GH & Expt.~\\cite{Trehan1990} \\\\\n\t\t\t\\hline\n\t\t\tB $1s$ & 175.7 & 177.7 & 192.5 & 191 \\\\\n\t\t\t\\hline\n\t\t\tN $1s$ & 376.9 & 378.9 & 400.5 & 398 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{GKS single-particle eigenvalues in eV of the B and N $1s$ core levels obtained by employing the LDA, LRCH and GH functionals (see table~\\ref{prsh} for functional parameters) are shown. Experimental XPS core-level binding energies in eV from reference~\\cite{Trehan1990} are shown in the last column}\\label{xps}\n\\end{table}\nThe in-plane component of $\\epsilon_2(\\omega)$ obtained from the time-dependent current is plotted in figures~\\ref{rt-bknk}(c) for the B and N K-edges. It is seen that the RT-LRCH result is very similar to the RT-LDA result showing only small changes in predicted peak intensities and no significant additional exciton binding. Note also that while ALDA is expected to underestimate excitation energies relative to experiment, the absolute energies for the onset of K-edge absorption from RT-LRCH are very similar to RT-LDA.  The absolute GKS eigenvalues relative to the vacuum level, of the N and B 1$s$ core states obtained from groundstate LDA and LRCH DFT are shown in table~\\ref{xps}. Clearly, also at the single-particle level, both the LDA and LRCH functionals underestimate the binding energy of 1$s$ core-states relative to X-ray photoemission spectroscopy (XPS) observations~\\cite{Trehan1990}. Therefore the LRCH functional that performs well for valence band gaps and exciton binding essentially fails for core-excitations. This is not very surprising given that excitonic effects involving core- and valence-excitations operate on somewhat different length-scales~\\cite{Shirley2006,Song2008,Besley2009,Imamura2015}. In core-excitons the highly-localized nature of the core-hole renders short-range screening in its immediate neighborhood very important~\\cite{Shirley2006} requiring significant amounts of short-range nonlocal exchange within a GKS description~\\cite{Song2008,Besley2009,Imamura2015}. The LRCH functional used in this study features zero HFX in the short-range and therefore is unable to produce significant core-exciton binding. The above phenomenology is already known from molecular core-level spectroscopy using TDDFT where specially designed  short-range-corrected hybrid functionals~\\cite{Besley2009} or multiply-range-separated functionals~\\cite{Song2008,Imamura2015} have been proposed as a means of improving absolute edge energies in X-ray absorption spectra (XAS). The situation nevertheless seems somewhat more severe in the case of solids. The bound unoccupied single-particle states that are generally responsible for near-edge XAS lineshapes in small molecular systems form discrete well-spaced energy levels regardless of the DFT functional employed and electron-hole attraction effects in small molecules usually manifest as energy shifts but without involving large spectral intensity changes. Therefore, in molecules, LRCH or global hybrid functionals useful for predicting valence electronic structure are also known to usually reproduce satisfactory XAS spectral lineshapes but with the absolute edge energies having to be offset rigidly on the order of $\\sim$10 eV to match experiment~\\cite{Zhang2015,Petrenko2008,Attar2017}. In solids on the other hand, the conduction band is a continuum of states and strongly bound core- or valence-excitons can lead to qualitatively different spectral lineshapes featuring large near-edge intensity changes and energy shifts with short-range HFX being important to capture either effect.   \n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale=0.43]{Fig8.pdf}\n\t\\caption{B (left) and N (right) K-edge XAS spectra in monolayer h-BN, obtained from LCAO based VG-RT-TDDFT using a global hybrid functional and a standard planewave CH-DFT method are compared. CH-DFT spectra which do not have an absolute energy scale, are shifted rigidly to match the first peak position from VG-RT-TDDFT in each case.}\n\t\\label{xch}\n\\end{figure}\nSimulating core- and valence-excitation spectra simultaneously, as would be relevant for pump-probe~\\cite{Schultze2014} or nonlinear~\\cite{Zhang2015} X-ray spectroscopies, requires GKS functionals that balance both long- and short-range HFX and efforts in this direction have been made in the context of molecules through modified range-separation schemes~\\cite{Song2008,Imamura2015}. Extension of the present solid-state LCAO framework to such functionals will be considered in the future. For the purposes of this article a global hybrid (GH) functional that features 50\\% of HFX is considered instead as a way of striking a compromise between long- and short-range HFX. The valence band gap and core-level single-particle eigenvalues obtained using this GH are reported in tables~\\ref{bgap} and~\\ref{xps} respectively. The valence band gap is overestimated by $\\sim$16\\% but the core-level eigenvalues are now in much better agreement with XPS data. The time-dependent current for an in-plane-polarized electric field perturbation and corresponding $\\epsilon_2(\\omega)$ obtained at the RT-GH level are plotted in figure~\\ref{rt-bknk}(a,b) for both the B and N K-edges. It is apparent from figure~\\ref{rt-bknk}(c) that RT-GH predicts X-ray absorption onsets that are higher in energy and furthermore, a strong increase in peak intensity derived from excitonic effects near the absorption onset is seen, markedly changing the overall spectral shape relative to RT-LDA and RT-LRCH.  To assess the overall performance of RT-GH TDDFT for XAS, the total $\\epsilon_2(\\omega)$ including both in-plane and out-of-plane core-level response to small delta-function electric field perturbations is calculated at both the B and N K-edges and plotted in figure~\\ref{xch}. The VG-RT-TDDFT results in this instance are compared against those from an established planewave based core-hole (CH) DFT approach for XAS simulations known as the XCH method~\\cite{Prendergast2006,Prendergast2009a}. Solid-state CH-DFT approaches~\\cite{Taillefumier2002,Prendergast2006,Prendergast2009a} which are widely used in core-level spectroscopy modeling employ modified  pseudopotentials to mimic core-excited atoms within a periodic supercell framework and rely on a DFT self-consistent-field procedure to generate Kohn-Sham eigenstates that are subsequently mapped onto XAS. Computational parameters relevant to CH-DFT are reported in table~\\ref{hbncomp}. As is apparent from~figure~\\ref{xch}, good agreement between LCAO based RT-GH TDDFT and planewave based CH-DFT is demonstrated for the spectral lineshapes at both the B and N K-edges. Therefore at the RT-GH level of theory, consistent improvements in both core-level single particle energies (table~\\ref{xps}) and excitonic effects in XAS (figures~\\ref{rt-bknk},\\ref{xch}) are realized demonstrating the potential utility of GKS VG-RT-TDDFT also as a method for soft X-ray core-level spectroscopy prediction in solids.  \n\n\n\\section{Conclusions and Outlook}\\label{conc}\nIn conclusion, an LCAO framework for enabling velocity-gauge RT-TDDFT simulations in solids at the generalized Kohn-Sham (GKS) level of theory has been presented. Groundstate band structures and linear optical response properties, the latter from velocity-gauge (VG) time-domain simulations, were calculated using the LCAO GKS framework and assessed in comparison with planewave basis set results in a variety of solid-state materials such as Si, LiF and 2D h-BN that feature very different dielectric properties. Both valence single-particle and optical-excitation energies predicted by the LCAO scheme were shown to be in agreement with planewave results and\/or experiment with deviations being limited to within a few percent. Furthermore, LCAO core-level spectra obtained at the GKS level of theory were shown to match predictions from established planewave based core-hole DFT techniques. The developments discussed here therefore represent a promising avenue towards routine GKS-RT-TDDFT simulations of excitonic effects in solids with relevance to a wide range of spectroscopic applications.   \n\n\n\\section*{Acknowledgements}\nThe author is grateful to Kazuhiro Yabana and Shunsuke Sato for useful discussions. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515 through TIMES at SLAC. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. Core-hole DFT simulations in this work made use of the Shirley-XAS method developed and provided by The Molecular Foundry at Lawrence Berkeley National Laboratory, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\baselineskip 14pt\n\n\nCharts are oriented labeled graphs\nin a disk \nwith three kinds of vertices\ncalled black vertices, crossings,\nand white vertices (see Section~\\ref{s:Prel} for the precise definition of charts, \nblack vertices, crossings, and white vertices).\nFrom a chart, we can construct an oriented closed surface \nembedded in $4$-space ${\\Bbb R}^4$ \n (see \\cite[Chapter 14, Chapter 18 and Chapter 23]{BraidBook}). \nA C-move \nis a local modification between two charts\nin a disk (see Section~\\ref{s:Prel}).\nA C-move between two charts induces \nan ambient isotopy between oriented closed surfaces \ncorresponding to the two charts.\nTwo charts are said to be {\\it C-move equivalent} \nif there exists\na finite sequence of C-moves \nwhich modifies one of the two charts \nto the other.\n\n\n\nWe will work in the PL or smooth category. \nAll submanifolds are assumed to be locally flat.\nA {\\it surface link} is a closed surface embedded in 4-space ${\\Bbb R}^4$. \nA {\\it $2$-link} is a surface link each of whose connected component is a $2$-sphere.\nA {\\it $2$-knot}\nis a surface link which is a $2$-sphere.\nAn orientable surface link is called a \n{\\it ribbon surface link}\nif there exists an immersion of a 3-manifold $M$\ninto ${\\Bbb R}^4$ sending the boundary of $M$ onto the surface link\nsuch that each connected component of $M$ is a handlebody\nand its singularity\nconsists of ribbon singularities,\nhere a ribbon singularity\nis a disk in the image of $M$\nwhose pre-image consists of \ntwo disks;\none of the two disks is a proper disk of $M$ \nand\nthe other is a disk in the interior of $M$.\nIn the words of charts,\na ribbon surface link is\na surface link corresponding to a {\\it ribbon chart}, \na chart C-move equivalent to \na chart\nwithout white vertices \\cite{BraidThree}.\nA chart is called a {\\it $2$-link chart}\nif a surface link corresponding to the chart is a $2$-link.\n\nIn this paper, \nwe denote the closure, the interior, \nthe boundary, and the complement of $(...)$ by \n$Cl(...)$, Int$(...)$, \n$\\partial(...)$, $(...)^c$ \nrespectively.\nAlso for a finite set $X$, \nthe notation $|X|$ denotes \nthe number of elements in $X$.\n\nAt the end of this paper \nthere is the index of new words and notations \nintroduced in this paper.\n\nKamada showed that \nany $3$-chart is a ribbon chart \n\\cite{BraidThree}.\nKamada's result was extended by Nagase and Hirota:\nAny $4$-chart with at most one crossing\nis a ribbon chart \\cite{NH}.\nWe showed that any $n$-chart with at most one crossing is a ribbon chart\n\\cite{OneCrossing}.\nWe also showed that any $2$-link chart \nwith at most two crossings\n is a ribbon chart  \\cite{TwoCrossingI},  \n \\cite{TwoCrossingII}.\n\nLet $\\Gamma$ be a chart.\nFor each label $m$, we define\n$$\\Gamma_m=\\text{ \nthe union of \nall the edges of label $m$ and \ntheir vertices in }\\Gamma.$$\n\n\n\nIn this paper\nwe investigate \nthe structure of minimal charts with two crossings\n(see Section~\\ref{s:Prel}\nfor the precise definition of a minimal chart),\nand give us an enumeration of the charts \nwith two crossings.\nThe enumeration is much complicated \nthan the one of \n$2$-bridge links in ${\\Bbb R}^3$, of course. \nWe enumerate charts with two crossings as follows\n(see Section~\\ref{s:NormalForm} and\n\\cite{StI}):\nFor any minimal $n$-chart $\\Gamma$ \nwith two crossings in a disk $D^2$,\nthere exist two labels \n$1\\le \\alpha<\\beta\\le n-1$ such that \n $\\Gamma_\\alpha$ and $\\Gamma_\\beta$\ncontain cycles $C_\\alpha$ and $C_\\beta$\nwith $C_\\alpha\\cap C_\\beta$ the two crossings\nand that\nfor any label $k$ with $k<\\alpha$ or $\\beta<k$,\nthe set $\\Gamma_k$ does not contain a white vertex.\nIf $\\Gamma_\\alpha$ or $\\Gamma_\\beta$\ncontains at least three white vertices,\nthen\nafter shifting all the free edges and simple hoops\ninto a regular neighbourhood of $\\partial D^2$\nby applying C-I-M1 moves and C-I-M2 moves, \nwe can find \nan annulus $A$ \ncontaining all the white vertices of $\\Gamma$ \nbut not intersecting \nany hoops nor free edge\nsuch that (see Fig.~\\ref{fig01}(a))\n\\begin{enumerate}\n\\item[(1)] \neach connected component of $Cl(D^2-A)$ \ncontains a crossing,\n\\item[(2)]\n$\\Gamma\\cap \\partial A=\n(C_\\alpha\\cup C_\\beta)\\cap \\partial A$, and \n$\\Gamma\\cap \\partial A$ \nconsists of eight points.\n\\end{enumerate}\nWe can show \nthe annulus $A$ can be split \ninto \nmutually disjoint four disks \n$D_1,D_2,D_3,D_4$ and \nmutually disjoint four disks \n$E_1,E_2,E_3,E_4$ such that\n\\begin{enumerate}\n\\item[(3)] \nfor each $i=1,3$ (resp. $i=2,4$)\nthe tangle $(\\Gamma\\cap D_i,D_i)$ \nis an IO-tangle of label $\\alpha$ \n(resp. label $\\beta$) \n(see \\cite[Theorem 1.3]{MinimalChart}, Theorem~\\ref{StIITheorem2},\nTheorem~\\ref{StIITheorem3} and Theorem~\\ref{StIITheorem4}),\n\\item[(4)] \nfor each $i=1,2,3,4$, \nthe tangle $(\\Gamma\\cap E_i,E_i)$\nis a net-tangle with $\\Gamma\\cap E_i\\subset \\cup^{\\beta-1}_{j=\\alpha+1}\\Gamma_j$\nas shown in Fig.~\\ref{fig01}(b)\n(see \\cite[Theorem~1.1 and \nTheorem~1.2]{StI}).\n\\end{enumerate}\nWe count the number of edges between edges with black vertices\nin Fig.~\\ref{fig01}(b) \nto enumerate charts with two crossings.\nAs important results, \nfrom the enumeration \nwe can calculate the fundamental group of \nthe exterior of the surface link represented by $\\Gamma$, \nand the braid monodromy of the surface braid \nrepresented by $\\Gamma$.\n\nThe purpose of our research\nis a classification of surface links by using charts.\nHowever we do not classify ribbon surface links.\nIt is known that any chart $\\Gamma$ decomposes a ribbon chart and a main part of a chart, ${\\rm Main}(\\Gamma)$\nby C-I-M1 moves and C-I-M2 moves.\nWe want to classify charts (modulo ribbon charts)\nup to C-move equivalent\nand we want to find a new minimal chart.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig01.pdf}\n\\end{center}\n\\caption{\\label{fig01}\n(b) A tangle $(\\Gamma\\cap E_i,E_i)$ with $\\Gamma\\cap E_i\\subset \\Gamma_2\\cup \\Gamma_3\\cup \\Gamma_4$\nfor the case $\\alpha=1$ and $\\beta=5$,\nhere all the free edges and simple hoops are in a regular neighbourhood of $\\partial D^2$.\n}\n\\end{figure}\n\n\n\nA closed edge without vertices is called\na {\\it hoop}. \nAn edge with two black vertices is called \na {\\it free edge}.\n\n\n\nLet $\\Gamma$ be a chart\ncontaining a white vertex. \nDefine\n\n$\\alpha(\\Gamma)=\\min\\{~i~|~\\Gamma_i$ \ncontains a white vertex$\\}$,\n\n$\\beta(\\Gamma)=\\max\\{~i~|~\\Gamma_i$ \ncontains a white vertex$\\}$.\n\n\n\nA chart with exactly two crossings is called\na {\\it $2$-crossing chart}.\n\nThe following lemma is \nthe first step to investigate \na 2-crossing minimal chart \n(see Section~\\ref{s:Prel} for \nthe definition of minimal charts).\n\n\\begin{lem}\n\\label{StIILemma1} \n{\\rm$($cf. \\cite[Lemma~6.3]{TwoCrossingII}$)$}\nLet $\\Gamma$ be a $2$-crossing minimal chart in a disk $D^2$.\nSet $\\alpha=\\alpha(\\Gamma),\n\\beta=\\beta(\\Gamma)$.\nThen there exists a minimal chart $\\Gamma'$ obtained from $\\Gamma$ by applying C-I-M1 moves and C-I-M2 moves\nsatisfying the following conditions.\n\\begin{enumerate}\n\\item[{\\rm (a)}]\nThere exist two cycles \n$C_\\alpha,C_\\beta$ with $C_\\alpha\\subset\\Gamma_\\alpha',C_\\beta\\subset\\Gamma_\\beta'$ such that \n$C_\\alpha\\cap C_\\beta$ consists of the two crossings.\n\\item[{\\rm (b)}]\nThere exists an annulus $A$ containing all the white vertices of $\\Gamma'$ \nbut not intersecting hoops nor free edges.\n\\item[{\\rm (c)}]\nEach connected component of $Cl(D^2-A)$ contains a crossing.\n\\item[{\\rm (d)}]\n $\\Gamma'\\cap \\partial A=(C_\\alpha\\cup C_\\beta)\\cap \\partial A$,\n$\\Gamma'\\cap \\partial A$ consists of eight points.\n\\item[{\\rm (e)}] \n$\\Gamma_\\alpha'\\cap A$ \nconsists of \ntwo connected components \n$X_1,X_3$ \nseparated by $C_\\beta$. \n\\item[{\\rm (f)}]\n$\\Gamma_\\beta'\\cap A$ \nconsists of \ntwo connected components \n$X_2,X_4$ \nseparated by $C_\\alpha$. \n\\end{enumerate}\n\\end{lem}\n\nLet $\\Gamma$ be a chart, and\n$D$ a disk.\nThe pair $(\\Gamma\\cap D,D)$ is called \na {\\it tangle} provided that \n\\begin{enumerate}\n\\item[(i)]\n$\\partial D$ does not contain\nany white vertices, \nblack vertices\nnor crossings of $\\Gamma$, \n\\item[(ii)]\nif an edge of $\\Gamma$ intersects $\\partial D$, then the edge intersects $\\partial D$ transversely, and\n\\item[(iii)] $\\Gamma\\cap D\\not=\\emptyset$.\n\\end{enumerate}\n\n\n\nLet $\\Gamma$ be a chart, \nand $E$ a disk. \nSuppose that an edge $e$ of $\\Gamma$ \ntransversely intersects $\\partial E$.\nLet $p$ be a point in \n$e\\cap \\partial E$, \nand \n$N$ a regular neighbourhood of $p$.\nThen the orientation of $e$\ninduces the one of the arc $e\\cap N$.\nThe edge $e$ is said to be \n{\\it locally inward} \n(resp. {\\it locally outward}) \nat $p$ \nwith respect to $E$\nif the oriented arc $e\\cap N$ is \noriented\nfrom a point outside \n(resp. inside) $E$ \nto a point inside \n(resp. outside) $E$.\nWe often say that \n$e$ is locally inward (resp. outward) at $p$ \ninstead of saying that\n$e$ is locally inward (resp. outward) at $p$ \n{\\it with respect to $E$}, \nif there is no confusion.\n\nAn edge of a chart $\\Gamma$ is called \na {\\it terminal edge}\nif it contains a white vertex and \na black vertex.\n\n\nFor a simple arc $X$, we set\\\\\n\\ \\ \\ $\n\\begin{array}{rl}\n\\partial X&= \\text{ the set of its two endpoints, \nand}\\\\\n{\\rm Int}~X&=~X-\\partial X.\n\\end{array}\n$\n\n\n\nLet $\\Gamma$ be a chart, and \n$m$ a label of the chart.\nA tangle $(\\Gamma\\cap D,D)$\nis called an \n{\\it IO-tangle of label $m$}\nprovided that\n\\begin{enumerate}\n\\item[(i)]\nno terminal edge nor free edge intersects $\\partial D$, \n\\item[(ii)]\nthere exists a label $k$ with $|m-k|=1$ and\n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$,\n\\item[(iii)]\nthere exist two arcs $L_I,L_O$ on $\\partial D$\nwith $L_I\\cap L_O=\\partial L_I\n=\\partial L_O=\\Gamma_m\\cap \\partial D$,\n\\item[(iv)]\nfor any point \n$p\\in\\Gamma\\cap$~Int~$L_I$,\nthere exists an edge \nof label $k$ \nlocally inward at $p$, \nand\\\\\nfor any point \n$p\\in\\Gamma\\cap$~Int~$L_O$,\nthere exists an edge \nof label $k$ \nlocally outward \nat $p$.\n\\end{enumerate}\nThe pair $(L_I,L_O)$ is called a {\\it boundary IO-arc pair} of the tangle.\nAn IO-tangle of label $m$ is said to be {\\it simple}\nif all the terminal edge in $D$ is of label $m$.\n\n\nLet $X_1,X_2,X_3,X_4$ be \nthe connected components in \nLemma~\\ref{StIILemma1} above.\nFor each $i=1,2,3,4$,\nlet $D_i^*$ be \na regular neighbourhood of \nthe SC-closure $SC(Cl(X_i))$ in \nthe annulus $A$ \n(see Section~\\ref{s:ProofLemma1} \nfor the definition of \nSC-closures).\nThen the tangle $(\\Gamma'\\cap D_i^*,D_i^*)$\nis called a {\\it fundamental tangle} of \nthe $2$-crossing chart.\n\nWe obtain the following theorem for fundamental tangles.\n\n\\begin{thm}\n\\label{StIITheorem2} \nLet $\\Gamma$ be \na $2$-crossing minimal chart, \nand $(\\Gamma\\cap D,D)$ \na fundamental tangle. \nIf $D$ contains \nat least two white vertices,\nthen the tangle\nis a simple IO-tangle.\n\\end{thm}\n\n\nA simple arc $\\ell$ in $E$ \nis called a {\\it proper} arc \nprovided that \n$\\ell\\cap \\partial E=\\partial\\ell$.\n\nNext we decompose a simple IO-tangle.\nLet $\\Gamma$ be a chart, and \n$m$ a label of $\\Gamma$.\nAn IO-tangle \n$(\\Gamma\\cap D,D)$ of label $m$ \nis called \na {\\it Type-I elementary IO-tangle \nof label $m$} \nprovided that\n\\begin{enumerate}\n\\item[(I)] \nthere exists a disk $E$ \ncontaining \nall the white vertices in $D$ \nwith $\\partial E\\subset\\Gamma_m\\cap D$.\n\\end{enumerate}\nAn IO-tangle \n$(\\Gamma\\cap D,D)$ of label $m$ \nis called \na {\\it Type-II elementary IO-tangle \nof label $m$} \nprovided that\n\\begin{enumerate}\n\\item[(II)]\n$\\Gamma_m\\cap D$ contains no cycle,\nbut there exists \na simple proper arc $L_D$ of $D$ \ncontaining \nall the white vertices in $D$ \nwith $L_D\\subset\\Gamma_m\\cap D$\n(hence \nthe arc $L_D$ connects\nthe two point \n$\\Gamma_m\\cap\\partial D$).\n\\end{enumerate}\nLet $D^\\dagger$ be a disk. \nA tangle \n$(\\Gamma\\cap D^\\dagger,D^\\dagger)$\nis called a {\\it trivial tangle} \nof label $m$\nif $\\Gamma\\cap D^\\dagger$ \nis a proper arc of $D^\\dagger$ \ncontained in the interior of \nan edge of label $m$. \nWe consider the trivial tangle \nas a Type-II elementary IO-tangle.\nNow Type-I elementary IO-tangles and\nType-II elementary IO-tangles \nare called {\\it elementary IO-tangles}.\n\nLet $\\Gamma$ be a chart, and \n$m$ a label of the chart.\nLet $(\\Gamma\\cap D,D)$ be \nan IO-tangle of label $m$.\nThe tangle is said to have an \n{\\it elementary IO-tangle decomposition}\n$(\\Gamma\\cap D_1,D_1)\\#(\\Gamma\\cap D_2,D_2)\\#\n\\cdots\\#(\\Gamma\\cap D_{2s+1},D_{2s+1})$\nprovided that \n(see Fig.~\\ref{fig14})\n\\begin{enumerate}\n\\item[(i)]\nfor each $i=1,2,\\cdots,2s+1$,\\\\\nif $i$ is even, then\n$(\\Gamma\\cap D_i,D_i)$ is \na Type-I elementary IO-tangle of label $m$,\\\\\nelse \n$(\\Gamma\\cap D_i,D_i)$ is \na Type-II elementary IO-tangle of label $m$,\n\\item[(ii)]\n$D=\\cup_{i=1}^{2s+1}D_i$,\n\\item[(iii)]\nthere exist $2s$ proper arcs \n$\\ell_1,\\ell_2,\\cdots,\\ell_{2s}$ \nof $D$\nsuch that \nfor each $i=1,2,\\cdots,2s$,\n$\\ell_i\\cap \\Gamma\n=\\ell_i\\cap \\Gamma_m\n=\\text{a point}$, \n\\item[(iv)] \nfor each $1\\le i<j\\le 2s+1$,\\\\\n$\nD_i\\cap D_j=\n\\left\\{\n\\begin{array}{lll}\n\\ell_i&\\ \\ \\ &\\text{if $j=i+1$},\\\\ \n\\emptyset& &\\text{otherwise}.\n\\end{array}\n\\right.\n$\n\\end{enumerate}\n\n\\begin{thm}\n\\label{StIITheorem3} \nLet $\\Gamma$ be \na minimal chart, and\n$m$ a label of the chart. \nThen any simple IO-tangle \n$(\\Gamma\\cap D,D)$\nof label $m$ \nhas an elementary IO-tangle decomposition.\n\\end{thm}\n\nLet $\\Gamma$ be a chart.\nIf an edge is oriented \nfrom a vertex $v_1$ \nto a vertex $v_2$,\nthen \nedge is said to be \n{\\it inward} at the vertex $v_2$, \nor \n{\\it outward} at the vertex $v_1$.\n\nLet $\\Gamma$ be a chart, and $m$ a label of $\\Gamma$. \nLet $k$ be a positive integer.\nA Type-I elementary IO-tangle \n$(\\Gamma\\cap D,D)$ of label $m$ \nwith a boundary IO-arc pair $(L_I,L_O)$ \nis said to be of {\\it Type-I$_k$}\nprovided that (see Fig.~\\ref{fig02})\n\\begin{enumerate}\n\\item[(i)] \nthere exists a disk $E$ \ncontaining \nall the white vertices in $D$ \nwith \n$\\partial E\\subset\\Gamma_m\\cap D$, \nbut \nInt~$E$ does not contain a white vertex,\n\\item[(ii)] \nthe closure of $\\Gamma_m\\cap (D-E)$ \nconsists of \ntwo terminal edges $\\tau_I,\\tau_O$ \nand two arcs such that \none of the two arcs is contained in \nan edge $e_I$ inward at \na vertex in $\\partial E$, and \nthe other in \nan edge $e_O$ outward at \na vertex in $\\partial E$,\n\n\\item[(iii)] \n$\\partial E-(e_I\\cup e_O)$ \nconsists of\ntwo components $J_I',J_O'$ with \n$J_I'\\cap \\tau_I\\neq\\emptyset,\nJ_O'\\cap \\tau_O\\neq\\emptyset$\nsuch that\neach of $J_I'$ and $J_O'$ contains \nexactly $k$ white vertices,\n\\item[(iv)] \none of the two components of \n$J_I'-\\tau_I$ \ndoes not contain a white vertex, \nand\\\\\none of the two components of \n$J_O'-\\tau_O$ \ndoes not contain a white vertex.\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig02.pdf}\n\\end{center}\n\\caption{\\label{fig02}\nType-I$_3$ elementary IO-tangles \nof label $m$.\nThe thick arcs are of label $m$.}\n\\end{figure}\n\n\nThe following theorem simplifies \nthe structure of \nfundamental tangles \nfor a 2-crossing minimal chart.\n\n\\begin{thm}\\label{StIITheorem4} \nLet $\\Gamma$ be a minimal chart, and \n$m$ a label of $\\Gamma$. \nThen \nany Type-I elementary IO-tangle \nof label $m$ is of Type-I$_k$ \nfor some integer $k$.\n\\end{thm}\n\nOur paper is organized as follows:\nIn Section~\\ref{s:Prel}, \nwe introduce \nthe definition of charts and its related words.\nIn Section~\\ref{s:ProofLemma1},\nwe shall prove Lemma~\\ref{StIILemma1}.\nIn Section~\\ref{s:Th2Th3},\nwe shall prove Theorem~\\ref{StIITheorem2}\nand Theorem~\\ref{StIITheorem3}.\nIn Section~\\ref{s:OneWayPath},\nwe investigate a path of label $m$\ncalled a one-way path.\nIn Section~\\ref{s:Th4},\nwe shall prove Theorem~\\ref{StIITheorem4}.\nIn Section~\\ref{s:Indices}, \nwe define indices for simple IO-tangles and \nN-tangles.\nIn Section~\\ref{s:NormalForm}, \nwe define a normal form for \n$2$-crossing minimal charts.\n\n\n\n\n\\section{Preliminaries}\n\\label{s:Prel}\n\nIn this section, \nwe introduce \nthe definition of charts and its related words.\n\nLet $n$ be a positive integer.\nAn $n$-{\\it chart}  \n(a braid chart of degree $n$ \\cite{KS}\nor a surface braid chart of degree $n$ \\cite{BraidBook}) \nis \nan oriented labeled graph in a disk,\nwhich may be empty \nor\nhave closed edges without vertices\nsatisfying the following four conditions\n(see Fig.~\\ref{fig03}):\n\\begin{enumerate}\n\\item[(i)] \nEvery vertex has degree $1$, $4$, or $6$.\n\\item[(ii)] \nThe labels of edges are \nin $\\{1,2,\\dots,n-1\\}$.\n\\item[(iii)]\nIn a small neighborhood of\neach vertex of degree $6$,\nthere are six short arcs,\nthree consecutive arcs are\noriented inward \nand\nthe other three are outward,\nand\nthese six are labeled $i$ and $i+1$\nalternately for some $i$,\nwhere the orientation and label of\neach arc are inherited from\nthe edge containing the arc.\n\\item[(iv)]\nFor each vertex of degree $4$,\ndiagonal edges have the same label\nand\nare oriented coherently,\nand the labels $i$ and $j$ of\nthe diagonals satisfy $|i-j|>1$.\n\\end{enumerate}\nWe call a vertex of degree $1$ a {\\it black vertex},\na vertex of degree $4$ a {\\it crossing}, and \na vertex of degree $6$ a {\\it white vertex}\nrespectively.\nAmong six short arcs\nin a small neighborhood of\na white vertex,\na central arc of each three consecutive arcs\noriented inward (resp. outward) \nis called a   \n{\\it middle arc} at the white vertex\n(see Fig.~\\ref{fig03}(c)).\nFor each white vertex $v$, \nthere are two middle arcs at $v$ \nin a small neighborhood of $v$.\nAn edge $e$ is said to be \n{\\it middle at} a white vertex $v$ \nif it \ncontains a middle arc at $v$.\n\n\n\n\\begin{figure}[htb]\n\\includegraphics{fig03.pdf}\n\\caption{ \\label{fig03} (a) A black vertex. (b) A crossing. (c) A white vertex. \nEach arc with three transversal short arcs is a middle arc at the white vertex. }\n\\end{figure}\n\n\n\n\n\nNow {\\it C-moves} are local modifications \nof charts as shown in Fig.~\\ref{fig04}\n(cf. \\cite{KS}, \n\\cite{BraidBook} and \\cite{Tanaka}).\nWe often use C-I-M2 moves, C-I-M3 moves, C-II moves\nand C-III moves. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics{fig04.pdf}\n\\end{center}\n\\caption{ \\label{fig04} For the C-III move, the edge containing the black vertex does not contain a middle arc at\na white vertex in the left figure. }\n\\end{figure}\n\n\nLet $m$ be a label of a chart $\\Gamma$.\nA simple closed curve in $\\Gamma_m$ \nis called a {\\it ring}, \nif it contains a crossing \nbut does not contain \na white vertex nor a black vertex.\n\n\nLet $\\Gamma$ be a chart. \nLet $e_1$ and $e_2$ be edges of $\\Gamma$\nwhich connect two white vertices $w_1$ and $w_2$\nwhere possibly $w_1=w_2$.\nSuppose that \nthe union $e_1\\cup e_2$ bounds \nan open disk $U$.\nThen $Cl(U)$ \nis called \na {\\it bigon} of $\\Gamma$\nprovided that\nany edge containing $w_1$ or $w_2$ \ndoes not intersect the open disk $U$\n(see Fig.~\\ref{fig05}).\nNote that neither $e_1$ nor $e_2$ contains a crossing.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig05.pdf}\n\\end{center}\n\\caption{ \\label{fig05} }\n\\end{figure}\n\n\nLet $\\Gamma$ be a chart.\nLet \n$w(\\Gamma),~f(\\Gamma),~c(\\Gamma)$, \nand \n$b(\\Gamma)$ be \nthe number of white vertices, \nthe number of free edges, \nthe number of crossings, \nand \nthe number of bigons of $\\Gamma$\nrespectively.\nThe 4-tuple $(c(\\Gamma),w(\\Gamma),-f(\\Gamma),-b(\\Gamma))$ is called a \n{\\it $c$-complexity} of the chart $\\Gamma$.\nThe 4-tuple $(w(\\Gamma),c(\\Gamma),-f(\\Gamma),-b(\\Gamma))$ is called a \n{\\it $w$-complexity} of the chart $\\Gamma$.\nThe 3-tuple $(c(\\Gamma)+w(\\Gamma),-f(\\Gamma),-b(\\Gamma))$ is called a \n{\\it $cw$-complexity} of the chart $\\Gamma$\n(see \\cite{BraidThree} \nfor complexities of charts).\n\n\n\n\nA chart $\\Gamma$ is said to be \n{\\it $c$-minimal $($resp. $w$-minimal or $cw$-minimal$)$} if\nits $c$-complexity (resp. $w$-complexity or $cw$-complexity) is minimal among the charts \nwhich are C-move equivalent to \nthe chart $\\Gamma$\nwith respect to \nthe lexicographical order of the \n4-tuple (or 3-tuple) of the integers.\nIf a chart is $c$-minimal, $w$-minimal or $cw$-minimal, \nthen we say that the chart is {\\it minimal}\nin this paper.\n\n\n\n\nA hoop is said to be {\\it simple} \nif one of the complementary domains\nof the hoop\ndoes not contain any white vertices.\n\n\nFor any chart in a disk $D^2$\nwe can move free edges and simple hoops into \na regular neighbourhood of $\\partial D^2$ \nby C-I-M2 moves and ambient isotopies of $D^2$\nas shown in Fig.~\\ref{fig06}.\nEven during argument,\nif free edges or simple hoops appear, \nwe immediately move them \ninto a regular neighbourhood of $\\partial D^2$.\nThus we assume the following\n\\cite{OneCrossing}, \\cite[Assumption 1]{MinimalChart}:\n\n\\begin{assumption}\n\\label{AssumptionFreeEdge}\n{\\it For any chart in a disk $D^2$, \nall the free edges and \nsimple hoops  \nare in a regular neighbourhood of \n$\\partial D^2$.}\n\\end{assumption}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig06.pdf}\n\\end{center}\n\\caption{ \\label{fig06} }\n\\end{figure}\n\n\nLet $\\Gamma$ be a chart in $D^2$, and \n$X$ the union of all the free edges \nand simple hoops.\nNow $X$ is in a regular neighbourhood $N$ of \n$\\partial D^2$ in $D^2$ \nby Assumption~\\ref{AssumptionFreeEdge}.\nWe say that $\\Gamma$ is a chart with a {\\it brim $N$} in $D^2$.\nDefine \n$${\\rm Main}(\\Gamma)=\\Gamma-X.$$\nLet $\\widehat D=Cl(D^2-N)$. \nThen $\\Gamma\\cap\\widehat D=$Main$(\\Gamma)$.\nHence $(\\Gamma\\cap\\widehat D,\\widehat D)$ is \na tangle without free edges \nand simple hoops.\n\n\\begin{assumption}\n\\label{AssumptionFreeEdgeSimpleHoop}\n{\\it In this paper, our arguments are done \nin the disk $\\widehat D$, \notherwise mentioned.}\n\\end{assumption}\n\n\n\n\n\\begin{remark}\n\\label{Assumption0}\n{\\rm \n(\\cite[Remark 2.2]{MinimalChart})\n Let $\\Gamma$ be a minimal chart. \nThen we have \nthe following:\n\\begin{enumerate}\n\\item[(1)]\n{ If an edge of $\\Gamma$ contains a black vertex, \nthen it is a terminal edge or a free edge.}\n\\item[$(2)$]\nAny terminal edge of $\\Gamma$ contains a middle arc at its white vertex.\n\\item[$(3)$]\nEach complementary domain of\nany ring must contain \nat least one white vertex. \n\\end{enumerate}\n}\n\\end{remark}\n\n\n\n\nLet $E$ be a disk, and\n$\\ell_1,\\ell_2,\\ell_3$ three arcs on $\\partial E$\nsuch that each of $\\ell_1\\cap \\ell_2$ and $\\ell_2\\cap \\ell_3$ is one point and $\\ell_1\\cap \\ell_3=\\emptyset$\n(see Fig.~\\ref{fig07}(a)),\nsay $p=\\ell_1\\cap \\ell_2$,\n$q=\\ell_2\\cap \\ell_3$.\nLet $\\Gamma$ be a chart in a disk $D^2$.\nLet $e_1$ be a terminal edge of \n $\\Gamma$. \nA triplet $(e_1,e_2,e_3)$ of \nmutually different edges of $\\Gamma$\nis called \na {\\it consecutive triplet}\nif there exists\na continuous map $f$ from the disk $E$ \nto the disk $D^2$ such that (see Fig.~\\ref{fig07}(b) and (c))\n\\begin{enumerate}\n\\item[(i)] the map $f$ is injective on $E-\\{p,q\\}$,\n\\item[(ii)] \n$f(\\ell_3)$ is an arc in $e_3$, and $f({\\rm Int}~E)\\cap\\Gamma=\\emptyset$,\n$f(\\ell_1)=e_1$,\n$f(\\ell_2)=e_2$,\n\\item[(iii)]\n$f(p)$ and $f(q)$ are white vertices.\n\\end{enumerate}\nIf the label of $e_3$ is different\nfrom the one of $e_1$ \nthen the consecutive triplet is said to be\n{\\it admissible}.\n\n\n\n\\begin{remark}\n{\\rm Let $(e_1,e_2,e_3)$\nbe a consecutive triplet. \nSince $e_2$ is an edge of $\\Gamma$, \nthe edge $e_2$ MUST NOT contain a crossing.}\n\\end{remark}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig07.pdf}\n\\end{center}\n\\caption{ \\label{fig07} }\n\\end{figure}\n\n\n\\begin{lem}\n\\label{ConsecutiveTripletLemma} \n{\\rm [Consecutive Triplet Lemma]}\n{\\rm $($\\cite[Lemma 1.1]{OneCrossing}, \\cite[Lemma 3.1]{MinimalChart}$)$}\n{\\it Any consecutive triplet \nin a minimal chart is admissible.}\n\\end{lem}\n\n\n\nLet $\\Gamma$ be a chart.\nA tangle $(\\Gamma\\cap D,D)$ is \ncalled an {\\it NS-tangle of label $m$} (new significant tangle) \nprovided that\n\\begin{enumerate}\n\\item[(i)] if $i\\neq m$, \nthen $\\Gamma_i\\cap \\partial D$ is \nat most one point,\n\\item[(ii)] \n$\\Gamma\\cap D$ contains at least one white vertex, and \n\\item[(iii)]\nfor each label $i$, \nthe intersection $\\Gamma_i\\cap D$ contains \nat most one crossing.\n\\end{enumerate}\n\n\\begin{lem} \n{\\rm $($\\cite[Theorem 1.2]{MinimalChart}$)$ }\n\\label{LemNS-Tangle}\nIn a minimal chart, \nthere does not exist \nan NS-tangle of any label.\n\\end{lem}\n\n\n\n\\begin{lem}\\label{BoundaryConditionLemma}\n{\\rm [Boundary Condition Lemma]\n$($\\cite[Lemma 4.1]{TwoCrossingI}, \n\\cite[Lemma 11.1]{MinimalChart}$)$}\nLet $(\\Gamma\\cap D,D)$ be a tangle \nin a minimal chart $\\Gamma$ \nsuch that \n$D$ does not contain any crossing,\nfree edge nor simple hoop.\nLet $a=\\min\\{~i~|~\\Gamma_i\\cap\\partial D\\not=\\emptyset\\}$ and\n$b=\\max\\{~i~|~\\Gamma_i\\cap\\partial D\\not=\\emptyset\\}$.\nThen\n $\\Gamma_i\\cap D=\\emptyset$ \n except for $a\\le i \\le b$.\n\\end{lem}\n\n\n\nLet $\\Gamma$ be a chart, and \n$m$ a label of $\\Gamma$. \nA simple closed curve in $\\Gamma_m$ is \ncalled a {\\it cycle of label $m$}.\n\nLet $\\Gamma$ be a chart,\nand $m$ a label of $\\Gamma$.\nLet $C$ be a cycle of label $m$ in $\\Gamma$ \nbounding a disk $E$.\nThen an edge $e$ of label $m$ \nis called\nan {\\it outside edge for $C$} provided that \n\\begin{enumerate}\n\\item[(i)]\n$e\\cap C$ consists of one white vertex or two white vertices, and\n\\item[(ii)]\n$e\\not\\subset E$.\n\\end{enumerate}\nFor a cycle $C$ of label $m$, \nwe define\n$$\\begin{array}{ll}\n{\\mathcal{W}}(C)&= \\{ w \\ | \\text{ $w$ is a white vertex in $C$} \\},\\\\\n{\\mathcal{W}}_O^{{\\rm Mid}}(C,m)&= \\{ w\\in \\mathcal{W}(C)  \\ | \\text{ there exists an outside edge for $C$ {\\it middle} at $w$} \\}.\n\\end{array}$$\n\nThe following lemma will be used in\nthe proof of Lemma~\\ref{Lemma4}.\n\n\n\\begin{lem} \n{\\rm $($\\cite[Lemma 2.8]{StI}$)$} \n\\label{LemTwoColorTangle}\nLet $\\Gamma$ be a minimal chart, and \n$m,k$ labels of $\\Gamma$ with $|m-k|=1$. \nLet $(\\Gamma\\cap D,D)$ be \na tangle\nwith $\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$\nbut without free edges nor simple hoops. \nThen for any cycle $C$ of label $m$ in $D$, \nwe have \n$|{\\mathcal W}_O^{{\\rm Mid}}(C,m)|\\ge 2$.\n\\end{lem}\n\n\nLet $\\Gamma$ be a chart, and \n$E$ a disk.\nAn edge $e$ of the chart $\\Gamma$\nis called \nan {\\it I-edge $($resp. O-edge$)$} \nfor $E$ provided that (see Fig.~\\ref{fig08})\n\\begin{enumerate}\n\\item[(i)] \nthe edge $e$ possesses two white vertices,\none is in Int~$E$  \nand \nthe other in $E^c$,\n\\item[(ii)] \nthe edge $e$ intersects $\\partial E$ \nby exactly one point, and\n\\item[(iii)] \nthe edge $e$ is inward (resp. outward) at\nthe vertex in Int~$E$.\n\\end{enumerate}\nWe often say just {\\it an I-edge} \ninstead of {\\it an I-edge for $E$}\nif there is no confusion.\nSimilarly we often say just {\\it an O-edge} \ninstead of {\\it an O-edge for $E$}.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig08.pdf}\n\\end{center}\n\\caption{ \\label{fig08}\nThe gray area is a disk $E$.\nThe edge $e_1$ is an I-edge for $E$, and\nthe edge $e_2$ is an O-edge for $E$.\n}\n\\end{figure}\n\nIf \na terminal edge is inward\nat its black vertex,\nthen the edge \nis called an {\\it I-terminal edge},\notherwise \nthe edge is called an {\\it O-terminal edge}.\n\nLet $\\Gamma$ be a chart.\nFor each tangle $(\\Gamma\\cap D,D)$, \nwe define\n\n$E_I(D)=$ the number of I-edges for $D$,\n\n$E_O(D)=$ the number of O-edges for $D$,\n\n$T_I(D)=$ the number of I-terminal edges \nin $D$,\n\n$T_O(D)=$ the number of O-terminal edges \nin $D$.\n\n\nThe following lemma will be used in \nthe proof of Theorem~\\ref{StIITheorem4}.\n\n\\begin{lem}\n{\\rm $($\\cite[Lemma 7.1]{StI}$)$ }\n\\label{EI+TO=EO+TI}\nLet $\\Gamma$ be \na minimal chart, \nand \n$(\\Gamma\\cap D,D)$ a tangle without crossing. \nIf $\\partial D$ does not intersect \nany terminal edge,\nthen we have\n\\center{$E_I(D)+T_O(D)=E_O(D)+T_I(D)$}.\n\\end{lem}\n\n\n\n\n\n\n\\section{Proof of Lemma~1.1}\n\\label{s:ProofLemma1}\n\n\nLet $\\Gamma$ be a chart with a brim $N$ in a disk $D^2$.\nBy Assumption~\\ref{AssumptionFreeEdge}\nand Assumption~\\ref{AssumptionFreeEdgeSimpleHoop},\nall the simple hoops and free edges of $\\Gamma$\nare in the brim $N$.\nSet $C^*=\\partial N-\\partial D^2$.\nLet $e$ be an edge in Main($\\Gamma$) \nof label $m$ \nsuch that \nthere exists an arc $\\ell$ in $Cl(D^2-N)$ \nconnecting \na point $p$ in Int~$e$ and \na point $q$ in $C^*$\nwith $\\ell\\cap{\\rm Main}(\\Gamma)=p$ \n(see Fig.~\\ref{fig09}(a)).\n\nWe construct a chart from $\\Gamma$\nby a C-I-M1 move and C-I-M2 moves as follows.\nFirst we create a simple hoop $H$ of label $m$ \nsurrounding the point $q$\nby a C-I-M1 move (see Fig.~\\ref{fig09}(b)),\nwhere $H$ is oriented such that \nwe can apply a C-I-M2 move \nbetween $e$ and $H$.\nNext apply a C-I-M2 move to \nthe hoop $H$ along $C^*$\n(see Fig.~\\ref{fig09}(c) and (d)).\nThen we obtain two simple hoops \nparallel to $C^*$;\none hoop $H_1$ is in the brim $N$ and \nthe other hoop $H_2$ is in $D^2-N$.\nFinally apply a C-I-M2 move \nbetween the edge $e$ and $H_2$\nalong the arc $\\ell$ \nto get a new edge $e^*$ of label $m$ \n(see Fig.~\\ref{fig09}(e)). \nLet\n$$\\Gamma^*=(\\Gamma-e)\\cup e^*\\cup H_1.$$\nThen $\\Gamma^*$ is a chart \nC-move equivalent to $\\Gamma$.\nWe say that\nthe chart $\\Gamma^*$ is obtained from \n$\\Gamma$ by\n{\\it a double hoops trick $($DH-trick$)$} for the edge $e$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig09.pdf}\n\\end{center}\n\\caption{ \\label{fig09} The dark gray area $N$ is a brim in $D^2$.}\n\\end{figure}\n\n\n\n\\begin{lem}\\label{Lemma3-1}\nLet $\\Gamma$ be a minimal chart \nin a disk $D^2$.\nIf $c(\\Gamma)\\le3$,\nthen any hoop is simple.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that there exists \na non-simple hoop $C$.\nThen each complementary domains \nof the hoop contains a white vertex.\nSince there exist at most three crossings, \none of the two complementary domains contains \nat most one crossing,\nsay $U$.\nSince $\\partial U=C$,\nthe closure $Cl(U)$ is a disk or an annulus containing $\\partial D^2$.\n\nIf $Cl(U)$ is a disk, then \nlet $D$ be \na regular neighbourhood of $Cl(U)$.\nSince $\\Gamma\\cap \\partial D=\\emptyset$,\nthe tangle $(\\Gamma\\cap D,D)$ \nis an NS-tangle. \nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\nIf $Cl(U)$ is an annulus,\nthen take an arc $\\ell$ connecting a point $p$ in $C$ and a point in $\\partial D^2$\nwith $\\ell \\cap C=p$,\nand apply the chart $\\Gamma$ by a DH-trick for each edge of ${\\rm Main}(\\Gamma)$ intersecting $\\ell$ one by one from the outside.\nThen the hoop $C$ is modified to a hoop bounding a disk with at least one white vertex and \nwith at most one crossing.\nSimilarly we can find an NS-tangle.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\\end{Proof}\n\n\n\\begin{lem}\\label{Lemma3-2}\nLet $\\Gamma$ be a minimal chart \nin a disk $D^2$.\nIf $c(\\Gamma)\\le3$,\nthen there exists no ring.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that there exists a ring $C$.\nLet $A$ be a regular neighbourhood of $C$.\nSince there exist at most three crossings, \none of a complementary domain of $A$\ncontains at most one crossing.\nLet $D$ be the closure of the complementary domain of $A$.\nBy DH-tricks,\nwe can assume that $D$ is a disk.\nSince each complementary domain of $C$ \ncontains a white vertex \nby Remark~\\ref{Assumption0}(3), \nand \nsince there are at most three crossings on $C$,\nthe tangle $(\\Gamma\\cap D,D)$ is an NS-tangle.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\\end{Proof}\n\n\nLet $\\Gamma$ be a minimal chart.\nFor a subset $X$ of $\\Gamma$,\nlet\n\\begin{enumerate}\n\\item[]\n$B(X)=$ the union of all the disk \nbounded by a cycle in $X$, and\n\\item[]\n$T(X)=$ the union of all the terminal edge \nintersecting $X\\cup B(X)$.\n\\end{enumerate}\nThe set $X\\cup B(X)\\cup T(X)$ is called\nthe SC-{\\it closure} of $X$ and denoted by $SC(X)$.\n\nLet $\\Gamma$ be a chart, and \n$m$ a label of the chart. \nLet $\\mathcal W$ be the set of \nall the white vertices of $\\Gamma$. \nThe closure of a connected component of $\\Gamma_m-\\mathcal W$ \nis called an {\\it internal} edge of label $m$ \nif it contains a white vertex \nbut does not contain any black vertex, \nhere we consider $\\Gamma_m$ as a topological set. \n\n\nLet $G$ be a subgraph of a chart $\\Gamma$. \nAn internal edge $e$ in $G$ is called a {\\it cut-edge} for $G$ \nif $G-e$ is not connected.\n\n\n\n\\begin{lem} \n\\label{NoCutEdge}\nLet $\\Gamma$ be a minimal chart \nin a disk $D^2$. \nLet $\\alpha=\\alpha(\\Gamma)$, and \n$\\beta=\\beta(\\Gamma)$. \nIf $c(\\Gamma)\\le 3$, \nthen neither $\\Gamma_\\alpha$ \nnor $\\Gamma_\\beta$ \ncontains an internal cut-edge.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that $\\Gamma_\\alpha$ contains \nan internal cut-edge $\\overline{e}^*$. \nBy DH-tricks, we can assume that \n\\begin{enumerate}\n\\item[(1)] \nthere exists an arc $L$ connecting \na point $p$ in Int~${\\overline e}^*$ and \na point in the brim \nwith $L\\cap\\Gamma_\\alpha=p$ \n(see Fig.~\\ref{fig10}(a)).\n\\end{enumerate}\nLet $X_1,X_2$ be the \nconnected components of \n$Cl(\\Gamma_\\alpha-{\\overline e}^*)$ \nsuch that\n\\begin{enumerate}\n\\item[(2)] \neach of $X_1\\cap {\\overline e}^*$ and \n$X_2\\cap {\\overline e}^*$ \nconsists of exactly one point.\n\\end{enumerate}\nFurther, the arc $L$ of Statement (1) \nassures us that  \nthe SC-closures of $X_1,X_2$ \ndo not intersect each other, \ni.e. \n$SC(X_1)\\cap SC(X_2)=\\emptyset$.\nFurthermore, \n$c(\\Gamma)\\le 3$ implies that \n\\begin{enumerate}\n\\item[(3)] \none of $SC(X_1)$ and $SC(X_2)$ contains \nat most one crossing, say $SC(X_1)$.\n\\end{enumerate}\nIf $X_1$ does not contain a crossing, \nlet $E$ be a regular neighbourhood of $SC(X_1)$ \nin $D^2$. \nSince $SC(X_1)$ contains at most one crossing,\nthe disk $E$ contains at most one crossing.\nHence $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\nSuppose that $X_1$ contains a crossing $v$ \nin $\\Gamma_\\alpha\\cap\\Gamma_k$ \nfor some label $k$ with $1<|\\alpha-k|$. \nThen there exists an internal edge ${\\overline e}$ of \nlabel $\\alpha$ containing the crossing $v$. \nIf ${\\overline e}$ is not an internal cut-edge, \nlet $E$ be a regular neighbourhood of $SC(X_1)$ \nin $D^2$. \nThen $E$ contains a white vertex of ${\\overline e}$.\nHence by Statement $(3)$, \nthe tangle $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\nIf ${\\overline e}$ is an internal cut-edge, \nlet $N$ be a regular neighbourhood of \nthe internal edge of label $k$ \ncontaining the crossing $v$. \nThen  $X_1-N$ consists of \ntwo connected components.\nBy Statement (2), \none of the connected component\ndoes not intersect the edge ${\\overline e}^*$, \nsay $X$. \nLet $E$ be a regular neighbourhood of \n$SC(X)$. \nThen $\\Gamma_\\alpha\\cap\\partial E$ \nconsists of one point.\nFurther, $E$ does not contain a crossing \nby Statement $(3)$.\nHence the tangle $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nThus $\\Gamma_\\alpha$ does not contain \nan internal cut-edge.\n\nSimilarly we can show that \n$\\Gamma_\\beta$ does not contain \nan internal cut-edge.\nThus Lemma~\\ref{NoCutEdge} holds.\n\\end{Proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig10.pdf}\n\\caption{\\label{fig10}}\n\\end{center}\n\\end{figure}\n\nLet $\\Gamma$ be a chart.\nFor each label $m$,\nwe define\n ${\\rm Main}(\\Gamma_m)=\\Gamma_m\\cap {\\rm Main}(\\Gamma)$. \n \n\n\\begin{lem} \n\\label{Gamma1Connected}\nLet $\\Gamma$ be a minimal chart \nwith a brim in a disk $D^2$. \nLet $\\alpha=\\alpha(\\Gamma)$, and \n$\\beta=\\beta(\\Gamma)$. \nIf $c(\\Gamma)\\le 3$, \nthen ${\\rm Main}(\\Gamma_\\alpha)$ and \n${\\rm Main}(\\Gamma_\\beta)$ \nare connected.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that ${\\rm Main}(\\Gamma_\\alpha)$ is not connected. \nLet $X_1,X_2$ be \nconnected components of ${\\rm Main}(\\Gamma_\\alpha)$. \nBy Lemma~\\ref{Lemma3-1} and Lemma~\\ref{Lemma3-2},\neach of $X_1,X_2$ contains at least one white vertex.\nBy DH-tricks, we can assume that \ntheir SC-closures do not intersect, \ni.e. \n$SC(X_1)\\cap SC(X_2)=\\emptyset$ \n(see Fig.~\\ref{fig10}(b)). \nThen $c(\\Gamma)\\le 3$ implies that \n\\begin{enumerate}\n\\item[$(1)$] \none of $SC(X_1)$ and $SC(X_2)$ contains \nat most one crossing, say $SC(X_1)$.\n\\end{enumerate}\nIf $X_1$ does not contain a crossing, \nlet $E$ be a regular neighbourhood of $SC(X_1)$ \nin $D^2$. \nThen Statement~$(1)$\nimplies that $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\nSuppose that $X_1$ contains a crossing $v$ \nin $\\Gamma_\\alpha\\cap\\Gamma_k$ \nfor some label $k$ with $\\alpha<k\\le \\beta$. \nThen there exists an internal edge $\\overline e$ of \nlabel $\\alpha$ containing the crossing $v$. \nBy Lemma~\\ref{NoCutEdge}, \nthe edge $\\overline e$ is not an internal cut-edge. \nLet $E$ be a regular neighbourhood of $SC(X_1)$ \nin $D^2$. \nThen Statement~$(1)$ implies that \n$E$ contains exactly one crossing $v$\nand $\\Gamma_k\\cap\\partial E$ consists of exactly one point.\nHence $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nThus ${\\rm Main}(\\Gamma_\\alpha)$ is connected. \n\nSimilarly we can show that \n${\\rm Main}(\\Gamma_\\beta)$ is connected.\nThus Lemma~\\ref{Gamma1Connected} holds.\n\\end{Proof}\n\n\n\n\\begin{lem} \n\\label{ContainCrossing}\nLet $\\Gamma$ be a minimal chart \nwith a brim in a disk $D^2$. \nLet $\\alpha=\\alpha(\\Gamma)$, and \n$\\beta=\\beta(\\Gamma)$. \nIf $c(\\Gamma)\\le 3$, \nthen each of $\\Gamma_\\alpha$ \nand $\\Gamma_\\beta$ \ncontains a crossing.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that $\\Gamma_\\alpha$ does not \ncontain a crossing.\n\nSuppose that \nthere exists a complementary domain $U$ \nof ${\\rm Main}(\\Gamma_\\alpha)$ \ncontaining exactly one crossing. \nBy DH-tricks, we can assume that \n\\begin{enumerate}\n\\item[(1)] \n$U$ does not intersect the brim.\n\\end{enumerate}\nLet $A$ be a regular neighbourhood of \n$\\partial U$ in $D^2$, and $E=Cl(U-A)$. \nThen Statement $(1)$ and \nLemma~\\ref{Gamma1Connected} \nimply that \n$E$ is a disk. \nSince $U$ contains \nexactly one crossing, so does $E$.\nHence $(\\Gamma\\cap E,E)$ \nis an NS-tangle of label $\\alpha+1$. \nThis contradicts Lemma~\\ref{LemNS-Tangle}.\n\nSuppose that \nthere exists a complementary domain $U$ of \n${\\rm Main}(\\Gamma_\\alpha)$ contains \nall the crossings. \nThen by DH-tricks, we can assume that \n\\begin{enumerate}\n\\item[(2)] \n$U$ intersects the brim.\n\\end{enumerate}\nLet $A$ be a regular neighbourhood of \n$\\partial U-\\partial D^2$ in $D^2$, \nand $E=(D^2-U)\\cup A$. \nThen Statement $(2)$ and \nLemma~\\ref{Gamma1Connected} \nimply that \n$E$ is a disk. \nSince $\\Gamma_\\alpha$ contains a white vertex, \nso does $E$.\nHence $(\\Gamma\\cap E,E)$\nis an NS-tangle of label $\\alpha+1$ \nwithout crossing. \nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nHence $\\Gamma_\\alpha$ contains a crossing.\n\nSimilarly we can show that \n$\\Gamma_\\beta$ contains a crossing.\nThus Lemma~\\ref{ContainCrossing} holds.\n\\end{Proof}\n\n\n\\begin{lem} \n\\label{ContainTwoCrossing}\nLet $\\Gamma$ be a minimal chart \nwith a brim in a disk $D^2$. \nLet $\\alpha=\\alpha(\\Gamma)$, and \n$\\beta=\\beta(\\Gamma)$. \nIf $c(\\Gamma)\\le 3$, \nthen there exist cycles $C_\\alpha$ \nand $C_\\beta$ with \n$C_\\alpha\\subset\\Gamma_\\alpha,\n C_\\beta\\subset\\Gamma_\\beta$ \neach of which contains two crossings.\n\\end{lem}\n\n\\begin{Proof}\nBy Lemma~\\ref{ContainCrossing}, \n$\\Gamma_\\alpha$ contains at least one crossing $v$.\nBy Lemma~\\ref{NoCutEdge},\nthere exist two complementary domains $U_1,U_2$ of ${\\rm Main}(\\Gamma_\\alpha)$\nwith $v\\in\\partial U_1\\cap\\partial U_2$. \nThen $c(\\Gamma)\\le 3$ implies that \none of $U_1,U_2$ contains at most one crossing,\nsay $U$. \nBy DH-tricks, \nwe can assume that\n$U$ does not intersect the brim. \nBy Lemma~\\ref{Gamma1Connected}, \n$U$ is an open disk.\nBy Lemma~\\ref{NoCutEdge},\nthe boundary $\\partial U$ is a simple closed curve. \nLet $C_\\alpha=\\partial U$.\n\nSuppose that $C_\\alpha$ contains exactly one crossing $v$.\nLet \n$A$ be a regular neighbourhood of \n$\\partial U$ in $D^2$, and \n$E=Cl(U-A)$.\nThen $E$ is a disk. \nNow $v\\in\\partial U=C_\\alpha$ \nimplies that\nthe disk $E$ contains a white vertex. \nThus $(\\Gamma\\cap E,E)$ is \nan NS-tangle of label $\\alpha+1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nHence $C_\\alpha$ contains \nat least two crossings.\n\nSimilarly we can show that \nthere exists a cycle \n$C_\\beta\\subset\\Gamma_\\beta$ \ncontains at least two crossing crossings.\nThus Lemma~\\ref{ContainTwoCrossing} holds.\n\\end{Proof}\n\n\n\n{\\bf Proof of Lemma~\\ref{StIILemma1}}.\nLet $\\Gamma$ be a \n$2$-crossing minimal chart \nwith two crossings $x_1,x_2$.\nBy Lemma~\\ref{ContainTwoCrossing},\nthere exist cycles $C_\\alpha$ \nand $C_\\beta$ with \n$C_\\alpha\\subset\\Gamma_\\alpha,\n C_\\beta\\subset\\Gamma_\\beta$ \nand $C_\\alpha\\cap C_\\beta=\\{ x_1,x_2 \\}$.\nThus Statement (a) holds.\n\nLet $N_1,N_2$ be a regular neighbourhood of $x_1,x_2$ in $D^2$ respectively.\nBy DH-tricks,\nwe can assume that there exists a simple arc $\\gamma$ connecting a point in $\\partial N_1$ and \na point in $\\partial D^2$\nwith $\\gamma\\cap{\\rm Main}(\\Gamma)=\\emptyset$.\nLet $N(\\gamma)$ be a regular neighbourhood of $\\gamma$ in $D^2$,\nand $N$ a brim.\nSet $A=Cl(D^2-(N_1\\cup N_2\\cup N(\\gamma)\\cup N))$.\nThen $A$ is an annulus\nand $\\Gamma\\cap \\partial A=\\Gamma\\cap(\\partial N_1\\cup\\partial N_2)$.\nThus $\\Gamma\\cap \\partial A=(C_\\alpha\\cup C_\\beta)\\cap \\partial A$ and\n$\\Gamma\\cap \\partial A$ is consists of eight points.\nHence Statement (d) holds.\n\nSince $N_1\\cup N_2\\cup N(\\gamma)\\cup N$\ndoes not contain any white vertex,\nthe annulus $A$ contains all the white vertices \nof $\\Gamma$.\n\nBy Assumption~\\ref{AssumptionFreeEdgeSimpleHoop},\n all simple hoops and free edges are in a brim $N$.\nBy Lemma~\\ref{Lemma3-1}, \nany hoop is simple.\nThus the annulus $A$ contains neither hoops \nnor free edges.\nHence Statement (b) holds.\n\nSince \n$N_2$ and $N_1\\cup N(\\gamma)\\cup N$ are\nconnected components of $Cl(D^2-A)$,\neach connected component of $Cl(D^2-A)$ \ncontains a crossing.\nThus Statement (c) holds.\n\nSince $C_\\beta\\cap \\Gamma_\\alpha$\nconsists of two points,\n$\\Gamma_\\alpha\\cap A$ separates two components \nby $C_\\beta$.\nSimilarly\n$\\Gamma_\\beta\\cap A $ separates two components \nby $C_\\alpha$.\nThus Statement (e) and Statement (f) hold.\nHence Lemma~\\ref{StIILemma1} holds. \n{\\hfill {$\\square$}\\vspace{1.5em}}\n\n\n\n\n\n\\section{Proofs of Theorem~1.2 and Theorem~1.3}\n\\label{s:Th2Th3}\n\n\n\nLet $\\Gamma$ be a chart.\nA tangle $(\\Gamma\\cap D,D)$ is said to be\n{\\it admissible} \nprovided that\n\\begin{enumerate}\n\\item[(i)] $D$ contains neither free edge nor simple hoop.\n\\item[(ii)] \nany edge intersecting $\\partial D$\nis contained in an internal edge, \n\\item[(iii)] \nif an internal edge $\\overline e$  \nintersects $\\partial D$,\nthen each connected component of \n$\\overline e\\cap D$ contains a white vertex.\n\\end{enumerate}\n\n\n\n\\begin{lem}\n{\\rm(\\cite[Theorem 1.3]{MinimalChart})}\n\\label{TwoColorGateTangle} \nIf $(\\Gamma\\cap D,D)$ is \nan admissible tangle \nin a minimal chart $\\Gamma$ such that \n\\begin{enumerate}\n\\item[{\\rm $($a$)$}] \n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_{m-1}$ \nor \n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_{m+1}$ \nfor some label $m$,\n\\item[{\\rm $($b$)$}] \n$\\Gamma_m\\cap \\partial D$ \nconsists of exactly two points, \nand \n\\item[{\\rm $($c$)$}] \n$\\Gamma_m\\cap D$ contains a cycle, \n\\end{enumerate}\nthen\nthe tangle $(\\Gamma\\cap D,D)$ is \na simple IO-tangle of label $m$.\n\\end{lem}\n\n\nLet $m$ be a label \nof a chart $\\Gamma$, \nand $(\\Gamma\\cap D,D)$ \na Type-II elementary IO-tangle \nof label $m$. \nLet $L_D$ be \na simple proper arc of $D$ \ncontained in $\\Gamma_m\\cap D$ \nand containing \nall the white vertices \n$v_1,v_2,\\cdots,v_t$ \nin $D$, \nsituated on $L_D$ in this order.\nLet $\\Delta_I,\\Delta_O$ \nbe the closures of \nconnected components \nof $D-L_D$. \nThen the tangle \n$(\\Gamma\\cap D,D)$ is\nsaid to be \nof {\\it Type-II$_t$} provide that \n(see Fig.~\\ref{fig11})\n\\begin{enumerate}\n\\item[(i)] \nfor each $i=1,2,\\cdots,t$, \nthere exists a terminal edge $\\tau_i$ \nof label $m$ \ncontaining $v_i$ and \n\\item[(ii)] \nfor each $i=1,2,\\cdots,t$, \nthe terminal edge $\\tau_i$ \nis contained in \n$\\Delta_I$ or $\\Delta_O$ alternately.\n\\end{enumerate}\n\n\\begin{figure}\n\\centerline{\\includegraphics{fig11.pdf}}\n\\caption{\\label{fig11}\nA Type-II$_5$ elementary IO-tangle of label $m$. The thick lines are of label $m$.}\n\\end{figure}\n\n\n\n\\begin{lem}\\label{lemEQ}\nLet $\\Gamma$ be a minimal chart, and \n$(\\Gamma\\cap D,D)$ a non-trivial admissible tangle\nsuch that\n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$ \nfor some labels $m,k$ with $|m-k|=1$.\nIf \n$\\Gamma_m\\cap D$ contains no cycle,\nthen the following conditions are equivalent.\n\\begin{enumerate}\n\\item[{\\rm (a)}] \nThe tangle \nis a Type-II elementary IO-tangle \nof label $m$.\n\\item[{\\rm (b)}] \nThe tangle is \na tangle with \n$\\Gamma_m\\cap \\partial D$ two points, and \nthere exists \na proper arc $L_D$ of $D$ in \n$\\Gamma_m\\cap D$ \ncontaining all the white vertices \nin $D$.\n\\item[{\\rm (c)}] \nThe tangle is \na Type-II$_t$ elementary IO-tangle \nof label $m$ \nfor some integer $t$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{Proof}\nClearly \nStatement (a) implies Statement (b).\n\nSuppose that \nStatement (b) holds.\nLet $v_1,v_2,\\cdots,v_t$ be \nall the white vertices in $L_D$\nsituated on $L_D$ in this order.\n\nNow for each $i=1,2,\\cdots,t$, \nthe vertex $v_i$ is contained in \na terminal edge $\\tau_i$\nof label $m$.\nFor, if not, let $e$ be \nthe edge of label $m$ \nwith $e\\cap L_D\\ni v_i$ and $e\\not\\subset L_D$ \nbut not a terminal edge. \nSince $\\Gamma_m\\cap \\partial D$ \nconsists of two points\n$\\partial L_D$, \nwe have $e\\cap\\partial D=\\emptyset$. \nSince the arc $L_D$ contains \nall the white vertices in $D$, \nthe union $e\\cup L_D$ contains \na cycle in $\\Gamma_m\\cap D$. \nThis is a contradiction.\nThus each vertex $v_i$ is contained in \na terminal edge $\\tau_i$.\n\nNow $D-L_D$ consists of \ntwo connected components.\nIf $\\tau_1$ is inward \n(resp. outward) at $v_1$,\nthen let $\\Delta_I$ (resp. $\\Delta_O$) \nbe \nthe closure of the one of \nthe two connected components \ncontaining $\\tau_1$, and\n$\\Delta_O$ (resp. $\\Delta_I$) \nthe closure of \nthe other connected component.\n\nSuppose that \nfor some $i\\in\\{1,2,\\cdots,t-1\\}$,\ntwo terminal edges \n$\\tau_i$ and $\\tau_{i+1}$ are \ncontained in $\\Delta_I$ or $\\Delta_O$\nsimultaneously.\nThen we can eliminate \nthe two white vertices $v_i,v_{i+1}$\nby two C-I-M2 moves and a C-I-M3 move\n(see Fig.~\\ref{fig12}).\nThis contradicts that $\\Gamma$ is \na minimal chart.\nHence the terminal edges \n$\\tau_1,\\tau_2,\\cdots,\\tau_t$\nalternately belong to \n$\\Delta_I$ and $\\Delta_O$.\n\nThus each terminal edge of label $m$ in $\\Delta_I$ \nis inward at a vertex of $L_D$, and\neach terminal edge of label $m$ in $\\Delta_O$\nis outward at a vertex of $L_D$.\nHence if an edge intersects Int~$\\Delta_I$ \n(resp. Int~$\\Delta_O$),\nthen the edge contains an inward (resp. outward)\narc in $\\Delta_I$ (resp. $\\Delta_O$) \nat a vertex in $L_D$.\nLet $L_I=Cl(\\partial \\Delta_I - L_D)$\nand $L_O=Cl(\\partial \\Delta_O - L_D)$.\nThen $L_I,L_O$ satisfy \nCondition (iii) and (iv) for\nan IO-tangle.\nThus the tangle $(\\Gamma\\cap D,D)$ \nis \na Type-II$_t$ elementary IO-tangle of\nlabel $m$. \nHence Statement (c) holds.\n\nIt is also clear that \nStatement (c) implies Statement (a).\nThis proves Lemma~\\ref{lemEQ}.\n\\end{Proof}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics{fig12.pdf}}\n\\caption{The thick lines are of label $m$.\n\\label{fig12}}\n\\end{figure}\n\n\n\nLet $\\Gamma$ be a chart, \n$v$ a white vertex, \nand \n$e$ a terminal edge inward \n(resp. outward) at $v$.\nThen the two edges inward \n(resp. outward) at $v$ \ndifferent from $e$ \nare called the {\\it sibling edges} of $e$.\n\n\\begin{lem}\n\\label{LemSimpleIO-Tangle1}\nIf $(\\Gamma\\cap D,D)$ \nis an admissible tangle \nin a minimal chart $\\Gamma$\nsuch that\n\\begin{enumerate}\n\\item[{\\rm $($a$)$}] \n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_k$ \nfor some labels $m,k$ with $|m-k|=1$,\n\\item[{\\rm $($b$)$}] \n$\\Gamma_m\\cap \\partial D$ consists of \nexactly two points, \nand\n\\item[{\\rm (c)}]\nthe disk \n$D$ contains at least two white vertices, \n\\end{enumerate}\nthen \n$(\\Gamma\\cap D,D)$ \nis a simple IO-tangle tangle\nof label $m$. \nFurther, if $D$ does not contain \na cycle of label $m$, then\nthe tangle is a Type-II elementary IO-tangle \nof label $m$.\n\\end{lem}\n\n\\begin{Proof}\nNeither terminal edge nor free edge is \nan internal edge.\nBy Condition (ii) for an admissible tangle,\nthe boundary $\\partial D$ intersects \nneither terminal edge nor free edge.\nThus the tangle $(\\Gamma\\cap D, D)$ \nsatisfies Condition (i) for an IO-tangle.\n\nCondition (a) is the same one of\nCondition (ii) for an IO-tangle.\n\nIf $\\Gamma_m\\cap D$ \ncontains a cycle,\nthen the tangle $(\\Gamma\\cap D,D)$ \nis a simple IO-tangle \nof label $m$ by \nLemma~\\ref{TwoColorGateTangle}.\n\nSuppose $\\Gamma_m\\cap D$ contains no cycle.\nWe claim that \n$\\Gamma_m\\cap D$ contains \na proper arc \nconnecting the two points \n$\\Gamma_m\\cap \\partial D$. \nFor, if not, let $X$ be \na connected component\nof $\\Gamma_m\\cap D$ containing a white vertex. \nThen $X$ contains at most \none point in $\\Gamma_m\\cap\\partial D$.\nLet $D^*$ be \na regular neighbourhood of \nthe SC-closure $SC(X)$ in $D$.\nThen $(\\Gamma\\cap D^*,D^*)$\nis an NS-tangle of label $k$.\nThis contradicts \nLemma~\\ref{LemNS-Tangle}.\nThus\n$\\Gamma_m\\cap D$ contains \na proper arc \nconnecting the two points \n$\\Gamma_m\\cap \\partial D$.\n\nLet $L$ be \na proper arc in $\\Gamma_m\\cap D$ \nconnecting \nthe two points $\\Gamma_m\\cap\\partial D$.\n\n{\\bf Claim.}\nThe arc $L$ contains \nall of the white vertices in $D$.\n\n{\\bf Proof of Claim}. \nSuppose that there exists \na white vertex in $D-L$. \nLet $X$ be the closure of \nthe connected component of \n$\\Gamma_m\\cap (D-L)$ containing the vertex. \nThen $X$ does not intersect $\\partial D$, \nbecause $L$ contains the two points \n$\\Gamma_m\\cap \\partial D$. \nIf $X\\cap L=\\emptyset$, \nlet $D'$ be a regular neighbourhood of\nthe SC-closure $SC(X)$ in $D$. \nThen the tangle $(\\Gamma\\cap D',D')$ \nis an NS-tangle of label $k$. \nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nIf $X\\cap L\\neq\\emptyset$, \nthen $X\\cap L$ consists of one vertex $v$, \nbecause there does not exist \na cycle of label $m$. \nLet $e$ be the edge in $X$ containing $v$. \nLet $X'=Cl(X-e)$, and \n$D'$ a regular neighbourhood of\nthe SC-closure $SC(X')$ in $D$. \nThen the tangle $(\\Gamma\\cap D',D')$ \nis an NS-tangle of label $k$. \nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nHence Claim holds.\n\nNow by Lemma~\\ref{lemEQ}, \nthe tangle $(\\Gamma\\cap D,D)$ \nis \na Type-II elementary IO-tangle of\nlabel $m$.\nSuppose that there exists a terminal edge $e$ of label $k$ in $D$.\nLet $e^*,e^{**}$ be the sibling edges of $e$.\nIf one of  $e^*,e^{**}$ is contained in the disk $D$,\nthen we can find a non-admissible consecutive triplet \nwhich contradicts\n Consecutive Triplet Lemma\n(Lemma~\\ref{ConsecutiveTripletLemma}).\nThus both of $e^*$ and $e^{**}$ \nintersect $\\partial D$. \nHence $L=(e^*\\cup e^{**})\\cap D$ contains exactly one white vertex.\nThis contradicts Condition (c).\nThus there does not exist any terminal edge of label $k$ in $D$.\nTherefore the tangle is a simple IO-tangle.\n\\end{Proof}\n\n\n\\begin{remark}\\label{remWhite1}\n{\\rm \nIf $(\\Gamma\\cap D,D)$ \nis an admissible tangle \nin a minimal chart $\\Gamma$\nsuch that\n\\begin{enumerate}\n\\item[{\\rm $($a$)$}] \n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_k$ \nfor some labels $m,k$ with $|m-k|=1$,\n\\item[{\\rm $($b$)$}] \n$\\Gamma_m\\cap \\partial D$ consists of \nexactly two points, \nand\n\\item[(c)]\nthe disk \n$D$ contains \nexactly one white vertex,\n\\end{enumerate}\nthen \n$(\\Gamma\\cap D,D)$ \nis one of the tangles shown in \nFig.~\\ref{fig13}. \nThe tangles shown in \nFig.~\\ref{fig13}(a),(b) \nare simple IO-tangles. \nThe tangle shown in \nFig.~\\ref{fig13}(c) \nis not a simple IO-tangle, but \nan N-tangle (see Section~\\ref{s:Indices} \nfor the precise definition of N-tangles).\n}\n\\end{remark}\n\n\\begin{figure}\n\\centerline{\\includegraphics{fig13.pdf}}\n\\caption{Admissible tangles with exactly one white vertex.\\label{fig13}}\n\\end{figure}\n\n\n{\\bf Proof of Theorem~\\ref{StIITheorem2}.} \nLet $(\\Gamma\\cap D,D)$ be \na fundamental tangle of label \n$m=\\alpha(\\Gamma)$ \nor $\\beta(\\Gamma)$.\nIf $m=\\alpha(\\Gamma)$, \nthen\nlet $k=\\alpha(\\Gamma)+1$,\nelse \nlet $k=\\beta(\\Gamma)-1$.\nSince $D$ is a regular neighbourhood of\na SC-closure of a set in $\\Gamma_m$,\nby Boundary Condition Lemma\n(Lemma~\\ref{BoundaryConditionLemma}) \nwe have\n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$.\nSince $D$ contains \nat least two white vertices,\nthe fundamental tangle is \na simple IO-tangle\nby Lemma~\\ref{LemSimpleIO-Tangle1}. \n{\\hfill {$\\square$}\\vspace{1.5em}}\n\n\nLet $\\Gamma$ be a chart. \nA tangle $(\\Gamma\\cap D,D)$ \nis said to be \n{\\it $2$-color} if \nthere exist two labels $m,k$ \nwith $|m-k|=1$ and\n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_k$.\n\n\\begin{lem}\n\\label{2ColorBeads}\n{\\rm(\\cite[Lemma 10.1]{MinimalChart})}\nLet $\\Gamma$ be a minimal chart, and\n$m$ a label of $\\Gamma$.\nLet $(\\Gamma\\cap D,D)$ be\na $2$-color admissible tangle with $|\\Gamma_m\\cap\\partial D|=2$.\nIf $\\Gamma_m\\cap D$ contains a cycle, \nthen \nthere exist disks $E_1,E_2,\\cdots, E_d$ in {\\rm Int}~$D$ and simple arcs $L_0,L_1,\\cdots,L_d$ in \n$Cl(D-\\cup^{d}_{i=1}E_i)$\n such that\n\\begin{enumerate}\n\\item[{\\rm (a)}] \n$\\partial E_i\\subset \\Gamma_m$ for each $i=1,2,\\cdots,d$ and $L_j\\subset \\Gamma_m$ for each $j=0,1,\\cdots,d$,\n\\item[{\\rm (b)}] \nfor each $j=1,\\cdots,d-1$, $L_j$ connects a vertex in $\\partial E_j$ and a vertex in $\\partial E_{j+1}$, \\\\\nthe arc $L_0$ connects a point in $\\partial D$ and a vertex in $\\partial E_{1}$, and\\\\\nthe arc $L_d$ connects a vertex in $\\partial E_{d}$ and a point in $\\partial D$,\n\\item[{\\rm (c)}]\nif an edge of label $m$ intersects \n$D-((\\cup_{i=1}^{d} E_i)\\cup (\\cup_{j=0}^{d} L_j))$, \nthen it is a terminal edge.\n\\end{enumerate}\n\\end{lem}\n\n\n{\\bf Proof of Theorem~\\ref{StIITheorem3}.}\nBy Condition (iii) for an IO-tangle,\nwe have $|\\Gamma_m\\cap \\partial D|=2$.\nLet $k$ be a label of $\\Gamma$\nwith \n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$. \n\nSuppose that $\\Gamma_m\\cap D$ \ncontains a cycle.\nBy Lemma~\\ref{2ColorBeads},\nthere exist disks $E_1,E_2,\\cdots, E_d$ in {\\rm Int}~$D$ and simple arcs $L_0,L_1,\\cdots,L_d$ in \n$Cl(D-\\cup^{d}_{i=1}E_i)$\nsatisfying conditions (a), (b) and (c) in \nLemma~\\ref{2ColorBeads}.\nHence $D-((\\cup^{d}_{i=1}E_i)\\cup(\\cup^{d}_{j=0}L_j))$ does not contain any white vertices.\nSince there is no terminal edge \nof label $k$\nby Condition for a simple IO-tangle,\nfor each $i=1,2,\\cdots,d$ \nthere exist \ntwo proper arcs $\\ell_{2i-1}$ and $\\ell_{2i}$ \nof $D$ \n(see Fig.~\\ref{fig14}(a))\n\n$\\ell_{2i-1}=$ the proper arc of $D$ \ncontained in $\\Gamma_k$ with \n$\\ell_{2i-1}\\cap\\Gamma_m=\nL_{i-1}\\cap E_i$, and\n\n$\\ell_{2i}=$ the proper arc of $D$ \ncontained in $\\Gamma_k$ with \n$\\ell_{2i}\\cap\\Gamma_m=E_i\\cap L_i$.\\\\\nThen proper arcs \n$\\ell_{2i-1}$ and $\\ell_{2i}$\nsplit the disk $D$ into three disks.\nLet $D_{2i}^*$ be the one of the three disks\ncontaining $E_i$\n(see Fig.~\\ref{fig14}(a)).\nLet $D_{2i}$ be \na regular neighbourhood of $D_{2i}^*$ \n(see Fig.~\\ref{fig14}(b)). \nNow $Cl(D-\\cup_{i=1}^dD_{2i})$\nconsists of $d+1$ disks.\nFor each $i=0,1,2,\\cdots,d$,\nlet $D_{2i+1}$ be the one of the disks\nintersecting the arc $L_i$ \n(see Fig.~\\ref{fig14}(b)).\nThen for each $i=2,4,\\cdots,2d$,\nthe tangle $(\\Gamma\\cap D_i,D_i)$ is \na Type-I elementary IO-tangle of label $m$.\nFurther for each $i=1,3,\\cdots,2d+1$,\nthe tangle $(\\Gamma\\cap D_i,D_i)$ is \na Type-II elementary IO-tangle of label $m$.\nHence the tangle has\nan elementary IO-tangle decomposition\n$(\\Gamma\\cap D_1,D_1)\\#(\\Gamma\\cap D_2,D_2)\\#\n\\cdots (\\Gamma\\cap D_{2d+1},D_{2d+1})$.\n\nIf $\\Gamma_m\\cap D$ \ndoes not contain a cycle,\nby Lemma~\\ref{LemSimpleIO-Tangle1} \nwe have that \nthe simple IO-tangle is \nan elementary IO-tangle.\nThis proves Theorem~\\ref{StIITheorem3}.\n{\\hfill {$\\square$}\\vspace{1.5em}}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics{fig14.pdf}}\n\\caption{(a) The gray area is the disk $D^*_{2i}$. \n(b) The gray areas are the disks $D_{2i},D_{2i+2}$.\nThe dark gray area is the disk $D_{2i+1}$.\n\\label{fig14}}\n\\end{figure}\n\n\n\n\n\\section{One-way paths}\n\\label{s:OneWayPath}\n\nIn this section\nwe investigate a path in $\\Gamma_m$ called a one-way path.\n\n\nLet $\\Gamma$ be a chart, \nand $m$ a label of $\\Gamma$.\nA simple arc $P$ in $\\Gamma$\nis called a {\\it path}\nprovided that \nthe endpoints of $P$ are vertices of $\\Gamma$.\nIn particular, \nif the path $P$ is in $\\Gamma_m$,\nthen $P$ \nis called a {\\it path of label $m$}.\nSuppose that \n$v_0,v_1,\\cdots,v_{p}$ \nare  \nall the vertices in a path $P$ \nsituated in this order on $P$.\nFor each $i=1,\\cdots,p$, \nlet $e_i$ be the edge in $P$ \nwith $\\partial e_i=\\{v_{i-1},v_{i}\\}$.\nThen the $(p+1)$-tuple \n$(v_0,v_1,\\cdots,v_{p})$\nis called \na {\\it vertex sequence} \nof $P$,\nand the $p$-tuple \n$(e_1,e_2,\\cdots,e_{p})$\nis called \nan {\\it edge sequence} \nof $P$.\nFor two integers $i,j$ \nwith $0\\le i<j\\le p$, \nwe denote the path \n$e_{i+1}\\cup e_{i+2}\\cup\\cdots \\cup e_{j}$ \nby $P[v_i,v_j]$.\n\n\n\nLet $m$ be a label of a chart $\\Gamma$, and \n$P$ a path of label $m$\nin $\\Gamma$ with \na vertex sequence $(v_0,v_1,\\cdots,v_p)$ and\nan edge sequence $(e_1,e_2,\\dots,e_p)$. \nIf for each $i=1,2,\\cdots,p$,\nthe edge $e_i$ is oriented \nfrom $v_{i-1}$ to $v_i$, \nthen $P$ \nis called a {\\it one-way path}.\nIf the edge $e_1$ is middle at $v_0$ \nand if \nthe edge $e_p$ is middle at $v_p$, \nthen $P$  \nis called an {\\it M$\\&$M path}. \nA path of label $m$ in a chart $\\Gamma$ \nis called \na {\\it dichromatic path} \nif there exists \na label $k$ with $|m-k|=1$ such that\nany vertex of the path \nis contained in $\\Gamma_m\\cap\\Gamma_k$.\n\n\\begin{lem}\n{\\rm (\\cite[Lemma~3.1]{StI})}\n \\label{LemNoM-M}\nLet $\\Gamma$ be a minimal chart. \nThen\nany dichromatic one-way path in $\\Gamma$ \nis not an M$\\&$M path. {\\hfill {$\\square$}\\vspace{1.5em}}\n\\end{lem}\n\n\nLet $\\Gamma$ be a chart, \nand $m$ a label of $\\Gamma$.\nA simple closed curve in $\\Gamma_m$ \nis called a {\\it loop} \nif it contains \nexactly one white vertex. \nLet $C$ be \na cycle of label $m$.\nLet  \n$v_0,v_1,\\cdots,v_{p-1}$ \nbe all the vertices in $C$, \nand \n$e_1,e_2,\\cdots,e_p$\nall the edges in $C$.\nThen the cycle $C$  \nis called \na {\\it one-way cycle} provided that \nfor each $i=1,2,\\cdots,p$,\nthe edge $e_i$ is oriented from $v_{i-1}$ to $v_i$,\nwhere $v_p=v_0$.\nWe consider a loop as a one-way cycle.\n\n\\begin{lem}\n{\\rm (\\cite[Lemma~3.2]{StI})} \n\\label{LemOneWayCycle}\nLet $\\Gamma$ be a minimal chart, \n$m,k$ labels of $\\Gamma$ with $|m-k|=1$, and \n$(\\Gamma\\cap D,D)$ a tangle\nwith\n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$\nbut without free edges nor simple hoops. \nThen $D$ does not contain any one-way cycle \nof label $m$. {\\hfill {$\\square$}\\vspace{1.5em}}\n\\end{lem} \n\nLet $\\Gamma$ be a chart, \nand $m$ a label of the chart.\nLet $e$ be an edge of label $m$, and\n$P$ a one-way path of label $m$ \nwith an edge sequence \n$(e_1,e_2,\\dots,e_p)$.\nIf $e=e_1$,\nthen the path $P$ is called \na one-way path {\\it starting from} $e$. \nAlso if $e=e_p$,\nthen the path $P$ is called \na one-way path {\\it leading to} $e$.\n\nLet $P$ be a one-way path \nin a chart, and  \n$e$ an edge containing a vertex $v$ \nin Int~$P$ but \n$e\\not\\subset P$. \nSuppose that $e$ is not a loop.\nThe edge $e$ is said to be \n{\\it locally right-side }\n(resp. {\\it locally left-side}) at $v$\nprovided that\nfor a regular neighbourhood $N$ \nof $v$,\nthe arc $e\\cap N$ \nis situated right (resp. left) side\nof $P$ with respect to the direction of $P$ \n(see Fig.~\\ref{fig15}).\nIf the edge $e$ is locally \nright-side at \na vertex $v$ \nand inward (resp. outward) at $v$, \nthen\nthe edge \nis called a \nlocally right-side edge {\\it inward} \n(resp. {\\it outward}) at $v$.\nSimilarly \nif the edge $e$ is locally \nleft-side at \na vertex $v$ \nand inward (resp. outward) at $v$, \nthen\nthe edge \nis called a \nlocally left-side edge {\\it inward} \n(resp. {\\it outward}) at $v$.\nIn Fig.~\\ref{fig15},\nthe edge $e_2$ is a locally right-side edge \ninward at $v_1$,\nthe edge $e_3$ is a locally right-side edge \noutward at $v_1$,\nthe edge $e_4$ is a locally left-side edge \ninward at $v_2$,\nthe edge $e_5$ is a locally left-side edge \noutward at $v_2$.\nBut $e_1$ is not \nlocally right-side at $v_0$\nnor $e_6$ is not \nlocally left-side at $v_3$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig15.pdf}\n\\end{center}\n\\caption{ \\label{fig15}\nThe path $P$ is a one-way path \nwith a vertex sequence\n$(v_0,v_1,v_2,v_3)$.}\n\\end{figure}\n\n\nLet $\\Gamma$ be a chart, \nand $m$ a label of the chart.\nA one-way path $P$ of label $m$ \nis said to be\n{\\it upward right-selective} \n(resp. {\\it upward left-selective})\nif any edge \nof label $m$\nlocally right-side \n(resp. left-side) \nat a vertex in Int~$P$\nis inward at the vertex\n(see Fig.~\\ref{fig16}(a) and (b)).\nA one-way path $P$ of label $m$\nis said to be\n{\\it downward right-selective} \n(resp. {\\it downward left-selective})\nif any edge \nof label $m$ \nlocally right-side (resp. left-side) \nat a vertex in Int~$P$\nis outward at the vertex.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig16.pdf}\n\\end{center}\n\\caption{\\label{fig16} \n(a) the thick line is an upward right-selective path, \n(b) the thick line is an upward left-selective path.}\n\\end{figure}\n\n\n\nLet $\\Gamma$ be a chart, \nand $m$ a label of $\\Gamma$.\nLet $P^*$ be \nan upward right-selective one-way path \nof label $m$\nwith a vertex sequence\n$(v^*_0,v^*_1,\\cdots,v^*_s)$,\nand $\\widetilde P$ \nan upward left-selective one-way path \nof label $m$\nwith a vertex sequence \n$(\\widetilde v_0,\\widetilde v_1,\\cdots,\n\\widetilde v_t)$\nequipped with \n$P^*\\cap \\widetilde P=v^*_s=\\widetilde v_t$.\nLet $E$ be a disk \nand $P^\\dagger$ an arc on $\\partial E$.\nThe disk $E$ is called a \n{\\it half spindle} for $\\Gamma$\nwith a path triplet \n$(\\widetilde P,P^\\dagger,P^*)$\nprovided that \n(see Fig.~\\ref{fig17}(a))\n\\begin{enumerate}\n\\item[(i)]\n$\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_h$\nfor some label $h$ with $|m-h|=1$,\n\\item[(ii)] \n$(P^*\\cup\\widetilde P)\\cap$~Int~$E=\\emptyset$,\nand\n$E\\not\\ni v^*_0,\\widetilde v_0$, \n\\item[(iii)]\nthe edge of label $m$ outward at $v^*_s$ \nis contained in $Cl(E^c)$,\n\\item[(iv)] \n$P^*\\cap E$ \nis an arc containing \n$P^*[v^*_1,v^*_s]$,\n$\\widetilde P\\cap E$\nis an arc containing \n$\\widetilde P[\\widetilde v_1,\\widetilde v_t]$,\nand\n$P^\\dagger=\nCl(\\partial E-(P^*\\cup\\widetilde P))$,\n\\item[(v)] \nthe three arcs \n$\\widetilde P\\cap E,P^\\dagger,P^*\\cap E$ \nare counterclockwise situated \non $\\partial E$ in this order,\n\\item[(vi)]\nif an edge $e$ \nintersects Int~$P^\\dagger$,\nthen it is an I-edge of label $m$ for $E$, \n\\item[(vii)]\n$s\\ge 2$ and $t\\ge 2$.\n\\end{enumerate}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig17.pdf}\n\\end{center}\n\\caption{\\label{fig17} \n(a) A half spindle \n(b) The edge $X_2$ is situated between $X_1$ and $X_3$ around the vertex $v$.}\n\\end{figure}\n\nThe following lemma will be used in\nthe proof of Lemma~\\ref{Lemma3}.\n\n\n\\begin{lem}\n{\\rm (\\cite[Lemma~6.3]{StI})} \n\\label{lemHalfSpindle}\nFor any minimal chart \nthere does not exist a half spindle. {\\hfill {$\\square$}\\vspace{1.5em}}\n\\end{lem}\n\n\n\\begin{lem} \\label{lemNoTerminal}\nLet $\\Gamma$ be a minimal chart, and \n$m,k$ labels of $\\Gamma$ with \n$|m-k|=1$.\nLet $E$ be a disk with \n$\\partial E\\subset\\Gamma$ and\n$\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$. \nThen $E$ does not contain a terminal edge.\n\\end{lem}\n\n\\begin{Proof} \nSince $\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$, \nthe disk $E$ does not contain a crossing.\nSuppose that $E$ contains a terminal edge $e$. \nWithout loss of generality \nwe can assume that $e$ is of label $m$.\nLet $e^*$ be a sibling of $e$. \nThen $e^*\\subset E$. \nHence there exists an edge $e^{**}$ of label $m$ \nso that the triplet $(e,e^*,e^{**})$ is \na non-admissible consecutive triplet. \nThis contradicts Consecutive Triplet Lemma \n(Lemma~\\ref{ConsecutiveTripletLemma}).\n\\end{Proof}\n\nLet $v$ be a white vertex \nof a chart $\\Gamma$, and \n$X_1,X_2,X_3$ edges or paths in $\\Gamma$ \nwith $v\\in\\partial X_i~~(i=1,2,3)$. \nIn a regular neighbourhood of $v$, \nthere are six short arcs at $v$. \nFor each $i=1,2,3$, let \n$\\gamma_i$ be the one \nof the six short arcs in $X_i$. \nIf the three short arcs \n$\\gamma_1,\\gamma_2,\\gamma_3$ are \nconsecutive among the six short arcs \nin this order, \nthen $X_2$ is said to be \n{\\it situated between \n$X_1$ and $X_3$ around $v$} \n(see Fig.~\\ref{fig17}(b)).\n\nLet $\\Gamma$ be a chart, \n$P$ a path of $\\Gamma$, and \n$E$ a disk. \nIf each edge in $P$ \nintersects Int~$E$, and \nif $P\\cap E$ is connected, \nthen we say that the path $P$ \n{\\it is dominated by} the disk $E$ or \nthe disk $E$ {\\it dominates} the path $P$.\n\n\nLet $\\Gamma$ be a chart, \nand $m$ a label of $\\Gamma$. \nLet $E$ be a disk, and\n$P$ a one-way path \nof label $m$ dominated by $E$\nwith an edge sequence \n$(e_1,e_2,\\cdots,e_p)$. \nIf $P$ is not contained in another \none-way path \nof label $m$ \nstarting from $e_1$ \ndominated by $E$,\nthen the path $P$ is said to be \n{\\it upward maximal} \nwith respect to $E$.\nSimilarly if $P$ is not contained in \nanother one-way path of label $m$ \nleading to $e_p$ \ndominated by $E$, \nthen the path $P$ is said to be \n{\\it downward maximal} \nwith respect to $E$.\n\nLet $\\Gamma$ be a chart, \nand $E$ a disk with $\\partial E\\subset \\Gamma$. \nAn edge $e$ of $\\Gamma$ is called \nan {\\it outside edge}\n(resp. {\\it inside edge}) for $E$ \nprovided that \n\\begin{enumerate}\n\\item[(i)] $e\\cap \\partial E$ consists of \none white vertex or two white vertices, and\n\\item[(ii)] $e\\not\\subset E$ (resp. $e\\subset E)$.\n\\end{enumerate}\n\n\n\n\\begin{lem}\\label{lemMaxAtBoundaryUp} \nLet $\\Gamma$ be a minimal chart, \nand $m,k$ labels of $\\Gamma$  \nwith $|m-k|=1$.\nLet $E$ be a disk with \n$\\partial E\\subset\\Gamma$ and \n$\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$. \nLet $P$ be \na one-way path \nof label $k$ \nupward maximal \nwith respect to $E$ \nwith a vertex sequence \n$(v_0,v_1,v_2,\\cdots,v_p)$ and\nan edge sequence \n$(e_1,e_2,\\cdots,e_p)$. \nLet $e^\\dagger_1,e^\\dagger_2,e^\\dagger_3$ \nbe the edges of label $k$ at $v_p$, and\n$e^*_1,e^*_2,e^*_3$ the edges \nof label $m$ at $v_p$, \nhere $e^\\dagger_1=e_p$. \nThen we have the following:\n\\begin{enumerate}\n\\item[{\\rm (a)}]\n$v_p\\in\\partial E$.\n\\item[{\\rm (b)}] \nSuppose that \n$e^*_1\\cup e^*_2\\subset \\partial E$. \nIf \nthe edge $e^*_1$ is inward at $v_p$, \nand if \nthe edge $e^*_2$ is outward at $v_p$,\nthen \n$e^\\dagger_2,e^\\dagger_3,e^*_3$ \nare outside edges for $E$, \nthe edge $e^*_3$ is outward at $v_s$, and\n$e^*_1$ is middle at $v_p$ \n$($see Fig.~\\ref{fig18}{\\rm(a))}.\n\\item[{\\rm (c)}] \nSuppose that \n$e^*_1\\cup e^*_2\\subset \\partial E$. \nIf $e^*_3$ is an inside edge for $E$, \nthen one of $e^\\dagger_2,e^\\dagger_3$ is \nan  inside edges for $E$, and\nthe other is an outside edge for $E$ \noutward and\nmiddle at $v_p$\n$($see Fig.~\\ref{fig18}{\\rm(b))}.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{Proof}\n{\\bf Statement (a)}. \nSince $\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$,\nthe disk $E$ does not contain \na terminal edge \nby Lemma~\\ref{lemNoTerminal}.\nThus if $v_p\\in$~Int~$E$, \nthen there exists \nan edge $e$ of label $k$\noutward at $v_p$. \nSince there does not exist \na one-way cycle of label $k$\nby Lemma~\\ref{LemOneWayCycle},\nwe have $P\\cap e=v_p$. \nHence \n$P\\cup e$ is \na one-way path of label $k$\ndominated by $E$. \nThis contradicts that\nthe path $P$ is upward maximal.\n\n{\\bf Statement (b)}.\nThe edge $e_1^\\dagger=e_p$ is an edge of label $k$ inward at $v_p$\nin $E$.\nSuppose that $e^*_3\\subset E$. \nThen $E$ contains one of two edges $e^\\dagger_2,e^\\dagger_3$\nof label $k$ at $v_p$, \nsay $e^\\dagger_2$. \nIf $e^\\dagger_2$ is inward at $v_p$, \n(see Fig.~\\ref{fig18}(c)), \nthen \nthe edge $e^*_3$ is inward at $v_p$,\nbecuase $e^*_3$ is situated \nbetween $e^\\dagger_1=e_p$ and $e^\\dagger_2$ \naround $v_p$. \nHence the four edges \n$e^\\dagger_1,e^\\dagger_2,e^*_1,e^*_3$ are \ninward at $v_p$. \nThis contradicts the definition of a chart. \nThus $e^\\dagger_2$ is outward at $v_p$. \nSince there does not exist \na one-way cycle of label $k$\nby Lemma~\\ref{LemOneWayCycle},\nwe have $P\\cap e^\\dagger_2=v_p$. \nHence \n$P\\cup e^\\dagger_2$ is \na one-way path of label $k$\ndominated by $E$. \nThis contradicts that\nthe path $P$ is upward maximal.\nThus $e^*_3\\not\\subset E$ \n(see Fig.~\\ref{fig18}(a)). \nThen $e^\\dagger_2,e^\\dagger_3$ \nare outside edges for $E$.\nSince $e_p=e^\\dagger_1$ is inward at $v_p$, \nthe edge $e^*_3$ is outward at $v_p$. \nThus $e^*_1$ is middle at $v_p$.\nHence Statement (b) holds.\n\n{\\bf Statement (c)}. \nSuppose that $e^*_3$ is an inside edge for $E$ \n$($see Fig.~\\ref{fig18}{\\rm(b))}. \nThen one of \n$e^\\dagger_2,e^\\dagger_3$ \nis an inside edge for $E$,\nsay $e^\\dagger_2$.\nThen $e^\\dagger_3$ is \nan outside edge for $E$. \nIf $e^\\dagger_2$ is outward at $v_p$, \nthen again $P\\cup e^\\dagger_2$ is \na one-way path of label $k$.\nThis contradicts that\nthe path $P$ is upward maximal.\nHence $e^\\dagger_2$ is \ninward at $v_p$. \nThus $e^\\dagger_3$ is \noutward and middle at $v_p$. \n\\end{Proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig18.pdf}\n\\end{center}\n\\caption{\\label{fig18} }\n\\end{figure}\n\n\nSimilarly we can show the following lemma.\n\n\\begin{lem}\\label{lemMaxAtBoundaryDown} \nLet $\\Gamma$ be a minimal chart, \nand $m,k$ labels of $\\Gamma$  \nwith $|m-k|=1$.\nLet $E$ be a disk with \n$\\partial E\\subset\\Gamma$ and \n$\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$. \nLet $P$ be \na one-way path \nof label $m$ \ndownward maximal \nwith respect to $E$ \nwith a vertex sequence \n$(v_0,v_1,v_2,\\cdots,v_p)$ and\nan edge sequence \n$(e_1,e_2,\\cdots,e_p)$. \nLet $e^\\dagger_1,e^\\dagger_2,e^\\dagger_3$ \nbe the edges of label $m$ at $v_0$, and\n$e^*_1,e^*_2,e^*_3$ the edges \nof label $k$  at $v_0$, \nhere $e^\\dagger_1=e_1$. \nThen we have the following:\n\\begin{enumerate}\n\\item[{\\rm (a)}]\n$ v_0\\in\\partial E$.\n\\item[{\\rm (b)}] \nSuppose that \n$e^*_1\\cup e^*_2\\subset \\partial E$.\nIf \nthe edge $e^*_1$ is inward at $v_0$, \nand if \nthe edge $e^*_2$ is outward at $v_0$, \nthen \n$e^\\dagger_2,e^\\dagger_3,e^*_3$ \nare outside edges for $E$, \nthe edge $e^*_3$ is inward at $v_0$, and\n$e^*_2$ is middle at $v_0$.\n\\item[{\\rm (c)}] \nSuppose that \n$e^*_1\\cup e^*_2\\subset \\partial E$.\nIf  $e^*_3$ is an inside edge for $E$, \nthen one of $e^\\dagger_2,e^\\dagger_3$ is \nan inside edge for $E$, \nand the other is \nan outside edge for $E$ \ninward and\nmiddle at $v_0$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem} \\label{lemBetweenMiddle}\nLet $\\Gamma$ be a minimal chart, \nand $m,k$ labels of $\\Gamma$\nwith $|m-k|=1$.\nLet $E$ be a disk  \nbounded by a cycle $C$ in $\\Gamma_m$ \nwith $\\Gamma\\cap E\\subset\\Gamma_m\\cup\\Gamma_k$.\nLet \n$e^\\dagger,e^*$ \nbe outside edges \nof label $k$ for $E$ \nmiddle at vertices $v^\\dagger,v^*$ in\n$C$ respectively. \nThen any arc $L$ on $C$\nwith $\\partial L=\\{v^\\dagger,v^*\\}$  \nintersects \nan outside edge of label $m$ \nfor $E$.\n\\end{lem}\n\n\n\\begin{Proof}\nIt is sufficient to show \nfor the case that \n\\begin{enumerate}\n\\item[(1)] \nno outside edge of label $k$ for $E$ \nis middle at a vertex in Int~$L$.\n\\end{enumerate}\nSuppose that \n$L$ does not intersect \nan outside edge of label $m$ for $E$. \nThen $L$ is a one-way path of label $m$ \nby (1). \nLet $(v_0,v_1,v_2,\\cdots,v_p)$ be \na vertex sequence of $L$. \nWithout loss of generality \nwa can assume that\n$e^\\dagger$ is outward at $v_0$ and\n$e^*$ is inward at $v_p$.\nHence there exist \nan outside edge of label $k$ for $E$ \noutward at $v_i$ and\nan outside edge of label $k$ for $E$ \ninward at $v_{i+1}$\nfor some integer $i$ with $0\\le i<p$. \nThen we can eliminate the two vertices \n$v_i,v_{i+1}$ \nby two C-I-M2 moves and one C-I-M3 move. \nThis contradicts that \nthe chart is minimal.\nThus the arc $L$ intersects \nan outside edge of label $m$ for $E$.\n\\end{Proof}\n\n\n\n\\section{Proof of Theorem~1.4}\n\\label{s:Th4}\n\nThroughout this section we assume that\n(see Fig.~\\ref{fig19})\n\\begin{enumerate}\n\\item[$\\bullet$] \n$\\Gamma$ is a  minimal chart, \nand $m$ is a label of $\\Gamma$, \n\\item[$\\bullet$] \n$(\\Gamma\\cap D,D)$ is \na Type-I elementary IO-tangle of label $m$\nwith a boundary IO-arc pair $(L_I,L_O)$,\n\\item[$\\bullet$] \n$k$ is a label of $\\Gamma$ with $|m-k|=1$\nand $\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$,\n\\item[$\\bullet$] \n$e_I,e_O$ are the edges of label $m$ \nintersecting $\\partial D$,\n\\item[$\\bullet$]\n$\\Delta$ is a disk \ncontaining \nall the white vertices in $D$\nwith $\\partial\\Delta\\subset \n\\Gamma_m\\cap D$, \n\\item[$\\bullet$]\n$\\Delta_I$ \n(resp. $\\Delta_O$) \nis the closure of \na connected component of \n$D-(\\Delta\\cup e_I\\cup e_O)$ with\n$L_I\\subset \\Delta_I$ \n(resp. $L_O\\subset \\Delta_O$), and\n\\item[$\\bullet$]\n$J_I$ (resp. $J_O$) is \nthe closure of \na connected component of \n$\\partial \\Delta-(e_I\\cup e_O)$ with\n$J_I\\subset \\Delta_I$ \n(resp. $J_O\\subset \\Delta_O$).\n\\end{enumerate}\nThen we have the following \nby the definition of a simple IO-tangle.\n\\begin{enumerate}\n\\item[$\\bullet$] \nany edge dominated by $\\Delta_I$ is \ninward at a vertex in $J_I$.\n\\item[$\\bullet$] \nany edge dominated by $\\Delta_O$ is \noutward at a vertex in $J_O$.\n\\item[$\\bullet$] \nany edge of label $m$ intersecting \nInt~$J_I$ is a terminal edge \ninward at a vertex in $J_I$.\n\\item[$\\bullet$] \nany edge of label $m$ intersecting \nInt~$J_O$ is a terminal edge \noutward at a vertex in $J_O$.\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig19.pdf}\n\\end{center}\n\\caption{ \\label{fig19} \nThick arcs are of label $m$.}\n\\end{figure}\n\n\n\n\n\n\\begin{lem}\\label{Lemma1}\nLet $\\tau$ be a terminal edge of label $m$ \nin $\\Delta_I$\ninward at a vertex $v_0\\in J_I$, and \n$P$ a one-way path \nof label $k$ \nupward maximal \nwith respect to  $\\Delta$ \nwith a vertex sequence \n$(v_0,v_1,v_2,\\cdots,v_p)$. \nThen\n\\begin{enumerate}\n\\item[{\\rm (a)}] \n$v_p\\in J_O$, and\n\\item[{\\rm (b)}] \nthe disk $\\Delta_O$ dominates \nan outside edge $e^\\dagger$ \nof label $k$ \nfor $\\Delta$ \noutward and middle at $v_p$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{Proof}\nLet \n$(e_1,e_2,\\cdots,e_p)$ be \nan edge sequence \nof the path $P$.\nSince the terminal edge $\\tau$ is \nmiddle at $v_0$, so is the edge $e_1$. \n\n{\\bf Statement (a)}. \nSuppose that  $v_p\\not\\in J_O$. \nSince $P$ is \nupward maximal with respect to $\\Delta$, \nwe have \n$v_p\\in\\partial \\Delta=J_I\\cup J_O$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(a). \nHence $v_p\\in J_I-J_O$. \nLet $e$ be the edge of label $m$ \nat $v_p$ with $e\\not\\subset J_I$.\nThen $e\\subset \\Delta$. \nFor, if not, \nthe edge $e$ is dominated by $\\Delta_I$.\nHence $e$ is \na terminal edge at $v_p$. \nThus $e_p$ is middle at $v_p$. \nSince $e_1$ is middle at $v_0$, \nthe path $P$ is \na dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}. \nHence $e\\subset\\Delta$. \n\nNow there exists an outside edge $e'$\nof label $k$ for $\\Delta$ outward at $v_p$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(c). \nSince $e'$ is dominated by $\\Delta_I$, \nthe edge $e'$ must be inward at $v_p$.\nThis is a contradiction.\nThus Statement (a) holds.\n\n{\\bf Statement (b)}. \nThere are two cases.\\\\\nCase 1. $v_p\\in \\partial J_O$.\\\\\nCase 2. $v_p\\in$~Int~$J_O$.\n\n{\\bf Case 1.} \nThere exist \ntwo edges \n$e^\\dagger,e'$ \nof label $k$  \nat $v_p$ \nsuch that\n$e^\\dagger$ is dominated by $\\Delta_O$ and \noutward at $v_p$, and that\n$e'$ is dominated by $\\Delta_I$ and\ninward at $v_p$. \nSince $e_p$ is inward at $v_p$, \nthe edge $e^\\dagger$ is middle at $v_p$.\nHence Statement (b) holds for this case.\n\n{\\bf Case 2.}\nLet $e$ be the edge \nof label $m$ at $v_p$ with \n$e\\not\\subset J_O$. \nWe claim that $e\\subset \\Delta$. \nFor, if $e\\not\\subset \\Delta$, \nthen $e$ is dominated by $\\Delta_O$. \nHence $e$ is a terminal edge.\nThus $e_p$ is middle at $v_p$. \nSince $e_1$ is middle at $v_0$, \nthe path $P$ is \na dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\nHence $e\\subset \\Delta$.\n\nSince $P$ is \na one-way path \nof label $k$ \nupward maximal \nwith respect to $\\Delta$\nand since $e$ is \nan inside edge \nof label $m$ for $\\Delta$,\nthere exists an outside edge of label $k$ \nfor $\\Delta$ outward and middle at $v_p$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(c).\nThus Statement (b) holds \nfor this case, too.\n\\end{Proof}\n\n\nThe path $P$ in Lemma~\\ref{Lemma1} \nis called \nan {\\it upward principal path} \nfor $\\tau$, \nand $e^\\dagger$ a {\\it corresponding edge} \nfor $\\tau$.\n\n\\begin{remark} \\label{remUpwardMaximal} \n{\\rm Since $e_1$ is middle at $v_0$, \nthe path $P$ is \nan upward right-selective and \nleft-selective one-way path \nstarting from $e_1$ \ndominated by $\\Delta$\nby Lemma~\\ref{LemNoM-M}.}\n\\end{remark}\n\nBy the similar way as \nthe one of Lemma~\\ref{Lemma1}\nwe can show the following lemma.\n\n\\begin{lem}\\label{Lemma2}\nLet $\\tau$ be a terminal edge of label $m$ \nin $\\Delta$ \noutward at a vertex $v^\\dagger\\in J_O$, and \n$P$ a one-way path \nof label $k$ \ndownward maximal \nwith respect to  $\\Delta$ \nwith a vertex sequence \n$(v_0,v_1,v_2,\\cdots,v_p)$, \nand $v_p=v^\\dagger$. \nThen\n\\begin{enumerate}\n\\item[{\\rm (a)}] \n$v_0\\in J_I$, and\n\\item[{\\rm (b)}] \nthe disk $\\Delta_I$ dominates \nan outside edge $e^*$ \nof label $k$ for $\\Delta$\ninward and middle at $v_0$. {\\hfill {$\\square$}\\vspace{1.5em}}\n\\end{enumerate}\n\\end{lem}\n\nThe path $P$ in Lemma~\\ref{Lemma2} is called \na {\\it downward principal path} \nfor $\\tau$, \nand $e^*$ a {\\it corresponding edge} \nfor $\\tau$.\n\n\nLet $P$ be a path of $\\Gamma$, and\n$e^\\dagger,e^*$ \nedges such that each of \n$e^\\dagger\\cap P$ and $e^*\\cap P$ \nis a vertex, say \n$v^\\dagger=e^\\dagger\\cap P,\nv^*=e^*\\cap P$. \nThe arc $P[v^\\dagger,v^*]$ is called \na {\\it connection arc} between \n$e^\\dagger$ and $e^*$ \nwith respect to $P$, \nand denoted by $P[e^\\dagger,e^*]$.\nLet $\\tau^\\dagger,\\tau^*$ be \nterminal edges intersecting $P$.\nIf \nInt~$P[\\tau^\\dagger,\\tau^*]$ \ndoes not intersect any terminal edge, \nthen $\\tau^\\dagger,\\tau^*$ are said \nto be {\\it adjacent} with respect to $P$.\n\n\\begin{lem}\\label{Lemma3}\nLet $\\tau^\\dagger,\\tau^*$ be \nterminal edges of label $m$ in $\\Delta_I$. \nThen two upward principal paths \nfor the two terminal edges do not intersect \neach other.\n\\end{lem}\n\n\\begin{Proof}\nSince each upward principal path \nfor a terminal edge \nsplits the disk $\\Delta$, \nit is sufficient to show \nfor the case that \n$\\tau^\\dagger$ and $\\tau^*$ are adjacent \nwith respect to $J_I$.\n\nLet $P^\\dagger,P^*$ be \nthe upward principal paths \nfor $\\tau^\\dagger,\\tau^*$, and\n$(v^\\dagger_0,v^\\dagger_1,v^\\dagger_2,\n\\cdots,v^\\dagger_p)$,\n$(v^*_0,v^*_1,v^*_2,\\cdots,v^*_q)$\nvertex sequences of \n$P^\\dagger,P^*$ respectively.\nSuppose that \n$P^\\dagger\\cap P^*\\neq\\emptyset$.\nLet \n\\begin{enumerate}\n\\item[] $s=\\min\\{i~|~v^\\dagger_i\\in P^*\\}$, $t=\\min\\{i~|~v^*_i\\in P^\\dagger\\}$.\n\\end{enumerate}\nLet $\\widetilde L$ be the connection arc \n$J_I[v^\\dagger_0,v^*_0]$. \nLet $e^\\dagger$ be a sibling of \n$\\tau^\\dagger$ \nsituated between \n$\\widetilde L$ and $\\tau^\\dagger$ \naround the vertex \n$v^\\dagger_0=\\tau^\\dagger\\cap J_I$, and \n$e^*$ a sibling of $\\tau^*$ \nsituated between $\\widetilde L$ and $\\tau^*$ \naround the vertex \n$v^*_0=\\tau^*\\cap J_I$. \nThen \n$e^\\dagger\\cup \nP^\\dagger[v^\\dagger_0,v^\\dagger_s]\n\\cup P^*[v^*_0,v^*_t]\n\\cup e^*$\nsplits the disk $D$ into disks. \nLet $E$ be the one of the disks \ncontaining $\\widetilde L$ \n(see Fig.~\\ref{fig20}).\nThen $E$ is a half spindle of label $k$ \nby Remark~\\ref{remUpwardMaximal}. \nThis contradicts Lemma~\\ref{lemHalfSpindle}. \nTherefore Lemma~\\ref{Lemma3} holds. \n\\end{Proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig20.pdf}\n\\end{center}\n\\caption{ \\label{fig20} \nRight gray is the half spindle $E$.}\n\\end{figure}\n\n\n\n\\begin{lem}\\label{Lemma4}\nEach of the disks \n$\\Delta_I,\\Delta_O$ contains \nexactly one terminal edge of label $m$.\n\\end{lem}\n\n\\begin{Proof} \nSuppose that \n$\\Delta_I$ contains \nat least two terminal edges of label $m$.\nLet $\\tau^\\dagger,\\tau^*$ be \nadjacent terminal edges with respect to $J_I$. \nLet $P^\\dagger,P^*$ be \nthe upward principal paths \nof label $k$\nfor $\\tau^\\dagger,\\tau^*$ \nrespectively, and\n$e^\\dagger,e^*$ \nthe corresponding edges of label $k$ \nfor $\\tau^\\dagger,\\tau^*$ \nrespectively. \nSet $v^\\dagger=e^\\dagger\\cap J_O, \nv^*=e^*\\cap J_O$.\n\nThere exists \nan outside edge $\\tau'$ \nof label $m$ for $\\Delta$\nat a vertex in the arc \n$J_O[v^\\dagger,v^*]$ \nby Lemma~\\ref{lemBetweenMiddle}.\nThen $\\tau'$ is a terminal edge.\nLet $P'$ be a downward principal path \nof label $k$ \nfor $\\tau'$ and\n$e'$ a corresponding edge of label $k$ for \n$\\tau'$. \nSet $v'_0=e'\\cap J_I$.\nWe claim that\n\\begin{enumerate}\n\\item[(1)] \n$P'\\cap (P^\\dagger\\cup P^*)=\\emptyset$,\n\\item[(2)] \nthe edge $\\tau'$ is the only one\noutside edge of label $m$ \nfor $\\Delta$\nat a vertex in the arc \n$J_O[v^\\dagger,v^*]$, \n\\item[(3)] \nthe disk $\\Delta_O$ does not dominate \nany edge of label $k$ middle at a vertex \nin Int~$J_O[v^\\dagger,v^*]$, and\n\\item[(4)] there exists one inside edge of label $m$ at $v_0'$ for $\\Delta$ and \nthere exist two inside edge of label $k$ at $v_0'$ for $\\Delta$.\n\\end{enumerate}\n\n{\\bf Statement (1)}. \nLet $(v'_0,v'_1,v'_2,\\cdots,v'_p),\n(v^\\dagger_0,v^\\dagger_1,\nv^\\dagger_2,\\cdots,v^\\dagger_q)$ be \nvertex sequences of $P',P^\\dagger$. \nIf $P'\\cap P^\\dagger\\neq\\emptyset$, \nthen $v'_i=v^\\dagger_j$ for some \n$1\\le i<p,1\\le j<q$. Then \n$P^\\dagger[v^\\dagger_0,v^\\dagger_j]\\cup\nP'[v'_i,v'_p]$ is \na dichromatic M$\\&$M one-way path of label $k$. \nThis contradicts Lemma~\\ref{LemNoM-M}.\nThus $P'\\cap P^\\dagger=\\emptyset$. \nSimilarly we can show $P'\\cap P^*=\\emptyset$. \nThus Statement (1) holds.\n\n{\\bf Statement (2)}. \nSuppose that \nthere exists \nanother outside edge $\\tau''$ \nof label $m$\nfor $\\Delta$\nat a vertex in the arc \n$L_O[v^\\dagger,v^*$]. \nLet $P''$ be a downward principal path \nof label $k$ \nfor $\\tau''$ and\n$e''$ a corresponding edge of label $k$ for \n$\\tau''$. \nThen \nby Lemma~\\ref{lemBetweenMiddle}\nthere exists an outside edge \n$e'''$ \nof label $m$ \nfor $\\Delta$ \nat a vertex in \n$J_I[e',e'']\\subset \nJ_I[\\tau^\\dagger,\\tau^*]$.\nThe edge $e'''$ is a terminal edge. \nThis contradicts the fact that \n$\\tau^\\dagger,\\tau^*$ \nare adjacent terminal edges \nwith respect to $J_I$. \nThus Statement (2) holds.\n\n{\\bf Statement (3)}. \nSuppose that \n$\\Delta_O$ dominates \nan edge of label $k$ \nmiddle at a vertex $\\widetilde v$\nin Int~$J_O[v^\\dagger,v^*]$. \nThen each of \n$J_O[v^\\dagger,\\widetilde v]$ and \n$J_O[\\widetilde v,v^*]$ \nintersects \nan edge of label $m$ \ndominated by $\\Delta_O$ \nby Lemma~\\ref{lemBetweenMiddle}. \nEach of the two edges is a terminal edge of label $m$.\nThis contradicts Statement (2).\nThus Statement (3) holds. \n\n{\\bf Statement (4)}. \nSince $\\tau^{\\dagger},\\tau^*$ are adjacent terminal edges with respect to $J_I$,\nthe disk $\\Delta_I$ does not dominate an edge of label $m$ at a vertex in ${\\rm Int}~J_I[\\tau^{\\dagger},\\tau^*]$.\nThus there does not exist an outside edge of label $m$ at $v_0'$ for $\\Delta$.\nHence there exists an inside edge of label $m$ at $v_0'$ for $\\Delta$,\nand there exist two inside edges of label $k$ at $v_0'$ for $\\Delta$.\nThus Statement (4) holds. \n\n\nNow the three paths $P^\\dagger,P',P^*$ split \nthe disk $\\Delta$ into disks.\nLet $\\Delta^\\dagger$ be \nthe one of the disks intersecting \nboth of $P^\\dagger$ and $P'$, and\n$\\Delta^*$ \nthe one of the disks intersecting \nboth of $P'$ and $P^*$ \n(see Fig.~\\ref{fig21}(a)).\nLet $(v'_0,v'_1,v'_2,\\cdots,v'_p)$ and \n$(e'_1,e'_2,\\cdots,e'_p)$ \nbe a vertex sequence and \nan edge sequence of $P'$, \nhere $v'_0\\in e'$ and \n$v'_p\\in\\tau'$.\n\nLet $e''$ be the edge \nof label $k$ at $v'_{p-1}$ \nnot contained in $P'\\cup e'$\n(possibly $v'_{p-1}=v'_0$). \nThen \nif $p>1$,\nthen $v_{p-1}'\\in{\\rm Int~\\Delta}$\nimplies $e''\\subset \\Delta$,\nand if $p=1$,\nthen  $e''\\subset \\Delta$ by (4).\nWe claim that $e''$ is outward at $v'_{p-1}$. \nFor, if $e''$ is inward at $v'_{p-1}$, \nthen $e'_p$ is \na dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\nLet $\\widehat e$ be the edge of label $m$ \noutward at $v'_{p-1}$\nsituated between $e''$ and $e'_{p}$ \naround $v'_{p-1}$. \n\\begin{enumerate}\n\\item[(5)]\nThe edge $\\widehat e$ is \nmiddle at $v'_{p-1}$.\n\\end{enumerate}\nWithout loss of generality \nwe can assume that \n$e'',\\widehat e\\subset \\Delta^\\dagger$. \nLet $\\widehat P$ be \na one-way path \nof label $m$\nstarting from $\\widehat e$ \nupward maximal \nwith respect to $\\Delta^\\dagger$ \nwith a vertex sequence \n$(\\widehat v_0,\\widehat v_1,\\widehat v_2,\n\\cdots,\\widehat v_s)$ and \nan edge sequence \n$(\\widehat e_1,\\widehat e_2,\n\\cdots,\\widehat e_s)$, \nhere $\\widehat v_0=v'_{p-1}$. \nThere are five cases:\n\\begin{enumerate}\n\\item[] Case 1. $\\widehat v_s\\in$~Int~$P^\\dagger$.\n\\item[] Case 2. $\\widehat v_s=v^\\dagger$.\n\\item[] Case 3. $\\widehat v_s\\in {\\rm Int}~J_I[v^{\\dagger}_0,v'_{0}]$.\n\\item[] Case 4. $\\widehat v_s\\in P'[v'_0,v'_{p-1}]$.\n\\item[] Case 5. \n$\\widehat v_s\\in$~Int~$J_O[v^\\dagger,v'_p]$.\n\\end{enumerate}\n\n{\\bf Case 1}. \nThere exists an outside edge \nof label $k$ \nfor $\\Delta^\\dagger$ \noutward at $\\widehat v_s$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(b).\nThen $P^\\dagger[v^\\dagger_0,\\widehat v_s]$ \nis a dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\n{\\bf Case 2}. \nThe edge $\\widehat e_s$ is \nmiddle at $\\widehat v_s$,\nbecause $e^{\\dagger}$ is middle at \n$v^{\\dagger}=\\widehat v_s$. \nThus the path $\\widehat P$ is a dichromatic \nM$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\n{\\bf Case 3}.\nThere exists an outside edge of label $k$ for $\\Delta^\\dagger$ outward at $\\widehat v_s$. \nThis contradicts the fact that any edge dominated by $\\Delta_I$ is inward at a vertex in $J_I$.\n\n{\\bf Case 4}. \nSince there does not exist \none-way cycle in $\\Delta$\nby Lemma~\\ref{LemOneWayCycle},\nwe have $\\widehat v_s=v'_i$ \nfor some $0\\le i< p-1$.\nLet $E$ be the disk \nbounded by the cycle \n$\\widehat P\\cup P'[v'_i,v'_{p-1}]$ \n(see Fig.~\\ref{fig21}(b)). \nLet $\\widetilde e$ be \nan edge of label $m$ \nat $v'_{p-1}$ situated between \n$e''$ and $e'_{p-1}$ around $v'_{p-1}$.\nSince $e'',\\widehat e,e'_p$ \nare outward at $v'_{p-1}$,\nthe edge $\\widetilde e$ is \ninward at $v'_{p-1}$. \nLet $\\widetilde P$ be \na one-way path \nof label $m$ \nleading to $\\widetilde e$ \ndownward maximal \nwith respect to $E$ \nwith a vertex sequence \n$(\\widetilde v_0,\\widetilde \nv_1,\\widetilde v_2,\\cdots,\n\\widetilde v_t)$.\nIf $\\widetilde v_0\\in \\widehat P$, \nthen we can find a one-way cycle \nin $\\widehat P\\cup \\widetilde P$.\nThis contradicts Lemma~\\ref{LemOneWayCycle}.\nThus $\\widetilde v_0\\in P'[v'_i,v'_{p-1}]$. \nNamely $\\widetilde v_0=v'_j$ \nfor some $i\\le j<p-1$.\nThen $e'_{j+1}$ is middle at $v'_j$ \nby Lemma~\\ref{lemMaxAtBoundaryDown}(b). \nHence $P'[v'_j,v'_p]$ is a dichromatic M$\\&$M-path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\n{\\bf Case 5}. \nLet $E$ be the disk bounded by \n$\\widehat P\\cup J_O[\\widehat v_s,v'_p]\n\\cup e'_p$ \n(see Fig.~\\ref{fig21}(c)).\nBy Statement (2) and (3) we have \n\\begin{enumerate}\n\\item[(6)] \nthe path \n$J_O[v^\\dagger,v'_p]$ \nis a one-way path \nof label $m$ \noriented from \n$v^\\dagger$ to $v'_p$.\n\\end{enumerate}\nLet $\\widetilde e$ be \nthe edge of label $k$\noutward at $\\widehat v_s$ in the disk $E$.\nLet $\\widetilde P$ be \na one-way path of label $k$ \nstarting from $\\widetilde e$\nupward maximal \nwith respect to $E$ \nwith a vertex sequence \n$(\\widetilde v_0,\\widetilde v_1,\n\\widetilde v_2,\\cdots,\\widetilde v_t)$, \nhere $\\widetilde v_0=\\widehat v_s$. \nNow $\\widetilde v_t\\neq v'_p$. \nIf $\\widetilde v_t\\in J_O[\\widehat v_s,v'_p]$, \nthen Statement (6) implies that\nthere exists an outside edge \nof label $m$ for $E$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(b). \nSince the outside edge \nof label $m$ is \ndifferent from $\\tau'$, \nthis contradicts Statement (2).\nHence $\\widetilde v_t\\in \\widehat P$. \nThen $\\widetilde v_t=\\widehat v_j$ \nfor some $1\\le j<s$.\nHence there exists \nan outside edge of label $m$ \nfor $E$ outward at $\\widehat v_j$ \nby Lemma~\\ref{lemMaxAtBoundaryUp}(b).\nThus $\\widehat e_j$ is \nmiddle at $\\widetilde v_j$. \nSince $\\widehat e_1=\\widehat e$ is \nmiddle at $\\widehat v_0$\nby Statement (5),\nthe path  $\\widehat P[\n\\widehat v_0,\\widehat v_j]$ \nis \na dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\nSince we get a contradiction for \neach of the five cases, \nthe disks $\\Delta_I$ \ncontains at most one terminal edge. \nSimilarly we can show that \nthe disks $\\Delta_O$ \ncontains at most one terminal edge. \nBy Lemma~\\ref{LemTwoColorTangle}, \nthere exist at least two terminal edges \nin $\\Delta_I\\cup\\Delta_O$. \nTherefore each of the disks \n$\\Delta_I,\\Delta_O$ \ncontains exactly one terminal edge.\nThus Lemma~\\ref{Lemma4} holds. \n\\end{Proof}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig21.pdf}\n\\end{center}\n\\caption{ \\label{fig21} (b), (c) The disk $E$ is right gray.}\n\\end{figure}\n\nThroughout this section \nfurther we assume that \n(see Fig.~\\ref{fig22}(a))\n\\begin{enumerate}\n\\item[$\\bullet$] \n$v_I=e_I\\cap\\partial\\Delta,~\nv_O=e_O\\cap\\partial\\Delta$.\n\\item[$\\bullet$]\n$\\tau_I$ is \nthe terminal edge \nin $\\Delta_I$, and\n$\\tau_O$ is \nthe terminal edge \nin $\\Delta_O$. \n\\item[$\\bullet$]\n$w_I=\\tau_I\\cap\\partial\\Delta$, and\n$w_O=\\tau_O\\cap\\partial\\Delta$.\n\\item[$\\bullet$] \n$P_I$ is an upward principal path \nof label $k$ for $\\tau_I$.\n\\item[$\\bullet$] \n$P_O$ is a downward principal path \nof label $k$ for $\\tau_O$.\n\\end{enumerate}\n\nThen we have the following:\n\\begin{enumerate}\n\\item[$\\bullet$] \nNeither $\\Delta_I$ nor $\\Delta_O$ \ndominates an outside edge \nof label $m$ for $\\Delta$ \nintersecting \n$\\partial\\Delta-\\{v_I,w_I,v_O,w_O\\}$.\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig22.pdf}\n\\end{center}\n\\caption{ \\label{fig22} \nThick arcs are of label $m$.}\n\\end{figure}\n\n\n\\begin{lem}\\label{Lemma5}\nThere does not exist \nan outside edge of label $k$ for $\\Delta$ \nmiddle at a vertex in \n$\\partial\\Delta-\\{v_I,v_O,w_I,w_O\\}$.\n\\end{lem}\n\n\\begin{Proof} \nSuppose that \nthere exists an outside edge \nof label $k$ for $\\Delta$ \nmiddle at a vertex in \nInt~$J_O[v_I,w_O]$. \nLet $v^\\dagger$ be the vertex  \nin Int~$J_O[v_I,w_O]$ \nsuch that \nan outside edge $e^\\dagger$\nof label $k$ for $\\Delta$ \nis middle at $v^\\dagger$ but \nno outside edge \nof label $k$ for $\\Delta$ \nis middle at a vertex \nin Int~$J_O[v_I,v^\\dagger]$. \nSince $\\Delta_O$ does not dominate \nan outside edge of label $m$\nintersecting Int~$J_O[v_I,w_O]$, \nand \nsince \nno outside edge \nof label $k$ for $\\Delta$ \nis middle at a vertex \nin Int~$J_O[v_I,v^\\dagger]$, \nthe path $J_O[v_I,v^\\dagger]$  \nis a one-way path. \nSince $e^\\dagger$ is outward at $v^\\dagger$, \nthe one-way path \n$J_O[v_I,v^\\dagger]$\nis oriented from $v^\\dagger$ to $v_I$. \nThus an edge $e$ \nin $J_O[v_I,v^\\dagger]$ with $v_I\\in e$\nis inward at $v_I$. \nHence by Condition (iii) \nof the definition of a chart,\nthe edge $e'$ \nbetween $e_I$ and $J_I[v_I,w_I]$ \naround $v_I$ \nis outward at $v_I$ \n(see Fig.~\\ref{fig22}(b)). \nBut the edge $e'$ is dominated by \n$\\Delta_I$. \nThus $e'$ must be \ninward at $v_I$. \nThis is a contradiction.\nThus there does not exist \nan outside edge \nof label $k$ for $\\Delta$ \nmiddle at a vertex in \nInt~$J_O[v_I,w_O]$. \n\nSimilarly we can show that \nthere does not exist \nan outside edge \nof label $k$ for $\\Delta$ \nmiddle at a vertex in \nInt~$J_O[w_O,v_O]\n\\cup$~Int~$J_I[v_O,w_I]\n\\cup$~Int~$J_I[w_I,v_I]$.\nThus Lemma~\\ref{Lemma5} holds. \n\\end{Proof}\n\n\n\\begin{lem}\\label{Lemma6}\nOne of $e_I,e_O$ is an I-edge and \nthe other is an O-edge.\n\\end{lem}\n\n\n\\begin{Proof}\nSuppose that the both of $e_I,e_O$ \nare I-edges.\nNamely $e_I$ is inward at $v_I$, and\n$e_O$ is inward at $v_O$. \nBy Lemma~\\ref{Lemma5}, \nthere does not exist \nan outside edge of label $k$ \nfor $\\Delta$ middle at \na vertex in \n$\\partial\\Delta-\\{v_I,v_O,w_I,w_O\\}$. \nSince $e_I,e_O$ are inward at \n$v_I,v_O$ respectively,\nthe disk $\\Delta_O$ does not dominate \nan edge of label $k$ middle at $v_I,v_O$. \nFurther the siblings of $\\tau_O$ are not \nmiddle at $w_O$. \nHence $\\Delta_O$ does not dominate \nan edge of label $k$ \nmiddle at a vertex in $J_O$. \nThus considering \nthe upward principal path $P_I$, \nwe have a contradiction \nby Lemma~\\ref{Lemma1}. \nThus one of $e_I$ and $e_O$ is an O-edge.\nSimilarly we can show that\none of $e_I$ and $e_O$ is an I-edge.\nHence Lemma~\\ref{Lemma6} holds. \n\\end{Proof}\n\n\n\nTherefore throughout this section \nwe can assume that \n(see Fig.~\\ref{fig23})\n\\begin{enumerate}\n\\item[$\\bullet$] \n$e_I$ is inward at $v_I$, and \n$e_O$ is outward at $v_O$.\n\\item[$\\bullet$] \n$J_I[w_I,v_I]$ is a one-way path of label $m$ \noriented from $w_I$ to $v_I$.\n\\item[$\\bullet$] \n$J_I[w_I,v_O]$ is \na one-way path of label $m$ \noriented from $w_I$ to $v_O$.\n\\item[$\\bullet$] \n$J_O[v_I,w_O]$ is a one-way path of label $m$ \noriented from $v_I$ to $w_O$.\n\\item[$\\bullet$] \n$J_O[v_O,w_O]$ is a one-way path of label $m$ \noriented from $v_O$ to $w_O$.\n\\end{enumerate}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig23.pdf}\n\\end{center}\n\\caption{ \\label{fig23} Thick arcs are of label $m$.}\n\\end{figure}\n\n\n\\begin{lem}\\label{Lemma7}\nLet $(v^*_0,v^*_1,v^*_2,\\dots,v^*_p)$ \nbe a vertex sequence of \nthe upward principal path $P_I$ for $\\tau_I$, \nand \n$(v^\\dagger_0,v^\\dagger_1,\nv^\\dagger_2,\\dots,v^\\dagger_q)$  \na vertex sequence of \nthe downward principal path $P_O$ \nfor $\\tau_O$.\nThen we have the following:\n\\begin{enumerate}\n\\item[{\\rm (a)}]\n$v^*_p=v_O,~v^\\dagger_0=v_I$.\n\\item[{\\rm (b)}] \n$P_I\\cap$~{\\rm Int}~$J_O=\\emptyset,~\nP_O\\cap$~{\\rm Int}~$J_I=\\emptyset$. \n\\item[{\\rm (c)}] \n$P_I-J_I[w_I,v_O]$ does not contain \nany white vertex, and \nneither does $P_O-J_O[w_O,v_I]$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{Proof} \nLet \n$(e^*_1,e^*_2,\\dots,e^*_p),\n(e^\\dagger_1,e^\\dagger_2,\\dots,\ne^\\dagger_q)$\nbe edge sequences of $P_I, P_O$ respectively.\n\n{\\bf Statement (a)}.\nBy Lemma~\\ref{Lemma1} \nand Lemma~\\ref{Lemma5}, we have \n$v^*_p=v_I,~v^*_p=w_O,$ or $v^*_p=v_O$.\nIf $v^*_p=w_O$, \nthen the path $P_I$ is \na dichromatic M$\\&$M one-way path. \nThis contracists Lemma~\\ref{LemNoM-M}. \nSince $e^*_p$ is inward at $v^*_p$, \nwe have $v^*_p=v_O$.\nSimilarly we can show that \n$v^\\dagger_0=v_I$.\nHence Statement (a) holds.\n\n{\\bf Statement (b)}. \nSuppose that \n$P_I\\cap$~Int~$J_O\\neq\\emptyset$.\nLet $s=\\min\\{i~|~v^*_i\\in$~Int~$J_O\\}$. \nThe path $P_I[v^*_0,v^*_s]$ \nsplits the disk $\\Delta$ into disks.\nLet $E$ be the disk \ncontaining $v_O$.\n\nIf $v^*_s\\in$~Int~$J_O[v_O,w_O]$, \nthen $E$ does not contain $v^*_{s+1}$, \nbecause $J_O[v_O,w_O]$ is oriented \nfrom $v_O$ to $w_O$ \n(see Fig.~\\ref{fig24}(a)). \nNow $v^*_p=v_O$ by Statement (a). \nThus \n$P_I[v^*_{s+1},v^*_p]$ \nintersects $P_I[v^*_0,v^*_s]$. \nHence $P_I$ contains a one-way cycle. \nThis contradicts \nLemma~\\ref{LemOneWayCycle}. \n\nIf $v^*_s\\in$~Int~$J_O[v_I,w_O]$, \nthen $E$ does not contain $v_I$ but \n$w_O$ \n(see Fig.~\\ref{fig24}(a)). \nSince $v^\\dagger_0=v_I$ \nby Statement (a), \nthe path $P_O$ \nintersects $P_I[v^*_0,v^*_s]$.  \nThus $v^*_i=v^\\dagger_j$ \nfor some $1\\le i<p,1\\le j<q$. \nThen $P_I[v^*_0,v^*_i]\\cup \nP_O[v^\\dagger_j,v^\\dagger_q]$ \nis \na dichromatic M$\\&$M one-way path. \nThis contradicts Lemma~\\ref{LemNoM-M}.\n\nThus we have \n$P_I\\cap$~Int~$J_O=\\emptyset$. \nSimilarly we can show that \n$P_O\\cap$~Int~$J_I=\\emptyset$. \nHence Statement (b) holds.\n\n{\\bf Statement (c)}. \nSince $e^*_1$ is middle at $v^*_0$, \nby the help of Lemma~\\ref{LemNoM-M} \nwe have that\n\\begin{enumerate}\n\\item[(1)] \nif an edge $e$ of label $k$ \nintersects Int~$P_I$ but $e\\not\\subset P_I$, \nthen the edge $e$ is inward at \na vertex of $P_I$.\n\\end{enumerate}\nSuppose that \nthere exists a white vertex \n$v^*_j \\in P_I-J_I[w_I,v_O]$ \nfor some $1\\le j<p$. \nNow $P_I$ splits the disk $\\Delta$ \ninto disks. \nLet $E$ be one of the disks \nwith $v^*_j\\in E$ and \n$\\partial E\\subset P_I\\cup J_I$. \nSince $P_I\\cap$~Int~$J_O=\\emptyset$ by \nStatement (b), \nStatement (1) implies that\n\\begin{enumerate}\n\\item[(2)] \nfor each vertex $v$ in \n$(P_I-J_I[w_I,v_O])\\cap E$, \nthe disk $E$ contains \n(see Fig.~\\ref{fig24}(b),(c))\n\\begin{enumerate}\n\\item[(i)] \nexactly one edge of label $m$ \noutward at $v$, or \n\\item[(ii)]\nexactly two edges of label $m$, \none is inward and \nthe other outward at $v$.\n\\end{enumerate}\n\\end{enumerate}\nHence the disk $E$ contains \nan edge $e'$ of label $m$ \noutward at $v^*_j$.\nLet $P'$ be \na one-way path of label $m$ starting from $e'$\nupward maximal \nwith respect to $E$ with \na vertex sequence \n$(v'_0,v'_1,v'_2,\\cdots,v'_t)$ and \nan edge sequence \n$(e'_1,e'_2,\\cdots,e'_t)$.\nThen Statement (2) implies \n$v'_t\\in$~Int~$J_I$. \nSince $e'_t$ is inward at $v'_t$, \nthere exists an outside edge $e''$ \nof label $k$ for $E$\noutward at $v'_t$. \nOn the other hand, \nthe edge $e''$ is \ndominated by $\\Delta_I$. \nThus $e''$ is inward at $v'_t$. \nThis is a contradiction. \nHence there does not exist \na white vertex \nin $P_I-J_I[w_I,v_O]$. \nSimilarly we can show that \nthere does not exist \na white vertex \nin $P_O-J_O[v_I,w_O]$. \nThus Statement (c) holds. \n\\end{Proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig24.pdf}\n\\end{center}\n\\caption{ \\label{fig24} Thick arcs are of label $m$.}\n\\end{figure}\n\n\\begin{lem}\\label{Lemma8}\nThere does not exist \na white vertex in \n{\\rm Int}~$J_I[w_I,v_I]$ nor \n{\\rm Int}~$J_O[v_O,w_O]$.\n\\end{lem}\n\n\\begin{Proof}\nSuppose that \nthere exists a white vertex $v$ in\nInt~$J_I[w_I,v_I]$. \nThen $\\Delta$ contains \nan edge $e$ of label $k$\noutward at $v$. \nLet $P'$ be a one-way path of label $k$ \nstarting from $e$ \nupward maximal \nwith respect to $\\Delta$ \nwith \na vertex sequenc \n$(v'_0,v'_1,v'_2,\\cdots,v'_t)$. \nSince $\\{v_I,v_O,w_I,w_O\\}\\subset  P_I\\cup  P_O$\nby Lemma~\\ref{Lemma7}(a),\nwe have $v'_t\\not\\in \\{v_I,v_O,w_I,w_O\\}$.\nSince neither $\\Delta_I$ nor $\\Delta_O$ dominates an outside edge of label $m$ for $\\Delta$\nintersecting $\\partial \\Delta-\\{v_I,v_O,w_I,w_O\\}$,\nwe have \n$v'_t\\not\\in \\partial \\Delta-\\{v_I,v_O,w_I,w_O\\}$ by Lemma~\\ref{lemMaxAtBoundaryUp}(b).\nTherefore \n$v'_t\\not\\in\\partial\\Delta$.\nThis contradicts \nLemma~\\ref{lemMaxAtBoundaryUp}(a). \nHence there does not exists \nany white vertex \nin Int~$J_I[w_I,v_I]$. \nSimilarly we can show that \nthere does not exists \nany white vertex \nin Int~$J_O[v_O,w_O]$. \nHence Lemma~\\ref{Lemma8} holds. \n\\end{Proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig25.pdf}\n\\end{center}\n\\caption{ \\label{fig25} \nThe disk $E$ is right gray. \nThick arcs are of label $m$.}\n\\end{figure}\n\n{\\bf Proof of Theorem~\\ref{StIITheorem4}.}\nBy the similar way \nas the one of Lemma~\\ref{Lemma8} \nwe can show that \nthere does not exist \nany white vertex in Int~$\\Delta$. \nNow the tangle is like \nthe one in Fig.~\\ref{fig25}. \nLet $s$ be \nthe number of white vertices \nin Int~$J_I[w_I,v_O]$, and \n$t$\nthe number of white vertices \nin Int~$J_O[v_I,w_O]$. \nThen $s=t$ by Lemma~\\ref{EI+TO=EO+TI}. \nThus $(\\Gamma\\cap D,D)$ is\na Type-$I_s$ elementary IO-tangle. \nThis proves Theorem~\\ref{StIITheorem4}. {\\hfill {$\\square$}\\vspace{1.5em}}\n\n\n\n\n\\section{Indices}\n\\label{s:Indices}\n\nLet $\\Gamma$ be a minimal chart, and \n$m$ a label of the chart.\nLet $(\\Gamma\\cap D,D)$ be \na simple IO-tangle \nof label $m$ \nwith a boundary IO-arc pair $(L_I,L_O)$. \nLet $k$ be a label of $\\Gamma$ with \n$|m-k|=1$ and \n$\\Gamma\\cap D\\subset\\Gamma_m\\cup\\Gamma_k$. \nLet $\\sigma_1,\\sigma_2,\\cdots,\\sigma_s$\nbe all the O-terminal edges \nof label $m$ in $D$\nand \n$\\widetilde\\sigma_1,\\widetilde\\sigma_2,\n\\cdots,\\widetilde\\sigma_t$\nall the I-terminal edges of \nlabel $m$ in $D$.\nFor each terminal edge $\\tau$ in $D$,\nlet $e^*,e^{**}$ be the sibling edges \nof the terminal edge.\nThe union $e^*\\cup e^{**}$ splits\nthe disk $D$ into two disks.\nLet $\\Delta(\\tau)$ be\nthe one of the two disks containing\nthe terminal edge.\nSet\\\\\n\\ \\ \\ $D^\\dagger=\nCl(D-\\cup_{i=1}^s\\Delta(\\sigma_i)\n-\\cup_{j=1}^t\\Delta(\\widetilde\\sigma_j))$ \n(see Fig.~\\ref{fig26}(b)).\\\\\nLet\n$A_0,A_1,\\cdots,A_s$ be the connected components of\n$\\partial D^\\dagger\\cap L_I$\nsituated counterclockwise on $\\partial D^\\dagger$\nin this order, and\n$B_0,B_1,\\cdots,B_t$ be the connected components of\n$\\partial D^\\dagger\\cap L_O$\nsituated clockwise on $\\partial D^\\dagger$\nin this order.\nLet $X_I$ be the union of \nall the I-edges \nof label $k$ for $D$, and\n$X_O$ the union of \nall the O-edges \nof label $k$ for $D$. \nFor each $i=0,1,\\cdots,s$ and \n$j=0,1,\\cdots,t$, \nlet\\\\\n\\ \\ \\ \\ $a_i=|A_i\\cap X_I|$, and\n$b_j=|B_j\\cap X_O|$.\\\\\nThen $(s+1)$-tuple\n${\\bf a}(\\Gamma,D)=\n(a_0,a_1,\\cdots,a_s)$ and\n$(t+1)$-tuple ${\\bf b}(\\Gamma,D)=\n(b_0,b_1,\\cdots,b_t)$\nare called \nthe {\\it I-index} and the {\\it O-index} \nof the simple IO-tangle respectively.\nThe pair Index$(\\Gamma,D)=\n({\\bf a}(\\Gamma,D),{\\bf b}(\\Gamma,D))$ \nis called the {\\it index} of the \nsimple IO-tangle. \nFor the simple IO-tangle as the one shown \nin Fig.~\\ref{fig26}(a),\nthe I-index is $(4,3,1)$ and \nthe O-index is $(2,5)$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig26.pdf}\n\\end{center}\n\\caption{ \\label{fig26}\n(a) The thick edges are of label $m$.\n(b) The thick edges are of label $m$, and\nthe thin edges are of label $k$.}\n\\end{figure}\n\nLet $\\Gamma$ be a chart.\nA tangle $(\\Gamma\\cap D,D)$ \nis called a {\\it net-tangle} \nprovided that\n\\begin{enumerate}\n\\item[(i)]\nthe disk $D$ contains no crossing, hoop, nor free edge\nbut a white vertex,\n\\item[(ii)]\nthere exist two labels $\\alpha,\\beta$ \nwith $\\alpha<\\beta$ and\n$\\Gamma\\cap D\n\\subset\n\\cup_{i=\\alpha}^{\\beta}\\Gamma_{i}$,\nand\n\\item[(iii)]\nthere exist two arcs \n$L_\\alpha,L_\\beta$ on $\\partial D$ \nwith $L_\\alpha\\cap L_\\beta$ two points \nsuch that\n\\begin{enumerate}\n\\item[(a)] \n$\\Gamma\\cap\\partial D=\n(\\Gamma_\\alpha\\cap {\\rm Int}~L_\\alpha)\n\\cup \n(\\Gamma_\\beta\\cap {\\rm Int}~L_\\beta)$,\n\\item[(b)]\nall the edges intersecting $L_\\alpha$ \nare I-edges of label $\\alpha$ or\\\\\nall the edges intersecting $L_\\alpha$ \nare O-edges of label $\\alpha$,\nand\n\\item[(c)]\nall the edges intersecting $L_\\beta$ \nare O-edges of label $\\beta$ or\\\\\nall the edges intersecting $L_\\beta$ \nare I-edges of label $\\beta$.\n\\end{enumerate}\n\\end{enumerate}\nThe pairs $(\\alpha,\\beta)$ and \n$(L_\\alpha,L_\\beta)$ are called \na {\\it label pair} \nand \na {\\it boundary arc pair} of the net-tangle\nrespectively.\nIf all the edges of labe $\\alpha$ \nintersecting $L_\\alpha$\nare I-edges (resp. O-edges), \nand\nif all the edges of label $\\beta$ \nintersecting $L_\\beta$\nare \nO-edges (resp. I-edges),\nthen the net-tangle is said to be \n{\\it upward} (resp. {\\it downward}).\nAn upward or downward \nnet-tangle with \na label pair $(\\alpha,\\alpha+1)$ \nis called an {\\it N-tangle}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig27.pdf}\n\\end{center}\n\\caption{ \\label{fig27}\nAn example of an upward net-tangle with \na label pair $(\\alpha,\\alpha+1)$.\nThe gray area is a disk $D$,\nthe thick edges are of label $\\alpha$, and\nthe thin edges are of label $\\alpha+1$.\n}\n\\end{figure}\n\nTo define an index of an N-tangle, \nwe need the following theorem \nestablished in \\cite{StI}.\n\n\\begin{thm}\n{\\rm (\\cite[Theorem 1.1]{StI})}\n\\label{StITheorem1}\nLet $\\Gamma$ be a minimal chart, and \n$(\\Gamma\\cap D,D)$ a net-tangle\nwith a label pair $(\\alpha,\\alpha+1)$. \nThen we have the following:\n\\begin{enumerate}\n\\item[{\\rm (a)}] \nThe tangle is an N-tangle.\n\\item[{\\rm (b)}] \nThe number of the edges of label $\\alpha$ \nintersecting $\\partial D$\nis equal to\nthe number of the edges of label $\\alpha+1$ \nintersecting $\\partial D$.\n\\item[{\\rm (c)}] \nThere exists a terminal edge in $D$.\n\\item[{\\rm (d)}]\nThe number of terminal edges of \nlabel $\\alpha$ in $D$ \nis equal to \nthe number of terminal edges of \nlabel $\\alpha+1$ in $D$.\n\\item[{\\rm (e)}]\nIf the tangle is upward $($resp. downward$)$, \nthen \nall the terminal edges of label $\\alpha$ in $D$ \nare I-terminal $($resp. O-terminal$)$ edges\nand \nall the terminal edges of label $\\alpha+1$ in $D$ \nare O-terminal $($resp. I-terminal$)$ edges\n\\end{enumerate}\n\\end{thm}\n\n\nNow we can define an index of the N-tangle\nas follows. \n\nLet $\\Gamma$ be a minimal chart.\nLet $(\\Gamma\\cap D,D)$ be an N-tangle \nwith \na label pair $(\\alpha,\\alpha+1)$ and a boundary arc pair \n$(L_{\\alpha},L_{\\alpha+1})$.\nLet $s_I$ be the label of the I-edges, and \n $s_O$ the label of the O-edges.\nFurther, we denote by $L_I$ (resp. $L_O$)\nthe one of the arcs \n$L_{\\alpha},L_{\\alpha+1}$\nwhich intersects I-edges (resp. O-edges).\nConsidering \nTheorem~\\ref{StITheorem1}(d),\nlet $k=T_O(D)=T_I(D)$,\nand $\\sigma_1,\\sigma_2,\\cdots,\\sigma_k$\n all the O-terminal edges in $D$\nand \n$\\widetilde\\sigma_1,\\widetilde\\sigma_2,\n\\cdots,\\widetilde\\sigma_k$\nall the I-terminal edges in $D$.\nSet\\\\\n\\ \\ \\ $D^\\dagger=\nCl(D-\\cup_{i=1}^k\\Delta(\\sigma_i)\n-\\cup_{j=1}^k\\Delta(\\widetilde\\sigma_j))$ \n(see Fig.~\\ref{fig28}(b)).\\\\\nLet\n$A_0,A_1,\\cdots,A_k$ be \nthe connected components of\n$\\partial D^\\dagger\\cap L_I$\nsituated counterclockwise \non $\\partial D^\\dagger$\nin this order, and\n$B_0,B_1,\\cdots,B_k$ be \nthe connected components of\n$\\partial D^\\dagger\\cap L_O$\nsituated clockwise \non $\\partial D^\\dagger$\nin this order.\nLet $X_I$ be the union of \nall the I-edges for $D$, and\n$X_O$ the union \nof all the O-edges for $D$.\nFor each $i=0,1,\\cdots,k$, let\\\\\n\\ \\ \\ \\ $a_i=|A_i\\cap X_I|$, and\n$b_i=|B_i\\cap X_O|$.\\\\\nThen $(k+1)$-tuples\n${\\bf a}(\\Gamma,D)=(a_0,a_1,\\cdots,a_k)$ and\n${\\bf b}(\\Gamma,D)=(b_0,b_1,\\cdots,b_k)$\nare called \nthe I-{\\it index} and O-{\\it index} \nof the N-tangle respectively.\nThe pair Index$(\\Gamma,D)=\n({\\bf a}(\\Gamma,D),{\\bf b}(\\Gamma,D))$\nis called the {\\it index} of the N-tangle. \nAccording to Theorem~\\ref{StITheorem1}(b),\nwe have $\\sum_{i=0}^ka_i=\\sum_{i=0}^kb_i$.\n\nFor the N-tangle as the one shown \nin Fig.~\\ref{fig28}(a),\nthe I-index is $(3,3,3,1)$ and \nthe O-index is $(2,3,4,1)$.\n\nLet ${\\bf x}=(x_1,x_2,\\cdots,x_k)$ \nbe an I-index or O-index of a tangle, \nthen\nthe sum of components, $\\sum_{i=1}^kx_i$, \nis called {\\it index sum}, and denoted by $||\\bf x||$. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig28.pdf}\n\\end{center}\n\\caption{\\label{fig28}\nThe thick edges are of label $s_I$, and\nthe thin edges are of label $s_O$.}\n\\end{figure}\n\n\\section{Normal forms}\n\\label{s:NormalForm}\n\n\nTo define a normal form \nfor 2-crossing minimal charts, \nwe need \none more theorem \nestablished in \\cite{StI}.\n\nThere exists a special C-move \ncalled a C-I-M2 move \n(see Fig.~\\ref{fig04} in Section~\\ref{s:Prel}).\nLet $\\Gamma$ and $\\widetilde\\Gamma$ be \nminimal charts,\nand $(\\Gamma\\cap D,D)$ a net-tangle with \na label pair $(\\alpha,\\beta)$.\nThen the chart $\\widetilde\\Gamma$ is said to be \nM2-{\\it related to $\\Gamma$ with respect to $D$}\nprovided that\n\\begin{enumerate}\n\\item[(i)]\n$\\Gamma\\cap D^c=\\widetilde\\Gamma\\cap D^c$, and\n\\item[(ii)]\nthe chart $\\widetilde\\Gamma$ is obtained from\nthe chart $\\Gamma$ by a finite sequence of \nC-I-M2 moves in $D$ each of which modifies \ntwo edges of label $m$ with $\\alpha<m<\\beta$.\n\\end{enumerate}\n\n\n\n\n\\begin{thm}\n{\\rm (\\cite[Theorem 1.2]{StI})}\n\\label{StITheorem2}\nLet $\\Gamma$ be a minimal chart, and \n$(\\Gamma\\cap D,D)$ a net-tangle\nwith a label pair $(\\alpha,\\beta)$. \nThen \nthere exists a minimal chart $\\widetilde\\Gamma$\nM2-related to $\\Gamma$ with respect to $D$ such that\nthere exist N-tangles \n$(\\widetilde\\Gamma\\cap D_{\\alpha},D_{\\alpha}),\n(\\widetilde\\Gamma\\cap D_{\\alpha+1},D_{\\alpha+1}),\n\\cdots,\n(\\widetilde\\Gamma\\cap D_{\\beta-1},D_{\\beta-1})$\nequipped with\n\\begin{enumerate}\n\\item[{\\rm (a)}] for each \n$i=\\alpha,\\alpha+1,\\cdots,\\beta-1$,\nthe tangle \n$(\\widetilde\\Gamma\\cap D_{i},D_{i})$ \nis an N-tangle with the label pair $(i,i+1)$,\n\\item[{\\rm (b)}] $D=\\cup_{i=\\alpha}^{\\beta-1}D_i$, \n\\item[{\\rm (c)}] \nfor each $i=\\alpha,\\alpha+1,\\cdots,\\beta-2$,\nthe intersection $D_i\\cap D_{i+1}$ \nis a proper arc of $D$,\n\\item[{\\rm (d)}]\nall the N-tangles are \nupward or downward\nsimultaneously.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{remark} \\label{remWhite2}\n{\\rm \nIf $(\\Gamma\\cap D,D)$ \nis an admissible tangle \nin a minimal chart $\\Gamma$\nsuch that\n\\begin{enumerate}\n\\item[{\\rm $($a$)$}] \n$\\Gamma\\cap D\\subset \\Gamma_m\\cup\\Gamma_k$ \nfor some labels $m,k$ with $|m-k|=1$,\n\\item[{\\rm $($b$)$}] \n$\\Gamma_m\\cap \\partial D$ consists of \nexactly two points, \nand\n\\item[{\\rm (c)}]\nthe disk \n$D$ contains \nexactly two white vertices.\n\\end{enumerate}\nThen $(\\Gamma\\cap D,D)$ is \nthe one shown in \nFig.~\\ref{fig29}(a). \nThe tangle shown in \nFig.~\\ref{fig29}(c) \nis a simple IO-tangle \npossessing a cycle and \nthe least number of \nI-edges and O-edges.\nFor each of tangles \nwith exactly one white vertex \nshown in Fig.~\\ref{fig13}(a),(b), \nthere exist at most two I-edges and \nat most two O-edges. \nFor each of tangles \nwith two white vertices \nshown in Fig.~\\ref{fig29}, \nthere exist at least three I-edges and \nat least three O-edges. \nThus } \nfor fundamental tangles \n$(\\Gamma\\cap D,D)$,\n\\begin{enumerate}\n\\item[{\\rm (i)}] \nif there exist \nat least three I-edges or \nat least three O-edges for $D$, \nthen \nthere exist \nat least two white vertices in $D$,\n\\item[{\\rm (ii)}] \nif there exist \nat least two white vertices in $D$, \nthen there exist \nat least three I-edges and \nat least three O-edges for $D$.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig29.pdf}\n\\end{center}\n\\caption{ \\label{fig29}\n(a) A simple IO-tangle \nwith exactly two white vertices. \n(b) An N-tangle \nwith exactly two white vertices.\n(c) A simple IO-tangle with a cycle.}\n\\end{figure}\n\n\n\nThroughout this section,\n$\\Gamma$ is a $2$-crossing minimal chart \nin a disk $D^2$ different from the chart \nin Fig.~\\ref{fig30}(a).\nSet\n$\\alpha=\\alpha(\\Gamma)$, and \n$\\beta=\\beta(\\Gamma)$.\nThen by Lemma~\\ref{StIILemma1},\nwe can assume that \n\\begin{enumerate}\n\\item[$\\bullet$]\nthere exist two cycles \n$C_\\alpha,C_\\beta$ \nwith \n$C_\\alpha\\subset\\Gamma_\\alpha,C_\\beta\n\\subset\\Gamma_\\beta$ such that \n$C_\\alpha\\cap C_\\beta$ consists of \nthe two crossings,\n\\item[$\\bullet$]\nthere exists an annulus $A$ containing all the white vertices of $\\Gamma$ \nbut not intersecting hoops nor free edges,\n\\item[$\\bullet$]\neach connected component of $Cl(D^2-A)$ contains a crossing,\n\\item[$\\bullet$]\n $\\Gamma\\cap \\partial A=(C_\\alpha\\cup C_\\beta)\\cap \\partial A$, \n$\\Gamma\\cap \\partial A$ consists of eight points,\n\\item[$\\bullet$] \n$\\Gamma_\\alpha\\cap A$ \nconsists of \ntwo connected components \n$X_1,X_3$ \nseparated by $C_\\beta$, \n\\item[$\\bullet$]\n$\\Gamma_\\beta\\cap A$ \nconsists of \ntwo connected components \n$X_2,X_4$ \nseparated by $C_\\alpha$. \n\\end{enumerate}\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig30.pdf}\n\\end{center}\n\\caption{ \\label{fig30}\n(a) the thin edges are of label $m+1$.\n(b) $A=D_1\\cup E_1\\cup D_2\\cup E_2\\cup\nD_3\\cup E_3\\cup D_4\\cup E_4$.}\n\\end{figure}\nFor each $i=1,2,3,4$, \nlet $D_i$ be a regular neighbourhood \nof the SC-closure $SC(X_i)$ in the annulus $A$.\nBy Boundary Condition Lemma \n(Lemma~\\ref{BoundaryConditionLemma}),\n\\begin{enumerate}\n\\item[$\\bullet$] $\\Gamma\\cap D_i\\subset\\Gamma_\\alpha\\cup\\Gamma_{\\alpha+1}$ for $i=1,3$,\n\\item[$\\bullet$] $\\Gamma\\cap D_i\\subset\\Gamma_\\beta\\cup\\Gamma_{\\beta-1}$ for $i=2,4$.\n\\end{enumerate}\nBy Theorem~\\ref{StIITheorem2} and Remark~\\ref{remWhite1},\n\\begin{enumerate}\n\\item[$\\bullet$]\nthe tangle $(\\Gamma\\cap D_i,D_i)$ is an IO-tangle for\n$i=1,2,3,4$.\n\\end{enumerate}\nWithout loss of generality\nwe can assume that\n(see Fig.~\\ref{fig30}(b))\n\\begin{enumerate}\n\\item[$\\bullet$]\n$Cl(A-(D_1\\cup D_2\\cup D_3\\cup D_4))$ consists of\nfour disks, say\n$E_1,E_2,E_3,E_4$,\n\\item[$\\bullet$]\n$D_1,E_1,D_2,E_2,\nD_3,E_3,D_4,E_4$\nare situated on the annulus $A$ \nin this order,\n\\item[$\\bullet$]\nan O-edge for $D_1$ is an I-edge for $E_1$.\n\\end{enumerate}\n\n\n\\begin{lem}\n\\label{FoundamentalTwo}\nFor each fundamental tangle \n$(\\Gamma\\cap D_i,D_i)$ $(i=1,2,3,4)$,\nthe disk $D_i$ contains at least two white vertices.\n\\end{lem}\n\n\\begin{Proof}\nFirst we shall show that \neach of $\\Gamma_\\alpha$ and $\\Gamma_\\beta$\ncontains at least two white vertices.\nBy Lemma~\\ref{Lemma3-2},\n$\\Gamma_\\alpha$ contains \nat least one white vertex.\nIf $\\Gamma_\\alpha$ contains exactly one white vertex,\nthen $\\Gamma_\\alpha$  contains a terminal edge \nnot middle at its white vertex. \nThen we can eliminate \nthe white vertex \nby a C-III move. \nThis contradicts that\nthe chart is minimal. \nThus $\\Gamma_\\alpha$ contains at least two white vertices.\nSimilarly we can show that $\\Gamma_\\beta$ contains at least two white vertices.\n\n\nNext we shall show that\none of $D_1,D_2,D_3,D_4$ \ncontains at least two white vertices.\nSuppose that \neach of $D_1,D_2,D_3,D_4$ \ncontains at most one white vertex.\nSince each of $\\Gamma_\\alpha$ and $\\Gamma_\\beta$\ncontains at least two white vertices,\neach of $D_1,D_2,D_3,D_4$  \ncontains exactly one white vertex.\nThus for each $i=1,2,3,4$ \nthe tangle $(\\Gamma\\cap D_i,D_i)$ is \none of the three tangles as shown in\nFig.~\\ref{fig13}\nby Remark~\\ref{remWhite1}.\nWithout loss of generality\nwe can assume that\n\n\\begin{enumerate}\n\\item[] $\\Gamma\\cap(D_1\\cap E_1)$ consists of at most one point.\n\\end{enumerate}\nThere are two cases: $\\beta-\\alpha>2$ or $\\beta-\\alpha=2$.\nSuppose $\\beta-\\alpha>2$.\nIf $E_1$ contains at least one white vertex,\nthen $(\\Gamma\\cap E_1,E_1)$ is an NS-tangle of label $\\beta-1$.\nThis contradicts Lemma~\\ref{LemNS-Tangle}.\nThus $E_1$ does not contain any white vertex.\nSince $\\beta-\\alpha>2$,\nwe have $\\Gamma\\cap(D_1\\cap E_1)=\\emptyset$.\nHence $(\\Gamma\\cap D_1,D_1)$\nis the tangle as shown in Fig.~\\ref{fig13}(c).\nMoreover\n$\\Gamma\\cap(D_2\\cap E_1)=\\emptyset$ and\n$(\\Gamma\\cap D_2,D_2)$\nis the tangle as shown in Fig.~\\ref{fig13}(c)\n(see Fig.~\\ref{fig31}(a)).\nSimilarly\nwe can show that\nthe chart is a chart as shown \nin Fig.~\\ref{fig31}(a).\nBy a C-II move, a C-I-M2 move and a C-III move,\nwe obtain the chart as shown in \nFig.~\\ref{fig31}(d). \nThis chart is not a minimal chart. \nThus the chart $\\Gamma$ is not minimal.\nThis is a contradiction.\n\nIf $\\beta-\\alpha=2$,\nthen we can show that the chart $\\Gamma$ is \nthe one shown in \nFig.~\\ref{fig30}(a).\nThis contradicts the fact that the chart $\\Gamma$ is different from \nthe one shown in \nFig.~\\ref{fig30}(a).\nTherefore one of $D_1,D_2,D_3,D_4$ \ncontains at least two white vertices.\n\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics{fig31.pdf}\n\\end{center}\n\\caption{ \\label{fig31}}\n\\end{figure}\n\nWithout loss of generality\nwe can assume that \n$D_1$ contains at least two white vertices.\nThen $(\\Gamma\\cap D_1,D_1)$ is \na simple IO-tangle\nby Theorem~\\ref{StIITheorem2}. \nHence \nthere exist\nat least three O-edges \nof label $\\alpha+1$ \nfor $D_1$ by \nRemark~\\ref{remWhite2}(ii). \nNamely there exist \nat least three I-edges \nof label $\\alpha+1$ \nfor $E_1$.\nSince $(\\Gamma\\cap D_1,D_1)$ is \nan IO-tangle of label $\\alpha$\nand since $(\\Gamma\\cap D_2,D_2)$ is \nan IO-tangle of label $\\beta$,\nthe thangle $(\\Gamma\\cap E_1,E_1)$ is \na net-tangle with \na label pair $(\\alpha+1,\\beta-1)$\nby Boundary Condition Lemma\n(Lemma~\\ref{BoundaryConditionLemma}).\nBy Theorem~\\ref{StITheorem1}\nand Theorem~\\ref{StITheorem2}, \nthere are \nat least three O-edges \nof label $\\beta-1$ \nfor $E_1$ \n(here possibly $\\alpha+1=\\beta-1$,  \nin this case, $E_1$ contains \njust parallel arcs). \nNamely there are \nat least three I-edges \nof label $\\beta-1$ \nfor $D_2$. \nHence by Remark~\\ref{remWhite2}(i), \nthe disk $D_2$ contains \nat least two white vertices.\n\nBy the similar way, \nwe can show that \neach of $D_1,D_2,D_3,D_4$ \ncontains at least two white vertices.\nThus Lemma~\\ref{FoundamentalTwo} holds.\n\\end{Proof}\n\nNow we have\n\\begin{enumerate}\n\\item[$\\bullet$] \nfor each of $E_1,E_3$, \nthere are \nat least three I-edges \nof label $\\alpha+1$ \nand \nat least three O-edges \nof label $\\beta-1$.\n\\item[$\\bullet$] \nfor each of $E_2,E_4$, \nthere are \nat least three I-edges \nof label $\\beta-1$ \nand \nat least three O-edges \nof label $\\alpha+1$.\n\\end{enumerate}\n\n\nLet $e_1^*$ be \nan O-edge for $D_1$.\nLet $C_0$ be a simple closed curve in Int~$A$\ncontaining the edge $e_1^*$ \nand intersecting \neach of the eight disks \n$D_1,E_1,D_2,E_2,D_3,$ $E_3,D_4,E_4$ \nby a proper arc.\nThe oriented edge $e_1^*$\ninduces the orientation of \nthe simple closed curve $C_0$.\nLet $\\ell$ be an arc with \n$\\ell \\cap A=E_1\\cap D_2$\nconnecting a point in $\\partial A$\nand a point in $\\partial D^2$.\n\nIf the simple closed curve $C_0$\nis oriented clockwise \n(see Fig.~\\ref{fig32}(a)),\nthen apply the chart $\\Gamma$ \nby a DH-trick \nfor each the edge of ${\\rm Main}(\\Gamma)$ \nintersecting $\\ell$ one by one from the outside\n(see Fig.~\\ref{fig32}(b) and (c)), \nwe can assume\n\n\\begin{enumerate}\n\\item[$\\bullet$]\nthe simple closed curve $C_0$\nis oriented counterclockwise \n(see Fig.~\\ref{fig33}).\n\\end{enumerate}\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics{fig32.pdf}\n\\end{center}\n\\caption{ \\label{fig32}}\n\\end{figure}\n\n\nLet $U$ be the connected component of $D^2-A$\ncontaining $\\partial D^2$.\nNow consider the cycles \n$C_\\alpha,C_\\beta$ \nas non-oriented simple closed curves. \nThe oriented edge $e_\\alpha$ of $\\Gamma_\\alpha$\ncontaining $\\Gamma_\\alpha\\cap U$ \ninduces \nthe orientation of \nthe simple closed curve $C_\\alpha$.\nSimilarly \nthe oriented edge $e_\\beta$ of $\\Gamma_\\beta$\ncontaining $\\Gamma_\\beta\\cap U$\ninduces \nthe orientation of \nthe simple closed curve $C_\\beta$.\n\nIf necessary \nwe apply the chart $\\Gamma$ by\na DH-trick for  \nthe edge $e_\\alpha$,\nwe can assume that \n\\begin{enumerate}\n\\item[$\\bullet$] the simple closed curve \n$C_\\alpha$ is \noriented counterclockwise.\n\\end{enumerate}\nIf necessary \nwe apply the chart $\\Gamma$ by\na DH-trick for \nthe edge $e_\\beta$ \n(see Fig.~\\ref{fig32}(c) and (d))\nand if necessary \nwe renumber $E_1,D_1,E_2,D_2,E_3,D_3,E_4,D_4$, \nwe can assume that\n(see Fig.~\\ref{fig33})\n\\begin{enumerate}\n\\item[$\\bullet$] \n$E_1$ does not intersect any of disks bounded by \n$C_\\alpha$ nor $C_\\beta$.\n\\end{enumerate}\nDefine\\\\\n$\n\\delta=\\left\\{\n\\begin{array}{ll}\n1&~~\\text{if the simple closed curve \n$C_\\beta$ \nis oriented counterclockwise,}\\\\\n2&~~{\\rm otherwise.}\n\\end{array}\n\\right.\n$\\vspace{2mm}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics{fig33.pdf}\n\\end{center}\n\\caption{ \\label{fig33}\n(a) $\\delta=1$ (b) $\\delta=2$.}\n\\end{figure}\n\n\n\n{\\bf Case 1}: $\\beta-\\alpha<2$.\nThen the chart is a ribbon chart.\n\n{\\bf Case 2}: $\\beta-\\alpha=2$.\nThen we can assume that\n\\begin{enumerate}\n\\item[(i)] \n$D_1\\cup D_2\\cup D_3\\cup D_4=A$,\n\\item[(ii)] for each $i=1,2,3,4$,\\\\\n$||{\\bf a}(\\Gamma,D_{i+1})||=\n||{\\bf b}(\\Gamma,D_{i})||$,\nhere $D_5=D_1$.\n\\end{enumerate}\nWe define the {\\it  normal form} for the chart $\\Gamma$\nby\\\\\n$F(\\Gamma)=\n((n,\\beta-\\alpha,\\alpha,\\delta),\n(||{\\bf b}(\\Gamma,D_1)||,\n||{\\bf b}(\\Gamma,D_2)||,\n||{\\bf b}(\\Gamma,D_3)||,\n||{\\bf b}(\\Gamma,D_4)||);\\\\\n{\\bf b}(\\Gamma,D_1),\n{\\bf a}(\\Gamma,D_2),\n{\\bf b}(\\Gamma,D_2),\n{\\bf a}(\\Gamma,D_3),\n{\\bf b}(\\Gamma,D_3),\n{\\bf a}(\\Gamma,D_4),\n{\\bf b}(\\Gamma,D_4),\n{\\bf a}(\\Gamma,D_1)\n).$\n\n{\\bf Case 3}: $\\beta-\\alpha\\ge 3$.\nBy Theorem~\\ref{StITheorem2},\nfor each $i=1,2,3,4$ and\n$j=\\alpha+1,\\alpha+2,\\cdots,\\beta-2$,\nthere exists \nan N-tangle \n$(\\Gamma\\cap D_i(j),D_i(j))$\nwith a label pair $(j,j+1)$\nsuch that  \n\\begin{enumerate}\n\\item[(i)]\n$A=(\\cup_{i=1}^4 D_i)\\cup(\n\\cup_{i=1}^4\\cup_{j=\\alpha+1}^{\\beta-2} D_i(j))$,\n\\item[(ii)] \n$E_i=\\cup_{j=\\alpha+1}^{\\beta-2} D_i(j)$\nfor each $i=1,2,3,4$,\n\\item[(iii)] by Theorem~\\ref{StITheorem1} \nfor each $i=1,2,3,4$,\\\\\n$\n||{\\bf a}(\\Gamma,D_{i+1})||=\n||{\\bf b}(\\Gamma,D_i)||=\n||{\\bf a}(\\Gamma,D_{i}(\\alpha+1))||=\n||{\\bf b}(\\Gamma,D_{i}(\\alpha+1))||\\\\=\n||{\\bf a}(\\Gamma,D_{i}(\\alpha+2))||=\n||{\\bf b}(\\Gamma,D_{i}(\\alpha+2))||=\n\\cdots=\n||{\\bf a}(\\Gamma,D_{i}(\\beta-2))||\\\\=\n||{\\bf b}(\\Gamma,D_{i}(\\beta-2))||$,\nhere $D_5=D_1$.  \n\\end{enumerate}\nWe define the {\\it  normal form} for the chart $\\Gamma$\nby $F(\\Gamma)=$\\\\ \n$((n,\\beta-\\alpha,\\alpha,\\delta),\n(||{\\bf b}(\\Gamma,D_1)||,\n||{\\bf b}(\\Gamma,D_2)||,\n||{\\bf b}(\\Gamma,D_3)||,\n||{\\bf b}(\\Gamma,D_4)||);\\\\\n{\\bf b}(\\Gamma,D_1),\n$Index$(\\Gamma,D_1(\\alpha+1)),\n$Index$(\\Gamma,D_1(\\alpha+2)),\n\\cdots,\n$Index$(\\Gamma,D_1(\\beta-2)),\n{\\bf a}(\\Gamma,D_2);\\\\\n{\\bf b}(\\Gamma,D_2),\n$Index$(\\Gamma,D_2(\\beta-2)),\n$Index$(\\Gamma,D_2(\\beta-3)),\n\\cdots,\n$Index$(\\Gamma,D_2(\\alpha+1)),\n{\\bf a}(\\Gamma,D_3);\\\\\n{\\bf b}(\\Gamma,D_3),\n$Index$(\\Gamma,D_3(\\alpha+1)),\n$Index$(\\Gamma,D_3(\\alpha+2)),\n\\cdots,\n$Index$(\\Gamma,D_3(\\beta-2)),\n{\\bf a}(\\Gamma,D_4);\\\\\n{\\bf b}(\\Gamma,D_4),\n$Index$(\\Gamma,D_4(\\beta-2)),\n$Index$(\\Gamma,D_4(\\beta-3)),\n\\cdots,\n$Index$(\\Gamma,D_4(\\alpha+1)),\n{\\bf a}(\\Gamma,D_1)\n).$\n\nFor example, \nthe normal form\nfor the 5-chart in Fig.~\\ref{fig34}\nis\\\\ \n$((5,3,1,2),(8,9,8,7);\\\\\n(1,3,4),(3,3,2),(2,3,3),(6,2);\\\\\n(1,3,5),(2,3,4),(4,3,2),(2,3,3,1);\\\\\n(2,4,2),(1,3,2,2),(1,2,4,1),(4,3,1);\\\\\n(2,5),(2,3,2),(2,3,2),(5,2))$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics{fig34.pdf}\n\\end{center}\n\\caption{ \\label{fig34}}\n\\end{figure}\n\n\nWe have the following:\n\\begin{enumerate}\n\\item[(1)]\nLet\n$\\Gamma$ and $\\Gamma^*$ be \n2-crossing minimal charts.\nIf $F(\\Gamma)=F(\\Gamma^*)$ and \n$\\Gamma- {\\rm Main}(\\Gamma)=\n\\Gamma^*- {\\rm Main}(\\Gamma^*)$, \nthen  \nthe two charts are C-move equivalent.\n\\item[(2)]\nThere does not exist \na 2-crossing $w$-minimal chart $\\Gamma$ with \n$\\alpha(\\Gamma)-\\beta(\\Gamma)\\ge 3$.\nHence c-minimality seems to be \ngood for classifying 2-crossing charts.\n\\item[(3)]\nThere does not exist\na 2-crossing minimal chart\nrepresenting a surface braid whose closure \nis a 2-knot \\cite[Theorem 1.2]{TwoCrossingII}.\n\\item[(4)]\nThere exists a 2-crossing minimal 4-chart (see Figure~$1$ in \\cite{4-chartShpere} and Fig.~\\ref{fig30}(a)).\n\\end{enumerate}\n\nBut we do not know \nif there exists a 2-crossing \n$c$-minimal chart with\n$\\alpha(\\Gamma)-\\beta(\\Gamma)\\ge 3$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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Scott Penberthy, George Ferguson,\nHans Guesgen, Francisco Cruz, Marc Pujol-Gonzalez)\n\/TemplateVersion (2021.1)\n}\n\\end{verbatim}\\end{scriptsize}\n\\end{quote}\n\n\\subsection{Preparing Your Paper}\n\nAfter the preamble above, you should prepare your paper as follows:\n\\begin{quote}\n\\begin{scriptsize}\\begin{verbatim}\n\\begin{document}\n\\maketitle\n\\begin{abstract}\n\\end{abstract}\\end{verbatim}\\end{scriptsize}\n\\end{quote}\n\n\\noindent You should then continue with the body of your paper. Your paper must conclude with the references, which should be inserted as follows:\n\\begin{quote}\n\\begin{scriptsize}\\begin{verbatim}\n\n\\section{\\textsc{ProLoNet} Forward Pass}\nAn algorithmic step-through of the forward pass for the \\textsc{ProLoNet} is provided in Algorithm \\ref{alg:prolo-forward}. The example from the main paper is included here:\n\\begin{example}[\\textsc{ProLoNet} Inference]\nConsider an example cart pole state, X=[2, 1, 0, 3]. Following the equation in Line 3 of Algorithm \\ref{alg:prolo-forward}, the network arrives at $\\sigma([1,0,0,0]*[2,1,0,3]-0)=0.88$ for $D_0$, meaning \"mostly true.\" This decision probability propagates to the two leaf nodes using their respective paths (Lines 9-15 in Algorithm \\ref{alg:prolo-forward}), making the output of the network a probability given by $(0.88 * [1, 0] + (1-0.88)*[0, 1]) = [0.88, 0.12]$. Accordingly, the agent selects the first action with probability 0.88 and the second action otherwise.\n\\end{example}\n\n\\begin{algorithm}[H]\n  \\caption{\\textsc{ProLoNet} Forward Pass}\n  \\label{alg:prolo-forward}\n\\begin{algorithmic}\n  \\STATE {\\bfseries Input:} Input Data $X$, \\textsc{ProLoNet} $P$\n  \\FOR{$d_n \\in D \\in P$}\n  \\STATE $\\sigma_n$ = $\\sigma [\\alpha(\\vec{w_n}^T * \\vec{X} - c_n)]$\n  \\ENDFOR\n  \\STATE $\\vec{A}_{OUT} = $ Output Actions\n  \\FOR{$\\vec{l_i} \\in L$}\n  \\STATE Path to $\\vec{l_i} = Z(L)$\n  \\STATE $z = 1$\n  \\FOR{$\\sigma_i \\in Z(L)$}\n  \\IF{$\\sigma_i \\,$ should be $\\, TRUE \\in Z(L)$}\n  \\STATE $z = z * \\sigma_i$\n  \\ELSE\n  \\STATE $z = z * (1 - \\sigma_i)$\n  \\ENDIF\n  \\ENDFOR\n  \\STATE $\\vec{A}_{OUT} = \\vec{A}_{OUT} + \\vec{l_i} * z$\n  \\ENDFOR\n  \\STATE {\\bfseries Return:} $\\vec{A}_{OUT}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Hyperparameters and Optimization Details}\nAll actors are updated with proximal policy optimization (PPO)~\\cite{schulman2017proximal}. Notably, for the two SC2 domains, we find that multiplying the PPO update by the Kullback-Leibler divergence between old and new policies yields superior performance. The critic's loss function is the mean-squared error between the output of the critic and the reward from the state-action pair. All approaches are trained with RMSProp~\\cite{tieleman2012lecture}. We set our reward discount factor to 0.99, learning rates to 1e-2 for Gym environments, and 1e-4 for the SC2 domains, following a hyperparameter search between 1-e2 and 1e-5. Update batch sizes dynamically grow as more replay experience is available. In all domains, the \\textsc{ProLoNet} $\\alpha$ parameter is initialized to 1.  Our agents utilize two separate networks: one for the actor and one for the critic. For our approach, the critic network is initialized as a copy of the actor as we do not solicit intelligent value predictions, only policies. Our dynamic growth hyperparameter $\\epsilon$ is set to $\\epsilon=0.1$ based upon experimental observation.\n\n\\section{Experimental Domain Details}\n\\subsection{Cart Pole}\n\n\nCart pole is an RL domain~\\cite{barto1983neuronlike} where the object is to balance an inverted pendulum on a cart that moves left or right. The state space is a 4D vector representing \\{\\textit{cart position, cart velocity, pole angle, pole velocity}\\}, and the action space is is \\{\\textit{left, right}\\}. We use the cart pole domain from the OpenAI Gym~\\cite{1606.01540}. \n\nFor the cart pole domain, we set all agent's learning rates to 0.01, the batch size is set to dynamically grow as there is more replay experience available, we initialized $\\alpha=1$, and each agent trains on all data gathered after each episode, then empties its replay buffer. All agents train on 2 simulations concurrently, pooling replay experience after each episode, and updating their policy parameters. For the $LOKI$ agent, we set $N$=200. All agents are updated according to the standard PPO loss function. We selected all parameters empirically to produce the best results for each method.\n\n\\subsection{Lunar Lander}\nLunar lander is the second domain we use from the OpenAI Gym~\\cite{1606.01540}, and is based on the classic Atari game of the same name. Lunar lander is a game where the player attempts to land a small ship (the lander) safely on the ground, keeping the lander upright and touching down slowly. The 8D state consists of the lander's \\{x, y\\} position and velocity, the lander's angle and angular velocity, and two binary flags which are true when the left or right legs have touched down.\n\nWe use the discrete lunar lander domain, and so the 4D action space contains \\{\\textit{do nothing, left engine, main engine, right engine}\\}. \nFor the lunar lander domain, we set most hyperparameters to the same values as in the cart pole domain. The two exceptions are the number of concurrent processes, which we set to 4, and the $LOKI$ agent's $N$, which is set to 300. All agents use the standard PPO loss function. \n\\subsection{FindAndDefeatZerglings}\n\nFindAndDefeatZerglings is a minigame from the SC2LE designed to challenge RL agents to learn how to effectively micromanage their individual attacking units in SC2. The agent controls three attacking units on a small, partially-observable map, and must explore the map while killing enemy units. The agent receives +1 reward for each enemy unit that is killed, and -1 for each allied unit that is killed. Enemy units respawn in random locations, and so the best agents are ones that continuously explore and kill enemy units until the three minute timer has elapsed.\n\nWe leverage the SC2 API \\footnote{https:\/\/github.com\/Blizzard\/s2client-api} to manufacture a 37D state which contains \\{\\textit{x\\_position, y\\_position, health, weapon\\_cooldown}\\} for three allied units, and \\{\\textit{x\\_position, y\\_position, health, weapon\\_cooldown, is\\_baneling}\\} the five nearest visible enemy units. Missing information is filled with -1. Our action space is 10D, containing move commands for north, east, south, west, attack commands for each of the five nearest visible enemies, and a ``do nothing'' command. For this problem, we assign an agent to each individual allied unit, which generates actions for only that unit. Experience from each agent stops accumulating when the unit dies. All experience is pooled for policy updates after each episode, and parameters are shared between agents.\n\nFor the SC2LE minigame, we set all agents' learning rates to 0.001, we again initialized $\\alpha=1$, and the batch size to 4. Each agent trains on replay data for 50 update iterations per episode, and pools experience from 2 concurrent processes. The $LOKI$ agent's $N$, is set to 500. The agents in this domain update according to the loss function in Equation \\ref{eqn:KLLoss}.\n\n\\begin{equation}\n\\label{eqn:KLLoss}\nL(a, s, \\pi_{new}, \\pi_{old})  = \\frac{(A*log(a|\\pi_{new}))}{KL(P(\\vec{a}|\\pi_{new}, s), P(\\vec{a}|\\pi_{old}, s))}\n\\end{equation}\n\nWhere $A$ is the advantage gained by taking action $a$ in state $s$, $\\pi_{new}$ is the current set of model parameters, and $\\pi_{old}$ is the set of model parameters used during the episode which generated this state-action pair. $\\vec{a}$ is the probability distribution over all actions that a policy $\\pi$ yields given state $s$. As in prior work, the advantage $A$ is calculated by subtracting the reward (obtained by taking action $a$ in state $s$) from the value prediction for taking action $a$ in state $s$, given by a critic network.\n\\subsection{SC2 Full Game}\nOur simplified StarCraft 2 state contains:\n\\begin{itemize}\n    \\item \\textit{Allied Unit Counts}: A 36x1 vector in which each index corresponds to a type of allied unit, and the value corresponds to how many of those units exist.\n\n    \\item \\textit{Pending Unit Counts}: As above, but for units that are currently in production and do not exist yet.\n\n    \\item \\textit{Enemy Unit Counts}: A 112x1 vector in which each index corresponds to a type of unit, and the value corresponds to how many of those types are visible.\n\n    \\item \\textit{Player State}: A 9x1 vector of specific player state information, including minerals, vespene gas, supply, etc. \n\\end{itemize}\n\nThe disparity between allied unit counts and enemy unit counts is due to the fact that we only play as the Protoss race, but we can play against any of the three races.\n\nThe number of actions in SC2 can be well into the thousands if one considers every individual unit's abilities. As we seek to encode a high-level strategy, rather than rules for moving every individual unit, we restrict the action space for our agent. Rather than using exact mouse and camera commands for individual units, we abstract actions out to simply: ``Build Pylon.'' As such, our agents have 44 available actions, including 35 building and unit production commands, 4 research commands, and 5 commands for attack, defend, harvest resources, scout, and do nothing.\n\nFor the full SC2 game, we set all agents' learning rates to 0.0001, we again initialized $\\alpha=1$, set the batch size to 4, and updates per episode to 8. We run 4 episodes between updates, and set the $LOKI$ $N$=1000. Agents train for as long as necessary to achieve a $>80\\%$ win-rate against the easiest AI, then move up to successive levels of difficulty as they achieve $>80\\%$ win-rates. The agents in this domain update according to the loss function in Equation \\ref{eqn:KLLoss}.\n\n\\subsection{User Study Domain: Wildfire Tracking}\nThe objective in the wildfire tracking domain is to keep two drones on top of two fire centroids as they progress through the map. The task is complicated by the fact that the two drones do not communicate, and do not have complete access to the state of the world. Instead, they have access to a 6D vector containing \\{ $D_N(F_1)$, $D_W(F_1)$, $D_N(F_2)$, $D_W(F_1)$, $C(F_1)$, $C(F_2)$ \\} where $D_N$ is the ``distance to the north`` function and $C(F_1)$ is the ``closer to fire 1'' boolean flag.\n\nThe actions available to the drones include move commands in four directions: north, east, south, and west.\n\n\\section{Initialization Heuristics in Experimental Evaluation}\n\\subsection{Cart Pole Heuristics}\n\\label{sec:cart-heuristics}\nWe use a simple set of heuristics for the cart pole problem, visualized in Figure \\ref{fig:cart-heuristic}. If the cart is close enough to the center, we move in the direction opposite to the lean of the pole, as long as that motion will not push us too far from the center. If the cart is close to an edge, the agent attempts to account for the cart's velocity and recenter the cart, though this is often an unrecoverable situation for the heuristic. The longest run we saw for a \\textsc{ProLoNet} with no training was about 80 timesteps.\n\n\\begin{figure*}[b]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/cart_heuristic.jpeg}\n    \\caption{Visualization of the heuristics used to initialize the cart pole \\textsc{ProLoNet}, and to train the LOKI agent}\n    \\label{fig:cart-heuristic}\n\\end{figure*}\n\n\\subsection{Lunar Lander Heuristics}\n\\label{sec:lunar-heuristics}\nFor the lunar lander problem, the heuristic rules are split into two primary phases. The first phase is engaged at the beginning of an episode while the lander is still high above the surface. In this phase, the lander focuses on keeping the lander's angle as close to 0 as possible.\nPhase two occurs when the lander gets closer to the surface, and the agent then focuses on keeping the y\\_velocity lower than 0.2. As is depicted in Figure \\ref{fig:lunar-heuristic}, there are many checks for both lander legs being down. We found that both $LOKI$ and \\textsc{ProLoNets} were prone to landing successfully, but continuing to fire their left or right boosters. In an attempt to ameliorate this problem, we added the extra ``legs down'' checks.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/lunar_heuristic.jpeg}\n    \\caption{Visualization of the heuristics used to initialize the lunar lander \\textsc{ProLoNet}, and to train the LOKI agent}\n    \\label{fig:lunar-heuristic}\n\\end{figure*}\n\n\n\\subsection{FindAndDefeatZerglings Heuristics}\n\\label{sec:micro-heuristics}\nFor the SC2LE minigame, the overall strategy of our heuristic is to stay grouped up and fight or explore as a group. As such, the first four checks are all in place to ensure that the marines are all close to each other. After they pass the proximity checks, they attack whatever is nearest. If nothing is nearby, they will move in a counter-clockwise sweep around the periphery of the map, searching for more zerglings. Our heuristic is shown in Figure \\ref{fig:micro-heuristic}.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/micro_heuristic.jpeg}\n    \\caption{Visualization of the heuristics used to initialize the FindAndDefeatZerglings \\textsc{ProLoNet}, and to train the LOKI agent}\n    \\label{fig:micro-heuristic}\n\\end{figure*}\n\n\n\\subsection{SC2 Full Game Heuristics}\n\\label{sec:macro-heuristics}\nThe SC2 full game heuristic first checks for important actions that should always be high priority, such as attacking, defending, harvesting resources, and scouting. Once initial checks for these are all passed, the heuristic descends into the build order, where it simply uses building or unit count checks to determine when certain units should be built or trained. After enough attacking units are trained, the heuristic indicates that it is time to attack. The SC2 full game heuristic is depicted in Figure. \\ref{fig:macro-heuristic}.\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/macro_heuristic.jpeg}\n    \\caption{Visualization of the heuristics used to initialize the SC2 full game \\textsc{ProLoNet}, and to train the LOKI agent}\n    \\label{fig:macro-heuristic}\n\\end{figure*}\n\n\n\\section{Architectures for Algorithms in Experimental Evaluation}\n\\label{sec:fc-lstm-arches}\nIn this section we briefly overview the $MLP$ and $LSTM$ action network information. The $LOKI$ agent maintained the same architecture as the $MLP$ agent.\n\n\\subsection{Cart Pole}\nThe cart pole $MLP$ network is a 3-layer network following the sequence:\n\n4x4 -- 4x4 -- 4x2.\n\nWe experimented with sizes ranging from 4-64 and numbers of hidden layers from 1 to 10, and found that the small network performed the best.\n\nThe $LSTM$ network for cart pole is the same as the $MLP$ network, though with an LSTM unit inserted between the first and second layers. The LSTM unit's hidden size is 4, so the final sequence is:\n\n4x4 -- LSTM(4x4) -- 4x4 -- 4x2.\n\nWe experimented with hidden-sizes for the LSTM unit from 4 to 64, though none were overwhelmingly successful, and we varied the number of layers after the LSTM unit from 1-10.\n\nThe \\textsc{ProLoNet} agent for this task used 9 decision nodes and 11 leaves. For the deepening experiment, we tested an agent with only a single node and 2 leaves, and found that it still solved the task very quickly. We tested randomly initialized architectures from 1 to 9 nodes and from 2 to 11 leaves, and we found that all combinations successfully solved the task.\n\n\\subsection{Lunar Lander}\nThe lunar lander $MLP$ network is a 3-layer network, following the sequence:\n\n8x8 -- 8x8 -- 8x4. \n\nWe again experimented with sizes from 8-64 and number of hidden layers from 2 to 11.\n\nThe $LSTM$ network for lunar lander mimics the architecture from cart pole. The LSTM unit's hidden size is 8, so the final sequence is:\n\n8x8 -- LSTM(8x8) -- 8x4.\n\nWe experimented with hidden-sizes for the LSTM unit from 8 to 64, and again we varied the number of layers succeeding the LSTM unit from 1 to 10.\n\nThe \\textsc{ProLoNet} agent for this task featured 14 decision nodes and 15 leaves. We experimented with intelligent initialization architectures ranging from 10 nodes to 14 and from 10 to 15 leaves, and found little difference between their performances. The additional nodes were an attempt to encourage the agent to ``do nothing'' once successfully landing, as the agent had a tendency to continue shuffling left-right after successfully touching down. \n\n\\subsection{FindAndDefeatZerglings}\nWe failed to find a $MLP$ architecture that succeeded in this task, and so we choose one that compromised between the depth of the \\textsc{ProLoNet} and the simplicity that $MLP$ agents seemed to prefer for toy domains. The final network is a 7-layer network with the following sequence:\n\n37x37 -- 37x37 -- 37x37 -- 37x37 -- 37x37 -- 37x37 -- 37x10.\n\nWe choose to keep the size to 37 after testing 37 and 64 as sizes, and deciding that trying to get as close to the \\textsc{ProLoNet} architecture was the best bet.\n\nThe $LSTM$ network for FindAndDefeatZerglings features more hidden layers than the $LSTM$ for lunar lander and cart pole. The hidden size is set to 37, and the LSTM unit is followed by 5 layers. The final sequence is:\n\n37x37 -- LSTM(37x37) -- 37x37 -- 37x37 -- 37x37 -- 37x37 -- 37x10.\n\nWe experimented with hidden-sizes for the LSTM unit from 37 to 64 and varied the number of successive layers from 4-10.\n\nThe \\textsc{ProLoNet} agent for FindAndDefeatZerglings featured 10 nodes and 11 leaves. We tested architectures from 6 to 15 nodes and from 7 to 13 leaves, and found that the initialized policy and architecture had more of an immediate impact for this task. The 7 node policy allowed agents to spread out too much, and they died quickly, whereas the 15 node policy had agents moving more than shooting, and they would walk around while being overrun. \n\\subsection{SC2 Full Game}\nWe again failed to find a $MLP$ architecture that succeeded in this task, and so used a similar architecture to that of the FindAndDefeatZerglings task. The 7-layer network is of the sequence:\n\n193x193 -- 193x193 -- 193x193 -- 193x193 -- 193x193 -- 193x193 -- 193x44.\n\nWe again experimented with a variety of shapes and number of layers, though none succeeded.\n\nAgain, the $LSTM$ network shadows the $MLP$ network for this task. As in the FindAndDefeatZerglings task, we experimented with a variety of LSTM hidden unit sizes, hidden layer sizes, and hidden layer numbers. The final architecture reflects the FindAndDefeatZerglings sequence:\n\n193x193 -- LSTM(193x193) -- 193x193 -- 193x193 -- 193x193 -- 193x193 -- 193x44.\n\nThe \\textsc{ProLoNet} agent for the SC2 full game featured 10 nodes and 11 leaves. We tested architectures from 10 to 16 nodes and from 1 to 17 leaves, and found that the initialized policy and architecture was not as important for this task as it was for the FindAndDefeatZerglings task. As long as we included a basic build order and the ``attack'' command, the agent would manage to defeat the VeryEasy in-game AI at least 10\\% of the time. We found that constraining the policy to fewer nodes and leaves provided less noise as updates progressed, and kept the policy close to initialization while also providing improvements. An initialization with too many parameters often seemed to degrade quickly, presumably due to small changes over many parameters having a larger impact than small changes over few parameters.\n\n\n\\subsection{User Study Domain: Wildfire Tracking}\nThe wildfire tracking domain has a similar state-action space to the lunar lander domain. Therefore, we reuse architecture specifics from the lunar lander architecture sweep. The $MLP$ agent's action network is a 3-layer network, following the sequence:\n\n6x6 -- 6x6 -- 6x4. \n\nThe $LSTM$ network for the wildfire tracking problem also mimics the architecture from the $LSTM$ agent on the lunar lander problem. The LSTM unit's hidden size is 6, so the final sequence is:\n\n6x6 -- LSTM(6x6) -- 6x4.\n\n\\textsc{ProLoNet} initializations varied substantially on this domain, though most produced initializations which eventually became nearly-optimal policies.\n\n\\section{Policy Divergence After Training}\nWe observe that finished policies are not rehashes of the originals; rather, they change and deviate from the original throughout training. In the figures below, we compare checkpointed models to the original initialization. The x-axis corresponds to how far along in the experiment the checkpoint is, and the y-axis corresponds to the average mean-squared error between the initialization and the checkpoint. As the mean-squared error for the weight vectors, comparator values, and leaf weights can be markedly different, we use a logarithmic scale on the y-axis so that the trends can clearly be seen regardless of the raw value. Results are shown in Figure \\ref{fig:divergence-results}. Note that we checkpoint after every 25\\% of training or when the agent has ``solved'' the domain by the OpenAI Gym standards (500 for cart pole, 200+ for lunar lander), accounting for the greater density of checkpoints in the cart pole domain.\n\n\n\\begin{figure*}[h]\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig\/divergence-legend.png}\n\\end{subfigure}\n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/cart-divergence.png}\n        \\caption{Cart Pole}\n        \\label{fig:cart-divergence}\n    \\end{subfigure}\n    ~~\n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/lunar-divergence.png}\n        \\caption{Lunar Lander}\n        \\label{fig:lunar-divergence}\n    \\end{subfigure}\n    ~~\n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/zerg-divergence.png}\n        \\caption{FindAndDefeatZerglings}\n        \\label{fig:micro-divergence}\n    \\end{subfigure}\n\\caption{A comparison of divergence of policies from initialization on cart pole, lunar lander, and FindAndDefeatZerglings.}\n\\label{fig:divergence-results}\n\\end{figure*}\n\n\\section{User Study Examples}\nOur user study includes soliciting natural language instructions from participants as a way to prime them to think through policies before interacting with our interface. We include excerpts below to demonstrate the variety of participants in our study.\n\n\\subsection{Before Using the User Interface}\n\\begin{flushenumerate}\n    \\item ``If it is the closet drone to the first fire, go to the first fire. \nElse, if it is the closet drone to the second fire, go to the second fire. \nElse, (it is not closet drone to the first fire and second fire), go to the second fire.''\n    \\item `` If you are the closest drone to the first fire:\n\tDetermine how far north, south, east, and west you are from the fire\n\tPick a direction to travel first (north\/south, east\/west) and see if the distance in that axis between you and the fire gets smaller. \nIf so, continue that direction but add some angle towards a direction in another axis (e.g. if you determined that traveling north is getting you closer to the fire, pick northwest or northeast to start driving towards). If the new axis addition is not getting you closer to the fire, then decrease your addition of that angle. If you reach zero, then switch to the other axis (i.e. if going northwest does not help, switch to northeast).\nIf your original direction (pure north, south, east, or west) does not get you closer to the fire, go the opposite direction in that axis and see if the distance in that axis decreases. \n\nIf you are the closest drone to the second fire:\n\tRun same instructions as for tracking the first fire.\n\nIf you are the closest drone to both fires:\n\tPick the fire that's the closest to you and run the same instructions as you would for tracking first fire.\n\nIf you're not the closest drone to either fire:\n\tMove randomly until you are the closest drone to one of the fires''\n\\end{flushenumerate}\n\n\n\n\n\\subsection{While Using the User Interface}\n\\begin{flushenumerate}\n    \\item ``If closest to fire 1, if fire 1 is south, move south. If fire 1 is north, move north. If fire 1 is east, move east. If fire 1 is west, move west. Otherwise, move east. If not closest to fire 1, if fire 2 is south, move south. If fire 2 is north, move north. If fire 2 is east, move east, otherwise just move west.''\n    \\item ``So let's start with fire 1. So if I'm the closest drone to fire 1, if fire 1 is to my south, move south. Else if fire 1 is to my north, move north. Else if fire 1 is to my east, move east. Else if fire 1 is to my west, move west. If I'm not the closest drone to fire 1, check if I'm the closest drone to fire 2. If fire 2 is to your south, move south. If fire 2 is to your north, move north. If fire 2 is to your east, move east. If fire 2 is to your west, move west. If not, just take a random action. If you're not closest to fire 1 or fire 2, move north.''\n\\end{flushenumerate}\n\n\\subsection{After Using the User Interface}\n\\begin{flushenumerate}\n    \\item ``If you are the closest to fire 1, check if the fire1 is north of you. If so, move north. If not, check if the drone is south of you. If so, move south. If not, check if the drone is east of you. If so, move east. If not, check if the drone is west of you. If so move east, otherwise do a random action.\nIf you are NOT the closest to fire 1, check if fire 2 is north of you. If so, move north. If not, check if the drone is south of you. If so, move south. If not, check if the drone is east of you. If so, move east. If not, check if the drone is west of you. If so move east, otherwise do a random action.''\n    \\item `` If it is the closet drone to the first fire, \nIf the first fire's north direction is positive, move north. \nIf the first fire's north direction is negative, move south. \nIf the first fire's east direction is positive, move east. \nIf the first fire's east direction is negative, move west. \nElse, if it is the closet drone to the second fire, go to the second fire. \nIf the second fire's north direction is positive, move north. \nIf the second fire's north direction is negative, move south. \nIf the second fire's east direction is positive, move east. \nIf the second fire's east direction is negative, move west. \nElse, (it is not closet drone to the first fire and second fire), go to the second fire. \n\tIf the first fire's north direction is positive, move north. \nIf the first fire's north direction is negative, move south. \nIf the first fire's east direction is positive, move east. \nIf the first fire's east direction is negative, move west. ''\n\\end{flushenumerate}\n\\section{Introduction}\n\\label{sec:intro}\nAs reinforcement learning (RL) is applied to increasingly complex domains, such as real-time strategy games\nor robotic manipulation\nRL and imitation learning (IL) approaches fail to quickly capture the wealth of expert knowledge that already exists for many domains. \nExisting approaches to using IL as a warm start require large datasets or tedious human labeling as the agent learns everything, from vision to control to policy, all at once.\nUnfortunately, these large datasets often do not exist, as collecting these data is impractical or expensive, and humans will not patiently label data for IL-based agents \\cite{amershi2014power}.\nWhile humans may not label enough state-action pairs to train IL-based agents , there is an opportunity to improve warm starts by soliciting expertise from a human once, and then leveraging this expertise to initialize an RL agent's neural network architecture and policy. With this approach, we circumvent the need for IL and instead directly imbue human expertise into an RL agent.\n\n\nTo achieve this blending of human domain knowledge with the strengths of RL, we propose Propositional Logic Nets (\\textsc{ProLoNets}), a new approach to directly encode domain knowledge as a set of propositional rules into a neural network, as depicted in Figure \\ref{fig:study-ui}. \nOur approach leverages decision tree policies from humans to directly initialize a neural network (Figure \\ref{fig:architecture}). \nWe use decision trees to allow humans to specify behaviors to guide the agent through a given domain, such as high-level instructions for keeping a pole balanced on the cart pole problem.\nImportantly, this policy specification does not require the human to demonstrate the balancing act in all possible states, nor does it require the human to label actions as being ``good'' or ``bad.'' \n\nBy directly imbuing logical propositions from the tree into neural network weights, an RL agent can immediately begin learning productive strategies.\nThis approach leverages readily available domain knowledge while still retaining the ability to learn and improve over time, eventually outperforming the expertise with which it was initialized. By exploiting the structural and logical rules inherent to many tasks to which RL is applied, we can bypass early random exploration and expedite an agent's learning in a new domain. \n\nWe demonstrate that our approach can outperform standard deep RL across two OpenAI gym domains \\cite{1606.01540} and two modified StarCraft II domains \\cite{vinyals2017starcraft}, and that our framework is superior to state-of-the-art, IL-based RL, even with observation of that same domain expert knowledge. Finally, in a wildfire simulation domain, we show that our framework can work with untrained human participants. \nOur three primary contributions include:\n\n\\begin{figure*}[t]\n\\centering\n        \\includegraphics[width=0.7\\linewidth]{fig\/pipeline_no_robotarium.jpeg}\n        \\caption{A visualization of our approach as it applies to our user study. Participants interact with a UI of state-checks and actions to construct a decision tree policy that is then used to directly initialize a \\textsc{ProLoNet}'s architecture and parameters. The \\textsc{ProLoNet} can then begin reinforcement learning in the given domain, outgrowing its original specification.}\n        \\label{fig:study-ui}\n\\end{figure*}\n\n\n\\begin{enumerate}\n    \\item We formulate a novel approach for capturing human domain expertise in a trainable RL framework via our architecture, \\textsc{ProLoNets}, which we show outperforms baseline RL approaches, including IL-based \\cite{cheng2018fast} and knowledge-based techniques \\cite{humbird2018deep}, obtaining $>100\\%$ more average reward on a StarCraft 2 mini-game.\n    \\item We introduce dynamic growth to \\textsc{ProLoNets}, enabling greater expressivity over time to surpass original initializations and yielding twice as much average reward in the lunar lander domain.\n    \\item We conduct a user study in which non-expert humans leveraged \\textsc{ProLoNets} to specify policies that resulted in higher cumulative rewards, both before and after training, relative to all baselines ($p < 0.05$).\n\\end{enumerate}\n\\section{Related work}\n\\label{sec:related-work}\n\nWarm starts have been used for RL ~\\cite{cheng2018fast,aaai2019lecture,zhu2017effective} as well as in supervised learning for many tasks \\cite{garcez2012neural,hu2016harnessing,kontschieder2015deep,wang2017combining}. While these warm start or knowledge-based systems have provided an interesting insight into the efficacy of warm starts or human-in-the-loop learning in various domains, these systems typically involve either large labeled datasets with tedious human labeling and feedback, or they require some automated oracle to label actions as ``good'' or ``bad.'' In highly challenging domains or problems, building such an oracle is rarely feasible. Moreover, it is not always possible to acquire a large labeled dataset for new domains.\nHowever, it is often possible to solicit a policy from a human in the form of a high-level series of if-then checks in critical states. These decisions can be collected as a decision tree. \nOur research seeks to convert decision tree into a neural network for RL.\n\nResearchers have previously sought to bridge the gap between decision trees and deep networks~\\cite{humbird2018deep,kontschieder2015deep,laptev2014convolutional}. This work has focused on either partitioning a subspace of the data for more efficient inference~\\cite{tanno2018adaptive}, to enable more explicit interpretability by visualizing a network's classification policy~\\cite{frosst2017distilling,silva2019optimization}, or for warm starting through supervised pre-training on labeled data. As discussed, this data may not be available thus creating a need for methods which can solicit this initialization tree directly from a human.\n\nMost closely related to our work is deep jointly-informed neural networks (DJINN) \\cite{humbird2018deep}, which is the latest in a long line of knowledge-based neural network research \\cite{francca2014fast,garcez2012neural,maclin1996creating,richardson2006markov,towell1994knowledge}. DJINN uses a decision tree learned over a training set in order to initialize the structure of a network's hidden layers and to route input data appropriately. However, DJINN does not explicitly initialize rules, nor does it leverage rules solicited from humans.\nThis distinction means that DJINN creates an architecture for routing information appropriately, but the decision-criteria in each layer must be learned from scratch.\nOur work, on the other hand, directly initializes both the structure \\emph{and} the rules of a neural network, meaning that the human's expertise is more completely leveraged for a more useful warm start in RL domains. We build on decades of research demonstrating the value of human-in-the-loop learning \\cite{towell1994knowledge,zhang2019leveraging} to leverage logical rules solicited from humans in the form of a decision tree to intelligently initialize the structure and rules of a deep network. \n\n\nOur work is related to IL and to knowledge-based or human-in-the-loop RL frameworks \\cite{zhang2019leveraging, aaai2019lecture,macglashan2017interactive} and apprenticeship learning and IRL \\cite{abbeel2004apprenticeship,knox2009interactively}. Importantly, however, our approach does not require demonstrations or datasets to mimic human behavior. While our approach directly initializes with a human-specified policy, IL methods require large labeled datasets \\cite{edwards2018imitating} or an oracle to label data before transitioning to RL, as in the LOKI \\cite{cheng2018fast} framework. \nOur approach translates human expertise directly into an RL agent's policy and begins learning immediately, sidestepping the IL and labeling phase.\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nWithin RL, we consider problems presented as a Markov decision process (MDP), which is a 5-tuple $\\langle S, A, T, R, \\lambda \\rangle$ where $s \\in S$ are states drawn from the state space or domain, $a \\in A$ are possible actions drawn from the action space, $T(s', a, s)$ is the transition function representing the likelihood of reaching a next state $s'$ by taking some action $a$ in a given state $s$, $R(s)$ is the reward function which determines the reward for each state, and $\\lambda$ is a discount factor. In this work, we examine discrete action spaces and semantically meaningful state spaces-- intelligent initialization for continuous outputs and unstructured inputs is left to future work. The goal of our RL agent is to find a policy, $\\pi(a|s)$, that selects actions in states to maximize the agent's expected long-term cumulative reward. IL approaches, such as ILPO~\\cite{edwards2018imitating}, operate under a similar framework, though they do not make use of the reward signal and instead perform supervised learning according to oracle data. \n\n\\section{Approach}\n\\label{sec:approach} \n\n\\begin{figure}[t]\n    \\centering\n         \\includegraphics[width=\\linewidth]{fig\/tree_to_prolo.png}\n    \\caption{A traditional decision tree and a \\textsc{ProLoNet}. Decision nodes become linear layers, leaves become action weights, and the final output is a sum of the leaves weighted by path probabilities.}\n    \\label{fig:architecture}\n\\end{figure}\n\nWe provide a visual overview of the \\textsc{ProLoNet} architecture in Figure \\ref{fig:architecture}. To intelligently initialize a \\textsc{ProLoNet}, a human user first provides a policy in the form of some hierarchical set of decisions. These policies are solicited through simple user interactions for specifying instructions, as in Section \\ref{sec:user-study}. The user's decision-making process is then translated into a set of weights $\\vec{w_n} \\in W$ and comparator values $c_n \\in C$ representing each rule, shown in Algorithm \\ref{alg:prolo-init}. Each weight $\\vec{w_n}$ determines which input features to consider, and, optionally, how to weight them, as there is a unique weight value for each input feature (i.e. $|\\vec{w_n}|==|S|$ for an input space $S$). The comparator $c_n$ is used as a threshold for the weighted features. \n\nEach decision node $D_n$ throughout the network is represented as $D_{n} = \\sigma [\\alpha(\\vec{w_n}^T * \\vec{X} - c_n)]$, where $\\vec{X}$ is the input data, $\\sigma$ is the sigmoid function, and $\\alpha$ serves to throttle the confidence of decision nodes. Less confidence in the tree allows for more uncertainty in decision making~\\cite{yuan1995induction}, leading to more exploration, even from an expert initialization. High values of $\\alpha$ emphasize the difference between the comparator and the weighted input, thus pushing the tree to be more boolean. Lower values of $\\alpha$ encourage a smoother tree, with $\\alpha=0$ producing uniformly random decisions. We allow $\\alpha$ to be a learned parameter.\n\n\\begin{example}[\\textsc{ProLoNet} Initialization]\n\\label{ex:init}\nAssume we are in the cart pole domain \\cite{barto1983neuronlike} and have solicited the following from a human: ``If the cart's $x$ position is right of center, move left; otherwise, move right,'' and that the user indicates $x\\_position$ is the first input feature of four and that the center is at 0. We therefore initialize our primary node $D_0$ with $\\vec{w_0} = [1, 0, 0, 0]$ and $c_0 = 0$, following lines 5-8 in Alg. \\ref{alg:prolo-init}. Following lines 11-13, we create a new leaf $\\vec{l_0} = [1, 0]$ (Move Left) and a new leaf $\\vec{l_1} = [0, 1]$ (Move Right). Finally, we set the paths $Z(\\vec{l_0}) = D_0$ and  $Z(\\vec{l_1}) = (\\neg \\, D_0)$. The resulting probability distribution over the agent's actions is a softmax over $(D_0*\\vec{l_0} + (1-D_0)*\\vec{l_1})$. \n\\end{example}\n\n\n\\begin{algorithm}[H]\n  \\caption{Intelligent Initialization}\n  \\label{alg:prolo-init}\n\n\\begin{algorithmic}[1]\n  \\STATE {\\bfseries Input:} Expert Propositional Rules $R_d$\n  \\STATE {\\bfseries Input:} Input Size $I_S$, Output Size $O_S$\n  \\STATE $W, C, L =$ \\{\\}\n  \\FOR{$r \\in R_d$}\n  \\IF{$r$ is a state check}\n  \\STATE $\\mathbf{s} =$ feature index in $r$\n  \\STATE $w = \\vec{0}^{I_S}$, $w[\\mathbf{s}] = 1$\n  \\STATE $c = $ comparison value in $r$\n  \\STATE $W = W \\cup w$, $C = C \\cup c$\n  \\ENDIF\n  \\IF{$r$ is an action}\n  \\STATE $\\mathbf{a} =$ action index in $r$\n  \\STATE $l = \\vec{0}^{O_S}$, $l[\\mathbf{a}] = 1$\n  \\STATE $L = L \\cup l$\n  \\ENDIF\n  \\ENDFOR\n  \\STATE {\\bfseries Return:} $W$, $C$, $L$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n  \\caption{Dynamic Growth}\n  \\label{alg:deepeninng}\n\n\\begin{algorithmic}[1]\n  \\STATE {\\bfseries Input:} \\textsc{ProLoNet} $P_d$\n  \\STATE {\\bfseries Input:} Deeper \\textsc{ProLoNet} $P_{d+1}$\n  \\STATE {\\bfseries Input:} $\\epsilon =$ minimum confidence \n  \\STATE $H(\\vec{l_i}) = $ Entropy of leaf $\\vec{l_i}$,\n  \\FOR{$l_i \\in L \\in P_d$}\n  \\STATE Calculate $H(l_i)$\n  \\STATE Calculate $H(l_{d1})$, $H(l_{d2})$ \\\\ for leaves under $l_i$ in $P_{d+1}$\n  \\IF{$H(l_i) > (H(l_{d1}) + H(l_{d2}) + \\epsilon)$}\n  \\STATE Deepen $P_d$ at $l_i$ using $l_{d1}$ and $l_{d2}$\n  \\STATE Deepen $P_{d+1}$ at $l_{d1}$ and $l_{d2}$ randomly\n  \\ENDIF\n  \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nAfter all decision nodes are processed, the values of $D_n$ from each node represent the likelihood of that condition being $TRUE$. In contrast, $(1-D_n)$ represents the likelihood of the condition being $FALSE$. With these likelihoods, the network then multiplies out the probabilities for different paths to all leaf nodes. Every leaf $\\vec{l} \\in L$ contains a path $z \\in Z$, a set of decision nodes which should be $TRUE$ or $FALSE$ in order to reach $\\vec{l}$, as well as a prior set of weights for each output action $a \\in \\vec{a}$. For example, in Figure \\ref{fig:architecture}, $z_1 = D_1 * D_2$, and $z_3 = (1-D_1)*D_3$. The likelihood of each action $a$ in leaf $\\vec{l_i}$ is determined by multiplying the probability of reaching leaf $\\vec{l_i}$ by the prior weight of the outputs within leaf $\\vec{l_i}$. After calculating the outputs for every leaf, the leaves are summed and passed through a softmax function to provide the final output distribution.\n\n\n\\begin{example}[\\textsc{ProLoNet} Inference]\n\\label{ex:inference}\n\nConsider an example cart pole state, X=[2, 1, 0, 3] passed to the \\textsc{ProLoNet} from Example \\ref{ex:init}. Following $D_{n} = \\sigma [\\alpha(\\vec{w_n}^T * \\vec{X} - c_n)]$, the network arrives at $\\sigma([1,0,0,0]*[2,1,0,3]-0)=0.88$ for $D_0$, meaning \"mostly true.\" This probability propagates to the two leaf nodes using their respective paths, making the output of the network a probability given by $(0.88 * [1, 0] + (1-0.88)*[0, 1]) = [0.88, 0.12]$. Accordingly, the agent selects the first action with probability 0.88 and the second action otherwise. An algorithmic expression of the forward-pass is provided in the supplementary material.\n\\end{example}\n\n\\paragraph{Dynamic Growth --}\n\\label{sub:deepening}\n\n\\textsc{ProLoNets} are able to follow expert strategies immediately, but they may lack the expressive capacity to learn more optimal policies once they are deployed into a domain. If an expert policy involves a small number of decisions, the network will have a small number of weight vectors and comparators to use for its entire existence. To enable the \\textsc{ProLoNet} architecture to continue to grow beyond its initial definition, we introduce a dynamic growth procedure, which is outlined in Algorithm \\ref{alg:deepeninng} and Figure \\ref{fig:example-deepen}. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/example_deepen.jpeg}\n    \\caption{The dynamic growth process with a deeper \\textsc{ProLoNet} shown in paler colors and dashed lines. When $H(L_3)+H(L_4) < H(L_1)$, the agent replaces $L_1$ with $D_2$, $L_3$, and $L_4$ and adds a new level to the deeper actor.}\n    \\label{fig:example-deepen}\n\\end{figure}\n\n\nUpon initialization, a \\textsc{ProLoNet} agent maintains two copies of its actor. The first is the shallower, unaltered initialized version, and the second is a deeper version in which each leaf is transformed into a randomly initialized decision node with two new randomly initialized leaves (line 1 of Alg. \\ref{alg:deepeninng}). This deeper agent has more parameters to potentially learn more complex policies, but at the cost of added randomness and uncertainty, reducing the utility of the intelligent initialization.\n\nAs the agent interacts with its environment, it relies on the shallower network to generate actions, as the shallow network represents the human's domain knowledge. After each episode, the off-policy update is run over the shallower and deeper networks. Finally, after the off-policy updates, the agent compares the entropy of the shallower actor's leaves to the entropy of the deeper actor's leaves and selectively deepens when the leaves of the deeper actor are less uniform than those of the shallower actor (lines 3-7). We find that this dynamic growth mechanism improves stability and average cumulative reward.\n\n\\begin{example}[\\textsc{ProLoNet} Dynamic Growth]\n\\label{ex:deepen}\n\nAssume the cart pole agent's shallower actor has found a local minimum with $l_1=[0.5, 0.5]$, while the deeper actor has $l_3=[0.9, 0.1]$ and $l_4=[0.1, 0.9]$. Seeing that $l_1$ is offering little benefit to the current policy, and $D_2$ in the deeper actor is able to make a decision about which action offers the most reward, the agent would dynamically deepen at $l_1$, copying over the deeper actor's parameters and becoming more decisive in that area of its policy. The deeper actor would also grow with a random set of new parameters, as shown in Figure \\ref{fig:example-deepen}.\n\\end{example}\n\n\\section{Experimental evaluation}\n\\label{sec:experiments}\n\nWe conduct two complementary evaluations of the \\textsc{ProLoNet} as a framework for RL with human initialization. The first is a controlled investigation with expert initialization in which an author designs heuristics for a set of domains with varying complexity; this allows us to confirm that our architecture is competitive with baseline learning frameworks. We also perform an ablation of intelligent initialization and dynamic growth in this set of experiments.  The second evaluation is a user study to support our claim that untrained users can specify policies that serve to improve RL. \n\nIn our first evaluation, we assess our algorithm in StarCraft II (SC2) for macro and micro battles as well as the OpenAI Gym \\cite{1606.01540} lunar lander and cart pole environments. Optimization details, hyperparameters, and code are all provided in the supplementary material. \n\n\n\nTo evaluate the impact of dynamic growth and intelligent initialization, we perform an ablation study and include results from these experiments in Table \\ref{tab:ablate}. For each $N$-mistake agent, weights, comparators, and leaves are randomly negated according to $N$, up to a maximum of $2N$ for each category.\n\n\\begin{figure*}[t]\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{fig\/exp_legend.png}\n\\end{subfigure} \n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/cart_prolo_vs_baselines_no_legend.png}\n        \\caption{Cart Pole}\n        \\label{fig:cart-results}\n    \\end{subfigure}\n    ~~\n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/lunar_prolo_vs_baselines_no_legend.png}\n        \\caption{Lunar Lander}\n        \\label{fig:lunar-results}\n    \\end{subfigure}\n    ~~\n    \\begin{subfigure}[b]{0.31\\textwidth}\n        \\includegraphics[width=\\textwidth]{fig\/FindAndDefeatZerglings_prolo_vs_baselines_no_legend.png}\n        \\caption{FindAndDefeatZerglings}\n        \\label{fig:micro-results}\n    \\end{subfigure}\n\\caption{A comparison of architectures on cart pole, lunar lander, and FindAndDefeatZerglings \\cite{vinyals2017starcraft}. As the domain complexity increases, we see that intelligent initialization is increasingly important, and \\textsc{ProLoNets} are the most effective method for leveraging domain expertise, and perform well even when domain expertise is unnecessary, as in cart pole.}\n\\label{fig:easy-results}\n\\end{figure*}\n\n\\subsection{Agent formulations}\n\\label{sub:bot-variants}\n\nWe compare several agents across our experimental domains. The first is a \\textit{\\textsc{ProLoNet}} agent as described above and with expert initialization. We also evaluate a multi-layer perceptron (\\textit{MLP}) agent and a long short-term memory (\\textit{LSTM})~\\cite{hochreiter1997long} agent, both using ReLU activations~\\cite{nair2010rectified}. We include comparisons to a \\textsc{ProLoNet} with random initialization (\\textit{Random \\textsc{ProLoNet}}) as well as the \\textit{Heuristic} used to initialize our agents. We compare to an IL agent trained with the \\textit{LOKI} framework, in which the agent imitates for the first N episodes~\\cite{cheng2018fast}, where N is a tuned hyperparameter, and then transitions to RL. The \\textit{LOKI} agent supervises with the same heuristic that is used to initialize the \\textit{\\textsc{ProLoNet}} agent. Finally, although the original DJINN framework \\cite{humbird2018deep} requires a decision tree learned over a labeled dataset, we extend the DJINN architecture to allow for initialization with a hand-crafted decision tree in order to compare to a \\textit{DJINN} agent that is initialized using the same heuristic as \\textit{LOKI} and \\textit{\\textsc{ProLoNet}}, but built with the DJINN architecture.\n\n\\subsection{Environments}\n\\label{sub:environment}\nWe consider four environments to empirically evaluate \\textsc{ProLoNets}: cart pole, lunar lander, the FindAndDefeatZerglings minigame from the SC2LE~\\cite{vinyals2017starcraft}, and a full game of SC2 against the in-game artificial intelligence (AI). These environments provide us with a steady increase in difficulty, from a toy problem to the challenging game of full SC2. These evaluations also showcase the ability of the \\textsc{ProLoNet} framework to compete with state-of-the-art approaches in simple domains and excel in more complex domains. For the SC2 and SC2LE problems, we use the SC2 API\\footnote{https:\/\/github.com\/Blizzard\/s2client-api} to manufacture 193D and 37D state spaces, respectively, and 44D and 10D action spaces, respectively. In the full SC2 domain, making the right parameter update is a significant challenge for RL agents. As such, we verify that the agent's parameter updates increase its probability of victory, and if a new update has decreased the agent's chances of success, then the update is rolled back, and the agent gathers experience for a different step, similar to the checkpointing approach in \\citet{hosu2016playing}. \n\n\\begin{table*}\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{*{7}{p{1.6cm}}}\n & & Random. & Shallow & & &\\\\\nDomain & \\textsc{ProLoNet} & \\textsc{ProLoNet} & \\textsc{ProLoNet} & N = 0.05 & N = 0.1 & N = 0.15 \\\\\n\\midrule\nCart Pole    & 449$\\pm$15& 401$\\pm$26& 415$\\pm$ 27& 426$\\pm$ 30& 369$\\pm$ 28& 424$\\pm$ 29 \\\\\nLunar & 86 $\\pm$ 33&  55$\\pm$19& 49$\\pm$ 20& 50$\\pm$ 22& 45$\\pm$ 22& 45$\\pm$ 22\\\\\nZerglings    & 8.9$\\pm$1.5& -1.3$\\pm$0.6& 8.8$\\pm$1.5& 5.1$\\pm$1.1& 5.9$\\pm$1.2 & 4.1$\\pm$1.1 \\\\\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{\\textsc{ProLoNet} ablation study of average cumulative reward. Units are in thousands.}\n\\label{tab:ablate}\n\\end{table*}\n\\paragraph{OpenAI Gym --}\n\\label{para:gym-results}\n\nAs depicted in Figure \\ref{fig:cart-results} and \\ref{fig:lunar-results}, \\textsc{ProLoNets} are able to either match or exceed performance of standard reinforcement and imitation learning based RL architectures. Furthermore, we find that the \\textsc{ProLoNet} architecture--even without intelligent-initialization--is competitive with baseline architectures in the OpenAI Gym. Running reward in these domains is averaged across five runs, as recommended by~\\citet{henderson2018deep}. \\textit{MLP} and \\textit{LSTM} agents use 1-layer architectures which maintain input dimension until the output layer. We find success with intelligent initializations using as few as three nodes for the cart pole domain and as few as 10 nodes for the lunar lander. These results show that \\textsc{ProLoNets} can leverage user knowledge to achieve superior results, and our ablation study results (Table \\ref{tab:ablate}) show that the architecture is robust to sub-optimal initialization in these domains.\n\nEven where intelligent initialization is not always necessary or where high-level instruction is difficult to provide, as in cart pole, it does not hinder RL from finding solutions to the problem. Further, while baselines appear unstable in these domains, potentially owing to missing implementation hacks and tricks \\cite{engstrom2019implementation}, we observe that the \\textsc{ProLoNet} approaches are able to succeed with the same PPO implementation and learning environment.\n \n\n\\paragraph{StarCraft II: FindAndDefeatZerglings --} For this problem, we assign an agent to each individual allied unit. The best-performing initialization in this domain has 6 decision nodes and 7 leaves. Running reward is depicted in Figure \\ref{fig:micro-results}, again averaged over 5 runs. Intelligent initialization is crucial in this more complex domain, and the \\textit{Random \\textsc{ProLoNet}} fails to find much success despite having the same architecture as the \\textit{\\textsc{ProLoNet}}. \\textit{LOKI} performs on par with the \\textit{Heuristic} used to supervise actions, but \\textit{LOKI} is unable to generalize beyond the \\textit{Heuristic}. \\textit{MLP} and \\textit{LSTM} agents use a 7-layer architecture after a hyperparameter search, and we extend this to the full game of SC2. Importantly, this result (Figure \\ref{fig:micro-results}) shows user-initialized \\textsc{ProLoNets} can outperform our baselines and that this initialization is key to efficient exploration and learning. The importance of the initialization policy is again shown in Table \\ref{tab:ablate}, where even negating 10\\% of the agent's parameters results in a significantly lower average reward.\n\n\n\n\n\n\n\\paragraph{StarCraft II: Full Game --} After 5,000 episodes, no agent other than the \\textit{\\textsc{ProLoNet}} is able to win a single game against the in-game AI at the easiest setting. Even the \\textit{LOKI} and \\textit{DJINN} agents, which have access to the same heuristics used by the \\textit{\\textsc{ProLoNet}}, are unable to win one game. The \\textit{\\textsc{ProLoNet}}, on the other hand, is able to progress to the ``hard\" in-game AI, achieving 100\\% win rates against easier opponents as it progresses. Even against the ``hard\" in-game AI, the \\textit{\\textsc{ProLoNet}} agent is able to double its win rate from initialization. This result demonstrates the importance of an intelligent initialization in complex domains, where only a very narrow and specific set of actions yield successful results. Access to oracle labeling (\\textit{LOKI}) or a knowledge-based architecture (\\textit{DJINN}) does not suffice; the agent requires the actual warm start of having intelligent rules built-in. Thus, we believe these results demonstrate that our novel formulation is singularly capable of harnessing domain knowledge. \n\n\n\\section{User study with non-experts}\n\\label{sec:user-study}\n\nOur second evaluation investigates the utility of our framework with untrained humans providing the expert initialization for \\textsc{ProLoNets}. As presented in Section \\ref{subsec:study-results}, our user study shows that untrained users can leverage \\textsc{ProLoNets} to train RL policies with superior performance. These results provide evidence that our approach can help democratize RL.\n\n\\begin{table}\n\\label{tab:macro-results}\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lccc}\n&\\textsc{ProLoNet}&\\textsc{ProLoNet} at& All \\\\\nAI Difficulty & (Ours) & Initialization & Others \\\\\n\\midrule\nVeryEasy & \\textbf{100\\%}  & 14.1 \\%& 0\\%\\\\\nEasy & \\textbf{100\\%}  & 10.9 \\%& 0 \\%\\\\\nMedium & \\textbf{82.2\\%} & 11.3 \\%& 0 \\%\\\\\nHard & \\textbf{26\\%} & 10.7 \\%& 0 \\%\\\\\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\caption{Win rates against the StarCraft II in-game AI. ``All Others'' includes all agents in Section \\ref{sub:bot-variants}.}\n\\end{table}\n\n\\paragraph{Hypotheses --}\n\\label{para:hypotheses}\nWe seek to investigate whether an untrained user can provide a useful initial policy for \\textsc{ProLoNets}. Hypothesis 1 (\\textbf{H1}): Expert initializations may be solicited from average users, requiring no particular training of the user, and these initializations are superior to random initializations. Hypothesis 2 (\\textbf{H2}): RL can improve significantly upon these initializations, yielding superior policies after training.\n\n\\paragraph{Metrics --}\n\\label{para:metrics}\nTo test \\textbf{H1}, we measure the reward over time for our best participant, all participants, and baseline methods. Testing \\textbf{H2}, we measure the average reward for the first 50 and final 50 episodes for all agents specified by participants and our strongest baseline. Our metrics allow us to effectively examine our hypotheses in the context of expert initialization in our study domain.\n\n\\paragraph{Domain: Wildfire Tracking --}\n\\label{subsec:fire-sim}\nWe develop a Python simulator for a domain that is both suited to RL and of relevance to a wider audience: wildfire tracking. \nWe randomly instantiate two fires and two drones in a 500x500 grid. The drones receive a 6D state as input, containing distances to fire centroids and Boolean flags for which a drone is the closest to each fire. The action space for drones is a 4D discrete decision of which cardinal direction to move into. Pre-made state checks include statements such as ``If I am the closest drone to Fire 2'' and ``If Fire 1 is to my west.'' The two drones are controlled by separate agents without communication, and network weights are shared. \nThe reward function is the negative distance between drones and fire centroids, encouraging drones to follow the fire as closely as possible.\n\n\\subsection{Study details}\n\\label{subsec:study-details}\nTo solicit policy specifications from users, we designed a user interface that enabled participants to select from a set of pre-made state checks and actions. Participants were first briefed on the domain and shown a visualization and then asked to talk-through a strategy for monitoring the fires with two independent drones. After describing a solution and seeing the domain, participants were presented with the UI to build out their policies. As the participant selected options, those rules were composed into a decision tree. Once participants completed the study, we leveraged their policy specifications to initialize the structure and parameters of a \\textsc{ProLoNet}. The \\textsc{ProLoNet} was then deployed to the wildfire domain, where it further improved through RL. Our results are presented in Figure \\ref{fig:firesim-results} and described below. We present both the highest performing participant (``Best''), as well as the median over all participants (``Median''), and compare against the agents presented in Section \\ref{sub:bot-variants}. \\textit{LOKI} and \\textit{DJINN} agents use the ``Best'' participant policy specification as a heuristic.\n\n\n\n\\subsection{Study results}\n\\label{subsec:study-results}\nOur IRB-approved study involved 15 participants (nine male, six female) between 21 and 29 years old ($M=24, SD=2$). The study took approximately 45 minutes, and participants were compensated for their time. Our pre-study survey revealed varying degrees of experience with robots and games, though we note that our participants were mostly computer science students. Importantly, we found that their prior experience with robots, learning from demonstration, or strategy games did not impact their ability to specify useful policies for our agents.\n \nNearly all participants provided policy specifications that were superior to random exploration. After performing RL over participant specifications, we can see in Figure \\ref{fig:firesim-results} that \\textbf{intelligent initialization yields the most successful RL agents,} even from non-experts. We compare to the best performing baseline, \\textit{Random \\textsc{ProLoNet}} in Figure \\ref{fig:fire-bars}. We can again see that the participants' initializations are not only better than random initialization, but are also better than the trained RL agent.\nA Wilcoxon signed-rank test shows that our participants' initializations (Median = -23, IQR = 19) were significantly better than a baseline initialization  (Median = -87, IQR = 26), $W(15)=1.0, p < .001$. Our participants' agents (Median = -7.9 , IQR = 29) were also significantly better than a baseline (Median = -52, IQR = 7.9) after training, $W(15)=15.0, p = 0.011$. These results are significant after applying a Bonferroni correction to test the relative performance both before and after training. \\textbf{This result supports hypothesis \\textbf{H1}}, showing that average users can specify useful policies for RL agents to explore more efficiently than random search and significantly outperform baselines.\n\nFurthermore, our participants' agents are significantly better post-training than at initialization, as shown by a Wilcoxon signed-rank test ($W(15)=4.0, p < 0.01$). \\textbf{This finding supports hypothesis \\textbf{H2}}, showing that RL improves on human specifications, not merely repeating what the humans have demonstrated. By combining human intuition and expertise with computation and optimization for long-term expected reward, we are able to produce agents that outperform both humans and traditional RL approaches.\n\nFinally, we qualitatively demonstrate the utility of intelligent initialization and the \\textsc{ProLoNet} architecture by deploying the top performing agents from each method to two drones with simulated fires to track. Videos of the top four agents are included as supplementary material.\n\n\n\\begin{figure}[t]\n\\centering\n        \\includegraphics[width=0.7\\linewidth]{fig\/dist_to_fire.png}\n        \\caption{Initial and final distance between drones and wildfire centroids in our user study domain, where lower distance is better. Participant initializations are significantly better at tracking fires than random, showing that untrained users can leverage our approach to provide useful warm starts.}\n        \\label{fig:fire-bars}\n    \\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.93\\linewidth]{fig\/FireSim_user_study.png}\n    \\caption{Wildfire tracking results, again demonstrating the importance of direct intelligent initialization (\\textsc{ProLoNet}) rather than IL or random initialization.}\n    \\label{fig:firesim-results}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe proposed two complementary evaluations of our proposed architecture, demonstrating the significance of our contribution. Through our first set of experiments on an array of RL benchmarks with a domain expert building heuristics, we empirically validated that \\textsc{ProLoNets} are competitive with baseline methods when initialized randomly and, with a human initialization, outperforms state-of-the-art imitation and RL baselines. As we see in Figure \\ref{fig:easy-results}, \\textsc{ProLoNets} are as fast or faster than baseline methods to learn an optimal policy over the same environments and optimization frameworks. In our more complex domains, we identify the importance of an intelligent initialization. While the IL baseline performs well in the FindAndDefeatZerglings minigame, \\textit{LOKI} cannot improve on the imitated policy. In the full game of SC2, no approach apart from our intelligently-initialized \\textsc{ProLoNet} wins even a single game. The ability to leverage domain knowledge to initialize rules as well as structure, rather than simple architecture and routing information, as in DJINN, is a key difference that enables the success of our approach.\n\nThrough our user study, we demonstrated the practicality of our approach and shown that average participants, even those with no prior experience in the given domain, can produce policy specifications which significantly exceed random initialization ($p<0.05$). Furthermore, we have demonstrated that RL can significantly improve upon these policies, learning to refine ``good enough'' solutions into optimal ones for a given domain. This result shows us that our participants did not simply provide our agents with optimal solutions iterated upon needlessly. Instead, our participants provided good but sub-optimal starting points for optimization. These starting policies were then refined into a solution that was more robust than either the human's solution or the best baseline solution. Our study confirms that our approach can leverage readily available human initializations for success in deep RL, and moreover, that the combination of human initialization and RL yields the best of both worlds.\n\n\n\\section{Conclusion}\n\nWe present a new architecture for deep RL agents, \\textsc{ProLoNets}, which permits intelligent initialization of agents. \\textsc{ProLoNets} grant agents the ability to grow their network capacity as necessary, and are surprisingly capable even with random initialization. We show that \\textsc{ProLoNets} permit initialization from average users and achieve a high-performing policy as a result of the blend of human instruction and RL. We demonstrate, first, that our approach is superior to imitation and reinforcement learning on traditional architectures and, second, that intelligent initialization allows deep RL agents to explore and learn in environments that are too complex for randomly initialized agents. Further, we have confirmed that we can solicit these useful warm starts from average participants and still develop policies superior to baseline approaches in the given domains, paving the way for reinforcement learning to become a more collaborative enterprise across a variety of complex domains.\n\n\n\n\n\n\n\n\n\n\\section*{Ethical Considerations}\nOur work is a contribution targeted at democratizing reinforcement learning in complex domains. The current state of the art in reinforcement learning in complex domains requires compute time and power beyond the capacity of many labs, hand-engineering which is rarely explained publicly, or large labeled datasets which are not always shared. By providing a means for intelligent initialization by practitioners and improved exploration in many domains, we attempt to lower the barrier to entry for research in reinforcement learning and to broaden the number of potential applications of reinforcement learning to more grounded, real-world problems. While there are risks with any technology being misused, we believe the benefits of democratizing RL outweigh the risks. We posit that giving everyone the ability to use RL rather than just large corporations and select universities is a positive contribution to society.\n\n\\paragraph{Beneficiaries --} Our work seeks to improve and simplify reinforcement learning research for all labs and to take steps toward democratizing reinforcement learning for non-experts. We feel that the computational and dataset savings of our work stand to benefit all researchers within reinforcement learning.\n\\paragraph{Negatively affected parties --} We do not feel that any group of people or research direction is negatively impacted by this work. Our work is complementary to other explorations within reinforcement learning, and insights from imitation learning translate naturally into insights on the qualities of useful or harmful intelligent initializations.\n\\paragraph{Implications of failure --} While our method seeks to simplify reinforcement learning, in the worst case the initialization falls back to random and the learning agent is again faced with an intractable random exploration problem. Adversarial agents using our approach would be able to instantiate a worse-than-random agent, though our results imply that it is possible to overcome such an initialization in simple domains.\n\\paragraph{Bias and fairness --} Our work does rely on the ``bias'' of its initialization--that is, it is biased towards the actions which a human has pre-specified. While this biased exploration may fail to accurately explore or understand the intricacies of a complex domain, the alternative (years of compute with random exploration) is simply unavailable to many researchers. This bias may be overcome through diversification of intelligent initializations which may lead to a diversity of final strategies. However, the unification of such diverse policies into a single agent and the thorough study of diverse initializations is left to future work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\n\n\nLet $X$ denote a compact Riemann surface of genus $g \\geq 2$. By the\nretrosection theorem of Koebe-Courant (e.g., \\cite{Bers}) $X$ can be\nrepresented by a Schottky group $\\Gamma$: we can write $$X=\\Gamma\n\\backslash(\\mathbf{P}^1(\\mathbf{C})-\\Lambda_\\Gamma),$$ where $\\Lambda=\\Lambda_\\Gamma$\nis the set of limit points of $\\Gamma$ and $\\Gamma$ is a free group of\nrank $g$, discrete in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. In complex analysis, it is well\nknown that the dynamics of the action of $\\Gamma$ on the limit set\nendowed with Patterson-Sullivan measure encodes a lot about the\nstructure of the Riemann surface. Our purpose is to show that this\naction can be conveniently encoded by a notion from non-commutative\ngeometry, namely a \\emph{spectral triple} (\\cite{ConnesLMP}) which\nprovides the non-commutative analogue of a Riemannian manifold. As it\nwill turn out, the spectral triple we will consider is\ncommutative. But, as has been observed frequently, ``even for\nclassical spaces, which correspond to commutative algebras, the new\npoint of view [of noncommutative geometry] will give new tools and\nresults.'' (Connes, \\cite{Connesbook}, p.\\ 1).  \nWe show that the isometry class of the boundary action is encoded in\nthe \\emph{zeta function} formalism of the spectral triple.  \n\n\\paragraph{Construction of the spectral triple} At the\ntopological level, we consider the commutative algebra $A=C(\\Lambda)$.\nWe also need the dense subalgebra $A_\\infty=C(\\Lambda,\\mathbf{Z})\\otimes_\\mathbf{Z}\n\\mathbf{C}$ of locally constant functions on $\\Lambda$. \n\nIt might be natural to consider the boundary operator algebra\n$C(\\Lambda) \\rtimes \\Gamma$ instead. However, by the non-amenability\nof the group $\\Gamma$, the hyperfiniteness result of Connes implies\nthat this algebra does not carry any finitely summable spectral triple\n(\\cite{Conneshyper}). As we will indicate at the end of the proof of\nthe main theorem, our construction can be extented to an AF-algebra\nthat is Morita equivalent to a large subalgebra of the boundary\noperator algebra. \n\nTo retrieve the actual conformal structure, we need to make the\noperator algebra act on a Hilbert space in a way compatible with a\nDirac operator. The Hilbert space $H$ is a particular\nGNS-representation of $A$. Its construction depends on chosing a\nstate, and we make this choice in such a way that it encodes the\nmetric action of $\\Gamma$ on $\\Lambda$, expressed via the\nPatterson-Sullivan measure. More specifically, on certain elements of\n$A$ related to words in a presentation of $\\Gamma$, it gives the\nmeasure of the subset of $\\Lambda$ reached from that word in  the\nrepresentation of the limit set of $\\Gamma$ via word group completion\n(Floyd \\cite{Floyd}). Finally, the Dirac operator $D$ is composed from\nprojection operators depending on the word length grading in a\npresentation for $\\Gamma$. Let ${\\mathcal S}_X$ denote the spectral triple so\nconstructed (see Section \\ref{s1} for details). \n\n\\begin{introprop} \\label{tA}\nIf $X$ is a compact Riemann surface of genus at least $2$, ${\\mathcal S}_X$ is a 1-summable spectral triple.\n\\end{introprop}\n\n\\paragraph{Zeta function rigidity} The theory of finitely summable\nspectral triples comes with an elegant framework of zeta functions:\nfor any $a \\in A_\\infty$, one has the spectral zeta function\n$\\zeta_{X,a}(s):=\\mathrm{tr} (a|D|^s)$. \n\n\\begin{introthm}\\label{tB}\nIf $X_1$ and $X_2$ are compact Riemann surfaces of genus at least $2$,\nsuch that  \n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for all $a \\in A_\\infty$, then\n$X_1$ and $X_2$ are conformally or anti-conformally equivalent as\nRiemann surfaces. In particular, the spectral triple ${\\mathcal S}_X$ encodes\nthe conformal\/anticonformal isomorphism type of $X$. \\end{introthm}  \n\nIn the theorem, equality of zeta functions should be understood as\nfollows: both the algebras $A_1$ and $A_2$ of $X_1$ and $X_2$,\nrespectively, have a unit. If the zeta functions for this unit are\nequal, we first deduce that the Riemann surfaces have the same\ngenus. This allows us to identify the algebras $A_1$ and $A_2$, along\nwith the corresponding dense subalgebras of locally constant\nfunctions. It then makes sense to interpret the expression\n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$. \n\nAbout the proof: by the analogue of Fenchel-Nielsen theory (cf.\\ Tukia\n\\cite{Tukia2}), the abstract isomorphism of Schottky groups of $X_1$\nand $X_2$ induces a unique homeomorphism of limit sets, equivariant\nwith respect to the group isomorphism (the \\emph{boundary map}). We\nshow that equality of zeta functions implies that this boundary map is\nabsolutely continuous. For this, one has to trace through the\nrepresentation of the limit set in the sense of Floyd (\\cite{Floyd}) to deduce fairly explicit expressions for various zeta functions. We do this by computing traces in an explicit orthonormal basis for $H$.\n\nWe then apply an ergodic rigidity theorem for Schottky uniformization\nthat we deduce from a theorem of Yue (cf.\\ \\cite{Yue}; from the long\nhistory we also mention the names Mostow, Kuusalo, Bowen, Sullivan,\nTukia).  This says that there are only two alternatives for the\nboundary map: either the Patterson-Sullivan measures are mutually\nsingular with respect to the boundary map, or the map extends to a\ncontinuous automorphism of $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Absolute continuity excludes\nthe first case.  \n\nWe will end the paper with a list  of open questions. \n\n\\paragraph{Can you hear the shape of a Riemann surface?} Theorem\n\\ref{tB} fits into the framework of isospectrality questions, as\ncoined by I.M.~Gelfand for compact Riemannian manifolds. Vign\\'eras (\\cite{V1}, \\cite{V2})  and Sunada\n(\\cite{Sunada}) constructed non-isometric surfaces with identical Laplace operator zeta function $\\mathop{\\mathrm{tr}}\\nolimits(\\Delta^s)$ (`isospectral'). The work of Sunada, in particular, transports an idea from algebraic number theory due to Gassmann (\\cite{Gassmann}), where the same phenomenon is visible: there exist non-isomorphic algebraic number fields with identical Dedekind zeta function.\n\n\n For the specific case of Riemann surfaces, Buser (\\cite{Buser}) obtained  a\n(finite) upper bound on the number of Riemann surfaces isospectral with a given Riemann surface, depending only on the genus of that surface, and work of Brooks, Gornet and Gustafson (\\cite{BGG})\nshows this bound is of the correct order of magnitude. \n\nAs for Dirac operators instead of Laplace operators, B\\\"ar has\nconstructed non-isometric space forms with the same Dirac spectrum\n(\\cite{Baer1}).  \n\nFor the case of planar domains, the problem of isospectrality was\ncoined by Bochner, to quote Kac --- quoting Lipman Bers ---  ``can you\nhear the shape of a drum?'' (\\cite{Kac},  solved by Gordon, Webb and\nWolpert \\cite{GWW}). \n\nIn this phrasing, Theorem \\ref{tB} says you can\nhear the complex analytic type of a compact Riemann surface from\nlistening to the noncommutative spectra of its associated spectral\ntriple (that is, to the collection of the associated zeta\nfunctions). A main difference with respect to the classical isospectrality\nquestion is that in this case you do have to listen to $\\mathop{\\mathrm{tr}}\\nolimits(aD^s)$ for a\ndense subset of operators $a \\in A$, and the eigenvalues of $D$\nthemselves are not so interesting. For example, at the unit $1 \\in A$, we find the innocent zeta function (cf.\\ Formula (\\ref{zeta1}))\n$$ \\zeta_{X,1}(s) = 1 +\\frac{2g-2}{2g-1} \\cdot \\frac{(2g)^{3s+1}}{1-(2g-1)^{3s+1}}. $$\n We note that in the completely\ndifferent construction of a  ``conformal'' spectral triple by B\\\"ar,\nthe eigenvalues themselves are uninteresting, too (\\cite{Baer}). \n\nThus, the main point of our discussion is that the ``spectral object'' ${\\mathcal S}_X$ determines the ``conformal object'' $X$, up to complex conjugation.\n\n\n\n\n\\section{A spectral triple associated to a Kleinian Schottky group} \\label{s1}\n\n\n\\begin{se}The aim of this section is to introduce a finitely summable spectral\ntriple ${\\mathcal S}_X:=(A,H,D)$ associated to a Schottky group $\\Gamma$ that\nuniformizes a compact Riemann surface $X$ of genus $g \\geq 2$.  Recall\nthat a spectral triple is a noncommutative analogue of a \nRiemannian spin manifold, \nwhere $A$ is a $C^*$-algebra, $H$ is a Hilbert space on which\n$A$ acts by bounded operators, and $D$ is an unbounded self\nadjoint operator on $H$ with compact resolvent $(D-z)^{-1}$ for\n$z\\notin \\mathbf{R}$, and such that the commutators $[D,a]$ are bounded\noperators for all $a$ in a dense involutive subalgebra $A_\\infty$ of\n$A$. Connes has shown \\cite{ConnesLMP} that if $(A,H,D)$ arises from a\nRiemannian spin manifold, then the distance element is encoded by the\ninverse of the Dirac operator.  \n\\end{se}\n\n\n\\begin{se}Let $\\Gamma$ denote a Schottky group of rank $g \\geq 2$.  As an abstract group, $\\Gamma$ is isomorphic to $F_g$, the free group on $g$ generators. \nWe think of $\\Gamma$ as being specified by an injective group\nhomomorphism $\\rho  :  F_g \\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Let\n$\\Lambda=\\Lambda_\\Gamma$ denote the limit set of the action of\n$\\Gamma$ on $\\mathbf{P}^1(\\mathbf{C})$. \n \\end{se}\n\n\\begin{se}[Group completion and limit set] We recall what we need from\nFloyd's relation between the group completion of $F_g$ and the limit\nset $\\Lambda$ of $\\Gamma$ (\\cite{Floyd}).  Let $Y_g$ denote the Cayley\ngraph (with unordered edges) of $F_g$ for a presentation of $F_g$ in a\nfixed alphabet on $g$ letters, and let $\\bar{Y}_g$ denote the\ncompletion of the Cayley graph as a metric space for the following\nmetric. Let $|w|$ denote the reduced word length of a word in the\ngenerators of $F_g$. The edge between two words $w_1$ and $w_2$ is\ngiven length $\\min \\{ |a|^{-2}, |b|^{-2} \\}$ (with $|e|^{-2}:=1$ for\nthe empty word $e$).  \nThe \\emph{group completion} of $F_g$ is by definition the space\n$\\bar{F}_g:=\\bar{Y}_g-Y_g$. It is a compact metric space. A different\n(finite) presentation for $F_g$ leads to a Lipshitz equivalent group\ncompletion. Since $F_g$ has no ``parabolic ends'' in the sense of\nFloyd, we have the following: \n\\end{se}\n\n\\begin{lem}[Floyd, \\cite{Floyd}, p.\\ 213--217] Given a point $x_0 \\in\n\\mathbf{P}^1(\\mathbf{C})$ and an embedding $\\rho  :  F_g \\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$\nas above, the following map is a continuous bijection: \n$$ \\begin{array}{lccc} \\iota_\\rho \\ : \\  & \\bar{F}_g & \\rightarrow &\n\\Lambda \\\\ &  \\lim\\limits_i w_i & \\mapsto & \\lim\\limits_i\n\\rho(w_i)(x_0). \\end{array} \\ \\ \\ \\ \\Box $$  \n\\label{Floyd} \\end{lem}\n\n\\begin{df} Given a reduced word $w$ in the generators of $F_g$, let\n$i(w)$ respectively $t(w)$ denote the initial, respectively terminal\nletter of $w$. \nFor two reduced words $w$ and $v$ (or $v$ a limit of such), we write $$w \\subseteq v \\mbox{ if }(\\exists w_0)(v=w \\cdot w_0) \\mbox{ with }t(w) \\neq i(w_0)^{-1}.$$\n We write $w \\subset v$ if $w \\subseteq v$ and $w \\neq v$. \n\nGiven $\\rho: F_g \\rightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and a word $w \\in F_g$,\ndefine the subset of $\\Lambda$ of \\emph{ends of $w$ with respect to\n$\\rho$} to be  \n$$ \\overrightarrow{w}_\\rho := \\{ \\iota_\\rho(v) \\ : \\ v \\in \\bar{F}_g \\mbox{ and } w \\subseteq v \\}. $$ \n\\end{df}\n\n\n\\begin{lem}\n$A=C(\\Lambda)$ is the closure of the span of the characteristic\nfunctions $\\chi_{\\overrightarrow{w}_\\rho}$ of the sets\n$\\overrightarrow{w}_\\rho$ for $w \\in F_g$.  \n\\end{lem}\n\n\\begin{proof}\nThis is immediate, since $\\Lambda$ is a totally disconnected compact\nHausdorff space, and the sets $\\overrightarrow{w}$ form a basis of\nclopen sets for its topology. \n\\end{proof}\n\nWe denote by $A_\\infty$ the dense involutive subalgebra of $A$ spanned\nby the characteristic functions $\\chi_{\\overrightarrow{w}_\\rho}$. \n\n\\begin{df} Let $\\mu_\\Lambda$ denote the Patterson-Sullivan measure on\n$\\Lambda$ (cf. \\cite{Patterson}, \\cite{SullivanIHES}). Its main property is scaling by the Hausdorff dimension\n$\\delta_H$ of $\\Lambda$: \\begin{equation*}\\label{PSmeas} \n(\\gamma^* d\\mu)(x)= |\\gamma^\\prime (x)|^{\\delta_H} \\, d\\mu(x), \\ \\ \\\n\\forall \\gamma\\in \\Gamma.\n\\end{equation*} We define a state $\\tau : A_\\infty \\rightarrow \\mathbf{R}$ by \n$$ \\tau(\\chi_{\\overrightarrow{w}_\\rho}) := \\int\\limits_{\\Lambda}\n\\chi_{\\overrightarrow{w}_\\rho} \\mathrm{d}\\mu_\\Lambda =\n\\mu_\\Lambda(\\overrightarrow{w}_\\rho). $$ \nThe above lemma shows that $\\tau$ extends uniquely to a state on\n$A$. We define the Hilbert space $H$ to be the GNS-representation of\n$A$ arising from this state $\\tau$, that \nis, the completion of $A$ with respect to the inner product $\\langle\na | b \\rangle := \\tau(b^*a)$.  \n\\end{df} \n\n\\begin{df} We now take our inspiration from the construction of\nAntonescu and Christensen in \\cite{AntChris}.  \nThe subalgebra $A_\\infty$ of $A=C(\\Lambda)$ is a limit of finite\ndimensional subspaces $A_\\infty =\n\\lim\\limits_{\\xrightarrow[]{}} A_n$ with $A_n$ the span of the\ncharacteristic functions of sets of ends of reduced words of length\n$\\leq n$. This filtration is inherited by  $H$. We denote by $H_n$ the\nterm of the filtration of $H$ corresponding to $A_n$ and we let\n$P_n$ denote the orthogonal projection operator onto $H_n$. \nWe define the Dirac operator to be \n$$ D:= \\lambda_0 P_0 + \\sum_{n \\geq 1} \\lambda_n (P_n-P_{n-1}), $$\nwhere $\\lambda_n = (\\dim A_n)^3$. Note that $Q_n:=P_n-P_{n-1}$ is the\nprojection onto the graded pieces, identified with the orthogonal\ncomplements $H_n \\ominus H_{n-1}$, which correspond to words of exact \nlength $n$. The choice of $\\lambda_n$ arises from the fact that we\nthen arrive at 1-summability (Proposition \\ref{tA}): \n\\end{df}\n\n\\begin{prop}\nThe triple ${\\mathcal S}_{X}=(A,H,D)$ is a 1-summable spectral\ntriple.\n\\end{prop}\n\\begin{proof} The $*$-operation is complex conjugation, and since $D$\nis real, it is self-adjoint. For $a\\in A_{n}$ and for any $m>n$, multiplication by $a$ maps $A_{m-1}$ and $A_m$ into itself. Therefore, $a$ commutes with the projections $P_m$ and $P_{m-1}$ and so $[Q_m,a]=0$. Hence \n$$ [D,a] = \\sum_{i=0}^n \\lambda_i [Q_i,a]$$\nis a finite sum (we set $P_{-1}=0$ for convenience). \nThus, the commutators of $D$ with elements in  \nthe dense subalgebra $A_{\\infty}$ of $A$ are bounded. \n\nMoreover, one has $\\dim A_{n} \\geq\nn+1$, hence the 1-summability (and compact resolvent): \n$$ \\mathop{\\mathrm{tr}}\\nolimits((1+D^2)^{-1\/2})=1 +\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2} (\\dim\nH_{n}-\\dim H_{n-1}) $$\n$$ \\leq 1+\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2}\\dim  H_{n}\n\\leq 1+\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2}\\dim  A_{n} $$\n$$ \\leq 1+\\sum_{n=1}^\\infty (\\dim  A_{n})^{-2} \\leq  \n1+\\sum_{n=1}^\\infty (n+1)^{-2} \\leq  2, $$ where we used $\\lambda_n =\n(\\dim A_n)^3$ in the second-to-last inequality. This proves the\nproposition. \n\\end{proof}\n\n\\begin{rem}\nA recent deep theorem of Rennie and V\\'arilly (\\cite{RV}) allows one\nto decide whether a given spectral triple is associated to an actual\ncommutative Riemannian spin manifold. For the purpose of this paper,\nsince we are mostly interested in the zeta functions, we do not\nconsider any additional structure on the spectral triple. In\nparticular, our Dirac operator is only considered up to sign, since\nthe sign does not play a role in the zeta functions, while for\n\\cite{RV}, the sign provides the essential information on the\n$K$-homology fundamental class. It is possible that our construction\nmay be refined to incorporate the further necessary properties of an\nabelian spectral triple to which the reconstruction theorem can be\napplied. In that case, it seems that the underlying metric geometry\nshould probably relate to the existence of quasi-circles of limit sets\nof Schottky groups as in  \\cite{Bowen} --- see also the next\nremark. \\end{rem} \n\n\\begin{rem}\nNotice that our construction provides a 1-summable spectral triple on\nthe limit set, regardless of the actual value of its Hausdorff\ndimension (which can be greater than one). Thus, the metric dimension\nseen from this construction will be in general different from the\nactual metric dimension of the limit set embedded in $\\mathbf{P}^1(\\mathbf{C})$. The\nexistence in all cases of a 1-summable spectral triple on the limit\nset reflects the fact that topologically $\\Lambda$ is always a Cantor\nset that can be embedded in a topologically 1-dimensional quasi-circle\n(Bowen \\cite{Bowen}). In the metric induced by the embedding in\n$\\mathbf{P}^1(\\mathbf{C})$, the quasi-circle need not be rectifiable (when the\nHausdorff dimension of the limit set exceeds 1), but the existence of\n1-summable spectral triples is compatible with the topological\ndimension being one in all cases.   \n\\end{rem}\n\n\\section{Boundary isometry from the\nspectral zeta function} \\label{boundary}\n\n\nIn this section we study the effect of equality of zeta functions on\nmetric properties of the limit sets. Since we are dealing with two\nRiemann surfaces $X_1, X_2$, we will now sometimes index symbols\n($H,D,\\zeta,\\lambda,\\dots)$ by the index of the corresponding Riemann\nsurface and will do so without further mention. If there is no index,\nwe refer to any of the two Riemann surfaces. \n\nAs was already observed by Connes for the spectral triple associated\nto a usual spin manifold, only the action of $A$ on $H$, or of $D$ on\n$H$, doesn't capture interesting (metrical\/conformal)  information\nabout the space, it is the  interaction of the action of $A$ and $D$\nthat is important (\\cite{Connesbook}, VI.1). For our purposes, this\ninteraction will be encoded in the framework of zeta functions of\nspectral triples. The zeta functions are $\\zeta_a(s):=\\mathop{\\mathrm{tr}}\\nolimits(a|D|^s)$, a\npriori defined for $\\mathrm{Re}(s)$ sufficiently negative, but then\nmeromorphically extended to the whole complex plane with poles at the\ndimension spectrum of the spectral triple (see \\cite{ConnesLMP}). \n\n\\begin{thm}\\label{zetalambda}\nLet $X_1$ and $X_2$ be compact Riemann surfaces of genus at least\n$2$. If $ \\zeta_{a,{X_1}}(s) = \\zeta_{a,X_2}(s)$ for all $a \\in\nA_\\infty$, then $g_1=g_2$ and $$ \\forall \\eta \\in F_g \\ : \\\n\\mu_1(\\overrightarrow{\\eta}_{\\rho_1}) =\n\\mu_2(\\overrightarrow{\\eta}_{\\rho_2}). $$ \\end{thm} \n\n\n\\begin{rem}\nAs was indicated in the introduction, equality of zeta functions\nshould be understood as follows: both the algebras $A_1$ and $A_2$ of\n$X_1$ and $X_2$, respectively, have a unit. If the zeta functions for\nthis unit are equal, then we will conclude from this that the Riemann\nsurfaces have the same genus. Therefore, the algebras $A_1$ and $A_2$\nare isomorphic via the homeomorphism $\\Phi: \\Lambda_1 \\rightarrow\n\\Lambda_2$ induced from the Floyd maps in the triangle \n$$ \\xymatrix{    & & \\Lambda_1 \\ar[dd]^{\\Phi} \\\\ \\bar{F}_g\n\\ar[urr]^{\\iota_{\\rho_1}} \\ar[drr]^{\\iota_{\\rho_2}} & \\\\ & &\n\\Lambda_2}$$ It then makes sense to interpret the expression\n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for elements $a\\neq 1$ in\n$A_\\infty$. \n\\end{rem}\n\n\n\n\\begin{proof} We make the convention that all words are reduced.\n\nLet $X$ be a Riemann surface of genus $g \\geq 2$ and ${\\mathcal S}_X$ its\nassociated spectral triple.  Suppose given an element $a=\\chi_U$ in\n$A_\\infty$. We can assume $U=\\overrightarrow{\\eta}$ for a given word\n$\\eta$ of length $|\\eta|=m$, since any  $a$ is a linear combination of such.  \n\nWe now construct an orthogonal basis for $H$.\nFirst, we prove a lemma about ends of words.\n\n\\begin{lem} \\label{lemwords} Let $w_1, w_2$ denote two words. If\n$\\overrightarrow{w_1} \\cap \\overrightarrow{w_2} \\neq \\emptyset$, then\n$\\overrightarrow{w}_1 \\subseteq \\overrightarrow{w}_2$ or conversely\n$\\overrightarrow{w}_2 \\subseteq \\overrightarrow{w}_1$. In particular, if we set $\\max\\{w,v\\}$ to be the largest of the words $w$ and $v$ (if they are comparable in the order $\\subseteq$) and $\\emptyset$ otherwise, we find that $$ \\overrightarrow{w_1} \\cap \\overrightarrow{w_2} = \\overrightarrow{\\max\\{w_1,w_2\\}}$$ with the convention \n$\\overrightarrow{\\emptyset}=\\emptyset$, see Figure \\ref{figure1}. \n\\begin{figure}[h] \\begin{center} \\input{maxlength.pstex_t}  \n\\end{center}\n\\caption{{Illustration of Lemma \\ref{lemwords}.}}\n\\label{figure1}\n\\end{figure}\n \n\\end{lem}\n\\begin{proof}   \n\nIf this were not the case, then there is an end that lands in\nthe nonempty intersection and starts from both segments $w_1$ and\n$w_2$, leading to a loop in the Cayley graph $Y_g$, but $Y_g$ is a\ntree, so this is impossible, see Figure \\ref{figure2} (as usual, we identify the limit set $\\Lambda$ topologically with $\\bar{F}_g$, by Floyd's Lemma).\n\n\\begin{figure}[h] \\begin{center} \\input{forbidden.pstex_t} \n\\end{center}\n\\caption{{ A forbidden situation.}}\n\\label{figure2}\n\\end{figure}\n\n\\end{proof} \n\nIt is then easy to find a basis for the individual spaces $H_n$. \n\n\\begin{lem}\\label{basis}\nThe functions $\\chi_w$ for $|w|=n$ for a linear basis for $H_n$, and $$ \\langle \\chi_w |  \\chi_v \\rangle = \n\\mu(\\overrightarrow{\\max\\{v,w\\}}). $$\n \\end{lem}\n\n\\begin{proof} The characteristic functions $\\chi_{\\overrightarrow{w}}$\nfor $|w|=n$ give a linear basis for $H_n$: they are\nlinearly independent as their supports are disjoint, and they generate the space since for any word $u$ of length $|u|< n$ one\nhas $$\\chi_{\\overrightarrow{u}} =\\sum_{\\substack{|w|=n \\\\u \\subset w}}\n\\chi_{\\overrightarrow{w}}.$$ Indeed, by the previous Lemma, all occuring $\\chi_{\\overrightarrow{w}}$ have disjoint support, and their union $\\bigcup \\overrightarrow{w}$ equals $\\overrightarrow u$.\n\nNow\n$$  \\langle \\chi_{\\overrightarrow{w}_1} |  \\chi_{\\overrightarrow{w}_2}\n\\rangle = \\tau(\\chi_{\\overrightarrow{w}_1}^*\n\\chi_{\\overrightarrow{w}_2}) =\\mu_X (\\overrightarrow{w}_1 \\cap\n\\overrightarrow{w}_2), $$\nand the previous lemma applies. \n\\end{proof}\n\n\\begin{lem} For all $n>0$, we have \n$$\\dim A_n=\\dim H_n = 2g(2g-1)^{n-1}.$$ \n\\end{lem}\n\n\\begin{proof} The space $A_{n}$ is spanned by the linear basis \n$\\chi_{\\overrightarrow{w}_\\rho}$ with $w$ a word of exact length $\nn$, since as in the proof of Lemma \\ref{basis}, all functions corresponding to shorter words are dependent on these functions. An easy count gives the result: we pick the first letter from the alphabet on $g$ letters or its inverses, and consecutive letters with the condition that they differ from the terminal letter of the word already constructed. \n\\end{proof}\nWe now construct a complete orthonormal basis for $H$ inductively, by adding to a basis of $H_{n}$ suitable elements of $H_{n+1}$ in the style of a Gram-Schmidt process. Initially, we set\n$| \\Psi_e \\rangle = \\chi_\\Lambda$ and \\begin{equation} \\label{length1}  |\\Psi_w \\rangle := \\frac{1}{\\sqrt{\\mu_X(\\overrightarrow{w})}}\\,\n\\chi_{\\overrightarrow{w}} \\ \\ \\ \\ (|w|=1) \\end{equation}\nfor $w$ running through a set $S$ of words of length one (viz., letters in the alphabet, and their inverses) unequal to one (arbitrarily chosen) letter. Set $I_1:=S \\cup \\{e\\}$; then $\\{|\\Psi_w\\rangle\\}_{w \\in I_1}$  is an orthonormal basis for $H_1$ by Lemma \\ref{basis}. \n\nNow suppose $$\\{|\\Psi_w\\rangle \\, : \\, w \\in I_n \\}$$ is our inductively constructed basis for $H_n$, where  $I_n$ is an index set. For every word $w$ of length $n$, choose a set $V_w$ of $2g-2$ letters\nfrom the alphabet and its inverses, that are unequal to $t(w)^{-1}$, the inverse of the terminal letter of the fixed $w$, i.e., leave out one arbitrarily chosen letter from the possible extensions of $w$ to an admissible word of length $n+1$. Let $$I_{n+1} = I_n \\cup \\bigcup\\limits_{|w|=n} V_w.$$ Figure \\ref{figure3} has an example in the length $\\leq 3$ words in the Cayley graph $Y_2$ for $g=2$.\n\n\\begin{figure}[h] \\begin{center} \\begin{picture}(120,120)\\includegraphics{basis3.eps}\\end{picture} \\end{center}\n\\begin{center} \\caption{{ Black dots form a possible $I_3 \\subseteq Y_2$}}\\end{center}\n\\label{figure3}\n\\end{figure}\n\n\nWe claim that $\\{\\chi_{\\overrightarrow{w}}\\}_{w \\in I_{n+1}-I_n}$ is a basis for $H_{n+1} \\ominus H_n$. The functions are linearly independent since their supports are disjoint. Hence it suffices to check dimensions. But $$\\dim (H_{n+1} \\ominus H_n) = 2g(2g-1)^{n-1}(2g-2) = |I_{n+1}-I_n|.$$\n\nWe define $\\{|\\Psi_w \\rangle\\}_{w \\in I_{n+1}}$ as the Gram-Schmidt orthonormalisation of $$\\{ |\\Psi_w\\rangle \\}_{w \\in I_n} \\, \\cup \\, \\{ \\chi_{\\overrightarrow{w}} \\}_{w \\in I_{n+1} - I_n}.$$ \nWe recall that this means we choose an enumeration  of the words in $I_{n+1}-I_n:$ $$ I_{n+1}-I_n = \\{w_1,\\dots,w_r\\}$$ and set inductively for $i=1,\\dots,r$\n\\begin{equation} \\label{defPsi}  |\\Psi_{w_i}\\rangle = \\frac{|\\phi_{w_i} \\rangle}{\\| \\phi_{w_i} \\|} \\end{equation}\nwith\n\\begin{equation} \\label{defphi}  |\\phi_{w_i} \\rangle :=\\chi_{\\overrightarrow{w_{i}}} -\\sum_{w \\in I_n \\cup\\{w_1,\\dots,w_{i-1}\\}} \n\\langle \\Psi_w |\\chi_{\\overrightarrow{w_{i}}}\\rangle |\\Psi_w\\rangle. \\end{equation}\nThen indeed,\n$ \\langle \\Psi_v |\\Psi_w\\rangle =\\delta_{v,w}, $\nfor all $|v|,|w|\\leq n+1$.  \n\n\nSet $$I_\\infty = \\bigcup_{n \\geq 0} I_n.$$ We use the complete basis $\\{ |\\Psi_w\\rangle \\}_{w \\in I_\\infty}$ of $H$ to compute the trace of a trace-class\noperator $T$ in the form\n$ \\mathop{\\mathrm{tr}}\\nolimits(T)=\\sum \\langle \\Psi_w | T \\Psi_w\\rangle. $\nWith $T=aD^s$, we have\n$$ \\mathop{\\mathrm{tr}}\\nolimits(aD^s)=1+\\sum_w \\langle \\Psi_w |a \\sum_{n \\geq 1} \\lambda_n^s\n(P_n-P_{n-1}) \\Psi_w \\rangle . $$\nNow the projector $P_n-P_{n-1}$ onto $H_n \\ominus H_{n-1}$ is $\\sum\\limits_{r \\in I_n - I_{n-1}}| \\Psi_r \\rangle \\langle \\Psi_r |$, so we get\n$$ (P_n - P_{n-1}) |\\Psi_w\\rangle =\\sum_{r \\in I_n-I_{n-1}}| \\Psi_r \\rangle \\langle \\Psi_r |\n\\Psi_w \\rangle = \\delta_{|w|,n} \\delta_{r,w} | \\Psi_r \\rangle =  \\delta_{|w|,n} | \\Psi_w \\rangle.$$\nThus, we rewrite the above as\n$$ \\mathop{\\mathrm{tr}}\\nolimits(aD^s)=1+\\sum_{n \\geq 1} \\sum_{w \\in I_n-I_{n-1}} \\lambda_n^s  \\langle \\Psi_w |\na \\Psi_w \\rangle . $$\nIf we denote by \n$$ c_n(a) =\\sum_{w \\in I_n-I_{n-1}} \\langle \\Psi_w |\na  \\Psi_w \\rangle, $$\nwe can write\n$$ \\zeta_{a,X}(s) = \\mathop{\\mathrm{tr}}\\nolimits(a D^s)=1+ \\sum\\limits_{n\\geq 1} \\lambda_{n}^s\\,\n c_{n}(a), \\ \\ \\mathrm{Re}(s) \\ll 0. $$ \n\n\\begin{lem} \\label{genusequal} If $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ for\n$a=1=\\chi_\\Lambda$ the identity of $A_1$, respectively $A_2$, then\n$g_1=g_2$. \\end{lem} \n\n\\begin{proof} We know that $\\lambda_{n}=(\\dim\nA_{n})^3=(2g)^3(2g-1)^{3n-3}. $ By orthnormality, we find that $$c_n(1)= \\sum_{|w| \\in I_n-I_{n-1}} \\langle \\Psi_w | \\Psi_w \\rangle = \\sum_{|w| \\in I_n-I_{n-1}} 1 \\, = \\, 2g(2g-1)^{n-2}(2g-2).$$\nHence we find for $a=1$ that $$ \\zeta_{1}(s) = 1 + \\sum_{n \\geq 1} \\lambda_n^s c_n(1) = 1+ (2g)^{3s+1} \\, \\frac{2g-2}{2g-1} \\, \\sum_{n \\geq 1} (2g-1)^{(3s+1)(n-1)},  $$  and thus \\begin{equation} \\label{zeta1} \\zeta_1(s) = 1 +\\frac{2g-2}{2g-1} \\cdot \\frac{(2g)^{3s+1}}{1-(2g-1)^{3s+1}}. \\end{equation} \nFor $a=1$, the \ncondition $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ is thus equivalent to \n$$\\frac{2g_1-2}{2g_1-1} \\cdot \\frac{2g_2-1}{2g_2-2} \\cdot  \\left( \\frac{g_1}{g_2} \\right)^{3s+1}  = \\frac{1-(2g_1-1)^{3s+1}}{1-(2g_2-1)^{3s+1}} \\mbox{ for } \\mathrm{Re}(s) \\ll 0 $$\nIf we let $s$ tend to $- \\infty$, the right hand side tends to $1$. However, unless $g_1=g_2$, the left hand side tends to zero. This finishes the proof that $g_1=g_2$. \n\\end{proof}\n\nAs mentioned in the remark above, we conclude from this lemma that the\nalgebras $A_1$ and $A_2$ are isomorphic via the induced Floyd\nhomeomorphism $\\Phi: \\Lambda_1 \\rightarrow \\Lambda_2$. \n This makes the condition $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ meaningful.\n \n\\begin{lem}\\label{equal} $c_{n,{1}}(a)=c_{n,{2}}(a)$ for all $a \\in A_\\infty$. \n\\end{lem}  \n\\begin{proof}\nThe equality $\\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ is equivalent to \n $$\\sum_{n \\geq 0} \\left( c_{n,1}(a)-c_{n,2}(a)\\right) \\lambda_n^{s} \\equiv 0$$ \nfor $\\mathrm{Re}(s) \\ll 0$. Here, $\\lambda_n$ is the same for the two Riemann surfaces, since it only depends on their genus and those have just been shown to be equal in Lemma \\ref{genusequal}. Now since all $\\lambda_n$ are \\emph{distinct}\npositive integers, we also have an identically zero Dirichlet series \n$$ \\sum_{N \\geq 0} \\til{c}_N N^s \\equiv 0 \\mbox{ for $\\mathrm{Re}(s) \\ll 0$} $$\nwith $\\til{c}_N = c_{n,1}(a)-c_{n,2}(a)$ if $N=\\lambda_n$ for some $n$, and\n$\\til{c}_N=0$ otherwise. Now clearly $\\til{c}_N=0$ for all $N$, by the identity theorem for Dirichlet series (e.g., \\cite{HW}, 17.1). \n\\end{proof} \n\n\\begin{lem} \\label{algfunc}\nFor $a = \\chi_{\\overrightarrow{\\eta}}$  and $w$ a word of length $n<|\\eta|$, we have that \n$$ \\langle \\Psi_w | a \\Psi_w \\rangle = \\mu(\\overrightarrow{\\eta}) \\cdot \\kappa$$\nwhere $\\kappa$ depends only on measures $\\mu(\\overrightarrow{v})$ of certain words $v$ of length $|v|<|\\eta|$.\n\\end{lem}\n\n\\begin{proof} This holds for $w$ a word of length one, since by definition (\\ref{length1}) and Lemma \\ref{basis}, we have\n$$  \\langle \\Psi_w | a \\Psi_w \\rangle = \\frac{\\mu(\\overrightarrow{w} \\cap \\overrightarrow{\\eta})}{\\mu(\\overrightarrow{w})}  =\\left\\{ \\begin{array}{ll}  \\frac{\\mu(\\overrightarrow{\\eta})}{\\mu(\\overrightarrow{w})} & \\mbox{if } w \\subset \\eta; \\\\ 0 & \\mbox{otherwise.} \\end{array} \\right. $$\nWe then use induction on the word length of $w$. By construction of $\\Psi_w$ (looking at the definitions in {Formul\\ae}  (\\ref{defphi}) and (\\ref{defPsi})) it suffices to prove that \nfor $w,u$ of length $ \\leq n$, we have that $\\langle\\chi_{\\overrightarrow{w}}|a\\chi_{\\overrightarrow{u}}\\rangle$ is of the required form $\\mu(\\overrightarrow{\\eta}) \\cdot \\kappa$\nwhere $\\kappa$ depends only on measures $\\mu(\\overrightarrow{v})$ of certain words $v$ of length $|v|<|\\eta|$: $\\Psi_w$ is a linear combination of such terms. Now\n$$\\langle\\chi_{\\overrightarrow{w}}|a\\chi_{\\overrightarrow{u}}\\rangle = \\mu(\\overrightarrow{w} \\cap \\overrightarrow{\\eta} \\cap \\overrightarrow{u}) =\\left\\{ \\begin{array}{ll}  \\mu(\\overrightarrow{\\eta}) & \\mbox{if } w,u \\subset \\eta \\\\ 0 & \\mbox{otherwise,} \\end{array} \\right. $$\nsince $\\eta$ is longer than $w$ and $u$. This proves the claim.\n\\end{proof} \n\nComputing $c_{m-1}(\\chi_{\\overrightarrow{\\eta}})$ as a linear combination of terms of the form $\\langle \\Psi_w | a \\Psi_w \\rangle,$ we find from Lemma \\ref{algfunc} (now indicating the representation $\\rho$, since we will soon vary it) :\n\\begin{equation} \\label{inductmu} c_{m-1}(\\chi_{\\overrightarrow{\\eta}_\\rho}) =  \\mu(\\overrightarrow{\\eta}_{\\rho})  \\cdot \\kappa, \\end{equation}\nwhere $\\kappa$ only depends on $\\mu(\\overrightarrow{v}_\\rho)$ for $|v|<m$. \n\nWe need one more technical observation, namely that $\\kappa \\neq 0$ in (\\ref{inductmu}) or, what is the same: \n\\begin{lem} $c_{m-1}(a) \\neq 0$ for $a=\\chi_{\\overrightarrow{\\eta}}$ with $|\\eta|=m$. \\end{lem}\n\\begin{proof}\nRecall $c_{m-1}(a) = \\sum\\limits_{w \\in I_{m-1}-I_{m-2}} \\langle \\Psi_w | a \\Psi_w \\rangle.$ The terms $$ \\langle \\Psi_w | a \\Psi_w \\rangle = \\int |\\Psi_w|^2 \\cdot \\chi_{\\overrightarrow{\\eta}} \\, \\, d\\mu \\geq 0$$ are all positive, but some might be zero. It therefore suffices to prove that at least one of them is non-zero, and for this, it suffices to find $w$ such that the support of $\\Psi_w$ intersects $\\overrightarrow{\\eta}$. But if $x \\in \\overrightarrow{\\eta}$, then there is a word $v$ of length $m-1$ such that $x \\in \\overrightarrow{v}$, too, since $$\\bigcup\\limits_{|v|=m-1} \\overrightarrow{v} = \\Lambda.$$ Then $\\chi_{\\overrightarrow{v}}(x)  \\neq 0$, but,  as $\\{\\Psi_w\\}_{w \\in I_{m-1}-I_{m-2}}$ is a basis for $H_{m-1} \\ominus H_{m-2}$, we have $$\\chi_{\\overrightarrow{v}}(x) = \\sum_{w \\in I_{m-1}-I_{m-2}} a_w \\Psi_w(x)$$ for some coefficients $a_w$, hence there exists $w$ of length $m-1$ such that $\\Psi_w(x) \\neq 0$. \n\\end{proof}\n\n\\begin{prop}\nFor all $\\eta \\in F_g$, $\\mu_1(\\overrightarrow{\\eta}_{\\rho_1}) =\n\\mu_2(\\overrightarrow{\\eta}_{\\rho_2})$.  \n\\end{prop}\n\n\\begin{proof} We prove this by induction on the word length of $\\eta \\in F_g$\nIf $|\\eta|=0$, we find $\\overrightarrow{\\eta}_{\\rho_i}=\\Lambda_i$ for $i=1,2$, so the identity holds. If it holds for all words of length $<m$, let $\\eta$ denote a word of length $m$. \n\nRecall that the map $\\Phi^*: A_1 \\rightarrow A_2$ is such that\n$\\Phi^*(\\chi_{\\overrightarrow{w}_{\\rho_1}} )=\n\\chi_{\\overrightarrow{w}_{\\rho_2}}$. \n\nWe apply the expression (\\ref{inductmu}) to both Riemann surfaces, substituting $\\rho=\\rho_1$ and $\\rho=\\rho_2$, respectively. Since $\\kappa \\neq 0$, this is a genuine formula for $\\mu(\\overrightarrow{\\eta}_\\rho)$. \nThe equality $c_{m-1,1}(\\chi_{\\overrightarrow{\\eta}_{{\\rho_1}}})=c_{m-1,2}(\\chi_{\\overrightarrow{\\eta}_{{\\rho_2}}})$ is our assumption, and for the second factor on the right hand side we can inductively assume that the measures on the occuring words  of length $<m$ agree in both representations. Hence we find indeed $\\mu(\\overrightarrow{\\eta_{\\rho_1}})=\\mu(\\overrightarrow{\\eta_{\\rho_2}})$.\n\\end{proof}\nThis proposition finishes the proof of Theorem \\ref{zetalambda}.\n\\end{proof}\n\n\\section{Rigidity from boundary isometry} \n\nWe now prove Theorem \\ref{tB} from the introduction: \n\n\\begin{thm}\nIf $X_1$ and $X_2$ are compact Riemann surfaces of respective genus\n$g_1, g_2 \\geq 2$, such that  \n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for all $a \\in A$, then $X_1$ and\n$X_2$ are conformally or anti-conformally equivalent as Riemann\nsurfaces.\\end{thm}  \n\n\\begin{proof} From Theorem \\ref{zetalambda}, we find that $X_1$ and\n$X_2$ have the same genus $g \\geq 2$.  \nWe consider the two Schottky groups $\\Gamma_1$ and $\\Gamma_2$\ncorresponding to $X_1$ and $X_2$, respectively. Let $\\rho_i :  F_g\n\\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ denote the corresponding embeddings of the\nabstract group $F_g$ (so $\\rho_i(F_g)=\\Gamma_i$), and let\n$\\alpha:=\\rho_2 \\circ \\rho_1^{-1}$ denote the induced group\nisomorphism $\\Gamma_1 \\rightarrow \\Gamma_2$. We consider the map $\\Phi\n: \\Lambda_1 \\rightarrow \\Lambda_2$ as in the diagram of the proof of\n\\ref{zetalambda}: $x\\in \\Lambda_1$ can be written as\n$\\iota_{\\rho_1}(\\lim w_i)$ for some Cauchy sequence $\\lim w_i$ in\n$Y_g$. We then define $\\Phi(x):=\\iota_{\\rho_2}(\\lim w_i)$. As was\nremarked before, by Floyd's Lemma \\ref{Floyd}, $\\Phi$ is a\nhomeomorphism of $\\Lambda_1$ onto $\\Lambda_2$.  \n\n\\begin{lem} $\\Phi$ is $\\alpha$-equivariant. \n\\end{lem}\n\\begin{proof} Given $\\gamma \\in \\Gamma_1$, we can find $g \\in F_g$\nsuch that $\\gamma=\\rho_1(g)$. For $x = \\iota_{\\rho_1}(\\lim w_i)$, we\nfind that $\\gamma \\cdot x = \\iota_{\\rho_1}(\\lim gw_i)$. Hence\n$$\\Phi(\\gamma \\cdot x) = \\iota_{\\rho_2}(\\lim gw_i) = \\iota_{\\rho_2}(g)\n\\cdot \\iota_{\\rho_2}(\\lim w_i) = \\rho_2(\\rho_1^{-1}(\\gamma)) \\cdot\n\\Phi(x) = \\alpha(\\gamma) \\Phi(x).$$ So we do find that $\\Phi$ is\n$\\alpha$-equivariant. \\end{proof}  \n\nThis means that $\\Phi$ is a boundary homeomorphism in the sense of\nFenchel-Nielsen, see Tukia \\cite{Tukia2}, 3C.  \n\nBy  Theorem \\ref{zetalambda}, the equality of zeta functions implies\nthat $\\Phi$ is an isometry. Indeed,  \n$$\\mu_2(\\Phi^*(\\chi_{\\overrightarrow{w}_{\\rho_1}})) =\n\\mu_2(\\chi_{\\Phi(\\overrightarrow{w}_{\\rho_1})})= \n\\mu_2(\\chi_{\\overrightarrow{w}_{\\rho_2}})=\n\\mu_1(\\chi_{\\overrightarrow{w}_{\\rho_1}}),$$\nwhere we use the definition of $\\Phi$ in the second equality and the\nproposition in the third. Thus, since\n$\\{\\overrightarrow{w}_{\\rho_i}\\}$ is a basis for $\\Lambda_i$, we find\nthat $\\mu_2 \\circ \\Phi^* = \\mu_1$.  \n\nNow recall the following ergodic rigidity theorem: \n\n\\begin{lem}[Chengbo Yue] Let $\\Gamma_1$ and $\\Gamma_2$ be\ngeometrically  finite subgroups in two simple, \nconnected and adjoint Lie groups $G_1$ and $G_2$ of real rank one,\nsuch that $\\Gamma_1$ \nis Zariski dense in $G_1$. Let  $\\alpha : \\Gamma_1 \\rightarrow\n\\Gamma_2$ be a type-preserving isomorphism. \nThen there exists a homeomorphism  $\\phi: \\Lambda_{\\Gamma_1}\n\\rightarrow \\Lambda_{\\Gamma_2}$ which is equivariant \nwith respect to  $\\alpha$. If $\\phi$ is absolutely continuous with\nrespect to the Patterson--Sullivan measure, then  $\\alpha$ can be\nextended to a continuous homomorphism $G_1 \\rightarrow G_2$.\\end{lem}\n\n\\begin{proof} This is literally Corollary B from  \\cite{Yue}, apart\nfrom the fact that the extended homomorphism $G_1 \\rightarrow G_2$ can\nbe assumed \\emph{continuous}, but this follows by looking at the\nstatement of Theorem A from which the corollary follows. \\end{proof} \n\nWe want to apply this corollary with $G_1=G_2=\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and\n$\\Gamma_i$ our Schottky groups, so let us check the conditions:  \nBoth Schottky groups are geometrically finite subgroups of $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$;\n$\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})=\\mathop{\\mathrm{PSL}}\\nolimits(2,\\mathbf{C})$ is a simple and connected adjoint\nreal-rank-one Lie group; \nand finally:\n\n\\begin{lem} A noncommutative Schottky group is Zariski dense in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. \n\\end{lem}\n\n\\begin{proof} Since the group operations on such a Schottky group are\ninduced from the algebraic operations on the algebraic group\n$\\mathop{\\mathrm{PGL}}\\nolimits(2)$, the Zariski closure $\\hat{\\Gamma}$ is itself an algebraic\nsubgroup of $\\mathop{\\mathrm{PGL}}\\nolimits(2)$. Assume $\\hat{\\Gamma}$ is a strict algebraic\nsubgroup of $\\mathop{\\mathrm{PGL}}\\nolimits(2)$. Let $\\hat{\\Gamma}_0$ denote its connected\ncomponent of the identity. The group of connected components\n$\\hat{\\Gamma}\/\\hat{\\Gamma}_0$ is finite, and $\\Gamma \\cap\n\\hat{\\Gamma}_0$ \nis a finite index subgroup of the free group $\\Gamma$, hence free of\nthe same rank $g$; and its Zariski closure is connected. It suffices\nthat this group has full Zariski closure, hence we can assume without\nloss of generality that $\\hat{\\Gamma}$ is connected. However, if\n$\\hat{\\Gamma}$ is connected of dimension $\\leq 2$, then it is solvable\n(cf.\\ \\cite{LAG}, IV.11.6), and a solvable group cannot contain a free\ngroup of rank $g \\geq 2$ (since the composition series of\n$\\hat{\\Gamma}$ would descend to one for $\\Gamma$). On the other hand,\nif $\\dim \\hat{\\Gamma} = 3$, then since $\\mathop{\\mathrm{PGL}}\\nolimits(2)$ is connected, we have\n$\\hat{\\Gamma}=\\mathop{\\mathrm{PGL}}\\nolimits(2)$. \\end{proof} \n\nSince $F_g$ has no parabolic points, we know that the equivariant\nboundary homeomorphism $\\Phi$ is unique and type-preserving (Tukia,\n\\cite{Tukia1}, p.\\ 426), hence it coincides with the boundary\nhomeomorphism $\\phi$ in Yue's result. Since all conditions are\nsatisfied, we can apply the result (replacing $\\phi$ by $\\Phi$) to\nboth the isometry $\\Lambda_1 \\rightarrow \\Lambda_2$ and its inverse,\nwe find that $\\alpha$ extends to a continuous group automorphism\n$\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C}) \\rightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Now recall that the automorphisms\nof $\\mathop{\\mathrm{PGL}}\\nolimits(2,k)$ over a field $k$ have been classified by Schreier and\nvan der Waerden (cf.\\ \\cite{SW}, see also the supplement to\n\\cite{Hua}): the outer automorphisms are induced from field\nautomorphisms of $k$. Now all  continuous field automorphisms of $\\mathbf{C}$\nfix $\\mathbf{R}$. \n\nWe conclude that there is an isomorphism $\\Gamma_1 \\rightarrow\n\\Gamma_2$ of the form $$\\gamma_1 \\rightarrow g \\gamma_1^\\sigma\ng^{-1}$$ for $g \\in \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and $\\sigma \\in \\mathop{\\mathrm{Aut}}\\nolimits(\\mathbf{C}\/\\mathbf{R})$, that is,\n$\\Gamma_1$ and $\\Gamma_2^\\sigma$ are conjugate in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Now\nnote that $\\Gamma_2^\\sigma$ uniformizes the curve $X_2^\\sigma$: \n$$ (\\mathbf{P}^1(\\mathbf{C})-\\Lambda_{\\Gamma_2^\\sigma})\/ \\Gamma_2^\\sigma = \\left(\n(\\mathbf{P}^1(\\mathbf{C})-\\Lambda_{\\Gamma_2})\/ \\Gamma_2\\right)^\\sigma = X_2^\\sigma.$$ \n Hence $X_1$ and $X_2^\\sigma$ are isomorphic Riemann surfaces, so\n$X_1$ and $X_2$ are conformally or anti-conformally equivalent, the\nformer case arising when $\\sigma$ is trivial, and the latter case\narising when $\\sigma$ is complex conjugation. \n\\end{proof}\n\n\n\\begin{rem}\nThe statement that two hyperbolic compact Riemann surfaces of the same\ngenus are isomorphic if and only if the boundary map $S^1 \\rightarrow\nS^1$ induced from the isomorphism of their fundamental groups, seen as\nFuchsian groups of the first kind, is absolutely continuous in\nLebesgue measure is originally part of Mostous rigidity theorem (cf.\\\nMostow \\cite{Mostow} 22.14, p.\\ 178). An easy proof for this case is\nin Kuusalo \\cite{Kuusalo}, and  Bowen has given another proof using\nGibbs measures in \\cite{Bowen}. For more general M\\\"obius groups\n(whose limit set is not necessarily the full boundary of the symmetric\nspace, such as Schottky groups, and in higher dimensions), there is\nthe work of Sullivan (\\cite{Sullivan}) and Tukia (\\cite{Tukia1},\n\\cite{Tukia2}). The typical ergodic rigidity theorem in this setting\nis that an absolutely continuous boundary map is identical to the\nrestriction of a M\\\"obius transformation \\emph{on the limit set}. What\nhappens outside the limit set, however, depends on other\nconsiderations (cf. \\cite{Tukia2}, Marden \\cite{Marden}). See, e.g.,\nTukia (\\cite{Tukia1}), Section 4D for a 3-dimensional example of an\nisomorphism of Schottky groups with absolutely continuous boundary\nmap, that does \\emph{not} extend to a M\\\"obius map outside the limit\nset.\\end{rem} \n\n\\begin{rem}\nThe construction cannot be extended to the crossed product boundary\noperator algebra $C(\\Lambda) \\rtimes \\Gamma$, since Connes\nhyperfiniteness result \\cite{Conneshyper} and the hyperbolic growth of\nthe group $\\Gamma$ ($g \\geq 2$) prevent this algebra from carrying\nfinitely summable spectral triples.  \nHowever, $C(\\Lambda) \\rtimes \\Gamma$ can \nbe identified with a Cuntz--Krieger algebra (\\cite{CK}), which has a\nstandard AF-subalgebra $\\til{A}$. This has a maximal abelian\nsubalgebra that can be identified with our algebra $A$ (\\cite{CK},\n2.5). One can construct a conditional expectation $E$ from $\\til{A}$\nto $A$. Thus,  the construction of the spectral triple extends to this\nAF-algebra $\\til{A}$ by using an \\emph{unfaithful} state $\\tau \\circ\nE$ that is zero outside $A$ and equals our state $\\tau$ on $A$. Since\nthis construction, however, is just a ``factorisation'' through the\nabove commutative spectral triple, it doesn't give a lot of new\ninformation. \n\\end{rem}\n\n\n\\section{Remarks and Questions} \n\n\\begin{se} \n An interesting question (e.g., in the light of Arakelov geometry) is\nhow to generalize the result to the case of $p$-adic Mumford curves\n(\\cite{GvdP}, \\cite{Mumford}). See \\cite{CMRV} for some results on\ntrees, inspired by earlier work on non-finitely summable spectral\ntriples for tree actions by Consani and Marcolli\n\\cite{ConsaniMarcolli}. \n\\end{se}\n\\begin{se}  Is there such a theory for Fuchsian uniformisation instead\nof Schottky uniformisation? In the noncompact case? \n\\end{se}\n\\begin{se} There are other ways of looking at Riemann surfaces from\nthe point of view of non-commutative geometry. For example, as a\ncommutative conformal manifold, it carries the non-commutative\ndifferential geometry of ``quantized calculus'', whose Fredholm module\ndetermines the conformal isomorphism type of the surface (Connes,\nDonaldson, Sullivan, N.~Teleman \\cite{Connesbook},\nIV.4.$\\alpha$). Also, Consani and Marcolli (\\cite{CMinf}) have\nconstructed $\\theta$-summable, but non finitely summable spectral\ntriples from the boundary action, where the Hilbert space is a\nsymmetrized version of the $L^2$-space of the boundary that should\nhave Hausdorff dimension $<1$. There is the work of B\\\"ar\n(\\cite{Baer}) mentioned before. All of these constructions are rather\ndifferent from the one in this paper. In particular, our spectral\ntriple is finitely summable, so better suited to tools such as the\nlocal index formula of Connes and Moscovici (\\cite{CoMo}).  \nNevertheless, the question arises whether the Consani-Marcolli\nspectral triple hears the conformal shape of the Riemann surface.  B\\\"ar's spectral tiple does enjoy this property.\n\\end{se}\n\\begin{se}  Does our spectral triple ${\\mathcal S}_X$ carry a real structure?\nDoes it then arise from a commutative spin manifold (\\cite{GVF},\n\\cite{RV})? Can one enhance ${\\mathcal S}_X$ by additional classical structure,\nso this structure determines the conformal structure of $X$ completely\n(not just conformal or anti-conformal)? \nNotice that in our construction we work only with the absolute value\nof the Dirac operator, and we use zeta functions that do not see the\nsign. In order to relate it to spectral triples coming from a\nRiemannian manifold one would need to first enrich it with a sign\n(which gives the fundamental class in $K$-homology) and then with a\ncompatible real structure. It seems that the information on the sign\nwill be needed to reconstruct completely the conformal structure. \n\\end{se}\n\n\\begin{se}  Since special values of the spectral zeta functions are\njust 0-Hochschild homology, it is interesting to compute higher\ncharacteristic classes of the spectral triple and relate them to the\nactual geometry of the original Riemann surface. This is especially\ntempting in arithmetically interesting cases, such as modular curves. \n\\end{se}\n\\begin{se}  Can Theorem \\ref{tB} be extended to an injective functor\nfrom the category of Riemann surfaces with conformal and\nanti-conformal morphisms to a category of spectral triples? \n\\end{se}\n\n\n\n\n\n\n\n\n\\begin{small}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{\\bf Introduction:} Being the heaviest particle in the Standard Model (SM), the top quark is\nconsidered to play a special role in the electroweak (EWK) symmetry breaking. Precise\nunderstanding of the top quark production processes is essential for the study of physics\nbeyond the SM since they are usually the most important background of these new physics.\nDue to the short lifetime of top quark, its spin is fully inherited by its decay products.\nThus, top quark can be used as a tool to study the spin information of the related processes.\nSimilarly, we study in this paper that the top quark can also be utilized to study\nthe color properties of the production processes. An efficient treatment is just to\nanalyze the shape of the b jet from the top decay, as the top quark color are fully\ncarried on by its decay product b quark.\n\n\nRecently, the anomaly of top quark forward-backward asymmetry~($A_{FB}$) has been a hot topic.\nIt was observed by the CDF and D0 Collaborations at the Tevatron and the CDF latest result\nstill supports that there is an excess in top $A_{FB}$~\\cite{Aaltonen:2012it}. Disappointingly,\nthe top $A_{FB}$~(or sometimes named as the charge asymmetry) is very difficult to be measured\nprecisely at the LHC due to the large symmetric gluon-gluon fusion component. Two of us\nhave introduced a so called color octet axial-vector-like boson to explain\nthis anomaly~\\cite{Xiao:2010ph}. Such a new particle is also called the light\naxigluon~\\cite{Gross:2012bz}. Recent studies showed the color octet axial-vector-like boson\ncan have the virtue to give the correct differential distributions, such as\n $d\\sigma\/d M_{t\\bar{t}}$, $d A_{FB}\/ d M_{t\\bar{t}}$, and $d A_{FB}\/d Y$, where $\\sigma$ is\nthe production cross section, $M_{t\\bar{t}}$ is the top pair invariant mass, and $Y$ is the\nrapidity of the top quark. However, the other explanations often have difficulties to give\nsuch a good fit~\\cite{Wang:2011hc}. A very important feature of the new boson is that it carries\noctet color and then can not be seen directly due to the color confinement. Interestingly,\nthe color flow in quark pair production does have been studied in Ref.~\\cite{Gallicchio:2010sw}.\nThe mediating particles carrying different colors (color singlet or color octet) can be\ndistinguished successfully. In this paper, we adopt the color flow method in the\ntop $A_{FB}$ analysis, especially in studying the color octet axial-vector-like boson explanation.\n\n\n\n\n{\\bf The Color Octet Axial-Vector-Like Boson:} The color octet axial-vector-like boson\n$Z_C$ is an effective description to explain the top $A_{FB}$ anomaly.\nThe effective lagrangian can be written as\n\\begin{equation}\n\\begin{array}{rl}\n{\\cal L} \\propto &- i g^Q_A \\bar{Q} \\gamma_\\mu \\gamma_5 T_a Q {Z_C}^\\mu_a\\\\\n& - i g^q_A \\bar{q} \\gamma_\\mu \\gamma_5 T_a q {Z_C}^\\mu_a~,\\\\\n\\end{array}\n\\end{equation}\nin which $Q=t, b$ and $q=u, d, c, s$.\nThe squared invariant amplitude with color and spin summed is\n\\begin{equation}\n \\begin{array}{l}\n \\sum\\limits_{\\text{Color, Spin}}|{\\cal M}|^2= \\frac{C_A C_F}{2}\\{4g_s^4 (1+c^2+4m^2)\\\\\\\\\n  +\\frac{8g_s^2 \\hat{s}(\\hat{s}-M_{Z_C}^2)}{(\\hat{s}-M_{Z_C}^2)^2+M_{Z_C}^2\\Gamma_{Z_C}^2}2g_A^q g_A^t c\\\\\\\\\n  +\\frac{4\\hat{s}^2}{(\\hat{s}-M_{Z_C}^2)^2+M_{Z_C}^2\\Gamma_{Z_C}^2}\n (g_A^qg_A^t)^2(1+c^2-4m^2)\\},\n \\end{array}\n\\label{eq.simp_axigluon}\n\\end{equation}\nwhere $C_A=3$~, $C_F=4\/3$, $g_s$ is the strong coupling constant, $m=m_t\/\\sqrt{\\hat s}$, ${\\hat s}$ is the parton level interacting energy, $\\beta=\\sqrt{1-4m^2}$, and $c=\\beta\\cos\\theta$. The decay width $\\Gamma_{Z_C}$ of $Z_C$ is given by\n\\begin{equation}\n\\Gamma_{Z_C}=\\sum\\limits_{i}^{2m_i<M_{Z_C}}\\frac{g_A^i}{4\\pi}\\frac{C_F}{8}\n(1-4m_i^2\/M_{Z_C}^2)^{\\frac{3}{2}}M_{Z_C}\n\\label{ZCwidth}\n\\end{equation}\nIn Eq.~(\\ref{eq.simp_axigluon}), the first term is the contribution from SM gluon mediation,\n the second term is the interference between $q\\bar{q}\\to g \\to t\\bar{t}$ and\n $q\\bar{q}\\to Z_C \\to t\\bar{t}$, and the last term is the self-conjugation of the $Z_C$\nmediating process. Some characters can be described for the $Z_C$ explanation:\n (1) The additional asymmetry comes from the interference between the\ngluon and $Z_C$ processes. This can only change the angular distribution of\nthe produced $t\\bar{t}$ events and has no contribution to\n the total $t\\bar{t}$ cross section; (2) The third term of $Z_C$ self-conjugation can\ncontribute to the $t\\bar{t}$ total cross section. However, this contribution can be\nsuppressed by adopting proper couplings;  (3) The Breit-Wigner propagator of $Z_C$\nshows that $Z_C$ can only impact on its near mass peak region. Actually,\nour original idea was to introduce a near top pair production threshold\nparticle to replace the usual heavy ($>1$~TeV) axigluon.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.23\\textwidth]\n{plot_afb_mass.eps}\n\\includegraphics[width=0.23\\textwidth]\n{plot_afb_dy.eps}\n\\includegraphics[width=0.23\\textwidth]\n{plot_xsec_mass.eps}\n\\end{center}\n\\caption{\\label{Zc-differential-distribution}The differential distributions with best fit $Z_C$ parameters. The experimental data comes from~\\cite{Aaltonen:2012it, Aaltonen:2009iz}.}\n\\end{figure}\n\n\n\n\nFig.~\\ref{Zc-differential-distribution} shows the best fit results by $Z_C$. The experimental data of $A_{FB}$ versus $M_{t\\bar{t}}$ and $\\Delta Y$, together with the differential distributions of the cross section $d\\sigma\/d M_{t\\bar{t}}$ are constructed in the $\\chi^2$ global fit. The free parameters to fit can be taken as $g_A^q$, $g_A^Q$ and $M_{Z_C}$, or equally as $g_A^q g_A^Q$, $\\Gamma_{Z_C}$ and $M_{Z_C}$. Table~\\ref{fitted-parameter} shows the consistent results in the two parametrization scenarios.\n\n\n\\begin{table}[htb]\n\\caption{\\label{fitted-parameter}The best fit parameters in two free parameter selection scenarios. The unit of the mass and width is GeV. }\n\\begin{tabular}{ccc|ccc}\n\\hline\\hline\n$g_A^q$ & $g_A^t$ & $M_{Z_C}$ & $g_A^q g_A^t$&$\\Gamma_{Z_C}$ &$M_{Z_C}$ \\\\\n\\hline\n 0.034&9.5 & 340 &0.32 &400&340 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\nIt can be seen that $Z_C$ has tiny couplings to the light quarks and has very strong couplings to the heavy quarks. People may argue that $g_A^Q$ is so large that it may destroy the perturbation of the theory. Our statement is that the color octet axial-vector-like particle $Z_C$ is just an effective explanation of the top $A_{FB}$ anomaly. It is not necessary to be taken as a fundamental particle.\n\nActually, the forward-backward asymmetry is just a variate to exhibit the angular distribution of the final produced top quarks. For large enough luminosity, we can study the angular distribution directly and more information can be reserved by avoiding the half plane angular integration. Recently, by analyzing the $9.4~\\mbox{fb}^{-1}$ data at the Tevatron, the CDF Collaboration gave the polar angular distribution of the top quark in the $t\\bar{t}$ rest frame in Ref.~\\cite{angular-distribution}.\nFig.~\\ref{Zc-angular-distribution} shows our result by taking the above best fit parameters. The introduction of an effective color octet axial-vector-like particle can give a pretty good agreement with the experimental data.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]\n{plot_TopCosTheta.eps}\n\\end{center}\n\\caption{\\label{Zc-angular-distribution}The normalized angular distributions of the top quark in $t\\bar{t}$ rest frame at the Tevatron. The experimental data and SM QCD NLO +EWK predicted values are taken from Ref.~\\cite{angular-distribution}. }\n\\end{figure}\n\n{\\bf The Color Flow Method Analysis:} In this paper, we take the method in Ref.~\\cite{Gallicchio:2010sw} to analyze the octet color of the $Z_C$. The color flow method is invented to distinguish the color of the mediate particle in quark pair production process. If we draw the transverse momentum $P_T$ of the particles of the quark pair jets in the rapidity-azimuthal angular $y-\\phi$ plane, the different color of the mediate particle will lead to different shapes of the jets. For a color singlet\nmediating particle, such as $\\gamma\/Z$, the two jet shape seems to attract to each other, while for\na color octet mediating particle, such as gluon, the two jet shape seems to repel to each other. Such a difference is caused by the different color charge flow in the Feynman diagrams. A vivid example can be found as Fig.~2 in Ref.~\\cite{Gallicchio:2010sw}.\n\nTo distinguish the color flow, a measurable variate named as the Pull\nis defined as~\\cite{Gallicchio:2010sw}\n\\begin{equation}\n\\vec{t}=\\sum\\limits_{i\\in {\\rm jet}} \\frac{P_T^i|r_i|}{P^{\\rm jet}_T}\\vec{r}_i,\n\\label{Pull}\n\\end{equation}\nwhere $\\vec{r}_i=(\\delta y_i,\\delta \\phi_i)=\\vec{c}_i-\\vec{J}$, in which\n $\\vec{J}=(y_J,\\phi_J)$ is the location of the jet and $\\vec{c}_i$ is\nthe position of a cell or particle with transverse momentum $P_T^i$.\nWe would like to emphasize that $\\vec{t}$ is a two dimensional vector\nand it can be written as $\\vec{t}=(t_y, t_\\phi)=(|\\vec{t}|\\cos \\theta_t|,|\\vec{t}|\\sin\\theta_t)$. $\\theta_t$ is the polar angle determined by the two components of $\\vec{t}$.\n\nWe adopt the Madgraph~\\cite{Alwall:2011uj} to make the Monte-Carlo (MC) simulation. The top pair can be produced through the SM processes and the $q{\\bar q}\\to Z_C \\to t\\bar{t}$ process. We use\nPythia~\\cite{Sjostrand:2006za} to simulate the top decay and the sequent hadronization to form the jets. Some special characters in the simulation are described as follows\n\\begin{itemize}\n\\item Bottom jet Pull is calculated to replace the ``top jets''. As top quark decays to bottom and the $W$ boson. Its color is fully carried by the bottom quark, and then the selection of bottom jet Pull can avoid the dilution caused by $W$ boson from the top quark.\n\\item When calculating the b jet Pull, a core $\\vec{J}=(y_J,\\phi_J)$ is needed and it is taken as the b quark from the top decay in the Monte-Carlo sample events. Such a b quark is actually invisible in the real data sample.\n\n\\item For the hadronization of the bottom quark from the top decay in the MC simulated events, the b quark firstly forms a string\/cluster together with a light quark. The light quark can be from the hadron beam in the color octet mediating case or from the vacuum in the color singlet case. The string\/cluster then decays into a B meson together with some light quarks which transforms to light hadrons. These light hadrons can generally be divided into two categories. The first category belongs to the beam remnant hadrons, which have large rapidity $y$ and small $P_T$. The second category is the other light hadrons which are likely to locate near to the B meson. The summed momentum of the B meson nearby hadrons and the B meson decay products are approximately equal to the momentum of the b quark from the top decay; the summed momentum of the beam line nearby hadrons is approximately equal to the initial momentum of the light quark which forms a string\/cluster together with the b quark.\n\n\\item In calculating the Pull, we adopt the b quark from top decay as the core, and all the string\/cluster decay products are calculated in the summation in Eq.~(\\ref{Pull}). Although the first category jets actually can not be recorded in the real experiments, their contributions to the Pull are small enough to be neglected for their small $P_T$. So the involving of all the decay products from the string\/cluster can still be a good approximation.\n\n\\item We find that the Pull diagrams are very sensitive to the selection of the core. For example, if we take the B meson to be the core and sum the B meson decay products in the Pull,  the color singlet or octet cases can not be distinguished. This shows that the distortion of the jet shape happens just in time of the B meson production, rather than in its decay. Once the colorless B meson is formed, the color flow is stopped. So its decay is independent of the previous color information.\n\n\\item Based on the above analysis, we can infer that the main contribution to the different Pull really comes from the light hadrons near the B meson, which maybe tagged as a part of the b jet. Experimentally, the b tagging usually requires seeing a secondary vertex from the B meson decay. Inevitably, the light quark jets will fall in the B meson cone,  are irreducible from the B meson decay products, and form a component of the b jet. Such quarks and their decay products carry the color information from the b quark, inherit from the top quark.\n\n\\end{itemize}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]\n{Tevatron_b_kernel_string.eps}\n\\end{center}\n\\caption{\\label{Tevatron_b_kernel_string}The two dimensional diagrams of the Pull $\\vec{t}$ for the different color mediate particles~(upper diagrams), and the histograms of the polar angles $\\theta_t$~(lower diagrams) at the Tevatron. To make the peaks just locate in the central region of the diagrams, we take the $\\bar{p} p$ collide mode in the simulation. The convolution of the parton distribution function is included. }\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.23\\textwidth]\n{Tevatron_b_kernel_string_pp_to_tt.eps}\n\\includegraphics[width=0.23\\textwidth]\n{LHC_b_kernel_string_pp_to_tt.eps}\n\\end{center}\n\\caption{\\label{collider_Pull_theta_t}The distribution of polar angle $\\theta_t$ of the Pull vector $\\vec{t}$ at the Tevatron and LHC. $\\epsilon$ is the overall event selection efficiency and ${\\cal L}$ is the integrated luminosity.}\n\\end{figure}\n\nFig.~\\ref{Tevatron_b_kernel_string} shows the Pull vector of the b jet and the histogram of $\\theta_t$ for different processes at the Tevatron. The upper plots are the two dimensional points of the Pull, and the lower plots are the histograms of the polar angle $\\theta_t$ of the Pull. It can be seen that the color octet~($g$ and $Z_C$) and singlet~(SM gauge boson $Z$) mediating particles can be distinguished clearly. For the color octet mediating particles, the b jet shape tends to repel to each other, or in other words, the color flow of the jets are connected to the remnants in the beam direction. Thus, the component of $\\vec{t}$ in the rapidity $y$ direction $t_y$ is larger than the component in the azimuthal angle $\\phi$ direction $t_\\phi$. Consequently, the polar angle $\\theta_t$ of $\\vec{t}$ accumulates more in the $\\pi$ or 2$\\pi$ region. For the top quark produced from the $q\\bar{q}$ initial state, it is more probably be attracted by the remnant beam where the $q$ comes from. That is to say, the b quark from the top decay is likely to form a string\/cluster with a quark from the proton beam. This explains why $\\theta_t$ accumulate more at $\\theta_t=\\pi$ than $\\theta_t=2\\pi$ in $q\\bar{q}\\to g\/Z_C\\to t\\bar{t}$ process. While for the $g g$ initial state processes, the b quark from top decay has equal probability to form a string\/cluster with a quark either from the proton, or from the antiproton. Thus, the peaks of the $\\theta_t$ at $\\pi$ or $2\\pi$ for the $g g$ initial state processes are nearly the same. For the gauge boson $Z$ mediating process, the peaks of $\\theta_t$ locates at $\\pi\/2$ and $3\\pi\/2$. This can be explained as follows: because the top quark is relatively heavy, the longitudinal boost of the top pair is small and then the top quark and the anti-top quark are likely to fly back to back. The b quark pair from the top decay are also nearly back to back\nand their rapidities are small. The $\\phi$ component of Pull $\\vec{t}$ can be significantly larger than the rapidity component $t_y$. Thus, the polar angle $\\theta_t$ accumulates more at $\\pi\/2$ and $3\\pi\/2$ region.\n\nFig.~\\ref{collider_Pull_theta_t} shows the distribution histograms of the $\\theta_t$ at the hadron colliders. For the Tevatron, it is a summation of the sub-processes as shown in Figure~\\ref{Tevatron_b_kernel_string}. As we introduce a color octet particle, its peak also locates at the $\\theta_t=\\pi$ region. However, the peak becomes much wider than the SM case. By measuring the $\\theta_t$ distributions at the hadron collider, the different shapes are distinguishable. This is especially useful at the LHC, in which the top asymmetry is too small to be measured precisely. Because the\nLHC is a proton-proton collider without preferred initial direction, the peaks of $\\theta_t$ at $\\pi$ and 2$\\pi$ have the same altitude.\nTo distinguish the two histograms in Fig.~\\ref{collider_Pull_theta_t}, it is necessary to require $sig=|N_1-N_2|\/(\\sqrt{N_1}+\\sqrt{N_2})>1$, where $N_1$ and $N_2$ are the event numbers in each bin\n for the two histograms. $sig$ gets its largest value at the $\\theta_t$ peak region and the least effective luminosity can be estimated as $\\epsilon{\\cal L}=0.008\/0.00002~\\mbox{fb}^{-1}$ with the selection efficiency $\\epsilon$ being about 0.001\/0.00001 for the Tevatron\/LHC~{\\cite{CDF-dilepton-efficiency,Aad:2011yb}}.\n\n\n\n\n\n{\\bf Conclusion:} In this paper, we study the color properties in the top quark pair production process. Firstly, we show that a color octet axial-vector like new particle with different coupling strengths to heavy and light quarks can provide an excellent effective explanation of\nthe top $A_{FB}$ anomaly discovered at the Tevatron. The color octet property of this new particle makes it very suitable to be studied by adopting the color-flow method. By defining the variate Pull vector, the jet substructure of the b quark from the top decay can be used to exploit the color of the mediating particle in the top pair production. The polar angle of the Pull vector can be measured precisely to distinguish from the SM predictions. This can be a first indirect cross check of the Tevatron top $A_{FB}$ excess at the LHC.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\bf Acknowledgements:} This research is supported in part\nby the Natural Science Foundation of China\nunder grant numbers 11075003, 10821504, 11075194, 11135003,\nand 11275246, and\nby the Postdoctoral Science Foundation of China under grant\nnumber 2012M510564 and 2012M520098.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHistorically, the first planetary system to be resolved around a main-sequence star was the debris disc imaged by \\citet{Smith84} around $\\beta$~Pic, a bright ($V=3.5$) and nearby ($d=19.3$~pc) A5V star. Since then, tremendous instrumental development has allowed the $\\beta$~Pic planetary system to be studied with constantly improving angular resolution and dynamic range. This effort culminated with the discovery of a giant planet orbiting at about 8--9~AU from its host star \\citep{Lagrange10}. This planet is now widely recognised to be responsible for the warp detected in the inner debris disc up to about 85~AU from the star, as first proposed by \\citet{Mouillet97}. The origin of the dynamical excitation in the system, however, remains unclear. No other massive companion seems to be present at larger orbital distance, with upper limits of a few Jupiter masses \\citep{Boccaletti09}. Additional planetary-mass objects were also searched for closer to the star using various observing techniques. So far, coronagraphic observations provided upper limits significantly higher than $10 M_{\\rm Jup}$ in the $0\\farcs2$ to $0\\farcs3$ region \\citep{Boccaletti09,Quanz10}, while interferometric observations ruled out the presence of objects with masses larger than about $50 M_{\\rm Jup}$ within $0\\farcs1$ \\citep{Absil10}. Complementary information was also obtained with radial velocity measurements of the host star, providing detection limits for orbital periods up to 1000 days, that is, a maximum angular separation of about $0\\farcs12$, with values in the planetary mass range for periods of up to 500 days \\citep{Lagrange12}.\n\nIn this Letter, we aim to improve upon these previous studies and provide the best detection limits achieved so far at angular separations ranging from $0\\farcs1$ to $0\\farcs4$, using the new $L'$-band Annular Groove Phase Mask \\citep[AGPM,][]{Mawet05} vector vortex coronagraph on VLT\/NACO \\citep{Mawet13}. The AGPM provides a compelling combination of an exquisite inner working angle (IWA), down to the diffraction limit of the telescope ($\\sim 0\\farcs1$ at $L'$ band), with an operating wavelength that is often regarded as a sweet spot for exoplanet imaging.\n\n\n\\section{Observations and data reduction}\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=7.5cm]{betpicb_frame_v5.png} & \\includegraphics[width=7.5cm]{betpicb_final_v2.png}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Left.} Illustration of an individual image of the cleaned cube (log scale), showing that the companion can be readily identified without further image processing (cf.\\ green circle). The region used for frame selection is located between the two brown circles. \\textit{Right.} Final reduced image obtained as the median of the de-rotated sPCA-processed cleaned cube (linear scale). North is up and east to the left in both images.}\n\\label{fig:image}\n\\end{figure*}\n\nOn 31 January 2013, $\\beta$ Pic was observed for about 3.5 hours at $L'$ band within the science verification observing run of NACO's new AGPM coronagraphic mode. The observations were obtained under fair seeing (${\\rm FWHM} \\sim 1\\farcs0$), but the turbulence was fast ($\\tau_0\\simeq2$~msec), resulting in frequent openings of the AO loop despite the use of the infrared wave-front sensor. The Strehl ratio at $L'$ band typically ranged between 70\\% and 75\\% during the observations. Despite the mediocre observing conditions, $\\beta$~Pic~b could be directly identified on the NACO real time display, thanks to the peak starlight extinction of about 50:1 provided by the AGPM (see Fig.~\\ref{fig:image}, left). Individual observing blocks (OBs) consisted of 200 successive frames of 0.2 sec each, with the detector windowed to a $768\\times768$ pixels region. Sequences of 10 or 20 OBs were obtained, each of them followed by three OBs of 50 frames on a nearby sky region and by re-centring the star under the AGPM mask to achieve maximal extinction. A total of 190 OBs were obtained on $\\beta$~Pic, resulting in an on-source integration time of 114 min, out of an overall observing time of 204 min. Observations were taken in pupil-tracking mode, with a parallactic angle ranging from $-15\\degr$ to $68\\degr$. The undersized Lyot stop APO165 was used as in \\citet{Mawet13}. Before and after the coronagraphic observations, we measured the non-coronagraphic point spread function (PSF) through the AGPM and the Lyot stop by placing the star away from the vortex centre. These observations also serve as photometric reference in our data analysis, where we take the airmass variation into account to properly normalise individual frames.\n\nAfter applying basic cosmetic treatment to individual frames (flat-fielding, bad pixel\/cosmic-ray removal), we started the data processing by performing frame selection to remove the frames affected by strong AO loop openings. We found that a convenient selection criterion is the standard deviation of the pixel intensity in the $1-4 \\lambda\/D$ region ($0\\farcs1 - 0\\farcs4$), which is most affected by residual starlight but does not contain the signal of $\\beta$~Pic~b (located beyond $0\\farcs4$ during our observations). To enhance the contribution of additional residual starlight in bad frames with respect to good ones, we performed a principal component analysis (PCA) of the whole image cube, and subtracted from each individual image its projection onto the first principal component (PC) before computing the standard deviation of the pixel intensity. A histogram of all measured standard deviations was built, and the threshold for frame selection was set at a $1\\sigma$ level above the median. The fraction of frames rejected by this selection process was about $15\\%$. We found this image-processing step to be critical in reaching the best possible image quality.\n\nThe second image-processing step consisted of accurately recentring the frames. We performed a (negative) Gaussian fit to the PSF centre, which resembles a dark hole surrounded by a bright doughnut (see Fig.~\\ref{fig:image}, left). This shape is mainly due to the combination of residual tip-tilt with the off-axis transmission profile of the AGPM. By recentring each image using the fitted Gaussian, we made sure that the centre of AGPM is placed at the exact same position in all individual frames, with a typical accuracy of 0.005 pixel (i.e., about 0.1~mas). We emphasize that this centring accuracy pertains to the position of the AGPM, not of the star itself. We then subtracted from each frame the estimated contribution of the sky based on the median of neighbouring sky observations. A new image cube was then created by averaging 40 successive frames (i.e., 8~sec of effective integration time), resulting in 612 individual images in the cleaned, recentred cube.\n\nFinally, we used our implementation of the KLIP algorithm \\citep{Soummer12} to produce a final image of the $\\beta$~Pic system based on the cleaned cube, taking advantage of angular differential imaging (ADI). In the KLIP algorithm, the whole ADI image sequence is used as a PSF library, to which a PCA treatment is applied. In the presence of an off-axis companion, the PC computed on the ADI cube are expected to contain (part of) the companion signal. To prevent the planet from being partly removed from the individual images when subtracting its projection onto the first $K_{\\rm klip}$ PC, we decided to implement a ``smart'' version of the PCA (or \\emph{sPCA}), where the image library is built only from images where the off-axis companion has rotated by $1\\lambda\/D$ or more with respect to the image under consideration. In this way, the PC will not contain any (or only a very small amount of) signal from the companion at its current position. At the angular separation of $\\beta$~Pic~b ($\\sim 0\\farcs45$), this translates into rejecting from the PSF library all images that have parallactic angles within about $15\\degr$ from the image under consideration. We performed the sPCA in a region of about $3\\arcsec$ in radius around the star, after masking out the region located within the IWA of the AGPM \\citep[$\\sim \\lambda\/D$,][]{Delacroix13}. The final image, obtained as the median of the de-rotated cube after sPCA processing, is shown in Fig.~\\ref{fig:image}. The number of PC kept in this analysis is $K_{\\rm klip}=30$ (out of 612), which is a good compromise to prominently reveal the planetary companion. The black spots on either sides of the planet are artefacts related to the rotation of the planet around the optical axis in the image sequence taken in pupil-tracking mode.\n\n\n\n\\section{Analysis of $\\beta$ Pic b}\n\n\t\\subsection{Photometry}\n\nEven when using sPCA, part of the planetary signal is self-subtracted during the stellar halo removal process. To retrieve the photometric information without bias, we used the negative fake companion technique \\citep{Lagrange10,Bonnefoy11}. The method proceeds as follows: (i) estimate the (biased) position and flux of the companion from the first reduced image; (ii) use the measured off-axis PSF as a template to remove this first estimate from the cleaned data cube before applying PCA; and (iii) iterate on the position ($x$, $y$) and flux until a well-chosen figure of merit -- here, the weighted sum of the squared pixel intensity in a pie chart aperture centred on the first estimate of the companion position, $2.44\\lambda\/D$ in radius and $6 \\times 1.22\\lambda\/D$ in azimuth -- is minimized. The minimization was performed with the simplex-amoeba optimization. An exploration of the figure of merit (equivalent to a $\\chi^2$) around the best-fit position is used to evaluate the statistical error bar on the photometry of the planet ($0.15$~mag at $1\\sigma$ assuming Gaussian noise). Adding the contribution of photometric variations ($0.05$~mag for a photometric night) to the error bar, and correcting the companion photometry for the off-axis transmission profile of the AGPM \\citep{Delacroix13}, we obtain a final contrast $\\Delta L' = 8.01 \\pm 0.16$~mag between the planet and the star. This result agrees within error bars with the contrast ($7.8 \\pm 0.3$~mag) found by \\citet{Lagrange10} in the same photometric band. The final photometry of $\\beta$~Pic~b is given in Table~\\ref{tab:planet}. With an absolute magnitude $M_{L'}=10.02\\pm0.16$, and assuming an age of $12^{+8}_{-4}$~Myr for the system as in \\citet{Bonnefoy13}, the estimated mass of $\\beta$~Pic~b amounts to $8.0^{+3.2}_{-2.1} M_{\\rm Jup}$, based on the BT-Settl models of \\citet{Allard11}. The recent revision of the age \\citep[$21\\pm4$~Myr,][]{Binks13} would lead to a mass of $10.6^{+1.2}_{-1.8} M_{\\rm Jup}$.\n\n\\begin{table}\n\\caption{Measured properties of the $\\beta$~Pic system}\n\\label{tab:planet}\n\\centering\n\\begin{tabular}{c c c}\n\\hline\\hline\nParameter & $\\beta$ Pic A & $\\beta$ Pic b \\\\\n\\hline\n$d$ (pc) & $19.44 \\pm 0.05$ & --- \\\\\n$L'$ (mag) & $3.454\\pm0.003$ & $11.46\\pm0.16$ \\\\\n$M_{L'}$ (mag) & $2.011\\pm0.006$ & $10.02 \\pm 0.16$ \\\\\nSeparation (mas) & --- & $452 \\pm 10$ \\\\\nPA (deg) & --- & $211.2 \\pm 1.3$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\t\\subsection{Astrometry}\n\nThe astrometry of the companion was evaluated using the negative fake companion technique, following the same procedure as for the photometry. The main limitation in the astrometric accuracy in high-contrast imaging generally comes from the imperfect knowledge of the star position in the images (either because it is masked by a coronagraph, or because the star is saturated in the images). In our case, the recentring procedure allowed us to make sure that the AGPM position is the same in all images with an accuracy of about 0.1~mas. This does not mean that the position of the star in the image is known with such an accuracy, however, as it is generally not perfectly centred on the AGPM. We can evaluate the misalignment between the star and the AGPM centre thanks to the instantaneous starlight rejection rate. Assuming that the residual starlight is entirely due to misalignment, we derive a mean upper limit of about 8.5~mas on the error on the position of the star with respect to the AGPM centre. This error is folded into our astrometric error budget. Although this error bar is not drastically improved with respect to saturated $L'$-band data sets \\citep{Chauvin12}, the more robust knowledge of the stellar position is still a clear advantage of the AGPM observing mode.\n\nThe other contributors to the astrometric errors are the statistical error on the companion position and the calibration errors. The statistical error bar is estimated by exploring the figure of merit around the best-fit solution obtained with the negative fake companion technique, yielding a $1\\sigma$ error of 4.5~mas. The calibration errors, related to the platescale, true north orientation and uncertainty on the NACO rotator offset, are computed as in \\citet{Chauvin12}, using an estimate platescale of $27.10 \\pm 0.04$~mas, a true north of $-0\\fdg45 \\pm 0\\fdg09$, and a rotator offset of $104\\fdg84 \\pm 0\\fdg01$. They are negligible in the final error budget. The final astrometry with error bars is given in Table~\\ref{tab:planet}. A comparison with the orbital position predictions of \\citet{Chauvin12} suggests that the planet is now starting to recess.\n\n\n\n\\section{Search for additional, inner companions}\n\nIn this section, we exploit the exquisite IWA of the AGPM to provide new constraints on the presence of planetary companions down to an angular distance of $1\\lambda\/D$ ($0\\farcs1$) at L' band. At the distance of $\\beta$~Pic, this represents a linear separation of about 2~AU. To derive unbiased contrast curves, we removed the contribution of $\\beta$~Pic~b from all individual images in the cube, using the best-fit photometry and astrometry found in the previous section and the off-axis PSF obtained during our observations. The noise level can then be computed as the standard deviation of the pixel intensity in concentric annuli, or equivalently as the azimuthal median of the noise computed locally in small square boxes, even at the location of the (removed) planet. However, the final contrast curve cannot be trusted within a circular region about $1.22 \\lambda\/D$ ($0\\farcs13$) in radius around the position of $\\beta$~Pic~b, because the presence of an additional companion within this zone would result in a partial PSF overlap and in an improper subtraction of $\\beta$~Pic~b from the image cube.\n\nWe implemented an annulus-wise version of the sPCA method, where the exclusion criterion on the parallactic angles to be included in the PSF library is computed separately in thin annuli $2\\lambda\/D$ in width. To assess the amount of self-subtraction of potential companions at any given angular separation from the star, we introduced fake companions in our cube, separated by a few $\\lambda\/D$ from each other and placed on three radial branches separated by $120\\degr$ in azimuth. The fake companions were injected at $20\\sigma$ above the noise computed after a first pass of the sPCA algorithm on the cube without fake companions. By measuring the photometry of the fake companions in the final reduced image after a second pass of the sPCA algorithm and comparing it to their input flux, we inferred the attenuation of the sPCA algorithm in ADI mode. As expected, the self-subtraction strongly depends on the width of the exclusion zone in terms of parallactic angle, but the final contrast curve is only weakly affected by this parameter. Taking self-subtraction into account, small exclusion zones provide slightly better detection limits at very small angular separations, because the frames that are more correlated (i.e., closer in time) to the current frame are then kept in the library. An exclusion zone of only $0.1\\lambda\/D$ is used in the following. With this criterion, and taking $K_{\\rm klip}=20$ in each ring, the companion self-subtraction is significant in the innermost parts of the search region, with only about 40\\% (resp.\\ 20\\%) of the signal making it through at $0\\farcs5$ (resp.\\ $0\\farcs25$).\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=9cm]{contrast_box.pdf}\n\\end{center}\n\\caption{$5\\sigma$ detectability limits in terms of contrast for point-like companions around $\\beta$~Pic (solid line). The dotted line shows the contrast curve that would be derived if self-subtraction encountered in the ADI-PCA data processing was not taken into account. The grey-shaded region recalls that the detection limits are not valid in a small region of $\\sim 0\\farcs13$ in radius around $\\beta$~Pic~b.}\n\\label{fig:contrast}\n\\end{figure}\n\nThe final contrast curves, computed as five times the standard deviation of the pixel intensities in $\\lambda\/D$-wide annuli after applying a median filter on a $\\lambda\/D$-wide moving box, are displayed in Fig.~\\ref{fig:contrast} before and after taking into account companion self-subtraction. Owing to the small number of independent resolution elements in $\\lambda\/D$-wide annuli at short angular separations, the confidence level ($1-3\\times 10^{-7}$) associated to a $5\\sigma$ detection limit for pure Gaussian noise is not preserved, as discussed in Mawet et al.\\ (in prep). Reducing the confidence level (i.e., increasing the false alarm probability) at small angles is, however, not a severe limitation to the validity of the detection limits, not only because no candidate companion is found in our case, but also because one can readily distinguish false positives (bright speckles) from true companions at a few $\\lambda\/D$ with follow-up observations (background point-like sources being very unlikely in such a small search region). \n\nThe detection limits presented in Fig.~\\ref{fig:contrast} are based on the debatable assumption of Gaussian noise. To address this point, which was shown by \\citet{Marois08} to be a potential source of bias on the sensitivity limits, we constructed the histogram of the pixel intensities in the final image (the median of the de-rotated individual sPCA-processed images in the cleaned cube), using annuli $\\lambda\/D$ in width. Using the Shapiro-Wilk test \\citep{Shapiro65}, we verified that for most annuli within the search region, the statistics of the pixel intensity can be considered as Gaussian, with p-values typically ranging from 10\\% to 90\\%.\n\nTo convert our detection limits into sensitivity limits in terms of planetary mass, we used the BT-Settl models of \\citet{Allard11}. The result is given as the solid curve in Fig.~\\ref{fig:detlim}, where the sensitivity limits of previous high-contrast imaging studies have been plotted for comparison using the same BT-Settl models. This figure illustrates the high gain in sensitivity at short angular separation enabled by the $L'$-band AGPM, which allows planets more massive than $5M_{\\rm Jup}$ to be ruled out down to about $0\\farcs2$ from $\\beta$~Pic. Detection limits within the planetary-mass regime are derived down to $0\\farcs1$, although the strong self-subtraction, the bright speckles, and the small number of independent resolution elements urge us to take the detection limit with caution in the $0\\farcs1 - 0\\farcs2$ region. The sensitivity to companions located farther away than $\\beta$~Pic~b is also excellent, down to about $1M_{\\rm Jup}$ beyond $1\\farcs5$. With 768 pixels in our images, the outer working angle is about $10\\arcsec$.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=9cm]{betPic_sensitivity_box.pdf}\n\\end{center}\n\\caption{Sensitivity limits in terms of mass derived from our data set (solid line), compared with the results of \\citet[dashed and dashed-dotted]{Boccaletti09} and \\citet[dotted]{Quanz10} using the same evolutionary models (see text). This figure cannot serve as a direct comparison between the sensitivity of these NACO observing modes, as ADI was not used in the three previously published data sets.}\n\\label{fig:detlim}\n\\end{figure}\n\n\n\n\\section{Concluding remarks}\n\nWe have presented the results of our search for additional planetary companions around $\\beta$~Pic. These results nicely complement those obtained at shorter orbital periods  using radial velocity measurements \\citep{Lagrange12}. Combined, these two studies can now exclude the presence of giant planets with masses similar to that of $\\beta$~Pic~b at most orbital distances (only in the 1--2~AU region are the detection limits still in the 10--20\\,$M_{\\rm Jup}$ range). The fact that we do not detect any additional, massive companion around $\\beta$~Pic agrees with the conclusion that $\\beta$~Pic~b is the sole body responsible for the warp in the inner debris disc, as proposed by \\citet{Lagrange12}. It suggests that the dynamical excitation in the $\\beta$~Pic inner system is most probably not directly related to gravitational interactions with other massive bodies ($>5M_{\\rm Jup}$) located in the inner system.\n\nThe excellent sensitivity limits that we obtained down to very short angular separations from $\\beta$~Pic illustrates the potential of the $L'$-band AGPM coronagraph recently installed on NACO, despite the mediocre seeing conditions. We expect that the NACO\/AGPM sensitivity can be further improved under better adaptive optics (AO) correction, with an improved calibration of static instrumental aberrations, and with an optimal Lyot stop. The AGPM mode on NACO would be very well suited to follow-up and characterise new companions found by upcoming near-infrared imagers aided by extreme AO systems. Although our observing sequence on $\\beta$~Pic spans a wide range of parallactic angles ($83\\degr$), we note that the sensitivity limit close to the IWA is still largely affected by the limited displacement of any candidate companion between individual images obtained in pupil tracking mode. We suggest that reference-star differential imaging, an observing strategy mostly dropped in favour of ADI during the past few years, may be the most appropriate way to unleash the full potential of phase masks such as the AGPM in terms of IWA, when operating under good and stable AO correction.\n\n\n\n\\begin{acknowledgements}\nThe research leading to these results has received funding from the European Research Council\nunder the FP7 through ERC Starting Grant Agreement no.\\ 337569, and from the Communaut\\'e fran\\c caise de Belgique -- Actions de recherche concert\\'ees -- Acad\\'emie universitaire Wallonie-Europe.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAccurately describing the behavior of interacting enzymes, proteins, and genes\nrequires spatially extended stochastic models.  However, such models are\ndifficult to implement and fit to data. Hence tractable reduced models are frequently used instead. \nIn many popular models of biological networks, a single ODE is used \nto describe each node, and sigmoidal functions to describe interactions between them.  Even such simplified ODEs are typically intractable, as the number of parameters and\n the potential dynamical complexity make it difficult to analyze the behavior of the system using purely  numerical methods.\n   Reduced models that capture the overall dynamics, or allow approximate solutions\ncan be of great help in this situation [\\cite{Verhulst_GSPT,Hek2010}].\n\nAnalytical treatments are possible in certain limits.  The\napproaches that have been developed  to analyze models of gene interaction\nnetworks can be broadly classified into three\ncategories~[\\cite{PolynikisHoganBernardo2009}]:  \\emph{Quasi Steady State Approximations} (QSSA), \\emph{Piecewise Linear Approximations} (PLA), and\n \\emph{discretization of continuous time\nODEs}. In particular, in certain limits interactions between network elements become\nswitch--like~[\\cite{kauffman69,Snoussi1989,Mochizuki2005, Alon2006,mendoza2006, DavitichBornholdt2008, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}].   For instance,\nthe Hill function,   $f(x) = x^n \/ (x^n + J^n )$,  approaches the Heaviside function, $H(x - J)$,\nin the limit of large $n$.\nIn this limit the domain on which the network is modeled\nis naturally split into subdomains: The threshold,\ncorresponding to the parameter $J$ in the Hill function, divides the domain into two subdomains within which\nthe Heaviside function is constant.  Within each subdomain a node is either\nfully expressed, or\nnot expressed at all.  When $n$ is large, the Hill function, $f(x)$, is approximately constant in each\nof the subdomains, and boundary layers occur when $x$ is\nclose to the threshold, $x \\approx J$~[\\cite{IroniPanzeriPlahteSimoncini2011}]. To simplify the system further, we can map values of $x$ below the threshold to 0, and the values above the threshold to 1 to obtain a Boolean network (BN); that is, a map\n\\[h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow \\{0,1\\}^N,\\]\nwhere each function $h_i$ describes how variable $i$ qualitatively depends on the other variables\n [\\cite{GlassKauffman1973,Snoussi1989,thomasbook,edwards2000,edwards2001}].  Such reduced systems are simpler to analyze, and share the  dynamical properties of the original system, if the reduction is done properly.\n\n\nThe reduced models obtained in the limit of a large Hill coefficient, $n,$ have a long and rich history. Piecewise linear\nfunctions of the form proposed in~[\\cite{GlassKauffman1973}] have been shown to\nbe well suited for the modeling of genetic regulatory networks, and can sometimes be justified rigorously [\\cite{de04}].  In particular,  singular perturbation\ntheory  can\nbe used to obtain reduced equations within each subdomain and the boundary\nlayers,\nand global approximations   within the entire\ndomain~[\\cite{IroniPanzeriPlahteSimoncini2011}]. On the other hand, although BNs have been used to model the dynamics of different biological systems, their relation to more complete models\nwas mostly demonstrated with case studies, heuristically or only for steady states [\\cite{GlassKauffman1973,Snoussi1989, thomasbook, albert2003, mendoza2006, DavitichBornholdt2008, p53, p53ode, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}].\n\nHere we again start\nwith the Hill function, $ x^n \/ (x^n + J^n )$, but instead of assuming that $n$ is large, we assume that $J$ is small. This case\n has a simple physical\ninterpretation: Consider the Hill function that occurs in the Michaelis-Menten\nscheme, which models the catalysis of the inactive form of some\nprotein to its active form in the presence of an enzyme. When $J$ is small\nthe total enzyme concentration is much smaller than the total protein\nconcentration.\nAlthough the subsequent results hold for any fixed $n$, for simplicity we\nassume $n = 1$. \n\n\n\nMore precisely, we consider a model biological \nnetwork where the activity at each of $N$ nodes is described by $u_i \\in [0,1]$, and evolves according to\n\\begin{align}\\label{eqn:ProblemEquation}\n\t\\frac{du_i}{dt} = A_i\\frac{1-u_i}{J_{i}^A+1-u_i}-I_i\\frac{u_i}{J_{i}^I+u_i},\n\\end{align}\nwhere $J_i^A, J_i^I>0$, and the functions $A_i=A_i(u)$, $I_i=I_i(u)$ are affine functions.\n\n\nThis type of equations have been used successfully in many models [\\cite{GoldbeterKoshland1981, Goldbeter1991, NovakPatakiCilibertoTyson2001, de02, NovakPatakiCilibertoTyson2003, IshiiSugaHagiyaWatanabeMoriYoshinoTomita2007, CilibertoFabrizioTyson2007, DavitichBornholdt2008, vanZwietenRoodaArmbrusterNagy2011}]. Here  $A_i$ and $I_i$  describe how the other variables affect $u_i$  and can represent activation\/phosphorylation\/ production and inhibition\/dephosphorylation\/decay, respectively. The variables $u_i$ can represent species such as protein concentrations, the active form of enzymes, or activation level of genes.  A simple example is provided by a  protein that can exist in an unmodified form, $W,$ and a modified form, $W^*,$ (e.g. proteases, and Cdc2, Cdc25, Wee1, and Mik1 kinases [\\cite{Goldbeter1991, Novak1998, NovakPatakiCilibertoTyson2001}]) where the conversion between the two forms is catalyzed by two enzymes, $E_1$ and $E_2$ [\\cite{GoldbeterKoshland1981, Goldbeter1991,Novak1998, NovakPatakiCilibertoTyson2001}] (See Appendix for details). \nHowever, note that the models of chemical reactions we consider can be rigorously derived from the Chemical Master Equation only in the case of a single reaction~[\\cite{KumarJosic2011}]. The models of networks of chemical reactions that we take as the starting point of \nour reduction should therefore be regarded as phenomenological. \n\nIt is easy to show that the region $0\\leq u_i\\leq 1, 1 \\leq i \\leq N$ is invariant so that Eq.~\\eqref{eqn:ProblemEquation} is a system of equations on $[0,1]^N$. Equations involving this special class of Hill functions are generally referred to as\nMichaelis-Menten type equations, and $J$  the Michaelis-Menten \nconstant~[\\cite{MichaelisMenten1913, GoldbeterKoshland1981, Goldbeter1991, NovakTyson1993, NovakPatakiCilibertoTyson2001, NovakPatakiCilibertoTyson2003, CilibertoFabrizioTyson2007, DavitichBornholdt2008, ChaoTang2009}]. \n\nThe  constants $J$ are frequently very small in practice [\\cite{NovakPatakiCilibertoTyson2001,DavitichBornholdt2008}], which motivates examining Eq.~\\eqref{eqn:ProblemEquation} \nwhen  $0 < J \\ll 0$. In this case, we discuss a two step reduction of the model \n$$\n\\text{full, nonlinear model $\\longrightarrow$ piecewise linear model  (PL) $\\longrightarrow$ Boolean Network (BN)}.\n$$ \nWe first illustrate this reduction using two standard examples, and then provide a general mathematical justification. We note  that the reduction obtained in the first step (see Eq.~\\eqref{eqn:mainReduced_RSTa}) is actually (algebraic) piecewise affine.\nHowever, it is customary  to refer to the equation and the associated model as \\emph{piecewise linear} [\\cite{GlassKauffman1973,Snoussi1989, thomasbook,edwards2000,de02}], and we follow this convention.\n\nThe main idea behind the piecewise linear (PL) reduction is simple:\nIf $J \\ll x$ then the Hill functions, $f(x) = x \/ (x + J ) \\approx 1$. However, when $x$ and $J$ are comparable, $x \\sim J,$ this is no longer true.  In this boundary layer, we  rescale variables by introducing $\\tilde{x} := x\/J$. A similar argument works for the function $(1-x) \/ (J+1-x )$ (see Appendix). We show that using this observation, the domain $[0,1]^N$ naturally decomposes  into a nested sequence of hypercubes.  The dynamics on each hypercube in the sequence is described by a solvable differential-algebraic system of equations.  The PL reduction therefore gives an \\emph{analytically tractable} approximate\nsolution to the original system.\n\nIn the next step of the reduction we obtain a Boolean Network (BN): The PL  approximation is used to divide $[0,1]^N$ into chambers.   Within nearly all of a chamber the rate of change of each element of the network is constant when $J \\ll 1$.   We use these chambers to define a BN. A similar approach was recently used to motivate a Boolean reduction of a model protein interaction\nnetwork~[\\cite{DavitichBornholdt2008}].\n\n\nThe mathematical justification also follows two steps.  We use Geometric Singular Perturbation Theory (GSPT) in Section \\ref{sec:math_PL} to justify the PL approximation. \nThe justification of the BN reduction is given in Section \\ref{sec:math_BN}.\nWe show that  there is a one-to-one correspondence between steady states (equilibrium solutions) of the BN and the full and PL system near the vertices of $[0,1]^N$. Futhermore, we show that this  one-to-one correspondence between steady states is actually global (up to a set of small measure in $[0,1]^N$). BNs have been used to study oscillatory behavior [\\cite{Li_cc_2004,p53}], and we prove in Section \\ref{sec:math_BN_trajectories} that under some conditions oscillations in a BN correspond to oscillations in the full system.\n\n\n\\section{Example problems}\\label{ExampleProblems}\n\nWe start by demonstrating the main idea of our approach using networks of\ntwo and three mutually repressing nodes. These nodes\ncan represent genes that mutually inhibit each other's\nproduction~[\\cite{GardnerCantorCollins2000,ElowitzLeibler2000}].  \nHowever, \nthe theory we develop applies whenever the heuristic model given in Eq.~\\eqref{eqn:ProblemEquation} is applicable.\nWe accompany these examples with a \nheuristic explanation of the different steps in the\nreduction. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale = .35]{toggleSwitch_Represscilator_4}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize {(a) Nodes $u_1$, $u_2$ inhibiting each others activity resulting in a switch. The node which starts out stronger suppresses the activity of the other. (b) Nodes $u_1$, $u_2$, and $u_3$ suppress each other in a cyclic fashion. Under certain conditions, this can lead to oscillations.}%\n}\\label{fig:toggle_repressillator}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{A network of two mutually inhibiting elements}\n\nWe start with the common \\emph{toggle switch} motif, \\emph{i.e} a network of\ntwo mutually repressing elements (see Fig.~\\ref{fig:toggle_repressillator}a) ~[\\cite{NovakPatakiCilibertoTyson2003,GardnerCantorCollins2000}]. Let\n$(u_1,u_2) \\in [0,1]^2$ represent the normalized levels of activity at the two nodes. \nTherefore, when $u_i = 1$ the $i^{\\text{th}}$\nnetwork element is maximally active (expressed). The activity of the two nodes in the system \ncan be modeled by\n\\begin{align}\\label{EquationForToggleSwitch}\n\\begin{split}\n\\frac{du_1}{dt} &= 0.5\\frac{1-u_1}{J+1-u_1}-u_2\\frac{u_1}{J+u_1},  \\\\\n\\frac{du_2}{dt} &= 0.5\\frac{1-u_2}{J+1-u_2}-u_1\\frac{u_2}{J +u_2},\n\\end{split}\n\\end{align}\nwhere $J$ is some positive constant. The structure of\nEq.~\\eqref{EquationForToggleSwitch} implies that the cube $[0,1]^2 = \\{(u_1, u_2)\n\\,|\\, 0 \\le u_1, u_2 \\le 1 \\}$ is invariant (see\nProposition~\\ref{prop:invariantCube}).\n\n\\subsubsection{Piecewise linear approximation}\n\nIn the limit of small $J$, Eq.~\\eqref{EquationForToggleSwitch}  can be\napproximated by a piecewise linear differential equation:\nIf $u_i$ is not too close to zero the expression $u_i\/(J+u_i)$ is approximately unity.\n  More precisely, we fix a small $\\delta > 0 $, which will be chosen to depend on\n$J$.  When $u_i > \\delta$ and $J$ is small then $u_i\/(J+u_i) \\approx1$. Similarly, when \n$u_i > 1-\\delta$ then  $(1-u_i)\/(J+1-u_i) \\approx 1$.\n\n\nWith this convention in mind we  break the cube $[0,1]^2$ into several\nsubdomains, and define\na different reduction of Eq.~\\eqref{EquationForToggleSwitch} within each. Let $\\mathcal{R}_S^T$ to denote the region where $S$ is the set of variables that\nare close to 0, and $T$ is the set of variables close to 1 (See Table~\\ref{table:2d} and Eq.~\\eqref{Rdef}). Also, we omit the curly brackets and commas in $\\mathcal{R}_S^T$ (e.g.  $\\mathcal{R}^{\\{\\}}_{\\{\\}}=\\mathcal{R}$ and $\\mathcal{R}_{\\{1\\}}^{\\{\\}}=\\mathcal{R}_1$) (see Fig.~\\ref{fig:subdomains}). \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .65]{subdomains}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nSubdomains $\\mathcal{R}_S^T$ for the unit square $[0,1]^2$. \n\\label{fig:subdomains}}\n}\n\\end{center}\n\\end{figure}\n\nWe first reduce Eq.~\\eqref{EquationForToggleSwitch} on each of the subdomains.\nThe interior of the domain $[0,1]^2$ consist of points where neither coordinate is close to 0 nor 1, and\n is defined by\n\\begin{align}\\label{R00}\n\\mathcal{R}:= \\{(u_1,u_2) \\in [0,1]^2 \\,|\\, \\delta \\le u_1 \\le 1-\\delta \\text{ and }\n\\delta \\le u_2 \\le 1-\\delta\\}.\n\\end{align}\n Eq.~\\eqref{EquationForToggleSwitch}, restricted to $\\mathcal{R}$ is\napproximated by the linear differential equation\n\\begin{align}\\label{interiorLinear2d}\n\\frac{du_1}{dt} = 0.5-u_2, \\quad\n\\frac{du_2}{dt} = 0.5-u_1.\n\\end{align}\n \nOn the other hand, if one of the coordinates is near the boundary, while the\nother is in the interior, the approximation is different. For instance, the\nregion\n\\begin{align}\\label{R20}\n\\mathcal{R}_2:= \\{(u_1,u_2) \\in [0,1]^2 \\,|\\,  u_2 < \\delta \\text{ and } \\delta \\le\nu_1 \\le 1-\\delta\\},\n\\end{align}\nforms a boundary layer where $u_2$ is of the same order as $J$. \n \nThe term $u_2\/(J + u_2)$ cannot be approximated by unity.  Instead the approximation takes the form\n\\begin{subequations}\\label{edgeLinear2d}\n\\begin{align}\n\\frac{du_1}{dt} &= 0.5-u_2, \\label{edgeLinear2da} \\\\\n\\frac{du_2}{dt} &= 0.5-u_1 \\frac{u_2}{J + u_2}. \\label{edgeLinear2db}\n\\end{align}\n\\end{subequations}\nThis equation can be simplified further.  Since $\\mathcal{R}_2$ is invariant (for $u_1>.5$), $\\frac{d u_2}{dt}$ must be small inside the boundary layer\n$\\mathcal{R}_2$ (see Fig.~\\ref{fig:twoDimension_nullclines}).  We therefore use the approximations $u_2 \\approx 0$ in \nEq.~\\eqref{edgeLinear2da} and $\\frac{d u_2}{dt}\n\\approx 0$ in Eq.~\\eqref{edgeLinear2db} to obtain\n\\begin{subequations}\\label{smallu}\n\\begin{align}\n\\frac{du_1}{dt} &= 0.5, \\label{smallua}\\\\\n0 &= 0.5-u_1 \\frac{u_2}{J + u_2}.\\label{smallub}\n\\end{align}\n\\end{subequations}\nNote that Eq.~\\eqref{smallua} is linear  and decoupled from Eq.~\\eqref{smallub},\nwhile Eq.~\\eqref{smallub} is an algebraic system which can be solved to obtain\n$u_2 \\approx J\/(2u_1 - 1)$.\nWithin $\\mathcal{R}_2$ we thus obtain the approximation \n\\begin{subequations}\\label{smallu2}\n\\begin{align}\nu_1(t) & = 0.5 t  + u_1(0)  \\label{E:smalleqa}\\\\\nu_2(t) & = \\frac{J}{ t + 2 u_1(0) - 1}  \\label{E:smalleqb}\n\\end{align}\n\\end{subequations}\n\nWe only have the freedom of specifying\nthe initial condition $u_1(0)$, since  $u_2(0)$ is determined by the solution\nof the algebraic equation~\\eqref{smallub}.  As we explain below, this algebraic\nequation defines a slow manifold within the subdomain $\\mathcal{R}_2$.  The\nreduction assumes that solutions are instantaneously attracted to this manifold. \n\nTable~\\ref{table:2d} shows how this approach can be extended to all of $[0,1]^2$. \nThere are  9  subdomains of the cube, one corresponding to the interior and four each to the \nedges and vertices.  On the latter eight subdomains, one or both variables are close to either 0 or 1.\nFollowing the preceding arguments, variable(s)\nclose to 0 or 1 can be described by an algebraic equation. The resulting algebraic-differential systems are given in the last column of\nTable~\\ref{table:2d}. \nFurthermore, by using the approximations $u_i(t)\\approx 0$ for $i\\in S$ and $u_i(t)\\approx 1$ for $i\\in T$, we obtain a simple approximation of the dynamics in each subdomain which is $0$-th order in $J$. For example, in $\\mathcal{R}_2$, we obtain the approximation $u_1(t) \\approx 0.5t + u_1(0), u_2(t) \\approx 0$.\n\n\nEach approximate solution can potentially exit the subdomain within\nwhich it is defined if at some time $u_i\\approx 0$ or $u_i\\approx 1$ and the $i$-th coordinate of the vector field is positive or negative, respectively. This can happen when the sign of some entry of the vector field changes; that is, solutions can exit subdomains when they reach a nullcline. Also, solutions can leave the subdomain if they started on the other side of the nullcline to begin with.\nThe global approximate solution of Eq.~\\eqref{EquationForToggleSwitch} is \nobtained by using the exit point from one subdomain as the initial condition for\nthe approximation in the next.  In\nsubdomains other than $\\mathcal{R}$ some of the initial conditions will be\nprescribed by the algebraic part of the reduced system.  The global\napproximation may therefore be discontinuous, as solutions entering a new\nsubdomain are assumed to instantaneously jump to the slow manifold defined by\nthe algebraic part of the reduced system.\n Fig.~\\ref{fig:twoDimension} shows that when $J$ is small, this approach\nprovides a good approximation. \n\n\n\\begin{table}[t] \n\\[\n\\begin{array}{c|c|c|rcl}\n\\hline\n\\text{Name of the subdomain}\t\t&  u_1 \t& \tu_2\t&\n\\multicolumn{3}{c}{\\text{Approximating system}}  \\\\\n\\hline\n\\multirow{2}{*}{$\\mathcal{R}$}\t&  \\multirow{2}{*}{$ \\delta \\le u_1 \\le 1-\\delta\n$}\t&  \\multirow{2}{*}{ $ \\delta \\le u_2 \\le 1-\\delta $}\t&   \t u_1'\t&=&\n0.5-u_2,\t\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n  \t\t &    u_2'\t\t&=& 0.5-u_1\t\t\t\\\\\n\\hline\n\\multirow{2}{*}{$\\mathcal{R}^1$}\t& \\multirow{2}{*}{$ u_1 > 1-\\delta $} \t\n& \\multirow{2}{*}{$ \\delta \\le u_2 \\le 1-\\delta $}\t\t&  \t           \t\n  0\t&=&  \\displaystyle 0.5\\frac{1-u_1}{J+1-u_1}-u_2,\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &  \t u_2'  &=& -0.5\t\t\t\t       \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}^2$}     & \\multirow{2}{*}{$ \\delta \\le u_1 \\le 1-\\delta\n$}\t\t& \\multirow{2}{*}{$ u_2 > 1-\\delta $}\t&  u_1'\t&=&\t -0.5,\t\n\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t  \t &                         0\t&=&\t\\displaystyle\n0.5\\frac{1-u_2}{J+1-u_2}-u_1\t\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_1 $}\t& \\multirow{2}{*}{$ u_1 < \\delta $} \n& \\multirow{2}{*}{$ \\delta \\le u_2 \\le 1-\\delta $}    \t& \t\t\t\n0 \t &=& \\displaystyle 0.5-u_2\\frac{u_1}{J+u_1},  \\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t   \t & \t\tu_2'&=& 0.5  \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_2\t$}\t& \\multirow{2}{*}{$ \\delta \\le u_1\n\\le 1-\\delta $}\t\t&   \\multirow{2}{*}{$u_2 <  \\delta $}\t& \tu_1'\t\n&=& 0.5 ,  \\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t\t\t0\t\t&=& \\displaystyle 0.5\n-u_1\\frac{u_2}{J +u_2} \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}^{12}$}& \\multirow{2}{*}{$ u_1 > 1-\\delta$} \t& \n\\multirow{2}{*}{$u_2 > 1-\\delta$} \t&\t0\t&=& \\displaystyle\n0.5\\frac{1-u_1}{J+1-u_1}-1,\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5\\frac{1-u_2}{J+1-u_2}-1\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_{12}$}&  \\multirow{2}{*}{$u_1 < \\delta$}\t & \n\\multirow{2}{*}{$u_2 < \\delta$}\t&\t0\t&=& \\displaystyle 0.5- J\n\\frac{u_1}{J+u_1},\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5- J\\frac{u_2}{J +u_2}\t\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_2^1\t$}\t& \\multirow{2}{*}{$ u_1 > 1-\\delta$}\n\t& \\multirow{2}{*}{$u_2 < \\delta $}\t\t&\t0\t&=&\n\\displaystyle  0.5\\frac{1-u_1}{J+1-u_1},\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5- \\frac{u_2}{J +u_2}\t\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_1^2\t$}\t& \\multirow{2}{*}{$ u_1 < \\delta $}\n&\\multirow{2}{*}{$  u_2 > 1-\\delta$}\t\t&\t0\t&=&\n\\displaystyle 0.5-\\frac{u_1}{J+u_1},\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5\\frac{1-u_2}{J+1-u_2}\n\\\\\n\\hline\n\\end{array}\n\\]\n\\begin{center}\n\\parbox{.8\\textwidth}{\n\\caption{\\footnotesize  List of differential--algebraic systems that approximate\nEq.~\\eqref{EquationForToggleSwitch} in different parts of the domain. The\nsubdomains are named so that the superscript (subscript)  lists the coordinates\nthat are close to $1$ (close to 0), with 0 denoting the empty set.  For example,\n$\\mathcal{R}_1^2$ denotes that subdomain with $u_1 \\approx 1$ and $u_2 \\approx\n0$, and $\\mathcal{R}^2$ the subdomain where $u_2$ is near $1$, but $u_1$ is\naway from the boundary. The middle column define the subdomain explicitly.   \nThe right column gives the differential-algebraic system that approximates\nEq.~\\eqref{EquationForToggleSwitch}  within the given subdomain. }\n\\label{table:2d}\n}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale = .45]{twoVariable4}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nComparison of the numerical solution of Eq.~\\eqref{EquationForToggleSwitch}\n(dashed black) and the solution of the approximate system as listed in\nTable~\\ref{table:2d} (colors) for two different values of $J$. We used \n$J = 10^{-2}$ in (a); and $J =10^{-4} $ in (b).  Different colors are used for \nthe solution of the reduced system in different subdomains.  Solution of the linear approximation started in the subdomain\n$\\mathcal{R}$ (Initial value: $u_1 = 0.6, u_2 = 0.4$), and as soon as $u_2$\ndecreased below $\\delta = 0.01$, we assumed that the solution entered subdomain  $\\mathcal{R}_2$. \nThe approximate solution is discontinuous since when $u_2 = \\delta$, the solution jumped\n(see inset) to the manifold, described by the algebraic part of the linear\ndifferential algebraic system prevalent in the subdomain $\\mathcal{R}_2$, Eq.~\\eqref{smallub}.\nThe solution finally stopped in the  subdomain $\\mathcal{R}_2^1$.  }\n\\label{fig:twoDimension}\n}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsubsection{Boolean approximation}\n\nWe now derive a Boolean approximation, $h=(h_1,h_2):\\{0,1\\}^2\\rightarrow\\{0,1\\}^2$, that captures certain qualitative features of  Eq.~\\eqref{EquationForToggleSwitch}. The idea is to project small values of $u_i$ to 0 and  large values of $u_i$ to 1, and map  the value of the $i$-th variable into 0 and 1 depending on whether $u_i$ is decreasing or increasing, respectively. We will show that the resulting BN can be used directly to detect steady states in the corner subdomains. \n\nNote that for a BN time is discrete; a time step in the Boolean approximation can be interpreted as the time it takes the original system to transition between chambers. Different transitions in the Boolean network may have different duration in the original system; so the time steps in the BN are only used to keep track of the sequence of events, but not their duration.\n\nThe reduction described in the previous section gives a linear ODE  in the interior region $\\mathcal{R}$ (Eq.~\\eqref{interiorLinear2d}), where $\\mathcal{R}$ approaches $[0,1]^2$ as $J\\rightarrow 0$. The approximating  linear system therefore  provides  significant information about the behavior of the full, nonlinear system for $J$ small.\n\nWe first examine the nullclines. In Fig.~\\ref{fig:twoDimension_nullclines} we can see that as $J$ decreases, in the interior of $[0,1]^2$ the nullclines of Eq.~\\eqref{EquationForToggleSwitch} approach the nullclines of Eq.~\\eqref{interiorLinear2d} given by $u_2=.5$ and $u_1=.5$ restricted to $[0,1]^2$.  These lines divide the domain into four chambers, which we denote\n\\begin{equation*}\n\\mathcal{C}_{12}:=[0,0.5)\\times[0,0.5),\\quad\n\\mathcal{C}_{1}^2:=[0,0.5)\\times(0.5,1],\\quad\n\\mathcal{C}_{2}^1 :=(0.5,1]\\times[0,0.5),\\quad\n\\mathcal{C}^{12}:=(0.5,1]\\times(0.5,1].\n\\end{equation*}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .55]{twoDimension_nullclines}\\\\\n\\includegraphics[scale = .55]{twoDimension_nullclinesvsmanifold}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nBehavior of nullclines as $J$ decreases. Top: Nullclines of Eq.~\\eqref{EquationForToggleSwitch} for $J=10^{-2}$ (left) and $J=10^{-4}$ (right). Bottom: Nullcline $\\frac{du_2}{dt}=0$ of Eq.~\\eqref{EquationForToggleSwitch} (black curve) and the manifold defined by Eq.~\\eqref{smallub} (red) for $J=10^{-2}$ (left), and $J=10^{-4}$ (right). \n\\label{fig:twoDimension_nullclines}}\n}\n\\end{center}\n\\end{figure}\n\n\n\nOn the other hand, the part of the nullclines inside the boundary subdomains are approximately the slow manifolds defined by equivalents of Eq.~\\eqref{E:smalleqb}. Here the slow manifolds converge to the nullclines  as $J \\rightarrow 0$  (See Fig.~\\ref{fig:twoDimension_nullclines}).\n\nAs a shorthand, we define the ``sign'' of a vector $v=(v_1\\ldots,v_N)$ as the vector composed by the signs of its components, $sign(v):=(sign(v_i),\\ldots,sign(v_N))$.\nNote that although the sign of the vector $(.5-u_2,.5-u_1)$ is constant in each chamber, the sign of the vector field of Eq.~\\eqref{EquationForToggleSwitch} may differ. For example, in chamber $\\mathcal{C}_2^1$, the sign of the vector field can take all possible values. However, this difference is small when  $J$ is small, because the regions between the nullclines approach the actual chambers (Fig.~\\ref{fig:twoDimension_nullclines}).  \n\nWe consider Eq.~\\eqref{EquationForToggleSwitch} in each chamber, starting with the first coordinate, $u_1(t)$. \nFor any solution with initial condition in $\\mathcal{C}_{12}$, the sign of $u'_1(0)$ is positive and $u_1(t)$ increases within the chamber.  \nWe use this observation to define $h_1(\\mathcal{C}_{12})=1$.  The formal definition of this function will be given below -- \nintuitively $h_i(\\cdot)$ maps a chamber to 1 if $u_i$ is increasing within the chamber, and to 0 otherwise. \nSimilarly, since $u_1(t)$ initially increases within $\\mathcal{C}_2^1$, we let $h_1(\\mathcal{C}_2^1)=1$. Similarly we  set $h_1(\\mathcal{C}^{12})=0,$ $h_1(\\mathcal{C}_1^2)=0,$ $h_2(\\mathcal{C}_{12})=1$, $h_2(\\mathcal{C}_1^2)=1$, $h_2(\\mathcal{C}_2^1)=0$,  and $h_2(\\mathcal{C}^{12})=0$.  The $i$-th variable is  ``discretized,'' \\emph{i.e.} mapped to 0 and 1 depending on whether $u_i$ is decreasing or increasing, respectively. \n\nMore formally, consider the set $\\{0,1\\}^2$, with each element identified with a chamber (e.g., the element $(0,1)$ represents the chamber $\\mathcal{C}_1^2$). Then $h_1$ and $h_2$ are defined as Boolean functions from $\\{0,1\\}^2$ to $\\{0,1\\}$  by setting $h_1(0,0)=0, h_1(0,1)=0$, $h_1(1,0)=1$, $h_1(1,1)=0$, and $h_2(0,0)=0, h_2(0,1)=1$, $h_2(1,0)=0$, $h_2(1,1)=0$. These two Boolean functions define a BN, $h=(h_1,h_2):\\{0,1\\}^2\\rightarrow\\{0,1\\}^2$.  The functions also define a dynamical system, $x(t+1)=h(x(t)), x \\in \\{0,1\\}^2$. However, other update schedules   can be used [\\cite{AracenaGolesMoreiraSalinas2009}].\n\n\n\n\n\nThe BN reduction can be obtained easily  from the sign of $(.5-u_2,.5-u_1)$ at the vertices of $[0,1]^2$,  since the sign of the vector field is constant within a chamber. To do so we use the the Heaviside  function, $H$, defined by $H(y)=0$ if $y<0$, $H(y)=1$ if $y>0$, and $H(0)=\\frac{1}{2}$. For example, in $\\mathcal{C}_{12}$, both entries increase. We  can see this by evaluating  $H(.5-u_2)=H(.5-u_1)=1$ for $u=(0,0)$.  Using the same argument in each chamber, we obtain the BN \n\\begin{equation}\\label{EquationForToggleSwitchBN}\nh(x)=H(.5-x_2,.5-x_1),\n\\end{equation}\nwhere we used the convention that $H$ acts entrywise on each component in the argument.\n\n\\subsubsection{Steady states of the BN and the PL approximation}\n\nWhile the BN gives information about which variables increase and decrease within a chamber, it is not yet\nclear how or if the dynamics of the BN in Eq.~\\eqref{EquationForToggleSwitchBN}, and the PL approximation in Table \\ref{table:2d} are related. \n\nWe next show that the steady states of the PL approximation near the vertices can be determined by the steady states of the BN.  The reduced equations in the corner subdomains  $\\mathcal{R}_{12},\\mathcal{R}^{12}, \\mathcal{R}_1^2,$ and $\\mathcal{R}_2^1$ are purely algebraic. When $J$ is small, some of these equations have a solution in $[0,1]^2$, indicating a stable fixed point near the corresponding corner (in this case $\\mathcal{R}_1^2$ and $\\mathcal{R}_2^1$). Others will not have a solution in $[0,1]^2$, indicating that approximate solutions do not enter the corresponding subdomain (here $\\mathcal{R}_{12}$ and $\\mathcal{R}^{12}$). To make the relationship between steady states less dependent on the actual parameters, consider the system\n\\begin{align}\n\\begin{split}\\notag\n\\frac{du_1}{dt} &= b_1^+\\frac{1-u_1}{J+1-u_1}-(u_2+b_1^-)\\frac{u_1}{J+u_1},  \\\\\n\\frac{du_2}{dt} &= b_2^+\\frac{1-u_2}{J+1-u_2}-(u_1+b_2^-)\\frac{u_2}{J +u_2},\n\\end{split}\n\\end{align}\nwhere $x^+=\\max{\\{x,0\\}}$ and $x^-=\\max\\{-x,0\\}$. In the previous example $b_1=b_2=.5$, $b_1^+=b_2^+=.5$ and $b_1^-=b_2^-=0$.  \n\nNow, at the corner subdomain $\\mathcal{R}_{12}$ we have the approximate equations,\n\\[\n0= b_1^+ -b_1^-\\frac{u_1}{J+u_1},  0= b_2^+ -b_2^-\\frac{u_2}{J +u_2},\\]\nor equivalently,\n\\[u_1 =\\frac{ -b_1^+ J}{b_1}, u_2 =\\frac{ -b_2^+ J}{b_2}.\\]\n\nThese equations have a solution in $\\mathcal{R}_{12}$ if and only if $b_1<0$ and $b_2<0$, or equivalently, if and only if $H(b_1,b_2)=(0,0)$. A similar analysis  leads to the following conditions for the existence of fixed points in each corner subdomain \n\\begin{align}\n\\begin{split}\\notag\n&\\textrm{On $\\mathcal{R}_{12}$ :}\\ \\ H(b_1,b_2)=(0,0),\\qquad\n\\textrm{On $\\mathcal{R}^{12}$ :}\\ \\ H(b_1-1,b_2-1)=(1,1),\\\\\n&\\textrm{On $\\mathcal{R}^2_1$ :}\\ \\  H(b_1-1,b_2)=(0,1),\\qquad\n\\textrm{On $\\mathcal{R}^1_2$ :}\\ \\  H(b_1,b_2-1)=(1,0).\n\\end{split}\n\\end{align}\nMore compactly, the condition is $H(b_1-x_2,b_2-x_1)=(x_1,x_2)$, where $x=(x_1,x_2)$ is the corner of interest. \nHence, the BN can also be used directly to detect which corner subdomains contain steady states. \n\nThe relationship between steady states in the full system at the corner subdomains and the steady states of the BN is straightforward. However, since there are many update schemes for BNs, the relationship between trajectories is more subtle. For example, using  synchronous update we obtain the transition $(0,0)\\rightarrow (1,1)$ which is not compatible with the solutions of Eq.~\\eqref{EquationForToggleSwitch} (See Fig.~\\ref{fig:twoDimension_ODEandBN}). On the other hand, using  asynchronous update we obtain the transitions $(0,0)\\rightarrow (1,0)$ and $(0,0)\\rightarrow (0,1),$ which are more representative of the solutions of Eq.~\\eqref{EquationForToggleSwitch}. Thus, we will focus on transitions that are independent of the update scheme, that is, transitions where only one entry changes. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .6]{twoDimension_ODEandBN}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nLeft: Solutions of Eq.~\\eqref{EquationForToggleSwitch} for $J=10^{-4}$. When a solution is close to the boundary regions of $\\mathcal{C}_1^2$ and $\\mathcal{C}_2^1$, they enter the invariant region as shown in Fig.~\\ref{fig:twoDimension}b. Right: Graphical representation of the Boolean transitions ($00\\rightarrow 11$, $11\\rightarrow 00$, $01\\rightarrow 01$, $10\\rightarrow 10$).  \n\\label{fig:twoDimension_ODEandBN}\n}} \n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{A network of three mutually inhibiting elements}\\label{sec:3nodes}\n\nThe same reduction can be applied to systems of arbitrary dimension.  As an\nexample consider\nthe\n\\emph{repressilator}~[\\cite{NovakPatakiCilibertoTyson2003,ElowitzLeibler2000}] (see Fig.~\\ref{fig:toggle_repressillator}a)\ndescribed by\n\\begin{eqnarray}\\label{threeNodeMM}\n \\frac{du_1}{dt} &=& 0.6\\frac{1- u_1}{J+1-u_1} -  u_3\\frac{ u_1}{J+u_1}, \\notag\n\\\\\n \\frac{du_2}{dt} &=& 0.4\\frac{1- u_2}{J+1-u_2} -  u_1\\frac{ u_2}{J+u_2}, \\\\\n \\frac{du_3}{dt} &=& 0.3\\frac{1- u_3}{J+1-u_3} -  u_2\\frac{ u_3}{J+u_3}. \\notag\n\\end{eqnarray}\nThe cyclic repression of the three elements in this network leads to oscillatory\nsolutions over a large range of values of $J$.  The domain of this system,\n$[0,1]^3$, can\nbe divided into 27 subdomains corresponding to 1 interior,\n6 faces, 12 edges, and 8 vertices.  \n\nWe can again  approximate Eq.~\\eqref{threeNodeMM} with solvable\ndifferential--algebraic equation within each  subdomain, to obtain a global\napproximate solution (See  Fig.~\\ref{fig:threeDimension}). Note that both the numerically obtained solution to  Eq.~\\eqref{threeNodeMM} and the solution to the piecewise linear equation exhibit oscillations, and that the approximation is discontinuous. Again, in the limit $J \\rightarrow 0$ we obtain a continuous 0-th order approximation.\n\nIn this singular limit, solutions can exit a subdomain when they reach a nullcline of the linear system. For example, when $u_2$ is close to 0 and a solution transitions from $u_1>.4$ to $u_1<.4$, the sign of the second entry of $(0.6-u_3,0.4-u_1,0.3-u_2)$ changes from negative to negative; so the second entry of the solution starts increasing (see Fig.~\\ref{fig:threeDimension}, panel (e)).  Solutions therefore leave the subdomain on which $u_2\\sim  J$ is small and enter the subdomain where $u_2\\gg J$. Similarly when $u_1$ is close to 1 solutions transitions from $u_3<.6$ to $u_3>.6$, and the sign of the first entry of $(0.6-u_3,0.4-u_1,0.3-u_2)$ changes from positive to negative.  Hence the first entry of the solution starts decreasing (see Fig.~\\ref{fig:threeDimension}, panel (f)), and solutions leaves the subdomain where $1-u_1\\sim  J$  and enter another where $1-u_1\\gg J$.\n\nThe BN corresponding to Eq.~\\eqref{threeNodeMM},  $h=(h_1,h_2,h_3):\\{0,1\\}^3\\rightarrow\\{0,1\\}^3,$ is given by $h(x)=H(0.6-x_3,0.4-x_1,0.3-x_2)$, where $H$ is the Heaviside function acting entry wise on the arguments.\nEq.~\\eqref{threeNodeMM} does not have steady states at the corner subdomains, and neither does the corresponding  BN. A subset of states belong to a periodic orbit of the BN: \n$$ (0,1,1) \\rightarrow (0,1,0)\\rightarrow(1,1,0)\\rightarrow(1,0,0)\\rightarrow(1,0,1)\\rightarrow(0,0,1).\n$$  \nNote that subsequent states in this orbit differ in a single entry. Thus, the transitions between the states have an unambiguous interpretation in the original system: The BN predicts that if the initial condition is in chamber $\\mathcal{C}_{12}^3$, then solutions of  Eq.~\\eqref{threeNodeMM}  will go to chamber $\\mathcal{C}_{1}^{23}$, then to $\\mathcal{C}_{13}^2$, and so on. Indeed, solutions of  Eq.~\\eqref{threeNodeMM}, are attracted to a periodic orbit that transitions between the chambers in this order.  The remaining two states form a period two orbit under synchronous update, $(1,1,1) \\leftrightarrow (0,0,0)$.   Here the BN does not give precise information about the dynamics of the original system. We will show that under certain conditions, orbits of the BN where only entry changes \nat each timestep, correspond to oscillations in the original system.\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width = .95\\textwidth]{3D_timeseries_small}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nComparison of the numerical solution of Eq.~\\eqref{threeNodeMM} (dashed black)\nand the solution of the approximate piecewise linear system (colors) for\ntwo different sets of $J$ and $\\delta$. For (a)-(c) $J = 10^{-2}, \\delta = 0.06$; for\n(e)-(g)  $J = 10^{-4}, \\delta = 0.01$.  The approximate solution changes color when\nswitching between different subdomains.  \nPanel (d) shows the time series for the solutions of Eq.~\\eqref{threeNodeMM} (top) and the PL system (bottom), corresponding to (a)-(c). Panel (h) shows the time series for the solutions of Eq.~\\eqref{threeNodeMM} (top) and the PL system (bottom), corresponding to (e)-(g). \\label{fig:threeDimension}\n}}\n\\end{figure}\n\n\n\n\n\n\n\\section{General reduction  of the model system}\\label{GeneralTheory}\n\nThe approximations described in the previous section can be extended to the more general model given in Eq.~\\eqref{eqn:ProblemEquation}: \n\\begin{align}\\notag\n\t\\frac{du_i}{dt} = A_i\n\\frac{1-u_i}{J_{i}^A+1-u_i}-I_i\\frac{u_i}{J_{i}^I+u_i},\n\\end{align}\nwhere $J_i^A,J_i^I$ are some positive constants.  Here $A_i$ and $I_i$ are\nactivation\/inhibition functions that capture the impact of other variables on\nthe evolution of $u_i$.   The initial conditions are assumed to satisfy $u_i(0)\n\\in [0,1]$ for all $i$.\n\n\n We assume that the activation and inhibition functions are both\naffine~[\\cite{NovakPatakiCilibertoTyson2001,de02}],\n \\begin{equation} \\label{activation\/Inhibition}\n  A_i := \\sum_{j=1}^N{w_{ij}^+u_j} +b_i^+,\n\\quad\n   I_i := \\sum_{j=1}^N{w_{ij}^-u_j} +b_i^-,\n \\end{equation}\t\n where  we use the convention $x^+ = \\text{max}\\{x,0\\}$ and $x^- =\n\\text{max}\\{-x,0\\}$.\nThe  $N\\times N$ matrix, $W = [w_{ij}]$ and the  $N \\times 1$ vector  $b = [\\\nb_1 \\  b_2 \\  ...\\  b_N\\ ]^t$ capture the connectivity and external input to the\nnetwork, respectively. In particular,  $w_{ij}$\ngives  the contribution  of the $j^{\\text{th}}$ variable to the growth rate of\n$i^{\\text{th}}$ variable. If $w_{ij} > 0 $, then $w_{ij}$  appears in the\nactivation function for $u_i$; and if $w_{ij}<0$ then $-w_{ij}$  appears in the\ninhibition function for $u_i$.\nThe intensity of the external input to the $i^{\\text{th}}$ element is $|b_i|$,\nand it contributes to the activation or the inhibition function, depending on\nwhether $b_i > 0$ or $b_i < 0$,  respectively. \n\n\\begin{proposition}\\label{prop:invariantCube}\nThe cube $[0,1]^N$ is invariant for Eq.~\\eqref{eqn:ProblemEquation}.\n\\end{proposition}\n\\begin{proof}\nIt will be enough to show that the vector field at any point on the boundary is not\ndirected outward. Since, $A_i \\geq 0$ and $I_i \\geq 0$, for any $i$,\n\\begin{align*}\n\t\\frac{du_i}{dt}\\bigg|_{u_i = 0} = A_i\\frac{1}{J_{i}^A+1} \\ge 0,\n\\quad\n\\text{and}\n\\quad\n\t\\frac{du_i}{dt}\\bigg|_{u_i = 1} = -I_i\\frac{1}{J_{i}^I+1} \\le 0.\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{The PL approximation}\nTo obtain a solvable reduction of Eq.~\\eqref{eqn:ProblemEquation} we follow the \nprocedure outlined in  Section~\\ref{ExampleProblems}.  We first present the results, and provide the mathematical justification in the next section.  For\nnotational convenience we let $J_i^A = J_i^I = J$, with  $0 < J \\ll 1$.   \nThe general case is equivalent.  We will use  $\\delta = \\delta(J) > 0 $  to define the thickness of the boundary layers.  We start with the subdivision of the\n$N$-dimensional cube, $[0,1]^N$.\n\nLet $T$ and $S$ be two disjoint subsets of  $\\{1,2,...,N\\}$, and let\n\\begin{align} \\label{Rdef} \n\\mathcal{R}^{T}_{S}:=\n\\Big\\{\n(u_1,u_2,...,u_N) \\in [0,1]^N  \\,\\Big|\\,\n u_{s} \t\t <  \\delta \\text{ for all }   s \\in S; \\quad\n    &u_{t}\t \t> 1-\\delta\\text{ for all }   t \\in T; \\notag \\\\\n\\text{and }\n \\delta \\le  &u_k\\le 1-\\delta\n  \\text{ for all }  \\quad k  \\notin S \\cup T\n\\Big \\}.\n\\end{align}\nWe extend the convention used in Table~\\ref{table:2d}, and in Eqs.~(\\ref{R00})\nand (\\ref{R20}) so that  $\\mathcal{R}^{T} :=\\mathcal{R}^{T}_{S}$ when $S$ is\nempty;  $\\mathcal{R}_S :=\\mathcal{R}^{T}_{S}$ when $T$ is empty; and\n $\\mathcal{R} :=\\mathcal{R}^{T}_{S}$  when $T$, $S$ are both  empty.\n\nWithin each subdomain $\\mathcal{R}_S^T$,  Eq.~\\eqref{eqn:ProblemEquation} can be\napproximated by a\n different linear differential--algebraic system. Following the reduction from\nEq.~\\eqref{EquationForToggleSwitch} to Eq.~\\eqref{edgeLinear2d}, for $i \\notin S\n\\cup T$ we obtain the linear system\n\\begin{subequations}\\label{equationInRST}\n\\begin{equation} \n \\frac{du_i}{dt} = \\sum_{j = 1}^N w_{ij}u_j +  b_i. \\label{four}\n\\end{equation}\nFor $s \\in S$ one of the nonlinear terms remains and we  obtain\n\\begin{equation} \n\\frac{du_s}{dt} = \\left(\\sum_{j = 1}^N \nw_{sj}^+u_j +  b_s^+\\right)-\\left(\\sum_{j= 1}^N \nw_{sj}^-u_j +  b_s^-\\right)\\frac{u_s}{J+u_s}, \\label{five}\n\\end{equation}\nwhile for $t \\in T$ we will have\n\\begin{equation} \n\\frac{d u_t}{dt} = \\left(\\sum_{j = 1}^N\nw_{tj}^+u_j + b_t^+\\right)\\frac{1-u_t}{J+1-u_t}\n-\\left(\\sum_{j = 1}^N w_{tj}^-u_j +  b_t^-\\right). \\label{six}\n\\end{equation}\n\\end{subequations}\nEq.~\\eqref{equationInRST} is simpler than Eq.~\\eqref{eqn:ProblemEquation}, but it is\nnot solvable yet.  Following the reduction from Eq.~\\eqref{edgeLinear2d} to\nEq.~\\eqref{smallu},  we now further reduce Eqs.(\\ref{five}--\\ref{six}).\nFirst we use the approximations $u_s \\approx 0$ and $u_t \\approx 1$ in the\nactivation and inhibition functions appearing in Eq.~\\eqref{equationInRST}.\nSecond, we assume\nthat $u_s$ for $s \\in S$ and $u_t$ for $t \\in T$ are in steady state.\n\nUnder these assumptions we obtain the reduction of Eq.~\\eqref{eqn:ProblemEquation}\nwithin any subdomain $\\mathcal{R}_S^T$\n\\begin{subequations}\\label{eqn:mainReduced_RST}\n\\begin{align}\n\\frac{du_i}{dt} &=  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij} \n+b_i\n& i \\notin S \\cup T;  \\label{eqn:mainReduced_RSTa} \\\\\n0 &=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{u_s}{J+u_s},\n& s \\in S; \\label{eqn:mainReduced_RSTb} \\\\\n0 &= -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j + \\sum_{j \\in\nT}w_{tj}^++b_t^+\\right)\\frac{1-u_t}{J+1-u_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n& t \\in T. \\label{eqn:mainReduced_RSTc}\n\\end{align}\n\\end{subequations}\n\n Eq.~(\\ref{eqn:mainReduced_RST}) is  solvable since Eq.~(\\ref{eqn:mainReduced_RSTa}) is\ndecoupled, and   Eqs.(\\ref{eqn:mainReduced_RSTb}) and\n(\\ref{eqn:mainReduced_RSTc}) are solvable for ${u}_s$ and ${u}_t$, respectively, as\nfunctions of the solution of Eq.~(\\ref{eqn:mainReduced_RSTa}).\n\n\nNote that in the singular limit $J=0$ we obtain the $0$-th order approximations:\n\\begin{subequations}\\notag\n\\begin{align}\n\\frac{du_i}{dt} &= \\sum_{j\\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij}+b_i\n& i \\notin S \\cup T;  \\notag \\\\\nu_s &=  0, & s \\in S; \\notag \\\\\nu_t &= 1, & t \\in T. \\notag\n\\end{align}\n\\end{subequations}\n\n\\subsection{Boolean approximation}\n\nTo obtain the Boolean approximation we follow the process described in Section \\ref{ExampleProblems}.  We consider the chambers determined by the complement of the union of the $N$ hyperplanes $\\sum_{j=1}^N w_{ij} u_j +b_i= 0$ (restricted to $[0,1]^N$)  where $i=1,\\ldots, N$. We denote with $\\Omega$ the set of all chambers $\\Omega:=\\{\\mathcal{C}: \\mathcal{C} \\textrm{ is a chamber}\\}$; alternatively, $\\Omega$ is the set of connected components of $[0,1]^N\\setminus \\cup_{i=1}^N\\{u:\\sum_{j=1}^N w_{ij} u_j +b_i=0\\}$. We assume that $\\sum_{j=1}^N w_{ij} x_j +b_i\\neq 0$ for all $i=1,\\ldots, N$ and for all $x\\in\\{0,1\\}^N$. This guarantees that each corner of $[0,1]^N$ belongs to a chamber. The set of parameters excluded by this assumption has measure zero. \n\n\n\nLet $S$ and $T$ be two disjoint subsets of $\\{1, 2, \\ldots, N\\}$ such that $S\\cup T=\\{1, 2, \\ldots, N\\}$ and let $x\\in\\{0,1\\}^N$ be the corner that belongs to the corner subdomain $\\mathcal{R}_S^T$. Note that $x_i=0$ for $i\\in S$ and $x_i=1$ for $i\\in T$. The chamber $\\mathcal{C} \\in \\Omega$ that contains the corner in subdomain $\\mathcal{R}_S^T$\nwill be denoted by  $\\mathcal{C}_S^T$.  We do not name the remaining chambers.\n\nIn general, the chambers can be more complex than in the examples of  Section \\ref{ExampleProblems}. Chambers do not have to be hypercubes,  different corners  may belong to the same chamber, and some chambers may not even contain a corner of $[0,1]^N$, as illustrated in Fig.~\\ref{fig:extra_examples_hyperplanes}.  In the first example, $(0,1)$ and $(1,1)$ belong to the same chamber, that is, $\\mathcal{C}_1^2=\\mathcal{C}^{12}$, and neither  $\\mathcal{C}_{12}$ containing $(0,0),$ nor $\\mathcal{C}_2^1$ containing $(1,0)$ are rectangles. Also, $\\Omega$ has three elements: $\\mathcal{C}_{12}$, $\\mathcal{C}_2^1$, and $\\mathcal{C}_1^2=\\mathcal{C}^{12}$. In the second example, two chambers do not contain any corner of $[0,1]^2$ and are not named. Hence, $\\Omega$ has four elements: $\\mathcal{C}_{12}=\\mathcal{C}_1^2$, $\\mathcal{C}_2^1=\\mathcal{C}^{12}$, and two unnamed chambers that contain no corners. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .5]{extra_examples_hyperplanes}\n\\parbox{.9\\textwidth}{\n\\caption[fig]{ \\footnotesize\nChambers for $W=\\left[\\begin{array}{rr} -1 & -1 \\\\ 0 & -1 \\end{array}\\right]$ and \n$b=\\left[\\begin{array}{r} -0.5 \\\\ -0.6 \\end{array}\\right]$ (left);  \n$W=\\left[\\begin{array}{rr} 2 & -1 \\\\ 2 & 1 \\end{array}\\right]$ and\n$b=\\left[\\begin{array}{r} -0.5 \\\\ -1.5 \\end{array}\\right]$ (right).  \n\\label{fig:extra_examples_hyperplanes}\n}} \n\\end{center}\n\\end{figure}\n\nTo define the BN, $h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow \\{0,1\\}^N$ at $x\\in\\{0,1\\}^N$, we need to find the signs of the components of the vector field $Wu+b$ on the chamber that contains $x$. Consider  $x \\in \\mathcal{R}_S^T$. Within $\\mathcal{C}_S^T$ the signs of the components of $Wu+b$ do not change and are equal to the signs of the components of $Wx+b$. If the sign of the $i$-th component is negative, we let $h_i(x)=0,$ and if the sign is positive we let $h_i(x)=1$. In general, we can write \n\\begin{equation}\\label{eqn:mainReduced_BN}\nh_i(x) =  H\\left(\\sum_{j=1}^N w_{ij}x_j+b_i\\right),   \\qquad \\textrm{ or in vector form}  \\qquad\nh(x)=H(Wx+b).\n\\end{equation}\nHence the value of the BN at a corner $x\\in\\mathcal{R}_S^T$ is given by the  Heaviside function, applied entrywise  to $Wu+b$. Note that corners that are in the same chamber get mapped to the same point. \n\n\nImportantly, using Eq.~\\eqref{eqn:mainReduced_BN} we can compute the BN directly from Eq.~\\eqref{eqn:ProblemEquation}. For example, for Eq.~\\eqref{EquationForToggleSwitch} we have  $h(x_1,x_2)=H(0.5-x_2,0.5-x_1)$; and for  Eq.~\\eqref{threeNodeMM} we have $h(x_1,x_2,x_3)=H(0.6-x_3,0.4-x_1,0.3-x_2)$.\n\nBelow we show that up to a set of small measure, the BN preserves information about the steady states of the original system.  We will also show that under some conditions, ``regular'' trajectories of a BN correspond to trajectories in the original system.\n\n\\section{Mathematical justification}\\label{sec:math_justification}\n\n We next justify the different approximations made above: In Section \\ref{sec:math_PL} we use Geometric Singular Perturbation Theory (GSPT) to justify the PL approximation. In Section \\ref{sec:math_BN} we show that steady state information is preserved by the BN and that, under certain conditions, the BN also provides qualitative information about the global dynamics of the original system.\n\n\\subsection{Piecewise linear approximation}\\label{sec:math_PL}\n\nTo obtain the reduced equations at the boundary of $[0,1]^N$, we define the following rescaled variables \n\\begin{equation} \\label{seven}\n \\tilde{u}_s := \\frac{u_s}{J}  \\quad   \\text{ for }  s \\in S, \\text{ and } \n\\qquad\n \\tilde{u}_t  := \\frac{1-u_t}{J} \\,\\,\\quad \\text{ for }  t \\in T.\n\\end{equation}\nUsing Eq.~ (\\ref{seven}) in Eq.~(\\ref{equationInRST}) we get for $i \\notin S\n\\cup T$\n\\begin{subequations}\\label{equationInRST_scaled}\n\\begin{equation} \\label{eight}\n\\frac{du_i}{dt} =  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij}\n  + J\\left(\\sum_{s \\in S}w_{is}\\tilde{u}_s-\\sum_{t \\in\nT}w_{it}\\tilde{u}_t\\right) +b_i,\n\\end{equation}\nand for $s \\in S$,\n\\begin{align} \\label{nine}\nJ\\frac{d\\tilde{u}_s}{dt} = & \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in\nT}w_{st}^+\n  +J\\left(\\sum_{j \\in S}w_{sj}^+\\tilde{u}_j-\\sum_{t \\in\nT}w_{st}^+\\tilde{u}_t\\right)+b_s^+ \\nonumber\\\\\n&-\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s}\n  -J\\left(\\sum_{j \\in S}w_{sj}^+\\tilde{u}_j-\\sum_{t \\in\nT}w_{st}^+\\tilde{u}_t5\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n\\end{align}\nand similarly, for $t \\in T$,\n\\begin{align} \\label{ten}\nJ\\frac{d\\tilde{u}_t}{dt} = &-\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j +\n\\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n -J\\left(\\sum_{s \\in S}w_{ts}^+\\tilde{u}_s-\\sum_{j \\in\nT}w_{tj}^+\\tilde{u}_j\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t} \\nonumber \\\\\n&+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-\n+J\\left(\\sum_{s \\in S}w_{ts}^+\\tilde{u}_s-\\sum_{j \\in\nT}w_{tj}^+\\tilde{u}_j\\right).\n\\end{align}\n\\end{subequations}\n\n\nWhen $J$ is small, we can apply Geometric Singular Perturbation Theory (GSPT) to Eq. (\\ref{equationInRST_scaled}) ~[\\cite{Hek2010,Kaper1998}].  The GSPT posits that, under a normal hyperbolicity condition which we verify below, Eq.~(\\ref{equationInRST_scaled}) can be further simplified by assuming that $J = 0$. \nThis yields a \ndifferential-algebraic system\n\\begin{subequations}\\label{mainReduced}\n\\begin{align}\n\\frac{du_i}{dt} &=  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij} \n+b_i,\n& i \\notin S \\cup T;  \\label{eleven} \\\\\n0 &=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n& s \\in S; \\label{twelve} \\\\\n0 &= -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j + \\sum_{j \\in\nT}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n& t \\in T. \\label{thirteen}\n\\end{align}\n\\end{subequations}\nwhich is equivalent to Eq.~\\eqref{eqn:mainReduced_RST} after rescaling.  This\nconclusion\nwill be justified if the manifold defined by Eqs.~(\\ref{twelve}) and\n(\\ref{thirteen}) is normally hyperbolic and\nstable~[\\cite{Fenichel1979,Kaper1998,Hek2010}].  We verify this condition next.\n\nLet $ \\hat{u} =  \\{ u_{i_1},...,u_{i_m} \\}$ where $\\{ i_1,...,i_m \\} = \\{\n1,2,...,N \\}\\backslash (S \\cup T) $, be the coordinates of $u$ which are away\nfrom the boundary, and denote the right hand side of Eq.~(\\ref{twelve}) by\n$F_s(\\hat{u}, \\tilde{u}_{i_s})$, for all  $s \\in S$, so that\n\\begin{align*}\n F_s(\\hat{u}, \\tilde{u}_{i_s}) :=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t\n\\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n\\end{align*}\nand\n\\begin{align*}\n \\frac{\\partial F_s}{\\partial \\tilde{u}_{i_s}} =  -\\left( \\sum_{j \\notin S \\cup\nT }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- + b_s^-\\right) \\left(\n\\frac{1}{1+\\tilde{u}_s}\\right)^2 \n< 0,\n\\end{align*}\nfor all $s \\in S$.\nSimilarly, by denoting the right hand side of Eq.~(\\ref{thirteen}) by\n$G_t(\\hat{u}, \\tilde{u}_{i_t})$, for all  $t \\in T$. \\emph{i.e.}\n\\begin{align*}\n G_t(\\hat{u}, \\tilde{u}_{i_t}) := -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j +\n\\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n\\end{align*}\nwe see that\n\\begin{align*}\n \\frac{\\partial G_t}{\\partial \\tilde{u}_{i_t}} = -\\left( \\sum_{j \\notin S \\cup T\n}w_{tj}^+u_j + \\sum_{j \\in T}w_{tj}^++b_t^+\\right) \\left(\n\\frac{\\tilde{u}_t}{1+\\tilde{u}_t} \\right)^2\n< 0.\n\\end{align*}\nHence, the manifold defined by Eqs.~(\\ref{twelve}) and (\\ref{thirteen}) is\nnormally hyperbolic and stable.  This completes the proof that the reduction of\nthe non-linear system~(\\ref{eqn:ProblemEquation}) to the solvable system given in Eq.~(\\ref{eqn:mainReduced_RST}) is justified for small $J$.\n\n\\subsection{Boolean approximation}\\label{sec:math_BN}\n\nHere we formally show that the steady states of the BN given in Eq.~\\eqref{eqn:mainReduced_BN} are in a one-to-one correspondence with the steady states of the system  given in Eq.~\\eqref{eqn:ProblemEquation}. We also show that under some conditions trajectories in the BN correspond to trajectories of the  system  Eq.~\\eqref{eqn:ProblemEquation}.\n\n\\subsubsection{Steady state equivalence at the corner subdomains}\\label{sec:math_BN_ss_corner}\n\nFirst we prove the one-to-one correspondence only at the corner subdomains using the PL approximation. We do this by showing that Eq.~\\eqref{eqn:mainReduced_RST} has a steady state at a corner subdomain $\\mathcal{R}_S^T$ if and only if the BN has a steady state at the corner $x\\in\\{0,1\\}^N$ contained in $\\mathcal{R}_S^T$.\n\nWe proceed from Eq.~(\\ref{eqn:mainReduced_RST}) for a corner subdomain $\\mathcal{R}_S^T$ so that $S\\cup T=\\{1,\\ldots,N\\}$. We obtain the equations\n\n\\[0=\\sum_{t \\in T}w_{st}^+  +b_s^+ -\\left( \\sum_{t \\in T}w_{st}^- +b_s^-\\right)\\frac{u_s}{J+u_s},\\qquad  s \\in S \\] and \n\\[0= -\\left( \\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{1-u_t}{J+1-u_t} +\\sum_{j \\in T}w_{tj}^- + b_t^-, \\qquad  t \\in T.\\]\n\nFor the sets $S$ and $T$, consider $x\\in\\{0,1\\}^N$ such that $x_s=0$ for all $s\\in S$ and $x_t=1$ for all $t\\in T$. Then, we can write the equations above in a more compact form\n\n\\[0=A_s(x) -I_s(x)\\frac{u_s}{J+u_s}, \\quad s \\in S, \\qquad \\text{and,} \\qquad\n0=-A_t(x)\\frac{1-u_t}{J+1-u_t} +I_t(x), \\quad t \\in T.\\]\n\nSolving these equations for $u_s$ and $u_t$, respectively, we obtain\n\\begin{equation} \\label{E:fp_equation}\nu_s=-\\frac{A_s(x)J}{A_s(x)-I_s(x)}, \\quad s \\in S, \\qquad  \\text{and,}  \n\\qquad \nu_t=1-\\frac{I_t(x)J}{A_t(x)-I_t(x)}, \\quad t \\in T.\n\\end{equation}\n\nNow, let $\\epsilon>0$ small such that $\\left|\\frac{I_t(x)J}{A_t(x)-I_t(x)}\\right|, \\left| \\frac{I_t(x)J}{A_t(x)-I_t(x)} \\right|\\leq 1$. \nFor all $J$ such that $0<J<\\epsilon$, Eq.~\\eqref{E:fp_equation} has a solution inside $[0,1]^N$ if and only if \n$A_s(x)-I_s(x)<0, s \\in S,$ and $A_t(x)-I_t(x)>0, t \\in T$,\nor more compactly if and only if\n$H(A_i(x)-I_i(x))=x_i \\textrm{ for all } i=1,\\ldots,N.$\nThus, a steady state appears in the corner subdomain corresponding to $x$ if and only if $x$ is a steady state of the BN $h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow\\{0,1\\}^N$ given by Eq.~\\eqref{eqn:mainReduced_BN}. \nWe have proved,\n\n\\begin{theorem}\\label{thm:localss}\nThere is an $\\epsilon>0$ such that for all $0<J<\\epsilon$,  Eq.~\\eqref{eqn:mainReduced_RST} has a steady state at a corner subdomain containing $x\\in\\{0,1\\}^N$ if and only if the BN in Eq.~\\eqref{eqn:mainReduced_BN} has a steady state at $x$.\n\\end{theorem}\n\nAlso, since we showed that the PL system given by  Eq.~\\eqref{eqn:mainReduced_RST} is a valid approximation of the full system given in Eq.~\\eqref{eqn:ProblemEquation}, we have the following corollary.\n\n\\begin{corollary}\\label{cor:localss}\nThere is an $\\epsilon>0$ such that for all $0<J<\\epsilon$, the system in Eq.~\\eqref{eqn:ProblemEquation} has a steady state at a corner subdomain containing $x\\in\\{0,1\\}^N$ if and only if the BN in Eq.~\\eqref{eqn:mainReduced_BN} has a steady state at $x$.\n\\end{corollary}\n\n\n\n\\subsubsection{Global equivalence of steady states}\n\\label{sec:math_BN_ss_global}\n\nWe proved that, for $J$ small, the steady states at the corner subdomains of the system in Eq.~(\\ref{eqn:ProblemEquation}) are in a one-to-one correspondence with the steady states of the BN. However, the corner subdomains only cover a small portion of $[0,1]^N$.  We next show that the steady state correspondence is global.\n\nRecall that $\\Omega=\\{\\mathcal{C}: \\mathcal{C} \\textrm{ is a chamber}\\}$ and that  each chamber is  a connected component of $[0,1]^N\\setminus \\cup_{i=1}^N\\{u:\\sum_{j=1}^N w_{ij} u_j +b_i=0\\}$. The chambers are bounded convex open subsets of $[0,1]^N$ (with the topology inherited from $[0,1]^N$). Also note that $[0,1]^N\\setminus \\cup_{\\mathcal{C} \\in\\Omega}\\mathcal{C}$ is a finite union of hyperplanes and hence has measure zero. We denote by $\\mathcal{C}(x)$ the chamber that contains the point $x \\in [0,1]^N$. \n\n\n\\begin{theorem}\\label{thm:one2one}\nLet $K$ be a compact subset of $\\cup_{\\mathcal{C}\\in\\Omega}\\mathcal{C}$ such that $K\\cap \\mathcal{C}$ is convex for any $\\mathcal{C}\\in\\Omega$, and such that $K\\cap \\mathcal{C}(x)$ contains a neighborhood of $x$ for each $x\\in \\{0,1\\}^N$. \n\nThen, there is an $\\epsilon_K>0$ such that for all $0<J<\\epsilon_K$,  \nif $x^*\\in\\{0,1\\}^N$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, then the ODE in Eq.~\\eqref{eqn:ProblemEquation} has a steady state $u^*\\in \\mathcal{C}(x^*)$; and if $u^*\\in K$ is a steady state of the ODE, then $u^*\\in \\mathcal{C}(x^*)$ for some steady state $x^*$ of the BN. \n\nFurthermore, if $x^*$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, then the steady state of the ODE in Eq.~\\eqref{eqn:ProblemEquation} is unique in $K\\cap \\mathcal{C}(x^*)$, converges to $x^*$ as $J\\rightarrow 0$, and is asymptotically stable.\n\\end{theorem}\n\\begin{proof}\nSee Appendix.\n\\end{proof} \n\nWe can make the set $K$ in Theorem \\ref{thm:one2one} as close to $[0,1]^N$ as desired. For example, for each chamber $\\mathcal{C}$ and for $r>0$, denote $K_\\mathcal{C}:=\\{u\\in\\mathcal{C}:|\\sum_{j=1}^N w_{ij}u_j+b_i|\\geq r, \\forall i\\}$. By using $r$ small, and denoting Lebesgue measure by $\\mu$, we can make $\\mu([0,1]^N\\setminus \\cup_{\\mathcal{C}\\in\\Omega}K_{\\mathcal{C}})=\\mu( \\cup_{\\mathcal{C}\\in\\Omega} (\\mathcal{C}\\setminus K_{\\mathcal{C}}))= \\sum_{\\mathcal{C}\\in\\Omega} \\mu(\\mathcal{C}\\setminus K_{\\mathcal{C}})$  as small as desired. Hence, we have the following corollary.\n\n\n\\begin{corollary}\\label{cor:one2one}\nFor any $\\epsilon>0$, there is a set $K\\subseteq [0,1]^N$ satisfying $\\mu([0,1]^N\\setminus K)<\\epsilon$ and a number $\\epsilon_K$ such that for all $0<J<\\epsilon_K$, there is a one-to-one correspondence between the steady states of the BN in Eq.~\\eqref{eqn:mainReduced_BN} and the steady states in $K$ of the ODE in Eq.~\\eqref{eqn:ProblemEquation}. Furthermore, the steady states of the ODE in $K$ are asymptotically stable.\n\\end{corollary}\n\n \nNote that the set $K$ does not include the nullclines.  Hence,  steady states  outside $K$ are possible, and they could be stable. Such steady states can be studied using the PL approximation in Eq.~\\eqref{eqn:mainReduced_RSTa}.\n\n\\subsubsection{Equivalence of trajectories}\n\\label{sec:math_BN_trajectories}\n\nWe next examine under which conditions  the trajectories of the BN in Eq.~(\\ref{eqn:mainReduced_BN}) correspond trajectories of\nthe ODE given in Eq.~(\\ref{eqn:ProblemEquation}). The main assumption in the rest of this section is that the hyperplanes divide $[0,1]^N$ into $2^N$ chambers, and each chamber contains a corner. \n\nWe say that the solutions of the system in Eq.~(\\ref{eqn:ProblemEquation})  \\emph{transition from $K\\subseteq[0,1]^N$ to $K'\\subseteq[0,1]^N$} if for any solution of the system, $u(t)$, with initial condition $u(0)\\in K$, there exists $\\hat{t}$ such that $u(\\hat{t})\\in K'$. The \\emph{Hamming distance} between $x,y\\in\\{0,1\\}^N$ is defined as $d(x,y):=|\\{i\\in\\{1,\\ldots,N\\}:x_i\\neq y_i\\}|$. We denote a transition $h(x)=y$ (i.e. $H(Wx+b)=y$) with $x\\rightarrow y$. A trajectory is a sequence of transitions and is denoted similarly.\nWe call a transition $x\\rightarrow y$ \\emph{regular} if either (1) $x=y$ or (2) $d(x,y)=1$ and $h_j(y)=y_j$ for the index $j$ for which $x_i\\neq y_i$. \nIn other words, a transition  $x\\rightarrow y$ in the BN is regular if $x$ is a steady state or if $x$ transitions to $y$ by changing only one coordinate and this coordinate does not change back when transitioning from $y$ to $h(y)$. For example, if $000\\rightarrow 100\\rightarrow 111$, then $000\\rightarrow 100$ is a regular transition ($j=1$); on the other hand, if $000\\rightarrow 100\\rightarrow 010$, then $000\\rightarrow 100$ is not a regular transition. A trajectory is regular if each component transition is regular.\n\n\n\\begin{theorem}\\label{thm:transition}\nSuppose that the hyperplanes divide $[0,1]^N$ into $2^N$ chambers and consider a regular transition of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, $x\\rightarrow h(x)$. Then, there is a neighborhood $K$ of $x$, and an $\\epsilon_K>0$ such that for all $0<J<\\epsilon_K$ the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(h(x))$. Also, if $x$ is a steady state, the neighborhood $K$ can be chosen to be invariant.\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\nNext for a steady state $x$ define $B_1(x)=\\{y\\in\\{0,1\\}^N:h(y)=x\\textrm{ and } d(x,y)\\leq 1\\}$; that is, $B_1(x)$ is the set of states in the basin of attraction of $x$ with Hamming distance at most 1 from $x$. The following corollary of Theorem \\ref{thm:transition} states that this part of the basin of attraction of $x$ corresponds to part of the basin of attraction of the steady state in the ODE in Eq.~(\\ref{eqn:ProblemEquation})  corresponding to $x$.\n\\begin{corollary}\nSuppose that $x$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}. Then, for every $y\\in B_1(x)$, there is a neighborhood $K$ of $y$  and $\\epsilon_K>0$, such that for all $0<J<\\epsilon_K$  the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to  $\\mathcal{C}(x)$.\n\\end{corollary}\n\n\nNote that Theorem \\ref{thm:transition} implies that for a regular trajectory of the BN in Eq.~(\\ref{eqn:mainReduced_BN}),  $x\\rightarrow h(x)\\rightarrow h^2(x)\\rightarrow \\ldots \\rightarrow h^m(x)$, the solutions of the ODE in Eq.~(\\ref{eqn:ProblemEquation}) will transition from a neighborhood of $x$ to $\\mathcal{C}(h(x))$, from a neighborhood of $h(x)$ to $\\mathcal{C}(h^2(x))$ and so on. \nTo guarantee that a neighborhood of $x$ will reach a neighborhood of $h^m(x)$ (that is, to guarantee that the result is ``transitive''), we need the additional assumption that each hyperplane is orthogonal to some coordinate axis. Note that the example given in Section \\ref{sec:3nodes} satisfies this condition.\n\n\\begin{theorem}\\label{thm:trajectory}\nSuppose that  each hyperplane is orthogonal to some coordinate axis and let  $x\\rightarrow h(x)\\rightarrow \\ldots\\rightarrow h^m(x)$ be a regular trajectory of the BN in Eq.~(\\ref{eqn:mainReduced_BN}). Then, for any compact set $K\\subset \\mathcal{C}(x)$ there is $\\epsilon_{K}>0$ such that for all $0<J<\\epsilon_{K}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(h^m(x))$ following the order of the regular trajectory.\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\nFor a steady state of the BN define $B(x)=\\{y:\\textrm{ there is a regular trajectory from $y$ to $x$}\\}$. The following corollary of Theorem~\\ref{thm:trajectory} implies that some states in the basin of attraction of a steady state of the BN in Eq.~(\\ref{eqn:mainReduced_BN}) correspond to chambers in the basin of attraction of a steady state of the ODE in Eq.~(\\ref{eqn:ProblemEquation}).\n\n\n\\begin{corollary} \nSuppose that  each hyperplane is orthogonal to some coordinate axis and let $x$ be a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}. Consider $y\\in B(x)$. Then, for any compact set $K\\subseteq\\mathcal{C}(y)$, there exists $\\epsilon_{K}>0$ such that for all $0<J<\\epsilon_{K}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(x)$.\n\\end{corollary}\n\nSimilarly, we obtain the following corollary for oscillatory behavior. \n\\begin{corollary}\\label{cor:oscilation}\nSuppose that  each hyperplane is orthogonal to some coordinate axis and let  $x^1\\rightarrow x^2\\rightarrow \\ldots\\rightarrow x^p\\rightarrow x^1$ be a regular periodic orbit of the BN in Eq.~(\\ref{eqn:mainReduced_BN}). Then, for any compact set $K\\subseteq\\mathcal{C}(x^1)$ and any positive integer $m$, there exists $\\epsilon_{K,m}>0$ such that for all $0<J<\\epsilon_{K,m}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition between the chambers (starting at $K$) in the order $\\mathcal{C}(x^1), \\mathcal{C}(x^2),\\ldots,\\mathcal{C}(x^p),\\mathcal{C}(x^1)$, $m$ times.\n\\end{corollary}\n\n\nNote that the example in Section \\ref{sec:3nodes} satisfies the hypothesis of this last corollary. In general, Corollary \\ref{cor:oscilation} does not guarantee that the solution is periodic.\n\n\nFinally, we note that the requirement that there are $2^N$ chambers, each containing a corner is necessary.  Even if we have a transition where only one variable changes (e.g. $h(1,0)=(0,0)$), having an intermediate chamber can change the behavior of the solutions before they reach the chamber predicted by the BN. In the example shown in Fig.~\\ref{fig:twoDimension_counterexample_regular} the signs of the vector field of the approximating linear system imply that the BN transitions from $(1,0)$ to $(0,0)$.  However, solutions can transition from the chamber that contains $(1,0)$ to the bottom middle chamber and never reach the chamber that contains $(0,0)$. In summary, even having a transition where only one variable changes may not be sufficient to guarantee that the Boolean transition corresponds to a similar transition in the original system.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .35]{twoDimension_counterexample_regular}\n\\parbox{.9\\textwidth}{\n\\caption[]{\\footnotesize\nChambers and signs of vector field for the linear system given by $W=\\left[\\begin{array}{rr} -2 & -1 \\\\ -2 & 1 \\end{array}\\right]$ and $b=\\left[\\begin{array}{r} 1.5 \\\\ 0.5 \\end{array}\\right]$.\n\\label{fig:twoDimension_counterexample_regular}}\n}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion}\\label{Discussion}\n\nModels of biological systems are frequently nonlinear and difficult to analyze mathematically. In addition, accurate models \nfrequently contain numerous parameters whose exact values are not known. Thus, studying which parameters have a large impact on dynamics, and how a  model can be simplified, is crucial in finding the features of biological systems that determine their behavior and responses. Reduction techniques that preserve key dynamical properties are essential in this endeavor.\n\nWe studied a special class of non-linear differential equation  models of biological networks where interactions between nodes are described using Hill functions. When the Michaelis-Menten constants are sufficiently small, the behavior of the  system is captured by an approximate piecewise linear system and a Boolean Network. \nIn this case the domain of the full system naturally decomposes into nested hypercubes.  These hypercubes define subdomains within which a solvable linear--algebraic system approximates the original system. The Boolean Network is obtained from a decomposition of the domain into chambers and describes how solutions evolve between them. \n\n\nThe proposed reductions have a  number of advantages: The piecewise linear approximation is not only  easier to solve than the original system analytically, but also numerically (the original system becomes stiff for small $J$). When  one is interested in qualitative behavior such as steady state analysis, the Boolean approximation can be very useful, especially when the dimension of the system is large. Also, the Boolean framework has been used to model many biological systems where it is assumed that interactions are switch-like and variables can be discretized. It is therefore important to know when, and in what sense such reduced systems can be justified, and in particular, what dynamical properties of the full system are capture by a reduction. \n\nAlthough the case of large exponent $n$ in the Hill function  has been studied in the past [\\cite{GlassKauffman1973,Snoussi1989, thomasbook,mendoza2006, DavitichBornholdt2008, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}], the case of small $J$ has been studied only recently and heuristically [\\cite{DavitichBornholdt2008}]. In this manuscript we have shown that the PL and the BN approximations are also valid for the case of small $J$, and have given explicit formulas for their computation. The BN approximation preserves steady state behavior and under further restrictions, it can also be used to  infer the basins of attractions and oscillations in the original system.  Note that the Boolean functions in the Boolean approximation are threshold functions, as used in earlier models [\\cite{Li_cc_2004,DavitichBornholdt2008b}].  Our results show that such BN scan indeed appear when  approximating more detailed models, such as those described by Eq.~\\eqref{eqn:ProblemEquation}. \nIn summary, our results for the limit $J\\rightarrow 0$ complement previous results for the limit $n\\rightarrow \\infty$,  providing a useful framework for reducing nonlinear systems to PL systems and BNs.\n\n\n\nA potential limitation in our arguments is that we have an approximation \nvalid only in an asymptotic limit.  It is unknown when and how the approximation breaks down.\nHowever, the approximation is still valid as $J$ increases until we reach a bifurcation point, which can happen for relative large $J$ or for values of $J$ that are biologically relevant. Also, we have not provided a systematic relationship between the thickness of the boundary, $\\delta$, and the Michaelis-Menten constant, $J$. Numerical tests suggest that the relationship is between $J = \\mathcal{O}(\\delta)$ and $J = \\mathcal{O}(\\delta^2)$. Another limitation of our analysis is that, in the general case, it is not known when the transitions or cycles in a BN will correspond to similar transitions or cycles in the original ODE.\n\n\\section*{Acknowledgment}\n\nWe thank Matthew Bennett for help in preparing the manuscript. This work was funded by the NIH, through the joint NSF\/NIGMS Mathematical Biology Program Grant No. R01GM104974.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nTorsion pairs in abelian categories were introduced by Dickson~\\cite{dickson66}\n(see the introduction to~\\cite{br} for further details). They play an important\nrole in the study of localisation (see e.g.~\\cite{br}) and in tilting theory\n(see e.g.~\\cite{ass,br}). Also, torsion pairs in the case of triangulated\ncategories, as defined in~\\cite[2.2]{iy}, have been considered recently by a\nnumber of authors, e.g.~\\cite{hjr,iy,kz,nakaoka,ng,zz}.\n\nThe main object of study of this article is the collection of torsion pairs in\na tube category, which is a hereditary abelian category.\nSuch categories arise as full subcategories of module\ncategories over tame hereditary algebras, and are so-called because their\nAuslander-Reiten quivers have the shape of a tube.\n\nA geometric model for tube categories has been given in~\\cite{bama,warkentin}\n(see also~\\cite{bz}). The indecomposable objects are parametrized by a\ncollection of oriented arcs in an annulus with $n$ marked points on one of its\nboundary components. The dimension of the $\\operatorname{Ext}$-group between two\nindecomposable objects coincides with the negative intersection number of\nthe corresponding pair of arcs.\n\nIn this article, we classify the torsion pairs in a tube category $\\mathbf T$. We build\non results in~\\cite{bk1} giving a bijection between torsion pairs in a tube category and\ncotilting objects in the category obtained by taking arbitrary direct limits of modules in the tube.\nWe show that all torsion pairs in the tube category arise in this way or via a dual construction.\nThus they are in bijection with maximal rigid objects in the category $\\overline{\\mathbf T}$\nobtained from the tube by taking arbitrary direct or inverse limits of objects in the tube.\nWe give an explicit description of this bijection and its inverse.\n\nWe further show that the geometric model referred to above can be extended to the indecomposable\nobjects in $\\overline{\\mathbf T}$, i.e.\\ to include Pr\\\"{u}fer and adic modules associated to the tube. These\nextra objects are represented by certain infinite arcs in the annulus which spiral in towards the\ninner boundary component. The result concerning the dimensions of $\\operatorname{Ext}$-groups extends to this case also.\nThis enables us to give a characterisation of maximal rigid objects in $\\mathbf T$ in the geometric model. We also\ngive a characterisation of torsion\npairs in the geometric model in terms of certain closure properties corresponding to closure properties\nof the subcategories in a torsion pair. In particular, the collection of arcs corresponding to a\nsubcategory in a torsion pair must form an \\emph{oriented Ptolemy diagram}, an oriented version in the\nannulus case of the Ptolemy diagrams appearing in~\\cite{hjr,ng}, as well as satisfying additional criteria.\nWe also give a geometric description of the above bijection and its inverse.\n\nIn order to give the geometric interpretation, we first give a similar model\nfor the linearly oriented quiver in type A\n(note that M. Warkentin~\\cite{warkentin} also suggests such a model) and\nshow how to interpret tilting modules and torsion pairs in this model.\n\nWe remark that, in independent work, T. Holm, P. J\\o rgensen and\nM. Rubey~\\cite{holmjorgensenrubey} have classified the torsion pairs in the\ncluster category associated to a tube (as opposed to the tube itself,\nwhich we study here). The torsion pairs in the cluster case are different,\nalthough we note that unoriented Ptolemy diagrams play a role in the cluster\ntube case, while oriented Ptolemy diagrams appear here.\n\nIn Section 1, we recall the definition and some of the properties of torsion pairs in abelian categories.\nIn Section 2, we discuss the type A case. In Section 3, we recall the geometric model of the tube and show how it can be extended to include Pr\\\"{u}fer and\nadic modules. We also discuss a certain reflection map (from~\\cite{bk1})\non the indecomposable objects of the tube, and its properties.\nIn Section 4 we classify the torsion pairs in the tube and prove that\nthey are in bijection with maximal rigid objects in $\\overline{\\mathbf T}$,\ngiving an explicit description of the bijection and its inverse.\nIn Section 5, we give a geometric interpretation of maximal rigid objects\nin $\\mathbf T$ and torsion pairs in $\\mathbf T$ and the bijection between them.\n\n\\section{Preliminaries} \\label{s:preliminaries}\n\nWe shall adopt the convention throughout that all subcategories considered\nare strictly full (i.e.\\ full and closed under isomorphism). We shall also\nconsider modules up to isomorphism.\nLet $\\mathcal A$ be an abelian category. If $\\mathcal X$ is a subcategory of $\\mathcal A$,\nwe define ${}^{{\\perp}_{\\scriptscriptstyle H}} \\mathcal X$ (respectively, ${}^{{\\perp}_{\\scriptscriptstyle E}} \\mathcal X$) to be\nthe additive subcategory of $\\mathcal A$ consisting of the objects\n$Y$ satisfying $\\operatorname{Hom}_{\\mathcal A}(Y,X)=0$ (respectively,\n$\\operatorname{Ext}^1_{\\mathcal A}(Y,X)$=0) for all $X$ in $\\mathcal X$.\nFor an object $X$ of $\\mathcal A$ we define ${}^{{\\perp}_{\\scriptscriptstyle H}}\nX={}^{{\\perp}_{\\scriptscriptstyle H}} \\operatorname{add} X$, and ${}^{{\\perp}_{\\scriptscriptstyle E}} X={}^{{\\perp}_{\\scriptscriptstyle E}} \\operatorname{add} X$,\nwhere $\\operatorname{add} X$ denotes the additive subcategory of $\\mathcal A$ whose\nobjects are finite direct sums of direct summands of $X$. We\nsimilarly define $\\mathcal X^{{\\perp}_{\\scriptscriptstyle H}}$, $\\mathcal X^{{\\perp}_{\\scriptscriptstyle E}}$, $X^{{\\perp}_{\\scriptscriptstyle H}}$ and\n$X^{{\\perp}_{\\scriptscriptstyle E}}$. For additive subcategories $\\mathcal X_i$, $i=1,2, \\ldots ,m$,\nof $\\mathcal A$, we write $\\coprod_{i=1}^m \\mathcal X_i$ for the smallest additive\nsubcategory of $\\mathcal A$ containing the $\\mathcal X_i$.\n\nIf $\\mathcal X$ is an additive subcategory of $\\mathcal A$, we write $\\operatorname{Gen} \\mathcal X$ for the\nsubcategory consisting of objects which are quotients of objects of $\\mathcal X$ and,\nfor an object $X$, $\\operatorname{Gen} X=\\operatorname{Gen}(\\operatorname{add} X)$. Similarly, we write $\\operatorname{Cogen} \\mathcal X$ for\nthe subcategory consisting of objects which are subobjects of objects of $\\mathcal X$\nand $\\operatorname{Cogen} X=\\operatorname{Cogen}(\\operatorname{add} X)$. We write $\\operatorname{ind} \\mathcal X$ for the subcategory of $\\mathcal X$\nwhose objects are the indecomposable objects in $\\mathcal X$.\n\nWe next recall some of the theory of torsion pairs. Recall that\na pair $(\\mathcal T,\\mathcal F)$ of subcategories of\n$\\mathcal A$ is said to be a \\emph{torsion pair} \\cite[Sect.\\ 1]{dickson66}\nif $\\operatorname{Hom}(T,F)=0$ for all objects $T$ in $\\mathcal T$ and $F$ in $\\mathcal F$ and\nfor every object $A$ in $\\mathcal A$ there is a short exact sequence\n\\begin{equation}\n0 \\rightarrow {T} \\rightarrow A \\rightarrow {F} \\rightarrow 0\n\\label{e:linkingsequence}\n\\end{equation}\nwith $T$ in $\\mathcal T$ and $F$ in $\\mathcal F$.\nWe say that $\\mathcal T$ is the \\emph{torsion} part and $\\mathcal F$ is the\n\\emph{torsion-free} part of the pair.\n\nRecall that an abelian category $\\mathcal A$ is said to be \\emph{finite\nlength} if it is skeletally small and every object in it is artinian\nand noetherian. Equivalently, it is skeletally small and every object\nhas a finite length composition series.\n\nThus, for example, the category $\\operatorname{mod} A$ of finite dimensional modules over a finite-dimensional algebra is a finite length category.\nParts (a), (b) of the following can be proved using arguments as\nin~\\cite[Thm.\\ 2.1]{dickson66} (see also~\\cite[Lemma 1.7]{bk1}); part (c) is\na well-known corollary.\n\n\\begin{lemma} \\label{l:tpclosure}\nSuppose that $\\mathcal A$ is a finite length abelian category.\n\\begin{enumerate}\n\\item[(a)]\nLet $\\mathcal T$ be a subcategory of $\\mathcal A$ and let $\\mathcal F=\\mathcal T^{{\\perp}_{\\scriptscriptstyle H}}$.\nThen $(\\mathcal T,\\mathcal F)$ is a torsion pair in $\\mathcal A$ if and only if\n$\\mathcal T$ is closed under quotients and extensions.\n\\item[(b)]\nLet $\\mathcal F$ be a subcategory of $\\mathcal A$ and let $\\mathcal T={}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F$.\nThen $(\\mathcal T,\\mathcal F)$ is a torsion pair in $\\mathcal A$ if and only if $\\mathcal F$ is closed under\nsubobjects and extensions.\n\\item[(c)] A pair $(\\mathcal T,\\mathcal F)$\nof subcategories of $\\mathcal A$ is a torsion pair if and only if\n$\\mathcal T^{{\\perp}_{\\scriptscriptstyle H}}=\\mathcal F$ and ${}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F=\\mathcal T$.\n\\end{enumerate}\n\\end{lemma}\n\nRecall that a finite length abelian category is said to be \\emph{serial} if\nevery object is a direct sum of indecomposable uniserial objects. We recall\nthe following well-known result.\n\n\\begin{lemma} \\label{l:uniserialclosure}\nLet $\\mathcal A$ be a serial finite length abelian category. Then\n\\begin{enumerate}\n\\item[(a)] A morphism from an indecomposable object of $\\mathcal A$ to a direct sum of indecomposable\nobjects is a monomorphism if and only if at least one of its components is a monomorphism.\n\\item[(b)] Let $\\mathcal X$ be a subcategory of $\\operatorname{ind} \\mathcal A$.\nThen we have $\\operatorname{ind} \\operatorname{Gen}(\\operatorname{add} \\mathcal X)=\\operatorname{ind} \\operatorname{Gen} \\mathcal X$ and $\\operatorname{ind} \\operatorname{Cogen}(\\operatorname{add} \\mathcal X)=\\operatorname{ind} \\operatorname{Cogen} \\mathcal X$.\n\\end{enumerate}\n\\end{lemma}\n\nNote that the dual of Lemma~\\ref{l:uniserialclosure}(a) also holds.\nWe fix an algebraically-closed field $K$ and denote by $D$ the vector space duality $\\operatorname{Hom}_K(-,K)$.\nNext, consider a finite dimensional $K$-algebra $\\Lambda$. We denote\nby $\\operatorname{Mod} \\Lambda$ the category of all left $\\Lambda$-modules, and by\n$\\operatorname{mod} \\Lambda$ the subcategory of all finitely generated $\\Lambda$-modules.\nRecall that a module ${U}$ in $\\operatorname{mod} \\Lambda$ is called \\emph{tilting} if\n\\begin{itemize}\n\\item[(a)] The projective dimension of ${U}$ is at most 1;\n\\item[(b)] $\\operatorname{Ext}^1({U},{U}) = 0$;\n\\item[(c)] There is a short exact sequence\n$$0\\rightarrow \\Lambda \\rightarrow {U}_0 \\rightarrow {U}_1 \\rightarrow 0$$\nwith ${U}_0$ and ${U}_1$ in $\\operatorname{add} {U}$.\n\\end{itemize}\nCotilting modules in $\\operatorname{mod} \\Lambda$ are defined dually. We only\nconsider basic tilting or cotilting modules, i.e.\\ we assume that\n${U}= \\amalg {U}_i$, with ${U}_i$ indecomposable and ${U}_i \\not\\simeq {U}_j$\nfor $i \\neq j$.\n\n\\section{Dynkin type A}\n\\label{s:Dynkin}\n\nWe consider a linearly oriented quiver $Q$ of type $\\text{A}_n$:\n$$\n\\xymatrix{\n1 & 2 \\ar[l] & 3 \\ar[l] & \\cdots\\ar[l] & n-1 \\ar[l] & n \\ar[l]\n}\n$$\nWe write $S_i$, $i=1,\\dots,n$, for the simple $KQ$-modules.\nLet $M_{ij}$ denote the indecomposable $KQ$-module with composition factors\n$S_{i+1},\\dots,S_{j-1}$ (starting from the socle).\nIf $i$ lies in $\\{j-1,j\\}$ we regard $M_{ij}$ as zero.\nNote that $\\operatorname{mod} KQ$ is serial.\n\n\\subsection{Geometric model}\n\\label{s:geometricmodel}\n\nWe now describe a geometric model for the indecomposable $KQ$-modules. Note that\na similar such model is also suggested in~\\cite[Remark 4.28]{warkentin}.\nHere we will indicate how this model can incorporate torsion theories\nin $\\operatorname{mod} KQ$ --- later we will indicate how this can be done for tubes. We\nalso give some more explicit information as preparation for this.\nWe consider a line segment $\\ell_n$ with marked points $0,1,\\dots,n+1$:\n\\[\n\\begin{picture}(220.00,20.00)\n\\drawline(10.00,15.00)(210.00,15.00)\n\\put(10.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(10.00,05.00){\\makebox(0,0)[cc]{\\tiny $0$}}\n\\put(40.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(40.00,05.00){\\makebox(0,0)[cc]{\\tiny $1$}}\n\\put(70.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(70.00,05.00){\\makebox(0,0)[cc]{\\tiny $2$}}\n\\put(150.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(150.00,05.00){\\makebox(0,0)[cc]{\\tiny $n-1$}}\n\\put(180.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(180.00,05.00){\\makebox(0,0)[cc]{\\tiny $n$}}\n\\put(210.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(210.00,05.00){\\makebox(0,0)[cc]{\\tiny $n+1$}}\n\\end{picture}\n\\]\nand associate the module $M_{ij}$ with the arc $[i,j]$ above $\\ell_n$\nfrom $i$ to $j$ oriented towards $j$, $0\\le i<j-1\\le n$.\nThis gives a bijection between indecomposable $KQ$-modules and the\nset $\\mathcal A(\\ell_n)$ of arcs up to isotopy between marked points of\n$\\ell_n$, above $\\ell_n$, which are not isotopic to boundary arcs.\n\nIt is easy to check that, for $[i,j]$, $[i',j']\\in \\mathcal A(\\ell_n)$,\nwe have $\\operatorname{Ext}^1(M_{ij},M_{i'j'})\\cong K$ if there is a negative crossing between\n$[i,j]$ and $[i',j']$ (see Figure~\\ref{fig:neg}) and is zero otherwise.\nIn the former case, the non-split extension takes the form:\n$$\n0\\to M_{i'j'}\\to \\begin{array}{c} M_{i'j} \\\\ \\amalg \\\\ M_{ij'} \\end{array} \\to M_{ij} \\to 0\n$$\nup to equivalence, if $j'>i+1$, while if $j'=i+1$, it takes the form\n$$\n0\\to M_{i'j'}\\to M_{i'j} \\to M_{ij} \\to 0,\n$$\nThis is interpreted geometrically in Figure~\\ref{fig:split-sum}\nwhere the indecomposable summands of the middle term\nof the short exact sequence are indicated by dotted lines.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{i}{\\tiny $i$}\n\\psfrag{i'}{\\tiny $i'$}\n\\psfrag{j}{\\tiny $j$}\n\\psfrag{j'}{\\tiny $j'$}\n\\includegraphics[scale=.5]{neg.eps}\n\\caption{A negative crossing between $[i,j]$ and $[i',j']$.}\\label{fig:neg}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{i}{\\tiny $i$}\n\\psfrag{i'}{\\tiny $i'$}\n\\psfrag{j}{\\tiny $j$}\n\\psfrag{j'}{\\tiny $j'$}\n\\psfrag{j'=i+1}{\\tiny $j'=i+1$}\n\\subfigure[Case $j'>i+1$]{\n\\includegraphics[scale=.6]{splitsum.eps}}\n\\ \\ \\ \\ \\ \\\n\\subfigure[Case $j'=i+1$]{\n\\includegraphics[scale=.6]{splitsum2.eps}}\n\\caption{Non-split extension}\\label{fig:split-sum}\n\\end{figure}\n\nThe AR-translation moves an arc one step to the left (or gives zero\nif this is not defined). Furthermore, the indecomposable quotients\nof a module correspond to moving the starting point weakly to the right,\nwhile submodules correspond to moving the ending point weakly to the left.\nWe call the former arcs {\\em left-shortenings} of $[i,j]$ and the latter arcs\n{\\em right-shortenings} of $[i,j]$.\n\nBy~\\cite{bo}, a $KQ$-module is tilting if and only if it is\na maximal rigid $KQ$-module.\nSince $M_{0,n+1}$ is projective-injective, it is a summand of every tilting module.\nIt follows that in the above model, tilting modules are in bijection with\ntriangulations of a polygon with $n+2$ sides.\n\nMotivated by~\\cite{ng,hjr} and the above description of extensions,\nwe call a collection of arcs in $\\mathcal A(\\ell_n)$ an {\\em oriented Ptolemy\ndiagram} if, whenever $[i,j]$ and $[i',j']$ lie in the collection, with\n$i'<i<j'<j$, we have that the arcs $[i,j']$ (provided $j'>i+1$) and $[i',j]$\nalso lie in the collection.\n\n\\subsection{Torsion pairs}\n\nBy Lemma~\\ref{l:tpclosure} and Lemma~\\ref{l:uniserialclosure},\na collection of arcs in $\\mathcal A(\\ell_n)$ corresponds to the torsion (respectively, torsion-free) part of a torsion pair in $\\operatorname{mod} KQ$ if and only if it forms an oriented Ptolemy diagram and is closed under left-shortening (respectively, right-shortening).\n\nGiven a tilting $KQ$-module ${U}$, the pair\n$(\\operatorname{Gen} {U}, \\operatorname{Cogen}\\tau {U})=({U}^{{\\perp}_{\\scriptscriptstyle E}},{U}^{{\\perp}_{\\scriptscriptstyle H}})$\nis known to form a torsion pair (see~\\cite[VI.2]{ass}). A torsion pair arises\nin this way if and only if $\\mathcal T$ contains all the indecomposable injective\nmodules, if and only if $\\mathcal F$ contains no non-zero injective module (see~\\cite[VI.6]{ass}\nfor the first equivalence; the second is easy to check). The first equivalence\nholds for an arbitrary finite dimensional hereditary algebra of finite representation type, and the\nequivalence of the second two statements holds for any finite dimensional hereditary algebra.\n\nNoting that a $KQ$-module is tilting if and only if it is cotilting, we have:\n\n\\begin{corollary}\\label{cor:Gen-Cogen}\nThe map:\n$$\n{U} \\mapsto (\\operatorname{Gen} {U},\\operatorname{Cogen}\\tau {U})=({U}^{{\\perp}_{\\scriptscriptstyle E}},{U}^{{\\perp}_{\\scriptscriptstyle H}})\n$$\ngives a bijection between tilting $KQ$-modules and\ntorsion pairs $(\\mathcal T,\\mathcal F)$ for which $\\mathcal T$ contains all the\nindecomposable injective modules. The map:\n$$\n{U} \\mapsto (\\operatorname{Gen}\\tau^{-1}{U},\\operatorname{Cogen} {U})=({}^{{\\perp}_{\\scriptscriptstyle H}}{U},{}^{{\\perp}_{\\scriptscriptstyle E}}{U})\n$$\ngives a bijection between tilting $KQ$-modules and\ntorsion pairs $(\\mathcal T,\\mathcal F)$ for which for which $\\mathcal F$ contains all\nthe indecomposable projective modules.\n\\end{corollary}\n\nUsing Lemma~\\ref{l:uniserialclosure}, the first map\nin Corollary~\\ref{cor:Gen-Cogen} can be interpreted in the geometric model:\n$\\operatorname{Gen} {U}$ is obtained from ${U}$ by closure under left shortening and\n$\\operatorname{Cogen}\\tau {U}$ is obtained from ${U}$ by shifting to the left one step (deleting arcs starting at $0$) and then closing under right shortening (in both cases we then\ntake the additive closure).\n\nConversely, if $(\\mathcal T,\\mathcal F)$ is a torsion pair of the kind considered in\nCorollary~\\ref{cor:Gen-Cogen}, then ${U}$ can be recovered as the direct\nsum of the indecomposable Ext-projectives in $\\mathcal T$, i.e.\\ the objects\n\\begin{equation}\n\\{X\\in \\operatorname{ind}(\\mathcal T): \\operatorname{Ext}^1(X,{T})=0\\ \\text{\\ for all\\ } {T}\\in \\mathcal T\\},\n\\label{e:extprojectives}\n\\end{equation}\nby~\\cite[VI.2.5]{ass}.\nGeometrically, this means taking all of the arcs $X$ in\n$\\operatorname{ind} \\mathcal T$ which do not have a negative crossing with an arc in $\\operatorname{ind} \\mathcal T$.\n\nThere is also a geometric description of the second map in Corollary~\\ref{cor:Gen-Cogen}\nand its inverse. We leave the details to the reader.\n\n\\section{Tubes}\n\n\\subsection{Categorical description}\n\nFix an integer $n\\geq 1$. Consider the quiver $Q$\n\\begin{equation}\\label{quivertilde}\n\\xymatrix@R=0.3cm{\n& 2 \\ar[dl] & 3 \\ar[l] & \\cdots \\ar[l] & n \\ar[l] & \\\\\n1 & & & & & n+1 \\ar[lllll] \\ar[ul]\n}\n\\end{equation}\nof Euclidean type $\\widetilde{\\text{A}}_{1,n}$. The path algebra $\\Lambda= KQ$ is tame hereditary, and the module category $\\operatorname{mod} KQ$ has an\nextension closed subcategory $\\mathbf T_n$, which can be realized as the extension closure of the modules $L, S_2, \\dots, S_{n}$,\nwhere $S_i$ denotes the simple corresponding to vertex $i$, and $L$ denotes the unique indecomposable module\nwith composition factors $S_1$ and $S_{n+1}$. The category $\\mathbf T_n$ is called a {\\em tube} of rank $n$. Note that the indecomposables of $\\mathbf T$ form a standard component\n(see e.g.~\\cite{ass}) of the AR-quiver of $KQ$.\n\nActually $\\mathbf T_n$ is a hereditary finite length abelian category with\n$n$ simple objects, and equivalent categories appear in various\nsettings in representation theory and algebraic geometry.\nFor each pair of objects $X,Y$ in $\\mathbf T_n$, the\nspaces $\\operatorname{Hom}(X,Y)$ and $\\operatorname{Ext}^1(X,Y)$ have finite $K$-dimension, and\nthere is an autoequivalence $\\tau$ on $\\mathbf T_n$, induced by the\nAuslander-Reiten translate on $\\operatorname{mod} \\Lambda$, with the property\nthat $\\operatorname{Hom}(Y, \\tau X) \\simeq D\\operatorname{Ext}^1(X,Y)$. Let\n$\\sigma:\\mathbb{Z}\\rightarrow \\mathbb{Z}$ be the map taking $i$ to\n$i+n$. From now on we denote the simples in $\\mathbf T = \\mathbf T_n$ by\n$M_{i,i+2}$ for $i=0, \\dots, n-1$, in such a way that $\\tau\nM_{i,i+2} = M_{i-1,i+1}$,\nwhere we regard $M_{\\sigma^k(i),\\sigma^k(j)}$ as equal to $M_{i,j}$\nfor any integer $k$.\nThe category $\\mathbf T_n$ is serial; thus the indecomposable objects in\n$\\mathbf T_n$ are uniserial and uniquely determined by their simple socle\nand length (in $\\mathbf T_n$).\nWe denote by $M_{i,i+l+1}$ an indecomposable with socle $M_{i,i+2}$\nand length $l$. Then $\\tau M_{i,i+l+1}  = M_{i-1,i+l}$,\nand the AR-quiver of the tube $\\mathbf T_n$ is as in Figure~\\ref{fig:ARquiverTn}\n(with the columns on the left- and right-hand sides identified).\nNote that each indecomposable object has a unique name $M_{ij}$\n(with $j-i\\geq 2$) if we insist that $i$ lies in $\\{0,1,\\ldots ,n-1\\}$.\n\n\\begin{figure}[ht]\n$$\\xymatrix@R=5pt@C=4pt{\n&&&&&&&&&& \\\\\n&& \\vdots && && \\vdots && \\vdots  \\\\\nM_{n-1,3} \\ar@{--}[rr] \\ar@{.}[dd] \\ar@{.}[uu] \\ar[dr] && M_{0,4} && \\cdots && M_{n-3,1} \\ar@{--}[rr] \\ar[dr] && M_{n-2,2} \\ar@{--}[rr] \\ar[dr] && M_{n-1,3} \\ar@{.}[dd] \\ar@{.}[uu] \\\\\n& M_{0,3} \\ar@{--}[l] \\ar@{--}[r] \\ar[ur] \\ar[dr] && && && \\ar@{--}[l] M_{n-2,1} \\ar@{--}[rr] \\ar[ur] \\ar[dr] && M_{n-1,2} \\ar@{--}[r] \\ar[ur] \\ar[dr] & \\\\\nM_{0,2} \\ar@{--}[rr] \\ar[ur] && M_{1,3} && \\cdots && M_{n-2,0} \\ar@{--}[rr] \\ar[ur] && M_{n-1,1} \\ar@{--}[rr] \\ar[ur] && M_{0,2}\n}$$\n\\caption{The AR-quiver of $\\mathbf T_n$}\n\\label{fig:ARquiverTn}\n\\end{figure}\n\nAn additive subcategory of $\\mathbf T$ is said to be of {\\em finite type} if it contains only finitely many indecomposable objects.\nOtherwise it is said to be of {\\em infinite type}.\nSome particular subcategories of $\\mathbf T$ are important\nfor this paper.\nFor each fixed $i$ in $\\{0, \\dots n-1 \\}$, we consider the additive subcategory\nwhose objects are the indecomposable objects $M_{i,i+t}$, for all $t>1$.\nThis is called a {\\em ray}, and is denoted $\\mathcal R_i$.\nDually, for each  $i$ in $\\{0, \\dots n-1\\}$, we consider the\nadditive subcategory whose indecomposable objects are all indecomposable\nobjects $M_{i-u,i}$, for all $u>1$. This is called a {\\em coray}, and\ndenoted $\\mathcal C_i$.\n\nFor each $i$ in $\\{0, \\dots, n-1\\}$, and each\n$t>1$, the {\\em wing} $\\mathcal W_{i,i+t}$ is the\nadditive subcategory of $\\mathbf T$ whose indecomposable objects are the\n$M_{j,j+u}$ with $u\\geq 2$, $i \\leq j \\leq i+t-2$ and\n$j+u \\leq i+t$. It contains a unique indecomposable object $M_{i,i+t}$\nof maximal length. The objects $M_{i,i+2}$ and $M_{i+t-2,i+t}$ lie at the\nbottom left and bottom right corners of the wing, respectively, in the\ncollection of vertices corresponding to the indecomposable objects of the wing in the\nAR-quiver of $\\mathbf T$.\nFor $t \\leq 1$ we let $\\mathcal W_{i,i+t}$ be the zero subcategory.\nWe also denote the wing of an indecomposable object $X$ by $\\mathcal W_X$.\n\nDue to the following well-known fact, we can apply\nresults from the previous section in our analysis of $\\mathbf T$.\n\n\\begin{lemma}\\label{wingsareA}\nFor $u \\leq n+1$ the wing $\\mathcal W_{i,i+u}$ in $\\mathbf T_n$ is equivalent to\nthe module category $\\operatorname{mod} KQ$, where $Q$ is a linearly\noriented quiver of Dynkin type $A_{u-1}$. If $u\\leq n$ the equivalence\nis exact.\n\\end{lemma}\n\nNote that if $u=n+1$, the object corresponding to the projective-injective\nindecomposable module in $\\operatorname{mod} KQ$ under the above equivalence is not $\\operatorname{Ext}$-projective in the wing.\n\nWe denote by $\\operatorname{Mod} \\Lambda$ the category of all left\n$\\Lambda$-modules. Let $\\varinjlim \\mathbf T$ be the subcategory of $\\operatorname{Mod} \\Lambda$\nwhose objects are direct limits of filtered direct systems of objects in $\\mathbf T$.\nNote that $\\varinjlim \\mathbf T$ contains the  {\\em Pr\\\"{u}fer modules}, i.e.\\ the modules $M_{i,\\infty}$, $i=0,2,\\ldots n-1$ obtained as direct limits\nof the indecomposable objects in the rays, i.e.\\\n$$M_{i,\\infty} = \\varinjlim (M_{i,i+2} \\to M_{i,i+3} \\to M_{i,i+4} \\to \\cdots).$$\nLet $\\varprojlim \\mathbf T$ be the subcategory of $\\operatorname{Mod} \\Lambda$\nwhose objects are inverse limits of filtered inverse systems of objects in $\\mathbf T$.\nThis category contains the {\\em adic} modules, which are\nobtained as inverse limits along a coray:\n$$M_{-\\infty,i} = \\varprojlim (\\cdots \\to M_{i-4,i} \\to M_{i-3,i} \\to\nM_{i-2,i} ).$$\n\nLet $\\overline{\\mathbf T}$ be the subcategory of $\\operatorname{Mod} \\Lambda$ whose\nobjects are all filtered direct limits or filtered inverse limits of\nobjects in $\\mathbf T$. This category clearly contains $\\varinjlim \\mathbf T$\nand $\\varprojlim \\mathbf T$.\nWe extend the definition of $\\sigma$ to Pr\\\"{u}fer and adic modules\nwith the convention that $\\sigma(\\pm \\infty)=\\pm \\infty$: note that any Pr\\\"{u}fer\nmodule has a unique name $M_{i,\\infty}$ if we take $i$ in $\\{0,1,\\ldots ,n-1\\}$, and\nsimilarly for adic modules.\n\nRecall that a $\\Lambda$-module $M$ is \\emph{pure-injective} if\nthe canonical map $M \\rightarrow DDM$ is a split monomorphism.\nFor background on pure-injective modules, and other definitions,\nsee e.g.~\\cite{jensenlenzing89} or~\\cite{prest09}.\nIt can be shown (see \\cite{bk1}) that the category $\\overline{\\mathbf T}$\nhas the following properties:\n\n\\begin{itemize}\n\\item[$\\bullet$] All objects are pure-injective as $\\Lambda$-modules.\n\\item[$\\bullet$] Any object is determined by its indecomposable direct summands.\n\\item[$\\bullet$] The indecomposables in $\\overline{\\mathbf T}$ are exactly the indecomposables\n  $M_{ij}$ in $\\mathbf T$, the Pr{\\\"u}fer modules $M_{i,\\infty}$ and the\n  adic modules $M_{-\\infty,i}$.\n\\end{itemize}\n\nA module $M$ in $\\overline{\\mathbf T}$ is called {\\em rigid} if\n$\\operatorname{Ext}^1_{\\Lambda}(M,M) = 0$. Note that since all objects in\n$\\overline{\\mathbf T}$ are pure-injective, this definition is equivalent to having\n$\\operatorname{Ext}^1_{\\Lambda}(M',M'') = 0$ for all indecomposable direct summands $M',M''$ of\n$M$. Now, a rigid module $M$ in $\\overline{\\mathbf T}$ is called {\\em maximal\n  rigid}, if $\\operatorname{Ext}^1_{\\Lambda}(M \\amalg X,M \\amalg X) = 0$ for an\nindecomposable $X$ in $\\overline{\\mathbf T}$ implies\nthat $X$ is isomorphic to a direct summand in $M$.\n\n\\subsection{Geometric model}\\label{cm}\n\nWe now give a geometric model for $\\overline{\\mathbf T}$. This extends the\nknown geometric model for $\\mathbf T$~\\cite{bama,warkentin}.\nConsider an annulus $\\mathbb A(n)$ with $n$ marked points on the outer boundary.\nThe points are labelled $0, 1, \\dots, n-1$, and arranged anticlockwise\n(see Figure~\\ref{fig:annulus}).\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$\\scriptstyle 0$}\n\\psfrag{1}{$\\scriptstyle 1$}\n\\psfrag{2}{$\\scriptstyle 2$}\n\\psfrag{n-1}{$\\scriptstyle n-1$}\n\\includegraphics[width=2.5cm]{annulus.eps}\n\\caption{An annulus with $n$ marked points on its outer boundary.}\n\\label{fig:annulus}\n\\end{figure}\n\nLet $\\mathbb U(n)$ denote the universal cover of $\\mathbb A(n)$, with marked\npoints corresponding to $\\mathbb{Z}$ (and with $0,1, \\dots, n-1$\nlying in a fundamental domain). See Figure~\\ref{fig:universalcover}.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{-1}{$\\scriptstyle -1$}\n\\psfrag{0}{$\\scriptstyle 0$}\n\\psfrag{1}{$\\scriptstyle 1$}\n\\psfrag{2}{$\\scriptstyle 2$}\n\\psfrag{n-1}{$\\scriptstyle n-1$}\n\\psfrag{n}{$\\scriptstyle n$}\n\\includegraphics[height=2cm]{universalcover.eps}\n\\caption{The universal cover of the annulus in Figure~\\ref{fig:annulus}.}\n\\label{fig:universalcover}\n\\end{figure}\n\nFor integers $i,j$ with $i+2 \\leq j$, let $[i,j]$ denote the arc in\n$\\mathbb U(n)$ with starting point $i$ and ending point $j$, oriented from $i$ to $j$.\nWe also allow arcs which have only one end-point: arcs of the form $[i,\\infty]$\n(respectively, $[-\\infty,j]$ which start\nat $i$ (respectively, end at $j$) and are oriented in the positive $x$ direction.\nSee Figure~\\ref{fig:infinitearc1}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i}{$i$}\n\\psfrag{iinf}{$[i,\\infty]$}\n\\psfrag{i-inf}{$[-\\infty,i]$}\n\\includegraphics[width=12cm]{infinitearc1.eps}\n\\caption{Infinite arcs in $\\mathbb U(n)$.}\n\\label{fig:infinitearc1}\n\\end{figure}\n\nLet $\\pi_n([i,j])$ denote the corresponding arc in $\\mathbb A(n)$ and\nlet $\\widetilde{A}=\\widetilde{\\mathcal A}(\\mathbb A(n))$ denote the set of (isotopy classes\nof) such arcs. It contains the set $\\mathcal A=\\mathcal A(\\mathbb A(n))$ of arcs of the form $\\pi_n([i,j])$\nwith $i,j$ finite. The map $\\psi:\\widetilde{\\mathcal A}\\rightarrow \\operatorname{ind} \\overline{\\mathbf T}$\nsending $\\pi_n([i,j])$ to $M_{ij}$ is a bijection.\nThe infinite arcs are displayed in Figure~\\ref{fig:infinitearc2}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i}{$i$}\n\\psfrag{iinf}{$\\pi_n([i,\\infty])$}\n\\psfrag{i-inf}{$\\pi_n([-\\infty,i])$}\n\\includegraphics[height=3cm]{infinitearc2.eps}\n\\caption{Infinite arcs in $\\mathbb A(n)$.}\n\\label{fig:infinitearc2}\n\\end{figure}\n\nDefine a quiver with vertices given by the elements in $\\mathcal A$ and arrows:\n$$\\pi_n([i,j]) \\to \\pi_n([i,j+1])$$ and $$\\pi_n([i,j]) \\to\n\\pi_n([i+1,j]) \\text{ (if $j \\neq i+2)$} $$\nDefining a translate using the formula $\\tau(\\pi_n([i,j])) = \\pi_n([i-1,j-1])$,\nthis becomes a translation quiver. We call this the \\emph{(translation) quiver\nof $\\mathcal A(\\mathbb A(n))$}.\n\nBy~\\cite[Lemma 2.5]{bama},~\\cite[4.18]{warkentin} (or, using unoriented arcs,~\\cite[\\S3.4]{bz},~\\cite{gehrig}), we have:\n\n\\begin{proposition}\nThe restriction of $\\psi$ to $\\mathcal A$ gives an isomorphism between\nthe translation quiver of $\\mathcal A(\\mathbb A(n))$ and the AR-quiver of $\\mathbf T_n$.\n\\end{proposition}\n\nNote that the convention that $M_{\\sigma^k(i),\\sigma^k(j)}=M_{ij}$ corresponds exactly\nto the fact $\\pi_n([\\sigma^k(i),\\sigma^k(j)])=\\pi_n([i,j])$ for any integer $k$.\n\nFor arcs $\\alpha, \\beta$ in $\\mathcal A(\\mathbb A(n))$, let\n$I(\\alpha, \\beta)$ be the minimum number of intersections between arcs in the isotopy classes $\\alpha$ and\n$\\beta$, not allowing non-transverse or multiple intersections. Similarly we let $I^+(\\alpha, \\beta)$ (resp. $I^-(\\alpha, \\beta)$) denote the number of\npositive (resp. negative) crossings between $\\alpha$ and $\\beta$ (see Figure~\\ref{fig:neg} for an example of a negative crossing).\nWe will now prove the following result:\n\n\\begin{theorem}\\label{exts}\nGiven indecomposable objects $M_{ij}$ and $M_{i'j'}$ in\n$\\overline{\\mathbf T}$. Then:\n$$\\operatorname{Ext}^1(M_{ij},M_{i'j'}) \\cong \\prod_{I^-(\\pi_n([i,j]),\n  \\pi_n([i',j']))} K .$$\n\\end{theorem}\n\nIn the case where $i,i',j,j'$ are all finite, the result is proved in\n\\cite[Thm.\\ 3.7]{bama},~\\cite[Thm. 4.23]{warkentin}.\nSee also \\cite{bz} for further results in this direction.\nWe first recall some results we will need:\n\n\\begin{lemma}\\label{limits}\nLet $X,Y$ be arbitrary $\\Lambda$-modules,\n$(X_j)_j$ an arbitrary filtered direct system of modules and\n$(Y_j)_j$ an arbitrary filtered inverse system of modules.\n\\begin{itemize}\n\\item[(a)] $\\operatorname{Hom}(\\varinjlim X_j, Y) \\simeq \\varprojlim \\operatorname{Hom}(X_j,Y)$.\n\\item[(b)] If $X$ is finitely generated, then $\\operatorname{Hom}(X,\\varinjlim Y_j)\\simeq \\varinjlim \\operatorname{Hom}(X,Y_j)$\n\\item[(c)] If the $Y_j$ are finitely generated, then\n$\\varprojlim Y_j\\simeq D\\varinjlim DY_j$.\n\\item[(d)] If $Y$ is pure-injective, then\n$\\operatorname{Ext}^1(\\varinjlim X_j, Y) \\simeq \\varprojlim \\operatorname{Ext}^1(X_j,Y)$.\n\\item[(e)] If the $Y_j$ are finitely generated, then\n$\\operatorname{Ext}^1(X, \\varprojlim Y_j) \\simeq \\varprojlim \\operatorname{Ext}^1(X,Y_j)$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} For (a) see, for example,~\\cite{trlifaj03}.\nFor (b), see~\\cite[Lemma 1.6]{krausesolberg03}\nor~\\cite[Sect. 1.5]{crawleyboevey98}.\nFor (c), we have, using part (a):\n\\begin{align*}\nD\\varinjlim DY_j &= \\operatorname{Hom}(\\varinjlim \\operatorname{Hom}(Y_j,K),K) \\\\\n&\\simeq \\varprojlim \\operatorname{Hom}(\\operatorname{Hom}(Y_j,K),K)\n\\simeq \\varprojlim DDY_j \\simeq \\varprojlim Y_j,\n\\end{align*}\nas required.\nPart (d) is proved in~\\cite[Prop.\\ I.10.1]{auslander78}.\nFor (e), we recall that $\\operatorname{Ext}^1(X,DY)\\simeq \\operatorname{Ext}^1(Y,DX)$ for all modules\n$X$ and $Y$. Using parts (c) and (d)\nand the fact~\\cite[Prop.\\ 4.3.29]{prest09} that $DX$ is pure-injective for\nany module $X$, we have:\n\\begin{align*}\n\\operatorname{Ext}^1(X,\\varprojlim Y_j) &\\simeq \\operatorname{Ext}^1(X,D(\\varinjlim DY_j))\n\\simeq \\operatorname{Ext}^1(\\varinjlim DY_j,DX) \\\\\n&\\simeq \\varprojlim \\operatorname{Ext}^1(DY_j,DX)\n\\simeq \\varprojlim \\operatorname{Ext}^1(X,DDY_j) \\\\\n&\\simeq \\varprojlim \\operatorname{Ext}^1(X,Y_j),\n\\end{align*}\nand (e) is shown.\n\\end{proof}\n\nWe also need the following (see e.g.~\\cite[Sect.\\ 3.1]{crawleyboevey98}).\n\n\\begin{lemma} \\label{ARformula}\nFor modules $X$ and $Y$ with $X$ finitely generated, we have\n$D\\operatorname{Ext}^1(X,Y)\\cong \\operatorname{Hom}(Y,\\tau X)$ and $\\operatorname{Ext}^1(Y,X)\\cong D\\operatorname{Hom}(\\tau^{-1}X,Y)$.\n\\end{lemma}\n\nNote that if $X,Y$ are finitely generated, then the first formula can also\nbe written $\\operatorname{Ext}^1(X,Y)\\cong D\\operatorname{Hom}(Y,\\tau X)$. We recall the following\n(see, for example,~\\cite[p46]{ringel00}).\n\n\\begin{lemma} \\label{homvanishing}\n\\begin{itemize}\n\\item[(a)] If $X$ is a Pr\\\"{u}fer module and $Y$ is a finitely\ngenerated module then $\\operatorname{Hom}(X,Y)=0$.\n\\item[(b)] If $X$ is a finitely generated module and $Y$ is an adic\nmodule then $\\operatorname{Hom}(X,Y)=0$.\n\\end{itemize}\n\\end{lemma}\n\nWith arguments as in \\cite{bama}, the crossing numbers can now be computed as follows.\nRecall that $\\sigma \\colon \\mathbb{Z} \\to  \\mathbb{Z}$ is the function $i \\mapsto i+n$.\n\n\\begin{proposition}\\label{crossingnumbers}\nWe have the following:\n\\begin{itemize}\n\\item[(a)]  $I^-(\\pi_n([i,\\infty]), \\pi_n([a,b])) = \\mid \\{m \\in\n  \\mathbb{Z}  \\colon a < \\sigma^m(i)< b \\} \\mid$;\n\\item[(b)] $I^-(\\pi_n([a,b]), \\pi_n([i,\\infty])) = I^+(\\pi_n([i,\\infty]), \\pi_n([a,b])) = 0$;\n\\item[(c)] $I^-(\\pi_n([-\\infty,j]), \\pi_n([a,b]))= 0$;\n\\item[(d)] $I^- (\\pi_n([a,b]),\\pi_n([-\\infty,j])) =\nI^+(\\pi_n([-\\infty,j]), \\pi_n([a,b]))\n= \\mid \\{m \\in  \\mathbb{Z}  \\colon a < \\sigma^m(i)< b \\} \\mid$;\n\\item[(e)] $I^-(\\pi_n([i,\\infty]), \\pi_n([-\\infty, i'])) = \\aleph_0$,\n  for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(f)] $ I^-(\\pi_n([i,\\infty]),\n    \\pi_n([i',\\infty])) = 0$ for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(g)] $I^-(\\pi_n([-\\infty, i]), \\pi_n([i',\\infty])) =  0$ for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(h)] $I^-(\\pi_n([-\\infty,\ni]), \\pi_n([-\\infty, i'])) = 0$ for all  $i,i'$ in $\\{0,\\dots, n-1\\}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{exts}]\nWe need to compute $\\operatorname{Ext}^1(X,Y)$ for all pairs of indecomposables\n$X,Y$ in $\\overline{\\mathbf T}$, and compare these with the crossing-numbers\nfrom Proposition \\ref{crossingnumbers}.\nWe first determine $\\operatorname{Ext}^1(M_{i,\\infty}, M_{ab})$. By Lemma~\\ref{ARformula},\n$$\\operatorname{Ext}^1(M_{i,\\infty},M_{ab})\\cong D\\operatorname{Hom}(M_{a+1,b+1},M_{i,\\infty}).$$\nAny morphism $f:M_{a+1,b+1}\\rightarrow M_{i,\\infty}$ must factor through the\nunique submodule of $M_{i,\\infty}$ of length $b-a-1$, which is isomorphic to\n$M_{i,i+b-a}$.\nHence $\\operatorname{Hom}(M_{a+1,b+1},M_{i,\\infty})\\cong \\operatorname{Hom}(M_{a+1,b+1},M_{i,i+b-a})$.\nThus $\\operatorname{Ext}^1(M_{i,\\infty},M_{ab})$ equals the dimension of this last space,\ni.e.\\ the number of times the simple top $M_{b-1,b+1}$ of $M_{a+1,b+1}$\nappears as a composition factor in $M_{i,i+b-a}$, which, using arguments\nas in~\\cite{bama}, is given by\n$$|\\{n\\in\\mathbb{Z}\\,:\\,a<\\sigma^n(i)<b\\}|.$$\n\nUsing a dual argument, we obtain:\n$$\\dim \\operatorname{Ext}^1(M_{i,-\\infty}, M_{ab}) =\n\\mid \\{n \\in \\mathbb{Z} \\colon a < \\sigma^n(i) <b \\} \\mid.$$\n\nBy Lemma~\\ref{ARformula},\n$$D\\operatorname{Ext}^1(M_{ab}, M_{i, \\infty}) \\simeq \\operatorname{Hom}(M_{i,\\infty},M_{a-1, b-1}).$$\nWe see that $\\operatorname{Ext}^1(M_{ab},M_{i,\\infty})=0$ by Lemma~\\ref{homvanishing}(a).\n\nSimilarly, by Lemma~\\ref{ARformula}, we have that\n$$\\operatorname{Ext}^1(M_{-\\infty,i}, M_{ab}) \\simeq D\\operatorname{Hom}(\\tau^{-1}M_{ab},M_{-\\infty,i})=0,$$\nusing Lemma~\\ref{homvanishing}(b).\n\nWe next compute $\\operatorname{Ext}^1(M_{i,\\infty}, M_{-\\infty,i'})$. Using Lemma\n\\ref{limits}(e), we have that\n$$\\operatorname{Ext}^1(M_{i,\\infty}, M_{-\\infty,i'}) = \\operatorname{Ext}^1(M_{i,\\infty},\n\\varprojlim M_{j,i'}) \\simeq \\varprojlim \\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'}),$$\nwhere $j$ is the dummy variable in the limit.\nThe maps\n$$\\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'}) \\to \\operatorname{Ext}^1(M_{i,\\infty},\nM_{j-1,i'})$$\nare surjective. As $j$ tends to $-\\infty$, the dimension of\n$\\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'})$ is unbounded (by the above formula for it).\nTherefore, this limit evaluates to $\\prod_{\\aleph_0} K$.\n\nIt is shown in \\cite[Lemma 2.7]{bk1}, that, for all $i,i'$\nin $\\{0,\\dots ,n-1\\}$, all of \\linebreak\n$\\operatorname{Ext}^1(M_{-\\infty, i}, M_{i',\\infty})$, \n$\\operatorname{Ext}^1(M_{i,\\infty},M_{i',\\infty})$ and $\\operatorname{Ext}^1(M_{-\\infty, i}, M_{-\\infty,i'})$\nvanish.\n\nComparing the crossing numbers with the dimensions of the corresponding\nExt-groups concludes the proof of the theorem.\n\\end{proof}\n\n\\subsection{Reflection}\n\nRecall (see \\cite{bk1}) that there is a map  $M \\mapsto M^{\\vee}$, which gives a bijection\non the indecomposable objects in $\\overline {\\mathbf T}$. With our notation,\nthe map is given by $M_{ij} \\mapsto M_{-j,-i}$ for $i,j$ in\n$\\mathbb{Z} \\cup \\{ \\pm \\infty \\}$.\n\nFor a subcategory $\\mathcal X$ of $\\overline{\\mathbf T}$, we let $\\mathcal X^{\\vee}$ denote the\nsubcategory $\\{X^{\\vee} \\mid X \\in \\mathcal X \\}$. The map has the following properties.\n\\begin{lemma}\\label{properties}\n\\begin{itemize}\n\\item[(a)] $\\dim \\operatorname{Hom}(X,Y) = \\dim \\operatorname{Hom}(Y^{\\vee}, X^{\\vee})$ for\n  $X,Y$ in $\\overline{\\mathbf T}$;\n\\item[(b)] $\\operatorname{Ext}^1(X,Y) = 0$ if and only if $\\dim \\operatorname{Ext}^1(Y^{\\vee},\n  X^{\\vee}) = 0$ for\n  $X,Y$ in $\\overline{\\mathbf T}$;\n\\item[(c)]$(\\tau X)^{\\vee} = \\tau^{-1} X^{\\vee}$ for all\n  indecomposables $X$ in $\\mathbf T$.\n\\item[(d)] For each object $X$ in $\\overline \\mathbf T$, we have\n$(X^{{\\perp}_{\\scriptscriptstyle H}})^{\\vee} = {^{{\\perp}_{\\scriptscriptstyle H}}(X^{\\vee})}$ and\n$(X^{\\vee})^{{\\perp}_{\\scriptscriptstyle E}} = (^{{\\perp}_{\\scriptscriptstyle E}}X)^{\\vee}$.\n\\end{itemize}\n\\end{lemma}\n\nWe will also need the following.\n\n\\begin{lemma} \\label{l:twistgen}\nFor a maximal rigid object $U$ in $\\overline{\\mathbf T}$, we have\n\\begin{itemize}\n\\item[(a)] $(\\operatorname{Gen} {U}\\cap \\mathbf T)^{\\vee} = \\operatorname{Cogen}({U}^{\\vee})\\cap \\mathbf T$;\n\\item[(b)] $(\\operatorname{Cogen} {U}\\cap \\mathbf T)^{\\vee} = \\operatorname{Gen}({U}^{\\vee})\\cap \\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nThis follows directly from the definition of the map using\nLemma~\\ref{homvanishing} and the fact that\nan indecomposable $X$ in $\\mathbf T$ is generated by a maximal\nrigid object ${U}$ if and only if it is generated by an indecomposable direct\nsummand in ${U}$ (see also \\cite{bk2}).\n\\end{proof}\n\n\\section{Torsion pairs and maximal rigid objects}\n\nIn this section we give an improvement of a result from\n\\cite{bk1}, and discuss a combinatorial interpretation.\nThe idea is to link maximal rigid objects in $\\overline{\\mathbf T}$ with\ntorsion pairs in $\\mathbf T$.\n\n\\subsection{A bijection}\nA rigid object in $\\overline{\\mathbf T}$ is said to be of {\\em Pr\\\"{u}fer type}\nif it has a Pr\\\"{u}fer module as a direct summand, and it is said to be\nof {\\em adic type} if it has an adic module as a direct summand.\n\nAn object ${U}$ in $\\varinjlim \\mathbf T$ is maximal rigid if and only if it is\ncotilting in the category $\\varinjlim \\mathbf T$, in the sense of Colpi~\\cite{colpi99}.\nThis is proved in~\\cite{bk2} (see~\\cite[1.10]{bk2} and note that,\nby~\\cite[Lemma 1.2]{bk2}, a finitely presented $\\tilde{\\Lambda}_n$-module\nis maximal rigid if and only if it is a tilting module, where\n$\\tilde{\\Lambda}_n$ denotes the completion of the path algebra of an\noriented $n$-cycle).\n\nWe consider equivalence classes of maximal rigid\nobjects in $\\overline{\\mathbf T}$, where two maximal rigid objects are considered to\nbe equivalent if they have the same indecomposable direct summands.\nThe following is proved in~\\cite{bk1}.\n\n\\begin{proposition}\\label{propbk1}\nLet ${U}$ be a rigid object in $\\overline{\\mathbf T}$.\n\\begin{itemize}\n\\item[(a)] ${U}$ is maximal rigid in $\\overline{\\mathbf T}$ if and only if it has $n$ pairwise nonisomorphic indecomposable direct summands.\n\\item[(b)] If ${U}$ is maximal rigid, it is either of Pr{\\\"u}fer type or of adic type, but not both.\n\\item[(c)] A rigid object of Pr{\\\"u}fer type lies in the subcategory\n$\\varinjlim \\mathbf T \\subset \\overline{\\mathbf T}$.\n\\end{itemize}\n\\end{proposition}\n\nWe will give a parallel result concerning torsion pairs in $\\mathbf T$.\nA torsion pair $(\\mathcal T,\\mathcal F)$ in $\\mathbf T$ is said to be of {\\em ray type} if\n$\\mathcal F$ contains at least one ray of $\\mathbf T$.  It is said to be of {\\em coray type} if\n$\\mathcal T$ contains at least one coray of $\\mathbf T$.\nA class of objects (or subcategory) $\\mathcal X$ of a category $\\mathcal C$ is said to be {\\em generating},\nif for each map $f$ in $\\mathcal C$, there is an object $X$ in $\\mathcal X$ with\n$\\operatorname{Hom}(X,f) \\neq 0$. Dually, one can define {\\em cogenerating} classes\nof objects, i.e.\\ $\\mathcal X$ is cogenerating if, for each map $f$ in $\\mathcal C$,\nthere is an object $X$ in $\\mathcal X$ with $\\operatorname{Hom}(f,X)\\neq 0$.\n\n\\begin{lemma}\\label{onefinite}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$. Then it is not the case that\nboth $\\mathcal T$ and $\\mathcal F$ are of finite type.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\mathcal T$ and $\\mathcal F$ are both of finite type.\nBy the definition of a torsion pair, every object $M$ in $\\mathbf T$\nmust be the middle term of an exact sequence\n\\begin{equation*}\n0 \\to T \\to M \\to F \\to 0,\n\\end{equation*}\nwhere $T\\in \\mathcal T$ and $F\\in \\mathcal F$.\nSince $\\mathbf T$ is serial, if $M$ is indecomposable then so are\n$T$ and $F$. It follows that $\\mathbf T$ itself is of finite\ntype, a contradiction.\nHence, we cannot have that both $\\mathcal T$ and $\\mathcal F$ are of finite type.\n\\end{proof}\n\nWe next prove that the torsion pairs with $\\mathcal T$ of infinite type\nare exactly those of coray type. For an indecomposable object $X$ in $\\mathbf T$\nwe write $\\mathcal C_X$ for the coray containing $X$ and $\\mathcal R_X$ for the ray containing\n$X$.\n\n\\begin{lemma}\\label{coray-type}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$, where $\\mathbf T$ has rank $n$.\nAssume $\\mathcal T$ is of infinite type. Then the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal T$ contains an indecomposable object $X$, with $l(X) = n$.\n\\item[(ii)] $\\mathcal F$ is of finite type.\n\\item[(iii)] $\\mathcal T$ contains the coray $\\mathcal C_X$.\n\\item[(iv)] $\\mathcal T$ cogenerates $\\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFirst note that since $\\mathcal T$ is of infinite type, there is no limit on the length of the indecomposable objects\nin $\\mathcal T$. Since $\\mathcal T$ is closed under factor objects, it must therefore contain an indecomposable object with $l(X) = n$.\nHence (i) holds.\nSince $\\mathcal T$ is closed under factor objects, the\nindecomposable objects in the coray $\\mathcal C_X$ below $X$ are contained in\n$\\mathcal T$; more precisely $\\operatorname{ind}(\\mathcal C_X) \\cap \\{Y \\mid l(Y) \\leq n \\} \\subset \\mathcal T$.\nAssume first that $n>1$.\nLet $X'$ be the (uniquely defined) indecomposable object in $\\mathbf T$ such that there is an irreducible\nmonomorphism $X' \\to X$.\nSince $l(X) = n$, we have that $\\operatorname{Hom}(X,Y) = 0$ for an indecomposable object\n$Y$ in $\\mathbf T$ if and only if $Y$ is in the wing\n$\\mathcal W_{X'}$. By the definition of a torsion pair, we have that\n$\\mathcal F$ is contained in $\\mathcal W_{X'}$ so is of finite type.\nIf $n=1$, we have $\\operatorname{Hom}(X,Y)\\neq 0$ for any indecomposable\nobject $Y$ in $\\mathbf T$, so $\\mathcal F$ is the zero subcategory and (ii) holds.\nBy Lemma~\\ref{l:tpclosure}, (iii) holds, while (iv) is a direct consequence of (iii).\n \\end{proof}\n\nWe state the dual version of Lemma \\ref{coray-type}.\n\n\\begin{lemma}\\label{ray-type}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$, where $\\mathbf T$ has rank $n$.\nAssume $\\mathcal F$ is of infinite type. Then the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal F$ contains an indecomposable object $X$, with $l(X) = n$.\n\\item[(ii)] $\\mathcal T$ is of finite type.\n\\item[(iii)] $\\mathcal F$ contains the ray $\\mathcal R_X$.\n\\item[(iv)] $\\mathcal F$ generates $\\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\nCombining Lemma \\ref{onefinite}, \\ref{coray-type} and  \\ref{ray-type}, we obtain the following direct consequence.\n\n\\begin{corollary}\\label{cortorsion}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$.\n\\begin{itemize}\n\\item [(a)]  The following are equivalent\n\\begin{itemize}\n\\item[(i)] $\\mathcal T$ is of infinite type;\n\\item[(ii)] $\\mathcal T$ contains a coray;\n\\item[(iii)] $\\mathcal T$ cogenerates $\\mathbf T$;\n\\item[(iv)] $\\mathcal F$ is of finite type.\n\\end{itemize}\n\\item[(b)]\n$({\\mathcal T, \\mathcal F})$ is either of ray or of coray type (and not both).\n\\end{itemize}\n\\end{corollary}\n\nMoreover, by Lemma~\\ref{l:tpclosure} and a direct application of Lemma \\ref{properties},\nwe obtain the following.\n\n\\begin{lemma}\nThe map $M \\to M^{\\vee}$,  maps a torsion pair $(\\mathcal T, \\mathcal F)$ to a\ntorsion pair $(\\mathcal F^{\\vee}, \\mathcal T^{\\vee})$. Moreover if  $(\\mathcal T, \\mathcal F)$ is of\nray-type, then  $(\\mathcal F^{\\vee}, \\mathcal T^{\\vee})$ is of coray-type and vice-versa.\n\\end{lemma}\n\nFor a maximal rigid object ${U}$ of Pr{\\\"u}fer type, consider the\nsubcategory $\\mathcal F_{U} = {^{{\\perp}_{\\scriptscriptstyle E}}{U}} \\cap \\mathbf T$. We set $\\mathcal T_{U} =\n{^{{\\perp}_{\\scriptscriptstyle H}}(\\mathcal F_{U})} \\cap \\mathbf T.$\nFor a maximal rigid object ${U}$ of\nadic type, we define\n$\\mathcal T_{U} ={U} ^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T$ and $\\mathcal F_{U} =\n(\\mathcal T_{U})^{{\\perp}_{\\scriptscriptstyle H}} \\cap \\mathbf T$.\nWe have the following reformulation of a result of~\\cite{bk2}:\n\n\\begin{theorem}\\label{propbk2}\nThe map ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ gives\na one-to-one correspondence between equivalence classes of\nmaximal rigid objects in $\\varinjlim \\mathbf T$ and\ntorsion pairs in $ \\mathbf T$ with the property that $\\mathcal F$ generates $\\mathbf T$.\n\\end{theorem}\n\nNow, using Lemma \\ref{properties}, we obtain a commutative square (*):\n$$\n\\xymatrix{\n\\{\\text{maximal rigid objects of Pr{\\\"u}fer type in } \\overline{\\mathbf T} \\}\n\\ar[r]  \\ar[d] & \\{\\text{torsion pairs of ray-type in } \\mathbf T \\} \\ar[d] \\\\\n\\{\\text{maximal rigid objects of adic type in } \\overline{\\mathbf T} \\}\n\\ar[r]  & \\{\\text{torsion pairs of coray-type in } \\mathbf T  \\}\n}\n$$\nwhere the horizontal maps are given by ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ and\nthe vertical maps are induced by $M \\mapsto M^{\\vee}$.\n\nAs a direct consequence of Lemma~\\ref{properties} (as\nalso observed in \\cite{bk1}), we have that the left vertical map in (*)\nis a bijection. We have already observed that the right vertical map\nis a bijection. The upper horizontal map is bijective by Theorem~\\ref{propbk2},\ncombining with Proposition~\\ref{propbk1} and Lemma~\\ref{ray-type},\nand it follows that the lower horizontal map is bijective.\n\nCombining the commutative diagram of bijections (*) with Corollary \\ref{cortorsion}\nwe obtain the following improvement of Theorem \\ref{propbk2}.\n\n\\begin{theorem} \\label{t:mainbijection}\nThe map ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ gives a bijection between\n\\begin{itemize}\n\\item[$\\bullet$] Equivalence classes of maximal rigid objects in $\\overline{\\mathbf T}$\n\\item[$\\bullet$] Torsion pairs $(\\mathcal T,\\mathcal F)$ in $\\mathbf T$, and\n\\end{itemize}\n\\end{theorem}\n\n\\begin{corollary}\nThe number of torsion pairs in $\\mathbf T$ is $2\\binom{2n-1}{n-1}$.\n\\end{corollary}\n\n\\begin{proof}\nBy~\\cite[2.4, 5.2 and B.1]{bk2},\nthe number of cotilting objects in $\\varinjlim \\mathbf T$\nis $\\binom{2n-1}{n-1}$. The result then follows from Proposition~\\ref{propbk1},\nthe above diagram of maps, and Theorem~\\ref{t:mainbijection}.\n\\end{proof}\n\nNote that the above results hold in the case $n=1$: in this case there are two maximal\nrigid objects: the unique Pr\\\"{u}fer and adic modules, and two corresponding torsion pairs,\n$(0,\\mathbf T)$ and $(\\mathbf T,0)$ respectively.\n\n\\subsection{Alternative and explicit descriptions of $\\mathcal T_{U}$ and $\\mathcal F_{U}$}\n\nWe give alternative and more explicit descriptions for the subcategories\n$\\mathcal T_{U}$ and $\\mathcal F_{U}$ corresponding to a maximal rigid object ${U}$ in\n$\\overline{\\mathbf T}$.\n\nThe following observation is useful.\n\n\\begin{lemma}\\label{perps}\nLet $\\mathbf T$ be a tube of rank $n$. Then we have:\n\\begin{itemize}\n\\item[(a)]$^{{\\perp}_{\\scriptscriptstyle E}} M_{i,\\infty} \\cap \\mathbf T = \\mathbf T$;\n\\item[(b)]$M_{i,\\infty}^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T = \\mathcal W_{i,i+n}$\n\\end{itemize}\n\\end{lemma}\n\nWe now give an explicit description of the torsion pair $(\\mathcal T_{U},\\mathcal F_{U})$\ncorresponding to a  maximal rigid module ${U}$. We first of\nall give a combinatorial lemma concerning wings, which is easy to\ncheck.\n\n\\begin{lemma} \\label{l:wings}\nLet $0 \\leq i_0 < i_1 < \\cdots < i_{k-1} \\leq n-1$ be integers.\nThen\n\\begin{itemize}\n\\item[(i)] We have:\n$$\\bigcap_{r=0}^{k-1} \\mathcal W_{i_r,i_r + n} = \\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1} }$$\n(where the subscripts are interpreted modulo $k$ and, for $r=k-1$, we\ninterpret $i_{r+1}=i_0$ as $i_0+n$).\n\\item[(ii)] The wing $\\mathcal W_{i_r,i_{r+1} }$ is zero if and only if\n$i_{r+1} - i_{r}= 1$.\n\\item[(iii)] The wings $\\mathcal W_{i_r,i_{r+1}}$ do not overlap,\ni.e.\\ each indecomposable object in $\\mathbf T$ belongs to at most one $\\mathcal W_{i_r,i_{r+1}}$.\n\\item[(iv)] If $X$ and $Y$ lie in different wings among the\n  $\\mathcal W_{i_r,i_{r+1} } $, then $\\operatorname{Ext}^1(X,Y) = 0 = \\operatorname{Ext}^1(Y,X)$.\n\\end{itemize}\n\\end{lemma}\n\nFor an example illustrating Lemma~\\ref{l:wings}(i),\nwith $n=10$, $k=4$, $i_0=0$, $i_1=4$, $i_2=7$ and $i_3=8$, see\nFigure~\\ref{fig:wingsexample}.\n\n\\begin{figure}[hb]\n$$\\xymatrix@!0@=0.5cm{\n&&&&&&&&&& &&&&&&&&&& \\\\\n&& \\scriptstyle M_{7,17} && \\scriptstyle M_{8,18} &&&& \\scriptstyle M_{0,10} && &&&&&& \\scriptstyle M_{4,14} &&&& \\\\\n\\circ && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && \\circ && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && \\circ && \\circ && \\circ && *=0{\\circ} \\ar@{-}[rrrrdddd]+0 && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n*=0{\\circ} \\ar@{-}[rruu]+0 && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n*=0{\\circ} \\ar@{-}[rrrruuuu]+0 \\ar@{-}[rrrrdddd]+0 && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\\\\n\\circ && \\bullet & \\scriptstyle M_{0,4} & \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\bullet && \\bullet && \\circ && \\circ && \\bullet & \\scriptstyle M_{4,7} & \\circ && \\circ && \\circ && \\circ && \\circ && \\\\\n*=0{\\bullet} \\ar@{.}[uuuuuuuuuu]+0 \\ar@{-}[rrrrrrrruuuuuuuu]+0 && \\bullet && \\bullet && \\circ && *=0{\\bullet} \\ar@{-}[rrrrrrrruuuuuuuu]+0 && \\bullet && \\circ && *=0{\\circ} \\ar@{-}[rrrrrruuuuuu]+0 &&\n*=0{\\bullet} \\ar@{-}[rrrruuuu]+0 && \\circ && *=0{\\bullet} \\ar@{.}[uuuuuuuuuu]+0 \\\\\n\\scriptstyle M_{0,2} && \\scriptstyle M_{1,3} && \\scriptstyle M_{2,4} && \\scriptstyle M_{3,5} && \\scriptstyle M_{4,6} && \\scriptstyle M_{5,7} && \\scriptstyle M_{6,8} && \\scriptstyle M_{7,9} && \\scriptstyle M_{8,10} && \\scriptstyle M_{9,11} && \\scriptstyle M_{0,2}\n}$$\n\\caption{The indecomposable objects in the intersection of the four wings\n$\\mathcal W_{0,10},\\mathcal W_{4,14},\\mathcal W_{7,17}$ and $\\mathcal W_{8,18}$ in a tube of rank\n$10$ corresponding to a maximal rigid object of Pr\\\"{u}fer\ntype. The objects lying in the intersection are indicated by filled-in circles.\nThe intersection coincides with $\\mathcal W_{0,4}\\amalg \\mathcal W_{4,7}\\amalg \\mathcal W_{8,10}$\n(note that the wing $\\mathcal W_{i_3,i_4}=\\mathcal W_{7,8}$ is zero).}\n\\label{fig:wingsexample}\n\\end{figure}\n\nIn the following proposition and the sequel, we adopt the same convention\nfor the wings as in Lemma~\\ref{l:wings}(i).\n\n\\begin{proposition} \\label{p:ftgt}\nLet ${U}$ be maximal rigid in $\\overline{\\mathbf T}$.\n\\begin{itemize}\n\\item[(a)] If ${U}$ is of Pr{\\\"u}fer type, then\n$$\n(\\mathcal T_{U}, \\mathcal F_{U})  = ( {^{{\\perp}_{\\scriptscriptstyle H}}{U}} \\cap \\mathbf T, {^{{\\perp}_{\\scriptscriptstyle E}}{U}} \\cap \\mathbf T)\n  = (\\tau^{-1} (\\operatorname{Gen} {U} \\cap \\mathbf T), \\operatorname{Cogen} {U} \\cap \\mathbf T).\n$$\n\\item[(b)]\nAssume ${U}$ is of Pr{\\\"u}fer type with Pr{\\\"u}fer summands\n$M_{i_r,\\infty}$\nfor $r=0, \\dots, k-1$ where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = (\\coprod_{r=0}^{k-1}\n  \\mathcal T_r, (\\coprod_{r=0}^{k-1}\\mathcal F_r) \\amalg \\mathcal F_{\\infty})),$$ where\n$(\\mathcal T_r, \\mathcal F_r)$ is a torsion pair in $\\mathcal W_{i_r, i_{r+1}+1}$ with\n$\\mathcal F_r$ containing all of the projective objects in $\\mathcal W_{i_r,i_{r+1}+1}$\nand $\\mathcal F_{\\infty} = \\coprod_{r=0}^{k-1} \\mathcal R_{i_r}$.\n\n\\item[(c)]  If ${U}$ is of adic type, then\n$$(\\mathcal T_{U}, \\mathcal F_{U})  = ({U}\n  ^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T, {U} ^{{\\perp}_{\\scriptscriptstyle H}} \\cap \\mathbf T) =\n  (\\operatorname{Gen} {U} \\cap \\mathbf T, \\tau(\\operatorname{Cogen} {U} \\cap \\mathbf T)).$$\n\\item[(d)] Assume ${U}$ is of adic type with adic summands $M_{i_r,\\infty}$\n  for $r=0, \\dots, k-1$,  where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = ((\\coprod_{r=0}^{k-1}\n  \\mathcal T_r) \\amalg \\mathcal T_{\\infty}, \\coprod_{r=0}^{k-1}\\mathcal F_r),$$ where\n$(\\mathcal T_r, \\mathcal F_r)$ is a torsion pair in $\\mathcal W_{i_r, i_{r+1}+1}$ with\n$\\mathcal T_r$ containing all of the injective objects in $\\mathcal W_{i_r,i_{r+1}+1}$ and\n$\\mathcal T_{\\infty} = \\coprod_{r=0}^{k-1} \\mathcal C_{i_{r+1}+1}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nWe give the details for (a) and (b), while statements (c) and (d) can\nbe proved similarly (or using Lemmas~\\ref{properties} and ~\\ref{l:twistgen}).\nLet ${U}$ be maximal rigid of Pr{\\\"u}fer type in $\\overline{\\mathbf T}$.\nOur aim is to compute $\\mathcal F_{U}={}^{{\\perp}_{\\scriptscriptstyle E}} {U}\\cap \\mathbf T$ and then\n$\\mathcal T_{U}={}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F_{U}\\cap \\mathbf T$. We use the Pr\\\"{u}fer direct summands\nof ${U}$ in order to compute $\\mathcal F_{U}$ more precisely in terms of a set\nof wings in $\\mathbf T$. We then use this to compute $\\mathcal T_{U}$ using the theory of\ntorsion pairs in type A (see Section~\\ref{s:Dynkin}).\n\nLet ${U}_{\\mathbf T}$ be the direct sum of all indecomposable\ndirect summands of ${U}$ which are finitely generated.\nLet $M_{i_0,\\infty},\\ldots ,M_{i_{k-1},\\infty}$\nwith $0\\leq i_0<i_1<\\cdots <i_{k-1}\\leq n-1$ be the\nindecomposable Pr{\\\"u}fer summands of ${U}$.\n\nBy Lemma \\ref{perps}, we then have that the finite part ${U}_{\\mathbf T}$\nlies in $\\cap_{r=0}^{k-1} \\mathcal W_{i_r,i_r + n}$, which coincides with\n$\\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1}}$ by Lemma~\\ref{l:wings}(i).\nWe draw attention to the fact that the wings\nin the statement of the proposition are slightly larger than these ---\nthis will become clearer later in the proof.\n\nHence, we consider the decomposition ${U}_{\\mathbf T} =\\amalg_{r=0}^{k-1} {U}_r$, where each ${U}_r$ is\nrigid in $\\mathcal W_{i_r,i_{r+1}}$ (note that some of the ${U}_r$ then\nmight be $0$, i.e.\\ in the case when $i_{r+1}-i_r=1$).\nThen it follows easily from Lemma~\\ref{l:wings}(iv) that ${U}_r$ is\nmaximal rigid in $\\mathcal W_{i_r,i_{r+1}}$.\nSince ${U}_r$ is maximal rigid, its restriction to the abelian\ncategory $\\mathcal W_{i_r,i_{r+1}}$ is also cotilting. Our aim is to use this\ndecomposition of ${U}$ to compute $\\mathcal F_{U}$.\n\nIf $\\mathcal W_{i_r,i_{r+1}}$ is nontrivial, $M_{i_r,i_{r+1}}$\nis necessarily an indecomposable direct summand of ${U}_r$.\nIt is then straightforward to check that\n$$^{{\\perp}_{\\scriptscriptstyle E}}(\\amalg_{r=0}^{k-1} M_{i_r,i_{r+1}} \\cap \\mathbf T) =\n(\\coprod_{r=0}^{k-1}\\mathcal W_{i_r,i_{r+1}}) \\amalg \\coprod_{r=0}^{k-1} \\mathcal R_{i_r},$$\nand from this it follows that $\\mathcal F \\subset (\\coprod_{r=0}^{k-1}\\mathcal W_{i_r,i_{r+1}})\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r})$.\n\nWe also have that\n$^{{\\perp}_{\\scriptscriptstyle E}}{U}_{\\mathbf T}  \\cap \\mathcal W_{i_r,i_{r+1}} = {^{{\\perp}_{\\scriptscriptstyle E}}{U}_r}  \\cap \\mathcal W_{i_r,i_{r+1}}$,\nwhich gives us the following description of $\\mathcal F$:\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} (^{{\\perp}_{\\scriptscriptstyle E}}{U}_r)\\cap \\mathbf T)\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r}).$$\nSince ${U}_r$ is cotilting in $\\mathcal W_{i_r,i_{r+1}}$, we have that\n$^{{\\perp}_{\\scriptscriptstyle E}}{U}_r  \\cap \\mathcal W_{i_r,i_{r+1}} = \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r$.\nFrom this it follows that\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r)\n\\amalg \\coprod_{r=0}^{k-1} \\mathcal R_{i_r}.$$\nNoting that $\\operatorname{Cogen} M_{i,\\infty} \\cap \\mathbf T = \\mathcal R_i$,\nwe see that $\\mathcal F = \\operatorname{Cogen} {U} \\cap \\mathbf T$ as claimed in (a).\n\nNext, we consider slightly larger wings, $\\mathcal W_{i_r,i_{r+1}+1}$ in order to\nobtain the description of $\\mathcal F$ in (b).\nFor each $r$, define $\\widetilde{{U}}_r = {U}_r \\amalg M_{i_r, i_{r+1}+1}$. Then\n$\\widetilde{{U}}_r$ is a cotilting module in $\\mathcal W_{i_r, i_{r+1}+1}$.\nNote that if $U$ has exactly one Pr\\\"{u}fer summand, we need to\nregard $\\mathcal W_{i_r,i_{r+1}+1}=\\mathcal W_{i_0,i_0+n+1}$ as being a type $A$ category non-exactly embedded in the tube, since the image of the projective-injective object is not rigid.\n\nNote that\n$$\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\widetilde{{U}}_r=(\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}}{U}_r)\\amalg\n\\operatorname{add} M_{i_r,i_{r+1}}.$$\nWe can thus rewrite $\\mathcal F$ as follows:\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r)\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r}).$$\nSo, setting $\\mathcal F_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r$, we obtain a\ndescription of $\\mathcal F$ as claimed in (b).\n\nNext, we compute $\\mathcal T$.\nBy definition, we have that $\\mathcal T= {^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F} \\cap \\mathbf T$. Furthermore,\n$$^{{\\perp}_{\\scriptscriptstyle H}}( \\amalg_{r=0}^{k-1} M_{i_r,i_{r+1}}) \\cap \\mathbf T= (\\coprod_{r=0}^{k-1}\n\\tau^{-1} \\mathcal W_{i_r,i_{r+1}}) \\amalg (\\coprod_{r_0}^{k-1} \\mathcal C_{i_{r+1} +1}).$$\n\nNoting that\n$\\operatorname{Hom}(M_{l,i_{r+1}+1}, M_{i_r,i_{r+1}+1}) \\not= 0$ for $l \\leq i_r$, we see\nthat\n$$\\mathcal T \\subset \\coprod_{r=0}^{k-1} \\tau^{-1} (\\mathcal W_{i_r,i_{r+1}}) \\subset \\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1}+1}.$$\nFrom this, it follows that\n$\\mathcal T = \\coprod_{r=0}^{k-1}(^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F_r \\cap \\mathcal W_{i_r,i_{r+1}+1})$, noting\nthat $$\\mathcal F_r= \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r.$$\nUsing the fact that $\\widetilde{{U}}_r$ is cotilting in\n$\\mathcal W_{i_r,i_{r+1}+1}$, combined with Corollary~\\ref{cor:Gen-Cogen},\nwe see that\n$\\mathcal T = \\coprod_{r=0}^{k-1}\\mathcal T_r$, where\n$$\\mathcal T_r = {^{{\\perp}_{\\scriptscriptstyle H}}\\widetilde{{U}}_r} \\cap\n\\mathcal W_{i_r,i_{r+1}+1} = \\operatorname{Gen}_{\\mathcal W_{i_r, i_{r+1}+1}} \\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} \\widetilde{{U}}_r$$\nis the torsion part of the torsion pair in $\\mathcal W_{i_r,i_{r+1}+1}$ with torsion-free\npart $\\mathcal F_r$.\nWe have that $^{{\\perp}_{\\scriptscriptstyle H}}(\\amalg_{r=0}^{k-1} M_{i_r, \\infty}) =\n\\coprod_{r=0}^{k-1} \\mathcal W_{i_r, i_{r+1}}$, and therefore $\\mathcal T= {^{{\\perp}_{\\scriptscriptstyle H}}{U}} \\cap\n\\mathbf T$.\n\nSince $\\operatorname{Gen} M_{i, \\infty} \\cap \\mathbf T$ is zero for all $i$ and\nsince we have\n$\\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} \\widetilde{{U}}_r  = \\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} {U}_r=\n\\tau^{-1} {U}_r$, we see that\n$\\mathcal T = \\tau^{-1} {\\operatorname{Gen} {U} \\cap \\mathbf T}$.\n\nSee Figure~\\ref{fig:wing} for an illustration of one of the wings\n$\\mathcal W_{i_r,i_{r+1}+1}$. We have that $\\mathcal F_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\widetilde{{U}}_r$.\nThe indecomposable objects in $\\mathcal F_r$ apart from $M_{i_r,i_{r+1}+1}$ lie\nin the left hand shaded triangle. Also, $\\mathcal T_r=\\operatorname{Gen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1}\\widetilde{{U}}_r$,\nand the indecomposable objects in $\\mathcal T_r$ lie in the right hand shaded triangle.\n\\begin{figure}\n\\psfragscanon\n\\psfrag{M1}{$\\scriptstyle M_{i_r,i_r+2}$}\n\\psfrag{M2}{$\\scriptstyle M_{i_r+1,i_r+3}$}\n\\psfrag{M3}{$\\scriptstyle M_{i_{r+1}-2,i_{r+1}}$}\n\\psfrag{M4}{$\\scriptstyle M_{i_{r+1}-1,i_{r+1}+1}$}\n\\psfrag{M5}{$\\scriptstyle M_{i_r,i_{r+1}}$}\n\\psfrag{M6}{$\\scriptstyle M_{i_r,i_{r+1}+1}$}\n\\psfrag{M7}{$\\scriptstyle M_{i_r,i_{r+1}+1}$}\n\\includegraphics[width=7cm]{wing.eps}\n\\caption{A wing $\\mathcal W_{i_r,i_{r+1}+1}$.}\n\\label{fig:wing}\n\\end{figure}\n\\end{proof}\n\n\\subsection{The maximal rigid module corresponding to a torsion pair}\nIn this section we give an explicit description of the inverse of the bijection\n${U}\\mapsto (\\mathcal T_{U},\\mathcal F_{U})$ in Theorem~\\ref{t:mainbijection} between maximal rigid\nobjects in $\\overline{\\mathbf T}$ and torsion pairs in $\\mathbf T$.\n\n\\begin{lemma} \\label{l:limG}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$. Then an indecomposable object\nin $\\varinjlim(\\operatorname{ind} \\mathcal F)$ either lies in $\\operatorname{ind}\\mathcal F$\nor it is a Pr\\\"{u}fer module which is the direct limit along a ray in\n$\\operatorname{ind} \\mathcal F$.\n\\end{lemma}\n\\begin{proof}\nIt is clear that any indecomposable in $\\mathcal F$ or Pr\\\"{u}fer module which\nis the direct limit along a ray in $\\operatorname{ind} \\mathcal F$ lies in\n$\\operatorname{ind}(\\varinjlim \\mathcal F)$.\nSo suppose that $X=\\varinjlim X_j$ is an indecomposable\nin $\\overline{\\mathbf T}$, where each $X_j$ is in $\\operatorname{ind} \\mathcal F$.\nFirstly note that $X$ cannot be an adic module, since the adic modules\ndo not lie in $\\varinjlim \\mathbf T$.\nNext, for any object ${T}$ in $\\mathcal T$, we have\n$$\\operatorname{Hom}({T},X)=\\operatorname{Hom}({T},\\varinjlim X_j)\\simeq \\varinjlim \\operatorname{Hom}({T},X_j)=0,$$\nusing Lemma~\\ref{limits}(b), since each $X_j$ lies in $\\mathcal F$.\nIt follows that if $X$ is in $\\operatorname{ind} \\mathbf T$, we have that $X$ lies in $\\operatorname{ind}\\mathcal F$.\nThe only other possibility is if $X$ is a Pr\\\"{u}fer module. If $Y$ is\nany indecomposable object in the corresponding ray in $\\mathbf T$,\nwe have an embedding $Y\\rightarrow X$. Hence $\\operatorname{Hom}({T},Y)=0$ for any object\n${T}$ in $\\mathcal T$, so $Y$ lies in $\\mathcal F$. The result follows.\n\\end{proof}\n\n\\begin{proposition} \\label{p:TfromFG}\n\\begin{enumerate}\n\\item[(a)] Let $(\\mathcal T,\\mathcal F)$ be a torsion pair of ray type. Then $(\\mathcal T,\\mathcal F)$ can\nbe written in the form $(\\mathcal T_{U},\\mathcal F_{U})$, where ${U}$\nis the direct sum of the indecomposable objects in\n\\begin{equation}\n\\varinjlim (\\operatorname{ind} \\mathcal F) \\cap (\\varinjlim (\\operatorname{ind} \\mathcal F))^{{\\perp}_{\\scriptscriptstyle E}}.\n\\label{e:rayinverse}\n\\end{equation}\nCompare~\\cite[Thm.\\ 1.5]{bk1}.\n\\item[(b)] Let $(\\mathcal T,\\mathcal F)$ be a torsion pair of coray type. Then $(\\mathcal T,\\mathcal F)$ can\nbe written in the form $(\\mathcal T_{U},\\mathcal F_{U})$, where ${U}$\nis the direct sum of the indecomposable objects in\n\\begin{equation}\n\\varprojlim (\\operatorname{ind} \\mathcal T) \\cap {}^{{\\perp}_{\\scriptscriptstyle E}}(\\varprojlim (\\operatorname{ind} \\mathcal T)).\n\\label{corayinverse}\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nWe only consider (a); the proof of (b) is similar.\nBy Theorem~\\ref{t:mainbijection} and Proposition~\\ref{p:ftgt}(b) and its proof,\nthere is a maximal rigid object ${U}$ in $\\overline{\\mathbf T}$ of Pr\\\"{u}fer type such\nthat $(\\mathcal T,\\mathcal F)=(\\mathcal T_{U},\\mathcal F_{U})$ and $\\mathcal T_{U}$, $\\mathcal F_{U}$ have the following description.\nLet the Pr{\\\"u}fer summands of ${U}$ be $M_{i_r,\\infty}$\nfor $r=0, \\dots, k-1$ where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = (\\coprod_{r=0}^{k-1}\n  \\mathcal T_r, (\\coprod_{r=0}^{k-1}\\mathcal F'_r) \\amalg \\mathcal F_{\\infty}),$$ where\n$\\mathcal T_r,\\mathcal F_{\\infty}$ are as in Proposition~\\ref{p:ftgt}(b) and\n$\\mathcal F'_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r$, where ${U}|_{\\mathbf T}=\\amalg_{r=0}^{k-1} {U}_r$\nwith ${U}_r$ in $\\mathcal W_{i_r,i_{r+1}}$.\n\nBy~\\eqref{e:extprojectives} in Section~\\ref{s:Dynkin},\n\\begin{align*}\n\\operatorname{ind} {U}_r &=  \\{X\\in \\operatorname{ind}\\mathcal F'_r\\,:\\, \\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\mathcal F'_r\\} \\\\\n&= \\{X\\in \\operatorname{ind}\\mathcal F'_r \\,:\\, \\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\operatorname{ind}\\mathcal F\\}.\n\\end{align*}\nNote that $\\mathcal F_{\\infty}=\\coprod_{r=0}^{k-1} \\mathcal R_{i_r} \\subset \\operatorname{ind}\\mathcal F$.\nIt follows that any indecomposable object $X$ in $\\mathbf T$ which satisfies $\\operatorname{Ext}^1(Y,X)=0$\nfor all $Y$ in $\\operatorname{ind} \\mathcal F$ must lie inside some wing $\\mathcal W_{i_r,i_{r+1}}$,\nand hence in some $\\mathcal F'_r$. Therefore\n\\begin{align*}\n\\operatorname{ind} {U}|_{\\mathbf T} &= \\operatorname{ind}(\\amalg_{r=0}^{k-1} {U}_r) \\\\\n&= \\{X\\in \\operatorname{ind} \\mathcal F\\,:\\,\\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\operatorname{ind}\\mathcal F\\}.\n\\end{align*}\n\nNext, by Lemma~\\ref{l:limG}, an indecomposable object in\n$\\varinjlim(\\operatorname{ind} \\mathcal F)$ either lies in $\\operatorname{ind}\\mathcal F$ or is the direct limit\nalong a ray in $\\mathcal F$, i.e.\\ it is a Pr\\\"{u}fer summand $M_{i_r,\\infty}$ of ${U}$.\nIf $Y$ is one of the latter summands and $X$ lies in $\\operatorname{ind}\\mathcal F$, we have\n$D\\operatorname{Ext}^1(Y,X)\\simeq \\operatorname{Hom}(X,\\tau Y)=0$,\nby Lemmas~\\ref{ARformula} and~\\ref{homvanishing}. Since the Pr\\\"{u}fer summands\nthemselves satisfy\n$\\operatorname{Ext}^1(M_{i_r,\\infty},M_{i_{r'},\\infty})=0$\nfor all $r,r'$, the result follows.\n\\end{proof}\n\n\\section{Geometric interpretation in the tube case}\n\\label{s:tubegeometric}\n\\subsection{Short exact sequences}\n\nLet $[i,j],[i',j']$ be arcs in $\\mathcal A(\\mathbb U(n))$, giving\nrise to arcs $\\pi_n([i,j]),\\pi_n([i',j'])$ in $\\mathcal A(\\mathbb A(n))$ with corresponding\nindecomposable objects $M_{ij},M_{i'j'}$ in ${U}$.\nSuppose that $I^-_{\\mathbb U(n)}([i,j],[i'+kn,j'+kn])$ (i.e.\\ the intersection\nnumber between arcs $[i,j]$ and $[i'+kn,j'+kn]$ in $\\mathbb U(n)$) is equal to $1$ for some\n$k$ in $\\mathbb{Z}$. Then, if $j'+kn>i+1$, we have four objects in the Auslander-Reiten quiver of $\\mathbf T$ as shown in Figure~\\ref{fig:diamond}(a); if $j'+kn=i+1$, we have three\nobjects as shown in Figure~\\ref{fig:diamond}(b).\n\n\\begin{figure}\n\\centering\n\\subfigure[Case $j'+kn>i+1$.]{\n$$\\begin{xy} *!D\\xybox{\n\\xymatrix@!0@=1.2cm{\n& M_{i'+kn,j} \\ar[dr] & \\\\\nM_{i'j'}=M_{i'+kn,j'+kn} \\ar[dr] \\ar[ur] && M_{ij} \\\\\n& M_{i,j'+kn} \\ar[ur]\n}}\\end{xy}$$}\n\\subfigure[Case $j'+kn=i+1$.]{\n$$$$\\begin{xy} *!D\\xybox{\n\\xymatrix@!0@=1.2cm{\n& M_{i'+kn,j} \\ar[dr] & \\\\\nM_{i'j'}=M_{i'+kn,j'+kn} \\ar[ur] && M_{ij} \\\\\n&\n}}\\end{xy}$$}\n\\caption{Objects and paths in the Auslander-Reiten quiver of $\\mathbf T$ corresponding\n to an intersection in $\\mathbb U(n)$ between arcs $[i,j]$ and $[i'+kn,j'+kn]$.}\n\\label{fig:diamond}\n\\end{figure}\n\nIn the case $j'+kn>i+1$, this corresponds to a non-split short exact sequence:\n\\begin{equation}\n0 \\rightarrow M_{i',j'} \\overset{f}{\\longrightarrow} M_{i'+kn,j}\\amalg M_{i,j'+kn}\n\\overset{g}{\\rightarrow} M_{ij}\\rightarrow 0,\n\\label{e:diamondses}\n\\end{equation}\nand in the case $j'+kn=i+1$, it corresponds to a short exact sequence:\n\\begin{equation}\n0 \\rightarrow M_{i',j'} \\overset{f}{\\longrightarrow} M_{i'+kn,j}\\overset{g}{\\rightarrow} M_{ij}\\rightarrow 0\n\\label{e:diamondsesb}\n\\end{equation}\nin $\\mathbf T$. As in~\\cite[Remark 4.25]{warkentin},\nthese can be interpreted geometrically in $\\mathbb U(n)$: see Figure~\\ref{fig:tubesplit-sum}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i'+kn}{$\\scriptstyle i'+kn$}\n\\psfrag{i}{$\\scriptstyle i$}\n\\psfrag{j'+kn}{$\\scriptstyle j'+kn$}\n\\psfrag{j}{$\\scriptstyle j$}\n\\subfigure[Case $j'+kn>i+1$.]{\\includegraphics[width=3.2cm]{tubesplitsum.eps}}\n\\quad \\quad \\quad\n\\subfigure[Case $j'+kn=i+1$.]{\\includegraphics[width=3.25cm]{tubesplitsum2.eps}}\n\\caption{Non-split extensions in $\\mathbf T$ represented geometrically.}\n\\label{fig:tubesplit-sum}\n\\end{figure}\n\nIf $j'+kn>i+1$, write $f=\\begin{pmatrix} f_1 \\\\ f_2 \\end{pmatrix}$ and\n$g=\\begin{pmatrix} g_1 & g_2 \\end{pmatrix}$.\nThen $f_1$ and $g_2$ are monomorphisms and $f_2$ and $g_1$ are epimorphisms.\nThe $f_i$ and $g_i$ are uniquely determined up to a choice\nof scalars. If $j'+kn=i+1$, $f$ is a monomorphism and $g$ is an epimorphism,\nagain uniquely determined up to a choice of scalars.\n\n\\begin{lemma}\nAny non-split short exact sequence with first term $M_{i'j'}$ and last\nterm $M_{ij}$ has the same form as~\\eqref{e:diamondses} or~\\eqref{e:diamondsesb}.\n\\end{lemma}\n\n\\begin{proof}\nLet\n\\begin{equation}\n0\\rightarrow M_{i'j'} \\overset{u}{\\rightarrow} E \\overset{v}{\\rightarrow} M_{ij} \\rightarrow 0\n\\label{e:typicalses}\n\\end{equation}\nbe an arbitrary non-split short exact sequence with first term $M_{i'j'}$ and\nlast term $M_{ij}$.\nBy Lemma~\\ref{l:uniserialclosure} we may write $E=E_1\\amalg E_2$ where $E_1$ is\nindecomposable and such that we have that, decomposing $u=\\begin{pmatrix} u_1 \\\\ u_2 \\end{pmatrix}$ and $v=(v_1,v_2)$, we have that $u_1$ a monomorphism.\nSince the sequence is not split, $u_1$ is not an isomorphism, so, denoting the length\nof an object $M$ in $\\mathbf T$ by $\\ell(M)$, we have\n$$\\ell(E_2)=\\ell(M_{ij})+\\ell(M_{i'j'})-\\ell(E_1)<\\ell(M_{ij}).$$\nSince $v_2$ is not an epimorphism, $v_1$ must be an epimorphism, again using\nLemma~\\ref{l:uniserialclosure}.\n\n\\emph{Case (i)}: Suppose first that $v_1u_1\\not=0$.\nIf an integer $k$ is such that $i'+kn\\leq i$ and $i+2\\leq j'+kn\\leq j$,\nthere is a homomorphism in $\\mathbf T$ from $M_{i'j'}$ to $M_{ij}$ obtained by\ncomposing an epimorphism from $M_{i'j'}$ to $M_{i,j'+kn}$ with a monomorphism\nfrom $M_{i,j'+kn}$ to $M_{ij}$ (i.e.\\ the two maps in the lower edges of the\ndiamond in Figure~\\ref{fig:diamond}).\n\nIt is easy to check that the homomorphisms of this kind (allowing $k$ to vary)\nform a basis of $\\operatorname{Hom}(M_{i'j'},M_{ij})$. Since $vu=0$ and $v_1u_1\\not=0$ and\n$v_1u_1$ is a scalar multiple of such a basis element, there must be an indecomposable\nsummand $X$ of $E_2$ such that $v_Xu_X$ is a scalar multiple of $v_1u_1$\n(where $u_X$, $v_X$ are the corresponding components of $u_2$, $v_2$).\nBut\n\\begin{equation}\n\\ell(X)\\leq \\ell(M_{ij})+\\ell(M_{i'j'})-\\ell(E_1),\n\\label{e:lowpoint}\n\\end{equation}\nand there is a unique path from $M_{i'j'}$ to $M_{ij}$ through\nsuch an $X$ giving rise to $g_1f_1$\n(i.e.\\ with $X=M_{i,j'+kn}$) from which it follows that\nwe have equality in~\\eqref{e:lowpoint} and thus that $E_2=X$ is\nindecomposable and $u_2$ is an epimorphism and $v_2$ is a monomorphism.\nIt follows that the short exact sequence~\\eqref{e:typicalses} is of\nthe form~\\eqref{e:diamondses} up to a choice of scalars.\n\n\\emph{Case (ii)}: Now assume that $v_1u_1=0$. This implies that\n$\\ell(E_1)\\geq \\ell(M_{ij})+\\ell(M_{i'j'})$,\nbut we also have $\\ell(E_1)\\leq \\ell(E_1\\amalg E_2)=\\ell(M_{ij})+\\ell(M_{i'j'})$,\nso we must have equality and $E=E_1$ is indecomposable.\nIt follows that~\\eqref{e:typicalses} is of the form~\\eqref{e:diamondsesb} up\nto a choice of scalars. The proof is complete.\n\\end{proof}\n\n\\begin{definition}\nWe call a collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ an \\emph{oriented Ptolemy\ndiagram} (in $\\mathbb A(n)$) if, whenever $\\pi_n([i,j])$ and $\\pi_n([i',j'])$ lie in\n$\\SS$ with $i'<i<j'<j$ then $\\pi_n([i,j'])$ (when $j'>i+1$) and $\\pi_n([i',j])$\nalso lie in $\\SS$ (see related definitions in Section~\\ref{s:geometricmodel} and~\\cite{ng,hjr}).\n\\end{definition}\n\nWe note that the additive closure of a collection of indecomposable objects in $\\mathbf T$ is closed\nunder extensions if and only if the corresponding collection of arcs is an oriented Ptolemy diagram in $\\mathbb A(n)$. It is also easy to check that if $\\pi_n([i,j])$\nis an arc in $\\mathcal A(\\mathbb A(n))$, then the indecomposable quotients of $M_{ij}$ are the $M_{i'j}$ where\n$i\\leq i'\\leq j-2$, i.e.\\ arcs in $\\mathbb A(n)$ corresponding to arcs in $\\mathbb U(n)$ with the same ending point and with starting point weakly to the right of $i$.\nCall these the \\emph{left-shortenings} of $\\pi_n([i,j])$. Similarly, submodules\nare given by right-shortenings.\n\nWe make the following remark, which follows from Proposition~\\ref{crossingnumbers}.\nRecall that $\\widetilde{\\mathcal A}(\\mathbb A(n))$ denotes $\\mathcal A = \\mathcal A(\\mathbb A(n))$ extended to include the homotopy classes of the arcs $\\pi_n([i,\\infty])$ and $\\pi_n([-\\infty,i])$.\n\n\\begin{remark}\nThe bijection $\\psi:\\pi_n([i,j])\\mapsto M_{ij}$ between $\\widetilde{\\mathcal A}(\\mathbb A(n))$ and $\\operatorname{ind}(\\overline{\\mathbf T})$\ninduces a bijection between maximal noncrossing collections of arcs in $\\mathbb A(n)$ (including\nthe infinite arcs) and maximal rigid objects in $\\overline{\\mathbf T}$.\n\\end{remark}\n\nWe can now describe the conditions on collections of arcs appearing in torsion\npairs in $\\mathbf T$, using the above and Lemmas~\\ref{l:tpclosure} and~\\ref{l:uniserialclosure}.\n\n\\begin{proposition}\n\\begin{enumerate}\n\\item[(a)]\nA collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ corresponds to the torsion part\nof a torsion pair in $\\mathbf T$ if and only if\n$\\SS$ is an oriented Ptolemy diagram in $\\mathbb A(n)$ and $\\SS$ is closed under left-shortening.\n\\item[(b)]\nA collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ corresponds to the torsion-free part\nof a torsion pair in $\\mathbf T$ if and only if $\\SS$ is an oriented Ptolemy diagram in $\\mathbb A(n)$\nand $\\SS$ is closed under right-shortening.\n\\end{enumerate}\n\\end{proposition}\n\nWe remark that, if ${U}$ is of Pr\\\"{u}fer type, by Proposition~\\ref{p:ftgt},\n$\\psi(\\operatorname{ind}\\mathcal T_{U})$ can be obtained by taking the closure of the set of arcs\ncorresponding to finitely generated indecomposable summands of ${U}$ under left\nshortening and rotating all resulting arcs one step to the right.\nWe obtain $\\psi(\\operatorname{ind}\\mathcal F_{U})$ by taking the closure of the set of arcs\ncorresponding to \\emph{all} of the summands of ${U}$ under right shortening.\n\nSimilarly, by Proposition~\\ref{p:TfromFG}, if $(\\mathcal T,\\mathcal F)$ is a torsion pair where\n$\\mathcal F$ generates $\\mathbf T$ (i.e.\\ of ray type), then $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$ can be obtained from\n$\\psi(\\operatorname{ind}\\mathcal F)$ by first adding any infinite arc $\\pi_n([i,\\infty])$ for which all\narcs $\\pi_n([i,j])$ for $j\\geq a$ for some $a$ lie in $\\psi(\\operatorname{ind}\\mathcal F)$.\nThen ${U}$ is the direct sum of the indecomposable objects corresponding to the\narcs $\\alpha$ in $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$ such that the pair\n$(\\beta,\\alpha)$ of arcs has no negative intersections for all\n$\\beta$ in $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$.\n\nSimilar descriptions can be given in the adic\/coray type case.\n\nFinally, we give an example in a tube of rank $n=14$ to illustrate Proposition~\\ref{p:ftgt} and the results in this section.\nThe arcs corresponding to the indecomposable direct summands of ${U}$ are\ndisplayed in Figure~\\ref{fig:tubeex}\n(only the beginnings of the infinite arcs are shown). Note that the Pr\\\"{u}fer modules\nwhich are indecomposable summands of ${U}$ are $M_{0,\\infty},M_{6,\\infty},M_{10,\\infty}$ and\n$M_{13,\\infty}$, so $i_0=0$, $i_1=6$, $i_2=10$ and $i_3=13$.\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeex.eps}\n\\caption{The maximal rigid object ${U}$}\n\\label{fig:tubeex}\n\\end{figure}\nThe arcs corresponding to the indecomposable objects in $\\mathcal T_{U}$ are displayed in\nFigure~\\ref{fig:tubeF}.\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeF.eps}\n\\caption{The torsion part, $\\mathcal T_{U}=\\tau^{-1}(\\operatorname{Gen} {U}\\cap \\mathbf T)$, of the torsion pair corresponding to ${U}$.\nThe arcs corresponding to indecomposable objects not in $\\tau^{-1}(\\operatorname{add} {U}\\cap \\mathbf T)$ are drawn in blue.}\n\\label{fig:tubeF}\n\\end{figure}\n\nThe arcs corresponding to the indecomposable objects in the $\\mathcal F_{U}\\cap \\mathcal W_{i_r,i_{r+1}+1}$\nare displayed in Figure~\\ref{fig:tubeG}\n(with dotted arcs indicating the indecomposable\nsummands of ${U}$ which are not in $\\mathbf T$ (or $\\operatorname{ind}\\mathcal F$)).\nNote that there are infinitely many additional arcs not displayed, corresponding to indecomposables in $\\mathcal F_{\\infty}$ but not in any of the $\\mathcal F_r$.\nThe missing arcs are $\\pi_{14}([0,j])$ for $j\\geq 8$, $\\pi_{14}([6,j])$ for $j\\geq 12$,\n$\\pi_{14}([10,j])$ for $j\\geq 15$ and $\\pi_{14}([13,j])$ for $j\\geq 16$.\nNote that, as indicated by Proposition~\\ref{p:ftgt}, the intersections of\n$\\mathcal T_{U}$ and $\\mathcal F_{U}$ with the wings $\\mathcal W_{i_r,i_{r+1}+1}$ (which are\n$\\mathcal W_{0,7},\\mathcal W_{6,11},\\mathcal W_{10,14}$ and $\\mathcal W_{13,15}$) are torsion pairs.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeG.eps}\n\\caption{The torsion-free part, $\\mathcal F_{U}=\\operatorname{Cogen} {U}\\cap \\mathbf T$, of the torsion pair corresponding to ${U}$.\nThe arcs not in $\\operatorname{add} {U}$ are drawn in red. The dashed arcs are the indecomposable summands of ${U}$\nwhich are not of finite length (and thus not in $\\mathcal F_{U}$).\nThe arcs $\\pi_{14}([0,j])$ for $j\\geq 8$, $\\pi_{14}([6,j])$ for $j\\geq 12$,\n$\\pi_{14}([10,j])$ for $j\\geq 15$ and $\\pi_{14}([13,j])$ for $j\\geq 16$ have been omitted\nfor clarity.}\n\\label{fig:tubeG}\n\\end{figure}\nIn Figure~\\ref{fig:tubeAR}, we show the indecomposable summands of $\\mathbf T$ and the indecomposable objects in $\\mathcal T_{U}$ and $\\mathcal F_{U}$ in the AR-quiver of the tube.\n\\begin{figure}[ht]\n$$\\xymatrix@!0@=0.35cm{\n&& \\ast &&&&&& \\ast && \\ast &&&&&&&&&&&& \\ast &&&&&& \\\\\n&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\\\\n\\ar@{--}[uurr] &&&&&&&&&&&&&&&&&&&&&&&&&&&& \\\\\n&\\cdot && \\cdot && \\circ \\ar@{--}[uuurrr] && \\circ \\ar@{--}[uuurrr] && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ \\ar@{--}[uuurrr] && \\cdot && \\cdot && \\cdot && \\circ \\ar@{-}[ur] & \\\\\n\\cdot && \\cdot && \\circ && \\circ && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\\\\n&\\cdot && \\circ && \\circ && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot & \\\\\n\\cdot && \\circ && \\bullet && \\Box && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\\\\n& \\circ && \\circ && \\cdot && \\Box && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot & \\\\\n\\circ && \\circ && \\cdot && \\bullet && \\Box && \\cdot && \\cdot && \\bullet && \\Box && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\circ && \\\\\n&\\circ && \\cdot && \\circ && \\cdot && \\Box && \\cdot && \\circ && \\bullet && \\Box && \\cdot && \\bullet && \\Box && \\cdot && \\circ & \\\\\n*=0{\\bullet}  \\ar@{.}[uuuuuuuuu]+0 && \\Box && \\bullet && \\Box && \\bullet && \\Box && \\circ && \\bullet && \\Box && \\Box && \\bullet && \\Box && \\Box && \\circ && *=0{\\bullet} \\ar@{.}[uuuuuuuuu]+0 &&\n}$$\n\\caption{The AR-quiver of the tube, showing the indecomposable summands of ${U}$ ($\\bullet$ or $\\ast$), $\\operatorname{ind}(\\mathcal T_{U})$ ($\\Box$) and $\\operatorname{ind}(\\mathcal F_{U})$ ($\\circ$ or $\\bullet$). The Pr\\\"{u}fer direct summands of ${U}$ are shown (symbolically) at the top of the diagram as asterisks.}\n\\label{fig:tubeAR}\n\\end{figure}\n\n\\noindent \\textbf{Acknowledgements}\nAll three authors would like to thank the referee for helpful comments on an\nearlier version of this manuscript, and the Mathematics Research Institute in\nOberwolfach for its support during a conference in February 2011.\nABB would also like to thank Karin Baur and the FIM at ETH for their support\nand kind hospitality during a visit in May 2011.\nKB would like to thank Aslak Buan and the NTNU for their kind hospitality\nduring a visit in December 2010.\nRJM would like to thank Karin Baur and the FIM at the ETH, Zurich, for their\nsupport and kind hospitality during a visit in Spring 2011,\nand would like to thank Andrew Hubery for a helpful conversation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe inner parts of some spiral galaxies show higher star formation\nrates than usual and this star formation is frequently arranged in a\nring or pseudo-ring pattern around their nuclei. \nIn general, Circumnuclear Star\nForming Regions (CNSFR), also referred to as ``hotspots'', are alike\nluminous and large disk HII regions, but look more compact and show\nhigher peak surphace brightness  (Kennicut et al. 1989). In many cases\nthey contribute substantially to the UV emission of the entire nuclear\nregion ({\\it e.g.} Colina et al. 2002). Their H$\\alpha$ luminosities overlap with those of HII galaxies being typically higher than 10$^{39}$ erg s$^{-1}$ which points to relatively massive ionizing star clusters. \nThese regions are expected to show a high metallicity as corresponds to their position \nnear the galactic bulge. They have considerable weight in the determination of abundance gradients, which in turn are widely used to constrain chemical evolution models, and constitute excellent\nlaboratories to study how star formation proceeds in high metallicity environments.\n\n\n\\section{Observations and reductions}\n12 CNSFRs were observed with the 4.2m WHT at the Roque de los Muchachos Observatory\nusing the ISIS double spectrograph, with the EEV12 and TEK4 detectors in \nthe blue and red arm respectively with the dichroic \nat $\\lambda$7500 \\AA\\ .  Gratings  R300B in the blue\narm and R600R in the red arm were used, covering  the spectral ranges  $\\lambda$3650\n- $\\lambda$7000 {\\AA} in the blue and  ($\\lambda$8850 -$\\lambda$9650) in the near IR, yielding spectral  dispersions of 1.73 {\\AA} pixel$^{-1}$ in the blue arm and  0.79 {\\AA} pixel$^{-1}$ in the red arm.  A slit width of 1 arcesec was used.  \nThe nominal spatial sampling was 0.4 arcsec pixel$^{-1}$ in each frame and the average seeing for this night was {$\\sim$}1.2 arcsec. \n\n\n\\begin{figure}\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF1a.eps}\n\\hfill\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF1b.eps}\n\\caption{{\\it Left panel}: Blue spectrum of region 4 in NGC~2903  {\\it Right panel}: Red spectrum of region 3 in NGC~3351.}\n\\label{fig}\n\\end{figure}\n\n\n The data were reduced using the IRAF (Image Reduction and Analysis Facility) package following standard procedures. The high spectral dispersion used in the near infrared allowed the almost complete elimination of the night-sky OH\nemission lines and, in fact, the observed  $\\lambda$9532\/$\\lambda$9069 ratio is close to the theoretical value of 2.48 in all cases. Telluric absorptions have been removed from the spectra \nof the regions by dividing by a relatively featureless continuum of a subdwarf star \nobserved in the same night. \n\n\n\\section{Results and discussion}\nExamples of blue and red spectra are shown in Figure 1.\nEmission line fluxes were measured using the IRAF SPLOT software package.\nThe presence of a conspicuous underlying stellar population, more evident in the\nblue spectra, in most observed regions complicates the measurements. An example of underlying absorption can be seen in the inset to the left panel of Figure 1. A two-component (emission and absorption) gaussian fit was performed in order to correct the Balmer lines for underlying absorption. An example of this procedure is shown in Figure 2.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=1.5in]{angelesdiazF2.eps}\n\\caption{Example of the fitting procedure used in order to correct the Balmer emission line intensities for the presence of the underlying stellar population.}\n\\label{fig}\n\\end{center}\n\\end{figure}\n\n\nThe low excitation of the regions, as evidenced by the weakness of the [OIII] $\\lambda$ 5007 \\AA\\ line (see left panel of Figure 1), precludes the detection and measurement of the auroral [OIII] $\\lambda$ 3463 \\AA\\ necessary for the derivation of the electron temperature. It is therefore impossible to obtain a direct determination of the oxygen abundances. Empirical calibrations have to be used instead. \nIn the left panel of Figure 3 we show the calibration of oxygen abundance by the commonly used $O_{23}$ parameter defined as $([OII]\\lambda 3727,29 + [OIII]\\lambda 4959,5007)\/H_\\beta$ (Pagel et al. 1979). Data on HII galaxies, disc HII regions and CNSFR are shown. The HII region sample ( P\\'erez-Montero \\& D\\'\\i az 2005) has been divided in under-solar (open triangles)  and over-solar (filled triangles) according to the empirical criterion given by D\\'\\i az \\& P\\'erez-Montero (2000), {\\it i. e.} $O_{23} \\leq$ 0.47 and -0.5$\\leq S_{23} \\leq$ 0.28 \\footnote{The sulphur abundance parameters $S_{23}$ is defined as  $([SII] + [SIII])\/H_\\beta$\\\\}. The HII galaxy data (filled squares) come from P\\'erez-Montero \\& D\\'\\i az (2003). CNSFR are represented by circles. Solid ones for our observed objects and open ones for regions in NGC~3310 and NGC~7714, known to have under-solar abundances (Pastoriza et. 1993; Gonz\\'alez-Delgado et al. 1995). As can be seen the calibration is two-folded, shows a considerable scatter and its high abundance end is not well sampled. The position of the observed CNSFR is indicated. Their observed  $O_{23}$ values, lower than those of the lowest abundance galaxy known (IZw18), indicates that CNSFR belong to the high abundance branch of the calibration showing possibly the highest metallcities shown by HII region like objects. In the right panel of Figure 3 the position of the regions is indicated in the $N2$ ([NII]\/ H$_\\alpha$) abundance calibration diagram (Denicol\\'o et al. 2002), which shows a linear behaviour. Again CNSFR appear to show the highest oxygen abundances.\n\n\n\\begin{figure}\n\\includegraphics[width=2.0in,height=2in]{angelesdiazF3a.eps}\n\\hfill\n\\includegraphics[width=2.0in,height=2in]{angelesdiazF3b.eps}\n\\caption{{\\it Left panel:} The $O_{23}$ abundance parameter calibration. {\\it Right panel:} The $N2$ abundance parameter calibration. The location of CNSFR is indicated}\n\\label{fig}\n\\end{figure}\n\n\n\nThe left panel of Figure 4 shows the [NII]\/[OII] ratio {\\it versus} the $N2$ abundance parameter. Since a good correlation has been found to exist between the [NII]\/[OII] ratio and the N$^+$\/O$^+$ ionic abundance ratio, which in turn can be assumed to measure the N\/O ratio, this graph is the observational equivalent of the N\/O {\\it vs} O\/H diagram. We can see that a very tight correlation exists for all represented objects: HII galaxies, low and high abundance HII regions and CNSFR. Again our observed regions show the highest N\/O ratios of the sample.\n\n\n\\begin{figure}\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF4a.eps}\n\\hfill\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF4b.eps}\n\\caption{{\\it Left panel:} Empirical version of the N\/O {\\it vs} O\/H relation.{\\it Right panel:} The $N2$ abundance parameter as a funtion of excitation, as measured by the [SII]\/[SIII] ratio.  \\vspace*{0.5cm}}\n\\label{fig}\n\\end{figure}\n\n\n\nThe right panel of Figure 4 shows the run of the excitation degree with metallicity for the observed regions through the [SII]\/[SIII] ratio, which has been shown to be a good ionization parameter indicator (D\\'\\i az et al. 1991), and the $N2$ parameter. It can be seen that the observed CNSFR show the lowest excitation of the sample.\n\nA hint on the ionizing temperature of the regions can be obtained through the use of the $\\eta$' parameter, which is a measure of the softness of the ionizing radiation (see V\\'\\i lchez \\& Pagel 1988) and increases with decreasing ionizing temperature. The left panel of Figure 5  shows the run of $\\eta$' with metallicity as parametrized by $N2$. Unexpectedly, CNSFR show values of $\\eta$' higher than those of over-solar disc HII regions. This is better appreciated in the right panel of the figure where CNSFR are seen to segregate from disc HII regions in the [OII]\/[OIII] {\\it vs} [SII]\/[SIII] diagram. The former cluster around the value of log$\\eta$' = 0.0 (T${ion}\\sim$ 40,000 K) while the latter cluster around log $\\eta$' = 0.8 (T${ion}\\sim$ 35,000 K).\n\nThis work has been supported by Spanish projects DGICYT- AYA-2004-08262-C03-03 and CM-ASTROCAM. \n\n\n\n\\begin{figure}\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF5a.eps}\n\\hfill\n\\includegraphics[width=2.55in,height=2.0in]{angelesdiazF5b.eps}\n\\caption{{\\it Left panel:} The $N2$ {\\it vs} $\\eta$' relation. {\\it Right panel:} Logarithmic relation between the [OII]\/[OIII] and [SII]\/[SIII] line ratios}\\label{fig}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\\label{intro}\nIon-molecule reactions represent a specific class of chemical reactions that differ from neutral-neutral reactions in several important aspects \\cite{bowers79a,ng92a,clary90a,gerlich92a,anderson01a,smith11a,oka13a}: Firstly, many important ion-molecule reactions are barrierless and exothermic, and are therefore fast, even at low temperatures. Secondly, the attraction forces between a nonpolar molecule and an ion are dominated, at long range, by the ion--induced-dipole interaction. The interaction potential scales with the intermolecular distance $R$ as $R^{-4}$ and extends to much larger distances than the potential between two neutral species. Finally, the centrifugal barriers in the intermolecular potential associated with different collisional partial waves are lowered by the long-range attraction so that at a given collisional energy more partial waves contribute to an ion-neutral collision than to the corresponding neutral-neutral collision. Ion-neutral collisions can thus often be described by classical dynamics at temperatures where quantum mechanical effects dominate the corresponding neutral-neutral collision.\n\nStudies of chemical processes involving ions at low temperature are motivated in part by the need to understand and model chemical reaction cycles in interstellar molecular clouds \\cite{herbst73a,gerlich06a,smith11a,oka13a}, which are characterized, depending on the nature of the cloud, by temperatures down to the cosmic background temperature of 2.7~K. They are also motivated by the desire to explore the regime of cold and ultracold chemistry where the reactivity is influenced by quantum phenomena~\\cite{krems09a,bell09a,meerakker12a,narevicius12a,heazelwood15a}.\n\nIon-molecule reactions can often be described, at low temperatures, by capture models (see, e.g., \\cite{gioumousis58a,bowers79a,su82a,clary85a,troe87a,clary90a,troe96a,dashevskaya05a,gao11a,dashevskaya16a}). Such models rely on the assumption that the rate coefficients only depend on the long-range electrostatic interactions between the charge $q$ of the ion and the induced (polarizability $\\alpha$) or permanent electric dipoles, quadrupoles, etc., of the neutral species and do not consider the details of the chemical transformation, which are determined by short-range interactions. Capture models usually express the rate coefficients $k$ as ratios $\\frac{k}{k_{\\rm L}}$ to the classical Langevin-capture rate coefficient \\cite{langevin05a} (in SI units)\n\\begin{equation}\\label{eq:langevin}\nk_{\\rm L}=\\sqrt{\\frac{\\alpha q^2}{4\\epsilon_0^2 \\mu}}=v_{\\rm rel}\\sigma_{\\rm L},\n\\end{equation}\nwhich is temperature independent and known to provide a good description of exothermic, barrier-free reactions between ions and polarizable molecules (e.g., H$_2$ or N$_2$) down to very low temperatures \\cite{mackenzie94a,glenewinkelmeyer97a,sanzsanz15a}, even below 1~K \\cite{dashevskaya05a}. In Eq.~\\eqref{eq:langevin}, $\\alpha$, $q$, $\\epsilon_0$, $\\mu$, $v_{\\rm rel}$, and $\\sigma_{\\rm L}$ are the polarizability of the neutral reactant, the charge of the ionic reactant, the electric constant, the reduced mass and the relative velocity of the collision partners, and the Langevin cross section, respectively. In the zero-collision-energy limit, the capture rate coefficients must deviate from the classical Langevin behavior and reach their quantum (q) s-wave-scattering values~$k_{{\\rm q}}$ \\cite{landau77a}. For collisions of ions with neutral atoms or molecules without permanent moments, for instance, $\\frac{k_{{\\rm q}}}{k_{\\rm L}}=2$ \\cite{vogt54a,fabrikant01a,dashevskaya05a,gao11a}.\n\nAs the collisional temperature rises above 0 K, the number of partial waves contributing to the scattering increases until the relative motion of the reactants can be described within the classical approximation. The transition from quantum ($k\/k_{\\rm L}=2$) to classical ($k\/k_{\\rm L}=1$) capture takes place in the sub-Kelvin range, even for ion-molecule reactions involving the lightest species, and has not been observed experimentally so far. For collisions of ions with neutral diatomic molecules, the rotational degrees of freedom of the neutral molecule get gradually excited in the range from 0.1 to 20 K and unlock themselves from, but remain strongly perturbed by, the anisotropic intermolecular potential and by Coriolis interactions \\cite{dashevskaya05a,maergoiz09a}. In this range, which also remains unexplored experimentally, strongly enhanced and quantum-state-specific rate coefficients are predicted theoretically \\cite{clary85a,troe87a,clary90a,wickham93a,troe96a,auzinsh08a,klippenstein10a,auzinsh13a,auzinsh13b,dashevskaya16a}.\n\nWe present here a measurement of the energy dependence of the cross section of the reaction\n\\begin{equation}\\label{eq:reaction}\n{\\rm H}_2^+ + {\\rm H}_2\\rightarrow {\\rm H}_3^+ + {\\rm H}\n\\end{equation}\nin the range of collision energies from $k_\\mathrm{B}\\cdot\\unit[10]{K}$ to $k_\\mathrm{B}\\cdot\\unit[300]{mK}$ ($k_\\mathrm{B}$ is Boltzmann's constant). When the collision energy is reduced below $k_\\mathrm{B}\\cdot\\unit[1]{K}$, we observe an increase of the rate coefficient and a clear deviation from the classical capture-rate coefficient. At a collision energy of $k_\\mathrm{B}\\cdot\\unit[1]{K}$, partial waves up to $\\ell=5$ contribute to the collision. On the basis of the predictions of Ref.~\\cite{dashevskaya05a} and the results presented in the accompanying article \\cite{dashevskaya16a}, we attribute the observed enhancement to the joint effects of the ion-quadrupole and Coriolis interactions in collisions involving ortho-H$_2$ molecules in the $j = 1$ rotational level, which make up 75\\% of the beam intensity in our experiments.\n\n\\section{Experimental procedure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{Fig1.pdf}\n\\caption{\\label{fig:setup}Schematic representation of the merged-beam apparatus used to study ion-molecule reactions at low collision energies, with the two skimmed supersonic beams initially propagating at an angle of 10$^\\circ$, the Rydberg-Stark deflector made of a bent printed circuit board (PCB) and used to merge the beams after laser excitation, the reaction zone located within an electrode stack (gray), which constitutes the linear time-of-flight spectrometer used to detect reactants and products separately. (MCP1) and (MCP2) Microchannel-plate detectors to monitor the flight times of Rydberg H$_2$ molecules and the ion-time-of-flight spectra, respectively.}\n\\end{center}\n\\end{figure}\n\nThe experiments are performed with a merged-beam apparatus designed for studies of ion-molecule reactions at low collision energies ~\\cite{allmendinger16a}. The experimental setup is depicted schematically in Fig.~\\ref{fig:setup}. Two supersonic molecular beams of H$_2$ are created by expansion of H$_2$ from liquid-nitrogen-cooled pulsed valves and initially propagate at an angle of 10$^\\circ$. The molecules in one of these two beams are excited from the $\\mathrm{X}\\, ^1\\Sigma_{\\mathrm g}^+ \\, (v=0,j=0)$ ground state to long-lived $nkm$ Rydberg-Stark states (designated H$_2^*$ below) with principal quantum number $n=22$ and the ion core in the $\\mathrm{X}^+\\, ^2\\Sigma_{\\mathrm g}^+\\,(v^+=0,N^+=0)$ state using a resonant three-photon excitation sequence~\\cite{seiler11b}. The two beams are then merged by deflecting the Rydberg molecules with a surface-electrode Rydberg-Stark deflector and accelerator (RYSDAC)~\\cite{allmendinger13a}.\n\nOnce the two beams are merged, they co-propagate through the reaction zone (see Fig.~\\ref{fig:setup}), where the reaction \n\\begin{equation}\\label{rydberg-reaction}\n{\\rm H}_2^* + {\\rm H}_2\\rightarrow {\\rm H}_3^+ + {\\rm H} + {\\rm e}^-\n\\end{equation}\nis observed by monitoring the yield of H$_3^+$ ions after extraction with a pulsed electric field and detection at a microchannel-plate detector located at the end of a time-of-flight mass-spectrometer (MCP2). The cross sections of Reaction~(\\ref{rydberg-reaction}) are equivalent, within the sensitivity limits of our experiment, to the cross section of Reaction~\\eqref{eq:reaction}, which can be regarded as taking place within the orbit of the distant Rydberg electron without being affected by it, \nas was demonstrated previously~\\cite{pratt94a,matsuzawa10a,allmendinger16a}. This equivalence, which has also been exploited in studies of the H$^+ + {\\rm H}_2$ reaction \\cite{wrede05a,dai05a}, can be rationalized by comparing the maximal impact parameter $r_L=\\sqrt{\\sigma_{\\rm L}\/\\pi}$ of the ion-neutral reaction to the classical Rydberg-orbit radius $\\langle r \\rangle_n =  a_0n^2$. Even at the lowest collision energies ($k_\\mathrm{B}\\cdot\\unit[300]{mK}$) investigated here, $\\langle r \\rangle_{n=22}$ is more than ten times larger than $r_L$. Moreover, in the long-lived Rydberg molecules that survive the time interval of more than 50 $\\mu$s between initial preparation by laser excitation and the reaction, the Rydberg-electron density close to the ion core is negligible. Consequently, the Rydberg electron does not influence the reaction but acts like a Faraday cage, which shields the charged reactants from heating by external stray fields and allows the control of the collision energy with the  precision achievable in neutral-neutral reactions~\\cite{henson12a,shagam15a,jankunas14a}. This control is achieved by varying the temperature of the pulsed valve generating the beam of ground-state molecules while adjusting the valve trigger time for optimal overlap of the two merged packets (\\textit{i.e.}, for maximal number $N_{\\mathrm{H}_3^+}^\\mathrm{max}$ of detected H$_3^+$ ions), or by acceleration of the H$_2^*$ molecules using the RYSDAC. \n\nThe distribution of collision energies is determined from the time- and position-dependent velocity distributions of the two beams. The packet of ground-state H$_2$ molecules has dispersed strongly when reaching the reaction zone. This dispersion, which leads to a strong correlation between spatial position and velocity, results from the short opening time of the pulsed valve (13~$\\mu$s) and the long distance of about 80~cm between the orifice of the valve and the reaction zone. The packet of Rydberg molecules, in contrast, is strongly localized when released from the RYSDAC and disperses only weakly, corresponding to a translational temperature of approximately 300~mK. The H$_2^*$ sample thus remains strongly localized in space as it propagates through the sample of ground-state H$_2$ molecules in the reaction zone. At very low mean collision energies, the distribution of collision energies, and thus the energy resolution of the measurement, is limited by the translational temperature of the H$_2^*$ Rydberg molecules. For higher relative velocities of the two beams, the energy resolution depends on the time during which the two beams are allowed to interact before the product ions are extracted. To adjust this time, the H$_3^+$ ion yield is measured by using either a one-pulse or a two-pulse extraction-field sequence, the latter permitting a higher collision-energy resolution, as explained in Ref.~\\cite{allmendinger16a}.\n\nUnder conditions where the opening time of the ground-state H$_2$ valve is minimized, we observe a weak, second gas pulse originating from the rebounce of the valve plunger. This rebounce pulse has been fully characterized  (see Ref.~\\cite{allmendinger16a}). For all experimental results presented below, we have either fully included its effects in the analysis or made sure that its effects are negligible.\n\nThe peak number density of H$_2$ molecules greatly exceeds that of the H$_2^*$ Rydberg molecules. The reaction probability per Rydberg molecule is, however, much smaller than one. The reaction thus follows pseudo-first-order kinetics and the number of formed H$_3^+$ ions is directly proportional to the rate coefficient $k$, the number $N_{\\mathrm{H}_2}$ of H$_2$ molecules and $N_{\\mathrm{H}_2^*}$ of H$_2^*$ molecules, and to the product, averaged over time ($t$) and volume ($V$),\nof the two normalized density distributions, which represents an overlap integral over the reactant distributions\n\\begin{equation}\\label{eq:Nh3}\nN_{\\mathrm{H}_3^+} = k \\, N_{\\mathrm{H}_2} \\, N_{\\mathrm{H}_2^*} \\langle \\rho_{\\mathrm{H}_2} \\rho_{\\mathrm{H}_2^*}\\rangle_{t,V,\\gamma}. \\;\n\\end{equation}\nThe relative number of H$_2$ molecules as a function of the valve temperature is determined in a separate measurement by electron-impact ionization directly in the reaction zone, while the relative number of H$_2^*$ molecules is obtained from the pulsed-field-ionization signal monitored at MCP2 which is recorded together with the H$_3^+$  ion signal (see Fig.~\\ref{fig:setup} and Ref.~\\cite{allmendinger16a}).  The overlap factor $\\langle \\rho_{\\mathrm{H}_2} \\rho_{\\mathrm{H}_2^*}\\rangle_{t,V,\\gamma}$ depends on the experimental parameters (indicated by the index $\\gamma$) and is obtained, together with the mean collision energy and energy resolution, by Monte-Carlo particle-trajectory simulations of the two beams. The simulations are based on the experimentally measured velocity and density distributions of the H$_2$ ground-state beam, the well-defined electric-potential functions applied to the RYSDAC, and the exact geometry and timings of the experiment, as described in Ref.~\\cite{allmendinger16a}. Additionally, the efficiency with which a product ion is detected depends on the center-of-mass velocity of the reactants, the velocity and propagation direction of the product ion, and the time interval between formation and extraction of the product ion.\nThe detection efficiency $D_\\gamma$ is also determined by Monte-Carlo simulations of the reaction and detection process~\\cite{allmendinger16a}. The relative rate coefficient of Reaction~(\\ref{eq:reaction}) is then obtained by dividing the measured number $N_{\\mathrm{H}_3^+}^\\mathrm{max}$ of H$_3^+$ ions by the product $N_{\\mathrm{H}_2} \\, N_{\\mathrm{H}_2^*} \\langle \\rho_{\\mathrm{H}_2} \\rho_{\\mathrm{H}_2^*}\\rangle_{t,V} D_\\gamma$ (see Eq.~\\eqref{eq:Nh3}).\n\n\\section{Experimental Results}\n\nIn a previous set of experiments \\cite{allmendinger16a}, we measured the relative cross section of Reaction~(\\ref{eq:reaction}) over the range of collision energies between $k_{\\rm B}\\cdot 60$~K and $k_{\\rm B}\\cdot 5$~K, typical for molecular clouds in the interstellar medium, under conditions where the velocity of the Rydberg-H$_2$ beam was always lower than that of the ground-state H$_2$ beam. To study the range of collision energies below $k_{\\rm B}\\cdot 5$~K in the present study, we slightly accelerated the Rydberg-H$_2$ beam to a velocity of 1800~m\/s and varied the collision energies by tuning the temperature of the ground-state H$_2$ valve from $-135\\ ^\\circ$C, at which point the velocity of the ground state molecules is lower than that of the Rydberg molecules, to $-105\\ ^\\circ$C, where it is higher. The relative mean velocity of the beams passes through zero at a ground-state-H$_2$-beam valve temperature of $-123\\ ^\\circ$C. The range of collision energies probed in this experiment is depicted in Fig.~\\ref{fig2}b, where the dots and the vertical bars indicate the mean value of the collision energies and the full widths of the distributions of collision energies, respectively, which we know precisely from separate measurements and from the full numerical simulation of the propagation of both beams, as described in detail in Ref.~\\cite{allmendinger16a}. When the relative mean velocity of the two beams passes through zero, the mean collision energy is only $k_{\\rm B}\\cdot 300$~mK, limited by the velocity distribution of the Rydberg H$_2$ beam, as discussed above. We estimate the systematic uncertainty in the determination of the valve temperature at which the relative velocity of the two beams crosses zero to be about 1~K, leading to a negligible uncertainty in the determination of the collision energy (see Fig.~\\ref{fig2}b).\n\nGiven that the distribution of collision energies is known and that the numbers of reactant molecules ($N_{\\mathrm{H}_2}$ and  $N_{\\mathrm{H}_2^*}$) and the overlap factor $\\langle \\rho_{\\mathrm{H}_2} \\rho_{\\mathrm{H}_2^*}\\rangle_{t,V,\\gamma}$ (see Eq.~(\\ref{eq:Nh3})) are independently determined up to a global scaling factor, one can predict the relative yield of H$_3^+$ products for an assumed energy dependence of the reaction rate. If, for instance, one assumes that the reaction rate is independent of the collision energy, as would be the case for a classical Langevin-capture behavior (see Eq.~(\\ref{eq:langevin})), the H$_3^+$ product yield depicted as dashed line in Fig.~\\ref{fig2}a is obtained. The weak oscillations in this curve have their origin in the small fluctuations of the particle densities and velocities measured experimentally. The fact that this dashed curve significantly  drops below 1 (by about 6\\%) at the lowest valve temperatures is the consequence of a reduced detection efficiency of the H$_3^+$ product ions. Because the velocity of the Rydberg-H$_2$ beam is kept constant during the measurement, the mean center-of-mass velocity of the reactants decreases as the ground-state H$_2$ beam velocity is reduced. The H$_3^+$ product ions that are emitted against the propagation direction of the merged beams, and which make the largest part of the H$_3^+$ signal, are detected slightly less efficiently as the center-of-mass velocity is reduced because the fastest H$_3^+$ ions emitted in the backward direction are lost from the detection volume before the product-extraction electric field is applied.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10 cm]{Fig2}\n\\caption{\\label{fig2} a) Normalized peak number of H$_3^+$ ions $N_\\mathrm{H_3^+}^\\mathrm{max}$ as a function of the temperature of the ground-state-H$_2$ valve: (red dots) experimental data points for H$_2^*$ Rydberg beam central velocity $v$(H$_2^*$)\\,=\\,1800~m\/s (ion extraction with a two-pulse sequence with $\\Delta t = 7 \\mu\\mathrm{s}$), scaled to 1.0 at -105$^\\circ$~C; (dashed line) simulation based on an energy-independent rate coefficient; (full line) simulation based on the rate coefficients predicted by Dashevskaya {\\it et al.} and displayed as green line in Fig. 2 of Ref. \\cite{dashevskaya16a}. b) Ranges of collision energies (vertical bars) and of mean collision energies (dots) probed experimentally as a function of the temperature of the ground-state-H$_2$ valve when the mean velocity of the Rydberg-H$_2$ beam selected by the RYSDAC is 1800~m\/s.}\n\\end{center}\n\\end{figure}\n\nThe H$_3^+$-product-ion yields we measure under these conditions are presented as red dots in Fig.~\\ref{fig2}a and systematically deviate from the dashed line, the deviation being largest at the valve temperatures for which the collision energy is smallest, i.e. around $-123\\ ^\\circ$C where the mean collision energy is $k_{\\rm B}\\cdot 300$~mK, and are smallest at the valve temperatures for which the collision energy is largest, i.e., around $-105\\ ^\\circ$C where the mean collision energy is $k_{\\rm B}\\cdot 1.05$~K. At this latter collision energy, the observed reaction rate does not deviate, within the uncertainty of our measurements, from the behavior predicted on  the basis of an energy-independent reaction rate coefficient. The largest deviation between our experimental data and the dashed line is \nabout 15\\% and unambiguously indicates that the reaction rate coefficient increases as the collision energy decreases towards zero.\n\nThe energy dependence of the ratio $k(E) \/ (\\kappa k_\\mathrm{L})$ is extracted from the data by dividing the measured number of H$_3^+$ ions (red dots in Fig.~\\ref{fig2}\\,a)) by the simulated product-ion yield assuming a constant reaction rate coefficient. The constant $\\kappa$ is chosen so that the experimental results match the theoretical results above 1 K (dashed black line in Fig.~\\ref{fig2}a)). The experimental data are binned in classes of collision energy. The results of this procedure are depicted by the green dots in Fig.~\\ref{fig3}, which show that the rate coefficients rise by about 15\\% when the mean collision energy is reduced from $k_{\\rm B}\\cdot\\unit[1.05]{K}$ to $k_{\\rm B}\\cdot\\unit[0.3]{K}$. The vertical bars in Fig.~\\ref{fig3} correspond to one standard deviation.\n\n\n\\begin{center}\n\\begin{figure}[tb]\n\\includegraphics[width=12 cm]{Fig3}\n\\caption{Comparison of the energy-dependence of the measured relative rate coefficients $k(E) \/ k_L$ (color dots) to the calculation by Dashevskaya \\textit{et al.} \\cite{dashevskaya16a} for normal H$_2$ (75\\% H$_2$ in $j=1$ and 25\\% H$_2$ in $j=0$) at fixed collision energies (solid line) and for collision energies averaged over the simulated experimental energy distributions (black dots, gray bars indicate one standard deviation). Green dots: two-pulse sequence ($\\Delta t = 7 \\mu\\mathrm{s}$) and H$_2^*$ Rydberg beam central velocity $v$(H$_2^*$)\\,=\\,1800~m\/s. Red dots: single-pulse sequence for $v$(H$_2^*$)\\,=\\,1700~m\/s. Blue dots: single-pulse sequence, $v$(H$_2^*$)\\,=\\,1540~m\/s. The absolute scaling of each experimental data set was chosen to minimize the deviation from the simulation. The simulation (black dots) is based on the experimental parameters of the two-pulse measurement (green dots), but the result is very similar for the other measurements.}\n\\label{fig3}\n\\end{figure}\n\\end{center}\nIn a second set of measurements, we reduced the mean velocity of the H$_2$ Rydberg molecules to 1700~m\/s by adapting the potentials applied to the RYSDAC and repeated the measurement by varying the temperature of the valve used to generate the ground-state H$_2$ beam over the corresponding range. The results of this measurement, analyzed in the same way as described above, are depicted by the red dots in Fig.~\\ref{fig3}. Although the experimental conditions of the two sets of measurements are different, the energy dependence of the extracted ratios $k(E) \/ k_\\mathrm{L}$ agrees within the experimental uncertainties, which illustrates the robustness and reliability of our measurement and analysis procedures. The results of a third set of measurements, recorded with a sample of H$_2$ Rydberg molecules prepared at a mean velocity of 1540~m\/s and depicted by blue dots in Fig.~\\ref{fig3}, does not reveal any systematic energy dependence of the rate coefficients at collision energies higher than $k_{\\rm B}\\cdot 1$~K and confirms the results previously reported in Ref.~\\cite{allmendinger16a}.\n\nIn the accompanying article, Dashevskaya  {\\it et al.} \\cite{dashevskaya16a} report a prediction of the energy-dependent rate coefficients of the reaction ${\\rm H}_2^+ + {\\rm H}_2\\rightarrow {\\rm H}_3^+ + {\\rm H}$ for a ground-state H$_2$ sample consisting of a mixture of 25\\% para-H$_2$ molecules in the $j=0$ rotational level and 75\\% ortho-H$_2$ molecules in the $j=1$ rotational level corresponding to the gas sample used in our experiments (see the green curve labeled $\\bar{\\chi}$ in their figure 2, which is reproduced as full black line in our Fig.~\\ref{fig3}). To see whether their data are compatible with our experimental results, we have calculated the H$_3^+$ ion yield from their data using our simulation program and the velocity and density distributions corresponding to our experiments (i.e., using the same input as used to generate the dashed line in Fig.~\\ref{fig2} except the reaction rate coefficient). This procedure resulted in the black dots presented in Fig.~\\ref{fig3}, where the horizontal gray bars indicate the range of collision energies sampled.  \n\nThe simulation indicates that the observed increase of the reaction rate at low collision energy is consistent with the energy dependence of the rate coefficient for the reaction ${\\rm H}_2^+ + {\\rm H}_2\\rightarrow {\\rm H}_3^+ + {\\rm H}$ calculated by Dashevskaya  {\\it et al.} \\cite{dashevskaya16a} when averaged over the experimental distribution of collision energies (black dots and gray bars in Fig.~\\ref{fig3}). Our experimental results thus reveal for the first time a pronounced and rather sudden departure of the reaction rate of the ${\\rm H}_2^+ + {\\rm H}_2\\rightarrow {\\rm H}_3^+ + {\\rm H}$ from the behavior predicted on the basis of the classical Langevin-capture model at low-collision energies.\n \n\\section{Conclusions}\n\nThe good agreement between experimental observations and simulations based on the energy-dependent rate coefficients calculated by Dashevskaya  {\\it et al.} \\cite{dashevskaya16a} leads to the conclusion that the mechanism responsible for the observed enhancement of the rate coefficient for low collision energies has its origin in the interaction between the charge of H$_2^+$ and the rotational quadrupole moment of the ground state of ortho-H$_2$ ($j=1$).  This interaction, which scales with the intermolecular separation $R$ as $1\/R^3$, leads to an anisotropic modification of the long-range scattering potential which is dominated by the isotropic charge--induced-dipole coupling (falling of as $1\/R^4$). \\textit{A priori} all orientations of the rotating H$_2$ molecule (or, equivalently, all values of the projection quantum number $\\omega = 0, \\pm 1$ of $j$ onto the collision axis) have equal probabilities. At large collision energies, no (re)locking of the ground-state H$_2$ intrinsic angular momentum takes place, and the anisotropic contributions average out. At low collision energies, however, the collision complex can follow the minimum energy trajectory adiabatically, leading to a ``locking'' of the intrinsic rotation of the H$_2$ molecule to the collision axis and an enhanced rate coefficient (see Fig.~\\ref{fig3}). In the limit of zero collision energy, which is not probed yet with sufficient precision in our experiments, the rate constant should approach the Bethe-Wigner limit of about 3.6~$k_\\mathrm{L}$, as given by $\\frac{1}{4}\\cdot 2+\\frac{3}{4}\\cdot 4.18$ for a $\\frac{1}{4}$-$\\frac{3}{4}$ para-ortho H$_2$ mixture at low temperature~\\cite{dashevskaya16a}.\n\nSeveral aspects of the low-collision-energy behavior predicted theoretically by Dashevskaya  {\\it et al.} \\cite{dashevskaya16a} remain untested by our experiments, such as the weak oscillations of the reaction rate coefficient at low collision energies, which provide information on the contributions of individual partial waves, and the magnitude of the reaction rate at the lowest energies, which is dominated by the relocking of the angular momentum of the ground-state H$_2$ molecules as the capture rate approaches the s-wave-scattering limit.\nWe expect that improvements of the signal-to-noise ratio and of the energy resolution of our measurements will make the observation of the predicted oscillations of the rate coefficient possible in the future and will enable us to observe an even larger departure from the classical Langevin-capture model.\n\n\\section*{acknowledgments}\nWe thank Hansj{\\\"u}rg Schmutz and Josef A. Agner for their help in the development of the experimental setup, and Professor J. Troe and Professor E. Nikitin for making the content of the accompanying article available to us prior to submission. This work is supported financially by the Swiss National Science Foundation under Project Nr.~200020-159848 and by the NCCR QSIT. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nConsider a compact metric space $\\Omega$ and a homeomorphism $T : \\Omega \\to \\Omega$. Such a pair $(\\Omega, T)$ will be called a \\emph{dynamical system} in this paper.\\footnote{It would be more accurate to call it a topological dynamical system, but we hope this slight abuse of language does not lead to any confusion. Given that we are interested in topological notions and quantities, this is the natural setting for us to work in.} We will freely use standard concepts from the theory of dynamical systems such as minimality and unique ergodicity; see, for example, the textbook \\cite{Wal}.\n\nThe set of real $2\\times 2$ matrices with determinant equal to one is denoted by ${\\mathrm{SL}}(2,{\\mathbb R})$. The elements of\n$$\nC(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}) := \\{ A : \\Omega \\to {\\mathrm{SL}}(2,{\\mathbb R}) : A \\mbox{ continuous}\\}\n$$\nare referred to as \\emph{continuous} ${\\mathrm{SL}}(2,{\\mathbb R})$ \\emph{cocycles}.\n\nAny continuous cocycle $A \\in C(\\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ gives rise to the skew-product\n$$\n(T,A) : \\Omega \\times {\\mathbb R}^2 \\to \\Omega \\times {\\mathbb R}^2, \\; (\\omega,v) \\mapsto (T \\omega, A(\\omega)v).\n$$\nFor $n \\in {\\mathbb Z}$, define $A_n : \\Omega \\to {\\mathrm{SL}}(2,{\\mathbb R})$ by $(T,A)^n = (T^n,A_n)$.\n\nA cocycle $A$  is called \\emph{uniformly hyperbolic} if there exists $L > 0$ with\n$$\n\\liminf_{n \\to \\infty} \\frac{1}{n}\\log\\|A_n (\\omega)\\| \\ge L\n$$\nuniformly in $\\omega \\in \\Omega$.\n\nOne says that $A$ is \\emph{uniform} if there is a number $L(A)$ such that\n$$\n\\lim_{n \\to \\infty} \\frac1n \\log \\|A_n(\\cdot)\\| = L(A)\n$$\nuniformly. Clearly, any uniform cocycle $A$  with $L(A)>0$ is uniformly hyperbolic.\n\nA cocycle may or may not be uniform. However, by the subadditive ergodic theorem, once an ergodic measure $\\mu$ is chosen, there is always a ($\\mu$-dependent) $L_\\mu(A)$ such that\n$$\n\\lim_{n \\to \\infty} \\frac1n \\log \\|A_n(\\omega)\\| = L_\\mu(A) \\text{ for $\\mu$-almost every } \\omega \\in \\Omega.\n$$\nThe numbers $L(A)$ and $L_\\mu(A)$ are called \\emph{Lyapunov exponents}.\n\n\nFor our actual considerations, a further uniformity property of\ncocycles will be relevant. A cocycle $A \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}))$ is\nsaid to have \\textit{uniform behavior} if it is either uniformly\nhyperbolic,  or\n$$\n\\limsup_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| = 0\n$$\nuniformly in $\\omega \\in \\Omega$. Note that this latter condition can also be written as  $\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| = 0$\nuniformly in $\\omega \\in\\Omega$ (as $\\|B\\|\\geq 1$ for any $B \\in {\\mathrm{SL}}(2,{\\mathbb R})$, and hence $\\log \\|A_n (\\omega)\\| \\geq 0$).\n\n\n\\begin{remark}\\label{r.uniformvanishing}\nIt is known that the property\n$$\n\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| = 0 \\text{ uniformly in } \\omega \\in\\Omega\n$$\nis equivalent to the simultaneous vanishing of the Lyapunov exponent for all ergodic Borel probability measures $\\mu$,\n$$\n\\sup \\{ L_\\mu(A) : \\mu \\text{ ergodic} \\} = 0;\n$$\ncompare \\cite[Proposition~1]{AB07}, \\cite[Theorem~1]{S98}, and\n\\cite[Theorem~1.7]{SS00}. See also \\cite{F97} for the special case\nwhere there is only one ergodic measure and \\cite{BN14} for related\nwork.\n\\end{remark}\n\n\nLet us briefly discuss the relationship between these uniformity notions, see Appendix~\\ref{a.UH} for more details. For uniquely ergodic dynamical systems, a continuous cocycle is uniformly hyperbolic if and only if it is uniform with $L(A) > 0$. From this we immediately conclude that for uniquely ergodic dynamical systems, a continuous cocycle is uniform if and only if it has uniform behavior. For general dynamical systems, it is obviously true that a uniform cocycle has uniform behavior. However, the converse does not hold. Indeed, for any non-uniquely ergodic system, there exist uniformly hyperbolic continuous cocycles that are not uniform.\n\nWe will be interested in one-parameter families of cocycles. This is\npartly motivated by the application of our general results,\npresented below, to the case of Schr\\\"odinger cocycles, which\nnaturally depend on the energy parameter. Let $I \\subseteq {\\mathbb R}$ be an\ninterval in ${\\mathbb R}$ and equip $C(I\\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ with the topology of local uniform convergence. Define $ W(I, \\Omega)$ to be the set\n$$\n\\{ A \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})) : A(E,\\cdot)\\text{ has uniform behavior for each $E\\in I$} \\}.\n$$\nThen we have the following result:\n\n\\begin{theorem}\\label{t:general-general}\nLet $(\\Omega,T)$ be a dynamical system and let $I \\subseteq {\\mathbb R}$ be\nan interval. Then, $W(I,\\Omega)$ is a $G_\\delta$-set.\n\\end{theorem}\n\n\\begin{remark}\\label{r:after-main-theorem}\n(a) Of course the theorem can be applied with $I$ being just one\npoint. This gives that the set of $A \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}))$ with\nuniform behavior is a $G_\\delta$-set. This particular case was\nknown; see the first paragraph of the proof of\n\\cite[Theorem~1]{AB07}. In fact, under suitable assumptions on\n$(\\Omega,T)$, Avila and Bochi even show that it is a \\emph{dense}\n$G_\\delta$-set \\cite[Theorem~1]{AB07}.\n\\\\[1mm]\n(b) An inspection of the proof shows that $I$ could be chosen as any topological space that is a countable union of compact subspaces.\n\\end{remark}\n\n\nAs pointed out above, for uniquely ergodic dynamical systems, a\ncocycle is uniform if and only if it has uniform behavior, but in\ngeneral the set of uniform cocycles may be strictly smaller than the\nset of cocycles with uniform behavior. This naturally raises the\nquestion whether the set of uniform cocycles is a $G_\\delta$-set in\ngeneral. Thus, let us consider\n$$\nU(I, \\Omega) := \\{ A \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})) : A(E,\\cdot) \\text{ is uniform for each } E \\in I \\}.\n$$\n\nThe following theorem answers the question affirmatively.\n\n\\begin{theorem}\\label{t:main-abstract}\nLet $(\\Omega,T)$ be a dynamical system and let $I \\subset {\\mathbb R}$ be an\ninterval. Then, $U(I,\\Omega)$ is a $G_\\delta$-set.\n\\end{theorem}\n\n\\begin{remark}\n(a) With the obvious modifications, parts (a) and (b) of Remark~\\ref{r:after-main-theorem} apply here as well.\n\\\\[1mm]\n(b) For equicontinuous systems, it is known that the set of uniform\ncocycles is a dense $G_\\delta$-set (see Furman \\cite{F97}).\n\\end{remark}\n\nThe results above are relevant in the study of spectral properties of discrete one-dimensional Schr\\\"odinger operators with dynamically defined potentials. Operators of this kind arise as follows. The set of continuous $f : \\Omega\\longrightarrow {\\mathbb R}$ is denoted by $C(\\Omega,{\\mathbb R})$. Any choice of an $f\\in C(\\Omega,{\\mathbb R})$, commonly referred to as a \\emph{sampling function}, gives rise to \\emph{potentials} $V_\\omega(n) = f(T^n \\omega)$, $\\omega \\in \\Omega$, $n \\in {\\mathbb Z}$, and the associated \\emph{Schr\\\"odinger operators}\n$$\n[H_\\omega \\psi] (n) = \\psi(n+1) + \\psi(n-1) + V_\\omega(n) \\psi(n)\n$$\nin $\\ell^2({\\mathbb Z})$. The spectral theory of such operators has been reviewed in \\cite{D17} and it will be discussed in full detail in the forthcoming monographs \\cite{DF21a, DF21b}. We refer to these works for details on the concepts and results discussed next.\n\nIf $\\mu$ is a $T$-ergodic Borel probability measure, then the spectral properties of $H_\\omega$ are $\\mu$-almost surely independent of $\\omega \\in \\Omega$. For example, there are sets $\\Sigma, \\Sigma_\\mathrm{ac}, \\Sigma_\\mathrm{sc}, \\Sigma_\\mathrm{pp}$ such that $\\sigma(H_\\omega) = \\Sigma$ and $\\sigma_\\bullet(H_\\omega) = \\Sigma_\\bullet$, $\\bullet \\in \\{ \\mathrm{ac}, \\mathrm{sc}, \\mathrm{pp} \\}$ for $\\mu$-almost every $\\omega \\in \\Omega$.\n\nSeveral recent works have investigated the question of which\nspectral properties are generic. One usually fixes the base dynamics\n$(\\Omega, T)$ and studies the set of $f \\in C(\\Omega,{\\mathbb R})$ for which\na certain spectral phenomenon occurs. For example, Avila and Damanik\nshowed in \\cite{AD05} that $\\{ f \\in C(\\Omega,{\\mathbb R}) :\n\\Sigma_\\mathrm{ac} = \\emptyset \\}$ is  a dense $G_\\delta$-set for\nany ergodic $\\mu$, provided that $T$ is not periodic.\\footnote{By\nthe standard theory of periodic Schr\\\"odinger operators, the result\nclearly fails if the assumption is dropped.} A companion result was\nobtained by Boshernitzan and Damanik in \\cite{BD08}: $\\{ f \\in\nC(\\Omega,{\\mathbb R}) : \\Sigma_\\mathrm{pp} = \\emptyset \\}$ is residual (i.e.,\nit contains a dense $G_\\delta$-set), provided that $(\\Omega,T,\\mu)$\nhas the metric repetition property. See \\cite{BD08, BD09} for the\ndefinition of this property and many examples, including shifts and\nskew-shifts on tori.\n\nThe proofs of the results in \\cite{AD05, BD08} just mentioned rely on approximation of $f$ by functions taking finitely many values. Realizing that the absence of point spectrum, as well as the absence of absolutely continuous spectrum, are phenomena that are quite well understood in the setting of sampling functions taking finitely many values, the results in \\cite{AD05, BD08} then appear to be somewhat\nnatural.\\footnote{It should be noted, however, that they were both initially quite surprising as one had previously expected the presence of absolutely continuous spectrum for small quasi-periodic potentials, and the presence of point spectrum for operators generated by the standard skew-shift $T(\\omega_1,\\omega_2) =\n(\\omega_1 + \\alpha, \\omega_1 + \\omega_2)$.}\n\n\n\n\nLet us discuss some key concepts underlying the general theory and the results just mentioned. A cocycle of the form\n$$\n\\omega \\mapsto \\begin{pmatrix} g(\\omega) & -1 \\\\ 1 & 0 \\end{pmatrix}\n$$\nwith $g \\in C(\\Omega,{\\mathbb R})$ is called a \\emph{Schr\\\"odinger cocycle} and denoted by $A^g$. Given an operator family $\\{ H_\\omega \\}_{\\omega \\in \\Omega}$ as introduced above, the associated one-parameter family of Schr\\\"odinger cocycles $\\{ A^{E - f} \\}_{E \\in {\\mathbb R}}$ is intimately related to the study of the solutions of the associated difference equation\n$$\nu(n+1) + u(n-1) + V_\\omega(n) u(n) = E u(n)\n$$\nand hence provides important information. The parameter $E$ is referred to as the \\textit{energy} in this context.\n\n\n\nWe write\n\\begin{align*}\n\\mathcal{UH} & = \\{ E \\in {\\mathbb R} : A^{E-f} \\text{ is uniformly hyperbolic} \\}, \\\\\n\\mathcal{Z} & = \\{ E \\in {\\mathbb R} : L_\\mu(A^{E-f}) = 0 \\}, \\\\\n\\mathcal{NUH} & = {\\mathbb R} \\setminus (\\mathcal{UH} \\cup \\mathcal{Z} ).\n\\end{align*}\nNote that $\\mathcal{Z}$ and $\\mathcal{NUH}$ depend on the choice of ergodic measure $\\mu$, while $\\mathcal{UH}$ does not. This provides a ($\\mu$-dependent) partition of the energy axis: ${\\mathbb R} = \\mathcal{UH} \\sqcup \\mathcal{NUH}  \\sqcup \\mathcal{Z} $.\n\nLet us now relate the Lyapunov exponents with the spectra mentioned earlier. The Johnson-Lenz theorem \\cite{J86, L02} states that\n\\begin{equation}\\label{e.JL}\n\\mathcal{Z} \\subseteq \\Sigma \\subseteq \\mathcal{Z} \\cup \\mathcal{NUH}.\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{e.JL2}\n\\supp \\mu = \\Omega \\quad \\Rightarrow \\quad \\Sigma = \\mathcal{Z} \\cup \\mathcal{NUH}.\n\\end{equation}\n\nRecall that the essential closure of a measurable set $M \\subseteq {\\mathbb R}$ is given by $\\overline{M}^\\mathrm{ess} = \\{ E \\in {\\mathbb R} : {\\mathrm{Leb}}(M\n\\cap (E - \\varepsilon, E + \\varepsilon) > 0 \\text{ for every } \\varepsilon > 0 \\}$. The Ishii-Pastur-Kotani theorem\n\\cite{I73, K84, P80} (see also \\cite{D07, K97} for an exposition) states that\n\\begin{equation}\\label{e.IPK}\n\\Sigma_\\mathrm{ac} = \\overline{\\mathcal{Z}}^\\mathrm{ess}.\n\\end{equation}\nFinally, if the potentials $\\{ V_\\omega \\}$ take finitely many values and are $\\mu$-almost surely aperiodic, then by Kotani \\cite{K89}, we have\n\\begin{equation}\\label{e.K}\n{\\mathrm{Leb}}(\\mathcal{Z}) = 0,\n\\end{equation}\nwhich by \\eqref{e.IPK} implies that $\\Sigma_\\mathrm{ac} =  \\emptyset$. The very general result \\eqref{e.K} was alluded to in the discussion above as one of the general spectral phenomena in the setting of potentials taking finitely many values, and it forms the basis of the generic $C^0$ result from \\cite{AD05} also mentioned above.\n\nNote that under the assumption $\\supp \\mu = \\Omega$ (which holds, e.g., when $T$ is minimal) \\eqref{e.JL2} shows that $\\Sigma = \\mathcal{Z}$ if and only if $\\mathcal{NUH} =\\emptyset$. Now, for uniquely ergodic  dynamical systems uniform behavior is equivalent to uniformity, see appendix,  and $L_\\mu(A) =0$ if and\nonly if $A$ is uniform with $L(A) =0$. Thus, for minimal uniquely ergodic dynamical systems we have\n\\begin{equation}\\label{e.Zero-uniform}\n\\Sigma = \\mathcal{Z} \\Longleftrightarrow \\mathcal{NUH}=\\emptyset\n\\Longleftrightarrow A^{E-f} \\text{ is uniform for all $E\\in {\\mathbb R}$}.\n\\end{equation}\nFor general systems it follows from the definitions and Remark~\\ref{r.uniformvanishing} that\n$$\n\\{ E \\in {\\mathbb R} : A^{E-f} \\text{ has uniform behavior} \\} = \\mathcal{UH}\n\\cup \\bigcap_{\\mu \\text{ ergodic}} \\mathcal{Z}_\\mu.$$\nIn other\nwords, uniform behavior fails for $A^{E-f}$ precisely when\n$$\nE\\in \\bigcup_{\\mu \\text{ ergodic}} \\mathcal{NUH}_\\mu.\n$$\nThis gives\n\\begin{eqnarray*} \\mathcal{NUH}_\\mu = \\emptyset  \\text{\nfor each ergodic $\\mu$ } &\\Longleftrightarrow & A^{E-f} \\text{ has\nuniform behavior for all } E\\in {\\mathbb R}  \\\\\n&\\Longleftrightarrow & \\Sigma_\\mu = \\mathcal{Z}_{\\mathcal{U}}\\text{\nfor each ergodic $\\mu$}.\n\\end{eqnarray*}\nHere, $$\\mathcal{Z}_\\mathcal{U}:=\\{ E\\in {\\mathbb R} : A^{E-f} \\text{\nis uniform with } L(A^{E-f}) = 0\\} =  \\bigcap_{\\mu \\text{ ergodic}}\n\\mathcal{Z}_\\mu.\n$$\n\nIn any case, if the potentials $\\{ V_\\omega \\}$ take finitely many\nvalues, then \\eqref{e.K} implies zero-measure spectrum whenever one\ncan show that $\\mathcal{NUH} = \\emptyset$. Thus, pursuing a proof of\nthe absence of non-uniformity is a natural approach to zero-measure\nspectrum whenever a property such as \\eqref{e.K} is known. This\napproach is implemented in \\cite{DL06a, L02}, as well as in the\npresent paper.\n\nLet us mention that the zero-measure spectrum property has been investigated extensively for sampling functions taking finitely many values. From the classical results for the Fibonacci Hamiltonian \\cite{S89} or the more general class of operators with Sturmian potentials \\cite{BIST89} through numerous results for operators with potentials generated by substitutions to the general result\n\\cite{DL06a} by Damanik and Lenz, which covers many examples \\cite{DL06b}, this is a spectral statement that is quite ubiquitous in this setting.\n\nIt has therefore been a very natural open problem to find conditions on the base dynamics $T : \\Omega \\to \\Omega$ such that $\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma) = 0 \\}$ is residual. The paper\n\\cite{ADZ14} by Avila, Damanik, and Zhang discusses this question in\nthe particular case $T : {\\mathbb R}\/{\\mathbb Z} \\to {\\mathbb R}\/{\\mathbb Z}$, $\\omega \\mapsto \\omega +\n\\alpha$, $\\alpha \\not\\in {\\mathbb Q}$, but fails to answer it. Instead,\n\\cite{ADZ14} proves the weaker result that the singularity of the\nintegrated density of states is generic in this setting.\n\nNot only is the problem open in the case of irrational circle rotations, it is open in \\emph{any} setting and hence one of our goals is to exhibit the first class of base dynamics $T : \\Omega \\to \\Omega$ for which $\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma) = 0 \\}$ is a dense $G_\\delta$-set. At the same time we  will provide the\nfirst class of aperiodic base dynamics for which $\\{ f \\in C(\\Omega,{\\mathbb R}) : \\mathcal{NUH} = \\emptyset\\}$ or, equivalently, $\\{f\\in C(\\Omega,{\\mathbb R}) : \\Sigma = \\mathcal{Z}\\}$ is a dense $G_\\delta$-set. This is of interest as the equality $\\Sigma = \\mathcal{Z}$ is known in the periodic case and, hence, aperiodic dynamics giving this feature deserve particular attention.\n\nWe will work with aperiodic subshifts that satisfy the Boshernitzan\ncondition. Recall that a \\emph{subshift} is a closed shift-invariant\nsubset $\\Omega$ of $A^{\\mathbb Z}$, where $A$ is a finite set carrying the\ndiscrete topology and $A^{\\mathbb Z}$ is endowed with the product topology.\nThe map $T : \\Omega \\to \\Omega$ is given by the shift $(T \\omega)_n\n= \\omega_{n+1}$, and it is clearly a homeomorphism. We say that a\nsubshift $\\Omega$ satisfies the \\emph{Boshernitzan condition} (B) if\nit is minimal and there is a $T$-invariant Borel probability measure\n$\\mu$ such that\n$$\n\\limsup_{n \\to \\infty} n \\cdot \\min \\{ \\mu([w]) : w \\in \\Omega_n \\} > 0.\n$$\nHere $\\Omega_n = \\{ \\omega_1 \\ldots \\omega_n : \\omega \\in \\Omega \\}$ is the set of words of length $n$ that occur in elements of $\\Omega$ and $[w]$ is the cylinder set $[w] = \\{ \\omega \\in \\Omega : \\omega_1 \\ldots \\omega_n = w \\}$. This condition was introduced by Boshernitzan in \\cite{B92} as a sufficient condition for unique ergodicity.\n\n\\begin{theorem}\\label{t.main}\nSuppose $\\Omega$ is a subshift that satisfies the Boshernitzan\ncondition $\\mathrm{(B)}$. Then, the following holds:\n\n{\\rm (a)} The set\n$$\n\\{ f \\in C(\\Omega,{\\mathbb R}) : \\mathcal{NUH} = \\emptyset \\} = \\{f\\in C(\\Omega,{\\mathbb R}) : \\Sigma = \\mathcal{Z}\\}\n$$\nis a dense $G_\\delta$-set.\n\n{\\rm (b)} If $\\Omega$ is furthermore aperiodic, then the set\n$$\n\\{f\\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma) =0\\}\n$$\nis a dense $G_\\delta$-set.\n\\end{theorem}\n\n\nThe theorem has the following immediate consequence.\n\n\\begin{coro}\\label{c.main}\nSuppose $\\Omega$ is an aperiodic subshift that satisfies the\nBoshernitzan condition $\\mathrm{(B)}$. Then, zero-measure spectrum\ngiven by the vanishing set of the Lyapunov exponent  is generic,\nthat is, $$\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma) = 0 \\mbox{ and }\n\\Sigma = \\mathcal{Z} \\}$$\n is a dense $G_\\delta$-set.\n\\end{coro}\n\n\\begin{remark}\\label{r.afterCorollary1.7}\n(a) It is well known that the spectrum is always closed and, in the\ndynamically defined setting we consider, it never contains any\nisolated points. Thus, Corollary~\\ref{c.main} shows that Cantor\nspectrum of zero Lebesgue measure is generic when the base dynamics\nis given by an aperiodic subshift that satisfies $\\mathrm{(B)}$.\n\\\\[1mm]\n(b) As pointed out above, if the subshift $\\Omega$ satisfies (B),\nthen it is uniquely ergodic by \\cite[Theorem~1.2]{B92}. For this\nreason there is no ambiguity in writing $\\Sigma$ without specifying\n$\\mu$. On the other hand, the minimality of $\\Omega$ and the\ncontinuity of the sampling functions $f$ in question also imply the\nindependence of the spectrum of $\\omega$, so that in the setting of\nTheorem~\\ref{t.main}, $\\sigma(H_\\omega) = \\Sigma$ for every $\\omega\n\\in \\Omega$, not merely for $\\mu$-almost every $\\omega \\in \\Omega$.\n\\\\[1mm]\n(c) Many important classes of subshifts satisfy (B); see\n\\cite{DL06b} for a detailed discussion.\n\\\\[1mm]\n(d) It remains very interesting to clarify whether zero-measure\nspectrum is ($C^0$-) generic for quasi-periodic potentials, or at\nleast for one-frequency quasi-periodic potentials.\n\\end{remark}\n\n\nFinally, we note that our general result, Theorem~\\ref{t:main-abstract}, can also be seen in the context of a question of Walters on existence of non-uniform cocycles.\nSpecifically, Walters asks in \\cite{Wal1} whether every uniquely ergodic dynamical system (with non-atomic invariant measure) allows for a non-uniform cocycle. Walters discusses some examples,  where the answer is affirmative.  The question in general  seems to still be open with further partial results contained in \\cite{F97}. In\nthis situation, the following consequence of (the proof of) our spectral results may be of interest.\n\n\\begin{coro}\\label{c:walters}\nSuppose $\\Omega$ is an aperiodic subshift that satisfies the Boshernitzan condition $\\mathrm{(B)}$. Then, the set of uniform cocycles is a dense $G_\\delta$-set.\n\\end{coro}\n\n\n\\begin{remark}\nBased on these considerations we feel that aperiodic Boshernitzan subshifts are the best candidates for a potential negative answer to Walters' question, but at this time we are unable to extend the uniformity result to all continuous cocycles over a Boshernitzan subshift.\n\\end{remark}\n\n\nThe paper is organized as follows. We prove\nTheorem~\\ref{t:general-general} in Section~\\ref{s.3} and\nTheorem~\\ref{t:main-abstract} in Section~\\ref{s.4}. We then provide\na result on semicontinuity of the measure of the spectrum for\ngeneral dynamical systems in Section~\\ref{s.6} and a result on\ndenseness of cocycles for subshifts in Section \\ref{s.7}.  In\nSection~\\ref{s.5} we then  derive Theorem~\\ref{t.main} from results\nin the earlier sections. That section contains also the  proof of\nCorollary~\\ref{c:walters}.  Finally, there are two appendices, one\ndiscussing the relationships between the uniformity notions we\nconsider, and one discussing a consequence of the Avalanche\nPrinciple that we need in the earlier sections.\n\n\n\n\\section*{Acknowledgment}\n\nOur original version of part (b) of  Theorem~\\ref{t.main} was based on Theorem~\\ref{t.main2}. We are indebted to Jake Fillman for pointing out that the ideas in \\cite{DFL17} should make the proof possible that we now present.\n\n\n\n\n\n\n\n\n\n\n\\section{Cocycles With Uniform Behavior as a $G_\\delta$-Set}\\label{s.3}\n\nIn this section we prove Theorem~\\ref{t:general-general}. That is, we show that the set of cocycles with uniform behavior is a $G_\\delta$-set, and in fact we prove this result for families of cocycles depending on one real parameter.\n\nWe start with  a simple observation.\n\n\\begin{lemma}\\label{l:aux-zwei}\nLet $(\\Omega,T)$ be a dynamical system and $A \\in C(\\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$. If there exist $L > 0$ and $k \\in {\\mathbb N}$ with $M := \\max \\{ \\frac{1}{k} \\log \\|A_k(\\omega)\\| : \\omega \\in \\Omega\\} < L$, then\n$$\n\\frac{1}{n} \\log\\|A_n (\\omega)\\| < L\n$$\nfor all $\\omega \\in \\Omega$ and\n$$\nn \\geq \\frac{2 k \\max \\{ \\log \\|A(\\omega)\\| : \\omega\\in \\Omega\\}}{L- M}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSet  $N := \\frac{2  k  \\max \\{ \\log \\|A(\\omega)\\| : \\omega \\in \\Omega\\}}{L- M}$. By definition of $N$, we have\n\\begin{equation}\\label{hilfe}\n\\frac{1}{N} \\log \\|A_r(\\omega)\\| \\leq  \\frac{L-M}{2}\n\\end{equation}\nfor all $\\omega \\in \\Omega$, $r = 0, \\ldots, k$. Clearly, this estimate continues to hold if $N$ is replaced by any $n\\geq N$.\n\nConsider now an $n \\in {\\mathbb N}$ with $n \\geq N$. Of course,  $n$ can be uniquely written in the form $n = s k + r$ with $s \\in {\\mathbb N}\\cup \\{0\\}$ and $0 \\leq r < k$. By construction of the cocycle, we obtain\n$$\nA_{n}(\\omega) = A_r (T^{s k} \\omega) A_k (T^{(s-1)k} \\omega) \\ldots A_k (T^k\\omega) A_k(\\omega).\n$$\nTaking logarithms and using submultiplicativity of the matrix norm and additivity of the logarithm, we find\n\\begin{eqnarray*} \\frac{1}{n}\\log\\|A_n (\\omega)\\| &\\leq &\n\\frac{1}{n} \\log\\|A_r (T^{sk}\\omega)\\| + \\frac{1}{n}\\sum_{j=0}^{s-1}\n\\log\\|A_k (T^{jk} \\omega)\\|\\\\\n\\eqref{hilfe} &\\leq& \\frac{L-M}{2} + \\frac{1}{n}\\sum_{j=0}^{s-1} k\n\\frac{\\log\\|A_k (T^{jk} \\omega)\\|}{k}\\\\\n(\\mbox{definition of $M$}) &\\leq & \\frac{L-M}{2} +\n\\frac{1}{n}\\sum_{j=0}^{s-1} k M\\\\\n(sk \\leq n) &=& \\frac{L-M}{2} + M \\\\\n(M<L) &< & L\n\\end{eqnarray*}\nThis finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:general-general}]\nWe first consider a compact interval $I$.\n\nFor $\\varepsilon > 0$, we define $W_\\varepsilon$ to be the set of $A \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ such that for each $E \\in I$, the cocycle $A(E,\\cdot)$ is uniformly hyperbolic or there exists a $k \\in {\\mathbb N}$ with $\\frac{1}{n} \\log \\|A_n (\\omega)\\| < \\varepsilon$ for all $\\omega \\in \\Omega$ and $n \\geq k$. Clearly,\n$$\nW = \\bigcap_{m\\in{\\mathbb N}} W_{\\frac{1}{m}}.\n$$\nThus it suffices to show that $W_\\varepsilon$ is open for any $\\varepsilon > 0$. To do so, we consider $E \\in I$ arbitrary. There are two cases:\n\n\\smallskip\n\n\\textit{Case 1: $A(E,\\cdot)$ is uniformly hyperbolic.} As is well-known, the set of uniformly hyperbolic cocycles is open (see, e.g., \\cite{Z19}). As $A$ is continuous in the first variable, there exists a $\\delta > 0$ such that any $B \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ close enough to $A$ will have the property that $B(E',\\cdot)$ is uniformly hyperbolic for all $E' \\in (E-\\delta,E+\\delta) \\cap I$.\n\n\\smallskip\n\n\\textit{Case 2:  $A(E,\\cdot)$ satisfies $\\frac{1}{k} \\log \\|A_k (E,\\omega)\\| < \\varepsilon$ for all $\\omega \\in \\Omega$ for some $k \\in {\\mathbb N}$.} By continuity of $A$ and compactness of $\\Omega$, there exists a $\\delta > 0$ with\n$$\n\\sup_{\\omega \\in \\Omega, E' \\in (E-\\delta, E+\\delta) \\cap I} \\frac{1}{k} \\log \\|A_k (E',\\omega)\\| < \\varepsilon.\n$$\nThis same inequality will then also hold for any $B \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ sufficiently close to $A$. By Lemma~\\ref{l:aux-zwei} there exists then an $N \\in {\\mathbb N}$ with\n$$\n\\frac{1}{n} \\log \\|B_n (E',\\omega)\\| < \\varepsilon\n$$\nfor all $\\omega \\in \\Omega$, $n \\geq N$ and $E'\\in (E - \\delta , E + \\delta) \\cap I$ for all such $B$.\n\n\\smallskip\n\nSo, in both  of these two cases there is an open neighborhood $(E - \\delta, E + \\delta) \\cap I$ of $E$ such that any $B$ sufficiently close to $A$ shares the respective property of $A(E,\\cdot)$ for all $E'$ in this neighborhood. As $I$ is compact, the openness of $W_\\varepsilon$ then follows by standard reasoning.\n\n\\smallskip\n\nWe now consider an arbitrary interval $I$ in ${\\mathbb R}$. We can write $I$ as a countable union of compact intervals $I_n$, i.e.\\ $I = \\bigcup_{n \\in {\\mathbb N}}  I_n$. By what we have shown already, $W(I_n,\\Omega)$ is a $G_\\delta$-set for each $n \\in {\\mathbb N}$. For any $n \\in {\\mathbb N}$, there is the canonical embedding  $j_n : I_n \\times \\Omega \\longrightarrow I \\times \\Omega, (E,\\omega) \\mapsto (E,\\omega)$, and the associated restriction map\n$$\nR_n : C(I\\times \\Omega,{\\mathrm{SL}}(2,{\\mathbb R})) \\longrightarrow C(I_n \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})), \\; A \\mapsto A \\circ j_n.\n$$\nThen, $R_n$ is continuous. Hence, $R_n^{-1} (W(I_n,\\Omega))$ is a $G_\\delta$-set for each $n \\in {\\mathbb N}$  (as the inverse image of a $G_\\delta$-set under a continuous map) and so is then\n$$\nW(I,\\Omega) = \\bigcap_{n \\in {\\mathbb N}} R_n^{-1} (W(I_n,\\Omega)).\n$$\nThis finishes the proof.\n\\end{proof}\n\n\n\\section{Uniform Cocycles are a $G_\\delta$-Set}\\label{s.4}\n\nIn this section we prove Theorem~\\ref{t:main-abstract}. A pertinent\nidea is that for a uniquely ergodic dynamical system $(\\Omega,T)$, a\ncontinuous $B : \\Omega \\to {\\mathrm{SL}}(2,{\\mathbb R})$ is uniform if and only if\n\\begin{equation}\\label{e.uniformitycondition}\n\\lim_{n\\to \\infty} \\frac{1}{n} \\sup_{\\omega, \\varrho \\in \\Omega} \\left| \\log \\|B_n (\\omega)\\| - \\log \\|B_n (\\varrho)\\| \\right| = 0.\n\\end{equation}\n\nIndeed, it is clear that any uniform $B$ will satisfy \\eqref{e.uniformitycondition}. Conversely, any $B$ satisfying \\eqref{e.uniformitycondition} must be uniform as there exists (by the subadditive ergodic theorem) an $\\omega_0 \\in \\Omega$ with $\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\|B_n (\\omega_0)\\| = L(B)$. Some additional work will be needed to deal with the dependence on the parameter.\n\nWe start with two  auxiliary statements. For the convenience of the\nreader we include sketches of the proofs.\n\n\\begin{lemma}\\label{l:aux}\nLet $(\\Omega,T)$ be a dynamical system and $A \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R})$ be arbitrary.\n\\begin{itemize}\n\n\\item[(a)] If there exist $L > 0$ and $N \\in {\\mathbb N}$ with $\\frac{1}{k} \\log \\|A_k(\\omega)\\| < L$ for all $\\omega \\in \\Omega$ and $k = N,\\ldots, 2N$, then\n$$\n\\frac{1}{n} \\log\\|A_n (\\omega)\\| < L\n$$\nfor all $\\omega \\in \\Omega$ and $n \\geq N$.\n\n\\item[(b)] Let $c := \\max_{\\omega \\in \\Omega} \\{ \\log \\|A(\\omega)\\|, \\log \\|A^{-1}(\\omega)\\| \\}$. Then, for any $n \\in {\\mathbb N}$,\n$$\n\\left| \\frac{\\log \\|A_{n+1}(\\omega)\\|}{n+1} - \\frac{\\log\\|A_n (\\omega)\\|}{n}\\right| \\leq \\frac{1}{n+1} \\frac{\\log\\|A_n (\\omega)\\|}{n} + \\frac{c}{n+1}.\n$$\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\n(a) Consider $n \\geq N$. Then, we can uniquely write $n$ in the form $n = k N + r$ with $k \\in {\\mathbb N} \\cup \\{0\\}$ and $N \\leq r \\leq 2 N-1$. Now, the proof follows similar lines as the proof of Lemma~\\ref{l:aux-zwei}.\n\n\\smallskip\n\n(b) For invertible matrices $C,B$, we clearly have $\\|B C\\| \\leq \\|B\\| \\|C\\|$ and $\\|C\\| = \\|B^{-1} B C\\| \\leq \\|B^{-1}\\| \\|B C\\|$. Applying this with $C = A_n (\\omega)$ and $B = A(T^n \\omega)$, we infer (b) after a short computation.\n\\end{proof}\n\n\\begin{remark}\nIt follows from part (a) of the lemma that for any $L > 0$ and $N \\in {\\mathbb N}$, the set of $A \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}))$ with $\\sup_{\\omega \\in \\Omega, n \\geq N} \\frac{1}{n} \\log\\|A_n (\\omega)\\| < L$ is open.\n\\end{remark}\n\n\\smallskip\n\nWe now  show that the pointwise uniformity of the $A(E,\\cdot)$ appearing in the definition of $U(I,\\Omega)$ can be replaced by a uniform uniformity when $I$ is compact. This is the content of the next proposition.\n\n\\begin{prop}\\label{p:inclusion-one}\nLet $(\\Omega,T)$ be a  dynamical system. Let $I \\subset {\\mathbb R}$ be a compact interval. Consider $A \\in U(I, \\Omega)$. Then, for any $\\varepsilon > 0$, there exists $N \\in {\\mathbb N}$ with\n$$\n\\frac{1}{n}| \\log \\|A_n(E,\\omega)\\| - \\log \\|A_n (E,\\varrho)\\| | < \\varepsilon\n$$\nfor all $\\omega, \\varrho \\in \\Omega$, $E \\in I$, and $n \\geq N$.\n\\end{prop}\n\n\\begin{proof}\n\nAs $I$ is compact, it suffices to find for each $E \\in I$, a $\\delta > 0$ such that the desired estimate holds in $(E - \\delta, E + \\delta) \\cap I$. We consider two cases:\n\n\\smallskip\n\n\\textit{Case 1: $A(E,\\cdot)$ is uniform with $L(A(E,\\cdot)) > 0$}. The proof follows from Lemma~\\ref{l:consequence-avalanche} in the following way: Assume without loss of generality $\\frac{\\varepsilon}{46 L} < \\frac{1}{12}$. By uniformity of $A(E,\\cdot)$, there exists $N \\in {\\mathbb N}$ such that the assumptions of Lemma~\\ref{l:consequence-avalanche} will be satisfied with $L = L(A(E,\\cdot))$, $\\ell = N$ and $\\frac{\\varepsilon}{ 47 L}< \\frac{1}{12}$ instead of $\\varepsilon$. Now, as discussed in part (c) of Remark~\\ref{r:nach-avalanche-lemma}, the assumptions are open assumptions in the following sense: If they are satisfied for the cocycle $A(E,\\cdot)$ with $\\frac{\\varepsilon}{ 47   L}$, then for any $\\frac{1}{12} > \\varepsilon' > \\frac{\\varepsilon}{47 L}$, any $B$ sufficiently close to $A(E,\\cdot)$ will satisfy the assumptions as well with $\\varepsilon'$ instead of $\\frac{\\varepsilon}{47 L}$ and the same $L$ and $\\ell$. So, the conclusion of the lemma will hold for such $B$. With $\\varepsilon' = \\frac{\\varepsilon }{46 L}$, the conclusion of the lemma gives\n$$\nL \\Big( 1 - \\frac{44}{46 L}\\varepsilon \\Big) \\leq \\frac{1}{n} \\log\\|B_n (\\omega)\\| \\leq L \\Big( 1 + \\frac{\\varepsilon}{46 L} \\Big)\n$$\nfor all $n \\geq \\ell$ and $\\omega \\in \\Omega$ for any such $B$. This in turn implies\n$$\n\\frac{1}{n}| \\log \\|B_n(\\omega)\\| - \\log \\|B_n (\\varrho)\\| | < \\varepsilon\n$$\nfor all $\\omega \\in \\Omega$ and $n \\geq \\ell = N$ for any such $B$. By continuity of $A$ (in the first variable), there exists $\\delta > 0$ such that each $A(E',\\cdot)$ with $E' \\in (E - \\delta, E + \\delta) \\cap I$ is such a $B$. This gives the desired statement.\n\n\\smallskip\n\n\\textit{Case 2:  $A(E,\\cdot)$ is uniform with $L(A(E,\\cdot))=0$.} In this case, there exists $N \\in {\\mathbb N}$ with\n$$\n\\frac{1}{k} \\log \\|A_k (E,\\omega)\\| < \\varepsilon \/ 3\n$$\nfor all $\\omega \\in \\Omega$ and $k \\geq N$. By continuity of $A$, there exists a $\\delta > 0$ with\n$$\n\\frac{1}{k} \\log \\|A_k (E',\\omega)\\| < \\varepsilon \/ 2\n$$\nfor all $E' \\in (E - \\delta, E + \\delta) \\cap I$,  $\\omega \\in \\Omega$ and $k =N,\\ldots, 2N$. By (a) of Lemma~\\ref{l:aux}, we find\n$$\n\\frac{1}{n} \\log \\|A_n(E',\\omega)\\| < \\varepsilon \/ 2\n$$\nfor all $\\omega \\in \\Omega$, $E' \\in (E - \\delta, E + \\delta) \\cap I$, and $n \\geq N$, and this easily gives the desired statement in this case.\n\\end{proof}\n\nWhenever $(\\Omega,T)$ is a dynamical system and $I$ is a compact interval, we define for $n \\in {\\mathbb N}$,\n$$\n{\\widetilde{\\mathrm{Var}}}_n : C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})) \\longrightarrow [0,\\infty)\n$$\nby\n$$\n{\\widetilde{\\mathrm{Var}}}_n (A) := \\sup_{E \\in I} \\sup_{\\varrho, \\omega \\in \\Omega} \\{ \\left| \\log \\|A_n(E,\\omega)\\| - \\log \\|A_n(E,\\varrho)\\| \\right| \\}.\n$$\n\nBy the preceding proposition, any $A \\in U(I,\\Omega)$ satisfies\n$$\n\\lim_{n \\to \\infty} \\frac{1}{n} {\\widetilde{\\mathrm{Var}}}_n (A) = 0.\n$$\nIn fact, also an even stronger converse holds. This is the content of the next proposition.\n\n\\begin{prop}\\label{p:inclusion-two}\nLet $(\\Omega,T)$ be a dynamical system. Let $I \\subset {\\mathbb R}$ be a compact interval. Then, any $A \\in C(I \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ with\n$$\n\\liminf_{n \\to \\infty} \\frac{1}{n} {\\widetilde{\\mathrm{Var}}}_n (A) = 0\n$$\nbelongs to $U(I,\\Omega)$.\n\\end{prop}\n\n\\begin{proof}\nChoose $E \\in I$ arbitrary and write $A$ instead of $A(E,\\cdot)$. By the assumption on $A$, we can find $n_k \\in {\\mathbb N}$ with\n\\begin{equation}\n\\tag{*} \\delta_k := \\frac{1}{n_k} {\\widetilde{\\mathrm{Var}}}_{n_k} (A) \\to 0, \\; k \\to \\infty.\n\\end{equation}\n\nWe consider two cases:\n\n\\smallskip\n\n\\textit{Case 1: There exists $\\omega_0\\in\\Omega$ with $\\liminf_{k \\to \\infty} \\frac{1}{n_k} \\log \\|A_{n_k} (\\omega_0)\\| = 0$.} Without loss of generality we can assume $\\lim_{k \\to \\infty} \\frac{1}{n_k} \\log\\|A_{n_k} (\\omega)\\| = 0$. By $(*)$ this gives $\\lim_{k \\to \\infty} \\frac{1}{n_k} \\log \\|A_{n_k} (\\omega)\\| = 0$ uniformly in $\\omega \\in \\Omega$. By  Lemma~\\ref{l:aux-zwei}, we infer $\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| = 0$ uniformly.\n\n\\smallskip\n\n\\textit{Case 2:  There exists $\\omega_0 \\in \\Omega$ with $L := \\liminf_{k \\to \\infty} \\frac{1}{n_k} \\log \\|A_{n_k} (\\omega_0)\\| > 0 $.} Assume without loss of generality that\n$$\nL_k := \\frac{1}{n_k} \\log\\|A_{n_k} (\\omega_0)\\| \\to L, \\; k \\to \\infty.\n$$\n\nBy $(*)$ and the definition of ${\\widetilde{\\mathrm{Var}}}_n$, we then have for all $\\omega \\in \\Omega$,\n\\begin{equation}\\label{twostar}\n\\tag{**} L_k - \\delta_k \\leq \\frac{1}{n_k} \\log\\|A_{n_k}\\| (\\omega)\n< L_k + \\delta_k\n\\end{equation}\nwith $\\delta_k \\to 0$, $k \\to \\infty$, and $L_k \\to L$, $k \\to \\infty$. By (b) of Lemma~\\ref{l:aux}, we can assume without loss of generality that each $n_k$ is even (as we could otherwise replace $n_k$ by $n_k +1 $).\n\nBy $n_k \\to \\infty$,   Lemma~\\ref{l:aux-zwei} and the upper bound in \\eqref{twostar}, there exist $\\delta_k' > 0$ with $\\delta_k' \\to 0$, $k \\to \\infty$ and\n$$\n\\frac{1}{n} \\log \\|A_{n} (\\omega)\\| \\leq L + \\delta_k'\n$$\nfor all $n \\geq n_k\/2$. Also, by $n_k \\to \\infty$, we clearly have\n$\\frac{3}{4} L \\frac{n_k}{2} \\geq \\lambda_0$ (with $\\lambda_0$ from\nLemma~\\ref{l:consequence-avalanche}) for all sufficiently large $k$.\n\nFrom these considerations we see that for arbitrary $\\varepsilon <\n\\frac{1}{12}$, the assumptions of\nLemma~\\ref{l:consequence-avalanche} are satisfied with $\\ell =\n\\frac{n_k}{2}$, provided that $k$ is sufficiently large. The\nstatement of the lemma then gives the desired uniformity of $A$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:main-abstract}]\nIt suffices to consider a compact interval $I$ (compare the proof of Theorem~\\ref{t:general-general}). Set\n$$\nU_{n,\\varepsilon} := \\Big\\{ A \\in C(I \\times \\Omega,{\\mathrm{SL}}(2,{\\mathbb R})) : \\frac{1}{n} {\\widetilde{\\mathrm{Var}}}_n (A) < \\varepsilon \\Big\\}.\n$$\nBy continuity of $A$, the set $U_{n,\\varepsilon}$ is open. Hence,\n$$\n\\widetilde{U}_{N,\\varepsilon} := \\bigcup_{n \\geq N} U_{n,\\varepsilon}\n$$\nis open as well. Thus,\n$$\nW := \\bigcap_{N, k \\in {\\mathbb N}} \\widetilde{U}_{N, \\frac{1}{k}}\n$$\nis a $G_\\delta$-set.\n\n\\smallskip\n\nIt remains to show $W = U(I,\\Omega)$. To show this, we prove two inclusions:\n\n\\smallskip\n\n$U(I,\\Omega) \\subset W$: This is a direct consequence of Proposition~\\ref{p:inclusion-one}.\n\n\\smallskip\n\n$W \\subset U(I,\\Omega)$: It is not hard to see that\n$$\nW = \\Big\\{  A \\in C(I \\times \\Omega,{\\mathrm{SL}}(2,{\\mathbb R})) : \\liminf_{n \\to \\infty} \\frac{1}{n} {\\widetilde{\\mathrm{Var}}}_n (A) = 0 \\Big\\}.\n$$\nNow, the inclusion follows from Proposition~\\ref{p:inclusion-two}.\n\\end{proof}\n\n\n\\section{Upper Semicontinuity of the Measure of the Spectrum}\\label{s.6}\n\nIn this section we consider a dynamical system $(\\Omega,T)$ and the\nassociated Schr\\\"odinger operators and note that the map\n\\begin{equation}\\label{e.msigmadef}\nM_\\Sigma: C(\\Omega,{\\mathbb R}) \\to [0,\\infty), \\quad f \\mapsto\n{\\mathrm{Leb}}(\\Sigma_f)\n\\end{equation}\nis upper semi-continuous.  The proof uses variations of ideas\ndeveloped in \\cite{DFL17} in the context of continuum limit-periodic\nSchr\\\"odinger operators and was suggested to us by Jake Fillman.\nThis will then imply that $\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma_f) =\n0 \\}$ is a $G_\\delta$-set.\n\n\n\\begin{prop}\\label{p.semicont}\nThe map $M_\\Sigma$ defined in \\eqref{e.msigmadef} is upper\nsemi-continuous, that is, for every $\\delta > 0$, we have that\n$M_\\Sigma(\\delta) := \\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma_f) < \\delta\n\\}$ is open.\n\\end{prop}\n\n\\begin{proof}\nLet $\\delta > 0$ be given, and let us consider $f \\in\nM_\\Sigma(\\delta)$. We have to show that there exists $\\varepsilon >\n0$ such that every $g \\in C(\\Omega,{\\mathbb R})$ with $\\|f - g\\|_\\infty <\n\\varepsilon$ belongs to $M_\\Sigma(\\delta)$ as well.\n\nBy assumption, we have $\\varepsilon' := \\delta - {\\mathrm{Leb}}(\\Sigma_f) >\n0$. By basic properties of the Lebesgue measure, we can choose\nfinitely many open intervals $I_1, \\ldots, I_m$ with\n$$\n\\Sigma_f \\subset \\bigcup_{j = 1}^m I_j \\quad \\text{and} \\quad\n\\sum_{j = 1}^m |I_j| < {\\mathrm{Leb}}(\\Sigma_f) + \\frac{\\varepsilon'}{2}.\n$$\n\nLet us set $\\varepsilon := \\frac{\\varepsilon'}{4m} > 0$. By\nwell-known properties of the spectrum of a Schr\\\"odinger operator\nwith respect to $\\ell^\\infty$ perturbations of the potential, if\n$\\|f - g\\|_\\infty < \\varepsilon$, then $\\Sigma_g \\subset\nB_\\varepsilon(\\Sigma_f)$ (where the latter notation denotes the\n$\\varepsilon$ neighborhood).\n\nPutting these two ingredients together, we obtain\n$$\n\\Sigma_g \\subset B_\\varepsilon \\left( \\bigcup_{j = 1}^m I_j \\right),\n$$\nand hence\n$$\n{\\mathrm{Leb}}(\\Sigma_g) \\le {\\mathrm{Leb}} \\left( B_\\varepsilon \\left( \\bigcup_{j =\n1}^m I_j \\right) \\right) \\le 2m \\varepsilon + \\sum_{j = 1}^m |I_j| <\n2m \\varepsilon + {\\mathrm{Leb}}(\\Sigma_f) + \\frac{\\varepsilon'}{2} = \\delta,\n$$\nas desired. This completes the proof.\n\\end{proof}\n\n\\begin{remark} The statement of the proposition  can also be\nunderstood as follows: Let $\\mathcal{K}$ be  set of all compact\nsubsets of ${\\mathbb R}$ equipped with the the Hausdorff metric $d_H$ and let\n$S(\\ell^2 ({\\mathbb Z}))$ be the set of bounded self-adjoint operators\nequipped with the operator norm $\\|\\cdot\\|$. Then, the map\n$S(\\ell^2({\\mathbb Z}))\\ni A\\mapsto \\sigma(A)\\in \\mathcal{K}$, mapping a\nbounded self-adjoint operator to its spectrum is continuous and,\nactually, satisfies  $d_H(\\sigma(A),\\sigma(B))\\leq \\|A-B\\|$, by\nwell-known perturbation theory of self-adjoint operators. Moreover,\nthe map $\\mathcal{K}\\ni K\\mapsto {\\mathrm{Leb}}(K)\\in [0,\\infty)$ is upper\nsemi-continuous, as is certainly well-known (and can also be seen\nfrom the proof above). Altogether, we find that the map $S(\\ell^2\n({\\mathbb Z}))\\longrightarrow [0,\\infty), A\\mapsto {\\mathrm{Leb}} (\\sigma(A)),$ is\nupper semi-continuous. The statement of the proposition then follows\nby composition as the map $C(\\Omega,{\\mathbb R})\\longrightarrow\nS(\\ell^2({\\mathbb Z}))$, $f\\mapsto H_\\omega^f$, is continuous with\n$\\|H_\\omega^f - H_\\omega^g\\|\\leq \\|f-g\\|_\\infty$ for each\n$\\omega\\in\\Omega$.\n\\end{remark}\n\n\\begin{coro}\\label{c.gdeltaset} Let $(\\Omega,T)$ be a dynamical\nsystem. Then, the set $\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma_f) = 0\n\\}$ is a $G_\\delta$-set.\n\\end{coro}\n\n\\begin{proof}\nSimply write\n$$\n\\{ f \\in C(\\Omega,{\\mathbb R}) : {\\mathrm{Leb}}(\\Sigma_f) = 0 \\} = \\bigcap_{n \\in {\\mathbb N}}\nM_\\Sigma \\left( \\tfrac1n \\right)\n$$\nand use the fact that each $M_\\Sigma(1\/n)$ is open by\nProposition~\\ref{p.semicont}.\n\\end{proof}\n\n\n\n\n\n\\section{Denseness of Locally Constant Cocycles}\\label{s.7}\nIn this section we consider subshifts. Clearly, the set of locally\nconstant cocycles is dense in the set of continuous cocycles over a\nsubshift. Here, we show that a similar result holds for\none-parameter families of cocycles.\n\n\\begin{lemma}\nLet $(\\Omega,T)$ be a subshift and let $I$ be an interval\nin ${\\mathbb R}$. Then, the set\n$$\n\\{A\\in C(I \\times \\Omega,{\\mathrm{SL}}(2,{\\mathbb R})) : A(E,\\cdot) \\mbox{ is locally\nconstant for each $E\\in I$}\\}\n$$\nis dense in $C(I\\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$.\n\\end{lemma}\n\n\\begin{proof}\nWe consider ${\\mathrm{SL}}(2,{\\mathbb R})$ as a subspace of the space ${\\mathrm{M}}(2,{\\mathbb R})$ of real $2\\times 2$-matrices with metric induced by the standard norm on these matrices.\n\nLet $A : I\\times \\Omega \\longrightarrow {\\mathrm{SL}}(2,{\\mathbb R})$ continuous,\n$\\varepsilon >0$, and $J\\subset I$ compact be given.\n\nWe will construct a continuous $A' : J\\times \\Omega\\longrightarrow\n{\\mathrm{SL}}(2,{\\mathbb R})$ such that $A'(E,\\cdot)$ is locally constant for each\n$E\\in J$ and\n$$\n\\|A(E,\\omega)- A'(E,\\omega)\\|\\leq  \\varepsilon\n$$\nholds for all $E\\in J$ and $\\omega \\in \\Omega$.  This $A'$ can then be extended to a continuous function $A^* : I\\times \\Omega\\longrightarrow {\\mathrm{SL}}(2,{\\mathbb R})$, which is locally constant in the second argument,  by extending it constantly outside of the compact $J$. Specifically, with $J=[E_{\\min},E_{\\max}]$  we define $A^*(E,\\omega):= A'(E_{\\max},\\omega)$ for $E\\geq E_{\\max}$ and $A^*(E,\\omega) = A'(E_{\\min},\\omega)$ for $E\\leq E_{\\min}$.\n\nAs $A$ is continuous,  the set  $A(J\\times \\Omega)\\subset {\\mathrm{M}}(2,{\\mathbb R})$\nis compact. Hence, as the determinant is a continuous function on\n${\\mathrm{M}}(2,{\\mathbb R})$ and\n$$\n\\det C = 1 \\quad \\text{and} \\quad  \\frac{1}{\\sqrt{\\det C}} C - C = 0\n$$\nfor any $C \\in A(J\\times \\Omega)$ (since $A(J\\times \\Omega) \\subseteq {\\mathrm{SL}}(2,{\\mathbb R})$), there exists a $\\delta>0$ such that\n$$\n\\det C > 0 \\quad \\text{and} \\quad \\left\\| \\frac{1}{\\sqrt{\\det C}}  C - C \\right\\| \\leq \\frac{\\varepsilon}{2}\n$$\nfor any $C\\in {\\mathrm{M}}(2,{\\mathbb R})$ with distance from $A(J\\times \\Omega)$\nsmaller than $\\delta$. Without loss of generality we assume $\\delta\n\\leq \\frac{\\varepsilon}{2}$.\n\nBy continuity of $A$ again, we can find finitely many open sets $I_1,\\ldots, I_N$ in $I$ with $$J\\subset \\bigcup_k I_k$$ such that\n$$\n\\|A(E,\\omega) - A(E',\\omega)\\|< \\frac{\\delta}{2}\n$$\nfor all $\\omega\\in \\Omega$ whenever $E',E$ belong to the same $I_k$. As locally constant cocycles are dense in $C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}))$ we can then choose for each  $k=1,\\ldots, N$ a locally constant $B_k \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R})$ with\n$$\n\\|B_k(\\omega)- A(E,\\omega)\\| < \\delta\n$$\nfor any $\\omega\\in \\Omega$ and $E\\in I_k$.\n\nLet $\\varphi_k$, $k=1,\\ldots, N$,   be a partition of\nunity subordinate to $I_1,\\ldots, I_N$.  This means that each\n$\\varphi_k$ is a continuous non-negative function on $I$  with compact\nsupport contained in $I_k$  and\n$$\\sum_{k} \\varphi_k (E) =1$$\n for each $E\\in J$.  Define\n$$A_k :J\\times \\Omega\\longrightarrow {\\mathrm{M}}(2,{\\mathbb R}), (E,\\omega)\\mapsto\n\\varphi_k (E) B_k (\\omega).$$ Then, each $A_k$ is a continuous\nfunction and $A_k (E,\\cdot)$ is locally constant for each $E\\in J$. Hence,\n$$\n\\widetilde{A}:=\\sum_k A_k\n$$\nis a continuous function on $J\\times \\Omega$ and $\\widetilde{A}(E,\\cdot)$ is locally constant for each $E\\in J$. A short computation invoking $A(E,\\omega) = \\sum_k \\varphi_k(E) A(E,\\omega)$ for all $E\\in J$ and $\\omega\\in\\Omega$ shows\n$$\n\\|\\widetilde{A}(E,\\omega) - A(E,\\omega)\\| \\leq \\sum_k \\varphi_k\n(E) \\|B_k(\\omega) - A(E,\\omega)\\|<  \\sum_k  \\varphi_k(E)\\delta  =\n\\delta\n$$\nfor all $E\\in J$. Hence  by our choice of $\\delta$ we infer\n$$\n\\det \\widetilde{A}(E,\\omega) > 0 \\quad \\text{and} \\quad \\left\\| \\frac{1}{\\sqrt{\\det \\widetilde{A}(E,\\omega)}} \\widetilde{A}(E,\\omega) - \\widetilde{A}(E,\\omega) \\right\\| \\leq \\frac{\\varepsilon}{2}\n$$\nfor all $E\\in J$ and $\\omega\\in\\Omega$.  Define $A'$ on $J\\times\n\\Omega$ by\n$$\nA'(E,\\omega) := \\frac{1}{\\sqrt{\\det \\widetilde{A}(E,\\omega)}} \\widetilde{A}(E,\\omega).\n$$\nThen, $A'$ is continuous with values in ${\\mathrm{SL}}(2,{\\mathbb R})$ and $A'(E,\\cdot)$ is locally constant (as the determinant of the locally constant $\\widetilde{A}(E,\\cdot)$ is locally constant). By construction we find\n\\begin{eqnarray*}\n\\|A'(E,\\omega) - A(E,\\omega)\\| & \\leq & \\|A'(E,\\omega) - \\widetilde{A}(E,\\omega)\\| + \\|\\widetilde{A}(E,\\omega) - A(E,\\omega)\\|\\\\\n&\\leq &  \\frac{\\varepsilon}{2} + \\delta\\\\\n& \\leq& \\varepsilon\n\\end{eqnarray*}\nand the proof is finished.\n\\end{proof}\n\n\n\\begin{remark}\nThe proof carries over directly to  any compact topological space\n$I$.\n\\end{remark}\n\nFrom the preceding lemma and our main results we immediately obtain the following corollary.\n\n\\begin{coro}\\label{c:abstract-general}\nLet $(\\Omega,T)$ be a subshift over a finite alphabet.\n\n{\\rm (a)}  If all locally constant cocycles on $\\Omega$  have uniform\nbehaviour, then  for any interval $I$ the set $U(I,\\Omega)$ is a\ndense $G_\\delta$-set.\n\n{\\rm (b)}  If all locally constant cocycles on $\\Omega$ are uniform, then\nfor any interval $I$ the set $U(I,\\Omega)$ is a dense\n$G_\\delta$-set.\n\\end{coro}\n\n\\begin{proof}\n(a)  This follows from the preceding lemma and\nTheorem~\\ref{t:general-general}.\n\n(b) This follows from the preceding lemma and Theorem~\\ref{t:main-abstract}.\n\\end{proof}\n\n\n\n\n\n\\section{Generic Absence of Non-Uniform Hyperbolicity for Schr\\\"odinger Operators Over Boshernitzan Subshifts}\\label{s.5}\n\nIn this section we show that for a generic continuous sampling function over an aperiodic subshift satisfying the Boshernitzan condition, the associated Schr\\\"odinger cocycles are uniform for all energies and the associated spectrum is a Cantor set of Lebesgue measure zero equal to the vanishing set of the Lyapunov exponent. That is, we prove Theorem~\\ref{t.main} (and its corollary). We then\nalso point out a generalization.\n\n\n\n\nOur proof of Theorem~\\ref{t.main} relies on what we have shown in earlier sections  together with the  the following crucial feature of subshift satisfying (B).\n\n\\begin{lemma}[\\cite{DL06a,DL06b}]\\label{l:input-omega}\nLet $(\\Omega,T)$ be a subshift satisfying the Boshernitzan condition {\\rm (B)}. Then any locally constant cocycle is uniform. In particular, if $(\\Omega,T)$ is additionally assumed to be aperiodic, then  $\\Sigma =\\mathcal{Z}$ is a Cantor set of\nLebesgue measure zero for each Schr\\\"odinger operator associated to a locally constant $f\\in C(\\Omega,{\\mathbb R})$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem~\\ref{t.main}.]\n(a)   Clearly, the map\n$$\nS : C(\\Omega) \\longrightarrow C({\\mathbb R} \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})), \\; f\n\\mapsto \\left( (E,\\omega) \\mapsto A^{E - f}(\\omega) \\right)\n$$\nis continuous.  Hence, the inverse image under $S$ of any $G_\\delta$-set in $C({\\mathbb R} \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ is a $G_\\delta$-set in $C(\\Omega)$. Thus, the set $\\mathcal{G}$ consisting of $f \\in C(\\Omega)$ with $A(E, \\cdot) := A^{E -\nf(\\cdot)} \\in U({\\mathbb R}, \\Omega)$ is a $G_\\delta$-set by\nTheorem~\\ref{t:main-abstract}.\n\nMoreover, for subshifts satisfying (B), it is known that any locally constant $f \\in C(\\Omega)$ yields a one-parameter family $A(E, \\cdot) := A^{E - f(\\cdot)} \\in U({\\mathbb R},\\Omega)$; see Lemma~\\ref{l:input-omega}. As locally constant $f\\in C(\\Omega)$ are\ndense in $C(\\Omega)$, we infer that the set $\\mathcal{G}$ is dense as well. Altogether, this shows that $\\mathcal{G}$ is a dense $G_\\delta$-set.\n\n\nFinally, as mentioned already, any subshift satisfying (B) is\nuniquely ergodic and minimal. Hence, by the discussion in the\nintroduction, and in particular, by \\eqref{e.Zero-uniform}, the\nSchr\\\"odinger operator associated to $f\\in C(\\Omega,{\\mathbb R})$ satisfies\n$\\Sigma = \\mathcal{Z}$ if and only if $\\mathcal{NUH} =\\emptyset$\nholds, and this is the case if and only if the associated\nSchr\\\"odinger cycle is uniform for all $E\\in{\\mathbb R}$, i.e. if and only if\n$f\\in \\mathcal{G}$. As $\\mathcal{G}$ is a $G_\\delta$-set, this\nproves (a).\n\n\\smallskip\n\n(b) By Corollary~\\ref{c.gdeltaset}, the set  $\\{ f \\in C(\\Omega,{\\mathbb R})\n: {\\mathrm{Leb}}(\\Sigma_f) = 0 \\}$ is a $G_\\delta$-set. Moreover, by\naperiodicity and (B)  this set  is dense by\nLemma~\\ref{l:input-omega}. This shows (b).\n\\end{proof}\n\n\nAs a by-product of the considerations in the preceding proof we now deal with our result concerning the question of Walters.\n\n\\begin{proof}[Proof of Corollary~\\ref{c:walters}.]\nAny  locally constant cocycle on a subshift satisfying (B) is uniform, see Lemma~\\ref{l:input-omega}. Now, the corollary is immediate from (b) of Corollary~\\ref{c:abstract-general} (applied with an interval $I$ consisting of one point).\n\\end{proof}\n\n\nInvoking \\cite{AD05} we can give  also a variant of\nTheorem~\\ref{t.main}. This variant deals with  a more general\nsetting. We formulate it mainly as a reference point for potential\nfuture generalizations.\n\n\\begin{theorem}\\label{t.main2}\nLet $(\\Omega,T)$ be an aperiodic dynamical system. Assume that the set\n$$\n\\{ f \\in C(\\Omega,{\\mathbb R}) : A^{E-f} \\mbox{ has uniform behavior for all } E \\in {\\mathbb R} \\}\n$$\nis dense in $C(\\Omega,{\\mathbb R})$. Then, for any ergodic measure $\\mu$ on\n$\\Omega$, the set of $f \\in C(\\Omega,{\\mathbb R})$ for which we have that\n$\\mathcal{NUH} = \\emptyset$ {\\rm (}and hence $\\Sigma = \\mathcal{Z}${\\rm )} and\n$\\Sigma$ is a Cantor set of Lebesgue measure zero is residual {\\rm (}i.e., it contains a dense $G_\\delta$-set{\\rm )}.\n\\end{theorem}\n\n\\begin{proof}\nClearly, the map\n$$\nS : C(\\Omega) \\longrightarrow C({\\mathbb R} \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R})), \\; f \\mapsto \\left( (E,\\omega) \\mapsto A^{E - f}(\\omega) \\right)\n$$\nis continuous.\n\nHence, the inverse image under $S$ of any $G_\\delta$-set in $C({\\mathbb R} \\times \\Omega, {\\mathrm{SL}}(2,{\\mathbb R}))$ is a $G_\\delta$-set in $C(\\Omega)$. Thus, the set $\\mathcal{G}$ consisting of $f \\in C(\\Omega)$ with $A(E, \\cdot) := A^{E - f(\\cdot)} \\in W({\\mathbb R}, \\Omega)$ is a $G_\\delta$-set by Theorem~\\ref{t:general-general}. Moreover, by assumption $\\mathcal{G}$ is dense. Hence, $\\mathcal{G}$ is a dense $G_\\delta$-set and for each $f \\in \\mathcal{G}$, we have $\\mathcal{NUH} = \\emptyset$.  By \\eqref{e.JL}, for all $f \\in \\mathcal{G}$ we then have $\\Sigma = \\mathcal{Z}$. Thus, the set of of $f$ such that $\\mathcal{NUH} = \\emptyset$ and $\\Sigma =\\mathcal{Z}$ holds is\nresidual.\n\nMoreover, by \\cite{AD05} and our aperiodicity assumption, the set of $f$ with ${\\mathrm{Leb}}(\\mathcal{Z}) = 0$ is a dense $G_\\delta$-set and hence residual.\n\nSince the intersection of two residual sets is residual and\n${\\mathrm{Leb}}(\\Sigma) = 0$ implies that $\\Sigma$ is a Cantor set by general\nprinciples (cf.~Remark~\\ref{r.afterCorollary1.7}.(a)), we are done.\n\\end{proof}\n\n\n\\begin{remark}\\label{r:method}\n(a) Our proof of Theorem~\\ref{t.main} works for  all uniquely ergodic minimal  subshifts for which locally constant cocycles are uniform as for these subshifts the conclusions of Lemma~\\ref{l:input-omega} hold by \\cite{DL06a}. Recent results show\nthe uniformity of locally constant cocycles for all simple Toeplitz subshifts \\cite{GLNS, S19} (see \\cite{LQ11} for related earlier work as well). All simple Toeplitz subshifts are minimal and uniquely ergodic.  Hence, the statement of Theorem~\\ref{t.main} will hold for these subshifts as well. Note that the class of simple Toeplitz subshifts contains examples not satisfying the Boshernitzan\ncondition. A characterization of those simple Toeplitz subshifts satisfying the Boshernitzan condition is contained in \\cite{LQ11}.\n\\\\[1mm]\n(b) Theorem~\\ref{t.main2} does not require the dynamical system to be a subshift, nor does it require unique ergodicity or minimality. It can be applied to general ergodic dynamical systems. However, so far, the necessary denseness condition is only known for classes of uniquely ergodic minimal subshifts.\n\\\\[1mm]\n(c) Theorem~\\ref{t.main2}  gives a slightly weaker conclusion than\nTheorem~\\ref{t.main} in that the involved sets are shown to be\nresidual rather then dense $G_\\delta$-sets. The reason is that in\nthe  first part of the proof we obtain  the implication $f \\in\n\\mathcal{G}\\Longrightarrow \\mathcal{NUH} =\\emptyset$ but do not know\nthe converse (as we are dealing with general dynamical systems). If\nadditionally the condition of minimality and unique ergodicity is\nimposed on the dynamical system, the converse holds and we can\nconclude that the sets in question are $G_\\delta$-sets (compare the\nproof of Theorem~\\ref{t.main} as well).\n\\\\[1mm]\n(d) The corresponding results hold for Jacobi operators. The alert reader may point out that in this more general setting, the standard transfer matrices are not given by ${\\mathrm{SL}} (2,{\\mathbb R})$ cocycles, but rather by ${\\mathrm{GL}}(2,{\\mathbb R})$ cocycles. However, it is not difficult (see, e.g., \\cite{BP13,DKS10}) to identify an affiliated family of\n${\\mathrm{SL}} (2,{\\mathbb R})$ cocycles whose study via the results above yields the desired conclusions.\n\\\\[1mm]\n(e) A similar remark applies in the setting of CMV matrices with dynamically defined Verblunsky coefficients. The necessary tools to adapt the present work to that setting are discussed in \\cite{DL07}. The CMV analog of \\cite{AD05}, as well as the adaptation of Corollary~\\ref{c.main}, have been worked out in \\cite{DFG}.\n\\end{remark}\n\n\n\n\n\\begin{remark} We  note that our proof of Theorem \\ref{t:general-general} allows\nfor a (semi-) explicit construction of a potential with infinitely\nmany values (and arbitrarily close to any given potential) whose\ncocycles are uniform for all energies whenever the underlying\ndynamical system is a subshift $(\\Omega,T)$ satisfying (B). The\npoint of the construction is that any finite sum of locally constant\nfunctions $f : \\Omega \\to {\\mathbb R}$ is locally constant again. Here are\nthe details:\n\nLet $g_0$ be a locally constant function. Let $I$ be a compact interval containing an open neighborhood of the range of $g_0$. Let $g_n$, $n\\in{\\mathbb N}$, be an arbitrary\nsequence of locally constant functions on $\\Omega$ with $\\|g_n\\|\n=1$ for each $n$. Let $\\varepsilon_n\\to 0$.\n\nWe now use the $W_\\varepsilon$ from the proof of Theorem~\\ref{t:general-general}. Consider $g_0$. Clearly $g_0$ belongs to $W_{\\varepsilon_1}$ (as $g_0$ is locally constant).  As $W_{\\varepsilon_1}$ is open, there exists a $\\delta_1 > 0$ such that\nany perturbation  of $g_0$ with sup norm not exceeding $\\delta_1$ belongs to $W_{\\varepsilon_1}$ as well. Without loss of generality, $\\delta_1 < 1$.  Consider $g_0 + \\frac{\\delta_1}{2} g_1$. Clearly, this belongs to $W_{\\varepsilon_2}$ (as it is locally constant). As $W_{\\varepsilon_2}$ is open, there exists a $\\delta_2>0$ such that any perturbation of $g_0 + \\frac{\\delta_1}{ 2} g_1$ with sup norm not exceeding $\\delta_2$ belongs to $W_{\\varepsilon_2}$. Without loss of generality $\\delta_2 < \\delta_1 \/ 2$. Inductively, we can then construct for each $N\\in{\\mathbb N}$ a  $\\delta_N$ with $\\delta_{N+1} < \\delta_{N}\/2$ such that any perturbation of $g_0 +\n\\frac{1}{2}\\left(\\sum_{j=1}^N \\delta_j g_j\\right)$ with sup norm not exceeding $\\delta_{N+1}$ belongs to $W_{\\varepsilon_{N+1}}$. Consider\n$$\ng:=g_0 + \\lim_{N \\to \\infty} \\Big( g_1 + \\frac {1}{2}\\sum_{j=1}^N \\delta_j g_{j+1} \\Big).\n$$\nBy construction $g$ belongs to $W_{\\varepsilon_j}$ for any $j\\in{\\mathbb N}$.\nNow, the intersection of the $W_{\\varepsilon_j}$ is $W(I,\\Omega)$\n(by definition of $W_\\varepsilon$). This easily gives the desired\nstatement. As our choice of $g_0$ is arbitrary and $\\sum \\delta_j$\ncan be made arbitrarily small by making $\\delta_1$ as small as\nnecessary, the function $g$ can be made arbitrarily close to any\ngiven continuous function on $\\Omega$.\n\\end{remark}\n\n\n\n\n\n\n\\begin{appendix}\n\n\\section{Notions of Uniform Hyperbolicity}\\label{a.UH}\n\nIn this section we discuss various notions of uniform hyperbolicity in the context of continuous ${\\mathrm{SL}}(2,{\\mathbb R})$ cocycles and the relationships between them. Related discussions can be found in \\cite{DFLY16, V14, Y04, Z19}.\n\n\\medskip\n\nLet $(\\Omega,T)$ be a dynamical system. Denote the projective space over ${\\mathbb R}^2$  consisting of lines through the origin by ${\\mathbb R} \\mathbb{P}^1$. This is a topological space in a natural way. Then any $B\\in {\\mathrm{SL}} (2,{\\mathbb R})$ can be considered as a map on ${\\mathbb R} \\mathbb{P}^1$ as it maps lines through the origin to lines through the origin. This map on ${\\mathbb R} \\mathbb{P}^1$ will be denoted by $B$ as well.\n\nLet us consider the following three conditions for a continuous cocycle $A : \\Omega\\longrightarrow {\\mathrm{SL}}(2,{\\mathbb R})$:\n\n\\begin{itemize}\n\n\\item[$\\mathrm{(UH1)}$] There exists $L > 0$ with $\\liminf_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| \\geq L$ uniformly in $\\omega \\in \\Omega$.\n\n\\item[$\\mathrm{(UH2)}$] There exists continuous maps $u, s : \\Omega \\longrightarrow {\\mathbb R} \\mathbb{P}^1$ as well as $\\lambda > 1$ and $C > 0$ with\n\n\\begin{itemize}\n\n\\item[$(\\alpha)$] $A (\\omega) u(\\omega) = u(T\\omega) \\mbox{ and } A(\\omega) s(\\omega) = s(T\\omega)$ for all $\\omega \\in \\Omega$;\n\n\\item[$(\\beta)$] $\\| A_n (\\omega) U \\|, \\|A_{-n} S(\\omega)\\| \\leq C \\lambda^{-n}$ for all $n \\in {\\mathbb N}$, $\\omega \\in \\Omega$ whenever $U \\in u(\\omega)$ and $S \\in s(\\omega)$ are normalized.\n\n\\end{itemize}\n\n\\item[$\\mathrm{(UH3)}$] There exists $L > 0$ with $\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\|A_n (\\omega)\\| = L$ uniformly in $\\omega \\in \\Omega$.\n\n\\end{itemize}\n\n\\begin{prop}\\label{p.uhimplications}\n\\begin{itemize}\n\n\\item[(a)] The conditions $\\mathrm{(UH1)}$ and $\\mathrm{(UH2)}$ are equivalent.\n\n\\item[(b)] $\\mathrm{(UH3)}$ implies $\\mathrm{(UH1)}$.\n\n\\item[(c)] If $(\\Omega,T)$ is uniquely ergodic, then $\\mathrm{(UH1)}$ is equivalent to  $\\mathrm{(UH3)}$.\n\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\n(a) This is well-known; see, for example, \\cite[Theorem~1.2]{DFLY16}, \\cite[Proposition~2.5]{V14}, \\cite[Proposition~2]{Y04}, and \\cite[Corollary~1]{Z19}.\n\n\\smallskip\n\n(b) This is obvious.\n\n\\smallskip\n\n(c) By (a) and (b) it suffices to show $\\mathrm{(UH2)} \/ \\mathrm{(UH1)} \\Longrightarrow \\mathrm{(UH3)}$. This follows by standard methods as discussed, for example, in \\cite{F97, L04}. More specifically, \\cite[Theorem~3]{L04} shows that $\\mathrm{(UH3)}$ follows from $\\mathrm{(UH1)}$ under an additional minimality assumption. This minimality assumption is only used in the proof to ensure $(\\beta)$ of $\\mathrm{(UH2)}$. Hence,  the proof carries over\nto our case.\n\\end{proof}\n\n\\begin{remark}\\label{r.example}\nIt is not hard to see that the implication $\\mathrm{(UH1)} \\Longrightarrow \\mathrm{(UH3)}$ fails whenever the system is not uniquely ergodic. Indeed consider a non-uniquely ergodic dynamical system $(\\Omega,T)$. Then, there exists a continuous $f : \\Omega \\longrightarrow {\\mathbb R}$ such that\n$$\n\\frac{1}{n} \\sum_{k=0}^{n-1} f(T^k \\omega)\n$$\ndoes not converge uniformly in $\\omega \\in \\Omega$. Without loss of generality we can assume $f \\geq 1$ (otherwise replace $f$ by $f + 1 + \\|f\\|_\\infty$). Set $h := \\exp (f)$ and let $A : \\Omega \\longrightarrow {\\mathrm{SL}}(2,{\\mathbb R})$ be given by\n$$\nA(\\omega) = \\begin{pmatrix} h(\\omega) & 0 \\\\ 0 & 1\/h(\\omega) \\end{pmatrix}.\n$$\nAs $f \\geq 1$, the cocycle $A$ clearly satisfies $\\mathrm{(UH1)}$ with $L = 1$.  However, we have\n$$\n\\frac{1}{n} \\log \\|A_n(\\omega)\\| =\\frac{1}{n} \\sum_{k=0}^{n-1} f(T^k \\omega),\n$$\nwhich does not converge uniformly, and therefore $\\mathrm{(UH3)}$ fails.\n\\end{remark}\n\n\\begin{coro}\nLet $(\\Omega,T)$ be uniquely ergodic. Then, an $A \\in C(\\Omega,{\\mathrm{SL}}(2,{\\mathbb R}))$ is uniform if and only if it has uniform behavior.\n\\end{coro}\n\n\n\\section{A Consequence of the Avalanche Principle}\n\nThe Avalanche Principle deals with products of ${\\mathrm{SL}}(2,{\\mathbb R})$ matrices $A_N \\ldots A_1$. Roughly stated, it asserts that the norm of this product is large once the norm of each $A_j$ and of the products $A_{j+1} A_{j}$ of consecutive matrices are large. It was introduced by Goldstein-Schlag in \\cite{GS01} and then extended by Bourgain-Jitomirskaya in \\cite{BJ02}. Subsequently, various further variations and extensions have been found; see, for example, \\cite{B05, DK14, DK16, S13}. For us, the following consequence, essentially taken from \\cite{DL06a} and based on \\cite{BJ02}, will be relevant.\n\n\\begin{lemma}\\label{l:consequence-avalanche}\nThere exist constants $\\kappa > 0$ and $\\lambda_0 > 0$ such that the\nfollowing holds. Let $(\\Omega,T)$ be a dynamical system and $A :\n\\Omega \\longrightarrow {\\mathrm{SL}}(2,{\\mathbb R})$. Let $0 < \\varepsilon <\n\\frac{1}{12}$ be arbitrary. Assume that there exist $\\ell \\in {\\mathbb N}$\nand $L\n>0$ with\n\\begin{itemize}\n\n\\item[(a1)] $\\frac{1}{n} \\log\\|A_n (\\omega)\\|\\leq L(1 + \\varepsilon)$ for all $\\omega \\in \\Omega$ and $n\\geq l$,\n\n\\item[(a2)] $L (1 - \\varepsilon) \\leq \\frac{1}{2l}\\log \\|A_{2l} (\\omega)\\|$ for all $\\omega \\in \\Omega$,\n\n\\item[(a3)] $ \\frac{3}{4} L \\ell \\geq \\lambda_0$,\n\n\\item[(a4)] $\\frac{1}{\\ell} \\frac{2 \\kappa}{ \\exp(\\lambda_0)} < \\varepsilon L$.\n\n\\end{itemize}\n\nThen,\n$$\nL (1- 44\\varepsilon) \\leq \\frac{1}{n} \\log\\|A_n (\\omega)\\| \\leq L (1 + \\varepsilon)\n$$\nfor all $\\omega \\in \\Omega$ and $n \\geq \\ell$.\n\\end{lemma}\n\n\\begin{proof}\nThe assumptions (a1), (a2), (a3) and (a4) of the lemma are just the conditions (I), (II), (III), (IV) appearing in the proof of \\cite[Theorem~1]{DL06a}. The lower bound given in the conclusion of the lemma then follows by following this proof verbatim. The upper bound is obvious from the assumptions.\n\\end{proof}\n\n\\begin{remark} \\label{r:nach-avalanche-lemma}\n(a) Let us emphasize that the number $L$ appearing in the lemma is not required to be the Lyapunov exponent of $A$. It suffices that it is sufficiently close to the actual Lyapunov exponent. This is\nrelevant for an application to families of cocycles.\n\\\\[1mm]\n(b) It may be instructive to discuss the assumptions appearing in the lemma: The assumptions (a3) and (a4) are independent of $A$. For given $L > 0$ and $\\varepsilon > 0$, they will be satisfied for all large enough $\\ell$. For uniquely ergodic systems, the assumption (a1) is automatically satisfied for any given $\\varepsilon$ if $L = L(A)$ and $\\ell$ is large enough. So, in this sense for uniquely ergodic dynamical systems, the crucial condition is the second assumption (a2).\n\\\\[1mm]\n(c) Note  that the assumptions of the lemma are open conditions in the following sense: Consider an $A$ satisfying the assumptions for $\\ell \\in {\\mathbb N}$, $L > 0$ and $\\varepsilon > 0$.  Now, let $\\varepsilon' > \\varepsilon$ (with $\\varepsilon' < 1\/12$) be given. Then, any $B$ sufficiently close to $A$ will satisfy the assumptions of the lemma with the same $\\ell$ and $L$ and $\\varepsilon$ replaced by $\\varepsilon'$. Indeed, the last two assumptions (a3) and (a4) do not depend on $A$ and are then clearly satisfied for $B$. The second assumption (a2) is satisfied for $B$ sufficiently close to $A$ due to $\\varepsilon' > \\varepsilon$. Similarly, the first assumption (a1) is satisfied for $B$ sufficiently close to $A$ by part (a) of Lemma~\\ref{l:aux}.\n\\end{remark}\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany observational and theoretical works suggest that planetary systems with only rocky \nplanets are the most common in the Universe. In particular, \\cite[Miguel et al. (2011)]{Miguel.et.al.2011} indicated that \na planetary system with only small rocky planets is the most common outcome obtained from a low-mass disk \n($\\lesssim$ 0.03 $M_{\\odot}$) for different surface density profiles. In general, most of the studies of terrestrial-type planet \nformation typically use ad hoc initial conditions (\\cite[Kokubo \\& Ida, 1998]{Kokubo.et.al.1998}; \\cite[Raymond et al. 2005]{Raymond.et.al.2005}). Here, \nwe complement the results of \n\\emph{N-body high-resolution simulations} performed by \\cite[Ronco \\& de El\\'{\\i}a (2014)]{Ronco \\& de El\\'{\\i}a2014} \nstarting from a semi-analytical model developed by \\cite[Brunini \\& Benvenuto (2008)]{Brunini.and.Benvenuto2008} and \n\\cite[Guilera et al. (2010)]{Guilera.et.al.2010} wich simulates the evolution of the protoplanetary disk during the gas phase. \nWe analyze the formation of terrestrial planets and water delivery without gas giants and compare\nthe results with the previous work.\n\nThe semi-analytical model is used to calculate the formation of several embryos between 0.5~AU and 5~AU (as in \\cite[de El\\'{\\i}a et al. 2013]{deElia.et.al.2013}). These embryos were \nseparated by 10 mutual Hill radii and their initial masses correspond to the transition mass between runaway and oligarchic \ngrowth (\\cite[Ida \\& Makino, 1993]{Ida.and.Makino1993}). In our previous work we adopted three different values for the exponent $\\gamma$ that characterize the slope of the surface density ($\\gamma = 0.5$, $1$ and $1.5$). The planetary systems formed \nwith $\\gamma = 1.5$ were the most distinctive ones from an astrobiological point of view. Thus, in this new work we developed three \nN-body simulations with new initial conditions given by the semi-analytical model, only for $\\gamma = 1.5$. For this \nprofile we found the same proportion for both populations: half the mass in embryos and half the mass in planetesimals after the gas \nis completely dissipated in 3~Myr.\n\n\\section{Results}\n\nThese new planetary systems are globally similar to those found in our previous work. Each simulation formed \nbetween 1 and 2 planets in the habitable zone (HZ) (between 0.8~AU and 1.5~AU) after 200~Myr of evolution (Fig. \\ref{fig1}). Their masses \nrange between $1.18M_\\oplus$ and $2M_\\oplus$ and their water content between 7.5\\% and 24.3\\% by mass, which represent between 427 and 1671 \nEarth oceans (1 Earth ocean $=$ $2.8\\times 10^{-4}M_\\oplus$). We also found planets with masses from $1M_\\oplus$ to $2.36M_\\oplus$ \nnear the snow line (located at 2.7~AU), which can be discovered by the microlensing technique. The masses of the planets in the HZ \nare large enough to retain an atmosphere and to sustain plate tectonics, and as we also formed in the previous simulations, this profile \nformed \\emph{water worlds} that come from beyond the snow line. Thus, the planets that remain in the HZ present the characteristics \nto be potencially habitable.\n\n\\begin{figure}[t]\n\\vspace*{-0.5 cm}\n\\begin{center}\n \\includegraphics[width=2.03in]{Ronco-fig1.eps} \n\\includegraphics[width=2.8in]{Ronco-fig2.eps} \n \\vspace*{-0.1 cm}\n \\caption{Left: a) Distributions of embryos used to start the N-body simulations. The squares represent the distribution \nof embryos used by \\cite[Ronco \\& de El\\'{\\i}a (2014)]{Ronco \\& de El\\'{\\i}a2014} and the circles represent the final results obtained \nwith the semi-analytical model. b) Surface density profiles used to distribute 1000 planetesimals to start the N-body simulations. \nThe dashed line represents the surface density used in \\cite[Ronco \\& de El\\'{\\i}a (2014)]{Ronco \\& de El\\'{\\i}a2014} and the solid line \nrepresents the final results obtained with the semi-analytical model. Right: Final configuration of the simulations obtained in \n\\cite[Ronco \\& de El\\'{\\i}a (2014)]{Ronco \\& de El\\'{\\i}a2014} and the new ones obtained with the \nsemi-analytical model. The color scale represent the water content and the shaded region, the HZ. The excentricity of \neach planet is shown over it, by its radial movement over an orbit. Color figure only available in the electronic version.}\n \\label{fig1}\n\\end{center}\n\\vspace*{-0.1 cm}\n\\end{figure}\n\nWe therefore conclude that the results are globally similar to those found by \\cite[Ronco \\& de El\\'{\\i}a (2014)]{Ronco \\& de El\\'{\\i}a2014}. These planetary systems do not seem to be sensitive to the particular initial distribution of embryos and planetesimals and we suggest that the strong dependence on the final results would go with the initial mass proportion used in both populations. However, these more realistic initial conditions allowed us to find more reliable results concerning the water delivery and the global dinamics of the planetary systems.\n\n\\vspace*{-0.35 cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nStochastic Multi-Armed Bandits (MAB) \\cite{robbins1952,gittins89} constitute the most fundamental sequential decision problems with an exploration vs. exploitation trade-off. In such problems, the decision maker selects an arm in each round, and observes a realization of the corresponding unknown reward distribution. Each decision is based on past decisions and observed rewards. The objective is to maximize the expected cumulative reward over some time horizon by balancing exploitation (arms with higher observed rewards should be selected often) and exploration (all arms should be explored  to learn their average rewards). Equivalently, the performance of a decision rule or algorithm can be measured through its expected {\\it regret}, defined as the gap between the expected reward achieved by the algorithm and that achieved by an oracle algorithm always selecting the best arm. MAB problems have found many fields of application, including sequential clinical trials, communication systems, economics, see e.g. \\cite{CesaBianchi06,bubeck12}. \n\nIn their seminal paper \\cite{lai1985}, Lai and Robbins solve MAB problems where the successive rewards of a given arm are i.i.d., and where the expected rewards of the various arms are not related. They derive an asymptotic (when the time horizon grows large) lower bound of the regret satisfied by any algorithm, and present an algorithm whose regret matches this lower bound. This initial algorithm was quite involved, and many researchers have tried to devise simpler and yet efficient algorithms. The most popular of these algorithms are UCB \\cite{auer2002} and its extensions, e.g. KL-UCB \\cite{garivier2011,cappe2012} (note that KL-UCB algorithm was initially proposed in \\cite{lai1987}, see (2.6)). When the expected rewards of the various arms are not related \\cite{lai1985}, the regret of the best algorithm is essentially of the order $O(K\\log(T))$ where $K$ denotes the number of arms, and $T$ is the time horizon. When $K$ is very large or even infinite, MAB problems become more challenging. Fortunately, in such scenarios, the expected rewards often exhibit some structural properties that the decision maker can exploit to design efficient algorithms. Various structures have been investigated in the literature, e.g., Lipschitz \\cite{agrawal95, kleinberg2008, bubeck08}, linear \\cite{dani08}, convex \\cite{kalai05}. \n\nWe consider bandit problems where the expected reward is a unimodal function over partially ordered arms as in \\cite{yu2011}. The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. This unimodal structure occurs naturally in many practical decision problems, such as sequential pricing \\cite{yu2011} and bidding in online sponsored search auctions \\cite{edelman05}.   \n\n{\\bf Our contributions.} We mainly investigate unimodal bandits with finite sets of arms, and are primarily interested in cases where the time horizon $T$ is much larger than the number of arms $K$. \n\n(a) For these problems, we derive an asymptotic regret lower bound satisfied by any algorithm. This lower bound does not depend on the structure of the graph, nor on its size: it actually corresponds to the regret lower bound in a classical bandit problem \\cite{lai1985}, where the set of arms is just a neighborhood of the best arm in the graph. \n\n(b) We propose OSUB (Optimal Sampling for Unimodal Bandits), a simple algorithm whose regret matches our lower bound, i.e., it optimally exploits the unimodal structure. The asymptotic regret of \\ouralgo does not depend on the number of arms. This contrasts with LSE (Line Search Elimination), the algorithm proposed in \\cite{yu2011} whose regret scales as $O(\\gamma D\\log(T))$ where $\\gamma$ is the maximum degree of vertices in the graph and $D$ is its diameter. We present a finite-time analysis of OSUB, and derive a regret upper bound that scales as $O(\\gamma \\log(T) +K)$. Hence OSUB offers better performance guarantees than LSE as soon as the time horizon satisfies $T \\ge \\exp(K\/\\gamma D)$. Although this is not explicitly mentioned in \\cite{yu2011}, we believe that LSE was meant to address bandits where the number of arms is not negligible compared to the time horizon.\n\n(c) We further investigate OSUB performance in non-stationary environments where the expected rewards smoothly evolve over time but keep their unimodal structure. \n\n(d) We conduct numerical experiments and show that \\ouralgo significantly outperforms LSE and other classical bandit algorithms when the number of arms is much smaller than the time horizon. \n\n(e) Finally, we briefly discuss systems with a continuous set of arms. We show that using a simple discretization of the set of arms, UCB-like algorithms are order-optimal, and actually outperform more advanced algorithms such as those proposed in \\cite{yu2011}. This result suggests that in discrete unimodal bandits with a very large number of arms, it is wise to first prune the set of arms, so as to reduce its size to a number of the order of $\\sqrt{T}\/\\log(T)$. \n\n\\section{Related work}\n\nUnimodal bandits have received relatively little attention in the literature. They are specific instances of bandits in metric spaces \\cite{Kleinberg2004, kleinberg2008, bubeck08}. In this paper, we add unimodality and show how this structure can be optimally exploited. Unimodal bandits have been specifically addressed in \\cite{cope09, yu2011}. In \\cite{cope09}, bandits with a continuous set of arms are studied, and the author shows that the Kiefer-Wolfowitz stochastic approximation algorithm achieves a regret of the order of $O(\\sqrt{T})$ under some strong regularity assumptions on the reward function. In \\cite{yu2011}, for the same problem, the authors present LSE, an algorithm whose regret scales as $O(\\sqrt{T}\\log(T))$ without the need for a strong regularity assumption. The LSE algorithm is based on Kiefer's golden section search algorithm. It iteratively eliminates subsets of arms based on PAC-bounds derived after appropriate sampling. By design, under LSE, the sequence of parameters used for the PAC bounds is pre-defined, and in particular does not depend of the observed rewards. As a consequence, LSE may explore too much sub-optimal parts of the set of arms. For bandits with a continuum set of arms, we actually show that combining an appropriate discretization of the decision space (i.e., reducing the number of arms to $\\sqrt{T}\/\\log(T)$ arms) and the UCB algorithm can outperform LSE in practice (this is due to the adaptive nature of UCB). Note that the parameters used in LSE to get a regret of the order $O(\\sqrt{T}\\log(T))$ depend on the time horizon $T$.\n\nIn \\cite{yu2011}, the authors also present an extension of the LSE algorithm to problems with discrete sets of arms, and provide regret upper bounds of this algorithm. These bounds depends on the structure of the graph defining unimodal structure, and on the number of arms as mentioned previously. LSE performs better than classical bandit algorithms only when the number of arms is very large, and actually becomes comparable to the time horizon. Here we are interested in bandits with relatively small number of arms. \n\nNon-stationary bandits have been studied in \\cite{hartland06,garivier08, slivkins08, yu2011}. Except for \\cite{slivkins08}, these papers deal with environments where the expected rewards and the best arm change abruptly. This ensures that arms are always well separated, and in turn, simplifies the analysis. In \\cite{slivkins08}, the expected rewards evolve according to independent brownian motions. We consider a different, but more general class of dynamic environments: here the rewards smoothly evolve over time. The challenge for such environments stems from the fact  that, at some time instants, arms can have expected rewards arbitrarily close to each other.\n\nFinally, we should mention that bandit problems with structural properties such as those we address here can often be seen as specific instances of problems in the control of Markov chains, see \\cite{graves1997}. We leverage this observation to derive regret lower bounds. However, algorithms developed for the control of generic Markov chains are often too complex to implement in practice. Our algorithm, OSUB, is optimal and straightforward to implement. \n\n\n\\section{Model and Objectives}\\label{sec:model}\n\nWe consider a stochastic multi-armed bandit problem with $K \\geq 2$ arms. We discuss problems where the set of arms is continuous in Section \\ref{sec:continuous}. Time proceeds in rounds indexed by $n=1,2,\\ldots$. Let $X_k(n)$ be the reward obtained at time $n$ if arm $k$ is selected. For any $k$, the sequence of rewards $(X_{k}(n))_{n\\ge 1}$ is i.i.d. with distribution and expectation denoted by $\\nu_k$ and $\\mu_k$ respectively. Rewards are independent across arms. Let $\\mu=(\\mu_1,\\ldots,\\mu_K)$ represent the expected rewards of the various arms. At each round, a decision rule or algorithm selects an arm depending on the arms chosen in earlier rounds and their observed rewards. We denote by $k^\\pi(n)$ the arm selected under $\\pi$ in round $n$. The set $\\Pi$ of all possible decision rules consists of policies $\\pi$ satisfying: for any $n\\ge 1$, if ${\\cal F}_n^\\pi$ is the $\\sigma$-algebra generated by $(k^\\pi(t),X_{k^\\pi(t)}(t))_{1\\le t\\le n}$, then $k^\\pi(n+1)$ is ${\\cal F}_{n}^\\pi$-measurable. \n\n\\subsection{Unimodal Structure}\n\nThe expected rewards exhibit a {\\it unimodal} structure, similar to that considered in \\cite{yu2011}. More precisely, there exists an undirected graph $G= (V,E)$ whose vertices correspond to arms, i.e., $V = \\{1,\\dots,K\\}$, and whose edges characterize a partial order (initially unknown to the decision maker) among expected rewards. We assume that there exists a unique arm $k^\\star$ with maximum expected reward $\\mu^\\star$, and that from any sub-optimal arm $k\\neq k^\\star$, there exists a path $p=(k_1=k,\\ldots,k_m=k^\\star)$ of length $m$ (depending on $k$) such that for all $i=1,\\ldots,m-1$, $(k_i,k_{i+1})\\in E$ and $\\mu_{k_i} < \\mu_{k_{i+1}}$. We denote by ${\\cal U}_G$ the set of vectors $\\mu$ satisfying this unimodal structure.\n\nThis notion of unimodality is quite general, and includes, as a special case, classical unimodality (where $G$ is just a line). Note that we assume that the decision maker knows the graph $G$, but ignores the best arm, and hence the partial order induced by the edges of $G$. \n\t\n\\subsection{Stationary and non-stationary environments}\n\nThe model presented above concerns stationary environments, where the expected rewards for the various arms do not evolve over time. In this paper, we also consider non-stationary environments where these expected rewards could evolve over time according to some deterministic dynamics. In such scenarios, we denote by $\\mu_k(n)$ the expected reward of arm $k$ at time $n$, i.e., $\\mathbb{E}[X_k(n)]=\\mu_k(n)$, and $(X_k(n))_{n\\ge 1}$ constitutes a sequence of independent random variables with evolving mean. In non-stationary environments, the sequences of rewards are still assumed to be independent across arms. Moreover, at any time $n$, $\\mu(n)=(\\mu_1(n),\\ldots \\mu_K(n))$ is unimodal with respect to some fixed graph $G$, i.e., $\\mu(n)\\in {\\cal U}_G$ (note however that the partial order satisfied by the expected rewards may evolve over time). \n\n\\subsection{Regrets}\n\nThe performance of an algorithm $\\pi\\in \\Pi$ is characterized by its {\\it regret} up to time $T$ (where $T$ is typically large). The way regret is defined differs depending on the type of environment.\n\n{\\it Stationary Environments.} In such environments, the regret $R^\\pi(T)$ of algorithm $\\pi\\in \\Pi$ is simply defined through the number of times $t_k^\\pi(T) = \\sum_{1 \\leq n \\leq T} \\indic \\{k^\\pi(n) = k \\}$ that arm $k$ has been selected up to time $T$:\n$\nR^{\\pi}(T) = \\sum_{k=1}^K (\\mu^{\\star} - \\mu_k) \\EE[ t_k^\\pi(T) ].\n$\nOur objectives are (1) to identify an asymptotic (when $T\\to\\infty$) regret lower bound satisfied by {\\it any} algorithm in $\\Pi$, and (2) to devise an algorithm that achieves this lower bound.\n\n{\\it Non-stationary Environments.} In such environments, the regret of an algorithm $\\pi\\in \\Pi$ quantifies\t how well $\\pi$ tracks the best arm over time.  Let $k^\\star(n)$ denote the optimal arm with expected reward $\\mu^\\star(n)$ at time $n$. The regret of $\\pi$ up to time $T$ is hence defined as:\n$\nR^{\\pi}(T) = \\sum_{n=1}^T \\left( \\mu^\\star(n) - \\EE[\\mu_{k^\\pi(n)}(n)]\\right).\n$ \n\n\n\\section{Stationary environments}\\label{sec:stationary}\n\nIn this section, we consider unimodal bandit problems in stationary environments. We derive an asymptotic lower bound of regret when the reward distributions belong to a parametrized family of distributions, and propose OSUB, an algorithm whose regret matches this lower bound.\n\n\\subsection{Lower bound on regret}\n\nTo simplify the presentation, we assume here that the reward distributions belong to a parametrized family of distributions. More precisely, we define a set of distributions ${\\cal V} = \\{\\nu(\\theta) \\}_{\\theta \\in [0,1]}$ parametrized by $\\theta\\in [0,1]$. The expectation of $\\nu(\\theta)$ is denoted by $\\mu(\\theta)$ for any $\\theta\\in [0,1]$. $\\nu(\\theta)$ is absolutely continuous with respect to some positive measure $m$ on $\\RR$, and we denote by $p(x,\\theta)$ its density. The Kullback-Leibler (KL) divergence number between $\\nu(\\theta)$ and $\\nu(\\theta')$ is:\n$\nKL(\\theta,\\theta^{\\prime}) = \\int_{\\RR}  \\log(  p(x,\\theta) \/ p(x,\\theta^{\\prime}) ) p(x,\\theta) m(dx).\n$\nWe denote by $\\theta^\\star$ a parameter (it might not be unique) such that $\\mu(\\theta^\\star)=\\mu^\\star$, and we define the minimal divergence number between $\\nu(\\theta)$ and $\\nu(\\theta^\\star)$ as:\n$\nI_{\\min}(\\theta, \\theta^{\\star}) = \\inf_{ \\theta\\in [0,1]: \\mu( \\theta^{\\prime} ) \\geq \\mu^{\\star}} KL(\\theta,\\theta^{\\prime}).\n$\n\nFinally, we say that arm $k$ has parameter $\\theta_k$ if $\\nu_k =  \\nu(\\theta_k)$, and we denote by $\\Theta_G$ the set of all parameters $\\theta=(\\theta_1, \\dots, \\theta_K)\\in [0,1]^K$ such that the corresponding expected rewards are unimodal with respect to graph $G$: $\\mu=(\\mu_1,\\ldots,\\mu_K)\\in {\\cal U}_G$. Of particular interest is the family of Bernoulli distributions: the support of $m$ is $\\{0,1\\}$, $\\mu(\\theta)=\\theta$, and $I_{\\min}(\\theta,\\theta^\\star) = I(\\theta,\\theta^\\star)$ where $I(\\theta,\\theta^\\star)=\\theta\\log({\\theta\\over\\theta^\\star})+(1-\\theta)\\log({1-\\theta\\over 1-\\theta^\\star})$ is KL divergence number between Bernoulli distributions of respective means $\\theta$ and $\\theta^\\star$.\n\nWe are now ready to derive an asymptotic regret lower in parametrized unimodal bandit problems as defined above. Without loss of generality, we restrict our attention to so-called uniformly good algorithms, as defined in \\cite{lai1985} (uniformly good algorithms exist as shown later on). We say that $\\pi\\in\\Pi$ is uniformly good if for all $\\theta \\in \\Theta_G$, we have that $R^{\\pi}(T) = o(T^{a})$ for all $a > 0$. \n\n\\begin{theorem}\\label{th:graves_lai}\nLet $\\pi\\in \\Pi$ be a uniformly good algorithm, and assume that $\\nu_k = \\nu(\\theta_k) \\in {\\cal V}$ for all $k$. Then for any $\\theta\\in \\Theta_G$,\n\\eq{\n\\lim \\inf_{T \\to +\\infty} \\frac{R^{\\pi}(T)}{ \\log(T)}  \\geq c(\\theta) = \\sum_{(k,k^*) \\in E} \\frac{ \\mu^{\\star} - \\mu_k}{I_{\\min}(\\theta_k, \\theta^{\\star})}.\n}\n\\end{theorem}\n\nThe above theorem is a consequence of results in optimal control of Markov chains \\cite{graves1997}. All proofs are presented in appendix. As in classical discrete bandit problems, the regret scales at least logarithmically with time (the regret lower bound derived in \\cite{lai1985} is obtained from Theorem \\ref{th:graves_lai} assuming that $G$ is the complete graph). We also observe that the unimodal structure, if optimally exploited, can bring significant performance improvements: the regret lower bound does not depend on the size $K$ of the decision space. Indeed $c(\\theta)$ includes only terms corresponding to arms that are neighbors in $G$ of the optimal arm (as if one could learn without regret that all other arms are sub-optimal). \n\nIn the case of Bernoulli rewards, the lower regret bound becomes $\\log(T)\\sum_{(k,k^*) \\in E} \\frac{ \\mu^{\\star} - \\mu_k}{I(\\theta_k, \\theta^{\\star})}$. Note that LSE and GLSE, the algorithms proposed in \\cite{yu2011}, have performance guarantees that do not match our lower bound: when $G$ is a line, LSE achieves a regret bounded by $41\/\\Delta^2\\log(T)$, whereas in the general case, GLSE incurs a regret of the order of $O(\\gamma D\\log(T))$ where $\\gamma$ is the maximal degree of vertices in $G$, and $D$ is its diameter. The performance of LSE critically depends on the graph structure, and the number of arms. Hence there is an important gap between the performance of existing algorithms and the lower bound derived in Theorem \\ref{th:graves_lai}. In the next section, we close this gap and propose an asymptotically optimal algorithm.\n\n\\subsection{The OSUB Algorithm}\n\nWe now describe OSUB, a simple algorithm whose regret matches the lower bound derived in Theorem of~\\ref{th:graves_lai} for Bernoulli rewards, i.e., OSUB is asymptotically optimal. The algorithm is based on KL-UCB proposed in \\cite{lai1987, cappe2012}, and uses KL-divergence upper confidence bounds to define an {\\it index} for each arm. OSUB can be readily extended to systems where reward distributions are within one-parameter exponential families by simply modifying the definition of arm indices as done in \\cite{cappe2012}. In OSUB, each arm is attached an index that resembles the KL-UCB index, but the arm selected at a given time is the arm with maximal index within the neighborhood in $G$ of the arm that yielded the highest empirical reward. Note that since the sequential choices of arms are restricted to some neighborhoods in the graph, OSUB is not an index policy. To formally describe OSUB, we need the following notation. For $p \\in [0,1]$, $s \\in \\NN$, and $n \\in \\NN$, we define:\n\\begin{align}\nF(p,s,n) =& \\sup \\{ q \\geq p : \\nonumber\\\\\n& sI(p,q) \\leq \\log(n) + c \\log( \\log (n) ) \\},\n\\end{align}\nwith the convention that $F(p,0,n) = 1$, and $F(1,s,n) = 1$, and where $c>0$ is a constant. Let $k(n)$ be the arm selected under OSUB at time $n$, and let $t_k(n)$ denote the number of times arm $k$ has been selected up to time $n$. The empirical reward of arm $k$ at time $n$ is $\\hat \\mu_k(n) = {1\\over t_k(n)} \\sum_{t=1}^{n} \\indic \\{ k(t)=k  \\} X_k(t)$, if $t_k(n) > 0$ and $\\hat \\mu_k(n) = 0$ otherwise. We denote by $L(n) = \\arg \\max_{1 \\leq k \\leq K} \\hat \\mu_k(n)$ the index of the arm with the highest empirical reward (ties are broken arbitrarily). Arm $L(n)$ is referred to as the {\\it leader} at time $n$. Further define $l_k(n) = \\sum_{t=1}^n  \\indic \\{ L(t) = k  \\}$ the number of times arm $k$ has been the leader up to time $n$. Now the index of arm $k$ at time $n$ is defined as: \n\\eqs{\nb_k(n) = F(\\hat \\mu_k(n),t_k(n),l_k( L(n)) ).\n}\nFinally for any $k$, let $N(k)=\\{k':(k',k)\\in E\\}\\cup\\{k\\}$ be the neighborhood of $k$ in $G$. \nThe pseudo-code of OSUB is presented below.\n\n\\begin{separation}\n\\vspace{-0.2cm}\n    {\\bf Algorithm} OSUB \n\\vspace{-0.6cm}\\separator\n\\vspace{-0.4cm}\nInput: graph $G=(V,E)$\\\\\nFor $n \\ge 1$, select the arm $k(n) $ where:\n$$ k(n) = \\begin{cases}  L(n)  & \\text{if } {l_{L(n)}(n) - 1 \\over\\gamma+1} \\in \\NN, \\\\\n\t  \\displaystyle \\arg\\max_{k\\in N(L(n))} b_k(n) & \\text{otherwise,}\n\t  \\end{cases} $$\nwhere $\\gamma$ is the maximal degree of nodes in $G$ and ties are broken arbitrarily.\n\\vspace{-0.2cm}\n\\end{separation}\nNote that OSUB forces us to select the current leader often: $L(n)$ is chosen when $l_{L(n)}(n)-1$ is a multiple of $\\gamma+1$. This ensures that the number of times an arm has been selected is at least proportional to the number of times this arm has been the leader. This property significantly simplifies the regret analysis, but it could be removed.\n\n\\subsection{Finite-time analysis of OSUB}\n\nNext we provide a finite time analysis of the regret achieved under OSUB. Let $\\Delta$ denote the minimal separation between an arm and its best adjacent arm: $\\Delta = \\min_{1 \\leq k \\leq K} \\max_{k^\\prime: (k,k^\\prime) \\in E} \\mu_{k^\\prime} - \\mu_{k}$. Note that $\\Delta$ is not known a priori.\n\n\\begin{theorem}\\label{th:kluucb_finite}\nAssume that the rewards lie in [0,1] (i.e., the support of $\\nu_k$ is included in $[0,1]$, for all $k$), and that $(\\mu_1,\\ldots,\\mu_K)\\in {\\cal U}_G$. The number of times suboptimal arm $k$ is selected under OSUB satisfies: for all $\\epsilon>0$ and all $T\\ge 3$,\n\\als{\n\\EE[ t_k(T) ] &\\leq   \\begin{cases} (1 + \\epsilon) \\frac{ \\log(T) + c \\log(\\log(T))}{ I(\\mu_k,\\mu^*)}  \n & \\text{ if }  (k,k^{\\star}) \\in E, \\\\\n\\;\\;\\;\\; +  C_1 \\log \\log (T) + \\frac{C_2}{T^{\\beta(\\epsilon)}}  & \\\\ \n\\frac{C_3}{\\Delta^2} & \\text{ otherwise,}  \\end{cases} \n}\nwhere $\\beta(\\epsilon) > 0$, and $0<C_1 < 7$, $C_2>0$, $C_3>0$ are constants.\n\\end{theorem}\n\nTo prove this upper bound, we analyze the regret accumulated (i) when the best arm $k^\\star$ is the leader, and (ii) when the leader is arm $k\\neq k^\\star$. (i) When $k^\\star$ is the leader, the algorithm behaves like KL-UCB restricted to the arms around $k^\\star$, and the regret at these rounds can be analyzed as in \\cite{cappe2012}. (ii) Bounding the number of rounds where $k\\neq k^\\star$ is not the leader is more involved. To do this, we decompose this set of rounds into further subsets (such as the time instants where $k$ is the leader and its mean is not well estimated), and control their expected cardinalities using concentration inequalities. Along the way, we establish Lemma~\\ref{lem:concentr}, a new concentration inequality of independent interest. \n\n\\begin{lemma}\\label{lem:concentr}\nLet $\\{ Z_t \\}_{t \\in \\mathbb{Z}}$ be a sequence of independent random variables with values in $[0,B]$. Define ${\\cal F}_n$ the $\\sigma$-algebra generated by $\\{ Z_t \\}_{t \\leq n}$ and the filtration ${\\cal F} = ( {\\cal F}_n )_{n \\in \\mathbb{Z}} $. Consider $s \\in \\NN$, $n_0 \\in \\mathbb{Z}$ and $T \\geq n_0$. We define $S_n = \\sum_{t=n_0}^n B_t (Z_t - \\EE[Z_t])$, where $B_t \\in \\{0,1\\}$ is a ${\\cal F}_{t-1}$-measurable random variable. Further define $t_n = \\sum_{t=n_0}^n B_t$. Define $\\phi \\in \\{n_0,\\dots,T+1\\}$ a ${\\cal F}$-stopping time such that either $t_{\\phi} \\geq s$ or $\\phi = T+1$. \n\n Then we have that:\n $\n \\PP[ S_{\\phi}  \\geq t_{\\phi} \\delta \\;,\\;  \\phi \\leq T  ] \\leq \\exp( -2 s \\delta^2 B^{-2}).\n $\n As a consequence:\n $\n \\PP[ | S_{\\phi} | \\geq t_{\\phi} \\delta \\;,\\;  \\phi \\leq T  ] \\leq 2 \\exp( -2 s \\delta^2 B^{-2}).\n $\n\\end{lemma}\nLemma~\\ref{lem:concentr} concerns the sum of products of i.i.d. random variables and of a previsible sequence, evaluated at a stopping time (for the natural filtration). We believe that concentration results for such sums can be instrumental in bandit problems, where typically, we need information about the empirical rewards at some specific random time epochs (that often are stopping times). Refer to the appendix for a  proof. A direct consequence of Theorem \\ref{th:kluucb_finite} is the asymptotic optimality of OSUB in the case of Bernoulli rewards:\n\n\\begin{corollary}\nAssume that rewards distributions are Bernoulli (i.e for any $k$, $\\nu_k \\sim \\text{Bernoulli}(\\theta_k)$), and that $\\theta\\in \\Theta_G$. Then the regret achieved under $\\pi$=OSUB satisfies:\n$\n\\lim \\sup_{T \\to +\\infty} R^{\\pi}(T) \/ \\log(T) \\leq c(\\theta).\n$\n\\end{corollary}\n\n\n\\section{Non-stationary environments}\\label{sec:non_stationary}\n\nWe now consider time-varying environments. We assume that the expected reward of each arm varies smoothly over time, i.e., it is Lipschitz continuous: for all $n, n' \\ge 1$ and $1 \\leq k \\leq K$: $|\\mu_k(n) - \\mu_k(n^\\prime)  | \\leq \\sigma |n - n^\\prime|$.\n\nWe further assume that the unimodal structure is preserved (with respect to the same graph $G$): for all $n\\ge 1$, $\\mu(n)\\in {\\cal U}_G$. Considering smoothly varying rewards is more challenging than scenarios where the environment is abruptly changing. The difficulty stems from the fact that the rewards of two or more arms may become arbitrarily close to each other (this happens each time the optimal arm changes), and in such situations, regret is difficult to control. To get a chance to design an algorithm that efficiently tracks the best arm, we need to make some assumption to limit the proportion of time when the separation of arms becomes too small. Define for $T \\in \\NN$, and $\\Delta > 0$:\n\\eqs{\nH(\\Delta,T) = \\sum_{n=1}^{T} \\sum_{ (k,k^\\prime) \\in E } \\indic \\{ | \\mu_k(n) - \\mu_{k^\\prime}(n) | < \\Delta  \\}.\n}\n\\begin{assumption}\\label{ass:smooth}\nThere exists a function $\\Phi$ and $\\Delta_0$ such that for all $\\Delta < \\Delta_0$:\n$\n\\lim \\sup_{T \\to +\\infty} H(\\Delta,T)\/T \\leq \\Phi(K) \\Delta.\n$\n\\end{assumption}\n\n\t\t\n\\subsection{\\ouralgo with a Sliding Window}\n\t\nTo cope with the changing environment, we modify the \\ouralgo algorithm, so that decisions are based on past choices and observations over a time-window of fixed duration equal to $\\tau+1$ rounds. The idea of adding a sliding window to algorithms initially designed for stationary environments is not novel \\cite{garivier08}; but here, the unimodal structure and the smooth evolution of rewards make the regret analysis more challenging.\n\nDefine: $t^{\\tau}_k(n) = \\sum_{t =n-\\tau}^{n} \\indic\\{ k(t) = k \\}$; $\\hat\\mu^{\\tau}_k(n) = (1\/t^{\\tau}_k(n)) \\sum_{t =n-\\tau}^{n} \\indic\\{ k(t) = k \\} X_k(t)$ if $t^{\\tau}_k(n) > 0$ and $\\hat\\mu^{\\tau}_k(n) = 0$ otherwise; $L^{\\tau}(n) = \\arg \\max_{1 \\leq k \\leq K} \\hat\\mu^{\\tau}_k(n)$;          $l^{\\tau}_k(n) = \\sum_{t=n-\\tau}^{n} \\indic\\{ L^{\\tau}(t) = k \\}$. The index of arm $k$ at time $n$ then becomes: \n$\nb^{\\tau}_k(n) = F(\\hat\\mu^{\\tau}_k(n), t_k^\\tau(n), l_k^\\tau(L^\\tau(n))).\n$\nThe pseudo-code of \\ouralgosw is presented below.\n\\begin{separation}\n\\vspace{-0.2cm}\n    {\\bf Algorithm} \\ouralgosw\n\\vspace{-0.6cm}\\separator\n\\vspace{-0.4cm}\nInput: graph $G=(V,E)$, window size $\\tau+1$\\\\\nFor $n \\ge 1$, select the arm $k(n) $ where:\n$$ k(n) = \\begin{cases}  L^\\tau(n)  & \\text{if } {l_{L^\\tau(n)}^\\tau(n) - 1\\over\\gamma+1} \\in \\NN, \\\\\n\t  \\displaystyle \\arg\\max_{k\\in N(L^\\tau(n))} b_k^\\tau(n) & \\text{otherwise.}\n\t  \\end{cases} $$\n\\vspace{-0.2cm}\n\\end{separation}\n\n\n\\subsection{Regret Analysis}\n\nIn non-stationary environments, achieving sublinear regrets is often not possible. In \\cite{garivier08}, the environment is subject to abrupt changes or breakpoints. It is shown that if the density of breakpoints is strictly positive, which typically holds in practice, then the regret of any algorithm has to scale linearly with time. We are interested in similar scenarios, and consider smoothly varying environments where the number of times the optimal arm changes has a positive density. The next theorem provides an upper bound of the regret per unit of time achieved under SW-OSUB. This bound holds for any non-stationary environment with $\\sigma$-Lipschitz rewards. \n\n\\begin{theorem}\t\\label{th:nonstat}\nLet $\\Delta$: $2\\tau\\sigma < \\Delta < \\Delta_0$. Assume that for any $n\\ge 1$, $\\mu(n)\\in {\\cal U}_G$ and $\\mu^\\star(n)\\in [a,1-a]$ for some $a>0$. Further suppose that $\\mu_k(\\cdot)$ is $\\sigma$-Lipschitz for any $k$. The regret per unit time under $\\pi=$\\ouralgosw with a sliding window of size $\\tau+1$ satisfies: if $a>\\sigma\\tau$, then for any $T\\ge 1$,\n\\als{\t\n\\frac{R^{\\pi}(T)}{T} &\\leq  \\frac{H(\\Delta,T)}{T}(1 + \\Delta)  +  \\frac{C_1 K \\log(\\tau)}{\\tau (\\Delta - 4 \\tau \\sigma)^2} \\sk\n & +\\gamma \\Lp 1 + g_0^{-1\/2} \\Rp \\frac{\\log(\\tau) + c \\log(\\log(\\tau)) + C_2}{2 \\tau (\\Delta - 2 \\tau \\sigma)^2} ,\n}\nwhere $C_1,C_2$ are positive constants and $g_0=(a - \\sigma\\tau)(1-a + \\sigma\\tau)\/2$.\n\\end{theorem}\n\n\\begin{corollary} Assume that for any $n\\ge 1$, $\\mu(n)\\in {\\cal U}_G$ and $\\mu^\\star(n)\\in [a,1-a]$ for some $a>0$, and that $\\mu_k(\\cdot)$ is $\\sigma$-Lipschitz for any $k$. Set $\\tau = \\sigma^{-3\/4} \\log(1\/\\sigma) \/ 8 $. The regret per unit of time of $\\pi=$\\ouralgosw with window size $\\tau+1$ satisfies: \n$$\n\\lim \\sup_{T\\to\\infty} \\frac{R^{\\pi}(T)}{T}  \\leq C \\Phi(K) \\sigma^{\\frac{1}{4}} \\log \\Lp \\frac{1}{\\sigma} \\Rp ( 1 + K j(\\sigma) ),\n$$\nfor some constant $C>0$, and some function $j$ such that $\\lim_{\\sigma\\to 0^+} j(\\sigma)  =0.$\n\\end{corollary}\n\nThese results state that the regret per unit of time achieved under \\ouralgosw decreases and actually vanishes when the speed at which expected rewards evolve decreases to 0. Also observe that the dependence of this regret bound in the number of arms is typically mild (in many practical scenarios, $\\Phi(K)$ may actually not depend on $K$).\n\nThe proof of Theorem \\ref{th:nonstat} relies on the same types of arguments as those used in stationary environments. To establish the regret upper bound, we need to evaluate the performance of the KL-UCB algorithm in non-stationary environments (the result and the corresponding analysis are presented in appendix).\n\n\\section{Continuous Set of Arms}\\label{sec:continuous}\n \nIn this section, we briefly discuss the case where the decision space is continuous. The set of arms is $[0,1]$, and the expected reward function $\\mu:[0,1] \\to \\RR$ is assumed to be Lipschitz continuous, and unimodal: there exists $x^\\star\\in [0,1]$ such that $\\mu(x^\\prime) \\geq \\mu(x)$ if $x^\\prime \\in [x ,x^{\\star}]$ or $x^\\prime \\in [x^{\\star} , x]$. Let $\\mu^\\star=\\mu(x^\\star)$ denote the highest expected reward. A decision rule selects at any round $n\\ge 1$ an arm $x$ and observes the corresponding reward $X(x,n)$. For any $x\\in [0,1]$, $(X(x,n))_{n\\ge 1}$ is an i.i.d. sequence. We make the following additional assumption on function $\\mu$.\n\n\\begin{assumption}\\label{ass:22}\nThere exists $\\delta_0>0$ such that (i) for all $x , y$ in $[x^\\star,x^\\star+\\delta_0]$ (or in $[x^\\star-\\delta_0,x^\\star]$), $C_1 |x-y|^\\alpha \\leq |\\mu(x) - \\mu(y)  |$; (ii) for $\\delta \\leq \\delta_0$, if $|x - x^*| \\leq \\delta$, then $| \\mu(x^*) - \\mu(x) | \\leq C_2 \\delta^\\alpha$. \n\\end{assumption}\n\nThis assumption is more general than that used in \\cite{yu2011}. In particular it holds for functions with a {\\it plateau} and a {\\it peak}: $\\mu(x)=\\max(1-|x-x^\\star|\/\\epsilon,0)$. Now as for the case of a discrete set of arms, we denote by $\\Pi$ the set of possible decision rules, and the regret achieved under rule $\\pi\\in \\Pi$ up to time $T$ is:\n$\nR^{\\pi}(T) = T \\mu^{\\star} - \\sum_{n=1}^T  \\mathbb{E}[\\mu(x^\\pi(n))],\n$\nwhere $x^\\pi(n)$ is the arm selected under $\\pi$ at time $n$. \n\nThere is no known precise asymptotic lower bound for continuous bandits. However, we know that for our problem, the regret must be at least of the order of $O(\\sqrt{T})$ up to logarithmic factor. In \\cite{yu2011}, the authors show that the LSE algorithm achieves a regret scaling as $O(\\sqrt{T}\\log(T))$, under more restrictive assumptions. We show that combining discretization and the UCB algorithm as initially proposed in \\cite{Kleinberg2004} yields lower regrets than LSE in practice (see Section \\ref{sec:numerical}), and is order-optimal, i.e., the regret grows as $O(\\sqrt{T}\\log(T))$.  \n\nFor $\\delta > 0$, we define a discrete bandit problem with $K = \\ceil{1\/\\delta}$ arms, and where the rewards of $k$-th arm are distributed as $X((k-1)\/\\delta,n)$. The expected reward of the $k$-th arm is $\\mu_k = \\mu((k-1)\/\\delta)$. Let $\\pi$ be an algorithm running on this discrete bandit problem. The regret of $\\pi$ for the initial continuous bandit problem is at time $T$:\\\\\n$ \nR^{\\pi}(T) = T \\mu^{\\star} - \\sum_{k=1}^{ \\ceil{1\/\\delta} } \\mu_k \\EE[t_k^\\pi(T)].\n$\nWe denote by UCB($\\delta$) the UCB algorithm \\cite{auer2002} applied to the discretized bandit. In the following proposition, we show that when $\\delta= (\\log(T)\/\\sqrt{T} )^{1\/\\alpha}$, UCB($\\delta$) is order-optimal. In practice, one may not know the time horizon $T$ in advance. In this case, using the ``doubling trick'' (see e.g. \\cite{CesaBianchi06})  would incur an additional logarithmic multiplicative factor in the regret.\n\t\n\\begin{proposition}\\label{th:unimodal_cont}\nConsider a unimodal bandit on $[0,1]$ with rewards in $[0,1]$ and satisfying Assumption \\ref{ass:22}. Set $\\delta= (\\log(T)\/\\sqrt{T} )^{1\/\\alpha}$. The regret under UCB($\\delta$) satisfies:\n\\eqs{ \\lim \\sup_{T\\to\\infty} {R^{\\pi}(T) \\over \\sqrt{T} \\log(T)} \\leq C_2 3^{\\alpha} + 16 \/C_1.}\n\\end{proposition}\n\t\n\n\\section{Numerical experiments}\\label{sec:numerical}\n\n\\subsection{Discrete bandits}\n\nWe compare the performance of our algorithm to that of KL-UCB \\cite{cappe2012}, LSE \\cite{yu2011}, UCB \\cite{auer2002}, and UCB-U. The latter algorithm is obtained by applying UCB restricted to the arms which are adjacent to the current leader as in OSUB. We add the prefix \"SW\" to refer to Sliding Window versions of these algorithms.\n\n{\\it Stationary environments.} In our first experiment, we consider $K=17$ arms with Bernoulli rewards of respective averages $\\mu=(0.1, 0.2, .... , 0.9 , 0.8,  \\dots, 0.1)$. The rewards are unimodal (the graph $G$ is simply a line). The regret achieved under the various algorithms is presented in Figure~\\ref{fig:compare_stationary} and Table~\\ref{table:regret_log}. The parameters in LSE algorithm are chosen as suggested in Proposition 4.5 \\cite{yu2011}. Regrets are calculated averaging over $50$ independent runs. \\ouralgo significantly outperforms all other algorithms. The regret achieved under LSE is not presented in Figure~\\ref{fig:compare_stationary}, because it is typically much larger than that of other algorithms. This poor performance can be explained by the non-adaptive nature of LSE, as already discussed earlier. LSE can beat UCB when the number of arms is not negligible compared to the time horizon (e.g. in Figure 4 in \\cite{yu2011}, $K=250.000$ and the time horizon is less than $3K$): in such scenarios, UCB-like algorithms perform poorly because of their initialization phase (all arms have to be tested once).\n\nIn Figure \\ref{fig:stat2}, the number of arms is 129, and the expected rewards form a triangular shape as in the previous example, with minimum and maximum equal to 0.1 and 0.9, respectively. Similar observations as in the case of 17 arms can be made. We deliberately restrict the plot to small time horizons: this corresponds to scenarios where LSE can perform well.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{llll}\n\\hline\n T  & 1000 & 10000 & 100000\\\\\n\\hline\nUCB & 30.1 & 35.1 & 39\\\\\nKL-UCB & 18.8 & 21.4 & 23\\\\\nUCB-U & 8.5 & 11.7 & 13.9\\\\\nOSUB & 5.8 & 5.9 & 6\\\\\nLSE & 36.3 & 271.5 & 999.1\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ $R^{\\pi}(T) \/ \\log(T)$ for different algorithms -- 17 arms.}\n\\label{table:regret_log}\n\\end{table}\n\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{compare_stationary}\n\t\\caption{Regret vs. time in stationary environments -- $K=17$ arms.}\n\t\\label{fig:compare_stationary}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{compare_stationary_129-2}\n\t\\caption{Regret vs. time in stationary environments -- $K=129$ arms.}\n\t\\label{fig:stat2}\n\\end{center}\n\\end{figure}\n\n\n{\\it Non-stationary environments.} We now investigate the performance of \\ouralgosw in a slowly varying environment. There are $K=10$ arms whose expected rewards form a moving triangle: for $k=1,\\ldots,K$, $\\mu_k(n) = (K-1)\/K - | w(n) - k |\/K$, where  $w(n) = 1 + (K - 1)( 1 +  \\sin( n \\sigma) )\/2$. Figure~\\ref{fig:compare_sw_regret_time} presents the regret as a function of time under various algorithms when the speed at which the environment evolves is $\\sigma= 10^{-3}$. The window size are set as follows for the various algorithms: $\\tau =\\sigma^{-4\/5}$ for SW-UCB and SW-KL-UCB (the rationale for this choice is explained in appendix), $\\tau = \\sigma^{-3\/4} \\log(1\/\\sigma)\/8$ for SW-UCB-U and OSUB. In Figure~\\ref{fig:compare_sw_regret_speed}, we show how the speed $\\sigma$ impacts the regret per time unit. \\ouralgosw provides the most efficient way of tracking the optimal arm.\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{compare_sw_regret_time}\n\t\\caption{Regret vs. time in a slowly varying environment -- $K=10$ arms, $\\sigma = 10^{-3}$.}\n\t\\label{fig:compare_sw_regret_time}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{compare_sw_regret_speed}\n\t\\caption{Regret per unit of time $R^{\\pi}(T)\/T$ vs. speed $\\sigma$ -- $K=10$ arms.}\n\t\\label{fig:compare_sw_regret_speed}\n\\end{center}\n\\end{figure}\n\n\\subsection{Continuous bandits}\n\nIn Figure \\ref{fig:compare_continuous}, we compare the performance of the LSE and UCB($\\delta$) algorithms when the set of arms is continuous. The expected rewards form a triangle: $\\mu(x) = 1\/2 - |x - 1\/2|$ so that $\\mu^\\star = 1\/2$ and $x^\\star = 1\/2$. The parameters used in LSE are those given in \\cite{yu2011}, whereas the discretization parameter $\\delta$ in UCB($\\delta$) is set to $\\delta = \\log(T) \/ \\sqrt{T}$. UCB($\\delta$) significantly outperforms LSE at any time: an appropriate discretization of continuous bandit problems might actually be more efficient than other methods based on ideas taken from classical optimization theory.\n\t\nFigure \\ref{fig:number_arms} compares the regret of the discrete version of LSE (with optimized parameters), and of OSUB as the number of arms $K$ grows large, $T=50,000$. The average rewards of arms are extracted from the triangle used in the continuous bandit, and we also provide the regret achieved under UCB($\\delta$). OSUB outperforms UCB($\\delta$) even if the number of arms gets as large as 7500! OSUB also beats LSE unless the number of arms gets bigger than $0.6\\times T$. \n\t\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{compare_continuous}\n\t\\caption{Regret vs. time for a continuous set of arms.}\n\t\\label{fig:compare_continuous}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=0.7\\columnwidth]{number_arms}\n\t\\caption{Normalized regret vs. $K\/T$, $T = 5.10^{4}$ for a continuous set of arms.}\n\t\\label{fig:number_arms}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nIn this paper, we address stochastic bandit problems with a unimodal structure, and a finite set of arms. We provide asymptotic regret lower bounds for these problems and design an algorithm that asymptotically achieves the lowest regret possible. Hence our algorithm optimally exploits the unimodal structure of the problem. Our preliminary analysis of the continuous version of this bandit problem suggests that when the number of arms become very large and comparable to the time horizon, it might be wiser to prune the set of arms before actually running any algorithm. \n\n\\clearpage\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec1}\n\nThe space of flattenings $F(L)$ associated to a simplicial $k$-sphere $L$ is the space of all simplicial embeddings of the cone $cL$ of $L$ in $\\mathbb{R}^{k+1}$. The space of flateenings of simplicial spheres was first studied by S.S Cairns \\cite{cairns} in the early 1940s. In \\cite{cairns}, Cairns proved that the space of all flattenings of a simplicial 2-sphere which have an orientation preserving isomorphism onto a given triangulation is path connected. \n\n\nThe space of flattenings associated to a simplicial 1-sphere  is homotopy equivalent to the orthogonal group $O(2)$ as we will prove in Theorem \\ref{my_flat} \\cite{kayman}. The only known positive result about the topology of $F(L)$ when $\\dim(L) > 2$ is the theorem of N.H Kuiper \\cite{kuiper:nh} where he proved that $F(L)$ has the homotopy type of $O(n+1)$ when $L$ is the boundary of the $(n+1)-$simplex. In \\cite{milin}, Milin showed that there exist a simplicial sphere $L$ of dimension $3$ whose subset of $F(L)$ consisting of flattenings which have an orientation preserving isomorphism onto a given triangulation is not path connected.\n\n\nLet $\\Delta^n$ denote an $n$-simplex. We will establish that the space of flattenings associated to $\\partial\\Delta^n \\ast \\partial\\Delta^1$ has the homotopy type of an orthogonal group. \n\nLet $L$ be a simplicial k-sphere and CF(L) denote the quotient of $F(L)$ by $\\mathrm{GL}_{k+1}$ the invertible $(k+1)\\times (k+1)$ matrices. To establish the above statement, we will first prove that $CF(L)$ is contractible. In Section \\ref{or:flat}, we will introduce a poset $P(L)$ called \\textit{The poset of oriented matroid flattenings} of $L$ and a poset stratification map $\\pi: CF(L) \\rightarrow P(L)$. We will prove that $P(L)$ is contractible when $L$ is either a simplicial 1-sphere, $\\partial\\Delta^{n+1}$ the boundary of an $(n+1)$-simplex or $\\partial \\Delta^1 \\ast \\partial\\Delta^{n+1}$\n\nIn Section \\ref{top:flat}, we will prove for the above simplicial spheres that $\\{\\overline{\\pi^{-1}(M)} : M \\in P(L)\\}$ is a totally normal cellular decomposition of $CF(L)$. Theorem \\ref{thm:total} will then conclude that $\\|P(L)\\|$ can be embedded in $CF(L)$ as a deformation retract.\n\n\n\n\\section{Oriented Matroids} \\label{sec:back}\n\nWe will view elements of $\\mathbb{R}^n$ as  $1\\times n$ row vectors so that $X$ is the rowspace of a $r \\times n$ matrix. Suppose $X \\in \\mathrm{Gr}(r, \\mathbb{R}^n)$ so that $X = \\mathrm{Rowspace}(v_1\\; v_2 \\; v_3 \\ldots \\; v_n)$. We consider the following function $\\chi : [n]^r \\rightarrow \\{+, - , 0\\}$  associated to $X$\n\t$$\\chi(i_1, i_2, \\ldots , i_r) = \\mathrm{sign}(\\det(v_{i_1} \\; v_{i_2} \\cdots v_{i_r}) )$$\nThe collection $\\{\\pm \\chi\\}$ is independent of the choice of basis vectors for $X$. The resulting functions $(\\pm \\chi)$ defines a rank $r$ \\textit{oriented matroid}. \n\t\n\t\nIn general, an oriented matroid can be obtained from an arrangement of pseudospheres as evident by the following theorem. Figure \\ref{pseudo} illustrates an arrangement of pseudospheres.\n\n\\begin{thm}(\\cite{jim:law}){The Topological Representation Theorem (Folkman-Lawrence 1978)}\n\tThe rank $r$ oriented matroids are exactly the sets $(E,\\mathcal{V}^*)$ arising from essential {\\em arrangements of pseudospheres} in $S^{r-1}$.\n\t \\end{thm}\n\n\n\n\\begin{figure}[htb]\n\\begin{tikzpicture}\n\t\\begin{scope}[scale=.7]\n\t\t\\draw (0,0) circle(3);\n\t\n\t\t\\draw[->](64:2.8)--(56:2.8);\n\t\t\\draw[->](111:2.8)--(118:2.8);\n\t\t\\draw[->, rotate=32](62:2.8)--(55:2.8);\n\t\t\\draw[->, rotate=121](62:2.8)--(55:2.8);\n\t\t\\draw[->, rotate=140](62:2.9)--(55:2.8);\n\t\t\\draw[rotate=120] (60:3)  arc[radius = 5, start angle= 110, end angle= 184] node[above] at (0:3){$3$};\n\t\t\\draw[rotate=240] (60:3)  arc[radius = 5, start angle= 110, end angle= 184] node[left] at (300:3){$4$};\n\t\t\\draw[rotate=30] (60:3)  arc[radius = 5, start angle= 110, end angle= 184] node[above] at (60:3){$2$};\n\t\t\\draw(195:3) arc[radius=3.15, start angle=210, end angle =355] node[left] at (200:3){$5$};\n\t\t\n\t\t\\draw (60:3)..controls (100:2.5) and (120:1.5)..(120:1)..controls(-60:.9) and (-80:1)..(240:3) node[left] at (60:3.5){$1$};\n\t\n\\draw[] (160:3)..controls (170:2.5) and (200:1.5)..(200:1)..controls(195:.9) and (30:1)..(20:1)..controls(22:1.2) and (-10:2.5)..(-20:3) node[left] at (160:3){$6$};\n\\draw[->] (162:2.8)--(155:2.8);\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Arrangement of Pseudospheres}\n\\label{pseudo}\n\\end{figure}\n\nA detailed introduction to the theory of oriented matroids can be found in the in the book \\cite{anders:bjo}. Associated to a rank $r$ oriented matroid on $n$ elements are the functions $\\pm \\chi : [n]^r \\rightarrow \\{+, - , 0\\}$ called the chirotopes. Let $\\{+, -, 0\\}$ be a poset with the partial order $0 < -$ and $0< +$.\n\n\\begin{defn}(\\cite{macp:rob})\n    Let $\\mathcal{N} = (\\pm \\chi_1)$ and $\\mathcal{M} = (\\pm \\chi_2)$ be two rank $r$ oriented matroids. We say that $\\mathcal{N} \\leq \\mathcal{M}$ if and only if $\\chi_1 \\leq \\chi_2$ or $\\chi_1 \\leq -\\chi_2$. The oriented matroid $\\mathcal{M}$ is said to {\\em weak map} to $\\mathcal{N}$. \n\\end{defn}\n\n\n\n \\begin{defn}(\\cite{macp:rob})\n $\\mathrm{MacP}(p,n)$ denotes the poset of all rank $p$ oriented matroids on elements $\\{1,2,\\ldots, n\\}$, with weak map as the partial order. The poset is called the \\textit{MacPhersonian} ~\\cite{macp:rob}.\n \\end{defn}\n\nWe have explained how to obtain a rank $r$ oriented matroid on $n$ elements from a rank $r$ subspace of $\\mathbb{R}^n$. That is, there is  a function $\\mu: \\mathrm{Gr}(r, \\mathbb{R}^n) \\rightarrow \\mathrm{MacP}(r,n): X \\to (\\pm \\chi_X)$.\n\nThe following Proposition and Theorem are from the work of the author in \\cite{kay:ab}, \\cite{kayman}.\n\n\t\\begin{prop}(\\cite{kay:ab})\\label{ref1}\n\t\tLet $M \\in \\mbox{MacP}(2,n)$. Then $\\partial \\overline{\\mu^{-1}(M)} = \\bigcup_{N < M} \\mu^{-1}(N)$\n\t\\end{prop}\n\n\\begin{thm}(\\cite{kay:ab})\\label{ref2} $\\{\\overline{\\mu^{-1}(M)} : M \\in \\; \\mbox{MacP}(2, n)\\}$ is a regular cell decomposition of $Gr(2,\\mathbb{R}^n)$.\n\t\\end{thm}\n\t\n\t\n\n\\section{Cellular stratified spaces }\\label{cellular}\n \n\\begin{defn}(\\cite{Dai:Tam})\n     A \\textit{globular} $n$-cell is a subset $D$ of $D^n$ containing $H= \\mathrm{Int}(D^n)$. We call $D \\cap \\partial D^n$ the \\textit{boundary} of $D$ and denote it by $\\partial D$. The number $n$ is called the \\textit{globular dimension} of $D$. \n\\end{defn}\n\nA globular $n$-cell was introduced by Tamaki \\cite{Dai:Tam} as an extension of closure of $n$-cells to non-closed cells.\n\n\n\\begin{defn} (\\cite{Dai:Tam})\n    Let $X$ be a topological space. For a non-negative integer $n$, an $n$-cell structure on a subspace $e \\subset X$ is a pair $(D, \\varphi)$ of a globular $n$-cell $D$ and a continuous map $$\\varphi: D \\rightarrow X$$\n    satsifying the following conditions:\n    \\begin{itemize}\n        \\item $\\varphi(D) = \\bar{e}$ and $\\varphi : D \\rightarrow \\bar{e}$ is a quotient map.\n        \n        \\item The restriction $\\varphi$ : $H \\rightarrow e$ is a homeomorphism.\n    \\end{itemize}\n    \n\\end{defn}\n\n\\begin{defn}(\\cite{Dai:Tam})\nLet $X$ be a topological space and $P$ be a poset with the Alexandroff topology. A stratification of $X$ indexed by $P$ is an open continuous map \n$$\\pi : X \\rightarrow P$$\nsatisfying the condition that for each $\\lambda \\in P$, $e_\\lambda = \\pi^{-1}(\\lambda)$ is connected and locally closed. $X$ is called a \\textit{cellular stratified space} if each $e_\\lambda$ is homeomorphic to an open ball.\n\\end{defn}\n\n\n\\begin{defn}(\\cite{Dai:Tam}, \\cite{Fur:Muk})\n    Let $X$ be a cellular stratifed space. $X$ is called totally normal if for each globular $n$-cell $(D_\\lambda, \\varphi)$,  and $e_\\lambda = \\varphi(\\mathrm{Int}(D_\\lambda))$\n    \\begin{enumerate}[(i)]\n        \\item If $e_\\lambda \\cap \\overline{e_\\mu} \\neq \\emptyset $, then $e_\\lambda \\subseteq \\overline{e_\\mu}$. \n        \\item There exists a structure of a regular cell complex on $S^{n-1}$ containing $\\partial D_\\lambda$ as a cellular stratified subspace of $S^{n-1}$.\n        \\item For each cell $e$ in the cellular stratification on $\\partial D_\\lambda$, there exists a cell $e_\\eta$ in $X$  and a map $b: D_\\eta \\rightarrow \\partial D_\\lambda$ such that $b(\\mathrm{Int}(D_\\eta)) = e$ and $\\varphi_\\lambda \\circ b = \\varphi_\\eta$.\n    \\end{enumerate}\n\\end{defn}\n\n\\begin{thm}\\label{thm:total} (\\cite{Dai:Tam})\nFor a totally  normal cellular stratified space $X$ with stratification $\\pi : X \\rightarrow P$, there is an embedding of $\\|P\\|$ as a strong deformation retract of $X$. \n\\end{thm}\n\n\n\n\n\n\n\n\\section{Flattenings} \\label{def:flattenings}\n\n\\begin{defn} (\\cite{losik:1}, \\cite{milin})\n\tLet $L$ be a triangulation of a $k$-sphere, and let $cL$ be a simplicial cone over $L$. A flattening of $L$ is an embedding $\\psi : cL \\rightarrow \\mathbb{R}^{k+1}$ that maps the cone vertex to the origin and it is linear on simplices of $cL$.  \n\\end{defn}\n\n\n\\begin{nota}\nLet $L$ be a simplicial $k$-sphere. We denote as in \\cite{losik:1} by $F(L)$ the space of all flattenings of $L$. Also, the group $GL_{k+1}$ of invertible $(k+1) \\times (k+1)$ matrices acts on $F(L)$; the quotient space denoted by $CF(L)$ is the configuration space of $L$.\n\\end{nota}\n\nThe space $F(L)$ is an open subset of $\\mathbb{R}^{(k+1)|\\mbox{Vert}(L)|}$, and so has a natural smooth manifold structure. The space of flattenings comes up in the problem of existence and uniqueness of differentiable structures on triangulated manifolds (see \\cite{losik:2}, \\cite{kuiper:nh}). \n\nWe will show that $CF(L)$ is contractible  when $L$ is a simplicial $1$-sphere, and so, $F(L)$ has the homotopy type of $O(2)$. Some few other non-trivial results that are known about the topology of $CF(L)$ and $F(L)$ are as follows.\n\n\\begin{thm}(\\cite{cairns})\n\tLet $L$ be a triangulated $2$-sphere. Then $CF(L)$ is path connected. \n\\end{thm}\n\nFor $\\dim(L) \\geq 3$, Cairns \\cite{cairns} also showed that $CF(L)$ can be empty. When the dimension of $L$  is greater than $2$,  Milin \\cite{milin} obtained the following negative result about the topology of $CF(L)$.\n \n \\begin{thm}\\cite{milin}\n \tThere exists a $3$ dimensional simplicial sphere $L$ such that $CF(L)$ is disconnected.\n \\end{thm}\n \nSo far, for $n > 2$ the only known positive result about the homotopy type of $F(L)$ is the following result of Kuiper.\n\n\\begin{thm}(\\cite{kuiper:nh})\\label{kuip}\n\tLet $\\partial \\Delta^{n+1}$ be the boundary of an $(n+1)$-simplex. Then $F(\\partial \\Delta^{n+1})$ has the homotopy type of $O(n+1)$. \n\\end{thm}\n\\begin{cor}(\\cite{kuiper:nh})\n\tLet $\\partial \\Delta^{n+1}$ be the boundary of an $(n+1)$-simplex. Then any two smoothings of $\\partial \\Delta^{n+1}$ are diffeomorphic. \n\\end{cor}\n\nAs in Theorem \\ref{kuip}, we also obtain the following positive result for the simplicial sphere $\\partial \\Delta^1 \\ast \\partial \\Delta^{n+1}$.\n\n\\begin{thm}\\label{my_flat}\nLet $\\partial \\Delta^{n+1}$ be the boundary of an $(n+1)$-simplex . Then $F(\\partial \\Delta^1 \\ast \\partial \\Delta^{n+1})$ has the homotopy type of $O(n+2)$. Let $L$ be a simplicial $1$-sphere. Then $F(L)$ has the homotopy type of $O(2)$. \n\\end{thm}\n\n\\begin{cor}\\label{cor_flat}\n\tLet $\\partial \\Delta^{n+1}$ be the boundary of an $(n+1)$-simplex. Then any two smoothings of $\\partial \\Delta^1 \\ast \\partial \\Delta^{n+1}$ are diffeomorphic. \n\\end{cor}\n\n\n\\section{Oriented matroid flattenings} \\label{or:flat}\nLet $L$ be a triangulated of a $k$-sphere, and $\\psi : cL \\rightarrow \\mathbb{R}^{k+1}$ a flattening of $L$. Then the arrangement of vectors $(\\psi(v): v \\in \\mathrm{Vert}(L))$ determines a rank $k+1$ oriented matroid $M$. Definition \\ref{comb_abstr} gives a combinatorial abstraction for oriented matroids obtained from flattenings of a simplicial sphere.\n\n\n\\begin{figure}[htb]\n\\centering\n\\begin{subfigure}[t]{0.35\\textwidth}\n    \\begin{tikzpicture}[line join=bevel,z=-5.5]\n    \\coordinate (A1) at (-0.2,-2);\n    \\coordinate (A2) at (-2,0);\n    \\coordinate (A3) at (-0.67,2);\n    \\coordinate (A5) at (0,0);\n    \\coordinate (A4) at (2,0.67);\n    \\draw (A1) -- (A2) -- (A3) ;\n    \\draw (A1) -- (A4) -- (A3);\n    \\path[->] (0,0) edge node[at end, left]{$3$} (-0.22, -2.2);\n    \\path[->] (0,0) edge node[at end, left]{$4$} (-2.08, 0);\n    \\path[->] (0,0) edge node[at end, left]{$1$} (-0.707, 2.06);\n    \\path[->] (0,0) edge node[at end, above]{$2$} (2.1, 0.7);\n    \\foreach \\Coor\/\\Texto\/\\Pos in \n    {A5\/0\/below\n    }\n    \\node[circle,draw,inner sep=1.5pt,fill=black,label={\\Pos:$\\Texto$}] \n    at (\\Coor) {};\n    \\end{tikzpicture}\n\\end{subfigure}\n\n~\n\n\\begin{subfigure}[t]{0.35\\textwidth}\n    \\begin{tikzpicture}[line join=bevel,z=-5.5]\n    \\coordinate (A1) at (-0.2,-2);\n    \\coordinate (A2) at (-2,0);\n    \\coordinate (A3) at (0.6,2);\n    \\coordinate (A5) at (0,0);\n    \\coordinate (A4) at (2,0);\n    \\draw (A1) -- (A2) -- (A3) ;\n    \\draw (A1) -- (A4) -- (A3);\n    \\path[->] (0,0) edge node[at end, left]{$3$} (-0.22, -2.2);\n    \\path[->] (0,0) edge node[at end, left]{$4$} (-2.08, 0);\n    \\path[->] (0,0) edge node[at end, left]{$1$} (0.613, 2.133);\n    \\path[->] (0,0) edge node[at end, above]{$2$} (2.093, 0);\n    \\foreach \\Coor\/\\Texto\/\\Pos in \n    {A5\/0\/below\n    }\n    \\node[circle,draw,inner sep=1.5pt,fill=black,label={\\Pos:$\\Texto$}] \n    at (\\Coor) {};\n    \\end{tikzpicture}\n\\end{subfigure}\n\\caption{Flattenings of a simplicial $1$-sphere.}\n\\end{figure}\n\n\n\\begin{defn}\\label{comb_abstr}\n    Let $L$ be a simplicial sphere of dimension $k$. An oriented matroid flattening of $L$ is a rank $k+1$ oriented matroid $\\mathcal{M}$ satisfying the following:\n    \\begin{enumerate}[(i)]\n        \\item The elements of $\\mathcal{M}$ are the vertices of $L$.\n        \\item The set of vertices in a simplex are independent.\n        \\item The set of vertices in a simplex has no other elements in its convex hull.\n        \n    \\end{enumerate}\n\\end{defn}\n\n \\begin{nota}\n The poset of all oriented matroid flattenings of $L$ is denoted by $P(L)$. \n\\end{nota}\n\n\\begin{prop}\\label{contr_prop}\nLet $L$ be a simplicial sphere and $\\partial \\Delta^n$ the boundary of an $n$-simplex. Then $\\|P(L)\\|$ is contractible when $L$ is either a simplicial $1$-sphere, $\\partial \\Delta^n$ or $\\partial \\Delta^1 \\ast \\partial \\Delta^n$. \n\\end{prop}\n\n\\begin{proof}\nThe poset $P(\\partial \\Delta^n)$ consists of a point. Let $P(\\partial \\Delta^n) = \\{\\mathcal{M}_n\\}$. For the sphere $\\partial\\Delta^1 \\ast \\partial \\Delta^n$, $P(\\partial\\Delta^1 \\ast \\partial \\Delta^n)$ has a minimum; given by the join $\\mathcal{M}_1 \\oplus \\mathcal{M}_n$ of two oriented matroids.\n\nIn the case when $L$ is a simplicial $1$-sphere, this will follow by induction on the number of vertices in $\\mathrm{Vert}(L)$. Let $L_n$ denote a simplical $1$-sphere on $n$ vertices. We know that $P(L_3)$ consists of a point say $M_0 = (\\pm \\chi_0)$. In the following argument, we will consider chirotopes with positive value on  the basis $\\{1,2\\}$ \n\nLet $\\Sigma^{n+1}$ denote a subposet of $P(L_{n+1})$ consisting of $\\mathcal{M}'$ such that $\\mathcal{M}'\\setminus \\{n+1\\}$ is an element of $P(L_n)$. An oriented matroid in $\\Sigma^{n+1}$ is thus an extension of an oriented matroid $\\mathcal{M}$ in $P(L_n)$ by an element ${n+1}$, with $n+1$ lying in the convex hull of $\\{1,n\\}$.\n\nThere is a poset map $P(L_{n+1}) \\rightarrow \\Sigma^{n+1}$ obtained as composition of some poset maps  as given below.\nLet $f_0 : P(L_{n+1}) \\rightarrow P(L_{n+1})$ defined as:\n$$f_0(\\chi)(B) = \\left\\{\\begin{array}{ccc}\n    \\chi(B) & \\mbox{if} & B \\neq (n, 1)  \\\\\n     0 &  \\mbox{if} & B = (n,1) \\; \\mbox{and}\\; \\chi(n, 1) \\in \\{0, -\\}\\\\\n     + & \\mbox{if} & B = (n,1) \\; \\mbox{and} \\; \\chi(n,1) = +\n\\end{array}\\right\\}$$\nLet $P_0 = f_0(P(L_{n+1}))$. The poset map $f_0$ is a lowering homotopy, and so $\\|P_0\\|$ is homotopy equivalent to $\\|P(L_{n+1})\\|$. We again consider another poset map $f_1: P_0 \\rightarrow P_0$ defined as:\n\n$$f_1(\\chi)(B) = \\left\\{\\begin{array}{ccc}\n    \\chi(B) & \\mbox{if} & B \\neq (n,1)  \\\\\n     + & \\mbox{if} & B = (n,1) \n\\end{array}\\right\\}$$\n\nThe image of $f_1$ is denoted is given by $f_1(P_0) = \\Sigma^{n+1}$. The poset map $f_1 : P_0 \\rightarrow P_0$ is a raising homotopy, and so $\\|P_0\\|$ is homotopy equivalent to $\\|\\Sigma^{n+1}\\|$. The poset map $\\Sigma^{n+1} \\rightarrow P(L_n)$ induces a homotopy equivalence between $\\|\\Sigma^{n+1}\\|$ and $\\|P(L_n)\\|$.\n\\end{proof}\n\nFor a simplicial sphere $L$, there is a stratification map $\\mu_0: CF(L) \\rightarrow P(L)$. \n\n\\begin{conj}\n\tLet $L$ be a simplicial sphere of dimension at least $2$. Then $\\|P(L)\\|$ is contractible.\n\\end{conj}\n\n\n\n\n\n\n\n\\section{Topology of space of flattenings of some spheres} \\label{top:flat}\nLet $\\mu' : \\mathrm{Gr}(r, \\mathbb{R}^{r+2}) \\rightarrow \\mathrm{MacP}(r, r+2)$ and $\\mu: \\mathrm{Gr}(2, \\mathbb{R}^n) \\rightarrow \\mathrm{MacP}(2,n) $.  Let $\\mu_0 : CF(L) \\rightarrow P(L)$ be the restriction of $\\mu'$ to $CF(L)$ when $L = \\partial \\Delta^1 \\ast \\partial \\Delta^{r-1}$ or the restriction of $\\mu$ when $L$ is a simplicial 1-sphere on $n$ vertices.\n\n\nThe stratification map $\\mu_0: CF(L) \\rightarrow P(L)$ gives a decomposition of $CF(L)$ into semi-algebraic sets $\\{\\mu_0^{-1}(M) : M \\in P(L)\\}$. When $L$ is a simplicial 1-sphere or $L = \\partial \\Delta^1 \\ast \\partial \\Delta^n$, we will show that the decomposition is a totally normal cellular decomposition. \n\n\nWe have the following commutative diagram\n\n\\begin{tikzcd}\n\\mathrm{Gr}(r, \\mathbb{R}^{r+2}) \\arrow[r, \"\\mu'\"] \\arrow[d, \"V \\mapsto V^{\\perp}\", labels = left ] & \\mathrm{MacP}(r, r+2) \\arrow[d, \"M\\mapsto M^*\"]\\\\\n\\mathrm{Gr}(2, \\mathbb{R}^{r+2}) \\arrow[r, \"\\mu\"] & \\mathrm{MacP}(2, r+2)\n\\end{tikzcd}\n\nThe commutativity of the diagram follows from the fact that  \n\n$V = (I_r | A) \\in \\mathrm{Gr}(r, \\mathbb{R}^{r+2})$ if and only if $V^{\\perp} = \\mathrm{Rowspace}(-A^T|I_2) \\in \\mathrm{Gr}(2, \\mathbb{R}^{r+2})$. The oriented matroid $M^*$ is called the dual of $M$.\n\nThe map $\\mathrm{Gr}(r, \\mathbb{R}^{r+2}) \\rightarrow \\mathrm{Gr}(2, \\mathbb{R}^{r+2}) : V \\mapsto V^{\\perp}$ is a homeomorphism and the poset map $\\mathrm{MacP}(r, r+2) \\rightarrow \\mathrm{MacP}(2, r+2) : M \\mapsto M^*$ is a poset isomorphism.\n\n\nThe  following result thus follows  from Theorem \\ref{ref2}\nand the  commutativity of the diagram described above.\n\n\n\\begin{thm}\\label{thm:reg}\n\tLet $M \\in \\mbox{MacP}(r, r+2)$ be a rank $r$ oriented matroid on $r+2$ elements, and $\\mu' : \\mathrm{Gr}(r, \\mathbb{R}^{r+2}) \\rightarrow \\mbox{MacP}(r,r+2) $. Then $\\{\\overline{(\\mu')^{-1}(M)}: M \\in \\mathrm{MacP}(r, r+2)\\}$ is a regular cell decomposition of $\\mathrm{Gr}(r, \\mathbb{R}^{r+2})$.\n\\end{thm}\n\n\n\\begin{prop}\\label{tot_nor}\nLet $L$ be a simplicial sphere and  $\\mu_0 : CF(L) \\rightarrow P(L)$ a stratification map. If $L$ is a simplicial $1$-sphere or $L = \\partial \\Delta^1 \\ast \\partial \\Delta^n$, then the decomposition $\\{\\mu_0^{-1}(M) : M \\in P(L)\\}$ is a totally normal cellular decomposition of $CF(L).$\n\\end{prop}\n\n\\begin{proof}\n\\begin{enumerate}[(i)]\n    \\item Suppose $L$ is as given above. It was proven in Proposition \\ref{ref1} that if $N, M \\in P(L)$ such that $N < M$, then $\\mu_0^{-1}(N) \\subseteq \\overline{\\mu_0^{-1}(M)}$. So, the decomposition  $\\{\\mu_0^{-1}(M) : M \\in P(L)\\}$ is normal.\n    \n    \\item In Theorem \\ref{ref2}, it was proven that $\\{\\overline{\\mu^{-1}(M)} : M \\in \\; \\mbox{MacP}(2, |\\mathrm{Vert}(L)|)\\}$ is a regular cell decomposition of $Gr(2,\\mathbb{R}^{|\\mathrm{Vert}(L)|})$. Similarly, we have in Theorem \\ref{thm:reg} that $\\{\\overline{(\\mu')^{-1}(M)} : M \\in \\; \\mbox{MacP}(r, r+2)\\}$ is a regular cell decomposition of $Gr(r,\\mathbb{R}^{r+2})$.\n    \n    If $L$ is a simplicial $1$-sphere, and $M \\in P(L)$, then $\\partial \\overline{\\mu^{-1}(M)}$ is a regular cellular cell complex  homeomorphic to a sphere. Let $\\overline{\\mu_0^{-1}(M)}$ denote the closure of $\\mu_0^{-1}(M)$ in $CF(L)$. Then $\\partial \\overline{\\mu^{-1}(M)}$ contains $\\partial \\overline{\\mu_0^{-1}(M)}$ as a cellular stratified subspace. Similarly when $L = \\partial \\Delta^1 \\ast \\partial \\Delta^n$ and $M \\in P(L)$, $\\partial \\overline{(\\mu')^{-1}(M)}$ contains $\\partial \\overline{\\mu_0^{-1}(M)}$ as a cellular stratified subspace.\n    \n    \\item $D_M = \\overline{\\mu_0^{-1}(M)}$, and let $\\varphi_M$ be the restriction to $D_M$ of the characteristic map of the cell $\\overline{\\mu^{-1}(M)}$ if $L$ is a simplicial $1$-sphere or  restriction of the characteristic map of $\\overline{(\\mu')^{-1}(M)}$ if $L = \\partial \\Delta^1 \\ast \\partial \\Delta^n$.\n    \n    For a cell $e$ in the boundary of $D_M$, there exists an oriented matroid $N$ in $P(L)$ such that $N < M$ and $\\mu_0^{-1}(N) = e$. The map $b : D_N \\rightarrow \\partial D_M$ is given by $b = (\\varphi_M)^{-1} \\circ \\varphi_N.$ \n    \n    \\end{enumerate}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{my_flat}]\nSuppose $L$ is a simplicial $1$-sphere or $L= \\partial \\Delta^1 \\ast \\partial \\Delta^n$. The decomposition $\\{\\mu_0^{-1}(M): M \\in P(L)\\}$ is a totally normal cellular decomposition of $CF(L)$ by Proposition \\ref{tot_nor}. It thus follows from Theorem \\ref{thm:total} that $\\|P(L)\\|$ is a deformation retract of $CF(L)$. We know from  Proposition \\ref{contr_prop} that $\\|P(L)\\|$ is contractible. Hence, $CF(L)$ is contractible.\n\nSuppose $L$ is a simplicial $1$-sphere. We know that $F(L)|_H \\cong \\mathrm{GL}_2(\\mathbb{R}) \\times CF(L)$. Hence $F(L)$ has the homotopy type of $O(2)$. Similarly, if $L = \\partial \\Delta^1 \\ast \\partial \\Delta^n$, then $F(L) \\cong \\mathrm{GL}_{n+1}(\\mathbb{R}) \\times CF(L)$. Hence $F(L)$ has the homotopy type of $O(n+1)$.\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","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction }\n\nThe development of clever fabrication techniques in semiconductors has\nbrought the reduction of the effective dimension of electronic states\nfrom their usual three-dimensional character in bulk materials, to \n``zero-dimensional'' states in quantum dots.\\cite{1}  The quantum effects\nof these lower-dimensional systems have attracted much attention in\nrecent years, due in part to possible applications which include\nelectronic devices based on parallel and perpendicular transport,\nquantum well lasers, and optical devices. \\cite{2}  Two-dimensional\nquantum-well or quantum-film structures, which provide confinement in\none space dimension, have been well investigated, and quantum exciton\neffects observable even at room temperature have been studied.\\cite{3} \nThe confinement of excitons has also been shown to result in very large\nelectro-optical shifts of the absorption peaks, producing the so-called\nquantum-confined Stark effect.\\cite{3,4}\n                                            \nIn quasi-zero-dimensional quantum dot systems, the additional  quantum\nconfinement dramatically changes the optical and electronic properties,\ncompared to those in higher-dimensional structures, as the whole\nsingle-particle spectrum is now discrete.  Correspondingly, the\nexcitonic spectrum is expected to be strongly affected.  The properties\nof excitons confined in quantum boxes were first analyzed theoretically\nby Bryant,\\cite{5} who used variational and configuration-interaction\nrepresentations.  Later on, excitons and biexcitons have been studied,\n\\cite{6,7} as well as excitons in the presence of a strong magnetic\nfield,\\cite{8} using numerical matrix diagonalization schemes.   \n\nOn the experimental side, interband optical spectroscopies, such as\nphotoluminescence, have been used to study various quantum dot systems\n--- such as those produced in the GaAs\/Al$_x$Ga$_{1-x}$As structure, with its\nbandgap modulation.\\cite{9,10,11}  More recently, fascinating studies\non so-called ``self-assembled'' quantum dots, such as \nInAs and In$_x$Ga$_{1-x}$As\nclusters on GaAs substrates, have also been reported.\\cite{12,13,14,15}\n Most of the theoretical investigations are based on the assumption\nthat the shape of quantum dots is a simple sphere or box, having a\ngreat deal of symmetry, both because  it simplifies calculations and\nbecause quantities such as the exciton binding energy scale very well\nwith the overall dot size.   However, realistic dot shapes are probably\nmuch less symmetrical, as well as being typically flat and more\ntwo-dimensional in shape.\\cite{12,13,14,15}  \n                                           \nHere, we consider the effect that less symmetric structures, namely\nflat quantum dots with elliptical cross sections, or ``elliptical\nquantum disks'', have on the excitonic optical properties. To date,\nlittle work has been reported on the properties of nonsymmetric\nquantum dots, probably because this system has more complicated\nsolutions.\\cite{13,15}  Our studies within the effective mass\napproximation yield some very interesting consequences of the\nelliptical asymmetry: apart from the expected blue shift of the first\nexcitonic transition for dots with the same overall area but different\naxes, we find a rearrangement of the oscillator strength which\ncharacterizes individual dot shapes.  In particular, since elliptical\ncross-section dots have less symmetry, some of the accidental\ndegeneracies in circular dots giving rise to stronger and fewer peaks\nin the imaginary part of the optical susceptibility are split.  This\ngives rise to a more monotonically decreasing peak intensity for higher\nenergy features in the susceptibility of noncircular dots.  This\nbehavior can in turn be used to structurally characterize specific dots\nfrom their photoluminescence excitation response.  \n                                   \nThe remainder of the paper is organized as follows.   We introduce the\ntheoretical method in Sec. II.   Here, we outline the effective mass\nHamiltonian approach and introduce the various basis function\nrepresentations which allow us to use numerical methods to calculate\nthe eigenvalues and  eigenfunctions of excitons in these quantum dots. \nA great deal of care is needed to assure that the solutions obtained\nare well behaved and converged with a finite computational effort.  We\ndiscuss in this section how this is accomplished.  In Sec. III, we\ndiscuss the main geometrical effects on various exciton\ncharacteristics, such as the exciton binding energy, electron-hole\nseparation, and the linear optical susceptibility.  Solutions for\nexcitons in quantum dots with both circular and elliptical\ncross sections are shown, using large enough basis sets, and a set of\noptimized basis functions, which improve the accuracy of the solutions\nat a modest computational cost. Finally, we summarize our conclusions\nin Sec. IV.  The Appendix contains an outline of the derivation of\nthe Coulomb matrix element with these basis functions.  The analytical\nexpression presented there greatly simplifies our calculations.\n\n\n\\section{ Theoretical Method }\n\nFor concreteness, and to simulate recent quantum dot\nsystems,\\cite{12,13,14} we assume quantum dots with an oblate\nspheroidal profile where the lateral $xy$ confinement is much weaker\n(or larger size) than that along the $z$ direction.  Correspondingly,\nthe electrons and holes are assumed confined in an effectively\ntwo-dimensional potential with a constant $z$ profile, $V_z$.  We\nassume $V_z$ to be a hard-wall confinement potential, so that the\n$z$ component of the energy is $\\hbar^2 \\pi^2\/2mL_z^2$, with $L_x, L_y\n\\gg L_z$.  Further, we approximate the single $z$ wavefunction in the\nproblem as a $\\delta$ function centered at the origin, so that the problem\ncan be described by a separable Hamiltonian in two dimensions.  The\nlateral confinement is modeled via harmonic potentials with two\ndifferent frequencies $\\omega_x$ and $\\omega_y$, which yield the\nelliptical cross section of the dots with axes ratio given by $L_x \/\nL_y = \\sqrt{\\omega_y\/\\omega_x}$, for both electrons and holes.   The\nsmoothly varying potential should mimic well the situation in\nexperiments  where the dots are effectively embedded in a dielectric\nmatrix.\\cite{1,12}\n\nThe effective-mass parabolic-band Hamiltonian for an electron-hole pair\nis given by $H = H_e + H_h + H_{e-h}$, where the subscripts $e$ and $h$\nrepresent electron and hole, and \n \\begin{equation}\nH_e = \\frac{p^2}{2m_e} +  \\frac{1}{2}\nm_e \\omega_x^2 x_e^2 + \\frac{1}{2} m_e \\omega_y^2 y_e^2 + V_{ze} \\, ,\n \\end{equation}\n with a similar expression for the Hamiltonian of the hole,\n$H_h$.\\cite{NOTE}  The Coulomb interaction between electron and hole is\nscreened by a background dielectric constant $\\epsilon$, so that\n$H_{e-h} = -e^2\/\\epsilon r_{e-h}$.  \n                                               \nWe rewrite the Hamiltonian into relative and center of mass\ncoordinates, described by ${\\bf r} = {\\bf r}_e - {\\bf r}_h$, and\n${\\bf R} = (m{_e}{\\bf r}_e + m_h{\\bf r}_h)\/M$.\n The total and reduced masses are given by $M = m{_e} + m{_h}$, and\n$\\mu = m{_e}m{_h}\/M$, respectively.  The total Hamiltonian of this\nsystem can then be written in the form \n$H = H{_{c.m.}} + H{_{rel}}$,\nwith the expected expressions  \n \\begin{equation}\nH_{c.m.} = \\frac{P^2}{2M} + \\frac{1}{2}M\\omega_x^2\nX^2  + \\frac{1}{2}M\\omega_y^2 Y^2 + V_Z \\, ,\n \\end{equation}\n and \n \\begin{equation}\nH{_{rel}} = \\frac{p{{^2}}}{2\\mu} + \\frac{1}{2}{\\mu}\\omega_x^2 x{^2}\n + \\frac{1}{2}{\\mu}\\omega_y^2 y{^2}\n - \\frac{e{^2}}{\\epsilon \\sqrt{x{^2} + y{^2}}} + V_z \\, .\n \\end{equation} \n The Hamiltonian of the center of mass is obviously a two-dimensional\nharmonic oscillator in the $XY$ plane, with wavefunction  $\\Psi_{N_X\nN_Y} = \\phi_{N_X}(X)\\, \\phi_{N_Y}(Y)$, and energy $E_{c.m.}$, where\n \\begin{equation} \n \\phi{_{N}}(X)  = \\left(\\frac{{\\alpha_M}}{{\\pi^{1\/2}}2{^{N}}N!}\\right)^{1\/2} \\,\ne^{-\\alpha_M^2 X^2\/2} H_N \\left( \\alpha_M X \\right) \\, , \n \\end{equation} \n$\\alpha_M = \\sqrt{M \\omega_x \/ \\hbar} \\, ,$ and \n \\begin{equation}\nE_{c.m.} = \\left(\nN_X + \\frac{1}{2} \\right) \\hbar \\omega_x + \\left( N_Y + \\frac{1}{2} \\right)\n\\hbar \\omega_y + \\frac{\\hbar^2 \\pi^2}{2ML_z^2} \\, .\n \\end{equation}\n  Here, $N{_X}$ and $N{_Y}$ are quantum numbers for the center of mass\ncoordinate, and $H{_N}$ is a Hermite polynomial.\\cite{17}\n\nThe physics of the problem is determined to a great extent by the ratio\nbetween the effective Bohr radius, $a_B^* = \\hbar^2 \\epsilon \/ \\mu\ne^2$, and the size of the dot, $L=\\sqrt{L_x L_y}$, where\n$L_i=\\sqrt{\\hbar \/\\mu \\omega_i}$.  The strong confinement limit for $L\n\\leq a_B^*$ is characterized by a weak electron-hole correlation and by\nthe Coulomb term being a small perturbation of the single-particle\nconfined-level energy.  On the other hand, the weak-confinement limit\nfor $L \\geq a_B^*$ reduces asymptotically to the problem of a free\ntwo-dimensional exciton for large $L$, where the Coulomb interaction \ndominates the state of the exciton \\cite{1}.   \n                                          \nWith this in mind, the effects of the Coulomb term $H_1$ in $H_{rel} =\nH_0 + H_1$, are treated by using the solutions of $H_0$ as the basis\nset in the diagonalization of $H_{rel}$. The unperturbed Hamiltonian of\nthe relative coordinate $H_0$ is also a two-dimensional harmonic\noscillator, so that the wavefunction of the interacting electron-hole\npair is described by a linear combination of wavefunctions,  $\\psi_{n_x\nn_y}=\\phi_{n_x}(x) \\phi_{n_y}(y)$, with the $\\phi$'s  satisfying a\nsimilar expression to Eq.\\ (4), and correspondingly  \n \\begin{equation}\nE{_{rel}^0} =\n\\left( n_x + \\frac{1}{2} \\right) \\hbar \\omega_x + \\left( n_y +\n\\frac{1}{2} \\right) \\hbar \\omega_y + \\frac{\\hbar^2 \\pi^2}{2\\mu (2L_z)^2} \\, ,\n \\end{equation} \n where $n_x$ and $n_y$ are quantum numbers for the relative coordinate.\n(Notice $z$ confinement length for this coordinate is $2L_z$.)\n \nWith this basis set, the interaction matrix elements of the\nelectron-hole pair can be calculated analytically and expressed in\nterms of hypergeometric functions as outlined in the Appendix.  This\nanalytical expression greatly simplifies the calculation, as most of\nthe computational time is spent on the calculation of the matrix\nelements rather than on the diagonalization of the matrix.   The\nresulting Hamiltonian matrix is real, symmetric, and sparse.  The\nenergies and eigenfunctions are calculated from the numerical\ndiagonalization of the matrix, for a given size of the basis.  The\ndiagonalization is repeated with larger basis sets until the desired\nconvergence is achieved (see below).\n \n\\subsection {Circular limit}  \n As an additional test of our numerical procedures we use the resulting\nradial equation of the relative Hamiltonian for the circular dot case,\n$\\omega_x = \\omega_y = \\omega$, which is then given by \n \\begin{equation}\n H_{rel} = \\frac{p^2}{2\\mu} + \\frac{1}{2} \\mu \\omega^2 r^2 - \n \\frac{e^2}{\\epsilon r} \\, .\n \\label{neweq7}\n \\end{equation}\n The resulting one-dimensional equation can be directly integrated\nnumerically, as reported by Que.\\cite{7}  Furthermore, the problem can\nalso be solved using a harmonic basis set of radial states as those\ndescribed above.  For a large enough basis set, this approach yields\nthe same results as those obtained by direct numerical\nintegration.\\cite{7}  Notice also that the large dot limit $(\\omega\n\\approx 1\/L \\approx 0)$ is easily solved as an expansion in terms of\nthe Laguerre polynomial-based solutions of the free exciton\nproblem.\\cite{7}  This allows one to obtain very accurate solutions for\n$1\/L \\approx 0$ with little computation.  These results, moreover,\nallow us to test the convergence of the irreducible two-dimensional\nnonsymmetric problem $(\\omega_x \\neq \\omega_y)$, as we discuss below.\n                                          \n\\subsection {Optimized basis}\n  Solutions for these elliptic cylinderlike quantum dots are also\ncarried out using an optimized basis set, as the solution method\ndiscussed above requires a rather large number of states for full\nconvergence, especially for large dots.  Here, one notices that for\nlarge dots it is the Coulomb interaction that dominates the exciton\nstates (since confinement becomes less important).  Correspondingly, we\nchoose a set of optimized frequencies, ${\\Omega}_x$ and ${\\Omega}_y$, \nwhich are larger than the original frequencies,  ${\\omega}_x$ and\n${\\omega}_y$.   The values of the $\\Omega$'s are determined\nvariationally, and allow one to consider $H$-matrix systems that are\nmuch smaller than those required when one uses the $\\omega$ basis.  \nThe physical reason for this is that as the dot size increases, the\nexciton size converges to $a_B^{2D}$ (the radius of the free\ntwo-dimensional exciton), and one needs a large number of\n$\\omega$ states to describe the small-scale structure of the exciton.\n\nNotice that the harmonic $\\Omega$ basis allows also an easy calculation\nof  the $H$-matrix elements in this case, so that for example,\n \\begin{eqnarray}\n  \\langle n_x{^{\\prime}} \\, n_y{^{\\prime}} \\, &| &H_{0} | n_x \\, n_y\n\\, \\rangle_\\Omega =  \\left[{\\hbar}\\Omega{_x}(n{_x}+\\frac{1}{2}) \\, + \\,\n{\\hbar}\\Omega{_y}(n{_y}+\\frac{1}{2}) \\right. \\nonumber  \\\\ \\,\n & & - \\, \\left.\n\\frac{\\hbar}{2\\Omega{_x}}(\\Omega_x^2-\\omega_x^2)(n{_x}+\\frac{1}{2})\n\\, - \\,\n\\frac{\\hbar}{2\\Omega{_y}}(\\Omega_y^2-\\omega_y^2)(n{_y}+\\frac{1}{2})\n\\right] \\, \\delta{_{n{_{x^\\prime}},n{_x}}} \\,\n\\delta_{n_{y^\\prime},n_{y}} \\nonumber \\\\ \n & &- \\,\n\\frac{\\hbar}{4\\Omega{_x}}(\\Omega_x^2- \\omega_x^2) \\left[\n\\sqrt{n_x(n{_x}-1)} \\, \\delta{_{n{_{x^\\prime}},n{_x}-2}} +\n\\sqrt{(n_x+1)(n{_x}+2)} \\, \\delta{_{n{_x^{\\prime}},n{_x}+2}} \\right]\n \\, \\delta{_{n_{y^\\prime},n_y}} \n \\nonumber \\\\ \n & & - \\,\n\\frac{\\hbar}{4\\Omega{_y}}(\\Omega_y^2-\\omega_y^2) \\left[\n\\sqrt{n_y(n{_y}-1)} \\, \\delta{_{n{_{y^\\prime}},n_y-2}} +\n\\sqrt{(n_y+1)(n{_y}+2)} \\, \\delta{_{n{{_{y^{\\prime}}}},n{_y}+2}}\n\\right] \\, \\delta{_{n{{_{x^\\prime}}},n{_x}}}  \\,\\,  .\n \\label{oldeq7}\n \\end{eqnarray}\n Notice this reduces to the obvious diagonal matrix for $\\Omega\n\\rightarrow \\omega$.  This expression allows one to evaluate the\nHamiltonian matrix rather conveniently, even for this other basis set.\n\n\\subsection {Exciton characteristics}             \n The wavefunctions of the relative coordinate problem can then be\nwritten as $| \\psi \\rangle  = \\sum_{n{_x}n{_y}}  a{_{n{_x}n{_y}}} |\nn{_x} , n{_y} \\rangle$, with either the $\\omega$ or $\\Omega$ basis\nstates, which can be used to study various characteristic properties of\nthe exciton system.  For example, the mean electron-hole separation\n$r_s$, is given by\n \\begin{eqnarray}\n r_s^2 &=& \\langle \\psi | r^2 | \\psi \\rangle  =  \\sum_{n{_x}n_y}\n \\left[\\frac{\\hbar}{{\\mu}\\Omega{_y}}\n(n{_y}+\\frac{1}{2})+\\frac{\\hbar}{{\\mu}\\Omega{_x}}(n{_x}+\\frac{1}{2})\\right]\n|a{_{n{_x},n{_y}}}|^2 \\, \\nonumber \\\\ \n & &+\n \\, \\frac{1}{2}\\frac{\\hbar}{{\\mu}\\Omega{_y}}\\left[\\sqrt{(n{_y}+2)(n{_y}+1)}\n \\, a{{^*}{_{n{_x},n{_y}+2}}} +\n\\sqrt{n{_y}(n{_y}-1)}\n \\, a{{^*}{_{n{_x},n{_y}-2}}}\\right]a{_{n{{_x}},n{{_y}}}} \\nonumber \\\\\n & &+ \\,\n\\frac{1}{2}\\frac{\\hbar}{{\\mu}\\Omega{_x}}\\left[\\sqrt{(n{_x}+2)(n{_x}+1)}\n  \\, a{{^*}{_{n{_x}+2,n{_y}}}} +\n\\sqrt{n{_x}(n{_x}-1)}\n  \\, a{{^*}{_{n{_x}-2,n{_y}}}}\\right]a{_{n{{_x}},n{{_y}}}} \\, ,\n \\end{eqnarray}\n which gives an idea of the exciton size.\n\n One can also use the diagonalization results to calculate\ndirectly measurable properties, such as the linear optical\nsusceptibility of the quantum dot\/disk.  The linear optical\nsusceptibility is proportional to the dipole matrix elements between\none electron-hole pair j state and the vacuum state, $\\langle  0 | P |1 \n\\rangle_j$.\\cite{5}  These in turn are proportional to the interband\nmatrix element, $p_{cv}$,\\cite{18} which is the matrix element formed\nbetween an electron and hole in the conduction and valence bands,\nrespectively.  The form of the dipole matrix elements for a single\nexciton in the envelope function approximation is given by\\cite{5,7}  \n \\begin{equation}\n|\\langle 0 | P |1 \n\\rangle|^2 = |p_{cv}|^2 {|{\\psi}(0)|}^2 \\left|{\\int}{\\int} {\\Psi}(X_e,Y_e)\n \\, dX_e \\, dY_e \\right|^2 \\, .\n \\end{equation}\n  Here, the wavefunction for the relative coordinate is given as\nabove, so that \n \\begin{equation}\n |\\psi(0)|^2 = (\\mu\/\\hbar \\pi) \\sqrt{w_ x w_y} \\, \\left|\\sum_{n_xn_y}\n(2^{n_x+n_y}n_x!n_y!)^{-1\/2} \\, a_{n_xn_y}\\right|^2 \\, ,\n \\end{equation}\n where the $ a_{n_xn_y}$ coefficients are obtained from the diagonalization\nof the relative-coordinate Hamiltonian, and\n \\begin{equation}\n \\left|{\\int}{\\int} {\\Psi}(X_e,Y_e) dX_e dY_e \\right|^2 = 4 \\pi^2 \\hbar\nN_X! \\, N_Y! \\, \\left[\\pi M \\sqrt{w_xw_y} \\,\\, 2^{N_X+N_Y} \\,\n(N_X\/2)!^2 \\, (N_Y\/2)!^2 \\right]^{-1} \\, , \n \\end{equation}\n with $N_X= {\\rm even}$ and $N_Y = {\\rm even}$, for nonzero matrix\nelements.  Finally, the dipole matrix elements have the form  \n \\begin{equation} \n |\\langle0| P |1\\rangle|^2 = 4 |p_{cv}|^2 \\frac{\\mu}{M} N_X! \\, N_Y! \\,\n\\left[2^{N_X+N_Y} \\, (N_X\/2)!^2 \\, (N_Y\/2)!^2 \\, \\right]^{-1} \\, \\left|\n\\sum_{n_x n_y}  (2^{n_x+n_y}n_x! \\, n_y!)^{-1\/2} \\, a_{n_x n_y} \\,\n\\right|^2 \\, .\n \\end{equation}\n The linear optical susceptibility can then be calculated from\n \\begin{equation}\n\\chi ( \\omega ) = \\sum_{j} |\\langle 0 | P | 1 \\rangle_j |^2 (\\hbar\n\\omega -E_j -i \\hbar \\Gamma)^{-1} \\, ,\n \\end{equation}\n where $\\Gamma$ is introduced as a phenomenological level broadening\nconstant.\n  \nNotice also that since experimental systems are typically configured\nto analyze a large collection of nearby dots, one should in principle\nbe concerned by the effect of local fields.   However, in typical\nexperimental systems so far, where the separation between dots can be\nseveral microns, it is valid to assume that dots are basically\nindependent.  In the case of higher-dot densities, however, the\ndynamical response of the system may be affected by the local fields\nproduced by neighboring dots, and one can obtain that response from the\nindividual microscopic polarizabilities.\\cite{16} \n       \n\n\\section { Results }\n\nAs an interesting example of a typical system, we use parameters to\ndescribe GaAs quantum dots, so that the dielectric constant is\n$\\epsilon=13.1$, and the carrier masses are $m{_e}=0.067m{_0}$, and (the\nheavy hole effective mass) $m_{hh}=0.37m{_0}$.\\cite{NOTE}  We present\nthe numerical results for heavy-hole excitons in GaAs quantum dots with\nelliptical and circular cross sections.  The solutions can be\ncalculated using a sufficiently large basis set and\/or an optimized\nbasis set, as described above.  Results obtained from these methods and\ndifferent basis sets are shown in Fig.\\ 1.  The exciton binding energy\nand normalized electron-hole separation are shown as a function of\nquantum dot size ranging from 2 to 100 nm.\n                     \nIn the insets, results are shown for {\\em circular} dots, with dot,\ndashed, and dot-dashed curves showing results for basis sets with\n$M=30, 100$, and $500$ wavefunctions, respectively.  Here, states with\n$n$ and $n{^{\\prime}}$ from $0$ to $29$, $99$, and $499$ are used in\nEq.\\ (\\ref{neweq7}).  (The matrix size is obviously $M \\times M$, and\nis diagonalized by a QL decomposition technique.\\cite{19})  The results\nof the one-dimensional radial equation in the weak-confinement limit\nare shown with the solid line for comparison, and represent the exact\nquantity (both $E_b$ and $r_s$) for large $L$. The transition between\nthe strong and weak confinement regimes comes appropriately when the\nsize of the quantum dot is near the effective Bohr radius, $L \\approx\na_B^* =12.2$ nm.  Notice that it is for $L \\approx 15$ nm that the\n$M=100$ curve (dashed) departs from the exact result (solid line), and\nthat one requires larger $M$ values as $L$ increases to achieve better\nconvergence.  The $M=500$ basis set  (dot-dashed curve) yields the\nconvergent solutions with acceptable accuracy and execution time for a\nlarger range of $L$ values ($\\leq 60$ nm).  Similar behavior can be\nseen in the electron-hole separation in the ground state [inset in\npanel (b)].\n\nThe inset in 1(a) also shows the difference between the $\\omega$ basis\nand $\\Omega$ basis  results.   Diamonds show results for the $M=400$\nbasis set with optimized $\\Omega$ frequency, while triangles show\nresults for $M=400$ with the original $\\omega$ basis set.   For both\ncases, the states with $n_x$, $n_y$, $n_x{^{\\prime}}$, and\n$n_y{^{\\prime}}$ from $0$ to $19$, respectively, are included for the\n$M=400$  basis set (see Eq.\\ \\ref{oldeq7}).  These states are used as\nthe basis set for elliptical quantum dots.  The results with the\noptimized basis set are essentially identical to the convergent\nsolutions.   The optimized frequencies used are also given as a\nfunction of dot size in the  inset 1(a) (with $E_\\Omega = \\hbar\n\\Omega$) as a long-dashed curve.  For dot sizes $30$ nm and larger, the\noptimized frequency $\\Omega$ converges to 80 meV, corresponding to a\ndot size $L_{\\Omega}=\\sqrt{\\hbar\/{\\mu}{\\Omega}} =4.1$ nm.  This latter\nvalue is close to the two-dimensional effective Bohr radius\n($a^{2D}_B=a^*_B\/2\\sqrt{2}=4.3$ nm), as one would expect.  Notice that\nfor most of the range of $L$'s shown, one is in the weak-confinement\nregime, where $L \\ge a^{2D}_B$.   This is why the optimized\nbasis set gives very reasonable results with minimal effort.  It is\nalso interesting that the agreement continues also for smaller dot\nsizes, completing the range from $2$ to $100$ nm.  The exciton ground\nstate binding energies and the normalized electron-hole  separation\nobtained with the optimized $\\Omega$ basis approach are basically exact\nto the fully converged results.\n\nThe main panels in Fig.\\ 1 (a) and (b) show the geometrical confinement\neffects of excitons with several different size axis ratios in the\n$xy$ plane.  The plots show results versus $L=\\sqrt{L_xL_y}$, the\neffective size of the dot, for ${\\omega_x}$\/${\\omega_y}=1$, $4$, and\n$9$ ($L_y\/L_x=1$, $2$, and $3$) with diamonds, pluses, and triangles,\nrespectively.  \nThe exciton binding energy increases for a small $L$ as the axis \nratio is increased; meanwhile, the normalized electron-hole separation \nis basically unchanged, \nexcept for small $L$ values, where\nthe confinement energy dominates.  Notice that as the axis ratio\nincreases, the single-particle and exciton states move up in energy but \nthe {\\em binding} energy increases.  \nThis increase in binding energy for\nthe elliptical dots is then related to the increase of the Coulomb\nenergy {\\em relative} to the confinement contribution to the total\nenergy.  (In fact, since $r_s\/L$ is basically unchanged with geometry,\nthe Coulomb interaction energy is nearly constant in all these cases.)\n\nTo better explore the geometrical confinement effects on excitons,  we\nshow the linear optical susceptibility of the GaAs quantum disks with\nelliptical cross sections and lateral mean size of $L=\\sqrt{L_xL_y}=5$\nnm (Fig.\\ 2), and 10 nm (Fig.\\ 3).  The dot thickness along the $z$-\ndirection is kept constant at $L_z=3$ nm, and several size ratios for\neach axis in the $xy$ plane are shown.  We use here also a value for\nthe optical bandgap of $E_g = 1.51$ eV.  The results presented here\nwere obtained using the optimized $\\Omega$ basis set approach discussed\nabove.  Notice that since this function represents all of the possible\ntransitions of this excitonic system, its features would be measurable\nvia photoluminescence excitation measurements.  On the other hand,\nthe photoluminescence response would correspond to the first (lower\nenergy) feature in these traces, associated with the ground state of\nthe excitonic system.\n\nFigure 2 shows the imaginary part of the linear optical susceptibility\nas a function of frequency for a dot with $L = \\sqrt{L_x L_y} = 5$ nm\n(a broadening of $\\Gamma=2$ meV is used).  The bottom trace is for a\ncircular dot, so that $L_x = L_y = 5$ nm.  The upper two traces show\nresults for elliptical dots with a size ratio  $L_y:L_x = 2:1$\n($L_y=7.0$ nm, and $L_x=3.5$ nm), and $3:1$ ($L_y=8.7$ nm, $L_x=2.9$\nnm), respectively, all having the same mean size $L=5$ nm.  [Notice\nthat  although the peak heights are in arbitrary units, the ratio\nbetween different peaks or traces is real, reflecting the different\ndipole matrix elements involved.]~  One obvious difference among traces\nis that the first transition energy shifts to higher values by $\\approx\n26$ and 69 meV, as the size ratio changes from 1:1 to 2:1 and 3:1,\nrespectively.  Notice that while increasing the size ratio, the exciton\nbinding energies {\\em increase} from 47 meV to 48 and 49 meV for each\nratio.  It is then clear that the larger blue shifts are due mostly to\nthe increasing confinement as the disk becomes more elliptical, and not\ndue to the binding energy between electron and hole.  In\nall the cases shown, the transition involving the ground state of the\nexciton is dominant, as the excited states appear only with smaller\noscillator strength.  Notice further that for the larger length ratio,\nthe spectrum is understandably sparser, as the levels associated with\nthe narrow dimension are quickly pushed upwards in energy.  Finally,\nsince the elliptical dots have lower symmetry, accidental degeneracies\nare fewer and the transition peaks show nearly monotonically decreasing\nintensities (unlike the circular dot case). [Notice these all the\nsusceptibility traces include transitions between different\ncenter-of-mass states up to the $N_X=N_Y=10$ levels.  Additional $N_X$\nand $N_Y$ values would yield higher energy structure.]\n\nFigure 3 shows also the imaginary part of the susceptibility but for a\ndot size of 10 nm, and here with level broadening of 1 meV.\\@ The first\ntransition energy in the circular quantum dot is 1.78 eV, while the\nvalues in the elliptical dots are 1.79 and 1.80 eV, for ratios $2:1$\nand $3:1$.  The transition energies result here to be lower than in\nFig.\\ 2  since the confinement is not as strong, reducing the effective\ngap energy.   In this set of curves, the first transition energy\n(involving the ground state of the exciton) is shifted upwards for\nelliptical dots, due to\nthe increased confinement, although the shift is not as large as in\nFig.\\ 2, since the overall lengths are larger.  The first excited state\nappears split because the two-fold degeneracy of the excited state is\nbroken as the dot becomes elliptical.\n\nAs dot size is increased, it is apparent that the geometrical effects\nare not as prominent, producing only a small shift of the spectrum of\ntransitions.  Incidentally, the onset of transitions for larger size\ndots ($L \\approx 100$ nm) compares qualitatively well with experimental\nphotoluminescence spectra in dots with similar disk geometry,\\cite{10}\nwhere features appear in the energy range 1.73 -- 1.74 eV, for dots\nwith radius thought to be in the range 150--200 nm.  According to the\nexperimental results, the additional confinements give the observed\nblue shift, compared to two-dimensional quantum-well exciton case.\nSimilarly, a blue shift appears due to the elliptical shape, although\nfor $L \\approx 100$ nm they are only $\\leq 10$ meV, as the size ratio\nincreases to 2:1 and 3:1.\n\nAs an example of the effects for different materials, Fig.\\ 4 shows the\nimaginary part of the susceptibility of InAs quantum dots with lateral\nmean size of $12$ nm and thickness of $2.8$ nm --- having a dielectric\nconstant $\\epsilon=14.6$, $m{_e}=0.026m{_0}$, and heavy hole effective\nmass $m_{hh}=0.41m{_0}$.  Here, the energy gap is taken as 0.43 eV, and\nwe also use the optimized $\\Omega$ basis set to obtain the results\nshown.  The convergent optimized size, $L_{\\Omega}=10.5$ nm, is close\nto the two dimensional effective Bohr radius ($a^{2D}_B=10.54$ nm).  In\nthis case, the first transition energy shifts to higher values by\n$\\approx 11$ and 28 meV, as the size ratio changes from 1:1 to 2:1 and\n3:1, respectively.  Notice the similarity with Fig.\\ 2, although the\nenergy scale and size (12 nm) are completely different here. This\nsimilarity is due to the scaling of the problem in terms of $a^*_B$.\nFor these InAs parameters, $a^*_B=29.8$ nm, so that $a^*_B\/L \\approx\n2.5$, comparable to the value in Fig.\\ 2 for GaAs, where $a^*_B=12.2$,\n$L=5$, and $a^*_B\/L \\approx  2.4$.\n\n\n\\section{ Conclusions}\n \nWe have demonstrated that strong geometrical confinement effects appear\non excitons in GaAs and InAs quantum dots with elliptical\ncross sections. The solutions have been obtained using sufficiently\nlarge basis sets as well as with an optimized basis set. The results\nobtained with the optimized basis sets, such as exciton binding energy\nand normalized electron-hole separation, are extremely close to the\nbest converged results, and only with a relatively modest computational\neffort.\n\nThe linear optical susceptibilities are calculated for several\ndifferent lateral size ratios of each axis ($x$ and $y$). Strong blue\nshifts in the susceptibilities are observed as the size ratio is\nincreased and the shifts due to the geometrical shape effects are\nespecially important for the smaller dot sizes ($ \\leq 25$ nm).  The\nshifts are due mostly to the increasing confinement as the dot becomes\nmore elliptical, and not due to the interaction energy between\nelectron and hole.  A splitting of the first few excited states appears\nin the elliptical cross section cases since the symmetry-related\ndegeneracy of the excited states in the circular dot is broken.  This\ngives also rise to a more monotonic decrease of the peak intensities\nseen as the energy of the transition increases.\n\n\n   \\acknowledgments\n\nWe would like to thank R.L. Cappelletti and D.A. Drabold for helpful\ndiscussions, and the support of the US Department of Energy through\ngrant no.\\ DE--FG02--91ER45334.  Calculations were partially performed\nat the Cray Y\/MP of the Ohio Supercomputer Center.  S.E.U. acknowledges\nsupport of the A. v. Humboldt Foundation.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nRecent years have seen significant advances in the capacity of Artificial Intelligence (AI), which is growing in sophistication, complexity and autonomy. A continuously veritable and explosive growth of data with a rapid iteration of computing hardware advancement provides a \\emph{turbo boost} for the development of AI. \n\nAI is a generic concept and an umbrella term that implies the use of a machine with limited human interference to model intelligent actions. It covers a broad range of research studies from machine intelligence for computer vision, robotics, natural language processing to more theoretical machine learning algorithms design and recently re-branded and thrived \\emph{deep learning} development (Figure \\ref{fig:fig1}).\n\n\\subsection{Born of AI}\n\nAI changes almost every sector globally, e.g., enhancing (digital) healthcare (e.g., making diagnosis more accurate, allowing improved disease prevention), accelerating drug\/vaccine development and repurposing, raising agricultural productivity, leading to mitigation and adaptation in climate change, improving the efficiency of manufacturing processes by predictive maintenance, supporting the development of autonomous vehicles and programming more efficient transport networks, and in many other successful applications, which make significant positive socio-economic impact. Besides, AI systems are being deployed in highly-sensitive policy fields, such as facial recognition in the police or recidivism prediction in the criminal justice system, and in areas where diverse social and political forces are presented. Therefore, nowadays, AI systems are incorporated into a wide variety of decision-making processes. As AI systems become integrated into all kinds of decision-making processes, the degree to which people who develop AI, or are subject to an AI-enabled decision, can understand how the resulting decision-making mechanism operates and why a specific decision is reached, has been increasingly debated in science and policy communities.\n\nA collection of innovations, which are typically correlated with human or animal intelligence, is defined as the term \"artificial intelligence\". John McCarthy, who coined this term in 1955, described it as \"the scientific and technical expertise in the manufacture of intelligent machines\", and since then many different definitions have been endowed.\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=\\linewidth]{.\/figures\/Figure1.pdf} \n   \n    \\caption{Left: Terminology and historical timeline of AI, machine learning and deep learning. Right: We are still at the stage of narrow AI, a concept used to describe AI systems that are capable of handling a single or limited task. General AI is the hypothetical wisdom of AI systems capable of comprehending or learning any intelligent activity a human being might perform. Super AI is an AI that exceeds human intelligence and skills.}\n    \\label{fig:fig1}\n\\end{center}\n\\end{figure}\n\n\\subsection{Growth of Machine Learning}\n\nMachine learning is a subdivision of AI that helps computer systems to intelligently execute complex tasks. Traditional AI methods, which specify step by step how to address a problem, are normally based on hard-coded rules. Machine learning framework, by contrast, leverages the power of a large amount of data (as examples and not examples) for the identification of characteristics to accomplish a pre-defined task. The framework then learns how the target output will be better obtained. Three primary subdivisions of machine learning algorithms exist:\n\n\\begin{itemize}\n\t\\item A machine learning framework, which is trained using labelled data, is generally categorised as supervised machine learning. The labels of the data are grouped into one or more classes at each data point, such as \"cats\" or \"dogs.\" The supervised machine learning framework exploits the nature from these labelled data (i.e., training data), and forecasts the categories of the new or so called test data.\n\t\n    \\item Learning without labels is referred to as unsupervised learning. The aim is to identify the mutual patterns among data points, such as the formation of clusters and allotting data points to these clusters.\n    \n    \\item Reinforcement learning on the other hand is about knowledge learning, i.e., learning from experience. In standard reinforcement learning settings, an agent communicates with its environment, and is given a reward function that it tries to optimise. The purpose of the agent is to understand the effect of its decisions, and discover the best strategies for maximising its rewards during the training and learning procedure.\n    \n\\end{itemize}\n\nIt is of note that some hybrid methods, e.g., semi-supervised learning (using partially labelled data) and weakly supervised (using indirect labels), are also under development.\n\nAlthough not achieving the human-level intelligence often associated with the definition of the general AI, the capacity to learn from knowledge increases the amount and sophistication of tasks that can be tackled by machine learning systems (Figure \\ref{fig:fig1}). A wide variety of technologies, many of which people face on a daily basis, are nowadays enabled by rapid developments in machine learning, contributing to current advancements and dispute about the influence of AI in society. Many of the concepts that frame the existing machine learning systems are not new. The mathematical underpinnings of the field date back many decades, and since the 1950s, researchers have developed machine learning algorithms with varying degrees of complexity. In order to forecast results, machine learning requires computers to process a vast volume of data. How systems equipped with machine learning can handle probabilities or uncertainty in decision-making is normally informed by statistical approaches. Statistics, however, often cover areas of research that are not associated with the development of algorithms that can learn to make forecasts or decisions from results. Although several key principles of machine learning are rooted in data science and statistical analysis, some of the complex computational models do not converge with these disciplines naturally. Symbolic approaches, compared to statistical methods, are also used for AI. In order to create interpretations of a problem and to reach a solution, these methods use logic and inference.\n\n\\subsection{Boom of Deep Learning}\n\nDeep learning is a relatively recent congregation of approaches that have radically transformed machine learning. Deep learning is not an algorithm per se, but a range of algorithms that implements neural networks with deep layers. These neural networks are so deep that they can only be implemented on computer node clusters --- modern methods of computing --- such as graphics processing units (GPUs), are needed to train them successfully. Deep learning functions very well for vast quantities of data, and it is never too difficult to engineer the functionality even if a problem is complex (for example, due to the unstructured data). When it comes to image detection, natural language processing, and voice recognition, deep learning can always outperform the other types of algorithms. Deep learning assisted disease screening and clinical outcome prediction or automated driving, which were not feasible using previous methods, are well manifested now. Actually, the deeper the neural network with more data loaded for training, the higher accuracy a neural network can produce. The deep learning is very strong, but there are a few disadvantages to it. The reasoning of how deep learning algorithms reach to a certain solution is almost impossible to reveal clearly. Although several tools are now available that can increase insights into the inner workings of the deep learning model, this black-box problem still exists. Deep learning often involves long training cycles, a lot of data and complex hardware specifications, and it is not easy to obtain the specific skills necessary to create a new deep learning approach to tackle a new problem. \n\nAlthough acknowledging that AI includes a wide variety of scientific areas, this paper uses the umbrella word 'AI' and much of the recent interest in AI has been motivated by developments in machine learning and deep learning. More importantly, we should realise that there is not one algorithm, though, that will adapt or solve all issues. Success normally depends on the exact problem that needs to be solved and the knowledge available. A hybrid solution is often required to solve the problem, where various algorithms are combined to provide a concrete solution. Each issue involves a detailed analysis into what constitutes the best-fit algorithm. Transparency of the input size, capabilities of the deep neural network and time efficiency should also be taken into consideration, since certain algorithms take a long time to train.\n\n\\subsection{Stunt by the Black-box and Promotion of the Explainable AI}\n\nAny of today's deep learning tools are capable of generating extremely reliable outcomes, but they are often highly opaque, if not fully invisible, making it difficult to understand their behaviours. For even skilled experts to completely comprehend these so-called 'black-box' models may be still difficult. As these deep learning tools are applied on a wide scale, researchers and policymakers can challenge whether the precision of a given task outweighs more essential factors in the decision-making procedure.\n\nAs part of attempts to integrate ethical standards into the design and implementation of AI-enabled technologies, policy discussions around the world increasingly involve demands for some form of Trustable AI, which includes Valid AI, Responsible AI, Privacy-Preserving AI, and Explainable AI (XAI), in which the XAI want to address the fundamental question about the rationale of the decision making process including both human level XAI and machine level XAI (Figure \\ref{fig:fig2}). For example, in the UK, such calls came from the AI Committee of the House of Lords, which argued that the development of intelligible AI systems is a fundamental requirement if AI will be integrated as a trustworthy tool for our society. In the EU, the High-Level Group on AI has initiated more studies on the pathway towards XAI (Figure \\ref{fig:fig2}). Similarly, in the USA, the Defence Advanced Research Projects Agency funds a new research effort aiming at the development of AI with more explainability. These discussions will become more urgent as AI approaches are used to solve problems in a wide variety of complicated policy making areas, as experts increasingly work alongside AI-enabled decision-making tools, for example in clinical studies, and as people more regularly experience AI systems in real life when decisions have a major impact. Meanwhile, research studies in AI continue to progress at a steady pace. XAI is a vigorous area with many on-going studies emerging and several new strategies evolving that make a huge impact on AI development in various ways.\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=\\linewidth]{.\/figures\/Figure2.pdf} \n   \n    \\caption{Left: Trustable AI or Trustworthy AI includes Valid AI, Responsible AI, Privacy-Preserving AI, and Explainable AI (XAI). Right: EU General Data Protection Regulation (GDPR) highlights the Fairness, Privacy, Transparency and Explainability of the AI.}\n    \\label{fig:fig2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n    \\includegraphics[width=\\linewidth]{.\/figures\/Figure3.pdf} \n   \n    \\caption{Schema of the added explainable surrogate module for the normal machine or deep learning procedure that can achieve a more transparent and trustworthy model.}\n    \\label{fig:fig3}\n\\end{center}\n\\end{figure}\n\nWhile the usage of the term is inconsistent, \"XAI\" refers to a class of systems that have insight into how an AI system makes decisions and predictions. XAI explores the reasoning for the decision-making process, presents the positives and drawbacks of the system, and offers a glimpse of how the system will act in the future. By offering accessible explanations of how AI systems perform their study, XAI can allow researchers to understand the insights that come from research results. For example, in Figure \\ref{fig:fig3}, an additional explainable surrogate module can be added to the learnt model to achieve a more transparent and trustworthy model. In other words, for a conventional machine or deep learning model, only generalisation error has been considered while adding an explainable surrogate, both generalisation error and human experience can be considered and a verified prediction can be achieved. In contrast, a learnt black-box model without an explainable surrogate module will cause concerns for the end-users although the performance of the learnt model can be high. Such a black-box model can always cause confusions like ``Why did you do that?\", ``Why did you not do that?\", ``When do you succeed or fail?\", ``How do I correct an error?\", and ``Can I trust the prediction?\". The XAI powered model, on the other hand, can provide clear and transparent predictions to reassure ``I understand why.\", ``I understand why not.\", ``I know why you succeed or fail.\", ``I know how to correct an error.\", and ``I understand, therefore I trust\". A typical feedback loop of the XAI development can be found in Figure \\ref{fig:fig4}, which includes seven steps from training, quality assurance (QA), deployment, prediction, split testing (A\/B test), monitoring, and debugging.   \n\n\\begin{figure}[!htbp]\n\\begin{center}\n    \\includegraphics[width=0.8\\linewidth]{.\/figures\/Figure4.pdf} \n   \n    \\caption{A typical feedback loop of the XAI development that includes seven steps from training, quality assurance (QA), deployment, prediction, split testing (A\/B test), monitoring, and debugging.}\n    \\label{fig:fig4}\n\\end{center}\n\\end{figure}\n\nA variety of terms are used to define certain desired characteristics of an XAI system in research, public, and policy debates, including:\n\n\\begin{itemize}\n\t\\item Interpretability: it means a sense of knowing how the AI technology functions.\n\t\n    \\item Explainability: it provides an explanation for a wider range of users that how a decision has been drawn.\n    \n    \\item Transparency: it measures the level of accessibility to the data or model.\n\n    \\item Justifiability: it indicates an understanding of the case to support a particular outcome.\n \n    \\item Contestability: it implies how the users can argue against a decision. \n    \n\\end{itemize}\n\n\\begin{figure}[!htb] \n\\begin{center}\n    \\includegraphics[width=0.7\\linewidth]{.\/figures\/Figure5.pdf} \n   \n    \\caption{Model explainability vs. model performance for widely used machine learning and deep learning algorithms. The ideal solution should have both high explainability and high performance. However, existing linear models, rule-based models and decision trees are more transparent, but with lower performance in general. In contrast, complex models, e.g., deep learning and ensembles, manifest higher performance while less explainability can be obtained. HBN: Hierarchical Bayesian Networks; SLR: Simple Linear Regression; CRF: Conditional Random Fields; MLN: Markov Logic Network; SVM: Support Vector Machine; AOG: Stochastic And-Or-Graphs; XGB: XGBoost; CNN: Convolutional Neural Network; RNN: Recurrent Neural Network; and GAN: Generative Adversarial Network.}\n    \\label{fig:fig5}\n\\end{center}\n\\end{figure}\n\nComprehensive surveys on general XAI can be found elsewhere, e.g., \\cite{arrieta2020explainable,adadi2018peeking,samek2019towards,rai2020explainable}; therefore, here we provide an overview of most important concepts of the XAI. Broadly speaking, XAI can be categorised into model-specific or model-agnostic based approaches. Besides, these methods can be classified into local or global methods that can be either intrinsic or post-hoc \\cite{arrieta2020explainable}. Essentially, there are many machine learning models that are intrinsically explainable, e.g., linear models, rule-based models and decision trees, which are also known as transparent models or white-box models. However, these relatively simple models may have a relatively lower performance (Figure \\ref{fig:fig5}). For more complex models, e.g, support vector machines (SVM), convolutional neural networks (CNN), recurrent neural networks (RNN) and ensemble models, we can design model-specific and post-hoc XAI strategies for each of them. For example, commonly used strategies include explanation by simplification, architecture modification, feature relevance explanation, and visual explanation \\cite{arrieta2020explainable}. Clearly these more complex models can achieve better performance while the explainability becomes lower (Figure \\ref{fig:fig5}). \n\nRecently, model-agnostic based approaches attract great attention that rely on a simplified surrogate function to explain the predictions \\cite{samek2019towards}. Model-agnostic approaches are not attached to a specific machine learning model. This class of techniques, in other words, distinguishes prediction from explanation. Model-agnostic representations are usually post-hoc that are generally used to explain deep neural networks with interpretable surrogates that can be local or global \\cite{adadi2018peeking}. Below is some summary for XAI in more complex deep learning based models.\n\n\\subsubsection{Model-Specific Global XAI}\n\nBy integrating interpretability constraints into the procedure of deep learning, these model-specific global XAI strategies can improve the understandability of the models. Structural restrictions may include sparsity and monotonicity, where fewer input features are leveraged or the correlation between features and predictions is confined as monotonic). Semantic prior knowledge can also be impelled to restrict the higher-level abstractions derived from the data. For instance, in a CNN based brain tumour detection\/classification model using multimodal MRI data fusion, constraints can be imposed by forcing disengaged representations that are recognisable to each MRI modality (e.g., T1, T1 post-contrast and FLAIR), respectively. In doing so, the model can identify crucial information from each MRI modality and distinguish brain tumours and sub-regions into necrotic, more or less infiltrative that can provide vital diagnosis and prognosis information. On the contrary, simple aggregation based information fusion (combining all the multimodal MRI data like a sandwich) would not provide such explainability.\n\n\\subsubsection{Model-Specific Local XAI}\n\nIn a deep learning model, a model-specific local XAI technique offers an interpretation for a particular instance. Recently, novel attention mechanisms have been proposed to emphasise the importance of different features of the high-dimensional input data to provide an explanation of a representative instance. Consider a deep learning algorithm that encodes an X-ray image into a vector using a CNN and then use an RNN to produce a clinical description for the X-ray image by using the encoded vector. For the RNN, an attention module can be applied to explain to the user what image fragments the model focuses on to produce each substantive term for the clinical description. For example, the attention mechanism will represent the appropriate segments of the image corresponding to the clinical key words derived by the deep learning model when a clinician is baffled to link the clinical key words to the regions of interest in the X-ray image.\n\n\\subsubsection{Model-Agnostic Global XAI}\n\nIn model-agnostic global XAI, a surrogate representation is developed to approximate an interpretable module for the black-box model. For instance, an interpretable decision tree based model can be used to approximate a more complex deep learning model on how clinical symptoms impact treatment response. A clarification of the relative importance of variables in affecting treatment response to clinical symptoms can be given by the IF-THEN logic of the decision tree. Clinical experts can analyse these variables and are likely to believe the model to the extent that particular symptomatic factors are known to be rational and confounding noises can be accurately removed. Diagnostic methods can also be useful to produce insights into the significance of individual characteristics in the predictions of the model. Partial dependence plots can be leveraged to determine the marginal effects of the chosen characteristics vs. the performance of the forecast, whereas individual conditional expectation can be employed to obtain a granular explanation of how a specific feature affects particular instances and to explore variation in impacts throughout instances. For example, a partial dependency plot can elucidate the role of clinical symptoms in reacting favourably to a particular treatment strategy, as observed by a computer-aided diagnosis system. On the other hand, individual conditional expectation can reveal variability in the treatment response among subgroups of patients.\n\n\\subsubsection{Model-Agnostic Local XAI}\n\nFor this type of XAI approaches, the aim is to produce model-agnostic explanations for a particular instance or the vicinity of a particular instance. Local Interpretable Model-Agnostic Explanation (LIME) \\cite{ribeiro2016lime}, a well-validated tool, can provide an explanation for a complex deep learning model in the neighbourhood of an instance. Consider a deep learning algorithm that classifies a physiological attribute as a high-risk factor for certain diseases or cause of death, for which the clinician requires a post-hoc clarification. The interpretable modules are perturbed to determine how the predictions made by the change of those physiological attributes. For this perturbed dataset, a linear model is learnt with higher weights given to the perturbed instances in the vicinity of the physiological attribute. The most important components of the linear model can indicate the influence of a particular physiological attribute that can suggest a high-risk factor or the contrary can be implied. This can provide comprehensible means for the clinicians to interpret the classifier. \n\n\\section{Related Studies in AI for Healthcare and XAI for Healthcare}\n\n\\subsection{AI in Healthcare}\n\nAI attempts to emulate the neural processes of humans, and it introduces a paradigm change to healthcare, driven by growing healthcare data access and rapid development in analytical techniques. We survey briefly the present state of healthcare AI applications and explore their prospects. For a detailed up to date review, the readers can refer to Jiang et al. \\cite{jiang2017artificial}, Panch et al. \\cite{panch2019inconvenient}, and Yu et al. \\cite{yu2018artificial} on general AI techniques for healthcare and Shen et al. \\cite{shen2017deep}, Litjens et al. \\cite{litjens2017survey} and Ker et al. \\cite{ker2017deep} on medical image analysis.\n\nIn the medical literature, the effects of AI have been widely debated \\cite{dilsizian2014artificial,patel2009coming,jha2016adapting}. Sophisticated algorithms can be developed using AI to 'read' features from a vast amount of healthcare data and then use the knowledge learnt to help clinical practice. To increase its accuracy based on feedback, AI can also be fitted with learning and self-correcting capabilities. By presenting up-to-date medical knowledge from journals, manuals and professional procedures to advise effective patient care, an AI-powered device \\cite{strickland2019ibm} will support clinical decision making. Besides, in human clinical practice, an AI system may help to reduce medical and therapeutic mistakes that are unavoidable (i.e., more objective and reproducible) \\cite{dilsizian2014artificial,patel2009coming,strickland2019ibm,weingart2000epidemiology, graber2005diagnostic,winters2012diagnostic,lee2013cognitive}. In addition, to help render real-time inferences for health risk warning and health outcome estimation, an AI system can handle valuable knowledge collected from a large patient population \\cite{neill2013using}. \n\nAs AI has recently re-emerged into the scientific and public consciousness, AI in healthcare has new breakthroughs and clinical environments are imbued with novel AI-powered technologies at a breakneck pace. Nevertheless, healthcare was described as one of the most exciting application fields for AI. Researchers have suggested and built several systems for clinical decision support since the mid-twentieth century \\cite{miller1994medical,musen2014clinical}. Since the 1970s, rule-based methods had many achievements and have been seen to interpret ECGs \\cite{kundu2000knowledge}, identify diseases \\cite{de1972computer}, choose optimal therapies \\cite{shortliffe1975computer}, offer scientific logic explanations \\cite{barnett1987dxplain} and assist doctors in developing diagnostic hypotheses and theories in challenging cases of patients \\cite{miller1986internist}. Rule-based systems, however, are expensive to develop and can be unstable, since they require clear expressions of decision rules and, like any textbook, require human-authored modifications. Besides, higher-order interactions between various pieces of information written by different specialists are difficult to encode and the efficiency of the structures is constrained by the comprehensiveness of prior medical knowledge \\cite{berner1994performance}. To narrow down the appropriate psychological context, prioritise medical theories, and prescribe treatment, it was also difficult to incorporate a method that combines deterministic and probabilistic reasoning procedures \\cite{szolovits1978categorical,szolovits1994categorical}.\n\nRecent AI research has leveraged machine learning approaches, which can account for complicated interactions \\cite{deo2015machine}, to recognise patterns from the clinical results, in comparison to the first generation of AI programmes, which focused only on the curation of medical information by experts and the formulation of rigorous decision laws. The machine learning algorithm learns to create the correct output for a given input in new instances by evaluating the patterns extracted from all the labelled input-output pairs \\cite{yu2016omics}. Supervised machine learning algorithms are programmed to determine the optimal parameters in the models in order to minimise the differences between their training case predictions and the effects observed in these cases, with the hope that the correlations found are generalisable to cases not included in the dataset of training. The model generalisability can be then calculated using the test dataset. For supervised machine learning models, grouping, regression and characterisation of the similarity between instances with similar outcome labels are among the most commonly used tasks. For the unlabelled dataset, unsupervised learning infers the underlying patterns for discovering sub-clusters of the original dataset, for detecting outliers in the data, or for generating low-dimensional data representations. However, it is of note that in a supervised manner, the recognition of low-dimensional representations for labelled dataset may be done more effectively. Machine-learning approaches allow the development of AI applications that promote the exploration of previously unrecognised data patterns without the need to define decision-making rules for each particular task or to account for complicated interactions between input features. Machine learning has therefore been the preferred method for developing AI utilities \\cite{deo2015machine,roberts2017biomedical,rogers2020radiomics}.\n\nThe recent rebirth of AI has primarily been motivated by the active implementation of deep learning---which includes training a multi-layer artificial neural network (i.e., a deep neural network) on massive datasets---to wide sources of labelled data \\cite{goodfellow2016deep}. Existing neural networks are getting deeper and typically have $>$100 layers. Multi-layer neural networks may model complex interactions between input and output, but may also require more data, processing time, or advanced architecture designs to achieve better performance. Modern neural networks commonly have tens of millions to hundreds of millions of parameters and require significant computing resources to perform the model training \\cite{yu2018artificial}. Fortunately, recent developments in computer-processor architecture have empowered the computing resources required for deep learning \\cite{wang2019benchmarking}. However, in labelled instances, deep-learning algorithms are incredibly 'data hungry.' Huge repositories of medical databases that can be integrated into these algorithms have only recently become readily available, due to the establishment of a range of large-scale research (in particular the Cancer Genome Atlas \\cite{tomczak2015cancer} and the UK Biobank \\cite{sudlow2015uk}), data collection platforms (e.g., Broad Bioimage Benchmark Collection \\cite{ljosa2012annotated} and the Image Data Resources \\cite{williams2017image}) and the Health Information Technology for Economic and Clinical Health (HITECH) Act, which has promised to provide financial incentives for the use of electronic health records (EHRs) \\cite{desroches2008electronic,hsiao2013office}. In general, deep learning based AI algorithms have been developed for image-based classification \\cite{hu2020weakly}, diagnosis \\cite{litjens2016deep,zhang2019deep,cao2020multiparameter} and prognosis \\cite{cheerla2019deep,roberts2020machine}, genome interpretation \\cite{zou2019primer}, biomarker discovery \\cite{waldstein2020unbiased,li2020atrial}, monitoring by wearable life-logging devices \\cite{lane2015early}, and automated robotic surgery \\cite{chen2020deeprobotic} to enhance the digital healthcare \\cite{esteva2019guide}. The rapid explosion of AI has given rise to the possibilities of using aggregated health data to generate powerful models that can automate diagnosis and also allow an increasingly precise approach to medicine by tailoring therapies and targeting services with optimal efficacy in a timely and dynamic manner. A non-exhaustive map of possible applications is showing in Figure \\ref{fig:fig6}. \n\n\\begin{figure}[!htb] \n\\begin{center}\n    \\includegraphics[width=0.75\\linewidth]{.\/figures\/Figure6.pdf} \n   \n    \\caption{A non-exhaustive map of the AI in healthcare applications.}\n    \\label{fig:fig6}\n\\end{center}\n\\end{figure}\n\nWhile AI is promising to revolutionise medical practice, several technological obstacles lie ahead. Because deep learning based approaches rely heavily on the availability of vast volumes of high-quality training data, caution must be taken to collect data that is representative of the target patient population. For example, data from various healthcare settings, which include different forms of bias and noise, may cause a model trained in the data of one hospital to fail to generalise to another \\cite{obermeyer2016predicting}. Where the diagnostic role has an incomplete inter-expert agreement, it has been shown that consensus diagnostics could greatly boost the efficiency of the training of the deep learning based models \\cite{krause2018grader}. In order to manage heterogeneous data, adequate data curation is important. However, achieving a good quality gold standard for identifying the clinical status of the patients requires physicians to review their clinical results independently and maybe repeatedly, which is prohibitively costly at a population scale. A silver standard \\cite{rebholz2010calbc} that used natural-language processing methods and diagnostic codes to determine the true status of patients has recently been proposed \\cite{kirby2016phekb}. Sophisticated algorithms that can handle the idiosyncrasies and noises of different datasets can improve the efficiency and safety of prediction models in life-and-death decisions.\n\nMost of the recent advancement in neural networks has been limited to well-defined activities that do not require data integration across several modalities. Approaches for the application of deep neural networks to general diagnostics (such as analysis of signs and symptoms, prior medical history, laboratory findings and clinical course) and treatment planning are less simple. While deep learning has been effective in image detection \\cite{zhao2019object}, translation \\cite{singh2017machine}, speech recognition \\cite{amodei2016deep,nassif2019speech}, sound synthesis \\cite{purwins2019deep} and even automated neural architecture search \\cite{elsken2019neural}, clinical diagnosis and treatment tasks often need more care (e.g., patient interests, beliefs, social support and medical history) than the limited tasks that deep learning can be normally adept. Moreover, it is unknown if transfer learning approaches will be able to translate models learnt from broad non-medical datasets into algorithms for the study of multi-modality clinical datasets. This suggests that more comprehensive data-collection and data-annotation activities are needed to build end-to-end clinical AI programmes.\n\nThe design of a computing system for the processing, storage and exchange of EHRs and other critical health data remains a problem \\cite{lee2009ethical}. Privacy-preserving approaches, e.g., via federated learning, can allow safe sharing of data or models across cloud providers \\cite{narayan2010privacy}. However, the creation of interoperable systems that follow the requirement for the representation of clinical knowledge is important for the broad adoption of such technology \\cite{dolin2006hl7}. Deep and seamless incorporation of data across healthcare applications and locations remains questionable and can be inefficient. However, new software interfaces for clinical data are starting to show substantial adoption through several EHR providers, such as Substitutable Medical Applications and Reusable Technologies on the Fast Health Interoperability Resources platform \\cite{mandl2012escaping,mandel2016smart}. Most of the previously developed AI in healthcare applications were conducted on retrospective data for the proof of concept \\cite{topol2019high}. Prospective research and clinical trials to assess the efficiency of the developed AI systems in clinical environments are necessary to verify the real-world usefulness of these medical AI systems \\cite{yu2019framing}. Prospective studies will help recognise the fragility of the AI models in real-world heterogeneous and noisy clinical settings and identify approaches to incorporate medical AI for existing clinical workflows.\n\nAI in medicine would eventually result in safety, legal and ethical challenges \\cite{miller2019medical} with respect to medical negligence attributed to complicated decision-making support structures, and have to face the regulation hurdles \\cite{challen2019artificial}. If malpractice lawsuits involving medical AI applications occur, the judicial system will continue to provide specific instructions as to which agency is responsible. Health providers with malpractice insurance have to be clear on coverage as health care decisions are taken in part by the AI scheme \\cite{yu2018artificial}. With the deployment of automatic AI for particular clinical activities, the criteria for diagnostic, surgical, supporting and paramedical tasks will need to be revised and the functions of healthcare practitioners will begin to change as different AI modules are implemented into the quality of treatment, and the bias needs to be minimised while the patient satisfaction must be maximised \\cite{decamp2020latent,esmaeilzadeh2020use}.\n\n\n\\subsection{XAI in Healthcare}\n\nDespite deep learning based AI technologies will usher in a new era of digital healthcare, challenges exist. XAI can play a crucial role, as an auxiliary development (Figure \\ref{fig:fig6}), for potentially solving the small sample learning by filter out clinically meaningless features. Moreover, many high-performance deep learning models produce findings that are impossible for unaided humans to understand. While these models can produce better-than-human efficiency, it is not easy to express intuitive interpretations that can justify model findings, define model uncertainties, or derive additional clinical insights from these computational 'black-boxes.' With potentially millions of parameters in the deep learning model, it can be tricky to understand what the model sees in the clinical data, e.g., radiological images \\cite{england2019artificial}. For example, research investigation has explicitly stated that being a black box is a \"strong limitation\" for AI in dermatology since it is not capable of doing a personalised evaluation by a qualified dermatologist that can be used to clarify clinical facts \\cite{gomolin2020artificial}. This black-box design poses an obstacle for the validation of the developed AI algorithms. It is necessary to demonstrate that a high-performance deep learning model actually identifies the appropriate area of the image and does not over-emphasise unimportant findings. Recent approaches have been developed to describe AI models including the visualisation methods. Some widely used levers include occlusion maps \\cite{zeiler2014visualizing}, salience maps \\cite{Simonyan14a}, class activation maps \\cite{selvaraju2017grad}, and attention maps \\cite{zhang2017mdnet}. Localisation and segmentation algorithms can be more readily interpreted since the output is an image. Model understanding, however, remains much more difficult for deep neural network models trained on non-imaging data other than images that is a current open question for ongoing research efforts \\cite{ribeiro2016lime}.\n\nDeep learning-based AI methods have gained popularity in the medical field, with a wide range of work in automatic triage, diagnosis, prognosis, treatment planning and patient management \\cite{jiang2017artificial}. We can find many open questions in the medical field that have galvanised clinical trials leveraging deep learning and AI approaches (e.g., from grand-challenge.org). Nevertheless, in the medical field, the issue of interpretability is far from theoretical development. More precisely, it is noted that interpretabilities in the clinical sectors include considerations not recognised in other areas, including risk and responsibilities \\cite{croskerry2017diagnosis,panch2019inconvenient}. Life may be at risk as medical responses are made, and leaving those crucial decisions to AI algorithms that without explainabilities and accountabilities will be irresponsible \\cite{quinn2020three}. Apart from legal concerns, this is a serious vulnerability that could become disastrous if used with malicious intent. \n\nAs a result, several recent studies \\cite{zhang2017mdnet,pmlr-v106-tonekaboni19a,holzinger2019causability} have been devoted to the exploration of explainability in medical AI. More specifically, specific analyses have been investigated, e.g., chest radiography \\cite{kallianos2019far}, emotion analysis in medicine \\cite{zucco2018explainable}, COVID-19 detection and classification \\cite{hu2020weakly}, and the research encourages understanding of the importance of interpretability in the medical field \\cite{langlotz2019roadmap}. Besides, the exposition argues \\cite{london2019artificial} that a certain degree of opaqueness is appropriate, that is, it would be more important for us to deliver empirically checked reliable findings than to dwell too hard on how to unravel the black-box. It is advised that readers consider these studies first, at least for an overview of interpretability in medical AI. \n\nAn obvious XAI approach has been taken by many researchers is to provide their predictive models with interpretability. These methods depend primarily on maintaining the interpretability of less complicated AI models while improving their performance by techniques of refinement and optimisation. For example, as Figure \\ref{fig:fig5} shows, decision tree based methods are normally interpretable, research studies have been done using automated pruning of decision trees for various classifications of illnesses \\cite{stiglic2012comprehensive} and accurate decision trees focused on boosting patient stratification \\cite{valdes2016mediboost}. However, such model optimisation is not always straightforward and it is not a trivial task. \n\nPrevious survey studies on XAI in healthcare can be found elsewhere, e.g., Tjoa and Guan \\cite{tjoa2020survey} in medical XAI and Payrovnaziri et al. \\cite{payrovnaziri2020explainable} in XAI for EHR. For specific applications, e.g., digital pathology, the readers can refer to Pocevivciute et al. \\cite{pocevivciute2020survey} and Tosun et al. \\cite{tosun2020histomapr}. The research studies in XAI and medical XAI have been increased exponentially especially after 2018 alongside increasingly development of multimodal clinical information fusion (Figure \\ref{fig:fig7}). In this mini-review, we only surveyed the most recent studies that were not covered by previous more comprehensive review studies. In this mini-review, we classified XAI in medicine and healthcare into five categories, which synthesised the approach by Payrovnaziri et al. \\cite{payrovnaziri2020explainable}, including (1) XAI via dimension reduction, (2) XAI via feature importance, (3) XAI via attention mechanism, (4) XAI via knowledge distillation, and (5) XAI via surrogate representations (Table \\ref{tab:tab1}).\n\n\\subsubsection{XAI via Dimension Reduction}\n\nDimension reduction methods, e.g., using principal component analysis (PCA) \\cite{wold1987principal}, independent component analysis (ICA) \\cite{comon1994independent}, and Laplacian Eigenmaps \\cite{belkin2001laplacian} and other more advanced techniques, are commonly and conventionally used approaches to decipher AI models by representing the most important features. For example, by integrating multi-label k-nearest neighbour and genetic algorithm techniques, Zhang et al. \\cite{zhang2015predicting} developed a model for drug side effect estimation based on the optimal dimensions of the input features. Yang et al. \\cite{yang2015manifold} proposed a nonlinear dimension reduction method to improve unsupervised classification of the $^1$H MRS brain tumour data and extract the most prominent features using Laplacian Eigenmaps. Zhao and Bolouri \\cite{zhao2016object} stratified stage-one lung cancer patients by defining the most insightful examples via a supervised learning scheme. In order to recognise a group of \"exemplars\" to construct a \"dense data matrix,\" they introduced a hybrid method for dimension reduction by combining pattern recognition with regression analytics. Then they used examples in the final model that are the most predictive for the outcome. Based on domain knowledge, Kim et al. \\cite{kim2016opening} developed a deep learning method to extract and rank the most important features based on their weights in the model, and visualised the outcome for predicting cell-type-specific enhancers. To explore the gene pathways and their associations in patients with the brain tumour, Hao et al. \\cite{hao2018pasnet} proposed a pathway-associated sparse deep learning method. Bernardini et al. \\cite{bernardini2019discovering} used the least absolute shrinkage and selection operator (LASSO) to prompt sparsity for SVMs for the early diagnosis of type 2 diabetes.\n\nSimplifying the information down to a small subset using dimension reduction methods can make the underlying behaviour of the model understandable. Besides, with potentially more stable regularised models, they are less prone to overfitting, which may also be beneficial in general. Nevertheless, the possibility of losing crucial features, which may still be relevant for clinical predictions on a case-by-case basis, can be common and these important features may be neglected unintentionally by the dimensional reduced models. \n\n\n\\subsubsection{XAI via Feature Importance}\n\nResearchers have leveraged the feature importance to explain the characteristics and significance of the extracted features and the correlations among features and between features and the outcomes for providing interpretability for AI models \\cite{carvalho2019machine,adadi2018peeking,linardatos2021explainable}. Ge et al. \\cite{ge2018interpretable} used feature weights to rank the top ten extracted features to predict mortality of the intensive care unit. Suh et al. \\cite{suh2020development} developed a risk calculator model for prostate cancer (PCa) and clinically significant PCa with XAI modules that used Shapley value to determine the feature importance \\cite{lundberg2020local}. Sensitivity analysis of the extracted features can represent the feature importance, and essentially the more important features are those for which the output is more sensitive \\cite{montavon2018methods}. Eck et al. \\cite{eck2017interpretation} defined the most significant features of a microbiota-based diagnosis task by roughly marginalising the features and testing the effect on the model performance.\n\nShrikumar et al. \\cite{shrikumar17a} implemented the Deep Learning Important FeaTures (DeepLIFT)---a backpropagation based approach to realise interpretability. Backpropagation approaches measure the output gradient for input through the backpropagation algorithm to report the significance of the feature. Zuallaert et al. \\cite{zuallaert2018splicerover} developed the DeepLIFT based method to create interpretable deep models for splice site prediction by measuring the contribution score for each nucleotide. A recent comparative study of different models of XAI, including DeepLIFT \\cite{shrikumar17a}, Guided backpropagation (GBP) \\cite{DB15a}, Layer wise relevance propagation (LRP) \\cite{bach2015pixel}, SHapley Additive exPlanations (SHAP) \\cite{Chen2021} and others, was conducted for ophthalmic diagnosis \\cite{singh2020interpretation}.\n\nXAI, by the extraction of feature importance, can not only explain essential feature characteristics, but may also reflect their relative importance to clinical interpretation; however, numerical weights are either not easy to understand or maybe misinterpreted.\n\n\\subsubsection{XAI via Attention Mechanism}\n\nThe core concept behind the attention mechanism \\cite{BahdanauCB14} is that the model \"pays attention\" only to the parts of the input where the most important information is available. It was originally proposed for tackling the relation extraction task in machine translation and other natural language processing problems. Because certain words are more relevant than others in the relation extraction task, the attention mechanism can assess the importance of the words for the purpose of classification, generating a meaning representation vector. There are various types of attention mechanisms, including global attention, which uses all words to build the context, local attention, which depends only on a subset of words, or self-attention, in which several attention mechanisms are implemented simultaneously, attempting to discover every relation between pairs of words \\cite{putelli2019applying}. The attention mechanism has also been shown to contribute to the enhancement of interpretability as well as to technical advances in the field of visualisation \\cite{mascharka2018transparency}.\n\nKaji et al. \\cite{kaji2019attention} demonstrated particular occasions when the input features have mostly influenced the predictions of clinical events in ICU patients using the attention mechanism. Shickel et al. \\cite{shickel2019deepsofa} presented an interpretable acuity score framework using deep learning and attention-based sequential organ failure assessment that can assess the severity of patients during an ICU stay. Hu et al. \\cite{hu2019deephint} provided \"mechanistic explanations\" for the accurate prediction of HIV genome integration sites. Zhang et al. \\cite{zhang2018patient2vec} also built a method to learn how to represent EHR data that could document the relationship between clinical outcomes within each patient. Choi et al. \\cite{choi2016retain} implemented the Reverse Time Attention Model (RETAIN), which incorporated two sets of attention weights, one for visit level to capture the effect of each visit and the other at the variable-level. RETAIN was a reverse attention mechanism intended to maintain interpretability, to replicate the actions of clinicians, and to integrate sequential knowledge. Kwon et al. \\cite{kwon2018retainvis} proposed a visually interpretable cardiac failure and cataract risk prediction model based on RETAIN (RetainVis). The general intention of these research studies is to improve the interpretability of deep learning models by highlighting particular position(s) within a sequence (e.g., time, visits, DNA) in which those input features can affect the prediction outcome.\n\nClass activation mapping (CAM) \\cite{zhou2016learning} method and its variations have been investigated for XAI since 2016, and have been subsequently used for digital healthcare, especially the medical image analysis areas. Lee et al. \\cite{lee2019explainable} developed an XAI algorithm for the detection of acute intracranial haemorrhage from small datasets that is one of the most famous studies using CAM. Kim et al. \\cite{kimartificial2020} summarised AI based breast ultrasonography analysis with CAM based XAI. Zhao et al. \\cite{zhao2018respond} reported a Respond-CAM method that offered a heatmap-based saliency on 3D images obtained from cryo-tomography of cellular electrons. The region where macromolecular complexes were present was marked by the high intensity in the heatmap. Izadyyazdanabadi et al. \\cite{izadyyazdanabadi2018weakly} developed a multilayer CAM (MLCAM), which was used for brain tumour localization. Coupling with CNN, Couture et al. \\cite{couture2018multiple} proposed a multi-instance aggregation approach to classify breast tumour tissue microarray for various clinical tasks, e.g., histologic subtype classification, and the derived super-pixel maps could highlight the area where the tumour cells were and each mark corresponded to a tumour class. Rajpurkar et al. \\cite{rajpurkar2020appendixnet} used Grad-CAM for the diagnosis of appendicitis from a small dataset of CT exams using video pretraining. Porumb et al. \\cite{porumb2020precision} combined CNN and RNN for electrocardiogram (ECG) analysis and applied Grad-CAM for the identification of the most relevant heartbeat segments for the hypoglycaemia detection. In Hu et al. \\cite{hu2020weakly}, a COVID-19 classification system was implemented with multiscale CAM to highlight the infected areas. By the means of visual interpretability, these saliency maps are recommended. The clinician analysts who examine the AI output can realise that the target is correctly identified by the AI model, rather than mistaking the combination of the object with the surrounding as the object itself.\n\nAttention based XAI methods do not advise the clinical end user specifically of the response, but highlight the areas of greater concern to facilitate easier decision-making. Clinical users can, therefore, be more tolerant of imperfect precision. However, it might not be beneficial to actually offer this knowledge to a clinical end user because of the major concerns, including information overload and warning fatigue. It can potentially be much more frustrating to have areas of attention without clarification about what to do with the findings if the end user is unaware of what the rationale of a highlighted segment is, and therefore the end user can be prone to ignore non-highlighted areas that could also be critical.\n\n\\subsubsection{XAI via Knowledge Distillation and Rule Extraction}\n\nKnowledge distillation is one form of the model-specific XAI, which is about eliciting knowledge from a complicated model to a simplified model---enables to train a student model, which is usually explainable, with a teacher model, which is hard to interpret. For example, this can be accomplished by model compression \\cite{polino2018model} or tree regularisation \\cite{wu2018beyond} or through a coupling approach of model compression and dimension reduction \\cite{carvalho2019machine}. Research studies have investigated this kind of technique for several years, e.g., Hinton et al. \\cite{Hinton44873}, but has recently been uplifted along with the development of AI interpretability \\cite{Hinton46495,yang2020auto,li2020survey}. Rule extraction is another widely used XAI method that is closely associated with knowledge distillation and can have a straightforward application for digital healthcare, for example, decision sets or rule sets have been studied for interpretability \\cite{lage2019evaluation} and Model Understanding through Subspace Explanations (MUSE) method \\cite{lakkaraju2019faithful} has been developed to describe the projections of the global model by considering the various subgroups of instances defined by user interesting characteristics that also produces explanation in the form of decision sets.\n\nChe et al. \\cite{che2016interpretable} introduced an interpretable mimic-learning approach, which is a straightforward knowledge-distillation method that uses gradient-boosting trees to learn interpretable structures and make the baseline model understandable. The approach used the information distilled to construct an interpretable prediction model for the outcome of the ICU, e.g., death, ventilator usage, etc. A rule-based framework that could include an explainable statement of death risk estimation due to pneumonia was introduced by Caruana et al. \\cite{caruana2015intelligible}. Letham et al. \\cite{letham2015interpretable} also proposed an XAI model named Bayesian rule lists, which offered certain stroke prediction claims. Ming et al. \\cite{ming2018rulematrix} developed a visualisation approach to derive rules by approximating a complicated model via model induction at different tasks such as diagnosis of breast cancer and the classification of diabetes. Xiao et al. \\cite{xiao2018readmission} built a deep learning model to break the dynamic associations between readmission to hospital and possible risk factors for patients by translating EHR incidents into embedded clinical principles to characterise the general situation of the patients. Classification rules were derived as a way of providing clinicians interpretable representations of the predictive models. Davoodi and Moradi \\cite{davoodi2018mortality} developed a rule extraction based XAI technique to predict mortality in ICUs and Das et al. \\cite{das2019interpretable} used a similar XAI method for the diagnosis of Alzheimer's disease. In the LSTM-based breast mass classification, Lee et al. \\cite{lee2019generation} incorporated the textual reasoning for interpretability. For the characterisation of stroke and risk prediction, Prentzas et al. \\cite{prentzas2019integrating} implemented the argumentation theory for their XAI algorithm training process by extracting decision rules. \n\nXAI approaches, which rely on knowledge distillation and rule extraction, are theoretically more stable models. The summarised representations of complicated clinical data can provide clinical end-users with the interpretable results intuitively. However, if the interpretation of these XAI results could not be intuitively understood by clinical end-users, then the representations are likely to make it much harder for the end-users to comprehend.\n\n\\subsubsection{XAI via Surrogate Representation}\n\nAn effective application of XAI in the medical field is the recognition of individual health-related factors that lead to disease prediction using the local interpretable model-agnostic explanation (LIME) method \\cite{ribeiro2016lime} that offers explanations for any classifier by approximating the reference model with a surrogate interpretable and \"locally faithful\" representation. LIME disrupts an instance, produces neighbourhood data, and learns linear models in the neighbourhood to produce explanations \\cite{liang2021explaining}. \n\nPan et al. \\cite{pan2019development} used LIME to analyse the contribution of new instances to forecast central precocious puberty in children. Ghafouri-Fard et al. \\cite{ghafouri2019application} have applied a similar approach to diagnose autism spectrum disorder. Kovalev et al. \\cite{KOVALEV2020106164} proposed a method named SurvLIME to explain AI base survival models. Meldo et al. \\cite{meldo2020natural} used a local post-hoc explanation model, i.e., LIME, to select important features from a special feature representation of the segmented lung suspicious objects. Panigutti et al. \\cite{Panigutti2020} developed the \"Doctor XAI\" system that could predict the readmission, diagnosis and medications order for the patient. Similar to LIME, the implemented system trained a local surrogate model to mimic the black-box behaviour with a rule-based explanation, which can then be mined using a multi-label decision tree. Lauritsen et al. \\cite{lauritsen2020explainable} tested an XAI method using Layer-wise Relevance Propagation \\cite{montavon2019layer} for the prediction of acute critical illness from EHR.\n\nSurrogate representation is a widely used scheme for XAI; however, the white-box approximation must accurately describe the black-box model to gain trustworthy explanation. If the surrogate models are too complicated or too abstract, the  clinician comprehension might be affected.\n\n\n\\begin{sidewaystable}\n\n\\resizebox{0.55\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\n\\begin{tabular}{@{}lllllll@{}}\n\\toprule\nXAI Category           & Reference                                                  & Method                           & Intrinsic\/Post-hoc & Local\/Global  & Model-specific\/Model-agnostic & Application                                                     \\\\ \\midrule\nDimension Reduction    & Zhang et al. \\cite{zhang2015predicting}                   & Optimal feature selection        & Intrinsic         & Global        & Model-specific                & Drug side effect estimation                                     \\\\\n                       & Yang et al. \\cite{yang2015manifold}                       & Laplacian Eigenmaps              & Intrinsic         & Global        & Model-specific                & Brain tumour classification using MRS                           \\\\\n                       & Zhao and Bolouri \\cite{zhao2016object}                    & Cluster analysis and LASSO       & Intrinsic         & Global        & Model-agnostic                & Lung cancer patients stratification                             \\\\\n                       & Kim et al. \\cite{kim2016opening}                          & Optimal feature selection        & Intrinsic         & Global        & Model-agnostic                & Cell-type specific enhancers prediction                         \\\\\n                       & Hao et al. \\cite{hao2018pasnet}                           & Sparse deep learning             & Intrinsic         & Global        & Model-agnostic                & Long-term survival prediction for glioblastoma multiforme       \\\\\n                       & Bernardini et al. \\cite{bernardini2019discovering}        & Sparse-balanced SVM              & Intrinsic         & Global        & Model-agnostic                & Early diagnosis of type 2 diabetes                              \\\\ \\midrule\nFeature Importance     & Eck et al. \\cite{eck2017interpretation}                   & Feature marginalisation          & Post-hoc           & Global, Local & Model-agnostic                & Gut and skin microbiota\/inflammatory bowel diseases diagnosis   \\\\\n                       & Ge et al. \\cite{ge2018interpretable}                      & Feature weighting                & Post-hoc           & Global        & Model-agnostic                & ICU mortality prediction (all-cause)                            \\\\\n                       & Zuallaert et al. \\cite{zuallaert2018splicerover}          & DeepLIFT                         & Post-hoc           & Global        & Model-agnostic                & Splice site detection                                           \\\\\n                       & Suh et al. \\cite{suh2020development}                      & Shapley value                    & Post-hoc           & Global, Local & Model-agnostic                & Decision-supporting for prostate cancer                         \\\\\n                       & Singh et al. \\cite{singh2020interpretation}               & DeepLIFT and others              & Post-hoc           & Global, Local & Model-agnostic                & Ophthalmic diagnosis                                            \\\\ \\midrule\nAttention Mechanism    & Kwon et al. \\cite{kwon2018retainvis}                      & Attention                        & Intrinsic         & Global, Local & Model-specific                & Clinical risk prediction (cardiac failure\/cataract)             \\\\\n                       & Zhang et al. \\cite{zhang2018patient2vec}                  & Attention                        & Intrinsic         & Local         & Model-specific                & EHR based future hospitalisation prediction                     \\\\\n                       & Choi et al. \\cite{choi2016retain}                         & Attention                        & Intrinsic         & Local         & Model-specific                & Heart failure prediction                                        \\\\\n                       & Kaji et al. \\cite{kaji2019attention}                      & Attention                        & Intrinsic         & Global, Local & Model-specific                & Predictions of clinical events in ICU                           \\\\\n                       & Shickel et al. \\cite{shickel2019deepsofa}                 & Attention                        & Intrinsic         & Global, Local & Model-specific                & Sequential organ failure assessment\/in-hospital mortality       \\\\\n                       & Hu et al. \\cite{hu2019deephint}                           & Attention                        & Intrinsic         & Local         & Model-specific                & Prediction of HIV genome integration site                       \\\\ \\cmidrule{2-7}\n                       & Izadyyazdanabadi et al. \\cite{izadyyazdanabadi2018weakly} & MLCAM                            & Intrinsic         & Local         & Model-specific                & Brain tumour localisation                                       \\\\\n                       & Zhao et al. \\cite{zhao2018respond}                        & Respond-CAM                      & Intrinsic         & Local         & Model-specific                & Macromolecular complexes                                        \\\\\n                       & Couture et al. \\cite{couture2018multiple}                 & Super-pixel maps                 & Intrinsic         & Local         & Model-specific                & Histologic tumour subtype classification                        \\\\\n                       & Lee et al. \\cite{lee2019explainable}                      & CAM                              & Intrinsic         & Local         & Model-specific                & Acute intracranial haemorrhage detection                        \\\\\n                       & Kim et al. \\cite{kimartificial2020}                       & CAM                              & Intrinsic         & Local         & Model-specific                & Breast neoplasm ultrasonography analysis                        \\\\\n                       & Rajpurkar et al. \\cite{rajpurkar2020appendixnet}          & Grad-CAM                         & Intrinsic         & Local         & Model-specific                & Diagnosis of appendicitis                                       \\\\\n                       & Porumb et al. \\cite{porumb2020precision}                  & Grad-CAM                         & Intrinsic         & Local         & Model-specific                & ECG based hypoglycaemia detection                               \\\\\n                       & Hu et al. \\cite{hu2020weakly}                             & Multiscale CAM                   & Intrinsic         & Local         & Model-specific                & COVID-19 classification                                         \\\\ \\midrule\nKnowledge Distillation & Caruana et al. \\cite{caruana2015intelligible}             & Rule-based system                & Intrinsic         & Global        & Model-specific                & Prediction of pneumonia risk and 30-day readmission forecast    \\\\\n                       & Letham et al. \\cite{letham2015interpretable}              & Bayesian rule lists              & Intrinsic         & Global        & Model-specific                & Stroke prediction                                               \\\\\n                       & Che et al. \\cite{che2016interpretable}                    & Mimic learning                   & Post-hoc           & Global, Local & Model-specific                & ICU outcome prediction (acute lung injury)                      \\\\\n                       & Ming et al. \\cite{ming2018rulematrix}                     & Visualization of rules           & Post-hoc           & Global        & Model-specific                & Clinical diagnosis and classification (breast cancer, diabetes) \\\\\n                       & Xiao et al. \\cite{xiao2018readmission}                    & Complex relationships distilling & Post-hoc           & Global        & Model-specific                & Prediction of the heart failure caused hospital readmission     \\\\\n                       & Davoodi and Moradi \\cite{davoodi2018mortality}            & Fuzzy rules                      & Intrinsic         & Global        & Model-specific                & In-hospital mortality prediction (all-cause)                    \\\\\n                       & Lee et al. \\cite{lee2019generation}                       & Visual\/textual justification     & Post-hoc           & Global, Local & Model-specific                & Breast mass classification                                      \\\\\n                       & Prentzas et al. \\cite{prentzas2019integrating}            & Decision rules                   & Intrinsic         & Global        & Model-specific                & Stroke Prediction                                               \\\\ \\midrule\nSurrogate Models       & Pan et al. \\cite{pan2019development}                      & LIME                             & Post-hoc           & Local         & Model-agnostic                & Forecast of central precocious puberty                          \\\\\n                       & Ghafouri-Fard et al. \\cite{ghafouri2019application}       & LIME                             & Post-hoc           & Local         & Model-agnostic                & Autism spectrum disorder diagnosis                              \\\\\n                       & Kovalev et al. \\cite{KOVALEV2020106164}                   & LIME                             & Post-hoc           & Local         & Model-agnostic                & Survival models construction                                    \\\\\n                       & Meldo et al. \\cite{meldo2020natural}                      & LIME                             & Post-hoc           & Local         & Model-agnostic                & Lung lesion segmentation                                        \\\\\n                       & Panigutti et al. \\cite{Panigutti2020}                     & LIME like with rule-based XAI    & Post-hoc           & Local         & Model-agnostic                & Prediction of patient readmission, diagnosis and medications    \\\\\n                       & Lauritsen et al. \\cite{lauritsen2020explainable}          & Layer-wise relevance propagation & Post-hoc           & Local         & Model-agnostic                & Prediction of acute critical illness from EHR                   \\\\ \\bottomrule\n\\end{tabular}\n\n\\end{minipage}}\n\\caption{Summary of various XAI methods in digital healthcare and medicine including their category (XAI via dimension reduction, feature importance, attention mechanism, knowledge distillation, and surrogate representations), reference, key idea, type (Intrinsic or Post-hoc, Local or Global, and Model-specific or Model-agnostic) and specific clinical applications.}\n    \\label{tab:tab1}\n\n\\end{sidewaystable}\n\n\n\\begin{figure}[!htb] \n\\begin{center}\n    \\includegraphics[width=0.8\\linewidth]{.\/figures\/Figure7.pdf} \n   \n    \\caption{Publication per year for XAI and medical XAI (top) and percentage for two categories of research (bottom). Data retrieved from Scopus\u00ae (Jan 8th, 2021) by using these commands when querying this database---XAI: (ALL(\"Explainable AI\") OR ALL(\"Interpretable AI\") OR ALL(\"Explainable Artificial Intelligence\") OR ALL(\"Interpretable Artificial Intelligence\") OR ALL(\"XAI\")) AND PUBYEAR = 20XX; Medical XAI: (ALL(\"Explainable AI\") OR ALL(\"Interpretable AI\") OR ALL(\"Explainable Artificial Intelligence\") OR ALL(\"Interpretable Artificial Intelligence\") OR ALL(\"XAI\")) AND (ALL(\"medical\") OR ALL(\"medicine\")) AND PUBYEAR = 20XX, in which XX represents the actual year.}\n    \\label{fig:fig7}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Proposed Method}\n\n\\subsection{Problem Formulation}\n\nIn this study, we have demonstrated two typical but important applications of using XAI, which have been developed for classification and segmentation---two mostly widely discussed problems in medical image analysis and AI-powered digital healthcare. Our developed XAI techniques have been manifested using CT images classification for COVID-19 patients and segmentation for hydrocephalus patients using CT and MRI datasets.\n\n\\subsection{XAI for Classification}\n\nIn this subsection, we provide a practical XAI solution for explainable COVID-19 classification that is capable of alleviating the domain shift problem caused by multicentre data collected for distinguishing COVID-19 patients from other lung diseases using CT images. The main challenge for multicentre data is that hospitals are likely to use different scanning protocols and parameters for CT scanners when collecting data from patients leading to distinct data distribution. Moreover, it can be observed that images obtained from various hospitals are visually different although they are imaging the same organ. If a machine learning model is trained on data from one hospital and tested on the data from another hospital (i.e., another centre), the performance of the model often degrades drastically. Besides, another challenge is that only patient-level annotations are available commonly but image-level labels are not since it would take a large amount of time for radiologists to annotate them \\cite{lin2005emergency}. Therefore, we propose a weakly supervised learning based classification model to cope with these two problems. Besides, an explainable diagnosis module in the proposed model can also offer the auxiliary diagnostic information visually for radiologists. The overview of our proposed model is illustrated in Figure \\ref{fig:cls_network}.\n\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=\\linewidth]{.\/figures\/cls_network.pdf} \n   \n    \\caption{The overview of our proposed model. $P(c \\given S_i)$ denotes the probability of the Section $S_i$, and $P(c \\given \\mathcal{P})$ represents the probability of the patient who is COVID-19 infected or not. $Q\\in \\mathbb{R}^{2\\times 2 \\times C}$ indicates the noise transaction from the probability of the true label $P(y_c \\given \\mathcal{I})$ to the noise label $P(z_c \\given \\mathcal{I})$. Besides, $\\phi(\\cdot)$ is a non-linear feature transformation function, which projects the feature into embedding space.}\n    \\label{fig:cls_network}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Explainable Diagnosis Module (EDM)}\n\nAs the predicting process of deep learning models is in a black-box, it is desirable to develop an explainable technique in medical image diagnosis, which provides an explainable auxiliary tool for radiologists. For common practice, CAM can generate the localisation maps for the prediction through the weighted sum of feature maps from the backbone networks such as ResNet \\cite{he2016resnet}. Suppose $F^{k} \\in \\mathbb{R}^{H'\\times W'}$ is the $k$-th feature map with the shape of $H' \\times W'$, and $W^{fc} \\in \\mathbb{R}^{K \\times C}$, where $K$ is the number of feature maps. Therefore, the class score for class $c$ can be computed as\n\\begin{align}\n    s_c = \\sum_{k=1}^K W_{k,c}^{fc} \\left( \\frac{1}{H'W'} \\sum_{i=1}^{H'}\\sum_{j=1}^{W'}F_{i,j}^k \\right).\n\\label{eq:cls_cam}\n\\end{align}\nTherefore, the activation map $A_c^{fc}$ for class $c$ can be defined by \n\\begin{align}\n    (A_c^{fc})_{i,j} = \\sum_{k=1}^K W_{k,c}^{fc} F_{i,j}^k.\n\\end{align}\n\nHowever, generating CAMs is not an end-to-end process, in which the network should be firstly trained on the dataset and utilises the weights of the last fully connected layer to compute the CAMs, bringing extra computation. To tackle this drawback, in our explainable diagnosis module (EDM), we replace the fully connected layer with a $1\\times 1$ convolutional layer of which the weight $W^{conv}$ shares the same mathematical form as $W^{fc}$. So we can reformulate Eq.(\\ref{eq:cls_cam}) as\n\\begin{align}\n    s_c =  \\frac{1}{H'W'} \\sum_{i=1}^{H'}\\sum_{j=1}^{W'} \\left( \\sum_{k=1}^K W_{k,c}^{conv}F_{i,j}^k \\right) = \\frac{1}{H'W'} \\sum_{i=1}^{H'}\\sum_{j=1}^{W'} (A_c^{conv})_{i,j},\n\\end{align}\nwhere $A_c^{conv}$ is the activation map for class $c$ that can be learnt adaptively during the training procedure. The activation map produced by the EDM can not only accurately indicate the importance of the region from CT images and locate the infected parts of the patients, but can also offer the explainable results which are able to account for the prediction.\n\n\\subsubsection{Slice Integration Module (SIM)}\n\nIntuitively, each COVID-19 patient case has a different severity. Some patients are severely infected with large lesions, while most of the positive cases can be mild of which only a small portion of the CT volume is infected. Therefore, if we directly apply the patient level annotations as the labels for the image slices, the data would be extremely noisy leading to poor performance as the consequence. To overcome this problem, instead of relying on single images, we propose a slice integration module (SIM) and use the joint distribution of the image slices to model the probability of the patient being infected or not. In our SIM, we assume that the lesions are consecutive and the distribution of the lesion positions is consistent. Therefore, we adopt a section based strategy to handle this problem and fit this into a Multiple Instance Learning (MIL) framework \\cite{zhou2004mil}. In the MIL, each sample is regarded as a bag, which is composed of a set of instances. A positive bag contains at least one positive instance, while the negative bag solely consists of negative instances. In our scenario, only patient annotations (bag labels) are provided, and the sections can be regarded as instances in the bags.\n\nGiven a patient $\\mathcal{P} = [\\mathcal{I}_1, \\mathcal{I}_2, \\cdots, \\mathcal{I}_n]$ with $n$ CT slices, we divide them into disjoint sections $\\mathcal{P} = \\{S_i\\}_{i=1}^{|S|}$, where $|S|$ is the total amount of sections for patient $\\mathcal{P}$, that is\n\\begin{align}\n    |S| = \\max \\left(1, \\left\\lfloor\\frac{n}{l_s}\\right\\rfloor \\right).\n\\end{align}\n\nHere $l_s$ is the section length, which is a designed parameter. Then we integrate the probability of each section as the probability of the patient, that is\n\\begin{align}\n    P(c\\givenbase \\mathcal{P}) = P(c \\givenbase \\{S_i\\}_{i=1}^{|S|}) = \\frac{1}{1+ \\prod_{i=1}^{|S|} (\\frac{1}{P(c\\givenbase S_i)} - 1)}, \n\\end{align}\nwhere $P(c\\givenbase S_i)$ is the probability of the $i$-th section $S_i$ that belongs to class $c$. By taking the $k$-max probability of the images for each class to compute the section probability, we can mitigate the problem that some slices may contain few infections, which can hinder the prediction for the section. The $k$-max selection method can be formulated as\n\\begin{align}\n    P(c \\given S_i) = \\sigma\\left( \\frac{1}{k} \\max_{\\substack{s^{(j)} \\in M}} \\sum_{j=1}^k s^{(j)}_c \\right), \\nonumber \\\\ s.t. \\quad M \\subset S_i, |M| = k.\n\\end{align}\nwhere $s^{(j)}_c$ is the top $j$-th class score of the slice in the $i$-th section for the class $c$, and $\\sigma (\\cdot)$ represents the Sigmoid function. Then we apply the patient annotations $\\textbf{y}$ to compute the classification loss, which can be formulated as\n\\begin{align}\n    \\mathcal{L}_{cls} = -\\sum_{c = 0}^1 \\left[y_c\\log P(c \\given \\mathcal{P}) + (1-y_c)\\log (1-P(c \\given \\mathcal{P})) \\right].\n\\label{eq:loss_cls}\n\\end{align}\n\n\\subsubsection{Noisy Correction Module (NCM)}\n\nIn real-world applications, radiologists would only diagnose the disease from one image. Therefore, it is also significant for improving the prediction accuracy on single images. However, the image-level labels are extremely noisy since only patient-level annotations are available. To further alleviate the negative impact of patient-level annotations, we propose a noisy correction module (NCM). Inspired by \\cite{bekker2016training}, we model the noise transaction distribution $P(z_c = i \\given y_c = j, \\mathcal{I})$, which transforms the true posterior distribution $P(y_c \\given \\mathcal{I})$ to the noisy label distribution $P(z_c \\given \\mathcal{I})$ by \n\\begin{align}\n    P(z_c = i \\given \\mathcal{I}) = \\sum_j P(z_c = i \\given y_c = j, \\mathcal{I}) P(y_c = j \\given \\mathcal{I}).\n\\label{eq:noise_prob_simple}\n\\end{align}\n\nIn practice, we estimate the noise transaction distribution $Q^c_{ij} = P(z_c = i \\given y_c = j, \\mathcal{I})$ for the class $c$ via\n\\begin{align}\n    Q^c_{ij} = P(z_c = i \\given y_c = j, \\mathcal{I}) = \\frac{\\exp(w^c_{ij} \\phi(\\mathcal{I}) + b^c_{ij})}{\\sum_i\\exp(w^c_{ij} \\phi(\\mathcal{I}) + b^c_{ij})},\n\\label{eq:noise_trans}\n\\end{align}\nwhere $i, j \\in \\{0,1\\}$; $\\phi(\\cdot)$ is a nonlinear mapping function implemented by convolution layers; $w^c_{ij}$ and $b^c_{ij}$ are the trainable parameters. The noise transaction score $T^c_{ij} = w^c_{ij} \\phi(\\mathcal{I}) + b^c_{ij}$ represents the confidence score of the transaction from the true label $i$ to the noise label $j$ for the class $c$. Therefore, Eq.(\\ref{eq:noise_prob_simple}) can be reformulated as\n\\begin{align}\n    P(z_c = i \\given \\mathcal{I}) = \\sum_j Q^c_{ij} P(y_c = j \\given \\mathcal{I}).\n\\label{eq:noise_prob}\n\\end{align}\n\nBy estimating the noisy label distribution $P(z_c \\given \\mathcal{I})$ for patient $\\mathcal{P}$, the noisy classification loss can be computed by\n\\begin{align}\n    \\mathcal{L}_{noisy} = -\\frac{1}{N} \\sum_{n=1}^N\\sum_{c = 0}^1 [y_c^n\\log P(z_c = 1 \\given \\mathcal{I}_n) \\nonumber \\\\+ (1-y_c^n)\\log P(z_c = 0 \\given \\mathcal{I}_n)].\n\\label{eq:loss_noisy}\n\\end{align}\n\nBy combining Eq. (\\ref{eq:loss_cls}) and Eq. (\\ref{eq:loss_noisy}), we can obtain the total loss function for our XAI solution of an explainable COVID-19 classification, that is\n\\begin{align}\n    \\mathcal{L} = \\mathcal{L}_{cls} + \\lambda \\mathcal{L}_{noisy},\n\\label{eq:loss}\n\\end{align}\nwhere $\\lambda$ is a hyper-parameter to balance the classification loss $\\mathcal{L}_{cls}$ and the noisy classification loss $\\mathcal{L}_{noisy}$.\n\n\n\n\\subsection{XAI for Segmentation}\n\nIn this subsection, we introduce an XAI model that is applicable for the explainable brain ventricle segmentation using multimodal MRI data acquired from the hydrocephalus patients. Previous methods \\cite{qian2017objective, cherukuri2017learning} have conducted experiments using images with a slice thickness of less than 3 mm. This is because the smaller of the image thickness, the more images could be obtained, which helps improve the representation power of the model. However, in a real-world scenario, it is not practical for clinicians to use these models because labelling these image slices is extremely labour-intensive and time-consuming. Therefore, it is more common for the annotations of images with larger slice thicknesses, which are easily available while those images with smaller slice thickness are not. Besides, models trained only on thick-slice images have poor generalisation on thin-slice images. To alleviate these problems, we proposed a thickness agnostic image segmentation model, which can be applicable for both thick-slice and thin-slice images, but only requires the annotations of thick-slice images during the training procedure.\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=\\linewidth]{.\/figures\/seg_network.pdf} \n   \n    \\caption{Overview of our proposed XAI model for explainable segmentation. Here ResBlock represents the residual block proposed in the ResNet \\cite{he2016resnet}. }\n    \\label{fig:seg_network}\n\\end{center}\n\\end{figure}\n\n\nSuppose we have a set of thick-slice images $\\mathcal{D}_\\mathcal{S} = \\{(x_s, y_s) | x_s \\in \\mathbb{R}^{H\\times W \\times 3}, y_s \\in \\mathbb{R}^{H \\times W}\\}$ and a set of thin-slice images $\\mathcal{D}_\\mathcal{T} = \\{ x_t | x_t \\in \\mathbb{R}^{H\\times W \\times 3}\\}$. The main idea of our model is to utilise the unlabelled thin-slice images $\\mathcal{D}_\\mathcal{T}$ to minimise the model performance gap between thick-slice and thin-slice images while a post-hoc XAI can also be developed. \n\n\\subsubsection{Segmentation Network}\n\nWith the wide applications of deep learning methods, the encoder-decoder based architectures are usually adopted in automated high accuracy medical image segmentation. The workflow of our proposed segmentation network is illustrated in Figure \\ref{fig:seg_network}. Inspired by the U-Net \\cite{ronneberger2015unet} model, we replace the original encoder with ResNet-50 \\cite{he2016resnet} pre-trained on ImageNet dataset \\cite{deng2009imagenet} since it can provide better feature representation for the input images. In addition, the decoder of the U-Net has at least a couple of drawbacks: 1) the increase of low-resolution feature maps can bring a large amount of computational complexity, and 2) interpolation methods \\cite{dong2015bilinear} such as bilinear interpolation and bicubic interpolation do not bring extra information to improve the segmentation. Instead, the decoder of our model adopts sub-pixel convolution for constructing segmentation results. The sub-pixel convolution can be represented as \n\\begin{align}\n    F^{L} = SP(W_L * F^{L-1} + b_L),\n\\end{align}\nwhere $SP(\\cdot)$ operator transforms and arranges a tensor shaped in $H \\times W \\times C \\times r^2$ into a a tensor with the shape of $rH \\times rW \\times C$, and $r$ is the scaling factor. $F^{L-1}$ and $F^L$ are the input feature maps and output feature maps. $W_L$ and $b_L$ are the parameters of the sub-pixel convolution operators for the layer $L$.\n\n\\subsubsection{Multimodal Training}\n\nAs aforementioned, the thick-slice images with annotations are available. Therefore, in order to minimise the performance gap between thick-slice images and thin-slice images. We apply a multimodal training procedure to jointly optimise for both types of images. Overall, the objective function of our proposed multimodal training can be computed as\n\\begin{align}\n    \\mathcal{L}(x_t, x_s) = \\mathcal{L}_\\mathcal{S}(p_s, y_s) + \\beta \\mathcal{L}_\\mathcal{T}(p_t),\n\\end{align}\nwhere $\\beta$ is a hyper-parameter for weighting the impact of $\\mathcal{L}_\\mathcal{S}$ and $\\mathcal{L}_\\mathcal{T}$. $p_s$ and $p_t$ are the prediction of the segmentation probability maps shaped in $H\\times W\\times C$ for thick-slice images and thin-slices images, respectively. In particular, $\\mathcal{L}_\\mathcal{S}$ is the cross-entropy loss defined as follows\n\\begin{align}\n    \\mathcal{L}_\\mathcal{S} (p_s, y_s) = -\\frac{1}{HWC} \\sum_{n=1}^{HW} \\sum_{c=1}^C y_s^{n,c} \\log p_s^{n,c}.\n\\end{align}\n\nFor the unlabelled thin-slice images, we assume that $\\mathcal{L}_\\mathcal{T}$ can push the features away from the decision boundary of the feature distributions of the thick-slice images, thus achieving distribution alignment. Besides, according to \\cite{grandvalet2005semi}, minimising the distance between the prediction distribution $p$ and the uniform distribution $\\mathcal{U} = \\frac{1}{C}$ can diminish the uncertainty of the prediction. To measure the distance of these two distributions, the objective function $\\mathcal{L}_\\mathcal{T}$ can be modelled by the $f$-divergence, that is\n\\begin{align}\n    \\mathcal{L}_\\mathcal{T} (p_t) = -\\frac{1}{HWC} \\sum_{n=1}^{HW} \\sum_{c=1}^C D_f(p_t^{n,c} ||  \\mathcal{U}) = -\\frac{1}{HWC} \\sum_{n=1}^{HW} \\sum_{c=1}^Cf(Cp_t^{n,c}).\n\\end{align}\n\nMost existing methods \\cite{grandvalet2005semi, vu2019advent} tend to choose $f(x) = x\\log x$, which is alternatively named as KL-divergence. However, one of the main obstacle is that when adopting $f(x) = x\\log x$, the gradient of $\\mathcal{L}_\\mathcal{T}$ would be extremely imbalanced. To be more specific, it can assign a large gradient to the easily classified samples, while assigning a small gradient to hardly classified samples. Therefore, in order to mitigate the unbalancing problem during the optimisation, we incorporate Pearson $\\chi^2$-divergence (i.e., $f(x) = x^2-1$) rather than using the KL-divergence for $\\mathcal{L}_\\mathcal{T}$, that is \n\\begin{align}\n    \\mathcal{L}_\\mathcal{T} (p_t) = -\\frac{C}{HW} \\sum_{n=1}^{HW} \\sum_{c=1}^C(p_t^{n,c})^2.\n\\end{align}\n\nAfter applying the Pearson $\\chi^2$-divergence, the gradient imbalanced issue can be mitigated since the slope of the gradient is constant, which can be verified by taking the second order derivative of $\\mathcal{L}_\\mathcal{T}$.\n\nDuring the training procedure, $\\mathcal{L}(x_t, x_s)$ is optimised alternatively for both thick-slice and thin-slice images.\n\n\\subsubsection{Latent Space Explanation}\n\nOnce the model is trained using multimodal datasets, the performance of the network can be quantitatively evaluated by volumetric or regional overlapping metrics, e.g., Dice scores. However, the relation between network performance and input samples remains unclear. In order to provide information about the characteristics of data and their effect on model performance, through which users can set their expectations accordingly, we investigate the feature space and their correlation with the model performance. For feature space visualisation, we extract the outputs of the encoder module of our model, and then decompose them into a two-dimensional space via Principal Component Analysis (PCA). For estimating the whole space, we use a multi-layer perceptron to fit the decomposed samples and their corresponding Dice scores, which can provide an understanding of Dice scores for particular regions of interests in the latent space where there is no data available. Therefore, through analysing the characteristics of the samples in the latent space, we can retrieve the information about the relationships between samples and their prediction power.\n\n\\subsection{Implementation Details}\nFor both our classification and segmentation tasks, we used ResNet-50 \\cite{he2016resnet} as the backbone network pre-trained on ImageNet \\cite{deng2009imagenet}. For classification, we resized these images into a spatial resolution of $224 \\times 224$. During the training procedure, we set $\\lambda = 1 \\times 10^{-4}$, the dropout rate as $0.7$, and the $L_2$ weight decay coefficient as $1\\times 10^{-5}$. Besides, $l_s$ was set to 16, and $k$ was set to 8 for the sake of computing the patient-level probability. For segmentation, we set $\\beta = 1 \\times 10^{-2}$ for balancing the impact of supervised loss and unsupervised loss. During the training, Adam \\cite{kingma2014adam} optimiser was utilised with a learning rate $1 \\times 10^{-3}$. The training procedure is terminated after 4,000 iterations with batch size 8. All of the experiments were conducted on a workstation with 4 NVIDIA RTX GPUs using PyTorch framework with version 1.5. \n\n\\section{Experimental Settings and Results}\n\n\\subsection{Showcase I: Classification for COVID-19} \n\n\\subsubsection{Datasets}\nWe collected CT data from four different local hospitals in China and removed the personal information to ensure data privacy. The information of our collected data is summarised in Table \\ref{fig:cls_dataset}. In total, there were 380 CT volumes of the patients who tested COVID-19 positive (reverse transcription polymerase chain reaction test confirmed) and 424 COVID-19 negative CT volumes. For a fair comparison, we trained the model on the cross-centre datasets collected from hospital A, B, C, and D. For an unbiased independent testing, CC-CCII data \\cite{zhang2020ccii}, a publicly available dataset, which contained 2,034 CT volumes with 130,511 images, was adopted to verify the effectiveness of the trained models.\n\n\\begin{figure}[!ht]\n\\begin{center}\n    \\subfigure[Patient-level Statistics]{\n        \\includegraphics[width=0.9\\linewidth]{.\/figures\/number_of_patient.pdf} \n    }\n    \\subfigure[Image-level Statistics]{\n        \\includegraphics[width=0.9\\linewidth]{.\/figures\/number_of_images.pdf} \n    }\n   \n    \\caption{Class distribution of the collected CT data. The numbers in the sub-figures (a) and (b) represent the counts for the patient-level statistics and image-level statistics, respectively. The data collected from several clinical centres can result in great challenges in learning discriminative features from those class-imbalanced centres.}\n    \\label{fig:cls_dataset}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsubsection{Data Standardisation, Pre-Processing and Augmentation}\nFollowing the protocol described in \\cite{zhang2020ccii}, we used the U-Net segmentation network \\cite{ronneberger2015unet} to segment the CT images. Then, we randomly cropped a rectangular region whose aspect ratio was randomly sampled in $[3\/4, 4\/3]$, the area was randomly sampled in $[90\\%, 100\\%]$, and the region was then resized into $224 \\times 224$. Meanwhile, we randomly flipped the input volumes horizontally with 0.5 probability. The input data would be a set of CT volumes, which were composed of consecutive CT image slices.\n\n\\subsubsection{Quantitative Results}\nWe compared our proposed classification model with several state-of-the-art COVID-19 CT classification models \\cite{he2016resnet, wang2020covid, li2020artificial, ouyang2020dual}. Table \\ref{tb:sota} summarises the experimental results of COVID-19 classification on the CC-CCII data. For image-level annotations, ResNet-50 \\cite{he2016resnet} and COVID-Net \\cite{wang2020covid} simply treated patient-level labels as the image labels. Different from methods proposed by \\cite{he2016resnet, wang2020covid, li2020artificial}, VBNet \\cite{ouyang2020dual} utilised the 3D residual convolutional neural network to train with patient-level annotations on the whole CT volumes rather than single slices. Besides, COVNet \\cite{li2020artificial} extracted prediction scores from each slice in the CT volumes with ResNet and aggregated the prediction scores via a max-pooling operator to get the patient-level probability. \n\nIn Table \\ref{tb:sota}, we can find that our method achieved the best performance among these SOTA methods. In particular, our method obtained a better performance by 7.2\\% on AUC compared to VB-Net \\cite{ouyang2020dual} on the patient-level indicating that our method can be applicable for the real-world scenario. This also verified the benefit of modelling section information in the CT volumes via our proposed SIM, which we believe is also vital to the improvement of the classification performance. Besides, our method significantly outperformed other methods by at least 40\\% with respect to the specificity while maintaining high sensitivity, which is also a crucial indication for diagnosing COVID-19. In addition, models trained on patient-level annotations could achieve better performance compared to those trained on image-level labels. This is because the noise in the image labels could have a negative impact during the training, which might degrade the representation ability of the model. According to \\cite{geirhos2018imagenet}, models trained on images may rely on learning the textures of images that were highly discriminative among multiple centres. Therefore, these trained models might be overfitted and biased to the texture features of the images collected from different centres, which could explain the phenomenon that these methods (i.e., \\cite{he2016resnet, wang2020covid}) were poorly performed on the unseen centres.\n\nIn another aspect, for CT volumes, the sequential ordering of CT image slices is also informative. COVID-Net \\cite{li2020artificial} took the most discriminative slice as the representation of the whole CT volume, which ignored the encoding of adjacent slices. This would enforce the model only detect the most discriminative slice, leading to the bias towards positive cases, which could impede the detecting of negative cases that resulted in a low specificity. On the contrary, VBNet proposed by Ouyang et al. \\cite{ouyang2020dual} preserved the sequential information by training on the whole CT volumes. In contrast, we partitioned the CT volume into several sections in order to preserve the sequential information to some extent. Besides, VB-Net was trained with stronger supervision that it utilised additional masks for its supervised training. For our method, we only used patient-level annotations that were much more efficient. More importantly, our method achieved better performance on both AUC and accuracy compared to VBNet \\cite{ouyang2020dual} and COVNet \\cite{li2020artificial}.\n\n\\begin{table*}\n\\begin{center}\n\\resizebox{1.0\\linewidth}{!}{%\n\\begin{tabular}{c|c|c|c|c|c|c}\n\\hline\n\\textbf{Annotation}            & \\textbf{Method}                      & \\textbf{Patient Acc. (\\%)}  & \\textbf{Precision (\\%)} & \\textbf{Sensitivity (\\%)} & \\textbf{Specificity (\\%)} & \\textbf{AUC (\\%)} \\\\ \\hline\n\\multirow{5}{*}{Patient-level} & ResNet-50 \\cite{he2016resnet}        & 53.44                      & 64.45                   & 63.03                     & 35.71             & 53.24       \\\\\n                               & COVID-Net \\cite{wang2020covid}       & 57.13                      & 62.53                   & 84.70                     & 6.16             & 49.58      \\\\\n                               & COVNet \\cite{li2020artificial}       & 69.96                      & 70.20                   & \\textbf{93.33}            & 26.75             & 81.61     \\\\\n                               & VB-Net \\cite{ouyang2020dual}         & 76.11                      & 75.84                   & 92.73                     & 45.38             & 88.34      \\\\\n                               & Ours                                 & \\textbf{89.97}             & \\textbf{92.99}          & 91.44                     & \\textbf{87.25}    & \\textbf{95.53}      \\\\ \\hline\n\\multirow{4}{*}{Image-level}   & ResNet-50 \\cite{he2016resnet}        & 52.56                      & 61.60                   & 71.27                     & 18.06     &   50.19              \\\\\n                               & COVID-Net \\cite{wang2020covid}       & 60.03                      & 64.81                   & \\textbf{83.91}            & 15.98     &   58.39            \\\\\n                               & COVNet \\cite{li2020artificial}       & 75.55                      & 79.90                   & 83.24                     & 61.37     &   79.48            \\\\\n                               & Ours                                 & \\textbf{80.41}             & \\textbf{88.56}          & 80.15                     & \\textbf{80.89}  &  \\textbf{86.06}      \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{Comparison results of our method vs. state-of-the-art methods performed on the CC-CCII dataset.}\n\\label{tb:sota}\n\\end{table*}\n\nIn addition, we also provided the Precision-Recall (PR) and Receiver Operating Characteristic (ROC) curves to compare different methods on patient-level annotations and image-level annotations (Figure \\ref{fig:pr} and \\ref{fig:roc}). From the figure, we can observe that models trained on image-level annotations (e.g., ResNet \\cite{he2016resnet} and COVID-Net \\cite{wang2020covid}) were poorly performed since their AUCs were close to 50\\% which indicated a random guess. In contrast, models trained on patient-level was more reliable since their AUCs were greater than 50\\%. In particular, we found that overall our proposed method remained the best-performed algorithm with an AUC of 95.53\\% at the patient-level and 86.06\\% at the image-level. These results verified our assumption that for mild COVID-19 cases, most of the image slices are disease-free.\n\n\n\\begin{figure}\n\\begin{center}\n    \\subfigure[Patient-level Annotation]{\n        \\includegraphics[width=0.46\\linewidth]{.\/figures\/patient_roc.pdf} \n    }\n    \\quad\n    \\subfigure[Image-level Annotation]{\n        \\includegraphics[width=0.46\\linewidth]{.\/figures\/image_roc.pdf} \n    }\n   \n    \\caption{The Receiver Operating Characteristic (ROC) curves of different compared methods.}\n    \\label{fig:roc}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n    \\subfigure[Patient-level Annotation]{\n        \\includegraphics[width=0.46\\linewidth]{.\/figures\/patient_pr.pdf} \n    }\n    \\quad\n    \\subfigure[Image-level Annotation]{\n        \\includegraphics[width=0.46\\linewidth]{.\/figures\/image_pr.pdf} \n    }\n   \n    \\caption{The Precision-Recall (PR) curves of different compared methods.}\n    \\label{fig:pr}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsubsection{Qualitative Results}\n\nIn order to make the prediction to be more explainable, we used the trained model to visualise the CAMs and bounding boxes generated by our EDM as described above. Figure \\ref{fig:cls_cam} shows the visualisation results of the derived CAMs (i.e., $A^{conv}$). In this figure, we can clearly observe that our method tended to pay more attention to the discriminative part of the images so as to make the predictions. For example, in the first column, the lower left part of the lung was seriously infected and had a large area of lesions. Therefore, our method would make the predictions that the image was classified as COVID-19 positive, demonstrating the capability of our XAI model to make explainable predictions.\n\nIn addition, based on the results of the derived CAMs, we also extracted the lesion bounding boxes from the CAMs. It can be found that our method was capable of yielding accurate bounding boxes from the salient part of the CAMs, as illustrated in Figure \\ref{fig:cls_cam}, which further confirmed that our XAI method was applicable to be an auxiliary diagnosis tool for the clinicians. \n\n\n\\begin{figure}\n\\begin{center}\n\n    \\includegraphics[width=\\linewidth]{.\/figures\/new_cams.pdf} \n   \n    \\caption{Examples of the CAMs $A^{conv}$ generated by our proposed EDM for classifying COVID-19 positive patients. The first row contains the original CT-scan image slices, and the second row illustrates the heatmaps of CAMs $A^{conv}$ with bounding boxes confined to the infected areas.}\n    \\label{fig:cls_cam}\n\\end{center}\n\\end{figure}\n\nTo further illustrate the learnt features from our proposed method, we extracted the feature from the backbone network of our architecture, and used T-SNE \\cite{maaten2008tsne} visualisation technique to transform the features extracted from the backbone network of our proposed model, and visualised the distribution of the classified images as shown in Figure \\ref{fig:t_sne}. In this figure, we can find the distinctive visual characteristics of the CT images from different hospitals (i.e., Hospital A, B, C, and D). Besides, it can be observed that the COVID-19 positive images were mostly clustered together, and negative images were mainly distributed in another cluster. More interestingly, in the cluster of the negative images, we can find several positive images in this cluster since these images were scanned from patients who were tested COVID-19 positive. In our intuition, we assume that for some mild cases, lesions were not presented in all of the CT slices. Therefore, there were indeed disease-free CT slices that could be falsely labelled as COVID-19 positive, which verified our assumption.\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/figures\/TSNE.pdf}\n\\caption{T-SNE visualisation \\cite{maaten2008tsne} of the learnt features from CT images. Original images are sampled from four different hospitals and represented in the figure. Besides, a falsely annotated image is drawn from the negative cluster.}\n\\label{fig:t_sne}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{.\/figures\/lime.pdf}\n\\caption{Visualisation of the super-pixels that are positively contributed to the predictions via the LIME method \\cite{ribeiro2016lime}.}\n\\label{fig:lime}\n\\end{figure*}\n\n\nAdditionally, in order to explain each individual prediction, we adopted the LIME method \\cite{ribeiro2016lime} to investigate the contribution of each pixel for the prediction. Instead of using the individual pixel, we divided an image into super-pixels, which were composed of interconnected pixels with similar imaging patterns. Figure \\ref{fig:lime} shows the explanations via LIME for COVID-19 positive images. In each pair of images, we visualised the super-pixels that contributed to the COVID-19 positive prediction results. We can observe that the lesion parts would explain for the positive prediction, which is reasonable to our deep learning model.\n\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/figures\/shap.pdf}\n\\caption{The SHAP values for different super-pixels of the sampled images. We computed the SHAP values through the Kernel SHAP method \\cite{lundberg2017shap}. The super-pixel with positive SHAP value indicates the positive impact to the positive prediction, while the negative value means that the super-pixel contributes to the negative prediction.}\n\\label{fig:shap}\n\\end{figure*}\n\nHowever, the LIME method could only quantitatively estimate the importance according to how close the combination of super-pixels was to the original instance. It discarded the global view of the individual feature contributed by the super-pixels. To overcome this drawback, we further leveraged Kernel SHapley Additive exPlanations (Kernel SHAP) method \\cite{lundberg2017shap} to estimate the contribution of each super-pixel quantitatively by the SHAP value. Samples explained by the Kernel SHAP are demonstrated in Figure \\ref{fig:shap}. We can observe that the super-pixels contained lesion areas positively contributed to the positive prediction, while those super-pixels related to the backgrounds or disease-free areas would reflect the contribution to negative prediction.\n\n\n\n\\subsection{Showcase II: Segmentation for Hydrocephalus} \n\n\\subsubsection{Datasets}\nThe studied cohort included 20 normal elderly people, 20 patients with cerebral atrophy, 64 patients with normal pressure hydrocephalus, and 51 patients with acquired hydrocephalus (caused by subarachnoid haemorrhage, brain trauma or brain tumour). CT scans of the head were performed using two CT instruments, one of which was the SOMATOM Definition Flash from Siemens, Germany, and the other was the SOMATOM Emotion 16 from Siemens, Germany. Secondly, MRI examinations were conducted using a 1.5T MR scanner(Avanto, Siemens, Erlangen, Germany) and a 3.0T MRI scanner(Prisma, Siemens, Erlangen, Germany). The slice thickness of the CT images includes: 0.5mm, 1.0mm, 1.5mm, 2.0mm, 4.8mm, 5.0mm. The slice thickness of the MRI images includes: 1.0mm, 7.8mm, 8.0mm. For experiments, we randomly split the thick-slice and thin-slice images into training, validation and testing sets. The details of the dataset are summarised in Table \\ref{tb:seg_dataset}.\n\n\n\\begin{table}[]\n\\begin{center}\n\\resizebox{1.0\\linewidth}{!}{%\n\\begin{tabular}{l|c|c|c|c|cc}\n\\hline\n\\multirow{2}{*}{Modality} & \\multicolumn{2}{c|}{Training Set} & \\multicolumn{2}{c|}{Validation Set} & \\multicolumn{2}{c}{Test Set}                  \\\\ \\cline{2-7} \n                          & Thick-slice      & Thin-Slice     & Thick-slice       & Thin-Slice      & \\multicolumn{1}{c|}{Thick-slice} & Thin-Slice \\\\ \\hline\nMRI                       & 810              & 1,303          & 203               & 326             & \\multicolumn{1}{c|}{189}         & 982        \\\\\nCT                        & 2,088            & 2,076          & 523               & 519             & \\multicolumn{1}{c|}{309}         & 492        \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{The number of thick-slice and thin-slice images used in our study.}\n\\label{tb:seg_dataset}\n\\end{table}\n\n\n\\subsubsection{Data Standardisation, Pre-Processing and Augmentation}\nFor the pre-processing of these data, we normalised images using the $z$-score normalisation scheme, which was done by subtracting its mean then divided by its standard deviation. For anomaly pixels, we clipped them within the range of 1-quantile and 99-quantile. For data augmentation, we resized the images using a bicubic interpolation method and resized masks with the nearest interpolation. Then we flipped the images horizontally with 0.5 probability, and scaled the hue, saturation, and brightness with coefficients uniformly drawn from $[0.8, 1.2]$. \n\n\\subsubsection{Quantitative Results}\nTable \\ref{tb:Dice_seg} shows the segmentation performance of various compared models on different modalities. All of the models were trained on the thick-slice images with annotations and the unlabelled thin-slice images. We can observe that our proposed method outperformed all of the compared state-of-the-art methods by a large margin on the mixed datasets (i.e., the mixture of thick-slice and thin-slice images) with at least 4.4\\% of the Dice scores. It is of note that all three models achieved similar segmentation performance on thick-slice images. However, our proposed method gained a significant improvement on the thin-slice images for both MRI and CT scans. The primary reason is that our model could diminish the uncertainty of these thin-slice images while achieving the distribution alignment between thick-slice and thin-slice images, which could enhance the representation and generalisation capabilities of our model. Besides, we also investigated the effectiveness of $\\mathcal{L}_\\mathcal{S}$ and $\\mathcal{L}_\\mathcal{T}$ in Table \\ref{tb:seg_ablation}. In the table, we can find that when only training on thick-slice images, the model performed perfectly on thick-slice images while performing poorly on thin-slice images, since the distribution of these two kinds of slices could vary. Moreover, the performance of the models trained only on unlabelled thin-slice images degraded sharply because of the lack of annotations to guide the segmentation. In the Exp.3 as shown in Table \\ref{tb:seg_ablation}, our model could gain significant improvement on the thin-slice images while preserving good performance on the thick-slice images, which demonstrated that our trained model was applicable for both types of images for both CT and MRI modalities.\n\n\n\\begin{table}[]\n\\begin{center}\n\\begin{tabular}{l|c|c|c|ccc}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{3}{c|}{MRI}                                                                & \\multicolumn{3}{c}{CT}                                                                 \\\\ \\cline{2-7} \n                        & Thick                       & Thin                        & Mixed                       & \\multicolumn{1}{c|}{Thick}  & \\multicolumn{1}{c|}{Thin}   & Mixed                      \\\\ \\hline\nU-Net \\cite{ronneberger2015unet}                   & 0.9226                      & 0.7665                      & 0.8353                      & \\multicolumn{1}{c|}{0.9351} & \\multicolumn{1}{c|}{0.7987} & 0.8513                     \\\\\nU-Net++ \\cite{zhou2018unet++}                & \\multicolumn{1}{l|}{0.9159} & \\multicolumn{1}{l|}{0.8495} & \\multicolumn{1}{l|}{0.8602} & \\multicolumn{1}{l|}{\\textbf{0.9421}} & \\multicolumn{1}{l|}{0.7797} & \\multicolumn{1}{l}{0.8424} \\\\\nOurs                    & \\textbf{0.9323}                  & \\textbf{0.9056}                      & \\textbf{0.9099}                      & \\multicolumn{1}{c|}{0.9365} & \\multicolumn{1}{c|}{\\textbf{0.8697}} & \\textbf{0.8954}                     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison results (Dice scores) of our method vs. other state-of-the-art methods. Mixed represents the test set containing both thick-slice and thin-slice images.}\n\\label{tb:Dice_seg}\n\\end{table}\n\n\n\\begin{table}[]\n\\begin{center}\n\\begin{tabular}{l|c|c|c|c|c|ccc}\n\\hline\n\\multirow{2}{*}{\\textbf{Exp.}} & \\multirow{2}{*}{$\\mathcal{L}_\\mathcal{S}$} & \\multirow{2}{*}{$\\mathcal{L}_\\mathcal{T}$} & \\multicolumn{3}{c|}{\\textbf{MRI}}                   & \\multicolumn{3}{c}{\\textbf{CT}}                                                               \\\\ \\cline{4-9} \n                               &                             &                             & Thick           & Thin            & Mixed           & \\multicolumn{1}{c|}{Thick}           & \\multicolumn{1}{c|}{Thin}            & Mixed           \\\\ \\hline\n1                              & $\\surd$                     &                             & \\textbf{0.9390} & 0.8199          & 0.8391          & \\multicolumn{1}{c|}{\\textbf{0.9438}} & \\multicolumn{1}{c|}{0.8345}          & 0.8767          \\\\\n2                              &                             & $\\surd$                     & 0.0034          & 0.0108          & 0.0110          & \\multicolumn{1}{c|}{0.0109}          & \\multicolumn{1}{c|}{0.0006}          & 0.0069          \\\\\n3                              & $\\surd$                     & $\\surd$                     & 0.9323          & \\textbf{0.9056} & \\textbf{0.9099} & \\multicolumn{1}{c|}{0.9365}          & \\multicolumn{1}{c|}{\\textbf{0.8697}} & \\textbf{0.8954} \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Dice scores comparison for verifying the effectiveness of each loss term. Mixed represents the test set containing both thick-slice and thin-slice images.}\n\\label{tb:seg_ablation}\n\\end{table}\n\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/figures\/ious.pdf}\n\\caption{The visualisation of the Dice scores of the projected images. The plane was computed by smoothing the Dice scores. It is of note that images sharing similar characteristics were clustered together. On the left-hand side and right-hand side, samples from different regions of the plane are presented.}\n\\label{fig:ious}\n\\end{figure*}\n\nBesides, in order to interpret the black-box segmentation model, we extracted the lowest bottom features and projected them into a 2D latent space using the PCA technique. We then computed the Dice score for each sample and visualised it in Figure \\ref{fig:ious}. In this figure, we can observe that slices sampled from the orange circle all contained a small region of ventricle where the model could not perform well. However, images from the green and yellow circle had multiple ventricles, which took a large proportion of the images. Therefore, these images could be well-predicted by our model.\n\n\n\\subsubsection{Qualitative Results}\nTo qualitatively examine the performance of our model and other state-of-the-art models, we presented some visualisation results of the CT and MRI images with thin-slices in Figure \\ref{fig:seg_results}, and computed the Dice scores for each segmentation result. For MRI images, our model and U-Net++ \\cite{zhou2018unet++} were able to segment four ventricles in the brain. In particular, our model could predict the third ventricle in the brain more completely compared to the prediction generated by the U-Net++ \\cite{zhou2018unet++} due to the informative feature representation by the pre-trained encoder. However, for CT images, the performance varied among different models. The primary reason is that original CT volumes contained the skull which could cause the brain to be visually unclear, after removing the skull, the contrast of the images could be largely distinct. More concretely, for those images with low contrast, (e.g., the row 1 and row 5 in Figure \\ref{fig:seg_results}), all of the three compared methods were capable of predicting the left lateral and right lateral ventricles. However, for those images with high contrast (e.g., the row 2 and row 4 in Figure \\ref{fig:seg_results}), our proposed method could predict most of the ventricle part in the brain while U-Net and U-Net++ failed.\n\n\\begin{sidewaysfigure}\n\\begin{center}\n\n    \\includegraphics[width=\\linewidth]{.\/figures\/results_seg.pdf} \n   \n    \\caption{The visualisation of the 3D brain ventricles segmentation results using different compared models. The right lateral ventricle is coloured in red; the left lateral ventricle is coloured in green; the yellow coloured region represents the third ventricle; and the blue region represents the fourth ventricle.}\n    \\label{fig:seg_results}\n\\end{center}\n\\end{sidewaysfigure}\n\nIn addition, we used the segmentation results generated by compared models to reconstruct the 3D images of each ventricle. The example is illustrated in Figure \\ref{fig:seg_3d_results}. We can observe that U-Net \\cite{ronneberger2015unet} could hardly predict the ventricles on thin-slice images, while U-Net++ \\cite{zhou2018unet++} was able to segment the left lateral and right lateral ventricles by taking advantage of dense connections of the intermediate feature maps. In contrast, our proposed method could not only predict the two ventricles mentioned above, but could also segment the third ventricle and the fourth ventricle well. One limitation of our model is that it could not predict the connection region between the third ventricle and fourth ventricle because the area is too small to be distinguished.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\n    \\includegraphics[width=\\linewidth]{.\/figures\/3d_visual.pdf} \n   \n    \\caption{Three-dimensional visualisation of the predictions on thin-slice MRI images for each ventricle segmented by different comparison models. The 3D segmentation results were visualised from the axial plane, the coronal plane, and the sagittal plane. Colouring scheme is consistent with Figure \\ref{fig:seg_results}.}\n    \\label{fig:seg_3d_results}\n\\end{center}\n\\end{figure}\n\n\\subsection{Discussions}\n\nIn missions increasingly vital to human healthcare, AI is being deployed. Automated decisions should be explainable in order to create trust in AI and prevent an algorithm-based totalitarian society. This is not just a human right, for example, enshrined in the European GDPR, but an ultimate goal for algorithm developers who want to know if the necessary clinical characteristics are captured by the decision support systems. XAI should be possible to provide explanations in a systematic manner in order to make the explainability scalable. To construct a surrogate while-box model for the black-box model used to make a prediction, a typical solution is to use simpler, more intuitive decision algorithms. There is a chance, though, that the surrogate model is too complicated or too abstract for it to be truly understandable for humans. \n\nIn this study, we have firstly provided a mini-review for XAI methods and their specific applications in medicine and digital healthcare that is followed by two example showcases that we have developed. From our two showcases, we have explored the classification model and segmentation model in terms of sensitivity (i.e., LIME \\cite{ribeiro2016lime} and Kernel SHAP \\cite{lundberg2017shap}) and decomposition (i.e., T-SNE \\cite{maaten2008tsne} and CAMs). For LIME and Kernel SHAP methods, the individual sample can be analysed and interpreted with each super-pixel, which is useful for individual diagnosis. These methods can provide a straightforward view of how local explanations affect the final predictions. \n\nOn the other hand, T-SNE provides us with an insight into the strength and weakness of our proposed models. For example, in Figure \\ref{fig:ious}, the distribution of the decomposed image features has an association with the prediction performance, which indicates the weakness of the black-box segmentation models. Meanwhile, the distribution of decomposed image features also reveals the clustered characteristics of the raw inputs (Figure \\ref{fig:t_sne}), which can help us to find the reason why a model would make such predictions.\n\nIn consequence, these methods can also be classified into two categories named as perceptive interpretability and mathematical interpretability. When visual evidence is not useful or erroneous, the mathematical evidence can be used as the complement for interpretability. Therefore, various methods should be applied simultaneously for the sake of providing reliable interpretability.\n\nNevertheless, a significant drawback of the current studies on XAI is that the interpretations are focused on the intuition of experts rather than from the demands of the end-users \\cite{du2019techniques}. Current local explanations are typically provided in a feature-importance vector format, which is a full causal attribution and a low-level interpretation. This format would be satisfactory if the description viewers were the developers and analysts, since they could use the mathematical study of the distribution of features to debug the models. However, this type of XAI is less accommodating if the description receivers are lay-users of the AI. XAI can explain the complete judgement logic of the model, which includes a large amount of repetitive knowledge which can confuse the lay-users. The presentation of the XAI algorithms should be further improved to increase customer satisfaction.\n\nThe poor abstraction level of explanations is another drawback. For example, despite XAI derived heatmaps can indicate that individual pixels are important, there is normally no correlation computed between these significance regions to more abstract principles such as the anatomical or pathological regions shown in the images. More importantly, the explanations ought to be understood by humans to make sense of them and to grasp the understandable actions of the model. It is indeed desirable to provide meta-explanations that can integrate evidence from these low-level heatmaps to describe the behaviour of the model at a more abstract, more humanly understandable level. However, this level of understanding can be hard and erroneous. Previously proposed methods have recently been suggested to aggregate low-level explanations and measure the semantics of neural representations. Thus, a constructive topic for future study is the development of more advanced meta-explanations that leverages multimodal information fusion.\n\nBecause the audiences of XAI results are essentially human users, an important future research direction is the use of XAI in human-machine interaction; therefore, research studies in XAI need to explore human factors. A prerequisite for good human-machine interaction is to construct explanations for the right user focus, for instance, develop XAI to ask the correct questions in the proper manner, which is crucial in the clinical environment. Optimisation of the reasoning procedure for optimal human use, however, is still a problem that demands more research. Eventually, a broad open gap in XAI is the use of interpretabilities beyond using visualisation techniques. Future studies will demonstrate how to incorporate XAI into a broader optimisation mechanism in order to, e.g., boost the efficiency of the model and reduce the model complexity.\n\n\\section{Conclusion}\n\nThe recent confluence of large-scale annotated clinical databases, the innovation of deep learning approaches, open-source software packages, and inexpensive and rapidly increasing computing capacity and cloud storage has fuelled the recent exponential growth in AI. This foretells to change the landscape of medical practice in the near future. AI systems have specialised success in certain clinical activities that are more able to assess patient prognosis compared to doctors, and can help in surgical procedures. If deep learning models continue to advance, there is a growing chance that AI could revolutionise medical practice and redefine the role of clinicians in the process. Our mini-review has demonstrated the research trends towards the trustable AI or trustworthy AI, which promotes the XAI globally, and XAI methods in medicine and digital healthcare are highly in demand. Additionally, our two showcases have shown promising XAI results for the two most widely investigated classification and segmentation problems in medical image analysis. We can envisage further development of XAI in medicine and digital healthcare by integrating information fusion from cross-modalities imaging and non-imaging clinical data can be a stepping stone toward a more general acceptance of AI in clinical practice. Ultimately, the trustable AI will promote confidence and openness of its deployment in the clinical arena and also make it easier to comply with the legislation of the GDPR and regulations of the NHS$^X$ in the UK, CE-mark in the EU, FDA in the USA, and NMPA in China. \n\n\\section*{Acknowledgement}\n\nThis work was supported in part by the European Research Council Innovative Medicines Initiative on Development of Therapeutics and Diagnostics Combatting Coronavirus Infections Award 'DRAGON: rapiD and secuRe AI imaging based diaGnosis, stratification, fOllow-up, and preparedness for coronavirus paNdemics' [H2020-JTI-IMI2 101005122], in part by the British Heart Foundation [PG\/16\/78\/32402], in part by the AI for Health Imaging Award 'CHAIMELEON: Accelerating the Lab to Market Transition of AI Tools for Cancer Management' [H2020-SC1-FA-DTS-2019-1 952172], in part by the Hangzhou Economic and Technological Development Area Strategical Grant [Imperial Institute of Advanced Technology], in part by the Project of Shenzhen International Cooperation Foundation (GJHZ20180926165402083), and in part by the Clinical Research Project of Shenzhen Health and Family Planning Commission (SZLY2018018).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThere has been a large interest in generating automatic text description \\cite{mckeown1992text} of tabular data -- for example, prior work has sought to generate biographies from tables of biographical information \\cite{lebret2016neural}, and generating descriptions from structured meaning representations \\cite{gardent2017webnlg}. \nHowever, in many of these tasks, the main focus is on designing systems that are able to \\emph{select entries} from tabular or equivalent data during generation by using neural attention mechanisms.\nIn many naturally occurring descriptions of tabular data, humans often refer to higher-level patterns, for example in the description of stock index pricing over the week in Fig. \\ref{fig:pull}, \nthe speaker refers to how the stock price peaks towards the ending. Some recent work has looked into setups that require non-trivial inference \\cite{wiseman2017challenges,chen2020logical}. However, they typically don't involve inference about numerical patterns in time series data. \nMoreover, much recent prior work on identifying more complex patterns in data for captioning has relied on deep neural networks, often employing neural encoders and attention mechanisms. \nHowever, such approaches often fail to generate faithful responses and lack interpretability \\cite{DBLP:journals\/corr\/abs-1910-08684,dhingra2019handling,parikh2020totto}. \n\n\\begin{figure}[t]\n    \\includegraphics[width=0.95\\textwidth]{figures\/pull.png}\n    \\vspace{-0.1\\abovedisplayskip}\n    \\caption{\\footnotesize We propose a neural truth-conditional model for high precision and diverse time series caption generation. \n    }\n    \\label{fig:pull}\n    \\vspace{-3mm}\n\\end{figure}\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n    \\includegraphics[width=0.80\\textwidth]{figures\/model_overview.png}\n    \\vspace{-0.3\\abovedisplayskip}\n    \\caption{\\footnotesize Method Overview: We present a truth-conditional model for time series captioning, which first identifies patterns (composed of simpler modules) that hold true for a given data point. Decoder conditions only on a sampled program $z$ (and not on input $x$), generating high precision outputs. \n    } \n    \\label{fig:ts_method_overview}\n    \\end{center}\n    \\vspace{-2mm}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\begin{center}\n    \\includegraphics[width=0.95\\textwidth]{figures\/example7.png}\n    \\vspace{-0.3\\abovedisplayskip}\n    \\caption{\\footnotesize \n    A program $z=(z_P,z_L)$ operates on an input time series $x$ to given final output score $s_z(x)$. The module instances are learned from scratch during training. \n    } \n    \\label{fig:vizoutput}\n    \\end{center}\n    \\vspace{-2mm}\n\\end{figure}\n\nWe present a novel neural\ntruth-conditional model for time series captioning, which learns to identify patterns that hold true for the input time series (Figure \\ref{fig:ts_method_overview}). \nWe first sample a latent program from the space of learned neural operators. Each program produces a soft truth-value. Then, with probability proportional to each program's truth-value, a language decoder generates a caption. Thus, programs that yield low truth values, do not produce captions. Critically, the decoder takes \\textit{an encoding of the program itself}, rather than the time series, in order to determine output text. Overall, this approach allows for both: (a) precision in generated output through explicit truth conditioning, and explicit program structure as a representation of time series trends, and (b) diversity in caption generation through the sampling process.\n\n\nWhile some of the patterns in data are complex, they can be considered to have been constructed by composing simpler concepts such as slope (rate of change of value) or comparisons (between values at give points). As such, our programs are constructed by composing simpler operations\/modules.\nSuch a modular design enables sharing of modules across multiple programs, leading to more data efficient learning of module parameters, and also providing better generalization to unseen compositions of modules. \nWe consider a relatively simple space of three module types, using which our model is able to capture a significant fraction of the patterns present in data. The module types could be expanded in future to capture more complex patterns.\nOur model treats the choice of composed computation graph of programs as a latent variable, learned using natural language descriptions as the only supervision. \nIn this respect, our approach is related to neural module networks used in \\citet{andreas2016learning,andreas2016neural}, which condition on a question to generate a program, which then operates on an image or other data to predict an answer. \nIn our case, the constructed computation graph operates and identifies salient patterns in the source data directly, without being guided by an input question. \n\n\n\n\nOur main contributions are as follows:\nWe propose a novel method for time series captioning which first induces useful patterns via composing simpler modules, identifies the programs which hold true, and finally generates text describing the selected program.\nTowards this end, we collect and release two datasets consisting of time series data with accompanying English language description of salient patterns.\nWe observe that the proposed method is able to learn useful patterns, exhibits compositionality and interpretability, and generates outputs that are \nmuch more faithful to the input \ncompared to strong traditional neural baselines.\n\\footnote{Data and code can be found at \\url{https:\/\/github.com\/harsh19\/TRUCE}.}\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\n\n\\noindent \\textbf{Time-Series Numerical Data and Natural Language}\n\\newcite{andreas2014grounding} worked on grounding news headlines to stock time series data by aligning sub-trees in sentence parses to segments of time series. \n\\newcite{DBLP:conf\/acl\/MurakamiWMGYTM17} generate stock data commentary using encoders such as convolutional and recurrent neural networks, similar to the baselines used in our experiments. \n\\newcite{sowdaboina2014learning} focus on the task of describing wind speed and direction. \nTime series data in the form of charts has been utilized in some prior work in figure question answering \\cite{DBLP:conf\/iclr\/KahouMAKTB18,DBLP:journals\/corr\/abs-1906-02850}. \n\n\n\nPast work has explored ways to handle numerical data in a variety of input data domains using neural networks.\n\\newcite{trask2018neural} propose neural logic unit for tasks such as counting objects in images. \nPrior work has investigated handling of numeracy in question answering datasets \\cite{dua2019drop,andor2019giving,DBLP:conf\/iclr\/GuptaLR0020}, typically using a predefined set of executable operations or using specific distributions for number prediction \\cite{DBLP:conf\/emnlp\/Berg-Kirkpatrick20,DBLP:conf\/naacl\/ThawaniPIS21}.\n\n\\noindent \\textbf{Neuro-Symbolic Methods:}\n\\citet{andreas2016neural} proposed to use neural modular networks for visual question answering. Since then, similar approaches have been used for several other tasks such as referring expression comprehension \\cite{DBLP:conf\/aaai\/CirikBM18}, image captioning \\cite{DBLP:conf\/iccv\/YangZC19}, and text question answering {\\cite{andreas2016learning,DBLP:conf\/naacl\/KhotKRCS21}}. %\nCompared to such past efforts, we induce the latent numerical and temporal detection operations, pick a high-scoring program, and condition only on a program encoding to generate the output description. \nIn this respect, our work is also related to prior work on neural discrete representation learning \\cite{DBLP:conf\/nips\/OordVK17,DBLP:conf\/acl\/EskenaziLZ18}, though none of these past works explore utilizing such techniques for data to text problems. \nOur proposed model abstracts the numerical pattern detection from text generation. Related ideas have been explored in the past in other domains and tasks \\cite{DBLP:conf\/emnlp\/GehrmannDR18,DBLP:conf\/emnlp\/JhamtaniB18,DBLP:conf\/icml\/AmizadehPPHK20}. \n \n\n\n\\noindent \\textbf{Data to Text:}\nTabular or structured data to text generation has been explored in prior work \\cite{lebret2016neural,DBLP:conf\/sigdial\/NovikovaDR17,wiseman2017challenges,jhamtani2018chess,DBLP:journals\/corr\/abs-2102-01672}. \nThe Rotowire dataset \\cite{wiseman2017challenges} is comprised of sports summaries for tabular game data which may require modeling of numerical operations and trends.\nHowever, much of the past work has relied on neural models with attention mechanisms, without explicit and interpretable notions of numerical operations.\nFidelity to the input in the context of neural text generation has received a lot of attention lately \\cite{DBLP:conf\/aaai\/CaoWLL18}. \nPrior work has approached the aspect of fidelity to input through changes in model training and\/or decoding methods \\cite{DBLP:journals\/corr\/abs-1910-08684,DBLP:conf\/acl\/KangH20,DBLP:conf\/acl\/MajumderBMJ20,DBLP:conf\/naacl\/GoyalD21,DBLP:conf\/aaai\/0001ZCS21}. \nWe explore a different approach that increases fidelity through conditional independence structure and model parameterization.\n\n\n\n\n\n\n\\section{Experiment Setup}\n\\section{Experiments with Synthetic Data}\n\n\\begin{table}[t]\n    \\centering\n    \\footnotesize\n    \\begin{tabular}{@{}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}lll@{}}\n        \\toprule\n        \\bf Method & \\bf COR & \\bf PPL & \\bf Bleu-3\/4 & \\bf Cider & \\bf Rouge  & \\bf BERT  \\\\ \\midrule\n        \\textsc{TRUCE}{} & $\\bf 92\\%$ & $13.9$ & $0.61\/0.46$ & $1.40$ & $0.74$ & $0.77$   \\\\  \n        \\textsc{FcEnc}{} & $39\\%$ & $16.7$ & $0.45\/0.28$ & $0.81$ & $0.61$ & $0.65$ \\\\  \n        \\textsc{LstmEnc}{} & $45\\%$ & $11.2$ & $0.43\/0.28$ & $0.87$ & $0.62$ & $0.63$  \\\\ \n        \\textsc{ConvEnc}{} & $53\\%$ & $11.0$  &  $0.47\/0.32$ & $1.00$ & $0.66$ & $0.67$  \\\\  \n       \n        \\textsc{FftEnc}{} & $39\\%$ & $22.7$ & $0.38\/0.22$ & $0.67$ & $0.58$ & $0.54$  \\\\  %\n        \\textsc{NearNbr}{} & $71\\%$ & NA & $0.28\/0.14$ & $0.60$ & $0.40$ & $0.48$  \\\\ \n        \\bottomrule\n        \\end{tabular}\n        \\caption{\\footnotesize\n        Results on test split of SYNTH dataset: Human evaluation for correctness (COR) and various automated metrics. \\textsc{TRUCE}{} performs much better than baselines as per correctness evaluation. \n        }\n        \\label{tab:synth6_results}\n\\end{table}\n\n\n\n\\subsection{Methods}\nFor SYNTH data, we consider several baselines listed below (More detailed descriptions are provided in the Appendix). Note that all non-retrieval baselines use the same LSTM decoder architecture as our model. \n(1) \\textbf{\\textsc{NearNbr}{}:} The ground-truth caption of the closest matching training data instance is used as the prediction. The closest matching instance is identified via L2 distance between input time series.\n(2) \\textbf{\\textsc{FcEnc}{}}: Encodes the input time series sequence using a multi-layer feed-forward encoder.\n(3) \\textbf{\\textsc{LstmEnc}{}}: Encodes the input time series sequence using a LSTM recurrent neural network.\n(4) \\textbf{\\textsc{ConvEnc}{}}: Encodes time series using a multi layer convolutional neural network. \n(5) \\textbf{\\textsc{FftEnc}{}}: Encodes time series using Fourier transform features of the input. \n\n\n\n\n\n\n\n\n\n\\subsection{Results}\n\nFor \\textsc{TRUCE}{}, we pick the highest scoring program, according to the prior, for description generation. \nWe generate captions (using greedy decoding) from each of the methods for the test split.\n\\\\\n\\textbf{Automated metrics} measure overlap between model generated caption and the reference ground truth captions. We report Perplexity (\\textbf{PPL}), BLEU-3\/4 \\shortcite{papineni2002bleu}, METEOR \\cite{banerjee2005meteor}, ROUGE-L (\\textbf{Rouge}) \\cite{lin2004rouge}, and BertScore-Precision (\\textbf{BERT}) \\cite{DBLP:conf\/iclr\/ZhangKWWA20}. The proposed \\textsc{TRUCE}{} method gets favorable scores as per various automated metrics on the test split of SYNTH (Table \\ref{tab:synth6_results}).\n\n\\noindent \\textbf{Human Evaluations for Correctness:} \nAutomated metrics may not correlate well with actual quality of the generated output in text generation tasks \\cite{celikyilmaz2020evaluation}.\nAs such, we report human evaluation results as well. We recruit human annotators who are requested to provide a binary label on factual correctness \\textbf{(COR)} of the captions for the test split. Each caption is annotated by three annotators, and the majority label is used. The proposed method is able to achieve a high correctness score of $92\\%$, which is much better than the baselines. This demonstrates the usefulness of the proposed truth-conditional model in generating highly faithful captions. \nOutput samples are provided in the Appendix.\n \n\n\n\n\\begin{SCtable}[]\n   \n    \\centering\n    \\footnotesize\n    \\begin{tabular}{@{}ll@{}}\n        \\toprule\n        \\bf Method & \\bf COR  \\\\ \\midrule\n        \\textsc{TRUCE}{} &  $\\bf 97\\%$    \\\\  \n        \\textsc{FcEnc}{} &  $38\\%$ \\\\  \n        \\textsc{LstmEnc}{} &  $50\\%$ \\\\  \n        \\textsc{ConvEnc}{} & $59\\%$ \\\\\n        \\textsc{FftEnc}{} &  $39\\%$ \\\\  \n        \\textsc{NearNbr}{} &  $72\\%$ \\\\  \n        \\bottomrule\n    \\end{tabular}\n    \\label{tab:synthetic_clf_transfer}\n    \\caption{\\footnotesize Models trained on SYNTH data (where each time series has T=12 values) are tested on another synthetic data with T=24 without any fine-tuning.\n    }\n\\end{SCtable}\n\n\n\n\\subsection{Analysis}\n\\noindent \\textbf{Generalization to different time series duration:} SYNTH data consists of time series instances with T=12 sequence of values. We experiment the extent to which models trained on SYNTH can accurately detect patterns in time series data of different lengths without any fine-tuning. For this, we evaluate results on a separate synthetic data consisting of 100 time series with T'=24 values per time series (dataset created in the same manner as SYNTH and consists of the same set of 6 classes as in SYNTH).  \n\nWe observe that \\textsc{TRUCE}{} retains high correctness of the output captions (Table \\ref{tab:synthetic_clf_transfer}), whereas some of the high performing baseline show significant reduction in correctness.\nNote that some of the employed methods like \\textsc{NearNbr}{} and \\textsc{FcEnc}{} cannot work directly on inputs of length different than present in the training data. For such models, we first adjust length of series. For example, for length 24 input, we consider alternate values only, thereby reducing the series to length 12 (same as in the training data).  \n\\vspace{2mm}\n\n\n\n\n\\begin{table}\n\\footnotesize\n\\begin{tabular}{l@{\\hskip 0.1in}l}\n\\toprule\n\\bf Module  & \\bf Most freq. words associated \\\\\n\\bf id & \\bf with learned modules   \\\\ \\midrule\npattern-1 &  increases, rises   \\\\\npattern-2 &  decreases, decline, dips \\\\  \nlocate-1 &  end, late  \\\\  \nlocate-2 &  beginning , start, initial \\\\\nlocate-3 &  middle, halfway  \\\\\n\\bottomrule\n\\end{tabular}\n  \\caption{\\footnotesize Some of the most frequent words associated with some of the learned module instances for SYNTH data.\n  \\label{tab:module_analysis}\n}\n\\end{table}\n\n\\noindent \\textbf{Analyzing Learned Modules:}\nWe analyze the characteristics of the learned modules by identifying the top words (excluding stop words) associated with each learned module. To do so, for a given series, we find program with highest score, and associate the annotations for that series to corresponding modules in that program. Finally, we collect the most frequent words in annotations associated with each module. We show a summary in the Table \\ref{tab:module_analysis}. The two trend modules seem to be getting activated for increase and decrease patterns respectively.\n\\vspace{2mm}\n\n\n\n\n\n\\noindent \\textbf{Compositionality of Learned Modules}\nWe analyze if the proposed model uses its compositional parameterization effectively.\nTo do so,\nwe conduct a simple analysis as follows:\nWe train \\textsc{TRUCE}{} on a subset of synthetic data consisting of only the following 4 patterns: increase-beginning, decreases-end, increase-middle, decreases-middle. \nWe examine this trained model's behavior on test data points consisting of the two unseen patterns: increase-end and decrease-beginning. More specifically, we analyze the argmax program prediction as per the conditional prior. Based on manual inspection of modules (similar to what we discussed for analysis in Table \\ref{tab:module_analysis}), we know before hand the program which should be selected for these patterns. Model's prediction is considered to be correct if, for example, for an input with `decrease-beginning' pattern, model assigns highest score to the program composed using modules corresponding to `decrease' and `beginning'.\nWe observe that the highest scoring program is the correct\/expected program for 92\\% of the cases in the test split. \n\n\n\\section{Experiments with STOCK Dataset}\n\n\n\\subsection{Posterior Regularization:}\nIn the initial experiments with STOCK dataset, we observe that our model suffers from model collapse, and degenerates into learning a single program only. \nThis is perhaps because randomly initialized modules do not \nhave much guidance to begin with. To mitigate such mode collapse issues, prior work has used mutual posterior divergence (MPD) regularization  \\cite{ma2019mae} \n$-E_{y_i,y_j} KL(q(z|y_i)||q(z|y_j)) $,\nwhere $y_i$ and $y_j$ captions for two randomly chosen data points. \n\nHowever, we note that MPD term enforces the divergence in an indiscriminate manner -- divergence is encouraged even if captions are paraphrases of each other. An alternate way to encourage divergence in the inference network prediction is to encourage divergence only when two captions $y_i$ and $y_j$ represent different programs or patterns. However, such information is not available in the training data. \nInstead, we use an approximation as follows: \nWe identify the $M$ most frequently occurring words excluding stop-words (list available in Appendix) in the captions and are manually labelled to to represent pattern or locate or neither. Each of the words labelled to be of type pattern or locate is assigned a unique \\emph{pattern} or \\emph{locate} module id respectively. \nThe corresponding captions thus get tagged with some heuristic (but potentially noisy) labels for module ids.\nOnly those captions are tagged which have exactly one `locate' word and one `pattern' word.\nThis leads to about 31\\% of the captions being assigned such heuristic labels, while the remaining data stays unlabelled. \n\nThe above procedure does involve a small human-in-the-loop component.  However, we note that it is a pretty light-weight involvement. For example, the system presents M(=10) most frequent pairs of words (excluding stopwords) in captions, and a person spends a couple of minutes labeling their type (locate or pattern).\n\n\n\n\n\n\\subsection{Results}\nWe now report results with STOCK dataset.\nAs mentioned above,\nwe utilize heuristic labels as an auxiliary loss when training the proposed method. Thus, for a fair comparison,\nthe baselines \\textbf{\\textsc{Lstm-Multi}{}}, \\textbf{\\textsc{Conv-Multi}{}} and \\textbf{\\textsc{FcEnc}{}} also use the same set of heuristic labels via a classification loss on the encoded representation in a multi-task learning setup.\n \nThe proposed method \\textsc{TRUCE}{} produces high precision captions as judged by human annotators (Table \\ref{tab:stock_results}). \nWe additionally report automated text overlap scores against reference captions, though the automated metrics seem only mildly correlated with human judgement ratings. \nInterestingly, some of the baselines show large differences in performance in STOCK vs SYNTH datasets. For example, \\textsc{NearNbr}{} performs well on SYNTH but rather poorly on STOCK dataset, perhaps because of variety in time series instances in SYNTH being small, while the same being large in STOCK.\n\\vspace{2mm}\n\n\n\\noindent \\textbf{Diversity and Coverage:}\nIdeally, we want models which can  identify all the interesting patterns present in an input time series. Correctness results discussed earlier are indicative of faithful generation but do not necessarily capture coverage of patterns.\nWe compute coverage of various models via the following procedure. First, we collect L(=12) samples per data point from the model. Next, we recruit human annotators to rate whether a human written reference annotations for that data point is covered by the set of L generated captions or not.\nFor \\textsc{TRUCE}{}, we perform sampling at the program selection stage, while baselines admit sampling only at the token generation stage. \n\n\n\\begin{table}[]\n    \\centering\n    \\footnotesize\n\\begin{tabular}{@{}l@{\\hskip 0.05in}l@{\\hskip 0.09in}l@{\\hskip 0.09in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}ll@{}}\n\\toprule\n\\textbf{Method} & \\textbf{COR} & \\textbf{Bleu-3\/4} & \\textbf{Cider} & \\textbf{Rouge} & \\textbf{BERT}  \\\\\n\\midrule\n\\textsc{TRUCE}{}(Ours) & \\bf 88.4\\% & 0.35 \/ 0.19 & 0.36 & 0.50 & 0.57  \\\\\n\\textsc{FcEnc}{} & $64.2\\%$ & 0.32 \/ 0.19 & 0.43 & 0.47 & 0.56  \\\\\n\\textsc{Lstm-Multi}{} & $65.5\\%$ & 0.35 \/ 0.21 & 0.41 & 0.50 & 0.61 \\\\\n\\textsc{Conv-Multi}{} & $65.9\\%$  & 0.33 \/ 0.18 & 0.41 & 0.49 & 0.59 \\\\\n\\textsc{FftEnc}{} & 61.8\\% &  0.34 \/ 0.19 & 0.39 & 0.49 & 0.58 \\\\\n\\textsc{NearNbr}{} & 47.2\\%  & 0.12 \/ 0.06 & 0.14 & 0.28 & 0.35  \\\\\n\\bottomrule\n\\end{tabular}\n    \\caption{\\footnotesize Results with STOCK data: Proposed method \\textsc{TRUCE}{} scores the best on  correctness evaluation. The best performing baseline scores $20\\%$ less on correctness evaluation. Greedy decoding was used for all the methods.}\n    \\label{tab:stock_results}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n    \\includegraphics[width=0.65\\textwidth]{figures\/coverage.png}\n    \\vspace{-0.5\\abovedisplayskip}\n    \\caption{\\footnotesize Coverage and Correctness of model outputs at different sampling settings. In general, settings with higher coverage of human written captions have lower precision of generated captions. \\textsc{TRUCE}{} achieves much higher correctness scores compared to baselines for similar coverage values.}\n    \\label{fig:coverage}\n    \\end{center}\n    \\vspace{-3mm}\n\\end{figure}\nNote that this makes the coverage score depend on the settings used in the sampling process (e.g. top-p value in nucleus sampling), which will also affect the correctness of the generated captions. In Figure \\ref{fig:coverage}, we demonstrate coverage and correctness values of \\textsc{TRUCE}{} and two of the baseline models under different sampling conditions. In general, restricting samples to a low value of top-p leads to lower coverage but higher correctness. \nOverall, \\textsc{TRUCE}{} behaves in a more favorable manner. For example, comparing \\textsc{TRUCE}{} against \\textsc{ConvEnc}{}, for roughly same level of coverage (e.g. ~50\\%), correctness is much higher for \\textsc{TRUCE}{} (~83\\% against ~45\\% for \\textsc{ConvEnc}{}). \nHowever, there still seems to be a gap in the coverage of patterns, and can perhaps be addressed by incorporating more module types. \n\\vspace{2mm}\n\n\n\n\n\n\n\\subsection{Analysis}\n\n\n\n\\noindent \\textbf{Direct conditioning on the input:}\nOur decoder conditions only an encoding of a sampled program. We hypothesize that such an approach creates a bottleneck discouraging the decoder from learning spurious correlations between the input time series and the output text. \nTo inspect the usefulness of the proposed abstraction, we consider an alternative model wherein the decoder conditions on the input time series as well -- by providing output of a convolutional encoder (same as in  \\textsc{ConvEnc}{}) to the decoder. More specifically, the program representation and the encoder representation are concatenated before being fed to the decoder. Lets refer to such a model with decoder having direct access to the input as \\textsc{TRUCE}\\textsc{-D}. For STOCK data, \\textsc{TRUCE}\\textsc{-D} gets correctness of $69\\%$ compared to $88\\%$ for \\textsc{TRUCE}{}.  \\\\\n\n\n\n\\noindent \\textbf{Analysis of Inference Network:}\nWe analyze the predictions of the inference network at the end of model training.  Particularly, we associate the set of ground truth annotations in validation split to module-ids present in the argmax program prediction from the inference network. Next, we identify the most frequently occurring tokens present for each module-id\/module-instance. We observe that the inference network seems to be associating semantically similar words to the same module instance (\\autoref{tab:inference}). \n\\begin{table}[]\n    \\centering\n    \\footnotesize\n    \\begin{tabular}{l@{\\hskip 0.1in}l}\n    \\toprule\n    \\bf Module id  & \\bf Most frequently associated words\\\\\n    \\midrule\n    pattern-1 &  increases, rises, gains \\\\\n    pattern-3 &  stays, remains, flat  \\\\\n    pattern-4 &  bottoms, out, decline, dips \n    \\\\  \n    loc-1 & start, beginning, initially  \\\\  \n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\footnotesize Inference Network Analysis: Analyzing words frequently present in captions when the argmax program prediction from inference network comprises of a give module-id.}\n    \\label{tab:inference}\n\\end{table}\n\n\n\n\n\n\\subsection{Model}\n\nOur goal is to generate a text caption $y$\ndescribing a salient pattern in an input time series $x$. Our model's generative process is depicted in Figure \\ref{fig:ts_method_overview} and operates as follows: Conditioned on an input time series $x$, we first sample a program $z$ from a learned prior, $p(z|x)$. \nThe latent program $z$ is composed of several operations\/modules composed together, and outputs a truth value score. The prior is governed by the truth-values of corresponding programs so that we are likely to sample programs with high truth values.\nNext, we sample caption $y$ conditioning \\emph{only} on the encoding of sampled program $z$ to generate the final text -- i.e. $y$ is independent of $x$ given $z$. Intuitively, if the latent program encodes sufficient information to describe the pattern it detects, caption needs to only depend on the program itself. \n\n\nThe set of latent `programs' in our model are learned from data. On executing a program $z$ on the input time series data $x$, we obtain an output score $s_z(x)$ (between 0 and 1, both inclusive). Score $s_z(x)$ represents the model's confidence about whether the pattern corresponding to the program holds true for the given input time series. Note that $s_z(x)$ does \\emph{not} represent the prior probability of program $z$ -- since multiple programs can be true for a given time series, and $\\sum_z s_z(x) \\neq 1$.  \nWe provide our model with a set of building blocks\/modules, which combine to form programs. The composition of modules into programs as well as the module parameters are unobserved in data and are learned during model training. The compositionality in the program space enables modules to be shared across programs, leading to more efficient learning. \nThe programs we consider will prove quite effective in experiments, but are actually relatively simple, being composed of only three module types. Our framework is extensible, however, and future work might consider larger program spaces.\nWe refer to our proposed method as \\textsc{TRUCE}{} (\\textbf{TRU}th \\textbf{C}onditional g\\textbf{E}neration).\n\n\n\n\\subsection{Programs and Modules}\nAs previously mentioned, each program $z$ in our model is composed of several learnable operations\/modules. \nFollowing prior work on neural modular networks \\cite{andreas2016neural}, we consider multiple module types, and incorporate inductive biases in their architecture to learn useful numerical patterns. In the current study, however, we limit to three simple types of patterns: \\emph{pattern}, \\emph{locate}, and \\emph{combine}, leaving extensions to the module space as a future direction.\nThese modules are composed together into programs that operate on the input time series (Figure \\ref{fig:ts_method_overview})\n\nThe module types \\emph{pattern}  and \\emph{locate}, output a vector of the same length as the input vector. Both of them output a temporally localized vector, with each value between 0 and 1 (achieved by applying a sigmoid activation function), representing the degree of confidence that the pattern it represents is present at the corresponding position on the temporal axis. For example, as shown in Figure \\ref{fig:vizoutput}, the output of a learned \\emph{locate} module is a vector with high values in the middle part, and the output of the \\emph{pattern} module is high on those positions where there is a decrease in the value in the input time series.\n\nFor the current study, we restrict the space of programs to consist of one \\emph{pattern} ($z_P$) module instance and one \\emph{locate} ($z_L$) module instance. Outputs from the two modules are combined using a \\emph{combine} module, which carries out position-wise multiplication of outputs from $z_P$ and $z_L$, followed by a feed-forward layer and a sigmoid non-linearity. \n\n\\emph{Pattern} modules are aimed at learning patterns such as peaks, dips, increasing trend, and so on. \nWe realize \\emph{pattern} modules through multi layer 1-D convolutions. We argue that 1D convolutions provide an appropriate architecture to induce aspects such as slopes, and compose them to identify patterns such as peaks. \nThe \\emph{locate} module types are realized though a mixture model of K fixed Gaussians placed at equal intervals on the temporal axis of given length $T$. The weights of the components represent learnable parameters for such types of modules. \nThe \\emph{combine} module type learns to transform the position-wise multiplied outputs to a real-valued score, which is then passed through a sigmoid function. \n\n\n\\subsection{Prior}\nAs discussed above, the output of each program $z$ is a real-valued score between 0 and 1. We define prior over the set of programs $Z$ as $p(z) \\propto e^{\\lambda s(z)}$, where $\\lambda$ is a hyperparameter.\nThis formulation makes an implicit assumption that a program $z$ being true for an input time series will make other programs less probable through conservation of probability mass. Such an assumption is necessary, as otherwise directly trying to optimize the likelihood without normalizing across programs will lead to trivial solutions, wherein each program will output a high score for every input. \nNote that an alternative formulation could directly use softmax on an unrestricted real-value output from modules -- such a formulation loses out on the semantics of soft truth output from the programs, and also fared worse in our preliminary experimental evaluations in comparison with the proposed formulation. \n\n\n\n\\subsection{Decoder}\nAs mentioned previously, our decoder conditions only on the program $z$ sampled from  the prior $p(z|x)$ to generate final text. \nTo achieve this, we need to pass a program representation to the decoder. We consider an auto-regressive neural decoder such as LSTM or Transformer. At every step, the decoder considers embedding of the previous token as well as the input program representation. \n\nA straightforward approach to obtain program representation is to associate each unique program with a low dimension embedding vector. However, such an approach will not fully exploit the program structures and shared modules. \nInstead, we first associate each module with an embedding. Next, the representation of a program is constructed by appending the embeddings of the corresponding modules (using a fixed pre-determined order of module types). Such a representation achieves sharing of module embeddings across programs. Moreover, it enables obtaining the representation of a new (unseen) program composed using the same set of modules. \n\n\n\\section{Datasets}\n\\label{sec:data}\n\nWe are interested in modeling numerical patterns and trends in time series data. However, there is a lack of existing data sources with time series data paired with natural language descriptions. \nSome prior work on weather forecasting data (such as Sumtime-Mausam \\cite{sripada2003sumtime}) are typically small (only 1045 data instances), and are limited in the scope of patterns they encompass.\nToTTo dataset \\cite{parikh2020totto} contains a small fraction of descriptions based on numerical reasoning and patterns - however, the main challenge is to find the correct value(s) by identifying the relevant row and column in a table.\nLOGIC-NLG \\cite{chen2020logical} consists of 37K tables and corresponding natural language descriptions, some of which require comparisons of cells in a table. \nIn contrast, we focus on trends and patterns in time series data. \nThus, we construct a new dataset where natural language descriptions are collected for naturally occurring stock price time series data (Section \\ref{sec:data:stock}). Additionally, we collect natural language descriptions for a synthetically constructed set of time series to evaluate and analyse our models in a more controlled setup (Section \\ref{sec:data:synth}).\n\n\n\\subsection{STOCK Dataset}\n\\label{sec:data:stock}\n\nWe collect naturally occurring time series data in the form of stock prices. We utilize the Google Finance API to collect stock prices of 7 randomly chosen technology companies over a period of 20 years. We collect weekly (beginning of week) as well as and daily stock price values. \nWe sub-select a total of 1900 instances, each of consists of sequence of T(=12) values. \nEach instance is sampled from the stock data as follows: (1) we pick one of the companies uniformly at random (2) we randomly pick weekly or daily series with equal probability, (3) we pick a sequence of values of given length T, ensuring no overlap with any previously selected time series. (4) Additionally, since different company stocks can be in very different range of values, we normalize such that all the values are between 0 and 100: $v' = 100*(v-min)\/(max-min) $ . However, normalizing this way directly would create undesirable biases in the dataset since each time series would necessarily cover entire range 0-100. Instead, to compute \\emph{max} and \\emph{min}, we additionally consider 10 values (chosen based on manual inspection) just before and just after the currently selected range. \\vspace{2mm}\n\n\\noindent \\textbf{Annotation collection:}\nWe collect 3 natural language annotations for each of the 1900 data points, leading to a total of 5700 paired time-series with natural language descriptions. We split the 1900 unique time series and associated captions into train, dev, and test splits with ratio 8:1:1. \n\n\n\\noindent \\textbf{Annotator description:}\nWe use Amazon Mechanical Turk as a crowd-sourcing platform.\nWe limit to annotators from Anglophone countries, with HIT (Human Intelligence Task) acceptance rates of more than $90\\%$, and minimum number of accepted HITs as 100. Annotators were paid 25 cents for each annotation (which comes to average hourly rate of over USD 23).\n\n\n\n\\noindent \\textbf{Quality Control:}\nBased on initial pilot studies, we found it useful to show annotators plots instead of tables of values, as we are interested in high level patterns rather than specific values. We do not label the plot lines with actual stock names to remove any potential biases one may have about specific company stocks. Finally, we restrict annotations to a maximum of 9 words, so that one annotation reflects only one pattern.\nEach HIT is labelled by 3 different annotators. We manually inspected at least one annotation from each unique annotator, and ruled out (but still paid) annotations for about 7\\% annotators for being poor quality.\n\n\n\n\\noindent \\textbf{Encouraging Lexical Diversity:}\nWe encouraged annotators (through instructions) to not limit themselves to words shown in examples. Additionally, we limit each annotator to a maximum of 10 HITs to increase diversity in annotations. \n\n\n\n\\noindent \\textbf{Dataset Statistics:}\nThere are a total of 861 unique words across the 5700 captions. Most annotation sentences follow a simple syntactic structure. Additionally, we picked a random subset of 100 data points, and manually classified most of them into following major buckets: trend (increase\/decrease trends: 48\\%) superlative(max\/min values; peaks and troughs: 20\\%); comparisons(comparison of start and end values: 10\\%); volatility (flat\/smooth; irregular: 12\\%).  \n\n\n\n\n\\subsection{Synthetic Time Series (SYNTH)}\n\\label{sec:data:synth}\nTo develop and test models in a more controlled setup, we synthetically construct time series data. \nOur synthetic time series data is constructed such that each time series has exactly one of the following 6 patterns: increases-in-beginning, increases-in-middle, increases-in-end, decreases-in-beginning, decreases-in-middle, decreases-in-end. \nThe resulting dataset consists of a total of paired 720 time series - natural language annotations.  \n\n\nEach synthetic time series is generated as follows: \nFirst, the trend is chosen: increase or decrease. A trend is realized through a straight line of length $L<=T\/3$, with randomly chosen intercept and slope within a range based on the trend selected. \nNext, we randomly select one of the 3 temporal locations : begin, middle, end -- and based on the choice, the pattern is placed in first 40 percentile, 30-70 percentile, or 60-100 percentile respectively, of the entire length T. The region outside the trend is flat. \nFinally, small noise is added to each point. The setup is such that the resulting values are always in (0,100) range. Examples and more specific details can be found in Appendix.\n\n\\section{Conclusion}\nWe present a truth-conditional neural model for time series captioning. Our model composes learned operations\/modules to identify patterns which hold true for a given input. Outputs from the proposed model demonstrate higher precision and diversity compared to various baselines. Further, the proposed model (and some of the baselines) successfully generalize, to some extent, to multiple input sizes. We release two new datasets (in English) for the task of time series captioning. Future work might expand to a broader set of module types to cover more numerical patterns. \n\n\\section*{Acknowledgements}\nWe thank anonymous EMNLP reviewers for insightful comments and feedback. We thank Nikita Duseja for useful discussions.\n\n\n\n\n\n\\section{Truth-Conditional Natural Language Description} \n\nOur goal is to learn models for describing salient patterns in time series data. The main research challenge involved is to learn the types of patterns that humans find salient in time series data, using natural language descriptions as the only source of supervision during training.\nBased on the novel dataset we collect (described in Section \\ref{sec:data} \n, we find that the patterns humans identify tend to describe increasing or decreasing trends, volatility, comparisons of start and end values, presence of peaks and dips. They also mention the temporal location of patterns, such as `at the beginning'\nof the time series.\nThus, our model should be able to learn patterns such as `increase' or `ends with higher value compared to start', and temporal aspects such as `begin' or `end'. \n\n\nOne way to operationalize this process is through the lens of formal logic: e.g. an increasing trend at the beginning of a time series $x$ \ncan be represented trough the logic $z$: $\\big[ \\exists_i$ s.t. \\textsc{increase}($x_i$) AND \\textsc{begin}($i$) $\\big]$ \nThereafter, if the program returns \\texttt{true} on the input, one can condition on only the logical program $z$ to generate output text that describes this pattern via a decoder, $p(y|z)$. However, this still requires learning or defining modules for patterns and temporal location. Inspired by neural module networks \\cite{andreas2016learning,andreas2016neural}, we propose to use functions parameterized by neural networks (Figure \\ref{fig:ts_method_overview}) as modules, incorporating inductive bias through architecture design. However, unlike past work, we condition only on an encoding of sampled programs that return \\texttt{true} to generate output text. \n\\section{Additional Details on Data Sets}\n\n\n\nA downloadable json file for each of the two datasets is provided in the github repository \\footnote{\\url{https:\/\/github.com\/harsh19\/TRUCE}}.\n\n\n\\subsection{Synthetic Data}\n\\label{appendix:sec:synth}\n\nOur synthetic time series data is constructed such that each time series has exactly one of the following 6 patterns: increases-in-beginning, increases-in-middle, increases-in-end, decreases-in-beginning, decreases-in-middle, decreases-in-end. \nThe position in which the pattern is placed is based on the temporal choice (begin\/middle\/end). i.e. L must lie withing first one-third of the time-series (0,T\/3) in case of `begin' pattern, should lie in middle one-third for `middle', and last one third for `end' respectively. We consider equation a*x+b of a line, where `a' represents the slope and `b' represents the y-axis intercept.  We pick a random slope value between 0 and 2, and a random intercept value between 1 and 20. Finally, we pick $|L|$ random integral values for x such that ax+b point lies between 0 and 1. The points in the time series outside the pattern are fixed to be same as the nearest point in the patter. Finally, small noise is added to each point using U(-2,2).  \n\nSome random data samples are shown in Fig. \\ref{fig:syntheg1}. The text corresponding to `HUMAN' marker represents one of the collected annotations for the corresponding time series data. \n\n\n\n\n\\subsection{STOCK data}\n\nFigures \\ref{fig:stockeg1} show data samples for STOCK dataset. The text corresponding to `HUMAN' marker represents one of the collected annotations for the corresponding time series data. \nThe total number of unique words (considering train and validation splits) are 861, out of which only 560 words occur more than once in the dataset.\n\n\n\n\n\n\\section{Additional Results}\n\n\n\\subsection{SYNTH: Generated Samples}\n\\label{appendix:sec:synthgensamples}\nAdditional examples are provided in Figure \\ref{fig:syntheg1}.\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{samples\/egall.png}\n    \\caption{SYNTH: Data and Generated Samples. The captions marked in red were judged as incorrect by human annotators. \\textsc{TRUCE}{} achieves very high precision of 95\\% on outputs for the test split of SYNTH dataset. }\n    \\label{fig:syntheg1}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{STOCK: Generated Samples}\n\\label{appendix:sec:stockgensamples}\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{samples\/egstockall.png}\n    \\caption{STOCK: Data and Generated Samples. The captions marked in red were judged as incorrect by human annotators. (Best viewed in color)}\n    \\label{fig:stockeg1}\n\\end{figure*}\nFigure \\ref{fig:stockeg1} shows some generated samples on STOCK dataset. \n\n\n\n\n\n\n\n\\subsection{Validation Split Results}\nTables \\ref{tab:synth6_results_val} and \\ref{tab:stock_results_val} show automated metrics on the validation split.\n\n\\begin{table}[]\n    \\centering\n    \\footnotesize\n    \\begin{tabular}{@{}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}lll@{}}\n        \\toprule\n        \\bf Method & \\bf PPL & \\bf Bleu-3\/4 & \\bf Cider & \\bf Rouge  & \\bf BERT  \\\\ \\midrule\n        \\textsc{TRUCE}{}  & $9.02$ & $0.61\/0.50$ & $1.92$ & $0.74$ & $0.76$   \\\\  \n        \\textsc{FcEnc}{}  & $9.66$ & $0.41\/0.34$ & $1.17$ & $0.63$ & $0.57$ \\\\  \n        \\textsc{LstmEnc}{}  & $7.5$ & $0.43\/0.35$ & $1.39$ & $0.63$ & $0.63$  \\\\  \n        \\textsc{ConvEnc}{}  & $7.6$  &  $0.63\/0.53$ & $1.99$ & $0.73$ & $0.71$  \\\\  \n        \\textsc{FftEnc}{}  & $15.7$ & $0.39\/0.29$ & $1.26$ & $0.61$ & $0.62$  \\\\  %\n        \\textsc{NearNbr}{} & NA & $0.32\/0.19$ & $0.68$ & $0.50$ & $0.48$  \\\\ \n        \\bottomrule\n        \\end{tabular}\n        \\caption{\\footnotesize\n        Results on validation split for SYNTH dataset. \n        }\n        \\label{tab:synth6_results_val}\n\\end{table}\n\n\n\n\n\\begin{table}[]\n    \\centering\n    \\footnotesize\n\\begin{tabular}{@{}l@{\\hskip 0.05in}l@{\\hskip 0.09in}l@{\\hskip 0.09in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}l@{\\hskip 0.05in}ll@{}}\n\\toprule\n\\textbf{Method} & \\textbf{Bleu-3\/4} & \\textbf{Cider} & \\textbf{Rouge} & \\textbf{BERT}  \\\\\n\\midrule\n\\textsc{TRUCE}{}(Ours) &  0.36 \/ 0.22 & 0.40 & 0.50 & 0.58  \\\\\n\\textsc{FcEnc}{}  & 0.32 \/ 0.20 & 0.38 & 0.47 & 0.56  \\\\\n\\textsc{Lstm-Multi}{} & 0.34 \/ 0.18 & 0.33 & 0.51 & 0.61 \\\\\n\\textsc{Conv-Multi}{}  & 0.34 \/ 0.17 & 0.35 & 0.50 & 0.60 \\\\\n\\textsc{FftEnc}{}  &  0.32 \/ 0.18 & 0.36 & 0.48 & 0.56 \\\\\n\\textsc{NearNbr}{}  & 0.11 \/ 0.05 & 0.11 & 0.27 & 0.37  \\\\\n\\bottomrule\n\\end{tabular}\n    \\caption{\\footnotesize Results on validation split of STOCK data.\n    }\n    \\label{tab:stock_results_val}\n\\end{table}\n\n\n\\subsection{Analyzing Learned Modules}\nFigure \\ref{fig:loc_module_viz} shows visualization of a learned \\emph{locate} module when model is trained on SYNTH data.\n\n\\begin{figure*}[]\n    \\includegraphics[width=0.75\\textwidth]{figures\/temporal_synth_begin.png}\n    \\caption{Visualizing a learned 'locate' module. Our locate modules are weighted mixtures of equally spaced Gaussians. The module's weight on each of these components is shown, along with the resulting distribution -- the module being visualized seems to have learned to focus on middle part of the time series.}\n    \\label{fig:loc_module_viz}\n\\end{figure*}\n\n\n\\subsection{Additional Ablation Studies}\nWe consider following ablations for the \\textsc{TRUCE}{}:\n(1) \\textsc{TRUCE}\\textsc{-NoInf}: Train \\textsc{TRUCE}{} without the use of inference network\n(2) \\textsc{TRUCE}\\textsc{-NoHeur}: Train \\textsc{TRUCE}{} without the use of heuristic labels\n\n\n\n\n\n\n\n\\section{Additional Training Details}\n\nWe code our models in Pytorch library. \n\n\\subsection{Heuristic Labels}\nList of the keywords selected for use in constructing heuristic labels: \\\\\n--- `locate':[`beginning',`middle',`end',`throughout'], \\\\ \n--- `pattern':[`increase',`decrease',`peak',`flat',`dip']\n\n\n\\subsection{Optimizer}\nWe use Adam optimizer with initial learning rate of $1e-4$.\n\n\\subsection{Infrastructure}\nWe use GeForce RTX 2080 GPUs for training models.\n\n\n\\subsection{Additional method details} \nWhile the automated metrics are only moderately correlated with quality, we found it reasonable to select best model configurations based on the Bleu-4 scores on validation split. \nThe model configurations, when using STOCK dataset, are as follows:\n\\begin{itemize}\n    \\item LSTM Decoder: Token embedding size and  hidden size are varied from the set \\{32,64,128,256\\}. \n    \\item Weight for the classification loss term (in case of multitask objective in baselines): Following three weights of classification loss (i.e. the weight of the classification term which is present in addition to the conditional language modeling objective) are tried: 0.3,1.0,3.0. \n    \\item \\textsc{TRUCE}{}: Program embedding encoding size. Number of module instantiations are varied in following ranges:\n    \\begin{itemize}\n        \\item LOCATE: 4-7 instantiations of each of locate \n        \\item PATTERN: 6-10 instantiations of each of trend\n        \\item COMBINE: 1 instantiation\n    \\end{itemize}\n    - Module embedding is varied in the set \\{9,18,36,72\\}. Final module embedding size is 18. \\\\\n    - Number of trainable parameters: 466K (excluding inference network parameters since inference network is used only at training and not at prediction time)\n    \\item \\textsc{FftEnc}{}: \n    - Number of trainable parameters: 462K\n    - Construct features based on numpy:fft:rfft functions, using real as well as imaginary components from the transformation.\n    \\item \\textsc{ConvEnc}{}: \n    Number of trainable parameters: 463K\n    \\item \\textsc{LstmEnc}{}: \n    - Representation: A single LSTM step involves feeding an embedding of the input and using the previous step's hidden state. To construct an input embedding of size $h$ for a given number $x_t$, we simply repeat the number $x_t$ for $h$ times. \\\\\n    - Number of trainable parameters: 464K \n     \\item \\textsc{NearNbr}{}: \n     We experiment with L2 distance and L1 distance, and observed former to perform better in terms of automated as well as human evaluations. \n\\end{itemize}\n\n\n\n\\section{Learning and Inference}\n\nThe log probability of observing a natural language description $y$ of the time series $x$ under the model can be written as follows:\n\\begin{equation*}\n    \\log p(y|x) = \\log \\sum_{z \\in \\mathcal{Z}} p_\\phi(z|x)p_\\theta(y|z)\n\\end{equation*}\nwhere $\\mathcal{Z}$ is the set of all possible programs, and  $\\theta$ and $\\phi$ are learnable model parameters.\nThe model is trained to maximize the log likelihood of the the observed descriptions conditioned on the corresponding time series data. Since the programs $z$ are unobserved at training, we must marginalize over all possible values of $z$.\n\\vspace{2mm}\n\n\n\n\\noindent \\textbf{Inference Network:}\nThe space of programs we currently employ is relatively small (about 20-60 number of programs), which makes it feasible to marginalize over the program space. However, any future work expanding the space of programs might run into feasibility issues when computing the exact likelihood. In such cases, we can perhaps resort to variational learning to optimize a lower bound to the likelihood by drawing samples from an inference network. \n\nAdditionally, use of inference networks can provide a useful inductive bias by using the observed text descriptions to guide the model learning. For example, words `increase' and `begin' in a caption could inform the inference network about a high chance of the presence of an increase pattern in the initial duration of the time series. \nWe observe that training with inference networks results in models which can better capture the patterns in data. \nNote that the inference network is used only for model training. At test time, we sample from the learned prior and decoder without regard to the inference network.\n\n\nWe use amortized variational learning by introducing an inference network $q_\\gamma$, and train the model to maximize the following evidence lower-bound (ELBO):\n\\begin{align*}\n    \\mathbb{E}_{z \\sim q_\\gamma(z|y)} [\\log p_\\theta(y|z)] - \\text{KL}(q_\\gamma(z|y)||p_\\phi(z|x))\n\\end{align*}\n\n\n\nWe use a BiLSTM encoder to encode the caption $y$, followed by a classification layer to predict the approximate posterior $q_\\gamma(z|y)$ over the programs.\nWe also considered fine-tuning of a pre-trained BERT model instead of BiLSTM, but did not observe any improvement in the model performance during the initial experiments.\n\\vspace{2mm}\n\n\n\n\\noindent \\textbf{Optimization:}\n$\\theta$, $\\phi$ and $\\gamma$\nare learned through directly optimizing the ELBO term. \nWe compute the exact reconstruction and the KL-terms -- the number of programs in our case is small enough to enable this exact computation (typically we consider 6-10 instances each of \\emph{pattern} and \\emph{locate} module). \n\n\n\\section*{Ethics Statement}\nWe collect natural language annotations from a crowd-sourcing platform. We do not collect or store any person identifiable information. We did not observe any toxic or hateful language in our dataset -- though researchers working on the dataset in future are advised due caution since the annotations are crowd-sourced, and might reflect certain biases.\nOur work primarily performs experiments on text generation in English language. \nOur method generates high precision text output -- much higher than all the baselines considered. However, it is still not perfect, and must be used cautiously in any real world deployment.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe spin group $Spin(n)$ is the universal cover of the special orthogonal\ngroup $SO(n)$. The spin$^{c}$ group $Spin^{c}(n)$ is the central extension \nSpin(n)\\times _{\\mathbb{Z}_{2}}U(1)$ of $SO(n)$ by the circle group $U(1)$.\nIn the first part of the paper we introduce a pair $F=(\\alpha ,\\gamma )$ of\nsecondary cohomology operations, which is applied to construct the integral\ncohomology rings of the classifying spaces $B_{Spin^{c}(n)}$ and \nB_{Spin(n)} $.\n\nThe $\\func{mod}2$ cohomology of the space $B_{Spin(n)}$ has been determined\nby Borel \\cite{B1} for $n\\leq 10$, and completed by Quillen in \\cite{Q}.\nConcerning the integral cohomology $H^{\\ast }(B_{Spin(n)})$ partial\ninformation are known \\cite{Web,Web1}. In \\cite{Th} Thomas calculated the\ncohomology $H^{\\ast }(B_{Spin(n)})$ in the stable range $n=\\infty $, but his\nresult relies on two sequences $\\{\\Phi _{i}\\},\\{\\Psi _{i}\\}$ of\nindeterminacies. Another inspiring approach is due to Benson and Wood. By\ncomputing with the Weyl invariants a partial presentation of the ring \nH^{\\ast }(B_{Spin(n)})$ is formulated in \\cite[Theorem 11.1]{BW}, where the\ndetermination of explicit generators and relations is noted to be a rather\ndaunting task. For the difficulties that one encounters when computing with\nthe cohomologies of the classifying space $B_{G}$ of a Lie group $G$, we\nrefer to Feshbach \\cite[Final remarks]{Fe}. In our approach the pair \nF=(\\alpha ,\\gamma )$ of cohomology operators will make the structure of the\nring $H^{\\ast }(B_{Spin(n)})$ appearing in a new light, see Remarks 8.6 and\n9.5.\n\nKnowing the integral cohomology of the classifying space $B_{G}$ of a Lie\ngroup $G$ has direct consequences in geometry and invariant theory. In\nparticular, assuming that a minimal system $\\{q_{1},\\cdots ,q_{m}\\}$ of\ngenerators of the ring $H^{\\ast }(B_{G})$ has been specified, one can\nintroduce the characteristic classes for a principle $G$ bundle $\\xi $ over\na space $X$ by letting\n\n\\begin{quote}\n$q_{r}(\\xi ):=f_{\\xi }^{\\ast }(q_{r})\\in H^{\\ast }(X)$, $1\\leq r\\leq m$,\n\\end{quote}\n\n\\noindent where $f_{\\xi }:X\\rightarrow B_{G}$ is the classifying map of the\nbundle $\\xi $. One obtains also the basic Weyl invariants of the group $G$\nby setting\n\n\\begin{quote}\n$d_{r}:=B_{t}^{\\ast }(q_{r})\\in H^{\\ast }(B_{T})$, $1\\leq r\\leq m$,\n\\end{quote}\n\n\\noindent where $T$ is a maximal torus on $G$, and the map \nB_{t}:B_{T}\\rightarrow B_{G}$ is induced by the inclusion $T$ $\\subset G$.\nFor the classical groups $G=U(n),SO(n)$ and $Sp(n)$ these stories have been\nwell understood by the 1950's \\cite{B,BH}. In the second part of the paper\nwe complete the projects for the spinor groups $Spin(n)$ and $Spin^{c}(n)$.\n\nIn mathematical physics the Postnikov tower anchored by the classifying\nspace $B_{SO(n)}$ is\n\n\\begin{quote}\n$\\cdots \\rightarrow $ $B_{Fivebrane(n)}\\rightarrow B_{String(n)}\\rightarrow\nB_{Spin(n)}\\rightarrow B_{SO(n)}$,\n\\end{quote}\n\n\\noindent indicating that the calculation of ring $H^{\\ast }(B_{Spin(n)})$\nis a necessary step towards the integral cohomologies of the further spaces \nB_{String(n)}$ and $B_{Fivebrane(n)}$ in the tower. In addition, the\nintegral cohomology $H^{\\ast }(B_{Spin(n)})$ is essential to form the\ncohomology of the Thom spectrum $\\{M_{Spin(n)},n\\geq 3\\}$ \\cite{ABP}, is\nexplicitly requested by the study of the complex cobordism of the\nclassifying space $B_{Spin(n)}$ \\cite[Corollary 1.4]{T}, and by the\nunderstanding of the total Chern class of the complex spin presentation of\nthe group $Spin(n)$ \\cite{ABS,Q}.\n\n\\bigskip\n\n\\noindent \\textbf{Remark 1.1.} If $3\\leq n\\leq 6$ there hold the following\ngroup isomorphisms\n\n\\begin{quote}\n$Spin(3)=SU(2)$, $Spin(4)=SU(2)\\times SU(2)$,\n\n$Spin(5)=Sp(2)$, $Spin(6)=SU(4)$,\n\\end{quote}\n\n\\noindent where $SU(k)$ is the special unitary group of rank $k$, $Sp(2)$ is\nthe symplectic group of rank $2$. Therefore, we may assume $n\\geq 7$ to\nexclude these cases.$\\square $\n\n\\section{The main results}\n\nFor a topological space $X$ let $Sq^{k}$ be the Steenrod squares on the \n\\func{mod}2$ cohomology algebra $H^{\\ast }(X;\\mathbb{Z}_{2})$, and denote by \n$\\delta _{m}$ the Bockstein homomorphism from the $\\func{mod}m$ cohomology \nH^{r}(X;\\mathbb{Z}_{m})$ into the integral cohomology $H^{r+1}(X)$. For the\nhomomorphisms of coefficients groups\n\n\\begin{quote}\n$\\theta :\\mathbb{Z}_{2}\\rightarrow \\mathbb{Z}_{4}$ by $\\theta (1)=2$, and \n\\rho _{m}:\\mathbb{Z}\\rightarrow \\mathbb{Z}_{m}$ by $\\rho _{m}(1)=1$,\n\\end{quote}\n\n\\noindent the same notion are applied to denote their induced maps on the\ncohomologies.\n\nLet $\\mathcal{B}:$ $H^{2r}(X;\\mathbb{Z}_{2})\\rightarrow H^{4r}(X;\\mathbb{Z\n_{4})$ be the Pontryagin square \\cite{BT}. For an even degree class $u\\in\nH^{2r}(X;\\mathbb{Z}_{2})$ there holds the following universal relation\n\n\\begin{enumerate}\n\\item[(2.1)] $\\delta _{2}(u\\cup u)=2\\delta _{4}\\mathcal{B}(u)$ on \nH^{4r+1}(X)$ (see (3.1)).\n\\end{enumerate}\n\n\\noindent \\textbf{Definition 2.1.} The space $X$ is called $\\delta _{2}\n\\textsl{--formal} if $\\delta _{2}(u\\cup u)=0$ for every $u\\in H^{2r}(X\n\\mathbb{Z}_{2})$, $r\\geq 1$.$\\square $\n\n\\bigskip\n\nIt follows from (2.1) that, if $X$\\ is a space whose integral cohomologies \nH^{\\ast }(X)$ in degrees $4r+1$ have no torsion element of order $4$, then \nX $\\ is $\\delta _{2}$--formal. In particular, all the $1$--connected Lie\ngroups, the classifying spaces $B_{SO(n)}$, $B_{Spin(n)}$ and \nB_{Spin^{c}(n)}$ of our concern, as well as the Thom spectrum \n\\{M_{Spin(n)},n\\geq 3\\}$, are all examples of $\\delta _{2}$--formal spaces.\n\nTo introduce the promised operations observe that the Bockstein operator \nSq^{1}=\\rho _{2}\\circ \\delta _{2}$ defines the following decomposition on\nthe $\\mathbb{Z}_{2}$ space $H^{\\ast }(X;\\mathbb{Z}_{2})$\n\n\\begin{quote}\n$H^{\\ast }(X;\\mathbb{Z}_{2})=\\ker Sq^{1}\\oplus S_{2}^{\\ast }(X)$ with \nS_{2}^{\\ast }(X)=H^{\\ast }(X;\\mathbb{Z}_{2})\/\\ker Sq^{1}$.\n\\end{quote}\n\n\\noindent \\textbf{Theorem A.} \\textsl{For any }$\\delta _{2}$\\textsl{--formal\nspace }$X$\\textsl{\\ there exists a unique pair of cohomological operations}\n\n\\begin{quote}\n$F:H^{2r}(X;\\mathbb{Z}_{2})\\rightarrow H^{4r}(X;\\mathbb{Z}_{4})\\times\nS^{4r}(X;\\mathbb{Z}_{2})$\\textsl{,}\n\\end{quote}\n\n\\noindent \\textsl{written }$F(u)=(\\alpha (u),\\gamma (u))$\\textsl{,} $u\\in\nH^{2r}(X;\\mathbb{Z}_{2})$\\textsl{, that is characterized by the following\nthree properties:}\n\n\\begin{quote}\n\\textsl{i) }$\\alpha (u)\\in \\func{Im}\\rho _{4}$\\textsl{;}\n\n\\textsl{ii)} $\\mathcal{B}(u)=\\alpha (u)+\\theta (\\gamma (u))$\\textsl{;}\n\n\\textsl{iii)} $Sq^{1}(\\gamma (u))=Sq^{2r}Sq^{1}(u)+u\\cup Sq^{1}(u)$\\textsl{.}\n\\end{quote}\n\nThe uniqueness assertion of Theorem A implies that the properties i), ii)\nand iii) can be taken as \\textsl{an axiomatic definition} of the pair \nF=(\\alpha ,\\gamma )$ of operators. In particular, since $Sq^{1}$ injects on \nS_{2}^{\\ast }(X)$ while $\\gamma (u)\\in S_{2}^{\\ast }(X)$, the operator \n\\gamma $ is determined uniquely by the relation iii). Thus, applying $Sq^{1}$\nto both sides verifies at once the following equalities useful to evaluate \n\\gamma $.\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 2.2.} \\textsl{For any }$\\delta _{2}$\\textsl\n--formal space }$X$\\textsl{\\ and }$u_{1}$\\textsl{,}$u_{2}\\in H^{\\ast }(X\n\\mathbb{Z}_{2})$ \\textsl{with} $\\deg u_{i}=2r_{i}$\\textsl{,} \\textsl{one has}\n\n\\begin{quote}\n\\textsl{i)} $\\gamma (u_{1}+u_{2})\\equiv \\gamma (u_{1})+\\gamma\n(u_{2})+u_{1}\\cup u_{2}\\func{mod}\\ker Sq^{1}$\n\n\\textsl{ii)} $\\gamma (u_{1}\\cup u_{2})\\equiv u_{1}^{2}{}\\cup \\gamma\n(u_{2})+u_{2}^{2}{}\\cup \\gamma (u_{1})+u_{1}\\cup Sq^{1}u_{1}\\cup\nSq^{2r_{2}-1}u_{2}$\n\n$\\qquad +u_{2}\\cup Sq^{1}u_{2}\\cup Sq^{2r_{1}-1}u_{1}\\func{mod}\\ker Sq^{1}$.\n\\square $\n\\end{quote}\n\nThe operator $\\gamma $ can be iterated to yield the following notion.\n\n\\bigskip\n\n\\noindent \\textbf{Definition 2.3. }For an even degree cohomology class $u\\in\nH^{2r}(X;\\mathbb{Z}_{2})$ of a $\\delta _{2}$--formal space $X$, the sequence \n$\\left\\{ u^{(0)},u^{(1)},u^{(2)},\\cdots \\right\\} $ on $H^{\\ast }(X;\\mathbb{Z\n_{2})$ defined recursively by $u^{(0)}=u$, $u^{(k+1)}=\\gamma (u^{(k)})$ is\ncalled \\textsl{the derived sequence} \\textsl{of} $u$.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 2.4.} Recall that $H^{\\ast }(B_{SO(n)};\\mathbb{Z\n_{2})=\\mathbb{Z}_{2}[w_{2},\\cdots ,w_{n}]$, where $w_{i}$ is the $i^{th}$\nStiefel--Whitney class of the canonical real $n$--bundle on $B_{SO(n)}$. For \n$u=w_{2r}$ solving the equation iii) of Theorem A using\ncoefficient--comparison yields\n\n\\begin{enumerate}\n\\item[(2.2)] $\\gamma (w_{2r})=w_{4r}+w_{2}w_{4r-2}+\\cdots +w_{2r-2}w_{2r+2}$.\n\\end{enumerate}\n\n\\noindent Since the $Sq^{k}$ action on $H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$\nis determined by the Wu--formula, the formula (2.2), together with the\nalgorithms given by Corollary 2.2, suffices to evaluate the $\\gamma $ action\non $H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$. For example when $u=w_{2}$ we get\nthat\n\n\\begin{quote}\n$w_{2}^{(1)}=w_{4},$\n\n$w_{2}^{(2)}=w_{8}+w_{2}w_{6},$\n\n\nw_{2}^{(3)}=w_{16}+w_{2}w_{14}+w_{4}w_{12}+w_{6}w_{10}+w_{2}w_{6}w_{8}+w_{4}w_{6}^{2}+w_{2}w_{7}^{2} \n$\n\n$\\qquad\n+w_{3}^{2}(w_{10}+w_{2}w_{8}+w_{4}w_{6})+w_{2}^{2}(w_{12}+w_{2}w_{10}+w_{4}w_{8}), \n$\n\\end{quote}\n\n\\noindent and in general, if $2^{k}\\leq n$, that\n\n\\begin{quote}\n$w_{2}^{(k)}=w_{2^{k}}+w_{2}w_{2^{k}-2}+\\cdots +w_{2^{k-1}-2}w_{2^{k-1}+2}+$\nhigher terms.\n\\end{quote}\n\n\\noindent In contrast to the structure of the algebra $H^{\\ast }(B_{SO(n)}\n\\mathbb{Z}_{2})$ as a module over the Steenrod algebra \\cite[Lemma 3.2]{PW},\nthese calculation reveal a striking property about the operator $\\gamma $:\nmodulo the decomposable elements the derived sequence of $w_{2}$ consists of\nall the $2$--power Stiefel--Whitney classes:\n\n\\begin{quote}\n$\\{w_{2}^{(0)},w_{2}^{(1)},w_{2}^{(2)},\\cdots \\}\\equiv \\{w_{2},w_{4},\\cdots\n,w_{2^{l(n)}},0,\\cdots \\}$, $l(n)=\\left[ \\ln n\\right] $.\n\\end{quote}\n\n\\noindent This sequence will play a central role in the construction and\ncalculation of this paper.$\\square $\n\n\\bigskip\n\nAn essential point of the operator $\\alpha $ is that it always admits an\nintegral lift by i) of Theorem A. That is, it can be factored into $\\rho\n_{4}\\circ f_{\\alpha }$ for some $f_{\\alpha }:H^{2r}(X;\\mathbb{Z\n_{2})\\rightarrow H^{4r}(X)$. In the case $X=B_{SO(n)}$ of our interest a\ncanonical choice of such an integral lift $f_{\\alpha }$ can be made\nexplicitly. Recall from Brown \\cite{Br} and Feshbach \\cite{Fe} that the\nintegral cohomology ring of $B_{SO(n)}$ is\n\n\\begin{enumerate}\n\\item[(2.3)] $H^{\\ast }(B_{SO(n)})=\\left\\{ \n\\begin{tabular}{l}\n$\\mathbb{Z}[p_{1},p_{2},\\cdots ,p_{\\left[ \\frac{n-1}{2}\\right]\n},e_{n}]\\oplus \\tau (B_{SO(n)})$ if $n$ is even; \\\\ \n$\\mathbb{Z}[p_{1},p_{2},\\cdots ,p_{\\left[ \\frac{n-1}{2}\\right] }]\\oplus \\tau\n(B_{SO(n)})$ if $n$ is odd\n\\end{tabular\n\\right. $\n\\end{enumerate}\n\n\\noindent where $\\tau (X)$ denotes the torsion ideal of the integral\ncohomology $H^{\\ast }(X)$ of a complex $X$, $2\\tau (B_{SO(n)})=0$, and where \n$p_{i}$ (resp. $e_{n}$) is the $i^{th}$ Pontryagin class (resp. the Euler\nclass) of the canonical real $n$--bundle on $B_{SO(n)}$. With respect to\n(2.3) Thomas introduced in \\cite[\\S 3]{Th} an operator $f:H^{r}(B_{SO(n)}\n\\mathbb{Z}_{2})\\rightarrow H^{2r}(B_{SO(n)})$, i.e. \\textsl{the integral\nrepresentation}, by the following practical rules:\n\n\\begin{enumerate}\n\\item[(2.4)] $f(u):=\\left\\{ \n\\begin{tabular}{l}\n$p_{r}$, $\\delta _{2}(Sq^{2r}w_{2r+1})$ or $e_{n}^{2}$ if $u=w_{2r},w_{2r+1}$\nor $w_{n}$ if $n$ is even; \\\\ \n$f(w_{i_{1}})\\cdots f(w_{i_{k}})$ if $u=w_{i_{1}}\\cdots w_{i_{k}}$ is a\nmonomial; \\\\ \n$f(u_{1})+\\cdots +f(u_{k})$ if $u=$ $u_{1}+\\cdots +u_{k}$\n\\end{tabular\n\\right. $\n\\end{enumerate}\n\n\\noindent where $2r,2r+1<\\left[ \\frac{n-1}{2}\\right] $, and where the $u_{i}\n's are distinct monomials in $w_{2},\\cdots ,w_{n}$. Based on Theorem A we\nshall show that\n\n\\bigskip\n\n\\noindent \\textbf{Theorem B.} \\textsl{The pair }$(f,\\gamma )$\\textsl{\\ of\noperators on }$H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$ \\textsl{satisfies,} \n\\textsl{for any }$u\\in S_{2}^{\\ast }(B_{SO(n)})$\\textsl{, that}\n\n\\begin{quote}\n\\textsl{i)} $\\mathcal{B}(u)=\\rho _{4}(f(u))+\\theta (\\gamma (u))$\\textsl{,}\n\n\\textsl{ii)} $Sq^{1}(\\gamma (u))=Sq^{2r}Sq^{1}(u)+u\\cup Sq^{1}(u)$.\n\\end{quote}\n\n\\noindent \\textbf{Example 2.5.} For $u=w_{2r}\\in S_{2}^{\\ast }(B_{SO(n)})$\nwe have $f(w_{2r})=p_{r}$ by the definition of $f$. Substituting this and\n(2.2) into i) of Theorem B we obtain the formula\n\n\\begin{quote}\n$\\mathcal{B}(w_{2r})=\\rho _{4}(p_{r})+\\theta (w_{4r}+w_{2}w_{4r-2}+\\cdots\n+w_{2r-2}w_{2r+2})$\n\\end{quote}\n\n\\noindent implying that the $\\func{mod}4$ reductions of the Pontryagin\nclasses of a manifold are homotopy invariants. This formula was originally\nobtained by W. Wu \\cite{Wu} by computing with the Schubert cells on \nB_{SO(n)}$. S.S. Chern suggested a different approach which was implemented\nby Thomas in \\cite[Theorem C]{Th1}.\n\nConcerning this topic property ii) of Theorem A may be called \\textsl{the} \n\\textsl{generalized Wu--formula }on $\\delta _{2}$--formal spaces.$\\square $\n\n\\bigskip\n\nTurning to our main concerns the classifying spaces $B_{Spin(n)}$ and \nB_{Spin^{c}(n)}$ fit into the fibered sequences\n\n\\begin{enumerate}\n\\item[(2.5)] $\\mathbb{C}P^{\\infty }\\overset{i}{\\rightarrow }B_{Spin^{c}(n)\n\\overset{\\pi }{\\rightarrow }B_{SO(n)}$ and\n\n\\item[(2.6)] $U(1)\\rightarrow B_{Spin(n)}\\overset{\\psi }{\\rightarrow \nB_{Spin^{c}(n)}\\overset{\\iota }{\\rightarrow }\\mathbb{C}P^{\\infty }$,\n\\end{enumerate}\n\n\\noindent where the maps $\\pi $ and $\\iota $ are induced by the obvious\nepimorphisms\n\n\\begin{quote}\n$Spin(n)\\times _{\\mathbb{Z}_{2}}U(1)\\rightarrow SO(n)$ and $Spin(n)\\times _\n\\mathbb{Z}_{2}}U(1)\\rightarrow U(1)$,\n\\end{quote}\n\n\\noindent respectively. Let $\\{w_{2},w_{2}^{(1)},w_{2}^{(2)},\\cdots \\}$ be\nthe derived sequence of the second Stiefel Whitney class $w_{2}$. Applying\nthe operator $f$ gives rise to the sequence $\\{f(w_{2}),$ \nf(w_{2}^{(1)}),f(w_{2}^{(2)}),\\cdots \\}$ of integral cohomology classes on \nB_{SO(n)}$. By examining the $\\pi ^{\\ast }$ images of these two sequences in\nthe cohomology of $B_{Spin^{c}(n)}$ we single out a set of new generators of\nthe ring $H^{\\ast }(B_{Spin^{c}(n)})$ in the following result, where $x$\\\ndenotes the Euler class of the Hopf line bundle $\\lambda $\\ on $\\mathbb{C\nP^{\\infty }$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem C. }\\textsl{There exists a unique sequence }\n\\left\\{ q_{r}\\QTR{sl}{,\\ }r\\geq 0\\right\\} $ \\textsl{of} \\textsl{integral} \n\\textsl{cohomology classes on }$B_{Spin^{c}(n)}$\\textsl{, }$\\deg\nq_{r}=2^{r+1}$\\textsl{,} \\textsl{that satisfies the following system}\n\n\\begin{quote}\n\\textsl{i)} $q_{0}=\\iota ^{\\ast }(x)$\\textsl{,}\n\n\\textsl{ii)} $\\rho _{2}(q_{r})=\\pi ^{\\ast }(w_{2}^{(r)})$\\textsl{; }\n\n\\textsl{iii)} $2q_{r+1}+q_{r}^{2}=\\pi ^{\\ast }f(w_{2}^{(r)})$\\textsl{, }\nr\\geq 0$\\textsl{.}\n\\end{quote}\n\nFor an integer $n\\geq 7$ (see Remark 1.1) we set $h(n)=\\left[ \\frac{n-1}{2\n\\right] $, and let $\\theta _{n}\\in H^{\\ast }(B_{Spin^{c}(n)})$ be the Euler\nclass of the complex bundle $\\xi _{n}$ associated to the complex spin\npresentation $Spin^{c}(n)\\rightarrow U(2^{h(n)})$ (\\cite{ABS,HK}). Regarding\nthe cohomology $H^{\\ast }(B_{Spin^{c}(n)})$ as a module over its subring \n\\pi ^{\\ast }H^{\\ast }(B_{SO(n)})$, our main result presents the cohomology \nH^{\\ast }(B_{Spin^{c}(n)})$ by the unique sequence $\\left\\{ q_{r},r\\geq\n0\\right\\} $ obtained by Theorem C, together with the Euler class $\\theta\n_{n} $.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem D. }\\textsl{The cohomology of }$B_{Spin^{c}(n)}\n\\textsl{\\ has the presentation}\n\n\\begin{enumerate}\n\\item[(2.6)] $H^{\\ast }(B_{Spin^{c}(n)})=\\pi ^{\\ast }H^{\\ast\n}(B_{SO(n)})\\otimes \\mathbb{Z}[q_{0},q_{1},\\cdots ,q_{h(n)-1},\\theta\n_{n}]\/R_{n}$,\n\\end{enumerate}\n\n\\noindent \\textsl{in which }$R_{n}$ \\textsl{denotes the ideal generated by\nthe following elements}\n\n\\begin{quote}\n\\textsl{i)} $2q_{r+1}+q_{r}^{2}-\\pi ^{\\ast }f(w_{2}^{(r)})$\\textsl{,\\quad }\n0\\leq r\\leq h(n)-2\\QTR{sl}{;}$\n\n\\textsl{ii)} $\\pi ^{\\ast }\\delta _{2}(z)\\cup q_{r}-\\pi ^{\\ast }\\delta\n_{2}(z\\cup w_{2}^{(r)}),$ $0\\leq r\\leq h(n)-1$\\textsl{;}\n\n\\textsl{iii)} $4(-1)^{h(n)}\\theta _{n}+q_{h(n)-1}^{2}-a_{n}$\\textsl{,}\n\\end{quote}\n\n\\noindent \\textsl{where} $z\\in H^{\\ast }(B_{SO(n)},\\mathbb{Z}_{2})$\\textsl{,\nand where }$a_{n}\\in \\pi ^{\\ast }H^{+}(B_{SO(n)})\\otimes \\mathbb{Z\n(q_{0},q_{1},\\cdots ,q_{h(n)-1})$ \\textsl{is an} \\textsl{element to be\nspecified in Lemma 5.3.}\n\n\\bigskip\n\nLeaving out $\\otimes $ for notational simplicity the relations i) and ii) of\nTheorem D are inherited from the properties iii) and ii) of Theorem C. They\nexpress the relationship between the two sequences $\\left\\{ q_{r},r\\geq\n0\\right\\} $ and $\\left\\{ \\pi ^{\\ast }p_{r},r\\geq 1\\right\\} $ of integral\ncohomology classes on $B_{Spin^{c}(n)}$, and deal with the product between\nthe free part and the torsion ideal $\\tau (B_{Spin^{c}(n)})$ of the ring \nH^{\\ast }(B_{Spin^{c}(n)})$, respectively.\n\nAn outline of the paper is as follows. Sections \\S 3 to \\S 5 are devoted to\nshow Theorems A--D. The calculation is extended in Section \\S 6 to obtain a\nsimilar presentation of the ring $H^{\\ast }(B_{Spin(n)})$ in Theorem D'. The\nremaining sections constitute applications of Theorem D'. We determine in \\S\n8 the ring of integral Weyl invariants of the group $Spin(n)$; introduce in \n\\S 9 the spin characteristic classes for the spin vector bundles. In spin\ngeometry the Spin characteristic classes can play roles that may not be\nreplaced by the regular characteristic classes. Section \\S 10 is devoted to\npresent such examples.\n\nEffective computability with characteristic classes is essential in geometry\nand invariant theory. To illustrate the algorithmic nature of the operators\nand results developed in this paper a number of examples are included. In\nparticular, in a concrete situation Theorems D and D' are directly\napplicable to present the rings $H^{\\ast }(B_{Spin^{c}(n)})$ and $H^{\\ast\n}(B_{Spin(n)})$ by a minimal system of explicit generators and relations,\nsee in Examples 5.4 and 6.5; Theorems 7.7 and 7.8 are ready to deduce the\nintegral Weyl invariants of the groups $Spin^{c}(n)$ and $Spin(n)$, which\nare shown in Examples 7.6 and 7.9.\n\n\\section{The cohomology operation $F=(\\protect\\gamma ,\\protect\\alpha )$}\n\nFor a $\\func{mod}2$ cohomology class $u\\in H^{2r}(X;\\mathbb{Z}_{2})$ of a CW\ncomplex $X$ the Pontryagin square $\\mathcal{B}(u)\\in H^{4r}(X;\\mathbb{Z\n_{4}) $ can be defined by the formula\n\n\\begin{quote}\n$\\mathcal{B}(u)\\equiv \\rho _{4}(\\widetilde{u}\\cup _{0}\\widetilde{u}+\\delta \n\\widetilde{u})\\cup _{1}\\widetilde{u})$ (see \\cite{BT}),\n\\end{quote}\n\n\\noindent where $\\widetilde{u}$ is an integral lift of $u$ in the cochain\ncomplex $C^{\\ast }(X;\\mathbb{Z})$ associated to $X$, and $\\cup _{i}$ denotes\nthe $i^{th}$ cup product on $C^{\\ast }(X;\\mathbb{Z})$. Based on this formula\na co--chain level calculation verifies the following universal relations\n\n\\begin{enumerate}\n\\item[(3.1)] $\\delta _{2}(u\\cup u)=2\\delta _{4}(\\mathcal{B}(u))$ in \nH^{4r+1}(X)$;\n\n\\item[(3.2)] $\\rho _{2}\\delta _{4}\\mathcal{B}(u)=Sq^{2r}Sq^{1}u+u\\cup\nSq^{1}u $ in $H^{4r+1}(X;\\mathbb{Z}_{2})$.\n\\end{enumerate}\n\n\\noindent \\textbf{Proof of Theorem A.} Let $X$ be a $\\delta _{2}$ formal\nspace. For each $u\\in H^{2r}(X;\\mathbb{Z}_{2})$ the property $\\delta\n_{2}(u\\cup u)=0$ implies that $\\delta _{4}(\\mathcal{B}(u))\\in \\func{Im\n\\delta _{2}$ by (3.1). In view of the isomorphism $\\delta _{2}:S_{2}^{\\ast\n}(X)\\cong \\func{Im}\\delta _{2}$ there exists a unique $u_{1}\\in\nS_{2}^{4r}(X) $ so that\n\n\\begin{enumerate}\n\\item[(3.3)] $\\delta _{2}(u_{1})=\\delta _{4}(\\mathcal{B}(u))$.\n\\end{enumerate}\n\n\\noindent We can now formulate the desired operation $F=(\\alpha ,\\gamma\n):H^{2r}(X;\\mathbb{Z}_{2})\\rightarrow H^{4r}(X;\\mathbb{Z}_{4})\\times\nS^{4r}(X;\\mathbb{Z}_{2})$ by setting\n\n\\begin{quote}\n$\\gamma (u):=u_{1}$ and $\\alpha (u):=\\mathcal{B}(u)-\\theta (\\gamma (u))$.\n\\end{quote}\n\nApplying $\\rho _{2}$ to both sides of (3.3) we get by (3.2) that\n\n\\begin{quote}\n$Sq^{1}\\gamma (u)=Sq^{2r}Sq^{1}u+u\\cup Sq^{1}u$,\n\\end{quote}\n\n\\noindent showing property iii) of Theorem A. From $\\delta _{4}\\circ \\theta\n=\\delta _{2}$ and (3.3) we obtain\n\n\\begin{quote}\n$\\delta _{4}\\alpha (u)=\\delta _{4}(\\mathcal{B}(u)-\\theta (\\gamma\n(u)))=\\delta _{4}(\\mathcal{B}(u))-\\delta _{2}(\\gamma (u))=0$,\n\\end{quote}\n\n\\noindent implying $\\alpha (u)\\in \\func{Im}\\rho _{4}$ (i.e. property i) of\nTheorem A).\n\nSummarizing, the pair $F=(\\gamma ,\\alpha )$ of operators fulfills the\nproperties i), ii) and iii) of Theorem A, whose uniqueness comes directly\nfrom its definition.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Proof of Theorem B.} Let $f:H^{\\ast }(B_{SO(n)};\\mathbb{Z\n_{2})\\rightarrow H^{\\ast }(B_{SO(n)})$ be the map entailed in (2.4). It has\nbeen shown by Thomas \\cite[Lemma (3.9)]{Th} that for any $u\\in S_{2}^{\\ast\n}(B_{SO(n)})$ there exists a unique element $v\\in S_{2}^{\\ast }(B_{SO(n)})$\nso that\n\n\\begin{enumerate}\n\\item[(3.5)] $\\mathcal{B}(u)=\\rho _{4}(f(u))+\\theta (v)$.\n\\end{enumerate}\n\n\\noindent It suffices for us to show that $v=\\gamma (u)$. Since the space \nB_{SO(n)}$ is $\\delta _{2}$--formal applying $\\delta _{4}$ to both sides of\n(3.5) we get by (3.3) that\n\n\\begin{quote}\n$\\delta _{2}(\\gamma (u))=\\delta _{4}(\\theta (v))=\\delta _{2}(v)$ (since \n\\delta _{4}\\circ \\theta =\\delta _{2}$).\n\\end{quote}\n\n\\noindent With $\\gamma (u),v\\in S_{2}^{\\ast }(B_{SO(n)})$ while $\\delta _{2}$\ninjects on $S_{2}^{\\ast }(B_{SO(n)})$, we obtain $v=$ $\\gamma (u)$.$\\square $\n\n\\section{The proof of Theorem C}\n\nApplying the Serre spectral sequence to the fibration\n\n\\begin{enumerate}\n\\item[(4.1)] $\\mathbb{C}P^{\\infty }\\overset{i}{\\rightarrow }B_{Spin^{c}(n)\n\\overset{\\pi }{\\rightarrow }B_{SO(n)}$ (see (2.5))\n\\end{enumerate}\n\n\\noindent the cohomologies of the total space $B_{Spin^{c}(n)}$ with fields\ncoefficients have been computed. We begin by recalling the relevant results\ndue to Borel, Hirzebruch, Quillen, Harada and Kono. Let $\\mathbb{Z}_{0}$ be\nthe field of rationals and denote by $\\rho _{0}$ be the cohomology\nhomomorphism induced by the inclusion $\\mathbb{Z}\\subset \\mathbb{Z}_{0}$.\nSet $q_{0}:=\\iota ^{\\ast }(x)$, where $x$ is the Euler class of the Hopf\nline bundle $\\lambda $ on $\\mathbb{C}P^{\\infty }$.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 4.1.} \\textsl{If either }$p=0$\\textsl{\\ or }$p\\geq 3\n\\textsl{\\ is a prime, then }$H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{p})\n\\textsl{\\ is a free polynomial algebra on the generators}\n\n\\begin{enumerate}\n\\item[(4.2)] $\\rho _{p}(q_{0}),\\rho _{p}(\\pi ^{\\ast }p_{1}),\\cdots ,\\rho\n_{p}(\\pi ^{\\ast }p_{\\left[ \\frac{n-1}{2}\\right] }),$ \\textsl{and} $\\rho\n_{p}(\\pi ^{\\ast }e_{n})$ \\textsl{if} $n\\equiv 0\\func{mod}2$.\n\\end{enumerate}\n\n\\textsl{In addition, the map }$\\rho :H^{\\ast }(B_{Spin^{c}(n)})\\rightarrow\nH^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{0})\\times H^{\\ast }(B_{Spin^{c}(n)}\n\\mathbb{Z}_{2})$ \\textsl{by} $\\rho (z)=(\\rho _{0}(z),\\rho _{2}(z))$\\textsl{\\\ninjects.}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} Since the composition $\\iota \\circ i:\\mathbb{C\nP^{\\infty }\\rightarrow \\mathbb{C}P^{\\infty }$ (see (2.5) and (2.6)) is of\ndegree $2$ the class $i^{\\ast }(q_{0})=2x$ generates the algebra $H^{\\ast }\n\\mathbb{C}P^{\\infty };\\mathbb{Z}_{p})=\\mathbb{Z}_{p}[x]$ for all $p\\neq 2$.\nThe first assertion follows from the Leray--Hirsch property \\cite[p.231]{Hus}\nof the fibration (4.1) with $\\mathbb{Z}_{p}$ coefficients.\n\nAccording to Borel and Hirzebruch \\cite[30.6.]{BH} if $X$ is a space with \n2\\tau (X)=0$, then the map $\\rho :H^{\\ast }(X)\\rightarrow H^{\\ast }(X\n\\mathbb{Z}_{0})\\times H^{\\ast }(X;\\mathbb{Z}_{2})$ by $\\rho (z)=(\\rho\n_{0}(z),\\rho _{2}(z))$ injects. The second assertion follows from $2\\tau\n(B_{Spin^{c}(n)})=0$ by Harada and Kono \\cite[Theorem 3.7]{HK}.$\\square $\n\n\\bigskip\n\nTurning to the algebra $H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})$ the\ntransgression $\\sigma $ in the fibration (4.1) clearly satisfies $\\sigma\n(\\rho _{2}(x))=w_{3}$. Since $\\sigma $ commutes with the Steenrod squares\nthe standard relation $Sq^{2^{k}}x_{k}=x_{k+1}$ on $H^{\\ast }(\\mathbb{C\nP^{\\infty };\\mathbb{Z}_{2})$ implies that\n\n\\begin{enumerate}\n\\item[(4.3)] $\\sigma (x_{k+1})=Sq^{2^{k}}\\sigma (x_{k})$, $k\\geq 0$, where \nx_{k}:=(\\rho _{2}(x))^{2^{k-1}}$.\n\\end{enumerate}\n\n\\noindent For a subset $\\{a_{1},\\cdots ,a_{r}\\}$ of an algebra $A$ denote by \n$\\left\\langle a_{1},\\cdots ,a_{r}\\right\\rangle $ the ideal generated by \na_{1},\\cdots ,a_{r}$, and let $A\/\\left\\langle a_{1},\\cdots\n,a_{r}\\right\\rangle $ be the quotient algebra.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 4.2.} \\textsl{Let }$J=\\left\\langle \\sigma\n(x_{1}),\\cdots ,\\sigma (x_{h(n)})\\right\\rangle $\\textsl{, }$h(n)=\\left[ \n\\frac{n-1}{2}\\right] $\\textsl{, and let }$\\theta _{n}$\\textsl{\\ be the Euler\nclass of the presentation }$Spin^{c}(n)\\rightarrow U(2^{h(n)})$\\textsl{. The\n}\n\n\\begin{enumerate}\n\\item[(4.4)] $H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})=H^{\\ast }(B_{SO(n)}\n\\mathbb{Z}_{2})\/J\\otimes \\mathbb{Z}_{2}[\\theta _{n}]$\\textsl{.}\n\\end{enumerate}\n\n\\textsl{In particular, the torsion ideal }$\\tau (B_{Spin^{c}(n)})$\\textsl{\\\nof the ring }$H^{\\ast }(B_{Spin^{c}(n)})$\\textsl{\\ is}\n\n\\begin{enumerate}\n\\item[(4.5)] $\\tau (B_{Spin^{c}(n)})=\\pi ^{\\ast }\\tau (B_{SO(n)})\\otimes \n\\mathbb{Z}[\\theta _{n}]$.\n\\end{enumerate}\n\n\\noindent \\textbf{Proof.} Formula (4.4) goes to Harada and Kono \\cite\nTheorem 3.5]{HK} (see also Quillen \\cite[Theorem 6.5]{Q}). With $2\\tau\n(B_{SO(n)})=2\\tau (B_{Spin^{c}(n)})=0$ the map $\\pi $ induces the\ncommutative diagram\n\n\\begin{quote}\n\\begin{tabular}{llll}\n$0\\rightarrow $ & $\\tau (B_{SO(n)})$ & $\\overset{\\rho _{2}}{\\rightarrow }$ & \n$H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$ \\\\ \n& $\\pi ^{\\ast }\\downarrow $ &  & $\\pi ^{\\ast }\\downarrow $ \\\\ \n$0\\rightarrow $ & $\\tau (B_{Spin^{c}(n)})$ & $\\overset{\\rho _{2}}\n\\rightarrow }$ & $H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})\n\\end{tabular}\n\\end{quote}\n\n\\noindent in which both $\\rho _{2}$ inject. Since $\\tau (B_{Spin^{c}(n)})$\nis an ideal the map\n\n\\begin{quote}\n$h:\\pi ^{\\ast }\\tau (B_{SO(n)})\\otimes \\mathbb{Z}[\\theta _{n}]\\rightarrow\n\\tau (B_{Spin^{c}(n)})$ by $h(\\pi ^{\\ast }x\\otimes \\theta _{n})=\\pi ^{\\ast\n}x\\cup \\theta _{n}$\n\\end{quote}\n\n\\noindent is well defined, and gives rise to the isomorphism (4.5).\n\nIndeed, by (4.4) the composition $\\rho _{2}\\circ h$ injects, hence $h$\ninjects, too. On the other hand for any $x\\in \\tau (B_{Spin^{c}(n)})$ there\nexists $y\\in H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})$ so that $\\delta\n_{2}(y)=x$. By formula (4.4) the map $h$ also surjects.$\\square $\n\n\\bigskip\n\nAs the space $B_{SO(n)}$ is $\\delta _{2}$--formal the derived sequence \n\\{w_{2},w_{2}^{(1)},\\cdots \\}$ of the Stiefel Whitney class $w_{2}$ is\ndefined, see Example 2.4. Its relationship with the sequence $\\{\\sigma\n(x_{1}),\\sigma (x_{2}),\\cdots \\}$ defined by (4.3) is stated in the\nfollowing result.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 4.3.} \\textsl{In }$H^{\\ast }(B_{SO(n)};\\mathbb{Z\n_{2})$\\textsl{\\ let }$J_{n,k}=$\\textsl{\\ }$\\left\\langle \\sigma\n(x_{1}),\\cdots ,\\sigma (x_{k})\\right\\rangle $\\textsl{, }$k\\geq 0$\\textsl{.\nThen}\n\n\\begin{enumerate}\n\\item[(4.6)] $Sq^{1}(w_{2}^{(k-1)})=\\sigma (x_{k})+\\beta _{k-1}$ \\textsl\nwith }$\\beta _{k-1}\\in J_{n,k-1}$\\textsl{, }$k\\geq 1$\\textsl{.}\n\\end{enumerate}\n\n\\noindent \\textbf{Proof.} If $k=1$ the formula (4.6) is verified by $\\sigma\n(x_{1})=w_{3}=Sq^{1}(w_{2})$. Assume next that it holds for some $k\\geq 1$.\nThen\n\n\\begin{quote}\n$Sq^{1}(w_{2}^{(k)})=Sq^{1}(\\gamma (w_{2}^{(k-1)}))$ (by $w_{2}^{(k)}=\\gamma\n(w_{2}^{(k-1)})$)\n\n$=Sq^{2^{k}}Sq^{1}(w_{2}^{(k-1)})+w_{2}^{(k-1)}\\cup Sq^{1}(w_{2}^{(k-1)})$\n(by iii) of Theorem A)\n\n$=Sq^{2^{k}}(\\sigma (x_{k})+\\beta _{k-1})+w_{2}^{(k-1)}\\cup (\\sigma\n(x_{k})+\\beta _{k-1})$ (by induction)\n\n$=\\sigma (x_{k+1})+Sq^{2^{k}}\\beta _{k-1}+w_{2}^{(k-1)}\\cup (\\sigma\n(x_{k})+\\beta _{k-1})$ (by (4.3)).\n\\end{quote}\n\n\\noindent The inductive procedure showing (4.6) is completed by taking\n\n\\begin{quote}\n$\\beta _{k}:=Sq^{2^{k}}\\beta _{k-1}+w_{2}^{(k-1)}\\cup (\\sigma (x_{k})+\\beta\n_{k-1})$.$\\square $\n\\end{quote}\n\nWe proceed to a constructive proof of Theorem C. Since $\\pi ^{\\ast }\\circ\n\\sigma =0$ we get by (4.5) that $Sq^{1}(\\pi ^{\\ast }(w_{2}^{(k)}))=0$. With\nthe space $B_{Spin^{c}(n)}$ being $\\delta _{2}$--formal we obtain further \n\\delta _{2}(\\pi ^{\\ast }(w_{2}^{(k)}))=0$, $k\\geq 0$. In view of the\nBockstein exact sequence\n\n\\begin{center}\n$\\cdots \\rightarrow H^{r}(B_{Spin^{c}(n)})\\overset{\\rho _{2}}{\\rightarrow \nH^{r}(B_{Spin^{c}(n)};\\mathbb{Z}_{2})\\overset{\\delta _{2}}{\\rightarrow \nH^{r+1}(B_{Spin^{c}(n)})\\rightarrow \\cdots $\n\\end{center}\n\n\\noindent this implies that the classes $\\pi ^{\\ast }(w_{2}^{(k)})$ admit\nintegral lifts\n\n\\begin{enumerate}\n\\item[(4.7)] $\\rho _{2}(q_{k}^{\\prime })=\\pi ^{\\ast }(w_{2}^{(k)})$ for some \n$q_{k}^{\\prime }\\in H^{2^{k+1}}(B_{Spin^{c}(n)})$, $k\\geq 0$.\n\\end{enumerate}\n\n\\noindent In particular,\n\n\\begin{quote}\n$\\mathcal{B}(\\pi ^{\\ast }(w_{2}^{(k)}))=\\rho _{4}(q_{k}^{\\prime }\\cup\nq_{k}^{\\prime })$, $\\theta (\\pi ^{\\ast }(w_{2}^{(k)}))=$ $2\\rho\n_{4}(q_{k}^{\\prime })$.\n\\end{quote}\n\n\\noindent Thus, applying $\\pi ^{\\ast }$ to $u=w_{2}^{(k)}$ the relation i)\nof Theorem B becomes\n\n\\begin{quote}\n$\\rho _{4}(q_{k}^{\\prime }\\cup q_{k}^{\\prime })=\\rho _{4}(\\pi ^{\\ast\n}f(w_{2}^{(k)}))+2\\rho _{4}(q_{k+1}^{\\prime })$ on $H^{\\ast\n}(B_{Spin^{c}(n)};\\mathbb{Z}_{4})$,\n\\end{quote}\n\n\\noindent implying that there exist integral class $v_{k+1}\\in H^{\\ast\n}(B_{Spin^{c}(n)})$ so that\n\n\\begin{enumerate}\n\\item[(4.8)] $2q_{k+1}^{\\prime }+q_{k}^{\\prime }\\cup q_{k}^{\\prime }=\\pi\n^{\\ast }f(w_{2}^{(k)})+4v_{k+1}$.\n\\end{enumerate}\n\n\\noindent \\textbf{Proof of Theorem C. }Since the class $q_{0}=\\iota ^{\\ast\n}(x)$ generates $H^{2}(B_{Spin^{c}(n)})=\\mathbb{Z}$ with $\\rho\n_{2}(q_{0})=\\pi ^{\\ast }(w_{2})$ we can take in (4.7) that $q_{0}^{\\prime\n}=q_{0}$, and define in term of (4.8) that $q_{1}:=q_{1}^{\\prime }-2v_{1}$.\nThen, the relations ii) and iii) of Theorem C for the case $r=1$ are\nverified by\n\n\\begin{quote}\n$\\rho _{2}(q_{1})=\\rho _{2}(q_{1}^{\\prime })=\\pi ^{\\ast }(w_{2}^{(1)})$ (by\n(4.7));\n\n$2q_{1}+q_{0}\\cup q_{0}=\\pi ^{\\ast }f(w_{2}^{(0)})$ (by (4.8)).\n\\end{quote}\n\nAssume next that a sequence $q_{0},\\cdots ,q_{r}$ of classes satisfying the\nproperties i), ii) and iii) of Theorem C has been obtained for some $r\\geq 1\n. Take in (4.7) that $q_{r}^{\\prime }=q_{r}$ and define in term of (4.8)\nthat $q_{r+1}:=q_{r+1}^{\\prime }-2v_{r+1}$. Then\n\n\\begin{quote}\n$\\rho _{2}(q_{r+1})=\\pi ^{\\ast }(w_{2}^{(r+1)})$, $2q_{r+1}+q_{r}\\cup\nq_{r}=\\pi ^{\\ast }f(w_{2}^{(r)})$.\n\\end{quote}\n\n\\noindent This completes the inductive construction of a sequence \n\\{q_{r},r\\geq 0\\}$ fulfilling the system i)--iii) of Theorem C.\n\nTo see the uniqueness of the sequence $\\{q_{r},r\\geq 0\\}$ we make use of the\ninjection $\\rho $ in Lemma 4.1. Note that the properties i), ii) and iii) of\nTheorem C suffices to decide the $\\rho $--image of $q_{r}$ as $\\rho\n(q_{r})=(g_{r},\\pi ^{\\ast }w_{2}^{(r)})$, $r\\geq 1$, where $g_{r}\\in H^{\\ast\n}(B_{Spin^{c}(n)};\\mathbb{Z}_{0})$ is the unique polynomial (with rational\ncoefficients) in the generators (4.2) defined recurrently by the relation\niii) of Theorem C as\n\n\\begin{quote}\n$\\rho _{0}(q_{1})=g_{1}:=\\frac{1}{2}(\\rho _{0}(\\pi ^{\\ast }p_{1})-\\rho\n_{0}(q_{0})^{2})$\n\n$\\rho _{0}(q_{2})=g_{2}:=\\frac{1}{2}(\\rho _{0}(\\pi ^{\\ast }p_{2})-g_{1}^{2})\n, $\\cdots $,\n\n$\\rho _{0}(q_{r})=g_{r}:=\\frac{1}{2}(\\rho _{0}(\\pi ^{\\ast\n}f(w_{2}^{(r-1)}))-g_{r-1}^{2}))$, $r\\geq 2$.\n\\end{quote}\n\n\\noindent The injectivity of $\\rho $ implies that, if $\\left\\{ q_{r}^{\\prime\n},1\\leq r\\right\\} $ is a second sequence satisfying the system i)--iii) of\nTheorem C, then $q_{r}^{\\prime }=q_{r}$, as required.$\\square $\n\n\\bigskip\n\nWe conclude this section with two applications of Theorem C. Firstly, let \nH^{+}(B_{SO(n)})$ be the subring of $H^{\\ast }(B_{SO(n)})$ consisting of\nelements in the positive degrees. The induced action of the fiber inclusion \ni$ on cohomology is determined in the following result.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 4.4. }\\textsl{The map }$i^{\\ast }$\\textsl{\\\nsatisfies that }$i^{\\ast }\\circ \\pi ^{\\ast }=0$\\textsl{\\ on }\nH^{+}(B_{SO(n)})$\\textsl{, and that}\n\n\\begin{enumerate}\n\\item[(4.9)] $i^{\\ast }(q_{r})=2(-1)^{r}x^{2^{r}}$, $r\\geq 0$; $\\quad\ni^{\\ast }(\\theta _{n})=x^{2^{h(n)}}$.\n\\end{enumerate}\n\n\\noindent \\textbf{Proof.} As the composition $\\pi \\circ i$ is null--homotopy\nwe get $i^{\\ast }\\circ \\pi ^{\\ast }=0$. Consequently, applying $i^{\\ast }$\nto the relation iii) of Theorem C yields\n\n\\begin{quote}\n$2i^{\\ast }(q_{r+1})+i^{\\ast }(q_{r})^{2}=0$, $r\\geq 0$.\n\\end{quote}\n\n\\noindent Inputting $i^{\\ast }(q_{0})=2x$ we obtain the first relation in\n(4.9) by induction on $r$. The second one is verified by the geometric fact \ni^{\\ast }\\xi _{n}=\\lambda \\oplus \\cdots \\oplus \\lambda $ ($2^{h(n)}$ copies)\n$\\square $\n\n\\bigskip\n\nRecall that the $\\func{mod}2$ Bockstein cohomology $H_{\\beta }^{\\ast }(X)$\nof a space $X$ is the kernel modulo the image of the operation $\\beta\n=Sq^{1} $ on $H^{\\ast }(X;\\mathbb{Z}_{2})$. Granted with Theorem C we can\nexpress the Bockstein $H_{\\beta }^{\\ast }(B_{Spin^{c}(n)})$ by the mod $2$\nreductions of explicit integral cohomology classes on $B_{Spin^{c}(n)}$. For\na set $\\{y_{1},\\cdots ,y_{r}\\}$ of graded elements let $\\Delta (y_{1},\\cdots\n,y_{r})$ be the graded free $\\mathbb{Z}$ module with the basis\n\n\\begin{quote}\n$\\{1,y_{i_{1}}y_{i_{2}}\\cdots y_{i_{k}}$, $1\\leq i_{1}<\\cdots <i_{k}\\leq r\\}$\n\\end{quote}\n\n\\noindent (i.e. the square free monomials in the $y_{1},\\cdots ,y_{r}$). For\na prime $p$ we use the notion $\\Delta _{p}(y_{1},\\cdots ,y_{r})$ to simplify\nthe tensoring $\\mathbb{Z}_{p}\\otimes \\Delta (y_{1},\\cdots ,y_{r})$.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 4.5.} \\textsl{Let }$\\{q_{r},r\\geq 0\\}$ \\textsl{be\nthe unique sequence of integral cohomology classes on }$B_{Spin^{c}(n)}$\\ \n\\textsl{obtained in Theorem C. Then}\n\n\\begin{enumerate}\n\\item[(4.10)] $H_{\\beta }^{\\ast }(B_{Spin^{c}(n)})=\\pi ^{\\ast }H_{\\beta\n}^{\\ast }(B_{SO(n)})\\otimes \\Delta _{2}(q_{0},\\cdots ,q_{h(n)-1})\\otimes \n\\mathbb{Z}_{2}[\\theta _{n}]$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{where, for }$n=2k+1$\\textsl{\\ or }$2k$\\textsl{,}\n\n\\begin{quote}\n$\\pi ^{\\ast }H_{\\beta }^{\\ast }(B_{SO(n)})=\\mathbb{Z}_{2}[w_{2}^{2},\\cdots\n,w_{2k}^{2}]$ \\textsl{or} $\\mathbb{Z}_{2}[w_{2}^{2},\\cdots\n,w_{2(k-1)}^{2},w_{2k}]$\\textsl{,}\n\\end{quote}\n\n\\noindent \\textsl{and where, with }$\\mathbb{Z}_{2}$ \\textsl{coefficients we\ncan write }$q_{r}$\\textsl{, }$\\theta _{n}$\\textsl{\\ in places of }$\\rho\n_{2}(q_{r})$\\textsl{, }$\\rho _{2}(\\theta _{n})$\\textsl{.}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} A sequence $\\{a_{1},\\cdots ,a_{r}\\}$ of elements\nin the algebra $H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$ is called \\textsl\nregular} if for each $i\\geq 1$ the residue class $\\left[ a_{i}\\right] $ in\nthe quotient algebra\n\n\\begin{quote}\n$H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})\/\\left\\langle a_{1},\\cdots\n,a_{i-1}\\right\\rangle $\n\\end{quote}\n\n\\noindent is not a zero divisor. According to Quillen \\cite{Q} the sequence \n\\left\\{ \\sigma (x_{1}),\\cdots ,\\sigma (x_{h(n)-1})\\right\\} $ is regular in \nH^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$ (see Remark 4.6 below). From formula\n(4.6) we can conclude further that\n\na) the sequence $\\{Sq^{1}(w_{2}^{(0)}),\\cdots ,Sq^{1}(w_{2}^{(h(n)-1)})\\}$\nis regular;\n\nb) $J_{n,k}=I_{n,k}$, $1\\leq k\\leq h(n)-1$, where $I_{n,k}=\\left\\langle\nSq^{1}(w_{2}^{(0)}),\\cdots ,Sq^{1}(w_{2}^{(k-1)})\\right\\rangle $.\n\n\\noindent Combining formula (4.4) with property b) we get the more useful\npresentation\n\n\\begin{quote}\n$H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})=H^{\\ast }(B_{SO(n)};\\mathbb{Z\n_{2})\/I_{n,h(n)}\\otimes \\mathbb{Z}_{2}[\\theta _{n}]$,\n\\end{quote}\n\n\\noindent of the algebra $H^{\\ast }(B_{Spin^{c}(n)};\\mathbb{Z}_{2})$, on\nwhich the operator $\\beta $ is determined by\n\n\\begin{quote}\n$\\beta (\\pi ^{\\ast }(w_{2r}))=\\pi ^{\\ast }(w_{2r+1})$, $\\beta (\\pi ^{\\ast\n}(w_{2r+1}))=0$, $\\beta (\\rho _{2}(\\theta _{n}))=0$,\n\n$\\beta (Sq^{1}(w_{2}^{(k)}))=Sq^{1}Sq^{1}w_{2}^{(k)}=0$, $0\\leq k\\leq h(n)-1\n.\n\\end{quote}\n\n\\noindent As a result, if we introduce the quotient algebras $C_{k}=H^{\\ast\n}(B_{SO(n)};\\mathbb{Z}_{2})\/I_{n,k}$, $0\\leq k\\leq h(n)-1$, then the pairs \n\\left\\{ C_{k},\\beta \\right\\} $ become $\\beta $--complexes that fit into the\nshort exact sequences\n\n\\begin{quote}\n$0\\rightarrow \\left\\{ C_{k},\\beta \\right\\} \\overset{\\cup Sq^{1}(w_{2}^{(k)})\n{\\rightarrow }\\left\\{ C_{k},\\beta \\right\\} \\rightarrow \\left\\{ C_{k+1},\\beta\n\\right\\} \\rightarrow 0,$\n\\end{quote}\n\n\\noindent where the first map $\\cup Sq^{1}(w_{2}^{(i)})$ injects (because \nSq^{1}(w_{2}^{(k)})\\in C_{k}$ is not a zero divisor by property a)), and\nwhere the second map is the obvious quotient. Direct calculation shows that\nits induced long exact sequences in cohomologies break into the short exact\nsequences\n\n\\begin{quote}\n$0\\rightarrow H_{\\beta }^{\\ast }(C_{k})\\rightarrow H_{\\beta }^{\\ast\n}(C_{k+1})\\rightarrow \\rho _{2}(q_{k})\\cup H_{\\beta }^{\\ast\n}(C_{k})\\rightarrow 0$,\n\\end{quote}\n\n\\noindent implying that $H_{\\beta }^{\\ast }(C_{k+1})$ is a free module over \nH_{\\beta }^{\\ast }(C_{k})$ with basis $\\{1,\\rho _{2}(q_{k})\\}$. Arguing\ninductively we obtain the additive presentation\n\n\\begin{quote}\n$H_{\\beta }^{\\ast }(C_{h(n)})=H_{\\beta }^{\\ast }(C_{0})\\otimes \\Delta\n_{2}(q_{0},\\cdots ,q_{h(n)-1})$\n\\end{quote}\n\n\\noindent in which $H_{\\beta }^{\\ast }(C_{0})=\\pi ^{\\ast }H_{\\beta }^{\\ast\n}(B_{SO(n)})$ by the definition. Finally, we obtain (4.10) by the K\\\"{u\nnneth formula\n\n\\begin{quote}\n$H_{\\beta }^{\\ast }(C_{h(n)}\\otimes \\mathbb{Z}_{2}[\\theta _{n}])=H_{\\beta\n}^{\\ast }(C_{h(n)})\\otimes \\mathbb{Z}_{2}[\\theta _{n}]$.$\\square $\n\\end{quote}\n\n\\bigskip\n\n\\noindent \\textbf{Remark 4.6. }In \\cite[(6.2)]{Q} Quillen defined the\nfunction $h$ on the positive integers whose values are given by the\nfollowing table\n\n\\begin{center}\n\\begin{tabular}{l||llllllll}\n\\hline\n$n$ & $8l+1$ & $8l+2$ & $8l+3$ & $8l+4$ & $8l+5$ & $8l+6$ & $8l+7$ & $8l+8$\n\\\\ \\hline\n$h(n)$ & $4l$ & $4l+1$ & $4l+2$ & $4l+2$ & $4l+3$ & $4l+3$ & $4l+3$ & $4l+3$\n\\\\ \\hline\n\\end{tabular\n.\n\\end{center}\n\n\\noindent He stated in \\cite[Theorem 6.3]{Q} that the sequence $w_{2},\\sigma\n(x_{1}),\\cdots ,\\sigma (x_{h(n)})$ is regular on $H^{\\ast }(B_{SO(n)}\n\\mathbb{Z}_{2})$. This statement should be corrected as\n\n\\begin{quote}\n\"the sequence $w_{2},\\sigma (x_{1}),\\cdots ,\\sigma (x_{h(n)-1})$ is regular\nin $H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})$.\"\n\\end{quote}\n\n\\noindent Indeed, direct computation shows that\n\n\\begin{quote}\n$\\sigma (x_{1})=w_{3},$\n\n$\\sigma (x_{2})=Sq^{2}(\\sigma (x_{1}))=w_{5}+w_{2}w_{3}$\n\n$\\sigma (x_{3})=Sq^{4}(\\sigma\n(x_{2}))=w_{9}+w_{2}w_{7}+w_{3}w_{6}+w_{3}^{3}+w_{2}^{2}(w_{5}+w_{2}w_{3})$\n\\end{quote}\n\n\\noindent confirming that \\cite[Theorem 6.3]{Q} already fails for $n=5,6,7,8\n.$\\square $\n\n\\section{The cohomology $H^{\\ast }(B_{Spin^{c}(n)})$}\n\nFor a topological space $X$ the quotient ring $H^{\\ast }(X)\/\\tau (X)$ can be\nidentified with \\textsl{the free part} of the integral cohomology $H^{\\ast\n}(X)$, and will be denoted by $\\mathcal{F}(X)$. If $X=B_{SO(n)}$ and in\naccordance to $n\\equiv 0$ or $1$ we get from (2.3) that\n\n\\begin{quote}\n$\\mathcal{F}(B_{SO(n)})=\\mathbb{Z}[p_{1},p_{2},\\cdots ,p_{\\left[ \\frac{n-1}{\n}\\right] },e_{n}]$ or $\\mathbb{Z}[p_{1},p_{2},\\cdots ,p_{\\left[ \\frac{n-1}{2\n\\right] }]$,\n\\end{quote}\n\n\\noindent on which the map $\\pi ^{\\ast }$ injects by Lemma 4.1.\n\nConsider the graded free $\\mathbb{Z}$--module over the ring $\\pi ^{\\ast \n\\mathcal{F}(B_{SO(n)})$\n\n\\begin{enumerate}\n\\item[(5.1)] $A:=\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta\n(q_{0},\\cdots ,q_{h(n)-1})\\otimes \\mathbb{Z}[\\theta _{n}]$,\n\\end{enumerate}\n\n\\noindent and let $\\varphi :A\\rightarrow H^{\\ast }(B_{Spin^{c}(n)})$ be the\nadditive homomorphism induced by the inclusions $\\theta _{n},q_{r}\\in\nH^{\\ast }(B_{Spin^{c}(n)})$.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 5.1.} \\textsl{The map }$\\varphi $ \\textsl{carries }\nA $\\textsl{\\ isomorphically onto }$\\mathcal{F}(B_{Spin^{c}(n)})$\\textsl{.}\n\n\\bigskip\n\n\\noindent \\textbf{Proof. }For a prime $p$ denote by $\\varphi _{p}$ the\ninduced map\n\n\\begin{quote}\n$\\varphi _{p}:=\\varphi \\otimes id:A\\otimes \\mathbb{Z}_{p}\\rightarrow H^{\\ast\n}(B_{Spin^{c}(n)})\\otimes \\mathbb{Z}_{p}$.\n\\end{quote}\n\n\\noindent in which we already have, by $2\\tau (B_{Spin^{c}(n)})=0$, that\n\n\\begin{enumerate}\n\\item[(5.2)] $H^{\\ast }(B_{Spin^{c}(n)})\\otimes \\mathbb{Z}_{2}=H_{\\beta\n}^{\\ast }(B_{Spin^{c}(n)})\\oplus \\tau (B_{Spin^{c}(n)})$,\n\n\\item[(5.3)] $H^{\\ast }(B_{Spin^{c}(n)})\\otimes \\mathbb{Z}_{p}=H^{\\ast\n}(B_{Spin^{c}(n)};\\mathbb{Z}_{p})$\\textsl{\\ }if\\textsl{\\ }$p\\neq 2$.\n\\end{enumerate}\n\n\\noindent It suffices to show that $\\varphi _{p}$ injects for $p=2$, and is\nan isomorphism for $p\\neq 2$.\n\nIf $p=2$ we get from $\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\mathbb{Z\n_{2}=\\pi ^{\\ast }H_{\\beta }^{\\ast }(B_{SO(n)})$ and (4.10) that\n\n\\begin{quote}\n$A\\otimes \\mathbb{Z}_{2}=H_{\\beta }^{\\ast }(B_{Spin^{c}(n)})$.\n\\end{quote}\n\n\\noindent Comparing this with (5.2) shows that $\\varphi _{2}$ injects.\n\nAssume next that\\textbf{\\ }$p\\neq 2$. With $2\\tau (B_{SO(n)})=0$ we have \n\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\mathbb{Z}_{p}=\\pi ^{\\ast }H^{\\ast\n}(B_{SO(n)};\\mathbb{Z}_{p})$ and consequently\n\n\\begin{quote}\n$A\\otimes \\mathbb{Z}_{p}=\\pi ^{\\ast }H^{\\ast }(B_{SO(n)};\\mathbb{Z\n_{p})\\otimes \\Delta _{p}(q_{0},\\cdots ,q_{h(n)-1})\\otimes \\mathbb{Z\n_{p}[\\theta _{n}]$.\n\\end{quote}\n\n\\noindent Since the composition $i^{\\ast }\\circ \\psi _{p}$ carries the\nfactor $\\Delta _{p}(q_{0},\\cdots ,q_{h(n)-1})\\otimes \\mathbb{Z}_{p}[\\theta\n_{n}]$ isomorphically onto $H^{\\ast }(\\mathbb{C}P^{\\infty };\\mathbb{Z}_{p})$\nby Lemma 4.4, we obtain\n\n\\begin{quote}\n$A\\otimes \\mathbb{Z}_{p}=\\pi ^{\\ast }H^{\\ast }(B_{SO(n)},\\mathbb{Z\n_{p})\\otimes H^{\\ast }(\\mathbb{C}P^{\\infty };\\mathbb{Z}_{p})=H^{\\ast\n}(B_{Spin^{c}(n)};\\mathbb{Z}_{p})$,\n\\end{quote}\n\n\\noindent where the second equality follows from Lemma 4.1. Comparing this\nwith (5.3) shows that $\\psi _{p}$ is an isomorphism for $p\\neq 2$.$\\square $\n\n\\bigskip\n\nCombining Lemma 5.1 with formula (4.4) we can express $H^{\\ast\n}(B_{Spin^{c}(n)})$ in term of its free part $\\mathcal{F}(B_{Spin^{c}(n)})$\nand the torsion ideal $\\tau (B_{Spin^{c}(n)})$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 5.2. }\\textsl{The cohomology }$H^{\\ast\n}(B_{Spin^{c}(n)})$\\textsl{\\ has the additive presentation}\n\n\\begin{enumerate}\n\\item[(5.4)] $\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta (q_{0},\\cdots\n,q_{h(n)-1})\\otimes \\mathbb{Z}[\\theta _{n}]\\oplus \\pi ^{\\ast }\\tau\n(B_{SO(n)})\\otimes \\mathbb{Z}[\\theta _{n}]$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{where the cup products between the free part and torsion\nideal are given by}\n\n\\begin{quote}\n$\\pi ^{\\ast }(p_{i})\\cup \\pi ^{\\ast }(z)=\\pi ^{\\ast }(w_{2i}^{2}\\cup z)$ \n\\textsl{(by }$\\rho _{2}(p_{i})=w_{2i}^{2}$\\textsl{) }\n\n$q_{r}\\cup \\pi ^{\\ast }(z)=\\pi ^{\\ast }(z\\cup w_{2}^{(r)})$ \\textsl{(by }\n\\rho _{2}(q_{r})=w_{2}^{(r)}$\\textsl{),}\n\\end{quote}\n\n\\noindent \\textsl{and} \\textsl{where }$z\\in \\tau (B_{SO(n)})$, $1\\leq i\\leq\n\\left[ \\frac{n-1}{2}\\right] $, $0\\leq r\\leq h(n)-1$.$\\square $\n\n\\bigskip\n\nBy virtue of Theorem 5.2 we clarify a crucial relation relating the class \nq_{h(n)-1}$ with the Euler class $\\theta _{n}$.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 5.3. }\\textsl{There exists an element}\n\n\\begin{quote}\n$\\alpha _{n}\\in \\pi ^{\\ast }H^{\\ast }(B_{SO(n)})\\otimes \\Delta (q_{0},\\cdots\n,q_{h(n)-1})$\\textsl{, }$\\deg \\alpha _{n}=2^{h(n)+1}$\\textsl{,\\ }\n\\end{quote}\n\n\\noindent \\textsl{so that the following relation\\ holds on the ring }\nH^{\\ast }(B_{Spin^{c}(n)})$\n\n\\begin{enumerate}\n\\item[(5.5)] $4(-1)^{h(n)}\\theta _{n}+q_{h(n)-1}^{2}=a_{n}$\\textsl{.}\n\\end{enumerate}\n\n\\noindent \\textsl{In particular, the class }$a_{n}$\\textsl{\\ is\ncharacterized uniquely by}\n\n\\begin{quote}\n$\\rho (a_{n})=(4(-1)^{h(n)}\\rho _{0}(\\theta\n_{n})+g_{h(n)-1}^{2},(w_{2}^{(h(n)-1)})^{2})$\\textsl{\\ (see Lemma 4.1).}\n\\end{quote}\n\n\\noindent \\textbf{Proof. }Since in degree $2^{h(n)+1}$ the quotient of $A$\nby its subgroup $\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta\n(q_{0},\\cdots ,q_{h(n)-1})\\otimes 1$ is isomorphic to $\\mathbb{Z}$ with the\nbasis $\\{\\theta _{n}\\}$, formula (5.4) implies that the integral class \nq_{h(n)}\\in H^{\\ast }(B_{Spin^{c}(n)})$ admits the expression\n\n\\begin{quote}\n$q_{h(n)}=k\\cdot \\theta _{n}+\\varphi (\\beta _{n})$ for some $\\beta _{n}\\in\n\\pi ^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta (q_{0},\\cdots ,q_{h(n)-1})$\n\\end{quote}\n\n\\noindent and some integer $k\\in \\mathbb{Z}$. Applying $i^{\\ast }$ we get by\nLemma 4.4 that\n\n\\begin{quote}\n$2(-1)^{h(n)}x^{2^{h(n)}}=k\\cdot x^{2^{h(n)}}$ on $H^{2^{h(n)+1}}(\\mathbb{C\nP^{\\infty })$,\n\\end{quote}\n\n\\noindent showing $k=2(-1)^{h(n)}$. On the other hand, by iii) of Theorem C\nwe have\n\n\\begin{quote}\n$2q_{h(n)}+q_{h(n)-1}^{2}=\\pi ^{\\ast }f(w_{2}^{(h(n)-1)})\\in \\pi ^{\\ast\n}H^{+}(B_{SO(n)})$.\n\\end{quote}\n\n\\noindent We obtain (5.5) by taking $a_{n}=\\pi ^{\\ast\n}f(w_{2}^{(h(n)-1)})\\otimes 1-2\\beta _{n}$.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Proof of Theorem D.} Set $\\widetilde{A}:=\\pi ^{\\ast\n}H^{\\ast }(B_{SO(n)})\\otimes \\mathbb{Z}[q_{0},q_{1},\\cdots\n,q_{h(n)-1},\\theta _{n}]$, and let $\\widetilde{\\varphi }:\\widetilde{A\n\\rightarrow H^{\\ast }(B_{Spin^{c}(n)})$ be induced by the inclusions \nq_{r},\\theta _{n}\\in H^{\\ast }(B_{Spin^{c}(n)})$. Then by the properties ii)\nand iii) of Theorem C the map $\\widetilde{\\varphi }$ satisfies that\n\n\\begin{enumerate}\n\\item[(5.6)] $\\widetilde{\\varphi }(2\\otimes q_{r+1}+1\\otimes q_{r}^{2}-\\pi\n^{\\ast }f(w_{2}^{(r)})\\otimes 1)=0$, $0\\leq r\\leq h(n)-2$,\n\n\\item[(5.7)] $\\widetilde{\\varphi }(\\pi ^{\\ast }\\delta _{2}(x)\\otimes\nq_{r}-\\pi ^{\\ast }\\delta _{2}(x\\cup w_{2}^{(r)})\\otimes 1)=0$.\n\\end{enumerate}\n\n\\noindent In addition, we have by the relation (5.5) that\n\n\\begin{enumerate}\n\\item[(5.8)] $\\widetilde{\\varphi }(4(-1)^{h(n)}\\otimes \\theta _{n}+1\\otimes\nq_{h(n)-1}^{2}-a_{n})=0$.\n\\end{enumerate}\n\n\\noindent Therefore $R_{n}\\subseteq \\ker \\widetilde{\\varphi }$, where $R_{n}$\nis the ideal on $\\widetilde{A}$ stated in Theorem D. It remains to show that\nthe map $\\widetilde{\\varphi }$, when factoring through the quotient \n\\widetilde{A}\\func{mod}R_{n}$, yields an additive isomorphism onto $H^{\\ast\n}(B_{Spin^{c}(n)})$.\n\nPutting in transparent forms the relations (5.6) and (5.8) tell that\n\n\\begin{quote}\n$q_{r}^{2}=-2q_{r+1}+\\pi ^{\\ast }f(w_{2}^{(r)})$, $0\\leq r\\leq h(n)-2;$\n\n$q_{h(n)-1}^{2}=4(-1)^{h(n)+1}\\theta _{n}+a_{n}$,\n\\end{quote}\n\n\\noindent respectively. These are the relations required to reduce each\nmonomial\n\n\\begin{quote}\n$q_{0}^{r_{0}}q_{1}^{r_{1}}\\cdots q_{h(n)-1}^{r_{h(n)-1}}\\in \\widetilde{A}$\nwith $r_{i}\\geq 2$ for some $i$\n\\end{quote}\n\n\\noindent to an element of the $\\pi ^{\\ast }H^{\\ast }(B_{SO(n)})$ module\n\n\\begin{quote}\n$A_{1}:=\\pi ^{\\ast }H^{\\ast }(B_{SO(n)})\\otimes \\Delta (q_{0},\\cdots\n,q_{h(n)-1})\\otimes \\mathbb{Z}[\\theta _{n}]$.\n\\end{quote}\n\n\\noindent Moreover, combining $H^{\\ast }(B_{SO(n)})=\\mathcal{F\n(B_{SO(n)})\\oplus \\tau (B_{SO(n)})$ with Lemma 5.1 we get the decomposition\n\n\\begin{quote}\n$A_{1}=\\mathcal{F}(B_{Spin^{c}(n)})\\oplus \\pi ^{\\ast }\\tau\n(B_{SO(n)})\\otimes \\Delta (q_{0},\\cdots ,q_{h(n)-1})\\otimes \\mathbb{Z\n[\\theta _{n}]$.\n\\end{quote}\n\n\\noindent in which the relation (5.7) tells, for any $z\\in H^{\\ast\n}(B_{SO(n)};\\mathbb{Z}_{2})$, that\n\n\\begin{quote}\n$\\widetilde{\\varphi }(\\pi ^{\\ast }\\delta _{2}(z)\\otimes q_{r})=\\pi ^{\\ast\n}\\delta _{2}(z)\\cup q_{r}=\\pi ^{\\ast }\\delta _{2}(z\\cup w_{2}^{(r)})$ (by \n\\rho (q_{r})=w_{2}^{(r)}$),\n\\end{quote}\n\n\\noindent showing\n\n\\begin{quote}\n$\\widetilde{\\varphi }(\\pi ^{\\ast }\\tau (B_{SO(n)})\\otimes \\Delta\n(q_{0},\\cdots ,q_{h(n)-1})\\otimes \\mathbb{Z}[\\theta _{n}])$ $=\\tau\n(B_{Spin^{c}(n)})$\n\\end{quote}\n\n\\noindent by (4.5). The proof of Theorem D has now been completed by Theorem\n5.2.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 5.4. }In concrete situations Theorem D is directly\napplicable to present the ring $H^{\\ast }(B_{Spin^{c}(n)})$ by a minimal\nsystem of explicit generators and relations. For instance if $n=8$ we get by\nLemma 5.1 that\n\n\\begin{quote}\n$\\mathcal{F}(B_{Spin^{c}(8)})=\\mathbb{Z}[p_{1},p_{2},p_{3},e_{8},\\theta\n_{8}]\\otimes \\Delta (q_{0},q_{1},q_{2})$\n\n$=\\mathbb{Z}[q_{0},q_{1},q_{2},p_{3},e_{8},\\theta _{8}]\/\\left\\langle 4\\theta\n_{8}-q_{2}^{2}-a_{8}\\right\\rangle $,\n\\end{quote}\n\n\\noindent where the second equality is obtained by subtracting the\ngenerators $p_{1},p_{2}$ using the relations\n\n\\begin{quote}\n$p_{1}=2q_{1}+q_{0}^{2}$, $p_{2}=2q_{2}+q_{1}^{2}$ by (5.6).\n\\end{quote}\n\n\\noindent Furthermore, by (4.5), as well as the obvious relation on $\\tau\n(B_{Spin(8)})$\n\n\\begin{center}\n$q_{0}^{r_{1}}\\cup q_{1}^{r_{2}}\\cup q_{2}^{r_{3}}\\cup p_{3}^{r_{4}}\\cup\ne_{8}^{r_{5}}\\cup \\delta _{2}(w_{6})=\\delta _{2}(w_{2}^{r_{1}}\\cup\nw_{4}^{r_{2}}\\cup w_{6}^{2r_{2}}\\cup w_{8}^{r_{3}+r_{5}}\\cup w_{6})$,\n\\end{center}\n\n\\noindent we find that $\\tau (B_{Spin(8)})$ is the ideal on $H^{\\ast\n}(B_{Spin(8)})$ generated by the single integral class $\\delta _{2}(w_{6})$\nwith degree $7$. Summarizing we get that\n\n\\begin{enumerate}\n\\item[(5.9)] $H^{\\ast }(B_{Spin^{c}(8)})=\\mathbb{Z\n[q_{0},q_{1},q_{2},p_{3},e_{8},\\theta _{8},\\delta _{2}(w_{6})]\/\\left\\langle\n2\\delta _{2}(w_{6}),4\\theta _{8}-q_{2}^{2}-a_{8}\\right\\rangle $\n\\end{enumerate}\n\n\\noindent where computing with the invariant $\\theta _{8}$ shows that\n\n\\begin{quote}\n$a_{8}=e_{8}^{2}-2e_{8}q_{2}-q_{0}^{2}p_{3}+2e_{8}q_{0}^{2}q_{1}$.$\\square $\n\\end{quote}\n\n\\section{The cohomology $H^{\\ast }(B_{Spin(n)})$}\n\nThe method developed in the preceding sections \\S 4--5 applies equally well\nto formulate the ring $H^{\\ast }(B_{Spin(n)})$. To clarify this consider the\nfibration induced by the two sheets covering $Spin(n)\\rightarrow SO(n)$\n\n\\begin{enumerate}\n\\item[(6.1)] $\\mathbb{R}P^{\\infty }\\overset{i}{\\rightarrow }B_{Spin(n)\n\\overset{\\overline{\\pi }}{\\rightarrow }B_{SO(n)}$,\n\\end{enumerate}\n\n\\noindent where $\\overline{\\pi }=\\pi \\circ \\psi $ in the notion of (2.5) and\n(2.6).\n\n\\bigskip\n\n\\noindent \\textbf{Theorem C'. }\\textsl{There exists a unique sequence }\n\\left\\{ \\overline{q}_{r}\\QTR{sl}{,\\ }r\\geq 1\\right\\} $ \\textsl{of} \\textsl\nintegral} \\textsl{cohomology classes on }$B_{Spin(n)}$\\textsl{, }$\\deg \n\\overline{q}_{r}=2^{r+1}$\\textsl{,} \\textsl{that} \\textsl{satisfies the\nfollowing system}\n\n\\begin{quote}\n\\textsl{i)} $\\rho _{2}(\\overline{q}_{r})=\\overline{\\pi }^{\\ast }w_{2}^{(r)}\n\\textsl{;}\n\n\\textsl{ii)} $2\\overline{q}_{1}=\\overline{\\pi }^{\\ast }p_{1}$\\textsl{,} $\n\\overline{q}_{r+1}+\\overline{q}_{r}^{2}=\\overline{\\pi }^{\\ast\n}f(w_{2}^{(r)}) $\\textsl{, }$r\\geq 1$\\textsl{.}\n\\end{quote}\n\n\\noindent \\textbf{Proof.} The existence of a sequence $\\left\\{ \\overline{q\n_{r}\\QTR{sl}{,\\ }r\\geq 0\\right\\} $ fulfilling the properties i) and ii) is\nobtained from ii) and iii) of Theorem C by setting $\\overline{q}_{r}:=$ \n\\psi ^{\\ast }(q_{r})$. Note that $\\overline{q}_{0}=0$, because $q_{0}$\nbelongs to the image of the transgression in the Serre spectral sequence of\nthe $U(1)$ fibration (2.6).\n\nFor the uniqueness of the sequence $\\left\\{ \\overline{q}_{r},\\QTR{sl}{\\ \nr\\geq 1\\right\\} $ so obtained we make use of the map $\\rho $ in Lemma 4.2\nfor the space $B_{Spin(n)}$. With $2\\tau (B_{Spin(n)})=0$ it injects by \\cit\n[30.6.]{BH}. On the other hand, the relations i) and ii) suffice to express\nthe images $\\rho (\\overline{q}_{r})$, $r\\geq 1$. This competes the proof.\n\\square $\n\n\\bigskip\n\nLet $\\eta _{n}$ be the oriented real vector bundle on $B_{Spin(n)}$\nassociated to the real spin representation of the group $Spin(n)$, $\\dim _\n\\mathbb{R}}\\eta _{n}=$ $2^{k(n)+1}$\\cite{ABS}, and denote by $\\overline\n\\theta }_{n}\\in H^{\\ast }(B_{Spin(n)})$ its Euler class, where\n\n\\begin{quote}\n$k(n)=\\left[ \\frac{n-1}{2}\\right] $ if $n\\func{mod}8\\neq 0,1,7$, or $\\left[ \n\\frac{n-1}{2}\\right] -1$ otherwise.\n\\end{quote}\n\n\\noindent It is known that (see \\cite[Theorem 8.3]{BW} or \\cite[Theorem 6.5\n{Q}):\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 6.1.} \\textsl{The maps }$\\psi $ \\textsl{in (2.6) and \n}$i$ \\textsl{in (6.1)} \\textsl{satisfy, respectively, that}\n\n\\begin{enumerate}\n\\item[(6.2)] $\\psi ^{\\ast }(\\theta _{n})=\\overline{\\theta }_{n}$ \\textsl{or \n$\\overline{\\theta }_{n}^{2}$\\textsl{, in accordance to }$n\\func{mod}8\\neq\n0,1,7$,\\textsl{\\ or otherwise.}\n\n\\item[(6.3)] $i^{\\ast }(\\overline{\\theta }_{n})=z^{2^{k(n)+1}}$\\textsl{,} \n\\textsl{where }$z$\\textsl{\\ is the generator of} $H^{1}(\\mathbb{R}P^{\\infty\n};\\mathbb{Z}_{2})=\\mathbb{Z}_{2}$.$\\square $\n\\end{enumerate}\n\nLet $\\{\\overline{q}_{r},r\\geq 1\\}$ be the unique sequence\\ obtained in\nTheorem C', and let $\\overline{\\sigma }$ be the transgression in the\nfibration (6.1). The counterparts of the Lemmas 4.2, 4.5 are summarized in\nthe following result.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 6.2. }\\textsl{For }$X=B_{Spin(n)}$\\textsl{\\ the\nalgebras} $H^{\\ast }(X;\\mathbb{Z}_{2})$, $H_{\\beta }^{\\ast }(X)$ \\textsl{and \n}$\\tau (X)$\\textsl{\\ have the following presentations}\n\n\\begin{enumerate}\n\\item[(6.4)] $H^{\\ast }(B_{Spin(n)};\\mathbb{Z}_{2})=\\overline{\\pi }^{\\ast\n}H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})\\otimes \\mathbb{Z}_{2}[\\overline{\\theta \n_{n}]$\n\n\\item[(6.5)] $H_{\\beta }^{\\ast }(B_{Spin(n)})=\\overline{\\pi }^{\\ast\n}H_{\\beta }^{\\ast }(B_{SO(n)})\\otimes \\Delta _{2}(\\overline{q}_{1},\\cdots \n\\overline{q}_{k(n)-1})\\otimes \\mathbb{Z}_{2}[\\overline{\\theta }_{n}]$\\textsl\n;}\n\n\\item[(6.6)] $\\tau (B_{Spin(n)})=\\overline{\\pi }^{\\ast }\\tau\n(B_{SO(n)})\\otimes \\mathbb{Z}_{2}[\\overline{\\theta }_{n}]$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{respectively}, \\textsl{where the map }$\\overline{\\pi }$ \n\\textsl{in (6.1)} \\textsl{induces the isomorphisms}\n\n\\begin{quote}\n$\\overline{\\pi }^{\\ast }H^{\\ast }(B_{SO(n)};\\mathbb{Z}_{2})=H^{\\ast\n}(B_{SO(n)};\\mathbb{Z}_{2})\/\\left\\langle \\overline{\\sigma }(z),\\overline\n\\sigma }(z^{2}),\\cdots ,\\overline{\\sigma }(z^{2^{k(n)-1}})\\right\\rangle \n\\textsl{;}\n\n$\\overline{\\pi }^{\\ast }H_{\\beta }^{\\ast }(B_{SO(n)})=H_{\\beta }^{\\ast\n}(B_{SO(n)})\/(w_{2}^{2})$\\textsl{.}$\\square $\n\\end{quote}\n\n\\bigskip\n\n\\noindent \\textbf{Remark 6.3. }Formula\\textbf{\\ }(6.4) is due to Quillen \n\\cite[Theorem 6.5]{Q}\n\nIn\\textbf{\\ }\\cite[Proposition 6.1]{ABP} Anderson, Brown, and Peterson\nobtained a formula for the Bockstein $H_{\\beta }^{\\ast }(B_{Spin(n)})$ in\nthe stable range $n=\\infty $. The unstable instances has been calculated by\nKono \\cite[\\S 3]{Ko}, Benson and Wood \\cite[\\S 9]{BW}. Taking the advantage\nof Theorem C' our formula (6.5) makes the structure of $H_{\\beta }^{\\ast\n}(B_{Spin(n)})$ transparent, whose proof is similar to that of (4.10).\n\\square $\n\n\\bigskip\n\nTo formulate an additive presentation of the integral cohomology $H^{\\ast\n}(B_{Spin(n)})$, and to take the care of the overlap $\\overline{q}_{1}=\n\\overline{\\pi }^{\\ast }p_{1}$ by ii) of Theorem C', we introduce the graded\nquotient group\n\n\\begin{quote}\n$A:=\\overline{\\pi }^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta (\\overline{q\n_{1},\\cdots ,\\overline{q}_{k(n)-1})\\otimes \\mathbb{Z}[\\overline{\\theta }_{n}\n\\func{mod}(\\overline{q}_{1}-2\\overline{\\pi }^{\\ast }p_{1})$,\n\\end{quote}\n\n\\noindent and let $\\varphi :A\\rightarrow H^{\\ast }(B_{Spin(n)})$ the\nadditive map induced by the inclusions $\\overline{\\theta }_{n},\\overline{q\n_{r}\\in H^{\\ast }(B_{Spin(n)})$. The same argument used in the proof of\nLemma 5.1 is applicable to show that $\\varphi $ is an isomorphism onto the\nfree part $\\mathcal{F}(B_{Spin(n)})$ of the ring $H^{\\ast }(B_{Spin(n)})$.\nCombining this with formula (6.6) we obtain the following result, which is\ncompatible with Theorem 5.2.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 6.4.} \\textsl{The cohomology }$H^{\\ast\n}(B_{Spin(n)})$\\textsl{\\ has the additive presentation}\n\n\\begin{enumerate}\n\\item[(6.7)] $A\\oplus \\overline{\\pi }^{\\ast }\\tau (B_{SO(n)})\\otimes \\mathbb\nZ}[\\overline{\\theta }_{n}]$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{where the cup products between the free part and torsion\nideal are given by}\n\n\\begin{quote}\n$\\overline{\\pi }^{\\ast }(p_{i})\\cup \\overline{\\pi }^{\\ast }(z)=\\overline{\\pi \n}^{\\ast }(w_{2i}^{2}\\cup z)$\\textsl{\\ (by }$\\rho _{2}(p_{i})=$\\textsl{\\quad \n$w_{2i}^{2}$\\textsl{)}\n\n$\\overline{q}_{r}\\cup \\overline{\\pi }^{\\ast }(z)=\\overline{\\pi }^{\\ast\n}(z\\cup w_{2}^{(r)})$ \\textsl{(by }$\\rho _{2}(\\overline{q}_{r})=$\\textsl\n\\quad }$\\overline{\\pi }^{\\ast }(w_{2}^{(r)})$\\textsl{),}\n\\end{quote}\n\n\\noindent \\textsl{and} \\textsl{where }$z\\in \\tau (B_{SO(n)})$, $1\\leq i\\leq\n\\left[ \\frac{n-1}{2}\\right] $, $0\\leq r\\leq k(n)-1$.$\\square $\n\n\\bigskip\n\nConsider the integral class $\\overline{q}_{k(n)}\\in H^{\\ast }(B_{Spin(n)})$.\nSince in degree $2^{k(n)+1}$ the quotient of $A$ by its subgroup\n\n\\begin{quote}\n$A_{1}:=\\overline{\\pi }^{\\ast }\\mathcal{F}(B_{SO(n)})\\otimes \\Delta \n\\overline{q}_{1},\\cdots ,\\overline{q}_{k(n)-1})\\func{mod}(\\overline{q}_{1}-\n\\overline{\\pi }^{\\ast }p_{1})$\n\\end{quote}\n\n\\noindent is isomorphic to $\\mathbb{Z}$ with the basis $\\{\\overline{\\theta \n_{n}\\}$, we must have the expression\n\n\\begin{enumerate}\n\\item[(6.8)] $\\overline{q}_{k(n)}=\\kappa \\cdot \\overline{\\theta \n_{n}+\\varphi (b_{n})$ for some $\\kappa \\in \\mathbb{Z}$ and $b_{n}\\in A_{1}$\n\\end{enumerate}\n\n\\noindent by Theorem 6.4. However, in contrast to the proof of Lemma 5.3 the\nfiber inclusion $i$ in (6.1) fails to be help to evaluate $\\kappa $.\nNevertheless, resorting to the canonical subgroup $U(k)\\subset Spin(2k)$ we\nshall show in Lemma 7.10 that $\\kappa =2(-1)^{k(n)-1}$. Thus, combining\n(6.8) with the relation by Theorem C'\n\n\\begin{quote}\n$2\\overline{q}_{k(n)}+\\overline{q}_{k(n)-1}^{2}=\\overline{\\pi }^{\\ast\n}f(w_{2}^{(k(n)-1)})\\in $ $\\overline{\\pi }^{\\ast }H^{+}(B_{SO(n)})$\n\\end{quote}\n\n\\noindent yields the following relation relating the class $\\overline{q\n_{k(n)-1}$ with the Euler class $\\overline{\\theta }_{n}$\n\n\\begin{enumerate}\n\\item[(6.9)] $4(-1)^{k(n)-1}\\overline{\\theta }_{n}+\\overline{q\n_{k(n)-1}^{2}=\\varphi (\\overline{b}_{n})$, $\\overline{b}_{n}\\in \\overline\n\\pi }^{\\ast }\\overline{H}^{+}(B_{SO(n)})\\otimes \\Delta (\\overline{q\n_{1},\\cdots ,\\overline{q}_{k(n)-1})$.\n\\end{enumerate}\n\nGranted with Theorem C', Theorem 6.4 and formula (6.9), the same argument\nused in the proof of Theorem D is applicable to obtain the following result,\nwhose proof can therefore be omitted.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem D'. }\\textsl{The ring }$H^{\\ast }(B_{Spin(n)})\n\\textsl{\\ has the presentation}\n\n\\begin{enumerate}\n\\item[(6.10)] $H^{\\ast }(B_{Spin(n)})=\\overline{\\pi }^{\\ast }H^{\\ast\n}(B_{SO(n)})\\otimes \\mathbb{Z}[\\overline{q}_{1},\\cdots ,\\overline{q\n_{k(n)-1},\\overline{\\theta }_{n}]\/K_{n}$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{in which }$K_{n}$ \\textsl{denotes the ideal generated by\nthe following elements}\n\n\\begin{quote}\n\\textsl{i)} $2\\overline{q}_{1}-\\overline{\\pi }^{\\ast }p_{1}$\\textsl{,} $\n\\overline{q}_{r+1}+\\overline{q}_{r}^{2}-\\overline{\\pi }^{\\ast\n}f(w_{2}^{(r)}) $\\textsl{,\\quad }$1\\leq r\\leq k(n)-2$\\textsl{;}\n\n\\textsl{ii)} $\\overline{\\pi }^{\\ast }\\delta _{2}(y)\\cup \\overline{q}_{r}\n\\overline{\\pi }^{\\ast }\\delta _{2}(y\\cup w_{2}^{(r)}),$ $1\\leq r\\leq k(n)-2\n\\textsl{;}\n\n\\textsl{iii)} $4(-1)^{k(n)-1}\\overline{\\theta }_{n}+\\overline{q\n_{k(n)-1}^{2}-\\overline{b}_{n}$\\textsl{,}\n\\end{quote}\n\n\\noindent \\textsl{where }$y\\in H^{\\ast }(B_{SO(n)},\\mathbb{Z}_{2})$\\textsl{,}\n$\\overline{b}_{n}\\in \\overline{\\pi }^{\\ast }\\overline{H}^{+}(B_{SO(n)}\n\\otimes \\Delta (\\overline{q}_{1},\\cdots ,\\overline{q}_{k(n)-1})$\\textsl{.}\n\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 6.5. }If $n=8$ we get from formula (6.7) that\n\n\\begin{quote}\n$\\mathcal{F}(B_{Spin(8)})=\\mathbb{Z}[p_{1},p_{2},p_{3},e_{8},\\overline\n\\theta }_{8}]\\otimes \\Delta (\\overline{q}_{1})\\func{mod}(\\overline{q}_{1}-\n\\overline{\\pi }^{\\ast }p_{1})$\n\n$\\qquad =\\mathbb{Z}[\\overline{q}_{1},p_{3},e_{8},\\overline{\\theta }_{8}]$\n\\end{quote}\n\n\\noindent where the second equality follows from\n\n\\begin{quote}\n$p_{1}=2\\overline{q}_{1}$ (by i) of Theorem D'),\n\n$p_{2}=4\\overline{\\theta }_{8}+2e_{8}-\\overline{q}_{1}^{2}$ (by iii) of\nTheorem D').\n\\end{quote}\n\n\\noindent Furthermore, by formula (6.6), together with the relation on \nH^{\\ast }(B_{Spin(8)})$\n\n\\begin{quote}\n$\\overline{q}_{1}^{r_{1}}\\cup p_{3}^{r_{2}}\\cup e_{8}^{r_{3}}\\cup \\delta\n_{2}(w_{6})=\\delta _{2}(w_{4}^{r_{1}}\\cup w_{6}^{2r_{2}}\\cup\nw_{8}^{r_{3}}\\cup w_{6})$,\n\\end{quote}\n\n\\noindent the ideal $\\tau (B_{Spin(8)})$ is generated by $\\delta _{2}(w_{6})\n. Summarizing we get\n\n\\begin{enumerate}\n\\item[(6.12)] $H^{\\ast }(B_{Spin(8)})=\\mathbb{Z}[\\overline{q\n_{1},p_{3},e_{8},\\overline{\\theta }_{8},\\delta _{2}(w_{6})]\/\\left\\langle\n2\\delta _{2}(w_{6})\\right\\rangle $.\n\\end{enumerate}\n\nSimilarly, one can show that\n\n\\begin{enumerate}\n\\item[(6.13)] $H^{\\ast }(B_{Spin(12)})=\\frac{\\mathbb{Z}[\\overline{q}_{1}\n\\overline{q}_{2},\\overline{q}_{3},\\overline{q}_{4},p_{3},p_{5},e_{12}\n\\overline{\\theta }_{12},\\delta _{2}(w_{6}),\\delta _{2}(w_{10})]}\n\\left\\langle 2\\delta _{2}(w_{6}),2\\delta\n_{2}(w_{10}),R_{32},R_{64}\\right\\rangle },$\n\\end{enumerate}\n\n\\noindent where\n\n\\begin{quote}\n$R_{32}=2\\overline{q}_{4}+\\overline{q}_{3}^{2}-\\overline{\\pi }^{\\ast\n}f(w_{2}^{(3)});$\n\n$R_{64}=4\\overline{\\theta }_{12}+\\overline{q}_{4}^{2}-\\overline{b}_{8}$,\n\\end{quote}\n\n\\noindent and where the deficiencies $\\overline{\\pi }^{\\ast }f(w_{2}^{(3)})$\nand $\\overline{b}_{8}$ can be computed explicitly (they are too lengthy to\nbe presented).$\\square $\n\n\\section{The subgroup $U(k)\\subset Spin(2k)$}\n\nThe unitary group $U(k)$ of order $k$ is contained in the special orthogonal\ngroup $SO(2k)$ as the centralizer of the circle subgroup $diag\\{z,\\cdots\n,z\\} $ with $k$ copies of $z\\in SO(2)$. It is also contained in the group \nSpin(2k)$ as the centralizer of the circle subgroup\n\n\\begin{center}\n$U(1)=\\{\\underset{1\\leq i\\leq k}{\\Pi }(\\cos t+\\sin t\\cdot e_{2i-1}e_{2i})\\in\nSpin(2k),t\\in \\mathbb{R}\\}$ (see \\cite[p.173]{BD}, \\cite{DL}).\n\\end{center}\n\n\\noindent Let $\\lambda _{0}:U(k)\\rightarrow SO(2k)$ and $\\lambda\n:U(k)\\rightarrow Spin(2k)$ be the inclusions, respectively, and define \n\\lambda ^{c}$ to be the homomorphism\n\n\\begin{quote}\n$\\lambda ^{c}:=\\lambda \\times _{\\mathbb{Z}_{2}}id:U^{c}(k):=U(k)\\times _\n\\mathbb{Z}_{2}}U(1)\\rightarrow Spin^{c}(2k)$.\n\\end{quote}\n\n\\noindent These homomorphisms then fit into the commutative diagram\n\n\\begin{enumerate}\n\\item[(7.1)] \n\\begin{array}{ccccc}\nU(k) & \\overset{a^{\\prime }}{\\rightarrow } & U^{c}(k) & \\overset{b^{\\prime }\n{\\rightarrow } & U(k) \\\\ \n\\lambda \\downarrow &  & \\lambda ^{c}\\downarrow &  & \\lambda _{0}\\downarrow\n\\\\ \nSpin(2k) & \\overset{a}{\\rightarrow } & Spin^{c}(2k) & \\overset{b}\n\\rightarrow } & SO(2k\n\\end{array\n$,\n\\end{enumerate}\n\n\\noindent where $a$ (resp. $a^{\\prime }$) denotes the inclusion onto the\nkernel of the epimorphism\n\n\\begin{quote}\n$Spin(2k)\\times _{\\mathbb{Z}_{2}}U(1)\\rightarrow U(1)$ (resp. $U(k)\\times _\n\\mathbb{Z}_{2}}U(1)\\rightarrow U(1)$),\n\\end{quote}\n\n\\noindent and where $b$ (resp. $b^{\\prime }$) is the obvious epimorphism\n\n\\begin{quote}\n$Spin(2k)\\times _{\\mathbb{Z}_{2}}U(1)\\rightarrow SO(2k)$ (resp. $U(k)\\times\n_{\\mathbb{Z}_{2}}U(1)\\rightarrow U(k)$).\n\\end{quote}\n\n\\noindent It is easy to see that\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 7.1.} \\textsl{The composition }$b\\circ a$\\textsl{\\\nis the universal covering on }$SO(2k)$\\textsl{.}\n\n\\textsl{The composition }$b^{\\prime }\\circ a^{\\prime }$\\textsl{\\ is the two\nsheets covering of }$U(k)$\\textsl{\\ by its self with associated }$\\func{mod\n2 $\\textsl{\\ Euler class the generator of }$H^{1}(U(k);\\mathbb{Z}_{2})\n\\mathbb{Z}_{2}$\\textsl{.}$\\square $\n\n\\bigskip\n\nThe homomorphisms $\\lambda $ and $\\lambda ^{c}$ are particularly relevant to\nthe understanding of the relationship between spinors and complex structures\non vector bundles \\cite{ABS}. The purpose of this section is to determine\ntheir induced maps on the cohomologies of the relevant classifying spaces.\nThe results will bring us direct recurrences to produce the integral Weyl\ninvariants of the groups $Spin^{c}(n)$ and $Spin(n)$, by the algorithms\nillustrated in Examples 7.6 and 7.9.\n\n\\bigskip\n\n\\textbf{7.1. The conventions.} For a homomorphism $h:H\\rightarrow G$ of Lie\ngroups denote by $B_{h}:B_{H}\\rightarrow B_{G}$ its induced map on the\nclassifying spaces. The diagram (7.1) then gives rise to the commutative\ndiagram of induced ring maps\n\n\\begin{enumerate}\n\\item[(7.2)] \n\\begin{array}{ccccc}\nH^{\\ast }(B_{U(k)}) & \\overset{B_{a^{\\prime }}^{\\ast }}{\\leftarrow } & \nH^{\\ast }(B_{U^{c}(k)}) & \\overset{B_{b^{\\prime }}^{\\ast }}{\\leftarrow } & \nH^{\\ast }(B_{U(k)}) \\\\ \nB_{\\lambda }^{\\ast }\\uparrow &  & B_{\\lambda ^{c}}^{\\ast }\\uparrow &  & \nB_{\\lambda _{0}}^{\\ast }\\uparrow \\\\ \nH^{\\ast }(B_{Spin(2k)}) & \\overset{\\psi ^{\\ast }}{\\longleftarrow } & H^{\\ast\n}(B_{Spin^{c}(2k)}) & \\overset{\\pi ^{\\ast }}{\\longleftarrow } & H^{\\ast\n}(B_{SO(2k)}\n\\end{array\n$\n\\end{enumerate}\n\n\\noindent where $\\psi =B_{a}$, $\\pi =B_{b}$ in the notation of (2.5) and\n(2.6). By the naturality of characteristic classes, and for the sake to\nsimplify notation, the following convention is adopted throughout this\nsection.\n\n\\bigskip\n\n\\textbf{Convention a).} Let $1+c_{1}+\\cdots +c_{k}$ be the total Chern class\nof the canonical bundle on $B_{U(k)}$, and recall that $H^{\\ast }(B_{U(k)})\n\\mathbb{Z}[c_{1},\\cdots ,c_{k}]$. The notion $c_{r}\\in H^{\\ast\n}(B_{U^{c}(k)})$ will be used to simplify $B_{b^{\\prime }}^{\\ast }(c_{r})$.\n\n\\bigskip\n\n\\textbf{Convention b). }Let $p_{r}\\in H^{\\ast }(B_{SO(2k)})$ be the $r^{th}$\nPontryagin class if $r<k$, or $e_{2k}^{2}$ if $r=k$. The same notion $p_{r}$\nwill also be reserved for both of the classes\n\n\\begin{quote}\n$\\pi ^{\\ast }p_{r}\\in H^{\\ast }(B_{Spin^{c}(2k)})$ and $\\psi ^{\\ast }\\pi\n^{\\ast }p_{r}\\in H^{\\ast }(B_{Spin(2k)})$,\n\\end{quote}\n\n\\noindent as well as the polynomial $B_{\\lambda _{0}}^{\\ast }(p_{r})\\in $ \nH^{\\ast }(B_{U(k)})$ and $B_{\\lambda ^{c}}^{\\ast }(p_{r})\\in $ $H^{\\ast\n}(B_{U^{c}(k)})$ whose canonical expression is (where $c_{i}=0$\\ if $i$\\ $>k\n)\n\n\\begin{enumerate}\n\\item[(7.3)] $p_{r}=c_{r}^{2}-2c_{r-1}c_{r+1}+\\cdots\n+2(-1)^{r-1}c_{1}c_{2r-1}+2(-1)^{r}c_{2r}$ (\\cite[p.177]{MS}).\n\\end{enumerate}\n\n\\textbf{Convention c).} The notion $e_{2k}$ for the Euler class of the\ncanonical bundle on $B_{SO(2k)}$ is also applied to denote either\n\n\\begin{quote}\n$\\pi ^{\\ast }e_{2k}\\in H^{\\ast }(B_{Spin^{c}(2k)})$ or $\\psi ^{\\ast }\\pi\n^{\\ast }e_{2k}\\in H^{\\ast }(B_{Spin(2k)})$.\n\\end{quote}\n\n\\noindent These convention will not cause any confusion, because in each\ncircumstance the cohomologies or the homomorphisms involved will be clearly\nstated.\n\n\\bigskip\n\n\\textbf{7.2. The operators }$(\\psi ,\\delta )$\\textbf{\\ on the} \\textbf\ncohomology }$H^{\\ast }(B_{U^{c}(k)})$\\textbf{.} The group $U^{c}(k)$ has two\nobvious $1$--dimensional unitary representations\n\n\\begin{quote}\n$U(k)\\times _{\\mathbb{Z}_{2}}U(1)\\rightarrow U(k)\\overset{\\det }{\\rightarrow \n}U(1)$ and $U(k)\\times _{\\mathbb{Z}_{2}}U(1)\\rightarrow U(1)$\n\\end{quote}\n\n\\noindent whose Euler classes are $c_{1}$ and $B_{\\lambda ^{c}}^{\\ast\n}(q_{0})$, respectively. By the commutivity of the second diagram in (7.2)\nwe have\n\n\\begin{quote}\n$\\rho _{2}(B_{\\lambda ^{c}}^{\\ast }(q_{0})+c_{1})=B_{\\lambda ^{c}}^{\\ast\n}\\circ B_{b}^{\\ast }(w_{2})+B_{b^{\\prime }}^{\\ast }\\circ B_{\\lambda\n_{0}}^{\\ast }(w_{2})=0$,\n\\end{quote}\n\n\\noindent implying that the sum $B_{\\lambda ^{c}}^{\\ast }(q_{0})+c_{1}\\in\nH^{2}(B_{U^{c}(k)})$ is divisible by $2$. This brings us the integral class\n\n\\begin{quote}\n$y:=\\frac{1}{2}(B_{\\lambda ^{c}}^{\\ast }(q_{0})+c_{1})\\in\nH^{2}(B_{U^{c}(k)}) $\n\\end{quote}\n\n\\noindent by which following result becomes transparent by Lemma 7.1.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 7.2.} \\textsl{We have }$H^{\\ast }(B_{U^{c}(k)})\n\\mathbb{Z}[y,c_{1},\\cdots ,c_{k}]$\\textsl{\\ so that}\n\n\\begin{quote}\n\\textsl{i)} $B_{b^{\\prime }}^{\\ast }(c_{r})=c_{r},1\\leq r\\leq k$;\n\n\\textsl{ii) }$B_{a^{\\prime }}^{\\ast }(z)=c_{1},2c_{1}$\\textsl{\\ or }$c_{r}\n\\textsl{\\ for }$z=y,c_{1}$\\textsl{\\ or }$c_{r}$\\textsl{\\ with }$r\\geq 2\n\\textsl{;}\n\n\\textsl{iii) }$B_{\\lambda ^{c}}^{\\ast }(q_{0})=2y-c_{1}$\\textsl{.}$\\square $\n\\end{quote}\n\nFor each ordered sequence $\\lambda =(\\lambda _{1},\\cdots ,\\lambda _{k})$ of \nk$ non-negative integers define\n\n\\begin{quote}\n$c_{\\lambda }:=c_{1}^{\\lambda _{1}}\\cdots c_{k}^{\\lambda _{k}}\\in H^{\\ast\n}(B_{U^{c}(k)})$.\n\\end{quote}\n\n\\noindent Since all the monomials $y^{r}c_{\\lambda }$, $r\\geq 0$, form an\nadditive basis of the cohomology $H^{\\ast }(B_{U^{c}(k)})$ every element \nu\\in H^{\\ast }(B_{U^{c}(k)})$ has the unique expansion\n\n\\begin{quote}\n$u=\\underset{(r,\\lambda )}{\\Sigma }u_{(r,\\lambda )}\\cdot y^{r}c_{\\lambda }$, \n$u_{(r,\\lambda )}\\in \\mathbb{Z}$.\n\\end{quote}\n\n\\noindent We may then introduce the operator $\\psi $ on the ring $H^{\\ast\n}(B_{U^{c}(k)})$ by\n\n\\begin{quote}\n$\\psi (u):=\\underset{(r,\\lambda )}{\\Sigma }\\rho _{2}(u_{(r,\\lambda )})\\cdot\ny^{2r}p_{\\lambda },$ $u\\in H^{\\ast }(B_{U^{c}(k)})$,\n\\end{quote}\n\n\\noindent where $p_{\\lambda }=p_{1}^{\\lambda _{1}}\\cdots p_{k}^{\\lambda\n_{k}} $ with $p_{i}$ the polynomials in the Chern classes $c_{r}$ given by\n(7.3). Alternatively,\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 7.3.} \\textsl{The operator }$\\psi $\\textsl{\\ is\ncharacterized uniquely by the following algorithmic properties, where }\nu,v\\in H^{\\ast }(B_{U^{c}(k)})$\\textsl{, }$b\\in \\mathbb{Z}$\\textsl{,}\n\n\\begin{quote}\n\\textsl{i) }$\\psi (u+b\\cdot y^{r}c_{\\lambda })=\\psi (u)+\\rho _{2}(b)\\cdot\ny^{2r}p_{\\lambda }$\\textsl{\\ if }$\\rho _{2}(u_{(r,\\lambda )})=0;$\n\n\\textsl{ii) }$\\psi (u\\cup v)=\\psi (u)\\cup \\psi (v)$\\textsl{.}$\\square $\n\\end{quote}\n\nIt follows from $p_{\\lambda }\\equiv c_{\\lambda }^{2}\\func{mod}2$ by (7.3)\nthat the operator $\\psi $ satisfies also the relation $\\psi (u)\\equiv u^{2\n\\func{mod}2$, which implies that there exists a unique operator $\\delta $ on\nthe ring $H^{\\ast }(B_{U^{c}(k)})$ that is related to $\\psi $ by the formula\n\n\\begin{quote}\n$\\psi (u)=u^{2}+2\\delta (u).$\n\\end{quote}\n\n\\noindent \\textbf{Definition 7.4. }For a polynomial $u\\in\nH^{2n}(B_{U^{c}(k)})$ the sequence $\\left\\{ u,\\delta (u),\\delta\n^{2}(u),\\cdots \\right\\} $ on $H^{\\ast }(B_{U^{c}(k)})$ defined inductively\nby $\\delta ^{r}(u)=\\delta (\\delta ^{r-1}(u)$ will be called \\textsl{the\nderived sequence} \\textsl{of} \\textsl{the initial polynomial} $u$.$\\square $\n\n\\bigskip\n\nSince the ring $H^{\\ast }(B_{U^{c}(k)})$ is torsion free we get\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 7.5. }\\textsl{For any }$u\\in H^{\\ast\n}(B_{U^{c}(k)})$\\textsl{\\ the derive sequence }$\\left\\{ u,\\delta (u),\\delta\n^{2}(u),\\cdots \\right\\} $\\textsl{\\ can be computed by }$\\psi $ \\textsl{via\nthe following recurrence relations}\n\n\\begin{quote}\n$\\delta ^{0}(u)=u$\\textsl{, }$2\\delta ^{r+1}(u)+\\delta ^{r}(u)^{2}=\\psi\n(\\delta ^{r}(u))$, $r\\geq 0$.$\\square $\n\\end{quote}\n\n\\bigskip\n\n\\noindent \\textbf{Example 7.6. }Corollaries 7.3 and 7.5 indicate a direct\nand effective recurrence to produce the sequence $\\left\\{ u,\\delta\n(u),\\delta ^{2}(u),\\cdots \\right\\} $ from the initial one $u$. For example\nwe take $u=2y-c_{1}\\in H^{2}(B_{U^{c}(k)})$. Then\n\n\\begin{quote}\n$\\delta ^{1}(u)=-c_{2}+2yc_{1}-2y^{2}$;\n\n$\\delta\n^{2}(u)=c_{4}-c_{1}c_{3}-2y^{2}c_{1}^{2}-2y^{4}+2yc_{1}c_{2}-2y^{2}c_{2}+4y^{3}c_{1} \n$;\n\n$\\delta\n^{3}(u)=c_{_{8}}-c_{_{1}}c_{_{7}}+c_{2}c_{6}-c_{3}c_{5}+(c_{1}^{2}-2c_{2})(-c_{2}c_{4}+c_{1}c_{5}-c_{6}) \n$\n\n$\\qquad\n-c_{2}c_{3}^{2}-c_{1}c_{3}c_{4}-2(-y^{2}c_{1}^{2}-y^{4}+yc_{1}c_{2}-y^{2}c_{2}+2y^{3}c_{1})^{2} \n$\n\n$\\qquad\n-2(c_{4}-c_{1}c_{3})(-y^{2}c_{1}^{2}-y^{4}+yc_{1}c_{2}-y^{2}c_{2}+2y^{3}c_{1}) \n$, $\\cdots $.\n\\end{quote}\n\n\\noindent In addition, it can be shown that\n\n\\begin{enumerate}\n\\item[(7.4)] $\\psi (\\delta ^{r}(u))\\in \\left\\langle p_{2},\\cdots\n,p_{k}\\right\\rangle $, $r>1$.$\\square $\n\\end{enumerate}\n\n\\textbf{7.3. The ring map }$B_{\\lambda ^{c}}^{\\ast }:H^{\\ast\n}(B_{Spin^{c}(2k)})\\rightarrow H^{\\ast }(B_{U^{c}(k)})$\\textbf{.} By the\nconventions in \\textbf{7.1} and with respect to the presentation\n\n\\begin{quote}\n$H^{\\ast }(B_{Spin^{c}(2k)})=\\pi ^{\\ast }H^{\\ast }(B_{SO(2k)})\\otimes \n\\mathbb{Z}[q_{0},q_{1},\\cdots ,q_{h(2k)-1},\\theta _{2k}]\/R_{2k}$,\n\\end{quote}\n\n\\noindent by Theorem D, partial information on the ring map $B_{\\lambda\n^{c}}^{\\ast }$ has already known. Precisely we have\n\n\\begin{quote}\n$B_{\\lambda ^{c}}^{\\ast }(e_{2k})=c_{k}$, \\textsl{\\ }$B_{\\lambda ^{c}}^{\\ast\n}(p_{r})=p_{r}$, $B_{\\lambda ^{c}}^{\\ast }(\\tau (B_{Spin^{c}(2k)}))=0$,\n\\end{quote}\n\n\\noindent where the third relation follows from the fact that the target\nring $H^{\\ast }(B_{U^{c}(k)})$ is torsion free. In addition, since $\\theta\n_{n}\\in H^{\\ast }(B_{Spin^{c}(n)})$ is the Euler class of the the complex\nspin presentation of the group $Spin^{c}(n)$, the polynomials $B_{\\lambda\n^{c}}^{\\ast }(\\theta _{n})$ can be computed in representation theory \\cit\n{ABS}. As examples we have\n\n\\begin{quote}\n$B_{\\lambda ^{c}}^{\\ast }(\\theta _{4})=y^{2}-yc_{1};$\n\n$B_{\\lambda ^{c}}^{\\ast }(\\theta\n_{6})=y^{4}-2y^{3}c_{1}+y^{2}(c_{2}+c_{1}^{2})-y(c_{1}c_{2}-c_{3})$;\n\n$B_{\\lambda ^{c}}^{\\ast }(\\theta\n_{8})=y^{8}-4y^{7}c_{1}+y^{6}(2c_{2}+6c_{1}^{2})-y^{5}(6c_{3}+4c_{1}^{3}+6c_{1}c_{2}) \n$\n\n$\\\n+y^{4}(c_{1}^{4}+c_{2}^{2}+6c_{1}^{2}c_{2}+c_{1}c_{3}-4c_{4})-y^{3}(2c_{1}c_{2}^{2}-8c_{1}c_{4}+2c_{1}^{2}c_{3}+2c_{1}^{3}c_{2}) \n$\n\n\\ \\ \n+y^{2}(c_{1}^{3}c_{3}-22c_{2}c_{4}+c_{1}c_{2}c_{3}-c_{3}^{2}-5c_{1}^{2}c_{4}+\\allowbreak c_{1}^{2}c_{2}^{2})-y(c_{1}^{2}c_{2}c_{3}-c_{1}^{3}c_{4}-c_{1}c_{3}^{2}) \n$.\n\\end{quote}\n\n\\noindent Summarizing, the determination of the ring map $B_{\\lambda\n^{c}}^{\\ast }$ is reduced to express the sequence $\\{B_{\\lambda ^{c}}^{\\ast\n}(q_{r}),0\\leq r\\leq h(n)-1\\}$ as explicit polynomials in $H^{\\ast\n}(B_{U^{c}(k)})$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 7.7.} \\textsl{The sequence }$\\{B_{\\lambda\n^{c}}^{\\ast }(q_{r}),0\\leq r\\leq h(n)-1\\}$\\textsl{\\ is the first }$h(n)$ \n\\textsl{terms of the} \\textsl{derived sequence }$\\left\\{ u,\\delta (u),\\delta\n^{2}(u),\\cdots \\right\\} $ \\textsl{of} $u=c_{1}-2y$\\textsl{\\ (see in Example\n7.6).}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} The proof serves also the purpose to bring a\npassage from the operators $\\{f,\\gamma \\}$ on $H^{\\ast }(B_{SO(2k)};\\mathbb{\n}_{2})$ (Theorem B) to the ones $\\{\\psi ,\\delta \\}$ on $H^{\\ast\n}(B_{U^{c}(k)})$ via the map $B_{\\lambda ^{c}}^{\\ast }$. In view of the\npresentation\n\n\\begin{quote}\n$H^{\\ast }(B_{U^{c}(k)};\\mathbb{Z}_{2})=\\mathbb{Z}_{2}[\\rho _{2}(y),\\rho\n_{2}(c_{1}),\\cdots ,\\rho _{2}(c_{k})]$\n\\end{quote}\n\n\\noindent by Lemma 7.2 define the operator $R:H^{2k}(B_{U^{c}(k)};\\mathbb{Z\n_{2})\\rightarrow H^{4k}(B_{U^{c}(k)})$ by\n\n\\begin{enumerate}\n\\item[(7.5)] $R(u):=\\left\\{ \n\\begin{tabular}{l}\n$y^{2r}p_{\\lambda }$ if $u=\\rho _{2}(y^{r}c_{\\lambda })$ is a monomial; \\\\ \n$R(u_{1})+\\cdots +R(u_{m})$ if $u=$ $u_{1}+\\cdots +u_{m}\n\\end{tabular\n\\right. $\n\\end{enumerate}\n\n\\noindent where the $u_{i}$'s are distinct monomials in the $\\rho\n_{2}(y),\\rho _{2}(c_{1}),\\cdots ,\\rho _{2}(c_{k})$. Then, in addition to the\nobvious factorization\n\n\\begin{enumerate}\n\\item[(7.6)] $\\psi =R\\circ \\rho _{2}$,\n\\end{enumerate}\n\n\\noindent comparison with the formula (2.4) of the operator $f$ tells that\n\n\\begin{enumerate}\n\\item[(7.7)] $R\\circ B_{\\lambda ^{c}}^{\\ast }\\circ \\pi ^{\\ast }=B_{\\lambda\n^{c}}^{\\ast }\\circ \\pi ^{\\ast }\\circ f$ (where $\\pi ^{\\ast }=B_{b}^{\\ast }$).\n\\end{enumerate}\n\n\\noindent On the other hand, by the second diagram in (7.2), applying the\nring map $B_{\\lambda ^{c}}^{\\ast }$ to the relation ii) and iii) of Theorem\nC gives rise to\n\n\\begin{enumerate}\n\\item[(7.8)] $\\rho _{2}\\circ B_{\\lambda ^{c}}^{\\ast }(q_{r})=B_{\\lambda\n^{c}}^{\\ast }\\circ \\pi ^{\\ast }(w_{2}^{(r)})$ on $H^{\\ast }(B_{U^{c}(k)}\n\\mathbb{Z}_{2})$, and\n\n\\item[(7.9)] $2B_{\\lambda ^{c}}^{\\ast }(q_{r+1})+B_{\\lambda ^{c}}^{\\ast\n}(q_{r})^{2}=B_{\\lambda ^{c}}^{\\ast }\\circ \\pi ^{\\ast }f(w_{2}^{(r)})$, \nr\\geq 0$,\\textsl{\\ }on $H^{\\ast }(B_{U^{c}(k)})$,\n\\end{enumerate}\n\n\\noindent respectively. Setting $u_{r+1}:=B_{\\lambda ^{c}}^{\\ast }(q_{r})$\nthese imply that\n\n\\begin{quote}\n$2u_{r+1}+u_{r}^{2}=B_{\\lambda ^{c}}^{\\ast }\\circ \\pi ^{\\ast }\\circ\nf(w_{2}^{(r)})$ (by (7.9))\n\n$=R\\circ B_{\\lambda ^{c}}^{\\ast }\\circ \\pi ^{\\ast }(w_{2}^{(r)})$ (by (7.7))\n\n$=R\\circ \\rho _{2}(u_{r})$ (by (7.8))\n\n$=\\psi (u_{r})$ (by (7.6)).\n\\end{quote}\n\n\\noindent With $u_{1}=2y-c_{1}$ by iii) of Lemma 7.2 we obtain the result by\nCorollary 7.5.$\\square $\n\n\\bigskip\n\n\\textbf{7.4. The ring map }$B_{\\lambda }^{\\ast }:H^{\\ast\n}(B_{Spin(2k)})\\rightarrow H^{\\ast }(B_{U(k)})$\\textbf{. }Carrying on\ndiscussion from Example 7.6 let $\\left\\{ u,\\delta (u),\\delta ^{2}(u),\\cdots\n\\right\\} $ be the derived sequence of the polynomial $u=2y-c_{1}$. As\nelements in the polynomial ring $H^{\\ast }(B_{U^{c}(k)})$ we can write for\neach $r\\geq 1$ that\n\n\\begin{quote}\n$\\delta ^{r}(u)=u^{(r)}(y,c_{1},c_{2},\\cdots ,c_{k})$.\n\\end{quote}\n\n\\noindent Applying the ring map $B_{a^{\\prime }}^{\\ast }$ to the polynomials \n$p_{r},\\delta ^{r}(u)$, $\\psi (\\delta ^{r}(u))\\in H^{\\ast }(B_{U^{c}(k)})$\nwe obtain by ii) of Lemma 7.2 the following polynomials in $H^{\\ast\n}(B_{U(k)}):$\n\n\\begin{enumerate}\n\\item[(7.10)] $g_{r}:=B_{a^{\\prime }}^{\\ast\n}(p_{r})=p_{r}+2(-1)^{r-1}c_{1}c_{2r-1}$ (see (7.3) for $p_{r}$);\n\n\\item[(7.11)] $\\alpha _{r}:=B_{a^{\\prime }}^{\\ast }(\\delta\n^{r}(u))=u^{(r)}(c_{1},2c_{1},c_{2},\\cdots ,c_{k})$,\n\n\\item[(7.12)] $f_{r}:=B_{a^{\\prime }}^{\\ast }(\\psi (\\delta\n^{r}(u))=u^{(r)}(g_{1},2g_{1},g_{2},\\cdots ,g_{k})$\n\\end{enumerate}\n\n\\noindent On the other hand, according to Theorem D' and by the conventions\nin 7.1, the ring $H^{\\ast }(B_{Spin(2k)})$ is generated multiplicatively by\nthe integral classes\n\n\\begin{quote}\n$\\{p_{r},$ $1\\leq r\\leq k-1\\}$; $\\{\\overline{q}_{r},1\\leq r\\leq h(n)-1\\}$; \ne_{2k}$, $\\overline{\\theta }_{n}$,\n\\end{quote}\n\n\\noindent together with the ideal $\\tau (B_{Spin^{c}(n)})$. Thus, we obtain\nfrom Theorem 7.7, the relation $\\overline{q}_{r}=\\psi ^{\\ast }(q_{r})$ by\nTheorem C', as well as the commutivity of the first diagram in (7.2), the\nfollowing result.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 7.8.} \\textsl{The map }$B_{\\lambda }^{\\ast\n}:H^{\\ast }(B_{Spin(2k)})\\rightarrow H^{\\ast }(B_{U(k)})$\\textsl{\\ is\ndetermined by}\n\n\\begin{quote}\n\\textsl{i)} $B_{\\lambda }^{\\ast }(p_{r})=g_{r},$ $1\\leq r\\leq k-1$;\n\n\\textsl{ii)} $B_{\\lambda }^{\\ast }(\\overline{q}_{r})=\\alpha _{r},1\\leq r\\leq\nh(n)-1$,\n\\end{quote}\n\n\\noindent \\textsl{and by }$B_{\\lambda }^{\\ast }(e_{2k})=c_{k},$ $B_{\\lambda\n}^{\\ast }(\\tau (B_{Spin^{c}(n)}))=0$\\textsl{.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 7.9. }We shall show in Section \\S 8 that the\nsequence $\\{\\alpha _{r},1\\leq r\\leq h(n)-1\\}$ (as well as $\\{p_{r},1\\leq\nr\\leq k-1\\}$) of polynomials in the Chern classes are integral Weyl\ninvariants of the group $Spin(n)$. We emphasize at this stage that these\npolynomials can be effectively produced by the simple algorithm indicated by\nCorollary 7.5, together with the formula (7.11). As examples, combining\nresults in Example 7.6 with formula (7.11) we obtain that\n\n\\begin{quote}\n$\\alpha _{1}=-c_{2}+2c_{1}^{2}$;\n\n$\\alpha _{2}=c_{4}-2c_{1}c_{3}+2c_{1}^{2}c_{2}-2c_{1}^{4}$;\n\n$\\alpha\n_{3}=-c_{8}+2c_{1}c_{7}-c_{2}c_{6}+c_{3}c_{5}-(2c_{1}^{2}-c_{2})(c_{3}^{2}-2c_{2}c_{4}+4c_{1}c_{5}-2c_{6}) \n$\n\n$\\qquad\n-2c_{4}(c_{1}c_{3}-c_{1}^{2}c_{2}+c_{1}^{4})+2(c_{1}c_{3}-c_{1}^{2}c_{2}+c_{1}^{4})^{2},\\cdots \n${\\small .}$\\square $\n\\end{quote}\n\n\\bigskip\n\n\\textbf{7.5.} We conclude this section with the following result which has\nplayed a role in showing Theorem D'.\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 7.10.} \\textsl{The constant }$\\kappa $\\textsl{\\ in\nthe formula (6.9) is }$2(-1)^{k(n)-1}$\\textsl{.}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} Let $D$ be the ideal on $H^{\\ast }(B_{U(k)})\n\\mathbb{Z}[c_{1},\\cdots ,c_{k}]$ generated by the $c_{r}$ with $r\\geq 2$,\nand consider the ring map $e:H^{\\ast }(B_{U(k)})\\rightarrow \\mathbb{Z}$\ndefined by\n\n\\begin{quote}\n$e(c_{1})=1$, $e(c_{r})=0$, $2\\leq r\\leq k$.\n\\end{quote}\n\n\\noindent That is, for every $u\\in H^{2k}(B_{U(k)})$,\n\n\\begin{quote}\ni) $u=e(u)\\cdot c_{1}^{k}+h(u)$ with $h(u)\\in D$;\n\nii) $e(u)=0$ if and only if $u\\in D$.\n\\end{quote}\n\n\\noindent In particular, applying $e$ to the relations\n\n\\begin{quote}\n$2\\alpha _{r+1}+\\alpha _{r}^{2}=f_{r}(g_{1},\\cdots ,g_{k-1})$ (by Theorem D'\nand (7.4))\n\\end{quote}\n\n\\noindent on $H^{\\ast }(B_{U(k)})$ we get from $e(\\alpha _{1})=2$ (by\nExample 7.9) that\n\n\\begin{enumerate}\n\\item[(7.13)] $e(\\alpha _{r})=2(-1)^{r-1}$.\n\\end{enumerate}\n\nOn the other hand, since $B_{\\lambda }^{\\ast }(\\overline{\\theta }_{n})$ is\nthe Euler class of the induced bundle $B_{\\lambda _{n}}^{\\ast }\\eta _{n}$ on \n$B_{U(k)}$ we have\n\n\\begin{quote}\n$e(B_{\\lambda }^{\\ast }(\\overline{\\theta }_{n}))=1$ (i.e. $B_{\\lambda\n}^{\\ast }(\\overline{\\theta }_{n})=c_{1}^{2^{k(n)}}+\\alpha $ for some $\\alpha\n\\in D$).\n\\end{quote}\n\n\\noindent Thus, applying the ring map $e\\circ B_{\\lambda }^{\\ast }$ to the\nequation (6.8) and noting that $B_{\\lambda }^{\\ast }(\\varphi (\\overline{b\n_{n}))\\in D$ (since $\\overline{b}_{n}\\in A_{1}$), we obtain \n2(-1)^{k(n)-1}=\\kappa $ as required.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Remark 7.10. }For a $CW$ complex $X$ the maps $B_{\\lambda\n_{0}}$, $B_{\\lambda }$ and $B_{\\lambda ^{c}}$ induce, respectively, the\ncorrespondences between homotopy sets\n\n\\begin{quote}\n$B_{\\lambda _{0\\ast }}:[X,B_{U(k)}]\\rightarrow \\lbrack X,B_{SO(2k)}]$ by \nB_{\\lambda _{0}\\ast }[g]=[B_{\\lambda _{0}}\\circ g]$,\n\n$B_{\\lambda _{\\ast }}:[X,B_{U(k)}]\\rightarrow \\lbrack X,B_{Spin(2k)}]$ by \nB_{\\lambda _{\\ast }}[g]=[B_{\\lambda }\\circ g]$,\n\n$B_{\\lambda _{\\ast }^{c}}:[X,B_{U^{c}(k)}]\\rightarrow \\lbrack\nX,B_{Spin^{c}(2k)}]$ by $B_{\\lambda _{\\ast }^{c}}[g]=[B_{\\lambda ^{c}}\\circ\ng]$,\n\\end{quote}\n\n\\noindent in which $B_{\\lambda _{0\\ast }}$ is well known to be \\textsl{the\nreal reduction} on the complex bundles \\cite[p.155]{MS}. Likewise, the maps \nB_{\\lambda _{\\ast }}$ and $B_{\\lambda _{\\ast }^{c}}$ can be regarded as the \n\\textsl{spin} and the \\textsl{spin}$^{c}$\\textsl{\\ reduction} of complex\nbundles, respectively. In this connection, the formulae in Theorem 7.8\nexpress the Spin characteristic classes (see in \\S 9) of the spin reduction\nof a complex vector bundle by its Chern classes.\n\nThe map $B_{\\lambda }$ fits also into the fibration\n\n\\begin{quote}\n$Spin(2k)\/U(k)\\hookrightarrow B_{U(k)}\\rightarrow B_{Spin(2k)}$\n\\end{quote}\n\n\\noindent that is also of geometric significances \\cite{D,D1}: the fiber\nmanifold $Spin(2k)\/U(k)$ acts as the Grassmanian of complex structures on\nthe $2k$ dimensional Euclidean space $\\mathbb{R}^{2k}$, and can be\nidentified with the classifying space of the complex $k$ bundles whose real\n(resp. $Spin$) reductions are trivial.$\\square $\n\n\\section{The Weyl invariants of the groups $Spin(n)$}\n\nLet $G$ be a compact connected Lie group with a maximal torus $T$, and the\nWeyl group $W=N_{G}(T)\/T$. The canonical $W$ action on $T$ induces an action\non the integral cohomology $H^{\\ast }(B_{T})$. Denote by $H^{\\ast\n}(B_{T})^{W}$ the subring consisting of all the $W$ invariants, and let \nB_{t}:B_{T}\\rightarrow B_{G}$ be induced by the inclusion $t:T\\rightarrow G\n. A classical result of Borel \\cite{B2} states that\n\n\\bigskip\n\n\\noindent \\textbf{Lemma 8.1.} \\textsl{The ring map }$B_{t}^{\\ast }:H^{\\ast\n}(B_{G})\\rightarrow H^{\\ast }(B_{T})$\\textsl{\\ annihilates the torsion ideal \n}$\\tau (B_{G})$\\textsl{, and induces an injection }$H^{\\ast }(B_{G})\/\\tau\n(B_{G})\\rightarrow H^{\\ast }(B_{T})^{W}$\\textsl{.}$\\square $\n\n\\bigskip\n\nThe fundamental problem of the invariant theory of Weyl groups is to present\nthe ring $H^{\\ast }(B_{T})^{W}$ by explicit generators and relations \\cit\n{Gu,We}. A closely related problem in topology is to decide the subring \n\\func{Im}B_{t}^{\\ast }\\subseteq H^{\\ast }(B_{T})^{W}$. For the Weyl group $W$\nof a semi--simple Lie group $G$ Chevalley has shown that\n\n\\begin{quote}\n$H^{\\ast }(B_{T})^{W}\\otimes \\mathbb{Z}_{0}=H^{\\ast }(B_{G})\\otimes \\mathbb{\n}_{0}=\\mathbb{Z}_{0}[P_{1},\\cdots ,P_{n}]$, $n=\\dim T$,\n\\end{quote}\n\n\\noindent where \\textsl{the basic (rational) polynomial} \\textsl{invariants \n$\\left\\{ P_{1},\\cdots ,P_{n}\\in H^{\\ast }(B_{T})^{W}\\otimes \\mathbb{Z\n_{0}\\right\\} $ have been made explicitly by Mehta \\cite{Me}. In addition,\nfor a prime $p$ methods to calculate the algebras $H^{\\ast\n}(B_{T})^{W}\\otimes \\mathbb{Z}_{p}$ have been developed from the\nperspectives of algebraic topology \\cite{KM,Sm}, combinatorics \\cite{St},\nand algorithm \\cite{Bt}. However, apart from Borel's classical result \\cite\n\\S 3. Examples]{Fe1}\n\n\\begin{quote}\n$\\func{Im}B_{t}^{\\ast }=H^{\\ast }(B_{T})^{W}=H^{\\ast }(B_{G})$ for $G=SU(n)$\nor $Sp(n)$,\n\\end{quote}\n\n\\noindent complete information on the ring $H^{\\ast }(B_{T})^{W}$ are not\nknown for most other semi--simple Lie groups. In particular, the types of\nthe Weyl groups of $G=Spin(n)$ consist of $B_{k}$ ($n=2k+1$) or $D_{k}$ (\nn=2k$), and partial information has been obtained by Borel, Feshbach,\nTotaro, Benson and Wood \\cite{B2,BW,Fe1,T}.\n\nFor $G=Spin(n)$ our approach to $H^{\\ast }(B_{T})^{W}$ begins with the\nsubring $\\func{Im}B_{t}^{\\ast }$. For an integer $n>6$ (see Remark 1.1) we\nset $k=\\left[ \\frac{n}{2}\\right] $, and let $h$ be the inclusion of the\ndiagonal subgroup $T=U(1)\\times \\cdots \\times U(1)$ ($k$ copies) into $U(k)\n. Then a convenient maximal torus on $Spin(n)$ is\n\n\\begin{quote}\n$t=\\lambda \\circ h:T\\rightarrow U(k)\\rightarrow Spin(2k)\\subseteq Spin(n)$,\n\\end{quote}\n\n\\noindent where $\\lambda $ is the inclusion given in table (7.1).\nFurthermore, with respect to the canonical presentation\n\n\\begin{enumerate}\n\\item[(8.1)] $H^{\\ast }(B_{T})=\\mathbb{Z}[x_{1},\\cdots ,x_{k}]$, $\\deg\nx_{i}=2$,\n\\end{enumerate}\n\n\\noindent the ring map $B_{h}^{\\ast }:$ $H^{\\ast }(B_{U(k)})\\rightarrow\nH^{\\ast }(B_{T})$ is given by\n\n\\begin{quote}\n$B_{h}^{\\ast }(c_{r})=e_{r}(x_{1},\\cdots ,x_{k})$, $1\\leq r\\leq k$,\n\\end{quote}\n\n\\noindent where $e_{r}$ is the $r^{th}$ elementary symmetric function in the \n$x_{i}$'s. It follows that $B_{h}^{\\ast }$ carries $H^{\\ast }(B_{U(k)})$\\\nisomorphically onto the subring $Sym[x_{1},\\cdots ,x_{k}]$\\ of symmetric\nfunctions, while $B_{\\lambda }^{\\ast }$\\ induces an injection from the\nquotient ring $H^{\\ast }(B_{Spin(n)})\/\\tau (B_{Spin(n)})$ into $H^{\\ast\n}(B_{U(k)})$. Thus, applying the ring map $B_{t}^{\\ast }$ to the formula\n(6.10) of the ring $H^{\\ast }(B_{Spin(n)})$ we obtain by Theorem 7.8 the\nfollowing characterization of the subring $\\func{Im}B_{t}^{\\ast }\\subseteq\nH^{\\ast }(B_{T})^{W}$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 8.2.} \\textsl{The subring }$\\func{Im}B_{t}^{\\ast }\n\\textsl{\\ has the presentations:}\n\n\\begin{enumerate}\n\\item[(8.2)] $\\func{Im}B_{t}^{\\ast }=\\left\\{ \n\\begin{tabular}{l}\n$\\mathbb{Z}[g_{1},\\cdots ,g_{k-1},c_{k},\\alpha _{1},\\cdots ,\\alpha\n_{k(n)-1},B_{t}^{\\ast }(\\overline{\\theta }_{n})]\/D_{n}$ \\textsl{if} $n=2k$;\n\\\\ \n$\\mathbb{Z}[g_{1},\\cdots ,g_{k},\\alpha _{1},\\cdots ,\\alpha\n_{k(n)-1},B_{t}^{\\ast }(\\overline{\\theta }_{n})]\/D_{n}$ \\textsl{if} $n=2k+1$\n\\end{tabular\n\\right. $\n\\end{enumerate}\n\n\\noindent \\textsl{where }$D_{n}$\\textsl{\\ denotes the ideal generated by the\nfollowing relations}\n\n\\begin{quote}\n\\textsl{i)} $2\\alpha _{1}-g_{1}$\\textsl{,} $2\\alpha _{r+1}+\\alpha\n_{r}^{2}-f_{r}(g_{1},\\cdots ,g_{k})$\\textsl{,} $1\\leq r\\leq k(n)-2$\\textsl{,}\n\n\\textsl{ii)} $(-1)^{k(n)-1}\\cdot 4\\cdot B_{t}^{\\ast }(\\overline{\\theta \n_{n})+\\alpha _{k(n)-1}^{2}-\\beta _{n}$\\textsl{,}\n\\end{quote}\n\n\\noindent \\textsl{where }$\\beta _{n}:=B_{t}^{\\ast }(\\overline{b}_{n})\n\\textsl{,} \\textsl{and where the invariants }$g_{r}$\\textsl{,} $\\alpha _{r}$ \n\\textsl{and} $f_{r}(g_{1},\\cdots ,g_{k})$ \\textsl{are given by formulae\n(7.10), (7.11) and (7.12), respectively.}$\\square $\n\n\\bigskip\n\nIn term of (8.2) define $\\func{Im}\\overline{B}_{t}^{\\ast }\\subset $\\ $\\func\nIm}B_{t}^{\\ast }$ to be the subring generated by\n\n\\begin{quote}\n$g_{2},\\cdots ,g_{\\left[ \\frac{n-1}{2}\\right] },\\alpha _{1},\\cdots ,$ \n\\alpha _{k(n)-2}$,\\ and $c_{k}$\\ if $n=2k$.\n\\end{quote}\n\n\\noindent Then, for the degree reason, the relation $2\\alpha\n_{k(n)-1}+\\alpha _{k(n)-2}^{2}=f_{k(n)-2}(g_{1},\\cdots ,g_{k})$ on $\\func{Im\nB_{t}^{\\ast }$ implies that\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 8.3. }\\textsl{The quotient group }$\\func{Im\nB_{t}^{2^{k(n)}}\/\\func{Im}\\overline{B}_{t}^{2^{k(n)}}$\\textsl{\\ is\nisomorphic to }$\\mathbb{Z}_{2}$\\textsl{\\ with generator }$\\alpha _{k(n)-1}\n\\textsl{.}$\\square $\n\n\\bigskip\n\nFor $G=Spin(n)$ the extension problem from $\\func{Im}B_{t}^{\\ast }$ to the\nring $H^{\\ast }(B_{T})^{W}$ was raised by Borel \\cite[1954]{B2}, studied by\nFeshbach \\cite[1981]{Fe1}, and has been solved by Benson and Wood in the\nremarkable work \\cite[1995]{BW}. Efforts to bring the relevant constructions\nand calculations taking place in \\cite{BW} into our context gives rise to\nthe following results.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 8.4 (Benson and Wood).} \\textsl{Assume that }\nG=Spin(n)$ \\textsl{with} $n>6$\\textsl{.}\n\n\\textsl{i) If }$n\\neq 3,4,5\\func{mod}8$\\textsl{\\ then }$\\func{Im}B_{t}^{\\ast\n}=H^{\\ast }(B_{T})^{W}$\\textsl{.}\n\n\\textsl{ii) If} $n$\\textsl{\\ }$=3,4,5\\func{mod}8$\\textsl{,\\ then }$H^{\\ast\n}(B_{T})^{W}$ \\textsl{is} \\textsl{generated by its subring }$\\func{Im\nB_{t}^{\\ast }$\\textsl{, together with an additional invariant }$\\omega\n_{k(n)-1}\\in H^{2^{k(n)}}(B_{T})^{W}$\\textsl{\\ that is related to the known\ninvariants }$B_{t}^{\\ast }(\\overline{\\theta }_{n})$ \\textsl{and} $\\alpha\n_{k(n)-1}$ \\textsl{by the relations}\n\n\\begin{quote}\n\\textsl{a) }$\\omega _{k(n)-1}^{2}=B_{\\lambda }^{\\ast }(\\overline{\\theta \n_{n})$\\textsl{; }\n\n\\textsl{b) }$2\\omega _{k(n)-1}-\\alpha _{k(n)-1}=l_{n}$\\textsl{, }$l_{n}\\in \n\\func{Im}\\overline{B}_{t}^{2^{k(n)}}$\\textsl{.}\n\\end{quote}\n\n\\noindent \\textbf{Proof. }With respect to $n=2k+1$ or $n=2k$ consider the\nsequence of cohomology classes introduced in \\cite[\\S 4]{BW}\n\n\\begin{quote}\n$\\left\\{ \\eta _{j}\\in H^{\\ast }(B_{T})\\text{, }1\\leq j\\leq k\\right\\} $ or \n\\left\\{ \\mu _{j}\\in H^{\\ast }(B_{T})\\text{, }1\\leq j\\leq k\\right\\} $.\n\\end{quote}\n\n\\noindent Then by \\cite[Table 2]{BW} we have\n\n\\begin{enumerate}\n\\item[(8.3)] $B_{\\lambda }^{\\ast }(\\overline{\\theta }_{n})=\\eta _{k(n)}$ or \n\\mu _{k(n)\\text{ }}$ if $n\\equiv 3,5\\func{mod}8$ or $n\\equiv 4\\func{mod}8$.\n\\end{enumerate}\n\n\\noindent Let us put\n\n\\begin{quote}\n$\\omega _{k(n)-1}:=\\eta _{k(n)-1}$ if $n\\equiv 3,5\\func{mod}8$, or $\\mu\n_{k(n)-1}$ if $n\\equiv 4\\func{mod}8$.\n\\end{quote}\n\n\\noindent Then Benson and Wood \\cite[Proposition 4.1]{BW} has shown that\n\n\\begin{enumerate}\n\\item[(8.4)] $\\omega _{k(n)-1}\\in H^{\\ast }(B_{T})^{W}$ and $\\omega\n_{k(n)-1}^{2}=B_{t}^{\\ast }(\\overline{\\theta }_{n})$.\n\\end{enumerate}\n\n\\noindent Now, apart from for the relation b) all the statements of the\ntheorem are verified by comparing \\cite[Theorem 7.1]{BW} with \\cite[Theorem\n10.2]{BW}.\n\nConsider the sequence $\\left\\{ q_{r},r\\geq 1\\right\\} $ on $H^{\\ast\n}(B_{T})^{W}$ constructed in the proof of \\cite[Proposition 3.3]{BW}. By \n\\cite[Corollary 7.2]{BW}\n\n\\begin{enumerate}\n\\item[(8.5)] $2\\omega _{k(n)-1}-q_{k(n)-1}=l_{n}$ for some $l_{n}\\in \\func{I\n}\\overline{B}_{t}^{2^{k(n)}}$.\n\\end{enumerate}\n\n\\noindent On the other hand, by ii) and iii) of \\cite[Theorem 10.2]{BW} the\nclass $q_{k(n)-1}$ generates also the quotient group\n\n\\begin{quote}\n$\\func{Im}B_{t}^{2^{k(n)}}\/\\func{Im}\\overline{B}_{t}^{2^{k(n)}}=\\mathbb{Z\n_{2}$.\n\\end{quote}\n\n\\noindent Comparing this with Corollary 8.3 we can replace in (8.5) the\nclass $q_{k(n)-1}$ by our $\\alpha _{k(n)-1}$ to obtain the desired relation\nb).$\\square $\n\n\\bigskip\n\nAssume that $n$\\textsl{\\ }$=3,4,5\\func{mod}8$. The relations a) and b) of\nTheorem 8.4 imply that the generators $B_{t}^{\\ast }(\\overline{\\theta }_{n})\n\\textsl{\\ }and\\textsl{\\ }$\\alpha _{k(n)-1}$ of $\\func{Im}B_{t}^{\\ast }$ can\nbe expressed as polynomials in the elements\n\n\\begin{quote}\n$\\omega _{k(n)-1}$, $g_{2},\\cdots ,g_{\\left[ \\frac{n-1}{2}\\right] },\\alpha\n_{1},\\cdots ,\\alpha _{k(n)-2}$, and $c_{k}$\\ if $n=2k$.\n\\end{quote}\n\n\\noindent In addition, combining the relation b) with the relation on $\\func\nIm}B_{t}^{\\ast }$\n\n\\begin{quote}\n$2\\alpha _{k(n)-1}+\\alpha _{k(n)-2}^{2}=f_{k(n)-2}(g_{1},\\cdots ,g_{k})$ (by\nTheorem 8.2)\n\\end{quote}\n\n\\noindent one gets\n\n\\begin{quote}\n$4\\omega _{k(n)-1}-\\alpha _{k(n)-2}^{2}=\\varepsilon _{n}$ for some \n\\varepsilon _{n}\\in \\func{Im}\\overline{B}_{t}^{\\ast }$.\n\\end{quote}\n\n\\noindent Thus, putting Theorems 8.2 and Theorem 8.4 together we obtain that\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 8.5.} \\textsl{Assume that }$G=Spin(n)$ \\textsl{wit\n} $n>6$\\textsl{\\ (see Remark 1.1).} \\textsl{Then }\n\n\\begin{enumerate}\n\\item[(8.6)] $H^{\\ast }(B_{T})^{W}=\\left\\{ \n\\begin{tabular}{l}\n$\\func{Im}B_{t}^{\\ast }\\text{ \\textsl{if} }n\\QTR{sl}{\\ }\\neq 3,4,5\\func{mod}\n\\text{;}$ \\\\ \n$\\func{Im}\\overline{B}_{t}^{\\ast }\\otimes \\mathbb{Z}[\\omega\n_{k(n)-1}]\/\\left\\langle h_{n}\\right\\rangle \\text{ \\textsl{if} }n\\QTR{sl}{\\ \n\\equiv 3,4,5\\func{mod}8\\text{,}\n\\end{tabular\n\\right. $\n\\end{enumerate}\n\n\\noindent \\textsl{where }$h_{n}=4\\cdot \\omega _{k(n)-1}-\\alpha\n_{k(n)-2}^{2}-\\varepsilon _{n}$\\textsl{\\ with }$\\varepsilon _{n}\\in \\func{Im\n\\overline{B}_{t}^{\\ast }.\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Remarks 8.6. }Theorems 8.2 and 8.5 present both of the\nrings $\\func{Im}B_{t}^{\\ast }$ and $H^{\\ast }(B_{T})^{W}$ by the explicit\ngenerators (e.g. (7.10) and (7.11))\n\n\\begin{quote}\n$g_{1},\\cdots ,g_{k-1},c_{k},\\alpha _{1},\\cdots ,\\alpha\n_{k(n)-1},B_{t}^{\\ast }(\\overline{\\theta }_{n})$\n\\end{quote}\n\n\\noindent together with the invariant $\\omega _{k(n)-1}$ given by Benson and\nWood \\cite[\\S 4]{BW}.\n\nIn \\cite[Theorem 7.1]{BW} Benson and Wood obtained a presentation of the\nring $H^{\\ast }(B_{T})^{W}$ without specifying the relations among their\ngenerators. In our approach the recurrence relations\n\n\\begin{quote}\n$2\\alpha _{1}-g_{1}$, $2\\alpha _{r+1}+\\alpha _{r}^{2}-f_{r}(g_{1},\\cdots\n,g_{k}),$ $1\\leq r\\leq k(n)-2$,\n\\end{quote}\n\n\\noindent are originated from property ii) of Theorem C', which are useful\nto produce the sequence $\\{\\alpha _{1},\\alpha _{2},\\cdots \\}$ of invariants\nfrom the initial one $\\alpha _{1}=-c_{2}+2c_{1}^{2}$, see Examples 7.6 and\n7.9.$\\square $\n\n\\section{The Spin characteristic classes}\n\nFor the groups $SO=\\cup _{n=2}^{\\infty }SO(n)$ and $Spin=\\cup _{n=2}^{\\infty\n}Spin(n)$ in the stable range we have by formulae (2.3) and (6.10) that\n\n\\begin{quote}\n$H^{\\ast }(B_{SO})=\\mathbb{Z}[p_{1},p_{2},\\cdots ]\\oplus \\tau (B_{SO})$ with \n$2\\cdot \\tau (B_{SO})=0$,\n\n$H^{\\ast }(B_{Spin})=\\overline{\\pi }^{\\ast }H^{\\ast }(B_{SO})\\otimes \\mathbb\nZ}[\\overline{q}_{1},\\overline{q}_{2},\\cdots ]\/K_{\\infty }$,\n\\end{quote}\n\n\\noindent respectively, where the Euler classes $\\overline{\\pi }^{\\ast\n}(e_{n})$, $\\overline{\\theta }_{n}$ are disappeared at $n=\\infty $. In view\nof these formulae we introduce the sequence $\\left\\{ Q_{k},\\text{ }k\\geq\n1\\right\\} $, $\\deg Q_{k}=4k$, of integral cohomology classes on $B_{Spin}$\nby setting\n\n\\begin{quote}\n$Q_{k}:=\\left\\{ \n\\begin{tabular}{l}\n$\\overline{\\pi }^{\\ast }p_{k}$ if $k>1$ is not a power of $2$; \\\\ \n$\\overline{q}_{r}$ if $k=2^{r}$, $r\\geq 0$\n\\end{tabular\n\\right. $\n\\end{quote}\n\n\\noindent Then the relation ii) of Theorem C' implies the formulae\n\n\\begin{quote}\n$2Q_{1}=\\overline{\\pi }^{\\ast }p_{1}$, $2Q_{2^{r}}+Q_{2^{r-1}}^{2}=\\overline\n\\pi }^{\\ast }f(w_{2}^{(r)})$\n\\end{quote}\n\n\\noindent in which by Example 2.4 and by the formula (2.4) of $f$\n\n\\begin{quote}\n$f(w_{2}^{(r)})=p_{2^{r}}+p_{1}p_{2^{r}-2}+\\cdots\n+p_{2^{r-1}-2}p_{2^{r-1}+2}+$ higher terms.\n\\end{quote}\n\n\\noindent This implies that\\ every $2$--power Pontryagin class $\\overline\n\\pi }^{\\ast }p_{2^{r}}$\\ can be expressed as a polynomial in the $Q_{k}$'s,\nplus some torsion element. Summarizing, taking $n=\\infty $ in Theorem D' we\ncan eliminate\n\n\\begin{quote}\n$\\overline{\\pi }^{\\ast }(e_{n})$, $\\overline{\\theta }_{n}$, $\\overline{\\pi \n^{\\ast }p_{2^{r}}$ and $2Q_{2^{r}}+Q_{2^{r-1}}^{2}=\\overline{\\pi }^{\\ast\n}f(w_{2}^{(r)})$\n\\end{quote}\n\n\\noindent form the sets of generators and relations to obtain the following\nresult from Theorem C', as well as the formula (6.6) of $\\tau (B_{Spin(n)})$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 9.1.} \\textsl{The integral cohomology of }\nB_{Spin} $\\textsl{\\ has the presentation}\n\n\\begin{enumerate}\n\\item[(9.1)] $H^{\\ast }(B_{Spin})=\\mathbb{Z}[Q_{1},Q_{2},Q_{3},\\cdots\n]\\oplus \\overline{\\pi }^{\\ast }\\tau (B_{SO})$\\textsl{, }$\\deg Q_{k}=4k\n\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{where} \\textsl{the generators }$Q_{k}$\\textsl{\\ are\ncharacterized uniquely by the following properties:}\n\n\\textsl{i) if }$k>1$\\textsl{\\ is not a power of }$2$\\textsl{, then }$Q_{k}\n\\overline{\\pi }^{\\ast }p_{k}$\\textsl{;}\n\n\\textsl{ii) if }$k=2^{r}$\\textsl{\\ with }$r\\geq 0$ \\textsl{then}\n\n\\begin{enumerate}\n\\item[(9.2)] $\\rho _{2}(Q_{k})=\\overline{\\pi }^{\\ast }(w_{2}^{(k+1)})\n\\textsl{,}\n\n\\item[(9.3)] $2Q_{1}=\\overline{\\pi }^{\\ast }(p_{1})$\\textsl{,} \n2Q_{2k}+Q_{k}^{2}=\\overline{\\pi }^{\\ast }f(w_{2}^{(k+1)})$\\textsl{, }$k\\geq\n1 $\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{In particular, the cup product between the free part and\nthe torsion ideal of the ring }$H^{\\ast }(B_{Spin})$ \\textsl{is given by}\n\n\\begin{center}\n$Q_{k}\\cup \\overline{\\pi }^{\\ast }(\\delta _{2}(x))=\\left\\{ \n\\begin{tabular}{l}\n$\\overline{\\pi }^{\\ast }(\\delta _{2}(x\\cup w_{2k}^{2}))\\text{ }$\\textsl{if }\nk>1$\\textsl{\\ is not a power of} $2$\\textsl{;} \\\\ \n$\\overline{\\pi }^{\\ast }(\\delta _{2}(x\\cup w_{2}^{(r+1)}))\\text{ \\textsl{if} \n}k=\\text{ }2^{r}\\text{.}\\square \n\\end{tabular\n\\right. $\n\\end{center}\n\nIn the formula (9.1) the torsion ideal $\\overline{\\pi }^{\\ast }\\tau (B_{SO})$\nis fashioned from $\\tau (B_{SO})$, hence contributes nothing essentially\nnew. It is the generators $\\{Q_{k},k\\geq 1\\}$ of the free part, together\nwith their uniqueness property, are just what requested for us to define the\ncharacteristic classes for the Spin vector bundles.\n\nLet $\\xi $ be an oriented real bundle over a connected $CW$--complex $X$,\ninduced by a map $f_{\\xi }$ from $X$ to $B_{SO}$, and suppose that \nw_{2}(\\xi )=0$ (i.e. $\\xi $ is Spin). Then $f_{\\xi }$ can be factored into a\ncomposition\n\n\\begin{quote}\n$X\\overset{g_{\\xi }}{\\rightarrow }B_{Spin}\\overset{\\overline{\\pi }}\n\\rightarrow }B_{SO}$,\n\\end{quote}\n\n\\noindent where the map $g_{\\xi }$ is unique up to homotopy.\n\n\\bigskip\n\n\\noindent \\textbf{Definition 9.2.} The cohomology class $q_{k}(\\xi ):=g_{\\xi\n}^{\\ast }(Q_{k})\\in H^{4k}(X)$, $k\\geq 1$, (resp. the sum $q(\\xi\n):=1+q_{1}(\\xi )+q_{2}(\\xi )+\\cdots \\in H^{\\ast }(X)$) is called the $k^{th}$\n\\textsl{Spin\\ characteristic class} (resp. \\textsl{the total Spin\\\ncharacteristic class}) of the bundle $\\xi $.$\\square $\n\n\\bigskip\n\nThe Spin characteristic classes so defined possesses the naturality property.\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 9.3.} \\textsl{For any map }$g:Y\\rightarrow X\n\\textsl{\\ between CW--complexes one has}\n\n\\begin{quote}\n$q_{k}(g^{\\ast }\\xi )=g^{\\ast }q_{k}(\\xi )$\\textsl{, }$k\\geq 1$\\textsl{.}\n\\end{quote}\n\n\\noindent \\textsl{In particular, the generator }$Q_{2^{r}}$\\textsl{\\ of }\nH^{\\ast }(B_{Spin})$ \\textsl{restricts to} \\textsl{the generator }$\\overline\nq}_{r}$\\textsl{\\ of }$H^{\\ast }(B_{Spin(n)})$\\textsl{\\ (see (6.10)) via the\ninclusion }$B_{Spin(n)}\\subset B_{Spin}$\\textsl{.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 9.4. }The relations (9.2) and (9.3) have\nnon--trivial implications. Let $\\xi $ be a spin vector bundle over a space \nX $, $\\dim \\xi =n$, and let\n\n\\begin{quote}\n$w(\\xi )=1+w_{4}(\\xi )+$ $\\cdots +w_{n}(\\xi )\\in H^{\\ast }(X;\\mathbb{Z}_{2})$\nor\n\n$p(\\xi )=1+p_{1}(\\xi )+p_{2}(\\xi )+$ $\\cdots +p_{\\left[ \\frac{n-1}{2}\\right]\n}(\\xi )\\in H^{\\ast }(X)$\n\\end{quote}\n\n\\noindent be its total Stiefel--Whitney or Pontryagin classes, respectively.\nThen (9.2) turns out\n\n\\begin{quote}\n$\\rho _{2}(q_{1}(\\xi ))=w_{4}(\\xi )$,\n\n$\\rho _{2}(q_{2}(\\xi ))=w_{8}(\\xi ),$\n\n$\\rho _{2}(q_{4}(\\xi ))=w_{16}(\\xi )+w_{4}(\\xi )w_{12}(\\xi )+w_{6}(\\xi\n)w_{10}(\\xi )+w_{4}(\\xi )w_{6}^{2}(\\xi )$, $\\cdots $\n\\end{quote}\n\n\\noindent by Example 2.4. Since these polynomials admit integral lifts,\napplying $Sq^{1}$ to both sides yields the following universal relations\namong the Stiefel--Whitney classes of a spin bundle $\\xi $:\n\n\\begin{quote}\n$w_{5}(\\xi )=0$, $w_{9}(\\xi )=0$,\n\n$w_{17}(\\xi )+w_{4}(\\xi )w_{13}(\\xi )+w_{7}(\\xi )w_{10}(\\xi )+w_{6}(\\xi\n)w_{11}(\\xi )=0$, $\\cdots $.\n\\end{quote}\n\nIgnoring the elements of order $2$ the relation (9.3) allows us to express\nthe Pontryagin classes $p_{i}(\\xi )$'s as polynomials in the Spin ones \nq_{i}(\\xi )$'s, such as\n\n\\begin{enumerate}\n\\item[(9.4)] $p_{1}=2q_{1}$, $p_{2}=2q_{2}+q_{1}^{2}$, $p_{3}=q_{3}$, \np_{4}=2q_{4}+q_{2}^{2}-2q_{1}q_{3}$, $\\cdots $,\n\\end{enumerate}\n\n\\noindent by Example 2.4 and formula (2.4). In Section \\S 10 these\ntransition functions will be applied to simplify various formulae of spin\nmanifolds.$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Remark 9.5.} The spinors were first discovered by E.\nCartan in 1913 in his investigations of the representation theory of\ntopological groups, and had subsequently found significant and wide\napplications to geometry and mathematical physics. However, a precise\ndefinition of spin structure was possible only after the notion of fiber\nbundle was introduced. Notably, Haefliger (1956) proved\\ that the second\nStiefel--Whitney class $w_{2}(M)$ is the only obstruction to the existence\nof a spin structure on an oriented Riemannian manifold $M$. This was soon\nextended by Borel and Hirzebruch (1958) to the cases of vector bundles over\nCW--complexes.\n\nThe idea of \\textsl{Spin characteristics} was initiated by Thomas. He \\cite\nTheorem (1.2)]{Th} described the integral cohomology $H^{\\ast }(B_{Spin})$\nusing a squence $\\{Q_{j}\\}$ of generators, but that is subject to two\nsequences $\\{\\Phi _{j}\\}$ and $\\{\\Psi _{j}\\}$ of indeterminacies, where he\nasked also for axioms with geometric significance by which the uniqueness of\nsuch a sequence $\\{Q_{j}\\}$ can be secured.\n\nGranted with the two sequences $\\{w_{2},w_{2}^{(1)},\\cdots \\}$ and \n\\{f(w_{2}),f(w_{2}^{(1)}),\\cdots \\}$ of cohomology classes depending on the\nonly obstruction $w_{2}$ to the existence of spin structure, Theorem C\n(resp. Theorem C') amounts to an axiomatic characterization of our\ngenerators $\\left\\{ q_{r},r\\geq 0\\right\\} $ on $H^{\\ast }(B_{Spin^{c}(n)})$\n(resp. $\\left\\{ \\overline{q}_{r},r\\geq 0\\right\\} $ on $H^{\\ast\n}(B_{Spin(n)}) $). Not surprisingly, the Spin characteristic classes so\nobtained are better adapted with topics of spin geometry, see in Section \\S\n10.$\\square $\n\n\\section{Applications to spin geometry}\n\nFor a smooth manifold $M$ its total Pontryagin class (resp. total Stiefel\nWhitney class) is defined to be that of the tangent bundle $TM$ of $M$, and\nis denoted by\n\n\\begin{quote}\n$p(M):=1+p_{1}+\\cdots +p_{k}$, $k=\\left[ \\frac{n}{4}\\right] $\n\n(resp. $w(M):=1+w_{1}+\\cdots +w_{n}$, $n=\\dim M$).\n\\end{quote}\n\n\\noindent Similarly, if $M$\\ is spin (i.e. $w_{2}(M)=0$), then its total\nSpin characteristic class is also defined, and will be written as\n\n\\begin{quote}\n$q(M):=1+q_{1}+\\cdots +q_{k}$, $k=\\left[ \\frac{\\dim M}{4}\\right] $.\n\\end{quote}\n\nIn the topological approach to spin geometry, the Spin characteristic\nclasses can play roles that may not be replaced by the regular\ncharacteristic classes. In this section we provide such initial evidences.\nSubject to the main theme of this paper the examples and calculations will\nbe restricted to relatively lower dimensional cases.\n\n\\bigskip\n\n\\textbf{10.1. The tangent invariants of Spin manifolds.} According to C.T.C.\nWall \\cite[Theorem 5]{Wa1} for each integer $b$ there exists a unique $6\n--dimensional smooth manifold $\\mathbb{C}P_{b}^{3}$ homotopy equivalent to\nthe $3$--dimensional complex projective space $\\mathbb{C}P^{3}$, whose first\nPontryagin class is $p_{1}=4(1+6b)x^{2}$, where $x$ denotes a generator of \nH^{2}(\\mathbb{C}P_{b}^{3})=\\mathbb{Z}$. Since the total Stiefel--Whitney\nclass of a manifold is a homotopy invariant we must have $w(\\mathbb{C\nP_{b}^{3})=w(\\mathbb{C}P^{3})=1$. In particular, the manifold $\\mathbb{C\nP_{b}^{3}$ is spin with\n\n\\begin{quote}\n$q_{1}(\\mathbb{C}P_{b}^{3})=\\frac{1}{2}p_{1}(\\mathbb{C\nP_{b}^{3})=2(1+6b)x^{2}$ (by (9.4)).\n\\end{quote}\n\nLet $\\pi :M_{b}^{7}\\rightarrow \\mathbb{C}P_{b}^{3}$ be the oriented circle\nbundle on $\\mathbb{C}P_{b}^{3}$ with Euler class $e=4(1+6b)x$. From the\nGysin sequence of $\\pi $ \\cite[p.157]{MS} one sees that the groups \nH^{2r}(M_{b}^{7})$ is cyclic of order $4(1+6b)$ with generators $\\pi ^{\\ast\n}(x^{r})$, where $r=1,2,3$. In view of the decomposition $TM_{b}^{7}=\\pi\n^{\\ast }T\\mathbb{C}P_{b}^{3}\\oplus \\varepsilon $ on the tangent bundle of \nM_{b}^{7}$ we get by the naturality of the characteristic classes that\n\n\\begin{quote}\n$w(M_{b}^{7})=1$,\\textsl{\\ }$p(M_{b}^{7})=1$,\\textsl{\\ }but \nq_{1}(M_{b}^{7})=2(1+6b)\\pi ^{\\ast }(x^{2})\\neq 0$,\n\\end{quote}\n\n\\noindent where $\\varepsilon $ denotes the $1$--dimensional trivial bundle\non $M_{b}^{7}$. This shows that:\n\n\\bigskip\n\n\\noindent \\textbf{Proposition 10.1.} \\textsl{The family }$\\{M_{b}^{7},b\\in \n\\mathbb{Z}\\}$ \\textsl{of} $7$\\textsl{--dimensional} \\textsl{smooth} \\textsl\nSpin manifolds satisfies that }$w(M_{b}^{7})=1$\\textsl{, }$p(M_{b}^{7})=1,\n\\textsl{\\ but }$q(M_{b}^{7})\\neq 1$\\textsl{.}$\\square $\n\n\\bigskip\n\n\\textbf{10.2. The integral lifts of Wu--classes.} For a sufficiently large \nn $ let $v_{r}\\in H^{r}(B_{Spin(n)};\\mathbb{Z}_{2})$ be the $r^{th}$\nWu--class of the canonical real $n$--bundle on $B_{Spin(n)}$. It is well\nknown that $v_{r}=0$ unless $r\\equiv 0\\func{mod}4$, and that all the classes \n$v_{4k}$ admit integral lifts (see \\cite{ABP} or \\cite[Lemma E1]{HS}). In \n\\cite{HS} Hopkins and Singer constructed a stable exponential characteristic\nclass $v_{t}^{Spin}$ with values in the integral cohomology $H^{\\ast\n}(B_{Spin(n)})$, whose mod $2$--reduction is the total Wu--class.\nRationally, in terms of Pontryagin classes, the first four terms are\n\n\\begin{quote}\n$v_{4}^{Spin}=-\\frac{1}{2}p_{1}$,\n\n$v_{8}^{Spin}=\\frac{1}{2^{3}}(20p_{2}-9p_{1}^{2})$,\n\n$v_{12}^{Spin}=-\\frac{1}{2^{4}}(80p_{3}+60p_{1}p_{2}-17p_{1}^{3})$,\n\n$v_{16}^{Spin}=\\frac{1}{2^{7}}(2^{6}\\cdot 29p_{4}-2^{4}\\cdot\n33p_{2}^{2}+2^{3}\\cdot 147p_{1}^{2}p_{2}-277p_{1}^{4})$.\n\\end{quote}\n\n\\noindent Substituting the Pontryagin classes by the Spin characteristic\nclasses using the transition (9.4) shows that\n\n\\bigskip\n\n\\noindent \\textbf{Proposition 10.2.} \\textsl{In term of the Spin\ncharacteristic classes, the first four Wu--classes }$v_{4k}$, $i=1,2,3$ \n\\textsl{or} $4,$\\textsl{\\ admit the integral lifts:}\n\n\\begin{quote}\n$\\widetilde{v}_{4}=q_{1}$\\textsl{, }\n\n$\\widetilde{v}_{8}=q_{2}$\\textsl{, }\n\n$\\widetilde{v}_{12}=q_{3}+q_{1}q_{2}+q_{1}^{3}$\\textsl{,}\n\n$\\widetilde{v}_{16}=q_{4}+q_{1}q_{3}+q_{1}^{2}q_{2}$\\textsl{.}$\\square $\n\\end{quote}\n\nFor possible applications of these formulae in geometry we refer to Wilson \n\\cite{Wi}, Landweber and Stong \\cite{LS}, where the following subtle\nrelations are found for the spin manifolds of dimension $8k+2$\n\n\\begin{enumerate}\n\\item[(10.1)] $Sq^{3}v_{4k}=0$, $w_{4}w_{8k-2}=v_{4k}Sq^{2}v_{4k}$.\n\\end{enumerate}\n\n\\textbf{10.3. The Eells--Kuiper invariant.} For a closed, smooth, oriented \n(4k-1)$ manifold $M$ furnished with a spin coboundary $W$ that satisfies so\ncalled $\\mu $--conditions, Eells and Kuiper \\cite{EK} introduced a\ndifferential invariant $\\mu _{k}(M)$ of $M$ in terms of Pontryagin numbers\nand the signature $\\sigma $ of the coboundary $W$. For the small values \nk=2,3,4$ the formulae of $\\mu _{k}(M)$ reads\n\n\\begin{quote}\n$\\mu _{2}(M)\\equiv \\frac{(p_{1}^{2}-4\\sigma )[W]}{2^{7}\\cdot 7}\\func{mod}1$\n\n$\\mu _{3}(M)\\equiv \\frac{(4p_{1}p_{2}-3p_{1}^{3}-24\\sigma )[W]}{2^{11}\\cdot\n3\\cdot 31}\\func{mod}1$\n\n$\\mu _{4}(M)\\equiv \\frac\n(12096p_{1}p_{3}+5040p_{2}^{2}-22680p_{1}^{2}p_{2}+9639p_{1}^{4}-18144\n\\sigma )[W]}{2^{15}\\cdot 3^{4}\\cdot 5\\cdot 7\\cdot 127}\\func{mod}1$.\n\\end{quote}\n\n\\noindent Since the coboundary $W$ is spin we can use the Spin\ncharacteristic classes to replace the Pontryagin classes by the transition\n(9.4) to get the following simpler expressions.\n\n\\bigskip\n\n\\noindent \\textbf{Proposition 10.3.} \\textsl{In term of the Spin\ncharacteristic classes, the Eells and Kuiper invariants }$\\mu _{k}(M)\n\\textsl{,} $k=2,3$ \\textsl{or} $4,$ \\textsl{have the following expressions}\n\n\\begin{quote}\n$\\mu _{2}(M)\\equiv \\frac{(q_{1}^{2}-\\sigma )[W]}{2^{5}\\cdot 7}\\func{mod}1$\n\n$\\mu _{3}(M)\\equiv \\frac{(2(q_{1}q_{2}-q_{1}^{3})-3\\sigma )[W]}{2^{7}\\cdot\n(2^{5}-1)\\cdot 3}\\func{mod}1$\n\n$\\mu _{4}(M)\\equiv \\frac\n(6q_{1}q_{3}+5q_{2}^{2}-40q_{1}^{2}q_{2}+17q_{1}^{4}-45\\sigma )[W]}\n2^{9}(2^{7}-1)}\\func{mod}1$.$\\square $\n\\end{quote}\n\n\\textbf{10.4. The Rohlin type formula.} For an $4m$ dimensional oriented\nmanifold $M$ denote by $\\sigma _{M}$ the signature of the intersection form \nI_{M}:H^{2m}(M)\\rightarrow \\mathbb{Z}$, $I_{M}(x)=\\left\\langle\nx^{2},[M]\\right\\rangle $, on the middle dimensional integral cohomology.\n\nBy the year 1950 there had no example of topological manifolds that are not\nsmoothable. Nevertheless, Whitehead \\cite[1949]{Wh} had constructed for each\nintegral unimodular symmetric matrix $A$ a simply--connected $4$ dimensional\ntopological manifold $M$, whose the intersection form $I_{M}$ is precisely\ngiven by $A$. In contrast, the following result of Rokhlin \\cite{Ro,Fr}\nsingles out a severe gap between the topological and smooth categories,\ndetectable by the elementary and topological invariant $\\sigma _{M}$.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.4 (Rokhlin, 1952).} \\textsl{The signature }\n\\sigma _{M}$ \\textsl{of a }$4$\\textsl{--dimensional smooth spin manifold }$M$\n\\textsl{must be divisible by }$16$\\textsl{.}$\\square $\n\n\\bigskip\n\nTo extend Rokhlin's result to the higher dimensional settings we resort to\nthe $\\widehat{A}$ genus $\\alpha _{m}$ and the $L$ genus $\\tau _{m}$ of $4m$\ndimensional smooth oriented manifolds $M^{4m}$, which are certain\npolynomials in the Pontryagin classes $p_{1},\\cdots ,p_{m}$ of $M$ with\nhomogeneous degree $4m$\n\n\\begin{quote}\n$\\alpha _{m}=a_{m}\\cdot p_{m}+l_{m}(p_{1},\\cdots ,p_{m-1})$ and\n\n$\\tau _{m}=b_{m}\\cdot p_{m}+k_{m}(p_{1},\\cdots ,p_{m-1})$ \\cite[\\S 19]{MS},\n\\end{quote}\n\n\\noindent where $a_{m}$ and $b_{m}$ are certain non--zero rationals, and\nwhere $l_{m}$ and $k_{m}$ are certain polynomials in $p_{1},\\cdots ,p_{m-1}$\nwith rational coefficients. These allow us to eliminate the top degree\nPontryagin class $p_{m}$ to obtain the following expression of $\\tau _{m}$\nwithout involving $p_{m}$:\n\n\\begin{enumerate}\n\\item[(10.2)] $\\tau _{m}=$ $\\frac{b_{m}}{a_{m}}(\\alpha\n_{m}-l_{m}(p_{1},\\cdots ,p_{m-1}))+k_{m}(p_{1},\\cdots ,p_{m-1})$.\n\\end{enumerate}\n\n\\noindent For the geometric implications of the polynomials $\\alpha _{m}$\nand $\\tau _{m}$ we recall the following classical results.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.5 (Hirzebruch }\\cite{BH}\\textbf{) }\\textsl{The \n$L$\\textsl{--genus }$\\tau _{m}$\\textsl{\\ of }$M^{4m}$\\textsl{\\ equals to the\nsignature }$\\sigma _{M}$ \\textsl{of the intersection form on }\nH^{2m}(M^{4m}) $\\textsl{.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.6 (Borel--Hirzebruch \\cite{BH})} \\textsl{The }\n\\widehat{A}$ \\textsl{genus} $\\alpha _{m}$\\textsl{\\ of a spin manifold }\nM^{4m}$ \\textsl{is an integer, and is an even integer when }$m$\\textsl{\\ is\nodd.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.7 (Gromov, Lawson and Stolz \\cite{L,S0}).} \n\\textsl{If }$M^{4m}$\\textsl{\\ is a simply connected spin manifold with }$m>1\n\\textsl{, then }$M^{4m}$\\textsl{\\ admits a metric with positive scalar\ncurvature if and only if }$\\alpha _{m}=0$\\textsl{.}$\\square $\n\n\\bigskip\n\nPrecisely, for $1\\leq m\\leq 4$ the polynomials $\\alpha _{m}$ and $\\tau _{m}$\nare, respectively,\n\n\\begin{quote}\n$\\alpha _{1}=-\\frac{1}{24}p_{1}$;\n\n$\\alpha _{2}=\\frac{1}{2^{7}\\cdot 3^{2}\\cdot 5}(-4p_{2}+7p_{1}^{2});$\n\n$\\alpha _{3}=\\frac{1}{2^{10}\\cdot 3^{3}\\cdot 5\\cdot 7\n(-16p_{3}+44p_{2}p_{1}-31p_{1}^{3});$\n\n$\\alpha _{4}=\\frac{1}{2^{15}\\cdot 5^{2}\\cdot 3^{4}\\cdot 7\n(-192p_{4}+512\\cdot p_{1}p_{3}+208p_{2}^{2}-904p_{1}^{2}p_{2}+381p_{1}^{4}),$\n\\end{quote}\n\n\\noindent and\n\n\\begin{quote}\n$\\tau _{1}=\\frac{1}{3}p_{1}$;\n\n$\\tau _{2}=\\frac{1}{3^{2}\\cdot 5}(7p_{2}-p_{1}^{2})$;\n\n$\\tau _{3}=\\frac{1}{3^{3}\\cdot 5\\cdot 7}(62p_{3}-13p_{2}p_{1}+2p_{1}^{3})$;\n\n$\\tau _{4}=\\frac{1}{3^{4}\\cdot 5^{2}\\cdot 7}(381p_{4}-71\\cdot\np_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{1}^{4})$.\n\\end{quote}\n\n\\noindent Assume now that our manifold $M$ is spin (i.e. $w_{2}(M)=0$). Then\nthe formulae in (9.4) is applicable to replace the Pontryagin classes \np_{1},\\cdots ,p_{m-1}$ in (10.2) to yield the following simpler formulae of\nthe signature $\\sigma _{M}=\\tau _{m}$ in the Spin characteristic classes.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.8. }\\textsl{In accordance to }$m=1,2,3$\\textsl\n\\ and }$4$\\textsl{,\\ the signature }$\\sigma _{M}$\\textsl{\\ of a smooth spin\nmanifold }$M^{4m}$ \\textsl{is given, respectively, by}\n\n\\begin{quote}\n$\\sigma _{M}=-2^{3}\\cdot \\alpha _{1}$;\n\n$\\sigma _{M}=q_{1}^{2}-2^{5}\\cdot (2^{3}-1)\\cdot \\alpha _{2}$;\n\n$\\sigma _{M}=\\frac{2}{3}(q_{1}q_{2}-q_{1}^{3})-2^{7}\\cdot (2^{5}-1)\\cdot\n\\alpha _{3}$;\n\n$\\sigma _{M}=\\frac{2}{3\\cdot 5}q_{1}q_{3}+\\frac{1}{3^{2}}q_{2}^{2}-\\frac\n2^{3}}{3^{2}}q_{1}^{2}q_{2}+\\frac{17}{3^{2}\\cdot 5}q_{1}^{4}-2^{9}(2^{7}-1\n\\cdot \\alpha _{4}$,\n\\end{quote}\n\n\\noindent \\textsl{where }$\\alpha _{1}\\equiv \\alpha _{3}\\equiv 0\\func{mod}2\n\\textsl{.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Example 10.9.} For a smooth spin manifold $M$ one has by\nTheorems 10.5, 10.6 and 10.8 that\n\n\\begin{quote}\ni) if $\\dim M=4$, then $\\sigma _{M}\\equiv 0\\func{mod}2^{4}$;\n\nii) if $\\dim M=8$, then $\\sigma _{M}\\equiv q_{1}^{2}\\func{mod}2^{5}\\cdot\n(2^{3}-1)$\\textsl{.}\n\niii) if $\\dim M=12$, then $\\sigma _{M}\\equiv \\frac{2}{3\n(q_{1}q_{2}-q_{1}^{3})\\func{mod}2^{8}\\cdot (2^{5}-1)$\\textsl{,}\n\\end{quote}\n\n\\noindent where assertion i) is identical to Theorem 10.4. For this reason\nwe may call the formulae in Theorem 10.8 \\textsl{the Rokhlin type formulae}\nof spin manifolds.\n\nSince the string group $String(n)$ is the $3$--connected cover of $Spin(n)$,\na spin manifold is \\textsl{string} \\cite{S} if and only if its first spin\ncharacteristic class $q_{1}$ vanishes. In this case the second spin\ncharacteristic class $q_{2}$ has been shown to be divisible by $3$ \\cite{LD\n. Thus, for a smooth string manifold $M$ we have by Theorem 10.8 the\nfollowing Rokhlin type formulae.\n\n\\begin{quote}\na) if $\\dim M=8$, then $\\sigma _{M}\\equiv 0\\func{mod}2^{5}\\cdot (2^{3}-1)\n\\textsl{.}\n\nb) if $\\dim M=12$, then $\\sigma _{M}\\equiv 0\\func{mod}2^{8}\\cdot (2^{5}-1)\n\\textsl{,}\n\nc) if $\\dim M=16$, then $\\sigma _{M}\\equiv (\\frac{1}{3}q_{2})^{2}\\func{mod\n2^{9}\\cdot (2^{7}-1)$.\n\\end{quote}\n\n\\noindent In addition, by Theorem 10.7 if $M$\\ is a simply connected, then \nM $\\ admits a metric with positive scalar curvature if and only if\n\n\\begin{quote}\n$\\sigma _{M}=0,0$ or $(\\frac{1}{3}q_{2})^{2}$ in accordance to $\\dim M=8,12$\nor $16$.$\\square $\n\\end{quote}\n\n\\textbf{10.5. The existence of smooth structure on triangulable manifolds.}\nWithout involving the top degree Pontryagin class $p_{m}$ the Rokhlin type\nformulae in Theorem 10.8 is ready to apply to study of the existence problem\nof smooth structures on certain $4m$ dimensional triangulable manifolds. To\nprovide such examples in the case $m=2$ we need the following notation.\n\n\\bigskip\n\n\\noindent \\textbf{Definition 10.10.} For a unimodular symmetric integral\nmatrix $A=(a_{ij})_{n\\times n}$ of rank $n$, and a sequence $b=(b_{1},\\cdots\n,b_{n})$ of integers with length $n$, the pair $(A,b)$ is called a\\textsl{\\\nWall pair} if the following congruences are satisfied\n\n\\begin{enumerate}\n\\item[(10.3)] $a_{ii}\\equiv b_{i}\\func{mod}2,1\\leq i\\leq n$.$\\square $\n\\end{enumerate}\n\n\\bigskip\n\nLet $D^{8}$ be the unit disk on the $8$ dimensional Euclidean space $\\mathbb\nR}^{8}$. The following result is due to C.T.C. Wall \\cite{Wa}.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.11. }\\textsl{For each Wall pair }$(A,b)$\\textsl\n\\ with }$A=(a_{ij})_{n\\times n}$ \\textsl{and }$b=(b_{1},\\cdots ,b_{n})\n\\textsl{,} \\textsl{there exists a closed }$8$\\textsl{\\ dimensional} \\textsl\ntopological manifold }$M$\\textsl{\\ that satisfies the following properties}\n\n\\textsl{i) }$M$\\textsl{\\ admits a decomposition }$M=W\\cup _{h}D^{8}$\\textsl\n, where }$W$ \\textsl{is a }$3$\\ \\textsl{connected} \\textsl{smooth manifold\nwith boundary }$\\partial W$ \\textsl{a homotopy }$7$\\textsl{--sphere, and\nwhere }$h:\\partial W\\rightarrow \\partial D^{8}$ \\textsl{is a homeomorphism;}\n\n\\textsl{ii) there is a basis }$\\left\\{ x_{1},\\cdots ,x_{n}\\right\\} $\\textsl\n\\ on }$H^{4}(M)$\\textsl{\\ so that }$x_{i}\\cup x_{j}=a_{i,j}\\cdot \\omega _{M}\n\\textsl{, where }$\\omega _{M}\\in H^{8}(M)$ \\textsl{is an orientation class o\n} $M$\\textsl{;}\n\n\\textsl{iii) the first Spin characteristic class }$q_{1}$\\textsl{\\ of }$M\n\\textsl{\\ is well defined (by i)), and is determined by }$b$\\textsl{\\ as }\n\n\\begin{quote}\n$\\qquad q_{1}=b_{1}x_{1}+\\cdots +b_{n}x_{n}\\in H^{4}(M)$\\textsl{.}\n\\end{quote}\n\n\\textsl{Furthermore, if }$(A^{\\prime },b^{\\prime })$\\textsl{\\ is a second\nWall pair, then the associated manifold }$M^{\\prime }$\\textsl{\\ is\ncombinatorially homeomorphic to }$M$\\textsl{\\ (in the sense of \\cite{Wa}) if\nand only if there exists an integer matrix }$P=(p_{i,j})_{n\\times n}$\\textsl\n\\ so that }\n\n\\begin{quote}\n$P^{\\tau }AP=A^{\\prime }$ \\textsl{and} $bP=b^{\\prime }$\\textsl{, }\n\\end{quote}\n\n\\noindent \\textsl{where }$P^{\\tau }$\\textsl{\\ denotes the transpose of the\nmatrix }$P$\\textsl{.}$\\square $\n\n\\bigskip\n\n\\noindent \\textbf{Remark 10.12.} In \\cite{Wa} Wall classified the\ncombinatorial homeomorphism types of all the $(n-1)$ connected $2n$\ndimensional manifolds $M$ that are smooth off one point $o\\in M$. The result\nin Theorem 10.11 corresponds to the case $n=4$.\n\nIt is known that for a $4$--dimensional real vector bundle $\\xi $ on the $4$\ndimensional sphere $S^{4}$ the difference $2e(\\xi )-p_{1}(\\xi )$ (resp. \ne(\\xi )-q_{1}(\\xi )$) is divisible by $4$ (resp. by $2$), where $e(\\xi )$ is\nthe Euler class of $\\xi $ \\cite[Lemma 20.10]{MS}. In Theorem 10.11 the\nnecessity of the Wall condition (10.3) is governed by the following\ngeometric fact. According to Haefliger \\cite{Ha}, for the manifold $W$ in i)\nof Theorem 10.11, there exist $n$ smooth embeddings\n\n\\begin{quote}\n$\\iota _{i}:S^{4}\\rightarrow W$, $1\\leq i\\leq n$,\n\\end{quote}\n\n\\noindent so that the Kronecker dual of the cycle classes $\\iota _{i\\ast\n}[S^{4}]\\in H_{4}(M)$ is the basis $\\left\\{ x_{1},\\cdots ,x_{n}\\right\\} \n\\textsl{\\ }on\\textsl{\\ }$H^{4}(M)$. Then, the matrix $A$ is the intersection\nform on $H^{4}(M)$ corresponding to the basis, while the normal bundle \n\\gamma _{i}$ of the embedding $\\iota _{i}$ is related to the pair $(A,b)$ by\nthe relations\n\n\\begin{quote}\n$(e(\\gamma _{i}),q_{1}(\\gamma _{i}))=(a_{ii}\\cdot \\omega ,b_{i}\\cdot \\omega\n) $, $1\\leq i\\leq n,$\n\\end{quote}\n\n\\noindent where $\\omega $ is the orientation class on $S^{4}$ that\ncorresponds $x_{i}$ via $\\iota _{i}$.\n\nFor an $8$--dimensional manifold $M$ associated to a Wall pair $(A,b)$\nproperties ii) and iii) of Theorem 10.9 imply, respectively, that\n\n\\begin{enumerate}\n\\item[(10.4)] $\\sigma _{M}=sign(A)$ and $q_{1}(M)^{2}=bAb^{\\tau }$,\n\\end{enumerate}\n\n\\noindent where $b^{\\tau }$\\textsl{\\ }denotes the transpose of the row\nvector $b$.$\\square $\n\n\\bigskip\n\nConcerning the manifold $M$ associated to a Wall pair $(A,b)$ a natural\nquestion is whether there exists a smooth structure that extends the given\none on $W$. For the special case $A=(1)_{1\\times 1}$ this question has been\nstudied by Milnor \\cite{M}, Eells and Kuiper \\cite[\\S 6]{EK} in their\ncalculation on the group $\\Theta _{7}$ of homotopy $7$ spheres. We extend\ntheir calculations in the following results.\n\n\\bigskip\n\n\\noindent \\textbf{Theorem 10.13. }\\textsl{Let }$M^{8}$\\textsl{\\ be the\nmanifold associated to a Wall pair }$(A,b)$\\textsl{. There exists a smooth\nstructure} \\textsl{on }$M^{8}$\\textsl{\\ extending the one on }$W$\\textsl{\\\nif and only if}\n\n\\begin{enumerate}\n\\item[(10.5)] $sign(A)\\equiv bAb^{\\tau }\\func{mod}2^{5}\\cdot (2^{3}-1)\n\\textsl{.}\n\\end{enumerate}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} The necessity of (10.5) comes from ii) of Example\n10.9. The sufficiency is verified by computing with the Eells--Kuiper $\\mu $\ninvariant \\cite[formula (11)]{EK} of the boundary $\\partial W$, which by\nProposition 10.3 reads\n\n\\begin{quote}\n$\\mu (\\partial W)\\equiv \\frac{4bAb^{\\tau }-4sign(A)}{2^{7}\\cdot (2^{3}-1)\n\\equiv \\frac{bAb^{\\tau }-sign(A)}{2^{5}\\cdot (2^{3}-1)}\\func{mod}1$.$\\square \n$\n\\end{quote}\n\nTheorem 10.13 has several direct, but notable consequences. A theorem of\nKervaire states that there exist a $10$ dimensional manifold which do not\nadmit any smooth structure \\cite{K}. Eells and Kuiper provided further\nexamples which have the same cohomology ring as that of the projective plane \n\\cite{EK1}. Theorem 10.13 implies that\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 10.14. }\\textsl{If }$(A,b)$\\textsl{\\ is a Wall\npair so that }\n\n\\begin{quote}\n$sign(A)\\neq bAb^{\\tau }\\func{mod}2^{5}\\cdot (2^{3}-1)$\\textsl{, }\n\\end{quote}\n\n\\noindent \\textsl{then the corresponding manifold} $M$ \\textsl{does not\nadmit any smooth structure.}$\\square $\n\n\\bigskip\n\nConversely, for those $M$ which do admit smooth structures, their total Spin\ncharacteristic class $q(M)$ can be determined completely.\n\n\\bigskip\n\n\\noindent \\textbf{Corollary} \\textbf{10.15.} \\textsl{If the manifold} $M$ \n\\textsl{associated to a Wall pair }$(A,b)$ \\textsl{admits a smooth\nstructure, then its total Spin characteristic class is}\n\n\\begin{enumerate}\n\\item[(10.6)] $q(M)=1+(b_{1}x_{1}+\\cdots +b_{1}x_{n})+\\frac{3(15\\cdot\nsign(A)-bAb^{\\tau })}{2\\cdot (2^{3}-1)}\\cdot \\omega _{M}$\\textsl{,}\n\\end{enumerate}\n\n\\noindent \\textsl{where }$2\\cdot (2^{3}-1)$\\textsl{\\ divides }$15\\cdot\nsign(A)-bAb^{\\tau }$\\textsl{\\ by ii) of Example 10.9..}\n\n\\bigskip\n\n\\noindent \\textbf{Proof.} With $\\tau _{2}=\\sigma _{M}=sign(A)$ and \np_{1}^{2}=4bAb^{\\tau }$ the formula $\\tau _{2}=\\frac{7p_{2}-p_{1}^{2}}\n3^{2}\\cdot 5}$ implies that\n\n\\begin{quote}\n$p_{2}=\\frac{45\\cdot sign(A)+4bAb^{\\tau }}{7}\\cdot \\omega _{M}$.\n\\end{quote}\n\n\\noindent From $p_{2}=2q_{2}+q_{1}^{2}$ by (9.4) we get $q_{2}=\\frac\n3(15\\cdot sign(A)-bAb^{\\tau })}{2\\cdot (2^{3}-1)}\\cdot \\omega _{M}$.$\\square \n$\n\n\\bigskip\n\nIn view of the formula of $q_{2}$ in (10.6) Theorem 10.7 implies that\n\n\\bigskip\n\n\\noindent \\textbf{Corollary 10.16. }\\textsl{For} \\textsl{a manifold} $M$ \n\\textsl{associated to a Wall pair }$(A,b)$\\textsl{\\ the following statements\nare equivalent:}\n\n\\begin{quote}\n\\textsl{i) }$M$\\textsl{\\ is smoothable and has a metric with positive scalar\ncurvature;}\n\n\\textsl{ii)} $sign(A)=bAb^{\\tau }$\\textsl{;}\n\n\\textsl{iii)} $q_{2}=3sign(A)\\cdot \\omega _{M}$\\textsl{.}$\\square $\n\\end{quote}\n\n\\bigskip\n\n\\noindent \\textbf{Example 10.17. }Corollary 10.16 reduces the problem of\nfinding all the $3$--connected and $8$-- dimensional smooth manifolds that\nhave a metric with positive scalar curvature to the arithmetic problem of\nfinding those Wall pairs $(A,b)$ satisfying the quadratic equation\n\n\\begin{enumerate}\n\\item[(10.7)] $sign(A)=bAb^{\\tau }$.\n\\end{enumerate}\n\nConsider a Wall pair $(A,b)$ with $A$ the identity matrix $I_{n}$ of rank $n\n, and with $b=(2k_{1}+1,\\cdots ,2k_{n}+1)$, $k_{i}\\in \\mathbb{Z}$ (see\n(10.3)). The equation (10.7) is\n\n\\begin{quote}\n$n=(2k_{1}+1)^{2}+\\cdots +(2k_{n}+1)^{2}$.\n\\end{quote}\n\n\\noindent It implies that the corresponding $M$\\textsl{\\ }is smooth and\nadmits a metric with positive scalar curvature, if and only if $M$ is\ncombinatorially homeomorphic \\cite{Wa}\\textsl{\\ }to $\\mathbb{H}P^{2}\\#\\cdots\n\\#\\mathbb{H}P^{2}$, the connected sum of $n$ copies of the projective plan \n\\mathbb{H}P^{2}$.\n\nConsider next a Wall pair $(A,b)$ with $A=$ $\\left( \n\\begin{array}{cc}\n0 & 1 \\\\ \n1 & \n\\end{array\n\\right) $ and $b=(2k_{1},2k_{2})$, $k_{i}\\in \\mathbb{Z}$ (see (10.3)). The\nequation (10.7) turns to be $k_{1}\\cdot k_{2}=0$. It implies that the\ncorresponding manifold $M$\\textsl{\\ }is smooth and admits a metric with\npositive scalar curvature, if and only if $M$ is combinatorially\nhomeomorphic to the spherical bundle $S(\\xi )$ of a $5$ dimensional\nEuclidean bundle $\\xi $ over $S^{4}$. Such manifolds $S(\\xi )$ are\nclassified by the homotopy group $\\pi _{3}(SO(5))$.\n\nConcerning this topic general cases will be studied in the sequel work \\cit\n{DL}.$\\square $\n\n\\bigskip\n\n\\textbf{Acknowledgement.} The author would like to thank Fei Han, Ruizhi\nHuang for discussion on the integral lifts of Wu--classes \\cite{HS,DHH} (see\nSection \\S 9.2), and to Yang Su for informing him the recent questions in\nmathoverflow \\cite{Web,Web1} about the integral cohomology of the\nclassifying space $B_{Spin(n)}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nDiagnostic assays are an integral part of clinical practice. Whole blood is one of the most common sample types used in diagnostics \\citep{cui15_sampleprep_review} and is an important material for point-of-care applications as it can be collected relatively easily in small volumes. Blood is made up primarily of blood plasma (about 55 \\% by volume). The remainder is cellular content, the vast majority of which comprises erythrocytes (red blood cells), which outnumber both the slightly larger white blood cells and the smaller platelets that together make up only about 1\\% of blood volume. Blood plasma comprises about 92\\% by weight of water with the remaining consisting of a dissolved proteins (around 7\\%), sugars and salts. \n\n\nThe demand for sample analysis at low cost and high accuracy has made microfluidic techniques increasingly popular in recent years. Much attention has been focused on separating plasma from whole blood for diagnostics \\citep{ker13_microcalesep_review}, in particular because traditional centrifugal fractionation of blood can be labor intensive and expensive. Recent studies have sought to address these challenges by developing microfluidic plasma filtration devices using a host of techniques including membrane-based \\citep{hom12_plasmasep,lu18_separation_paper} or channel-based microfilters \\citep{fai06_geometrical_separation,wu12_blood_filtration_microfilter, kuo18_channelfilter}. Other have used external fields to apply forces to cells and separate them from plasma, including electrokinetic \\citep{min02_electrokinetic}, magnetic \\citep{jun08_magnetic, kim12_removal, tas15_levitation_magnet}, acoustic \\citep{len09_acoustic}, and inertial \\citep{mac10_inertial_filtration} forces. Notable within the category of separation devices using external fields are those that exploit the gravitational sedimentation of blood cells \\citep{dim11_bloodanalysis}, with some designs additionally making use of fluid flow \\citep{zha12_gravitational}.  Others have developed separation devices on centrifugal microfluidic chips by incorporating microvalves \\citep{li10_separation} or by employing centrifugal forces in curved or branched microchannels  \\citep{zha08_separation}. There has also been an interest in cell-scale processes \\citep{fre14_annurev,tom14_biomechanical}, including the effects of erythrocyte morphology and deformability on blood rheology \\citep{for11_multiscale_RBC_dynamics,lan16_morphology_viscosity}, and the apparent dependence of blood viscosity on the radius of the capillary (the F\\r{a}hr{\\ae}us--Lindqvist effect) \\citep{sec17_blood_annurev}. Cell deformability has also been used to develop microfluidic cell sorters \\citep{fai06_geometrical_separation, guo14_deformability_microfluidic} and is important to the life cycle of erythrocytes \\citep{piv16_spleen}. \n\n\nA typical blood sample in diagnostics (extracted, for example, by a pin-prick) has a volume of about $50$ $\\mu$L. In contrast, the volume that can be contained in a microfluidic device is much smaller (typically a few $\\mu$L). Thus, for most diagnostic techniques using microfluidics, it is usually only necessary for a small amount of the original sample to be processed, i.e. the original sample must be ``subsampled'' for the diagnostic assay. The cellular constituents of blood are denser than the surrounding plasma and will therefore sediment over time due to an external force such as gravity or a centrifugal force. For a typical cell volume fraction -- or hematocrit -- of about 45 \\%, the sedimentation speeds of erythrocytes under gravity are about $0.2$--$0.3$ mm\/min. Sedimentation rates are faster for lower cell fractions \\citep{mil83_sediment, jac87_sed_hem}. Thus, with processing times on the order of several minutes, sedimentation is appreciable in millimeter-sized geometries: over time, the top of the sample continuously becomes more dilute, while cells aggregate near the bottom of the sample. As discussed above and in previous studies \\citep{dim11_bloodanalysis, zha12_gravitational,sun12_blood_sep_twophase_sedimentation}, this is a useful property  for separating plasma from cells. It is, however, counterproductive if the goal is to maintain the composition of the sample throughout the sampling time.\n\n\nHere, we present experiments and a corresponding model for the sedimentation of erythrocytes in a sample blood volume. Our primary goal is to use the predictions of this model to identify design considerations that extract a representative subsample in spite of the continuous sedimentation of the blood sample. We identify the region of the design space, which involves the device geometry and the sampling (or filling) time, for which representative subsampling is possible. Furthermore, we are able to tune the design parameters so that the collected subsample has a prescribed concentration of cells different from that of the original sample, allowing us to design subsampling protocols to collect either a more dilute or more concentrated extract than the original sample. The remainder of the paper is organized as follows: Sec. \\ref{SecModel} introduces the experimental procedure and a corresponding model that quantifies blood sedimentation. Section \\ref{SecSubsampling} discusses solutions of the model in the context of extracting either representative or non-representative subsamples. In Sec. \\ref{SecSideChannel} we discuss the process of subsampling by a microfluidic channel (or network of channels) throughout the sedimentation process, after which we conclude in Sec. \\ref{SecConcl}.\n\n\\section{Sedimentation and design considerations}  \\label{SecModel}\n\n\\begin{figure}[t!]\n    \\includegraphics[scale=1]{Sketch.pdf}\n\\caption{A sketch of the setup showing a reservoir filled with a blood sample up to a height $h$. Connected to the reservoir at a height $d$ is a narrow side channel that continuously draws a small volume of fluid from the reservoir; the fluid collected within the side channel forms the subsample. Over time the suspension in the reservoir sediments and develops a moving front $z = z_f(t)$, such that the reservoir contains only plasma (cell volume fraction $\\phi = 0$) between the front and the fill line $(z = 0)$. As the sedimentation proceeds, the opening of the side channel is presented with suspension properties that vary in time. An applied horizontal pressure gradient pumps fluid into the side channel, causing it to be filled to a length $\\ell(t)$ that increases with time.}\n\\label{FigSketch}\n\\end{figure}\nWe are interested in subsampling a volume $V_s$ of blood from a larger sample of volume $V > V_s$. The original sample has a volume fraction of red blood cells (erythrocytes) $\\phi \\leq 1$, often referred to as the hematocrit (and expressed as a percentage). The volume fraction is typically around $0.45$, though it may be much higher (up to $0.7$) or lower (as low as $0.1$) depending on physiological or pathological conditions \\citep{bri99_clinica_ESR}. White blood cells and platelets together make up only about 1\\% of blood volume and so their contribution to the dynamics discussed below will be neglected.\n\n\n\nHere we discuss design criteria that ensure a representative subsampling for the entire range of physiologically relevant hematocrit $(0<\\phi\\lesssim 0.75)$. A sketch of the gravitational sedimentation setup considered in this work is shown in Fig. \\ref{FigSketch}. We consider a reservoir that is filled with the original sample volume $V$, connected to which is a side channel that continuously draws in suspension from the reservoir over time. The fill height of the sample in the reservoir is $h = V\/A$, where $A$ is the constant area of cross-section of the main container. The side channel is located at a distance $d$ from the bottom of the reservoir. We are interested in the properties of the fluid drawn into the side channel -- the subsample -- relative to those of the original sample placed in the reservoir. \n\n\\subsection{Sedimentation Experiments}\nThe sedimentation rate $v$ of erythrocytes depends on several factors including the mean hematocrit \\citep{rou30_sediment}, the protein content of the plasma \\citep{rop39_sed_plasma}, the age of the sample, pathological and physiological conditions, and the presence of additives \\citep{bri99_clinica_ESR}. Here, we focus on the dependence of $v$ on $\\phi$, which is particularly important as the local hematocrit of a given sample of blood changes during the sedimentation process. Despite the many variables controlling the sedimentation rate it is agreed that the sedimentation rate decreases with hematocrit, provided the geometry is large compared with the size of the cell and cell-aggregates \\citep{fab87_aggregation_sedimentation}. \n\nWe quantify the dependence of sedimentation rate on hematocrit by direct measurements. Unspun whole blood with tri-potassium ethylenediaminetetraacetic acid (EDTA, an anti-coagulant) was received from Biological Specialty Corporation (Colmar, PA). The blood was centrifuged at 3000 rpm for 10 minutes to separate the red blood cells from the plasma; the blood, as received, was 45\\% hematocrit. Then, 10 mL each of different hematocrit samples were prepared by mixing different volume ratios of cells and plasma in 15 mL centrifuge tubes (inner diameter = 1.4 cm). Prior to sedimentation, the tubes were inverted repeatedly and gently by hand to mix the suspensions, and then placed upright in Styrofoam racks. Images were captured every minute during sedimentation (Nikon D5100, Camera Control Pro). The images were analyzed using ImageJ software \\citep{sch12_ImageJ} to track the location of the sedimentation front over time (see Fig. \\ref{FigSketch}). Consistent with previous studies \\citep{rou30_sediment,fab87_aggregation_sedimentation} we observe that the sedimentation front moves at a roughly constant speed over a range of times. We determine the maximum sedimentation rate for each hematocrit by analyzing this part of the displacement versus time curve (see \\citep{rou30_sediment}).  Sedimentation experiments were conducted simultaneously in four tubes per hematocrit at room temperature (21--22$^\\circ$C).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1]{fluxfig.pdf}\n\\caption{(a) Experimentally measured sedimentation rates as a function of the hematocrit: filled circles and open squares indicate two sets of present experimental results (whole blood samples from two different anonymous donors) and triangles are the data of \\citep{rou30_sediment}. Error bars indicate one standard deviation. Curves are best fits of the sedimentation law \\eqref{SedimentationLaw} with $\\phi_m = 0.75$; a fit to the current experiments (circles) yields $v_0 \\approx 1.76$ mm\/min, $k \\approx 2.75$ and the fit to the data of \\citep{rou30_sediment} results in $v_0 \\approx 3.81$ mm\/min, $k \\approx 3.03$. The inset plots the same data but with the sedimentation rates represented by squares and triangles being scaled by a uniform factors of $0.7$ and $0.5$, respectively, showing that all three data sets are similar up to scale. (b) Dimensionless sedimentation velocity $V(\\Phi) = (1-\\Phi)^k$ (solid; left axis), flux $J(\\Phi) = \\Phi V$ (solid; right axis) and wave speed $U(\\Phi) = \\dd{(\\Phi V)}{\\Phi}$ (solid; left axis) with  $k = 3$. The flux is maximum when $\\Phi = (1 + k)^{-1}$.}\n\\label{FigFlux}\n\\end{figure}\n\nFig. \\ref{FigFlux}(a) plots sedimentation speed as a function of the sample hematocrit, extracted from our measurements (solid circles), alongside the measurements of Rourke and Ernstene \\citep{rou30_sediment} (triangles). In both cases, we find that the data are well fit by a sedimentation law of the form \n\\begin{equation} \\label{SedimentationLaw}\n    v(\\phi) = v_0 \\left(1 - \\frac{\\phi}{\\phi_m}\\right)^k,\n\\end{equation}\nwhere $v_0$ is the sedimentation speed in the dilute limit, $\\phi_m$ is the hematocrit at which no sedimentation can be observed and $k$ is a dimensionless exponent. With $\\phi_m = 0.75$, a best fit to our data gives $v_0 \\approx 1.76$ mm\/min and the exponent $k = 2.75$, while a fit to the data of \\citep{rou30_sediment} yields $v_0 \\approx 3.81$ mm\/min  and $k \\approx 3.03$. \n\nTo further validate our data, we performed separate measurements with a different whole blood sample (obtained from a different anonymous donor) of volume $\\approx 60\\,\\mu$L in a narrow capillary tube (inner diameter of 1.1--1.2 mm). Sedimentation rates were measured using a similar procedure as the one described above. The results of these measurements are indicated as open squares in Fig. \\ref{FigFlux}(a) and are largely consistent with the data obtained in the wider tubes (circles). Both datasets corresponding to the present experiments suggest slower sedimentation compared with \\citep{rou30_sediment}, which we speculate may be due to factors such as differing physiological or pathological conditions that we do not control (cf. \\citep{fab87_aggregation_sedimentation}). Interestingly, within the range of measurement, the data sets differ by a scale factor between $0.5$ and $0.7$ but are nearly identical in shape, as indicated in the inset of Fig. \\ref{FigFlux}(a). As noted above, the detailed values of $v_0$ and $k$ may depend on factors other than the hematocrit and will not be directly relevant to the discussion below. The more important feature is that \\eqref{SedimentationLaw} provides a good representation of the dependence of sedimentation rate on hematocrit.  \n\nEquation \\eqref{SedimentationLaw} may also interpreted in terms of a Krieger--Dougherty law \\citep{kri59_suspensions} for the viscosity of a suspension $\\eta(\\phi) = \\eta_0 \\left(1 - \\frac{\\phi}{\\phi_m}\\right)^{-k}$ by writing $v(\\phi)=  \\frac{g(\\Delta m)}{6 \\pi a \\eta(\\phi)}$, where $a$ is a length scale characterizing Stokes drag on an erythrocyte, $\\Delta m$ is its buoyant mass, $g$ is the gravitational acceleration, and $\\eta_0$ is the plasma viscosity. We recognize that this description is only approximate since blood is a complex fluid that is shear-thinning, slightly viscoelastic and has a small yield stress (1--10 mPa) \\citep{hor18_loas}. However, its rheology is often well described by a Casson law \\citep{apo14_rheology_shear,tom16_casson}, which, in the limit of a small yield stress, reduces to an Newtonian effective-viscosity description such as the one used above.\n\n\n\\subsection{Sedimentation model}\nWe describe the sedimentation of the suspension using a one-dimensional Kynch model \\citep{kyn52_sedimentation}, where the sedimentation speed depends only on the local volume fraction of cells, which in turn evolves in space and time. We take $z = 0$ at the top of the sample and $z = h$ at the bottom of the reservoir (Fig. \\ref{FigSketch}). Writing $\\phi = \\phi(z,t)$, conservation of cell number in the reservoir containing the original sample can be expressed as  \n\\begin{align}\n\\pp{\\phi}{t} + \\pp{(\\phi v(\\phi))}{z} = 0.\n\\end{align}\nNote that we have neglected the loss of cells (and plasma) into the subsampling side channels, which implicitly assumes that the subsample volume is much smaller than that of the original sample ($V_s \\ll V$). We also assume that the side channel is sufficiently narrow to be able to sample the container at a single height $d$ from the bottom, and that the flow into the side channel does not influence the sedimentation in the main container. This loss of sample into the side channel may be modeled within the present framework by a point sink of material placed at a distance $z = h-d$, though we will neglect it in our discussion below. \nFor a one-dimensional model, the physically relevant boundary condition is that of zero cell flux $\\phi v(\\phi)$ at the top and bottom boundaries.\n\n\n We also assume that the reservoir is instantaneously filled with a uniform sample, $\\phi(0<z<h,\\, t=0) = \\phi_0$. Introducing dimensionless variables $Z = z\/h$, $T = t v_0\/h$, $\\Phi = \\phi\/\\phi_m$, and defining\n\\begin{align} \\label{Vnondim}\nV(\\Phi) = \\frac{v(\\phi)}{v_0} = (1-\\Phi)^{k},\n\\end{align}\nsedimentation in the reservoir is governed by\n\\begin{subequations} \\label{SedGE}\n\\begin{align}\n    \\pp{\\Phi}{T} + \\pp{(\\Phi V(\\Phi))}{Z} &= 0 \\qmbox{with} \\\\\n    \\Phi V(\\Phi) &= 0 \\qmbox{at} Z = 0 \\qmbox{and} Z = 1 \\\\\n    \\Phi(Z,T=0) &= \\Phi_0 \\equiv \\frac{\\phi_0}{\\phi_m} \\leq 1\\,.\n\\end{align}\n\\end{subequations}\nThe sedimentation flux  $J (\\Phi) \\equiv \\Phi V(\\Phi)$ is zero both when $\\Phi = 0$  and $\\Phi = 1$, and the wave speed $U= \\dd{J(\\Phi)}{\\Phi} = (1 -(1+k)\\Phi)(1-\\Phi)^{k-1}$ can be either positive or negative. We note that it is appropriate to state two boundary conditions in $Z$ despite (\\ref{SedGE}a) being first order in $Z$ since the problem is hyperbolic and information may propagate into the domain at either boundary depending on the sign of $U$ \\citep{whitha74_book}. In Fig. \\ref{FigFlux}(b) we plot $U(\\Phi)$, $J(\\Phi)$ and $V(\\Phi)$ according to the sedimentation model \\eqref{Vnondim}, with $k = 3$. The side channel, which is located at the dimensionless distance $D = d\/h$ from the base of the reservoir, continuously samples the fluid at $Z = 1 -D$, and thus may encounter a range of suspension properties over an interval of time. It is useful to define the time-averaged hematocrit $\\overline{\\Phi}$ at a location $Z$ during a time interval $T$ by\n\\begin{align}\n\\overline{\\Phi}(Z,T) \\equiv \\frac{1}{T} \\int_0^{T} \\Phi(Z,T') \\rmd T',\n\\end{align}\nwhich is a useful measure of the average properties encountered at the entrance of the side channel, with $Z = 1-D$. We note that while $v_0$ is the natural velocity scale, it is an extrapolated value for the $\\phi \\rightarrow 0$ limit and is thus an order of magnitude greater than the sedimentation rate of typical blood $(\\phi \\approx 0.45)$; see Fig. \\ref{FigFlux}(a).\n\n\n\n\n\\subsection{Numerical solutions} \n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[scale=1]{ConcvsTime.pdf}\n\\caption{Results of a typical simulation with initial condition $\\Phi(Z,T=0) = \\Phi_0 = 0.6$ ($\\phi_0 = 0.45$, $\\phi_m = 0.75$, $k = 3$; typical for blood) showing profiles of (a) the instantaneous hematocrit $\\Phi(Z,T)$ and (b) the mean hematocrit $\\overline{\\Phi}(Z,T)$ as functions of the spatial coordinate $Z$, at times $T = \\{0, 1, 2, 3, 5, 100\\}$. Dashed arrows indicate profiles at increasing $T$. A sedimentation front propagates towards $Z = 1$ (the base), while a compression wave propagates towards $Z = 0$, until the two meet. }\n\\label{FigProfiles}\n\\end{figure}\n\n We solve the hyperbolic problem defined by equations (\\ref{SedGE}a--c) using a first-order conservative Godunov-type scheme with upwind differencing in the direction of the local wavespeed $U$ \\citep{har83_upstream}. Here and below, we use $k = 3$, consistent with experimental data (see Fig\\ref{FigFlux}, \\citep{rou30_sediment}). Instantaneous and time-averaged concentration profiles obtained from a typical simulation with an initially uniform concentration $\\Phi_0$ are plotted in Figs. \\ref{FigProfiles}(a,b), respectively. As is typical of sedimenting suspensions, the system evolves by the formation of a front with position $Z = Z_f(T)$ that propagates towards the bottom of the container (the positive $Z$ direction). Upstream of the front, the suspension contains only fluid $(\\Phi = 0)$ while downstream of the front the concentration of the suspension is either greater than or equal to its initial value $\\Phi_0$. Additionally, a compression wave propagates rearward (towards $Z= 0$) from the no-flux boundary at $Z = 1$ (the base of the reservoir), as shown in Fig. \\ref{FigProfiles}. The concentration increases behind this compression wave (as measured in its direction of propagation), corresponding to cells settling at the bottom of the container.  Between the forward traveling sedimentation front and the rearward propagating compression wave is a ``quiet'' region where $\\Phi = \\Phi_0$. This region shrinks and eventually vanishes as the two waves meet (around $T = 3$ in Fig. \\ref{FigProfiles}). This process determines the largest time for which the subsample is representative of the original sample, which we discuss in detail in Sec. \\ref{SecSubsampling}.\n\nThe speed of the sedimentation front is given by the Rankine--Hugoniot condition\n \\begin{equation}\n     \\dd{Z_f}{T} = \\frac{\\left[J\\right]_{Z_f}}{\\left[\\Phi\\right]_{Z_f}},\n \\end{equation}\nwhere $[f]_Z = f|_{Z^+} - f_{Z^-}$ is the jump of a quantity $f$ around the point $Z$. We note that in a semi-infinite system, $\\Phi(Z>Z_f) = \\Phi_0$ and $\\Phi(Z<Z_f) = 0$, so that $\\dd{Z_f}{T} = V(\\Phi_0)$.  Under confinement, the system eventually reaches a steady state which, due to mass conservation, is given by $\\Phi(Z,T\\rightarrow\\infty) = \\Theta[Z-(1-\\Phi_0)]$, where $\\Theta$ is the Heaviside step function, with the front approaching a fixed position $Z_f \\rightarrow 1-\\Phi_0$ (Fig. \\ref{FigProfiles}). \n\n\\section{Subsampling the sedimenting suspension} \\label{SecSubsampling}\nSolutions of equation \\eqref{SedGE} may be analyzed further using the method of characteristics though this is not the primary focus of the present work. Instead we are interested in sampling the sedimenting suspension -- whose properties evolve in both space and time -- in a way that is predictable and therefore of potential value to diagnostics. For applications it is often necessary that the side channel subsamples the sedimentation suspension for a prescribed sampling time $t_s$, which defines a sampling window $T \\in [0, T_s \\equiv \\frac{t_s v_0}{h}]$. Typically, $t_s$ is set by the application and $v_0$ is a property of the suspension, with $h$ being a design parameter. Thus, we have three dimensionless parameters: the hematocrit of the sample $\\Phi_0$, the sampling time $T_s$ (related to the fill height), and the location of the subsampling port $D = d\/h$; $\\Phi_0$ is an input  while $T_s$ and $D$ are design parameters. \n\nWe are interested in finding the region of the design space ($T_s$, $D$) for which the side channel samples a target hematocrit to within a specified tolerance. We introduce two variants of this problem: \n\\begin{enumerate}[(i)]\n     \\item the side channel instantaneously samples only the original sample hematocrit throughout the sampling time window, i.e. $\\Phi(1-D,0<T<T_s) = \\Phi_0$, and\n     \\item the side channel samples a target mean hematocrit $\\Phi^*$ (generally $\\neq \\Phi_0$) over the sampling window, i.e. $\\overline{\\Phi}(1-D,T_s) = \\Phi^*$. \n \\end{enumerate}  \n In both cases we specify a tolerance $\\delta \\Phi$. Note that a solution to (i) automatically solves (ii) if $\\Phi^* = \\Phi_0$. Due to the finite ``quiet'' region downstream of the sedimentation front as discussed above (cf. Fig. \\ref{FigProfiles}), solutions to problem (i) generally result in finite regions in the ($T_s$, $D$) design space even for arbitrary small tolerance. By contrast a solution of (ii) seeking a target mean concentration $\\overline{\\Phi}(1-D,T_s) = \\Phi^* \\neq \\Phi_0$ will result in solution regions whose ``size'' scales with the tolerance $\\delta \\Phi$.\n\n\\subsection{Representative subsampling} \\label{SecRepresentative}\n\\begin{figure}[t!]\n    \\includegraphics[scale=1]{DesignSpacePlot.pdf}\n\\caption{Combinations of the dimensionless side-channel height $D$ and the dimensionless sampling time $T_s$, using $k = 3$ in \\eqref{Vnondim}, for which the subsample is representative of the original sample, indicated as shaded regions for different values of $\\Phi_0$, with a tolerance $\\delta \\Phi = 0.001$. The subsample is not representative of the original sample outside the shaded region for a particular $\\Phi_0$. The regions are roughly triangular in shape. Dashed lines indicate the fastest forward and rearward propagating waves, which intersect at the point $\\left(T_s = \\frac{4}{5}, D = \\frac{1}{5} \\right)$. Along with the $T_s = 0$ axis, this point defines the triangular region of the design space (hatched) that ensures a representative subsampling for any hematocrit $\\Phi_0$ [see equation \\eqref{MinTriangle}].}\n\\label{FigSol1}\n\\end{figure}\nWe first consider the case where we demand that the subsample collected in the side channel is representative of the original sample hematocrit $\\Phi_0$ throughout the sampling interval $0 < T < T_s$, corresponding to problem (i) introduced above. Thus, for a specified $\\Phi_0$, we solve the system \\eqref{SedGE} to obtain $\\Phi(Z,T)$ and then identify combinations of sampling times $T = T_s$ and side-channel locations $Z = 1-D$ for which \n\\begin{align} \\label{SolCriterion1}\n\\Phi_0 - \\delta \\Phi \\leq \\Phi(1-D,T_s) \\leq  \\Phi_0 + \\delta \\Phi.\n\\end{align}\nAs noted above we may in principle obtain solution regions with finite size in phase space for $\\delta \\Phi = 0$, although doing so numerically is nontrivial. For the results below we use a small tolerance of $\\delta \\Phi = 0.001$. \n\nAdmissible solutions ($T_s,\\,D$) of \\eqref{SedGE} and \\eqref{SolCriterion1} are indicated as shaded regions in Fig. \\ref{FigSol1} for different values of $\\Phi_0$. These solution regions in the phase plane are of roughly triangular shape with details that depend on $\\Phi_0$. In the limit of arbitrarily short sampling times, the concentration at any location $D$ is identical to the initial concentration $\\Phi_0$, thus identifying one of the solution boundaries with the $T_s = 0$ axis. Furthermore, we observe that $\\Phi = \\Phi_0$ just downstream of the sedimentation front for short times (Fig. \\ref{FigProfiles}) identifying another boundary of the triangular solution region with the trajectory of the front, $Z = V(\\Phi_0) T$ or $D = 1 - V(\\Phi_0) T$. The third side is set by the speed of the backward propagating compression wave [cf. Fig. \\ref{FigFlux}(b)] that starts at $Z = 1$ ($D = 0$), although this speed of propagation is not trivial to obtain as a function of $\\Phi_0$. \n\n\n\nAs Fig. \\ref{FigSol1} shows, solution regions ($T_s,\\,D$) depend strongly on the initial hematocrit $\\Phi_0$. However, it may be important in an application to obtain a representative subsample of the original suspension over a range of initial concentrations $\\Phi_0$. Designing such an application that is viable for any value $\\Phi_0$ therefore requires us to identify the region of intersection of solutions over the entire range of $\\Phi_0 \\in (0,1)$. We estimate the size of this intersection region by considering the fastest sedimenting front and the fastest (rearward moving) compression wave. The fastest sedimentation wave moves with a velocity $V_{\\rm max} = \\mathrm{max}_{\\Phi_0} \\left\\{V(\\Phi_0)\\right\\} = 1$ corresponding to a trajectory $Z = T$ (or $D = 1 - T_s$; Fig. \\ref{FigSol1}). The fastest rearward propagating waves are determined by the fastest negative wavespeeds in the system [cf. Fig. \\ref{FigFlux}(b)]. The maximum negative wavespeed is $U_{\\rm max}^{-} = U(\\Phi_{\\rm max}^{-}) < 0$ where $\\Phi_{\\rm max}^{-}$ is identified by $\\dd{U(\\Phi)}{\\Phi} = 0$; for example $\\Phi_{\\rm max}^{-} = \\frac{1}{2}$ and $U_{\\rm max}^{-} = -\\frac{1}{4}$ for $k = 3$. Thus, backward propagating waves are bounded by the trajectory $Z = 1 + U_{\\rm max}^{-} T$ (or $D = -U_{\\rm max}^{-} T_s$; Fig. \\ref{FigSol1}). These limiting forward and backward waves intersect at $T_s = T_s^c = \\left(1 - U_{\\rm max}^{-}\\right)^{-1}$ and $D = D^c = 1 - T_s^c$.  Thus, representative subsampling is guaranteed for any $\\Phi_0$ within the triangular design-space region defined by the points \n\\begin{align} \\label{MinTriangle}\n(T_s, D) = \\left\\{(0,0);\\quad  (0,1) ; \\quad \\left(T_s^c = \\frac{1}{1 - U_{\\rm max}^{-}},\\, D^c = \\frac{-U_{\\rm max}^{-}}{1 - U_{\\rm max}^{-}} \\right)\\right\\},\n\\end{align}\nwhere we recall that $U_{\\rm max}^{-} \\leq 0$ and therefore both $D^c$ and $T_s^c$ are smaller than unity. This region is hatched in Fig. \\ref{FigSol1}, showing good agreement with numerical results. Thus, placing the side channel at height $D = D^c$ allows the largest sampling time $T_s = T_s^c$ for any $\\Phi_0$; for $k = 3$, $T_s = T_s^c = \\frac{4}{5}$ and $D = D^c = \\frac{1}{5}$. \n\n\\begin{figure}[t!]\n    \\includegraphics[scale=1]{DesignHemPlot.pdf}\n\\caption{Map of the permissible range of $D$ and sample hematocrit $\\Phi_0$ for representative sampling (tolerance $\\delta \\Phi = 0.001$), indicated as shaded regions for different sampling times $T_s$, using $k = 3$ in \\eqref{Vnondim}. The subsample is not representative of the original sample outside the shaded regions for a prescribed $T_s$. The data here are equivalent to those of Fig. \\ref{FigSol1}. The dotted horizontal line represents the limiting value $D \\approx 0.2$ that is marginally within the solution space for all $\\Phi_0$ and occurs for $T_s \\approx 0.8$, in agreement with the prediction of equation \\eqref{MinTriangle}.\n}\n\\label{FigSol2}\n\\end{figure}\nWe note that in practice the physical sampling time $t_s$ and the range of hematocrit is often prescribed so it may be necessary to choose $h$, or equivalently the dimensionless quantity $T_s = t_s v_0\/h$, as a design parameter. A slightly different but equivalent visualization of Fig. \\ref{FigSol1} is shown in Fig. \\ref{FigSol2}, where the permissible range of $D$ is plotted against $\\Phi_0$ as shaded regions for different values of $T_s$. For small $T_s$, the range of allowable $D$ for representative sampling is finite across the entire range of $\\Phi_0$ values (e.g. $T_s \\leq 0.6$ in Fig. \\ref{FigSol2}). As $T_s$ increases up to a critical value, the range of allowable $D$ decreases to a point: at $T_s = 0.8$, only a single choice of $D \\approx 0.21$ (dotted line) can guarantee a representative subsampling over all $\\Phi_0$ values, consistent with the theoretical predictions $D= D^c$, $T_s=T_s^c$ in \\eqref{MinTriangle}. For larger sampling times $T_s^c < T_s \\leq 1$, no single value of $D$ is suitable for all $\\Phi_0$ although individual solutions $D(\\Phi_0)$ exist for any choice of $\\Phi_0$. For even larger sampling times $T_s > 1$, representative subsampling is only possible for sufficiently large hematocrit (sufficiently slow sedimentation), as indicated by the solution for $T_s = 1.5$ in Fig. \\ref{FigSol2}: no representative subsampling solutions $D(\\Phi_0; T_s = 1.5)$ exist for $\\Phi_0 \\lesssim 0.2$.\n\n\\subsection{Oversampling and undersampling}\nAs we have shown above it is possible to obtain a representative subsample of the original sample, but only within a finite region of the dimensionless design space $(T_s, D)$. Outside of this solution region the collected subsample will on average differ in hematocrit from that of the original sample $\\Phi_0$. This feature presents the possibility of developing a design that is capable of controlled undersampling ($\\overline{\\Phi}(Z,T) < \\Phi_0$) or oversampling ($\\overline{\\Phi}(Z,T) > \\Phi_0$) the suspension on average in a systematic way. Such a design may be useful to extract a more dilute or concentrated version of the original sample without the need for an extra separation or mixing step.\n\nWe denote the target mean hematocrit of the subsample by $\\Phi^*$, so that we seek to design a subsampling protocol that achieves\n\\begin{align} \\label{SolCriterion2}\n\\Phi^* - \\delta \\Phi \\leq \\overline{\\Phi}(1-D,T_s) \\leq  \\Phi^* + \\delta \\Phi.\n\\end{align}\nWe are particularly interested in cases where $\\Phi^* \\neq \\Phi_0$ since for $\\Phi^* = \\Phi_0$ (within the tolerance $\\delta \\Phi$) the solution to this problem is the same as that of section \\ref{SecRepresentative}. \n\nAs discussed earlier in Sec. \\ref{SecSubsampling}, the size of the solution region for $\\Phi^* \\neq \\Phi_0$ will strongly depend on both $\\Phi^*$ and the tolerance $\\delta \\Phi$. Thus, we have a 5-parameter space in general ($D, T_s, \\Phi_0, \\Phi^*, \\delta \\Phi$), which is difficult to map out in its entirely. Some insight can be obtained from the resulting map of $\\overline{\\Phi}(1-D, T_s)$ over the phase space $(D, T_s)$ for different initial hematocrit $\\Phi_0$. Figure \\ref{FigSolMean} shows contours of $\\overline{\\Phi}(1-D, T_s)$ for three values of the initial hematocrit $\\Phi_0$ (0.2, 0.5 and 0.7). Contours are spaced $\\delta \\Phi = 0.05$ apart. For each panel of Fig. \\ref{FigSolMean}, we note the presence of a roughly triangular region with one side along the $T_s = 0$ axis. Within the tolerance, this region corresponds with the shaded triangular regions in Fig. \\ref{FigSol1}, where $\\Phi(1-D,T_s)$, and therefore $\\overline{\\Phi}(1-D, T_s)$, is identical to the initial value $\\Phi_0$. Outside of this region a range of $\\overline{\\Phi}$ is obtained for different combinations of $D$ and $T_s$, corresponding to the possibility of achieving different target $\\Phi^*$. \n\nAs expected, subsampling close to the top of the container $(D \\approx 1)$ results in a more dilute suspension ($\\overline{\\Phi} < \\Phi_0$), whereas subsampling near the bottom of the container $(D \\approx 0)$ leads to a more concentrated suspension ($\\overline{\\Phi} > \\Phi_0$). As $T \\rightarrow \\infty$ the system approaches the static solution $\\Phi(Z,T) = \\Theta[Z-(1-\\Phi_0)]$ as discussed in Sec. \\ref{SecModel}, so that the side channel samples $\\overline{\\Phi} \\sim 0$ if $D > \\Phi_0$, and $\\overline{\\Phi} \\sim 1$ if $D < \\Phi_0$. For relatively small $\\Phi_0 \\lesssim 0.5$ (e.g. Fig. \\ref{FigSolMean}a), dilution of the subsample to a prescribed $\\Phi^* < \\Phi_0$ can be achieved for a larger regions of the $(T_s, D)$ design space, while precise concentration to $\\Phi^* > \\Phi_0$ restricts solutions to a narrower range of the design space. This behavior is reversed for a more concentrated initial suspension -- it is easier to further concentrate it (oversample) to a precise value than to dilute it (undersample) with similar precision (e.g. Fig. \\ref{FigSolMean} c). For intermediate values $\\Phi_0 \\approx 0.5$, both dilution and concentration to a prescribed precision allow similarly sized regions of the design space (e.g. Fig. \\ref{FigSolMean} b). We recall that typically $\\Phi_0 \\approx 0.6$ for blood, suggesting that using sedimentation it is somewhat easier to concentrate than to dilute a sample with precision. \n\n\\begin{figure}\n    \\includegraphics[scale=1]{TsDPhase_Mean.pdf}\n\\caption{Map of the mean hematocrit $\\overline{\\Phi}(1-D,T_s)$ obtained for a given side channel height $D$ and sampling time $T_s$ for different initial hematocrit values $\\Phi_0$: (a) 0.2, (b) 0.5 and (c) 0.7. Contours are $\\delta \\Phi = 0.05$ apart. Colors correspond to $\\overline{\\Phi}(1-D,T_s)$, with the scale indicated by the color bar on the right. The triangular regions attached to the $T_s = 0$ axis are comparable with the shaded regions of Fig. \\ref{FigSol1} and correspond to representative subsampling. The remainder of the ($D$, $T_s$) space represents either undersampling $(\\overline{\\Phi} < \\Phi_0)$ or oversampling $(\\overline{\\Phi} > \\Phi_0)$, corresponding to contours terminating at $D = 1$ or $D = 0$, respectively. The sample can be diluted for a larger region of the design space for small $\\Phi_0$ (e.g. panel (a)) whereas the opposite is true for a large $\\Phi_0$ (e.g. panel (c)).}\n\\label{FigSolMean}\n\\end{figure}\n\n\\section{Filling the side channel} \\label{SecSideChannel}\nThe implicit assumption so far is that the suspension drawn into the side channel is well represented by the properties of the suspension in the reservoir at the vertical location $Z = 1-D$. However, the uptake of fluid into the side channel is itself a dynamic process that depends on several factors including the time-dependent viscosity $\\eta$ of the suspension at the mouth of the side channel [a result of the time dependent hematocrit $\\Phi(Z,T)$], the elapsed time and the mechanism by which fluid is pumped into the side channel.\n\nTo illustrate this point, we assume that the fluid is driven into the side channel due to an imposed pressure drop $\\Delta p$ that may be time dependent. For clarity, we denote the hematocrit in the side channel by $\\psi(x,t)$, where ($x=0$, $z = h-d$) represents the mouth of the side channel where it meets the reservoir (see Fig. \\ref{FigSketch}). The side channel is initially devoid of any fluid. Over time, the sample is drawn into the side channel, occupying a length $\\ell(t)$ as shown in Fig. \\ref{FigSketch}. For an incompressible flow, and a side channel with uniform cross-sectional area $A_s$, $\\dd{\\ell}{t}$ is equal to the mean fluid velocity $\\overline{u}$ at any position $x \\leq \\ell(t)$ in the side channel. \n\n\nWe neglect the motion of cells relative to that of the fluid, which may be important in flows of suspensions in narrow channels and may cause particle accumulation \\citep{hol11_particle_imbibition}. Instead, we assume that cells are advected passively with the flow, and that the local viscosity of the suspension depends only on the local volume fraction $\\psi(x,t)$. Then,  Darcy's law for flow in the channel gives\n\\begin{align} \\label{Darcy}\n\\overline{u}(t) = \\dd{\\ell}{t} = - \\frac{\\kappa}{\\eta(\\psi(x,t))} \\pp{p}{x},\n\\end{align}\nwhere $\\kappa$ is the permeability of the channel and depends on its cross-sectional shape and $p(x,t)$ is the pressure. The volumetric flux through the side channel is $q = \\overline{u} A_s$. \nIntegrating \\eqref{Darcy} yields\n\\begin{align} \\label{DelP}\n\\Delta p = \\frac{1}{\\kappa} \\dd{\\ell}{t} \\int_0^{\\ell} \\eta(\\psi(x,t)) \\rmd x.\n\\end{align}\nAssuming that cells are passively advected by the mean fluid flow, the transport equation for $\\psi$ in the side channel is  \n\\begin{align}\\label{SideChannelTransport}\n\\pp{\\psi}{t} + \\dd{\\ell}{t}\\pp{\\psi}{x} = 0,\n\\end{align}\nwith the initial condition $\\psi(x = 0,t) = \\phi(z = h-d,t) \\equiv \\psi_0(t)$. The governing equation is satisfied by a general solution of the form $\\psi = f(\\ell(t) - x)$, where $f$ is an arbitrary function. Using the initial condition determines $f$, so that  \n\\begin{align} \\label{PhiSideSol}\n\\psi(x,t) = \\psi_0\\left(\\ell^{-1}(\\ell(t) - x)\\right)\n\\end{align}\nwhere $\\ell^{-1}$ is the inverse function defined such that $\\ell(t) = x \\iff \\ell^{-1}(x) = \\ell^{-1}(\\ell(t)) = t$. Physically, $\\ell^{-1}(\\ell(t) - x)$ is the time $t^*$ required to fill the side channel up to a length $\\ell^* = \\ell(t) - x$, i.e. $\\ell(t^*) = \\ell^*$. Combining \\eqref{PhiSideSol} with \\eqref{DelP} yields an integro-differential equation for $\\ell(t)$,\n\\begin{align} \\label{DlDtIntegroDiff}\n\\dd{\\ell}{t} \\int_0^{\\ell} \\eta \\left[\\psi_0\\left(\\ell^{-1}(\\ell(t) - x)\\right)\\right)]\\, \\rmd x = \\kappa \\Delta p,\n\\end{align}\nwhere we note that $\\Delta p$ may be time-dependent.\n\n\\begin{figure}[t!]\n    \\includegraphics[scale=1]{ChannelFill.pdf}\n\\caption{Transport in a side channel at a constant pressure difference across the channel for $\\Phi_0 = 0.5$. (a1)--(a3) Profiles of rescaled hematocrit $\\Psi(X,T)$ versus $X$ for different times (indicated by dashed arrows or labels) with (a1) $D=0.3$, (a2) $D = 0.6$ and (a3) $D = 0.8$. (b) Filling length in the side channel $L(T)$ for different $D$, showing that $L(T) \\sim \\sqrt{2 T V(\\Phi_0)}$ for small times. (c) Mean concentrations $\\overline{\\Phi}(1-D,T)$ (symbols) and $\\left<\\Psi\\right> (T)$ for a constant pressure drop across the side channel (curves), plotted versus time $T$, for different channel heights $D$. Both averages are equal to each other and to $\\Phi_0$ for a range of times corresponding to shaded regions in Figs. \\ref{FigSol1} and \\ref{FigSol2} (representative subsampling). Beyond this time $\\overline{\\Phi} \\neq \\langle \\Psi\\rangle \\neq \\Phi_0$, with $\\langle \\Psi\\rangle$ deviating comparatively less from $\\Phi_0$ than $\\overline{\\Phi}$. }\n\\label{FigSideChannel}\n\\end{figure}\n\nBelow, we consider the case of a constant applied $\\Delta p$. We rescale time as before ($T = t v_0\/h$) since the initial condition $\\psi_0(t)$ has features set by sedimentation in the reservoir. We use a Krieger--Dougherty viscosity law consistent with \\eqref{SedimentationLaw}, to write $\\eta(\\psi) = \\eta_0(1 - \\psi\/\\phi_m)^{-k}$, as discussed in Sec. \\ref{SecModel}. Next, we use \\eqref{DlDtIntegroDiff} to scale horizontal distances by $\\ell_{\\Delta p} = \\left\\{\\kappa h \\Delta p \/(\\eta_0 v_0) \\right\\}^{1\/2}$ and define $X = x\/\\ell_{\\Delta p}$ and $L(T) = \\ell(t)\/\\ell_{\\Delta p}$. Defining $\\Psi(X,T) = \\psi(x,t)\/\\phi_m$ so that $\\eta(\\psi) = \\eta_0\/V(\\Psi)$ [see \\eqref{Vnondim}], \\eqref{DlDtIntegroDiff} rescales as \n\\begin{align} \\label{DlDtIntegroDiff_ND}\n\\dd{L}{T} \\int_0^{L} \\frac{1}{V \\left[\\Psi_0\\left(L^{-1}(L(T) - X)\\right)\\right)]}\\, \\rmd X = 1,\n\\end{align}\nwhere $\\Psi_0(T) = \\Psi(0,T) = \\Phi(1-D,T)$ and $L^{-1}(L(T)) = T$. Then, the spatially averaged dimensionless hematocrit of the subsample in the side channel is \n\\begin{align}\n\\left<\\Psi\\right>(T) = \\frac{1}{L(T)}\\int_0^{L(T)}\\Psi(X,T)\\,\\rmd X.\n\\end{align}\n\n\n If $\\Psi_0(T) = \\Psi_0$ is constant over the time of interest, we find \n \\begin{align}\n L(T) = L_0(T) \\equiv \\left(2 T V(\\Psi_0)\\right)^{1\/2}\n \\end{align}\n or $\\ell(t) = \\left(2 t \\kappa \\Delta p\/ \\eta(\\psi_0) \\right)^{1\/2}$, which is a result similar to capillary imbibition of wetting fluids. This result corresponds to subsampling within the shaded regions of the design space in Figs. \\ref{FigSol1} and \\ref{FigSol2}, for which the hematocrit at the opening of side channel ($z = h-d$, $x = 0$) is equal to that of the  original sample (representative subsampling). Consequently the subsample in the side channel is spatially homogeneous and is representative of the original sample in terms of hematocrit, i.e. $\\Psi(X,T) = \\left<\\Psi\\right>(T) = \\Phi(1-D,T) = \\Phi_0$. \n\n\n\nOutside of these solution regions, we solve \\eqref{DlDtIntegroDiff_ND} numerically by first-order Euler integration, using $L(T) \\sim L_0(T) =\\left(2 T V(\\Psi_0)\\right)^{1\/2}$ as an asymptotic result as $T \\rightarrow 0$.  In this case, the subsample is spatially inhomogeneous since the conditions presented to the inlet of the side channel change over time. Figure \\ref{FigSideChannel}(a1--a3) shows profiles of $\\Psi$ versus $X$ at different times and for different sampling heights. For $D< \\Phi_0$ [Fig. \\ref{FigSideChannel}(a1)] the concentration at the mouth of the side channel increases in time due to sedimentation in the reservoir. The opposite is true for $D> \\Phi_0$ [Fig. \\ref{FigSideChannel}(a2--a3)] where the side channel eventually samples $\\Phi = 0$ fluid. For some values of $D \\gtrsim \\Phi_0$, the concentration first increases and then decreases to zero at the inlet $X =0$ [Fig. \\ref{FigSideChannel}(a2)]. \n\nThe growth of $L(T)$ is plotted in Fig. \\ref{FigSideChannel}(b) for different values of $D$. In all cases $L \\sim L_0(T)$ initially, with late time dynamics depending on the location $D$ of the side port. We recall the steady state sedimentation solution $\\Phi(Z,T \\rightarrow \\infty) \\rightarrow \\Theta[Z - (1-\\Phi_0)]$. Thus, for $D < \\Phi_0$, the side channel samples increasingly more concentrated suspension, resulting in $L(T)$ approaching a constant value. For any $D > \\Phi_0$, $L(T)$ grows indefinitely and may either be faster or slower than $L_0(T)$ at any finite time $T$, though it is guaranteed to be asymptotically faster than this early-time growth law as $T \\rightarrow \\infty$ since the side channel takes up only (low viscosity) plasma  beyond a certain time. \n\nIt is reasonable to expect the mean hematocrit of the subsample $\\left<\\Psi \\right>$ to be close to $\\overline{\\Phi}$, as discussed earlier. However, there are two sources of systematic deviation from this value. First, the flux into the side channel is greater for lower hematocrit (due to lower viscosity) resulting in a bias toward lower average cell concentrations over the entire side channel. Second, the flux is greater at earlier times when $\\ell(t)$ is small (due to greater $|\\pp{p}{x}| = \\frac{\\Delta p}{\\ell(t)}$), so the mean subsample hematocrit is biased towards earlier parts of the inlet condition $\\phi_0(t)$. The spatially averaged subsample hematocrit $\\left<\\Psi \\right>$ is plotted in Fig. \\ref{FigSideChannel}(c) for different $D$, showing the time average at the mouth of the side channel, $\\overline{\\Phi}(1-D,T)$, for comparison. As expected, the two averages are identical for sufficiently short times, for which both quantities are equal to $\\Phi_0$ (representative sampling). At a critical time that depends on $D$ (see Fig. \\ref{FigSol1}) both quantities deviate from $\\Phi_0$ and from each other. As discussed above, we observe a tendency for $\\langle \\Psi\\rangle$ to stay closer to the initial value $\\Phi_0$ than $\\overline{\\Phi}$. We also note that the deviations between $\\langle \\Psi\\rangle$ and $\\overline{\\Phi}$ are larger for $\\overline{\\Phi} > \\Phi_0$, consistent with the argument that high-concentration fluid has a greater resistance to flow and therefore less of it is drawn into the channel. \n\nFinally, we note that in practice the pressure may be modulated so that the suspension is drawn into the side channel at constant volumetric flux (e.g. using a syringe pump). We retain rescaled variables as before, with the caveat that lengths are rescaled by a reference length $\\ell_q = \\frac{q h}{v_0 A_s}$ instead of $\\ell_{\\Delta p}$. At constant flux through a channel with constant cross-sectional area, $\\dd{L}{T}$ is constant, so that \\eqref{SideChannelTransport} admits the solution $\\Psi(X,T) = \\Psi_0\\left(T\\left(1-\\frac{ X}{L}\\right)\\right)$. Consequently, the space-averaged hematocrit in the side channel is equal to the time-averaged hematocrit at its inlet, i.e. $\\left<\\Psi\\right>(T) = \\overline{\\Phi}(1-D,T)$. This result for constant-flux pumping into the side channel is in contrast with the case of constant-pressure pumping, as discussed above and plotted in Fig. \\ref{FigSideChannel}(c). \n\nThus, the transport of the suspension within the channel adds an additional layer of complexity to sampling, over features arising from sedimentation in the reservoir, particularly when this transport is pressure-controlled. We note that the presence of viscosity gradients in the channel may produce additional flow features such as instabilities. Further, interactions of cells with channel walls or with the fluid velocity distribution may cause the cells to either lag or accumulate near the moving front in the side-channel. These effects are beyond the focus of this study but may be important in applications.\n\n\n\n\n\n\n\\section{Concluding remarks} \\label{SecConcl}\nWe have developed design considerations to subsample sedimenting suspensions, with a particular focus on blood, which may be relevant to some classes of diagnostic devices. Using a Krieger--Dougherty law along with a one-dimensional Kynch sedimention model we identify combinations of design parameters that ensure representative subsampling; these parameters relate to (i) the geometry of the container holding the original suspension, and (ii) the location of the side channel where subsampling occurs. Further, we explore the possibility of extracting a subsample that is either more or less concentrated than the original sample. We find that doing so with precision depends on several factors including the process by which fluid is pumped into the channel in which the subsample is collected. We find important distinctions between flux-controlled and pressure-controlled pumping into the side channel, particularly outside the solution region for representative sampling. \n\nIn the present work it is implicit that the suspension may be treated as a continuum both in the reservoir and the subsampling channel. This assumption may need to be modified particularly if the side channel has transverse dimensions comparable with that of a single cell. At such scales the deformability of erythrocytes becomes an important feature. Furthermore, it has also been shown that flowing suspensions in narrow channels may result in further particle separation or instabilities, neither of which is modeled here. Nonetheless, our present work provides systematic guiding principles to design devices that extract subsamples from a larger volume of a suspension, such as blood, in ways that either exploit or compensate for sedimentation.\n\n\n\n\n\\begin{acknowledgments}\nThe authors thank Abbott Point of Care Inc. and the National Science Foundation, via grant CBET-1702693, for partial support of this work. \n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the human and various animal brain, information processing involves inference based on inputs from the external world through the sensory systems, which obtains information with uncertainty due to noise. \nPrevious studies suggested that animals such as humans and monkeys process inputs according to a Bayesian inference framework to deal with such uncertainty\\citep{KNILL2004712, ANGELAKI2009452, HAEFNER2016649, Ernst2002, FRISTON20121230, Merfeld1999, Doya2007-yy, probabilistic_brain, Beck15310, illusion_perception, doi:10.1073\/pnas.1918143117}.\n\nBayesian inference is performed by calculating the posterior from the prior, which refers to the information possessed in advance about the signal, and the likelihood estimated by observing the input signal. \nHence, it is believed that the prior must first be represented in the brain, but how prior information is shaped in the brain remains unclear.\nIn previous studies, the prior has often been treated as a given value\\citep{Echeveste:2020aa}, and the mechanism for shaping the prior by learning has not been considered. \nEvolutionary acquisition of the prior has been proposed\\citep{universal_darwinism, evolution_bayesian}, whereas it is naturally expected that such information should be shaped within one generation through observing and learning time-dependent signals.\nExperimental results suggest that the prior and the likelihood for Bayesian inference are encoded in different brain areas\\citep{differential_representation, Chan7817, dAcremont10887}. \nStill, the validity and the mechanisms underlying the results remain controversial, and how area differentiation is relevant to the accuracy of Bayesian inference is not well understood.\nA simulation\\citep{population_codes_of_prior_knowledge} suggested that a gain of the activation function encodes the prior.\nHowever, because the prior was fixed in this study, how shaping occurs when the prior varies over time was not considered.\n\nIn general, to obtain the prior, it is necessary to estimate the prior distribution based on previous observations, and the population of neurons that represents the prior must integrate observed inputs over time.\nOne possible mechanism for achieving such integration may be two neural modules functioning at distinct time scales: a downstream neuron population with slower activity changes separated from an upstream neuron population that processes input information.\nIn this structure, the slow module that does not directly receive inputs may facilitate integration. \nSome experimental reports have suggested that the time scale of neural activities in downstream areas of the brain that do not directly receive external input is slow\\citep{hierarchy_of_intrinsic_timescale, diversity_intrinsic_timescale, brain_and_its_time}. \nOn this basis, we evaluated recurrent neural networks (RNNs) with two modules; a main module with direct connection to the input-output layer and a sub-module with a direct connection to the main module and no connections to the input-output layer (i.e., a hierarchical structure)(Fig.\\ref{fig:hierarchical_rnn}). \nThen, we examined the role of modular structure and the relevance of the time scale difference between the main and sub-modules for the prior representation for Bayesian inference.\n\nWe found that RNNs with a modular structure shape the prior more appropriately than regular RNNs. \nFurther, Bayesian inference is more accurate when the time scale of the sub-module is appropriately slow. \nWhen the time scale is uniform, prior information is maintained in both the main module and sub-module. \nIn contrast, when the time scales are different, prior information is represented by the slow sub-module. \nComparing these two cases revealed that the coded variance of prior on the neural manifold was easier to decode in the time scale difference model, which facilitated the distinction of the average input change from noise. \n\nIn addition, we examined if the modular structure with distinct time scales would emerge from a homogeneous neural network.\nWe trained the network in a Bayesian inference task where the time scale of each neuron varied in time. \nAs the training progressed, we observed that the time scales of neurons differentiated into slower and faster scales.\nA modular structure arose in which slow neurons were separated from the input\/output layers, which were predominantly connected to the fast neurons, and a sub-module with slow neurons represented the prior information.\n\nThese results are crucial for understanding the prior representation mechanism in Bayesian inference and provide insight into the relationships between neural network structure, neural dynamics\\citep{AmuntsENEURO.0316-21.2022, MASTROGIUSEPPE2018609, Computation_Through_Neural_Population_Dynamics, Shaping_dynamics}, and time scales\\citep{timescale_in_cognitive_neuroscience} underlying information processing in the brain, which is considered the central issue of computational neuroscience.\n\n\n\n\\section{Model}\n\\subsection{Recurrent Neural Networks with\/without modular structure}\n\nTo investigate the effect of structure and time scale on Bayesian inference, we considered the following RNNs\\citep{versatile}.\n\nFirst, we established a regular RNN consisting of an input layer, a recurrent(hidden) layer, and an output layer, as shown in Fig.\\ref{fig:hierarchical_rnn}(a).\nThe following equation represents the dynamics of the recurrent layer:\n\n\\begin{equation}\n    {\\bf x}(t+1) = ({\\bf I}-{\\boldsymbol \\alpha}){\\bf x}(t) + {\\boldsymbol \\alpha} {\\rm ReLU}(W_{in}{\\bf u}(t)+W{\\bf x}(t)) + \\sqrt{\\boldsymbol \\alpha}{\\boldsymbol \\xi},\n    \\label{eq:regular_network_dynamics}\n\\end{equation}\nwhere ${\\boldsymbol \\alpha} = (\\alpha_1, \\alpha_2, ..., \\alpha_{200})^\\mathsf{T}$ represents a vector to introduce the time scale of the neurons as \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=12cm]{rnn.png}\n  \\caption{Schematic of RNN. (a) Standard RNN without modular structure (b) RNN with modular structure}\n  \\label{fig:hierarchical_rnn}\n\\end{figure}\n\n\\begin{equation}\n    \\alpha_i  =  \\left\\{  \\begin{array}{ll}\n        \\alpha_m  & (1 \\leq i \\leq 150)\\\\\n        \\alpha_s  &  (150 \\leq i \\leq 200), \\\\\n        \\end{array}\n\t\\right.\n    \\label{eq:alpha_separation}\n\\end{equation}\nwhere the standard homogeneous network is given by $\\alpha_s=\\alpha_m$; the case with $\\alpha_s < \\alpha_m$ was also studied to investigate the effect of time scale difference.\nAlthough we mainly studied the systems with 150 fast, and 50 slow neurons, the results to be discussed are not altered, as long as both the numbers are sufficient (say  100 vs 50, 150 vs 150 for fast and slow neurons).\nHere, ${\\bf u}(t)$ is the input signal, and ${\\bf x}$ is the state of the neurons in the recurrent layer. \nWe adopted the activation function ReLU(${\\rm RELU}(z)=0$ for $z \\leq 0$ and $=0$ for $z > 0$)\\citep{ReLU}. \nThen, the output of the RNN was determined by the linear combination of the internal states as follows.\n\n\\begin{equation}\n    {\\bf y}(t) = W_{out}{\\bf x}(t)\n    \\label{output_calculation}\n\\end{equation}\nIn Eq.\\ref{eq:regular_network_dynamics}, ${\\boldsymbol \\xi}$ was used to account for noise in dynamics given by a random variable that follows a normal distribution with mean $0$ and standard deviation $0.05$.\n\nNext, we introduced a modular structure to the above RNN to ensure the distinction of main and sub-modules(Fig.\\ref{fig:hierarchical_rnn}(b)). \nOnly the main module was connected to the input\/output layers. Thus, the dynamics of the recurrent layer are given by \n\n\\begin{equation}\n        {\\bf x}_m(t+1) = (1-\\alpha_m){\\bf x}_m(t) + \\alpha_m {\\rm ReLU}(W_{in}{\\bf u}(t)+W_{main}{\\bf x}_m(t)+W_{s\\rightarrow m}{\\bf x}_s(t)) + \\sqrt{\\alpha_m}{\\boldsymbol \\xi}_m\n    \\label{eq:modular_network_main}\n\\end{equation}\n\\begin{equation}\n        {\\bf x}_s(t+1) = (1-\\alpha_s){\\bf x}_s(t) + \\alpha_s {\\rm ReLU}(W_{sub}{\\bf x}_s(t)+W_{m\\rightarrow s}{\\bf x}_m(t)) + \\sqrt{\\alpha_s}{\\boldsymbol \\xi}_s,\n    \\label{eq:modular_network_sub}\n\\end{equation}\nwhere ${\\bf x}_m and {\\bf x}_s$ represent the firing rate of neurons in the main and sub-modules, respectively. \nHere, $\\alpha_m and \\alpha_s$ represent the time scale of the main and the sub-module, respectively. $\\alpha_m$ is fixed at $1$, while we varied $\\alpha_s$ from $1$ to $0.01$ to examine the effect of the time scale difference. \nThe RNN output was determined by the linear combination of internal states of the main module.\n\n\\begin{equation}\n    {\\bf y}(t) = W_{out}{\\bf x}_m(t)\n    \\label{eq:output_calculation_modular}\n\\end{equation}\n\n\\subsection{Task}\nIn this study, we considered a task in which Bayesian inference improves estimation accuracy. Specifically, the RNN was tasked with estimating the true value from an observed signal with noise. \nWe generated the external input as follows:\nFirst, the true value $y_{true}$ was randomly sampled from a generator(cause) distribution, given by the normal distribution with mean $\\mu_g$ and variance $\\sigma_g^2$. \nNext, the observed signal $s$ was generated from $y_{true}$ by adding noise so that the input is given by the normal distribution with mean $y_{true}$ and variance $\\sigma_s^2$.\nThe generator did not remain constant: It changed with probability $p_t$ over time. \nWhen the generator changed, $\\mu_g, \\sigma_g$ were sampled uniformly from $\\mu_g \\in [-0.5, 0.5], \\sigma_g \\in [0, 0.8]$ respectively.\n\nAs mentioned in the Introduction, the prior distribution needed for Bayesian estimation must be estimated from the observed signal so that it is close to the generator distribution.\nThen, ${\\bf u}(t)$ for Eq.\\ref{eq:regular_network_dynamics} (or \\ref{eq:modular_network_main},\\ref{eq:modular_network_sub}) is given by using the Probabilistic Population Code (PPC), which has been proposed as the neural basis for Bayesian inference\\citep{Ma2006}.\nPPC assumes that the information in a signal is encoded by a population of neurons with a position-based preferred stimulus that fires probabilistically according to a Poisson distribution. \nIt has been shown that neural networks with a population of neurons following PPC as the input layer can learn probabilistic inference effectively\\citep{Orhan2017}. \nTherefore, in this study, we also assumed that the activity ${\\bf u}$ of the input-layer neurons encoding the observed signal followed the PPC model. \n${\\bf u}$ was sampled from the following Poisson distribution\\citep{PPC_sampling}:\n\n\\begin{equation}\n    p({\\bf u}|s) = \\prod_i \\frac{e^{-f_i(s)}f_i(s)^{u_i}}{u_i!}\n    \\label{eq:poisson}\n\\end{equation}\n\nHere, $s$ is the observed signal generated from $y_{true}$ by adding noise, and $f_i$ is the tuning curve of the neurons. \nThis selective firing occurs in proportion to the gain when the observed signal is generated.\nThis gain is inversely proportional to the noise variance as $g=1\/\\sigma_l^2$, and corresponds to signal clarity. \nNamely, the gain  decreases and noise increases due to uncertainty in observations\\citep{TOLHURST1983775}.\nConsidering the gain, we obtain:\n\n\\begin{equation}\n    f_i(s)=g\\exp \\biggl(\\frac{-(s-\\phi_i)^2}{2\\sigma_{\\rm PPC}^2}\\biggr),\n    \\label{eq:tuning_function}\n\\end{equation}\nwhere $\\phi_i$ represents the preferred stimuli of neurons in the input layer. \nIt was assumed that $\\phi_i$ follows an arithmetic sequence for $i$ ($\\phi_i=-1\/2+i\/m$ when the number of neurons in the input layer is $m$)\\citep{tuning_curve}. \nAlso, $\\sigma_{\\rm PPC}^2$ is a constant that represents the ease of firing and was set as $\\sigma_{\\rm PPC}^2=1\/2$ in this study.\n\nIn this task, the true value $y_{true}$ was to be estimated based on the input signal ${\\bf u}$. \nTherefore, training was performed to minimize the mean squared error (MSE) between the neural network output $y(t)$ and the true value $y_{true}(t)$.  \nNote that the loss function was not based on the Bayesian optimal value calculated from the generator distribution and the noise in the observed signal but only calculated based on the true value.\n\n\\begin{equation}\n    L = \\frac{1}{T}\\sum_t (y(t)-y_{true}(t))^2\n    \\label{eq:loss_function}\n\\end{equation}\n\nTraining was performed by the backpropagation method \\citep{backpropagation_original, bptt}. \nAn efficient Stochastic Gradient Descent method, Adam\\citep{adam}, was used for optimization.\nThe batch size of training samples was set to 50, and the weight decay rate was set to 0.0001; training was performed for 6000 iterations(See Table.\\ref{table:hyperparameters} for the hyperparameters used in the experiment).\n\n\\begin{table}\n    \\caption{Hyperparameters}\n\t\\begin{center}\n            \\begin{tabular}{c|c}  \\hline\\hline\n                Attribute  &  Value  \\\\  \\hline\\hline\n\t        Range  of  $\\mu_p$  &  $-0.5\\leq  \\mu_p \\leq  0.5$    \\\\\n                Range  of  $\\sigma_p$  &  $0  \\leq  \\sigma_p \\leq  0.8$  \\\\\n                Range  of  $\\sigma_l$  &  $\\sqrt{1\/5}  \\leq  \\sigma_l \\leq  1$  \\\\\n                Switching probability of prior & $p_t=0.03$ \\\\\n                Length  of  ${\\bf  u}(t)$  &  $100$  \\\\\n                $\\sigma_{\\rm  PPC}$  &  0.5  \\\\\n                Lasting  time  of  ${\\bf  u(t)}$  &  $T=120$  \\\\\n                \\#Neurons  in  the  main module  &  150  \\\\\n                \\#Neurons  in  the  submodule  &  50  \\\\\n                $\\alpha_m$  &  1  \\\\\n                $\\alpha_s$  &  1, 0.5, 0.2, 0.1, 0.05, 0.01  \\\\\n                Batch  size  &  50  \\\\\n                Optimization  algorithm  &  Adam  \\\\\n                Learning  rate  &  0.001  \\\\\n                Iteration  &  6000  \\\\\n                Weight  decay  &  0.0001  \\\\  \\hline\n                \\end{tabular}\n\t\\end{center}\n    \\label{table:hyperparameters}\n\\end{table}\n\n\n\n\\section*{Results1: Fixed structure and time scales}\n\\subsection*{Bayesian optimality}\nBecause the generated signal $s$ was observed under noise, the neural network was required to estimate the true value sampled from the generator. \nIf the information from the generator was known, $y_{true}$ would be estimated by minimizing the long-term MSE, which reveals the optimal $y$ value as follows (maximum a posteriori(MAP) estimation \\citep{PRML}).\n\n\\begin{equation}\n    y_{opt}=\\frac{\\sigma_g^2}{\\sigma_g^2+\\sigma_s^2}s+\\frac{\\sigma_s^2}{\\sigma_g^2+\\sigma_g^2}\\mu_g\n    \\label{eq:y_opt}\n\\end{equation}\n\nHowever, as described in the \"Task\" section, the information from the generator was not explicitly given to the neural network, so it must be estimated from observed signals as a prior distribution.\nFirst, we examined whether the neural network could achieve this prior-based estimation.\n\nThe output $y$ of RNN with modular structure trained with $\\alpha_s=0.1$, when given an observed signal $s$, is shown Fig.\\ref{fig:output}. \n$s$ was sampled from the prior with $\\mu_g=0.5, \\sigma_g=0.5$, and $\\sigma_s=\\sqrt{1\/5}$ of noise was added.\nThe green points represent the estimation based on the maximum likelihood estimation $y_{ML}$, which is that with the highest accuracy when no prior information is available. \nHere, this estimation is equal to the observed signal $s$.\nThe blue points represent $y_{opt}$ when estimated according to the MAP estimation, and the orange points represent the actual neural network output $y$.\nFig.\\ref{fig:output} shows that the output of RNN is closer to the blue points $y_{opt}$ rather than to the green points, indicating that approximate Bayesian inference (Near-optimal Bayesian inference) with a well-estimated prior is achieved (the mean squared error between $y$ and $y_{ML}$ is $0.15$, and the mean squared error between $y$ and $y_{opt}$ is $0.019$, the latter being smaller).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=10cm]{bayesian_estimation_modular_1.png}\n  \\caption{The output $y$ of RNN against the observed signal value $s$. \n  Before $s$ is input, the time series signal, which is sampled from the normal distribution with the mean $\\mu_g=0.5$ and the standard deviation $\\sigma_g=0.5$ and then the noise with the standard deviation $\\sigma_s=\\sqrt{1\/5}$ is added in the input.\n  The accuracy can be increased by estimating prior based on the signal input before $s$ and performing Bayesian inference.\n  Blue points represent $y_{opt}=\\frac{\\sigma_g^2}{\\sigma_g^2+\\sigma_s^2}s+\\frac{\\sigma_s^2}{\\sigma_g^2+\\sigma_g^2}\\mu_g$ value, orange points represent the output of RNN $y$, and green points represent estimation based on maximum likelihood estimation $y_{ML}=s$. The result is for a model with $\\alpha_s=0.1$.}\n  \\label{fig:output}\n\\end{figure}\n\nNext, we examined the optimality of the Bayesian estimation for networks with and without modular structures and time scale differences. \nFig.\\ref{fig:bayesian_optimality}(a) shows the MSE between $y$ and $y_{opt}$ by the RNN trained under each condition. \nThis result shows that the modular structure improved the accuracy of Bayesian estimation, which was further increased when $\\alpha_s$ decreased to an appropriate degree. \nIn fact, we found the optimal time scale $\\alpha_s=0.1\\sim 0.2$, at which maximum accuracy was achieved.\nEven without modular structure, the time scale difference contributed to inference accuracy, but the accuracy increased significantly with both the modular structure and time scale difference.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=15cm]{accuracy.png}\n  \\caption{\n    (a) MSE between the optimal value $y_{opt}(t)$ and the output of RNN $y(t)$, plotted against the time scale $\\alpha_s$. $\\cdot$ with modular and $\\times$ without modular structure.\n    RNNs with a modular structure is more accurate. \n    In addition, those with $\\alpha_s \\sim (0.1\\sim 0.2)$ have optimal error.\n    (b) MSE between the true value $y_{true}(t)$ and the output of RNN $y(t)$ for the network with $\\alpha_s=1(\\cdot)$ and $\\alpha_s=0.1(\\times)$. The value increases as $p_t$ increases, but the model with $\\alpha_s=0.1$ is always more accurate.\n  }\n  \\label{fig:bayesian_optimality}\n\\end{figure}\n\n\n\\subsection*{Adjustability to rapid generator switching}\nSo far, we studied the performance of Bayesian inference models under a fixed generator to compare the accuracy of Bayesian inference itself.\nNext, we examined their performance when the generator changes in time.\nTo perform Bayesian inference for a rapidly changing input, it was necessary for the model to quickly approach the new optimal value $y_{opt}$ to yield a good estimation. \nTo verify the accuracy of the RNN in this case, we compared the MSE between $y_{true}(t)$ generated by the generator and the output $y(t)$ of RNN under various $p_t$(Fig.\\ref{fig:output}(b)). \nThe model with $\\alpha_s=0.1$ was found to be more accurate for all values of $p_t$.\n\nAs a special case, we considered a setting where the input moves back and forth between two generators, A and B. \nThen we examined whether the prior distribution estimated by the RNN was closer to the distribution of either generator.\nSpecifically, we adopted the generator A with $(\\mu_g, \\sigma_g^2)=(\\mu_A, \\sigma_A^2)$ and the generator B with $(\\mu_g, \\sigma_g^2)=(\\mu_B, \\sigma_B^2)$ and computed the following values when the Bayesian optimal estimates under each generators were $y_{opt}^A, y_{opt}^B$.\n\\begin{equation}\n    a(t) = \\frac{y_{opt}^B-y(t)}{y_{opt}^B-y_{opt}^A}\n    \\label{eq:adjustability}\n\\end{equation}\nWhen $a(t)$ is close to 1, the model's prior is closer to generator B, and when $a(t)$ is close to -1, it is closer to prior A.\n\nComparing the change in $a(t)$ between the model with $\\alpha_s=0.1$ and the model with $\\alpha_s=1$, we found that the model with $\\alpha_s=0.1$ was more adjustable to the generator change as shown in Fig.\\ref{fig:adjustability}(a). \nThis result shows that the model with $\\alpha_s=0.1$ was more responsive to the changes of the generators and recognized the generator change more quickly in all runs. \nThe difference between the two models was especially pronounced in the extreme case in which the two generators switched every time(Fig.\\ref{fig:adjustability}(b)). \nIntuitively, having a population of slow neurons would seem to be a disadvantage in responding to rapid environmental changes, but the results showed the opposite.  \nThe network with $\\alpha_s=1$ could not follow rapid input changes, whereas that with $\\alpha_s=0.1$ could estimate the input prior effectively.\nWe discuss the importance of slow neurons in responding to rapid changes below.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=15cm]{adjustability.png}\n  \\caption{\n    Adjustability to rapid generator change. \n    (a) $a(t)$ for the case where generator A($(\\mu_A, \\sigma_A^2)=(-0.5, 0.04)$) and generator B($(\\mu_A, \\sigma_A^2)=(0.5, 0.04)$) switch alternately with probability $p_t=0.2$. \n    The model with $\\alpha_s=0.1$ adjusted more quickly to the generator change.\n    The thin line represents $a(t)$ when the output was fully adjusted to generator switching.\n    (b) $a(t)$ for the case where generator A($(\\mu_A, \\sigma_A^2)=(-0.5, 0.04)$) and generator B($(\\mu_A, \\sigma_A^2)=(0.5, 0.04)$) switch every time(periodic switch).}\n  \\label{fig:adjustability}\n\\end{figure}\n\n\\subsection*{Representation of the prior}\nWe investigated how the slow sub-module facilitated improved prior representation for Bayesian inference. \nBeginning from the hypothesis that a group of downstream slow neurons represent the prior by integrating the observed signal over time, we investigated which side of the main\/sub-module was responsible for the prior information in the modular RNN.\n\nHere, by using the prior information, the estimated value was shifted from the observed signal $s$ to an appropriate value $y_{opt}$(Eq.\\ref{eq:y_opt}).\nIn other words, even given the same signal input $s$, the output varied depending on which time series signal was input before $s$ (because the prior estimation changed). \nEven if one module returned to its original state, the output shifted from $s$ because the prior information remained in the other module. \nThe scale of this change is considered to represent the degree to which the module utilizes the estimated prior information.\nTherefore, it is possible to estimate the extent to which each module plays a role in prior information processing by examining the change in the output $y(t)$ when the internal state of each main and sub-modules is changed to the value corresponding to a different prior.\n\nFirst, let ${\\bf x}_m(\\mu_g, \\sigma_g), {\\bf x}_s(\\mu_g, \\sigma_g)$ be the internal states of the main and sub-module, respectively when the input signal $s$ from a generator $(\\mu_g, \\sigma_g)$ is applied for a certain period. \nBecause the output $y$ is determined by the internal states of two modules and the input signal, it can be written as $y({\\bf x}_m(\\mu_g, \\sigma_g); {\\bf x}_s(\\mu_g, \\sigma_g), s, \\sigma_s)$.\nFrom this, the change in output $y$ is computed by fixing one of the two modules and varying the other to a different internal state ${\\bf x}_i(\\mu_g, \\sigma_g) \\rightarrow {\\bf x}_i(\\mu_g', \\sigma_g')$.\nThe degree of change in $y$ represents the impact on the output of each module reflecting the prior information.\nHence, by comparing the above variances of $y$ by ${\\bf x}_m$(or ${\\bf x}_s$) with fixed ${\\bf x}_s$(or ${\\bf x}_m$) respectively, it is possible to estimate how much each module is responsible for the prior representation.\nSpecifically, we fixed one of the modules at $\\mu_g=0$, $\\sigma_g=0.4$ (These values are set to the median of the range of values $-0.5 \\leq \\mu_g \\leq 0.5$, $0\\leq \\sigma_p \\leq 0.8$), i.e. ${\\bf x}_i(0, 0.4)$, while for the other module $\\mu_g$ and $\\sigma_g$ are changed as ${\\bf x}_g(\\mu_g, \\sigma_g)$.\nThen, we calculated the variance of $y$ as\n\n\\begin{equation}\n    V_m = \\langle {\\rm Var}[y({\\bf x}_m(\\mu_p, \\sigma_p), {\\bf x}_s(0, 0.4), s, \\sigma_l)]_{(\\mu_p, \\sigma_p)}\\rangle _{(s, \\sigma_l)}\n\\end{equation}\n\\begin{equation}\n    V_s = \\langle {\\rm Var}[y({\\bf x}_m(0, 0.4), {\\bf x}_s(\\mu_p, \\sigma_p), s, \\sigma_l)]_{(\\mu_p, \\sigma_p)}\\rangle _{(s, \\sigma_l)},\n\\end{equation}\nwhere ${\\rm Var}[\\ ]_{(\\mu_p, \\sigma_p)}$ denotes the variance over the changes of $(\\mu_p, \\sigma_p)$, and $\\langle \\rangle_{(s, \\sigma_l)}$ denotes the average over the changes of $(s, \\sigma_l)$.\nThe larger $V_s$ or $V_m$ indicates that the sub-module or main module strongly reflects the difference in the prior distribution to the difference in output, respectively.\n\nDependencies of $V_s$ and $V_m$ on different $\\alpha_s$ are shown in Fig.\\ref{fig:differentiation_of_prior_representation}.\nThis result shows that when $\\alpha_s=1$ (i.e., the time scale is uniform), both the main and sub-modules contribute to the representation of prior distribution to the same degree. \nConversely, when $\\alpha_s=0.1\\sim 0.5$, $V_s$ is much larger than $V_m$, meaning that the sub-module selectively contributes to the representation of the prior. In particular, when $\\alpha_s=0.1$ and $0.2$, the differentiation of representation between the main and sub-modules is more pronounced. \nNote that the contribution of the main module is large when $\\alpha_s=0.01$, probably because the time scale of the sub-module is too slow to code the information of the prior. \nComparing of Fig.\\ref{fig:differentiation_of_prior_representation} and Fig.\\ref{fig:bayesian_optimality} shows that the highly accurate Bayesian inference is achieved when the prior distribution information is localized in the sub-module.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{division_of_roles.png}\n  \\caption{\n    Division of roles for representing prior distribution. \n    $V_s, V_m$ defined in the text Eqs. (12,13) plotted for different values of $\\alpha_s$ computed over $1000$ samples of data. \n    $V_s$ and $V_m$ represent the degree to which the sub-module and the main module are responsible for prior-based information processing. When $\\alpha_s=0.2, 0.1$, the sub-module selectively contributes to the representation of the prior.\n    }\n  \\label{fig:differentiation_of_prior_representation}\n\\end{figure}\n\nNext, we investigated how the prior is represented by the main and sub-modules by visualizing the neural activity by principal component analysis(PCA)\\citep{Mante2013,short-term-memory}.\nFirst, ${\\bf x}_m(\\mu_g, \\sigma_g)$ and ${\\bf x}_s(\\mu_g, \\sigma_g)$ were computed for various $(\\mu_g, \\sigma_g)$ in a model with $\\alpha_s =0.1$, and made PCA.\nThe results were projected on a plane using the first and second principal components and color-coded according to $\\mu_g$ and $\\sigma_g$ (Fig.\\ref{fig:pca_internal_states}(a,b)).\nThe neural activity in the main module was loosely distributed on a one-dimensional manifold, represented by the first principal component(PC1). \nThis PC1 approximately corresponded to the $\\mu_g$ value, although the distinction was not clear.\nIn contrast, the activity in the sub-module was clearly represented by 2-dimensional manifolds, as in Fig.\\ref{fig:pca_internal_states}(b2), where PC1 corresponds to $\\mu_g$ and PC2 corresponds to $\\sigma_g$, rather well.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=15cm]{pca_internal_states-2.png}\n  \\caption{\n    The neural activities of the main module ${\\bf x}_m(\\mu_g, \\sigma_g)$ and sub-module ${\\bf x}_s(\\mu_g, \\sigma_g)$ were plotted by the first and second principal component spaces. \n    (a,b) is the result of $\\alpha=0.1$, and (c,d) is of $\\alpha_s=1$. \n    (a1,c1) Main module, color-coded by $\\mu_g$. \n    (a2, c2) Main module, color-coded by $\\sigma_g$. \n    (b1,d1) Sub-module, color-coded by $\\mu_g$. \n    (b2, d2) Sub-module, color-coded by $\\sigma_g$. \n    300 data are plotted.\n    }\n  \\label{fig:pca_internal_states}\n\\end{figure*}\n\nThen, we performed the same analysis on the model with $\\alpha_s=1$ (Fig.\\ref{fig:pca_internal_states}(c,d)).\nIn this case, the manifolds of neural activities for the main and sub-modules did not change significantly. \nBoth were represented in a one-dimensional manifold corresponding to $\\mu_g$; there was no axis corresponding to $\\sigma_g$. \nThe decodability of $\\sigma_g$ achieved in the internal states of sub-module with $\\alpha_s=0.1$ was not observed for $\\alpha_s=1$. \nIn fact, the coefficient of determination when $\\sigma_g$ was calculated by Ridge regression from the internal state of the sub-module with $\\alpha_s=0.1$ was $0.68$, while that using the sub-module with $\\alpha_s=1$ is $-0.03$.\nThis suggests that the model with $\\alpha_s \\sim 0.1$ can better distinguish the input's variance from noise to accurately perform Bayesian inference.\n\nWhen the generator changed rapidly, the variance of the prior was larger than the variance of the generator, as shown in the SI for the case with $\\alpha_s=0.1$.\nWhen $\\sigma_g$ was large, as seen from Eq.\\ref{eq:y_opt}, the influence of the observed signal $s$ was larger than that of $\\mu_g$, allowing the model to \"keep up\" with large changes in the observed signal. \nThis explains the higher adjustability to rapid generator changes as seen in Fig.\\ref{fig:adjustability}.\n\n\\subsection*{Effects of different time scales}\nTo examine the impact of $\\alpha_s$ differences on Bayesian inference accuracy in detail, we considered how each model with $\\alpha_s=1$ and $\\alpha_s=0.1$ represents prior as a function of the input signal. \nAs seen in Fig.\\ref{fig:pca_internal_states}, when the generator is constant, the internal state of the neural network corresponds with the state of the generator $(\\mu_g, \\sigma_g)$. \nConversely, when the generator changes, the internal state at a certain time does not necessarily correspond to the state of the generator at that time because some time is needed to estimate the state of the prior after the generator switches. \nLet $\\mu_p$ and $\\sigma_p$ be the mean and standard deviation of prior used by the neural network to compute $y(t)$.\nHence, $\\mu_p$ must memorize the input $s(t)$ for a certain time in the form of\n\n\\begin{equation}\n    \\mu_p \\simeq \\sum a_k s(t-k).\n    \\label{eq:approximation_of_mu_p}\n\\end{equation}\n\nTo examine how many past steps $k$ are memorized, $\\mu_p$ must be estimated.\nThis can be achieved by estimating $\\mu_p$ from the internal state ${\\bf x}(t)$.\n\nFirst, we calculated the internal state ${\\bf x}$ for the observed signal with a fixed generator instead of a time-varying case. \nThen, we found the transformation matrix $W_{\\mu_p}$ from the internal state ${\\bf x}$ to the recognized prior $\\mu_p$ by assuming that $\\mu_p$ can be represented by linear transformations of the internal state as $\\mu_p \\simeq W_{\\mu_p}{\\bf x}$.\nThis transformation matrix $W_{\\mu_p}$ was obtained by a pseudo-inverse method\\citep{reservoir_computing}(SI).\n\nNext, we obtained ${\\bf x}(t)$ against the time-varying signal with a probability of $p_t=0.03$. \nBy applying the above transformation matrix, $W_{\\mu_p}$ to ${\\bf x}(t)$ obtained at this time, the prior $\\mu_p$ was estimated accordingly.\nThe state of the prior was thus obtained for the time series of the observed signal $s(t)$. \n\nThen, $a_k$ in Eq.\\ref{eq:approximation_of_mu_p} was obtained to minimize the difference between the two sides of Eq.\\ref{eq:approximation_of_mu_p}. \nBecause the obtained coefficients correspond to the contribution of the signal before $k$ time steps, we could estimate the extent to which the neural network uses past information when estimating the prior.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=12cm]{a_k.png}\n  \\caption{\n    $a_k$ defined by Eq.\\ref{eq:approximation_of_mu_p} is plotted against $t$, for the model with $\\alpha_s=1$ and $\\alpha_s=0.1$ using 3000 data points.\n    }\n  \\label{fig:recognize_alpha}\n\\end{figure}\n\nThe estimated coefficients of Eq.\\ref{eq:approximation_of_mu_p} were plotted against $k$(Fig.\\ref{fig:recognize_alpha}), revealing that the model with $\\alpha_s=0.1$ used more past information in estimating prior information than the model with $\\alpha_s=1$.\nThis difference in time windows leads to a difference in accuracy for prior encoding.\n\n\\section*{Results2: Modular structure organization and time-scale separation by learning}\nSo far, we investigated neural networks with fixed and modular structures along fixed time scales and demonstrated that those with fast and slow modules effectively represented the prior distribution.\nThen, we investigated whether such a structure would emerge by training a neural network to predict $y_{true}$.\nWe again used the same neural network model as the normal RNN.\n\n\\begin{equation}\n    {\\bf x}(t+1)=({\\bf I}-{\\boldsymbol \\alpha}){\\bf x}(t)+{\\boldsymbol \\alpha}{\\rm ReLU}(W_{in}{\\bf u}(t)+W_{rec}{\\bf x}(t)) + \\sqrt{\\boldsymbol \\alpha}{\\boldsymbol \\xi}, \n\\end{equation}\nwhere ${\\boldsymbol \\alpha}$ represents a vector of time scales of neurons consisting of $\\alpha_i$.\nThese ${\\boldsymbol \\alpha}$ values, as well as elements of $W$, change by training to start from initial values set randomly according to $\\mathcal{N}(0.5, 0.1)$. \nDuring training, each matrix $W$ and ${\\boldsymbol \\alpha}$ are optimized according to the gradient descent method\\citep{neural_heterogeneity} at each step.\nThe number of neurons in the recurrent layer of the neural network was set to 80.\n\nThe change in ${\\boldsymbol \\alpha}$ distribution during the learning task is shown in Fig.\\ref{fig:trainable_alpha}(a).\nAs shown, ${\\boldsymbol \\alpha}$ split into two groups over the learning period: one with large values close to 1 and the other with small values near 0.1.\n\nNext, we measured the contribution of prior representation as examined in the \"Representation of the prior\" section for groups of neurons with large values ${\\boldsymbol \\alpha}$ (neurons with $\\alpha_i>0.8$) and groups of neurons with small values ${\\boldsymbol \\alpha}$(neurons with $\\alpha_i<0.2$) for three epochs in the learning process(Fig.\\ref{fig:trainable_alpha}(b)).\nWe found that after 10000 epochs, the slow neurons were responsible for the representation of prior distribution, as in the model with $\\alpha_s=0.1$ in the fixed time scale setting.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=15cm]{trainable_alpha-3.png}\n  \\caption{\n  RNN features obtained by learning when ${\\boldsymbol \\alpha}$ is variable by learning. \n  (a) Frequency distribution of ${\\boldsymbol \\alpha}$ for all neurons at 200, 1000, and 10000 learning epochs. At 10000 epochs, the learning process was complete.\n  (b) Division of roles $V_{slow}$ and $V_{fast}$. \n  See the \"Representation of the prior\" section for definitions of $V_{slow}$ and $V_{fast}$; $V_{slow}\\ (V_{fast})$ was computed for neurons with $\\alpha < 0.2\\ (\\alpha > 0.8)$ respectively. \n  (c) The average degree of RNN connections of 10000 epochs. Connections between input layer, recurrent neurons with $\\alpha < 0.2$, $0.2\\leq \\alpha \\leq 0.8$, $\\alpha > 0.8$, and output layer. \n  Each was normalized so that the maximum value was 1.\n  }\n  \\label{fig:trainable_alpha}\n\\end{figure*}\n\nFinally, we investigated the neural network structure shaped by training. \nIn Fig.\\ref{fig:trainable_alpha}(a), the recurrent layer neurons of the network of epoch 10000 was split into the three groups, divided by the magnitude of $\\alpha_i$, slow neurons with $\\alpha_i<0.2$, fast neurons with $\\alpha_i > 0.8$, and $0.2\\leq \\alpha_i \\leq 0.8$ neurons as the others. \nThe average connectivity between the input layer, each group, and the output layer is shown in Fig.\\ref{fig:trainable_alpha}(c)\\citep{task-representation}.\nThe connection from the input layer to the group of fast neurons and that from the fast neurons to the output layer were distinctively larger than those to or from the slow neurons.\nAmong connections within the recurrent layer, those between the fast and slow neurons were larger than others.\nIn summary, a modular structure, shown in Fig.\\ref{fig:hierarchical_rnn}(b), emerged through learning alone.\n\n\\section*{Discussion}\nIn this study, we demonstrated that neural networks with slow and fast activity modules play an essential role in the prior representation for Bayesian inference.\nWe set up a task to predict a time-varying signal under noise that could be estimated by Bayesian inference and trained RNNs with or without modular structure and with or without time scale differences.\n\nThe RNN could learn to approximate Bayesian inference using prior(approximating the generator distribution) in all conditions tested.\nHowever, the accuracy was higher in the modular RNN; further, the accuracy was significantly higher when the time scale of the sub-module was moderately slower than that of the main module.\nIn addition, the increase in accuracy was pronounced against a rapidly varying input, for which it was necessary to generate a prior that changes quickly.\nTo achieve such accuracy with a slow sub-module, the sub-module was found to dominantly represent the prior, indicating role differentiation between representation of the prior and representation of the observed signal (likelihood).\nOf note, such functional differentiation is caused by differences in time scales.\nThis result is consistent with experimental observations in the brain in which areas that code the prior and likelihood in Bayesian inference are different\\citep{differential_representation, Chan7817, dAcremont10887}.\nFinally, it was shown that a modular structure with distinct time scales was spontaneously organized in the RNN by learning. \n\nIt is important to note that a relatively slow time scale of the neuron population encoding the prior is required, but the difference between fast and slow neurons should not be excessive.\nIf the time scale is too small, the accuracy is decreased (Fig.\\ref{fig:bayesian_optimality}) in which case the sub-module is not responsible for representing the prior (Fig.\\ref{fig:differentiation_of_prior_representation}).\nThis is because prior construction requires a larger time span to address changes in external input for a neural network with such a slow time scale.\nTherefore, we suggest that there is an optimal time scale for the slow sub-module.\nFuture research should investigate how this optimal time scale depends on the time scale of environmental changes.\n\nIt has been suggested that the time scale of neurons slows down hierarchically from the area where the signal is directly applied to the area where information is proceed\\citep{hierarchy_of_intrinsic_timescale, diversity_intrinsic_timescale, brain_and_its_time}.\nThis time scale hierarchy with a modular structure\\citep{hierarchical_task} is suggested to be relevant to information processing\\citep{multi-timescales, hierarchical_task, multi_scale_dynamics}.\nOur study showed that modular structures with two-level time scales could deal with slowly changing inputs.\nA deeper modular structure with multiple time scales may be necessary to deal with further complex changes in environments. \nWith such a structure, Bayesian inference against complex temporal changes could be achieved by extrapolating the results of this study.\nFurther research verifying this finding will elucidate the significance of hierarchical structuring in the brain. \nIt is noteworthy that the time scale separation was not only found to be influential for accurate Bayesian inference but also emerged from learning in our simulation. \nConsidering these findings, a similar process may be expected in evolution\\citep{doi:10.1063\/5.0019116}.\n\nThe modular network with slow\/fast time scales could integrate out noise and distinguish the average change in the inputs from fast noise.\nIn fact, the network could effectively predict temporal changes in the input, even under rapidly changing conditions.\nThe brain must adapt to time-varying, noisy inputs; hence, the performance of Bayesian inference by the network design reported herein is considered relevant to brain information processing.\n\nWe adopted a simple RNN and trained it using backpropagation.\nred{Backpropagation} is often believed to be different from the learning algorithm implemented in the brain\\citep{biologically_plausible1, biologically_plausible2}, so care should be taken when generalizing our results.\nHowever, previous studies have also suggested that neural networks obtained by backpropagation can show similar behavior to that of the actual brain\\citep{deeplearning-neuroscience1, deeplearning-neuroscience2, Mante2013, BARAK2013214, j.2018emergence, goal_driven_deep_learning, HAESEMEYER20191123}. \nWith these considerations, our findings are considered to be relevant to the brain's learning processes despite the potential limitation of backpropagation.\n\nUnraveling the relationship between the structure of neural networks, neural dynamics, and the information processing performed by the brain is a primary goal in computational neuroscience\\citep{MASTROGIUSEPPE2018609, role_of_population_structure, Computation_Through_Neural_Population_Dynamics, Shaping_dynamics}. \nIn this study, the relevance of modular structure and time scale difference in neural dynamics to the representation of the prior in Bayesian inference is demonstrated, as well as their formation by learning\\citep{LORENZ2011129, spontaneous_evolution_of_modularity}, which will support ongoing research in the field.\n\n\\subsection*{Data Availability}\nSource codes for these models can be found at \\href{https:\/\/github.com\/tripdancer0916\/slow-reservoir}{https:\/\/github.com\/tripdancer0916\/slow-reservoir}\n\n\n\\section*{Appendix}\n\\subsection*{Appendix A: Adjustment to rapid environment changes}\nThe trajectories of the internal state ${\\bf x}_{sub}(t)$ are plotted by the first and second principal components in Fig. 6 for the cases in which generators A and B switch every 2 time steps and every 30 time steps. \nGenerators A and B both have $\\sigma_g=0.04$. \nIn the case of switching every 30 time steps, they were located in the region taken by the internal state when $\\sigma_g$ was small. In the case of switching every 2 time steps, they were located in the region taken by the internal state when $\\sigma_g$ was large (See Fig. S1). \nThis occurred because the generators switched so rapidly that the RNN recognized that the signal was created by a generator with a large variance. \nThis made it possible to switch $y(t)$ quickly because the information of the observed signal $s$ was prioritized over the prior information when calculating the output $y$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=10cm]{adjust_to_fast_environment_changes.png}\n\\label{fig:adjust}\n\\caption{Trajectory of the internal state ${\\bf x}_{sub}(t)$ of the sub-module when generator A($(\\mu_A, \\sigma_A^2)=(-0.5, 0.04)$) and generator B($(\\mu_A, \\sigma_A^2)=(0.5, 0.04)$) switch alternately.}\n\\end{figure}\n\n\\subsection*{Appendix B: Results of the RNN with $\\alpha_m=\\alpha_s=0.1$}\nWe argue that the slower time scale of the sub-module relative to the main module is important for accurate Bayesian inference. \nHere, to investigate whether the difference in time scale or the slower time scale itself were more influential, we trained an RNN with $\\alpha_s=\\alpha_m=0.1$ and examined its accuracy.\nWe found that when $(\\alpha_m, \\alpha_s)=(0.1,0.1)$, the MSE was larger, and the accuracy was worse than that in the cases with $(\\alpha_m, \\alpha_s)=(1,1)$ and $(\\alpha_m, \\alpha_s)=(1,0.1)$, as shown in Fig. S2. \nTherefore, it is not simply the slower time scale of the neurons but the time scale difference between the main and sub-modules that facilitate accurate Bayesian inference.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{bayesian_optimality_comparison_p_t=0.0.png}\n\\label{fig:comparison}\n\\caption{Mean squared error between the optimal value $y_{opt}(t)$ and the output of RNN $y(t)$, plotted against the setting $(\\alpha_m, \\alpha_s)=(1,1),(1,0.1),(0.1,0.1)$.}\n\\end{figure}\n\n\\subsection*{Appendix C: Detailed analysis of the time scale difference effect}\nTo obtain the transformation matrix $W_{\\mu_p}$, we created a data vector $M_g$ that arranges $\\mu_g$ and a data matrix $X$ that arranges the internal state $\\bf x$ obtained when fixed $\\mu_g$ was input as follows:\n\n\\begin{equation*}\n    M_g = (\\mu_g^1, \\mu_g^2, ...)^{\\mathsf T}, \\ X = ({\\bf x}^1, {\\bf x}^2, ...)\n\\end{equation*}\nThen, we attempted to find matrix $W_{\\mu_p}$ such that $M_g \\simeq W_{\\mu_p}X$.\nUsing the Moore-Penrose pseudo inverse, we can find the best-fit matrix as $W_{\\mu_p}=M_g X^\\dag$\\citep{penrose_1955}.\nLet $\\mu_p$ be the result of the transformation by $W_{\\mu_p}$. \nAs Fig. S3(a) shows, $\\mu_g\\simeq \\mu_p$ is valid.\nTo find $a_k$, we calculated ${\\bf x}(t)$ in the case that $\\mu_g$ varies randomly using $W_{\\mu_p}$ and obtained $\\mu_p$. \nThen, we created a data vector $M_p$ that arranges $\\mu_p$ and a data matrix $S$ that arranges ${\\bf s}=(s(t-1), s(t-2), ..., s(t-K))^{\\mathsf T}$.\nUsing the Moore-Penrose pseudo inverse, we found ${\\bf a}=(a_1,a_2,...,a_K)^{\\mathsf T}$.\nAs Fig. S3(a) shows, $\\mu_p \\simeq \\sum_k^K a_k s(t-k)$ was valid.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=14cm]{decoding_mu_p.png}\n\\label{fig:decoding}\n\\caption{(a)Comparison between the estimated mean of prior $\\mu_p$ and the mean of generator $\\mu_g$.\n(b)Comparison between the linear weighted sum of past signals $s(t-k)$ and the estimated mean of prior $\\mu_p$.}\n\\end{figure}\n\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nVisibility, and hence, visibility representations of graphs are\ncentral to many areas, such as architecture, sensor networks, robot\nmotion planning, and surveillance and security.  There is a long\nhistory of research on characterizing and recognizing visibility\ngraphs in various settings; see the related work below.  Despite\ntremendous efforts characterizations and efficient recognition\nalgorithms are only known for very restricted\ncases~\\cite{ec-rvgsp-90,ec-nrcvg-95}.  Recently, Alpert et\nal.~\\cite{akl-ong-10} introduced obstacle representations of graphs,\nwhich generalize many previous visibility variants, such as\npolygon--vertex visibility.  In this paper, we study \\emph{plane\n  outside-obstacle representations}, where the visibility segments may\nnot cross, and a single obstacle is located in the outer face of the\nrepresentation.  We characterize the biconnected graphs admitting such\na representation and give a linear-time recognition algorithm.  This\nis one of the first results that characterizes such a class of graphs\nand gives an efficient recognition algorithm.  In the following we\nfirst give some basic definitions.  Afterwards, we present an overview\nof related work and describe our contribution in more detail.\n\nAn \\emph{obstacle representation} of a graph~$G=(V,E)$ consists of a\nset of polygonal obstacles and a distinct point for each vertex\nin~$V$.  The representation is such that two points see each other\nif and only if the corresponding vertices are adjacent.  The\n\\emph{obstacle number} of~$G$ is the smallest number of obstacles in\nany obstacle representation of~$G$.\n\nIn an \\emph{outside-obstacle representation} all obstacles are in the\nunbounded face of the representation, i.e., they are contained in the\nunbounded face of the corresponding straight-line drawing of the\ngraph.  Outside-obstacle representations are a recent generalization\nof classical \\emph{polygon--vertex visibility} graphs, where the\nobstacle is a simple polygon, the points are the vertices of the\npolygon and visibility segments have to lie inside the polygon.  The\ncorresponding characterization problem and the complexity of the recognition\nproblem are long-standing open questions.\n\nFigure~\\ref{fig:examples-oor} shows examples of outside-obstacle\nrepresentations.  We note that, in the case of outside-obstacle\nrepresentations, it can be assumed that there exists only a single\nobstacle that surrounds the outer face of the representation, as\nillustrated in Fig.~\\ref{fig:example-oor-1}.  In particular, graphs\nwith an outside-obstacle representation have obstacle number at\nmost~1.  In our setting, given a straight-line drawing of a graph, we\nhence do not make the outside obstacle explicit.  Instead, we require\nfor an outside-obstacle representation of a graph~$G$ that any\nnon-edge of~$G$ intersects the outer face of~$G$.  It is then not\ndifficult to construct a corresponding obstacle.  Throughout this\ndocument, we assume that points and vertices of obstacles are in\n\\emph{general position}, so no three of them are collinear.\n\n\n\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.45\\textwidth}\n    \\centering\n    \\includegraphics[page=5]{fig\/examples-oor}\n    \\caption{}\\label{fig:example-oor-1}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.45\\textwidth}\n    \\centering\n    \\includegraphics[page=1]{fig\/examples-oor}\n    \\caption{}\\label{fig:example-oor-2}\n  \\end{subfigure}\n  \\caption{(a) A non-planar outside-obstacle representation of the\n    octahedron; the obstacle is shown in gray.  (b) A plane outside-obstacle representation of\n    Figure~\\ref{fig:example-3}.}\n  \\vspace{-3ex}\n  \\label{fig:examples-oor}\n\\end{figure}\n\n\\myparagraph{Related Work.}\nGraphs with an outside-obstacle representation are equivalent to\nvisibility graphs of a pointset within a simple polygon. Therefore,\npolygon--vertex visibility graphs form an important subclass of our\ngraphs class, where the pointset coincides with the corners of the\nsurrounding simple polygon.  Such graphs have been extensively studied\ndue to their many applications, e.g., in gallery\nguarding~\\cite{r-agta-87}.\n\nPolygon--vertex visibility graphs were first introduced in 1983 by\nAvis and ElGindy~\\cite{ae-caps-83} and are most studied in the field\nof visibility problems~\\cite{gg-upvgpsp-12}.  One of the first results\non the topic was that maximal outerplanar graphs are polygon--vertex\nvisibility graphs~\\cite{e-hdpa-85}.  Ghosh~\\cite{g-rcvgs-97} gives a\nset of four necessary conditions for polygon--vertex visibility\ngraphs, which he conjectured to be also sufficient. However,\nStreinu~\\cite{s-npvg-05} constructed a counterexample.  As pointed out\nin~\\cite{gg-upvgpsp-12}, two of Ghosh's necessary\nconditions~\\cite{g-rcvgs-97} imply the conditions of a\ncharacterization attempt by Abello and Kumar in terms of oriented\nmatroids~\\cite{ak-vgom-02}, which hence cannot be sufficient.\n\nSo far, characterizations have only been achieved for polygon--vertex\nvisibility graphs of restricted polygons.  Everett and\nCorneil~\\cite{ec-rvgsp-90,ec-nrcvg-95} give a characterization of\nvisibility graphs in spiral and 2-spiral polygons -- polygons that\nhave exactly one and two chains of reflex vertices, respectively.\nThey are characterized as interval graphs and perfect graphs.  The\nbest known complexity result about the recognition and reconstruction\nproblem of polygon--vertex visibility graphs is that they are in\nPSPACE~\\cite{e-vgr-90}.\n\nA generalization of these graphs are induced subgraphs of\npolygon--vertex visibility graphs, or \\emph{induced visibility graphs}\nfor short.  Spinrad~\\cite{s-egr-03} considers this graph class the\nnatural generalization of polygon--vertex visibility graphs, which is\nhereditary with respect to induced subgraphs.\nEverett and Corneil~\\cite{ec-nrcvg-95} show that there is no finite\nset of forbidden induced subgraphs in polygon--vertex visibility\ngraphs.\n\n\n\nCoullard and Lubiw show that 3-connected polygon--vertex visibility\ngraphs admit a 3-clique ordering~\\cite{cl-dvg-91}.\nAbello et al.~\\cite{alp-vgsp-92} prove that every 3-connected planar\npolygon--vertex visibility graph is maximal planar and that every\n4-connected such graph cannot be planar.  Their conjecture that\nHamiltonian maximal planar graphs with a 3-clique ordering are\npolygon--vertex visibility graphs was disproven by Chen and\nWu~\\cite{cw-dcpvg-01}.  According to O'Rourke~\\cite{o-opcvi-98}\nnecessary and sufficient conditions for a polygon--vertex visibility\ngraph to be planar are known~\\cite{lc-pvg-94}, but do not lead to a\npolynomial recognition algorithm.\n\n\nFor the more general question of obstacle numbers, Alpert et\nal.~\\cite{akl-ong-10} give a construction for graphs with large\nobstacle number and small example graphs that have obstacle number\ngreater than~1.  They further show that every outerplanar graph admits\na (non-planar) outside-obstacle representation, i.e., they are\nvisibility graphs of pointsets inside simple polygons.  Subsequent\npapers extend the results on obstacle numbers.  Pach and\nSar\u0131\u00f6z~\\cite{ps-sglon-11} construct small graphs with obstacle\nnumber~2 and show that bipartite graphs with arbitrarily large\nobstacle number exist.  Mukkamala et al.~\\cite{mpp-lbong-12} show that\nthere are graphs on $n$ vertices with obstacle number\n$\\Omega(n\/\\log{n})$. It is an open question whether any graph with\nobstacle number~1 admits an outside-obstacle representation.\n\n\\myparagraph{Contribution and Outline.}\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.1\\textwidth}\n    \\centering\n    \\includegraphics[page=1]{fig\/examples}\n    \\caption{}\\label{fig:example-1}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n    \\centering\n    \\includegraphics[page=2]{fig\/examples}\n    \\caption{}\\label{fig:example-2}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n    \\centering\n    \\includegraphics[page=3]{fig\/examples}\n    \\caption{}\\label{fig:example-3}\n  \\end{subfigure}\n  \\caption{Graphs admitting (c) \/ not admitting (a,b) a\n    plane outside-obstacle representation.}  \\vspace{-3ex}\n  \\label{fig:examples}\n\\end{figure}\nIn this paper, we study \\emph{plane outside-obstacle representations},\nwhere the drawing of~$G$, without the obstacles, is free of crossings;\nsee Fig.~\\ref{fig:example-oor-2} for an example.  Consider the graphs\nshown in Fig.~\\ref{fig:examples}.  We will see that the first two\ngraphs do not admit a plane outside-obstacle representation, whereas\nthe last example has one.  Note that the drawing in\nFig.~\\ref{fig:example-1} is a (non-planar) outside-obstacle\nrepresentation.  Our main results are the following.\n\\begin{inparaenum}[(1)]\n\\item Every outerplanar graph whose inner faces are triangles admits a\n  plane outside-obstacle representation.\n\\item A characterization of the biconnected graphs that admit a plane\n  outside-obstacle representation.\n\\item A linear-time algorithm for testing whether a biconnected graph\n  admits a plane outside-obstacle representation.\n\\end{inparaenum}\nAs a side result, we obtain a simple combinatorial proof of ElGindy's\nclassical result that maximal outerplanar graphs are polygon--vertex\nvisibility graphs~\\cite{e-hdpa-85}.\n\nOur paper is structured as follows.  First, we derive a simple\nnecessary condition on the structure of biconnected graphs that admit\na plane outside-obstacle representation in\nSection~\\ref{sec:inner-chordal}.  This restricts the class of graphs\nwe have to consider and we derive some useful structural results about\nsuch graphs.  Afterwards, in Section~\\ref{sec:characterization}, we\ngive a local description of plane outside-obstacle representations\nand, based on this, we derive a combinatorial characterization of the\nbiconnected planar graphs that admit a plane outside-obstacle\nrepresentation in terms of an orientation of a certain subset of\nedges.  Using this characterization, we prove our main results in\nSection~\\ref{sec:main}.\n\n\n\n\n\\section{Inner-Chordal Plane Graphs}\n\\label{sec:inner-chordal}\n\nA graph with a fixed planar embedding is \\emph{inner-chordal plane} if\nany cycle~$C$ of length at least~4 has a chord that is embedded in the\nbounded region of~$\\mathbb{R}^2 \\setminus C$; see\nFig.~\\ref{fig:inner-chordal}.  We first show that we can restrict our\nanalysis to inner-chordal plane graphs.\n\n\\begin{lemma}\\label{lem:plane-oor-chordal}\n  Graphs with a plane outside-obstacle representation are\n  inner-chordal plane.\n\\end{lemma}\n\\begin{proof}\n  Let~$G$ be a graph with a plane outside-obstacle representation, and\n  assume it is not inner-chordal.  Hence, there exists a cycle~$C$ of\n  length at least~4, whose interior does not contain a chord.  Note\n  that the obstacle lies outside of~$C$ by definition.  The cycle~$C$\n  is embedded as the boundary of a simple polygon~$P$ on at least four\n  vertices.  Since~$P$ can be triangulated, there exists a pair of\n  non-adjacent vertices~$u$ and~$v$ on~$C$ such that the segment~$uv$\n  is completely contained in~$P$.  Hence the obstacle cannot\n  intersect~$uv$, and thus~$\\{u,v\\} \\in E(G)$ by definition,\n  contradicting our choice of~$u$ and~$v$.\n\\end{proof}\n\n\\begin{figure}[tb]\n  \\centering\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\includegraphics[page=1]{inner-chordal}\n      \\caption{}   \n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\centering\n    \\includegraphics[page=3]{inner-chordal}\n    \\caption{}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\centering\n    \\includegraphics[page=2]{inner-chordal}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{(a) An inner-chordal graph. (b), (c) Chordal, but not inner-chordal, plane graphs.}\n  \\vspace{-2ex}\n  \\label{fig:inner-chordal}\n\\end{figure}\n\nNote that Lemma~\\ref{lem:plane-oor-chordal} shows immediately that the\ngraph from Fig.~\\ref{fig:example-1} does not admit a plane outside-obstacle\nrepresentation.  Although this graph is chordal, it does not have a\nplanar embedding that is inner-chordal.\nIn the following, we consider only inner-chordal plane graphs.  Note\nthat, in particular, every inner face of an inner-chordal plane graph\nis a triangle.  Moreover, an outerplanar graph is inner-chordal if and\nonly if it is chordal, which is the case if and only if every inner\nface is a triangle.\n\n\\begin{lemma}\n  \\label{lem:inner-chordal-degree}\n  Let~$G$ be an inner-chordal plane graph.  Then, every inner vertex\n  of~$G$ has degree~3 and no two inner vertices are adjacent.\n\\end{lemma}\n\n\\begin{proof}\n  Since~$G$ is inner-chordal, every inner face is necessarily a\n  triangle.  This implies that the neighbors of any inner vertex form\n  a cycle.  This cycle does not have an inner chord, and hence,\n  since~$G$ is simple and inner-chordal, it must have length~3.  This\n  implies that any inner vertex has degree~3.  Moreover, it follows\n  that the neighbors of any inner vertex~$v$ must be incident to the\n  outer face as they already have degree~3 in the neighborhood of~$v$.\n\\end{proof}\n\n\n\nThis description also gives rise to a certain tree that is associated\nwith every biconnected inner-chordal graph.  Let~$G$ be a biconnected\ninner-chordal plane graph.  A \\emph{chord} of~$G$ is an edge that is\nnot incident to the outer face but whose endpoints are incident to the\nouter face.  Lemma~\\ref{lem:inner-chordal-degree} implies a\ndecomposition of~$G$ along its chords.  Namely, we first remove all\ninner vertices.  Each such removal transforms a 4-clique of~$G$ into a\ntriangular face; we mark each triangle that results from such a\nremoval.  The resulting graph~$G'$ is outerplanar and every inner face\nis a triangle.  Now the weak dual~$T'$ of~$G'$ is a tree, where each\nnode corresponds to a triangle of~$G'$.  By marking the nodes of~$T'$\nthat correspond to a marked triangle, we obtain the construction\ntree~$T$ of~$G$, denoted~$T(G)$.  Note that each marked node of~$T(G)$\ncorresponds to a 4-clique of~$G$, whereas an unmarked node corresponds\nto a triangular face of~$G$.  We refer to these nodes as~$K_4$-\nand~$K_3$-nodes, respectively.  The edges of~$T(G)$ correspond\nbijectively to the chords of~$G$.  We refer to the vertices of~$T(G)$\nas nodes to distinguish them from the vertices of~$G$.  For a\nnode~$\\tau$ of~$T(G)$, we denote by~$V_\\tau$ the vertices of the\ncorresponding triangle or 4-clique.  \n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics{fig\/construction-trees}\n  \\caption{The construction tree of graphs~(b) and~(c) in\n    Fig.~\\ref{fig:examples}; $K_3$-nodes are empty and $K_4$-nodes are\n    filled.  Structurally, the only difference between the two graphs\n    is how the two subgraphs of~$G$ are attached to the triangle\n    corresponding to the $K_3$-node~$\\tau$.}\n  \\vspace{-3ex}\n\\end{figure}\n\nObserve that, if we store with each node of~$T(G)$ the corresponding\nedges and use the bijection of the edges of~$T(G)$ with the chords\nof~$G$ to find the shared chord of adjacent nodes, we can reobtain~$G$\nfrom~$T(G)$ by merging triangles and 4-cliques that are adjacent\nin~$T(G)$ along their shared chords.  Then~$T(G)$ is essentially the\nSPQR-tree of~$G$~\\cite{dt-omtc-96}.  We decided to avoid the\ntechnical machinery associated with SPQR-trees and rather work with\nthe construction tree, which is more tailored to our needs.\n\n\\section{Characterization of Plane Visibility Representations}\n\\label{sec:characterization}\n\nIn this section we devise a combinatorial characterization of the\nbiconnected inner-chordal graphs that admit a plane outside-obstacle\nrepresentation.  This is done in two steps.  First, we show that,\naside from being free of crossings, the property of being a plane\noutside-obstacle representation depends only on local features in the\ndrawing, namely, for each chord of a graph~$G$, its neighbors must be\nembedded in certain regions.  In a second step, we show that this\nessentially induces a binary choice for each chord.  In this way, an\noutside-obstacle orientation induces an orientation of the chords\nof~$G$, and we will characterize the existence of a plane\noutside-obstacle representation in terms of existence of a suitable\nchord orientation.\n\n\\boldparagraph{Local Description of Plane Visibility Representations.}\nNext we aim to understand better which planar straight-line drawings\nare outside-obstacle representations.  As a first observation,\nconsider two triangles~$D$ and~$D'$ sharing a common edge~$e=\\{u,v\\}$,\nwhich then forms a chord.  Let~$w$ and~$w'$ denote the tips of~$D$\nand~$D'$ with respect to base~$e$, respectively.  For an\noutside-obstacle representation it is a necessary condition that the\nnon-edge~$\\{w,w'\\}$ intersects the outer face.  We thus have to\nposition the tips in such a way that the segments~$ww'$ does not lie\ninside the drawing of~$D$ and~$D'$.  We use the following definition;\nsee Fig.~\\ref{fig:region} for an illustration.\n\n\\begin{definition}\\label{dfn:regions1}\n  Let $D$ be a triangle and $u$ a vertex of $D$.  Then $R_D(u)$\n  denotes the intersection of the half-planes defined by the sides\n  of~$D$ incident to~$u$ not containing~$D$.\n\\end{definition}\n\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.5\\textwidth}\n    \\centering\n    \\includegraphics{regions3}\n    \\caption{}\\label{fig:region}\n  \\end{subfigure}\\hfil\n\\begin{subfigure}[b]{.45\\textwidth}\n  \\centering\n    \\includegraphics[page=1]{fig\/regions4}\n    \\caption{}\\label{fig:attachment}\n  \\end{subfigure}\n  \\caption{(a) Regions of triangle~$D$ at edge~$e=\\{u,v\\}$ and (b)\n    attachment of two triangles~$D$ and~$D'$ along chord~$\\{u,v\\}$.}\n  \\label{fig:proof-regions2}\n  \\vspace{-2ex}\n\\end{figure}\n\nTo ensure that the segment~$ww'$ intersects the outer face, it is\nclearly necessary that~$w' \\in R_D(u) \\cup R_D(v)$ or~$w \\in R_{D'}(u)\n\\cup R_{D'}(v)$.  The former ensures that the segment~$ww'$ does not\nintersect the interior of~$D$ and the letter ensures the same property\nfor~$D'$.  These intersections behaviors are not independent.  It is\nin fact readily seen that~$w' \\in R_D(x)$ if and only if~$w \\in\nR_{D'}(x)$ for~$x \\in \\{u,v\\}$; see Fig.~\\ref{fig:attachment}.  More generally, the\nsame observations also hold for a chord~$e= \\{u,v\\}$ that is shared by\n\\begin{inparaenum}[(a)]\n  (a) two triangles~$\\tau$ and~$\\tau'$, (b) a triangle~$\\tau$ and a\n  4-clique~$\\tau'$ and (c) by two 4-cliques~$\\tau$ and~$\\tau'$.\n\\end{inparaenum}\nTo see this, note that the regions~$R_{D}(u)$ and~$R_{D'}(u)$ in\nFig.~\\ref{fig:attachment} do not change if~$D$ and\/or $D'$ are part of\na 4-clique.  More formally, let~$W$ and~$W'$ be the vertices of~$\\tau$\nand~$\\tau'$ that are distinct from~$u$ and~$v$, respectively.  Let~$D$\nand~$D'$ denote the triangles incident to~$e$.  Then the following\ncondition is necessary\n\\vspace{-1ex}\n\\begin{align}\n  \\label{eq:1}\n  W' \\subseteq R_D(u) \\text{\\quad or \\quad }  W' \\subseteq R_D(v)\\,\\,.\\tag{*}\n\\end{align}\n\\vspace{-4ex}\n\n\\noindent Again it holds that $W' \\subseteq R_D(x)$ if and only if~$W \\subseteq\nR_{D'}(x)$ for~$x \\in \\{u,v\\}$.\n\nGiven a planar straight-line drawing of a graph~$G$, we say that a\nchord~$e$ is \\emph{good} if its adjacent triangles or 4-cliques\nsatisfy condition~\\eqref{eq:1}.  This notion gives us a more local\ncriterion to decide whether a given planar straight-line drawing is an\noutside-obstacle representation.\n\n\\begin{lemma}\n  \\label{lem:nonlocal-crossing}\n  Let~$G$ be a biconnected inner-chordal plane graph and let~$\\Gamma$\n  be a planar straight-line drawing of~$G$.  Then~$\\Gamma$ is a\n  (plane) outside-obstacle representation if and only if each chord is\n  good.\n\\end{lemma}\n\n\n\\begin{proof}\n  The condition that each chord is good is necessary.  For sufficiency\n  we show that, in a drawing where each chord is good, every non-edge\n  intersects the outer face.\n\n  \\begin{figure}[tb]\n    \\centering\n    \\includegraphics{fig\/non-local-crossing}\n    \\caption{Illustration of the proof of\n      Lemma~\\ref{lem:nonlocal-crossing}.}\n    \\label{fig:non-local-crossing}\n  \\vspace{-2ex}\n  \\end{figure}\n\n  Suppose for the sake of contradiction that~$u$ and~$v$ are two\n  non-adjacent vertices of~$G$ such that the segment~$uv$ does\n  \\emph{not} intersect the outer face.  Then there is a minimal series\n  $D_1, \\dotsc, D_n$ of adjacent triangles of $G$ such that the\n  segment~$uv$ is completely contained in the union of these\n  triangles.  Clearly,~$n\\ge2$ and, without loss of generality,~$u \\in\n  V(D_1)$ and~$v \\in V(D_n)$.  We consider the subdrawing induced\n  by~$D_1,\\dots, D_n$.  Since~$uv$ intersects the triangle~$D_1$, it\n  intersects the edge of~$D_1$ opposite of~$u$, which implies that~$v$\n  is contained inside the cone~$C$ defined by~$D_1$ with base~$u$.  We\n  show that this contradicts statement $2$.  If triangle~$D_i$ for~$1\n  \\le i \\le n-1$ has the following properties: (i) its points are\n  outside (or on the boundary) of~$C$, (ii) the edge shared by~$D_i$\n  and~$D_{i+1}$ cuts across the cone~$C$, and (iii) the line defined\n  by the two points of~$D_i$ that lie on the same side of~$C$ slope\n  away from~$C$ in the direction towards~$v$, then the very same\n  properties hold for~$D_{i+1}$ due to the chords being good; see\n  Fig.~\\ref{fig:non-local-crossing}.  By definition the property holds\n  for~$D_1$, and hence it also holds for~$D_n$.  But this implies that\n  the tip of~$D_n$, which is~$v$, must be placed outside of~$C$,\n  contradicting the assumption.\n\\end{proof}\n\nUnfortunately, it is not always possible to place the vertices inside\nthe regions such that all chords become good as this placement may\nrequire crossings.\n\n\\boldparagraph{Chord Orientations and Outside-Obstacle Representations.}\nNext, we introduce a certain type of orientations of the chords of\nbiconnected inner-chordal graphs.  Let~$G$ be a biconnected\ninner-chordal graph and let~$\\Gamma$ be a plane outside-obstacle\nrepresentation of~$G$.  Let~$e=\\{u,v\\}$ be a chord of~$G$, which\nexists, unless~$G$ is~$K_3$ or~$K_4$.  Let~$D$ and~$D'$ denote the two\ntriangles incident to~$e$, and let~$w$ and~$w'$ denote the tips of~$D$\nand~$D'$, respectively.  Due to Lemma~\\ref{lem:nonlocal-crossing},\neach chord satisfies condition~\\eqref{eq:1}.  Hence, either~$w \\in\nR_{D'}(u)$ and~$w' \\in R_{D}(u)$ or~$w \\in R_{D'}(v)$ and~$w' \\in\nR_{D}(v)$.  We direct the chord~$e$ towards~$u$ in the former case and\ntowards~$v$ in the latter case.  In this way, we obtain an orientation\nof the chords of~$G$.  Note that outer edges and inner edges of\n4-cliques remain undirected.  The following lemma shows two crucial\nproperties of such an orientation.\n\n\\begin{lemma}\n  \\label{lem:chord-orientation}\n  Let~$G$ be a biconnected inner-chordal graph with plane\n  outside-obstacle representation~$\\Gamma$.  The chord orientation\n  determined by~$\\Gamma$ satisfies the following properties.\n\n  \\begin{compactenum}[(i)]\n  \\item Each vertex has in-degree at most~2.\n  \\item If vertex~$v$ has in-degree~2, then its two incoming edges share a face.\n  \\end{compactenum}\n\\end{lemma}\n\n\\begin{proof}\n  Consider an orientation according to~$\\Gamma$.  Let~$e=(u,v)$ be a\n  directed chord with incident triangles~$D$ and~$D'$, whose tips with\n  respect to the base~$e$ are~$w$ and~$w'$, respectively.  Due to the\n  direction of~$e$, we have that~$w \\in R_{D'}(v)$ and~$w' \\in\n  R_D(v)$.  It is readily seen, e.g., in Fig.~\\ref{fig:attachment},\n  that the two angles at~$v$ incident to~$e$ sum up to more\n  than~$\\pi$.\n\n  Let~$e_1,\\dots,e_k$ be chords that are directed towards~$v$.\n  Without loss of generality assume that these chords are numbered in\n  the order of counterclockwise occurrence around~$v$, starting from\n  the outer face.  Since the angles at~$v$ incident to each of these\n  edges sum up to more than~$\\pi$, it follows that some of these\n  angles must coincide.  Due to the ordering, it follows that the\n  angle at~$v$ right of~$e_i$ (with respect to the orientation\n  towards~$v$) coincides with the left angle of~$e_{i+1}$\n  for~$i=1,\\dots,k-1$.  By planarity and since~$v$ is an outer vertex,\n  no other angles may coincide.  For~$i=1,\\dots,k$, let~$\\alpha_i$\n  denote the angle left of~$e_i$ and let~$\\alpha_{k+1}$ denote the\n  angle right of~$e_k$.  By the above observation, we\n  have~$\\alpha_i+\\alpha_{i+1} > \\pi$ for~$i=1,\\dots,k$.\n\n  For~$k\\ge 3$, the sum of inner angles incident to~$v$ is at\n  least~$\\alpha_1 + \\alpha_2 + \\alpha_3+ \\alpha_4 > 2\\pi$; a\n  contradiction.  For~$k=2$ the shared angle~$\\alpha_2$ implies\n  property~(ii).\n\\end{proof}\n\nBy virtue of Lemma~\\ref{lem:chord-orientation}, we call any\norientation of the chords of a biconnected inner-chordal graph that\nsatisfies the properties~(i) and~(ii) an \\emph{outside-obstacle\n  orientation}.  \n\nLemma~\\ref{lem:chord-orientation} finally allows us to give a concise\nargument why the graph from Fig.~\\ref{fig:example-2} does not admit a\nplane outside-obstacle representation.  We argue that it does not\nadmit an outside-obstacle orientation.  It follows from the conditions\nof such an orientation that, for each 4-clique that is incident to\nthree chords, these chords must be oriented such that they form a\ncycle.  Consider the middle 4-clique in Fig.~\\ref{fig:example-2}.  If\nwe orient it clockwise, then the lower left edge may not have\nadditional incoming chords, which prevents us from orienting the\nchords of the left 4-clique as a cycle.  Symmetrically, choosing a\ncounterclockwise orientation for the middle 4-clique prevents correct\norientation of the right 4-clique.  The graph in\nFig.~\\ref{fig:example-3}, however, does admit an outside-obstacle\norientation, which is indicated in the figure.\n\nOur next goal is to prove that the existence of an outside-obstacle\norientation is equivalent to the existence of a plane outside-obstacle\nrepresentation.  In particular, this shows our claim that the graph in\nFig.~\\ref{fig:example-3} indeed admits a plane outside-obstacle\nrepresentation, e.g., the one shown in Fig.~\\ref{fig:example-oor-2}.\n\n\\begin{theorem}\n  \\label{thm:orientation-representation}\n  Let~$G$ be a biconnected inner-chordal plane graph.  Then~$G$ admits\n  a plane outside-obstacle representation if and only if it admits an\n  outside-obstacle orientation.\n\\end{theorem}\n\n\\begin{proof}\n  The ``only if''-part holds due to Lemma~\\ref{lem:chord-orientation}.\n  Let~$G$ be a biconnected inner-chordal graph with an\n  outside-obstacle orientation and let~$T(G)$ be its construction\n  tree.  We construct a plane outside-obstacle representation of~$G$.\n\n  For a subtree~$T' \\subseteq T$, we denote by~$G(T')$ the subgraph\n  of~$G$ corresponding to~$T'$.  Note that~$G(T) = G$.\n  Let~$\\tau_1,\\dots,\\tau_k$ denote the nodes of~$T$ in breadth-first\n  order starting at an arbitrary node~$\\tau_1$.  For~$j=1,\\dots,k$,\n  let~$T_j$ be the subtree of~$T$ consisting of\n  nodes~$\\tau_1,\\dots,\\tau_j$, and let~$G_j = G(T_j)$ the\n  corresponding subgraph of~$G$.  We inductively construct a sequence\n  of plane outside-obstacle representations~$\\Gamma_1,\\dots,\\Gamma_k$\n  of~$G_1,\\dots, G_k$.  Then~$\\Gamma_k$ is the desired plane\n  outside-obstacle representation of~$G=G_k$.\n\n  Consider the orientation of~$G_i$ induced by~$G$ (note that some\n  edges remain undirected).  We call a directed edge \\emph{active} if\n  it is incident to the outer face of~$G_i$ and \\emph{inactive}\n  otherwise.  An outer vertex~$v$ is \\emph{active} if it is the target\n  of an active edge.  It is \\emph{inactive} otherwise.  The\n  \\emph{inactive degree} of~$v$ in~$G_i$ is the number of inactive\n  edges with target~$v$.\n\n  Throughout steps $i=1,\\dots,k$, we maintain the following\n  properties:\n  \\begin{compactenum}[(i)]\n  \\item The outer angle of vertices with inactive degree~0 is convex.\n  \\item For an active vertex~$v$ with inactive degree~1, removing the\n    unique active in-edge incident to~$v$ results in a convex outer\n    angle.\n  \\end{compactenum}\n  For~$G_1$ any plane outside-obstacle representation~$\\Gamma_1$ satisfies\n  these properties.  We now show how to proceed from~$G_i$\n  to~$G_{i+1}$.  Let~$e=(u,v)$ be the directed chord determined by\n  adding~$\\tau_{i+1}$ to~$T(G_i)$, let~$D$ be the inner triangle\n  of~$G_i$ bounded by~$e$ and let~$e'=(u',v)$ denote the other edge\n  of~$D$ incident to~$v$.\n\n  We aim to place the vertices in~$V(G_{i+1}) \\setminus V(G_i)$ inside\n  the region~$R_D(v)$, which is consistent with the orientation\n  of~$e$.  We first show that this is possible without creating\n  intersections.  If~$v$ has inactive degree~0, then~$v$ is convex,\n  and hence the intersection of~$R_D(v)$ with a suitably\n  small~$\\varepsilon$-ball around~$v$ is disjoint from any vertices\n  and edges of~$\\Gamma_i$.  Similarly, if~$v$ is active but has\n  inactive degree~1, then, after removing~$(u,v)$,~$v$ is convex by\n  property~(ii).  In this case the subcone of~$R_D(v)$ defined by~$e'$\n  and the other outer edge incident to~$v$ intersected with a suitably\n  small~$\\varepsilon$-ball is empty; see Fig.~\\ref{fig:construction-1}.\n\n  \\begin{figure}[tb]\n    \\centering\n    \\begin{subfigure}[b]{.4\\textwidth}\n      \\centering\n      \\includegraphics[page=1]{fig\/construction}\n      \\caption{}\\label{fig:construction-1}\n   \\end{subfigure}\\hfil\n   \\begin{subfigure}[b]{.3\\textwidth}\n      \\centering\n      \\includegraphics[page=3]{fig\/construction}\n      \\caption{}\\label{fig:construction-2}\n   \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n      \\centering\n      \\includegraphics[page=2]{fig\/construction}\n      \\caption{}\\label{fig:construction-3}\n   \\end{subfigure}\\hfil\n   \\caption{Construction of a plane outside-obstacle representation. (a)\n     Attaching at a vertex~$v$ with inactive degree~1. The region\n     where the new points are placed is shaded in dark gray. (b), (c)\n     show that adding an edge preserves the properties required by\n     the construction.  The shaded region is the inner part of the\n     drawing after removing the active edge; the outer angle at~$v$ is\n     convex.}\n    \\label{fig:construction}\n    \\vspace{-2ex}\n  \\end{figure}\n\n  We show that, by placing the new vertices in these regions suitably\n  close to~$v$, properties~(i) and~(ii) can be established for the\n  resulting plane outside-obstacle representation~$\\Gamma_{i+1}$.  First\n  observe that placing the new vertices close enough to~$v$ avoids\n  touching or crossing any vertices and edges of~$G_i$, i.e.,\n  $\\Gamma_{i+1}$ is plane.  Moreover, the addition only changes angles\n  at~$u$ and~$v$, and hence all other vertices satisfy properties~(i)\n  and~(ii) by virtue of the induction hypothesis.\n\n  Consider vertex~$v$.  In~$G_{i+1}$ it has inactive degree at least~1\n  since~$(u,v)$ is an inner edge.  If the inactive degree of~$v$ is~2,\n  there is nothing to prove as~$v$ must be inactive, since\n  outside-obstacle orientations have in-degree at most~2.  Hence\n  assume that~$v$ has inactive degree~1 and it is active.  The\n  properties of outside-obstacle orientations imply that there is a\n  unique active edge directed towards~$v$ in~$G_{i+1}$ and it must be\n  a neighbor of~$e$.  This edge is either the edge~$e'$ or the newly\n  added outer edge~$e''$ incident to~$v$.\n\n  If~$e'$ is the incoming active edge at~$v$, the outer angle\n  at~$v$ was convex in~$\\Gamma_i$, and hence any point in~$R_D(v)$\n  results in an outer angle of less than~$\\pi$ after removing~$e'$;\n  see Fig.~\\ref{fig:construction-2}.\n  If~$e''$ is the incoming active edge at~$v$, the outer angle\n  at~$v$ in~$\\Gamma_{i+1}$ after removing~$e''$ is the outer angle\n  of~$v$ in~$\\Gamma_i$, which is convex by the induction hypothesis;\n  see Fig.~\\ref{fig:construction-3}.\n\n  In all cases vertex~$v$ satisfies properties~(i) and~(ii).  We show\n  that, by positioning the new vertices close enough to~$v$, we can\n  also satisfy properties~(i) and~(ii) for $u$.  First note that the\n  inactive degree of~$u$ does not change.  If the inactive degree\n  of~$u$ is~2, there is nothing to prove.  If the inactive degree\n  of~$u$ is~0 or~1, by placing the new vertices close to the line\n  through~$u$ and~$v$, the angle between~$e$ and the new outer edge\n  incident to~$u$ can be made arbitrarily small.  Thus, if~$u$ was\n  convex in~$\\Gamma_i$, it remains so in~$\\Gamma_{i+1}$.  And, by the\n  same argument, if~$u$ was convex in~$\\Gamma_i$ after removing the\n  active edge incident to~$u$, it remains so in~$\\Gamma_{i+1}$.\n  Hence~$\\Gamma_{i+1}$ satisfies the induction hypothesis.\n\\end{proof}\n\n\n\n\\section{Characterization and Decision Algorithm}\n\\label{sec:main}\n\nIn this section, we prove characterizations of graphs that admit a\nplane outside-obstacle representation and we present a linear-time\nalgorithm that decides whether a given graph admits a plane\noutside-obstacle representation.\n\n\\boldparagraph{Characterization of Outerplanar Graphs.}\nFor biconnected outerplanar graphs\nTheorem~\\ref{thm:orientation-representation} immediately implies a\ncomplete characterization of the graphs that admit a plane\noutside-obstacle representation.\n\n\\begin{theorem}\n  \\label{thm:bico-outer}\n  A biconnected outerplanar graph admits a plane outside-obstacle\n  representation if and only if it is chordal.\n\\end{theorem}\n\n\\begin{proof}\n  Being chordal is a necessary condition due to\n  Lemma~\\ref{lem:plane-oor-chordal}.  Conversely, if an outerplanar\n  graph is chordal, it is obviously inner-chordal.  We show that every\n  biconnected inner-chordal outerplane graph admits an outside\n  obstacle orientation.\n\n  Recall that a biconnected outerplanar graphs contains a vertex of\n  degree at most~2.  We iteratively construct an orientation by\n  directing the incident edges of a vertex with degree at most~2\n  towards it and removing it from the graph.  In this way, we obtain\n  an orientation with the properties that each vertex has in-degree at\n  most~2, and moreover, if a vertex has in-degree~2, then the two\n  incoming edges share an inner face.  Undoing the orientations of the\n  outer edges, we obtain an outside-obstacle orientation.  Now the\n  claim follows from Theorem~\\ref{thm:orientation-representation}.\n\\end{proof}\n\nThis result can easily be strengthened in two ways.  First, if the\nouterplanar graph is not biconnected but chordal, then it can easily\nbe augmented such that it becomes biconnected but remains\n(inner-)chordal and outerplanar and hence satisfies the conditions of\nTheorem~\\ref{thm:bico-outer}, yielding a plane outside-obstacle\nrepresentation of the augmented graph.  By iteratively removing\naugmentation edges that are incident to the outer face we obtain a plane\noutside-obstacle representation of the original graph.\n\n\\begin{corollary}\n  \\label{cor:outer}\n  An outerplanar graph admits a plane outside-obstacle representation\n  if and only if it is chordal.\n\\end{corollary}\n\nAnother observation is that the construction of the orientation in the\nproof of Theorem~\\ref{thm:bico-outer} essentially consists of a\nbottom-up traversal of the construction tree of the graph with respect\nto the root node, which is removed last.  It is then readily seen that\nwe can also remove~$K_4$-nodes that are leaves, provided they have\ndegree at most~2 in~$T(G)$.  A $K_4$ with degree~3 requires that its\nchords are oriented to form a cycle, which cannot be ensured by the\nconstruction.  It can, however, always be achieved it the $K_4$ of\ndegree~3 is the root of the tree.  We thus have the following\ncorollary.\n\n\\begin{corollary}\n  \\label{cor:bico-onek4deg3}\n  Every biconnected inner-chordal graph that contains at most one\n  $K_4$ for which all outer edges are chords admits a plane\n  outside-obstacle representation.\n\\end{corollary}\n\nNote that an augmentation as in the proof of\nCorollary~\\ref{cor:bico-onek4deg3} may increase the number of\n$K_4$-nodes with degree~3.  Hence the result does not extend to\nnon-biconnected graphs.\n\n\n\\boldparagraph{Decision Algorithm for General Graphs.}\nNext, we devise a linear-time algorithm to decide whether a\nbiconnected graph admits a plane outside-obstacle representation.  Of\ncourse it is not difficult to test whether a graph is inner-chordal\nand plane in linear time, and we assume in the following that our\ninput graph has these properties.\n\nDue to Theorem~\\ref{thm:orientation-representation}, deciding the\nexistence of a plane outside-obstacle representation is equivalent to\ndeciding the existence of an outside-obstacle orientation.  To test\nwhether a biconnected inner-chordal plane graph~$G$ admits an outside\nobstacle orientation, we use dynamic programming on its construction\ntree~$T(G)$, rooted at an arbitrary node.  \n\nFor each node~$\\tau$ with parent edge~$\\{u,v\\}$ with orientation~$uv$\nand binary flags~$d_{\\tau,u}$ and~$d_{\\tau,v}$, we are\n  interested whether the subtree of~$T(G)$ with root~$\\tau$ admits an\n  outside-obstacle orientation such that\n\\begin{compactenum}\n\\item $\\{u,v\\}$ is oriented as $uv$,\n\\item $u$ has incoming edges if and only if~$d_{\\tau,u} = 1$, and\n\\item $v$ has incoming edges distinct from~$uv$ if and only\n  if~$d_{\\tau,v} = 1$.\n\\end{compactenum}\n\nWe store this information in a 4-dimensional\ntable~$T[\\tau,e,d_{\\tau,u},d_{\\tau,v}]$ of boolean variables.\nNote that, for each node~$\\tau$, table~$T$ contains only~$2^3 = O(1)$\nentries.  We now show how to fill the entries of this table in linear\ntime.  Initially, we set all entries to \\emph{false}.\n\nFor a leaf node~$\\tau$ with parent edge~$\\{u,v\\}$, we\nset~$T[\\tau,uv,0,0] = T[\\tau,vu,0,0] =$ \\emph{true}, which models the\nfact that we can choose any orientation of~$\\{u,v\\}$ and neither~$u$\nnor~$v$ has incoming edges distinct from~$\\{u,v\\}$ in the subtree\nconsisting only of the leaf.  Let~$\\tau$ be a node with\nchildren~$\\tau'$ and~$\\tau''$ and corresponding chords~$\\{u,w\\},\n\\{v,w\\}$ that connect them to~$\\tau$.  We can easily check whether the\nentries can be combined to an entry of~$\\tau$.  Namely, try both\npossible orientations of~$\\{u,v\\}$ and use the orientations\nof~$\\{u,w\\}$ and~$\\{v,w\\}$ determined by the entries of the children\nand the flags~$d_{\\tau',u}$, $d_{\\tau',w}$, $d_{\\tau'',v}$,\nand~$d_{\\tau'',w}$ of the children to check that~$u,v$ and~$w$ satisfy\nthe constraints of the orientation.  If this is the case, we can\neasily compute the two flags~$d_{\\tau,u}$ and~$d_{\\tau,v}$ from the\norientations of~$uw$, $vw$ and the flags~$d_{\\tau',u}$\nand~$d_{\\tau'',v}$.  A simple induction shows that, in this way, we\nset exactly the correct entries~$T[\\tau,\\cdot,\\cdot,\\cdot]$ to\n\\emph{true}.\n\nCombining two entries takes~$O(1)$ time.  Since there are only~$2^3 =\nO(1)$ entries per node, we can compute all entries of a node~$\\tau$\nfrom all combinations of entries of its at most two children in~$O(1)$\ntime.  Since there are~$O(n)$ nodes, the overall algorithm runs\nin~$O(n)$ time.  At the root we may have to combine up to three\nchildren, but the checks remain essentially the same.  Thus, the\noverall algorithm runs in~$O(n)$ time.\n\n\\begin{theorem}\n  There is a linear-time algorithm that decides whether a given\n  biconnected graph admits a plane outside-obstacle representation.\n\\end{theorem}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nInspired by obstacle representations introduced by Alpert et\nal.~\\cite{akl-ong-10}, we studied plane outside-obstacle\nrepresentations of graphs.  We characterized the biconnected graphs\nthat admit such a representation as the inner-chordal graphs that\nadmit a certain type of orientation of their chords.  Based on this,\nwe gave a combinatorial proof that every chordal outerplanar graph\nadmits a plane outside-obstacle representation.  We further derived a\nlinear-time algorithm for deciding whether a given biconnected graph\nadmits a plane outside-obstacle representation.\n\nOur main open question are the following.  Can our characterization\nand testing algorithm can be extended to general (inner-chordal)\ngraphs that are not necessarily biconnected?  Which graphs admit a\nplane representation with a single obstacle?\n\n\\smallskip\n\n\\noindent\\textbf{Acknowledgments}\nPart of this work has been done while Alexander Koch participated in\nthe academic exchange program of T\\=ohoku University and KIT.  AK\nthanks Prof.~Dorothea Wagner and Prof.~Takeshi Tokuyama for their\nsupport and Prof.~Yota \\=Otachi from JAIST for helpful comments on the topic.\n\n\\bibliographystyle{abbrv}\n\n\\section{Introduction}\n\nVisibility, and hence, visibility representations of graphs are\ncentral to many areas, such as architecture, sensor networks, robot\nmotion planning, and surveillance and security.  There is a long\nhistory of research on characterizing and recognizing visibility\ngraphs in various settings; see the related work below.  Despite\ntremendous efforts characterizations and efficient recognition\nalgorithms are only known for very restricted\ncases~\\cite{ec-rvgsp-90,ec-nrcvg-95}.  Recently, Alpert et\nal.~\\cite{akl-ong-10} introduced obstacle representations of graphs,\nwhich generalize many previous visibility variants, such as\npolygon--vertex visibility.  In this paper, we study \\emph{plane\n  outside-obstacle representations}, where the visibility segments may\nnot cross, and a single obstacle is located in the outer face of the\nrepresentation.  We characterize the biconnected graphs admitting such\na representation and give a linear-time recognition algorithm.  This\nis one of the first results that characterizes such a class of graphs\nand gives an efficient recognition algorithm.  In the following we\nfirst give some basic definitions.  Afterwards, we present an overview\nof related work and describe our contribution in more detail.\n\nAn \\emph{obstacle representation} of a graph~$G=(V,E)$ consists of a\nset of polygonal obstacles and a distinct point for each vertex\nin~$V$.  The representation is such that two points see each other\nif and only if the corresponding vertices are adjacent.  The\n\\emph{obstacle number} of~$G$ is the smallest number of obstacles in\nany obstacle representation of~$G$.\n\nIn an \\emph{outside-obstacle representation} all obstacles are in the\nunbounded face of the representation, i.e., they are contained in the\nunbounded face of the corresponding straight-line drawing of the\ngraph.  Outside-obstacle representations are a recent generalization\nof classical \\emph{polygon--vertex visibility} graphs, where the\nobstacle is a simple polygon, the points are the vertices of the\npolygon and visibility segments have to lie inside the polygon.  The\ncorresponding characterization problem and the complexity of the recognition\nproblem are long-standing open questions.\n\nFigure~\\ref{fig:examples-oor} shows examples of outside-obstacle\nrepresentations.  We note that, in the case of outside-obstacle\nrepresentations, it can be assumed that there exists only a single\nobstacle that surrounds the outer face of the representation, as\nillustrated in Fig.~\\ref{fig:example-oor-1}.  In particular, graphs\nwith an outside-obstacle representation have obstacle number at\nmost~1.  In our setting, given a straight-line drawing of a graph, we\nhence do not make the outside obstacle explicit.  Instead, we require\nfor an outside-obstacle representation of a graph~$G$ that any\nnon-edge of~$G$ intersects the outer face of~$G$.  It is then not\ndifficult to construct a corresponding obstacle.  Throughout this\ndocument, we assume that points and vertices of obstacles are in\n\\emph{general position}, so no three of them are collinear.\n\n\n\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.45\\textwidth}\n    \\centering\n    \\includegraphics[page=5]{fig\/examples-oor}\n    \\caption{}\\label{fig:example-oor-1}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.45\\textwidth}\n    \\centering\n    \\includegraphics[page=1]{fig\/examples-oor}\n    \\caption{}\\label{fig:example-oor-2}\n  \\end{subfigure}\n  \\caption{(a) A non-planar outside-obstacle representation of the\n    octahedron; the obstacle is shown in gray.  (b) A plane outside-obstacle representation of\n    Figure~\\ref{fig:example-3}.}\n  \\vspace{-3ex}\n  \\label{fig:examples-oor}\n\\end{figure}\n\n\\myparagraph{Related Work.}\nGraphs with an outside-obstacle representation are equivalent to\nvisibility graphs of a pointset within a simple polygon. Therefore,\npolygon--vertex visibility graphs form an important subclass of our\ngraphs class, where the pointset coincides with the corners of the\nsurrounding simple polygon.  Such graphs have been extensively studied\ndue to their many applications, e.g., in gallery\nguarding~\\cite{r-agta-87}.\n\nPolygon--vertex visibility graphs were first introduced in 1983 by\nAvis and ElGindy~\\cite{ae-caps-83} and are most studied in the field\nof visibility problems~\\cite{gg-upvgpsp-12}.  One of the first results\non the topic was that maximal outerplanar graphs are polygon--vertex\nvisibility graphs~\\cite{e-hdpa-85}.  Ghosh~\\cite{g-rcvgs-97} gives a\nset of four necessary conditions for polygon--vertex visibility\ngraphs, which he conjectured to be also sufficient. However,\nStreinu~\\cite{s-npvg-05} constructed a counterexample.  As pointed out\nin~\\cite{gg-upvgpsp-12}, two of Ghosh's necessary\nconditions~\\cite{g-rcvgs-97} imply the conditions of a\ncharacterization attempt by Abello and Kumar in terms of oriented\nmatroids~\\cite{ak-vgom-02}, which hence cannot be sufficient.\n\nSo far, characterizations have only been achieved for polygon--vertex\nvisibility graphs of restricted polygons.  Everett and\nCorneil~\\cite{ec-rvgsp-90,ec-nrcvg-95} give a characterization of\nvisibility graphs in spiral and 2-spiral polygons -- polygons that\nhave exactly one and two chains of reflex vertices, respectively.\nThey are characterized as interval graphs and perfect graphs.  The\nbest known complexity result about the recognition and reconstruction\nproblem of polygon--vertex visibility graphs is that they are in\nPSPACE~\\cite{e-vgr-90}.\n\nA generalization of these graphs are induced subgraphs of\npolygon--vertex visibility graphs, or \\emph{induced visibility graphs}\nfor short.  Spinrad~\\cite{s-egr-03} considers this graph class the\nnatural generalization of polygon--vertex visibility graphs, which is\nhereditary with respect to induced subgraphs.\nEverett and Corneil~\\cite{ec-nrcvg-95} show that there is no finite\nset of forbidden induced subgraphs in polygon--vertex visibility\ngraphs.\n\n\n\nCoullard and Lubiw show that 3-connected polygon--vertex visibility\ngraphs admit a 3-clique ordering~\\cite{cl-dvg-91}.\nAbello et al.~\\cite{alp-vgsp-92} prove that every 3-connected planar\npolygon--vertex visibility graph is maximal planar and that every\n4-connected such graph cannot be planar.  Their conjecture that\nHamiltonian maximal planar graphs with a 3-clique ordering are\npolygon--vertex visibility graphs was disproven by Chen and\nWu~\\cite{cw-dcpvg-01}.  According to O'Rourke~\\cite{o-opcvi-98}\nnecessary and sufficient conditions for a polygon--vertex visibility\ngraph to be planar are known~\\cite{lc-pvg-94}, but do not lead to a\npolynomial recognition algorithm.\n\n\nFor the more general question of obstacle numbers, Alpert et\nal.~\\cite{akl-ong-10} give a construction for graphs with large\nobstacle number and small example graphs that have obstacle number\ngreater than~1.  They further show that every outerplanar graph admits\na (non-planar) outside-obstacle representation, i.e., they are\nvisibility graphs of pointsets inside simple polygons.  Subsequent\npapers extend the results on obstacle numbers.  Pach and\nSar\u0131\u00f6z~\\cite{ps-sglon-11} construct small graphs with obstacle\nnumber~2 and show that bipartite graphs with arbitrarily large\nobstacle number exist.  Mukkamala et al.~\\cite{mpp-lbong-12} show that\nthere are graphs on $n$ vertices with obstacle number\n$\\Omega(n\/\\log{n})$. It is an open question whether any graph with\nobstacle number~1 admits an outside-obstacle representation.\n\n\\myparagraph{Contribution and Outline.}\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.1\\textwidth}\n    \\centering\n    \\includegraphics[page=1]{fig\/examples}\n    \\caption{}\\label{fig:example-1}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n    \\centering\n    \\includegraphics[page=2]{fig\/examples}\n    \\caption{}\\label{fig:example-2}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n    \\centering\n    \\includegraphics[page=3]{fig\/examples}\n    \\caption{}\\label{fig:example-3}\n  \\end{subfigure}\n  \\caption{Graphs admitting (c) \/ not admitting (a,b) a\n    plane outside-obstacle representation.}  \\vspace{-3ex}\n  \\label{fig:examples}\n\\end{figure}\nIn this paper, we study \\emph{plane outside-obstacle representations},\nwhere the drawing of~$G$, without the obstacles, is free of crossings;\nsee Fig.~\\ref{fig:example-oor-2} for an example.  Consider the graphs\nshown in Fig.~\\ref{fig:examples}.  We will see that the first two\ngraphs do not admit a plane outside-obstacle representation, whereas\nthe last example has one.  Note that the drawing in\nFig.~\\ref{fig:example-1} is a (non-planar) outside-obstacle\nrepresentation.  Our main results are the following.\n\\begin{inparaenum}[(1)]\n\\item Every outerplanar graph whose inner faces are triangles admits a\n  plane outside-obstacle representation.\n\\item A characterization of the biconnected graphs that admit a plane\n  outside-obstacle representation.\n\\item A linear-time algorithm for testing whether a biconnected graph\n  admits a plane outside-obstacle representation.\n\\end{inparaenum}\nAs a side result, we obtain a simple combinatorial proof of ElGindy's\nclassical result that maximal outerplanar graphs are polygon--vertex\nvisibility graphs~\\cite{e-hdpa-85}.\n\nOur paper is structured as follows.  First, we derive a simple\nnecessary condition on the structure of biconnected graphs that admit\na plane outside-obstacle representation in\nSection~\\ref{sec:inner-chordal}.  This restricts the class of graphs\nwe have to consider and we derive some useful structural results about\nsuch graphs.  Afterwards, in Section~\\ref{sec:characterization}, we\ngive a local description of plane outside-obstacle representations\nand, based on this, we derive a combinatorial characterization of the\nbiconnected planar graphs that admit a plane outside-obstacle\nrepresentation in terms of an orientation of a certain subset of\nedges.  Using this characterization, we prove our main results in\nSection~\\ref{sec:main}.\n\n\n\n\n\\section{Inner-Chordal Plane Graphs}\n\\label{sec:inner-chordal}\n\nA graph with a fixed planar embedding is \\emph{inner-chordal plane} if\nany cycle~$C$ of length at least~4 has a chord that is embedded in the\nbounded region of~$\\mathbb{R}^2 \\setminus C$; see\nFig.~\\ref{fig:inner-chordal}.  We first show that we can restrict our\nanalysis to inner-chordal plane graphs.\n\n\\begin{lemma}\\label{lem:plane-oor-chordal}\n  Graphs with a plane outside-obstacle representation are\n  inner-chordal plane.\n\\end{lemma}\n\\begin{proof}\n  Let~$G$ be a graph with a plane outside-obstacle representation, and\n  assume it is not inner-chordal.  Hence, there exists a cycle~$C$ of\n  length at least~4, whose interior does not contain a chord.  Note\n  that the obstacle lies outside of~$C$ by definition.  The cycle~$C$\n  is embedded as the boundary of a simple polygon~$P$ on at least four\n  vertices.  Since~$P$ can be triangulated, there exists a pair of\n  non-adjacent vertices~$u$ and~$v$ on~$C$ such that the segment~$uv$\n  is completely contained in~$P$.  Hence the obstacle cannot\n  intersect~$uv$, and thus~$\\{u,v\\} \\in E(G)$ by definition,\n  contradicting our choice of~$u$ and~$v$.\n\\end{proof}\n\n\\begin{figure}[tb]\n  \\centering\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\includegraphics[page=1]{inner-chordal}\n      \\caption{}   \n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\centering\n    \\includegraphics[page=3]{inner-chordal}\n    \\caption{}\n  \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.2\\textwidth}\n    \\centering\n    \\includegraphics[page=2]{inner-chordal}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{(a) An inner-chordal graph. (b), (c) Chordal, but not inner-chordal, plane graphs.}\n  \\vspace{-2ex}\n  \\label{fig:inner-chordal}\n\\end{figure}\n\nNote that Lemma~\\ref{lem:plane-oor-chordal} shows immediately that the\ngraph from Fig.~\\ref{fig:example-1} does not admit a plane outside-obstacle\nrepresentation.  Although this graph is chordal, it does not have a\nplanar embedding that is inner-chordal.\nIn the following, we consider only inner-chordal plane graphs.  Note\nthat, in particular, every inner face of an inner-chordal plane graph\nis a triangle.  Moreover, an outerplanar graph is inner-chordal if and\nonly if it is chordal, which is the case if and only if every inner\nface is a triangle.\n\n\\begin{lemma}\n  \\label{lem:inner-chordal-degree}\n  Let~$G$ be an inner-chordal plane graph.  Then, every inner vertex\n  of~$G$ has degree~3 and no two inner vertices are adjacent.\n\\end{lemma}\n\n\\begin{proof}\n  Since~$G$ is inner-chordal, every inner face is necessarily a\n  triangle.  This implies that the neighbors of any inner vertex form\n  a cycle.  This cycle does not have an inner chord, and hence,\n  since~$G$ is simple and inner-chordal, it must have length~3.  This\n  implies that any inner vertex has degree~3.  Moreover, it follows\n  that the neighbors of any inner vertex~$v$ must be incident to the\n  outer face as they already have degree~3 in the neighborhood of~$v$.\n\\end{proof}\n\n\n\nThis description also gives rise to a certain tree that is associated\nwith every biconnected inner-chordal graph.  Let~$G$ be a biconnected\ninner-chordal plane graph.  A \\emph{chord} of~$G$ is an edge that is\nnot incident to the outer face but whose endpoints are incident to the\nouter face.  Lemma~\\ref{lem:inner-chordal-degree} implies a\ndecomposition of~$G$ along its chords.  Namely, we first remove all\ninner vertices.  Each such removal transforms a 4-clique of~$G$ into a\ntriangular face; we mark each triangle that results from such a\nremoval.  The resulting graph~$G'$ is outerplanar and every inner face\nis a triangle.  Now the weak dual~$T'$ of~$G'$ is a tree, where each\nnode corresponds to a triangle of~$G'$.  By marking the nodes of~$T'$\nthat correspond to a marked triangle, we obtain the construction\ntree~$T$ of~$G$, denoted~$T(G)$.  Note that each marked node of~$T(G)$\ncorresponds to a 4-clique of~$G$, whereas an unmarked node corresponds\nto a triangular face of~$G$.  We refer to these nodes as~$K_4$-\nand~$K_3$-nodes, respectively.  The edges of~$T(G)$ correspond\nbijectively to the chords of~$G$.  We refer to the vertices of~$T(G)$\nas nodes to distinguish them from the vertices of~$G$.  For a\nnode~$\\tau$ of~$T(G)$, we denote by~$V_\\tau$ the vertices of the\ncorresponding triangle or 4-clique.  \n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics{fig\/construction-trees}\n  \\caption{The construction tree of graphs~(b) and~(c) in\n    Fig.~\\ref{fig:examples}; $K_3$-nodes are empty and $K_4$-nodes are\n    filled.  Structurally, the only difference between the two graphs\n    is how the two subgraphs of~$G$ are attached to the triangle\n    corresponding to the $K_3$-node~$\\tau$.}\n  \\vspace{-3ex}\n\\end{figure}\n\nObserve that, if we store with each node of~$T(G)$ the corresponding\nedges and use the bijection of the edges of~$T(G)$ with the chords\nof~$G$ to find the shared chord of adjacent nodes, we can reobtain~$G$\nfrom~$T(G)$ by merging triangles and 4-cliques that are adjacent\nin~$T(G)$ along their shared chords.  Then~$T(G)$ is essentially the\nSPQR-tree of~$G$~\\cite{dt-omtc-96}.  We decided to avoid the\ntechnical machinery associated with SPQR-trees and rather work with\nthe construction tree, which is more tailored to our needs.\n\n\\section{Characterization of Plane Visibility Representations}\n\\label{sec:characterization}\n\nIn this section we devise a combinatorial characterization of the\nbiconnected inner-chordal graphs that admit a plane outside-obstacle\nrepresentation.  This is done in two steps.  First, we show that,\naside from being free of crossings, the property of being a plane\noutside-obstacle representation depends only on local features in the\ndrawing, namely, for each chord of a graph~$G$, its neighbors must be\nembedded in certain regions.  In a second step, we show that this\nessentially induces a binary choice for each chord.  In this way, an\noutside-obstacle orientation induces an orientation of the chords\nof~$G$, and we will characterize the existence of a plane\noutside-obstacle representation in terms of existence of a suitable\nchord orientation.\n\n\\boldparagraph{Local Description of Plane Visibility Representations.}\nNext we aim to understand better which planar straight-line drawings\nare outside-obstacle representations.  As a first observation,\nconsider two triangles~$D$ and~$D'$ sharing a common edge~$e=\\{u,v\\}$,\nwhich then forms a chord.  Let~$w$ and~$w'$ denote the tips of~$D$\nand~$D'$ with respect to base~$e$, respectively.  For an\noutside-obstacle representation it is a necessary condition that the\nnon-edge~$\\{w,w'\\}$ intersects the outer face.  We thus have to\nposition the tips in such a way that the segments~$ww'$ does not lie\ninside the drawing of~$D$ and~$D'$.  We use the following definition;\nsee Fig.~\\ref{fig:region} for an illustration.\n\n\\begin{definition}\\label{dfn:regions1}\n  Let $D$ be a triangle and $u$ a vertex of $D$.  Then $R_D(u)$\n  denotes the intersection of the half-planes defined by the sides\n  of~$D$ incident to~$u$ not containing~$D$.\n\\end{definition}\n\n\\begin{figure}[tb]\n  \\centering  \n  \\begin{subfigure}[b]{.5\\textwidth}\n    \\centering\n    \\includegraphics{regions3}\n    \\caption{}\\label{fig:region}\n  \\end{subfigure}\\hfil\n\\begin{subfigure}[b]{.45\\textwidth}\n  \\centering\n    \\includegraphics[page=1]{fig\/regions4}\n    \\caption{}\\label{fig:attachment}\n  \\end{subfigure}\n  \\caption{(a) Regions of triangle~$D$ at edge~$e=\\{u,v\\}$ and (b)\n    attachment of two triangles~$D$ and~$D'$ along chord~$\\{u,v\\}$.}\n  \\label{fig:proof-regions2}\n  \\vspace{-2ex}\n\\end{figure}\n\nTo ensure that the segment~$ww'$ intersects the outer face, it is\nclearly necessary that~$w' \\in R_D(u) \\cup R_D(v)$ or~$w \\in R_{D'}(u)\n\\cup R_{D'}(v)$.  The former ensures that the segment~$ww'$ does not\nintersect the interior of~$D$ and the letter ensures the same property\nfor~$D'$.  These intersections behaviors are not independent.  It is\nin fact readily seen that~$w' \\in R_D(x)$ if and only if~$w \\in\nR_{D'}(x)$ for~$x \\in \\{u,v\\}$; see Fig.~\\ref{fig:attachment}.  More generally, the\nsame observations also hold for a chord~$e= \\{u,v\\}$ that is shared by\n\\begin{inparaenum}[(a)]\n  (a) two triangles~$\\tau$ and~$\\tau'$, (b) a triangle~$\\tau$ and a\n  4-clique~$\\tau'$ and (c) by two 4-cliques~$\\tau$ and~$\\tau'$.\n\\end{inparaenum}\nTo see this, note that the regions~$R_{D}(u)$ and~$R_{D'}(u)$ in\nFig.~\\ref{fig:attachment} do not change if~$D$ and\/or $D'$ are part of\na 4-clique.  More formally, let~$W$ and~$W'$ be the vertices of~$\\tau$\nand~$\\tau'$ that are distinct from~$u$ and~$v$, respectively.  Let~$D$\nand~$D'$ denote the triangles incident to~$e$.  Then the following\ncondition is necessary\n\\vspace{-1ex}\n\\begin{align}\n  \\label{eq:1}\n  W' \\subseteq R_D(u) \\text{\\quad or \\quad }  W' \\subseteq R_D(v)\\,\\,.\\tag{*}\n\\end{align}\n\\vspace{-4ex}\n\n\\noindent Again it holds that $W' \\subseteq R_D(x)$ if and only if~$W \\subseteq\nR_{D'}(x)$ for~$x \\in \\{u,v\\}$.\n\nGiven a planar straight-line drawing of a graph~$G$, we say that a\nchord~$e$ is \\emph{good} if its adjacent triangles or 4-cliques\nsatisfy condition~\\eqref{eq:1}.  This notion gives us a more local\ncriterion to decide whether a given planar straight-line drawing is an\noutside-obstacle representation.\n\n\\begin{lemma}\n  \\label{lem:nonlocal-crossing}\n  Let~$G$ be a biconnected inner-chordal plane graph and let~$\\Gamma$\n  be a planar straight-line drawing of~$G$.  Then~$\\Gamma$ is a\n  (plane) outside-obstacle representation if and only if each chord is\n  good.\n\\end{lemma}\n\n\n\\begin{proof}\n  The condition that each chord is good is necessary.  For sufficiency\n  we show that, in a drawing where each chord is good, every non-edge\n  intersects the outer face.\n\n  \\begin{figure}[tb]\n    \\centering\n    \\includegraphics{fig\/non-local-crossing}\n    \\caption{Illustration of the proof of\n      Lemma~\\ref{lem:nonlocal-crossing}.}\n    \\label{fig:non-local-crossing}\n  \\vspace{-2ex}\n  \\end{figure}\n\n  Suppose for the sake of contradiction that~$u$ and~$v$ are two\n  non-adjacent vertices of~$G$ such that the segment~$uv$ does\n  \\emph{not} intersect the outer face.  Then there is a minimal series\n  $D_1, \\dotsc, D_n$ of adjacent triangles of $G$ such that the\n  segment~$uv$ is completely contained in the union of these\n  triangles.  Clearly,~$n\\ge2$ and, without loss of generality,~$u \\in\n  V(D_1)$ and~$v \\in V(D_n)$.  We consider the subdrawing induced\n  by~$D_1,\\dots, D_n$.  Since~$uv$ intersects the triangle~$D_1$, it\n  intersects the edge of~$D_1$ opposite of~$u$, which implies that~$v$\n  is contained inside the cone~$C$ defined by~$D_1$ with base~$u$.  We\n  show that this contradicts statement $2$.  If triangle~$D_i$ for~$1\n  \\le i \\le n-1$ has the following properties: (i) its points are\n  outside (or on the boundary) of~$C$, (ii) the edge shared by~$D_i$\n  and~$D_{i+1}$ cuts across the cone~$C$, and (iii) the line defined\n  by the two points of~$D_i$ that lie on the same side of~$C$ slope\n  away from~$C$ in the direction towards~$v$, then the very same\n  properties hold for~$D_{i+1}$ due to the chords being good; see\n  Fig.~\\ref{fig:non-local-crossing}.  By definition the property holds\n  for~$D_1$, and hence it also holds for~$D_n$.  But this implies that\n  the tip of~$D_n$, which is~$v$, must be placed outside of~$C$,\n  contradicting the assumption.\n\\end{proof}\n\nUnfortunately, it is not always possible to place the vertices inside\nthe regions such that all chords become good as this placement may\nrequire crossings.\n\n\\boldparagraph{Chord Orientations and Outside-Obstacle Representations.}\nNext, we introduce a certain type of orientations of the chords of\nbiconnected inner-chordal graphs.  Let~$G$ be a biconnected\ninner-chordal graph and let~$\\Gamma$ be a plane outside-obstacle\nrepresentation of~$G$.  Let~$e=\\{u,v\\}$ be a chord of~$G$, which\nexists, unless~$G$ is~$K_3$ or~$K_4$.  Let~$D$ and~$D'$ denote the two\ntriangles incident to~$e$, and let~$w$ and~$w'$ denote the tips of~$D$\nand~$D'$, respectively.  Due to Lemma~\\ref{lem:nonlocal-crossing},\neach chord satisfies condition~\\eqref{eq:1}.  Hence, either~$w \\in\nR_{D'}(u)$ and~$w' \\in R_{D}(u)$ or~$w \\in R_{D'}(v)$ and~$w' \\in\nR_{D}(v)$.  We direct the chord~$e$ towards~$u$ in the former case and\ntowards~$v$ in the latter case.  In this way, we obtain an orientation\nof the chords of~$G$.  Note that outer edges and inner edges of\n4-cliques remain undirected.  The following lemma shows two crucial\nproperties of such an orientation.\n\n\\begin{lemma}\n  \\label{lem:chord-orientation}\n  Let~$G$ be a biconnected inner-chordal graph with plane\n  outside-obstacle representation~$\\Gamma$.  The chord orientation\n  determined by~$\\Gamma$ satisfies the following properties.\n\n  \\begin{compactenum}[(i)]\n  \\item Each vertex has in-degree at most~2.\n  \\item If vertex~$v$ has in-degree~2, then its two incoming edges share a face.\n  \\end{compactenum}\n\\end{lemma}\n\n\\begin{proof}\n  Consider an orientation according to~$\\Gamma$.  Let~$e=(u,v)$ be a\n  directed chord with incident triangles~$D$ and~$D'$, whose tips with\n  respect to the base~$e$ are~$w$ and~$w'$, respectively.  Due to the\n  direction of~$e$, we have that~$w \\in R_{D'}(v)$ and~$w' \\in\n  R_D(v)$.  It is readily seen, e.g., in Fig.~\\ref{fig:attachment},\n  that the two angles at~$v$ incident to~$e$ sum up to more\n  than~$\\pi$.\n\n  Let~$e_1,\\dots,e_k$ be chords that are directed towards~$v$.\n  Without loss of generality assume that these chords are numbered in\n  the order of counterclockwise occurrence around~$v$, starting from\n  the outer face.  Since the angles at~$v$ incident to each of these\n  edges sum up to more than~$\\pi$, it follows that some of these\n  angles must coincide.  Due to the ordering, it follows that the\n  angle at~$v$ right of~$e_i$ (with respect to the orientation\n  towards~$v$) coincides with the left angle of~$e_{i+1}$\n  for~$i=1,\\dots,k-1$.  By planarity and since~$v$ is an outer vertex,\n  no other angles may coincide.  For~$i=1,\\dots,k$, let~$\\alpha_i$\n  denote the angle left of~$e_i$ and let~$\\alpha_{k+1}$ denote the\n  angle right of~$e_k$.  By the above observation, we\n  have~$\\alpha_i+\\alpha_{i+1} > \\pi$ for~$i=1,\\dots,k$.\n\n  For~$k\\ge 3$, the sum of inner angles incident to~$v$ is at\n  least~$\\alpha_1 + \\alpha_2 + \\alpha_3+ \\alpha_4 > 2\\pi$; a\n  contradiction.  For~$k=2$ the shared angle~$\\alpha_2$ implies\n  property~(ii).\n\\end{proof}\n\nBy virtue of Lemma~\\ref{lem:chord-orientation}, we call any\norientation of the chords of a biconnected inner-chordal graph that\nsatisfies the properties~(i) and~(ii) an \\emph{outside-obstacle\n  orientation}.  \n\nLemma~\\ref{lem:chord-orientation} finally allows us to give a concise\nargument why the graph from Fig.~\\ref{fig:example-2} does not admit a\nplane outside-obstacle representation.  We argue that it does not\nadmit an outside-obstacle orientation.  It follows from the conditions\nof such an orientation that, for each 4-clique that is incident to\nthree chords, these chords must be oriented such that they form a\ncycle.  Consider the middle 4-clique in Fig.~\\ref{fig:example-2}.  If\nwe orient it clockwise, then the lower left edge may not have\nadditional incoming chords, which prevents us from orienting the\nchords of the left 4-clique as a cycle.  Symmetrically, choosing a\ncounterclockwise orientation for the middle 4-clique prevents correct\norientation of the right 4-clique.  The graph in\nFig.~\\ref{fig:example-3}, however, does admit an outside-obstacle\norientation, which is indicated in the figure.\n\nOur next goal is to prove that the existence of an outside-obstacle\norientation is equivalent to the existence of a plane outside-obstacle\nrepresentation.  In particular, this shows our claim that the graph in\nFig.~\\ref{fig:example-3} indeed admits a plane outside-obstacle\nrepresentation, e.g., the one shown in Fig.~\\ref{fig:example-oor-2}.\n\n\\begin{theorem}\n  \\label{thm:orientation-representation}\n  Let~$G$ be a biconnected inner-chordal plane graph.  Then~$G$ admits\n  a plane outside-obstacle representation if and only if it admits an\n  outside-obstacle orientation.\n\\end{theorem}\n\n\\begin{proof}\n  The ``only if''-part holds due to Lemma~\\ref{lem:chord-orientation}.\n  Let~$G$ be a biconnected inner-chordal graph with an\n  outside-obstacle orientation and let~$T(G)$ be its construction\n  tree.  We construct a plane outside-obstacle representation of~$G$.\n\n  For a subtree~$T' \\subseteq T$, we denote by~$G(T')$ the subgraph\n  of~$G$ corresponding to~$T'$.  Note that~$G(T) = G$.\n  Let~$\\tau_1,\\dots,\\tau_k$ denote the nodes of~$T$ in breadth-first\n  order starting at an arbitrary node~$\\tau_1$.  For~$j=1,\\dots,k$,\n  let~$T_j$ be the subtree of~$T$ consisting of\n  nodes~$\\tau_1,\\dots,\\tau_j$, and let~$G_j = G(T_j)$ the\n  corresponding subgraph of~$G$.  We inductively construct a sequence\n  of plane outside-obstacle representations~$\\Gamma_1,\\dots,\\Gamma_k$\n  of~$G_1,\\dots, G_k$.  Then~$\\Gamma_k$ is the desired plane\n  outside-obstacle representation of~$G=G_k$.\n\n  Consider the orientation of~$G_i$ induced by~$G$ (note that some\n  edges remain undirected).  We call a directed edge \\emph{active} if\n  it is incident to the outer face of~$G_i$ and \\emph{inactive}\n  otherwise.  An outer vertex~$v$ is \\emph{active} if it is the target\n  of an active edge.  It is \\emph{inactive} otherwise.  The\n  \\emph{inactive degree} of~$v$ in~$G_i$ is the number of inactive\n  edges with target~$v$.\n\n  Throughout steps $i=1,\\dots,k$, we maintain the following\n  properties:\n  \\begin{compactenum}[(i)]\n  \\item The outer angle of vertices with inactive degree~0 is convex.\n  \\item For an active vertex~$v$ with inactive degree~1, removing the\n    unique active in-edge incident to~$v$ results in a convex outer\n    angle.\n  \\end{compactenum}\n  For~$G_1$ any plane outside-obstacle representation~$\\Gamma_1$ satisfies\n  these properties.  We now show how to proceed from~$G_i$\n  to~$G_{i+1}$.  Let~$e=(u,v)$ be the directed chord determined by\n  adding~$\\tau_{i+1}$ to~$T(G_i)$, let~$D$ be the inner triangle\n  of~$G_i$ bounded by~$e$ and let~$e'=(u',v)$ denote the other edge\n  of~$D$ incident to~$v$.\n\n  We aim to place the vertices in~$V(G_{i+1}) \\setminus V(G_i)$ inside\n  the region~$R_D(v)$, which is consistent with the orientation\n  of~$e$.  We first show that this is possible without creating\n  intersections.  If~$v$ has inactive degree~0, then~$v$ is convex,\n  and hence the intersection of~$R_D(v)$ with a suitably\n  small~$\\varepsilon$-ball around~$v$ is disjoint from any vertices\n  and edges of~$\\Gamma_i$.  Similarly, if~$v$ is active but has\n  inactive degree~1, then, after removing~$(u,v)$,~$v$ is convex by\n  property~(ii).  In this case the subcone of~$R_D(v)$ defined by~$e'$\n  and the other outer edge incident to~$v$ intersected with a suitably\n  small~$\\varepsilon$-ball is empty; see Fig.~\\ref{fig:construction-1}.\n\n  \\begin{figure}[tb]\n    \\centering\n    \\begin{subfigure}[b]{.4\\textwidth}\n      \\centering\n      \\includegraphics[page=1]{fig\/construction}\n      \\caption{}\\label{fig:construction-1}\n   \\end{subfigure}\\hfil\n   \\begin{subfigure}[b]{.3\\textwidth}\n      \\centering\n      \\includegraphics[page=3]{fig\/construction}\n      \\caption{}\\label{fig:construction-2}\n   \\end{subfigure}\\hfil\n  \\begin{subfigure}[b]{.3\\textwidth}\n      \\centering\n      \\includegraphics[page=2]{fig\/construction}\n      \\caption{}\\label{fig:construction-3}\n   \\end{subfigure}\\hfil\n   \\caption{Construction of a plane outside-obstacle representation. (a)\n     Attaching at a vertex~$v$ with inactive degree~1. The region\n     where the new points are placed is shaded in dark gray. (b), (c)\n     show that adding an edge preserves the properties required by\n     the construction.  The shaded region is the inner part of the\n     drawing after removing the active edge; the outer angle at~$v$ is\n     convex.}\n    \\label{fig:construction}\n    \\vspace{-2ex}\n  \\end{figure}\n\n  We show that, by placing the new vertices in these regions suitably\n  close to~$v$, properties~(i) and~(ii) can be established for the\n  resulting plane outside-obstacle representation~$\\Gamma_{i+1}$.  First\n  observe that placing the new vertices close enough to~$v$ avoids\n  touching or crossing any vertices and edges of~$G_i$, i.e.,\n  $\\Gamma_{i+1}$ is plane.  Moreover, the addition only changes angles\n  at~$u$ and~$v$, and hence all other vertices satisfy properties~(i)\n  and~(ii) by virtue of the induction hypothesis.\n\n  Consider vertex~$v$.  In~$G_{i+1}$ it has inactive degree at least~1\n  since~$(u,v)$ is an inner edge.  If the inactive degree of~$v$ is~2,\n  there is nothing to prove as~$v$ must be inactive, since\n  outside-obstacle orientations have in-degree at most~2.  Hence\n  assume that~$v$ has inactive degree~1 and it is active.  The\n  properties of outside-obstacle orientations imply that there is a\n  unique active edge directed towards~$v$ in~$G_{i+1}$ and it must be\n  a neighbor of~$e$.  This edge is either the edge~$e'$ or the newly\n  added outer edge~$e''$ incident to~$v$.\n\n  If~$e'$ is the incoming active edge at~$v$, the outer angle\n  at~$v$ was convex in~$\\Gamma_i$, and hence any point in~$R_D(v)$\n  results in an outer angle of less than~$\\pi$ after removing~$e'$;\n  see Fig.~\\ref{fig:construction-2}.\n  If~$e''$ is the incoming active edge at~$v$, the outer angle\n  at~$v$ in~$\\Gamma_{i+1}$ after removing~$e''$ is the outer angle\n  of~$v$ in~$\\Gamma_i$, which is convex by the induction hypothesis;\n  see Fig.~\\ref{fig:construction-3}.\n\n  In all cases vertex~$v$ satisfies properties~(i) and~(ii).  We show\n  that, by positioning the new vertices close enough to~$v$, we can\n  also satisfy properties~(i) and~(ii) for $u$.  First note that the\n  inactive degree of~$u$ does not change.  If the inactive degree\n  of~$u$ is~2, there is nothing to prove.  If the inactive degree\n  of~$u$ is~0 or~1, by placing the new vertices close to the line\n  through~$u$ and~$v$, the angle between~$e$ and the new outer edge\n  incident to~$u$ can be made arbitrarily small.  Thus, if~$u$ was\n  convex in~$\\Gamma_i$, it remains so in~$\\Gamma_{i+1}$.  And, by the\n  same argument, if~$u$ was convex in~$\\Gamma_i$ after removing the\n  active edge incident to~$u$, it remains so in~$\\Gamma_{i+1}$.\n  Hence~$\\Gamma_{i+1}$ satisfies the induction hypothesis.\n\\end{proof}\n\n\n\n\\section{Characterization and Decision Algorithm}\n\\label{sec:main}\n\nIn this section, we prove characterizations of graphs that admit a\nplane outside-obstacle representation and we present a linear-time\nalgorithm that decides whether a given graph admits a plane\noutside-obstacle representation.\n\n\\boldparagraph{Characterization of Outerplanar Graphs.}\nFor biconnected outerplanar graphs\nTheorem~\\ref{thm:orientation-representation} immediately implies a\ncomplete characterization of the graphs that admit a plane\noutside-obstacle representation.\n\n\\begin{theorem}\n  \\label{thm:bico-outer}\n  A biconnected outerplanar graph admits a plane outside-obstacle\n  representation if and only if it is chordal.\n\\end{theorem}\n\n\\begin{proof}\n  Being chordal is a necessary condition due to\n  Lemma~\\ref{lem:plane-oor-chordal}.  Conversely, if an outerplanar\n  graph is chordal, it is obviously inner-chordal.  We show that every\n  biconnected inner-chordal outerplane graph admits an outside\n  obstacle orientation.\n\n  Recall that a biconnected outerplanar graphs contains a vertex of\n  degree at most~2.  We iteratively construct an orientation by\n  directing the incident edges of a vertex with degree at most~2\n  towards it and removing it from the graph.  In this way, we obtain\n  an orientation with the properties that each vertex has in-degree at\n  most~2, and moreover, if a vertex has in-degree~2, then the two\n  incoming edges share an inner face.  Undoing the orientations of the\n  outer edges, we obtain an outside-obstacle orientation.  Now the\n  claim follows from Theorem~\\ref{thm:orientation-representation}.\n\\end{proof}\n\nThis result can easily be strengthened in two ways.  First, if the\nouterplanar graph is not biconnected but chordal, then it can easily\nbe augmented such that it becomes biconnected but remains\n(inner-)chordal and outerplanar and hence satisfies the conditions of\nTheorem~\\ref{thm:bico-outer}, yielding a plane outside-obstacle\nrepresentation of the augmented graph.  By iteratively removing\naugmentation edges that are incident to the outer face we obtain a plane\noutside-obstacle representation of the original graph.\n\n\\begin{corollary}\n  \\label{cor:outer}\n  An outerplanar graph admits a plane outside-obstacle representation\n  if and only if it is chordal.\n\\end{corollary}\n\nAnother observation is that the construction of the orientation in the\nproof of Theorem~\\ref{thm:bico-outer} essentially consists of a\nbottom-up traversal of the construction tree of the graph with respect\nto the root node, which is removed last.  It is then readily seen that\nwe can also remove~$K_4$-nodes that are leaves, provided they have\ndegree at most~2 in~$T(G)$.  A $K_4$ with degree~3 requires that its\nchords are oriented to form a cycle, which cannot be ensured by the\nconstruction.  It can, however, always be achieved it the $K_4$ of\ndegree~3 is the root of the tree.  We thus have the following\ncorollary.\n\n\\begin{corollary}\n  \\label{cor:bico-onek4deg3}\n  Every biconnected inner-chordal graph that contains at most one\n  $K_4$ for which all outer edges are chords admits a plane\n  outside-obstacle representation.\n\\end{corollary}\n\nNote that an augmentation as in the proof of\nCorollary~\\ref{cor:bico-onek4deg3} may increase the number of\n$K_4$-nodes with degree~3.  Hence the result does not extend to\nnon-biconnected graphs.\n\n\n\\boldparagraph{Decision Algorithm for General Graphs.}\nNext, we devise a linear-time algorithm to decide whether a\nbiconnected graph admits a plane outside-obstacle representation.  Of\ncourse it is not difficult to test whether a graph is inner-chordal\nand plane in linear time, and we assume in the following that our\ninput graph has these properties.\n\nDue to Theorem~\\ref{thm:orientation-representation}, deciding the\nexistence of a plane outside-obstacle representation is equivalent to\ndeciding the existence of an outside-obstacle orientation.  To test\nwhether a biconnected inner-chordal plane graph~$G$ admits an outside\nobstacle orientation, we use dynamic programming on its construction\ntree~$T(G)$, rooted at an arbitrary node.  \n\nFor each node~$\\tau$ with parent edge~$\\{u,v\\}$ with orientation~$uv$\nand binary flags~$d_{\\tau,u}$ and~$d_{\\tau,v}$, we are\n  interested whether the subtree of~$T(G)$ with root~$\\tau$ admits an\n  outside-obstacle orientation such that\n\\begin{compactenum}\n\\item $\\{u,v\\}$ is oriented as $uv$,\n\\item $u$ has incoming edges if and only if~$d_{\\tau,u} = 1$, and\n\\item $v$ has incoming edges distinct from~$uv$ if and only\n  if~$d_{\\tau,v} = 1$.\n\\end{compactenum}\n\nWe store this information in a 4-dimensional\ntable~$T[\\tau,e,d_{\\tau,u},d_{\\tau,v}]$ of boolean variables.\nNote that, for each node~$\\tau$, table~$T$ contains only~$2^3 = O(1)$\nentries.  We now show how to fill the entries of this table in linear\ntime.  Initially, we set all entries to \\emph{false}.\n\nFor a leaf node~$\\tau$ with parent edge~$\\{u,v\\}$, we\nset~$T[\\tau,uv,0,0] = T[\\tau,vu,0,0] =$ \\emph{true}, which models the\nfact that we can choose any orientation of~$\\{u,v\\}$ and neither~$u$\nnor~$v$ has incoming edges distinct from~$\\{u,v\\}$ in the subtree\nconsisting only of the leaf.  Let~$\\tau$ be a node with\nchildren~$\\tau'$ and~$\\tau''$ and corresponding chords~$\\{u,w\\},\n\\{v,w\\}$ that connect them to~$\\tau$.  We can easily check whether the\nentries can be combined to an entry of~$\\tau$.  Namely, try both\npossible orientations of~$\\{u,v\\}$ and use the orientations\nof~$\\{u,w\\}$ and~$\\{v,w\\}$ determined by the entries of the children\nand the flags~$d_{\\tau',u}$, $d_{\\tau',w}$, $d_{\\tau'',v}$,\nand~$d_{\\tau'',w}$ of the children to check that~$u,v$ and~$w$ satisfy\nthe constraints of the orientation.  If this is the case, we can\neasily compute the two flags~$d_{\\tau,u}$ and~$d_{\\tau,v}$ from the\norientations of~$uw$, $vw$ and the flags~$d_{\\tau',u}$\nand~$d_{\\tau'',v}$.  A simple induction shows that, in this way, we\nset exactly the correct entries~$T[\\tau,\\cdot,\\cdot,\\cdot]$ to\n\\emph{true}.\n\nCombining two entries takes~$O(1)$ time.  Since there are only~$2^3 =\nO(1)$ entries per node, we can compute all entries of a node~$\\tau$\nfrom all combinations of entries of its at most two children in~$O(1)$\ntime.  Since there are~$O(n)$ nodes, the overall algorithm runs\nin~$O(n)$ time.  At the root we may have to combine up to three\nchildren, but the checks remain essentially the same.  Thus, the\noverall algorithm runs in~$O(n)$ time.\n\n\\begin{theorem}\n  There is a linear-time algorithm that decides whether a given\n  biconnected graph admits a plane outside-obstacle representation.\n\\end{theorem}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nInspired by obstacle representations introduced by Alpert et\nal.~\\cite{akl-ong-10}, we studied plane outside-obstacle\nrepresentations of graphs.  We characterized the biconnected graphs\nthat admit such a representation as the inner-chordal graphs that\nadmit a certain type of orientation of their chords.  Based on this,\nwe gave a combinatorial proof that every chordal outerplanar graph\nadmits a plane outside-obstacle representation.  We further derived a\nlinear-time algorithm for deciding whether a given biconnected graph\nadmits a plane outside-obstacle representation.\n\nOur main open question are the following.  Can our characterization\nand testing algorithm can be extended to general (inner-chordal)\ngraphs that are not necessarily biconnected?  Which graphs admit a\nplane representation with a single obstacle?\n\n\\smallskip\n\n\\noindent\\textbf{Acknowledgments}\nPart of this work has been done while Alexander Koch participated in\nthe academic exchange program of T\\=ohoku University and KIT.  AK\nthanks Prof.~Dorothea Wagner and Prof.~Takeshi Tokuyama for their\nsupport and Prof.~Yota \\=Otachi from JAIST for helpful comments on the topic.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDespite its complexity, language seems to be organized with a few structural principles~\\cite{hau02}.\nFor example, every language has a lexicon of thousands of words. These are basic elements with a particular meaning\nwhich can be combined in utterances to transmit a full idea. Therefore, the potential number of\nword combinations can be overwhelming large. However, a statistical analysis of lexical frequencies\nshows a scaling behavior (Zipf's law~\\cite{zipf}) that establishes an inverse proportion with respect to word rankings.\nThis probability distribution holds for large corpora and many different languages~\\cite{pia14} and has been linked\nto a cognitive principle of least effort in human communication~\\cite{mandelbrot,fer03}.\n\nYet, the Zipf's law yields no information on the selection rules that govern grammatical arrangements\nwithin a sentence. Indeed, words with the highest frequencies often operate with a purely syntactic purpose, such as\ndeterminers (e.g., \\textit{the} in English), prepositions (\\textit{of}), conjunctions (\\textit{and}) or pronouns (\\textit{I}),\nbut unigram distributions like the Zipf's law cannot provide insight into the deep relationships formed\nbetween function and content words to produce meaningful sentences. What is desirable, thus,\nis to investigate distributions of bigrams, trigrams, etc.~\\cite{ha09} to have a complete picture of the statistical\npatterns that underlie human language.\n\nAt first sight the task looks formidable. If $N$ is the vocabulary cardinality, the number of distinct $n$-grams\nis $N^n$. For a rough estimate of $N=10^4$, the possible combinations become exceedingly large already\nfor $n=3$ and cannot hence be statistically analyzed with the largest available resources\n(e.g., the Google Books corpus~\\cite{mic11} includes around $10^{11}$  tokens).\nEven if one takes into account syntactic rules\nthat forbid certain combinations, the number would continue to be enormous. Here, we take an approach that\nsignificantly simplifies the problem while revealing at the same time interesting linguistic patterns.\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, clip]{chesterton.pdf}\n\\end{center}\n \\caption{The text seen as a time series. The frequency count for the tokens that appear in\n \\textit{The Man Who Was Thursday} establishes a frequency ranking for all word types therein. For illustrative purposes, we show the beginning of the novel.}\n\\label{fig_ches}\n\\end{figure}\n\nOur approach is based on an ordinal analysis~\\cite{bra02,zanin}.\nA text is viewed as a time series where the time dimension\ncorresponds to the discrete position of the word inside the text. This perspective is accurate because language, with very few exceptions, is linear:\none word comes after the other. Let us consider the beginning of the \\textit{The Man Who Was Thursday},\na 1908 novel by G. K. Chesterton: ``A cloud was on the mind of men and wailing went the weather\\ldots\". In Fig.~\\ref{fig_ches} we plot the ranking of these words calculated from their absolute\nfrequencies within the novel as a function of position. It follows that \\textit{the} is the top word type and appears\nat the bottom of the time series while content words (\\textit{cloud}, \\textit{mind}, \\textit{men}) possess\na much lower occurrence and come into the high part of series. As a consequence, any text portion in the book\nconsists of a succession of ups and downs as the story unfolds. Our aim is to study this dynamics rather than\nthe particular ranking value as the Zipf's law does. Below, we show that the distribution of\nincreasing and decreasing patterns contains extremely valued information not only about the language itself\nbut also about its history and the speaker (or the writer) who generates the speech.\n\n\\section{Method}\\label{sec:method}\nLet $W$ be the number of words in a given text. We rank its words according to their absolute frequency\nand render the text a sequence of rankings: $\\mathcal{S}_r=\\{r_1,r_2,\\ldots,r_W\\}$. This way, the $i$-th word\nin the sequence is replaced with its frequency ranking $r_i$.\nThe rankings are calculated from each text separately. This guarantees that each word is assigned with a ranking.\nAnother possibility is to use a common ranking for all works under consideration (see the Supplementary Information (SI)), but our results are not significantly altered because $W\\gg 1$ is large for the texts considered in this work.\nA word of caution is necessary for rare words~\\cite{tan16} since it may be that two words with very low frequency share the same ranking. Whenever this happens we modify randomly the rankings of the affected words to make sure that in $\\mathcal{S}_r$ two neighboring terms are never equal. In the SI we give details of this procedure and prove that this modification does not affect the final results.\n\nOur objective is to obtain the pattern distribution for the text.\nDepending on the embedding dimension\n$D$ in the time series~\\cite{tso92} there exist $D!$ ordinal patterns.\nFor instance, if $D=2$ we have either an increasing\nor a decreasing pattern between two consecutive words\nwith rankings $r_i<r_{i+1}$ in the first case and $r_{i+1}<r_i$ in the second case.\nWe plot a sketch of these in the top-left panel of Fig.~\\ref{fig_pat}.\nThen, for the data in Fig.~\\ref{fig_ches} the pattern sequence becomes $\\mathcal{S}_p=\\{ 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2,1\\}$,\nwhere we have assigned the symbols $1$ ($2$) to the increasing (decreasing) pattern.\nFor $D=3$ we have six possibilities, namely, $1$ ($r_i<r_{i+1}<r_{i+2})$,\n$2$ ($r_i<r_{i+2}<r_{i+1}$), $3$ ($r_{i+1}<r_i<r_{i+2}$), $4$ ($r_{i+2}<r_{i}<r_{i+1}$),\n$5$ ($r_{i+1}<r_{i+2}<r_{i}$) and $6$ ($r_{i+2}<r_{i+1}<r_i$),\nsee top-right panel in Fig.~\\ref{fig_pat}. As a result,\nthe sequence in Fig.~\\ref{fig_ches} can be symbolized as $\\mathcal{S}_p=\\{2,6,6,3,2,5,2,3,2,6,5\\}$.\nThe procedure can be straightforwardly generalized to higher dimensions\n(see bottom panel in Fig.~\\ref{fig_pat} for the 24 ordinal patterns with $D=4$).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, trim=5.cm 2.5cm 6.5cm 2.cm, clip]{fig2.pdf}\n\\end{center}\n \\caption{Ordinal patterns employed in this work for embedding dimensions $D=2$ (top left),\n $D=3$ (top right) and $D=4$ (bottom). Each dot represents a value\n in a word ranking diagram such as the one in Fig.~\\ref{fig_ches}.}\n\\label{fig_pat}\n\\end{figure}\n\nAdditionally, one may consider the embedding delay $\\tau \\in \\mathbb{N}$ that defines the time separation between the elements~\\cite{tso92}. In the remainder of this paper, we take $\\tau = 1$, which implies consecutive data, thus fulfilling the linear property of language. As a consequence, the embedding dimension agrees with the number of items\nin an $n$-gram. Another remark is in order. The pattern sequences $\\mathcal{S}_p$\nare generated allowing for overlaps between frequency rankings. Linguistically,\nthis implies that our method probes the phrase structure of the sentence.\nThis can be understood as follows. Quite generally, the pattern distributions\nshow text correlations between segments of length $(D-1)\\tau$. We have checked\nthat for $\\tau =3$ the results do not differ from a random sequence\nobtained by shuffling the text words.\nIt follows that the method is sensitive to short-ranged correlations,\nwhich occur at the phrase (syntagmatic) level. Below, we provide further evidence for this.\n\nWe perform an analysis in three different levels. In the macroscale, we contrast\nthe different pattern distributions across major language families~\\cite{ethnologue}:\nIndo-European (English, Spanish, French, German, Latin and Russian),\nAfro-Asiatic (Somali), Niger-Congo (Xhosa), Turkic (Turkish), Austroasiatic (Vietnamese)\nand Austronesian (Tagalog).\nOur choice also allows for a broad variety of linguistic typologies. Since word order\nplays a crucial role in our findings (see Sec.~\\ref{sec:results}),\nwe concentrate on the most common subject-verb-object (SVO) arrangements found in human language:\nSVO (English, Spanish, French, Russian, Vietnamese, Xhosa), SOV (German, Latin, Somali, Turkish)\nand VSO (Tagalog), which amount to 96\\% of the existing typologies.\nWe only exclude East Asian families (Sino-Tibetan, Japonic, Koreanic)\nbecause word boundaries are not clearly depicted in their written samples\nand linear ordering is not fulfilled.\nHowever, the number of selected languages suffices to support our findings.\nTo avoid possible differences due to the distinct nature of the analyzed texts\nand allow for a fair comparison,\nwe need a single work, long enough, translated to the previously mentioned\nlanguages. The Bible fulfills all these requirements, is publicly available\nfor natural language processing purposes~\\cite{chr15}\nand has already been employed in quantitative linguistics~\\cite{meh17}.\n\nIn the mesoscale, we focus on one language (English) and examine the pattern distribution\nalong time. For definiteness, we bring our attention to the four periods into which\nscholars traditionally divide the history of English: Old English, Middle English, Early Modern English\nand Modern English~\\cite{history}. We pick representative works for each period\n(see Table~S1 in the SI). It is worth mentioning here that Sigaki~\\textit{et al.}~\\cite{sigaki2018} have recently shown that physics-inspired measures estimated from ordinal patterns distributions are able to capture relevant information about paintings, their style and their temporal evolution. Moreover, these measures can be consistently connected with qualitative canonical concepts proposed by art historians to distinguish artworks.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, trim=1.7cm 0cm 1.5cm 0.5cm, clip]{Macroscale_Fig3_D2.png}\n\\end{center}\n \\caption{Relative frequency for pattern~1, as defined in Fig.~\\ref{fig_pat} for embedding dimension $D=2$, as a function of the text position\n for the Bible and the languages indicated above.\n  Each frequency is calculated for a temporal window of $10^4$ words. Then, the window is shifted $10^3$ time units until the text finishes. Labels in the horizontal axis indicate the different, consecutive, window series. Whereas the blue curves correspond to the original text,\n the red dots indicate a shuffled (random) realization generated\n by randomly varying the token positions.}\n\\label{fig_dyn}\n\\end{figure}\n\n\nFinally, the microscale is concerned with individual authors. We fix both the language\nand the period (Modern English) and analyze a literary corpus~\\cite{gutenberg} corresponding to\nfour writers: G.~K.~Chesterton, A.~C.~Doyle, H.~P.~Lovecraft and E.~A.~Poe.\nNotice that the two most important varieties of English (British and American)\nare equally represented with these authors. In Table~S1 in the SI we quote the five book titles for each of these writers employed for our microscale analysis.\n\n\\section{Results}\\label{sec:results}\n\nWe start our analysis by showing with blue curves in Fig.~\\ref{fig_dyn}\nthe normalized frequencies of the first $D=2$ pattern for the different Bibles.\n(The frequency for the second pattern can be simply derived from probability normalization.)\nEach frequency is calculated for a temporal window of $10^4$ words. Then, the window is shifted $10^3$\ntime units until the book finishes. For all languages, the signals appear stochastic\nbut clearly differ from a random sequence obtained by shuffling all the words (red curves).\nIn the latter case, the series also fluctuate but their mean is $0.5$ as should be.\nIn contrast, the expectation value for the original text is above or below $0.5$, depending\non the language, suggesting that the stationary probabilities contain correlations\nentirely due to the word ordering dictated by the syntactic rules that operate in human language.\nNote that we depict two sequences for German, each\ncorresponding to a different Bible translation, showing that their dynamical behavior\ndo not show significant changes.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, trim=1.7cm 0cm 1.5cm 0.5cm, clip]{Macroscale_Fig4_D4.png}\n\\end{center}\n \\caption{Pattern probability distributions calculated at $D=4$ for the Bible in the indicated languages. In the $x$-axis we label the $D!=24$ possible patterns ordered as indicated in Fig.~\\ref{fig_pat}. Similarly to\n Fig.~\\ref{fig_dyn}, the blue dots correspond to the original texts while the red curves show  results for an ensemble of 100 shuffled realizations.}\n\\label{fig_bib}\n\\end{figure}\n\nWe now compare in Fig.~\\ref{fig_bib} (blue curves) the probability distributions for the observed stationary \npatterns. Here, we choose a representative value for $D$\n($D=4$ but the same conclusion is achieved for any other value\nprovided that $D! \\ll W$).\nQuite remarkably, we find that every language has its own fingerprint.\nAdmittedly, a few languages display almost equal histograms, such as French and Spanish\n(both Romance languages), but this should not make us think that the distributions are determined\nby the linguistic family. For instance, English and German are both Germanic languages\nand show distinct probability functions. On the other hand, Xhosa (and possibly Turkish)\nshows a uniform distribution close to the shuffled case, the latter shown as a red band with $3\\sigma$ limits obtained after $100$ independent realizations. We attribute this result to the fact that\nXhosa is a strongly agglutinative language where articles and prepositions are not typically\nindependent words but morphemes that join to root words. We further discuss this\nparticular effect in the SI. Finally, we do not observe any connection between the SVO order\nand the pattern probabilities. The linguistic reason underlying the divergences\nmust be sought for somewhere else.\n\nTo gain further insight, we include in Table~\\ref{tab_big} the most frequent bigrams and their $D=2$\nassociated pattern for both English and Spanish. Whereas in English the pattern 2 is more common,\nin Spanish the pattern with the highest probability is 1. What is the rationale for this difference?\nIf we examine the top bigrams we find that their parsing is preposition + determiner\n(\\textit{of the} in English or \\textit{de la} in Spanish)\nor determiner + noun (\\textit{the Lord} in English or \\textit{la tierra} in Spanish).\nTherefore, their deep structures (in the generative grammar language sense~\\cite{cry97}) do not differ.\nIt is instead the surface structure that determines the mean values for each pattern.\nSince \\textit{the} is the top word type in English, we find more instances of pattern 2\ncorresponding to the group preposition + determiner. Contrarily, this structure is built\nin Spanish with the preposition \\textit{de}, which is the top word type in this language\nand, as a consequence, pattern 1 appears more often.\nIt is therefore not surprising that, as compared with Spanish, we derive an almost\nequal distribution in French, which employ similar words for these functions\nand with similar frequencies. Thus, the concrete pattern distributions\nare not only caused by the syntactic rules but also by the diverse strategies that languages\nemploy to express these rules with the vocabulary at their disposal.\nImportantly, our results do no invalidate the existence of a universal grammar.\nWe will unveil in Sec.~\\ref{sec:discussion} a true universal feature\nthat is not observable in the patterns' mean values but surprisingly in their statistical\ndistribution through short-ranged correlations.\n\n\\begin{table}\n\\subfloat{%\n\\begin{tabular}{|l|r|c|}\n\\hline\nBigram & Counts & Pattern \\\\\n\\hline\n \\textit{of the} & 11545 & 2 \\\\\n \\textit{the lord} & 7036 & 1 \\\\\n \\textit{and the} & 6278 & 2 \\\\\n \\textit{in the} & 5031 & 2 \\\\\n \\textit{and he} & 2794 & 1\\\\\n \\hline\n\\end{tabular}\n}\n\\hfill\n\\subfloat{%\n\\begin{tabular}{|l|r|c|}\n\\hline\nBigram & Counts & Pattern \\\\\n\\hline\n \\textit{de la} & 4250 & 1 \\\\\n \\textit{de los} & 3966 & 1 \\\\\n \\textit{en el} & 2494 & 2 \\\\\n \\textit{a los} & 2331 & 1\\\\\n \\textit{la tierra} & 2202 & 1\\\\\n \\hline\n\\end{tabular}\n}\n\\caption{Bigrams with the largest frequencies in the English (left) and Spanish (right) Bibles. We also include their $D=2$ ordinal pattern as labeled in Fig.~\\ref{fig_pat}.}\n\\label{tab_big}\n\\end{table}\n\n\n\\begin{figure}\n\n\\subfloat{\\includegraphics[width=0.24\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{Mesoscale_Fig5_D4_A.png}}\n\\hfill\n\\subfloat{\\includegraphics[width=0.24\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{Mesoscale_Fig5_D4_B.png}}\n\\newline\n\\subfloat{\\includegraphics[width=0.24\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{Mesoscale_Fig5_D4_C.png}}\n\\hfill\n\\subfloat{\\includegraphics[width=0.24\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{Mesoscale_Fig5_D4_D.png}}\n\n \\caption{Ordinal pattern probability distributions calculated for embedding dimension $D=4$\n and different historical periods: a) Old English, b) Middle English, c) Early-Modern English and d) Modern English.\n Each curve corresponds to a single work or collection of works as listed in Table~S1 of the SI. In the $x$-axis we label the $D!=24$ possible patterns ordered as indicated in Fig.~\\ref{fig_pat}.}\n\\label{fig_his}\n\\end{figure}\n\nLet us now discuss the mesoscale level.\nIt is well known that language changes with time. Then, we expect that pattern distributions will evolve along history. We illustrate this phenomenon in Fig.~\\ref{fig_his}.\nWe take representative works for each historical period. In the Old English case [Fig.~\\ref{fig_his}a)],\nwe plot the probability distribution function for \\textit{Andreas} (curve 1), \\textit{Anglo-Saxon Chronicle} (2),\n\\textit{Beowulf} (3), \\textit{Christ} (4), \\textit{Genesis} (5) and \\textit{Guthlac} (6).\nDespite the fact that these texts are quite short,\nthe distributions appear similar (with fluctuations due to the different lengths and genre).\nFor the Middle English [Fig.~\\ref{fig_his}b)] we use \\textit{Layamon's Brut} (to have texts of similar lengths we split this work in curves 7 and 8),\n\\textit{Canterbury Tales} (9, 10 and 11), \\textit{Confession Amantis} (12, 13 and 14),\n\\textit{Book of the Knight of La Tour-Landry} (15) and \\textit{Mandeville's Travels} (16).\nIt is nice to see the evolution in the patterns from the Old [Fig.~\\ref{fig_his}a)] to the Middle periods [Fig.~\\ref{fig_his}b)]. The latter distributions appear more uniform because of both the smaller time range of their period and a higher language standardization, a process that began in the Late Middle Ages.\nThe Early-Modern English corpus [Fig.~\\ref{fig_his}c)] comprises works of Ben Jonson and those of Christopher Marlowe, Milton's \\textit{Paradise Lost}, and Shakespeare's \\textit{Tragedies} and \\textit{Comedies}\n(see Table~S1 in the SI for their number identification). Finally, those chosen authors living in the Modern English period [Fig.~\\ref{fig_his}d)]\nwere aforementioned in Sec.~\\ref{sec:method} and their works are also enumerated in Table~S1 of the SI.\nClearly, there is an overall coherence among patterns belonging to the same time, which suggests that the traditional classification in periods has a lexical support.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, trim=1.3cm 0cm 1.5cm 0.5cm,clip]{Mesoscale_Fig6_D4.png}\n\\end{center}\n \\caption{Permutation Jensen-Shannon distance between pairwise probability distributions calculated in Fig.~\\ref{fig_his} for $D=4$.\n Darker (lighter) colors indicate a small (large) distance,\n which quantifies the lexical (in the ordinal analysis language)\n difference among literary works belonging to distinct English\n historical periods.}\n\\label{fig_his2}\n\\end{figure}\n\nThis is better seen when one calculates the permutation Jensen-Shannon\ndistance~\\cite{zunino2022} between distributions and plots this distance for $D=4$ as in Fig.~\\ref{fig_his2}.\nWe observe four dark areas that well correspond to the four historical periods.\nThe distinction is clear between Old, Modern and the cluster formed by\nMiddle English and Early-Modern English, between which the transition\nis less clear. This is because the Early-Modern English spans a period\nbetween the Renaissance, when the medieval forms were still popular,\nand the 17th century, when English conventions were approaching those of the\nModern period. Interestingly, there exist individual deviations from the historical\npattern. For instance, \\textit{Anglo-Saxon Chronicle} (work 2) appears to be close to the Middle English\ncluster whereas \\textit{Paradise Lost} (work 22) would be more suitable\nto be classified in the previous stage (Middle English), probably due to Milton's intentionally\narchaic style. Another exception is Jonson (works 17, 18 and 19), whose style is better categorized\nwithin the Modern period. We highlight that despite the method's simplicity\nwe are not only able to correctly place literary works in their composition period but also detect singular departures assignable to particular style features.\n\nPrevious results are encouraging because they show that on top of a common background\nthat characterizes written works in a given language there may exist fluctuations large enough\nto allow us to determine the author of a set of texts. In fact, a few subclusters with smaller distances can be distinguished in Fig.~\\ref{fig_his2} for works of the same writers. This is particularly evident for Layamon's, Gower's, Jonson's and Shakespeare's works. We now pursue this idea by further analyzing the last historical period (microscale). In Fig.~\\ref{fig_fluc} we depict the ordinal\npatterns and their probability frequencies for $D=4$. Fluctuations are seen\nin the slight differences within the histogram. Then, we assess the pairwise distribution distance and plot the resulting matrix in Fig.~\\ref{fig_matrix}.\nTo obtain a more efficient discrimination between the texts, thus amplifying\nthe fluctuations shown in Fig.~\\ref{fig_fluc}, we set the embedding dimension to $D=6$, which allows for 720~patterns.\nStrikingly, we observe in Fig.~\\ref{fig_matrix} that each writer forms a cluster of his own, with the largest distance taking place between\nPoe (works numbered between 16 and 20) and the rest, perhaps due to Poe's highly mixed style.\nHere, we add an important caveat: the microscale is the most sensitive situation and\nFig.~\\ref{fig_matrix} is only a proof of concept. To use this technique in author\nattribution tasks~\\cite{gri07} would require better refined analyses that fall beyond the scope of the present work.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth, trim=1.3cm 0cm 1.5cm 0.5cm,clip]{Microscale_Fig7_D4.png}\n\\end{center}\n \\caption{Histograms showing $D=4$ pattern probability distributions for four different authors from the same historical period (Modern English):\n G. K. Chesterton (blue), A. C. Doyle (black), H. P. Lovecraft (red) and E. A. Poe (green). On average the distributions are similar with fluctuations caused by author stylistic differences.}\n\\label{fig_fluc}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe have seen that the analysis of ordinal patterns is a powerful method that allows to identify\n(i) language, (ii) historical period and (iii) single authors. These results have their origin in the greatly diverse\nprocedures with which human languages embody abstract formal relationships.\nWe now discuss a universal trait, common to all languages studied in this work, which is unearthed\nafter a closer look at the dynamical properties of the pattern series.\n\nLet $x$ denote an ordinal pattern, which as discussed in Sec.~\\ref{sec:method} can take on $D!$ possible values. The probability\nto find a block of size $n\\ge 1$ in the pattern sequence $\\mathcal{S}_p$ is\n$P(x_{t+n-1}=y_k,\\ldots,x_{t+1}=y_j,x_t=y_i)$ where $t$ is the time (or position) index and $y$\ndesignates the pattern values with $i,j,k$ running from 1 to $D!$. For a sufficiently long text ($W\\gg 1$)\nwe can faithfully compute these block probabilities from their occurrences as long as $n\\ll W$. Notably, $P$ contains the complete\ninformation on the possible correlations between adjacent patterns. In Sec.~\\ref{sec:results}\nwe calculated the expectation $E(x_t)$ which in the long term becomes stationary and consequently\nindependent of $t$. However, to investigate pattern-pattern correlations one needs to find the covariance \n$E(x_t,x_{t+1})$. The issue is that this autocorrelation function depends on the employed symbolization for $D>2$.\n(We discuss this in more detail in the SI). To overcome this important difficulty, we resort to the approach\ndiscussed in Ref.~\\cite{juan}, where a method is presented to determine the order of a Markov chain.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.52\\textwidth, clip]{Microscale_Fig8_D6_shift.png}\n\\end{center}\n \\caption{Permutation Jensen-Shannon distance between the $D=6$ pattern probability distributions for the works of four English Modern authors (Chesterton, Conan Doyle, Lovecraft and Poe). The axes correspond to the numerical identification in parentheses of Table~S1 of the SI.}\n\\label{fig_matrix}\n\\end{figure}\n\nThat language can be modeled as a Markov chain is an old subject. Markov himself proposed this technique~\\cite{markov}\nfor his pattern analysis of vowels and consonants in A. Pushkin's novel \\textit{Eugene Onegin}.\nIn a Markovian system with a discrete space of states, the probability to find at time $t$ the random variable $x$ in state $y$\ndepends on the state at time $t-1$. This correlation is captured by the conditional probability.\nIn general, a Markov chain has memory (or order) $m$ if the conditional probabilities satisfy\n\\begin{align} \\label{eq_P}\nP(x_{t}&=y_k | x_{t-1}=y_j,\\ldots,x_{t-m} = y_{i},\\ldots, x_0 = y_l) \\nonumber \\\\\n&= P(x_{t}=y_k | x_{t-1}=y_j,\\ldots,x_{t-m} = y_{i})\\,.\n\\end{align}\nThis means that one would need to check $(D!)^{m+2}$ equalities even\nassuming that memory decreases with time~\\cite{par06}. Luckily, there is an equivalent, much shorter scheme, which consists of calculating the block entropy\n\\begin{align}\\label{eq_H}\nH_n&=-\\sum_{i,j,\\ldots,k=1}^{D!}\nP(x_{t+n-1}=y_k,\\ldots,x_{t+1}=y_j,x_t=y_i) \\nonumber \\\\\n&\\times\\log P(x_{t+n-1}=y_k,\\ldots,x_{t+1}=y_j,x_t=y_i)\\,,\n\\end{align}\nwhere $H_n$ is independent of $t$ due to time homogeneity.\nApplying the condition given by Eq.~\\eqref{eq_P}, the entropy becomes\na linear function of $n$~\\cite{grassberger}:\n\\begin{align}\\label{eq_Hlinear}\nH_\\mu (n) = (H_{\\mu+1}-H_\\mu) (n-\\mu) + H_\\mu\\,,\n\\end{align}\nfor $n\\ge\\mu>0$ and $H_0 (n) = n H_1$ for $n\\ge 1$\nwith $\\mu\\in\\mathbb{N}_0 $ a trial memory. Therefore,\nthe parameter\n\\begin{equation}\n\\Delta_\\mu = \\frac{1}{n_{\\rm max}-\\mu +1}\\sum_{n=\\mu}^{n_{\\rm max}} (H_\\mu (n)-H_n)^2\n\\end{equation}\nvanishes if $\\mu\\ge m$ for all values of $n_{\\rm max}$. Thus, the memory $m$\ncan be found from the condition\n\\begin{equation}\nm = \\min(\\mu : \\Delta_\\mu = 0)\\,.\n\\end{equation}\nIn practice, we define a small threshold for the vanishing of the order parameter.\nThis method can then provide the $m$ value that better fits the sequences $\\mathcal{S}_p$\nfor the languages considered in Sec.~\\ref{sec:results}.\n\n\\begin{figure}\n\n\\subfloat{\\includegraphics[width=0.24\\textwidth, clip]{English_D2.pdf}}\n\\hfill\n\\subfloat{\\includegraphics[width=0.24\\textwidth, clip]{English_D3.pdf}}\n\n \\caption{Block entropies as a function of the block size $n$ with embedding dimensions a) $D=2$ and b) $D=3$ for the pattern sequence extracted from the English Bible (red), its shuffled realization (green)  and the Markov model with memory $m=2$ (blue). Note that the red and blue  curves are overlapping. Insets: order parameter $\\Delta$ as a function of the trial memory $\\mu$. In both cases we find that the Markov chain order is 2.}\n\\label{fig_ent}\n\\end{figure}\n\nIn Fig.~\\ref{fig_ent}a) and b) we plot the entropy as a function of the block size for\nthe English Bible at $D=2$ and $D=3$, respectively.\nWe find in both cases that the sequences $\\mathcal{S}_p$ can be described with a Markov model of memory $m=2$\nsince the data (red dots) are well fitted with Eq.~\\eqref{eq_Hlinear}\nfor $\\mu =2$ (blue curve overlapping with the red line).\nInterestingly, the entropy is smaller than that of the shuffled realization\n(green line). Therefore, the deviation with the random case\nis entirely due to correlations among words.\n\nWe also show in the insets of Fig.~\\ref{fig_ent} the behavior of $\\Delta_\\mu$ for a set of trial memories. For the two $D$ values,\nthe parameter $\\Delta_\\mu$ attains 0 at $\\mu=2$ (within a tiny numerical error). We have checked that this is true for all the studied languages. Naturally, each language has its own\ntransition probabilities given by Eq.~\\eqref{eq_P}. What is remarkable is that every lexical pattern distribution\ncan be modeled with a Markov chain of order 2 independently of the considered language.\n\nEven further, the deviations of these sequences from an\n$m=1$ model are all of the same order, as can be seen in Table~\\ref{tab_dev}, where we present the Markovity parameter $\\delta$\ndefined as\n\\begin{equation}\\label{eq_delta}\n\\delta = \\frac{|H_1-\\mathcal{H}(1)|}{a}\\,.\n\\end{equation}\nHere, $\\mathcal{H}(n)=a n + b$ with coefficients $a$ and $b$ obtained from linear regression. This fitting is done for $n>1$ in order that $H_1$ and $\\mathcal{H}(1)$ be independent.\nThen, $\\delta$ quantifies the departure of the entropy curves from a pure Markovian model of memory $m=1$.\nIn all cases, $\\delta$ enhances as $D$ increases because the number of possible blocks grows as $(D!)^n$ and the calculation becomes\nless reliable due to finite size effects. Strikingly enough, $\\delta$ is approximately the same for all languages.\nTherefore, a $m=2$ Markov model appears to be an accurate modelization for ordinal sequences\nof human language, the deviations from $m=1$ being worldwide established.\nIn the SI, we provide a further proof for this finding using an alternative method based on autocorrelation functions.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l||r|r|}\n\\hline\n   & $D=2$ & $D=3$ \\\\\n\\hline\n\\hline\n English & 0.0201 & 0.0443 \\\\\n French & 0.0187 & 0.0421 \\\\\n German 1 & 0.0187 & 0.0470 \\\\\n German 2 & 0.0181 & 0.0459 \\\\\n Latin & 0.0152 & 0.0411 \\\\\n Russian & 0.0136 & 0.0411 \\\\\n Somali & 0.0144 & 0.0420 \\\\\n Spanish & 0.0206 & 0.0414 \\\\\n Tagalog & 0.0189 & 0.0480 \\\\\n Turkish & 0.0156 & 0.0361 \\\\\n Vietnamese & 0.0209 & 0.0362 \\\\\n Xhosa & 0.0148 & 0.0365 \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Deviations of the Bible in the indicated languages from a pure Markov model as defined by the Markovity parameter given by Eq.~\\eqref{eq_delta}. We consider ordinal pattern sequences \nfor embedding dimensions $D=2$ (middle column) and $D=3$ (right column).}\n\\label{tab_dev}\n\\end{table}\n\nWe emphasize that our results are not in contradiction with recent claims that point to long-range correlations\nin texts~\\cite{sch93,ebe94,alv06,mon11,alt12}.\nRecalling that our ordinal analysis replaces a huge number of words with a few ranking-based patterns,\nit is natural to expect that pattern-pattern correlations quickly vanish after a few Markov steps. \nThis is consistent with our method being responsive to interrelations inside a phrase, which are typically\nmuch shorter than a sentence, and whose syntactic structure puts constraints on the parts of speech\nto which its elements may belong. We quantify that these constrains act between maximum 3 adjacent\npatterns and that this is a universal feature of human communication, possibly due to the few grammar\nrules (in comparison with the number of words) that govern all languages.\nIn the SI we recalculate the pattern statistical distributions shuffling the sentences and obtain exactly\nthe same distributions, which is another indication that pattern correlations are established above single words\nbut below the sentence level.\n\n\\section{Conclusions}\\label{sec:conclusion}\n\nIn short, we have demonstrated that every language has a characteristic fingerprint in terms of a statistical\ndistribution for symbolic patterns. The observed patterns emerge from a combination of the syntactic rules\nthat shape each language and the way that this language articulate those rules.\nA careful view of the pattern distribution provides useful information on the historical\nperiod when the text was produced and the author that wrote it. These findings bode well for possible applications\nof our method. We envisage implementations in stylometry studies that seek a correct authorship attribution,\nas mentioned above,\nor in forensic linguistics for legal cases where linguistic data play a decisive role. Another interesting\napplication would aim at the detection of speech impairments in individuals.\n\nThe procedure discussed here has obvious limitations, the most important of which concerns semantics.\nSince every word is replaced with its ranking value in a table of frequencies, the symbolic patterns are agnostic\nwith regard to meaning. However, this is the same limitation that takes place in all information-theoretic\napproaches to language, like Shannon's theory of communications~\\cite{shannon}. This does not preclude our analysis\nfrom being capable of finding a novel linguistic universal~\\cite{gree63} in relation with a common memory value \nthat determines short-ranged correlations in phrase structures.\n\n\\acknowledgments\nWe thank M. Zanin for useful comments. \nFinancial support has been received from MCIN\/AEI\/10.13039\/501100011033 and the Fondo Europeo de Desarrollo Regional (FEDER, UE) through project PACSS (RTI2018-093732-B-C21) and the Mar{\\'\\i}a de Maeztu Program for units of Excellence in R\\&D, grant MDM-2017-0711. L.Z. acknowledges the financial support from Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas (CONICET), Argentina.\n\n\n\n\\section{Common ranking}\nIn this section, we calculate the ranking sequences differently.\nWe consider a large corpus and arrange its words based on their\noccurrences. The corresponding rankings are then used\nto determine the ordinal patterns. The advantage of this approach\nis that all literary works are symbolized using the same ranking.\nThe limitation is that word types that do not appear in the corpus\ncannot be assigned to a definite ranking and therefore\nnot all patterns consist of consecutive words. However,\nwe do not see a significant difference with the method employed\nin the main text.\n\nIt suffices to illustrate this fact with a single language (e.g., English). We have checked\nthat our conclusions are unaltered for different languages.\nThe English word frequency list contains the 1\/3 million most frequent\nwords~\\cite{nor09} built from the Google Books Corpus (GBC)~\\cite{google}.\nIn Fig.~\\ref{fig_GC}a) we depict the symbol dynamics for $D=2$\nobtained from the GBC ranking, comparing with the original series,\nwhich is reproduced in Fig.~\\ref{fig_GC}b) from top left panel in Fig.~3 of the main text. We find that the dynamical patterns resemble each other\nalthough the peak amplitudes differ. This is expected because the strength of the fluctuations depend on the word frequencies, which in turn\nare calculated from different corpora. However, the probability\ndistributions are almost unaltered. We show this in\nFig.~\\ref{fig_GC}c) for the GBC side by side with\nFig.~\\ref{fig_GC}d), which is replicated from top left panel in Fig.~4 of the main text. This demonstrates the robustness of our method\nfor alternative corpora provided that the size of the corpus is sufficiently large.\n\n\\begin{figure}[h]\n\n\\subfloat{\\includegraphics[width=0.46\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{EnglishBible-GC-D2-Dynamic.png}}\n\\hfill\n\\subfloat{\\includegraphics[width=0.46\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{EnglishBible-OC-D2-Dynamic.png}}\n\\newline\n\\subfloat{\\includegraphics[width=0.46\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{EnglishBible-GC-D4.png}}\n\\hfill\n\\subfloat{\\includegraphics[width=0.46\\textwidth, trim=0.cm 0cm 2.cm 0.cm, clip]{EnglishBible-OC-D4.png}}\n\n \\caption{Dynamical behavior of the first $D=2$ pattern (blue curves) for the English Bible when the ranking sequences are generated by using a) a common and b) its own corpus. The corresponding ordinal pattern probability distributions for $D=4$ are displayed in c) and d), respectively. Results obtained when words are shuffled (red curves) are also included only for reference purpose. As in the main text, in the dynamical panels we only include a single shuffling realization to avoid finite size effects but in the distribution panels the red curves are indeed bands with $3\\sigma$ limits calculated after 100 realizations. }\n\\label{fig_GC}\n\\end{figure}\n\n\\section{Sequences with equal rankings}\nIf a sequence of $k$ words have the same ranking $r_{i}<r_{i+1}=r_{i+2}=\\dots=r_{i+k}<r_{i+k+1}$, then we modify randomly those rankings by adding to each of $r_{i+1},\\dots,r_{i+k}$ a uniform random number in the interval $(-(r_{i+1}-r_i),r_{i+k+1}-r_{i+k})$.\nHere, we provide evidence that our procedure of breaking ranking\nties does not affect the main results except for a case that deserves\nattention. In Fig.~\\ref{fig_noise} we plot with solid blue lines\nthe pattern distributions when two\nconsecutive words are allowed to have the same ranking.\nIf this happens, an ordinal pattern cannot form but this will occur\nmostly for rare words. In addition, in Fig.~\\ref{fig_noise} we reproduce\nwith red dashed lines the case as in Fig.~4 of the main text.\nClearly, adding a small random number preserves the general structure of the\nprobability distributions with the advantage that the number of\ngenerated patterns is the maximum possible. The case\nwhere the two distributions seriously differ is Xhosa\n(and, to a smaller degree, Turkish). This is caused\nby the large number of word types that have occurrence 1 or 2\nin the text. Therefore, it is likely that two consecutive words\nhave the same ranking and adding a small random number is now not negligible.\nWe ascribe this effect to the agglutinative nature of Xhosa.\nUnlike, e.g., English, which expresses most of its syntactic functions\nwith isolated words, Xhosa displays agglutinated morphological\ncomplexes. It is thus natural to expect in the Xhosa Bible\na large amount of hapax legomena. Any random shift will then represent\na significant perturbation to the original series, as observed in our data.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth, clip]{noise.png}\n\\end{center}\n \\caption{Distribution of linguistic ordinal patterns for embedding dimension $D=4$ for the original text sequences (blue lines) and when a small random number is added to break ranking equalities (dashed red lines).}\n\\label{fig_noise}\n\\end{figure}\n\n\n\\section{List of literary works}\n\nTable~\\ref{tab_list} shows the full list of literary works\nemployed in the mesoscale\nand microscale analyses in Sec.~III of the main text.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|r||l|l|}\n\\hline\nNumber & Title & Author \\\\\n\\hline\n1 & Andreas & Anonymous \\\\\n2 & Anglo-Saxon Chronicle & Anonymous \\\\\n3 & Beowulf & Anonymous \\\\\n4 & Christ & Anonymous and Cynewulf \\\\\n5 & Genesis & Anonymous \\\\\n6 & Guthlac & Anonymous \\\\\n7 & Brut I & Layamon \\\\\n8 & Brut II & Layamon \\\\\n9 & Canterbury I & Chaucer \\\\\n10 & Canterbury II & Chaucer \\\\\n11 & Canterbury III & Chaucer \\\\\n12 & Confessio Amantis I & Gower \\\\\n13 & Confessio Amantis II & Gower \\\\\n14 & Confessio Amantis III & Gower \\\\\n15 & The Book of the Knight of La Tour-Landry & Caxton \\\\\n16 & The Travels of Sir John Mandeville & Mandeville \\\\\n17 & Every Man in His Humor. The Poetaster & Jonson \\\\\n18 & Epicoene. Cynthia's Revels & Jonson \\\\\n19 & Bartholomew Fair. The Alchemist & Jonson \\\\\n20 & Tamburlaine the Great. Hero and Leander & Marlowe \\\\\n21 & The Jew of Malta. The Massacre at Paris & Marlowe \\\\\n22 & Paradise Lost & Milton \\\\\n23 & Tragedies I & Shakespeare \\\\\n24 & Tragedies II & Shakespeare \\\\\n25 & Comedies I & Shakespeare \\\\\n26 & Comedies II & Shakespeare \\\\\n27 & The Innocence of Father Brown (1) & Chesterton \\\\\n28 & The Man Who Know Too Much (2) & Chesterton \\\\\n29 & The Napoleon of Notting Hill (3) & Chesterton \\\\\n30 & The Man Who Was Thursday (4) & Chesterton \\\\\n31 & The Wisdom of Father Brown (5) & Chesterton \\\\\n32 & Memoirs of Sherlock Holmes (6) & Conan Doyle \\\\\n33 & The Return of Sherlock Holmes (7) & Conan Doyle \\\\\n34 & The Sign of Four (8) & Conan Doyle \\\\\n35 & The Hound of the Baskervilles (9) & Conan Doyle \\\\\n36 & The Adventures of Sherlock Holmes (10) & Conan Doyle \\\\\n37 & The Randolph Carter Stories (11) & Lovecraft \\\\\n38 & The Dream Cycle (12)  & Lovecraft \\\\\n39 & Twenty-Nine Tales (13) & Lovecraft \\\\\n40 & Twenty-Nine Collaborative Stories  (14) & Lovecraft et al. \\\\\n41 & At the Mountains of Madness (15)  & Lovecraft \\\\\n42 & The Works of Edgar Allan Poe I  (16) & Poe \\\\\n43 & The Works of Edgar Allan Poe II  (17) & Poe \\\\\n44 & The Works of Edgar Allan Poe III  (18) & Poe \\\\\n45 & The Works of Edgar Allan Poe IV  (19) & Poe \\\\\n46 & The Works of Edgar Allan Poe V  (20) & Poe \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Literary works considered in the mesoscale and microscale analyses. The numerical identification of the left column is used in the axes of Figs.~5 and~6 of the main text while the numbers in parentheses are employed for the axes of Fig.~8 of the main text.}\n\\label{tab_list}\n\\end{table}\n\n\\section{Autocorrelation analysis}\n\\label{sec:auto}\nLet us consider a two-state system with values $z_i\\in\\{1,2\\}$.\nThis corresponds to the $D=2$ case for ordinal patterns discussed\nin the main text.\nA Markov process or order $m=2$ is defined by the following $8$ (of which $4$ are independent) transition probabilities\n\\begin{align}\\label{eq:pijdef}\n p_{ij}&=\\text{Prob}(z_{n+2}=1|z_{n}=i,z_{n+1}=j),&\\hspace{10pt}i,j=1,2\\\\\n q_{ij}&=\\text{Prob}(z_{n+2}=2|z_{n}=i,z_{n+1}=j)=1-p_{ij},&i,j=1,2\n\\end{align}\nGiven a sequence $\\{z_0,z_1,z_2.\\dots\\}$ we can form a new series by concatening two consecutive values $x_0=(z_0,z_1)$, $x_1=(z_1,z_2)$, etc. For instance the series in the $z$ variable $\\{1,1,2,1,2,1,1,2,2\\}$ yields the series\\\\ $\\{11,12,21,12,21,11,12,22\\}$ in the $x$ variable. Note that the length of the $x$ series is one unit less than the original length of the $z$ variable series. The key point is that in the new $x$ variable space the process is Markovian with the matrix of transition probabilities $\\mathbb{W}_{ij}=P(x_n=X_i|x_{n-1}=X_j)$ with $X_1=11,\\,X_2=12,\\,X_3=21,\\,X_4=22$:\n\\begin{equation}\n\\mathbb{W}=\\begin{pmatrix}p_{11} & 0 & p_{21}& 0\\\\ q_{11} & 0 & q_{21}& 0\\\\ 0 & p_{12} & 0& p_{22}\\\\ 0& q_{12} & 0 & q_{22} \\end{pmatrix}\n\\end{equation}\nIf we denote by $\\vec{P}_n=\\begin{pmatrix}P(x_n=11)\\\\ P(x_n=12)\\\\ P(x_n=21)\\\\ P(x_n=22)\\end{pmatrix}$ the probability vector whose components are the probability that the value of $x_n$ at time step $n$ is equal to $ij,\\, i,j=1,2$, we can write down the evolution equation $\\vec{P}_n=\\mathbb{W}\\vec{P}_{n-1}$, of general solution $\\vec{P}_{n+\\ell}=\\mathbb{W}^n\\vec{P}_\\ell$ valid for all values of $n$ and $\\ell$. The steady state $\\vec{P}_\\text{st}=\\lim_{n\\to\\infty}\\vec{P}_n$ satisfies\n\\begin{equation}\\label{Pst}\n \\begin{pmatrix}P_\\text{st}(x=11)\\\\ P_\\text{st}(x=12) \\\\ P_\\text{st}(x=21) \\\\ P_\\text{st}(x=22)\\end{pmatrix}=\\frac{1}{q_{11}q_{12}+(2q_{11}+p_{21})p_{22}}\\begin{pmatrix}p_{21}p_{22}\\\\ q_{11} p_{22} \\\\ q_{11}p_{22} \\\\ q_{11}q_{12},\\end{pmatrix}\n\\end{equation}\nwhere the right hand side is the, conveniently normalized, eigenvector associated to the eigenvalue $\\lambda_0=1$ of the matrix of transition probabilities $\\mathbb{W}$. Note that $P_\\text{st}(x=12)=P_\\text{st}(x=21)$, as in a given $\\{z_n\\}$ series there are always as many transitions $1\\to 2$ than $2\\to 1$ with the possible exception arising from the last term of the series being equal to the first one. In addition to $\\lambda_0=1$, the matrix $\\mathbb{W}$ has three further eigenvalues, $\\lambda_1,\\lambda_2,\\lambda_3$, different from $1$ that can be obtained as the roots of the characteristic equation:\n\\begin{equation}\\label{lambda3}\n \\lambda^3+(p_{22}-p_{11})\\lambda^2+(p_{22}q_{11}-p_{12}q_{21})\\lambda+(p_{12}-p_{22})(p_{11}-p_{21})=0,\n\\end{equation}\nwhose explicit solutions are too cumbersome to be written down here.\n\nAll statistical properties of the $z$ series can be derived from those of the series in the $x$ variables. For example,\n\\begin{align}\n P(z_n=i)&=\\sum_{j=1,2}P(x_n=ij),\\label{P1}\\\\\n P(z_n=i,z_{n+1}=j)&=P(x_n=ij)\\label{P2}\\\\\n P(z_n=k,z_{n+2}=i)&=\\sum_{j=1,2}P(z_n=k,z_{n+1}=j,z_{n+2}=i)\\nonumber\\\\\n P(z_n=k,z_{n+1}=j,z_{n+2}=i)&=P(z_{n+2}=i|z_n=k,z_{n+1}=j)P(z_n=k,z_{n+1}=j)\\nonumber\\\\\n &=P(z_{n+2}=i|z_n=k,z_{n+1}=j)P(x_n=kj)\\nonumber \\\\\n &=\\begin{cases}p_{kj}P(x_n=kj) & i=1\\\\ q_{kj}P(x_n=kj), &i=2\\end{cases},\\label{P3}\\\\\n P(z_n=i,z_{n+\\ell}=j)&=\\sum_{k_1=1,2,k_2=1,2}P(x_{n-1}=ik_1,x_{n+\\ell}=jk_2), \\quad \\ell>2\\label{P4}\n\\end{align}\nvalid for all values $i,j,k=1,2$, etc.\n\nWe are interested in the correlation function $C_\\ell$ of the original $\\{z_n\\}$ series as a function of the time lag $\\ell$ between values in the series. This is defined as\n\\begin{equation}\\label{eq:Ckdef}\n C_\\ell=\\frac{\\langle z_{n+\\ell}z_n\\rangle_\\text{st}-\\langle z_n\\rangle^2_\\text{st}}{\\langle z_{n}^2\\rangle_\\text{st}-\\langle z_n\\rangle^2_\\text{st}}.\n\\end{equation}\nAll averages are performed in the steady-state where all dependence on the initial condition has been lost. Note that $C_0=1$, and that this definition is independent on the actual values assigned to the variables $z_n$ since any linear transformation $z_n\\to az_n+b$ with arbitrary $a,b$ would give the same result. This is a particular result valid only for a series whose elements take only two values. For an ordinal analysis with $D>2$ we would have to face the issue that different assignments to the values of the $z$ variable would yield different numerical values to the autocorrelation function.\n\nThe averages in the steady state are computed as:\n\\begin{align}\\label{eq_avz}\n \\langle z_{n}\\rangle_\\text{st}&=\\sum_{z_n=1,2}z_nP_\\text{st}(z_n),\\\\\n \\label{eq_corrz}\n \\langle z_{n+\\ell}z_n\\rangle_\\text{st}&=\\sum_{z_n=1,2; z_{n+\\ell}=1,2}z_nz_{n+\\ell}P_\\text{st}(z_n,z_{n+\\ell}),\n\\end{align}\nusing Eqs.~(\\ref{P1}-\\ref{P4}) and the steady-state solution given by Eq.~\\eqref{Pst}. \n \nBased on Eqs.~(\\ref{eq_avz}-\\ref{eq_corrz}) and the general solution $\\vec{P}_{n+\\ell}=\\mathbb{W}^\\ell\\vec{P}_n$ of the recurrence relation, we can write the correlation function in terms of the eigenvalues of the matrix $\\mathbb{W}$ as:\n\\begin{equation}\\label{eq:Ck}\n C_\\ell=\\alpha \\lambda_1^\\ell+\\beta\\lambda_2^\\ell+(1-\\alpha-\\beta)\\lambda_3^\\ell,\n\\end{equation}\n(note the fulfilment of the condition $C_0=1$).\n\nUsing the definition given by Eq.(\\ref{eq:Ckdef}) and the previous relations, a long but straightforward calculation yields $C_1$ and $C_2$ in terms of the transition probabilities:\n\\begin{align}\\label{c1c21}\n C_1&=Z^{-1}(p_{21}q_{12}-q_{11}p_{22}),& Z=(q_{11}+p_{21})(q_{12}+p_{22})\\\\\n C_2&=Z^{-1}(p_{21}q_{21}-p_{22}q_{11}+(p_{12}-p_{21})(-p_{21}q_{22}+q_{11}(2p_{22}+q_{12}))\\label{c1c22}.\n\\end{align}\nSetting $\\ell=1,2$ in Eq.~\\eqref{eq:Ck} we can connect $\\alpha$ and $\\beta$ with the values of $C_1$ and $C_2$:\n\\begin{align}\\label{ab1}\n \\alpha=&\\frac{C_2-C_1(\\lambda_2+\\lambda_3)+\\lambda_2\\lambda_3}{(\\lambda_1-\\lambda_2)(\\lambda_1-\\lambda_3)},\\\\ \\beta=&\\frac{C_2-C_1(\\lambda_1+\\lambda_3)+\\lambda_1\\lambda_3}{(\\lambda_2-\\lambda_1)(\\lambda_2-\\lambda_3)}\\label{ab2}.\n\\end{align}\n\nIn practise, we proceed as follows: Given the series $\\{z_n\\}$ obtained from the lexical pattern analysis explained in the main text for $D=2$, we numerically compute the probabilities $p_{11},\\,p_{12},\\,p_{21},\\,p_{22}$ using Eq.~\\eqref{eq:pijdef}, i.e., counting the frequency with which a value $z_n=1$ follows a pair $z_{n-2}=i,z_{n-1}=j$. Once these values have been obtained, we compute the correlation function $C_\\ell$ using Eq.~\\eqref{eq:Ck} with values of $\\lambda_1,\\lambda_2,\\lambda_3$ following from Eq.~(\\ref{lambda3}) and values of $\\alpha,\\beta$ from Eqs.~(\\ref{c1c21},\\ref{c1c22},\\ref{ab1},\\ref{ab2}). We then compare the obtained values of $C_\\ell$ with the ones computed numerically from the original series.\n\nThe results for the analysis of the Bible in English, French, German and Spanish are displayed in Fig.~\\ref{fig_gen_markov}. It can be seen that the autocorrelation function $C_\\ell$ for $\\ell\\ge 0$ is rather well represented by the memory-2 Markov model introduced here\nand is consistent with the results shown in Fig.~9 of the main text. The agreement is much better for the French and Spanish versions of the Bible, while the English and, mostly, the German versions show some discrepancy for large values of $\\ell$. \n\nIf we had used a purely Markovian model of memory $m=1$, then the transition probabilities for the $z$ variables are\n\\begin{align}\n p_1&=P(z_{n+1}=1|z_n=1), \\quad q_1=P(z_{n+1}=2|z_n=1)=1-p_1,\\\\\n p_2&=P(z_{n+1}=1|z_n=2), \\quad q_2=P(z_{n+1}=2|z_n=2)=1-p_2.\n\\end{align}\nA standard analysis concludes that the correlation function decays exponentially, $C_\\ell=(\\lambda)^\\ell$, where $\\lambda=p_1-p_2$ is the eigenvalue different from $1$ of the transition probability matrix. This functional form is totally excluded from the numerical data.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth, clip]{Markov_generalized_comparison.png}\\\\\n\\includegraphics[width=0.95\\textwidth, clip]{Markov_generalized_comparison_II.png}\\end{center}\n \\caption{Autocorrelation function (ACF), defined in Eq.~\\eqref{eq:Ckdef}, of the series generated from the lexical pattern analysis of the Bible in four languages, as discussed in the main text for $D=2$. Blue dots with error bars show the numerical values while the red line is the theoretical expression given by Eq.~\\eqref{eq:Ck}, which is obtained after computing the transitions probabilities $p_{11}, p_{12}, p_{21}, p_{22}$ from the series. Four top panels show that an $m=2$ Markov chain correctly captures the original series correlations. Four bottom panels are zoomed versions that point to small discrepancies when the time lag $\\ell$ is large.}\n\\label{fig_gen_markov}\n\\end{figure}\n\n\n\\section{Shuffled sentences}\nThe shuffled realizations of Figs.~3 and 4 in the main paper\nand Fig.~S1 of this SI\nare obtained by randomly shuffling all the words in the original\ntext. A different shuffled realization shuffles the sentences\ninstead of the individual words. Remarkably, our results obtained\nfor shuffled sentences are the same than those obtained for the original sequences.\nFor definiteness, we select four languages and plot in Fig.~\\ref{fig_shuffdyn} both the original pattern dynamics for $D=2$ (blue dots),\nreproduced from Fig.~3 of the main text, and the ordinal\npattern when the sentences are shuffled (red dots).\nObviously, the dynamics do not agree because the relative frequencies\nare calculated over time windows and these windows contain texts\nwith totally different sentences in both cases. However,\nthe stationary values and their probability distributions\nare not \nmodified. This is shown in Fig.~\\ref{fig_shuffpat} for $D=4$,\nwhere one can note that there is exact match between\nthe original Bible (blue lines as replicated\nfrom Fig.~4 of the main text) and the Bible with shuffled sentences\n(red dots). Since the short memory encountered in our analysis is based\non these statistical distributions, we can safely conclude that\nour method detects short-ranged correlations\nthat typically occur inside a sentence.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth, clip]{ShuffledSentences-D2-Dynamic.png}\n\\end{center}\n \\caption{Dynamical behavior for $D=2$ as in Fig.~3 of the main text but comparing the original Bible (blue dots) and the Bible with shuffled sentences (red dots). In both cases, the curves differ from the case with shuffled words (the red curve in Fig.~3 of the main text), which corresponds to the trivial dynamics.}\n\\label{fig_shuffdyn}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth, clip]{ShuffledSentencesD4.png}\n\\end{center}\n \\caption{Pattern probability distributions for $D=4$ for the original Bible (blue lines as in Fig.~4 of the main text) and the Bible with shuffled sentences (red dots).}\n\\label{fig_shuffpat}\n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\input{intro}\n\n\\section{Background: Trade-off in Data Augmentation for Semantic Parsing}\n\\input{background}\n\n\\section{Hierarchical Neural Synthesis Pipeline}\n\\input{method}\n\n\n\n\\section{Experiments}\n\\input{experiments}\n\n\\section{Conclusion}\n In this work, we propose a data augmentation method for the cross-domain text-to-SQL task. Being a fully neural method, our approach avoids burdensome grammar engineering, while yielding more performance gain than previous data augmentation methods. While designed for semantic parsing, we believe our hierarchical synthesis approach coupled with a teacher-student framework could be applied more broadly in other problems such as machine reading comprehension, question answering over knowledge base, and visual question answering.  \n\n\n\\clearpage\n\n\n\n\\subsection{Data set and evaluation}\nTo evaluate {\\sc H-NeurSyn} we apply it to the popular and challenging\nSpider cross-domain text-to-SQL benchmark data set \\cite{yu2018spider}.\nThe data set contains $10,181$ questions and $5,693$ queries covering 200 databases in 138 domains. As with many previous works \\cite{shi2020learning,zhao2021gp}, on the test set, we just consider the Exact Match (EM) evaluation, which only requires models to predict the logical form with table and columns correctly, but not the values. On the dev set, in addition to Exact Match (EM), we can also evaluate Execution Accuracy (EX) by using the ground-truth values. EX is less noisy as it accepts equivalent queries that execute to the same results. \n\n\\subsection{Implementation details}\n\nFor both the teacher and student semantic parser models, we follow the model and hyperparameter setup in \\citet{xu2021optimizing}. \nThe model consists of a deep transformer plus relation-aware transformer module that encodes the question and schema jointly. Our backbone semantic parser model \\citet{xu2021optimizing} is comparable to the state-of-the-art method at the time of writing \nlisted in the Spider leaderboard. \n\nThe entity sampler is also an encoder-decoder neural net, where the encoder for schema representation is the same as the one used in the semantic parser.\nFor the decoder, we use a single layer LSTM with pointer net output over the available columns of the domain, i.e.\\ the column prediction part of the semantic parser used in \\citet{xu2021optimizing, xu2021turing}. The sampler is trained from scratch, with the same training hyperparameter as the parser.\n\nWe fine-tune a T5-base \\cite{raffel2020exploring} as the question generator without warm-up steps with \na learning rate $3e-4$, a batch size of $8$ for $3$ epochs, and gradient clipping with a threshold $1.0$. \n\nFor comparisons, we benchmark GAZP \\cite{zhong-etal-2020-grounded}  and L2S \\cite{wang-etal-2021-learning-synthesize}, both in their original setup, as well as using their augmented data with our (stronger) baseline semantic parser for a fairer comparison. For L2S, we use the augmented data set publicly released by the authors.\nFor GAZP, we use the publicly released code\\footnote{https:\/\/github.com\/vzhong\/gazp} to generate the augmented datasets. For the cycle consistency check required\nby GAZP, we implement our own DT-Fixup-based parser. We apply our deduplication filtering for both GAZP and L2S too. \n\n\\subsection{Main Results}\n\\paragraph{SOTA and Near-SOTA performance}\nFrom Table \\ref{tab:spider}, the first important observation is that our method, {\\sc H-NeurSyn}, achieves state-of-the-art dev accuracy and near state-of-the-art test accuracy. Because the Spider test set has only $2147$ examples, the gap of $0.1\\%$ and $.2\\%$ to \\citet{scholak2021picard} and \\citet{cao2021lgesql} only represent $2$ and $4$ extra mistakes by our method.\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{l l l l l}\n\\hline\n\n & \\multicolumn{2}{c}{\\textbf{Dev Set}} & \\multicolumn{2}{c}{\\textbf{Test Set}} \\\\\n\\bf Model & \\bf EM \\bf  & \\bf EX & \\bf EM   \\\\ \\hline\n\\citet{wang2019rat} & $69.7$   & - & $65.6$  \\\\\n\\citet{yu2020grappa} & $73.4$  &- & $69.6$ \\\\\n\\citet{shi2020learning} & $71.8$  &- & $69.7$  \\\\\n\\citet{zhao2021gp} & $72.8$  & -& $69.8$ \\\\\n\\citet{cao2021lgesql}  & $75.1$  & -& $\\mathbf{72.0}$ \\\\ \n\\citet{scholak2021picard} & $75.5$ & -& $71.9$  \\\\ \n\\hdashline\nL2S & $71.9$   & $72.5$&-   \\\\\nGAZP & $59.1$   & $59.2$  &  $53.3$  \\\\\n\\hline\nDT-Fixup  & $75.0$ & $74.6$ & ${70.9}$ \\\\ \n + L2S(train)  & $75.1$& $73.5$&-  \\\\\n + GAZP(dev)  & $74.8$& $74.0$ &- \\\\\n + GAZP(train+dev)  & $76.0$& $74.9$ &-  \\\\\n  + Ours(train)  & $76.4$& - &- \\\\\n + Ours(train+dev) & $\\mathbf{77.2}$ & $\\mathbf{76.1}$ & $71.8$\\\\\\hline\n\\end{tabular}\n\\caption{Accuracy on the Spider development and test sets, as compared to the other approaches at the top of the Spider leaderboard as of November $28$th, 2021. }  \n\\label{tab:spider}\n\\end{table}\n\n\\begin{table}[t!]\n\\centering\n\\small\n\\begin{tabular}{l c c c c c}\n\\hline\n\\bf \\# Train & $1694$ &  $2777$ & $1461$ & $1068$ & $7000$\\\\\n\\bf \\# Test & $248$ &  $446$ & $174$ & $166$ & $1034$\\\\ \\hline\n & \\bf Easy & \\bf Medium & \\bf Hard & \\bf Extra & \\bf All \\\\ \\hline\n\\bf DT-Fixup & $91.9$ & $80.9$ & $60.3$ & $48.8$ & $75.0$ \\\\ \n\\bf +Ours & $\\mathbf{92.7}$ & $\\mathbf{82.3}$ & $\\mathbf{65.5}$ & $\\mathbf{52.4}$ & $\\mathbf{77.2}$ \\\\\\hline\n\\hline\n\\end{tabular}\n\\caption{Accuracy on Spider by hardness levels.}\n\\label{tab:breakdown}\n\\end{table}\n\n\nOn the developement set, without zero-shot augmentation, we achieve $76.4$ (``Ours(train)\"), while $77.2$ with it (``Ours(train+dev)\"). Note that, our number on the test set could be even higher if we apply the zero-shot version because the test domain schema is available to the inference model. In practice, due to the compute-time limitation of the evaluation script, we cannot apply zero-shot augmentation. However, we believe it is relevant for bootstrapping semantic parsers in real-world applications, so the improvements on the developement set holds practical significance. \n\n\\paragraph{Improvement from data augmentation}\nAlthough the GAZP, L2S each produced improvements over their baseline parsers, with our stronger parser, they do not yield significant improvements. The largest is from GAZP applied in the zero-shot setting with augmentation from both training and dev sets, leading to an improvement of $1.0$ (EM) and $0.3$ (EX). On the other hand, using the same number of augmented points as GAZP, {\\sc H-NeurSyn} produced an improvement of $2.2$ (EM) and $1.5$ (EX).\n\n\nFurthermore, as shown by Table \\ref{tab:breakdown}, our augmentation method leads to gains across all difficulty levels, where the levels are defined by \\citet{yu2018spider}, with more boost in the hard and extra hard categories ($~6$-points on average).  \nThis is encouraging, as training data is particularly limited in those hard categories, while manual curation or simple heuristics are not feasible to generate more difficult examples. \n\n    \\begin{table*}[ht]\n        \\small\n        \\centering\n        \\begin{tabular}{p{0.16\\linewidth} p{0.25\\linewidth} p{0.5\\linewidth}}\n        \\toprule\n        \\textbf{Sampled entities} & \\textbf{Generated Question} & \\textbf{Self-labeld SQL} \\\\ \n        \\toprule\n        \\texttt{perpetrator. location}\n        & What are the different locations for perpetrators?\n         & \n        \\texttt{SELECT DISTINCT perpetrator.location FROM perpetrator}\\\\\n       \\cline{2-3}\n       & List the locations of perpetrators in ascending alphabetical order.\n       &\n       \\texttt{SELECT perpetrator.location FROM perpetrator ORDER BY perpetrator.location}\\\\ \n        \\cline{2-3}\n    \n        \\midrule\n        \\texttt{captain.name, captain.age}\n        & What are the names and ages of all captains?\n        & \n        \\texttt{SELECT captain.name, captain.age FROM captain}\\\\\n        \\cline{2-3}\n       & What is the name of the youngest captain?\n       &\n       \\texttt{SELECT captain.name FROM captain ORDER BY captain.age}\\\\ \n       \\bottomrule\n        \\end{tabular}\n        \\caption{Effect of beam search in the question generator}\n        \\label{tab:beam}\n    \\end{table*}\n\n\n\n\\subsection{Analysis of synthesized data}\n\\label{sec:sample}\n\nTo gain insights about why {\\sc H-NeurSyn}'s augmented data improves semantic parsing, let us inspect some statistics of the SQL sketches and the columns. \n\nFirst, we convert the SQL queries to  \"sketches\" by masking off the table and column names (values are already not being considered in the target SQL queries). Second, we use a handcrafted rule system to split complex queries into simpler parts, like the template system of \\cite{elgohary2020speak} or the shallow grammar of \\cite{xu2021turing}. \n\nThen treating the deconstructed SQL sketches as discrete categories, we can compute the normalized entropy of sketches for both the original and augmented datasets. Similarly, we can compute the same statistic for the table sets and columns sets occurring in the queries. The normalized entropy $\\tilde{H}_{P}$ of distribution $P$ is the ratio of its entropy over that of a uniform distribution $U$ over the same space, i.e.\\ $\\tilde{H}_{P} = H(P)\/H(U)$. \n\nFurthermore, we can evaluate the normalized mutual information of SQL sketches with the table or column entities, where the normalized mutual information is defined as\n$\\tilde{I}_{X:Y}= {2 I(X:Y) } \/ {(H(X) + H(Y))}$.\n\nThe normalized entropy gives a measure of diversity, whereas the normalized mutual information reflects statistical association. Both are standardized so that they can be averaged across the different databases on Spider, which is shown in Table~\\ref{tab:stats}. \nWe see that the augmentation improves the diversity of columns $\\tilde{H}_{Col}$ by a large margin, while slightly reducing the diversity of sketches $\\tilde{H}_{Sketch}$. More importantly, the normalized mutual information between sketch and entities are reduced ($\\tilde{I}_{DB:Sketch}$ and $\\tilde{I}_{Col:Sketch}$), suggesting that the augmented data potentially can help break some of the spurious correlation that occurs in the original dataset between the logical form and entity occurrences. \n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{l c c | c c}\n\\hline\n\\bf Model & \\bf Train & \\bf Train$_{Aug}$ & \\bf Dev & \\bf Dev$_{Aug}$\\\\\\hline\n\\# instances & 7000 &  \\textbf{44785} & 1007 & \\textbf{6736}\\\\\n\\# unique sketch & 267 & \\textbf{343} & 106 & \\textbf{166} \\\\ \n\\# unique col set & 1251 & \\textbf{11398} & 181 & \\textbf{1748} \\\\ \\hline\n$\\tilde{H}_{DB} \\uparrow$ & 0.961 &\\textbf{0.996} & 0.942 & \\textbf{0.995} \\\\ \n$\\tilde{H}_{Col} \\uparrow$ & 0.483 & \\textbf{0.962} & 0.512 & \\textbf{0.958} \\\\ \n$\\tilde{H}_{Sketch} \\uparrow$ & \\textbf{0.747} & 0.644 & \\textbf{0.820} & 0.702  \\\\ \n$\\tilde{I}_{DB:Sketch} \\downarrow$ & {0.290} & \\textbf{0.108} & {0.307} & \\textbf{0.108} \\\\ \n$\\tilde{I}_{Col:Sketch} \\downarrow$ & {0.510} & \\textbf{0.396} & {0.531} & \\textbf{0.438}  \\\\ \n\\hline\n\\end{tabular}\n\\caption{Statistics of datasets in Spider and Spider-aug.}\n\\label{tab:stats}\n\\end{table}\n\nQualitatively, Table \\ref{tab:beam} shows some samples from the question generator. Given a fixed entity set, there are various ways to generate questions with different logical forms. We find that in most cases, rather than paraphrasing, the generator learns to \neffectively sweep through the possible logical forms given the entities, which\nleads to a reduction of the correlation between entities and SQL sketches, as shown by $\\tilde{I}_{DB:Sketch}$ and $\\tilde{I}_{Col:Sketch}$ of Table~\\ref{tab:stats}.\n\n\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{p{0.65\\linewidth} p{0.2\\linewidth} }\n\\hline\n\\bf Model & \\bf Dev EM\\\\ \\hline\nBackbone (DT-Fixup w\/ small encoder)  & $72.7 \\pm 0.4 $\\\\\n\\hline\n \\makecell[l]{\\textbf{Ours} \\\\ 1. learnable(train+dev) \\\\ 2. $s_1$=80, $s_2$=20 \\\\ 3. $\\alpha_{train}$=0.3, $\\alpha_{dev}$=0.1 \\\\4. w\/ \\texttt{DEDUP} filtering} & \\bm{$75.2 \\pm 0.3$} \\\\\\hline\n\\hline\n\\it  Augmentation Baselines \\\\ \nL2S(train) w\/o \\texttt{DEDUP} filtering & $72.3 \\pm 0.2$  \\\\\nGAZP(dev) w\/o \\texttt{DEDUP} filtering & $73.1 \\pm 0.4$  \\\\\nL2S(train) & $72.5 \\pm 0.3$  \\\\\nGAZP(dev) & $73.2 \\pm 0.5$  \\\\\nGAZP(train+dev) & $74.2 \\pm 0.1$\\\\ \\hline\n\\it Sampling Strategies \\\\ \nOurs-random(train) & $73.0 \\pm 0.2$\\\\\nOurs-learnable(train) & $75.0 \\pm 0.3$  \\\\\nOurs-learnable(dev) & $74.4 \\pm 0.3$\\\\\\hline\n\\it Question Diversity  \\\\ \n\n$s_1$=80, $s_2$=1 & $74.0 \\pm 0.5$\\\\\n$s_1$=40, $s_2$=20 & $75.0 \\pm 0.4$\\\\\n$s_1$=80, $s_2$=40 & $75.1 \\pm 0.2$\\\\\n\n\\hline\n\\it Joint Training \\\\ \n\n$\\alpha_{train}$=0.3, $\\alpha_{dev}$=0.3 & $74.7 \\pm 0.1$\\\\\n$\\alpha_{train}$=0.1, $\\alpha_{dev}$=0.1 & $74.6 \\pm 0.0$\\\\\nPretrain-Finetune & $73.6\\pm 0.1$\\\\\nCombine & $73.6\\pm 0.4$\\\\\n\\hline\n\\it Filtering \\\\ \n\nw\/o \\texttt{DEDUP} filtering & $73.7 \\pm 0.6$  \\\\\nrandom down-sampling & $74.3 \\pm 0.3$  \\\\\n\\hline\n\n\\end{tabular}\n\\caption{Alation Study}\n\\label{tab:ablation}\n\n\\end{table}\n\n\n\\subsection{Ablation study}\n\nWe analyze to demonstrate the importance of each aspect of our approach in Table \\ref{tab:ablation}.\nIn all experiments, we use the smaller encoder due to computing limitations: RoBERTa-base with $8$ relation-aware transformer (RAT) layers \\cite{wang2019rat}, instead of RoBERTa-large with $24$ RAT layers. \n\nOn sampling strategy, we compare to uniform-randomly sampling \nentities from the schema, instead of the learned sampler. \nUnsurprisingly, the learned sampler is superior to random sampling, as it captures a more natural combination of entities. Table \\ref{tab:random-learnable} gives more insights about why learned sampling is crucial: it is often impossible to formulate a natural sentence that makes sense from randomly sampled entities, as shown in Table \\ref{tab:random-learnable}.\n\nThe hyper-parameters $s_1$ and $s_2$ control the diversity of entities $E$ and logics $L$. From Table \\ref{tab:ablation} we can see tuning them at a reasonable range does not affect the augmentation performance much. However, the performance drops dramatically when $s_2$=1 (i.e. no beam search for question generation). \n\nTuning the student training hyper-parameters $\\alpha_{train}$ and $\\alpha_{dev}$ moderately affects the performance, while the best performance is achieved when  $\\alpha_{train}$=0.3 and  $\\alpha_{dev}$=0.1. This is because the quality of augmented examples on the dev set databases is relatively lower than those on the training set databases due to sampling error and self-labeling error, as neither sampler nor parser is trained on the dev set domains. Besides, we also compare the training strategies (Pretrain-Finetune and Combine) employed by GAZP and L2S, and the results show neither of them works for our data augmentation method. Note that both the main results in Table \\ref{tab:spider} and ``Augmentation Baselines\" in Table \\ref{tab:ablation} confirm that our augmentation method itself is superior to GAZP and L2S even after controlling for other aspects of improvements. \n\n\n\n\n\\begin{table}[t]\n    \\small\n    \\centering\n    \\begin{tabular}{p{0.1\\linewidth} p{0.43\\linewidth} p{0.36\\linewidth}}\n    \\toprule\n    \\textbf{} & \\textbf{Sampled entities} & \\textbf{Generated questions} \\\\ \n    \\toprule\n    Random\n    & \\texttt{people.name, church.church_id, wedding.male_id,  church.organized_by}\n     & Show the names of people and the organized by the church they belong to .\n    \\\\ \\midrule\n    Learned\n    & \\texttt{people.people_id, people.name, people.is_male, wedding.male_id}\n    & \n    What are the names of people who are male and have never attended a wedding ?\n    \\\\ \\midrule\n    Learned\n    & \\texttt{church.church_id, church.name, wedding.church_id, wedding.year}\n    & \n    What is the name of the church that hosted the largest number of weddings in year 2010 ?\n    \\\\ \\bottomrule\n    \\end{tabular}\n    \\caption{Random v.s. learned entity sampler}\n    \\label{tab:random-learnable}\n\\end{table}\n\n\n\n\n\n\n\\subsection{Problem Setup}\nIn cross-domain Text-to-SQL semantic parsing, we want to parse a natural language question $q$ into its corresponding SQL query $r$, given the underlying domain $D$ (the database).\n\nWe are given a set of domains \n$\\mathcal{D}_{train}=\\{D_i=(\\mathcal{A}_d, {G}_d)\\}_{d=1}^{N_{train}}$, where each domain has a set of annotated pairs of natural language questions and SQL queries $\\mathcal{A}_d=\\{(q^d_i, r^d_i)\\}_{i=1}^{\\lvert \\mathcal{A}_d \\rvert}$, and a schema ${G}_d$.\n\nThe schema consists of entities $\\mathcal{E}_d=\\{e^d_k\\}_{k=1}^{\\lvert \\mathcal{E}_d \\rvert}$ belonging to the domain (represented by ``table.column\" for example), as well as relationships (links) $\\mathcal{L}_d = \\{(e^d_k, e^d_l, t)\\}$, where $t$ denotes the type of relationship, such as a foreign key or primary key. And each query $r^d_i$ contains one or more entities denoted by $\\pmb{e}^d_i$ which is a sequence of entities $e^d_k$'s concatenated in the order they appear in the query $r^d_i$. The corresponding question also implies the same set of entities, although possibly written differently. \n\nThe goal is to train on $\\mathcal{D}_{train}$ and generalize to questions on unseen domains $\\mathcal{D}_{test}$ given only their schema. \n\nWe first train a baseline semantic parser $P_{\\theta}^{\\text{teacher}}$ on the training set via maximum likelihood, following the setup of \\citet{xu2021optimizing}. Then we look to improve it further using data augmentation.\n\n\\subsection{Method Details}\n\n\n\\begin{algorithm}[t]\n\\caption{{\\sc H-NeurSyn}}\n\\begin{algorithmic}[1]\n    \\Require training domains $\\mathcal{D}_{train}$ and sample sizes $s_1$, $s_2$.\n    \\LineComment{Phase-I: traing synthesizer components}\n    \\State Train entity sampler $P_{\\phi}(\\pmb{e}|G)$ on $\\mathcal{D}_{train}$\n    \\State Finetune a T5 as question generator $P_{\\omega}(q|\\pmb{e})$ on entities-question pairs extracted from $\\mathcal{D}_{train}$\n    \\State Train a base parser $P_{\\theta}(r|q, G)$ on $\\mathcal{D}_{train}$\n    \\LineComment{Phase-II: synthesize}\n    \\State $D_{aug} \\leftarrow \\varnothing$\n    \\For {$G_d \\in D_{train}$}\n        \\State $\\{\\pmb{e}_j\\}_{j=1}^{s_1} \\sim p_\\phi(\\pmb{e}|G_d)$\n\n        \\For {$\\pmb{e}_j \\in \\{\\pmb{e}_j\\}_j$}\n            \\State $\\{q_l\\}_l \\!\\leftarrow\\! \\texttt{BEAM-SEARCH}(P_{\\omega}(.|\\pmb{e}_j))$  \n            \\State $\\{r_l\\}_l\\!\\leftarrow\\! \\texttt{PRED}(P_{\\theta},\\!\\{q_l\\}_l, G_d)$\n            \\State $\\{(q_l,\\!r_l)\\}_{l=1}^{s_2} \\!\\leftarrow\\! \\texttt{DEDUP}(\\{(q_l,\\! r_l)\\}_l, \\mathcal{A}_d)$\n            \\State $D_{aug} \\leftarrow D_{aug} \\cup \\{(q_l, r_l)\\}_{l=1}^{s_2})$\n\n        \\EndFor\n   \\EndFor\n\\State $\\texttt{return} \\;\\; D_{aug}$\n\\end{algorithmic}\n\\label{alg:highlevel}\n\\end{algorithm}\n\n\n\n\nAs described earlier, we hierarchically decompose the synthesis, starting with sampling entities given the domain first. \nOverall, for a given domain described by its schema $G$, to synthesize data, our method can be described as the following auto-regressive factorization:\n$P(q, r, \\pmb{e} | G) = P(\\pmb{e}|G) P(q|\\pmb{e}, G) P(r|q, \\pmb{e}, G)$, from which we can sample $(q, r, \\pmb{e})$ and keep only the $(q, r)$-pairs for data augmentation. For brevity, we dropped the domain index $d$ henceforth.\n\nFurthermore, we make two conditional independence assumptions to simplify the factorization: $P(q|\\pmb{e}, G) = P(q|\\pmb{e})$ and $P(r|q, \\pmb{e}, G)=P(r|q, G)$. \n\nThe first independence is a good approximation because we can include the domain name in the entity sequence to inform the context of the question. Together with the sequence of table and column names being sampled for the question $q$, there is little additional information in the schema $G$ that is relevant to the $q$. This allows us to fine-tune a pre-trained seq2seq model $P_{\\omega}$ for $P(q|\\pmb{e})$. \n\nThe second assumption is reasonable as the entity information is expected to be captured explicitly in $q$ or implicitly via the interplay of $q$ and $G$ together. This allows us to use the baseline teacher parser $P_{\\theta}$ as a proxy for $P(r|q, G)$. Hence, overall our hierarchical synthesis model is $P(q, r, \\pmb{e} | G) \\approx P_{\\phi, \\omega, \\theta} (q, r, \\pmb{e} | G) = P_{\\phi}(\\pmb{e}|G) P_{\\omega}(q|\\pmb{e}) P_{\\theta}(r|q, G)$.\n\nOur synthesis procedure {\\sc H-NeurSyn} is detailed in Algorithm~\\ref{alg:highlevel}. The process starts by first training each of $P_{\\phi}(\\pmb{e}|G), P_{\\omega}(q|\\pmb{e}), P_{\\theta}(r|q, G)$. Then generate in order $G \\rightarrow \\pmb{e} \\rightarrow q \\rightarrow r$. \nFor each entity set, we perform beam search to get a large number of questions (line $9$). For each of those questions, we keep only the top-$1$ scoring prediction from beam-search that can successfully execute ($\\texttt{PRED}$ line $10$).\nAfterward, the examples are deduplicated ($\\texttt{DEDUB}$) both within the sampled set as well as with respect to the training data. Finally, for any questions $q$'s with the same logical form $r$, we keep only one question, i.e.\\ removing paraphrases via $\\texttt{NO-PARA}$. \n\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{p{0.52\\linewidth} p{0.4\\linewidth}}\n    \\toprule\n   \\textbf{Input} & \\textbf{Output} \\\\ \n    \\toprule\n   \\texttt{department management : head name text | head age number | head born state text}\n     & List the name, born state and age of the heads of departments ordered by age.\n    \\\\ \\midrule\n    \\texttt{culture company : movie year number | movie director text}\n    & \n    Which directors had a movie in either 1999 or 2000?\n    \\\\ \\bottomrule\n    \\end{tabular}\n    \\caption{Input and Output of the Generator}\n    \\label{tab:ent_q}\n\\end{table*}\n\n\n\n\n\\paragraph{Entity Sampler}\n\n\nThe entity sampler is an autoregressive model given the domain schema: \n\\begin{equation}\n    p_\\phi(\\pmb{e}|G) = \\prod_{t}p_\\phi(e_t| e_{<t}, G) \n\\end{equation}%\nwhere $\\pmb{e} = [\\text{<DB_NAME>},\\!e_1\\!\\ldots\\!e_t\\!\\ldots\\!,\\text{<EOS>}]$, with $e_1\\!\\ldots\\!e_t\\!\\ldots$ denoteing the sequence of entities as they appear in the SQL query; prefixed by the domain\/database name $e_0=\\text{<DB_NAME>}$, and with termination marked by the special $\\text{<EOS>}$ token. We model the entities as a sequence rather than set because the order they appear in the query sometimes carries some information about the input question. Given schema $G$, the domain name is deterministically given, so $P(e_0|G)$ is $1$ if $e_0$ is the databse name corresponding to $G$, and $0$ otherwise.\n\nThe model is trained via standard maximum likelihood with teacher-forcing. The training examples are entity sequences extracted from ground-truth training queries on each domain, with repeating entities in the same query deduplicated.  \n\nAs the model is conditioned on the domain schema $G$, and the parameters $\\phi$ are re-used across different domains, \nthe sampler can learn to sample entities on both seen and unseen domains. \n\n\n\\paragraph{Question Generator}\nFor the question generator, we fine-tune a T5 on a small dataset consisting of entity sequences and questions from the training set. \nWe use the human-friendly version for the entity names (with underscores removed). See Table \\ref{tab:ent_q} for examples of input and outputs.   \n\n\n\\subsection{Training with augmented data}\n\nTo combine training on the augmented data and the original training set, we initialize a new parser (the student) and \ntrain it with the following multi-task loss with \nhyper-parameter $\\alpha$:\n$\\mathcal{L}_{\\text{MLE}}(\\theta) = \\mathcal{L}_{\\text{MLE}}^{train}(\\theta)  + \\alpha \\times \\mathcal{L}_{\\text{MLE}}^{aug}(\\theta)$.  \nAdditionally, for each augmented sample, we prepend a special $[\\text{AUG}]$ token, inspired by tagged back-translation \\cite{caswell2019tagged}.\n\n\\paragraph{Zero-shot augmentation}\nFurthermore, because all component models of {\\sc H-NeurSyn} can generalize across domains, the overall data augmentation can also be applied in the zero-shot setting: given an unseen domain with only its schema $G_{\\text{new}}$ available, not the questions, we can synthesis new examples by following the {\\sc H-NeurSyn} pipeline.\nIn this transductive case, \nwe can optionally modify the \nloss function by weighting the test data term differently with $\\alpha_{new}$ from the training data term with $\\alpha_{train}$. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nTHE AENEID\n\nPUBLIUS VERGILIUS MARO was born in 70 BC near Mantua in the north of Italy, where his parents owned a farm. He had a good education and went to perfect it in Rome. There he came under the influence of Epicureanism and later joined an Epicurean colony on the Gulf of Naples where he was based for the rest of his life. In 42 BC he began to write the _Eclogues_ , which he completed in 37 BC, the year in which he accompanied Horace to Brindisi. The _Georgics_ were finished in 29 BC, and he devoted the rest of his life to the composition of the _Aeneid_. In his last year he started on a journey to Greece; meeting Augustus at Athens, he decided to travel back with him but he fell ill at Megara. He died in 19 BC on reaching Brindisi.\n\nDAVID WEST is an Aberdonian, educated at the local grammar school and university and then at Sidney Sussex College, Cambridge. He has taught in the universities of Sheffield and Edinburgh and was Professor of Latin at Newcastle upon Tyne from 1969 to 1992. He notes that no such career would now be possible, since the departments of Classics at Aberdeen and Sheffield are now both defunct. His publications include _Reading Horace_ (1967), _The Imagery and Poetry of Lucretius_ (1969) and _Horace_ : _The Complete Odes and Epodes_ (1997). He has also produced editions with text, translation and commentary of the first three books of Horace's Odes (1995, 1998 and 2002). He is now working on a commentary on Shakespeare's Sonnets.\n\nVIRGIL\n\n# The Aeneid\n\n_Translated and with an Introduction by_  \nDAVID WEST\n\nREVISED EDITION\n\nPENGUIN BOOKS\n\nPENGUIN BOOKS\n\nPublished by the Penguin Group  \nPenguin Books Ltd, 80 Strand, London WC2R 0RL, England  \nPenguin Putnam Inc., 375 Hudson Street, New York, New York 10014, USA  \nPenguin Books Australia Ltd, 250 Camberwell Road, Camberwell, Victoria 3124, Australia  \nPenguin Books Canada Ltd, 10 Alcorn Avenue, Toronto, Ontario, Canada M4v 3B2  \nPenguin Books India (P) Ltd, 11, Community Centre, Panchsheel Park, New Delhi \u2013 110 017, India  \nPenguin Books (NZ) Ltd, Cnr Rosedale and Airborne Roads, Albany, Auckland, New Zealand  \nPenguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank 2196, South Africa\n\nEISBN: 978\u20130\u2013140\u201344932\u20137\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nwww.penguin.com\n\nFirst published 1990  \nPublished in Penguin Classics 1991  \nReissued with a revised Introduction and new Further Reading 2003  \n1\n\nTranslation and Introduction copyright \u00a9 David West, 1990, 2003\n\nAll rights reserved\n\nThe moral right of the translator has been asserted\n\nExcept in the United States of America, this book is sold subject   \nto the condition that it shall not, by way of trade or otherwise, be lent,   \nre-sold, hired out, or otherwise circulated without the publisher's   \nprior consent in any form of binding or cover other than that in   \nwhich it is published and without a similar condition including this   \ncondition being imposed on the subsequent purchaser\n\nEISBN: 978\u20130\u2013140\u201344932\u20137\n\n## Contents\n\nAcknowledgements\n\nIntroduction\n\nFurther Reading\n\nNote on the Translation\n\nTHE AENEID\n\nAppendix I: The Parade of Future Romans in the Underworld (Book 6, lines 756\u2013892)\n\nAppendix II: The Shield of Aeneas (Book 8, lines 626\u2013728)\n\nAppendix III: Genealogical Trees\n\nThe Julian Family\n\nThe House of Priam\n\nThe House of Anchises\n\nMaps, Gazetteer and Select Index\n\nThe Voyages of Aeneas\n\nRome during the Reign of Augustus\n\nGazetteer\n\nSelect Index\n\n## Acknowledgements\n\nThis translation is of course based on such of the vast scholarly literature as I have been able to read. Previous translations have been plundered. Standard commentaries have been consulted, notably R. G. Austin on Books 1, 2, 4 and 6; R. D. Williams on 3 and 5; C. J. Fordyce on 7 and 8. Particularly valuable have been E. Norden on 6, P. T. Eden on 8 and Stephen Harrison on 10. The _Aeneidea_ of James Henry have been an inspiration.\n\nRosemary Burton and E. L. Harrison criticized the whole translation. Stephen Harrison, James Morwood and Nicholas Horsfall commented on whole books or extended passages. Pamela West, Janet Watson and Jane Curran were shrewd and generous consultants. To all of these I owe a debt that cannot be paid, as I do to my wonderful colleagues in the best of all imaginable university departments of Classics.\n\n_To the great dead who will not die_\n\n## Introduction\n\n### A POEM FOR OUR TIME\n\nThe _Aeneid_ is the story of a man who lived three thousand years ago in the city of Troy in the north-west tip of Asia Minor. What has that to do with us?\n\nTroy was besieged and sacked by the Greeks. After a series of disasters Aeneas met and loved a woman, Dido, queen of Carthage, but obeyed the call of duty to his people and his gods and left her to her death. Then, after long years of wandering, he reached Italy, fought a bitter war against the peoples of Latium and in the end formed an alliance with them which enabled him to found his city of Lavinium. From these beginnings, 333 years later, in 753 BC, the city of Rome was to be founded. The Romans had arrived in Italy.\n\nThe _Aeneid_ is still read and still resonates because it is a great poem. Part of its relevance to us is that it is the story of a human being who knew defeat and dispossession, love and the loss of love, whose life was ruled by his sense of duty to his gods, his people and his family, particularly to his beloved son Ascanius. But it was a hard duty and he sometimes wearied of it. He knew about war and hated the waste and ugliness of it, but fought, when he had to fight, with hatred and passion. After three millennia, the world is still full of such people. While we are of them and feel for them we shall find something in the _Aeneid_. The gods have changed, but for human beings there is not much difference:\n\nPitiless Mars was now dealing grief and death to both sides with impartial hand. Victors and vanquished killed and were killed and neither side thought of flight. In the halls of Jupiter the gods pitied the futile anger of the two armies and grieved that men had so much suffering...\n\n10.755\u20139\n\nBut the _Aeneid_ is not simply a contemplation of the general human predicament. It is also full of individual human beings behaving as human beings still do. Take the charm and humour of Dido putting the Trojans at their ease at 1.562\u201378; the grief of Andromache when she meets the Trojan youth who is the same age as her son Astyanax would have been if he had been allowed to live \u2013 we do not need to be told that Astyanax is the name on the second altar at 3.305; the cunning of Acestes and Aeneas as they shame the great old champion back into the ring at 5.389\u2013408; the childish joke of Iulus at 7.116 and its momentous interpretation; the aged hero feasting his eyes on his old friend's son at 8.152 or realizing at 8.560 that he can do nothing now except talk; the native's abuse of the foreigners from 9.598; the lying harridans at the beginning of Book 10 or the death of Mezentius and his horse from 10.858; the growling of Aeneas and the fussing and fumbling of the doctor as he plies his mute, inglorious art from 12.387.\n\nThe _Aeneid_ presents a heroic view of the life of man in all its splendour and anguish, but it is also full of just observation of the details of individual behaviour. It is not yet out of date.\n\n### THE _AENEID_ IN ITS OWN TIME\n\nVirgil was born seventy years before Christ. In 44 BC, after a century of civil war and disorder, Julius Caesar was assassinated by Brutus and Cassius in the name of liberty. His heir was his nineteen-year-old grand-nephew and adopted son, Octavian, astute, ruthless and determined. In 42 BC at Philippi Brutus and Cassius were defeated and the fortunes of Virgil were at their lowest ebb. His family estates at Mantua were confiscated by the victors to provide land for their soldiers to settle on. But he won the patronage of Maecenas, one of the two chief aides of Octavian, and published his pastoral _Eclogues_ in 37 BC. In 29 BC, after Octavian had made himself master of the known world by defeating Antony and Cleopatra at Actium, Virgil finished what John Dryden called 'the best poem of the best poet', the _Georgics_ , on the agriculture of Italy. Throughout the twenties Virgil was at work on his _Aeneid_ , a poem in imitation of Homer's _Iliad_ and _Odyssey_ and in praise of Augustus, the name Octavian had taken on 16 January 27 BC. Virgil died before finishing it, on his way back from Athens with Augustus in 19 BC. To qualify for membership of the Senate, a Roman had to be extremely wealthy. When Virgil died, he owned property ten times that requirement. He left instructions that the _Aeneid_ was to be burned. These instructions were countermanded by Augustus.\n\nIt is therefore clear that Virgil wrote and wrote acceptably in praise of his patron, the ruler of Rome.\n\nIt would be easy to despise or dislike the poem for that. But wrong, for the following reasons:\n\n**(1)** Rome had endured a century of violence, discord, corruption and insecurity of life and property. Augustus, after intense effort and suffering, notably in his disastrous campaign in Sicily in 37 BC, by his victory at Actium promised peace, order, prosperity and moral regeneration. He even, according to Suetonius ( _Life of Augustus_ 89), fostered the talents of his generation in every possible way. It was the promise of a Golden Age, and in this euphoria Virgil and his friend Horace, another client of Maecenas and Augustus, wrote their great patriotic poems. At that time it was not foolish to hope and to believe.\n\n**(2)** Although Virgil wrote in praise of Augustus and the ideal of empire, he was no Chauvin. He loved country people and country ways, their traditions and their stubborn independence. He responded to human love, between man and woman, between father and son, between men and their homes (consider only 6.450 ff., 12.435 ff., 10.779 ff.), and he knew that empire had to be bought with the coin of human suffering and deprivation. He also knew the other side \u2013 the hard work and danger, the dedication and sacrifice which empire demanded of those who had made it and who maintained it, notably Augustus. Virgil does not solve the problems inherent in all this. He does not even pose them. The _Aeneid_ is a story. But behind that story we have all the issues which would have moved a contemporary Roman, and may still move us.\n\n**(3)** Praise is one thing. Flattery is another, and the _Aeneid_ is not flattery. The action of the epic is set a thousand years before Augustus and it praises him in two ways: first, by telling the story of his great ancestor, the first founder of Rome, in such a way as resembles the story of Augustus himself, its third founder. The resemblances are not pointed out. The reader is left to observe and ponder them for himself if he wishes. The second mode of praise is direct allusion to Augustus in prophecies and visions, notably near the beginning and end of the poem, in the descent of Aeneas to consult his father in the Underworld at the end of Book 6, and on the great shield of Aeneas at the end of Book 8.\n\nThe _Aeneid_ is, among other things, a search for a vision of peace and order for Rome and for humanity. To see its outlines through the mists of time nothing is more helpful than the family tree of the Julians on page 295. Allusions to these names in the _Aeneid_ are often to be heard as praise of Augustus, the contemporary Julian.\n\n### THE _AENEID_ BOOK BY BOOK\n\n#### Background\n\n_Paris, son of Priam, king of Troy, judged Venus to be more beautiful than Juno and Pallas Athene, and claimed his reward, Helen, wife of Menelaus, king of Sparta. The Greeks gathered an army and sacked the city of Troy after a ten years' siege. Aeneas escaped with his father, Anchises, and his son, Ascanius Iulus. Driven by the jealous hatred of Juno, he wandered across the Mediterranean for six years, trying to found a new city. Atthe opening of the poem, his father has just died in Sicily and Aeneas is sailing for Italy._\n\n### BOOK 1  \nSTORM AND BANQUET\n\n_Juno sends a fearful storm which wrecks the Trojan ships on the coast of Libya, near Carthage. There the Trojans are hospitably received by Dido, queen of Carthage. Venus, mother of Aeneas, anxious for the safety of her son, contrives that Dido should fall in love with him._\n\n#### _Virgil and Homer_\n\nThe poems that set the benchmark for all future epics were Homer's _Iliad_ , the story of Achilles at the siege of Troy, and his _Odyssey_ , the story of Odysseus' wanderings and homecoming from Troy to his native Ithaca. The first words of the _Aeneid_ are 'I sing of arms and of the man...' ( _arma virumque cano_ ). Since the _Iliad_ is the epic of war, and the first word in the _Odyssey_ is 'man', Virgil has begun by announcing that he is writing an epic in the Homeric style. The 'man' is Aeneas, the legendary first founder of Rome, who escaped from the sack of Troy and wandered the seas for six years looking for a place to found a new city. The 'arms' are the battles he fought at the fall of Troy as described in the second book of the _Aeneid_ and also, in the last four books, the war he fought against the Latin peoples as he tried to establish his city in Italy.\n\n#### _Virgil and Augustus_\n\nAeneas was victorious. He founded his city of Lavinium and ruled it for three years. After thirty years his son Ascanius _Iulus_ , moved from Lavinium to Alba Longa, where the Alban kings ruled for three hundred years, until the birth of Romulus and Remus. It was Romulus, son of the priestess _Ilia_ and Mars, who founded the city of Rome and gave it its name in 753 BC, according to the traditional dating. When Virgil was writing the _Aeneid_ in the twenties BC, Rome was ruled by Augustus, the adopted son of _Julius_ Caesar. The _Julian_ family, therefore, still ruled Rome, and in describing how Aeneas, father of the _Julians_ , suffered in founding his city, Virgil is paying tribute to the contemporary Julian in his palace on the Palatine Hill in Rome.\n\n#### _Aeneas and the Gods_\n\nFor six years Aeneas and the remnants of his people were driven across the Mediterranean by the anger of the goddess Juno, and yet as early as the tenth line of the poem we learn that Aeneas had done no wrong, but on the contrary was famous for his piety. This introduces the divine machinery which so enriches the poem. At a lowly level it unfolds the comedy of manners of the divine family. But more seriously, it raises insoluble problems about the relationship between man and god, between Juno, queen of the gods, and Jupiter their king, and between ineluctable Fate and the will of omnipotent Jupiter; and, crucially, about the function of the will of human beings whom the gods seem to control and, when they wish, destroy. 'Can there be so much anger in the hearts of the heavenly gods?' asks Virgil in the eleventh line of the _Aeneid_ , and the poem is, among other things, a meditation on that problem, which, in one formulation or another, is still with us.\n\nWhen the narrative begins after a short preamble, the Trojan ships are caught in a storm and driven ashore on the Syrtes. These were sandbanks on the north coast of Africa, east of the new city of Carthage, just founded by Phoenicians who had come from Sidon on the eastern seaboard of the Mediterranean. Venus sees this and with tears flooding her eyes pleads with her father, Jupiter, to put an end to her son's suffering and to honour his promise that Aeneas would live to found the Roman race. Jupiter smiles at his daughter and assures her that his will has not changed. Romulus, son of Ilia (and therefore a Julian), will indeed found the city of Rome and give his name to his people, on whom will be imposed no limits of time or space. And in time to come another Julian will conquer the world and give it peace. Praise of Augustus thus appears in a prophecy of the king of the gods, uttered a millennium before Augustus was born.\n\n#### _Aeneas Meets Dido_\n\nVenus descends in disguise, teases her son, wraps him in a mist of invisibility and guides him to Carthage. There he gazes at the new temple of Juno with its representations of the Trojan War including a depiction of himself in the confusion of battle, and weeps to see that all men knew what Troy had suffered. 'Here too,' he says, 'there are tears for suffering and men's hearts are touched by what man has to bear' (462) ( _sunt lacrimae rerum et mentem mortalia tangunt_ ). Dido then arrives and Aeneas sees the comrades whom he had assumed to be drowned coming to ask her assistance. When she responds graciously, Venus dissolves the cloud in which she has concealed Aeneas, and Dido and Aeneas meet.\n\nThe book ends with a description of the banquet which Dido gives in honour of her Trojan guests. But Venus suspects that Juno, the goddess of Carthage, may do her son some mischief while he is in the city. To protect him she decides to make Dido love him, and effects this by sending her rascally young son Cupid to drive her insane with love. As the men drink their wine, the doomed queen begs Aeneas to tell the story of the fall and sack of his city.\n\n#### _The_ Aeneid _and Carthage_\n\nThe _Aeneid_ tells the tale of a legendary hero, but it also casts a long shadow over a thousand years of Roman history. Rome's greatest danger had been the three Punic Wars fought against Carthage from 264 to 146 BC, in the second of which Hannibal had destroyed Roman armies and overrun the Italian peninsula. The end came in 146 BC when Carthage was razed to the ground and ploughed with salt. The first and fourth books of the _Aeneid_ contain pre-echoes of that traumatic conflict. We sense the dramatic irony as Aeneas describes in such detail the building of Carthage \u2013 ' _Their_ walls are already rising!' he says enviously (437). We know that his Romans were to destroy them. When Aeneas offers Dido his heartfelt gratitude and promises that she will be praised for all time in every land to which he is called, we know that his descendants will destroy, not praise, her descendants. When she prays that her people should always remember the day of the banquet, we know how they will remember it, and as she invokes kindly Juno, the goddess of marriage and of Carthage, we know that the goddess of Carthage will use a false marriage to destroy its queen.\n\n### BOOK 2  \nTHE FALL OF TROY\n\n_This book takes the form of a flashback, as Aeneas tells the banqueters the story of the fall of Troy. The Greeks had erected a huge wooden horse and persuaded the Trojans to drag it into the city. In the dead of night Greek soldiers pour from the horse and open the gates to their comrades. The Trojans put up a fierce but hopeless resistance, and Aeneas escapes from the city with his father and his son._\n\n#### _The Deception of the Trojans_\n\nAfter ten years of hard fighting around Troy, the Greeks act as though they are giving up the siege. They build a huge wooden horse outside the walls, fill it with their best soldiers and sail away, pretending that it is an offering for their safe return to Greece. But they go only as far as the offshore island of Tenedos and leave Sinon behind to persuade the Trojans to take the horse into the city. Laocoon, the priest of Neptune, warns the Trojans not to trust the Greeks. 'I am afraid of Greeks,' he says, 'even when they bear gifts' (49). But Sinon appears and the Trojans are persuaded. This speech of Sinon's is at once an expos\u00e9 of the decadence of contemporary Greeks in Roman eyes, and a satire on the corruption of ancient rhetoric, a satire sharpened by several interjections by a naive and gullible audience. (The nearest thing in English is Antony's funeral oration in Shakespeare's _Julius Caesar_ with the inane interjections of the crowd.) Once again Laocoon protests, but the gods are against the Trojans. Two serpents come out of the sea and kill the priest of Neptune and his two sons. The Trojans breach their walls and drag in the horse.\n\n#### _The Courage of Aeneas_\n\nIn all of this book Virgil has a difficulty. His hero is the leading Trojan warrior and he has survived the sack of his city. Since Aeneas himself is speaking, he cannot blatantly advertise his own courage, but at every point in his speech Virgil is careful to give him words which leave no possibility that he could be thought guilty of cowardice or even of misjudgement. The first example of this is that Aeneas is not said to be one of the Trojans who ignored the warnings of Laocoon or were duped by Sinon. He does not enter the stage until a third of the way through the book, when Hector, appearing to him as he sleeps, tells him that Troy is doomed and orders him, as only Hector could, to abandon Troy and carry its gods to a new city across the sea. Ignoring these orders, Aeneas plunges into a hopeless battle where the only safety for the defeated is to hope for none. A few Trojans gather around him and they try the stratagem of carrying Greek shields emblazoned with Greek insignia. But although this wins them their only moment of success, the leader in this dubious tactic is not Aeneas, not even a Trojan, but Coroebus, who had arrived in Troy only a few days before. Inevitably their ruse is detected and they are overwhelmed. Aeneas is swept by the tide of battle to the palace of King Priam, the last centre of resistance. Here he joins the few surviving Trojans on the roof in levering down a tower, and rolling beams and gilded ceilings down on the heads of the Greeks. From there he sees Priam's wounded son, Polites, come rushing into the palace pursued by Pyrrhus and die at his father's feet. Aged as he is, Priam challenges Pyrrhus and is killed. Here we might have asked why Aeneas saw this and lived to tell the tale. We might have asked why Aeneas did not come down off the roof and try to avenge his king. Virgil has forestalled that thinking by the very next words of Aeneas: 'There came into my mind the image of my own dear father, as I looked at the king who was his equal in age breathing out his life with that cruel wound. There came into my mind also my wife Creusa...and the fate of young Iulus' (560\u201363). His divine mother now strips the mortal mist from his eyes and shows him a fearful vision of the Olympian gods tearing his city apart. Resistance now would be absurd. Venus escorts him to his home and he asks his father to leave Troy with him. Anchises refuses. In despair, Aeneas puts on his armour again and is rushing out to die in battle when fire is suddenly seen playing around Iulus' head. As _paterfamilias_ , father and priest of the family, Anchises prays to the gods for confirmation of the portent, and they see a star falling from the sky and ploughing its fiery path on Mount Ida. Anchises accepts the will of the gods and agrees to leave the city.\n\nAt this moment, the beginning of the history of Rome, Aeneas lifts his father up on his shoulders, takes his son in his left hand and his sword in his right, and with Creusa walking behind he passes through the burning city, starting at every breath of wind. When they gather with a few other fugitives outside the walls there comes what for Aeneas was the cruellest thing he saw in all the sack of the city. Creusa is lost. He girds on his armour and rushes back into the captured city calling out her name at the top of his voice. Creusa appears to him and assures him that it is not the will of the gods that she should stay with him. She has no part to play in the great future that lies before him. Aeneas is to go with her blessing and never fail in his love for their son.\n\nAeneas has done all that a man could do. He goes back to the tattered remains of the people of Troy, hoists his father on to his shoulders and leads the way into the mountains.\n\n### BOOK 3  \nTHE WANDERINGS\n\n_The flashback continues as Aeneas now gives an account of the wanderings of the Trojans after the fall of their city. After six years of hardship and failure, guided and misguided by prophecies and dreams, they arrive at Epirus in north-west Greece and are welcomed by another group of Trojan refugees, the priest-king Helenus and his wife, Andromache, once the wife of Hector. They had built a small-scale replica of Troy, but that was never going to be the solution for Aeneas, whosedestiny was to found a great new city. Aeneas and his little fleet set sail again, and as they approach Sicily they follow the directions of Helenus and veer away south to circumnavigate it rather than go through the strait guarded by Scylla and Charybdis. At last they put in at Drepanum on the north-west tip of the island, where Anchises dies. So, at the banquet given by Dido, Aeneas ends his story of the fall of Troy._\n\n### BOOK 4  \nDIDO\n\n_Dido now loves Aeneas and Juno arranges a kind of marriage in order to keep him with Dido and prevent him from founding the city which was fated to destroy her beloved Carthage. Jupiter reminds Aeneas of his destiny and orders him to leave Dido. She senses that he is going to abandon her and builds a great pyre, ostensibly to cure herself of love by burning the relics of Aeneas' stay. She curses Aeneas, calls upon her Carthaginians to wage eternal war against his people and dies in the flames._\n\n#### _Dido's Guilt?_\n\nThis book has gripped the imagination of readers for two millennia as a love story and as such it needs little comment. Part of its power may come from the eternal questions it raises and does not answer: the suffering of the innocent and the deceived, the conflict between love and duty, and the relationship between free will and irresistible fate.\n\nThe case against Dido could not be put more harshly than she puts it herself in her first speech and at line 552. When her husband died, she swore an oath that she would never love another man, and broke it to love Aeneas. Against that self-condemnation a substantial defence could be erected. Would it not be inhuman to hold a wife to such an oath taken in the moment of bereavement? It would certainly be harsh to condemn her to death for breaking it. Would any widow be condemned for marrying again? Certainly not in Virgil's Rome. This case can be supported by the personal and political arguments in favour of marriage put so persuasively by Dido's own sister.\n\nBut the clinching consideration is probably the unscrupulous cynicism of the two goddesses who engineer Dido's destruction for their own ends. To protect her son Aeneas, Venus has already driven Dido into madness. Now, to block his destiny to found a city, Juno proposes that Aeneas should settle in Carthage as Dido's husband. Venus, the daughter of Jupiter, has already been told by Jupiter himself that all this is totally contrary to his will, but she dissembles and urges Juno, the wife of Jupiter, to go and put this proposal to her husband. The two shrews play out their charade, each pursuing her own ends. Juno sets up a false marriage with herself as matron of honour, nymphs howling the wedding hymn and the fires of heaven's lightning instead of marriage torches. The powerless human being is crushed between two goddesses.\n\nThis is to read the interview between them as a comedy of manners, a family squabble in Olympus. But the divine machinery allows us to hold in our minds a different view of Dido's motivation. The quarrel between the goddesses could be seen as a dramatization of her emotions, the internal turmoil between love for Aeneas, longing for marriage, loyalty to her dead husband and duty to the city of which she is queen.\n\nBe that as it may, the case against her is not strong. We are left bewildered and Virgil means us to be. At line 172 he says explicitly that she is guilty, she 'called it marriage, using the word to cover her guilt'. On the other hand Juno, showing consideration at last, cuts short Dido's death agony because her death is undeserved. Virgil knows better than to propose solutions to problems that can never be solved.\n\n#### _Aeneas' Love_\n\nAeneas loved Dido. We have this from Virgil after each of her first two appeals to him. But when Jupiter sends his messenger, Aeneas instantly decides to leave her. Once again the divine machinery provides double motivation. We have heard the voice of Jupiter in all his majesty and seen the brilliant flight of Mercury. At another level we could sense this as a dramatization of a sudden victory of duty over desire in Aeneas' heart. Modern susceptibilities are offended, not least by his decision not to tell Dido \u2013 yet. This is a shrewd observation by Virgil of the sort of thing men do, and may well increase our sympathy for Dido. Aeneas is condemned also for the cold formality of his response to Dido's appeals. On this count, however, it is more difficult to fault him. Her speeches are passionate, yet full of tight logic. At their first meeting after Dido divines that he is going to leave her, she hurls argument after argument. Given that he has taken an irreversible decision to leave her, he answers the points to which answer is possible in the best imaginable way. It all comes down to his statement that it is not by his will that he goes to Italy. Modern views of his behaviour tend to be severe. But it does not make sense that Aeneas, founder of the Roman race and ancestor of Augustus, should behave contemptibly in this Roman epic written by Virgil in praise of his patron. True, Aeneas' decision not to tell Dido the truth immediately, shows him in a moment of weakness, and his replies to her are cold and feeble. But Aeneas is the hero of the poem, and his weakness and misery in this book are a measure of Virgil's human understanding, not a demolition of the character of the hero of his epic.\n\nThese are the problems that linger after a reading of this book. The _Aeneid_ would be a weaker poem if they could be solved. Dido's fault, if fault there was, did not merit the punishment she received. Why then did she receive it? Aeneas put duty before love at the behest of the gods, and Dido and others have despised him for it. Was he then despicable? The goddesses are spiteful and heartless, but can we not imagine that Dido would have behaved as she did in a godless world, and that Aeneas would have left her even if Mercury had never swooped down from Mount Atlas to a roof in Carthage? All these questions are set in the context of Roman history. In one of Dido's last speeches, for instance, she prophesies the Punic Wars and Hannibal's invasion of Italy although she could not know the name of the avenger who would arise from her dead bones (622\u20139). These Roman questions touch upon human life in any era.\n\n### BOOK 5  \nFUNERAL GAMES\n\n_On their way to Italy the Trojans are caught in another storm and run before the winds back to Sicily where Anchises had died precisely one year before. Aeneas celebrates rites in his honour and holds funeral games. Weary with their wanderings, the Trojan women fire the ships, and Aeneas decides to leave the women, children and old men in Sicily in a city ruled by Acestes, the Trojan who had been their host in Sicily. Aeneas' steersman Palinurus is lost overboard on the voyage to Italy_.\n\n#### _Roman Religion_\n\nThe tragedy of Book 4 is followed by the games of Book 5, but first Aeneas looks back at Carthage and sees the flames rising from the pyre on which Dido is dying. None of the Trojans knows what is causing the fire but their hearts are filled with foreboding, soon to be fulfilled by the storm which forces them to return to the place where Anchises had died. Here the piety of Aeneas shows in the scrupulous care with which he performs, for the first time in history, the rites of the _Parentalia_ , the Roman festival of the dead, in honour of his father, who now becomes a god. The _Aeneid_ is authenticating contemporary Roman religious practice by attributing its origins to the founder of the Julian family, and at the same time authenticating the stress upon the revitalization of Roman religion so dear to the heart of the contemporary Julian, Augustus.\n\n#### _Aeneas the Leader_\n\nThere are tears at the heart of things, _sunt lacrimae rerum_ , and for the Victorians Virgil was often seen as a sad presence brooding on the griefs of humanity. On the other hand, throughout these funeral games Aeneas is cheerful, inspiriting, active, efficient, statesmanlike, and a sensitive leader of his men. He sets up the branch on an island to mark the turning point for the boat-race. He gives munificent prizes to every competitor, even to Sergestus when his ship limps home last. He is amused by the effrontery of Nisus and skilfully defuses a nasty situation when Nisus and Salius squabble over the prizes. He tries with a joke to tempt a challenger into the ring with the formidable Dares. When this fails, he conspires with Acestes to tempt the old champion Entellus to put on his gloves again, and when Entellus is on the rampage in this great boxing match, it is Aeneas who saves the life of Dares and shows supreme tact in consoling him for his defeat. He shows his statesman-like vision in acknowledging the blessing of the gods on his Trojan host, Acestes. When the competitive events are over he allows no gap. He has seen to everything. All he has to do to set in motion the grand cavalry display of the Trojan boys is to whisper a word in the ear of a young friend of Ascanius. Throughout, Father Aeneas cares like a father for his people, grieving when he is persuaded that it is the the will of the gods and the wisest course that he should leave the women and children in Sicily in the new city of Segesta he founds for them under Acestes. Once again, the _Aeneid_ looks forward from the legendary past to more recent events. (In the Punic Wars Segesta was to side with Rome.)\n\nThroughout the poem Aeneas is said to be _pius_. But Roman _pietas_ is not the same as our piety. It is not simply a matter of respecting the gods. Pietas requires that a man should do what is due and right not only by his gods, but also for his city, his family, his friends and his enemies. Apart from his lapse in Book 4, Aeneas is its embodiment, and it shows vividly here. Perhaps this is part of the explanation of Montaigne's view that the fifth book of the _Aeneid_ seems to be the most perfect ('le cinquiesme livre de l'Aeneide me semble le plus parfaict', _Essays_ 2.10).\n\n### BOOK 6  \nTHE UNDERWORLD\n\n_Aeneas arrives in Italy at last, landing at Cumae just north of the Bay of Naples. There he consults the Sibyl, begging her to allow him to go down to the Underworld to see his father Anchises. She agrees to escort him on condition that he finds a golden branch in a dark tree and buries the body of Misenus, a comrade who has been drowned. These tasks he achieves andin the Underworld they meet, in reverse order of their deaths, Palinurus, Dido and heroes who had died at Troy. They proceed to the place of eternal torture of the damned and to the Fields of the Blessed where they find Anchises, who explains the creation of the universe and the origin of life, and takes them to see a parade of great Romans of the future marching up family by family towards the light of life._\n\n#### _Why the Underworld?_\n\nWhy did Virgil send his hero down into the Underworld? In Virgil there is often more than one answer to a question. The simple explanation is that this allows him the emotional intensity of the scenes where Aeneas meets dead friends and enemies \u2013 his pilot Palinurus drowned in the crossing to Cumae, Dido ignoring his tears and words of love, Trojans who had died at the sack of the city, Greeks fleeing at his approach. This episode is also a watershed in the plot. In the Underworld Aeneas faces his memories and is given a view of the future. From this time forth he is looking towards the destiny of Rome. Another factor in Virgil's decision must have been the Homeric model. Virgil is writing a Latin epic to stand beside the great epics of the Greeks. Odysseus had conversed with the shades over a trench filled with blood; Aeneas, too, will converse with the dead. The resemblances are obvious, but the differences are profound. There are two eloquent silences in classical epic. In the _Odyssey_ Ajax, the great rival of Odysseus, stood aloof and would not speak, but went to join the other souls of the dead in Erebus. In the _Aeneid_ Dido refuses to speak to Aeneas, but rushes off into a dark wood to rejoin Sychaeus who had been her husband. Virgil plunders Homer, and refashions what he takes.\n\nThe descent to the Underworld has also a philosophical dimension. Virgil puts on the lips of Anchises an explanation of the creation of the world and of the nature of life and death. Just as Plato ends _The Republic_ with the Myth of Er, who tells how he died in battle and saw the souls of the dead waiting to rise again to rebirth, so Anchises shows to Aeneas the procession of his descendants moving up towards the light of life. The end of Book 6 is philosophy in epic.\n\nIt is also politics. Almost nine-tenths of the heroes represented in this parade are members of the Julian family. In a Roman funeral the masks of the ancestors were carried through the streets to their tombs while fathers would retail to their sons the achievements of their forefathers. In Virgil's pageant of the heroes, the dead go in procession by families, not to their tombs along the Appian Way, but up to glorious rebirth while Anchises predicts their great achievements to his son. This book therefore ends with a funeral in reverse, culminating in a eulogy of the Julian family of Augustus and an obituary of his nephew, son-inlaw and heir designate, young Marcellus; it is so powerful that Marcellus' mother swooned when she heard Virgil speak it. The _Aeneid_ is a poem set in the distant heroic past. To make it a political poem relevant to his own times, one of Virgil's strategies is to include praise of Augustus in prophecies like the great speeches of Jupiter near the beginning and end of the poem, the history of the wars of Rome depicted on the prophetic shield of Aeneas at the end of Book 8 and here in the Parade of Future Romans, the prophecy which Anchises delivers to embolden his son with this vision of the destiny which lies before his family.\n\nThis is all fiction. The pageant is invented by Virgil. We do not know what Virgil's beliefs were about the creation of the world or the transmigration of souls. Just as Plato's myths are not meant to be taken as the literal truth but as stories resembling truth, so, after what started as a narrative of a journey and ends as a dream, Aeneas leaves the Underworld not by the Gate of Horn, the gate of true shades, but by the Gate of Ivory which sends up false dreams towards the heavens. At the beginning of the first century BC Meleager, in introducing the epigrams included in his _Garland_ , had given Plato a golden branch to carry as his emblem. Perhaps the Golden Bough and the Gate of Ivory in the _Aeneid_ are there to give us notice that the philosophy at the end of this book and the Parade of Future Romans are, like the Platonic myths, falsehoods resembling the truth.\n\nFor an explanation of the details in the Parade of Future Romans in the underworld, see Appendix I.\n\n### BOOK 7  \nWAR IN LATIUM\n\n_Aeneas and his fleet sail into the mouth of the River Tiber and build a camp on its banks. Latinus, the king of Latium, welcomes them and offers Aeneas his daughter, Lavinia, in marriage. Seeing this, Juno sends down her agent Allecto to stir up resentment against Aeneas. She persuades Queen Amata to oppose Aeneas' marriage and whips up Turnus, a neighbouring Latin prince, to go to war against the Trojans. She then engineers a skirmish between the local people of Latium and a Trojan hunting party led by Ascanius. War has begun_.\n\n#### _Turnus and Allecto_\n\nTurnus, prince of Ardea, had hopes of marriage to Latinus' daughter and succession to his throne, and Queen Amata supported him. But when Allecto, disguised as an aged priestess, visited him in his sleep and urged him to war, he rebuffed her: 'Leave peace and war to men. War is the business of men' (444). Enraged, she threw a burning torch into his heart, and he woke sweating with terror and roaring for his armour. So much for the mythical narrative. At another level this could be read as an account of how a man's rational assessment was overturned in the small hours by patriotic passion and rankling sexual jealousy. The narrative has treble power: as a vision of the supernatural, as an account of an emotional experience and as a dramatic scene between an old woman (who is more than a woman) and a tactless, passionate and impressionable young man.\n\n#### _The Catalogue of Italian Allies_\n\nJust as Homer provides in the second book of the _Iliad_ a catalogue of the Greek ships that sailed against Troy, so here Virgil supplies a catalogue of the Italians who fought against the Trojans. To us it may read as an arid, largely alphabetical list of anthropological curiosities and meaningless place names: Caeculus, found as a baby on a burning hearth at Praeneste, Abellans with their boomerangs, a snake-charming priest from Marruvium, etc. But this list would have struck Virgil's audience quite differently. Many Romans had ties with the country districts of Italy, and would have been moved by this as a celebration of their local cultures, their links with Greece, the myths of Italy, local dress styles, armour, religion, even landscape, as in the twins of Tibur\/Tivoli like Centaurs plunging down a steep forest in Greece, which is not unlike the tree-clad cliff on which their city of Tibur stands.\n\nItaly was a crucial part of Augustus' power base, and at 8.678 Virgil visualizes Augustus leading the men of Italy against the forces of the East under Antony and Cleopatra. 'Of its own free will,' claims the Julian Augustus himself in his official obituary ( _Res Gestae_ 25), 'the whole of Italy swore allegiance to me and demanded me as leader for the war in which I was victorious at Actium.' Although the Italians go to war against the Julian Aeneas, they are never slighted in the _Aeneid_. At the end the stock of Rome is to be 'made mighty by the manly courage of Italy' (12.821\u20137). At a political level this catalogue of the peoples of Italy is a hymn to the indigenous peoples of Italy, and it accords with the stated policy of Augustus.\n\n### BOOK 8  \nAENEAS IN ROME\n\n_With the blessing of the god of the River Tiber, Aeneas goes to the village of Pallanteum, on what is later known as the Palatine, one of the seven hills of Rome. Here King Evander describes how Hercules had saved them from the ravages of the monster Cacus and tells the story of Mezentius, a brutal Etruscan despot who has been dethroned by his subjects and is being harboured by Turnus. Evander tells Aeneas of a prophecy which forbids the Etruscans to be led by an Italian, and advises him to go with a detachment of cavalry led by his son Pallas, to claim leadership of all the armies opposed to the Latins. Venus, concerned for her son's safety against these formidable enemies, persuades Vulcan to make new armour for Aeneas, including a prophetic shield depicting the future wars of Rome._\n\n#### _The Politics_\n\nThis is not a book of intense dramatic incidents or heroic deeds, but it is vital to the argument of the _Aeneid_. On the face of it the Trojans are invaders in a foreign country, seizing land and power from the rightful inhabitants. But these aggressors are the ancestors of the Romans, and their leader Aeneas is the founder of the Julian family. A vital part of Augustus' policy was his claim to be the beneficent leader of Italians as well as Romans against the barbarian East, and yet here at the dawn of Roman history his ancestor Aeneas is leading Orientals, that is the Trojans, against the native peoples of Italy. Book 8 tackles this difficulty and provides justification not only for Aeneas but also for Augustus' rule over Italy.\n\nThe Romans loved their river, and Virgil's first step is to show Father Tiber welcoming Aeneas to Latium. The opening of the book makes it clear, on the evidence of the god of the river, that Latium, in the centre of Italy, is the home decreed by the gods for Aeneas and his people. The second step is to provide historical warrant for the presence of the Trojans on Italian soil. This is achieved when Aeneas visits the future site of Rome, Pallanteum, a settlement of Greeks from Arcadia, and points out to its king, Evander, that Dardanus, father of the Trojan people, had been born in Italy, and that Evander and himself were both descended from the god Atlas. Evander in turn recognizes Aeneas as the son of Anchises whom he had known and admired in his youth, and explains that the two families are therefore linked by the sacred tie of guest-friendship. Hence the lengthy genealogical discussions when Aeneas first meets Evander (pp. 296\u2013).\n\nWe have seen that Virgil expresses contemporary issues in his legendary tale by means of prophecies and visions, but there is another subtler technique at work in this book. Hercules had saved the settlement of Pallanteum from the ravages of the monster Cacus and had deigned to accept Evander's hospitality. Now, on the very day of Hercules' festival, arrives Aeneas who has also saved his people, will also stoop to enter that same little hut and will go on to found a city which will move to Pallanteum and become the city of Rome. There is a third saviour involved in this story. Rome was again saved, in Virgil's day, by Augustus, who returned to Rome after the defeat of Antony and Cleopatra at Actium on 12 August 29 BC, the first day of the Festival of Hercules, and who now lives simply and modestly in his house in what was Pallanteum and is now the Palatine Hill. 'You...must have the courage to despise wealth,' says Evander as he invites Aeneas to enter his simple little hut. 'You must mould yourself to be worthy of the god' (364\u20135). The god is Hercules. Aeneas himself will become a god. But now Augustus, famed for the simplicity of his daily life and another saviour of Rome, is dwelling on that same spot, and he, too, will be a god.\n\nAn important part of the story of Rome is the long series of wars by which she subdued the peoples of Italy, culminating in the fierce and bloody Social War of 90\u201388 BC. Just as the defeat of Cacus is a pre-enactment of the defeat of Antony and Cleopatra, and the arrival of Aeneas at Pallanteum is a pre-enactment of the return of Augustus, so the war in Italy in the second half of the _Aeneid_ is a pre-enactment of the Social War. This is why the Latins who confront Aeneas are presented as courageous and virtuous peoples, eventually defeated but never disgraced. This why they are put in the wrong not for any vices of their own, but by the malice of Juno and the fact that Turnus, prince of the Latin city of Ardea, is harbouring Mezentius, a tyrant whose vices would attract adverse comment even in our own day. The other Etruscans are baying for his blood, but they are waiting for a leader and a prophecy has said that they must not be led by any man of Italy. So the scene is set. Aeneas has an ancestor who came from Italy; he has a guest-friend and relative in Evander to justify his presence in Italy; he has allies in Etruria who have just cause to go to war and need a leader. Aeneas' presence and position in Italy are therefore legitimated. This has implications for the whole Julian family, and in particular for its contemporary representative who rules Italy and the whole known world from his house on the hill which had been Pallanteum.\n\n#### _The Humanity_\n\nThis discussion has moved into the politics of the epic, but the first thing to grasp about the _Aeneid_ is its humanity. In this part of the poem we may be struck by two recurring motifs: the beauty of youth and the depth of the love between parent and child. Pallas, son of Evander, is an important figure. We meet him for the first time when the masts of Aeneas' ships are seen gliding through the trees on the banks of the Tiber, and we can gauge his ardour and courage as he leaps up to confront these formidable strangers. Evander in his young days had known Anchises, and the joy with which he recognizes his old friend's son testifies to the warmth of his admiration. Then later, when he explains that he is too old to go to war, and gives Aeneas charge of young Pallas on his first campaign, we are left in no doubt of the intensity of Evander's love for his son and the solemnity of the responsibility he lays upon Aeneas.\n\nThere is another very different manifestation of parental affection, when Venus, alarmed by the formidable Italians whom Aeneas is about to confront in battle, persuades her husband Vulcan, the god of fire, to make a shield for the son she bore to her mortal lover Anchises. When Venus persuades, she seduces. Vulcan then sleeps and rises early to go to work in his foundry, and his rising is compared to the early rising of a virtuous peasant woman who goes to work in order to keep chaste her husband's bed and bring her young sons to manhood. It is impossible to feel secure about the tone of this astonishing episode. It is probably a contribution to the comedy of the divine in the _Aeneid_ , but it certainly is also a demonstration of Venus' motherly concern for her son, and a tribute to the courage and prowess of the people of Italy, and therefore a part of the politics of the _Aeneid_.\n\n#### _Art Described in Epic_\n\nThere never was such a shield as Virgil describes, but he does his best to make us believe in it. There are repeated references to colours, like the silver geese in the golden portico and the golden torques on the milk-white (does that suggest ivory?) necks of the Gauls scaling the Capitol in their striped cloaks. There are suggestions of texture in the she-wolf bending back her neck to lick the twin babies into shape, in matrons in cushioned carriages, in blood dripping from bramble bushes or reddening the furrows of Neptune's fields. There are vivid scenes: the rape of the Sabine women, Augustus at Actium with the Julian Star shining over his head, the River Araxes furious at being bridged. There are sound effects, as so often in descriptions of works of art in classical epic: when we hear at the Battle of Actium the barking of the dog-headed god Anubis; the cracking of the bloody whip of Bellona; the babel of all the tongues of the earth in the triumphal procession in Rome. There is also serial narration depicting successive episodes of a narrative all within the same frame, as when Cleopatra's fleet advances, Apollo draws his bow, Cleopatra pays out the sail ropes for flight, runs before the wind for Egypt, and at the last the Nile, with grief in every lineament of his body, beckons his defeated people into his blue-grey breast and secret waters.\n\nThis is a vivid description of an imaginary work of art. It is also praise of Augustus. Three-fifths of this depiction of 'the story of Italy and the triumphs of the Romans' (626) are devoted to Augustus' defeat of Antony and Cleopatra at the Battle of Actium, and in line with Augustan propaganda the name of Antony is never mentioned. Civil war is presented as though it were a conflict between the barbarian East and the civilized world of the West. Augustus also received a shield, the Shield of Valour, presented to him by the Senate and People of Rome to honour his courage, clemency, justice and piety.\n\nFor an explanation of the details of the Shield of Aeneas, see Appendix II.\n\n### BOOK 9  \nNISUS AND EURYALUS\n\n_When Aeneas and Pallas are on their mission to the Etruscans, the Trojan camp is attacked by Turnus and his Rutulians. In accordance with the strict instructions given by Aeneas, theTrojans close the gates and decline battle. Nisus and Euryalus die on a night foray and Ascanius kills Numanus. The siege continues and Turnus breaks into the Trojan camp. In his fury and folly he slaughters Trojans instead of opening the gates, and eventually is forced to withdraw and swim the Tiber fully armed to return to his men._\n\n#### _Nisus and Euryalus_\n\nVirgil was moved by the glory and the grief of the deaths of the young in battle. His story of Nisus and Euryalus is also a delicate portrayal of the passionate love between two young men. Less obviously, it is a negative example. By their blunders and their impetuosity, by their neglect of the disciplines of war and above all by their failure to show respect to the gods, they are standing exemplars of what Aeneas is not.\n\nThe crucial mistake by Nisus is to take young Euryalus with him on this perilous mission. In a similar situation in Homer's _Iliad_ , Diomede chose as his companion Odysseus, the cleverest of the Greeks\u2013'the skill of his mind is with out equal'\u2013and Odysseus justified the choice. Here Nisus does not want Euryalus to go with him, but allows the younger man to take the crucial decision. It is Euryalus who wakes sentries to keep guard for Nisus and himself when they go to tell the council of their plan.\n\nThe council of chosen Trojan warriors is also at fault. The original plan suggested by Nisus was to take a message to Aeneas, but now the young heroes propose to set an ambush, kill large numbers of the enemy and come back laden with booty. Aletes, though 'heavy with years and mature in judgement' (246), approves this madcap scheme, and young Ascanius enthusiastically welcomes it, promising all manner of extravagant rewards, including the horse of Turnus, the enemy leader.\n\nThey set out, enter the Rutulian camp and slaughter their sleeping enemies where they lie. Nisus eventually realizes that daylight is coming and checks Euryalus, but still allows him to put on armour he had plundered from the dead \u2013 medallions, a gold-studded belt, a helmet with gorgeous plumes. The helmet is their undoing. A passing detachment of three hundred cavalry catches sight of it glinting in the moonlight. Nisus escapes but Euryalus is captured, hampered by the booty he is carrying. Nisus sees him being carried off by the enemy and breaks cover in a hopeless attempt at rescue. Whenever Aeneas begins an undertaking, he prays to the great gods, to Jupiter, Juno, Apollo, Mars or to his mother. But here Ascanius swears by his own head, and Nisus by chance, Vesta, his household gods, the sky and the stars. At the end, when his beloved Euryalus is in mortal danger, Nisus prays at last, but prays only to Diana, the moon goddess, who had just betrayed them.\n\nThere are no doubts about their ardour or their courage or their love, and Virgil steps out of his role as anonymous narrator to salute them and rejoice in their immortality, but he has already made it plain that the weaknesses of youth, lack of judgement, of discipline and of piety are not the stuff of which Roman leaders are made. Aeneas is a different kind of man.\n\n#### _Ascanius Kills Numanus_\n\nBefore his return Ascanius will have had his baptism of fire. A young Latin warrior, husband of the sister of Turnus, Numanus Remulus speaks up for the Latins against these effeminate incomers from the East. The Latins are a race of hardy sons of toil, and these 'Phrygians' from Troy are effete, with their saffron and purple robes and their sleeved and beribboned bonnets. They are women, not men, playing tambourines and flutes in their dubious women's rites on Mount Ida. This is the case against the Trojans and it has to be answered because the Trojans are the ancestors of the Romans. Ascanius gives the only possible answer, and Apollo instantly withdraws him from the battle, but not before prophesying the glory of his descendants. 'This is the way,' he tells Iulus, 'that leads to the stars. You are born of the gods and will live to be the father of gods' (642), and Virgil's audience would have taken the point. At Caesar's funeral games a comet appeared, which was hailed by the common people as proof that Caesar had been received among the gods. We have already had sightings of this Julian Star at critical moments in Julian history, at 2.694 when Anchises consents to leave Troy and at 8.681 on Octavian's helmet at Actium. It was also generally understood in the twenties BC that Augustus, his adoptive son, would be deified. Finally, the peace which Apollo proceeds to prophesy is the _Pax Augusta_ , the peace which Augustus was promising to bring to the whole Roman world, coming not from Troy, but from a much greater city. As Apollo says, 'Troy is not large enough for you' (644). The honour of the Julians is thus vindicated by Ascanius Iulus, and his descendants are cleared of the imputations levelled by Numanus.\n\n### BOOK 10  \nPALLAS AND MEZENTIUS\n\n_Aeneas returns at the head of the Etruscan armies. Turnus kills Pallas and tears the belt off his dead body. As Aeneas slaughters the Latins in an orgy of revenge, Juno saves Turnus from his fury by spiriting him from the battlefield. Mezentius takes his place, and in battle with Aeneas his life is saved by the intervention of his young son Lausus. Aeneas kills Lausus, and the wounded Mezentius challenges him and dies in single combat._\n\n#### _The Council of the Gods_\n\nJupiter opens the debate of the council of the gods by asking why Italians are at war with Trojans against his express will. Strange. After all he is omniscient \u2013 he knows the answer to all questions, and he is omnipotent \u2013 his will is the unalterable decree of fate. That is the theology, but in epic theology does not always apply. Sometimes Jupiter is not the all-powerful lord of the universe, but the father of a rowdy family where there is constant trouble between jealous wife and unruly daughter. The gods in epic sweep the action to the heights, as at the beginning and end of his episode. They also pull it down to the level of domestic comedy, as when Venus and Juno wrangle in council like a pair of rhetorically trained fishwives.\n\nVenus complains that after all these years her son is still homeless and his people are under siege again, this time on Italian soil; Juno says that if they are suffering, it is by their own choice. Venus pretends to believe that the destiny of empire pronounced by Jupiter at the beginning of the epic is being altered; Juno's reply is that the Trojans are not fulfilling their destiny, but obeying the prophecies of a madwoman, Priam's daughter Cassandra. Venus objects to the storm Juno raised against Aeneas in Book 1; Juno wilfully misunderstands and says that Aeneas' voyage back from Etruria is none of her doing. In Venus' view Turnus is swollen with his success in war; for Juno he is taking his stand in defence of his native land. Venus grumbles because she is at risk from the violence of mere mortals; Juno's reply sketches Turnus' descent from the gods of Italy. Venus tries to rouse pity for the Trojans because of the absence of Aeneas; Juno advises him to stay away. It is an established device of ancient oratory to appeal for clemency by bringing in the children of the defendant at the end of a speech. Venus brings in Ascanius, and begs to be allowed, if all else is lost, to take him to safety in one of her beautiful sanctuaries in Amathus, Paphos, Cythera or Idalium; Juno taunts her by telling her to be content with Paphos, Idalium and Cythera and to keep away from these rough Italians. Point by point Juno has stripped down Venus' arguments, offering two lies for every one by Venus and adding half-a-dozen new ones of her own.\n\nThe speeches of Sinon in Book 2 were a satirical attack upon Roman rhetoric, the technical study of the arts of persuasion on which Roman education was based. This clash between Venus and Juno is the _coup de gr\u00e2ce_. Why should Virgil launch these attacks upon the false values of Roman rhetoric? An obvious approach to this question would be to connect it with the political conditions of the day. In the first century BC the Roman republic was torn apart by the rivalries of ambitious men, fought out not only on battlefields but also in political debates in the Senate and in political trials in the courts. In both arenas, lies, calumny, melodrama, confrontational debate, all the vices of rhetoric, had been common coin. The Augustan settlement took the power from these arenas and lodged it with the _princeps_ , and the style of government changed. Augustus had no love for the liberties which had destroyed the republic and had no intention of allowing them to weaken his own position. We may remember that Anchises in the Underworld started his litany of the areas in which Greeks would surpass Romans by saying 'Others will plead cases better' (6.849), a calculated obliteration of the memory of Rome's greatest orator. Augustus had connived at the killing of Cicero in 43 BC. He would also have enjoyed Virgil's demolition of rhetoric.\n\n#### _The Death of Mezentius_\n\nAccording to an ancient commentator the _Aeneid_ is written to imitate Homer and to praise Augustus with respect to his family. But panegyric is raised to poetry by Virgil's deep sense that victory has its price. The Latin warriors, we have seen, are courageous and upright, and they and their women suffer the cruelty of war. Dido is a noble queen who died a death she did not deserve, and Virgil so told her story that for over two millennia men have grieved for her. Turnus is the great enemy of the hero of the epic, but by the end of it he has claims to our admiration and pity. Mezentius is a villain through and through, a monster of cruelty to his subjects and a scorner of the gods, but when he stands alone against all his enemies we begin to admire him. When he refuses to cut down Orodes from the rear and manoeuvres to meet him face to face, we know we are in the presence of a hero. The most revealing moment comes with his answer to Orodes' dying taunt: 'Die now. As for me, that will be a matter for the Father of the Gods and the King of Men' (743\u20134). The scorner of the gods is now admitting and accepting the supremacy of Jupiter. It is almost as though Virgil had not the heart to let the villain die a villain. When the balance of Mezentius' life is about to swing from wickedness to tragedy, Virgil's sympathies reach out towards him.\n\nSoon Mezentius is wounded by Aeneas, and would have been killed had not his son Lausus so loved his father, that, lightly armed as he was, he threw himself between the combatants. Aeneas kills him, and when he sees his dying face and features, the face 'strangely white', he is reminded of his love for his own father (821\u20132) and we too are reminded of it when Virgil here refers to Aeneas by his patronymic, _Anchisiades_ , son of Anchises. Our sympathies are divided. Then, while Mezentius is trying to recover from his wound on the banks of the Tiber, he hears the wailing in the distance and knows the truth, and bursts into a paroxysm of grief and self-hate. Before Mezentius goes to fight his last battle, like Achilles in the _Iliad_ , he addresses his horse, and each man's utterance is a testimony to human and animal courage and the obstinacy of affection. Nothing in Mezentius' life becomes him like the leaving it.\n\nCrude panegyric is unrelieved, direct praise with no regard for truth. The panegyric of the _Aeneid_ praises Augustus, intermittently and often obliquely, and it is always based upon a genuine and intelligent response by the poet to the contemporary political situation. It also takes in a great sweep of human experience. While saluting the victor and acclaiming his victories, Virgil records the sufferings of the defeated and of the innocent. He also acknowledges the cost to the victors in the persons of Aeneas and Augustus.\n\n### BOOK 11  \nDRANCES AND CAMILLA\n\n_Pallas is mourned and his funeral rites conducted. The Latins send an embassy to Aeneas to beg a truce in order to gather up their dead. He consents and makes it clear that the war was not of his choosing. Turnus could have met him in single combat and only one man would have died. The Latins engage in fierce debate, Drances abusing Turnus and pleading for an end to the war, Turnus returning the abuse and offering to meet Aeneas in single combat. Despite that, when news comes that Aeneas is approaching the city, Turnus immediately rouses his forces for battle. The maiden Camilla volunteers to confront the enemy cavalry while Turnus waits in ambush for Aeneas in a pass in the hills. Camilla is killed, and Turnus gives up his ambush. A moment later Aeneas enters the pass, and both armies move towards the city of Latinus within sight and sound of each other._\n\nThis book, like all the books of the _Aeneid_ , can be divided into three sections; here, the funerals, the debate, the cavalry engagement. In each of these the dice are weighted against Turnus and to the credit of Aeneas. In the first Aeneas' great grief at Pallas' death was partly because he had failed to protect the young man in his first battle, but Latinus insists that Aeneas is in no way to be blamed for his son's death. In his dealings with the Latins (100\u201321), Aeneas behaves with clemency and consideration. At the debate in the Latin assembly a report is received by an embassy which had been sent to ask help from Diomede, whom Aeneas had called the 'bravest of the Greeks' (1.96). Diomede had refused: 'We have faced each other, spear against deadly spear, and closed in battle. Believe me, for I have known it, how huge he rises behind his shield' (282\u20134). At the end of the assembly King Latinus blamed himself for the war by his failure to give full support to Aeneas. And in the cavalry engagement, a question may hang over Turnus' military judgement in granting such an important battle role to Camilla, and in his own impotence in sitting in ambush far from the battlefield and leaving the position at precisely the wrong moment: 'this is what the implacable will of Jupiter decreed' (901).\n\n### BOOK 12  \nTRUCE AND DUEL\n\n_Turnus now demands to meet Aeneas in battle, and Aeneas and Latinus strike a treaty agreeing that the victor will receive Lavinia in marriage, and that if Aeneas is defeated, the Trojans will withdraw peacefully and settle with Evander in Pallanteum. But Juno suborns Turnus' divine sister Juturna to engineer a violation of the treaty. In the m\u00eal\u00e9e which follows Aeneas is wounded by an arrow shot by an unknown assailant. He is healed by the intervention of Venus and returns to battle. Once again Turnus is rescued from the wrath of Aeneas \u2013 this time by Juturna \u2013 but when Aeneas attacks the city of Latinus, Turnus realizes his responsibilities and returns to the field. Jupiter and Juno are reconciled, and Juno gives up her opposition to the destiny of Rome. Aeneas wounds Turnus and kills him as he begs for mercy._\n\n#### _The Death of Turnus_\n\n'I sing of arms and of the man' is how Virgil began his epic, and nowhere does he sing more intensely of Aeneas than in the last book. It opens with bold words from Turnus as he steels himself for battle, taunting Aeneas and issuing a ringing challenge: 'Let the Trojan and Rutulian armies be at peace. His blood, or mine, shall decide this war' (78\u20139). While he dons his splendid armour and girds on his sword (the wrong one, as shall emerge), roaring like a bull and lashing himself into a fury, Aeneas, too, is rousing himself to anger, but is also reassuring his allies, comforting his son, accepting the challenge and laying down the terms of the peace that will follow the duel.\n\nThe steadiness and maturity of Aeneas are thus shown by means of a contrast with the wildness of Turnus. This technique of tacit contrast is also used by Virgil when the armies meet to ratify the treaty. Day has dawned with the most glorious epic sunrise, and the first witness Aeneas then calls upon is the Sun, a courteous compliment to Latinus since the Sun is his grandfather, but that address is followed immediately by an invocation of the great Olympians, Jupiter, Juno and Mars: Jupiter, since the golden rule is always to begin with him; Juno, because Aeneas is remembering the instructions he received from the god Tiber at the beginning of Book 8; and Mars, as god of battle and later to be the father of Romulus. This is theologically correct, and a striking contrast to the ragbag of divinities addressed by Latinus, ending, contrary to the golden rule, with Jupiter. The contrast demonstrates Aeneas' piety towards the gods.\n\nThe next display of character by tacit contrast comes after the Rutulians, egged on by Juturna, have violated the treaty in the very moment of its ratification. In the battle which follows, Aeneas, unhelmeted, tries to control his allies, insisting that a treaty has been made and that by its terms no one is allowed to fight except Turnus and himself. But when the arrow comes whirring from an unknown hand and Aeneas is led wounded from the field, Turnus seizes his opportunity. Clapping on his armour he launches into a fierce and bloodthirsty attack upon the Trojan forces. The contrast demonstrates Aeneas' sense of justice.\n\nSome readers have found Aeneas an unsympathetic character, cold and inhibited. This notion is nowhere more thoroughly refuted than in the episode which follows. As he is taken back to the camp bleeding from his wound, he is in a fury of impatience, tugging at the broken arrowhead and ordering his comrades to hack it out of his flesh. There he stands in the camp growling savagely while the doctor plies his mute, inglorious art, and the enemy are heard fighting their way nearer and nearer to the camp. No sooner has Venus healed the wound than he is throwing on his armour and storming back to battle. But first he takes his leave of Ascanius, whom he loves. Those who do not admire Aeneas are amazed that he does not take off his helmet to kiss his son. Others will listen to his words and see in Aeneas a heroic ideal in the Roman mould.\n\nTurnus had cut a swathe of slaughter through the Trojan ranks, but when Aeneas now routs the Rutulians he ignores the fugitives. He is stalking Turnus, and only Turnus, and he would certainly have caught him, had not Juturna seized the reins of Turnus' chariot and driven him off to kill stragglers in remote parts of the battlefield.\n\nBetrayed, wounded and now thwarted, Aeneas erupts in an orgy of killing. Here we notice no difference between Aeneas and Turnus: in the heat of battle neither is a 'verray parfit gentil knight'. Each is driven by uncontrollable passions of hatred, contempt, rivalry and revenge, and each taunts his wounded enemies and kills his suppliants. This is not a diminution of the individuals, but a fact of war, and part of the power of these last books is that Virgil does not flinch from fact. Until the mid twenties BC when Virgil was in his mid-forties, Rome had been in a continual state of war. He did not romanticize it. He knew as well as his contemporaries, and as well as John Hampden, quoted by Macaulay, that 'the essence of war is violence, and that moderation in war is imbecility'.\n\nAeneas' attempt to end the war by single combat has failed. Turnus is not to be seen and full-scale battle is raging. At this desperate point Aeneas orders his men to break off the fighting and follow him to attack Latinus' undefended city. His sole purpose is to smoke out Turnus, to bring him to combat, but even so, this is scarcely an act of high chivalry. At this point we see Virgil's determination to preserve the character of his hero. The plan to attack an undefended city is not in origin his own: 'At that moment Aeneas' mother, loveliest of the goddesses, put it into his mind...to lead his army' (554\u20135) against the walls of the city. We have already seen double motivation in action, for example when Dido fell in love as a woman, while at the same time Venus and Cupid manoeuvred her into the madness of love. There the double motivation made the event more complex and more profound. Here it is put to ingenious use. When the hero thinks of a course of action which does him little credit, any stain on his character is lessened by a narrative which attributes the motive force to a god, who by definition cannot be resisted.\n\nThe ruse works. Turnus hears the sounds of despair from the city and realizes that his sister has misled him. In a speech of great nobility he accepts the truth and resolves to return and confront Aeneas. The moment Aeneas hears the name of Turnus he abandons his attack on the city. The armies part to clear a space. The gods leave the field and what we see at the last is two men fighting. Turnus is wounded and begs for mercy for the sake of his father. At this Aeneas wavers, no doubt remembering his own father and also how he suffered when he killed Lausus, but then he catches sight of the belt which Turnus had plundered from the dead body of Pallas, the boy who had been given into his charge, and in a blaze of raging anger he plunges his sword into the breast of his defenceless enemy. Revenge is part of war, as Augustus knew. As a boy he had won the support of the legions by promising to avenge their beloved Caesar, and over the years he had hunted down every last one of the conspirators, formally recording his revenge at the beginning of his _Res Gestae_. Virgil passes no judgement on Aeneas. He describes it as it would have been.\n\n#### _The Solution_\n\nMeanwhile Juno, the greatest liar in the _Aeneid_ , has not been idle. It is she who had suborned Juturna to go to the aid of Turnus in a speech which begins, as usual in rhetoric, with flattery, proceeds to self-justification and ends by urging Juturna into action while offering her no hope. But because Juno is trying to avoid responsibility, her instructions are so deviously expressed that Juturna barely understands them. Juno then loses patience and has to tell her straight out to go to rescue her brother or else stir up a war to block the signing of the treaty. When the arrow wounds Aeneas, no man knows who shot it, but we know who was responsible, and so does Jupiter, as at the end of the _Aeneid_ he smiles at his wife's evasions.\n\nThis final interview between Juno and Jupiter is the solution to a central problem of the _Aeneid_ , how the Roman empire is to be established against the opposition of Juno. The settlement is arranged in the final act of the divine comedy which has run through the whole poem. Although Juno has told Juturna that she cannot bear to watch the battle, Jupiter sees her doing so. He speaks affectionately to her, and then teases her gently: 'What do you hope to achieve by perching there in those chilly clouds?' He knows precisely what, and she knows that he knows. He then changes tack and pleads with her in loving terms: 'Do not let this great sorrow gnaw at your heart in silence, and do not make me listen to grief and resentment for ever streaming from your sweet lips.' He then reminds her of what she has achieved. At the last, after the affection and the praise, the command: 'I forbid you to go further' (791\u2013806).\n\nJuno submits, but not before a flood of bluster, face-saving and self-justification: 'I, Juno, yield and quit these battles which I so detest' (818). Having yielded, she now lays down her stipulations. Her essential point is that she will allow these Trojan men to settle in Italy and marry Italian wives, but only on condition that they forfeit all trace of their Trojan origins. Now we understand why the Trojan women had to be left in Sicily at the end of Book 5. Now we understand how the repeated slur of effeminacy is to be erased from the reputation of these incomers from the East. The Trojans are to lose their name and become Latins. They are to dress in the Italian style and give up their Oriental flounces, so mocked by Numanus Remulus in Book 9. The Alban kings are to rule from generation to generation, and we see that the wheel has come full circle. At the opening of the poem we were told that the _Aeneid_ would reveal the origins of the Alban fathers. Now we remember that the Alban kings, like Augustus, are Julians, descended from Iulus. Juno's last stipulation is the final cleansing of the bloodstock of the Trojans. Rome is to be made mighty by the manly virtue of Italy, _sit Romana potens Itala virtute propago_. _Vir_ is the Latin for 'man', and _virtute_ is the Latin for manly virtue ('manly courage' in the text, 827), so this blend of blood will finally erase all trace of Oriental effeminacy from the founders of Rome. 'Troy has fallen. Let it lie, Troy and the name of Troy' (828).\n\n'He who devised mankind and all the world smiled', and, remarkably, he goes on to remind Juno of their double relationship, brother and sister, husband and wife. He accepts her stipulations and adds his own details. The language of the new people will not be Trojan, but Latin. The overtones of Jupiter's formulation are important. Latin was superseding the native tongues of Italy as the _lingua franca_ of commerce, law and government. When Jupiter says that Ausonia (an ancient name for Italy) will keep the tongue of its fathers, he is suggesting some sort of justification for Latin against the languages which it is supplanting all over Italy. Throughout this dialogue of the gods Virgil is making his legend more plausible by linking it to known contemporary facts.\n\nJupiter will also provide ritual and modes of worship, another ingenious element. At the fall of Troy, Aeneas had been given a solemn charge to establish the Trojan gods in a new city. But Virgil does not wish to argue that the gods of Augustan Rome came from the East. Nor does he want Aeneas to negotiate away the gods which were his sacred responsibility, and capitulate to the Latins in a matter of such central importance in the _Aeneid_. The ingenuity of Virgil's solution to this problem lies in the fact that Aeneas capitulates not to any man but to Jupiter, the supreme god of the Romans. No one could object to a religious ordinance imposed by Jupiter Best and Greatest. The discussion between Jupiter and Juno ends with his assurance that the Romans will surpass all men in piety and also all gods, a prophecy which is less astonishing than it seems, if we recollect that obedience to just authority is part of _pietas_ , and that the gods have not always excelled in that virtue. In particular \u2013 his last assurance \u2013 no other race will be the equals of the Romans in doing honour to Juno.\n\nJupiter has the last word. Juno seems to have the last gesture. The Latin, like all Latin, is untranslatable, literally, 'Rejoicing, she twisted back her mind' (841). Juno then did in the end change her mind, but clearly, she found it a bitter-sweet experience. The domestic dispute is thus resolved. Turnus will be killed. Aeneas will marry Lavinia and found Lavinium, and world history will proceed according to the decisions of this humorous discussion between a god and his wife.\n\nDivine machinery is an obsolete literary device, but it gives a great sweep of human interest to the _Aeneid_ and as a dramatic representation of ordinary human relations and of the unpredictable in life, the place of justice in the world, the limits of human effort and understanding and the inscrutable splendour of the universe, it is not a bad model.\n\n## Further Reading\n\n### BIBLIOGRAPHICAL SURVEY\n\nP. Hardie, _Virgil_ , New Surveys in the Classics 28 (Oxford University Press for the Classical Association, 1998)\n\n### INTRODUCTORY\n\nW. S. Anderson, _The Art of the_ Aeneid (reprinted Bristol Classical Press, 1994)\n\nW. A. Camps, _An Introduction to Virgil's_ Aeneid (Oxford University Press, 1969)\n\nK. W. Gransden, _Virgil's_ Iliad (Cambridge University Press, 1984)\n\nJ. Griffin, _Virgil_ (Oxford University Press, 1986)\n\nR. Jenkyns, _Classical Epic: Homer and Virgil_ (Bristol Classical Press, 1992)\n\n### COMPANIONS\n\nN. Horsfall (ed.), _A Companion to the Study of Virgil_ (Brill, 1995)\n\nC. Martindale (ed.), _The Cambridge Companion to Virgil_ (Cambridge University Press, 1997)\n\n### BACK GROUND\n\nK. Galinsky, _Augustan Culture_ (Princeton University Press, 1996)\n\nP. Zanker, _The Power of Images in the Age of Augustus_ , tr. A. Shapiro (Michigan University Press, 1988)\n\n### COLLECTIONS\n\nS. Commager (ed.), _Virgil: A Collection of Critical Essays_ (from studies published 1945\u201364) (Prentice-Hall, 1966)\n\nP. Hardie (ed.), _Virgil: Critical Assessments of Classical Authors_ (1901\u201395), 4 vols. (Routledge, 1999)\n\nS. J. Harrison (ed.), _Oxford Readings in Vergil's_ Aeneid (1933\u201387) (Oxford University Press, 1990)\n\nI. McAuslan and P. Walcot (eds.), _Virgil_ (1972\u201386) (Oxford University Press for the Classical Association, 1990)\n\nH.-P. Stahl (ed.), _Vergil's_ Aeneid: _Augustan Epic and Political Context_ (Conference Proceedings) (Duckworth, 1998)\n\n### CRITICISM\n\nD. L. Drew, _The Allegory of the_ Aeneid (Blackwell, 1927)\n\nR. Heinze, _Virgil's Epic Technique_ , tr. H. and D. Harvey and F. Robertson (Bristol Classical Press, 1993)\n\nE. Henry, _The Vigour of Prophecy_ (Southern Illinois University Press, 1989)\n\nR. O. A. M. Lyne, _Words and the Poet_ (Oxford University Press, 1989)\n\nK. Quinn, _Virgil's_ Aeneid: A _Critical Description_ (Routledge and Kegan Paul, 1968)\n\nG. Williams, _Technique and Ideas in the_ Aeneid (Yale University Press, 1983)\n\n## Note on the Translation\n\nThe text used, with very few exceptions, is the Oxford Classical Text by Sir Roger Mynors. The numbers in the margin refer to the line numbers of the Latin. Latin being a very compact language, ten lines of Virgil (given in the margin) have often required more than ten in the translation.\n\nReceived wisdom, as represented by _The Proceedings of the Virgil Society_ 19(1988), 14, states that 'to translate poetry into prose is always a folly'. I believe that this view does less than justice to the range, power and music of contemporary English prose. As written by our best novelists and journalists and even sometimes by ordinary letter-writers, it daily moves us towards pity, terror or laughter, and does so more than the voices of contemporary poets. Further \u2013 this is ungentle but the argument requires that it be said \u2013 the English poets who have translated the _Aeneid_ since Dryden have not done well. We may accept that poetic translation need not be true to the tone or detail of the original. A poet's first concern is with his own poem. But if we grant this freedom, we must then judge their works as poems, and as such the poetic translations of the _Aeneid_ are low in interest and inspiration.\n\nThe ruling prose version is Jackson Knight's Penguin Classic of 1956. This is lovingly faithful to the author's vision of Virgil but the language is dated. It would be difficult to disagree with Sandbach's judgement in _The Proceedings of the Virgil Society_ 10(1970\u201371), 35 (reprinted in _Meminisse Iuvabit_ (1989), ed. F. Robertson): '...too often the attempt to grasp and represent each of Virgil's words has pushed aside the need to give the sentence rhythm and cohesion and the emphasis that goes with form'.\n\nThe object of this translation has been to write readable English which does honour to the richness and sublimity of Virgil's language \u2013 ebullient, for example in the utterances of Aeneas at the games in Book 5, charged with grief for the death of Marcellus at the end of Book 6 and ringing with the courage and cruelty of war in the four great last books. Another impossible task. But if it is to be attempted, the translator must be ready to jettison the idiom of Latin and search for the English words that will carry as much as possible of the spirit of the Latin.\n\n### THE AENEID\n\n### BOOK 1  \nSTORM AND BANQUET\n\nI sing of arms and of the man, fated to be an exile, who long  \nsince left the land of Troy and came to Italy to the shores of  \nLavinium; and a great pounding he took by land and sea at the  \nhands of the heavenly gods because of the fierce and unforgetting  \nanger of Juno. Great too were his sufferings in war before he  \ncould found his city and carry his gods into Latium. This was  \nthe beginning of the Latin race, the Alban fathers and the high  \nwalls of Rome. Tell me, Muse, the causes of her anger. How did  \nhe violate the will of the Queen of the Gods? What was his  \n10 offence? Why did she drive a man famous for his piety to such  \nendless hardship and such suffering? Can there be so much  \nanger in the hearts of the heavenly gods?\n\nThere was an ancient city held by colonists from Tyre, opposite  \nItaly and the distant mouth of the river Tiber. It was a city  \nof great wealth and ruthless in the pursuit of war. Its name was  \nCarthage, and Juno is said to have loved it more than any other  \nplace, more even than Samos. Here the goddess kept her armour.  \nHere was her chariot, and this was the city she had long  \n20 favoured, intending to give it sovereignty over the peoples of  \nthe earth, if only the Fates would allow it. But she had heard  \nthat there was rising from the blood of Troy a race of men who  \nin days to come would overthrow this Tyrian citadel; a people  \nproud in war and rulers of a great empire would come to sack  \nthe land of Libya; this is the destiny the Fates were unrolling.  \nThese were the fears of the daughter of Saturn, and she had not  \nforgotten the war she had fought long since at Troy for her  \nbeloved Argos, nor had her bitter resentment and the reasons  \nfor it ever left her mind. There still rankled deep in her heart the  \njudgement of Paris and the injustice of the slight to her beauty,  \nher loathing for the whole stock of Dardanus and her fury at  \nthe honours done to Ganymede, whom her husband Jupiter had  \ncarried off to be his cup-bearer. With all this fuelling her anger  \n30 she was keeping the remnants of the Trojans, those who had  \nescaped the savagery of Achilles and the Greeks, far away from  \nLatium, driven by the Fates to wander year after year round all  \nthe oceans of the world. So heavy was the cost of founding the  \nRoman race.\n\nThe Trojans were in high spirits. They were almost out of  \nsight of Sicily and heading for the open sea with the wind astern  \nand their bronze prows churning the salt sea to foam, as Juno  \nbrooded, still nursing the eternal wound deep in her breast: 'Am  \nI to admit defeat and give up my attempt to keep the king of the  \nTrojans away from Italy? So the Fates do not approve! Yet  \n40 Pallas Athene could fire the fleet and drown my own Argives in  \nthe sea because of the guilt of one man, the mad passion of Ajax,  \nson of Oileus. With her own hand she threw the consuming fire  \nof Jupiter from the clouds, shattering his ships and sending  \nwinds to churn up the level sea. Then, as he breathed out flame  \nfrom his breast where the thunderbolt had pierced it, she caught  \nhim up in a whirlwind and impaled him on a jagged rock. But  \nhere am I, the Queen of the Gods, the sister of Jupiter and his  \nwife, and I have waged war all these years against a whole race  \nof men! Is there no one left who worships the godhead of Juno?  \nWill there be no one in the future to pray to me and lay an  \noffering on my altars?'\n\n50 These are the thoughts the goddess turned over in her burning  \nheart as she came to Aeolia, the home of the clouds, a place  \nteeming with the raging winds of the south. Here Aeolus is king  \nand here in a vast cavern he keeps in subjection the brawling  \nwinds and howling storms, chained and bridled in their prison.  \nThey murmur in loud protest round bolted gates in the mountainside  \nwhile Aeolus sits in his high citadel, holding his sceptre,  \nsoothing their spirits and tempering their angry passions. But  \nfor him they would catch up the sea, the earth and the deeps of  \nthe sky and sweep them along through space. In fear of this, the  \n60 All-powerful Father banished them to these black caverns with  \nmassive mountains heaped over them, and gave them under a  \nfixed charter a king who knew how to hold them in check or,  \nwhen ordered, to let them run with free rein. It was to him that  \nJuno made supplication in these words: 'I come to you, Aeolus,  \nbecause the Father of the Gods and King of Men has given you  \nthe power to calm the waves of the sea or raise them by your  \nwinds. A race of men hateful to me is sailing the Tyrrhenian sea  \ncarrying Ilium to Italy, along with the Penates, their defeated  \ngods. Whip up your winds. Overwhelm their ships and sink  \n70 them. Drive their fleet in all directions and scatter their bodies  \nover the sea. I have fourteen nymphs of the rarest beauty and  \nthe loveliest of them all is Deiopea. I shall make her yours and  \njoin you in lawful wedlock. If you do me this service, she shall  \nspend all her years with you and make you the father of beautiful  \nchildren.'\n\nTo this Aeolus made answer: 'Your task, O queen, is to decide  \nyour wishes; my duty is to carry out your orders. It is thanks to  \nyou that I rule this little kingdom and enjoy this sceptre and the  \nblessing of Jupiter. Through you I have a couch to lie on at the  \n80 feasts of the gods, and my power over cloud and storm comes  \nfrom you.'\n\nAt these words he struck the side of the hollow mountain  \nwith the butt of his spear and the winds seemed to form a  \ncolumn and pour out through an open gate to blow a hurricane  \nover the whole earth. The east wind and the south and the  \nsouth-west with squall upon squall fell upon the sea at once,  \nwhipping it up from its bottom-most depths and rolling huge  \nwaves towards its shores. Men shouted, ropes screamed, clouds  \nsuddenly blotted out the light of the sky from the eyes of the  \nTrojans and black night brooded on the sea as the heavens  \n90 thundered and lightning flashed again and again across the sky.  \nWherever the Trojans looked, death stared them in the face. A  \nsudden chill went through Aeneas and his limbs grew weak.  \nGroaning, he lifted his hands palms upward to the stars and  \ncried: 'Those whose fate it was to die beneath the high walls of  \nTroy with their fathers looking down on them were many, many  \ntimes more fortunate than I. O Diomede, bravest of the Greeks,  \nwhy could I not have fallen to your right hand and breathed out  \nmy life on the plains of Troy, where fierce Hector fell by the  \n100 sword of Achilles, where great Sarpedon lies and where the river  \nSimois caught up so many shields and helmets and bodies of  \nbrave men and rolled them down its current?'\n\nEven as he threw out these words, a squall came howling  \nfrom the north, catching his sail full on and raising the waves to  \nthe stars. The oars broke, the prow was wrenched round, and  \nas they lay beam on to the seas, there came towering over them  \na sheer mountain of water. Some of the ships were hanging on  \nthe crests of the waves; for others the waters opened and in the  \ntroughs could be seen the sea-bed and the seething sand. Three  \nof them were caught by the south wind and driven off course  \non to a reef hidden in mid-ocean \u2013 Italians know it as the Altars  \n110 \u2013 a huge spine of rock just under the surface; three of them the  \nsoutheaster took and carried helplessly from the high sea on to  \nthe sandbanks of the Syrtes, ran them aground and blocked  \nthem in with walls of sand; before the very eyes of Aeneas, the  \nship that carried the faithful Orontes and his Lycians was struck  \non the stern by a great sea and the helmsman was swept away  \nhead first into the water. Three times she spun round on the  \nsame spot till the swift whirlpool sucked her down. Here and  \nthere men could be seen swimming in the vast ocean, and with  \nthem in the waves their armour, spars of wood and the treasures  \n120 of Troy. One by one the stout ships of Ilioneus and brave  \nAchates, then Abas and old Aletes, succumbed to the storm.  \nThe fastenings of the ships' sides were loosened, the deadly  \nwater poured in and the timbers sprang.\n\nNeptune, meanwhile, observed the loud disturbance of the  \nocean, the rampaging of storms, the draining of his deepest  \npools, and was moved to anger. Rising from the depths, he lifted  \nhis head high above the crests of the waves and looked serenely  \nout over the sea at Aeneas' fleet scattered over the face of the  \nwaters and the Trojans overwhelmed by the waves and by the  \n130 rending of the sky. He recognized at once the anger and the  \ncunning of his sister Juno and instantly summoned the east wind  \nand the west and spoke to them in these words: 'Is it your noble  \nbirth that has made you so sure of yourselves? Do you winds  \nnow dare to move heaven and earth and raise these great masses  \nof water without my divine authority? I could take you now and  \n...but first I must still the waves you have stirred up. For any  \ncrimes you commit in the future, you will pay a dearer price.  \nAway with you and take this message to your king: \"He is not  \nthe one who has jurisdiction over the sea or holds the trident  \nthat knows no pity. That is my responsibility, given to me by  \n140 lot. His domain, O Eurus, wind of the east, is the huge crags  \nwhere you have your home. That is where Aeolus can do his  \nswaggering, confining his rule to the closed walls of the prison  \nof the winds.\" '\n\nThese were his words, and before he had finished speaking,  \nhe was calming the swell, dispersing the banked clouds and  \nbringing back the sun. Triton and the sea nymph Cymothoe  \nheaved and strained as they pushed the ships off jagged rocks,  \nwhile Neptune himself lifted them out of the sandbanks with  \nhis trident and opened up the vast Syrtes, restraining the sea as  \nhe skimmed along with his chariot wheels touching the crests of  \nthe waves. As when disorder arises among the people of a great  \ncity and the common mob runs riot, wild passion finds weapons  \n150 for men's hands and torches and rocks start flying; at such a  \ntime if people chance to see a man who has some weight among  \nthem for his goodness and his services to the state, they fall  \nsilent, standing and listening with all their attention while his  \nwords command their passions and soothe their hearts \u2013 so did  \nall the crashing of the sea fall silent and Father Neptune, looking  \nout over the waves, drove the horses of his chariot beneath a  \nclear sky and gave them rein to fly before the wind.\n\nAeneas and his men were exhausted, and making what speed  \nthey could for the nearest land, they set course for the coast of  \n160 Libya. There is a place where a harbour is formed by an island  \nblocking the mouth of a long sound. As the waves come in from  \nthe open sea and break on the sides of this island, they are divided  \ninto the deep inlets of the bay. Rock cliffs are everywhere. A  \ngreat pinnacle threatens the sky on either side, and beneath all  \nthis the broad water lies still and safe. At the end of the bay  \nthere rises a backcloth of shimmering trees, a dark wood with  \nquivering shadows, looming over the water, and there, at the  \nfoot of this scene, is a cave of hanging rocks, a home for the  \nnymphs, with fresh spring water inside it and seats in the virgin  \nrock. Here there is no need of chains to moor the weary ships,  \n170 or of anchors with hooked teeth to hold them fast. This is where  \nAeneas put in with seven ships gathered from all the Trojan fleet,  \nand great was their longing for the land as they disembarked and  \nstepped at last on to the shore and threw their sea-wasted bodies  \ndown on the sand. First of all Achates struck a spark from the  \nflint, caught it in some leaves, fed the flame by putting dry twigs  \nround it and set the fire going with brushwood. Then weary as  \nthey were after all their labours, they laid out their corn, the gift  \nof the goddess Ceres, all tainted with salt, and the goddess's  \nown implements and set themselves to scorch with flame this  \ngrain they had saved from the sea and to grind it on stone.\n\n180 Meanwhile Aeneas climbed a rock to get a view over the  \nwhole breadth of the ocean and see if there was any trace of the  \nstorm-tossed Antheus or of the double-banked Trojan galleys,  \nCapys perhaps, or Caicus' armour high on the poop. There was  \nnot a ship to be seen, but he did see three stags wandering about  \nthe shore with all their herd behind them grazing the low ground  \nin a long line. He stopped in his tracks and snatched his bow  \n190 and swift arrows from the trusty Achates. First he took down  \nthe three leaders with their high heads of branching antlers. The  \nwhole of the rest of the herd scattered into the leafy cover of the  \nwood, but not before he succeeded in stretching seven huge  \ncarcasses on the ground, one for each of the ships. He then made  \nfor the harbour and gave them out to all his men. Last of all he  \nshared out the wine the good Acestes with a hero's generosity  \nhad poured into casks for them as they left the shores of Sicily.  \nThen, as they mourned, he comforted them, saying: 'My friends,  \nthis is not the first trouble we have known. We have suffered  \n200 worse before, and this too will pass. God will see to it. You have  \nbeen to Scylla's cave and heard the mad dogs howling in the  \ndepths of it. You have even survived rocks thrown by the  \nCyclops. So summon up your courage once again. This is no  \ntime for gloom or fear. The day will come, perhaps, when it will  \ngive you pleasure to remember even this. Whatever chance may  \nbring, however many hardships we suffer, we are making for  \nLatium, where the Fates show us our place of rest. There it is  \nthe will of God that the kingdom of Troy shall rise again. Your  \ntask is to endure and save yourselves for better days.' These  \nwere his words, but he was sick with all his cares. He showed  \nthem the face of hope and kept his misery deep in his heart.\n\n210 His men went briskly to work preparing the coming feast.  \nThey flayed the hide off the ribs and exposed the flesh. Some cut  \nit into quivering slices and speared it on spits. Others laid out  \ncauldrons of water on the shore and lit fires. Then at last they  \nate, and recovered their strength, lying on the grass and taking  \ntheir fill of old wine and rich venison. When their hunger was  \nsatisfied and the remains of the feast removed, they talked at  \nlength about their missing comrades, not knowing whether to  \nhope or fear, wondering whether they were still alive or whether  \nat that very moment they were drawing their last breath and  \n220 beyond all calling. Most of all did Aeneas, who loved his men,  \nmourn to himself the loss of eager Orontes and Amycus and the  \ncruel death of Lycus, then brave Gyas, and brave Cloanthus.\n\nNow the feast was ended and Jupiter was looking down from  \nthe height of heaven on the sea flying with sails and the land far  \nbeneath him, on the shores of the seas and the far-spread  \npeoples, when suddenly he stopped in his survey at the highest  \npoint of the sky and fixed his eyes upon the kingdom of Libya.  \nEven as he was turning over in his mind all the suffering that he  \nsaw, his daughter Venus came to him, her shining eyes brimming  \n230 with tears, and spoke with a sadness greater than his own: 'You  \nwho rule the affairs of gods and men with your eternal law and  \nat whose lightning we are all afraid, what great harm has my  \nson Aeneas been able to do to you? What crime have the Trojans  \ncommitted that they should suffer all this loss of life and the  \nwhole world be closed to them for the sake of Italy? Did you  \nnot promise that with the rolling years there would come a time  \nwhen from this stock the Romans would arise? From this blood  \nof Teucer, recalled to Italian soil, there would come leaders of  \nmen who would hold power over every land and sea. O father,  \nfather, has some argument changed your mind? As for me, I  \nused to console myself with this for the cruel fall and sack of  \n240 Troy, by weighing one destiny against another. But unrelieved  \nmisfortune is now hounding these men from disaster to disaster.  \nO great king, what end do you set to their labours? The Greeks  \nwere all around Antenor, but he escaped them, made his way  \nsafely into the Illyrian Gulf and the heartlands of the kingdom  \nof the Liburnians, and then went beyond the mouth of the  \nTimavus. From there with a great roar from inside the mountain,  \na sea of water bursts out of nine mouths and covers the fields  \nwith a sounding ocean. But in this place he founded the city of  \nPatavium as a home for his Trojans and gave them a name.  \nThere he dedicated the arms with which he fought at Troy and  \n250 there he now lives in settled peace and quiet. But as for us, your  \nown children, to whom you grant a place in the citadel of  \nheaven, we lose our ships. It is unspeakable. We are betrayed  \nand kept far away from the shores of Italy because there is one  \nwho hates us. Is this our reward for piety and obedience? Is this  \nhow you bring us to our kingdom?'\n\nThe Father of Gods and Men, looking at his daughter with  \nthe smile that clears the sky and dispels the storms, kissed her  \nlightly on the lips, and said: 'Spare yourself these fears, my  \nlady from Cythera. You can be sure that the destiny of your  \ndescendants remains unchanged. You will see the city of Lavinium  \n260 and its promised walls. You will take great-hearted Aeneas  \nup to the stars of heaven. No argument changes my mind. But  \nnow, since this anxiety is gnawing at you, I shall tell you more,  \nunrolling for you the secrets of the scroll of the Fates. He will  \nwage a great war in Italy and crush its fierce tribes. He will build  \nwalls for his people and establish their way of life, until a third  \nsummer has seen him reigning in Latium and a third winter has  \npassed after the subjection of the Rutulians. But the reign of his  \nson Ascanius, who now receives the second name Iulus (it was  \nIlus while the kingdom of Ilium still stood), shall last while the  \n270 months of thirty long years revolve, and he shall transfer his  \nkingdom from its seat at Lavinium and build a city with powerful  \nfortifications at Alba Longa. Here the rule of the race of  \nHector will last for three hundred long years until Ilia the  \npriestess queen, heavy with the seed of Mars, shall give birth to  \ntwin sons. Then Romulus shall receive the people, wearing with  \njoy the tawny hide of the wolf which nursed him. The walls he  \nbuilds will be the walls of Mars and he shall give his own name  \nto his people, the Romans. On them I impose no limits of time  \nor place. I have given them an empire that will know no end.  \n280 Even angry Juno, who is now wearying sea and land and sky  \nwith her terrors, will come to better counsel and join with me  \nin cherishing the people of Rome, the rulers of the world, the  \nrace that wears the toga. So it has been decreed. There will come  \na day, as the years glide by, when the house of Assaracus will  \nreduce Achilles' Pthia and glorious Mycenae to slavery and will  \nconquer and rule the city of Argos. From this noble stock there  \nwill be born a Trojan Caesar to bound his empire by Oceanus  \nat the limits of the world, and his fame by the stars. He will be  \ncalled Julius, a name passed down to him from the great Iulus.  \n290 In time to come, have no fear, you will receive him in the sky,  \nladen with the spoils of the East. He too will be called upon in  \nprayer. Then wars will be laid aside and the years of bitterness  \nwill be over. Silver-haired Truth and Vesta, and Romulus Quirinus  \nwith his brother Remus, will sit dispensing justice. The  \ndread Gates of War with their tight fastenings of steel will then  \nbe closed, and godless Strife will sit inside them on his murderous  \narmour roaring hideously from bloody mouth, hands shackled  \nbehind his back with a hundred bands of bronze.'\n\nSo spoke Jupiter, and he sent down Mercury, the son of Maia,  \nto make the lands and the citadel of the new city of Carthage  \nhospitable to the Trojans, in case Dido, in her ignorance of  \n300 destiny, should bar her country to them. Through the great  \nexpanse of air he flew, wielding his wings like oars, and soon  \nalighted on the shores of Libya. There he lost no time in carrying  \nout the commands of Jupiter, and in accordance with the divine  \nwill the Carthaginians laid aside their fiery temper. Most of all  \nthe queen took into her heart a feeling of quiet and kindness  \ntowards the Trojans.\n\nBut all that night the dutiful Aeneas was turning many things  \nover in his mind. As soon as life-giving morning came, he decided  \nto go out and explore this new land and bring back to his men  \na true account of the shores to which the winds had driven him,  \nand the beasts and men who lived there, if there were any men,  \n310 for he saw no signs of cultivation. So, leaving his ships hidden  \nin the wooded cove under the overhanging rocks, and shut in  \non every side by trees and quivering shade, he set out alone with  \nAchates, gripping two broad-bladed steel spears in his hand. As  \nhe walked through the middle of the wood, his mother came to  \nmeet him looking like a Spartan girl out hunting, wearing the  \ndress of a Spartan girl and carrying her weapons, or like the  \nThracian Harpalyce, as she wearies horses with her running and  \noutstrips the swift current of the river Hebrus. She had a light  \nbow hanging from her shoulders in hunting style, her hair was  \n320 unbound and streaming in the wind and her flowing dress was  \ncaught up above the knee. 'Hey there, soldiers,' she called out  \nto them, 'do you happen to have seen one of my sisters wandering  \nabout here or in full cry after the foaming boar? She was  \nwearing a spotted lynx skin and had a quiver hanging from  \nher belt.'\n\nSo spoke Venus, and Venus' son so began his reply: 'I have  \nneither seen nor heard any of your sisters. But how am I to  \naddress a girl like you? Your face is not the face of a mortal,  \nand you do not speak like a human being. Surely you must be a  \ngoddess? Are you Diana, sister of Apollo? Are you one of the  \n330 sister nymphs? Be gracious to us, whoever you may be, and  \nlighten our distress. Tell us what sky this is we find ourselves at  \nlast beneath. What shore of the world is this on which we now  \nwander, tossed here by the fury of wind and wave? We do not  \nknow the place. We do not know the people. Tell us and many  \na victim will fall by my right hand before your altars.'\n\nVenus replied: 'I am sure I deserve no such honour. Tyrian  \ngirls all carry the quiver and wear purple boots with this high  \nankle binding. This is a Phoenician kingdom you are looking at.  \nWe are Tyrians. This is the city of the people of Agenor, but the  \nland belongs to the Libyans, a race not easy to handle in war.  \n340 Dido, who came from the city of Tyre to escape her brother,  \nholds sway here. There was a crime long ago. It is a long and  \nwinding story, but I shall trace its outlines for you. Her father  \nhad given her in marriage to Sychaeus, the wealthiest of the  \nPhoenicians. They were joined with all the due rites of a first  \nmarriage and great was the love the poor queen bore for him.  \nBut the kingdom of Tyre was ruled by her brother Pygmalion,  \nthe vilest of criminals. A mad passion came between the two  \nmen. In blind lust for his gold the godless Pygmalion attacked  \n350 him without warning, ambushing him at the altar. With no  \nthought for his sister's love he killed Sychaeus and for a long  \ntime concealed what he had done. Dido was sick with love and  \nhe deceived her with false hopes and empty pretences. But one  \nnight there appeared to her in a dream the very ghost of her  \nunburied husband. He lifted up his face, pale with the strange  \npallor of the dead, and, baring the sword wounds on his breast,  \nhe pointed to the altar where he had been killed and revealed  \nthe whole horror of the crime that had been hidden in their  \nhouse. He then urged her to escape with all speed from their  \nnative land, and to help her on her wanderings he showed  \nher where to find an ancient treasure buried in the earth, an  \n360 incalculable weight of silver and gold. This moved Dido to plan  \nher escape and gather followers, men driven by savage hatred  \nor lively fear of the tyrant. They seized some ships which happened  \nto be ready for sea. They loaded them with the gold and  \nsailed away with the wealth Pygmalion had coveted. The woman  \nled the whole undertaking. When they arrived at the place where  \nyou will now see the great walls and rising citadel of the new  \ncity of Carthage, they bought a piece of land called the \"Byrsa\",  \nthe animal's hide, as large an area as they could include within  \nthe hide of a bull. But now tell me, who are you? What country  \n370 have you sailed from? Where are you making for?'\n\nIn reply to her questions Aeneas drew a great sigh from the  \nbottom of his heart and said: 'O goddess, if I were to start at the  \nbeginning and retrace our whole story, and if you had the time  \nto listen to the annals of our suffering, before I finish the doors  \nof Olympus would close and the Evening Star would lay the day  \nto rest. We come from the ancient city of Troy, if the name of  \nTroy has ever reached your ears. We have sailed many seas and  \nby the chance of the winds we have been driven ashore here in  \nLibya. I am Aeneas, known for my devotion. I carry with me on  \nmy ships the gods of my home, the Penates, wrested from my  \nenemies, and my fame has reached beyond the skies. I am  \n380 searching for my fatherland in Italy. My descent is from highest  \nJupiter. With my goddess mother to show the way, I embarked  \nupon the Phrygian sea with twenty ships, following the destiny  \nwhich had been given to me, and now a bare seven of them  \nremain, and these torn to pieces by wind and wave. I am a  \nhelpless stranger, driven out of Europe and out of Asia, tramping  \nthe desert wastes of Libya.'\n\nVenus could listen to no more. She broke in on the tale of his  \nsufferings, saying: 'Whoever you are I do not believe you are  \nhated by the gods: you live and breathe and have reached this  \n390 Tyrian city. Go on now from here to the queen's door. I can tell  \nyou that your comrades are restored and your fleet returned to  \nyou. The winds have veered to the north and blown them safe  \nto shore. All this is true unless my parents have failed in their  \nefforts to teach me to interpret the flight of birds. Look at these  \ntwelve swans flying joyfully in formation. The eagle of Jupiter  \nwas swooping down on them from the heights of heaven and  \nscattering them over the open sky, but now look at them in  \ntheir long column. Some are reaching land. Some have already  \nreached it and are looking down on it. Just as they have come  \nto their home and their flock has circled the sky in play, singing  \nas they fly with whirring wings, so your ships and your warriors  \n400 are either already in port or crossing the bar in full sail. Go on  \nnow, and follow where the road takes you.'\n\nWhen she had finished speaking and was turning away, her  \nneck shone with a rosy light and her hair breathed the divine  \nodour of ambrosia. Her dress flowed free to her feet and as she  \nwalked he knew she was truly a goddess. As she hastened away,  \nhe recognized her as his mother and called after her: 'Why do  \nyou so often mock your own son by taking on these disguises?  \nYou too are cruel. Why am I never allowed to take your hand  \nin mine, to hear your true voice and speak to you as you really are?'\n\n410 With these reproaches he took the road that led to the city,  \nbut Venus hedged them about with a thick mist as they walked.  \nThe goddess spread a great veil of cloud over them so that no  \none could see them or touch them or cause any delay or ask the  \nreason for their coming. She herself soared high into the sky and  \ndeparted for Paphos, returning happily to her beloved home  \nwhere she has her temple, and a hundred altars steam with the  \nincense of Sheba and breathe the fragrance of fresh-cut flowers.\n\nMeanwhile Aeneas and Achates hurried on their way, following  \n420 the track, and they were soon climbing the great hill which  \ntowered over the city and looked down upon the citadel opposite.  \nAeneas was amazed by the size of it where recently there  \nhad been nothing but shepherds' huts, amazed too by the gates,  \nthe paved streets and all the stir. The Tyrians were working with  \na will: some of them were laying out the line of walls or rolling  \nup great stones for building the citadel; others were choosing  \nsites for building and marking them out with the plough; others  \nwere drawing up laws and electing magistrates and a senate  \nwhom they could revere; on one side they were excavating a  \nharbour; on the other laying deep foundations for a theatre and  \nquarrying huge columns from the rock to make a handsome  \n430 backdrop for the stage that was to be. They were like bees at  \nthe beginning of summer, busy in the sunshine all through the  \nflowery meadows, bringing out the young of the race, just come  \nof age, or treading the oozing honey and swelling the cells with  \nsweet nectar, or taking the loads as they come in or mounting  \nguard to keep the herds of idle drones out of their farmstead.  \nThe hive seethes with activity and the fragrance of honey  \nflavoured with thyme is everywhere. 'How fortunate they are!'  \ncried Aeneas, now looking up at the high tops of the buildings.  \n' _Their_ walls are already rising!' and he moved on through the  \n440 middle of the people, hedged about by the miraculous cloud,  \nand no one saw him.\n\nThere was a wooded grove which gave abundant shade in the  \nmiddle of the city. When first the Phoenicians had been driven  \nthere by wind and wave, Juno, the Queen of the Gods, had led  \nthem to this spot where they had dug up the head of a spirited  \nstallion. This was a sign that from generation to generation they  \nwould be a race glorious in war and would have no difficulty in  \nfinding fields to graze. Here Sidonian Dido was building for  \nJuno a huge temple rich with offerings and rich, too, with the  \npresence of the goddess. It was a raised temple, and at the top  \nof its steps the threshold was of bronze, the beams were jointed  \nwith bronze and the bronze doors grated as they turned in their  \n450 sockets. Here in this grove Aeneas saw a strange sight which for  \nthe first time allayed his fears. Here for the first time he dared  \nto hope, and despite all the calamities of the past to have  \nbetter confidence in the future. While waiting for the queen and  \nstudying everything there was to see under the roof of this huge  \ntemple, as he marvelled at the good fortune of the city, the skill  \nof the workmen and all the works of their hands, he suddenly  \nsaw, laid out in order, depictions of the battles fought at Troy.  \nThe Trojan War was already famous throughout the world. The  \ntwo sons of Atreus were there, and Priam, and Achilles who  \nhated both sides. Aeneas stopped, and wept, and said to Achates:  \n460 'Is there anywhere now on the face of this earth that is not  \nfull of the knowledge of our misfortunes? Look at Priam. Here  \ntoo there is just reward for merit, there are tears for suffering  \nand men's hearts are touched by what man has to bear. Forget  \nyour fears. We are known here. This will give you some hope  \nfor the future.'\n\nAs he spoke these words, he was feeding his spirit with the  \nempty images and groaning, and rivers of tears washed down  \nhis cheeks as he gazed at the fighting round the walls of Troy.  \nOn one side Greeks were in flight with Trojan warriors hard on  \ntheir heels; on the other Trojans were retreating and Achilles  \n470 with his crested helmet was pursuing them in his chariot. He  \nwept, too, when he recognized the white canvas of the tents of  \nRhesus nearby. It was the first sleep of the night. The tents had  \nbeen betrayed, and were being torn down by Diomede, red with  \nthe blood of all the men he had slaughtered. He stole the fiery  \nhorses and took them back to the Greek camp before they could  \ncrop the grass of Troy or drink the water of the Xanthus. In  \nanother part of the picture poor Troilus, a mere boy and no  \nmatch for Achilles, had lost his armour and was in full flight.  \nHis horses had run away with the chariot and he was being  \ndragged along helpless on his back behind it, still holding on to  \nthe reins. His neck and hair were trailing along the ground and  \nthe end of his spear was scoring the dust behind him. The women  \nof Troy, meanwhile, were going in supplication to the temple  \n480 of Pallas Athene, but the goddess was hostile to them. Their hair  \nwas unbound, and they were carrying a robe to offer her, beating  \ntheir breasts in grief, but her head was turned from them and  \nher eyes were fixed upon the ground. There too was Achilles.  \nHe had dragged Hector three times round the walls of Troy,  \nand now was selling his dead body for gold. Aeneas groaned  \nfrom the depths of his heart to see the armour stripped off him,  \nthe chariot, the corpse of his dear friend and Priam stretching  \nout his feeble hands. Aeneas even recognized himself in the  \nconfusion of battle, with the leaders of the Greeks all around  \n490 him. There were the warriors of the East, the armour of Memnon  \nand his dark skin. The Amazons were there in their thousands  \nwith crescent shields and their leader Penthesilea in the middle  \nof her army, ablaze with passion for war. There, showing her  \nnaked breast supported by a band of gold, was the warrior  \nmaiden, daring to clash with men in battle.\n\nWhile Trojan Aeneas stood gazing, rooted to the spot and  \nlost in amazement at what he saw, queen Dido in all her beauty  \narrived at the temple with a great crowd of warriors around her.  \nShe was like Diana leading the dance on the banks of the Eurotas  \nor along the ridges of Mount Cynthus with a thousand mountain  \n500 nymphs thronging behind her on either side. She carries her  \nquiver on her shoulder, and as she walks, she is the tallest of all  \nthe goddesses. Her mother Latona does not speak, but a great  \njoy stirs her heart at the sight of her. Dido was like Diana, and  \nlike Diana she bore herself joyfully among her people, urging  \non their work for the kingdom that was to be. Then in the  \ndoorway of the goddess, under the middle of the vault of the  \ntemple, she took her seat with her armed guards about her.  \nThere, as she was giving laws and rules of conduct to her people,  \nand dividing the work that had to be done in equal parts or  \nallocating it by lot, Aeneas suddenly saw a great throng approaching,  \n510 Antheus, Sergestus, brave Cloanthus and the other  \nTrojans who had been scattered over the sea by the dark storm  \nand swept away to distant shores. He was astounded, and  \nAchates, too, was stunned with joy and fear. They burned with  \nlonging to clasp the hands of their comrades, but were at a  \nloss because they did not understand what they saw. They did  \nnothing, but stayed hidden in their cloak of cloud, waiting to  \nlearn how Fortune had dealt with their comrades. On what  \nshore had they left their fleet? Why were they here? For these  \nwere picked men coming from each of the ships to plead their  \ncase, and they were now walking to the temple with shouting  \nall about them.\n\n520 They came in and were allowed to address the queen. Ilioneus,  \nthe oldest of them, made this appeal: 'You are a queen whom  \nJupiter has allowed to found a new city and curb proud peoples  \nwith your justice; we are the unhappy men of Troy, blown by  \nthe winds over all the oceans of the world, and we come to you  \nas suppliants. Save our ships from the impious threat of fire. We  \nare god-fearing men. Take pity on us. Look more closely at us  \n\u2013 we have not come to Libya to pillage your homes and their  \ngods, to take plunder and drive it down to the shore. Such  \nviolence and arrogance are not to be found in the hearts of the  \ndefeated.\n\n530 'There is a place which Greeks know by the name Hesperia.  \nIt is an ancient land, strong in war and rich in the fertility of its  \nsoil. It was once tilled by Oenotrians, but now we believe  \ntheir descendants have called themselves Italians after their king  \nItalus. This is where we were steering when suddenly Orion rose  \nin cloud and tempest and drove us on to hidden shallows, the  \nsea overwhelmed us and fierce southerly squalls scattered us far  \nand wide among breakers and uncharted rocks. A few of us  \ndrifted ashore here to your land. What manner of men are these?  \n540 Is this a country of barbarians that allows its people to act in  \nthis way? Sailors have a right to the shore and we are refused it.  \nThey make war on us and will not let us set foot on land. You  \nmay be no respecters of men. You may fear no men's arms, but  \nthink of the gods, who see right and wrong and do not forget.  \nOur king was Aeneas. He had no equal for his piety and his care  \nfor justice, and no equal in the field of battle. If the Fates still  \nprotect him, if he still breathes the air of heaven, if he is not  \neven now laid low among the merciless shades, you would have  \nnothing to fear or to regret by taking the lead in a contest of  \n550 kindness. In the land of Sicily we have arms and cities and the  \ngreat Acestes, sprung from Trojan blood. Allow us to draw up  \nour storm-battered ships, to hew timbers in your woods and  \nshape new oars, so that we can make for Italy and Latium with  \njoy in our hearts, if indeed we go to Italy with our comrades  \nand our king; but if they are lost, if you, great Father of the  \nTrojans, are drowned in the sea off Libya, and there are no  \nhopes left in Iulus, then we can at least go back to where we  \ncame from across the Sicilian sea, to the place that is prepared  \n560 for us, and return to king Acestes.' So spoke Ilioneus and all the  \nTrojans to a man murmured in agreement.\n\nThen Dido looked down at them and made a brief answer:  \n'Have no fear, men of Troy. Put every anxious thought out of  \nyour hearts. This is a new kingdom, and it is harsh necessity  \nthat forces me to take these precautions and to post guards on  \nall our frontiers. But who could fail to know about the people  \nof Aeneas and his ancestry, about the city of Troy, the valour of  \nits men and the flames of war that engulfed it? We here in  \nCarthage are not so dull in mind as that. The sun does spare a  \nglance for our Tyrian city when he yokes his horses in the  \nmorning. Whether you choose to go to great Hesperia and the  \n570 fields of Saturn, or to the land of Eryx and king Acestes, you will  \nleave here safe under my protection, and I shall give you supplies  \nfor your voyage. Or do you wish to settle with me on an equal  \nfooting, even here in this kingdom of Carthage? The city which  \nI am founding is yours. Draw up your ships on the beach. Trojan  \nand Tyrian shall be as one in my eyes. I wish only that your king  \nAeneas had been driven by the same wind, and were here with  \nyou now. But what I can, I shall do. I shall send men whom I  \ncan trust all along the coast, and order them to cover every  \nfurthest corner of Libya, in case he has been shipwrecked and is  \nwandering in any of the woods or cities.'\n\n580 The brave Achates and Father Aeneas had long been impatient  \nto break out of the cloud, and at Dido's words their eagerness  \nincreased. 'Aeneas,' said Achates, 'son of the goddess, what  \nthoughts are now rising in your heart? You see there is no  \ndanger. Our ships are safe. Our comrades are rescued. Only one  \nof them is missing, and we saw him with our own eyes founder  \nin mid-ocean. Everything else is as your mother Venus said it  \nwould be.'\n\nHe had scarcely finished speaking when the cloud that was  \nall about them suddenly parted and dissolved into the clear sky.  \nAeneas stood there resplendent in the bright light of day with  \nthe head and shoulders of a god. His own mother had breathed  \n590 upon her son and given beauty to his hair and the sparkle of joy  \nto his eyes, and the glow of youth shone all about him. It was  \nas though skilled hands had added embellishments to ivory or  \napplied gilding to silver or Parian marble. Then suddenly, to the  \nsurprise of all, he addressed the queen in these words: 'The man  \nyou are looking for is standing before you. I am Aeneas the  \nTrojan, saved from the Libyan sea, and you, Dido, alone have  \npitied the unspeakable griefs of Troy. We are the remnants left  \nby the Greeks. We have suffered every calamity that land and  \n600 sea could inflict upon us, and have lost everything. And now  \nyou offer to share your city and your home with us. It is not  \nwithin our power to repay you as you deserve, nor could whatever  \nsurvives of the Trojan race, scattered as it is over the face  \nof the wide earth. May the gods bring you the reward you  \ndeserve, if there are any gods who have regard for goodness, if  \nthere is any justice in the world, if their minds have any sense of  \nright. What happy age has brought you to the light of life? What  \nmanner of parents have produced such a daughter? While rivers  \nrun into the sea, while shadows of mountains move in procession  \nround the curves of valleys, while the sky feeds the stars, your  \nhonour, your name, and your praise will remain for ever in  \n610 every land to which I am called.' As he spoke, he put out his  \nright hand to his friend Ilioneus and his left to Serestus, then  \ngreeted the others, brave Gyas, and brave Cloanthus.\n\nDido of Sidon was amazed at her first sight of him and then  \nat the thought of the ill fortune he had endured. 'What sort of  \nchance is this,' she exclaimed, 'that hounds the son of a goddess  \nthrough all these dangers? What power has driven you to these  \nwild shores? Are you that Aeneas whom the loving goddess  \nVenus bore to Dardanian Anchises in Phrygia by the river waters  \nof the Simois? I myself remember the Greek Teucer coming to  \n620 Sidon after being exiled from his native Salamis. He was looking  \nto found a new kingdom, and was helped by my father Belus,  \nwho in those days was laying waste the wealth of Cyprus. He  \nhad conquered the island and it was under his control. From  \nthat day on I knew all the misfortunes of the city of Troy. I  \nknew your name and the names of the Greek kings. Teucer  \nhimself, your enemy, held the Teucrians, the people of Troy, in  \nhighest respect and claimed descent from an ancient Teucrian  \nfamily. This is why I now invite your warriors to come into my  \nhouse. I, too, have known ill fortune like yours and been tossed  \nfrom one wretchedness to another until at last I have been  \n630 allowed to settle in this land. Through my own suffering, I am  \nlearning to help those who suffer.'\n\nWith these words she led Aeneas into her royal palace, and  \nas she went she appointed sacrifices to be offered in the temples  \nof the gods. Nor at that moment did she forget Aeneas' comrades  \non the shore, but sent down to them twenty bulls, a hundred  \ngreat bristling hogs' backs and a hundred fat lambs with their  \nmothers, rich gifts to celebrate the day. Meanwhile the inside  \nof her palace was being prepared with all royal luxury and  \nsplendour. They were laying out a banquet in the central hall  \nand the draperies were of proud purple, richly worked. The  \n640 silver was massive on the tables, with the brave deeds of their  \nancestors embossed in gold, a long tradition of feats of arms  \ntraced through many heroes from the ancient origins of the race.\n\nBut a father's love allowed Aeneas' mind no rest, and he asked  \nAchates to go quickly ahead to the ships to take the news to  \nAscanius and bring him back to the city. All his thoughts  \nwere on his dear son Ascanius. He also told Achates to bring  \nback with him as gifts for Dido some of the treasures that  \nhad been rescued from the ruins of Troy, a cloak stiff with  \ngold-embroidered figures and a dress with a border woven of  \nyellow acanthus flowers. These miracles of workmanship had  \n650 been given to Helen of Argos by her mother Leda, and she had  \ntaken them from Mycenae when she came to Troy for her illicit  \nmarriage with Paris. There was also the sceptre which had once  \nbeen carried by Ilione, the eldest daughter of Priam, a necklace  \nof pearls and a double gold coronet set with jewels. Achates set  \noff for the ships in great haste to carry out his instructions.\n\nBut Venus meanwhile was turning over new schemes in her  \nmind and devising new plans. She decided to change the form  \nand features of Cupid, and send him in place of the lovely  \n660 young Ascanius to inflame the heart of the queen, driving her to  \nmadness by the gifts and winding the fire of passion round  \nher bones. For Venus was afraid of the treacherous house of  \nCarthage and the double-tongued people of Tyre. The thought  \nof the bitterness of Juno's hatred burned in her heart, and as  \nnight began to fall and her anxiety kept returning, she spoke to  \nthe winged god of love in these words: 'My dear son, you are  \nthe source of my power. You are my great strength. Only you,  \nmy son, can laugh at the thunderbolts which my father, highest  \nJupiter, hurled against the Giant Typhoeus. To you I come for  \nhelp. I am your suppliant, begging the aid of your divine power.  \nYou well know how Juno's bitter hatred is tossing your own  \nbrother from shore to shore round all the seas of the world and  \n670 you have often grieved to see me grieving. Now he is in the  \nhands of the Phoenician Dido, who is delaying him with honeyed  \nwords, and I am afraid of Juno's hospitality and what it may  \nbring. She will not stand idle when the gate of the future is  \nturning. That is why I am resolved to act first, taking possession  \nof the queen by a stratagem and surrounding her with fire, so  \nthat no power in heaven may change her, but she will be held  \nfast, as I am, in love for Aeneas. As for how you are to achieve  \nthis, listen now and I shall tell you my mind. Aeneas has sent  \nfor his son, whom I so love, and the young prince is preparing  \nto go to the city of Carthage, bringing gifts which have survived  \n680 the hazards of the sea and the burning of Troy. I shall put him  \ninto a deep sleep and hide him in one of my sacred shrines above  \nIdalium or the heights of Cythera, so that he will not know of  \nmy scheme or suddenly arrive to interrupt it. You will have to  \nuse your cunning and take on his appearance for just one night.  \nHe is a boy like yourself and you know him, so put on his  \nfeatures, and when the royal table is flowing with wine that  \nbrings release, and Dido takes you happily on to her lap and  \ngives you sweet kisses, you can then breathe fire and poison into  \nher and she will not know.'\n\n690 Cupid obeyed his beloved mother. He took off his wings and  \nstrutted about copying Iulus' walk and laughing. But the goddess  \npoured quiet and rest into all the limbs of Ascanius, and holding  \nhim to the warmth of her breast, she lifted him into the high  \nIdalian woods, where the soft amaracus breathed its fragrant  \nshade and twined its flowers around him.\n\nNow Cupid was obeying his instructions and was amused to  \nbe escorted by Achates as he took the royal gifts to the Tyrians.  \nWhen he came in, the queen was already seated under a rich  \nawning on a golden couch in the middle of the palace. Presently  \n700 Father Aeneas and after him the men of Troy arrived and  \nreclined on purple coverlets. Attendants gave them water for  \ntheir hands, plied them with bread from baskets and brought  \nthem fine woollen napkins with close-cut nap. Inside were fifty  \nserving-women, whose task it was to lay out the food in order  \nin long lines and honour the Penates by tending their fires. There  \nwere a hundred other female slaves and a hundred men, all of  \nthe same age, to load the tables for the banquet and set out the  \ndrinking cups. The Tyrians, too, came thronging through the  \ndoors, and the palace was full of joy as they took their appointed  \nplaces on the embroidered couches. They admired the gifts  \n710 Aeneas had given. They admired Iulus, the glowing face of the  \ngod and his false words, the cloak and the dress embroidered  \nwith yellow acanthus flowers. But most of all the unfortunate  \nDido, doomed to be the victim of a plague that was yet to come,  \ncould not have her fill of gazing, and as she gazed, moved by  \nthe boy as much as by the gifts, the fire within her grew. After  \nhe had embraced Aeneas and hung on his neck to satisfy the  \ngreat love of his father who was not his father, he went to the  \nqueen. She fixed her eyes and her whole heart on him and  \nsometimes dandled him on her knee, without knowing what a  \ngreat god was sitting there marking her out to suffer. But he was  \n720 remembering his mother, the goddess of the Acidalian spring,  \nand he began gradually to erase the memory of Sychaeus, trying  \nto turn towards a living love a heart that had long been at peace  \nand long unused to passion.\n\nAs soon as the first pause came in the feasting and the tables  \nwere cleared away, they set up great mixing bowls full of wine  \nand garlanded them with flowers. The palace was ringing with  \nnoise and their voices swelled through the spacious hall. Lamps  \nwere lit and hung from the gold-coffered ceilings and the flame  \nof torches routed the darkness. The queen now asked for a  \ngolden bowl heavy with jewels, and filled it with wine unmixed  \n730 with water. From this bowl Belus had drunk, and all the royal  \nline descended from Belus. Then there was silence in the hall as\n\nDido spoke: 'Jupiter, to you we pray, since men say that you  \nordain the laws of hospitality. Grant that this day may be a day  \nof happiness for the Tyrians and the men from Troy, and may  \nour descendants long remember it. Let Bacchus, giver of good  \ncheer, be among us, and kindly Juno, and you, Tyrians, celebrate  \nthis gathering with welcome in your hearts.'\n\nAt these words she poured a libation of wine on the table to  \nhonour the gods, and having poured it, she took it first and just  \ntouched it to her lips. She then passed it to Bitias with a smile  \nand a challenge. Nothing loth, he took a great draught from the  \n740 golden bowl foaming to the brim, and bathed himself in wine.  \nThe other leaders of the Carthaginians did the same after him.  \nLong-haired Iopas, the pupil of mighty Atlas, then sang to his  \ngilded lyre of the wanderings of the moon and the labours of  \nthe sun, the origin of the human race and of the animals, the  \ncauses of rain and of the fires of heaven, of Arcturus, of  \nthe Hyades, bringers of rain, of the two Triones, the oxen of the  \nPlough; why the winter suns are so eager to immerse themselves  \nin the ocean, and what it is that slows down the passage of the  \nnights. The Tyrians applauded again and again and the Trojans  \nfollowed their lead.\n\nSo the doomed Dido was drawing out the night with all  \nmanner of talk, drinking long draughts of love as she asked  \n750 question after question about Priam and Hector, what armour  \nMemnon, son of the Dawn, was wearing when he came, what  \nkind of horses did Diomede have, how tall was Achilles. 'But  \nno,' she said, 'come tell your hosts from the beginning about  \nthe treachery of the Greeks, the sufferings of your people and  \nyour own wanderings, for this is now the seventh summer that  \nhas carried you as a wanderer over every land and sea.'\n\n## BOOK 2  \nTHE FALL OF TROY\n\nThey all fell silent, gazing at Father Aeneas, and he began to  \nspeak from his raised couch: 'O queen, the sorrow you bid  \nme bring to life again is past all words, the destruction by the  \nGreeks of the wealth of Troy and of the kingdom that will be  \nmourned for ever, and all the horrors I have seen, and in which  \nI played a large part. No man could speak of such things and  \nnot weep, none of the Myrmidons of Achilles or the Dolopians  \nof Neoptolemus, not even a follower of Ulixes, a man not prone  \nto pity. Besides, the dewy night is already falling fast from the  \n10 sky and the setting stars are speaking to us of sleep. But if you  \nhave such a great desire to know what we suffered, to hear in  \nbrief about the last agony of Troy, although my mind recoiled  \nin anguish when you asked and I shudder to remember, I shall  \nbegin:\n\nYear after year the leaders of the Greeks had been broken in  \nwar and denied by the Fates, until, with the aid of the divine  \nskill of Pallas Athene, they built a horse the size of a mountain,  \ncutting pine trees to weave into it for ribs. They pretended it  \nwas a votive offering for their safe return to Greece, and that  \nwas the story on men's lips. Then they chose some men by lot  \nfrom their best warriors and shut them up in the darkness of its  \n20 belly, filling the vast cavern of its womb with armed soldiers.\n\nWithin sight of the mainland is the island of Tenedos, famous  \nin story. While the kingdom of Priam stood, it was rich and  \nprosperous, but now there is only a bay giving a none too safe  \nanchorage for ships. The Greeks sailed here and took cover on  \nits lonely shore. We thought they had left us and sailed for  \nMycenae with favouring winds. The whole of Troy then shook  \nitself free of its long sorrow. The gates were thrown open and  \nthe people went out rejoicing to see the Greek encampment, the  \ndeserted shore and all the places abandoned by the enemy. Here  \nwas the Dolopian camp and here fierce Achilles had his tent.  \n30 This was where the fleet was drawn up. This was where they  \nused to fight their battles. Some gazed at the fatal offering to the  \nvirgin goddess Minerva and marvelled at the huge size of the  \nhorse. Thymoetes was the first to urge them to drag it inside  \ntheir walls and set it on their citadel, whether it was treachery  \nthat made him speak, or whether the Fates of Troy were already  \nmoving towards that end. But Capys, and those of sounder  \njudgement, did not trust this offering. They thought it was some  \ntrick of the Greeks and should be thrown into the sea, or set fire  \nto and burned, or that they should bore holes in its hollow belly  \nand probe for hiding places. The people were uncertain and  \ntheir passions were divided.\n\n40 Then suddenly at the head of a great throng Laocoon came  \nrunning down in a blaze of fury from the heights of the citadel,  \nshouting from a distance as he came: 'O you poor fools! Are  \nyou out of your minds, you Trojans? Do you seriously believe  \nthat your enemies have sailed away? Do you imagine Greeks  \never give gifts without some devious purpose? Is this all you  \nknow about Ulixes? I tell you there are Greeks hiding in here,  \nshut up in all this wood, or else it is a siege engine designed for  \nuse against our walls, to spy on our homes and come down on  \nthe city from above, or else there is some other trick we cannot  \nsee. Do not trust the horse, Trojans. Whatever it is, I am afraid  \nof Greeks, even when they bear gifts.'\n\n50 With these words he threw a great spear with all his strength  \ninto the beast's side, into the curved timbers of its belly. It stuck  \nthere vibrating, the creature's womb quivered and the hollow  \ncaverns boomed and groaned. If divine Fate, if the minds of the  \ngods had not been set against us, Laocoon would surely have  \nforced us to tear open the hiding places of the Greeks with our  \nswords, Troy would still be standing and the high citadel of  \nPriam would still be in its place.\n\nWhile this was going on, there was a sudden outcry, and some  \nTrojan shepherds came before the king, dragging a man with  \nhis hands tied behind his back. They knew nothing about him.  \n60 They had come upon him and he had given himself up. This was  \nall part of his scheme. His purpose was to open Troy to the  \nGreeks. He knew exactly what he wanted to do, and he was  \nready for either outcome, to spin his web or to meet certain  \ndeath if he failed. In their eagerness to see the prisoner, Trojan  \nsoldiers came running up from all sides, and gathered round to  \njoin in jeering at him. Listen now to this story of Greek treachery,  \nand from this one indictment, learn the ways of a whole people.  \nDishevelled and defenceless, he stood there with every eye upon  \nhim, looking all round him at the warriors of Troy, and said  \n70 with a great sigh: 'There is nowhere for me now on sea or land.  \nThere is nothing left for a man like me, who has no place among  \nthe Greeks, and now here are my enemies the Trojans, baying  \nfor my blood.'\n\nHe groaned. We had a change of heart, and all our passions  \nwere checked. We fell to asking him what his family was, and  \nwhat he had come to tell us. We wanted to hear why he had  \nallowed himself to be taken prisoner.\n\n'O king Priam,' he replied, 'I am the sort of man who will  \nconfess the whole truth to you, whatever it may be. First of all,  \n80 I am a Greek from Argos, and I will not deny it. Fortune may  \nhave made Sinon an object of pity, but for all her malice, she  \nwill never make him a cheat or a liar. You may perhaps have  \nheard tell of the name of Palamedes, son of Belus, and the great  \nglory that was his. Although he was innocent, false information  \nwas infamously laid against him. His offence was that he  \nobjected to the war, and the Greeks put him to death. They  \nmurdered him and now they mourn him. This Palamedes was  \nmy comrade and my kinsman. My father was a poor man, and  \nsent me here to the war to be with him from my earliest years.  \nWhile Palamedes was secure in his kingship and had authority  \n90 in the council of the kings, we too had some standing and some  \ncredit. But after he left the shores of this upper world, the victim  \nof the jealousy of Ulixes and his smooth tongue (you all know  \nabout Ulixes), I was prostrate and dragged out my life in darkness  \nand grief, brooding to myself over the downfall of my  \ninnocent friend, till, like a madman, I broke my silence and  \npromised that I would miss no chance of revenge if ever I came  \nback in victory to our native Argos. My words roused his bitter  \nhatred. This was my first step on a slippery path. From this  \nmoment on, Ulixes kept me in a constant state of fear by one  \nnew accusation after another. From this moment on he spread  \nvague rumours about me among the common soldiers. He knew  \nhe was guilty and was looking for weapons to use against me.  \n100 Nor did he rest until with Calchas the priest as his lackey...  \nbut why do I waste time? Why go over this sordid story to no  \npurpose? If in your eyes all Greeks are the same, and all you have  \nto know is that a man is a Greek, then give me my punishment. It  \nis long overdue. This would please Ulixes, our friend from  \nIthaca, and Agamemnon and Menelaus would pay you well  \nfor it.'\n\nBy this time we were burning to ask questions and find out  \nwhy all this had happened. We had never met villainy on this  \nscale before. We were not familiar with the arts of Greece. He  \nwent on with his lies, cringing with fear as he spoke:\n\n'The Greeks have often wanted to make their escape from  \nhere and leave Troy far behind them, abandoning this long and  \n110 weary war. And oh how I wish they had done so! But again and  \nagain rough seas here kept them in port or the south wind  \nalarmed them as they were setting sail. And most of all, when  \nthis construction of interwoven maple beams, this horse, was at  \nlast in position here, the black clouds thundered all round the  \nsky. We were at a loss and sent Eurypylus to consult the oracle  \nof Phoebus Apollo, and this is the grim response he brought  \nback from the shrine: \"When you Greeks first came to Troy you  \nkilled a virgin and appeased the winds with her blood. With  \nblood you must find a way to return. You must sacrifice a Greek  \n120 life.\" When this answer came to people's ears, they did not  \nknow where to turn, and the cold fear ran through the marrow  \nof their bones. For whom were they to prepare death? Whom  \ndid Apollo want? At this point there was a great uproar, and  \nthe Ithacan dragged out the prophet Calchas into the middle of  \nus and demanded to know what was the will of the gods. Many  \npeople could detect even then the ruthless hand of the schemer  \ndirected against me. They saw what was to come and held their  \npeace. For ten days Calchas gave no answer, concealing himself  \nand refusing to say the word that would betray a man and send  \nhim to his death. But at long last, all according to plan, he  \nallowed the clamour raised by the Ithacan to force him to break  \n130 his silence and mark me out for the altar. They all agreed. They  \nhad all been afraid, but now one man was doomed, and this  \nthey could endure.\n\n'The day of the abomination was soon upon us. The sacred  \nrites were all prepared for me. The salted meal was sprinkled  \nand the sacrificial ribbons were round my head. I escaped from  \ndeath, I admit it, I broke my bonds, and lay hidden all night in  \nthe reeds of a marsh, waiting for them to set sail, and wondering  \nif they had. I have no hope now of seeing the land which was  \nonce my home, or my beloved children, or my father whom I  \n140 have so often longed for. Perhaps they will be punished for my  \nescape, and wash away this guilt of mine with their own helpless  \nblood. But I beg of you by the gods who know the truth, by any  \nhonesty that may survive unsullied between men, pity me in my  \ngreat suffering. I know in my heart I have not deserved it.'\n\nHe wept. We spared him and and even began to pity him.  \nPriam spoke first and ordered him to be freed from the manacles  \nand the ropes that tied him, and spoke these friendly words:  \n'Whoever you are, from this moment on forget the Greeks  \nwhom you have lost. You will be one of us. But now give full  \n150 and truthful answers to the questions I ask you: why have they  \nset up this huge monster of a horse? Who proposed it? What is  \nthe purpose of it? Does it have some supernatural power? Is it  \nan engine of war?'\n\nSinon was ready with all his Greek arts and stratagems.  \nRaising to the skies the hands we had just freed from their  \nshackles, he cried: 'I call upon you, eternal fires of heaven and  \nyour inviolable godhead. I call upon the altars and the impious  \nswords from which I have escaped. I call upon the sacred ribbons  \nwhich I wore as sacrificial victim. It is no sin for me to break my  \nsacred oaths of allegiance to the Greeks. It is no sin for me to  \nhate these men and bring all their secrets out into the open. I  \n160 am no longer subject to the laws of my people. Only you must  \nstand by your promises. If I keep Troy safe, Troy must keep its  \nword and save me, if what I say is true, and what I offer is a full  \nand fair exchange.\n\n'All the hopes and confidence of the Greeks in this war they  \nstarted have always depended upon the help of Pallas Athene.  \nBut ever since the impious Diomede and Ulixes, the schemer  \nbehind all their crimes, took it upon themselves to tear the  \nfateful Palladium, the image of the goddess, from her own sacred  \ntemple in Troy, ever since they slew the guards on the heights  \nof the citadel and dared to touch the sacred bands on the head  \nof the virgin goddess with blood on their hands, from that  \n170 moment their hopes turned to water and ebbed away from them,  \ntheir strength was broken and the mind of the goddess was set  \nagainst them. Tritonian Pallas gave clear signs of this by sending  \nportents that could not be doubted. No sooner had they laid  \ndown the image in the Greek camp, than its eyes glared and  \nflashed fire, the salt sweat streamed over its limbs and by some  \nmiracle the image of the goddess leapt three times from the  \nground with her shield and spear quivering. Calchas declared  \nthat they had to take to instant flight across the sea, and prophesied  \nthat Troy could not be sacked by Argive weapons unless  \nthey first took the omens again in Argos, and then brought back  \nto Troy the divine image which they have now carried away  \n180 across the sea on their curved ships. So now they have set sail  \nfor their native Mycenae to rearm and to muster their gods to  \ncome with them and they will soon remeasure the ocean and be  \nback here when you least expect them. This is how Calchas  \ninterprets the omens, and on his advice they have set up this  \neffigy of a horse to atone for the violation of the Palladium and  \nthe divinity of Pallas, and for their deadly sin of sacrilege. But  \nhe told them to make it an immense structure of interlaced  \ntimbers soaring to the sky, so that it could not be taken through  \nthe gates and brought into the city or protect the people should  \nthey receive it with their traditional piety. For if your hands  \n190 violate this offering to Minerva, then total destruction shall fall  \nupon the empire of Priam and the Trojans (and may the gods  \nrather send that on his own head). But if your hands raise it up  \ninto your city, Asia shall come unbidden in a mighty war to the  \nwalls of Pelops, and that is the fate in store for our descendants.'\n\nThe trap was laid. These were the arts of the liar Sinon, and  \nwe believed it all. Cunning and false tears had overcome the  \nmen who had not been subdued by Diomede, son of Tydeus,  \nnor Achilles of Larisa, not by ten years of siege nor a thousand  \nships.\n\n200 And now there came upon this unhappy people another and  \nyet greater sign, which caused them even greater fear. Their  \nhearts were troubled and they could not see what the future  \nheld. Laocoon, the chosen priest of Neptune, was sacrificing a  \nhuge bull at the holy altar, when suddenly there came over the  \ncalm water from Tenedos (I shudder at the memory of it), two  \nserpents leaning into the sea in great coils and making side by  \nside for the shore. Breasting the waves, they held high their  \nblood-stained crests, and the rest of their bodies ploughed the  \nwaves behind them, their backs winding, coil upon measureless  \ncoil, through the sounding foam of the sea. Now they were on  \n210 land. Their eyes were blazing and flecked with blood. They  \nhissed as they licked their lips with quivering tongues. We grew  \npale at the sight and ran in all directions, but they made straight  \nfor Laocoon. First the two serpents seized his two young sons,  \ntwining round them both and feeding on their helpless limbs.  \nThen, when Laocoon came to the rescue with his sword in his  \nhand, they seized him and bound him in huge spirals, and soon  \ntheir scaly backs were entwined twice round his body and twice  \n220 round his throat, their heads and necks high above him as he  \nstruggled to prise open their coils, his priestly ribbons befouled  \nby gore and black venom, and all the time he was raising horrible  \ncries to heaven like the bellowing of a wounded bull shaking  \nthe ineffectual axe out of its neck as it flees from the altar. But  \nthe two snakes escaped, gliding away to the highest temples  \nof the city and making for the citadel of the heartless Pallas, the  \nTritonian goddess, where they sheltered under her feet and  \nunder the circle of her shield.\n\nAt that moment a new fear crept into all their trembling  \n230 hearts. They said that Laocoon had been justly punished for his  \ncrime. He had violated the sacred timbers by hurling his sinful  \nspear into the horse's back, and they all shouted together that  \nit should be taken to a proper place and prayers offered up to  \nthe goddess. We breached the walls and laid open the buildings  \nof our city. They all buckled to the task, setting wheels to roll  \nbeneath the horse's feet and stretching ropes of flax to its neck.  \nThe engine of Fate mounted our walls, teeming with armed  \nmen. Unmarried girls and boys sang their hymns around it  \n240 and rejoiced to have a hand on the rope. On it came, gliding  \nsmoothly, looking down on the heart of the city. O my native  \nland! O Ilium, home of the gods! O walls of the people of  \nDardanus, famous in war! Four times it stopped on the very  \nthreshold of the gate, and four times the armour clanged in its  \nwomb. But we paid no heed and pressed on blindly, madly, and  \nstood the accursed monster on our consecrated citadel. Even at  \nthis last moment Cassandra was still opening her lips to foretell  \nthe future, but God had willed that these were lips the Trojans  \nwould never believe. This was the last day of a doomed people  \nand we spent it adorning the shrines of the gods all through the  \ncity with festal garlands.\n\n250 Meanwhile the sky was turning and night was rushing up  \nfrom the Ocean to envelop in its great shadow the earth, the sky  \nand the treachery of the Greeks, while the Trojans were lying  \nquiet in their homes, their weary bodies wrapped in sleep. The  \nGreek fleet in full array was already taking the army from  \nTenedos through the friendly silence of the moon and making  \nfor the shore they knew so well, when the royal flagship raised  \nhigh the fire signal and Sinon, preserved by the cruelty of the  \ndivine Fates, stealthily undid the pine bolts of the horse and  \n260 freed the Greeks from its womb. The wooden horse was open,  \nand the Greeks were pouring gratefully out of its hollow chambers  \ninto the fresh air, the commanders Thessandrus and  \nSthenelus and fierce Ulixes sliding down the rope they had  \nlowered, and with them Acamas, Thoas, Neoptolemus of the  \nline of Peleus, Machaon, who came out first, Menelaus and  \nEpeos himself, the maker of the horse that tricked the Trojans.  \nThey moved into a city buried in wine and sleep, slaying the  \nguards and opening the gates to let in all their waiting comrades  \nand join forces as they had planned.\n\nIt was the time when rest, the most grateful gift of the gods,  \nwas first beginning to creep over suffering mortals, when Hector  \n270 suddenly appeared before my eyes in my sleep, full of sorrow  \nand streaming with tears. He looked as he did when he had been  \ndragged behind the chariot, black with dust and caked with  \nblood, his feet swollen where they had been pierced for the  \nleather thongs. What a sight he was! How changed from the  \nHector who had thrown Trojan fire on to the ships of the Greeks  \nor come back clad in the spoils of Achilles. His beard was filthy,  \nhis hair matted with blood, and he had on his body all the  \n280 wounds he had received around the walls of his native city. In  \nmy dream I spoke to him first, forcing out my words, and I too  \nwas weeping and full of sorrow: 'O light of Troy, best hope and  \ntrust of all Trojans, what has kept you so long from us? Long  \nhave we waited for you, Hector. From what shores have you  \ncome? With what eyes do we look upon you in our weariness  \nafter the death of so many of your countrymen, after all the  \nsufferings of your people and your city? What has so shamefully  \ndisfigured the face that was once so serene? What wounds are  \nthese I see?'\n\nThere was no reply. He paid no heed to my futile questions,  \nbut heaved a great groan from the depths of his heart and said:  \n'You must escape, son of the goddess. You must save yourself  \n290 from these flames. The enemy is master of the walls and Troy is  \nfalling from her highest pinnacle. You have given enough to  \nyour native land and to Priam. If any right hand could have  \nsaved Troy, mine would have saved it. Into your care she now  \ncommends her sacraments and her household gods. Take them  \nto share your fate. Look for a great city to establish for them  \nafter long wanderings across the sea.' These were his words,  \nand he brought out in his own hands from her inmost shrine the  \nmighty goddess Vesta with the sacred ribbons on her head and  \nher undying flame.\n\nMeanwhile the city was in utter confusion and despair.  \n300 Although the house of my father Anchises stood apart and was  \nscreened by trees, the noise was beginning to be heard and the  \ndin of battle was coming closer and closer. I shook the sleep  \nfrom me and climbed to the top of the highest gable of the roof,  \nand stood there with my ears pricked up like a shepherd when  \na furious south wind is carrying fire into a field of grain, or a  \nmountain river whirls along in spate, flattening all the fields, the  \ngrowing crops and all the labour of oxen, carrying great trees  \nheadlong down in its floods while the shepherd stands stupefied  \non the top of the rock, listening to the sound without knowing  \n310 what it is. Then in that moment I knew the truth. The treacherous  \nscheming of the Greeks was there to see. Soon the great  \nhouse of Deiphobus yielded to the flames and fell in ruins. Soon  \nhis neighbour Ucalegon was burning and the broad waters of  \nthe strait of Sigeum reflected the flames. The clamour of men  \nand the clangour of trumpets rose to high heaven. Mindlessly I  \nput on my armour, for reason had little use for armour, but my  \nheart was burning to gather comrades for battle and rush to the  \ncitadel with them. Frenzy and anger drove me on and suddenly  \nit seemed a noble thing to die in arms.\n\nI now caught sight of Panthus, just escaped from the weapons  \nof the Greeks, Panthus, son of Othrys, priest of Apollo and of  \n320 the citadel. He was carrying in his hands the sacraments and the  \ndefeated gods from the temple, and dragging his young grandson  \nalong behind him in a mad rush to the door of my father's  \nhouse. 'Where is our strong-point? Where are we rallying?' I  \nhad scarcely time to speak before he replied, groaning: 'The last  \nday has come for the people of Dardanus. This is the hour they  \ncannot escape. The Trojans are no more. Ilium has come to an  \nend and with it the great glory of the race of Teucer. Pitiless  \nJupiter has given everything over to Argos. The Greeks are  \nmasters of the burning city. The horse stands high in the heart  \n330 of it, pouring out its armed men, and Sinon is in triumph,  \nspreading the flames and gloating over us. The great double  \ngates are open and Greeks are there in their thousands, as many  \nas ever came from great Mycenae. Others have blocked the  \nnarrow streets with their weapons levelled. Their lines are drawn  \nup and the naked steel is flashing, ready for slaughter. Only the  \nfirst few guards on the gates are trying to fight and offering blind  \nresistance.'\n\nI went where I was driven by the words of Panthus and the  \nwill of the gods, into the fighting and the flames, where the grim  \nFury of war called me, where I could hear the din of battle  \nand the shouts rising to heaven. I came across Rhipeus in the  \n340 moonlight and Epytus, huge in his armour, and they threw in  \ntheir lot with me. Hypanis and Dymas too came to my side, and  \nso did Coroebus, son of Mygdon. He had happened to come to  \nTroy just in these last few days, burning with mad love for  \nCassandra, and was fighting as son-in-law on the side of Priam  \nand the Trojans. It was his misfortune not to heed the advice his  \nbride had given him in her prophetic frenzy.\n\nWhen I saw them standing shoulder to shoulder and spoiling  \nfor battle, I addressed them in these words: 'You are the bravest  \n350 of all our warriors, and your bravery is in vain. If your desire is  \nfixed to follow a man who fights to the end, you see how things  \nstand with us. All the gods on whom this empire once depended  \nhave left their shrines and their altars. You are rushing to defend  \na burning city. Let us die. Let us rush into the thick of the  \nfighting. The one safety for the defeated is to have no hope of  \nsafety.'\n\nThese words added madness to their courage. From that  \nmoment, like wolves foraging blindly on a misty night, driven  \nout of their lairs by a ravening hunger that gives them no rest  \nand leaving their young behind to wait for them with their  \nthroats all dry, we ran the gauntlet of the enemy to certain  \n360 death, holding our course through the middle of the city, with  \nthe hollow blackness of dark night hanging over us. Who could  \nunfold the horrors of that night? Who could speak of such  \nslaughter? Who could weep tears to match that suffering? It was  \nthe fall of an ancient city that had long ruled an empire. The  \nbodies of the dead lay through all its streets and houses and the  \nsacred shrines of its gods. Nor was it only Trojans who paid  \ntheir debts in blood; sometimes valour came back even to the  \nhearts of the defeated and Greeks were cut down in their hour  \nof triumph. Bitter grief was everywhere. Everywhere there was  \nfear, and death in many forms.\n\n370 The first of the Greeks to come to meet us was Androgeos,  \nand he had a large contingent of men with him. Not knowing  \nwho we were, but thinking we were allies, he called out first to  \nus: 'Move along there, friends! Why are you so slow? What is  \nkeeping you back? The citadel is on fire, and everyone else is  \npillaging and plundering. Have you just arrived from your tall  \nships?' He spoke, and when no convincing answer came, he  \ninstantly realized that he had fallen amongst enemies. He was  \nstupefied and started backwards without another word. He was  \n380 like a man going through rough briers who steps on a snake  \nwith all his weight without seeing it, and starts back in sudden  \npanic as it raises its wrath and puffs up its blue-green neck: that  \nis how Androgeos recoiled in terror at the sight of us. We fell  \nupon them and surrounded them with a wall of weapons. They  \ndid not know the ground, and were stricken with fear, so we  \ncut them down wherever we caught them. Fortune gave us a  \nfair wind for our first efforts, and Coroebus, his spirits raised  \nby our success, cried out: 'Come comrades, let us take the first  \nroad Fortune shows us to safety, and go where she shows that  \n390 she approves. Let us change shields with the Greeks and put on  \ntheir insignia. Is this treachery or is it courage? Who would ask  \nin dealing with an enemy? The Greeks themselves will provide  \nour armour.'\n\nHe spoke, and then put on the plumed helmet of Androgeos  \nand his richly blazoned shield, and buckled the Greek sword to  \nhis side. Rhipeus cheerfully followed suit, then Dymas himself  \nand the whole band. Every man armed himself with the spoils  \nhe had just taken, and, moving through the city, we mingled  \nwith the Greeks and fought many battles under gods not our  \nown, clashing blindly in the night, and many a Greek did we send  \ndown to Orcus. Some scattered towards their ships, running for  \n400 the safety of the shore. Some climbed back in abject fear into  \nthe huge horse, and hid themselves in its familiar belly.\n\nBut no man can put trust in gods who are opposed to him.  \nSuddenly there was Cassandra, the maiden daughter of Priam,  \nbeing dragged from the temple of Minerva, from her very sanctuary,  \nwith hair streaming and her burning eyes raised in vain to  \nheaven, but only her eyes \u2013 they had tied her gentle hands.  \nCoroebus could not endure the sight of this, but a wild frenzy  \ntook him and he hurled himself into the middle of the enemy to  \nhis death. We all went after him and ran upon their spears where  \n410 they were thickest. First we were attacked by our own men and  \noverwhelmed by their missiles thrown from the high gable of  \nthe temple roof, and the sight of our armour and the confusion  \ncaused by our Greek crests brought pitiable slaughter on us.  \nThen the Greeks raised furious alarm at the rescue of Cassandra  \nand gathered from every quarter to attack us, Ajax fiercest of  \nthem all, the two sons of Atreus and the whole army of the  \nDolopians. It was as though a whirlwind had burst and opposing  \nwinds were clashing, the west, the south, and the east wind  \nglorying in the horses of the morning, with woods wailing and  \n420 wild Nereus churning up the sea from its depths. Then also  \nappeared all those Greeks who had been routed by our stratagem  \nin the darkness of the night and scattered through the city. They  \nrealized that our shields and weapons were not our own and  \ndid not accord with the words on our lips. In an instant they  \noverwhelmed us by the sheer weight of their numbers. Coroebus  \nwas the first to die. He fell by the right hand of Peneleus and lay  \nthere face down on the altar of Minerva, goddess mighty in  \narms. Rhipeus also fell. Of all the Trojans he was the most  \nrighteous, the greatest lover of justice. But the gods took their  \nown decision. Hypanis and Dymas were cut down by their  \n430 fellow-Trojans, and as for you, Panthus, you found as you fell  \nthat your great devotion and the ribbon you wore as priest of  \nApollo were no protection. I call to witness the ashes of Troy. I  \ncall upon the flames in which my people died. In the hour of  \nyour fall I did not flinch from the weapons of the Greeks or  \nfrom anything they could do. If it had been my fate to fall, my  \nright hand fully earned it.\n\nFrom here we were swept along in the fighting, Iphitus and  \nPelias with me. Iphitus was no longer young, and Pelias had  \nbeen slowed by a wound he had received from Ulixes. The noise  \nof shouting drew us straight to Priam's palace and there we  \nfound the fighting so heavy that it seemed there were no battles  \nanywhere else, that this was the only place in the city where men  \n440 were dying. We saw Mars, the irresistible God of War, Greeks  \nrushing to the palace, men with shields locked over their backs  \npacking the threshold, ladders hooked to the walls and men  \nstruggling to climb them right against the very doorposts, thrusting  \nup their shields on their left arms to protect themselves while  \ntheir right hands gripped the top of the walls. The Trojans for  \ntheir part were tearing down their towers and the roofs of all  \ntheir buildings. They saw the end was near, and these were the  \nweapons they were preparing to defend themselves with in the  \nvery moment of death, rolling down on the heads of their  \nenemies the gilded beams and richly ornamented ceilings of  \ntheir ancestors. Down on the ground others were standing  \n450 shoulder to shoulder with drawn swords blocking the doorway.  \nMy spirit was renewed and I rushed to bring relief to the  \npalace of my king, to help its defenders, to put heart into men  \nwho were defeated.\n\nThere was a forgotten entrance at the rear, a secret doorway  \nentering into a passage which joined the different parts of  \nPriam's palace. While the kingdom of Troy still stood, poor  \nAndromache often used to come this way unattended to visit  \nHector's parents, taking her son Astyanax to see his grandfather.  \nI slipped through this door and climbed to the highest gable of  \nthe roof, from where the doomed Trojans were vainly hurling  \n460 missiles. There was a tower rising sheer towards the stars from  \nthe top of the palace roof, from which we used to look out over  \nthe whole of Troy, the Greek fleet and the camp of the Achaeans.  \nWe set about this tower and worked round it with iron bars  \nwhere there was a join we could open up above the top floor of  \nthe palace. Having loosened it from its deep bed in the walls,  \nwe rocked it and suddenly sent it toppling, spreading instant  \ndestruction and crushing great columns of Greeks. But others  \nstill came on and the hail of rocks and other missiles never  \nslackened.\n\nIn the portico in front of the palace, on the very threshold,  \n470 Pyrrhus, son of Achilles, whom men also call Neoptolemus, was  \nrampaging and the light flashed on the bronze of his weapons.  \nHe was like a snake which has fed on poisonous herbs and  \nhidden all winter in the cold earth, but now it emerges into the  \nlight, casts its slough and is renewed. Glistening with youth, it  \ncoils its slithering back and lifts its breast high to the sun with  \nits triple tongue flickering from its mouth. Huge Periphas was  \nwith him, and Automedon, the charioteer and armour-bearer  \nof Achilles. With him too were all the young warriors of Scyros  \ncoming to attack the palace and throwing firebrands on to the  \n480 roof. Pyrrhus himself at their head seized a double-headed axe  \nand with it smashed the hard stone of the threshold, wrenching  \nthe bronze-plated doorposts from their sockets. He then hacked  \na panel out of the mighty timbers of the door and broke a gaping  \nhole which gave them a view into the house. There before their  \neyes were the long colonnades and the inner chambers. There  \nbefore their eyes was the heart of the palace of Priam and the  \nancient kings. They saw armed men standing in the doorway,  \nbut inside all was confusion and lamentation, and deep into the  \nhouse the hollow chambers rang with the wailing of women,  \nand their cries rose to strike the golden stars. Frightened mothers  \n490 were wandering through the great palace, clinging to the doorposts  \nand kissing them. But Pyrrhus pressed on with all the  \nviolence of his father Achilles, and no bolts or guards could hold  \nhim. The door gave way under repeated battering and the posts  \nhe had dislodged from their sockets fell to the ground. Brute  \nforce made the breach and the Greeks went storming through,  \nbutchering the guards who stood in their way and filling the  \nwhole house with soldiers. No river foaming in spate was ever  \nlike this, bursting its banks and leaving its channel to overwhelm  \neverything in its path with its swirling current, as it bears down  \nfuriously on ploughed fields in a great wave, and cattle and their  \n500 pens are swept all over the plains. I myself saw Neoptolemus in  \nan orgy of killing and both the sons of Atreus on the threshold.  \nI saw Hecuba with a hundred women, her daughters and the  \nwives of her sons. I saw Priam's blood all over the altar, polluting  \nthe flame which he himself had sanctified. Down fell the fifty  \nbedchambers with all the hopes for generations yet to come,  \nand down came the proud doorposts with their spoils of barbaric  \ngold. Everything not claimed by fire was now held by Greeks.\n\nPerhaps you may also ask how Priam died. When he saw the  \ncapture and fall of his city, the doors of his palace torn down  \n510 and his enemy in the innermost sanctuary of his home, although  \nhe could achieve nothing, the old man buckled his armour long  \nunused on shoulders trembling with age, girt on his feeble sword  \nand made for the thick of the fight, looking for his death. In the  \nmiddle of the palace, under the naked vault of heaven, there  \nstood a great altar, and nearby an ancient laurel tree leaning  \nover it and enfolding the household gods in its shade. Here,  \nvainly embracing the images of the gods, Hecuba and her daughters  \nwere sitting flocked round the altar, like doves driven down  \nin a black storm. When Hecuba saw that Priam had now put on  \n520 his youthful armour, 'O my poor husband,' she cried, 'this is  \nmadness. Why have you put on this armour? Where can you  \ngo? This is not the sort of help we need. You are not the defender  \nwe are looking for. Not even my Hector, if he were here now  \n...Just come here and sit by me. This altar will protect us all,  \nor you will die with us.' As she spoke she took the old man to  \nher and led him to a place by the holy altar.\n\nSuddenly Polites, one of Priam's sons, came in sight. He had  \nescaped death at the hands of Pyrrhus and now, wounded and  \nwith enemy weapons on every side, he was running through the  \nlong porticos of the palace and across the empty halls with  \n530 Pyrrhus behind him in full cry, almost within reach, pressing  \nhim hard with his spear and poised to strike. As soon as he  \nreached his father and mother, he fell and vomited his life's  \nblood before their eyes. There was no escape for Priam. Death  \nwas now upon him, but he did not check himself or spare the  \nanger in his voice. 'As for you,' he cried, 'and for what you have  \ndone, if there is any power in heaven that cares for such things,  \nmay the gods pay you well. May they give you the reward you  \nhave deserved for making me see my own son dying before my  \n540 eyes, for defiling a father's face with the murder of his son. You  \npretend that Achilles was your father, but this is not how Achilles  \ntreated his enemy Priam. He had respect for my rights as a  \nsuppliant and for the trust I placed in him. He gave me back the  \nbloodless body of Hector for burial and allowed me to return  \nto the city where I was king.' With these words the old man  \nfeebly threw his harmless spear. It rattled on the bronze of  \nPyrrhus' shield and hung there useless sticking on the surface of  \nthe central boss. Pyrrhus then made his reply. 'In that case you  \nwill be my messenger and go to my father, son of Peleus. Let  \nhim know about my wicked deeds and do not forget to tell  \nhim about the degeneracy of his son Neoptolemus. Now, die.'  \n550 As he spoke the word, he was dragging Priam to the very altar,  \nhis body trembling as it slithered through pools of his son's  \nblood. Winding Priam's hair in his left hand, in his right he  \nraised his sword with a flash of light and buried it to the hilt  \nin Priam's side.\n\nSo ended the destiny of Priam. This was the death that fell to  \nhis lot. He who had once been the proud ruler over so many  \nlands and peoples of Asia died with Troy ablaze before his eyes  \nand the citadel of Pergamum in ruins. His mighty trunk lay  \nupon the shore, the head hacked from the shoulders, a corpse  \nwithout a name.\n\nThen for the first time I knew the horror that was all about  \n560 me. What was I to do? There came into my mind the image of  \nmy own dear father, as I looked at the king who was his equal  \nin age breathing out his life with that cruel wound. There came  \ninto my mind also my wife Creusa whom I had left behind, the  \nplundering of my home and the fate of young Iulus. I turned to  \nlook at the men fighting by my side. Exhausted, they had all  \ndeserted me and thrown themselves from the roof or given their  \nsuffering bodies to the flames.\n\nNow that I was alone, I caught sight of Helen keeping watch  \non the doors of the temple of Vesta where she was staying quietly  \n570 in hiding. The fires gave a bright light and I was gazing all  \naround me wherever I went. This Helen, this Fury sent to be the  \nscourge both of Troy and of her native Greece, was afraid of the  \nTrojans, who hated her for the overthrow of their city. She was  \nafraid the Greeks would punish her and afraid of the wrath of  \nthe husband she had deserted, so, hated by all, she had gone  \ninto hiding and was sitting there at the altar. The passion flared  \nin my heart and I longed in my anger to avenge my country even  \nas it fell and to exact the penalty for her crimes. 'So this woman  \nwill live to set eyes on Sparta and her native Mycenae again,  \nand walk as queen in the triumph she has won? Will she see her  \n580 husband, her father's home and her children and be attended  \nby women of Troy and Phrygian slaves, while Priam lies dead  \nby the sword, Troy has been put to the flames and the shores of  \nthe land of Dardanus have sweated so much blood? This will  \nnot be. Although there is no fame worth remembering to be  \nwon by punishing a woman and such a victory wins no praise,  \nnevertheless I _shall_ win praise for blotting out this evil and  \nexacting a punishment which is richly deserved. I shall also take  \npleasure in feeding the flames of vengeance and appeasing the  \nashes of my people.'\n\n590 As I ran towards her ranting and raving, my loving mother  \nsuddenly appeared before my eyes. I had never before seen her  \nso clearly, shining in perfect radiance through the darkness of  \nthe night. She revealed herself as a goddess as the gods in heaven  \nsee her, in all her majesty of form and stature. As she caught my  \nright hand and held me back, she opened her rosy lips and spoke  \nto me \u2013 'O my son, what bitterness can have been enough to stir  \nthis wild anger in you? Why this raging passion? Where is all  \nthe love you used to have for me? Will you not first go and see  \nwhere you have left your father, crippled with age, and find  \nwhether your wife Creusa is still alive, and your son Ascanius?  \n600 The whole Greek army is prowling all around them and they  \nwould have been carried off by the flames or slashed by the  \nswords of the enemy if my loving care were not defending them.  \nIt is not the hated beauty of the Spartan woman, the daughter  \nof Tyndareus, that is overthrowing all this wealth and laying  \nlow the topmost towers of Troy, nor is it Paris although you all  \nblame him, it is the gods, the cruelty of the gods. Look, for I  \nshall tear away from all around you the dank cloud that veils  \nyour eyes and dulls your mortal vision. You are my son, do not  \nbe afraid to do what I command you, and do not disobey me.  \n610 Here where you see shattered masonry, stone torn from stone,  \nand waves of dust-laden smoke, Neptune has loosened the  \nfoundations with his great trident and is shaking the walls,  \ntearing up your whole city from the place where it is set. Here  \ntoo is Juno, cruellest of all, the first to seize the Scaean Gate,  \nstanding there sword in hand, and furiously calling up the  \nsupporting columns from the ships. Now look behind you,  \nTritonian Pallas is already sitting on top of your citadel shining  \nout of the cloud with her terrible Gorgon, while the Father of  \nthe Gods himself puts heart into the Greeks and gives them  \nstrength. It is Jupiter himself who is rousing the gods against  \nthe armies of Troy. Escape, my son, escape with all haste. Put  \n620 an end to your struggle, I shall not leave your side till I see you  \nsafely standing on the threshold of your father's door.' She  \nfinished speaking and melted into the dense shadows of that  \nnight, and there before my eyes I saw the dreadful vision of the  \ngods in all their might, the enemies of Troy.\n\nAt that moment I seemed to see the whole of Ilium settling  \ninto the flames and Neptune's Troy toppling over from its  \nfoundations like an ancient ash tree high in the mountains which  \nfarmers have hacked with blow upon blow of their double axes,  \nlabouring to fell it; again and again it threatens to fall, its foliage  \n630 shudders and its head trembles and nods until at last it succumbs  \nto its wounds and breaks with a dying groan, spreading ruin  \nalong the ridge. I came down from the roof and with the god to  \nlead me, a way opened through fire and sword. The weapons  \nparted and the flames drew back before me.\n\nWhen at last I had reached the door of my father's house and  \nour ancient home, my first wish was to find my father and take  \nhim into the high mountains, but he refused to go on living now  \nthat Troy had been levelled to the earth. He would not hear of  \nexile, but cried: 'Those of you with young blood still thick in  \nyour veins, those of you whose strength is sound and unimpaired,  \n640 you are the ones who must busy yourselves with escaping.  \nIf the gods in heaven had wished me to go on living, they  \nwould have preserved this place for me. I have already seen one  \nsack of the city and survived its capture, and that is more than  \nenough. Here I lie and here I stay. Take your farewells and leave  \nme. My own right hand will earn me my death. The enemy will  \ntake pity on me. They will  \nbe looking for spoils. I shall have no  \ntomb, but that is an easy loss to bear. For long years, ever since  \nthe Father of the Gods and King of Men blew the wind of his  \nthunderbolt upon me and touched me with its fire, I have been  \nlingering here hated by the gods and useless to men.'\n\n650 As he said these words he stood there rooted and no power  \ncould move him. Streaming with tears, my wife Creusa, Ascanius,  \nall of us begged him not to bring everything down on his  \nown head: when Fate batters a house, the father should not add  \nhis weight to the blows. But he still refused. He stood by his  \ndecision and stayed where he was. I rushed to take up arms  \nagain in complete despair. Death was the only thing I could  \nhope for. What course could I follow? What fate was in store  \nfor us? 'Did you think I could run away and leave my father  \nhere?' I exclaimed. 'How did such a sacrilege escape my father's  \n660 lips? If the gods above decree that nothing of this great city is to  \nsurvive, if your mind is fixed and it is your pleasure to add  \nyourself and those you love to the destruction of the city, the  \ndoor is open and the deaths you want will come. Pyrrhus will  \nsoon be here, soaked in the blood of Priam. He is the one who  \nmurders the son before the face of the father, and the father at  \nthe altar. O my loving mother, is this why you took me through  \nfire and sword, so that I could see my enemy in the innermost  \nsanctuary of my home, and Ascanius and my father and my wife  \nCreusa with them lying sacrificed in each other's blood? Bring  \nme my armour, comrades. Bring it here. This is the last light we  \n670 shall see and it is calling the defeated. Give me back to the  \nGreeks. Let me go back and rejoin the battle. Today we die. But  \nnot all of us shall die unavenged.'\n\nI buckled on my sword again and was fixing my left arm into  \nthe shield. But as I was leaving Creusa suddenly threw herself  \nat my feet in the doorway and held me, stretching out our little  \nson Iulus towards me. 'If you are going to your death,' she cried,  \n'take us with you to share your fate, whatever it is. But if you  \nhave reason to put any hope in arms, your first duty is to guard  \nthis house. If you leave us here, what fate is waiting for little  \nIulus, for your father and for the woman who used to be called  \nyour wife?'\n\n680 Her cries of anguish were filling the whole house, when suddenly  \nthere was a great miracle. At the very moment when we  \nwere both holding Iulus and he was there between our sorrowing  \nfaces, a light began to stream from the top of the pointed cap he  \nwas wearing and the flame seemed to lick his soft hair and feed  \nround his forehead without harming him. We took fright and  \nrushed to beat out the flames in his hair and quench the holy  \nfire with water, but Father Anchises, looking joyfully up to the  \nstars of heaven and raising his hands palms upward, lifted his  \nvoice in prayer: 'O All-powerful Jupiter, if ever you yield to  \n690 prayers, look down upon us, that is all we ask, and if we deserve  \nanything for our devotion, give us help at last, Father Jupiter,  \nand confirm this omen.'\n\nScarcely had he spoken when a sudden peal of thunder rang  \nout on the left and a star fell from the sky, trailing a great torch  \nof light in its course through the darkness. We watched it glide  \nover the topmost pinnacles of the house and bury itself, still  \nbright, in the woods of Mount Ida, leaving its path marked out  \nbehind it, a broad furrow of light, and the whole place smoked  \nall around with sulphur. Now at last my father was truly convinced.  \n700 He rose up and addressed the gods, praying to the sacred  \nstar: 'There is now no more delay. Now I follow, O gods of my  \nfathers. Wherever you lead, there am I. Preserve this house.  \nPreserve my grandson. This is your sign. Troy is in your mighty  \nhands. Anchises yields. I am willing to go with you, my son, and  \nbe your companion.'\n\nHe had spoken. The noise of the fires was growing louder and  \nlouder through the city and the tide of flame was rolling nearer.  \n'Come then, dear father, up on my back. I shall take you on my  \n710 shoulders. Your weight will be nothing to me. Whatever may  \ncome, danger or safety, it will be the same for both of us. Young  \nIulus can walk by my side and my wife can follow in my footsteps  \nat a distance. And you, the slaves of our house, must pay  \nattention to what I am saying. As you leave the city there is a  \nmound with a lonely old temple of Ceres. Near it is an ancient  \ncypress preserved and revered for many long years by our ancestors.  \nWe shall go to that one place by different routes. You,  \nfather, take in your arms the sacraments and the ancestral gods  \nof our home. I am fresh from all the fighting and killing and it  \n720 is not right for me to touch them till I have washed in a running  \nstream.'\n\nWhen I had finished speaking, I put on a tawny lion's skin as  \na covering for my neck and the breadth of my shoulders and  \nthen I bowed down and took up my burden. Little Iulus twined  \nhis fingers in my right hand and kept up with me with his short  \nsteps. Creusa walked behind us and we moved along, keeping  \nto the shadows. This was the man who had been unmoved by  \nall the missiles of the Greeks and had long faced their serried  \nranks without a tremor, but now every breath of wind frightened  \nme and I started at every sound, so anxious was I, so afraid both  \nfor the man I carried and for the child at my side.\n\nI was now coming near the gates and it seemed that our  \n730 journey was nearly over and we had escaped, when I suddenly  \nthought I heard the sound of many marching feet and my father  \nlooking out through the darkness cried: 'Run, my son, run. They  \nare coming this way. I can see the flames reflected on their shields  \nand the bronze glinting.' At that moment some hostile power  \nconfused me and robbed me of my wits. I ran where there was  \nno road, leaving the familiar area of the streets. Then it was that  \nmy wife Creusa was torn from me by the cruelty of Fate \u2013  \n740 whether she stopped or lost her way or sat down exhausted, no  \none can tell. I never saw her again. Nor did I look behind me or  \nthink of her or realize that she was lost till we arrived at the  \nmound and the ancient sanctuary of Ceres. But when at last  \neveryone had gathered there, she was the only one who was not  \nwith us and neither her companions nor her son nor her husband  \nknew how she had been lost. I stormed and raged and blamed  \nevery god and man that ever was. This was the cruellest thing I  \nsaw in all the sack of the city. Leaving Ascanius, my father and  \nthe gods of Troy with my companions and hiding them all away  \nin a winding valley, I put on my flashing armour and went back  \n750 to the city, resolved to face all its dangers again, to go back  \nthrough the whole of Troy and once more put my life at peril.  \nFirst I went back to the walls and the dark gateway by which I  \nhad left the city. I found my route and retraced it, gazing all  \naround me through the darkness. Horror was everywhere and  \nthe very silence chilled the blood. Then I went on to our house,  \nthinking it was possible, just possible, that she had gone there.  \nThe Greeks had come flooding in and were everywhere. Consuming  \nflames, fanned by the winds, were soon rolling to the  \n760 top of the roof and leaping above it as their hot breath raged at  \nthe sky. From there I went on to Priam's palace and the citadel  \nwhere Phoenix and the terrible Ulixes, who had been chosen to  \nkeep watch, were already guarding the loot in the empty porticos  \nof the shrine of Juno. Here Greeks were piling up the treasures  \nof Troy, pillaged from all the burning temples \u2013 the tables of the  \ngods, mixing bowls of solid gold and all the robes they had  \nplundered. Children and frightened mothers stood around in  \nlong lines. I even dared to call her name into the darkness, filling  \n770 the streets with my shouts. Grief-stricken, I called her name  \n'Creusa! Creusa!' again and again, but there was no answer. I  \nwould not give up the search but was still rushing around the  \nhouses of the city when her likeness appeared in sorrow before  \nmy eyes, her very ghost, but larger than she was in life. I was  \nparalysed. My hair stood on end. My voice stuck in my throat.  \nThen she spoke to me and comforted my sorrow with these  \nwords: 'O husband that I love, why do you choose to give  \nyourself to such wild grief? These things do not happen without  \nthe approval of the gods. It is not their will that Creusa should  \ngo with you when you leave this place. The King of High  \n780 Olympus does not allow it. Before you lies a long exile and a  \nvast expanse of sea to plough before you come to the land of  \nHesperia where the Lydian river Thybris flows with smooth  \nadvance through a rich land of brave warriors. There prosperity  \nis waiting for you, and a kingdom and a royal bride. Wipe away  \nthe tears you are shedding for Creusa whom you loved. I shall  \nnot have to see the proud palaces of the Myrmidons and Dolopians.  \nI am a daughter of Dardanus and my husband was the son  \nof Venus, and I shall never go to be a slave to any matron of  \nGreece. The Great Mother of the Gods keeps me here in this  \nland of Troy. Now fare you well. Do not fail in your love for  \nour son.'\n\n790 She spoke and faded into the insubstantial air, leaving me  \nthere in tears and longing to reply. Three times I tried to put my  \narms around her neck. Three times her phantom melted in my  \narms, as weightless as the wind, as light as the flight of sleep.\n\nBy now the night was over. I returned to my comrades without  \nher. Here I found that new companions had streamed in and I  \nwas amazed at the numbers of them, men and women, an army  \ncollected for exile, a pitiable crowd. They had come from all  \ndirections ready to follow me with all their resources and all  \n800 their hearts to whatever land I should wish to lead them. And  \nnow Lucifer was rising above the ridges of Mount Ida and  \nbringing on the day. The Greeks were on guard at the gates and  \nthere was no hope of helping the city. I yielded. I lifted up my  \nfather and set out for the mountains.\n\n##  BOOK 3  \nTHE WANDERINGS\n\nWhen the gods had seen fit to lay low the power of Asia and the  \ninnocent people of Priam, when proud Ilium had fallen and all  \nNeptune's Troy lay smoking on the ground, we were driven by  \nsigns from heaven into distant exile to look for a home in some  \ndeserted land. There, hard by Antandros under the Phrygian  \nmountain range of Ida, we were mustering men and building a  \nfleet without knowing where the Fates were leading us or where  \nwe would be allowed to settle. The summer had barely started  \nand Father Anchises was bidding us hoist sail and put ourselves  \n10 in the hands of the Fates. I wept as I left the shores of my native  \nland and her harbours and the plains where once had stood the  \ncity of Troy. I was an exile taking to the high seas with my  \ncomrades and my son, with the gods of our house and the great  \ngods of our people.\n\nAt some distance from Troy lay the land of Mars, a land  \nof vast plains farmed by Thracians, once ruled by the savage  \nLycurgus. This people had ancient ties with Troy, while the  \nfortunes of Troy remained, and our household gods were linked  \nin alliance. Here I sailed, and using the name Aeneadae, formed  \nafter my own, I laid out my first walls on the curved shore. But  \nthe Fates frowned on these beginnings. I was worshipping my  \n20 mother Venus, the daughter of Dione, and the gods who preside  \nover new undertakings, and sacrificing a gleaming white bull to  \nthe Most High King of the Heavenly Gods. Close by there  \nhappened to be a mound on top of which there grew a thicket  \nbristling with spears of cornel and myrtle wood. I had gone  \nthere and was beginning to pull green shoots out of the ground  \nto cover the altar with leafy branches, when I saw a strange and  \nhorrible sight. As soon as I broke the roots of a tree and was  \npulling it out of the ground dark gouts of blood dripped from it  \n30 and stained the earth with gore. The horror of it chiled me to  \nthe bone, I trembled and my blood congealed with fear.\n\nI went on, pulling up more tough shoots from another tree,  \nsearching for the cause, however deep it might lie, and the dark  \nblood flowed from the bark of this second tree. With my mind  \nin turmoil I began to pray to the country nymphs and to Father  \nMars Gradivus who rules over the fields of the Getae, begging  \nthem to turn what I was seeing to good and to make the omen  \nblessed, but after I had set about the spear-like shoots of a third  \nshrub with greater vigour and was on my knees struggling to  \n40 free it from the sandy soil (shall I speak? Or shall I be silent?) I  \nheard a heart-rending groan emerge from deep in the mound  \nand a voice rose into the air: 'Why do you tear my poor flesh,  \nAeneas?' it cried. 'Take pity now on the man who is buried here  \nand do not pollute your righteous hands. I am no stranger to  \nyou. It was Troy that bore me and this is no tree that is oozing  \nblood. Escape, I beg you, from these cruel shores, from this land  \nof greed. It is Polydorus that speaks. This is where I was struck  \ndown and an iron crop of weapons covered my body. Their  \nsharp points have rooted and grown in my flesh.' At this, fear  \nand doubt oppressed me. My hair stood on end with horror and  \nthe voice stuck in my throat.\n\n50 This was the Polydorus the doomed Priam had once sent in  \nsecret with a great mass of gold, to be brought up by the king  \nof Thrace, when at last he was losing faith in the arms of  \nTroy and saw his city surrounded by besiegers. When Fortune  \ndeserted the Trojans and their wealth was in ruins, the king  \nwent over to the side of the victors and joined the armies  \nof Agamemnon. Breaking all the laws of God, he murdered  \nPolydorus and seized the gold. Greed for gold is a curse. There  \nis nothing to which it does not drive the minds of men. When  \nthe fear had left my bones, I told the chosen leaders of the people  \nand first of all my father about this portent sent by the gods and  \n60 asked what should be done. They were of one mind. We must  \nleave this accursed land where the laws of hospitality had been  \nviolated and let our ships run before the wind. So we gave  \nPolydorus a second burial, heaping the earth high in a mound  \nand raising to his shade an altar dark with funeral wreaths and  \nblack cypress, while the women of Troy stood all around with  \ntheir hair unbound in mourning. With offerings of foaming cups  \nof warm milk and bowls of sacrificial blood we committed his  \nsoul to the grave and lifted up our voices to call his name for  \nthe last time.\n\nThen as soon as we could trust ourselves to the waves, when  \n70 the winds had calmed the swell and a gentle breeze was rattling  \nthe rigging to call us out to sea, my comrades drew the ships  \ndown to the water and crowded the shore. We sailed out of the  \nharbour, and the land and its cities soon fell away behind us. In  \nthe middle of the ocean lies a beautiful island dear to Aegean  \nNeptune and the mother of the Nereids. It used to float from  \nshore to shore until in gratitude the Archer God Apollo moored  \nit to Gyaros and high Myconos, allowing it to stand firm and  \nbe inhabited and mock the winds. Here I sailed, and in this  \npeaceful haven of Delos we came safe to land, weary from the  \nsea. We went ashore and were admiring Apollo's city when its  \n80 king Anius, king of men and priest of the god, came to meet  \nus, his forehead garlanded with ribbons and the sacred laurel.  \nRecognizing Anchises as an old friend, he gave us his hand in  \nhospitality and we entered his house.\n\nThere I gazed in reverence at the god's temple built high  \nof ancient stone and made this prayer to Apollo: 'O god of  \nThymbra, grant us a home of our own. We are weary. Grant us  \nwalls and descendants and a city that will endure. Preserve these  \nremnants that have escaped the Greeks and pitiless Achilles, to  \nbe a second citadel for Troy. Whom are we to follow? Where  \ndo you bid us go? Where are we to settle? Send us a sign, O  \nfather, and steal into our hearts.'\n\n90 I had scarcely spoken when everything seemed to begin to  \ntremble. The threshold of the doors of the god, his laurel  \ntree, and all the mountain round about were shaken. The sanctuary  \nopened and a bellowing came from the bowl on the sacred  \ntripod. We threw ourselves to the ground and these were the  \nwords that came to our ears: 'O much-enduring sons of Dardanus,  \nthe land which first bore you from your parents' stock  \nwill be the land that will take you back to her rich breast. Seek  \nout your ancient mother. For that is where the house of Aeneas  \nand his sons' sons and their sons after them will rule over the  \nwhole earth.'\n\n100 So spoke Phoebus Apollo, and a great joy and tumult arose  \namong us, all asking what city this was, where Apollo was  \ndirecting us in our wanderings, what this land was to which we  \nwere to return. Then spoke my father Anchises who had been  \nturning over in his mind what he had heard from the men of  \nold: 'Listen,' he said, 'you leaders of Troy, and learn what you  \nhave to hope for. In the middle of the ocean lies Crete, the island  \nof great Jupiter, where there is a Mount Ida, the cradle of our  \nrace, and where the Cretans live in a hundred great cities, the  \nrichest of kingdoms. If I remember rightly what I have heard,  \nour first father Teucer sailed from there to Asia, landing at Cape  \nRhoeteum, and chose that place to found his kingdom. Troy  \n110 was not yet standing, nor was the citadel of Pergamum, and  \nthey lived low down in the valleys. This is the origin of the Great  \nMother of Mount Cybele, the bronze cymbals of the Corybants,  \nour grove of Ida, the inviolate silence of our worship and the  \nyoked lions that draw the chariot of the mighty goddess. Come  \nthen, let us follow where we are led by the bidding of the gods.  \nLet us appease the winds and set forth for the kingdoms of  \nCnossus. It is not far to sail. If only Jupiter is with us, the third  \nday will see our ships on the shores of Crete.' So he spoke, and  \n120 made due sacrifice on the altars, a bull to Neptune and a bull to  \nfair Apollo, a black lamb to the storms and a white lamb to  \nfavouring breezes.\n\nRumour as she flew told the tale of the great Idomeneus, how  \nhe had been forced to leave his father's kingdom and how the  \nshores of Crete were now deserted. Here was a place empty of  \nour enemies, their homes abandoned, waiting for us. We left the  \nharbour of Ortygia and flew over the sea to Naxos where  \nBacchants dance on the mountain ridges and to green Donusa,  \nto Olearos, to Paros marble-white and the Cyclades scattered  \non the face of the sea, skimming over an ocean churned up by  \nthe coasts of a hundred islands. The sailors raised all manner of  \nshouts as they vied with one another in their rowing and my  \ncomrades kept urging me to make for Crete and go back to the  \n130 home of their ancestors. The wind rising astern sped us on our  \nway and we came to shore at last on the ancient land of the  \nCuretes. Impatiently I set to work on walls for the city we all  \nlonged for. I called it Pergamea and the people rejoiced in the  \nname. I urged them to love their hearths and homes and raise a  \ncitadel to protect them.\n\nOur ships were soon drawn up on dry land, our young men  \nwere busy with marrying and putting new land under plough  \nand I was giving them homes and laws to live by, when suddenly  \nfrom a polluted quarter of the sky there came a cruel, suppurating  \nplague upon our bodies and upon the trees and crops. It was  \n140 a time of death. Men were losing the lives they loved or dragging  \naround their sickly bodies. The Dogstar burned the fields and  \nmade them barren, the grass dried, the crops were infected and  \ngave us no food. My father bade me retrace our course back  \nacross the sea to Phoebus Apollo and his oracle at Ortygia, to  \npray for his gracious favour and ask when he would put an end  \nto our toil, where we were to look for help in our adversity and  \nwhat course we were to steer.\n\nIt was night and sleep held in its grasp all living things upon  \nthe earth. There as I lay, the holy images of the gods, the  \n150 Phrygian Penates whom I had rescued from the thick of the  \nflames of the burning city of Troy, seemed to be standing bathed  \nin clear light before my eyes, where the full moon streamed in  \nthrough the unshuttered windows. At last they spoke to me and  \ncomforted my sorrow with these words: 'Apollo here speaks the  \nprophecy he will give you if you sail back to Ortygia. By his  \nown will he has sent us here and we stand at your door. We  \nfollowed you and your arms when Troy was burned to ashes.  \nWith you to lead us we have sailed across unmeasured tracts of  \nswelling seas, and in time to come we shall raise your sons to  \n160 the stars and give dominion to your city. Your task is to build  \ngreat walls to guard this great inheritance. You must never flag  \nin the long toil of exile, and you must leave this place. Delian  \nApollo did not send you to these shores. Crete is not where he  \ncommanded you to settle. There is a place \u2013 Greeks call it  \nHesperia \u2013 an ancient land, strong in arms and in the richness  \nof her soil. The Oenotrians lived there, but the descendants of  \nthat race are now said to have taken the name of their king  \nItalus and call themselves Italians. This is our true home. This  \nis where Dardanus sprang from and his father Iasius from whom  \nour race took its beginning. Rise then with cheerful heart and  \n170 pass on these words to Anchises your father, and let him be in  \nno doubt. He must look for Corythus and the lands of Ausonia.  \nJupiter forbids you the Dictaean fields of Crete.'\n\nI was astounded by this vision and by the words of the gods.  \nThis was no sleep. I seemed to be face to face with them and to  \nrecognize their features and the garlands on their heads, and at  \nthe sight my whole body was bathed in cold sweat. Leaping  \nfrom my bed, I raised my hands palms upward to the sky and  \nlifted up my voice in prayer, making pure offerings at the hearth.  \nHaving performed these rites, I went with joyful heart to  \n180 Anchises and told him everything in order. He remembered that  \nour race had two founders, Dardanus and Teucer, a double  \nancestry. He realized that he had fallen into a new mistake about  \nthese ancient places. 'O my son,' he said, 'you who have been  \nso tested by the Fates of Troy, only Cassandra made such a  \nprophecy to me. Now I remember how she used to foretell that  \nthis is what Fate had in store for us and she kept talking about  \nHesperia and about the kingdoms of Italy. But who would have  \nbelieved that Trojans would land on the shores of Hesperia?  \nWho in those days would have believed the prophecies of Cassandra?  \nLet us yield to Phoebus Apollo. We have been advised.  \nLet us follow the better course.' We all accepted his command  \n190 with cries of joy and abandoned this second settlement, leaving  \nonly a few of our number behind, and set sail upon our hollow  \nships to run before the wind over the vast ocean.\n\nWhen we were out at sea and no longer in sight of land, and  \nall around was sky and all around was sea, I saw a dark cloud  \ncome over our heads bringing storm and black night, and the  \nwaves shivered in the darkness. The wind soon whipped up a  \ngreat swell and the storm rose and scattered us all over the  \nocean. A pall of cloud obscured the light, rain fell from a sky  \nwe could not see, and lightning tore the clouds, flash upon flash.  \n200 We were thrown off course and drifted blindly in the waves.  \nUnder that sky even Palinurus said he had lost his bearings in  \nmid-ocean and could not tell day from night. For three long  \ndays, if days they were, of darkness, and three starless nights we  \nran before the storm, until at last on the fourth day we saw the  \nfirst land rising before us and there opened a clear view of  \ndistant mountains and curling smoke. Down came the sails and  \nwe sprang to the oars. The sailors were not slow to sweep the  \n210 blue sea and churn it into foam. I was saved from the ocean and  \nthe shores of the Strophades were the first to receive me.\n\nThis is the Greek name for islands in the great Ionian sea.  \nThis is where the deadly Celaeno and the other Harpies have  \nlived ever since the house of Phineus was barred to them and  \nthey were frightened away from the tables where they used to  \nfeed. These are the vilest of all monsters. No plague or visitation  \nof the gods sent up from the waves of the river Styx has ever  \nbeen worse than these. They are birds with the faces of girls,  \nwith filth oozing from their bellies, with hooked claws for hands  \nand faces pale with a hunger that is never satisfied.\n\nAs soon as we reached the Strophades and entered the harbour,  \n220 there we saw on every side rich herds of cattle on the level  \nground and flocks of goats unguarded on the grass. We drew  \nour swords and rushed upon them, calling on the gods and on  \nJupiter himself to share our plunder. Then we raised couches  \nalong the shore of the bay and were feasting on this rich fare  \nwhen suddenly the Harpies were among us, swooping down  \nfrom the mountains with a fearful clangour of their wings,  \ntearing the food to pieces and polluting everything with their  \nfoul contagion. The stench was rank, and through all this we  \n229 heard their hideous screeching. Once again, in a sheltered spot  \nfar back under an overhanging rock, we relaid our tables and  \nrelit the altar fires. Once again the noisy flock came from some  \nhidden roost in a different quarter of the sky and fluttered round  \ntheir prey, clutching it in their hooked claws and fouling it in  \ntheir mouths. Then it was I ordered my men to arm themselves  \nto make war against this fearsome tribe. They did as ordered,  \nhiding swords and shields here and there in the grass. And so  \nwhen Misenus in his high lookout heard the sound of them  \nswooping down along the whole curved shore of the bay, he  \n240 raised the alarm by blowing on the hollow bronze of his trumpet  \nand my comrades attacked. This was a new kind of battle \u2013  \nswords against filthy sea birds. But these were feathers that felt  \nno violence and backs that could receive no wounds. They  \nsoared in swift flight up towards the stars, leaving behind them  \nthe half-eaten food and their filthy droppings, all but one who  \nremained, perched high on a pinnacle of rock (Celaeno was her  \nname), and from her breast there burst this dire prophecy: 'Is it  \nwar you offer us now, sons of Laomedon, for the slaughter of  \nour bullocks and the felling of our oxen? Is it your plan to make  \nwar against the innocent Harpies and drive us from the kingdom  \n250 of our ancestors? Listen to what I have to say and fix it in your  \nminds. These words were spoken by the Almighty Father of the  \nGods to Phoebus Apollo, and Phoebus Apollo spoke them to  \nme, and now I, the greatest of the Furies, speak them to you.  \nYou are calling upon the winds and trying to sail to Italy. To  \nItaly you will go and you will be allowed to enter its harbours,  \nbut you will not be given a city, and you will not be allowed to  \nbuild walls around it before a deadly famine has come upon  \nyou, and the guilt of our blood drives you to gnaw round the  \nedges of your tables, to put them between your teeth and eat  \nthem.'\n\nWith these words she rose on her wings and flew into the  \n260 forest. In that instant the blood of my comrades was congealed  \nwith fear. Their spirits fell and they lost all desire for fight,  \ntelling me to plead and pray to the creatures for peace, whether  \nthey were goddesses or foul and deadly birds. Then Father  \nAnchises stood on the shore and raised his hands palms upward  \nto heaven, calling upon the great gods and pledging to pay them  \nall the honours that were their due. 'O you gods,' he cried, 'let  \nnot this threat be fulfilled. O gods, turn away this fate from us  \nand graciously preserve your devoted people.' He then gave  \norders to pull in the cables, undo the sail-ropes and let them  \nrun. The south wind filled the canvas, and wind and helmsman  \neach set the same course for us as we flew over the foaming  \n270 waves. Soon there appeared in mid-ocean the woods of  \nZacynthus, and Dulichium, Same and the stone cliffs of Neritos.  \nWe raced away from the rocks of Ithaca, the kingdom of Laertes,  \nand cursed the land that had nurtured the villain Ulixes. In no  \ntime there rose before us the cloudy cap of Mount Leucas and  \nApollo's temple, the terror of sailors. Being weary we set course  \nfor it and came to land at the little city. The anchors ran out  \nfrom the prows and our ships stood to the shore.\n\nSo at last our feet were on dry land again \u2013 more than we had  \ndared to hope for. We performed rites of purification to Jupiter  \n280 and lit altar fires in fulfilment of our vows, crowding the shores  \nof Actium with our Trojan games. My comrades stripped and  \nmade their bodies slippery with oil and wrestled in the style of  \ntheir fathers, as we celebrated our escape and safe voyage past  \nso many Greek cities, right through the middle of our enemies.\n\nIn due course the sun rolled on round the great circle of the  \nyear. Icy winter came and the north winds were roughening the  \nseas. I then took a concave shield of bronze, the armour once  \ncarried by great Abas, and nailed it on the doors of the temple  \nwhere all could see, proclaiming the dedication of it with this  \ninscription:\n\n### AENEAS DEDICATES THESE ARMS  \nTAKEN FROM THE CONQUERING GREEKS\n\nThen I gave orders to leave port and told the rowers to sit to  \n290 their benches. They vied with one another to strike the sea and  \nsweep the surface of it with their oars. We had soon put the  \ncloud-capped citadels of Phaeacia down below the horizon and  \nwe coasted along Epirus until we entered the harbour of Chaonia  \nand then walked up to the lofty city of Buthrotum.\n\nHere there came to our ears a story almost beyond belief, that  \nHelenus, a son of Priam, was king over these Greek cities of  \nEpirus, having succeeded to the throne and the bed of Pyrrhus,  \nson of Achilles and descendant of Aeacus. Andromache, once  \nwife of Hector, had for a second time taken a husband from her  \nown people. I was astounded and the heart within me burned  \nwith love for the man and longing to meet him and find out  \n300 about these great events. I was walking away from the harbour,  \nleaving ships and shore behind me, when I caught sight of  \nAndromache, offering a ritual meal and performing rites to the  \ndead in a grove in front of a city on the banks of a river Simois,  \nbut not the true Simois of Troy. She was pouring a libation to  \nthe ashes of her husband Hector, calling on his shade to come  \nto the empty tomb, a mound of green grass on which she had  \nconsecrated two altars. There she used to go and weep. When  \nshe saw me approaching with armed Trojans all about me, she  \nwas beside herself, numb with fear the moment she saw this  \ngreat miracle, and the warmth of life went out of her bones. She  \nfainted, and only after a long time was she at last able to speak  \n310 to me: 'Is this a true vision? Is it a true messenger that comes to  \nme, son of the goddess? Are you alive? If the light of life has left  \nyou, why are you here? Where is Hector?' As she spoke she  \nburst into tears and her cries filled all the grove. I could hardly  \nfind an answer to these wild words, but stammered a few broken  \nphrases. 'I am indeed alive. After all that has happened I still go  \non living. Do not doubt it. What you see is true. But tell me,  \nwhat fate has overtaken you since you were deprived of such a  \nhusband? What has fallen to the lot of Hector's Andromache?  \nAre you still the wife of Pyrrhus?'\n\n320 She answered, and her voice was low and her eyes downcast:  \n'The happiest of all Trojan women was the virgin daughter of  \nPriam who was made to die by the tomb of her enemy Achilles  \nunder the high walls of Troy. Polyxena did not have to endure  \nthe casting of lots or live to be the slave of a conqueror and lie  \nin a master's bed! But we saw our home burned and sailed over  \nmany seas. We submitted to the arrogance of the house of  \nAchilles and the insolence of his son and bore him a child in  \nslavery. In due course he turned his attention to marrying a  \nSpartan, Hermione, granddaughter of Leda, giving his slave  \nAndromache over to his slave Helenus. But Orestes loved Hermione  \n330 and had hoped to marry her. Incensed at losing her and  \ndriven on by the madness brought upon him by his own crimes,  \nhe caught Pyrrhus where Pyrrhus least expected him and slaughtered  \nhim on the altar he had raised to his father Achilles. At his  \ndeath some of the kingdom he had ruled over came into the  \npossession of Helenus, who then called the plains the Chaonian  \nplains and the whole district Chaonia after Chaon of Troy. He  \nthen built a Pergamum, this Trojan citadel on the ridge. But  \nwhat winds and what fates have given you passage here? Is it  \nsome god that has driven you to these shores that you did not  \nknow were ours? What about your boy Ascanius? Is he alive  \n340 and breathing the air? If he were with you now in Troy...But  \ndoes he ever think of the mother he has lost? Does the old  \ncourage and manliness ever rise in him at the thought of his  \nfather Aeneas and his uncle Hector?'\n\nShe was weeping her useless tears and sobbing bitterly as  \nthese words poured from her when the hero Helenus, son of  \nPriam, arrived from the walls of the city with a great escort. He  \nrecognized his own people and took us gladly to his home. He  \ntoo was weeping and could speak only a few broken words to  \nus between his tears. As I walked I recognized a little Troy, a  \n350 citadel modelled on great Pergamum and a dried-up stream they  \ncalled the Xanthus. There was the Scaean Gate and I embraced  \nit. Nor were my Trojans slow to enjoy this Trojan city with  \nme. The king received them in a broad colonnade and in the  \nmiddle of the courtyard they poured libations of the wine of  \nBacchus and fed off golden dishes and every man had a goblet  \nin his hand.\n\nDay after day wore on with breezes tempting our sails and  \nthe canvas filling and swelling in the south wind, until I went to  \nthe prophet Helenus with this request: 'You are Trojan born.  \n360 You can read the signs sent by the gods. You understand the  \nwill of Phoebus Apollo of Claros, his tripods and his laurels.  \nYou know the meaning of the stars, the cries of birds and the  \nomens of their flight. Come tell me \u2013 for every sign I have  \nreceived from heaven has spoken in favour of this journey, and  \nI am persuaded by all the divine powers to set course for Italy  \nand try to find that distant land. Only the Harpy Celaeno has  \nprophesied a strange and monstrous portent, threatening us  \nwith her deadly anger and all the horrors of famine \u2013 come  \ntell me now, what dangers am I to avoid as I start upon this  \njourney? And as it goes on, what must I do to overcome such  \nadversities?'\n\n370 Before replying Helenus first performed a ritual slaughter of  \nbullocks and asked for the blessing of the gods. He then loosened  \nthe ribbons from his consecrated head, and taking my hand, he  \nled me in anxious expectation into the mighty presence of the  \ngod. In due course he spoke as priest and this was the prophecy  \nthat came from his hallowed lips. 'O son of the goddess, the  \nproof is full and clear that the highest auspices favour your  \nvoyage. This is the fate allotted to you by the King of the Gods.  \nThis is how your fortune rolls and this is the order of its turning.  \nMy words will tell you a small part of all there is to know so  \nthat you may trust yourself more safely to cross the seas that  \nare waiting to receive you, and come to harbour in Ausonia.  \n380 The Fates do not allow Helenus to know the rest and Saturnian  \nJuno forbids it to be spoken. First, you are wrong to imagine  \nthat it is a short voyage to Italy and that there are harbours  \nclose at hand for you to enter. Far and pathless are the ways  \nthat lie between you and that far distant land. You must first  \nbend the oar in the waves of Sicilian seas, then cross the ocean  \nof Ausonia and the lakes of the underworld, and pass Aeaea,  \nthe island of Circe, before you can come to the land which will  \nbe safe for the founding of your city. I shall give you a sign and  \nyou must keep it deep within your heart: when in an hour of  \nperplexity by the flowing waters of a lonely river you find under  \n390 some holm-oaks on the shore a great sow with the litter of thirty  \npiglets she has farrowed, lying there on her side all white, with  \nher young all white around her udders, that will be the place for  \nyour city. There you will find the rest ordained for all your  \nlabours. Nor is there any need for you to shudder at the thought  \nof eating your tables. The Fates will find a way. Call upon  \nApollo and he will come. But you must quickly leave this land  \nof ours and keep well clear of the shore of Italy that lies nearest  \nus bathed by the tide of our sea, for hostile Greeks live in all  \nthese cities. Here Locrians from Narycum have built their walls  \n400 and the army of the Cretan Idomeneus of Lyctos has seized the  \nSallentine plains in Calabria. Here too is the little town of Petelia  \nperching on the wall built for it by Philoctetes, leader of the  \nMeliboeans. And when you have passed all these and your  \nships are moored across the sea, when you have raised altars  \non the shore to fulfil your vows, do not forget to veil your  \nhead in purple cloth so that when the altar fires are burning to  \nhonour the gods, no enemy presence can intrude and spoil the  \nomens. Your comrades and you yourself must keep this mode  \nof sacrifice and your descendants must maintain this purity of  \nworship for ever.\n\n410 'But when you sail on and the wind carries you near the shore  \nof Sicily, and the close-set barriers of Pelorus open before you,  \nmake for the land to the south and the sea to the south, taking  \nthe long way round Sicily and keeping well clear of the breakers  \non the coast to starboard. Men say these lands were originally  \none but were long ago convulsed by some great upheaval and  \ntorn apart. Such changes can occur in the long ageing of time.  \nThe waves of the sea burst in between them and cut Sicily  \nloose from the flank of the land of Hesperia, putting coastlines  \nbetween their fields and cities and flowing in between them in a  \n420 narrow tide. On your right waits Scylla in ambush and on your  \nleft the insatiable Charybdis. Three times a day with the deep  \nvortex of her whirlpool Charybdis sucks great waves into the  \nabyss and then throws them upwards again to lash the stars.  \nBut Scylla lurks in the dark recesses of her cave and shoots out  \nher mouths to seize ships and drag them on to the rocks. She  \nhas a human face and as far as the groin she is a girl with lovely  \nbreasts, but below she is a monstrous sea creature, her womb  \n430 full of wolves, each with a dolphin's tail. It is better to lose time  \nby taking the long course round Cape Pachynus rather than set  \neyes on the hideous Scylla deep in her cave or see those rocks  \nloud with the barking of dogs as blue as the sea.\n\n'One thing more: if the prophet Helenus has any insight into  \nthe future, if there is any reason to believe what I say, if Apollo  \nfills my mind with the truth, there is one prophecy I shall make  \nto you above all others, one counsel I shall repeat to you again  \nand again \u2013 worship the godhead of great Juno first and foremost  \nin your prayers, of your own free will submit your vows to Juno  \nand win over the mighty Queen of Heaven with your offerings  \n440 as you pray. If you do this you will at last leave Sicily behind  \nyou and succeed in reaching the shores of Italy. When you have  \nlanded and come to the city of Cumae and the sacred lakes of  \nAvernus among their sounding forests, there deep in a cave in  \nthe rock you will see a virgin priestess foretelling the future in  \nprophetic frenzy by writing signs and names on leaves. After  \nshe has written her prophecies on these leaves she seals them all  \nup in her cave where they stay in their appointed order. But the  \nleaves are so light that when the door turns in its sockets the  \nslightest breath of wind dislodges them. The draught from  \n450 the door throws them into confusion and the priestess never  \nmakes it her concern to catch them as they flutter round her  \nrocky cave and put them back in order or join up the prophecies.  \nSo men depart without receiving advice and are disappointed in  \nthe house of the Sibyl. No matter how impatient your comrades,  \nno matter how the winds may cry out to your sails to take to sea,  \nthough you know that you could fill the canvas with favouring  \nbreezes, you must not begrudge the time but must stay to visit  \nthe priestess. Approach her oracle with prayers and beg her by  \nher own gracious will to prophesy to you herself, opening her  \n460 lips and speaking to you in her own voice. She will tell you of  \nthe peoples of Italy and the wars that are to come, and how you  \nare to escape or endure all the labours that lie before you. If  \nyou do her reverence she will give you a prosperous voyage.  \nThis is as much as my voice may utter to give you guidance.  \nNow go forward and by your actions raise the greatness of Troy  \nto the skies.'\n\nAfter the prophet Helenus had told us these things in the  \nfriendliness of his heart, he then ordered his people to carry gifts  \nof solid gold and carved ivory down to our ships and stowed a  \ngreat quantity of silver in their hulls with cauldrons from  \nJupiter's temple at Dodona, a breastplate of chain mail interwoven  \nwith triple threads of gold and a noble helmet with crest  \nand streaming plumes once worn by Neoptolemus. There were  \n470 other gifts for my father, and he also gave us horses and leaders  \nof men, rowers to make up the crews and arms for my comrades.\n\nMeanwhile Anchises was ordering us to fit out the ships with  \ntheir sails and not lose the following winds when the priest of  \nApollo addressed him in deep respect: 'Anchises, the gods love  \nyou. You have been thought worthy of the highest of all honours,  \nthe love of Venus. You have been twice rescued from the ruins  \nof Troy, and now before you, look, the land of Ausonia. Sail  \nthere and take possession of it. But you must sail past the  \nopposite coast. The part of Ausonia which Apollo reveals to  \n480 you is far from here. Go then, Anchises, fortunate in the devotion  \nof your son. There is no more to say. Why do I keep you talking  \nwhen the wind is rising?'\n\nAndromache also grieved at this parting that was to be our  \nlast and brought us robes embroidered with gold thread and a  \nPhrygian cloak for Ascanius. She was as generous as Helenus  \nhad been, heaping the gifts of her weaving upon him and saying:  \n'Take these too, my boy, and I hope the work of my hands may  \nremind you of Andromache, wife of Hector, and be a token of  \nmy long-enduring love for you. Accept them. They are the last  \ngifts you will receive from your own people. You are the only  \n490 image left to me of my own son Astyanax. He had just those  \neyes, and just those hands. His face was just like yours. He  \nwould have been growing up now, the same age as yourself.'\n\nThe tears were starting to my eyes as I was leaving them, and  \nI spoke these words. 'Live on and enjoy the blessing of heaven.  \nYour destiny has been accomplished. But we are called from  \nfate to fate. Your rest is won. You do not need to plough tracts  \nof ocean searching for the ever-receding Ausonian fields. You  \nhave before your eyes an image of the river Xanthus and a Troy  \nmade by your own hands, more fortunate, I pray, than the Troy  \n500 that was, and less of a stumbling-block to the Greeks. If ever I  \nreach the river Thybris and the fields through which the Thybris  \nflows and see my people with their own city walls, we shall in  \nsome future age unite our cities and the peoples of Hesperia and  \nEpirus, for we are kith and kin, the same Dardanus is our  \nfounder and the same destiny attends us. We shall make them  \nboth one Troy in spirit. Let that be a duty for our descendants.'\n\nDown the coast we sailed near the Ceraunian rocks where the  \ncrossing to Italy is shortest, and as we sailed the sun set and  \nshadow darkened the mountains. At last we lay down by the  \nwaves of the sea in the lap of earth, and after allotting the next  \n510 day's order of rowing, we took our ease all along the dry beach  \nand sleep washed into our weary limbs.\n\nNight in its chariot drawn by the Hours was not yet coming  \nup to the middle of the sky, but there was no more sleep for  \nPalinurus. He rose from his bed and studied all the winds,  \npricking up his ears to test the air and marking the path of every  \nstar gliding in the silent sky, Arcturus and the rainy Hyades and  \nthe two Triones, the oxen of the Plough, and he looked round  \nto the south at Orion armed in gold, and saw that the whole  \nsky was serene and settled. Clear came his signal from the high  \n520 stern. We broke camp, started our voyage and spread the wings  \nof our sails.\n\nThe stars had been put to flight and dawn was reddening in  \nthe sky when we sighted in the far distance the dim hills and  \nplains of Italy. 'Italy!' \u2013 the first shout was from Achates \u2013 and  \n'Italy!' \u2013 the men took up the cry in cheerful salute. Then Father  \nAnchises, standing on the high stern, garlanded a great mixing  \nbowl, filled it with unwatered wine and called upon the gods:  \n'O you who rule sea, land and storm, give us an easy wind for  \nour voyage. Blow kindly upon us.'\n\n530 His prayer was answered. The breeze freshened and a harbour  \nopened up before us, growing nearer and nearer till we could  \nsee the temple of Minerva on the citadel. My comrades furled  \ntheir sails and pointed their prows to the shore. The harbour  \nwas shaped like a bow, curving away from the swell which came  \nin from the east. The rocks at the mouth were foaming with salt  \nspray but the harbour lay tucked away behind. Towering rocks  \non either side stretched down their arms to form a double wall  \nand the temple stood well back from the shore. The first omen  \nI saw here was four horses white as snow cropping the grass on  \n540 a broad plain and my father Anchises interpreted it: 'This land  \nthat receives us is promising us war! Men arm horses for war  \nand so this troop of horses means threat of war. Yet at other  \ntimes they are harnessed to chariots and accept reins under the  \nyoke in harmony. There is hope of peace also.'\n\nAt that moment we prayed to the sacred godhead of Pallas,  \nclasher of arms, the first goddess to welcome us in this hour of  \nour joy. Standing at the altar we veiled our heads with Phrygian  \ncloth, and in accordance with the instructions which Helenus  \nhad told us to follow before all others, duly paid the prescribed  \nhonour to Juno of Argos with our burnt offerings.\n\nWe did not linger there but as soon as we had performed the  \nrites in due order we raised our sails, swung the yards round  \n550 and left behind us this home of Greeks, this land we could not  \ntrust. Next we saw the bay of Tarentum, the city of Hercules if  \nthe story is true, and over against it rose the temple of the  \ngoddess Juno at Lacinium, the citadel of Caulon and the bay of  \nScylaceum, that great breaker of ships. Then from far out at sea  \nwe sighted Mount Etna in Sicily and heard a loud moaning of  \nwaters and grinding of rocks and the voice of breakers beating  \non the shore, as the sea began to rise and swirl the sand in  \nits surge. Father Anchises cried out: 'This must be the deadly  \nCharybdis. These are the cliffs Helenus warned us against. These  \n560 are the terrible rocks. Use all your strength to save yourselves,  \ncomrades. Keep well in time and rise to the oar.' They did as  \nthey were bidden. Palinurus was the first to wrench his ship to  \nport and out to sea with a loud creaking of the bow, and the  \nwhole fleet with every sail and oar steered to port with him. A  \ngreat arching wave came and lifted us to the sky and a moment  \nlater as the wave was sucked down we plunged into the abyss  \nof hell. Three times the cliffs roared between their hollow rocks.  \nThree times we saw the foam shoot up and spatter the stars.  \nMeanwhile the sun had set, the wind had fallen and we were  \nweary and lost, drifting towards the shore of the Cyclopes.\n\n570 The harbour there is out of the wind. It is still and spacious  \nbut close by Mount Etna thunders and hurls down its deadly  \ndebris. Sometimes it shoots a pitch-black cloud of swirling  \nsmoke and glowing ashes into the sky and tosses up balls of  \nflame to lick the stars. Sometimes it belches boulders, tearing  \nout the bowels of the mountain and throwing molten rock up  \ninto the air, seething and groaning in its very depths. The story  \ngoes that the body of Enceladus, half-consumed by the fire of  \nthe thunderbolt, is crushed under this great mass. Mighty Etna  \n580 lies on top of him breathing fire from its shattered furnaces and  \nevery time he turns over from one weary flank to another the  \nwhole of Sicily trembles and murmurs and wreathes the sky  \nwith smoke. We hid in the woods and lived through a night of  \nhorror, not seeing what was making these monstrous sounds.  \nThe fire of the stars was quenched and the dark bowl of heaven  \nwas denied their radiance. Clouds darkened the sky and  \nunbroken night obscured the moon.\n\nAt last the Morning Star appeared and the next day was  \n590 beginning to rise. The Goddess of the Dawn had dispersed the  \ndank mists from the sky when suddenly we saw a strange sight.  \nComing out of the woods was a man we did not know, in  \npitiable plight and half-dead with hunger, coming towards us  \non the shore with his hands stretched out in supplication. We  \nstared at him. The filth on his body was indescribable. He had  \na straggling beard and the rags he wore were pinned together  \nby thorns, but for all that he was a Greek, one of those who had  \nbeen sent to Troy bearing the arms of his country. When still at  \na distance he saw our Trojan clothes and Trojan armour, he  \nchecked his stride and stood in terror at the sight of us. But he  \n600 soon rushed down to the shore weeping and pleading: 'I beg  \nyou, Trojans, by all the stars, by the gods above, by the bright  \nair of heaven which we breathe, take me aboard your ships.  \nTake me anywhere. That is all I ask. I know I was one of those  \nwho sailed with the Greek fleet. I admit I made war against the  \ngods of your homes in Troy. If that offence is so great, tear me  \nlimb from limb, scatter the pieces on the waves and let them  \nsink into the vastness of the sea. If I am to die, I shall be pleased  \nto die at the hands of men.'\n\nWhen he had spoken he clasped our knees, he grovelled on  \nhis knees, and would not rise. We urged him to explain who he  \n610 was, what family he came from and what misfortune was driving  \nhim to this. Father Anchises himself was not slow to offer his  \nright hand and that assurance gave him courage. He laid aside  \nhis fear and told his story: 'My native land is Ithaca. I am a  \ncomrade of the unfortunate Ulixes. My name is Achaemenides.  \nMy father Adamastus being poor, I went to Troy \u2013 cursed be  \nthe day! My comrades, distraught with fear, forgot me and left  \nme here in the vast cave of the Cyclops when they crossed that  \ncruel threshold to safety. This huge cavern is his home, deep  \n620 and dark and filthy with the gore of his feasts. He himself is so  \ntall that his head knocks against the stars \u2013 O you gods, relieve  \nthe earth of all such monsters. No one dares to look at him or  \nspeak to him. He feeds on the flesh of his victims and drinks the  \nblack blood. I have seen him with my own eyes lolling in the  \nmiddle of his cave with two of our men in one huge hand,  \nbashing their bodies on the rock till the threshold was swimming  \nwith blood. I have seen him chewing arms and legs with black  \ngore oozing from them and the warm limbs twitching between  \nhis teeth. But he met his punishment. The man from Ithaca  \n630 did not submit to this. Whatever happened Ulixes was always  \nUlixes. As soon as the Cyclops had his fill and was sunk in a  \ndrunken stupor, lying there with his head back and his neck  \nexposed, sprawling all over the cave and belching blood and  \nwine and pieces of flesh as he slept, we prayed to the great gods  \nand after casting lots spread ourselves out all round him. Then,  \ntaking a sharp weapon, we drilled the one huge eye that lay, like  \nan Argive shield or the lamp of Apollo's sun, deep set in that  \ndreadful forehead. That was how in the end we took sweet  \nrevenge for the death of our comrades. But you are in danger.  \n640 You must escape and escape now. Cut your moorings and put  \nto sea. You know what Polyphemus is and how huge he is,  \nkeeping his woolly sheep penned there in his hollow cave and  \nsqueezing the milk from their udders, but there are a hundred  \nother horrible Cyclopes living together near this shore and  \nroving the high mountains. This is now the third time I have  \nseen the horns of the moon filling with light as I have dragged  \nout my existence in the woods alone among the dens and lairs  \nof wild beasts, climbing rocks to keep watch on the giant  \n650 Cyclopes and trembling at the sound of their voices and the  \ntread of their feet. My food is miserable. The trees yield me  \nsome berries and the fruit of the cornel, hard as stone, and I tear  \nup herbs by the root and eat them. I have kept constant watch  \nbut this is the first time I have seen ships coming near this shore.  \nI have put myself in your hands, and would have done so  \nwhoever you had been. It is enough for me to escape from this  \nunspeakable people. You can take this life of mine by whatever  \nmeans you please.'\n\nScarcely had he finished speaking when we saw the shepherd  \nPolyphemus himself high up on the mountain among his sheep,  \nheaving his vast bulk down towards the shore he knew so well.  \nHe was a terrifying sight, huge, hideous, blinded in his one eye  \nand using the trunk of a pine tree to guide his hand and give  \n660 him a firm footing. His woolly sheep were coming with him.  \nThey were the only pleasure he had left, his sole consolation in  \ndistress. As soon as he felt the waves deepening and reached the  \nlevel ocean, he washed away with sea water the blood that was  \nstill trickling from his gouged-out eye, grinding his teeth and  \nmoaning, and as he strode now in mid-ocean, the waves still did  \nnot wet his towering flanks.\n\nWe were terrified and lost no time in taking the fugitive  \naboard \u2013 he had suffered enough \u2013 and making our escape.  \nKeeping silence as we cut the cables we churned the surface of  \nthe sea, leaning forward and straining at the oars. He heard us,  \n670 and whirled round in the direction of our voices, but he had no  \nchance of laying a hand on us or keeping up with the current of  \nthe Ionian sea, so he raised a great clamour which set the ocean  \nand all its waves shivering. The whole land of Italy trembled  \nwith fear and the bellowing boomed in the hollow caverns of  \nMount Etna. The tribe of Cyclopes was roused and came rushing  \ndown from their woods and high mountains to the harbour and  \nfilled the shore. We saw the brotherhood of Etna standing there  \n680 helpless, each with his one eye glaring and head held high in the  \nsky, a fearsome gathering, standing like high-topped mountain  \noaks or cone-bearing cypresses in Jupiter's soaring forest or the  \ngrove of Diana. With terror driving us along we let the sheets  \nfull out and filled our sails with whatever wind was blowing.  \nThis is what Helenus had told us not to do. He had advised us  \nthat it was a narrow passage between Scylla and Charybdis with  \ndeath on either side if I did not hold a steady course. I resolved  \nto turn about, and sure enough the north wind came to our  \nrescue and blew down the narrow strait from Cape Pelorus. I  \nsailed south past the mouth of the river Pantagias with its  \nharbour of natural rock, past the bay of Megara and low-lying  \n690 Thapsus. Achaemenides pointed out such places to us as we took  \nhim back along the shores he had once sailed in his wanderings as  \na comrade of the unfortunate Ulixes.\n\nAt the entrance to the bay of Syracuse, opposite the wave-beaten  \nheadland of Plemyrium, there stands an island which  \nmen of old called Ortygia. The story goes that the river-god  \nAlpheus of Elis forced his way here by hidden passages under  \nthe sea and now mingles with Sicilian waters at the mouth of  \nArethusa's fountain. Obeying the instructions we had received,  \nwe worshipped the great gods of the place and I then sailed on  \nleaving behind the rich lands around the marshy river Helorus.  \n700 From here we rounded Cape Pachynus, Keeping close in to its  \njutting cliffs of rock, and Camerina came in to view in the  \ndistance, the place the Fates forbade to move, and then the  \nGeloan plains and Gela itself, called after its turbulent river.  \nThen in the far distance appeared the great walls of Acragas on  \nits crag, once famous for the breeding of high-mettled horses.  \nNext the winds carried me past Selinus, named after the parsley  \nit gave to crown the victors in Greek games, and I steered past  \nthe dangerous shoals and hidden rocks of Lilybaeum.\n\nI then put into port at Drepanum, but had little joy of that  \n710 shore. This was the place where weary as I was with all these  \nbatterings of sea and storm, to my great grief I lost my father  \nAnchises who had been my support in every difficulty and  \ndisaster. This is where you left me, O best of fathers, whom I  \nrescued from so many dangers and all to no purpose. Neither  \nHelenus for all his fearsome predictions nor the Harpy Celaeno  \ngave me any warning of this sorrow. This was the last of my  \nlabours. With this my long course was run. From here I sailed,  \nand God drove me upon your shores.'\n\nIn these words did Father Aeneas recount his wanderings and  \nthe fates the gods had sent him, and they all listened. At last he  \nwas silent. Here he made an end and was at peace.\n\n## BOOK 4  \nDIDO\n\nBut the queen had long since been suffering from love's deadly  \nwound, feeding it with her blood and being consumed by its  \nhidden fire. Again and again there rushed into her mind thoughts  \nof the great valour of the man and the high glories of his line.  \nHis features and the words he had spoken had pierced her heart  \nand love gave her body no peace or rest. The next day's dawn  \nwas beginning to traverse the earth with the lamp of Phoebus'  \nsunlight and had moved the dank shadow of night from the sky  \nwhen she spoke these words from the depths of her affliction to  \n10 her loved and loving sister: 'O Anna, what fearful dreams I have  \nas I lie there between sleeping and waking! What a man is this  \nwho has just come as a stranger into our house! What a look on  \nhis face! What courage in his heart! What a warrior! I do believe,  \nand I am sure it is true, he is descended from the gods. If there  \nis any baseness in a man, it shows as cowardice. Oh how cruelly  \nhe has been hounded by the Fates! And did you hear him tell  \nwhat a bitter cup of war he has had to drain? If my mind had  \nnot been set and immovably fixed against joining any man in  \nthe bonds of marriage ever since death cheated me of my first  \nlove, if I were not so utterly opposed to the marriage torch and  \n20 bed, this is the one temptation to which I could possibly have  \nsuccumbed. I will admit it, Anna, ever since the death of my  \npoor husband Sychaeus, since my own brother spilt his blood  \nand polluted the gods of our home, this is the only man who  \nhas stirred my feelings and moved my mind to waver: I sense  \nthe return of the old fires. But I would pray that the earth open  \nto its depths and swallow me or that the All-powerful Father of  \nthe Gods blast me with his thunderbolt and hurl me down to  \nthe pale shades of Erebus and its bottomless night before I go  \nagainst my conscience and rescind its laws. The man who first  \njoined himself to me has carried away all my love. He shall keep  \nit for himself, safe in his grave.'\n\n30 The tears came when she had finished speaking, and streamed  \ndown upon her breast. But Anna replied: 'O sister, dearer to me  \nthan the light of life, are you going to waste away, living alone  \nand in mourning all the days of your youth, without knowing  \nthe delight of children and the rewards of love? Do you believe  \nthis is what the dead care about when they are buried in the  \ngrave? Since your great sadness you have paid no heed to any  \nman in Libya, or before that in Tyre. You have rejected Iarbas  \nand other chiefs bred in Africa, this rich home of triumphant  \nwarriors. Will you now resist even a love your heart accepts?  \nHave you forgotten what sort of people these are in whose land  \n40 you have settled? On the one side you are beset by invincible  \nGaetulians, by Numidians, a race not partial to the bridle, and  \nthe inhospitable Syrtes; on the other, waterless desert and fierce  \nraiders from Barca. I do not need to tell you about the war being  \nraised against you in Tyre and your brother's threats. I for my  \npart believe that it is with the blessing of the gods and the favour  \nof Juno that the Trojan ships have held course here through the  \nwinds. Just think, O my sister, what a city and what a kingdom  \nyou will see rising here if you are married to such a man! To  \nwhat a pinnacle of glory will Carthage be raised if Trojans are  \n50 marching at our side! You need only ask the blessing of the gods  \nand prevail upon them with sacrifices. Indulge your guest. Stitch  \ntogether some reasons to keep him here while stormy seas and  \nthe downpours of Orion are exhausting their fury, while his  \nships are in pieces and it is no sky to sail under.'\n\nWith these words Anna lit a fire of wild love in her sister's  \nbreast. Where there had been doubt she gave hope and Dido's  \nconscience was overcome. First they approached the shrines and  \nwent round the altars asking the blessing of the gods. They  \npicked out yearling sheep, as ritual prescribed, and sacrificed  \nthem to Ceres the Lawgiver, to Phoebus Apollo, to Bacchus the  \n60 Releaser and above all to Juno, the guardian of the marriage  \nbond. Dido in all her beauty would hold a sacred dish in her  \nright hand and would pour wine from it between the horns of  \na white cow or she would walk in state to richly smoking  \naltars before the faces of the gods, renewing her offerings all  \nday long, and when the bellies of the victims were opened she  \nwould stare into their breathing entrails to read the signs. But  \npriests, as we know, are ignorant. What use are prayers and  \nshrines to a passionate woman? The flame was eating the soft  \nmarrow of her bones and the wound lived quietly under her  \nbreast. Dido was on fire with love and wandered all over the  \n70 city in her misery and madness like a wounded doe which a  \nshepherd hunting in the woods of Crete has caught off guard,  \nstriking her from long range with steel-tipped shaft; the arrow  \nflies and is left in her body without his knowing it; she runs  \naway over all the wooded slopes of Mount Dicte, and sticking  \nin her side is the arrow that will bring her death.\n\nSometimes she would take Aeneas through the middle of  \nCarthage, showing him the wealth of Sidon and the city waiting  \nfor him, and she would be on the point of speaking her mind to  \nhim but checked the words on her lips. Sometimes, as the day  \nwas ending, she would call for more feasting and ask in her  \ninfatuation to hear once more about the sufferings of Troy and  \n80 once more she would hang on his lips as he told the story. Then,  \nafter they had parted, when the fading moon was dimming her  \nlight and the setting stars seemed to speak of sleep, alone and  \nwretched in her empty house she would cling to the couch  \nAeneas had left. There she would lie long after he had gone and  \nshe would see him and hear him when he was not there for her  \nto see or hear. Or she would keep back Ascanius and take him  \non her knee, overcome by the likeness to his father, trying to  \nbeguile the love she could not declare. The towers she was  \nbuilding ceased to rise. Her men gave up the exercise of war and  \nwere no longer busy at the harbours and fortifications making  \nthem safe from attack. All the work that had been started, the  \nthreatening ramparts of the great walls and the cranes soaring  \nto the sky, all stood idle.\n\n90 As soon as Saturnian Juno, the dear wife of Jupiter, realized  \nthat Dido was infected by this sickness and that passion was  \nsweeping away all thought for her reputation, she went and  \nspoke to Venus: 'You are covering yourselves with glory. These  \nare the supreme spoils you are bringing home, you and that boy  \nof yours \u2013 and what a noble and notable specimen of the divine  \nhe is \u2013 one woman has been overthrown by the arts of two gods!  \nI do not fail to see that you have long been afraid of our walls  \nand looked askance at the homes of lofty Carthage. But how is  \nthis going to end? Where is all this rivalry going to lead us now?  \n100 Why do we not instead agree to arrange a marriage and live at  \npeace for ever? You have achieved what you have set your whole  \nheart on: Dido is passionately in love and the madness is working  \nthrough her bones. So let us make one people of them and share  \nauthority equally over them. Let us allow her to become the  \nslave of a Phrygian husband and to hand over her Tyrians to  \nyou as a dowry!'\n\nVenus realized this was all pretence in order to divert the  \nempire of Italy to the shores of Libya, and made this response  \nto the Queen of Heaven: 'Who would be so insane as to reject  \nsuch an offer and choose instead to contend with you in war? If  \n110 only a happy outcome could attend the plan you describe! But  \nI am at the mercy of the Fates and do not know whether Jupiter  \nwould wish there to be one city for the Tyrians and those who  \nhave come from Troy or whether he would approve the merging  \nof their peoples and the making of alliances. You are his wife.  \nIt could not be wrong for you to approach him with prayers  \nand test his purpose. You proceed and I shall follow.'\n\n'That will be my task,' replied Juno. 'But now listen and I  \nshall explain in a few words how the first part of the plan may  \nbe carried out. Aeneas and poor Dido are preparing to go  \nhunting together in the forest as soon as tomorrow's sun first  \nrises and the rays of the Titan unveil the world. When the beaters  \n120 are scurrying about and putting nets round copses, I shall pour  \ndown a dark storm of rain and hail on them and shake the  \nwhole sky with thunder. Their companions will run away and  \nbe lost to sight in a pall of darkness. Dido and the leader of the  \nTrojans will both take refuge in the same cave. I shall be there,  \nand if your settled will is with me in this, I shall join them in  \nlasting marriage and make her his. This will be their wedding.'  \nThis was what Juno asked and Venus of Cythera did not refuse  \nher but nodded in assent. She saw through the deception and  \nlaughed.\n\nMeanwhile Aurora rose from the ocean and when her light  \n130 came up into the sky, a picked band of men left the gates  \nof Carthage carrying nets, wide-meshed and fine-meshed, and  \nbroad-bladed hunting spears, and with them came Massylian  \nhorsemen at the gallop and packs of keen-scented hounds. The  \nqueen was lingering in her chamber and the Carthaginian leaders  \nwaited at her door. There, resplendent in its purple and gold,  \nstood her loud-hoofed, high-mettled horse champing its foaming  \nbit. She came at last with a great entourage thronging round  \nher. She was wearing a Sidonian cloak with an embroidered  \nhem. Her quiver was of gold. Gold was the clasp that gathered  \nup her hair and her purple tunic was fastened with a golden  \n140 brooch. Nor was the Trojan company slow to move forward,  \nAscanius with them in high glee. Aeneas himself marched at  \ntheir head, the most splendid of them all, as he brought his men  \nto join the queen's. He was like Apollo leaving his winter home  \nin Lycia and the waters of the river Xanthus to visit his mother  \nat Delos, there to start the dancing again, while all around the  \naltars gather noisy throngs of Cretans and Dryopes and painted  \nAgathyrsians; the god himself strides the ridges of Mount  \nCynthus, his streaming hair caught up and shaped into a soft  \ngarland of green and twined round a band of gold, and the  \n150 arrows sound on his shoulders \u2013 with no less vigour moved  \nAeneas and his face shone with equal radiance and grace. When  \nthey had climbed high into the mountains above the tracks of  \nmen where the animals make their lairs, suddenly some wild  \ngoats were disturbed on the top of a crag and came running  \ndown from the ridge. Then on the other side there were deer  \nrunning across the open plain. They had gathered into a herd  \nand were raising the dust as they left the high ground far behind  \nthem. Down in the middle of the valley young Ascanius was  \nriding a lively horse and revelling in it, galloping past the deer  \nand the goats and praying that among these flocks of feeble  \ncreatures he could come across a foaming boar or that a tawny  \nlion would come down from the mountains.\n\n160 While all this was happening a great rumble of thunder began  \nto stir in the sky. Down came the rain and the hail, and Tyrian  \nhuntsmen, men of Troy and Ascanius of the line of Dardanus  \nand grandson of Venus, scattered in fright all over the fields,  \nmaking for shelter as rivers of water came rushing down the  \nmountains. Dido and the leader of the Trojans took refuge  \ntogether in the same cave. The sign was first given by Earth and  \nby Juno as matron of honour. Fires flashed and the heavens  \nwere witness to the marriage while nymphs wailed on the mountain  \n170 tops. This day was the beginning of her death, the first cause  \nof all her sufferings. From now on Dido gave no thought to  \nappearance or her good name and no longer kept her love as a  \nsecret in her own heart, but called it marriage, using the word  \nto cover her guilt.\n\nRumour did not take long to go through the great cities of  \nLibya. Of all the ills there are, Rumour is the swiftest. She thrives  \non movement and gathers strength as she goes. From small and  \ntimorous beginnings she soon lifts herself up into the air, her  \nfeet still on the ground and her head hidden in the clouds. They  \n180 say she is the last daughter of Mother Earth who bore her in  \nrage against the gods, a sister for Coeus and Enceladus. Rumour  \nis quick of foot and swift on the wing, a huge and horrible  \nmonster, and under every feather of her body, strange to tell,  \nthere lies an eye that never sleeps, a mouth and a tongue that  \nare never silent and an ear always pricked. By night she flies  \nbetween earth and sky, squawking through the darkness, and  \nnever lowers her eyelids in sweet sleep. By day she keeps watch  \nperched on the tops of gables or on high towers and causes fear  \nin great cities, holding fast to her lies and distortions as often as  \n190 she tells the truth. At that time she was taking delight in plying  \nthe tribes with all manner of stories, fact and fiction mixed in  \nequal parts: how Aeneas the Trojan had come to Carthage and  \nthe lovely Dido had thought fit to take him as her husband; how  \nthey were even now indulging themselves and keeping each  \nother warm the whole winter through, forgetting about their  \nkingdoms and becoming the slaves of lust. When the foul goddess  \nhad spread this gossip all around on the lips of men, she  \nthen steered her course to king Iarbas to set his mind alight and  \nfuel his anger.\n\nJupiter had ravished a Garamantian nymph and Iarbas was  \n200 his son. Over his broad realm he had erected a hundred huge  \ntemples to the god and set up a hundred altars on which he  \nhad consecrated ever-burning fires to keep undying holy vigil,  \nenriching the earth with the blood of slaughtered victims and  \ndraping the doors with garlands of all kinds of flowers. Iarbas,  \nthey say, was driven out of his mind with anger when he heard  \nthis bitter news. Coming into the presence of the gods before  \ntheir altars in a passion of rage, he offered up prayer upon  \nprayer to Jupiter, raising his hands palms upward in supplication:  \n'Jupiter All-powerful, who now receive libations of wine  \nfrom the Moorish people feasting on their embroidered couches,  \ndo you see all this? Or are we fools to be afraid of you, Father,  \n210 when you hurl your thunderbolts? Are they unaimed, these fires  \nin the clouds that cow our spirits? Is there no meaning in the  \nmurmur of your thunder? This woman was wandering about  \nour land and we allowed her at a price to found her little city.  \nWe gave her a piece of shore to plough and laid down the laws  \nof the place for her and she has spurned our offer of marriage  \nand taken Aeneas into her kingdom as lord and master, and  \nnow this second Paris, with eunuchs in attendance and hair  \ndripping with perfume and Maeonian bonnet tied under his  \nchin, is enjoying what he has stolen while we bring gifts to  \ntemples we think are yours and keep warm with our worship  \nthe reputation of a useless god.'\n\n220 As Iarbas prayed these prayers with his hand on the altar, the  \nAll-powerful god heard him and turned his eyes towards the  \nroyal city and the lovers who had lost all recollection of their  \ngood name. Then he spoke to Mercury and gave him these  \ninstructions: 'Up with you, my son. Call for the Zephyrs, glide  \ndown on your wings and speak to the Trojan leader who now  \nlingers in Tyrian Carthage without a thought for the cities  \ngranted him by the Fates. Take these words of mine down to  \nhim through the swift winds and tell him that this is not the  \nman promised us by his mother, the loveliest of the goddesses.  \nIt was not for this that she twice rescued him from the swords  \n230 of the Greeks. She told us he would be the man to rule an Italy  \npregnant with empire and clamouring for war, passing the high  \nblood of Teucer down to his descendants and subduing the  \nwhole world under his laws. If the glory of such a destiny does  \nnot fire his heart, if he does not strive to win fame for himself,  \nask him if he grudges the citadel of Rome to his son Ascanius.  \nWhat does he have in mind? What does he hope to achieve  \ndallying among a hostile people and sparing not a thought for  \nthe Lavinian fields and his descendants yet to be born in  \nAusonia? He must sail. That is all there is to say. Let that be  \nour message.'\n\nJupiter had finished speaking and Mercury prepared to obey  \nthe command of his mighty father. First of all he fastened on his  \n240 feet the golden sandals whose wings carry him high above land  \nand sea as swiftly as the wind. Then, taking the rod which  \nsummons pale spirits out of Orcus or sends them down to  \ngloomy Tartarus, which gives sleep and takes it away and opens  \nthe eyes of men in death, he drove the winds before him and  \nfloated through the turbulent clouds till in his flight he saw the  \ncrest and steep flanks of Atlas whose rocky head props up the  \nsky. This is the Atlas whose head, covered in pine trees and  \n250 beaten by wind and rain, never loses its dark cap of cloud. The  \nsnow falls upon his shoulders and lies there, then rivers of water  \nroll down the old man's chin and his bristling beard is stiff with  \nice. This is where Mercury the god of Mount Cyllene first  \nlanded, fanning out his wings to check his flight. From here he  \nlet his weight take him plummeting to the wave tops, like a bird  \nskimming the sea as it flies along the shore, among the rocks  \nwhere it finds the fish. So flew the Cyllenian god between earth  \nand sky to the sandy beaches of Libya, cleaving the winds as he  \nswooped down from the mountain that had fathered his own  \nmother, Maia.\n\nAs soon as his winged feet touched the roof of a Carthaginian  \n260 hut, he caught sight of Aeneas laying the foundations of the  \ncitadel and putting up buildings. His sword was studded with  \nyellow stars of jasper, and glowing with Tyrian purple there  \nhung from his shoulders a rich cloak given him by Dido into  \nwhich she had woven a fine cross-thread of gold. Mercury  \nwasted no time: 'So now you are laying foundations for the high  \ntowers of Carthage and building a splendid city to please your  \nwife? Have you entirely forgotten your own kingdom and your  \n270 own destiny? The ruler of the gods himself, by whose divine will  \nthe heavens and the earth revolve, sends me down from bright  \nOlympus and bids me bring these commands to you through  \nthe swift winds. What do you have in mind? What do you hope  \nto achieve by idling your time away in the land of Libya? If the  \nglory of such a destiny does not fire your heart, spare a thought  \nfor Ascanius as he grows to manhood, for the hopes of this Iulus  \nwho is your heir. You owe him the land of Rome and the  \nkingdom of Italy.'\n\nNo sooner had these words passed the lips of the Cyllenian  \ngod than he disappeared from mortal view and faded far into  \n280 the insubstantial air. But the sight of him left Aeneas dumb and  \nsenseless. His hair stood on end with horror and the voice stuck  \nin his throat. He longed to be away and leave behind him this  \nland he had found so sweet. The warning, the command from  \nthe gods, had struck him like a thunderbolt. But what, oh what,  \nwas he to do? What words dare he use to approach the queen  \nin all her passion? How could he begin to speak to her? His  \nthoughts moved swiftly now here, now there, darting in every  \npossible direction and turning to every possible event, and as he  \npondered, this seemed to him a better course of action: he called  \nMnestheus, Sergestus and brave Serestus and ordered them to  \nfit out the fleet and tell no one, to muster the men on the shore  \n290 with their equipment at the ready, and keep secret the reason  \nfor the change of plan. In the meantime, since the good queen  \nknew nothing and the last thing she expected was the shattering  \nof such a great love, he himself would try to make approaches  \nto her and find the kindest time to speak and the best way to  \nhandle the matter. They were delighted to receive their orders  \nand carried them out immediately.\n\nBut the queen \u2013 who can deceive a lover? \u2013 knew in advance  \nsome scheme was afoot. Afraid where there was nothing to fear,  \nshe was the first to catch wind of their plans to leave, and while  \nshe was already in a frenzy, that same wicked Rumour brought  \nword that the Trojans were fitting out their fleet and preparing  \n300 to sail away. Driven to distraction and burning with passion,  \nshe raged and raved round the whole city like a Bacchant stirred  \nby the shaking of the sacred emblems and roused to frenzy when  \nshe hears the name of Bacchus at the biennial orgy and the  \nshouting on Mount Cithaeron calls to her in the night. At last  \nshe went to Aeneas, and before he could speak, she cried: 'You  \ntraitor, did you imagine you could do this and keep it secret?  \nDid you think you could slip away from this land of mine and  \nsay nothing? Does our love have no claim on you? Or the pledge  \nyour right hand once gave me? Or the prospect of Dido dying a  \n310 cruel death? Why must you move your fleet in these winter  \nstorms and rush across the high seas into the teeth of the north  \nwind? You are heartless. Even if it were not other people's fields  \nand some home unknown you were going to, if old Troy were  \nstill standing, would any fleet set sail even for Troy in such  \nstormy seas? Is it me you are running away from? I beg you, by  \nthese tears, by the pledge you gave me with your own right hand  \n\u2013 I have nothing else left me now in my misery \u2013 I beg you by  \nour union, by the marriage we have begun \u2013 if I have deserved  \nany kindness from you, if you have ever loved anything about  \nme, pity my house that is falling around me, and I implore you,  \n320 if it is not too late for prayers, give up this plan of yours. I am  \nhated because of you by the peoples of Libya and the Numidian  \nkings. My own Tyrians are against me. Because of you I have  \nlost all conscience and self-respect and have thrown away the  \ngood name I once had, my only hope of reaching the stars. My  \nguest is leaving me to my fate and I shall die. \"Guest\" is the only  \nname I can now give the man who used to be my husband. What  \nam I waiting for? For my brother Pygmalion to come and raze  \nmy city to the ground? For the Gaetulian Iarbas to drag me off  \nin chains? Oh if only you had given me a child before you  \nabandoned me! If only there were a little Aeneas to play in my  \npalace! In spite of everything his face would remind me of yours  \n330 and I would not feel utterly betrayed and desolate.'\n\nShe had finished speaking. Remembering the warnings of  \nJupiter, Aeneas did not move his eyes and struggled to fight  \ndown the anguish in his heart. At last he spoke these few words:  \n'I know, O queen, you can list a multitude of kindnesses you  \nhave done me. I shall never deny them and never be sorry to  \nremember Dido while I remember myself, while my spirit still  \ngoverns this body. Much could be said. I shall say only a little.  \nIt never was my intention to be deceitful or run away without  \nyour knowing, and do not pretend that it was. Nor have I ever  \n340 offered you marriage or entered into that contract with you. If  \nthe Fates were leaving me free to live my own life and settle all  \nmy cares according to my own wishes, my first concern would  \nbe to tend the city of Troy those of my dear people who survive.  \nA lofty palace of Priam would still be standing and with my  \nown hands I would have built a new citadel at Pergamum for  \nthose who have been defeated. But now Apollo of Gryneum has  \ncommanded me to claim the great land of Italy and \"Italy\" is  \nthe word on the lots cast at his Lycian oracle. That is my love,  \nand that is my homeland. You are a Phoenician from Asia and  \nyou care for the citadel of Carthage and love the very sight of  \n350 this city in Libya; what objection can there be to Trojans settling  \nin the land of Ausonia? How can it be a sin if we too look for  \ndistant kingdoms? Every night when the earth is covered in mist  \nand darkness, every time the burning stars rise in the sky, I see  \nin my dreams the troubled spirit of my father Anchises coming  \nto me with warnings and I am afraid. I see my son Ascanius and  \nthink of the wrong I am doing him, cheating him of his kingdom  \nin Hesperia and the lands the Fates have decreed for him. And  \nnow even the messenger of the gods has come down through  \nthe swift winds \u2013 I swear it by the lives of both of us \u2013 and  \nbrought commands from Jupiter himself. With my own eyes I  \nhave seen the god in the clear light of day coming within the  \nwalls of your city. With my own ears I have listened to his voice.  \n360 Do not go on causing distress to yourself and to me by these  \ncomplaints. It is not by my own will that I search for Italy.'\n\nAll the time he had been speaking she was turned away from  \nhim, but looking at him, speechless and rolling her eyes, taking  \nin every part of him. At last she replied on a blaze of passion:  \n'You are a traitor. You are not the son of a goddess and Dardanus  \nwas not the first founder of your family. It was the Caucasus  \nthat fathered you on its hard rocks and Hyrcanian tigers offered  \nyou their udders. Why should I keep up a pretence? Why should  \nI hold myself in check in order to endure greater suffering in the  \nfuture? He did not sigh when he saw me weep. He did not even  \n370 turn to look at me. Was he overcome and brought to tears? Had  \nhe any pity for the woman who loves him? Where can I begin  \nwhen there is so much to say? Now, after all this, can mighty  \nJuno and the son of Saturn, the father of all, can they now look  \nat this with the eyes of justice? Is there nothing we can trust in  \nthis life? He was thrown helpless on my shores and I took him  \nin and like a fool settled him as partner in my kingdom. He had  \nlost his fleet and I found it and brought his companions back  \nfrom the dead. It drives me to madness to think of it. And now  \nwe hear about the augur Apollo and lots cast in Lycia and now  \nto crown all the messenger of the gods is bringing terrifying  \ncommands down through the winds from Jupiter himself, as  \n380 though that is work for the gods in heaven, as though that is an  \nanxiety that disturbs their tranquillity. I do not hold you or  \nbandy words with you. Away you go. Keep on searching for  \nyour Italy with the winds to help you. Look for your kingdom  \nover the waves. But my hope is that if the just gods have any  \npower, you will drain a bitter cup among the ocean rocks,  \ncalling the name of Dido again and again, and I shall follow you  \nnot in the flesh but in the black fires of death and when its cold  \nhand takes the breath from my body, my shade shall be with  \nyou wherever you may be. You will receive the punishment you  \ndeserve, and the news of it will reach me deep among the dead.'\n\nAt these words she broke off and rushed indoors in utter  \n390 despair, leaving Aeneas with much to say and much to fear. Her  \nattendants caught her as she fainted and carried her to her bed  \nin her marble chamber. But Aeneas was faithful to his duty.  \nMuch as he longed to soothe her and console her sorrow, to  \ntalk to her and take away her pain, with many a groan and with  \na heart shaken by his great love, he nevertheless carried out the  \ncommands of the gods and went back to his ships.\n\nBy then the Trojans were hard at work. All along the shore  \nthey were hauling the tall ships down to the sea. They set the  \nwell-caulked hulls afloat and in their eagerness to be away they  \nwere carrying down from the woods unworked timber and  \n400 green branches for oars. You could see them pouring out of  \nevery part of the city, like ants plundering a huge heap of  \nwheat and storing it away in their home against the winter, and  \ntheir black column advances over the plain as they gather  \nin their booty along a narrow path through the grass, some  \nputting their shoulders to huge grains and pushing them along,  \nothers keeping the column together and whipping in the stragglers,  \nand the whole track seethes with activity. What were your  \n410 feelings, Dido, as you looked at this? Did you not moan as you  \ngazed out from the top of your citadel and saw the broad shore  \nseething before your eyes and confusion and shouting all over  \nthe sea? Love is a cruel master. There are no lengths to which it  \ndoes not force the human heart. Once again she had recourse to  \ntears, once again she was driven to try to move his heart with  \nprayers, becoming a suppliant and making her pride submit to  \nher love, in case she should die in vain, leaving some avenue  \nunexplored. 'You see, Anna, the bustle all over the shore. They  \nare all gathered there, the canvas is calling for the winds, the  \nsailors are delighted and have set garlands on the ships' sterns.  \n420 I was able to imagine that this grief might come; I shall be able  \nto endure it. But Anna, do this one service for your poor sister.  \nYou are the only one the traitor respected. To you he entrusted  \nhis very deepest feelings. You are the only one who knew the  \nright time to approach him and the right words to use. Go to  \nhim, sister. Kneel before our proud enemy and tell him I was  \nnot at Aulis and made no compact with the Greeks to wipe out  \nthe people of Troy. I sent no fleet to Pergamum. I did not tear  \nup the ashes of his dead father Anchises. Why are his cruel ears  \nclosed to what I am saying? Where is he rushing away to? Ask  \nhim to do this last favour to the unhappy woman who loves him  \n430 and wait till there is a following wind and his escape is easy. I  \nam no longer begging for the marriage which we once had and  \nwhich he has now betrayed. I am not pleading with him to do  \nwithout his precious Latium and abandon his kingdom. What I  \nam asking for is some time, nothing more, an interval, a respite  \nfor my anguish, so that Fortune can teach me to grieve and to  \nendure defeat. This is the last favour I shall beg. O Anna, pity  \nyour sister. I shall repay it in good measure at my death.'\n\nThese were Dido's pleas. These were the griefs her unhappy  \nsister brought and brought again. But no griefs moved Aeneas.  \n440 He heard but did not heed her words. The Fates forbade it and  \nGod blocked his ears to all appeals. Just as the north winds off  \nthe Alps vie with one another to uproot the mighty oak whose  \ntimber has hardened over long years of life, blowing upon it  \nfrom this side and from that and howling through it; the trunk  \nfeels the shock and the foliage from its head covers the ground,  \nbut it holds on to the rocks with roots plunged as deep into the  \nworld below as its crown soars towards the winds of heaven \u2013  \njust so the hero Aeneas was buffeted by all this pleading on this  \nside and on that, and felt the pain deep in his mighty heart but  \nhis mind remained unmoved and the tears rolled in vain.\n\n450 Then it was that unhappy Dido prayed for death. She had  \nseen her destiny and was afraid. She could bear no longer to  \nlook up to the bowl of heaven, and her resolve to leave the  \nlight was strengthened when she was laying offerings on the  \nincense-breathing altars and saw to her horror the consecrated  \nmilk go black and the wine, as she poured it, turn to filthy gore.  \nNo one else saw it and she did not tell even her sister. There  \nwas more. She had in her palace a marble shrine dedicated to  \nSychaeus, who had been her husband. This she used to honour  \nabove all things, hanging it with white fleeces and sacred  \n460 branches. When the darkness of night covered the earth, she  \nthought she heard, coming from this shrine, the voice of her  \nhusband and the words he uttered as he called to her, and all  \nthe while the lonely owl kept up its long dirge upon the roof,  \ndrawing out its doleful song of death. And there was more.  \nShe kept remembering the predictions of ancient prophets that  \nterrified her with their dreadful warnings, and as she slept  \nAeneas himself would drive her relentlessly in her madness, and  \nshe was always alone and desolate, always going on a long road  \nwithout companions, looking for her Tyrians in an empty land.  \nShe would be like Pentheus in his frenzy when he was seeing  \n470 columns of Furies and a double sun and two cities of Thebes; or  \nlike Orestes, son of Agamemnon, driven in flight across the stage  \nby his own mother armed with her torches and black snakes,  \nwhile the avenging Furies sat at the door.\n\nAnd so Dido was overwhelmed by grief and possessed by  \nmadness. She decided to die and planned in her mind the time  \nand the means. She went and spoke to her sorrowing sister with  \nher face composed to conceal her plan and her brow bright with  \nhope. 'My dear Anna, rejoice with your sister. I have found a  \n480 way to bring him back to me in love or else to free me from him.  \nNear Oceanus and the setting of the sun is the home of the  \nEthiopians, the most distant part of our earth, where mightiest  \nAtlas turns on his shoulders the axis of the sky, studded with its  \nburning stars. From here, they say, there comes a Massylian  \npriestess who was the guardian of the temple of the Hesperides.  \nShe used to keep watch over the branches of the sacred tree and  \nbring rich foods for the serpent, spreading the oozing honey and  \nsprinkling the sleep-bringing seeds of the poppy. She undertakes  \nto free by her spells the mind of anyone she wishes and to send  \ncruel cares to others, to stop the flow of rivers and turn stars  \n490 back in their courses. At night she raises the spirits of the dead  \nand you will see the ash trees coming down from the mountains  \nand hear the earth bellow beneath your feet. I call the gods and  \nyour own sweet self to witness, O my dearest sister, that it is  \nnot by my own will that I have recourse to magic arts. Go now,  \ntelling no one, and build up a pyre under the open sky in the  \ninner courtyard of the palace and lay on it the armour this  \ntraitor has left hanging on the walls of my room, everything  \nthere is of his remaining, and the marriage bed on which I was  \ndestroyed. I want to wipe out everything that can remind me of  \nsuch a man and that is what the priestess advises.'\n\n500 She spoke, and spoke no more. Her face grew pale, but Anna  \ndid not understand that these strange rites were a pretence and  \nthat her sister meant to die. She had no inkling that such madness  \nhad seized Dido, no reason to fear that she would suffer more  \nthan she had at the death of Sychaeus. She did what she was  \nasked.\n\nBut the queen knew what the future held. As soon as the pine  \ntorches and the holm-oak were hewn and the huge pyre raised  \nunder the open sky in the very heart of the palace, she hung the  \nplace with garlands and crowned the pyre with funeral branches.  \nThen she laid on a bed an effigy of Aeneas with his sword and  \neverything of his he had left behind. There were altars all around  \n510 and the priestess with hair streaming called with a voice of  \nthunder upon three hundred gods, Erebus, Chaos, triple Hecate  \nand virgin Diana of the three faces. She had also sprinkled water  \nto represent the spring of Lake Avernus. She also sought out  \npotent herbs with a milk of black poison in their rich stems and  \nharvested them by moonlight with a bronze sickle. She found,  \ntoo, a love charm, torn from the forehead of a new-born foal  \nbefore the mare could bite it off. Dido herself took meal in her  \nhands and worshipped, standing by the altars with one foot  \nfreed from all fastenings and her dress unbound, calling before  \n520 she died to gods and stars to be witnesses to her fate and praying  \nto whatever just and mindful power there is that watches over  \nlovers who have been betrayed.\n\nIt was night and weary living things were peacefully taking  \ntheir rest upon the earth. The woods and wild waves of ocean  \nhad been stilled. The stars were rolling on in mid-course. Silence  \nreigned over field and flock and all the gaily coloured birds were  \nlaid to sleep in the quiet of night, those that haunt broad lakes  \nand those that crowd the thickets dotted over the countryside.  \n530 But not Dido. Her heart was broken and she found no relief in  \nsleep. Her eyes and mind would not accept the night, but her  \ntorment redoubled and her raging love came again and again in  \ngreat surging tides of anger. These are the thoughts she dwelt  \nupon, this is what she kept turning over in her heart: 'So then,  \nwhat am I to do? Shall I go back to those who once wooed me  \nand see if they will have me? I would be a laughing stock. Shall  \nI beg a husband from the Numidians after I have so often  \nscorned their offers of marriage? Shall I then go with the Trojan  \nfleet and do whatever the Trojans ask? I suppose they would be  \ndelighted to take me after all the help I have given them! They  \nare sure to remember what I have done and be properly grateful!  \n540 No: even if I were willing to go with them, they will never allow  \na woman they hate to come aboard their proud ships. There is  \nnothing left for you, Dido. Do you not know, have you not yet  \nnoticed, the treacheries of the race of Laomedon? But if they did  \nagree to take me, what then? Shall I go alone into exile with a  \nfleet of jubilant sailors? Or shall I go in force with all my Tyrian  \nbands crowding at my side? It was not easy for me to uproot  \nthem from their homes in the city of Sidon. How can I make  \nthem take to the sea again and order them to hoist sail into the  \nwinds? No, you must die. That is what you have deserved. Let  \nthe sword be the cure for your suffering. You could not bear,  \nAnna, to see your sister weeping. When the madness was taking  \nme, you were the first to lay this load upon my back and put me  \n550 at the mercy of my enemy. I was not allowed to live my life  \nwithout marriage, in innocence, like a wild creature, and be  \nuntouched by such anguish as this \u2013 I have not kept faith with  \nthe ashes of Sychaeus.'\n\nWhile these words of grief were bursting from Dido's heart,  \nAeneas was now resolved to leave and was taking his rest on the  \nhigh stern of his ship with everything ready for sailing. There,  \nas he slept, appeared before him the shape of the god, coming  \nto him with the same features as before and once again giving  \nadvice, in every way like Mercury, the voice, the radiance, the  \n560 golden hair, the youthful beauty of his body: 'Son of the goddess,  \nhow can you lie there sleeping at a time like this? Do you not  \nsee danger all around you at this moment? Have you lost your  \nwits? Do you not hear the west wind blowing off the shore?  \nHaving decided to die, she is turning her schemes over in her  \nmind and planning some desperate act, stirring up the storm  \ntides of her anger. Why do you not go now with all speed  \nwhile speed you may? If morning comes and finds you loitering  \nhere, you will soon see her ships churning the sea and deadly  \ntorches blazing and the shore seething with flames. Come  \nthen! No more delay! Women are unstable creatures, always  \nchanging.'\n\n570 When he had spoken he melted into the blackness of night  \nand Aeneas was immediately awake, terrified by the sudden  \napparition. There was no more rest for his men, as he roused  \nthem to instant action: 'Wake up and sit to your benches,' he  \nshouted. 'Let out the sails and quick about it. A god has been  \nsent down again from the heights of heaven \u2013 I have just seen  \nhim \u2013 spurring us on to cut our plaited ropes and run from here.  \nWe are following you, O blessed god, whoever you are. Once  \nagain we obey your commands and rejoice. Stand beside us and  \ngraciously help us. Put favouring stars in the sky for us.'\n\n580 As he spoke he drew his sword from its scabbard like a flash  \nof lightning and struck the mooring cables with the naked steel.\n\nIn that instant they were all seized by the same ardour and set  \nto, hauling and hustling. The shore was emptied. The sea could  \nnot be seen for ships. Bending to the oars they whipped up the  \nfoam and swept the blue surface of the sea.\n\nAurora was soon leaving the saffron bed of Tithonus and  \nbeginning to sprinkle new light upon the earth. The queen saw  \nfrom her high tower the first light whitening and the fleet moving  \nout to sea with its sails square to the following winds. She saw  \nthe deserted shore and harbour and not an oarsman in sight.  \n590 Three times and more she beat her lovely breasts and tore her  \ngolden hair, crying, 'O Jupiter! Will this intruder just go, and  \nmake a mockery of our kingdom? Why are they not running to  \narms and coming from all over the city to pursue him? And  \nothers should be rushing ships out of the docks. Move! Bring  \nfire and quick about it! Give out the weapons! Heave on the  \noars! \u2013 What am I saying? Where am I? What madness is this  \nthat changes my resolve? Poor Dido, you have done wrong and  \nit is only now coming home to you. You should have thought  \nof this when you were offering him your sceptre. So much for  \nhis right hand! So much for his pledge, the man who is supposed  \nto be carrying with him the gods of his native land and to have  \n600 lifted his weary old father up on to his shoulders! Could I not  \nhave taken him and torn him limb from limb and scattered the  \npieces in the sea? Could I not have put his men to the sword,  \nand Ascanius, too, and served his flesh at his father's table? I  \nknow the outcome of a battle would have been in doubt. So it  \nwould have been in doubt! Was I, who am about to die, afraid  \nof anyone? I would have taken torches to his camp and filled  \nthe decks of his ships with fire, destroying the son and the father  \nand the whole Trojan people before throwing myself on the  \nflames. O heavenly Sun whose fires pass in review all the works  \nof this earth, and you, Juno, who have been witness and party  \nto all the anguish of this love, and Hecate whose name is heard  \nin nightly howling at crossroads all over our cities, and the  \n610 avenging Furies and you, the gods of dying Dido, listen to these  \nwords, give a hearing to my sufferings, for they are great, and  \nheed my prayers. If that monster of wickedness must reach  \nharbour, if he must come to shore and that is what the Fates of  \nJupiter demand, if the boundary stone is set and may not be  \nmoved, then let him be harried in war by a people bold in arms;  \nmay he be driven from his own land and torn from the embrace  \nof Iulus; may he have to beg for help and see his innocent people  \ndying. Then, after he has submitted to the terms of an unjust  \npeace, let him not enjoy the kingdom he longs for or the life he  \n620 longs to lead, but let him fall before his time and lie unburied  \non the broad sand. This is my prayer. With these last words I  \npour out my life's blood. As for you, my Tyrians, you must  \npursue with hatred the whole line of his descendants in time to  \ncome. Make that your offering to my shade. Let there be no  \nlove between our peoples and no treaties. Arise from my dead  \nbones, O my unknown avenger, and harry the race of Dardanus  \nwith fire and sword wherever they may settle, now and in the  \nfuture, whenever our strength allows it. I pray that we may  \nstand opposed, shore against shore, sea against sea and sword  \nagainst sword. Let there be war between the nations and between  \ntheir sons for ever.'\n\n630 Even as she spoke Dido was casting about in her mind how  \nshe could most quickly put an end to the life she hated. She then  \naddressed these few words to Sychaeus' nurse, Barce, for the  \nblack ashes of her own now lay far away in her ancient homeland:  \n'My dear nurse, send my sister Anna quickly to me, telling  \nher to sprinkle her body with river water and take with her the  \nanimals and the other offerings as instructed. That is how she is  \nto come, and your own forehead must be veiled with a sacred  \nribbon. I have prepared with due care offerings to Jupiter of the  \nStyx and I am now of a mind to complete them and put an end  \n640 to the pain of love by giving the pyre of this Trojan to the  \nflames.'\n\nThe old woman bustled away leaving Dido full of wild fears  \nat the thought of what she was about to do. Her cheeks trembling  \nand flecked with red, her bloodshot eyes rolling, she was pale  \nwith the pallor of approaching death. Rushing through the door  \ninto the inner courtyard, she climbed the high pyre in a frenzy  \nand unsheathed the Trojan sword for which she had asked \u2013  \nthough not for this purpose. Then her eyes lit on the Trojan  \nclothes and the bed she knew so well, and pausing for a moment  \n650 to weep and to remember, she lay down on the bed and spoke  \nthese last words: 'These are the possessions of Aeneas which I  \nso loved while God and the Fates allowed it. Let them receive  \nmy spirit and free me from this anguish. I have lived my life and  \ncompleted the course that Fortune has set before me, and now  \nmy great spirit will go beneath the earth. I have founded a  \nglorious city and lived to see the building of my own walls. I  \nhave avenged my husband and punished his enemy who was my  \nbrother. I would have been happy, more than happy, if only  \nTrojan keels had never grounded on our shores.' She then buried  \nher face for a moment in the bed and cried: 'We shall die  \n660 unavenged. But let us die. This, this, is how it pleases me to go  \ndown among the shades. Let the Trojan who knows no pity  \ngaze his fill upon this fire from the high seas and take with him  \nthe omen of my death.'\n\nSo she spoke and while speaking fell upon the sword. Her  \nattendants saw her fall. They saw the blood foaming on the  \nblade and staining her hands, and filled the high walls of the  \npalace with their screaming. Rumour ran raving like a Bacchant  \nthrough the stricken city. The palace rang with lamentation and  \ngroaning and the wailing of women and the heavens gave back  \nthe sound of mourning. It was as though the enemy were within  \n670 the gates and the whole of Carthage or old Tyre were falling  \nwith flames raging and rolling over the roofs of men and gods.  \nAnna heard and was beside herself. She came rushing in terror  \nthrough the middle of the crowd, tearing her face and beating  \nher breast, calling out her sister's name as she lay dying: 'So this  \nis what it meant? It was all to deceive your sister! This was the  \npurpose of the pyre and the flames and the altars! You have  \nabandoned me. I do not know how to begin to reproach you.  \nDid you not want your sister's company when you were dying?  \nYou could have called me to share your fate and we would both  \n680 have died in the same moment of the same grief. To think it was  \nmy hands that built the pyre, and my voice that called upon the  \ngods of our fathers, so that you could be so cruel as to lay  \nyourself down here to die without me. It is not only yourself  \nyou have destroyed, but also your sister and your people, their  \nleaders who came with you from Sidon and the city you have  \nbuilt. Give me water. I shall wash her wounds and catch any  \nlast lingering breath with my lips.'\n\nSaying these words, she had climbed to the top of the pyre  \nand was now holding her dying sister to her breast and cherishing  \nher, sobbing as she dried the dark blood with her own  \ndress. Once more Dido tried to raise her heavy eyes, but failed.  \n690 The wound hissed round the sword beneath her breast. Three  \ntimes she raised herself on her elbow. Three times she fell back  \non the bed. With wavering eyes she looked for light in the heights  \nof heaven and groaned when she found it.\n\nAll-powerful Juno then took pity on her long anguish and  \ndifficult death and sent Iris down from Olympus to free her  \nstruggling spirit and loosen the fastenings of her limbs. For since  \nshe was dying not by the decree of Fate or by her own deserts  \nbut pitiably and before her time, in a sudden blaze of madness,  \nProserpina had not yet taken a lock of her golden hair or  \n700 consigned her to Stygian Orcus. So Iris, bathed in dew, flew  \ndown on her saffron wings, trailing all her colours across the  \nsky opposite the sun, and hovered over Dido's head to say: 'I  \nam commanded to take this lock of hair as a solemn offering to  \nDis, and now I free you from your body.'\n\nWith these words she raised her hand and cut the hair, and  \nas she cut, all warmth went out of Dido's body and her life  \npassed into the winds.\n\n## BOOK 5  \nFUNERAL GAMES\n\nMeanwhile Aeneas, without slackening in his resolve, kept his  \nfleet on course in mid-ocean, as he cut through waves darkened  \nby the north wind and looked back at the walls of Carthage,  \nglowing now in the flames of poor Dido's pyre. No one understood  \nwhat had lit such a blaze, but since they well knew what  \nbitter suffering is caused when a great love is desecrated and  \nwhat a woman is capable of when driven to madness, the minds  \nof the Trojans were filled with dark foreboding. The ships were  \nnow in mid-ocean, with no land in sight. All around was sky  \n10 and all around was sea, when there came a cloud like lead and  \nstood over Aeneas bringing storm and black night and the waves  \nshivered in the darkness. Even Palinurus himself called out from  \nthe high stern: 'What can be the meaning of these great clouds  \nfilling the sky? What have you in mind for us, Father Neptune?'  \nNot till then did he give orders to shorten sail and bend to the  \nstout oars. Then, setting the canvas aslant to the winds, he  \nturned to Aeneas and said: 'Great-hearted Aeneas, not if Jupiter  \nhimself gave me his guarantee, would I expect to reach Italy  \n20 under a sky like this. The wind has changed and is freshening,  \nhowling across us from the west where the sky is black. We  \ncannot struggle against it or make any real headway. Since  \nFortune is too strong for us to resist, let us follow her. Let us  \nchange course and go where she calls. I do not think we are far  \nfrom the safety of the shores of your brother Eryx and the  \nharbours of Sicily, if only my memory serves me right, and I  \nplot our course back by the stars I observed on the way out.'\n\nThe good Aeneas then replied: 'That is what the wind wants.  \nI have seen it myself for some time and watched you fighting it  \nto no effect. Change course then and adjust the sails. There is  \nno land that would please me more, nowhere I would rather put  \n30 in with our weary ships, than the place that gives a home to the  \nTrojan Acestes and holds the bones of my father Anchises in the  \nlap of earth.' As soon as this was said they set course for harbour  \nand the wind blew from astern and stretched their sails. The  \nfleet raced over the sea and the sailors were delighted to have  \ntheir prows pointing at last towards a beach they knew.\n\nFar away, on the top of a high mountain, Acestes saw his  \nfriends' ships arriving and was amazed. He came down to meet  \nthem bristling with javelins and the shaggy fur of a Libyan  \nshe-bear. Acestes had been born of a Trojan mother to the  \nriver-god Crinisus and he had not forgotten his ancestry, but  \n40 welcomed the returning Trojans and gladly received them with  \nall the treasures of the countryside, comforting their weariness  \nwith his loving care.\n\nAs soon as the next day had risen bright in the east and put  \nthe stars to flight, Aeneas called his men from all along the shore  \nto a council and addressed them from a raised mound: 'Great  \nsons of Dardanus, who draw your high blood from the gods,  \nthe months have passed and the cycle of the year is now complete  \nsince we laid in the ground the bones that were all that remained  \nof my divine father and consecrated an altar of mourning. This  \nis now the day, if I am right, which I shall always find bitter and  \n50 always hold in honour, for so the gods have willed. If I were  \nspending this day as an exile in the Syrtes among the Gaetulians,  \nor if I had been caught in Greek waters and were a prisoner in  \nthe city of Mycenae, I would still offer up these annual vows,  \nperform these processions in ritual order and lay due offerings  \non altars. Today we find ourselves near the very place where the  \nbones and ashes of my father lie (I for one do not believe this is  \nwithout the wish and will of the gods), and the sea has taken us  \ninto this friendly harbour. Come then, let us all celebrate these  \n60 rites with joy. Let us ask for favouring winds and may it be his  \nwill that we found a city and offer him this worship in it every  \nyear in temples dedicated to his name. Trojan-born Acestes is  \ngiving you two head of oxen for each ship. Call to your feast  \nthe Penates, the gods of your ancestral home, and those of your  \nhost Acestes. After all this, when in nine days the dawn, god  \nwilling, lifts up her life-giving light among men and the round  \nearth is revealed in her rays, I shall hold games for the Trojans,  \nfirst a race for the ships, then for those who are fleet of foot,  \nand a contest for those who take the arena in the boldness of  \ntheir strength to compete with the javelin or the flying arrow,  \n70 for those too who dare to do battle in rawhide gauntlets. Let  \nthem all come and see who wins the prizes of victory. Keep  \nholy silence, all of you, and crown your heads with shoots of  \nliving green.'\n\nWhen he had spoken he shaded his temples with a garland of  \nhis mother's myrtle. So did Helymus. So did old Acestes. So did  \nthe boy Ascanius and all the men, while Aeneas, and many  \nthousands with him, left the council and walked to the tomb in  \nthe middle of this great escort. Here he offered a libation, duly  \npouring two goblets of unmixed wine upon the ground with  \ntwo of fresh milk and two of sacrificial blood. Then, scattering  \n80 red flowers, he spoke these words: 'Once more I greet you, my  \ndivine father. I come to greet your sacred ashes, the spirit and  \nthe shade of a father rescued in vain. Without you I must search  \nfor the land of Italy, for the fields decreed by Fate and for the  \nThybris of Ausonia, whatever that may be.'\n\nWhen he had finished speaking, a snake slithered from under  \nthe shrine. Moving gently forward in seven great curves and  \nseven great coils, it glided between the altars and twined itself  \nround the tomb, its back flecked with blue and its scales flashing  \nmottled gold like the thousand different colours cast by a rainbow  \n90 on the clouds opposite the sun. Aeneas was struck dumb  \nat the sight. At last it dragged its long length among the polished  \nbowls and goblets and tasted the offerings, then, harming no  \none, it left the altars where it had fed and went back under the  \ntomb. Encouraged by this, Aeneas renewed the rites he had  \nbegun for his father, not knowing whether to think of the snake  \nas the genius of the place or as his father's attendant spirit. He  \nslew a pair of yearling sheep as ritual prescribed, two swine,  \nand as many black-backed bullocks, pouring wine from bowls  \nand calling repeatedly upon the spirit of great Anchises and his  \n100 shade released from Acheron. His comrades, too, each brought  \nwhat gifts he could and gladly offered them. They heaped the  \naltars and slaughtered bullocks while others laid out bronze  \nvessels in due order, and all over the grass there was lighting of  \nfires under spits and roasting of flesh.\n\nThe long-awaited day had come and the horses of Phaethon  \nwere now drawing the ninth dawn through a cloudless sky.  \nRumour and the famous name of Acestes had brought out all  \nthe surrounding peoples and a joyful crowd had filled the shore,  \nsome coming only to see Aeneas and his men, some also to  \n110 compete. First the prizes were displayed before their eyes in the  \nmiddle of the arena, sacred tripods, crowns of green, palm leaves  \nfor the victors, arms, purple-dyed garments and talents of silver  \nand gold. The trumpet gave the signal from a mound of earth  \nin the middle. The games had started.\n\nThe first event was for four heavy-oared ships of the same  \nclass picked out of the fleet. The _Pristis_ was a fast ship with a  \nkeen crew commanded by Mnestheus. He was soon to become  \nthe Italian Mnestheus, from whom the family of the Memmii  \ntake their name. The huge _Chimaera_ was a great hulk of a ship  \n120 the size of a city, commanded by Gyas, and to drive her through  \nthe water the Trojans sat in three tiers and plied three banks of  \noars one above the other. Sergestus sailed the great _Centaur_ (he  \nit was who gave his name to the Sergii), and Cloanthus, the  \nfounder of the Roman Cluentii, was in the blue-green _Scylla_.\n\nWell out to sea off a wave-beaten shore there stands a rock  \nwhich in winter, when the north-westerly winds are darkening  \nthe stars, is often submerged and battered by the swell. But in  \ncalm weather all is quiet and the level top of it stands up from  \n130 a glassy sea and gulls love to bask on it. Here Father Aeneas set  \nup a green branch of holm-oak as a mark round which the  \nsailors would know they had to turn to begin the long row  \nhome. They then drew lots for their starting positions, and the  \ncaptains stood on the high sterns gleaming in the splendour of  \npurple and gold. The crews wore garlands of poplar leaves and  \nthe oil they had poured on their shoulders glistened on the naked  \nskin. There they sat at the thwarts, straining their arms at  \nthe oars and their ears to hear the starting signal. They were  \nshuddering with fear and their hearts were leaping and pumping  \nthe blood for the sheer love of glory. When the shrill trumpet  \n140 sounded, in that one instant the ships all surged forward from  \nthe line and the shouting of the sailors rose and struck the  \nheavens. Their arms drew the oars back and the water was  \nchurned to foam. Side by side they ploughed their furrows and  \ntore open the whole sea to its depths with their oars and triple  \nbeaks, like two-horse chariots streaming full-pelt from the starting  \ngates and racing over the ground, or like charioteers at full  \ngallop cracking the rippling reins on their horses' backs and  \nhanging forward over them to use the whip. All the woods  \nresounded with the din and cheers and roars of encouragement.  \n150 The echo of the shouting rolled round the curve of the shore  \nand bounced back off the hills.\n\nIn all this noise and excitement Gyas shot out in front and  \ntook the lead over the first stretch of water. Cloanthus was next.  \nHis rowers were better but he was slowed down by the weight of  \nhis ship. Behind them the _Pristis_ and the _Centaur_ were contesting  \nthird place. Now the _Pristis_ has it. Now the huge _Centaur_ moves  \ninto the lead, and now they are level, bow by bow, ploughing  \nthe salt sea with their long keels. They were soon getting near  \n160 the rock, almost at the turning point, when Gyas, still in the  \nlead at this half-way stage, called out to his helmsman: 'Where  \nare you going, Menoetes? Who told you to steer to starboard?  \nYour line is over here, to port! Hug the shore. The oars on the  \nport side should be scraping the rocks. Leave the deep water to  \nthe others!' These were his orders, but Menoetes was afraid of  \nhidden rocks and pulled the bows round to the open sea. 'You're  \noff course!' shouted Gyas, correcting his line. 'Where do you  \nthink you're going? Make for the rocks, Menoetes!' and even  \nas he was shouting, he saw Cloanthus close behind him and  \n170 cutting in, just scraping past on the port side between Gyas' ship  \nand the roaring rocks. He was past in a moment, safe in clear  \nwater and sailing away from the mark. Young Gyas was  \nincensed. The rage burned in his bones and tears ran down his  \ncheeks. Without a thought for his own dignity or the safety of  \nhis crew he took the sluggard Menoetes and threw him off the  \nhigh stern head first into the sea. He then took over the tiller  \nhimself and became his own helmsman, urging on the rowers  \nand pulling the rudder round to make for the shore. Menoetes  \nwas no lightweight and was no longer young. He went straight  \n180 to the bottom and it was some time before he surfaced. At last  \nhe climbed to the top of the dry rock and sat there with the  \nwater streaming out of his clothes. The Trojans had laughed as  \nhe fell and as he swam and they laughed as he spewed up waves  \nof salt water from his stomach.\n\nSergestus and Mnestheus in the last two boats were both  \ndelighted that Gyas was losing time and both saw a hope of  \novertaking him. Sergestus took the lead as they came up to the  \nrock, but not by a whole ship's length. His bow was out in front  \nbut the _Pristis_ was pressing him hard and her beak was ahead  \nof his stern. Her captain Mnestheus was pacing the gangway  \nbetween the rowers, urging them on on either side: 'Now is the  \n190 time!' he cried. 'Now you must rise to your oars. You are the men  \nwho stood with Hector. You are the men I chose as comrades in  \nthe last hours of Troy. Now let us see the courage and the heart  \nyou showed off Gaetulia in the shoals of the Syrtes and in the  \nIonian sea when the waves were driving us on to Cape Malea. I  \nam no longer hoping to be first. It is not victory that Mnestheus  \nis fighting for, though who knows?...But let victory go to  \nwhom Neptune has given it. The disgrace would be to be last.  \nPrevent that shame, my fellow-Trojans, and that will be our  \nvictory.' At this they bent to the oars and strove with all their  \nmight. The bronzed ship shuddered at their great thrusts and the  \nsurface of the water sped away beneath them. Their breathing  \n200 quickened, chests heaved, mouths dried and the sweat poured  \noff their bodies in rivers. It was pure chance that brought them  \nthe honour they longed for. Sergestus was desperately forcing  \nthe bow of his ship close to the rocks and cutting inside into  \ndangerous water when all ended in disaster as he ran aground  \non a projecting reef. The rock quivered at the impact, the flailing  \noars grated on its jagged edges and the shattered prow was left  \nhanging in mid-air. The crew leapt up and stood there shouting.  \nSome busied themselves with iron-tipped poles and their pointed  \nboat-hooks. Some were salvaging broken oars from the surf.  \n210 Mnestheus was exultant and success only made him more determined.  \nThe oars pulled fast and true. He called upon the winds  \nand as he set course for the homeward stretch and ran shoreward  \nover the open sea, he was like a dove startled out of the cave  \nwhere it has its home and its beloved nestlings in the secret  \nhoneycombs of the rock; it flies off in terror to the fields with a  \ngreat explosion of wings inside the cave, but it soon swoops  \ndown through the quiet air and glides along in the bright light;  \nits wings are swift but they scarcely move \u2013 just so was Mnestheus.  \nJust so was the _Pristis_ as she cut through the last stretch  \nof water. Just so did she fly along under her own impetus.\n\n220 First Mnestheus left Sergestus struggling behind him, stuck  \non his rock high out of the water. There he was in the shallows,  \nshouting in vain for help and learning how to row with broken  \noars. Next Mnestheus went after Gyas and the huge _Chimaera_  \nwhich soon fell behind for lack of its helmsman. Now, at  \nthe very end of the race, only Cloanthus was in front of him.  \nHe took up the pursuit and pressed him hard, straining every  \nnerve.\n\nThe shouting grew twice as loud. They all cheered him on as  \nhe gave chase and the heavens rang with the noise. Cloanthus  \nand his men on the _Scylla_ saw the honour as theirs by right.  \n230 They had already won the victory and had no intention of giving  \nit up. They would rather have lost their lives than lose the glory.  \nMnestheus and his men on the _Pristis_ were feeding on success.  \nThey could win because they thought they could. They drew  \nlevel and would perhaps have taken the prize if Cloanthus had  \nnot stretched out his arms to the sea, pouring out his prayers  \nand calling on the gods to witness his vows: 'O you gods who  \nrule the sea and over whose waters I now race, this is my vow  \nand gladly will I keep it: I shall come to your altars on this shore  \nwith a gleaming white bull. On the salt waves of the sea I shall  \n240 scatter its entrails and pour streams of wine.' He spoke and was  \nheard by the sea nymph Panopaea and all the dancing bands of  \nthe Nereids and of Phorcys. As he sailed on, Father Portunus  \npushed the ship with his own great hand and it flew landward  \nswifter than the wind from the south or the flight of an arrow,  \ntill it arrived safe in the deep waters of the harbour.\n\nThen the son of Anchises called them all together in due order  \nand bade the herald loudly proclaim Cloanthus the victor, and  \nveiled his head with the green leaves of the laurel. For each ship  \nthere was a gift of wine, three bullocks of their choice and a  \ngreat talent of silver. In addition the captains were singled out  \n250 for special honours. The victor received a cloak embroidered  \nwith gold round which there ran a broad double meander of  \nMeliboean purple, and woven into it was the royal prince running  \nwith his javelin and wearying the swift stags on the leafy  \nslopes of Mount Ida. There he was, eager and breathless, so it  \nseemed, and down from Ida plunged the bird that carries the  \nthunderbolt of Jupiter and carried him off in its hooked talons  \nhigh into the heavens while the old men who were there as his  \nguards stretched their hands in vain towards the stars and the  \ndogs barked furiously up into the air. To Mnestheus, whose  \n260 courage had in the end won him second place, Aeneas gave a  \nbreastplate interwoven with burnished mail and triple threads  \nof gold, which he had stripped with his own hands from the  \ndefeated Demoleos on the banks of the swift Simois under the  \nhigh walls of Troy. For Mnestheus this was to be a proud  \npossession and his protection in battle. His attendants Phegeus  \nand Sagaris hoisted it up on to their shoulders, all the many  \nlayers of it, but they could hardly carry it away, yet Demoleos  \nused to wear it while running all over the battlefield in pursuit  \nof Trojans. The third prize was a pair of bronze drinking cauldrons  \nand some embossed drinking cups of solid silver.\n\nAt last they had all received rich gifts and were glorying in  \nthem as they walked, their foreheads bound with purple ribbons,  \nwhen Sergestus appeared, taking in the boat that was the object  \n270 of all their laughter and had missed all the honours. He had  \nprised her off the cruel rock with great difficulty and no mean  \nskill, but she had lost oars and was limping in with only one  \nbank of them. Like a snake caught crossing a raised road, as  \nthey often are, and run over by a bronze wheel or battered by a  \ntraveller with a heavy stone and left mangled and half-dead, it  \ntries in vain to escape by twisting its body into long curves, part  \nof it still fierce, the blazing eyes, the hissing, high-uplifted head,  \nbut the wounded part holds it back as it writhes and coils and  \n280 twines itself into knots \u2013 this is how the _Centaur_ moved, rowing  \nslowly along. But she put up sails and came into the harbour  \nmouth under full canvas. Aeneas, delighted that Sergestus had  \nsaved his ship and brought his men to port, gave him a prize, as  \npromised, the Cretan slave woman Pholoe, good with her hands  \nand with two sons at the breast.\n\nAfter the boat race, dutiful Aeneas strode to a piece of grassy  \nlevel ground. All around it stood wooded hills and in the middle  \nof the valley there was a circle for a theatre. When he reached  \n290 this place \u2013 and many thousands went with him \u2013 Aeneas sat  \ndown on a raised platform in the middle of the concourse. Here  \nhe offered prizes for any men who might wish to take part in a  \nfoot race, whetting their ambition with rewards, and Trojans  \nand Sicanians flocked in from all sides. Nisus and Euryalus were  \nfirst, Euryalus standing out for the bloom of his youthful beauty  \nand Nisus for the loving care he showed to him. Then came  \nDiores, a prince of the noble line of Priam, and after him Salius  \nand Patron together, one an Acarnanian, the other an Arcadian  \n300 of Tegean stock. Then came two young Sicilians, Helymus and  \nPanopes, men of the woods, attendants of old Acestes, and many  \nmore whose names are buried in oblivion. When they had  \ngathered, Aeneas spoke in the middle of them: 'Give your minds  \nto what I have to say. Mark it well and be of good cheer. No  \nman of you will leave without winning a prize from my hand.  \nTwo Cretan arrows I shall give, their steel tips burnished and  \ngleaming, and a two-headed axe embossed with silver. These  \nrewards will be the same for all of you, but there will be other  \nprizes for the first three in the race and crowns woven of golden  \n310 olive for their heads. The winner will have a horse with splendid  \ntrappings, the second an Amazonian quiver full of Thracian  \narrows, slung on a belt with a broad gold band and the clasp  \nthat fastens it is a polished jewel. The third can leave the field  \ncontent with an Argive helmet.'\n\nWhen he had finished speaking, they took their places, the  \nsignal sounded and they were off, streaming away from the  \nstarting-point in one great cloud. But as soon as they came in  \nsight of the finish, Nisus shot out a long way in front of all of  \nthem, swifter than the wind and the wings of the lightning.  \n320 Second, but a long way behind, was Salius. Then, after a gap,  \ncame Euryalus in third place. Behind him was Helymus, then,  \nimmediately behind him and hard on his heels, was Diores  \nleaning over his shoulder, and if there had been more course to  \nrun, he would have overtaken and passed him or they would  \nhave run a dead heat.\n\nThey were soon almost at the end of the course and tiring as  \nthey came up to the line, when the unlucky Nisus slid and fell  \n330 on a slippery patch of blood that had been spilt where they had  \nkilled bullocks and wet the earth and the green grass that grew  \nupon it. Here, as he pounded the track exulting in the very  \nmoment of victory, he lost his footing and fell on his face in the  \nfilthy dung and blood from the sacrifice. But he was not the man  \nto forget Euryalus and the love he bore him. He rose from the  \nslime and threw himself in the path of Salius and knocked him  \nhead over heels, sprawling on the hard-packed sand. Euryalus  \nflashed past. Thanks to his friend he was in the lead and speeding  \nalong to loud applause and cheers, Helymus behind him with  \n340 Diores now winning the third prize. But Salius stood up before  \nthe faces of the fathers in the front rows and filled the whole  \nbowl of the huge assembly with loud clamour, demanding the  \nhonour of which he had been cheated. On the side of Euryalus  \nwere the favour in which he was held, his beauty as he stood  \nthere weeping and the manly spirit growing in that lovely body.  \nOn his side too was Diores, protesting at the top of his voice.  \nHe had come in third but there would be no third prize for him  \nif the first were to be given to Salius. Father Aeneas then spoke:  \n'You young men will all keep your prizes. The awards have been  \n350 made and no one changes that. Let it be my task to offer  \nconsolation to our friend for the downfall he did nothing to  \ndeserve.' With these words he gave Salius the hide of a huge  \nGaetulian lion, weighed down with gilded claws and mane. This  \nwas too much for Nisus, who burst out: 'If losers win prizes like  \nthis and you take pity on people who fall, what gift will be  \nenough to give to Nisus? I would have won the victor's crown  \nof glory and deserved it if the same bad luck as brought down  \nSalius had not disposed of me,' and as he spoke he pointed to  \nthe filthy wet dung on his face and body. Good Father Aeneas  \nlaughed and ordered them to bring out a shield made by the  \n360 hand of Didymaon which had been dedicated to Neptune and  \ntaken down from the doorposts of his temple by Greeks, and he  \ngave this superb gift to the noble young Nisus.\n\nThe race was over and the prizes finally awarded. Then spoke  \nAeneas: 'If there is any courage here, any man with a heart in  \nhis breast, now is the time for him to come forward with gloves  \non his hands and his guard up,' and he set out two prizes for the  \nfight, for the victor a bullock with its head shadowed by ribbons  \nand its horns plated with gold, and a sword and splendid helmet  \nas a consolation prize for the loser. Dares did not hesitate.  \nImmediately that great face of his appeared and all his mighty  \nstrength, and the people murmured as he hoisted himself to his  \n370 feet. He had been the only man who used to stand against Paris.  \nHe was the man who had felled the huge Butes and stretched  \nhim out to die on the yellow sand by the mound where great  \nHector lay, when Butes came as champion from the Bebrycian  \nrace of Amycus. This was the Dares who stood there with his  \nhead held high to begin the battle, flexing his shoulders, throwing  \nlefts and rights and thrashing the air. They looked around  \nfor an opponent, but no one in all that company dared go near  \n380 him or put on the gloves. Thinking that no one was challenging  \nhim for the prize, he went straight up to Aeneas and stood there  \nin front of him. Without more ado he took one of the bull's  \nhorns in his left hand and said: 'Son of the goddess, if no one  \ndares trust himself to battle, how long are we going to stand  \nhere? What is the point of keeping me waiting? Tell them I can  \ntake away my prize,' and all the Trojans to a man murmured  \nand told Aeneas to award the prize as promised.\n\nAt this Acestes had hard words for Entellus, sitting next him  \non a bank of green turf. 'Entellus,' he said, 'I have seen the day  \nwhen you were the bravest of the heroes. Is it all in the past?  \n390 Are you going to sit there meekly when a prize like this is lifted  \nand no opposition offered? Tell me, where is Eryx now, the god  \nthey say was once your teacher? Has all that come to nothing?  \nWhat about that reputation of yours that used to ring round  \nthe whole island of Sicily? And what about the great trophies  \nhanging in your house?' 'I am not afraid,' replied Entellus. 'I  \nhave still my pride and my love of honour. But old age is slowing  \nme down. The blood is cold and sluggish. My strength is gone  \nand my body is worn out. But if I were what I once was, if I had  \nthe youth that makes that puppy so full of himself, prancing  \nabout there, I would not have needed the reward of a pretty  \n400 bullock to bring me to my feet. I am not interested in prizes.' At  \nthese last words he threw into the middle the pair of prodigiously  \nheavy gauntlets in which Eryx used to raise his guard, carrying  \nthem into battle with the hard leather stretched over his forearms.  \nThey were amazed. The hides of seven huge oxen were  \nthere, stiffened by lead and iron sewn into them. Dares was  \nmore amazed than anyone and stood well back at the sight of  \nthem, but the great-hearted son of Anchises picked them up and  \nfelt their weight, turning over the great folds of the jointed  \nhides from one hand to another. Then spoke old Entellus, his  \n410 voicedeep in his chest: 'What would you have thought, any of  \nyou, if you had seen the gauntlets that were the armour of  \nHercules himself and the cruel battle these two fought on this  \nvery shore? This, Aeneas, is the armour your brother Eryx used  \nto wear. You see it is still caked with blood and spattered brains.  \nWith these he stood that day against great Hercules. With these  \nI used to fight while there was still good blood in me to give me  \nstrength, before old age came to tangle with me and sprinkled  \nboth my temples with grey. But if Trojan Dares recoils from this  \narmour of ours, and if good Aeneas is satisfied and my patron  \nAcestes approves, let us level the odds. There's nothing to be  \n420 afraid of, Dares. For you I give up the boxing leathers of Eryx,  \nand you take off your Trojan gauntlets,' and as he spoke he  \nthrew the double cloak off his shoulders and stripped to show  \nthe great joints of his limbs, the great bones and muscles on his  \narms, and stood there a giant in the middle of the arena.\n\nThen the son of Anchises took out two matching pairs of  \ngauntlets, and tied armour of equal weight on the hands of both  \nmen. There was no more delay. Each man took up his stance,  \npoised on his toes, stretching to his full height, guard held high  \nin the air and no sign of fear. They kept their towering heads  \nwell back from the punches and fist struck fist as they warmed  \n430 to their work. Dares had youth on his side and speed of foot.  \nEntellus had the reach and the weight, but his knees were going.  \nHe was slow and shaky and his whole huge body heaved with  \nthe agony of breathing. Blow upon blow they threw at each  \nother and missed. Blow upon blow drummed on the hollow rib  \ncage, boomed on the chest and showered round the head and  \nears, and the cheekbones rattled with the weight of the punches.  \nEntellus, being the heavier man, held firm in his stance, keeping  \nwatchful eyes on his opponent and swaying away from the  \n440 bombardment. For Dares it was like attacking some massive  \nhigh-built city or besieging a mountain fortress. This way and  \nthat he tried, covering all the ground in his manoeuvres, pressing  \nhard with all manner of assaults and all to no avail. Then  \nEntellus drew himself up and showed his right hand raised for  \nthe blow, but Dares was quick to see it coming down and backed  \naway smartly. Entellus' full force was in the blow and it met the  \nempty air. Great was his weight and great was the fall of that  \nhuge body. He fell as a hollow pine tree falls, torn up by  \n450 the roots on great Mount Ida or on Erymanthus. Trojans and  \nSicilians leapt to their feet as one man in their excitement and  \nthe shouting rose to high heaven. Acestes was the first to run to  \ncomfort his old friend and help him from the ground. But the  \nhero Entellus did not slow down or lose heart because of a fall.  \nHe returned to the fray with his ferocity renewed and anger  \nrousing him to new heights of violence. His strength was kindled  \nby shame at his fall and pride in his prowess, and in a white  \nheat of fury he drove Dares before him all over the arena,  \nhammering him with rights and lefts and allowing him no rest  \nor respite. Like hailstones from a dark cloud rattling down on  \n460 roofs, Entellus battered Dares with a shower of blows from  \nboth hands and sent him spinning.\n\nAt this point Father Aeneas did not allow the anger of Entellus  \nto go any further but checked his savage passion and put an end  \nto the fight. As he rescued the exhausted Dares he comforted  \nhim with these words: 'Unlucky Dares, what madness is this  \nthat has taken possession of you? Do you not see that your  \nstrength is not as his and the divine will has turned against you?  \nYield to God.' He spoke and his voice parted the combatants,  \nand Dares was led back to the ships by his faithful comrades,  \ndragging his weary legs, shaking his head from side to side and  \n470 spitting out a mixture of gore and teeth. His men were then  \ncalled and given the helmet and the sword, leaving the palm of  \nvictory and the bull to Entellus. Then spoke the victor in all his  \npride of spirit, glorying in the bull he had won: 'Son of the  \ngoddess, know this, and you too, men of Troy: this is the  \nstrength there used to be in my body when I was in my prime  \nand this is the death from which you have rescued Dares.' With  \nthese words he took up his stance in front of the bullock's head  \nas it stood there as the prize of battle, then, drawing back his  \n480 right hand and rising to his full height, he swung the brutal  \ngauntlet straight down between its horns, shattering the brains  \nand grinding them into the bone. The ox fell and lay full out on  \nthe ground, dead and twitching, and these are the words Entellus  \nspoke and spoke them from the heart: 'The life of this ox is  \nworth more than the life of Dares, and with it, Eryx, I pay my  \ndebt to you in full, and here and now in the moment of victory,  \nI lay down my gauntlets and my art.'\n\nAeneas immediately summoned all those who wished to take  \npart in an archery contest and announced the prizes. With his  \ngreat hand he set up the mast taken from Serestus' ship and put  \na cord round a fluttering dove to hang it from the top of the  \n490 mast as a target for the steel-tipped arrows. The contestants  \ngathered. Lots were thrown into a bronze helmet, and the first  \nto leap out, to loud acclaim, gave the first place to Hippocoon,  \nson of Hyrtacus. Next came Mnestheus, fresh from his triumph  \nin the boat race, Mnestheus with the green olive binding his  \nhair. Third was Eurytion, brother of the famous Pandarus who  \nin days long past had been ordered to break the truce, and had  \nbeen the first to shoot an arrow into the middle of the Greeks.  \nLast of all, at the bottom of the helmet, was Acestes. He too  \n500 dared to try his hand at the test of warriors. Soon they were  \nbending their bows with all their strength and taking the arrows  \nout of their quivers. A string twanged and the first arrow, from  \nyoung Hippocoon, cut through the breezes of heaven to strike  \nhome full in the wood of the mast. The mast quivered, there  \nwas a flash of wings from the frightened bird and all around  \nrang out the loud applause. Next the eager Mnestheus took his  \nstand and drew, aiming high, straining both eye and bow, but  \n510 to his dismay he failed to hit the bird, cutting the knot in the  \nlinen cords which bound her feet as she hung there at the top  \nof the mast. She made off, flying south towards some dark  \nclouds. Eurytion lost no time (his bow had long been bent and  \nhis arrow at the ready), but called upon his brother Pandarus as  \nhe prayed, and took aim at the dove now glorying in the freedom  \nof the sky. As she beat her wings just beneath the black cloud,  \nthe arrow struck her and she fell dead, leaving her life among  \nthe stars of heaven and bringing back as she fell the arrow that  \nhad pierced her.\n\nFather Acestes alone remained and the victor's palm was lost  \n520 to him, but he aimed an arrow high into the breezes of the air  \nto display his old skill and let the sound of his bow be heard. At  \nthis a sudden miracle appeared before their eyes, a mighty sign  \nof what the future held in store. In times to come was the  \ngreat fulfilment revealed and awesome prophets interpreted the  \nomens to future ages. As it flew through the vaporous clouds,  \nthe arrow burst into flames and marked its path with fire till it  \nwas consumed and faded into thin air, like those stars that leave  \ntheir appointed places and race across the sky trailing their  \n530 blazing hair behind them as they fly. Sicilians and Trojans stood  \nstock still in amazement, praying to the gods above, but the  \nmighty Aeneas welcomed the omen and embraced the exultant  \nAcestes, heaping great gifts on him and saying these words:  \n'Accept these, Father Acestes, for the Great King of Olympus  \nhas shown by this sign that he has willed you to receive honours  \nbeyond the lot of other men. Here is a gift from my old father  \nAnchises himself, a mixing bowl engraved with figures which  \nhe once received as a great tribute from Thracian Cisseus to be  \na memorial and pledge of his love.' With these words he put a  \n540 wreath of green laurel round Acestes' temples and declared him  \nfirst victor above all the others. Nor did good Eurytion grudge  \nhim the highest honour although he alone had brought down  \nthe dove from the heights of heaven. Next in order for the prizes  \ncame the archer who had cut the cord, and last the one who had  \npierced the mast with his flying arrow.\n\nBut before the end of the archery contest Father Aeneas was  \nalready calling to his side Epytides, the trusty comrade and  \nguardian of young Iulus, to speak a word in his ear: 'Go now,  \nand if Ascanius has with him his troop of boys all ready and the  \n550 horses drawn up and prepared to move, tell him to lead on his  \nsquadrons in honour of his grandfather and show himself in  \narms.' The people had all flooded into the circus, so Aeneas  \nordered them to clear the whole long track and leave the level  \nground free. Then came the boys, riding in perfect order on their  \nbridled mounts, resplendent in full view of their parents, and all  \nthe men of Sicily and of Troy murmured in admiration as they  \nrode. They wore their hair close bound in trimmed garlands in  \nceremonial style and each carried a pair of cornel-wood spears  \ntipped with steel. Some of them had polished quivers hanging  \nfrom their shoulders with circlets of twisted gold round neck  \n560 and chest. They spread out into three separate squadrons of  \nhorse, each with its own leader at the head of a dozen boys in  \ntwo separate files of six, each squadron with its own trainer, all  \nof them gleaming in the sunlight. The first of these three squadrons  \nof young warriors was led in triumph by a little Priam, the  \nnoble son of Polites who bore the name of his grandfather and  \nwas destined to give increase to the Italian race. His horse was  \na piebald Thracian with white above its hooves and a white  \nforehead carried high. The second squadron was led by Atys,  \nthe founder of the Atii of Latium. Young Atys was a dear friend  \n570 of the boy Iulus, and Iulus was last and comeliest of them all,  \nriding on a Sidonian horse given to him by the lovely Dido as a  \nmemorial and pledge of her love. The other youngsters rode  \nSicilian mounts presented by old Acestes. They were daunted  \nby the praise they received as the Trojans feasted their eyes  \nupon them, tracing in their features the features of their distant  \nancestors.\n\nAfter they had paraded happily on horseback round the whole  \ngathering and shown themselves to their loved ones, when they  \nwere all ready, Epytides, standing at a distance, gave the signal  \n580 with a loud call and a crack of his whip and the warriors wheeled  \napart into two separate sections, each of the three troops dividing  \nits ranks equally. At a second command the two new formations  \nturned and advanced on each other with spears at the  \nlevel. All over the arena they charged and turned and charged  \nagain, winding in circles now in one direction now in the other,  \nfighting out in full armour the very image of a battle, now  \nexposing their backs in flight, now turning to point their spears  \nat the enemy and now when peace is made riding along side by  \nside. They say there was a labyrinth once in the hills of Crete  \nwhere the way weaved between blind walls and lost itself in a  \n590 thousand treacherous paths; there was no following of tracks in  \nthis maze, no finding of a way and no retracing of steps \u2013 such  \nwas the pattern woven by the paths of the sons of the Trojans  \nas they wound their movements of mock battle and retreat, like  \ndolphins swimming in the waters of the sea, cleaving the waves  \noff Carpathos or Libya. The tradition of these manoeuvres and  \nbattles was first renewed by Ascanius, who taught the native  \nLatins to celebrate it as he was building his walls round Alba  \n600 Longa. The Albans taught their sons to do as Ascanius himself  \nand the Trojans had done with him when they were boys. In  \ndue course great Rome itself received this tradition from Alba  \nand preserved it. It is now called 'Troy' and the boys are called  \n'the Trojan Troop'. Here ended the games held in honour of the  \ndivine father of Aeneas.\n\nAt this moment Fortune first changed and turned against  \nthem. While they were paying to the tomb the solemn tribute of  \nall these games, Saturnian Juno sent Iris down from the sky to  \nthe Trojan fleet and breathed favouring winds upon her as she  \nwent. Juno had many schemes in her mind and her ancient  \n610 bitterness remained unsatisfied. Unseen by human eye the virgin  \ngoddess ran her swift course down her bow of a thousand  \ncolours till she came within sight of the great assembly. She  \nthen passed along the shore and saw the empty harbour and  \nunattended ships. But there, far apart on the deserted beach,  \nwere the women of Troy, weeping for the loss of Anchises and  \nweeping, all of them, as they looked out over the unfathomable  \nsea. How weary they were, how numberless the breakers and  \nhow vast the sea that still remained for them to cross! These  \nwere the words on all their lips. What they were praying for was  \na city \u2013 they were heart sick of toiling with the sea. Iris knew  \nhow to cause mischief. She rushed into the middle of them,  \n620 laying aside her divine form and dress and appearing as Beroe,  \nthe aged wife of Doryclus of Tmaros, a woman of good birth,  \nwho had borne sons and been held in high regard. In this guise  \nshe mingled with the mothers of Troy and spoke these words:  \n'Our sadness is that Greek hands did not drag us off to our  \ndeaths in war under the walls of our native city. O my unhappy  \npeople, for what manner of destruction is Fortune preserving  \nyou? This is the seventh summer since the fall of Troy that we  \nhave been driven by the winds and have measured every sea and  \nland, every inhospitable rock and every angry star, rolling for  \never on the waves as we search the mighty ocean for an Italy  \n630 that ever recedes. Here we are in the land of our brother Eryx  \nand Acestes is our host. Who is to prevent us from laying down  \nthe foundations of walls and giving a city to our people? I call  \nupon our native land and household gods snatched from the  \nhands of our enemies to no purpose, tell us, will there never  \nagain be walls that will be called the walls of Troy? Shall I never  \nsee a place with the rivers that Hector knew, the Xanthus and  \nthe Simois? It is too much to endure. Come with me now and set  \nfire to these accursed ships and destroy them. I have seen in a  \ndream the image of the priestess Cassandra putting blazing  \ntorches in my hands and saying: \"This is your home. This is  \nwhere you must find your Troy.\" Now is the time to act. Portents  \n640 like these brook no delay. Look at these four altars of Neptune.  \nThe god himself is giving us the torches and the courage.' While  \nstill speaking she took the lead and snatched up the deadly fire,  \nbrandished it in her right hand and threw it with all her force.  \nThe minds of the women of Troy were roused and their hearts  \nwere bewildered, but one of the many, the oldest of them all,  \nPyrgo, who had been royal nurse to all the sons of Priam, called  \nout: 'This is not Beroe speaking to you, women of Troy. This is  \nnot the wife of Doryclus from Rhoeteum. Look at the marks of  \ndivine beauty, the blazing eyes. Look at her proud bearing, her  \n650 features, the sound of her voice, her walk. I have just left Beroe  \nsick and fretting because she was the only one who could not  \ncome to this ceremony and would not be paying due honour to  \nAnchises.'\n\nThese were the words of Pyrgo and at first the women were  \nat a loss, looking at the ships with loathing in their eyes, torn  \nbetween their pitiable desire to stay where they were on land,  \nand the kingdom to which destiny was calling them, when the  \ngoddess soared through the heavens on poised wings, cutting in  \nher flight a great rainbow beneath the clouds. This portent  \n660 overwhelmed them. Driven at last to madness they began to  \nscream and snatch flames from the innermost hearths of the  \nencampment or rob the altar fires, hurling blazing branches and  \nbrushwood and torches. The God of Fire raged with unbridled  \nfury over oars and benches and the fir wood of the painted  \nsterns.\n\nIt was Eumelus who brought the news to the Trojans while  \nthey were still in the wedge-shaped blocks of seats in the theatre  \nnear the tomb of Anchises, and they could see for themselves  \nthe dark ash flying in a cloud. Ascanius was happily leading the  \ncavalry manoeuvres, so he made off to the troubled camp at full  \ngallop although the breathless trainers tried in vain to hold him  \n670 back. 'What strange madness is this?' he cried. 'Where, oh where  \nis this leading you, you unhappy women of Troy? This is not  \nthe camp of your Greek enemies. What you are burning is your  \nown hopes for the future! Look at me! I am your own Ascanius!'  \nHe had been wearing a helmet as he stirred the images of war  \nin the mock battle and now he took it off and threw it on the  \nground at his feet. At this moment Aeneas came rushing up and  \ncolumns of Trojans with him, but the women took to flight and  \nscattered all over the shore making for the woods and caves in  \nthe rocks, wherever they could hide. They were ashamed of  \nwhat they had done and ashamed to look upon the light of day.  \nTheir wits were restored now and they recognized their own  \npeople. Juno was cast out of their hearts.\n\n680 But that did not cause the fire and flame to abate their  \nunquenchable fury. The pitch was still smouldering beneath the  \nwet timbers, oozing slow smoke, and a consuming heat was  \ncreeping along the hulls. The canker was sinking deep into the  \nbodies of the ships and all the exertions of men and the pouring  \non of water were achieving nothing. This was when the devout  \nAeneas tore the cloak off his shoulders and called upon the gods  \nfor help, stretching out his hands and praying: 'All-powerful  \nJupiter, if you do not yet abhor the whole race of Trojans, if  \nyour loving-kindness still looks as of old on the labours of men,  \n690 grant now, O Father, that our fleet escape the flames. Save from  \ndestruction what little remains to the Trojans, or else with your  \nown angry thunder cast the remnants of us down to death and,  \nif that is what I deserve, overwhelm us here with your own right  \nhand.' Scarcely had he spoken, when a black deluge of torrential  \nrain came lashing down, mountain peak and plain trembled at  \nthe thunder and from the whole sky streamed the wild tempest  \nof rain, dark with the cloud-bearing winds of the south. It  \npoured down and filled the ships and soaked the charred timbers  \ntill all the fire was quenched and, except for four that were lost,  \nall the ships were saved from destruction.\n\n700 But this was a bitter blow for Aeneas, and his heart was heavy  \nas he turned his thoughts this way and that, wondering whether  \nhe should forget about his destiny and settle in the fields of  \nSicily, or whether he ought to make for the shores of Italy. Then  \nspoke old Nautes. He was the one man Tritonian Pallas had  \nchosen to instruct and make pre-eminent in his art, providing  \nhim with responses to explain what the great anger of the gods  \nportended and what the settled order of the Fates demanded.  \nThese were the words of comfort he now began to address to  \nAeneas: 'Son of the goddess, let us follow the Fates, whether  \n710 they lead us on or lead us back. Whatever fortune may be ours,  \nwe must at all times rise above it by enduring it. Acestes is by  \nyour side and he is a Trojan, offspring of the gods. Take him  \ninto your counsels. Be one with him. He is willing. Hand over  \ninto his care the people from the ships that are lost and those who  \nare heart-weary of your great enterprise and destiny. Choose the  \nold men, the women who are worn out by the sea, all of your  \ncompany who are frail and have no stomach for danger, and  \nweary as they are, here in this land let them have their city.  \nAcestes will give them his name and they will call it Acesta.'\n\n720 Aeneas was fired by these words from his old friend, but his  \nheart was divided between all his cares as never before. Dark  \nnight had risen in her chariot to command the vault of heaven,  \nwhen suddenly there appeared the form of his father Anchises  \ngliding down from the sky and these were the words that came  \npouring from him: 'O my son, dearer to me than life itself in the  \ndays when life remained to me, O my son, who has been tested  \nby the Fates of Troy, I come here in fulfilment of the command  \nof Jupiter. He it was who drove the fire from your ships and has  \nat last looked down from the sky and pitied you. Follow now  \nthis most wise advice which old Nautes is giving you and choose  \nwarriors from your people, the bravest hearts among them, to  \n730 take to Italy. There in Latium is a wild and hardy people whom  \nyou have to overcome in war. But first you must come to the  \nhome of Dis in the underworld and go through the depths of  \nhell to seek a meeting with me. I am not confined in the grim  \nshades of impious Tartarus but live in Elysium in the radiant  \ncouncils of the just. A chaste Sibyl will lead you to this place,  \nshedding the blood of many black cattle in sacrifice. Then you  \nwill learn about all the descendants who will come after you  \nand the city walls you are to be given. But now farewell. The  \ndewy night is turning her chariot in mid-course. The cruel sun  \nis beginning to rise in the east and I have felt the breath of his  \n740 panting horses.' As he finished speaking he fled into thin air like  \nsmoke dissolving. 'Where are you going in such haste? Who are  \nyou escaping from? Who is there to keep you from my arms?'  \nSo cried Aeneas, and he stirred the smouldering ashes of the fire  \nto worship the Lar of Pergamum and the shrine of white-haired  \nVesta with a ritual offering of coarse meal and incense from a  \nfull censer.\n\nImmediately then he called his allies, Acestes first of all, and  \nexplained the command of Jupiter, the instructions of his own  \ndear father and the resolve now firm in his own mind. There  \nwas no time lost in words and no dissent from Acestes. They  \n750 transferred the mothers to the city and put ashore those who  \nwished it, those spirits that felt no need for glory, while they  \nthemselves repaired the rowing benches, replaced the charred  \ntimbers and fitted out the ships with oars and ropes. They were  \na small band but their hearts were high for war. Meanwhile,  \nAeneas was ploughing the city bounds and allotting homes to  \nhis people. This was to be Ilium, and this was to be Troy. Trojan  \nAcestes was delighting in his kingdom, choosing a site for his  \nforum, summoning a senate and laying down a code of laws.  \n760 Then they founded a temple to Venus of Ida, soaring to the stars  \non the peak of Mount Eryx, and appointed a priest to tend the  \ntomb of Anchises, consecrating to his name a great grove all  \naround it.\n\nAnd now the whole people had feasted for nine days and  \nperformed their rites at the altars. A gentle breeze had calmed  \nthe waves and the breath of a steady south wind was calling  \nthem again to sea. Loud was the weeping along the curved shore  \nof the bay as they lingered for a night and a day in their  \nlast embraces. Even the women, even the men who had been  \nshuddering at the sight of the sea and unable to face its god,  \nwere now eager to sail and endure to the end the whole agony  \n770 of exile, but good Aeneas comforted them with words of love  \nand wept as he entrusted them to their kinsman Acestes. At last  \ncame the command to sacrifice three calves to Eryx and a lamb  \nto the Storms and to cast off their moorings in due order. There  \nstood Aeneas alone on the prow, his head bound with a wreath  \nof trimmed olive leaves and holding a goblet in his hands as he  \nscattered the sacrificial entrails and poured the streaming wine  \ninto the salt sea. His men vied with one another to strike the  \nwaves, sweeping them with their oars as a freshening wind from  \nastern helped them on their way.\n\nBut Venus, never resting all this time from her cares, went to  \n780 Neptune and poured out to him these words of complaint from  \nher heart: 'It is the deadly anger of Juno, her implacable fury,  \nthat forces me to use every prayer I can. No man's piety can  \nsoften her, nor does the long passage of time. Her will is not  \nbroken by the Fates nor by the command of Jupiter and she  \nknows no rest. In black hatred she has eaten the city of the  \nPhrygians out of the heart of their race and dragged the Trojans  \nwho survive through every form of suffering, but she is still not  \nsatisfied. She is still persecuting the dead bones and ashes of the  \ncity she has destroyed. She alone can understand her reasons for  \n790 this terrible rage. You yourself, I know, were a witness of the  \nturmoil she has just created in the waves of the Libyan ocean,  \nstirring up sea and sky to no avail with the help of Aeolus'  \nwinds. To think she took all this upon herself in your kingdom!  \nAnd now this! Look how she has driven the mothers of the  \nTrojans to wrong-doing. It is her cruelty that has burned out  \ntheir ships, lost them their fleet and forced them to abandon  \ntheir own dear ones in a strange land. As for what is to come, if  \nwhat I am asking is readily conceded, if the Fates are giving  \nthem a city in that land, I beg of you to allow them a safe  \ncrossing and let them reach the Laurentine Thybris.'\n\nThen Neptune, son of Saturn and master of the ocean depths,  \n800 answered in these words: 'O Venus of Cythera, it is wholly right  \nthat you should put your trust in the sea, which is my kingdom,  \nfor you are born from it. I also have deserved your trust, for I  \nhave often checked the wild fury of the sea and sky and my care  \nfor your Aeneas has been no less on land \u2013 I call the rivers  \nXanthus and Simois to testify to this. During Achilles' pursuit  \nof the broken army of Troy, when he was driving them against  \ntheir own walls and killing them in their thousands, when the  \nrivers were choked and groaning with corpses and Xanthus  \ncould find no way to roll down to the sea, there was Aeneas  \nstanding against the might of Achilles, his strength not equal to  \n810 it and the gods opposed, and it was I who caught him up in a  \nhollow cloud, although my own desire was to take these walls  \nthat I had built with my own hands for the treacherous Trojans  \nand turn them over from top to bottom. As my mind was then,  \nso is it even now. Put away your fears. He will arrive safely  \nwhere you wish, at the harbour of Avernus. One only will be  \nlost. One only will you look for in vain upon the sea, and that  \none life will be given for many.' When these words had soothed  \nand gladdened the heart of the goddess, Father Neptune put a  \ngolden yoke on the necks of his horses and bits between their  \nwild and foaming jaws and gave them full rein. As his blue-green  \n820 chariot skimmed the surface of the sea, the waves were stilled,  \nthe swell subsided beneath his thundering axle and the rain  \nclouds fled from the vast vault of heaven. Then all his retinue  \nappeared, the huge sea beasts, Glaucus and his band of ageing  \ndancers, Palaemon, son of Ino, the swift Tritons and all the  \nranks of Phorcys' army, while there on the left was Thetis with  \nMelite and the maiden Panopaea, Nisaee and Spio, Thalia and  \nCymodoce.\n\nNow all indecision was past and it was the turn of glad joy to  \n830 capture the heart of Aeneas. Instantly he ordered all masts to be  \nput up and canvas stretched from the yard-arms. As one man  \nthey all set their sails, letting them out in time, first to port and  \nthen to starboard. As one man they swung round the high ends  \nof the yard-arms and swung them round again as fair winds  \ncarried the fleet on its way. They were sailing close, in line ahead  \nwith Palinurus in the lead, and their orders were to make all  \nspeed and take their course from him.\n\nThe dank night was near the mid-point of the sky. The sailors  \nwere taking their rest in peace and quiet, stretched out under  \ntheir oars along the hard benches, when the God of Sleep,  \nparting the dark and misty air, came gliding lightly down from  \n840 the stars of heaven. He was coming to you, Palinurus, bringing  \ndeadly dreams you did not deserve. The god took the shape of  \nPhorbas and sat on the high poop pouring these soft words into  \nthe ears of Palinurus: 'Son of Iasius, the sea is carrying the ships  \nalong itself. The breeze is gentle and steady. This is an hour for  \nsleep. Put down your head and steal a little time from your  \nlabours to rest your tired eyes. I'll take over a short watch for  \nyou myself.'\n\nScarcely lifting his eyes, Palinurus replied: 'Are you asking me  \nto forget what I know about the calm face of the sea and quiet  \nwaters? There is a strange power in the sea and I would never  \n850 rely on it. Winds are liars and, believe me, I would never trust  \nthem with Aeneas, I who have so often been betrayed by a clear  \nsky.' This was his answer, and he stood by the tiller, gripping it  \nwith no intention of letting it go or taking his eyes off the stars.  \nBut look! The god takes a branch dripping with the water of  \nLethe for forgetfulness and the water of Styx for sleep. He shakes  \nit over Palinurus, first one temple, then the other, and for all his  \nstruggles it closes his swimming eyes. As soon as this sudden  \nsleep came upon him and his limbs began to relax, the god  \nleaned over him, broke off a part of the poop, tiller and all, and  \n860 threw him with it into the waves of the sea. Down fell Palinurus,  \ncalling again and again on his comrades, but they did not hear.  \nThe god then rose on his wings and flew off into the airy breezes,  \nwhile the ships sped on their way none the worse, sailing safely  \non in accordance with the promises of Father Neptune.\n\nThey were soon coming near the Sirens' rocks, once a difficult  \ncoast and white with the bones of drowned men, and at that  \nmoment sounding far with the endless grinding of breaker upon  \nrock, when Father Aeneas sensed that he was adrift without a  \nhelmsman. In mid-ocean in the dead of night he took control of  \nthe ship himself, and grieving to the heart at the loss of his  \n870 friend, he cried out: 'You trusted too much, Palinurus, to a clear  \nsky and a calm sea, and your body will lie naked on an unknown  \nshore.'\n\n## BOOK 6  \nTHE UNDERWORLD\n\nSo spoke Aeneas, weeping, and gave the ships their head and at  \nlong last they glided to land at the Euboean colony of Cumae.  \nThe prows were turned out to sea, the teeth of the anchors held  \nand they moored with their curved sterns fringing the shore.  \nGleaming in the sun, an eager band of warriors rushed out on  \nto the shore of the land of Hesperia, some searching for the  \nseeds of flame hidden in the veins of flint, some raiding the dense  \nwoods, the haunts of wild beasts, and pointing the way to rivers  \nthey had found. But the devout Aeneas made for the citadel  \n10 where Apollo sits throned on high and for the vast cave standing  \nthere apart, the retreat of the awesome Sibyl, into whom Delian  \nApollo, the God of Prophecy, breathes mind and spirit as he  \nreveals to her the future. They were soon coming up into the  \ngrove of Diana Trivia and Apollo's golden shrine.\n\nThey say that when Daedalus was fleeing from the kingdom  \nof Minos, he dared to trust his life to the sky, floating off on  \nswiftly driving wings towards the cold stars of the north, the  \nGreater and Lesser Bears, by a route no man had ever gone  \nbefore, until at last he was hovering lightly in the air above the  \ncitadel of Chalcidian Cumae. Here he first returned to earth,  \ndedicating to Phoebus Apollo the wings that had oared him  \n20 through the sky, and founding a huge temple. On its doors were  \ndepicted the death of Androgeos, son of Minos, and then the  \nAthenians, the descendants of Cecrops, ordered to pay a cruel  \npenalty and yield up each year the living bodies of seven of their  \nsons. The lots are drawn and there stands the urn. Answering  \nthis on the other door are Cnossus and the land of Crete rising  \nfrom the sea. Here can be seen the loving of the savage bull and  \nPasiphae laid out to receive it and deceive her husband Minos.  \nHere too is the hybrid offspring, the Minotaur, half-man and  \nhalf-animal, the memorial to a perverted love, and here is its  \nhome, built with such great labour, the inextricable Labyrinth.  \nBut Daedalus takes pity on the great love of the princess Ariadne  \n30 and unravels the winding paths of his own baffling maze, guiding  \nthe blind steps of Theseus with a thread. You too, Icarus, would  \nhave taken no small place in this great work had the grief of  \nDaedalus allowed it. Twice your father tried to shape your fall  \nin gold and twice his hands fell helpless. The Trojans would  \nhave gone on gazing and read the whole story through, but  \nAchates, who had been sent ahead, now returned bringing with  \nhim Deiphobe, the daughter of Glaucus, priestess of Phoebus  \nand Trivia, who spoke these words to the king: 'This is no time  \nfor you to be looking at sights like these. Rather at this moment  \nyou should be sacrificing seven bullocks from a herd the yoke  \nhas never touched and seven yearling sheep as ritual prescribes.'  \n40 So she addressed Aeneas. Nor were the Trojans slow to obey,  \nand when the sacrifices were performed she called them into the  \nlofty temple.\n\nThis rocky citadel had been colonized by Chalcidians from  \nEuboea, and one side of it had been hollowed out to form a  \nvast cavern into which led a hundred broad shafts, a hundred  \nmouths, from which streamed as many voices giving the  \nresponses of the Sibyl. They had reached the threshold of the  \ncavern when the virgin priestess cried: 'Now is the time to ask  \nyour destinies. It is the god. The god is here.' At that moment,  \nas she spoke in front of the doors, her face was transfigured, her  \ncolour changed, her hair fell in disorder about her head and she  \nstood there with heaving breast and her wild heart bursting in  \n50 ecstasy. She seemed to grow in stature and speak as no mortal  \nhad ever spoken when the god came to her in his power and  \nbreathed upon her. 'Why are you hesitating, Trojan Aeneas?'  \nshe cried. 'Why are you so slow to offer your vows and prayers?  \nUntil you have prayed the great mouths of my house are dumb  \nand will not open.' She spoke and said no more. A cold shiver  \nran through the very bones of the Trojans and their king poured  \nout the prayers from the depths of his heart: 'Phoebus Apollo,  \nyou have always pitied the cruel sufferings of the Trojans. You  \nguided the hands of Trojan Paris and the arrow he sent into the  \nbody of Achilles. You were my leader as I set out upon all  \nthe oceans that lap the great lands of the earth and reached the  \n60 far-flung peoples of Massylia and the fields that lie out to sea in  \nfront of the Syrtes. Now at long last we lay hold upon the shores  \nof Italy that have so often receded before us. I pray that from  \nthis moment the fortunes of Troy may follow us no further. You  \ntoo, you gods and goddesses who could not endure Troy and  \nthe great glory of the race of Dardanus, it is now right that you  \nshould have mercy upon the people of Pergamum. And you, O  \nmost holy priestess, you who know in advance what is to be,  \ngrant my prayer, for the kingdom I ask for is no more than what  \nis owed me by the Fates, and allow the Trojans and their  \n70 homeless and harried gods to settle in Latium. Then I shall  \nfound a temple of solid marble to Phoebus and Trivia, and holy  \ndays in the name of Phoebus. And for you too there will be a  \ngreat shrine in our kingdom. Here I shall establish your oracle  \nand the riddling prophecies you have given my people and I  \nshall dedicate chosen priests to your gracious service, only do not  \nconsign your prophecies to leaves to be confused and mocked by  \nevery wind that blows. Sing them in your own voice, I beg of  \nyou.' He said no more.\n\nBut the priestess, not yet submissive, was still in wild frenzy  \nin her cave. The more she tried to shake her body free of the  \n80 great god the harder he strained upon her foaming mouth,  \ntaming that wild heart and moulding her by his pressure. And  \nnow the hundred huge doors of her house opened of their own  \naccord and gave her answer to the winds: 'At long last you have  \ndone with the perils of the ocean, but worse things remain for  \nyou to bear on land. The sons of Dardanus shall come into their  \nkingdom in Lavinium (put that fear out of your mind), but it is  \na coming they will wish they had never known. I see wars,  \ndeadly wars, I see the Thybris foaming with torrents of blood.  \nThere you will find a Simois and a Xanthus. There, too, will be  \na Greek camp. A second Achilles is already born in Latium, and  \n90 he too is the son of a goddess. Juno too is part of Trojan destiny  \nand will never be far away when you are a suppliant begging in  \ndire need among all the peoples and all the cities of Italy. Once  \nagain the cause of all this Trojan suffering will be a foreign  \nbride, another marriage with a stranger. You must not give way  \nto these adversities but must face them all the more boldly  \nwherever your fortune allows it. Your road to safety, strange as  \nit may seem, will start from a Greek city.'\n\nWith these words from her shrine the Sibyl of Cumae sang  \nher fearful riddling prophecies, her voice booming in the cave  \n100 as she wrapped the truth in darkness, while Apollo shook the  \nreins upon her in her frenzy and dug the spurs into her flanks.  \nThe madness passed. The wild words died upon her lips, and  \nthe hero Aeneas began to speak: 'O virgin priestess, suffering  \ncannot come to me in any new or unforeseen form. I have  \nalready known it. Deep in my heart I have lived it all before.  \nOne prayer I have. Since they say the gate of the king of the  \nunderworld is here and here too in the darkness is the swamp  \nwhich the tide of Acheron floods, I pray to be allowed to go and  \nlook upon the face of my dear father. Show me the way and  \n110 open the sacred doors for me. On these shoulders I carried him  \naway through the flames and a hail of weapons and rescued  \nhim from the middle of his enemies. He came on my journey  \nwith me over all the oceans and endured all the threats of sea  \nand sky, feeble as he was but finding a strength beyond his years.  \nBesides, it was my father himself who begged and commanded  \nme to come to you as a suppliant and approach your doors. Pity  \nthe father, O gracious one, and pity the son, I beg of you. All  \nthings are within your power and Hecate had her purpose in  \ngiving you charge of the grove of Avernus. Was not Orpheus  \n120 allowed to summon the shade of his wife with the sound of the  \nstrings of his Thracian lyre? And since Pollux was allowed to  \nredeem his brother by sharing his death, does he not often travel  \nthat road and often return? Do I need to speak of Theseus? Or  \nof great Hercules? I too am descended from highest Jupiter.'\n\nWhile he was still speaking these words of prayer with his  \nhand upon the altar, the prophetess began her answer: 'Trojan,  \nson of Anchises, sprung from the blood of the gods, it is easy to  \ngo down to the underworld. The door of black Dis stands open  \nnight and day. But to retrace your steps and escape to the upper  \nair, that is the task, that is the labour. Some few have succeeded,  \n130 sons of the gods, loved and favoured by Jupiter or raised to the  \nheavens by the flame of their own virtue. The middle of that  \nworld is filled with woods and the river Cocytus glides round  \nthem, holding them in its dark embrace. But if your desire is so  \ngreat, if you have so much longing to sail twice upon the pools  \nof Styx and twice to see black Tartarus, if it is your pleasure to  \nindulge this labour of madness, listen to what must first be done.  \nHidden in a dark tree, there is a golden bough. Golden are its  \nleaves and its pliant stem and it is sacred to Proserpina, the Juno  \nof the underworld. A whole grove conceals it and the shades of  \n140 a dark, encircling valley close it in. But no man may enter the  \nhidden places of the earth before plucking the golden foliage  \nand fruit from this tree. The beautiful Proserpina has ordained  \nthat this is the offering that must be brought to her. When one  \ngolden branch has been torn from that tree, another comes to  \ntake its place and the stem puts forth leaves of the same metal.  \nSo then, lift up your eyes and look for it, and when in due time  \nyou find it, take it in your hand and pluck it. If you are a man  \ncalled by the Fates, it will come easily of its own accord. But if  \nnot, no strength will prevail against it and hard steel will not be  \n150 able to hack it off. Besides, you have a friend lying dead. Of this  \nyou know nothing, but his body is polluting the whole fleet  \nwhile you linger here at our door asking for oracles. First you  \nmust carry him to his place of rest and lay him in a tomb. Then  \nyou must bring black cattle to begin the purification. When all  \nthis is done, you will be able to see the groves of Styx and the  \nkingdom where no living man may set his foot.' So she spoke  \nand no other word would cross her lips.\n\nWith downcast eyes and sorrowing face Aeneas walked from  \nthe cave, revolving in his mind the fulfilment of these dark  \nprophecies. With him stride for stride went the faithful Achates,  \n160 and his heart was no less heavy. Long did they talk and many  \ndifferent thoughts they shared. Who was this dead comrade of  \nwhom the priestess spoke? Whose body was this that had to be  \nburied? And when they came to the shore, there above the tide  \nline they found the body of Misenus, who had died a death he  \nhad not deserved. Misenus, son of Aeolus, who had no equal at  \nsummoning the troops with his trumpet and kindling the God  \nof War with his music, had been the comrade of great Hector,  \nand by Hector's side had borne the brunt of battle, excelling not  \nonly with the trumpet but also with the spear. But after Achilles  \nhad defeated Hector and taken his life, the brave Misenus had  \n170 found no less a hero to follow by joining Aeneas of the stock of  \nDardanus. Then one day in his folly he happened to be blowing  \ninto a sea shell, sending the sound ringing over the waves, and  \nchallenged the gods to play as well as he. At this his rival Triton,  \nif the tale is to be believed, had caught him up and drowned him  \nin the surf among the rocks. So then they raised around his  \nbody a loud noise of lamentation, not least the dutiful Aeneas.  \nWithout delay they hastened, still weeping, to obey the commands  \nof the Sibyl, gathering trees to build an altar which would  \nbe his tomb and striving to raise it to the skies. Into the ancient  \n180 forest they went among the deep lairs of wild beasts. Down  \ncame the pines. The ilex rang under the axe. Beams of ash and  \noak were split along the grain with wedges, and they rolled great  \nmanna ashes down from the mountains.\n\nAeneas took the lead in all this work, urging on his comrades  \nand carrying at his side the same tools as they, but he was always  \ngloomily turning one thought over in his mind as he looked at  \nthe measureless forest and he chanced to utter it in this prayer:  \n'If only that golden bough would now show itself to us in this  \ngreat grove, since everything the priestess said about Misenus  \n190 has proved only too true.' No sooner had he spoken than two  \ndoves chanced to come flying out of the sky and settle there on  \nthe grass in front of him. Then the great Aeneas knew they were  \nhis mother's birds and he was glad. 'Be my guides,' he prayed,  \n'if there is a way, and direct your swift flight through the air  \ninto the grove where the rich branch shades the fertile soil.  \nAnd you, goddess, my mother, do not fail me in my time of  \nuncertainty.' So he spoke and waited to see what signs they  \nwould give and in what direction they would move. They flew  \n200 and fed and flew again, always keeping in sight of those who  \nfollowed. Then, when they came to the evil-smelling throat of  \nAvernus, first they soared and then they swooped down through  \nthe clear air and settled where Aeneas had prayed they would  \nsettle, on the top of the tree that was two trees, from whose  \ngreen there gleamed the breath of gold along the branch. Just as  \nthe mistletoe, not sown by the tree on which it grows, puts out  \nfresh foliage in the woods in the cold of winter and twines its  \nyellow fruit round slender tree trunks, so shone the golden  \nfoliage on the dark ilex, so rustled the golden foil in the gentle  \n210 breeze. Aeneas seized the branch instantly. It resisted, but he  \nbroke it off impatiently and carried it into the house of the  \npriestess, the Sibyl.\n\nAll this time the Trojans on the shore did not cease to weep  \nfor Misenus and pay their last tributes to his ungrateful ashes.  \nFirst they built a huge pyre with rich pine torches and oak logs,  \nand wove dark-leaved branches into its sides, setting up funeral  \ncypresses in front of it and crowning it with his shining armour.  \nSome prepared hot water in cauldrons and when it was seething  \nover the flames, they washed and anointed the cold body and  \n220 raised their lament. When they had wept their fill, they placed  \nhim on the bier and draped him in his familiar purple robes.  \nOthers then performed their sad duty of carrying the bier and  \nheld their torches to the bottom of the pyre with averted faces,  \nafter the practice of their ancestors. Then all the heaped-up  \nofferings burned \u2013 the incense, the sacrificial food, the bowls  \nfilled with oil. After the embers had collapsed and the flames  \ndied down, they washed with wine the thirsty ashes that were  \nall that remained of him and Corynaeus collected his bones and  \nsealed them in a bronze casket. Three times he carried them in  \n230 solemn ritual round the comrades of Misenus and sprinkled the  \nheroes lightly with pure water from the branch of a fruitful olive  \ntree, uttering words of farewell as he performed the lustration.  \nBut dutiful Aeneas raised a great mound as a tomb and set on it  \nthe hero's arms, the oars he rowed with and the trumpet he had  \nblown, there near the airy top of Mount Misenus which bears  \nhis name now and for ever through all years to come.\n\nAs soon as this was done he hastened to carry out the commands  \nof the Sibyl. There was a huge, deep cave with jagged  \npebbles underfoot and a gaping mouth guarded by dark woods  \n240 and the black waters of a lake. No bird could wing its flight over  \nthis cave and live, so deadly was the breath that streamed out  \nof that black throat and up into the vault of heaven. Hence the  \nGreek name, 'Aornos', 'the place without birds'. Here first of  \nall the priestess stood four black-backed bullocks and poured  \nwine upon their foreheads. She then plucked the bristles from  \nthe peak of their foreheads between their horns to lay upon the  \naltar fires as a first offering and lifted up her voice to call on  \nHecate, mighty in the sky and mighty in Erebus. Attendants put  \n250 the knife to the throat and caught the warm blood in bowls.  \nAeneas himself took his sword and sacrificed a black-fleeced  \nlamb to Night, the mother of the Furies, and her sister Earth,  \nand to Proserpina a barren cow. Then he set up a night altar for  \nthe worship of the Stygian king and laid whole carcasses of bulls  \non its flames and poured rich oil on the burning entrails. Then  \nsuddenly, just before the sun had crossed his threshold in the  \nsky and begun to rise, the earth bellowed underfoot, the wooded  \nridges quaked and dogs could be heard howling in the darkness.  \nIt was the arrival of the goddess. 'Stand apart, all you who are  \nunsanctified,' cried the priestess. 'Stand well apart. The whole  \n260 grove must be free of your presence. You, Aeneas, must enter  \nupon your journey. Draw your sword from the sheath. Now  \nyou need your courage. Now let your heart be strong.' With  \nthese words she moved in a trance into the open cave and step  \nfor step Aeneas strode fearlessly along behind her.\n\nYou gods who rule the world of the spirits, you silent shades,  \nand Chaos, and Phlegethon, you dark and silent wastes, let it be  \nright for me to tell what I have been told, let it be with your  \ndivine blessing that I reveal what is hidden deep in the mists  \nbeneath the earth.\n\nThey walked in the darkness of that lonely night with shadows  \nall about them, through the empty halls of Dis and his desolate  \n270 kingdom, as men walk in a wood by the sinister light of a fitful  \nmoon when Jupiter has buried the sky in shade and black night  \nhas robbed all things of their colour. Before the entrance hall of  \nOrcus, in the very throat of hell, Grief and Revenge have made  \ntheir beds and Old Age lives there in despair, with white-faced  \nDiseases and Fear and Hunger, corrupter of men, and squalid  \nPoverty, things dreadful to look upon, and Death and Drudgery  \nbesides. Then there are Sleep, Death's sister, perverted Pleasures,  \n280 murderous War astride the threshold, the iron chambers of the  \nFuries and raving Discord with blood-soaked ribbons binding  \nher viperous hair. In the middle a huge dark elm spreads out its  \nancient arms, the resting-place, so they say, of flocks of idle  \ndreams, one clinging under every leaf. Here too are all manner  \nof monstrous beasts, Centaurs stabling inside the gate, Scyllas \u2013  \nhalf-dogs, half-women \u2013 Briareus with his hundred heads, the  \nHydra of Lerna hissing fiercely, the Chimaera armed in fire,  \n290 Gorgons and Harpies and the triple phantom of Geryon. Now  \nAeneas drew his sword in sudden alarm to meet them with  \nnaked steel as they came at him, and if his wise companion had  \nnot warned him that this was the fluttering of disembodied  \nspirits, a mere semblance of living substance, he would have  \nrushed upon them and parted empty shadows with steel.\n\nHere begins the road that leads to the rolling waters of  \nAcheron, the river of Tartarus. Here is a vast quagmire of boiling  \nwhirlpools which belches sand and slime into Cocytus, and  \nthese are the rivers and waters guarded by the terrible Charon  \n300 in his filthy rags. On his chin there grows a thick grey beard,  \nnever trimmed. His glaring eyes are lit with fire and a foul cloak  \nhangs from a knot at his shoulder. With his own hands he plies  \nthe pole and sees to the sails as he ferries the dead in a boat the  \ncolour of burnt iron. He is no longer young but, being a god,  \nenjoys rude strength and a green old age. The whole throng of  \nthe dead was rushing to this part of the bank, mothers, men,  \ngreat-hearted heroes whose lives were ended, boys, unmarried  \n310 girls and young men laid on the pyre before the faces of their  \nparents, as many as are the leaves that fall in the forest at the  \nfirst chill of autumn, as many as the birds that flock to land  \nfrom deep ocean when the cold season of the year drives them  \nover the sea to lands bathed in sun. There they stood begging to  \nbe allowed to be the first to cross and stretching out their arms  \nin longing for the further shore. But the grim boatman takes  \nsome here and some there, and others he pushes away far back  \nfrom the sandy shore.\n\nAeneas, amazed and distressed by all this tumult, cried out:  \n'Tell me, virgin priestess, what is the meaning of this crowding  \n320 to the river? What do the spirits want? Why are some pushed  \naway from the bank while others sweep the livid water with  \ntheir oars?' The aged Sibyl made this brief reply: 'Son of  \nAnchises, beyond all doubt the offspring of the gods, what you  \nare seeing is the deep pools of the Cocytus and the swamp of  \nthe Styx, by whose divine power the gods are afraid to swear  \nand lie. The throng you see on this side are the helpless souls of  \nthe unburied. The ferryman there is Charon. Those sailing the  \nwaters of the Styx have all been buried. No man may be ferried  \nfrom fearful bank to fearful bank of this roaring current until  \nhis bones are laid to rest. Instead they wander for a hundred  \n330 years, fluttering round these shores until they are at last allowed  \nto return to the pools they have so longed for.' The son of  \nAnchises checked his stride and stood stock still with many  \nthoughts coursing through his mind as he pitied their cruel fate,  \nwhen there among the sufferers, lacking all honour in death, he  \ncaught sight of Leucaspis, and Orontes, the captain of the Lycian  \nfleet, men who had started with him from Troy, sailed the  \nwind-torn seas and been overwhelmed by gales from the south  \nthat rolled them in the ocean, ships and crews.\n\nNext he saw coming towards him his helmsman Palinurus  \nwho had fallen from the ship's stern and plunged into the sea  \nwhile watching the stars on the recent crossing from Libya.  \n340 Aeneas recognized this sorrowing figure with difficulty in the  \ndark shadow and was the first to speak: 'What god was it,  \nPalinurus, that took you from us and drowned you in mid-ocean?  \nCome tell me, for this is the one response of Apollo  \nthat has misled me. I have never found him false before. He  \nprophesied that you would be safe upon the sea and would  \nreach the boundaries of Ausonia. Is this how he has kept his  \npromise?' 'O great leader, son of Anchises,' replied Palinurus,  \n'the bowl on the tripod of Apollo has not deceived you and no  \ngod drowned me in the sea. While I was holding course and  \n350 gripping the tiller which it was my charge to guard, it was  \nbroken off by some mighty force and I dragged it down with me  \nas I fell. I swear by the wild sea that I felt no fear for myself to  \nequal my fear that your ship might come to grief, stripped of its  \nsteering and with its pilot pitched into the sea and that great  \nswell rising. Three long winter nights the wind blew hard from  \nthe south and carried me over seas I could not measure, till,  \nwhen light came on the fourth day, and a wave lifted me to its  \ncrest, I could just make out the land of Italy. I swam slowly to  \nshore and was on the point of reaching safety when a tribe of  \nruffians set upon me with their knives, weighed down as I was  \n360 by my wet clothes and clinging by my finger tips to the jagged  \nrocks at the foot of a cliff. Knowing nothing of me they made  \nme their plunder, and now I am at the mercy of the winds, and  \nthe waves are turning my body over at the water's edge. But I  \nbeg of you, by the joyous light and winds of heaven, by your  \nfather, by your hopes of Iulus as he grows to manhood, you  \nwho have never known defeat, rescue me from this anguish.  \nEither throw some earth on my body \u2013 you can do that. Just  \nsteer back to the harbours of Velia. Or else if there is a way and  \nthe goddess who gave you life shows it to you \u2013 for I do not  \nbelieve you are preparing to sail these great rivers and the swamp  \n370 of the Styx unless the blessing of the gods is with you \u2013 take pity  \non me, give me your right hand, take me aboard and carry me  \nwith you over the waves, so that in death at least I can be at  \npeace in a place of quiet.' These were the words of Palinurus  \nand this was the reply of the Sibyl: 'How did you conceive this  \nmonstrous desire, Palinurus? How can you, who are unburied,  \nhope to set eyes on the river Styx and the pitiless waters of the  \nFuries? How can you come near the bank unbidden? You must  \ncease to hope that the Fates of the gods can be altered by prayers.  \nBut hear my words, remember them and find comfort for your  \nsad case. The people who live far and wide in all their cities  \nround the place where you died, will be driven by signs from  \n380 heaven to consecrate your bones. They will raise a burial mound  \nfor you and to that mound will pay their annual tribute and the  \nplace will bear the name of Palinurus for all time to come.' At  \nthese words his sorrows were removed and the grief was driven  \nfrom that sad heart for a short time. He rejoiced in the land that  \nwas to bear his name.\n\nAnd so they carried on to the end of the road on which they  \nhad started, and at last came near the river. When the boatman,  \nnow in mid-stream, looked ashore from the waves of the Styx  \nand saw them coming through the silent wood towards the  \nbank, he called out and challenged them: 'You there, whoever  \nyou are, making for our river with a sword by your side, come  \ntell us why you are here. Speak to us from where you stand.  \n390 Take not another step. This place belongs to the shades, to Sleep  \nand to Night, the bringer of Sleep. Living bodies may not be  \ncarried on the boat that plies the Styx. It gave me little enough  \npleasure to take even Hercules aboard when he came, or  \nTheseus, or Pirithous, although they said they were born of gods  \nand their strength was irresistible. It was Hercules whose hand  \nput chains on the watchdog of Tartarus and dragged him shivering  \nfrom the very throne of our king. The others had taken it  \nupon themselves to steal the queen, my mistress, from the  \nchamber of Dis.' The answer of the Amphrysian Sibyl was brief:  \n'Here there are no such designs. You have no need for alarm.  \n400 These weapons of his bring no violence. The monstrous keeper  \nof the gate can bark in his cave and frighten the bloodless shades  \ntill the end of time and Proserpina can stay chaste behind her  \nuncle's doors. Trojan Aeneas, famous for his devotion and his  \nfeats of arms, is going down to his father in the darkest depths  \nof Erebus. If the sight of such devotion does not move you, then  \nlook at this branch,' she said, showing the branch that had been  \nhidden in her robes, 'and realize what it is.' At this the swelling  \nanger subsided in his heart. No more words were needed. Seeing  \nit again after a long age, and marvelling at the fateful branch,  \n410 the holy offering, he turned his dark boat and steered towards  \nthe bank. He then drove off the souls who were on board with  \nhim sitting all along the cross benches, and cleared the gangways.  \nIn the same moment he took the huge Aeneas into the hull of  \nhis little boat. Being only sewn together, it groaned under his  \nweight, shipping great volumes of stagnant water through the  \nseams, but in the end it carried priestess and hero safely over  \nand landed them on the foul slime among the grey-green reeds.\n\nThe kingdom on this side resounded with barking from the  \nthree throats of the huge monster Cerberus lying in a cave in  \nfront of them. When the priestess was close enough to see the  \n420 snakes writhing on his neck, she threw him a honey cake steeped  \nin soporific drugs. He opened his three jaws, each of them rabid  \nwith hunger, and snapped it up where it fell. The massive back  \nrelaxed and he sprawled full length on the ground, filling his  \ncave. The sentry now sunk in sleep, Aeneas leapt to take command  \nof the entrance and was soon free of the bank of that river  \nwhich no man may recross.\n\nIn that instant they heard voices, a great weeping and wailing  \nof the souls of infants who had lost their share of the sweetness  \nof life on its very threshold, torn from the breast on some black  \n430 day and drowned in the bitterness of death. Next to them were  \nthose who had been condemned to death on false charges, but  \nthey did not receive their places without the casting of lots and  \nthe appointment of juries. Minos, the president of the court,  \nshakes the lots in the urn, summoning the silent dead to act as  \njurymen, and holds inquiry into the lives of the accused and the  \ncharges against them. Next to them were those unhappy people  \nwho had raised their innocent hands against themselves, who  \nhad so loathed the light that they had thrown away their own  \nlives. But now how they would wish to be under high heaven,  \nenduring poverty and drudgery, however hard! That cannot be,  \nfor they are bound in the coils of the hateful swamp of the  \nwaters of death, trapped in the ninefold windings of the river  \n440 Styx. Not far from here could be seen what they call the Mourning  \nPlains, stretching away in every direction. Here are the  \nvictims of unhappy love, consumed by that cruel wasting sickness,  \nhidden in the lonely byways of an encircling wood of  \nmyrtle trees, and their suffering does not leave them even in  \ndeath. Here Aeneas saw Phaedra, and Procris, and Eriphyle in  \ntears as she displayed the wounds her cruel son had given her.  \nHere he saw Evadne and Pasiphae with Laodamia walking by  \ntheir side, and Caeneus, once a young man, but now a woman  \nrestored by destiny to her former shape.\n\n450 Wandering among them in that great wood was Phoenician  \nDido with her wound still fresh. When the Trojan hero stopped  \nbeside her, recognizing her dim form in the darkness, like a man  \nwho sees or thinks he has seen the new moon rising through the  \nclouds at the beginning of the month, in that instant he wept  \nand spoke sweet words of love to her: 'So the news they brought  \nme was true, unhappy Dido? They told me you were dead and  \nhad ended your life with the sword. Alas! Alas! Was I the cause  \nof your dying? I swear by the stars, by the gods above, by  \n460 whatever there is to swear by in the depths of the earth, it was  \nagainst my will, O queen, that I left your shore. It was the stern  \nauthority of the commands of the gods that drove me on, as it  \ndrives me now through the shades of this dark night in this foul  \nand mouldering place. I could not have believed that my leaving  \nwould cause you such sorrow. Do not move away. Do not leave  \nmy sight. Who are you running from? Fate has decreed that I  \nshall not speak to you again.' With these words Aeneas, shedding  \ntears, tried to comfort that burning spirit, but grim-faced  \n470 she kept her eyes upon the ground and did not look at him. Her  \nfeatures moved no more when he began to speak than if she had  \nbeen a block of flint or Parian marble quarried on Mount  \nMarpessus. Then at last she rushed away, hating him, into  \nthe shadows of the wood where Sychaeus, who had been her  \nhusband, answered her grief with grief and her love with love.  \nAeneas was no less stricken by the injustice of her fate and long  \ndid he gaze after her with tears, pitying her as she went.\n\nFrom here they continued on their appointed road and they  \nwere soon on the most distant of these fields, the place set  \n480 apart for brave warriors. Here Tydeus came to meet him, and  \nParthenopaeus, famous for his feats of arms, and the pale phantom  \nof Adrastus. Here he saw and groaned to see standing in  \ntheir long ranks all the sons of Dardanus who had fallen in  \nbattle and been bitterly lamented in the upper world, Glaucus,  \nMedon and Thersilochus, the three sons of Antenor, and  \nPolyboetes, the consecrated priest of Ceres, and Idaeus still  \nkeeping hold of Priam's chariot, still keeping hold of his armour.  \nThe shades crowded round him on the right and on the left and  \nit was not enough just to see him, they wished to delay him, to  \nwalk with him, to learn the reasons for his coming. But when the  \nGreek leaders and the soldiers of Agamemnon in their phalanxes  \n490 saw the hero and his armour gleaming through the shadows, a  \nwild panic seized them. Some turned and ran as they had run  \nonce before to get back to their ships, while others lifted up  \ntheir voices and raised a tiny cry, which started as a shout from  \nmouth wide open, but no shout came.\n\nHere too he saw Deiphobus, son of Priam, his whole body  \nmutilated and his face cruelly torn. The face and both hands  \nwere in shreds. The ears had been ripped from the head. He was  \nnoseless and hideous. Aeneas, barely recognizing him as he tried  \nfrantically to hide the fearsome punishment he had received,  \nwent up to him and spoke in the voice he knew so well:  \n500 'Deiphobus, mighty warrior, descended from the noble blood  \nof Teucer, who could have wished to inflict such a punishment  \nupon you? And who was able to do this? I was told that on that  \nlast night you wore yourself out killing the enemy and fell on a  \nhuge pile of Greek and Trojan dead. At that time I did all I  \ncould do, raising an empty tomb for you on the shore of Cape  \nRhoeteum and lifting up my voice to call three times upon your  \nshade. Your name and your arms mark the place but you I could  \nnot find, my friend, to bury your body in our native land as I  \nwas leaving it.'\n\nTo this the son of Priam answered: 'You, my friend, have left  \n510 nothing undone. You have paid all that is owed to Deiphobus  \nand to his dead shade. It is my own destiny and the crimes of  \nthe murderess from Sparta that have brought me to this. These  \nare reminders of Helen. You know how we spent that last night  \nin false joy. It is our lot to remember it only too well. When the  \nhorse that was the instrument of Fate, heavy with the brood of  \narmed men in its belly, leapt over the high walls of Pergamum,  \nHelen was pretending to be worshipping Bacchus, leading the  \nwomen of Phrygia around the city, dancing and shrieking their  \nritual cries. There she was in the middle of them with a huge  \ntorch, signalling to the Greeks from the top of the citadel, and  \n520 all the time I was sleeping soundly in our accursed bed, worn  \nout by all I had suffered and sunk in a sleep that was sweet and  \ndeep and like the peace of death. Meanwhile this excellent wife  \nof mine, after moving all my armour out of the house and taking  \nthe good sword from under my head, called in Menelaus and  \nthrew open the doors, hoping no doubt that her loving husband  \nwould take this as a great favour to wipe out the memory of her  \npast sins. You can guess the rest. They burst into the room,  \ntaking with them the man who had incited them to their crimes,  \ntheir comrade Ulixes \u2013 they say he is descended from Aeolus.  \n530 You gods, if the punishment I ask is just, grant that a fate like  \nmine should strike again and strike Greeks. But come, it is now  \ntime for you to tell me what chance has brought you here alive.  \nIs it your sea wanderings that have taken you here? Are you  \nunder the instructions of the gods? What fortune is dogging  \nyou, that you should come here to our sad and sunless homes  \nin this troubled place?'\n\nWhile they were speaking to one another, Dawn's rosy chariot  \nhad already run its heavenly course past the mid-point of the  \nvault of the sky, and they might have spent all the allotted  \ntime in talking but for Aeneas' companion. The Sibyl gave her  \nwarning in few words: 'Night is running quickly by, Aeneas,  \n540 and we waste the hours in weeping. This is where the way  \ndivides. On the right it leads up to the walls of great Dis. This is  \nthe road we take for Elysium. On the left is the road of punishment  \nfor evil-doers, leading to Tartarus, the place of the  \ndamned.' 'There is no need for anger, great priestess,' replied  \nDeiphobus. 'I shall go to take my place among the dead and  \nreturn to darkness. Go, Aeneas, go, great glory of our Troy,  \nand enjoy a better fate than mine.' These were his only words,  \nand as he spoke he turned on his heel and strode away.\n\nAeneas looked back suddenly and saw under a cliff on his left  \n550 a broad city encircled by a triple wall and washed all round by  \nPhlegethon, one of the rivers of Tartarus, a torrent of fire and  \nflame, rolling and grinding great boulders in its current. There  \nbefore him stood a huge gate with columns of solid adamant so  \nstrong that neither the violence of men nor of the heavenly gods  \nthemselves could ever uproot them in war, and an iron tower  \nrose into the air where Tisiphone sat with her blood-soaked  \ndress girt up, guarding the entrance and never sleeping, night or  \nday. They could hear the groans from the city, the cruel crack  \n560 of the lash, the dragging and clanking of iron chains. Aeneas  \nstood in terror, listening to the noise. 'What kinds of criminal are  \nhere? Tell me, virgin priestess, what punishments are inflicted on  \nthem? What is this wild lamentation in the air?' The Sibyl  \nreplied: 'Great leader of the Trojans, the chaste may not set foot  \nupon the threshold of that evil place, but when Hecate put me  \nin charge of the groves of Avernus, she herself explained the  \npunishments the gods had imposed and showed me them all.  \nHere Rhadamanthus, king of Cnossus, holds sway with his  \nunbending laws, chastising men, hearing all the frauds they have  \npractised and forcing them to confess the undiscovered crimes  \nthey have gloated over in the upper world \u2013 foolishly, for they  \n570 have only delayed the day of atonement till after death. Immediately  \nthe avenging Tisiphone leaps upon the guilty and flogs  \nthem till they writhe, waving fearful serpents over them in her  \nleft hand and calling up the cohorts of her savage sisters, the  \nFuries. Then at last the gates sacred to the gods below shriek in  \ntheir sockets and open wide. You see what a watch she keeps,  \nsitting in the entrance? What a sight she is guarding the  \nthreshold? Inside, more savage still, the huge, black-throated,  \nfifty-headed Hydra has its lair. And then there is Tartarus itself,  \nstretching sheer down into its dark chasm twice as far as we  \n580 look up to the ethereal Olympus in the sky. Here, rolling in the  \nbottom of the abyss, is the ancient brood of Earth, the army of  \nTitans, hurled down by the thunderbolt. Here too I saw the  \nhuge bodies of the twin sons of Aloeus who laid violent hands  \non the immeasurable sky to wrench it from its place and tear  \ndown Jupiter from his heavenly kingdom. I saw too Salmoneus  \nsuffering cruel punishment, still miming the flames of Jupiter and  \nthe rumblings of Olympus. He it was who, riding his four-horse  \nchariot and brandishing a torch, used to go in glory through the  \npeoples of Greece and the city of Olympia in the heart of Elis,  \n590 laying claim to divine honours for himself \u2013 fool that he was to  \ncopy the storm and the inimitable thunderbolt with the rattle of  \nthe horn of his horses' hooves on bronze. Through the thick  \nclouds the All-powerful Father hurled his lightning \u2013 no smoky  \nlight from pitchy torches for him \u2013 and sent him spinning deep  \ninto the abyss. Tityos too I could see, the nurseling of Earth,  \nmother of all, his body sprawling over nine whole acres while a  \nhuge vulture with hooked beak cropped his immortal liver and  \n600 the flesh that was such a rich supplier of punishment. Deep in  \nhis breast it roosts and forages for its dinners, while the filaments  \nof his liver know no rest but are restored as soon as they are  \nconsumed. I do not need to speak of the Lapiths, of Ixion or  \nPirithous, over whose heads the boulder of black flint is always  \nslipping, always seeming to be falling. The gold gleams on the  \nhigh supports of festal couches and a feast is laid in regal  \nsplendour before the eyes of the guilty, but the greatest of the  \nFuries is reclining at table and allows no hand to touch the food,  \nbut leaps up brandishing a torch and shouting with a voice of  \nthunder. Immured in this place and waiting for punishment  \nare those who in life hated their brothers, beat their fathers,  \n610 defrauded their dependants, found wealth and brooded over it  \nalone without setting aside a share for their kinsmen \u2013 these are  \nmost numerous of all \u2013 men caught and killed in adultery, men  \nwho took up arms against their own people and did not shrink  \nfrom abusing their masters' trust. Do not ask to know what  \ntheir punishments are, what form of pain or what misfortune  \nhas engulfed them. Some are rolling huge rocks, or hang spreadeagled  \non the spokes of wheels. Theseus is sitting there dejected,  \nand there he will sit until the end of time, while Phlegyas, most  \nwretched of them all, shouts this lesson for all men at the top of  \n620 his voice in the darkness: \"Learn to be just and not to slight the  \ngods. You have been warned.\" Here is the man who has sold  \nhis native land for gold, and set a tyrant over it, putting up  \ntablets with new laws for a price and for a price removing them.  \nHere is the man who forced his way into his daughter's bed  \nand a forbidden union. They have all dared to attempt some  \nmonstrous crime against the gods and have succeeded in their  \nattempt. If I had a hundred tongues, a hundred mouths and a  \nvoice of iron, I could not encompass all their different crimes or  \nspeak the names of all their different punishments.'\n\nWhen the aged priestess of Apollo had finished her answer,  \nshe added these words: 'But come now, you must take the road  \n630 and complete the task you have begun. Let us hasten. I can see  \nthe high walls forged in the furnaces of the Cyclopes and the  \ngates there in front of us in the arch. This is where we have been  \ntold to lay the gift that is required of us.' After these words they  \nwalked the dark road together, soon covering the distance and  \ncoming close to the doors. There Aeneas leapt on the threshold,  \nsprinkled his body with fresh water and fixed the bough full in  \nthe doorway.\n\nWhen this rite was at last performed and his duty to the  \ngoddess was done, they entered the land of joy, the lovely glades  \n640 of the fortunate woods and the home of the blest. Here a broader  \nsky clothes the plains in glowing light, and the spirits have their  \nown sun and their own stars. Some take exercise on grassy  \nwrestling-grounds and hold athletic contests and wrestling  \nbouts on the golden sand. Others pound the earth with dancing  \nfeet and sing their songs while Orpheus, the priest of Thrace,  \naccompanies their measures on his seven-stringed lyre, plucking  \nthe notes sometimes with his fingers, sometimes with his ivory  \nplectrum. Here was the ancient line of Teucer, the fairest of  \n650 all families, great-hearted heroes born in a better time, Ilus,  \nAssaracus and Dardanus, the founder of Troy. Aeneas admired  \nfrom a distance their armour and empty chariots. Their swords  \nwere planted in the ground and their horses wandered free on  \nthe plain cropping the grass. Reposing there below the earth,  \nthey took the same joy in their chariots and their armour as  \nwhen alive, and the same care to feed their sleek horses. Then  \nsuddenly he saw others on both sides of him feasting on the  \ngrass, singing in a joyful choir their paean to Apollo all through  \na grove of fragrant laurels where the mighty river Eridanus rolls  \n660 through the forest to the upper world. Here were armies of men  \nbearing wounds received while fighting for their native land,  \npriests who had been chaste unto death and true prophets whose  \nwords were worthy of Apollo; then those who have raised  \nhuman life to new heights by the skills they have discovered and  \nthose whom men remember for what they have done for men.  \nAll these with sacred ribbons of white round their foreheads  \ngathered round Aeneas and the Sibyl, and she addressed these  \nwords to them, especially to Musaeus, for the whole great  \nthrong looked up to him as he stood there in the middle, head  \nand shoulders above them all: 'Tell me, blessed spirits, and you,  \n670 best of poets, which part of this world holds Anchises? Where  \nis he to be found? It is because of Anchises that we have come  \nhere and crossed the great rivers of Erebus.' The hero returned  \na short answer: 'None of us has a fixed home. We live in these  \ndensely wooded groves and rest on the soft couches of the river  \nbank and in the fresh water-meadows. But if that is the desire  \nof your hearts, come climb this ridge and I shall soon set you on  \nan easy path.' So saying, he walked on in front of them to a  \nplace from where they could see the plains below them bathed  \nin light, and from that point Aeneas and the Sibyl came down  \nfrom the mountain tops.\n\nFather Anchises was deep in a green valley, walking among  \n680 the souls who were enclosed there and eagerly surveying them  \nas they waited to rise into the upper light. It so happened that  \nat that moment he was counting the number of his people,  \nreviewing his dear descendants, their fates and their fortunes,  \ntheir characters and their courage in war. When he saw Aeneas  \ncoming towards him over the grass, he stretched out both hands  \nin eager welcome, with the tears streaming down his cheeks,  \nand these were the words that broke from his mouth: 'You have  \ncome at last,' he cried. 'I knew your devotion would prevail  \nover all the rigour of the journey and bring you to your father.  \nAm I to be allowed to look upon your face, my son, to hear the  \n690 voice I know so well and answer it with my own? I never  \ndoubted it. I counted the hours, knowing you would come, and  \nmy love has not deceived me. I understand how many lands you  \nhave travelled and how many seas you have sailed to come to  \nme here. I know the dangers that have beset you. I so feared the  \nkingdom of Libya would do you harm.' 'It was my vision of  \nyou,' replied Aeneas, 'always before my eyes and always stricken  \nwith sorrow, that drove me to the threshold of this place. The  \nfleet is moored in the Tyrrhenian sea on the shores of Italy.  \nGive me your right hand, father. Give it me. Do not avoid my  \nembrace.' As he spoke these words his cheeks were washed with  \n700 tears and three times he tried to put his arms around his father's  \nneck. Three times the phantom melted in his hands, as weightless  \nas the wind, as light as the flight of sleep.\n\nAnd now Aeneas saw in a side valley a secluded grove with  \ncopses of rustling trees where the river Lethe glided along past  \npeaceful dwelling houses. Around it fluttered numberless races  \nand tribes of men, like bees in a meadow on a clear summer  \nday, settling on all the many-coloured flowers and crowding  \nround the gleaming white lilies while the whole plain is loud  \n710 with their buzzing. Not understanding what he saw, Aeneas  \nshuddered at the sudden sight of them and asked why this was,  \nwhat was that river in the distance and who were all those  \ncompanies of men crowding its banks. 'These are the souls to  \nwhom Fate owes a second body,' replied Anchises. 'They come  \nto the waves of the river Lethe and drink the waters of serenity  \nand draughts of long oblivion. I have long been eager to tell you  \nwho they are, to show them to you face to face and count the  \ngenerations of my people to you so that you could rejoice the  \nmore with me at the finding of Italy.' 'But are we to believe,'  \n720 replied Aeneas to his dear father, 'that there are some souls who  \nrise from here to go back under the sky and return to sluggish  \nbodies? Why do the poor wretches have this terrible longing for  \nthe light?' 'I shall tell you, my son, and leave you no longer in  \ndoubt,' replied Anchises, and he began to explain all things in  \ndue order.\n\n'In the beginning Spirit fed all things from within, the sky and  \nthe earth, the level waters, the shining globe of the moon and  \nthe Titan's star, the sun. It was Mind that set all this matter in  \nmotion. Infused through all the limbs, it mingled with that great  \nbody, and from the union there sprang the families of men and  \nof animals, the living things of the air and the strange creatures  \n730 born beneath the marble surface of the sea. The living force  \nwithin them is of fire and its seeds have their source in heaven,  \nbut their guilt-ridden bodies make them slow and they are dulled  \nby earthly limbs and dying flesh. It is this that gives them their  \nfears and desires, their griefs and joys. Closed in the blind  \ndarkness of this prison they do not see out to the winds of air.  \nEven when life leaves them on their last day of light, they are  \nnot wholly freed from all the many ills and miseries of the body  \nwhich must harden in them over the long years and become  \ningrained in ways we cannot understand. And so they are put  \n740 to punishment, to pay the penalty for all their ancient sins. Some  \nare stretched and hung out empty to dry in the winds. Some  \nhave the stain of evil washed out of them under a vast tide of  \nwater or scorched out by fire. Each of us suffers his own fate in  \nthe after-life. From here we are sent over the broad plains of  \nElysium and some few of us possess these fields of joy until the  \ncircle of time is completed and the length of days has removed  \ningrained corruption and left us pure ethereal sense, the fire of  \nelemental air. All these others whom you see, when they have  \nrolled the wheel for a thousand years, are called out by God to  \n750 come in great columns to the river of Lethe, so that they may  \nduly go back and see the vault of heaven again remembering  \nnothing, and begin to be willing to return to bodies.'\n\nWhen he had finished speaking, Anchises led his son and the  \nSibyl with him into the middle of this noisy crowd of souls, and  \ntook up his stance on a mound from which he could pick them  \nall out as they came towards him in a long line and recognize  \ntheir faces as they came.\n\n'Come now, and I shall tell you of the glory that lies in store  \nfor the sons of Dardanus, for the men of Italian stock who will  \nbe our descendants, bright spirits that will inherit our name,  \n760 and I shall reveal to you your own destiny. That young warrior  \nyou see there leaning on the sword of valour, to him is allotted  \nthe place nearest to the light in this grove, and he will be the  \nfirst of us to rise into the ethereal air with an admixture of Italic  \nblood. He will be called Silvius, an Alban name, and he will be  \nyour son, born after your death. You will live long, but he will  \nbe born too late for you to know, and your wife Lavinia will  \nrear him in the woods to be a king and father of kings and found  \nour dynasty to rule in Alba Longa. Next to him is Procas, glory  \nof the Trojan race, and Capys, and Numitor, and the king who  \n770 will renew your name, Silvius Aeneas, your equal in piety and  \nin arms if ever he succeeds to his rightful throne in Alba. What  \nwarriors they are! Look at the strength of them! Look at the  \noak wreaths, the Civic Crowns, that shade their foreheads!  \nThese are the men who will build Nomentum for you, and  \nGabii, and the city of Fidenae. They will set Collatia's citadel  \non the mountains, and Pometia too, and Castrum Inui, and Bola  \nand Cora. These, my son, will be the names of places which are  \nat this moment places without names. And Romulus, son of  \nMars, will march at his grandfather's side. He will be of the  \nstock of Assaracus, and his mother, who will rear him, will be  \nIlia. Do you see how the double crest stands on his head and the  \n780 Father of the Gods himself already honours him with his own  \nemblem? Look at him, my son. Under his auspices will be  \nfounded Rome in all her glory, whose empire shall cover the  \nearth and whose spirit shall rise to the heights of Olympus. Her  \nsingle city will enclose seven citadels within its walls and she  \nwill be blest in the abundance of her sons, like Cybele, the  \nMother Goddess of Mount Berecyntus riding in her chariot  \nturret-crowned through the cities of Phrygia, rejoicing in her  \ndivine offspring and embracing a hundred descendants, all of  \nthem gods, all dwellers in the heights of heaven.\n\n'Now turn your two eyes in this direction and look at this  \nfamily of yours, your own Romans. Here is Caesar, and all the  \n790 sons of Iulus about to come under the great vault of the sky.  \nHere is the man whose coming you so often hear prophesied,  \nhere he is, Augustus Caesar, son of a god, the man who will  \nbring back the golden years to the fields of Latium once ruled  \nover by Saturn, and extend Rome's empire beyond the Indians  \nand the Garamantes to a land beyond the stars, beyond the  \nyearly path of the sun, where Atlas holds on his shoulder the  \nsky all studded with burning stars and turns it on its axis. The  \nkingdoms round the Caspian sea and Lake Maeotis are even  \n800 now quaking at the prophecies of his coming. The seven mouths  \nof the Nile are in turmoil and alarm. Hercules himself did not  \nmake his way to so many lands though his arrow pierced the  \nhind with hooves of bronze, though he gave peace to the woods  \nof Erymanthus and made Lerna tremble at his bow. Nor did  \ntriumphing Bacchus ride so far when he drove his tiger-drawn  \nchariot down from the high peak of Nysa, and the reins that  \nguided the yoke were the tendrils of the vine. And do we still  \nhesitate to extend our courage by our actions? Does any fear  \ndeter us from taking our stand on the shore of Ausonia?\n\n'But who is this at a distance resplendent in his crown of olive  \nand carrying holy emblems? I know that white hair and beard.  \n810 This is the man who will first found our city on laws, the Roman  \nking called from the little town of Cures in the poor land of the  \nSabines into a mighty empire. Hard on his heels will come Tullus  \nto shatter the leisure of his native land and rouse to battle men  \nthat have settled into idleness and armies that have lost the habit  \nof triumph. Next to him, and more boastful, comes Ancus, too  \nfond even now of the breath of popular favour. Do you wish to  \nsee now the Tarquin kings, the proud spirit of avenging Brutus  \n820 and the rods of office he will retrieve? He will be the first to be  \ngiven authority as consul and the stern axes of that office. When  \nhis sons raise again the standards of war, it is their own father  \nthat will call them to account in the glorious name of liberty.  \nHe is not favoured by Fortune, however future ages may judge  \nthese actions \u2013 love of his country will prevail with him and his  \nlimitless desire for glory. Look too at the Decii and the Drusi  \nover there and cruel Torquatus with his axe and Camillus carrying  \nback the standards. Those two spirits you see gleaming there  \nin their well-matched armour are in harmony now while they  \nare buried in night, but if once they reach the light of life, what  \na terrible war they will stir up between them! What battles!  \n830 What carnage when the father-in-law swoops from the ramparts  \nof the Alps and his citadel of Monaco and his son-in-law leads  \nagainst him the embattled armies of the East! O my sons, do not  \nharden your hearts to such wars. Do not turn your strong hands  \nagainst the flesh of your motherland. You who are sprung from  \nOlympus, you must be the first to show clemency. Throw down  \nyour weapons. O blood of my blood! Here is the man who will  \ntriumph over Corinth, slaughtering the men of Achaea, and will  \nride his chariot in triumph to the hill of the Capitol. Here is  \nthe man who will raze Argos and Agamemnon's Mycenae to  \nthe ground, and will kill Perseus the Aeacid, descendant of the  \n840 mighty warrior Achilles, avenging his Trojan ancestors and  \nthe violation of the shrine of Minerva. Who would leave you  \nunmentioned, great Cato? Or you, Cossus? Who would be  \nwithout the Gracchi? Or the two Scipios, both of them thunderbolts  \nof war, the bane of Libya? Or Fabricius, who will find  \npower in poverty? Or you, Serranus, sowing your seed in the  \nfurrow? Where are you rushing that weary spirit along to, you  \nFabii? You there are the great Fabius Maximus, the one man  \nwho restores the state by delaying. Others, I do not doubt it,  \nwill beat bronze into figures that breathe more softly. Others  \nwill draw living likenesses out of marble. Others will plead cases  \n850 better or describe with their rod the courses of the stars across  \nthe sky and predict their risings. Your task, Roman, and do not  \nforget it, will be to govern the peoples of the world in your  \nempire. These will be your arts \u2013 and to impose a settled pattern  \nupon peace, to pardon the defeated and war down the proud.'\n\nAeneas and the Sibyl wondered at what they heard, and Father  \nAnchises continued: 'Look there at Marcellus marching in glory  \nin spoils torn from the enemy commander he will fight and  \ndefeat. There he is, victorious and towering above all others.  \nThis is the man who will ride into battle and quell a great  \nuprising, steadying the ranks of Rome and laying low the  \nCarthaginian and the rebellious Gaul. He will be the third to  \ndedicate the supreme spoils to Father Quirinus.'\n\n860 At this Aeneas addressed his father, for he saw marching with  \nMarcellus a young man, noble in appearance and in gleaming  \narmour, but his brow was dark and his eyes downcast. 'Who is  \nthat, father, marching at the side of Marcellus? Is it one of his  \nsons or one of the great line of his descendants? What a stir his  \nescort makes! And himself, what a presence! But round his head  \nthere hovers a shadow dark as night.'\n\nThen his father Anchises began to speak through his tears: 'O  \nmy son, do not ask. This is the greatest grief that you and yours  \n870 will ever suffer. Fate will just show him to the earth \u2013 no more.  \nThe gods in heaven have judged that the Roman race would  \nbecome too powerful if this gift were theirs to keep. What a  \nnoise of the mourning of men will come from the Field of Mars  \nto Mars' great city. What a corte\u00e8ge will Tiber see as he glides  \npast the new Mausoleum on his shore! No son of Troy will ever  \nso raise the hopes of his Latin ancestors, nor will the land of  \nRomulus so pride itself on any of its young. Alas for his goodness!  \nAlas for his old-fashioned truthfulness and that right hand  \n880 undefeated in war! No enemies could ever have come against  \nhim in war and lived, whether he was armed to fight on foot or  \nspurring the flanks of his foaming warhorse. Oh the pity of it!  \nIf only you could break the harsh laws of Fate! You will be  \nMarcellus. Give lilies from full hands. Leave me to scatter red  \nroses. These at least I can heap up for the spirit of my descendant  \nand perform the rite although it will achieve nothing.'\n\nSo did they wander all over the broad fields of air and saw all  \nthere was to see, and after Anchises had shown each and every  \nsight to his son and kindled in his mind a love for the glory that  \n890 was to come, he told them then of the wars he would in due  \ncourse have to fight and of the Laurentine peoples, of the city of  \nLatinus and how he could avoid or endure all the trials that lay  \nbefore him.\n\nThere are two gates of sleep: one is called the Gate of Horn  \nand it is an easy exit for true shades; the other is made all in  \ngleaming white ivory, but through it the powers of the underworld  \nsend false dreams up towards the heavens. There on that  \nnight did Anchises walk with his son and with the Sibyl and  \nspoke such words to them as he sent them on their journey  \nthrough the Gate of Ivory.\n\n900 Aeneas made his way back to his ships and his comrades, then  \nsteered a straight course to the harbour of Caieta. The anchors  \nwere thrown from the prows and the ships stood along the  \nshore.\n\n## BOOK 7  \nWAR IN LATIUM\n\nYou too, Caieta, nurse of Aeneas, have given by your death  \neternal fame to our shores; the honour paid you there even now  \nprotects your resting-place, and your name marks the place  \nwhere your bones lie in great Hesperia, if that glory is of any  \nvalue.\n\nGood Aeneas duly performed the funeral rites and heaped up  \na barrow for the tomb, and when there was calm on the high  \nseas, he set sail and left the port behind him. A fair breeze kept  \nblowing as night came on, the white moon lit their course and  \n10 the sea shone in its shimmering rays. Keeping close inshore, they  \nskirted the land where Circe, the daughter of the Sun, lives  \namong her riches. There she sets the untrodden groves ringing  \nwith never-ending singing and burns the fragrant cedar wood  \nin her proud palace to lighten the darkness of the night as her  \nsounding shuttle runs across the delicate warp. From her palace  \ncould be heard growls of anger from lions fretting at their chains  \nand roaring late into the night, the raging of bristling boars and  \npenned bears and howling from huge creatures in the shape of  \n20 wolves. These had all been men, but with her irresistible herbs  \nthe savage goddess had given them the faces and hides of wild  \nbeasts. To protect the devout Trojans from suffering these monstrous  \nchanges, Neptune kept them from sailing into the harbour  \nor coming near that deadly shore. He filled their sails with  \nfavouring winds and took them past the boiling breakers to  \nsafety.\n\nAnd now the waves were beginning to be tinged with red  \nfrom the rays of the sun and Aurora on her rosy chariot glowed  \nin gold from the heights of heaven, when of a sudden the wind  \nfell, every breath was still and the oars toiled in a sluggish sea.  \n30 Here it was that Aeneas, still well off shore, sighted a great forest  \nand the river Tiber in all its beauty bursting through it into the  \nsea with its racing waves and their burden of yellow sand.  \nAround it and above it all manner of birds that haunted the  \nbanks and bed of the river were flying through the trees and  \nsweetening the air with their singing. Aeneas gave the order to  \nchange course and turn the prows to the land, and he came into  \nthe dark river rejoicing.\n\n40 Come now, Erato, and I shall tell of the kings of ancient  \nLatium, of its history, of the state of this land when first the  \narmy of strangers beached their ships on the shores of Ausonia.  \nI shall recall too, the cause of the first battle \u2013 come, goddess,  \ncome and instruct your prophet. I shall speak of fearsome fighting,  \nI shall speak of wars and of kings driven into the ways of  \ndeath by their pride of spirit, of a band of fighting men from  \nEtruria and the whole land of Hesperia under arms. For me this  \nis the birth of a higher order of things. This is a greater work I  \nnow set in motion.\n\nKing Latinus was by this time an old man and he had reigned  \nover the countryside and the cities for many peaceful years. We  \nare told that he was the son of Faunus and the Laurentine nymph  \nMarica. The father of Faunus was Picus, and the father claimed  \nby Picus was Saturn. Saturn then was the first founder of the  \n50 line. By divine Fate Latinus had no male offspring. His son had  \nbeen snatched from him as he was rising into the first bloom of  \nhis youth. An only daughter tended his home and preserved  \nthe succession for this great palace. She was now grown to  \nwomanhood and at the age for marriage and many were seeking  \nher hand from great Latium and the whole of Ausonia, Turnus  \nthe handsomest of them all, his claim supported by the long line  \nof his forebears. The queen Amata longed above all things to  \nsee him married to her daughter, but many frightening portents  \nfrom the gods forbade it.\n\n60 Deep in the innermost courtyard of the palace there stood a  \nlaurel tree. Its foliage was sacred and it had been preserved and  \nheld in awe for many years, ever since Father Latinus himself  \nhad found it, so the story went, when he was building his first  \ncitadel, and dedicated it to Phoebus Apollo, naming the settlers  \nafter it, the Laurentines. To this tree there came by some miracle  \na cloud of bees, buzzing loudly as they floated through the liquid  \nair till suddenly they formed a swarm and settled on its very top,  \nhanging there from a leafy branch with their feet intertwined. A  \nprophet thus interpreted: 'What we see is a stranger arriving,  \n70 and an army coming from the same direction, making for the  \nsame place and gaining mastery over the heights of the citadel.'  \nThen again when Lavinia was standing by her father's side  \ntending the altar with her chaste torches, another fearful sight  \nwas seen. Her long hair caught fire and all its adornment was  \ncrackling in the flames. The princess's hair was blazing, her  \ncrown with all its lovely jewels was blazing, and soon she was  \nwrapped in smoke and a yellow glare, and scattering fire all  \nover the palace. The horror and miracle of it were on everyone's  \n80 lips, and it was prophesied that her own fate and fame would  \nbe bright, but that a great war would come upon the people.\n\nTroubled by such portents, the king consulted the oracle of  \nhis prophetic father Faunus, visiting the grove under Mount  \nAlbunea, a huge forest sounding with the waters of its sacred  \nfountain and breathing thick clouds of sulphurous vapour. Here  \nthe Italian tribes and the whole land of Oenotria came to consult  \nthe oracle in their times of doubt. Here the priest brought his  \nofferings, and when he lay down to sleep in the silence of the  \nnight on a bed of the fleeces of slaughtered sheep, he would see  \n90 many strange fleeting visions, hear all manner of voices, enjoy  \nthe converse of the gods and speak to Acheron in the depths of  \nAvernus. Here too on that day Father Latinus himself came to  \nconsult the oracle, and after sacrificing a hundred unshorn  \nyearling sheep as ritual prescribes, he was lying propped on a  \nbed of their hides and fleeces, when suddenly a voice was heard  \nfrom the depths of the forest: 'Do not seek to join your daughter  \nin marriage to a Latin. O my son, do not place your trust in  \nany union that lies to hand. Strangers will come to be your  \nsons-in-law and by their blood to raise our name to the stars.  \n100 The descendants of that stock will see the whole world turning  \nunder their feet and guided by their will, from where the rising  \nSun looks down on the streams of Ocean to where he sees them  \nas he sets.' This was the reply of his father Faunus, the warning  \nthat came in the silence of the night. Latinus did not keep it  \nlocked in his heart, and Rumour as she flew had already spread  \nit far and wide through the cities of Ausonia when the young  \nwarriors from Laomedon's Troy tied up their ships to the grassy  \nramparts of the river bank.\n\nAeneas, the leading captains of Troy and lovely Iulus had lain  \ndown on the grass under the branches of a tall tree and were  \n110 starting to eat a meal, setting out their banquets on wheaten  \ncakes \u2013 for Jupiter himself had so advised them \u2013 and heaping  \ncountry fruits on these foundations, the gift of Ceres, the Goddess  \nof Grain. When the fruit had all been eaten and the sparseness  \nof the diet had driven them to sink their teeth into Ceres'  \nbounty, scant as it was, to violate with bold hand and jaw the  \nfateful circles of crust and show no mercy to the flat quartercircles  \nof bread, suddenly Iulus said, as a joke: 'Look! We  \nare eating even our tables!' That was all. This was the first  \nannouncement they had received of the end of their sufferings.  \nAstounded by the presence of the divine, Aeneas seized upon  \nhis son's first words while he was still speaking and made him  \n120 be silent. In that instant he lifted up his voice and cried out:  \n'Hail to the land owed to me by the Fates, and hail to the  \nhousehold gods of Troy who have kept faith with me! This is  \nour home. This is our own land. For now I remember it, my  \nfather Anchises left me this riddle of the Fates. \"When you sail  \nto an unknown shore and your food is so scanty that hunger  \nforces you to eat your tables, that is the time, weary as you are,  \nto hope for a home. This is where you must with your own hand  \nlay down the foundations of your first buildings and raise a  \nrampart round them.\" This is the hunger of which he spoke.  \nThis is the last hunger we had to endure and it will put an end  \n130 to our calamities. Come then, with joy in your hearts, and at  \nthe first light of the sun let us all go in different directions from  \nthe harbour to explore this place and find out who are the men  \nthat live here and where their cities are. And now pour libations  \nfrom your goblets to Jupiter, call upon my father Anchises with  \nyour prayers and set the wine in due order on the tables.'\n\nAt these words he wound a branch of living green round his  \nforehead and offered up prayers to the Genius of the place and  \nto Earth, the first of gods, to nymphs and rivers not yet known,  \nthen to Night and the stars of Night then rising, to Jupiter of  \nMount Ida and the Phyrygian Mother in due order, to his mother  \n140 in the heavens and his father in Erebus. In reply the All-powerful  \nFather thundered clear three times from the heights of the sky  \nand with his own hand he displayed in heaven a burning cloud,  \nquivering with rays of golden light. In that instant the word  \nspread through the Trojan ranks that the day had come for them  \nto found their promised city. Eagerly they renewed their feast,  \nand delighting in this great omen, they set up their mixing bowls  \nand crowned the wine with garlands.\n\nWhen the next day first rose and began to traverse the earth  \n150 with its lamp, they set out in different directions to explore the  \ncity and the boundaries and shores of this people. Here were  \nthe pools where the river Numicus springs, here was the river  \nTiber and here were the homes of the stalwart Latins. Then  \nAeneas, son of Anchises, ordered one hundred men chosen as  \nspokesmen from every rank of his people to go to the sacred  \nwalls of king Latinus all bearing branches of Pallas Athene's  \nolive wreathed in wool, carrying gifts and asking for peace for  \nthe Trojans. They made no delay, but hastened with all speed  \nto do as they were bidden, while Aeneas himself was marking  \nout the line of his walls with a shallow ditch and beginning to  \nbuild on the site, surrounding this first settlement on the shore  \n160 with a stockade and rampart as though it were a camp. The  \nwarriors, meanwhile, their long journey ended, were within  \nsight of the towers and high roofs of the city of the Latins and  \ncame up to the wall. There in front of the city boys and young  \nmen in the first flower of their age were exercising with their  \nhorses, training chariot teams in clouds of dust, bending the  \nspringy bow, spinning the stiff-shafted javelin, racing and sparring,  \nwhen a messenger riding ahead of the Trojans brought to  \nthe ear of the old king the news that huge men in strange costume  \nhad arrived. Latinus ordered them to be summoned into his  \npalace while he took his seat in the middle on his ancestral  \nthrone.\n\n170 A sacred building, massive and soaring to the sky with a  \nhundred columns, stood on the highest point of the city. This  \nwas the palace of Laurentine Picus, a building held in great awe  \nbecause of an ancestral sense of the presence of the divine in the  \ngrove that surrounded it. Here the omens declared that kings  \nshould receive their sceptres and take up the rods of office for  \nthe first time. This temple was their senate-house, this the hall  \nin which they held their sacred banquets and here the elders  \nwould sacrifice a ram and sit down to feast at long tables. Here  \ntoo, carved in old cedar wood, stood in order in the forecourt  \nthe statues of their ancestors from time long past: Italus and  \nFather Sabinus, planter of the vine, still holding in effigy his  \n180 curved pruning knife, old Saturn, the image of Janus with his  \ntwo faces, all the other kings since the foundation of the city  \nand with them the men who had been wounded while fighting  \nto defend their native land. Many too were the weapons hung  \non the posts of the temple doors, captured chariots, curved axes,  \ncrests of helmets, great bolts from the gates of cities, spears,  \nshields and beaks broken off the prows of ships. Here too, with  \nhis short toga, and the augural staff of Quirinus in his left hand,  \nsat the Horse-Tamer, Picus himself, whose wife Circe, possessed  \n190 by lust, struck him with her rod of gold and changed him with  \nher potions into a bird, sprinkling colours on his wings.\n\nSuch was the temple of the gods where Latinus sat in the seat  \nof his fathers and called the Trojans to him in his palace. When  \nthey entered he was the first to speak, addressing them in these  \nkindly words: 'Tell me, sons of Dardanus \u2013 you see we know  \nyour city and your family and had heard about you before you  \nset your course here \u2013 what are you searching for? What has  \ntaken your ships over all the blue waters of ocean to the shore  \nof Ausonia? What need has brought you here? Whether you  \nhave lost your way or been driven off course by the storms that  \n200 sailors have to endure so often on the high seas, you have now  \nsailed between the banks of our river and are sitting in harbour.  \nDo not refuse the guest-friendship we offer you and do not  \nforget that we Latins are Saturn's people, righteous not because  \nof laws and restraints but holding of our own free will to the  \nway of life of our ancient god. Besides, I myself remember that  \nthe Auruncan elders used to say \u2013 the story is dimmed by the  \nmists of time \u2013 that Dardanus was born in these fields and went  \nfar away to the cities of Ida in Phrygia and the Thracian island  \nof Samos now known as Samothrace. He set out from here,  \n210 from his Tyrrhenian home in Corythus, and now sits on a throne  \nin the palace of gold in the starry sky, and his altars add a name  \nto the roll of the gods.'\n\nHe spoke these words, and these were the words in which  \nIlioneus made answer: 'Great king, son of Faunus, it is not black  \nstorms and heavy seas that have driven us to this land of yours,  \nnor have we lost our way by mistaking a star or a coastline. It  \nis by design and with willing hearts that we all sail to this city,  \ndriven from our own kingdom which was once the greatest the  \njourneying Sun could see from the highest part of the heavens.  \n220 Our race begins with Jupiter. The warriors of Dardanus' Troy  \nrejoice in Jupiter as their ancestor. Their king, Aeneas himself,  \nis descended from Jupiter's exalted stock, and Trojan Aeneas  \nhas sent us to your door. The storm that gathered in merciless  \nMycenae and swept across the plains beneath Mount Ida, and  \nthe fate that drove the worlds of Europe and of Asia to collide,  \nthese are known to all men, those who live far to the north  \nwhere the ends of the earth beat back the stream of Oceanus,  \nand those who are separated from us by the zone of the cruel  \nsun whose expanse covers the middle zone of five. Since that  \ncataclysm we have sailed all those desolate seas, and now we  \n230 ask for a little piece of land for our fathers' gods, for harmless  \nrefuge on the beach, for the air and sea which are there for all  \nmen. We shall not bring discredit on your kingdom. Great fame  \nwill be yours, and our gratitude for such a service will never  \nfade. The men of Ausonia will never regret taking Troy to their  \nhearts. I swear by the destiny of Aeneas and his right arm, strong  \nin the truth to all who have tested it, and strong in war and the  \nweapons of war, that many nations have asked to enter into  \nalliance with us. Do not despise us because we choose to come  \nto you with words of supplication and olive branches wreathed  \nin wool in our hands. Many races have wished to be joined to  \n240 ours, but the commands of divine destiny have driven us to seek  \nout your country. This was the first home of Dardanus. This is  \nthe land to which Apollo calls us back, and urges us with his  \nmighty decrees towards the Tyrrhenian Thybris and the sacred  \nshallows of the fountain of Numicus. These gifts, besides,  \nAeneas offers you, some small relics of his former fortunes  \nrescued from the flames of Troy. From this gold cup his father  \nAnchises used to pour libations at the altar. This was the sceptre  \nPriam would hold in his hand as he gave solemn judgement  \nbefore the concourse of the nations, and here are his sacred  \nhead-dress and the vestments woven for him by the women  \nof Troy.'\n\n250 When Ilioneus had finished speaking, Latinus kept his gaze  \nfixed upon the ground and did not move. He never raised his  \nburning eyes but they were never still. As a king he was moved  \nto see the sceptre of Priam and his embroidered purple but much  \nmore was he moved by the thought of a marriage and a husband  \nfor his daughter, and long did he ponder in his heart the prophecy  \nof old Faunus. So this was the fulfilment of the portents sent  \nby the Fates! So this was the son-in-law who would come from  \na distant land and be called to share his kingdom with equal  \nauspices. This was the man whose descendants would excel in  \nvalour and whose power would win the whole world. He spoke  \nat last, and joyfully: 'May the gods give their blessing to what  \n260 we begin today and to their own prophecies! You will receive  \nwhat you ask, Trojan, and I do not refuse your gifts. While  \nLatinus is king, you will have rich land to farm and you will  \nnever feel the lack of the wealth of Troy. Only Aeneas must  \ncome here himself if he is so eager and impatient to join us in  \nfriendship and be called our ally. He has no need to recoil from  \nthe face of his friends. It will be a condition of the peace I offer  \nthat I must clasp the hand of your king. But now I charge you  \nto take back this answer to him. Tell him I have a daughter, and  \n270 the oracles from my father's shrine agree with all the signs from  \nheaven in forbidding me to join her in marriage to any man of  \nour people. Strangers will come from a foreign land to be my  \nsons-in-law \u2013 this is what is in store for Latium according to the  \nprophecies \u2013 and by their blood they shall raise our name to the  \nstars. This Aeneas is the man the Fates demand. This I believe,  \nand this is my will, if my mind has any true insight into the  \nfuture.' After these words, Father Latinus made a choice from his  \nwhole stable where three hundred well-groomed horses stood in  \ntheir high-built stalls, ordering one to be brought out instantly  \nfor each of the Trojans in due order. Their hooves were swift as  \nwings, their saddle-cloths were of embroidered purple. Gold  \nmedallions hung at their breasts, their caparisons were of gold  \n280 and they champed bright golden bits between their teeth. For  \nAeneas in his absence, he chose a chariot and pair of heavenly  \ndescent breathing fire from their nostrils. They were sprung  \nfrom a stock which cunning Circe had crossbred by stealing one  \nof the stallions of her father the Sun to mate with a mare. With  \nthese gifts Aeneas' men returned, riding high in the saddle and  \nbringing messages of peace.\n\nBut at that very moment fierce Juno, wife of Jupiter, was  \ncoming back from Argos, city of Inachus, holding her course  \nthrough the winds of the air, when from far away in the heavens,  \nas far as Cape Pachynus in Sicily, she caught sight of the jubilant  \n290 Aeneas and his Trojan fleet. When she saw that they were  \nalready at work on their buildings, having abandoned their  \nships and committed themselves to the land, she stopped in  \nmid-flight, pierced by bitter resentment. Then, shaking her head,  \nshe poured out these words from the depths of her heart: 'A  \ncurse on that detested race of Phrygians and on their destiny, so  \nopposed to our own! Could they not have died on the Sigean  \nplains? They were defeated. Why could they not accept defeat?  \nTroy was set alight. Could they not have burned with it? But  \nno! They found a way through the press of the battle and the  \nthick of the flames. They must think my divine powers are  \nexhausted and discredited, or that I have glutted my appetite  \nfor hatred and am now at peace. After all, when they were cast  \n300 out of their native land, I dared to hound them over the waves  \nand wherever they ran across the face of the ocean I was there  \nand set my face against them. I have used every resource of  \nsea and sky against these Trojans, and what use have the Syrtes  \nbeen to me? Or Scylla? Or the bottomless Charybdis? The  \nTrojans are where they wanted to be in the valley of the Thybris,  \nsafe from the sea and safe from me. Mars had the strength to  \ndestroy the monstrous race of Lapiths. The Father of the Gods  \nhimself handed over the ancient kingdom of Calydon to the  \nwrath of Diana, and what great crime had the Lapiths or  \nCalydon committed? But here am I, great Juno, wife of Jupiter,  \nthwarted, though I have tried everything that could be tried.  \n310 Nothing has been too bold for me. And I am being defeated by  \nAeneas! But if my own resources as a goddess are not enough, I  \nam not the one to hesitate. I shall appeal to whatever powers  \nthere are. If I cannot prevail upon the gods above, I shall move  \nhell. I cannot keep him from his kingdom in Latium: so be it.  \nThe decree of the Fates will stand and he will have Lavinia to  \nwife. But I shall be able to delay it all and drag it out, I shall be  \nable to cut the subjects of both those kings to pieces. This will  \nbe the cost of the meeting between father-in-law and son-in-law,  \nand their peoples will bear it. Your dowry, Lavinia, will be the  \nblood of Rutulians and Trojans, and your matron-of-honour  \nwill be the Goddess of War herself, Bellona. Hecuba, daughter  \n320 of Cisseus, was pregnant with a torch and gave birth to the  \nmarriage torches of Paris and Helen. But she is not alone. Venus,  \ntoo, has a son, a second Paris, and torches will again be fatal,  \nfor this second Troy.'\n\nWith these words the fearsome goddess flew down to the  \nearth and roused Allecto, bringer of grief, from the infernal  \ndarkness of her home among the Furies. Dear to her heart were  \nthe horrors of war, anger, treachery and vicious accusations.  \nHer own father Pluto hated his monstrous daughter. Her own  \nsisters in Tartarus loathed her. She had so many faces and such  \nfearsome shapes, and her head crawled with so many black  \n330 serpents. This was the creature Juno now roused to action with  \nthese words: 'Do this service for me, O virgin daughter of Night.  \nIt is a task after your own heart. See to it that my fame and the  \nhonour in which I am held are not impaired or slighted, and see  \nto it that Aeneas and his men do not win Latinus over with their  \noffers of marriage and are not allowed to settle on Italian soil.  \nYou can take brothers who love each other and set them at each  \nother's throats. You can turn a house against itself in hatred and  \nfill it with whips and funeral torches. You have a thousand  \nnames and a thousand ways of causing hurt. Your heart is  \nteeming with them. Shake them all out. Shatter this peace they  \nhave agreed between them and sow the seeds of recrimination  \n340 and war. Make their young men long for weapons, demand  \nthem, seize them!'\n\nIn that moment Allecto, gorged with the poisons of the Gorgons,  \nwent straight to Latium and the lofty palace of the king  \nof the Laurentines and settled on the quiet threshold of the  \nchamber of Amata. There the queen was seething with womanly  \nanger and disappointment at the arrival of the Trojans and the  \nloss of the wedding with Turnus. Taking one of the snakes from  \nher dark hair, the goddess Allecto threw it on Amata's breast to  \nenter deep into her heart, a horror driving her to frenzy and  \n350 bringing down her whole house in ruin. It glided between her  \ndress and her smooth breasts and she felt no touch of its coils.  \nWithout her knowing it, it breathed its viper's breath into her  \nand made her mad. The serpent became a great necklace of  \ntwisted gold round her neck. It became the trailing end of a long  \nribbon twined round her hair. It slithered all over her body.  \nWhile the first infection of the liquid venom was still oozing  \nthrough all her senses and winding the fire about her bones,  \nbefore her mind in her breast had wholly consumed the fever of  \nit, she spoke with some gentleness, as a mother might, and wept  \nbitterly over the marriage of her daughter to a Phrygian: 'Is  \n360 Lavinia being given in marriage to these Trojan exiles? You are  \nher father. Have you no feelings for your daughter or her mother  \nor yourself? When the first wind blows from the north, that  \nlying brigand will take to the high seas and carry off my daughter,  \nleaving me desolate. Is this not how the Phrygian shepherd  \nwormed his way into Sparta and carried Leda's daughter Helen  \noff to the cities of Troy? Where is your sacred word of honour?  \nWhere is the care you used to have for your kinsmen? And what  \nof all the pledges you have given Turnus, your own flesh and  \nblood? But if you are searching for a son-in-law among strangers  \nand that is decided, if the commands of your father Faunus  \n370 weigh so heavily upon you, then I maintain that all peoples who  \nare not subject to our sceptre are strangers. That is what the  \ngods are saying. Besides, if you were to trace the house of Turnus  \nback to its first beginnings, his forefathers were Inachus and  \nAcrisius of Argos and his home is in the heart of Mycenae.'\n\nWhen with these words she had tried in vain to move Latinus  \nand seen that he held firm, when the maddening poison of the  \nserpent had soaked deep into her flesh and oozed all through  \nher body, the unhappy Amata, driven out of her mind by her  \nmonstrous affliction, raged in a wild frenzy through the length  \nand breadth of the city like a spinning top flying under the  \n380 plaited whip when boys are engrossed in their play and make it  \ngo in great circles round an empty hall; the whip drives it on its  \ncurved course and the boys look down, puzzled and fascinated  \nas they lash the spinning boxwood into life \u2013 as swift as any top  \nAmata ran through the middle of the cities of the fierce Latian  \npeople. Not content with this, she flew into the forests, pretending  \nthat she was possessed by Bacchus, and rose to greater  \nimpieties and greater madness by hiding her daughter in the  \nleafy woods, hoping to cheat the Trojans out of the marriage or  \n390 delay the lighting of the torches. 'Euhoe, Bacchus!' she screamed.  \n'Only you are worthy of the virgin. For you she takes up the  \nsoft-leaved thyrsus. Round you she moves in ritual dance. She  \ngrows her hair to consecrate it to you.' Rumour flew fast. The  \nsame passion kindled in the hearts of all the mothers of Latium  \nand drove them out to search for new homes. They left their  \nhouses, their throats bare and their hair streaming in the winds.  \nOthers, clad in animal skins and carrying vine shoots sharpened  \ninto spears, made the heavens ring with whimpering and wailing.  \nAmata herself, in the fever of her madness, held high a  \nburning torch in the midst of them and sang a wedding hymn  \nfor Turnus and her daughter, rolling her bloodshot eyes. Suddenly  \n400 she gave a dreadful cry: 'Io, Io, all you mothers of Latins  \nwherever you may be, if in your faithful hearts there remains  \nany regard for unhappy Amata, if your minds are troubled by  \nthe thought of what is due to a mother, untie the ribbons of  \nyour hair and take to the secret rites with me.' This, then, was  \nthe queen whom Allecto drove with the lash of Bacchus through  \nthe forests and the desolate haunts of wild beasts.\n\nAfter she saw that this first madness was well under way, and  \nthat she had subverted Latinus' plans and all his house, the  \ndeadly goddess rose on her dark wings and flew straight to the  \nwalls of the bold prince of the Rutulians. Danae is said to  \nhave been driven on to this coast by southern gales and to have  \n410 founded this city for settlers who were subjects of her father  \nAcrisius, king of Argos. Our ancestors long ago gave it the name  \nof Ardea, and Ardea still keeps its great name though its fortune  \nlies in the past. Here in his lofty palace in the darkness of  \nmidnight Turnus was lying deep in sleep. Allecto changed her  \nappearance. No longer wild and raving, she took on the face of  \nan old woman, with her brow furrowed by horrible wrinkles  \nand her white hair tied in a sacred ribbon and bound in a chaplet  \nof olive leaves. She became Calybe, the aged priestess of Juno  \n420 and her temple, and appeared before the eyes of young Turnus  \nsaying: 'Are you going to stand by and see all your labours go  \nfor nothing, Turnus, and your crown made over to these  \nincomers from Troy? The king is refusing to give you the marriage  \nand the dowry you have earned in blood and is searching  \nfor a stranger to inherit his kingdom. So now, Turnus, go and  \nexpose yourself to danger! Your reward is to be laughed at. Go  \nand cut down these Etruscans in their battle lines! Go and cover  \nthe Latins with the shield of peace. These are the very words  \nwhich the daughter of Saturn, All-powerful Juno, has commanded  \nme to say and say clearly to you as you lie in the peace  \n430 of night. So up with you, and with a light heart prepare to arm  \nyour young warriors and move them from inside the city gates  \nand out to the fields to burn the Phrygian captains and their  \npainted ships where they have made themselves at home on our  \nlovely river. The mighty power of heaven demands it. If king  \nLatinus does not agree to obey this command and allow you  \nthis marriage, he must learn, he must in the end face Turnus  \nwith his armour on.'\n\nTurnus was laughing as he made his reply to the priestess:  \n'You are wrong. The report has not failed to reach my ears. I  \nknow a fleet has sailed into the waters of the Thybris. Do not  \ninvent these fears for me. Royal Juno has not entirely forgotten  \n440 us. It is old age and decay that cause you all this futile agitation  \nand distress and make you barren of truth, taking a prophetess  \namong warring kings and making a fool of her with false fears.  \nYour duty is to guard the statues of the gods and their temples.  \nLeave peace and war to men. War is the business of men.'\n\nWhen she heard the warrior's words Allecto burst into blazing  \nanger, and while he was still replying, a sudden trembling came  \nover his limbs and the eyes stared in his head as the Fury revealed  \nherself in her full size and set all her hydras hissing. As he  \n450 faltered and tried to go on speaking, she flung him back with  \nher eyes flashing fire, two snakes stood up on her head and she  \ncracked her whips as she spoke again from her now maddened  \nlips: 'So I am old and decayed and barren of truth and old age  \nis taking me among warring kings and making a fool of me with  \nfalse fears! Have a look at these! I come here from the home of  \nthe dread Furies, my sisters, and in my hands I carry war and  \ndeath.'\n\nWith these words she threw a burning torch at the warrior  \nand it lodged deep in his heart, smoking with black light. A  \ngreat terror burst in upon his sleep, and the sweat broke out all  \n460 over his body and soaked him to the bone. In a frenzy of rage  \nhe roared for his armour. 'My armour!' he shouted, ransacking  \nhis bed and the whole palace for it. The lust for battle raged  \nwithin him, the criminal madness of war and, above all, anger.  \nIt was as though a heap of brushwood were crackling and  \nburning under the sides of a bronze vessel, making the water  \nseethe and leap up, a great river of it raging in the pot, with  \nboiling foam spilling over and dense steam flying into the air.  \nThe peace was violated. Turnus gave orders to the leaders of his  \narmy to march to king Latinus, to prepare for war, to defend  \nItaly and thrust the enemy out of its borders. When he arrived,  \n470 that would be enough for the Trojans, and enough for the  \nLatins. These were his words and he called upon the gods to  \nwitness them. The eager Rutulians urged each other to arms,  \nsome of them inspired by the rare grace of his youthful beauty,  \nsome by the long line of kings that were his ancestors, some by  \nhis brilliant feats of arms.\n\nWhile Turnus was filling the hearts of the Rutulians with  \nboldness, Allecto flew off with all speed to the Trojans on her  \nwings of Stygian black. Here, spying out the ground where  \nlovely Iulus was hunting along the shore, trapping and coursing,  \nshe hatched a new plot. Into his hounds the virgin goddess of  \n480 Cocytus put a sudden fit of madness by touching their nostrils  \nwith the familiar scent of a stag and sending them after it in full  \ncry. This was the first cause of all the suffering. It was this that  \nkindled the zeal for war in the hearts of the country people. It  \nwas a huge and beautiful stag with a fine head of antlers, which  \nhad been torn from the udders of its mother and fed by Tyrrhus  \nand his young sons \u2013 Tyrrhus looked after the royal herds and  \nwas entrusted with the wardenship of the whole broad plain.  \nSilvia, the boys' sister, had given this wild creature every care  \nand trained it to obey her. She would weave soft garlands for  \n490 its horns, combing and washing it in clear running water. It  \nbecame tame to the hand and used to come to its master's table.  \nIt would wander through the woods and come back home of its  \nown accord to the door it knew so well, no matter how late the  \nnight. This is the creature that was roaming far from home,  \nfloating down a river, cooling itself in the green shade of the  \nbank when it was startled by the maddened dogs of the young  \nhuntsman Iulus. He himself, Ascanius, burning with a passionate  \nlove of glory, bent his bow and aimed the arrow. The god  \nwas with him and kept his hand from erring. The arrow flew  \nwith a great hiss and passed straight through the flank into the  \n500 belly. Fleeing to the home it knew so well, the wounded stag  \ncame into its pen moaning, and stood there bleeding and filling  \nthe house with its cries of anguish, as though begging and  \npleading. Silvia was the first to call for help. She beat her own  \narms in grief and summoned the country people, who came long  \nbefore she expected them, for savage Allecto was lurking in the  \nsilent woods. Some came armed with stakes burned to a point  \nin the fire; some with clubs made from knotted tree trunks; each  \nman searched for what he could find and anger taught him how  \nto make a weapon of it. Tyrrhus was calling up the troops. He  \n510 had been driving in wedges to split an oak into four and he  \nsnatched up his axe, breathing furiously.\n\nThe cruel goddess saw from her vantage point that this was a  \nmoment when harm might be done and, flying to the top of the  \nfarm roof, from the highest gable she sounded the herdsman's  \nsignal with a loud call on the curved horn, and its voice was the  \nvoice of Tartarus. The trees shivered at the noise and the whole  \nforest rang to its very depths. Far away the lake of Trivia heard  \nit. The white sulphur-laden streams of the river Nar heard it  \nand its springs in Lake Velinus, and terrified mothers pressed  \n520 their babies to their breasts. Swift to answer the call of that  \ndread horn, the hardy countrymen snatched up their weapons  \nand gathered from every side. The Trojans, for their part, opened  \nthe gates of their camp and streamed out to help Ascanius. They  \ndrew up in line of battle, and this was no longer a village brawl  \nwith knotted clubs and stakes sharpened in the fire. They fought  \nwith two-edged steel, and a dark crop of drawn swords sprouted  \nall over the field while bronze gleamed in answer to the challenge  \nof the sun and threw its light up to the clouds, like the sea  \nwhitening at the first breath of wind and slowly stirring itself,  \n530 raising its waves higher and higher till it reaches from the depths  \nof the sea-bed to the heights of heaven. Suddenly there was the  \nhiss of an arrow and a young man standing out in front of the  \nleading line of battle fell to the ground. It was Almo, the eldest  \nson of Tyrrhus. The shaft had stuck deep in his throat, blocking  \nthe moist passage of the voice and closing off the narrow channel  \nof his life in blood. The bodies of slain men soon lay around  \nhim, among them old Galaesus, who died when he stepped  \nbetween the armies to make peace. He was the justest man in  \nthe broad fields of Ausonia in these far days, and the richest.  \nFive flocks of sheep and five herds of cattle came back at evening  \nto his stalls and he turned the soil with a hundred ploughs.\n\n540 While the battle was evenly poised on the plain, the mighty  \ngoddess, having fulfilled her promise when the first blood was  \nspilt in war and the first clash of arms had led to death, left  \nHesperia and returned through the breezes of the sky to address  \nJuno in these words of proud triumph: 'You asked and I have  \ngiven. Discord is made perfect in the horror of war. Now tell  \nthem to come together and form alliances when I have sprinkled  \nthe Trojans with Italian blood! And I shall do more than this, if  \nsuch be still your will for me. I shall spread rumours to draw  \n550 the neighbouring cities into the war. I shall set their hearts ablaze  \nwith a mad lust for battle and they will come from all sides to  \njoin in the fray. I shall sow a crop of weapons in all their fields.'  \nJuno gave her answer: 'There is enough terror and lying. The  \ncauses of war are established. They are fighting at close quarters  \nand fresh blood is staining whatever weapons chance first puts  \ninto their hands. Let this be the wedding they will celebrate, the  \nnoble son of Venus and great king Latinus. Let this be their  \nwedding hymn. The Father of the Gods, the ruler of high  \nOlympus, would not wish you to rove too freely over the breezes  \nof heaven. You must withdraw. Should there be any need for  \n560 further effort, I shall take the guidance into my own hands.' No  \nsooner had the daughter of Saturn spoken these words than  \nAllecto lifted up her wings, hissing with snakes, and flew down  \nto her home on the banks of the Cocytus, leaving the steeps of  \nthe sky. At the foot of high mountains in the middle of Italy,  \nthere is a well-known place, whose fame has spread to many  \nlands, the valley of Amsanctus. A dark forest presses in upon it  \nfrom both sides with its dense foliage and in the middle a  \ncrashing torrent roars over the rocks, whipping up crests of  \nfoam. Here they point to a fearful cave which is a vent for the  \nbreath of Dis, the cruel god of the underworld. Into this cave  \n570 bursts Acheron and here a vast whirlpool opens its pestilential  \njaws, and here the loathsome Fury disappeared, lightening  \nheaven and earth by her absence.\n\nBut none the less the Queen of the Gods, the daughter of  \nSaturn, was at that moment putting the finishing touches to  \nthe war. A whole crowd of herdsmen came rushing from the  \nbattlefield into the city, carrying the bodies of young Almo and  \nGalaesus with his face mutilated. They were all imploring the  \nhelp of the gods and appealing to Latinus. Turnus was there,  \nand when the fire of their fury and the accusations of murder  \nwere at their height, he heaped fear upon fear by claiming that  \nthe Trojans were being invited to take a share in the kingdom;  \ntheir own Latin blood would be adulterated by Phrygians while  \nhe was being turned from the door. At this there gathered from  \nall sides, wearying Mars with their clamour for war, those whose  \n580 mothers had been crazed by Bacchus and were now dancing in  \nwild rout in the pathless forests \u2013 the name of Amata had great  \nweight with them. In an instant they were all demanding this  \nwicked war against all the omens, against divine destiny and  \ncontrary to the will of the gods. They rushed to besiege the  \npalace of king Latinus, who stood unmoved like a rock in the  \nocean, like a solid rock in the ocean pounded by breakers,  \nstanding fast with the waves howling round it, while reefs and  \n590 foam-soaked scars roar in helpless anger and the seaweed is  \nforced against its side, then streams back with the undertow.  \nBut there was no resisting the counsels of blind folly. All things  \nwere taking their course according to the nod of savage Juno.  \nAgain and again the king, the father of his people, called upon  \nthe gods and the empty winds to witness: 'We are caught in the  \ngale of Fate,' he cried. 'Our ship is breaking under us. You, my  \npoor people, will pay for this sacrilege with your blood. You  \nare the guilty one, Turnus, and a grim punishment lies in store  \nfor you. You will supplicate the gods but your prayers will be  \ntoo late. I have already reached calm water and here at the  \nharbour mouth I lose all the happiness I might have had in the  \nhour of my death.' He said no more, but shut himself away in  \n600 his palace and gave up the reins of power.\n\nIn Hesperia, in the lands of Latium, there was a custom, later  \ninherited and revered in the cities of Alba, and now observed by  \nRome, the greatest of the great, when men first rouse Mars for  \nbattle, whether they are preparing to bring the sorrows of war  \nto the Getae, the Hyrcani or the Arabs, or whether they are  \nheading for India and the rising of the sun and reclaiming the  \nstandards from the Parthians. There are two gates known as the  \nGates of War, sanctified by religion and the fear of savage Mars.  \nThese gates are closed by a hundred bolts of bronze and the  \n610 everlasting strength of iron, nor does their sentry Janus ever  \nleave the threshold. When the Fathers are resolved on war, the  \nconsul himself, conspicuous in the short toga of Quirinus girt  \nabout him in the Gabine manner, unbars the doors. They grind  \nin their sockets and he summons war. The whole army takes up  \nthe call and the bronze horns breathe their shrill assent. So too  \nin those days Latinus was bidden to declare war upon the men  \nof Aeneas by opening these grim gates. The old king, father of  \nhis people, would not lay his hand upon them, but recoiled from  \nthis wickedness and refused to perform the task, shutting himself  \n620 up in the darkness away from the sight of men. At this, the  \nQueen of the Gods came down from the sky and struck the  \nstubborn doors, bursting the iron-bound Gates of War and  \nturning them in their sockets. Till that moment Ausonia had  \nbeen at peace and unalarmed, but now the foot-soldiers  \nmustered on the plain and high in the saddle came the excited  \nhorsemen stirring up the dust. Every man was looking for  \nweapons, polishing shields with rich fat till they were smooth,  \nburnishing spears till they shone and grinding axes on the whetstone.  \nWhat joy to raise the standards and hear the trumpets  \n630 sound! Five great cities, no less, set up anvils to forge new  \nweapons, mighty Atina, proud Tibur, Ardea, Crustumerium  \nand Antemnae with its towers. They hollowed out helmets to  \nprotect the heads of warriors. They wove frames of willow  \nshoots to form shields. They made bronze breastplates and  \nsmooth shields of ductile silver. This is what had become of all  \ntheir regard for the sickle and the share. This is what had become  \nof all their love for the plough \u2013 the swords of their fathers were  \nnow retempered in the furnace. Now the trumpets blew and out  \nwent the signal that called them to war. In high excitement they  \ntore down their helmets from the roof, yoked their trembling  \n640 horses to the chariot, buckled on their shields and their breastplates  \nof triple-woven gold and girt their trusty swords about  \nthem.\n\nNow goddesses, it is time to open up Mount Helicon, to set  \nyour songs in motion and tell what kings were roused to war,  \nwhat armies followed each of them to fill the plains, the heroes  \nthat flowered and the weapons that blazed in those far-off days  \nin the bountiful land of Italy. You are the divine Muses. You  \nremember, goddesses, and can utter what you remember. Our  \nears can barely catch the faintest whisper of the story.\n\nThe first to enter upon the war and arm his columns was cruel  \nMezentius from Etruria, scorner of the gods. At his side was his  \n650 son Lausus, who for his beauty was second to none but the  \nLaurentine Turnus. Lausus was a tamer of horses and a hunter  \nof wild beasts, and he was at the head of a thousand men who  \nhad followed him and followed him in vain from the city of  \nAgylla. He deserved a father whom it would have been more  \nof a joy to obey, a father other than Mezentius.\n\nBehind them, driving over the grassland and displaying his  \nvictorious horses and his chariot which proudly bore the palm  \nof victory, came Aventinus, son of Hercules, fair son of a fair  \nfather, and on his shield he carried his father's blazon, the Hydra  \nand its snakes, the hundred snakes encircling it. His mother, the  \n660 priestess Rhea, had given birth to him in secret, bringing him  \ninto the land of light in the wood on the Aventine hill. She had  \nlain with Hercules, a woman with a god, when he had come in  \ntriumph to the land of the Laurentines, the hero of Tiryns who  \nhad slain Geryon and washed the cattle of Spain in the river of  \nthe Etruscans. His men carried javelins and fearsome pikes  \ninto battle and used the Sabine throwing spear with its round  \ntapering point. He himself was on foot, swinging a great lion  \nskin about him as he walked. It was matted and bristling, and  \nhe had put it with its white teeth over his head and a fearsome  \nsight he was as he came up to the palace with his father's garb  \ntied round his shoulders.\n\n670 Next came two bold Argive warriors, the twin brothers Catillus  \nand fierce Coras, leaving the walls of Tibur, which took its  \nname from their brother Tiburtus. They would charge out in  \nfront of the first line of battle through showers of missiles, like  \ntwo cloud-born Centaurs plunging down in wild career from the  \nsnow-clad tops of Mount Homole or Mount Othrys, crashing  \nthrough the trees as the great forest opens to let them pass.\n\nThe founder of the city of Praeneste was also there, a king  \nwho ruled among the herds and flocks of the countryside. Men  \n680 have always believed that he was the son of Vulcan, Caeculus,  \nfound as a baby on the burning hearth. His rustic legion came  \nfrom far and wide to follow him: from Praeneste on its hilltop;  \nfrom the fields round Juno's city of Gabii, from the icy waters  \nof the Anio and the streaming river rocks of the Hernici; men  \nnurtured by the rich city of Anagnia and by your river, Father  \nAmasenus. Not all of these came into battle with shields and  \narms and chariots sounding: most of them showered acorns of  \nblue lead from slings; some carried a pair of hunting spears in  \none hand and wore on their heads tawny caps made from the  \n690 hides of wolves, their left foot leaving a naked print while a  \nrawhide boot protected the right.\n\nNow Messapus, breaker of horses, son of Neptune, whom  \nneither fire nor steel might lay low, suddenly took up his sword  \nagain and called to arms tribes that had long lived at ease and  \narmies that had lost the habit of war. These were the men who  \ncame from the ridges of Fescennium, from Aequum Faliscum,  \nfrom the citadel of Soracte and the Flavinian fields, from the  \nlake of Ciminius and its mountain and the groves of Capena.  \nThey marched in regular formations singing the praises of their  \n700 king like white swans flying back from their feeding grounds  \nthrough wisps of cloud and pouring out the measured music  \nfrom their long necks while far and wide the echo of their singing  \nbeats back from the river and the Asian marsh. This great  \nmingled swarm of men seemed not like a bronze-clad army, but  \nan aery cloud of clamorous birds on the wing, straining in from  \nthe high seas to the shore.\n\nThere comes Clausus of the blood of the ancient Sabines,  \nleading a great army, and a great army in himself. From Clausus  \nare descended the tribe and family of the Claudii, spread all  \nover Latium ever since the Sabines were given a share in Rome.  \n710 With him came a large contingent from Amiternum and the first  \nQuirites, all the troops from Eretum and from olive-bearing  \nMutusca, all who lived in the city of Nomentum and the Rosean  \nplains round Lake Velinus, on the bristling rocks of Tetrica and  \nits gloomy mountain, in Casperia and Foruli and on the banks  \nof the Himella, men who drank the Tiber and the Fabaris, men  \nsent by chilly Nursia, levies from Orta, tribes from old Latium  \nand the peoples whose lands are cut by the Allia, that river of  \nill-omened name. They were as many as the waves that roll in  \nfrom the Libyan ocean when fierce Orion is sinking into the  \n720 winter sea, or as thick as the ears of corn scorched by the  \nearly sun on the plain of Hermus or the golden fields of Lycia.  \nTheir shields clanged and the earth quaked under the beat of  \ntheir feet.\n\nHalaesus next, one of Agamemnon's men and an enemy of  \nall things Trojan, yoked his horses to his chariot and rushed a  \nthousand fierce tribes to join Turnus: men whose mattocks turn  \nthe rich Massic soil for Bacchus; Auruncans sent by their fathers  \nfrom their high hills; men sent from the nearby plains of Sidicinum;  \nmen who come from Cales and the banks of the Volturnus,  \nriver of many fords, and with them the tough Saticulan and  \n730 bands of Oscans. Their weapon was the aclys, a light spear, and  \nit was their practice to attach a supple thong to it. A leather  \nshield protected their left side and for close fighting they used  \nswords shaped like sickles.\n\nNor will you, Oebalus, go unmentioned in our song. Men say  \nyou were the son of Telon by the nymph of the river Sebethus,  \nborn when Telon was already an old man and ruling over  \nCapreae, the island of the Teleboae. But the son no more than  \nthe father had been content with the lands he had inherited and  \nby now he had long held sway over the tribes of the Sarrastes,  \nthe plains washed by the river Sarnus, men who lived in Rufrae,  \n740 Batulum and the fields of Celemna and those on whom the walls  \nof apple-bearing Abella look down. Their missile was the catei  \na, a weapon thrown like the Teuton boomerang. Their heads were  \nprotected by helmets of bark stripped from the cork oak. They  \ncarried gleaming half-moon shields of plated bronze and their  \nswords too were of gleaming bronze.\n\nYou too, Ufens, famous for your feats of arms, were sent into  \nbattle from the mountains of Nersae. These Aequi live in a hard  \nland and are the most rugged of races, schooled in hunting the  \nforests. They work the soil with their armour on. Their delight  \nis always to bring home fresh plunder and live off what they  \ntake.\n\n750 Then came a priest from Marruvium, his helmet decorated  \nby a sprig of fruitful olive, the bravest of men, Umbro by name,  \nsent by king Archippus. By his spells and the touch of his hand  \nhe knew well how to sow the seed of sleep on nests of vipers  \nand on water-snakes, for all their deadly breath. His arts could  \ncharm their anger and soothe their bites, but he had no antidote  \nfor the sting of a Trojan sword and not all his lullabies and  \nherbs gathered in the Marsian hills could help him with his  \nwounds. For you wept the grove of the goddess Angitia. For  \n760 you wept the glassy waves and clear pools of Lake Fucinus.\n\nThere too, sent by his mother Aricia, glorious Virbius came  \nto the war, the lovely son of Hippolytus. He had grown to  \nmanhood in the grove of Egeria around the dank lake-shores by  \nthe altar where rich sacrifices win the favour of Diana. For after  \nHippolytus had been brought to his death by the wiles of his  \nstepmother Phaedra, torn to pieces by bolting horses and paying  \nwith his blood the penalty imposed by his father, men say he  \ncame back under the stars of the sky and the winds of heaven,  \n770 restored by healing herbs and the love of Diana. Then the  \nAll-powerful Father was enraged that any mortal should rise  \nfrom the shades below into the light of life and with his own  \nhand he took the inventor of those healing arts, Asclepius, son  \nof Apollo, and hurled him with his thunderbolt down into the  \nwave of the river Styx. But Diana Trivia, in her loving care,  \nfound a secret refuge for Hippolytus and consigned him to the  \nnymph Egeria and her grove, where, alone and unknown, his  \nname changed to Virbius, he might live out his days. Thus it is  \nthat horn-hooved horses are not admitted to the sacred grove  \n780 of the temple of Trivia because in their terror at the monsters of  \nthe deep the horses of Hippolytus had overturned his chariot  \nand thrown him on the shore. But none the less his son was  \ndriving fiery horses across the level plain as he rushed to the  \nwars in a chariot.\n\nThere, looking around him and moving among the leaders,  \nwas Turnus himself, in full armour, the fairest of them all, and  \ntaller by a head than all the others. On the towering top of his  \ntriple-plumed helmet there stood a Chimaera breathing from its  \nthroat a fire like Etna's, and the fiercer and bloodier the battle,  \nthe more savagely she roared and belched the deadly flames.  \nThe blazon on his polished shield showed a mighty theme, a  \n790 golden figure of Io, raising her horned head, with rough hair on  \nher hide, already changed into a heifer. And there was Argus,  \nguarding her, and her father Inachus pouring his river from an  \nurn embossed on the shield. Behind Turnus came a cloud of  \nfoot-soldiers and the whole plain was crowded with columns of  \nmen bearing shields, the youth of Argos, bands of Auruncans,  \nRutulians, Sicani, that ancient race, Sacrani in battle order and  \nLabici with their painted shields; men who ploughed the Tiber  \nvalley and the sacred banks of the Numicus; men whose ploughshare  \nworked the Rutulian hills and the ridge of Circeii; men  \n800 from the fields ruled by Jupiter of Anxur and the goddess Feronia  \ndelighting in her greenwood grove, and men from the black  \nswamps of Satura where the icy river Ufens threads his way  \nalong his valley bottom to lose himself in ocean.\n\nLast of all came Camilla, the warrior maiden of the Volsci,  \nleading a cavalry squadron flowering in bronze. Not for her  \ngirlish hands the distaff and wool-basket of Minerva. She was a  \nmaid inured to battle, of a fleetness of foot to race the winds.  \nShe could have skimmed the tops of a standing crop without  \ntouching them and her passage would not have bruised the  \n810 delicate ears of grain. She could have run over the ocean, hovered  \nover the swell and never wet her foot in the waves. Young men  \nstreamed from house and field and mothers came thronging to  \ngaze at her as she went, lost in wonderment at the royal splendour  \nof the purple veiling the smoothness of her shoulders, her  \nhair weaving round its gold clasp, her Lycian quiver and the  \nshepherd's staff of myrtle wood with the head of a lance.\n\n## BOOK 8  \nAENEAS IN ROME\n\nWhen Turnus raised the flag of war above the Laurentine citadel  \nand the shrill horns blared, when he whipped up his eager horses  \nand clashed his sword on his shield, there was instant confusion.  \nIn that moment the whole of Latium rose in a frenzy to take the  \noath and young warriors were baying for blood. Their great  \nleaders Messapus and Ufens and the scorner of the gods Mezentius  \nwere levying men everywhere, stripping the fields of those  \nwho tilled them. They also sent Venulus to the city of great  \n10 Diomede to ask for help and to let him know that Trojans were  \nsettling in Italy, that Aeneas had arrived with a fleet bringing  \nthe defeated household gods of Troy, claiming that he was being  \ncalled by the Fates to be king; the tribes were flocking to join  \nthis Trojan, this descendant of Dardanus, and his name was on  \nthe lips of men all over Latium; what all this was leading up to,  \nwhat Aeneas hoped to gain from the fighting if Fortune smiled  \nupon him, Diomede himself would know better than king  \nTurnus or than king Latinus.\n\nThis is what was happening in Latium. The Trojan hero,  \ndescendant of Laomedon, saw it all and great tides of grief  \n20 flowed in his heart. His thoughts moved swiftly, now here, now  \nthere, darting in every possible direction and turning to every  \npossible event, like light flickering from water in bronze vessels  \nas it is reflected from the sun or its image the moon, now flying  \nfar and wide in all directions, now rising to strike the high  \ncoffers of a ceiling.\n\nIt was night, and over the whole earth the weary animals, all  \nmanner of birds and all manner of flocks, were already deep in  \nsleep before Father Aeneas, on the bank of the river, under the  \n30 cold vault of the sky, heart sick at the sadness of war, lay down  \nat last and gave rest to his body. There on that lovely river he  \nsaw in his sleep the god of the place, old Tiber himself, rising  \namong the leaves of the poplars. He was veiled in a blue-green  \ncloak of fine-spun flax and dark reeds shaded his hair. He then  \nspoke to Aeneas and lightened his sadness with these words: 'O  \nyou who are born of the race of the gods, who are bringing back  \nto us the city of Troy saved from its enemies, who are preserving  \nits citadel Pergamum for all time, long have we waited for you  \nin the land of the Laurentines and the fields of Latium. This is  \nthe home that is decreed for you. This is the home decreed for  \n40 the gods of your household. Do not give it up. Do not be  \nintimidated by the threat of war. All the angry passions of the  \ngods are now spent. But come now, so that you may not think  \nwhat you are seeing is an empty dream, I tell you that you will  \nfind a great sow with a litter of thirty piglets lying beneath ilex  \ntrees on a shore. There she will lie all white on the ground and  \nthe young around her udders will be white. This will be a sign  \nthat after three times ten years revolve, Ascanius will found the  \ncity of Alba, white in name and bright in glory. What I prophesy  \n50 will surely come to pass. Attend now and I shall teach you in  \nfew words how you may triumphantly resolve the difficulties  \nthat lie before you.\n\n'The Arcadians are a race descended from Pallas. They came  \nto these shores following the standards of their king Evander,  \nchose a site here and established in these hills a city called  \nPallanteum after their founder Pallas. This people wages continual  \nwar with the Latin race. Welcome them into your camp as  \nyour allies. Make a treaty with them. I will take you to them  \nstraight up my river between these banks and you will be able  \nto row upstream into the current. Up with you then, son of the  \n60 goddess, for the first stars are beginning to set. Offer due prayers  \nto Juno and overcome her angry threats with vows and supplications.  \nTo me you will give honour and make repayment when  \nyou are victorious. I am that full river whom you see scouring  \nthese banks and cutting through the rich farmland. I am the  \nriver Thybris, blue as the sky and favoured of heaven. Here is  \nmy great home. My head waters rise among lofty cities.'\n\nSo spoke the river-god and plunged to the bottom of a deep  \npool. The night was over and so was Aeneas' sleep. As he rose  \nhe looked up to the light of the sun rising in the sky, took up  \n70 water from the river in cupped hands and poured out these  \nwords of prayer to the heavens: 'O you Laurentine nymphs,  \nnymphs who are the mothers of rivers, and you, Father Thybris  \nwith your holy stream, receive Aeneas, and now after all his  \nsuffering keep him safe from peril. In whichever of your pools  \nyou may be, at whichever of your sources, you who pity our  \nmisfortunes, in whatever land you emerge in all your splendour,  \nI will always pay you honour and always make offerings to you,  \nO horne\u00e8d river, king of all the waters of Hesperia, only be with  \nme and by your presence confirm your divine will.' So speaking  \n80 he picked out two biremes from the fleet, manned them with  \nrowers and at the same time put some of his comrades on board  \nin full armour.\n\nNow suddenly before his astonished eyes there appeared a  \nportent. There through the trees he caught sight of a white sow  \nwith offspring of the same colour, lying on the green shore. This  \nsow devout Aeneas offered to you as a sacrifice, even to you, O  \ngreatest Juno, leading her to your altar with all her young. And  \nall that long night the Thybris calmed his flood, reversing his  \ncurrent, and was as still and silent as a peaceful lake or quiet  \nmarsh. There were no ripples on the surface of his waters, and  \n90 no toiling for the oar. Thus they began their journey and made  \ngood speed, raising a cheerful noise as the caulked hulls glided  \nover the water. The waves were amazed and the woods were  \nfull of wonder at the unaccustomed sight of far-glinting shields  \nof warriors and painted prows floating on the river. So did they  \nwear out the night and the day with rowing and mastered all  \nthe long windings of the river, moving under the shade of all  \nmanner of trees and cleaving green woods in smooth water. The  \nfiery sun had climbed to the middle of the vault of heaven when  \nthey saw in the distance walls and a citadel and the roofs of  \n100 scattered houses. What Roman power has now raised to the  \nheights of the sky, in those days was a poor land ruled by  \nEvander. Quickly they turned their prows to the bank and  \nsteered for the city.\n\nIt so happened that on that day the Arcadian king Evander  \nwas performing yearly rites in honour of the mighty Hercules,  \nson of Amphitryon, and was sacrificing to the gods in a grove  \noutside the city. His son Pallas was with him, and with him also  \nwere all the leading warriors and the senators, poor men as they  \nwere. They were offering incense and warm blood was smoking  \non the altars. When they saw the tall ships and saw them gliding  \nthrough the dense grove with men bending to the oars in silence,  \n110 they were seized with sudden fright and rose in a body, abandoning  \nthe sacred tables. Not so Pallas. Boldly he told them not  \nto disturb their holy feast, and seizing a weapon he rushed off  \nto face the strangers by himself. 'What is it, warriors, that has  \ndriven you to try these new paths?' he called out from the top  \nof a mound while he was still at a distance. 'Where are you  \ngoing? What race are you? Where is your home? Is it peace you  \nare bringing us or war?' Then Father Aeneas replied from the  \nhigh poop of his ship, holding out in his hand the olive branch  \nof peace: 'We are of the Trojan race. These weapons you see are  \nfor use against our enemies the Latins. It is they who have driven  \nus here, exiles as we are, with all the insolence of war. We are  \nlooking for Evander. Tell him of this. Say to him that the chosen  \n120 leaders of the race of Dardanus have come to ask him to be their  \nally in battle.' At this great name Pallas was dumbfounded.  \n'Whoever you may be,' he cried, 'leave your ship and come and  \nspeak with my father face to face. Come as a guest into our  \nhouse.' With these words he took Aeneas by the right hand in a  \nlong clasp, and they moved forward into the grove, leaving the  \nriver behind them.\n\nThen Aeneas addressed the king with words of friendship: 'O  \nnoblest of the race of the Greeks, Fortune has willed that I  \nshould come to you as a suppliant with an olive branch draped  \nwith wool. I was not alarmed at the thought that you are a  \n130 leader of Greeks, an Arcadian and joined by blood to the two  \nsons of Atreus, for I am joined to you by my courage and by the  \nholy oracles of the gods, by our fathers who were kinsmen and  \nby your fame which is known throughout the world. All these  \nhave driven me here by the command of the Fates, and I have  \nwillingly obeyed. Dardanus, the first founder and father of the  \ncity of Troy, sailed to our Teucrian land. According to the  \nGreeks he was the son of Electra, and that same Electra was the  \ndaughter of Atlas, the mighty Atlas who carries the circle of  \n140 the heavens on his shoulder. On your side you are the son  \nof Mercury and he was the son of Maia, conceived and born on  \nthe snow-clad top of Mount Cyllene. But the father of Maia, if  \nwe put any trust in what we hear, was Atlas, that same Atlas  \nwho supports the stars of the sky. And so we are of one blood,  \ntwo branches of the same family. Trusting in this, I have not  \nsent emissaries or made trial of you in advance by any form of  \nsubterfuge, but have come in person as a suppliant to your door,  \nand laid my life before you. The same race harries us both in  \nbitter war, the Rutulians of king Daunus, and they are persuaded  \nthat if they were to drive us away, nothing would prevent them  \nfrom putting all the heartlands of Italy under their yoke and  \n150 becoming masters of the Tyrrhenian sea to the south and the  \nAdriatic to the north. Take the right hand of friendship I offer  \nand give me yours. Our hearts are strong in war. Our spirits are  \nhigh. Our fighting men are tried and proved.'\n\nSo spoke Aeneas. All the time he was speaking, Evander had  \nbeen gazing at his face and his eyes and his whole body. He then  \nreplied in these few words: 'Bravest of the Trojans, I welcome  \nyou with great joy, and with great joy I recognize who you are.  \nOh how well do I recall the words of your father, the very voice  \nand features of the great Anchises! For I remember that when  \nPriam, son of Laomedon, was on a visit to his sister Hesione in  \nthe kingdom of Salamis, he came on to visit us in the cold lands  \n160 of Arcadia. In those days the first bloom of youth was still  \ncovering my cheeks, and I was full of admiration for the leaders  \nof Troy. Priam himself, too, I admired, but taller than them all  \nwalked Anchises. With all a young man's ardour, I longed to  \nspeak with him and put my right hand in his, so I approached  \nhim and led him with full heart to the walls of Pheneus. When  \nhe was leaving he gave me a wonderful quiver filled with Lycian  \narrows, a soldier's cloak interwoven with gold thread and a pair  \nof golden bridles which now belong to my son Pallas. So then,  \nthe right hand of friendship for which you ask has already been  \n170 given in solemn pledge, and as soon as tomorrow's sun returns  \nto the earth, I shall send you on your way and you will not be  \ndisappointed with the reinforcements and supplies I shall give  \nyou. Meanwhile, since you are here as friends, come favour  \nthese annual rites of ours which it would be sinful to postpone,  \nby celebrating them with us. It is time you began to feel at home  \nat the tables of your allies.'\n\nThe food and drink had been cleared away, but as soon as he  \nwas finished speaking, he ordered them to be replaced, and the  \nking himself showed the Trojans to seats on the grass, but took  \nAeneas apart to a couch of maple wood and seated him on a  \nrough lion skin for a cushion. Then the priest of the altar and  \n180 some chosen warriors served with great good will the roast flesh  \nof bulls, loaded into baskets the grain which is the gift of Ceres  \nworked by the hand of man, and poured out the juice of Bacchus.  \nAeneas and the warriors of Troy then feasted together on the  \nwhole chine and entrails of the sacrificial ox.\n\nAfter their hunger was relieved and their appetite satisfied,  \nking Evander spoke as follows: 'This annual rite, this set feast  \nand this altar to a great divinity have not been imposed upon us  \nby any vain superstition working in ignorance of our ancient  \ngods. It is because we have been saved from desperate dangers,  \nmy Trojan friend, that we perform this worship and renew it  \nyearly in honour of one who has well deserved it.\n\n190 'First of all, look at this vaulted cavern among the rocks. You  \nsee how this great massive home inside the mountain has been  \ntorn apart and is now abandoned, with boulders lying everywhere  \nin ruins. Here, deep in the vast recesses of the rock, was  \nonce a cave which the rays of the sun never reached. This was  \nthe home of a foul-featured, half-human monster by the name  \nof Cacus. The floor of the cave was always warm with freshly  \nshed blood, and the heads of men were nailed to his proud doors  \nand hung there pale and rotting. The father of this monster was  \nVulcan, and it was his father's black fire he vomited from his  \nmouth as he moved his massive bulk. Long did we pray and in  \n200 the end we too were granted the help and the presence of a god.  \nFor the great avenger was at hand. Exulting in the slaughter  \nof the triple-bodied Geryon and the spoils he had taken, the  \nvictorious Hercules was driving the huge bulls through our land  \nand the herd was grazing the valley and drinking the water of  \nthe river. But Cacus was a robber, and thinking in the savagery  \nof his heart not to leave any crime or treachery undared or  \nunattempted, he stole from pasture four magnificent bulls and  \nas many lovely heifers. So that there would be no hoof prints  \n210 pointing forwards in the direction of the cave, he dragged them  \nin by their tails to reverse the tracks, and was now keeping his  \nplunder hidden deep in the darkness of the rock. There were no  \ntracks leading to the cave for any searcher to see.\n\n'Meanwhile, when his herd had grazed its fill, and the son of  \nAmphitryon was moving them out of pasture and preparing to  \ngo on his way, the cows began to low plaintively at leaving  \nthe place, filling the whole grove with their complaints, and  \nbellowing to the hills they were leaving behind them. Then, deep  \nin the cave, a single cow lowed in reply. Cacus had guarded her  \nwell, but she thwarted his hopes. At this Hercules blazed up in  \n220 anger. The black bile of his fury rose in him, and snatching up  \nhis arms and heavy knotted club, he made off at a run for  \nthe windswept heights of the mountain. Never before had our  \npeople seen Cacus afraid. Never before had there been terror in  \nthese eyes. He turned and fled, running to his cave with the  \nspeed of the wind, fear lending wings to his feet. There he shut  \nhimself up, dropping a huge rock behind him and breaking the  \niron chains on which it had been suspended by his father's art,  \nso that its great mass was jammed against the doorposts and  \nblocked the entrance. There was Hercules in a passion, trying  \n230 every approach, turning his head this way and that and grinding  \nhis teeth. Three times he went round the whole of Mount  \nAventine in his anger. Three times he tried to force the great rock  \ndoorway without success. Three times he sat down exhausted in  \nthe valley.\n\n'Above the ridge on top of the cave, there stood a sharp needle  \nof flint with sheer rocks falling away on either side. It rose to a  \ndizzy height and was a favourite nesting-place of carrion birds.  \nHercules put his weight on the right-hand side of it where it  \nleaned over the ridge towards the river on its left. He rocked it,  \nloosened it, wrenched it free from its deep base and then gave a  \nsudden heave, a heave at which the great heavens thundered,  \n240 the banks of the river leapt apart and the river flowed backwards  \nin alarm. The cave and whole huge palace of Cacus were unroofed  \nand exposed to view and his shadowy caverns were  \nopened to all their depths. It was as though the very depths of the  \nearth were to gape in some cataclysm and unbar the chambers of  \nthe underworld, the pale kingdom loathed by the gods, so that  \nthe vast abyss could be seen from above with the shades of the  \ndead in panic as the light floods in.\n\n'So Cacus was caught in the sudden rush of light and trapped  \nin his cavern in the rock, howling as never before, while Hercules  \n250 bombarded him from above with any missile that came to hand,  \nbelabouring him with branches of trees and rocks the size of  \nmillstones. There was no escape for him now, but he vomited  \nthick smoke from his monstrous throat and rolled clouds of it  \nall round his den to blot it from sight. Deep in his cave he  \nchurned out fumes as black as night and the darkness was shot  \nthrough with fire. Hercules was past all patience. He threw  \nhimself straight down, leaping through the flames where the  \nsmoke spouted thickest and the black cloud boiled in the vast  \ncavern. There, as Cacus vainly belched his fire in the darkness,  \n260 Hercules caught him in a grip and held him, forcing his eyes out  \nof their sockets and squeezing his throat till the blood was dry  \nin it. Then, tearing out the doors and opening up the dark house  \nof Cacus, he brought into the light of heaven the stolen cattle  \nwhose theft Cacus had denied, and dragged the foul corpse out  \nby the feet. No one could have enough of gazing at his terrible  \neyes and face, at the coarse bristles on his beastly chest and the  \nthroat charred by fires now dead.\n\n'Ever since that time we have honoured his name and succeeding  \ngenerations have celebrated this day with rejoicing. This  \n270 altar was set up in its grove by Potitius, the first founder of these  \nrites of Hercules, and by the Pinarii, the guardians of the rites.  \nWe shall always call it the Greatest Altar, and the greatest altar  \nit will always be. Come then warriors, put a crown of leaves  \naround your hair in honour of this great exploit, and hold out  \nyour cups in your right hands. Call upon the god who is a god  \nfor all of us and offer him wine with willing hearts.' No sooner  \nhad he spoken than his head was shaded by a wreath and  \npendant of the green-silver leaves of Hercules' poplar woven  \ninto his hair, and the sacred goblet filled his hand. Soon they  \nwere all pouring their libations on the table and praying to  \nthe gods.\n\n280 Meanwhile the Evening Star was drawing nearer as the day  \nsank in the heavens and there came a procession of priests led by  \nPotitius, wearing their ritual garb of animal skins and carrying  \ntorches. They were starting the feast again with a second course  \nof goodly offerings, and they heaped the altar with loaded  \ndishes. Then the Salii, the priests of Mars, their heads bound  \nwith poplar leaves, came to sing around the altar fires. On one  \nside was a chorus of young warriors, on the other a chorus of  \nold men, hymning the praise of Hercules and his great deeds:  \nhow he seized the two snakes, the first monsters sent against  \nhim by his stepmother, and throttled them, one in each hand;  \n290 how too he tore stone from stone the cities of Troy and Oechalia,  \nfamous in war; how he endured a thousand labours under king  \nEurystheus to fulfil the fate laid upon him by the cruel will of  \nJuno. 'O unconquered Hercules,' they sang, 'you are the slayer  \nof the half-men born of the cloud, the Centaurs Hylaeus and  \nPholus; of the monstrous Cretan bull and the huge lion of Nemea  \nin its rocky lair; the pools of the Styx trembled at your coming,  \nand the watchdog of Orcus cringed where he lay in his cave  \nweltering in blood on heaps of half-eaten bones. But nothing  \nyou have seen has ever made you afraid, not even Typhoeus  \n300 himself, rising up to heaven with his weapons in his hands. Nor  \ndid reason fail you when the hundred heads of the Lernaean  \nHydra hissed around you. Hail, true son of Jupiter, the latest  \nlustre added to the company of the gods, come to us now, to  \nyour own holy rite, and bless us with your favouring presence.'  \nTo end their hymn they sang of the cave of Cacus, and Cacus  \nhimself breathing fire, till the whole grove rang and all the hills  \nre-echoed.\n\nAs soon as the sacred rites were completed, they all returned  \nto the city. The king, weighed down with age, kept Aeneas and  \nhis son Pallas by his side as he walked, and made the way  \n310 seem shorter by all the things he told them. Aeneas was lost in  \nadmiration and his eyes were never still as he looked about him  \nenthralled by the places he saw, asking questions about them  \nand joyfully listening to Evander's explanations of all the relics  \nof the men of old. This is what was said that day by Evander,  \nthe founder of the citadel of Rome: 'These woods used to be the  \nhaunt of native fauns and nymphs and a race of men born from  \nthe hard wood of oak-tree trunks. They had no rules of conduct  \nand no civilization. They did not know how to yoke oxen for  \nploughing, how to gather wealth or husband what they had,  \nbut they lived off the fruit of the tree and the harsh diet of  \n320 huntsmen. In those early days, in flight from the weapons of  \nJupiter, came Saturn from heavenly Olympus, an exile who had  \nlost his kingdom. He brought together this wild and scattered  \nmountain people, gave them laws and resolved that the name of  \nthe land should be changed to Latium, since he had _lain_ hidden  \nwithin its borders. His reign was what men call the Golden Age,  \nsuch was the peace and serenity of the people under his rule.  \nBut gradually a worse age of baser metal took its place and with  \nit came the madness of war and the lust for possessions. Then  \nbands of Ausonians arrived and Sicanian peoples, and the land  \n330 of Saturn lost its name many times. Next there were kings,  \namong them the cruel and monstrous Thybris, after whom we  \nItalians have in later years called the river Thybris, and the old  \nriver Albula has lost its true name. I had been driven from my  \nnative land and was setting course for the most distant oceans  \nwhen Fortune, that no man can resist, and Fate, that no man  \ncan escape, set me here in this place, driven by fearsome words  \nof warning from my mother, the nymph Carmentis, and by the  \nauthority of the god Apollo.'\n\nHe had just finished saying this and moved on a little, when  \nhe pointed out the Altar of Carmentis and the Carmental Gate,  \nas the Romans have called it from earliest times in honour of  \n340 the nymph Carmentis. She had the gift of prophecy and was the  \nfirst to foretell the future greatness of the sons of Aeneas and  \nthe future fame of Pallanteum. From here he pointed out the  \ngreat grove which warlike Romulus set up as a sanctuary \u2013 he  \nwas to call it the Asylum \u2013 and also the Lupercal there under its  \ncool rock, then called by Arcadian tradition they had brought  \nfrom Parrhasia, the cave of Pan Lycaeus, the wolf god. He also  \npointed out the grove of the Argiletum, and, calling upon that  \nconsecrated spot to be his witness, he told the story of the killing  \nof his guest Argus.\n\nFrom here he led the way to the house of Tarpeia and the  \nCapitol, now all gold, but in those distant days bristling with  \n350 rough scrub. Even then a powerful sense of a divine presence in  \nthe place caused great fear among the country people, even then  \nthey went in awe of the wood and the rock. 'This grove,' said  \nEvander, 'this leafy-topped hill, is the home of some god, we  \nknow not which. My Arcadians believe they have often seen  \nJupiter himself shaking the darkening aegis in his right hand to  \ndrive along the storm clouds. And then here are the ruined walls  \nof these two towns. What you are looking at are relics of the  \nmen of old. These are their monuments. One of these citadels  \nwas founded by Father Janus; the other by Saturn. This one  \nused to be called the Janiculum; the other, Saturnia.'\n\n360 Talking in this way they were coming up to Evander's humble  \nhome, and there were cattle everywhere, lowing in the Roman  \nForum and the now luxurious district of the Carinae. When  \nthey arrived at his house, Evander said: 'The victorious Hercules  \nof the line of Alceus stooped to enter this door. This was a  \npalace large enough for him. You are my guest, and you too  \nmust have the courage to despise wealth. You must mould  \nyourself to be worthy of the god. Come into my poor home and  \ndo not judge it too harshly.' With these words he led the mighty  \nAeneas under the roof-tree of his narrow house and set him  \ndown on a bed of leaves covered with the hide of a Libyan bear.  \nNight fell and its dark wings enfolded the earth.\n\n370 But his mother Venus was terrified, and with good reason, by  \nthe threats of the Laurentines and the savagery of the fighting,  \nso she spoke to her husband Vulcan. Coming to him in his  \ngolden bedroom and breathing divine love into her voice, she  \nsaid: 'When the citadel of Troy was being ravaged in war by the  \nkings of Greece, it was owed to Fate and was doomed to fall in  \nthe fires lit by its enemies, but I asked for nothing for those who  \nsuffered. I did not call upon the help of your art to make arms  \n380 for them. Although I owed much to the sons of Priam and had  \noften wept at the sufferings endured by Aeneas, I did not wish,  \nO my dearest husband, that you should exert yourself to no  \npurpose. But now, in obedience to the commands of Jupiter,  \nAeneas is standing on Rutulian soil and so now I come to you  \nas a suppliant. I approach that godhead which I so revere, and  \nas a mother, I ask you to make arms for my son. You yielded to  \nThetis, the daughter of Nereus, you yielded to the wife of  \nTithonus when they came and wept to you. Look at all the  \nnations gathering. Look at the walled cities that have closed  \ntheir gates and are sharpening their swords against me to destroy  \nthose I love.' She had finished speaking and he was hesitating.  \nThe goddess took him gently in her white arms and caressed  \nhim, and caressed him again. Suddenly he caught fire as he  \n390 always did. The old heat he knew so well pierced to the marrow  \nof his bones and coursed through them till they melted, as in a  \nthunderstorm when a fiery-flashing rift bursts the clouds and  \nruns through them in dazzling brightness. His wife knew and  \nwas pleased. She was well aware of her beauty and she knew  \nhow to use it. Father Vulcan, bound to her by eternal love, made  \nthis reply: 'You need not delve so deep for arguments. Where is  \nthat trust, O goddess, which you used to have in me? If your  \ncare for Aeneas was then as it is now, it would have been right  \nfor us even then to arm the Trojans. Neither the All-powerful  \nFather nor the Fates were forbidding Troy to stand and Priam  \n400 to go on living for ten more years. And now if you are preparing  \nfor war and this is what you wish, whatever care I can offer you  \nin the exercise of my skill, whatever can be done by melting iron  \nor electrum, anything that fire and bellows can achieve, you do  \nnot have to pray to me. You need not doubt your power.' At  \nthese words he gave his wife the embraces so much desired, and  \nthen, relaxed upon her breast, he sought and found peace and  \nrepose for all his limbs.\n\nWhen the night had passed the middle of its course, when  \nVulcan's first sleep was over and there was no more rest, just  \n410 when the ashes are first stirred to rouse the slumbering fire by a  \nwoman whose task it is to support life by the humble work of  \nspinning thread on a distaff; taking time from the night for her  \nlabours, she sets her slave women going by lamplight upon their  \nlong day's work, so that she can keep her husband's bed chaste  \nand bring her young sons to manhood \u2013 with no less zeal than  \nsuch a woman and not a moment later did the God of Fire rise  \nfrom his soft bed and go to work at his forge.\n\nBetween Lipari in the Aeolian Islands and the flank of Sicily,  \nan island of smoking rocks rises sheer from the sea. Deep within  \nit is a great vault, and in that vault caves have been scooped out  \nlike those under Etna to serve as forges for the Cyclopes. The  \n420 noise within them is the noise of thunder. Mighty blows can be  \nheard booming on the groaning anvils, the caves are filled with  \nthe sound of hissing as the Chalybes plunge bars of white-hot  \npig-iron into water and all the time the fires are breathing in the  \nfurnaces. This is the home of Vulcan, and Vulcania is the name  \nof the island. Into these depths the God of Fire descended from  \nthe heights of heaven.\n\nThe Cyclopes were forging steel, working naked in that vast  \ncavern, Brontes, Sterope and Pyracmon. In their hands was a  \nthunderbolt which they had roughed out, one of those the Father  \nof the Gods and Men hurls down upon the earth in such numbers  \nfrom every part of the sky. Some of it was already burnished,  \nsome of it unfinished. They had attached three shafts of lashing  \n430 rain to it, three shafts of heavy rainclouds, three of glowing fire  \nand three of the south wind in full flight. They were now adding  \nto the work the terrifying lightning and the sound of thunder,  \nthen Fear and Anger with their pursuing flames. In another  \npart of the cave they were working for Mars, busy with the  \nwing-wheeled chariot in which he stirs up men and cities to war.  \nOthers were hard at work polishing the armour worn by Pallas  \nAthene when roused, the fearsome aegis and its weaving snakes  \nwith their reptilian scales of gold, even the Gorgon rolling her  \neyes in the bodiless head on the breast of the goddess. 'Put all  \nthis away!' he cried. 'Whatever work you have started, you  \n440 Cyclopes of Etna, lay it aside and give your attention here.  \nArmour has to be made for a brave hero. You need strength and  \nquick hands now. Now you need all your arts to guide you. Let  \nnothing stand in your way.' He said no more, but instantly they  \nall bent to the work, dividing it equally between them. The  \nbronze was soon flowing in rivers. The gold ore and iron, the  \ndealer of death, were molten in a great furnace. They were  \nshaping one great shield to be a match for all the weapons of  \nthe Latins, fastening the seven thicknesses of it circle to circle.  \n450 Bellows were taking in air and breathing it out again. Bronze  \nwas being plunged into troughs of water and hissing. The cave  \nboomed with the anvils standing on its floor while the Cyclopes  \nraised their arms with all their strength in time with one another  \nand turned the ore in tongs that did not slip.\n\nWhile Father Vulcan, the god of Lemnos, was pressing on  \nwith this work in the Aeolian Islands, Evander was roused from  \nsleep in his humble hut by the life-sustaining light of day and  \nthe dawn chorus of the birds under his eaves. The old man rose,  \nput on his tunic and bound Etruscan sandals on the soles of his  \nfeet. He then girt on a Tegean sword with its baldric over the  \n460 shoulder and threw on a panther skin to hang down on his left  \nside. Nor did the sentinels from his high threshold fail to precede  \nhim \u2013 his two dogs went with their master \u2013 as the hero walked  \nto the separate quarters of his guest Aeneas, remembering their  \ntalk and remembering the help he had promised to give. Aeneas  \nwas up and about just as early, walking with Achates. Evander  \nhad his son Pallas with him. They met, clasped right hands, and  \nsitting there in the middle of Evander's house, they were at last  \nable to discuss affairs of state.\n\n470 The king spoke first: 'Great leader of the Trojans, while you  \nare alive I shall never accept that Troy and its kingdom are  \ndefeated. Beside your mighty name, the power we have to help  \nyou in this war is as nothing. On one side we are hemmed in by  \nthe Tuscan river, on the other the Rutulians press us hard and  \nwe can hear the clang of their weapons round our walls. But I  \nhave a plan to join vast peoples and the armies of wealthy  \nkingdoms to your cause. A chance that no man could have  \nforeseen is showing us the path to safety. Fate was calling you  \nwhen you came to this place.\n\n'Not far from here is the site of Agylla, founded long ago on  \n480 its ancient rock by the warlike Lydians who once settled there  \non the ridges of the Etruscan mountains. After this city had  \nflourished for many years, Mezentius eventually took it under  \nhis despotic rule as king and held it by the ruthless use of armed  \nforce. I shall not speak of the foul murders and other barbaric  \ncrimes committed by this tyrant. May the gods heap equal  \nsuffering upon his own head and the heads of his descendants!  \nHe even devised a form of torture whereby living men were  \nroped to dead bodies, tying them hand to hand and face  \nto face to die a lingering death oozing with putrefying flesh in this  \ncruel embrace. But at last his subjects reached the end of their  \nendurance and took up arms against him. Roaring and raging  \n490 he was besieged in his palace, his men were butchered and fire  \nwas thrown on his roof. In all this bloodshed he himself escaped  \nand took refuge in the land of the Rutulians under the protection  \nof the armies of his guest-friend, Turnus. At this the whole of  \nEtruria rose in righteous fury and has now come in arms to  \ndemand that Mezentius be given up for punishment. They have  \nthousands of troops and I shall put you at their head. Their  \nships are massed all along the shore, clamouring for the signal  \nfor battle, but they are held in check by this warning from an  \naged prophet: \"O you chosen warriors from Lydian Maeonia,  \n500 flower of the chivalry of an ancient race, it is a just grievance  \nthat drives you to war, and Mezentius deserves the anger that  \nblazes against him, but it is not the will of heaven that such a  \nrace as the Etruscans should ever obey an Italian. You must  \nchoose your leaders from across the seas.\"\n\n'At this the Etruscan army has settled down again on the  \nplain, held back by fear of these divine warnings. Tarchon  \nhimself has sent envoys to me with crown and sceptre, and  \noffers me the royal insignia of Etruria if I agree to come to their  \ncamp and take over the kingdom. But my powers have passed  \nwith the passing of the generations. Age has taken the speed  \nfrom my feet and the warmth from my blood. I am too old for  \n510 command and no longer have the strength for battle. I would  \nbe urging my son to go, but he is of mixed stock through his  \nSabine mother and is therefore part Italian. It is you who are  \nfavoured of the Fates for your years and your descent. You are  \nthe man the gods are asking for. Go then, O bravest leader of  \nall the men of Troy and Italy, and I shall send with you this my  \nson Pallas, our hope and our comfort. Let him be hardened to  \nthe rigours of war under your leadership. Let him daily see your  \nconduct and admire you from his earliest years. Two hundred  \nhorsemen I shall give him, the flower of our fighting men, and  \nPallas will give you two hundred more in his own name.'\n\n520 He had scarcely finished speaking, and Aeneas, son of  \nAnchises, and his faithful Achates were still looking sadly down  \nat the ground, and long would they have pondered in the anguish  \nof their hearts, had Venus not given a sign from the clear sky.  \nThere came from the heavens a sudden flash of lightning and a  \nrumble of thunder and the whole sky seemed to be crashing  \ndown upon them with the blast of an Etruscan trumpet shrilling  \nacross the heavens. They looked up and again and again great  \npeals broke over their heads and in bright sky in a break between  \nthe clouds they saw armour glowing red and heard it thunder  \n530 as it clashed. The others were all astonished but the hero of  \nTroy understood the sound and knew this was the fulfilment of  \nthe promise of his divine mother. At last he spoke: 'There is no  \nneed, my friend, no need to ask what these portents mean. This  \nis heaven asking for me. The goddess who is my mother told me  \nshe would send this sign if war were threatening, and bring  \narmour made by Vulcan down through the air to help me. Alas!  \nWhat slaughter waits upon the unhappy Laurentines! What a  \npunishment Turnus will endure at my hands! How many shields  \nand helmets and bodies of brave men will Father Thybris roll  \n540 down beneath his waves. Now let the Laurentines ask for war!  \nNow let them break their treaties!'\n\nWhen he had said this, he rose from his high throne. First of  \nall he stirred the fires smouldering on the altar of Hercules and  \napproached with joy the humble gods of home and hearth whom  \nhe had worshipped on the day before, and then Evander and  \nthe warriors of Troy made sacrifice together of duly chosen  \nyearling sheep. When this was done Aeneas went back from  \nEvander's house to his ships and his comrades, from whom he  \nchose men of outstanding courage to follow him to war. The  \nrest sailed downstream, floating effortlessly on the current, to  \n550 bring Ascanius news of his father and tell him what had happened.  \nThe Trojans going to Etruria were given horses. The  \nmount picked out for Aeneas was caparisoned in one great  \ntawny lion skin with gleaming gold claws.\n\nSwiftly round the little city flew the rumour that they were  \nriding to the gates of the king of Etruria. Frightened mothers  \nheaped prayer upon prayer, their fear increasing with the  \napproach of danger, and the vision of Mars loomed ever larger  \nbefore them. As they left, Evander took the right hand of his  \n560 son Pallas and clung to it inconsolably: 'If only Jupiter would  \ngive me back the years that are past,' he cried, 'when I laid low  \nthe front rank of the enemy's battle line under the very walls of  \nPraeneste, heaping up their shields and burning them to celebrate  \nmy victory, with this right hand sending down to Tartarus  \ntheir king Erulus, whose mother Feronia had given him three  \nlives at birth \u2013 I shudder to remember it \u2013 three sets of armour  \nto carry into battle, and three times I had to lay him dead on the  \nground, but in those days this one right hand was able to take  \nall his lives and strip him of all those sets of armour...no  \npower on earth would be tearing me from your arms, O my  \nbeloved son, and Mezentius would never have been able to  \n570 trample upon his neighbour, putting so many of my countrymen  \nto the sword and emptying the city of so many of its people. But  \nO you gods above, and you, Greatest Jupiter, ruler of the gods,  \nI beseech you, take pity on an Arcadian king, and hear a father's  \nprayers. If your divine powers and the Fates are keeping Pallas  \nsafe for me, if I am going to live to see him again and be with  \nhim again, then I pray for life and harden my heart to endure  \nany suffering. But if Fortune has some horror in store, let me  \ndie now, let me break off this cruel life here and now, before I  \n580 can put a name to my sorrow, before I know what the future  \nwill bring and while I still hold you in my arms, O my dear son,  \nmy only source of joy, given to me so late in life. I want no grim  \nnews to come and wound my ears.' These are the words that  \npoured from the lips of Evander at his last parting with his son.  \nWhen he had uttered them, he collapsed and was carried into  \nhis house by his attendants.\n\nAnd now the gates had been opened and the horsemen had  \nridden out, Aeneas among the first of them and his faithful  \nAchates with him, then the other Trojan commanders with  \nPallas conspicuous in the middle of the column in his Greek  \nmilitary cloak and brightly coloured armour. He was like the  \n590 Morning Star, which Venus loves above all other starry fires, as  \nhe leaves his ocean bath and lifts up his holy face into the sky  \nto scatter the darkness. Mothers stood on the city walls, full of  \ndread and following with their eyes the cloud of dust and the  \nglint of bronze from the squadrons. They were riding in their  \narmour by the shortest route over rough scrub and their shouts  \nrose to the sky as the four-hoofed beat of the galloping column  \ndrummed on the dusty plain. Near Caere's cold river there was  \na wide glade, revered for generations as a holy place by peoples  \nnear and far. It was enclosed on every side by a ring of hills clad  \nin black firs. The story is told that the ancient Pelasgians, who  \nin days long past were the first inhabitants of Latium, consecrated  \n600 this grove and a holy day to be observed in it to Silvanus,  \nthe god of field and flock. Not far from here Tarcho and the  \nEtruscans were occupying a strong position and their whole  \narmy could be seen from the heights of the hills, encamped on  \nthe broad fields. Aeneas and his chosen warriors had come down  \nto the camp and, weary from the ride, were seeing to their horses  \nand refreshing themselves.\n\n610 But the goddess Venus, bringing her gifts, was at hand, shining  \namong the clouds of heaven. When she saw her son at some  \ndistance from the others, alone in a secluded valley across the  \nicy river, she spoke to him, coming unasked before his eyes:  \n'Here now are the gifts I promised you, perfected by my husband's  \nskill. When the time comes you need not hesitate, my  \nson, to face the proud Laurentines or challenge fierce Turnus to  \nbattle.' With these words the goddess of Cythera came to her  \nson's embrace and laid the armour in all its shining splendour  \nbefore him under an oak tree.\n\nAeneas rejoiced at these gifts from the goddess and at the  \nhonour she was paying him and could not have his fill of gazing  \n620 at them. He turned them over in his hands, in his arms, admiring  \nthe terrible, crested, fire-spurting helmet, the death-dealing  \nsword, the huge, unyielding breastplate of blood-red bronze like  \na dark cloud fired by the rays of the sun and glowing far across  \nthe sky, then the polished greaves of richly refined electrum and  \ngold, the spear and the fabric of the shield beyond all words to  \ndescribe. There the God of Fire, with his knowledge of the  \nprophets and of time that was to be, had laid out the story of  \nItaly and the triumphs of the Romans, and there in order were  \nall the generations that would spring from Ascanius and all the  \nwars they would fight.\n\n630 He had made, too, a mother wolf stretched out in the green  \ncave of Mars with twin boys playing round her udders, hanging  \nthere unafraid and sucking at her as she bent her supple neck  \nback to lick each of them in turn and mould their bodies into  \nshape with her tongue.\n\nNear this he had put Rome and the violent rape of the Sabines  \nat the great games in the bowl of the crowded Circus, and a new  \nwar suddenly breaking out between the people of Romulus and  \nthe stern Sabines from Cures led by their aged king Tatius. Then,  \n640 after these same kings had put an end to their conflict, they  \nstood in their armour before the altar of Jupiter with sacred  \nvessels in their hands, sacrificing a sow to ratify the treaty.\n\nClose by, four-horse chariots had been driven hard in opposite  \ndirections and had torn Mettus in two \u2013 the man of Alba should  \nhave stood by his promises \u2013 and Tullus was dragging the  \ndeceiver's body through a wood while a dew of blood dripped  \nfrom the brambles.\n\nThere too was Porsenna ordering the Romans to take Tarquin  \nback after they had expelled him, and mounting a great siege  \nagainst the city while the descendants of Aeneas were running  \n650 upon the drawn swords of the enemy in the name of liberty.  \nThere you could see him as though raging and blustering because  \nHoratius Cocles was daring to tear the bridge down and Cloelia  \nhad broken her chains and was swimming the river.\n\nAt the top of the shield Manlius, the keeper of the citadel on  \nthe Tarpeian rock, stood in front of the temple and kept guard  \non the heights of the Capitol. The new thatch stood out rough  \non the roof of Romulus' palace, and here was a silver goose  \nfluttering through the golden portico, honking to announce that  \nthe Gauls were at the gates. There were the Gauls close by,  \namong the thorn bushes, climbing into the citadel under the  \ncover of darkness on that pitch-black night. Their hair was gold,  \n660 their clothing was gold, their striped cloaks gleamed and their  \nmilk-white necks were encircled by golden torques. In each right  \nhand there glinted two heavy Alpine spears and long shields  \nprotected their bodies. Here too Vulcan had hammered out the  \nleaping Salii, the priests of Mars, and the naked Luperci, the  \npriests' conical hats tufted with wool, the figure-of-eight shields  \nwhich had fallen from heaven and chaste matrons leading sacred  \nprocessions through the city in cushioned carriages.\n\nAt some distance from these scenes he added the habitations  \nof the dead in Tartarus, the tall gateway of Dis and the punishments  \nof the damned, with Catiline hanging from his beetling  \ncrag and shivering at the faces of the Furies. There too were the  \n670 righteous, in a place apart, and Cato administering justice.\n\nBetween all these there ran a representation of a broad  \nexpanse of swelling sea, golden, but dark blue beneath the white  \nfoam on the crests of the waves, and all round it in a circle swam  \ndolphins picked out in silver, cleaving the sea and feathering its  \nsurface with their tails.\n\nIn the middle were the bronze-armoured fleets at the battle of  \nActium. There before your eyes the battle was drawn up with  \nthe whole of the headland of Leucas seething and all the waves  \ngleaming in gold. On one side was Augustus Caesar, leading the  \nmen of Italy into battle alongside the Senate and the People of  \n680 Rome, its gods of home and its great gods. High he stood on  \nthe poop of his ship while from his radiant forehead there  \nstreamed a double flame and his father's star shone above his head.  \nOn the other wing, towering above the battle as he led his  \nships in line ahead, sailed Agrippa with favouring winds and  \nfavouring gods, and the beaks of captured vessels flashed from  \nthe proud honour on his forehead, the Naval Crown. On the  \nother side, with the wealth of the barbarian world and warriors  \nin all kinds of different armour, came Antony in triumph from  \nthe shores of the Red Sea and the peoples of the Dawn. With  \nhim sailed Egypt and the power of the East from as far as distant  \nBactria, and there bringing up the rear was the greatest outrage  \nof all, his Egyptian wife! On they came at speed, all together,  \n690 and the whole surface of the sea was churned to foam by the  \npull of their oars and the bow-waves from their triple beaks.  \nThey steered for the high sea and you would have thought that  \nthe Cycladic Islands had been torn loose again and were floating  \non the ocean, or that mountains were colliding with mountains,  \nto see men in action on those ships with their massive, turreted  \nsterns, showering blazing torches of tow and flying steel as the  \nfresh blood began to redden the furrows of Neptune's fields. In  \nthe middle of all this the queen summoned her warships by  \nrattling her Egyptian timbrels \u2013 she was not yet seeing the two  \nsnakes there at her back \u2013 while Anubis barked and all manner  \n700 of monstrous gods levelled their weapons at Neptune and Venus  \nand Minerva. There in the eye of battle raged Mars, engraved  \nin iron, the grim Furies swooped from the sky and jubilant  \nDiscord strode along in her torn cloak with Bellona at her heels  \ncracking her bloody whip. But high on the headland of Actium,  \nApollo saw it all and was drawing his bow. In terror at the sight  \nthe whole of Egypt and of India, all the Arabians and all the  \nShebans were turning tail and the queen herself could be seen  \ncalling for winds and setting her sails by them. She had untied  \nthe sail-ropes and was even now paying them out. There in all  \n710 the slaughter the God of Fire had set her, pale with the pallor of  \napproaching death, driven over the waves by the Iapygian winds  \nblowing off Calabria. Opposite her he had fashioned the Nile  \nwith grief in every line of his great body, opening his robes and  \nwith every fold of drapery beckoning his defeated people into  \nhis blue-grey breast and the secret waters of his river.\n\nBut Caesar was riding into Rome in triple triumph, paying  \nundying vows to the gods of Italy and consecrating three hundred  \ngreat shrines throughout the city. The streets resounded  \nwith joy and festivities and applause. There was a chorus of  \nmatrons at every temple, at every temple there were altars and  \nthe ground before the altars was strewn with the bodies of  \n720 slaughtered bullocks. He himself was seated at the white marble  \nthreshold of gleaming white Apollo, inspecting the gifts brought  \nbefore him by the peoples of the earth and hanging them high  \non the posts of the doors of the temple, while the defeated  \nnations walked in long procession in all their different costumes  \nand in all their different armour, speaking all the tongues of the  \nearth. Here Mulciber, the God of Fire, had moulded the Nomads  \nand the Africans with their streaming robes; here, too, the  \nLelegeians and Carians of Asia and the Gelonians from Scythia  \nwith their arrows. The Euphrates was now moving with a  \nchastened current, and here were the Gaulish Morini from the  \nends of the earth, the two-horned Rhine, the undefeated Dahae  \nfrom beyond the Caspian and the river Araxes chafing at his  \nbridge.\n\nSuch were the scenes spread over the shield that Vulcan made  \n730 and Venus gave to her son. Marvelling at it, and rejoicing at the  \nthings pictured on it without knowing what they were, Aeneas  \nlifted on to his shoulder the fame and the fate of his descendants.\n\n## BOOK 9  \nNISUS AND EURYALUS\n\nWhile this was happening far away in Etruria, Juno, daughter  \nof Saturn, sent Iris down from the sky to bold Turnus, who  \nchanced at that moment to be sitting in a grove sacred to his  \nancestor Pilumnus. These were the words that came to him from  \nthe rosy lips of Iris, daughter of Thaumas: 'There, Turnus, time  \nin its ever-rolling course has brought you unasked what none  \nof the gods would have dared to promise you if you had prayed  \nfor it \u2013 Aeneas has left his city, his allies and his fleet, and gone  \n10 to visit the royal seat of Evander on the Palatine. And as though  \nthat were not enough, he has travelled as far as the remotest  \ncities of Corythus and is arming a band of Lydians, some country  \npeople he has collected. What are you waiting for? This is the  \nmoment to call for your horses and chariots. Do not allow any  \ndelay. Make a surprise attack on their camp and seize it.' At  \nthese words she soared into the sky on poised wings, cutting in  \nher flight a great rainbow under the clouds. The warrior knew  \nher, and raising his hands palms upward to the stars, he called  \nout to her as she flew: 'Iris, glory of the sky, who has sent you  \nhere to me, riding the clouds down to the earth? Why this  \n20 sudden brightness in the air? I see the heights of heaven parting  \nand stars wandering through the vault of the sky. I follow this  \ngreat sign, whoever you are that call me to arms.' When he had  \nspoken these words, he walked to the river's edge and scooped  \nup in his hands the water from its surface as he offered up prayer  \nupon prayer to the gods and burdened heaven with his vows.\n\nThe whole army was soon moving across the open plain, rich  \nin its horses, rich in embroidered apparel, rich in gold. The  \nvanguard was controlled by Messapus, the rear by the sons of  \nTyrrhus, while Turnus, the chief commander, was in the middle  \n30 of the column. It was like the Ganges fed by the steady flow of  \nits seven rivers and silently rising, or like the fertile waters of the  \nNile when it withdraws from the plains and settles back at last  \ninto its own channel. The Trojans saw this distant cloud of  \nblack dust suddenly gathering and the darkness rising on the  \nplain. Caicus was on the rampart on that side and he was the  \nfirst to raise the alarm: 'What is that ball of dark dust rolling  \nalong the plain? Fetch your weapons, fellow-citizens, and fetch  \nthem now! Give out missiles! Mount the walls! The enemy is  \nupon us. To your posts!' With a great clamour the Trojans  \n40 streamed in by all the gates to man the walls, for these were the  \norders they had received from Aeneas, the greatest of warriors,  \nas he left them: if anything should happen in his absence, they  \nwere not to dare take up position for a pitched battle or trust  \nthemselves to the plain, but only to stay on the ramparts and  \ndefend the camp and the walls. So, though shame and anger  \nurged them to join battle, they nevertheless obeyed orders and  \nclosed the gates against the enemy, waiting for them in full  \narmour inside their towers.\n\nBy this time Turnus had taken wing and gone on ahead of the  \nslow-moving column. With twenty picked horsemen he arrived  \n50 at the city before he was expected, riding a piebald Thracian  \ncharger and wearing his gold helmet shaded by red plumes. 'Is  \nthere any man among you, my friends, will come with me and  \nbe first upon the enemy? There!' he cried, and sent his javelin  \nspinning into the air as a signal for battle, then, rising in the  \nsaddle he charged across the plain. His comrades took up the  \ncry and followed him with blood-curdling shouts. They were  \namazed at the faint-heartedness of the Trojans. Why did they  \nnot commit themselves to a fair fight on the level plain? They  \nwere men. Why did they huddle in their camp and not meet  \narms with arms? Turnus in a fury prowled round the walls this  \nway and that, searching for an approach where there was none,  \n60 like a wolf in the dead of night, lying in wait in all the wind and  \nrain by a pen full of sheep, and growling at the gaps in the  \nfence, while the lambs keep up their bleating, safe beneath their  \nmothers; beside himself with anger he storms and rages but  \ncannot reach them; he is worn out by the ravening hunger he  \nhas been so long in gathering and many a day has passed since  \nblood wet his throat \u2013 so did the Rutulian blaze with anger as  \nhe surveyed the walls of the Trojan camp and the pain burned  \nhim to the bone. How could he try to come at them? What  \n70 device could shake out the Trojans shut up there behind their  \nrampart and spill them on to the plain? Ah! The fleet! There it  \nwas moored in a sheltered position along the side of the camp,  \nprotected by the water of the river, and to the landward by  \nramparts. There he made his attack. Burning with fury himself  \nhe demanded fire from his exultant comrades and took up a  \ngreat blazing pine torch in his hand. At this they all bent to the  \ntask, with Turnus there to urge them on. They plundered what  \nfires they could find, and their reeking torches smouldered with  \na pitchy light as Vulcan whirled to the stars dense clouds of  \nsmoke shot through with sparks.\n\nTell me, Muses, what god turned these fierce flames away  \nfrom the Trojans and drove such fire from their ships. The tale  \nwas told in times long past but the fame of it will live for ever.  \n80 When Aeneas was first building his fleet on Mount Ida in Phrygia  \nand preparing to take to the high seas, Berecyntian Cybele  \nherself, the Mother of the Gods, is said to have addressed these  \nwords to great Jupiter: 'O my son, grant my prayer. Now that  \nOlympus is subdued, grant what your dear mother asks of you.  \nOn top of my citadel I had a wood of pine trees which I had  \nloved for many years, a dark grove of black pine and maple  \nwhere men would bring their offerings. These trees I gladly gave  \nto the Trojan warrior when he needed a fleet, but now my heart  \n90 is seized by anxiety and dread. Put all my fears at rest and  \nanswer your mother's prayer. Grant that my ships should not  \nbe wrecked on any of their voyages or overwhelmed by any  \nsquall of wind. Let it stand to their favour that they were born  \non our mountains.' Her son, who turns the stars of heaven in  \ntheir courses, made this reply to his mother: 'What is this you  \nare calling on the Fates to do? What do these words of yours  \nmean? Are ships made by mortal hands to have immortal rights?  \nIs Aeneas to face all his doubts and dangers and never know  \nuncertainty? Is there any god to whom such a privilege has been  \ngranted? No. But when the ships have done their duty, when in  \ndue course they reach the end of their voyaging and are safe in  \nharbour in Ausonia, each one to survive the sea and reach the  \n100 Laurentine fields with the Trojan leader will lose its mortal  \nshape. I shall order all of them to become goddesses of the great  \nocean, like Galatea and Doto, daughters of Nereus, whose  \nbreasts cleave the foam of the waves of the sea.' Jupiter had  \nspoken, ratifying his words by the waters of the Styx, his  \nbrother's river, by the banks and dark whirlpools of that pitch-black  \ntorrent, and at his nod the whole of Olympus shook.\n\nAnd so the promised day had come and the Fates had completed  \nthe allotted time, when the violent attack of Turnus  \nwarned the Mother Goddess to defend her sacred ships from  \n110 these burning brands. A strange light now shone before men's  \neyes and a great cloud seemed to cross the sky from the east,  \nbearing with it votaries of the goddess from Mount Ida. A  \nfearsome voice then fell from the air and filled the ears of Trojans  \nand Rutulians in their armed ranks: 'Do not trouble, Trojans,  \nto defend my ships. Do not take your weapons in your hands.  \nTurnus will burn the sea dry before he can burn these sacred  \npine trees. Go then! You are freed. Go, you goddesses of the  \nsea! The Mother of the Gods commands.' In an instant every  \n120 ship burst the ropes that moored it to the bank, and they plunged  \nlike dolphins, beak first to the bottom. When they returned to  \nthe surface, they were miraculously changed, each one a nymph  \nswimming in the sea.\n\nThe Rutulians were astonished. Messapus himself was afraid  \nand his horses reared. Even Tiber checked his flow with a harsh  \nroaring of his waters as he called back his current from the sea.  \nBut the boldness and confidence of Turnus never wavered.  \nWithout hesitation he set about haranguing his men and whipping  \nup their spirits: 'These portents strike at the Trojans: they  \nmean that Jupiter has taken from them the help they have  \n130 become accustomed to. The ships did not wait to taste Rutulian  \nfire and sword! So now the seas are barred to the Trojans and  \nthey have no hope of escape. By this they have lost one half of  \nthe world, and the land is already in our hands, so many thousands  \nof men are marching under arms from all the races of  \nItaly. This Phrygian talk of destiny and the oracles of the gods  \ndoes not dismay me. Destiny and Venus were satisfied the  \nmoment Trojans set foot on the fertile fields of Italy. I too have  \na destiny, of a different sort \u2013 to cut down with the sword this  \nvicious people that has robbed me of my bride. The sons of  \nAtreus are not the only ones who have suffered, and the people  \nof Mycenae are not the only men who can take up arms. Let  \n140 them not imagine it is enough to have been destroyed once! It  \nshould have been enough for them to sin once. They had no  \nneed to show loathing and contempt for every woman in the  \nworld. Look at them now, all courage and confidence because  \nof this rampart that keeps us from them and these ditches they  \nhave dug to hold us back. This is no sort of barrier to stand  \nbetween them and death. Did they not see the walls of Troy  \nsettling into the flames? And those were fashioned by the hands  \nof Neptune. You are my chosen few. Which one of you is ready  \nto cut through their rampart with the sword and rush into that  \ncamp of cowards? To fight Trojans I do not need the armour  \nVulcan made for Achilles. I do not need a thousand ships, not  \n150 if every man in Etruria went and joined them as allies this  \ninstant. Nor do they need to be frightened of the dark. We shall  \nnot be creeping up on them like cowards to kill the guards all  \nover their citadel and steal their Palladium. We shall not be  \nhiding in the blind belly of a horse. Our plan is to come in  \ndaylight in full view and gird their walls with fire. I shall soon  \nmake sure they realize it is not Greeks they have to deal with or  \nthe army of Pelasgians Hector held off into a tenth year. But the  \nbest part of the day is already spent. For what remains of it you  \ncan now rest yourselves. You have done well. Be of good cheer,  \nin high hopes that we can bring them to battle.' Meanwhile  \n160 Messapus was given the task of blockading the gates with a  \nnight guard and ringing the walls with watch-fires. Fourteen  \nRutulians were chosen to keep watch on the walls, each commanding  \na hundred men with purple crests on their helmets and  \ngleaming with gold. They dispersed, some going to their various  \nduties, others lying out on the grass, enjoying their wine and  \ntipping up the bronze mixing bowls. The watch-fires burned  \nand the guards kept awake by gaming the night away.\n\nThe Trojans looked out on all this from the top of their  \n170 rampart and kept armed guards on all the high points while  \nanxiously checking the gates, building bridges to their outlying  \nfortlets, and bringing up missiles. Mnestheus and the zealous  \nSerestus never relaxed their vigilance. They were the men Father  \nAeneas had appointed to take over the command of the troops  \nand the government of the people should adversity require it.  \nThe whole legion was on the alert along the walls. Lots had been  \ncast for posts of danger and each man was taking his turn to  \nstand guard.\n\nNisus, son of Hyrtacus, was keeper of a gate. This  \nformidable warrior, swift to throw the spear or send the arrow flying, had  \nbeen sent by Ida, the hunters' mountain, to be the comrade of  \n180 Aeneas, and with him came his own comrade, Euryalus, a boy  \nwith the first signs of manhood on cheeks as yet unshaven. There  \nwas no lovelier youth among the people of Aeneas, and no  \nlovelier youth ever put on Trojan armour. They were one in  \nlove, and side by side they used to charge into battle. So now  \ntoo, they were sharing guard duty on the gate, when Nisus said  \nto Euryalus: 'Is it the gods who put this ardour into our minds,  \nor does every man's irresistible desire become his god? My mind  \nis not content to rest in peace and quiet but has long been driving  \nme to rush into battle or into some great enterprise. You see the  \nRutulians there with just a few scattered lights piercing the  \n190 darkness, how sure they are of everything, lying sunk in sleep  \nand wine, and silence everywhere. Just listen to what I am  \nthinking and to the plan beginning to form in my mind. The  \npeople and the fathers, they are all clamouring for Aeneas to be  \nsummoned and messengers sent to tell him exactly what is  \nhappening. If they promise to give you what I ask \u2013 all I want is  \ncredit for the deed \u2013 I think I can find a way round the foot of  \nthat hill to the city of Pallanteum.' Euryalus was overcome,  \npierced to the heart with a great love of glory, and in an instant  \nhe replied in these words to his ardent friend: 'So you do not  \n200 want me as your comrade on this great expedition, and I am to  \nlet you go alone into dangers like this? This is not how I was  \nbrought up by my father Opheltes during the Greek terror and  \nour sufferings at Troy, and he knew all about war. Nor is this  \nhow I have conducted myself with you, in following to the end  \nthe Fates of great-hearted Aeneas. I have here a heart that  \ndespises the light, that would gladly spend life to buy the honour  \nyou are striving for.' To this Nisus replied: 'So may great Jupiter,  \nor whatever god looks with favour on this undertaking, bring  \nme back to you in triumph, I swear I never had any such fears  \n210 about you. That would have been a sin. But if some chance or  \nsome god were to lead me into disaster \u2013 and you know how  \nmany things can happen in dangerous affairs like this \u2013 I would  \nwish you to go on living. You are young and your claim on life  \nis greater than mine. There would then be someone to consign  \nmy body to the earth if it is rescued from the battlefield or  \nrecovered by ransom, or if some fortune forbids that \u2013 and we  \nknow her ways \u2013 to make offerings for me here and honour me  \nwith an empty tomb. Besides, let me not be the cause of such  \nheartbreak to your mother, who of all the mothers of Troy is  \nthe only one who has dared to follow her son here with never a  \nthought for the walls of great Acestes.' 'One feeble argument  \nafter another,' replied Euryalus, 'and all to no purpose. My  \n220 mind is made up and you have done nothing to change it. Let  \nus go, and quickly.' So saying, he woke sentries to take over and  \nkeep guard for Nisus and himself. They left their post and  \nmarched off side by side to look for prince Ascanius.\n\nOver the whole world the creatures of the earth were relaxed  \nin sleep, all resting from their cares, and their hearts had forgotten  \ntheir labours; but the chosen warriors who were the great  \nleaders of the Trojans were holding a council on matters of the  \nhighest importance to the kingdom. What were they to do now?  \n230 Who would go as a messenger to Aeneas? As they stood there  \non the level ground in the middle of the camp, leaning on  \ntheir long spears and carrying their shields, Nisus and Euryalus  \nsuddenly arrived in great haste and asked to be admitted, saying  \nthat their business was urgent and well worth listening to. Seeing  \ntheir excitement, Iulus was the first to welcome them and invited  \nNisus to speak. These were the words of the son of Hyrtacus:  \n'Give us a fair hearing, sons of Aeneas. Do not judge what is  \nsaid by the age of the speakers. The Rutulians have fallen quiet,  \ndeep in their drunken sleep, and we have seen a place for an  \nambush, some open ground where the two roads meet by the  \ngate nearest the sea. There the ring of watch-fires is broken and  \n240 the smoke is rising black to the stars. If you allow us to take  \nthis opportunity to go and look for Aeneas and the city of  \nPallanteum, you will soon see us coming back laden with booty  \nand much slaughter done. We have no doubts about the way to  \ngo. We always hunt there and have seen the first houses of the  \ncity in the dark valleys. We have explored the whole river.'\n\nIt was Aletes, heavy with years and mature in judgement, who  \nnow replied: 'O gods of our fathers, in whose divine hands Troy  \nstill remains, in spite of all, it is not your will utterly to destroy  \nthe Trojans, if you have put such firmness of mind and heart  \n250 into our young warriors,' and as he spoke he clasped the right  \nhands of both of them and laid his hands on their shoulders while  \nthe tears ran down his cheeks and face: 'Can any recompense be  \nfound for you?' he cried. 'Can anything match the glorious  \ndeeds you propose? The first and richest reward will come from  \nthe gods and from your own virtue, but the others will soon  \nfollow from a grateful Aeneas, and young Ascanius for the rest  \nof his life will never forget such a service.' 'More than that,'  \ninterposed Ascanius, 'my whole life hangs upon the return of  \n260 my father and I call upon you both to witness, by the great  \nPenates and Lar of Assaracus, and the shrine of white-haired  \nVesta, I now place all my fortunes and all my hopes for the  \nfuture in your hands, Nisus. Call back my father. Bring him  \nback to my sight. If he is restored there can be no cause for grief.  \nI shall give you two solid silver embossed cups which he took at  \nthe fall of Arisba, and with them a pair of tripods, two great  \ntalents of gold and an ancient mixing bowl given him by Dido  \nof Sidon. But if he succeeds in taking Italy and winning the  \ncrown, while he is presiding over the distribution of booty in  \nhis hour of victory \u2013 you have seen the horse that Turnus rides,  \n270 you have seen him all golden in his armour \u2013 I shall exclude  \nfrom the lot that horse, the shield and the scarlet plumes, and  \nthese will now be yours, Nisus, as your reward. In addition my  \nfather will give you twelve chosen matrons and twelve prisoners  \nof war, each with his armour, and all the lands on the plain now  \nheld by king Latinus. But as for you, Euryalus, although you  \nare a boy and not so far ahead of myself in the race of life, I  \nrevere you and take you wholly into my heart, embracing you  \nas my comrade, whatever may lie before us. Whatever I may do,  \nI shall look for no glory that is not shared with you. In war or  \n280 in peace, whatever I say or do, my whole trust will be placed  \nin you.'\n\nTo this Euryalus replied: 'The day shall never come when I  \nshall be found unequal to acts of courage like this, if only the  \nfall of fortune is in our favour tonight, and not against us. But  \none thing I ask of you, more precious than any gifts: I grieve for  \nmy mother of the ancient line of Priam. The land of Troy could  \nnot hold her when she came away with me, nor did the walls of  \nking Acestes. As I now leave her, she knows nothing of the  \ndanger I am entering upon, whether it be great or small, and I  \nhave taken no farewell of her because \u2013 and I swear it by the  \nNight and your own right hand \u2013 I could not bear to see my  \n290 mother weep. But comfort her in her helplessness, I beg you,  \nand support her in her desolation. Let me take with me the hope  \nthat you will do this and I shall go all the more boldly into  \nwhatever dangers lie before me.' The Trojans were overcome  \nand wept, the fair Iulus most of all, as this image of his love for  \nhis own father touched his heart, and he replied: 'You can be  \ncertain that everything I do will be worthy of your great enterprise.  \nYour mother will be my mother in everything but the  \nname Creusa. The woman who gave birth to such a son will  \nreceive no ordinary gratitude. I have promised you rewards  \nwhen you return in triumph. Whatever the outcome of your  \n300 bravery, I swear by this head of mine, by which my father used  \nto swear, that these same promises will hold good for your  \nmother and your kin.' So he spoke, weeping, and in that moment  \nhe took from his shoulder a gilded sword that Lycaon of Cnossus  \nhad fashioned with consummate art and fitted in an ivory scabbard  \nto hang perfectly at his side, while Mnestheus gave Nisus  \na rough hide stripped from a lion, and trusty Aletes changed  \nhelmets with him. As soon as they were armed they marched  \n310 off, and all the leading Trojans, young and old, escorted them  \nto the gates with their prayers. Foremost among them was the  \nfair Iulus, bearing beyond his years a man's load of cares and a  \nman's spirit. He gave them many commissions to bear to his  \nfather, but they were all futile. The wind scattered them among  \nthe clouds.\n\nThey moved off and crossed the ditch, making their way  \nunder cover of night to the camp that would be their death, but  \nnot before they had brought death to many others. They could  \nsee men sprawling in drunken sleep all over the grass and  \nchariots standing along the river bank with their poles in the air  \nand a tangle of men's bodies and armour and wine vessels  \n320 among the reins and wheels. Nisus was the first to speak: 'Now,  \nEuryalus,' he said, 'my right hand must show its mettle. The  \nhour calls out for it. Our road goes this way. You keep guard to  \nthe rear in case a party of men creeps up on us from behind, and  \nlook well into the distance. I shall make havoc here and clear a  \nbroad path for you.' So he spoke and then had done with words.  \nWith sword drawn he made for proud Rhamnes who happened  \nto be propped up there on a deep pile of rugs, his whole chest  \nheaving as he slept. A king he was, and a prophet cherished by  \na king, by Turnus. But not all his prophesying could drive from  \nhim the plague of death. Nisus then caught three of Rhamnes'  \nattendants lying in a heap among their weapons, then the  \n330 armour-bearer of Remus and his charioteer among the hooves  \nof the horses. Their heads were lolling. He cut them off. Next  \nhe removed the head of their master Remus and left the blood  \ngurgling out of his trunk and warming the ground as the black  \ngore soaked through the bedding. Lamyrus also he slew, and  \nLamus and young Serranus, a handsome youth who had  \ngambled late into the night. There he lay overcome by all the  \nwine of Bacchus he had drunk. He would have been happy if he  \ncould have made his gambling last the night and kept it up till  \n340 daylight. Nisus was like a lion driven mad with hunger and  \nravening through pens full of sheep, dumb with fear, while he  \ngrowls from jaws dripping with blood as he mauls and champs  \ntheir soft flesh.\n\nMeanwhile there was no less slaughter from the hand of  \nEuryalus. He too was in a blazing frenzy as he crept up on a  \ngreat crowd of nameless warriors lying unconscious in his path,  \nFadus and Herbesus, Rhoetus and Abaris. Rhoetus was awake  \nand saw it all, so hid in panic behind a great mixing bowl. But  \nwhen Euryalus came near him, he rose and Euryalus plunged  \nhis sword to the hilt in his chest. When he withdrew it, the  \nwhole life of Rhoetus flooded out after it. As he lay there dying,  \n350 still vomiting his crimson life's breath and bringing up wine and  \ngore together, Euryalus was already prowling on, hot for blood.  \nHe was soon making for Messapus and his comrades, where he  \nsaw the dying embers of the watch-fires and the horses tethered  \nin good order cropping the grass, when Nisus had a few words  \nto say to him \u2013 for he noticed that Euryalus was being carried  \naway by bloodlust and greed: 'Let us make an end,' he said.  \n'Daylight is no friend of ours and it will soon be here. Our  \nenemies have taken enough punishment and we have cut our  \npath through the middle of them.' They left behind them many  \npieces of men's armour wrought in solid silver, and mixing  \nbowls besides, and lovely rugs, but Euryalus took Rhamnes'  \n360 medallions and his gold-studded belt. Long ago the wealthy  \nCaedicus had sent them from his home as gifts to Remulus of  \nTibur to form a guest-friendship with him. When Remulus was  \ndying, he gave them to his grandson, and after his death they  \npassed to the Rutulians as spoils of war. Euryalus now snatched  \nthem up and put them round his brave shoulders, but little good  \nwere they to do him. He also put on the helmet of Messapus  \nwith its gorgeous plumes, and they left the camp and made  \nfor safety.\n\nAt this moment, while the rest of the Latin army was waiting  \nin battle order on the plain, a detachment of cavalry had been  \nsent out from their city and was now on its way with dispatches  \n370 to Turnus, three hundred of them, all carrying shields, under  \nthe command of Volcens. They were approaching the camp and  \ncoming up to its ramparts when they saw Nisus and Euryalus  \nin the distance, veering off along the road to the left. Euryalus  \nhad forgotten about the helmet, and its glittering betrayed him,  \nreflecting the rays of the moon in the dim shadows of the night.  \nThe enemy saw and did not fail to act. 'Halt there, you men!'  \nshouted Volcens from the head of his column. 'Why are you on  \nthe road? Who are you? Why are you armed? Where are you  \ngoing?' They offered no reply, but ran off into the trees, putting  \ntheir trust in the darkness of the night. The horsemen spread  \n380 out along each side of the wood they knew so well, blocking the  \ntracks that led in, and putting guards on every approach. It was  \na rough wood full of dense undergrowth and dark ilex trees, all  \nof it choked with thick brambles, and the path glimmered only  \nhere and there among the faint tracks left by animals. Euryalus  \nwas held back by the darkness under the trees and by the weight  \nof his booty, and in his fright he lost his way. But Nisus escaped.  \nWithout knowing it he had come through the enemy and the  \narea later to be known as Alban, taking its name from the  \ncity of Alba, but in those days king Latinus had high-fenced  \nenclosures there for his cattle. He now stopped and looked back  \n390 for his friend, but could not see him. 'Poor Euryalus,' he cried.  \n'Where have I left you? Where can I look for you?' and even as  \nhe spoke, he was beginning to go back over his path through  \nthe wood with all its deceptive twists and turns, retracing every  \nremembered step as he wandered through the silent undergrowth.  \nHe heard horses. He heard the noise of the pursuers  \nand their signals, and in no time shouts reached his ears and he  \nsaw Euryalus. Lost in the treacherous darkness of the wood and  \nconfused by the sudden tumult, he had been caught by the whole  \nenemy troop and was now being carried off, still struggling  \ndesperately against all the odds. What was Nisus to do? How  \ncould he rescue his young friend? How should he attack? What  \n400 weapons could he use? Should he throw himself into the thick  \nof their swords and rush through wound upon wound to a  \nglorious death? In that instant he drew back his arm, and  \nbrandishing his throwing spear, he looked up to the moon in  \nheaven and prayed in these words: 'O goddess, daughter of  \nLatona, O glory of the stars and guardian of the groves, be with  \nme now and help me in my hour of trouble. If ever my father  \nHyrtacus has offered gifts for me at your altars, if ever I myself  \nhave enriched them with the spoils of my hunting, hanging my  \nofferings in the dome of your temple or nailing them on your  \nholy gables, guide my weapons through the air and grant that I  \n410 may throw this troop of my enemies into confusion.' When he  \nhad spoken, he hurled his spear with the whole force of his  \nbody. Parting the shadows of the night it flew towards Sulmo,  \nwhose back was turned, and there it struck and broke, sending  \na splinter through his diaphragm. He rolled over, vomiting a  \nstream of warm blood from his chest in the chill of death, and  \nheaving his flanks in deep-drawn agonies. While the enemy were  \nlooking round in all directions, there was Nisus, emboldened  \nby his success, with another shaft ready by his ear, poised to  \naim. They were still in tumult when the spear came whistling  \nand caught Tagus in the middle of the forehead, went through  \n420 the brain, and stuck there, growing warm. Volcens was wild  \nwith rage, but nowhere could he see the thrower and he could  \nnot decide where to direct the fury of his assault. 'Never mind!'  \nhe shouted. 'For the moment, you and your warm blood will  \npay me for both of them!' and he drew his sword and rushed at  \nEuryalus. This was too much for Nisus. Out of his mind with  \nterror and unable to endure his anguish, he broke cover, shouting  \nat the top of his voice: 'Here I am! Here I am! I am the one  \nwho did it! Aim your weapons at me, you Rutulians! The whole  \nscheme was mine. He is innocent. He could not have done it. I  \nswear by this sky above me and the stars who know the truth,  \n430 his only offence is to have loved the wrong friend too much!'  \nHe was still speaking as the sword was driven through the ribs  \nof Euryalus, full force, shattering his white breast. He rolled on  \nthe ground in death, the blood flowed over his beautiful body,  \nhis neck grew limp and the head drooped on his shoulders, like  \na scarlet flower languishing and dying when its stem has been  \ncut by the plough, or like poppies bowing their heads when the  \nrain burdens them and their necks grow weary. But Nisus rushed  \ninto the thick of the enemy, looking only for Volcens. Volcens  \n440 was the only thought in his mind. The Rutulians gathered round  \ntheir leader and in close fighting threw Nisus back again and  \nagain as he came at them from one side after another, but he  \nbore on none the less, whirling a sword like lightning till he met  \nthe Rutulian face to face and buried it in his mouth as he opened  \nit to shout. So, in the moment of his own dying, he cut off the  \nbreath of his enemy. Then, pierced through and through, he  \nhurled himself on the dead body of his friend and rested there  \nat last in the peace of death.\n\nFortune has favoured you both! If there is any power in my  \npoetry, the day will never come when time will erase you from  \nthe memory of man, while the house of Aeneas remains by the  \nimmovable rock of the Capitol and the Father of the Romans  \nstill keeps his empire.\n\n450 The victorious Rutulians had collected their booty and their  \nspoils and carried the body of Volcens to their camp, weeping  \nas they went. There was no less sorrow waiting for them there,  \nwhen they found Rhamnes dead, and with him Serranus and  \nNuma and all their other leaders who had been killed in that  \none night of slaughter. A great crowd gathered round the dead  \nand dying heroes and the ground was running with rivers of  \nnewly shed blood, still warm and foaming. Between them they  \nrecognized the spoils, the shining helmet of Messapus, and the  \nmedallions which had cost so much sweat to recover.\n\n460 By now Aurora was just leaving the saffron bed of Tithonus  \nand sprinkling her new light upon the world. The sun was soon  \nstreaming over the earth and soon all things stood revealed in  \nits light. Turnus, in full armour himself, was rousing his men to  \narms, and each of the leaders was taking his own troop into  \nbattle in ranks of bronze, whipping up their anger with different  \naccounts of the night's work. They even stuck the heads of  \nEuryalus and Nisus on spears \u2013 what a sight that was! \u2013 and  \nparaded along behind them shouting. Aeneas' men, long-enduring,  \ndrew up in battle order to face them on the walls on  \ntheir left flank \u2013 the right was guarded by the river \u2013 and they  \n470 manned their great ditches and stood on their high towers  \nstricken with grief and shocked by the sight of the heads of the  \ncomrades they knew so well, impaled on spears and dripping  \nblack gore.\n\nMeanwhile Rumour flew with the news on her swift wings  \nthrough the whole terrified city of the Trojans, and came gliding  \ninto the ears of the mother of Euryalus. In that instant the  \nwarmth left her very bones, the shuttle was dashed from her  \nfingers and its thread unwound. Crazed with grief she rushed  \nout, and wailing as women do and tearing her hair, she made  \nfor the front ranks of the army on the walls. With no thought  \n480 for the presence of men, with no thought of the danger of flying  \nweapons, she stood there on the ramparts and filled heaven with  \nher cries of mourning: 'Is this you I am looking at, Euryalus?  \nHow could you leave me alone, so cruelly, you who were the  \nlast comfort of my old age? Could not your poor mother have  \nbeen allowed a few last words with you, before you went on  \nthat dangerous expedition? So now you lie in a strange land,  \nand your body is food for the dogs and the birds of Latium! I  \nam your mother and did not walk before you at your funeral;  \nnor close your eyes, nor wash your wounds, nor cover you with  \nthe robe I have been weaving for you day and night with what  \nspeed I could, finding in my loom some solace for the cares of  \n490 age. Where am I to go to look for you, my son? What piece of  \nearth holds your mutilated body and dismembered limbs? Is this  \nhead all you bring back to me? Is that what I have followed over  \nland and sea? Strike me, you Rutulians, if you have any human  \nfeelings! Throw all your spears at me! Let me be the first to die.  \nOr will you take pity on me, Great Father of the Gods, and blast  \nmy detested body into Tartarus with your lightning, since I can  \nfind no other way to end this bitter life?' Sorrow like this was  \ntoo much for the Trojans to bear. The sound of mourning was  \nheard all through the army. Their strength was broken. They  \nwere losing their appetite for battle and her presence was fanning  \n500 the flames of their grief. At a word from Ilioneus and the bitterly  \nweeping Iulus, Idaeus and Actor came and took her between  \nthem back into her house.\n\nThe ringing bronze of the trumpet gave out its shrill and  \nterrible note from close at hand. The shouting rose and the  \nheavens bellowed in reply. The Volsci all at once rushed the  \nwalls with their shields locked in tortoise formation and tried  \nto fill in the ditches and tear down the rampart. Some were  \nlooking for a point of access and putting up scaling ladders  \nwhere the line of defenders was strung out along the walls, and  \nlight could be seen in the breaks between them. From their side  \n510 the Trojans showered down missiles of every kind, and pushed  \nthe ladders off with stout poles \u2013 in their long war they had  \nlearned how to defend walls \u2013 and they rolled great heavy rocks  \ndown on the enemy to try to break their armoured formations,  \nbut in their close-packed tortoise they cheerfully endured whatever  \nfell on them. But they still did not succeed. For where a  \nsolid mass of Rutulians was threatening the walls, the Trojans  \nrolled along a huge block of stone and sent it crashing down on  \nthem to loosen their interlocking shields and cut a great swathe  \nthrough them. After this the bold Rutulians no longer cared to  \nfight blind under cover of their shields but strove to clear the  \n520 defenders off the ramparts with a barrage of missiles. At another  \nsection of the wall Mezentius was brandishing a torch of Etruscan  \npine and a fearful sight he was as he came at them with fire  \nand smoke. Messapus, son of Neptune and tamer of horses, was  \ncutting a way through the rampart and shouting for scaling  \nladders.\n\nI pray to you, Calliope, and to your sister Muses, to breathe  \nupon me as I sing of the death and destruction wrought by the  \nsword of Turnus and to tell who sent down to Orcus each  \nwarrior that died. Unroll with me now the mighty scroll of war.\n\n530 There was a tower, well placed and of commanding height,  \nwith high connecting bridges. The Latins were trying to take it  \nby main force, striving with all their powers to bring it down,  \nwhile the Trojans packed inside tried to defend it by throwing  \nrocks and sending a hail of weapons through the loopholes.  \nTurnus, who was leading the attack, hurled a blazing torch  \nwhich set fire to the side of the tower. Fanned by the wind, the  \nflames took hold of the planking and ate into the upright posts.  \nInside all was confusion, terror and desperate attempts to escape  \nthe heat. As everyone crowded together to take refuge on the  \n540 side away from the flames, all at once the whole sky seemed to  \nthunder and the tower toppled over with the weight, and men  \nplunged to the ground in their death throes with the massive  \nfabric following them down, impaling them on their own  \nweapons and driving the broken timbers through their breasts.  \nOnly Helenor and Lycus were able to escape. Helenor was a  \nyoung man, son of the king of Maeonia and the slave girl  \nLicymnia, who had reared him in secret and sent him to Troy  \nunder arms although this had been forbidden. His equipment  \nwas light, a sword with no scabbard and an inglorious shield of  \nplain white, and he found himself caught in the middle of the  \n550 thousands of men who fought with Turnus, looking at the battle  \nlines of the Latins drawn up on all sides of him, like a wild beast  \ntrapped in a dense ring of hunters; it rages against the steel, and  \nwith full understanding it hurls itself to its death by springing  \non to the hunting spears \u2013 just so did young Helenor leap into  \nthe middle of his enemies, rushing to his death where he saw the  \nsteel was thickest. But Lycus was far fleeter of foot. He ran the  \ngauntlet of the enemy and their weapons as far as the wall.  \nThere as he was trying to take hold of the top of the outworks  \nand reach the outstretched hands of his comrades, Turnus, who  \n560 had been pursuing him with his javelin, came to gloat over him:  \n'You fool! Did you think you could escape my hands?' and even  \nas he shouted, he seized hold of him where he hung and tore  \nhim down, taking a great section of the wall with him, like the  \neagle, the armour-bearer of Jupiter, seizing in his hooked talons  \na hare or the white body of a swan and soaring into the air with  \nit; or like the wolf of Mars tearing a lamb out of the sheep pen,  \nand loud and long will be the bleating of its mother, as she looks  \nfor it.\n\nThe shouting rose on every side. The attackers levelled the  \nrampart, filled in the ditch and tossed blazing torches high on  \n570 to the roofs. Lucetius, who was coming to set fire to a gate, was  \nlaid low by a rock thrown by Ilioneus, a huge block torn out of  \na mountain. Liger felled Emathion with a javelin; Asilas brought  \ndown Corynaeus with an arrow he never saw in all its long flight.  \nCaeneus slew Ortygius; Turnus slew the victorious Caeneus;  \nTurnus also slew Itys and Clonius, Dioxippus and Promolus,  \nthen Sagaris and Idas, who was standing out in front of the  \nhighest towers. Privernus was killed by Capys: Themillas had  \nfirst grazed him with a light spear and the fool had thrown his  \nshield away to put his hand to the wound. So the winged arrow  \n580 flew and, plunging deep into his left side, it broke the passages  \nof his life's breath with a mortal wound. The son of Arcens  \nstood there in gorgeous armour, resplendent in his embroidered  \ncloak and Spanish purple, a noble sight to see. He had been sent  \nto war by his father, who had reared him in his mother's grove  \non the banks of the river Symaethus where the people of Sicily  \nmade their offerings at the rich altar of the mild god Palicius.  \nMezentius laid down his spears. Then, whirling his sling three  \ntimes round his head, he shot the hissing bolt and struck the son  \nof Arcens full in the middle of the forehead. Melting in its flight,  \nthe lead bullet split his skull and stretched him full length on  \nthe sand.\n\n590 It was then, men say, that Ascanius first shot in war the swift  \narrow which till this time had only driven wild animals to terror  \nand flight, and his was the hand that laid the brave Numanus  \nlow. This was a warrior whose family name was Remulus, and  \nnot long before he had been joined in marriage to the younger  \nsister of Turnus. His heart was swollen with pride at the royal  \nrank he had newly acquired, and he stepped out in front of the  \nbattle line, swaggering and shouting abuse, some fit and some  \nunfit to be repeated: 'You have been sacked twice already, you  \nPhrygians! Are you not ashamed to be cooped up again in a  \nsiege behind ramparts with only a wall between yourselves and  \n600 death! Are you the men who came here to fight us for our brides?  \nIs it some god that has driven you to Italy? Or some madness?  \nYou will not find here the sons of Atreus or the fictions and fine  \nwords of Ulixes! We are men of a hardy stock. We take our  \nbabies down to the river the moment they are born and harden  \nthem in the icy water. Our boys stay awake all night and weary  \nthe woods with their hunting. For games they ride horses and  \nstretch the bow to the arrow. Our men endure hard labour and  \nlive spare, subduing the land with the mattock and shaking the  \ntowns of their enemies with war. We are worn hard by iron all  \n610 our lives and turn our spears to goad our oxen. There is no  \nsluggish old age for us to impair the strength and vigour of our  \nminds. We crush our grey hair into the helmet, and our delight  \nis always to bring home new plunder and live off what we take.  \nBut you like your clothes dyed with yellow saffron and the  \nbright juice of the purple fish. Your delight is in dancing and  \nidleness. You have sleeves to your tunics and ribbons to keep  \nyour bonnets on. You are Phrygian women, not Phrygian men!  \nAway with you over the heights of Mount Dindymus, where  \nyou can hear your favourite tunes on the double pipe. The  \ntambourines are calling you and the boxwood fifes of the Berecyntian  \n620 Mother of Mount Ida. Leave weapons to the men. Make  \nway for the iron of our swords.'\n\nSo he hurled his abuse and threats till Ascanius could endure  \nit no longer. Turning to face him, he drew his bow and stretched  \nthe horsegut string, and as he stood there with his arms straining  \nwide apart, he prayed first to Jupiter with this vow: 'All powerful  \nJupiter, bless now this my first trial of arms, and with  \nmy own hands I shall bring yearly offerings to your temple and  \nset before your altar a milk-white bullock, with gilded horns,  \nholding its head as high as its mother's, already butting with its  \n630 horns and kicking up the sand with its hooves.' The Father  \nheard and thundered on the left from a clear sky, and the sound  \nof the death-dealing bow of Ascanius mingled with the sound  \nof the thunder. The arrow had been drawn back, and it flew  \nwith a fearful hiss straight through the head of Remulus, its iron  \npoint piercing his hollow temples. 'Go, Remulus!' he cried, 'and  \nmock brave men with proud words! This is the reply to the  \nRutulians from the twice-sacked Phrygians!' Ascanius said no  \nmore than this, but the Trojans followed it with a shout of joy,  \ntheir spirits raised to the skies.\n\nAt that moment Apollo, the youthful god, whose hair is never  \ncut, chanced to be seated on a cloud, looking down from the  \n640 expanse of heaven on the armies and cities of Italy, and he  \naddressed these words to the victorious Iulus: 'You have become  \na man, young Iulus, and we salute you! This is the way that  \nleads to the stars. You are born of the gods and will live to be  \nthe father of gods. Justice demands that all the wars that Fate  \nwill bring will come to an end under the offspring of Assaracus.  \nTroy is not large enough for you.' At these words he plunged  \ndown from the heights of heaven, parting the breathing winds,  \nand made for Ascanius, taking on the features of old Butes.  \nButes had once been armour-bearer to the Dardan Anchises and  \nthe trusted guard of his door, and Aeneas had then appointed  \nhim as companion to his son Ascanius. This was the guise in  \n650 which Apollo came, the old man Butes to the life \u2013 voice,  \ncolouring, white hair, weapons grimly clanking \u2013 and these were  \nthe words he spoke to Iulus in the flush of his victory: 'Let that  \nbe enough, son of Aeneas. Numanus has fallen to your arms  \nand you are unhurt. Great Apollo has granted you this first taste  \nof glory and does not grudge you arrows as sure as his own.  \nYou must ask for no more, my boy, in this war.' So began  \nApollo, but while speaking, he left the sight of men, fading  \nfrom their eyes into the insubstantial air. The Trojan leaders  \n660 recognized the god. They knew his divine arrows and the quiver  \nthat sounded as he flew. So, although Ascanius was thirsting for  \nbattle, they held him back, urging upon him the words of  \nPhoebus Apollo and the will of the god. But they themselves  \nwent back into battle and put their lives into naked danger. The  \nshouting rang round the ramparts all along the walls. They bent  \ntheir deadly bows and twisted their spear thongs till the ground  \nwas strewn with missiles. Shield and round helmet rang with  \nthe blows as fiercer and fiercer raged the battle. It was like a  \ngreat shower from the west drumming on the earth in the rainy  \nseason when the Kids are rising, or like hailstones dropping  \n670 from the clouds into the sea when the south wind is blowing  \nand Jupiter hurls down squalls of rain in his fury and bursts the  \nhollow thunderclouds in the sky.\n\nPandarus and Bitias, sons of Alcanor of Mount Ida, had been  \nbrought up by the wood nymph Iaera in the grove of Jupiter  \nand they were built like the pines and mountains of their fatherland.  \nSo sure were they of their weapons that they now flung  \nopen the gate that had been entrusted to them by their leader's  \ncommands, and took it upon themselves to invite the enemy to  \ncome within the walls. They themselves stood inside at the  \nready, like twin towers, one on the right and one on the left,  \narmed in steel, with their crests flashing high on their heads.  \nThey were like a pair of tall oaks by a flowing river, on the  \n680 banks of the Po or by the lovely Adige, holding their unshorn  \nheads up to the sky with their high tops nodding in the breeze.  \nAs soon as they saw the gate open, the Rutulians came bursting  \nin. Quercens and Aquiculus in splendid armour, impetuous  \nTmarus and Haemon, son of Mars, but instantly with all their  \nmen they either turned and ran or gave up their lives on the very  \nthreshold of the gate. The fury mounted in all their hearts as  \nthey fought. Trojans now came crowding to the spot and not  \n690 only joined in the fray but also dared to sally out further and  \nfurther in front of the gate.\n\nMeanwhile Turnus, the Rutulian commander, was raging and  \nstorming and creating havoc in another part of the field, when  \na message arrived to say that the enemy were hot with the  \nRutulian blood they were now spilling and that open gates were  \non offer. Turnus instantly abandoned the work he had in hand  \nand rushed to the Trojan gate in a savage rage to meet these  \narrogant brothers. The first man to fall to his javelin was Antiphates  \n\u2013 for he was the first to confront him. Antiphates was the  \nbastard son of great Sarpedon by a Theban mother. The spear  \nof Italian cornel wood flew through the unresisting air, went in  \n700 by his belly and twisted upwards deep into his chest. A wave of  \nfrothing blood welled out of the black hole of the wound, and  \nthe steel grew warm where it had lodged in the lung. Then  \nErymas and Meropes fell to his hand; then Aphidnus; then Bitias  \nhimself for all the fire that flashed from his eyes and the roaring  \nfury of his heart. No javelin for him. He was not the man to  \nyield his life to a javelin. It was an artillery spear with an iron  \nhead a cubit long and a ball of lead at its butt which came rifling  \nthrough the air with a loud hiss and the force of a thunderbolt.  \nThe two bull-hides of his shield did not resist it, nor did his  \ntrusty breastplate with its overlapping scales of gold. His huge  \nbody collapsed and fell. The earth groaned and the mighty shield  \n710 thundered as it came down on top of him. It was like the fall of  \na stone pile by the shore at Euboean Baiae; men first build it to  \nits massive height and then they let it down into the sea, and it  \nspreads ruin all along its length, grinding the sea-bed as it settles  \nin the shallows; the water boils, the black sand rises, the high  \nrock of Procida is shaken, and Inarime with it, the hard bed laid  \nfor Typhoeus at Jupiter's command.\n\nNow Mars, mighty in war, put new spirit and strength into  \nthe Latins and twisted a sharp goad into their flesh, while  \n720 sending Flight and black Fear upon the Trojans. Now that their  \nchance had come to fight, the Latins gathered from all sides and  \nthe God of War stormed their hearts. When Pandarus saw his  \nbrother stretched out in death and knew how his fortunes stood  \nand the turn events were taking, he put his broad shoulder to  \nthe gate with all his force and heaved it shut on its hinges,  \nleaving many of his own people cut off outside the walls with a  \nhard battle to fight, but taking in those who came running, and  \nshutting them in with himself. Fool that he was! He did not see  \nthe Rutulian king bursting into the city in the middle of the  \n730 press. By his own act he penned him in like a great tiger among  \nhelpless cattle. In that instant a new light shone from the eyes  \nof Turnus. He clashed his armour with a fearsome noise, the  \nblood-red crest trembled on his head, his shield flashed lightning.  \nSuddenly Aeneas' men recognized him \u2013 the hated face, the huge  \nbody \u2013 and were thrown into confusion. But the giant Pandarus  \nleapt forward to confront him, burning with anger at the death  \nof his brother: 'This is not your bridal chamber in the palace of  \nAmata!' he shouted. 'Turnus is not safe in the middle of Ardea  \nbehind his father's walls. This is the camp of your enemies and  \n740 there is no way out.' Turnus replied, smiling calmly: 'If there is  \nany courage in you, then come and fight. You will soon be able  \nto tell Priam that here too you found an Achilles!' At these  \nwords Pandarus took a spear of rough, knotted wood with its  \nbark unplaned and hurled it with all his force. As it flew to  \nwound Turnus, the winds caught it, Juno deflected it and it  \nlodged in the gate. 'You will not escape this weapon of mine,'  \ncalled out Turnus, 'which I brandish here in my right hand. This  \nsword is wielded by a different arm, and gives a deeper wound.'  \nWith these words he lifted it above his head, rising with it, and  \n750 struck Pandarus between the temples. The blade went straight  \nthrough the middle of the forehead and parted the smooth,  \nyoung cheeks. The wound was hideous. He fell with a crash and  \nthe ground shook with the weight of him. As he lay dying he  \nstrewed around his nerveless limbs and armour blooded with  \nbrains, and the two halves of his head hung on his two shoulders.\n\nThe Trojans turned and ran in terror. If at that moment the  \nvictor had thought of breaking the bolts and letting his comrades  \nin through the gates, that would have been the end of the war  \n760 and the end of the Trojan race, but instead his mad lust for  \nblood drove him upon his enemies in an ecstasy of passion. First  \nhe caught Phaleris and Gyges, slitting his hamstrings. He then  \ntook their spears, and with Juno lending him strength and spirit,  \nhe hurled them into the backs of the retreating enemy. Next he  \nsent Halys to keep them company and Phegeus, the spear passing  \nthrough his shield; then Alcander, Halius, Noemon and Prytanis,  \nwho were on the walls in the thick of battle and did not  \nknow he was inside. Now Lynceus was coming at him and  \ncalling on his comrades for help. Turnus from the rampart on  \n770 his right stopped him short with one flashing stroke of his sword,  \na blow from close range that severed the head and sent it flying  \nfar from the body, helmet and all. Next he brought down  \nAmycus, that mighty hunter and slayer of wild beasts \u2013 no man  \nbetter to charge the spear-point with poison or smear the tip of  \nthe arrow; then Clytius, son of Aeolus, and Cretheus, that dear  \ncompanion of the Muses, Cretheus, a great lover of song and of  \nthe lyre, a great setter of poems to the strings, always singing  \nof horses and armour and the battles of heroes.\n\nAt last the Trojan leaders, Mnestheus and the bold Serestus,  \nhearing of the slaughter of their men, came on the scene to find  \n780 their allies scattering and the enemy within the walls. 'Where  \nare you running to now, citizens?' cried Mnestheus. 'Where is  \nthere to go? What other walls have you? What other defences  \nwhen you leave these? Can one man, and one man hemmed in  \non every side by your ramparts, cause all this slaughter and send  \nso many of your best fighting men to their deaths all over your  \ncity, and still live? Have you no spirit? Have you no shame? No  \nthought for your fatherland in its anguish, for your ancient gods  \nor for great Aeneas?' These words fired them. They rallied and  \nheld fast in close formation while Turnus gradually began to  \n790 disengage, making for the river and the part of the camp in the  \nbend of the river. Seeing this the Trojans laid on all the harder,  \nshouting at the top of their voices and crowding him like a pack  \nof huntsmen with levelled spears pressing hard on a savage lion;  \nthe lion is afraid and gives ground, but he is still dangerous, still  \nglaring at his attackers; his anger and his courage forbid him to  \nturn tail, and though he would dearly love to, he cannot charge  \nthrough the wall of steel and the press of men \u2013 just so did  \nTurnus give ground, uncertain but unhurried, and his mind was  \n800 boiling with rage. Twice he even hurled himself into the middle  \nof his enemies, breaking their ranks and sending them flying  \nalong the walls, but a whole army came together in a rush  \nagainst him from the camp, and Juno, daughter of Saturn, did  \nnot dare to renew his strength to withstand them, for Jupiter  \nsent Iris down from the sky bearing stern commands through  \nthe air for his sister Juno if Turnus did not withdraw from the  \nhigh walls of the Trojans. So sword-arm and shield were of no  \navail. The warrior could no longer stand his ground in the hail  \nof weapons that overwhelmed him from every side. The helmet  \nrang and rang again on his hollow temples and the solid bronze  \n810 was cracked by rocks. The plumes were torn from his head and  \nthe boss of his shield gave way under the blows. The Trojans  \ndoubled their barrage and the spear of Mnestheus was like the  \nlightning. Sweat poured off the whole body of Turnus like a  \nriver of pitch and he was given no breathing space. His lungs  \nwere heaving. He was shaking and sick with weariness. Then,  \nand only then, he dived head first into the river in full armour.  \nThe Tiber took him when he came into his yellow tide, bore him  \nup in his soft waves, washing away the blood of slaughter, and  \ngave him back in high heart to his comrades.\n\n## BOOK 10  \nPALLAS AND MEZENTIUS\n\nMeanwhile the house of All-powerful Olympus was thrown  \nopen and the Father of Gods and King of Men summoned a  \ncouncil to his palace among the stars, from whose steep heights  \nhe looked down upon all the lands of the earth, upon the Trojan  \ncamp and the peoples of Latium. The gods sat in their chamber  \nopen east and west to the light, and Jupiter began to speak: 'O  \ngreat dwellers in the sky, why have you gone back on your  \nword? Why do you contend with such bitterness of heart? I had  \nforbidden Italy to clash with the Trojans. Why is there discord  \n10 against my express command? What has made them afraid and  \ninduced them to take up arms and make each other draw the  \nsword? The time will come for war \u2013 there is no need to hasten  \nit \u2013 when barbarous Carthage will let destruction loose upon  \nthe citadels of Rome, opening up the Alps and sending them  \nagainst Italy. That will be the time for pillaging, and for hate to  \nvie with hate. But now let it be. A treaty has been decided upon.  \nAccept it, and be content.'\n\nThese were the few words spoken by Jupiter, but when golden  \nVenus replied, her words were not few: 'O father, imperishable  \npower over men and over all the world \u2013 how could there be  \n20 any other to whom we might address our prayers? \u2013 you see the  \nRutulians rampant and Turnus riding in glory in the midst of  \nthem, swollen with the success of his arms. A closed ring of  \nfortifications no longer offers protection to the Trojans. They  \nnow have to fight hand to hand inside their gates, even on the  \nramparts of their walls, and their ditches are swimming with  \nblood. Aeneas is far away and knows nothing of this. Will you  \nnever allow them to be free of besiegers? Even as Troy is being  \nreborn, a new enemy is threatening its walls with a new army  \nbehind him, and from Arpi the Aetolian Diomede is once more  \nrising against the Trojans. I suppose I shall soon be wounded  \n30 again \u2013 after all, mortals are at war and your daughter stands in  \ntheir way!\n\n'If the Trojans have come to Italy without your approval, in  \ndefiance of your heavenly will, they must be punished for their  \nsins and you must not raise a finger to help them. But if they  \nhave obeyed all the commands they have received from the gods  \nabove and the shades below, how can anyone overturn what  \nyou have ordered or fashion a new destiny? You have seen their  \nships burned on the shores of my own son Eryx. You have seen  \nthe king of the storms and his raging winds roused out of their  \nAeolian island. You have seen Iris driven down from the clouds.  \nAnd now she even turns to the one remaining part of the world  \n40 and stirs up the powers below \u2013 Allecto has suddenly been let  \nloose upon the earth and has run wild through all the cities in  \nthe middle of Italy! I no longer give a thought to empire. That  \nwas our hope, as you well know, while our fortunes remained.  \nBut those who must prevail are those you wish to prevail. If  \nthere is no region on earth that your cruel queen could concede  \nto the Trojans, I beg of you, father, by the smoking ruins of the  \nsacked city of Troy, allow me to take Ascanius safely out of the  \nwar. Allow my grandson to live. As for Aeneas, let him be tossed  \nby storms in unknown waters and go the road that Fortune  \n50 gives him, but grant me the power to protect Ascanius and take  \nhim out of this fearful battle. I have Amathus. I have lofty  \nPaphos, and Cythera, and my palace at Idalium. Let him lay  \ndown his arms and there live out his life in obscurity, while you  \ngive the order for Italy to be crushed beneath the mighty empire  \nof Carthage. The cities of Tyre will have nothing to fear from  \nAscanius. What good has it done him to escape the plague of  \nwar and come safe through the middle of all the fires of the  \nGreeks, to have drained the cup of danger over all the vast earth  \nand sea while the Trojans have been searching for Latium and  \na new Pergamum? Would it not have been better for them to  \n60 settle on the dead ashes of their native land, on the soil that was  \nonce Troy? Take pity on them, I beg you, and if the wretched  \nTrojans must live again the fall of Troy, give them back their  \nXanthus and their Simois.'\n\nAt this Juno, Queen of Heaven, burst out, wild with rage:  \n'Why do you force me to break my deep silence? The scars have  \nformed over my wounds. Why do you make me speak and  \nreopen them? Neither man nor god compelled Aeneas to choose  \nthe ways of war and confront king Latinus as an enemy. We are  \ntold he has the authority of the Fates for coming to Italy. The  \nFates, indeed! He was goaded into it by the ravings of Cassandra!  \nAnd did we urge him to abandon his camp or put his life at  \n70 the mercy of the winds? Did we advise him to entrust his  \nfortifications and the whole management of the war to a boy?  \nTo disturb the loyalty of the Etruscans and stir up a peaceful  \npeople? Was it a god that drove him to dishonesty? Was it some  \ncruel power of mine? Where is Juno in all this? Where is Iris  \nsent down from the clouds? It is wrong, we hear, for Italians to  \nring Troy with fire at the moment of its birth, and for Turnus  \nto take his stand in the land of his fathers, Turnus, whose  \ngrandfather was Pilumnus and whose mother was the goddess  \nVenilia. Why then is it right for Trojans to raise the blacksmoking  \ntorches of war against Latins, to put other men's lands  \nunder their yoke, to carry off plunder, to pick and choose who  \nare to be their fathers-in-law, to tear brides from their mothers'  \n80 laps and to hold out the olive branch of peace with their weapons  \nfixed on the high sterns of their ships? You can steal Aeneas  \naway from the hands of the Greeks, and where there was a man  \nyou can spread a cloud with empty winds. You can change ships  \ninto sea nymphs. Is it an impiety if we in our turn have given  \nsome help to the Rutulians? Aeneas, you tell us, is far away and  \nknows nothing of all this. Keep him in ignorance and let him  \nstay away! You have Paphos and Idalium. You have the heights  \nof Cythera. Why do you concern yourself with those roughhearted  \nItalians and their city teeming with war? You claim  \nwe are trying to overturn from the foundations the tottering  \nfortunes of these Phrygians from Troy. No! Who was it who  \n90 put your wretched Trojans at the mercy of the Greeks? What  \ncaused Europe and Asia to rise in arms and betray the sacred  \nties of friendship? Was I in the lead when the Trojan adulterer  \nstormed the walls of Sparta? Did I hand him his weapons? Was  \nit I who kindled the fires of war with lust? That was when you  \nshould have feared for your people. Now, when it is too late,  \nyou get to your feet with these complaints and lies, and hurl this  \nempty abuse.'\n\nAs Juno was making her plea, all the gods began to murmur  \nin support or in dissent. It was like the murmuring of a storm  \nwhen the first breeze is caught in a wood and the rustling rolls  \nthrough the trees unseen, warning sailors that winds are on the  \n100 way. Then the All-powerful Father, the highest power in all the  \nuniverse, began to speak, and at his voice the lofty palace of  \nthe gods fell silent, the earth trembled to its foundations and the  \nheights of heaven were hushed. The winds in that moment were  \nstilled and the sea kept its waves at peace. 'So be it,' he said.  \n'Hear my words and lay them to your hearts. Since you have  \nnot allowed the people of Ausonia to be joined in a treaty with  \nthe Trojans, and since there is no end to this discord of yours,  \nthis day let each man face his own fortune and set his course by  \nhis own hopes. Trojan and Rutulian I shall treat alike. Whether  \n110 this camp is blockaded by the destiny of Italy or because of the  \nfolly and wickedness of the Trojans and false prophecies they  \nhave received, as each man has set up his loom, so will he endure  \nthe labour and the fortune of it \u2013 I do not exempt the Rutulians.  \nJupiter is the same king to all men. The Fates will find their  \nway.' Then, swearing an oath by the waves of the Styx, his  \nbrother's river, by the banks and dark whirlpools of that pitch-black  \ntorrent, he nodded and his nod shook the whole of  \nOlympus. There were no more words. He rose from his golden  \nthrone, and the heavenly gods thronged around him and  \nescorted him to the threshold.\n\nThe Rutulians meanwhile were fighting hard round each of  \nthe gates to bring down their enemies in blood and ring their  \n120 walls with fire, while Aeneas' legion was trapped inside its own  \nramparts with no hope of escape. Helpless and desperate, they  \nstood on their high towers and manned the circle of their walls  \nwith a thin line of defenders. Asius, son of Imbrasus, Thymoetes,  \nson of Hicetaon, the two Assaraci and old Thymbris alongside  \nCastor were there in the forefront of the battle, and the two  \nbrothers of Sarpedon were with them, Clarus and Thaemon  \nfrom the mountains of Lycia. Acmon of Lyrnesus, as great a  \nwarrior as his father Clytius or his brother Mnestheus, was  \nputting out all his strength to carry a boulder, no small part of  \n130 a mountain, while they strove to defend their camp by throwing  \nrocks and javelins, or hurling fire and fitting arrows to the string.  \nThere in the middle of them, with his noble head bared, stood  \nthe boy Ascanius for whom the goddess Venus cares above all  \nothers, and rightly cares. He was like a gem sparkling in its gold  \nsetting, an adornment for a head or neck, or like glowing ivory  \nskilfully inlaid in boxwood or Orician terebinth, and his long  \nhair lay on his milk-white neck, held in place by a circlet of soft  \ngold. There too was Ismarus. The warriors of those great-hearted  \n140 peoples could see him tipping his arrows with poison  \nand aiming them at the enemy. He was the offshoot of a noble  \nhouse in Maeonia where men worked the rich lands and the  \nriver Pactolus watered them with gold. Mnestheus also was  \nthere, raised to the heights of glory for his recent repulse of  \nTurnus out of the ring of the walls; Capys, too, who gives his  \nname to the city of Capua in Campania.\n\nThese were the men who clashed that day in bitter fighting.  \nIn the middle of the night that followed, Aeneas was ploughing  \nthe waves of the ocean. After leaving king Evander, he had  \nentered the Etruscan camp and gone to their king to tell him his  \n150 name and nation, what he wanted, what he offered and what  \narmed forces Mezentius was winning to his support. He told  \nhim too of the violent passions of Turnus and reminded him  \nthat in human affairs there is no room for certainty, and to all  \nthis he added his appeal for help. Tarchon instantly joined forces  \nwith him and made a treaty. Then these Etruscans, these men  \nof Lydian stock, having paid their debts to destiny, put to sea  \nand committed themselves to a foreign leader in accordance  \nwith the will of the gods. Aeneas' ship took the lead. Phrygian  \nlions were yoked to it for a beak, and above them the figurehead  \nwas Mount Ida, a sight most dear to the Trojan exiles. Here sat  \n160 great Aeneas, turning over in his mind the varied chances of  \nwar, and all the while young Pallas stayed close by his left side,  \nasking him now about the stars and the course they were steering  \nthrough the darkness of the night, now about all he had suffered  \nby land and sea.\n\nNow goddesses, it is time to open up Mount Helicon, to set  \nyour songs in motion and tell of the army which came that night  \nwith Aeneas from the shores of Etruria, to say who fitted out  \nthe ships and who sailed in them across the ocean.\n\nMassicus was the first, cutting through the water on the  \nbronze-plated _Tiger_. Under him sailed a band of a thousand  \nwarriors who had left behind them the walls of Clusium and the  \ncity of Cosae. Their weapons were arrows carried in light quivers  \non their shoulders, and death-dealing bows.\n\n170 With them sailed grim Abas, whose whole troop shone in  \nbrilliant armour, and a gilded Apollo gleamed on the stern of  \nhis ship. Populonia, his motherland, had given him six hundred  \nfighting men, skilful in the wars, while three hundred came from  \nIlua, the island of the Chalybes, teeming with its inexhaustible  \nores.\n\nThe third ship was sailed by Asilas, the great mediator  \nbetween gods and men, master of the stars of the sky and the  \nentrails of the beasts of the field, of bird cries and the prescient  \nfires of lightning. He sped along leading a thousand men in close  \nformation with their spears bristling. Pisa put them under his  \n180 command, a city on Etruscan soil but founded by men from the  \nAlpheus, the river of Olympia.\n\nNext in line sailed fair Astyr, whose trust was in his horse and  \nhis iridescent armour. To him were joined three hundred men,  \nand all were as one in their zeal to follow him, men whose home  \nwas Caere, men from the fields of Minio, from ancient Pyrgi  \nand the unwholesome swamps of Graviscae.\n\nNor could I pass over Cunarus, so brave in war, the leader of  \nthe Ligurians, nor Cupavo with his small band of fighting men.  \nHigh above his head tossed the swan feathers that were a token  \nof his father's change of form \u2013 all the fault of the God of Love.  \nThey say that Cycnus sought comfort from the Muse for the  \nsadness of his love, by singing of the loss of his dear Phaethon  \n190 in the green shade of the poplars that had been Phaethon's  \nsisters. There, when he grew old, he put on soft white plumage  \nand rose from the earth, singing as he flew towards the stars. It  \nwas his son who now commanded the huge _Centaur_ , driving it  \nalong under oar, and with him in his fleet he took a throng  \nof his peers. The Centaur figurehead loomed over the water,  \nthreatening to hurl down a massive rock into the waves from its  \ndizzy height, and the long keel ploughed its furrow deep in the  \nsea.\n\nThere too was Ocnus, driving on an army from his fatherland.  \nHe was the son of Manto the prophetess and the Tuscan river  \n200 Tiber. To you, Mantua, he gave your walls and the name of his  \nmother \u2013 Mantua, rich in the roll of its forefathers, and not all  \nof one race, but of three, and in each race four peoples. Of all  \nthese peoples Mantua is the head, and its strength comes from  \nits Etruscan blood. From here too, Mezentius had roused five  \nhundred men to fight against him, and these the river Mincius,  \nveiled in blue-green reeds, led down to the sea in their ships of  \nwar from his father, Lake Benacus. There sailed Aulestes, heavy  \nin the water, but rising as his hundred oars thrashed the waves  \nand churned the marble of the sea to foam. He sailed the  \n210 monstrous _Triton_ , which terrified the blue sea with its horn. As  \nit swam along, its figurehead showed a shaggy front like a man  \nas far as its flanks, but its belly ended in a monster of the deep,  \nwhile under the breast of this creature, half-man half-beast, the  \nwaves foamed and murmured.\n\nThese were the chosen leaders who went to the help of Troy  \nin their thirty ships, and ploughed the plains of salt with bronze.\n\nBy now the day had left the sky and Phoebe, the kindly  \nGoddess of the Moon, was pounding the middle of Olympus  \nwith the hooves of her night-wandering horses. Duty allowed  \nno rest to the limbs of Aeneas. As he sat controlling the tiller  \n220 and seeing to the sails, a band of his old comrades came suddenly  \ntowards him in mid-voyage. They were nymphs, the nymphs  \ninto whom his ships had been changed at the bidding of the  \nkindly Mother Goddess Cybele, and they now held divine power  \nover the sea. There they were, swimming in line, as many of  \nthem now cleaving the waves as had then stood to the shore  \nwith bronze-plated prows. They recognized their king from a  \ndistance and danced around him in the water, and Cymodocea,  \nthe best speaker among them, came behind his ship and putting  \nher right hand on its stern, raised her back out of the water,  \nwhile her left hand was below the surface, oaring silently along.  \nAeneas was still bewildered when she began to speak to him:  \n'Are you awake, Aeneas,' she asked, 'son of the gods? Wake  \n230 then and let out the sail-ropes. We are the pines from the sacred  \ntop of Mount Ida, now sea nymphs. We are your fleet. When  \nthe treacherous Rutulian was pressing us hard with fire and  \nsword, against our wishes we had to break the moorings you  \ngave us, and now we have been looking for you all over the  \nocean. Mother Cybele took pity on us and gave us this new  \nform, allowing us to become goddesses and spend our lives  \nbeneath the waves. But the boy Ascanius is trapped behind a  \nwall and ditches, surrounded by missiles and by Latins bristling  \nwith war. The Arcadian cavalry from Pallanteum are now in  \ntheir places as ordered, along with the brave Etruscans, and  \n240 Turnus has firmly resolved to prevent them joining forces with  \nthe Trojan camp by taking up position between them with his  \nown troops. Up with you then, and at the first coming of dawn,  \norder your allies to arms and then take up the invincible shield  \nwith its rim of gold given you by the God of Fire himself.  \nTomorrow's light, unless you think these are empty words of  \nmine, will see the field of battle heaped high with Rutulian  \ndead.' So she spoke, and as she left him she gave the high stern  \na push with her right hand \u2013 and well she knew the art of it. The  \nship flew through the waves faster than a javelin or wind-swift  \narrow, and the others sped along behind it. The leader of the  \n250 Trojans, the son of Anchises, was struck dumb with bewilderment,  \nbut his heart lifted at the omen, and looking up to the  \nvault of heaven, he uttered this short prayer: 'Kindly Mother of  \nthe Gods, dweller on Ida, who takes delight in Mount  \nDindymus, in cities crowned with towers and in the lion pair  \nresponsive to your chariot reins, be now my leader in this battle.  \nBring near to us the due fulfilment of your omen. Stand by the  \nside of your Phrygians and give us your divine blessing.' These  \nwere his words, and even as he spoke them the revolving day  \nwas already rushing back in its full brightness and had put the  \ndarkness to flight. His first thought was to order his allies to  \nfollow the standards, to fit their minds for the use of their  \nweapons and prepare themselves for battle.\n\n260 And now, as soon as Aeneas, standing high on the stern of  \nhis ship, could see the Trojans and his own camp, at that moment  \nhe lifted the shield on his left arm and made it flash. The Trojans  \non the wall raised a shout to heaven, fresh hope renewing their  \nanger, and they hurled their spears, like cranes from the river  \nStrymon in Thrace giving out their signals under the black  \nclouds, trumpeting as they cross the sky and flying before the  \nstorm winds with exultant cries. The Rutulian king and the  \nleaders of Italy were amazed until they turned round and saw a  \nfleet making for the shore and a whole sea of ships gliding in  \n270 towards them. On the head of Aeneas there blazed a tongue of  \nfire, baleful flames poured from the top of his crest and the  \ngolden boss of his shield belched great streams of fire, like the  \ngloomy, blood-red glow of a comet on a clear night, or the  \ndismal blaze of Sirius the Dog-star shedding its sinister light  \nacross the sky and bringing thirst and disease to suffering  \nmortals.\n\nBut the bold confidence of Turnus never wavered as he quickly  \ntook up position on the shore to repel the landing. 'This is the  \nanswer to your prayer,' he cried, 'now is the time to break them.  \n280 Brave men have the God of War in their own right arms. Each  \nof you must now think of his own wife and his own home and  \nremember the great deeds which brought glory to our fathers.  \nLet us go down to the sea to meet them while they are still in  \nconfusion and finding their feet after landing. Fortune favours  \nthe bold.' So he spoke and pondered in his mind who could be  \nled against the fleet and who could be trusted to keep up the  \nsiege of the walls.\n\nMeanwhile Aeneas was landing his allies by gang-planks from  \nthe high sterns. Many waited for the spent waves to be sucked  \n290 back and then took a leap into the shallow water. Others were  \nclambering down the oars. Tarchon, who had been looking out  \nfor a stretch of shore where there seemed to be no shoals and  \nno grumbling of broken water, where the swelling tide could  \ncome in without obstruction, suddenly swung his ship round  \nand appealed to his comrades: 'Now, my chosen band, now  \nbend to your stout oars. Up with your ships out of the water.  \nTake the weight of them. Split with your rams this land that we  \nhate, and let each keel plough its own furrow. I do not care if  \nmy ship is wrecked by such a mooring, if only we take possession  \nof this land.' When Tarchon had spoken, his comrades rose to  \n300 their oars and drove their ships foaming at the prow, hard on  \nto the soil at Latium, till their beaks struck home on dry land  \nand their keels were safely settled. But not yours, Tarchon.  \nYou ran aground on a shoal and hung there see-sawing on a  \ndangerous ridge of rock, till at last the waves were weary of you  \nand your ship broke up, throwing your men into the sea to be  \ntangled in smashed oars and floating thwarts, as the undertow  \nof the waves kept taking the feet from them.\n\nTurnus was no sluggard. Wasting no time he eagerly led his  \nwhole force to face the Trojans and drew them up at the ready  \n310 on the shore. The trumpets sounded, and Aeneas was the first  \nto move against the army of the country people of Latium and  \nlay them low. This was an omen of the battle that was to come.  \nTheron was the first to fall. He was the tallest of their warriors,  \nand had taken it upon himself to attack Aeneas. Through the  \nmesh of his chain mail of bronze, through his tunic stiffened  \nwith threads of gold, Aeneas tore a huge gash with his sword in  \nthe flesh of his side. He then struck Lichas. His mother was  \nalready dead when Lichas was cut from her womb and dedicated  \nto Phoebus Apollo, the God of Healing. Little good did it do the  \nbaby to escape the hazard of steel at birth. Next Aeneas saw  \nhuge Gyas and tough Cisseus felling the embattled Trojans with  \ntheir clubs, and sent them down to death. Nothing could help  \n320 them now: not the weapons of Hercules, nor the strength of  \ntheir hands, nor their father Melampus, who had stood by the  \nside of Hercules as long as the earth supplied him with heavy  \nlabours to perform. There was Pharus, hurling his empty threats,  \ntill Aeneas spun the javelin and planted it in his throat even as  \nhe shouted. You too, Cydon, desperately following your latest  \nbeloved Clytius, with the first gold down on his cheeks, would  \nhave forgotten the young men you were always in love with.  \nYou would have fallen by the right hand of a Trojan and lain  \nthere for men to pity, had not Aeneas been confronted by seven  \n330 brothers in serried ranks, the sons of Phorcus, hurling their  \nseven spears. Some rebounded harmlessly from his helmet or  \nhis shield. Others his loving mother Venus deflected so that they  \nonly grazed his body, and Aeneas addressed his faithful Achates:  \n'Pile up some javelins for me. No weapon that has stood in the  \nbody of a Greek on the plains of Troy will spin in vain from my  \nright hand against Rutulians!' He then caught up a great spear  \nand hurled it. Flying through the air it beat through the bronze  \nof Maeon's shield and shattered in one instant the breastplate  \nand the breast. Alcanor came to help him as he fell, a brother's  \n340 right hand to support a brother. Through Alcanor's arm went  \nthe spear of Aeneas and flew on its way dripping with his blood,  \nwhile the dying arm hung by its tendons from the shoulder.  \nAnother brother, Numitor, snatched the weapon from Maeon's  \nbody and aimed at Aeneas in return, but was not allowed to  \nstrike him, only to graze the thigh of great Achates. Then came  \nClausus of Cures in all the pride of his youthful strength and  \nwith a long-range cast of his unbending spear he struck Dryops  \nfull force under the chin. It went straight through his throat and  \ntook from him in one moment, even as he spoke, his voice and  \nhis life's breath. His forehead struck the ground and his mouth  \n350 vomited great gouts of blood. Then Aeneas laid three Thracians  \nlow, men from the exalted stock of Boreas, then three more sent  \nby their father Idas from their fatherland Ismara, all by different  \nforms of death. Halaesus came running to the spot with his  \nAuruncans; Messapus too, son of Neptune, whose horses drew  \nevery eye. Trojans and Latins were battling on the very threshold  \nof Italy, each striving to dislodge the other, like opposing winds  \nfighting their wars in the great reaches of the sky, equal in spirit  \nand equal in strength; they do not give way to one another,  \nneither the winds themselves nor the clouds nor the sea, but  \nlong rages the fight, undecided, and they all stand locked in  \n360 battle \u2013 just so clashed the armies of Troy and the armies of  \nLatium, foot planted against foot, and man face to face with  \nman.\n\nIn another part of the battle, where a torrent had rolled down  \nboulders and trees uprooted from its banks and strewn them  \neverywhere, Pallas saw his Arcadians, who had for once  \nadvanced on foot, now retreating with Latins in hot pursuit \u2013  \nthe floods had so roughened the ground that they had decided  \nto abandon their horses. One course alone remained \u2013 to fire  \nthe valour of his men by appeals and bitter reproaches: 'Where  \nare you running to, comrades? I beg you by your pride in  \n370 yourselves, by your bravery in time past, by the name of Evander  \nyour leader, by the wars you have won, by the hopes rising in  \nme to gain glory like my father's, this is no time to trust to your  \nfeet! It is swords you need, to cut your way through the enemy.  \nThere, where the moil is thickest, where the attack is fiercest,  \nthat is where your proud fatherland requires you and your  \nleader Pallas to go. These are not gods who are pressing you so  \nhard; they are mortals pursuing mortals. Like us they have two  \nhands, and like us they have one life to lose. Look about you!  \nThe great barrier of ocean closes us in. There is no more land to  \nrun to. Shall we take to the sea? Shall we set course for Troy?'  \nWith these words he threw himself into the thick of his enemies.\n\n380 The first man to meet him, drawn there by an unkindly fate,  \nwas Lagus. While he was trying to tear loose a great heavy rock,  \nPallas hurled his spear and struck him in the middle of the back  \nwhere the spine divides the ribs. Pallas was pulling out the  \nweapon, which had wedged between the bones, when Hisbo  \nswooped on him, hoping to take him by surprise, but Pallas  \ncaught him first in the fury of his charge, made reckless by the  \ncruel death of his comrade. Hisbo's lungs were swollen. Pallas  \nburied his sword in them. He then turned on Sthenius; then on  \nAnchemolus of the ancient stock of Rhoetus, who had shamefully  \n390 debauched his own stepmother. You too fell on the  \nRutulian fields, Larides and Thymber, sons of Daucus, identical  \ntwins, a source of confusion and delight to your parents. But  \nPallas made a grim difference between you: with the sword of  \nhis father Evander he removed the head of Thymber, and cut  \noff the hand of Larides. As it lay there, it groped for its owner  \nand the fingers twitched, still half alive, and kept clutching at  \nthe sword. The Arcadians were stung by Pallas' reproaches, and  \nas they watched his glorious feats, remorse and shame armed  \nthem against their enemies.\n\n400 Then Pallas put a spear through Rhoeteus as he fled past on  \nhis two-horse chariot, and gave that much respite and reprieve  \nto Ilus. For it was against Ilus that Pallas had aimed a long throw  \nwith his mighty spear, but Rhoeteus had come between them  \nand taken the blow while fleeing from great Teuthras and his  \nbrother Tyres. He rolled from his chariot, and died with his  \nheels drumming on the Rutulian ploughland. Just as a shepherd  \nfires a wood at different points when the summer winds get up  \nat last, and suddenly all the flames merge in the middle to make  \none bristling battle-front of fire stretching over the broad plain,  \nand there he sits in triumph looking down on the exulting blaze  \n410 \u2013 just so, Pallas, did the valour of your men all come together in  \none, and put joy in your heart. But Halaesus was a fierce warrior,  \nand he made straight for the enemies that stood in front of him,  \ngathering all his strength behind his weapons. Ladon and Pheres  \nand Demodocus he slew, and his flashing sword ripped off the  \nright hand of Strymonius as it was poised to lunge at his throat.  \nThoas he struck with a rock in the face, shattering the bones  \nand grinding them into the blood-soaked brains. Halaesus was  \nnext. His father, foreseeing the future, had hidden him in the  \nwoods, but when the father grew old and his whitening eyes  \ndissolved in death, the Fates laid a hand on the son and consecrated  \n420 him to Evander's spear. This was the prayer of Pallas  \nbefore he attacked: 'Grant now, O Father Thybris, that the  \nspear I am holding poised to throw may reach the mark and go  \nthrough the stout breast of Halaesus, and I shall strip these arms  \nof his from his body and hang them on your sacred oak as  \nspoils.' The god heard his prayer. As the hapless Halaesus  \nprotected Imaon, he left his breast exposed to the Arcadian  \nspear.\n\nBut Lausus, who was bearing the brunt of the battle, did not  \nallow his men to be dismayed by all this slaughter done by  \nPallas. First of all he slew Abas as he stood before him, the very  \nknot and stumbling-block of war. The youth of Arcadia were  \n430 laid low and the Etruscans fell beside them, and you too,  \nTrojans, who had faced the Greeks unscathed. The armies  \nclashed, equal in their leaders and in their strength, and the  \nwings of the battle line were forced into the centre so that men  \ncould not raise a hand or a weapon in the crowd. On the one  \nside Pallas thrust and pressed, on the other Lausus. They were  \nalmost of an age, and noble in appearance, but Fortune had  \ndenied each of them a homecoming. Yet the ruler of high  \nOlympus did not yet allow their paths to cross, reserving for  \neach his own death at the hand of a stronger enemy.\n\nMeanwhile, after Juturna had advised her dear brother  \n440 Turnus to take the place of Lausus, he cut through the middle  \nof the ranks of warriors on his swift chariot, and as soon as he  \nsaw his allies he called out: 'Time now to stand down from the  \nfighting. I am the only one who attacks Pallas. Pallas is mine,  \nand mine alone. I wish his father were here to see it.' So he  \nspoke and his allies left the ground clear as ordered. When the  \nRutulians withdrew, Pallas marvelled at these proud commands  \nand stood amazed at the sight of Turnus, running his eyes all  \nover that mighty body, his grim stare taking it in part by part  \nfrom where he stood, and these were the words he hurled in  \nreply to the words of the insolent prince: 'I shall win rich  \nrenown today, either for stripping the corpse of the leader of  \n450 my country's enemies, or else for a glorious death. My father  \nwill bear the one fate as easily as the other. Do not waste your  \nthreats on me.' With these words he strode on to the level  \nground in the middle of the battlefield, and the blood of the  \nArcadians froze in their breasts. Turnus leapt down from his  \nchariot and prepared to come to close quarters on foot, flying  \nat him like a lion which has seen from some high vantage point  \na bull practising for combat far away on the plain \u2013 this is how  \nTurnus appeared as he came on. Pallas made the first attack,  \njudging that Turnus would be within range of a spear-cast and  \nhoping that Fortune would favour the weaker for his daring.  \nLifting up his voice to the wide expanse of heaven, he cried: 'I  \n460 call upon you, Hercules of the stock of Alceus, by my father's  \ntable and by the friendship he offered you when you came as a  \nstranger to his home, stand at my side now as I set my hand to  \nthis great task. May Turnus as he dies see me tear the blood-stained  \narmour off his body, and may the last sight he endures  \nbe the face of the man who has defeated him!' Hearing the  \nyoung warrior, Hercules checked the great groan rising from  \nthe depths of his heart and the helpless tears streamed from his  \neyes. Then Father Jupiter spoke these loving words to his son:  \n'Each man has his allotted day. All life is brief and time once  \npast can never be restored. But the task of the brave man is to  \n470 enlarge his fame by his actions. So many sons of gods fell under  \nthe high walls of Troy, and with them fell also my son Sarpedon.  \nTurnus too is called by his own destiny and has reached the  \nlimits of the time he has been given.' So he spoke and instantly  \nturned his eyes away from the Rutulian fields.\n\nBut Pallas hurled his spear with all his strength and tore his  \nbright sword from its enclosing scabbard. The spear flew and  \nfell where the armour stood highest on the shoulder of Turnus,  \nforcing its way through the edge of the shield and grazing at last  \n480 the skin of that huge body. Then Turnus took long aim at Pallas  \nwith his steel-pointed hardwood spear and threw it saying:  \n'Now see whether mine is any better at piercing!' With a shuddering  \nblow it beat through the middle of the shield, through all  \nthe plates of iron and of bronze and all the ox-hides that covered  \nit, and unchecked by the breastplate, it bored through that  \nmighty breast. In desperation Pallas tore the warm blade out of  \nthe wound, and blood and life came out together after it, both  \nby the same channel. He fell forward on the wound, his armour  \nringing on top of his body, and as he died his bleeding mouth  \n490 bit the soil of his enemies. Turnus stood over him and said:  \n'Take this message of mine to Evander, you Arcadians, and do  \nnot forget it: I am sending him back the Pallas he deserves.  \nWhatever honour there is in a tomb, and any comfort he finds  \nin burying him, these I gladly give him. His hospitality to Aeneas  \nwill cost him dear!' With these words he planted his left foot on  \nthe dead body, and tore off the huge, heavy baldric. On this  \ngreat belt an abominable crime was embossed, how in one night,  \nthe night of their marriage, a band of young men were foully  \nslain, and their marriage chambers bathed in blood, all worked  \nby Clonus, son of Eurytus, in a wealth of gold. This was the  \n500 spoil in which Turnus now exulted and he gloried in the taking  \nof it. The mind of man has no knowledge of what Fate holds in  \nstore, and observes no limit when Fortune raises him up. The  \ntime will come when Turnus would gladly pay, and pay richly,  \nto see Pallas alive and unharmed. He will bitterly regret this  \nspoil and the day he took it. A throng of Pallas' comrades laid  \nhim on his shield and carried him back with tears and groans. O  \nPallas, a great grief and a great glory are coming home to your  \nfather! This one day gave you to war, and now takes you from it,  \nand yet you leave behind you huge piles of Rutulian dead.\n\n510 First a rumour of this calamity came flying to Aeneas and  \nthen a reliable messenger, to tell him his men were on the very  \nedge of destruction; the Trojans were in retreat; now was the  \ntime to help them. Everything that stood before him he harvested  \nwith the sword, cutting a broad swathe through the enemy  \nranks, and burning with rage as he looked for this Turnus  \nflushed with slaughter. Before his eyes he could see Pallas,  \nEvander, everything, the table he had sat down to that day when  \nhe first came to their house, and the right hands of friendship  \nthey had given him. Four warrior sons of Sulmo he now captured  \nalive and four reared by Ufens, to sacrifice them as offerings to  \n520 the shade of Pallas and pour their captive blood on the flames  \nof his pyre. Next he aimed his deadly spear from long range at  \nMagus, who cleverly ran under it. The quivering spear flew over  \nhis head and he clasped the knees of Aeneas with this prayer:  \n'By the shade of your own father and the hopes you have of  \nIulus as he grows to manhood, I beg you to spare this life of  \nmine for the sake of my son and my father. Our home is a  \nhigh-built palace, and buried deep within it I have talents of  \nengraved silver and great weights of gold, both worked and  \nunworked. A Trojan victory does not depend on me. My one  \n530 life will not make so great a difference.' This was Aeneas' reply:  \n'Keep for your children all those talents of silver and gold you  \ntalk about. Turnus put an end to such war-trading the moment  \nhe murdered Pallas. So judges the shade of my father Anchises.  \nAnd so judges Iulus.' When he had spoken he took Magus'  \nhelmet in his left hand, and bending back his neck when he was  \nstill begging for mercy, he drove the sword home to the hilt.  \nNot far away was Haemonides, priest of Phoebus Apollo and  \nDiana Trivia, his temples bound by a headband of sacred wool,  \n540 all shining white in his white robes and insignia. Aeneas closed  \nwith him, drove him across the plain, stood over him when he  \nfell, darkening the whiteness with his great shadow, and took  \nhim as his victim. Serestus collected the spoils and carried them  \nback on his shoulders as a trophy to Mars Gradivus.\n\nCaeculus of the stock of Vulcan renewed the battle, and  \nUmbro from the Marsian mountains with him. Aeneas confronted  \nthem in all his fury. His sword had already struck off  \nthe left hand of Anxur \u2013 a stroke of the blade had sent the whole  \ncircle of his shield to the ground. He had uttered some great  \nthreat, imagining that the strength would be there to make it  \ngood. It seemed he was trying to raise his spirits to the skies,  \nand had promised himself that he would live to enjoy grey hairs  \n550 and a long life. Next Aeneas in his fury was faced by Tarquitus,  \nglorying in his shining armour, the son of Faunus, God of the  \nWoods, and the nymph Dryope. Drawing back his spear, Aeneas  \nthrew and pinned the great heavy shield to the breastplate.  \nWhile he was still begging for mercy, and still had much to say,  \nAeneas smashed his head to the ground, and as he set the warm  \ntrunk rolling, these were the words he spoke with hatred in his  \nheart: 'Lie there now, you fearsome warrior. Your good mother  \nwill not bury you in the earth or burden your body with the  \nfamily tomb. You will be left for the wild birds, or thrown into  \n560 the sea to be carried away by the waves, and the hungry fish will  \ncome and lick your wounds!' Next he pursued and caught  \nAntaeus and Lucas, the front rank of Turnus, then brave Numa  \nand yellow-haired Camers, son of great-hearted Volcens, who  \nwas richest in land of all the men of Italy and ruled over silent  \nAmyclae. Aeneas was like Aegaeon, who they say had a hundred  \narms and a hundred hands, with fire flaming from fifty breasts  \nand mouths, and fifty was the number of swords he drew against  \nthe lightning of Jupiter, fifty the number of identical shields he  \nclashed \u2013 so seemed Aeneas, raging victorious all over the plain,  \n570 when once his sword blade had warmed to the work. Imagine  \nhim next bearing down on the chariot of Niphaeus, with the  \nfour horses showing their chests as they stood to meet him, but  \nwhen they saw Aeneas' great stride and heard his fearsome roar,  \nthey wheeled in panic and bolted, throwing their master out of  \nthe chariot and stampeding to the shore.\n\nMeanwhile Lucagus was coming into the middle of battle on  \na chariot drawn by two white horses. With him was his brother  \nLiger, handling the reins and controlling the horses while  \nLucagus whirled his naked sword about him. Aeneas could not  \nendure to see such fury and such fervour, but rushed forward  \n580 and loomed huge before them with his levelled spear. It was  \nLiger who spoke: 'These are not the horses of Diomede you are  \nlooking at, or the chariot of Achilles. These are not the plains  \nof Troy. Here in this land today there will be an end to your  \nwars and to your life.' Far flew these wild words of Liger. The  \nTrojan was preparing a reply to his enemy, but it was not in  \nwords \u2013 it was his javelin he hurled. Lucagus had been leaning  \nforward over his horses to urge them on by beating them with  \nthe flat of his spear. Now, when he had planted his left foot to  \nthe front and was preparing for battle, through the bottom rim  \nof his shining shield came the spear of Aeneas and pierced his  \n590 left groin. He was pitched from his chariot and as he lay dying  \non the ground, good Aeneas addressed these bitter words to  \nhim: 'It is not the panic of your horses, Lucagus, that has brought  \nyour chariot to grief. They did not shy away from the shadow  \nof their enemy. It is your own doing, leaping off the car and  \nabandoning your team!' With these words Aeneas caught the  \nhorses' bridles. The wretched brother of Lucagus fell from the  \nchariot and stretched out his helpless hands to Aeneas: 'Great  \nTrojan, I implore you by your own self and by the parents who  \nbrought such a man as you into the world, spare this life of mine  \nand take pity on a suppliant.' Aeneas cut short his appeal. 'This  \n600 is not what I heard you say a moment ago. Die now. A brother's  \nplace is with his brother.' And as he spoke the point of his sword  \nopened the breast of Liger, the hiding place of his soul. So did  \nthe Trojan leader deal out death all over the plain like a raging  \ntorrent of water or a storm of black wind, until at last the young  \nAscanius and his warriors sallied forth and left the camp. The  \nsiege was lifted.\n\nIn the meanwhile Jupiter came to Juno and said to her: 'O my  \ntrue sister and most pleasing of wives, you are right, it is Venus,  \nas you thought, who is maintaining the strength of the Trojans,  \n610 not the warlike vigour of their right arms nor their fierce  \nand danger-hardened spirit.' Humbly Juno replied: 'O finest of  \nhusbands, why do you cause me anguish when I am in despair  \nand in terror of your harsh commands? If your love for me had  \nthat power which once it had, and should have still, you who  \ncan do all things would not be refusing me this. I should be able  \nto withdraw Turnus from the battle and keep him safe for his  \nfather Daunus. But as things are, let him die. Let him pay the  \npenalty to the Trojans with his righteous blood. Nevertheless  \nhe is descended from our stock, Pilumnus was his ancestor in  \n620 the fourth generation and his generous hand has often weighed  \ndown your threshold with abundant gifts.' The King of  \nHeavenly Olympus made brief reply: 'If what you ask is a stay  \nof the death that is upon him and respite for a young man who  \nmust die, and if you accept that this is what I ordain, then rescue  \nTurnus. Let him flee. Snatch him from the Fates that tread upon  \nhis heels. There is room for me to grant you indulgence thus far.  \nBut if there is some deeper thought of mercy underlying these  \nappeals of yours, and if you believe that the whole course of the  \nwar can be affected or its outcome changed, the hopes which  \nyou nourish are empty.' Juno replied, weeping as she spoke:  \n'What if your heart wished to give what your words refuse?  \n630 What if you listened to me and let Turnus live? As it is, although  \nhe is innocent, a cruel death is waiting for him, unless I am wide  \nof the mark and there is no truth in me. But oh how I wish my  \nfears were false and I were deluded! How I wish you would  \nrecast your plans, for you can do so, and choose a better course!'\n\nAs soon as the goddess had finished speaking, she flew down  \nfrom the heights of heaven swathed in cloud and driving a great  \nstorm before her towards the battle line of the Trojans and the  \nLaurentine camp. Then she fashioned out of empty vapour an  \neffigy in the form of Aeneas, a weird sight, a shade without  \nstrength or substance, armed with Trojan weapons. She copied  \nhis shield and the crest on his godlike head and gave the phantom  \n640 power to speak its empty words. Sound without thought she  \ngave it, and moulded its strides as it moved. It was like the  \nflitting shapes which men say are the ghosts of the dead, or like  \nthe dreams which delude our sleeping senses. There in high glee  \nin front of the first line of warriors pranced this apparition  \nand goaded Turnus by brandishing weapons and shouting  \nchallenges. Turnus attacked, throwing his whirring spear from  \nlong range. The apparition turned tail and fled. At that moment  \nTurnus believed that Aeneas had turned his back on him and  \nwas running away. Taking a wild draught from the empty cup  \nof hope, he cried: 'Where are you running to, Aeneas? You must  \nnot leave. Your marriage is arranged. This is the land you  \n650 crossed the seas to find and my right hand will give it to you!'  \nShouting such taunts, he went in pursuit with his sword drawn  \nand flashing and did not see that all his exultation was scattering  \nto the winds.\n\nThe ship which king Osinius had sailed from the land of  \nClusium happened to be moored to a high shelf of rock, with  \nher ladders and gangway out. Here the panic-stricken phantom  \nof Aeneas fled and hid itself, with Turnus hard behind it. Nothing  \ncould delay him. He leapt across the gangways, high above  \nthe water, and scarcely had he set foot on the prow when  \n660 Saturnian Juno tore the ship from her moorings, breaking the  \nropes, and took her quickly out to sea on the ebbing tide. But  \nby this time the phantom was no longer looking for a place to  \nhide. It had flown high into the air and melted into a black  \ncloud. Meanwhile, Aeneas was calling on Turnus to fight, and  \nthere was no Turnus, but every man who crossed his path he  \nsent down to death, and all the time the wind was blowing  \nTurnus round and round in mid-ocean. Looking back to the  \nshore in bewilderment and thanking no one for his safety, he  \nraised his arms in prayer and lifted up his voice to the stars of  \nheaven: 'All-powerful Father, have you decided that I deserve  \nthis disgrace? Have you decreed that I must endure this punishment?  \n670 Where am I being taken? What have I left behind me?  \nHow can I go back after running away? What sort of Turnus  \nwould that be? Shall I ever see my camp and the walls of the  \nLaurentines again? And what about that band of great warriors  \nwho have followed me and followed my sword? The horror of  \nit \u2013 I have left them all to die! I see them wandering about  \nwithout a leader. I hear them groaning as they fall. What am I  \nto do? If only the earth could open deep enough to swallow me!  \nOr rather I pray to the winds, and pray to them from my heart,  \nto take pity on me and drive my ship on to the rocks and cliffs,  \nor run it aground on some shoal of deadly sand, where there  \nwill be no Rutulian and no word of my shame can follow me.'  \n680 Even as he spoke, his mind was tossed this way and that, in  \ndespair at his disgrace. Should he fall on his sword and drive  \nthe raw steel through his ribs? Should he throw himself into the  \nsea and try to swim from mid-ocean back into the curve of the  \nbay to face the weapons of the Trojans once again? Three times  \nhe tried each way, and three times mighty Juno held him back,  \npitying the young man in her heart, and would not let him move.  \nCutting the deep water, he floated on a favouring tide and  \nfollowing waves, and came to land in the ancient city of his  \nfather Daunus.\n\nBut Mezentius meanwhile, by the promptings of Jupiter, took  \n690 the place of Turnus in the battle and fell furiously on the triumphant  \nTrojans. Instantly all the Etruscan troops converged on  \nhim alone, united in their hatred, and pressed him hard under a  \nhail of weapons. He stood like a rock jutting out into the ocean  \nwastes, exposed to the threats and fury of wind and wave and  \nbearing all the violence of sea and sky, unmoved. He felled  \nHebrus, son of Dolichaon, and Latagus with him, and Palmus  \nas he ran. Latagus he stopped by hitting him full in the face and  \nmouth with a rock, a huge block broken off a mountain, but he  \n700 cut the hamstrings of Palmus and left him rolling helpless on  \nthe ground. His armour he gave to Lausus to put on his shoulders,  \nand his crest to fix on his helmet. Then it was the turn of  \nEuanthes the Phrygian, and Mimas, the same age as Paris and  \nhis comrade in war. In one night Theano, wife of Amycus,  \nbrought him into the light of life, while Hecuba, daughter of  \nCisseus, pregnant with a torch, was giving birth to Paris. Paris  \nfell in the city of his fathers, but Mimas lies a stranger on the  \nLaurentine shore. Like the wild boar who has long kept his  \ncitadel among the pines of Mount Vesulus, and long have the  \n710 Laurentine marshes fed him in the reed beds of the forest; when  \nthe great beast is driven down from the mountains with the dogs  \nsnapping at him, and is caught between the nets, he stands at  \nbay snorting, and the bristles rise on his shoulders and no one  \nhas the courage to clash with him or go near him, but they  \nattack from a safe distance with javelins and shouts, while he  \nstands his ground unafraid and wondering in which direction  \nto charge, grinding his teeth and shaking the spears out of his  \nback \u2013 even so, none of those men who had just cause of anger  \nagainst Mezentius was minded to draw the sword and run upon  \nhim, but instead they stood well back and bombarded him with  \nmissiles and deafening shouts.\n\n720 Acron was a Greek who had come from the ancient land  \nof Corythus, driven into exile while waiting to be married.  \nMezentius saw him from a distance causing havoc in the middle  \nof the battle line in the purple feathers and purple cloak given  \nhim by his promised bride. Just as a ravening lion scouring the  \ndeep lairs of wild beasts, driven mad by the pangs of hunger, if  \nhe sights a frightened she-goat, or sees a stag's antlers rising, he  \nopens his great jaws in delight, his mane bristles, and he springs  \nand fastens on the flesh with foul gore washing his pitiless mouth  \n\u2013 just so did Mezentius charge hot-haste into the thick of the  \n730 enemy and felled the unlucky Acron, who breathed out his life  \ndrumming the black earth with his heels and blooding the  \nweapons broken in his body. Orodes fled, but Mezentius did  \nnot deign to cut him down as he ran, or deal him a wound,  \nunseen, from the back, but came to bar his way and meet him  \nface to face, proving himself the better man by strength in arms  \nand not by stealth. He then put his foot on his prostrate enemy  \nand leaned on his spear, calling out: 'Here, comrades, lies no  \nsmall part of their battle strength, Orodes, that stood so tall.'  \nHis men shouted their glad paean of victory after him, but with  \nhis dying breath Orodes replied: 'Whoever you are that have  \n740 conquered me, I shall be revenged. You will not enjoy your  \nvictory for long. The same fate is looking out for you, and we  \nshall soon be lying in the same fields.' Half smiling, half in anger,  \nMezentius replied: 'Die now. As for me, that will be a matter  \nfor the Father of the Gods and the King of Men,' and at these  \nwords he drew his spear out of the body of Orodes. A cruel rest  \nthen came to him, and an iron sleep bore down upon his eyes  \nand closed them in everlasting night.\n\nCaedicus cut down Alcathous, Sacrator Hydaspes; Rapo  \nkilled Parthenius and Orses, a strong and hardy warrior. Messapus  \nput an end to Clonius and Erichaetes, son of Lycaon,  \n750 Erichaetes being on foot, but Clonius lying on the ground,  \nhaving lost his reins and fallen from his horse. On foot also was  \nAgis the Lycian, who had come out in front of the battle line,  \nbut Valerus had some spark of his family's courage and overthrew  \nhim. Thronius was killed by Salius, and Salius by Nealces,  \nfamed for his javelin and far-shot arrows.\n\nPitiless Mars was now dealing grief and death to both sides  \nwith impartial hand. Victors and vanquished killed and were  \nkilled and neither side thought of flight. In the halls of Jupiter  \nthe gods pitied the futile anger of the two armies and grieved  \n760 that men had so much suffering, Venus looking on from one  \nside and Saturnian Juno from the other, while in the thick of all  \nthe thousands raged the Fury Tisiphone, pale as death.\n\nThen came Mezentius storming over the plain, brandishing a  \nhuge spear, and as tall as Orion who walks in mid-ocean cleaving  \nhis path through its deepest pools with his shoulders rising clear  \nof the waves, or strides along carrying an ancient ash from the  \nmountain tops with his feet on the ground and his head hidden  \nin the clouds \u2013 so did Mezentius advance in his massive armour.  \n770 Aeneas had picked him out in the long ranks of men in front of  \nhim and was going to meet him. Mezentius held his ground,  \nunafraid, and the huge bulk of him stood fast waiting to receive  \nhis great-hearted enemy. Measuring a spear-cast with his eye,  \nhe cried: 'Let the right hand which is my god not fail me now,  \nnor the spear which I brandish to throw. My vow is to strip the  \narmour from that brigand's body and clothe you with it, Lausus.  \nMy trophy over Aeneas will be my own son!' With these words  \nhe threw his spear from long range. Hissing as it flew, it bounced  \noff Aeneas' shield and struck the noble Antores as he stood  \nsome distance away, entering his body between flank and groin.  \nAntores had been a comrade of Hercules. He had come from  \n780 Argos but attached himself to Evander, settling with him in his  \ncity in Italy. And so, falling cruelly by a wound intended for  \nanother, he looked up at the sky and remembered his beloved  \nArgos as he died.\n\nThen the devout Aeneas hurled his spear. Through the circle  \nof Mezentius' convex shield it flew, the triple bronze, the  \nlayers of linen, the three stitched bull-hides, and it stuck low in  \nMezentius' groin, but it had lost its force. Exultant at the sight  \nof the Etruscan's blood, Aeneas tore the sword from the scabbard  \n790 at this thigh. Seeing Mezentius in distress and Aeneas  \nbearing down on him in hot fury, Lausus moaned bitterly for  \nthe father whom he loved and the tears rolled down his face.  \nNow Lausus, I shall tell of your cruel death and glorious deeds  \nin the hope that the distance of time may lead men to believe  \nyour great exploit. Never will it be my wish to be silent about  \nyou, Lausus \u2013 you are a warrior who does not deserve to be  \nforgotten. Mezentius was falling back, defenceless and encumbered,  \ndragging his enemy's spear behind him, stuck in his  \nshield, when young Lausus leapt forward and threw himself  \nbetween them. Just as Aeneas was standing to his full height and  \nraising his arm to strike, he came in beneath the sword blade,  \nblocking Aeneas and checking his advance. Lausus' comrades  \nraised a great shout and supported him by bombarding Aeneas  \n800 and harassing him with their missiles from long range, till the  \nfather could withdraw protected by the shield of the son. Aeneas,  \nenraged, kept under cover. Just as when the clouds descend in a  \nsudden storm of hail, and all the ploughmen and all the workers  \nin the fields scatter across the open ground and the traveller  \nfinds a sure fortress to hide in under a river bank or the arch of  \nsome high-vaulted rock till the rain stops falling on the earth,  \nso that they can continue to do the work of the day when the  \nsunshine is restored \u2013 just so Aeneas, overwhelmed by missiles  \n810 from all sides, weathered the storm of war till the last roll of its  \nthunder, and then it was Lausus he challenged, and Lausus he  \nthreatened: 'Why are you in such a haste to die? Why do you  \ntake on tasks beyond your strength? You are too rash. Your  \nlove for your father is deceiving you.' But Lausus was in full cry  \nand his madness knew no check. At this the anger rose even  \nhigher in the heart of the leader of the Trojans and the Fates  \ngathered up the last threads for Lausus. Aeneas drove his mighty  \nsword through the middle of the young man's body, burying it  \nto the hilt, the point going straight through his light shield, no  \nproper armour to match the threats he had uttered. It pierced,  \ntoo, the tunic his mother had woven for him with a soft thread  \nof gold and filled the folds of it with blood. Then did his life  \n820 leave his body and go in sorrow through the air to join the  \nshades.\n\nBut when Aeneas, son of Anchises, saw the dying face and  \nfeatures, the face strangely white, he groaned from his heart in  \npity and held out his hand, as there came into his mind the  \nthought of his own devoted love for his father, and he said:  \n'What will the devout Aeneas now give to match such merit?  \nWhat gift can he give that will be worthy of a heart like yours?  \nTake your armour, that gave you so much pleasure. Now I  \nreturn you to the shades and the ashes of your ancestors, if that  \nis any comfort for you. In your misfortune you will have one  \n830 consolation for your cruel death, that you fell by the hand of  \nthe great Aeneas.' At this he turned on Lausus' comrades, railing  \nat them as they hung back, while he lifted Lausus off the ground  \nwhere he was soiling his carefully tended hair with blood.\n\nMeanwhile by the bank of the river Tiber Lausus' father was  \nstaunching his wounds with water and leaning against the trunk  \nof a tree to rest. Nearby, his bronze helmet hung from the  \nbranches and his weighty armour lay quiet on the grass. About  \nhim stood his chosen warriors as he bathed his neck, gasping  \nwith pain, and his great beard streamed down his chest. Again  \nand again he asked about Lausus, and kept sending men to  \n840 recall him and take him orders from his anxious father. But  \nLausus was dead and his weeping comrades were carrying him  \nback on his shield, a mighty warrior laid low by a mighty wound.  \nMezentius had a presentiment of evil. He heard the wailing in  \nthe distance and knew the truth. Then, fouling his grey hair with  \ndust, he raised both hands to heaven and flung himself on his  \nson's body: 'Was I so besotted with the pleasure of living that I  \nallowed my own son to take my place under my enemy's sword?  \nIs the father to be saved by the wounds of the son? Have you  \n850 died so that I might live? Now for the first time is death bitter  \nto me! Now for the first time does a wound go deep. And I have  \neven stained your name, my son, by my crimes. Men hated me  \nand drove me from the throne and sceptre of my fathers. I owed  \na debt to my country and my people who detested me, and I  \nwould to heaven I had paid it with this guilty life of mine by  \nevery death a man can die! But I am still alive. I have still not  \nleft the world of men and the light of day. But leave it I shall!'  \nEven as he was speaking, he was raising himself on his wounded  \nthigh, and slow as he was with the violence of the pain deep  \nin his wound, his spirit was unsubdued. He ordered his horse to  \nbe brought. This was his glory and his comfort, and on it he had  \n860 ridden home victorious from all his wars. Seeing it pining, he  \nspoke to it in these words: 'We have lived a long time, Rhaebus,  \nif any mortal life is long. Either you will be victorious today and  \ncarry back the head of Aeneas with the blood-stained spoils  \nstripped from his body, and you and I shall avenge the sufferings  \nof Lausus; or else, if that road is barred and no force can open  \nit, we shall fall together. I do not think, with courage like yours,  \nthat you will accept instructions from any other man or take  \nkindly to Trojan masters.' With these words Mezentius mounted  \nand Rhaebus took on his back the weight of the rider he knew  \nso well. Both his hands were laden with sharp-pointed javelins  \nand on his head he wore his helmet of gleaming bronze with its  \n870 shaggy horsehair crest. So armed, he galloped into the thick of  \nbattle, fierce shame, frenzy and grief all seething together in his  \nheart. Three times he shouted the name of Aeneas. Aeneas knew  \nhis voice and offered up this joyful prayer: 'Let this be the will  \nof the Father of the Gods. Let this be the will of high Apollo.  \nStand and fight with me.' He said no more, but made for  \nMezentius with spear at the ready. Mezentius replied: 'Now  \nthat you have taken my son, you savage, you need not try to  \nfrighten me. That was the only way you could have found to  \n880 destroy me. Death holds no terrors for us and we give not a  \nthought for the gods. Enough words. I have come here to die.  \nBut first I have these gifts for you.' He spoke and hurled a  \nspear at his enemy, then another and another, planting them in  \nAeneas' shield as he flew round him in a great circle, but the  \ngolden boss of the shield held fast. Aeneas stood there and  \nMezentius rode round him three times hurling his spears and  \nkeeping Aeneas on his left side. Three times the Trojan pivoted  \nwith him, turning his huge bronze shield, with its bristling forest  \nof bronze spears. Then, weary of all the delay, weary of plucking  \njavelins out of his shield and hard-pressed in this unequal battle,  \n890 Aeneas, after turning many plans over in his mind, at last burst  \nforward and threw his spear, catching Mezentius' warhorse in  \nthe hollow between its temples. Up it reared thrashing the air  \nwith its hooves and throwing its rider. Then as it came down  \nwith all its weight, dislocating its shoulder, it fell head first on  \ntop of Mezentius and pinned him to the ground. The sky blazed  \nwith the shouts of Trojans and Latins as Aeneas rushed up  \ntearing his sword from the sheath and crying: 'Where is the bold  \nMezentius now? Where is that fierce spirit of his?' The Etruscan  \nlooked up, drinking in the bright air of heaven as he came back  \n900 to his senses, and replied: 'You are my bitter enemy. Why jeer  \nat me and threaten me with death? There is no sin in killing. I  \ndid not come into battle on those terms and my son Lausus  \nstruck no such bargain with you on my behalf. One thing I ask,  \nif the defeated can ask favours from their enemies, to let my  \nbody be buried in the earth. I know the bitter hatred of my  \npeople is all about me. Protect me, I beg you, from their fury  \nand let me lie in the grave with my son.' These were his last  \nwords. He then took the sword in the throat with full knowledge  \nand poured out his life's breath in wave upon wave of blood all  \nover his armour.\n\n## BOOK 11  \nDRANCES AND CAMILLA\n\nMeanwhile the Goddess of the Dawn had risen from Ocean,  \nand anxious and eager as Aeneas was to give time to burying  \nhis comrades, distraught as he was in mind at their deaths, at  \nfirst light the victor was paying his vows to the gods. Cutting all  \nthe branches off a huge oak, he set it up on a mound as a trophy  \nto the great god mighty in war, and clothed it in the shining  \narmour he had stripped from the body of the enemy leader  \nMezentius. There he set the hero's crest dripping its dew of  \n10 blood, the broken spears and the breastplate struck and pierced  \nthrough in twelve places. On the left he bound the bronze shield  \nand from the neck he hung the ivoried sword. He then addressed  \nhis comrades (for all the Trojan leaders were pressing close  \naround him), and these were the words he spoke to urge them  \non in their hour of triumph: 'The greatest part of our work is  \ndone, my friends. In what remains there is nothing to fear. These  \nare spoils I have taken from a proud king, the first fruits of this  \nwar. This is Mezentius, and my hands have set him in this place.  \nOur way now lies towards the king of the Latins and the walls  \nof their city. Make ready your weapons. Fill your minds and  \nyour hopes with the thought of war, so that no man shall hesitate  \n20 or not know what to do when the gods permit us to pull up our  \nstandards and lead the army out of camp. When that time  \ncomes, there must be no faintheartedness or sluggishness in our  \nthoughts to slow us down. In the meanwhile, let us consign the  \nunburied bodies of our comrades to the earth, for that is the  \nonly honour a man has in the underworld. Go,' he said, 'and  \ngrace these noble spirits with their last rites, for they have shed  \ntheir blood to win this land for us. But first let Pallas be sent  \nback to the stricken city of Evander. This was a warrior who  \ndid not fail in courage when his black day took him from us and  \ndrowned him in the bitterness of death.'\n\nSo he spoke, weeping, and made his way back to his own  \n30 threshold where the body of Pallas lay guarded by old Acoetes.  \nAcoetes had once been the armour-bearer of Arcadian Evander,  \nbut the auspices were no longer so favourable when he was  \nappointed as companion to his dear ward, Pallas. About them  \nstood the whole throng of their attendants and all the Trojans  \nand the women of Troy with their hair unbound in mourning  \nafter the manner of their people. But when Aeneas entered his  \nhigh doorway, they beat their breasts and raised their wild  \nlament to the sky till the palace rang with the sound of their  \n40 grief. When he himself saw the head of Pallas cushioned there  \nand his white face, and the open wound torn in that smooth  \nbreast by the Italian spear, the tears welled up and he spoke  \nthese words: 'Oh the pity of it! Fortune came to me with smiles,  \nbut took you from me while you were still a boy, and would not  \nlet you live to see us in our kingdom, or to ride back in triumph  \nto your father's house. This is not what I promised Evander for  \nhis son, when he took me in his arms as I left him, and sent me  \nout to take up this great command, warning me with fear in his  \nheart that these were fierce warriors, that this was a hardy race  \n50 I had to meet in battle. Even now, deluded by vain hopes, he  \nmay be making vows and heaping altars with offerings, while  \nwe bring him with tears and useless honours a young warrior  \nwho owes no more debts to any heavenly power. With what  \neyes will you look at the dead body of your son? Is this how we  \nreturn from war? Are these the triumphs expected of us? Is this  \nmy great pledge? But you will not see a wound on him, Evander,  \nof which you need to be ashamed. You will not be a father who  \nhas the terrible wish that his son who is alive were dead. The  \nland of Italy has lost a great bulwark, and great too is your loss,  \nIulus.'\n\n60 After he had his fill of weeping, he ordered them to take up  \nthe pitiable corpse, and from the whole army he sent a thousand  \nchosen men as escort to pay a last tribute and join their tears  \nwith those of Evander, a small comfort for a great sorrow, but  \na debt that was owed to the stricken father. Others were not  \nslow to weave a soft wickerwork bier of arbutus and oak shoots  \nto make a raised couch, shaded by a canopy of green, where  \nthey laid the young warrior high on his bed of country straw.  \nThere he lay like a flower cut by the thumbnail of a young girl,  \n70 a soft violet or drooping lily, still with its sheen and its shape,  \nthough Mother Earth no longer feeds it and gives it strength.  \nThen Aeneas brought out two robes stiffened with gold and  \npurple threads which Sidonian Dido had long since made for  \nhim with her own hands, picking out the warp in fine gold, and  \nthe work had been a joy to her. With grief in his heart he put  \none of these on the young man's body as his last tribute and in  \na fold of it he veiled the hair that would soon be burned.  \nThen he gathered a great heap of spoil from the battle on the  \nLaurentine fields and ordered it to be brought to the pyre in a  \n80 long procession, adding to it the horses and weapons he had  \ntaken from the enemy. Then came the captives, whose hands he  \nhad bound behind their backs to send them as offerings to the  \nshades of the dead and sprinkle the funeral pyre with the blood  \nof their sacrifice. He also commanded the leaders of the army  \nto carry in their own arms tree trunks draped with weapons  \ncaptured from the enemy and inscribed with their hated names.  \nAcoetes, worn out with age, was led along in the procession,  \nbeating his breast with clenched fists and tearing his face with  \nhis nails, but he collapsed and lay all his length on the ground.  \nChariots were drawn along drenched with Rutulian blood, and  \nthen came Pallas' warhorse Aethon, stripped of all its trappings  \n90 with the tears rolling down in great drops and soaking its face.  \nThere were men to carry his spear and his helmet. The victorious  \nTurnus had the rest. A great phalanx of mourners followed, all  \nthe Trojans and the Etruscans and the Arcadians with their arms  \nreversed. After this procession of all the comrades of Pallas had  \nmarched well clear of the camp, Aeneas halted, and with a deep  \ngroan he spoke these words: 'The same grim destiny of war calls  \nus away from here to weep other tears. For ever hail, great  \nPallas, and farewell for ever.' He said no more but set off  \ntowards his high-built fortifications and marched back into  \ncamp.\n\n100 And now envoys appeared from the city of the Latins bearing  \nolive branches wreathed in wool and asking for a truce. The  \nbodies of their dead were all over the plain where the steel had  \nlaid them, and they begged Aeneas to give them back and let  \nthem go to their graves in the earth, for he could have no quarrel  \nwith men who were defeated and had lost the light of life; he  \nmust show mercy to those who had once been called his hosts  \nand the kinsmen of his bride. Good Aeneas could not refuse this  \npetition. He honoured the envoys, granted what they asked and  \nadded these words: 'What cruel Fortune is this, men of Latium,  \nthat has embroiled you in war and made you run away from us,  \n110 who are your friends? You ask me for peace for the dead, whose  \ndestiny has been to die in battle: I for my part would have been  \nwilling to grant them peace when they were still alive. Nor  \nwould I ever have come to this land if the Fates had not offered  \nme a place here to be my home. I do not wage war with your  \npeople. It was your king who abandoned our sworn friendship  \nand preferred to put his trust in the weapons of Turnus. It is not  \nthese men who should have risked their lives but Turnus. If it is  \nhis plan to put an end to this war by the strength of his arm,  \nand drive out the Trojans, he should have faced me and these  \nweapons of mine in battle. One of us would have lived. God or  \nour own right hands would have seen to that. Go now and light  \n120 fires beneath the bodies of your unfortunate citizens.' Aeneas  \nhad spoken. They were astonished and stood looking at each  \nother in silence.\n\nThen Drances, an older man who had always hated the young  \nwarrior Turnus, and spoken against him, began to make his  \nreply: 'O Trojan great in fame, and greater still in arms, what  \nwords of mine could raise you to the skies? What shall I first  \npraise? Your justice, or your labours in war? Gratefully shall we  \ncarry these words of yours back to our native city, and if Fortune  \nshows us a way, we shall reconcile you to our king Latinus.  \n130 Turnus can make his own treaties. We shall do more. We shall  \ndelight to raise the massive walls Fate has decreed for you and  \nlift up the building stones of Troy on our shoulders!'\n\nAll to a man they murmured in agreement when he had  \nfinished speaking. Twelve days they decided on, and during that  \ntime, with peace as mediator between them, Trojans and Latins  \nwere together in the hills and wandered the woods, and no man  \nharmed another. The iron axe rang upon tall ash trees and  \nbrought down skyward-thrusting pines. They never rested from  \ntheir labours, splitting the oak and fragrant cedar with wedges  \nand carrying down the ash trees on carts from the mountains.\n\n140 But Rumour was already on the wing, overwhelming Evander  \nand the house and city of Evander with the first warnings of  \nanguish. The talk was no longer of Pallas, conqueror of Latium.  \nThe Arcadians rushed to the gates, snatching up funeral torches  \naccording to their ancient practice. The road was lit by a long  \nline of flames which showed up the fields far on either side.  \nNearer and nearer came the throng of Trojans till it joined the  \ncolumns of mourners. When the mothers of Pallanteum saw  \nthem entering the walls, the stricken city was ablaze with their  \ncries. No power on earth could restrain Evander. Coming into  \nthe middle of the throng where the bier had been laid on the  \n150 ground, he threw himself on the body of Pallas and clung to it  \nweeping and moaning until at last grief freed a path for his  \nvoice: 'O Pallas, this is not what you promised your father! You  \nsaid you would not be too rash in trusting yourself to the cruel  \nGod of War. I well knew the glory of one's first success in arms,  \nthe joy above all other joys of one's first battle. These are bitter  \nfirst fruits for a young man. A hard schooling it has been in war,  \nand you did not have far to go for it. None of the gods listened  \nto my vows and prayers. O my dear wife, most blessed of  \nwomen, you were fortunate in your death, in not living to see  \n160 this day. But I have outstayed my time. A father should not  \nsurvive his son. If only I had followed our Trojan allies into  \nbattle and the Rutulians had buried me under their spears! If  \nonly I had given up my own life and this procession was bringing  \nhome my body and not the body of Pallas. I would not wish to  \nblame you, Trojans, nor our treaties, nor regret the joining of  \nour right hands in friendship. The death of my son was a debt I  \nwas fated to pay in my old age. But if an early death was his  \ndestiny, I shall rejoice to think that first he killed thousands of  \nVolscians and fell while leading the Trojans into Latium. Nor  \nwould I wish you any other funeral than this, Pallas, given you  \n170 by good Aeneas and the great men from Phrygia, the leaders of  \nthe Etruscans and all the soldiers of Etruria, bearing the great  \ntrophies of the warriors your right hand has sent to their deaths.  \nAnd you too, Turnus, would now be standing in the fields, a  \nhuge headless trophy, had Pallas been your equal in age, had  \nthe years given you both equal strength. But why does my grief  \nkeep the Trojans from their arms? Go now, take this charge to  \nyour king and do not forget it. If I drag out my hated life now  \nthat Pallas is killed, the reason, Aeneas, lies in your right arm.  \nYou know it owes the life of Turnus to the son and to the father.  \nThis is the one field where you must put your courage and your  \n180 fortune to the test. I seek no joy in life \u2013 that is not what the  \ngods have willed \u2013 only to take this satisfaction down to my son  \namong the dead.'\n\nAurora meanwhile had lifted up her life-giving light for miserable  \nmortals, bringing back their toil and sufferings. Both Tarchon  \nand Father Aeneas soon built funeral pyres on the curving  \nshore and carried there the bodies of their dead, each after the  \nfashion of their fathers. They then set black-burning torches to  \nthe fires and the heights of heaven were plunged into pitchy  \n190 darkness. Three times they ran round the blazing pyres in gleaming  \narmour. Three times they rode in solemn procession round  \nthe fires of the dead with wails of lamentation. Tears fell upon  \ntheir armour and fell upon the earth beneath. The clamour of  \nmen and the clangour of trumpets rose to heaven as some threw  \ninto the flames spoils torn from the corpses of the Latins, their  \nsplendid swords and helmets, the bridles of horses and scorching  \nchariot wheels, while others burned the familiar possessions of  \ntheir dead friends, the shields and spears which Fortune had not  \nblessed. All around, oxen were being sacrificed and their bodies  \noffered to the God of Death, while bristling swine and flocks  \ncarried off from the fields were slaughtered over the fires. All  \n200 along the shore they watched the bodies of their comrades burn  \nand tended the dying flames, nor would they be torn away till  \ndank Night turned over the heavens and showed a sky studded  \nwith burning stars.\n\nThe mourning Latins too had built countless pyres some  \ndistance apart from the Trojans. Many bodies of men they  \nburied in the earth; many they took up and carried back to the  \ncity or to their homes nearby in the countryside. The rest they  \nburned uncounted and unhonoured, a huge pile of jumbled  \ncorpses, and all the wide land on every side was lit by fire upon  \n210 fire, each brighter than the other. When the third day had risen  \nand dispersed the chill darkness of the sky, the mourners levelled  \non the pyres the deep ash in which the bones of the dead were  \nmingled, and weighed it down with mounds of warm earth.  \nThat day in their homes in the city of king Latinus, famous for  \nhis wealth, the noise of grief was at its loudest. That day their  \nlong mourning reached its height. Here were the mothers and  \nheart-broken wives of the dead. Here were loving sisters beating  \ntheir breasts, and children who had lost their fathers, all cursing  \nthis deadly war and Turnus' marriage; he was the man who  \nshould be deciding this matter with his own sword and shield  \nsince he was the man who was claiming the kingdom of Italy  \n220 and the highest honours for himself. The bitter Drances heaped  \nfuel on the fire and swore that Turnus was the only man whose  \nname was being called; nobody else was being asked to fight.  \nBut at the same time many voices were raised for Turnus and  \nmuch was said on his behalf. The great name of the queen cast  \nits protecting shadow and also in his favour was all the fame  \nand all the trophies he had won in his wars.\n\nIn the middle of this disturbance, while the dispute was still  \nraging, to crown all, the envoys suddenly arrived back with a  \ngloomy answer from the city of Diomede. They had achieved  \nnothing for all the efforts they had expended; their gifts, their  \n230 gold, their earnest prayers had failed; the Latins would have to  \nlook elsewhere for reinforcements or plead for peace with the  \nTrojan king. At this bitter blow even king Latinus lost heart.  \nAeneas was chosen by Fate and brought there by the express  \nwill of heaven \u2013 this was what the anger of the gods was telling  \nthem; this was the message of these tombs newly raised before  \ntheir eyes. With such thoughts in mind he summoned a great  \ncouncil, commanding the leaders of his people to come within  \nhis lofty doors. They duly gathered, filling the streets as they  \nstreamed to the royal palace. Greatest in age and first of those  \nwho carried the sceptre, Latinus sat in the middle with sadness  \n240 on his brow and asked the envoys who had returned from the  \ncity of the Aetolians to tell what reply they brought, demanding  \nto hear every detail in due order. The assembly was called to  \nsilence. Venulus obeyed the command and began to speak:  \n'Fellow-citizens, we have seen Diomede and the Argive camp.  \nWe have paced out the road and lived through all the chances  \nof the journey. We have touched the hand that brought down  \nthe land of Ilium. There in the fields near Mount Garganus, in  \nthe Apulian kingdom of Iapyx, the victorious Diomede was  \nfounding his city called Argyripa after the home of his fathers  \nat Argos. After we were admitted to his presence and given leave  \nto speak, we offered our gifts, telling him our names and the  \n250 land from which we came, who had brought war among us and  \nwhat had taken us to Arpi. He heard us out and made this reply  \nin words of peace:\n\n' \"The peoples of your land are blest by Fortune. Yours are the  \nkingdoms of Saturn, the ancient Ausonians, but what Fortune is  \nit that disturbs your peace and persuades you to stir up wars  \nyou do not understand? Those of us whose swords violated the  \nfields of Ilium \u2013 let me not speak of all we endured as we fought  \nbeneath her walls or of our men drowned in her river Simois \u2013  \nwe are scattered over the round earth, paying unspeakable  \npenalties and suffering all manner of punishment for our crimes.  \nWe are a band of men that even Priam might pity. The deadly  \n260 star of Minerva knows us well. So do the rocks of Euboea and  \nCaphereus, the cape of vengeance. From that campaign we have  \nbeen washed up on many a different shore: Menelaus, son of  \nAtreus, is in exile in distant Egypt at the pillars of Proteus; Ulixes  \nhas seen the Cyclopes on Etna; shall I speak of the kingdom of  \nNeoptolemus in Epirus? Of the new home of Idomeneus in  \nCalabria? Of Locrians living on the shores of Libya? Even the  \nleader of the great Achivi from Mycenae was struck down by  \nthe hand of his evil queen the moment he stepped over his own  \nthreshold! The adulterous lover had been waiting for Asia to  \nfall. To think that the envious gods forbade me to return to the  \n270 altars of my fathers or to see the wife I longed for and my  \nbeautiful homeland of Calydon. Even now I am pursued by the  \nsight of hideous portents. My lost comrades have taken to the  \nsky on wings. They have become birds and haunt the rivers \u2013 so  \ncruelly have my people been punished \u2013 weeping till the rocks  \nring with the sound of their voices. From that moment of madness  \nwhen I attacked the body of a goddess and my spear defiled  \nthe hand of Venus, I should have known that this was bound to  \ncome. Do not, I beg you, do not urge me to take part in any  \nsuch battle. I have had no quarrel with the Trojans since the  \n280 uprooting of their citadel of Pergamum, and I do not remember  \nold wrongs or take any pleasure in them. As for the gifts you  \nbring me from your country, give them rather to Aeneas. We  \nhave faced each other, spear against deadly spear, and closed in  \nbattle. Believe me, for I have known it, how huge he rises behind  \nhis shield, with what a whirr he spins his javelin. If the land of  \nIlium had borne two other such heroes, the Trojan would have  \ncome in war to the cities of the Greek, the Fates would have  \nchanged and Greece would now be in mourning. As for all the  \nlong delay before the stubborn walls of Troy, it was the hands  \nof Hector and Aeneas \u2013 both men noble in their courage, noble  \n290 in their skill in arms, but Aeneas the greater in piety \u2013 that held  \nback the victory of the Greeks and did not let it come till the  \ntenth year. Let your hands join in a treaty of peace while the  \nchance is offered, but take care not to let your weapons clash  \non his!\"\n\n'You have heard, O best of kings, the answer of a king. You  \nhave heard his judgement on this great war.'\n\nThe envoys had scarcely finished before a confused roar was  \nrunning through the troubled ranks of the Italians, as when  \nrocks resist a river in spate and the trapped waters eddy and  \ngrowl while the banks on either side roar with the din of the  \n300 waves. As soon as calm returned to their minds and the words  \nof fear were stilled on their lips, the king on his high throne  \naddressed the gods and then began. 'For my part, O men of  \nLatium, I would have wished, and it would have been better so,  \nto have decided this great issue long since, and not be summoning  \na council at a time like this with the enemy sitting by our  \nwalls. We are fighting a misguided war, fellow-citizens, against  \nunconquerable heroes and the sons of gods. Battle does not  \nweary them, and even in defeat they cannot take their hands  \nfrom the sword. If you had any hope of recruiting the Aetolians  \nas your allies, lay it aside. To everyone his own hopes, but you  \n310 can see how feeble this one is. All other resource is shattered  \nand lies in ruins. You can see this with your own eyes. The  \nwhole truth is there at your finger tips. I accuse no one. Courage  \nhas done all that courage could do. The whole body of the  \nkingdom has fought this fight. But now the time has come for  \nme to express an opinion which has formed in my doubting  \nmind. Give me your attention, and I shall tell it in a few words.  \nNear the Tuscan river Tiber I have long owned some land which  \nstretches away to the west beyond the land of the Sicani. Here  \nAuruncans and Rutulians sow their seeds, wearying the stony  \n320 hills with the plough and grazing the roughest of them. Let this  \nwhole area with the pine forests clothing its high mountains be  \ngiven to the Trojans as a token of our friendship, and let us  \npropose a treaty in just terms, inviting them to become partners  \nin our kingdom. Let them settle here, if their hearts are so set  \non it, and build their walls. But if it is their wish to go elsewhere  \nand seize the land of some other nation, and if it is within their  \npower to leave this country of ours, let us weave the timbers of  \ntwenty ships in Italian oak, or more if they can man them. The  \nwood is all lying on the shore. Let them say what ships they  \nwant and how many, and we can provide the bronze, the dockyards  \n330 and the hands to do the work. I propose also that a  \nhundred envoys, men of the highest rank in the Latin race, be  \nsent to carry this message and conclude this treaty, holding out  \nthe branches of peace in their hands and bearing gifts, talents of  \ngold and ivory, and the throne and robe which are the emblems  \nof our royal power. Consider this together, and rescue our  \ncrippled fortunes.'\n\nThen rose Drances, hostile as ever, who always looked askance  \nat Turnus' great reputation and was goaded by bitter  \njealousy. He was generous with his wealth and readier still with  \nhis tongue, but his hand did not warm to battle. His voice had  \n340 some weight in council and was always a force for discord. His  \nmother's breeding gave him pride of rank; his father's origins  \nwere unknown. These were the words he spoke to add force  \nand substance to their anger: 'What you propose, good king  \nLatinus, is clear to all and needs no words of mine to support  \nit. Everyone knows, and admits that he knows, what Fortune  \nhas in store for the people, but they are all afraid to utter it. It  \nis time for the man whose auspices the gods reject to blow a  \nlittle less hard and give us freedom to speak. It is because of his  \nfatal recklessness \u2013 I, for one, shall not be silent though he draw  \nhis sword and threaten me with death \u2013 we have seen so many  \nof our leaders, who have been the lights of our people, extinguished,  \n350 and the whole of our city now slumped in grief, while he  \nstorms the Trojan camp and frightens the sky with his weapons,  \nknowing he can save his own life by taking to his heels. There  \nis still one thing you must add, O best of kings, to all those  \nmany proposals and gifts you tell us to send to the sons of  \nDardanus, one thing only, and no man's violence should be able  \nto overrule your right as a father to give your own daughter to  \na noble husband in a marriage that will be worthy of her, sealing  \nthis peace in a treaty for all time. But if our hearts and minds  \nare so beset with fear of the man, let us beg and beseech him to  \ngive her up and restore to his king and to his fatherland the  \n360 rights which are their due. Why do you keep throwing your  \nunfortunate fellow-citizens into the jaws of danger, Turnus, you  \nwho are the single source and cause of all these sufferings of  \nLatium? War will never save us. We are all asking you for peace,  \nand the one inviolable pledge of that peace. I am the first to  \ncome to you as a suppliant \u2013 you imagine I am your enemy and  \nthat causes me no distress \u2013 look at me! I beg you to pity your  \npeople and lay down your pride. You are defeated. You must  \nleave the field. We have been routed often enough and have seen  \nenough funerals. We have stripped our wide fields bare. But if  \nfame drives you on, if you have the strength in your heart, if  \nyou have such a yearning to receive a palace as a dowry, then  \n370 be bold, have the confidence to go and stand face to face with  \nyour enemy. So that Turnus can get himself a royal bride, our  \nlives are cheap. We, the rank and file, are to litter the fields,  \nunburied and unwept. But you too, if there is any strength in  \nyou, if you have any of the fighting spirit of your fathers, stand  \nup to your challenger and look him in the face.'\n\nAt this, Turnus groaned, and blazed up into a violent rage.\n\nThe words burst from the depths of his heart: 'You have always  \na good supply of words, Drances, when war calls for action.  \n380 When the senate is summoned, you are the first to appear. But  \nthis is no time for filling the council chamber with talk, for  \npouring out high-flown speeches in comfort while our walls and  \nramparts are all that keep the enemy from us, and we are waiting  \nfor the ditches to fill with blood. By all means, Drances, you can  \nthunder out your eloquence in your usual style and accuse me  \nof cowardice, when your right hand has heaped up as many  \nTrojan corpses as mine has and all the fields are studded with  \nyour trophies. But now is our chance to test our vigour and our  \nvalour. We do not have to look too far for enemies \u2013 they are  \nstanding all round the walls. Shall we advance to meet them?  \nYou hesitate? Where is your martial spirit? Will it always be in  \n390 your long tongue and nimble feet? You say I have been defeated.  \nYou scum of the earth, who can say I am defeated when he sees  \nthe Thybris rising, swollen with Trojan blood, the house of  \nEvander destroyed root and branch and the Arcadians stripped  \nof their arms? This is not how great Pandarus and Bitias found  \nme, nor the thousand men I sent down to Tartarus on my day  \nof victory when I was trapped inside the walls and rampart of  \nthe enemy. You say that war will never save us. That prophecy  \n400 is for the Trojan and for yourself, you fool. But go on, stirring  \nup panic everywhere and praising to the skies the strength of a  \nrace of men who have been twice defeated. Go on insulting the  \narmies of Latinus. Now, it seems, the leaders of the Myrmidons  \nare afraid of Phrygian weapons! Now it seems that Diomede  \nand Achilles of Larisa are taking fright, and the river Aufidus is  \nflowing backwards in full retreat from the waves of the Adriatic!  \nDrances even pretends to be terrified when I speak \u2013 a rogue's  \ntrick! The fear is a pretence to add sting to his charges against  \nme. But there is no need for you to be alarmed. My hand will  \nnever take the breath of life from a man like you. It is welcome  \nto stay where it is in that breast of yours.\n\n410 'But now, father, I come to you and to your great plan. If you  \nno longer hold out any hope for our arms, if we are left to  \nfight on utterly alone, if after one setback we are completely  \ndestroyed, and Fortune has abandoned us never to return, let us  \nstretch out our defenceless arms and sue for peace. But if only  \nthere were a spark of our old courage left in us! Any man who  \nhas fallen and bitten the dust of death rather than live to see  \nsuch a thing, I count him fortunate in his life's labours, the  \nnoblest spirit amongst us! Surely we still have untapped resources  \nand warriors who have not yet engaged and there are  \n420 still cities and peoples in Italy to help us? And surely the Trojans  \nhave paid a heavy price in blood for the glory they have won!  \nThey too have had their funerals. The same storm has fallen on  \nall of us. Why then do we disgrace ourselves by stumbling on  \nthe threshold? Why do our knees start shaking before we hear  \nthe trumpet? Many things change for the better with the passing  \nof the days and the ever-varying workings of time. Fortune  \ncomes and goes. She has mocked many a man, and then set his  \nfeet back on solid ground. So the Aetolian Diomede and his city  \nof Arpi will not help us. But Messapus will, and Tolumnius,  \n430 blessed by the gods, and all the leaders who have come to us  \nfrom so many peoples, and great will be the glory for the chosen  \nmen of Latium and the Laurentine fields. We have Camilla too,  \nfrom the noble Volscian race, leading her mounted column and  \nher squadrons flowering with bronze. But if I am the only one  \nthe Trojans want to meet in battle, if that is your will and I am  \nsuch a great obstacle to the good of all, then the Goddess of  \nVictory has not entirely abandoned me, nor is she so ill-disposed  \nto these hands of mine that I should refuse any undertaking for  \nwhich I have such hopes. I shall go and face him with my spirits  \n440 high were he mightier than Achilles and with armour the equal  \nof his, made like his by the hands of Vulcan. To all of you, and  \nto Latinus, father of my bride, I, Turnus, second in courage to  \nnone of those who have gone before me, have offered up my  \nlife. Is Aeneas challenging me, and me alone? Let him challenge.  \nIt is the answer to my prayer. If this is the anger of the gods I  \nwould not have Drances appease it; if it is a moment for courage  \nand glory, I would not give it to Drances.'\n\nSo they disputed among themselves in deep uncertainty.  \nAeneas, meanwhile, had struck camp and was moving his army.  \nSuddenly there came a messenger rushing wildly through the  \nroyal palace and causing panic all over the city: the Trojans,  \n450 drawn up in line of battle, the Etruscan squadron with them,  \nwere coming down the valley of the Tiber and filling the whole  \nplain. There was instant confusion and dismay among the people  \nand hearts were roused by the sharp spur of anger. With wild  \ngestures the young men asked for arms. 'Arms!' they shouted,  \nwhile their fathers wept and murmured. On every side a great  \nclamour of dissenting voices rose to the winds like the sound of  \nflocks of birds settling in groves of tall trees, or swans whose  \nharsh calls ring across the chattering pools of the river Padusa,  \nso rich in fish. 'Do not disturb yourselves, citizens!' shouted  \n460 Turnus, seizing the moment. 'Convene your council and sit there  \npraising peace while your enemies invade your kingdom with  \nswords in their hands.' These were his only words to them as he  \nleapt to his feet and rushed from the lofty palace shouting: 'You,  \nVolusus, tell the Volscian contingents to arm! And take the  \nRutulians with you! Deploy the cavalry, Messapus, and you  \ntoo Coras with your brother, in battle array over the whole  \nplain! Some of you reinforce the approaches to the city and man  \nthe towers. The rest of you come and advance with me where I  \norder.'\n\nIn an instant they poured on to the walls from all over the  \n470 city. Father Latinus himself left the council and abandoned his  \nhigh designs till a later time, in deep distress at the troubles of  \nthe hour. Again and again he blamed himself for not eagerly  \nwelcoming Trojan Aeneas and taking him into the city as his  \nson-in-law. Meanwhile men were digging pits in front of the  \ngates and bringing up rocks and stakes. The shrill trumpet blew  \nthe signal for bloody battle and mothers and sons went to make  \na motley ring round the walls of the city. Their last labour called  \nthem and they came. The queen too, with a great retinue of the  \nmothers of the city, rode in her carriage to bring offerings to the  \ntemple of Pallas on the heights of the citadel. With her went the  \n480 maiden Lavinia, the cause of all this suffering, her lovely eyes  \ndowncast. The mothers followed them and filled the temple  \nwith the smoke of incense, pouring out their sad prayers from  \nits high threshold: 'Mighty in arms, ruler of the battle, Tritonian  \nmaiden, break with your hand the spear of the Phrygian pirate  \nand throw him to the ground. Spread out his body beneath your  \nhigh gates.' Turnus in a fury was eagerly arming himself for  \nbattle, and soon had on his breastplate glowing red with bristling  \nscales of bronze, and his golden greaves. His head was still bare,  \n490 but the sword was girt to his side as he ran down from the  \nheights of the citadel in a blaze of gold, ardent and exulting and  \nalready grappling with the enemy in hope and expectation. He  \nwas like a stallion that has broken his tether and burst from his  \nstall; free at last he gains the open plain and runs to the fields  \nwhere the herds of mares are pastured or gallops off to bathe in  \nthe river which he used to know so well, tossing high his head  \nand whinnying with delight while the mane streams over his  \nneck and flanks.\n\nThe princess Camilla came to meet him with her Volscians in  \nbattle order. Under the very gates of the city she leapt down  \n500 from her horse, and all her squadron followed her example,  \ndismounting in one flowing movement. These were her words:  \n'Turnus, if the brave are right to have faith in themselves, I dare  \nto meet the Trojan cavalry \u2013 this is my undertaking \u2013 and go  \nalone against the horsemen of Etruria. Give me leave to try the  \nfirst hazard of war, while you stay on foot by the walls and  \nguard the city.'\n\nAt these words Turnus fixed his eyes on this formidable  \nwarrior maiden and replied: 'O Camilla, glory of Italy, I cannot  \n510 hope to express my gratitude in words or deeds. But now, since  \nthat spirit of yours knows no limits, come share with me the  \nheat of battle. According to a firm report my scouts have brought  \nme, that scoundrel Aeneas has sent his light-armed cavalry  \nahead to scour the plains, while he himself is coming to the city  \nalong a ridge in deserted mountain country. I am planning an  \nambush where there is a sunken path through a wood, and shall  \npost armed men where the road enters and where it leaves the  \ngorge. You go to meet the Etruscan cavalry and engage them.  \nBold Messapus will be with you with the horsemen of Latium  \nand the squadron of Tiburtus, and you will have the task of  \n520 leading them.' So he spoke and with like words urged Messapus  \nand the leaders of his allies into battle, while he went to meet  \nhis enemy.\n\nThere is a winding valley well suited to stealth and stratagem  \nin war. Hemmed in on both sides, it is darkened by the dense  \nfoliage of trees, and a narrow path leads into it making a  \ntreacherous approach through a ravine. Above this valley,  \namong the viewpoints on the hilltop, there lies a little-known  \nplateau which gives safe cover whether you wish to engage the  \nenemy on your right flank or on your left or stand on the  \n530 ridges rolling down great boulders. Marching by paths he knew,  \nTurnus took up position here and settled into ambush in this  \ndangerous forest.\n\nMeanwhile in the palace of the heavens Diana, daughter of  \nLatona, spoke to swift Opis, one of the sacred company of girls  \nwho were her companions, and these were her sad words:  \n'Camilla is going to a cruel war. Dear as she is to me above all  \nothers, she has put on our armour, and it will avail her nothing.  \nThis is no new love, believe me, that has come to move the heart  \n540 of Diana with sudden sweetness. When Metabus, hated by his  \npeople for his arrogant use of power, was driven from his throne,  \nhe left the ancient city of Privernum and took his infant daughter  \nwith him through all his wars and battles, to be his companion  \nin exile. He called her Camilla, changing part of her mother's  \nname, Casmilla. Carrying her in his arms, he made for the long  \nridges and the lonely woods, cruel spears pressing him hard on  \nevery side and Volscian soldiers on the move all about him.  \nSuddenly he found his way blocked by the river Amasenus in  \nfull spate, foaming to the top of its banks \u2013 such a deluge of rain  \nhad burst from the clouds. He was about to leap into the water  \n550 to swim across, but checked himself out of love for his child and  \nfear for the burden he so loved. As he pondered all the dangers,  \na painful resolve soon formed in his mind. He took the warrior's  \nspear he chanced to have in his hand, a mighty weapon of  \nsolid, knotted, well-seasoned wood, and wrapping the baby in  \ncork-tree pith and bark, he lashed her tightly to the middle of  \nthe spear. Then brandishing it in his mighty hand, he cried out  \nto heaven: \"To you, kindly maiden, lover of woods and daughter  \nof Latona, I dedicate my daughter as your handmaiden. She is  \nyour suppliant, and as she flies through the air to escape her  \n560 enemies, the first weapon she holds is yours. O goddess, I  \nsolemnly pray, receive her as your own as I now commit her to  \nthe hazard of the winds.\" At these words he drew back his arm  \nand sent the weapon spinning. The waters rang with the sound  \nas helpless Camilla flew over the wild river on the whistling  \njavelin. But by now a great throng of his enemies was pressing  \nMetabus even closer, and he threw himself into the water. Then,  \nin triumph on the other side, he wrenched from the turf spear  \nand the maiden with it, his dedication to Diana.\n\n'No cities took him under their roofs or within their walls \u2013  \nhe himself was too savage to have submitted to them \u2013 but he  \n570 spent his whole life on the lonely mountains among the herdsmen.  \nThere in the scrub among the rough dens of beasts he fed  \nhis daughter with milk from the udders of wild brood-mares,  \nputting the teats to her soft lips, and as soon as she had taken  \nthe first steps on her infant feet, he put a keen-edged javelin in  \nher hand and slung a bow and arrows from her little shoulder.  \nInstead of gold in her hair and a long cloak to cover her, a tiger  \nskin hung from her head all down her back. While her hand was  \nstill soft, she was spinning her baby javelins and whirling the  \n580 sling round her head on its tapering thong to shoot the white  \nswan or crane from the river Strymon. Many a mother in the  \ntowns of Etruria longed in vain to see her married to her son,  \nbut all she cared for was Diana. Undefiled, she preserved a  \nconstant love for her weapons and her chastity. If only she had  \nnever been caught up in such a war as this, daring to challenge  \nthe Trojans! I would have loved her and she would now have  \nbeen one of my companions. But come now, since a bitter fate  \nis closing in on her, glide down from the sky, Opis my nymph,  \nand visit the land of Latium, where a dreadful battle is being  \n590 fought and all the omens are adverse. Take these weapons, and  \ndraw an avenging arrow from my quiver. Then, with that same  \nshaft, whoever violates that sacred body with a wound, be he  \nTrojan or Italian, must pay to me an equal penalty in blood.  \nThen I shall put a cloud round her poor body and her armour  \nand take them undespoiled to lie in a tomb in her own country.'  \nThe goddess spoke, and Opis, veiled in a dark storm, glided  \nlightly down through the breezes of the sky, whirring as she  \nflew.\n\nBut all this time nearer and nearer to the walls came the  \nTrojan column, the Etruscan leaders and the whole cavalry  \n600 army drawn up in regular squadrons. Horses were prancing and  \nsnorting all over the plain, fretting at the reins that held them in  \nand plunging to one side after another. Far and wide the field  \nbristled with the steel of the spears, and all the land was a blaze  \nof light from uplifted weapons. There too, coming to oppose  \nthem, appeared Messapus and the swift Latins, Coras with his  \nbrother, and the squadron of Camilla. Their right arms were  \ndrawn back, their lances thrust forward with tips quivering.  \nMen were arriving. Horses were neighing. The whole plain was  \nablaze. They had now come within a spear-cast of each other  \nand stopped. Then, with a sudden shout, they galloped forward,  \n610 urging their horses to frenzy, and showering weapons thick as  \nsnow till the sky was curtained with shadow. Tyrrhenus and  \nbold Aconteus were first to charge each other, riding full force  \nwith levelled spears, and great was the din and fearful the fall  \nas they crashed their warhorses against each other, smashing  \nbreast on breast. Aconteus was thrown forward a great distance  \nand fell like a thunderbolt, or a rock hurled from a catapult,  \nscattering his life's breath into the breezes.\n\nIn that instant the battle lines were thrown into disorder.  \nPutting their shields on to their backs, the Latins turned and  \n620 rode back towards the city walls driven by the Trojan squadrons  \nunder Asilas. But when they were almost at the gates, they raised  \nanother shout and pulled round the supple necks of their horses,  \nwhile the Trojans fled in their turn, galloping with slack reins in  \na long retreat. As the sea advances wave by wave, now rushing  \nto the land, throwing foam over the rocks and soaking the edge  \nof the sand in the bay; now turning and hurrying back, sucking  \ndown the stones and rolling them along in its undertow while  \nthe shallows retreat and the shore is left dry \u2013 just so the  \nEtruscans twice turned and drove the Rutulians to the city walls,  \n630 and twice they were repulsed and had to cover their backs with  \ntheir shields and look over their shoulders at their enemies. But  \nwhen they clashed in battle for the third time, and all the ranks  \nwere embroiled together, each man singled out his own enemy,  \nand then the groans of the dying could be heard, weapons  \nand bodies lay deep in blood, half-dead horses rolled about  \nentangled with the corpses of men, and ever fiercer and fiercer  \ngrew the battle. Orsilochus did not dare go near Remulus, but  \nhurled his spear at his horse and its steel point stuck under its  \near. Maddened by the blow, it reared, heaving its chest high and  \nlashing its hooves, unable to endure the pain of the wound.  \n640 Remulus was thrown and sent rolling on the ground, Catillus  \nfelled Iollas and then Herminius, great in stature, in spirit, and  \nin arms. His head of golden hair was bare, his shoulder bare,  \nand he had no fear of wounds, so vast he stood and open to the  \nweapons of his enemies. Catillus' spear drove right through  \nhim and stood out between his broad shoulders quivering, and  \nHerminius doubled up in agony. Black blood was flowing everywhere  \nas they dealt out slaughter with the steel, searching for  \ndeath and glory among the wounds.\n\nThere in the middle of all this bloodshed, exulting in it, was  \nthe Amazon Camilla with the quiver on her shoulder, and one  \n650 side bared for battle. Sometimes the pliant spears came thick  \nfrom her hand; sometimes, unwearied, she caught up her mighty  \ndouble axe, and the golden bow and arrows of Diana rang on  \nher shoulder. Whenever she was forced to retreat, she turned  \nher bow and aimed her arrows while still in flight. The girls she  \nhad chosen as her companions were all about her, Larina, Tulla,  \nand Tarpeia brandishing her bronze axe, all of them daughters  \nof Italy, chosen by the servant of the gods Camilla to do her  \nhonour by their beauty and to be her own trusted attendants in  \npeace and war. They were like the Amazons of Thrace whose  \n660 horses' hooves drum on the frozen waters of the river Thermodon  \nwhen they fight round Hippolyte in their brightly coloured  \narmour, or when Penthesilea, daughter of Mars, rides home in  \nher chariot and her army of women with their crescent shields  \nexult in a great howling tumult.\n\nWhom first did your spear bring down from his horse? Whom  \nlast, fierce warrior maiden? How many bodies of dying men did  \nyou strew on the ground? Eunaeus, son of Clytius, was the first.  \nWhen he stood face to face with Camilla and she drove the long  \npine shaft of her spear through his unprotected chest, he vomited  \nrivers of blood and champed the gory earth with his teeth,  \n670 twisting himself round his wound as he died. Then she brought  \ndown Liris and Pagasus on top of him: Liris when he was trying  \nto collect the reins after his wounded horse had reared and  \nthrown him, Pagasus when he came and stretched out an undefended  \nright hand to support Liris as he fell; but they both  \nwent flying head over heels. Then she sent Amastrus, the son of  \nHippotas, to join them, and raced after Tereus and Harpalycus,  \nDemophoon and Chromis, pressing them hard even at long  \nrange with her spear, and for every dart that flew from her hand,  \na Trojan hero fell. The huntsman Ornytus was rushing past in  \nstrange armour, mounted on his horse Iapyx. This was a warrior  \n680 who wore on his broad shoulders the hide of a bullock, while  \nhis head was encased in the huge gaping jaws of a wolf, complete  \nwith cheekbones and white teeth. A country spear shaped like  \na sickle armed his hand as he moved in the middle of the press,  \ntaller by a head than them all. She caught him \u2013 it was not  \ndifficult, for the whole column had turned and run \u2013 and when  \nshe had pierced him through, she spoke these bitter taunts over  \nhim: 'So you thought you were driving game in the woods, my  \nEtruscan friend? The day has come when you have been proved  \nwrong by a woman's weapons! But it is no mean name you will  \nbe taking to your fathers when you tell them you fell by the  \nspear of Camilla.'\n\n690 Instantly then she struck Orsilochus and Butes, the two tallest  \nof the Trojans. Butes was turned away from her and the tip of  \nher spear went in between helmet and breastplate where his  \nneck shone white as he sat in the saddle with the shield hanging  \nloose on his left arm. She fled from Orsilochus, but after he had  \ndriven her in a great circle, she cut inside the arc and began to  \npursue her pursuer. Then, rising above him, she struck again  \nand again with her mighty axe, hacking through his armour and  \nhis bones as he begged and pleaded with her and the axe-blows  \n700 spilt the hot brains down his face. The warrior son of Aunus of  \nthe Apennines then came upon her and stood stock still in  \nsudden terror at the sight. He was not the least of the Ligurians  \nwhile the Fates gave him leave to tell his lies. So, when he saw  \nthat it was too late to save himself by running away, and that  \nthe princess was upon him and would not be deflected, he began  \nto play his tricks, using all his cunning and calculation. 'What  \nis so wonderful,' he said, 'if a woman depends on the courage  \nof a horse? Give up your chance of running away, and risk your  \nlife in close combat with me on level ground. Gird yourself to  \nfight on foot and you will soon discover that the winds are  \nblowing you only the illusion of glory.' These words stung  \n710 Camilla to a burning fury of resentment. Handing her horse to  \na companion, she stood there to face him without a trace of  \nfear, armed like her enemy with a naked sword and a plain light  \nshield. The moment he thought his ruse had succeeded, the  \nwarrior took to his heels himself. Jerking the reins around, he  \nmade off, driving his horse to the gallop with steel spurs. 'You  \nLigurian fool!' she cried. 'You are the one who has been carried  \naway by the empty winds of pride! You have taken to the  \nslippery arts of your ancestors, but little good will they do you.  \nTrickery will not bring you safe back home to your treacherous  \nfather Aunus.' These were her words, and on nimble feet she  \nran as swift as fire in front of the horse and stood full in its path.  \n720 Then, seizing the reins, she exacted punishment from her enemy  \nin blood, as easily as the sacred falcon flies from his crag to  \npursue a dove high in the clouds, catches it, holds it and rips out  \nits entrails with hooked claws while blood and torn feathers  \nfloat down from the sky.\n\nBut the Father of Gods and Men was not blind to this as he  \nsat high above on the top of Olympus, and he roused Tarchon  \nthe Etruscan to bitter battle, laying on him the sharp goad of  \n730 anger. So Tarchon rode among the slaughter in the ranks of his  \nretreating squadrons, whipping them up with all manner of  \ncries, calling on each man by name and rallying the routed to  \ndo battle: 'What are you afraid of, you Etruscans? Will you  \nnever know shame? Will you always be so spiritless? This is  \nrank cowardice! One woman has turned this whole army and is  \nscattering you to all points of the compass! What are weapons  \nfor? Why do we carry swords in our hands and not use them?  \nYou are not so sluggish when it comes to lovemaking and night  \ncampaigns, or when the curved pipe calls you up to the dancing  \nchorus of Bacchus! Wait, then, for feasts and goblets from  \ngroaning tables. That is what you love. That is what you care  \nabout. Do nothing till the soothsayer gives his blessing and  \n740 announces the festival and the fat victim calls you into the deep  \ngroves.' When this harangue was over, he spurred his horse into  \nthe thick of the enemy \u2013 he too was willing to die \u2013 and made a  \nwild charge at Venulus. Tearing him off his horse and clasping  \nhim in his right arm, he rode off at full gallop with his enemy  \nheld in front of him. A shout rose to the sky and all the Latins  \nturned to look as Tarchon flew like fire across the plain carrying  \nman and armour with him. Then he broke off the steel head of  \nVenulus' spear and with it probed for exposed flesh where  \n750 he could give the fatal wound. Venulus fought back to keep  \nTarchon's hand from his throat, pitting strength against violence,  \njust as when a tawny eagle has seized a snake and flown  \nup into the sky, winding its talons round it and digging in its  \nclaws; meanwhile the wounded serpent writhes in sinuous coils,  \nits scales stiff and rough, and hisses as it reaches up with its  \nhead; but for all its struggles, the eagle never stops tearing at it  \nwith its great hook of a beak, beating the air all the time with  \nits wings \u2013 just like such an eagle did the victorious Tarchon  \ncarry off his prey from the Tiburtine ranks. Following their  \nleader's example, and seeking like success, the Etruscans, the  \nmen from Maeonia, rushed into battle. Then Arruns, whose life  \n760 was owed to the Fates, circled round Camilla to find where  \nFortune would offer the easiest approach. She was swift of foot,  \nbut he was more than her equal with the javelin and far superior  \nin cunning. Wherever she went on her wild forays through the  \nthick of battle, Arruns was behind her, quietly following in her  \ntracks. Wherever she went as she returned in triumph and  \nwithdrew from her enemies, Arruns pulled on his swift reins  \nand kept out of sight. Round a whole circle he went, trying now  \none approach, now another, brandishing the fatal spear that  \nnever missed its mark.\n\nIt then so chanced that Chloreus appeared, a man who had  \nbeen consecrated to Cybele on her mountain, and in days long  \npast had been a priest. She saw him a long way off, resplendent  \n770 in his Phrygian armour and spurring his foaming warhorse. The  \nhorse-cloth was of hide with gold stitching and overlapping  \nbrass scales in the shape of feathers. He himself shone with  \nexotic indigo and purple. The arrows he shot from his Lycian  \nbow were from Gortyn in Crete and the bow hanging from his  \nshoulder was of gold. Gold too was the helm on the head of the  \npriest, and on that day he had gathered the rustling linen folds  \nof his saffron-yellow cloak into a knot with a golden brooch. He  \nwore an embroidered tunic and barbaric embroidered trousers  \ncovered his legs. Whether her intention was to nail his Trojan  \narmour to the temple doors or to sport captive gold on her  \n780 hunting expeditions, she picked him out in the press of battle,  \nand blind to all else and unthinking, she tracked him through  \nthe whole army, burning with all a woman's passion for spoil  \nand plunder. At last the lurking Arruns saw his moment and  \nhurled his spear, offering up this prayer to heaven: 'O highest  \nof the gods, guardian of the holy mountain of Soracte, Apollo,  \nwe are the first to worship you. We heap up the wood of the  \npine to feed your flames, and in your holy rites, sure in our faith,  \nwe walk on fire, sinking our feet deep in the hot ash. Grant  \n790 now, All-powerful Father, that our arms be wiped clean of this  \ndisgrace. My mind is not set on spoils won from a girl or a  \ntrophy set up for routing her or for any form of booty. My fame  \nwill come from my other feats of arms. But let this deadly  \nscourge be defeated and fall to my spear, and I shall go back to  \nthe cities of my fathers and claim no credit.'\n\nPhoebus Apollo heard, and part of his prayer he decided to  \nanswer, part he scattered to the swift breezes of air. He granted  \nhis prayer to surprise Camilla and lay her low in death, but did  \nnot allow the mountains of his native land to see him ever again.  \nA sudden squall took these words and blew them far away to  \nthe winds of the south. So, when the spear that left his hand  \nwent whirring through the air and the Volscians, all of them,  \n800 turned their minds and eyes intently to their queen, she was not  \nthinking of whirring or of air or of weapons coming out of the  \nsky, and the shaft struck home beneath her naked breast and  \nlodged there drinking deep of her virgin blood. Her companions  \nrushed in panic to support their falling queen, and Arruns fled,  \nmore terrified than anyone, joy mixed with his fear. He had lost  \nhis faith in his spear and was afraid to face the weapons of the  \nwarrior maiden. As when a wolf has killed a shepherd or a great  \n810 ox, and goes at once to hide high in the trackless hills before the  \navenging spears can come to look for him; he knows what he  \nhas done, and takes fright, comforting his quivering tail by  \ntucking it under his belly as he makes for the woods \u2013 just so  \ndid Arruns disappear from sight in wild confusion, happy to  \nescape and mingle in the press of battle. Camilla was dying. She  \ntried to pull out the spear, but its steel point stood deep in the  \nwound between the bones of her ribs. She was swooning from  \nloss of blood, her eyes dimming in the chill of death, and the  \n820 flush had faded from her cheeks. With her dying breath she  \nspoke to Acca, alone of all her young friends. She was her most  \nfaithful companion and to her alone she used to open her heart.  \n'I can do no more, Acca, my sister. This cruel wound is taking  \nall my strength, and everything is going dark around me. Run  \nfrom this place and take my last commands to Turnus. He must  \ncome into battle and keep the Trojans away from the city. And  \nnow, farewell.' Even as she was speaking she was losing her  \nhold on her reins and in spite of all her efforts she slid to the  \nground. Then, growing cold, she little by little freed herself  \n830 from her body. Her neck drooped and she laid down her head,  \nyielding to death and letting go her weapons, as her life left her  \nwith a groan and fled in anger down to the shades. At this a  \nmeasureless clamour rose and struck the golden stars. Now that  \nCamilla had fallen, the battle raged as never before. Charging  \nin one solid mass came the whole army of the Trojans, the  \nEtruscan nobles and the Arcadian squadrons of Evander.\n\nOpis, Diana's sentinel, had long been at her post high in the  \nmountains, watching the fighting and knowing no fear. But  \nwhen, far beneath her in the press of warriors shouting in the  \nfrenzy of battle, she saw Camilla receive the bitter stroke of  \n840 death, she groaned and spoke these words from the depths of  \nher heart: 'Alas, Camilla! You have paid too cruel a price for  \ndaring to challenge the Trojans in war, nor has it profited you  \nthat alone in the wild woods you have worshipped Diana and  \nworn our quiver on your shoulder. But your queen has not left  \nyou unhonoured now at your last hour. This death of yours will  \nnot be forgotten among the peoples of this earth, and no one  \nshall say that you have died unavenged. Whoever has desecrated  \nyour body with a wound will pay just penalty with his life.'\n\nAt the foot of a high mountain there was a huge mound of  \n850 earth shaded by dense ilex trees. It was the tomb of Dercennus,  \nan ancient king of the Laurentines. Here the lovely goddess first  \nalighted on her swift flight, keeping watch for Arruns from the  \nhigh mound. When she saw him gleaming in his armour and  \nswollen with empty pride, she called out: 'Why are you leaving?  \nTurn round and come in this direction. Come here and die! You  \nmust receive your reward for Camilla. Come, even a man can die  \nby the weapons of Diana!' When she had spoken, the Thracian  \n860 nymph took a winged arrow from her gilded quiver and drew  \nher deadly bow. Far back she stretched the string until the  \ncurved horns of the bow were close together, her hands level,  \nthe left on the steel point of the arrow, the right holding the  \nstring against her breast. Arruns heard the hiss of the arrow and  \nthe whirr in the air, and in that same moment the steel was  \nplanted in his flesh. His comrades paid no heed. They left him  \nbreathing his last and groaning in some place unknown in the  \ndust of the plain, while Opis soared on her wings to heavenly  \nOlympus.\n\nThe light-armed squadron of Camilla were the first to flee  \n870 when they lost their queen; then the Rutulians in a rout; then  \nbold Asilas and all the scattered leaders and leaderless columns  \nmade for safety, wheeling their horses and galloping for the  \nwalls. No weapon could check the deadly onset of the Trojans  \nand no one could stand against them. Back rode the Latins with  \nslack bowstrings on slumped shoulders, and the four-hooved  \nbeat of their galloping horses drummed on the crumbling plain.  \nAs the black cloud of swirling dust rolled up to the walls, the  \nmothers stood on the watch-towers beating their breasts and  \nthe wailing of women rose to the stars in the sky. The first Latins  \n880 to burst into the open gates were pressed hard by a pursuing  \ncolumn of enemies mingled with friends and did not escape a  \npitiable death. There, on the very threshold, within the walls of  \ntheir native city and in the safe refuge of their own homes, their  \nbodies were pierced and they breathed out their life's breath.  \nSome shut the gates and dared not open them to take their own  \npeople within the walls for all their pleading, and there was  \npiteous slaughter of the armed men guarding the approaches  \nand of men rushing to death on their weapons. Of those who  \nwere shut out before the weeping eyes of their own parents,  \nsome rolled headlong down into the ditches with the weight of  \nthe rout behind them, while others came on blindly at full gallop  \n890 and crashed into the massive gates with their firm-set posts.  \nEven the mothers strove their utmost \u2013 the true love of their  \nnative land showed them the way and Camilla was their example.  \nWildly they hurled missiles from the walls and rushed to  \ndo the work of steel with stumps and stakes of oak wood  \nhardened in the fire, longing to be the first to die in defence of  \nthe walls of their city.\n\nMeanwhile the warrior Turnus was still in the wood when  \nthe bitter news came and filled his heart to overflowing. The  \nwords of Acca brought him great turmoil of spirit: the battle  \nforces of the Volscians were destroyed; Camilla had fallen;  \n900 the enemy were attacking fiercely and had carried everything  \nirresistibly before them; panic was already reaching the city  \nwalls. In a frenzy \u2013 and this is what the implacable will of Jupiter  \ndecreed \u2013 he came down from the hills where he had kept his  \nambush and left the wild woods behind him. Scarcely was he  \nout of sight and moving on to the plains when Father Aeneas  \nentered the open pass, came over the ridge and then emerged  \nfrom the woods. So then they were both making for the walls at  \nspeed, with their whole armies marching not many paces from  \neach other. Aeneas saw the Laurentine columns and the long  \nline of dust smoking on the plains at one and the same moment  \n910 as Turnus recognized Aeneas advancing relentlessly under arms  \nand heard the drumming of approaching hooves and the  \nbreathing of horses. They would have joined battle instantly  \nand tried the fortunes of war if the rose-red sun had not been  \ndipping its weary horses in the Iberian sea, drawing down the  \nlight of day and bringing on the night. They both encamped  \nbefore the city and built stockades on their ramparts.\n\n## BOOK 12  \nTRUCE AND DUEL\n\nWhen Turnus saw the line of the Latins broken, the battle going  \nagainst them and their spirits flagging, when he realized that the  \ntime had come to honour his promises and that all eyes were  \nupon him, no more was needed. He burned with implacable  \nrage and his courage rose within him. Just as a lion in the fields  \nround Carthage, who does not move into battle till he has  \nreceived a great wound in his chest from the hunters, and then  \nrevels in it, shaking out the thick mane on his neck; fearlessly  \nhe snaps off the shaft left in his body by the ruffian that threw  \nit, and opens his gory jaws to roar \u2013 just so did the violent  \n10 passion rise in Turnus. At last he spoke these wild words to the  \nking: 'Turnus keeps no man waiting. There is no excuse for  \nAeneas and his cowards to go back on their word or fail to keep  \ntheir agreement. I am coming to meet them. Bring out the  \nsacraments, father, and draw up the terms of the treaty. Either  \nthis right hand of mine will send this Trojan who has deserted  \nAsia down into Tartarus \u2013 the Latins can sit and watch \u2013 and  \none man's sword shall refute a charge brought against a whole  \npeople, or else he can rule over those he has defeated and have  \nLavinia as his wife.'\n\n20 Latinus answered him, and his voice was calm: 'You are a  \ngreat-hearted young warrior. The more you excel in fierce courage,  \nthe more urgent is my duty to take thought, to weigh all  \npossible chances and to be afraid. You have the kingdom of  \nyour father Daunus. You have all the cities your right hand has  \ntaken. I too, Latinus, have some wealth and some generosity of  \nspirit. In Latium and the Laurentine fields there are other women  \nfor you to marry, and of the noblest families. This is not easy to  \nsay. Allow me to speak openly and honestly, and as you listen,  \nlay these words to your heart. For me it would have been wrong  \nto unite my daughter with any of those who came to ask for her  \nin the past. It was forbidden by all the prophecies of gods and  \n30 men. But I gave way to my love for you. I gave way to the  \nkinship of blood and to the grief and tears of my wife. Breaking  \nall the ties that bound me, I seized Lavinia from the man to  \nwhom she had been promised and took up arms in an unjust  \ncause. From that moment you see the calamities of war that fall  \nupon me, and the suffering that you bear more than any other.  \nTwice we have been crushed in great battles, and we can scarcely  \nprotect within our city the future hopes of Italy. The current of  \nthe Thybris is even now warm with our blood and the broad  \nplains white with our bones. Why do I always give way? Why  \ndo I change my resolve? What folly this is! I am ready to accept  \nthem as allies if Turnus is killed; why not put an end to the war  \n40 while he is still alive? What will your kinsmen the Rutulians,  \nwhat will the whole of the rest of Italy say if I betray you and  \nsend you to your death \u2013 which Fortune forbid \u2013 when you are  \nasking to marry my daughter? Remember the many accidents  \nof war and take pity on your old father waiting with heavy heart  \nfar away in your native Ardea.' These words had no effect on  \nTurnus. The violence of his fury mounted. The healing only  \nheightened the fever. As soon as he could bring himself to speak,  \nout came his reply: 'This concern you are so kind as to show for  \nmy sake, I beg of you for my sake, forget it, and allow me to  \n50 barter my life for glory. We too have weapons, father. We too  \nhave some strength in our right arm to throw the steel around,  \nand when we strike a man, the blood flows from the wound.  \nHis mother the goddess will not be at hand with her woman's  \ntricks, lurking in the treacherous shadows and trying to hide  \nhim in a cloud when he turns tail!'\n\nTerrified by this new turn in the fortunes of battle, queen  \nAmata began to weep. Seeing her own death before her, she  \ntried to check the frenzy of Turnus, the man she had chosen to  \nbe the husband of her daughter: 'By these tears, Turnus, by any  \n60 respect for me that touches your heart, Amata begs of you this  \none thing. You are the one hope and the one relief of my old  \nage. In your hands rest the honour and the power of Latinus.  \nOur whole house is falling and you are its one support. Do not  \npersist in meeting the Trojans in battle. Whatever fate awaits  \nyou in that encounter, waits also for me. If you die, I too will  \nleave the light I loathe. I shall never live to be a captive and see  \nAeneas married to Lavinia.' When Lavinia heard these words  \nof her mother, her burning cheeks were bathed in tears and the  \ndeep flush glowed and spread over her face. As when Indian  \nivory has been stained with blood-red dye, or when white lilies  \nare crowded by roses and take on their red, such were the  \n70 colours on the maiden's face. Turnus was distraught with love  \nand fixed his eyes on Lavinia. Burning all the more for war, he  \nthen spoke these few words to Amata: 'Do not, I beg of you,  \nmother, send me to the harsh encounters of war with tears and  \nwith such an evil omen. Turnus is not free to hold back the day  \nof his death. Go as my messenger, Idmon, and take these words  \nof mine to the leader of the Phrygians, and little pleasure will  \nthey give him: when tomorrow's dawn reddens in the sky, borne  \non the crimson wheels of Aurora's chariot, let him not lead  \nTrojans against Rutulians. Let the Trojan and Rutulian armies  \n80 be at peace. His blood, or mine, shall decide this war. This is  \nthe field where the hand of Lavinia shall be won.'\n\nWhen he had finished speaking and rushed back into the  \npalace, he called for his horses and it gladdened his heart to see  \nthem standing there before him neighing. Orithyia, wife of  \nBoreas, had given them to Turnus' grandfather Pilumnus to  \nhonour him, and they were whiter than the snow and swifter  \nthan the winds. The impatient charioteers stood round them,  \ndrumming on the horses' chests with cupped hands and combing  \ntheir streaming manes. Then Turnus himself drew over his  \nshoulders the breastplate with scales of gold and pale copper  \nand fitted on his sword and shield and his helmet with its red  \n90 crests in horned sockets. The God of Fire himself had made the  \nsword for Turnus' father Daunus, dipping it white-hot in the  \nwaters of the Styx. Then instantly he snatched up his mighty  \nspear which was leaning there against a great column in the  \nmiddle of the palace, spoil taken from Actor the Auruncan, and  \nbrandished it till it quivered, shouting: 'You, my spear, have  \nnever failed me when I have called upon you. Now the time is  \nhere. Mighty Actor once wielded you. Now it is the right of  \nTurnus. Grant me the power to bring down that effeminate  \nPhrygian, to tear the breastplate off his body and rend it with  \n100 my bare hands, to foul in the dust the hair he has curled with  \nhot steel and steeped in myrrh!' Such was the blazing fury that  \ndrove him on. Sparks flew from his whole face and his piercing  \neyes flashed fire. He was like a bull coming into his first battle,  \nbellowing fearfully and gathering his anger into his horns by  \ngoring a tree trunk and slashing the air, pawing the sand and  \nmaking it fly as he rehearses for battle.\n\nAeneas meanwhile, arrayed in the arms his mother had given  \nhim, was no less ferocious. He too was sharpening his spirit and  \nrousing himself to anger, rejoicing that the war was being settled  \n110 by the treaty he had proposed. He then reassured his allies and  \ncomforted the fears and anxieties of Iulus, telling of the future  \nthat had been decreed, ordering envoys to return a firm answer  \nto Latinus and lay down the conditions for peace.\n\nThe next day had scarcely risen, sprinkling the mountain tops  \nwith brightness. When the horses of the Sun first reared up from  \nthe deep sea and raised their nostrils to breathe out the light,  \nthe Rutulians and Trojans were measuring a field for the duel  \nunder the walls of the great city, setting out braziers between  \nthe two armies and building altars of turf to the gods they shared.  \n120 Others, wearing sacrificial aprons, their foreheads bound with  \nholy leaves, brought fire and spring water. The Ausonian legion  \nadvanced, armed with javelins, filling the gateways as they  \nstreamed out of their city in serried ranks. On the other side the  \nwhole Trojan and Etruscan army came at the run in all their  \nvaried armour, drawn up with weapons at the ready as though  \nit were the bitter business of battle that was calling them out.  \nThere too, in the middle of all these thousands, the leaders  \nhovered in the pride of purple and gold, Mnestheus of the  \nline of Assaracus, brave Asilas and Messapus, tamer of horses,  \nson of Neptune. The signal was given. They all withdrew to  \ntheir places, planting their spears in the ground and propping  \n130 their shields against them. Then in a sudden rush the  \nmothers, those who could not bear arms and the weak old men  \ntook up their seats on the towers and roofs of the city or stood  \nhigh on the gates.\n\nBut Juno looked out from the top of what is now the Alban  \nMount \u2013 in those days it had neither name nor honour nor glory  \n\u2013 and saw the plain, the two armies of Laurentines and Trojans,  \nand the city of Latinus. Immediately the goddess Juno addressed  \n140 the goddess who was the sister of Turnus, the ruler of lakes and  \nroaring rivers, an honour granted by Jupiter the High King of  \nHeaven as the price of her ravished virginity: 'Nymph, pride of  \nall rivers, dearest to our heart, you know how I have favoured  \nyou above all the other women of Italy who have mounted the  \nungrateful bed of magnanimous Jupiter, and have gladly set you  \nin your place in the skies, learn now the grief which is yours,  \nJuturna, and do not lay the blame on me. As long as Fortune  \nseemed to permit it, as long as the Fates allowed all to go well  \nwith Latium, I have protected the warrior Turnus and your  \nwalls. But now I see he is confronting a destiny to which he is  \n150 not equal. The day of the Fates and the violence of his enemy  \nare upon him. My eyes cannot look at this battle or at this  \ntreaty. If you dare to stand closer and help your brother, go. It  \nis right and proper. You suffer now. Perhaps a better time will  \ncome.' She had scarcely spoken when the tears flooded from  \nJuturna's eyes, and three times and more she beat her lovely  \nbreasts. 'This is no time for tears,' said Juno, daughter of Saturn.  \n'Go quickly and if you can find a way, snatch your brother from  \ndeath or else stir up war and dash from their hands this treaty  \nthey have drawn up. You dare. I sanction.' With these words  \n160 she urged her on, then left her in doubt and confusion and  \nwounded to the heart.\n\nMeanwhile the kings arrived, Latinus mighty in his four-horse  \nchariot, with twelve gold rays encircling his shining temples,  \nproof of his descent from his grandfather the God of the Sun.  \nTurnus was in his chariot drawn by two white horses, gripping  \ntwo broad-bladed spears in his hand. From the other side,  \nadvancing from the camp, came Father Aeneas, the founder of  \nthe Roman race, with his divine armour blazing and his shield  \nlike a star. Beside him were Ascanius, the second hope for the  \nfuture greatness of Rome, and a priest arrayed in pure white  \n170 vestments, driving to the burning altars a yearling ewe as yet  \nunshorn and the young of a breeding sow. Turning their eyes  \ntowards the rising sun, the leaders stretched out their hands  \nwith offerings of salted meal, marked the peak of their victims'  \nforeheads with their blades and poured libations on the altars  \nfrom their goblets.\n\nThen devout Aeneas drew his sword and prayed: 'I now call  \nthe Sun to witness, and this land for which I have been able to  \nendure such toil; I call upon the All-powerful Father of the  \nGods, and you his wife, Saturnian Juno \u2013 and I pray you,  \ngoddess, from this moment look more kindly on us \u2013 and you,  \n180 glorious Mars, under whose sway all wars are disposed; I call  \nupon springs and rivers; I call upon all the divinities of high  \nheaven and all the gods of the blue sea: if victory should chance  \nto fall to Ausonian Turnus, it is agreed that the defeated withdraw  \nto the city of Evander. Iulus will leave these lands, and  \nafter this the people of Aeneas will not rise again in war, or  \nbring their armies here, or disturb this kingdom with the sword.  \nBut if Victory grants the day to us and to our arms \u2013 as I believe  \nshe will, and may the gods so rule \u2013 I shall not order Italians to  \n190 obey Trojans, nor do I seek royal power for myself. Both nations  \nshall move forward into an everlasting treaty, undefeated, and  \nequal before the law. I shall give the sacraments and the gods.  \nLatinus, the father of my bride, will have the armies and solemn  \nauthority in the state. For me the Trojans will build the walls of  \na city and Lavinia will give it her name.'\n\nSo prayed Aeneas, and Latinus followed him, looking up and  \nstretching his right hand towards the sky: 'I too swear, Aeneas,  \nby the same: by earth and sea and stars; by the two children of  \nLatona and by two-browed Janus; by the divine powers beneath  \n200 the earth and the holy house of unyielding Dis; and let the Father  \nhimself, who sanctions treaties by the flash of his lightning, hear  \nthese my words. I touch his altar. I call to witness the gods and  \nthe fires that stand between us. The day shall not come when  \nmen of Italy shall violate this treaty or break this peace, whatever  \nchance will bring. This is my will and no power will set it aside,  \nnot if it dissolve the earth in flood and pour it into the sea, not  \nif it melt the sky into Tartarus, just as this sceptre' \u2013 at that  \nmoment he was holding his sceptre in his hand \u2013 'will never  \nsprout green or cast a shadow from delicate leaves, now that it  \nhas been cut from the base of its trunk in the forest, leaving its  \nmother tree and losing its limbs and leafy tresses to the steel.  \n210 What was once a tree, skilled hands have now clad in the beauty  \nof bronze and given to the fathers of Latium to bear.' With such  \nwords they sealed the treaty between them in full view of the  \nleaders of the peoples. Then, taking the duly consecrated victims,  \nthey cut their throats on to the altar fires, and, tearing the  \nentrails from them while they still lived, they heaped the altars  \nfrom laden platters.\n\nBut it had long seemed to the Rutulians that this was not an  \neven contest and their hearts were still more confused and  \ndismayed when the two men appeared before their eyes and  \nthey saw at close range the difference in their strength. Their  \n220 fears were increased by the sight of Turnus stepping forward  \nquietly with downcast eyes to worship at the altar like a suppliant.  \nHis cheeks were like a boy's and there was a pallor over all  \nhis youthful body. As soon as his sister Juturna saw that such  \ntalk was spreading and that men's minds were weakening and  \nwavering, she came into the battle lines in the guise of Camers,  \nwhose family had been great from his earliest ancestors, whose  \nfather had won fame for his courage, and who himself was the  \nboldest of the bold in the use of arms. Into the middle of the  \nbattle lines she advanced, well knowing what she had to do, and  \nthere with these words she sowed the seeds of many different  \n230 rumours: 'Is it not a disgrace, Rutulians, to sacrifice the life of  \none man for all of us? Are we not their equals in numbers and  \nin strength? Look, these few here are all they have, the Trojans,  \nArcadians and the army sent by Fate \u2013 the Etruscans who hate  \nTurnus! We are short of enemies, even if only half our number  \nwere to engage them in battle. As things are, the fame of Turnus  \nwill rise to the gods on whose altars he now dedicates himself,  \nand he will live on the lips of men, but if we lose our native land,  \nwe shall be forced to obey proud masters, who now sit here  \nidling in our fields!'\n\nBy such words she more and more inflamed the minds of the  \n240 warriors, and murmurs crept through their ranks. Even the  \nLaurentines had a change of heart, even the Latins, and men  \nwho a moment ago were longing for a rest from fighting and  \nsafety for their people, now wanted their weapons and prayed  \nthat the treaty would come to nothing, pitying Turnus and the  \ninjustice of his fate. At this moment Juturna did even more and  \nshowed a sign high in the sky, the most powerful portent that  \never confused and misled men of Italy. The tawny eagle of  \nJupiter was flying in the red sky of morning, putting to clamorous  \nflight the winged armies of birds along the shore, when he  \n250 suddenly swooped down to the waves and seized a noble swan  \nin his pitiless talons. The men of Italy thrilled at the sight, the  \nbirds all shrieked and \u2013 a wonder to behold \u2013 they wheeled in  \ntheir flight, darkening the heavens with their wings, and formed  \na cloud to mob their enemy high in the air until, exhausted by  \ntheir attacks and the weight of his prey, he gave way, dropping  \nit out of his talons into the river below and taking flight far  \naway into the clouds.\n\nThe Rutulians greeted the portent with a shout and their  \nhands were quick to their swords. Tolumnius, the augur, was  \n260 the first to speak: 'At last!' he cried. 'At last! This is what I have  \nso often prayed to see. I accept the omen and acknowledge the  \ngods. It is I who will lead you. Now take up your arms, O my  \npoor countrymen, into whose hearts the pitiless stranger strikes  \nthe terror of war. You are like the feeble birds and he is attacking  \nand plundering your shores. He will take to flight and sail far  \naway over the sea, but you must all be of one mind, mass your  \nforces into one flock and fight to defend your king whom he has  \nseized.' When he had spoken he ran forward and hurled his  \ncornel-wood spear at the enemy standing opposite. It whirred  \nthrough the air and flew unerringly. In that moment a great  \nshout arose. In that moment all the ranks drawn up in wedge  \nformation were thrown into disorder, and in the confusion  \n270 men's hearts blazed with sudden passion. The spear flew on. By  \nchance nine splendid brothers had taken their stand opposite  \nTolumnius, all of them sons borne by the faithful Tyrrhena to  \nher Arcadian husband Gylippus. It struck one of these in the  \nwaist where the sewn belt chafed the belly and the buckle bit  \nthe side-straps. He was noble in his looks and in the brilliance  \nof his armour, and the spear drove through his ribs and stretched  \nhim on the yellow sand. Burning with grief, his brothers, a whole  \nphalanx of spirited warriors, drew their swords or snatched up  \n280 their throwing spears and rushed blindly forward. The ranks of  \nthe Laurentines ran to meet them while from the other side the  \nmassed Trojans came flooding up with Etruscans from Agylla  \nand Arcadians in their brightly coloured armour. One single  \npassion drove them on \u2013 to settle the matter by the sword. They  \ntore down the altars and a wild storm of missiles filled the whole  \nsky and fell in a rain of steel. The mixing bowls and braziers  \nwere removed, and now that the treaty had come to nothing  \neven Latinus took to flight with his rejected gods. Some bridled  \nthe teams of their chariots; some leapt on their horses and stood  \nat the ready with drawn swords.\n\n290 Messapus, eager to wreck the treaty, rode straight at the  \nEtruscan Aulestes, a king wearing the insignia of a king, and the  \ncharging horse drove him back in terror. He fell as he retreated,  \nand crashed violently head and shoulders into the altar behind  \nhim. Riding furiously, Messapus flew to him and, towering over  \nhim with a lance as long as a housebeam, he struck him his  \ndeath blow even as he poured out prayers for mercy. 'So much  \nfor Aulestes!' cried Messapus. 'This is a better victim to offer to  \nthe great gods!' and the men of Italy ran to strip the body while  \nit was still warm. Corynaeus came to meet them, snatching a  \nhalf-burnt torch from an altar. Ebysus made for him, but before  \n300 he could strike a blow, Corynaeus filled his face with fire.  \nHis great beard flared up and gave off a stench as it burned.  \nCorynaeus pressed his attack and, clutching the hair of his  \nhelpless enemy in his left hand, he forced him to the ground,  \nkneeling on him with all his weight, and sunk the hard steel  \nin his flank. Meanwhile Podalirius had been following the  \nshepherd Alsus as he rushed through the hail of missiles in the  \nfront line of battle and was now poised over him with the naked  \nsword. But, drawing back his axe, Alsus struck him full in  \nthe middle of the forehead and split it to the chin, bathing all  \nhis armour in a shower of blood. It was a cruel rest then for  \n310 Podalirius. An iron sleep bore down upon him and closed his  \neyes in everlasting night.\n\nBut true to his vow Aeneas, unhelmeted, stretched out his  \nweaponless right hand and called to his allies: 'Where are you  \nrushing? What is this sudden discord rising among you? Control  \nyour anger! The treaty is already struck and its terms agreed. I  \nalone have the right of conflict. Leave me to fight and forget  \nyour fears. We have a treaty, and my right hand will make it  \ngood. The rituals we have performed have made Turnus mine.'  \nWhile he was still speaking, while words like these were still  \npassing his lips, an arrow came whirring in its flight and struck  \n320 him, unknown the hand that shot it and the force that spun it  \nto its target, unknown what chance or what god brought such  \nhonour to the Rutulians. The shining glory of the deed is lost in  \ndarkness, and no man boasted that he had wounded Aeneas.\n\nWhen Turnus saw him leaving the field and the leaders of the  \nallies in dismay, a sudden fire of hope kindled in his heart.  \nHorses and arms he demanded both at once, and in a flash he  \nleapt on his chariot with spirits soaring and gathered up the  \nreins. Then many a brave hero he sent down to death as he flew  \n330 along, and many half-dead bodies he sent rolling on the ground,  \ncrushing whole columns of men under his chariot wheels as he  \ncaught up their spears and showered them on those who had  \ntaken to flight. Just as Mars, spattered with blood, charges along  \nthe banks of the icy river Hebrus, clashing sword on shield and  \ngiving full rein to his furious horses as he stirs up war; they fly  \nacross the open plain before the winds of the south and the  \nwest, till Thrace roars to its furthest reaches with the drumming  \nof their hooves as his escort gallops all round him, Rage, Treachery  \nand the dark faces of Fear \u2013 just so did bold Turnus lash his  \nhorses through the thick of battle till they smoked with sweat,  \nand as he trampled the pitiable bodies of his dead enemies, the  \n340 flying hooves scattered a dew of blood and churned the gore  \ninto the sand. Sthenelus he sent to his death with a throw from  \nlong range; then Thamyrus and Pholus, both in close combat.  \nFrom long range, too, he struck down the Imbrasidae, Glaucus  \nand Lades, whom their father Imbrasus himself had brought up  \nin Lycia, and gave them armour that equipped them either to  \ndo battle or to outstrip the winds on horseback.\n\nIn another part of the field, Eumedes was charging into the  \nfray. He was a famous warrior, son of old Dolon, bearing his  \ngrandfather's name, but his spirit and his hand for war were his  \n350 father's. It was Dolon who dared to ask for the chariot of  \nAchilles as a reward for going to spy on the camp of the Greeks.  \nBut Diomede provided a different reward for his daring, and he  \nsoon ceased to aspire to the horses of Achilles. When Turnus  \ncaught sight of Eumedes far off on the open plain, he struck him  \nfirst with a light javelin thrown over the vast space that lay  \nbetween. Then, halting the two horses that drew his chariot, he  \nleapt down and stood over his dying enemy with his foot on his  \nneck. He wrenched the sword out of Eumedes' hand, and it  \nflashed as he dipped it deep in his throat, saying: 'There they  \n360 are, Trojan. These are the fields of Hesperia you tried to take  \nby war. Lie there and measure them! This is my reward for those  \nwho test me by the sword. This is how they build their cities.'  \nNext, with a throw of his javelin, he sent Asbytes to join him,  \nthen Chloreus, Sybaris, Dares, Thersilochus and Thymoetes,  \nwhose horse had fallen and thrown him over its head. Just as  \nwhen the breath of Thracian Boreas sounds upon the deep  \nAegean as he pursues the waves to the shore, and wherever the  \nwinds put out their strength the clouds take to flight across the  \nsky, just so, wherever Turnus cut his path, the enemy gave way  \nbefore him, their ranks breaking and running, and his own  \n370 impetus carried him forward with the plumes on his helmet  \ntossing as he drove his chariot into the wind. Phegeus could not  \nendure this onslaught of Turnus and his wild shouting, but leapt  \nin front of the chariot and pulled round the horses' heads as  \nthey galloped at him, foaming at their bits. Then, as he was  \ndragged along hanging from the yoke, the broad blade of  \nTurnus' lance struck his unprotected side, piercing and breaking  \nthe double mesh of his breastplate and grazing the skin of his  \nbody. He put up his shield and was twisting round to face his  \n380 enemy when he fell and was caught by the flying wheel and axle  \nand stretched out on the ground. Turnus, following up, struck  \nhim between the bottom of the helmet and the top edge of the  \nbreastplate, cutting off his head and leaving the trunk on  \nthe sand.\n\nWhile the victorious Turnus was dealing death on the plain,  \nAeneas was taken into the camp by Mnestheus and faithful  \nAchates. Ascanius was with them. Aeneas was bleeding and  \nleaning on his long spear at every other step. He was in a fury,  \ntugging at the arrowhead broken in the wound and demanding  \nthat they should take the quickest way of helping him, make a  \n390 broad cut with the blade of a sword, slice open the flesh where  \nthe arrow was embedded and get him back into battle. But now  \nthere came Iapyx, son of Iasus, whom Phoebus Apollo loved  \nabove all other men. Overcome by this fierce love, Apollo had  \nlong since offered freely and joyfully to give him all his arts and  \nall his powers, prophecy, the lyre, the swift arrow, but, in order  \nto prolong the life of his dying father, Iapyx chose rather to ply  \na mute, inglorious art and know the virtues of herbs and the  \n400 practice of healing. There, with the grieving Iulus, in the middle  \nof a great crowd of warriors, stood Aeneas, growling savagely,  \nleaning on his great spear and unmoved by their tears. The old  \nman, with his robe caught up and tied behind him after the  \nfashion of Apollo Paeon, tried anxiously and tried in vain all he  \ncould do with his healing hands and the potent herbs of Apollo.  \nIn vain his right hand worked at the dart. In vain the forceps  \ngripped the steel. Fortune did not show the way and his patron  \nApollo gave no help. And all the time the horror of battle grew  \nfiercer and fiercer on the plain, and nearer and nearer drew  \nthe danger. They soon could see a wall of dust in the sky. The  \ncavalry rode up, and showers of missiles were falling into  \nthe middle of the camp. A hideous noise of shouting rose to  \n410 the heavens as young men fought and fell under the iron hand  \nof Mars.\n\nAt this Venus, dismayed by her son's undeserved suffering,  \npicked some dittany on Mount Ida in Crete. The stalk of this  \nplant has a vigorous growth of leaves and its head is crowned  \nby a purple flower. It is a herb which wild goats know well and  \nfeed on when arrows have flown and stuck in their backs. This  \nVenus brought down, veiled in a blinding cloud, and with it  \ntinctured the river water they had poured into shining bowls,  \nimpregnating it secretly and sprinkling in it fragrant panacea  \n420 and the health-giving juices of ambrosia. Such was the water  \nwith which old Iapyx, without knowing it, bathed the wound,  \nand suddenly, in that moment, all the pain left Aeneas' body  \nand the blood was staunched in the depths of the wound. Of its  \nown accord the arrow came away in the hand of Iapyx and fresh  \nstrength flowed into Aeneas, restoring him to his former state.  \nIt was Iapyx who was the first to fire their spirits to face the  \nenemy. 'Bring the warrior his arms, and quickly!' he cried. 'Why  \nstand there? This cure was not effected by human power, nor  \nby the guidance of art. It is not my right hand that saved you,  \nAeneas. Some greater power, some god, is driving you and  \n430 sending you back to greater deeds.' Aeneas was hungry for  \nbattle. He had already sheathed his calves in his golden greaves  \nand was brandishing his flashing spear, impatient of delay. When  \nthe shield was fitted to his side and the breastplate to his back,  \nhe took Ascanius in an armed embrace and kissed him lightly  \nthrough the helmet, saying: 'From me, my son, you can learn  \ncourage and hard toil. Others will teach you about Fortune. My  \nhand will now defend you in war and lead you where the prizes  \nare great. I charge you, when in due course your years ripen and  \nyou become a man, do not forget, but as you go over in your mind  \n440 the examples of your kinsmen, let your spirit rise at the thought  \nof your father Aeneas and your uncle Hector.'\n\nWhen he had finished speaking, he moved through the gates  \nin all his massive might, brandishing his huge spear, and there  \nrushed with him in serried ranks Antheus and Mnestheus and  \nall his escort, streaming from the camp. A blinding dust then  \ndarkened the plain. The very earth was stirred and trembled  \nunder the drumming of their feet. As they advanced, Turnus  \nsaw them from the rampart opposite. The men of Ausonia also  \nsaw them and cold tremors of fear ran through the marrow of  \ntheir bones. But before all the Latins, Juturna heard the sound  \n450 and knew its meaning. She fled, trembling, but Aeneas came  \nswiftly on, leading his dark army over the open plain. Just as  \nwhen a cloud blots out the sun and begins to move from mid-ocean  \ntowards the land; long-suffering farmers see it in the far  \ndistance and shudder to the heart, knowing what it will bring,  \nthe ruin of trees, the slaughter of their crops and destruction  \neverywhere; the flying winds come first, and their sound is first  \nto reach the shore \u2013 just so the Trojan leader from Rhoeteum  \ndrove his army forward against the enemy in wedge formation,  \neach man shoulder to shoulder with his neighbour. Fierce Osiris  \nwas struck by the sword of Thymbraeus. Mnestheus cut down  \n460 Arcetius, Achates Epulo, and Gyas Ufens. Tolumnius himself  \nfell, the augur who had been the first to hurl a spear against his  \nenemies. The shouting rose to the sky and now it was the  \nRutulians who turned and fled over the fields, raising the dust  \non their backs. Aeneas did not think fit to cut down men who  \nhad turned away from him, nor did he go after those who stood  \nto meet him in equal combat or carried spears. He was looking  \nfor Turnus, and only Turnus, tracking him through the thick  \nmurk. Turnus was the only man he asked to fight.\n\nSeeing this and being stricken with fear, the warrior maiden  \n470 Juturna threw out Metiscus, the driver of Turnus' chariot, from  \nbetween the reins and left him lying where he fell, far from the  \nchariot pole. She herself took over the reins and whipped them  \nup to make them ripple, the very image of Metiscus in voice and  \nform and armour, like a black swallow flying through the great  \nhouse of some wealthy man, and collecting tiny scraps of food  \nand dainties for her young chattering on the nest; sometimes  \nher twittering is heard in empty colonnades, sometimes round  \nmarshy pools \u2013 just so did Juturna ride through the middle of  \nthe enemy and the swift chariot flew all over the field. Now  \nhere, now there she gave glimpses of her brother in triumph,  \n480 but then she would fly off and not allow him to join in the battle.  \nBut Aeneas was no less determined to meet him and followed  \nhis every twist and turn, tracking him and calling his name at  \nthe top of his voice all through the scattered lines of battle.  \nEvery time he caught sight of his enemy, he tried to match the  \nspeed of his wing-footed horses, and every time Juturna swung  \nthe chariot round and took to flight. What was Aeneas to do?  \nConflicting tides seethed in his mind, but no answer came, and  \ndifferent passions drove him to opposing thoughts. Then the  \nnimble Messapus, who was running with two pliant steel-tipped  \n490 javelins in his left hand, aimed one of them at Aeneas and hurled  \nit true. Aeneas checked himself and crouched on one knee behind  \nhis shield, but the flying spear sheared off the peak of his helmet  \nand carried away the plumes from the top of it. At this his anger  \nrose. Treachery had given him no choice. When he saw Turnus'  \nhorses pull the chariot round and withdraw, again and again he  \ncalled upon Jupiter and the altars of the broken treaty, and then,  \nand not till then, he plunged into the middle of his enemies. He  \nwas terrible in his might and Mars was aiding him. Sparing no  \nman, he roused himself to savage slaughter and gave full rein to  \nhis anger.\n\n500 What god could unfold all this bitter suffering for me? What  \ngod could express in song all the different ways of death for  \nmen and for their leaders, driven back and forth across the  \nplain, now by Turnus, now by Trojan Aeneas? Was it your will,  \nO Jupiter, that peoples who were to live at peace for all time  \nshould clash so violently in war?\n\nAeneas met Sucro the Rutulian \u2013 this was the first clash to  \ncheck the Trojan charge \u2013 but Sucro did not detain them long.  \nAeneas caught him in the side and drove the raw steel through  \nthe cage of the ribs to the breast where death comes quickest.  \n510 Turnus, now on foot, met Diores and his brother Amycus who  \nhad been unhorsed. As Diores rode at him he struck him with  \nhis long spear; Amycus he dispatched with his sword. Then,  \ncutting off both their heads, he hung them from his chariot and  \ncarried them along with him, dripping their dew of blood.  \nAeneas sent Talos, Tanais and brave Cethegus to their deaths,  \nall three in one encounter, then the gloomy Onites, who bore a  \nname linked with Echion of Thebes and whose mother was  \nPeridia. Turnus killed the brothers who came from the fields of  \nApollo in Lycia, then young Menoetes, who hated war \u2013 but  \nthat did not save him. He was an Arcadian who had plied his  \nart all round the rivers of Lerna, rich in fish. His home was poor  \n520 and he never knew the munificence of the great. His father  \nsowed his crops on hired land. Like fires started in different  \nplaces in a dry wood or in thickets of crackling laurel; or like  \nfoaming rivers roaring as they run down in spate from the high  \nmountains to the sea, sweeping away everything that lies in their  \npath \u2013 no more sluggish were Aeneas and Turnus as they rushed  \nover the field of battle. Now if ever did the anger seethe within  \nthem; now burst their unconquerable hearts and every wound  \nthey gave, they gave with all their might.\n\n530 Murranus was sounding the names of his father's fathers and  \ntheir fathers before them, his whole lineage through all the kings  \nof Latium, when Aeneas knocked him flying from his chariot  \nwith a rock, a huge boulder he sent whirling at him, and  \nstretched him out on the ground. The wheels rolled him forward  \nin a tangle of yoke and reins and his galloping horses had  \nno thought for their master as they trampled him under their  \nclattering hooves. Hyllus made a wild charge, roaring hideously,  \nbut Turnus ran to meet him and spun a javelin at his gilded  \nforehead. Through the helmet it went and stuck in his brain. As  \nfor you, Cretheus, bravest of the Greeks, your right hand did  \nnot rescue you from Turnus; nor was Cupencus protected by  \n540 his gods when Aeneas came near, but his breast met the steel  \nand the bronze shield did not hold back the moment of his  \ndeath. You too, Aeolus. The Laurentine plains saw you fall, and  \nyour back cover a broad measure of their ground. The Greek  \nbattalions could not bring you down, nor could Achilles who  \noverturned the kingdom of Priam, but here you lie. This was the  \nfinishing line of your life. Your home was in the hills below  \nMount Ida, a home in the hills of Lyrnesus, but your grave is in  \nLaurentine soil. The two armies were now wholly turned to face  \none another. All the Latins and all the Trojans \u2013 Mnestheus and  \n550 bold Serestus, Messapus, tamer of horses, and brave Asilas \u2013  \nthe battalion of Etruscans and the Arcadian squadrons of  \nEvander were striving each man with all his resources of strength  \nand will, waging this immense conflict with no rest and no  \nrespite.\n\nAt that moment Aeneas' mother, loveliest of the goddesses,  \nput it into his mind to go to the city, to lead his army instantly  \nagainst the walls and throw the Latins into confusion at this  \nsudden calamity. Turning his eyes this way and that as he  \ntracked down Turnus through all the different battle lines,  \nhe noticed the city, untouched by this great war, quiet and  \n560 unharmed, and his spirit was fired by the sudden thought of a  \ngreater battle he could fight. Calling the leaders of the Trojans  \ntogether, Mnestheus, Sergestus and the brave Serestus, he took  \nup position on some rising ground and the whole of the Trojan  \nlegion joined them there in close formation without laying down  \ntheir shields or spears. Aeneas addressed them standing in the  \nmiddle of a high mound of earth: 'There must be no delay in  \ncarrying out my commands. Jupiter is on our side. No man must  \ngo to work half-heartedly, because my plan is new to him. The  \ncity is the cause of this war. It is the very kingdom of Latinus,  \nand if they do not this day agree to submit to the yoke, to accept  \ndefeat and to obey, I shall root it out and level its smoking roofs  \n570 to the ground. Am I to wait until Turnus thinks fit to stand up  \nto me in battle and consents to meet the man who has already  \ndefeated him? O my fellow-citizens, this city is the head and  \nheart of this wicked war. Bring your torches now and we shall  \nclaim our treaty with fire!'\n\nWhen he had finished speaking, they formed a wedge, all of  \nthem striving with equal resolve in their hearts, and moved  \ntowards the walls in a solid mass. Ladders suddenly appeared.  \nFire came to hand. They rushed the gates and cut to pieces the  \nfirst guards that met them. They spun their javelins and darkened  \nthe heavens with steel. Aeneas himself, standing among the  \n580 leaders under the city wall with his right hand outstretched,  \nlifted up his voice to accuse Latinus, calling the gods to witness  \nthat this was the second time he had been forced into battle;  \ntwice already the Italians had shown themselves to be his  \nenemies; this was not the first treaty they had violated. Alarm  \nand discord rose among the citizens. Some wanted the city to be  \nopened up and the gates thrown wide to receive the Trojans and  \nthey even dragged the king himself on to the ramparts; others  \ncaught up their weapons and rushed to defend the walls: just as  \nwhen a shepherd tracks some bees to their home, shut well away  \ninside a porous rock, and fills it with acrid smoke; the bees,  \n590 alarmed for their safety, rush in all directions through their  \nwax-built camp, sharpening their wrath and buzzing fiercely;  \nthen as the black stench rolls through their chambers, the inside  \nof the rock booms with their blind complaints and the smoke  \nflies to the empty winds.\n\nWeary as they were, a new misfortune now befell the Latins  \nand shook their whole city to its foundations with grief. As  \nsoon as the queen, standing on the palace roof, saw the enemy  \napproaching the city, the walls under attack, fire flying up to the  \nroofs, no Rutulian army anywhere to confront the enemy and  \nno sign of Turnus' columns, she thought in her misery that he  \nhad been killed in the cut and thrust of battle. In that instant  \n600 her mind was deranged with grief and she screamed that she  \nwas the cause, the guilty one, the fountainhead of all these evils.  \nPouring her heart out in sorrow and madness, she resolved to  \ndie. Her hand rent her purple robes, and she died a hideous  \ndeath in the noose of a rope tied to a high beam. When the  \nunhappy women of Latium heard of this, her daughter Lavinia  \nwas the first to tear her golden hair and rosy cheeks. The  \nwhole household was wild with grief around her, and their  \nlamentations rang all through the palace. From there the report  \nspread through the whole city and gloom was everywhere.  \n610 Latinus went with his garments torn, dazed by the death of his  \nwife and the downfall of his city, fouling his grey hair with  \nhandfuls of dirt and dust.\n\nMeanwhile, on a distant part of the plain, the warrior Turnus  \nwas chasing a few stragglers. He was less vigorous now,  \nand less and less delighted with the triumphant progress of his horses,  \nwhen the wind carried to him this sound of shouting and of  \nunexplained terror. He pricked up his ears. It was a confused  \n620 noise from the city, a murmuring with no hint of joy in it. 'What  \nis this?' he cried in wild dismay, pulling on the reins to stop the  \nchariot. 'Why such grief and distress on the walls and all this  \nclamour streaming from every part of the city?' His sister, who  \nwas driving the chariot in the shape of Metiscus and had control  \nof the horses and the reins, protested: 'This way, Turnus. Let us  \ngo after these Trojans. This is where our first victories showed  \nus the way. There are others whose hands can defend the city.  \nAeneas is bearing hard on Italians in all the confusion of battle;  \n630 we too can deal out death without pity to Trojans. You will kill  \nas many as he does and not fall short in the honours of war.'\n\nTurnus made his reply: 'O my sister, I recognized you some  \ntime ago when first you shattered the treaty with your scheming  \nand engaged in this war, and you do not deceive me now,  \npretending not to be a goddess. But whose will is it that you  \nhave been sent down from Olympus to endure this agony? Was  \nit all to see the cruel death of your pitiable brother? For what  \nam I to do? What stroke of Fortune could grant me safety now?  \nNo one is left whom I love as much as I loved Murranus, and I  \n640 have seen him before my own eyes calling for me as he fell, a  \nmighty warrior laid low by a mighty wound. The luckless Ufens  \nhas died rather than look on my disgrace, and the Trojans have  \nhis body and his arms. Shall I stand by and see our homes  \ndestroyed? This is the one indignity that remained. And shall I  \nnot lift my hand to refute the words of Drances? Shall I turn  \ntail? Will this land of Italy see Turnus on the run? Is it so bad a  \nthing to die? Be gracious to me, you gods of the underworld,  \nsince the gods above have turned their faces from me. My spirit  \nwill come down to you unstained, knowing nothing of such  \ndishonour and worthy of my great ancestors to the end.'\n\n650 Scarcely had he finished speaking when Saces suddenly came  \ngalloping up on his foaming horse having ridden through the  \nmiddle of the enemy with an arrow wound full in his face. On  \nhe rushed, calling the name of Turnus and imploring him: 'You  \nare our last hope of safety, Turnus. You must take pity on your  \npeople. The sword and spear of Aeneas are like the lightning  \nand he is threatening to throw down the highest citadels of Italy  \nand give them over to destruction. Firebrands are already flying  \nto the roofs. Every Latin face, every Latin eye, is turned to you.  \nThe king himself is at a loss. Whom should he choose to marry  \n660 our daughters? What treaties should he turn to? And then the  \nqueen, who placed all her trust in you, has taken her own life.  \nFear overcame her and she fled the light of day. Alone in front  \nof the gates Messapus and bold Atinas are holding the line and  \nall round them on every side stand the battalions of the enemy  \nin serried ranks. Their drawn swords are a crop of steel bristling  \nin the fields. And you are out here wheeling your chariot in the  \ndeserted grasslands.'\n\nTurnus was thunderstruck, bewildered by the changing shape  \nof his fortune, and stood there dumb and staring. In that one  \nheart of his there seethed a bitter shame, a grief shot through  \nwith madness, love driven on by fury, and a consciousness of  \nhis own courage. As soon as the shadows lifted from his mind  \n670 and light returned, he forced his burning eyes round towards  \nthe walls, looking back in deep dismay from his chariot at the  \ngreat city. There, between the storeys of a tower, came a tongue  \nof flame, rolling and billowing to the sky. It was taking hold of  \nthe tower, which he had built himself, putting the wheels under  \nit and fitting the long gangways. 'Sister,' he said, 'the time has  \ncome at last. The Fates are too strong. You must not delay them  \nany longer. Let us go where God and cruel Fortune call me. I  \nam resolved to meet Aeneas in battle. I am resolved to suffer  \nwhat bitterness there is in death. You will not see me put to  \n680 shame again. This is madness, but before I die, I beg of you, let  \nme be mad.' No sooner had he spoken than he leapt to the  \nground from his chariot and dashed through all his enemies and  \ntheir weapons, leaving his sister behind him to grieve as his  \ncharge broke through the middle of their ranks. Just as a boulder  \ncomes crashing down from the top of a mountain, torn out by  \ngales, washed out by flood water or loosened by the stealthy  \npassing of the years; it comes down the sheer face with terrific  \nforce, an evil mountain of rock, and bounds over the plain,  \n690 rolling with it woods and flocks and men \u2013 so did Turnus crash  \nthrough the shattered ranks of his enemies towards the walls of  \nthe city where all the ground was wet with shed blood and the  \nair sang with flying spears. There he made a sign with his hand,  \nand in the same moment he called out in a loud voice: 'Enough,  \nRutulians! Put up your weapons, and you too, Latins! Whatever  \nFortune brings is mine. It is better that I should be the one man  \nwho atones for this treaty for all of you, and settles the matter  \nwith the sword.' At these words the armies parted and left a  \nclear space in the middle between them.\n\nBut when Father Aeneas heard the name of Turnus, he abandoned  \nthe walls and the lofty citadel, sweeping aside all delay  \n700 and breaking off all his works of war. He leapt for joy and  \nclashed his armour with a noise as terrible as thunder. Huge he  \nwas as Mount Athos or Mount Eryx or Father Appenninus  \nhimself roaring when the holm-oaks shimmer on his flanks and  \ndelighting to raise his snowy head into the winds. Now at last  \nthe Rutulians and the Trojans and all the men of Italy, the  \ndefenders guarding the high ramparts and the besiegers  \npounding the base of the walls with their rams, they all turned  \ntheir eyes eagerly to see and took the armour off their shoulders.\n\nKing Latinus himself was amazed at the sight of these two huge  \nheroes born at opposite sides of the earth coming together to  \n710 decide the issue by the sword. There, on a piece of open ground  \non the plain, they threw their spears at long range as they  \ncharged, and when they clashed the bronze of their shields rang  \nout and the earth groaned. Blow upon blow they dealt with  \ntheir swords as chance and courage met and mingled in confusion.  \nJust as two enemy bulls on the great mountain of Sila or  \non top of Taburnus bring their horns to bear and charge into  \nbattle; the herdsmen stand back in terror, the herd stands silent  \nand afraid, and the heifers low quietly together waiting to see  \nwho is to rule the grove, who is to be the leader of the whole  \n720 herd; meanwhile the bulls are locked together exchanging blow  \nupon blow, gouging horn into hide till their necks and shoulders  \nare awash with blood and all the grove rings with their lowing  \nand groaning \u2013 just so did Aeneas of Troy and Turnus son of  \nDaunus rush together with shields clashing and the din filled the  \nheavens. Then Jupiter himself lifted up a pair of scales with the  \ntongue centred and put the lives of the two men in them to  \ndecide who would be condemned in the ordeal of battle, and  \nwith whose weight death would descend.\n\nTurnus leapt forward thinking he was safe, and lifting his  \n730 sword and rising to his full height, he struck with all his strength  \nbehind it. The Trojans shouted and the Latins cried out in their  \nanxiety, while both armies watched intently. But in the height  \nof his passion the treacherous sword broke in mid-blow and left  \nhim defenceless, had he not sought help in flight. Faster than  \nthe east wind he flew, when he saw his own right hand holding  \nnothing but a sword handle he did not recognize. The story goes  \nthat when his horses were yoked and he was mounting his  \nchariot in headlong haste to begin the battle, he left his father's  \nsword behind and caught up the sword of his charioteer  \nMetiscus. For some time, while the Trojans were scattered and  \nin flight, that was enough. But when it met the divine armour  \n740 made by Vulcan, the mortal blade was brittle as an icicle and  \nshattered on impact, leaving its fragments glittering on the  \ngolden sand. At this Turnus fled in despair and tried to escape  \nto another part of the plain, weaving his uncertain course now  \nto this side now to that, for the Trojans formed a dense barrier  \nround him, hemming him in between a huge marsh and the  \nhigh walls.\n\nNor did Aeneas let up in his pursuit. Slowed down as he was  \nby the arrow wound, his legs failing him sometimes and unable  \nto run, he still was ablaze with fury and kept hard on the heels  \n750 of the terrified Turnus, like a hunting dog that happens to trap  \na stag in the bend of a river or in a ring of red feathers used as a  \nscare, pressing him hard with his running and barking; the stag  \nis terrified by the ambush he is caught in or by the high river  \nbank; he runs and runs back a thousand ways, but the untiring  \nUmbrian hound stays with him with jaws gaping; now he has  \nhim; now he seems to have him and the jaws snap shut, but he  \nis thwarted and bites the empty air; then as the shouting rises  \nlouder than ever, all the river banks and pools return the sound  \nand the whole sky thunders with the din. As he ran Turnus kept  \nshouting at the Rutulians, calling each of them by name and  \n760 demanding the sword he knew so well. Aeneas on the other  \nhand was threatening instant death and destruction to anyone  \nwho came near. Much as that alarmed them, he terrified them  \neven more by threatening to raze their city to the ground, and  \nthough he was wounded he did not slacken in his pursuit. Five  \ntimes round they ran in one direction, five times they rewound  \nthe circle. For this was no small prize they were trying to win at  \ngames. What they were competing for was the lifeblood of  \nTurnus.\n\nIt so chanced that a bitter-leaved wild olive tree had stood on  \nthis spot, sacred to Faunus and long revered by sailors. On it  \nmen saved from storms at sea used to nail their offerings to the  \nLaurentine god, and dedicate the clothes they had vowed for  \n770 their safety. But the Trojans, making no exception for the sacred  \ntree trunk, had removed it to clear space for the combat. In this  \nstump the spear of Aeneas was now embedded. The force of his  \nthrow had carried it here and lodged it fast in the tough wood  \nof the root. He strained at it and tried to pull it out so that he  \ncould hunt with a missile the quarry he could not catch on foot.  \nWild now with fear, Turnus cried: 'Pity me, I beg of you, Faunus,  \nand you, good Mother Earth, hold on to that spear, if I have  \nalways paid you those honours which Aeneas and his men have  \n780 profaned in war.' So he prayed and he did not call for the help  \nof the god in vain. Aeneas was long delayed struggling with the  \nstubborn stump and no strength of his could prise open the bite  \nof the wood. While he was heaving and straining with all his  \nmight, the goddess Juturna, daughter of Daunus, changed once  \nmore into the shape of the charioteer Metiscus and ran forward  \nto give Turnus his sword. Venus was indignant that the nymph  \nwas allowed to be so bold, so she came and wrenched out  \nAeneas' spear from deep in the root. Then these glorious warriors,  \ntheir weapons and their spirits restored to them, one  \nrelying on his sword, the other towering and formidable behind  \n790 his spear, stood there breathing hard, ready to engage in the  \ncontest of war.\n\nMeanwhile the King of All-powerful Olympus saw Juno  \nwatching the battle from a golden cloud and spoke these words  \nto her: 'O my dear wife, what will be the end of this? What is  \nthere left for you to do? You yourself know, and admit that you  \nknow, that Aeneas is a god of this land, that he has a right to  \nheaven and is fated to be raised to the stars. What are you  \nscheming? What do you hope to achieve by perching there in  \nthose chilly clouds? Was it right that a god should suffer violence  \nand be wounded by the hand of a mortal? Was it right that  \nTurnus should be given back the sword that was taken from  \nhim? For what could Juturna have done without your help?  \n800 Why have you put strength into the arm of the defeated? The  \ntime has come at last for you to cease and give way to our  \nentreaties. Do not let this great sorrow gnaw at your heart in  \nsilence, and do not make me listen to grief and resentment for  \never streaming from your sweet lips. The end has come. You  \nhave been able to harry the Trojans by sea and by land, to light  \nthe fires of an unholy war, to soil a house with sorrow and mix  \nthe sound of mourning with the marriage song. I forbid you to  \ngo further.'\n\nThese were the words of Jupiter. With bowed head the goddess  \nJuno, daughter of Saturn, made this reply: 'Because I have  \nknown your will, great Jupiter, against my own wishes I have  \n810 abandoned Turnus and abandoned the earth. But for your will,  \nyou would not be seeing me sitting alone in mid-air on a cloud,  \nsuffering whatever is sent me to suffer. I would be clothed in  \nfire, standing close in to the line of battle and dragging Trojans  \ninto bloody combat. It was I, I admit it, who persuaded Juturna  \nto come to the help of her unfortunate brother, and with my  \nblessing to show greater daring for the sake of his life, but not  \nto shoot arrows, not to stretch the bow. I swear it by the  \nimplacable fountainhead of the river Styx, the one oath which  \nbinds the gods of heaven. And now I, Juno, yield and quit these  \n820 battles which I so detest. But I entreat you for the sake of Latium  \nand the honour of your own kin, to allow what the law of Fate  \ndoes not forbid. When at last their marriages are blessed \u2013 I  \noffer no obstruction \u2013 when at last they come together in peace  \nand make their laws and treaties together, do not command the  \nLatins to change their ancient name in their own land, to become  \nTrojans and be called Teucrians. They are men. Do not make  \nthem change their voice or native dress. Let there be Latium.  \nLet the Alban kings live on from generation to generation and  \nthe stock of Rome be made mighty by the manly courage of  \nItaly. Troy has fallen. Let it lie, Troy and the name of Troy.'\n\nHe who devised mankind and all the world smiled and replied:  \n830 'You are the true sister of Jupiter and the second child of Saturn,  \nsuch waves of anger do you set rolling from deep in your heart.  \nBut come now, lay aside this fury that arose in vain. I grant  \nwhat you wish. I yield. I relent of my own free will. The people  \nof Ausonia will keep the tongue of their fathers and their ancient  \nways. As their name is, so shall it remain. The Trojans will join  \nthem in body only and will then be submerged. Ritual I will give  \nand the modes of worship, and I will make them all Latins,  \nspeaking one tongue. You will see that the people who arise  \nfrom this admixture of Ausonian blood will be above all men,  \n840 above the gods, in devotion and no other race will be their equals  \nin paying you honour.' Juno nodded in assent. She rejoiced and  \nforced her mind to change, leaving the cloud behind her and  \nwithdrawing from the sky.\n\nThis done, the Father of the Gods pondered another task in  \nhis mind and prepared to dismiss Juturna from her brother's  \nside. There are two monsters named Dirae born to the goddess  \nof the dead of night in one and the same litter with Megaera of  \nTartarus. The heads of all three she bound with coiling snakes  \n850 and gave them wings to ride the wind. These attend the throne  \nof savage Jupiter in his royal palace, and sharpen the fears of  \nsuffering mortals whenever the King of the Gods sets plagues or  \nhideous deaths in motion or terrifies guilty cities by the visitation  \nof war. One of these Jupiter sent swiftly down from the heights  \nof heaven with orders to confront Juturna as an omen. She flew  \nto earth, carried in a swift whirlwind. Like an arrow going  \nthrough a cloud, spun from the bowstring of a Parthian who  \nhas armed the barb with a virulent poison for which there is no  \ncure, a Parthian, or a Cretan from Cydonia; and it whirrs as it  \n860 flies unseen through the swift darkness \u2013 so flew the daughter  \nof Night, making for the earth. When she saw the Trojan battle  \nlines and the army of Turnus, she took in an instant the shape  \nof the little bird which perches on tombs and the gables of empty  \nhouses and sings late its ill-omened song among the shades of  \nnight. In this guise the monster flew again and again at Turnus'  \nface, screeching and beating his shield with her wings. A strange  \nnumbness came over him and his bones melted with fear. His  \nhair stood on end and the voice stuck in his throat.\n\n870 His sister Juturna recognized the Dira from a long way off by  \nthe whirring of her wings, and grieved. She loosened and tore  \nher hair. She scratched her face and beat her breast, crying:  \n'What can your sister do to help you now, Turnus? Much have  \nI endured but nothing now remains for me, and I have no art  \nthat could prolong your life. How can I set myself against such  \na portent? At last, at last, I leave the battle. Do not frighten me,  \nyou birds of evil omen. I am already afraid. I know the beating  \nof your wings and the sound of death. I do not fail to understand  \nthe proud commands of great-hearted Jupiter. Is this his reward  \nfor my lost virginity? For what purpose has he granted me  \n880 eternal life? Why has he deprived me of the state of death? But  \nfor that I could at least have put an end to my suffering and  \nborne my poor brother company through the shades. So this is  \nimmortality! Will anything that is mine be sweet to me without  \nyou, my brother? Is there no abyss that can open deep enough  \nto take a goddess down to the deepest of the shades?' At these  \nwords, covering her head in a blue-green veil and moaning  \nbitterly, the goddess plunged into the depths of her own river.\n\nAeneas kept pressing his pursuit with his huge spear flashing,  \nas long as a tree, and these were the words he spoke in his anger:  \n'What is the delay now? Why are you still shirking, Turnus?  \n890 This is not a race! It is a fight with dangerous weapons at close  \nquarters. Turn yourself into any shape you like. Scrape together  \nall your resources of spirit and skill. Pray to sprout wings and  \nfly to the stars of heaven, or shut yourself up and hide in a hole  \nin the ground!' Turnus replied, shaking his head: 'You are fierce,  \nAeneas, but wild words do not frighten me. It is the gods that  \ncause me to fear, the gods and the enmity of Jupiter.' He said  \nno more but looked round and saw a huge rock, a huge and  \nancient rock which happened to be lying on the plain, a boundary  \n900 stone put there to settle a dispute about land. Twelve  \npicked men like those the earth now produces could scarcely lift it up  \non to their shoulders, but he caught it up in his trembling  \nhands and, rising to his full height and running at speed, he  \nhurled it at his enemy. But he had no sense of running or going,  \nof lifting or moving the huge rock. His knees gave way. His  \nblood chilled and froze and the stone rolled away under its own  \nimpetus over the open ground between them, but it did not go  \nthe whole way and it did not strike its target. Just as when we  \nare asleep, when in the weariness of night rest lies heavy on our  \n910 eyes, we dream we are trying desperately to run further and not  \nsucceeding, till we fall exhausted in the middle of our efforts;  \nthe tongue is useless; the strength we know we have fails our  \nbody; we have no voice, no words to obey our will \u2013 so it was  \nwith Turnus. Wherever his courage sought a way, the dread  \ngoddess barred his progress. During these moments, the  \nthoughts whirled in his brain. He gazed at the Rutulians and  \nthe city. He faltered with fear. He began to tremble at the death  \nthat was upon him. He could see nowhere to run, no way to  \ncome at his enemy, no chariot anywhere, no sister to drive it.\n\n920 As he faltered the deadly spear of Aeneas flashed. His eyes  \nhad picked the spot and he threw from long range with all his  \nweight behind the throw. Stones hurled by siege artillery never  \nroar like this. The crash of the bursting thunderbolt is not so  \nloud. Like a dark whirlwind it flew carrying death and destruction  \nwith it. Piercing the outer rings of the sevenfold shield and  \nlaying open the lower rim of the breastplate, it went whistling  \nthrough the middle of the thigh. When the blow struck, down  \nwent great Turnus, bending his knee to the ground. The Rutulians  \nrose with a groan which echoed round the whole mountain,  \nand far and wide the high forests sent back the sound of their  \n930 voices. He lowered his eyes and stretched out his right hand to  \nbeg as a suppliant. 'I have brought this upon myself,' he said,  \n'and for myself I ask nothing. Make use of what Fortune has  \ngiven you, but if any thought of my unhappy father can touch  \nyou, I beg of you \u2013 and you too had such a father in Anchises \u2013  \ntake pity on the old age of Daunus, and give me back to my  \npeople, or if you prefer it, give them back my dead body. You  \nhave defeated me, and the men of Ausonia have seen me defeated  \nand stretching out my hands to you. Lavinia is yours. Do not  \ncarry your hatred any further.'\n\n940 There stood Aeneas, deadly in his armour, rolling his eyes,  \nbut he checked his hand, hesitating more and more as the words  \nof Turnus began to move him, when suddenly his eyes caught  \nthe fatal baldric of the boy Pallas high on Turnus' shoulder with  \nthe glittering studs he knew so well. Turnus had defeated and  \nwounded him and then killed him, and now he was wearing his  \nbelt on his shoulder as a battle honour taken from an enemy.  \nAeneas feasted his eyes on the sight of this spoil, this reminder  \nof his own wild grief, then, burning with mad passion and  \nterrible in his wrath, he cried: 'Are you to escape me now,  \nwearing the spoils stripped from the body of those I loved? By  \nthis wound which I now give, it is Pallas who makes sacrifice of  \nyou. It is Pallas who exacts the penalty in your guilty blood.'  \n950 Blazing with rage, he plunged the steel full into his enemy's  \nbreast. The limbs of Turnus were dissolved in cold and his life  \nleft him with a groan, fleeing in anger down to the shades.\n\n## Appendix I: The Parade of Future  \nRomans in the Underworld  \n(Book 6, lines 756\u2013892)\n\n**_Silvius_** : According to Jupiter's prophecy at 1.257-77, Rome is to be founded in four stages. Aeneas will build his city at Lavinium and live for three years. His son Ascanius Iulus will reign for thirty years and transfer the city to Alba Longa. After their descendants, the Alban kings, rule for three hundred years, Romulus (Quirinus), son of Mars and Ilia, will found his city at Rome. But here at 6.763, where Aeneas begins his survey of the Alban kings waiting in the Underworld, Ascanius, being still alive, is not in the parade, and the first to be mentioned is Silvius, a son of Aeneas not yet born.\n\n**_Alban kings_ :** Virgil offers five names to cover the years from about 1053 to 753 BC.\n\n**_Romulus_ :** Romulus restored his grandfather Numitor to the throne which Numitor's younger brother had usurped. Romulus then founded Rome in 753 BC.\n\n**_Caesar_** : Julius Caesar, 102\u201344 BC, adopted his grand-nephew Octavian as his son and heir.\n\n**_Augustus_ :** Name adopted by Octavian in 27 BC.\n\n( ** _Numa_** ): From the village of Cures, he gave Rome religion and laws. His traditional dates are 715\u2013673 BC.\n\n**_Tullus_** : Tullius Hostilius, the warrior king, 673\u2013642 BC.\n\n**_Ancus_** : Ancus Marcius, 642\u2013617 BC, here only appears as a king who courted popular favour.\n\n**_Tarquins_** : L. Tarquinius Priscus, 616\u2013579 BC, and L. Tarquinius Superbus, 534\u2013510 BC.\n\n**_Brutus_** : L. Junius Brutus led a rising against Tarquinius Superbus to avenge the rape of Lucretia. Later, as one of the first two consuls of Rome, in 510 BC, he executed his own two sons who tried to restore the Tarquins. The rods and axes carried by the consuls signified their right to flog and execute. This passage alludes also to the other avenging Brutus who assassinated Julius Caesar in 44 BC.\n\n**_Decii_** : P. Decius Mus, father and son of the same name, were famous for self-immolation, each taking his own life to secure victory for Roman armies, the father in 340 BC in the Latin War and the son in 295 BC in battle against the Samnites.\n\n**_Drusi_** : Livia, wife of Augustus from 38 BC till his death in ad 14, was a member of this notable Roman family.\n\n**_Torquatus_** : T. Manlius Torquatus led the Romans against the Gauls in 361 BC, and in 340 BC in the Latin War he executed his own son for disobeying orders in engaging and defeating an enemy champion.\n\n**_Camillus_** : M. Furius Camillus recovered not gold, but the standards said to have been the price of the Gaulish withdrawal from Rome in 390 BC. This passage may also be read as an oblique tribute to Augustus, who, after long negotiations, recovered in 20 BC the standards lost to the Parthians at Carrhae in 53 BC.\n\n( ** _Pompey_** ): Gnaeus Pompeius and Julius Caesar are the two spirits in gleaming armour. Caesar defeated Pompey at the battle of Pharsalus in 48 BC.\n\n( ** _Mummius_** ): L. Mummius sacked Corinth in 146 BC.\n\n( ** _Paullus_** ): L. Aemilius Paullus is here credited with the conquest of Greece for his defeat of Pyrrhus, king of Epirus, at the battle of Pydna in 168 BC.\n\n**_Cato_** : M. Porcius Cato, Cato the Elder, 234\u2013149 BC, was famed as the custodian of traditional Roman virtues.\n\n**_Cossus_** : A. Cornelius Cossus defeated Tolumnius, king of the Veientes, in single combat, perhaps in 246 BC.\n\n**_Gracchi_** : Tiberius Sempronius Gracchus (died 133 BC), and his brother Gaius Sempronius Gracchus (died 121 BC), the two reforming tribunes, were members of this famous Roman family.\n\n**_Scipios_** : Scipio Africanus Maior defeated Hannibal at Zama in 202 BC. Scipio Africanus Minor destroyed Carthage in 146 BC.\n\n**_Fabricius_** : Gaius Fabricius Luscinus fought against Pyrrhus, king of Epirus, in 80\u201379 BC. The power he found in poverty is an allusion to his rejection of Pyrrhus' gifts.\n\n**_Serranus_** : Gaius Atilius Regulus was sowing seed ( _serere_ : to sow) on his farm when he was called to the consulship in 257 BC. He therefore acquired the name Serranus.\n\n**_Fabii_** : Anchises at 6.845 calls out to his friends the members of the great Fabian family to ask why they are all in such a hurry to reach the light of life that they are hustling one weary spirit along with them, and then he realizes that the problem is not weariness. This is the great Q. Fabius Maximus Cunctator ( _cunctator_ : delayer) who used Fabian tactics against Hannibal in 217\u2013216 BC in the Second Punic War. He is not tired. It is his nature to delay!\n\n**_Marcellus_** : M. Claudius Marcellus, consul five times, killed the Gaulish chieftain Viridomarus in single combat in 222 BC, thus becoming the third Roman, after Romulus and Cossus, to win the Supreme Spoils ( _Spolia Opima_ ). Augustus was eager to make sure that there would not be a fourth (see Livy 4.20.5). The younger M. Claudius Marcellus (42\u201323 BC) was the son of Augustus' sister Octavia, and was adopted by Augustus in 25 BC. An ancient life of Virgil ( _Vita Donati_ 32) describes how, when Virgil was reading this passage to Octavia and Augustus, Octavia swooned when he reached line 882.\n\n## Appendix II: The Shield of Aeneas  \n(Book 8, lines 626\u2013728)\n\nMost of the scenes on the shield are incidents from Italian wars (see lines 626 and 678), all depicted with vivid evocation of the colours, textures and materials used in this imaginary work of art and the sounds evoked by it.\n\nAround the outside of the circle are six scenes described in forty-one lines:\n\n**(i)** The wolf suckling Romulus and Remus, who are to found the city in 753 BC.\n\n**(ii)** The rape of the Sabine women as planned by Romulus and the subsequent war and reconciliation.\n\n**(iii)** The punishment of Mettus Fufetius, dictator of Alba Longa who will make a treaty with Tullus Hostilius, king of Rome 673\u2013642 BC, and then desert him in battle.\n\n**(iv)** Two famous scenes from the Etruscan attack on Rome in 508 BC.\n\n**(v)** At the top of the shield the attack of the Gauls in 390 BC and the origin of some traditional features of Roman religion. The matrons of Rome were permitted to drive in carriages to the games and temples in return for giving their gold and jewels to enable Camillus to build a temple to Apollo after the defeat of Veii in 396 BC.\n\n**(vi)** Presumably at the bottom of the shield, scenes in the Underworld showing Catiline whose conspiracy was put down by Cicero in 63 BC and M. Porcius Cato who fought for the Republican cause against Caesar and committed suicide after his defeat at Thapsus in 46 BC. Like his great ancestor Cato the Elder (6.841) he was regarded as a model of the uncompromising Republican virtues.\n\nIn the centre of the shield, in a ring of silver dolphins feathering with white foam the silver sea and its golden waves, is depicted Augustus' victory over Antony and Cleopatra at Actium in 31 BC and his triple triumph of 29 BC (Dalmatian, Actian and Alexandrian). To this Augustan theme Virgil devotes fifty-four lines.\n\n## Appendix III: Genealogical Trees\n\n### THE JULIAN FAMILY\n\n1. Anchises' grandfather Assaracus seems to be mentioned in a Julian connection at 1.284, 6.778, 9.259, 643.\n\n2. This gap is variously filled (see S. Weinstock, _Divus Julius_ , p. 183 n. I.).\n\n3. Augustus was born C. Octavius in 63 BC. He was adopted as Julius Caesar's son by Caesar's will in 44 BC under the name of C. Iulius Caesar Octavianus (called Octavian in English), and took the name of Augustus in January 27 BC.\n\n### THE HOUSE OF PRIAM\n\n### THE HOUSE OF ANCHISES\n\n## Maps, Gazetteer and Select Index\n\n_Rome during the reign of Augustus_\n\n### GAZETTEER\n\nI started to compile a glossary of mythological terms in the _Aeneid_ , but soon decided that it was not necessary. Such is Virgil's command of narration that the poem usually explains itself as it goes along. Where this is not so, explanations have been added to the text, for example at the beginning of Book 6 where there is an unusual concentration of such difficulties. Here, the modern reader needs to be told that the Chalcidian citadel is the Chalcidian colony of Cumae; that Phoebus in line 18 is the same god as Apollo in line 9; that Androgeos was the son of Minos and that the Athenians were held to be the descendants of Cecrops. The _Aeneid_ is first and foremost a narrative, and narratives do not thrive on interruptions. A glossary would drive readers to the end of the book. Even footnotes would take the eye to the foot of the page and the mind to scholarly furniture. It is a regrettable interference with the text of Virgil, but I have preferred to add such information to the body of the work where it is necessary rather than check the flow of the narrative.\n\nGeography is another matter. The ancients knew their Mediterranean world better than we do. I have therefore supplied maps and an index which are meant to give topographical information which may be helpful for understanding the poem. These therefore omit peoples and places whose locality is sufficiently indicated by the context, for example the lists of the Latin enemies of Aeneas at the end of Book 7 and his Etruscan allies at 10.163\u2013214.\n\nVirgil has many equivalent or nearly equivalent geographical terms at his disposal. Greeks are called Achaeans, Argives, Graians, and Pelasgians; Troy is Dardania; Ilium, Pergamum (strictly its citadel), and its people are Phrygians, Teucrians, even Laomedontiadae, as well as Trojans; Etruscans are also Lydians, Tuscans and Tyrrhenians. Where Virgil seems to be using these terms purely for metrical convenience, the translation speaks of Greeks, Trojans and Etruscans. But the variants are preserved where they are used to some effect, rhetorical at 2.324\u20136, for example, or emotive (the term 'Phrygian' usually carries a contemptuous allusion to the alleged effeminacy of the Trojans). In particular Italy is variously referred to as Ausonia, Oenotria, Hesperia (the Western Land), and sometimes these terms are used in prophecies not understood by those who hear them. This oracular obscurity is preserved in the translation since the progressive revelation of the divine will is an important aspect of the plot of the poem. The Tiber, for instance, is called the Lydian Thybris at 2.781\u20132 and Aeneas can  \nhave no idea what is meant. The Italian river is always referred to by this Greek form of its name until 6.873.\n\nIn the index these equivalents will be noted but they will not occur on the maps. So too rivers and mountains appear in the list, but normally not on the maps.\n\n### SELECT INDEX\n\nNames in brackets do not appear on the maps; names with map references appear on the map 'The Voyages of Aeneas'; other names appear on the map of Pallanteum\/Rome.\n\nAcarnania 5G\n\n(Achaeans \u2013 Greeks)\n\nAcrages 6B\n\nActium 5F\n\nAeneadae 3J\n\nAeolia 5C\n\nAgathyrsians 1GHJ\n\nAlba Longa 3B\n\n(Albunea \u2013 fountain at Tibur)\n\n(Alpheus \u2013 river in Elis)\n\n(Amasenus \u2013 river in Latium)\n\n(Amathus \u2013 town in Cyprus)\n\nAmyclae 6G\n\nAntandros 4K\n\n(Appenninus \u2013 mountain in Italy)\n\nApulia 3D\n\nAra Maxima \u2013 Greatest Altar\n\n(Araxes \u2013 river in Armenia)\n\nArcadia 5G\n\nArdea 3B\n\n(Arethusa \u2013 fountain at Syracuse)\n\nArgiletum\n\n(Argives \u2013 Greeks)\n\nArgos 5H\n\nArisba 4J\n\nArpi 3D\n\n(Asian Marsh \u2013 on coast of Asia opposite Samos)\n\nAsylum\n\n(Athos \u2013 mountain in Macedonia)\n\n(Atlas \u2013 mountain in Mauretania)\n\n(Aufidus \u2013 river in Apulia)\n\nAulis 5H\n\n(Auruncans \u2013 ancient people of central Italy)\n\n(Ausonia \u2013 Italy)\n\nAventine Mount\n\n(Avernus \u2013 lake near Cumae)\n\n(Bactrians \u2013 people east of Caspian)\n\nBaiae 4C\n\n(Bebrycians \u2013 people south of Caspian)\n\nBenacus 1A\n\n(Berecyntus \u2013 mountain in Phrygia)\n\nBoeotia 5H\n\nButhrotum 4F\n\nCaere 3B\n\nCaieta 3C\n\nCamerina 6C\n\nCampus Martius\n\nCaphereus 5J\n\nCapitol\n\nCarinae\n\nCarmental Gate\n\nCarpathos 6K\n\nCarthage 6A\n\nCaspian Sea 2K\n\nCaulonia 5D\n\nChalcis 5H\n\nChaonia 4F\n\n(Charybdis \u2013 whirlpool off Scylaceum)\n\n(Cithaeron \u2013 mountain north of Athens)\n\nClaros 5K\n\nClusium 2B\n\nCorinth 5H\n\nCorythus 2B\n\nCrete 6HJK\n\n(Crinisus \u2013 Sicilian river)\n\nCumae 3C\n\nCures 3B\n\n(Cybelus \u2013 mountain near Corinth)\n\n(Cynthus \u2013 mountain on Delos)\n\n(Cyprus \u2013 island in Eastern Mediterranean)\n\nCythera 6H\n\nDacia 1GHJ\n\n(Dahae \u2013 people east of Caspian)\n\nDaunia 3D\n\nDelos 5J\n\n(Dicte \u2013 mountain in Crete)\n\n(Dindymus \u2013 mountain in Phrygia)\n\nDodona 4F\n\nDolopians 4G\n\nDonusa 5J\n\nDrepanum 5B\n\nDryopes 4G\n\n(Dulichium \u2013 island near Ithaca)\n\nElis 5G\n\nEpirus 4F\n\n(Erymanthus \u2013 mountain in Arcadia)\n\nEryx 5B\n\nEtna 5C\n\nEtruria 2AB, 3B\n\nEuboea 5H\n\n(Euphrates \u2013 river of Mesopotamia)\n\n(Eurotas \u2013 Spartan river)\n\nForum Boarium\n\n(Gaetulians \u2013 people of the Sahara)\n\n(Garamantians \u2013 people of the Sahara)\n\n(Garganus \u2013 mountain in Apulia)\n\n(Geloni \u2013 Scythian people)\n\nGetae 2HJ\n\nGreatest Altar \u2013 see Ara Maxima\n\nGortyn 6J\n\nGryneum 4K\n\nGyaros 5J\n\n(Haemus \u2013 mountain in Thrace)\n\n(Hebrus \u2013 river in Thrace)\n\n(Helicon \u2013 mountain in Boeotia)\n\nHelorus 6C\n\n(Hermus \u2013 river in Lydia)\n\n(Hesperia \u2013 the Western Land, Italy)\n\n(Homole \u2013 mountain in Thessaly)\n\n(Hyrcanians \u2013 people near the Caspian Sea)\n\n(Ida \u2013 mountain in Crete)\n\n(Ida \u2013 mountain near Troy)\n\n(Idalium \u2013 mountain in Cyprus)\n\n(Ilium \u2013 Troy)\n\nIthaca 5F\n\nJaniculum\n\nLacinium 5D\n\nLarisa 4G\n\nLatium 3B\n\n(Laurentines \u2013 people on the coast of Latium)\n\nLavinium 3B\n\nLemnos 4J\n\nLerna 5H\n\nLeucas 5F\n\nLiburnia 1D\n\n(Libya \u2013 land east of the Syrtes)\n\nLiguria 1A\n\nLilybaeum 5B\n\nLipari 5C\n\nLocri 5D\n\nLupercal\n\n(Lycia \u2013 land on south coast of Asia Minor)\n\nLydia 5K\n\nLyrnessus 4K\n\nMacedonia 3G\n\nMaeonia 4K\n\n(Maeotians \u2013 people on north shore of Caspian)\n\nMalea 6H\n\nMantua 1A\n\n(Marpessa \u2013 mountain on Paros)\n\nMarsians 3C\n\n(Massylians \u2013 people west of Carthage)\n\nMausoleum of Augustus\n\nMegara 5H\n\nMeliboea 4G\n\n(Misenum \u2013 cape south of Cumae)\n\n(Morini \u2013 Belgian people)\n\nMycenae 5H\n\nMyconos 5J\n\nMyrmidons 4G\n\n(Nar \u2013 river in Umbria)\n\nNarycum 5H\n\nNaxos 5J\n\nNemea 5H\n\n(Neritos \u2013 island near Ithaca)\n\nNumidians \u2013 people west of Carthage\n\nOechalia 5G\n\n(Oenotria \u2013 Italy)\n\nOlearos 5J\n\n(Orthrys \u2013 mountain in Thessaly)\n\n(Ortygia \u2013 another name for Delos)\n\n(Ortygia \u2013 island in the bay of Syracuse)\n\nPachynus 6C\n\n(Pactolus \u2013 river in Lydia)\n\n(Padus \u2013 one of the mouths of the river Po)\n\nPalinurus 4C\n\nPallanteum 3B\n\n(Pantagias \u2013 river in Sicily)\n\n(Paphos \u2013 town in Cyprus)\n\nParos 5J\n\nParrhasia 5G\n\nPatavium 1B\n\n(Pelasgians \u2013 ancient north Aegean people)\n\nPelorus 5D\n\n(Pergamum \u2013 Troy, strictly its citadel)\n\nPetelia 4D\n\nPheneus 5G\n\n(Phoenicia \u2013 land on eastern seaboard of Mediterranean)\n\nPhrygia 4K\n\nPlemyrium 6C\n\nPraeneste 3B\n\nPrivernum 3B\n\nProchyta 4C\n\nPthia 5G\n\nRhoeteum 4J\n\nRutulians 3B\n\nSabines 3B\n\nSalamis 5H\n\nSallentine Plains 4E\n\nSame 5F\n\nSamnium 3C\n\nSamos 5K\n\nSamothrace 3J\n\nSaturnia\n\nScylaceum 5D\n\n(Scythia \u2013 people north of Caspian)\n\nScyros 4H\n\nSelinus 5B\n\n(Shebans \u2013 Sabaeans, Arabian people)\n\n(Sicanians \u2013 people who moved from Central Italy to Sicily)\n\n(Sidon \u2013 Phoenician city)\n\nSila 5D\n\n(Simois \u2013 Trojan river)\n\n(Soracte \u2013 mountain in Etruria)\n\nStrophades 5G\n\nSyracuse 6C\n\nSyrtes 6A\n\n(Taburnus \u2013 mountain in Samnium)\n\nTarentum 4D\n\nTarpeian Rock\n\n(Tetrica \u2013 mountain in Sabine country)\n\n(Teucrians \u2013 Trojans)\n\nThapsus 6C\n\nThebes 5H\n\nThrace 2GHJ\n\nThymbra 4J\n\nTiber 2B\n\nTibur 3B\n\nTimavus 1B\n\nTiryns 5H\n\n(Trinacria \u2013 Sicily)\n\n(Troad \u2013 the region around Troy)\n\nTroy 4J\n\n(Tuscans \u2013 Etruscans)\n\n(Tyre \u2013 Phoenician city)\n\n(Tyrrhenians \u2013 Etruscans)\n\nUmbria 2B\n\n(Velinus \u2013 lake in Sabine country)\n\n(Vesulus \u2013 mountain in Liguria)\n\nVolsci 3BC\n\n(Xanthus \u2013 river in the Troad)\n\nZacynthus 5F\n _For lines 756\u2013892_ , _seeAppendix I_.\n _For lines 626\u2013728_ , _seeAppendix II_.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n* * *\n\nMy appreciation goes out to all the pet owners and breeders for allowing me to love, mingle with, and photograph some of the most magnificent creatures in the world. The time and effort put forth by their owners and the willingness of these cats to be displayed and exhibited at cat shows throughout the world has allowed me to be a part of their lives and capture very special moments. Many thanks go out to my wife, Susan and my family for putting up with my artistry over all these years and years to come.\n\n* * *\nCopyright \u00a9 2018 by Larry Johnson\n\nAll rights reserved.\n\nAll photographs by the author unless otherwise noted.\n\nPublished by:\n\nAmherst Media, Inc., P.O. Box 538, Buffalo, N.Y. 14213\n\nwww.AmherstMedia.com\n\nPublisher: Craig Alesse\n\nSenior Editor\/Production Manager: Michelle Perkins\n\nEditors: Barbara A. Lynch-Johnt and Beth Alesse\n\nAcquisitions Editor: Harvey Goldstein\n\nAssociate Publisher: Katie Kiss\n\nEditorial Assistance from: Ray Bakos, Rebecca Rudell, Jen Sexton, Carey Miller\n\nBusiness Manager: Sarah Loder\n\nMarketing Associate: Tonya Flickinger\n\nISBN-13: 978-1-68203-311-1\n\nLibrary of Congress Control Number: 2017949330\n\n10 9 8 7 6 5 4 3 2 1\n\nNo part of this publication may be reproduced, stored, or transmitted in any form or by any means, electronic, mechanical, photocopied, recorded or otherwise, without prior written consent from the publisher.\n\nNotice of Disclaimer: The information contained in this book is based on the author's experience and opinions. The author and publisher will not be held liable for the use or misuse of the information in this book.\n\nwww.facebook.com\/AmherstMediaInc\n\nwww.youtube.com\/AmherstMedia\n\nwww.twitter.com\/AmherstMedia\nContents\n\nAbout the Author\n\nFaces\n\nThe Studio\n\nSession Sample\n\nBackgrounds\n\nBlack and White Cats\n\nBlack on Black\n\nWhite on White\n\nBlue Cats\n\nEars\n\nEyes\n\nBob Tails\n\nLong Tails\n\nNo Tails\n\nPuffy Tails\n\nBig Hair\n\nCurly Hair\n\nWavy Hair\n\nKinky Hair\n\nNo Hair\n\nPlush Coat\n\nHousehold Pets\n\nThe Standard\n\nHangers\n\nHigh Fives\n\nHolidays\n\nIn Motion\n\nGoofy\n\nGroups\n\nIn the Air\n\nKittens\n\nLarge Cats\n\nPairs\n\nProfiles\n\nProps\n\nSitting Up\n\nStanding\n\nStretching\n\nThe Wild Look\n\nTalking\n\nUp\n\nAwards\n\nAgility\n\nCombo Imagery\n\nFinal Thoughts\n\nIndex\nAbout the Author\n\nLarry's love for photography has allowed him to work with the finest cats on Earth, taking him to cities, countries, and world-class events. Larry's thirty years of experience in handling and photographing animals have gained him world renown. As a member of the Professional Photographers of America, he provides the highest quality images and professionalism.\n\nLarry was born in Chicago. He studied music and art in college. Upon graduating he taught music in the Miami-Dade Schools. Larry has attended workshops and lectures, read extensively, and studied photography. He upgraded his equipment and added a darkroom in his house where he developed skills in both black & white and color printing. In November of 1979, Larry photographed his first cat show in Miami, Florida. As he studied the images and the breed standards of purebred cats, Larry became acquainted with different breeds and their characteristics, learning how to photograph the most flattering look for each breed. In addition to cat show photography, he photographed at dog shows, grooming salons, and the Fort Lauderdale Humane Society.\n\nCurrently, he travels with a portable studio to shows throughout the U.S. and abroad. In the past year, Larry has visited over 35 different major cities in the U.S., Europe, China, and Japan, as well as working in his studio in Baton Rouge, Louisiana.\n\nHis primary subjects are ever-captivating cats, but he also photographs dogs, birds, horses, and even people. He has also produced travel, cat, and dog calendars.\n\n**FOR MORE INFORMATION:**\n\n\u2022 JohnsonAnimalPhoto.com\n\n\u2022 Instagram.com\/JohnsonPhoto951\n\n\u2022 Facebook.com\/Larry-Johnson-Photography-120318871864\nFaces\n\nWelcome to my world of cats\u2014expressive, inquisitive, cunning, cute, whimsical, curious, and playful. Their faces speak to me. I look for the moment to capture their thoughts.\n\nThe Studio\n\nMy job as a cat show photographer at cat shows spans the globe, so I need to be portable and secure for my subjects. Thus my setup needs to be easily compacted for travel by plane as well as by car to the locations. Signage and visibility are important but need not be extensive. A display consisting of hanging images and several portfolios will suffice. My reputation precedes me. Wherever I travel, my clients will seek me out. My area is ususally not often in the mainstream of the busy show. A quiet corner or a separate room allows my subjects to relax and allows me to feel at ease and to concentrate on capturing great images.\n\nSession Sample\n\nA typical session will be limited in time. Many sessions do not last more than fifteen minutes per cat. On occasion it may take longer or even shorter. I need to capture great images and a variety of poses in a short period of time (posed or unposed and without any set sequence). I need to be ready at a moment's notice to see their behavior, eyes, ears and body form (as per breed) and make that moment real.\n\nHere is a sample of one typical session: fifteen total final images. I have learned to shoot when the time is right and to waste few shots. While working with the cat with teasers and other toys, I can see and predict what poses are working for the cat.\n\nBackgrounds\n\nThe second most important question to my clients is which background to use. Much depends upon the owner's preference, and then, also the cat's color, eyes, and fur texture. Many times I will use my best judgment. The background can enhance, detract from, or blend with the cat. Eye color is important; coat color is important; and, contrast is important\u2014both low contrast to high contrast.\n\nI ask many questions prior to the session to determine the best background for the cat, the owner, and what is most important to emphasize. Many times the cat's best features will determine the background color or texture. Some backgrounds are unsuitable for certain cat colors. A helpful photographer's expertise is necessary. There are several options for most cat colors and patterns. Many of my clients already have a preferred color or one that will match their websites. Some clients like to experiment.\n\nPhotographs are important to my clients, so I want to have the best options available for them. Many times it is a single opportunity to photograph when the cats are at their best (bathed, groomed, fluffed, and so on). I would hate to disappoint them by using a background that is not suited for their cat. My traveling set-up has numerous available backgrounds. On many occasions, I will use only three or four, but I prefer to have options available for particular clients.\n\nBlack and White Cats\n\nBlack and white cats are sometimes referred to as tuxedo cats, which describes their patterns. Black and white refers to the cat's color. It is not a breed.\n\nBlack on Black\n\nPhotographing black cats on a black background is a dramatic way to show both the intensity of the coat color and the eye color.\n\nWhite on White\n\nSome are reluctant to photograph their white cats on white background for fear they will disappear and details will not be visible. This is erroneous. Here are some examples.\n\nBlue Cats\n\nSo many people are unaware that blue cats are not always Russian Blues. Moreover, grey cats are designated as blue. Several breeds are exclusively blue in color, as the Charteux and Korat. Blue is the most popular British Shorthair color, but they do come in a vast amount of other colors. The Korat has a blue coat but with upright ears and green eyes. The Korat was discovered in Ampur Pimai of the Korat province in Thailand. The Korat is known as a good-luck cat of Thailand. The head contains three heart-shapes. Looking straight on at the Korat, you can see the Valentine-shaped heart of the head. The second heart can be seen by looking down over the top of the Korat's head. The third heart associated with the head is the front of its nose. An additional heart can be found in the muscular area of the chest when the cat is in a sitting position. Of all the breeds the Korat adheres most closely to its original look.\n\nThe Russian Blue has green eyes, a silvery short coat, and a rounded but V shape to the head and ears. The Russian Blue is a natural breed originating from northern Russia. Some are the descendants of the cats kept by the Russian Czars. The coat color is an even, plush, bright blue, and each guard hair appears as if dipped in silver. Russian Blues are registered in only one color: blue. The Russian Blue has large, rounded, wide-set eyes that are vivid green. Its large ears are wide at the base and set rakishly toward the side of the head. The Russian Blue is a medium-sized, fine-boned, and a muscular cat.\n\nThe British Shorthair is predominately in the solid blue color, but other colors are accepted. It is a very robust cat with a rounded body and head, jowls, gold eyes, a plush coat, and smallish ears set wide on the head. These are sturdy, dense-coated, teddy bear cats with large round eyes. The oldest English breed of cat, the British Shorthair can be traced back to the domestic cats of Rome.\n\nPictured above is a Chartreux. It is a native cat from France. It has a beautiful coat not as short as the Russian Blue, but not as plush as the British Shorthair. It has brilliant gold eyes and very small upright ears. It has balance of a pear, so the smaller head and larger back body is normal.\n\nChartreux probably arrived from the Middle East to the French monasteries with knights returning from the Crusades. According to legend is that the Chartreux lived with, and were named for, the Carthusian monks of France\u2014and perhaps even shared a tipple or two of their famous Chartreuse liqueur! Nevertheless, the breed of cat was noted in documents as early as the 16th century. The Chartreux is also known for its smile. France has adopted the breed as its national cat.\nEars\n\nThe size and shape of ears is one of the most prominent features of several breeds.\n\nOriental Shorthair: The Oriental Shorthairs are closely related to the Siamese and have large ears. They come in many color patterns as well as solid colors.\n\nScottish Fold: The ears on a Scottish Fold are folded forward to cup around the head. This is a natural occurrence and not created by surgical methods.\n\nTurkish Angora: The cat has high, large ears, and its most popular color is white.\n\nAmerican Curl: The American Curl is another natural occurring mutation like the Scottish Fold. Instead their ears curl backwards with a pleasant furl and a more moderate body and head structure.\n\nSiamese: The Siamese has large ears conforming to a perfect triangle from the chin to the tip of the ears.\n\nSphynx: This is a cat without any hair (or very little hair) with very large ears.\n\nMaine Coon Cat: This is one of the largest cats and has very large erect ears.\nEyes\n\nThe eyes are the key to the cat's intellect. They are so expressive.\n\nMore photographs of eyes, which capture wonderful expressions and color.\n\nBob Tails\n\nSome breeds are distinguished by bobbed tails.\n\nAmerican Bobtail\n\nKurilian Bobtail\n\nPixiebob\n\nJapanese Bobtail Shorthair\n\nJapanese Bobtail Longhair\nLong Tails\n\nThe tail is the cat's means of balance. Cats also rely on their tails to sweep counter tops, pose, and jump to high places. Both the long-haired cats and short-haired cats show off their plumes and length.\n\nNo Tails\n\nThe Manx is the only cat that has no tail. It originated on the Isle of Man and was one of the first show cats shown in England. Manx come in both shorthair and longhair. It has a round body, round face and a smooth rounded rear with larger hind quarters, which give it its balance.\n\nPuffy Tails\n\nSeveral breeds have very full coated tails as well as full-coated bodies. The Maine Coon, the Persian (especially), the Ragdoll, and the Somali are some of these breeds. Most of these longhair cats have a fuller coats during the winter months and shed to a shorter length (but not shorthair or bald) during the summer. Most of my clients with Persians will wait to have photographs taken until the cat's coat is at its fullest.\n\nBig Hair\n\nThe bigger the coat, the better the coat. Big, full coats are the predominant trait of the Persian. Such coats do not come easily. It takes hours and hours of bathing, combing each and every day to maintain a cat in full show coat. Some coats even reach the floor when the cat is standing.\n\nCurly Hair\n\nThe Selkirk Rex has curly hair which looks much like a lamb's wool coat. Forget the comb and straightener: the coat needs to be left alone for the most part. This coat originates from a different mutation than the Cornish Rex, Devon Rex and the LaPerm. Jokingly the phrase \"bad hair day\" prevails at the shows. The coat feels silky and has both long and short hair variants. The body structure is round and full, and not too long or too short.\n\nOwners like to give their cats names that reflect their uniqueness. Like the dog \"Spot,\" the hairy and lack of hair spark creativity. Sphynx owners like bald or naked names such as Belle-of-the-Bald, Nudie Garland, Liza Skinnelli. Scottish Fold owners like names reflecting folded ears such as Foldilocks, Scottie, Tipsy, and Masqu'ear'ade. And Selkirk Rex owners use names like Twisted Sister, Curleone, and Oliver Twist.\n\nWavy Hair\n\nThe Cornish Rex has a Marseille wave to its coat, like a washboard effect. The coat is super silky and lies close to the body. This cat has large ears, long legs, and a long tail. It can leap and create havoc flying through the air. They are sweet and personable cats, but they are also very active and agile. Posing the Cornish Rex is a challenge, as I must display the waves, the ears, eyes, and the arch (as its backbone is not flat). A Cornish Rex is a fun cat to photograph due to its playfulness.\n\nKinky Hair\n\nThree breeds have the strangest coats; kinky is probably be the best general description. These are the Devon Rex, LaPerm, and the American Wirehair. The Devon Rex is the most popular of these three, and none are genetically related. They all have different coat and body structures. The Devon Rex is a small cat and all its fur has kinks as does its whiskers. It has a plushier coat than the Cornish Rex, with a silky feel. The LaPerm is a medium sized cat with a thin wild curly coat. The American Wirehair resembles an American Shorthair in stature, but the hairs have a wiry feel to them, as if they were brittle and ready to break. This cat is not combed as their hairs would break, so very little grooming is needed.\n\nNo Hair\n\nVisitors always want to see these extraordinary cats with no hair. They are called ugly and alien, and visitors will often comment, \"Is it a dog? and \"Look at its feet!\"\n\nThe Sphynx has become very popular and has a unique body and coat. The body is pudgy with wrinkled skin. The Sphynx comes in many colors but the color is on its skin and not in its hair like other cats. They appear to be pinkish as their blood vessels are super close to the skin. Without any hair (or very little hair) they can easily get scratched. They are super friendly cats that feel like a hot water bottle, a warm chamois, or a peach. If you look closely, you might see hairs behind their ears and at the base of their tail.\n\nThe Bambino is a man-made variety which resulted from mating a Sphynx with a Munchkin, which is a short legged cat. It is more popular at cat shows in Europe than in the United States.\n\nThe Peterbald is a bald to semi-hairless cat that looks like an Oriental Shorthair in body, ears, and eyes. It is also a man-made breed which originated from breeding Sphynx with Oriental Shorthairs or Siamese. The legs are long, as are their long whippy tail. The Peterbald comes in several coat types and is a most interesting cat.\n\nPlush Coat\n\nThese cats have a coat that you want to dig your fingers into. The density and texture gives them a teddy bear look. The densest coat is the British Shorthair. The Exotic Shorthair has the same physical characteristics as a Persian without the long dripping coat. These are my mushy cats and are fairly easy to photograph. With these breeds I am looking to emphasize the eyes, coat pattern, color, and their body structure.\n\nAmerican Shorthair\n\nScottish Fold\n\nBirman\n\nExotic Shorthair\n\nBritish Shorthair\nHousehold Pets\n\nHousehold pets include domestic short and longhairs, rescues, and anything else that is not a purebred (although some purebreds are shown as household pets because they lack the registration papers). They have all sorts of personalities, shapes, and colors patterns. Their origins may be unknown: some are from feral colonies, some from rescues, some from cat shelters, and some from the neighboring cats who had kittens. These cats can be shown as household pets, are neutered and spayed\u2014and do not roam the streets at night. Declawing is frowned upon. They can also stay at home and be loved.\n\nBreed Standards\n\nAbyssinian\n\nAmerican Bobtail\n\nAmerican Curl\n\nAmerican Wirehair\n\nAmerican Shorthair\n\nBalinese\n\nBengal\n\nBirman\n\nBombay\n\nBritish Longhair\n\nBritish Shorthair\n\nBurmese\n\nBurmilla\n\nChartreux\n\nChaussie\n\nCornish Rex\n\nDevon Rex\n\nColor Point Shorthair\n\nEgyptian Mau\n\nExotic Shorthair\n\nHavana Brown\n\nHimalayan Persian\n\nLykoi\n\nJapanese Bobtail\n\nJapanese Bobtail Longhair\n\nKheo Manee\n\nKorat\n\nKurilian Bobtail\n\nLaPerm\n\nMaine Coon\n\nManx\n\nManx Longhair\n\nOcicat\n\nMunchkin\n\nNorwegian Forest Cat\n\nOriental Shorthair\n\nOriental Longhair\n\nPersian, Solid White\n\nPersian, Tabby\n\nPersian, Solid Black\n\nPeterbald\n\nPersian Shaded Silver\n\nPersian, Bi-Color\n\nPixiebob\n\nRagamuffin\n\nRagdoll\n\nRussian Blue\n\nSavannah\n\nSelkirk Rex\n\nSiamese\n\nSiberian\n\nSingapura\n\nSnowshoe\n\nSomali\n\nSphynx\n\nThai\n\nTonkinese\n\nToyger\n\nTurkish Angora\n\nTurkish Van\nHangers\n\nI love my paws! So many cats hang over the counter and other places. They do not fear the edge. With a little cajoling, I can position their paws out or over the edge.\n\nCats have an amazing sense of balance when they extend their paws. They know exactly what they are doing. It makes them look attentive, playful and cute. It is a pose that can anticipate action or even no action when they are sound asleep in this precarious pose.\n\nThe Birman has four matching white paws. It is necessary to see two or even all four paws in the photograph.\n\nThe Abyssinian and the Russian Blue shows off their fine boning and length of their legs.\n\nHigh Fives\n\nIf a cat wants something but is too lazy to jump for it, they will use their paws to reach out a paw to try to catch it. The \"paw up\" pose is one of the poses I seek, but occasionally they are stretching to their limits.\n\nHolidays\n\nAs you may have noticed, I rarely use props for my images. The true reason is that my clients prefer to see all of the cat and not a prop or gimmick. Some cats will be willing to do many things but the true sense of each cat is their expression. To add a prop, distracts from that focus. Sometimes I will use some holiday props as some clients like to send holiday cards. I can make them happy with a minimal amount of clutter.\n\nIn Motion\n\nCats' movements are fluid. Capturing them in motion can be a challenge. The best time is when the cat is willing to move around my area but is still being attentive to my toys. I cannot allow them to run around the photographing area. Photographs taken while a cat is chasing my toys accentuates the attributes of certain breeds.\n\nCats are innate hunters so they \"stalk\" their prey (my toys, teasers, and feathers). Encouraging them to stalk demonstrates their sleekness, agility and ability to move around the photographing area without destroying the set. Some cats become so intense that they forget where they are and leap off or even up and over the posing area. This is when it is good that I am photographing in an enclosed tent.\n\nGoofy\n\nCats can be characters. Inside out or upside down\u2014they do not need much to entertain themselves.\n\nGroups\n\nEvery cat is a new challenge, but challenges multiply when there are more than one. So wrangling together more than two cats is a great accomplishment. No photos shown here were digitally altered in Photoshop to achieve these images. When arranging these images, I like to have all my ducks in a row. Sweat or no sweat, it is what keeps me going. I do not charge extra for more than one cat. As long as the cats get along, I can get the shot. The image of the Abyssinian litter had to have some props to keep a couple of the shyer babies feeling comfortable.\n\nWrangling cats takes expertise and timing. I work directly with each cat or group as I need them to focus on me and what I am doing. I elicit the owner(s) or my wife to assist by just being on each side of my work area table to avoid the cats leaving the scene or calming them down. Cats have a limited attention span, so I need to move quickly and be totally ready to shoot when the time is right. It may only be a split second where everyone is acting together.\n\nIn the Air\n\nWhen working with show-off cats, timing is of the essence. I use split-second shots but not rapid fire. My eyes are focused on their positions, their heads, ears, and eyes. I love the dancer. Many cats like to do the \"praying\" pose or the \"prairie dog\" pose. The challenge is to keep eye contact, because as the cats rise they will look up too high, which distorts the face and is unusable.\n\nKittens\n\nThere are kittens at shows but they are not baby kittens. Show rules restrict competition to kittens over the age of four months old. Some associations allow slightly younger kittens to be present in the show hall but not competing. Kittens compete in the kitten class until they are eight months old. At eight months old, they begin to compete as adults. I have photographed many kittens younger than four months old, but not at cat shows.\n\nLarge Cats\n\nThe largest cat in competition is the Maine Coon cat. Fully grown Maine Coons can weigh up to thirty pounds and can span tip to tail three foot or more. Its medium length shaggy coat, long tail, high ears, and squared off muzzle shows its ability to hunt in the low brush without getting snagged. Their feet have tufts of fur and allow them to walk undetected. The Maine Coon cat is known to descend from the cats that came over on the Mayflower from England. They were allowed to roam freely after landing, hence the Maine in their name.\n\nPairs\n\nExhibitors do not bring two cats to the show for the purpose of breeding. These cats are buddies, litter mates, or traveling companions. I will rarely put two cats together that do not know each other.\n\nIt thrills me when the cats enjoy and interact with each other. Having a good time is the most important objective for them to experience at the shows. The shows are a foreign environment: lots of new smells, people, crowds, kids, strollers, wheelchairs, loud speakers, and over a two-to-three day period, being handled by many judges. My goal is for the cats to enjoy their time with me and to capture the moments.\n\nProfiles\n\nProfiles are important in breed standards. No matter how good the rest of the image is, my clients will not use the images if the cat appears to have an improper profile in the image.\n\nI listen carefully to the owners when they talk about their cats. Many times they give me clues as to what to emphasize and what not to emphasize.\n\nProps\n\nI occasionally utilize props but like to keep it to a minimum and be subtle in their use. Kittens interact best with these objects. I have a selection of velvet chairs, lounges, baskets, and more. If I am asked to bring something special, I will. Since I fly to most of my shows, I usually do not bring any props. If the owners bring their own props, I will not object to photographing it with their cat.\n\nSitting Up\n\nThe \"praying\" and \"prairie dog\" poses are the cutest poses. To achieve the look and intensity, it takes a sharp eye and split second reaction. These images cannot be easily planned. If the cat is prone to balancing in this way, I will add it to the selection of poses.\n\nStanding\n\nA full stance usually occurs during the early part of the session. Full body poses are mostly needed in breeds where the coat pattern, color, and length are most important.\n\nStretching\n\nThe stretch is another pose that requires a split second reaction on my part. The cat is usually just placed on the table area and begins to stretch. It may only last a second or two. It is a totally unplanned spontaneous action. This adds a fun aspect to the work I offer my clients. My clients use many of these types of photographs on their web pages as headers or boarders. The majority of the session will consist of more properly posed images reflecting the characteristics of the breed.\n\nThe Wild Look\n\nThree of these four cat breeds have wild blood ancestry. All four are hybrid cats.\n\nThe Ocicat is a mix between an Abyssinian, Siamese, and American Shorthair. It has thumb size spots, and it resembles a wild looking cat.\n\nThe Bengal resulted from an outcross between Asian leopard cats and domestic shorthairs in the 1950s. The Bengal is the only domestic cat breed that can have rosettes like the markings on Leopards, Jaguars, and Ocelots.\n\nThe Toyger is a man-made breed a striped coat pattern. It resulted from a cross between a Bengal and a Domestic Shorthair to acquire the pattern.\n\nThe Savannah is an outcross between a Serval, Bengals, Oriental Shorthairs, and Egyptian Maus resulting in a large cat with high ears and a spotted pattern.\n\nTalking\n\nOn a rare occasion, the cat will \"talk\" to me or the teaser. Many cats like to chatter at some of the toys as if they were talking to the birds through a window. They are not ready to bite. They are just responding.\n\nUp\n\nTiming is everything. If the cat has a sense of balance and is relaxed enough, it can go up or even \"lift\" off the posing area. The Bengal shows off its belly spots that would not be visible in other poses. I love their expressions.\n\nAwards\n\nI rarely pose cats with their awards. Many awards are not as lavish as they were twenty years ago. The larger awards come at the end of the show year and are handed out at fancy banquets, where the cats are not present. Occasionally, a client will bring the award specifically for a image or two. I like encouraging the cats to interact with their awards.\n\nAgility\n\nAn agility course is offered at some shows and the goal is to lead your cat through the bars, hoops and tunnels for the fastest over all time. Several breeds are more agile that others and are more focused on the teasers than others. Turkish Angoras, American Shorthairs, Ocicats, Bengals, and Maine Coon cats tend to have the highest rate of success. It is open for all breeds and non-breeds to compete.\n\nThe owner is a leader for the cat and attempts to keep the cat attentive to their lead throughout the course. On many occasion the cat sees a distraction and just stops mid-course or wanders off. The timing to run the course has varies from nine seconds to several minutes. Agility is now scored on a national basis, and it is a wonderful event to watch. Many shows offer trial runs to see how the cat reacts and will have practice animals for kids to try their hand at course. The Turkish Angora here is with its favorite teaser toy.\n\nCombo Imagery\n\nAfter a session has been delivered, I offer my client specialty combo images. The owner selects images that express their cat at its best. My Photoshop skill allows me to produce a unique image that is printed and hangs on the client's wall. Most are very special award-winning cats. These images also make a very nice display on their website and Facebook pages.\n\nI will place ads in various show catalogues to ensure my visibility to the exhibitors. These are some samples.\n\nFinal Thoughts\n\nA big thank to all the cats that have allowed me to photograph them. It is time to say farewell for now and to take a catnap. See you again soon.\n\nFor more information about cat shows and their organizations, check out: CFA, TICA, WCF, ACFA, CCA, ACA, FIFE, GCCF, and many other associations including those in Australia and South Africa. Many groups are worldwide registries. One can find information about specific breeds by searching online for these breeds. Please remember to visit your local shelters and rescue groups for some of the most awesome cat companions.\n\nIndex\n\n**A**\n\nAbyssinian breed, , , ,\n\nAmerican Bobtail breed, ,\n\nAmerican Curl breed, ,\n\nAmerican Wirehair breed, , ,\n\nAmerican Shorthair breed, , , , ,\n\nagility, , 118\u201319\n\naward-winning cats,\n\nawards, 116\u201317\n\n**B**\n\nbackgrounds, 12\u201313, ,\n\nBalinese breed,\n\nbehavior,\n\nBengal breed, , , ,\n\nBirman breed, , ,\n\nblack cats, 16\u201317,\n\nblack and white cats, 14\u201315\n\nblue cats, 20\u201323, ,\n\nBombay breed,\n\nbreed standards, , 5\u201365,\n\nBritish Longhair breed,\n\nBritish Shorthair breed, , , ,\n\nBurmese breed,\n\nBurmilla breed,\n\n**C**\n\ncat shows, , , , ,\n\nclients, , , , , , , ,\n\ncolor, fur, , , 20\u201323, , , , ,\n\ncompanions, 94\u201395,\n\ncontrast,\n\nChartreux breed, ,\n\nChaussie breed,\n\nColor Point Shorthair breed,\n\nCornish Rex breed, , , ,\n\n**D**\n\nDevon Rex breed, , ,\n\n**E**\n\nears, , , , , , 24\u201325, , , , , , ,\n\nEgyptian Mau breed, ,\n\nexpression, , , ,\n\neye color, ,\n\nExotic Shorthair breed, ,\n\n**F**\n\nfaces, 6\u20137, ,\n\n**G**\n\ngroups, 84\u201385,\n\n**H**\n\nhair, curly, 40\u201341\n\nhair, longhair, , , , , , ,\n\nhair, none, , 46\u201347\n\nhair, shorthair, , , , , , , , , , , , , , , , ,\n\nHavana Brown breed,\n\nHimalayan Persian breed,\n\nholidays, 74\u201375\n\n**J**\n\nJapanese Bobtail breed, ,\n\nJapanese Bobtail Longhair breed, ,\n\n**K**\n\nKheo Manee breed,\n\nkittens, , 90\u201391,\n\nKorat breed, ,\n\nKurilian Bobtail breed, ,\n\n**L**\n\nLaPerm breed, , ,\n\nlarge cats, 92\u201393,\n\nlonghair, , , , , , ,\n\nLykoi breed,\n\n**M**\n\nmotion, 76\u201377\n\nMaine Coon breed, , , , ,\n\nManx breed, ,\n\nManx Longhair breed,\n\nMunchkin breed, ,\n\n**N**\n\nNorwegian Forest Cat breed,\n\n**O**\n\nOcicat breed, , ,\n\norganizations,\n\nOriental Longhair breed,\n\nOriental Shorthair breed, , , ,\n\n**P**\n\nPersian, Solid White breed,\n\nPersian, Tabby breed,\n\nPersian, Solid Black breed,\n\npets, household, 50\u201351\n\nphotographer's expertise, 12\u201313,\n\nphotographing, , , ,\n\npositions, , ,\n\n**S**\n\nstudio, , 8\u20139\n\nshorthair, , , , , , , , , , , , , , , , ,\n\nsession, photography, 10\u201311, , , ,\n\n**T**\n\ntails, , , ,\n\ntails, bobtails, 30\u201331\n\ntails, long, 32\u201333,\n\ntails, none, 34\u201335\n\ntails, puffy, 36\u201337\n\ntalking, 112\u201313\n\ntoys and teasers, , , ,\n\ntravel, , , ,\n\ntuxedo cats, 14\u201315\n\n**W**\n\nwhite cats,\n\nwhite, , , , \nTable of Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Contents\n  5. About the Author\n  6. Faces\n  7. The Studio\n  8. Session Sample\n  9. Backgrounds\n  10. Black and White Cats\n  11. Black on Black\n  12. White on White\n  13. Blue Cats\n  14. Ears\n  15. Eyes\n  16. Bob Tails\n  17. Long Tails\n  18. No Tails\n  19. Puffy Tails\n  20. Big Hair\n  21. Curly Hair\n  22. Wavy Hair\n  23. Kinky Hair\n  24. No Hair\n  25. Plush Coat\n  26. Household Pets\n  27. The Standard\n  28. Hangers\n  29. High Fives\n  30. Holidays\n  31. In Motion\n  32. Goofy\n  33. Groups\n  34. In the Air\n  35. Kittens\n  36. Large Cats\n  37. Pairs\n  38. Profiles\n  39. Props\n  40. Sitting Up\n  41. Standing\n  42. Stretching\n  43. The Wild Look\n  44. Talking\n  45. Up\n  46. Awards\n  47. Agility\n  48. Combo Imagery\n  49. Final Thoughts\n  50. Index\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Title Page\n\n  1. \n  2. \n  3. \n  4. \n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nBegin Reading\n\nTable of Contents\n\nAbout the Author\n\nCopyright Page\n\n**Thank you for buying this**\n\n**St. Martin's Press ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. **Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at:us.macmillanusa.com\/piracy.**\nfor the weirdos & the dreamers\n\n# ONE\n\nOn the White House roof, tucked into a corner of the Promenade, there's a bit of loose paneling right on the edge of the Solarium. If you tap it just right, you can peel it back enough to find a message etched underneath, with the tip of a key or maybe a stolen West Wing letter opener.\n\nIn the secret history of First Families\u2014an insular gossip mill sworn to absolute discretion about most things on pain of death\u2014there's no definite answer for who wrote it. The one thing people seem certain of is that only a presidential son or daughter would have been daring enough to deface the White House. Some swear it was Jack Ford, with his Hendrix records and split-level room attached to the roof for late-night smoke breaks. Others say it was a young Luci Johnson, thick ribbon in her hair. But it doesn't matter. The writing stays, a private mantra for those resourceful enough to find it.\n\nAlex discovered it within his first week of living there. He's never told anyone how.\n\nIt says:\n\n> RULE #1: DON'T GET CAUGHT\n\nThe East and West Bedrooms on the second floor are generally reserved for the First Family. They were first designated as one giant state bedroom for visits from the Marquis de Lafayette in the Monroe administration, but eventually they were split. Alex has the East, across from the Treaty Room, and June uses the West, next to the elevator.\n\nGrowing up in Texas, their rooms were arranged in the same configuration, on either side of the hallway. Back then, you could tell June's ambition of the month by what covered the walls. At twelve, it was watercolor paintings. At fifteen, lunar calendars and charts of crystals. At sixteen, clippings from _The Atlantic,_ a UT Austin pennant, Gloria Steinem, Zora Neale Hurston, and excerpts from the papers of Dolores Huerta.\n\nHis own room was forever the same, just steadily more stuffed with lacrosse trophies and piles of AP coursework. It's all gathering dust in the house they still keep back home. On a chain around his neck, always hidden from view, he's worn the key to that house since the day he left for DC.\n\nNow, straight across the hall, June's room is all bright white and soft pink and minty green, photographed by _Vogue_ and famously inspired by old '60s interior design periodicals she found in one of the White House sitting rooms. His own room was once Caroline Kennedy's nursery and, later, warranting some sage burning from June, Nancy Reagan's office. He's left up the nature field illustrations in a neat symmetrical grid above the sofa, but painted over Sasha Obama's pink walls with a deep blue.\n\nTypically, the children of the president, at least for the past few decades, haven't lived in the Residence beyond eighteen, but Alex started at Georgetown the January his mom was sworn in, and logistically, it made sense not to split their security or costs to whatever one-bedroom apartment he'd be living in. June came that fall, fresh out of UT. She's never said it, but Alex knows she moved in to keep an eye on him. She knows better than anyone else how much he gets off on being this close to the action, and she's bodily yanked him out of the West Wing on more than one occasion.\n\nBehind his bedroom door, he can sit and put Hall & Oates on the record player in the corner, and nobody hears him humming along like his dad to \"Rich Girl.\" He can wear the reading glasses he always insists he doesn't need. He can make as many meticulous study guides with color-coded sticky notes as he wants. He's not going to be the youngest elected congressman in modern history without earning it, but nobody needs to know how hard he's kicking underwater. His sex-symbol stock would plummet.\n\n\"Hey,\" says a voice at the door, and he looks up from his laptop to see June edging into his room, two iPhones and a stack of magazines tucked under one arm, and a plate in her hand. She closes the door behind her with her foot.\n\n\"What'd you steal today?\" Alex asks, pushing the pile of papers on his bed out of her way.\n\n\"Assorted donuts,\" June says as she climbs up. She's wearing a pencil skirt with pointy pink flats, and he can already see next week's fashion columns: a picture of her outfit today, a lead-in for some sponcon about flats for the professional gal on the go.\n\nHe wonders what she's been up to all day. She mentioned a column for _WaPo,_ or was it a photoshoot for her blog? Or both? He can never keep up.\n\nShe's dumped her stack of magazines out on the bedspread and is already busying herself with them.\n\n\"Doing your part to keep the great American gossip industry alive?\"\n\n\"That's what my journalism degree's for,\" June says.\n\n\"Anything good this week?\" Alex asks, reaching for a donut.\n\n\"Let's see,\" June says. \" _In Touch_ says I'm... dating a French model?\"\n\n\"Are you?\"\n\n\"I wish.\" She flips a few pages. \"Ooh, and they're saying you got your asshole bleached.\"\n\n\"That one is true,\" Alex says through a mouthful of chocolate with sprinkles.\n\n\"Thought so,\" June says without looking up. After riffling through most of the magazine, she shuffles it to the bottom of the stack and moves on to _People_. She flips through absently\u2014 _People_ only ever writes what their publicists tell it to write. Boring. \"Not much on us this week... oh, I'm a crossword puzzle clue.\"\n\nFollowing their tabloid coverage is something of an idle hobby of hers, one that in turns amuses and annoys their mother, and Alex is narcissistic enough to let June read him the highlights. They're usually either complete fabrications or lines fed from their press team, but sometimes it's just funny. Given the choice, he'd rather read one of the hundreds of glowing pieces of fan fiction about him on the internet, the up-to-eleven version of himself with devastating charm and unbelievable physical stamina, but June flat-out refuses to read those aloud to him, no matter how much he tries to bribe her.\n\n\"Do _Us Weekly,_ \" Alex says.\n\n\"Hmm...\" June digs it out of the stack. \"Oh, look, we made the cover this week.\"\n\nShe flashes the glossy cover at him, which has a photo of the two of them inlaid in one corner, June's hair pinned on top of her head and Alex looking slightly over-served but still handsome, all jawline and dark curls. Below it in bold yellow letters, the headline reads: FIRST SIBLINGS' WILD NYC NIGHT.\n\n\"Oh yeah, that was a wild night,\" Alex says, reclining back against the tall leather headboard and pushing his glasses up his nose. \"Two whole keynote speakers. Nothing sexier than shrimp cocktails and an hour and a half of speeches on carbon emissions.\"\n\n\"It says here you had some kind of tryst with a 'mystery brunette,'\" June reads. \"'Though the First Daughter was whisked off by limousine to a star-studded party shortly after the gala, twenty-one-year-old heartthrob Alex was snapped sneaking into the W Hotel to meet a mystery brunette in the presidential suite and leaving around four a.m. Sources inside the hotel reported hearing amorous noises from the room all night, and rumors are swirling the brunette was none other than... _Nora Holleran,_ the twenty-two-year-old granddaughter of Vice President Mike Holleran and third member of the White House Trio. Could it be the two are rekindling their romance?'\"\n\n\"Yes!\" Alex crows, and June groans. \"That's less than a month! You owe me fifty dollars, baby.\"\n\n\"Hold on. _Was_ it Nora?\"\n\nAlex thinks back to the week before, showing up at Nora's room with a bottle of champagne. Their thing on the campaign trail a million years ago was brief, mostly to get the inevitable over with. They were seventeen and eighteen and doomed from the start, both convinced they were the smartest person in any room. Alex has since conceded Nora is 100 percent smarter than him and definitely too smart to have ever dated him.\n\nIt's not his fault the press won't let it go, though; that they _love_ the idea of them together as if they're modern-day Kennedys. So, if he and Nora occasionally get drunk in hotel rooms together watching _The West Wing_ and making loud moaning noises at the wall for the benefit of nosy tabloids, he can't be blamed, really. They're simply turning an undesirable situation into their own personal entertainment.\n\nScamming his sister is also a perk.\n\n\"Maybe,\" he says, dragging out the vowels.\n\nJune swats him with the magazine like he's an especially obnoxious cockroach. \"That's cheating, you dick!\"\n\n\"Bet's a bet,\" Alex tells her. \"We said if there was a new rumor in a month, you'd owe me fifty bucks. I take Venmo.\"\n\n\"I'm not paying,\" June huffs. \"I'm gonna kill her when we see her tomorrow. What are you wearing, by the way?\"\n\n\"For what?\"\n\n\"The wedding.\"\n\n\"Whose wedding?\"\n\n\"Uh, the _royal wedding,_ \" June says. \"Of England. It's literally on every cover I just showed you.\"\n\nShe holds _Us Weekly_ up again, and this time Alex notices the main story in giant letters: PRINCE PHILIP SAYS I DO! Along with a photograph of an extremely nondescript British heir and his equally nondescript blond fianc\u00e9e smiling blandly.\n\nHe drops his donut in a show of devastation. \"That's _this_ weekend?\"\n\n\"Alex, we leave in the morning,\" June tells him. \"We've got two appearances before we even go to the ceremony. I can't believe Zahra hasn't climbed up your ass about this already.\"\n\n\"Shit,\" he groans. \"I know I had that written down. I got sidetracked.\"\n\n\"What, by conspiring with my best friend against me in the tabloids for fifty dollars?\"\n\n\"No, with my research paper, smart-ass,\" Alex says, gesturing dramatically at his piles of notes. \"I've been working on it for Roman Political Thought all week. And I thought we agreed Nora is _our_ best friend.\"\n\n\"That can't possibly be a real class you're taking,\" June says. \"Is it possible you willfully forgot about the biggest international event of the year because you don't want to see your archnemesis?\"\n\n\"June, I'm the son of the President of the United States. Prince Henry is a figurehead of the British Empire. You can't just call him my 'archnemesis,'\" Alex says. He returns to his donut, chewing thoughtfully, and adds, \"'Archnemesis' implies he's actually a rival to me on any level and not, you know, a stuck-up product of inbreeding who probably jerks off to photos of himself.\"\n\n\"Woof.\"\n\n\"I'm just saying.\"\n\n\"Well, you don't have to like him, you just have to put on a happy face and not cause an international incident at his brother's wedding.\"\n\n\"Bug, when do I ever not put on a happy face?\" Alex says. He pulls a painfully fake grin, and June looks satisfyingly repulsed.\n\n\"Ugh. Anyway, you know what you're wearing, right?\"\n\n\"Yeah, I picked it out and had Zahra approve it last month. I'm not an animal.\"\n\n\"I'm still not sure about my dress,\" June says. She leans over and steals his laptop away from him, ignoring his noise of protest. \"Do you think the maroon or the one with the lace?\"\n\n\"Lace, obviously. It's England. And why are you trying to make me fail this class?\" he says, reaching for his laptop only to have his hand swatted away. \"Go curate your Instagram or something. You're the worst.\"\n\n\"Shut up, I'm trying to pick something to watch. Ew, you have _Garden State_ on your watch list? Wow, how's film school in 2005 going?\"\n\n\"I hate you.\"\n\n\"Hmm, I know.\"\n\nOutside his window, the wind stirs up over the lawn, rustling the linden trees down in the garden. The record on the turntable in the corner has spun out into fuzzy silence. He rolls off the bed and flips it, resetting the needle, and the second side picks up on \"London Luck, & Love.\"\n\n* * *\n\nIf he's honest, private aviation doesn't really get old, not even three years into his mother's term.\n\nHe doesn't get to travel this way a lot, but when he does, it's hard not to let it go to his head. He was born in the hill country of Texas to the daughter of a single mother and the son of Mexican immigrants, all of them dirt poor\u2014luxury travel is still a luxury.\n\nFifteen years ago, when his mother first ran for the House, the Austin newspaper gave her a nickname: the Lometa Longshot. She'd escaped her tiny hometown in the shadow of Fort Hood, pulled night shifts at diners to put herself through law school, and was arguing discrimination cases before the Supreme Court by thirty. She was the last thing anybody expected to rise up out of Texas in the midst of the Iraq War: a strawberry-blond, whip-smart Democrat with high heels, an unapologetic drawl, and a little biracial family.\n\nSo, it's still surreal that Alex is cruising somewhere over the Atlantic, snacking on pistachios in a high-backed leather chair with his feet up. Nora is bent over the _New York Times_ crossword opposite him, brown curls falling across her forehead. Beside her, the hulking Secret Service agent Cassius\u2014Cash for short\u2014holds his own copy in one giant hand, racing to finish it first. The cursor on Alex's Roman Political Thought paper blinks expectantly at him from his laptop, but something in him can't quite focus on school while they're flying transatlantic.\n\nAmy, his mother's favorite Secret Service agent, a former Navy SEAL who is rumored around DC to have killed several men, sits across the aisle. She's got a bulletproof titanium case of crafting supplies open on the couch next to her and is serenely embroidering flowers onto a napkin. Alex has seen her stab someone in the kneecap with a very similar embroidery needle.\n\nWhich leaves June, next to him, leaning on one elbow with her nose buried in the issue of _People_ she's inexplicably brought with them. She always chooses the most bizarre reading material for flights. Last time, it was a battered old Cantonese phrase book. Before that, _Death Comes for the Archbishop._\n\n\"What are you reading in there now?\" Alex asks her.\n\nShe flips the magazine around so he can see the double-page spread titled: ROYAL WEDDING MADNESS! Alex groans. This is definitely worse than Willa Cather.\n\n\"What?\" she says. \"I want to be prepared for my first-ever royal wedding.\"\n\n\"You went to prom, didn't you?\" Alex says. \"Just picture that, only in hell, and you have to be really nice about it.\"\n\n\"Can you believe they spent $75,000 just on the cake?\"\n\n\"That's depressing.\"\n\n\" _And_ apparently Prince Henry is going sans date to the wedding and everyone is freaking out about it. It says he was,\" she affects a comical English accent, \"'rumored to be dating a Belgian heiress last month, but now followers of the prince's dating life aren't sure what to think.'\"\n\nAlex snorts. It's insane to him that there are legions of people who follow the intensely dull dating lives of the royal siblings. He understands why people care where he puts his own tongue\u2014at least _he_ has personality.\n\n\"Maybe the female population of Europe finally realized he's as compelling as a wet ball of yarn,\" Alex suggests.\n\nNora puts down her crossword puzzle, having finished it first. Cassius glances over and swears. \"You gonna ask him to dance, then?\"\n\nAlex rolls his eyes, suddenly imagining twirling around a ballroom while Henry drones sweet nothings about croquet and fox hunting in his ear. The thought makes him want to gag.\n\n\"In his dreams.\"\n\n\"Aw,\" Nora says, \"you're blushing.\"\n\n\"Listen,\" Alex tells her, \"royal weddings are trash, the princes who have royal weddings are trash, the imperialism that allows princes to exist at all is trash. It's trash turtles all the way down.\"\n\n\"Is this your TED Talk?\" June asks. \"You do realize America is a genocidal empire too, right?\"\n\n\"Yes, _June,_ but at least we have the decency not to keep a monarchy around,\" Alex says, throwing a pistachio at her.\n\nThere are a few things about Alex and June that new White House hires are briefed on before they start. June's peanut allergy. Alex's frequent middle-of-the-night requests for coffee. June's college boyfriend, who broke up with her when he moved to California but is still the only person whose letters come to her directly. Alex's long-standing grudge against the youngest prince.\n\nIt's not a grudge, really. It's not even a rivalry. It's a prickling, unsettling annoyance. It makes his palms sweat.\n\nThe tabloids\u2014the world\u2014decided to cast Alex as the American equivalent of Prince Henry from day one, since the White House Trio is the closest thing America has to royalty. It has never seemed fair. Alex's image is all charisma and genius and smirking wit, thoughtful interviews and the cover of _GQ_ at eighteen; Henry's is placid smiles and gentle chivalry and generic charity appearances, a perfectly blank Prince Charming canvas. Henry's role, Alex thinks, is much easier to play.\n\nMaybe it is technically a rivalry. Whatever.\n\n\"All right, MIT,\" he says, \"what are the numbers on this one?\"\n\nNora grins. \"Hmm.\" She pretends to think hard about it. \"Risk assessment: FSOTUS failing to check himself before he wrecks himself will result in greater than five hundred civilian casualties. Ninety-eight percent probability of Prince Henry looking like a total dreamboat. Seventy-eight percent probability of Alex getting himself banned from the United Kingdom forever.\"\n\n\"Those are better odds than I expected,\" June observes.\n\nAlex laughs, and the plane soars on.\n\n* * *\n\nLondon is an absolute spectacle, crowds cramming the streets outside Buckingham Palace and all through the city, draped in Union Jacks and waving tiny flags over their heads. There are commemorative royal wedding souvenirs everywhere; Prince Philip and his bride's face plastered on everything from chocolate bars to underwear. Alex almost can't believe this many people care so passionately about something so comprehensively dull. He's sure there won't be this kind of turnout in front of the White House when he or June get married one day, nor would he even want it.\n\nThe ceremony itself seems to last forever, but it's at least sort of nice, in a way. It's not that Alex isn't into love or can't appreciate marriage. It's just that Martha is a perfectly respectable daughter of nobility, and Philip is a prince. It's as sexy as a business transaction. There's no passion, no drama. Alex's kind of love story is much more Shakespearean.\n\nIt feels like years before he's settled at a table between June and Nora inside a Buckingham Palace ballroom for the reception banquet, and he's irritated enough to be a little reckless. Nora passes him a flute of champagne, and he takes it gladly.\n\n\"Do either of y'all know what a viscount is?\" June is saying, halfway through a cucumber sandwich. \"I've met, like, five of them, and I keep smiling politely as if I know what it means when they say it. Alex, you took comparative international governmental relational things. Whatever. What are they?\"\n\n\"I think it's that thing when a vampire creates an army of crazed sex waifs and starts his own ruling body,\" he says.\n\n\"That sounds right,\" Nora says. She's folding her napkin into a complicated shape on the table, her shiny black manicure glinting in the chandelier light.\n\n\"I wish I were a viscount,\" June says. \"I could have my sex waifs deal with my emails.\"\n\n\"Are sex waifs good with professional correspondence?\" Alex asks.\n\nNora's napkin has begun to resemble a bird. \"I think it could be an interesting approach. Their emails would be all tragic and wanton.\" She tries on a breathless, husky voice. \"'Oh, please, I beg you, take me\u2014take me to lunch to discuss fabric samples, you beast!'\"\n\n\"Could be weirdly effective,\" Alex notes.\n\n\"Something is wrong with both of you,\" June says gently.\n\nAlex is opening his mouth to retort when a royal attendant materializes at their table like a dense and dour-looking ghost in a bad hairpiece.\n\n\"Miss Claremont-Diaz,\" says the man, who looks like his name is probably Reginald or Bartholomew or something. He bows, and miraculously his hairpiece doesn't fall off into June's plate. Alex shares an incredulous glance with her behind his back. \"His Royal Highness Prince Henry wonders if you would do him the honor of accompanying him for a dance.\"\n\nJune's mouth freezes halfway open, caught on a soft vowel sound, and Nora breaks out into a shit-eating grin.\n\n\"Oh, she'd _love_ to,\" Nora volunteers. \"She's been hoping he'd ask all evening.\"\n\n\"I\u2014\" June starts and stops, her mouth smiling even as her eyes slice at Nora. \"Of course. That would be lovely.\"\n\n\"Excellent,\" Reginald-Bartholomew says, and he turns and gestures over his shoulder.\n\nAnd there Henry is, in the flesh, as classically handsome as ever in his tailored three-piece suit, all tousled sandy hair and high cheekbones and a soft, friendly mouth. He holds himself with innately impeccable posture, as if he emerged fully formed and upright out of some beautiful Buckingham Palace posy garden one day.\n\nHis eyes lock on Alex's, and something like annoyance or adrenaline spikes in Alex's chest. He hasn't had a conversation with Henry in probably a year. His face is still infuriatingly symmetrical.\n\nHenry deigns to give him a perfunctory nod, as if he's any other random guest, not the person he beat to a _Vogue_ editorial debut in their teens. Alex blinks, seethes, and watches Henry angle his stupid chiseled jaw toward June.\n\n\"Hello, June,\" Henry says, and he extends a gentlemanly hand to June, who is now blushing. Nora pretends to swoon. \"Do you know how to waltz?\"\n\n\"I'm... sure I could pick it up,\" she says, and she takes his hand cautiously, like she thinks he might be pranking her, which Alex thinks is way too generous to Henry's sense of humor. Henry leads her off to the crowd of twirling nobles.\n\n\"So is that what's happening now?\" Alex says, glaring down at Nora's napkin bird. \"Has he decided to finally shut me up by wooing my sister?\"\n\n\"Aw, little buddy,\" Nora says. She reaches over and pats his hand. \"It's cute how you think everything is about you.\"\n\n\"It should be, honestly.\"\n\n\"That's the spirit.\"\n\nHe glances up into the crowd, where June is being rotated around the floor by Henry. She's got a neutral, polite smile on her face, and he keeps looking over her shoulder, which is even more annoying. June is amazing. The least Henry could do is pay attention to her.\n\n\"Do you think he actually likes her, though?\"\n\nNora shrugs. \"Who knows? Royals are weird. Might be a courtesy, or\u2014oh, there it is.\"\n\nA royal photographer has swooped in and is snapping a shot of them dancing, one Alex knows will be leaked to _Hello_ next week. So, that's it, then? Using the First Daughter to start some idiotic dating rumor for attention? God forbid Philip gets to dominate the news cycle for one week.\n\n\"He's kind of good at this,\" Nora remarks.\n\nAlex flags down a waiter and decides to spend the rest of the reception getting systematically drunk.\n\nAlex has never told\u2014will never tell\u2014anyone, but he saw Henry for the first time when he was twelve years old. He only ever reflects upon it when he's drunk.\n\nHe's sure he saw his face in the news before then, but that was the first time he really _saw_ him. June had just turned fifteen and used part of her birthday money to buy an issue of a blindingly colorful teen magazine. Her love of trashy tabloids started early. In the center of the magazine were miniature posters you could rip out and stick up in your locker. If you were careful and pried up the staples with your fingernails, you could get them out without tearing them. One of them, right in the middle, was a picture of a boy.\n\nHe had thick, tawny hair and big blue eyes, a warm smile, and a cricket bat over one shoulder. It must have been a candid, because there was a happy, sun-bright confidence to him that couldn't be posed. On the bottom corner of the page in pink and blue letters: PRINCE HENRY.\n\nAlex still doesn't really know what kept drawing him back, only that he would sneak into June's room and find the page and touch his fingertips to the boy's hair, as if he could somehow feel its texture if he imagined it hard enough. The more his parents climbed the political ranks, the more he started to reckon with the fact that soon the world would know who he was. Then, sometimes, he'd think of the picture, and try to harness Prince Henry's easy confidence.\n\n(He also thought about prying up the staples with his fingers and taking the picture out and keeping it in his room, but he never did. His fingernails were too stubby; they weren't made for it like June's, like a girl's.)\n\nBut then came the first time he met Henry\u2014the first cool, detached words Henry said to him\u2014and Alex guessed he had it all wrong, that the pretty, flung-open boy from the picture wasn't real. The real Henry is beautiful, distant, boring, and closed. This person the tabloids keep comparing him to, whom he compares _himself_ to, thinks he's _better_ than Alex and everyone like him. Alex can't believe he ever wanted to be anything like that.\n\nAlex keeps drinking, keeps alternating between thinking about it and forcing himself not to think about it, disappears into the crowd and dances with pretty European heiresses about it.\n\nHe's pirouetting away from one when he catches sight of a lone figure hovering near the cake and the champagne fountain. It's Prince Henry yet again, glass in hand, watching Prince Philip and his bride spinning on the ballroom floor. He looks politely half-interested in that obnoxious way of his, like he has somewhere else to be. And Alex can't resist the urge to call his bluff.\n\nHe picks his way through the crowd, grabbing a glass of wine off a passing tray and downing half of it.\n\n\"When you have one of these,\" Alex says, sidling up to him, \"you should do two champagne fountains instead of one. Really embarrassing to be at a wedding with only one champagne fountain.\"\n\n\"Alex,\" Henry says in that maddeningly posh accent. Up close, the waistcoat under his suit jacket is a lush gold and has about a million buttons on it. It's horrible. \"I wondered if I'd have the pleasure.\"\n\n\"Looks like it's your lucky day,\" Alex says, smiling.\n\n\"Truly a momentous occasion,\" Henry agrees. His own smile is bright white and immaculate, made to be printed on money.\n\nThe most annoying thing of all is Alex _knows_ Henry hates him too\u2014he _must,_ they're naturally mutual antagonists\u2014but he refuses to outright act like it. Alex is intimately aware politics involves a lot of making nice with people you loathe, but he wishes that once, just once, Henry would act like an actual human and not some polished little windup toy sold in a palace gift shop.\n\nHe's too perfect. Alex wants to poke it.\n\n\"Do you ever get tired,\" Alex says, \"of pretending you're above all this?\"\n\nHenry turns and stares at him. \"I'm sure I don't know what you mean.\"\n\n\"I mean, you're out here, getting the photographers to chase you, swanning around like you hate the attention, which you clearly don't since you're dancing with my sister, of all people,\" Alex says. \"You act like you're too important to be anywhere, ever. Doesn't that get exhausting?\"\n\n\"I'm... a bit more complicated than that,\" Henry attempts.\n\n_\"Ha.\"_\n\n\"Oh,\" Henry says, narrowing his eyes. \"You're drunk.\"\n\n\"I'm just saying,\" Alex says, resting an overly friendly elbow on Henry's shoulder, which isn't as easy as he'd like it to be since Henry has about four infuriating inches of height on him. \"You could try to act like you're having fun. Occasionally.\"\n\nHenry laughs ruefully. \"I believe perhaps you should consider switching to water, Alex.\"\n\n\"Should I?\" Alex says. He pushes aside the thought that maybe the wine is what gave him the nerve to stomp over to Henry in the first place and makes his eyes as coy and angelic as he knows how. \"Am I offending you? Sorry I'm not obsessed with you like everyone else. I know that must be confusing for you.\"\n\n\"Do you know what?\" Henry says. \"I think you are.\"\n\nAlex's mouth drops open, while the corner of Henry's turns smug and almost a little mean.\n\n\"Only a thought,\" Henry says, tone polite. \"Have you ever noticed I have never once approached you and have been _exhaustively_ civil every time we've spoken? Yet here you are, seeking me out again.\" He takes a sip of his champagne. \"Simply an observation.\"\n\n\"What? I'm not\u2014\" Alex stammers. \"You're the\u2014\"\n\n\"Have a lovely evening, Alex,\" Henry says tersely, and turns to walk off.\n\nIt drives Alex _nuts_ that Henry thinks he gets to have the last word, and without thinking, he reaches out and pulls Henry's shoulder back.\n\nAnd then Henry turns, suddenly, and almost does push Alex off him this time, and for a brief spark of a moment, Alex is impressed at the glint in his eyes, the abrupt burst of an actual personality.\n\nThe next thing he knows, he's tripping over his own foot and stumbling backward into the table nearest him. He notices too late that the table is, to his horror, the one bearing the massive eight-tier wedding cake, and he grabs for Henry's arm to catch himself, but all it does is throw both of them off-balance and send them crashing together into the cake stand.\n\nHe watches, as if in slow motion, as the cake leans, teeters, shudders, and finally tips. There's absolutely nothing he can do to stop it. It comes crashing down onto the floor in an avalanche of white buttercream, some kind of sugary $75,000 nightmare.\n\nThe room goes heart-stoppingly silent as momentum carries him and Henry through the fall and down, down onto the wreckage of the cake on the ornate carpet, Henry's sleeve still clutched in Alex's fist. Henry's glass of champagne has spilled all over both of them and shattered, and out of the corner of his eye, Alex can see a cut across the top of Henry's cheekbone beginning to bleed.\n\nFor a second, all he can think as he stares up at the ceiling while covered in frosting and champagne is that at least Henry's dance with June won't be the biggest story to come out of the royal wedding.\n\nHis next thought is that his mother is going to murder him in cold blood.\n\nBeside him, he hears Henry mutter slowly, \"Oh my fucking Christ.\"\n\nHe registers dimly that it's the first time he's ever heard the prince swear, before the flash from someone's camera goes off.\n\n# TWO\n\nWith a resounding smack, Zahra slaps a stack of magazines down on the West Wing briefing room table.\n\n\"This is just what I saw on the way here this morning,\" she says. \"I don't think I need to remind you I live two blocks away.\"\n\nAlex stares down at the headlines in front of him.\n\n> THE $75,000 STUMBLE\n> \n> BATTLE ROYAL: Prince Henry and FSOTUS Come to Blows at Royal Wedding\n> \n> CAKEGATE: Alex Claremont-Diaz Sparks Second English-American War\n\nEach one is accompanied by a photo of himself and Henry flat on their backs in a pile of cake, Henry's ridiculous suit all askew and covered in smashed buttercream flowers, his wrist pinned in Alex's hand, a thin slice of red across Henry's cheek.\n\n\"Are you sure we shouldn't be in the Situation Room for this meeting?\" Alex attempts.\n\nNeither Zahra nor his mother, sitting across the table, seems to find it funny. The president gives him a withering look over the top of her reading glasses, and he clamps his mouth shut.\n\nIt's not exactly that he's afraid of Zahra, his mom's deputy chief of staff and right-hand woman. She has a spiky exterior, but Alex swears there's something soft in there somewhere. He's more afraid of what his mother might do. They grew up made to talk about their feelings a lot, and then his mother became president, and life became less about feelings and more about international relations. He's not sure which option spells a worse fate.\n\n\"'Sources inside the royal reception report the two were seen arguing minutes before the... _cake-tastrophe,_ '\" Ellen reads out loud with utter disdain from her own copy of _The Sun._ Alex doesn't even try to guess how she got her hands on today's edition of a British tabloid. President Mom works in mysterious ways. \"'But royal family insiders claim the First Son's feud with Henry has raged for years. A source tells _The Sun_ that Henry and the First Son have been at odds ever since their first meeting at the Rio Olympics, and the animosity has only grown\u2014these days, they can't even be in the same room with each other. It seems it was only a matter of time before Alex took the American approach: a violent altercation.'\"\n\n\"I really don't think you can call tripping over a table a 'violent'\u2014\"\n\n\"Alexander,\" Ellen says, her tone eerily calm. \"Shut up.\"\n\nHe does.\n\n\"'One can't help but wonder,'\" Ellen reads on, \"'if the bitterness between these two powerful sons has contributed to what many have called an icy and distant relationship between President Ellen Claremont's administration and the monarchy in recent years.'\"\n\nShe tosses the magazine aside, folding her arms on the table.\n\n\"Please, tell me another joke,\" Ellen says. \"I want so badly for you to explain to me how this is funny.\"\n\nAlex opens his mouth and closes it a couple of times.\n\n\"He started it,\" he says finally. \"I barely touched him\u2014he's the one who pushed me, and I only grabbed him to try and catch my balance, and\u2014\"\n\n\"Sugar, I cannot express to you how much the press does not give a fuck about who started what,\" Ellen says. \"As your mother, I can appreciate that maybe this isn't your fault, but as the president, all I want is to have the CIA fake your death and ride the dead-kid sympathy into a second term.\"\n\nAlex clenches his jaw. He's used to doing things that piss his mother's staff off\u2014in his teens, he had a penchant for confronting his mother's colleagues with their voting discrepancies at friendly DC fund-raisers\u2014and he's been in the tabloids for things more embarrassing than this. But never in quite such a cataclysmically, internationally terrible way.\n\n\"I don't have time to deal with this right now, so here's what we're gonna do,\" Ellen says, pulling a folder out of her padfolio. It's filled with some official-looking documents punctuated with different colors of sticky tabs, and the first one says: AGREEMENT OF TERMS.\n\n\"Um,\" Alex says.\n\n\"You,\" she says, \"are going to make nice with Henry. You're leaving Saturday and spending Sunday in England.\"\n\nAlex blinks. \"Is it too late to take the faking-my-death option?\"\n\n\"Zahra can brief you on the rest,\" Ellen goes on, ignoring him. \"I have about five hundred meetings right now.\" She gets up and heads for the door, stopping to kiss her hand and press it to the top of his head. \"You're a dumbass. Love you.\"\n\nThen she's gone, heels clicking behind her down the hallway, and Zahra settles into her vacated chair with a look on her face like she'd prefer arranging his death for real. She's not technically the most powerful or important player in his mother's White House, but she's been working by Ellen's side since Alex was five and Zahra was fresh out of Howard. She's the only one trusted to wrangle the First Family.\n\n\"All right, here's the deal,\" she says. \"I was up all night conferencing with a bunch of uptight royal handlers and PR pricks and the prince's fucking _equerry_ to make this happen, so you are going to follow this plan to the letter and not fuck it up, got it?\"\n\nAlex still privately thinks this whole thing is completely ridiculous, but he nods. Zahra looks deeply unconvinced but presses on.\n\n\"First, the White House and the monarchy are going to release a joint statement saying what happened at the royal wedding was a complete accident and a misunderstanding\u2014\"\n\n\"Which it was.\"\n\n\"\u2014and that, despite rarely having time to see each other, you and Prince Henry have been close personal friends for the past several years.\"\n\n\"We're _what_?\"\n\n\"Look,\" Zahra says, taking a drag from her massive stainless steel thermos of coffee. \"Both sides need to come out of this looking good, and the only way to do that is to make it look like your little slap-fight at the wedding was some homoerotic frat bro mishap, okay? So, you can hate the heir to the throne all you want, write mean poems about him in your diary, but the minute you see a camera, you act like the sun shines out of his dick, and you make it convincing.\"\n\n\"Have you met Henry?\" Alex says. \"How am I supposed to do that? He has the personality of a cabbage.\"\n\n\"Are you really not understanding how much I don't care at all how you feel about this?\" Zahra says. \"This is what's happening so your stupid ass doesn't distract the entire country from your mother's reelection campaign. Do you want her to have to get up on the debate stage next year and explain to the world why her son is trying to destabilize America's European relationships?\"\n\nWell, no, he doesn't. And he knows, in the back of his mind, that he's a better strategist than he's been about this, and that without this stupid grudge, he probably could have come up with this plan on his own.\n\n\"So Henry's your new best friend,\" Zahra continues. \"You will smile and nod and not piss off anyone while you and Henry spend the weekend doing charity appearances and talking to the press about how much you love each other's company. If somebody asks about him, I want to hear you gush like he's your fucking prom date.\"\n\nShe slides him a page of bulleted lists and tables of data so elaborately organized he could have made it himself. It's labeled: HRH PRINCE HENRY FACT SHEET.\n\n\"You're going to memorize this so if anybody tries to catch you in a lie, you know what to say,\" she says. Under HOBBIES, it lists polo and competitive yachting. Alex is going to set himself on fire.\n\n\"Does he get one of these for me?\" Alex asks helplessly.\n\n\"Yep. And for the record, making it was one of the most depressing moments of my career.\" She slides another page over to him, this one detailing requirements for the weekend.\n\n> Minimum two (2) social media posts per day highlighting England\/visit thereof.\n> \n> One (1) on-air interview with _ITV This Morning,_ lasting five (5) minutes, in accordance with determined narrative.\n> \n> Two (2) joint appearances with photographers present: one (1) private meeting, one (1) public charity appearance.\n\n\"Why do I have to go over there? He's the one who pushed me into the stupid cake\u2014shouldn't he have to come here and go on _SNL_ with me or something?\"\n\n\"Because it was the _royal wedding_ you ruined, and _they're_ the ones out seventy-five grand,\" Zahra says. \"Besides, we're arranging his presence at a state dinner in a few months. He's not any more excited about this than you are.\"\n\nAlex pinches the bridge of his nose where a stress headache is already percolating. \"I have class.\"\n\n\"You'll be back by Sunday night, DC time,\" Zahra tells him. \"You won't miss anything.\"\n\n\"So there's really no way I'm getting out of this?\"\n\n\"Nope.\"\n\nAlex presses his lips together. He needs a list.\n\nWhen he was a kid, he used to hide pages and pages of loose leaf paper covered in messy, loopy handwriting under the worn denim cushion of the window seat in the house in Austin. Rambling treatises on the role of government in America with all the _G_ s written backward, paragraphs translated from English to Spanish, tables of his elementary school classmates' strengths and weaknesses. And lists. Lots of lists. The lists help.\n\nSo: Reasons this is a good idea.\n\nOne. His mother needs good press.\n\nTwo. Having a shitty record on foreign relations definitely won't help his career.\n\nThree. Free trip to Europe.\n\n\"Okay,\" he says, taking the file. \"I'll do it. But I won't have any fun.\"\n\n\"God, I hope not.\"\n\n* * *\n\nThe White House Trio is, officially, the nickname for Alex, June, and Nora coined by _People_ shortly before the inauguration. In actuality, it was carefully tested with focus groups by the White House press team and fed directly to _People._ Politics\u2014calculating, even in hashtags.\n\nBefore the Claremonts, the Kennedys and Clintons shielded the First Offspring from the press, giving them the privacy to go through awkward phases and organic childhood experiences and everything else. Sasha and Malia were hounded and picked apart by the press before they were out of high school. The White House Trio got ahead of the narrative before anyone could do the same.\n\nIt was a bold new plan: three attractive, bright, charismatic, marketable millennials\u2014Alex and Nora are, technically, just past the Gen Z threshold, but the press doesn't find that nearly as catchy. Catchiness sells, coolness sells. Obama was cool. The whole First Family could be cool too; celebrities in their own right. _It's not ideal,_ his mother always says, _but it works._\n\nThey're the White House Trio, but here, in the music room on the third floor of the Residence, they're just Alex and June and Nora, naturally glued together since they were teenagers stunting their growth with espresso in the primaries. Alex pushes them. June steadies them. Nora keeps them honest.\n\nThey settle into their usual places: June, perched on her heels at the record collection, foraging for some Patsy Cline; Nora, cross-legged on the floor, uncorking a bottle of red wine; Alex, sitting upside down with his feet on the back of the couch, trying to figure out what he's going to do next.\n\nHe flips the HRH PRINCE HENRY FACT SHEET over and squints at it. He can feel the blood rushing to his head.\n\nJune and Nora are ignoring him, caught in a bubble of intimacy he can never quite penetrate. Their relationship is something enormous and incomprehensible to most people, including Alex on occasion. He knows them both down to their split ends and nasty habits, but there's a strange girl bond between them he can't, and knows he isn't supposed to, translate.\n\n\"I thought you were liking the _Post_ gig?\" Nora says. With a dull pop, she pulls the cork out of the wine and takes a swig directly from the bottle.\n\n\"I was,\" June says. \"I mean, I _am._ But, it's not much of a gig. It's, like, one op-ed a month, and half my pitches get shot down for being too close to Mom's platform, and even then, the press team has to read anything political before I turn it in. So it's like, email in these fluff pieces, and know that on the other side of the screen people are doing the most important journalism of their careers, and be okay with that.\"\n\n\"So... you don't like it, then.\"\n\nJune sighs. She finds the record she's looking for, slides it out of the sleeve. \"I don't know what else to _do,_ is the thing.\"\n\n\"They wouldn't put you on a beat?\" Nora asks her.\n\n\"You kidding? They wouldn't even let me in the building,\" June says. She puts the record on and sets the needle. \"What would Reilly and Rebecca say?\"\n\nNora tips her head and laughs. \"My parents would say to do what they did: ditch journalism, get really into essential oils, buy a cabin in the Vermont wilderness, and own six hundred LL Bean vests that all smell like patchouli.\"\n\n\"You left out the investing in Apple in the nineties and getting stupid-rich part,\" June reminds her.\n\n\"Details.\"\n\nJune walks over and places her palm on the top of Nora's head, deep in her nest of curls, and leans down to kiss the back of her own fingers. \"I'll figure something out.\"\n\nNora hands over the bottle, and June takes a pull. Alex heaves a dramatic sigh.\n\n\"I can't believe I have to learn this garbage,\" Alex says. \"I _just_ finished midterms.\"\n\n\"Look, you're the one who has to fight everything that moves,\" June says, wiping her mouth on the back of her hand, a move she'd only do in front of the two of them. \"Including the British monarchy. So, I don't really feel bad for you. Anyway, he was totally fine when I danced with him. I don't get why you hate him so much.\"\n\n\"I think it's amazing,\" Nora says. \"Sworn enemies forced to make peace to settle tensions between their countries? There's something totally Shakespearean about it.\"\n\n\"Shakespearean in that hopefully I'll get stabbed to death,\" Alex says. \"This sheet says his favorite food is mutton pie. I literally cannot think of a more boring food. He's like a cardboard cutout of a person.\"\n\nThe sheet is filled with things Alex already knew, either from the royal siblings dominating the news cycle or hate-reading Henry's Wikipedia page. He knows about Henry's parentage, about his older siblings Philip and Beatrice, that he studied English literature at Oxford and plays classical piano. The rest is so trivial he can't imagine it'll come up in an interview, but there's no way he'll risk Henry being more prepared.\n\n\"Idea,\" Nora says. \"Let's make it a drinking game.\"\n\n\"Ooh, yes,\" June agrees. \"Drink every time Alex gets one right?\"\n\n\"Drink every time the answer makes you want to puke?\" Alex suggests.\n\n\"One drink for a correct answer, two drinks for a Prince Henry fact that is legitimately, objectively awful,\" Nora says. June has already dug two glasses out of the cabinet, and she hands them to Nora, who fills both and keeps the bottle for herself. Alex slides down from the couch to sit on the floor with her.\n\n\"Okay,\" she goes on, taking the sheet out of Alex's hands. \"Let's start easy. Parents. Go.\"\n\nAlex picks up his own glass, already pulling up a mental image of Henry's parents, Catherine's shrewd blue eyes and Arthur's movie-star jaw.\n\n\"Mother: Princess Catherine, oldest daughter of Queen Mary, first princess to obtain a doctorate\u2014English literature,\" he rattles off. \"Father: Arthur Fox, beloved English film and stage actor best known for his turn as James Bond in the eighties, deceased 2015. Y'all drink.\"\n\nThey do, and Nora passes the list to June.\n\n\"Okay,\" June says, scanning the list, apparently looking for something more challenging. \"Let's see. Dog's name?\"\n\n_\"David,\"_ Alex says. \"He's a beagle. I remember because, like, _who does that_? Who names a dog _David_? He sounds like a tax attorney. Like a dog tax attorney. Drink.\"\n\n\"Best friend's name, age, and occupation?\" Nora asks. \"Best friend other than _you,_ of course.\"\n\nAlex casually gives her the finger. \"Percy Okonjo. Goes by Pez or Pezza. Heir to Okonjo Industries, Nigerian company leading Africa in biomedical advancements. Twenty-two, lives in London, met Henry at Eton. Manages the Okonjo Foundation, a humanitarian nonprofit. Drink.\"\n\n\"Favorite book?\"\n\n\"Uh,\" Alex says. \"Um. Fuck. Uh. What's the one\u2014\"\n\n\"I'm sorry, Mr. Claremont-Diaz, that is incorrect,\" June says. \"Thank you for playing, but you lose.\"\n\n\"Come on, what's the answer?\"\n\nJune peers down at the list. \"This says... _Great Expectations_?\"\n\nBoth Nora and Alex groan.\n\n\"Do you see what I mean now?\" Alex says. \"This dude is reading Charles Dickens... _for pleasure._ \"\n\n\"I'll give you this one,\" Nora says. \"Two drinks!\"\n\n\"Well, I think\u2014\" June says as Nora glugs away. \"Guys, it's kinda nice! I mean, it's pretentious, but the themes of _Great Expectations_ are all like, love is more important than status, and doing what's right beats money and power. Maybe he relates\u2014\" Alex makes a long, loud fart noise. \"Y'all are such assholes! He seems really nice!\"\n\n\"That's because you are a nerd,\" Alex says. \"You want to protect those of your own species. It's a natural instinct.\"\n\n\"I am helping you with this out of the goodness of my heart,\" June says. \"I'm on _deadline_ right now.\"\n\n\"Hey, what do you think Zahra put on my fact sheet?\"\n\n\"Hmm,\" Nora says, sucking her teeth. \"Favorite summer Olympic sport: rhythmic gymnastics\u2014\"\n\n\"I'm not ashamed of that.\"\n\n\"Favorite brand of khakis: Gap.\"\n\n\"Listen, they look best on my ass. The J. Crew ones wrinkle all weird. And they're not _khakis,_ they're _chinos._ Khakis are for _white people_.\"\n\n\"Allergies: dust, Tide laundry detergent, and shutting the fuck up.\"\n\n\"Age of first filibuster: nine, at SeaWorld San Antonio, trying to force an orca wrangler into early retirement for, quote, 'inhumane whale practices.'\"\n\n\"I stood by it then, and I stand by it now.\"\n\nJune throws her head back and laughs, loud and unguarded, and Nora rolls her eyes, and Alex is glad, at least, that he'll have this to come back to when the nightmare is over.\n\n* * *\n\nAlex expects Henry's handler to be some stout storybook Englishman with tails and a top hat, probably a walrus mustache, definitely scurrying to place a velvet footstool at Henry's carriage door.\n\nThe person who awaits him and his security team on the tarmac is very much not that. He's a tall thirty-something Indian man in an impeccably tailored suit, roguishly handsome with a neatly trimmed beard, a steaming cup of tea, and a shiny Union Jack on his lapel. Well, okay then.\n\n\"Agent Chen,\" the man says, extending his free hand to Amy. \"Hope the flight was smooth.\"\n\nAmy nods. \"As smooth as the third transatlantic flight in a week can be.\"\n\nThe man half-smiles, commiserative. \"The Land Rover is for you and your team for the duration.\"\n\nAmy nods again, releasing his hand, and the man turns his attention to Alex.\n\n\"Mr. Claremont-Diaz,\" he says. \"Welcome back to England. Shaan Srivastava, Prince Henry's equerry.\"\n\nAlex takes his hand and shakes it, feeling a bit like he's in one of Henry's dad's Bond movies. Behind him, an attendant unloads his luggage and carries it off in the direction of a sleek Aston Martin.\n\n\"Nice to meet you, Shaan. Not exactly how we thought we'd be spending our weekend, is it?\"\n\n\"I'm not as surprised at this turn of events as I'd like to be, sir,\" Shaan says coolly, with an inscrutable smile.\n\nHe pulls a small tablet from his jacket and pivots on his heel toward the waiting car. Alex stares at his back, speechless, before hastily refusing to be impressed by a grown man whose job is handling the prince's schedule, no matter how cool he is or how long and smooth his strides are. He shakes his head a little and jogs to catch up, sliding into the back seat as Shaan checks the mirrors.\n\n\"Right,\" Shaan says. \"You'll be staying in the guest quarters at Kensington Palace. Tomorrow you'll do the _This Morning_ interview at nine\u2014we've arranged for a photo call at the studio. Then it's children with cancer all afternoon and off you go back to the land of the free.\"\n\n\"Okay,\" Alex says. He very politely does not add, _could be worse._\n\n\"For now,\" Shaan says, \"you're to come with me to chauffeur the prince from the stables. One of our photographers will be there to photograph the prince welcoming you to the country, so do try to look pleased to be here.\"\n\nOf course, there are _stables_ the prince needs to be _chauffeured_ from. He was briefly worried he'd been wrong about what the weekend would look like, but this feels a lot more like it.\n\n\"If you'll check the seat pocket in front of you,\" Shaan says as he reverses, \"there are a few papers for you to sign. Your lawyers have already approved them.\" He passes back an expensive-looking black fountain pen.\n\nNONDISCLOSURE AGREEMENT, the top of the first page reads. Alex flips through to the last page\u2014there are at least fifteen pages of text\u2014and a low whistle escapes his lips.\n\n\"This is...\" Alex says, \"a thing you do often?\"\n\n\"Standard protocol,\" Shaan says. \"The reputation of the royal family is too valuable to risk.\"\n\n> The words \"Confidential Information,\" as used in this Agreement, shall include the following:\n> \n> 1. Such information as HRH Prince Henry or any member of the Royal Family may designate to the Guest as \"Confidential Information\";\n> \n> 2. All proprietary and financial information regarding HRH Prince Henry's personal wealth and estate;\n> \n> 3. Any interior architectural details of Royal Residences including Buckingham Palace, Kensington Palace, etc., and personal effects found therein;\n> \n> 4. Any information regarding or involving HRH Prince Henry's personal or private life not previously released by official Royal documents, speeches, or approved biographers, including any personal or private relationship the Guest may have with HRH Prince Henry;\n> \n> 5. Any information found on HRH Prince Henry's personal electronic devices...\n\nThis seems... excessive, like the kind of paperwork you get from some perverted millionaire who wants to hunt you for sport. He wonders what the most mind-numbingly wholesome public figure on earth could possibly have to hide. He hopes it's not people-hunting.\n\nAlex is no stranger to NDAs, though, so he signs and initials. It's not like he would have divulged all the boring details of this trip to anyone anyway, except maybe June and Nora.\n\nThey pull up to the stables after another fifteen minutes, his security close behind them. The royal stables are, of course, elaborate and well-kept and about a million miles from the old ranches he's seen out in the Texas panhandle. Shaan leads him out to the edge of the paddock, and Amy and her team regroup ten paces behind.\n\nAlex rests his elbows on the lacquered white fence boards, fighting back the sudden, absurd feeling he's underdressed for this. On any other day, his chinos and button-down would be fine for a casual photo op, but for the first time in a long time, he's feeling distinctly out of his element. Does his hair look awful from the plane?\n\nIt's not like Henry is going to look much better after polo practice. He'll probably be sweaty and disgusting.\n\nAs if on cue, Henry comes galloping around the bend on the back of a pristine white horse.\n\nHe is definitely not sweaty or disgusting. He is, instead, bathed dramatically in a sweeping and resplendent sunset, wearing a crisp black jacket and riding pants tucked into tall leather boots, looking every inch an actual fairy-tale prince. He unhooks his helmet and takes it off with one gloved hand, and his hair underneath is just attractively tousled enough to look like it's supposed to be that way.\n\n\"I'm going to throw up on you,\" Alex says as soon as Henry is close enough to hear him.\n\n\"Hello, Alex,\" Henry says. Alex really resents the extra few inches of height Henry has on him right now. \"You look... sober.\"\n\n\"Only for you, Your Royal Highness,\" he says with an elaborate mock-bow. He's pleased to hear a little bit of ice in Henry's voice, finally done pretending.\n\n\"You're too kind,\" Henry says. He swings one long leg over and dismounts from his horse gracefully, removing his glove and extending a hand to Alex. A well-dressed stable hand basically springs up out of the ground to whisk the horse away by the reins. Alex has probably never hated anything more.\n\n\"This is idiotic,\" Alex says, grasping Henry's hand. The skin is soft, probably exfoliated and moisturized daily by some royal manicurist. There's a royal photographer right on the other side of the fence, so he smiles winningly and says through his teeth, \"Let's get it over with.\"\n\n\"I'd rather be waterboarded,\" Henry says, smiling back. The camera snaps nearby. His eyes are big and soft and blue, and he desperately needs to be punched in one of them. \"Your country could probably arrange that.\"\n\nAlex throws his head back and laughs handsomely, loud and false. \"Go fuck yourself.\"\n\n\"Hardly enough time,\" Henry says. He releases Alex's hand as Shaan returns.\n\n\"Your Highness,\" Shaan greets Henry with a nod. Alex makes a concentrated effort not to roll his eyes. \"The photographer should have what he needs, so if you're ready, the car is waiting.\"\n\nHenry turns to him and smiles again, eyes unreadable. \"Shall we?\"\n\n* * *\n\nThere's something vaguely familiar about the Kensington Palace guest quarters, even though he's never been here before.\n\nShaan had an attendant show him to his room, where his luggage awaited him on an ornately carved bed with spun gold bedding. Many of the rooms in the White House have a similar hauntedness, a sense of history that hangs like cobwebs no matter how pristine the rooms are kept. He's used to sleeping alongside ghosts, but that's not it.\n\nIt strikes further back in his memory, around the time his parents split up. They were the kind of married lawyer couple who could barely order Chinese takeout without legally binding documents, so Alex spent the summer before seventh grade shuttled back and forth from home to their dad's new place outside of Los Angeles until they could strike a long-term arrangement.\n\nIt was a nice house in the valley, a clear blue swimming pool and a back wall of solid glass. He never slept well there. He'd sneak out of his thrown-together bedroom in the middle of the night, stealing Helados from his dad's freezer and standing barefoot in the kitchen eating straight from the quart, washed blue in the pool light.\n\nThat's how it feels here, somehow\u2014wide awake at midnight in a strange place, duty-bound to make it work.\n\nHe wanders into the kitchen attached to his guest wing, where the ceilings are high and the countertops are shiny marble. He was allowed to submit a list to stock the kitchen, but apparently it was too hard to get Helados on short notice\u2014all that's in the freezer is UK-brand packaged ice cream cones.\n\n\"What's it like?\" Nora's voice says, tinny over his phone's speaker. On the screen, her hair is up, and she's poking at one of her dozens of window plants.\n\n\"Weird,\" Alex says, pushing his glasses up his nose. \"Everything looks like a museum. I don't think I'm allowed to show you, though.\"\n\n\"Ooh,\" Nora says, wiggling her eyebrows. \"So secretive. So fancy.\"\n\n\"Please,\" Alex says. \"If anything, it's creepy. I had to sign such a massive NDA that I'm convinced I'm gonna drop through a trapdoor into a torture dungeon any minute.\"\n\n\"I bet he has a secret lovechild,\" Nora says. \"Or he's gay. Or he has a secret gay lovechild.\"\n\n\"It's probably in case I see his equerry putting his batteries back in,\" Alex says. \"Anyway, this is boring. What's going on with you? Your life is so much better than mine right now.\"\n\n\"Well,\" Nora says, \"Nate Silver won't stop blowing up my phone for another column. Bought some new curtains. Narrowed down the list of grad school concentrations to statistics or data science.\"\n\n\"Tell me those are both at GW,\" Alex says, hopping up to sit on one of the immaculate countertops, feet dangling. \"You can't leave me in DC to go back to MIT.\"\n\n\"Haven't decided yet, but astonishingly, it will not be based on you,\" Nora tells him. \"Remember how we sometimes talk about things that are not about you?\"\n\n\"Yeah, weirdly. So is the plan to dethrone Nate Silver as reigning data czar of DC?\"\n\nNora laughs. \"No, what I'm gonna do is silently compile and process enough data to know exactly what's gonna happen for the next twenty-five years. Then I'm gonna buy a house on the top of a very tall hill at the edge of the city and become an eccentric recluse and sit on my veranda. Watch it all unfold through a pair of binoculars.\"\n\nAlex starts to laugh, but cuts off when he hears rustling down the hall. Quiet footsteps approaching. Princess Beatrice lives in a different section of the palace, and so does Henry. The PPOs and his own security sleep on this floor, though, so maybe\u2014\n\n\"Hold on,\" Alex says, covering the speaker.\n\nA light flicks on in the hallway, and the person who comes padding into the kitchen is none other than Prince Henry.\n\nHe's rumpled and half awake, shoulders slumping as he yawns. He's standing in front of Alex wearing not a suit, but a heather-gray T-shirt and plaid pajama bottoms. He has earbuds in, and his hair is a mess. His feet are bare.\n\nHe looks, alarmingly, human.\n\nHe freezes when his eyes fall on Alex perched on the countertop. Alex stares back at him. In his hand, Nora begins a muffled, \"Is that\u2014\" before Alex disconnects the call.\n\nHenry pulls out his earbuds, and his posture has ratcheted back up straight, but his face is still bleary and confused.\n\n\"Hello,\" he says, hoarse. \"Sorry. Er. I was just. Cornettos.\"\n\nHe gestures vaguely toward the refrigerator, as if he's said something of any meaning.\n\n\"What?\"\n\nHe crosses to the freezer and extracts the box of ice cream cones, showing Alex the name _Cornetto_ across the front. \"I was out. Knew they'd stocked you up.\"\n\n\"Do you raid the kitchens of all your guests?\" Alex asks.\n\n\"Only when I can't sleep,\" Henry says. \"Which is always. Didn't think you'd be awake.\" He looks at Alex, deferring, and Alex realizes he's waiting for permission to open the box and take one. Alex thinks about telling him no, just for the thrill of denying a prince something, but he's kind of intrigued. He usually can't sleep either. He nods.\n\nHe waits for Henry to take a Cornetto and leave, but instead he looks back up at Alex.\n\n\"Have you practiced what you'll say tomorrow?\"\n\n\"Yes,\" Alex says, bristling immediately. This is why nothing about Henry has ever intrigued him before. \"You're not the only professional here.\"\n\n\"I didn't mean\u2014\" Henry falters. \"I only meant, do you think we should, er, rehearse?\"\n\n\"Do you need to?\"\n\n\"I thought it might help.\" Of course, he thinks that. Everything Henry's ever done publicly has probably been privately rehearsed in stuffy royal quarters like this one.\n\nAlex hops down off the counter, swiping his phone unlocked. \"Watch this.\"\n\nHe lines up a shot: the box of Cornettos on the counter, Henry's hand braced on the marble next to it, his heavy signet ring visible along with a swath of pajamas. He opens up Instagram, slaps a filter on it.\n\n\"'Nothing cures jet lag,'\" Alex narrates in a monotone as he taps out a caption, \"'like midnight ice cream with @PrinceHenry.' Geotag Kensington Palace, and posted.\" He holds the phone for Henry to see as likes and comments immediately pour in. \"There are a lot of things worth overthinking, believe me. But this isn't one of them.\"\n\nHenry frowns at him over his ice cream.\n\n\"I suppose,\" he says, looking doubtful.\n\n\"Are you done?\" Alex asks. \"I was on a call.\"\n\nHenry blinks, then folds his arms over his chest, back on the defensive. \"Of course. I won't keep you.\"\n\nAs he leaves the kitchen, he pauses in the doorframe, considering.\n\n\"I didn't know you wore glasses,\" he says finally.\n\nHe leaves Alex standing there alone in the kitchen, the box of Cornettos sweating on the counter.\n\n* * *\n\nThe ride to the studio for the interview is bumpy but mercifully quick. Alex should probably blame some of his queasiness on nerves but chooses to blame it all on this morning's appalling breakfast spread\u2014what kind of garbage country eats bland beans on white toast for breakfast? He can't decide if his Mexican blood or his Texan blood is more offended.\n\nHenry sits beside him, surrounded by a cloud of attendants and stylists. One adjusts his hair with a fine-toothed comb. One holds up a notepad of talking points. One tugs his collar straight. From the passenger seat, Shaan shakes a yellow pill out of a bottle and passes it back to Henry, who readily pops it into his mouth and swallows it dry. Alex decides he doesn't want or need to know.\n\nThe motorcade pulls up in front of the studio, and when the door slides open, there's the promised photo line and barricaded royal worshippers. Henry turns and looks at him, a little grimace around his mouth and eyes.\n\n\"Prince goes first, then you,\" Shaan says to Alex, leaning in and touching his earpiece. Alex takes one breath, two, and turns it on\u2014the megawatt smile, the All-American charm.\n\n\"Go ahead, Your Royal Highness,\" Alex says, winking as he puts on his sunglasses. \"Your subjects await.\"\n\nHenry clears his throat and unfolds himself, stepping out into the morning and waving genially at the crowd. Cameras flash, photographers shout. A blue-haired girl in the crowd lifts up a homemade poster that reads in big, glittery letters, GET IN ME, PRINCE HENRY! for about five seconds until a member of the security team shoves it into a nearby trash can.\n\nAlex steps out next, swaggering up beside Henry and throwing an arm over his shoulders.\n\n\"Act like you like me!\" Alex says cheerfully. Henry looks at him like he's trying to choose between a million choice words, before tipping his head to the side and offering up a well-rehearsed laugh, putting his arm around Alex too. \"There we go.\"\n\nThe hosts of _This Morning_ are agonizingly British\u2014a middle-aged woman named Dottie in a tea dress and a man called Stu who looks as if he spends weekends yelling at mice in his garden. Alex watches the introductions backstage as a makeup artist conceals a stress pimple on his forehead. _So, this is happening_. He tries to ignore Henry a few feet to his left, currently getting a final preening from a royal stylist. It's the last chance he'll get to ignore Henry for the rest of the day.\n\nSoon Henry is leading the way out with Alex close behind. Alex shakes Dottie's hand first, smiling his Politics Smile at her, the one that makes a lot of congresswomen and more than a few congressmen want to tell him things they shouldn't. She giggles and kisses him on the cheek. The audience claps and claps and claps.\n\nHenry sits on the prop couch next to him, perfect posture, and Alex smiles at him, making a show of looking comfortable in Henry's company. Which is harder than it should be, because the stage lights suddenly make him uncomfortably aware of how fresh and handsome Henry looks for the cameras. He's wearing a blue sweater over a button-down, and his hair looks soft.\n\nWhatever, fine. Henry is annoyingly attractive. That's always been a thing, objectively. It's fine.\n\nHe realizes, almost a second too late, that Dottie is asking him a question.\n\n\"What do you think of _jolly old England,_ then, Alex?\" Dottie says, clearly ribbing him. Alex forces a smile.\n\n\"You know, Dottie, it's gorgeous,\" Alex says. \"I've been here a few times since my mom got elected, and it's always incredible to see the history here, and the beer selection.\" The audience laughs right on cue, and Alex shakes out his shoulders a little. \"And of course, it's always great to see this guy.\"\n\nHe turns to Henry, extending his fist. Henry hesitates before stiffly bumping his own knuckles against Alex's with the heavy air of an act of treason.\n\n* * *\n\nAlex's whole reason for wanting to go into politics, when he knows so many past presidential sons and daughters have run away screaming the minute they turned eighteen, is he genuinely cares about people.\n\nThe power is great, the attention fun, but the people\u2014the people are everything. He has a bit of a caring-too-much problem about most things, including whether people can pay their medical bills, or marry whomever they love, or not get shot at school. Or, in this case, if kids with cancer have enough books to read at the Royal Marsden NHS Foundation Trust.\n\nHe and Henry and their collective hoard of security have taken over the floor, flustering nurses and shaking hands. He's trying\u2014really trying\u2014not to let his hands clench into fists at his sides, but Henry's smiling robotically with a little bald boy plugged full of tubes for some bullshit photograph, and he wants to scream at this whole stupid country.\n\nBut he's legally required to be here, so he focuses on the kids, instead. Most of them have no idea who he is, but Henry gamely introduces him as the president's son, and soon they're asking him about the White House and does he know Ariana Grande, and he laughs and indulges them. He unpacks books from the heavy boxes they've brought, climbs up onto beds and reads out loud, a photographer trailing after him.\n\nHe doesn't realize he's lost track of Henry until the patient he's visiting dozes off, and he recognizes the low rumble of Henry's voice on the other side of the curtain.\n\nA quick count of feet on the floor\u2014no photographers. Just Henry. Hmm.\n\nHe steps quietly over to the chair against the wall, right at the edge of the curtain. If he sits at the right angle and cranes his head back, he can barely see.\n\nHenry is talking to a little girl with leukemia named Claudette, according to the board on her wall. She's got dark skin that's turned sort of a pale gray and a bright orange scarf tied around her head, emblazoned with the Alliance Starbird.\n\nInstead of hovering awkwardly like Alex expected, Henry is squatting at her side, smiling and holding her hand.\n\n\"... Star Wars fan, are you?\" Henry says in a low, warm voice Alex has never heard from him before, pointing at the insignia on her headscarf.\n\n\"Oh, it's my absolute favorite,\" Claudette gushes. \"I'd like to be just like Princess Leia when I'm older because she's so tough and smart and strong, and she gets to kiss Han Solo.\"\n\nShe blushes a little at having mentioned kissing in front of the prince but fiercely maintains eye contact. Alex finds himself craning his neck farther, watching for Henry's reaction. He definitely does not recall Star Wars on the fact sheet.\n\n\"You know what,\" Henry says, leaning in conspiratorially, \"I think you've got the right idea.\"\n\nClaudette giggles. \"Who's your favorite?\"\n\n\"Hmm,\" Henry says, making a show of thinking hard. \"I always liked Luke. He's brave and good, and he's the strongest Jedi of them all. I think Luke is proof that it doesn't matter where you come from or who your family is\u2014you can always be great if you're true to yourself.\"\n\n\"All right, Miss Claudette,\" a nurse says brightly as she comes around the curtain. Henry jumps, and Alex almost tips his chair over, caught in the act. He clears his throat as he stands, pointedly not looking at Henry. \"You two can go, it's time for her meds.\"\n\n\"Miss Beth, Henry said we were mates now!\" Claudette practically wails. \"He can stay!\"\n\n\"Excuse you!\" Beth the nurse tuts. \"That's no way to address the prince. Terribly sorry, Your Highness.\"\n\n\"No need to apologize,\" Henry tells her. \"Rebel commanders outrank royalty.\" He shoots Claudette a wink and a salute, and she positively melts.\n\n\"I'm impressed,\" Alex says as they walk out into the hallway together. Henry cocks an eyebrow, and Alex adds, \"Not impressed, just surprised.\"\n\n\"At what?\"\n\n\"That you actually have, you know, feelings.\"\n\nHenry is beginning to smile when three things happen in rapid succession.\n\nThe first: A shout echoes from the opposite end of the hall.\n\nThe second: There's a loud pop that sounds alarmingly like gunfire.\n\nThe third: Cash grabs both Henry and Alex by the arms and shoves them through the nearest door.\n\n_\"Stay down,\"_ Cash grunts as he slams the door behind them.\n\nIn the abrupt darkness, Alex stumbles over a mop and one of Henry's legs, and they go crashing down together into a clattering pile of tin bedpans. Henry hits the floor first, facedown, and Alex lands in a heap on top of him.\n\n\"Oh God,\" Henry says, muffled and echoing slightly. Alex thinks hopefully that his face might be in a bedpan.\n\n\"You know,\" he says into Henry's hair, \"we have got to stop ending up like this.\"\n\n\"Do you _mind_?\"\n\n\"This is _your_ fault!\"\n\n\"How is this _possibly_ my fault?\" Henry hisses.\n\n\"Nobody ever tries to shoot me when I'm doing presidential appearances, but the minute I go out with a fucking royal\u2014\"\n\n\"Will you shut up before you get us both killed?\"\n\n\"Nobody's going to kill us. Cash is blocking the door. Besides, it's probably nothing.\"\n\n\"Then at least _get off me._ \"\n\n\"Stop telling me what to do! You're not the prince of me!\"\n\n\"Bloody hell,\" Henry mutters, and he pushes hard off the ground and rolls, knocking Alex onto the floor. Alex finds himself wedged between Henry's side and a shelf of what smells like industrial-strength floor cleaner.\n\n\"Can you move over, Your Highness?\" Alex whispers, shoving his shoulder against Henry's. \"I'd rather not be the little spoon.\"\n\n\"Believe me, I'm trying,\" Henry replies. \"There's no room.\"\n\nOutside, there are voices, hurried footsteps\u2014no signs of an all-clear.\n\n\"Well,\" Alex says. \"Guess we better make ourselves comfortable.\"\n\nHenry exhales tightly. \"Fantastic.\"\n\nAlex feels him shifting against his side, arms crossed over his chest in an attempt at his typical closed-off stance while lying on the floor with his feet in a mop bucket.\n\n\"For the record,\" Henry says, \"nobody's ever made an attempt on my life either.\"\n\n\"Well, congratulations,\" Alex says. \"You've officially made it.\"\n\n\"Yes, this is exactly how I always dreamed it would be. Locked in a cupboard with your elbow inside my rib cage,\" Henry snipes. He sounds like he wants to punch Alex, which is probably the most Alex has ever liked him, so he follows an impulse and drives his elbow into Henry's side, hard.\n\nHenry lets out a muffled yelp, and the next thing Alex knows, he's been yanked sideways by his shirt and Henry is halfway on top of him, pinning him down with one thigh. His head throbs where he's clocked it against the linoleum floor, but he can feel his lips split into a smile.\n\n\"So you _do_ have some fight in you,\" Alex says. He bucks his hips, trying to shake Henry off, but he's taller and stronger and has a fistful of Alex's collar.\n\n\"Are you _quite_ finished?\" Henry says, sounding strangled. \"Can you perhaps stop putting your sodding life in danger now?\"\n\n\"Aw, you do care,\" Alex says. \"I'm learning all your hidden depths today, sweetheart.\"\n\nHenry exhales and slumps off him. \"I cannot believe even mortal peril will not prevent you from being the way you are.\"\n\nThe weirdest part, Alex thinks, is that what he said was true.\n\nHe keeps getting these little glimpses into things he never thought Henry was. A bit of a fighter, for one. Intelligent, interested in other people. It's honestly disconcerting. He knows exactly what to say to each Democratic senator to make them dish about bills, exactly when Zahra's running low on nicotine gum, exactly which look to give Nora for the rumor mill. Reading people is what he does.\n\nHe really doesn't appreciate some inbred royal baby upending his system. But he did rather enjoy that fight.\n\nHe lies there, waits. Listens to the shuffling of feet outside the door. Lets minutes go by.\n\n\"So, uh,\" he tries. \"Star Wars?\"\n\nHe means it in a nonthreatening, offhanded way, but habit wins and it comes out accusatory.\n\n\"Yes, Alex,\" Henry says archly, \"believe it or not, the children of the crown don't only spend their childhood going to tea parties.\"\n\n\"I assumed it was mostly posture coaching and junior polo league.\"\n\nHenry takes a deeply unhappy pause. \"That... may have been part of it.\"\n\n\"So you're into pop culture, but you act like you're not,\" Alex says. \"Either you're not allowed to talk about it because it's unseemly for the crown, or you choose not to talk about it because you want people to think you're _cultured._ Which one?\"\n\n\"Are you psychoanalyzing me?\" Henry asks. \"I don't think royal guests are allowed to do that.\"\n\n\"I'm trying to understand why you're so committed to acting like someone you're not, considering you just told that little girl in there that greatness means being true to yourself.\"\n\n\"I don't know what you're talking about, and if I did, I'm not sure that's any of your concern,\" Henry says, his voice strained at the edges.\n\n\"Really? Because I'm pretty sure I'm legally bound to pretend to be your best friend, and I don't know if you've thought this through yet, but that's not going to stop with this weekend,\" Alex tells him. Henry's fingers go tense against his forearm. \"If we do this and we're never seen together again, people are gonna know we're full of shit. We're stuck with each other, like it or not, so I have a right to be clued in about what your deal is before it sneaks up on me and bites me in the ass.\"\n\n\"Why don't we start...\" Henry says, turning his head to squint at him. This close Alex can just make out the silhouette of Henry's strong royal nose. \"... with you telling me why exactly you hate me so much?\"\n\n\"Do you really want to have that conversation?\"\n\n\"Maybe I do.\"\n\nAlex crosses his arms, recognizes it as a mirror to Henry's tic, and uncrosses them.\n\n\"Do you really not remember being a prick to me at the Olympics?\"\n\nAlex remembers it in vivid detail: himself at eighteen, dispatched to Rio with June and Nora, the campaign's delegation to the summer games, one weekend of photo ops and selling the \"next generation of global cooperation\" image. Alex spent most of it drinking caipirinhas and subsequently throwing caipirinhas up behind Olympic venues. And he remembers, down to the Union Jack on Henry's anorak, the first time they met.\n\nHenry sighs. \"Is that the time you threatened to push me into the Thames?\"\n\n_\"No,\"_ Alex says. \"It was the time you were a _condescending prick_ at the diving finals. You really don't remember?\"\n\n\"Remind me?\"\n\nAlex glares. \"I walked up to you to introduce myself, and you stared at me like I was the most offensive thing you had ever seen. Right after you shook my hand, you turned to Shaan and said, 'Can you get rid of him?'\"\n\nA pause.\n\n\"Ah,\" Henry says. He clears his throat. \"I didn't realize you'd heard that.\"\n\n\"I feel like you're missing the point,\" Alex says, \"which is that it's a douchey thing to say either way.\"\n\n\"That's... fair.\"\n\n\"Yeah, so.\"\n\n\"That's all?\" Henry asks. \"Only the Olympics?\"\n\n\"I mean, that was the start.\"\n\nHenry pauses again. \"I'm sensing an ellipsis.\"\n\n\"It's just...\" Alex says, and as he's on the floor of a supply closet, waiting out a security threat with a Prince of England at the end of a weekend that has felt like some very specific ongoing nightmare, censoring himself takes too much effort. \"I don't know. Doing what we do is fucking hard. But it's harder for me. I'm the son of the first female president. And I'm not white like she is, can't even pass for it. People will _always_ come down harder on me. And you're, you know, _you,_ and you were born into all of this, and everyone thinks you're Prince fucking Charming. You're basically a living reminder I'll always be compared to someone else, no matter what I do, even if I work twice as hard.\"\n\nHenry is quiet for a long while.\n\n\"Well,\" Henry says when he speaks at last. \"I can't very well do much about the rest. But I can tell you I was, in fact, a prick that day. Not that it's any excuse, but my father had died fourteen months before, and I was still kind of a prick every day of my life at the time. And I am sorry.\"\n\nHenry twitches one hand at his side, and Alex falls momentarily silent.\n\nThe cancer ward. Of course, Henry chose a cancer ward\u2014it was right there on the fact sheet. _Father: Famed film star Arthur Fox, deceased 2015, pancreatic cancer._ The funeral was televised. He goes back over the last twenty-four hours in his head: the sleeplessness, the pills, the tense little grimace Henry does in public that Alex has always read as aloofness.\n\nHe knows a few things about this stuff. It's not like his parents' divorce was a pleasant time for him, or like he runs himself ragged about grades for fun. He's been aware for too long that most people don't navigate thoughts of whether they'll ever be good enough or if they're disappointing the entire world. He's never considered Henry might feel any of the same things.\n\nHenry clears his throat again, and something like panic catches Alex. He opens his mouth and says, \"Well, good to know you're not perfect.\"\n\nHe can almost hear Henry roll his eyes, and he's thankful for it, the familiar comfort of antagonism.\n\nThey're silent again, the dust of the conversation settling. Alex can't hear anything outside the door or any sirens on the street, but nobody has come to get them yet.\n\nThen, unprompted, Henry says into the stretching stillness, \" _Return of the Jedi._ \"\n\nA beat. \"What?\"\n\n\"To answer your question,\" Henry says. \"Yes, I do like Star Wars, and my favorite is _Return of the Jedi._ \"\n\n\"Oh,\" Alex says. \"Wow, you're wrong.\"\n\nHenry huffs out the tiniest, most poshly indignant puff of air. It smells minty. Alex resists the urge to throw another elbow. \"How can I be wrong about my own favorite? It's a personal truth.\"\n\n\"It's a personal truth that is wrong and bad.\"\n\n\"Which do you prefer, then? Please show me the error of my ways.\"\n\n\"Okay, _Empire._ \"\n\nHenry sniffs. \"So _dark,_ though.\"\n\n\"Yeah, which is what makes it _good,_ \" Alex says. \"It's the most thematically complex. It's got the Han and Leia kiss in it, you meet Yoda, Han is at the top of his game, fucking _Lando Calrissian,_ and _the_ best twist in cinematic history. What does _Jedi_ have? Fuckin' Ewoks.\"\n\n\"Ewoks are _iconic._ \"\n\n\"Ewoks are _stupid._ \"\n\n\"But _Endor._ \"\n\n\"But _Hoth._ There's a reason people always call the best, grittiest installment of a trilogy the _Empire_ of the series.\"\n\n\"And I can appreciate that. But isn't there something to be valued in a happy ending as well?\"\n\n\"Spoken like a true Prince Charming.\"\n\n\"I'm only saying, I like the resolution of _Jedi._ It ties everything up nicely. And the overall theme you're intended to take away from the films is hope and love and... er, you know, all that. Which is what _Jedi_ leaves you with a sense of most of all.\"\n\nHenry coughs, and Alex is turning to look at him again when the door opens and Cash's giant silhouette reappears.\n\n\"False alarm,\" he says, breathing heavily. \"Some dumbass kids brought fireworks for their friend.\" He looks down at them, flat on their backs and blinking up in the sudden, harsh light of the hallway. \"This looks cozy.\"\n\n\"Yep, we're really bonding,\" Alex says. He reaches a hand out and lets Cash haul him to his feet.\n\n* * *\n\nOutside Kensington Palace, Alex takes Henry's phone out of his hand and swiftly opens a blank contact page before he can protest or sic a PPO on him for violating royal property. The car is waiting to take him back to the royals' private airstrip.\n\n\"Here,\" Alex says. \"That's my number. If we're gonna keep this up, it's going to get annoying to keep going through handlers. Just text me. We'll figure it out.\"\n\nHenry stares at him, expression blankly bewildered, and Alex wonders how this guy has any friends.\n\n\"Right,\" Henry says finally. \"Thank you.\"\n\n\"No booty calls,\" Alex tells him, and Henry chokes on a laugh.\n\n# THREE\n\n> FROM AMERICA, WITH LOVE: Henry and Alex Flaunt Friendship\n> \n> NEW BROMANCE ALERT? Pics of FSOTUS and Prince Henry\n> \n> PHOTOS: Alex's Weekend in London\n\nFor the first time in a week, Alex isn't pissed off scrolling through his Google alerts. It helps they've given _People_ an exclusive\u2014a few generic quotes about how much Alex \"cherishes\" his friendship with Henry and their \"shared life experience\" as sons of world leaders. Alex thinks their main shared life experience is probably wishing they could set that quote adrift on the ocean between them and watch it drown.\n\nHis mother doesn't want him fake-dead anymore, though, and he's stopped getting a thousand vitriolic tweets an hour, so he counts it as a win.\n\nHe dodges a starstruck freshman gawking at him and exits the hall onto the east side of campus, draining the last cold sip of his coffee. First class today was an elective he's taking out of a combination of morbid fascination and academic curiosity: The Press and the Presidency. He's currently jet-lagged to all hell from trying to keep the press from _ruining_ the presidency, and the irony isn't lost on him.\n\nToday's lecture was on presidential sex scandals through history, and he texts Nora: numbers on one of us getting involved in a sex scandal before the end of second term?\n\nHer response comes within seconds: 94% probability of your dick becoming a recurring personality on face the nation. btw, have you seen this?\n\nThere's a link attached: a blog post full of images, animated GIFs of himself and Henry on _This Morning._ The fist bump. Shared smiles that pass for genuine. Conspiratorial glances. Underneath are hundreds of comments about how handsome they are, how nice they look together.\n\nomfg, one commenter writes, make out already.\n\nAlex laughs so hard he almost falls in a fountain.\n\n* * *\n\nAs usual, the day guard at the Dirksen Building glares at him as he slides through security. She's certain he was the one who vandalized the sign outside one particular senator's office to read BITCH MCCONNELL, but she'll never prove it.\n\nCash tags along for some of Alex's Senate recon missions so nobody panics when he disappears for a few hours. Today, Cash hangs back on a bench, catching up on his podcasts. He's always been the most indulgent of Alex's antics.\n\nAlex has had the layout of the building memorized since his dad first got elected to the Senate. It's where he's picked up his encyclopedic knowledge of policy and procedure, and where he spends more afternoons than he's supposed to, charming aides and trawling for gossip. His mom pretends to be annoyed but slyly asks for intel later.\n\nSince Senator Oscar Diaz is in California speaking at a rally for gun control today, Alex punches the button for the fifth floor instead.\n\nHis favorite senator is Rafael Luna, an Independent from Colorado and the newest kid on the block at only thirty-nine. Alex's dad took him under his wing back when he was merely a promising attorney, and now he's the darling of national politics for (A) winning a special election and a general in consecutive upsets for his Senate seat, and (B) dominating _The Hill_ 's 50 Most Beautiful.\n\nAlex spent summer 2018 in Denver on Luna's campaign, so they have their own dysfunctional relationship built on tropical-flavored Skittles from gas stations and all-nighters drafting press releases. He sometimes feels the ghost of carpal tunnel creeping back, a fond ache.\n\nHe finds Luna in his office, horn-rimmed reading glasses doing nothing to detract from his usual appearance of a movie star who tripped and fell sideways into politics. Alex has always suspected the soulful brown eyes and perfectly groomed stubble and dramatic cheekbones won back any votes Luna lost by being both Latino and openly gay.\n\nThe album playing low in the room is an old favorite Alex remembers from Denver: Muddy Waters. When Luna looks up and sees Alex in his doorway, he drops his pen on a haphazard pile of papers and leans back in his chair.\n\n\"Fuck you doing here, kid?\" he says, watching him like a cat.\n\nAlex reaches into his pocket and pulls out a packet of Skittles, and Luna's face immediately softens into a smile.\n\n\"Atta boy,\" he says, scooping the bag up as soon as Alex drops it on his blotter. He kicks the chair in front of the desk out for him.\n\nAlex sits, watching Luna rip open the packet with his teeth. \"Whatcha working on today?\"\n\n\"You already know more than you're supposed to about everything on this desk.\" Alex does know\u2014the same health care reform as last year, the one stalled out since they lost the Senate in midterms. \"Why are you really here?\"\n\n\"Hmm.\" Alex hooks a leg over one armrest of the chair. \"I resent the idea I can't come visit a dear family friend without ulterior motives.\"\n\n\"Bullshit.\"\n\nHe clutches his chest. \"You _wound_ me.\"\n\n\"You exhaust me.\"\n\n\"I enchant you.\"\n\n\"I'll call security.\"\n\n\"Fair enough.\"\n\n\"Instead, let's talk about your little European vacation,\" Luna says. He fixes Alex with shrewd eyes. \"Can I expect a joint Christmas present from you and the prince this year?\"\n\n\"Actually,\" Alex swerves, \"since I'm here, I do have a question for you.\"\n\nLuna laughs, leaning back and lacing his hands together behind his head. Alex feels his face flash hot for half a second, a zip of good-banter adrenaline that means he's getting somewhere. \"Of course you do.\"\n\n\"I wondered if you had heard anything about Connor,\" Alex asks. \"We could really use an endorsement from another Independent senator. Do you think he's close to making one?\"\n\nHe kicks his foot innocently where it's dangling over the armrest, like he's asking something as innocuous as the weather. Stanley Connor, Delaware's kooky and beloved old Independent with a social media team stacked with millennials, would be a big get down the line in a race projected to be this close, and they both know it.\n\nLuna sucks on a Skittle. \"Are you asking if he's close to endorsing, or if I know what strings need to be pulled to get him to endorse?\"\n\n\"Raf. Pal. Buddy. You know I'd never ask you anything so unseemly.\"\n\nLuna sighs, swivels in his chair. \"He's a free agent. Social issues would push him your way usually, but you know how he feels about your mom's economic platform. You probably know his voting record better than I do, kid. He doesn't fall on one side of the aisle. He might go for something radically different on taxes.\"\n\n\"And as for something you know that I don't?\"\n\nHe smirks. \"I know Richards is promising Independents a centrist platform with big shake-ups on non-social issues. And I know part of that platform might not line up with Connor's position on healthcare. Somewhere to start, perhaps. Hypothetically, if I were going to engage with your scheming.\"\n\n\"And you don't think there's any point in chasing down leads on Republican candidates who aren't Richards?\"\n\n\"Shit,\" Luna says, the set of his mouth turning grim. \"Chances of your mother facing off against a candidate who's not the fucking anointed messiah of right-wing populism and heir to the Richards family legacy? Highly fucking unlikely.\"\n\nAlex smiles. \"You complete me, Raf.\"\n\nLuna rolls his eyes again. \"Let's circle back to you,\" he says. \"Don't think I didn't notice you changing the subject. For the record, I won the office pool on how long it'd take you to cause an international incident.\"\n\n\" _Wow,_ I thought I could _trust_ you.\" Alex gasps, mock-betrayed.\n\n\"What's the deal there?\"\n\n\"There's no _deal,_ \" Alex says. \"Henry is... a person I know. And we did something stupid. I had to fix it. It's fine.\"\n\n\"Okay, okay,\" Luna says, holding up both hands. \"He's a looker, huh?\"\n\nAlex pulls a face. \"Yeah, I mean, if you're into, like, fairy-tale princes.\"\n\n\"Is anyone not?\"\n\n\" _I'm_ not,\" Alex says.\n\nLuna arches an eyebrow. \"Right.\"\n\n\"What?\"\n\n\"Just thinking about last summer,\" he says. \"I have this really vivid memory of you basically making a Prince Henry voodoo doll on your desk.\"\n\n\"I did not.\"\n\n\"Or was it a dartboard with a photo of his face on it?\"\n\nAlex swings his foot back over the armrest so he can plant both feet on the floor and fold his arms indignantly. \"I had a magazine with his face on it at my desk, once, because I was in it and he happened to be on the cover.\"\n\n\"You stared at it for an hour.\"\n\n\"Lies,\" Alex says. \"Slander.\"\n\n\"It was like you were trying to set him on fire with your mind.\"\n\n\"What is your point?\"\n\n\"I think it's interesting,\" he says. \"How fast the times they are a-changin'.\"\n\n\"Come on,\" Alex says. \"It's... politics.\"\n\n\"Uh-huh.\"\n\nAlex shakes his head, doglike, as if it's going to disperse the topic from the room. \"Besides, I came here to talk about endorsements, not my embarrassing public relations nightmares.\"\n\n\"Ah,\" Luna says slyly, \"but I thought you were here to pay a family friend a visit?\"\n\n\"Of course. That's what I meant.\"\n\n\"Alex, don't you have something else to do on a Friday afternoon? You're twenty-one. You should be playing beer pong or getting ready for a party or something.\"\n\n\"I do all of those things,\" he lies. \"I just also do this.\"\n\n\"Come on. I'm trying to give you some advice, from one old man to a much younger version of himself.\"\n\n\"You're thirty-nine.\"\n\n\"My liver is ninety-three.\"\n\n\"That's not my fault.\"\n\n\"Some late nights in Denver would beg to differ.\"\n\nAlex laughs. \"See, this is why we're friends.\"\n\n\"Alex, you need other friends,\" Luna tells him. \"Friends who _aren't in Congress._ \"\n\n\"I have friends! I have June and Nora.\"\n\n\"Yes, your sister and a girl who is also a supercomputer,\" Luna deadpans. \"You need to take some time for yourself before you burn out, kid. You need a bigger support system.\"\n\n\"Stop calling me 'kid,'\" Alex says.\n\n\"Ay.\" Luna sighs. \"Are you done? I do have some actual work to do.\"\n\n\"Yeah, yeah,\" Alex says, gathering himself up from his chair. \"Hey, is Maxine in town?\"\n\n\"Waters?\" Luna asks, crooking his head. \"Shit, you really have a death wish, huh?\"\n\n* * *\n\nAs political legacies go, the Richards family is one of the most complex bits of history Alex has tried to unravel.\n\nOn one of the Post-it notes stuck to his laptop he's written: KENNEDYS + BUSHES + BIZARRO MAFIA OLD MONEY SITH POWERS = RICHARDSES? It's pretty much the thesis of what he's dug up so far. Jeffrey Richards, the current and supposedly only frontrunner to be his mother's opponent in the general, has been a senator for Utah nearly twenty years, which means plenty of voting history and legislation that his mother's team has already gone over. Alex is more interested in the things harder to sniff out. There are so many generations of Attorney General Richards and Federal Judge Richards, they'd be able to bury anything.\n\nHis phone buzzes under a stack of files on his desk. A text from June: Dinner? I miss your face. He loves June\u2014truly, more than anything in the world\u2014but he's kind of in the zone. He'll respond when he hits a stopping point in like thirty minutes.\n\nHe glances at the video of a Richards interview pulled up in a tab, checking the man's face for nonverbal cues. Gray hair\u2014natural, not a piece. Shiny white teeth, like a shark's. Heavy Uncle Sam jaw. Great salesman, considering he's blatantly lying about a bill in the clip. Alex takes a note.\n\nIt's an hour and a half later before another buzz pulls him out of a deep dive into Richards's uncle's suspicious 1986 taxes. A text from his mother in the family group chat, a pizza emoji. He bookmarks his page and heads upstairs.\n\nFamily dinners are rare but less over-the-top than everything else that happens in the White House. His mother sends someone to pick up pizzas, and they take over the game room on the third floor with paper plates and bottles of Shiner shipped in from Texas. It's always amusing to catch one of the burly suits speaking in code over their earpieces: \"Black Bear has requested extra banana peppers.\"\n\nJune's already on the chaise and sipping a beer. A stab of guilt immediately hits when he remembers her text.\n\n\"Shit, I'm an asshole,\" he says.\n\n\"Mm-hmm, you are.\"\n\n\"But, technically... I am having dinner with you?\"\n\n\"Just bring me my pizza,\" she says with a sigh. After Secret Service misread an olive-based shouting match in 2017 and almost put the Residence on lockdown, they now each get their own pizzas.\n\n\"Sure thing, Bug.\" He finds June's\u2014margherita\u2014and his\u2014pepperoni and mushroom.\n\n\"Hi, Alex,\" says a voice from somewhere behind the television as he settles in with his pizza.\n\n\"Hey, Leo,\" he answers. His stepdad is fiddling with the wiring, probably rewiring it to do something that'd make more sense in an _Iron Man_ comic, like he does with most electronics\u2014eccentric millionaire inventor habits die hard. He's about to ask for a dumbed-down explanation when his mother comes blazing in.\n\n\"Why did y'all let me run for president?\" she says, tapping too forcefully at her phone's keyboard in little staccato stabs. She kicks off her heels into the corner, throwing her phone after them.\n\n\"Because we all knew better than to try to stop you,\" Leo's voice says. He peeks his bearded, bespectacled head out and adds, \"And because the world would fall apart without you, my radiant orchid.\"\n\nHis mother rolls her eyes but smiles. It's always been like that with them, ever since they first met at a charity event when Alex was fourteen. She was the Speaker of the House, and he was a genius with a dozen patents and money to burn on women's health initiatives. Now, she's the president, and he's sold his companies to spend his time fulfilling First Gentleman duties.\n\nEllen releases two inches of zipper on the back of her skirt, the sign she's officially done for the day, and scoops up a slice.\n\n\"All right,\" she says. She does a scrubbing gesture in the air in front of her face\u2014president face off, mom face on. \"Hi, babies.\"\n\n\"'Lo,\" Alex and June mumble in unison through mouthfuls of food.\n\nEllen sighs and looks over at Leo. \"I did that, didn't I? No goddamn manners. Like a couple of little opossums. This is why they say women can't have it all.\"\n\n\"They are masterpieces,\" Leo says.\n\n\"One good thing, one bad thing,\" she says. \"Let's do this.\"\n\nIt's her lifelong system for catching up on their days when she's at her busiest. Alex grew up with a mother who was a sometimes baffling combination of intensely organized and committed to lines of emotional communication, like an overly invested life coach. When he got his first girlfriend, she made a PowerPoint presentation.\n\n\"Mmm.\" June swallows a bite. \"Good thing. Oh! Oh my God. Ronan Farrow tweeted about my essay for _New York_ magazine _,_ and we totally engaged in witty Twitter repartee. Part one of my long game to force him to be my friend is underway.\"\n\n\"Don't act like this isn't all part of your extra-long game of abusing your position to murder Woody Allen and make it look like an accident,\" Alex says.\n\n\"He's just so frail; it'd only take one good push\u2014\"\n\n\" _How many times_ do I have to tell y'all not to discuss your murder plots in front of a sitting president?\" their mother interrupts. \" _Plausible deniability._ Come on.\"\n\n\" _Anyway,_ \" June says. \"One bad thing would be, uh... well, Woody Allen's still alive. Your turn, Alex.\"\n\n\"Good thing,\" Alex says, \"I filibustered one of my professors into agreeing a question on our last exam was misleading so I would get full credit for my answer, which was correct.\" He takes a swig of beer. \"Bad thing\u2014Mom, I saw the new art in the hall on the second floor, and I need to know why you allowed a George W. Bush terrier painting in our home.\"\n\n\"It's a bipartisan gesture,\" Ellen says. \"People find them endearing.\"\n\n\"I have to walk past it whenever I go to my room,\" Alex says. \"Its beady little eyes follow me everywhere.\"\n\n\"It's staying.\"\n\nAlex sighs. \"Fine.\"\n\nLeo goes next\u2014as usual, his bad thing is somehow also a good thing\u2014and then Ellen's up.\n\n\"Well, my UN ambassador fucked up his _one job_ and said something idiotic about Israel, and now I have to call Netanyahu and personally apologize. But the good thing is it's two in the morning in Tel Aviv, so I can put it off until tomorrow and have dinner with you two instead.\"\n\nAlex smiles at her. He's still in awe, sometimes, of hearing her talk about presidential pains in the ass, even three years in. They lapse into idle conversation, little barbs and inside jokes, and these nights may be rare, but they're still nice.\n\n\"So,\" Ellen says, starting on another slice crust-first. \"I ever tell you I used to hustle pool at my mom's bar?\"\n\nJune stops short, her beer halfway to her mouth. \"You did what now?\"\n\n\"Yep,\" she tells them. Alex exchanges an incredulous look with June. \"Momma managed this shitty bar when I was sixteen. The Tipsy Grackle. She'd let me come in after school and do my homework at the bar, had a bouncer friend make sure none of the old drunks hit on me. I got pretty good at pool after a few months and started betting the regulars I could beat them, except I'd play dumb. Hold the stick the wrong way, pretend to forget if I was stripes or solid. I'd lose one game, then take them double or nothing and get twice the payout.\"\n\n\"You've got to be kidding me,\" Alex says, except he can totally picture it. She has always been scary-good at pool and even better at strategy.\n\n\"All true,\" Leo says. \"How do you think she learned to get what she wants from strung-out old white men? The most important skill of an effective politician.\"\n\nAlex's mother accepts a kiss to the side of her square jaw from Leo as she passes by, like a queen gliding through a crowd of admirers. She sets her half-eaten slice down on a paper towel and selects a cue stick from the rack.\n\n\"Anyway,\" she says. \"The point is, you're never too young to figure out your skills and use them to get shit accomplished.\"\n\n\"Okay,\" Alex says. He meets her eyes, and they swap appraising looks.\n\n\"Including...\" she says thoughtfully, \"a job on a presidential reelection campaign, maybe.\"\n\nJune puts down her slice. \"Mom, he's not even out of college yet.\"\n\n\"Uh, yeah, that's the point,\" Alex says impatiently. He's been _waiting_ for this offer. \"No gaps in the resume.\"\n\n\"It's not only for Alex,\" their mother says. \"It's for both of you.\"\n\nJune's expression changes from pinched apprehension to pinched dread. Alex makes a shooing motion in June's direction. A mushroom flies off his pizza and hits the side of her nose. \"Tell me, tell me, tell me.\"\n\n\"I've been thinking,\" Ellen says, \"this time around, y'all\u2014the 'White House Trio.'\" She puts it in air quotes, as if she didn't sign off on the name herself. \"Y'all shouldn't only be faces. Y'all are more than that. You have skills. You're smart. You're talented. We could use y'all not only as surrogates, but as staffers.\"\n\n\"Mom...\" June starts.\n\n\"What positions?\" Alex interjects.\n\nShe pauses, drifts back over to her slice of pizza. \"Alex, you're the family wonk,\" she says, taking a bite. \"We could have you running point on policy. This means a lot of research and a lot of writing.\"\n\n\"Fuck yes,\" Alex says. \"Lemme romance the hell out of some focus groups. I'm in.\"\n\n\"Alex\u2014\" June starts again, but their mom cuts her off.\n\n\"June, I'm thinking communications,\" she goes on. \"Since your degree is mass comm, I was thinking you can come handle some of the day-to-day liaising with media outlets, working on messaging, analyzing the audience\u2014\"\n\n\"Mom, I have a job,\" she says.\n\n\"Oh, yeah. I mean, of course, sugar. But this could be full-time. Connections, upward mobility, real experience in the field doing some amazing work.\"\n\n\"I, um...\" June rips a piece of crust off her pizza. \"Don't remember ever saying I wanted to do anything like that. That's, uh, kind of a big assumption to make, Mom. And you realize if I go into campaign communications now, I'm basically shutting down my chances of ever being a journalist, because, like, journalistic neutrality and everything. I can barely get anyone to let me write a column as it is.\"\n\n\"Baby girl,\" their mom says. She's got that look on her face she gets when she's saying something with a fifty-fifty chance of pissing you off. \"You're so talented, and I know you work hard, but at some point, you have to be realistic.\"\n\n\"What's _that_ supposed to mean?\"\n\n\"I just mean... I don't know if you're happy,\" she says, \"and maybe it's time to try something different. That's all.\"\n\n\"I'm not y'all,\" June tells her. \"This isn't _my_ thing.\"\n\n\"Juuuuune,\" Alex says, tilting his head back to look at her upside down over the arm of his chair. \"Just think about it? I'm doing it.\" He looks back at their mom. \"Are you offering a job to Nora too?\"\n\nShe nods. \"Mike is talking to her tomorrow about a position in analytics. If she takes it, she'll start ASAP. You, mister, are not starting until after graduation.\"\n\n\"Oh man, the White House Trio, riding into battle. This is awesome.\" He looks over at Leo, who has abandoned his project with the TV and is now happily eating a slice of cheesy bread. \"They offer you a job too, Leo?\"\n\n\"No,\" he says. \"As usual, my duties as First Gentleman are to work on my tablescapes and look pretty.\"\n\n\"Your tablescapes are really coming along, baby,\" Ellen says, giving him a sarcastic little kiss. \"I really liked the burlap placemats.\"\n\n\"Can you believe the decorator thought velvet looked better?\"\n\n\"Bless her heart.\"\n\n\"I don't like this,\" June says to Alex while their mother is distracted talking about decorative pears. \"Are you sure you want this job?\"\n\n\"It's gonna be fine, June,\" he tells her. \"Hey, if you wanna keep an eye on me, you can always take the offer too.\"\n\nShe shakes him off, returning to her pizza with an unreadable expression. The next day there are three matching sticky notes on the whiteboard in Zahra's office. CAMPAIGN JOBS: ALEX-NORA-JUNE, the board reads. The sticky notes under his and Nora's names read YES. Under June's, in what is unmistakably her own handwriting, NO.\n\n* * *\n\nAlex is taking notes in a policy lecture when he gets the first text.\n\nThis bloke looks like you.\n\nThere's a picture attached, an image of a laptop screen paused on Chief Chirpa from _Return of the Jedi_ : tiny, commanding, adorable, pissed off.\n\nThis is Henry, by the way.\n\nHe rolls his eyes, but adds the new contact to his phone: HRH Prince Dickhead. Poop emoji.\n\nHe's honestly not planning to respond, but a week later he sees a headline on the cover of _People_ \u2014PRINCE HENRY FLIES SOUTH FOR WINTER\u2014complete with a photo of Henry artistically posed on an Australian beach in a pair of sensible yet miniscule navy swim trunks, and he can't stop himself.\n\nyou have a lot of moles, he texts, along with a snap of the spread. is that a result of the inbreeding?\n\nHenry's retort comes two days later by way of a screenshot of a _Daily Mail_ tweet that reads, _Is Alex Claremont-Diaz going to be a father?_ The attached message says, But we were ever so careful, dear, which surprises a big enough laugh out of Alex that Zahra ejects him from her weekly debriefing with him and June.\n\nSo, it turns out Henry can be funny. Alex adds that to his mental file.\n\nIt also turns out Henry is fond of texting when he's trapped in moments of royal monotony, like being shuttled to and from appearances, or sitting through meandering briefings on his family's land holdings, or, once, begrudgingly and hilariously receiving a spray tan.\n\nAlex wouldn't say he _likes_ Henry, but he does enjoy the quick rhythm of arguments they fall into. He knows he talks too much, hopeless at moderating his feelings, which he usually hides under ten layers of charm, but he ultimately doesn't care what Henry thinks of him, so he doesn't bother. Instead, he's as weird and manic as he wants to be, and Henry jabs back in sharp flashes of startling wit.\n\nSo, when he's bored or stressed or between coffee refills, he'll check for a text bubble popping up. Henry with a dig at some weird quote from his latest interview, Henry with a random thought about English beer versus American beer, a picture of Henry's dog wearing a Slytherin scarf. (i don't know WHO you think you're kidding, you hufflepuff-ass bitch, Alex texts back, before Henry clarifies his dog, not him, is a Slytherin.)\n\nHe learns about Henry's life through a weird osmosis of text messages and social media. It's meticulously scheduled by Shaan, with whom Alex is slightly obsessed, especially when Henry texts him things like, Did I tell you Shaan has a motorbike? or Shaan is on the phone with Portugal.\n\nIt's quickly becoming apparent the HRH Prince Henry Fact Sheet either omitted the most interesting stuff or was outright fabricated. Henry's favorite food isn't mutton pie but a cheap falafel stand ten minutes from the palace, and he's spent most of his gap year thus far working on charities around the world, half of them owned by his best friend, Pez.\n\nAlex learns Henry's super into classical mythology and can rattle off the configurations of a few dozen constellations if you let him get going. Alex hears more about the tedious details of operating a sailboat than he would ever care to know and sends back nothing but: cool. Eight hours later. Henry hardly ever swears, but at least he doesn't seem to mind Alex's filthy fucking mouth.\n\nHenry's sister, Beatrice\u2014she goes by Bea, Alex finds out\u2014pops up often, since she lives in Kensington Palace as well. From what he gathers, the two of them are closer than either are to their brother. They compare notes on the trials and tribulations of having older sisters.\n\ndid bea force you into dresses as a child too?\n\nHas June also got a fondness for sneaking your leftover curry out of the refrigerator in the dead of night like a Dickensian street urchin?\n\nMore common are cameos by Pez, a man who cuts such an intriguing and bizarre figure that Alex wonders how someone like him ever became best friends with someone like Henry, who can drone on about Lord Byron until you threaten to block his number. He's always either doing something insane\u2014BASE jumping in Malaysia, eating plantains with someone who might be Jay-Z, showing up to lunch wearing a studded, hot-pink Gucci jacket\u2014or launching a new nonprofit. It's kind of incredible.\n\nHe realizes that he's shared June and Nora too, when Henry remembers June's Secret Service codename is Bluebonnet or jokes about how eerie Nora's photographic memory is. It's weird, considering how fiercely protective Alex is of them, that he never even noticed until Henry's Twitter exchange with June about their mutual love of the 2005 _Pride & Prejudice_ movie goes viral.\n\n\"That's not your emails-from-Zahra face,\" Nora says, nosing her way over his shoulder. He elbows her away. \"You keep doing that stupid smile every time you look at your phone. Who are you texting?\"\n\n\"I don't know what you're talking about, and literally no one,\" Alex tells her. From the screen in his hand, Henry's message reads, In world's most boring meeting with Philip. Don't let the papers print lies about me after I've garroted myself with my tie.\n\n\"Wait,\" she says, reaching for his phone again, \"are you watching videos of Justin Trudeau speaking French again?\"\n\n\"That's not a thing I do!\"\n\n\"That is a thing I have caught you doing at least twice since you met him at the state dinner last year, so yeah, it is,\" she says. Alex flips her off. \"Wait, oh my God, is it fan fiction about yourself? And you didn't _invite me_? Who do they have you boning now? Did you read the one I sent you with Macron? I _died._ \"\n\n\"If you don't stop, I'm gonna call Taylor Swift and tell her you changed your mind and want to go to her Fourth of July party after all.\"\n\n\"That is _not_ a proportional response.\"\n\nLater that night, once he's alone at his desk, he replies: was it a meeting about which of your cousins have to marry each other to take back casterly rock?\n\nHa. It was about royal finances. I'll be hearing Philip's voice saying the words \"return on investment\" in my nightmares for the rest of time.\n\nAlex rolls his eyes and sends back, the harrowing struggle of managing the empire's blood money.\n\nHenry's response comes a minute later.\n\nThat was actually the crux of the meeting\u2014I've tried to refuse my share of the crown's money. Dad left us each more than enough, and I'd rather cover my expenses with that than the spoils of, you know, centuries of genocide. Philip thinks I'm being ridiculous.\n\nAlex scans the message twice to make sure he's read it correctly.\n\ni am low-key impressed.\n\nHe stares at the screen, at his own message, for a few seconds too long, suddenly afraid it was a stupid thing to say. He shakes his head, puts the phone down. Locks it. Changes his mind, picks it up again. Unlocks it. Sees the little typing bubble on Henry's side of the conversation. Puts the phone down. Looks away. Looks back.\n\nOne does not foster a lifelong love of Star Wars without knowing an \"empire\" isn't a good thing.\n\nHe would really appreciate it if Henry would stop proving him wrong.\n\n* * *\n\n> HRH Prince Dickhead\n\nOct 30, 2019, 1:07 PM\n\n> i hate that tie\n> \n> HRH Prince Dickhead  \n> \n> What tie?\n> \n> the one in that instagram you just posted\n> \n> HRH Prince Dickhead  \n> \n> What's wrong with it? It's only grey.\n> \n> exactly. try patterns sometime, and stop frowning at your phone like i know you're doing rn\n> \n> HRH Prince Dickhead  \n> \n> Patterns are considered a \"statement.\" Royals aren't supposed to make statements with what we wear.\n> \n> do it for the gram\n> \n> HRH Prince Dickhead  \n> \n> You are the thistle in the tender and sensitive arse crack of my life.\n> \n> thanks!\n\nNov 17, 2019, 11:04 AM\n\n> HRH Prince Dickhead  \n> \n> I've just received a 5-kilo parcel of Ellen Claremont campaign buttons with your face on them. Is this your idea of a prank?\n> \n> just trying to brighten up that wardrobe, sunshine\n> \n> HRH Prince Dickhead  \n> \n> I hope this gross miscarriage of campaign funds is worth it to you. My security thought it was a bomb. Shaan almost called in the sniffer dogs.\n> \n> oh, definitely worth it. even more worth it now. tell shaan i say hi and i miss that sweet sweet ass xoxoxo\n> \n> HRH Prince Dickhead  \n> \n> I will not.\n\n# FOUR\n\n\"It's public knowledge. It's not my problem you just found out,\" his mother is saying, pacing double-time down a West Wing corridor.\n\n\"You mean to tell me,\" Alex half shouts, jogging to keep up, \"every Thanksgiving, those stupid turkeys have been staying in a luxury suite at the Willard on the taxpayers' dime?\"\n\n\"Yes, Alex, they do\u2014\"\n\n_\"Gross government waste!\"_\n\n\"\u2014and there are two forty-pound turkeys named Cornbread and Stuffing in a motorcade on Pennsylvania Avenue right now. There is no time to reallocate the turkeys.\"\n\nWithout missing a beat, he blurts out, \"Bring them to the house.\"\n\n\"Where? Are you hiding a turkey habitat up your ass, son? Where, in our historically protected house, am I going to put a couple of turkeys until I pardon them tomorrow?\"\n\n\"Put them in my room. I don't care.\"\n\nShe outright laughs. \"No.\"\n\n\"How is it different from a hotel room? Put the turkeys in my room, Mom.\"\n\n\"I'm not putting the turkeys in your room.\"\n\n\"Put the turkeys in my room.\"\n\n\"No.\"\n\n\"Put them in my room, put them in my room, put them in my room\u2014\"\n\nThat night, as Alex stares into the cold, pitiless eyes of a prehistoric beast of prey, he has a few regrets.\n\nTHEY KNOW, he texts Henry. THEY KNOW I HAVE ROBBED THEM OF FIVE-STAR ACCOMMODATIONS TO SIT IN A CAGE IN MY ROOM, AND THE MINUTE I TURN MY BACK THEY ARE GOING TO FEAST ON MY FLESH.\n\nCornbread stares emptily back at him from inside a huge crate next to Alex's couch. A farm vet comes by once every few hours to check on them. Alex keeps asking if she can detect a lust for blood.\n\nFrom the en suite, Stuffing releases another ominous gobble.\n\nAlex was going to get things accomplished tonight. He really was. Before he learned of exorbitant turkey expenditures from CNN, he was watching the highlights of last night's Republican primary debate. He was going to finish an outline for an exam, then study the demographic engagement binder he convinced his mother to give him for the campaign job.\n\nInstead, he is in a prison of his own creation, sworn to babysit these turkeys until the pardoning ceremony, and is just now realizing his deep-seated fear of large birds. He considers finding a couch to sleep on, but what if these demons from hell break out of their cages and murder each other during the night when he's supposed to be watching them? BREAKING: BOTH TURKEYS FOUND DEAD IN BEDROOM OF FSOTUS, TURKEY PARDON CANCELED IN DISGRACE, FSOTUS A SATANIC TURKEY RITUAL KILLER.\n\nPlease send photos, is Henry's idea of a comforting response.\n\nHe drops onto the edge of his bed. He's grown accustomed to texting with Henry almost every day; the time difference doesn't matter, since they're both awake at all ungodly hours of the day and night. Henry will send a snap from a seven a.m. polo practice and promptly receive one of Alex at two a.m., glasses on and coffee in hand, in bed with a pile of notes. Alex doesn't know why Henry never responds to his selfies from bed. His selfies from bed are always hilarious.\n\nHe snaps a shot of Cornbread and presses send, flinching when the bird flaps at him threateningly.\n\nI think he's cute, Henry responds.\n\nthat's because you can't hear all the menacing gobbling\n\nYes, famously the most sinister of all animal sounds, the gobble.\n\n\"You know what, you little shit,\" Alex says the second the call connects, \"you can hear it for yourself and then tell me how you would handle this\u2014\"\n\n\"Alex?\" Henry's voice sounds scratchy and bewildered across the line. \"Have you really rung me at three o'clock in the morning to make me listen to a turkey?\"\n\n\"Yes, obviously,\" Alex says. He glances at Cornbread and cringes. \"Jesus Christ, it's like they can see into your _soul._ Cornbread knows my sins, Henry. Cornbread knows what I have done, and he is here to make me atone.\"\n\nHe hears a rustling over the phone, and he pictures Henry in his heather-gray pajama shirt, rolling over in bed and maybe switching on a lamp. \"Let's hear the cursed gobble, then.\"\n\n\"Okay, brace yourself,\" he says, and he switches to speaker and gravely holds out the phone.\n\nNothing. Ten long seconds of nothing.\n\n\"Truly harrowing,\" Henry's voice says tinnily over the speaker.\n\n\"It\u2014okay, this is not representative,\" Alex says hotly. \"They've been gobbling all fucking night, I swear.\"\n\n\"Sure they were,\" Henry says, mock-gently.\n\n\"No, hang on,\" Alex says. \"I'm gonna... I'm gonna get one to gobble.\"\n\nHe hops off the bed and edges up to Cornbread's cage, feeling very much like he is taking his life into his own hands and also very much like he has a point to prove, which is an intersection at which he finds himself often.\n\n\"Um,\" he says. \"How do you get a turkey to gobble?\"\n\n\"Try gobbling,\" Henry says, \"and see if he gobbles back.\"\n\nAlex blinks. \"Are you serious?\"\n\n\"We hunt loads of wild turkeys in the spring,\" Henry says sagely. \"The trick is to get into the mind of the turkey.\"\n\n\"How the hell do I do that?\"\n\n\"So,\" Henry instructs. \"Do as I say. You have to get quite close to the turkey, like, physically.\"\n\nCarefully, still cradling the phone close, Alex leans toward the wire bars. \"Okay.\"\n\n\"Make eye contact with the turkey. Do you have it?\"\n\nAlex follows Henry's instructions in his ear, planting his feet and bending his knees so he's at Cornbread's eye level, a chill running down his spine when his own eyes lock on the beady, black little murder eyes. \"Yeah.\"\n\n\"Right, now hold it,\" Henry says. \"Connect with the turkey, earn the turkey's trust... befriend the turkey...\"\n\n\"Okay...\"\n\n\"Buy a summer home in Majorca with the turkey...\"\n\n\"Oh, I _fucking_ hate you!\" Alex shouts as Henry laughs at his own idiotic prank, and his indignant flailing startles a loud gobble out of Cornbread, which in turn startles a very unmanly scream out of Alex. \" _Goddammit!_ Did you hear that?\"\n\n\"Sorry, what?\" Henry says. \"I've been stricken deaf.\"\n\n\"You're such a _dick,_ \" Alex says. \"Have you ever even _been_ turkey hunting?\"\n\n\"Alex, you can't even hunt them in Britain.\"\n\nAlex returns to his bed and face-plants into a pillow. \"I hope Cornbread does kill me.\"\n\n\"No, all right, I did hear it, and it was... proper frightening,\" Henry says. \"So, I understand. Where's June for all this?\"\n\n\"She's having some kind of girls' night with Nora, and when I texted them for backup, they sent back,\" he reads out in a monotone, \"'hahahahahahahaha good luck with that,' and then a turkey emoji and a poop emoji.\"\n\n\"That's fair,\" Henry says. Alex can picture him nodding solemnly. \"So what are you going to do now? Are you going to stay up all night with them?\"\n\n\"I don't know! I guess! I don't know what else to do!\"\n\n\"You couldn't just go sleep somewhere else? Aren't there a thousand rooms in that house?\"\n\n\"Okay, but, uh, what if they escape? I've seen _Jurassic Park._ Did you know birds are directly descended from raptors? That's a scientific fact. Raptors in my bedroom, Henry. And you want me to go to sleep like they're not gonna bust out of their enclosures and take over the island the minute I close my eyes? Okay. Maybe your white ass.\"\n\n\"I'm really going to have you offed,\" Henry tells him. \"You'll never see it coming. Our assassins are trained in discretion. They will come in the night, and it will look like a humiliating accident.\"\n\n\"Autoerotic asphyxiation?\"\n\n\"Toilet heart attack.\"\n\n\"Jesus.\"\n\n\"You've been warned.\"\n\n\"I thought you'd kill me in a more personal way. Silk pillow over my face, slow and gentle suffocation. Just you and me. Sensual.\"\n\n\"Ha. Well.\" Henry coughs.\n\n\"Anyway,\" Alex says, climbing fully up onto the bed now. \"It doesn't matter because one of these goddamn turkeys is gonna kill me first.\"\n\n\"I really don't think\u2014 _Oh, hello there._ \" There's rustling over the phone, the crinkling of a wrapper, and some heavy snuffling that sounds distinctly doglike. \" _Who'za good lad, then?_ David says hello.\"\n\n\"Hi, David.\"\n\n\"He\u2014 Oi! _Not_ for you, Mr. Wobbles! Those are _mine_!\" More rustling, a distant, offended meow. \" _No,_ Mr. Wobbles, you bastard!\"\n\n\"What in the fuck is a Mr. Wobbles?\"\n\n\"My sister's idiot cat,\" Henry tells him. \"The thing weighs a ton and is still trying to steal my Jaffa Cakes. He and David are mates.\"\n\n\"What are you even doing right now?\"\n\n\"What am _I_ doing? I was trying to _sleep._ \"\n\n\"Okay, but you're eating Jabba Cakes, so.\"\n\n\" _Jaffa_ Cakes, my _God,_ \" Henry says. \"I'm having my entire life haunted by a deranged American Neanderthal and a pair of turkeys, apparently.\"\n\n\"And?\"\n\nHenry heaves another almighty sigh. He's always sighing when Alex is involved. It's amazing he has any air left. \"And... don't laugh.\"\n\n\"Oh, yay,\" Alex says readily.\n\n\"I was watching _Great British Bake Off._ \"\n\n\"Cute. Not embarrassing, though. What else?\"\n\n\"I, er, might be... wearing one of those peely face masks,\" he says in a rush.\n\n\"Oh my God, I knew it!\"\n\n\" _Instant_ regret.\"\n\n\"I knew you had one of those crazy expensive Scandinavian skin care regimens. Do you have that, like, eye cream with diamonds in it?\"\n\n\"No!\" Henry pouts, and Alex has to press the back of his hand against his lips to stifle his laugh. \"Look, I have an appearance tomorrow, all right? I didn't know I'd be _scrutinized._ \"\n\n\"I'm not scrutinizing. We all gotta keep those pores in check,\" Alex says. \"So you like _Bake Off,_ huh?\"\n\n\"It's just so soothing,\" Henry says. \"Everything's all pastel-colored and the music is so relaxing and everyone's so lovely to one another. And you learn so much about different types of biscuits, Alex. So much. When the world seems awful, such as when you're trapped in a Great Turkey Calamity, you can put it on and vanish into biscuit land.\"\n\n\"American cooking competition shows are nothing like that. They're all sweaty and, like, dramatic death music and intense camera cuts,\" Alex says. \" _Bake Off_ makes _Chopped_ look like the fucking Manson tapes.\"\n\n\"I feel like this explains loads about our differences,\" Henry says, and Alex gives a small laugh.\n\n\"You know,\" Alex says. \"You're kind of surprising.\"\n\nHenry pauses. \"In what way?\"\n\n\"In that you're not a totally boring asshole.\"\n\n\"Wow,\" Henry says with a laugh. \"I'm honored.\"\n\n\"I guess you have your depths.\"\n\n\"You thought I was a dumb blond, didn't you?\"\n\n\"Not exactly, just, _boring,_ \" Alex says. \"I mean, your dog is named David, which is pretty boring.\"\n\n\"After Bowie.\"\n\n\"I\u2014\" Alex's head spins, recalibrating. \"Are you serious? What the hell? Why not call him Bowie, then?\"\n\n\"Bit on the nose, isn't it?\" Henry says. \"A man should have some element of mystery.\"\n\n\"I guess,\" Alex says. Then, because he can't stop it in time, lets out a tremendous yawn. He's been up since seven for a run before class. If these turkeys don't end him, exhaustion will.\n\n\"Alex,\" Henry says firmly.\n\n\"What?\"\n\n\"The turkeys are not going to _Jurassic Park_ you,\" he says. \"You're not the bloke from _Seinfeld._ You're Jeff Goldblum. Go to sleep.\"\n\nAlex bites down a smile that feels bigger than the sentence has truly earned. \"You go to sleep.\"\n\n\"I will,\" Henry says, and Alex thinks he hears the weird smile returned in Henry's voice, and honestly, this whole night is really, really weird, \"as soon as you get off the phone, won't I?\"\n\n\"Okay,\" Alex says, \"but, like, what if they gobble again?\"\n\n\"Go sleep in June's room, you numpty.\"\n\n\"Okay,\" Alex says.\n\n\"Okay,\" Henry agrees.\n\n\"Okay,\" Alex says again. He's suddenly very aware they've never spoken on the phone before, and so he's never had to figure out how to hang up the phone with Henry before. He's at a loss. But he's still smiling. Cornbread is staring at him like he doesn't get it. _Me fuckin' too, buddy._\n\n\"Okay,\" Henry repeats. \"So. Good night.\"\n\n\"Cool,\" Alex says lamely. \"Good night.\"\n\nHe hangs up and stares at the phone in his hand, as if it should explain the static electricity in the air around him.\n\nHe shakes it off, gathers up his pillow and a bundle of clothes, and crosses the hall to June's room, climbing up into her tall bed. But he can't stop thinking there's some end left loose.\n\nHe takes his phone back out. i sent pics of turkeys so i deserve pics of your animals too.\n\nA minute and a half later: Henry, in a massive, palatial, hideous bed of white and gold linens, his face looking slightly pink and recently scrubbed, with a beagle's head on one side of his pillow and an obese Siamese cat curled up on the other around a Jaffa Cake wrapper. He's got faint circles under his eyes, but his face is soft and amused, one hand resting above his head on the pillow while the other holds up the phone for the selfie.\n\nThis is what I must endure, he says, followed by, Good night, honestly.\n\n> HRH Prince Dickhead\n\nDec 8, 2019, 8:53 PM\n\n> yo there's a bond marathon on and did you know your dad was a total babe\n> \n> HRH Prince Dickhead  \n> \n> I BEG YOU TO NOT\n\n* * *\n\nEven before Alex's parents split, they both had a habit of calling him by the other's last name when he exhibited particular traits. They still do. When he runs his mouth off to the press, his mom calls him into her office and says, \"Get your shit together, Diaz.\" When his hard-headedness gets him stuck, his dad texts him, \"Let it go, Claremont.\"\n\nAlex's mother sighs as she sets her copy of the _Post_ down on her desk, open to an inside page article: SENATOR OSCAR DIAZ RETURNS TO DC FOR HOLIDAYS WITH EX-WIFE PRESIDENT CLAREMONT. It's almost weird how much it isn't weird anymore. His dad is flying in from California for Christmas, and it's fine, but it's also in the _Post._\n\nShe's doing the thing she always does when she's about to spend time with his father: pursing her lips and twitching two fingers of her right hand.\n\n\"You know,\" Alex says from where he's kicked back on an Oval Office couch with a book, \"somebody can go get you a cigarette.\"\n\n\"Hush, Diaz.\"\n\nShe's had the Lincoln Bedroom prepared for his dad, and she keeps changing her mind, having housekeeping undecorate and redecorate. Leo, for his part, is unfazed and mollifies her with compliments between fits of tinsel. Alex doesn't think anyone but Leo could ever stay married to his mother. His father certainly couldn't.\n\nJune is in a state, the perpetual mediator. His family is pretty much the only situation where Alex prefers to sit back and let it all unfold, occasionally poking when it's necessary or interesting, but June takes personal responsibility for making sure nobody breaks any more priceless White House antiques like last year.\n\nHis dad finally arrives in a flurry of Secret Service agents, his beard impeccably groomed and his suit impeccably tailored. For all June's anxious preparations, she almost breaks an antique vase herself catapulting into his arms. They disappear immediately to the chocolate shop on the ground floor, the sound of Oscar raving about June's latest blog post for _The Atlantic_ fading around the corner. Alex and his mother share a look. Their family is so predictable sometimes.\n\nThe next day, Oscar gives Alex the follow-me-and-don't-tell-your-mother look and pulls him out to the Truman Balcony.\n\n\"Merry fuckin' Christmas, mijo,\" his dad says, grinning, and Alex laughs and lets himself be hauled into a one-armed hug. He smells the same as ever, salty and smoky and like well-treated leather. His mom used to complain that she felt like she lived in a cigar bar.\n\n\"Merry Christmas, Pa,\" Alex says back.\n\nHe drags a chair close to the railing, putting his shiny boots up. Oscar Diaz loves a view.\n\nAlex considers the sprawling, snowy lawn in front of them, the sure line of the Washington Monument stretching up, the jagged French mansard roofs of the Eisenhower Building to the west, the same one Truman hated. His dad pulls a cigar from his pocket, clipping it and lighting up in the careful ritual he's done for years. He takes a puff and passes it over.\n\n\"It ever make you laugh to think how much this pisses assholes off?\" he says, gesturing to encompass the whole scene: two Mexican men putting their feet up on the railing where heads of state eat croissants.\n\n\"Constantly.\"\n\nOscar does laugh, then, enjoying his brazenness. He is an adrenaline junkie\u2014mountain climbing, cave diving, pissing off Alex's mother. Flirting with death, basically. It's the flip side of the way he approaches work, which is methodical and precise, or the way he approaches parenting, which is laid-back and indulgent.\n\nIt's nice, now, to see him more than he ever did in high school, since Oscar spends most of his year in DC. During the busiest congressional sessions, they'll convene Los Bastardos\u2014weekly beers in Oscar's office after hours, just him, Alex, and Rafael Luna, talking shit. And it's nice that proximity has forced his parents through the era of mutually assured destruction to now, where they have one Christmas instead of two.\n\nAs the days go by, Alex catches himself remembering sometimes, just for a second, how much he misses having everyone under one roof.\n\nHis dad was always the cook of the family. Alex's childhood was perfumed with simmering peppers and onions and stew meat in a cast iron pot for caldillo, fresh masa waiting on the butcher block. He remembers his mom swearing and laughing when she opened the oven for her guilty-pleasure pizza bagels only to find all the pots and pans stored there, or when she'd go for the tub of butter in the fridge and find it filled with homemade salsa verde. There used to be a lot of laughter in that kitchen, a lot of good food and loud music and parades of cousins and homework done at the table.\n\nExcept eventually there was a lot of yelling, followed by a lot of quiet, and soon Alex and June were teenagers and both their parents were in Congress, and Alex was student body president and lacrosse cocaptain and prom king and valedictorian, and, very intentionally, it stopped being a thing he had time to think about.\n\nStill, his dad's been in the Residence for three days without incident, and one day Alex catches him in the kitchens with two of the cooks, laughing and dumping peppers into a pot. It's just, you know, sometimes he thinks it might be nice if it could be like this more often.\n\nZahra's heading to New Orleans to see her family for Christmas, only at the president's insistence, and only because her sister had a baby and Amy threatened to stab her if she didn't deliver the onesie she knitted. Which means Christmas dinner is happening on Christmas Eve so Zahra won't miss it. For all her late nights cursing their names, Zahra is family.\n\n\"Merry Christmas, Z!\" Alex tells her cheerfully in the hall outside the family dining room. For holiday flare, she's wearing a sensible red turtleneck; Alex is wearing a sweater covered in bright green tinsel. He smiles and presses a button on the inside of the sleeve, and \"O Christmas Tree\" plays from a speaker near his armpit.\n\n\"I can't wait to not see you for two days,\" she says, but there's real affection in her voice.\n\nThis year's dinner is small, since his dad's parents are on vacation, so the table is set for six in glittering white and gold. The conversation is pleasant enough that Alex almost forgets it's not always like this.\n\nUntil it shifts to the election.\n\n\"I was thinking,\" Oscar says, carefully cutting his filet, \"this time, I can campaign with you.\"\n\nAt the other end of the table, Ellen puts her fork down. \"You can what?\"\n\n\"You know.\" He shrugs, chewing. \"Hit the trail, do some speeches. Be a surrogate.\"\n\n\"You can't be serious.\"\n\nOscar puts down his own fork and knife now on the cloth-covered table, a soft thump of _oh, shit._ Alex glances across the table at June.\n\n\"You really think it's such a bad idea?\" Oscar says.\n\n\"Oscar, we went through all of this last time,\" Ellen tells him. Her tone is instantly clipped. \"People don't like women, but they like mothers and wives. They like _families._ The last thing we need to do is remind them that I'm divorced by parading my ex-husband around.\"\n\nHe laughs a little grimly. \"So, you'll pretend he's their dad then, eh?\"\n\n\"Oscar,\" Leo speaks up, \"you know I'd never\u2014\"\n\n\"You're missing the _point,_ \" Ellen interrupts.\n\n\"It could help your approval ratings,\" he says. \"Mine are quite high, El. Higher than yours ever were in the House.\"\n\n\"Here we go,\" Alex says to Leo next to him, whose face remains pleasantly neutral.\n\n\"We've done _studies,_ Oscar! Okay?\" Ellen's voice has risen in volume and pitch, her palms planted flat on the table. \"The data shows, I track worse with undecided voters when they're reminded of the divorce!\"\n\n\"People know you're divorced!\"\n\n\"Alex's numbers are high!\" she shouts, and Alex and June both wince. \"June's numbers are high!\"\n\n\"They're not _numbers_!\"\n\n\"Fuck off, I know that,\" she spits, \"I never said they were!\"\n\n\"You think sometimes you use them like they are?\"\n\n\"How _dare_ you, when you don't seem to have any problem trotting them out every time you're up for reelection!\" she says, slicing one hand through the air beside her. \"Maybe if they were just Claremonts, you wouldn't have so much luck. It'd sure as hell be less confusing\u2014it's the name everybody knows them by anyway!\"\n\n\"Nobody's taking any of our names!\" June jumps in, her voice high.\n\n_\"June,\"_ Ellen says.\n\nTheir dad pushes on. \"I'm trying to help you, Ellen!\"\n\n\"I don't need your help to win an election, Oscar!\" she says, hitting the table so hard with her open palm that the dishes rattle. \"I didn't need it when I was in Congress, and I didn't need it to become president the first time, and I don't need it now!\"\n\n\"You need to get serious about what you're up against! You think the other side is going to play fair this time? Eight years of Obama, and now you? They're angry, Ellen, and Richards is out for blood! You need to be ready!\"\n\n\"I will be! You think I don't have a team on all this shit already? I'm the President of the United fucking States! I don't need you to come here and\u2014and\u2014\"\n\n\"Mansplain?\" Zahra offers.\n\n\"Mansplain!\" Ellen shouts, jabbing a finger across the table at Oscar, eyes wide. \"This presidential race to me!\"\n\nOscar throws his napkin down. \"You're still so _fucking_ stubborn!\"\n\n\"Fuck you!\"\n\n\"Mom!\" June says sharply.\n\n\"Jesus Christ, are you kidding me?\" Alex hears himself shout before he even consciously decides to say it. \"Can we not be civil for one fucking meal? It's _Christmas,_ for fuck's sake. Aren't y'all supposed to be running the country? Get your shit together.\"\n\nHe pushes his chair back and stalks out of the dining room, knowing he's being a dramatic asshole and not really caring. He slams his bedroom door behind him, and his stupid sweater plays a few depressingly off-key notes when he yanks it off and throws it at the wall.\n\nIt's not that he doesn't lose his temper often, it's just... he doesn't usually lose it with his family. Mostly because he doesn't usually _deal_ with his family.\n\nHe digs an old lacrosse T-shirt out of his dresser, and when he turns and catches his reflection in the mirror by the closet, he's right back in his teens, caring too much about his parents and helpless to change his situation. Except now he doesn't have any AP classes to enroll in as a distraction.\n\nHis hand twitches for his phone. His brain is a two-passenger minimum ride as far as he's concerned\u2014alone and busy or thinking with company.\n\nBut Nora's doing Hanukkah in Vermont, and he doesn't want to annoy her, and his best friend from high school, Liam, has barely spoken to him since he moved to DC.\n\nWhich leaves...\n\n\"What could I possibly have done to have brought this upon myself now?\" says Henry's voice, low and sleepy. It sounds like \"Good King Wenceslas\" is playing in the background\n\n\"Hey, um, sorry. I know it's late, and it's Christmas Eve and everything. You probably have, like, family stuff, I'm just realizing. I don't know why I didn't think of it before. Wow, this is why I don't have friends. I'm a dick. Sorry, man. I'll, uh, I'll just\u2014\"\n\n\"Alex, Christ,\" Henry interrupts. \"It's fine. It's half two here, everyone's gone to bed. Except Bea. Say hi, Bea.\"\n\n\"Hi, Alex!\" says a clear, giggly voice on the other end of the line. \"Henry's got his candy-cane jim-jams on\u2014\"\n\n\"That's quite enough,\" Henry's voice comes back through, and there's a muffled sound like maybe a pillow has been shoved in Bea's direction. \"What's happening, then?\"\n\n\"Sorry,\" Alex blurts out, \"I know this is weird, and you're with your sister and everything, and, like, argh. I kind of didn't have anyone else to call who would be awake? And I know we're, uh, not really friends, and we don't really talk about this stuff, but my dad came in for Christmas, and he and my mom are like fucking tiger sharks fighting over a baby seal when you put them in the same room together for more than an hour, and they got in this huge fight, and it shouldn't _matter,_ because they're already divorced and everything, and I don't know why I lost my shit, but I wish they could give it a rest for _once_ so we could have one single normal holiday, you know?\"\n\nThere's a long pause before Henry says, \"Hang on. _Bea, can I have a minute? Hush. Yes, you can take the biscuits._ All right, I'm listening.\"\n\nAlex exhales, wondering faintly what the hell he's doing, but plows onward.\n\nTelling Henry about the divorce\u2014those weird, tumultuous years, the day he came home from a Boy Scout camp-out to discover his dad's things moved out, the nights of Helados ice cream\u2014doesn't feel as uncomfortable as it probably should. He's never bothered to filter himself with Henry, at first because he honestly didn't care what Henry thought, and now because it's how they are. Maybe it should be different, bitching about his course load versus spilling his guts about this. It isn't.\n\nHe doesn't realize he's been talking for an hour until he finishes retelling what happened at dinner and Henry says, \"It sounds like you did your best.\"\n\nAlex forgets what he was going to say next.\n\nHe just... Well, he gets told he's great a lot. He just doesn't often get told he's good enough.\n\nBefore he can think of a response, there's a soft triple knock on the door\u2014June.\n\n\"Ah\u2014okay, thanks, man, I gotta go,\" Alex says, his voice low as June eases the door open.\n\n\"Alex\u2014\"\n\n\"Seriously, um. Thank you,\" Alex says. He really does not want to explain this to June. \"Merry Christmas. Night.\"\n\nHe hangs up and tosses the phone aside as June settles down on the bed. She's wearing her pink bathrobe, and her hair is wet from the shower.\n\n\"Hey,\" she says. \"You okay?\"\n\n\"Yeah, I'm fine,\" he says. \"Sorry, I don't know what's up with me. I didn't mean to lose it. I've been... I don't know. I've been kind of... off... lately.\"\n\n\"It's okay,\" she says. She tosses her hair over her shoulder, flicking droplets of water onto him. \"I was a total basket case for the last six months of college. I would lose it at anybody. You know, you don't have to do everything all the time.\"\n\n\"It's fine. I'm fine,\" he tells her automatically. June tilts an unconvinced look at him, and he kicks at one of her knees with his bare foot. \"So, how did things go after I left? Did they finish cleaning up the blood yet?\"\n\nJune sighs, kicking him back. \"Somehow it shifted to the topic of how they were a political power couple before the divorce and how good those times were, Mom apologized, and it was whiskey and nostalgia hour until everybody went to bed.\" She sniffs. \"Anyway, you were right.\"\n\n\"You don't think I was out of line?\"\n\n\"Nah. Though... I kind of agree with what Dad was saying. Mom can be... you know... Mom.\"\n\n\"Well, that's what got her where she is now.\"\n\n\"You don't think it's ever a problem?\"\n\nAlex shrugs. \"I think she's a good mom.\"\n\n\"Yeah, to you,\" June says. There's no accusation behind it, just observation. \"The effectiveness of her nurturing kind of depends on what you need from her. Or what you can do for her.\"\n\n\"I mean, I get what she's saying, though,\" Alex hedges. \"Sometimes it still sucks that Dad decided to pack up and move just to run for the seat in California.\"\n\n\"Yeah, but, I mean, how is that different from the stuff Mom's done? It's all politics. I'm just saying, he has a point about how Mom pushes us without always giving us the other Mom stuff.\"\n\nAlex is opening his mouth to answer when June's phone buzzes from her robe pocket. \"Oh. Hmm,\" she says when she slides it out to eye the screen.\n\n\"What?\"\n\n\"Nothing, uh.\" She thumbs open the message. \"Merry Christmas text. From Evan.\"\n\n\"Evan... as in ex-boyfriend Evan, in California? Y'all still text?\"\n\nJune's biting her lip now, her expression a little distant as she types out a response. \"Yeah, sometimes.\"\n\n\"Cool,\" Alex says. \"I always liked him.\"\n\n\"Yeah. Me too,\" June says softly. She locks her phone and drops it on the bed, blinking a couple times as if to reset. \"Anyway, what'd Nora say when you told her?\"\n\n\"Hmm?\"\n\n\"On the phone?\" she asks him. \"I figured it was her, you never talk to anyone else about this crap.\"\n\n\"Oh,\" Alex says. He feels inexplicable, traitorous warmth flash up the back of his neck. \"Oh, um, no. Actually, this is gonna sound weird, but I was talking to Henry?\"\n\nJune's eyebrows shoot up, and Alex instinctively scans the room for cover. \"Really.\"\n\n\"Listen, I know, but we kind of weirdly have stuff in common and, I guess, similar weird emotional baggage and neuroses, and for some reason I felt like he would get it.\"\n\n\"Oh my God, Alex,\" she says, lunging at him to yank him into a rough hug, \"you made a friend!\"\n\n\"I have friends! Get off me!\"\n\n\"You made a friend!\" She is literally giving him a noogie. \"I'm so proud of you!\"\n\n\"I'm gonna murder you, _stop it,_ \" he says, alligator-rolling out of her clutches. He lands on the floor. \"He's not my friend. He's someone I like to antagonize all the time, and _one_ time I talked to him about something real.\"\n\n\"That's a friend, Alex.\"\n\nAlex's mouth starts and stops several silent sentences before he points to the door. \"You can leave, June! Go to bed!\"\n\n\"Nope. Tell me everything about your new best friend, who is a _royal._ That is so bougie of you. Who would have guessed it?\" she says, peering over the edge of the bed at him. \"Oh my God, this is like all those romantic comedies where the girl hires a male escort to pretend to be her wedding date and then falls in love with him for real.\"\n\n\"That is _not at all_ what this is like.\"\n\n* * *\n\nThe staff has barely finished packing up the Christmas trees when it starts.\n\nThere's the dance floor to set up, menu to finalize, Snapchat filter to approve. Alex spends the entire 26th holed up in the Social Secretary's office with June, going over the waivers they've gotten for everyone to sign after a daughter of a Real Housewife fell down the rotunda stairs last year; Alex remains impressed that she didn't spill her margarita.\n\nIt's time once more for the Legendary Balls-Out Bananas White House Trio New Year's Eve Party.\n\nTechnically, the title is the Young America New Year's Eve Gala, or as at least one late-night host calls it, the Millennial Correspondents' Dinner. Every year, Alex, June, and Nora fill up the East Room on the first floor with three hundred or so of their friends, vague celebrity acquaintances, former hookups, potential political connections, and otherwise notable twenty-somethings. The party is, officially, a fund-raiser, and it generates so much money for charity and so much good PR for the First Family that even his mom approves of it.\n\n\"Um, excuse me,\" Alex is saying from a first-floor conference table, one hand full of confetti samples\u2014do they want a metallic color palette or a more subdued navy and gold?\u2014while staring at a copy of the finalized guest list. June and Nora are stuffing their faces with cake samples. \"Who put Henry on here?\"\n\nNora says through a mouthful of chocolate cake, \"Wasn't me.\"\n\n\"June?\"\n\n\"Look, you should have invited him yourself!\" June says, by way of admission. \"It's really nice you're making friends who aren't us. Sometimes when you get too isolated, you start to go a little crazy. Remember last year when Nora and I were both out of the country for a week, and you almost got a tattoo?\"\n\n\"I still think we should have let him get a tramp stamp.\"\n\n\"It wasn't going to be a _tramp stamp,_ \" Alex says hotly. \"You were in on this, weren't you?\"\n\n\"You know I love chaos,\" Nora tells him serenely.\n\n\"I have friends who aren't y'all,\" Alex says.\n\n\"Who, Alex?\" June says. \"Literally who?\"\n\n\"People!\" he says defensively. \"People from class! Liam!\"\n\n\"Please. We all know you haven't talked to Liam in a year,\" June says. \"You need friends. And I know you like Henry.\"\n\n\"Shut up,\" Alex says. He brushes a finger under his collar and finds his skin damp. Do they always have to crank the heat up this high when it's snowing outside?\n\n\"This is interesting,\" Nora observes.\n\n\"No, it's not,\" Alex snaps. \"Fine, he can come. But if he doesn't know anybody else, I'm not babysitting him all night.\"\n\n\"I gave him a plus-one,\" June says.\n\n\"Who is he bringing?\" Alex asks immediately, reflexively. Involuntarily. \"Just wondering.\"\n\n\"Pez,\" she says. She's giving him a weird look he can't parse, and he decides to chalk it up to June being confusing and strange. She often works in mysterious ways, organizes and orchestrates things he never sees coming until all the threads come together.\n\nSo, Henry is coming, he guesses, confirmed when he checks Instagram the day of the party and sees a post from Pez of him and Henry on a private jet. Pez's hair has been dyed pastel pink for the occasion, and beside him, Henry is smiling in a soft-looking gray sweatshirt, his socked feet up on the windowsill. He actually looks well-rested for once.\n\nUSA bound! #YoungAmericaGala2019 Pez's caption reads.\n\nAlex smiles despite himself and texts Henry.\n\nATTN: will be wearing a burgundy velvet suit tonight. please do not attempt to steal my shine. you will fail and i will be embarrassed for you.\n\nHenry texts back seconds later.\n\nWouldn't dream of it.\n\nFrom there everything speeds up, and a hairstylist is wrangling him into the Cosmetology Room, and he gets to watch the girls transform into their camera-ready selves. Nora's short curls are swept to one side with a silver pin shaped to match the sharp geometric lines on the bodice of her black dress; June's gown is a plunging Zac Posen number in a shade of midnight blue that perfectly complements the navy-and-gold color palette they chose.\n\nThe guests start arriving around eight, and the liquor starts flowing, and Alex orders a middle-shelf whiskey to get things going. There's live music, a pop act that owed June a personal favor, and they're covering \"American Girl\" right now, so Alex grabs June's hand and spins her onto the dance floor.\n\nFirst arrivals are always the first-time political types: a small gaggle of White House interns, an event planner for Center for American Progress, the daughter of a first-term senator with a punk rock\u2013looking girlfriend who Alex makes a mental note to introduce himself to later. Then, the wave of politically strategic invites chosen by the press team, and lastly, the fashionably late\u2014minor to mid-range pop stars, teen soap actors, children of major celebrities.\n\nHe's just wondering when Henry's going to make his appearance, when June appears at his side and yells, \"Incoming!\"\n\nAlex's gaze is met by a bright burst of color that turns out to be Pez's bomber jacket, which is a shiny silk thing in such an elaborate, colorful floral print that Alex almost has to squint. The colors fade slightly, though, when his eyes slide to the right.\n\nIt's the first time Alex has seen Henry in person since the weekend in London and the hundreds of texts and weird in-jokes and late-night phone calls that came after, and it almost feels like meeting a new person. He knows more about Henry, understands him better, and he can appreciate the rarity of a genuine smile on the same famously beautiful face.\n\nIt's a weird cognitive dissonance, Henry present and Henry past. That must be why something feels so restless and hot somewhere beneath his sternum. That and the whiskey.\n\nHenry's wearing a simple dark blue suit, but he's opted for a bright coppery-mustard tie in a narrow cut. He spots Alex, and his smile broadens, giving Pez's arm a tug.\n\n\"Nice tie,\" Alex says as soon as Henry is close enough to hear over the crowd.\n\n\"Thought I might be escorted off the premises for anything less exciting,\" Henry says, and his voice is somehow different than Alex remembers. Like very expensive velvet, something moneyed and lush and fluid all at once.\n\n\"And _who_ is this?\" June asks from Alex's side, interrupting his train of thought.\n\n\"Ah yes, you've not officially met, have you?\" Henry says. \"June, Alex, this is my best mate, Percy Okonjo.\"\n\n\"Pez, like the sweets,\" Pez says cheerfully, extending his hand to Alex. Several of his fingernails are painted blue. When he redirects his attention to June, his eyes grow brighter, his grin spreading. \"Please do smack me if this is out of line, but you are the most exquisite woman I have ever seen in my life, and I would like to procure for you the most lavish drink in this establishment if you will let me.\"\n\n\"Uh,\" Alex says.\n\n\"You're a charmer,\" June says, smiling indulgently.\n\n\"And you are a goddess.\"\n\nHe watches them disappear into the crowd, Pez a blazing streak of color, already spinning June in a pirouette as they go. Henry's smile has gone sheepish and reserved, and Alex understands their friendship at last. Henry doesn't want the spotlight, and Pez naturally absorbs what Henry deflects.\n\n\"That man has been begging me to introduce him to your sister since the wedding,\" Henry says.\n\n_\"Seriously?\"_\n\n\"We've probably just saved him a tremendous amount of money. He was going to start pricing skywriters soon.\"\n\nAlex tosses his head back and laughs, and Henry watches, still grinning. June and Nora had a point. He does, against all odds, really like this person.\n\n\"Well, come on,\" Alex says. \"I'm already two whiskeys in. You've got some catching up to do.\"\n\nMore than one conversation drops out as Alex and Henry pass, mouths hanging open over entremets. Alex tries to imagine what they must look like: the prince and the First Son, the two leading heartthrobs of their respective countries, shoulder to shoulder on their way to the bar. It's intimidating and thrilling, living up to that kind of rich, untouchable fantasy. That's what people _see,_ but none of them know about the Great Turkey Calamity. Only Alex and Henry do.\n\nHe scores the first round and the crowd swallows them up. Alex is surprised how pleased he is by the physical presence of Henry next to him. He doesn't even mind having to look up at him anymore. He introduces Henry to some White House interns and laughs as they blush and stutter, and Henry's face goes pleasantly neutral, an expression Alex used to mistake as unimpressed but can now read for what it is: carefully concealed bemusement.\n\nThere's dancing, and mingling, and a speech by June about the immigration fund they're supporting with their donations tonight, and Alex ducks out of an aggressive come-on by a girl from the new Spider-Man movies and into a haphazard conga line, and Henry actually seems to have fun. June finds them at some point and steals Henry away to gab at the bar. Alex watches them from afar, wondering what they could possibly be talking about that has June nearly falling off her barstool laughing, until the crowd overtakes him again.\n\nAfter a while, the band breaks and a DJ takes over with a mix of early 2000s hip-hop, all the greatest hits that came out when Alex was a child and were somehow still in rotation at dances in his teens. That's when Henry finds him, like a man lost at sea.\n\n\"You don't dance?\" he says, watching Henry, who is very visibly trying to figure out what to do with to do with his hands. It's endearing. Wow, Alex is drunk.\n\n\"No, I do,\" Henry says. \"It's just, the family-mandated ballroom dancing lessons didn't exactly cover this?\"\n\n\"C'mon, it's, like, in the hips. You have to loosen up.\" He reaches down and puts both hands on Henry's hips, and Henry instantly tenses under the touch. \"That's the opposite of what I said.\"\n\n\"Alex, I don't\u2014\"\n\n\"Here,\" Alex says, moving his own hips, \"watch me.\"\n\nWith a grave gulp of champagne, Henry says, \"I am.\"\n\nThe song crossfades into another _buh-duh dum-dum-dum, dum-duh-dum duh-duh-dum\u2014_\n\n_\"Shut up,\"_ Alex yells, cutting off whatever else Henry was saying, \"shut your dumb face, this is my _shit_!\" He throws his hands up in the air as Henry stares at him blankly, and around them, people start cheering too, hundreds of shoulders shimmying to the shouty, Lil Jon\u2013flavored nostalgia of \"Get Low.\"\n\n\"Did you seriously never go to an awkward middle school dance and watch a bunch of teenagers dry hump to this song?\"\n\nHenry is holding on to his champagne for dear life. \"You absolutely must know I did not.\"\n\nAlex flails one arm out and snatches Nora from a nearby huddle, where she's been flirting with Spider-Man girl. \"Nora! _Nora!_ Henry has never watched a bunch of teenagers dry hump to this song!\"\n\n_\"What?\"_\n\n\"Please tell me nobody is going to _dry hump_ me,\" Henry says.\n\n\"Oh my God, Henry,\" Alex yells, seizing Henry by one lapel as the music pounds on, \"you have to dance. You _have to_ dance. You need to understand this formative American coming-of-age experience.\"\n\nNora grabs Alex, pulling him away from Henry and spinning him around, her hands on his waist, and starts grinding with abandon. Alex whoops and Nora cackles and the crowd jumps around and Henry just gawks at them.\n\n\"Did that man just say ' _sweat drop down my balls_ '?\"\n\nIt's _fun_ \u2014Nora against his back, sweat on his brow, bodies pushing in around him. To one side, a podcast producer and that guy from _Stranger Things_ are hitting the Kid 'n Play, and to the other, Pez is literally bending over to the front and touching his toes as instructed. Henry's face is shocked and confused, and it's hilarious. Alex accepts a shot off a passing tray and drinks to the strange spark in his gut at the way Henry watches them. Alex pouts his lips and shakes his ass, and with extreme trepidation, Henry starts bopping his head a little.\n\n\"Fuck it up, vato!\" Alex yells, and Henry laughs despite himself. He even gives his hips a little shake.\n\n\"I thought you weren't going to babysit him all night,\" June stage-whispers in his ear as she twirls by.\n\n\"I thought _you_ were too busy for guys,\" Alex replies, nodding significantly at Pez in the periphery. She winks at him and disappears.\n\nFrom there, it's a series of crowd-pleasers until midnight, the lights and music blasting at full capacity. Confetti, somehow blasting into the air. Did they arrange for confetti cannons? More drinks\u2014Henry starts drinking directly from a bottle of Mo\u00ebt & Chandon. Alex likes the look on Henry's face, the sure curl of his hand around the neck of the bottle, the way his lips wrap around the mouth of it. Henry's willingness to dance is directly proportionate to his proximity to Alex's hands, and the amount of giddy warmth bubbling under Alex's skin is directly proportionate to the cut of Henry's mouth when he watches him with Nora. It's an equation he is not nearly sober enough to parse.\n\nThey all huddle up at 11:59 for the countdown, eyes blurry and arms around one another. Nora screams \"three, two, one\" right in his ear and slings her arm around his neck as he yells his approval and kisses her sloppily, laughing through it. They've done this every year, both of them perpetually single and affectionately drunk and happy to make everyone else intrigued and jealous. Nora's mouth is warm and tastes horrifying, like peach schnapps, and she bites his lip and messes up his hair for good measure.\n\nWhen he opens his eyes, Henry's looking back at him, expression unreadable.\n\nHe feels his own smile grow wider, and Henry turns away and toward the bottle of champagne clutched in his fist, from which he takes a hearty swig before disappearing into the crowd.\n\nAlex loses track of things after that, because he's very, very drunk and the music is very, very loud and there are very, very many hands on him, carrying him through the tangle of dancing bodies and passing him more drinks. Nora bobs by on the back of some hot rookie NFL running back.\n\nIt's loud and messy and wonderful. Alex has always loved these parties, the sparkling joy of it all, the way champagne bubbles on his tongue and confetti sticks to his shoes. It's a reminder that even though he stresses and stews in private rooms, there will always be a sea of people he can disappear into, that the world can be warm and welcoming and fill up the walls of this big old house he lives in with something bright and infectiously alive.\n\nBut somewhere, beneath the liquor and the music, he can't stop noticing that Henry has disappeared.\n\nHe checks the bathrooms, the buffet, the quiet corners of the ballroom, but he's nowhere. He tries asking Pez, shouting Henry's name at him over the noise, but Pez just smiles and shrugs and steals a snapback off a passing yacht kid.\n\nHe's... worried isn't exactly the word. Bothered. Curious. He was having fun watching everything he did play out on Henry's face. He keeps looking, until he trips over his own feet by one of the big windows in the hallway. He's pulling himself up when he glances outside, down into the garden.\n\nThere, under a tree in the snow, exhaling little puffs of steam, is a tall, lean, broad-shouldered figure that can only be Henry.\n\nHe slips out onto the portico without really thinking about it, and the instant the door closes behind him, the music snuffs out into silence, and it's just him and Henry and the garden. He's got the hazy tunnel vision of a drunk person when they lock eyes on a goal. He follows it down the stairs and onto the snowy lawn.\n\nHenry stands quietly, hands in his pockets, contemplating the sky, and he'd almost look sober if not for the wobbly lean to the left he's doing. Stupid English dignity, even in the face of champagne. Alex wants to push his royal face into a shrub.\n\nAlex trips over a bench, and the sound catches Henry's attention. When he turns, the moonlight catches on him, and his face looks softened in half shadows, inviting in a way Alex can't quite work out.\n\n\"What're you doing out here?\" Alex says, trudging up to stand next to him under the tree.\n\nHenry squints. Up close, his eyes go a little crossed, focused somewhere between himself and Alex's nose. Not so dignified after all.\n\n\"Looking for Orion,\" Henry says.\n\nAlex huffs a laugh, looking up to the sky. Nothing but fat winter clouds. \"You must be really bored with the commoners to come out here and stare at the clouds.\"\n\n\"'m not bored,\" Henry mumbles. \"What are _you_ doing out here? Doesn't America's golden boy have some swooning crowds to beguile?\"\n\n\"Says Prince fucking Charming,\" Alex answers, smirking.\n\nHenry pulls a very unprincely face up at the clouds. \"Hardly.\"\n\nHis knuckle brushes the back of Alex's hand at their sides, a little zip of warmth in the cold night. Alex considers his face in profile, blinking through the booze, following the smooth line of his nose and the gentle dip at the center of his lower lip, each touched by moonlight. It's freezing and Alex is only wearing his suit jacket, but his chest feels warmed from the inside with liquor and something heady his brain keeps stumbling over, trying to name. The garden is quiet except for the blood rushing in his ears.\n\n\"You didn't really answer my question, though,\" Alex notes.\n\nHenry groans, rubbing a hand across his face. \"You can't ever leave well enough alone, can you?\" He leans his head back. It thumps gently against the trunk of the tree. \"Sometimes it gets a bit... much.\"\n\nAlex keeps looking at him. Usually, there's something about the set of Henry's mouth that betrays a bit of friendliness, but sometimes, like right now, his mouth pinches in the corner instead, pins his guard resolutely in place.\n\nAlex shifts, almost involuntarily, leaning back against the tree too. He nudges their shoulders together and catches that corner of Henry's mouth twitching, sees something move featherlight across his face. These things\u2014big events, letting other people feed on his own energy\u2014are rarely too much for Alex. He's not sure how Henry feels, but some part of his brain that is likely soaked in tequila thinks maybe it would be helpful if Henry could take what he can handle, and Alex could handle the rest. Maybe he can absorb some of the \"much\" from the place where their shoulders are pressed together.\n\nA muscle in Henry's jaw moves, and something soft, almost like a smile, tugs at his lips. \"D'you ever wonder,\" he says slowly, \"what it's like to be some anonymous person out in the world?\"\n\nAlex frowns. \"What do you mean?\"\n\n\"Just, you know,\" Henry says. \"If your mum weren't the president and you were just a normal bloke living a normal life, what things might be like? What you'd be doing instead?\"\n\n\"Ah,\" Alex says, considering. He stretches one arm out in front of him, makes a dismissive gesture with a flick of his wrist. \"Well, I mean, obviously I'd be a model. I've been on the cover of _Teen Vogue_ twice. These genetics transcend all circumstance.\" Henry rolls his eyes again. \"What about you?\"\n\nHenry shakes his head ruefully. \"I'd be a writer.\"\n\nAlex gives a little laugh. He thinks he already knew this about Henry, somehow, but it's still kind of disarming. \"Can't you do that?\"\n\n\"Not exactly seen as a worthwhile pursuit for a man in line for the throne, scribbling verses about quarter-life angst,\" Henry says dryly. \"Besides, the traditional family career track is military, so that's about it, isn't it?\"\n\nHenry bites his lip, waits a beat, and opens his mouth again. \"I'd date more, probably, as well.\"\n\nAlex can't help laughing again. \"Right, because it's so hard to get a date when you're a prince.\"\n\nHenry cuts his eyes back down to Alex. \"You'd be surprised.\"\n\n\"How? You're not exactly lacking for options.\"\n\nHenry keeps looking at him, holding his gaze for two seconds too long. \"The options I'd like...\" he says, dragging the words out. \"They don't quite seem to be _options_ at all.\"\n\nAlex blinks. \"What?\"\n\n\"I'm saying that I have... people... who interest me,\" Henry says, turning his body toward Alex now, speaking with a fumbling pointedness, as if it means something. \"But I shouldn't pursue them. At least not in my position.\"\n\nAre they too drunk to communicate in English? He wonders distantly if Henry knows any Spanish.\n\n\"I don't know what the hell you're talking about,\" Alex says.\n\n\"You don't?\"\n\n\"No.\"\n\n\"You really don't?\"\n\n\"I really, really don't.\"\n\nHenry's whole face grimaces in frustration, his eyes casting skyward like they're searching for help from an uncaring universe. \"Christ, you are as thick as it gets,\" he says, and he grabs Alex's face in both hands and kisses him.\n\nAlex is frozen, registering the press of Henry's lips and the wool cuffs of his coat grazing his jaw. The world fuzzes out into static, and his brain is swimming hard to keep up, adding up the equation of teenage grudges and wedding cakes and two a.m. texts and not understanding the variable that got him here, except it's... well, surprisingly, he really doesn't mind. Like, at all.\n\nIn his head, he tries to cobble a list together in a panic, gets as far as, _One, Henry's lips are soft,_ and short-circuits.\n\nHe tests leaning into the kiss and is rewarded by Henry's mouth sliding and opening against his, Henry's tongue brushing against his, which is, _wow._ It's nothing like kissing Nora earlier\u2014nothing like kissing anyone he's ever kissed in his life. It feels as steady and huge as the ground under their feet, as encompassing of every part of him, as likely to knock the wind out of his lungs. One of Henry's hands pushes into his hair and grabs it at the roots at the back of his head, and he hears himself make a sound that breaks the breathless silence, and\u2014\n\nJust as suddenly, Henry releases him roughly enough that he staggers backward, and Henry's mumbling a curse and an apology, eyes wide, and he's spinning on his heel, crunching off through the snow at double time. Before Alex can say or do anything, he's disappeared around the corner.\n\n\"Oh,\" Alex says finally, faintly, touching one hand to his lips. Then: \"Shit.\"\n\n# FIVE\n\nSo, the thing about the kiss is, Alex absolutely cannot stop thinking about it.\n\nHe's tried. Henry and Pez and their bodyguards were long gone by the time Alex made it back inside. Not even a drunken stupor or the next morning's pounding hangover can scrub the image from his brain.\n\nHe tries listening in on his mom's meetings, but they can't hold his attention, and Zahra bans him from the West Wing. He studies every bill trickling through Congress and considers making rounds to sweet-talk senators, but can't muster the enthusiasm. Not even starting a rumor with Nora sounds enticing.\n\nHe starts his last semester, goes to class, sits with the social secretary to plan his graduation dinner, buries himself in highlighted annotations and supplemental readings.\n\nBut beneath it all, there's the Prince of England kissing him under a linden tree in the garden, moonlight in his hair, and Alex's insides feel positively _molten,_ and he wants to throw himself down the presidential stairs.\n\nHe hasn't told anyone, not even Nora or June. He has no idea what he'd even say if he _did._ Is he even technically allowed to tell anyone, since he signed an NDA? Was this _why_ he had to sign it? Is this something Henry always had in mind? Does that mean Henry has _feelings_ for him? Why would Henry have acted like a tedious prick for so long if he liked him?\n\nHenry's not offering any insights, or anything at all. He hasn't answered a single one of Alex's texts or calls.\n\n\"Okay, that's it,\" June says on a Wednesday afternoon, stomping out of her room and into the sitting room by their shared hallway. She's in her workout clothes with her hair tied up. Alex hastily shoves his phone back into his pocket. \"I don't know what your problem is, but I have been trying to write for two hours and I can't do it when I can hear you pacing.\" She throws a baseball cap at him. \"I'm going for a run, and you're coming with me.\"\n\nCash accompanies them to the Reflecting Pool, where June kicks the back of Alex's knee to get him going, and Alex grunts and swears and picks up the pace. He feels like a dog that has to be taken on walks to get his energy out. Especially when June says, \"You're like a dog that has to be taken on walks to get his energy out.\"\n\n\"I hate you sometimes,\" he tells her, and he shoves his earbuds in and cranks up Kid Cudi.\n\nHe thinks, as he runs and runs and runs, the stupidest thing of all is that he's straight.\n\nLike, he's pretty sure he's straight.\n\nHe can pinpoint moments throughout his life when he thought to himself, _See, this means I can't possibly be into guys._ Like when he was in middle school and he kissed a girl for the first time, and he didn't think about a guy when it was happening, just that her hair was soft and it felt nice. Or when he was a sophomore in high school and one of his friends came out as gay, and he couldn't imagine ever doing anything like that.\n\nOr his senior year, when he got drunk and made out with Liam in his twin bed for an hour, and he didn't have a sexual crisis about it\u2014that had to mean he was straight, right? Because if he were into guys, it would have felt scary to be with one, but it wasn't. That was just how horny teenage best friends were sometimes, like when they would get off at the same time watching porn in Liam's bedroom... or that one time Liam reached over, and Alex didn't stop him.\n\nHe glances over at June, at the suspicious quirk of her lips. Can she hear what he's thinking? Does she know, somehow? June always knows things. He doubles his pace, if only to get the expression on her mouth out of his periphery.\n\nOn their fifth lap, he thinks back over his hormonal teens and remembers thinking about girls in the shower, but he also remembers fantasizing about a boy's hands on him, about hard jawlines and broad shoulders. He remembers pulling his eyes off a teammate in the locker room a couple times, but that was, like, an objective thing. How was he supposed to know back then if he wanted to look like other guys, or if he _wanted_ other guys? Or if his horny teenage urges actually even meant anything?\n\nHe's a son of Democrats. It's something he's always been around. So, he always assumed if he weren't straight, he would just _know,_ like how he knows that he loves cajeta on his ice cream or that he needs a tediously organized calendar to get anything done. He thought he was smart enough about his own identity that there weren't any questions left.\n\nThey're rounding the corner for their eighth lap now, and he's starting to see some flaws in his logic. Straight people, he thinks, probably don't spend this much time convincing themselves they're straight.\n\nThere's another reason he never cared to examine things beyond the basic benchmark of being attracted to women. He's been in the public eye since his mom became the favored 2016 nominee, the White House Trio the administration's door to the teen and twenty-something demographic almost as long. All three of them\u2014himself, June, and Nora\u2014have their roles.\n\nNora is the cool brainy one, the one who makes inappropriate jokes on Twitter about whatever sci-fi show everyone's watching, a bar trivia team ringer. She's not straight\u2014she's never been straight\u2014but to her, it's an incidental part of who she is. She doesn't worry about going public with it; feelings don't consume her the way his do.\n\nHe looks at June\u2014ahead of him now, caramel highlights in her swinging ponytail catching the midday sun\u2014and he knows her place too. The intrepid _Washington Post_ columnist, the fashion trendsetter everyone wants to have at their wine-and-cheese night.\n\nBut Alex is the golden boy. The heartthrob, the handsome rogue with a heart of gold. The guy who moves through life effortlessly, who makes everyone laugh. Highest approval ratings of the entire First Family. The whole point of him is that his appeal is as universal as possible.\n\nBeing... whatever he's starting to suspect he might be, is definitely not universally appealing to voters. He has a hard enough time being half-Mexican.\n\nHe wants his mom to keep her approval ratings up without having to manage a complication from her own family. He wants to be the youngest congressman in US history. He's absolutely sure that guys who kissed a Prince of England and liked it don't get elected to represent Texas.\n\nBut he thinks about Henry, and, _oh._\n\nHe thinks about Henry, and something twists in his chest, like a stretch he's been avoiding for too long.\n\nHe thinks about Henry's voice low in his ear over the phone at three in the morning, and suddenly he has a name for what ignites in the pit of his stomach. Henry's hands on him, his thumbs braced against his temples back in the garden, Henry's hands other places, Henry's mouth, what he might do with it if Alex let him. Henry's broad shoulders and long legs and narrow waist, the place his jaw meets his neck and the place his neck meets his shoulder and the tendon that stretches the length between them, and the way it looks when Henry turns his head to shoot him a challenging glare, and his impossibly blue eyes\u2014\n\nHe trips on a crack in the pavement and goes tumbling down, skinning his knee and ripping his earbuds out.\n\n\"Dude, what the hell?\" June's voice cuts through the ringing in his ears. She's standing over him, hands on her knees, brow furrowed, panting. \"Your brain could not be more clearly in another solar system. Are you gonna tell me or what?\"\n\nHe takes her hand and lets her pull him and his bloody knee up. \"It's fine. I'm fine.\"\n\nJune sighs, shooting him another look before finally dropping it. Once he's limped back home behind her, she disappears to shower and he stems the bleeding with a Captain America Band-Aid from his bathroom cabinet.\n\nHe needs a list. So: Things he knows right now.\n\nOne. He's attracted to Henry.\n\nTwo. He wants to kiss Henry again.\n\nThree. He has maybe wanted to kiss Henry for a while. As in, probably this whole time.\n\nHe ticks off another list in his head. Henry. Shaan. Liam. Han Solo. Rafael Luna and his loose collars.\n\nSidling up to his desk, he pulls out the binder his mother gave him: DEMOGRAPHIC ENGAGEMENT: WHO THEY ARE AND HOW TO REACH THEM. He drags his finger down to the LGBTQ+ tab and turns to the page he's looking for, titled with mother's typical flair: THE B ISN'T SILENT: A CRASH COURSE ON BISEXUAL AMERICANS.\n\n* * *\n\n\"I wanna start now,\" Alex says as he slams into the Treaty Room.\n\nHis mother lowers her glasses to the tip of her nose, eyeing him over a pile of papers. \"Start what? Getting your ass beat for barging in here while I'm working?\"\n\n\"The job,\" he says. \"The campaign job. I don't wanna wait until I graduate. I already read all the materials you gave me. Twice. I have time. I can start now.\"\n\nShe narrows her eyes at him. \"You got a bug up your butt?\"\n\n\"No, I just...\" One of his knees is bouncing impatiently. He forces it to stop. \"I'm ready. I've got less than one semester left. How much more could I possibly need to know to do this? Put me in, Coach.\"\n\nWhich is how he finds himself out of breath on a Monday afternoon after class, following a staffer who's managed to surpass even him in the caffeination department, on a breakneck tour of the campaign offices. He gets a badge with his name and photo on it, a desk in a shared cubicle, and a WASPy cubicle mate from Boston named Hunter with an extremely punchable face.\n\nAlex is handed a folder of data from the latest focus groups and told to start drafting policy ideas for the end of the following week, and WASPy Hunter asks him five hundred questions about his mom. Alex very professionally does not punch him. He just gets to work.\n\nHe's definitely not thinking about Henry.\n\nHe's not thinking about Henry when he puts in twenty-three hours in his first week of work, or when he's filling the rest of his hours with class and papers and going for long runs and drinking triple-shot coffees and poking around the Senate offices. He's not thinking about Henry in the shower or at night, alone and wide awake in his bed.\n\nExcept for when he is. Which is always.\n\nThis usually works. He doesn't understand why it's not working.\n\nWhen he's in the campaign offices, he keeps gravitating over to the big, busy whiteboards of the polling section, where Nora sits every day enshrined in graphs and spreadsheets. She's made easy friends with her coworkers, since competence translates directly to popularity in the campaign social culture, and nobody's better at numbers than her.\n\nHe's not jealous, exactly. He's popular in his own department, constantly cornered at the Keurig for second opinions on people's drafts and invited to after-work drinks he never has time for. At least four staffers of various genders have hit on him, and WASPy Hunter won't stop trying to convince him to come to his improv shows. He smiles handsomely over his coffee and makes sarcastic jokes and the Alex Claremont-Diaz Charm Initiative is as effective as ever.\n\nBut Nora makes _friends,_ and Alex ends up with acquaintances who think they know him because they've read his profile in _New York_ magazine _,_ and perfectly fine people with perfectly fine bodies who want to take him home from the bar. None of it is satisfying\u2014it never has been, not really, but it never mattered as much as it does now that there's the sharp counterpoint of Henry, who _knows_ him. Henry who's seen him in glasses and tolerates him at his most annoying and still kissed him like he wanted him, singularly, not the idea of him.\n\nSo it goes, and Henry is there, in his head and his lecture notes and his cubicle, every single stupid day, no matter how many shots of espresso he puts in his coffee.\n\n* * *\n\nNora would be the obvious choice for help, if not for the fact that she's neck deep in polling numbers. When she gets into her work like this, it's like trying to have a meaningful conversation with a high-speed computer that loves Chipotle and makes fun of what you're wearing.\n\nBut she's his best friend, and she's sort of vaguely bisexual. She never dates\u2014no time or desire\u2014but if she did, she says it'd be an even distribution of the intern pool. She's as knowledgeable about the topic as she is about everything else.\n\n\"Hello,\" she says from the floor as he drops a bag of burritos and a second bag of chips with guacamole on the coffee table. \"You might have to put guacamole directly into my mouth with a spoon because I need both hands for the next forty-eight hours.\"\n\nNora's grandparents, the Veep and Second Lady, live at the Naval Observatory, and her parents live just outside of Montpelier, but she's had the same airy one-bedroom in Columbia Heights since she transferred from MIT to GW. It's full of books and plants she tends to with complex spreadsheets of watering schedules. Tonight, she's sitting on her living room floor in a glowing circle of screens like some kind of Capitol Hill s\u00e9ance.\n\nTo her left, her campaign laptop is open to an indecipherable page of data and bar graphs. To her right, her personal computer is running three news aggregators at the same time. In front of her, the TV is broadcasting CNN's Republican primary coverage, while the tablet in her lap is playing an old episode of _Drag Race._ She's holding her iPhone in her hand, and Alex hears the little whoosh of an email sending before she looks up at him.\n\n\"Barbacoa?\" she says hopefully as Alex drops onto the couch.\n\n\"I've met you before today, so, obviously.\"\n\n\"There's my future husband.\" She leans over to pull a burrito out of the bag, rips off the foil, and shoves it into her mouth.\n\n\"I'm not going to have a marriage of convenience with you if you're always embarrassing me with the way you eat burritos,\" Alex says, watching her chew. A black bean falls out of her mouth and lands on one of her keyboards.\n\n\"Aren't you from Texas?\" she says through her mouthful. \"I've seen you shotgun a bottle of barbecue sauce. Watch yourself or I'm gonna marry June instead.\"\n\nThis might be his opening into \"the conversation.\" _Hey, you_ _know how you're always joking about dating June? Well, like, what if I dated a guy?_ Not that he wants to date Henry. At all. Ever. But just, like, hypothetically.\n\nNora goes off on a data nerd tangent for the next twenty minutes about her updated take on whatever the fuck the Boyer\u2013Moore majority vote algorithm is and variables and how it can be used in whatever work she's doing for the campaign, or something. Honestly, Alex's concentration is drifting in and out. He's just working on summoning up courage until she talks herself into submission.\n\n\"Hey, so, uh,\" Alex attempts as she takes a burrito break. \"Remember when we dated?\"\n\nNora swallows a massive bite and grins. \"Why yes, I do, Alejandro.\"\n\nAlex forces a laugh. \"So, knowing me as well as you do\u2014\"\n\n\"In the biblical sense.\"\n\n\"Numbers on me being into dudes?\"\n\nThat pulls Nora up short, before she cocks her head to the side and says, \"Seventy-eight percent probability of latent bisexual tendencies. One hundred percent probability this is not a hypothetical question.\"\n\n\"Yeah. So.\" He coughs. \"Weird thing happened. You know how Henry came to New Year's? He kinda... kissed me?\"\n\n\"Oh, no shit?\" Nora says, nodding appreciatively. \"Nice.\"\n\nAlex stares at her. \"You're not surprised?\"\n\n\"I mean.\" She shrugs. \"He's gay, and you're hot, so.\"\n\nHe sits up so quickly he almost drops his burrito on the floor. \"Wait, wait\u2014what makes you think he's gay? Did he tell you he was?\"\n\n\"No, I just... like, you know.\" She gesticulates as if to describe her usual thought process. It's as incomprehensible as her brain. \"I observe patterns and data, and they form logical conclusions, and he's just gay. He's always been gay.\"\n\n\"I... what?\"\n\n\"Dude. Have you met him? Isn't he supposed to be your best friend or whatever? He's gay. Like, Fire-Island-on-the-Fourth-of-July gay. Did you really not know?\"\n\nAlex lifts his hands helplessly. \"No?\"\n\n\"Alex, I thought you were supposed to be smart.\"\n\n\"Me too! How can he\u2014how can he spring a kiss on me without even telling me he's gay first?\"\n\n\"I mean, like,\" she attempts, \"is it possible he assumed you knew?\"\n\n\"But he goes on dates with girls all the time.\"\n\n\"Yeah, because princes aren't allowed to be gay,\" Nora says as if it's the most obvious thing in the world. \"Why do you think they're always photographed?\"\n\nAlex lets that sink in for half a second and remembers this is supposed to be about _his_ gay panic, not Henry's. \"Okay, so. Wait. Jesus. Can we go back to the part where he kissed me?\"\n\n\"Ooh, yes,\" Nora says. She licks a glob of guacamole off the screen of her phone. \"Happily. Was he a good kisser? Was there tongue? Did you like it?\"\n\n\"Never mind,\" Alex says instantly. \"Forget I asked.\"\n\n\"Since when are you a prude?\" Nora demands. \"Last year you made me listen to every nasty detail about going down on Amber Forrester from June's internship.\"\n\n\"Do _not,_ \" he says, hiding his face behind the crook of his elbow.\n\n\"Then spill.\"\n\n\"I seriously hope you die,\" he says. \"Yes, he was a good kisser, and there was tongue.\"\n\n\"I fucking knew it,\" she says. \"Still waters, deep dicking.\"\n\n_\"Stop,\"_ he groans.\n\n\"Prince Henry is a biscuit,\" Nora says, \"let him sop you up.\"\n\n\"I'm _leaving._ \"\n\nShe throws her head back and cackles, and seriously, Alex has _got_ to get more friends. \"Did you like it, though?\"\n\nA pause.\n\n\"What, um,\" he starts. \"What do you think it would mean... if I did?\"\n\n\"Well. Babe. You've been wanting him to dick you down forever, right?\"\n\nAlex almost chokes on his tongue. _\"What?\"_\n\nNora looks at him. \"Oh, shit. Did you not know that either? Shit. I didn't mean to, like, tell you. Is it time for this conversation?\"\n\n\"I... maybe?\" he says. \"Um. What?\"\n\nShe puts her burrito down on the coffee table and shakes her fingers out like she does when she's about to write a complicated code. Alex suddenly feels intimidated at having her undivided attention.\n\n\"Let me lay out some observations for you,\" she says. \"You extrapolate. First, you've been, like, Draco Malfoy\u2013level obsessed with Henry for years\u2014 _do not interrupt me_ \u2014and since the royal wedding, you've gotten his phone number and used it not to set up any appearances but instead to long-distance flirt with him all day every day. You're constantly making big cow eyes at your phone, and if somebody asks you who you're texting, you act like you got caught watching porn. You know his sleep schedule, he knows your sleep schedule, and you're in a noticeably worse mood if you go a day without talking to him. You spent the entire New Year's party straight-up ignoring the who's who of hot people who want to fuck America's most eligible bachelor to literally watch Henry stand next to the croquembouche. And he kissed you\u2014with tongue!\u2014and you liked it. So, objectively. What do you think it means?\"\n\nAlex stares. \"I mean,\" he says slowly. \"I don't... know.\"\n\nNora frowns, visibly giving up, resumes eating her burrito, and returns her attention to the newsfeed on her laptop. \"Okay.\"\n\n\"No, okay, look,\" Alex says. \"I know, like, objectively, on a fucking graphing calculator, it sounds like a huge embarrassing crush. But, ugh. I don't know! He was my sworn enemy until a couple months ago, and then we were friends, I guess, and now he's kissed me, and I don't know what we... _are._ \"\n\n\"Uh-huh,\" Nora says, very much not listening. \"Yep.\"\n\n\"And, still,\" he barrels on. \"In terms of, like, sexuality, what does that make me?\"\n\nNora's eyes snap back up to him. \"Oh, like, I thought we were already there with you being bi and everything,\" she says. \"Sorry, are we not? Did I skip ahead again? My bad. Hello, would you like to come out to me? I'm listening. Hi.\"\n\n\"I don't know!\" he half yells, miserably. \"Am I? Do you think I'm bi?\"\n\n\"I can't tell you that, Alex!\" she says. \"That's the whole point!\"\n\n\"Shit,\" he says, dropping his head back on the cushions. \"I need someone to just tell me. How did you know you were?\"\n\n\"I don't know, man. I was in my junior year of high school, and I touched a boob. It wasn't very profound. Nobody's gonna write an Off-Broadway play about it.\"\n\n\"Really helpful.\"\n\n\"Yup,\" she says, chewing thoughtfully on a chip. \"So, what are you gonna do?\"\n\n\"I have no idea,\" Alex says. \"He's totally ghosted me, so I guess it was awful or a stupid drunk mistake he regrets or\u2014\"\n\n\"Alex,\" she says. \"He _likes_ you. He's freaking out. You're gonna have to decide how you feel about him and do something about it. He's not in a position to do anything else.\"\n\nAlex has no idea what else to say about any of it. Nora's eyes drift back to one of her screens, where Anderson Cooper is unpacking the latest coverage of the Republican presidential hopefuls.\n\n\"Any chance someone other than Richards gets the nomination?\"\n\nAlex sighs. \"Nope. Not according to anybody I've talked to.\"\n\n\"It's almost cute how hard the others are still trying,\" she says, and they lapse into silence.\n\n* * *\n\nAlex is late, again.\n\nHis class is reviewing for the first exam today, and he's late because he lost track of time going over his speech for the campaign event he's doing in fucking _Nebraska_ this weekend, of all godforsaken places. It's Thursday, and he's hauling ass straight from work to the lecture hall, and his exam is next Tuesday, and he's going to _fail_ because he's missing the _review._\n\nThe class is Ethical Issues in International Relations. He really has got to stop taking classes so painfully relevant to his life.\n\nHe gets through the review in a haze of half-distracted shorthand and books it back toward the Residence. He's pissed, honestly. Pissed at everything; a crawling, directionless bad mood that's carrying him up the stairs toward the East and West Bedrooms.\n\nHe throws his bag down at the door of his room and kicks his shoes into the hallway, watching them bounce crookedly across the ugly antique rug.\n\n\"Well, good afternoon to you too, honey biscuit,\" June's voice says. When Alex glances up, she's in her room across the hall, perched on a pastel-pink wingback chair. \"You look like shit.\"\n\n\"Thanks, asshole.\"\n\nHe recognizes the stack of magazines in her lap as her weekly tabloid roundup, and he's just decided he doesn't want to know when she chucks one at him.\n\n\"New _People_ for you,\" she says. \"You're on page fifteen. Oh, and your BFF's on page thirty-one.\"\n\nHe casually extends her the finger over his shoulder and retreats into his room, slumping down onto the couch by the door with the magazine. Since he has it, he might as well.\n\nPage fifteen is a picture of him the press team took two weeks ago, a nice, neat little package on him helping the Smithsonian with an exhibit about his mom's historic presidential campaign. He's explaining the story behind a CLAREMONT FOR CONGRESS '04 yard sign, and there's a brief write-up alongside it about how dedicated he is to the family legacy, blah blah blah.\n\nHe turns to page thirty-one and almost swears out loud.\n\nThe headline: WHO IS PRINCE HENRY'S MYSTERY BLONDE?\n\nThree photos: the first, Henry out at a cafe in London, smiling over coffees at some anonymously pretty blond woman; the second, Henry, slightly out of focus, holding her hand as they duck behind the cafe; the third, Henry, halfway obscured by a shrub, kissing the corner of her mouth.\n\n\"What the _fuck_?\"\n\nThere's a short article accompanying the photos that gives the girl's name, Emily something, an actress, and Alex was generally pissed before, but now he's very singularly pissed, his entire shitty mood funneled down to the point on the page where Henry's lips touch somebody's skin that's not _his._\n\nWho the fuck does Henry think he is? How fucking\u2014how entitled, how aloof, how _selfish_ do you have to be, to spend months becoming someone's friend, let them show you all their weird gross weak parts, kiss them, make them question _everything,_ ignore them for _weeks,_ and go out with someone else and _put it in the press_? Everyone who's ever had a publicist knows the only way anything gets into _People_ is if you want the world to know.\n\nHe throws the magazine down and lunges to his feet, pacing. _Fuck_ Henry. He should never have trusted the silver-spoon little shit. He should have listened to his gut.\n\nHe inhales, exhales.\n\nThe thing is. The thing. Is. He doesn't know if, beyond the initial rush of anger, he actually believes Henry would do this. If he takes the Henry he saw in a teen magazine when he was twelve, the Henry who was so cold to him at the Olympics, the Henry who slowly came unraveled to him over months, and the Henry who kissed him in the shadow of the White House, and he adds them up, he doesn't get this.\n\nAlex has a tactical brain. A politician's brain. It works fast, and it works in many, many directions at once. And right now, he's thinking through a puzzle. He's not always good at thinking: _What if you were him? How would your life be? What would you have to do?_ Instead, he's thinking: _How do these pieces slot together?_\n\nHe thinks about what Nora said: \"Why do you think they're always photographed?\"\n\nAnd he thinks about Henry's guardedness, the way he carries himself with a careful separation from the world around him, the tension at the corner of his mouth. Then he thinks: _If there was a prince, and he was gay, and he kissed someone, and maybe it mattered, that prince might have to run a little bit of interference._\n\nAnd in one great mercurial swing, Alex is not just angry anymore. He's sad too.\n\nHe paces back over to the door and slides his phone out of his messenger bag, thumbs open his messages. He doesn't know which impulse to follow and wrestle into words that he can say to someone and make something, _anything,_ happen.\n\nFaintly, under it all, it occurs to him: This is all a very not-straight way to react to seeing your male frenemy kissing someone else in a magazine.\n\nA little laugh startles out of him, and he walks over to his bed and sits on the edge of it, considering. He considers texting Nora, asking her if he can come over to finally have some big epiphany. He considers calling Rafael Luna and meeting him for beers and asking to hear all about his first gay sexual exploits as an REI-wearing teenage antifascist. And he considers going downstairs and asking Amy about her transition and her wife and how she knew she was different.\n\nBut in the moment, it feels right to go back to the source, to ask someone who's seen whatever is in his eyes when a boy touches him.\n\nHenry's out of the question. Which leaves one person.\n\n\"Hello?\" says the voice over the phone. It's been at least a year since they last talked, but Liam's Texas drawl is unmistakable and warm in Alex's eardrum.\n\nHe clears his throat. \"Uh, hey, Liam. It's Alex.\"\n\n\"I know,\" Liam says, desert-dry.\n\n\"How, um, how have you been?\"\n\nA pause. The sound of quiet talking in the background, dishes. \"You wanna tell me why you're really calling, Alex?\"\n\n\"Oh,\" he starts and stops, tries again. \"This might sound weird. But, um. Back in high school, did we have, like, a thing? Did I miss that?\"\n\nThere's a clattering sound on the other side of the phone, like a fork being dropped on a plate. \"Are you seriously calling me right now to talk about this? I'm at lunch with my boyfriend.\"\n\n\"Oh.\" He didn't know Liam had a boyfriend. \"Sorry.\"\n\nThe sound goes muffled, and when Liam speaks again, it's to someone else. \"It's Alex. Yeah, him. I don't know, babe.\" His voice comes back clear again. \"What exactly are you asking me?\"\n\n\"I mean, like, we messed around, but did it, like, mean something?\"\n\n\"I don't think I can answer that question for you,\" Liam tells him. If he's still anything like Alex remembers, he's rubbing one hand on the underside of his jaw, raking through the stubble. He wonders faintly if, perhaps, his clear-as-day memory of Liam's stubble has just answered his own question for him.\n\n\"Right,\" he says. \"You're right.\"\n\n\"Look, man,\" Liam says. \"I don't know what kind of sexual crisis you're having right now, like, four years after it would have been useful, but, well. I'm not saying what we did in high school makes you gay or bi or whatever, but I can tell you _I'm_ gay, and that even though I acted like what we were doing wasn't gay back then, it super was.\" He sighs. \"Does that help, Alex? My Bloody Mary is here and I need to talk to it about this phone call.\"\n\n\"Um, yeah,\" Alex says. \"I think so. Thanks.\"\n\n\"You're welcome.\"\n\nLiam sounds so long-suffering and tired that Alex thinks about all those times back in high school, the way Liam used to look at him, the silence between them since, and feels obligated to add, \"And, um. I'm sorry?\"\n\n\"Jesus _Christ,_ \" Liam groans, and hangs up.\n\n# SIX\n\nHenry can't avoid him forever.\n\nThere's one part of the post-royal wedding arrangement left to fulfill: Henry's presence at a state dinner at the end of January. England has a relatively new prime minister, and Ellen wants to meet him. Henry's coming too, staying in the Residence as a courtesy.\n\nAlex smooths out the lapels on his tux and hovers close to June and Nora as the guests roll in, waiting at the north entrance near the photo line. He's aware that he's rocking anxiously on his heels but can't seem to stop. Nora smirks but says nothing. She's keeping it quiet. He's still not ready to tell June. Telling his sister is irreversible, and he can't do that until he's figured out what exactly this is.\n\nHenry enters stage right.\n\nHis suit is black, smooth, elegant. Perfect. Alex wants to rip it off.\n\nHis face is reserved, then downright ashen when he sees Alex in the entrance hall. His footsteps stutter, as if he's thinking of making a run for it. Alex is not above a flying tackle.\n\nInstead, he keeps walking up the steps, and\u2014\n\n\"All right, photos,\" Zahra hisses over Alex's shoulder.\n\n\"Oh,\" Henry says, like an idiot. Alex hates how much he likes the way that one stupid vowel curls in his accent. He's not even into British accents. He's into _Henry's_ British accent.\n\n\"Hey,\" Alex says under his breath. Fake smile, handshake, cameras flashing. \"Cool to see you're not dead or anything.\"\n\n\"Er,\" Henry says, adding to the list of vowel sounds he has to show for himself. It is, unfortunately, also sexy. After all these weeks, the bar is low.\n\n\"We need to talk,\" Alex says, but Zahra is physically shoving them into a friendly formation, and there are more photos until Alex is being shepherded off with the girls to the State Dining Room while Henry is hauled into photo ops with the prime minister.\n\nThe entertainment for the night is a British indie rocker who looks like a root vegetable and is popular with people in Alex's demographic for reasons he can't even begin to understand. Henry is seated with the prime minister, and Alex sits and chews his food like it's personally wronged him and watches Henry from across the room, seething. Every so often, Henry will look up, catch Alex's eye, go pink around the ears, and return to his rice pilaf as if it's the most fascinating dish on the planet.\n\nHow _dare_ Henry come into Alex's house looking like the goddamn James Bond offspring that he is, drink red wine with the prime minister, and act like he didn't slip Alex the tongue and ghost him for a month.\n\n\"Nora,\" he says, leaning over to her while June is off chatting with an actress from _Doctor Who_. The night is starting to wind down, and Alex is over it. \"Can you get Henry away from his table?\"\n\nShe slants a look at him. \"Is this a diabolical scheme of seduction?\" she asks. \"If so, yes.\"\n\n\"Sure, yes, that,\" he says, and he gets up and heads for the back wall of the room, where the Secret Service is stationed.\n\n\"Amy,\" he hisses, grabbing her by the wrist. She makes a quick, aborted movement, clearly fighting a hardwired takedown reflex. \"I need your help.\"\n\n\"Where's the threat?\" she says immediately.\n\n\"No, no, Jesus.\" Alex swallows. \"Not like that. I need to get Prince Henry alone.\"\n\nShe blinks. \"I don't follow.\"\n\n\"I need to talk to him in private.\"\n\n\"I can accompany you outside if you need to speak with him, but I'll have to get it approved with his security first.\"\n\n\"No,\" Alex says. He scrubs a hand across his face, glancing back over his shoulder to confirm Henry's where he left him, being aggressively talked at by Nora. \"I need him _alone._ \"\n\nThe slightest of expressions crosses over Amy's face. \"The best I can do is the Red Room. You take him any farther and it's a no-go.\"\n\nHe looks over his shoulder again at the tall doors across the State Dining Room. The Red Room is empty on the other side, awaiting the after-dinner cocktails.\n\n\"How long can I have?\" he says.\n\n\"Five min\u2014\"\n\n\"I can make that work.\"\n\nHe turns on his heel and stalks over to the ornamental display of chocolates, where Nora has apparently lured Henry with the promise of profiteroles. He plants himself between them.\n\n\"Hi,\" he says. Nora smiles. Henry's mouth drops open. \"Sorry to interrupt. Important, um. International. Relations. Stuff.\" And he seizes Henry by the elbow and yanks him bodily away.\n\n\"Do you mind?\" Henry has the nerve to say.\n\n\"Shut your face,\" Alex says, briskly leading him away from the tables, where people are too busy mingling and listening to the music to notice Alex frog-marching an heir to the throne out of the dining room.\n\nThey reach the doors, and Amy is there. She hesitates, hand on the knob.\n\n\"You're not going to kill him, are you?\" she says.\n\n\"Probably not,\" Alex tells her.\n\nShe opens the door just enough to let them through, and Alex hauls Henry into the Red Room with him.\n\n\"What on God's earth are you doing?\" Henry demands.\n\n\"Shut _up,_ shut all the way up, oh my God,\" Alex hisses, and if he weren't already hell-bent on destroying Henry's infuriating idiot face with his mouth right now, he would consider doing it with his fist. He's focused on the burst of adrenaline carrying his feet over the antique rug, Henry's tie wrapped around his fist, the flash in Henry's eyes. He reaches the nearest wall, shoves Henry against it, and crushes their mouths together.\n\nHenry's too shocked to respond, mouth falling open slackly in a way that's more surprise than invitation, and for a horrified moment Alex thinks he calculated all wrong, but then Henry's kissing him back, and it's _everything._ It feels as good as\u2014better than\u2014he remembered, and he can't recall why they haven't been doing this the whole time, why they've been running belligerent circles around each other for so long without doing anything about it.\n\n\"Wait,\" Henry says, breaking off. He pulls back to look at Alex, wild-eyed, mouth a vivid red, and Alex could fucking scream if he weren't worried dignitaries in the next room might hear him. \"Should we\u2014\"\n\n\"What?\"\n\n\"I mean, er, should we, I dunno, slow down?\" Henry says, cringing so hard at himself that one eye closes. \"Go for dinner first, or\u2014\"\n\nAlex is actually going to kill him.\n\n\"We just had dinner.\"\n\n\"Right. I meant\u2014I just thought\u2014\"\n\n\"Stop thinking.\"\n\n\"Yes. Gladly.\"\n\nIn one frantic motion, Alex knocks the candelabra off the table next to them and pushes Henry onto it so he's sitting with his back against\u2014Alex looks up and almost breaks into deranged laughter\u2014a portrait of Alexander Hamilton. Henry's legs fall open readily and Alex crowds up between them, wrenching Henry's head back into another searing kiss.\n\nThey're really moving now, wrecking each other's suits, Henry's lip caught between Alex's teeth, the portrait's frame rattling against the wall when Henry's head drops back and bangs into it. Alex is at his throat, and he's somewhere between angry and giddy, caught up in the space between years of sworn hate and something else he's begun to suspect has always been there. It's white-hot, and he feels crazy with it, lit up from the inside.\n\nHenry gives as good as he gets, hooking one knee around the back of Alex's thigh for leverage, delicate royal sensibilities nowhere in the cut of his teeth. Alex has been learning for a while Henry isn't what he thought, but it's something else to feel it this close up, the quiet burn in him, the pent-up person under the perfect veneer who tries and pushes and wants.\n\nHe drops a hand onto Henry's thigh, feeling the electrical pulse there, the smooth fabric over hard muscle. He pushes up, up, and Henry's hand slams down over his, digging his nails in.\n\n\"Time's up!\" comes Amy's voice through a crack in the doors.\n\nThey freeze, Alex falling back onto his heels. They can both hear it now, the sounds of bodies moving too close for comfort, wrapping up the night. Henry's hips give one tiny push up into him, involuntary, surprised, and Alex swears.\n\n\"I'm going to die,\" Henry says helplessly.\n\n\"I'm going to kill you,\" Alex tells him.\n\n\"Yes, you are,\" Henry agrees.\n\nAlex takes an unsteady step backward.\n\n\"People are gonna be coming in here soon,\" Alex says, reaching down and trying not to fall on his face as he scoops up the candelabra and shoves it back onto the table. Henry is standing now, looking wobbly, his shirt untucked and his hair a mess. Alex reaches up in a panic and starts patting it back into place. \"Fuck, you look\u2014 _fuck._ \"\n\nHenry fumbles with his shirt tail, eyes wide, and starts humming \"God Save the Queen\" under his breath.\n\n\"What are you doing?\"\n\n\"Christ, I'm trying to make it\"\u2014he gestures inelegantly at the front of his pants\u2014\" _go away._ \"\n\nAlex very pointedly does not look down.\n\n\"Okay, so,\" Alex says. \"Yeah. So here's what we're gonna do. You are gonna go be, like, five hundred feet away from me for the rest of the night, or else I am going to do something that I will deeply regret in front of a lot of very important people.\"\n\n\"All right...\"\n\n\"And then,\" Alex says, and he grabs Henry's tie again, close to the knot, and draws his mouth up to a breath away from Henry's. He hears Henry swallow. He wants to follow the sound down his throat. \"And then you are going to come to the East Bedroom on the second floor at eleven o'clock tonight, and I am going to do very bad things to you, and if you fucking ghost me again, I'm going to get you put on a fucking no-fly list. Got it?\"\n\nHenry bites down on a sound that tries to escape his mouth, and rasps, \"Perfectly.\"\n\n* * *\n\nAlex is. Well, Alex is probably losing his mind.\n\nIt's 10:48. He's pacing.\n\nHe threw his jacket and tie over the back of the chair as soon as he returned to his room, and he's got the first two buttons of his dress shirt undone. His hands are twisted up in his hair.\n\nThis is fine. It's fine.\n\nIt's definitely a terrible idea. But it's fine.\n\nHe's not sure if he should take anything else off. He's unsure of the dress code for inviting your sworn-enemy-turned-fake-best-friend to your room to have sex with you, especially when that room is in the White House, and especially when that person is a guy, and especially when that guy is a prince of England.\n\nThe room is dimly lit\u2014a single lamp, in the corner by the couch, washing the deep blues of the walls neutral. He's moved all his campaign files from the bed to the desk and straightened out the bedspread. He looks at the ancient fireplace, the carved details of the mantel almost as old as the country itself, and it may not be Kensington Palace, but it looks all right.\n\nGod, if any ghosts of Founding Fathers are hanging around the White House tonight, they must really be suffering.\n\nHe's trying not to think too hard about what comes next. He may not have experience in practical application, but he's done research. He has diagrams. He can do this.\n\nHe really, really wants to do this. That much he's sure about.\n\nHe closes his eyes, grounds himself with his fingertips on the cool surface of his desk, the feathery little edges of papers there. His mind flashes to Henry, the smooth lines of his suit, the way his breath brushed Alex's cheek when he kissed him. His stomach does some embarrassing acrobatics he plans to never tell anyone about, ever.\n\nHenry, the prince. Henry, the boy in the garden. Henry, the boy in his bed.\n\nHe doesn't, he reminds himself, even have feelings for the guy. Really.\n\nThere's a knock on the door. Alex checks his phone: 10:54.\n\nHe opens the door.\n\nAlex stands there and exhales slowly, eyes on Henry. He's not sure he's ever let himself just _look._\n\nHenry is tall and gorgeous, half royalty, half movie star, red wine lingering on his lips. He's left his jacket and tie behind, and the sleeves of his shirt are pushed up to his elbows. He looks nervous around the corners of his eyes, but he smiles at Alex with one side of his pink mouth and says, \"Sorry I'm early.\"\n\nAlex bites his lip. \"Find your way here okay?\"\n\n\"There was a very helpful Secret Service agent,\" Henry says. \"I think her name was Amy?\"\n\nAlex smiles fully now. \"Get in here.\"\n\nHenry's grin takes over his entire face, not his photograph grin, but one that is crinkly and unguarded and infectious. He hooks his fingertips behind Alex's elbow, and Alex follows his lead, bare feet nudging between Henry's dress shoes. Henry's breath ghosts over Alex's lips, their noses brushing, and when he finally connects, he's smiling into it.\n\nHenry shuts and locks the door behind them, sliding one hand up the nape of Alex's neck, cradling it. There's something different about the way he's kissing now\u2014it's measured, deliberate. _Soft._ Alex isn't sure why, or what to do with it.\n\nHe settles for pulling Henry in by the sway of his waist, pressing their bodies flush. He kisses back, but lets himself be kissed however Henry wants to kiss him, which right now is exactly how he would have expected Prince Charming to kiss in the first place: sweet and deep and like they're standing at sunrise in the fucking moors. He can practically feel the wind in his hair. It's ridiculous.\n\nHenry breaks off and says, \"How do you want to do this?\"\n\nAnd Alex remembers, suddenly, this is not a sunrise-in-the-moors type of situation. He grabs Henry by his loosened collar, pushes a little, and says, \"Get on the couch.\"\n\nHenry's breath hitches and he complies. Alex moves to stand over him, looking down at that soft pink mouth. He feels himself standing at a very tall, very dangerous precipice, with no intention of backing away. Henry looks up at him, expectant, hungry.\n\n\"You've been dodging me for _weeks,_ \" Alex says, widening his stance so his knees bracket Henry's. He leans down and braces one hand against the back of the couch, the other grazing over the vulnerable dip of Henry's throat. \"You went out with a _girl._ \"\n\n\"I'm gay,\" Henry tells him flatly. One of his broad palms flattens over Alex's hip, and Alex inhales sharply, either at the touch or at hearing Henry finally say it out loud. \"Not something wise to pursue as a member of the royal family. And I wasn't sure you weren't going to murder me for kissing you.\"\n\n\"Then why'd you do it?\" Alex asks him. He leans into Henry's neck, dragging his lips over the sensitive skin just behind his ear. He thinks Henry might be holding his breath.\n\n\"Because I\u2014I hoped you wouldn't. Murder me. I had... suspicions you might want me too,\" Henry says. He hisses a little when Alex bites down lightly on the side of his neck. \"Or I thought, until I saw you with Nora, and then I was... jealous... and I was drunk and an idiot who got sick of waiting for the answer to present itself.\"\n\n\"You were _jealous,_ \" Alex says. \"You _want_ me.\"\n\nHenry moves abruptly, heaving Alex off balance with both hands and down into his lap, eyes blazing, and he says in a low and deadly voice Alex has never heard from him before, \"Yes, you preening arse, I've wanted you long enough that I won't have you tease me for another _fucking_ second.\"\n\nTurns out being on the receiving end of Henry's royal authority is an extreme fucking turn-on. He thinks, as he's hauled into a bruising kiss, that he'll never forgive himself for it. So, like, fuck the moors.\n\nHenry gets a grip on Alex's hips and pulls him close, so Alex is properly straddling his lap, and he kisses hard now, more like he had in the Red Room, with teeth. It shouldn't work so perfectly\u2014it makes absolutely no _sense_ \u2014but it does. There's something about the two of them, the way they ignite at different temperatures, Alex's frenetic energy and Henry's aching sureness.\n\nHe grinds down into Henry's lap, grunting as he's met with Henry already half-hard under him, and Henry's curse in response is buried in Alex's mouth. The kisses turn messy, then, urgent and graceless, and Alex gets lost in the drag and slide and press of Henry's lips, the sweet liquor of it. He pushes his hands into Henry's hair, and it's as soft as he always imagined when he would trace the photo of Henry in June's magazine, lush and thick under his fingers. Henry melts at the touch, wraps his arms around Alex's waist and holds him there. Alex isn't going anywhere.\n\nHe kisses Henry until it feels like he can't breathe, until it feels like he's going to forget both of their names and titles, until they're only two people tangled up in a dark room making a brilliant, epic, unstoppable mistake.\n\nHe manages to get the next two buttons on his shirt undone before Henry grabs it by the tails and pulls it off over his head and makes quick work of his own. Alex tries not to be in awe of the simple agility of his hands, tries not to think about classical piano or how swift and smooth years of polo have trained Henry to be.\n\n\"Hang on,\" Henry says, and Alex is already groaning in protest, but Henry pulls back and rests his fingertips on Alex's lips to shush him. \"I want\u2014\" His voice starts and stops, and he's looking like he's resolving not to cringe at himself again. He gathers himself, stroking a finger up to Alex's cheek before jutting his chin out defiantly. \"I want you on the bed.\"\n\nAlex goes fully silent and still, looking into Henry's eyes and the question there: _Are you going to stop this now that it's real?_\n\n\"Well, come on, Your Highness,\" Alex says, shifting his weight to give Henry a last tease before he stands.\n\n\"You're a dick,\" Henry says, but he follows, smiling.\n\nAlex climbs onto the bed, sliding back to prop himself up on his elbows by the pillows, watching as Henry kicks off his shoes and regains his bearings. He looks transformed in the lamplight, like a god of debauchery, painted gold with his hair all mussed up and his eyes heavy-lidded. Alex lets himself stare; the whipcord muscle under his skin, lean and long and lithe. The spot right at the dip of his waist below his ribs looks impossibly soft, and Alex might die if he can't fit his hand into that little curve in the next five seconds.\n\nIn an instant of sudden, vivid clarity, he can't believe he ever thought he was straight.\n\n\"Quit stalling,\" Alex says, pointedly interrupting the moment.\n\n\"Bossy,\" Henry says, and he complies.\n\nHenry's body settles over him with a warm, steady weight, one of his thighs sliding between Alex's legs and his hands bracing on the pillows, and Alex feels the points of contact like a static shock at his shoulders, his hips, the center of his chest.\n\nOne of Henry's hands slides up his stomach and stops, having encountered the old silver key on the chain resting over his sternum.\n\n\"What's this?\"\n\nAlex huffs impatiently. \"The key to my mom's house in Texas,\" he says, winding a hand back into Henry's hair. \"I started wearing it when I moved here. I guess I thought it would remind me of where I came from or something\u2014did I or did I not tell you to quit stalling?\"\n\nHenry looks up into his eyes, speechless, and Alex tugs him down into another all-consuming kiss, and Henry bears down on him fully, pressing him into the bed. Alex's other hand finds that dip of Henry's waist, and he swallows a sound at how devastating it feels under his palm. He's never been kissed like this, as if the feeling could swallow him up whole, Henry's body grinding down and covering every inch of his. He moves his mouth from Henry's to the side of his neck, the spot below his ear, kisses and kisses it, and bares his teeth. Alex knows it'll probably leave a mark, which is against rule number one of clandestine hookups for political offspring\u2014and probably royals too. He doesn't care.\n\nHe feels Henry find the waistband of his pants, the button, the zipper, the elastic of his underwear, and then everything goes very hazy, very quickly.\n\nHe opens his eyes to see Henry bringing his hand demurely up to his elegant royal mouth to _spit_ on it.\n\n\"Oh my fucking God,\" Alex says, and Henry grins crookedly as he gets back to work. \"Fuck.\" His body is moving, his mouth spilling words. \"I can't believe\u2014God, you are the most insufferable goddamn bastard on the face of the planet, do you know that\u2014fuck\u2014you're infuriating, you're the worst\u2014you're\u2014\"\n\n\"Do you _ever_ stop talking?\" Henry says. \"Such a _mouth_ on you.\" And when Alex looks again, he finds Henry watching him raptly, eyes bright and smiling. He keeps eye contact and his rhythm at the same time, and Alex was wrong before, Henry's going to be the one to kill him, not the other way around.\n\n\"Wait,\" Alex says, clenching his fist in the bedspread, and Henry immediately stills. \"I mean, _yes,_ obviously, _oh my God,_ but, like, if you keep doing that I'm gonna\"\u2014Alex's breath catches\u2014\"it's, that's just\u2014that's not _allowed_ before I get to see you naked.\"\n\nHenry tilts his head and smirks. \"All right.\"\n\nAlex flips them over, kicking off his pants until only his underwear is left slung low on his hips, and he climbs up the length of Henry's body, watching his face grow anxious, eager.\n\n\"Hi,\" he says, when he reaches Henry's eye level.\n\n\"Hello,\" Henry says back.\n\n\"I'm gonna take your pants off now,\" Alex tells him.\n\n\"Yes, good, carry on.\"\n\nAlex does, and one of Henry's hands slides down, leveraging one of Alex's thighs up so their bodies meet again right at the hard crux between them, and they both groan. Alex thinks, dizzily, that it's been nearly five years of foreplay, and enough is enough.\n\nHe moves his lips down to Henry's chest, and he feels under his mouth the beat Henry's heart skips at the realization of what Alex intends. His own heartbeat is probably falling out of rhythm too. He's in so far over his head, but that's good\u2014that's pretty much his comfort zone. He kisses Henry's solar plexus, his stomach, the stretch of skin above his waistband.\n\n\"I've, uh,\" Alex begins. \"I've never actually done this before.\"\n\n\"Alex,\" Henry says, reaching down to stroke at Alex's hair, \"you don't have to, I'm\u2014\"\n\n\"No, I want to,\" Alex says, tugging at Henry's waistband. \"I just need you to tell me if it's awful.\"\n\nHenry is speechless again, looking as if he can't believe his fucking luck. \"Okay. Of course.\"\n\nAlex pictures Henry barefoot in a Kensington Palace kitchen and the little sliver of vulnerability he got to see so early on, and he thrills at Henry now, in his bed, spread out and naked and wanting. This can't be really happening after everything, but miraculously, it is.\n\nIf he's going by the way Henry's body responds, by the way Henry's hand sweeps up into his hair and clutches a fistful of curls, he guesses he does okay for a first try. He looks up the length of Henry's body and is met with burning eye contact, a red lip caught between white teeth. Henry drops his head back on the pillow and groans something that sounds like \"fucking _eyelashes._ \" He's maybe a little bit in awe of how Henry arches up off the mattress, at hearing his sweet, posh voice reciting a litany of profanities to the ceiling. Alex is living for it, watching Henry come undone, letting him be whatever he needs to be while alone with Alex behind a locked door.\n\nHe's surprised to find himself hauled up to Henry's mouth and kissed hungrily. He's been with girls who didn't like to be kissed afterward and girls who didn't mind it, but Henry revels in it, based on the deep and comprehensive way he's kissing him. It occurs to him to make a comment about narcissism, but instead\u2014\n\n\"Not awful?\" Alex says between kisses, resting his head on the pillow next to Henry's to catch his breath.\n\n\"Definitely adequate,\" Henry answers, grinning, and he scoops Alex up against his chest greedily as if he's trying to touch all of him at once. Henry's hands are huge on his back, his jaw sharp and rough with a long day's stubble, his shoulders broad enough to eclipse Alex when he rolls them over and pins Alex to the mattress. None of it feels anything like anything he's felt before, but it's just as good, maybe better.\n\nHenry's kissing him aggressively once more, confident in a way that's rare from Henry. Messy earnestness and rough focus, not a dutiful prince but any other twenty-something boy enjoying himself doing something he likes, something he's good at. And he is _good_ at it. Alex makes a mental note to figure out which shadowy gay noble taught Henry all this and send the man a fruit basket.\n\nHenry returns the favor happily, hungrily, and Alex doesn't know or care what sounds or words come out of his mouth. He thinks one of them is \"sweetheart\" and another is \"motherfucker.\" Henry is one talented bastard, a man of many hidden gifts, Alex muses half-hysterically. A true prodigy. God Save the Queen.\n\nWhen he's done, he presses a sticky kiss in the crease of Alex's leg where he'd slung it over his shoulder, managing to come off polite, and Alex wants to drag Henry up by the hair, but his body is boneless and wrecked. He's blissed out, dead. Ascended to the next plane, merely a pair of eyes floating through a dopamine haze.\n\nThe mattress shifts, and Henry moves up to the pillows, nuzzling his face into the hollow of Alex's throat. Alex makes a vague noise of approval, and his arms fumble around Henry's waist, but he's helpless to do much else. He's sure he used to know quite a lot of words, in more than one language, in fact, but he can't seem to recall any of them.\n\n\"Hmm,\" Henry hums, the tip of his nose catching on Alex's. \"If I had known this was all it took to shut you up, I'd have done it ages ago.\"\n\nWith a feat of Herculean strength, he summons up two whole words: \"Fuck you.\"\n\nDistantly, through a slowly clearing fog, through a messy kiss, Alex can't help marveling at the knowledge that he's crossed some kind of Rubicon, here in this room that's almost as old as the country it's in, like Washington crossing the Delaware. He laughs into Henry's mouth, instantly caught up in his own dramatic mental portrait of the two them painted in oils, young icons of their nations, naked and shining wet in the lamplight. He wishes Henry could see it, wonders if he'd find the image as funny.\n\nHenry rolls over onto his back. Alex's body wants to follow and tuck into his side, but he stays where he is, watching from a few safe inches away. He can see a muscle in Henry's jaw flexing.\n\n\"Hey,\" he says. He pokes Henry in the arm. \"Don't freak out.\"\n\n\"I'm not _freaking out,_ \" he says, enunciating the words.\n\nAlex wriggles an inch closer in the sheets. \"It was fun,\" Alex says. \"I had fun. You had fun, right?\"\n\n\"Definitely,\" he says, in a tone that sends a lazy spark up Alex's spine.\n\n\"Okay, cool. So, we can do this again, anytime you want,\" Alex says, dragging the back of his knuckles down Henry's shoulder. \"And you know this doesn't, like, change anything between us, right? We're still... whatever we were before, just, you know. With blowjobs.\"\n\nHenry covers his eyes with one hand. \"Right.\"\n\n\"So,\" Alex says, changing tracks by stretching languidly, \"I guess I should tell you, I'm bisexual.\"\n\n\"Good to know,\" Henry says. His eyes flicker down to Alex's hip, where it's bared above the sheet, and he says as much to himself as to Alex, \"I am very, very gay.\"\n\nAlex watches his small smile, the way it wrinkles the corners of his eyes, and very deliberately does not kiss it.\n\nPart of his brain keeps getting stuck on how strange, and strangely wonderful, it is to see Henry like this, open and bare in every way. Henry leans across the pillow to Alex and presses a soft kiss to his mouth, and Alex feels fingertips brush over his jaw. The touch is so gentle he has to once again remind himself not to care too much.\n\n\"Hey,\" Alex tells him, sliding his mouth closer to Henry's ear, \"you're welcome to stay as long as you want, but I should warn you it's probably in both of our best interests if you go back to your room before morning. Unless you want the PPOs to lock the Residence down and come requisition you from my boudoir.\"\n\n\"Ah,\" Henry says. He pulls away from Alex and rolls back over, looking up to the ceiling again like a man seeking penance from a wrathful god. \"You're right.\"\n\n\"You can stay for another round, if you want to,\" Alex offers.\n\nHenry coughs, scrubs a hand through his hair. \"I rather think I'd\u2014I'd better get back to my room.\"\n\nAlex watches him fish his boxers from the foot of the bed and start pulling them back on, sitting up and shaking out his shoulders.\n\nIt's for the best this way, he tells himself; nobody will get any wrong ideas about what exactly this arrangement is. They're not going to spoon all night or wake up in each other's arms or eat breakfast together. Mutually satisfying sexual experiences do not a relationship make.\n\nEven if he did want that, there are a million reasons why this will never, ever be possible.\n\nAlex follows him to the door, watching him turn to hover there awkwardly.\n\n\"Well, er...\" Henry attempts, looking down at his feet.\n\nAlex rolls his eyes. \"For fuck's sake, man, you just had my dick in your mouth, you can kiss me good-night.\"\n\nHenry looks back up at him, his mouth open and incredulous, and he throws his head back and _laughs,_ and it's only him, the nerdy, neurotic, sweet, insomniac rich guy who constantly sends Alex photos of his dog, and something slots into place. He leans down and kisses him fiercely, and then he's grinning and gone.\n\n* * *\n\n\"You're doing _what_?\"\n\nIt's sooner than either of them expected\u2014only two weeks since the state dinner, two weeks of wanting Henry back under him as soon as possible and saying everything short of that in their texts. June keeps looking at him like she's going to throw his phone in the Potomac.\n\n\"An invitation-only charity polo match this weekend,\" Henry says over the phone. \"It's in...\" He pauses, probably referring back to whatever itinerary Shaan has given him. \"Greenwich, Connecticut? It's $10,000 a seat, but I can have you added to the list.\"\n\nAlex almost fumbles his coffee all over the south entryway. Amy glares at him. \"Jesus _fuck._ That is _obscene,_ what are you raising money for, monocles for babies?\" He covers the mouthpiece of the phone with his hand. \"Where's Zahra? I need to clear my schedule for this weekend.\" He uncovers the phone. \"Look, I guess I'll _try_ to make it, but I'm really busy right now.\"\n\n* * *\n\n\"I'm sorry, Zahra said you're bailing on the fund-raiser this weekend because you're going to a _polo match_ in _Connecticut_?\" June asks from his bedroom doorway that night, almost startling another cup of coffee out of his hands.\n\n\"Listen,\" Alex tells her, \"I'm trying to keep up a geopolitical public relations ruse here.\"\n\n\"Dude, people are writing _fan fiction_ about y'all\u2014\"\n\n\"Yeah, Nora sent me that.\"\n\n\"\u2014I think you can give it a _rest._ \"\n\n\"The crown wants me to be there!\" he lies quickly. She seems unconvinced and leaves him with a parting look he'd probably be concerned about if he cared more about things that aren't Henry's mouth right now.\n\nWhich is how he ends up in his J. Crew best on a Saturday at the Greenwich Polo Club, wondering what the hell he's gotten himself into. The woman in front of him is wearing a hat with an entire taxidermied pigeon on it. High school lacrosse did not prepare him for this kind of sporting event.\n\nHenry on horseback is nothing new. Henry in full polo gear\u2014the helmet, the polo sleeves capped right at the bulge of his biceps, the snug white pants tucked into tall leather boots, the intricately buckled leather knee padding, the leather gloves\u2014is familiar. He has seen it before. Categorically, it should be boring. It should not provoke anything visceral, carnal, or bodice-ripping in nature in him at all.\n\nBut Henry urging his horse across the field with the power of his thighs, his ass bouncing hard in the saddle, the way the muscles in his arms stretch and flex when he swings, looking the way he does and wearing the things he's wearing\u2014it's a lot.\n\nHe's sweating. It's February in Connecticut, and Alex is sweating under his coat.\n\nWorst of all, Henry is _good_. Alex doesn't pretend to care about the rules of the game, but his primary turn-on has always been competence. It's too easy to look at Henry's boots digging into the stirrups for leverage and conjure up a memory of bare calves underneath, bare feet planted just as firmly on the mattress. Henry's thighs open the same way, but with Alex between them. Sweat dripping down Henry's brow onto his throat. Just, uh... well, just like that.\n\nHe wants\u2014God, after all this time ignoring it, he wants it again, now, _right now._\n\nThe match ends after a circle-of-hell amount of time, and Alex feels like he'll pass out or scream if he doesn't get his hands on Henry soon, like the only thought possible in the universe is Henry's body and Henry's flushed face and every other molecule in existence is just an inconvenience.\n\n\"I don't like that look,\" Amy says when they reach the bottom of the stands, peering into his eyes. \"You look... sweaty.\"\n\n\"I'm gonna go, uh,\" Alex says. \"Say hi to Henry.\"\n\nAmy's mouth settles into a grim line. \"Please don't elaborate.\"\n\n\"Yeah, I know,\" Alex says. \"Plausible deniability.\"\n\n\"I don't know what you could possibly mean.\"\n\n\"Sure.\" He rakes a hand through his hair. \"Yep.\"\n\n\"Enjoy your summit with the English delegation,\" she tells him flatly, and Alex sends up a vague prayer of thanks for staff NDAs.\n\nHe legs it toward the stables, limbs already buzzing with the steady knowledge of Henry's body getting incrementally closer to his. Long, lean legs, grass stains on pristine, tight pants, why does this sport have to be so completely _repulsive_ while Henry looks so damn _good_ doing it\u2014\n\n\"Oh shit\u2014\"\n\nHe barely stops himself from running headfirst into Henry in the flesh, who has rounded the corner of the stables.\n\n\"Oh, hello.\"\n\nThey stand there staring at each other, fifteen days removed from Henry swearing at the ceiling of Alex's bedroom and unsure how to proceed. Henry is still in his full polo regalia, gloves and all, and Alex can't decide if he is pleased or wants to brain him with a polo stick. Polo bat? Polo club? Polo... mallet? This sport is a travesty.\n\nHenry breaks the silence by adding, \"I was coming to find you, actually.\"\n\n\"Yeah, hi, here I am.\"\n\n\"Here you are.\"\n\nAlex glances over his shoulder. \"There's, uh. Cameras. Three o'clock.\"\n\n\"Right,\" Henry says, straightening his shoulders. His hair is messy and slightly damp, color still high in his cheeks from exertion. He's going to look like goddamn Apollo in the photos when they go to press. Alex smiles, knowing they'll sell.\n\n\"Hey, isn't there, uh, a thing?\" Alex says. \"You needed to. Uh. Show me?\"\n\nHenry looks at him, glances at the dozens of millionaires and socialites milling around, and back at him. \"Now?\"\n\n\"It was a four-and-a-half-hour car ride up here, and I have to go back to DC in an hour, so I don't know when else you're expecting to show it to me.\"\n\nHenry takes a beat, his eyes flickering to the cameras again before he switches on a stage smile and a laugh, cuffing Alex on the shoulder. \"Ah, yes. Right. This way.\"\n\nHe turns on his boot heel and leads the way around the back of the stables, veering right into a doorway, and Alex follows. It's a small, windowless room attached to the stables, fragrant with leather polish and stained wood from floor to ceiling, the walls lined with heavy saddles, riding crops, bridles, and reins.\n\n\"What in the rich-white-people-sex-dungeon hell?\" Alex wonders aloud as Henry crosses behind him. He whips a thick leather strap off a hook on the wall, and Alex almost blacks out.\n\n\"What?\" Henry says offhandedly, bypassing him to bind the doors shut. He turns around, sweet-faced and unbelievable. \"It's called a tack room.\"\n\nAlex drops his coat and takes three swift steps toward him. \"I don't actually care,\" he says, and grabs Henry by the stupid collar of his stupid polo and kisses his stupid mouth.\n\nIt's a good kiss, solid and hot, and Alex can't decide where to put his hands because he wants to put them everywhere at once.\n\n_\"Ugh,\"_ he groans in exasperation, shoving Henry backward by the shoulders and making a disgusted show of looking him up and down. \"You look _ridiculous._ \"\n\n\"Should I\u2014\" He steps back and puts a foot up on a nearby bench, moving to undo his kneepads.\n\n\"What? No, of course not, keep them on,\" Alex says. Henry freezes, standing there all artistically posed with his thighs apart and one knee up, the fabric straining. \"Oh my God, what are you doing? I can't even look at you.\" Henry frowns. \"No, Jesus, I just meant\u2014I'm so _mad_ at you.\" Henry gingerly puts his boot back on the floor. Alex wants to die. \"Just, come here. _Fuck._ \"\n\n\"I'm quite confused.\"\n\n\"Me fucking too,\" Alex says, profoundly suffering for something he must have done in a previous life. \"Listen, I don't know why, but this whole _thing_ \"\u2014he gestures at Henry's entire physical presence\u2014\"is... really doing it for me, so, I just need to.\" Without any further ceremony, he drops to his knees and starts undoing Henry's belt, tugging at the fastenings of his pants.\n\n\"Oh, God,\" Henry says.\n\n\"Yeah,\" Alex agrees, and he gets Henry's boxers down.\n\n\"Oh, _God,_ \" Henry repeats, this time with feeling.\n\nIt's all still so new to Alex, but it's not difficult to follow through on what's been playing out in elaborate detail in his head for the past hour. When he looks up, Henry's face is flushed and transfixed, his lips parted. It almost hurts to look at him\u2014the athlete's focus, all the dressings of aristocracy laid wide open for him. He's watching Alex, eyes blown dark and hazy, and Alex is watching him right back, every nerve in both bodies narrowed down to a single point.\n\nIt's fast and dirty and Henry is swearing up a storm, which is still disarmingly sexy, but this time it's punctuated by the occasional word of praise, and somehow that's even hotter. Alex isn't prepared for the way \"that's good\" sounds in Henry's rounded Buckingham vowels, or for how luxury leather feels when it strokes approvingly down his cheek, a gloved thumb brushing the corner of his mouth.\n\nAs soon as Henry's finished, he's got Alex on the bench and is putting his kneepads to use.\n\n\"I'm still fucking mad at you,\" Alex says, destroyed, slumped forward with his forehead resting on Henry's shoulder.\n\n\"Of course you are,\" Henry says vaguely.\n\nAlex completely undermines his point by pulling Henry into a deep and lingering kiss, and another, and they kiss for an amount of time he decides not to count or think about.\n\nThey sneak out quietly, and Henry touches Alex's shoulder at the gate near where his SUV waits, presses his palm into the wool of his coat and the knot of muscle.\n\n\"I don't suppose you'll be anywhere near Kensington anytime soon?\"\n\n\"That shithole?\" he says with a wink. \"Not if I can help it.\"\n\n\"Oi,\" Henry says. He's grinning now. \"That's disrespect of the crown, that is. Insubordination. I've thrown men in the dungeons for less.\"\n\nAlex turns, walking backward toward the car, hands in the air. \"Hey, don't threaten me with a good time.\"\n\n> Paris?\n\n* * *\n\n> A <agcd@eclare45.com> 3\/3\/20 7:32 PM\n> \n> to Henry\n> \n> His Royal Highness Prince Henry of Whatever,\n> \n> Don't make me learn your actual title.\n> \n> Are you going to be at the Paris fund-raiser for rainforest conservation this weekend?\n> \n> Alex\n> \n> First Son of Your Former Colony\n> \n> Re: Paris?\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 3\/4\/20 2:14 AM\n> \n> to A\n> \n> Alex, First Son of Off-Brand England:\n> \n> First, you should know how terribly inappropriate it is for you to intentionally botch my title. I could have you made into a royal settee cushion for that kind of l\u00e8se-majest\u00e9. Fortunately for you, I do not think you would complement my sitting room decor.\n> \n> Secondly, no, I will not be attending the Paris fund-raiser; I have a previous engagement. You shall have to find someone else to accost in a cloakroom.\n> \n> Regards,\n> \n> His Royal Highness Prince Henry of Wales\n> \n> Re: Paris?\n\n* * *\n\n> A <agcd@eclare45.com> 3\/4\/20 2:27 AM\n> \n> to Henry\n> \n> Huge Raging Headache Prince Henry of Who Cares,\n> \n> It is amazing you can sit down to write emails with that gigantic royal stick up your ass. I seem to remember you really enjoying being \"accosted.\"\n> \n> Everyone there is going to be boring anyway. What are you doing?\n> \n> Alex\n> \n> First Son of Hating Fund-raisers\n> \n> Re: Paris?\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 3\/4\/20 2:32 AM\n> \n> to A\n> \n> Alex, First Son of Shirking Responsibilities:\n> \n> A royal stick is formally known as a \"scepter.\"\n> \n> I've been sent to a summit in Germany to act as if I know anything about wind power. Primarily, I'll be getting lectured by old men in lederhosen and posing for photos with windmills. The monarchy has decided we care about sustainable energy, apparently\u2014or at least that we want to appear to. An utter romp.\n> \n> Re: fund-raiser guests, I thought you said I was boring?\n> \n> Regards,\n> \n> Harangued Royal Highness\n> \n> Re: Paris?\n\n* * *\n\n> A <agcd@eclare45.com> 3\/4\/20 2:34 AM\n> \n> to Henry\n> \n> Horrible Revolting Heir,\n> \n> It's recently come to my attention you're not quite as boring as I thought. Sometimes. Namely when you're doing the thing with your tongue.\n> \n> Alex\n> \n> First Son of Questionable Late Night Emails\n> \n> Re: Paris?\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 3\/4\/20 2:37 AM\n> \n> to A\n> \n> Alex, First Son of Inappropriately Timed Emails When I'm in Early Morning Meetings:\n> \n> Are you trying to get fresh with me?\n> \n> Regards,\n> \n> Handsome Royal Heretic\n> \n> Re: Paris?\n\n* * *\n\n> A <agcd@eclare45.com> 3\/4\/20 2:41 AM\n> \n> to Henry\n> \n> His Royal Horniness,\n> \n> If I were trying to get fresh with you, you would know it.\n> \n> For example: I've been thinking about your mouth on me all week, and I was hoping I'd see you in Paris so I could put it to use.\n> \n> I was also thinking you might know how to pick French cheeses. Not my area of expertise.\n> \n> Alex\n> \n> First Son of Cheese Shopping and Blowjobs\n> \n> Re: Paris?\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 3\/4\/20 2:43 AM\n> \n> to A\n> \n> Alex, First Son of Making Me Spill My Tea in Said Early Morning Meeting:\n> \n> Hate you. Will try to get out of Germany.\n> \n> x\n\n# SEVEN\n\nHenry does get out of Germany, and he meets Alex near a herd of cr\u00eape-eating tourists by Place du Tertre, wearing a sharp blue blazer and a wicked smile. They stumble back to his hotel after two bottles of wine, and Henry sinks to his knees on the white marble and looks up at Alex with big, blue, bottomless eyes, and Alex doesn't know a word in any language to describe it.\n\nHe's so drunk, and Henry's mouth is so soft, and it's all so fucking French that he forgets to send Henry back to his own hotel. He forgets they don't spend the night. So, they do.\n\nHe discovers Henry sleeps curled up on his side, his spine poking out in little sharp points that are actually soft if you reach out and touch them, very carefully so as not to wake him because he's actually sleeping for once. In the morning, room service brings up crusty baguettes and sticky tarts filled with fat apricots and a copy of _Le Monde_ that Alex makes Henry translate out loud.\n\nHe vaguely remembers telling himself they weren't going to do things like this. It's all a little hazy right now.\n\nWhen Henry's gone, Alex finds the stationery by the bed: _Fromagerie Nicole Barth\u00e9l\u00e9my._ Leaving your clandestine hookup directions to a Parisian cheese shop. Alex has to admit: Henry really has a solid handle on his personal brand.\n\nLater, Zahra texts him a screencap of a _BuzzFeed_ article about his \"best bromance ever\" with Henry. It's a mix of photos: the state dinner, a couple of shots of them grinning outside the stables in Greenwich, one picked up from a French girl's Twitter of Alex leaning back in his chair at a tiny cafe table while Henry finishes off the bottle of red between them.\n\nBeneath it, Zahra has begrudgingly written: Good work, you little shit.\n\nHe guesses this is how they're going to do this\u2014the world is going to keep thinking they're best friends, and they're going to keep playing the part.\n\nHe knows, objectively, he should pace himself. It's only physical. But Perfect Stoic Prince Charming laughs when he comes, and texts Alex at weird hours of the night: You're a mad, spiteful, unmitigated demon, and I'm going to kiss you until you forget how to talk. And Alex is kind of obsessed with it.\n\nAlex decides not to think too hard. Normally they'd only cross paths a few times a year; it takes creative schedule wrangling and a little sweet-talking of their respective teams to see each other as often as their bodies demand. At least they've got a ruse of international public relations.\n\nTheir birthdays, it turns out, are less than three weeks apart, which means, for most of March, Henry is twenty-three and Alex is twenty-one. (\"I knew he was a goddamn Pisces,\" June says). Alex happens to have a voter registration drive at NYU at the end of March, and when he texts Henry about it, he gets a brisk response fifteen minutes later: Have rescheduled visit to New York for nonprofit business to this weekend. Will be in the city ready to carry out birthday floggings &c.\n\nThe photographers are readily visible when they meet in front of the Met, so they clasp each other's hands and Alex says through his big on-camera smile, \"I want you alone, now.\"\n\nThey're more careful in the States, and they go up to the hotel room one at a time\u2014Henry through the back flanked by two tall PPOs, and later, Alex with Cash, who grins and knows and says nothing.\n\nThere's a lot of champagne and kissing and buttercream from a birthday cupcake Henry's inexplicably procured smeared around Alex's mouth, Henry's chest, Alex's throat, between Henry's hips. Henry pins his wrists to the mattress and swallows him down, and Alex is drunk and fucking transported, feeling every moment of twenty-two years and not a single day older, some kind of hedonistic youth of history. Birthday head from another country's prince will do that.\n\nIt's the last time they see each other for weeks, and after a lot of teasing and maybe some begging, he convinces Henry to download Snapchat. Henry mostly sends tame, fully clothed thirst traps that make Alex sweat in his lectures: a mirror shot, mud-stained white polo pants, a sharp suit. On a Saturday, the C-SPAN stream on his phone gets interrupted by Henry on a sailboat, smiling into the camera with the sun bright on his bare shoulders, and Alex's heart goes so fucking weird that he has to put his head in his hands for a full minute.\n\n(But, like. It's fine. It's not a whole thing.)\n\nBetween it all, they talk about Alex's campaign job, Henry's nonprofit projects, both of their appearances. They talk about how Pez is now proclaiming himself fully in love with June and spends half his time with Henry rhapsodizing about her or begging him to ask Alex if she likes flowers (yes) or exotic birds (to look at, not to own) or jewelry in the shape of her own face (no).\n\nThere are a lot of days when Henry is happy to hear from him and quick to respond, a fast, cutting sense of humor, hungry for Alex's company and the tangle of thoughts in Alex's head. But sometimes, he's taken over by a dark mood, an unusually acerbic wit, strange and vitrified. He'll withdraw for hours or days, and Alex comes to understand this as grief time, little bouts of depression, or times of \"too much.\" Henry hates those days completely. Alex wishes he could help, but he doesn't particularly mind. He's just as attracted to Henry's cloudy tempers, the way he comes back from them, and the millions of shades in between.\n\nHe's also learned that Henry's placid demeanor is shattered with the right poking. He likes to bring up things he knows will get Henry going, including:\n\n\"Listen,\" Henry is saying, heated, over the phone on a Thursday night. \"I don't give a damn what _Joanne_ has to say, Remus John Lupin is gay as the day is long, and I won't hear a word against it.\"\n\n\"Okay,\" Alex says. \"For the record, I agree with you, but also, tell me more.\"\n\nHe launches into a long-winded tirade, and Alex listens, amused and a little awed, as Henry works his way to his point: \"I just think, as the prince of this bloody country, that when it comes to Britain's _positive_ cultural landmarks, it would be nice if we could not throw our own marginalized people under the proverbial bus. People sanitize Freddie Mercury or Elton John or Bowie, who was shagging Jagger up and down Oakley Street in the seventies, I might add. It's just not the _truth._ \"\n\nIt's another thing Henry does\u2014whipping out these analyses of what he reads or watches or listens to that confronts Alex with the fact that he has both a degree in English literature and a vested interest in the gay history of his family's country. Alex has always _known_ his gay American history\u2014after all, his parents' politics have been part of it\u2014but it wasn't until he figured himself out that he started to _engage_ with it like Henry.\n\nHe's starting to understand what swelled in his chest the first time he read about Stonewall, why he ached over the SCOTUS decision in 2015. He starts catching up voraciously in his spare time: Walt Whitman, the Laws of Illinois 1961, The White Night Riot, _Paris Is Burning._ He's pinned a photo over his desk at work, a man at a rally in the '80s in a jacket that says across the back: IF I DIE OF AIDS\u2014FORGET BURIAL\u2014JUST DROP MY BODY ON THE STEPS OF THE F.D.A.\n\nJune's eyes stick on it one day when she drops by the office to have lunch with him, giving him the same strange look she gave him over coffee the morning after Henry snuck into his room. But she doesn't say anything, carries on through sushi about her latest project, pulling all her journals together into a memoir. Alex wonders if any of this stuff would make it into there. Maybe, if he tells her soon. He should tell her soon.\n\nIt's weird that the thing with Henry could make him understand this huge part of himself, but it does. When he sinks into thoughts of Henry's hands, square knuckles and elegant fingers, he wonders how he never realized it before. When he sees Henry next at a gala in Berlin, and he feels that gravitational pull, chases it down in the back of a limo, and binds Henry's wrists to a hotel bedpost with his own necktie, he knows himself better.\n\nWhen he shows up for a weekly briefing two days later, Zahra grabs his jaw with one hand and turns his head, peering closer at the side of his neck. \"Is that a _hickey_?\"\n\nAlex freezes. \"I... um, no?\"\n\n\"Do I look stupid to you, Alex?\" Zahra says. \"Who is giving you hickeys, and why have you not gotten them to sign an NDA?\"\n\n\"Oh my God,\" he says, because really, the last person Zahra needs to be concerned about leaking sordid details is Henry. \"If I needed an NDA, you would know. Chill.\"\n\nZahra does not appreciate being told to chill.\n\n\"Look at me,\" she says. \"I have known you since you were still leaving skid marks in your drawers. You think I don't know when you're lying to me?\" She jabs a pointy, polished nail into his chest. \"However you got that, it better be somebody off the approved list of girls you are allowed to be seen with during the election cycle, which I will email to you again as soon as you get out of my sight in case you have misplaced it.\"\n\n\"Jesus, okay.\"\n\n\"And to remind you,\" she goes on, \"I will chop my own tit off before I let you pull some idiotic stunt to cause your mother, our first female president, to be the first president to lose reelection since H fucking W. Do you understand me? I will lock you in your room for the next year if I have to, and you can take your finals by fucking smoke signal. I will staple your dick to the inside of your leg if that keeps it in your fucking pants.\"\n\nShe returns to her notes with smooth professionalism, as if she has not just threatened his life. Behind her, he can see June at her place at the table, very clearly aware that he's lying too.\n\n* * *\n\n\"Do you have a last name?\"\n\nAlex has never actually offered a greeting when calling Henry.\n\n\"What?\" The usual bemused, elongated, one-syllable response.\n\n\"A last name,\" Alex repeats. It's late afternoon and stormy outside the Residence, and he's on his back in the middle of the Solarium, catching up on drafts for work. \"That thing I have two of. Do you use your dad's? Henry Fox? That sounds fucking dope. Or does royalty outrank? Do you use your mom's name, then?\"\n\nHe hears some shuffling over the phone and wonders if Henry's in bed. They haven't been able to see each other in a couple weeks, so his mind is quick to supply the image.\n\n\"The official family name is Mountchristen-Windsor,\" Henry says. \"Hyphenate, like yours. So my full name is... Henry George Edward James Fox-Mountchristen-Windsor.\"\n\nAlex gapes up at the ceiling. \"Oh... my God.\"\n\n\"Truly.\"\n\n\"I thought Alexander Gabriel Claremont-Diaz was bad.\"\n\n\"Is that after someone?\"\n\n\"Alexander after the founding father, Gabriel after the patron saint of diplomats.\"\n\n\"That's a bit on the nose.\"\n\n\"Yeah, I didn't have a chance. My sister got Catalina June after the place and the Carter Cash, but I got all the self-fulfilling prophecies.\"\n\n\"I did get both of the gay kings,\" Henry points out. \"There's a prophecy for you.\"\n\nAlex laughs and kicks his files for the campaign away. He's not coming back to them tonight. \"Three last names is just mean.\"\n\nHenry sighs. \"In school, we all went by Wales. Philip is Lieutenant Windsor in the RAF now, though.\"\n\n\"Henry Wales, then? That's not too bad.\"\n\n\"No, it's not. Is this the reason you phoned?\"\n\n\"Maybe,\" Alex says. \"Call it historical curiosity.\" Except the truth is closer to the slight drag in Henry's voice and the half step of hesitation before he speaks that's been there all week. \"Speaking of historical curiosity, here's a fun fact: I'm sitting in the room Nancy Reagan was in when she found out Ronald Reagan got shot.\"\n\n\"Good Lord.\"\n\n\"And it's also where ol' Tricky Dick told his family he was gonna resign.\"\n\n\"I'm sorry\u2014who or what is a _Tricky Dick_?\"\n\n\" _Nixon!_ Listen, you're undoing everything this country's crusty forefathers fought for and deflowering the darling of the republic. You at least need to know _basic_ American history.\"\n\n\"I hardly think deflowering is the word,\" Henry deadpans. \"These arrangements are supposed to be with virgin brides, you know. That certainly didn't seem to be the case.\"\n\n\"Uh-huh, and I'm sure you picked up all those skills from books.\"\n\n\"Well, I did go to uni. It just wasn't necessarily the reading that did it.\"\n\nAlex hums in suggestive agreement and lets the rhythm of banter fall out. He looks across the room\u2014the windows that were once only gauzy curtains on a sleeping room for Taft's family on hot nights, the corner now stacked with Leo's old comic book collectibles where Eisenhower used to play cards. The stuff underneath the surface. Alex has always sought those things out.\n\n\"Hey,\" he says. \"You sound weird. You good?\"\n\nHenry's breath catches and he clears his throat. \"I'm fine.\"\n\nAlex doesn't say anything, letting the silence stretch in a thin thread between them before he cuts it. \"You know, this whole arrangement we have... you can tell me stuff. I tell you stuff all the time. Politics stuff and school stuff and nutso family stuff. I know I'm, like, not the paragon of normal human communication, but. You know.\"\n\nAnother pause.\n\n\"I'm not... historically great at talking about things,\" Henry says.\n\n\"Well, I wasn't historically great at blowjobs, but we all gotta learn and grow, sweetheart.\"\n\n_\"Wasn't?\"_\n\n_\"Hey,\"_ Alex huffs. \"Are you trying to say I'm still not good at them?\"\n\n\"No, no, I wouldn't dream of it,\" Henry says, and Alex can hear the small smile in his voice. \"It was just the first one that was... Well. It was enthusiastic, at least.\"\n\n\"I don't remember you complaining.\"\n\n\"Yes, well, I'd only been fantasizing about it for _ages._ \"\n\n\"See, there's a thing,\" Alex points out. \"You just told me that. You can tell me other stuff.\"\n\n\"It's hardly the same.\"\n\nHe rolls over onto his stomach, considers, and very deliberately says, \"Baby.\"\n\nIt's become a thing: _baby._ He knows it's become a thing. He's slipped up and accidentally said it a few times, and each time, Henry positively melts and Alex pretends not to notice, but he's not above playing dirty here.\n\nThere's a slow hiss of an exhale across the line, like air escaping through a crack in a window.\n\n\"It's, ah. It's not the best time,\" he says. \"How did you put it? Nutso family stuff.\"\n\nAlex purses his lips, bites down on his cheek. There it is.\n\nHe's wondered when Henry would finally start talking about the royal family. He makes oblique references to Philip being wound so tight as to double as an atomic clock, or to his grandmother's disapproval, and he mentions Bea as often as Alex mentions June, but Alex knows there's more to it than that. He couldn't tell you when he started noticing, though, just like he doesn't know when he started ticking off the days of Henry's moods.\n\n\"Ah,\" he says. \"I see.\"\n\n\"I don't suppose you keep up with any British tabloids, do you?\"\n\n\"Not if I can help it.\"\n\nHenry offers the bitterest of laughs. \"Well, the _Daily Mail_ has always had a bit of an affinity for airing our dirty laundry. They, er, they gave my sister this nickname years ago. 'The Powder Princess.'\"\n\nA ding of recognition. \"Because of the...\"\n\n\"Yes, the cocaine, Alex.\"\n\n\"Okay, that does sound familiar.\"\n\nHenry sighs. \"Well, someone's managed to bypass security to spray paint 'Powder Princess' on the side of her car.\"\n\n\"Shit,\" Alex says. \"And she's not taking it well?\"\n\n\"Bea?\" Henry laughs, a little more genuinely this time. \"No, she doesn't usually care about those things. She's fine. More shaken up that someone got past security than anything. Gran had an entire PPO team sacked. But... I dunno.\"\n\nHe trails off, and Alex can guess.\n\n\"But you care. Because you want to protect her even though you're the little brother.\"\n\n\"I... yes.\"\n\n\"I know the feeling. Last summer I almost punched a guy at Lollapalooza because he tried to grab June's ass.\"\n\n\"But you didn't?\"\n\n\"June had already dumped her milkshake on him,\" Alex explains. He shrugs a little, knowing Henry can't see it. \"And then Amy Tased him. The smell of burnt strawberry milkshake on a sweaty frat guy is really something.\"\n\nHenry laughs fully at that. \"They never do need us, do they?\"\n\n\"Nope,\" Alex agrees. \"So you're upset because the rumors aren't true.\"\n\n\"Well... they are true, actually,\" Henry says.\n\n_Oh,_ Alex thinks.\n\n\"Oh,\" Alex says. He's not sure how else to respond, reaching into his mental store of political platitudes and finding them all clinical and intolerable.\n\nHenry, with a little trepidation, presses on. \"You know, Bea has only ever wanted to play music,\" he starts. \"Mum and Dad played too much Joni Mitchell for her growing up, I think. She wanted guitar lessons; Gran wanted violin since it was more proper. Bea was allowed to learn both, but she went to uni for classical violin. Anyway, her last year of uni, Dad died. It happened so... quickly. He just _went._ \"\n\nAlex shuts his eyes. \"Fuck.\"\n\n\"Yeah,\" Henry says, voice rough. \"We all went round the bend a bit. Philip just _had_ to be the man of the family, and I was an arsehole, and Mum didn't leave her rooms. Bea just stopped seeing the point in anything. I was starting uni when she finished, and Philip was deployed halfway round the globe, and she was out every single night with all the posh London hipsters, sneaking out to play guitar at secret shows and doing mountains of cocaine. The papers _loved_ it.\"\n\n\"Jesus,\" Alex hisses. \"I'm sorry.\"\n\n\"It's fine,\" Henry says, steadiness rising in his voice as if he's stuck out his chin in that stubborn way he does sometimes. Alex wishes he could see it. \"In any event, the speculation and paparazzi photos and the goddamn nickname got to be too much, and Philip came home for a week, and he and Gran literally put her in a car and had her driven to rehab and called it a _wellness retreat_ to the press.\"\n\n\"Wait\u2014sorry,\" Alex says before he can stop himself. \"Just. Where was your mom?\"\n\n\"Mum hasn't been involved in much since Dad died,\" Henry says on an exhale, then stops short. \"Sorry. That's not fair. It's... the grief has been total for her. It was paralyzing. It _is_ paralyzing. She was such a spitfire. I dunno. She still listens, and she tries, and she wants us to be happy. But I don't know if she has it in her anymore to be a part of anyone's happiness.\"\n\n\"That's... horrible.\"\n\nA pause, heavy.\n\n\"Anyway, Bea went,\" Henry goes on, \"against her will, and didn't think she had a problem at all, even though you could see her bloody ribs and she'd barely spoken to me in months, when we grew up inseparable. Checked herself out after six hours. I remember her calling me that night from a club, and I lost it. I was, what, eighteen? I drove there and she was sitting on the back steps, high as a kite, and I sat down next to her and cried and told her she wasn't allowed to kill herself because Dad was gone and I was gay and I didn't know what the hell to do, and that was how I came out to her.\n\n\"The next day, she went back, and she's been clean ever since, and neither of us has ever told anyone about that night. Until now, I suppose. And I'm not sure why I've said all this, I just, I've never really said any of it. I mean, Pez was there for most of it, so, and I\u2014I don't know.\" He clears his throat. \"Anyway, I don't think I've ever said this many words out loud in a row in my entire life, so please feel free to put me out of my misery any time now.\"\n\n\"No, no,\" Alex says, stumbling over his own tongue in a rush. \"I'm glad you told me. Does it feel better at all to have said it?\"\n\nHenry goes silent, and Alex wants so badly to see the shadows of expressions moving across his face, to be able to touch them with his fingertips. Alex hears a swallow across the line, and Henry says, \"I suppose so. Thank you. For listening.\"\n\n\"Yeah, of course,\" Alex tells him. \"I mean, it's good to have times when it's not all about me, as tedious and exhausting as it may be.\"\n\nThat earns him a groan, and he bites back a smile when Henry says, \"You are a _wanker._ \"\n\n\"Yeah, yeah,\" Alex says, and he takes the opportunity to ask a question he's been wanting to ask for months. \"So, um. Does anybody else know? About you?\"\n\n\"Bea's the only one in the family I've told, though I'm sure the rest have suspected. I was always a bit different, never quite had the stiff upper lip. I think Dad knew and never cared. But Gran sat me down the day I finished my A levels and made it abundantly clear I was not to let anyone know about any deviant desires I might be beginning to harbor that might reflect poorly upon the crown, and there were appropriate channels to maintain appearances if necessary. So.\"\n\nAlex's stomach turns over. He pictures Henry, a teenager, back-broken with grief and told to keep it and the rest of him shut up tight.\n\n\"What the fuck. Seriously?\"\n\n\"The wonders of the monarchy,\" Henry says loftily.\n\n\"God.\" Alex scrubs a hand across his face. \"I've had to fake some shit for my mom, but nobody's ever outright told me to _lie_ about who I am.\"\n\n\"I don't think she sees it as lying. She sees it as doing what must be done.\"\n\n\"Sounds like bullshit.\"\n\nHenry sighs. \"Hardly any other options, are there?\"\n\nThere's a long pause, and Alex is thinking about Henry in his palace, Henry and the years behind him, how he got here. He bites his lip.\n\n\"Hey,\" Alex says. \"Tell me about your dad.\"\n\nAnother pause.\n\n\"Sorry?\"\n\n\"I mean, if you don't\u2014if you want to. I was just thinking I don't know much about him except that he was James Bond. What was he like?\"\n\nAlex paces the Solarium and listens to Henry talk, stories about a man with Henry's same sandy hair and strong, straight nose, someone Alex has met in shadows that pass through the way Henry speaks and moves and laughs. He hears about sneaking out of the palace and joyriding around the countryside, learning to sail, being propped up in director's chairs. The man Henry remembers is both superhuman and heartbreakingly flesh and blood, a man who encompassed Henry's entire childhood and charmed the world but was also simply a man.\n\nThe way Henry talks about him is a physical feat, drifting up in the corners with fondness but sagging in the middle under the weight. He tells Alex in a low voice how his parents met\u2014Princess Catherine, dead set on being the first princess with a doctorate, mid-twenties and wading through Shakespeare. How she went to see _Henry V_ at the RSC and Arthur was starring, how she pushed her way backstage and shook off her security to disappear into London with him and dance all night. How the Queen forbid it, but she married him anyway.\n\nHe tells Alex about growing up in Kensington, how Bea sang and Philip clung to his grandmother, but they were happy, buttoned up in cashmere and knee socks and whisked through foreign countries in helicopters and shiny cars. A brass telescope from his father for his seventh birthday. How he realized by the time he was four that every person in the country knew his name, and how he told his mother he didn't know if he wanted them to, and how she knelt down and told him she'd let nothing touch him, not ever.\n\nAlex starts talking too. Henry already hears nearly everything about Alex's current life, but talking about how they grew up has always been some invisible line of demarcation. He talks about Travis County, making campaign posters with construction paper for fifth-grade student council, family trips to Surfside, running headlong into the waves. He talks about the big bay window in the house where he grew up, and Henry doesn't tell him he's crazy for all the things he used to write and hide under there.\n\nIt starts to grow dark outside, a dull and soggy evening around the Residence, and Alex makes his way down to his room and his bed. He hears about the assortment of guys from Henry's university days, all of them enamored with the idea of sleeping with a prince, almost all of them immediately alienated by the paperwork and secrecy and, occasionally, Henry's dark moods about the paperwork and secrecy.\n\n\"But of course, er,\" Henry says, \"nobody since... well, since you and I\u2014\"\n\n\"No,\" Alex says, faster than he expects, \"me neither. Nobody else.\"\n\nHe hears words coming out of his mouth, ones he can't believe he's saying out loud. About Liam, about those nights, but also how he'd sneak pills out of Liam's Adderall bottle when his grades were slipping and stay awake for two, three days at a time. About June, the unspoken knowledge that she only lives here to watch out for him, the quiet sense of guilt he carries when he can't tear himself away. About how much some of the lies people tell about his mother hurt, the fear she'll lose.\n\nThey talk for so long Alex has to plug his phone in to keep the battery from dying. He rolls onto his side and listens, trails the back of his hand across the pillow next to him and imagines Henry lying opposite in his own bed, two parentheses enclosing 3,700 miles. He looks at his chewed-up cuticles and imagines Henry there under his fingers, speaking into only inches of distance. He imagines the way Henry's face would look in the bluish-gray dark. Maybe he would have a faint shadow of stubble on his jaw, waiting for a morning shave, or maybe the circles under his eyes would wash out in the low light.\n\nSomehow, this is the same person who had Alex so convinced he didn't care about anything, who still has the rest of the world convinced he's a mild, unfettered Prince Charming. It's taken months to get here: the full realization of just how wrong he was.\n\n\"I miss you,\" Alex says before he can stop himself.\n\nHe instantly regrets it, but Henry says, \"I miss you too.\"\n\n* * *\n\n\"Hey, wait.\"\n\nAlex rolls his chair back out of his cubicle. The woman from the after-hours cleaning crew stops, her hand on the handle of the coffeepot. \"I know it looks disgusting, but would you mind leaving that? I was gonna finish it.\"\n\nShe gives him a dubious look but leaves the last burnt, sludgy vestiges of coffee where they are and rolls off with her cart.\n\nHe peers down into his CLAREMONT FOR AMERICA mug and frowns at the almond milk that's pooled in the middle. Why doesn't this office keep normal milk around? This is why people from Texas hate Washington elites. Ruining the goddamn dairy industry.\n\nOn his desk, there are three stacks of papers. He keeps staring at them, hoping if he recites them enough times in his head, he'll figure out how to feel like he's doing enough.\n\nOne. The Gun File. A detailed index of every kind of insane gun Americans can own and state-by-state regulations, which he has to comb through for research on a new set of federal assault rifle policies. It's got a giant smudge of pizza sauce on it because it makes him stress-eat.\n\nTwo. The Trans-Pacific Partnership File, which he knows he needs to work on but has barely touched because it's mind-numbingly boring.\n\nThree. The Texas File.\n\nHe's not supposed to have this file. It wasn't given to him by the policy chief of staff or anyone on the campaign. It's not even about policy. It's also more of a binder than a file. He guesses he should call it: The Texas Binder.\n\nThe Texas Binder is his baby. He guards it jealously, stuffing it into his messenger bag to take home with him when he leaves the office and hiding it from WASPy Hunter. It contains a county map of Texas with complex voter demographic breakdowns, matched up with the populations of children of undocumented immigrants, unregistered voters who are legal residents, voting patterns over the last twenty years. He's stuffed it with spreadsheets of data, voting records, projections he had Nora calculate for him.\n\nBack in 2016, when his mother squeezed out a victory in the general election, the bitterest sting was losing Texas. She was the first president since Nixon to win the presidency but lose her own state of residence. It wasn't exactly a surprise, considering Texas had been polling red, but they were all secretly holding out for the Lometa Longshot to take it in the end. She didn't.\n\nAlex keeps coming back to the numbers from 2016 and 2018 precinct by precinct, and he can't shake this nagging feeling of hope. There's something there, something shifting, he swears it.\n\nHe doesn't mean to be ungrateful for the policy job, it's just... not what he thought it was going to be. It's frustrating and slow-moving. He should stay focused, give it more time, but instead, he keeps coming back to the binder.\n\nHe plucks a pencil out of WASPy Hunter's Harvard pencil cup and starts sketching lines on the map of Texas for the millionth time, redrawing the districts old white men drew years ago to force votes their way.\n\nAlex has this spark at the base of his spine to do the most good he can, and when he sits here in his cubicle for hours a day and fidgets under all the minutiae, he doesn't know if he is. But if he could only figure out a way to make Texas' vote reflect its soul... he's nowhere near qualified to single-handedly dismantle Texas' iron curtains of gerrymandering, but what if he\u2014\n\nAn incessant buzzing snaps him present, and he digs out his phone from the bottom of his bag.\n\n\"Where are you?\" June's voice demands over the line.\n\nFuck. He checks the time: 9:44. He was supposed to meet June for dinner over an hour ago.\n\n\"Shit, June, I'm so sorry,\" he says, jumping up from his desk and shoving his things into his bag. \"I got caught up at work\u2014I, I completely forgot.\"\n\n\"I sent you like a million texts,\" she says. She sounds like she's vision-boarding his funeral.\n\n\"My phone was on silent,\" he says helplessly, booking it for the elevator. \"I'm seriously so sorry. I'm a complete jackass. I'm leaving now.\"\n\n\"Don't worry about it,\" she says. \"I got mine to go. I'll see you at home.\"\n\n\"Bug.\"\n\n\"I'm gonna need you to _not_ call me that right now.\"\n\n\"June\u2014\"\n\nThe call drops.\n\nWhen he gets back to the Residence, she's sitting on her bed, eating pasta out of a plastic container, with _Parks &_ _Recreation_ playing on her tablet. She pointedly ignores him when he comes to her doorway.\n\nHe's reminded of when they were kids\u2014around eight and eleven years old. He recalls standing next to her at the bathroom mirror, looking at the similarities between their faces: the same round tips of their noses, the same thick, unruly brows, the same square jaw inherited from their mother. He remembers studying her expression in the reflection as they brushed their teeth, the morning of the first day of school, their dad having braided June's hair for her because their mom was in DC and couldn't be there.\n\nHe recognizes the same expression on her face now: carefully tucked-away disappointment.\n\n\"I'm sorry,\" he tries again. \"I honestly feel like complete and total shit. Please don't be mad at me.\"\n\nJune keeps chewing, looking steadfastly at Leslie Knope chirping away.\n\n\"We can do lunch tomorrow,\" Alex says desperately. \"I'll pay.\"\n\n\"I don't care about a stupid meal, Alex.\"\n\nAlex sighs. \"Then what do you want me to do?\"\n\n\"I want you not to be Mom,\" June says, finally looking up at him. She closes her food container and gets up off her bed, pacing across the room.\n\n\"Okay,\" Alex says, raising both hands, \"is that what's happening right now?\"\n\n\"I\u2014\" She takes a deep breath. \"No. I shouldn't have said that.\"\n\n\"No, you obviously meant it,\" Alex says. He drops his messenger bag and steps into the room. \"Why don't you say whatever it is you need to say?\"\n\nShe turns to face him, arms folded, her spine braced against her dresser. \"You really don't see it? You never sleep, you're always throwing yourself into something, you're willing to let Mom use you for whatever she wants, the tabloids are always after you\u2014\"\n\n\"June, I've always been this way,\" he interrupts gently. \"I'm gonna be a politician. You always knew that. I'm starting as soon as I graduate... in a month. This is how my life is gonna be, okay? I'm choosing it.\"\n\n\"Well, maybe it's the wrong choice,\" June says, biting her lip.\n\nHe rocks back on his heels. \"Where the hell is this coming from?\"\n\n\"Alex,\" she says, \"come on.\"\n\nHe doesn't know what the hell she's getting at. \"You've always backed me up until now.\"\n\nShe flings one arm out emphatically enough to upset an entire potted cactus on her dresser and says, \"Because until now you weren't _fucking the Prince of England_!\"\n\nThat effectively snaps Alex's mouth shut. He crosses to the sitting area in front of the fireplace, sinking down into an armchair. June watches him, cheeks bright scarlet.\n\n\"Nora told you.\"\n\n\"What?\" she says. \"No. She wouldn't do that. Although it kinda sucks you told her and not me.\" She folds her arms again. \"I'm sorry, I was trying to wait for you to tell me yourself, but, Jesus, Alex. How many times was I supposed to believe you were volunteering to take those international appearances we always found excuses to get out of? And, like, did you forget I've lived across the hall from you for almost my entire life?\"\n\nAlex looks down at his shoes, June's perfectly curated midcentury rug. \"So you're mad at me because of Henry?\"\n\nJune makes a strangled noise, and when he looks back up, she's digging through the top drawer of her dresser. \"Oh my God, how are you so smart and so dumb at the same time?\" she says, pulling a magazine out from underneath her underwear. He's about to tell her he's not in the mood to look at her tabloids when she throws it at him.\n\nAn ancient issue of _J14,_ opened to a center page. The photograph of Henry, age thirteen.\n\nHe glances up. \"You knew?\"\n\n\"Of course I knew!\" she says, flopping dramatically into the chair opposite him. \"You were always leaving your greasy little fingerprints all over it! Why do you always assume you can get away with things?\" She releases a long-suffering sigh. \"I never really... got what he was to you, until I _got_ it. I thought you had a crush or something, or that I could help you make a friend, but, Alex. We meet so many people. I mean, thousands and thousands of people, and a lot of them are morons, and a lot of them are incredible, unique people, but I almost never meet somebody who's a match for you. Do you know that?\" She leans forward and touches his knee, pink fingernails on his navy chinos. \"You have so much in you, it's almost impossible to match it. But he's your match, dumbass.\"\n\nAlex stares at her, trying to process what she's said.\n\n\"I feel like this is your starry-eyed romantic thing projecting onto me,\" is what he decides to say, and she immediately withdraws her hand from his leg and returns to glaring at him.\n\n\"You know Evan didn't break up with me?\" she says. \"I broke up with him. I was gonna go to California with him, live in the same time zone as Dad, get a job at the fucking _Sacramento Bee_ or something. But I gave all that up to come _here,_ because it was the right thing to do. I did what Dad did\u2014I went where I was most needed, because it was my responsibility.\"\n\n\"And you regret it?\"\n\n\"No,\" she says. \"I don't know. I don't think so. But I\u2014I wonder. Dad wonders, sometimes. Alex, you don't have to wonder. You don't have to be our parents. You can keep Henry, and figure the rest out.\" Now she's looking at him evenly, steadily. \"Sometimes you have a fire under your ass for no good goddamn reason. You're gonna burn out like this.\"\n\nAlex leans back, thumbing the stitching on the armrest of the chair.\n\n\"So, what?\" he asks. \"You want me to quit politics and go become a princess? That's not very feminist of you.\"\n\n\"That's not how feminism works,\" she says, rolling her eyes. \"And that's not what I mean. I mean... I don't know. Have you ever considered there might be more than one path to use what you have? Or to get where you want to be to make the most difference in the world?\"\n\n\"I'm not sure I'm following.\"\n\n\"Well.\" She looks down at her cuticles. \"It's like the whole _Sac Bee_ thing\u2014it never actually would have worked out. It was a dream I had before Mom was president. The kind of journalism I wanted to do is the kind of journalism that being a First Daughter pretty much disqualifies you from. But the world is better with her where she is, and right now I'm looking for a new dream that's better too.\" Her big brown Diaz eyes blink up at him. \"So, I don't know. Maybe there's more than one dream for you, or more than one way to get there.\"\n\nShe gives a crooked shrug, tilting her head to look at him openly. June is often a mystery, a big ball of complex emotions and motivations, but her heart is honest and true. She's very much what Alex holds in his memory as the sanctified idea of Southerness at its best: always generous and warm and sincere, work-strong and reliable, a light left on. She wants the best for him, plainly, in an unselfish and uncalculating way. She's been trying to talk to him for a while, he realizes.\n\nHe looks down at the magazine and feels the corner of his mouth tug upward. He can't believe June kept it all these years.\n\n\"He looks so different,\" he says after a long minute, gazing down at the baby Henry on the page and his easy, unfledged sureness. \"I mean, like, obviously. But the way he carries himself.\" His fingertips brush the page in the same place they did when he was young, over the sun-gold hair, except now he knows its exact texture. It's the first time he's seen it since he learned where this version of Henry went. \"It pisses me off sometimes, thinking about everything he's been through. He's a good person. He really cares, and he _tries._ He never deserved any of it.\"\n\nJune leans forward, looking at the picture too. \"Have you ever told him that?\"\n\n\"We don't really...\" Alex coughs. \"I don't know. Talk like that?\"\n\nJune inhales deeply and makes an enormous fart noise with her mouth, shattering the serious mood, and Alex is so grateful for it that he melts onto the floor in a fit of hysterical laughter.\n\n\"Ugh! Men!\" she groans. \"No emotional vocabulary. I can't believe our ancestors survived centuries of wars and plagues and genocide just to wind up with your sorry ass.\" She throws a pillow at him, and Alex scream-laughs as it hits him in the face. \"You should try saying some of that stuff to _him._ \"\n\n\"Stop trying to Jane Austen my life!\" he yells back.\n\n\"Listen, it's not my fault he's a mysterious and retiring young royal and you're the tempestuous ing\u00e9nue that caught his eye, okay?\"\n\nHe laughs and tries to crawl away, even as she claws at his ankle and wallops another pillow at his head. He still feels guilty for blowing her off, but he thinks they're okay now. He'll do better. They fight for a spot on her big canopy bed, and she makes him spill what it's like to be secretly hooking up with a real-life prince. And so June knows; she knows about him and she hugs him and doesn't care. He didn't realize how terrified he was of her knowing until the fear is gone.\n\nShe puts _Parks_ back on and has the kitchen send up ice cream, and Alex thinks about how she said, \"You don't have to be our parents\"\u2014she's never mentioned their dad in the same context as their mom like that before. He's always known part of her resents their mom for the position they occupy in the world, for not having a normal life, for taking herself away from them. But he never really realized she felt the same sense of loss he does deep down about their dad, that it's something she dealt with and moved past. That the stuff with their mom is something she's still going through.\n\nHe thinks she's wrong about him, mostly\u2014he doesn't necessarily believe he has to choose between politics and this thing with Henry yet, or that he's moving too fast in his career. But... there's the Texas Binder, and the knowledge of other states like Texas and millions of other people who need someone to fight for them, and the feeling at the base of his spine, like there's a lot of fight in him that could be honed down to a more productive point.\n\nThere's law school.\n\nEvery time he looks at the Texas Binder, he knows it's a big fat case for him to go take the damn LSAT like he knows both his parents wish he would instead of diving headfirst into politics. He's always, always said no. He doesn't wait for things. Doesn't put in the time like that, do what he's told.\n\nHe's never given much thought to options other than a crow's path ahead of him. Maybe he should.\n\n\"Is now a good time to point out Henry's very hot, very rich best friend is basically in love with you?\" Alex says to June. \"He's like some kind of billionaire, genius, manic-pixie-dream philanthropist. I feel like you would be into that.\"\n\n\"Please shut up,\" she says, and she steals the ice cream back.\n\n* * *\n\nOnce June knows, their circle of \"knowing\" is up to a tight seven.\n\nBefore Henry, most of his romantic entanglements as FSOTUS were one-off incidents that involved Cash or Amy confiscating phones before the act and pointing at the dotted line on the NDA on the way out\u2014Amy with mechanical professionalism, Cash with the air of a cruise ship director. It was inevitable they be looped in.\n\nAnd there's Shaan, the only member of the royal staff who knows Henry is gay, excluding his therapist. Shaan ultimately doesn't care about Henry's sexual preferences as long as they're not getting him into trouble. He's a consummate professional parceled in immaculately tailored Tom Ford, ruffled by absolutely nothing, whose affection for his charge shows in the way he tends to him like a favorite houseplant. Shaan knows for the same reason Amy and Cash know: absolute necessity.\n\nThen Nora, who still looks smug every time the subject arises. And Bea, who found out when she walked in on one of their after-dark FaceTime sessions, leaving Henry capable of nothing but flustered British stammering and thousand-yard stares for the next day and a half.\n\nPez seems to have been in on the secret all along. Alex imagines he demanded an explanation when Henry literally made them flee the country under the cover of night after putting his tongue in Alex's mouth in the Kennedy Garden.\n\nIt's Pez who answers when Alex FaceTimes Henry at four a.m. DC time, expecting to catch Henry over his morning tea. Henry is holidaying in one of the family's country homes while Alex suffocates under his last week of college. He doesn't reflect on why his migraine demands soothing images of Henry looking cozy and picturesque, sipping tea by a lush green hillside. He just hits the buttons on the phone.\n\n\"Alexander, babes,\" Pez says when he picks up. \"How lovely for you to give your auntie Pezza a ring on this magnificent Sunday morning.\" He's smiling from what looks like the passenger seat of a luxury car, wearing a cartoonishly large sunhat and a striped pashmina.\n\n\"Hi, Pez,\" Alex says, grinning back. \"Where are y'all?\"\n\n\"We are out for a drive, taking in the scenery of Carmarthenshire,\" Pez tells him. He tilts the phone over toward the driver's seat. \"Say good morning to your strumpet, Henry.\"\n\n\"Good morning, strumpet,\" Henry says, glancing away from the road to wink at the camera. He's looking fresh-faced and relaxed, all rolled-up sleeves and soft gray linen, and Alex feels calmer knowing somewhere in Wales, Henry got a decent night's sleep. \"What's got you up at four in the morning this time?\"\n\n\"My fucking economics final,\" Alex says, rolling over onto his side to squint at the screen. \"My brain isn't working anymore.\"\n\n\"Can't you get one of those Secret Service earpieces with Nora on the other end?\"\n\n\"I can take it for you,\" Pez interjects, turning the camera back to himself. \"I'm aces with money.\"\n\n\"Yes, yes, Pez, we know there's nothing you can't do,\" says Henry's voice off-camera. \"No need to rub it in.\"\n\nAlex laughs under his breath. From the angle Pez is holding the phone, he can see Wales rolling by though the car window, dramatic and plunging. \"Hey, Henry, say the name of the house you're staying at again.\"\n\nPez turns the camera to catch Henry in a half smile. \"Llwynywermod.\"\n\n\"One more time.\"\n\n_\"Llwynywermod.\"_\n\nAlex groans. \"Jesus.\"\n\n\"I was _hoping_ you two would start talking dirty,\" Pez says. \"Please, do go on.\"\n\n\"I don't think you could keep up, Pez,\" Alex tells him.\n\n\"Oh _really_?\" The picture returns to Pez. \"What if I put my co\u2014\"\n\n_\"Pez,\"_ comes the sound of Henry's voice, and a hand with a signet ring on the smallest finger covers Pez's mouth. \"I beg of you. Alex, what part of 'nothing he cannot do' did you think was worth testing? Honestly, you are going to get us all killed.\"\n\n\"That's the goal,\" Alex says happily. \"So what are y'all gonna do today?\"\n\nPez frees himself by licking Henry's palm and continues talking. \"Frolic naked in the hills, frighten the sheep, return to the house for the usual: tea, biscuits, casting ourselves upon the Thighmaster of love to moan about Claremont-Diaz siblings, which has become tragically one-sided since Henry took up with you. It used to be all bottles of cognac and shared malaise and 'When will they notice us'\u2014\"\n\n_\"Don't tell him that!\"_\n\n\"\u2014and now I just ask Henry, 'What is your secret?' And he says, 'I insult Alex all the time and that seems to work.'\"\n\n\"I will _turn this car around._ \"\n\n\"That won't work on June,\" Alex says.\n\n\"Let me get a pen\u2014\"\n\nIt turns out they're spending their holiday workshopping philanthropy projects. Henry's been telling Alex for months about their plans to go international, and now they're talking three refugee programs around Western Europe, HIV clinics in Nairobi and Los Angeles, LGBT youth shelters in four different countries. It's ambitious, but since Henry still staunchly covers all his own expenses with his inheritance from his father, his royal accounts are untouched. He's determined to use them for nothing but this.\n\nAlex curls around his phone and his pillow as the sun comes up over DC. He's always wanted to be a person with a legacy in this world. Henry is undoubtedly, determinedly that. It's a little intoxicating. But it's fine. He's just a little sleep-deprived.\n\nAll in all, finals come and go with much less fanfare than Alex imagined. It's a week of cramming and presentations and the usual amount of all-nighters, and it's over.\n\nThe whole college thing in general went by like that. He didn't really have the experiences everyone else has, always isolated by fame or harangued by security. He never got a stamp on his forehead on his twenty-first birthday at The Tombs, never jumped in Dalhgren Fountain. Sometimes it's like he barely went to Georgetown, merely powered through a series of lectures that happened to be in the same geographical area.\n\nAnyway, he graduates, and the whole auditorium gives him a standing ovation, which is weird but kind of cool. A dozen of his classmates want to take a photo with him afterward. They all know him by name. He's never spoken to any of them before. He smiles for their parents' iPhones and wonders if he should have tried.\n\n_Alex Claremont-Diaz graduates summa cum laude from Georgetown University with a bachelor's degree in Government,_ his Google alerts read when he checks them from the back seat of the limo, before he's even taken his cap and gown off.\n\nThere's a huge garden party at the White House, and Nora is there in a dress and blazer and a sly smile, pressing a kiss to the side of Alex's jaw.\n\n\"The last of the White House Trio finally graduates,\" she says, grinning. \"And he didn't even have to bribe any professors with political or sexual favors to do it.\"\n\n\"I think some of them might finally manage to purge me from their nightmares soon,\" Alex says.\n\n\"Y'all do school weird,\" June says, crying a little.\n\nThere's a mixed bag of political power players and family friends in attendance\u2014including Rafael Luna, who falls under the heading of both. Alex spots him looking tired but handsome by the ceviche, involved in animated conversation with Nora's grandfather, the Veep. His dad is in from California, freshly tanned from a recent trek through Yosemite, grinning and proud. Zahra hands him a card that says, _Good job doing what was expected of you,_ and nearly shoves him into the punch bowl when he tries to hug her.\n\nAn hour in, his phone buzzes in his pocket, and June gives him a mild glare when he diverts his attention mid-sentence to check it. He's ready to brush it off, but all around him iPhones and Blackberries are coming out in a flurry of movement.\n\nIt's WASPy Hunter: Jacinto just called a presser, word is he's dropping out of the primary a.k.a. officially Claremont vs. Richards 2020.\n\n\"Shit,\" Alex says, turning his phone around to show June the message.\n\n\"So much for the party.\"\n\nShe's right\u2014in a matter of seconds, half the tables are empty as campaign staffers and congresspeople leave their seats to huddle together over their phones.\n\n\"This is a bit dramatic,\" Nora observes, sucking an olive off the end of a toothpick. \"We all knew he was gonna give Richards the nomination eventually. They probably got Jacinto in a windowless room and bench-clamped his dick to the table until he said he'd concede.\"\n\nAlex doesn't hear whatever Nora says next because a rush of movement at the doors of the Palm Room near the edge of the garden catches his eye. It's his dad, pulling Luna by the arm. They disappear into a side door, toward the housekeeper's office.\n\nHe leaves his champagne with the girls and weaves a circuitous path toward the Palm Room, pretending to check his phone. Then, after considering whether the scolding he'll get from the dry-cleaning crew will be worth it, he ducks into the shrubbery.\n\nThere's a loose windowpane in the bottom of the third fixture of the south-facing wall of the housekeeper's office. It's popped out of its frame slightly, enough that its bulletproof, soundproof seal isn't totally intact. It's one of three windowpanes like this in the Residence. He found them during his first six months at the White House, before June graduated and Nora transferred, when he was alone, with nothing better to do than these little investigative projects around the grounds.\n\nHe's never told anyone about the loose panes; he always suspected they might come in handy one day.\n\nHe crouches down and creeps up toward the window, soil rolling into his loafers, hoping he guessed their destination right, until he finds the pane he's looking for. He leans in, tries to get his ear as close to it as he can. Over the sound of the wind rustling the bushes around him, he can hear two low, tense voices.\n\n\"... hell, Oscar,\" says one voice, in Spanish. Luna. \"Did you tell her? Does she know you're asking me to do this?\"\n\n\"She's too careful,\" his father's voice says. He's speaking Spanish too\u2014a precaution the two of them occasionally take when they're concerned about being overheard. \"Sometimes it's best that she doesn't know.\"\n\nThere's the sound of a hissing exhale, weight shifting. \"I'm not going behind her back to do something I don't even want to do.\"\n\n\"You mean to tell me, after what Richards did to you, there's not a part of you that wants to burn all his shit to the ground?\"\n\n\"Of course there is, Oscar, Jesus,\" Luna says. \"But you and I both know it's not that fucking simple. It never is.\"\n\n\"Listen, Raf. I know you kept the files on everything. You don't even have to make a statement. You could leak it to the press. How many other kids do you think since\u2014\"\n\n\"Don't.\"\n\n\"\u2014and how many more\u2014\"\n\n\"You don't think she can win on her own, do you?\" Luna cuts across him. \"You still don't have faith in her, after everything.\"\n\n\"It's not about that. This time is different.\"\n\n\"Why don't you leave me and something that happened _twenty fucking years ago_ out of your unresolved feelings for your ex-wife and focus on winning this goddamn election, Oscar? I don't\u2014\"\n\nLuna cuts himself off because there's the sound of the doorknob turning, someone entering the offices.\n\nOscar switches to clipped English, making an excuse about discussing a bill, then says to Luna, in Spanish, \"Just think about it.\"\n\nThere are muffled sounds of Oscar and Luna clearing out of the office, and Alex sinks down onto his ass in the mulch, wondering what the hell he's missing.\n\n* * *\n\nIt starts with a fund-raiser, a silk suit and a big check, a nice white-tablecloth event. It starts, as it always does, with a text: Fund-raiser in LA next weekend. Pez says he's going to get us all matching embroidered kimonos. Put you down for a plus-two?\n\nHe grabs lunch with his dad, who flat-out changes the subject every time Alex brings up Luna, and afterward heads to the gala, where Alex gets to properly meet Bea for the first time. She's much shorter than Henry, shorter even than June, with Henry's clever mouth but their mom's brown hair and heart-shaped face. She's wearing a motorcycle jacket over her cocktail dress and has a slight posture he recognizes from his own mother as a reformed chainsmoker. She smiles at Alex, wide and mischievous, and he gets her immediately: another rebel kid.\n\nIt's a lot of champagne and too many handshakes and a speech by Pez, charming as always, and as soon as it's over, their collective security convenes at the exit and they're off.\n\nPez has, as promised, six matching silk kimonos waiting in the limo, each one embroidered across the back with a different riff on a name from a movie. Alex's is a lurid teal and says HOE DAMERON. Henry's lime-green one reads PRINCE BUTTERCUP.\n\nThey end up somewhere in West Hollywood at a shitty, sparkling karaoke bar Pez somehow knows about, neon bright enough that it feels spontaneous even though Cash and the rest of their security have been checking it and warning people against taking photos for half an hour before they arrive. The bartender has immaculate pink lipstick and stubble poking through thick foundation, and they rapidly line up five shots and a soda with lime.\n\n\"Oh, dear,\" Henry says, peering down into his empty shot glass. \"What's in these? Vodka?\"\n\n\"Yep,\" Nora confirms, to which both Pez and Bea break out into fits of giggles.\n\n\"What?\" Alex says.\n\n\"Oh, I haven't had vodka since uni,\" Henry says. \"It tends to make me, erm. Well\u2014\"\n\n\"Flamboyant?\" Pez offers. \"Uninhibited? _Randy?_ \"\n\n\"Fun?\" Bea suggests.\n\n\" _Excuse_ you, I am _loads_ of fun all the time! I am a _delight_!\"\n\n\"Hello, excuse me, can we get another round of these please?\" Alex calls down the bar.\n\nBea screams, Henry laughs and throws up a V, and it all goes hazy and warm in the way Alex loves. They all tumble into a round booth, and the lights are low, and he and Henry are keeping a safe distance, but Alex can't stop staring at how the special-effect beams keep hitting Henry's cheekbones, hollowing his face out in blues and greens. He's something else\u2014half-drunk and grinning in a $2,000 suit and a kimono, and Alex can't tear his eyes away. He waves over a beer.\n\nOnce things get going, it's impossible to tell how Bea is the one persuaded up to the stage first, but she unearths a plastic crown from the prop chest onstage and rips through a cover of \"Call Me\" by Blondie. They all wolf whistle and cheer, and the bar crowd finally realizes they've got two members of the royal family, a millionaire philanthropist, and the White House Trio crammed into one of the sticky booths in a rainbow of vivid silk. Three rounds of shots appear\u2014one from a drunk bachelorette party, one from a herd of surly butch chicks at the bar, and one from a table of drag queens. They raise a toast, and Alex feels more welcomed than he ever has before, even at his family's victory rallies.\n\nPez gets up and launches into \"So Emotional\" by Whitney Houston in a shockingly flawless falsetto that has the whole club on their feet in a matter of moments, shouting their approval as he belts out the glory notes. Alex looks over in giddy awe at Henry, who laughs and shrugs.\n\n\"I told you, there's nothing he can't do,\" he shouts over the noise.\n\nJune is watching the whole performance with her hands clapped to her face, her mouth hanging open, and she leans over to Nora and drunkenly yells, \"Oh, _no_... he's... so... hot...\"\n\n\"I know, babe,\" Nora yells back.\n\n\"I want to... put my fingers in his mouth...\" she moans, sounding horrified.\n\nNora cackles and nods appreciatively and says, \"Can I help?\"\n\nBea, who has gone through five different lime and sodas so far, politely passes over a shot that's been handed to her as Pez pulls June up on stage, and Alex throws it back. The burn makes his smile and his legs spread a little wider, and his phone is in his hand before he registers sliding it out of his pocket. He texts Henry under the table: wanna do something stupid?\n\nHe watches Henry pull his own phone out, grin, and arch a brow over at him.\n\nWhat could be stupider than this?\n\nHenry's mouth falls open into a very unflattering expression of drunken, bewildered arousal, like a hot halibut, at his reply several beats later. Alex smiles and leans back into the booth, making a show of wrapping wet lips around the bottle of his beer. Henry looks like his entire life might be flashing before his eyes, and he says, an octave too high, \"Right, well, I'll just\u2014nip to the loo!\"\n\nAnd he's off while the rest of the group is still caught up Pez and June's performance. Alex gives it to the count of ten before slipping past Nora and following. He swaps a glance with Cash, who's standing against one wall, gamely wearing a bright pink feather boa. He rolls his eyes but peels off to watch the door.\n\nAlex finds Henry leaning against the sink, arms folded.\n\n\"Have I mentioned lately that you're a _demon_?\"\n\n\"Yeah, yeah,\" Alex says, double-checking the coast is clear before grabbing Henry by the belt and backing into a stall. \"Tell me again later.\"\n\n\"You\u2014you know this is still not convincing me to sing, don't you?\" Henry chokes out as Alex mouths along his throat.\n\n\"You really think it's a good idea to present me with a challenge, sweetheart?\"\n\nWhich is how, thirty minutes and two more rounds later, Henry is in front of a screaming crowd, absolutely butchering \"Don't Stop Me Now\" by Queen while Nora sings backup and Bea throws glittery gold roses at his feet. His kimono is dangling off one shoulder so the embroidery across the back reads PRINCE BUTT. Alex does not know where the roses came from, and he can't imagine asking would get him anywhere. He also wouldn't be able to hear the answer because he's been screaming at the top of his lungs for two minutes straight.\n\n_\"I wanna make a supersonic woman of youuu!\"_ Henry shouts, lunging violently sideways, catching Nora by both arms. _\"Don't stop me! Don't stop me! Don't stop me!\"_\n\n_\"Hey, hey, hey!\"_ the entire bar yells back. Pez is practically on top of the table now, pounding the back of the booth with one hand and helping June up onto a chair with the other.\n\n_\"Don't stop me! Don't stop me!\"_\n\nAlex cups his hands around his mouth. _\"Ooh, ooh, ooh!\"_\n\nIn a cacophony of shouting and kicking and pelvic-thrusting and flashing lights, the song blasts into the guitar solo, and there's not a single person in the bar in their seat, not when a Prince of England is knee-sliding across the stage, playing passionate and somewhat erotic air guitar.\n\nNora has produced a bottle of champagne and starts spraying Henry with it, and Alex loses his _mind_ laughing, climbs on top of his seat and wolf whistles. Bea is absolutely beside herself, tears streaming down her face, and Pez actually is on top of the table now, June dancing beside him, with a bright fuschia smear of lipstick in his platinum hair.\n\nAlex feels a tug on his arm\u2014Bea, dragging him down to the stage. She grabs his hand and spins him in a ballerina twirl, and he puts one of her roses between his teeth, and they watch Henry and grin at each other through the noise. Alex feels somewhere, under the fifty layers of booze, something crystal clear radiating off her, a shared knowledge of how rare and wonderful this version of Henry is.\n\nHenry is yelling into the microphone again, stumbling to his feet, his suit and kimono stuck to him with champagne and sweat in a confusingly sexy mess. His eyes flick upward, hazy and hot, and unmistakably lock with Alex's at the edge of the stage, smiling broad and messy. _\"I wanna make a supersonic man outta youuuuu!\"_\n\nBy the end, there's a standing ovation awaiting him, and Bea, with a steady hand and a devilish smile, ruffling his champagne-sticky hair. She steers him into the booth and Alex's side, and he pulls her in after him, and the six of them fall together in a tangle of hoarse laughter and expensive shoes.\n\nHe looks at all of them. Pez, his broad smile and glowing joy, the way his white-blond hair flashes against smooth, dark skin. The curve of Bea's waist and hip and her punk-rock grin as she sucks on the rind of a lime. Nora's long legs, one of which is propped up on the table and the other crossed over one of Bea's, her thigh bare where her dress has ridden up. And Henry, flushed and callow and lean, elegant and thrown wide open, his face always turned toward Alex, his mouth unguarded around a laugh, willing.\n\nHe turns to June and slurs, \"Bisexuality is truly a rich and complex tapestry,\" and she screams with laughter and shoves a napkin in his mouth.\n\nAlex doesn't catch much of the next hour\u2014the back of the limo, Nora and Henry jostling for a spot in his lap, an In-N-Out drive-thru and June screaming next to his ear, \"Animal Style, did you hear me say Animal Style? Stop fucking laughing, Pez.\" There's the hotel, three suites booked for them on the very top floor, riding through the lobby on Cash's impossibly broad back.\n\nJune keeps shushing them as they stumble to their rooms with hands full of grease-soaked burger bags, but she's louder than any of them, so it's a zero-sum game. Bea, perpetually the lone sober voice of the group, picks one of the suites at random and deposits June and Nora in the king-size bed and Pez in the empty bathtub.\n\n\"I trust you two can handle yourselves?\" she says to Alex and Henry in the hallway, a glimmer of mischief in her eyes as she hands them the third key. \"I fully intend to put on a robe and investigate this french-fries-dipped-in-milkshake thing Nora told me about.\"\n\n\"Yes, Beatrice, we shall behave in a manner befitting the crown,\" Henry says. His eyes are slightly crossed.\n\n\"Don't be a tosser,\" she says, and quickly kisses them both on the cheek before vanishing around the corner.\n\nHenry's laughing into the curls at the nape of Alex's neck by the time Alex is fumbling the door open, and they stumble together into the wall, and then toward the bed, clothes dropping in their wake. Henry smells like expensive cologne and champagne and a distinctly Henry smell that never goes away, clean and grassy, and his chest encompasses Alex's back when he crowds up behind him at the edge of the bed, splaying his hands over his hips.\n\n_\"Supersonic man out of youuuu,\"_ Alex mumbles low, craning his head back into Henry's ear, and Henry laughs and kicks his knees out from under him.\n\nIt's a clumsy, sideways tumble into bed, both of them grabbing greedy handfuls of the other, Henry's pants still dangling from one ankle, but it doesn't matter because Henry's eyes are fluttered shut and Alex is finally kissing him again.\n\nHis hands start traveling south on instinct, sweet muscle memory of Henry's body against his, until Henry reaches down to stop him.\n\n\"Hold on, hold on,\" Henry says. \"I'm just realizing. All that earlier, and you haven't gotten off yet tonight, have you?\" He drops his head back on the pillow, regards him with narrowed eyes. \"Well. That just shall not do.\"\n\n\"Hmm, yeah?\" Alex says. He takes advantage of the moment to kiss the column of Henry's throat, the hollow at his collarbone, the knot of his Adam's apple. \"What are you gonna do about it?\"\n\nHenry pushes a hand into his hair and gives it a little pull. \"I shall just have to make it the best orgasm of your life. What can I do to make it good for you? Talk about American tax reform during the act? Have you got talking points?\"\n\nAlex looks up, and Henry is grinning at him. \"I hate you.\"\n\n\"Maybe some light lacrosse role-play?\" He's laughing now, arms coming up around Alex's shoulders to squeeze him to his chest. _\"O captain, my captain.\"_\n\n\"You're literally the worst,\" Alex says, and undercuts it by leaning up to kiss him once more, gently, then deeply, long and slow and heated. He feels Henry's body shifting beneath his, opening up.\n\n\"Hang on,\" Henry says, breaking off breathlessly. \"Wait.\" Alex opens his eyes, and when he looks down, the expression on Henry's face is a more familiar one: nervous, unsure. \"I do actually. Er. Have an idea.\"\n\nHe slides a hand up Henry's chest to the side of his jaw, ghosting over his cheek with one finger. \"Hey,\" he says, serious now. \"I'm listening. For real.\"\n\nHenry bites his lip, visibly searching for the right words, and apparently comes to a decision.\n\n\"C'mere,\" he says, surging up to kiss Alex, and he's putting his whole body into it now, sliding his hands down to palm at Alex's ass as he kisses him. Alex feels a sound tear itself from his throat, and he's following Henry's lead blindly now, kissing him deep into the mattress, riding a continuous wave of Henry's body.\n\nHe feels Henry's thighs\u2014those goddamn horseback-riding, polo-playing thighs\u2014moving around him, soft, warm skin wrapping around his waist, heels pressing into his back. When Alex breaks off to look at him, the intention on Henry's face is as plain as anything he's ever read there.\n\n\"You sure?\"\n\n\"I know we haven't,\" Henry says quietly. \"But, er. I have, before, so, I can show you.\"\n\n\"I mean, I'm familiar with the mechanics,\" Alex says, smirking a little, and he sees a corner of Henry's mouth quirk up to mirror him. \"But you want me to?\"\n\n\"Yeah,\" he says. He pushes his hips up, and they both make some unflattering, involuntary noises. \"Yes. Absolutely.\"\n\nHenry's shaving kit is on the nightstand, and he reaches over and fumbles blindly through it before finding what he's looking for\u2014a condom and a tiny bottle of lube.\n\nAlex almost laughs at the sight. Travel-size lube. He's had some experimental sex in his lifetime, but it never occurred to him to consider if such a thing existed, much less if Henry was jetting around with it alongside his dental floss.\n\n\"This is new.\"\n\n\"Yes, well,\" Henry says, and he takes one of Alex's hands in his and brings it to his own mouth, kissing his fingertips. \"We all must learn and grow, mustn't we?\"\n\nAlex rolls his eyes, ready to snark, except Henry sucks two fingers into his mouth, very effectively shutting him the hell up. It's incredible and baffling, the way Henry's confidence comes in waves like this, how he struggles so much to get through the asking for what he wants and then readily takes it the moment he's given permission, like at the bar, how the right push had him dancing and shouting as if he'd been waiting for someone to tell him he was allowed to do it.\n\nThey're not as drunk as they were, but there's enough alcohol in their systems, and it doesn't feel as daunting as it would otherwise, the first time, even as his fingers start to find their way. Henry's head falls back onto the pillows, and he closes his eyes and lets Alex take over.\n\nThe thing about sex with Henry is, it's never the same twice. Sometimes he moves easily, caught up in the rush, and other times he's tense and taut and wants Alex to work him loose and take him apart. Sometimes nothing gets him off faster than being talked back to, but other times they both want him to use every inch of authority in his blood, not to let Alex get there until he's told, until he begs.\n\nIt's unpredictable and it's intoxicating and it's _fun,_ because Alex has never met a challenge he didn't love, and he\u2014well, Henry is a challenge, head to toe, beginning to end.\n\nTonight, Henry's silly and warm and ready, his body quick and smooth to give Alex what he's looking for, laughing and incredulous at his own responsiveness to touch. Alex leans down to kiss him, and Henry murmurs into the corner of his mouth, \"Ready when you are, love.\"\n\nAlex takes a breath, holds it. He's ready. He thinks he's ready.\n\nHenry's hand comes up to stroke along his jaw, his sweaty hairline, and Alex settles himself between his legs, lets Henry lace the fingers of his right hand with Alex's left.\n\nHe's watching Henry's face\u2014he can't imagine looking at anything other than Henry's face right now\u2014and his expression goes so soft and his mouth so happy and astonished that Alex's voice speaks without his permission, a hoarse \"baby.\" Henry nods, so small that someone who didn't know all his tics might miss it, but Alex knows exactly what it means, so he leans down and sucks Henry's earlobe between his lips and calls him _baby_ again, and Henry says, \"Yes,\" and, \"Please,\" and tugs his hair at the root.\n\nAlex nips at Henry's throat and palms at his hips and sinks into the white-out bliss of being that impossibly close to him, of getting to share his body. Somehow it still amazes him that all this seems to be as unbelievably, singularly _good_ for Henry as it is for him. Henry's face should be illegal, the way it's turned up toward him, flushed and undone. Alex feels his own lips spreading into a pleased smile, awed and proud.\n\nAfterward, he comes back into his own body in increments\u2014his knees, still dug into the mattress and shaking; his stomach, slick and sticky; his hands, twisted up in Henry's hair, stroking it gently.\n\nHe feels like he's stepped outside of himself and returned to find everything slightly rearranged. When he pulls his face back to look at Henry, the feeling comes back into his chest: an ache in answer to the curve of Henry's top lip over white teeth.\n\n\"Jesus Christ,\" Alex says at last, and when he looks over at Henry again, he's squinting at him impishly out of one eye, smirking.\n\n\"Would you describe it as _supersonic_?\" he says, and Alex groans and slaps him across the chest, and they both dissolve into messy laughter.\n\nThey slide apart and make out and argue over who has to sleep in the wet spot until they pass out around four in the morning. Henry rolls Alex onto his side and burrows behind him until he's covering him completely, his shoulders a brace for Alex's shoulders, one of his thighs pressed on top of Alex's thighs, his arms over Alex's arms and his hands over Alex's hands, nowhere left untouched. It's the best Alex has slept in years.\n\nTheir alarms go off three hours later for their flights home.\n\nThey shower together. Henry's mood turns dark and sour over morning coffee at the harsh reality of returning to London so soon, and Alex kisses him dumbly and promises to call and wishes there was more he could do.\n\nHe watches Henry lather up and shave, put pomade in his hair, put on his Burberry for the day, and he catches himself wishing he could watch it every day. He likes taking Henry apart, but there's something incredibly intimate about sitting on the bed they wrecked the night before, the only one who watches him create Prince Henry of Wales for the day.\n\nThrough his throbbing hangover, he's got a suspicion all these feelings are why he held off on fucking Henry for so long.\n\nAlso, he might puke. It's probably unrelated.\n\nThey meet the others in the hallway, Henry passing for hungover but handsome, and Alex just doing his best. Bea is looking well-rested, fresh, and very smug about it. June, Nora, and Pez all emerge disheveled from their suite looking like the cats that caught the canaries, but it's impossible to tell who is a cat and who is a canary. Nora has a smudge of lipstick on the back of her neck. Alex doesn't ask.\n\nCash chuckles under his breath when he meets them at the elevators, a tray of six coffees balanced on one hand. Hangover tending isn't part of his job description, but he's a mother hen.\n\n\"So this is the gang now, huh?\"\n\nAnd through it all, Alex realizes with a start: He has friends now.\n\n# EIGHT\n\n> You are a dark sorcerer\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 6\/8\/20 3:23 PM\n> \n> to A\n> \n> Alex,\n> \n> I can't think of a single other way to start this email except to say, and I do hope you will forgive both my language and my utter lack of restraint: You are so fucking beautiful.\n> \n> I've been useless for a week, driven around for appearances and meetings, lucky if I've made a single meaningful contribution to any of them. How is a man to get anything done knowing Alex Claremont-Diaz is out there on the loose? I am driven to distraction.\n> \n> It's all bloody useless because when I'm not thinking about your face, I'm thinking about your arse or your hands or your smart mouth. I suspect the latter is what got me into this predicament in the first place. Nobody's ever got the nerve to be cheeky to a prince, except you. The moment you first called me a prick, my fate was sealed. O, fathers of my bloodline! O, ye kings of olde! Take this crown from me, bury me in my ancestral soil. If only you had known the mighty work of thine loins would be undone by a gay heir who likes it when American boys with chin dimples are mean to him.\n> \n> Actually, remember those gay kings I mentioned? I feel that James I, who fell madly in love with a very fit and exceptionally dim knight at a tilting match and immediately made him a gentleman of the bedchamber (a real title), would take mercy upon my particular plight.\n> \n> I'll be damned but I miss you.\n> \n> x\n> \n> Henry\n> \n> Re: You are a dark sorcerer\n\n* * *\n\n> A <agcd@eclare45.com> 6\/8\/20 5:02 PM\n> \n> to Henry\n> \n> H,\n> \n> Are you implying that you're James I and I'm some hot, dumb jock? I'm more than fantastic bone structure and an ass you can bounce a quarter on, Henry!!!!\n> \n> Don't apologize for calling me pretty. Because then you're putting me in a position where I have to apologize for saying you blew my fucking mind in LA and I'm gonna die if it doesn't happen again soon. How's that for lack of restraint, huh? You really wanna play that game with me?\n> \n> Listen: I'll fly to London right now and pull you out of whatever pointless meeting you're in and make you admit how much you love it when I call you \"baby.\" I'll take you apart with my teeth, sweetheart.\n> \n> xoxo\n> \n> A\n> \n> Re: You are a dark sorcerer\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 6\/8\/20 7:21 PM\n> \n> to A\n> \n> Alex,\n> \n> You know, when you go to Oxford to get a degree in English literature, as I have, people always want to know who your favorite English author is.\n> \n> The press team compiled a list of acceptable answers. They wanted a realist, so I suggested George Eliot\u2014no, Eliot was actually Mary Anne Evans under a pen name, not a strong male author. They wanted one of the inventors of the English novel, so I suggested Daniel Defoe\u2014no, he was a dissenter from the Church of England. At one point, I threw out Jonathan Swift just to watch the collective coronary they had at the thought of an Irish political satirist.\n> \n> In the end they picked Dickens, which is hilarious. They wanted something less fruity than the truth, but truly, what is gayer than a woman who languishes away in a crumbling mansion wearing her wedding gown every day of her life, for the drama?\n> \n> The fruity truth: My favorite English author is Jane Austen.\n> \n> So, to borrow a passage from Sense and Sensibility: \"You want nothing but patience\u2014or give it a more fascinating name, call it hope.\" To paraphrase: I hope to see you put your green American money where your filthy mouth is soon.\n> \n> Yours in sexual frustration,\n> \n> Henry\n\n* * *\n\nAlex feels like somebody has probably warned him about private email servers before, but he's a little fuzzy on the details. It doesn't feel important.\n\nAt first, like most things that require time when instant gratification is possible, he doesn't see the point of Henry's emails.\n\nBut when Richards tells Sean Hannity that his mother hasn't accomplished anything as president, Alex screams into his elbow and goes back to: The way you speak sometimes is like sugar spilling out of a bag with a hole in the bottom. When WASPy Hunter brings up the Harvard rowing team for the fifth time in one workday: Your arse in those trousers is a crime. When he's tired of being touched by strangers: Come back to me when you're done being flung through the firmament, you lost Pleiad.\n\nNow he gets it.\n\nHis dad wasn't wrong about how ugly things would get with Richards leading the ticket. Utah ugly, Christian ugly, ugliness couched in dog whistles and toothy white smiles. Right-wing think pieces about entitlement thrown in his and June's direction, reeking of: _Mexicans stole the First Family jobs too._\n\nHe can't allow the fear of losing in. He drinks coffee and brings his policy work on the campaign trail and drinks more coffee, reads emails from Henry, and drinks even more coffee.\n\nThe first DC Pride since his \"bisexual awakening\" happens while Alex is in Nevada, and he spends the day jealously checking Twitter\u2014confetti raining down on the Mall, grand marshal Rafael Luna with a rainbow bandana around his head. He goes back to his hotel and talks to his minibar about it.\n\nThe biggest bright spot in all the chaos is that his lobbying with one of the campaign chairs (and his own mother) has finally paid off: They're doing a massive rally at Minute Maid Park in Houston. Polls are shifting in directions they've never seen before. Politico's top story of the week: IS 2020 THE YEAR TEXAS BECOMES A TRUE BATTLEGROUND STATE?\n\n\"Yes, I will make sure everyone knows the Houston rally was your idea,\" his mother says, barely paying attention, as she goes over her speech on the plane to Texas.\n\n\"You should say 'grit,' not 'fortitude' there,\" June says, reading the speech over her shoulder. \"Texans like grit.\"\n\n\"Can y'all both go sit somewhere else?\" she says, but she adds a note.\n\nAlex knows a lot of the campaign is skeptical, even when they've seen the numbers. So when they pull up to Minute Maid and the line wraps around the block twice, he feels beyond gratified. He feels _smug._ His mom gets up to make her speech to thousands, and Alex thinks, _Hell yeah, Texas. Prove the bastards wrong._\n\nHe's still riding the high when he swipes his badge at the door of the campaign office the following Monday. He's been getting tired of sitting at a desk and going through focus groups again and again and again, but he's ready to pick the fight back up.\n\nThe fact that he rounds the corner into his cubicle to find WASPy Hunter holding the Texas Binder brings him right the fuck back down.\n\n\"Oh, you left this on your desk,\" WASPy Hunter says casually. \"I thought maybe it was a new project they were putting us on.\"\n\n\"Do I go on _your_ side of the cubicle and turn off your Dropkick Murphys Spotify station, no matter how much I want to?\" Alex demands. \"No, _Hunter,_ I don't.\"\n\n\"Well, you do kind of steal my pencils a lot\u2014\"\n\nAlex snatches the binder away before he can finish. \"It's private.\"\n\n\"What is it?\" WASPy Hunter asks as Alex shoves it back into his bag. He can't believe he left it out. \"All that data, and the district lines\u2014what are you doing with all that?\"\n\n\"Nothing.\"\n\n\"Is it about the Houston rally you pushed for?\"\n\n\"Houston was a good idea,\" he says, instantly defensive.\n\n\"Dude... you don't honestly think Texas can go blue, do you? It's one of the most backward states in the country.\"\n\n\"You're from _Boston,_ Hunter. You really want to talk about all the places bigotry comes from?\"\n\n\"Look, man, I'm just saying.\"\n\n\"You know what?\" Alex says. \"You think y'all are off the hook for institutional bigotry because you come from a blue state. Not every white supremacist is a meth-head in Bumfuck, Mississippi\u2014there are _plenty_ of them at Duke or UPenn on Daddy's money.\"\n\nWASPy Hunter looks startled but not convinced. \"None of that changes that red states have been red forever,\" he says, laughing, like it's something to joke about, \"and none of those populations seem to care enough about what's good for them to vote.\"\n\n\"Maybe _those populations_ might be more motivated to vote if we made an actual effort to campaign to them and showed them that we care, and how our platform is designed to help them, not leave them behind,\" Alex says hotly. \"Imagine if nobody who claims to have your interests at heart ever came to your state and tried to talk to you, man. Or if you were a felon, or\u2014fucking voter ID laws, people who can't access polls, who can't leave work to get to one?\"\n\n\"Yeah, I mean, it'd be great if we could magically mobilize every eligible marginalized voter in red states, but political campaigns have a finite amount of time and resources, and we have to prioritize based on projections,\" WASPy Hunter says, as if Alex, the First Son of the United States, is unfamiliar with how campaigns work. \"There just aren't the same number of bigots in blue states. If they don't want to be left behind, maybe people in red states should do something about it.\"\n\nAnd Alex has, quite frankly, had it.\n\n\"Did you forget that you're working on the campaign of someone Texas fucking created?\" he says, and his voice has officially risen to the point where staffers in the neighboring cubicles are staring, but he doesn't care. \"Why don't we talk about how there's a chapter of the Klan in every state? You think there aren't racists and homophobes growing up in Vermont? Man, I appreciate that you're doing the work here, but you're not special. You don't get to sit up here and pretend like it's someone else's problem. None of us do.\"\n\nHe takes his bag and his binder and storms out.\n\nThe minute he's outside the building, he pulls out his phone on impulse, opens up Google. There are test dates this month. He knows there are.\n\nLSAT washington dc area test center, he types.\n\n> 3 Geniuses and Alex\n\nJune 23, 2020, 12:34 PM\n\n> juniper\n> \n> BUG\n> \n> Not my name, not anyone's name, stop\n> \n> leading member of korean pop band bts kim nam-june\n> \n> BUG\n> \n> I'm blocking your number\n> \n> HRH Prince Dickhead  \n> \n> Alex, please don't tell me Pez has indoctrinated you with K-pop.\n> \n> well you let nora get you into drag race so\n> \n> irl chaos demon\n> \n> [latrice royale eat it.gif]\n> \n> BUG\n> \n> What did you want Alex????\n> \n> where's my speech for milwaukee? i know you took it\n> \n> HRH Prince Dickhead  \n> \n> Must you have this conversation in the group chat?\n> \n> BUG\n> \n> Part of it needed to be rewritten!!! I put it back with edits in the outside pocket of your messenger bag\n> \n> davis is gonna kill you if you keep doing this\n> \n> BUG\n> \n> Davis saw how well my tweaks to the talking points went over on Seth Meyers last week so he knows better\n> \n> why is there a rock in here too\n> \n> BUG\n> \n> That is a clear quartz crystal for clarity and good vibes do not @ me. We need all the help we can get right now\n> \n> stop putting SPELLS on my STUFF\n> \n> irl chaos demon\n> \n> BURN THE WITCH\n> \n> irl chaos demon\n> \n> hey what do we think of this #look for the college voter thing tomorrow\n> \n> irl chaos demon\n> \n> [Attached Image]\n> \n> irl chaos demon\n> \n> i'm going for, like, depressed lesbian poet who met a hot yoga instructor at a speakeasy who got her super into meditation and pottery, and now she's starting a new life as a high-powered businesswoman selling her own line of hand-thrown fruit bowls\n> \n>...\n> \n> HRH Prince Dickhead  \n> \n> Bitch, you took me there.\n> \n> alskdjfadslfjad\n> \n> NORA YOU BROKE HIM\n> \n> irl chaos demon\n> \n> lmaoooooo\n\n* * *\n\nThe invitation comes certified airmail straight from Buckingham Palace. Gilded edges, spindly calligraphy: THE CHAIRMAN AND COMMITTEE OF MANAGEMENT OF THE CHAMPIONSHIPS REQUEST THE PLEASURE OF THE COMPANY OF ALEXANDER CLAREMONT-DIAZ IN THE ROYAL BOX ON THE 6TH OF JULY, 2020.\n\nAlex takes a picture and texts it to Henry.\n\n1. tf is this? aren't there poor people in your country?\n\n2. i've already been in the royal box\n\nHenry sends back, You are a delinquent and a plague, and then, Please come?\n\nAnd here Alex is, spending his one day off from the campaign at Wimbledon, only to get his body next to Henry's again.\n\n\"So, as I've warned you,\" Henry says as they approach the doors to the Royal Box, \"Philip will be here. And assorted other nobility with whom you may have to make conversation. People named Basil.\"\n\n\"I think I've proven that I can handle royals.\"\n\nHenry looks doubtful. \"You're brave. I could use some of that.\"\n\nThe sun is, for once, bright over London when they step outside, flooding the stands around them, which have already mostly filled with spectators. He notices David Beckham in a well-tailored suit\u2014once again, how had he convinced himself he was straight?\u2014before David Beckham turns away and Alex sees it was Bea he was talking to, her face bright when she spots them.\n\n\"Oi, Alex! Henry!\" she chirps over the murmur of the Box. She's a vision in a lime-green, drop-waist silk dress, a pair of huge, round Gucci sunglasses embellished with gold honeybees perched on her nose.\n\n\"You look gorgeous,\" Alex says, accepting a kiss on his cheek.\n\n\"Why _thank_ you, darling,\" Bea says. She takes one of their arms in each of hers and whisks them off down the steps. \"Your sister helped me pick the dress, actually. It's McQueen. She's a genius, did you know?\"\n\n\"I've been made aware.\"\n\n\"Here we are,\" Bea says when they've reached the front row. \"These are ours.\"\n\nHenry looks at the lush green cushions of the seats topped with thick and shiny _WIMBLEDON 2020_ programs, right at the front edge of the box.\n\n\"Front and center?\" he says with a note of nervousness. \"Really?\"\n\n\"Yes, Henry, in case you have forgotten, you are a royal and this is the Royal Box.\" She waves down to the photographers below, who are already snapping photos of them, before leaning into them and whispering, \"Don't worry, I don't think they can detect the thick air of horn-town betwixt you two from the lawn.\"\n\n\"Ha-ha, Bea,\" Henry monotones, ears pink, and despite his apprehension, he takes his seat between Alex and Bea. He keeps his elbows carefully tucked into his sides and out of Alex's space.\n\nIt's halfway through the day when Philip and Martha arrive, Philip looking as generically handsome as ever. Alex wonders how such rich genetics conspired to make Bea and Henry both so interesting to look at, all mischievous smiles and swooping cheekbones, but punted so hard on Philip. He looks like a stock photo.\n\n\"Morning,\" Philip says as he takes his reserved seat to the side of Bea. His eyes track over Alex twice, and Alex can sense skepticism as to why Alex was even allowed. Maybe it's weird Alex is here. He doesn't care. Martha's looking at him weird too, but maybe she's simply holding a grudge about her wedding cake.\n\n\"Afternoon, Pip,\" Bea says politely. \"Martha.\"\n\nBeside him, Henry's spine stiffens.\n\n\"Henry,\" Philip says. Henry's hand is tense on the program in his lap. \"Good to see you, mate. Been a bit busy, have you? Gap year and all that?\"\n\nThere's an implication under his tone. _Where exactly have you been? What exactly have you been doing?_ A muscle flexes in Henry's jaw.\n\n\"Yes,\" Henry says. \"Loads of work with Percy. It's been mad.\"\n\n\"Right, the Okonjo Foundation, isn't it?\" he says. \"Shame he couldn't make it today. Suppose we'll have to make do with our American friend, then?\"\n\nAt that, he tips a dry smile at Alex.\n\n\"Yep,\" Alex says, too loud. He grins broadly.\n\n\"Though, I do suppose Percy would look a bit out of place in the Box, wouldn't he?\"\n\n_\"Philip,\"_ Bea says.\n\n\"Oh, don't be so dramatic, Bea,\" Philip says dismissively. \"I only mean he's a peculiar sort, isn't he? Those frocks he wears? A bit much for Wimbledon.\"\n\nHenry's face is calm and genial, but one of his knees has shifted over to dig into Alex's. \"They're called dashikis, Philip, and he wore one _once._ \"\n\n\"Right,\" Philip says. \"You know I don't judge. I just think, you know, remember when we were younger and you'd spend time with my mates from uni? Or Lady Agatha's son, the one that's always quail hunting? You could consider more mates of... similar standing.\"\n\nHenry's mouth is a thin line, but he says nothing.\n\n\"We can't all be best mates with the Count of Monpezat like you, Philip,\" Bea mutters.\n\n\"In any event,\" Philip presses on, ignoring her, \"you're unlikely to find a wife unless you're running in the right circles, aren't you?\" He chuckles a little and returns to watching the match.\n\n\"If you'll excuse me,\" Henry says. He drops his program in his seat and vanishes.\n\nTen minutes later, Alex finds him in the clubhouse by a gigantic vase of lurid fuschia flowers. His eyes are intent on Alex the moment he sees him, his lip chewed the same furious red as the embroidered Union Jack on his pocket square.\n\n\"Hello, Alex,\" he says placidly.\n\nAlex takes his tone. \"Hi.\"\n\n\"Has anyone shown you round the clubhouse yet?\"\n\n\"Nope.\"\n\n\"Well, then.\"\n\nHenry touches two fingers to the back of his elbow, and Alex obeys immediately.\n\nDown a flight of stairs, through a concealed side door and a second hidden corridor, there is a small room full of chairs and tablecloths and one old, abandoned tennis racquet. As soon as the door is closed behind them, Henry slams him up against it.\n\nHe gets right up in Alex's space, but he doesn't kiss him. He hovers there, a breath away, his hands at Alex's hips and his mouth split open in a crooked smirk.\n\n\"D'you know what I want?\" he says, his voice so low and hot that it burns right through Alex's solar plexus, right into the core of him.\n\n\"What?\"\n\n\"I want,\" he says, \"to do the absolute last thing I'm supposed to be doing right now.\"\n\nAlex juts out his chin, grinningly defiant. \"Then tell me to do it, sweetheart.\"\n\nAnd Henry, tonguing the corner of his own mouth, tugs hard to undo Alex's belt and says, \"Fuck me.\"\n\n\"Well,\" Alex grunts, \"when at Wimbledon.\"\n\nHenry laughs hoarsely and leans down to kiss him, open-mouthed and eager. He's moving fast, knowing they're on borrowed time, quick to follow the lead when Alex groans and pulls at his shoulders to change their positions. He gets Henry's back to his chest, Henry's palms braced against the door.\n\n\"Just so we're clear,\" Alex says, \"I'm about to have sex with you in this storage closet to spite your family. Like, that's what's happening?\"\n\nHenry, who has apparently been carrying his travel-size lube with him this entire time in his jacket, says, \"Right,\" and tosses it over his shoulder.\n\n\"Awesome, fuckin' love doing things out of spite,\" he says without a hint of sarcasm, and he kicks Henry's feet apart.\n\nAnd it should be\u2014it should be funny. It should be hot, stupid, ridiculous, obscene, another wild sexual adventure to add to the list. And it is, but... it shouldn't also feel like last time, like Alex might die if it ever stops. There's a laugh in his mouth, but it won't get past his tongue, because he knows this is him helping Henry get through something. Rebellion.\n\n_You're brave. I could use some of that._\n\nAfter, he kisses Henry's mouth fiercely, pushes his fingers deep into Henry's hair, sucks the air out of him. Henry smiles breathlessly against his neck, looking extremely pleased with himself, and says, \"I'm rather finished with tennis, aren't you?\"\n\nSo, they steal away behind a crowd, blocked by PPOs and umbrellas, and back at Kensington, Henry brings Alex up to his rooms.\n\nHis \"apartment\" is a sprawling warren of twenty-two rooms on the northwest side of the palace closest to the Orangery. He splits it with Bea, but there's not much of either of them in any of the high ceilings and heavy, jacquard furniture. What is there is more Bea than Henry: a leather jacket flung over the back of a chaise, Mr. Wobbles preening in a corner, a seventeenth-century Dutch oil painting on one landing literally called _Woman at her Toilet_ that only Bea would have selected from the royal collection.\n\nHenry's bedroom is as cavernous and opulent and insufferably beige as Alex could have imagined, with a gilded baroque bed and windows overlooking the gardens. He watches Henry shrug out of his suit and imagines having to live in it, wondering if Henry simply isn't allowed to choose what his rooms look like or if he never wanted to ask for something different. All those nights Henry can't sleep, just knocking around these endless, impersonal rooms, like a bird trapped in a museum.\n\nThe only room that really feels like both Henry and Bea is a small parlor on the second floor converted into a music studio. The colors are richest here: hand-woven Turkish rugs in deep reds and violets, a tobacco-colored settee. Little poufs and tables of knickknacks spring up like mushrooms, and the walls are lined with Stratocasters and Flying Vs, violins, an assortment of harps, one stout cello propped up in the corner.\n\nIn the center of the room is the grand piano, and Henry sits down at it and plucks away idly, toying with the melody of something that sounds like an old song by The Killers. David the beagle naps quietly near the pedals.\n\n\"Play something I don't know,\" Alex says.\n\nBack in high school in Texas, Alex was the most cultured of the jock crowd because he was a book nerd, a politics junkie, the only varsity letterman debating the finer points of Dred Scott in AP US History. He listens to Nina Simone and Otis Redding, likes expensive whiskey. But Henry's got an entirely different compendium of knowledge.\n\nSo he just listens and nods and smiles a little while Henry explains that _this_ is what Brahms sounds like, and _this_ is Wagner, and how they were on the two opposing sides of the Romantic movement. \"Do you hear the difference there?\" His hands are fast, almost effortless, even as he goes off into a tangent about the War of the Romantics and how Liszt's daughter left her husband for Wagner, _quel scandale._\n\nHe switches to an Alexander Scriabin sonata, winking over at Alex at the composer's first name. The andante\u2014the third movement\u2014is his favorite, he explains, because he read once that it was written to evoke the image of a castle in ruins, which he found darkly funny at the time. He goes quiet, focused, lost in the piece for long minutes. Then, without warning, it changes again, turbulent chords circling back into something familiar\u2014the Elton John songbook. Henry closes his eyes, playing from memory. It's \"Your Song.\" _Oh._\n\nAnd Alex's heart doesn't spread itself out in his chest, and he doesn't have to grip the edge of the settee to steady himself. Because that's what he would do if he were here in this palace to fall in love with Henry, and not just continuing this thing where they fly across the world to touch each other and don't talk about it. That's not why he's here. It's not.\n\nThey make out lazily for what could be hours on the settee\u2014Alex wants to do it on the piano, but it's a priceless antique or whatever\u2014and then they stagger up to Henry's room, the palatial bed. Henry lets Alex take him apart with painstaking patience and precision, moans the name of God so many times that the room feels consecrated.\n\nIt pushes Henry over some kind of edge, melted and overwhelmed on the lush bedclothes. Alex spends nearly an hour afterward coaxing little tremors out of him, in awe of his elaborate expressions of wonder and blissful agony, ghosting featherlight fingertips over his collarbone, his ankles, the insides of his knees, the small bones of the backs of his hands, the dip of his lower lip. He touches and touches until he brings Henry to another brink with only his fingertips, only his breath on the inside of his thighs, the promise of Alex's mouth where he'd pressed his fingers before.\n\nHenry says the same two words from the secret room at Wimbledon, this time dressed up in, \"Please, I need you to.\" He still can't believe Henry can talk like this, that he gets to be the only one who hears it.\n\nSo he does.\n\nWhen they come back down, Henry practically passes out on his chest without another word, fucked-out and boneless, and Alex laughs to himself and pets his sweaty hair and listens to the soft snores that come almost immediately.\n\nIt takes him hours to fall asleep, though.\n\nHenry drools on him. David finds his way onto the bed and curls up at their feet. Alex has to be back on a plane for DNC prep in a matter of hours, but he can't sleep. It's jet lag. It's just jet lag.\n\nHe remembers, as if from a million miles away, telling Henry once not to overthink this.\n\n* * *\n\n\"As your president,\" Jeffrey Richards is saying on one of the flat screens in the campaign office, \"one of my many priorities will be encouraging young people to get involved with their government. If we're going to hold our control of the Senate and take back the House, we need the next generation to stand up and join the fight.\"\n\nThe College Republicans of Vanderbilt University cheer on the live feed, and Alex pretends to barf onto his latest policy draft.\n\n\"Why don't you come up here, Brittany?\" A pretty blond student joins Richards at the podium, and he puts an arm around her. \"Brittany here was the main organizer we worked with for this event, and she couldn't have done a better job getting us this amazing turnout!\"\n\nMore cheers. A mid-level staffer lobs a ball of paper at the screen.\n\n\"It's young people like Brittany who give us hope for the future of our party. Which is why I'm pleased to announce that, as president, I'll be launching the Richards Youth Congress program. Other politicians don't want people\u2014especially discerning young people like you\u2014to get up close in our offices and see just how the sausage gets made\u2014\"\n\ni want to see a cage match between your grandmother and this fucking ghoul running against my mom, Alex texts Henry as he turns back to his cubicle.\n\nIt's the last days before the DNC, and he hasn't been able to catch the coffeepot before it's empty in a week. The policy inboxes are overflowing since they released the official platform two days ago, and WASPy Hunter has been firing off emails like his life depends on it. He hasn't said anything else to Alex about his rant from last month, but he has started wearing headphones to spare Alex his musical choices.\n\nHe types out another text, this one to Luna: can you please go on anderson cooper or something and explain that paragraph you ghostwrote on tax law for the platform so people will stop asking? ain't got the time, vato.\n\nHe's been texting Luna all week, ever since the Richards campaign leaked that they've tapped an Independent senator for his prospective cabinet. That old bastard Stanley Connor flat-out denied every last request for an endorsement\u2014by the end, Luna privately told Alex they were lucky Connor didn't try to primary them. Nothing's official, but everyone knows Connor is the one joining Richards's ticket. But if Luna knows when the announcement's coming, he's not sharing.\n\nIt's a _week._ The polls aren't great, Paul Ryan is getting sanctimonious about the Second Amendment, and there's some _Salon_ hot take going around, WOULD ELLEN CLAREMONT HAVE GOTTEN ELECTED IF SHE WEREN'T CONVENTIONALLY BEAUTIFUL? If it weren't for her morning meditation sessions, Alex is sure his mom would have throttled an aide by now.\n\nFor his part, he misses Henry's bed, Henry's body, Henry and a place a few thousand miles removed from the factory line of the campaign. That night after Wimbledon from a week ago feels like something out of a dream now, all the more tantalizing because Henry is in New York for a few days with Pez to do paperwork for an LGBT youth shelter in Brooklyn. There aren't enough hours in the day for Alex to find a pretense to get there, and no matter how much the world enjoys their public friendship, they're running out of plausible excuses to be seen together.\n\nThis time is nothing like their first breathless trip to the DNC in 2016. His dad had been the delegate to cast the votes from California that put her over, and they all cried. Alex and June introduced their mother before her acceptance speech, and June's hands were shaking but his were steady. The crowd roared, and Alex's heart roared back.\n\nThis year, they're all frizzy-haired and exhausted from trying to run the country and a campaign simultaneously, and even one day of the DNC is a stretch. On the second night of the convention, they pile onto Air Force One to New York\u2014it'd be Marine One, but they won't all fit in one helicopter.\n\n\"Have you run a cost-benefit analysis on this?\" Zahra is saying into her phone as they take off. \"Because you know I'm right, and these assets can be transferred at any time if you disagree. Yes. Yeah, I know. Okay. That's what I thought.\" A long pause, then, under her breath, \"Love you too.\"\n\n\"Um,\" Alex says when she's hung up. \"Something you'd like to share with the class?\"\n\nZahra doesn't even look up from her phone. \"Yes, that was my boyfriend, and no, you may not ask me any further questions about him.\"\n\nJune has shut her journal in sudden interest. \"How could you possibly have a boyfriend we don't know about?\"\n\n\"I see you more than I see clean underwear,\" Alex says.\n\n\"You're not changing your underwear often enough, sugar,\" his mother interjects from across the cabin.\n\n\"I go commando a lot,\" Alex says dismissively. \"Is this like a 'my Canadian girlfriend' thing? Does he\"\u2014he does very animated air quotes\u2014\"'go to a different school'?\"\n\n\"You really are determined to get shoved out of an emergency hatch one day, huh?\" she says. \"It's long distance. But not like that. No more questions.\"\n\nCash jumps in too, insisting he deserves to know as the resident love guru of the staff, and there's a debate about appropriate information to share with your coworkers, which is laughable considering how much Cash already knows about Alex's personal life. They're circling New York when June suddenly stops talking, focused again on Zahra, who has gone silent.\n\n\"Zahra?\"\n\nAlex turns and sees Zahra sitting perfectly still, such a departure from her usual constant motion that everyone else freezes too. She's staring at her phone, mouth open.\n\n\"Zahra,\" his mother echoes now, deadly serious. \"What?\"\n\nShe looks up finally, her grip on her phone tight. \"The _Post_ just broke the name of the Independent senator joining Richards's cabinet,\" she says. \"It's not Stanley Connor. It's Rafael Luna.\"\n\n* * *\n\n_\"No,\"_ June is saying. Her heels are dangling from her hand, her eyes bright in the warm light near the hotel elevator where they've agreed to meet. Her hair is coming out of its braid in angry spikes. \"You're damn lucky I agreed to talk to you in the first place, so you get this or you get nothing.\"\n\nThe _Post_ reporter blinks, fingers faltering on his recorder. He's been hounding June on her personal phone since the minute they landed in New York for a quote about the convention, and now he's demanding something about Luna. June is not typically an angry person, but it's been a long day, and she looks about three seconds from using one of those heels to stab the guy through the eye socket.\n\n\"What about you?\" the guy asks Alex.\n\n\"If she's not giving it to you, I'm not giving it to you,\" Alex says. \"She's much nicer than me.\"\n\nJune snaps her fingers in front of the guy's hipster glasses, eyes blazing. \"You don't get to speak to him,\" June says. \"Here is my quote: My mother, the president, still fully intends to win this race. We're here to support her and to encourage the party to stay united behind her.\"\n\n\"But about Senator Luna\u2014\"\n\n\"Thank you. Vote Claremont,\" June says tightly, slapping her hand over Alex's mouth. She sweeps him off and into the waiting elevator, elbowing him when he licks her palm.\n\n\"That goddamn fucking _traitor,_ \" Alex says when they reach their floor. \"Duplicitous fucking _bastard_! I\u2014I fucking helped him get elected. I canvassed for him for twenty-seven hours straight. I went to his sister's wedding. I memorized his goddamn _Five Guys order_!\"\n\n\"I fucking know, Alex,\" June says, shoving her keycard into the slot.\n\n\"How did that Vampire Weekend\u2013looking little shit even have your personal number?\"\n\nJune throws her shoes at the bed, and they bounce off onto the floor in different directions. \"Because I slept with him last year, Alex, how do you think? You're not the only one who makes stupid sexual decisions when you're stressed out.\" She drops onto the bed and starts taking off her earrings. \"I just don't understand what the point is. Like, what is Luna's endgame here? Is he some kind of fucking sleeper agent sent from the future to give me an ulcer?\"\n\nIt's late\u2014they got into New York after nine, hurtling into crisis management meetings for hours. Alex still feels wired, but when June looks up at him, he can see some of the brightness in her eyes has started to look like frustrated tears, and he softens a little.\n\n\"If I had to guess, Luna thinks we're going to lose,\" he tells her quietly, \"and he thinks he can help push Richards farther left by joining the ticket. Like, putting the fire out from inside the house.\"\n\nJune looks at him, eyes tired, searching his face. She may be the oldest, but politics is Alex's game, not hers. He knows he would have chosen this life for himself given the option; he knows she wouldn't have.\n\n\"I think... I need to sleep. For, like, the next year. At least. Wake me up after the general.\"\n\n\"Okay, Bug,\" Alex says. He leans down to kiss the top of her head. \"I can do that.\"\n\n\"Thanks, baby bro.\"\n\n\"Don't call me that.\"\n\n\"Tiny, miniature, itty-bitty, baby brother.\"\n\n\"Fuck off.\"\n\n\"Go to bed.\"\n\nCash is waiting for him out in the hallway, his suit abandoned for plainclothes.\n\n\"Hanging in there?\" he asks Alex.\n\n\"I mean, I kind of have to.\"\n\nCash pats him on the shoulder with one gigantic hand. \"There's a bar downstairs.\"\n\nAlex considers. \"Yeah, okay.\"\n\nThe Beekman is thankfully quiet this late, and the bar is low-lit with warm, rich shades of gold on the walls and deep-green leather on the high-backed barstools. Alex orders a whiskey neat.\n\nHe looks at his phone, swallowing down his frustration with the whiskey. He texted Luna three hours ago, a succinct: what the fuck? An hour ago, he got back: I don't expect you to understand.\n\nHe wants to call Henry. He guesses it makes sense\u2014they've always been fixed points in each other's worlds, little magnetic poles. Some laws of physics would be reassuring right now.\n\nGod, whiskey makes him maudlin. He orders another.\n\nHe's contemplating texting Henry, even though he's probably somewhere over the Atlantic, when a voice curls around his ear, smooth and warm. He's sure he must be imagining it.\n\n\"I'll have a gin and tonic, thanks,\" it says, and there's Henry in the flesh, sidled up next to him at the bar, looking a little tousled in a soft gray button-down and jeans. Alex wonders for an insane second if his brain has conjured up some kind of stress-induced sex mirage, when Henry says, voice lowered, \"You looked rather tragic drinking alone.\"\n\nDefinitely the real Henry, then. \"You're\u2014what are you doing here?\"\n\n\"You know, as a figurehead of one of the most powerful countries in the world, I do manage to keep abreast on international politics.\"\n\nAlex raises an eyebrow.\n\nHenry inclines his head, sheepish. \"I sent Pez home without me because I was worried.\"\n\n\"There it is,\" Alex says with a wink. He goes for his drink to hide what he suspects is a small, sad smile; the ice clacks against his teeth. \"Speak not the bastard's name.\"\n\n\"Cheers,\" Henry says as the bartender returns with his drink.\n\nHenry takes the first sip, sucking lime juice off his thumb, and fuck, he looks _good._ There's color in his cheeks and lips, the glow of Brooklyn summertime warmth that his English blood isn't accustomed to. He looks like something soft and downy Alex wants to sink into, and he realizes the knot of anxiety in his chest has finally slackened.\n\nIt's rare anyone other than June goes out of their way to check on him. It's by his own design, mostly, a barricade of charm and fitful monologues and hard-headed independence. Henry looks at him like he's not fooled by any of it.\n\n\"Get moving on that drink, Wales,\" Alex says. \"I've got a king-size bed upstairs that's calling my name.\" He shifts on his stool, letting one of his knees graze against Henry's under the bar, nudging them apart.\n\nHenry squints at him. \"Bossy.\"\n\nThey sit there until Henry finishes his drink, Alex listening to the placating murmur of Henry talking about different brands of gin, thankful that for once Henry seems happy to carry the conversation alone. He closes his eyes, wills the disaster of the day away, and tries to forget. He remembers Henry's words in the garden months ago: \"D'you ever wonder what it's like to be some anonymous person out in the world?\"\n\nIf he's some anonymous, normal person, removed from history, he's twenty-two and he's tipsy and he's pulling a guy into his hotel room by the belt loop. He's pulling a lip between his teeth, and he's fumbling behind his back to switch on a lamp, and he's thinking, _I like this person._\n\nThey break apart, and when Alex opens his eyes, Henry is watching him.\n\n\"Are you sure you don't want to talk about it?\"\n\nAlex groans.\n\nThe thing is, he _does,_ and Henry knows this too.\n\n\"It's...\" Alex starts. He paces backward, hands on his hips. \"He was supposed to be me in twenty years, you know? I was fifteen the first time I met him, and I was... in awe. He was everything I wanted to be. And he cared about people, and about doing the work because it was the right thing to do, because we were making people's lives better.\"\n\nIn the low light of the single lamp, Alex turns and sits down on the edge of the bed.\n\n\"I've never been more sure that I wanted to go into politics than when I went to Denver. I saw this young, queer guy who looked like me, sleeping at his desk because he wants kids at public schools in his state to have free lunches, and I was like, I could do this. I honestly don't know if I'm good enough or smart enough to ever be either of my parents. But I could be _that._ \" He drops his head down. He's never said the last part out loud to anyone before. \"And now I'm sitting here thinking, that son of a bitch sold out, so maybe it's all bullshit, and maybe I really am just a naive kid who believes in magical shit that doesn't happen in real life.\"\n\nHenry comes to stand in front of Alex, his thigh brushing against the inside of Alex's knee, and he reaches one hand down to still Alex's nervous fidgeting.\n\n\"Someone else's choice doesn't change who you are.\"\n\n\"I feel like it does,\" Alex tells him. \"I wanted to believe in some people being good and doing this job because they want to do good. Doing the right things most of the time and most things for the right reasons. I wanted to be the kind of person who believes in that.\"\n\nHenry's hands move, brushing up to Alex's shoulders, the dip of his throat, the underside of his jaw, and when Alex finally looks up, Henry's eyes are soft and steady. \"You still are. Because you still bloody care so much.\" He leans down and presses a kiss into Alex's hair. \"And you are good. Most things are awful most of the time, but you're good.\"\n\nAlex takes a breath. There's this way Henry has of listening to the erratic stream of consciousness that pours out of Alex's mouth and answering with the clearest, crystallized truth that Alex has been trying to arrive at all along. If Alex's head is a storm, Henry is the place lightning hits ground. He wants it to be true.\n\nHe lets Henry push him backward on the bed and kiss him until his mind is blissfully blank, lets Henry undress him carefully. He pushes into Henry and feels the tight cords of his shoulders start to release, like how Henry describes unfurling a sail.\n\nHenry kisses his mouth over and over again and says quietly, \"You are good.\"\n\n* * *\n\nThe pounding on his door comes much too early for Alex to handle loud noises. There's a sharpness to it he recognizes instantly as Zahra before she even speaks, and he wonders why the hell she didn't just call before he reaches for his phone and finds it dead. Shit. That would explain the missed alarm.\n\n\"Alex Claremont-Diaz, it is almost seven,\" Zahra shouts through the door. \"You have a strategy meeting in fifteen minutes and I have a key, so I don't care how naked you are, if you don't answer this door in the next thirty seconds, I'm coming in.\"\n\nHe is, he realizes as he rubs his eyes, extremely naked. A cursory examination of the body pressed up against his back: Henry, very comprehensively naked as well.\n\n\"Oh fuck me,\" Alex swears, sitting up so fast he gets tangled in the sheet and flails sideways out of bed.\n\n\"Blurgh,\" Henry groans.\n\n\"Fucking shit,\" says Alex, whose vocabulary is apparently now only expletives. He yanks himself free and scrambles for his chinos. \"Goddammit ass fucker.\"\n\n\"What,\" Henry says flatly to the ceiling.\n\n\"I can hear you in there, Alex, I swear to God\u2014\"\n\nThere's another sound from the door, like Zahra has kicked it, and Henry flies out of bed too. He is truly a picture, wearing an expression of bewildered panic and absolutely nothing else. He eyes the curtains furtively, as if considering hiding in them.\n\n\"Jesus tits,\" Alex continues as he fumbles to pull his pants up. He snatches a shirt and boxers at random from the floor, shoves them at Henry's chest, and points him toward the closet. \"Get in there.\"\n\n\"Quite,\" he observes.\n\n\"Yes, we can unpack the ironic symbolism later. _Go,_ \" Alex says, and Henry does, and when the door swings open, Zahra is standing there with her thermos and a look on her face that says she did not get a master's degree to babysit a fully grown adult who happens to be related to the president.\n\n\"Uh, morning,\" he says.\n\nZahra's eyes do a quick sweep of the room\u2014the sheets on the floor, the two pillows that have been slept on, the two phones on the nightstand.\n\n\"Who is she?\" she demands, marching over to the bathroom and yanking open the door like she's going to find some Hollywood starlet in the bathtub. \"You let her bring a _phone_ in here?\"\n\n\"Nobody, Jesus,\" Alex says, but his voice cracks in the middle. Zahra arches an eyebrow. \"What? I got kinda drunk last night, that's all. It's chill.\"\n\n\"Yes, it is so very, very chill that you're going to be hungover for today,\" Zahra says, rounding on him.\n\n\"I'm fine,\" he says. \"It's fine.\"\n\nAs if on cue, there's a series of bumps from the other side of the closet door, and Henry, halfway into Alex's boxers, comes literally tumbling out of the closet.\n\nIt is, Alex thinks half-hysterically, a very solid visual pun.\n\n\"Er,\" Henry says from the floor. He finishes pulling Alex's boxers up his hips. Blinks. \"Hello.\"\n\nThe silence stretches.\n\n\"I\u2014\" Zahra begins. \"Do I even want you to explain to me what the fuck is happening here? Literally how is he even _here,_ like, physically or geographically, and _why_ \u2014no, nope. Don't answer that. Don't tell me anything.\" She unscrews the top of her thermos and takes a pull of coffee. \"Oh my God, did _I_ do this? I never thought... when I set it up... oh my _God._ \"\n\nHenry has pulled himself off the floor and put on a shirt, and his ears are bright red. \"I think, perhaps, if it helps. It was. Er. Rather inevitable. At least for me. So you shouldn't blame yourself.\"\n\nAlex looks at him, trying to think of something to add, when Zahra jabs a manicured finger into his shoulder.\n\n\"Well, I hope it was _fun,_ because if anyone ever finds out about this, we're all fucked,\" Zahra says. She points at Henry. \"You too. Can I assume I don't have to make you sign an NDA?\"\n\n\"I've already signed one for him,\" Alex offers up, while Henry's ears turn from red to an alarming shade of purple. Six hours ago, he was sinking drowsily into Henry's chest, and now he's standing here half-naked, talking about the paperwork. He fucking hates paperwork. \"I think that covers it.\"\n\n\"Oh, wonderful,\" Zahra says. \"I'm so glad you thought this through. Great. How long has this been happening?\"\n\n\"Since, um. New Year's,\" Alex says.\n\n_\"New Year's?\"_ Zahra repeats, eyes wide. \"This has been going on for _seven months_? That's why you\u2014Oh my God, I thought you were getting into international relations or something.\"\n\n\"I mean, technically\u2014\"\n\n\"If you finish that sentence, I'm gonna spend tonight in jail.\"\n\nAlex winces. \"Please don't tell Mom.\"\n\n\" _Seriously?_ \" she hisses. \"You're literally putting your dick in _the leader of a foreign state,_ who is a _man,_ at _the biggest political event before the election,_ in a hotel full of _reporters,_ in a city full of _cameras,_ in a race close enough to fucking _hinge_ on some bullshit like this, like a manifestation of my fucking _stress dreams,_ and you're asking me _not_ to tell the president about it?\"\n\n\"Um. Yeah? I haven't, um, come out to her. Yet.\"\n\nZahra blinks, presses her lips together, and makes a noise like she's being strangled. \"Listen,\" she says. \"We don't have time to deal with this, and your mother has enough to manage without having to process her son's fucking quarter-life NATO sexual crisis, so\u2014I won't tell her. But once the convention is over, you have to.\"\n\n\"Okay,\" Alex says on an exhale.\n\n\"Would it make any difference at all if I told you not to see him again?\"\n\nAlex looks over at Henry, looking rumpled and nauseated and terrified at the corner of the bed. \"No.\"\n\n\"God fucking dammit,\" she says, rubbing the heel of her hand against her forehead. \"Every time I see you, it takes another year off my life. I'm going downstairs, and you better be dressed and there in five minutes so we can try to save this goddamn campaign. And _you_ \"\u2014she rounds on Henry\u2014\"you need to get back to fucking England now, and if anyone sees you leave, I will personally end you. Ask me if I'm afraid of the crown.\"\n\n\"Duly noted,\" he says in a faint voice.\n\nZahra fixes him with a final glare, turns on her heel, and stalks out of the room, slamming the door behind her.\n\n# NINE\n\n\"Okay,\" he says.\n\nHis mother sits across the table, hands folded, looking at him expectantly. His palms are starting to sweat. The room is small, one of the lesser conference rooms in the West Wing. He knows he could have asked her to lunch or something, but, well, he kind of panicked.\n\nHe guesses he should just do it.\n\n\"I've been, um,\" he starts. \"I've been figuring some stuff out about myself, lately. And... I wanted to let you know, because you're my mom, and I want you to be a part of my life, and I don't want to hide things from you. And also it's, um, relevant to the campaign, from an image perspective.\"\n\n\"Okay,\" Ellen says, her voice neutral.\n\n\"Okay,\" he repeats. \"All right. Um. So, I've realized I'm not straight. I'm actually bisexual.\"\n\nHer expression clears, and she laughs, unclasping her hands. \"Oh, that's it, sugar? God, I was worried it was gonna be something worse!\" She reaches across the table, covering his hand with hers. \"That's great, baby. I'm so glad you told me.\"\n\nAlex smiles back, the anxious bubble in his chest shrinking slightly, but there's one more bomb to drop. \"Um. There's something else. I kind of... met somebody.\"\n\nShe tilts her head. \"You did? Well, I'm happy for you, I hope you had them do all the paperwork\u2014\"\n\n\"It's, uh,\" he interrupts her. \"It's Henry.\"\n\nA beat. She frowns, her brow knitting together. \"Henry...?\"\n\n\"Yeah, Henry.\"\n\n\"Henry, as in... the prince?\"\n\n\"Yes.\"\n\n\"Of England?\"\n\n\"Yes.\"\n\n\"So, not another Henry?\"\n\n\"No, Mom. Prince Henry. Of Wales.\"\n\n\"I thought you hated him?\" she says. \"Or... now you're friends with him?\"\n\n\"Both true at different points. But uh, now we're, like, a thing. Have been. A thing. For, like, seven-ish months? I guess?\"\n\n\"I... see.\"\n\nShe stares at him for a very long minute. He shifts uncomfortably in his chair.\n\nSuddenly, her phone is in her hand, and she's standing, kicking her chair under the table. \"Okay, I'm clearing my schedule for the afternoon,\" she says. \"I need, uh, time to prepare some materials. Are you free in an hour? We can reconvene here. I'll order food. Bring, uh, your passport and any receipts and relevant documents you have, sugar.\"\n\nShe doesn't wait to hear if he's free, just walks backward out of the room and disappears into the corridor. The door isn't even finished closing when a notification pops up on his phone. CALENDAR REQUEST FROM MOM: 2 P.M. WEST WING FIRST FLOOR, INTERNATIONAL ETHICS & SEXUAL IDENTITY DEBRIEF.\n\nAn hour later, there are several cartons of Chinese food and a PowerPoint cued up. The first slide says: SEXUAL EXPERIMENTATION WITH FOREIGN MONARCHS: A GRAY AREA. Alex wonders if it's too late to swan dive off the roof.\n\n\"Okay,\" she says when he sits down, in almost exactly the same tone he used on her earlier. \"Before we start, I\u2014I want to be clear, I love you and support you always. But this is, quite frankly, a logistical and ethical clusterfuck, so we need to make sure we have our ducks in a row. Okay?\"\n\nThe next slide is titled: EXPLORING YOUR SEXUALITY: HEALTHY, BUT DOES IT HAVE TO BE WITH THE PRINCE OF ENGLAND? She apologizes for not having time to come up with better titles. Alex actively wishes for the sweet release of death.\n\nThe one after is: FEDERAL FUNDING, TRAVEL EXPENSES, BOOTY CALLS, AND YOU.\n\nShe's mostly concerned with making sure he hasn't used any federally funded private jets to see Henry for exclusively personal visits\u2014he hasn't\u2014and with making him fill out a bunch of paperwork to cover both their asses. It feels clinical and wrong, checking little boxes about his relationship, especially when half are asking things he hasn't even discussed with Henry yet.\n\nIt's agonizing, but eventually it's over, and he doesn't die, which is something. His mother takes the last form and seals it up in an envelope with the rest. She sets it aside and takes off her reading glasses, setting those aside too.\n\n\"So,\" she says. \"Here's the thing. I know I put a lot on you. But I do it because I trust you. You're a dumbass, but I trust you, and I trust your judgment. I promised you years ago I would never tell you to be anything you're not. So I'm not gonna be the president or the mother who forbids you from seeing him.\"\n\nShe takes another breath, waiting for Alex to nod that he understands.\n\n\"But,\" she goes on, \"this is a really, really big fucking deal. This is not just some person from class or some intern. You need to think really long and hard because you are putting yourself and your career and, above all, this campaign and this entire administration, in danger here. I know you're young, but this is a forever decision. Even if you don't stay with him forever, if people find out, that sticks with you forever. So you need to figure out if you feel forever about him. And if you don't, you need to cut it the fuck out.\"\n\nShe rests her hands on the table in front of her, and the silence hangs in the air between them. Alex feels like his heart is caught somewhere between his tonsils.\n\n_Forever._ It seems like an impossibly huge word, something he's supposed to grow into ten years from now.\n\n\"Also,\" she says. \"I am so sorry to do this, sugar. But you're off the campaign.\"\n\nAlex snaps back into razor sharp reality, stomach plummeting.\n\n\"Wait, no\u2014\"\n\n\"This is not up for debate, Alex,\" she tells him, and she does look sorry, but he knows the set of her jaw too well. \"I can't risk this. You're way too close to the sun. We're telling the press you're focusing on other career options. I'll have your desk cleaned out for you over the weekend.\"\n\nShe holds out one hand, and Alex looks down into her palm, the worried lines there, until the realization clicks.\n\nHe reaches into his pocket, pulls out his campaign badge. The first artifact of his entire career, a career he's managed to derail in a matter of months. And he hands it over.\n\n\"Oh, one last thing,\" she says, her tone suddenly businesslike again, shuffling something from the bottom of her files. \"I know Texas public schools don't have sex ed for shit, and we didn't go over this when we had the talk\u2014which is on me for assuming\u2014so I just wanted to make sure you know you still need to be using condoms even if you're having anal interc\u2014\"\n\n_\"Okay, thanks, Mom!\"_ Alex half yells, nearly knocking over his chair in his rush for the door.\n\n\"Wait, honey,\" she calls after him, \"I had Planned Parenthood send over all these pamphlets, take one! They sent a bike messenger and everything!\"\n\n> A mass of fools and knaves\n\n* * *\n\n> A <agcd@eclare45.com> 8\/10\/20 1:04 AM\n> \n> to Henry\n> \n> H,\n> \n> Have you ever read any of Alexander Hamilton's letters to John Laurens?\n> \n> What am I saying? Of course you haven't. You'd probably be disinherited for revolutionary sympathies.\n> \n> Well, since I got the boot from the campaign, there is literally nothing for me to do but watch cable news (diligently chipping away at my brain cells by the day), reread Harry Potter, and sort through all my old shit from college. Just looking at papers, thinking: Excellent, yes, I'm so glad I stayed up all night writing this for a 98 in the class, only to get summarily fired from the first job I ever had and exiled to my bedroom! Great job, Alex!\n> \n> Is this how you feel in the palace all the time? It fucking sucks, man.\n> \n> So anyway, I'm going through my college stuff, and I find this analysis I did of Hamilton's wartime correspondence, and hear me out: I think Hamilton could have been bi. His letters to Laurens are almost as romantic as his letters to his wife. Half of them are signed \"Yours\" or \"Affectionately yrs,\" and the last one before Laurens died is signed \"Yrs for ever.\" I can't figure out why nobody talks about the possibility of a Founding Father being not straight (outside of Chernow's biography, which is great btw, see attached bibliography). I mean, I know why, but.\n> \n> Anyway, I found this part of a letter he wrote to Laurens, and it made me think of you. And me, I guess:\n> \n> The truth is I am an unlucky honest man, that speak my sentiments to all and with emphasis. I say this to you because you know it and will not charge me with vanity. I hate Congress\u2014I hate the army\u2014I hate the world\u2014I hate myself. The whole is a mass of fools and knaves; I could almost except you...\n> \n> Thinking about history makes me wonder how I'll fit into it one day, I guess. And you too. I kinda wish people still wrote like that.\n> \n> History, huh? Bet we could make some.\n> \n> Affectionately yrs, slowly going insane,\n> \n> Alex, First Son of Founding Father Sacrilege\n> \n> Re: A mass of fools and knaves\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 8\/10\/20 4:18 AM\n> \n> to A\n> \n> Alex, First Son of Masturbatory Historical Readings:\n> \n> The phrase \"see attached bibliography\" is the single sexiest thing you have ever written to me.\n> \n> Every time you mention your slow decay inside the White House, I can't help but feel it's my fault, and I feel absolutely shit about it. I'm sorry. I should have known better than to turn up at a thing like that. I got carried away; I didn't think. I know how much that job meant to you.\n> \n> I just want to... you know. Extend the option. If you wanted less of me, and more of that\u2014the work, the uncomplicated things\u2014I would understand. Truly.\n> \n> In any event... Believe it or not, I have actually done a bit of reading on Hamilton, for a number of reasons. First, he was a brilliant writer. Second, I knew you were named after him (the pair of you share an alarming number of traits, by the by: passionate determination, never knowing when to shut up, &c &c). And third, some saucy tart once tried to impugn my virtue against an oil painting of him, and in the halls of memory, some things demand context.\n> \n> Are you angling for a revolutionary soldier role-play scenario? I must inform you, any trace of King George III blood I have would curdle in my very veins and render me useless to you.\n> \n> Or are you suggesting you'd rather exchange passionate letters by candlelight?\n> \n> Should I tell you that when we're apart, your body comes back to me in dreams? That when I sleep, I see you, the dip of your waist, the freckle above your hip, and when I wake up in the morning, it feels like I've just been with you, the phantom touch of your hand on the back of my neck fresh and not imagined? That I can feel your skin against mine, and it makes every bone in my body ache? That, for a few moments, I can hold my breath and be back there with you, in a dream, in a thousand rooms, nowhere at all?\n> \n> I think perhaps Hamilton said it better in a letter to Eliza:\n> \n> You engross my thoughts too intirely to allow me to think of any thing else\u2014you not only employ my mind all day; but you intrude upon my sleep. I meet you in every dream\u2014and when I wake I cannot close my eyes again for ruminating on your sweetness.\n> \n> If you did decide to take the option mentioned at the start of this email, I do hope you haven't read the rest of this rubbish.\n> \n> Regards,\n> \n> Haplessly Romantic Heretic Prince Henry the Utterly Daft\n> \n> Re: A mass of fools and knaves\n\n* * *\n\n> A <agcd@eclare45.com> 8\/10\/20 5:36 AM\n> \n> to Henry\n> \n> H,\n> \n> Please don't be stupid. No part of any of this will ever be uncomplicated.\n> \n> Anyway, you should be a writer. You are a writer.\n> \n> Even after all this, I still always feel like I want to know more of you. Does that sound crazy? I just sit here and wonder, who is this person who knows stuff about Hamilton and writes like this? Where does someone like that even come from? How was I so wrong?\n> \n> It's weird because I always know things about people, gut feelings that usually lead me in more or less the right direction. I do think I got a gut feeling with you, I just didn't have what I needed in my head to understand it. But I kind of kept chasing it anyway, like I was just going blindly in a certain direction and hoping for the best. I guess that makes you the North Star?\n> \n> I wanna see you again and soon. I keep reading that one paragraph over and over again. You know which one. I want you back here with me. I want your body and I want the rest of you too. And I want to get the fuck out of this house. Watching June and Nora on TV doing appearances without me is torture.\n> \n> We have this annual thing at my dad's lake house in Texas. Whole long weekend off the grid. There's a lake with a pier, and my dad always cooks something fucking amazing. You wanna come? I kind of can't stop thinking about you all sunburned and pretty sitting out there in the country. It's the weekend after next. If Shaan can talk to Zahra or somebody about flying you into Austin, we can pick you up from there. Say yes?\n> \n> Yrs,\n> \n> Alex\n> \n> P.S. Allen Ginsberg to Peter Orlovsky\u20141958:\n> \n> Tho I long for the actual sunlight contact between us I miss you like a home. Shine back honey & think of me.\n> \n> Re: A mass of fools and knaves\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 8\/10\/20 8:22 PM\n> \n> to A\n> \n> Alex,\n> \n> If I'm north, I shudder to think where in God's name we're going.\n> \n> I'm ruminating on identity and your question about where a person like me comes from, and as best as I can explain it, here's a story:\n> \n> Once, there was a young prince who was born in a castle. His mother was a princess scholar, and his father was the most handsome, feared knight in all the land. As a boy, people would bring him everything he could ever dream of wanting. The most beautiful silk clothes, ripe fruit from the orangery. At times, he was so happy, he felt he would never grow tired of being a prince.\n> \n> He came from a long, long line of princes, but never before had there been a prince quite like him: born with his heart on the outside of his body.\n> \n> When he was small, his family would smile and laugh and say he would grow out of it one day. But as he grew, it stayed where it was, red and visible and alive. He didn't mind it very much, but every day, the family's fear grew that the people of the kingdom would soon notice and turn their backs on the prince.\n> \n> His grandmother, the queen, lived in a high tower, where she spoke only of the other princes, past and present, who were born whole.\n> \n> Then, the prince's father, the knight, was struck down in battle. The lance tore open his armor and his body and left him bleeding in the dust. And so, when the queen sent new clothes, armor for the prince to parcel his heart away safe, the prince's mother did not stop her. For she was afraid, now: afraid of her son's heart torn open too.\n> \n> So the prince wore it, and for many years, he believed it was right.\n> \n> Until he met the most devastatingly gorgeous peasant boy from a nearby village who said absolutely ghastly things to him that made him feel alive for the first time in years and who turned out to be the most mad sort of sorcerer, one who could conjure up things like gold and vodka shots and apricot tarts out of absolutely nothing, and the prince's whole life went up in a puff of dazzling purple smoke, and the kingdom said, \"I can't believe we're all so surprised.\"\n> \n> I'm in for the lake house. I must admit, I'm glad you're getting out of the house. I worry you may burn the thing down. Does this mean I'll be meeting your father?\n> \n> I miss you.\n> \n> x\n> \n> Henry\n> \n> P.S. This is mortifying and maudlin and, honestly, I hope you forget it as soon as you've read it.\n> \n> P.P.S. From Henry James to Hendrik C. Andersen, 1899:\n> \n> May the terrific U.S.A. be meanwhile not a brute to you. I feel in you a confidence, dear Boy\u2013which to show is a joy to me. My hopes and desires and sympathies right heartily and most firmly, go with you. So keep up your heart, and tell me, as it shapes itself, your (inevitably, I imagine, more or less weird) American story. May, at any rate, tutta quella gente be good to you.\n\n* * *\n\n\"Do _not,_ \" Nora says, leaning over the passenger seat. \"There is a system and you must respect the system.\"\n\n\"I don't believe in systems when I'm on vacation,\" June says, her body folded halfway over Alex's, trying to slap Nora's hand out of the way.\n\n\"It's math,\" Nora says.\n\n\"Math has no authority here,\" June tells her.\n\n\"Math is _everywhere,_ June.\"\n\n\"Get off me,\" Alex says, shoving June off his shoulder.\n\n\"You're supposed to back me up on this!\" June yelps, pulling his hair and receiving a very ugly face in response.\n\n\"I'll let you look at one boob,\" Nora tells him. \"The good one.\"\n\n\"They're both good,\" June says, suddenly distracted.\n\n\"I've seen both of them. I can practically see both of them now,\" Alex says, gesturing at what Nora is wearing for the day, which is a ratty pair of short overalls and the most perfunctory of bra-like things.\n\n\"Hashtag vacation nips,\" she says. \"Pleeeeeease.\"\n\nAlex sighs. \"Sorry, Bug, but Nora did put more hours into her playlist, so she should get the aux cord.\"\n\nThere's a combination of girl sounds from the back seat, disgust and triumph, and Nora plugs her phone in, swearing she's developed some kind of foolproof algorithm for the perfect road trip playlist. The first trumpets of \"Loco in Acapulco\" by the Four Tops blast, and Alex finally pulls out of the gas station.\n\nThe jeep is a refurb, a project his dad took on when Alex was around ten. It lives in California now, but he drives it into Texas once a year for this weekend, leaves it in Austin so Alex and June can drive it in. Alex learned to drive one summer in the valley in this jeep, and the accelerator feels just as good under his foot now as he falls into formation with two black Secret Service SUVs and heads for the interstate. He hardly ever gets to drive himself anywhere anymore.\n\nThe sky is wide open and bluebonnet blue for miles, the sun low and heavy with an early morning start, and Alex has his sunglasses on and his arms bare and the doors and roof off. He cranks up the stereo and feels like he could throw anything away on the wind whipping through his hair and it would just float away like it never was, as if nothing matters but the rush and skip in his chest.\n\nBut it's all right behind the haze of dopamine: losing the campaign job, the restless days pacing his room, _Do you feel forever about him?_\n\nHe tips his chin up to the warm, sticky hometown air, catches his own eye in the rearview mirror. He looks bronzed and soft-mouthed and young, a Texas boy, the same kid he was when he left for DC. So, no more big thoughts for today.\n\nOutside the hangar are a handful of PPOs and Henry in a short-sleeved chambray, shorts, and a pair of fashionable sunglasses, Burberry weekender over one shoulder\u2014a goddamn summer dream. Nora's playlist has segued into \"Here You Come Again\" by Dolly Parton by the time Alex swings out of the side of the jeep by one arm.\n\n\"Yes, hello, hello, it's good to see you too!\" Henry is saying from somewhere inside a smothering hug from June and Nora. Alex bites his lip and watches Henry squeeze their waists in return, and then Alex has him, inhaling the clean smell of him, laughing into the crook of his neck.\n\n\"Hi, love,\" he hears Henry say quietly, privately, right into the hair above his ear, and Alex's breath forgets how to do anything but laugh helplessly.\n\n_\"Drums, please!\"_ erupts from the jeep's stereo and the beat on \"Summertime\" kicks in, and Alex whoops his approval. Once Henry's security team has fallen in with the Secret Service cars, they're off.\n\nHenry is grinning wide beside him as they cruise down 45, happily bopping his head along to the music, and Alex can't help glancing over at him, feeling giddy that Henry\u2014Henry the prince\u2014is _here,_ in Texas, coming home with him. June pulls four bottles of Mexican Coke out of the cooler under her seat and passes them around, and Henry takes the first sip and practically melts. Alex reaches over and takes Henry's free hand into his own, lacing their fingers together on the console between them.\n\nIt takes an hour and a half to get out to Lake LBJ from Austin, and when they start weaving their way toward the water, Henry asks, \"Why is it called Lake LBJ?\"\n\n\"Nora?\" Alex says.\n\n\"Lake LBJ,\" Nora says, \"or Lake Lyndon B. Johnson, is one of six reservoirs formed by dams on the Colorado River known as the Texas Highland Lakes. Made possible by LBJ enacting the Rural Electrification Act when he was president. And LBJ had a place out here.\"\n\n\"That's true,\" Alex says.\n\n\"Also, fun fact: LBJ was obsessed with his own dick,\" Nora adds. \"He called it Jumbo and would whip it out all the time. Like, in front of colleagues, reporters, anybody.\"\n\n\"Also true.\"\n\n\"American politics,\" Henry says. \"Truly fascinating.\"\n\n\"You wanna talk, Henry VIII?\" Alex says.\n\n_\"Anyway,\"_ Henry says airily, \"how long have you lot come out here?\"\n\n\"Dad bought it when he and Mom split up, so when I was twelve,\" Alex tells him. \"He wanted to have a place close to us after he moved. We used to spend so much time here in the summers.\"\n\n\"Aw, Alex, remember when you got drunk for the first time out here?\" June says.\n\n\"Strawberry daiquiris all _day._ \"\n\n\"You threw up _so much,_ \" she says fondly.\n\nThey pull into a driveway flanked by thick trees and drive up to the house at the top of the hill, the same old vibrant orange exterior and smooth arches, tall cactuses and aloe plants. His mom was never into the whole hacienda school of home decor, so his dad went all in when he bought the lake house, tall teal doors and heavy wooden beams and Spanish tile accents in pinks and reds. There's a big wrap-around porch and stairs leading down the hill to the dock, and all the windows facing the water have been flung open, the curtains drifting out on a warm breeze.\n\nTheir teams fall back to check the perimeter\u2014they're renting out the place next door for added privacy and the obligatory security presence. Henry effortlessly lifts June's cooler up onto one shoulder and Alex pointedly does not swoon about it.\n\nThere's the loud yell of Oscar Diaz coming around the corner, dripping and apparently fresh from a swim. He's wearing his old brown huaraches and a pair of swim trunks with parrots on them, both arms extended to the sun, and June is summarily scooped up into them.\n\n\"CJ!\" he says as he spins her around and deposits her on the stucco railing. Nora is next, and then a bone-crushing hug for Alex.\n\nHenry steps forward, and Oscar looks him up and down\u2014the Burberry bag, the cooler on his shoulder, the elegant smile, the extended hand. His dad had been confused but ultimately willing to roll with it when Alex asked if he could bring a friend and casually mentioned the friend would be the Prince of Wales. He's not sure how this will go.\n\n\"Hello,\" Henry says. \"Good to meet you. I'm Henry.\"\n\nOscar slaps his hand into Henry's. \"Hope you're ready to fucking party.\"\n\n* * *\n\nOscar may be the cook of the family, but Alex's mom was the one who grilled. It didn't always track in Pemberton Heights\u2014his Mexican dad in the house diligently soaking a tres leches while his blond mom stood out in the yard flipping burgers\u2014but it worked. Alex determinedly picked up the best from both of them, and now he's the only one here who can handle racks of ribs while Oscar does the rest.\n\nThe kitchen of the lake house faces the water, always smelling like citrus and salt and herbs, and his dad keeps it stocked with plump tomatoes and clay-soft avocados when they're visiting. He's standing in front of the big open windows now, three racks of ribs spread out on pans on the counter in front of him. His dad is at the sink, shucking ears of corn and humming along to an old Chente record.\n\nBrown sugar. Smoked paprika. Onion powder. Chili powder. Garlic powder. Cayenne pepper. Salt. Pepper. More brown sugar. Alex measures each one out with his hands and dumps them into the bowl.\n\nDown by the dock, June and Nora are embroiled in what looks like an improvised jousting match, charging at each other on the backs of inflatable animals with pool noodles. Henry is tipsy and shirtless and attempting to referee, standing on the dock with one foot on a piling and waving a bottle of Shiner around like a madman.\n\nAlex smiles a little to himself, watching them. Henry and his girls.\n\n\"So, you wanna talk about it?\" says his father's voice, in Spanish, from somewhere to his left.\n\nAlex jumps a little, startled. His dad has relocated to the bar a few feet down from him, mixing up a big batch of cotija and crema and seasonings for elotes.\n\n\"Uh.\" Has he been that obvious already?\n\n\"About Raf.\"\n\nAlex exhales, his shoulders dropping, and returns his attention to the dry rub.\n\n\"Ah. That motherfucker,\" he says. They've only broached the topic in passing obscenities over text since the news broke. There's a mutual sting of betrayal. \"Do you have any idea what he's thinking?\"\n\n\"I don't have anything kinder to say about him than you do. And I don't have an explanation either. But...\" He pauses thoughtfully, still stirring. Alex can sense him weighing out several thoughts at once, as he often does. \"I don't know. After all this time, I want to believe there's a reason for him to put himself in the same room as Jeffrey Richards. But I can't figure out what.\"\n\nAlex thinks about the conversation he overheard in the housekeeper's office, wondering if his dad is ever going to let him in on the full picture. He doesn't know how to ask without revealing that he literally climbed into a bush to eavesdrop on them. His dad's relationship with Luna has always been like that\u2014grown-up talk.\n\nAlex was at the fund-raiser for Oscar's Senate run where they first met Luna, Alex only fifteen and already taking notes. Luna showed up with a pride flag unapologetically stuck in his lapel; Alex wrote that down.\n\n\"Why'd you pick him?\" Alex asks. \"I remember that campaign. We met a lot of people who would've made great politicians. Why wouldn't you pick someone easier to elect?\"\n\n\"You mean, why'd I roll the dice on the gay one?\"\n\nAlex concentrates on keeping his face neutral.\n\n\"I wasn't gonna put it like that,\" he says, \"but yeah.\"\n\n\"Raf ever tell you his parents kicked him out when he was sixteen?\"\n\nAlex winces. \"I knew he had a hard time before college, but he didn't specify.\"\n\n\"Yeah, they didn't take the news so well. He had a rough couple of years, but it made him tough. The night we met him, it was the first time he'd been back in California since he got kicked out, but he was damn sure gonna come in to support a brother out of Mexico City. It was like when Zahra showed up at your mom's office in Austin and said she wanted to prove the bastards wrong. You know a fighter when you see one.\"\n\n\"Yeah,\" Alex says.\n\nThere's another pause of Chente crooning in the background while his dad stirs, before he speaks again.\n\n\"You know...\" he says. \"That summer, I sent you to work on his campaign because you're the best point man I got. I knew you could do it. But I really thought there was a lot you could learn from him too. You got a lot in common.\"\n\nAlex says nothing for a long moment.\n\n\"I gotta be honest,\" his dad says, and when Alex looks up again, he's watching the window. \"I thought a prince would be more of a candy-ass.\"\n\nAlex laughs, glancing back out at Henry, the sway of his back under the afternoon sun. \"He's tougher than he looks.\"\n\n\"Not bad for a European,\" his dad says. \"Better than half the idiots June's brought home.\" Alex's hands freeze, and his head jerks back to his dad, who's still stirring with his heavy wooden spoon, face impartial. \"Half the girls you've brought around too. Not better than Nora, though. She'll always be my favorite.\" Alex stares at him, until his dad finally looks up. \"What? You're not as subtle as you think.\"\n\n\"I\u2014I don't know,\" Alex sputters. \"I thought you might need to, like, have a Catholic moment about this or something?\"\n\nHis dad slaps him on the bicep with the spoon, leaving a splatter of crema and cheese behind. \"Have a little more faith in your old man than that, eh? A little appreciation for the patron saint of gender-neutral bathrooms in California? Little shit.\"\n\n\"Okay, okay, sorry!\" Alex says, laughing. \"I just know it's different when it's your own kid.\"\n\nHis dad laughs too, rubbing a hand over his goatee. \"It's really not. Not to me, anyway. I see you.\"\n\nAlex smiles again. \"I know.\"\n\n\"Does your ma know?\"\n\n\"Yeah, I told her a couple weeks ago.\"\n\n\"How'd she take it?\"\n\n\"I mean, she doesn't care that I'm bi. She kind of freaked out it was him. There was a PowerPoint.\"\n\n\"That sounds about right.\"\n\n\"She fired me. And, uh. She told me I need to figure out if the way I feel about him is worth the risk.\"\n\n\"Well, is it?\"\n\nAlex groans. \"Please, for the love of God, do not ask me. I'm on _vacation._ I want to get drunk and eat barbecue in peace.\"\n\nHis dad laughs ruefully. \"You know, in a lot of ways, your mom and me were a stupid idea. I think we both knew it wouldn't be forever. We're both too fucking proud. But God, that woman. Your mother is, without question, the love of my life. I'll never love anyone else like that. It was wildfire. And I got you and June out of it, best things that ever happened to an old asshole like me. That kind of love is rare, even if it was a complete disaster.\" He sucks his teeth, considering. \"Sometimes you just jump and hope it's not a cliff.\"\n\nAlex closes his eyes. \"Are you done with dad monologues for the day?\"\n\n\"You're such a shit,\" he says, throwing a kitchen towel at his head. \"Go put the ribs on. I wanna eat today.\" He calls after Alex's back, \"You two better take the bunk beds tonight! Santa Maria is watching!\"\n\nThey eat later that evening, big piles of elotes, pork tamales with salsa verde, a clay pot of frijoles charros, ribs. Henry gamely piles his plate with some of each and eyeballs it as if waiting for it to reveal its secrets to him, and Alex realizes Henry has never eaten barbecue with his hands before.\n\nAlex demonstrates and watches with poorly concealed glee as Henry gingerly picks up a rib with his fingertips and considers his approach, cheering as Henry dives in face-first and rips a hunk of meat off with his teeth. He chews proudly, a huge smear of barbecue sauce across his upper lip and the tip of his nose.\n\nHis dad keeps an old guitar in the living room, and June brings it out on the porch so the two of them can pass it back and forth. Nora, one of Alex's chambrays thrown on over her bikini, floats barefoot in and out, keeping all their glasses filled from a pitcher of sangria brimming with white peaches and blackberries.\n\nThey sit around the fire pit and play old Johnny Cash songs, Selena, Fleetwood Mac. Alex sits and listens to the cicadas and the water and his dad's rough ranger voice, and when his dad slumps off to bed, June's songbird one. He feels wrapped up and warm, turning slowly under the moon.\n\nHe and Henry drift to a swing at the edge of the porch, and he curls into Henry's side, buries his face in the collar of his shirt. Henry puts an arm around him, touches the hinge of Alex's jaw with fingers that smell like smoke.\n\nJune plucks away at \"Annie's Song,\" _you fill up my senses like a night in a forest,_ and the breeze keeps moving to meet the highest branches of the trees, and the water keeps rising to meet the bulkheads, and Henry leans down to meet Alex's mouth, and Alex is. Well, Alex is so in love he could die.\n\n* * *\n\nAlex falls out of bed the following morning with a low-grade hangover and one of Henry's swimsuits tangled around his elbow. They did, technically, sleep in separate bunks. They just didn't _start_ there.\n\nOver the kitchen sink, he chugs a glass of water and stares out the window, the sun blinding and bright on the lake, and there's an incandescent little stone of certainty at the bottom of his chest.\n\nIt's this place\u2014the absolute separation from DC, the familiar old smells of cedar trees and dried chile de \u00e1rbol, the sanity of it. The roots. He could go outside and dig his fingers into the springy ground and understand anything about himself.\n\nAnd he does understand, really. He loves Henry, and it's nothing new. He's been falling in love with Henry for years, probably since he first saw him in glossy print on the pages of _J14,_ almost definitely since Henry pinned Alex to the floor of a medical supply closet and told him to shut the hell up. That long. That much.\n\nHe smiles as he reaches for a frying pan, because he knows it's exactly the kind of insane risk he can't resist.\n\nBy the time Henry comes wandering into the kitchen in his pajamas, there's an entire breakfast spread on the long green table, and Alex is at the stove, flipping his dozenth pancake.\n\n\"Is that an _apron_?\"\n\nAlex flourishes toward the polka-dotted thing he's got on over his boxers with his free hand, as if showing off one of his tailored suits. \"Morning, sweetheart.\"\n\n\"Sorry,\" Henry says. \"I was looking for someone else. Handsome, petulant, short, not pleasant until after ten a.m.? Have you seen him?\"\n\n\"Fuck off, five-nine is average.\"\n\nHenry crosses the room with a laugh and nudges up behind him at the stove to peck him on the cheek. \"Love, you and I both know you're rounding up.\"\n\nIt's only a step on the way to the coffeemaker, but Alex reaches back and gets a hand in Henry's hair before he can move, pulling him into a kiss on the mouth this time. Henry huffs a little in surprise but returns it fully.\n\nAlex forgets, momentarily, about the pancakes and everything else, not because he wants to do absolutely filthy things to Henry\u2014maybe even with the apron still on\u2014but because he _loves_ him, and isn't that wild, to know that _that's_ what makes the filthy things so good.\n\n\"I didn't realize this was a jazz brunch,\" says Nora's voice suddenly, and Henry springs backward so fast he almost puts his ass in the bowl of batter. She sidles up to the forgotten coffeemaker, grinning slyly at them.\n\n\"That doesn't seem sanitary,\" June is saying with a yawn as she folds herself into a chair at the table.\n\n\"Sorry,\" Henry says sheepishly.\n\n\"Don't be,\" Nora tells him.\n\n\"I'm not,\" Alex says.\n\n\"I'm hungover,\" June says as she reaches for the pitcher of mimosas. \"Alex, you did all this?\"\n\nAlex shrugs, and June squints at him, bleary but knowing.\n\nThat afternoon, over the sounds of the boat's engine, Henry talks to Alex's dad about the sailboats that jut up from the horizon, getting into a complex discussion on outboard motors that Alex can't hope to follow. He leans back against the bow and watches, and it's so easy to imagine it: a future Henry who comes to the lake house with him every summer, who learns how to make elotes and ties neat cleat hitches and fits right into place in his weird family.\n\nThey go swimming, yell over one another about politics, pass the guitar around again. Henry takes a photo of himself with June and Nora, one under each arm and both in their bikinis. Nora is holding his chin in one hand and licking the side of his face, and June has her fingers tangled up in his hair and her head in the crook of his neck, smiling angelically at the camera. He sends it to Pez and receives anguished keysmashes and crying emojis in response, and they all almost piss themselves laughing.\n\nIt's good. It's really, really good.\n\nAlex lies awake that night, drunk on Shiner and way too many campfire marshmallows, and he stares at whorls in the wood panels of the top bunk and thinks about coming of age out here. He remembers when he was a kid, freckly and unafraid, when the world seemed like it was blissfully endless but everything still made perfect sense. He used to leave his clothes in a pile on the pier and dive headfirst into the lake. Everything was in its right place.\n\nHe wears a key to his childhood home around his neck, but he doesn't know the last time he actually thought about the boy who used to push it into the lock.\n\nMaybe losing the job isn't the worst thing that could have happened.\n\nHe thinks about roots, about first and second languages. What he wanted when he was a kid and what he wants now and where those things overlap. Maybe that place, the meeting of the two, is here somewhere, in the gentle insistence of the water around his legs, crude letters carved with an old pocket knife. The steady thrum of another person's pulse against his.\n\n\"H?\" he whispers. \"You awake?\"\n\nHenry sighs. \"Always.\"\n\nThey sneak through the grass in hushed voices past one of Henry's PPOs dozing on the porch, racing down the pier, shoving at each other's shoulders. Henry's laugh is high and clear, his sunburned shoulders bright pink in the dark, and Alex looks at him and something so buoyant fills up his chest that he feels like he could swim the length of the lake without stopping for air. He throws his T-shirt down at the end of the pier and starts to shuck his boxers, and when Henry arches an eyebrow at him, Alex laughs and jumps.\n\n\"You're a menace,\" Henry says when Alex breaks back to the surface. But he only hesitates briefly before he's stripping out of his clothes.\n\nHe stands naked at the edge of the pier, looking at Alex's head and shoulders bobbing in the water. The lines of him are long and languid in the moonlight, just skin and skin and skin lit soft and blue, and he's so beautiful that Alex thinks this moment, the soft shadows and pale thighs and crooked smile, should be the portrait of Henry that goes down in history. There are fireflies winking around his head, landing in his hair. A crown.\n\nHis dive is infuriatingly graceful.\n\n\"Can't you ever just do one thing without having to be so goddamn extra about it?\" Alex says, splashing him as soon as he surfaces.\n\n\"That is bloody rich coming from you,\" Henry says, and he's grinning like he does when he's drinking in a challenge, like nothing in the world pleases him more than Alex's antagonizing elbow in his side.\n\n\"I don't know what you're talking about,\" Alex says, kicking over to him.\n\nThey chase each other around the pier, race down to the lake's shallow bottom and shoot back up in the moonlight, all elbows and knees. Alex finally manages to catch Henry around the waist, and he pins him, slides his wet mouth over the thudding pulse of Henry's throat. He wants to stay tangled up in Henry's legs forever. He wants to match the new freckles across Henry's nose to the stars above them and make him name the constellations.\n\n\"Hey,\" he says, his mouth right up in a breath's space from Henry's. He watches a drop of water roll down Henry's perfect nose and disappear into his mouth.\n\n\"Hi,\" Henry says back, and Alex thinks, _Goddamn, I love him._ It keeps coming back to him, and it's getting harder to look into Henry's soft smiles and not say it.\n\nHe kicks out a little to turn them in a slow circle. \"You look good out here.\"\n\nHenry's grin goes crooked and a little shy, dipping down to brush against Alex's jaw. \"Yeah?\"\n\n\"Yeah,\" Alex says. He twists Henry's wet hair around his fingers. \"I'm glad you came this weekend,\" Alex hears himself say. \"It's been so intense lately. I... I really needed this.\"\n\nHenry's fingers give a little jab to his ribs, gently scolding. \"You carry too much.\"\n\nHis instinct has always been to shoot back, _No, I don't_ , or, _I_ _want to_ , but he bites it back and says, \"I know,\" and he realizes it's the truth. \"You know what I'm thinking right now?\"\n\n\"What?\"\n\n\"I'm thinking about, after inauguration, like next year, taking you back out here, just the two of us. And we can sit under the moon and not stress about anything.\"\n\n\"Oh,\" Henry says. \"That sounds nice, if unlikely.\"\n\n\"Come on, think about it, babe. Next year. My mom'll be in office again, and we won't have to worry about winning any more elections. I'll finally be able to breathe. Ugh, it'll be amazing. I'll cook migas in the mornings, and we'll swim all day and never put clothes on and make out on the pier, and it won't even matter if the neighbors see.\"\n\n\"Well. It will matter, you know. It will always matter.\"\n\nHe pulls back to find Henry's face indecipherable.\n\n\"You know what I mean.\"\n\nHenry's looking at him and looking at him, and Alex can't shake the feeling Henry's really seeing him for the first time. He realizes it's probably the only time he's ever invited love into a conversation with Henry on purpose, and it must be lying wide open on his face.\n\nSomething moves behind Henry's eyes. \"Where are you going with all this?\"\n\nAlex tries to figure out how the hell to funnel everything he needs to tell Henry into words.\n\n\"June says I have a fire under my ass for no good reason,\" he says. \"I don't know. You know how they always say to take it one day at a time? I think I take it ten years in the future. Like when I was in high school, it was all: Well, my parents hate each other, and my sister is leaving for college, and sometimes I look at other guys in the shower, but if I keep looking directly ahead, that stuff can't catch up to me. Or if I take this class, or this internship, or this job. I used to think, if I pictured the person I wanted to be and took all the crazy anxiety in my brain and narrowed it down to that point, I could rewire it. Use it to power something else. It's like I never learned how to just be where I am.\" Alex takes a breath. \"And where I am is here. With you. And I'm thinking maybe I should start trying to take it day by day. And just... feel what I feel.\"\n\nHenry doesn't say anything.\n\n\"Sweetheart.\" The water ripples quietly around him as he slides his hands up to hold Henry's face in both palms, tracing his cheekbones with the wet pads of his thumbs.\n\nThe cicadas and the wind and the lake are probably still making sounds, somewhere, but it's all faded into silence. Alex can't hear anything but his heartbeat in his ears.\n\n\"Henry, I\u2014\"\n\nAbruptly Henry shifts, ducking beneath the surface and out of his arms before he can say anything else.\n\nHe pops back up near the pier, hair sticking to his forehead, and Alex turns around and stares at him, breathless at the loss. Henry spits out lake water and sends a splash in his direction, and Alex forces a laugh.\n\n\"Christ,\" Henry says, slapping at a bug that's landed on him, \"what are these infernal creatures?\"\n\n\"Mosquitos,\" Alex supplies.\n\n\"They're awful,\" Henry says loftily. \"I'm going to catch an exotic plague.\"\n\n\"I'm... sorry?\"\n\n\"I just mean to say, you know, Philip is the heir and I'm the spare, and if that nervy bastard has a heart attack at thirty-five and I've got malaria, whither the spare?\"\n\nAlex laughs weakly again, but he's got a distinct feeling of something being pulled out of his hands right before he could grasp it. Henry's tone has gone light, clipped, superficial. His press voice.\n\n\"At any rate, I'm knackered,\" Henry is saying now. And Alex watches helplessly as he turns and starts hauling himself out of the water and onto the dock, pulling his shorts back up shivering legs. \"If it's all the same to you, I think I'll go to bed.\"\n\nAlex doesn't know what to say, so he watches Henry walk the long line of the dock, disappearing into the darkness.\n\nA ringing, scooped-out sensation starts behind his molars and rolls down his throat, into his chest, down to the pit of his stomach. Something's wrong, and he knows it, but he's too afraid to push back or ask. That, he realizes suddenly, is the danger of allowing love into this\u2014the acknowledgment that if something goes wrong, he doesn't know how he will stand it.\n\nFor the first time since Henry grabbed him and kissed him with so much certainty in the garden, the thought enters Alex's mind: What if it was never his decision to make? What if he got so wrapped up in everything Henry is\u2014the words he writes, the earnest heartsickness of him\u2014he forgot to take into account that it's just _how_ he is, all the time, with everyone?\n\nWhat if he's done the thing he swore he would never do, the thing he hates, and fallen in love with a prince because it was a fantasy?\n\nWhen he gets back to their room, Henry's already in his bunk and silent, his back turned.\n\n* * *\n\nIn the morning, Henry is gone.\n\nAlex wakes up to find his bunk empty and made up, the pillow tucked neatly beneath the blanket. He practically throws the door off its hinges running out onto the patio, only to find it empty as well. The yard is empty, the pier is empty. It's like he was never even there.\n\nHe finds the note in the kitchen:\n\n> Alex,\n> \n> Had to go early for a family matter. Left with the PPOs. Didn't want to wake you.\n> \n> Thank you for everything.\n> \n> X\n\nIt's the last message Henry sends him.\n\n# TEN\n\nHe sends Henry five texts the first day. Two the second. By day three, none. He's spent too much of his life talking, talking, talking not to know the signs when someone doesn't want to hear him anymore.\n\nHe starts forcing himself to only check his phone once every two hours instead of once an hour, makes himself hang on by his fingernails until the minutes tick down. A few times, he gets wrapped up in obsessively reading press coverage of the campaign and realizes he hasn't checked in hours, and every time he's hit with a hiccupping, desperate hope that there will be something. There never is.\n\nHe thought he was reckless before, but he understands now\u2014holding love off was the only thing keeping him from losing himself in this completely, and he's gone, stupid, lovesick, a fucking disaster. No work to distract him. The tripwire of \"Things Only People in Love Say and Do\" set off.\n\nSo, instead:\n\nA Tuesday night, hiding on the roof of the Residence, pacing so many furious laps that the skin on the backs of his heels splits open and blood soaks into his loafers.\n\nHis CLAREMONT FOR AMERICA mug, returned in a carefully marked box from his desk at the campaign office, a concrete reminder of what this already cost him smashed in his bathroom sink.\n\nThe smell of Earl Grey curling up from the kitchens, and his throat going painfully tight.\n\nTwo and a half different dreams about sandy hair wrapped around his fingers.\n\nA three-line email, an excerpt dug up from an archived letter, Hamilton to Laurens, _You should not have taken advantage of my sensibility to steal into my affections without my consent,_ drafted and deleted.\n\nOn day five, Rafael Luna makes his fifth campaign stop as a surrogate, the Richards campaign's token twofer minority. Alex hits a momentary emotional impasse: either destroy something or destroy himself. He ends up smashing his phone on the pavement outside the Capitol. The screen is replaced by the end of the day. It doesn't make any messages from Henry magically appear.\n\nOn the morning of day seven, he's digging in the back of his closet when he stumbles upon a bundle of teal silk\u2014the stupid kimono Pez had made for him. He hasn't taken it out since LA.\n\nHe's about to shove it back into the corner when he feels something in the pocket. He finds a small folded square of paper. It's stationery from their hotel that night, the night everything inside Alex rearranged. Henry's cursive.\n\n> Dear Thisbe,\n> \n> I wish there weren't a wall.\n> \n> Love, Pyramus\n\nHe fumbles his phone out so fast he almost drops it on the floor and smashes it again. The search tells him Pyramus and Thisbe were lovers in a Greek myth, children of rival families, forbidden to be together. Their only way to speak to each other was through a thin crack in the wall built between them.\n\nAnd that is, officially, too fucking much.\n\nWhat he does next, he's sure he'll have no memory of doing, simply a white-noise gap of time that got him from point A to point B. He texts Cash, what are you doing for the next 24 hours? Then he unearths the emergency credit card from his wallet and buys two plane tickets, first class, nonstop. Boarding in two hours. Dulles International to Heathrow.\n\n* * *\n\nZahra nearly refuses to secure a car after Alex \"had the goddamn nerve\" to call her from the runway at Dulles. It's dark and pissing down rain when they land in London around nine in the evening, and he and Cash are both soaked the second they climb out of the car inside the back gates of Kensington.\n\nClearly, someone has radioed for Shaan, because he's standing there at the door to Henry's apartments in an impeccable gray peacoat, dry and unmoved under a black umbrella.\n\n\"Mr. Claremont-Diaz,\" he says. \"What a treat.\"\n\nAlex has not got the damn time. \"Move, Shaan.\"\n\n\"Ms. Bankston called ahead to warn me that you were on the way,\" he says. \"As you might have guessed by the ease with which you were able to get through our gates. We thought it best to let you kick up a fuss somewhere more private.\"\n\n\"Move.\"\n\nShaan smiles, looking as if he might be genuinely enjoying watching two hapless Americans become slowly waterlogged. \"You're aware it's quite late, and it's well within my power to have security remove you. No member of the royal family has invited you into the palace.\"\n\n\"Bullshit,\" Alex bites out. \"I need to see Henry.\"\n\n\"I'm afraid I can't do that. The prince does not wish to be disturbed.\"\n\n\"Goddammit\u2014Henry!\" He sidesteps Shaan and starts shouting up at Henry's bedroom windows, where there's a light on. Fat raindrops are pelting his eyeballs. \"Henry, you motherfucker!\"\n\n\"Alex\u2014\" says Cash's nervous voice behind him.\n\n\"Henry, you piece of shit, get your ass down here!\"\n\n\"You are making a scene,\" Shaan says placidly.\n\n\"Yeah?\" Alex says, still yelling. \"How 'bout I just keep yelling and we see which of the papers show up first!\" He turns back to the window and starts flailing his arms too. \"Henry! Your Royal fucking Highness!\"\n\nShaan touches a finger to his earpiece. \"Team Bravo, we've got a situa\u2014\"\n\n\"For Christ's sake, Alex, what are you doing?\"\n\nAlex freezes, his mouth open around another shout, and there's Henry standing behind Shaan in the doorway, barefoot in worn-in sweats. Alex's heart is going to fall out of his ass. Henry looks unimpressed.\n\nHe drops his arms. \"Tell him to let me in.\"\n\nHenry sighs, pinching the bridge of his nose. \"It's fine. He can come in.\"\n\n\" _Thank_ you,\" he says, pointedly looking at Shaan, who does not seem to care at all if he dies of hypothermia. He sloshes into the palace, ditching his soaked shoes as Cash and Shaan disappear behind the door.\n\nHenry, who led the way in, hasn't even stopped to speak to him, and all Alex can do is follow him up the grand staircase toward his rooms.\n\n\"Really nice,\" Alex yells after him, dripping as aggressively as he can manage along the way. He hopes he ruins a rug. \"Fuckin' ghost me for a week, make me stand in the rain like a brown John Cusack, and now you won't even talk to me. I'm really just having a great time here. I can see why all y'all had to marry your fucking cousins.\"\n\n\"I'd rather not do this where we might be overheard,\" Henry says, taking a left on the landing.\n\nAlex stomps up after him, following him into his bedroom. \"Do what?\" he says as Henry shuts the door behind them. \"What are you gonna do, Henry?\"\n\nHenry turns to face him at last, and now that Alex's eyes aren't full of rainwater, he can see the skin under his eyes is papery and purple, rimmed pink at his eyelashes. There's a tense set to his shoulders Alex hasn't seen in months, not directed at him at least.\n\n\"I'm going to let you say what you need to say,\" Henry says flatly, \"so you can leave.\"\n\nAlex stares. \"What, and then we're over?\"\n\nHenry doesn't answer him.\n\nSomething rises in Alex's throat\u2014anger, confusion, hurt, bile. Unforgivably, he feels like he might cry.\n\n\"Seriously?\" he says, helpless and indignant. He's still dripping. \"What the _fuck_ is going on? A week ago it was emails about how much you missed me and meeting my fucking _dad,_ and that's it? You thought you could fucking _ghost me_? I can't shut this off like you do, Henry.\"\n\nHenry paces over to the elaborately carved fireplace across the room and leans on the mantelpiece. \"You think I don't _care_ as much as you?\"\n\n\"You're sure as hell acting like it.\"\n\n\"I honestly haven't got the time to explain to you all the ways you're wrong\u2014\"\n\n\"Jesus, could you stop being an obtuse fucking asshole for, like, twenty seconds?\"\n\n\"So glad you flew here to _insult me_ \u2014\"\n\n\" _I fucking love you, okay?_ \" Alex half yells, finally, irreversibly. Henry goes very still against the mantelpiece. Alex watches him swallow, watches the muscle that keeps twitching in his jaw, and feels like he might shake out of his skin. \"Fuck, I swear. You don't make it fucking easy. But I'm in love with you.\"\n\nA small _click_ cuts the silence: Henry has taken his signet ring off and set it down on the mantel. He holds his naked hand to his chest, kneading the palm, the flickering light from the fire painting his face in dramatic shadows. \"Do you have any idea what that means?\"\n\n\"Of course I do\u2014\"\n\n\"Alex, _please,_ \" Henry says, and when he finally turns to look at him, he looks wretched, miserable. \"Don't. This is the entire goddamned reason. I can't do this, and you _know_ why I can't do this, so _please_ don't make me say it.\"\n\nAlex swallows hard. \"You're not even gonna try to be happy?\"\n\n\"For Christ's sake,\" Henry says, \"I've been trying to be happy my entire idiot life. My birthright is a _country,_ not happiness.\"\n\nAlex yanks the soggy note out of his pocket, _I wish there wasn't a wall,_ and throws it at Henry viciously, watches him pick it up. \"Then what is _that_ supposed to mean, if you don't want this?\"\n\nHenry stares down at his words from months ago. \"Alex, Thisbe and Pyramus both _die_ at the end.\"\n\n\"Oh my _God,_ \" Alex groans. \"So, what, was this all never going to be anything real to you?\"\n\nAnd Henry snaps.\n\n\"You really are a _complete_ idiot if you believe that,\" Henry hisses, the note balled in his fist. \"When have I _ever,_ since the first instant I touched you, pretended to be anything less than in love with you? Are you so fucking self-absorbed as to think this is about you and whether or not I love you, rather than the fact I'm an heir to the fucking throne? You at least have the _option_ to not choose a public life eventually, but I will live and die in these palaces and in this family, so don't you dare come to me and question if I love you when it's the thing that could bloody well ruin everything.\"\n\nAlex doesn't speak, doesn't move, doesn't breathe, his feet rooted to the spot. Henry isn't looking at him, but staring at a point on the mantel somewhere, tugging at his own hair in exasperation.\n\n\"It was never supposed to be an issue,\" he goes on, his voice hoarse. \"I thought I could have some part of you, and just never say it, and you'd never have to know, and one day you'd get tired of me and leave, because I'm\u2014\" He stops short, and one shaking hand moves through the air in front of him in a helpless sort of gesture at everything about himself. \"I never thought I'd be stood here faced with a choice I can't make, because I never... I never imagined you would love me back.\"\n\n\"Well,\" Alex says. \"I do. And you _can_ choose.\"\n\n\"You know bloody well I can't.\"\n\n\"You can _try,_ \" Alex tells him, feeling as if it should be the simplest fucking truth in the world. \"What do you _want_?\"\n\n\"I want you\u2014\"\n\n\"Then fucking _have me._ \"\n\n\"\u2014but I don't want _this._ \"\n\nAlex wants to grab Henry and shake him, wants to scream in his face, wants to smash every priceless antique in the room. \"What does that even _mean_?\"\n\n\"I don't _want_ it!\" Henry practically shouts. His eyes are flashing, wet and angry and afraid. \"Don't you bloody see? I'm not _like_ you. I can't afford to be _reckless._ I don't have a family who will support me. I don't go about shoving who I am in everyone's faces and dreaming about a career in fucking _politics,_ so I can be _more_ scrutinized and picked apart by the entire godforsaken world. I can love you and want you and still not want that life. I'm allowed, all right, and it doesn't make me a liar; it makes me a man with some infinitesimal shred of self-preservation, unlike _you,_ and you don't get to come here and call me a coward for it.\"\n\nAlex takes a breath. \"I never said you were a coward.\"\n\n\"I.\" Henry blinks. \"Well. The point stands.\"\n\n\"You think _I_ want _your_ life? You think I want _Martha's_? Gilded fucking cage? Barely allowed to _speak_ in public, or have a goddamn opinion\u2014\"\n\n\"Then what are we even doing here? Why are we fighting, then, if the lives we have to lead are so incompatible?\"\n\n\"Because you don't want that either!\" Alex insists. \"You don't want any of this bullshit. You _hate_ it.\"\n\n\"Don't tell me what I want,\" Henry says. \"You haven't a clue how it feels.\"\n\n\"Look, I might not be a fucking royal,\" Alex says, crosses the horrible rug, moves into Henry's space, \"but I know what it's like for your whole life to be determined by the family you were born into, okay? The lives we want\u2014they're _not that different._ Not in the ways that matter. You want to take what you were given and leave the world better than you found it. So do I. We can\u2014we can figure out a way to do that together.\"\n\nHenry stares at him silently, and Alex can see the scales balancing in his head.\n\n\"I don't think I can.\"\n\nAlex turns away from him, falling back on his heels like he's been slapped. \"Fine,\" he finally says. \"You know what? Fucking fine. I'll leave.\"\n\n\"Good.\"\n\n\"I'll leave,\" he says, and he turns back and leans in, \"as soon as you tell me to leave.\"\n\n_\"Alex.\"_\n\nHe's in Henry's face now. If he's getting his heart broken tonight, he's sure as hell going to make Henry have the guts to do it right. \"Tell me you're done with me. I'll get back on the plane. That's it. And you can live here in your tower and be miserable forever, write a whole book of sad fucking poems about it. Whatever. Just say it.\"\n\n\"Fuck you,\" Henry says, his voice breaking, and he gets a handful of Alex's shirt collar, and Alex knows he's going to love this stubborn shithead forever.\n\n\"Tell me,\" he says, a ghost of a smile around his lips, \"to leave.\"\n\nHe feels before he registers being shoved backward into a wall, and Henry's mouth is on his, desperate and wild. The faint taste of blood blooms on his tongue, and he smiles as he opens up to it, pushes it into Henry's mouth, tugs at his hair with both hands. Henry groans, and Alex feels it in his spine.\n\nThey grapple along the wall until Henry physically picks him up off the floor and staggers backward, toward the bed. Alex bounces when his back hits the mattress, and Henry stands over him for several breaths, staring. Alex would give anything to know what's going through that fucking head of his.\n\nHe realizes, suddenly, Henry's crying.\n\nHe swallows.\n\nThat's the thing: he doesn't know. He doesn't know if this is supposed to be some kind of consummation, or if it's one last time. He doesn't think he could go through with it if he knew it was the latter. But he doesn't want to go home without having this.\n\n\"C'mere.\"\n\nHe fucks Henry slow and deep, and if it's the last time, they go down shivering and gasping and epic, all wet mouths and wet eyelashes, and Alex is a clich\u00e9 on an ivory bedspread, and he hates himself but he's so in love. He's in stupid, unbearable love, and Henry loves him too, and at least for one night it matters, even if they both have to pretend to forget in the morning.\n\nHenry comes with his face turned into Alex's open palm, his bottom lip catching on the knob of his wrist, and Alex tries to memorize every detail down to how his lashes fan across his cheeks and the pink flush that spreads all the way up to his ears. He tells his too-fast brain: _Don't miss it this time. He's too important._\n\nIt's pitch-black outside when Henry's body finally subsides, and the room is impossibly quiet, the fire gone out. Alex rolls over onto his side and touches two fingers to his chest, right next to where the key on the chain rests. His heart is beating the same as ever under his skin. He doesn't know how that can be true.\n\nIt's a long stretch of silence before Henry shifts in the bed beside him and rolls onto his back, pulling a sheet over them. Alex reaches for something to say, but there's nothing.\n\n* * *\n\nAlex wakes up alone.\n\nIt takes a moment for everything to reorient around the fixed point in his chest where last night settled. The elaborate gilded headboard, the heavy embroidered duvet, the soft twill blanket beneath that's the only thing in the room Henry actually chose. He slides his hand across the sheet, over to Henry's side of the bed. It's cool to the touch.\n\nKensington Palace is gray and dull in the early morning. The clock on the mantelpiece says it's not even seven, and there's a violent rain lashing against the big picture window, half-revealed by parted curtains.\n\nHenry's room has never felt much like Henry, but in the quiet of morning, he shows up in pieces. A pile of journals on the desk, the topmost splotched with ink from a pen exploding in his bag on a plane. An oversized cardigan, worn through and patched at the elbows, slung over an antique wingback chair near the window. David's leash hanging from the doorknob.\n\nAnd beside him, there's a copy of _Le Monde_ on the nightstand, tucked under a gigantic leather-bound volume of Wilde's complete works. He recognizes the date: Paris. The first time they woke up next to each other.\n\nHe squeezes his eyes shut, feeling for once in his life that he should stop being so damn nosy. It's time, he realizes, to start accepting only what Henry can give him.\n\nThe sheets smell like Henry. He knows:\n\nOne. Henry isn't here.\n\nTwo. Henry never said yes to any kind of future last night.\n\nThree. This could very well be the last time he gets to inhale Henry's scent on anything.\n\nBut, four. Next to the clock on the mantel, Henry's ring still sits.\n\nThe doorknob turns, and Alex opens his eyes to find Henry, holding two mugs and smiling a wan, unreadable smile. He's in soft sweats again, brushed with morning mist.\n\n\"Your hair in the mornings is truly a wonder to behold,\" is how he breaks the silence. He crosses and kneels on the edge of the mattress, offering Alex a mug. It's coffee, one sugar, cinnamon. He doesn't want to feel anything about Henry knowing how he likes his coffee, not when he's about to be dumped, but he does.\n\nExcept, when Henry looks at him again, watches him take the first blessed sip of coffee, the smile comes back in earnest. He reaches down and palms one of Alex's feet through the duvet.\n\n\"Hi,\" Alex says carefully, squinting over his coffee. \"You seem... less pissy.\"\n\nHenry huffs a laugh. \"You're one to talk. I wasn't the one who stormed the palace in a fit of pique to call me an 'obtuse fucking asshole.'\"\n\n\"In my defense,\" Alex says, \"you _were_ an obtuse fucking asshole.\"\n\nHenry pauses, takes a sip of his tea, and places it on the nightstand. \"I was,\" he agrees, and he leans forward and presses his mouth to Alex's, one hand steadying his mug so it doesn't spill. He tastes like toothpaste and Earl Grey, and maybe Alex isn't getting dumped after all.\n\n\"Hey,\" he says when Henry pulls back. \"Where were you?\"\n\nHenry doesn't answer, and Alex watches him kick his wet sneakers onto the floor before climbing up to sit between Alex's open legs. He places his hands on Alex's thighs, bracketing him with his full attention, and when he looks up into Alex's eyes, his are clear blue and focused.\n\n\"I needed a run,\" he says. \"To clear my head a bit, figure out... what's next. Very Mr. Darcy brooding at Pemberley. And I ran into Philip. I hadn't mentioned it, but he and Martha are here for the week while they're doing renovations on Anmer Hall. He was up early for some appearance or other, eating toast. Plain toast. Have you ever seen someone eat toast without anything on it? Harrowing, truly.\"\n\nAlex chews his lip. \"Where's this going, babe?\"\n\n\"We chatted for a bit. He didn't seem to know about your... visitation... last night, thankfully. But he was on about Martha, and land holdings, and the hypothetical heirs they have to start working on, even though Philip hates children, and suddenly it was as if... as if everything you said last night came back to me. I thought, God, that's it, isn't it? Just following the plan. And it's not that he's unhappy. He's fine. It's all very deeply fine. A whole lifetime of fine.\" He's been pulling at a thread on the duvet, but he looks back up, squarely into Alex's eyes, and says, \"That's not good enough for me.\"\n\nThere's a desperate stutter in Alex's heartbeat. \"It's not?\"\n\nHe reaches up and touches a thumb to Alex's cheekbone. \"I'm not... good at saying these things like you are, but. I've always thought... ever since I knew about me, and even before, when I could sense I was _different_ \u2014and, after everything the past few years, all the mad things my head does\u2014I've always thought of myself as a problem that deserved to stay hidden. Never quite trusted myself, or what I wanted. Before you, I was all right letting everything happen to me. I honestly have never thought I deserved to choose.\" His hand moves, fingertips brushing a curl behind Alex's ear. \"But you treat me like I do.\"\n\nThere's something painfully hard in Alex's throat, but he pushes past it. He reaches over and sets his mug down next to Henry's on the nightstand.\n\n\"You do,\" he says.\n\n\"I think I'm actually beginning to believe that,\" Henry says. \"And I don't know how long it would have taken if I didn't have you to believe for me.\"\n\n\"And there's nothing wrong with you,\" Alex tells him. \"I mean, aside from the fact that you're occasionally an obtuse fucking asshole.\"\n\nHenry laughs again, wetly, his eyes crinkling up in the corners, and Alex feels his heart lift into his throat, up to the embellished ceilings, pushing out to fill the whole room all the way to the glinting gold ring still sitting above the fireplace.\n\n\"I am sorry about that,\" Henry says. \"I\u2014I wasn't ready to hear it. That night, at the lake... it was the first time I let myself think you might actually say it. I panicked, and it was daft and unfair, and I won't do it again.\"\n\n\"You better not,\" Alex tells him. \"So, you're saying... you're in?\"\n\n\"I'm saying,\" Henry begins, and the knit of his brow is nervous but his mouth keeps speaking, \"I'm terrified, and my whole life is completely mad, but trying to give you up this week nearly killed me. And when I woke up this morning and looked at you... there's no trying to get by for me anymore. I don't know if I'll ever be allowed to tell the world, but I... I want to. One day. If there's any legacy for me on this bloody earth, I want it to be true. So I can offer you all of me, in whatever way you'll have me, and I can offer you the chance of a life. If you can wait, I want you to help me try.\"\n\nAlex looks at him, taking in the whole parcel of him, the centuries of royal blood sitting under an antique Kensington chandelier, and he reaches out to touch his face and looks at his fingers and thinks about holding the Bible at his mother's inauguration with the same hand.\n\nIt hits him, fully: the weight of this. How completely neither of them will ever be able to undo it.\n\n\"Okay,\" he says. \"I'm into making history.\"\n\nHenry rolls his eyes and seals it with a smiling kiss, and they fall back into the pillows together, Henry's wet hair and sweatpants and Alex's naked limbs all tangled up in the lavish bedclothes.\n\nWhen Alex was a kid, before anyone knew his name, he dreamed of love like it was a fairy tale, as if it would come sweeping into his life on the back of a dragon one day. When he got older, he learned about love as a strange thing that could fall apart no matter how badly you wanted it, a choice you make anyway. He never imagined it'd turn out he was right both times.\n\nHenry's hands on him are unhurried and soft, and they make out lazily for hours or days, basking in the rare luxury of it. They take breaks to finish their lukewarm coffee and tea, and Henry has scones and blackcurrant jam sent up. They waste away the morning in bed, watching Mel and Sue squawk over tea cakes on Henry's laptop, listening to the rain slow to a drizzle.\n\nAt some point, Alex disentangles his jeans from the foot of the bed and fishes out his phone. He's got three missed calls from Zahra, one ominous voicemail from his mother, and forty-seven unread messages in his group text with June and Nora.\n\nALEX, Z JUST TOLD ME YOU'RE IN LONDON???????\n\nAlex oh my god\n\nI swear to god if you do something stupid and get yourself caught, I'm gonna kill you myself\n\nBut you went after him!!! That's SO Jane Austen\n\nI'm gonna punch you in the face when you get back. I can't believe you didn't tell me\n\nHow did it go??? Are you with Henry now?????\n\nGONNA PUNCH YOU\n\nIt turns out forty-six out of forty-seven texts are June and the forty-seventh is Nora asking if either of them know where she left her white Chuck Taylors. Alex texts back: your chucks are under my bed and henry says hi.\n\nThe message has barely delivered before his phone erupts with a call from June, who demands to be put on speaker and told everything. After, rather than facing Zahra's wrath himself, he convinces Henry to call Shaan.\n\n\"D'you think you could, er, phone Ms. Bankston and let her know Alex is safe and with me?\"\n\n\"Yes, sir,\" Shaan says. \"And shall I arrange a car for his departure?\"\n\n\"Er,\" Henry says, and he looks at Alex and mouths, _Stay?_ Alex nods. \"Tomorrow?\"\n\nThere's a very long pause over the line before Shaan says, \"I'll let her know,\" in a voice like he'd rather do literally anything else.\n\nAlex laughs as Henry hangs up, but he returns to his phone again, to the voicemail waiting from his mother. Henry sees his thumb hovering over the play button and nudges his ribs.\n\n\"I suppose we do have to face the consequences at some point,\" he says.\n\nAlex sighs. \"I don't think I told you, but she, uh. Well, when she fired me, she told me that if I wasn't a thousand percent serious about you, I needed to break things off.\"\n\nHenry nuzzles his nose behind Alex's ear. \"A thousand percent?\"\n\n\"Yeah, don't let it go to your head.\"\n\nHenry elbows him again, and Alex laughs and grabs his head and aggressively kisses his cheek, smashing his face into the pillow. When Alex finally relents, Henry is pink-faced and mussed and definitely pleased.\n\n\"I was thinking about that, though,\" Henry says, \"the chance being with me is going to keep ruining your career. Congress by thirty, wasn't it?\"\n\n\"Come on. Look at this face. People love this face. I'll figure out the rest.\" Henry looks deeply skeptical, and Alex sighs again. \"Look, I don't know. I don't even exactly know, like, how being a legislator would work if I'm with a prince of another country. So, you know. There's stuff to figure out. But way worse people with way bigger problems than me get elected all the time.\"\n\nHenry's looking at him in the piercing way he has sometimes that makes Alex feel like a bug stuck under a shadowbox with a pushpin. \"You're really not frightened of what might happen?\"\n\n\"No, I mean, of course I am,\" he says. \"It definitely stays secret until after the election. And I know it'll be messy. But if we can get ahead of the narrative, wait for the right time and do it on our own terms, I think it could be okay.\"\n\n\"How long have you been thinking about this?\"\n\n\"Consciously? Since, like, the DNC. Subconsciously, in total denial? A long-ass time. At least since you kissed me.\"\n\nHenry stares at him from the pillow. \"That's... kind of incredible.\"\n\n\"What about you?\"\n\n\"What about _me_?\" Henry says. \"Christ, Alex. The whole bloody time.\"\n\n\"The whole time?\"\n\n\"Since the Olympics.\"\n\n\"The _Olympics_?\" Alex yanks Henry's pillow out from under him. \"But that's, that's like\u2014\"\n\n\"Yes, Alex, the day we met, nothing gets past you, does it?\" Henry says, reaching to steal the pillow back. \"'What about you,' he says, as if he doesn't _know_ \u2014\"\n\n\"Shut your _mouth,_ \" Alex says, grinning like an idiot, and he stops fighting Henry for the pillow and instead straddles him and kisses him into the mattress. He pulls the blankets up and they disappear into the pile, a laughing mess of mouths and hands, until Henry rolls onto his phone and his ass presses the button on the voicemail.\n\n\"Diaz, you insane, hopeless romantic little shit,\" says the voice of the President of the United States, muffled in the bed. \"It had better be forever. Be safe.\"\n\n* * *\n\nSneaking out of the palace without security at two in the morning was, surprisingly, Henry's idea. He pulled hoodies and hats out for both of them\u2014the incognito uniform of the internationally recognizable\u2014and Bea staged a noisy exit from the opposite end of the palace while they sprinted through the gardens. Now they're on the deserted, wet pavement of South Kensington, flanked by tall, red brick buildings and a sign for\u2014\n\n\"Stop, are you kidding me?\" Alex says. \" _Prince Consort Road?_ Oh my God, take a picture of me with the sign.\"\n\n\"Not there yet!\" Henry says over his shoulder. He gives Alex's arm another pull to keep him running. \"Keep moving, you wastrel.\"\n\nThey cross to another street and duck into an alcove between two pillars while Henry fishes a keyring with dozens of keys out of his hoodie. \"Funny thing about being a prince\u2014people will give you keys to just about anything if you ask nicely.\"\n\nAlex gawks, watching Henry feel around the edge of a seemingly plain wall. \"All this time, I thought _I_ was the Ferris Bueller of this relationship.\"\n\n\"What, did you think I was Sloane?\" Henry says, pushing the panel open a crack and yanking Alex into a wide, dark plaza.\n\nThe grounds are sloping, white tiles carrying the sounds of their feet as they run. Sturdy Victorian bricks tower into the night, framing the courtyard, and Alex thinks, _Oh_. The Victoria and Albert Museum. Henry has a key to the V&A.\n\nThere's a stout old security guard waiting at the doors.\n\n\"Can't thank you enough, Gavin,\" Henry says, and Alex notices the thick wad of cash Henry slips into their handshake.\n\n\"Renaissance City tonight, yeah?\" Gavin says.\n\n\"If you would be so kind,\" Henry tells him.\n\nAnd they're off again, hustling through rooms of Chinese art and French sculptures. Henry moves fluidly from room to room, past a black stone sculpture of a seated Buddha and John the Baptist nude and in bronze, without a single false step.\n\n\"You do this a lot?\"\n\nHenry laughs. \"It's, ah, sort of my little secret. When I was young, my mum and dad would take us early in the morning, before opening. They wanted us to have a sense of the arts, I suppose, but mostly history.\" He slows and points to a massive piece, a wooden tiger mauling a man dressed as a European soldier, the sign declaring: _TIPU'S TIGER._ \"Mum would take us to look at this one and whisper to me, 'See how the tiger is eating him up? That's because my great-great-great-great grandad _stole_ this from India. I think we should give it back, but your gran says no.'\"\n\nAlex watches Henry's face in quarter profile, the slight pain that moves under his skin, but he shakes it off quickly and takes Alex's hand back up. They're running again.\n\n\"Now, I like to come at night,\" he says. \"A few of the higher-up security guards know me. Sometimes I think I keep coming because, no matter how many places I've been or people I've met or books I read, this place is proof I'll never learn it all. It's like Westminster: You can look at every individual carving or pane of stained glass and know there's this wealth of stories there, that everything was put in a specific place for a reason. Everything has a meaning, an intention. There are pieces in here\u2014 _The Great Bed of Ware,_ it's mentioned in _Twelfth Night, Epicoene, Don Juan,_ and it's here. Everything is a story, never finished. Isn't it incredible? And the archives, God, I could spend hours in the archives, they\u2014 _mmph._ \"\n\nHe's cut off mid-sentence because Alex has stopped in the middle of the corridor and yanked him backward into a kiss.\n\n\"Hello,\" Henry says when they break apart. \"What was that for?\"\n\n\"I just, like.\" Alex shrugs. \"Really love you.\"\n\nThe corridor dumps them out into a cavernous atrium, rooms sprawling out in each direction. Only some of the overhead lighting has been left on, and Alex can see an enormous chandelier looming high in the rotunda, tendrils and bubbles of glass in blues and greens and yellows. Behind it, there's an elaborate iron choir screen standing broad and gorgeous on the landing above.\n\n\"This is it,\" Henry says, pulling Alex by the hand to the left, where light spills out of an immense archway. \"I called ahead to Gavin to make sure they left a light on. It's my favorite room.\"\n\nAlex has personally helped with exhibitions at the Smithsonian and sleeps in a room once occupied by Ulysses S. Grant's father-in-law, but he still loses his breath when Henry pulls him through the marble pillars.\n\nIn the half light, the room is alive. The vaulted roof seems to stretch up forever into the inky London sky, and beneath it the room is arranged like a city square somewhere in Florence, climbing columns and towering altars and archways. Deep basins of fountains are planted in the floor between statues on heavy pedestals, and effigies lie behind black doorways with the Resurrection carved into their slate. Dominating the entire back wall is a colossal, Gothic choir screen carved from marble and adorned with ornate statues of saints, black and gold and imposing, holy.\n\nWhen Henry speaks again, it's soft, as if he's trying not to break the spell.\n\n\"In here, at night, it's almost like walking through a real piazza,\" Henry says. \"But there's nobody else around to touch you or gawk at you or try to steal a photo of you. You can just _be._ \"\n\nAlex looks over to find Henry's expression careful, waiting, and he realizes this is the same as when Alex took Henry to the lake house\u2014the most sacred place he has.\n\nHe squeezes Henry's hand and says, \"Tell me everything.\"\n\nHenry does, leading him around to each piece in turn. There's a life-size sculpture of Zephyr, the Greek god of the west wind brought to life by Francavilla, a crown on his head and one foot on a cloud. Narcissus on his knees, mesmerized by his own reflection in the pool, once thought to be Michelangelo's lost Cupid but actually carved by Cioli\u2014\"Do you see here, where they had to repair his knuckles with stucco?\"\u2014Pluto stealing Proserpina away to the underworld, and Jason with his golden fleece.\n\nThey wind up back at the first statue, _Samson Slaying a Philistine_ , the one that knocked the wind out of Alex when they walked in. He's never seen anything like it\u2014the smooth muscles, the indentations of flesh, the breathing, bleeding life of it, all carved by Giambologna out of marble. If he could touch it, he swears the skin would be warm.\n\n\"It's a bit ironic, you know,\" Henry says, gazing up at it. \"Me, the cursed gay heir, standing here in Victoria's museum, considering how much she _loved_ those sodomy laws.\" He smirks. \"Actually... you remember how I told you about the gay king, James I?\"\n\n\"The one with the dumb jock boyfriend?\"\n\n\"Yes, that one. Well, his most beloved favorite was a man named George Villiers. 'The handsomest-bodied man in all of England,' they called him. James was completely besotted. Everyone knew. This French poet, de Viau, wrote a poem about it.\" He clears his throat and starts to recite: \"'One man fucks Monsieur le Grand, another fucks the Comte de Tonnerre, and it is well known that the King of England, fucks the Duke of Buckingham.'\" Alex must be staring, because he adds, \"Well, it rhymes in French. Anyway. Did you know the reason the King James translation of the Bible exists is because the Church of England was so displeased with James for flaunting his relationship with Villiers that he had the translation commissioned to appease them?\"\n\n\"You're kidding.\"\n\n\"He stood in front of the Privy Council and said, 'Christ had John, and I have George.'\"\n\n\"Jesus.\"\n\n\"Precisely.\" Henry's still looking up at the statue, but Alex can't stop looking at him and the sly smile on his face, lost in his own thoughts. \"And James's son, Charles I, is the reason we have dear Samson. It's the only Giambologna that ever left Florence. He was a gift to Charles from the King of Spain, and Charles gave it, this massive, absolutely priceless masterpiece of a sculpture, to Villiers. And a few centuries later, here he is. One of the most beautiful pieces we own, and we didn't even steal it. We only needed Villiers and his trolloping ways with the queer monarchs. To me, if there were a registry of national gay landmarks in Britain, Samson would be on it.\"\n\nHenry's beaming like a proud parent, like Samson is his, and Alex is hit with a wave of pride in kind.\n\nHe takes his phone out and lines up a shot, Henry standing there all soft and rumpled and smiling next to one of the most exquisite works of art in the world.\n\n\"What are you doing?\"\n\n\"I'm taking a picture of a national gay landmark,\" Alex tells him. \"And also a statue.\"\n\nHenry laughs indulgently, and Alex closes the space between them, takes Henry's baseball cap off and stands on his toes to kiss the ridge of his brow.\n\n\"It's funny,\" Henry says. \"I always thought of the whole thing as the most unforgivable thing about me, but you act like it's one of the best.\"\n\n\"Oh, yeah,\" Alex says. \"The top list of reasons to love you goes brain, then dick, then imminent status as a revolutionary gay icon.\"\n\n\"You are quite literally Queen Victoria's worst nightmare.\"\n\n\"And that's why _you_ love _me._ \"\n\n\"My God, you're right. All this time, I was just after the bloke who'd most infuriate my homophobic forebears.\"\n\n\"Ah, and we can't forget they were also racist.\"\n\n\"Certainly not.\" Henry nods seriously. \"Next time we shall visit some of the George III pieces and see if they burst into flame.\"\n\nThrough the marble choir screen at the back of the room is a second, deeper chamber, this one filled with church relics. Past stained glass and statues of saints, at the very end of the room, is an entire high altar chapel removed from its church. The sign explains its original setting was the apse of the convent church of Santa Chiara in Florence in the fifteenth century, and it's stunning, set deep into an alcove to create a real chapel, with statues of Santa Chiara and Saint Francis of Assisi.\n\n\"When I was younger,\" Henry says, \"I had this very elaborate idea of taking somebody I loved here and standing inside the chapel, that he'd love it as much as I did, and we'd slow dance right in front of the Blessed Mother. Just a... daft pubescent fantasy.\"\n\nHenry hesitates, before finally sliding his phone out of his pocket. He presses a few buttons and extends a hand to Alex, and, quietly, \"Your Song\" starts to play from the tiny speaker.\n\nAlex exhales a laugh. \"Aren't you gonna ask if I know how to waltz?\"\n\n\"No waltzing,\" Henry says. \"Never cared for it.\"\n\nAlex takes his hand, and Henry turns to face the chapel like a nervous postulant, his cheeks hollowed out in the low light, before pulling Alex into it.\n\nWhen they kiss, Alex can hear a half-remembered old proverb from catechism, mixed up between translations of the book: \"Come, hijo m\u00edo, de la miel, porque es buena, and the honeycomb, sweet to thy taste.\" He wonders what Santa Chiara would think of them, a lost David and Jonathan, turning slowly on the spot.\n\nHe brings Henry's hand to his mouth and kisses the little knob of his knuckle, the skin over the blue vein there, bloodlines, pulses, the old blood kept in perpetuity within these walls, and he thinks, _Father, Son, and Holy Spirit, amen._\n\n* * *\n\nHenry charters a private plane to get him back home, and Alex is dreading the dressing-down he's going to get the minute he's stateside, but he's trying not to think about it. At the airstrip, the wind whipping his hair across his forehead, Henry fishes inside his jacket for something.\n\n\"Listen,\" he says, pulling a curled fist out of his pocket. He takes one of Alex's hands and turns it to press something small and heavy into his palm. \"I want you to know, I'm sure. A thousand percent.\"\n\nHe removes his hand and there, sitting in the center of Alex's callused palm, is the signet ring.\n\n\"What?\" Alex's eyes flash up to search Henry's face and find him smiling softly. \"I can't\u2014\"\n\n\"Keep it,\" Henry tells him. \"I'm sick of wearing it.\"\n\nIt's a private airstrip, but it's still risky, so he folds Henry in a hug and whispers fiercely, \"I completely fucking love you.\"\n\nAt cruising altitude, he takes the chain off his neck and slides the ring on next to the old house key. They clink together gently as he tucks them both under his shirt, two homes side by side.\n\n# ELEVEN\n\n> Hometown stuff\n> \n> * * *\n> \n> A <agcd@eclare45.com> 9\/2\/20 5:12 PM\n> \n> to Henry\n> \n> H,\n> \n> Have been home for three hours. Already miss you. This is some bullshit.\n> \n> Hey, have I told you lately that you're brave? I still remember what you said to that little girl in the hospital about Luke Skywalker: \"He's proof that it doesn't matter where you come from or who your family is.\" Sweetheart, you're proof too.\n> \n> (By the way, in this relationship, I am absolutely the Han and you are absolutely the Leia. Don't try to argue because you'll be wrong.)\n> \n> I was also thinking about Texas again, which I guess I do a lot when I'm stressed about election stuff. There's so much stuff I haven't shown you yet. We haven't even done Austin! I wanna take you to Franklin Barbecue. You have to wait in line for hours, but that's part of the experience. I really wanna see a member of the royal family wait in line for hours to eat cow parts.\n> \n> Have you thought any more about what you said before I left? About coming out to your family? Obviously, you're not obligated. You just seemed kind of hopeful when you talked about it.\n> \n> I'll be over here, still quarantined in the White House (at least Mom didn't kill me for London), rooting for you.\n> \n> Love you.\n> \n> xoxoxoxoxo\n> \n> A\n> \n> P.S. Vita Sackville-West to Virginia Woolf\u20141927:\n> \n> With me it is quite stark: I miss you even more than I could have believed; and I was prepared to miss you a good deal.\n> \n> Re: Hometown stuff\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 9\/3\/20 2:49 AM\n> \n> to A\n> \n> Alex,\n> \n> It is, indeed, bullshit. It's all I can do not to pack a bag and be gone forever. Perhaps I could live in your room like a recluse. You could have food sent up for me, and I'll be lurking in disguise in a shadowy corner when you answer the door. It'll all be very dreadfully Jane Eyre.\n> \n> The Mail will write mad speculations about where I've gone, if I've offed myself or vanished to St. Kilda, but only you and I will know that I'm just sprawled in your bed, reading books and feeding myself profiteroles and making love to you endlessly until we both expire in a haze of chocolate sauce. It's how I'd want to go.\n> \n> I'm afraid, though, I'm stuck here. Gran keeps asking Mum when I'm going to enlist, and did I know Philip had already served a year by the time he was my age. I do need to figure out what I'm going to do, because I'm certainly closing in on the end of what's an acceptable amount of time for a gap year. Please do keep me in your\u2014what is it American politicians say?\u2014thoughts and prayers.\n> \n> Austin sounds brilliant. Maybe in a few months, after things settle down a bit? I could take a long weekend. Can we visit your mum's house? Your room? Do you still have your lacrosse trophies? Tell me you still have posters up. Let me guess: Han Solo, Barack Obama, and... Ruth Bader Ginsburg.\n> \n> (I'll agree with your assessment that you're the Han to my Leia in that you are, without doubt, a scruffy-looking nerf herder who would pilot us into an asteroid field. I happen to like nice men.)\n> \n> I have thought more about coming out to my family, which is part of why I'm staying here for now. Bea has offered to be there when I tell Philip if I want, so I think I will. Again, thoughts and prayers.\n> \n> I love you terribly, and I want you back here soon. I need your help picking a new bed for my room; I've decided to get rid of that gold monstrosity.\n> \n> Yours,\n> \n> Henry\n> \n> P.S. From Radclyffe Hall to Evguenia Souline, 1934:\n> \n> Darling\u2014I wonder if you realize how much I am counting on your coming to England, how much it means to me\u2014it means all the world, and indeed my body shall be all, all yours, as yours will be all, all mine, beloved.... And nothing will matter but just we two, we two longing loves at last come together.\n> \n> Re: Hometown stuff\n> \n> * * *\n> \n> A <agcd@eclare45.com> 9\/3\/20 6:20 AM\n> \n> to Henry\n> \n> H,\n> \n> Shit. Do you think you're going to enlist? I haven't done any research on it yet. I'm gonna ask Zahra to have one of our people put together a binder on it. What would that mean? Would you have to be gone a lot? Would it be dangerous??? Or is it just like, wear the uniform and sit at a desk? How did we not talk about this when I was there?????\n> \n> Sorry. I'm panicking. I somehow forgot this was a thing looming on the horizon. I'm there for whatever you decide you want to do, just, like, let me know if I need to start practicing gazing wistfully out the window, waiting for my love to return from the war.\n> \n> It drives me nuts sometimes that you don't get to have more say in your life. When I picture you happy, I see you with your own apartment somewhere outside of the palace and a desk where you can write anthologies of queer history. And I'm there, using up your shampoo and making you come to the grocery store with me and waking up in the same damn time zone with you every morning.\n> \n> When the election is over, we can figure out what we'll do next. I would love to be in the same place for a bit, but I know you have to do what you have to do. Just know, I believe in you.\n> \n> Re: telling Philip, sounds like a great plan. If all else fails, just do what I did and act like a huge jackass until most of your family figures it out on their own.\n> \n> Love you. Tell Bea hi.\n> \n> A\n> \n> P.S. Eleanor Roosevelt to Lorena Hickock\u20141933:\n> \n> I miss you greatly dear. The nicest time of the day is when I write to you. You have a stormier time than I do but I miss you as much, I think.... Please keep most of your heart in Washington as long as I'm here for most of mine is with you!\n> \n> Re: Hometown stuff\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 9\/4\/20 7:58 PM\n> \n> to A\n> \n> Alex,\n> \n> Have you ever had something go so horribly, horribly, unbelievably badly that you'd like to be loaded into a cannon and jettisoned into the merciless black maw of outer space?\n> \n> I wonder sometimes what is the point of me, or anything. I should have just packed a bag like I said. I could be in your bed, languishing away until I perish, fat and sexually conquered, snuffed out in the spring of my youth. Here lies Prince Henry of Wales. He died as he lived: avoiding plans and sucking cock.\n> \n> I told Philip. Not about you, precisely\u2014about me.\n> \n> Specifically, we were discussing enlistment, Philip and Shaan and I, and I told Philip I'd rather not follow the traditional path and that I hardly think I'd be useful to anyone in the military. He asked why I was so intent on disrespecting the traditions of the men of this family, and I truly think I dissociated straight (ha) out of the conversation, because I opened my blasted mouth and said, \"Because I'm not like the rest of the men of this family, beginning with the fact that I am very deeply gay, Philip.\"\n> \n> Once Shaan managed to dislodge him from the chandelier, Philip had quite a few words for me, some of which were \"confused or misguided\" and \"ensuring the perpetuity of the bloodline\" and \"respecting the legacy.\" Honestly, I don't recall much of it. Essentially, I gathered that he was not surprised to discover I am not the heterosexual heir I'm supposed to be, but rather surprised that I do not intend to keep pretending to be the heterosexual heir I'm supposed to be.\n> \n> So, yes, I know we discussed and hoped that coming out to my family would be a good first step. I cannot say this was an encouraging sign re: our odds of going public. I don't know. I've eaten a tremendous amount of Jaffa Cakes about it, to be frank.\n> \n> Sometimes I imagine moving to New York to take over launching Pez's youth shelter there. Just leaving. Not coming back. Maybe burning something down on the way out. It would be nice.\n> \n> Here's an idea: Do you know, I've realised I've never actually told you what I thought the first time we met?\n> \n> You see, for me, memories are difficult. Very often, they hurt. A curious thing about grief is the way it takes your entire life, all those foundational years that made you who you are, and makes them so painful to look back upon because of the absence there, that suddenly they're inaccessible. You must invent an entirely new system.\n> \n> I started to think of myself and my life and my whole lifetime worth of memories as all the dark, dusty rooms of Buckingham Palace. I took the night Bea left rehab and I begged her to take it seriously, and I put it in a room with pink peonies on the wallpaper and a golden harp in the center of the floor. I took my first time, with one of my brother's mates from uni when I was seventeen, and I found the smallest, most cramped little broom cupboard I could muster, and I shoved it in. I took my father's last night, the way his face went slack, the smell of his hands, the fever, the waiting and waiting and terrible waiting and the even worse not-waiting anymore, and I found the biggest room, a ballroom, wide open and dark, windows drawn and covered. Locked the doors.\n> \n> But the first time I saw you. Rio. I took that down to the gardens. I pressed it into the leaves of a silver maple and recited it to the Waterloo Vase. It didn't fit in any rooms.\n> \n> You were talking with Nora and June, happy and animated and fully alive, a person living in dimensions I couldn't access, and so beautiful. Your hair was longer then. You weren't even a president's son yet, but you weren't afraid. You had a yellow ip\u00ea-amarelo in your pocket.\n> \n> I thought, this is the most incredible thing I have ever seen, and I had better keep it a safe distance away from me. I thought, if someone like that ever loved me, it would set me on fire.\n> \n> And then I was a careless fool, and I fell in love with you anyway. When you rang me at truly shocking hours of the night, I loved you. When you kissed me in disgusting public toilets and pouted in hotel bars and made me happy in ways in which it had never even occurred to me that a mangled-up, locked-up person like me could be happy, I loved you.\n> \n> And then, inexplicably, you had the absolute audacity to love me back. Can you believe it?\n> \n> Sometimes, even now, I still can't.\n> \n> I'm sorry things didn't go better with Philip. I wish I could send hope.\n> \n> Yours,\n> \n> Henry\n> \n> P.S. From Michelangelo to Tommaso Cavalieri, 1533:\n> \n> I know well that, at this hour, I could as easily forget your name as the food by which I live; nay, it were easier to forget the food, which only nourishes my body miserably, than your name, which nourishes both body and soul, filling the one and the other with such sweetness that neither weariness nor fear of death is felt by me while memory preserves you to my mind. Think, if the eyes could also enjoy their portion, in what condition I should find myself.\n> \n> Re: Hometown stuff\n> \n> * * *\n> \n> A <agcd@eclare45.com> 9\/4\/20 8:31 PM\n> \n> to Henry\n> \n> H,\n> \n> Fuck.\n> \n> I'm so sorry. I don't know what else to say. I'm so sorry. June and Nora send their love. Not as much love as me. Obviously.\n> \n> Please don't worry about me. We'll figure it out. It just might take time. I've been working on patience. I've picked up all kinds of things from you.\n> \n> God, what can I possibly write to make this better?\n> \n> Here: I can't decide if your emails make me miss you more or less. Sometimes I feel like a funny-looking rock in the middle of the most beautiful clear ocean when I read the kinds of things you write to me. You love so much bigger than yourself, bigger than everything. I can't believe how lucky I am to even witness it\u2014to be the one who gets to have it, and so much of it, is beyond luck and feels like fate. Catholic God made me to be the person you write those things about. I'll say five Hail Marys. Muchas gracias, Santa Maria.\n> \n> I can't match you for prose, but what I can do is write you a list.\n> \n> AN INCOMPLETE LIST: THINGS I LOVE ABOUT HRH PRINCE HENRY OF WALES\n> \n> 1. The sound of your laugh when I piss you off.\n> \n> 2. The way you smell underneath your fancy cologne, like clean linens but somehow also fresh grass (what kind of magic is this?).\n> \n> 3. That thing you do where you stick out your chin to try to look tough.\n> \n> 4. How your hands look when you play piano.\n> \n> 5. All the things I understand about myself now because of you.\n> \n> 6. How you think Return of the Jedi is the best Star Wars (wrong) because deep down you're a gigantic, sappy, embarrassing romantic who just wants the happily ever after.\n> \n> 7. Your ability to recite Keats.\n> \n> 8. Your ability to recite Bernadette's \"Don't let it drag you down\" monologue from Priscilla, Queen of the Desert.\n> \n> 9. How hard you try.\n> \n> 10. How hard you've always tried.\n> \n> 11. How determined you are to keep trying.\n> \n> 12. That when your shoulders cover mine, nothing else in the entire stupid world matters.\n> \n> 13. The goddamn issue of Le Monde you brought back to London with you and kept and have on your nightstand (yes, I saw it).\n> \n> 14. The way you look when you first wake up.\n> \n> 15. Your shoulder-to-waist ratio.\n> \n> 16. Your huge, generous, ridiculous, indestructible heart.\n> \n> 17. Your equally huge dick.\n> \n> 18. The face you just made when you read that last one.\n> \n> 19. The way you look when you first wake up (I know I already said this, but I really, really love it).\n> \n> 20. The fact that you loved me all along.\n> \n> I keep thinking about that last one ever since you told me, and what an idiot I was. It's so hard for me to get out of my own head sometimes, but now I'm coming back to what I said to you the night in my room when it all started, and how I brushed you off when you offered to let me go after the DNC, how I used to try to act like it was nothing sometimes. I didn't even know what you were offering to do to yourself. God, I want to fight everyone who's ever hurt you, but it was me too, wasn't it? All that time. I'm so sorry.\n> \n> Please stay gorgeous and strong and unbelievable. I miss you I miss you I miss you I love you. I'm calling you as soon as I send this, but I know you like to have these things written down.\n> \n> A\n> \n> P.S. Richard Wagner to Eliza Wille, re: Ludwig II\u20131864 (Remember when you played Wagner for me? He's an asshole, but this is something.)\n> \n> It is true that I have my young king who genuinely adores me. You cannot form an idea of our relations. I recall one of the dreams of my youth. I once dreamed that Shakespeare was alive: that I really saw and spoke to him: I can never forget the impression that dream made on me. Then I would have wished to see Beethoven, though he was already dead. Something of the same kind must pass in the mind of this lovable man when with me. He says he can hardly believe that he really possesses me. None can read without astonishment, without enchantment, the letters he writes to me.\n\n# TWELVE\n\nThere's a diamond ring on Zahra's finger when she shows up with her coffee thermos and a thick stack of files. They're in June's room, scarfing down breakfast before Zahra and June leave for a rally in Pittsburgh, and June drops her waffle on the bedspread.\n\n\"Oh my God, Z, what is _that_? Did you get _engaged_?\"\n\nZahra looks down at the ring and shrugs. \"I had the weekend off.\"\n\nJune gapes at her.\n\n\"When are you going to tell us who you're dating?\" Alex asks. \"Also, _how_?\"\n\n\"Uh-uh, nope,\" she says. \" _You_ don't get to say shit to me about secret relationships in and around this campaign, princess.\"\n\n\"Point,\" Alex concedes.\n\nShe brushes past the topic as June starts wiping syrup off the bed with her pajama pants. \"We've got a lot of ground to cover this morning, so focus up, little Claremonts.\"\n\nShe's got detailed agendas for each of them, bullet-pointed and double-sided, and she dives right in. They're already on Thursday's voter registration drive in Cedar Rapids (Alex is pointedly not invited) when her phone pings with a notification. She picks it up, scrolling through the screen offhandedly.\n\n\"So I need both of you to be dressed and ready... by...\" She's looking more closely at the screen, distracted. \"By, uh...\" Her face is taken over with a horrified gasp. \"Oh, _fuck my ass._ \"\n\n\"What\u2014?\" Alex starts, but his own phone buzzes in his lap, and he looks down to find a push notification from CNN: LEAKED SURVEILLANCE FOOTAGE SHOWS PRINCE HENRY AT DNC HOTEL.\n\n\"Oh, shit,\" Alex says.\n\nJune reads over his shoulder; somehow, some \"anonymous source\" got the security camera footage from the lobby of the Beekman that night of the DNC.\n\nIt's not... explicitly damning, but it very clearly does show the two of them walking out of the bar together, shoulder to shoulder, flanked by Cash, and it cuts to footage from the elevator, Henry's arm around Alex's waist while they talk with Cash. It ends with the three of them getting off together at the top floor.\n\nZahra looks up at him, practically murderous. \"Can you explain to me why this one day of our lives will not stop haunting me?\"\n\n\"I don't know,\" Alex says miserably. \"I can't believe this is the one that's\u2014I mean, we've done riskier things than this\u2014\"\n\n\"That's supposed to make me feel better _how_?\"\n\n\"I just mean, like, who is leaking fucking elevator tapes? Who's checking for that? It's not like Solange was in there\u2014\"\n\nA chirp from June's phone interrupts him, and she swears when she looks at it. \"Jesus, that _Post_ reporter just texted to ask for a comment on the speculation surrounding your relationship with Henry and whether it\u2014whether it has to do with you leaving the campaign after the DNC.\" She looks between Alex and Zahra, eyes wide. \"This is really bad, isn't it?\"\n\n\"It ain't great,\" Zahra says. She's got her nose buried in her phone, furiously typing out what are probably very strongly worded emails to the press team. \"What we need is a fucking diversion. We have to\u2014to send you on a date or something.\"\n\n\"What if we\u2014\" June attempts.\n\n\"Or, fuck, send _him_ on a date,\" Zahra says. \"Send you _both_ on dates.\"\n\n\"I could\u2014\" June tries again.\n\n\"Who the fuck do I call? What girl is gonna want to wade into this shitstorm to fake date either of you at this point?\" Zahra grinds the heels of both hands against her eyes. \"Jesus, be a gay beard.\"\n\n\"I have an idea!\" June finally half shouts. When they both look at her, she's biting her lip, looking at Alex. \"But I don't know if you're gonna like it.\"\n\nShe turns her phone around to show them the screen. It's a photo he recognizes as one of the ones they took for Pez in Texas, June and Henry lounging on the dock together. She's cropped Nora out so it's just the two of them, Henry sporting a wide, teasing grin under his sunglasses and June planting a kiss on his cheek.\n\n\"I was on that floor too,\" she says. \"We don't have to, like, confirm or deny anything. But we can imply something. Just to take the heat off.\"\n\nAlex swallows.\n\nHe's always known June was one inch from taking a bullet for him, but this? He would never ask her to do this.\n\nBut the thing is... it would work. Their social media friendship is well documented, even if half of it is GIFs of Colin Firth. Out of context, the photo looks as couple-y as anything, like a nice, gorgeous, heterosexual couple on vacation together. He looks over to Zahra.\n\n\"It's not a bad idea,\" Zahra says. \"We'd have to get Henry on board. Can you do that?\"\n\nAlex releases a breath. He absolutely doesn't want this, but he's also not sure what other choice he has. \"Um. Yeah, I. Yeah, I think so.\"\n\n* * *\n\n\"This is kind of exactly what we said we didn't want to do,\" Alex says into his phone.\n\n\"I know,\" Henry tells him across the line. His voice is shaky. Philip is waiting on Henry's other line. \"But.\"\n\n\"Yeah,\" Alex says. \"But.\"\n\nJune posts the picture from Texas, and it immediately burns through her stats to become her new most-liked post.\n\nWithin hours, it's everywhere. _BuzzFeed_ puts up a comprehensive guide to Henry and June's relationship, leading off with that goddamn photo of them dancing at the royal wedding. They dig up photos from the night in LA, analyze Twitter interactions. \"Just when you thought June Claremont-Diaz couldn't get any more #goals,\" one article writes, \"has she secretly had her own Prince Charming all along?\" Another one speculates, \"Did HRH's best friend Alex introduce them?\"\n\nJune's relieved, only because she managed to find a way to protect him, even though it means the world is digging through _her_ life for answers and evidence, which makes Alex want to murder everyone. He also wants to grab people by the shoulders and shake them and tell them Henry is _his,_ you idiots, even though the whole point of this was for it to be believable. He shouldn't feel wronged deep in his gut. But that everyone seems enamored, when the only difference between the lie and the truth that would burn up Fox News is the gender involved... well, it fucking stings.\n\nHenry is quiet. He says enough for Alex to glean that Philip is apoplectic and Her Majesty is annoyed but pleased Henry has finally found himself a girlfriend. Alex feels horrible about it. The stifling orders, pretending to be someone he's not\u2014Alex has always tried to be a refuge for Henry from it all. It was never supposed to come from his side too.\n\nIt's bad. It's stomach-cramps, walls-closing-in, no-plan-B-if-this-fails bad. He was in London barely two weeks ago, kissing Henry in front of a Giambologna. Now, this.\n\nThere's another piece in their back pocket that'll sell it. The only relationship in his life that can get more mileage than any of this. Nora comes to him at the Residence wearing bright red lipstick and presses cool, patient fingers against his temples and says, \"Take me on a date.\"\n\nThey choose a college neighborhood full of people who'll sneak shots on their phones and post them everywhere. Nora slides her hand into his back pocket, and he tries to focus on the comfort of her physical presence against his side, the familiar frizz of her curls against his cheek.\n\nFor half a second, he allows a small part of him to think about how much easier things would be if this were the truth: sliding back into comfortable, easy harmony with his best friend, leaving greasy fingerprints along her waistline outside Jumbo Slice, laughing at her crass jokes. If he could love her like people wanted him to, and she loved him, and there wasn't any more to it than that.\n\nBut she doesn't, and he can't, and his heart is on a plane over the Atlantic right now, coming to DC to seal the deal over a well-photographed lunch with June the next day. Zahra sends him an email full of Twitter threads about him and Nora that night when he's in bed, and he feels sick.\n\nHenry lands in the middle of the night and isn't even allowed to come near the Residence, instead sequestered in a hotel across town. He sounds exhausted when he calls in the morning, and Alex holds the phone close and promises he'll try to find a way to see him before he flies back out.\n\n\"Please,\" Henry says, paper-thin.\n\nHis mother, the rest of the administration, and half of the press at this point are caught up for the day dealing with news of a North Korean missile test; nobody notices when June lets him climb into her SUV with her that morning. June holds onto his elbow and makes half-hearted jokes, and when they pull up a block from the cafe, she offers him an apologetic smile.\n\n\"I'll tell him you're here,\" she says. \"If nothing else, maybe that'll make it a little easier for him.\"\n\n\"Thanks,\" he says. Before she opens the door to leave, he catches her by the wrist and says, \"Seriously. Thank you.\"\n\nShe gives his hand a squeeze, and she and Amy are gone, and he's alone in a tiny, secluded alleyway with the second car of backup security and a twisted-up feeling in his stomach.\n\nIt takes all of an hour before June texts him, All done, followed by, Bringing him to you.\n\nThey worked it out before they left: Amy brings June and Henry back to the alley, they have him swap cars like a political prisoner. Alex leans forward to the two agents sitting silently in the front seats. He doesn't know if they've figured out what this really is yet, and he honestly doesn't care.\n\n\"Hey, can I have a minute?\"\n\nThey exchange a look but get out, and a minute later, there's another car alongside him and the door is opening, and he's there. Henry, looking tense and unhappy, but within arm's reach.\n\nAlex pulls him in by the shoulder on instinct, the door shutting behind him. He holds him there, and this close he can see the faint gray tinge to Henry's complexion, the way his eyes aren't connecting. It's the worst he's ever seen him, worse than a violent fit or the verge of tears. He looks hollowed-out, vacant.\n\n\"Hey,\" Alex says. Henry's gaze is still unfocused, and Alex shifts toward the middle of the seat and into his line of vision. \"Hey. Look at me. Hey. I'm right here.\"\n\nHenry's hands are shaking, his breaths coming shallow, and Alex knows the signs, the low hum of an impending panic attack. He reaches down and wraps his hands around one of Henry's wrists, feeling the racing pulse under his thumbs.\n\nHenry finally meets his eyes. \"I hate it,\" he says. \"I _hate_ this.\"\n\n\"I know,\" Alex says.\n\n\"It was... _tolerable_ before, somehow,\" Henry says. \"When there was never\u2014never the possibility of anything else. But, Christ, this is\u2014it's _vile._ It's a bloody farce. And June and Nora, what, they just get to be _used_? Gran wanted me to bring my own photographers for this. Did you know that?\" He inhales, and it gets caught in his throat and shudders violently on the way back out. \"Alex. I don't want to _do_ this.\"\n\n\"I know,\" Alex tells him again, reaching up to smooth out Henry's brow with the pad of his thumb. \"I know. I hate it too.\"\n\n\"It's not fucking _fair_!\" he goes on, his voice nearly breaking. \"My shit ancestors walked around doing a thousand times worse than any of this, and nobody _cared_!\"\n\n\" _Baby,_ \" Alex says, moving his hand to Henry's chin to bring him back down. \"I know. I'm so sorry, babe. But it won't be like this forever, okay? I promise.\"\n\nHenry closes his eyes and exhales through his nose. \"I want to believe you. I do. But I'm so afraid I'll never be allowed.\"\n\nAlex wants to go to war for this man, wants to get his hands on everything and everyone that ever hurt him, but for once, he's trying to be the steady one. So he rubs the side of Henry's neck gently until his eyes drift back open, and he smiles softly, tipping their foreheads together.\n\n\"Hey,\" he says. \"I'm not gonna let that happen. Listen, I'm telling you right now, I will physically fight your grandmother myself if I have to, okay? And, like, she's old. I know I can take her.\"\n\n\"I wouldn't be so cocky,\" Henry says with a small laugh. \"She's full of dark surprises.\"\n\nAlex laughs, cuffing him on the shoulder.\n\n\"Seriously,\" he says. Henry's looking back at him, beautiful and vital and heartsick and still, always, the person Alex is willing to risk ruining his life for. \"I hate this so much. I know. But we're gonna do it together. And we're gonna make it work. You and me and history, remember? We're just gonna fucking fight. Because you're it, okay? I'm never gonna love anybody in the world like I love you. So, I promise you, one day we'll be able to just _be,_ and fuck everyone else.\"\n\nHe pulls Henry in by the nape of his neck and kisses him hard, Henry's knee knocking against the center console as his hands move up to Alex's face. Even though the windows are tinted black, it's the closest they've ever come to kissing in public, and Alex knows it's reckless, but all he can think is a supercut of other people's letters they've quietly sent to each other. Words that went down in history. \"Meet you in every dream... Keep most of your heart in Washington... Miss you like a home... We two longing loves... My young king.\"\n\n_One day,_ he tells himself. _One day, us too._\n\n* * *\n\nThe anxiety feels like buzzing little wings in his ear in the silence, like a petulant wasp. It catches him when he tries to sleep and startles him awake, follows him on laps paced up and down the floors of the Residence. It's getting harder to brush off the feeling he's being watched.\n\nThe worst part is that there's no end in sight. They'll definitely have to keep it up at least until the election is over, and even then, there's the always looming possibility of the queen outright forbidding it. His idealistic streak won't let him fully accept it, but that doesn't mean it isn't there.\n\nHe keeps waking up in DC, and Henry keeps waking up in London, and the whole world keeps waking up to talk about the two of them in love with other people. Pictures of Nora's hand in his. Speculation about whether June will get an official announcement of royal courtship. And the two of them, Henry and Alex, like the world's worst illustration of the _Symposium_ : split down the middle and sent bleeding into separate lives.\n\nEven that thought depresses him because Henry's the only reason he's become a person who cites Plato. Henry and his classics. Henry in his palace, in love, in misery, not talking much anymore.\n\nEven with both of them trying as hard as they are, it's impossible to feel like it's not pulling them apart. The whole charade takes and takes from them, takes days that were sacred\u2014the night in LA, the weekend at the lake, the missed chance in Rio\u2014and records over the tape with something more palatable. The narrative: two fresh-faced young men who love two beautiful young women and definitely not ever each other.\n\nHe doesn't want Henry to know. Henry has a hard enough time as it is, looked at sideways by his whole family, Philip who knows and has not been kind. He tries to sound calm and whole over the phone when they talk, but he doesn't think it's convincing.\n\nWhen he was younger and the anxiety got this bad, when the stakes in his life were much, much lower, this would be the point of self-destruction. If he were in California, he'd sneak the jeep out and drive way too fast down the 101, doors off, blasting N.W.A., inches from being painted on the pavement. In Texas, he'd steal a bottle of Maker's from the liquor cabinet and get wasted with half the lacrosse team and maybe, afterward, climb through Liam's window and hope to forget by morning.\n\nThe first debate is in a matter of weeks. He doesn't even have work to keep him busy, so he stews and stresses and goes for long, punishing runs until he has the satisfaction of blisters. He wants to set himself on fire, but he can't afford for anyone to see him burn.\n\nHe's returning a box of borrowed files to his dad's office in the Dirksen Building after hours when he hears the faint sound of Muddy Waters from the floor above, and it hits him. There's one person he can burn down instead.\n\nHe finds Rafael Luna hunched at his office's open window, sucking down a cigarette. There are two empty, crumpled packs of Marlboros next to a lighter and an overflowing ashtray on the sill. When he turns around at the slam of the door, he coughs out a startled cloud of smoke.\n\n\"Those things are gonna fucking kill you,\" Alex says. He said the same thing about five hundred times that summer in Denver, but now he means, _I kinda wish they would._\n\n\"Kid\u2014\"\n\n\" _Don't_ call me that.\"\n\nLuna turns, stubbing out his cigarette in the ashtray, and Alex can see a muscle clenching in his jaw. As handsome as he always is, he looks like shit. \"You shouldn't be here.\"\n\n\"No shit,\" Alex says. \"I just wanted to see if you would have the balls to actually talk to me.\"\n\n\"You do realize you're talking to a United States senator,\" he says placidly.\n\n\"Yeah, big fucking man,\" Alex says. He's advancing on Luna now, kicking a chair out of the way. \"Important fucking job. Hey, how 'bout you tell me how you're serving the people who voted for you by being Jeffrey Richards's chickenshit little sellout?\"\n\n\"What the hell did you come here for, Alex, eh?\" Luna asks him, unmoved. \"You gonna fight me?\"\n\n\"I want you to tell me _why._ \"\n\nHis jaw clenches again. \"You wouldn't understand. You're\u2014\"\n\n\"I swear to God, if you say I'm too young, I'm gonna lose my shit.\"\n\n\"This isn't you losing your shit?\" Luna asks mildly, and the look that crosses Alex's face must be murderous because he immediately puts a hand up. \"Okay, bad timing. Look, I know. I know it seems shitty, but there's\u2014there are moving parts at work here that you can't even imagine. You know I'll always be indebted to your family for what you all have done for me, but\u2014\"\n\n\"I don't give a shit about what you _owe_ us. I _trusted_ you,\" he says. \"Don't condescend to me. You know as much as anyone what I'm capable of, what I've seen. If you told me, I would get it.\"\n\nHe's so close he's practically breathing Luna's reeking cigarette smoke, and when he looks into his face, there's a flicker of recognition at the bloodshot, blackened eyes and the gaunt cheekbones. It reminds him of how Henry looked in the back of the Secret Service car.\n\n\"Does Richards have something on you?\" he asks. \"Is he making you do this?\"\n\nLuna hesitates. \"I'm doing this because it's what needs to be done, Alex. It was my choice. Nobody else's.\"\n\n\"Then tell me why.\"\n\nLuna takes a deep breath and says, \" _No._ \"\n\nAlex imagines his fist in Luna's face and removes himself by two steps, out of range.\n\n\"You remember that night in Denver,\" he says, measured, his voice quavering, \"when we ordered pizza and you showed me pictures of all the kids you fought for in court? And we drank that nice bottle of scotch from the mayor of Boulder? I remember lying on the floor of your office, on the ugly-ass carpet, drunk off my ass, thinking, 'God, I hope I can be like him.' Because you were brave. Because you stood up for things. And I couldn't stop wondering how you had the nerve to get up and do what you do every day with everyone knowing what they know about you.\"\n\nBriefly, Alex thinks he's gotten through to Luna, from the way he closes his eyes and braces himself against the sill. But when he faces Alex again, his stare is hard.\n\n\"People don't know a damn thing about me. They don't know the half of it. And neither do you,\" he says. \"Jesus, Alex, please, don't be like me. Find another fucking role model.\"\n\nAlex, finally at his limit, lifts his chin and spits out, \"I already _am_ like you.\"\n\nIt hangs in the air between them, as physical as the kicked-over chair. Luna blinks. \"What are you saying?\"\n\n\"You know what I'm saying. I think you always knew, before I even did.\"\n\n\"You don't\u2014\" he says, stammering, trying to put it off. \"You're not like me.\"\n\nAlex levels his stare. \"Close enough. And you know what I mean.\"\n\n\"Okay, fine, kid,\" Luna finally snaps, \"you want me to be your fucking sherpa? Here's my advice: Don't tell anyone. Go find a nice girl and marry her. You're luckier than me\u2014you can do that, and it wouldn't even be a lie.\"\n\nAnd what comes out of Alex's mouth, comes so fast he has no chance to stop it, only divert it out of English at the last second in case it's overheard: \"Ser\u00eda una mentira, porque no ser\u00eda \u00e9l.\" It would be a lie, because it wouldn't be _him._\n\nHe knows immediately Raf has caught his meaning, because he takes a sharp step backward, his back hitting the sill again.\n\n\"You can't tell me this shit, Alex!\" he says, clawing inside his jacket until he finds and removes another pack of cigarettes. He shakes one out and fumbles with the lighter. \"What are you even _thinking_? I'm on the opponent's fucking campaign! I can't hear this! How can you possibly think you can be a politician like this?\"\n\n\"Who fucking decided that politics had to be about lying and hiding and being something you're not?\"\n\n\"It's _always_ been that, Alex!\"\n\n\"Since when did _you_ buy into it?\" Alex spits. \"You, me, my family, the people we run with\u2014we were gonna be the honest ones! I have absolutely zero interest in being a politician with some perfect veneer and two-point-five kids. Didn't we decide it was supposed to be about helping people? About the fight? What part of that is so fucking irreconcilable with letting people see who I really am? Who _you_ are, Raf?\"\n\n\"Alex, please. Please. Jesus Christ. You have to leave. I can't know this. You can't tell me this. You have to be more careful than this.\"\n\n\"God,\" Alex says, voice bitter, his hands on his hips. \"You know, it's worse than trust. I _believed_ in you.\"\n\n\"I know you did,\" Luna says. He's not even looking at Alex anymore. \"I wish you hadn't. Now, I need you to get out.\"\n\n\"Raf\u2014\"\n\n\"Alex. Get. Out.\"\n\nHe goes, slamming the door behind him.\n\nBack at the Residence, he tries to call Henry. He doesn't pick up, but he texts: Sorry. Meeting with Philip. Love you.\n\nHe reaches under the bed and gropes in the dark until he finds it: a bottle of Maker's. The emergency stash.\n\n\"Salud,\" he mutters under his breath, and he unscrews the top.\n\n> bad metaphors about maps\n> \n> * * *\n> \n> A <agcd@eclare45.com> 9\/25\/20 3:21 AM\n> \n> to Henry\n> \n> h,\n> \n> i have had whiskey. bear with me.\n> \n> there's this thing you do. this thing. it drives me crazy. i think about it all the time.\n> \n> there's a corner of your mouth, and a place that it goes. pinched and worried like you're afraid you're forgetting something. i used to hate it. used to think it was your little tic of disapproval.\n> \n> but i've kissed your mouth, that corner, that place it goes, so many times now. i've memorized it. topography on the map of you, a world i'm still charting. i know it. i added it to the key. here: inches to miles. i can multiply it out, read your latitude and longitude. recite your coordinates like la rosaria.\n> \n> this thing, your mouth, its place. it's what you do when you're trying not to give yourself away. not in the way that you do all the time, those empty, greedy grabs for you. i mean the truth of you. the weird, perfect shape of your heart. the one on the outside of your chest.\n> \n> on the map of you, my fingers can always find the green hills, wales. cool waters and a shore of white chalk. the ancient part of you carved out of stone in a prayerful circle, sacrosanct. your spine's a ridge i'd die climbing.\n> \n> if i could spread it out on my desk, i'd find the corner of your mouth where it pinches with my fingers, and i'd smooth it away and you'd be marked with the names of saints like all the old maps. i get the nomenclature now\u2014saints' names belong to miracles.\n> \n> give yourself away sometimes, sweetheart. there's so much of you.\n> \n> fucking yrs,\n> \n> a\n> \n> p.s. wilfred owen to siegfried sassoon\u20141917:\n> \n> And you have fixed my Life\u2014however short. You did not light me: I was always a mad comet; but you have fixed me. I spun round you a satellite for a month, but shall swing out soon, a dark star in the orbit where you will blaze.\n> \n> Re: Bad metaphors about maps\n> \n> * * *\n> \n> Henry <hwales@kensingtonemail.com> 9\/25\/20 6:07 AM\n> \n> to A\n> \n> From Jean Cocteau to Jean Marais, 1939:\n> \n> Thank you from the bottom of my heart for having saved me. I was drowning and you threw yourself into the water without hesitation, without a backward look.\n\n* * *\n\nThe sound of Alex's phone buzzing on his nightstand startles him out of a dead sleep. He falls halfway out of bed, fumbling to answer it.\n\n\"Hello?\"\n\n\" _What did you do?_ \" Zahra's voice nearly shouts. By the clicking of heels in the background and muttered swearing, she's running somewhere.\n\n\"Um,\" Alex says. He rubs his eyes, trying to get his brain back online. What _did_ he do? \"Be more specific?\"\n\n\"Check the fucking news, you horny little miscreant\u2014how could you possibly be _stupid enough to get photographed_? I swear to God\u2014\"\n\nAlex doesn't even hear the last part of what she says, because his stomach has just dropped all the way down through the floor and into the fucking basements two floors below.\n\n\"Fuck.\"\n\nHands shaking, he switches Zahra to speaker, opens up Google, and types his own name.\n\n> BREAKING: Photos Reveal Romantic Relationship Between Prince Henry and Alex Claremont-Diaz\n> \n> OMFG: FSOTUS and Prince Henry\u2014Totally Doing It\n> \n> THE ORAL OFFICE: READ FSOTUS'S STEAMY EMAILS TO PRINCE HENRY\n> \n> Royal Family Declines to Comment on Reports of Prince Henry's Relationship with First Son\n> \n> 25 GIFs That Perfectly Describe Our Reaction When We Heard About Prince Henry & FSOTUS\n> \n> DON'T LET FIRST SON GO DOWN ON ME\n\nA bubble of hysterical laughter emerges from his throat.\n\nHis bedroom door flies open, and Zahra slams on the light, a steely expression of rage barely concealing the sheer terror on her face. Alex's brain flashes to the panic button behind his headboard and wonders if the Secret Service will be able to find him before he bleeds out.\n\n\"You're on communications lockdown,\" she says, and instead of punching him, she snatches his phone out of his hand and shoves it down the front of her blouse, which has been buttoned wrong in her rush. She doesn't even blink at his state of half-nakedness, just dumps an armload of newspapers onto his bedspread.\n\nQUEEN HENRY! twenty copies of the _Daily Mail_ proclaim in gigantic letters. INSIDE THE PRINCE'S GAY AFFAIR WITH THE FIRST SON OF THE UNITED STATES!\n\nThe cover is splashed with a blown-up photo of what is undeniably himself and Henry kissing in the back seat of the car behind the cafe, apparently shot with a long-range lens through the windshield. Tinted windows, but he forgot about the fucking _windshield._\n\nTwo smaller photos are inset on the bottom of the page: one of the shots of them on the Beekman's elevator and a photo of them side by side at Wimbledon, him whispering something in Henry's ear while Henry smiles a soft, private smile.\n\nFucking shitting hell. He is so fucked. Henry is so fucked. And, Jesus Christ, his mother's campaign is fucked, and his political career is fucked, and his ears are ringing, and he's going to throw up.\n\n\" _Fuck,_ \" Alex says again. \"I need my phone. I have to call Henry\u2014\"\n\n\"No, you do fucking not,\" Zahra says. \"We don't know yet how the emails got out, so it's radio silence until we find the leak.\"\n\n\"The\u2014what? Is Henry okay?\" God, Henry. All he can think about is Henry's big blue eyes looking terrified, Henry's breathing coming shallow and quick, locked in his bedroom in Kensington Palace and desperately alone, and his jaw locks up, something burning in the back of his throat.\n\n\"The president is sitting down right now with as many members of the Office of Communications as we could drag out of bed at three in the morning,\" Zahra tells him, ignoring his question. Her phone is buzzing nonstop in her hand. \"It's about to be gay DEFCON five in this administration. For God's sake, put some clothes on.\"\n\nZahra disappears into Alex's closet, and he flips the newspaper open to the story, his heart pounding. There are even more photos inside. He glances over the copy, but there's too much to even begin to process.\n\nOn the second page, he sees them: printed and annotated excerpts of their emails. One is labeled: PRINCE HENRY: SECRET POET? It begins with a line he's read about a thousand times by now.\n\n_Should I tell you that when we're apart, your body comes back to me in dreams..._\n\n_\"Fuck!\"_ he says a third time, spiking the newspaper at the floor. That one was _his._ It feels obscene to see it there. \"How the fuck did they _get these_?\"\n\n\"Yep,\" Zahra agrees. \"You dirty did it.\" She throws a white button-down and a pair of jeans at him, and he pitches himself out of bed. Zahra gamely holds out an arm for him to steady himself while he pulls his pants up, and despite it all, he's struck with overwhelming gratitude for her.\n\n\"Listen, I need to talk to Henry as soon as possible. I can't even imagine\u2014 God, I need to talk to him.\"\n\n\"Get some shoes, we're running,\" Zahra tells him. \"Priority one is damage control, not feelings.\"\n\nHe grabs a pair of sneakers, and they take off while he's still pulling them on, running west. His brain is struggling to keep up, running through about five thousand possible ways this could go, imagining himself ten years down the road being frozen out of Congress, plummeting approval ratings, Henry's name scratched off the line of succession, his mother losing reelection on a swing state's disapproval of him. He's so screwed, and he can't even decide who to be the angriest with, himself or the _Mail_ or the monarchy or the whole stupid country.\n\nHe nearly crashes into Zahra's back as she skids to a stop in front of a door.\n\nHe pushes the door open, and the whole room goes silent.\n\nHis mother stares at him from the head of the table and says flatly, \"Out.\"\n\nAt first he thinks she's talking to him, but she cuts her eyes down to the people around the table with her.\n\n\"Was I not clear? Everyone, out, now,\" she says. \"I need to talk to my son.\"\n\n# THIRTEEN\n\n\"Sit down,\" his mother tells him, and Alex feels dread coil deep in his stomach. He has no clue what to expect\u2014knowing your parent as the person who raised you isn't the same as being able to guess their moves as a world leader.\n\nHe sits, and the silence hovers over them, his mother's hands folded in a considering pose against her lips. She looks exhausted.\n\n\"Are you okay?\" she says finally. When he looks up in surprise, there's no anger in her eyes.\n\nThe president stands on the edge of a career-ending scandal, measures her breaths evenly, and waits for her son to answer.\n\nOh.\n\nIt hits him with sudden clarity that he hasn't at all stopped to consider his own feelings. There simply hasn't been the time. When he reaches for an emotion to name, he finds he can't pin one down, and something shudders inside him and shuts down completely.\n\nHe doesn't often wish away his position in life, but in this moment, he does. He wants to be having this conversation in a different life, just his mother sitting across from him at the dinner table, asking him how he feels about his nice, respectable boyfriend, if he's doing okay with figuring his identity out. Not like this, in a West Wing briefing room, his dirty emails spread out between them on the table.\n\n\"I'm...\" he begins. To his horror, he hears something shake in his voice, which he quickly swallows down. \"I don't know. This isn't how I wanted to tell people. I thought we'd get a chance to do this right.\"\n\nSomething softens and resolves in her face, and he suspects he's answered a question for her beyond the one she asked.\n\nShe reaches over and covers one of his hands with her own.\n\n\"You listen to me,\" she says. Her jaw is set, ironclad. It's the game face he's seen her use to stare down Congress, to cow autocrats. Her grip on his hand is steady and strong. He wonders, half-hysterically, if this is how it felt to charge into war under Washington. \"I am your mother. I was your mother before I was ever the president, and I'll be your mother long after, to the day they put me in the ground and beyond this earth. You are my child. So, if you're serious about this, I'll back your play.\"\n\nAlex is silent.\n\n_But the debates,_ he thinks. _But the general._\n\nHer gaze is hard. He knows better than to say either of those things. She'll handle it.\n\n\"So,\" she says. \"Do you feel forever about him?\"\n\nAnd there's no room left to agonize over it, nothing left to do but say the thing he's known all along.\n\n\"Yeah,\" he says, \"I do.\"\n\nEllen Claremont exhales slowly, and she grins a small, secret grin, the crooked, unflattering one she never uses in public, the one he knows best from when he was a kid around her knees in a small kitchen in Travis County.\n\n\"Then, fuck it.\"\n\n> The Washington Post\n> \n> As details emerge about Alex Claremont-Diaz's affair with Prince Henry, White House goes silent\n> \n> * * *\n\nSeptember 27, 2020\n\n> \"Thinking about history makes me wonder how I'll fit into it one day, I guess,\" First Son Alex Claremont-Diaz writes in one of the many emails to Prince Henry published by the _Daily Mail_ this morning. \"And you too.\"\n> \n> It seems the answer to that question may have come sooner than any anticipated with the sudden exposure of the First Son's romantic relationship with Prince Henry, an arrangement with major repercussions for two of the world's most powerful nations, less than two months before the United States casts its vote on President Claremont's second term.\n> \n> As security experts within the FBI and the Claremont administration scramble to find the sources that provided the British tabloid with evidence of the affair, the usually high-profile First Family has shuttered, with no official statement from the First Son.\n> \n> \"The First Family has always and continues to keep their personal lives separate from the political and diplomatic dealings of the presidency,\" White House Press Secretary Davis Sutherland said in a brief prepared statement this morning. \"They ask for patience and understanding from the American people as they handle this very private matter.\"\n> \n> The _Daily Mail_ 's report this morning revealed that First Son Alex Claremont-Diaz has been involved romantically and sexually with Prince Henry since at least February of this year, according to emails and photographs obtained by the paper.\n> \n> The full email transcripts have been uploaded to WikiLeaks under the moniker \"The Waterloo Letters,\" seemingly named for a reference to the Waterloo Vase in the Buckingham Palace Gardens in one email composed by Prince Henry. The correspondence continues regularly up to Sunday night and appears to have been lifted from a private email server used by residents of the White House.\n> \n> \"Setting aside the ramifications for President Claremont's ability to be impartial on issues of both international relations and traditional family values,\" Republican presidential candidate Senator Jeffrey Richards said at a press conference earlier today, \"I'm extremely concerned about this private email server. What kind of information was being disseminated on this server?\"\n> \n> Richards added that he believes the American voters have a right to know everything else for which President Claremont's server may have been used.\n> \n> Sources close to the Claremont administration insist the private server is similar to the one set up during President George W. Bush's administration and used only for communication within the White House about day-to-day operations and personal correspondence for the First Family and core White House personnel.\n> \n> First rounds of examination of \"The Waterloo Letters\" by experts have yet to reveal any evidence of classified information or otherwise compromising content outside of the nature of the First Son's relationship with Prince Henry.\n\n* * *\n\nFor five endless, unbearable hours, Alex is shuffled from room to room in the West Wing, meeting with what seems to be every strategist, press staffer, and crisis manager his mother's administration has to offer.\n\nThe only moment he recalls with any clarity is pulling his mother into an alcove to say, \"I told Raf.\"\n\nShe stares at him. \"You told Rafael Luna that you're bisexual?\"\n\n\"I told Rafael Luna about Henry,\" he says flatly. \"Two days ago.\"\n\nShe doesn't ask why, just sighs grimly, and they both hover over the implication before she says, \"No. No, those pictures were taken before that. It couldn't have been him.\"\n\nHe runs through pro and con lists, models of different outcomes, fucking charts and graphs and more data than he has ever wanted to see about his own relationship and its ramifications for the world around him. _This is the damage you cause, Alex,_ it all seems to say, right there in hard facts and figures. _This is who you hurt._\n\nHe hates himself, but he doesn't regret anything, and maybe that makes him a bad person and a worse politician, but he doesn't regret Henry.\n\nFor five endless, unbearable hours, he's not allowed to even try to contact Henry. The press sec drafts a statement. It looks like any other memo.\n\nFor five hours, he doesn't shower or change his clothes or laugh or smile or cry. It's eight in the morning when he's finally released and told to stay in the Residence and stand by for further instructions.\n\nHe's handed his phone, at last, but there's no answer when he calls Henry, and no response when he texts. Nothing at all.\n\nAmy walks him through the colonnade and up the stairs, saying nothing, and when they reach the hallway between the East and West Bedrooms, he sees them.\n\nJune, her hair in a haphazard knot on the top of her head and in a pink bathrobe, her eyes red-rimmed. His mom, in a sharp, no-nonsense black dress and pointed heels, jaw set. Leo, barefoot in his pajamas. And his dad, a leather duffel still hanging off one shoulder, looking harried and exhausted.\n\nThey all turn to look at him, and Alex feels a wave of something so much bigger than himself sweep over him, like when he was a child standing bowlegged in the Gulf of Mexico, riptide sucking at his feet. A sound escapes his throat uninvited, something that he barely even recognizes, and June has him first, then the rest of them, arms and arms and hands and hands, pulling him close and touching his face and moving him until he's on the floor, the goddamn terrible hideous antique rug that he hates, sitting on the floor and staring at the rug and the threads of the rug and hearing the Gulf rushing in his ears and thinking distantly that he's having a panic attack, and that's why he can't breathe, but he's just staring at the rug and he's having a panic attack and knowing why his lungs won't work doesn't make them work again.\n\nHe's faintly aware of being shifted into his room, to his bed, which is still covered in the godforsaken fucking _newspapers,_ and someone guides him onto it, and he sits down and tries very, very hard to make a list in his head.\n\nOne.\n\nOne.\n\nOne.\n\n* * *\n\nHe sleeps in fits and starts, wakes up sweating, wakes up shivering. He dreams in short, fractured scenes that swell and fade erratically. He dreams of himself at war, in a muddy trench, love letter soaking red in his chest pocket. He dreams of a house in Travis County, doors locked, unwilling to let him in again. He dreams of a crown.\n\nHe dreams once, briefly, of the lake house, an orange beacon under the moon. He sees himself there, standing in water up to his neck. He sees Henry, sitting naked on the pier. He sees June and Nora, hands clasped together, and Pez on the grass between them, and Bea, digging pink fingertips into the wet soil.\n\nIn the trees next to them, he hears the snap, snap, snap of branches.\n\n\"Look,\" Henry says, pointing up at the stars.\n\nAnd Alex tries to say, _Don't you hear it?_ Tries to say, _Something's coming._ He opens his mouth: a spill of fireflies, and nothing.\n\nWhen he opens his eyes, June is sitting up against the pillows next to him, bitten nails pressed against her bottom lip, still in her bathrobe and keeping watch. She reaches down and squeezes his hand. He squeezes back.\n\n* * *\n\nBetween dreams he catches the sound of muffled voices in the hallway.\n\n\"Nothing,\" Zahra's voice is saying. \"Not a thing. Nobody is taking our calls.\"\n\n\"How can they not be taking our calls? I'm the goddamn president.\"\n\n\"Permission to do a thing, ma'am, slightly outside diplomatic protocol.\"\n\n* * *\n\nA comment: The First Family Has Been Lying To Us, The American People!!1 WHAT ELSE Are They Lying About??!?!\n\nA tweet: I KNEW IT I KNEW ALEX WAS GAY I TOLD YOU BITCHES\n\nA comment: My 12 y\/o daughter has been crying all day. She's dreamt of marrying Prince Henry since she was a little girl. She is heartbroken.\n\nA comment: Are we really supposed to believe that no federal funds were used to cover this up?\n\nA tweet: lmaoooo wait look at page 22 of the emails alex is such a hoe\n\nA tweet: OMFG DID YOU SEE somebody who went to uni with Henry posted some photos of him at a party and he is just like Profoundly Gay in them i'm screaming\n\nA tweet: READ\u2014My column with @WSJ on what the #WaterlooLetters say about the inner workings of the Claremont White House.\n\nMore comments. Slurs. Lies.\n\nJune takes his phone away and shoves it under a couch cushion. He doesn't bother protesting. Henry's not going to call.\n\n* * *\n\nAt one in the afternoon, for the second time in twelve hours, Zahra bursts through his bedroom door.\n\n\"Pack a bag,\" she says. \"We're going to London.\"\n\n* * *\n\nJune helps him stuff a backpack with jeans and a pair of shoes and a broken-in copy of _Prisoner of Azkaban,_ and he stumbles into a clean shirt and out of his room. Zahra is waiting in the hall with her own bag and a freshly pressed suit of Alex's, a sensible navy one that she has apparently decided is appropriate for meeting the queen.\n\nShe's told him very little, except that Buckingham Palace has shut down communication channels in and out, and they're just going to show up and demand a meeting. She seems confident Shaan will agree to it and willing to physically overpower him if not.\n\nThe feeling rolling around in his gut is bizarre. His mom has signed off on them going public with the truth, which is _incredible,_ but there's no reason to expect that from the crown. He could get marching orders to deny everything. He thinks he might grab Henry and run if it comes down to that.\n\nHe's almost completely sure Henry wouldn't go along with pretending it was all fake. He trusts Henry, and he believes in him.\n\nBut they were also supposed to have more time.\n\nThere's a secluded side entrance of the Residence that Alex can sneak out of without being seen, and June and his parents meet him there.\n\n\"I know this is scary,\" his mom says, \"but you can handle it.\"\n\n\"Give 'em hell,\" his dad adds.\n\nJune hugs him, and he shoves on his sunglasses and a hat and jogs out the door and toward whatever way this is all going to end.\n\nCash and Amy are waiting on the plane. Alex wonders briefly if they volunteered for the assignment, but he's trying to get his emotions back under control, and that's not going to help. He bumps his fist against Cash's as he passes, and Amy nods up from the denim jacket she's needling yellow flowers into.\n\nIt's all happened so quickly that now, knees curled up to his chin as they leave the ground, is the first time Alex is able to actually think about everything.\n\nHe's not, he thinks, upset people know. He's always been pretty unapologetic when it came to things like who he dates and what he's into, although those were never anything like this. Still, the cocky shithead part of him is slightly pleased to finally have a claim on Henry. Yep, the prince? Most eligible bachelor in the world? British accent, face like a Greek god, legs for days? _Mine._\n\nBut that's only a tiny, tiny fraction of it. The rest is a knot of fear, anger, violation, humiliation, uncertainty, panic. There are the flaws everyone's allowed to see\u2014his big mouth, his mercurial temper, his searing impulses\u2014and then there's this. It's like how he only wears his glasses when nobody's around: Nobody's supposed to see how much he needs.\n\nHe doesn't care that people think about his body and write about his sex life, real or imagined. He cares that they know, in his own private words, what's pumping out of his heart.\n\nAnd Henry. God, Henry. Those emails\u2014those _letters_ \u2014were the one place Henry could say what he was really thinking. There's nothing that wasn't laid out in there: Henry being gay, Bea going to rehab, the queen tacitly keeping Henry in the closet. Alex hasn't been a good Catholic in a long time, but he knows confession is a sacrament. They were supposed to stay safe.\n\nFuck.\n\nHe can't sit still. He tosses _Prisoner of Azkaban_ aside after four pages. He encounters a think piece on his own relationship on Twitter and has to shut down the whole app. He paces up and down the aisle of the jet, kicking at the bottoms of the seats.\n\n\"Can you _please_ sit down?\" Zahra says after twenty minutes of watching him twitch around the cabin. \"You're giving my ulcer an ulcer.\"\n\n\"Are you sure they're gonna let us in when we get there?\" Alex asks her. \"Like, what if they don't? What if they, like, call the Royal Guard on us and have us arrested? Can they do that? Amy could probably fight them. Will she get arrested if she tries to fight them?\"\n\n\"For fuck's sake,\" Zahra groans, and she pulls out her phone and starts dialing.\n\n\"Who are you calling?\"\n\nShe sighs, holding the phone up to her ear as it rings. \"Srivastava.\"\n\n\"What makes you think he'll answer?\"\n\n\"It's his personal line.\"\n\nAlex stares at her. \"You have his personal line and you haven't used it until now?\"\n\n\" _Shaan,_ \" Zahra snaps. \"Listen up, you fuck. We are in the air right now. FSOTUS is with me. ETA six hours. You will have a car waiting. We will meet the queen and whoever the fuck else we have to meet to hash this shit out, or so help me God I will personally make your balls into fucking earrings. I will scorched-earth your entire motherfucking life.\" She pauses, presumably to listen to him agree because Alex can't imagine him doing anything else. \"Now, put Henry on the phone, and do _not_ try to tell me he's not there, because I know you haven't let him out of your sight.\"\n\nAnd she shoves her phone at Alex's face.\n\nHe takes it uncertainly and lifts it to his ear. There's rustling, a confused noise.\n\n\"Hello?\"\n\nIt's Henry's voice, sweet and posh and shaky and confused, and relief knocks the wind out of him.\n\n_\"Sweetheart.\"_\n\nHe hears Henry's exhale over the line. \"Hi, love. Are you okay?\"\n\nHe laughs wetly, amazed. \"Fuck, are you kidding me? I'm fine, I'm fine, are _you_ okay?\"\n\n\"I'm... managing.\"\n\nAlex winces. \"How bad is it?\"\n\n\"Philip broke a vase that belonged to Anne Boleyn, Gran ordered a communications lockdown, and Mum hasn't spoken to anyone,\" Henry tells him. \"But, er, other than that. All things considered. It's, er.\"\n\n\"I know,\" Alex says. \"I'll be there soon.\"\n\nThere's another pause, Henry's breath shaky over the receiver. \"I'm not sorry,\" he says. \"That people know.\"\n\nAlex feels his heart climb up into his throat.\n\n\"Henry,\" he attempts, \"I...\"\n\n\"Maybe\u2014\"\n\n\"I talked to my mom\u2014\"\n\n\"I know the timing isn't ideal\u2014\"\n\n\"Would you\u2014\"\n\n\"I want\u2014\"\n\n\"Hang on,\" Alex says. \"Are we. Um. Are we both asking the same thing?\"\n\n\"That depends. Were you going to ask me if I want to tell the truth?\"\n\n\"Yeah,\" Alex says, and he thinks his knuckles must be white around the phone. \"Yeah, I was.\"\n\n\"Then, yes.\"\n\nA breath, barely. \"You want that?\"\n\nHenry takes a moment to respond, but his voice is level. \"I don't know if I would have chosen it yet, but it's out there now, and... I won't lie. Not about this. Not about you.\"\n\nAlex's eyelashes are wet.\n\n\"I fucking love you.\"\n\n\"I love you too.\"\n\n\"Just hold on until I get there; we're gonna figure this out.\"\n\n\"I will.\"\n\n\"I'm coming. I'll be there soon.\"\n\nHenry exhales a wet, broken laugh. \"Please, do hurry.\"\n\nThey hang up, and he passes the phone back to Zahra, who takes it wordlessly and tucks it back into her bag.\n\n\"Thank you, Zahra, I\u2014\"\n\nShe holds up one hand, eyes closed. \"Don't.\"\n\n\"Seriously, you didn't have to do that.\"\n\n\"Look, I'm only going to say this once, and if you ever repeat it, I'll have you kneecapped.\" She drops her hand, fixing him with a glare that manages to be both chilly and fond. \"I'm rooting for you, okay?\"\n\n\"Wait. Zahra. Oh my God. I just realized. You're... my friend.\"\n\n\"No, I'm not.\"\n\n\"Zahra, you're my _mean friend._ \"\n\n\"Am not.\" She yanks a blanket from her pile of belongings, turning her back to Alex and wrapping it around her. \"Don't speak to me for the next six hours. I deserve a fucking nap.\"\n\n\"Wait, wait, okay, wait,\" Alex says. \"I have one question.\"\n\nShe sighs heavily. \"What?\"\n\n\"Why'd you wait to use Shaan's personal number?\"\n\n\"Because he's my fianc\u00e9, asshole, but _some_ of us understand the meaning of discretion, so you wouldn't know about it,\" she tells him without even so much as looking at him, curled up against the window of the plane. \"We agreed we'd never use our personal numbers for work contact. Now shut up and let me get some sleep before we have to deal with the rest of this. I'm running on nothing but black coffee, a Wetzel's Pretzel, and a fistful of B12. Do not even breathe in my direction.\"\n\n* * *\n\nIt's not Henry but Bea who answers when Alex knocks on the closed door of the music room on the second floor of Kensington.\n\n\"I _told_ you to stay away\u2014\" Bea is saying as soon as the door is open, brandishing a guitar over her shoulder. She drops it as soon as she sees him. \"Oh, Alex, I'm so sorry, I thought you were Philip.\" She scoops him up with her free hand into a surprisingly bone-crushing hug. \"Thank God you're here, I was about to come get you myself.\"\n\nWhen she releases him, he's finally able to see Henry behind her, slumped on the settee with a bottle of brandy. He smiles at Alex, weakly, and says, \"Bit short for a stormtrooper.\"\n\nAlex's laugh comes out half sob, and it's impossible to know if he moves first or if Henry does, but they meet in the middle of the room, Henry's arms around Alex's neck, swallowing him up. If Henry's voice on the phone was a tether, his body is the gravity that makes it possible, his hand gripping the back of Alex's neck a magnetic force, a permanent compass north.\n\n\"I'm sorry,\" is what comes out of Alex's mouth, miserably, earnestly, muffled against Henry's throat. \"It's my fault. I'm so sorry. I'm so sorry.\"\n\nHenry releases him, hands on his shoulders, jaw set. \"Don't you dare. I'm not sorry for a thing.\"\n\nAlex laughs again, incredulous, looking into the heavy circles under Henry's eyes and the chewed-up bottom lip and, for the first time, seeing a man born to lead a nation.\n\n\"You're unbelievable,\" Alex says. He leans up and kisses the underside of his jaw, finding it rough from a full, fitful day without a shave. He pushes his nose, his cheek into it, feels some of the tension sap out of Henry at the touch. \"You know that?\"\n\nThey find their way onto the lush purples and reds of the Persian rugs on the floor, Henry's head in Alex's lap and Bea on a pouf, plucking away at a weird little instrument she tells Alex is called an autoharp. Bea pulls over a tiny table and sets out crackers and a little chunk of soft cheese and takes away the brandy bottle.\n\nFrom the sound of it, the queen is absolutely livid\u2014not just to finally have confirmation about Henry, but because it's via something as undignified as a tabloid scandal. Philip drove in from Anmer Hall the minute the news broke and has been rebuffed by Bea every time he tries to get near Henry for what he says \"will simply be a stern discussion about the consequences of his actions.\" Catherine has been by, once, three hours ago, stone-faced and sad, to tell Henry that she loves him and he could have told her sooner.\n\n\"And I said, 'That's great, Mum, but as long as you're letting Gran keep me trapped, it doesn't mean a fucking thing,'\" Henry says. Alex stares down at him, shocked and a little impressed. Henry rests an arm over his face. \"I feel awful. I was\u2014I dunno. All the times she should have been there the past few years, it caught up to me.\"\n\nBea sighs. \"Maybe it was the kick in the arse she needs. We've been trying to get her to do _anything_ for years since Dad.\"\n\n\"Still,\" Henry says. \"The way Gran is\u2014Mum isn't to blame for that. And she did manage to protect us, before. It's not fair.\"\n\n\"H,\" Bea says firmly. \"It's hard, but she needed to hear it.\" She looks down at the little buttons of the autoharp. \"We deserve to have one parent, at least.\"\n\nThe corner of her mouth pinches, so much like Henry's.\n\n\"Are you okay?\" Alex asks her. \"I know I\u2014I saw a couple articles...\" He doesn't finish the sentence. \"The Powder Princess\" was the fourth-highest Twitter trend ten hours ago.\n\nHer frown twitches into a half-smile. \"Me? Honestly, it's almost a relief. I've always said that the most comfortable I could be is everyone knowing my story upfront, so I don't have hear the speculations or lie to cover the truth\u2014or explain it. I'd rather it, you know, hadn't been this way. But here we are. At least now I can stop acting as if it's something to be ashamed of.\"\n\n\"I know the feeling,\" Henry says softly.\n\nThe quiet ebbs and flows after a while, the London night black and pressing in against the windowpanes. David the beagle curls up protectively at Henry's side, and Bea picks a Bowie song to play. She sings under her breath, \"I, I will be king, and you, you will be queen,\" and Alex almost laughs. It feels like how Zahra has described hurricane days to him: stuck together, hoping the sandbags will hold.\n\nHenry drifts asleep at some point, and Alex is thankful for it, but he can still feel tension in every part of Henry's body against him.\n\n\"He hasn't slept since the news,\" Bea tells him quietly.\n\nAlex nods slightly, searching her face. \"Can I ask you something?\"\n\n\"Always.\"\n\n\"I feel like he's not telling me something,\" Alex whispers. \"I believe him when he says he's in, and he wants to tell everyone the truth. But there's something else he's not saying, and it's freaking me out that I can't figure out what it is.\"\n\nBea looks up, her fingers stilling. \"Oh, love,\" she says simply. \"He misses Dad.\"\n\n_Oh._\n\nHe sighs, putting his head in his hands. Of course.\n\n\"Can you explain?\" he attempts lamely. \"What that's like? What I can do?\"\n\nShe shifts on her pouf, repositioning the harp onto the floor, and reaches into her sweater. She withdraws a silver coin on a chain: her sobriety chip.\n\n\"D'you mind if I go a bit sponsor?\" she asks with a smirk. He offers her a weak half smile, and she continues.\n\n\"So, imagine we're all born with a set of feelings. Some are broader or deeper than others, but for everyone, there's that ground floor, a bottom crust of the pie. That's the maximum depth of feeling you've ever experienced. And then, the worst thing happens to you. The very worst thing that could have happened. The thing you had nightmares about as a child, and you thought, it's all right because that thing will happen to me when I'm older and wiser, and I'll have felt so many feelings by then that this one worst feeling, the worst possible feeling, won't seem so terrible.\n\n\"But it happens to you when you're young. It happens when your brain isn't even fully done cooking\u2014when you've barely experienced anything, really. The worst thing is one of the first big things that ever happens to you in your life. It happens to you, and it goes all the way down to the bottom of what you know how to feel, and it rips it open and carves out this chasm down below to make room. And because you were so young, and because it was one of the first big things to happen in your life, you'll always carry it inside you. Every time something terrible happens to you from then on, it doesn't just stop at the bottom\u2014it goes all the way down.\"\n\nShe reaches across the tiny tea table and the sad little pile of water crackers and touches the back of Alex's hand.\n\n\"Do you understand?\" she asks him, looking right into his eyes. \"You need to understand this to be with Henry. He is the most loving, nurturing, selfless person you could hope to meet, but there is a sadness and a hurt in him that is tremendous, and you may very well never truly understand it, but you need to love it as much as you love the rest of him, because that's him. That is him, part and parcel. And he is prepared to give it all to you, which is far more than I ever, in a thousand years, thought I would see him do.\"\n\nAlex sits, trying for a long moment to absorb it, and says, \"I've never... I haven't been through anything like that,\" he says, voice rough. \"But I've always felt it, in him. There's this side of him that's... unknowable.\" He takes a breath. \"But the thing is, jumping off cliffs is kinda my thing. That's the choice. I love him, with all that, _because_ of all that. On purpose. I love him on purpose.\"\n\nBea smiles gently. \"Then you'll do fine.\"\n\nSometime around four in the morning, he climbs into bed behind Henry, Henry whose spine pokes out in soft points, Henry who has been through the worst thing and now the next worst thing and is still alive. He reaches out a hand and touches the ridge of Henry's shoulder blade, the skin where the sheet has slid off him, where his lungs stubbornly refuse to stop pulling air. Six feet of boy curled around kicked-in ribs and a recalcitrant heart.\n\nCarefully, his chest to Henry's back, he slots himself into place.\n\n* * *\n\n\"It's foolishness, Henry,\" Philip is saying. \"You're too young to understand.\"\n\nAlex's ears are ringing.\n\nThey sat down in Henry's kitchen this morning with scones and a note from Bea that she'd gone to meet with Catherine. And then suddenly, Philip was bursting through the door, suit askew, hair uncombed, shouting at Henry about the nerve to break the communications embargo, to bring Alex here while the palace is being watched, to keep embarrassing the family.\n\nPresently, Alex is thinking about breaking his nose with the coffee percolator.\n\n\"I'm _twenty-three,_ Philip,\" Henry says, audibly struggling to keep his voice even. \"Mum was barely more than that when she met Dad.\"\n\n\"Yes, and you think that was a _wise_ decision?\" Philip says nastily. \"Marrying a man who spent half our childhoods making films, who never served his country, who got sick and _left_ us and Mum\u2014\"\n\n\" _Don't,_ Philip,\" Henry says. \"I swear to God. Just because your obsession with family legacy didn't impress _him_ \u2014\"\n\n\"You clearly don't know the first fucking thing about what a legacy means if you can let something like this happen,\" Philip snaps. \"The only thing to do now is bury it and hope that somehow people will believe that none of it was real. That's your duty, Henry. It's the _least_ you can do.\"\n\n\"I'm sorry,\" Henry says, sounding wretched, but there's a bitter defiance rising in him too. \"I'm sorry that I'm such a _disgrace_ for being the way I am.\"\n\n\"I don't care if you're _gay,_ \" Philip says, dropping that big fat _if_ like Henry hasn't already specifically _told_ him. \"I care that you've made this choice, with _him_ \"\u2014he cuts his eyes sharply to Alex as if he finally exists in the same room as this conversation\u2014\"someone with a fucking target on his back, to be so stupid and naive and _selfish_ as to think it wouldn't completely fuck us all.\"\n\n\"I knew, Philip. Christ,\" Henry says. \"I knew it could ruin everything. I was _terrified_ of exactly this. But how could I have predicted? How?\"\n\n\"As I said, _naive,_ \" Philip tells him. \"This is the life we live, Henry. You've always known it. I've tried to tell you. I wanted to be a good brother to you, but you don't bloody _listen._ It's time to remember your place in this family. Be a man. Stand up and take responsibility. _Fix this._ For once in your life, don't be a coward.\"\n\nHenry flinches like he's been physically slapped. Alex can see it now\u2014this is how he was broken down over the years. Maybe not always as explicitly, but always there, always implied. _Remember your place._\n\nAnd he does the thing Alex loves so much: He sticks his chin out, steeling himself up. \"I'm not a coward,\" he says. \"And I don't want to fix it.\"\n\nPhilip slants a harsh, humorless laugh at him. \"You don't know what you're talking about. You can't possibly know.\"\n\n\"Fuck off, Philip, I love him,\" Henry says.\n\n\"Oh, you _love him,_ do you?\" It's so patronizing that Alex's hand twitches into a fist under the table. \"What exactly do you intend to do, then, Henry? Hmm? _Marry him?_ Make him the Duchess of Cambridge? The First Son of the United bloody States, fourth in line to be Queen of England?\"\n\n\"I'll fucking abdicate!\" Henry says, voice rising. \"I don't care!\"\n\n\"You wouldn't _dare,_ \" Philip spits back.\n\n\"We have a great uncle who abdicated because he was a _fucking Nazi,_ so it'd hardly be the worst reason anyone's done it, would it?\" Henry's yelling now, and he's out of his chair, hands shaking, towering over Philip, and Alex notices that he's actually taller. \"What are we even _defending_ here, Philip? What kind of legacy? What kind of _family,_ that says, we'll take the murder, we'll take the raping and pillaging and the colonizing, we'll scrub it up nice and neat in a museum, but oh no, you're a bloody poof? That's beyond our sense of decorum! I've bloody well _had it._ I've sat about long enough letting you and Gran and the weight of the damned world keep me pinned, and I'm finished. _I don't care._ You can take your legacy and your decorum and you can _shove it up your fucking arse,_ Philip. I'm _done._ \"\n\nHe huffs out an almighty breath, turns on his heel, and stalks out of the kitchen.\n\nAlex, mouth hanging open, remains frozen in his seat for a few seconds. Across from him, Philip is looking red-faced and queasy. Alex clears his throat, stands, and buttons his jacket.\n\n\"For what it's worth,\" he says to Philip, \"that is the bravest son of a bitch I've ever met.\"\n\nAnd he leaves too.\n\n* * *\n\nShaan looks like he hasn't slept in thirty-six hours. Well, he looks perfectly composed and groomed, but the tag is sticking out of his sweater and the strong smell of whiskey is emanating from his tea.\n\nNext to him, in the back of the incognito van they're taking to Buckingham Palace, Zahra has her arms folded resolutely. The engagement ring on her left hand glints in the muted London morning.\n\n\"So, uh,\" Alex attempts. \"Are you two in a fight now?\"\n\nZahra looks at him. \"No. Why would you think that?\"\n\n\"Oh. I just thought because\u2014\"\n\n\"It's fine,\" Shaan says, still typing on his iPhone. \"This is why we set rules about the personal-slash-professional lines at the outset of the relationship. It works for us.\"\n\n\"If you want a fight, you should have seen it when I found out he had known about you two all along,\" Zahra says. \"Why do you think I got a rock this big?\"\n\n\"It _usually_ works for us,\" Shaan amends.\n\n\"Yep,\" Zahra agrees. \"Plus, we banged it out last night.\"\n\nWithout looking up, Shaan meets her hand in a high five.\n\nShaan and Zahra's forces combined have managed to secure them a meeting with the queen at Buckingham Palace, but they've been told to take a winding, circumspect route to avoid the paparazzi. Alex can feel a buzzing static electricity in London this morning, millions of voices murmuring about him and Henry and what might happen next. But Henry's beside him, holding his hand, and he's holding Henry's hand back, so at least that's something.\n\nThere's a small, older woman with Bea's upturned nose and Henry's blue eyes waiting outside the conference room when they approach it. She's wearing thick glasses, a worn-in maroon sweater, and a pair of cuffed jeans, looking decidedly out of place in the halls of Buckingham Palace. She has a paperback tucked into her back pocket.\n\nHenry's mother turns to face them, and Alex watches her expression flutter through something pained to reserved to gentle when she lays eyes on them.\n\n\"Hi, my baby,\" she says as Henry draws up even with her.\n\nHenry's jaw is tight, but it's not anger, only fear. Alex can see on his face an expression he recognizes: Henry wondering if it's safe to accept the love offered to him, and wanting desperately to take it regardless. He puts his arm around her, lets her kiss his cheek.\n\n\"Mum, this is Alex,\" Henry says, and adds, as if it's not obvious, \"my boyfriend.\"\n\nShe turns to Alex, and he's honestly not sure what to expect, but she pulls him toward her and kisses his cheek too.\n\n\"My Bea has told me what you've done for my son,\" she says, her gaze piercing. \"Thank you.\"\n\nBea is behind her, looking tired but focused, and Alex can only imagine the come-to-Jesus talk she must have given her mother before they got to the palace. She locks eyes with Zahra as their little party assembles in the hall, and Alex feels like they couldn't possibly be in more capable hands. He wonders if Catherine is up to joining the ranks.\n\n\"What are you going to say to her?\" Henry asks his mother.\n\nShe sighs, touching the edge of her glasses. \"Well, the old bird isn't much moved by emotion, so I suppose I'll try to appeal to her with political strategy.\"\n\nHenry blinks. \"Sorry\u2014what are you saying?\"\n\n\"I'm saying that I've come to fight,\" she says, straightforward and plain. \"You want to tell the truth, don't you?\"\n\n\"I\u2014yeah, Mum.\" A light of hope has switched on behind his eyes. \"Yes, I do.\"\n\n\"Then we can try.\"\n\nThey take their seats around the long, ornately carved table in the meeting room, awaiting the queen's arrival in nervous silence. Philip is there, looking like he's about to chew through his tongue, and Henry can't stop fidgeting with his tie.\n\nQueen Mary glides in wearing slate-gray separates and a stony expression, her gray bob arranged with razor precision around the edges of her face. Alex is struck by how tall she is, straight-backed and fine-jawed even in her early eighties. She's not exactly beautiful, but there's a definite story in her shrewd blue eyes and angular features, the heavy creases of frowns around her mouth.\n\nThe temperature in the room drops as she takes her seat at the head of the table. A royal attendant fetches the teapot from the center of the table and pours into the pristine china, and the quiet hangs as she fixes her tea at a glacial pace, making them wait. The milk, poured with one gently tremoring, ancient hand. One cube of sugar, picked up with deliberate care with the tiny silver tongs. A second cube.\n\nAlex coughs. Shaan shoots him a look. Bea presses her lips together.\n\n\"I had a visit earlier this year,\" the queen says at last. She takes up her teaspoon and begins to stir slowly. \"The President of China. You'll forgive me if the name escapes me. But he told me the most fascinating story about how technology has advanced in different parts of the world for these modern times. Did you know, one can manipulate a photograph to make it appear as if the most outlandish things are real? Just a simple... program, is it? A computer. And any manner of unbelievable falsehood could be made actual. One's eyes could hardly detect a difference.\"\n\nThe silence in the room is total, except for the sound of the queen's teaspoon scraping circular motions in the bottom of her teacup.\n\n\"I'm afraid I am too old to understand how things are filed away in space,\" she goes on, \"but I have been told any number of lies can be manufactured and disseminated. One could... create files that never existed and plant them somewhere easy to find. None of it real. The most flagrant of evidence can be discredited and dismissed, just like that.\"\n\nWith the delicate tinkling of silver on porcelain, she rests her spoon on the saucer and finally looks at Henry.\n\n\"I wonder, Henry. I wonder if you think any of this had to do with these unseemly reports.\"\n\nIt's right on the table between them: an offer. Keep ignoring it. Pretend it was a lie. Make it all go away.\n\nHenry grits his teeth.\n\n\"It's real,\" he says. \"All of it.\"\n\nThe queen's face moves through a series of expressions, settling on a terse frown, as if she's found something unsightly on the bottom of one of her kitten heels.\n\n\"Very well. In that case.\" Her gaze shifts to Alex. \"Alexander. Had I known you were involved with my grandson, I would have insisted upon a more formal first meeting.\"\n\n\"Gran\u2014\"\n\n\"Do be quiet, Henry, dear.\"\n\nCatherine speaks up, then. \"Mum\u2014\"\n\nThe queen holds up one wizened hand to silence her. \"I thought we had been humiliated enough in the papers when Beatrice had her little _problem._ And I made myself clear, Henry, years ago, that if you were drawn in _unnatural_ directions, appropriate measures could be taken. Why you have chosen to undermine the hard work I've done to maintain the crown's standing is beyond me, and why you seem set on disrupting my efforts to restore it by demanding I summit with some... _boy_ \"\u2014here, a nasty lilt to her polite tone, under which Alex can hear epithets for everything from his race to his sexuality\u2014\"when you were told to await orders, is truly a mystery. Clearly you have taken leave of your senses. My position is unchanged, dear: Your role in this family is to perpetuate our bloodline and maintain the appearance of the monarchy as the ideal of British excellence, and I simply cannot allow anything less.\"\n\nHenry is looking down, eyes distant and cast toward the grain of the table, and Alex can practically feel the energy roiling up from Catherine across from him. An answer to the fury tight in his own chest. The princess who ran away with James Bond, who told her children to give back what their country stole, making a choice.\n\n\"Mum,\" she says evenly. \"Don't you think we ought to at least have a conversation about other options?\"\n\nThe queen's head turns slowly. \"And what options might those be, Catherine?\"\n\n\"Well, I think there's something to be said for coming clean. It could save us a great deal of face to treat it not as a scandal, but as an intrusion upon the privacy of the family and the victimization of a young man in love.\"\n\n\"Which is what it was,\" Bea chimes in.\n\n\"We could integrate this into our narrative,\" Catherine says, choosing her words with extreme precision. \"Reclaim the dignity of it. Make Alex an official suitor.\"\n\n\"I see. So your plan is to allow him to choose this life?\"\n\nHere, a slight tell. \"It's the only life for him that's honest, Mum.\"\n\nThe queen purses her lips. \"Henry,\" she says, returning to him, \"wouldn't you have a more pleasant go of it without all these unnecessary complications? You know we have the resources to find a wife for you and compensate her handsomely. You understand, I'm only trying to protect you. I know it seems important to you in this moment, but you really must think of the future. You do realize this would mean years of reporters hounding you, all sorts of allegations? I can't imagine people would be as eager to welcome you into children's hospitals\u2014\"\n\n\"Stop it!\" Henry bursts out. All the eyes in the room swivel to him, and he looks pale and shocked at the sound of his own voice, but he goes on. \"You can't\u2014you can't intimidate me into submission forever!\"\n\nAlex's hand gropes across the space between them under the table, and the moment his fingertips catch on the back of Henry's wrist, Henry's hand is gripping his, hard.\n\n\"I know it will be difficult,\" Henry says. \"I... It's terrifying. And if you'd asked me a year ago, I probably would have said it was fine, that nobody needs to know. But... I'm as much a person and a part of this family as you. I deserve to be happy as much as any of you do. And I don't think I ever will be if I have to spend my whole life pretending.\"\n\n\"Nobody's saying you don't deserve to be happy,\" Philip cuts in. \"First love makes everyone mad\u2014it's foolish to throw away your future because of one hormonal decision based on less than a year of your life when you were barely in your twenties.\"\n\nHenry looks Philip square in the face and says, \"I've been gay as a maypole since the day I came out of Mum, Philip.\"\n\nIn the silence that follows, Alex has to bite down very hard on his tongue to suppress the urge to laugh hysterically.\n\n\"Well,\" the queen eventually says. She's holding her teacup daintily in the air, eyeing Henry over it. \"Even if you're willing to submit to the flogging in the papers, it doesn't erase the stipulations of your birthright: You are to produce heirs.\"\n\nAnd Alex apparently hasn't been biting his tongue hard enough, because he blurts out, \"We could still do that.\"\n\nEven Henry's head whips around at that.\n\n\"I don't recall giving you permission to speak in my presence,\" Queen Mary says.\n\n_\"Mum\u2014\"_\n\n\"That raises the issue of surrogates, or donors,\" Philip jumps back in, \"and rights to the throne\u2014\"\n\n\"Are those details pertinent right now, Philip?\" Catherine interrupts.\n\n\" _Someone_ has to bear the stewardship for the royal legacy, Mum.\"\n\n\"I don't care for _that_ tone at all.\"\n\n\"We can entertain hypotheticals, but the fact of the matter is that anything but maintaining the royal image is out of the question,\" the queen says, setting down her teacup. \"The country simply will not accept a prince of his proclivities. I am sorry, dear, but to them, it's perverse.\"\n\n\"Perverse to them or perverse to you?\" Catherine asks her.\n\n\"That isn't fair\u2014\" Philip says.\n\n\"It's _my_ life\u2014\" Henry interjects.\n\n\"We haven't even gotten a chance yet to see how people will react.\"\n\n\"I have been serving this country for forty-seven years, Catherine. I believe I know its heart by now. As I have told you since you were a little girl, you must remove your head from the clouds\u2014\"\n\n\"Oh, will you all shut up for a second?\" Bea says. She's standing now, brandishing Shaan's tablet in one hand. \"Look.\"\n\nShe thunks it down on the table so Queen Mary and Philip can see it, and the rest of them stand to look too.\n\nIt's a news report from the BBC, and the sound is off, but Alex reads the scroll at the bottom of the screen: WORLDWIDE SUPPORT POURS IN FOR PRINCE HENRY AND FIRST SON OF US.\n\nThe room falls silent at the images on the screen. A rally in New York outside the Beekman, decked out in rainbows, with waving signs that say things like: FIRST SON OF OUR HEARTS. A banner on the side of a bridge in Paris that reads: HENRY + ALEX WERE HERE. A hasty mural on a wall in Mexico City of Alex's face in blue, purple, and pink, a crown on his head. A herd of people in Hyde Park with rainbow Union Jacks and Henry's face ripped out of magazines and pasted onto poster boards reading: FREE HENRY. A young woman with a buzz cut throwing two fingers up at the windows of the _Daily Mail._ A crowd of teenagers in front of the White House, wearing homemade T-shirts that all say the same thing in crooked Sharpie letters, a phrase he recognizes from one of his own emails: HISTORY, HUH?\n\nAlex tries to swallow, but he can't. He looks up, and Henry is looking back at him, mouth open, eyes wet.\n\nPrincess Catherine turns and crosses the room slowly, toward the tall windows on the east side of the room.\n\n\"Catherine, don't\u2014\" the queen says, but Catherine grabs the heavy curtains with both hands and throws them open.\n\nA burst of sunlight and color pushes the air out of the room. Down on the mall in front of Buckingham Palace, there's a mass of people with banners, signs, American flags, Union Jacks, pride pennants streaming over their heads. It's not as big as the royal wedding crowd, but it's huge, filling up the pavement and pressed up to the gates. Alex and Henry were told to come in through the back of the palace\u2014they never saw it.\n\nHenry has carefully approached the window, and Alex watches from across the room as he reaches out and grazes his fingertips against the glass.\n\nCatherine turns to him and says on a shaky sigh, \"Oh, my love,\" and pulls him into her chest somehow, even though he's nearly a foot taller. Alex has to look away\u2014even after everything, this feels too private for him to witness.\n\nThe queen clears her throat.\n\n\"This is... hardly representative of how the country as a whole will respond,\" she says.\n\n\"Jesus _Christ,_ Mum,\" Catherine says, releasing Henry and nudging him behind her on protective reflex.\n\n\"This is precisely why I didn't want you to see. You're too softhearted to accept the truth, Catherine, given any other option. The majority of this country still wants the ways of old.\"\n\nCatherine draws herself up, her posture ramrod straight as she approaches the table again. It's a product of royal breeding, but it comes off more like a bow being drawn. \"Of course they do, Mum. Of course the bloody Tories in Kensington and the Brexit fools don't want it. That's not the _point._ Are you so determined to believe nothing could change? That nothing _should_ change? We can have a real legacy here, of hope, and love, and _change._ Not the same tepid shite and drudgery we've been selling since World War II\u2014\"\n\n\"You will not speak to me this way,\" Queen Mary says icily, one tremulous, ancient hand still resting on her teaspoon.\n\n\"I'm sixty years old, Mum,\" Catherine says. \"Can't we eschew decorum at this point?\"\n\n\"No respect. Never an ounce of respect for the _sanctity_ \u2014\"\n\n\"Or, perhaps I should bring some of my concerns to Parliament?\" Catherine says, leaning in to lower her voice right in Queen Mary's face. Alex recognizes the glint in her eyes. He never knew\u2014he always assumed Henry got it from his dad. \"You know, I do think Labour is rather finished with the old guard. I wonder, if I were to mention those meetings you keep forgetting about, or the names of countries you can't quite keep straight, if they might decide that forty-seven is perhaps enough years for the people of Britain to expect you to serve?\"\n\nThe tremor in the queen's hand has doubled, but her jaw is steely. The room is deadly silent. \"You wouldn't dare.\"\n\n\"Wouldn't I, Mum? Would you like to find out?\"\n\nCatherine turns to face Henry, and Alex is surprised to see tears on her face.\n\n\"I'm sorry, Henry,\" she says. \"I've failed you. I've failed all of you. You needed your mum, and I wasn't there. And I was so frightened that I started to think maybe it was for the best, to let you all be kept behind glass.\" She turns back to her mother. \"Look at them, Mum. They're not props of a legacy. They're my _children._ And I swear on my life, and _Arthur's,_ I will take you off the throne before I will let them feel the things you made me feel.\"\n\nThe room hangs in suspense for a few agonizing seconds, then:\n\n\"I still don't think\u2014\" Philip begins, but Bea seizes the pot of tea from the center of the table and dumps it into his lap.\n\n\"Oh, I'm _terribly_ sorry, Pip!\" she says, grabbing him by the shoulders and shoving him, sputtering and yelping, toward the door. \"So _dreadfully_ clumsy. You know, I think all that _cocaine_ I did must have really done a job on my reflexes! Let's go get you cleaned up, shall we?\"\n\nShe heaves him out, throwing Henry a thumbs-up over her shoulder, and shuts the door behind them.\n\nThe queen looks over at Alex and Henry, and Alex sees it in her eyes at last: She's afraid of them. She's afraid of the threat they pose to the perfect Faberge veneer she's spent her whole life maintaining. They _terrify_ her.\n\nAnd Catherine isn't backing down.\n\n\"Well,\" Queen Mary says. \"I suppose. I suppose you don't leave me much choice, do you?\"\n\n\"Oh, you have a choice, Mum,\" Catherine says. \"You've always had a choice. Perhaps today you'll make the right one.\"\n\n* * *\n\nIn the corridor of Buckingham Palace, as soon as the door has shut behind them, they fall sideways into a tapestry on a wall, breathless and delirious and laughing, cheeks wet. Henry pulls Alex close and kisses him, whispers, \"I love you I love you I love you,\" and it doesn't matter, it _doesn't matter_ if anyone sees.\n\n* * *\n\nHe's on the way back to the airstrip when he sees it, emblazoned on the side of a brick building, a shock of color against a gray street.\n\n\"Wait!\" Alex yells up to the driver. \"Stop! Stop the car!\"\n\nUp close, it's beautiful. Two stories tall. He can't imagine how somebody was able to put together something like this so fast.\n\nIt's a mural of himself and Henry, facing each other, haloed by a bright yellow sun, depicted as Han and Leia. Henry in all white, starlight in his hair. Alex dressed as a scruffy smuggler, a blaster at his hip. A royal and a rebel, arms around each other.\n\nHe snaps a photo on his phone, and fingers shaking, types out a tweet: _Never tell me the odds._\n\n* * *\n\nHe calls June from the air over the Atlantic.\n\n\"I need your help,\" he says.\n\nHe hears the click of her pen cocking on the other end of the line. \"Whatcha got?\"\n\n# FOURTEEN\n\n> Jezebel   @Jezebel\n> \n> WATCH: DC Dykes on Bikes chase protesters from Westboro Baptist Church down Pennsylvania Avenue, and yes, it's as amazing as it sounds. bit.ly\/2ySPeRj\n> \n> 9:15 PM \u00b7 29 Sept 2020\n\n* * *\n\nThe very first time Alex pulled up to Pennsylvania Avenue as the First Son of the United States, he almost fell into a bush.\n\nHe can remember it vividly, even though the whole day was surreal. He remembers the interior of the limo, how he was still unused to the way the leather felt under his clammy palms, still green and jittery and pressed too close to the window to look at all the crowds.\n\nHe remembers his mother, her long hair pulled back from her face in an elegant, no-nonsense twist at the back of her head. She'd worn it down for her first day as mayor, her first day in the House, her first day as Speaker, but that day it was up. She said she didn't want any distractions. He thought it made her look tough, like she was ready for a brawl if it came down to it, as if she might have a razor in her shoe. She sat there across from him, going over the notes for her speech, a twenty-four-karat gold American flag on her lapel, and Alex was so proud he thought he'd throw up.\n\nThere was a changeover at some point\u2014Ellen and Leo escorted to the north entrance and Alex and June shuffled off in another direction. He remembers, very specifically, a handful of things. His cuff links, custom sterling silver X-wings. A tiny scuff in the plaster on a western wall of the White House, which he was seeing up close for the first time. His own shoelace, untied. And he remembers bending over to tie his shoe, losing his balance because of nerves, and June grabbing the back of his jacket to keep him from plunging face-first into a thorny rosebush in front of seventy-five cameras.\n\nThat was the moment he decided he wasn't going to allow himself nerves ever again. Not as Alex Claremont-Diaz, First Son of the United States, and not as Alex Claremont-Diaz, rising political star.\n\nNow, he's Alex Claremont-Diaz, center of an international political sex scandal and boyfriend of a Prince of England, and he's back in a limo on Pennsylvania Avenue, and there's another crowd, and the imminent barf feeling is back.\n\nWhen the car door opens, it's June, standing there in a bright yellow T-shirt that says: HISTORY, HUH?\n\n\"You like it?\" she says. \"There's a guy selling them down the block. I got his card. Gonna put it in my next column for _Vogue._ \"\n\nAlex launches himself at her, engulfing her in a hug that lifts her feet off the ground, and she yelps and pulls his hair, and they topple sideways into a shrub, as Alex was always destined to do.\n\nTheir mother is in a decathlon of meetings, so they sneak out onto the Truman Balcony and catch each other up over hot chocolates and a plate of donuts. Pez has been trying to play telephone between the respective camps, but it's only so effective. June cries first when she hears about the phone call on the plane, then again at Henry standing up to Philip, and a third time at the crowd outside Buckingham Palace. Alex watches her text Henry about a hundred heart emojis, and he sends her back a short video of himself and Catherine drinking champagne while Bea plays \"God Save the Queen\" on electric guitar.\n\n\"Okay, here's the thing,\" June says afterward. \"Nobody has seen Nora in two days.\"\n\nAlex stares at her. \"What do you mean?\"\n\n\"I mean, I've called her, Zahra's called her, Mike and her parents have all called her, she's not answering anyone. The guard at her apartment says she hasn't left this whole time. Apparently, she's 'fine but busy.' I tried just showing up, but she'd told the doorman not to let me in.\"\n\n\"That's... concerning. And also, uh, kind of shitty.\"\n\n\"Yeah, I know.\"\n\nAlex turns away, pacing over to the railing. He really could have used Nora's nonplussed approach in this situation, or, really, just his best friend's company. He feels somewhat betrayed she's abandoned him when he needs her most\u2014when he and June _both_ need her most. She has a tendency to bury herself in complex calculations on purpose when especially bad things happen around her.\n\n\"Oh, hey,\" June says. \"And here's the favor you asked for.\"\n\nShe reaches into the pocket of her jeans and hands him a folded-up piece of paper.\n\nHe skims the first few lines.\n\n\"Oh my God, Bug,\" he says. \"I\u2014 Oh my God.\"\n\n\"Do you like it?\" She looks a little nervous. \"I was trying to capture, like, who you are, and your place in history, and what your role means to you, and\u2014\"\n\nShe's cut off because he's scooped her up in another bear hug, teary-eyed. \"It's perfect, June.\"\n\n\"Hey, First Offspring,\" says a voice suddenly, and when Alex puts June down, Amy is waiting in the doorway connecting the balcony to the Oval Room. \"Madam President wants to see you in her office.\" Her attention shifts, listening to her earpiece. \"She says to bring the donuts.\"\n\n\"How does she always _know_?\" June mutters, scooping up the plate.\n\n\"I have Bluebonnet and Barracuda, on the move,\" Amy says, touching her earpiece.\n\n\"I still can't believe you picked that for your stupid code name,\" June says to him. Alex trips her on the way through the door.\n\n* * *\n\nThe donuts have been gone for two hours.\n\nOne, on the couch: June, tying and untying and retying the laces on her Keds, for lack of anything else to do with her hands. Two, against a far wall: Zahra, rapidly typing out an email on her phone, then another. Three, at the Resolute Desk: Ellen, buried in probability projections. Four, on the other couch: Alex, counting.\n\nThe doors to the Oval Office fly open and Nora comes careening in.\n\nShe's wearing a bleach-stained HOLLERAN FOR CONGRESS '72 sweatshirt and the frenzied, sun-blinded expression of someone who has emerged from a doomsday bunker for the first time in a decade. She nearly crashes into the bust of Abraham Lincoln in her rush to Ellen's desk.\n\nAlex is already on his feet. \"Where the fuck have you _been_?\"\n\nShe slaps a thick folder down on the desk and turns halfway to face Alex and June, out of breath. \"Okay, I know you're pissed, and you have every right to be, but\"\u2014she braces herself against the desk with both hands, gesturing toward the folder with her chin\u2014\"I have been holed up in my apartment for two days doing _this,_ and you are super not gonna be mad anymore when you see what it is.\"\n\nAlex's mother blinks at her, perturbed. \"Nora, honey, we're trying to figure out\u2014\"\n\n\" _Ellen,_ \" Nora practically yells. The room goes silent, and Nora freezes, realizing. \"Uh. Ma'am. Mom-in-law. Please, just. You need to read this.\"\n\nAlex watches her sigh and put down her pen before pulling the folder toward her. Nora looks like she's about to pass out on top of the desk. He looks across to June on the opposite couch, who appears as clueless as he feels, and\u2014\n\n\"Holy... _fucking_ shit,\" his mother says, a dawning mix of fury and bemusement. \"Is this\u2014?\"\n\n\"Yup,\" Nora says.\n\n\"And the\u2014?\"\n\n\"Uh-huh.\"\n\nEllen covers her mouth with one hand. \"How the hell did you _get_ this? Wait, let me rephrase\u2014how the hell did _you_ get this?\"\n\n\"Okay, so.\" Nora withdraws herself from the desk and steps backward. Alex has no idea what the fuck is happening, but it's something, something big. Nora is pacing now, both hands clutched to her forehead. \"The day of the leaks, I get an anonymous email. Obvious sockpuppet account, but untraceable. I tried. They sent me a link to a fucking massive file dump and told me they were a hacker and had obtained the contents of the Richards campaign's private email server in their entirety.\"\n\nAlex stares at her. \" _What?_ \"\n\nNora looks back at him. \"I know.\"\n\nZahra, who has been standing behind Ellen's desk with her arms folded, cuts in to ask, \"And you didn't report this to any of the proper channels because?\"\n\n\"Because I wasn't sure it was anything at first. And when it was, I didn't trust anybody else to handle it. They said they sent it specifically to me because they knew I was personally invested in Alex's situation and would work as fast as possible to find what they didn't have time to.\"\n\n\"Which is?\" Alex can't believe he still has to ask.\n\n\"Proof,\" Nora says. And her voice is shaking now. \"That Richards fucking set you up.\"\n\nHe hears, distantly, the sound of June swearing under her breath and getting up from the couch, walking off to a far corner of the room. His knees give out, so he sits back down.\n\n\"We... we suspected that maybe the RNC had somehow been involved with some of what happened,\" his mother says. She's coming around the desk now, kneeling on the floor in front of him in her starched gray dress, the folder held against her chest. \"I had people looking into it. I never imagined... the whole thing, straight from Richards's campaign.\"\n\nShe takes the folder and spreads it open on the coffee table in the middle of the room.\n\n\"There were\u2014I mean, just, hundreds of thousands of emails,\" Nora is saying as Alex climbs down onto the rug and starts staring at the pages, \"and I swear a third of them were from dummy accounts, but I wrote a code that narrowed it down to about three thousand. I went through the rest manually. This is everything about Alex and Henry.\"\n\nAlex notices his own face first. It's a photo: blurry, out of focus, caught on a long-range lens, only barely recognizable. It's hard to place where he is, until he sees the elegant ivory curtains at the edge of the frame. Henry's bedroom.\n\nHe looks above the photo and sees it's attached to an email between two people. _Negative. Nilsen says that's not nearly clear enough. You need to tell the P we're not paying for Bigfoot sightings._ Nilsen. Nilsen, as in Richards's campaign manager.\n\n\"Richards outed you, Alex,\" Nora says. \"As soon as you left the campaign, it started. He hired a firm that hired the hackers who got the surveillance tapes from the Beekman.\"\n\nHis mother is next to him with a highlighter cap already between her teeth, slashing bright yellow lines across pages. There's movement to his right: Zahra is there too, pulling a stack of papers toward her and starting in with a red pen.\n\n\"I\u2014I don't have any bank account numbers or anything but, if you look, there are pay stubs and invoices and requests of service,\" Nora says. \"Everything, guys. It's all through back channels and go-between firms and fake names but it's\u2014there's a digital paper trail for everything. Enough for a federal investigation, which could subpoena the financial stuff, I think. Basically, Richards hired a firm that hired the photographers who followed Alex and the hackers who breached your server, and then he hired another third party to buy everything and resell it to the _Daily Mail._ I mean, we're talking about having private contractors surveil a member of the First Family and infiltrate White House security to try to induce a sex scandal to win a presidential race, that is some fucked-up shi\u2014\"\n\n\"Nora, can you\u2014?\" June says suddenly, having returned to one of the couches. \"Just, please.\"\n\n\"Sorry,\" Nora says. She sits down heavily. \"I drank like nine Red Bulls to get through all of those and ate a weed gummy to level back out, so I'm flying at fasten-seat-belts right now.\"\n\nAlex closes his eyes.\n\nThere's so fucking much in front of him, and it's impossible to process it all right now, and he's pissed, _furious,_ but he can also put a name on it. He can do something about it. He can go outside. He can walk out of this office and call Henry and tell him: \"We're safe. The worst is over.\"\n\nHe opens his eyes again, looks down at the pages on the table.\n\n\"What do we do with this now?\" June asks.\n\n\"What if we just leaked it?\" Alex offers. \"WikiLeaks\u2014\"\n\n\"I'm not giving them shit,\" Ellen cuts him off immediately, not even looking up, \"especially not after what they did to you. This is real shit. I'm taking this motherfucker down. It has to stick.\" She finally puts her highlighter down. \"We're leaking it to the press.\"\n\n\"No major publication is going to run this without verification from someone on the Richards campaign that these emails are real,\" June points out, \"and that kind of thing takes months.\"\n\n\"Nora,\" Ellen says, fixing her with a steely gaze, \"is there anything you can do at all to trace the person who sent this to you?\"\n\n\"I tried,\" Nora says. \"They did everything to obscure their identity.\" She reaches down into her shirt and produces her phone. \"I can show you the email they sent.\"\n\nShe swipes through a few screens and places her phone face-up on the table. The email is exactly as she described, with a signature at the bottom that's apparently a random combination of numbers and letters: 2021 SCB. BAC CHZ GR ON A1.\n\n2021 SCB.\n\nAlex's eyes stop on the last line. He picks up the phone. Stares at it.\n\n\"Goddammit.\"\n\nHe keeps staring at the stupid letters. 2021 SCB.\n\n2021 South Colorado Boulevard.\n\nThe closest Five Guys to the office where he worked that summer in Denver. He still remembers the order he was sent out to pick up at least once a week. Bacon cheeseburger, grilled onions, A1 Sauce. Alex memorized the goddamn Five Guys order. He feels himself start to laugh.\n\nIt's code, for Alex and Alex only: _You're the only one I trust._\n\n\"This isn't a hacker,\" Alex says. \"Rafael Luna sent this to you. That's your verification.\" He looks at his mother. \"If you can protect him, he'll confirm it for you.\"\n\n> [MUSICAL INTRODUCTION: 15 SECOND INSTRUMENTAL FROM DESTINY'S CHILD'S 1999 SINGLE \"BILLS, BILLS, BILLS\"]\n> \n> VOICEOVER: This is a Range Audio podcast.\n> \n> You're listening to \"Bills, Bills, Bills,\" hosted by Oliver Westbrook, Professor of Constitutional Law at NYU.\n> \n> [END MUSICAL INTRODUCTION]\n> \n> WESTBROOK: Hi. I'm Oliver Westbrook, and with me, as always, is my exceedingly patient, talented, merciful, and lovely producer, Sufia, without whom I would be lost, bereft, floating on a sea of bad thoughts and drinking my own piss. We love her. Say hi, Sufia.\n> \n> SUFIA JARWAR, PRODUCER, RANGE AUDIO: Hello, please send help.\n> \n> WESTBROOK: And this is Bills, Bills, Bills, the podcast where I attempt every week to break down for you, in layman's terms, what's happening in Congress, why you should care, and what you can do about it.\n> \n> Well. I gotta tell you, guys, I had a very different show planned out a few days ago, but I don't really see the point in getting into any of it.\n> \n> Let's just, ah. Take a minute to review the story the Washington Post broke this morning. We've got emails, anonymously leaked, confirmed by an anonymous source on the Richards campaign, that clearly show Jeffrey Richards\u2014or at least high-ranking staffers at his campaign\u2014orchestrated this fucking diabolical plan to have Alex Claremont-Diaz stalked, surveilled, hacked, and outed by the Daily Mail as part of an effort to take down Ellen Claremont in the general. And then, about\u2014uh, what is it, Suf? Forty minutes?\u2014forty minutes before we started recording this, Senator Rafael Luna tweeted he was parting ways with the Richards campaign.\n> \n> So. Wow.\n> \n> I don't think there's any need to discuss a leak from that campaign other than Luna. It's obviously him. From where I sit, this looks like the case of a man who\u2014maybe he didn't really want to be there in the first place, maybe he was already having second thoughts. Maybe he even infiltrated the campaign to do something exactly like this\u2014Sufia, am I allowed to say that?\n> \n> JARWAR: Literally, when has that ever stopped you?\n> \n> WESTBROOK: Point. Anyway, Casper Mattresses is paying me the big sponsorship bucks to give you a Washington analysis podcast, so I'm gonna attempt to do that here, even though what has happened to Alex Claremont-Diaz\u2014and Prince Henry too\u2014over the past few days has been obscene, and it feels cheap and gross to even talk about it like this. But in my opinion, here are the three big things to take away from the news we've gotten today.\n> \n> First, the First Son of the United States didn't actually do anything wrong.\n> \n> Second, Jeffrey Richards committed a hostile act of conspiracy against a sitting president, and I am eagerly awaiting the federal investigation that is coming to him once he loses this election.\n> \n> Third, Rafael Luna is perhaps the unlikeliest hero of the 2020 presidential race.\n\n* * *\n\nA speech has to be made.\n\nNot just a statement. A speech.\n\n\"You wrote this?\" their mother says, holding the folded-up page June had handed Alex on the balcony. \"Alex told you to scrap the statement our press secretary drafted and write this whole thing?\" June bites her lip and nods. \"This is\u2014this is _good,_ June. Why the hell aren't you writing all our speeches?\"\n\nThe press briefing room in the West Wing is ruled too impersonal, so they've called the press pool to the Diplomatic Reception Room on the ground floor. It's the room where FDR once recorded his fireside chats, and Alex is going to walk in there and make a speech and hope the country doesn't hate him for the truth.\n\nThey've flown Henry in from London for the telecast. He'll be positioned right at Alex's shoulder, steady and sure, the emblematic politician's spouse. Alex's brain can't stop sprinting laps around it. He keeps picturing it: an hour from now, millions and millions of TVs across America simulcasting his face, his voice, June's words, Henry at his side. Everyone will know. Everyone already knows now, but they don't _know,_ not the right way.\n\nIn an hour, every person in America will be able to look at a screen and see their First Son and his boyfriend.\n\nAnd, across the Atlantic, almost as many will look up over a beer at a pub or dinner with their family or a quiet night in and see their youngest prince, the most beautiful one, Prince Charming.\n\nThis is it. October 2, 2020, and the whole world watched, and history remembered.\n\nAlex waits on the South Lawn, within view of the linden trees of the Kennedy Garden, where they first kissed. Marine One touches down in a cacophony of noise and wind and rotors, and Henry emerges in head-to-toe Burberry looking dramatic and windswept, like a dashing hero here to rip bodices and mend war-torn countries, and Alex has to laugh.\n\n\"What?\" Henry shouts over the noise when he sees the look on Alex's face.\n\n\"My life is cosmic joke and you're not a real person,\" Alex says, wheezing.\n\n\" _What?_ \" Henry yells again.\n\n\"I said, you look great, baby!\"\n\nThey sneak off to make out in a stairwell until Zahra finds them and drags Henry off to get camera-ready, and soon they're being shuffled to the Diplomatic Reception Room, and it's time.\n\nIt's time.\n\nIt's been one long, long year of learning Henry inside and out, learning himself, learning how much he still had to learn, and just like that, it's time to walk out there and stand at a podium and confidently declare it all as fact.\n\nHe's not afraid of anything he feels. He's not afraid of saying it. He's only afraid of what happens when he does.\n\nHenry touches his hand, gently, two fingertips against his palm.\n\n\"Five minutes for the rest of our lives,\" he says, laughing a grim little laugh.\n\nAlex reaches for him in return, presses one thumb into the hollow of his collarbone, slipping right under the knot of his tie. The tie is purple silk, and Alex is counting his breaths.\n\n\"You are,\" he says, \"the absolute worst idea I've ever had.\"\n\nHenry's mouth spreads into a slow smile, and Alex kisses it.\n\n> FIRST SON ALEXANDER CLAREMONT-DIAZ'S ADDRESS FROM THE WHITE HOUSE, OCTOBER 2, 2020\n> \n> Good morning.\n> \n> I am, and have been\u2014first, last, and always\u2014a child of America.\n> \n> You raised me. I grew up in the pastures and hills of Texas, but I had been to thirty-four states before I learned how to drive. When I caught the stomach flu in the fifth grade, my mother sent a note to school written on the back of a holiday memo from Vice President Biden. Sorry, sir\u2014we were in a rush, and it was the only paper she had on hand.\n> \n> I spoke to you for the first time when I was eighteen, on the stage of the Democratic National Convention in Philadelphia, when I introduced my mother as the nominee for president. You cheered for me. I was young and full of hope, and you let me embody the American dream: that a boy who grew up speaking two languages, whose family was blended and beautiful and enduring, could make a home for himself in the White House.\n> \n> You pinned the flag to my lapel and said, \"We're rooting for you.\" As I stand before you today, my hope is that I have not let you down.\n> \n> Years ago, I met a prince. And though I didn't realize it at the time, his country had raised him too.\n> \n> The truth is, Henry and I have been together since the beginning of this year. The truth is, as many of you have read, we have both struggled every day with what this means for our families, our countries, and our futures. The truth is, we have both had to make compromises that cost us sleep at night in order to afford us enough time to share our relationship with the world on our own terms.\n> \n> We were not afforded that liberty.\n> \n> But the truth is, also, simply this: love is indomitable. America has always believed this. And so, I am not ashamed to stand here today where presidents have stood and say that I love him, the same as Jack loved Jackie, the same as Lyndon loved Lady Bird. Every person who bears a legacy makes the choice of a partner with whom they will share it, whom the American people will hold beside them in hearts and memories and history books. America: He is my choice.\n> \n> Like countless other Americans, I was afraid to say this out loud because of what the consequences might be. To you, specifically, I say: I see you. I am one of you. As long as I have a place in this White House, so will you. I am the First Son of the United States, and I'm bisexual. History will remember us.\n> \n> If I can ask only one thing of the American people, it's this: Please, do not let my actions influence your decision in November. The decision you will make this year is so much bigger than anything I could ever say or do, and it will determine the fate of this country for years to come. My mother, your president, is the warrior and the champion that each and every American deserves for four more years of growth, progress, and prosperity. Please, don't let my actions send us backward. I ask the media not to focus on me or on Henry, but on the campaign, on policy, on the lives and livelihoods of millions of Americans at stake in this election.\n> \n> And finally, I hope America will remember that I am still the son you raised. My blood still runs from Lometa, Texas, and San Diego, California, and Mexico City. I still remember the sound of your voices from that stage in Philadelphia. I wake up every morning thinking of your hometowns, of the families I've met at rallies in Idaho and Oregon and South Carolina. I have never hoped to be anything other than what I was to you then, and what I am to you now\u2014the First Son, yours in actions and words. And I hope when Inauguration Day comes again in January, I will continue to be.\n\n* * *\n\nThe first twenty-four hours after the speech are a blur, but a few snapshots will stay with him for the rest of his life.\n\nA picture: the morning after, a new crowd gathered on the Mall, the biggest yet. He stays in the Residence for safety, but he and Henry and June and Nora and all three of his parents sit in the living room on the second floor and watch the live stream on CNN. In the middle of the broadcast: Amy at the front of the cheering crowd wearing June's yellow HISTORY, HUH? T-shirt and a trans flag pin. Next to her: Cash, with Amy's wife on his shoulders in what Alex can now tell is the jean jacket Amy was embroidering on the plane in the colors of the pansexual flag. He whoops so hard he spills his coffee on George Bush's favorite rug.\n\nA picture: Senator Jeffrey Richards's stupid Sam the Eagle face on CNN, talking about his grave concern for President Claremont's ability to remain impartial on matters of traditional family values due to the acts her son engages in on the sacred grounds of the house our forefathers built. Followed by: Senator Oscar Diaz, responding via satellite, that President Claremont's primary value is upholding the Constitution, and that the White House was built by slaves, not our forefathers.\n\nA picture: the expression on Rafael Luna's face when he looks up from his paperwork to see Alex standing in the doorway of his office.\n\n\"Why do you even have a staff?\" Alex says. \"Nobody has ever tried to stop me from walking straight in here.\"\n\nLuna has his reading glasses on, and he looks like he hasn't shaved in weeks. He smiles, a little apprehensive.\n\nAfter Alex decoded the message in the email, his mother called Luna directly and told him, no questions asked, she would grant him full protection from criminal charges if he helped her take Richards down. He knows his dad has been in touch too. Luna knows neither of his parents are holding a grudge. But this is the first time they've spoken.\n\n\"If you think I don't tell every hire on their first day that you have a free pass,\" he says, \"you do not have an accurate sense of yourself.\"\n\nAlex grins, and he reaches into his pocket and produces a packet of Skittles, lobbing them underhand onto Luna's desk.\n\nLuna looks down at them.\n\nThe chair is next to his desk these days, and he pushes it out.\n\nAlex hasn't gotten a chance to thank him yet, and he doesn't know where to start. He doesn't even feel like it's the first order of business. He watches Luna rip open the packet and dump the candy out onto his papers.\n\nThere's a question hanging in the air, and they can both see it. Alex doesn't want to ask. They just got Luna back. He's afraid of losing him again to the answer. But he has to know.\n\n\"Did you know?\" he finally says. \"Before it happened, did you know what he was going to do?\"\n\nLuna takes his glasses off and sets them down grimly on his blotter.\n\n\"Alex, I know I... completely destroyed your faith in me, so I don't blame you for asking me,\" he says. He leans forward on his elbows, his eye contact hard and deliberate. \"But I need you to know I would never, ever intentionally let something like that happen to you. Ever. I had no idea until it came out. Same as you.\"\n\nAlex releases a long breath.\n\n\"Okay,\" he says. He watches Luna lean back, looks at the fine lines on his face, slightly heavier than they were before. \"So, what happened?\"\n\nLuna sighs, a hoarse, tired sound in the back of his throat. It's a sound that makes Alex think about what his dad told him at the lake, about how much of Luna is still hidden.\n\n\"So,\" he says, \"you know I interned for Richards?\"\n\nAlex blinks. \"What?\"\n\nLuna barks a small, humorless laugh. \"Yeah, you wouldn't have heard. Richards made pretty damn sure to get rid of the evidence. But, yeah, 2000. I was nineteen. It was back when he was AG in Utah. One of my professors called in a favor.\"\n\nThere were rumors, Luna explains, among the low-level staffers. Usually the female interns, but occasionally an especially pretty boy\u2014a boy like him. Promises, from Richards: mentorship, connections, if \"you'd just get a drink with me after work.\" A strong implication that \"no\" was unacceptable.\n\n\"I had _nothing_ back then,\" Luna says. \"No money, no family, no connections, no experience. I thought, 'This is your only way to get your foot in the door. Maybe he means it.'\"\n\nLuna pauses, taking a breath. Alex's stomach is twisting uncomfortably.\n\n\"He sent a car, made me meet him at a hotel, got me drunk. He wanted\u2014he tried to\u2014\" Luna grimaces away from finishing the sentence. \"Anyway, I got away. I remember I got home that night, and the guy I was renting a room with took one look at me and handed me a cigarette. That's when I started smoking, by the way.\"\n\nHe's been looking down at the Skittles on his desk, sorting the reds from oranges, but here he looks up at Alex with a bitter, cutting smile.\n\n\"And I went back to work the next day like nothing happened. I made _small talk_ with him in the _break room,_ because I wanted it to be okay, and that's what I hated myself the most for. So the next time he sent me an email, I walked into his office and told him that if he didn't leave me alone, I'd take it to the paper. And that's when he pulled out the file.\n\n\"He called it an 'insurance policy.' He knew stuff I did as a teenager, how I got kicked out by my parents and a youth shelter in Seattle. That I have family who are undocumented. He told me that if I ever said a word about what happened, not only would I never have a career in politics, but he would ruin my life. He'd ruin my _family's_ lives. So, I shut the fuck up.\"\n\nLuna's eyes when they meet his again are ice cold, sharp. A window slammed shut.\n\n\"But I've never forgotten. I'd see him in the Senate chamber, and he'd look at me like I owed _him_ something, because he hadn't destroyed me when he could have. And I knew he was going to do whatever shady shit it took to win the presidency, and I couldn't let a fucking _predator_ be the most powerful man in the country if it was within my power to stop it.\"\n\nHe turns now, a tiny shake of his shoulders like he's dusting off a light snowfall, pivoting his chair to pluck up a few Skittles and pop them into his mouth, and he's trying for casual but his hands aren't steady.\n\nHe explains that the moment he decided was this summer, when he saw Richards on TV talking about the Youth Congress program. That he knew, with more access, he could find and leak evidence of abuse. Even if he was too old for Richards to want to fuck, he could play him. Convince him he didn't believe Ellen would win, that he'd get the Hispanic and moderate vote in exchange for power.\n\n\"I fucking hated myself every minute of working with that campaign, but I spent the whole time looking for evidence. I was close. I was so focused, so zeroed in that, that I... I never noticed if there were whispers about you. I had no idea. But when everything came out... I knew. I just couldn't prove it. But I had access to the servers. I don't know much, but I'd been around the block enough in my teenage anarchist days to know people who know how to do a file dump. Don't look at me like that. I'm not _that_ old.\"\n\nAlex laughs, and Luna laughs too, and it's a relief, like the air coming back in the room.\n\n\"Anyway, getting it straight to you and your mother was the fastest way to expose him, and I knew Nora could do that. And I... I knew you would understand.\"\n\nHe pauses, sucking on a Skittle, and Alex decides to ask.\n\n\"Did my dad know?\"\n\n\"About me going triple agent? No, nobody does. Half my staff quit because they didn't know. My sister hasn't spoken to me in months.\"\n\n\"No, about what Richards did to you?\"\n\n\"Alex, your father is the only other person alive I've ever told any of this to,\" he says. \"Your father took it upon himself to help me when I wouldn't let anyone else, and I'll never stop being grateful to him. But he wanted me to come forward with what Richards did to me, and I... couldn't. I said it was a risk I wasn't willing to take with my own career, but truthfully, I didn't think what happened to one gay Mexican kid twenty years ago would make a difference to his base. I didn't think anyone would believe me.\"\n\n\"I believe you,\" Alex says readily. \"I just wish you would have told me what you were doing. Or, like, anybody.\"\n\n\"You would have tried to stop me,\" Luna says. \"You all would have.\"\n\n\"I mean... Raf, it was a fucking crazy plan.\"\n\n\"I know. And I don't know if I'll ever be able to fix the damage I've done, but I honestly don't care. I did what I had to do. There was no way in hell I was going to let Richards win. My whole life has been about fighting. I fought.\"\n\nAlex thinks it over. He can relate\u2014it echoes the same deliberations he's been having with himself. He thinks of something he hasn't allowed himself to think about since all this started after London: his LSAT results, unopened and tucked away inside the desk in his bedroom. How do you do all the good you can do?\n\n\"I'm sorry, by the way,\" Luna says. \"For the things I said to you.\" He doesn't have to specify which things. \"I was... fucked up.\"\n\n\"It's cool,\" Alex tells him, and he means it. He forgave Luna before he ever walked into the office, but he appreciates the apology. \"I'm sorry too. But also, I hope you know that if you ever call me 'kid' again after all this, I am literally going to kick your ass.\"\n\nLuna laughs in earnest. \"Listen, you've had your first big sex scandal. No more sitting at the kids' table.\"\n\nAlex nods appreciatively, stretching in his chair and folding his hands behind his head. \"Man, it fucking sucks it has to be like this, with Richards. Even if you expose him now, straight people always want the homophobic bastards to be closet cases so they can wash their hands of it. As if ninety-nine out of a hundred aren't just regular old hateful bigots.\"\n\n\"Yeah, especially since I think I'm the only male intern he ever took to a hotel. It's the same as any fucking predator\u2014it has nothing to do with sexuality and everything to do with power.\"\n\n\"Do you think you'll say anything?\" Alex says. \"At this point?\"\n\n\"I've been thinking about it a lot.\" He leans in. \"Most people have kind of already figured out that I'm the leak. And I think, sooner or later, someone is going to come to me with an allegation that is within the statute of limitations. Then we can open up a congressional investigation. _Big-time._ And _that_ will make a difference.\"\n\n\"I heard a 'we' in there,\" Alex says.\n\n\"Well,\" Luna says. \"Me and someone else with law experience.\"\n\n\"Is that a hint?\"\n\n\"It's a suggestion,\" Luna says. \"But I'm not gonna tell you what to do with your life. I'm busy trying to get my own shit together. Look at this.\" He lifts his sleeve. \"Nicotine patch, bitch.\"\n\n\"No way,\" Alex says. \"Are you actually quitting for real?\"\n\n\"I am a changed man, unburdened by the demons of my past,\" Luna says solemnly, with a jerk-off hand gesture.\n\n\"You fucker, I'm proud of you.\"\n\n\"Hola,\" says a voice at the door of the office.\n\nIt's his dad, in a T-shirt and jeans, a six-pack of beer in one hand.\n\n\"Oscar,\" Luna says, grinning. \"We were just talking about how I've decimated my reputation and killed my own political career.\"\n\n\"Ay,\" he says, dragging an extra chair over to the desk and passing out beers. \"Sounds like a job for Los Bastardos.\"\n\nAlex cracks open his can. \"We can also discuss how I might cost Mom the election because I'm a one-man bisexual wrecking ball who exposed the vulnerability of the White House private email server.\"\n\n\"You think?\" his dad says. \"Nah. Come on. I don't think this election is gonna hinge on an email server.\"\n\nAlex arches a brow. \"You sure about that?\"\n\n\"Listen, maybe if Richards had more time to sow those seeds of doubt, but I don't think we're there. Maybe if it were 2016. Maybe if this weren't an America that already elected a woman to the highest office once. Maybe if I weren't sitting in a room with the three assholes responsible for electing the first openly gay man to the Senate in US history.\" Alex whoops and Luna inclines his head and raises his beer. \"But, nah. Is it gonna be a pain in your mom's ass for the second term? Shit, yeah. But she'll handle it.\"\n\n\"Look at you,\" Luna says over his beer. \"Answer for everything, eh?\"\n\n\"Listen,\" his dad says, \"somebody on this damn campaign has to keep their fucking cool while everyone else catastrophizes. Everything's gonna be fine. I believe that.\"\n\n\"And what about me?\" Alex says. \"You think I got a chance in politics after going supernova in every paper in the world?\"\n\n\"They got you,\" Oscar says, shrugging. \"It happens. Give it time. Try again.\"\n\nAlex laughs, but still, he reaches in and plucks up something deep down in his chest. Something shaped not like Claremont but Diaz\u2014no better, no worse, just different.\n\n* * *\n\nHenry gets his own room in the White House while he's in. The crown spared him for two nights before he returns to England for his own damage control tour. Once again, they're lucky to have Catherine back in the game; Alex doubts the queen would have been so generous.\n\nThis particularly is what makes it a little funny that Henry's room\u2014the customary quarters for royal guests\u2014is called the Queen's Bedroom.\n\n\"It's quite... aggressively pink, innit?\" Henry mutters sleepily.\n\nThe room is, really, aggressively pink, done up in the Federal style with pink walls and rose-covered rugs and bedding, pink upholstery on everything from the chairs and settee in the sitting area to the canopy on the four-poster bed.\n\nHenry's agreed to sleep in the room rather than Alex's \"because I respect your mother,\" as if every person who had a hand in raising Alex has not read in graphic detail the things they get up to when they share a bed. Alex has no such hang-ups and enjoys Henry's half-hearted grumblings when he sneaks in from the East Bedroom right down the hall.\n\nThey've woken up half-naked and warm, tucked in tight while the first autumn chill creeps in under the lacy curtains. Humming low in his chest, Alex presses the length of his body against Henry's under the blankets, his back to Henry's chest, the swell of his ass against\u2014\n\n\"Argh, hello,\" Henry mumbles, his hips hitching at the contact. Henry can't see his face, but Alex smiles anyway.\n\n\"Morning,\" Alex says. He gives his ass a little wiggle.\n\n\"Time's it?\"\n\n\"Seven thirty-two.\"\n\n\"Plane in two hours.\"\n\nAlex makes a small sound in the back of his throat and turns over, finding Henry's face soft and close, eyes only half-open. \"You sure you don't need me to come with you?\"\n\nHenry shakes his head without picking it up from the pillow, so his cheek squishes against it. It's cute. \"You're not the one who slagged off the crown and your own family in the emails that everybody in the world has read. I've got to handle that on my own before you come back over.\"\n\n\"That's fair,\" Alex says. \"But soon?\"\n\nHenry's mouth tugs into a smile. \"Absolutely. You've got the royal suitor photos to take, the Christmas cards to sign... Oh, I wonder if they'll have you do a line of skincare products like Martha\u2014\"\n\n\"Stop,\" Alex groans, poking him in the ribs. \"You're enjoying this too much.\"\n\n\"I'm enjoying it the perfect amount,\" Henry says. \"But, in all seriousness, it's... frightening but a bit nice. To do this on my own. I've not gotten to do that much, well, ever.\"\n\n\"Yeah,\" Alex says. \"I'm proud of you.\"\n\n\"Ew,\" Henry says in a flat American accent, and he laughs and Alex throws an elbow.\n\nHenry's pulling him and kissing him, sandy hair on a pink bedspread, long lashes and long legs and blue eyes, elegant hands pinning his wrists to the mattress. It's like everything he's ever loved about Henry in a moment, in a laugh, in the way he shivers, in the confident roll of his spine, in happy, unfettered sex in the well-furnished eye of a storm.\n\nToday, Henry goes back to London. Today, Alex goes back to the campaign trail. They have to figure out how to do this for real now, how to love each other in plain sight. Alex thinks they're up for it.\n\n# FIFTEEN\n\nnearly four weeks later\n\n\"Let me just get this hair, love.\"\n\n_\"Mum.\"_\n\n\"Soz, am I embarrassing you?\" Catherine says, her glasses on the tip of her nose as she rearranges Henry's thick hair. \"You'll thank me when you've not got a great cowlick in your official portrait.\"\n\nAlex has to admit, the royal photographer is being exceedingly patient about the whole thing, especially considering they waffled through three different locations\u2014Kensington Gardens, a stuffy Buckingham Palace library, the courtyard of Hampton Court Palace\u2014before they decided to screw it all for a bench in a locked-down Hyde Park.\n\n(\"Like a common vagrant?\" Queen Mary asked.\n\n\"Shut up, Mum,\" Catherine said.)\n\nThere's a certain need for formal portraits now that Alex is officially in \"courtship\" with Henry. He tries not to think too hard about his face on chocolate bars and thongs in Buckingham gift shops. At least it'll be next to Henry's.\n\nSome psychological math always goes into styling photos like these. The White House stylists have Alex in something he'd wear any day\u2014brown leather loafers, slim-fit chinos in a soft tan, a loose-collared Ralph Lauren chambray\u2014but in this context, it reads confident, roguish, decidedly American. Henry's in a Burberry button-down tucked into dark jeans and a navy cardigan that the royal shoppers squabbled over in Harrods for hours. They want a picture of a perfect, dignified, British intellectual, a loved-up boyfriend with a bright future as an academic and philanthropist. They even staged a little pile of books on the bench next to him.\n\nAlex looks over at Henry, who's groaning and rolling his eyes under his mother's preening, and smiles at how much closer this packaging is to the real, messy, complicated Henry. As close as any PR campaign is ever going to get.\n\nThey take about a hundred portraits just sitting on the bench next to each other and smiling, and part of Alex keeps stumbling over the disbelief he's actually here, in the middle of Hyde Park, in front of God and everybody, holding Henry's hand atop his own knee for the camera.\n\n\"If Alex from this time last year could see this,\" Alex says, leaning into Henry's ear.\n\n\"He'd say, 'Oh, I'm in love with Henry? That must be why I'm such a berk to him all the time,'\" Henry suggests.\n\n\"Hey!\" Alex squawks, and Henry's chuckling at his own joke and Alex's indignation, one arm coming up around Alex's shoulders. Alex gives into it and laughs too, full and deep, and that's the last hope for a serious tone for the day gone. The photographer finally calls it, and they're set loose.\n\nCatherine's got a busy day, she says\u2014three meetings before afternoon tea to discuss relocating into a royal residence more centrally located in London, since she's begun taking up more duties than ever. Alex can see the glint in her eye\u2014she'll be gunning for the throne soon. He's choosing not to say anything about it to Henry yet, but he's curious to see how it all plays out. She kisses them both and leaves them with Henry's PPOs.\n\nIt's a short walk over the Long Water back to Kensington, and they meet Bea at the Orangery, where a dozen members of her event-planning team are scurrying around, setting up a stage. She's tromping up and down rows of chairs on the lawn in a ponytail and rain boots, speaking very tersely on the phone about something called \"cullen skink\" and why on earth would she ever request cullen skink and even if she had in fact requested cullen skink in what universe would she ever need twenty bloody liters of cullen skink for anything, ever.\n\n\"What in the hell is a 'cullen skink'?\" Alex asks once she's hung up.\n\n\"Smoked haddock chowder,\" she says. \"Enjoy your first royal dog show, Alex?\"\n\n\"It wasn't too bad,\" Alex says, smirking.\n\n\"Mum is _beyond,_ \" Henry says. \"She offered to _edit my manuscript_ this morning. It's like she's trying to make up for five years of absentee parenting all at once. Which, of course, I love her very much, and I appreciate the effort, but, Christ.\"\n\n\"She's trying, H,\" Bea says. \"She's been on the bench for a while. Let her warm up a bit.\"\n\n\"I know,\" Henry says with a sigh, but his eyes are fond. \"How are things over here?\"\n\n\"Oh, you know,\" she says, waving her phone in the air. \"Just the maiden voyage of my very controversial fund upon which all future endeavors will be judged, so, no pressure at all. I'm only slightly cross with you for not making it a Henry Foundation\u2013Beatrice Fund double feature so I could unload half the stress onto you. All this fund-raising for sobriety is going to drive me to drink.\" She pats Alex on the arm. \"That's drunk humor for you, Alex.\"\n\nBea and Henry both had an October as busy as their mother's. There were a lot of decisions to be made in that first week: Would they ignore the revelations about Bea in the emails (no), would Henry be forced to enlist after all (after days of deliberation, no), and, above all, how could all this be made into a positive? The solution had been one Bea and Henry came up with together, twin philanthropic efforts under their own names. Bea's, a charity fund supporting addiction recovery programs all over the UK, and Henry's, an LGBT rights foundation.\n\nTo their right, the lighting trusses are going up quickly over the stage where Bea will be playing an \u00a38,000-a-ticket concert with a live band and celebrity guests tonight, her first solo fund-raiser.\n\n\"Man, I wish I could stay for the show,\" Alex says.\n\nBea beams. \"It's a shame Henry here was too busy signing papers with Auntie Pezza all week to learn some sheet music or we could have fired our pianist.\"\n\n\"Papers?\" Alex says, cocking an eyebrow.\n\nHenry shoots Bea a silencing glare. \"Bea\u2014\"\n\n\"For the youth shelters,\" she says.\n\n\" _Beatrice,_ \" Henry admonishes. \"It was going to be a _surprise._ \"\n\n\"Oh,\" Bea says, busying herself with her phone. \"Oops.\"\n\nAlex looks at Henry. \"What's going on?\"\n\nHenry sighs. \"Well. We were going to wait to announce it\u2014and to tell you, obviously\u2014until after the election, so as not to step on your moment. But...\" He puts his hands in his pockets, in that way he does when he's feeling proud of something but trying not to act like it. \"Mum and I agreed the foundation shouldn't just be national, that there was work to be done all over the world, and I specifically wanted to focus on homeless queer youth. So, Pez signed all our Okonjo Foundation youth shelters over.\" He bounces on his heels a little, visibly tamping down a broad smile. \"You're looking at the proud father of four worldwide soon-to-be shelters for disenfranchised queer teenagers.\"\n\n\"Oh my God, you _bastard,_ \" Alex practically yells, lunging at Henry and throwing his arms around his neck. \"That's amazing. I _stupid_ love you. _Wow._ \" He yanks back suddenly, stricken. \"Wait, oh my God, this means the one in Brooklyn too? Right?\"\n\n\"Yes, it does.\"\n\n\"Didn't you tell me you wanted to be hands-on with the foundation?\" Alex says, his pulse jumping. \"Don't you think maybe _direct supervision_ might be helpful while it gets off the ground?\"\n\n\"Alex,\" Henry tells him, \"I can't _move_ to New York.\"\n\nBea looks up. \"Why not?\"\n\n\"Because I'm the prince of\u2014\" Henry looks over at her and gestures at the Orangery, at Kensington, sputtering. \" _Here!_ \"\n\nBea shrugs, unmoved. \"And? It doesn't have to be permanent. You spent a month of your gap year talking to yaks in Mongolia, H. It's hardly unprecedented.\"\n\nHenry moves his mouth a couple times, ever the skeptic, and swivels back to Alex. \"Well, I'd still hardly see you, would I?\" he reasons. \"If you're in DC for work all the time, beginning your meteoric rise to the political stratosphere?\"\n\nAnd this, Alex has to admit, is a point. A point that after the year he's had, after everything, after the finally opened and perfectly passable LSAT scores sitting expectantly on his desk back home, feels less and less concrete every day.\n\nHe thinks about opening his mouth to say as much.\n\n\"Hello,\" says a polished voice from behind them, and they all turn to see Philip, starched and well groomed, striding across the lawn.\n\nAlex feels the slight flutter through the air of Henry's spine automatically straightening beside him. Philip came to Kensington two weeks ago to apologize to both Henry and Bea for the years since their father's death, the harsh words, the domineeringness, the intense scrutiny. For basically growing from an uptight people-pleaser into an abusive, self-righteous twat under the pressure of his position and the manipulation of the queen. \"He's fallen out with Gran,\" Henry had told Alex over the phone. \"That's the only reason I actually believe anything he says.\"\n\nYet, there's blood that can't be unshed. Alex wants to throw a punch every time he sees Philip's stupid face, but it's Henry's family, not his, so he doesn't get to make that call.\n\n\"Philip,\" Bea says coolly. \"To what do we owe the pleasure?\"\n\n\"Just had a meeting at Buckingham,\" Philip says. The meaning hangs in the air between them: a meeting with the queen because he's the only one still willing. \"Wanted to come by to see if I could help with anything.\" He looks down at Bea's Wellington boots next to his shiny dress shoes in the grass. \"You know, you don't have to be out here\u2014we've got plenty of staff who can do the grunt work for you.\"\n\n\"I know,\" Bea says haughtily, every inch a princess. \"I want to do it.\"\n\n\"Right,\" Philip says. \"Of course. Well, er. Is there anything I can help with?\"\n\n\"Not really, Philip.\"\n\n\"All right.\" Philip clears his throat. \"Henry, Alex. Portraits go all right?\"\n\nHenry blinks, clearly startled Philip would ask. Alex has enough diplomatic instincts to keep his mouth shut.\n\n\"Yeah,\" Henry says. \"Er, yes. It was all right. A bit awkward, you know, just having to sit there for ages.\"\n\n\"Oh, I remember,\" Philip says. \"When Mazzy and I did our first ones, I had this horrible rash on my arse from some idiotic poison-oak prank one of my uni friends had played on me that week, and it was all I could do to hold still and not rip my trousers off in the middle of Buckingham, much less try to take a nice photo. I thought she was going to murder me. Here's hoping yours turn out better.\"\n\nHe chuckles a little awkwardly, clearly trying to bond with them. Alex scratches his nose.\n\n\"Well, anyway, good luck, Bea.\"\n\nPhilip walks off, hands in his pockets, and all three of them watch his retreating back until it starts to disappear behind the tall hedges.\n\nBea sighs. \"D'you think I should have let him have a go at the cullen skink man for me?\"\n\n\"Not yet,\" Henry says. \"Give him another six months. He hasn't earned it yet.\"\n\n* * *\n\nBlue or gray? Gray or blue?\n\nAlex has never been so torn between two equally innocuous blazers in his entire life.\n\n\"This is stupid,\" Nora says. \"They're both boring.\"\n\n\"Will you please just help me pick?\" Alex tells her. He holds up a hanger in each hand, ignoring her judgmental look from where she's perched atop his dresser. The pictures from election night tomorrow, win or lose, will follow him for the rest of his life.\n\n\"Alex, seriously. I hate them both. You need something killer. This could be your fucking _swan song._ \"\n\n\"Okay, let's not\u2014\"\n\n\"Yes, okay, you're right, if the projections hold, we're fine,\" she says, hopping down. \"So, do you want to talk about why you're choosing to punt so hard on this particular moment in your career as a risk-taking fashion plate?\"\n\n\"Nope,\" Alex says. He waves the hangers at her. \"Blue or gray?\"\n\n\"Okay, so.\" She's ignoring him. \"I'll say it, then. You're nervous.\"\n\nHe rolls his eyes. \"Of course I'm nervous, Nora, it's a presidential election and the president gave birth to me.\"\n\n\"Try again.\"\n\nShe's giving him that look. The \"I've already analyzed all the data on how much shit you're full of\" look. He releases a hiss of a sigh.\n\n\"Fine,\" he says. \"Fine, yeah, I'm nervous about going back to Texas.\"\n\nHe tosses both the blazers at the bed. Shit.\n\n\"I always felt like Texas claiming me as their son was, you know, kind of conditional.\" He paces, rubbing the back of his neck. \"The whole half-Mexican, all Democrat thing. There's a very loud contingent there that does not like me and does not want me to represent them. And now, it's just. Not being straight. Having a boyfriend. Having a _gay sex scandal_ with a _European prince._ I don't know anymore.\"\n\nHe loves Texas\u2014he _believes_ in Texas. But he doesn't know if Texas still loves him.\n\nHe's paced all the way to the opposite side of the room from her, and she watches him and cocks her head to one side.\n\n\"So... you're afraid of wearing anything too flashy for your first post-coming-out trip home, on account of Texans' delicate hetero sensibilities?\"\n\n\"Basically.\"\n\nShe's looking at him now more like he's a very complex problem set. \"Have you looked at our polling on you in Texas? Since September?\"\n\nAlex swallows.\n\n\"No. I, uh.\" He scrubs his face with one hand. \"The thought, like... stresses me out? Like, I keep meaning to go look at the numbers, and then I just. Shut down.\"\n\nNora's face softens, but she doesn't move closer yet, giving him space. \"Alex. You could have asked me. They're... not bad.\"\n\nHe bites his lip. \"They're not?\"\n\n\"Alex, our base in Texas hasn't shifted on you since September, at all. If anything, they like you more. And a lot of the undecideds are pissed Richards came after a Texas kid. You're really fine.\"\n\n_Oh._\n\nAlex exhales a shaky breath, running one hand through his hair. He starts to pace back, away from the door, which he realizes he's gravitated near as some fight-or-flight reflex.\n\n\"Okay.\"\n\nHe sits down heavily on the bed.\n\nNora sits gingerly next to him, and when he looks at her, she's got that sharpness to her eyes like she does when she's practically reading his mind.\n\n\"Look. You know I'm not good at the whole, like, tactful emotional communication thing, but, uh, June's not here, so. I'm gonna. Fuckin'. Give it a go.\" She presses on. \"I don't think this is just about Texas. You were recently fucking traumatized in a big way, and now you're scared of doing or saying the kind of stuff you actually like and want to because you don't want to draw any more attention to yourself.\"\n\nAlex almost wants to laugh.\n\nNora is like Henry sometimes, in that she can cut right down to the truth of things, but Henry deals in heart and Nora deals in facts. It takes her razor's edge, sometimes, to get him to pull his head out of his ass.\n\n\"Uh, well, yeah. That's. Probably part of it,\" he agrees. \"I know I need to start rehabilitating my image if I want any chance in politics, but part of me is like... really? Right now? Why? It's weird. My whole life, I was hanging on to this imaginary future person I was gonna be. Like, the plan\u2014graduation, campaigns, staffer, Congress. That was it. Straight into the game. I was gonna be the person who could do that... who _wanted_ that. And now here I am, and the person I've become is... not that person.\"\n\nNora nudges their shoulders together. \"But do you like him?\"\n\nAlex thinks; he's different, for sure, maybe a little darker. More neurotic, but more honest. Sharper head, wilder heart. Someone who doesn't always want to be married to work, but who has more reasons to fight than ever.\n\n\"Yeah,\" he says finally. Firmly. \"Yeah, I do.\"\n\n\"Cool,\" she says, and he looks over to see her grinning at him. \"So do I. You're Alex. In all this stupid shit, that's all you ever needed to be.\" She grabs his face in both hands and squishes it, and he groans but doesn't push her off. \"So, like. You want to throw out some contingency plans? You want me to run some projections?\"\n\n\"Actually, uh,\" Alex says, slightly muffled from how Nora's still squishing his face between her hands. \"Did I tell you that I kind of... snuck off and took the LSAT this summer?\"\n\n\"Oh! Oh... _law school,_ \" she says, as simply as she said _dick you down_ all those months ago, the simple answer to where he's been unknowingly headed all along. She releases his face, shoving his shoulders instead, instantly excited. \"That's _it,_ Alex. Wait\u2014yes! I'm about to start applying for my master's; we can do it together!\"\n\n\"Yeah?\" he says. \"You think I can hack it?\"\n\n\"Alex. Yes. Alex.\" She's on her knees on the bed now, bouncing up and down. \"Alex, this is genius. Okay\u2014listen. You go to law school, I go to grad school, June becomes a speechwriter-slash-author Rebecca Traister\u2013Roxane Gay voice of a generation, I become the data scientist who saves the world, and you\u2014\"\n\n\"\u2014become a badass civil rights attorney with an illustrious Captain America-esque career of curb-stomping discriminatory laws and fighting for the disenfranchised\u2014\"\n\n\"\u2014and you and Henry become the world's favorite geopolitical power couple\u2014\"\n\n\"\u2014and by the time I'm Rafael Luna's age\u2014\"\n\n\"\u2014people are going to be _begging_ you to run for Senate,\" she finishes, breathless. \"Yeah. So, like, a lot slower than planned. But.\"\n\n\"Yeah,\" Alex says, swallowing. \"It sounds good.\"\n\nAnd there it is. He's been teetering on the edge of letting go of this specific dream for months now, terrified of it, but the relief is startling, a mountain off his back.\n\nHe blinks in the face of it, thinks of June's words, and has to laugh. \"Fire under my ass for no good goddamn reason.\"\n\nNora pulls a face. She recognizes the June-ism. \"You are... passionate, to a fault. If June were here, she would say taking your time is going to help you figure out how best to use that. But I'm here, so, I'm gonna say: You are great at hustling, and at policy, and at leading and rallying people. You are so fucking smart that most people want to punch you. Those are all skills that will only improve over time. So, like, you are gonna crush it.\"\n\nShe jumps to her feet and ducks into his closet, and he can hear hangers sliding around. \"Most importantly,\" she goes on, \"you have become an icon of something, which is, like, a very big deal.\"\n\nShe emerges with a hanger in her hand: a jacket he's never worn out before, one she convinced him to buy online for an obscene price the night they got drunk and watched _The West Wing_ in a hotel in New York and let the tabloids think they were screwing. It's fucking _Gucci,_ a midnight-blue bomber jacket with red, white, and blue stripes at the waistband and cuffs.\n\n\"I know it's a lot, but\"\u2014she slaps the jacket against his chest\u2014\"you give people hope. So, get back out there and be Alex.\"\n\nHe takes the jacket from her and tries it on, checks his reflection in the mirror. It's perfect.\n\nThe moment is split with a half scream from the hallway outside of his bedroom, and he and Nora both run to the door.\n\nIt's June, tumbling into Alex's bedroom with her phone in one hand, jumping up and down, her hair bouncing on her shoulders. She's clearly come straight from one of her runs to the newsstand because her other arm is laden with tabloids, but she dumps them unceremoniously on the floor.\n\n\"I got the book deal!\" she shrieks, waving her phone in their faces. \"I was checking my email and\u2014the memoir\u2014 _I got the fucking deal!_ \"\n\nAlex and Nora both scream too, and they haul her into a six-armed hug, whooping and laughing and stomping on one another's feet and not caring. They all end up kicking off their shoes and jumping on the bed, and Nora FaceTimes Bea, who finds Henry and Pez in one of Henry's rooms, and they all celebrate together. It feels complete, the gang, as Cash once called them. They've earned their own media nickname in the wake of everything: The Super Six. Alex doesn't mind it.\n\nHours later, Nora and June fall asleep against Alex's headboard, June's head in Nora's lap and Nora's fingers in her hair, and Alex sneaks off to the en suite to brush his teeth. He nearly slips on something on the way back, and when he looks down, he has to do a double take. It's an issue of _HELLO! US_ from June's abandoned stack of magazines, and the image dominating the cover is one of the shots from his and Henry's portrait session.\n\nHe bends down to pick it up. It's not one of the posed shots\u2014it's one he didn't even realize had been taken, one he definitely didn't think would be released. He should have given the photographer more credit. He managed to capture the moment right when Henry cracked a joke, a candid, genuine photo, completely caught up in each other, Henry's arm around him and his own hand reaching up to grasp for Henry's on his shoulder.\n\nThe way Henry's looking at him in the picture is so affectionate, so openly loving, that seeing it from a third person's perspective almost makes Alex want to look away, like he's staring into the sun. He called Henry the North Star once. That wasn't bright enough.\n\nHe thinks again about Brooklyn, about Henry's youth shelter there. His mom knows someone at NYU Law, right?\n\nHe brushes his teeth and climbs into bed. Tomorrow they find out, win or lose. A year ago\u2014six months ago\u2014it would have meant no sleep tonight. But he's a new kind of icon now, someone who laughs on even footing with his royal boyfriend on the cover of a magazine, someone willing to accept the years stretching ahead of him, to give himself time. He's trying new things.\n\nHe props a pillow up on June's knees, stretches his feet out over Nora's legs, and goes to sleep.\n\n* * *\n\nAlex tugs his bottom lip between his teeth. Scuffs the heel of his boot against the linoleum floor. Looks down at his ballot.\n\n> PRESIDENT and VICE PRESIDENT of the UNITED STATES\n> \n> Vote for One\n\nHe picks up the stylus chained to the machine, his heart behind his molars, and selects: _CLAREMONT, ELLEN and HOLLERAN, MICHAEL._\n\nThe machine chirps its approval, and to its gently humming mechanisms, he could be anybody. One of millions, a single tally mark, worth no more or less than any of the others. Just pressing a button.\n\n* * *\n\nIt's a risk, doing election night in their hometown. There's no _rule,_ technically, saying that the sitting president can't host their rally in DC, but it is customary to do it at home. Still, though.\n\n2016 was bittersweet. Austin is blue, deep blue, and Ellen won Travis County by 76 percent, but no amount of fireworks and champagne corks in the streets changed the fact that they lost the state they stood in to make the victory speech. Still, the Lometa Longshot wanted to come home again.\n\nThere's been progress in the past year: a few court victories Alex has kept track of in his trusty binder, registration drives for young voters, the Houston rally, the shifting polls. Alex needed a distraction after the whole tabloid nightmare, so he threw himself into an after-hours committee with a bunch of the campaign's Texas organizers, Skyping in to figure out logistics of a massive election day shuttle service throughout Texas. It's 2020, and Texas is a battleground state for the first time in years.\n\nHis last election night was on the wide-open stretch of Zilker Park, against the backdrop of the Austin skyline. He remembers everything.\n\nHe was eighteen years old in his first custom-made suit, corralled into a hotel around the corner with his family to watch the results while the crowd swelled outside, running with his arms open down the hallway when they called 270. He remembers it felt like his moment, because it was his mom and his family, but also realizing it was, in a way, not his moment at all, when he turned around and saw Zahra's mascara running down her face.\n\nHe stood next to the stage set into the hillside of Zilker and looked into eyes upon eyes upon eyes of women who were old enough to have marched on Congress for the VRA in '65 and girls young enough never to have known a president who was a white man. All of them looking at their first Madam President. And he turned and looked at June at his right side and Nora at his left, and he distinctly remembers pushing them out onto the stage ahead of him, giving them a full thirty seconds of soaking it in before following them into the spotlight.\n\nThe soles of his boots hit brown grass behind the Palmer Events Center like he's coming down from a much greater altitude than the back seat of a limo.\n\n\"It's early,\" Nora is saying, thumbing through her phone as she climbs out behind him in a plunging black jumpsuit and killer heels. \"Like, really early for these exit polls, but I'm pretty sure we have Illinois.\"\n\n\"Cool, that was projected,\" Alex says. \"We're on target so far.\"\n\n\"I wouldn't go that far,\" Nora tells him. \"I don't like how Pennsylvania looks.\"\n\n\"Hey,\" June says. Her own dress is carefully selected, off-the-rack J. Crew, white lace, girl-next-door. Her hair is braided down one shoulder. \"Can't we, like, have _one_ drink before y'all start doing this? I heard there are mojitos.\"\n\n\"Yeah, yeah,\" Nora says, but she's still staring down at her phone, brow furrowed.\n\n> HRH Prince Dickhead\n\nNov 3, 2020, 6:37 PM\n\n> HRH Prince Dickhead  \n> \n> Pilot says we're having visibility problems? May have to reroute and land elsewhere.\n> \n> HRH Prince Dickhead  \n> \n> Landing in Dallas? Is that far?? I've no bloody clue about American geography.\n> \n> HRH Prince Dickhead  \n> \n> Shaan has informed me this is, in fact, far. Landing soon. Will try to take off again once the weather clears.\n> \n> HRH Prince Dickhead  \n> \n> I'm sorry, I'm so sorry. How are things on your end?\n> \n> things are shit\n> \n> please get your ass here asap i'm stressing tf out\n\n> Oliver Westbrook   @BillsBillsBills\n> \n> Any GOPers still backing Richards after his actions toward a member of the First Family\u2014and, now, this week's rumors of sexual predation\u2014are going to have to reckon with their Protestant God tomorrow morning.\n> \n> 7:32 PM \u00b7 3 Nov 2020\n\n> 538 politics   @538politics\n> \n> Our projections had Michigan, Ohio, Pennsylvania, and Wisconsin all at a 70% or higher chance of going blue, but latest returns have them too close to call. Yeah, we're confused too.\n> \n> 8:04 PM \u00b7 3 Nov 2020\n\n> The New York Times   @nytimes\n> \n> #Election2020 latest: a bruising round of calls for Pres. Claremont brings the electoral tally up to 178 for Sen. Richards. Claremont lags behind at 113.\n> \n> 9:15 PM \u00b7 3 Nov 2020\n\n* * *\n\nThey've partitioned off the smaller exhibit hall for VIPs only\u2014campaign staff, friends and family, congresspeople. On the other side of the event center is the crowd of supporters with their signs, their CLAREMONT 2020 and HISTORY, HUH? T-shirts, overflowing under the architectural canopies and into the surrounding hills. It's supposed to be a party.\n\nAlex has been trying not to stress. He knows how presidential elections go. When he was a kid, this was his Super Bowl. He used to sit in front of the living room TV and color each state in with red and blue magic markers as the night went on, allowed to stay up hours past his bedtime for one blessed night at age ten to watch Obama beat McCain. He watches his dad's jaw in profile now, trying to remember the triumph in the set of it that night.\n\nThere was a magic, then. Now, it's personal.\n\nAnd they're losing.\n\nThe sight of Leo coming in through a side door isn't entirely unexpected, and June rises from her chair and meets them both in a quiet corner of the room on the same instinct. He's holding his phone in one hand.\n\n\"Your mother wants to talk to you,\" Leo says, and Alex automatically reaches out until Leo holds out a hand to stop him. \"No, sorry, Alex, not you. June.\"\n\nJune blinks. \"Oh.\" She steps forward, pushes her hair away from her ear. \"Mom?\"\n\n\"June,\" says the sound of their mother's voice over the little speaker. On the other end, she's in one of the arena's meeting rooms, a makeshift office with her core team. \"Baby. I need you to, uh. I need you to come in here.\"\n\n\"Okay, Mom,\" she says, her voice measured and calm. \"What's going on?\"\n\n\"I just. I need you to help me rewrite this speech for, uh.\" There's a considerable pause. \"Well. Just in case of concession.\"\n\nJune's face goes utterly blank for a second, and suddenly, vividly _furious._\n\n\"No,\" she says, and she grabs Leo by the forearm so she can talk directly into the speaker. \" _No,_ I'm not gonna do that, because you're not gonna lose. Do you hear me? You're not losing. We're gonna fucking do this for four more years, _all of us._ I am not writing you a _goddamn concession speech,_ ever.\"\n\nThere's another pause across the line, and Alex can picture their mother in her little makeshift Situation Room upstairs, glasses on, high heels still in the suitcase, staring at the screens, hoping and trying and praying. President Mom.\n\n\"Okay,\" she says evenly. \"Okay. Alex. Do you think you could get up and say something for the crowd?\"\n\n\"Yeah, yeah, sure, Mom,\" he says. He clears his throat, and it comes out as strong as hers the second time. \"Of course.\"\n\nA third pause, then. \"God, I love you both so much.\"\n\nLeo leaves, and he's quickly replaced by Zahra, whose sleek red dress and ever-present coffee thermos are the biggest comfort Alex has seen all night. Her ring flashes at him, and he thinks of Shaan and wishes desperately Henry was _here_ already.\n\n\"Fix your face,\" she says, straightening his collar as she shepherds him and June through to the main exhibit hall and into the back of the stage area. \"Big smiles, high energy, confidence.\"\n\nHe turns helplessly to June. \"What do I say?\"\n\n\"Little bit, ain't no time for me to write you anything,\" she tells him. \"You're a leader. Go lead. You got this.\"\n\nOh God.\n\n_Confidence._ He looks down at the cuffs of his jacket again, the red, white, and blue. _Be Alex,_ Nora said when she handed it to him. _Be Alex._\n\nAlex is\u2014two words that told a few million kids across America they weren't alone. A letterman jacket in APUSH. Secret loose panels in White House windows. Ruining something because you wanted it too badly and still getting back up and trying again. Not a prince. Something bigger, maybe.\n\n\"Zahra,\" he asks. \"Did they call Texas yet?\"\n\n\"No,\" she says. \"Still too close.\"\n\n_\"Still?\"_\n\nHer smile is knowing. \" _Still._ \"\n\nThe spotlight is almost blinding when he walks out, but he knows something. Deep down in his heart. They still haven't called Texas.\n\n\"Hey, y'all,\" he says to the crowd. His hand squeezes the microphone, but it's steady. \"I'm Alex, your First Son.\" The hometown crowd goes wild, and Alex grins and means it, leans into it. When he says what he says next, he intends to believe it.\n\n\"You know what's crazy? Right now, Anderson Cooper is on CNN saying Texas is too close to call. _Too close to call._ Y'all may not know this about me, but I'm kind of a history nerd. So I can tell you, the last time Texas was _too close to call_ was in 1976. In 1976, we went blue. It was Jimmy Carter, in the wake of Watergate. He just barely squeezed out fifty-one percent of our vote, and we helped him beat Gerald Ford for the presidency.\n\n\"Now, I'm standing here, and I'm thinking about it... A reliable, hardworking, honest, Southern Democrat versus corruption, and maliciousness, and hate. And one big state full of honest people, sick as hell of being lied to.\"\n\nThe crowd absolutely loses it, and Alex almost laughs. He raises his voice into the microphone, speaks up over the sound of cheers and applause and boots stomping on the floor of the hall. \"Well, it sounds a little familiar to me, is all. So, what do y'all think, Texas? \u00bfSe repetir\u00e1 la historia? Are we gonna make history repeat itself tonight?\"\n\nThe roar says it all, and Alex yells with them, lets the sound carry him off the stage, lets it wrap around his heart and squeeze back in the blood that's drained out of it all night. The second he steps backstage, there's a hand on his back, the achingly familiar gravity of someone else's body reentering his space before it even touches his, a clean, familiar scent light in the air between.\n\n\"That was _brilliant,_ \" Henry says, smiling, in the flesh, _finally._ He's gorgeous in a navy-blue suit and a tie that, upon closer inspection, is patterned with little yellow roses.\n\n\"Your tie\u2014\"\n\n\"Oh, yes,\" he says, \"yellow rose of Texas, is it? I read that was a thing. Thought it might be good luck.\"\n\nAll at once, Alex is in love all over again. He wraps the tie once around the back of his hand and reels Henry in and kisses him like he never has to stop. Which\u2014he remembers, and laughs into Henry's mouth\u2014he doesn't.\n\nIf he's talking about who he is, he wishes he'd been someone smart enough to have done this last year. He wouldn't have made Henry banish himself to a bunch of frozen shrubbery, and he wouldn't have just stood there while Henry gave him the most important kiss of his life. It would have been like this. He would have taken Henry's face in both hands and kissed him hard and deep and on purpose and said, \"Take anything you want and know you deserve to have it.\"\n\nHe pulls back and says, \"You're late, Your Highness.\"\n\nHenry laughs. \"Actually, I'm just in time for the upswing, it would seem.\"\n\nHe's talking about the latest round of calls, which apparently came in while Alex was onstage. Out in their VIP area, everyone's out of their seat, watching Anderson Cooper and Wolf Blitzer parse the returns on the big screens. Virginia: Claremont. Colorado: Claremont. Michigan: Claremont. Pennsylvania: Claremont. It almost fully makes up the difference in votes, with the West Coast still to go.\n\nShaan is here too, in one corner with Zahra, huddled with Luna and Amy and Cash, and Alex's head almost spins at the thought of how many nations could be brought to their knees by this particular gang. He grabs Henry's hand and pulls him into it all.\n\nThe magic comes in a nervous trickle\u2014Henry's tie, hopeful lilts in voices, a few stray bits of confetti that escape the nets laced through the rafters and get stuck in Nora's hair\u2014and then, all at once.\n\n10:30 brings the big rush: Richards steals Iowa, yes, and sews up Utah and Montana, but the West Coast comes storming in with California's fifty-five fucking electoral votes. \"Big damn heroes,\" Oscar crows when it's called to raucous cheers and nobody's surprise, and he and Luna slap their palms together. _West Side Bastardos._\n\nBy midnight, they've taken the lead, and it does, finally, feel like a party, even if they're not out of the woods yet. Drinks are flowing, voices are loud, the crowd on the other side of the partition is electric. Gloria Estefan wailing through the sound system feels fitting again, not a stabbing, sick irony at a funeral. Across the room, Henry's with June, making a gesture at her hair, and she turns and lets him fix a piece of her braid that came loose earlier in a fit of anxiety.\n\nAlex is so busy watching them, his two favorite people, he doesn't notice another person in his path until he collides with them headfirst, spilling their drink and almost sending them both stumbling into the massive victory cake on the buffet table.\n\n\"Jesus, sorry,\" he says, immediately reaching for a pile of napkins.\n\n\"If you knock over another expensive cake,\" says an extremely familiar whiskey-warm drawl, \"I'm pretty sure your mom is gonna disinherit you.\"\n\nHe turns to see Liam, almost the same as he remembers\u2014tall, broad-shouldered, sweet-faced, scruffy.\n\nHe's so mad he has such a specific type of dude and never even noticed it for so long.\n\n\"Oh my God, you came!\"\n\n\"Of course I did,\" Liam says, grinning. Beside him, there's a cute guy grinning too. \"I mean, it kind of seemed like the Secret Service were gonna come requisition me from my apartment if I didn't come.\"\n\nAlex laughs. \"Look, the presidency hasn't changed me _that_ much. I'm still as aggressive a party instigator as I ever was.\"\n\n\"I'd be disappointed if you weren't, man.\"\n\nThey both grin, and God, on tonight of all nights it's good to see him, good to clear the air, good to stand next to someone outside of family who knew him before all this.\n\nA week after he got outed, Liam texted him: 1. I wish we hadn't been such dumb assholes back then so we both could have helped each other out with stuff. 2. Jsyk, a reporter from some right-wing website called me yesterday to ask me about my history with you. I told him to go fuck himself, but I thought you'd want to know.\n\nSo yeah, of course he got a personal invitation.\n\n\"Listen, I,\" Alex starts, \"I wanted to thank you\u2014\"\n\n\"Do not,\" Liam interrupts him. \"Seriously. Okay? We're cool. We'll always be cool.\" He makes a dismissive gesture with one hand and nudges the cute, dark-eyed guy at his side. \"Anyway, this is Spencer, my boyfriend.\"\n\n\"Alex,\" Alex introduces himself. Spencer's handshake is strong, all farmboy. \"Good to meet you, man.\"\n\n\"It's an honor,\" Spencer says earnestly. \"My mom canvassed for your mom when she ran for Congress back in the day, so like, we go way back. She's the first president I ever voted for.\"\n\n\"Okay, Spence, be cool,\" Liam says, putting an arm around Spencer's shoulders. A beam of pride cuts through Alex; if Spencer's parents were Claremont volunteers, they're definitely more open-minded than he remembers Liam's being. \"This guy shit his pants on the bus on the way back from the aquarium in fourth grade, so like, he's not that big of a deal.\"\n\n\"For the _last time,_ you douchebag,\" Alex huffs, \"that was Adam Villanueva, not me!\"\n\n\"Yeah, I know what I saw,\" Liam says.\n\nAlex is just opening his mouth to argue when someone shouts his name\u2014a photo op or interview or something for _BuzzFeed_. \"Shit. I gotta go, but Liam, we have, like, a shitload to catch up on. Can we hang this weekend? Let's hang this weekend. I'm in town all weekend. Let's hang this weekend.\"\n\nHe's already walking away backward, and Liam is rolling his eyes in an annoyed but fond way, not in a this-is-why-I-stopped-talking-to-you way, so he keeps going. The interview is quick, cut off mid-sentence: Anderson Cooper's face looms on the screen overhead like a disgustingly handsome Hunger Games cannon, announcing they're ready to call Florida.\n\n\"Come on, you backyard-shooting-range motherfuckers,\" Zahra is muttering under her breath beside him when he falls in with his people.\n\n\"Did she just say backyard shooting range?\" Henry asks, leaning into Alex's ear. \"Is that a real thing a person can have?\"\n\n\"You really have a lot to learn about America, mijo,\" Oscar tells him, not unkindly.\n\nThe screen flashes red\u2014 _RICHARDS_ \u2014and a collective groan grinds through the room.\n\n\"Nora, what's the math?\" June says, rounding on her, a slightly frantic look in her eyes. \"I majored in nouns.\"\n\n\"Okay,\" Nora says, \"at this point we just need to get over 270 or make it impossible for Richards to get over 270\u2014\"\n\n\"Yes,\" June cuts in impatiently, \"I am familiar with how the electoral college works\u2014\"\n\n\"You asked!\"\n\n\"I didn't mean to remediate me!\"\n\n\"You're kinda hot when you get all indignant.\"\n\n\"Can we _focus_?\" Alex puts in.\n\n\"Okay,\" Nora says. She shakes out her hands. \"So, right now we can get over 270 with Texas or Nevada _and_ Alaska combined. Richards has to get all three of those. So nobody is out of the game yet.\"\n\n\"So, we _have_ to get Texas now?\"\n\n\"Not unless they call Nevada,\" Nora says, \"which never happens this early.\"\n\nShe barely has time to finish before Anderson Cooper is back onscreen with breaking news. Alex wonders briefly what it's going to be like to have future Anderson Cooper stress hallucinations. _NEVADA: RICHARDS._\n\n\"Are you _fucking_ kidding me?\"\n\n\"So, now it's essentially\u2014\"\n\n\"Whoever wins Texas,\" Alex says, \"wins the presidency.\"\n\nThere's a heavy pause, and June says, \"I'm gonna go stress eat the cold pizza the polling people have. Sound good? Cool.\" And she's gone.\n\nBy 12:30, nobody can believe it's down to this.\n\nTexas has never in history gone this long without being called. If it were any other state, Richards probably would have called to concede by now.\n\nLuna is pacing. Alex's dad is sweating through his suit. June is going to smell like pizza for a week. Zahra is on the phone, yelling into someone's voicemail, and when she hangs up, she explains that her sister is having trouble getting into a good daycare and agreed to put Zahra on the job as an outlet for her stress. Ellen, too tense to stay upstairs, is stalking through it all like a hungry lioness.\n\nAnd that's when June comes charging up to them, her hand on the arm of a girl Alex recognizes\u2014her college roommate, his brain supplies. She's got on a poll volunteer shirt and a broad smile.\n\n\"Y'all\u2014\" June says, breathless. \"Molly just\u2014she just came from\u2014fuck, just, tell them!\"\n\nAnd Molly opens her blessed mouth and says, \"We think you have the votes.\"\n\nNora drops her phone. Ellen steps over it to grab Molly's other arm. \"You think or you know?\"\n\n\"I mean, we're pretty sure\u2014\"\n\n\"How sure?\"\n\n\"Well, they just counted another 10,000 ballots from Harris County\u2014\"\n\n\"Oh my God\u2014\"\n\n\"Wait, _look_ \u2014\"\n\nIt's on the projection screen now. They're calling it. _Anderson Cooper, you handsome bastard._\n\nTexas is gray for five more seconds, before flooding beautiful, beautiful, unmistakable Lake LBJ blue.\n\nThirty-eight votes for Claremont, for a grand total of 301. And the presidency.\n\n\" _Four more years!_ \" Alex's mom outright screams, louder than he's heard her scream in _years._\n\nThe cheers come in a hum, in a rumble, and finally, in a storm, pressing from the other side of the partition, from the hills surrounding the arena and the city surrounding the streets, from the country itself. From, maybe, a few sleepy allies in London.\n\nFrom his side, Henry, whose eyes are wet, seizes Alex's face roughly in both hands and kisses him like the end of the movie, whoops, and shoves him at his family.\n\nThe nets are cut loose from the ceiling, and down come the balloons, and Alex staggers into a press of bodies and his father's chest, a delirious hug, into June, who is a crying disaster, and Leo, who is somehow crying _more._ Nora is sandwiched between both beaming, proud parents, screaming at the top of her lungs, and Luna is throwing Claremont campaign pamphlets in the air like a mafioso with hundred dollar bills. He sees Cash, severely testing the weight limits of the venue's chairs by dancing on one, and Amy, waving around her phone so her wife can see it all over FaceTime, and Zahra and Shaan, aggressively making out against a giant stack of CLAREMONT\/HOLLERAN 2020 yard signs. WASPy Hunter hoisting another staffer up on his shoulders, Liam and Spencer raising their beers in a toast, a hundred campaign staffers and volunteers crying and shouting in disbelief and joy. They did it. They _did_ it. The Lometa Longshot and a long-awaited blue Texas.\n\nThe crowd pushes him back into Henry's chest, and after absolutely everything, all the emails and texts and months on the road and secret rendezvous and nights of wanting, the whole accidentally-falling-in-love-with-your-sworn-enemy-at-the-absolute-worst-possible-time thing, they made it. Alex said they would\u2014he _promised._ Henry's smiling so wide and bright that Alex thinks his heart's going to break trying to hold the size of this entire moment, the completeness of it, a thousand years of history swelling inside his rib cage.\n\n\"I need to tell you something,\" Henry says, breathless, when Alex pulls back. \"I bought a brownstone. In Brooklyn.\"\n\nAlex's mouth falls open. \"You _didn't_!\"\n\n\"I did.\"\n\nAnd for a fraction of a second, a whole crystallized life flashes into view, a next term and no elections left to win, a schedule packed with classes and Henry smiling from the pillow next to him in the gray light of a Brooklyn morning. It drops right into the well of his chest and spreads, like how hope spreads. It's a good thing everyone else is already crying.\n\n\"Okay, people,\" says Zahra's voice through the rush of blood and love and adrenaline and noise in his ears. Her mascara is streaming, her lipstick smeared across her chin. Beside her, he can hear his mother on the phone with one finger jammed into her ear, taking Richards's concession call. \"Victory speech in fifteen. Places, let's go!\"\n\nAlex finds himself shuffled sideways, through the crowd and over to a little corral near the stage, behind the curtains, and then his mother's on stage, and Leo, and Mike and his wife, and Nora and her parents and June and their dad. Alex strides out after them, waving into the white glow of the spotlight, shouting a jumble of languages into the noise. He's so caught up that he doesn't realize at first Henry isn't at his side, and he turns back to see him hovering in the wings, just behind a curtain. Always hesitant to step on anyone's moment.\n\nThat's not going to fly anymore. He's family. He's part of it all now, headlines and oil paintings and pages in the Library of Congress, etched right alongside. And he's part of _them._ Goddamn forever.\n\n\"Come on!\" Alex yells, waving him over, and Henry spares a second to look panicked before he's tipping his chin up and buttoning his suit jacket and stepping out onto the stage. He gravitates to Alex's side, beaming. Alex throws one arm around him and the other around June. Nora presses in at June's other side.\n\nAnd President Ellen Claremont steps up to the podium.\n\n> EXCERPT: PRESIDENT ELLEN CLAREMONT'S VICTORY ADDRESS FROM AUSTIN, TEXAS, NOVEMBER 3, 2020\n> \n> Four years ago, in 2016, we stood at a precipice as a nation. There were those who would have seen us stumble backward into hatred and vitriol and prejudice, who wanted to reignite old embers of division within our country's very soul. You looked them square in the eye and said, \"No. We won't.\"\n> \n> You voted instead for a woman and a family with Texas dirt under their shoes, who would lead you into four years of progress, of carrying on a legacy of hope and change. And tonight, you did it again. You chose me. And I humbly, humbly thank you.\n> \n> And my family\u2014my family thanks you too. My family, made up of the children of immigrants, of people who love in defiance of expectations or condemnation, of women determined never to back down from what's right, a braid of histories that stands for the future of America. My family. Your First Family. We intend to do everything we can, for the next four years and the years beyond, to continue making you proud.\n\n* * *\n\nThe second round of confetti is still falling when Alex grabs Henry by the hand and says, \"Follow me.\"\n\nEveryone's too busy celebrating or doing interviews to see them slip out the back door. He trades Liam and Spencer the promise of a six-pack for their bikes, and Henry doesn't ask questions, just kicks the stand out and disappears into the night behind him.\n\nAustin feels different somehow, but it hasn't changed, not really. Austin is dried flowers from a homecoming corsage in a bowl by the cordless phone, the washed-out bricks of the rec center where he tutored kids after school, a beer bummed off a stranger on the spill of the Barton Creek Greenbelt. The nopales, the hipster cold brews. It's a weird, singular constant, the hook in his heart that's kept tugging him back to earth his whole life.\n\nMaybe it's just that _he's_ different.\n\nThey cross the bridge into downtown, the gray grids intersecting Lavaca, the bars overflowing with people yelling his mother's name, wearing his own face on their chests, waving Texas flags, American flags, Mexican flags, pride flags. There's music echoing through the streets, loudest when they reach the Capitol, where someone has climbed up the front steps and erected a set of loudspeakers blasting Starship's \"Nothing's Gonna Stop Us Now.\" Somewhere above, against the thick clouds: fireworks.\n\nAlex takes his feet off the pedals and glides past the massive, Italian Renaissance Revival fa\u00e7ade of the Capitol, the building where his mom went to work every day when he was a kid. It's taller than the one back in DC. Everything's bigger, after all.\n\nIt takes twenty minutes to reach Pemberton Heights, and Alex leads the Prince of England up onto the high curb of a neighborhood in Old West Austin and shows him where to throw his bike in the yard, spokes still spinning little shadow lines across the grass. The sounds of expensive leather soles on the cracked front steps of the old house on Westover don't sound any stranger than his own boots. Like coming home.\n\nHe steps back and watches Henry take it all in\u2014the butter-yellow siding, the big bay window, the handprints in the sidewalk. Alex hasn't been inside this house since he was twenty. They pay a family friend to look after it, wrap the pipes, run the water. They can't bear to let it go. Nothing's changed inside, just been boxed up.\n\nThere are no fireworks out here, no music, no confetti. Just sleeping, single-family homes, TVs finally switched off. Just a house where Alex grew up, where he saw Henry's picture in a magazine and felt a flicker of something, a start.\n\n\"Hey,\" Alex says. Henry turns back to him, his eyes silver in the wash of the streetlight. \"We _won._ \"\n\nHenry takes his hand, one corner of his mouth tugging gently upward. \"Yeah. We won.\"\n\nAlex reaches down into the front of his dress shirt and finds the chain with his fingers, pulls it out carefully. The ring, the key.\n\nUnder winter clouds, victorious, he unlocks the door.\n\n# ACKNOWLEDGMENTS\n\nI came up with the idea for this book on an I-10 off-ramp in early 2016, and I never imagined what it would turn out to be. I mean, at that point I couldn't imagine what _2016 itself_ would turn out to be. Yikes. For months after November, I gave up on writing this book. Suddenly what was supposed to be a tongue-in-cheek parallel universe needed to be escapist, trauma-soothing, alternate-but-realistic reality. Not a perfect world\u2014one still believably fucked up, just a little better, a little more optimistic. I wasn't sure I was up to the task. I hoped I was.\n\nWhat I hoped to do, and what I hope I have done with this book by the time you've finished it, my dear reader, is to be a spark of joy and hope you needed.\n\nI couldn't have done any of this without the help of so many. To my angel of an agent, Sara Megibow, thank you for driving this crazy bus. I went into this whole experience hoping to find one person who felt even half of what I feel for this book, and you matched me from the first moment we spoke. Thank you for being the champion this book needed and the reassurance always at my back. To Vicki Lame, my editor, the Texas girl who fought for this book and always saw in it what it could mean to people. Thank you for giving this your all, for forever being the person in the corner of the ring with the water bottle. You and the team at St. Martin's Griffin have literally made dreams come true. Thank you to my publicity team, DJ DeSmyter and Meghan Harrington, and to everyone else who threw themselves behind this book.\n\nMore thanks: Elizabeth Freeburg, who taught me more than I can ever give back to her, without whom I'd be half the writer I am today. Lena Barsky, who doula'd this entire novel, who was the first to love these characters as much as I do. Sasha Smith, my literary sherpa who believed in me most, without whom I would have been drowning before I was even out of the slip. Shanicka Anderson, the beta reader of my dreams, who loved this book even when it was 40,000 words too long. Lauren Heffker, the person who sat with me in a Taco Bell while I untangled this plot, who never didn't want to hear what I was thinking. Season Vining, who poured my wine and told me that my dream wasn't so unattainable. Leah Romero, my number-one fan and political inspiration, the reader I was always writing to impress. Tiffany Martinez, who read this book with care and love and gave it to me straight. Laura Marquez, who helped with translations. CJSR, who knows it all, whose sleepless nights this book happened in spite of. My FoCo fam, my new home.\n\nTo my family, who have done more for me over the years than any person deserves: You had no idea what you were signing on for when I told you I wrote a book, but y'all still cheered me on. Thank you for loving me as I am. Thank you for letting me be your weirdo baby. To Dad, my original storyteller: I know you always knew I had this in me. Thank you for helping me believe it. Big as the universe, over the clouds, forever. This is my best work to date.\n\nTo the sources that helped me with the mountains of research I did for this: WhiteHouseMuseum.org, the Royal Collection Online, _My Dear Boy_ by Rictor Norton, the V&A's extremely helpful website, countless others. To the country of Norway, literally, for the week that broke me out of the slump and made 110,000 words of the first draft happen. To \"Texas Reznikoff\" by Mitski.\n\nTo every person in search of somewhere to belong who happened to pick up this book, I hope you found a place in here, even if just for a few pages. You are loved. I wrote this for you.\n\nKeep fighting, keep making history, keep looking after one another.\n\nAffectionately yrs. Have a Shiner on me.\n\n# **ABOUT THE AUTHOR**\n\nCASEY MCQUISTON grew up in the swamps of Southern Louisiana, where she cultivated an abiding love for honey butter biscuits and stories with big, beating hearts. She studied journalism and worked in magazine publishing for years before returning to her first love: joyous, offbeat romantic comedies and escapist fiction. She now lives in the mountains of Fort Collins, Colorado, with a collection of caftans and her poodle mix, Pepper. You can sign up for email updates here.\n\n**Thank you for buying this**\n\n**St. Martin's Press ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\n\n# CONTENTS\n\n  1. Title Page\n  2. Copyright Notice\n  3. Dedication\n  4. Chapter One\n  5. Chapter Two\n  6. Chapter Three\n  7. Chapter Four\n  8. Chapter Five\n  9. Chapter Six\n  10. Chapter Seven\n  11. Chapter Eight\n  12. Chapter Nine\n  13. Chapter Ten\n  14. Chapter Eleven\n  15. Chapter Twelve\n  16. Chapter Thirteen\n  17. Chapter Fourteen\n  18. Chapter Fifteen\n  19. Acknowledgments\n  20. About the Author\n  21. Copyright\n\nThis is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously.\n\nRED, WHITE & ROYAL BLUE. Copyright \u00a9 2019 by Casey McQuiston. All rights reserved. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.\n\nwww.stmartins.com\n\nCover design by Kerri Resnick\n\nCover illustration by Colleen Reinhart\n\nThe Library of Congress has cataloged the print edition as follows:\n\nNames: McQuiston, Casey, author.\n\nTitle: Red, white & royal blue: a novel \/ Casey McQuiston.\n\nOther titles: Red, white and royal blue\n\nDescription: First edition. | New York: St. Martin's Griffin, 2019.\n\nIdentifiers: LCCN 2018055526 | ISBN 9781250316776 (trade pbk.) | ISBN 9781250316783 (ebook)\n\nClassification: LCC PS3613.C587545 R43 2019 | DDC 813\/.6\u2014dc23\n\nLC record available at <https:\/\/lccn.loc.gov\/2018055526>\n\neISBN 9781250316783\n\nOur ebooks may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, extension 5442, or by email at MacmillanSpecialMarkets@macmillan.com.\n\nFirst Edition: May 2019\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Dedication\n  4. Chapter One\n  5. Chapter Two\n  6. Chapter Three\n  7. Chapter Four\n  8. Chapter Five\n  9. Chapter Six\n  10. Chapter Seven\n  11. Chapter Eight\n  12. Chapter Nine\n  13. Chapter Ten\n  14. Chapter Eleven\n  15. Chapter Twelve\n  16. Chapter Thirteen\n  17. Chapter Fourteen\n  18. Chapter Fifteen\n  19. Acknowledgments\n  20. About the Author\n  21. Newsletter Sign-up\n  22. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n  3. Start of Content\n  4. Acknowledgments\n\n## Pagebreaks of the print version\n\n  1. Cover Page\n  2. iii\n  3. v\n  4. \n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. \n  373. \n  374. \n  375. \n  376. \n  377. \n  378. \n  379. \n  380. \n  381. \n  382. \n  383. \n  384. \n  385. \n  386. \n  387. \n  388. \n  389. \n  390. \n  391. \n  392. \n  393. \n  394. \n  395. \n  396. \n  397. \n  398. \n  399. \n  400. \n  401. \n  402. \n  403. \n  404. \n  405. \n  406. \n  407. \n  408. \n  409. \n  410. \n  411. \n  412. \n  413. \n  414. \n  415. \n  416. \n  417. \n  418. \n  419. \n  420. \n  421. \n  422. \n  423. \n  424. \n  425. \n  426. iv\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_A_ _Persian_ __ _Requiem_ is a powerful and evocative novel. Set in the southern Persian town of Shiraz in the last years of World War II, when the British army occupied the south of Persia, the novel chronicles the life of Zari, a traditional, anxious and superstitious woman whose husband, Yusef, is an idealistic feudal landlord. The occupying army upsets the balance of traditional life and throws the local people into conflict. Yusef is anxious to protect those who depend upon him and will stop at nothing to do so. His brother, on the other hand, thinks nothing of exploiting his kinsmen to further his own political ambitions. Thus a web of political intrigue and hostilities is created, which slowly destroys families. In the background, tribal leaders are in open rebellion against the government, and a picture of a society torn apart by unrest emerges.\n\nIn the midst of this turbulence, normal life carries on in the beautiful courtyard of Zari's house, in the rituals she imposes upon herself and in her attempt to keep the family safe from external events. But the corruption engendered by occupation is pervasive \u2013 some try to profit as much as possible from it, others look towards communism for hope, whilst yet others resort to opium. Finally even Zari's attempts to maintain normal family life are shattered as disaster strikes.\n\nAn immensely moving story, _A_ _Persian_ _Requiem_ is also a powerful indictment of the corrupting effects of colonization.\n\nA _Persian_ _Requiem_ (first published in 1969 in Iran under the title _Savushun),_ was the first novel written by an Iranian woman and, sixteen reprints and half a million copies later, it remains the most widely read Persian novel. In Iran it has helped shape the ideas and attitudes of a generation in its revelation of the factors that contributed to the Islamic Revolution in 1979. \nSimin Daneshvar's _A_ _Persian_ __ _Requiem_... goes a long way towards deepening our understanding of Islam and the events leading up to the 1979 Revolution... The central characters adroitly reflect different Persian attitudes of the time, attitudes that were eventually to harden into support for either the Ayatollah and his Islamic fundamentalism or, alternatively, for the corrupting Westernisation of the Shah. The value of the book lies in its ability to present these emergent struggles in human terms, in the day-to-day realities of small-town life... Complex and delicately crafted, this subtle and ironic book unites reader and writer in the knowledge that human weakness, fanaticism, love and terror are not confined to any one creed.\n\n_The_ _Financial_ _Times_\n\n_A_ _Persian_ _Requiem_ is not just a great Iranian novel, but a world classic.\n\n_The_ _Independent_ _on_ _Sunday_\n\n... it would be no exaggeration to say that all of Iranian life is there.\n\n_Spare_ _Rib_\n\nFor an English reader, there is almost an embarrassment of new settings, themes and ideas... Under the guise of something resembling a family saga \u2013 although the period covered is only a few months \u2013 _A_ _Persian_ __ _Requiem_ teaches many lessons about a society little understood in the West.\n\nRachel Billington, _The_ _Tablet_\n\nThis very human novel avoids ideological cant while revealing complex political insights, particularly in light of the 1979 Iranian revolution.\n\n_Publishers_ _Weekly_\n\n_A_ _Persian_ _Requiem,_ originally published [in Iran] in 1969, was a first novel by Iran's first woman novelist. It has seen sixteen reprints, sold over half a million copies, and achieved the status of a classic, literally shaping the ideas of a generation. Yet when asked about the specific appeal of the novel, most readers are at a loss to pinpoint a single, or even prominent aspect to account for this phenomenal success. Is it the uniquely feminine perspective, allowing the reader to travel freely between the microcosm of the family and the larger framework of society? Is it the actual plot which mimics so presciently the events of the Islamic Revolution? Or does it lie in the deftly woven anecdotes and fragments which add up to a descriptive whole? It is each and all of these, and perhaps more.\n\n_Feminist_ _Review_\n\nDaneshvar offers a fascinating, detailed view of what seems to Western eyes the complicated, rarified world of Iranian culture.\n\n_Belles_ _Lettres_\n\nIn addition to being an important literary document of historical events, [ _A_ _Persian_ __ _Requiem_ ] __ represents a pioneering attempt to probe the multi-faceted aspects of Iranian womanhood in a period of great social and political upheaval.\n\n_San_ _Francisco_ _Review_ _of_ _Books_\n\nDaneshvar combines creative vision with an exceptional talent for conveying atmosphere to give a powerful portrait of the struggles and dilemmas of ordinary individuals caught in the maelstrom of war and occupation.\n\n_Middle_ _East_ _International_\n\nThis is a colourful and accurate portrayal of Persian character and spirit, a beautifully evoked picture of traditional life in times of upheaval. Its popularity in Iran is eloquent of Persian perceptions not only of themselves but also of the role of the British in their country. Roxane Zand is to be thanked for giving the English reader the chance to enjoy this sensitive and important novel.\n\n_British_ _Journal_ _of_ _Middle_ _Eastern_ _Studies_\n\nA powerful portrait of a bygone era of Iranian social history.\n\n_The_ _Jerusalem_ _Post_\n\n\"...a revelation of freshness and vivacity...\"\n\nAnita Desai\n\n\"Not to be missed.\"\n\nShusha Guppy\n\n\"Beautifully translated, and many-layered, _A Persian Requiem_ challenges convention, of east and west.\"\n\nFred Halliday\n\n\"...a great work by a great Persian writer.\"\n\nHan Suyin\n\n# A PERSIAN REQUIEM\n\n_A Novel by_  \nSimin Daneshvar\n\n_Translated by_   \nRoxane Zand\n\n# Contents\n\nPraise\n\nTitle Page\n\nAbout the translator\n\nAcknowledgements\n\nMap\n\nGlossary\n\nAbout the Author\n\nCopyright\n\n# About the translator\n\nRoxane Zand was born in Tehran. She studied Comparative Literature at Harvard University, and Social History at Oxford University. She takes a strong interest in women's issues. \n\n# Acknowledgements\n\nI would like to thank the following for their generous help and involvement with this translation, ever since it was first undertaken, and throughout the many years it collected dust or met with misadventure: Dr. John Gurney, Keyvan Mahjour, Mohsen Ashtiani, the late Dr. Hamid Enayat, Aamer Hussein, Iradj Bagherzade and Ali Gheissari.\n\nA special thanks to Simin Daneshvar whose place in our hearts extends beyond that of artist and humanist to a particular kind of inspiration. My gratitude for her patience and loyal support.\n\nFinally, my love and thanks to Hamid who has journeyed with me through this book, and to my sons, Vahid and Karim.\n\nThis translation is dedicated to the memory of Amou Sarrafi, who first introduced me to it in 1969.\n\nRoxane Zand\n\n# Map\n\n# _1_\n\nIt was the wedding day of the Governor's daughter. The Shirazi bakers had got together to bake an impressive sangak loaf, the likes of which had never been seen before.\n\nGroups of guests filed into the marriage room just to admire the bread. Zari Khanom and Yusef Khan also managed to see it close up. The minute Yusef set eyes on it, he blurted out loud: \"Those fools! Licking the boots that kick them! And to waste so much at a time like this...\"\n\nThe guests nearby who overheard Yusef first edged away and then left the room. Zari, suppressing her admiration, caught Yusef's hand and implored him, \"For God's sake, Yusef, don't talk like that, not tonight.\"\n\nYusef laughed at his wife. He always tried to laugh her off. His full, well-defined lips parted to reveal teeth which had once sparkled, but were now yellow from pipe-smoking. Then he left, but Zari stayed behind to gaze at the bread. Bending over, she lifted the hand-printed calico tablecloth to reveal an improvised table made of two old doors. All around the table were trays of wild rue arranged in flowery patterns and pairs of lovers. And in the centre was the bread, baked the colour of burnished copper. A poppy-seed inscription read: \"Presented by the Bakers' Guild to our honourable Governor\" with \"congratulations\" written all around the edge.\n\n\"Where on earth did they find an oven big enough to bake it?\" Zari wondered silently. \"How much flour did it take? Yusef's right\u2014what a time for all this! A time when a loaf like that would make supper for a whole family, when getting bread from the bakery is a major feat. Only recently there was a rumour in town that the Governor had threatened to throw a baker into his own oven as an example to others because everyone who had eaten his bread had come down with stomach cramps and vomiting. They said the bread was black as ink from all the dirt and scraps mixed in it. But then, as Yusef says, how can you blame the bakers? All the town's provisions\u2014from wheat to onions\u2014have been bought up by the occupying army. And now... how on earth do I cover up for what Yusef has just said?\"\n\nSuddenly a voice broke into her thoughts.\n\n\"Salaam.\"\n\nShe looked up and saw the English missionary doctor, Khanom Hakim, standing in front of her with Captain Singer. They shook hands with her. Both spoke only broken Persian.\n\n\"How are being the twins?\" Khanom Hakim asked, adding to Captain Singer in the same clumsy language, \"All of her three children being delivered by me.\"\n\n\"I did not doubt it,\" replied Captain Singer.\n\nTurning back to Zari, she asked, \"The babies' dummy still being used?\" Struggling through a few more sentences in Persian, she finally tired of it and carried on in English. But Zari was too distracted to understand, even though she had studied at the English school and her late father was considered the best English teacher in town.\n\nIt was really Singer who captured her attention, and although Zari had heard about his transformation, she refused to believe it until she saw him with her own eyes. The present Captain Singer was none other than Mr Singer, the sewing machine salesman who had come to Shiraz seventeen years ago, and who treated anyone buying his sewing machines to ten free sewing lessons delivered by himself in his barely understandable Persian. He would squeeze his enormous bulk behind the sewing machine and teach the girls of Shiraz embroidery, lattice-work and pleating. It was a wonder he didn't laugh at the ridiculous figure he cut. But the girls, including Zari, learned well.\n\nZari had been told that overnight, as soon as war broke out, Mr Singer had donned a military uniform, complete with badges of rank. Now she could see that it really suited him. It must have taken a lot, she thought, to live as an impostor for seventeen years. To have a fake job, fake clothes\u2014to be a fraud in every respect. But what an expert he had been! How cunningly he had persuaded Zari's mother to buy a sewing machine\u2014Zari's mother, whose sole fortune was her husband's modest pension. Mr Singer had told her that all a young woman needed for her dowry was a Singer sewing machine. He had claimed that the owner of a sewing machine could always earn her own living, and had said that all the leading families in town had bought one from him for their daughters' dowry; as proof, he had produced a notebook containing a list of his influential customers.\n\nAt this moment, three Scottish officers, wearing kilts and what seemed like women's knee-length socks, broke Zari's train of thoughts as they came forward to join them. Behind them came McMahon, the Irishman, who was Yusef's friend. McMahon was a war correspondent and always carried a camera. He greeted Zari and asked her to tell him all about the wedding ceremony. Willingly she described all the details of the vase, the candlesticks, the silver mirror, and the reasons for the shawl, the ring wrapped in silk brocade and the symbolic meaning of the bread and cheese, the herbs and the wild rue.\n\nTwo large sugar cones, made at the Marvdasht Sugar Refinery especially for the wedding, were placed one at either end of the ceremonial table. One cone was decorated as a bride and the other as a groom, complete with top hat. In one corner of the room stood a baby's pram lined in pink satin and piled high with coins and sugar-plums. Zari pulled back the silk brocade cloth covering the traditional saddle and explained to McMahon, \"The bride sits on this so she can dominate her husband forever.\"\n\nA few people around them chuckled loudly and McMahon clicked away busily with his camera.\n\nJust then, Zari's glance fell on Gilan Taj, the Governor's younger daughter, who seemed to be beckoning to her. She excused herself and went over to the young girl. Gilan Taj was no more than ten or eleven, the same age as Zari's own son, with honey-coloured eyes and sleek, brown shoulder-length hair. She was wearing ankle socks and a short skirt.\n\n\"Mother says would you please lend her your earrings,\" Gilan Taj asked Zari. \"She wants the bride to wear them just for tonight. They'll be returned to you first thing tomorrow morning. It's Khanom Ezzat-ud-Dowleh's fault for bringing a length of green silk for the bride to put around her shoulders. She says it will bring good luck, but my sister isn't wearing anything green to match it.\" The young girl could have been repeating a lesson by heart.\n\nZari was dumb-struck. When had they spotted her emerald earrings,  let alone made plans for getting their clutches on them? In all the bustle, who could have spared the time to fuss over such minor details of the bride's dress? She said to herself, \"I bet it was that woman Ezzat-ud-Dowleh's doing. Those beady eyes of hers constantly keep track of what everyone has.\" Aloud she replied nervously, \"Those were a wedding present\u2014a special gift from Yusef's poor mother.\"\n\nHer mind flashed back to that night in the bridal chamber when Yusef had put the earrings on her himself. He was sweating profusely, and in all the hustle and bustle he had groped nervously under the women's scrutiny to find the small holes in her earlobes.\n\n\"They're playing the wedding tune,\" Gilan Taj prompted. \"Please hurry. Tomorrow morning then...\"\n\nZari took off the earrings.\n\n\"Be very careful,\" she warned, \"make sure the drops don't come off.\" In her heart she knew that the likelihood of ever seeing those earrings again was very remote indeed. Yet how could she refuse?\n\nAt this point the bride entered on Ezzat-ud-Dowleh's arm. \"Yes,\" thought Zari, \"that woman is never slow to become confidante and busybody to every new governor of the town.\" The bride was followed by five little girls each carrying a posy of flowers and wearing frilly dresses, and five boys in suits and ties. The room was now full, and the ladies started to clap. The British officers who were still there quickly followed suit. Clearly all the pomp and formality was for their benefit, but to Zari the wedding march seemed more like a mournful procession out of a Tazieh passion play.\n\nThe bride sat on the saddle, in front of the silver mirror and Ezzat-ud-Dowleh rubbed the sugar cones together over her head to ensure sweetness in the marriage. Then a woman holding a needle and red thread pretended to sew up the tongues of the groom's relatives. This raised a loud guffaw from the British officers. Next, a black nursemaid carrying a brazier of smoking incense suddenly appeared out of nowhere like a genie.\n\n\"All the villains of the Ta'zieh are here,\" Zari mused to herself. \"Marhab, Shemr and Yazid, the farangi, the unwanted Zeynab, the rapacious Hend, Aysheh, and last but not least Fezza!\" And for an instant it occurred to her that she was thinking just like Yusef.\n\nThe crowded room was noisy and stifling. The smell of incense mixed with the strong scent of tuberoses, carnations and gladioli which were displayed in large silver vases around the room but glimpsed only from time to time between the whirl of the ladies' dresses.\n\nZari missed the moment when the bride gave her consent. Suddenly she felt a hand on her arm.\n\n\"Mother is very grateful,\" whispered Gilan Taj; \"they really suit her...\"\n\nThe rest of her sentence was drowned in the commotion and blare of military music which followed the wedding tune. A booming which pulsated like the beating of battle drums...\n\nNow it was Ferdows, the wife of Ezzat-ud-Dowleh's manservant, who came in, threading her way past the guests to give her mistress her handbag. Ezzat-ud-Dowleh took out a pouch full of sugar-plums and coins which she showered over the bride's head. To save the foreign officers the trouble of scrambling for a coin, she handed one to each of them and one to Khanom Hakim. Until that moment Zari had not seen Ezzat-ud-Dowleh's son, Hamid Khan, in the wedding room, but she noticed him now speaking to the British officers.\n\n\"My dear mother has the Midas touch!\" she heard him saying. Turning to her abruptly, he said, \"Zari Khanom, please translate for them.\"\n\nZari ignored him.\n\n\"Not on your life!\" she retorted silently. \"My former suitor! I had more than enough of you and your ways that time when our history teacher took us sixteen-year-old girls to your home on the pretext of visiting an eighteenth-century house. You looked us over with your lecherous eyes, supposedly showing us the baths and the Zurkhaneh, boasting that your ancestor, the famous Sheriff, built the hall of mirrors and that Lutf-Ali Khan had done the painting on the mirrors. And then your mother had the nerve to come to the Shapuri public baths on our usual bath-day and barge her way into our cubicle just so she could size up my naked body. It was lucky Yusef had already asked for my hand, otherwise my mother and brother might well have been taken in by your extravagant life-style.\"\n\nThe ceremony over, celebrations got under way in the garden and on the front verandah. All the cypresses, palms and orange trees had been strung with light bulbs\u2014each tree a different colour. Large bulbs lit the larger trees, while small ones had been used for the smaller, twinkling like so many stars. Water flowed from two directions in a terraced stream into a pool, cascading over the red glow of rose-shaped lamps set inside each step. The main part of the garden had been spread with carpets for dancing. Zari assumed the wiring for the waterfall lights ran under the carpets. Around the edge of the pool they had alternated bowls full of different kinds of fruit, three-branched candlelabra and baskets of flowers. If a gust of wind blew out one of the candles, a servant would instantly relight it with a short-stemmed taper.\n\nThe Governor, a tall, heavy-set man with white hair and a white moustache, was standing by the pool welcoming even more guests. An English Colonel with a squint, walking arm in arm with Zari's former headmistress, was the last to arrive. Behind them came two Indian soldiers carrying a basket of carnations in the shape of a ship. When they reached the Governor, they placed it at his feet. At first the Governor didn't notice the flowers as he was busy kissing the English-woman's hand. But the headmistress must have drawn his attention to them because the Governor shook hands with the Colonel again before extending his hand to the Indian soldiers. They, for their part, merely clicked their heels together, saluted, about-turned, and withdrew.\n\nThen came the hired musicians. One played the zither, while his plump friend accompanied him on the tar and an attractive young boy sang a song. When the song was over, there was a dance followed by another song. The musicians then changed to a rhythmic beat and a group of men and women dressed as Qashqais did a sort of tribal dance. Zari had seen a lot of fake things in her time, but never fake Qashqais!\n\nNow it was the turn of the hired musicians brought over especially from Tehran. The noises sounded confused to Zari; even the sight of all those dishes piled high with sweets and dried fruit and nuts nauseated her. The sweets had probably been sent by the Confectioners' Guild and the fruit and the nuts by the Grocers' Guild, she thought cynically. The five-tiered wedding cake flown in by air had, she knew, been presented by the Supreme Command of the foreign armed forces. They had displayed it on a table on the verandah. On the top tier stood a bride and groom hand in hand, with a British flag behind them, each crafted skilfully out of icing.\n\nTo Zari it felt like watching a film. Especially with the foreign army in full regalia: Scottish officers in kilts, Indian officers in turbans... If she hadn't lost her earrings, thought Zari, it would have been possible to sit back and enjoy the show.\n\nThe bride and groom led the dancing. The bride's long train with its glittering rhinestones, sequins and pearls swept over the carpet like a trail of shooting stars. She was no longer wearing the length of green silk or her bridal veil, but the earrings were still there. The British Colonel had one dance with the bride; so did Captain Singer, in whose large arms the bride skipped about like a grasshopper. He even trod on her toes several times.\n\nThen the foreign officers sought out the other ladies. The Shirazi women in their colourful dresses danced in the arms of strangers while their men, perched on the edge of their seats, kept a nervous eye on them. Some of the men seemed particularly restless and agitated. Was it the light-hearted tempo of the music, or an inner fire kindled at the sight of strangers holding their wives so closely? It was impossible to know. At the end of the dance the officers carefully returned the ladies to their chairs, as if they were incapable of finding their own way back. They clicked their heels and kissed the lady's hand, at which the woman's own escort would nearly jump out of his seat and then settle back to try to compose himself. Not unlike a jack-in-the-box. The only person who didn't dance was McMahon. He took pictures instead.\n\nCaptain Singer came over to Zari. He clicked his heels smartly and said with a bow: \"Shall we dance?\"\n\nShe excused herself. Singer shrugged and moved on to ask Khanom Hakim. Zari looked over at Yusef who was sitting a few chairs away. His eyes were fixed on her, those eyes that seemed to her deeper in colour than the azure of spring skies. He winked at her, and she felt a pang in her heart. A faint teardrop always seemed to lurk in the depths of Yusef's eyes, making them glisten like two moist jewels\u2014like the emeralds of her earrings.\n\nNow the Colonel and Singer, either together or singly, began to accompany some of the men on a brief walk to the bottom of the garden. After a few minutes they would return and head straight for the bar, where they drank each other's health. Zari saw Singer whisper something in Yusef's ear, at which Yusef rose and set off with him down the garden path, with its border of illuminated cypresses and orange trees. But they were back almost immediately. This time they did not visit the bar. Zari saw Captain Singer make a sign to the Colonel, whose expression reflected his annoyance. Yusef came and sat next to Zari, his face flushed and his fair moustache trembling.\n\n\"Let's get up and leave quietly,\" he said.\n\nFlicking her hair forward to cover her bare ears, Zari said: \"As you like.\"\n\nShe was getting up to leave when McMahon appeared, drink in hand, and sat down next to them. He had drunk so much gin he could barely keep his eyes open. He spoke in English:\n\n\"You're at loggerheads with the big tailor again, Yusef?\" he asked. \"I must admit, it's even more difficult for you Persians to deal with the British than it is for us Irish... Did you like my poem that I recited for you earlier tonight? You did, didn't you? Now I'm thinking of composing a poem for your town...\"\n\nPointing to the slice of lime in his drink, he said: \"The lime with its light green delicate peel, its fragrance combining all the perfumes of the plain, and the cypress tree with its strength and restraint\u2014these are the things which grow in this region. People usually resemble the nature surrounding them; in this case, delicate and restrained. They've sent me to ask why you're not delicate and restrained, Yusef. I'm doing well you know, even though I'm blind drunk. Look how easily I've accomplished my mission!\" He turned to Zari. \"Cheers!\" he said, draining his glass and putting it on the table.\n\n\"Let's go and sit on the bench near that ship of flowers,\" he suggested. \"Zari, you come too\u2014the presence of a lovely woman is always inspiring. That warship laden with flowers is a gift from our Supreme Command.\" They moved across to the bench. \"That's better. Where's my glass? Zari, please pour us another drink.\n\n\"We are related, aren't we?\" he carried on, with a faraway look in his eyes. \"Iran and Ireland. Both lands of the Aryans. You the ancestors and we the descendants. O ancient, ancient ancestors, console us! Here am I a Catholic Irishman, a patriarch, a drunkard, bound to end up dying in a ditch one foul, rain-sodden day, or wandering around poor houses looking for some old woman to claim as my mother. I can see her now, knitting woollen socks with little patterns for her son at the front... like the ones I'm wearing. You see, my father was on air-raid duty; he knew that the planes were bombing our area, he knew that at any moment they would wipe out our home, and he knew that mother was there knitting patterned socks for her son at the front. When they pulled her out from underneath the rubble, she was still clutching the knitting needles\u2014and now my father has written me a letter. He has written to me to say he's sorry... he's sorry that...\"\n\nMcMahon's speech was becoming slurred and he broke off for a moment. Then he raised his hand in a grandly drunken gesture:\n\n\"Why did you, you home-loving Catholic family, wrapped in your traditions, with your confession and such nonsense... why did you uproot yourselves and move to London? If you had stayed to help put right and free your own poor, blighted Ireland you wouldn't have had to pay so dearly for that move.\n\n\"Away from home,\" he paused, \"I remember making up tales of Ireland, boasting to others of her countless poets, and sighing for my impoverished land. I remember saying that in our land the youth were innocent, uncorrupted, and people would ask me if I thought they were corrupt in London. We were all fooling ourselves. We'd forgotten Ireland's alcoholics. We'd forgotten the ships which arrived every week and loaded up their cargo\u2014the youth of Ireland\u2014and set sail for America. We ignored the fact that the convicts among them would be sent to the colonies\u2014like our tailor here. That big tailor has surely got it in for you, Yusef. He can't stand the sight of you; nor me, for that matter. I told the Consul yesterday to count you out. But the big tailor won't let him....\"\n\nHe half-drained his glass, then continued:\n\n\"Some people are like rare flowers; others resent their existence. They imagine that such a flower will use up all the earth's strength, all the sunshine and moisture in the air, taking up their space, leaving them no sunlight or oxygen. They envy it and wish it didn't exist. Either be like us, or don't be at all\u2014that's what they say. You Persians have the occasional rare flower among you, but also a lot of oleander to keep mosquitoes away, and then some plain grass which is only good for the sheep. Well,\" he rambled on, smiling, \"there's always a branch on every tree which is taller and leafier than others. And this taller branch has its eyes and ears open and can see everything clearly. But no one likes it that way. So they send the drunken Irish poet, the war correspondent, to mollify you, Yusef, and this reporter carries his father's letter here in his coat pocket; his father who'd written to say he's sorry that... well, if you give in, Yusef, it's all over.\" He took a long gulp. His eyes were barely open. Then he continued sorrowfully:\n\n\"O Ireland, O land of Aryan descent, I have composed a poem for a certain tree which must grow in your soil. The name of this tree is the 'Tree of Independence'. You must nurture it with blood, not with water. Yes, Yusef, you were right. If independence is good for me, it's good for you too. And that story you told me turned out to be so useful when I began to write. You said that in your folklore they talk of a tree whose leaves, when dried and put on the eyes, make you invisible, allowing you to do whatever you want. I wish there was one of these trees in Ireland and one here in your town.\"\n\nMcMahon fell silent. After a while he lit a cigarette and continued:\n\n\"All this mumbo-jumbo was just to keep you listening. When my father's letter arrived with the news... I sat and wrote a story for your Mina\u2014for your twins. Where's my story?\" He searched in his pockets. \"I thought I put it with my father's letter... you see, I want to build an airplane which drops toys for children... or else pretty stories. Ah, here it is!\"\n\nHe took out a notebook and began to read.\n\n\"Once upon a time there was a little girl called Mina. She always cried for the stars when she couldn't see them in the sky. When she was smaller, her mother would pick her up in her arms, show her the sky and say: 'Little little moon, pretty pretty stars, come to Mina' or something like that, which is why Mina fell in love with the stars. Now whenever it's cloudy at night, Mina cries for the stars. If only the maid would sweep the sky\u2014she's slapdash and brushes the dust away here and there, so on the nights she sweeps, at least some of the stars can be seen. But alas, if mother sweeps, she polishes the sky clean and gathers up all the stars and the moon and puts them in a sack. Then she sews up the sack, puts it in the cupboard and locks the door. But Mina found out what to do. She plotted with her sister to steal their mother's keys and now they sleep hugging the keys tightly. If they don't have the keys, they don't sleep a wink. I've never seen a little girl so in love with the stars, and I've never seen a town like yours where you can hide stars in its cupboards...\"\n\nHe took another sip of his drink and said: \"That's the end of Mina's story. Say bravo, Yusef! See what a yarn I've spun from odds and ends you've told me about your twins. You say the people of your town are born poets: well, the Irish are like that too...\"\n\nThen he became silent.\n\nZari was deep in thought when she noticed her brother-in-law, Abol-Ghassem Khan, approaching. McMahon stood up, picked up his glass and left. Abol-Ghassem Khan took his seat.\n\n\"Is that whisky?\" he asked.\n\n\"No, it's gin,\" Zari answered. \"Shall I pour you a glass?\"\n\nAbol-Ghassem Khan said quietly to Yusef: \"Listen brother, you're being as stubborn as a mule. After all they're guests in our country. They won't be staying here forever, you know. And if we don't give them what they want, they'll take it by force. They won't be put off by the locks and bolts on your store-rooms either. Besides, you know they'll pay. I sold the entire contents of my store-rooms in one go... I've already taken a down payment for the wheat before it's even sprouted. After all, they're the bosses.\"\n\n\"I'm all too well aware that they're unwelcome guests,\" Yusef told his brother dryly. \"But the worst thing is the feeling of inferiority that's taken hold of everyone; overnight they've turned all of you into their lackeys, go-betweens, and errand-boys. Why don't you let at least one person stand up to them so they can say to themselves that they've finally come across a man?\"\n\nBefore Abol-Ghassem Khan could reply, dinner was announced. The guests filed inside the house. Zari, her husband and her brother-in-law pretended to be on their way too, but lingered.\n\n\"Sister, say something,\" said Abol-Ghassem Khan, turning to Zari. \"Your husband is downright insulting to his elder brother.\"\n\n\"What can I say?\" Zari challenged.\n\nTurning back to Yusef, Abol-Ghassem Khan said: \"Now listen, brother, you're young and you don't understand. You're gambling with your life with this stubbornness of yours, and creating trouble for all of us as well. These foreigners have to feed a whole army. You know very well an army that big can't be kept hungry.\"\n\n\"But our own people can be!\" Yusef replied sharply. \"The peasants who have been expecting to survive on the provisions from my store-rooms can be kept hungry!\"\n\n\"Listen, last year and the year before you got away with not giving them anything and somehow we covered up for you and made up the amount. But this year it just won't work. Right now provisions and petrol are even more valuable to them than guns and ammunition.\"\n\nThey were still arguing when Gilan Taj came up to them and said: \"Mother says please come in for dinner.\"\n\nAs they walked in, Abol-Ghassem Khan whispered in Zari's ear: \"I hope he doesn't take it into his head not to come to their party tomorrow evening. They've even invited Khosrow. I'll pick you all up myself.\"\n\n\"But tomorrow's Thursday; it's a holy evening and I have a lot to do. You know the vow I made.\"\n\n\"Sister, I'm counting on you!\" Abol-Ghassem Khan pleaded.\n\nWhen they reached home, Zari sat on the bed. She only took off her shoes. Yusef was straightening out his trousers on the bed, ready for the hanger. When he had put on his night-clothes he went into the children's room next door. Zari could see him from where she was sitting, standing by the twins' bed watching them. Then he moved forward out of sight, but Zari knew he would be smoothing out their pillows, taking the keychain which they liked to hold at bedtime. She knew he would be kissing them and murmuring endearments to them. Then she heard a door open, and knew he had gone into their son Khosrow's room. He would be tucking him in, and whispering a few words of prayer for his future.\n\nYusef came back to their bedroom. Zari had not moved from the bed.\n\n\"Aren't you going to sleep?\" Yusef asked, handing her the keychain, adding with a laugh, \"The little twins are so funny!\"\n\nHe sat down next to his wife. \"I suppose you want me to undo your buttons. I'm sorry I didn't remember.\"\n\nWithout turning her back, Zari said: \"McMahon wrote such a pretty story about them.\"\n\n\"Did you understand all of it?\"\n\n\"Yes, I've got used to his Irish accent by now.\"\n\n\"Do you know what Mina told me today when I tossed her in the air and hugged her? She asked, 'Daddy, did mummy give you two stars? I can see them in your eyes'.\"\n\nZari laughed. \"The child is right. There always seem to be stars twinkling in your eyes.\"\n\nYusef began to undo the buttons of his wife's dress.\n\n\"My goodness, what are all these buttons for?\" he said. \"Early this evening I said some things to McMahon, and if ever Singer gets to hear about them I'm done for.\" He undid the buttons and Zari's dress fell around her waist. He began to unhook her bra.\n\n\"I told McMahon that the people of this town were born poets but their poetry has been stifled; their heroes have been castrated. There's no room left for them to fight back, so at least there could be some glory, or the honour of an open challenge. They've made this into a land with no heroes and this town into a graveyard; the liveliest neighbourhood is the Mordestan district.\"\n\nYusef unhooked Zari's bra, and putting his hands over her breasts, said: \"I feel sorry for your breasts; you bind them so tightly.\"\n\nZari felt her breasts responding. Her nipples gradually hardened. Yusef put his lips on his wife's shoulder. His lips were warm.\n\n\"Didn't he ask what the Mordestan district was?\" Zari asked.\n\n\"Yes, he did. I told him it's the neighbourhood where the residents are mainly pathetic women who earn a livelihood from painting up their faces, and whom those Indian soldiers are sent to. The officers are much better off in that respect. I told him, 'You've killed the poetry, but instead the cab-drivers, prostitutes and go-betweens have picked up a few words of English.' McMahon said there was no need to tell him any of this, he was heartily sick of the war himself.\"\n\nYusef reached forward and stroked his wife's hair. He was about to kiss the back of her neck when Zari turned around and, throwing her arms about his neck, began to cry. Yusef asked in surprise: \"Are you crying because of me? You know I can't be like the others. I can't see our people go hungry. Someone has to be man enough to stand up...\"\n\n\"Let them do whatever they want, but please don't let them bring this war into my home. What do I care if the whole town has turned into a red-light district? My town, my country is this household... but they're going to drag this war to my doorstep too.\"\n\nYusef held his wife's face in his hands and kissed away her tears.\n\n\"Go and wash your face,\" he said soothingly. \"It's not the time for this sort of talk. I swear to God, you're a thousand times prettier without make-up. Your face is like one of those they paint on tiles. Come on, my love. I want you tonight.\"\n\nZari undressed, and put out the light. She didn't want Yusef to see the 'geography map' on her stomach, as she called it. Even though Yusef always kissed the scars and said, \"You've suffered this for me.\" It was Khanom Hakim who had disfigured her belly with stitch-marks and puckered scars.\n\nShe climbed into bed, and when Yusef's warm, hairy legs touched her cold ones, and his large hand caressed her breast moving lower and lower down, she forgot everything\u2014the earrings, Captain Singer, Khanom Hakim, the bride, the military music, the drums, and the beady-eyed, squinting, bald wedding guests... she forgot it all. Instead, in her ears was the sound of water flowing gently over red flowers; and before her stood the image of a ship full of flowers, a ship that was not a warship. \n\n# _2_\n\nWhen Zari woke up on Thursday morning it was still half dark. She crept quietly out of the bedroom, and when she had finished washing, she joined her sister-in-law at the breakfast table in the parlour. Ameh Khanom was sitting behind the boiling samovar. The twins, Mina and Marjan, were chattering like two little sparrows as they hung around the breakfast table. It was for their safe delivery, and also in thanks for the birth of their brother Khosrow, that Zari had vowed to take bread and dates to the prisoners and the patients in the asylum.\n\nBecause of her slender build and narrow hips, Zari had had a difficult time at childbirth. With each pregnancy she had hoped for a home birth, making all the necessary arrangements with the best midwife in town, but in the end she found herself resorting to Khanom Hakim and the Missionary Hospital on the one hand, and to vows and prayers on the other. And of course, Khanom Hakim was a great one for the scalpel. She loved to cut and sew. Delirious with pain at the first delivery, Zari had pleaded with God, vowing, as an act of charity, to take home-baked bread and dates every week to the mental patients. Then, when she became pregnant again five years later, she was so frightened, she made a vow in advance to do the same, but this time for prisoners.\n\nAmeh Khanom poured her a glass of tea. \"Well, how was last night?\" she asked.\n\n\"You should have been there! I'm afraid there was yet another quarrel between the two heads of the family.\"\n\n\"I know my brother Abol-Ghassem, I know Yusef too. Abol-Ghassem Khan isn't straightforward. And since he's taken it into his head to become a parliamentary deputy, he's even less so.\"\n\n\"He made me promise faithfully to go to the foreigners' party. I don't know how I'm going to carry out my vow.\"\n\n\"Don't worry about that. I'll ask Haj Mohammad Reza, the dyer, to go to the asylum with Gholam. I'll go to the prison with Hossein Agha, the grocer. Sakineh is here stoking up the oven, and the dough has already risen. I looked in after finishing my prayers. I think the bread is setting. You go to the party, sister. I don't want any more quarrelling between those two.\"\n\nAt that moment Khosrow came into the parlour.\n\n\"Here's Khosrow!\" Mina shouted gleefully, clapping her hands together. \"He'll let me ride his horse, won't you, Khosrow?\" Marjan, who was a quarter of an hour younger, imitated and followed her sister in everything. She clung to Khosrow's leg and said to Mina: \"First you play with him, then me, all right?\"\n\n\"No time to play, I have to go to school now,\" Khosrow said, patting them both on the head hurriedly. Mina pulled at the tablecloth. The samovar tipped and nearly fell over, but Ameh Khanom steadied it just in time.\n\n\"They can really drive you mad with their mischief,\" she said, as she handed them each a sugar lump.\n\nKhosrow reached for the sugar bowl. \"Mother, may I? They're shoeing Sahar this afternoon,\" he said, taking five lumps and putting them in his pocket. Then he took some tea from his aunt, and reached out for two more lumps. As he put them in his pocket, his aunt said, \"Don't you want any sugar in your tea?\"\n\n\"No, I'll be late for school.\"\n\n\"Abol-Ghassem has sent Seyyid Moti-ud-Din, the mullah, a sackful of sugar and twenty packets of tea belonging to his own peasants and workers,\" Ameh added to Zari with a laugh. \"I've heard my dear brother stands right behind the mullah when he leads the prayers in the mosque. Abol-Ghassem, who's never in his life known which way to face when he prays!\"\n\n\"Auntie, I've seen Seyyid Moti-ud-Din, the mullah! I saw him the day we went to the bazaar with Gholam to buy Sahar a saddle,\" Khosrow exclaimed. \"He was riding a white donkey. He brought his hand out from his cloak and held it up like this in the air... like this...\" He waved his hand in imitation of the mullah, sitting astride his chair and rocking himself back and forth as if he were riding a donkey. \"Everyone who passed by kissed his hand; Gholam and I kissed it too. He had to bring it lower down for me because I was shorter.\"\n\nSuddenly there was a knock at the garden gate. Zari's heart leapt. Perhaps they had brought her earrings back from the Governor's house! But so early in the morning? The sun was just rising. She went out to the verandah. There she saw Gholam in his nightshirt, coming out of the stables at the bottom of the garden. As always he was wearing his felt hat to cover his baldness. He opened the gate to let in Abol-Ghassem Khan who walked in with a brisk air. Disappointed, Zari thought to herself, \"What if they send them back so late that Yusef is up and finds out... oh, how silly I am! What earrings? Who on earth is going to remember my earrings!\"\n\nShe returned to the parlour and sat down. When Abol-Ghassem Khan walked in, Ameh said: \"Talk of the devil. I was just singing your praises.\"\n\n\"You must have been saying that with all this running about, I'll finally make it as a deputy,\" he said. \"And I will. I've seen the Colonel and the Consul. The Governor has promised, too. Only the mullah is putting his spoke in my wheel. He flatters me in the mosque one day, and takes it all back the next.\"\n\n\"Maybe the sugar and tea you sent him didn't go down too well!\" Ameh remarked.\n\n\"Sister, what are you talking about? What tea and sugar?\" Abol-Ghassem Khan retorted sharply, throwing a look in Khosrow's direction.\n\n\"I'm the eldest amongst you, and I'm entitled to give you advice,\" Ameh said quietly. \"You have not chosen the right path, brother. And besides, Khosrow is not a stranger.\"\n\n\"So you think the path your precious brother Yusef has chosen is the right one?\" Abol-Ghassem Khan replied angrily. \"Taking sugar and clothing coupons from the government with one hand and passing them on to his peasants with the other? Well, what's the young fool getting out of it for himself? Whenever he goes to his village he takes medicine for the peasants. God alone knows that all the medicine in the world won't cure our peasants.\"\n\nAs Khosrow stood up to say goodbye, Abol-Ghassem Khan asked, \"Where's Yusef now?\"\n\n\"He's getting up,\" Zari replied. \"He'll be here soon.\" She busied herself making fresh tea.\n\n\"Always sleeping, always sleeping!\" Abol-Ghassem Khan complained. \"In his village too, he's either asleep or sitting under the mosquito net, reading a book. My heels are cracked, my face scorched and wrinkled from the sun, but his Lordship keeps himself wrapped up in cotton wool.\" Then he added emphatically: \"Peasants have to be afraid of their landlords. You must stand over them with a whip, like an elephant driver. You have to use the cane and the bastinado. Remember the old saying: peasants must be kept living from hand to mouth.\" He took some tea from Zari before going on.\n\n\"Yusef doesn't know about winter crops or the summer harvest. He can only keep his eyes glued to the sky, watching for rain. And if it doesn't rain he gets really upset; not for himself, of course, but for the peasants and their sheep. And when you try to set him straight, he only comes up with his favourite saying, 'What the peasant reaps belongs to him, even if the land doesn't'.\"\n\nAmeh interrupted, \"It's his way of being charitable. If he can't ensure his lot in this world, he will at least have his salvation hereafter. Besides, brother, why is it any of your business? It's not your money he's giving away.\"\n\nZari could hear Sahar, Khosrow's horse, neighing in the garden. She knew Khosrow must have gone to the stables before leaving for school and set Sahar loose in the garden. When he heard the neighing, Abol-Ghassem Khan stood up and looked out of the parlour window. His eyes followed the colt carefully.\n\n\"What a beauty he's become,\" he said. \"Glitters like gold! Look at him rolling on the cool grass! Now he's standing again. Wide-set eyes, broad forehead, good ears\u2014a perfect creature! Look at that golden mane and arched tail. He holds his head high too, just like his mother.\"\n\nSahar neighed again, revelling in his freedom. Abol-Ghassem Khan returned to his seat.\n\n\"Thank God you approve of one thing in this household,\" Ameh Khanom said with a sigh.\n\nAbol-Ghassem Khan laughed: \"Everything he does is so fanciful. Who keeps horses nowadays? Apart from my brother, that is, who's got three in his stables...\" Mimicking Yusef, he said, \"I like to go to the village on horseback. I ride the bay mare myself, my steward rides the roan, and the colt belongs to Khosrow.\"\n\nAt that moment Yusef came in. He was wearing a light cloak over his shoulders. He greeted everyone, and looked with surprise from his brother to his sister. Then he threw Zari an enquiring look, but she merely shook her head.\n\n\"Has Khosrow gone?\" he asked.\n\n\"Yes.\"\n\n\"Where are Mina and Marjan?\"\n\n\"They're watching Sakineh bake bread and probably chattering away as usual,\" Ameh replied.\n\nYusef sat down. \"Has something happened, God forbid?\" he asked his brother.\n\nAbol-Ghassem did not answer. Instead, he took a small book from his pocket and put it solemnly on the table. \"Swear on the holy Quran,\" he said, \"that you'll come tonight and that you won't stir up any trouble with your usual comments. Now, if you don't want to sell the surplus provisions from your village to the foreign army, don't. But you don't have to say so to them in so many words. Stall them somehow, until harvest time. You have to go to the lowlands in a few days anyway\u2014tell them you'll give it to them after the harvest. Who knows what'11 happen tomorrow? Maybe they'll be defeated by then and good riddance to them. They say Hitler is having a bomb made that will wipe out the world... now swear!\"\n\nYusef sighed. \"I never said I wasn't coming this evening,\" he said. \"There's no need for swearing. But as far as fooling them goes, I'm a straightforward person. I won't lie to save my skin.\"\n\n\"For God's sake, swear,\" Abol-Ghassem implored. \"I've never said this before, but now I will. Our father Haj Agha, God rest his soul, spent a great deal of money on your education, but not much on mine. When he was dividing his wealth he gave us equal shares even though I'm the older brother. Did I say anything then? Even when it came to marriage, you were the one who ended up winning the hand of Zari Khanom, Razieh Khanom's attractive daughter. Now that there's an opportunity for me at last, let me make something of my life too.\" He quoted a line from a Hafez poem: \"Of strangers I have no complaints. Alas, what I've suffered has been at the hands of my own kith and kin...\"\n\n\"Brother,\" interrupted Ameh, \"one thing I know for sure is that neither your father nor his father before him ever begged a favour of anyone. Not from the unclean foreigners, nor from our own social climbers. Haj Agha never once took off his mullah's turban. He remained a recluse all his life. In that assembly\u2014I forget the name... who cares what it was called, anyway\u2014he didn't vote for the man they'd all been told to vote for. If Yusef was his favourite, it was because they had a similar temperament and believed in the same things.\"\n\n\"Now I'm getting it from you, too!\" Abol-Ghassem Khan shouted angrily. \"If our Haj Agha had had a brain in his head, we would be rolling in money today. He spent everything he had on that Indian dancer, Soudabeh. My mother died heartbroken in a foreign land because of her. If he had had any brains at all he wouldn't have married you off to that imbecile, Mirza Miyur's son, who got himself killed on purpose, and you wouldn't have ended up as a servant in the house of...\"\n\nZari cut her brother-in-law short. \"Abol-Ghassem Khan, Ameh Khanom is the eldest among us and the most respected. If it weren't for her, I could never manage such a large place by myself. Besides, this house is her home.\"\n\n\"Yes, I know,\" he said. \"She manages well enough for herself, and stirs up trouble for everyone else besides.\" He got up and added in a surprisingly gentle tone: \"I hadn't intended to mention the dead and speak badly of our past first thing in the morning. On such a nice day, too. Well, it just happened. Don't take it to heart, sister. Goodbye.\"\n\nZari accompanied the two brothers to the garden gate. Sahar was grazing, but the moment he smelt a stranger, he stopped and lifted his head. His pink nostrils flared. Abol-Ghassem Khan stopped in front of him. The colt stepped back and neighed. His mother answered from the stable. When Yusef approached, Sahar nuzzled at his cloak and lifted his head, sniffing the familiar odour. Yusef caressed his neck and mane. Later, when husband and wife returned from seeing Abol-Ghassem Khan out, they found Sahar cantering from one side of the garden to the other.\n\n\"Zari, look! He's chasing the butterflies,\" Yusef said.\n\nSahar must have been getting hot, because he rolled over several times on the shaded part of the grass. Then he got up, and all of a sudden charged after a brown and yellow butterfly.\n\nWhen they reached the verandah, Yusef paused and looked at the garden.\n\n\"Your town is looking pretty,\" he said. \"It's a pity that it's summer again, and I won't have so much time for you or your town as I'll be at the village.\"\n\n\"My town?\" Zari asked.\n\n\"Didn't you say last night that this house was your town?\"\n\nZari laughed. \"Oh yes,\" she said dreamily. \"This is my town and I love every inch of it. The hill behind the garden, the verandah all around the house, the two streams on either side of the footpath, the two elms, the orange trees you planted with your own hands. That fruit tree to which you grafted a new fruit each year, the scent distillery next door, with its mounds of flowers and herbs in season, flowers and herbs whose very names make you happy... citron, willow, eglantine; and more than anything the orange blossoms and the scents which waft into our garden from over the wall. The sparrows and starlings and the crows, too, have made this their home. But the sparrows make me cross, you know. They build their nests above the windows, or in the trees, and their eggs are always falling and breaking all over the place. They're so careless, those birds.\"\n\n\"Your voice is as soft as velvet,\" Yusef said with a smile. \"Like a lullaby. Go on.\"\n\n\"What shall I talk about?\" Zari said. \"About the people in my town? About you? About the children and Ameh and our neighbours?\"\n\n\"About Haj Mohammad Reza, the dyer...\" Yusef added with a laugh.\n\n\"About Haj Mohammad the dyer, with the colourful fabrics he ties on sticks, and leaves in the street to dry in the sun; with his arms dyed purple up to the elbows. About Gholam and Hossein Agha the grocer around the corner, and Hassan Agha the corn chandler... about Khadijeh... that's enough now! You're not letting me get on with my work.\"\n\nShe was interrupted by the sound of tinkling bells. She knew it would be the donkeys arriving at the neighbour's.\n\n\"They've bought orange-flower blossoms next door. What a scent!\" Yusef exclaimed.\n\nZari couldn't tear herself away. She waited until the donkeys entered the neighbour's garden and unloaded their perfumed bundles. Only yesterday morning she had taken the twins to see the pile of orange-blossoms. Mina had clapped and said, \"Oh look how many stars there are!\"\n\nAnd Marjan had laid her head on the heap of flowers and said, \"I want to sleep right here.\"\n\nZari meanwhile had been engrossed in the actions of the old distiller and his three sons. The old man had knelt before the orange-blossoms and piled them into baskets that the boys put on their heads to carry into the store-room. The old man had nicknamed Marjan 'Nargessi', and Mina 'Narengi'. Zari had no idea why. And when his work was finished, he made Nargessi and Narengi a toy water-mill from an apple and four pieces of thin wood. He put the water-mill in the stream so that the running water turned it. The children were so happy\u2014as if they owned the greatest water-mill in the world. And Zari kept on wondering why the old man hadn't married his sons off. It was high time they were married.\n\nThen she thought to herself: \"Why should people who live with so many beautiful flowers need to get married anyway...\" \n\n# _3_\n\nWhen they had cleared the table, Zari brought the hookah for her husband. Khosrow had been restless at lunch and became more so as time passed. It even looked as though there were tears in his eyes which he was fighting back. Zari put the twins to bed for their afternoon rest and then returned to the parlour to take the pipe away. Khosrow was pacing around the room. His father's eyes followed his movements.\n\n\"Tell me, why have we gone through all these preparations?\" he asked his son.\n\n\"So he wouldn't be afraid,\" Khosrow answered sadly.\n\n\"It wasn't only for that,\" Yusef added.\n\nKhosrow sat down next to his father. \"Every time the blacksmith comes, I lift Sahar's foot myself,\" he said. \"In the beginning he was very frightened and he shied, especially when the smith put the nails in. Of course, he hammered very lightly at first but yesterday he hit very hard.\"\n\n\"Well,\" reassured Yusef, \"he did it so that when Sahar is being shod, he won't be frightened or pull away which might cause a nail to go into his foot. Now today, I'll hold up his foot myself, just as I once helped to deliver him.\" He turned to Zari who had come to sit by them. \"You've put the hookah in front of you, as if you wanted to smoke it yourself,\" he said.\n\nZari took a puff but gave up the moment she began to cough.\n\n\"Father, may I come and watch?\" Khosrow asked.\n\n\"Of course. Weren't you there when he was born?\"\n\n\"Yes! Do I remember! Sahar stood up right away. The mare chewed off the cord and began to lick and smell him. You threw your cloak on him so he wouldn't catch cold and you rubbed his body to keep him warm while Gholam fetched a blanket... But he's really naughty now, isn't he?\" he added laughingly. \"He bites his mother, then he changes his mind and licks her.\" Khosrow paused, then said, \"Father, why do I love Sahar so much? I want to talk about him all the time. When I'm sitting in class I keep praying for the bell to ring so I can rush home and play with him.\"\n\n\"There's nothing wrong with loving, my son. Loving lightens the heart, just as malice and hatred darken it. Learn to love now, and then when you grow up you'll be ready to love what's good and beautiful in the world. The heart is like a garden full of flowers in bud. If you water them, they'll open; if you feed them with hatred, they'll wither. Remember that malice and hatred are not for the beautiful and good but for the ugly, the dishonourable and the unjust. A hatred of these things means a love of justice and honour.\"\n\n\"Father, you're talking above my head again,\" Khosrow complained.\n\n\"Didn't you understand what I said?\"\n\n\"I think I understood. You said that there is nothing wrong in loving Sahar. Then you said I must water the flowers...\"\n\n\"We must have been miles away while father was lecturing!\" laughed Zari. \"If you ask me, you should go to your uncle's and visit your cousin Hormoz, and come back when they've finished with Sahar.\"\n\n\"No Zari,\" Yusef said. \"Khosrow has to learn that if Sahar is to be shod, he must put up with a few nails. He has got to realize that there's pain and suffering in this world.\"\n\n\"Father, will it hurt him very much?\"\n\n\"No. The important thing is to learn to endure things. We've trained him to stop playing around for a few minutes, long enough to put up with the shoeing. Whereas other horses...\"\n\n\"But father, that herd of wild horses you told me a story about,\" Khosrow interrupted, \"they didn't have bridles or shoes.\"\n\n\"What was the story?\" Zari asked.\n\n\"I don't remember it myself,\" Yusef said.\n\nKhosrow sprang up, exclaiming: \"Don't you remember? You told me the story the night Sahar was born. Afterwards, Gholam and I talked a lot about the herd of horses. Gholam said you made it all up so I'd stop crying.\"\n\nStifling a laugh, Zari asked, \"What was the story?\"\n\n\"Father, let me tell it... It was when father was invited to stay with the Qashqai tribe. One night when there was a moon and the air was as clear as can be, with the sky full of stars, they went hunting. Suddenly, in the middle of a very, very big plain, they saw a herd of wild horses. The stallions were standing in a really wide circle facing outwards, their backs to the centre, where a mare was giving birth. The stallions were too embarrassed to look, because a baby comes out from a very bad place. Father and the others didn't go any closer because the horses would have charged on them... well, I mean the stallions were standing like that to reassure the mare, otherwise she would have been scared. After all, some wild animal could have attacked the foal. And, oh yes, I forgot to say that an older mare stood by as a kind of midwife.\"\n\n\"Did I say the baby comes out from a very bad place?\" Yusef asked.\n\n\"No, father, Gholam said that.\"\n\nAt that moment Gholam came in, wearing his faithful old felt hat.\n\n\"Is the blacksmith here?\" Khosrow asked.\n\n\"His wife is here. She says he's got a fever,\" he replied, and turning to Yusef, \"he won't be coming.\"\n\nThat evening Gholam came back with two porters who could carry loads on their heads. Two copper trays, piled high with bread and dates covered with a calico table-cloth, had been put out by the pool in front of the house, ready for collection. Ameh, wearing her veil, was sitting next to one of the trays. Haj Mohammad Reza, the dyer, was pacing up and down outside the gate. But Hossein Agha, the grocer, had come inside and was admiring the orange blossoms in the grove.\n\nZari herself went to the prison and asylum on alternate weeks. But there was always someone who could help her out with her vow and go to the place she wasn't visiting that week. And when there was no volunteer, there were Hossein Agha and Haj Mohammad Reza to turn to\u2014they were good neighbours who would never leave a friend in the lurch.\n\nZari, Ameh and Khadijeh the maid had been busy all afternoon putting dates between pieces of bread. Now Zari stood in front of her dressing-table, applying a touch of make-up. From her bedroom window she could see the garden and listen to what was going on. She could hear Ameh asking one of the porters, \"Well, how much do you charge?\"\n\n\"Where do I have to go?\" he asked.\n\n\"The Karim Khan prison\u2014the dungeon,\" Ameh told him, to which the man replied, \"God bless you; I don't want any money. Give me some home-made bread instead.\"\n\n\"Where do I go?\" the other porter asked.\n\n\"You go to the mental asylum,\" Ameh told him.\n\n\"Pay me in bread too,\" he said.\n\nZari patted her face, smoothing out the powder. Then she walked on to the verandah.\n\n\"Sister, they're asking for bread instead of money,\" Ameh explained to her.\n\n\"All right,\" Zari replied. Turning to Gholam she said, \"Give them each ten loaves.\"\n\n\"I have further to go, but it doesn't matter,\" the first porter said. \"This fellow's child is ill. It's this disease they say the foreign army has brought with them. I've heard that the water in the Vakil reservoir has been contaminated.\"\n\n\"God protect us!\" Ameh exclaimed.\n\n\"As if their presence alone wasn't enough, they had to bring their diseases as well,\" Hossein Agha complained.\n\n\"You're giving charity to prisoners and madmen on the holy eve of Friday,\" the first porter said, \"but no one remembers the needy standing right in front of them.\"\n\n\"May God repay them for their charity anyway,\" said the second porter. \"Our God is generous too.\"\n\nGholam arrived with the bread. Both porters unwound the cloths they usually twisted into a tight coil to use as padding for their heads while carrying the trays. Then they carefully wrapped the bread inside these cloths and tied the bundles around their waists, bulging out in front like a pair of pregnant women.\n\n\"What will you carry on your head, then?\" Zari asked.\n\n\"If we don't do this,\" the first porter explained, \"someone may snatch the bread from us. Especially this home-made bread, so fresh and delicate. Just the smell of it makes your stomach growl! It's a good thing you've covered the trays with tablecloths.\"\n\n\"But you're taking a droshke. No-one is going to snatch the bread from you in the little way you'll need to walk.\"\n\n\"It looks like the lady isn't a native of this town!\"\n\n\"Gholam, go and get the master's and Khosrow's waistcloths from Khadijeh,\" Zari said, \"and coil them into pads. These men can't carry the trays on their bare heads.\"\n\nAs Gholam ran back inside, a car drew up at the garden gate, and sounded its horn. Zari saw Abol-Ghassem Khan and his son Hormoz come in. She thought, \"Oh my God! I'm not ready yet,\" and dashed inside. There, she quickly took off her house-dress, pulled on a woollen sweater and a skirt, and started looking for her shoes.\n\n\"Hello, everybody!\" she heard Abol-Ghassem say. \"Will you be long?\"\n\n\"Now don't rush her,\" Ameh's voice rose in reply. \"This is the first time she isn't carrying out her vow herself, and all for your sake.\"\n\n\"It's a long way and we must be there at five o'clock sharp,\" Abol-Ghassem Khan insisted.\n\n\"Isn't it near Seyyid Abol Vafa's shrine?\"\n\n\"No, sister, it's about four miles further on.\"\n\n\"Now why don't you do a good deed for a change and help these poor porters. While the others get ready for the party you can give the porter and me a lift in your car.\"\n\n\"What's the hurry? Will you be late for your opium?\"\n\nAs Zari quickly combed her hair, she prayed that the two of them would not start a quarrel again. She could hear Hormoz trying to patch things up. \"Auntie,\" he offered, \"if you like I can go. I like talking to the prisoners. I've been there three times with Hossein Agha. Isn't that so, Hossein Agha?\"\n\n\"What nonsense is this again?\" Abol-Ghassem Khan turned on him angrily. Then he walked up to the edge of the verandah and called out jokingly to Zari, \"Sister, how many hours have you been spending in front of the mirror? Where's my brother; where's Khosrow?\"\n\nZari didn't answer; she was listening to Ameh who was saying: \"Let's go. Hossein Agha, help him lift the tray to his head.\"\n\nAs he heaved the tray up, one of the porters said: \"God give me strength!\"\n\nWhen they arrived at the open-air party, Captain Singer was there to greet them in person. Together they walked past the fields of summer crops to where the marquees had been set up. Zari was feeling hot, but she knew it would be cooler in the evening. She was walking ahead with Abol-Ghassem Khan while Yusef and Singer followed behind, and Khosrow and Hormoz brought up the rear.\n\nThey passed a field of lettuces, caked with dust and sand, standing in rows like soldiers on parade. As they walked on, they passed other fields where the entire crop of cucumbers, eggplants, tomatoes and melons\u2014ripe and unripe\u2014lay exposed to the relentless sun.\n\n\"They need watering,\" Abol-Ghassem Khan observed.\n\nTo the left of the fields large tents had been pitched, in which soldiers and officers were sitting or standing. Their army vehicles were parked nearby. Zari heard Yusef recite a familiar yet very apt line of verse: \"Will this wine ever suffice to quench our thirst?\"\n\n\"What do you mean by that?\" Captain Singer challenged him.\n\nAbol-Ghassem Khan stopped abruptly and turned to face them. Zari also stopped. Abol-Ghassem Khan blinked, and said to Captain Singer, \"To be quite frank, your honour, what my brother means is that a glass of whisky wouldn't go amiss right now; even though a person can't get drunk on only one glass.\" After that he made a careful manoeuvre, changing places with Yusef, and falling into step with Singer.\n\nThe guests were ushered into the Supreme Command's huge marquee. Abol-Ghassem Khan had rushed them so much that they were now too early. They greeted Khanom Hakim and a Scottish officer. A map of Iran had been spread out on a table near the entrance. Khanom Hakim was pacing around the marquee looking as though she were trying to memorize something from a piece of paper in her hand. Zari glanced at the map; there were enough multi-coloured markers stuck on it to confuse even the expert. Yusef headed for the map with an agitated Abol-Ghassem at his heels.\n\nStaring at the familiar outline of his country, Yusef murmured, \"How they've disembowelled her!\"\n\nAbol-Ghassem Khan placed a hand on his brother's arm.\n\nAt that moment, Singer directed an Indian soldier who had just entered the marquee, carrying a tray of sherbets and various soft drinks, to the table where the map was displayed. Turning to Yusef, he said: \"Let's have something to drink.\"\n\nThe three men each took a drink. Then Singer, raising his glass in a toast, proclaimed in his usual broken Persian: \"To Iran, so much bigger than France; and to Tehran, bigger than... than Vichy!\"\n\nYusef raised his head from the map and looked straight at Singer.\n\n\"But unfortunately we didn't get a chance to fight!\" he said.\n\nAbol-Ghassem Khan mumbled nervously: \"Actually, Vichy mineral water does wonders for indigestion...\"\n\n\"Why say you unfortunately?\" Singer asked, cutting him short and staring at Yusef.\n\n\"Because we're suffering the consequences anyway, without ever having tasted victory or even an honourable defeat,\" Yusef replied.\n\n\"Then why did you not fight, if you were able?\" Singer demanded. \"How to find right word? Straw? Yes, that's it, straw. We only found stuffed dummy when we come here. When we ripped him apart, there was no blood, only straw... stuffed with straw.\"\n\nYusef gave a hollow laugh and put his hand on Singer's shoulder.\n\n\"My dear Singer, you knew yourself what the score was, and that's what makes all of this even more ugly and despicable. We were deprived the chance of an honourable defeat...\n\nSinger raised a hand to stop him. \"A-a-a-a... slow now, slow, so that I can follow what you say...\"\n\nAbol-Ghassem Khan, with an attack of his nervous blink, tried to mediate: \"That's all water under the bridge...\"\n\n\"You talk in proverbs and confuse me,\" Singer said irritably.\n\nA number of other officers, English, Scottish and Indian, and McMahon the Irishman, entered the marquee. Hormoz, who had been following the conversation, whispered in Zari's ear, \"If Mr Fotouhi, our teacher, were here, he would shake uncle's hand, and call him a real man. Mr Fotouhi's always bragging about his own background. If only he could see my uncle now!\"\n\nBut Zari's attention was fixed on Singer who had taken Abol-Ghassem Khan's arm and was saying, in his stilted Persian: \"Give your brother some good advice. God has given you so much resources in this country. Give some to us. It belongs to everyone, to all mankind. It is too much just for you. You don't need all.\"\n\n\"Just what British Petroleum is doing!\" Yusef said with a laugh.\n\nSinger looked taken aback. His face and neck reddened noticeably. He placed his drink on the map and blurted out, \"You didn't know how! We don't need you. We can take it out ourselves and give to those who need...\" And suddenly he became amiable again. Lifting his glass, he said, \"Cheers!\"\n\nThe Governor, the Colonel and the newly married couple with Gilan Taj in tow, now made their entry. The officers stood to attention while the Governor nodded to all of them. The army commander, the town's newspaper owners and the heads of various civic departments, all began to drift in with their wives. The marquee was soon crammed full of people, and the sickly smell of feet, sweat, perfume and alcohol filled the air. Three Indian soldiers were busy serving drinks.\n\nZari signalled to Hormoz and Khosrow, and together they went over to Gilan Taj. Zari had decided to summon up courage to slip in a reminder about her earrings. First she introduced Khosrow and Hormoz. The girl extended a hand, and flashed a dimpled smile. Then the bride, wearing a wide-brimmed straw hat and green sunglasses, came up to them.\n\n\"Zari, darling,\" she cooed. \"Thank you so much for your gift. I'll always treasure them, and when I wear them I'll think of you.\"\n\nZari looked at her in astonishment. Since when had she and the Governor's daughter become so intimate? In the three years since the Governor's posting to Shiraz, she had not seen the girl more than three times. Well, maybe four or five times, counting the wedding. Zari opened her mouth to say: \"What gift? I only lent them, as your sister here knows full well!\" But no sound escaped her lips. She cursed herself inwardly for her own ineptitude and cowardice. \"Spineless women like me deserve no better!\" she thought to herself.\n\nThe bride looked at Hormoz and Khosrow.\n\n\"Zari dear,\" she said, \"I never knew you had such grown-up sons. You, so young and pretty! Tell everyone they're your brothers, not sons.\"\n\nKhosrow was quick to chip in. \"Four-eyed Hormoz here is my cousin,\" he announced.\n\nHormoz blushed and removed his glasses. But Zari knew he wouldn't be able to see a thing without them. She felt like scolding Khosrow there and then. Four-eyed Hormoz indeed! Talking like that to an older cousin, and in front of such uppity people as the Governor's daughter! But the bride was too quick for her.\n\n\"Master Hormoz, aren't you Mirza Abol-Ghassem Khan's son?\" she asked. \"I have a great deal of respect for him. How kind he's been! What a sweet man he is, and so amusing! Please don't be shy, put on your glasses by all means. I wear glasses myself; even my sunglasses have prescription lenses. Last night I had a frightful time without them.\"\n\nA sudden flurry of trumpets and drums announced that it was time to leave the marquee for other events of the evening. The Colonel and the Governor led the way, followed by the guests, and Zari felt as if she were being taken to an execution. They reached a vast, open space, where chairs had been arranged in a horse-shoe. Already thousands of soldiers, mostly Indian, were seated.\n\nThe officer behind the Colonel gave an order and at once, as all the soldiers rose to attention, there was a deafening scraping of chairs. Around a platform which had been made out of a couple of old boards and covered with a carpet, five flags waved of which Zari recognized only one\u2014that of Great Britain. Khanom Hakim made her way across the creaking platform to the microphone. A hush ensued. Reading from the piece of paper in her hand, she greeted the guests in Persian. Her voice was a little unsteady at first until she gained confidence and warmed to her speech. In the light of the setting sun, her dull teeth looked decidedly yellow.\n\nFrom what Zari could gather, the gist of her speech was that in order to amuse the fighting boys of Great Britain now on leave in Shiraz, that sweet city of birds and flowers\u2014they had arranged some entertainment. This was to enable the soldiers to fight the monster Fascism with greater strength of spirit, sending that devil Hitler back to hell in the shortest time possible. She thanked the Iranians for their hospitality, for they had made the war against Satan\u2014meaning Hitler\u2014easier to bear. Then she finished by declaring that Hitler was like a virus, a cancer, which had to be torn out.\n\nNow Khanom Hakim was not only a midwife, but also a surgeon quite keen on using the knife. And in addition to these talents, as she said herself, she \"brought glad tidings and led the people to Christ\". Every night, as Zari remembered, she had the pregnant women, and the ones she had already cut up, as well as their relatives, queuing up to watch a film. A silent film, of course. She would hold a long stick in her hand, and point out the characters in the film, explaining in broken Persian:\n\n\"This be Jesus Christ... this be Mary Magdalen... this be Judas Escariot...\" Afterwards, in that same irritating patois, she would preach a sermon about Satan and hell-fire.\n\n\"Why should a midwife, surgeon and missionary all rolled into one, suddenly appear in a place like Shiraz?\" Zari thought to herself, as she continued to muse about Khanom Hakim. \"Maybe her Satan has some connection with the 'Satan' the fighting boys are trying to send back to hell? The boys are mostly Indian, anyway. And, to use Abol-Ghassem Khan's phrase, 'they manage well enough themselves, and stir up trouble for everyone else besides.' Yet our people have started to call this devil 'the Messiah'. I've heard it many times myself.\"\n\nMcMahon took over from Khanom Hakim on the stage, his presence adding a note of gaiety. He had thrown a red cloak over his shoulders, and was wearing a pair of black boots. It made him look like a famous film star, though Zari could not for the life of her remember the name. It was a pity he was fat. He spoke in English, and Zari didn't understand all of his jokes, but after two or three sentences, the sound of the soldiers' laughter filled the air. Even the Governor and the army commander laughed occasionally, but it was one of the newspaper owners who laughed the loudest of all. Could all this laughter be out of politeness, Zari wondered, since despite her own good English, she hardly understood any of it.\n\nThen, with all the appropriate gestures, McMahon told a story about a soldier serving abroad who seduced a girl, exploiting her for what he could get out of her. He wanted new shoes and a hat; he wanted this and that, until one day the girl said she was pregnant and he must marry her. He confessed then that he already had a wife and children back home. McMahon rocked an imaginary cradle, put his arms around an imaginary wife, and said, \"I have a wife and little ones\" in Persian. This time the audience laughed a little less heartily.\n\nAfter the story, he recited one of his poems\u2014the one about the Tree of Independence. It told of a strange tree nourished by blood and by the earth on which it stood, tended by a prophet-like gardener who loved his tree above all others. When it needed water, the gardener would call for blood and people would surge to open the veins on their arms, eager to nurture the tree beneath whose cool, shady branches they sat and unburdened their sorrows. If its leaves were crushed and powdered, and then rubbed on the eyes, it would endow the bearer with pride, hope and confidence, triumphing over cowardice and treachery to create a people of strength and courage.\n\nThen the show began. A bearded Indian, wearing a turban and dressed from head to toe in white, came and knelt on the platform. He lowered the microphone and began to play a pipe. From a hole in front of him, which Zari had not noticed before, a dark-skinned woman with a red dot between her eyebrows bobbed up her head several times. Finally, the woman emerged completely, all the while moving to the music and slowly approaching the man. She was wearing a yellow sari with a gold embroidered border. When she started to sing in her shrill, high-pitched voice, she could hardly be heard above the din of the Indian soldiers who were whistling and shouting to the music. Her bracelets jangled as she moved her arms.\n\nAt one stage of the dance, it suddenly seemed as though the woman had unlocked the muscles of her neck. Her head fell effortlessly on to her shoulders, and she kept rotating it to the left and right just like a snake. She also lifted her eyebrows one at a time, to the rhythm of the music. Zari was amazed to see the amount of kohl the woman had used to outline her eyes.\n\nGradually the dancer moved back to the hole from which she had emerged. Then, as the snake-charmer quickened his pace, a rubber hose with a snake's head glued on to it rose out of the hole, stiff as a rod. The woman reached down, and pulled out the rest of the hose. Then she coiled it like a long snake, in a corner of the platform.\n\nAt this point a thin man with bushy eyebrows and a mottled moustache, wearing top hat and tails and carrying an umbrella, stepped up on to the stage. The fluteplayer kept on playing. The woman reached into the hole and brought out some odds and ends\u2014boards, sticks, McMahon's red cloak, a conical hat, a box, a hammer and an air-pump. Then she helped the bushy-eyebrowed man to make a dummy out of the sticks. Taking the rubber hose, the man wrapped it around the frame. After securing a kind of snake's head in place, he threw a cloak over the dummy's body. Next, he placed the conical hat on the serpent-head, glued on a long moustache and, taking a swastika from the dancer, pinned it on to the cloak. Then he went to the air-pump, attached the nozzle to the scarecrow's foot and, to the beat of the music, began to pump it up. Zari watched as it grew bigger and bigger. Its head, body, hands and feet became inflated, swelling to an unbelievable size. It took up so much of the main part of the stage that the turbaned man had to step aside. A voice behind Zari murmured: \"It's Hitler!\"\n\nSuddenly drums began to roll. A fat man, with a cigar in the corner of his mouth, rushed on to the stage, followed by another dressed as 'Uncle Sam'. Then various officers, some in kilts, some with hammer and sickle armbands, one and all invaded the stage. Armed with bows and arrows, they first began to tease the scarecrow. One of their number kept holding them back saying: \"Nyet! Nyet!\" Finally he too gave in, and yelled: \"Good! Good!\"\n\nThe drumming reached a crescendo. Arrows flew at the scarecrow from all directions. Slowly it began to deflate until it sank to the ground with a loud hiss. The crowd cheered and applauded. And then there were other shows... \n\n# _4_\n\nOn Saturday afternoon, Sahar was shod by a new blacksmith. Khosrow was at school so he wasn't there to witness it. When he came home, he looked reproachfully at his father, who said, \"I had to do it, otherwise it would have been too late.\"\n\nThey then began talking of hunting and Yusef promised to take both Khosrow and Sahar along. From that moment until Thursday afternoon, when the riders actually set off, Khosrow's entire concentration was focused on hunting and whether or not Sahar would be able to manage it.\n\nThey had not been gone twenty-four hours when Zari began to miss them. She couldn't help worrying, thinking of all the things that might go wrong. Ameh finally scolded her: \"They're probably thoroughly enjoying their ride, in spite of the fact that you keep imagining the worst.\"\n\nZari instructed Gholam to sprinkle water over the brick paving in front of the house, and to put out the cane chairs around the pool. She was sure they would be back before sunset on Friday. Mina and Marjan, meanwhile, played around the pool, dipping their hands in the water the moment Zari's back was turned.\n\nSuddenly there was a knocking at the garden gate. Zari, certain it was the huntsmen, ran out to greet them. By the time she reached the gates, Gholam had opened them wide. A horse-drawn cab pulled in. Zari was taken aback; they had left on horseback! When the droshke reached Zari, it came to a halt. Two women stepped down. They were wearing heavy veils, drawn tightly over their faces. But what strapping women! They were wearing thick woven summer shoes, and their feet looked very large. They also seemed unusually tall and broad-shouldered beneath their veils. The women bowed their heads at Zari's greeting, and one of them extended a coarse, thickly-veined hand to pay the driver. Zari noticed she was wearing a man's watch on her wrist. Zari racked her brains to remember where she had seen them before. Maybe they were friends of Ameh Khanom, who at that moment was smoking opium on the verandah. \"Could they just be masculine-looking women, or are they gypsies?\" Zari wondered.\n\nHer attention was suddenly drawn to Mina and Marjan, who had plunged their arms up to the elbow in the water. \"Get away from there!\" she scolded.\n\nIndicating the chairs by the pool, she offered the strangers a seat. But they took no notice and walked towards the house. The shorter one was obviously laughing because her shoulders were shaking underneath the veil. Ameh, glancing at the women as she puffed away at her pipe, said: \"I don't recall having had the pleasure...\"\n\nThe women, ignoring her remark, crossed the verandah, opened the parlour door and walked in. Zari was totally bewildered. They were certainly not inmates of the asylum where she usually took bread and dates. But neither was it normal behaviour to arrive at someone's house, walk right in without a by your leave and make yourself at home.\n\nShe followed the women to the parlour.\n\n\"Please take a seat,\" she said, \"although, quite frankly, I can't remember having made your acquaintance.\"\n\n\"Where is Yusef Khan?\" one of them asked in a husky voice.\n\n\"He's gone hunting with Khosrow,\" Zari replied.\n\nIt was a man's voice and a familiar one at that. Someone was playing a practical joke. At that moment, the two 'women' simultaneously pushed their veils aside. Thick eyebrows, dark eyes, long eyelashes and a hooked nose set in a longish sallow face\u2014the spitting image of each other, except that one was younger, and the older one wore a moustache.\n\n\"Malek Rostam Khan!\" Zari exclaimed in astonishment. \"What kind of get-up is this? You half scared me to death!\"\n\nMalek Rostam put a finger to his lips: \"Hush! Be quiet. I'll sit here and wait for Yusef,\" he whispered.\n\nZari went out on to the verandah. There was no one but the twins watching Ameh smoke her pipe. She returned to the parlour with straw fans for Malek Rostam and his brother Malek Sohrab.\n\n\"You really had me fooled, you know,\" she said laughingly. \"Turning up like this after all these years.\"\n\n\"When is he returning from his trip?\" Malek Rostam asked her anxiously. \"Is there a chance he won't be back today?\"\n\n\"I'm expecting him any minute now. But why?\" Zari asked.\n\n\"I hear he's going to the village\u2014to the lowlands tomorrow... Why isn't he back yet?\" Malek Rostam asked again.\n\nZari brought sherbet drinks for the guests, and then fruit and some nuts. She opened the parlour door for more air. But they wouldn't let her put on the lights. She sat facing them.\n\n\"Well, how is it that you've finally come to visit us?\"\n\n\"Oh, Sohrab has come on behalf of my uncle,\" Malek Rostam answered, playing with his moustache. \"I came because I missed you both.\"\n\n\"I bet it was Sohrab Khan's idea to wear the veils,\" Zari said. \"He's still the same mischievous child at heart. Do you remember, Sohrab Khan, what antics you were always up to?\"\n\n\"How could I forget!\" he said with a laugh. \"But we wore the veils so we wouldn't be recognized. If they catch us they will tear us to bits.\"\n\n\"Gone are the good old days when nothing used to worry us!\" Zari sighed.\n\nHer mind went back to one such day; a day in the first year of her marriage. In that same year the government had captured the head of the Qashqai tribe and taken him to Tehran. The tribe itself was breaking camp to move on. When Zari and Yusef arrived, a group of them came out to greet them. They even cheered for them, but it was an empty cheer, as Yusef said. They were dusty and depressed, and clearly not their usual selves. By the time Yusef and Zari reached the chieftain's large tent, most of them had scattered. Malek Sohrab was sitting in the tribal chieftain's place. When he saw them, he declared, \"Welcome to this, our mobile capital!\"\n\nZari had never seen a more beautiful tent in her life than that wandering capital. What carpets and rugs! The inside was painted with designs of legendary Shahnameh heroes such as Rostam, Ashkabus, Esfandiar, Sohrab and other characters whom Zari didn't recognize. It was funny; Malek Sohrab had seemed both childish and mature at the same time. He had got up, showing Zari the picture of Sohrab, and said, \"This is me!\"\n\n\"God forbid! \"Zari had replied, because Sohrab was depicted with a dagger deep in his side. Then he had pointed to the picture of Rostam and said, \"This is Malek Rostam, the elder brother of our chief.\"\n\nZari had glanced at Malek Rostam who was busy whispering to Yusef. There was a wistful smile on his face. Then Malek Sohrab had pointed to the image of a severed head lying in a large basin full of blood. A black horse stood at the basin, smelling the tulips which grew all around it. Malek Sohrab had said, \"This is my own little brother to whom my mother, Bibi, hasn't yet given birth!\"\n\n\"You can't fool me,\" Zari had replied. \"I bet you anything it's John the Baptist.\"\n\nMalek Sohrab had laughed and said, \"All right, let's have a bet.\"\n\n\"What do you bet?\"\n\n\"A Brno rifle.\" Malek Sohrab called Yusef over and showed him the drawing.\n\n\"Your wife says this is John the Baptist.\"\n\n\"Please forgive her,\" Yusef had smiled. \"My wife married straight from the classroom. Her head's still full of the Gospel stories she was forced to read every morning at the Missionary school.\"\n\n\"I know!\" Zari had rushed to correct herself. \"It's the beheaded martyr, Imam Hossein... and that horse...\" But Yusef stopped her.\n\n\"My dear, don't embarrass me any more. That's Siavush.\"\n\nReturning to the chieftain's seat, Malek Sohrab had said, \"We number six thousand in this camp. Let them kill a hundred and fifty sheep a day... And you, Zari Khanom,\" he said after a pause, \"I hear you've brought your own bridal carpet as an offering to our chieftain. We cannot accept it. The very fibres of this carpet were a labour of love.\" And he asked eagerly: \"Did you see the layout of the tents? Did you see the gunmen standing ready? Do you hear the horns and drums? This military march is being played in your honour.\"\n\nThe words had hardly left his lips when Bibi Hamdam, their mother, came in. After the usual greetings, she turned to Malek Sohrab and said, \"Get up from that seat, child! Are you talking nonsense again? They've caught two hens. Run out and cut their throats before the sun goes down.\"\n\nAngrily, Malek Sohrab stood up, made a face at Bibi, and stalked out of the room. When he returned, he threw the dead hens into his mother's lap.\n\nZari remembered it all as if it were yesterday.\n\n\"Zari Khanom, you're deep in thought,\" Malek Rostam observed, breaking the train of Zari's reminiscences. \"Are we disturbing you?\"\n\n\"Oh goodness, no!\" Zari replied with a laugh. \"I was only thinking back to the first time I came to your chieftain's tent. It was the first year of our marriage.\" Turning to Malek Sohrab she said, \"Do you remember what you did to your poor mother in front of me, a newly-wed bride?\"\n\n\"I remember it well,\" he replied.\n\n\"You were a child, then,\" Zari said.\n\n\"I was not a child. I was stubborn and rebellious,\" Malek Sohrab replied.\n\n\"I remember Bibi Hamdam had to change her skirts,\" said Zari. \"I counted, she had eight of them on. Bibi had caught a cold... and you, Malek Sohrab, kept saying that a hardy tribal woman should never get sick.\"\n\n\"I remember very well. That same night I won a Brno rifle from you... and I'm still waiting for it,\" he added jokingly.\n\nAt that moment Khadijeh came in and took the keychain from Zari to give the twins so that they could go to sleep. She looked in amazement at the veiled men.\n\n\"You're sitting in the dark!\" she exclaimed. \"Shall I put the light on?\"\n\n\"No,\" came the reply.\n\n\"I remember,\" Malek Rostam said, joining in the reminiscing. \"It was the year I caught malaria and came to you for refuge. I was in bed for three months in your house, at a time when nobody even dared say hello to us in the street. A friend of a Qashqai was an enemy of the Shah. Yet you nursed me like a sister. I'll never forget. Once the washerwoman didn't come and you washed my clothes with your own delicate hands. Yusef even helped me with the bed-pan himself.\" Turning to Sohrab, he said, \"Sohrab, I'm going. I shouldn't have come here.\"\n\nSohrab answered him in Turkish, and for a while the two brothers spoke together in their native dialect. Since Zari couldn't understand a word, she started to think, and that led her to worry about Khosrow again.\n\nAt the sound of hooves on the gravel, Zari once again rushed outside. The lights were on in the garden. They had shot two deer, and a live fawn was tied to the saddle of the chestnut horse which their steward, Seyyid Mohammad, was riding. The riders dismounted.\n\n\"A good day's work!\" exclaimed Ameh Khanom, who had also come out to greet them.\n\nKhosrow couldn't tell his mother fast enough what had happened on the trip.\n\n\"Mother,\" he babbled, \"Sahar has been really naughty. He chased after the fawn and bit it on the back. Of course he fell down himself. He's hurt his knee and now I have to treat it with burnt hazelnut oil. Mother, do you have any hazelnuts?\"\n\n\"There's some on the table in the parlour,\" Zari said, adding quickly: \"but don't go in there now. We have some important guests.\"\n\nYusef, meanwhile, had gone over to the wooden bed on the other side of the pool. The twins were sleeping there under a mosquito net.\n\nAs they went into the parlour, Zari quickly warned Yusef about their unexpected visitors. He switched on the lights.\n\n\"I was expecting you,\" he said to Malek Rostam, \"but not today. Your visit is not only too late, it's badly timed as well. Today, I can't even say I'm pleased to see you. Why you of all people? Why should you have agreed to such things? After all those discussions we had...\"\n\nHe sat on the sofa and Zari knelt in front of him to remove his boots. Malek Rostam bent his head and chewed his moustache. Sohrab rolled up his veil, threw it in a corner, and sat bolt upright. Yusef continued:\n\n\"You've taken out your rusty, broken guns from the cracks and crevices in the mountain-side, oiled them and taken to looting and killing your fellow-men again. What more can you and I have to say to each other?\"\n\n\"Zari Khanom isn't a stranger,\" Sohrab said, \"and I'm not afraid of saying in front of her that we had to take our revenge. How long can we take it from the government? With that general pardon of theirs, which they later broke\u2014and how! What they promised us on the one hand, they took away with the other. There was only bribery, excuses, hatred, and executions. Their forced settlements turned out to be a total waste of money. They built a couple of mud-huts in dried-up areas, and told us to go and live in them. Instead of books, teachers, doctors, medicine and health care, they sent us soldiers armed with bayonets, guns and hostility. It's only natural that we've gone back to our old trade and taken revenge on them.\"\n\nWhile Sohrab was talking, Khadijeh brought a hookah and placed it in front of Yusef. Zari whispered to her, \"Take the boots and give them to Gholam for cleaning. Bring some tea, too.\"\n\nDrawing on the pipe, Yusef said, \"What can I say, Sohrab my friend; you've put your finger on it yourself. You say you've gone back to your 'trade'. In other words the tribe has become a kind of business for you. You use it to make deals.\"\n\n\"Believe me,\" Malek Rostam protested, \"they acted entirely without provocation at the beginning. I'm personally in favour of the idea of settlements. You know that yourself. But it's as if they themselves don't want us to prosper. Certain forces are at work against us. They want us either to rot away from the inside and ultimately destroy ourselves, or else to stay in our present state.\"\n\nYusef lifted the pipe to his lips again. \"You yourselves prefer this present state of affairs,\" he said. \"If you had been more willing, the settlements might have worked. But my friend, you tribal chiefs have become too accustomed to exploiting your tribesmen. For you they aren't human beings; they're no different from your sheep\u2014you sell them both in one go.\"\n\n\"Don't speak to me like that, Yusef,\" Malek Rostam retorted angrily. \"You're a close friend, we've been classmates and we've shared each other's hospitality many times, but...\"\n\n\"I don't know how to say what I have to say any other way,\" Yusef interrupted. \"You know me well enough. I don't stand on ceremony with anyone\u2014especially not with my closest friends.\"\n\nMalek Rostam replied quietly, \"I know, better than anyone, that tribal life with all its excitement and adventure is not the right way to live. You know that I would prefer to be a settled Qashqai rather than a nomadic one. I know it's not right for thousands of men, women and children to be led by their herds, wandering from the top of the Gulf to the other side of the mountains in search of grass and water. I realize that the lives of so many people should not be tied to cows, sheep and grazing land. But do you think it's up to me alone? Am I the chief? What can one person do?\"\n\nYusef put his pipe aside. \"If that one person really wants to,\" he said quietly, \"he can easily sway others. There are a great many people who are capable of understanding what's right and fair, and recognizing it when they hear it. But these people are scattered and you must join forces with them... Even if you don't do it yourself, your children and the children of others will do it in their time. They will pass through towns and villages, they'll see schools, mosques, public baths and hospitals. They'll grow to understand and want these things and finally do something about their lot.\"\n\n\"You know it's too late for that,\" Malek Rostam replied wearily.\n\nThere was a pause when Khosrow came in to take hazelnuts for Sahar. After he left, Yusef asked, \"What was all that about the Malek Abad Pass? I've heard a few things, but I want to hear about it from you.\"\n\n\"I swear to you, it wasn't anything much,\" replied Malek Rostam. \"The Ezhdehakosh clan disarmed a group of soldiers, chopped off a few heads, took a dozen rifles or so, some ammunition and about twenty horses\u2014that's all. And what's more, they did it without permission. The Farsi Madan clan brought it to my uncle's attention. My uncle doesn't agree with this kind of pilfering.\"\n\nMalek Sohrab, who had been silent for a while, spoke up: \"Brother, tell Yusef about the incident with the captain's pups.\"\n\nWhen Malek Rostam remained silent, Sohrab began to tell the story himself.\n\n\"The dog belonging to the captain in charge of the tribal settlement had just whelped,\" he began. \"A couple of mischievous kids from the Ezhdehakosh clan threw stones at it\u2014a purebred wolfhound, no less. Anyway, afraid that they would be found out, they stole the dog and got rid of it. Again the tell-tale Farsi Madans gave them away, and the captain forced three women from the Ezhdehakosh to breast-feed the puppies.\"\n\nZari felt sick, but Yusef merely smiled and said, \"My dear Sohrab, that story must be at least ten, twelve years old. And it's the third time you've told it to me.\"\n\n\"Then why let me tell it a fourth time?\" Malek Sohrab countered indignantly.\n\n\"I didn't recognize it at first, but it came back to me as you went on. Anyway, what do you think I am, some sort of saint? I'm a human being, like everyone else.\" Turning to Rostam, Yusef continued, \"Well, what do you want from me? We've been talking about this and that; let's get to the point.\"\n\n\"Please believe that I don't agree with everything my uncle does,\" Malek Rostam said. \"I was even against his sending me to you. I don't want to ruin our friendship. But in these sensitive times, I can't turn my back on him.\"\n\nZari could have sworn Malek Rostam had told her it was Malek Sohrab who was there on behalf of their uncle, and he himself had just come along for the visit.\n\n\"You still haven't said what you want of me,\" Yusef reminded him.\n\nMalek Rostam lowered his head, seemingly lost in thought. But Malek Sohrab stepped in.\n\n\"Help,\" he said, after a moment's silence.\n\n\"What sort of help?\"\n\n\"Sell us whatever provisions you have. We'll even buy the unharvested crop. You just name the price.\"\n\n\"Who has put you up to this?\" Yusef asked suspiciously. \"Singer? Up to now there's only been talk of surplus crops. Now it's the whole lot!\"\n\nThe two brothers exchanged a glance. Suddenly Yusef raised his voice: \"You want the provisions to sell to the foreign army in exchange for arms, so you can go on fighting and looting your fellow-countrymen and brothers! Don't you realize that if you give them an inch, they'll take a mile? Haven't you any brains? Those 'mysterious government forces' which you claim have prevented you from prospering in the settlement could've been used to your advantage at a time like this... So where's that spirit of adventure, that fight and dignity now?\" Yusef's moustache trembled in anger.\n\n\"Do you know government men stopped the tribe in Kam Firouz?\" Malek Sohrab pleaded. \"Do you realize they refused permission to migrate this summer? All around us we face the guns and bullets of our own countrymen. The green grass on the slopes of the mountains is drying up untouched, and our sheep are starving and dying of thirst.\"\n\n\"Now look here, Sohrab,\" Yusef retorted angrily, \"don't give me this nonsense, you young fool. You sold most of your sheep to the foreigners. They're frozen now, and being dutifully guarded in the cold-storage of the Ahwaz-Bandar Shah railway.\"\n\nRostam's eyes were glued to the designs on the carpet.\n\n\"If we hadn't sold them,\" his brother answered, \"they would have died on us. Believe me, our sheep couldn't even walk at the end. They had to be carried away in trucks.\"\n\n\"What did you do with the money\u2014buy weapons? Golden pitchers? Golden jars? Did you sew royal crowns on to your hats, and get a thrill when they started calling your uncle 'His Highness'?\" Yusef snapped.\n\nMalek Sohrab, unable to contain himself any longer, jumped to his feet.\n\n\"Yusef Khan, our friendship is all very fine, but everything has a limit!\" he shouted. \"What right have you to call me a young fool? To say that we have no brains? You are the one with no brains, because right now you should be the deputy, not your brother...\"\n\n\"Deputy for whom\u2014Singer? I spit on the deputation for which you, Malek Sohrab, have to act as go-between!\" Yusef said, his voice shaking with anger.\n\n\"What on earth are you talking about!\" Malek Sohrab shouted even more angrily. \"You say whatever comes to your mind without pausing to think that you may be the one who's wrong. Who uses me as a go-between? Why are you so self-righteous? Who on earth do you think you are? And besides, what mistakes? What do golden pitchers have to do with us? Why do you blame us for what Davoud Khan may have done? Why? What right have you got?\"\n\n\"You're all the same,\" Yusef sighed wearily.\n\nMalek Rostam turned to Sohrab, trying to calm him down. \"Sit down, boy,\" he said. \"I made you promise you wouldn't insult my friend.\" Then the two brothers started talking in Turkish. Rostam's voice gradually became harsher as Sohrab's tone softened, until he finally sat and apologized. Yusef pulled the hookah towards him.\n\n\"The charcoal has gone out, let me go and light it again,\" Zari said.\n\n\"There's enough fire within me,\" Yusef sighed, as he drew on the pipe.\n\n\"I didn't mean to offend you; I apologize again,\" Sohrab said, forcing a smile.\n\n\"My dear Sohrab, for once you've discovered my weakness and see what a fuss you made! But I like that, you have guts. Only you still don't see very far.\" He put the pipe aside and continued, \"You know, I was never happy about your playing around with the Germans, nor am I happy now that you've made a deal with their enemies. You're the ones who've turned Hitler into a 'Messiah' among our people. These tricks don't really work with us, and your political flirtations only gave these foreigners an added excuse to come here.\"\n\n\"Well brother, after all it's a war,\" Malek Sohrab said gently. \"They don't give out sweets in a war. These people have to stay around to protect the oil and the access to the Gulf. They would have come in any case, even if we weren't here. And anyhow, they only come to the town on sick leave or on holiday. The main camp is at Khorramshahr... they have no other alternative.\"\n\n\"Now you're defending them too, my friend?\" Yusef asked in a fatherly tone. \"Their war is their affair. What does it have to do with us? Hitler is from their continent. They created him themselves. Let them pay for that. Let them pay for everything, even the unhappiness they've brought on the ones who, according to Singer, 'have resources they don't know how to use'. The English never ask who's to blame for this ignorance.\"\n\nMalek Sohrab glanced at his watch. \"It's getting late,\" he said. \"I've got a headache. Do you have any aspirin? Make sure it's Bayer.\"\n\nZari got up and took away the hookah. When she returned with the Bayer aspirin and a glass of water, Yusef was saying, \"I assure you that it's so. In order to discourage their ally from the plan I just mentioned, they'll arrange a few skirmishes and manoeuvres with your aid, and quite a number of our people will be slaughtered at your hands. The British never say no to an ally. They simply confront him with a fait accompli so that he abandons his original plan himself. Mark my words: they will stain your hands with blood, while they themselves just sit and watch. A real massacre will take place amongst us.\"\n\n\"We must be going soon,\" Malek Sohrab pointed out anxiously, \"so let's get back to the point. You still haven't told us whether you'll sell the provisions or not.\"\n\n\"He hasn't said? Must he spell it out for you? He went to such lengths...\" Malek Rostam laughed, but still Malek Sohrab bargained: \"Please believe that we don't want to sell all of it. Our men are hungry; they're falling like flies from sickness and hunger.\"\n\n\"I'll take Rostam's word of honour,\" Yusef answered. \"If he promises to buy just enough for your own men, and to use my provisions for your people only, then I will agree. Tomorrow I'm going to Kavar... I know you've been stopped there too. Bring camels to load the provisions. But, remember, only for the tribe's use. Band Bahman is just a kilometre further away, and there you'll find water. I'll also provide the pasture for free.\"\n\n\"I can't cheat you,\" Malek Rostam said dejectedly.\n\n\"I know you can't.\" Yusef paused, then said with feeling, \"Rostam, try to turn away from this path you've chosen for yourself. Why don't you try, at least, to create a spark of faith somehow, somewhere. Teach your people skills. How many times have I told you! My unused lands are there just waiting for houses, schools, public baths, hospitals, mosques, pastures...\"\n\nMalek Sohrab cut Yusef short. \"These things you talk about are not in our nature,\" he said. \"We've lived free. Nature has always been within our reach. We've ridden horses in the mountains, rested on the plains, camped under the skies. We can't be imprisoned in houses.\"\n\n\"But it seems it can be done to us Khans,\" Yusef said bitterly. \"We used to have the best gardens in town, the best houses... and where are they now? At the disposal of the Supreme Command, that's where!\"\n\nMalek Sohrab, knowing what Yusef was about to say, stopped him gently. \"I promise you our people love the kind of life they lead. If they settle down their spirit will be broken.\"\n\n\"Because it's the only life they've ever known,\" Yusef argued. \"But Sohrab, my friend, when a man cultivates a piece of land, labours over its soil and reaps its harvest, he becomes attached to that land. In a village, nature is still within reach. When you're settled...\"\n\nSohrab finished the sentence for him: \"You become stupid, helpless, petty and cowardly.\" Then, as if to change the subject, he said, \"May I ask you a question? What will you do with all your corn and grain and dates? It's harvest time now in the lowlands. What will you do after the harvest? Will you hoard everything?\"\n\n\"I'll give my villagers their share to the last grain,\" Yusef replied. \"The rest I'll bring to town. Unlike those traitors who sold both their villagers' share and the food for the town to the foreign army. There are five of us who'll do this, and we're landowners of considerable means. Two of us are on the city council, and we've all sworn to take control of the town's provisions. We have the mayor on our side. I know you're not the sort to reveal this to anyone. I also want you to know that it's not in my nature to hoard. The hoarders have sent their own people's provisions to North Africa and...\"\n\nMalek Rostam interrupted him. \"Majid is probably with you too,\" he said sadly. \"God willing, I hope you manage to accomplish something.\"\n\n\"What will you do about the Governor?\" Malek Sohrab asked.\n\n\"The Governor is a human being,\" Yusef replied. \"He'll agree to end the food shortage in order to have this part of the country quieten down.\"\n\n\"I think the outlook is bleak,\" Malek Sohrab commented. \"It's a dangerous plan. So long as you only talk about it, they'll leave you alone. But the minute you put your words into action, they'll stop you by whatever means they can.\" He stood up and put on his veil.\n\n\"We'll do our utmost,\" Yusef assured him, adding: \"Stay for dinner.\"\n\n\"No, we'd better go,\" Malek Sohrab replied. \"They'll be worried about us; they may think we've been caught. Please ask someone to get us a droshke.\"\n\nMalek Rostam got up then and put on his veil, inside out. Zari laughed.\n\n\"You have it on the wrong way round,\" she said. \"The seams are showing.\"\n\n\"You stay,\" Yusef said, turning to Rostam. \"I'll take you back myself tomorrow morning before sunrise.\"\n\n\"All right,\" he agreed.\n\nThey went into the garden together, and sat on the cane chairs to wait for Sohrab's cab. The verandah lights were on. Zari, standing at the edge of the verandah, saw Khosrow squatting by Ameh's opium brazier. He was roasting hazelnuts in the frying pan while she ground more nuts on a flat stone. Sahar was on the verandah, with his bridle tied to a door handle. Zari heard Yusef say, \"Why didn't you take the bridle off the poor creature? Why did you bring him on to the verandah? Child, the animal is tired. Take him to the stables and leave your treatments till the morning.\"\n\nKhosrow got up. \"Father, please let me. The hazelnut oil is ready now. I'll rub it on his knee-cap and then I'll take him to the stables. I brought him to the verandah because he was playing around. He was chasing the fawn, who kept waking with a start and throwing himself against the branches and bushes out of fright. So I brought Sahar here with me.\"\n\nAmeh burned herself taking the hot hazelnuts out of the frying pan. Dropping the nuts, and blowing on her fingers, she said:\n\n\"Brother, tell Gholam to kill the fawn tomorrow. First of all, not everyone managed to get meat from the hunt and they're grumbling. Secondly, keeping deer brings bad luck. Come to think of it, I wish the men in this family would put hunting out of their minds once and for all. Only last year you shot a pregnant deer. The minute they opened her up and I saw that little one sleeping there in her mother's womb, I beat myself on the head. I knew it was a bad omen...\"\n\n\"Put your veil on. Those women in the garden are really men,\" Zari informed her sister-in-law quietly.\n\n\"God protect us!\" Ameh said, jumping up in astonishment. \"Heaven have mercy!\" Frenziedly she covered herself with her veil.\n\nWhen the droshke arrived, Malek Rostam stood up too.\n\n\"Allow me to leave also,\" he said to Yusef. \"I have to reach my uncle as soon as possible. I think you're right. My uncle has blindly worked himself into a tight corner.\"\n\nYusef only asked, \"Blindly?\"\n\n# _5_\n\nIt was ten days now since Yusef had left for the lowlands. Zari wandered about the garden with her gardening scissors, looking unsuccessfully for flowers to cut. To her, the heat there felt every bit as oppressive as in the lowlands. Summer always seemed to rush upon them in this way, brushing away the last signs of spring. Mina and Marjan followed their mother around from one rose-bush to another, chattering and giggling, while Gholam watered the brick paving in front of the house to cool off the garden. Along one border of the stream that ran by the brick paving were some tired-looking amaranthus, while along the other side a variety of snapdragons stooped under layers of dust, side by side with the humble-plants sleepily closing their petals to the approaching dusk. Zari's only hope lay in the tuberoses that Gholam had said would bloom with the full moon. The orange blossoms were scattered now, brown and withered like so many burnt stars beneath the trees. At least in winter the narcissi bloomed gaily by the small stream, surrendering their image to the water only to be carried away, unseen and lost forever, as the water tumbled into the pool. Even spring brought with it white and purple violets that coyly greeted the passing stream, nodding cheerfully at their own reflection. But nothing seemed able to resist the heat of the summer.\n\n\"When is father coming to throw me up in the air?\" Mina asked her mother. \"You never do that to me!\"\n\nMarjan pouted. Her lips were like little buds that seemed to Zari more beautiful than all the flowers in the world. \"We won't ever talk to you again,\" she said, adding her voice to her sister's. \"So there!\" And she pleaded, \"Now why don't you throw us in the air, just once?\"\n\nZari picked Mina up and tried to throw her into the air.\n\n\"You're too heavy\u2014I can't do it,\" she complained, slapping the child's chubby thigh.\n\n\"Father's hands are big and he can do it. Your hands are too small, so you can't. We'll wait till your hands grow up,\" Mina told her mother.\n\nAt that moment, Ameh came through the garden gate. She had been to the public baths and was holding a paper bag dripping with water. Mina and Marjan ran towards her, shouting: \"Auntie, what have you brought us?\"\n\n\"Fresh walnuts.\"\n\n\"Give us some then.\"\n\n\"I hope you enjoyed your bath,\" Zari greeted Ameh, taking the paper bag over the children's heads, so that she could go and wash the walnuts. When she returned, Khadijeh had brought in the bag that Ameh used for her trips to the public baths, and put it down on one of the cane chairs. Ameh took out her towels and hung them on the line. Mina was dashing about chasing Marjan, but when Zari put the plate of walnuts on the table, the children rushed over excitedly.\n\n\"Well, talk about having your prayers answered!\" Zari smiled.\n\n\"This town has turned into a zoo,\" Ameh complained. \"Everywhere you go, those dark little Indian men follow you about saying, 'Need woman, need woman!'\" She dipped her hand in the pool as if to wash away the obscenity of their suggestion. Holding out her wet hand, she sat on a chair and continued:\n\n\"The children in the street tried to chase away the pathetic Indian who followed me; they were teasing, and singing some nonsense at him. Then suddenly the man brings out this chain he had with him, swings it around in the air, stamps his foot, and shoos them all away in no time.\"\n\nKhadijeh appeared then, carrying Ameh's opium brazier with all the accessories, as well as some fresh tea. As Zari and the twins joined her on the verandah, Mina asked: \"Auntie, did the Indian cut the children's heads off?\"\n\n\"Oh yes! He put them over his knee and sawed their heads off, didn't he?\" Marjan said with rounded eyes.\n\n\"Our Khosrow's late,\" Ameh commented. \"Perhaps that difficult final exam didn't go too well, and that's why he's not home yet. I think we should send Gholam to fetch him, sister.\"\n\nBefore Zari could answer, they spotted Abol-Ghassem Khan coming up the garden path towards the verandah. He was muttering to himself and gesturing with his hands. Zari's heart sank at the sight of him. Lately she had begun to feel as though she were facing the prophet of doom every time she saw him. And each time he blinked, she imagined he would blink her whole life away. As he reached the edge of the verandah, Zari stood up.\n\n\"Please come in,\" she invited.\n\n\"No, I'll just stay here.\"\n\n\"Greetings!\" said Ameh Khanom, between two puffs on her opium pipe.\n\nShe put the pipe down next to the brazier and poured some tea, which she handed to her brother. Zari's eyes were on Abol-Ghassem Khan who put a lump of sugar in his mouth, then poured some tea into the saucer to cool it.\n\n\"Has something happened?\" she asked.\n\nAbol-Ghassem Khan put the saucer down on the edge of the verandah and asked: \"Any news of my brother?\"\n\n\"No, not yet.\"\n\n\"I don't really know how to tell you this,\" he said. Zari suddenly felt dizzy. She sat down and said faintly, \"God forbid, has something happened to Yusef?\"\n\n\"Out with it, let's hear the worst!\" Ameh cried out.\n\n\"This morning they called from the Governor's house,\" said Abol-Ghassem Khan. \"They told me: 'Miss Gilan Taj has heard a great deal about Khosrow's colt, and she's decided she would like it, so we're offering to buy. Send us the colt and we'll gladly pay any price you like.' God knows I've been in a state since this morning; I'm so distracted, I can hardly think.\"\n\nZari's eyes filled with tears. She looked at Ameh, with her braided hair and red scarf, flushed and tearful by now, hardly able to fix the piece of opium on her pipe for the trembling of her hand. Sure enough, the opium slipped from her grasp into the blazing brazier, raising a lot of smoke.\n\n\"A curse upon their household!\" she said. \"I'm so furious I could take this brazier with its burning charcoal and smash it over my own head! You didn't happen to mention, I suppose, that the boy's whole existence revolves around his horse? They cut off your tongue, did they?\"\n\nMina sidled up to Ameh and tried to offer her the one remaining walnut which she had held tightly in her little fist.\n\n\"Have it, auntie,\" she urged, as if to comfort her. \"I was saving it for Khosrow.\"\n\n\"Khadijeh!\" Zari called out. \"Come and take the children to Haj Mohammad Reza's house and show them the snake he caught yesterday.\"\n\n\"Has he taken the teeth out?\" Marjan asked. \"Has he?\"\n\n\"Yes, dear, don't be afraid,\" Zari reassured her.\n\nMina took Marjan's hand, saying: \"You play with him for a minute, and then I'll play with him for a minute, all right?\"\n\n\"They play with snakes?\" Abol-Ghassem Khan asked incredulously.\n\n\"No, she won't let the children touch the snake...\" Zari replied distractedly.\n\n\"Seeing that everything is done backwards in this house, I thought perhaps...\" Abol-Ghassem Khan began with a laugh, but he never finished his sentence. Instead, he asked gently: \"Did you find the snake here, in this house?\"\n\nZari, distraught at the thought of Khosrow parting with Sahar, felt the more she talked about the snake the better.\n\n\"Yes,\" she answered. \"Yesterday as we were sitting on the verandah, a female snake fell from the windowsill on to the paving in front of the house. Gholam happened to be watering at the time and he smashed the snake over the head with his watering-can. But it kept moving so Gholam had to finish it off with the shovel. He told us that the male snake would eventually come after its mate. So he called in Haj Mohammad the dyer who went to the roof, found the nest, and caught the male snake.\"\n\n\"Now I've dropped all my precious opium in the fire!\" Ameh complained.\n\nIn the end, it was Abol-Ghassem Khan who returned to the main issue.\n\n\"Please don't imagine that I want to hurt Khosrow,\" he said. \"I swear on my son Hormoz's life that Khosrow is very dear to me. I told the Governor's secretary over the telephone: 'This child is very attached to his horse; he doesn't leave its side for a minute. I'm prepared to go to the village and bring my best horses for the Governor's daughter,' I forget her name\u2014Gilan Taj, Milan Taj, or whatever. She said, 'Well, Miss Gilan Taj has had typhus... she's just recovering... and she's been hankering after your nephew's horse.'\"\n\nAmeh Khanom prepared her opium pipe again. She drew on it long and hard.\n\n\"Didn't you tell them his father had gone to the lowlands and to wait until he returned next week for his permission?\" she said. \"Don't you know that my sister-in-law doesn't move without Yusef's permission?\"\n\n\"As God is my witness, I did. The Governor's secretary said: 'Your brother's wife would refuse you a worthless horse? They'll pay for it, they don't want it for free, you know.'\"\n\nPutting the opium aside, Ameh poured tea for Zari and herself.\n\n\"I know this mess is all your doing,\" she said to her brother. \"To become deputy, you'll stoop to anything. How did that little minx find out that Khosrow has a horse? You've engineered this whole thing. And now you're stuck with it.\"\n\n\"By God Almighty and all the holy prophets!\" Abol-Ghassem Khan protested. \"I swear on the Holy Quran that I never mentioned the horse. Don't you know about Ezzat-ud-Dowleh? She's at their house from morning to evening, plotting and scheming behind everyone's back\u2014a right old busybody... Anyway, I tried to ignore the whole thing, but just before I left, the Governor himself called me to ask about the horse. I said: 'Your honour, my brother is away in the lowlands.' He said: 'Come, come, my daughter has just recovered from an illness. Send the horse over for a few days. When she gets tired of it, we'll send it back.'\"\n\nZari thought that maybe he was telling the truth. She looked at Ameh, who was poking the ashes with the tongs. There were tears in her eyes. \"This town has gone mad,\" Ameh said. \"I'm getting out of this place. I'll go and live in Karbala, the holy city, as my poor mother did.\"\n\n\"What will you do for a passport?\" Abol-Ghassem Khan shouted, his temper rising. \"And what about an exit permit? No wonder they say women are bird-brained! And all of this with a war going on... You think it's going to be easy to leave?\" Turning to Zari, he said: \"Tomorrow morning they're sending for Sahar.\"\n\n\"When you've given in once, they expect you to give in every time,\" Zari said, remembering the earrings. \"It's my fault, I've been too weak. But this time I'll stand up to them.\" Suddenly she felt something awaken in her. \"I'll go to the Governor myself,\" she declared. \"I'll tell him there's a limit to everything. Is his daughter the only one who's allowed to hanker after a horse? Can't he bear to see anyone else in this town with something precious? Mine, mine, mine, everything always mine!'\"\n\nAbol-Ghassem Khan couldn't believe his ears.\n\n\"Sister!\" he exploded. \"I would never have thought it of you! Now you're sounding like Yusef!\"\n\n\"If only more people were like Yusef, things would be very different,\" Zari said. \"Our men must learn to stand up for themselves. And if they're away, their wives should do it in their place. If more of our people had the courage to stand up for their rights, maybe one day we could achieve something.\"\n\nAbol-Ghassem Khan put his head between his hands.\n\n\"I swear to God, you've all gone mad!\" he moaned. \"That one says she's going to up and leave... this one says she going to stand up to them. See what kind of a corner they've worked me into! And all for a miserable horse...\"\n\nAt that moment Khosrow came out of the stables with Sahar. Zari watched as he let the colt loose in the garden, then walked on towards the verandah. His eyes travelled from his uncle to his mother; then from his mother to his aunt.\n\n\"What's happened?\" he asked, seeing their sullen expressions.\n\nAbol-Ghassem Khan laughed. \"I'm going hunting, and I'm taking you with me. Don't listen to what women say. They're all cowards.\"\n\n\"How did your exam go?\" his aunt asked.\n\n\"It went very well, auntie. I think I'll get top marks,\" Khosrow replied. Turning to his uncle, he asked, \"Can I take Sahar?\"\n\n\"No, son, we're going a long way. Captain Singer is coming too. I want to show them what fine young men you and Hormoz have become. You can ride all kinds of horses, you can shoot well...\"\n\n\"It's not possible,\" Zari interrupted. \"Khosrow's got exams.\"\n\n\"But mother,\" Khosrow replied in astonishment, \"you know very well my exams finished today. Please let me go.\" Turning to his uncle again, he said: \"If only I could bring Sahar along...\"\n\n\"Sister, let him come and see something of life, become a man, outgrow his fears. He'll be in good hands, I promise you.\"\n\nAmeh, who had been deep in thought till now, interrupted him:\n\n\"The man you want to make out of him is a far cry from what Yusef has in mind. Leave the child alone. All this lying and pretence\u2014\"\n\n\"Auntie, mother, please let me go! I'm old enough now,\" Khosrow begged.\n\n\"Go on, son, get ready for the trip. I'll let you use my own Brno\u2014that is, if it's not too heavy for you,\" Abol-Ghassem Khan said cheerfully.\n\n\"I have my own gun,\" Khosrow answered as he hurried off.\n\n\"Do you imagine it's possible to stand up to the Governor?\" Abol-Ghassem asked gently. \"Yusef is risking his life with the kinds of things he's been doing and saying lately. At least let me cover up for him. I've heard that Malek Rostam has been listening to my brother's high-flown nonsense and he went and pulled a gun on his own uncle. Now he's taken refuge with Yusef. They're even saying that Yusef has handed out provisions to thirty tribal families. Malek Rostam, who's even more out of his mind than my brother, and that demented Majid, have joined forces with him to build houses for these families, filling their heads with all sorts of dangerous ideas.\" Abol-Ghassem Khan paused.\n\n\"For instance,\" he sighed, \"they're building thorn-houses out of star-thistle. Pipes take water to the roof, the water drips down on to the walls, and the wind cools the place. Of course our dear Yusef is sitting there, whistling away, thoroughly enjoying himself! Doesn't the young fool realize that these tribal people don't need provisions? That they don't need thorn-houses? As far back as anyone can remember, they've been content with their acorns, mountain almonds and their own shelters. Why should they need houses? Their black tents are more than enough for them. These people are rebels against the government. Just a few days ago they disarmed a gendarmerie regiment in the Takab Pass. And Yusef has joined up with a bunch of dreamers like himself to take over the unlawful distribution of the town's provisions.\" Abol-Ghassem Khan paused again. \"So you see,\" he concluded, \"there's no harm in giving the Governor a small bribe now and again to soften him a bit and make him better disposed towards Yusef. I tell you, it's fatal to fall out with the Governor.\"\n\n\"You can't oppose the Governor,\" said Zari dejectedly, \"and you can't oppose Singer. They've become sworn brothers. This town becomes more and more like the Mordestan red-light district every day.\"\n\n\"By Almighty God!\" Abol-Ghassem Khan lashed out. \"More of Yusef's nonsense! Look Zari, don't argue with me. Don't I have any rights at all __ over my brother's worthless horse? I swear on my dead father's soul that I won't let Khosrow suffer... I'll take him hunting. I'll keep him in the village for a few days. Whichever of my colts he chooses, he can have. Get a receipt for the horse when they come early tomorrow morning to take it away. When we return, just say the horse died. It's the only way. While we're in the village, I'll find a way to tell Khosrow that the horse is sick. I'll say he mustn't let himself be so attached to worldly things if he's going to suffer so badly when he loses them.\"\n\nAmeh raised her head. \"Why don't you practise what you preach?\" she said bitterly.\n\nBefore Abol-Ghassem Khan could reply, Khadijeh came to take away the opium brazier.\n\n\"Where are the children?\" Zari asked.\n\n\"They're with Gholam watching some tribal men and women do a dance. These beggars from the tribes are in such a sorry state, poor souls!\" Khadijeh sighed.\n\n\"Good gracious!\" Abol-Ghassem Khan said with a laugh. \"Why does every member of this household have to be so concerned with the welfare of tribesmen, peasants and porters? I just don't understand it.\"\n\n\"Khadijeh, go and light a hookah and bring it here,\" Zari said. As Khadijeh walked out, Khosrow came on to the verandah.\n\n\"Mother, will you give me the key to the cupboard?\" he asked. \"I want to get my gun... Have you seen my hunting trousers?\" he added. \"I can't find them.\"\n\n\"Come without them, boy,\" Abol-Ghassem Khan said cheerfully. \"We have so many pairs of hunting trousers there, you wouldn't believe it!\"\n\nZari felt a lump in her throat. She took out the chain from her house-dress, and placed it on the rug on which she was sitting. When Khosrow left, she started to cry. \"So this is how they manage to corner you,\" she thought to herself. \"By making all their deals behind your back. But there's still time. Tomorrow morning when they come from the Governor, I can refuse to give the horse. I can tell the Governor's servant that Sahar died and that will be an end to it.\"\n\n\"Sister, as God is my witness, I can't bear to see you cry,\" Abol-Ghassem Khan began. But he stopped short because Khosrow walked back in just then carrying his travelling clothes, his rifle and saddle-bag.\n\n\"I'm ready,\" he announced. Zari, struggling to hold back her tears, bent her head. Khosrow kissed his aunt, then turned to his mother and put his arms around her neck. Kissing her wet face, he said, \"I'm not going away to China, you know... Mother, ask uncle to let me take Sahar with us.\"\n\n\"Come along, son,\" Abol-Ghassem Khan urged. \"Gholam will take care of Sahar.\" And he said goodbye and walked away.\n\n\"I can't refuse to go,\" Khosrow whispered in his mother's ear, \"because uncle will think I'm afraid of the shooting.\"\n\nSahar was standing quite still under the orange trees. He didn't move when Khosrow approached him. The boy took the colt's face between his hands and stroked his mane. Then he called over his shoulder: \"Mother, don't forget to give him some sugar lumps. Gholam knows the rest. He'll muck out the stable, give him clean hay and groom him.\"\n\nSahar lowered his head and dug at the soil underneath the orange tree. After Khosrow had gone, he came near the verandah and neighed loudly. His mother answered him from the stables. Zari looked at him through her tears. \"You poor beast!\" she thought. \"What sweet eyes you have. Why don't you look straight at me? Why lower your gaze? Why don't you call me a helpless woman who'll betray you tomorrow?\"\n\n\"I for one am leaving this place,\" Ameh Khanom announced. \"Why do I need a 'dashport', 'pashport' or whatever they call it? I'll get myself smuggled out. I'll buy gold coins with my money, and sew them into the lining of my coat. I'll just take one suitcase, get myself to Ahwaz, and it'll be easy from there. I'll go through the date-palm plantations, then find some Arabs, give them each a gold coin, and they'll put me in one of their boats to cross the Tigris. Then I'll be rid of all this. From then on, I won't be a burden and I won't let anyone impose on me. And it won't be my own country so I won't have to worry all the time about what happens to it.\" She clenched a fist to her bosom, praying:\n\n\"O Imam Hossein! Allow this poor creature of yours to come to you in Karbala!\"\n\n\"Did you want the hookah, Khanom?\" Khadijeh came in to ask, bringing one with her.\n\nZari took the pipe from Khadijeh, and drew deeply on it. It made her cough. Then she drew on it again and again. It made her feel sick, but she kept on inhaling.\n\n\"They can drive you to addiction, sister,\" Ameh warned. \"You mustn't smoke if you can help it. A habit is a terrible thing.\" She looked up at the sky and said with bitterness, \"O Lord, I'm not ungrateful, yet I've never known anything but sorrow in this world of yours. They hounded and harassed my husband to death. He couldn't take it any more, and smashed himself, on horseback, against the pillars of the British Consulate building. My only son died young. A boil grew in his throat, and he withered away before my very eyes. In all of this godforsaken town no one could give him the medicine he needed... O Lord, maybe you brought me all these sorrows to see whether I have the patience of Job. Well, I haven't, I haven't! Grant my only wish now. Let me make my pilgrimage!\"\n\nZari was in tears again. She brushed them away with the back of her hand. \"Ameh Khanom,\" she pleaded, \"don't make me so unhappy. Where do you want to get up and go to? At least this is your homeland. Your husband and son are buried here. Whenever you feel lonely, you can visit their graves. Whom will you turn to there?\"\n\n\"To Imam Hossein.\"\n\n\"It's hot there. The climate won't agree with you. There's a big garden here. My children are like your own. We live like sisters. Besides, how will they send you money?\"\n\n\"I'm only one person. I'm willing to live on bread and water. What makes me more special than Bibi, my mother?\"\n\nBefore Zari could reply, Khadijeh came out to the verandah: \"Khanom, the children are making a fuss. No matter what I say they refuse to eat their meal. They're driving me out of my mind.\"\n\n\"I'll come,\" Zari said, getting up and going to the parlour. She found Marjan sitting on the table, rubbing her eyes. Mina was standing by her, looking frightened and staring anxiously at the door. Catching sight of her mother, she laughed and stretched out her arms. Zari sat next to them and tried to put a spoonful of food in Mina's mouth, but the child pushed the spoon away. When she tried to feed Marjan, the same thing happened.\n\n\"I don't want any rice-pudding!\" Marjan cried.\n\n\"Why not?\" asked Zari.\n\n\"I don't like it!\" she shouted.\n\n\"All right, then just have some bread,\" Zari offered.\n\n\"That child who threw a stone at me said, 'Gimme some bread! Gimme some fruit from your tree!'\" said Marjan, rubbing her eyes.\n\n\"Which child?\"\n\n\"That child who didn't have any shoes. That one whose mama danced. The papa sat down and said: 'Ouch!' His foot was hurt bad,\" Marjan explained.\n\n\"See, that poor child had no bread to eat. But you won't even have your rice-pudding and honey.\"\n\n\"Gholam went and hit him,\" Marjan said.\n\nThey drove Zari to distraction before taking a few more spoonfuls. As she was taking them to bed, she saw that Ameh was still sitting quietly next to the opium brazier.\n\nThe children, unable to get to sleep, tossed about restlessly. Obviously, Abol-Ghassem Khan's disturbing afternoon visit had affected them too.\n\n\"If you close your eyes, I'll tell you a story,\" Zari promised.\n\n\"I'm scared,\" Mina whimpered.\n\nZari didn't know why she should suddenly think of McMahon and the story he had written for Mina and Marjan. That night, the night of the wedding, when she had gone to the dinner table, McMahon had managed to find a plate and cutlery for her, despite his drunken state. The room was so crowded, with everyone rushing to find a place at the table. No one moved away, and late-comers were not given a chance. Those people didn't know the meaning of real hunger, Zari reflected, but they certainly behaved as though they did. Their children didn't have to go around barefoot, begging for a lump of bread...\n\nZari remembered thanking McMahon. \"I really enjoyed your story,\" she told him. McMahon had laughed. His eyes were like slits in his face. She remembered him saying, \"I'll polish it up, and send it to a publisher of children's books.\"\n\nThe Governor had come out then, and invited McMahon to sit at an empty table reserved for foreigners only, where they would be served roast pork. But McMahon wasn't tempted, choosing to stay with his friend's wife. Again, Zari thanked him.\n\n\"I hope you succeed in building that airplane which drops toys to little children!\" she said.\n\nMcMahon sighed. \"But who will ever build an airplane which will shower consolation over sorrowful men... men who've lost their mothers...\"\n\nYusef made his way to them, bringing a plate of rice spiced with pistachio nuts and raisins.\n\n\"For all three of us,\" he had announced.\n\nMcMahon went on talking to Zari. \"When I think about it,\" he said, \"I realize that all of us, all our lives, we're just children who get our happiness from our toys. The day, alas, they take away those toys, or don't let us have new ones\u2014our children, our mothers, our philosophies, our religions\u2014we crumble.\"\n\n\"Have some of this now,\" Yusef had laughed. \"I've never seen anyone so blind drunk and so philosophical at the same time!\"\n\n\"I promise you I couldn't swallow a thing,\" McMahon replied. \"Anything more, and I'd burst!\"\n\nMarjan brought Zari back to the present. \"I'm scared!\" she cried out. \"Snake!\"\n\n\"Go to sleep, dear,\" Zari said reassuringly. \"There's no snake around. It's in Haj Mohammad Reza's yellow box. They've taken its teeth out, too, and the box is locked.\"\n\nThen she started to tell a story.\n\n\"Once upon a time there was a man who built a big plane. The plane carried only toys, story books, fruit, food and sweets for children...\"\n\n\"Mummy, was there a snake in the plane?\" Mina asked.\n\n\"No, dear,\" Zari answered, \"there wasn't a snake; the plane was loaded with things children like. This plane would fly over the towns to drop whatever toys the children wanted.\"\n\n\"But they'll break!\" Marjan exclaimed.\n\n\"No, the plane flew low over the houses, and the children held out their skirts underneath the plane. Then the pilot dropped whatever they wanted into their skirts.\"\n\n\"What about Khosrow?\" Marjan asked. \"Khosrow doesn't have a skirt.\"\n\n\"You're right,\" Zari smiled. \"But the pilot also gave toys and things to boys even though they don't wear skirts. Sometimes he stopped his plane on the roof and...\"\n\n\"Would he give toys to the child who was throwing stones?\" Marjan interrupted.\n\n\"Of course,\" said Zari.\n\n\"Oh good.\"\n\n\"Now where was I?\" Zari continued. \"Oh yes. The pilot stopped the plane on a rooftop and picked up the good children and took them to the sky with him. They flew past the stars, past the moon. They flew past them so closely they could reach out and gather the stars and put them in their lap.\"\n\n\"Tell him to bring his plane on our roof,\" Marjan piped up again, \"and give Sahar to Khosrow... all right?\"\n\n\"All right,\" Zari promised; \"now go to sleep.\" It occurred to her that if the twins were developing a memory even for recent events, they were no longer babies.\n\nAs soon as the children were asleep, Zari went out on to the verandah. Ameh was still sitting there, with her hand under her chin, staring at the cold brazier in front of her.\n\n\"Are you thinking of your journey?\" Zari asked.\n\nAmeh lifted her head. Zari was taken aback to see tears in her eyes.\n\n\"Yes, sister,\" Ameh answered. \"Even if my heart is sad and heavy, it doesn't mean that's all there is in the whole world. Now that it's too late for happiness in this life, I want at least to prepare for my peace afterwards. They say whoever is buried next to the Imam won't have to answer to Nakir and Monkir. There's no inquisition of the dead either. First the Imam Ali, and then Imam Hossein come to you. If you're a woman, Hazrate Fatemeh comes to you. Hand in hand with these holy ones, the dead are taken to God...\"\n\n\"It's strange,\" commented Zari, \"how Abol-Ghassem Khan disturbed us all with his news! Even the children felt it. They saw Haj Mohammad Reza's snake yesterday and they weren't afraid. But tonight they were frightened and couldn't sleep.\"\n\n\"You're right,\" Ameh said. \"It's been a long time since I went over the untimely death of my loved ones in my mind. Tonight all of them passed again before my eyes.\"\n\n\"I've been in your family for many years now,\" Zari said, \"but I'd never heard you mention your late husband or your child before. Tonight...\"\n\n\"I know. I've always kept my grief to myself,\" Ameh replied. \"I've never told anyone what I've suffered.\"\n\nZari sat down and took her sister-in-law's hand. \"You've always said yourself that a sorrow shared is a sorrow halved. You used to say that the Imam Ali would lean over a well and tell his sorrows to the water deep down which he couldn't see.\"\n\nAmeh nodded. \"Should I sigh for you in sorrow?\" she recited, as if to herself. \"Then as Ali I look into a well.\"\n\n\"Am I not as good as a well?\" Zari asked.\n\n\"You're young. I don't want to destroy your hopes in life with my unhappy tales.\"\n\n\"I've had my share of sorrows.\"\n\n\"I know.\" And so it was that Ameh began to tell all she had kept locked away inside her; stories which Zari had never heard before. \n\n# _6_\n\nThat night, Ameh began, I was sitting right next to this very brazier, in the same wretched darkness, stirring the ashes with these tongs. I was gazing at the brass figurines, holding hands all around the edge of the brazier. That night I counted thirty-two of them. They're still intact, those featureless little figures.\n\nIt was the night my child died. Soudabeh, my father's mistress, sat with me till dawn, shedding tears as I cried. When he died... at the grove... all alone... I knew he was dead, but I held him, my six-year-old, and ran to Sardazak. If I hadn't lost him, I wouldn't be obsessed with veils and religious modesty now, nor with opium and convulsing at the mere thought of not getting any.\n\nI rushed to my father's house. He and Soudabeh were sitting in the room with the sash windows next to this very brazier. Grief-stricken, I wished I could breathe out fire and turn everyone around me into ashes. Soudabeh stood up and took my child from me. She was shaken, but trying hard not to show it. What a woman! She went out and came back without the child. I asked her what she had done with him. She said, \"Who knows, maybe my brother, Mohammad Hossein, can bring him back to life with his healing touch.\" I said, \"Don't blaspheme, woman! Only God brings to life. It says in the Quran: I breathed life unto him of My Spirit.\" But Soudabeh wanted to give me hope.\n\nI had studied Arabic and Persian with my father, and geography and geometry with Mohammad Hossein, Soudabeh's brother. When my father came back from Tehran after completing his religious training, he shut himself up at home. He didn't go out to lead prayers anymore. He was forced to give up teaching at the Khan seminary too. He only taught at home, in the main room with the sash windows. Men would come in, kiss his hand and bring him questions on jurisprudence or theology. I used to sit in the next room and listen. When my father returned from his pilgrimage to holy Najaf, everyone in town went all the way on foot to Baj Gah to welcome him. That first day he led the communal prayers, and all the mullahs in town\u2014even the Imam Juma\u2014followed him as a mark of respect. When he spoke at the Vakil Mosque there was a packed audience.\n\nOh Lord! And I was quite a woman in those days too! I remember being daring enough to carry my father's secret anti-government letters in my bosom to the Shah Cheraq shrine where I would deliver them to someone waiting there. I can remember it as if it were yesterday... the meeting place used to be between the two lion statues at the front of the shrine.\n\nThen my father, of all people, fell for an Indian dancer. Mohammad Hossein and his sister Soudabeh had recently arrived from India. Despite my father's courtship, to the last Soudabeh refused to marry him. She used to say they were better off the way they were. Of course she broke up our home and caused Bibi, my mother, no end of grief. But what a woman she was! And what a dancer! I'll never forget the day my father asked Soudabeh to dance for his guests at the Rashk Behesht Gardens. She wasn't really pretty, quite short and very sallow. She had a dark beauty spot on her upper lip, and used to outline her large brown eyes with a lot of kohl. When she wasn't laughing she looked like an owl, but when she smiled it was as if the heavens had opened.\n\nAt that gathering, everyone\u2014men and women\u2014stood around the paving of the garden to watch her dance and to clap. I'd never seen her perform before. It was certainly out of the ordinary. She seemed naked at first glance, except for a few bits of jewellery. But in fact she was wearing a jewel-studded brassiere and a flesh-coloured body-stocking. She managed to move each and every part of her body: not only her shoulders, belly, eyes and eyebrows, but even her chin, nose, ears and pupils. First, she pretended to do a ritual dance over the corpse of a man. For the second dance, she wore a blue silk dress with a gold border, and had two live doves with dyed feathers perched on her breasts. She moved slowly and gently, as if afraid of disturbing the birds. When the dance was over, she let them fly away. By the end of the third dance, she was looking hot and flushed, so she went and sat by the pool, dressed in her pink satin dress. As she dipped her bare feet in the water, I saw my father, my Haj Agha\u2014the high clergyman of the town\u2014sit down before Soudabeh and meekly fan her.\n\nSo it came about that my father asked Mohammad Hossein to teach me at home. I used to study geometry and geography with him, drawing endless charts and maps. I was so wrapped up in my studies, I was often unaware of what was going on around me. Just imagine, the first day an airplane came to this town, everyone packed their rugs and took off at dawn to Baq Takht to watch its arrival. I was sitting on the roof of our house in the sunshine, drawing a map of India. The airplane flew right over my head and I didn't even lift my eyes to look at it. Oh God, a person like that shouldn't become an opium addict!\n\nMohammad Hossein was quite a character. He was a sun-worshipper. Every morning and evening he'd go on to the roof to watch the sun rise or set, until finally sunlight ruined his eyesight. He could also do conjuring tricks. He fried eggs in a felt hat floating on the pool. He could produce gold coins from bits of paper. He would swallow my Haj Agha's fob-watch and bring it back out of Abol-Ghassem's pocket. He dabbled in palmistry, too. Once he told my fortune, and said I would have twelve sons, all of whom would become ministers. I remember thinking that my family would make up the entire cabinet! My father used to say that Mohammad Hossein had spiritual powers. But the townspeople thought he dabbled in witchcraft and black magic. Whatever he was, the man took great pains over my lessons, God rest his soul.\n\nThat terrible night it was Mohammad Hossein who washed and buried my child. For a whole week he would make me sit in front of him, gaze into my eyes, and repeat, \"I shall put you to sleep, and in your sleep you will see your child, see how well and happy he is in his new place.\" But I couldn't be hypnotized. He said I resisted too much. He even painted my thumbnail black and told me to gaze at that. \"Your child will appear right now,\" he said. \"Can't you see him? Here he is. Here he is. Ask him what he wants. He wants something to eat.\" But no matter how hard I stared, I didn't see anything.\n\nHis sister Soudabeh, however, had charmed my father. She never did become his wife, but she had him under her spell. What a woman she was! The kind who could draw people to herself as if by magnetism... once seduced, you could never be free of her. It had nothing to do with beauty. It had more to do with charisma. Everyone around Haj Agha was amazed at his behaviour. Perhaps they even cursed him behind his back. One sly fellow\u2014we never found out who it was\u2014commissioned several lengths of hand-printed cloth from Isfahan, picturing the proverbial Sheikh San'an going to Europe with his followers. The Sheikh was shown as a besotted-looking old man, wearing a turban and cloak just like my Haj Agha. There was a train of followers behind him and a lewd woman languished in an upper chamber of the house. Those days wherever you went, they seemed to have hung up one of these cloths. People certainly know how to be vicious when they want to.\n\nAs for Haj Agha himself, he would say, \"They've taken away my teaching and preaching from me. Far be it from me to interfere in other worldly affairs. I gave it a try and suffered the consequences. After all, a person must do something greater in life than just the daily business of living. He must bring about changes. Now that there's nothing more left for me to do, I'll abandon myself to love.\" \"Love hath done more than steal your faith,\" he used to quote, \"A Sufi it can turn to Christian.\" And sometimes he would add, \"The pilgrim's destination is but the starting place for love.\" The mullahs in town even spread a rumour that he had turned into a heretic and a Babi. But since my father was always a generous host, and continued to solve their problems by telephone, he was never officially excommunicated. Besides, the clergy had lost much of its power, and most mullahs had exchanged their religious turban for the civilian hat.\n\nOur Haj Agha felt the time hadn't come for his beliefs. So he decided to retire. But he was never one to put up with injustice either. During the fighting between the police-chief and Massoud Khan, almost every household hung up a British flag to show their loyalty to the police-chief and to prevent raids on their homes. My father not only refused to put up the flag, but he even helped, side by side with the chief Rabbi, to carry the Jewish wounded from the poorest quarters to a doctor. He did his best, too, to prevent the armed men from plundering the Jewish quarter, but to no avail. Those men had been well paid.\n\nThey had shot a Jewish mother as she was nursing her baby. The baby was still suckling when she passed out. When Haj Agha arrived on the scene, he quickly tucked the baby under his cloak and rushed straight to Dr Scott, the European doctor at the Missionary  Hospital. And who was this Dr Scott? None other than the special physician to the police-chief and his family, who refused to visit those wounded by the police-chief's cronies. Single-handed, my father had the hospital closed down that day, forcing Dr Scott and several Armenian nurses to visit the wounded mother and other casualties in the Jewish district. The mother recovered. Do you know who she is? Our very own Tavuus Khanom who still comes to see us regularly and brings wine for Yusef.\n\nI remember Felfelli, a drummer with Musa's musicians, had been among the wounded. They brought him to our house and stretched him out at the entrance on the doorman's bench. Blood was gushing out of the wound in his thigh like an open fountain, covering the entire entrance. My father happened to be away and Bibi, my mother, fell sick at the sight of all that blood. I grabbed my veil and ran to Dr Abdollah Khan's office in the Arab quarter. I didn't stop for breath until I got there. Between you and me, they hadn't put up the British flag either.\n\nDr Abdollah Khan's father was the well-known Haj Hakimbashi, who was still alive then. He had four sons, three of them doctors and the youngest a pharmacist. They owned a pharmacy too. God rest their souls. Only Dr Abdollah Khan is still with us. In his office that day you could hardly move for all the wounded and dying. I resorted to tears and pleas before the doctor agreed to come with me. They used to say he was quite a healer, despite his youth. But as fate would have it, Felfelli was already dead by the time we arrived, and he had been covered with a bedspread. His relatives were crowding into our house and raising the roof with their wailing and mourning. Bibi had fainted. And now where do you think our house was? Right opposite Ezzat-ud-Dowleh's father's house. Ezzat had just married, and her husband\u2014none other than the police-chief's son-in-law\u2014had actually moved into her parents' home. All the trouble had been started by this very son-in-law. Now what if they had heard the din in our house?\n\nOf course, Ezzat-ud-Dowleh and I had taken an oath to be sisters, but in those troubled times people hardly thought of their real sisters, let alone their sisters by oath. No, it was respect for Haj Agha that prevented them from raiding our house, especially since they were afraid he might decree a holy war. All this happened well before my father became a recluse, you see.\n\nI'm sure Haj Agha had the kind of power it took to hypnotize people, if he wanted to. He would stare right at the space between your eyes, and who could resist him? Imagine a man like that letting himself be enslaved by an Indian dancer and break our Bibi's heart! Oh Lord, don't put us to the test! Bibi knew what was going on, but she never said a word. It's all over with now, but she never even confided in me, her own daughter. Haj Agha and Soudabeh were the talk of the town, but my mother, the only one that mattered, remained silent.\n\nAt least my father had the decency not to bring Soudabeh and Mohammad Hossein into the house until all the family had moved away. I was married first, then Abol-Ghassem Khan found a wife. Finally Bibi went off to Karbala. My husband was a textile merchant who traded with Egypt and India. He and his father imported a delicate fabric known as 'miyur'. It was even finer and more beautiful than silk, and quite often used for underwear or babies' clothes. Nowadays you can't find it anywhere. But my husband was an unhappy man and he committed suicide. One day, at sunset, he dashed himself on horseback against the pillars of the British Consulate building. Because of our son, and because of an unjust society that made life unbearable for him. You see, Haj Agha could retire when he felt the time wasn't right for his ideas. But my poor husband was still a young man. Just like Yusef, God forbid. Yusef is ahead of his time, too. That poor soul used to say, like Yusef, that we had to change the times. But he was just beating his head against a stone wall\u2014as he literally did in the end. Let's face it, these are times for double-dealers like my brother Abol-Ghassem Khan. When will it be time for people like Yusef, I wonder?\n\nI'll never forget, after my husband and child died, Yusef wrote me a letter telling me to stand on my own two feet. He said if I fell, no-one in the world would bother to help lift me up. One could only rely on oneself, he said.\n\nThank God Bibi was not around to see my unhappiness. When she made up her mind to leave, she invited the entire family to dinner. That night she kept staring at us as if to engrave our faces on her memory. Only Yusef wasn't there because he had been sent abroad for two years to finish his education. Actually, when Abol-Ghassem Khan complains that our father never spent any money on his education, he isn't telling the truth. Haj Agha wanted to send both of them away together and Abol-Ghassem Khan turned it down of his own free will. He asked my father to give him what his education would have cost in land, and that's what Haj Agha did.\n\nAnyhow, Bibi bid us farewell that evening, supposedly to go on a pilgrimage to the shrine of Hazrate Massoumeh in Qom, and from there to Mashad. She said she would be away for a month or two. But unbeknownst to us, she had arranged to have herself smuggled over the Iraqi border to Karbala. All she had in the way of worldly possessions was some money Haj Agha had given her, and some women's trinkets, a suitcase and her ewer. Her emerald earrings she left in my care, in case something should happen to her on the journey. I was to keep them for Yusef to give to his wife on their wedding night.\n\nA month later a letter arrived from her telling us not to worry, that she was in Karbala where she planned to stay permanently as part of a religious vow she had taken. Only much later did we discover\u2014and please don't let this be known\u2014that she had ended up being a maidservant there\u2014to a Khanom Fakhr-ol-Sharia. All the time she was in Karbala, Bibi never asked for money, nor did my Haj Agha offer to... no, perhaps three or four times on our insistence he did send her some in one way or another. Whether she received it or not, I never found out. She wouldn't write, you see. In that first letter she stipulated that we were not to write to her, since she wanted no distractions from her religious calling.\n\nBut I've been digressing. I was talking about that dreadful night, wasn't I? Yes, I was sitting right here by this brazier, prodding the ashes with my tongs. There wasn't much of a fire left. I was counting my sorrows in the dark, with Soudabeh sitting next to me the whole night. What a woman she was! A pity she broke our mother's heart.\n\nThat night I asked Soudabeh, \"I never understood why you, with a thousand admirers, should have chosen my father, driving my mother out of her own home?\" She said she couldn't help it; she knew she had ruined the reputation of a Shi'ite clergyman of the highest order, and made an innocent woman homeless. But it was out of her hands, she claimed. \"Sometimes, in a previous life,\" she said, \"you've lost a person you've been very close to. Once that has happened, you keep coming back to this world to find him. You bear the waiting, the separation. But when you finally find the person again, how can you possibly let go of them? It's like two intertwining plants at first, where one withers and dies, then in a later life they happen to be two migrating birds, who return once again as two loving deer\u2014and perhaps one is shot by a hunter\u2014and so on. They could be father and daughter, sister and brother... who knows? And when they find each other at last, they can no longer be separated.\" She often used to say things like that. She would say these things and yet she never agreed to marry our father. She just stayed with him until they grew old.\n\nAfter my husband's untimely death, I decided, as Yusef had advised me, to stand on my own feet and run the estate I had received from Haj Agha as my wedding gift. I'd straddle my horse in my breeches, cover the poppy fields from one end to the other on horseback. How old do you think I was then? Twenty-eight. I even used the bastinado on my peasants, God forgive my sins! Bibi had been gone then for about three years. The poor woman was only forty-four when she died. One day, Fakhr-ol-Sharia telegraphed my father to say Bibi was ill. To his credit, Haj Agha made every effort to get exit permits. He cabled Yusef to go to his mother at once, but decided not to say she was on the point of death. Which is why Yusef only arrived after we had buried Bibi in the shrine tomb. Abol-Ghassem Khan had gone to great lengths, paying quite a bit out of his own pocket, to get permission for the body to be placed in the shrine, even though we knew the moment our backs were turned they would take the corpse out to a public graveyard. Still, even one night in such a holy place was quite a blessing, and Bibi's wish had been fulfilled.\n\nO merciful Lord, what a tragedy it was! My Bibi in the throes of death in a room two feet square, on a torn straw-mat covered by a ragged quilt... she cried out from the heat, but there was no cool basement, no iced water for her. Fakhr-ol-Sharia would call on her for service, for a hookah, for this or that without the least consideration or respect. Oh Lord, no Khanom, no title! My mother's name was Fassih-ol-Zaman, meaning 'eloquent one'. An eloquence which never uttered a word of what had happened to her! Even the story of her becoming a housemaid was told us by Fakhr-ol-Sharia herself, who talked about Bibi as though she came from a long line of devoted servants. I've never said a word of this to anyone. Not even Yusef. There was no point. He was only twenty years old, he couldn't have taken it. Is he ready to bear it now, at forty? I doubt it.\n\nWe never did find out how Bibi got herself to Karbala. We merely heard that when she arrived, she fell into the clutches of a certain Sheikh Abbas Qomi who used to disguise himself as an Arab and frighten illegal pilgrims by threatening to denounce them to the authorities unless they bribed him. When he confronted my mother, she was so panicked she dropped her suitcase, grabbed her ewer and ran away! As it happened, her birth certificate was in the suitcase. With the hundred tomans Haj Agha had given her, she managed to obtain a dead person's birth certificate from a worker at the mortuary.\n\nKhadijeh came out to the verandah to ask: \"Aren't you having any dinner tonight?\"\n\n\"We'll call you when we're ready,\" Zari answered.\n\n\"No, that's enough for now,\" Ameh intervened. \"I've talked too much and I've given you a headache. Let Khadijeh bring us a bite to eat, then we can go to sleep and see what tomorrow will bring.\" \n\n# _7_\n\nEarly next morning, Zari instructed Gholam to tell anyone coming from the Governor for anything that Khanom was not at home, and that nothing could be given away in her absence. If the person still insisted and mentioned a horse, Gholam was to feign ignorance and say they had come to the wrong house\u2014they used to have a horse, but it died. At a pinch, he was to give them the chestnut horse.\n\nIt was watering-day for the garden, and Zari went outdoors to watch the trees and the grass thirstily drink in the water, sharing their refreshment and feeling revived herself by breathing in the smell of moist earth. Gholam and the gardener, shovels against their shoulders, trouser-legs rolled up, crossed the garden barefoot from one end to the other, opening or closing the flow of water in the narrow irrigation canals. Mina and Marjan wanted to stay around, but kept getting in the way. Finally Zari had to coax them into building a mudhouse under the big elm near the stables. She told them they could plant flowers in it, and have a wedding for their dolls. But she warned them that if they didn't stay in the shade, the sun would scorch their lovely soft skin.\n\nMina started to draw a plan for the house, making room for a little pool, a cupboard, and a cold furnace. Marjan completed the plan by adding the stables. Then, with a lot of squealing and fuss they caught a toad which they put in the stables, but it soon leapt away. Still, there were plenty more in the garden.\n\nGholam directed the flow of water towards the elm trees, and before long the children's mudhouse was flooded. Water ran into pools around their feet, and they squatted down in it. Zari called them away, all the while listening for a knock at the door so she could hide in time from the Governor's messenger. Mina shouted at Gholam for having ruined their mudhouse:\n\n\"You meanie!\"\n\n\"It was just a flood, sweetheart,\" was his reply.\n\nAll that day and the next there was no messenger from the Governor, and Zari felt reassured, thinking that they must have changed their minds. Even Ameh commented, \"Thank God! So much needless worrying. They must have just mentioned something in passing, and Abol-Ghassem Khan made them a promise to curry favour, as usual.\"\n\nBut early in the morning on the third day, Zari had just got out of bed when there was a knock at the door. Gholam went to answer while Zari kept watch from her hiding-place. She saw a gendarme greet Gholam and embrace him, handing him an envelope. Gholam brought the envelope to Zari.\n\n\"It looks as though you know him,\" she said.\n\n\"Yes, he's from my village,\" he replied. \"From Bardeh. He always wanted to become a gendarme, and now he has.\"\n\nZari went to the verandah and waited until Ameh Khanom had ended her prayers before opening the envelope. Then she read out loud the neat handwriting addressed to herself:\n\nMy dear Madam,\n\nIf I were not certain of Shirazi hospitality and of the generosity of your respected family, I would never make the following request of you. Recently, my daughter Gilan Taj was so badly afflicted with typhus that the doctors had given up hope. But God's mercy was with us, and my child has recovered. My daughter enjoys horse riding, and despite carefully searching this town, we have not been able to find a horse gentle enough for her use. I assure you the general sent us two of the best horses from the army stables, but they were large, headstrong animals not suited for a child who has just left the sick-bed. Our honoured friend, Abol-Ghassem Khan has promised to send us your son's colt. I hear he is away on a trip. I humbly beg you to loan us the young horse belonging to your respected son for a few days by means of this messenger. The moment Gilan Taj tires of horses and horse riding, we shall return it.\n\nYours sincerely\n\nSince the signature of the Governor's wife differed from the rest of the handwriting, Zari decided the letter must have been written by someone else.\n\n\"Now what am I to do?\" Zari turned to Ameh.\n\n\"They've taken us by surprise,\" she replied. \"We can't give the colt away, and yet we can't refuse either. If we give them the colt I know Yusef and Khosrow will be up in arms. If we don't, well, you remember Abol-Ghassem's outburst the other day? There will be endless quarrelling. If he isn't made a deputy one of these days he'll blame it on us and our pettiness.\"\n\n\"And now that they've stated their request clearly,\" Zari added, \"I can't even send them the chestnut. What should I do?\"\n\n\"Just sit in a corner and think, I suppose,\" said Ameh with a sigh.\n\nThey asked the messenger to come in and sit on a chair by the pool. Khadijeh brought him some breakfast which she placed on another chair. The gendarme took off his hat and put it on his knee. Zari watched him empty the sugar-lumps from the bowl into his pocket and gulp down his unsweetened tea on top of huge mouthfuis of food. Gholam was sitting opposite him on the edge of the pool.\n\n\"Are you the guard at the entrance to the Governor's estate?\" Zari asked him.\n\n\"Hmph!\" grunted the man with his mouth full, and then he quickly swallowed his food.\n\n\"Do you have a wife and children?\"\n\nGrinning widely, he answered in a thick accent: \"I wed me cousin last New Year's.\"\n\n\"When will you return the horse?\"\n\n\"The lieutenant gave me a mission,\" he said. \"His honour said I'm a good lad. But he didn't say anything about bringing the horse back.\" And again he grinned from ear to ear.\n\n\"But it's not mating season yet, brother,\" Gholam intervened.\n\nThe gendarme dug a hand into his tunic pocket and produced an envelope which he presented to Zari, saying: \"Agha Mirza, the governor's secretary, gave me this. He said it's eighty tomans.\"\n\nZari took the envelope, opened it and began to count. It really was eighty tomans. She whispered to Ameh: \"They imagine they've paid for it, too.\"\n\n\"Let him take Sahar away for now until we think of something,\" said Ameh.\n\n\"Gholam, go and bring Sahar out of the stables,\" Zari ordered.\n\n\"Khanom, I swear what they're doing is wrong,\" Gholam protested. \"Mating season is over now. Besides Sahar is too young...\"\n\n\"They don't want him for mating,\" Zari explained wearily, \"the Governor's daughter has taken a fancy to Sahar...\"\n\nGholam took off his felt hat. His bald head was flushed and sweaty. He said, \"Khosrow Khan has left Sahar in my care. Now you ask me to give him away to someone else? Never!\"\n\n\"Gholam, can't you see they've sent a gendarme?\" Ameh said.\n\n\"What makes you think this poor fellow's a gendarme?\" said Gholam. \"He's just a simple, honest lad.\" Turning to the gendarme, he continued, \"Listen brother, go and tell your master that the horse was dead. Khanom here will give you your tip.\"\n\nBut the gendarme was insistent.\n\n\"Aren't we from the same village?\" he pleaded. \"Don't make it so hard for me. The lieutenant ordered me to bring the horse back by whatever means. He gave me a mission. He said I'm a good lad. He said if I don't bring the colt back with me, I can resign my post and go straight back to Bardeh, back to my mother's apron! He said that himself.\"\n\nGholam put on his hat and said: \"Whoever wants to take Sahar has to go and bring him out of the stables himself\u2014if he dares. I'll knock him over so hard with my shovel, he really will have to run straight back to his mother's apron!\"\n\n\"I give the orders here,\" Zari intervened authoritatively. \"I'm the mistress of this house. Go and bring Sahar from the stables.\"\n\nBy now Mina and Marjan had woken up and come outside to the verandah, with Khadijeh trailing behind asking them to wash their faces first.\n\n\"Khanom, if you ask me, you shouldn't do this. Think of tomorrow when your son comes home\u2014 he'll be heartbroken. Think of later on when the master gets back... don't be afraid of these people. Just refuse, that's all. What can they do to you?\"\n\nThe gendarme started off towards the stables. \"Aren't we brothers, from the same village?\" he appealed.\n\n\"Where do you think you're going?\" Gholam asked.\n\n\"To the stables.\"\n\n\"We may be from the same village,\" Gholam threatened, \"but if you dare set foot in those stables...\"\n\n\"I haven't brought my gun\" countered the gendarme, \"I'm going to get it now.\"\n\nGholam grabbed him by the collar and shouted:\n\n\"Now you're showing off to me with your gun? Aren't you the same miserable urchin who used to sneak off at night to steal chickens? Did your lieutenant tell you to come threatening me with your gun too?\"\n\nThe man shook himself free and muttered: \"No, I swear it! But he said I could resign my post and go back to the village if I failed. How can I go back there?\"\n\nAmeh Khanom called Gholam over.\n\n\"Gholam, don't be stubborn,\" she said quietly. \"Abol-Ghassem Khan has already made them a promise. Let him take Sahar away for the time being. I've had a good idea. I think I can get him back before Khosrow returns.\"\n\nGholam fetched Sahar from the stables and gave the reins to the gendarme. As he tried to mount, Sahar gave a mighty kick, reared up on his hind legs, and neighed loudly. Both the mare and the chestnut horse answered from the stables. The man fell back and let go of the bridle. Sahar turned to Gholam and sniffed at his rolled-up sleeves. The gendarme made several more attempts to mount, sweating profusely all the time. He tried stroking Sahar's mane and patting his neck. He brought out a sugar-lump and held it in front of the horse's mouth. Finally he managed to grab the bridle. Zari handed him the money.\n\n\"Take that for yourself.\"\n\nThe gendarme's eyes shone. Putting the notes in his tunic pocket, he dragged Sahar away.\n\nThe twins watched in horror, ignoring all Khadijeh's pleas for them to have breakfast. To Zari, it felt as if the garden had been robbed of its life and lustre. Ameh Khanom roundly cursed the universe, before turning on Zari: \"Now why did you have to go and tip him?\"\n\nGholam stood there, watching his mistress whose eyes had filled with tears.\n\n\"Khanom, may the men return safely from hunting,\" said Khadijeh appeasingly. \"That's all that matters. The mare is young, and soon she'll give birth to another Sahar.\"\n\n\"I bet I shall have Sahar back in three days!\" said Ameh. \"It's just as well you returned their money.\"\n\nZari was not convinced, however. She ordered Gholam to dig a mock grave down by the stables. She told him to pull out the weeds, smooth over the soil and arrange some stones in a rectangle with a few pots of petunia around it.\n\n\"Take my word and be patient for a while,\" Ameh advised.\n\nBut Zari merely turned to Gholam and warned him not to breathe a word of what had happened to Khosrow.\n\nWhen they went into the sitting room, Ameh Khanom went straight to __ the telephone and invited Ezzat-ud-Dowleh for lunch in three days' time.\n\nWhen the day of the luncheon invitation came, Zari went to great lengths to receive Ezzat-ud-Dowleh, although she had never liked the woman much. When she arrived, Zari took her guest's head-scarf and white gloves and dark glasses and wrapped them neatly in a bundle. Then she gave her a fresh peach-coloured chador to replace the dusty outdoor one. Even though Ezzat-ud-Dowleh had brought along her favourite maidservant, Ferdows, Zari sent the girl to rest in Khosrow's room. And even though Zari had cooled the parlour since early morning by closing the windows and letting down the straw blinds to keep out the sun, she still provided Ezzat-ud-Dowleh with a fan. Trying to make herself pleasant, she complimented her guest, \"What a beautiful head of hair you have.\"\n\n\"God bless you,\" responded Ezzat-ud-Dowleh.\n\nAlthough it was well before lunch-time, she refused any sherbet drink or fruit. She asked for tea which, when she tried, she did not seem to like. She merely remarked: \"Ration tea is always stale.\"\n\nAt lunch she didn't eat much. She toyed with a few spoonfuls of rice and kebab which she then pushed aside, asking for sour-grape juice instead. To that she added some grated cucumber and bread-crumbs and onions, saying it was good for her leg pains. Unfortunately the sour-grape juice was last year's, too, like the tea.\n\nAfter lunch, Zari spread out a thin cotton sheet in the parlour and brought a pillow and a delicate coverlet for her guest's afternoon nap. Ezzat-ud-Dowleh stretched herself out, fan in hand, while Ferdows the maid massaged her legs. Ameh Khanom lay down on another cotton sheet beside her. Zari left the so-called sisters by themselves, and went to her own bedroom, leaving the door slightly ajar so that she could overhear their conversation. If Ezzat-ud-Dowleh was willing to cooperate, she was the one person who could get Sahar back. She could even have Zari's earrings returned, sp Zari's pains would not have gone unrewarded.\n\nEzzat-ud-Dowleh's voice could easily be heard through the door: \"God bless you, Ferdows. Rub harder. That's better. Have you said your prayers yet? No? Then get up, child, go and say your prayers...\"\n\nZari could tell Ferdows had left, because Ameh was chatting and laying the groundwork for the favours she wanted to ask later. It was a pity Zari could not hear every word. But Ezzat-ud-Dowleh, lying closer to the door, could easily be heard in a loud and flowing monologue. \n\n# _8_\n\nNow I understand, she was saying. When you called I asked myself, why has my sister suddenly remembered me after all this time? We only see each other on holy days or funerals. So you have some sort of problem, and I may be able to help.\n\nDid you say horse? No, as God is my witness I had no idea your nephew had a colt called Sahar. I had heard about your brother keeping horses. I thought, well, talk about showing off... but as for giving the Governor's daughter the idea of harping after your nephew's colt\u2014upon my word, never!\n\nTrue, I can't stand the sight of your brother. And if Zari hadn't become your sister-in-law, I wouldn't have hesitated to destroy her entire family. Yes, the whole thing goes back thirteen or fourteen years. But can I ever forget? A distinguished lady like me going to their slum to ask for her hand for my son! Their smoky little living-room the size of our prayer-chamber at home, and her mother like a living skeleton! I'd be ashamed to have my own servant looking like that, with her white hair and yellow complexion, her front teeth missing, wearing an old crumpled dress. I had to hold my breath for the smell of her sweat. You would've thought she could at least get some false teeth, comb her hair and dab a spot of rouge on that wrinkled face. After all, I'd come to ask for her daughter's hand. Such a distinguished lady as myself, too! It was a great stroke of luck for her that my innocent boy had chosen her daughter of all the girls available to him. A hundred times I asked my Hamid, \"Son, isn't it beneath you to marry the daughter of Mirza Ali Akbar Kafar, that unbeliever of an English teacher at the Shoaieh School?\"... Now don't you be offended, sister, I'm only telling the truth. Anyway, Hamid would say to me, \"I'm looking for something I don't have myself.\" I said to him, \"What does this girl have besides a nice pair of eyes?\" He said, \"She has gentleness, virtue and education.\" I'd say, \"But my love, my son, you can't live on gentleness, virtue and education.\" To cut a long story short, they turned down that piece of luck themselves. I sent Kal Abbas, the doorman, to their awful house for an answer and all they said was that they had consulted the Quran for an augury and the outcome was unfavourable. Since when had Mirza Ali Akbar Kafar's family believed in consulting the Quran?\n\nBut Hamid had set his heart on marrying Zari, and there I was having to lower myself to go to that ramshackle house again\u2014not once, but twice, three times! Until finally the mother admitted you had taken their daughter the customary shawl and ring, and they'd promised her to you. I thought of coming to dissuade you, telling you that the mother had cancer, telling you that a beggar will always remain a beggar at heart. But you had long since turned your back on our oath and sisterhood. Now, now, how quickly you take offence! It's true isn't it?\n\nNo, as God is my witness I didn't give the Governor's daughter the idea of taking your nephew's horse. And now... all right. I'll do what I can. I'll tell them it's a real shame, the boy is utterly heartbroken, and they should give the horse back to him. You did say you've sent back their money, or haven't you? Maybe I'd better persuade her to ride the horse\u2014it's sure to take off with her and gallop right back to its old stable. That will put riding out of her mind for a while! But as for talking to them about \"oppression\" and \"cruelty\" and saying that everyone is cursing the Governor behind his back, that I can't do. Unlike you, I won't hurt my friends. You insist? Well, all right. Just for your sake I'll do it. You know me. I bear grudges, that's true. But I also understand friendship and sisterhood.\n\nI'll tell you the truth about those emerald earrings of Zari's. The minute I walked into that wedding and set eyes on your sister-in-law looking so pretty and prosperous, I decided to get my own back by making her suffer the loss of her precious earrings. What, you didn't know? How's that? You mean she hasn't let on about them? Well, I could have told you she wouldn't be particularly honest, coming from that family!... Now it's no use getting offended, I'm only telling the truth. Yes, it was my doing. I sent Ferdows off at top speed to the haberdasher's bazaar to buy some green silk. I threw it around the bride's neck, and told them to go and borrow the emerald earrings belonging to Yusef Khan's wife, knowing full well they're not the sort to return earrings. Why are you getting in a state about it now? Let Zari do the worrying. Come now, sister, please don't look so upset. Well yes... I did know they were a special token... very well, I'll try to get the earrings back, too. You don't need to tell me how to do it; I know myself.\n\nLet's be sisters again like we used to be. Do you remember that celebration we had when we were children, and we brought over a mullah to swear us to sisterhood, and then they showered us with sugar-plums? But then you changed. Ever since you lost your little boy and your husband killed himself, you seem to have changed into another person altogether. Do you remember, when we were a bit older we both fell in love with Dr Marhamat Khan? He'd just come from Tehran, and they said he'd studied in Europe. I can't forget that day when we made ourselves up so no-one would recognize us, and went to the doctor's office. We counted eleven other girls there\u2014some from the town's best families\u2014who'd also made themselves up and were pretending to be ill. like us, they were really there to show off their faces and bodies to the doctor. Do you remember Etrat, who later became Etrat-Saltaneh, wearing her fancy starched kerchief? Oh Lord, those were the days! You'd pulled out a handful of your hair to say you were going bald, and I'd made up a story about a lump in my right breast which was sometimes there and sometimes not. He dabbed some tincture on your bald spot, and told me I was imagining things. He never married any of us, either. He went and brought a wife from Abadeh.\n\nThen each of us went our separate ways. I was married first, but we both met with tragedy. Maybe you were better off in the beginning, but your happiness didn't last. And I couldn't bring myself to confide my troubles to anyone, not even to you, my sworn sister. They say every ill-starred woman has at least forty days' grace in her husband's home, but I didn't even have that. Imagine my large dowry, my parents' house and wealthy life-style all falling into the hands of that no-good husband of mine! And a distinguished lady like me, the police chief's granddaughter... and he was a man whose forefathers had ruled our province like sultans, generation after generation...\n\nYou see, we'd only been married three days before we began to quarrel and my husband said, \"Don't play the police chief's granddaughter  with me; all your ancestors were traitors. Even your great-grandfather was a close aide to that ruthless Agha Mohammad Khan Qajar in return for a piece of cold-blooded treachery.\" He said, \"Don't you show off to me with your ancestral home either. Every one of its stones and bricks was laid over the body of an honest, hard-working person. Its clay plaster was mixed with the blood of our wise men...\" What, sister! Are you saying my husband was right? I'll show all of you what right is! Anyway, that same evening when my brother turned up, my husband was all sweetness and light again. You should have seen him with his yes-sir no-sir!\n\nIt was during the first month of our marriage that he fell in love with Nim-Taj, the wife of Massoud Khan. \"Major\" Massoud had been appointed by the government as chief of police here, if you remember. My uncle and my brothers didn't want the town to fall into his hands. From the day he arrived they gave him trouble, and finally they set off that famous riot. I watched my coward of a husband suddenly change and become the driving force of that fight, turning my house into a sort of headquarters for my uncle's armed men. I said to him, \"Didn't you say my brothers, fathers and ancestors were all traitors? How is it that now you're fighting their battle for them?\" I'd just found out that he'd fallen for Nim-Taj. May he never rest in peace!\n\nEventually Major Massoud realized he had no chance of surviving. Early one morning he ran away on foot to Seyyid Abol Vafa's shrine so he could take sanctuary there. My disgraceful husband chased him on horseback and caught up with him before he ever got there. He shot him in the back, and left the poor wretch rolling on the grass crying out for water. A crowd gathered to watch him in his death-throes. No-one dared give him a drop of water for fear of the armed men. Haj Agha, your father, appeared on the scene and took control of the situation. He shouted at the armed men, and told them they'd gone out of their minds, just like their master. He said they'd do penance for this killing right here in this world, and they'd always be haunted by the memory of the poor man's death-agonies. And he carried Massoud off in a droshke, but apparently the young man died then and there in your Haj Agha's arms. My husband was afraid of your father, you know. Several times they were about to raid your house but my husband stopped them, saying that Haj Agha would call for a holy war, and Solat the Qashqai chief would join him\u2014and no-one could resist that combination.\n\nBut I was impressed by Nim-Taj. That very night she went to Agha Sheikh Razi, and wouldn't budge from the house until they arranged to get her back to her parents. When my shameless husband went for her, the bird had flown.\n\nMay your soul never rest in peace, man! He never deserved a well-born lady like me! Whenever we quarrelled, he would say that I was cross-eyed, and he'd been forced to marry me. He would say he didn't love me but wouldn't leave me either because he didn't want people to insult our son by saying that his mother was a divorcee. And I, pathetic fool that I was, loved him to distraction. He knew exactly what to do to get his own way with me. I was always finding strands of blonde or black hair or sequins from women's dresses on the collar of his coat. Eventually he had the nerve to bring his women to the house. First he only brought them as far as the outer courtyard, and then he even brought them to the inner rooms.\n\nTowards the end, he loved to have \"hundred toman\" whores. He'd say it was too demeaning for a \"hundred toman\" whore to be taken to the outer courtyard. So they would sit on the wooden bed we placed over the pool in the inner courtyard while I sent them trays of drinks. I would soak his tobacco in spirits and prepare his hookah. That hypocrite! First he would say his prayers, then he would settle down to his drinking. \"Don't perform the holy prayers after imbibing drink,\" he would quote from the Quran. I would watch them through the stained-glass windows of the sitting-room till dawn.\n\nIn the morning he would kiss my hand, he would kiss my feet. He would say, \"What can I do, that's how I am. The minute I see the flutter of a woman's veil, any woman, I lose my senses.\" And I would cry floods of tears and tell him, \"Take my marriage portion and set me free, go away, leave me alone. This house and everything in it is mine anyway. I don't need a useless effigy to call a husband.\" I would swear by my one and only son, threaten to go to Haj Agha your father or to you, my sister, and take sanctuary. Haj Agha wasn't the kind of man whose word people took lightly. But he was always ready with an answer. \"Whose house did you say you're going to?\" he would sneer. \"Haj Agha himself is one of the great lovers of our time. He keeps a mistress living under his own roof!\" He declared he didn't care a hoot what the Almighty said, let alone Haj Agha. Believe me, he meant it; he had turned away from God. Around that time he stopped bothering to say his prayers altogether. When he rode on horseback and people greeted him, he wouldn't return their greetings. He would signal to the outrider to answer them. Yes, my sister, this is the first time I'm telling you all this. You see, when your husband and son died, you forgot about me, your sworn sister.\n\nThe incident with my maid Ferdows and her mother? I suppose you heard rumours about that and now you want to hear the truth from myself? Well sister, I have nothing to hide from you.\n\nOne night after my evening prayers, I was coming out of the door of the New Mosque, when I saw a little girl crying by the door, with a bundle next to her. The sight of her was so pathetic, it would have melted a heart of stone. When I asked her why she was crying, she said, \"My mistress threw me out of her house where I was a maid and I don't know how to get home to Baj Gah.\" I took the child in as an act of charity. The next morning I sent for the midwife to examine her. I thought someone might have taken advantage of her and then the blame would fall on my poor, innocent son Hamid.\n\nTo cut a long story short, sister, within a week either my husband or my son managed to take advantage of the girl. It never occurred to me that they wouldn't even pass up a wretched little peasant girl. Of course, I didn't find out which of them had done it. I scorched the girl with a hot iron but she wouldn't confess; her screams pierced me to my very bones, but there was no way I could ask Hamid himself. A mother can't talk to her son about things like that.\n\nFerdows grew into a woman in our house. When she got her period, she bloomed into a rosy-cheeked, dimpled lass, with such a twinkle in her eye! I was worried all right, and I looked around desperately for a solution, but sure enough before I could do anything her belly was out there and I didn't know who to blame\u2014my husband or my son?\n\nAnyway, I was forced to latch her on to Kal Abbas, our doorman. Before that, his mother used to go to the Jewish quarter once a month and buy him a little girl for three tomans, dress her up in pink satin and bring her home. By the time the dress had worn out, so had Kai Abbas's interest, at which point she would take the girl back to her family. But do you think Ferdows would consent to my plan for her? I locked her up for three days in our chilly basement in the middle of winter. She had no food but her own thoughts and tears... I said to her, \"You shameless wench, what do you want from me? Should I be sending you back to Baj Gah with your belly full like this?\" She said, \"I can go to the police station to lodge a complaint against you, and then your family's reputation will be ruined.\" That half-size peasant wench certainly knew how to play her cards! \"I'll give you whatever you want,\" I promised, \"just get out of my house!\" Obviously she had fixed all her hopes on that bastard in her belly. She said to me, \"The child is yours; his inheritance and wealth will be worth piles and piles of money.\" Finally I beat her as hard as I could. Fortunately she started to bleed and Khanom Hakim got rid of that loathsome thing in her belly. With the baby gone, she gave up all her trouble-making. She just settled for having her mother brought over from Baj Gah and I had her start work for me on six qaran a day. Nana Ferdows, the mother, is an able woman. She's hard-working, but too bold as servants go... \n\n# _9_\n\nKhosrow was back from his hunting trip, covered with sweat and dust. His gun was still hanging from his shoulder, and a few dead partridges dangled from his hand. He went to the howzkhaneh which had a small pool in it and which Zari was preparing for use during the hot summer days. He held up the partridges before his mother's eyes as she was smoothing out the carpet.\n\n\"Look, I shot them myself!\"\n\n\"I can see,\" Zari replied, without looking up.\n\n\"Aren't you pleased to see me?\" Khosrow asked.\n\n\"Of course I am,\" said his mother.\n\n\"I'll give one to Sahar. He won't eat it, he'll just play with it.\" Then he added, \"No-one's happy to see me back. Gholam was sitting in Haj Mohammad Reza's shop; he almost ducked when he saw me. I came to you first and you didn't even kiss me. It doesn't matter.\"\n\nZari bit her lip and said, \"Take the partridges to the kitchen and give them to the cook to pluck. It's warm weather and they'll spoil. Tell him to serve them with rice tonight. Raisin rice, your favourite.\"\n\nAs soon as Khosrow had gone, Zari cursed the whole universe\u2014she cursed herself and her ancestors and her fears; she cursed her English schooling and her cowardice and Ezzat-ud-Dowleh. When she said goodbye Ezzat-ud-Dowleh had promised Ameh Khanom to send Sahar back to his old stable within three days. So what had happened? Zari sat by the small pool and turned on the fountains. At first the water came out in short, muddy spurts, then it cleared and rose higher. Soon after, the twins came in. They both sat down by the pool and held their hands underneath the fountain while their mother reminded them for the thousandth time not to tell Khosrow who took Sahar, but to say instead that he was dead.\n\nWhen Khosrow came back he didn't even notice Mina and Marjan.\n\n\"Mother, where's Sahar?\" he asked.\n\nZari didn't answer. Instead she busied herself washing the children's faces with water from the fountain.\n\n\"My uncle was saying Sahar had caught the glanders disease,\" Khosrow blurted, \"and that glanders is dangerous. Is that true? Captain Singer said glanders has become epidemic. Mother, he even imitated father. I nearly hit him when he said to me, 'This disease is yet another gift from the foreign army, as your father would say!'\"\n\n\"Singer was with you all the time?\" Zari asked, carefully skirting the issue.\n\n\"No, only for the first few days. There was a woman with him, too, who spoke good Persian. But she was just like a man. She even had a small moustache and wore boots. She rode well. Now tell me where have you sent Sahar?\"\n\n\"Well, why did they leave?\"\n\n\"Who?\"\n\n\"Singer and that old woman.\"\n\n\"How should I know?\" Khosrow complained. \"Why are you interrogating me? Now you're probably going to ask me what we had for dinner, what we had for lunch... aren't you going to tell me where Sahar is?\"\n\n\"You went off and left us for so long. After all, you were the man of the house. Now that you're back, won't you tell your mother where you went? Who was with you? Whether you had a good time?\"\n\n\"Well, we went hunting,\" Khosrow answered impatiently. \"On the third day when we came back after sunset, another foreigner wearing dark glasses arrived and took Singer and the woman with him. Uncle sent three armed men and one of his guides along with them. They headed for the mountains. Four-eyed Hormoz said, 'You can be sure they're off to see the tribe.' Now tell me where Sahar is.\"\n\nZari bit her lip. \"God help us!\" she exclaimed.\n\nMina got up from the edge of the pool. \"Sahar was hurt and died!\" she blurted out.\n\n\"Died!\" Khosrow shrieked. \"But why? Is it true, mother?\" he asked through his tears. \"I guessed it myself. I saw the flowerpots on his grave.\"\n\n\"What could I do, my dear?\" Zari said with a sigh. \"It was his fate. Your uncle took you to the village on purpose so you wouldn't see him die. At least he had a peaceful end. We buried him at the bottom of the garden just for your sake.\"\n\nKhosrow squatted by the pool and said, \"I knew inside me right from the start that something was going to happen. I could tell from the way my uncle talked. He went on about how a person should be patient, and what you should do when you lose someone you love. And after that he kept talking about the glanders disease. That's funny, you know, I dreamt last night that I was riding after game. Uncle and Singer were there too. Singer had spread a map on his saddle, and at the same time he was looking through his long binoculars for game. The first day of the hunt he was doing that, you see, and my uncle kept saying, 'Look how these foreigners do everything with calculation, even their hunting'...\"\n\n\"Yes, especially when they're hunting people...\" Zari commented sadly.\n\n\"But I was riding Sahar, not uncle's horse. We were coming down the mountain. Suddenly Sahar reared up. His front legs and mane froze in the air, and there I was hanging in space on horseback. The earth looked like a nutshell under my feet. In the morning I told uncle my dream. He said, 'It probably means something has happened to Sahar. Now don't you get upset! It's not worth it. Pick out whichever of my colts you like.' I said, 'Uncle, that's impossible. When we left Sahar was perfectly healthy. How could it be? No other colt will ever take Sahar's place for me.'\" Khosrow broke off, sobbing. \"Now I remember. When we were leaving, Sahar was stamping his foot and digging at the soil with his hoof. Poor animal knew he wouldn't see me again, but stupid me, I didn't know. Mother, why is my stomach turning so? I feel as if someone's choking me.\"\n\nZari hugged and kissed her son.\n\n\"Wash your face with some cold water, my dear,\" she said, \"you'll feel better.\" Her own heart was brimming with sorrow. \"Why don't you invite your schoolfriends over this afternoon to a mourning ceremony for Sahar? I'll bring out some tea and sherbet drinks for you.\"\n\n\"Will you make some halva too?\" Khosrow asked.\n\n\"Certainly, if you want some.\" She paused and added, \"Yes, I'll make some halva. As soon as the smell of halva rises, Sahar's spirit will know we're thinking of him.\"\n\n\"Can we come too?\" Mina asked.\n\n\"No,\" Khosrow answered, kissing each of his sisters in turn. \"The ceremony is for men only.\"\n\nThat afternoon, Sahar's all-male 'mourning ceremony' really did take place in the garden. At least twenty children of various ages poured in. Gholam had swept over the make-believe grave, and covered it with a carpet. Watching from the verandah, Zari could see the children squatting silently by the grave. She noticed a small boy wearing a black mourning shirt, staring fixedly at something. When she looked more carefully, she realized that he was staring at his thumbnails. Probably to stop himself laughing. But finally he started to giggle and then burst out laughing. All the other children, besides Khosrow and Hormoz who was sitting next to him, joined in the laughter and the ceremony broke up. Zari couldn't bear it anymore. She went to the parlour. Seeing a lot of flies buzzing around, she took a fly-swatter and attacked them, killing them left and right. She could hear the children playing in the garden and looking out from the parlour window saw that they were going at the unripe fruit on the trees. But Khosrow and Hormoz were still sitting on the carpet while Gholam walked toward them with some coffee and Khadijeh put the trays of halva on the ground. Hormoz whispered something in Khosrow's ear and Khosrow slapped his forehead with a grown-up gesture, then covered his eyes.\n\nWhen the children had gone, Khosrow and Hormoz came to the parlour. Khosrow's eyes were red and Hormoz's glasses all fogged up.\n\n\"Cheer up, my dear,\" Zari comforted, \"it's not so bad after all. As Khadijeh says, the mare is young, she'll give birth to another Sahar for you.\" And she thought to herself, \"If, as Ameh Khanom said, he ever sees that wench riding Sahar, then all hell will break loose! How we end up lying to our children!\"\n\n\"I'm trying not to cry,\" said Khosrow, \"but I feel so unhappy...\"\n\nHormoz took off his glasses. He took out a handkerchief from his pocket and wiped them. His eyes were puffy.\n\n\"I keep telling Khosrow this is just the beginning,\" said Hormoz. \"We have a lot of ups and downs ahead of us. We mustn't give up so easily. Besides, look how many people die of typhus or starvation each day. What's a colt next to all these people?\"\n\nZari looked at Hormoz. She wasn't sure whether they were his own words, or he had learned them from someone else. In any case, he was four years older than Khosrow. She thought with bitterness, \"The real death of humans next to the fake death of a colt! Certainly there's no comparison.\"\n\nSuddenly her mind went back to that evening in the Missionary Hospital where her mother was spending the last hours of her life. Zari had had no idea how near the end it was, even though Khanom Hakim had told her, \"Now the cancer be overtaking the whole body, and there be nothing more the knife can do.\"\n\nHer mother had looked at Zari out of the corner of her eye.\n\n\"Stay with me tonight!\" she had said.\n\nBut how could she stay? Khosrow was only three years old and would not eat unless she fed him nor sleep unless she were next to him. Besides, they had guests. Yusef had invited a number of people.\n\n\"I have to go,\" she had said. \"We have guests. I'll be back tomorrow morning.\"\n\n\"Tomorrow?\" her mother had echoed. And didn't insist anymore. She merely asked for some sacred soil to be brought her by Ameh Khanom. By the time Zari had gone home and Ameh Khanom had finished her prayers and her opium-smoking, put on her outdoor dress with the long sleeves and her gloves and her scarf, the evening had drawn on and she was unable to go all the way to the hospital by herself. In any case, no-one would have thought a person who seemed so alert one minute would die the next.\n\nAbol-Ghassem Khan had arrived before all the other guests and when he found out about the situation, offered to accompany Ameh Khanom. But he had no car in those days, and they couldn't find a droshke. They managed to get there, nevertheless, though it was after eleven by the time they returned. Zari was serving dinner and Khosrow had not been put to bed yet. The guests were playing with him, taking turns holding him and listening to his sweet baby-talk. Zari didn't even get a chance to ask Abol-Ghassem Khan how her mother was. As for Ameh, she went straight to bed. Later, at dinner, Abol-Ghassem Khan drank so much vodka that he became completely drunk. Tears streamed down his face and he babbled on about his own mother. He smashed several glasses against the wall and then threw up violently, upsetting the other guests. Finally they took him to the bottom of the garden so he could vomit as much as he liked. When the guests had left, they told Zari her mother had died, that alas, she hadn't received the sacred soil she asked for, that no-one had been at her bedside, except a foreign nurse who didn't speak her language...\n\nAt that moment Mina and Marjan barged into the room, bringing Zari abruptly back to the present. Each of them was holding a doll.\n\n\"Uncle gave me this,\" Mina said.\n\nAbol-Ghassem Khan followed them into the parlour with Gholam in his wake, carrying two loaded sacks.\n\n\"It's our first picking of lemons,\" Abol-Ghassem Khan announced. At a sign from Zari, Gholam took the sacks to the storage-room. Abol-Ghassem Khan embraced Khosrow and said, \"Shall I send for that colt of mine you liked in the village?\"\n\n\"No, uncle, I don't want a horse at all.\"\n\nMina, still holding her doll, put a hand on her brother's knee.\n\n\"Have you seen my doll?\" she asked. \"Do you want to have it?\"\n\n# _10_\n\nThat week Zari finished early at the asylum, for typhus had reduced the number of patients to slightly over half compared to the week before. The warden, a short fellow with a dark complexion who received her every alternate Thursday, would only allow her to distribute bread and dates among the inmates after taking adequate payment for himself and his nurses. This week he told her that the epidemic had hit them hard and that his patients had been refused admission to the town's hospitals. \"He doesn't look too well himself,\" thought Zari, as she handed over his payment. Not that he ever looked particularly well, dealing as he did all the time with mental patients. His eyes had sunk into their sockets.\n\nWhen they entered the men's ward, Gholam put the tray of food on the floor, but unlike other weeks, no-one seemed to show any interest. Zari looked around at the men, with their shaved heads and soiled white gowns, sitting silently in the room. They seemed to be listening to sounds only they could hear and to which one or other of them would occasionally mumble a reply. They took the bread and dates from Gholam absent-mindedly. Zari felt depressed. It was as if today her vow had not been fulfilled since she hadn't made anyone happy. Downhearted, she began to distribute cigarettes and matches. One patient who claimed to be the Chief Commander of the World and who always asked for the Homa brand of cigarette, took an Oshno this time and without striking a match, listlessly put the cigarette to his lips. The sun poured in through the shutterless windows, and flies buzzed sleepily around the room, exploring every nook and cranny, as well as the untouched food in the patients' hands.\n\n\"Ali!\" summoned the head nurse loudly. Ali was Zari's favourite patient, a tall German-looking young man who had attempted three times to escape from the asylum. Twice his relatives had found him, each time in the neighbourhood of the high-school where he had finished five grades. The last time Gholam had found him on the hill overlooking Yusef's garden. Apparently Ali had followed Gholam like a lamb, allowing himself to be led back quietly to the asylum. Hunger had taken its toll. He had told Gholam:\n\n\"They tricked me. They whispered to me that the airplane is ready; please get inside it and go to Europe to your uncle. I came out and no matter how hard I searched, I couldn't find the airplane. Maybe it left without me. I have many enemies, you see.\" Later he confessed, \"I've been drinking water from the gutter and stealing bones and bread from dogs. Yesterday I grabbed a piece of raw meat from a dog, and ran away with it. I washed the meat in the gutter and ate it. My stomach turned, and now I have diarrhoea. There's blood in it too. I really looked everywhere, but I just couldn't find our house. I know my father made our house get lost on purpose so I wouldn't find it.\"\n\nFrom that day on, they chained Ali in the asylum basement. Zari would visit him there and take him bread and dates. He always smiled when he saw her. Once he had asked her for 'Essential English, Part III', and Zari had brought him one. Thereafter, he refused to speak a word in Persian, talking instead in a language no-one could understand.\n\nAli came in. He had lost so much weight that Zari felt distressed at the sight of him, and he did not recognize her. He threw her a blank look and, without using his invented language, proclaimed in Persian, \"An attack of pliers equals typhus + famine + cheating in an exam. O madmen of the world unite!\"\n\nThe Seyyid from the Arab Quarter was also sitting silently in a corner. Usually when he saw Zari he would reach under his belly and start scratching himself, saying, \"Burning, burning, I am burning!\" And then he would add, \"It's me Eilan-ud-Dowleh, it's me Veilan-ud-Dowleh.\" In exchange for Zari's gifts, he would give her bits of imaginary paper with prayers of love and affection on them, or magical and occult charms or talismans.\n\n\"Our account is clear,\" he would say, \"but do wash your shirt with water from the morgue. Spread it out on a deadman's grave then have him wear it the next morning. Tiger's whiskers and the brain of a black mule...\"\n\nThen there was another patient who tied his imaginary leg wounds with whatever bits of material he could get hold of, and would stretch out the leg and fan it. But today the fan had fallen away from his hand.\n\nAs Zari and Gholam, accompanied by the warden, were passing through the dried-up yard of the asylum, they saw a young woman stretched out on an old mattress under a pine tree. Hearing footsteps, the woman flicked open her eyes. Zari recognized her, even though her face had been drained of colour until it blended with the dust on the ground. It was the same woman who sometimes claimed to be the wife of God, and at other times God himself. Occasionally she would smear her cheeks and lips with some red petals from the Marvel of Peru flowers in the garden and say she was waiting for God. Apparently she would stare at the sky and repeat some mumbo-jumbo in a language resembling Arabic, saying God was waiting for her on the roof. But she herself wouldn't go to him; she was a woman, and a woman could never take the first step.\n\n'God's wife' was now stretched out under the pine tree, her face twitching and her lips blistered. \"She seems ready to join Him at any time,\" Zari thought. \"If only she would intercede with Him for the rest of her fellow sufferers...\"\n\nA sound escaped the woman's lips. \"Water!\" she moaned, as her blankly staring eyes slowly closed. Gholam ran for some water.\n\n\"Why is she lying here?\" Zari asked the warden.\n\n\"She's got typhus,\" he replied.\n\n\"Well, all of them catch that at one time or another...\"\n\n\"All the better! It will be a relief for them. Their relatives pray that they'll be released from their suffering. What's the use of keeping them like this?\"\n\nGholam came back in a rush, holding a glazed bowl full of water. He lowered the edge of the bowl to the woman's lips. \"Drink, sister,\" he coaxed, but she couldn't swallow. Zari took her handkerchief from her handbag, soaked it and rubbed it on the woman's face and lips. Then she wet it again and placed it on her forehead.\n\nThey walked on. The warden followed alongside, offering explanations, \"Three of our nurses caught typhus,\" he said, \"and are now sitting comfortably under the Tuba Tree in paradise. 'God's wife' will be on her way there too tonight.\" Then, seeing Zari looking at him disapprovingly, he continued in a different tone, \"It's amazing. When their fever goes up, their madness seems to disappear. If only we could save them from this second disease, maybe they'd be cured of their madness too! But what's the use? If they ever came to their senses, it would only be the beginning of their troubles. Their families have become used to their absence, and they would have no room or patience for them.\"\n\nIn the women's ward, Zari noticed the crippled woman who always managed to frighten her. \"You fucking whore,\" she would say, \"are you back again? What do you want from my life?\" This woman blamed Zari for her paralysis and Zari felt guilty at heart about it too. When the woman had had healthy legs, she had asked Zari for a pair of old slippers, or a sturdy pair of second-hand givehs.\n\n\"I'm a respectable woman,\" she had said, \"and I can't go to the toilet barefoot.\" Then, \"May God strike Khanom Essmat dead! If she had spent my marriage portion and inheritance on me instead of on that goddam cuckold who sleeps with her, I'd never be grovelling for your droppings, you whore from Mordestan!\"\n\nBut the following week Zari had been due to go to the prison, and the week after that she had forgotten all about it. By the time she remembered to buy the woman her new shoes, it was too late\u2014she was already paralysed. Of course everyone knew her paralysis had nothing to do with the givehs. But every time after that when she saw Zari she threw unspeakable insults in her direction. Still, the nurses said she hugged the new shoes tightly each night as she went to sleep.\n\nZari glanced around for the young teacher with the glass eye. This one wasn't particularly fond of her, either, and wouldn't let her come close. Zari always left her share of bread and dates on the sill. Sometimes when the teacher was in a good mood she would say things like, \"Look how much perfume this harlot's used! Ugh! How lucky you are, my little servant, to have got this far. You remember you were the daughter of our dressmaker? I knew you'd finally give in. With that cab driver who had a wife in every town...\" And she would put a finger under Zari's chin and say, \"You little coquette!\" Then suddenly she would get angry and shout, \"You've put rat poison inside these dates! You've taken out the pip and put ratsbane instead. What an offering!\"\n\nApparently she used to teach first graders. One day, sometime after the veil was banned, the school was inspected by the Governor, the army commander and the minister of education. The minister had found out that this teacher would punish children by squeezing a pencil between their little fingers and laugh when they hopped with pain. He had made quite a fuss, but only about the issue of corporal punishment, yet the young teacher had fainted from humiliation at the sight of all those important people. She was immediately hauled off to the principal's office where they revived her, but the shock had been too much. She had stared blankly at everyone, then calmly taken out her glass eye, holding it out in the palm of her hand for her bewildered audience to behold.\n\nOne day in the asylum she played the same trick on Zari. Until then, Zari hadn't known that the woman had a glass eye, although she had noticed that the right eye didn't move in its socket. The young teacher was agitated that day. When Zari came into the room she went over to her, reached out her hand, and said, \"Take it!\" Then she opened her fist into Zari's hand, and there was Zari holding a large, shiny glass eye.\n\nNow, on inquiring, Zari was told that the first fatality from typhus had been this very girl.\n\n\"At first we didn't know she had typhus,\" the warden explained. \"Of course her fever was very high and she was delirious. She imagined she was putting on her shroud. She tied anything she could find around herself saying it was her shroud, and began reciting the Quran by heart. She was superb. But instead of cursing the Devil, she cursed the Cardboard Man. I believe the Cardboard Man was that same minister of education who fired her from her job. Finally, she said her last prayers and threw herself into the pool. She died that night.\"\n\nAt the end of her rounds, Zari went to Khanom Fotouhi whose bed lay next to a window where she could constantly watch the yard, in the hope that her relatives would come and take her to the 'hundred and twenty-four thousand metre garden'. Zari knew the Fotouhi family. They were well off. At the beginning of Khanom Fotouhi's illness, they kept her at home. But when she finally drove them to desperation, they gave up hoping for her recovery, and passed her on to the asylum. Before the war, she had had a private room where she was visited regularly by her mother who would even take her home for a week or two sometimes. When she had had enough, she would drag her daughter back to the asylum, leave her in the reception office and disappear. But the mother had died years ago.\n\nKhanom Fotouhi's brother was the well-known history teacher in town and something of an idol for its youth. The most he could manage was to visit his sister once in a blue moon. Now it looked as if they had all really abandoned her at the asylum. But Khanom Fotouhi never despaired. She was still waiting for them to come and take her to the 'hundred and twenty-four thousand metre' garden.\n\nShe was a sallow-looking girl with thick eyebrows that joined in the middle, protruding teeth, and grey hair. She never accepted food from Zari, as if it were too demeaning to show interest in something which others would grab at with such greed. When the fruit in her garden ripened, Zari usually took woven baskets piled high with apricots, sour apples, cherries, peaches and pears to the prison and asylum. But Khanom Fotouhi wouldn't even look at these.\n\nOn several occasions Zari had prepared a special fruit basket for her and left it on the windowsill. But the nurses later said that the minute she had stepped outside, the other patients raided the basket. When they got to the sour apples they would split them in half, ask for some salt and sprinkle them until they were well 'seasoned' in the Shirazi way. It was enough to make anyone else's mouth water. But Khanom Fotouhi would merely stare out of the window at the yard, waiting for her relatives to take her away to the 'hundred and twenty-four thousand metre garden'. The other patients didn't even spare the apricot pips which they would either split open with their teeth or bang with stones on the floor to get at the little kernels. After all, as the warden said, how could any of the patients get real sustenance on their pitiful daily allowance? Most of them had gone mad from poor nutrition in the first place.\n\nWhen she had finished dividing the food, Zari would sit next to Khanom Fotouhi's bed and listen to her complaints. Khanom Fotouhi hated all the other patients and never spoke to any of them. They, in turn, had nicknamed her 'Princess'. The kinds of things Khanom Fotouhi used to ask for included the large-format _Iran_ newspaper which was mailed to Yusef twice a week from Tehran, lined notebooks and pencils which she would accept from Zari, saying, \"I have allowed you to contribute to the world of science and literature.\" She loved all the serial articles in the _Iran,_ and the notebooks she would use for writing her autobiography\u2014or so she claimed.\n\nEach time she finished a notebook, she would hand it ceremoniously to Zari. \"Rent a safe deposit box in the National Bank,\" she would say. \"Take the money for it from my brother and store my works there. We could have a fire here someday, and I don't want to have my works destroyed.\"\n\nThe first time Zari had believed her, and tried to read one of the notebooks only to discover that it was filled with some incoherent ideas written out in a language of the occult sciences. Wherever the handwriting became legible, it described a 'hundred and twenty-four thousand metre garden' with man-made waterfalls and lakes, blooming water-lilies, acacias and ash-trees. In one part she wrote about a well-built man with a wide forehead and white hair around the temples who hid behind an ash-tree while she herself, wearing a loose white chiffon dress which fluttered in the wind, stepped gracefully into the open air. Her shapely breasts and erect nipples showed through her dress, and the well-built man rushed out from behind the tree, capturing her in his arms and hugging and embracing her. At the end of her notebook she had written, \"Thus endeth the sorrowful tale of the Fotouhi maiden in the Nai prison,\" and underneath this sentence she had added, \"Some verses by the Fotouhi maiden:\n\nI was a fledgeling my mother died\n\nThe wet nurse took me, but she too died\n\nThey raised me on cow's milk\n\nI was so ill-starred, the cow then died.\"\n\nZari was quite certain these lines were not composed by the 'Fotouhi maiden' because Ameh Khanom had hummed them herself from time to time. Actually, in the days when the 'Fotouhi maiden' was in her right mind, she had been a good writer, producing articles in local papers about women's rights, and the injustices of male domination. She also brought out a magazine which aimed to raise women's consciousness.\n\nIn the days before her mental breakdown, Khanom Fotouhi was a woman to be reckoned with. She had been the first to abandon the black veil\u2014or black shroud, as she called it\u2014in favour of a roomier, more attractive blue veil. The lifting of the veil had not yet been announced officially before she even gave up the blue one also. On a good day, she would complain to Zari that it was too bad she had not been appreciated. \"A pity,\" she would say, \"that our men were not ready to accept a woman like me. At first they thought I could be taken advantage of, like a pot of honey you could dip your finger into. But when I smacked them on the fingers and sent them off packing, they humiliated me or ignored me.\" Then she would suddenly shout with tears in her eyes, \"They drove me mad! They drove me mad! I told them I wouldn't give in! I won't give you what you're after! And that's that! When will other women\u2014those silly little dolls\u2014ever understand who I was and what I stood up for!\"\n\nZari sat down by Khanom Fotouhi's bed and greeted her. Khanom Fotouhi turned her gaze from the yard to Zari and said hello. Zari reached into her bag and brought out four issues of _Iran_ for her, at the same time catching sight of a pillow next to which all the previous newspapers had been neatly stacked. Khanom Fotouhi opened the new issues one by one. She frowned at the changes in detail and the recently introduced small format.\n\n\"Didn't you give the newspapers to the other patients to read this time?\" Zari asked.\n\n\"No, most of them have been freed from prison,\" she replied nervously. \"Ali took two of your newspapers and ate them.\" Then she looked Zari up and down. She didn't seem to like the long, wide sleeves of Zari's shirt. \"You've wasted a hundred metres of good material just on those sleeves, haven't you?\" she asked. Then she crumpled up the new newspapers and threw them down beside the bed. She turned her attention to the old newspapers and started counting them. Then she rolled one up and suddenly hit Zari very hard on the head with it. \"They say Afsar Khanom, the daughter of the commander, is dead!\" she shouted. \"And she didn't even have a shroud!\" \n\n# _11_\n\nA Shirazi woman, trained as a midwife in Tehran, had recently opened an office in town. She had more patients than she could handle, but Zari had managed to get an appointment for seven o'clock Thursday evening after her rounds at the asylum. As soon as she was finished, she sent Gholam away and headed for the doctor's office, thinking all the while of the futility of her charities. She remembered Yusef's words, \"What's the use of your charity and goodwill? This society is rotten at the core.\" But no matter how hard she thought, Zari did not seem to come up with any ideas on how to improve a society at its core. The solutions which Yusef suggested always seemed so dangerous that they sent shivers down her spine.\n\nAt six o'clock, she arrived at the midwife's office. She was feeling queasy. There were two donkeys standing at the door with their bridles tied to the door knockers. In the small courtyard next to the office, two women were huddled on a bare wooden bed, with another stretched out behind them. One couldn't tell their age because the expression on their faces was so strained. A sick man was tossing about on yet another bed. Right next to the door of the waiting-room a woman was stretched out stiff as a rod. Her bare, henna-dyed feet protruded grotesquely from underneath the blue polka-dot veil with which she was covered. Her black trouser-legs had been pulled up to her knees. Zari was taken aback. Surely the woman was dead. Zari had seen enough in life to recognize death when she saw it. But it would seem the woman had no-one, since she was obviously abandoned even in death.\n\nInside the waiting-room all the seats had been taken. Only five of the patients were pregnant women\u2014recognizable by their round bellies and blotchy skin\u2014the others were either male or elderly. A young girl with blistered lips leaning her head on the shoulder of an older woman entered the waiting room just then. \"Oh, my heart! My heart!\" she moaned. A pregnant woman stood up and gave her place to the young girl, opening the window above her. But only a blast of hot air came in. The door of the doctor's office opened, letting out a pregnant woman, who slowly crossed the waiting-room as if the weight of her nine-month burden made it impossible to move any faster. A nurse with dishevelled hair followed her and announced:\n\n\"Forty-eight!\"\n\nZari managed to reach the nurse as she scanned the patients for number forty-eight.\n\n\"Forty-nine!\" the nurse said loudly.\n\n\"I have an appointment for seven,\" interrupted Zari.\n\n\"It's no use getting an appointment these days, dear. All sorts of patients are crowding in on us. Even the courtyard is packed. Didn't you see for yourself?\"\n\n\"Yes, I did. One of them was dead.\"\n\n\"I know,\" said the nurse coolly. \"By the time they're brought here on donkey from the villages, they've taken their last breath.\" Then turning to the other patients she shouted, \"Forty-nine isn't here? Fifty!\"\n\nAn old hunch-backed woman got up. Clutching her veil tightly across her face, she walked over with an odd shuffle. The nurse opened the office door for her. Zari reached into her handbag, and the nurse followed her movements with her eyes as she groped for a handkerchief. Finally she grew impatient.\n\n\"If there's nothing wrong with you and you're here only for pregnancy, I suggest you leave it for another time.\" And with that she disappeared into the inner room.\n\n\"She's right,\" Zari thought to herself. \"After all, I'm in no hurry. In any case, I'll probably end up at Khanom Hakim's yet again.\" She decided to go home and wash thoroughly, even boil her clothes. She wasn't going to let those delicate children touch her before she had disinfected herself. On the way home she stopped at the pharmacy and bought anti-flea powder, alcohol, soap and sulphur.\n\nBy the time Zari reached the garden-gate of her home, the sun had already set. A dark little boy with curly hair opened the gate. As soon as he saw Zari, he grinned widely at her. Zari recognized him.\n\n\"What are you doing here, Kolu?\" she asked.\n\n\"I've come back with the master.\"\n\n\"Is he back then?\" she exclaimed, rushing past him towards the house. Yusef, still dressed in his dusty travelling clothes, was sitting on the cane chair by the pool, smoking a hookah. His face lit up at the sight of his wife.\n\n\"Where have you been till now?\" he asked. \"I was waiting for you. I came all the way to... why are you standing so far away?\"\n\n\"You're back so early,\" Zari answered, \"but I'm glad you've returned. You mustn't touch me, though. I have to take a bath first. I'm full of germs. Oh, when you're here everything seems so much brighter!\" And she hurried inside.\n\nBathed and perfumed, she came back into the garden, but by that time it was nearly dark. Yusef was holding his head in his hands. She went to him and lifting his head, kissed him on the hair.\n\n\"Don't you feel well?\" she asked him.\n\nYusef pulled his wife on to his lap and the chair creaked beneath them. He kissed her neck and face and bare arms with soft, tender lips. Zari got up.\n\n\"Let me go and put the lights on,\" she said.\n\n\"Leave it,\" he said, pulling her by the hand.\n\n\"It's a heavy sky,\" Zari observed, glancing up. \"But it won't rain either to let us breathe.\"\n\n\"Not unlike my heart...\"\n\n\"Well, it's midsummer,\" said Zari. Her mind was on Sahar and how to prevent her husband from asking after the horse.\n\n\"The house felt really empty when I arrived. Where are the children?\"\n\n\"Amen Khanom took them to Mehri's house for the Rowzeh,\" replied Zari. \"Khosrow has gone out with Hormoz.\"\n\n\"You really shouldn't be sending the children to the Rowzeh.\"\n\n\"They insisted on going,\" said Zari. \"Besides, they don't pay any attention to all the mourning. They play with Mehri's children. Ameh Khanom has made them chadors, and they say their prayers standing next to her...\" she stopped in mid-sentence. \"Why are you back so early?\" she asked. \"And why did you bring Kolu with you?\"\n\n\"Send him to the baths tomorrow and give him some new clothes. I've adopted him as a son,\" Yusef said quietly. \"I killed his father, so I couldn't stay at the village any longer.\"\n\nZari's heart sank. \"I don't understand,\" she said. \"You killed Kolu's father? Our shepherd? You? Nonsense!\"\n\nYusef buried his head in his hands. \"Don't talk about it anymore,\" he said. \"My head is about to burst.\"\n\n\"But won't you tell me what happened?\"\n\n\"Well, that's why I came back so early. I just dropped everything I had to do and rushed back so I could confide in you, but you weren't here.\"\n\nZari took a seat next to her husband and let his head rest on her shoulder, stroking him soothingly.\n\n\"My love, how was I to know you would suddenly arrive? Tell me about it now and I'll listen. You'll feel better if you talk about it.\"\n\n\"Our shepherd was supposed to take the last of our flocks up to the mountains. Before he went, he killed two of our sheep, cured the flesh and stored it in a sheepskin. I don't know what suddenly possessed him to do such a thing. He's never been dishonest before.\"\n\n\"Well, you told me yourself that people are panicking because of the famine.\"\n\nYusef got up and started to pace about.\n\n\"Nothing escapes the notice of the village headman,\" he continued, ignoring Zari's comment. \"When I got there, he had to come out and tell me all about it in front of everyone. I wanted to ignore the whole thing, but the headman had no intention of dropping the matter. When the shepherd brought the flock back at sunset, the headman reminded me again. I was forced to interrogate the shepherd and ask him why two sheep were missing. He swore that a wolf had eaten them. The headman then told him to take an oath, and swear by the holy prophet Hazrate Abbas that he was telling the truth.\"\n\nYusef paused. Then he went on, \"I could see the poor soul shaking at the knees as he stepped forward to take the oath. There I was watching him, stupid fool that I am, and I did nothing to stop him. That night he came down with a stomach-ache. I went to his house\u2014or rather hovel. He looked at me with dumb eyes\u2014like a lamb\u2014and begged to be forgiven. I nearly shouted at him that I'd forgiven him all along. I told him that he should know me well enough. But it was no use. Tears were rolling down his face on to his dirty pillow. I tried giving him sweetened warm wine, but he refused it. He kept saying that he'd sinned more than his share and the holy prophet Hazrate Abbas would take his due. 'But I'm the owner of the sheep and I forgive you, man,' I said. And still he wouldn't listen. He just repeated, 'The prophet has struck me down. You can't do anything for me anymore. Give the flock to my brother and he'll care for them in my place.'\"\n\nYusef sat down by his wife and went on, \"He motioned to Massoumeh, Yarqoli's wife, who disappeared for a moment and came back with two sheepskins full of cured flesh. She threw them in front of me. I wanted the ground to open up and swallow me...\"\n\n\"My love,\" Zari said calmly, \"you know very well it wasn't your fault. It was that cruel headman who didn't know any better. The shepherd took a false oath, or maybe he'd just eaten something bad. Besides, why must you think the worst? He could have caught typhus. We know nothing of God's will. Perhaps his son was meant to get an education and have a bright future. How do we know?\"\n\nKhadijeh came out to the verandah and put the light on. Then she went to the garden to lay out the beds. She fixed up the twins' bedclothes on a wooden bed on the far side of the pool. Then she arranged the mosquito net over it. When she got to Khosrow's bed, she laid out the mattress but then it seemed that she had lost something when she got to the bedclothes.\n\n\"Khanom, have you put Khosrow Khan's blanket somewhere?\" she asked Zari.\n\n\"No. Maybe you used it for the ironing yourself,\" Zari replied from where she sat.\n\n\"I didn't, Khanom.\"\n\n\"So what's happened to it?\"\n\n\"Well, I don't know. Maybe the same clumsy thief who stole the clothesline, stole Khosrow Khan's blanket too.\"\n\nSuddenly Zari was filled with anxiety. Could it be that Khosrow was behind the disappearance of both items? But what for? Very early that morning, even before prayer-time, Zari had been woken up by a light footstep next to where she slept on the roof terrace. When she opened her eyes, she had seen Khosrow, looking stealthily all around him, tiptoe to the clothesline and untie its knot from the hook on the wall. Then he had gathered the entire length of rope around his arm and sneaked into his room with it. When he returned he crawled silently under the mosquito net and pretended to be asleep.\n\nKhosrow had been acting very strangely these past few days. His mind seemed to be elsewhere and from time to time Zari had caught him staring blankly into space. When he first heard of Sahar's death, he seemed heart-broken, the tears springing to his eyes at the slightest excuse. He hung around most of the time at the bottom of the garden by the grave, digging out the weeds and watering the flowerpots with his own hands. But recently he had changed. He didn't even glance at the grave anymore. He avoided his mother's gaze, and gave only short, confused answers to her questions.\n\nZari got up. She had a feeling he had also taken his gun, even though she remembered having locked it away in the cupboard and taken the key with her. Yusef's voice brought her back to herself. He was saying, \"Why are you standing like that? Sit down. Say something.\"\n\n\"What did you say?\" she said, as if roused from a daydream.\n\n\"I know I've upset you. You're disappointed in me too.\"\n\n\"You're wrong,\" Zari answered absently. \"It's not at all your fault. I saw the sick they brought in from the villages to Khanom Massihadem, the midwife. One of them was dead. Typhus has spread in all the villages; the town is full of it too.\"\n\n\"What were you doing at Khanom Massihadem's office?\" Yusef asked in amazement. \"Are you...\"\n\nZari felt completely flustered. It was as if they had been inhabiting two different worlds. How little one knows of what goes on in the mind of another person!\n\n\"Oh I just went to buy some anti-flea powder from the pharmacy and I passed by there,\" she said. \"The door was open so I took a look. Well, maybe that patient wasn't really dead... I was probably imagining it...\" She didn't know what she was saying anymore, so before Yusef could pin her down, she hurried to the bedroom. Without switching the light on she found her bag, took out her keys and groped around for the keyhole in the cupboard. Her hand was shaking and her stomach turned. No, thank God, the guns were still there. To reassure herself, she touched their long, cool barrels, the breech-blocks and heavy butts, leaning tall against the cupboard wall. She locked the cupboard door, closed the windows and doors of the parlour and went to the telephone. She asked the operator softly to connect her to Abol-Ghassem Khan's house. She couldn't be heard, so she had to ask a second time. Abol-Ghassem Khan himself answered at the other end. She asked whether Khosrow was there. He said no, and Hormoz wasn't either. She could hear Abol-Ghassem Khan asking around from others in the household. Apparently Hormoz had said he was having dinner at his uncle Yusef's house. He had said that Zari had invited him... \"Now why weren't we invited too?\" Abol-Ghassem Khan complained jokingly. \"Do you think we would have turned down a treat?\"\n\nZari's throat constricted. She mumbled something about God willing next time, and hung up. She was terrified. Both boys had lied, so there was no doubt they were up to something. They had also taken a rope and a blanket with them. She must go and tell Yusef everything.\n\nAs she was leaving the parlour the telephone rang. She went over and picked up the receiver. It was Abol-Ghassem Khan. He had been thinking about the boys and had become worried too. Zari pulled herself together and managed to say, \"Don't worry. I think they've gone off to the cinema or somewhere together. They'll come here for dinner, it's not too late yet. As soon as they're here, I'll tell them to call you.\"\n\nShe opened the doors and windows of the parlour again. She heard Mina's voice. The children had arrived. She went out to the garden. Both children were sitting on Yusef's lap, and he seemed a little more relaxed. Mina was saying, \"Mother won't let us. She says we'll get all burned on our skins and we'll have to stay inside.\"\n\nAmeh Khanom was sitting there, with her veil still on.\n\n\"Sister,\" she said, \"Mehri sends her regards but says she's cross with you because you didn't call in for the Rowzeh. She won't forget it, she said.\"\n\nMina clapped her hands together from where she was sitting. \"She's cross with you! She's cross with you!\" she chanted.\n\nThen she turned round to kiss her father under the chin, and struggled to climb down from his lap. Yusef hugged both children tightly. \"Well,\" he said, \"what else are you going to tell me about, my little dolls?\"\n\nZari, staring at the verandah lights and listening to the sounds in the garden, could not think where to begin. Like the patients at the asylum that afternoon, her mind was all in a jumble though she seemed composed on the outside. Mosquitoes, tiny moths and various kinds of dragonflies flitted around the verandah light, got stuck to it, and finally dropped off. In the garden, the crickets and the frogs were having a contest. There was no other sound or movement. If the boys were heading home, she would easily have heard their footsteps. She had to tell the others now and rouse them to some kind of action, make them comb the town to find her son. What if this were the shepherd's vengeance? What if the Lord had sent them the shepherd's son in exchange for their own son? She felt sick. The trees seemed to slumber under the heavy blanket of the sky. If only there was a breeze, or if she could, like a furious wind, whip the trees and everyone around her into action. If only the sky would clear so the stars, like a million eyes, could scour the earth for Khosrow, and the trees could whisper his whereabouts to her.\n\n\"Let's go and sit somewhere else,\" she said involuntarily.\n\nYusef was holding up Marjan's hair and kissing the nape of her neck. He laughed and said, \"What better place than right here?\"\n\n\"Let's go and find Khosrow,\" said Zari.\n\n\"Sister, Khosrow has gone to Abol-Ghassem Khan's with Hormoz,\" said Ameh Khanom.\n\nZari was unable to contain herself anymore. \"But he's not there!\" she sobbed. \"He's gone off with a rope and a blanket, though his gun is still here.\"\n\nYusef put the children down in amazement. \"What for?\" he demanded. \"Where could he have gone to?\"\n\n\"I don't know where he's gone,\" Zari replied through her tears. \"Let's go and find him. I know something has happened to my son. I realized it when I saw Kolu. It\u2014it must be God's revenge. God has sent Kolu to replace my son.\" And she broke down into loud sobs.\n\nYusef got up and held her by her shoulders. \"Your nerves have been under strain,\" he said. \"It's my fault for telling you everything that happens. Put these superstitions out of your mind. Call Abol-Ghassem Khan's house. Maybe he's there.\"\n\n\"I've already called.\"\n\n\"I'll put the twins to bed,\" Ameh Khanom volunteered. \"Go over the hill to the Governor's house. I've a feeling Khosrow and Hormoz are there.\"\n\n\"What's all this, sister?\" Yusef asked with a look. \"Have you turned clairvoyant?\"\n\n\"The sooner you leave the better,\" Ameh insisted. \"I'll call Abol-Ghassem and ask him to get there as soon as possible.\"\n\n\"I don't understand it at all,\" Yusef said wearily. Then he had an idea. \"They could have gone to Fotouhi's house. Hormoz's history teacher. But then Fotouhi's in Isfahan. I know he's not back yet.\"\n\n\"Come on, leave right away,\" Ameh Khanom urged. \"Zari will tell you everything on the way.\"\n\nZari and Yusef went out by the small door in the back wall of the garden which opened on to the foot of the hill behind their house. They headed towards the hill.\n\n\"What have you been up to, woman?\" Yusef demanded. \"What have you led Khosrow into? Maybe it's my own fault for not controlling my tongue... walk faster...\" He took such long strides that Zari had to run over the rocky terrain to keep up with him. By the time they reached the top of the hill, Zari had had enough. The Governor's estate, on the other side of the hill, looked wide awake with all its twinkling lights. Zari, panting hard, collapsed on a rock.\n\n\"Wait a minute,\" she said.\n\nHer pulse was racing, her stomach heaved. She retched and then vomited so violently she thought she would bring up her insides too. Yusef took her by the shoulders and massaged her neck.\n\n\"You're driving me mad!\" he begged. \"Why don't you tell me what's happened, for goodness' sake? What has brought us all the way here to look for the boys?\"\n\n\"You go on,\" Zari replied. \"I'll sit right here. If you don't bring Khosrow back with you, I'll die on this very spot. I'll lay my head on this rock and die. Abol-Ghassem Khan forced us to send Sahar for the Governor's daughter. I guess Khosrow's now gone to steal Sahar back from the Governor's house. That place is surrounded by gendarmes and guards! They've probably killed my son!\" And she sobbed hysterically.\n\nYusef slapped Zari. It was the first time he had ever done such a thing. Zari didn't know it would be the last time also.\n\n\"Shut your mouth!\" he said quietly. \"In my absence you're no better than a stuffed dummy!\"\n\nHe let go of her roughly and headed downhill. He was wild with rage. Zari got up despite herself, wiped her mouth on her skirt and began to run. She stumbled, and got up again. She had to reach him and calm him down. She could see his looming silhouette in the darkness approach the wall of the Governor's estate and stop. Thank God he had stopped. Somehow she managed to reach him with her last ounce of energy. By now she was fighting for breath. She grabbed his hand, but he only peered around, listening for noises.\n\n\"We'll go to the guard-post by the gate,\" he said. \"If we hear the boys' voices we'll go in. God help them if there's so much as a scratch on either one of the boys!\"\n\n\"Promise me you won't make a fuss if they're all right,\" Zari pleaded.\n\nThey knocked at the gatehouse and went in. Yes, the boys were there. A young lieutenant was sitting casually on a desk, the smoke curling up from a cigarette dangling from his lips, in imitation of movie-star officers. When he saw the husband and wife, he asked, \"What can I do for you? I suppose you've lost the way too?\"\n\nOn the desk was a half-eaten tray of food, and in front of it stood Khosrow and Hormoz. Two armed non-commissioned officers\u2014one of whom Zari immediately recognized as the man who had come to take Sahar away\u2014were searching the boys' pockets. Khosrow looked as if he had been crying. When he saw his father, a smile broke across his face, and Zari felt as if she could breathe again.\n\nGholam's friend extracted a few lumps of sugar from Khosrow's pocket. He put them on the table and stood to attention.\n\n\"Sugar-lumps, lieutenant!\" he announced.\n\n\"On what charge have my boys been brought here?\" Yusef demanded angrily.\n\nDisregarding his question, the lieutenant said, \"To be included in the file.\"\n\n\"Sir,\" Zari interrupted as calmly as she could, \"these boys go on scientific expeditions in the afternoons.\" Her eyes took in the rope and blanket on the table and the sack Hormoz was holding, inside which something seemed to be squirming. \"They collect stones and... and...\" she hesitated, unable to guess what was inside the sack. So she said, \"They collect insects, butterflies, field mice. They dry them later. They take a blanket to sit on and rest. Sometimes they take a rope and pretend they're Tarzan... or if they find suitable trees, they make a swing...\"\n\nThe young lieutenant was clearly becoming interested in Zari's face and voice. Zari continued, \"Tonight they were late, so we came to fetch them.\"\n\n\"It's true, sir,\" Hormoz confirmed. \"We've sworn it to you. We'd gone on an expedition, lost our way, and when we saw the lights we came here.\"\n\nThe lieutenant squashed his cigarette butt in the ashtray.\n\n\"Then why did you whistle?\" he inquired.\n\n\"We whistled so some kindly person like yourself could hear us and come to our rescue,\" answered Hormoz.\n\nYusef lost his temper again. \"What possible harm could these two defenceless young boys do with a couple of sugar-lumps in their pockets?\" he shouted.\n\nZari grasped her husband's arm. \"Please don't get angry, my dear,\" she pleaded. \"You can see the boys are perfectly safe and sound. There's just been a misunderstanding which we'll soon clear up.\"\n\n\"They're treating my children like criminals,\" Yusef shouted more angrily than before. \"Do you know why they came here...\"\n\nZari knew that if Yusef told the truth, there would be no end to the matter, and none of them would be allowed to leave. \"My husband has just returned from a journey,\" she interrupted, explaining to the lieutenant, \"he's very tired...\"\n\nThe lieutenant suddenly noticed the sack Hormoz was holding. \"What's in this sack?\" he queried.\n\n\"A snake, sir!\" Hormoz answered coolly.\n\n\"A snake?\" the lieutenant exclaimed.\n\nZari instantly realized that it was probably the snake Haj Mohammad Reza had found in their house. She remembered that the snake's fangs had been pulled.\n\n\"I told you they collect reptiles. This time they found a snake. But it's probably harmless.\"\n\n\"Would you like to see it, sir?\" Hormoz asked. And he emptied the contents of the sack on the floor.\n\nA brightly-spotted snake crawled out. At first it held its head high, looking straight at the lieutenant's shoe. Then it flashed its tongue and slithered under the desk. The lieutenant hastily lifted his feet out of the way.\n\n\"Kill it!\" he cried.\n\nGholam's friend went for the snake with his rifle butt, but it escaped.\n\n\"Threatening the life of an officer on duty with a snake...!\" shouted the lieutenant. But he never finished his sentence. Jumping down from the desk on which he had taken refuge a few seconds ago, he inadvertently stepped on the head of the snake. Meanwhile Gholam's friend was about to attack the snake again when the lieutenant suddenly stood to attention and did a military salute.\n\n\"Good evening, your honour!\" he said.\n\nZari turned to discover Abol-Ghassem Khan in the doorway. \"Sister,\" he chuckled, \"is this where you bring your newly-arrived husband?\"\n\nThe young lieutenant was stammering in confusion. His foot was still on the snake's head, while at the other end the tail wriggled grotesquely.\n\n\"Your honour,\" he said, \"I had no idea the gentleman was your honour's brother. Even though the resemblance of nobility can be detected in every feature... if I have given offence, please forgive me, I apologize...\" And turning to Yusef, he bowed and said, \"Why did you not inform me, sir?\" Indicating the other graded officers he added, \"I shall have these bastards thrown in jail.\" Giving the man closest at hand a slap across the face, he barked, \"Imbecile, you bring the son of the most respected man in town to this sentry post?\"\n\n\"Forgive them this time,\" Abol-Ghassem Khan said with measured coolness and dignity. \"My regards to His Excellency, the Governor. It's too late, otherwise we would have gone to convey our regards in person.\"\n\nYusef, Abol-Ghassem and the boys were climbing back up the hill, joking and chatting. The sons were telling their fathers all about it from the beginning. They took no notice of Zari. She no longer had the strength to follow them uphill so she turned into a side-street which led up to the main road, and walked off alone as quickly as possible. A few Indian soldiers were sitting by the stream along the road, another one urinating at the foot of a tree. When Zari passed him, he turned and flashed his naked body at her, saying, \"Need woman!\"\n\nZari quickened her step. A gendarme and a night-guard turned to look at her as they walked past. Deep down she was hoping that either her son or her husband would follow her, but when she turned into the small road that ran alongside their garden, she saw no one on her trail and felt it was just as well they hadn't even done her that favour.\n\nAs she went into the garden, she was surprised that the others had not arrived yet. The twins were sleeping peacefully under the mosquito net. Zari sank on to her knees by the pool and immersed her face in the water. Then she sat on the edge of the pool and soaked her feet in the surrounding overflow. The water was luke-warm. She placed her hand on the head of the stone figure by the pool. Whenever they needed to use the well for watering the garden, the cistern supply flowed out of that open stone mouth. Hossein Kazerouni, the labourer, would arrive with a little cushion which he placed on the ledge behind the treadwheel, and from morning till dusk, from that cushion-seat he would work the wheel with his feet, filling the water-bucket and bringing it up to the surface. His hands were free, except when the brimming bucket appeared. Then he would detach the bucket and empty it into the little reservoir which led in turn to the cistern. Alone, from morning till dusk, that was all he did. When he went to other houses, he did the same thing. He never even sang, and Zari used to think it was a wonder his mind didn't wither away. In order to keep him entertained, she would send the twins to watch him and talk to him. But how long could they be expected to stand there and watch?\n\nSuddenly Zari thought, \"That's the way I'm spending my whole life! Every day I've sat behind a wheel and made it turn. The wheel of our lives, nurturing my children, my flowers...\"\n\nAmeh Khanom called her from the roof, interrupting her thoughts. \"Did Abol-Ghassem Khan arrive on time?\" she asked.\n\nZari lifted her head and said, \"Ameh Khanom, please come down. I'm not in the mood for arguing with them by myself.\"\n\nThere was loud knocking at the garden gate. Gholam, lantern in hand, dressed in a nightshirt and his usual felt hat, opened the gate. They all came in. But Khosrow followed Gholam straight to the stables and stopped in front of Sahar's make-believe grave. Zari could only see his legs in the light of Gholam's lantern and she stood up despite herself to get a better look at what he was doing. The feet kicked over the flowerpots one by one, and then all of Khosrow could be seen squatting to dig out the stones arranged around the grave. He flung them around the garden, disturbing the birds in the trees. The others came to join Zari and sat on the cane chairs. By this time, Ameh Khanom had come down too. Her head was bare and she was wearing a long white nightdress.\n\n\"Sister, which way did you come?\" Abol-Ghassem Khan asked. \"Halfway up the hill, we realized you weren't with us. We followed you to the street...\" He took off his hat and wiped the sweat from his forehead. \"I suppose there's no whisky to be had in this house? Spare us a bottle of Tavuus Khanom's wine then, will you? As there's no Dutch cheese to be found either, we'll put up with some goat cheese and thyme. I'm not an ungrateful sort, after all!\"\n\nZari didn't move, watching for Khosrow as he approached them by the garden path. His footsteps could be heard on the gravel, but his body was enveloped in darkness. He walked up to his mother and flung the bundle he had in his hand at her feet. It was the sack, the rope and the blanket.\n\n\"Mother, why did you tell me so many lies?\" he shouted. \"Why?\" And turning to his father, he added, \"Father, you ask them why they all got together to fool me? Would they do something like that if you had been here?\"\n\n\"I've decided,\" Yusef sighed, \"that I'm incapable of changing anything. If I can't even influence my own wife...\"\n\n\"We were afraid you might do something rash and endanger your life to try to get Sahar back,\" Ameh told Khosrow, interrupting Yusef, \"which you did... and now don't shout so much, you'll wake the twins.\"\n\nBut Khosrow stubbornly raised his voice louder than before. \"Either the children are sleeping, or the ladies are afraid!\" he shouted. \"Women are either worrying or lying. All they can do is to dig graves, or sit around and cry!\"\n\n\"Sister, how about that wine?\" Abol-Ghassem Khan asked, blinking.\n\nZari looked at him; she looked at all of them. How strange and unfamiliar they all seemed! Abol-Ghassem Khan bit his lip and turned to Khosrow. \"I told you it was my fault, my boy, now don't argue so much with your mother...\" And to Zari he said, \"Sister, give us your wine, I want to drink the boys' health.\"\n\nZari walked off like a robot. She went to the cellar and fetched the wine. Khadijeh followed her with a tray of drinks and snacks. Zari could hear Ameh Khanom telling Hormoz, \"You're the older one, you should've had the sense to tell us. Poor Zari nearly died of fright tonight.\"\n\n\"But if we told you, you would've tried to stop us,\" said Hormoz.\n\n\"If they had seen you climbing that wall, they would have shot you!\"\n\n\"Well, they didn't, and no one shot us,\" said Hormoz. \"Our plan was to have me climb the wall first, then pull Khosrow up by the rope tied around his waist. We wanted to throw the blanket over Sahar's head and bring him out through the back gate. We were going to let the snake loose in the garden as our revenge...\"\n\nAbol-Ghassem Khan poured three glasses of wine. He handed one to Hormoz. \"Cheers!\" he said. \"Drink this stuff from now on and try to enjoy the world! I hope you won't turn out like your uncle who ruins life for himself and everyone around him by taking on a whole nation's burdens. Brother, why aren't you drinking? Lord knows this world isn't worth it; all your pleas for justice, your frustration and your self-destructive attitude. A man of the world like myself is clever enough to have his smuggled whisky always at hand! One must take advantage of these foreigners, you know. Besides, they're having the time of their lives behind your back and a good laugh at your expense. Actually, why don't I break the good news to all of you now? I've finally made it as deputy in parliament and my appointment has just been confirmed! The telegram of approval arrived from Tehran today.\"\n\nAnd he got up and did an absurd little dance of joy.\n\n\"Uncle, you'll probably go to Tehran and take Hormoz with you,\" Khosrow said sadly. \"We had so many plans together...\"\n\n\"Yes, my dear boy,\" Abol-Ghassem Khan replied, \"I'm certainly taking Hormoz. He's very lucky too. Here the two of you have been taken in like so many idiots by that man Fotouhi. The fool's gone to Isfahan to get a permit to start a Communist party here, and he's persuading the seamen down south in Bushehr to join him. Pah!\" Turning to Yusef he added, \"I hear his highness came to you first to try to enlist you, but thank God for once you had the sense to refuse. I don't believe in these political parties one bit. They'd invited me to join that Anglophile Baradaran party too. I didn't refuse, though, I just put them off for the time being.\" Then he chuckled and added, \"Actually, it wouldn't be so bad, would it? One brother flirting with the Russians, and the other with the British. When the going gets tough, one brother could come to the rescue of the other. Still, I guess you're not the kind to help out your own flesh and blood when it's needed...\" He lifted his glass again and said, \"Cheers!\"\n\nThere was a pause while he carefully rolled some meat patties, pickled eggplant and fresh herbs in a piece of bread and gave it to Khosrow. Then he continued, \"I was there when the man reported to the Governor about you and Fotouhi, telling us how well you'd spoken and stood up to them. I said well, don't take my brother here too lightly! It's not for nothing that he has a doctorate in agricultural economics from Manchester or Massagussets or whatever university it is...\" He laughed heartily at his own joke. Then he added, \"Actually, I'm making up these names right now. At the time I didn't mention the name of your university. I don't even remember the name. Anyway, our man said you told them you don't like being a slave\u2014either to an individual or to a group. You'd said you despise party discipline. Even though laziness was probably behind it all, I'm still proud that, for once in your life, you came out with the right thing...\"\n\nYusef shook his head bitterly. \"That person was obviously a bit of a hypocrite, and hadn't understood most of what I said, or didn't repeat it all because you were there...\"\n\n\"On the contrary,\" Abol-Ghassem Khan interrupted, defending the man. \"From the report he gave the Governor, it was clear he had been keeping his eyes and ears open.\"\n\n\"The main thing I said was that it wasn't as easy as they thought,\" explained Yusef. \"I said Marxism or even socialism is a difficult school of thought which requires careful training and education. I told them that adapting those ideals successfully to our way of life, attitudes and social fabric, requires a great deal of maturity, open-mindedness and sacrifice. I said I was afraid they were about to stage a play with inexperienced actors; that because of its novelty the play would draw large crowds for a while, but that soon both actors and audience would tire of it and despair. To achieve something for the people of this country, we need enlightened minds, intellectuals, and no outside interference.\"\n\n\"And what actors these are! Gorbeh Shah Cheraq, Masha Allah Qari, Fotouhi, Seyyid Agha with the long face, the son of Ghavam's wet-nurse... Hah!\"\n\n\"I didn't mean to insult anyone,\" Yusef replied sadly. \"These people are worth ten times the rest of the so-called Actors of our Golden Age...\"\n\nHormoz laughed uproariously. Abol-Ghassem Khan threw him a ferocious look. Hormoz lifted his glass clumsily to his lips, grimacing  as he swallowed.\n\n\"My uncle is right,\" he opined.\n\n\"Who asked you to air your views, you young parasite?\" his father retorted.\n\n\"Brother, let him have his say,\" Ameh Khanom interceded. \"Don't shut him up like this in front of everyone.\"\n\nHormoz stammered, \"This\u2014this very Masha Allah Qari has so far sold two of the houses he inherited in the weavers' quarter, and distributed food among the poor with the money.\"\n\n\"Don't tell so many fibs, boy!\" Abol-Ghassem Khan snapped. \"I've had enough of this nonsense. Let's go now, it's getting late. I was in such a rush, I forgot my night-pass. We'll be lucky if we don't get stopped under the curfew.\" He stood up and told Yusef, \"Do you imagine that the British are just going to sit quietly and watch while others carry on as they like down by the Gulf? You just wait and see how they'll buy off all these upstart Communists in one go. If they can't do that, they'll bribe the big shots and the leaders. Then all the pious, gullible, freedom-loving idiots better start watching out!\"\n\nAfter Abol-Ghassem Khan had left, Yusef turned to Khosrow. \"How many times have you been to Fotouhi's house?\" he asked.\n\n\"Four times.\"\n\n\"Did he give you the idea of stealing the horse?\"\n\n\"No, he said, just like you did tonight, to try and find the solution to the problem on my own. Hormoz said let's do a sit-in and protest. I said no, it's better if we just steal him.\"\n\n\"You should have told your mother where you were going.\"\n\n\"Told my mother?\" Khosrow sniggered. \"I'm not a baby anymore, I'm a man. Mother likes to cover up and stop you from doing things. The first thing Mr Fotouhi taught us was to burn the bridges behind us so there would be no way back. He said we were to memorize those words like a lesson.\"\n\n\"Well, bless my soul!\" Zari exclaimed angrily. \"You've got to have a reason to be burning bridges behind you! What reason do you have? What have you ever had but love and affection from your father and me? Have we neglected your lessons, your schooling, your clothes or your fun for an instant? If Fotouhi is at all sincere, he should look after his pathetic sister at the insane asylum, who's glued to the window, waiting for him to come and take her to some imaginary garden!\"\n\n\"But Mr Fotouhi says when society is reformed, no-one will go mad, and every place will be a garden!\" Khosrow said innocently.\n\n\"I'm certain a Fotouhi-type is just what we need to reform our society!\" Zari snapped back sarcastically.\n\n\"Can't he, father?\" Khosrow turned to Yusef.\n\n\"If Fotouhi and others like him can't,\" Yusef answered, \"at least they've offered our people the opportunity of sharing an important experience.\"\n\n\"I don't understand, father,\" Khosrow said helplessly. \"You're talking above my head again.\" Suddenly he grimaced at his mother and said, \"In any case, Mr Fotouhi doesn't lie, and he defends your rights behind your back!\"\n\n\"If I lied about Sahar,\" Zari said in a calm and motherly tone, \"it was on your uncle's orders. At any rate, I don't want you children to be brought up with fighting and quarrelling around you. I want our home to be peaceful, so...\"\n\nKhosrow finished his mother's sentence, \"So, as Mr Fotouhi says, we can all be blind calves who never see when we turn into cows. Just like...\"\n\n\"That's enough now,\" Yusef stopped him authoritatively.\n\n\"No, let him talk,\" Zari said with bitterness. \"He probably means a cow like me. Now listen here, the two of you, do you really want to hear the truth? You remember the day of the Governor's daughter's wedding? They came and took my emerald earrings as a loan, and never returned them. On the day of the foreigners' party, the Governor's daughter had the nerve to thank me for the present I gave her. Then they started talking about the horse. I'd decided to stand firm and not give in this time, in spite of Abol-Ghassem Khan's insistence. I knew myself that eventually I'd have to stand up to them. But I was afraid. Yes I was afraid of that gendarme who came to get the horse...\"\n\n\"But that stupid idiot was Gholam's friend!\" Khosrow broke in. \"You could've tricked him somehow, you're good at that!\" Turning to his father, he explained, \"He was that same man who followed us half-way up the hill after we left the guard-house and said I could come and ride Sahar in the mornings. He said the little mistress wouldn't mind. He said the poor animal had lost a lot of weight and wouldn't let the girl ride him at first, but now she can take a few turns around the garden. She doesn't dare go outside with him yet... he said he'd taught Sahar to trot. He said Gholam beat him up...\" His lips puckered and for a moment he became the same little boy whose plaything had been snatched away and given to another, not the lad burning with desire for manhood.\n\n\"That night I wanted to tell you about my earrings, Yusef,\" Zari continued, \"but you were already so angry, I didn't want to make things worse... it's always like that, to keep peace in the family...\"\n\n\"I always tell lies,\" Khosrow finished her sentence for her.\n\n\"When I said enough, I meant enough!\" Yusef reprimanded sharply. And he added thoughtfully, \"It's not your mother's fault. It's the way things work in this town; the best school is the British school, the best hospital the missionary hospital, and when a girl wants to learn embroidery, it has to be on a Singer sewing machine with Singer for a salesman. The teachers who've trained your mother have always tried to steer her away from reality, filling her instead with some etiquette and coquetry and embroidery. She can only talk about peace and quiet...\" And suddenly turning on Zari he shouted, \"Woman, what use is this peace and quiet when it's based on deception? Why shouldn't you have the courage to stand up to them and say those earrings are a wedding present from my husband, a keepsake from his late mother? After all, the poor woman died in poverty but she was still thinking of the bride her son would choose... How could you have given them up so easily? It's not their value that matters. It's the memory and the love behind them.\" He paused for breath. \"Woman, think a little bit. When you become too soft, everyone will bend you.\"\n\nAmeh Khanom who had been silent for a long time, decided she had had enough. \"What's all this about?\" she said. \"Why are father and son taking it out on this poor soul? Giving away the horse was not at all her fault. I was a witness. I even told her to give it away. As for the earrings, when I first heard the story from Ezzat-ud-Dowleh, I was very upset too, but after I thought it over, I decided she couldn't have refused them. What can you do when there are people who govern your possessions and your life, not just your town? Now do you want to know the truth, brother? She is soft, she gives bribes, so they leave you alone. And that's enough for one night. Eat your dinner and go to bed. Tomorrow morning it will all be water under the bridge. As for me I'm going to bed.\" With that she got up and left.\n\n\"I'm going to show you what I can do,\" Khosrow said, standing up. \"I'm not my father's son if I don't get Sahar out of their clutches. First I'll write a letter to the Governor himself and if he doesn't answer, I'll go to see him. My father and Mr Fotouhi are right. I have to solve my problem myself. If the Governor refuses to see me, I'll do my best not to get upset. No one's ever going to see me cry again. Mother, when they caught us, I was really crying for your sake because I knew you'd be worried about our being late. I hated crying in front of Hormoz, in front of the officers, but I couldn't help it because I know how afraid you are about me or father... Comrade Fotouhi...\"\n\n\"Yes dear,\" Zari said, \"according to you and your father and your teacher, I'm a coward, I'm helpless, I'm soft. I'm always afraid something may happen to one of you... I couldn't bear it. But when I was a young girl I too had a lot of courage...\" And turning to Yusef she asked, \"Wasn't it a mark of courage to walk off with you that day in the middle of a street riot... you, a total stranger... which girl would've...\" she bit her lip and tried to change the subject. \"But you're right about the rest. Our English headmistress constantly harassed us about manners and how to live. Singer was always doing us a favour by teaching us to sew, and Khanom Hakim had us convinced that our cures and medicines lay in her hands alone. I knew in my heart there was more to it than they were willing to tell. Something was wrong somewhere. I knew all of us, all the time, were losing something... but I didn't know what it was...\"\n\n\"And that was why I married you,\" said Yusef. \"Why have you changed so much?\"\n\n\"I've already told you; must I repeat it a hundred times? You're too outspoken and deep down I know it's dangerous to say the things you do. If I wanted to stand firm and put my foot down, I'd have to do it right here with you first, and then what kind of battle of wills would we have at home? Shall I tell you one more thing? You are the one who took my courage away from me... I've obeyed you for so long that subservience has become a habit with me.\"\n\n\"Me?\" Yusef shouted. \"Stand up against me? No matter how fierce I am outside this house, you know well that once I'm within these four walls I'm as meek as a lamb before you! I think your courage has been all show. Prompted by pure, unrefined instinct.\"\n\nZari thought silently that if she carried on any longer, they would have a real quarrel on their hands. She hesitated and then said, \"Who knows, maybe I was a coward from the start and I didn't know it. Time and again I stood up to that headmistress of mine without stopping to think whether I was committing an act of courage or rebellion. That day in Ramadan when she forced Mehri to break her fast... all the other girls abandoned the poor soul out of fear, but I stayed with her. I don't know, maybe in those days I had nothing to lose... and now...\" Without knowing quite what happened she lost all her patience and composure. She got up from her chair and slapping her belly hard, cried, \"I hope this one in my stomach will miscarry tonight... I've gone close to death and back for your sakes. Khanom Hakim has carved up my insides... etched a map on my belly and here I am on trial for courage!\"\n\nShe collapsed on the chair and burst out sobbing. It felt as though nowhere in the whole world was there a person as lonely and as tired as she. Yusef went to her and clasped her head in his arms. He kissed her hair and wiped her tears away with his fingers. He lifted her chin and looked her in the eye, fighting back his own tears.\n\n\"Don't cry, my love,\" he said. \"Why didn't you tell us all this earlier? I was completely taken by surprise.\"\n\n\"Today I damn well wanted to get rid of this one,\" she moaned, unable to hold back the tears. \"Wasn't I brave to keep it? When you bring a child into this world with such agony as I go through, you can't bear to lose him so easily. Every day I... I turn the wheels in this household to nurture you, my precious flowers. I can't bear to see people trample you. Like Hossein Kazerouni I don't do anything with my hands for myself... I... I have no experience, I don't know much of the world...\"\n\n\"My love,\" Yusef smiled, \"instead of going to the mental asylum and getting tired and nervous, you should go to the new Anglo-Iranian Council here and teach Essential English II! Can you believe Singer sent me this message via McMahon?\"\n\n\"You're making fun of me!\" said Zari from between her tears.\n\n\"You know I can't bear to see you cry,\" Yusef said gently. \"I wanted to make you laugh with the suggestion... But my love, if only you'd told me the truth right away, we wouldn't have gone on at you like this. You said you went to Khanom Massihadem's office, but you quickly covered up the real reason behind it. Why did you keep it a secret from me? Now I feel guilty about the way I jumped at you.\" Khosrow had sat down at his mother's feet. He was holding on to her leg and listening silently to her words.\n\nZari wiped away her tears. \"You had just got home from your journey,\" she said. \"You were tired and unhappy. I didn't want to make you feel even worse.\" She asked wearily and at a loss, \"What can I do to please you two? What can I do to become brave, as you say?\"\n\n\"I could teach you,\" Yusef said with a laugh. \"Your first lesson in bravery is this: whenever you're afraid to do something, if you feel you're in the right, then do it even if you're frightened. My sweet little kitten!\"\n\n\"For one thing I'm a person, not a sweet little kitten,\" Zari said pensively. \"What's more, you give a first lesson to someone who has to start from scratch.\"\n\nIn bed under the mosquito net, despite Yusef's cool hand caressing her warm abdomen, despite his kisses, Zari seemed to have forgotten all sexual response. Instead, she kept thinking about her past, and wondering whether she had always been a coward or whether she had become one. Was Yusef really to blame? For one instant she even concluded that marriage was wrong at its very basis. Why should a man be tied for a lifetime to a woman and half a dozen children... or conversely, for a woman to be so dependent emotionally and otherwise on one man and his children that she couldn't breathe freely for herself? It had to be wrong. Yet she knew that all the joys of her own life stemmed from these very attachments.\n\nShe couldn't sleep for remembering those carefree days of her girlhood. The memory of that day in Ramadan when the headmistress broke Mehri's fast came back to her as if it were yesterday.\n\nThat year Zari and Mehri were taking their sixth-grade exams. Four months before the final examinations a letter from the Ministry of Education arrived at the school stipulating that sixth-graders must be taught the Quran and religious laws. Zari realized that her mother's petitioning had finally worked. Because she couldn't afford a private teacher to instruct her daughter in religious matters, she had been pressing to have these taught as part of the school curriculum. Letter after letter and notice upon notice from the Ministry arrived on the headmistress's desk, upsetting her considerably. But Zari knew that behind this pressure lay her mother's insistence...\n\nThose days every lesson in 'Ethics' turned into a nagging session about the Ministry of Education. The headmistress would complain that the Ministry had agreed from the start to maintain a policy of non-interference. She went on about the impossibility of suddenly producing a suitable teacher in the middle of the scholastic year. She nagged about finding extra hours to fit in these lessons... and so on. She would say, \"Why don't you girls find an old mullah-baji somewhere on Sundays when the school is closed and learn your Quran and religious laws from her... or better still, ask your people to teach you at home?\" She would use the idioms correctly, since this one knew Persian well.\n\nMehri, whose uncle was the head of the Sufi dervishes, was well versed in both the Quran and in religious law. Unbeknownst to the headmistress, she agreed to give her classmates lessons in these subjects when they came back to school after the lunch-break. Zari struggled to pronounce the Arabic word \"Fassayakafikohomo'allah\" correctly, but she didn't always succeed. Still, Mehri was patient, being a year or two older than the rest. And then came the month of Ramadan. She had just been teaching them the Ayat Prayer for use in times of natural calamity, when that memorable incident __ took place. It seemed like yesterday.\n\nFasting was forbidden in the school, but Mehri was doing it anyway. When the headmistress found out, she stormed into the classroom and demanded that Mehri end her fast there and then by eating something. Mehri refused. The headmistress gave her a shove which sent her sprawling all over the classroom floor. Then she kneeled by her and holding the girl's head with one hand, roughly opened her mouth and attempted to pour some water down her throat. Mehri bit the woman's hand, at which the headmistress shouted at her and called her a pathetic wretch. Mehri sat up. \"The dirty hand of an unbeliever in my mouth was enough to break my fast,\" she said. \"Give me the water and I'll drink it to the last drop. The sin be on your head.\"\n\nThe headmistress slapped Mehri across the face, and again sent the girl on to the classroom floor. She then left Mehri and turned to the class to rebuke them. But the other girls were whispering anxiously and no-one paid attention. Even their Indian teacher was just standing, staring round-eyed at the scene.\n\n\"In this school,\" the headmistress shouted, \"there is no room for superstition. Leave fasting and religious mourning to your aunties and grannies! Ask your nursemaids about religious rules on menstruation and childbirth. Fasting weakens the body. Why did I buy parallel bars, a vaulting horse, and a basket-ball net? To strengthen your bodies, that's why! Now you want to ruin all my efforts by fasting? You don't deserve any of it!\" Then she barked again at the top of her voice, \"The bell has rung\u2014why don't you leave the classroom? Mehri's punishment is to stay right here on the floor till this evening. Come along now, girls! No-one is allowed to remain with her.\"\n\nThe headmistress marched out, and the Indian teacher, tossing her braid over her shoulder, followed her. The other girls filed out too. But Zari felt she could not leave. She bent over Mehri and gave her a hand to stand up. She dusted her off and sat her on the teacher's chair. Both of them searched for a handkerchief to wipe away Mehri's tears but neither of them had one. So Zari dried her friend's face with her fingers, and kissed her, saying, \"I don't think your fast is broken. You were forced to drink the water.\"\n\n\"There were only two or three hours left to the end of the fasting day,\" Mehri cried, \"and I had managed to fast for twelve days. I'd even fasted two extra days. This year I was determined to fast all thirty days of the holy month, because by next year I'll have my period and I'll never be so lucky again.\"\n\n\"Oh it's a long way to next year! Besides, you said yourself that a woman in her menopause doesn't get periods anymore. When you get to that age, you can fast all thirty days again.\"\n\nMehri laughed at that, and Zari was pleased to have made her laugh.\n\n\"I know who's been telling tales\u2014it must have been Taji. That stupid girl has turned Christian. I know my saviour Imam Ali will strike at her and she'll fail her exams! Tonight the dervishes are holding a chanting session for the Imam Ali and I hope my uncle curses her.\"\n\nThat night, they went home together. As they passed the Sufi monastery, they heard the rhythmic chanting of the dervishes, \"Ya Hu. Ya Haq. Ya Ali!\", as it drifted through the open doors of the house of Imam Ali. \n\n# _12_\n\nAll the quarrelling, reconciliations, and anxieties of the past days faded into insignificance when that very Friday morning Sahar walked back to the house on his own feet.\n\nIt all began like this.\n\nThey were sitting on the verandah at the back of the house which was protected from the morning sunlight, and which looked out on the hill Zari and Yusef had climbed with such fear and anxiety the night before.\n\nZari was using the breakfast table as an ironing board. The rugged hill lay bathed in sunlight, so still that it seemed hardly touched by the tread of human feet. Khosrow was sitting across from Zari, and had put a pen, paper and several books on the table in front of him. He was leafing through a book called _Principles_ _of_ _Letter-Writing_ and reading out loud: \"Write a letter to the head of an office and ask for a job. Write a letter to your uncle and ask him to... Write a letter to your friend and invite him for the Mab'ath holidays. With fondest regards and compliments... what joy to receive your latest missive... with reference to your letter of...\" He put the book down on the table and said, \"As Mr Fotouhi says, nothing more than begging and flattery!\" then took another and started to flip through its pages.\n\nThough still early in the day, it felt hot and there was no breeze to relieve the heat. Sweat trickled down Zari's spine, and she longed for a cool, refreshing drink like willow-water or betony, or perhaps a piece of crushed ice to crunch between her teeth. She remembered how in each of her pregnancies Ameh Khanom had gone to great lengths to provide her with whatever she craved. From Hassan Agha the grocer she would order Indian magnesia, which was crunchy and white as snow, and reputed to be good for the baby's bone structure. Other days it would be lamb's rumen, which Ameh would buy fresh and clean out herself, cooking it with nutmeg and making Zari take it because it tightened the belly. If a single raisin proved too sweet for Zari, Ameh would ply her with tamarind sherbet, and if one sour grape was too acid, she made hot syrups for her. But ever since Ameh Khanom's decision to follow her mother's footsteps to Karbala, she had become listless and depressed. She had no patience for anyone, not even the children. It was very noticeable, but Zari had decided not to say anything.\n\nKhosrow put down his book. \"What rubbish!\" he exclaimed. \"There's not a word in here about how you should ask for your rights!\" He took another book and leafed through it. \"I think I've found it...\" he said, \"what good sentences!\" He raised his head and asked Yusef who was facing the hill in his armchair, reading a book, \"Father, what does this mean? 'His deep-toned voice resembled that of a violoncello.'\"\n\n\"It means like the mooing of a cow,\" Yusef answered, without raising his eyes from his book, \"it won't do for Sahar. Listen, why don't you just write what comes into your mind?\"\n\nAmeh's voice could be heard ordering Mina to put down her coins, which were unclean from being passed around hundreds of hands. Ameh was sitting on a rug with her back to the hill, leaning against the verandah railings, sewing gold dinars into the lining of her coat. This had been her sole activity over the past few days and now she had started on her second coat.\n\nGholam came out to the verandah. \"Khanom, are Kolu's clothes ready?\" he inquired.\n\n\"They'll be ready in a minute.\"\n\n\"I know it's not my place to say this,\" Gholam commented, \"but why bother to iron them? Last night he only dreamt of cows and sheep. He kept waking up with a start to look for his kid goat. He kept me awake all night with his sighing and moaning. This morning he sobbed for an hour, asking for his mother, his sister, his brother... I don't see how he can last here.\"\n\nKhosrow chuckled as he laboured with his letter, and Marjan tried to build towers with Ameh's gold dinars which Mina would immediately scatter with a fling of her hand. As always, the initiative came from Mina who behaved as if she knew she had a headstart on her sister, having arrived fifteen minutes earlier into the world.\n\n\"Run along now!\" Ameh Khanom shouted at the twins. \"Money isn't for playing! Call Kolu and tell him to come here. Gholam, take the girls to the stables.\"\n\nThe twins pretended to cry and crawled under the table.\n\n\"Why don't you put on the chadors your aunt made for you and show your father,\" Zari said.\n\nMina emerged from under the table. \"Auntie, can we have a prayer-stone so we can say our prayers?\" Whenever Ameh stood up to say her prayers, they would also put on their chadors and bend or stand in imitation of her. When Ameh pronounced the 'amen' in Arabic, they would quickly put their foreheads to the ground to ask God for what they wanted. God alone knew what these little souls could be asking of Him... They would try hard to pronounce the Arabic 'Wala-z'alin' but they couldn't, so they would turn to Ameh and say, \"Now you say it.\"\n\nZari finished ironing Khosrow's old trousers and shirt which she had let out for Kolu and handed them to Gholam along with some socks, a vest and underpants. \"Put anti-flea powder on all of these,\" she said. \"Buy him a pair of givehs too.\" And she sat down on a chair. She was feeling parched, perhaps from all that ironing in the heat.\n\n\"The powder is finished,\" Gholam told her. \"I mixed the whole lot with water in the ewer to splash around the stables. They're infested with lice.\"\n\n\"Send Kolu here,\" Yusef ordered.\n\n\"Let him have his bath first,\" said Zari.\n\n\"Agha, he won't come,\" Gholam complained. \"This morning he was like a wild animal. He wanted to run off into the hills. He kept saying he was going to walk all the way back to his mother.\"\n\nAfter Gholam had gone, Ameh Khanom said, \"Brother, you can't keep the boy here. He's like that wild fawn we finally had to get rid of... still, it's none of my business. I'm only a guest in this house for a few more days.\"\n\nAt heart Zari agreed with her sister-in-law about Kolu. When she had seen him the day before, his eyes had looked to her just like those of the wild fawn\u2014large and outlined, with a shocked expression. Even though he had smiled at his mistress, deep down in his eyes lurked the fearfulness of a trapped animal.\n\n\"It really is too soon to take him away from his home,\" Zari observed. \"It's no use being kind to him. We're only making him unhappy, and his relatives angry...\"\n\n\"Once he's lived here comfortably for a few days, he'll feel at home with his new surroundings and he won't even mention his village anymore,\" Yusef said impatiently. \"Next year I'll send him to school.\"\n\nKhosrow stopped writing and giggled. \"Not unless you send him there with his hands and feet tied inside a sack,\" he said, \"he's too wild. And he's too old, anyway; they may not accept him.\"\n\n\"In a sack?\" Yusef asked absently, folding his newspaper.\n\n\"Yes, father. I saw Davoud Khan's son when they brought him to school from the tribe. They'd brought him straight there. I think I was in the second grade. At break-time we saw this tribal man with a big moustache arriving at school on a mule. He was wearing a felt hat and a slit tunic with a shawl wrapped around his waist. On his saddle was a big canvas sack tied up carefully at the top, with something wriggling inside it. The man got down from his mule and tied the bridle to the same tree I always use when I take Sahar to school. All this time he was holding the sack firmly with his other hand. He was being very careful with it. Then he hoisted the sack and brought it into the schoolyard. When he put it down and opened it, the Khan's son jumped out, wearing nothing but long black trousers! He did a few somersaults\u2014I don't know what for. Then he started to run all around the schoolyard. As if anyone could catch him!\"\n\nZari picked up her ironing and went to the pantry. She checked the cupboards. They'd completely run out of flower essence. In the kitchen she found Khadijeh, frying egg-plants on the stove. She was working stripped to __ the waist in the furnace-like heat, exposing sagging breasts and hairy armpits, while below the waist she wore her loose, flowery-patterned trousers. On seeing her mistress, Khadijeh grabbed her veil to cover herself.\n\nZari decided to pay a visit to her neighbours, the distillers. Maybe they could supply her with some essence. She went out the garden gate with her purse and two large pitchers. The neighbours' garden door was open, so she went in without knocking. There wasn't the usual pile of flowers on the paving in the middle of the garden, and the old distiller himself was nowhere in sight.\n\n\"Is anyone there?\" she shouted.\n\nShe approached the house, knowing that the distillery store-rooms were in the basement. She had an uncontrollable urge to fill a china bowl with betony extract, add syrup to it and mix in some crushed ice... she would stir the ice in her drink with her fingers and with a ladle that had a carved handle... ah, how refreshing that would be! Even if the distillers weren't there, she could go to the store-rooms, fill her pitchers and leave the money somewhere in sight.\n\nInside the house, she called out again, \"Anybody there?\"\n\nSuddenly the head of the old distiller appeared behind one of the basement windows. He peered at her through the ornate stone lattice. Then he came out to greet her, dressed only in his drawers.\n\n\"Khanom, why go to the trouble of coming here yourself? You could've sent one of the servants...\" And then he added, \"Please come to the store-rooms. Take whatever you want. We were waiting for the last picking of eglantine which hasn't arrived. The flowers will wilt. They say the whole town's been blocked because some horse has taken off with the Governor's daughter. They're not even letting goods deliveries come through.\"\n\nZari put the pitchers down.\n\n\"I'll be going down to the garden door,\" said the distiller. \"I've sent my sons to fetch the load and I want to see if they're here yet. I just know those flowers are going to wither. This town is turning into bedlam. Why does the girl have to go about riding a horse and getting herself into trouble? How can you make those fools understand that flowers don't have the patience of human beings? Especially eglantine. They have to be picked at dawn and piled inside the store-rooms by early morning. Flowers can't be kept waiting in this blazing sun!\"\n\nZari didn't know whether to be glad or upset. She felt for the child's mother: she was, after all, a mother herself. She knew Sahar was a noble horse and wouldn't throw his rider. But how terrified the girl must be! And how anxious the mother!\n\nShe took the pitchers and went to the basement. An intoxicating fragrance permeated the cool air of the cellars. The covers of the stone vats made especially for boiling flowers had been removed and leaned against the walls. The bamboo pipes leading from the vats to the tank were dry and, unlike the last time she had brought the twins to watch, were not dripping with thin streams of fragrant essences. Of the two tanks, one was full and the other half-full with rose-water. Flasks of rose-water were stacked neatly around the store-room. She opened a small door and went to an adjoining cellar. She dipped her pitcher in the first tank there and filled it with betony extract. How she longed to lie down right there on the cool moist earth of the store-rooms, next to the sweet aroma of those tanks!\n\nOn her return home, the first thing she noticed when she went to the verandah was the noise in the distance. The others were seemingly oblivious to it, Khosrow still writing his letter and Yusef leafing through his book, chuckling. The noises, however, seemed to be coming closer, a mixture of the sounds of a crowd and the hum of car engines. Zari glanced towards the hill. Not a soul in sight.\n\n\"Where are the twins?\" Yusef asked.\n\nNo-one answered.\n\nShe could see two cars now, one following the other at an angle to the slope. A voice rose, saying, \"He's heading for the hill!\" Several people started in the same direction.\n\n\"There! I've finished my letter,\" said Khosrow. \"Father, will you listen while I read it to you?\"\n\nYusef shut his book, got up from the armchair and looked out. \"What on earth is going on over there?\" he asked.\n\nKhosrow stood up too and went to the edge of the verandah. \"Look how many people there are at the foot of the hill!\" he exclaimed. \"Four... five cars!\"\n\nA voice in the distance shouted, \"Did you see? Right there!\" And another voice commanded, \"Don't shoot, you idiot!\" Someone screamed. The crowd at the foot of the hill was growing by the minute. A policeman and two gendarmes arrived. Two more cars passed by the slope. The first car was sounding its horn like an emergency siren, and raising a great trail of dust and gravel as it drove forward.\n\n\"Is there a war, father?\" Khosrow asked. Before Yusef could answer, another voice shouted, \"He's going up the hill!\" Other voices were lost in the din of the crowd, and the revving of car engines.\n\n\"I think it's to do with Sahar,\" Zari said. \"The distiller next door was saying that a horse had taken off with the Governor's daughter.\"\n\nYusef clapped a hand to his stomach and laughed heartily. \"What a war!\" he said, catching his breath. \"All this to catch a colt! There he is! Look, it's Sahar all right! He's standing at the summit. She'll be lucky if he doesn't throw her!\"\n\nAmeh Khanom, still sitting with her back to the hill, didn't even turn round. She was struggling with a thread and needle. \"It's just like threading a needle,\" she observed. \"If you aim the thread exactly at the needle's eye and your vision is good, then you get it right the first time. But if your eyes are like mine, on the blind side, you have to keep wetting the thread in your mouth, and guessing at the eye. The thread goes back and forth so many times until finally, by accident, it goes through the hole. Now Khosrow, your horse has come to you on his own feet by accident too. Go out there and let's see how well you thread your own needle.\"\n\nYusef put a hand on his son's shoulder. \"Your aunt is right, son,\" he said. \"Go ahead.\"\n\nKhosrow jumped down from the verandah and ran off. Zari, understanding Ameh's hint, knelt down and threaded her needle for her. \"But it can't always be helped, you know,\" Ameh commented. \"In life you're not always allowed to follow the right path, so only after a great many battles and a lot of failures do you finally make up for your mistakes.\"\n\n\"Sister, ever since you've decided to leave for the Holy City, you've become quite a philosopher,\" Yusef observed.\n\n\"Just a wise old owl,\" Ameh sighed.\n\nAt that moment Zari noticed a car struggling noisily up the hill. Sahar, at the summit, neighed and shifted nervously from side to side. The girl grabbed at his golden mane, shrieking above the noise of the crowd. The mare and the chestnut horse neighed in response from the stables.\n\n\"I knew the first day they tried to ride him outside the four walls of their estate, he'd head straight back home,\" Yusef said.\n\n\"A credit to that noble beast,\" said Ameh, still busy with her sewing.\n\nSuddenly a long black limousine drew up. The policeman saluted and the gendarmes presented arms. The driver jumped out to open the door, but the man in the back seat opened it himself and stepped out. Zari recognized the Governor. Then another limousine drew up behind the first. Singer stepped out, followed by two Indian soldiers. He and the Governor shook hands.\n\nThe crowd kept parting and re-assembling to allow for the random movement of the cars. The car which had driven up on to the hill backed down noiselessly as if afraid of causing Sahar to shy again.\n\nZari couldn't see her son as she strained to pick him out in the crowd. This was the time to act, so where was he? By now, the army commander's car had drawn up as well. Out stepped the commander and three more officers, slamming the door loudly. The car moved on, veering closely past the other two limousines. The army commander took in the scene around him. The officers, with swords dangling at their sides, headed straight for the hill. The Indian soldiers saluted and Singer started to do the same, but the army commander prevented him as if to emphasize their warm relations. Then the commander turned and saluted the Governor.\n\nYusef had meanwhile fetched his binoculars, and he and Zari took turns surveying the scene on the hill. Sahar neighed several times. The girl was clutching at his mane, lying full-length along his neck. Sahar slipped several times on the rocky terrain, veering first to the left and then to the right. The army commander, holding a short, thick baton in his hand, left the Governor and Singer behind, and headed uphill.\n\n\"Gilly dear,\" he shouted at her, \"take your feet out of the stirrups, sit sideways and try to jump down.\"\n\n\"I'm scared! I'm scared!\" came Gilan Taj's voice.\n\n\"What an ass!\" murmured Yusef.\n\nZari couldn't tell whether he meant the army commander or 'Gilly dear'.\n\nSahar seemed to notice the gendarmes all of a sudden. One of them uncoiled the rope he was carrying and threw the noose at him, in an attempt to lasso the horse and its rider. Sahar backed off, the girl screamed, and both disappeared down the other side of the ridge. The crowd surged towards the hill. The drivers of those cars who had room to manoeuvre, jumped behind their steering wheels, revved their engines and drove away to the other side.\n\n\"Get back, you half-wits!\" yelled the army commander. \"You've frightened the horse. He was standing perfectly calmly...\"\n\n\"If there was an ounce of brain in their heads,\" Yusef said, \"they would all go away and let Sahar bring the girl safe and sound back here.\"\n\nSuddenly Zari caught sight of Khosrow clambering up the hill. Her stomach began to churn. \"Amen Khanom, pray for him, pray for him!\" She turned to Ameh and begged her. Ameh looked towards the hill, and her lips moved in prayer: \"God's protection upon him; He is the most merciful of the merciful.\"\n\nKhosrow had nearly reached the top. He put two fingers in his mouth and let out the long whistle he always used for Sahar. Whenever he heard that sound, no matter where he was in the garden, Sahar would come to Khosrow and sniff at his sleeves. The crowd fell silent. Zari looked at her husband. Yusef's face was radiant with smiles and his green eyes were shining like two stars. Again Khosrow whistled. Sahar's head appeared in sight, looking to left and right.\n\n\"Here I am, Sahar!\" Khosrow shouted. \"Don't be scared,\" he reassured the girl, \"he won't throw you.\" The crowd was so silent, it was as if there had never been an uproar. Sahar neighed and slowly approached Khosrow. When he reached the boy, he lowered his head, as tamely as a household pet. Zari knew he would be sniffing at Khosrow's sleeves and pockets, taking in the familiar odour. She knew how closely the animal's existence was tied to familiar smells around him. Khosrow hugged Sahar's head, kissed him and patted his mane. Then he held his hand to Sahar's mouth, and Zari knew Khosrow had not forgotten the sugar-lumps.\n\nKhosrow helped the girl dismount. She was wearing riding boots and jodhpurs. As she touched the ground, she collapsed. Khosrow held the bridle as he bent over to tell the girl something. She sat up and screamed. Khosrow stood in front of the girl and was obviously talking to her. Finally he gave her a hand and lifted her up and the three of them descended the hill. Sahar had brought his ears forward, as if to listen to Khosrow's words. Near the foot of the hill, the girl left her companions and threw herself into the arms of her father, who had come forward to meet her. As the boy and his horse reached the crowd, people stood aside to make way for them. Then Khosrow mounted and galloped back home. \n\n# _13_\n\nThe mare was ready, saddled and bridled. Yusef was about to mount when Kolu dashed out of the stables and threw himself at his feet, begging to be taken back to the village. He was so altered after a haircut, a bath and some second-hand clothes! Or had he got thinner in the past few days? His dark eyes seemed sunken in his haggard face. Yusef tried to reason with him. \"Listen son,\" he said, \"you'll be staying in town, going to school, really be making something of yourself. You can learn a thousand things from Khosrow.\"\n\nBut Kolu was deaf to the master's words, uncomprehending, pleading only to be taken back to his mother and brother. Finally Yusef lost patience and boxed his ears. \"I'm not going to your village just now! I'm going to Zarqan.\" And he mounted. Kolu burst into tears and threw himself into the bushes, kicking and howling like a trapped animal. When Yusef bent over from the saddle to kiss Zari, he noticed tears in her eyes.\n\n\"Would you like me to take him back?\" he asked.\n\n\"No, I expect he'll settle down eventually,\" Zari answered. \"He can't know what's good for him, can he? Just remember, this time you're the one who's being charitable! What's the use of helping this one out and adopting him when there are thousands of other peasant children like him?\"\n\nOver Yusef's departing footsteps, Ameh Khanom splashed the customary water and orange blossom leaves from the crystal bowl she was holding, before going off to recite the An'am Surah for his protection and blowing it towards him with a symbolic gesture. What a curious creature a human being is! How easily a ray of hope or a happy event can renew his will to live! But when all around is oppression and despair, a person feels no more than a used-up shell, abandoned by the wayside. Ever since Sahar's return, Ameh had connected her life with the family's again and had stopped repeating that nothing concerned her anymore.\n\nZari went over to Kolu who had rolled away as far as the middle of the garden path. She knelt beside him and stroked his hair.\n\n\"Now look how you've dirtied your new clothes...\" she scolded him gently.\n\nKolu sat up and tore his shirt off angrily, screwing it up and throwing it in front of the master's wife.\n\n\"Listen,\" said Zari, \"if you're a good boy, I'll ask Khosrow to give you lessons from tomorrow. When you can read and write, I'll send you to your village to see your mother and show her you can read their letters and write letters for her, too.\"\n\nKolu had calmed down. Either he was paying attention or he had tired himself out. \"But no one writes my mama letters,\" he said.\n\n\"Get up, child,\" Zari urged, patting his sweaty back. \"Go and wash your hands and face. Shake the dust off your clothes and put them back on.\"\n\nAs Kolu didn't budge, she asked, \"What do you want me to buy you?\"\n\nKolu burst out crying again and sobbed, \"Send me home, mistress! I beg you on your children's lives, send me back to my mother and brother. My brother's sitting right now by the stream playing his flute. My mother's putting oil in the lamp. I'd laid some traps to catch a few goldfinches, and now they must be trapped and there's no one to get them out... I put my slingshot on the shelf\u2014my sister Massoumeh will take it and lose it. If I was there now, I'd have pinched a few walnuts and I'd be cracking them and eating them.\"\n\n\"Maybe the goldfinches will chirp a lot and someone will hear them and let them loose. I'll send someone to buy you some walnuts: and you can sit right here and crack them. I'll even get you some elastic and you can make yourself a slingshot.\"\n\n\"You make slingshots with leather cord, not with elastic,\" Kolu said with an unhappy smile.\n\n\"All right then, I'll send out for some leather.\"\n\nKolu's lips quivered again. \"No-one will go to the goldfinches. The traps are far away from the village.\"\n\nZari tried to distract him. \"Look,\" she began, \"the master is going to the village. Maybe he'll pass by the place you set your traps. He'll hear the chirping. He'll get down from his horse and take the goldfinches out of the traps and set them free.\"\n\n\"But the master isn't going to our village.\"\n\nKhadijeh's voice came from the verandah. \"Khanom!\" she called out. \"Telephone!\"\n\nZari stood up. \"Who is it?\"\n\n\"Khanom Ezzat-ud-Dowleh.\"\n\nWhat could she be wanting, Zari wondered. Probably the woman wants to say what a huge favour she did us, and that she was the one who sent the horse back! When Zari came to the parlour, she saw Khosrow sitting idly by the window, staring out at the garden.\n\n\"For heaven's sake, Khosrow,\" she said, \"go and play a bit with that poor orphan boy...\"\n\nHe didn't move. \"Mother, don't even think about my giving Kolu lessons,\" he said.\n\nZari went to the telephone. It appeared that the very minute Ezzat-ud-Dowleh had set foot in her own home after their luncheon together, she had come down with a bout of her usual leg pains, confining her to the house. She had heard about her sister's intended pilgrimage, and she longed to see all of them\u2014including the twins\u2014in the near future. They owed her a visit after all. In fact, fresh water was being brought for her private baths the next day, and Ezzat-ud-Dowleh wondered if they would honour her with their company for a bath and luncheon on Wednesday. Zari's many excuses and protests were firmly turned down, and the date was set.\n\nOn Tuesday morning, Kolu went down with a fever. Zari darkened the pantry using reed blinds, and set up a bed in there so she could have him close at hand. Kolu would open his eyes wide and hold his fingers in front of them, straining to see. You could tell he was trying to focus, but wasn't able to. Khosrow, Gholam and even Ameh Khanom were of the opinion that he should be sent to hospital. There was little doubt he had typhus, and that put them all at risk. But which hospital would take him? Even the town's best doctors were down with typhus, and rumour had it that Khanom Massihadem and the three head-nurses at the Nemazee Hospital were in a grave condition. Khadijeh had heard from Sakineh, the woman who came to bake bread for them, that Dr Abdullah Khan, the town's most skilled physician, refused to leave Khanom Massihadem's  bedside. He would soak two large white towels in ice-cold water, wring them out, and continuously cover the patient's naked body with them. Sakineh, who had gone to visit Khanom Massihadem, had thought that she was already dead and they had spread a shroud on her. Before anyone could stop her, Sakineh was beating her head and searching for mud in the garden to smear over her hair in mourning. When they finally calmed her down and explained everything to her, she had rushed to the shrine of Seyyid Mir Mohammad to light ten candles in thanksgiving.\n\nNor was Sakineh the only one so concerned with Khanom Massihadem's fate. Large numbers of men and women had covered their heads with the Quran at Mehri's Rowzeh as a mark of urgent prayer for the sick woman, and had recited the Amman Yujib prayer for her deliverance. Akbar Khordel had circumambulated her bed with a sheep which he then slaughtered for her sake and distributed the flesh amongst the poor. The skin he had taken to the well-known mountain dervish, Baba Kouhi, so the old man would pray for her too.\n\nAmeh Khanom made Zari call Khanom Hakim for a hospital bed. But Khanom Hakim merely said, \"Unfortunately the beds of the Missionary Hospital be for the foreign officers and soldiers only and all the beds be full and even there be no place in the corridors.\"\n\nZari hung up without saying goodbye. \"Obviously the hospital was built for their own needs, not for the townspeople,\" she told Ameh who was waiting to hear what the doctor would say.\n\nThey put their heads together and began their nursing. They gave him manna of Hedysarum, and they wrung towels in cold water and wrapped them around him. They plied him with watermelon juice which he accepted eagerly, being parched from the fever. They moistened fleawort, sewed it up in some thin cloth, and kept it immersed in cold water, to be dabbed from time to time on his blistered lips. Ameh Khanom resorted to the traditional rite of placing some item blessed at the Shah Cheraq Shrine next to the patient. In this case, she cut two hand-lengths of braided white cord from the shrine, tied it around Kolu's neck, and sat by his bedside to recite the Hadith-i Kasa prayer. But despite all these measures, it was clear Ameh Khanom's spirits were sinking again.\n\n\"Obviously the poor boy's had a fever for several days and we hadn't noticed it, putting it down as we did to homesickness,\" she had begun to criticize as soon as Zari noticed Kolu's high fever that morning. \"Yes, nothing can replace a mother's loving care.\"\n\nDespite trying all day, they could not even get a doctor to visit Kolu, let alone a hospital bed. The boy was now semi-conscious and delirious. \"Goldfinches in the trap... chirp, chirp. Chirp, chirp. Beak down and feet up... in the air... no water... no seeds...\"\n\nAt sunset, Zari pleaded with Khosrow to go with Gholam to Khanom Massihadem's and persuade Dr Abdullah Khan to drop by for a minute to visit their patient. But Khosrow refused. \"I want to take Sahar out for a ride, and then go to Mr Fotouhi's with Hormoz,\" he said. \"Father didn't say I couldn't go.\"\n\n\"What a stubborn child!\" Zari snapped, losing her temper. \"Fotouhi is as crazy as his sister. All he does is to mislead other people's children!\" She was about to say that he was a paedophile, but stopped herself in time. Instead, she lodged a silent complaint, \"May God forgive you, Yusef! Look what trouble you've landed me in! What'11 I do if this poor child dies on my hands?\" And she vowed to send Kolu back to the village as soon as he recovered, whether Yusef liked it or not.\n\nMeanwhile, she felt she had no choice but to turn to Abol-Ghassem Khan for help. Gholam had returned without much success from Dr Abdullah Khan who had said he was getting old and hoped the townspeople would allow him to retire. Zari resolved to go back to Khanom Massihadem's herself and beg the doctor to attend to their patient if Abol-Ghassem Khan was unable to help. Surely a doctor couldn't take refuge by one patient's bedside and tell all the others that he's stopped practising, even if that particular patient is very young and has served the townspeople.\n\nAbol-Ghassem Khan was at home. He picked up the telephone himself. \"Well, to what do we owe the honour, sister?\" He was in a chatty mood and didn't allow Zari to get in a word edgeways. \"I hear Sahar came back to Khosrow on his own feet! I wasn't in town that day. I had to escape to the countryside, away from my honourable constituents. Can you believe they actually think I'm about to represent them? They've already started with their petty requests. One of them wants to have a patient hospitalized; another wants to obtain his rights in a court of justice; one fellow wants to have his daughter registered at the Mehrain School for free, and so on. For heaven's sake, this position as deputy cost me all of seventy thousand tomans! Anyway, it seems Sahar's escapade was quite a spectacle. Singer said my nephew charged into the middle of the crowd like a real hero wearing nothing but a pair of givehs and his shirtsleeves. Now sister, why wasn't he dressed in some respectable clothes? Anyhow, Singer was saying that as soon as the horse spotted Khosrow, he came forward like a long-lost lover and started kissing and sniffing at the boy, nuzzling into his arms.\"\n\nWith an effort, Zari forced herself to say, \"Abol-Ghassem Khan, I beg you to help me. Kolu has come down with typhus, and I have him on my hands. I can't get a doctor or anyone to come to him. All of them are so busy.\"\n\n\"Which Kolu? Why does this brother of mine bring the village sick into town? And in his own house too! Has he no thought for his delicate children? Didn't he always say that things must be changed at the root and our charities were of no use? I heard him say that to you myself.\"\n\n\"That's right, but this Kolu is our shepherd's son and his father died recently. He didn't have a fever when he first came. He's fallen ill now.\" Zari knew if she said anything about Yusef adopting Kolu she would receive a one-hour lecture on how another man's son will never behave as one's own.\n\nFinally Abol-Ghassem Khan consented. \"For your sake, sister, and for the sake of the children, I'll arrange to have him admitted at the Missionary Hospital.\"\n\n\"I've already called the Missionary Hospital. They didn't have any room.\"\n\n\"They'll have room for me,\" Abol-Ghassem Khan said grandly.\n\nIt was eight o'clock in the evening when Khanom Hakim called. \"Why haven't you tell me it be Abol-Ghassem Khan's patient?\" she complained at first. Then she added, \"There be an empty bed ready in the corridor and this be separated from an Indian sick man by a screen. And the Indian man also be sick with typhus. I be setting aside some pills for the family of Abol-Ghassem Khan which those who contracted... contacted the patient must be taking.\"\n\nAt the hospital, tents had been put up in the grounds to house extra beds. A strong smell of phenic acid penetrated the nostrils. Most of the patients were fair-skinned and fair-haired. They could not have been typhus cases because they were either sitting upright in bed with bandages around their heads or their arms in slings, or else lying down with their legs in traction. Four men were sitting around a table playing cards. Their fair hair shone under the light of a lantern which hung from the tent-pole. They did not seem to be ailing or suffering in any way.\n\nGholam held Kolu all the way in the droshke and carried him to the bed prepared for him at the hospital. From behind the screen, the Indian patient could be heard crying, muttering words Zari couldn't understand. \"Seri rama! Seri rama! Krishna!\" The crying became louder and he repeated names which Zari guessed must be those of his relatives, \"Sandra! Sandra! Kitu!\"\n\nWhen Zari got home, Khosrow was still not back. At first she wanted to call Fotouhi, give him a piece of her mind and vent her anger. But she soon thought better of it. Why blame Fotouhi? These young boys were looking for a way to express their manhood. Fotouhi was merely a vehicle. She decided to wait until her son returned, and then interrogate him. She would be gentle at first, then give him a scolding, and finally raise such hell, he would have something to remember.\n\nBut when Khosrow came back, he was at his most charming, pre-empting any efforts at remonstrating or questioning. The minute he arrived, he threw his arms around her and kissed her, saying out of the blue, \"Mother, you're not an aristocrat, are you? I mean, your father was a worker from a... something class... oh no! I forget what you call that class... anyway, your father was a worker, right?\" The questions tumbled out of his mouth.\n\n\"Why do you ask?\"\n\n\"Well, the comrades were feeling sorry for Comrade Hormoz and me because we're branded as aristocrats, and it takes so long to get rid of that label.\"\n\nZari burst out laughing when Khosrow confessed that the comrades were even against well-ironed trousers, so he and Hormoz had decided to smear their trousers with dirt and rumple them up before going to the meetings. As for ties, well, they were completely out. Then he admitted to having cut a hole in his new grey trousers and fraying the threads around the hole to make the trousers look old and worn. He told her he had boasted to the comrades about his maternal grandfather who had been very, very poor. \"Mother,\" he said, \"I told them my mother's mother had nothing but dry bread to eat in the morning, which is why she had a broken front tooth. I told them my mother now takes bread to prisoners and mental patients every week in memory of the dry bread that broke her front tooth...\"\n\n\"You've learnt to lie, too,\" Zari interrupted.\n\n\"The comrades really liked it. Now tell me about the day you stood up to your English headmistress. You had quarrelled, I mean struggled, with her many times. You said so yourself the other night. Those struggles are very important to me.\"\n\nZari felt depressed. What struggles!\n\nShe remembered the day when a group of Englishmen, newly arrived from London, were due to visit the school on a tour of inspection. Classes had been suspended in the morning so that Nazar Ali Beg, the Indian janitor, could sweep out the classrooms. The headmistress had sent the girls home and told them to come back in the afternoon looking absolutely spick and span, insisting that they all wear a spotless white shirt under their uniforms. Zari's father had recently died, and she owned just the one black shirt which she wore in mourning under her black-and-white check school tunic. All the girls who went into mourning did the same: it wasn't against the rules. But how on earth was Zari to produce a white shirt in the two or three hours she had, and with no money?\n\nHer mother was ill in bed, complaining of sharp pains in her breast and little lumps the size of lentils in her armpit which she wouldn't let Zari touch in case they were contagious. Zari couldn't let her mother pawn the silver mouth-piece on her hookah, nor the family silver plate, at Deror's the Armenian silversmith. She couldn't sell them either, to buy white material for Zari. Besides, even if it were possible, how could the blouse be made up in time? Those were very hard times, the first few months after father's death, as her mother used to say. They weren't getting a pension then. Later on, the head of the Shoa'ieh School gave them the idea of writing a petition. He had called Zari's brother into his office and quietly made him understand that his family could apply for a pension, giving suggestions on how to write the letter and to whom it should be addressed. When Zari's brother had come home and related the incident, their mother had prostrated herself and kissed the ground in thanksgiving.\n\nOn the day of the inspection, Zari decided to take a risk. She washed and ironed her blouse and went to school. They wouldn't kill her for it, after all, she decided. But when the headmistress spotted her, she was so upset, she nearly hit her. \"You ugly little runt!\" she shouted. \"You've become quite disobedient, haven't you?\" Of all her compatriots, this one had learned Persian well.\n\n\"I'm in mourning,\" Zari replied. \"My father died less than a month ago.\"\n\n\"And you answer back, too! When did your father ever believe in such superstitions?\" Then she calmed down and said, \"Too bad your English is so much better than all the other students and I need you to welcome the guests in English, otherwise I would expel you. Perhaps I was wrong to exempt you from paying tuition fees.\"\n\nNow it was all out. Until that day none of Zari's classmates had known she didn't pay fees. How could she ever hold up her head again?\n\nSomehow within fifteen minutes, the headmistress had found a white blouse Zari's size which she handed to her and ordered her to wear.\n\nBut Zari decided to be stubborn. \"I'm in mourning,\" she insisted, \"my father has just died.\"\n\nThe headmistress got down to it herself. In front of all the other girls, she carefully removed Zari's uniform, then yanked off the black shirt, ripping a sleeve in the process. The white blouse she put on again with care.\n\nSinger arrived before the others and assembled all the girls about him in the garden where they were scattered. Most of them knew him since they had bought sewing machines from him. He looked them over critically, saying, \"Like so. They enter the hall, you pretty girls bow. These people pay money for school from own pocket. For the sake of Jesus they give large school.\" Then he called Zari over. \"Zari, you say welcome. Lady stretch hand to you. You kiss hand!\"\n\nThe assistant headmistress rang the bell and all the girls lined up and filed into the assembly hall of the school to wait for the guests. Singer walked in after a while followed by an assortment of ageing ladies and gentlemen, some stooped over, others stiff as a rod, some of average height, others short. Zari counted sixteen of them. Singer was being particularly respectful to one of the old women who was sporting a large hat with what looked like two sparrows buried in it. One was perched with open wings, ready for flight, the other's head merely peeped out.\n\nZari stepped forward and spoke her welcome. The headmistress had a smile on her thin lips. Singer's eyes were fixed on the old woman with the sparrow's nest. When the woman stretched out her hand, Zari shook it. Singer frowned, but it was too late.\n\nThen Zari joined the other girls in singing the hymn \"Christ in Heaven\", ending with a resounding \"Hallelujah!\" Their Indian teacher opened the Bible, tossed her braid over her shoulder, and began to read St Paul's letter to the Corinthians: \"Though I speak with the tongues of men and of angels...\" But when it was Zari's turn to recite a poem, she involuntarily launched into Milton's \"Samson Agonistes\" instead of Kipling's \"If\":\n\n\"O dark, dark, dark, amid the blaze of noon...\"\n\nWhen they were filing out of the hall, the headmistress squeezed Zari's arm hard, whispering, \"You little wretch!\" This one knew Persian well. She even knew expressions Zari and her friends had never heard of. \n\n# _14_\n\nKolu's illness and the confusion that went with it, caused Zari to forget all about Ezzat-ud-Dowleh's lunch invitation. But Ezzat-ud-Dowleh herself had not forgotten. That distinguished lady had probably gone to great lengths to make preparations, because she rang bright and early on Wednesday morning to double check, reminding them of the invitation. Now it was Ameh's turn to grumble.\n\n\"Why don't you all go, sister. I, for one, am not going. I went to the baths only the day before yesterday. And sister, you didn't say a word to stop me. Besides, I'm not in the mood for Ezzat-ud-Dowleh's fuss and ceremony. She spreads a feast from one end of the room to the other, but her crossed eyes follow your every mouthful. She watches the sugar-bowl to count the sugar-lumps you take! And probably sees double, too.\"\n\nZari had never felt so tired in all her life as she had over the past few days. \"Ameh Khanom, the lunch is in your honour,\" she said. \"In any case, Ezzat-ud-Dowleh is your friend.\" She nearly added, \"She is your sister-by-oath and your crony,\" but decided against it. Instead, she said, \"You know, lately you've been cutting yourself off from us, and I was thinking perhaps it's because you're preparing to leave us altogether.\"\n\n\"You're quite right. When I leave here on my pilgrimage, I don't want to feel your absence all the time. Besides, I don't want these poor children to keep asking for me as soon as I go away.\"\n\nBut finally Ameh Khanom consented. They took a droshke through the avenues, but walked the narrow back-streets. Khadijeh carried one twin while Zari gave a hand with the other, who was walking, helping her over the rock-strewn alleys. They passed the narrow Qahr-o-Ashti street, and on the right-hand side, just before Sardazak, they stopped in front of the enormous gates of Ezzat-ud-Dowleh's house. Khadijeh was out of breath. Ameh Khanom read the Quranic inscription on the mosaic over the gate: \"Lo! We have brought unto ye a great and glorious victory.\" She glanced at the house opposite Ezzat-ud-Dowleh's, the house in which she had grown up. \"What a ruin it's become!\" she commented.\n\nThe gates of Ezzat-ud-Dowleh's house were open. As they passed through the large, shady octagonal porch, the doorman was sitting idly on his wooden bench. He jumped to attention, as if roused from a dream. Taking off his felt hat, he greeted them and invited them in. At the entrance to the outer courtyard, an old black maidservant held out a crystal bowl. She removed the lid of the bowl and invited them to help themselves. The two women each took a jasmine-flavoured almond sweet. The black maid bent down to serve the twins, and then came round to Khadijeh. At the entrance of the inner courtyard, which was an orangery, Ezzat-ud-Dowleh's personal maid Ferdows, wearing a blue silk chador, offered them a platter of fragrant melon. She served them as the black maid had done. Zari placed the cool melon against her face, inhaling its mild scent as if every refreshing aroma in the world was to be found right there.\n\nIn the large, cool basement, the fountains of the indoor marble pools had been turned on. Ezzat-ud-Dowleh was dominating the room from her position at one end where she was sitting on a folded blanket. She apologized for not rising to greet them, explaining that her chronic rheumatism plagued her even in the middle of summer. She then welcomed them profusely.\n\nFerdows re-appeared carrying a square bundle of cashmere brocade which she placed before Ameh Khanom. Then Ferdows helped her take off her black outdoor chador which she carefully folded while Ameh Khanom unwrapped the bundle and examined the pile of different chadors, choosing a plain navy one. Ferdows opened it up and draped it on her. Then she wrapped up the bundle of chadors again, including Ameh's black one, inside the cashmere brocade and took them away.\n\nAfter this, they were brought fresh lime juice in a decorative china bowl with a matching ladle. The bowl was placed carefully before Ezzat-ud-Dowleh. On a silver tray, the old black maid brought some finely-cut crystal glasses and Ezzat-ud-Dowleh served the lime juice with deliberation and ceremony. Turning to Ameh she said, \"You're so fortunate, Qods-ol-Saltaneh. If I didn't have this rheumatism, I would have dearly liked to become a pilgrim to such an imam...\"\n\nZari had long forgotten Ameh Khanom's title.\n\n\"First of all, tell them to turn off those fountains,\" Ameh said. \"The damp does your leg pains no good.\" Ezzat-ud-Dowleh ignored this. Zari concluded that the leg pains were merely pretence and wished that she would get to the point, in other words, the reason for all the hospitality. In an effort to make conversation, Zari once again complimented Ezzat-ud-Dowleh on the colour of her hair. Ezzat-ud-Dowleh smiled and passed a hand over her garish hair.\n\n\"Acquaintances,\" she said, \"even the Governor's wife, kill themselves to get me to reveal the ingredients of this hair-dye. But I've refused to tell anyone so far. Everyone who sees me says, 'What beautiful hair!' And I say, 'Beauty is in the eye of the beholder. 'But Zari dear, I'll tell it to you. You're like my own daughter. Your mother, God rest her soul, and I were like one soul in two bodies. I so wanted you to become my daughter-in-law. My poor Hamid singled you out from amongst all those girls. Well, it was not to be. That is, you played hard to get. But your own chestnut shade is also very pretty. It hasn't turned grey yet, so it's a shame to dye it. When you dye hair, it starts to go grey before you know it.\"\n\n\"God bless you for your kindness,\" replied Zari, and to herself, \"Thank God I didn't marry your lecherous son!\"\n\n\"I'm going to tell it to you, but you must swear never to divulge it...\" confided Ezzat-ud-Dowleh, staring cross-eyed at her guests, \"it's been a family secret. Henna, coffee and cocoa, that's what it is! I added the cocoa myself. It softens the hair. Take one soup-spoonful of henna, cocoa and coffee at a time, add some chamomile and rub all over the hair. Then cover this with fresh walnut leaves and wrap your hair overnight or from morning till afternoon...\"\n\nZari had no interest in hair-colour secrets. If her poor mother had been alive, it might have meant something. Her mother had vowed, if she ever recovered from her illness, to take a set of silver dishes as a gift to the shrine of Hazrate Abbas and then come back and dye her hair just like Ezzat-ud-Dowleh. She used to say that she would get the secret ingredients out of Ezzat-ud-Dowleh by whatever means. But her mother was away from all this now. She began to pray that Ezzat-ud-Dowleh's breathtaking generosity was not building up to some impossible favour in return.\n\nWhen they went to the changing rooms outside the bath, the black maid was squatting there next to a His Master's Voice gramophone with a conical horn which she switched on the moment they walked in. \"You left me and broke your pledge...\" The lower half of the changing-room walls was made of marble, while the upper half and the ceiling were covered with frescoes. Zari had seen this very hammam and the Zurkhaneh behind it, on that school trip when the teacher had brought all the girls of marriageable age to Ezzat-ud-Dowleh's place on the pretext of visiting a historic old house. The building was one of the town's landmarks, nevertheless, and no important foreign visitor left Shiraz without seeing it.\n\nIt was easy to understand why Hamid Khan himself had taken on the role of tour-guide to the school visitors. The large reception room with sash windows did not have electric lights yet, and Hamid Khan had tried to show the girls the paintings on the ceiling with the aid of a kerosene lamp which he held high above his head. The reception room ceiling was lined from one end to the other with portraits of men and women next to each other. The women were depicted with tiny, pea-sized mouths, doe-like eyes, and long, wavy locks. The men were identical to the women, only they had forelocks and no earrings.\n\nThat day Zari had not really noticed Hamid Khan's ogling. But the following week, when Ezzat-ud-Dowleh intruded into their private cubicle at the hammam, squinting curiously at her naked body, Zari suddenly realized what was going on. The woman's stare sent shivers down Zari's spine. It was as if something was being stripped away from her. How impudently Ezzat-ud-Dowleh had tilted Zari's chin upward to catch the sunlight in the cubicle, muttering to herself, \"God protect her, never seen such a fair and delicate body! Just like fine porcelain! Eyes the colour of mahogany... never seen eyes this colour. God created you for His own heart. By all that's perfect! God knows if we weren't in a bath I would've thought it was make-up or something...\"\n\nZari had wanted to shove the woman's hand away from her chin. But after two hours of Etiquette and one hour of Conduct every day at school, how could she possibly do such a thing? Of course they always ended up reading the Bible instead of Conduct, but Etiquette was about manners... and Ezzat-ud-Dowleh was not going to give up. \"Pearly teeth, such a beautiful neck you'd think it's carved out of marble, what eyelids...\"\n\nThe twins brought her back to the present with their refusal to undress and their fascination with the paintings on the ceiling, especially one of a man on horseback staring at a naked girl combing her long hair. Zari remembered that on the day of the school-trip, Hamid Khan had purposely kept the girls for a long time in the changing-rooms to explain in detail about this very picture which was a scene from the famous Khosrow and Shirin love-story. The naked woman had huge breasts and was sitting next to a stream, combing her long, black hair. Some kind of screen separated the woman from the rider, who sported a thick moustache and a royal hat, and although the screen should have hidden the man's anatomy too, every detail of his body and that of his horse was visible. And the woman had nothing covering her genitals, either.\n\nZari promised the twins that if they let Khadijeh undress them, she would send them in the afternoon to see the Zurkhaneh next door which had pictures of the ancient warrior Rostam with his parted beard and tiger-skin garment, torn off the body of the monster, Akvan. They could also see Akvan being slaughtered and skinned.\n\nIn the bath, Ameh Khanom did an ablution, rinsed her body quickly and left. She couldn't bear the noise of the scratchy records. But Zari tried to linger as long as she could. She sat on the lowest step of the warm-water pool and let the hike-warm water engulf her body. Soon every part of her was feeling limp and relaxed. She closed her eyes and leaned her head against the edge of the pool. When she got out, she sat on a shiny white tray, and wrapped a large white embroidered cloth around her body. The black maid came in at that moment. She was stark naked, and brought in the water-melon on a tray which she set down on top of one of the empty copper bowls. The water-melon had been neatly cut with a zig-zag pattern along the edge. The twins gaped at the sight of the negress. Marjan was about to cry out in fear, but was stopped by Mina's loud question, \"But mother, this one has a skin! Didn't you say they've skinned her and that bearded man is wearing the skin?\"\n\nZari laughed, and the black maid said, \"God bless you, my sweet child! I'll go burn some incense to protect you against the evil eye.\"\n\nNana Seyyid, the best bath-masseuse in town, came in holding a shiny pitcher with prayers engraved all round the rim. She was taken aback to see Zari, but she greeted her politely. She was naked except for a red loincloth tied between her legs and held up at the waist with a thin red band. On that day too, this same Nana Seyyid had been in their cubicle at the Shapuri Hammam. She had come to wash Ezzat-ud-Dowleh but was ordered to wash Zari first. Chatting away pleasantly, Nana Seyyid had first washed Zari's right arm, but had given the left one such a harsh rub that Zari was forced to say, \"Gently!\"\n\nNana Seyyid had quickly taken offence. Removing her bath-glove, she had placed it in front of Zari and said, \"Do it yourself, if you know how.\" And how pleased Zari had been about that! They didn't have any money to hire or tip a bath-masseuse, anyway.\n\nNow Nana Seyyid went over to the warm-water pool with the pitcher which she filled and then emptied over Zari's shoulders. She sat on the floor in front of Zari, pulling forward the raised tray containing the bath-glove and other items for the bath. She took a pinch of salt from a small copper bowl and rubbed it on Zari's heels. Then she began to gently massage the heels with a delicately fashioned pumice-stone which had a silver cap. It tickled, but Zari didn't make a sound. Again, the black maid came in and circled around each one of them\u2014even Khadijeh and Nana Seyyid\u2014with a fistful of incense. Shortly after she left, the smell of burning incense from the changing rooms filled the bath.\n\nZari sat on the outside step of the warm-water pool while Nana Seyyid massaged her scalp with a shampoo mixture of mud and rose petals. It occurred to her that it was a pity to stain the shiny whiteness of the marble floor with mud from the shampoo. But she surrendered herself to the gentle kneading of the masseuse, thinking of all those wonderful fragrances still lingering in her senses: melon, jasmine, lime, incense, rose-petal... and she wished this euphoria could go on for a long time. \n\n# _15_\n\nBut Ezzat-ud-Dowleh did not get to the point till late that afternoon. Even then she built up to it with much preamble, explanations and beating about the bush. It was early evening and her guests were sitting around cross-legged on a large, twelve-segment wooden takht placed over the pool for cool air. The takht was covered with layers of carpets over which soft, striped sheets had been spread. Carpet-covered cushions had been arranged against the tall latticed railings of the takht. Ezzat-ud-Dowleh had taken up her usual place at the head of the takht, fanning herself. Ameh Khanom and Zari were seated on either side of her, but were not using fans.\n\nThe air had cooled. The blossoms of jasmine bushes, in large flower-pots around the pool, seemed to twinkle like so many stars at the reluctant sun, unwilling to set over the orangery. Ezzat-ud-Dowleh had managed to send Mina and Marjan off with Khadijeh, Ferdows and her children, to the police-chief's garden to watch the Pahlavan Kachalak puppet show.\n\nZari didn't even quite realize how the conversation turned to her charities at the prison and the asylum. She found herself explaining about the women's prison. \"It's not too crowded there,\" she said. \"They're not too restricted, either, because the crimes are generally not more serious then stealing a ewer. Yes, I'm allowed to sit privately with the prisoners on the little rugs their relatives bring them and listen to their complaints. But I don't see the men. I just take their food to the Karim Khani citadel, and deliver it at the warden's office. What happens to it after that, is a matter between God and the warden! But there's a belief among prison wardens that whoever steals from rations will be stricken with leprosy.\" She added, \"One day I insisted on taking the food to the male prisoners myself. That day they were cleaning out the Dosagkhaneh latrines which are in the hallway. The stench makes you want to die.\"\n\nThen the conversation turned to the madam of a 'hospice' who had recently been imprisoned.\n\n\"I wanted this woman imprisoned myself,\" Zari said, \"but I wasn't the one who reported her. It was the regional officer who'd accompanied us. Mahin Khanom and I had been on an inspection tour of the houses in the Mordestan District, on behalf of the Women's Society. No matter how long we knocked at this woman's house, no one would answer. The regional officer started kicking the door. Finally the madam herself let us in. It was getting dark. We inspected all the rooms. Mahin had them open up some of the beds and she ordered fresh pillow cases and sheets for the mattresses. In the end, when we had gone to the madam's room to give her a supply of anti-flea powder and disinfectant, I saw something wriggling under the sewing-machine stand in the corner of the room. First I thought it was a cat. Only a black little head was visible. I reached out and switched on the light, motioning for the regional officer to take a look. Sure enough he pulled out a seven-or eight-year-old girl from underneath the sewing-machine table. The little girl was wearing a glittery, wrinkled dress, and her breasts hadn't yet fully developed. She was shivering like a sparrow in snow. Despite my quiet nature, I lost my temper. I shouted at the madam and asked whether she wasn't ashamed to use children of this age for work like that. At first she swore frantically that the girl was her niece who was staying with her for the night, but then she broke down and confessed. 'Well, what can I do, Khanom?' she said. 'There are too many customers. One Indian sergeant major has been waiting some time for a young girl. You can't let the customers down. We're constantly being ordered from above to keep our customers satisfied, and now you're here criticizing us? What brings you here, anyway? Isn't it to clean up the place to ensure the satisfaction of the foreign customers? After all, I've been in this business for many a year and no one has ever come to inspect us for anything else.'\"\n\nZari stopped talking. But when she sensed her hearers' eagerness to know more, she went on.\n\n\"Later it transpired that the madam had had ten or twelve of these children working for her and that day she had sent them off to escape over the roof\u2014all except the little one who hadn't been able to get away in time. But what bracelets the madam herself was wearing! She had on at least ten pairs of gold bracelets.\"\n\n\"Shameless woman!\" exclaimed Ezzat-ud-Dowleh. \"May she pay hereafter for what she did to those innocent children!\" Then she added, \"They've got our maid Nana Ferdows in prison too. I expect you'll see her tomorrow when you go there.\"\n\n\"On what charge?\" Ameh asked.\n\nZari suddenly understood. She realized the favour needed of her somehow related to the women's prison and Nana Ferdows. She waited. But Ezzat-ud-Dowleh was taking her time.\n\n\"What I suffer because of this child of mine! My husband\u2014may he never rest in peace\u2014had no idea how to raise a child. He didn't even let Hamid do his compulsory military service. He faked the medical certificate by slipping pebbles in the boy's urine sample and bribing the doctor to diagnose a kidney stone condition. If they'd taken him for military service, maybe it would have done him some good. May he never rest in peace, my husband! He would go whoring with a fifteen-year-old boy, and my poor Hamid caught gonorrhoea at sixteen. His wife isn't capable of making a man out of him now. How I wished he'd married Zari! It was not to be, I suppose. Like father, like son. May he turn in his grave, my husband, may he never rest in peace!\"\n\n\"But I heard Hamid has given up his extravagant habits and settled down,\" said Ameh Khanom.\n\n\"Settled down? With all the money he throws away and that shrew of a wife? I kept insisting that he should do up this big house and come to live here, but he wouldn't listen. Or rather, his wife wouldn't think of it. The woman kept repeating that she would get depressed living in these back alleys and nothing would do but that she had to live on a main street.\"\n\nEzzat-ud-Dowleh fell silent for a while and fanned herself.\n\nThen she went on. \"There's an old saying that only children turn out either mad or crazy. When my boy was five, all he did was to fly kites with coloured paper-lanterns. At seven or eight, he became obsessed with pigeons. When a person is born under an unlucky star... even now as a grown man all he does is play with pigeons. He's made three hundred nests on his roof-top for them. Every evening he flies his pigeons, and he claims that when the birds fly up and away, his heart flutters to the rhythm of their wings, and only comes to rest with them when they've returned.\"\n\nAmeh sighed. \"He was playmates with my poor son,\" she said. \"When my child died, I couldn't bear to see your Hamid. But now, time has taken care of all that. I miss Hamid.\"\n\n\"He'll come to see you in a little while. I told him his aunt would be here and he said he'd come early this evening to pay his respects. He misses you very much too...\"\n\nThe black maid appeared just then, carrying a tray of afternoon refreshments which she placed in the middle of the takht. There were all sorts of seasonal fruits as well as a variety of imported biscuits. She also brought in a brazierful of hot coals standing in an ornate copper tray. This she put in front of Ameh for her opium-smoking. She made tea in a red china teapot with floral designs which matched the china bowl of the opium pipe. The flowers on the design were white poppies. The tongs and the pipe-rod gleamed like gold.\n\nEzzat-ud-Dowleh went on. \"How I've suffered because of that child! You probably know that he sends foreign officers and soldiers here on the pretext of seeing antiques. In reality they sell us whatever extra bits and pieces they may have like biscuits, soap, shoes, stockings, silk, and so forth. I sell the goods in turn through Nana Ferdows...\"\n\nAmeh interrupted her harshly, \"Come now, Ezzat, do you think no-one knows? It's hardly a secret that you, a distinguished lady as you say, have turned into a smuggler! I didn't want to mention it today, but at our house I tried to give you some hints. You kept evading the issue and I didn't insist. Your son's driver told the story of your Jahrom haul in front of everyone at the Do-Mil teahouse. He said you and Nana Ferdows looked as if you'd put on quite a bit of weight overnight! He said you spent two whole hours wrapping up your body in silk to hide the smuggled arms. Apparently you also packed two big canvas sacks full of goods in the boot of a car which could have cost you a twenty thousand toman fine. Why have you become so greedy? A little bit of self-respect and dignity go a long way, you know.\"\n\nEzzat-ud-Dowleh controlled herself. Only the corner of her mouth twitched as she said, \"That driver was probably the one who betrayed us. I kept telling Hamid not to dismiss him in this godforsaken summer with all the sickness and famine around. But he wouldn't listen. What I go through because of that boy! But then you know, as I'm sitting here by myself of an evening, he comes along with a special rice dish, or a plateful of best quality apricots or some large tangerines... he'll say, 'Mother, I was thinking of you.' Then he'll kiss my hand, my foot, lay his head on my bosom and with all this pampering, I know that the next day he'll get anything he wants out of me.\"\n\nAmeh Khanom opened the small jewel-studded case before her, took a piece of opium, and smelled it. \"What good quality!\" she said. She warmed the opium and stuck it to the pipe-bowl.\n\n\"Forgive me for being so bold,\" Zari said, \"but you have a great deal of assets and property.\"\n\n\"May he never rest in peace that husband of mine!\" Ezzat-ud-Dowleh exclaimed. \"What assets and property? He would steal the title-deeds of my land, cover his sister with a chador and take her to Sheikh Gheib Ali the notary and introduce her as his wife. He would sell my land, and have his sister\u2014well-hidden under her chador\u2014thumb-print the foot of the sale transaction as signature. All the money was spent on his women... and on that bedroom! His private room where he took the prostitutes, with that double-bed he brought over from India. He bought every pack of old playing cards to be found in this town so he could paste all the aces, queens and jokers on one wall of that room. He hired a painter to illustrate another wall with every imaginable kind of love-making position. Whatever money was left over, at the end when he was confined to the house, he smoked away in opium.\"\n\nAmeh Khanom took a puff and said, \"He left enough for your family to live on respectably for several generations. But if you're hinting at my addiction, too, let me just say I don't smoke away anyone else's money... it's my own. Besides, I've vowed to give it up the instant I set foot in the shrine of Imam Hossein. Right then and there, I'll break my opium-pipe in two. O Lord, please give me the strength to do it!\"\n\n\"Sister, why have you become small-minded?\" Ezzat-ud-Dowleh asked. \"And why so touchy? I swear by my only son that I meant no offence to you. As for giving up opium, I'm certain that you'll be able to do it. You're one of those people who can do whatever they want.\"\n\nAmeh Khanom took a long puff. \"What good opium! Where do you get it? It brings the scent of the poppy-fields right to my nostrils! How often I used to ride around those fields! Field after field of poppies, and each one a different shade... the scent of it at sunset intoxicated both me and my horse. When the flower-petals have fallen, the yellowish, moss-green seed-heads nod in the breeze as if to talk to you, and you're certain they're alive. They have something no other flower in the world has. At sunrise, they come to cut them. The dew is still sparkling on the seed-heads, and drop by drop the pretty sap oozes out.\"\n\n\"Since you like it so much, I'll tell them to prepare some more pieces from the same batch for you to take with you on your trip. You can think of me when you use it.\"\n\n\"Curse the devil! Even if it kills me I'm going to give it up. The beauty of the poppy-fields is quite a different thing from its poison.\"\n\nZari was beginning to feel anxious. She had planned to visit Kolu in hospital earlier in the evening, but it was too late now. She was worried about Khosrow, who had gone to join Hormoz so the two of them could go to Fotouhi's together in the evening. Khosrow had inadvertently mentioned the night before that although they might not be accepting him at any party branch because he was under-age, Mr Fotouhi had generously allowed him to join Hormoz and his friends as an 'independent observer'. This was the same group whose members pitied those with aristocratic blood.\n\nZari turned to Ezzat-ud-Dowleh and said, \"I'm beginning to understand now. Nana Ferdows was caught red-handed smuggling.\"\n\nEzzat-ud-Dowleh sighed. \"I wish it were that simple,\" she said. \"This time she was actually smuggling arms.\"\n\n\"By Allah, the Almighty!\" exclaimed Ameh, putting her pipe down next to the brazier.\n\n\"Yes. Two Brno guns, ten revolvers and a box of ammunition. God knows we were very careful, very cautious. Four times previously Nana Ferdows had delivered the same load safely to its destination. But this time she was caught. I'm certain it was the driver who gave us away and was probably paid well for it too. A curse upon him! Nana Ferdows was supposed to take the load at sunrise before the women's public baths opened, to the Khani Hammam and deliver them to the Mirza Agha Hennasab.\"\n\n\"Which Mirza Agha? The son of your own wet-nurse?\" Zari asked.\n\n\"Oh no. No-one knows where my wet-nurse's son is. They say he's joined the Communists...\"\n\n\"I see. Go on.\"\n\n\"Yes, she was supposed to deliver the load to the Mirza Agha Hennasab and tell him, 'Mirza Agha, these are Khanom's bath things. I'm leaving them in your care. When it's the women's hour at the baths, give them to the bath-keeper's wife.' And Mirza was supposed to call out casually to one of the errand boys and ask him to take the bundle to the back of the hammam for safe-keeping. I'd wrapped up the 'bath things' myself in the dead of night. Even Nana Ferdows didn't know what was in it. I packed the guns end to end and wrapped them tightly inside a small rug. And even though my fingers were pricked till they bled, I pinned both ends of the rug so the guns wouldn't slip out and the fringes of the rug would cover up any parts that were showing. I placed the rolled-up rug on the porter's tray myself and put the large copper bowl which had the box of bullets hidden inside, next to it. The revolvers I rolled up in bath towels and carefully wrapped that in a cashmere brocade. These I put inside the large copper bowl as well, with part of the brocade cloth showing. I even sat down and prayed for the safe delivery of the load.\"\n\n\"What things you pray to God for!\" muttered Ameh Khanom.\n\nIgnoring her, Ezzat-ud-Dowleh continued. \"At dawn with the help of Kal Abbas, Ferdows's husband, we managed to lift the tray and put it on Nana Ferdows's head. It was very heavy, but she didn't have that far to go. Again I prayed and blessed the load and Nana Ferdows. I made her leave through the door of the inner courtyard. Kal Abbas had checked the street to see if the coast was clear.\"\n\n\"How did you find out she'd been caught?\" Ameh asked.\n\n\"I was saying my morning prayers when there was a knock. My heart sank. Apparently, just before reaching the public baths. Nana Ferdows had come across a policeman and a gendarme. I imagine they must have stopped her and searched her load. They asked her who it belonged to and where she'd got it. Kal Abbas says when she came home and he opened the door to her, it was obvious she'd been beaten up and had been crying. Anyhow, she had spilled the beans, and brought them to my doorstep. But see how clever and loyal Kal Abbas is. At the door, the policeman asked him whether he knew Nana Ferdows. Kal Abbas replied, 'No sir, I do not.' Nana Ferdows instantly burst out crying, saying, 'I spit on you! My own son-in-law! You don't know me? Has the world come to an end? Have you lost your eyesight that you don't know me?' And Kal Abbas said, 'listen you shrew, why make up such lies at this time of day? How should I know you?'\"\n\n\"What a mess you've got yourself into!\" Ameh said, between puffs.\n\n\"Well, by this time I was glued to the door of the outer courtyard, eavesdropping and trembling from head to toe. No-one should ever live through such a thing! Nana Ferdows was wailing and screaming, swearing by the Quran that the goods had been brought from our house. 'I had no idea there were guns and things like that in it,' she was saying. 'And this bastard here is Kal Abbas, my son-in-law, who's siding with them and won't help me out, his own mother-in-law! I shut up once when they dishonoured my daughter, but now they want to dishonour me too! I spit on you, Kal Abbas! You're a traitor, you help them. You helped them the other time too...' She sobbed her heart out, and cursed with such bitterness that my hair was standing on end. She kept saying, 'O Lord, where are You? Are You blind?'\"\n\nEzzat-ud-Dowleh fell silent for a while, fanning herself. Ameh Khanom and Zari kept quiet the whole time. Zari was biting her thumbnail. She thought silently, \"And now what is it I can do for you?\"\n\nEzzat-ud-Dowleh went on. Obviously she was not going to get to the point until she had recounted all the details.\n\n\"Either the gendarme or the policeman shouted at Nana Ferdows to stop blaspheming, and ordered Kal Abbas to wake the master of the house so he could be questioned. Kal Abbas told them the master had died a long time ago, at which point the policeman asked to see the mistress. By this time I was feeling so faint I had to sit on the ground. Kal Abbas said, 'The mistress is away on a pilgrimage to the shrine of Imam Reza.' The policeman shouted at Nana Ferdows, 'Didn't you say these were Khanom's belongings that you were taking to the Khani Hammam?' Kal Abbas didn't let Nana Ferdows answer. He laughed and said, 'Sir, we have a private bath in this house. The mistress never uses the public baths. I can show it to you if you like.' Then he said, 'Please go and have your fight elsewhere. I have a thousand things to do.' When the policeman started to hustle her away, Nana Ferdows pleaded with them, 'Where are you taking me?' The policeman said to her, 'First to the lieutenant, who's going to lock you up.' The foolish woman kept screaming, 'Let me see my child first, and I'll go wherever you want.' But they took her away. It was a stroke of luck that Ferdows and her children were sleeping far away from the entrance and didn't hear all the noise. As for me, well! No one should ever have to live through such a thing! I was shivering as if I'd been struck down with a fever. I couldn't breathe. I sent Kal Abbas at top speed to the Mirza Agha Hennasab to inform him.\"\n\nThen turning to Zari, she said, \"But Zari, my dear, you hold the solution to my problem. We've made the necessary investigations indirectly. We know that Nana Ferdows is in the women's ward. Now I beg of you, when you visit the prison tomorrow, go and see Nana Ferdows. Talk to her. Beg her on my behalf not to mention our name under any circumstances. You see, Kal Abbas managed to nudge her foot at the last moment and make her understand that she must keep her mouth shut. It seems she's either caught on or simply tired out, because she's stopped talking for the time being. My dear Zari, please tell Nana Ferdows to say at her trial that the mistress was on a pilgrimage; that Kal Abbas had bought the goods from a few Indians, wrapped them up in the mistress's bath things and given them to her to sell at the bazaar, and that the Mirza Agha Hennasab had offered to buy the goods. If she doesn't stick to this story, my whole family will be ruined. So will our long-standing reputation. We'll be utterly undone.\"\n\n\"Is it all right for Kal Abbas's family to be ruined, then?\" asked Ameh Khanom cynically. \"Why implicate that poor Mirza Agha Hennasab? I don't want to criticize you, but... well, anyway, it's none of my business.\"\n\n\"Qods-ol-Saltaneh, this is no time to talk Zari out of helping me,\" Ezzat-ud-Dowleh pleaded with her. \"Doesn't our sisterhood mean anything to you? I swear I'll repent and give this up. Besides, neither Kal Abbas nor Mirza Agha's family will suffer. We've notified Mirza Agha in good time and he's escaped to the tribe. And I've persuaded Kal Abbas to cooperate. Tell her I've persuaded her son-in-law to cooperate. We've made enquiries and found out that if an ordinary citizen smuggles arms just for money and nothing else, the sentence is no more than a year or two in prison. They confiscate the arms and levy a fine twice their value. That's nothing to worry about; I'll take care of all the fines. I'll give Kal Abbas five thousand tomans reward when he gets out of prison. And I've promised to take good care of his wife and children in his absence. Tell her to ask for Mr Sharifabadi as her lawyer. I'll contact the judge and the public prosecutor for her. And I promise that this time I really will keep my word and send her on a pilgrimage to Karbala.\"\n\nShe reached under the cotton sheet and pulled out two envelopes and a small box which she handed to Zari. She shouted to the maid, \"Bazm Ara, put the lights on!\" The tall garden lights which looked just like carriage-lights were immediately switched on.\n\n\"Give her these two envelopes, my dear,\" Ezzat-ud-Dowleh continued. \"The first one contains a written request for a lawyer, and the second one has the details that I've been telling you. She can read\u2014she reads the Quran\u2014but she can't write. Make her press her finger in this ink-box and fingerprint the bottom of the first letter. Then give the letter to the warden's office and ask for a receipt. You can say you wrote this letter yourself as a form of counsel or kindness to the prisoner. Since everyone knows you as a generous, charitable woman, no-one will suspect you. But make sure you take both letters from her... whatever you do, don't leave them with her. I beg you in God's name to do this... will you? I've thought of sending her daughter Ferdows to her as a visitor, but I don't trust the girl. There's a strange glint in her eyes these days. I'm afraid mother and daughter will get up to something and land us in a real mess. Should they decide to take their revenge, what better opportunity than this?\"\n\nZari wondered which would take more courage: to accept or to refuse? Giving two envelopes to a prisoner, and talking and probably reasoning with her, having her finger-print the letter, waiting for her to read all that was written on the two sheets of paper with her minimal reading ability\u2014all this in front of other prisoners, especially that madam who held Zari responsible for her imprisonment, demanded courage enough. But she could be adventurous and do it. What kind of justice, however, would that be? She would be shielding the real criminal and allowing her to appear innocent, while an innocent person took the blame for a crime. Besides, she wasn't afraid of Ezzat-ud-Dowleh.\n\nBut what if she refused to cooperate? Would she be showing the courage that her husband and son expected of her? After all, if Ezzat-ud-Dowleh didn't succeed in using her, she would merely find some other way, buying and safeguarding her reputation through whatever means. And it probably didn't make much of a difference to Kal Abbas whether he was imprisoned in the entrance of Ezzat-ud-Dowleh's house or in a real jail. Nevertheless, why should she be a vehicle for injustice? The right thing to do would be to encourage Nana Ferdows to tell the truth, undaunted by Ezzat-ud-Dowleh or anyone else's reactions or conclusions. But then, couldn't Ezzat-ud-Dowleh crush the woman with her money and influence anyway, and destroy her family? In any case, Nana Ferdows had long been an accomplice. She had accepted the life they offered for many years now.\n\nEzzat-ud-Dowleh broke her train of thought.\n\n\"Zari my dear, what a long time you take to weigh up such a small thing!\"\n\nZari pushed the letters and the ink-box in front of Ezzat-ud-Dowleh and said, \"No, I won't do it, I'm sorry.\"\n\n\"You won't do it? But why?\" Ezzat-ud-Dowleh asked, stupefied.\n\nZari didn't reply. Ezzat-ud-Dowleh tried to cajole her like a child. \"What if I get your emerald earrings back from the Governor's daughter?\" she coaxed. \"Would you still not do it? I was just about to do something about your son's horse when this whole situation came up...\"\n\n\"My earrings are not that important to me anymore. It's better if you allow the truth to be known. You were saying yourself it was a pity Hamid Khan didn't do his military service. Well, this might prove to be a form of military service for him.\"\n\nAmeh Khan laughed so hard that she was seized by a fit of coughing. Ezzat-ud-Dowleh forced a nervous titter and said, \"You don't seem to understand the difference between Kal Abbas and Hamid. If Kal Abbas is convicted, his jail sentence is only for a year or two. But if our family name is mentioned, our whole livelihood will be at stake. They'll charge us with smuggling arms with intent to jeopardize national security. Sharifabadi was saying that according to article 171 of the penal code, the sentence for that would be execution, or at best life imprisonment. The maximum he can do is to settle the case on appeal for ten or fifteen years. No-one is going to believe that our living expenses are high and we did this for money.\"\n\nTears sprang to her eyes as she said, \"When you're born under an unlucky star... so much for friends and avowed sisters... they abandon you in times of need.\" And she shouted, \"Bazm Ara, bring me the drops for my heart.\" Then she continued, \"I know why you're refusing. You disliked us from the start. I don't know what we ever did to you. Or maybe you regret now that we didn't press you harder to marry Hamid. A beggar like you played so hard to get! I know. Now you want to take your revenge on us. With that crazy, temperamental husband of yours, I don't blame you. He's made more enemies than he can count!\"\n\nAt this moment Hamid Khan arrived. He looked plump and jolly, and greeted everyone effusively. He took his shoes off at the foot of the takht and stepped up in his socks. He hugged and embraced his 'aunt' over and over again. Zari noticed that Ezzat-ud-Dowleh hurriedly wiped away her tears and smiled at him. Her son literally bent down to kiss her feet. He kneeled down next to his mother and asked, \"How are you, how have you been, what news?\"\n\nHe went over to Ameh Khanom, leaned his head on her shoulder and touched her braided hair. Looking Zari over, he said, \"Khanom Zahra, touch wood, you remind me of first-rate wine! You constantly improve with age.\"\n\nHe held Ameh Khanom's hand affectionately, then kissed it and said, \"My dear aunt, how many years is it since we saw each other?\"\n\nAmeh Khanom didn't answer. She poured a cup of tea and placed it in front of him. Then she took up her opium pipe which she cleaned and prepared for fresh use. She asked him, \"Will you smoke if I fix you a pipe?\"\n\n\"What I've gone through in your absence, my dear aunt!\" replied Hamid Khan, trying to ingratiate himself. Turning to Zari, he said, \"I was never blessed with brothers and sisters, but God gave me two mothers instead.\"\n\nHe puffed on the opium pipe once, then several times, and became even more talkative, going over old times. He asked Ameh, \"Do you remember I used to sit on your lap, and even though I was three or four years old, I'd try and fondle your breasts and then ask you to nurse me. I loved you like a mother because you always looked after me. I remember that time when the other children threw stones at my prize pigeon and broke its leg. You'd come to visit my mother, and I was hugging my pigeon, shedding tears like a river as my dear mother would say, begging people to do something for it. The poor bird was making the most pathetic noises. It was worse than all the moaning in the world to me! I remember you soaked some crushed peas and mixed it with egg yolk and myrtle to make a sort of plaster for the pigeon's leg. When you finished, the pigeon was cooing peacefully again.\"\n\nZari felt as though she had nothing more to do there. She was restless and couldn't wait to excuse herself and leave. But Hamid was not ready to give up.\n\n\"Remember that night on the summer estate?\" he asked Ameh again. \"We'd all gone there for the day but ended up staying overnight. The musicians couldn't find a droshke to take them home, so they were forced to stay, too. When they spread out the bedclothes, there wasn't enough room for everyone. My mother never gave up the chance of sleeping next to my father if she could help it, so I was left alone. No-one else wanted to sleep next to me because I had some boils on my face and the one on my nose had become infected. Everyone knew those boils were usually contagious, especially since the garden buzzed with flies and mosquitoes which were carriers of that disease. I was left there wondering where to sleep, feeling really tired. It was very cold, too. Even though your own child was sleeping next to you, you called me over and said, 'Come my dear, come and sleep on my other side.' Then I cried and you wiped my tears. You even kissed my nose despite the infected boil. When your son died, I used to avoid you to spare you grief. One day in the Vakil bazaar, I saw a woman who looked just like you. I called her 'my dear aunt', and she turned around and slapped me one! 'Your dear aunt,' she said, 'I bet!'\"\n\nHe looked at Bazm Ara bringing a tray containing a small bottle of medicine and a cup of water to his mother. \"Mother dear,\" he asked, \"why are you taking medicine again?\" And he stole a meaningful glance at Zari.\n\n\"It's nothing,\" Ezzat-ud-Dowleh answered. \"I'm having some palpitations again.\"\n\nTurning to Zari, Hamid asked, \"Khanom Zahra, I've heard you're going to the prison tomorrow. Will you be visiting our prisoner?\"\n\nCounting the drops she was putting in the water, Ezzat-ud-Dowleh said, \"But not as we wanted.\" And she resumed her counting. Hamid frowned and looked a little nervous. He took the opium and began to smoke again.\n\n\"Why?\" he asked. \"I suppose you're afraid. Well, it is frightening for you.\" He put the pipe down clumsily next to the brazier and addressed Ameh affectionately. \"But my dear aunt is as brave as they come. She'll kiss my boil this time again, won't she? I'm sure you don't want to see me on the gallows. Anyone but you carrying out a plan like this would be suspect, you know.\"\n\nZari saw Kal Abbas passing through the orangery and coming toward the takht. As he came forward and greeted them, Ameh Khanom pulled on her chador. Kal Abbas stood by the takht and called to Hamid Khan who bent forward while he whispered something in his ear. Hamid put his shoes on in a hurry and rushed off. Zari suddenly felt anxious. What if something had happened to her little girls! Ezzat-ud-Dowleh was capable of anything. She could even kidnap the children and keep them as hostages somewhere while she forced Zari to consent. Why hadn't she thought of that earlier? Why did she let the children go to the police-chief's garden in the first place? And she thought with bitterness that the real 'show' had been taking place right here! Only it was too complicated for the children.\n\n\"It's late. Why aren't the twins back?\" she asked Ameh in a shaky voice. She was ready to give in to anything they proposed now. If she were to choose between courageousness and her children, she would clearly choose the children. Yes, Hamid would come now and make the first move. Ameh looked at her sharply and said, \"Don't worry. They'll turn up sooner or later.\"\n\nZari thought, \"I'll wait. I'm worrying needlessly. It was a good thing I sent Khadijeh with them.\" She remembered a line from a poem Yusef often recited, \"From naught but a thought comes their fear and dread...\" No, she had changed the poem. It really went like this:\n\n\"From naught but a thought their peace or war\n\nFrom naught but a thought their fame or disgrace.\"\n\nHamid soon returned with a tall, well-built woman who was tightly clutching her chador. They came and sat on the takht.\n\n\"Mother, do you want a guest?\" he chuckled.\n\nZari immediately recognized the 'woman'. She had received her wearing the same veil in her own house. \"Malek Sohrab Khan!\" she exclaimed involuntarily.\n\nSohrab sat down and took off his veil. His unshaven face seemed thin and haggard, and he was covered with dust. Ezzat-ud-Dowleh laughed so hard, tears ran down her cheeks. He turned to Zari with a faint smile and said, \"I went to your house first. No-one was there.\" He held his head in his hands. \"If only Yusef Khan was in town,\" he said. \"I should have listened to him.\"\n\nBazm Ara came in, carrying a brightly-polished ewer and bowl which shone like gold. A thick towel with floral patterns was folded over the maid's arm and the soap she held was shaped like a pear. Zari's soap in the bath had resembled an apple. Was it for these items of luxury that Ezzat-ud-Dowleh had run such risks? But Malek Sohrab's presence there at that time of night seemed to shed a different light on the whole situation.\n\n\"Sohrab, we've just had the bath water changed,\" Ezzat-ud-Dowleh said. \"Why don't you go and take a bath?\"\n\n\"Maybe they'll call,\" he answered. \"I'm hoping against hope that we'll be contacted and that the English haven't tricked us. I've come straight from the battlefield. I've been to that English Colonel who's just like the treacherous Yazid. He thinks this is the desert and he can play Lawrence of Arabia with us. He wouldn't see me. He sent a message saying he has a cold. A cold in the middle of summer? Then I went to that sly fox Singer who gave me a garbled answer about being too busy to receive me. The fool still hasn't learned Persian after all these years. If they've tricked us into fighting and looting without keeping their promise, we've shed our brother's blood for nothing. Still, he said he would call.\"\n\nEzzat-ud-Dowleh tried to signal to him, but since everyone else noticed, she merely said, \"Did you go to them wearing the chador too?\"\n\n\"No, I was wearing the uniform of a Captain Mohammad Kashmiri Kermani. His identity card was in the uniform pocket. First we stripped him down, then we put a bullet through his neck. It was ten against one. Afterwards I went over to Mirza Agha Hennasab's to change into the uniform.\"\n\nSuddenly the children's voices could be heard from the outer courtyard and Zari sighed with relief. Hurriedly she excused herself, saying she must leave, and Ezzat-ud-Dowleh, happy to oblige, called out, \"Ferdows, bring my sister's chador! Bring my prayer things too. Make sure your hands are clean.\"\n\n\"Have you heard anything in town about our fighting in the region?\" Sohrab asked.\n\n\"No,\" replied Zari. \"They haven't mentioned it in the newspapers.\"\n\n\"When do they ever write anything in the newspapers? There's been a rumour that the bodies of the officers killed in battle with our tribe are being brought to town for official burial.\" He added, \"But we shouldn't be blamed for the bloodshed, because we only fought for our ideals. After the way the cunning English have treated us the past few days, I felt so guilty about the slaughter that the dead man's uniform seemed to be choking me.\"\n\nFerdows brought the brocade wrapper containing Ameh Khanom's black chador and Ezzat-ud-Dowleh's prayer rug. Zari could not help noticing Hamid's expression. When the maid bent over to put the prayer rug before Ezzat-ud-Dowleh, his eyes sparkled as they swept over her body appraisingly. Following his gaze, Zari noticed for the first time the shapeliness of Ferdows's figure. Her legs clad in sheer stockings looked as if they had been chiselled out of fine marble. Her light-blue chiffon chador moved tantalizingly over a flowery crepe de chine dress which barely disguised the firm, well-proportioned curves of her body. It was hard to believe she had had three pregnancies and a miscarriage.\n\nBut Ameh Khanom didn't open the brocade wrapper. \"Khanom Ferdows,\" she said, \"take some food for the children and keep them in the outer courtyard until we come. Tell Bazm Ara to come and take away the brazier.\"\n\nFerdows busied herself piling two small plates with fruit and biscuits, oblivious to what had been going on and not understanding why the mistress was darting such poisonous looks in her direction. Ezzat-ud-Dowleh pushed aside the cotton sheet on which she was sitting and performed a ritual dry ablution with the dust on the carpet. With some difficulty she adjusted the starched headscarf over her head to cover up her gaudy hair for prayer-time. Only her face was now visible. But what a face! She looked as if she had just swallowed some bitter poison. At war even with the deity to whom she was praying, she tugged angrily at the prayer rug as she spread it out, and began her prayers in a seated position.\n\nDeep down, Zari was feeling quite pleased with herself for standing up to the woman. If only Yusef would hurry up and come back! She'd never had so much to tell him. Her experiences at the prison and the asylum were interesting enough, but not for Yusef. Often he would ask her to talk to him and cheer him up, and she had to rack her brains for something comforting or cheerful. It was a long time since she had been able to come up with things like that. She knew her stories had become quite repetitive of late, and Yusef seemed content just to be lulled by her voice. But now Zari had a chance to show her mettle and she couldn't wait to tell him about it.\n\nShe felt sure Ezzat-ud-Dowleh would prolong her communication with heaven just to annoy them, and that they would have to maintain a respectful silence for a while. But Malek Sohrab would soon tire of it and begin to talk. Her curiosity about the fighting had been aroused to such a degree that, if Ezzat-ud-Dowleh didn't actually ask them to leave, she knew she would wait until the whole story was told.\n\nThe black maid reappeared and took away the opium brazier. Zari broke the silence. \"Sohrab Khan, you were telling us about the fighting...\"\n\n\"Actually, I had decided to confess all the details of these recent events so that if I take off to the mountains and become an outlaw against the government, or if I disappear altogether from this place; if my tribal blood gets the better of me and I take my revenge on these foreigners, or even if everything is lost and me with it, my friends should know why I did the things I did. How I wish I had listened to Yusef Khan like my brother did! He knew. He's friends with McMahon. They translate poetry together\u2014poems of a revolutionary poet who's changed all the rules of our verse.\" He shook his head and recited,\" 'where in all the darkness of this black night, should I hang my shabby robe...'\" Suddenly he said, \"But Singer promised us! He told us to attack at Semirom, then at Shiraz, next at Isfahan, and finally Tehran. And what barbarities we committed! I'm the first to admit it\u2014what mistakes our brothers made. What an ugly war it was!\"\n\n\"Brother, maybe I'm the one who's confused,\" said Hamid, \"but I don't understand a word you're saying.\"\n\n\"Mark my words, McMahon must have known their intentions. He's a war correspondent.\"\n\n\"Come now, don't take it too hard. Be grateful you're alive and in one piece. It's all over with now. I think you should smoke some opium and forget about the whole thing. Shall I tell them to prepare it for you?\"\n\n\"I'm not the kind who can drown my sorrows with opium. If I can't atone for my sins, I'll do away with myself. Right now, I'm prepared to do anything.\"\n\nHamid baited him. \"What if the English do call? Then you'd even forget the cardinal sins, wouldn't you?\"\n\nZari thought that this was the only true thing he had said in his life.\n\nSohrab unwittingly confirmed Hamid's intended taunt with his next sentence. \"If they were really going to call, they would have done so by now. They trust you and your mother.\" And he continued, \"You see, the Russians had asked for thirty or forty Iranian soldiers, maybe more... some say they had requested as many as five divisions for logistics service. Soldiers, that is, armed with guns alone to guard ammunition stores and roads, to help with transportation or unloading cargo, admitting patients to rural hospitals, that sort of thing. Although Russia and England are allies for now, the British are obviously very reluctant to allow the formation of a 'communist nucleus', as Yusef Khan calls it, here in Iran amongst its soldiers. So they made excuses about the lack of training of the Iranian army... how worthless they are even in the face of a group of local upstarts. They staged our recent little skirmishes to demonstrate that point to the Russians.\"\n\n\"But if you knew all this, why did you go ahead and fight your own countrymen?\" Ameh asked.\n\nHamid laughed and said, \"My dear aunt, the Qashqai tribe loves to fight. Fighting gives them the same pleasure as hunting.\"\n\nIgnoring Hamid, Sohrab answered Ameh. \"Because I thought it was all rumours. Now I know better. You see, I've only just found out that there was a Russian inspector present at the Khoongah Pass to send back reports about the fighting. But the British were telling us to prepare our crowns as successors to the Achaemenid dynasty. They managed to get weapons to us by whatever means. For instance, twice we were instructed to raid their own shipment by previous arrangement. They had loaded our guns and ammunition in a civilian car and transported them as a shipment of coins from Khuzestan through the foot of the Bakhtiari mountains to the Shahi Bank in Isfahan. They did exactly the same thing during the First World War, only then they used mules. According to what we had been told to do, we ambushed the car, tied up and abandoned the driver and their agent at the roadside. The driver had been waiting for us, since he even signalled with his lights. But we took the car too.\"\n\n\"But I heard that you killed the manager of a bank and stole all the money,\" Zari said. \"Was that the same incident?\"\n\n\"No, that was another time. Well, it takes money to do things like this. Our friends helped too. Hamid and the others got weapons to us...\" Turning to Hamid, he added, \"These last bullets really came in handy, even though you overcharged us... and the revolvers too, although they seem heavier in the heat.\"\n\nEzzat-ud-Dowleh, whose prayers had come to an abrupt end, turned to Malek Sohrab and said, \"Must you say all this in front of strangers? The fact of the matter is, we've been caught too. They found out about Nana Ferdows. You can't rely on your sister for help... and you can't even trust your very own eyes.\"\n\n\"Who's Nana Ferdows?\" asked Sohrab. \"The mother of this pretty maidservant here?\"\n\nHamid laughed and said, \"She stole your heart, too? When I used to tell my dear mother that this girl literally sends off sparks which go straight to the heart, she wouldn't believe me!\"\n\nEzzat-ud-Dowleh invoked God's name out loud and hurriedly clapped her hands over her ears. Either she was going to say her evening prayers, or she was paying penance for her previous debts.\n\nSuddenly Sohrab remembered. \"Mirza Agha Hannasab's wife did tell me that one of your people had been caught, but I didn't know her name was Nana Ferdows. She also told me her husband had managed to get away in time. Before we make any other decisions, we must send Mirza Agha's wife and children to him at the tribe. Hamid, go to Singer first thing tomorrow morning or even tonight and tell him that it was while helping to carry out their plans that you were caught. You can tell him from me that they have only twenty-four hours to keep their word. If they don't deliver, they'll be risking their very necks. As God is my witness, I'm going to round up a few of my bravest men, and they know what we can do... to think we've done all these things just to protect their precious oil pipe-lines!\"\n\nZari was beginning to understand. If a crime was committed successfully, then it wasn't such a crime after all, but if it met with failure, it was a sinful thing and had to be paid for. She was about to voice her thoughts, but she stopped herself in time. Who would pay attention to her? Hamid had no interests in life besides women, whisky and pigeons. Sohrab was blind to everything but ambition. And Ezzat-ud-Dowleh was wrestling it out with God that instant. As for Ameh, all she dwelt on was her departure for Karbala or giving up her opium addiction.\n\nSo Zari merely advised, \"Sohrab Khan, it's not too late yet... why don't you go and join Yusef now like your brother Malek Rostam?\"\n\n\"Everyone has a different nature,\" replied Sohrab. \"My brother has a settled farmer's disposition, but I'm a nomad. I don't like being patient and attaching hopes to the distant future. I want to seize the future right now. I want to die in combat, with bullets and axes, not in bed. I want to be the last person to surrender. But not on my own feet. I want them to drag me out and shoot me point blank and chop me up with an axe. I want to stare my executioners in the eye so they can envy me and wonder at my indifference to life or death!\"\n\n\"It's his tribal blood again...\" Hamid said.\n\n\"You were always fearless,\" Zari said, \"even as a child. But you were quite a poet too. I remember for your first wife...\"\n\n\"And what we need now is a fearless poet,\" Sohrab interrupted. Turning to Hamid, he asked, \"I wonder if you've ever gone to Semirom from Shahreza in a south-westerly direction?\"\n\n\"No, but if you remember, once we crossed the north-westerly foot of the Denna Range to Semirom,\" Hamid answered. \"We were going to the wedding of Esfandiar Khan Khashkouli's son. I remember we stopped at the Semiron spring. A strikingly beautiful girl there gave the driver some water from her pitcher and poured some into the car radiator, too. The way she walked, that girl! Tall as a cypress, yet graceful as a deer... she seemed to bless the ground with each delicate footstep...\"\n\n\"Is this the time for that sort of thing?\" Sohrab asked.\n\n\"It's always time for 'that sort of thing'!\" Hamid replied. Then he sighed and turning to Ameh, said, \"My dear aunt, you really should not have sent Ferdows away. Call her. You call her. I'm dying for a glass of gin and lime.\" After a pause, Hamid looked at Sohrab and said, \"You know, brother, your tribal ambitions can only lead you to more trouble and bloodshed. Personally, whatever I do is for money so I can possess the beautiful things of this world: women, wine, the most exquisite Fastoni cloth from Manchester...\"\n\n\"Just a minute!\" Sohrab interrupted, placing a hand on Hamid's knee. \"Isn't that the telephone?\" He stood up. Someone must have answered because the ringing stopped and then Ferdows came into the garden. All eyes were on her. Sohrab was standing expectantly. Ferdows said, \"Khanom Zahra, Khosrow Khan wants to know whether you will be home for dinner or should they go ahead and eat?\"\n\n\"I'll be there right away,\" replied Zari, and turning to Ameh added, \"Would you mind if we go?\"\n\nSohrab sat down again on the edge of the takht and said, \"Those sly foxes are not going to get away with it!\"\n\n\"Now, now! A great man shouldn't bend under a straw,\" said Hamid.\n\nEzzat-ud-Dowleh shouted, \"Ferdows, bring my sister's chador! Are you deaf?\" She couldn't have dismissed her guests more obviously. Ameh's chador was in front of her in the wrapper.\n\n\"But the night is young,\" said Hamid. \"Why are you going so soon? I know we've depressed you with all our talk about killings and war. Let me tell you the story about Sohrab's famous fox hunt, it'll cheer you up.\"\n\nZari felt too embarrassed to mention that she had already heard the story several times from Malek Sohrab himself. So she waited patiently while Hamid told it with gusto once again.\n\nApparently they had wanted to catch a fox that was attacking Hamid's hens every night, but each time the fox had outwitted them. One winter night they put a dead hen on a mound of snow so that Malek Sohrab could get a good aim at the fox when it climbed to the top of the mound, and shoot it. But the fox, sensing a trap, didn't head straight for the hen. Instead, it burrowed its way through the mound of snow and grabbed the hen from underneath. Of course they only discovered the creature's trick later, when they saw that the fox had disappeared along with the hen.\n\nZari wondered all the way home why Hamid had been so insistent on telling that story. Was he trying to remind Sohrab that he would never succeed in outwitting the clever British foxes? And it occurred to her that while Hamid made every effort to appear the pleasure-loving simpleton, he was in fact a very shrewd and cunning fellow.\n\nWhen they reached home, Zari switched the radio on in the hope of hearing some news of the fighting. But although she kept trying until dinner-time, she was unable to tune into the Persian newscast of Radio Berlin. They had bought the radio recently, but because it was in the parlour where it was usually hot, they didn't listen to it very often. Besides, the set was too heavy to be moved about frequently. When Yusef was in town, he would always go into the parlour at this time regardless of the heat, and play around with the radio, making some earsplitting sounds until finally he managed to find the Berlin station and the voice that carried on a stream of insults at the regime. The voice accused all influential people of being Jews and, as Yusef said, cursed them so whole-heartedly you thought it had a personal grudge against them. In the mornings, Yusef would listen to Shir-Khoda and enjoy his readings from the Shahnameh. On Fridays when Yusef was in the village, Zari tried to engage the twins in listening to Sobhi's stories on the radio. But they were too restless to stay still for half an hour.\n\nThat night after dinner, she tuned in to Iran and the World programme for international and domestic news. There was no mention of an incident in the south. She tried searching the local newspapers, but the most significant items seemed to be the obituaries. She turned to a stack of the newspapers which were sent to them from Tehran and which she collected to take to Khanom Fotouhi every other week. She opened the first newspaper. The Ministry of Provisions will be dissolved', it read. Then another headline: 'Lump sugar and sugar rationing... henceforth the ration for lump and granulated sugar will be as follows: three hundred grams of lump sugar, four hundred grams of granulated sugar\n\nIn the second newspaper there was only one item of news which vaguely interested her: 'The Fars Society will be composed of Fars residents in Tehran', followed by 'Shutdown of _Man_ _of_ _Today_ newspaper' and many more such commonplace articles. But she didn't want to give up. So she continued to search carefully through the papers every day until finally, several days later, she came across a short news item on the third page of a recently published newspaper. It read:\n\n'Reinforcement of the Semirom and Abadeh Garrisons: According to some reports, Boyer-Ahmadi and Qashqai insurgents have raided trucks carrying provisions, ammunition and clothing which were despatched by the army for the Semiron garrison. The garrison itself was attacked on 29 June, and a number of officers and soldiers were killed. The matter is currently under investigation in Tehran, and fortification of the Semirom and Abadeh garrisons is being considered.' \n\n# _16_\n\nWhen Kolu left hospital, he was too weak to be sent back to his village as Zari had vowed. They had shaved off his hair, and hung a copper crucifix around his neck which now seemed barely strong enough to support his head. His eyes were deeply sunk into their sockets, and his legs wobbled. He had been discharged too early, so Zari confined him to bed at home.\n\nKolu kept talking about a bearded man with a long black robe who always carried a book with him, and who wore a 'charm' around his neck like the one he had given Kolu, except that the chain on his was much longer. He had appeared on the day Kolu's Indian neighbour was in the throes of death. He had passed by Kolu's bed, and then Kolu had heard him chanting out loud. Kolu understood neither the bearded man's chanting nor the Indian. Actually, there, no-one understood anyone else's language except\u2014yes, except that woman with the fang-like teeth and the bearded man when he wasn't reciting verses, who both understood Kolu's language.\n\nThe Indian had walked over to Kolu one night, kissing him and crying over him as if Kolu were his own son and had kept repeating \"Sandra! Kitu! Kitu!\" In fact all he could say was Sandra or Kitu. Or did he think Kolu was called Sandra or Kitu? On his last night, Kolu had tiptoed over to him as he lay snoring, and saw the man moving his eyes and jaws in the same way his father had done before he died.\n\nBut the bearded man in black seemed to be living at the hospital because he appeared every day. At first Kolu had thought he was the prophet Hazrate Abol Fazl come to cure the sick. But when his Indian neighbour died, he was sure the man was not the prophet. At any rate, it was he who gave Kolu the 'charm' and told him to kiss it every morning, and then to go and fetch his uncle from the village so that he could get a 'charm' too.\n\nThe man in black had read Kolu three stories from the book he always carried with him. Kolu only liked one of them, the one about a shepherd boy who played the reed, just like Kolu. That boy had been friends with the King's son and had killed a giant with a slingshot. The man in black kept repeating that Jesus was everywhere and he had paid for everybody's sins with his own blood. Then he had taken Kolu by the hand and led him to the house of Jesus, which was just a very big, dark room, and Kolu had been frightened. But no matter how hard Kolu had peered around, he had not found Jesus in the room. The man in black had shown him a picture of their host, and their host's mother. She was holding a baby in her arms and sort of looked like Goldusti, Kolu's aunt.\n\nKolu had really wanted to find Jesus. But when he discovered from the man in black that Jesus was a shepherd too and was looking for his lost lambs, he felt sure Jesus had gone off to the plains and it would take him an age to find those poor creatures!\n\nEarly on Wednesday morning Yusef returned from the village. When Zari heard the knocking, she never imagined it could be her husband at the door. But she remembered that just recently he had gone to great lengths to obtain a night-pass. As she stepped out of the mosquito net to welcome him, she saw him dismount and come towards her. He was not alone. There was a man sitting astride the chestnut horse, his eyes closed. Zari had to rub her eyes to make sure she wasn't dreaming. The man was wearing Yusef's coat over his naked body. At first he appeared to be dead, since they had tied him to the saddle with ropes. But after Gholam and Yusef loosened him and lowered him gently to the ground, it was obvious he wasn't since he opened his eyes and tried to focus with an unseeing look. Blood had clotted on his right temple and his unshaven beard was white with dust. His underpants had dark red stains on them.\n\n\"Is the bathwater hot?\" Yusef asked.\n\n\"No, but we'll soon heat it up,\" replied Zari.\n\nBy the time Gholam was ready to take the stranger to the bath, Yusef had examined his wounds in the changing room, washed them with soap and water, and applied some tincture. The wounds were superficial but the man kept his eyes closed all this time.\n\nWhen they sat down to breakfast on the back verandah, Yusef explained to Zari how he had come across the man at dawn by the stream next to the Zarqan city gates. \"There he lay naked, except for his underwear and a pair of torn socks. At first we thought he was an animal or something. But when I shone my torch, I realized it was a human being who'd probably been robbed by some bandit. I dismounted, and he immediately begged to be taken into town. He said he knew of me and was on his way to our house, but his legs had given way and he'd collapsed on the ground. I told him he could still travel to the house with Seyyid Mohammad, our steward, on the back of his saddle. Then he could leave for town when he felt better. But he kept on insisting that I should take him home myself. He said I would realize later why it was so important to take him to town myself, and that if I didn't want to do it I should just let him lie there until someone else would. Well, since I'd invited a few guests for this morning I agreed to take him. At first he galloped right alongside me. But by the time we got to Baj-Gah, he couldn't even hold the reins anymore and I had to tie him to the saddle. I think he's either very tired or very frightened. We'll be seeing a lot of this sort of thing these days. He kept talking about a truck which caught fire. Maybe he's a truck driver or something.\"\n\nKolu came up to greet the master and kiss his hand. His legs still seemed a little shaky, and Zari was hoping he wouldn't fall. Yusef absently patted him on the head, as if he didn't recognize him.\n\n\"This is Kolu,\" Zari reminded him. \"He's had a narrow brush with the Angel of Death!\"\n\nWhen Kolu left, Yusef said, \"I really didn't know him at first. He's lost so much weight! I guessed this child would catch typhus too because a messenger from Kowar told me all his family had caught it. You were right, Zari. Our shepherd had typhus. It's spread through all the villages in that area. Imagine it\u2014in this heat... The messenger said our village looks abandoned. But the people haven't gone away. They're just lying sick at home. As well as all the other things I have to do, I must get a doctor and medicine to them.\"\n\n\"I doubt if you'll be able to find a doctor,\" Ameh Khanom said.\n\n\"I'll get one of Dr Abdullah Khan's assistants,\" Yusef said. Then turning to Zari he said, \"Go and wake the children, dear, I want to see them. Bring the past two weeks' newspapers for me to read, too.\" As she was getting up to go, Yusef added, \"Zari, we have a few guests today. When they come, don't let anyone disturb us. Tell Gholam to leave the garden gates open. They're coming by car.\"\n\nPassing the pantry, Zari came across Gholam carrying a plateful of fresh pistachios and hazelnuts. The outer green skin of the pistachios had a rosy blush, while the fresh hazelnuts looked like little buds severed from their leaves. Gholam told her he had found them in the mare's saddlebag. She had guessed right away that Yusef was preoccupied, otherwise he would never have returned empty-handed from the village. Each time he would bring her a seasonal offering which, when he handed it to her himself, seemed to evoke the very scents of the village with its harvests, streams and orchards.\n\nShe could hardly wait for Yusef to ask her for news so she could tell him some of the stories she'd been saving up. She noticed that Yusef was cutting sections of the newspapers and putting them aside. Soon he would be coming across the 'Semiron and Abadeh garrison' news, and she hoped he would ask her something about it. But although Yusef saw the news item, he only cut it and put it aside, without asking anything.\n\nOn Yusef's instructions, Gholam took Mina and Marjan for a ride on the mare around the gardens of the Verdy Mosque, with Khosrow following on Sadar. Although Yusef insisted that Kolu should go too, Khosrow refused to make Sahar carry two riders, so Kolu, too weak to walk so far, was told to lie down on Gholam's bed in the stables and not come out unless he was called. Khadijeh was very busy that morning and was quite happy to let Khanom take care of the guests herself. Ameh disappeared into the howzkhaneh where she planned to finish stitching in the rest of her gold dinars inside her one remaining coat. As for the stranger, he was sleeping soundly in the pantry. From time to time, Yusef would look in and listen to the sound of his breathing or would send his wife to check on him. If he woke up, Zari was to give him some food and clothing and send him on his way.\n\nYusef was pacing about anxiously in the garden, glancing towards the gates at the slightest noise. Finally a green car drove up with its headlights on. Obviously the driver had forgotten to switch the lights off, for the sun had outstripped the guests and was already caressing the tree-tops. The car stopped in front of the house by the pool. The driver stepped out, but went back to turn his lights off as soon as he noticed they were on. Zari recognized him. It was Majid Khan, one of her husband's sworn companions in the plan to take over the town's bread supplies. The other passengers were a man and two women with black chadors. Zari recognized the man as Fotouhi because of his resemblance to his sister. The 'women' she recognized as soon as they climbed out of the car to greet her. Malek Rostam and Malek Sohrab were relying more and more on the protection of the veil these days.\n\nIt looked as if they all had some important business in hand. As Zari was bringing them some tea in the parlour, she heard them shouting at each other, and she could tell from their expressions while she served them that they were not going to come to an agreement soon, either. At first all five of them paused while they took their tea pensively and without thanking her. Sohrab and Rostam had thrown off their chadors in a bundle at their feet. She picked up the garments and began to fold them with deliberation so as to listen to their talk, putting the chadors on one of the seats. Sohrab was saying, \"Khanom Zahra was a witness. She knows what I went through that night. The massacre has turned into a real nightmare for me. Now I'm ready for anything. I'll pay for the blood we shed with my own blood. Isn't that enough? I'm prepared to go on a suicide mission and destroy one of their oil docks\u2014I'll swallow gunpowder and blow myself up with gasoline next to it. I'm not afraid of death. I'm just afraid of our plan failing. Yusef Khan, why don't you devise a plan that has at least a thirty percent chance of succeeding...\"\n\nTurning to his wife, Yusef said, \"Zari, will you look in on our new guest?\"\n\nZari realized she was being politely dismissed, even though she very much wanted to stay. She went out, but stood behind the door to listen. Malek Sohrab's voice could be heard pleading, \"My uncle is still hopeful. I'm even willing to trick him into giving us at least two hundred guns. But you, Fotouhi, you insulted me. You're just as dependent on others yourself. Otherwise why would you be so concerned about how they're getting on in Stalingrad and whether or not the Russians have received weapons?\"\n\nZari felt discouraged. With three children on her hands and one more on the way, what part could she possibly have in these schemes to be standing there, eavesdropping? The children had barely been gone an hour, and she was already worried about whether they had fallen off the mare, or whether they were getting sunstroke despite the shady paths of the Verdy Mosque gardens.\n\nWhile preparing the hookah for Yusef, she reflected that, regardless of her courage or cowardice, both her upbringing and her life-style made it impossible for her to participate in anything that would jeopardize life as she knew it. One had to be prepared, physically and mentally, for any action which smelled of danger. And she was ready only for those things which ran contrary to danger. She had neither the courage nor the endurance required. It might be different if she were not so attached to her husband and children. On the one hand were Yusef's caresses, the words and the loving looks; on the other, witnessing the miracle of her children... no, a person like that could never take risks. True, she turned the treadwheel of her household, endlessly, every day; and it was no less true that from morning to night she laboured like Hossein Kazerouni with her feet and did nothing for herself with her 'free' hands\u2014where had she read that \"hands were the means to all other means\"? But the smile, the look, the voice and feel of the people she loved was her reward. Each new tooth her children had, every new curl on their little heads, their voices chirping like birds, fashioning words which then trailed each other randomly into sentences; their angelic sleep, and the softness of their skin alone\u2014all these had been her gratification. No, there was really nothing she could do. Her only act of courage would be not to hinder others who wished to be brave, and allow them to accomplish things with their free minds and hands\u2014their means to all other means.\n\nIf only the world were run by women, Zari mused, women who have given birth and cherish that which they've created. Women who value patience, forbearance, the daily grind; who know what it is to do nothing for oneself... Perhaps men risked everything in order to feel as if they have created something, because in reality they are unable to create life. If the world were run by women, Zari wondered, would there be any wars? And if one loses the blessings one has, what then?\n\nShe remembered the time when Abol-Ghassem Khan first bought a car and they all went on a hunting trip. It was before the war, and the two brothers had not fallen out yet, although Abol-Ghassem Khan occasionally complained about Yusef's methods of management  as a landlord and that he let his peasants get away with too much. The driver accidentally ran into a fawn. The poor creature lay there like a pile of broken bones. They stopped and got out to drag that wretched pile to the side of the road. Suddenly the mother appeared with another fawn at her heels. She circled her dead baby several times and then rammed herself against the car, unaware that it was made of metal. She kept charging at Abol-Ghassem Khan and Yusef and Zari, dazed and confused, staggering about on those long hind legs and appealing to each one of them with her large, dark-rimmed eyes, as if to ask, \"But why? Why?\" Abol-Ghassem Khan began to cry. The game had walked up to them on its own feet. But they turned back.\n\nZari put the glowing coals on the hookah, and took a puff herself before taking it into the parlour. On the way she looked in on the stranger, who seemed to be sobbing in his sleep. She thought of waking him, but decided against it. He was a well-built man.\n\nIn the parlour the argument was still raging. From outside she could hear Sohrab urging Yusef, \"Now that I know what's going on and have decided what to do about it, why do you want to stop the others from helping me? Are you saying I'm ambitious and dangerous and you'd hate to see me succeed?\"\n\nZari entered the room and placed the hookah in front of her husband. The air in the parlour was hot and stifling with all the doors shut, and she could see the sweat-beads on the men's foreheads. Majid had removed his coat and opened his shirt collar. She went to the cupboard and took out some fans which she placed on the table in the middle of the room. Then she took out some side-plates and knives and forks and set them noiselessly on the table.\n\n\"I'll be the only one facing danger in this plan,\" Sohrab continued. \"I know my death will be just one step away. But if I don't do it, the nightmare of our massacre will drive me mad. You say this plan is yet another kind of show... my dear fellow, don't you see I'll be courting death of my own free will?\" He put a hand to his eyes and suddenly wept. Zari stared at him in amazement and offered him a fan which she put on his lap. Sohrab quickly composed himself and smiled at Zari, saying, \"Otherwise I'd have to wait for you every Thursday to bring me bread and dates in the asylum!\" Turning to the others, he added, \"Khanom Zahra is like my own sister. I revealed my plans to her before I told any of you. Unfortunately, apart from her and her sister-in-law, some undesirable people also heard. Still it's too late for all that now. Even if you don't help me, I'll go ahead and do it. My brother will have to provide me with gunmen, and Yusef Khan must give us provisions. I myself have thirty reliable men who are willing to risk their lives.\"\n\nStrangely enough, the two water-melons which Zari had just cut open were both yellow and unripe. She took this as a bad omen. The third water-melon wasn't too bad, and she was about to cut each slice in a zig-zag pattern when she decided that her guests were too preoccupied to notice. She placed the dish of melon slices next to the map of Iran which they had spread out on the table. They were all bending over it now and Malek Sohrab put his finger at a particular spot on the map.\n\n\"If we can reach Yasuj,\" he explained, \"it's not too far to Basht. Then we can go on to Gachsaran...\"\n\n\"It'll take a long time to get the locals on our side,\" Yusef said, \"but we have no choice. This is just a first step. Meanwhile Mr Fotouhi has to create some internal diversions...\" Then turning to Zari, he said, \"Please don't make so much noise.\" Zari realized she was being asked to leave again. As she was going out, she heard Majid's voice, \"I doubt, Fotouhi, if your army of comrades will approve of such a plan. If you agree to it yourself, that's a different thing.\"\n\nOn Yusef's instructions, she set the table for lunch in the parlour. They had all removed their coats and ties and were using their fans by now.\n\nAt lunch, Yusef asked for wine, and Zari brought out two bottles of red from the cupboard. She imagined they must have reached some sort of an agreement to be asking for wine in the mid-day heat. As she was pulling on the cork, half of it broke off and the other half fell into the bottle as a result of the pressure. Yusef must have been watching her since he told her not to worry and that the cork must have been rotten. She poured wine for everybody, and they all drank her health. But she could only think to herself, \"What use is health alone?\"\n\nThey were talking and joking together, ignoring her presence, her sole function being to pass the salt here, fill a glass there or make sure Majid got the giblets which she knew he liked best.\n\n\"It would've been easier for our fathers,\" said Yusef, \"but if we don't take action, it will be harder for our sons. Our fathers had to face one usurper who became Shah and unfortunately they gave in to him, so that now we have to face two usurpers. Tomorrow there will a third, and before we know it, even more the day after that... and they'll all be guests at this table...\"\n\n\"If we achieve nothing more than showing the way to our children, we will have done enough,\" Malek Rostam said.\n\n\"Even if it's me against a whole army, I won't show them my back...\" put in Malek Sohrab.\n\n\"And for thousands of years, everyone's blood will rise in our revenge, brother!\" Malek Rostam said, and added, draining his glass, \"To the blood of Siavush!\"\n\nYusef held out his glass to be filled, but Zari was seized with such fear of the things they were saying that the pitcher slipped from her hand and broke to pieces on the floor. As she bent over to clear away the glass, she felt her throat constrict from the tears she struggled to hold back.\n\n\"Oh Lord, what kind of men are these who know what they're doing is no use, but just to prove their existence and their manhood, and just so their children won't spit on their graves, go ahead and actually dig them\u2014God forbid\u2014with their own hands...\" She bit her lip.\n\nAnd what odd things women remember at the strangest moments, Zari thought, as her mind jumped back to one night when Yusef had sighed in his sleep, and she had woken up and put on the bedside lamp, only to gaze for the longest time at the soft down on his earlobe which had looked just like pink velvet brushed the wrong way...\n\nThe stranger slept until sunset, then came out into the garden wearing Yusef's pyjamas which they had given him that morning. He sat by the pool and washed his face, and then watched Majid and Yusef playing backgammon. The other guests had left earlier that afternoon despite the heat. It was obvious from the man's demeanour, his easy movements and his comments on the backgammon game, that he was no truck driver.\n\nKhadijeh brought him some food. He ate voraciously. By the time Zari brought him the spirits he had asked for, he had already finished his meal.\n\nThe stranger stood up and looked at the garden, saying, \"You have a nice life. But it's a pity you don't have any children. There should be at least ten or twelve of them running about in this garden.\"\n\n\"Do you have any children?\" Zari asked.\n\n\"I have two sons,\" the man sighed.\n\nIt was a long time before the man got round to talking about himself, confessing that he was a lieutenant in the army. Only slowly did he warm up to his tale of the events that had befallen him. In the middle of his story, the twins arrived. The man fell silent and looked at them with envy. Yusef kissed the children and ordered Khadijeh to take them to Ameh on the roof terrace, but to watch out that they didn't fall or touch the hot coals in the brazier. The man took up his tale again, by now more involved than his audience, and then finally became so engrossed that by the time Khosrow and Hormoz arrived, he barely replied to their greeting. \n\n# _17_\n\nI was the commander of a motorized convoy travelling from Shiraz to Abadeh. All in all, we had fourteen yellow provision trucks, forty-five soldiers and five non-commissioned officers for guarding the trucks. A third-lieutenant, just out of the academy, was my immediate subordinate. He was young\u2014no more than nineteen or twenty years old. We were carrying provisions in three of the trucks, and soldiers' uniforms, gasoline, and weapons in the others. We also had an ambulance. I had verbal orders to lead the convoy to Abadeh and wait there\u2014no one had given me any written instructions. In Abadeh I received a telegram telling me to clear the needs of the Isfahan division and then proceed to Tehran.\n\nAmong my men was a fellow called Rezvani-Nejad who had accompanied me on several other missions and whom I knew well. The poor man had fourteen mouths to feed, including his parents who were blind. His brother was with us too. Both of them were warrant officers.\n\nWe spent the night at Abadeh. Late at night when we were returning from a good time out on the town, I saw a light in one of the trucks, I got in and saw Rezvani-Nejad and his brother having a little tea and dry bread. I felt sorry for them. I gave them permission to go out together and have rice and kebab at the local inn. I told them they could get the best spirits there\u2014so pure you could set fire to it. But the man said, \"Sir, don't you think we'd thought of having a drink ourselves? We would have liked to have had a good time too. But we took on this mission just to earn a two-hundred toman bonus for our children, and take it home to them.\"\n\nIn the morning we started off again, and stopped on the banks of a river by noon. We were supposed to wait there for a tank. When I got out of the truck I noticed a few tribesmen nearby. They were wearing their felt hats and cloaks. Their unsaddled horses were being watered on the other bank of the river. As soon as one of them saw us, he jumped on his horse and galloped off in the direction of the mountains. We went to a nearby orchard to eat our lunch. A few more tribesmen were there with their felt hats, but these men were wearing thin cloaks. We didn't realize they were spies. We only found that out from the tank commander. When he arrived, he told us to get into battle-formation since we would be passing through a gorge surrounded by tribesmen. Up until then we had all imagined we were on a simple mission of delivering provisions, weapons and gasoline to the Semirom garrison and returning home safe and sound.\n\nI said to the tank commander, \"But my friend, we have only a handful of men! How can we possibly traverse a route surrounded by tribesmen?\"\n\n\"They won't attack in daylight, and that's when we'll be passing through. If we start off right now, we'll reach the garrison by late afternoon. We'll return then if we can, and if we can't, we'll just spend the night at the garrison and come back first thing in the morning.\"\n\nSo we started out. But we had no sooner taken the first bend to the left, than we realized the road had been sabotaged. Every twenty metres or so the trucks either fell into potholes and stalled, or else they got stuck in deep puddles. We weren't even doing five kilometres an hour. We didn't turn the headlights on and the trucks followed each other closely. Later we found out that the man responsible for planning the road obstructions had been one of our own officers. Sentenced to death by the government, he had deserted and taken refuge with the tribe. He had even drawn up their general strategy and combat-formation.\n\nEventually the tank engine overheated and after a few yards, it stalled. It was getting dark and we could see scattered bonfires high up on the mountain. Obviously they were Qashqai and Boyer Ahmadi entrenchments. We had just reached the Khorus Galoo Pass, and they were high up on either side of us, but they were leaving us alone for the time being. They only made shrill, frightening noises like a war cry.\n\nWe decided to open the hood of the tank to let it cool down. But as soon as we did, we realized the pump had sustained several holes. The decision was to repair it by the light of a lantern, so while this was being done, I gathered up the men to dig trenches around the trucks as a precaution, with some of the soldiers on guard and others patrolling. I told everyone that we neither had the right nor the possibility of turning back.\n\nAll our truck-drivers were sergeants. The senior sergeant came over and told me the soldiers were new conscripts and had no combat experience. I ordered him to distribute their weapons among the officers and sergeants. Each of us received one rifle and fifteen bullets. We left three light machine-guns to guard the tank: two on either side, and one at the rear.\n\nWe spread out in the individual trenches which the soldiers had dug, cocked our weapons and sat ready for the attack. I had a revolver hidden under my tunic. We had no food or water, the weather was very cold, and we could neither turn back nor advance. The tank commander and a few others were working on the pump, but they never managed to repair it. Meanwhile the tribesmen kept up their shrill cries until ten o'clock that night. But they were still leaving us alone. When the tank commander\u2014he was a lieutenant like me\u2014gave up on the pump, he became so frightened he got the runs.\n\n\"There must be two thousand of them!\" he said. \"A thousand Qashqais on this side of the Khorus Galoo and a thousand Boyer Ahmadis on the other. They're going to tear us into pieces! If only we could leave the tank...\"\n\nI didn't let him finish his sentence. \"You're ruining everyone's morale,\" I said. \"Get inside the tank for now and keep the hatch shut.\"\n\nIt was well after ten, and darkness engulfed us. There was no light, no moon, no lamps. We didn't dare strike a match. All we could see were their bonfires, dotting both sides of the mountain. I ordered the sergeants not to waste any bullets but to wait until their target came well into range. Maybe help would arrive in time from Semirom or Shahreza. When the tank commander had first joined us, he'd talked of another mobilized convoy leaving from Shahr Kurd for Shahreza.\n\nIt must have been after eleven when, from behind us, I heard the sound of horses trotting. There must have been ten or twelve riders. The senior sergeant said, \"Sir, here they are!\" From the sound of the horses, I estimated them to be about thirty metres away. \"Here they come!\" said the sergeant manning the machine-gun at the rear of the convoy. The darkness was vast, and so was the silence. They had come to test us\u2014to see if we were awake. Three or four of them let out a whooping cry as they fired a few shots that rang against the metal of the trucks. Our machine-gun fire drove them away.\n\nBy dawn I could see some horsemen appearing and disappearing on the skyline. Suddenly they started down the mountain. The soil inside the trenches still felt cold from the morning air. Shots rang out against the body of the trucks. I ordered the men not to fire back. \"Shoot only when they're all the way down the slopes,\" I told them.\n\nThe sergeant who drove the last truck was an old man. Suddenly he cried out and fell. I rushed to his side. An Isfahani sergeant called to me, \"Sir! Get down! Lie flat! They're still shooting!\" I dragged the old man towards the ambulance and stretched him out on the bed inside, hoping for the best. Three minutes later, another soldier was wounded in the shoulder, and then another in the stomach and in the thigh... the men dragged those two to the ambulance and stretched them out on the beds as well. There were only three beds in the ambulance\u2014it was one of those old brown Fords with the lion and sun emblem on it.\n\nThe tribesmen crawled and slid down the mountainside. They took up positions behind the brick wall of a garden about a hundred and fifty metres from our trenches. We in turn opened fire as soon as they came within range. One Turkoman sergeant who drove the first truck volunteered to take a short-cut up to the top of the mountain and check out the enemy's situation. I refused permission because it was too light. It must have been seven, seven-thirty in the morning.\n\nFrom the top of the mountain came the sound of about sixty or seventy of them whooping and chanting: \"Army men, weapons down! Hands up! Army men, weapons down, hands up!\"\n\n\"They can go to the devil!\" I said. \"We will not surrender.\" By around nine-thirty, twelve of our men had been wounded. We heard the shrill cries again, followed by \"Attack!\" and then they swarmed down the mountain. We jumped into our trucks, and two of the drivers desperately tried to turn round. There was no other choice. The tank had to be abandoned so we could at least try to save the trucks carrying fuel and weapons. I'm ashamed to say that we had to leave the wounded behind, even though some of them were crying out...\n\nThey charged. There were about a thousand, maybe more, of them. The Turkoman driver managed to slip out from behind the steering wheel in the nick of time, but the driver of the truck in front was shot so our way was blocked. We were forced to get out then. The tribesmen were crawling forwards on their bellies, firing away all the time. I had only one bullet left. Now they were just ten steps away. Rezvani-Nejad raised his head to shoot, and fell. He cried out, \"Khandan!\" as he rolled to the ground. I imagine it was his child's name. The poor man had fourteen mouths to feed. The bullet had blown his brains out\u2014I saw the white of his brain with my own eyes. His brother ran to help him, but they shot him too. The bullets seemed to nail the two brothers together. I kneeled and aimed with my one remaining bullet at the man who had killed them. I got him in the middle of the chest. His friend ran to him, wailing, \"Did he hurt you, Zargham?\"\n\nI crawled underneath the weapons truck, and gradually managed to pull myself into one of the trenches. The sergeant inside the trench was dead. I stretched myself out on top of the dead man like another bloodied corpse. The Boyer Ahmadis were coming at us at a gallop, and once or twice they jumped over my head, covering me with dust. Then the looting began. First they took our weapons, and then I could hear their women ululating and repeating the shrill war-cry. I heard that chant so many times, I learned it by heart:\n\n\"Up the pass, down the pass, there's a camp, Sohrab Khan, look ahead, look ahead, how many thousand are there?\"\n\nAnd:\n\n\"Drunken drunken through and through\n\nI hold the army in my hand.\n\nDrunken drunken through and through\n\nI hold a rifle in my hand.\"\n\nA Qashqai loomed over my head and dug his heel into my shoulder. \"You dog, you're alive! Get up, I saw you lie down. Give us the new gun, get up, get up!\" A short, dark Boyer Ahmadi arrived just then. As I handed my gun to the Qashqai, the two men began to fight each other for it until the Boyer Ahmadi killed the Qashqai and grabbed the gun. Again I stretched out on top of the dead man in the trench, close to passing out from thirst and fatigue, and trembling with anger. By this time the Qashqai and Boyer Ahmadi women had arrived and were throwing out the sacks of provisions from the trucks. They tore them open and poured the tea, sugar and rice, beans and peas into their own sacks. I saw a Boyer Ahmadi take the gun belonging to the third-lieutenant, the one who had just graduated from the academy, and make him undress. Stark naked. The boy grabbed a piece of canvas to cover his genitals, but one of the women immediately snatched the rag from him and used it to collect some onions. Finally the women and children of the nearby village arrived on donkeys and filled their saddlebags with whatever remained.\n\nWe had left three wounded sergeants in the ambulance which was clearly marked with the lion and sun emblem. But they didn't realize, and set the ambulance on fire. You could smell the burnt flesh for a long time. And then they set the fuel truck on fire.\n\nAgain a Qashqai came along to where I was lying and kicked me in the shoulder, saying, \"Get up! Take off your jacket!\" I gathered all my strength and threw him bodily on to the burning fuel truck. But almost immediately another rider came towards me. He was a thin, dark man, carrying a baton spiked with a knife. His gun was fastened to his belt. He too, wanted my uniform. He said he wouldn't kill me so the clothes wouldn't be bloodied. He took my uniform and gold medals and army boots. Then he ripped off my watch with his knife, and with the same knife cut loose the revolver at my waist. Finally, the tribesmen drove away in the two undamaged trucks which contained military uniforms and ammunition. Later I heard they used those uniforms as disguise for a surprise attack on the Semirom garrison.\n\nI ran off in the direction of the mountains. On the way, I heard a moaning in the distance. I decided I'd find the person and steal his clothes. It turned out to be one of our own sergeant-drivers. He was spattered with blood. I asked if he was shot, and he said he'd managed to escape in time by giving up his gun. \"Get up and come with me, then,\" I told him. He pleaded, \"Captain, I beg you, my suitcase, my souvenirs from Shiraz...\" I interrupted him, \"From now on, we're equals.\" And we started up the mountain. We passed the tribesmen's entrenchments, made of white stone and each taking four people, but now littered with empty cartridge shells.\n\nWe were heading towards Abadeh by way of a side-track, and we had just passed the mountain ridge when we noticed a Qashqai rider approaching us at a gallop. We threw ourselves on the ground beneath a bush. Before long he was standing over our heads and saying, \"Hey you army dogs! Get up! I saw you.\" Eyeing the sergeant he said, \"Is it you, Mirza Hassan, you bastard? Where's your gun?\" The sergeant sat up, and started to undress of his own accord. Standing in nothing but his underwear, he took off his army boots and handed everything in a neat bundle to the Qashqai.\n\n\"How fat you've become, Mirza Hassan, you bastard!\" said the Qashqai.\n\n\"You'll be wasting two bullets if you kill us,\" I told him. \"Don't shoot us. On the other side of the mountain they're looting truck-loads of goods\u2014rice, chick peas, beans, lump sugar, tea, onions, oil, military uniforms, ammunition and guns. If you hurry you'll get there in time.\"\n\n\"Is he telling the truth, Mirza Hassan?\" he asked the sergeant.\n\n\"Yes, brother.\"\n\nThe Qashqai took out a pair of delicate women's slippers from his saddlebag and said, \"This piece of softskin is for you, Mirza Hassan.\"\n\n\"Keep them, I can't use them. Give them to Sister Golabtoon and greet her for me.\"\n\n\"I'm taking a flowery tunic for Golabtoon. And a gold necklace and mirrors, I don't need these.\"\n\n\"Then hurry so you can take her some provisions too,\" I said to the tribesman.\n\nWhen he had left, I asked the sergeant, \"Are you related?\"\n\n\"Yes, we're cousins. But my name isn't Mirza Hassan. That's the name they give to thieves.\"\n\nBy then I think it was almost two o'clock in the afternoon. A government airplane buzzed over our heads and circled around the remains of the convoy. There were a few retaliating shots from the Qashqais and Boyer Ahmadis, and then it roared away again. So much for aerial military reinforcement!\n\nNow the two of us were left thirsty, hungry and barefoot, wearing nothing but underwear, and holding on to a pair of women's delicate sandals which didn't fit either of us. We made our way down the ridge of the mountain until we reached the valley where we found a spring and washed our faces in its muddy water. The sergeant announced that he couldn't go on anymore and lay down wearily right there. \"As you like,\" I told him. \"I'm carrying on without you.\" But I walked on very slowly. I hadn't gone a hundred metres before I heard him call me. \"Captain,\" he said, as he caught up with me, \"I wanted to go to sister Golabtoon's tent. It isn't too far from here. But to tell you the truth I felt too ashamed.\" I didn't say a word. Soon we had left the valley and we could see several villages ahead of us, with crowds of people milling about.\n\nWe caught up with an old man, a pedlar, following a child riding a donkey. He had a small piece of bread, and gave us half of it, but no water. We said we were truck drivers, that bandits had raided us and had set our trucks on fire. He told us that the river was only a kilometre away but that we should be careful because the Qashqais and Boyer Ahmadis had taken to the mountains, and had been fighting government men on the other side.\n\nIt was early evening when we reached the river and drank some water. I told the sergeant not to drink too much because he would get bloated. We rested for about ten minutes, and then waded across the river. On the other bank we saw two Boyer Ahmadis sitting around a fire, having some tea. They asked us who we were and where we were going. We told them the Qashqais had robbed us and that we were truck drivers. The sergeant asked one of them who was smoking a pipe to give him a puff. When we gave the pipe back to him there was nothing left in it, and the man dumped it on the ground. He gave us a drink from his water-skin, and then sent us on our way.\n\nWe joined a few peasants headed towards the village. Again, we were asked who we were, and again we told them we were truck drivers. After a long trek, we finally reached Abadeh at eight o'clock in the evening. We found the police-sergeant who'd been left in charge of the garrison. He told us the deputy chief was at the teahouse, but the garrison chief himself had gone to Shiraz. We were taken to the deputy chief at the teahouse, and I told him how they had set up a fine trap for us\u2014looting, killing and burning as they went. We had some sweetened tea before going to the garrison. There the deputy chief called his assistant and said, \"This is the lieutenant. Come and listen to what he has to say. It's not as simple as we were told\u2014it was worse than Judgement Day! Their soldiers didn't even know how to shoot, and they were crying from fear in front of their lieutenant.\" Then turning to me, he said, \"When your convoy left Abadeh I was relieved, thinking that the poor colonel at Semirom won't be begging for help behind his wireless anymore. You'd be taking them reinforcements. But now... God help them!\"\n\nHe instructed his assistant to bolt the tower door, issued orders for protective measures, and went behind the wireless himself to report the situation to the gendarmerie and ask for help. He was quite sure they would be attacked that night. They did their best to find some us clothes from here and there, and then scraped together some money to give us. Those old clothes and shoes felt like a great blessing to us. The deputy chief said, \"Wash yourselves and then go to the village headman's house for the night, but whatever you do, don't tell them you're officers. If they find out, they'll kill you before morning.\" A gendarme accompanied us, past the local sheep-fold, to the headman's house. The headman, who had a red beard, came out of his room and led us to a bare, mud-built room. He took two old quilts from the top of a wooden chest in the corner of the room and spread them on the floor. He asked us if we'd eaten and we said no. So his daughter brought us some dirty-looking milk in a black bowl, and two loaves of brown bread which she took out from the wooden chest that was kept under lock and key. We slept like logs till the morning. They never found out we were officers. In the morning they gave us more brown bread and hot tea before sending us back to the deputy chief at the garrison. He was even kinder than the night before, allowing us to wait around until noon while he tried to get me permission to go back to Tehran. By then, five more people, wounded and half-naked, had straggled into the garrison at Abadeh with the aid of some peasants. They were patrols from the Semirom garrison. They told us that the real battle had begun only yesterday evening.\n\nFinally the deputy chief managed to contact the gendarmerie. He was instructed to help us out, but we were all to return immediately to Shiraz. The deputy chief agreed to find us two or three donkeys so we could head off to Deh Bid, hitching a ride as soon as we found a car that would take us. We treated the wounds of the injured as much as we could, and the deputy chief found some civilian clothes for us to wear. He also gave me eighty tomans. Meanwhile our Turkoman driver showed up, riding a Qashqai mare. He was the only one to have escaped safe and sound. Apparently a Qashqai had taken his gun, then left his horse in his care to go off looting. As soon as the Qashqai's back was turned, the driver had jumped on the mare and galloped straight to Abadeh. He had spent the night in a safe place, and been given rice and stew and a yoghurt drink. He'd even gone to the baths in the morning and been regaled with a massage and refreshments.\n\nThe injured rode on the donkeys while we followed on foot, taking turns on the Qashqai mare which the Turkoman driver had brought us like an unexpected blessing. We had some bread and cheese and a jug of water with us, and managed to reach Deh Bid by ten o'clock that night. At the town gate, we came across an officer with a riding crop and high boots. I looked him over and told him I was an officer too. His crop, shiny boots and officer's uniform were all brand new. I told him briefly what had happened to us and asked him for a car to take us to Shiraz. He said, \"The whole area has been taken over by bandits. No cars can pass through.\" He took us to the gendarmerie, where their chief welcomed us and said he'd been expecting us since he'd had news from Abadeh. They served us roast chicken, yoghurt with cucumber, and spirits to drink. We had just sat down to our meal, and the chief had gone to use the wireless, when the tribesmen arrived. But this time they weren't Qashqais or Boyer Ahmadis, they were Doshman Ziaris. The chief was shot right there behind his wireless. If looting and raiding was profitable for two tribes, why not for a third too?\n\nI haven't seen the others since then. I managed to escape on my own from the back of the garrison tower, running down the mountainside until I reached an open field. After a while I came to a walnut tree, and I wanted to lie down right there to sleep, but it was cold and dark, and I could hear shooting going on all around me, so I decided to pace about or jog to keep awake. There was no moonlight, no stars, no lamp. I didn't have any matches, but I had the eighty tomans that the deputy chief had given me. If I'd had matches, I would have made a fire with the bills and gone to sleep next to it.\n\nIn the morning two shepherds came along with their flock. I greeted them and told them I was a truck driver, I'd been robbed and I was hungry. The shepherds made a fire and one of them milked a sheep and gave me the milk in a dirty bowl. His son, a seven-or eight-year-old, showed up just then holding a loaf of bread. He told his father, \"I ran all the way. It's still piping hot!\" He was right, the brown bread was still warm. Suddenly we heard shots being fired and a bullet pierced the milk bowl. It was the tribesmen from the night before. Some of them went for the sheep which they herded off, shooting the two sheep dogs on their way down the mountain. A few others came towards us and tied us up, though they left the child alone. They made us walk ahead of them all the way to their tents. The tribal chief was sitting on a chair in front of his tent.\n\nOn the way, I had whispered to the shepherd boy to throw himself, on our arrival, at the chieftain's feet and beg him on the life of his children to spare his father and uncles. I told him I'd give him a reward when they freed us. The boy did as he was told, and the tribesmen spared our lives, but we were held for six days and then they stripped us naked before letting us go. They had taken my eighty tomans the very first day, and again I found myself trudging along, on and on, until I managed to reach Zarqan, where you found me.\n\nJust imagine what happened to those poor bastards in Semirom! That's where the real massacre took place\u2014at the garrison and on the Semirom plain. Those patrol soldiers whom we bandaged at Abadeh, told us that they'd had only one day's ration for four days. I knew they weren't equipped to put up any kind of resistance for long. They had guns, but no bullets. And of course we never managed to get any to them. The same officer at the Deh Bid gate with the riding crop and the boots, told me that the Boyer Ahmadis and the Qashqais had sent a letter to the colonel at Semirom saying he'd been sentenced to death and that he should surrender. The colonel had written back that he would sooner die than do such a thing. The poor colonel had given up on the Isfahan division, and had resorted to the Abadeh garrison. Now the Semirom wireless was dead.\n\nOne of the fellows from Semirom, whose arm injury I treated myself, told me on the way to Deh Bid that they'd seen the approach of the tribesmen through their binoculars, spotting three mules carrying machine guns. I asked him whether he'd seen the military tank they'd stolen from us. He said, \"They'd set the tank on fire, and we could see it burning as we ran away. We warned our poor colonel of their approach. He first made us pitch tents on four sides of the stream, so we wouldn't be hard up for water. Then we dug trenches all around the tents. He'd planned a circular defence, you see. The poor man kept urging us to resist. He believed we could mow them down with the crack of our machine-gun fire. He was certain help was on its way since the Abadeh wireless had said that the convoy had set off in our direction. To those of us on patrol, he promised a good reward for sighting the first vehicles of the convoy. He said to us, These people have no heavy arms and their firing range isn't more than four hundred metres.' When we told him the tribesmen were advancing with three machine guns on their mules, he paled. He realized then that the reinforcement convoy had been attacked. As soon as he lost all hope of your arrival, he was forced to change his defence tactics. Guessing that they would probably approach by way of a back-road following the Khorus Galoo Pass, he ordered combat formation, with the soldiers taking up positions on top of two high promontories on either side of the back-road. But those poor soldiers only had one bullet each. On the promontories there was a half-decayed brush made up of thorn-bushes, almond and lotus trees. The soldiers lay in ambush under this shelter. At the foot of the hills, we rapidly set up first-aid and food tents. But what food and what first aid! One day's ration for four!\"\n\nThe fellow from Semirom described the attack for me. He said the tribesmen charged from three sides, with a blare of trumpets and drums which echoed awesomely in the mountains. The Boyer Ahmadis had headed down from the north-eastern parts of Semirom, while the Qashqais charged from the north-west, with another group descending from the heights of the Denna mountains. They had approached through the orchards and vineyards, gradually tightening their circle. \"A mounted captain, the first lieutenant of the artillery, and some other non-commissioned officers as well as myself, had gone inside the Semirom garrison tower to dissuade the colonel from fighting back. We wanted him to put up the white flag. But the colonel was obstinate. He just sat behind his desk, hand under his chin, and after hearing us out, merely shook his head and asked if anyone had a cigarette. The captain begged him, 'This isn't a battle anymore; we're just waiting to be butchered.'\n\n\"'Maybe help will arrive at the last minute,' the colonel had said sadly.\n\n\"'But sir, you've been trying for ten days behind that wireless\u2014where on earth can the reinforcements be? Why are you putting up such a brave front and getting us all killed in the process? For whom?'\n\n\"'I'm not forcing you to stay. I'm staying myself. But you must forget about the white flag.'\"\n\nNo sooner had those men come out of the garrison tower than the shooting began. One of them, the same fellow who told me all this, was wounded in the arm. He tied a handkerchief around the wound and managed to get himself to the village of Semirom. There he was told by the villagers that groups of Qashqais and Boyer Ahmadis were turning up all the time, picking up military uniforms. Apparently, the plan was for the disguised tribesmen to penetrate right into the garrison and mingle with the soldiers, who probably rejoiced for one short instant that the long-awaited help had finally arrived!\n\nWhen the lieutenant's story was finished, Majid stood up, yawned and said, \"What a small world it is!\"\n\n\"There is no escaping one's deeds...\" Zari said pensively.\n\n\"My dear, you're beginning to understand quite a lot of things, aren't you?\" Yusef observed with a laugh.\n\nThe next day, the stranger, who was no longer a stranger, left for the army headquarters, wearing Yusef's ill-fitting clothes. They heard no more of him until a week later, when his letter arrived from Tehran thanking them and telling of his forthcoming court-martial. There was a whole file of trumped-up charges against him and, he said, it was not unlike the story of the famous coppersmith in Shushtar having to pay penance for the crimes of the infamous blacksmith in Balkh. He was resolved to resign from the army and go to Switzerland by whatever means, with his wife and two sons. But he made no mention of the two hundred tomans he had borrowed from Yusef. \n\n# _18_\n\nOn Thursday afternoon Zari went to the asylum. The warden was not there, so she set out on her rounds with the head nurse. She knew Khanom Fotouhi would be angry when she saw that parts of the newspapers had been cut out. In the women's ward, only the paralysed woman, who hugged her givehs every night, and Khanom Fotouhi remained out of all the others. But there was no shortage of new patients. Four strangers were sitting on the other beds, and a folding screen hid another newcomer. In the middle of the room three patients sat around on a straw mat, playing a children's game called \"Away flies the crow\". As soon as Zari walked in, one of them said, \"Away fly the bread and dates!\" Zari smiled at them. Fortunately she had brought bread and fruit which Gholam placed on the floor. One of the women said, \"Away flies the princess!\" Then they started to fight amongst themselves and played another game.\n\nThe head nurse didn't let her go near the bed which was protected by the folding screen.\n\n\"This patient has already received a lot of flowers and fruit,\" she whispered. \"Only she can't swallow any food. Right now she's on a drip. They're setting up a private room for her. Her relatives say it's all the strain and overwork in this heat, but the doctor says it's both from stress and typhus. May the Lord cure her!\" She added,\n\n\"There's a woman who comes here late at night after everyone's gone to sleep, does her ablutions and says a special prayer to Hazrate Fatemeh for her. Ezzat, the nurse who was on night-duty last night, said the woman stayed praying with her forehead glued to the ground for so long, she became worried. Ezzat went closer and was relieved to hear the woman repeating, \"O Fatemeh save her! Save her!\" She repeated it fifty, a hundred times, pleading with God. The woman had to sleep here last night as it was well past the curfew. Now Dr Abdullah Khan has prescribed donkey's meat for the patient... if they manage to find it, that is. They have to make her meat patties for dinner tonight; maybe she'll be tempted to eat.\"\n\nWhen she had finished distributing the food, Zari went up to Khanom Fotouhi who was sitting with her back to the patients, staring out the window. Zari said hello to her, left the papers by the bed and stood a little distance away. She knew that the moment the woman opened the newspapers, the incident of two weeks before would be repeated. Khanom Fotouhi suddenly jumped up from her bed. \"My brother!\" she exclaimed. \"I had a feeling my brother would come and take me to our hundred and twenty-four thousand metre garden!\"\n\nZari looked out of the window, but could not see anyone. Khanom Fotouhi brushed past her and left the room saying, \"The rudeness of it all! You stupid fools, I'll show you what I mean!\"\n\nBefore long she was back in the room accompanied by her brother who had truly arrived this time. Khanom Fotouhi sat on the bed and started to cry.\n\n\"Why did you come alone, brother?\" she asked. \"Why didn't mother come? After all this time, you've come empty-handed!\"\n\nMr Fotouhi greeted Zari who was about to go out and leave the brother and sister by themselves.\n\n\"Khanom Zahra,\" he said, \"I have something to tell you.\"\n\n\"Tell her to get lost,\" shouted Khanom Fotouhi furiously. \"Every week she comes here with a lot of fuss and bother to show off for me!\" And she asked again, \"Why hasn't mother come? Take me to the hundred and twenty-four thousand metre garden... my heart is withering in this cage. What kind of brother are you? You should at least get a private room for your sister...\"\n\nShe clutched her brother's hand tightly, kissing it and rubbing her tearful eyes on the dark, veined skin, asking over and over again why her mother had not come. She worried about whether her enemies had confiscated their garden... those enemies who constantly put electric currents through her body, her hands, her feet, her heart, making her heart beat backwards. She placed his hand on her heart and said, \"You see!\"\n\nThe head nurse and Gholam and all the other patients were staring, even those who had been playing \"Away flies the crow\" a moment ago. Fotouhi kissed his sister on her fair, tousled hair, and said, \"My dear, you know very well our mother's dead. I've told you that a hundred times.\"\n\n\"But you see, brother, I know my mother isn't dead. She's tricked you. When you put her in her coffin, she slipped out quietly and went into hiding. All this time she's been hiding somewhere in the hundred and twenty-four thousand metre garden and you haven't even tried to find her.\" She swallowed and said, \"I swear to God they came in the middle of the night last night and dug out my liver with a knife and stuffed some straw in its place. Since this morning my mouth tastes like straw.\"\n\n\"My dear, since when do we have a hundred and twenty-four thousand metre garden?\" Fotouhi said impatiently.\n\n\"Take me away,\" Khanom Fotouhi begged. \"I'll be like a servant to you. We'll live together, all by ourselves. We'll plant wheat in that huge garden. We'll plant mulberry trees and cucumbers, keep beehives, and I'll bake bread myself. We'll keep hens and a rooster and hatch chicks. We won't let anyone in, either... we'll buy narcissus bulbs and wrap them in cotton wool in our Kashkouli pot...\"\n\nA nurse came in and whispered something to the head nurse, who then said out loud, \"All right, arrange the flowers around the room, and put the fruit on the table. No, come back. Help me take the patient out. Wait a minute, take the suitcases first.\"\n\nThe nurse went behind the screen and came out with two new suitcases. Gholam helped her with one of the suitcases, and they left the room together.\n\nZari and Mr Fotouhi went outside too. They stood under a dust-covered pine tree in the flowerless courtyard of the asylum.\n\n\"I came here to see you,\" Fotouhi said. \"Yusef Khan probably told you I would come today to inform everyone of my decision using you as intermediary.\"\n\nNo, Yusef had not told her anything. Perhaps he had wanted their meeting to look as natural as possible.\n\n\"Since yesterday, I've investigated every aspect of the plan,\" Fotouhi said, \"and this morning I discussed the matter at the party leaders' meeting. Of course without mentioning any names, and more as a suggestion of my own. Everyone opposed it.\" He seemed nervous, shifting from one foot to the other, and talking in clipped phrases. \"You know we haven't officially announced the existence of our party yet,\" he continued; \"we're waiting for the right moment. But how would it look if I were to leave the comrades and go south to Khuzestan with a group of like-minded friends in a plan the comrades oppose... you realize I'm responsible for my students too. In my group... with a group of young boys, what can I do?\"\n\n\"So they were right,\" Zari said bluntly, \"they shouldn't have asked you to join in their plan. You don't care about your own friends, any more than you care about your sister.\" She was amazed at her own harshness, although she had been harbouring resentment against Fotouhi for some time now. Yet Fotouhi answered her without the slightest appearance of being upset.\n\n\"We must build our society in such a way that no-one's sister ends up having a mental breakdown. My sister's condition is the symptom of a social disease. When we eventually organize the masses and come into power, we will see to it that justice is carried out.\" Then he added after a pause, \"In my opinion the time is not ripe for their plan, and the only result will be chaos and anarchy. It's not as easy as Malek Sohrab thinks. I don't believe they should allow themselves to be led by a hot-headed fellow like him. And I'm sure they won't. After all, Yusef Khan has more experience than any of us, and even he said that without a forty percent chance of success, running the risks they have in mind is tantamount to suicide.\"\n\nThe words had hardly left his lips when Khanom Fotouhi appeared, coming towards them wrapped up in a white sheet which kept tripping her as she walked.\n\n\"Kill me and let me have some peace!\" she shouted. \"Take out your pen-knife from your coat pocket and kill me! I've put on my shroud and I'm ready!\"\n\nShe let go of the sheet when she reached them, exposing her stark naked body underneath. The nurses immediately rushed to her, but she fought them off, hitting one nurse sharply with her elbow.\n\n\"You bastard!\" she shouted, shaking a fist at her brother. \"Meeting under the pine-tree, is it?\" She held her own against all the nurses as they struggled to pin her down. \"You stole my property! You sold my hundred and twenty-four thousand metre garden to pay for this whore...\" Then she ran around the empty pool in the courtyard, dodging everyone and screaming, \"People, I want you to know I'm the greatest woman of this nation! I'm a poet. I've composed fifty thousand verses. This whore has stolen my verses...\" She gasped for breath for a moment, then went on, \"This whore has given my verses to the _Red_ _Aurora_ newspaper under her own name. I'm the Prophet's daughter, Hazrate Fatemeh... I'm pure and chaste like Hazrate Fatemeh herself. My brother's stolen my possessions... he executed my mother and father... all these flowers you see... springing from their blood... put these flowers in a bunch on my grave\u2014\" And she cried with abandon. \"Oh the fatherless wretch that I am! How wretched...\" she sobbed, and she went on and on until she started frothing at the mouth and collapsed. The female nurses covered her naked body with a chador and a well-built fellow came forward and picked her up to take her to the office.\n\nZari was tired and her head was aching. Telling the head nurse so, she began to take her leave.\n\n\"Thank goodness our new patient, Khanom Massihadem, is feeling much better,\" the nurse said. \"We've transferred her to a private room, and she can have visitors now. Why don't you go and wait there while I bring you a pain-killer or something.\"\n\nSo the patient who was getting such special treatment was the new midwife! Zari knocked and went in. Khanom Massihadem was sitting on the bed and shaking her head from left to right, sending a mass of black curls around her head and letting them sweep her face from side to side. The room was filled with flowers. Some of the bouquets had obviously been arranged by the caring hands of a lady gardener, and some of the others were made up of rare wild flowers which someone must have searched for in distant fields or plains. Khanom Massihadem went on shaking her head from side to side, and paid no attention either to the flowers or to the crystal bowls arranged tastefully on the table in the middle of the room. The bowls were covered with lids, and Zari guessed that they must be filled with all kinds of home-made sweets, painstakingly prepared for the patient in that heat.\n\nEventually Khanom Massihadem tired of shaking her head. She noticed Zari for the first time. Zari said hello, while Khanom Massihadem stared at her with a vacant gaze. Despairing eyes, set in a young but skeletal face. Her collar-bones stuck out from underneath her thin white night-dress. Her breasts sagged, and her complexion had a jaundiced look, paler than the sunshine touching the last row of bricks on the opposite wall.\n\n\"I hear you're feeling much better,\" Zari said.\n\n\"I've heard this voice somewhere before!\" replied Khanom Massihadem, biting her nail and staring at her hard. Then suddenly she burst out laughing as recognition came into her eyes. \"I know you! I know you! You're Tal'at Khanom!\" She clasped a hand to her heart as she said, \"How frightened I was! So you're alive. I knew God would answer my prayers. I asked God to take six months off my life, but to keep you from dying at my hands. Come closer so I can see you with my own eyes.\"\n\nZari knew Khanom Massihadem was mistaken, but she kept quiet. If the poor woman could smile and her eyes brighten up at thought of some friend or sister or patient being alive, why should that joy be taken away from her? Zari sat next to her on the bed. Khanom Massihadem took Zari's hand in hers and pressed it. Then she explained in a surprisingly sane manner,\n\n\"When it's born, if its complexion is pink as a petunia, if it screams until the mother can hear it, or if it pisses\u2014\" she put her other hand to her mouth and suppressed an innocent giggle, \"then all the tiredness seems to go out of your body. And you feel so satisfied, as if you yourself created the baby! But when your child came out, your first one too, dear oh dear, he had no colour. There was no blood in the umbilical cord. I hit him, I hit him hard, but he wouldn't scream. I felt the weight of a mountain on my shoulders. It was the first still-born child I had delivered. Suddenly I noticed you weren't bleeding either. I knew the blood would be running somewhere into your stomach, filling it up until it stretched out like a drum. I palpated your belly. But oh God, your eyes turned up, your pulse disappeared, your heart stopped. I heard the front door slam. Your mother had gone out into the street. Your husband came in and said, 'You killed them both? You murderer!'\" And she pressed Zari's hand even harder, complaining, \"But if you hadn't died, why were you pretending? Why?\"\n\nZari didn't reply, and Khanom Massihadem continued, \"You know, we doctors have to get used to death. We mustn't be afraid of the signs. But I panicked. It was as though a storm was raging inside my head, tearing out all the wires of my nerves and brain and jumbling them up in a heap. It was as if my heart had sunk down to my feet. These people think I've gone mad, but I haven't. I'm just very, very unhappy.\"\n\nZari tried to get up, but the young woman would not let go of her hand.\n\n\"I saw the ceiling part with my own eyes,\" she was saying, \"and a black-robed, winged person came down and took you away under his wings. But they won't believe me. I begged that black-robed person to spare Tal'at and take away my life instead. But he said he was taking her to heaven, under the Tuba tree. 'Take me instead,' I said... Now, for goodness sake, Tal'at, tell me how come he brought you back? Do you mean to tell me there was no room in heaven?\"\n\nShe was squeezing Zari's hand very hard and carrying on rapidly, \"Now, will you do something for me? You know that I've promised to go away in your place?\"\n\n\"Of course.\"\n\n\"Buy a few grams of good opium,\" she whispered in Zari's ear, \"and crush it well. Bring it to me before this evening, before the old man comes. But don't tell anyone anything. If the old man is here when you come, just drop it quietly in my lap and go. All right?\"\n\nZari bit her lip. Khanom Massihadem burst into tears and said, \"When the sun starts going down, I get so depressed... it's as though they're piling a ton of steel on my heart.\"\n\nAgain she began to shake her head. The long hair brushed Zari's face as she tried to pull herself away and free her hand. But she couldn't manage it. And all this time, Zari felt her head was about to explode with pain.\n\nFinally a white-haired old man leaning on a cane entered the room. Zari guessed with relief that it must be Dr Abdullah Khan. The old man went to the patient and placed a hand on her shoulder.\n\n\"But my dear, you've started it again!\" His voice was not authoritative, but infinitely soothing. The patient stopped her head-shaking and smiled at him.\n\n\"I kept her here so you could see her with your own eyes,\" she said. \"Do you see? There wasn't any room, so they sent her back...\"\n\n\"Has she been talking a lot of nonsense?\" the old man asked Zari softly.\n\n\"On the contrary,\" Zari replied quietly, \"she made a lot of sense.\"\n\n\"You see how the old man is going senile?\" shouted Khanom Massihadem abruptly. \"Why don't you ask Tal'at what goes on on the other side? What happens after the end? Because she's been there and back, you know. I thought she'd disappeared into that pitcher of water and I was too frightened to drink. Or I'd think she's gone inside the flowers and I wouldn't look at them. It's all the rubbish you've been saying, you daft old man, and now my brain's out of order.\" Then she mimicked the old man, \"'Only death is true, the rest is a lie.' Tal'at, for God's sake tell him that Death had wings and took you away. He keeps telling me I've tired myself out and I'm just imagining things.\"\n\n\"I have to go now,\" Zari said.\n\nThe old man accompanied her to the door. \"I've brought a pair of scissors to cut her hair,\" he whispered; \"she can't stand to see her mother and relatives, and won't let them near her. Do you know how to cut hair? It really gets in her way.\"\n\n\"I know how to do it, but it's getting late and I'm expecting guests tonight.\"\n\n\"Can't you spare five minutes?\"\n\nMaybe the patient heard the old man's whisper or had guessed what he was saying. \"Have you gone mad?\" she screamed, clutching her hair tightly with both hands.\n\n\"Your hair will grow out thicker than ever in less than a month, my dear,\" he said. \"By then you'll be healthier yourself and have a little more weight on you. I want to throw sugar-plums over your head with my own hands at your wedding. But hurry up and get well, my dear. I'm an old man.\"\n\nWhat a soothing voice he has, Zari marvelled. He could tame anyone with that voice\u2014a person with delusions, a person in a hurry...\n\nKhanom Massihadem motioned to Zari, saying, \"Come closer, I want to tell you something in your ear.\"\n\nTurning to the old man, she said, \"You go to the end of the room and shut your ears.\"\n\nZari was forced to bring her head close to the woman while she whispered, \"When you cut my hair, plunge the sharp end of the scissor into my artery, will you?\"\n\nThen she sat obediently while Zari wet her hair, combed it, and cut it short like a boy's. When she'd finished, Zari handed the scissors back to the old man. For a minute their eyes met; Zari looked into his bright and lively gaze which belied his age. The old man nodded knowingly and Zari realized he had guessed her secret. The old man put the scissors back in his pocket and Zari said goodbye, not certain whether the sparkle in those eyes was somehow a reflection of the snowy-white eyebrows or whether it was from his new-found knowledge. Khanom Massihadem, who had been staring at them, suddenly shouted, \"Get lost! Go drown yourself. Go to the other side...\" And again she started to shake her head.\n\nZari was about to step out of the room when the head nurse arrived with a pill wrapped in some paper. She gave it to Zari.\n\n\"I had to go out and buy it for you,\" she said. \"The warden went to the Department of Health this morning for our supplies of medicine, and he's not back yet. We've no drugs at all. If we don't get some by tonight, with all these lunatics...\" She didn't finish her sentence, but walked over to the pitcher of water in the corner of the room. She took the glass from the top of the pitcher while Zari unwrapped the paper to take out the pill.\n\n\"Wait, Khanom Zahra. Pain-killers are not too good for pregnant women,\" said the doctor.\n\n\"Do you know me?\" Zari asked in amazement. After a pause, she said, \"I recognized you too. You're Dr Abdullah Khan.\"\n\nAnd again she stared at him. The man looked as if he had knowledge of all the secrets in the world. \"If only his fingers would touch my forehead...\" she thought, \"this is a man who's healed people all his life; he has comforted them, guarded their secrets and only brought them to their attention for their own good.\"\n\nBut Zari was in a hurry. She had to get home as quickly as possible. Her headache was getting worse, and her heart felt no lighter than Khanom Massihadem's. McMahon was coming to dinner, and she kept praying he hadn't arrived yet so she could at least rest for half an hour in a darkened room.\n\nOutside, Gholam was sitting in the droshke next to the driver, smoking a pipe. When he saw her, he jumped out, emptied his pipe, and helped her to get in. The droshke seemed to move along so slowly, the horses shying each time they passed a car, and Zari began to feel as if they would never get home. But they did, finally.\n\nYusef and McMahon were sitting in the cane chairs on the pavement in front of the house. Mina and Marjan were sitting on McMahon's lap, leaning over the table. With one hand he was holding the children and with the other he was turning the pages of a book they were looking at. When Zari reached the twins, they laughed and clapped their hands. The men and the children seemed very cheerful. But Zari knew that if she sat down next to them, some of the sadness in her heart would infect them too. With her splitting headache, she hadn't the strength to smile and put on a pleasant face. When McMahon saw her, he carefully lowered the children to the ground and rose to his feet. They shook hands.\n\n\"I'm sorry I'm late,\" Zari apologized. \"I'll go to the kitchen for a minute and then I'll be with you.\"\n\nShe went straight to the bedroom and threw herself on the bed fully dressed, burying her head in the pillow and with it the pain that was radiating from her eyes, ears and left jaw. \"If this pain doesn't go away,\" she thought, \"I'll ruin their evening.\" She decided for a moment to take two aspirins, then she remembered Dr Abdullah Khan's words and changed her mind. The old man had not spent a lifetime treating people for nothing! He was wise, and held the key to many a secret. How quickly he had managed to guess her condition with those bright eyes of his!\n\nSomeone came in and switched on the light.\n\n\"Put it off!\" Zari ordered.\n\n\"Are you sleeping?\" It was Yusef's voice.\n\n\"Please turn the light off.\"\n\nYusef did as he was told and went to her side, sitting on the floor.\n\n\"Has something happened?\" he asked.\n\n\"I have a headache,\" said Zari.\n\nYusef removed his wife's shoes and put them quietly on the floor. Then he came closer and massaged her neck and her temples.\n\n\"Would you like me to get you some vinegar to smell?\" he asked gently.\n\n\"You go to your guest. When I feel better I'll come too.\"\n\n\"I can ask him to leave.\"\n\n\"No. But I'll feel more comfortable if you go to him.\"\n\nYusef left, and it was a while before he came back again. Switching on the bedside lamp, he said, \"Turn your head towards me so I can begin my treatments. I bet you'll feel better.\"\n\nZari turned around. Yusef was holding a tray which he placed on the vanity stool. On the tray was a bowl of steaming hot water. He dipped a small towel into it, wrung it and put it on his wife's face. He repeated that several times, then holding her head in an embrace, tried to make her take some hot lemon and honey. He kissed her forehead, her eyes, her ears, and said soothingly, \"Close your eyes and go to sleep now.\" He put two cotton wool swabs moistened in rose-water on her eyes, and said, \"Why do you tire yourself out like this?\"\n\nZari suddenly burst into tears. \"Why should there be so much unhappiness?\" she sobbed.\n\nYusef picked up the wads of cotton wool which had fallen on the pillow, dipped them again in rose-water, squeezed them and placed them on Zari's eyes. \"You're not responsible for all the unhappiness, you know,\" he said.\n\nZari sat up abruptly and the cotton wool swabs fell into her lap again. \"And you're not, either!\" she exclaimed. \"So why do you put yourself in danger?\" And after a pause, \"I saw Fotouhi. He's decided against collaborating with you.\"\n\n\"Now I understand. That frightened you, and it gave you a headache.\"\n\n\"That wasn't all. His sister attacked me, Khanom Massihadem took me for one of her patients who died in childbirth... Oh God! So much misery! So much loneliness!\"\n\n\"Someone has to do something...\"\n\n\"If I beg you not to be that someone, will you agree?\"\n\n\"Listen my love, if you start getting restless and impatient, it will distract me from what I'm doing.\"\n\nZari threw herself into her husband's arms and said, \"We have three children and one more on the way. I'm so frightened, Yusef!\"\n\n\"Would you like me to read you a Hafez poem and see what he predicts for us?\"\n\n\"No!\"\n\n\"Would you like me to bring the radio in this room and play you some music?\"\n\n\"No. Just promise me you won't be the one person to change things. I know you people want to go down to Khuzestan and do something dangerous there.\"\n\n\"I have a good idea. McMahon's story has been published. I'll ask him to come here and read it to you. I know it will make you feel better.\"\n\n\"All right,\" Zari agreed. \"Prop up the pillows behind me. I'll sit up. I feel better already.\" But she was only pretending.\n\nKhadijeh came in first. She had come to take away the tray. \"May all your troubles be on my head!\" she exclaimed. \"The master was frightened out of his mind, thinking you'd caught this disease that's going round the town.\" She went away for a few minutes and reappeared with a small round table from the parlour on which she arranged some glasses and drinks. Zari had given her instructions for everything that morning. She had even prepared the stuffing for the chicken herself that afternoon before going to the asylum, telling Khadijeh to leave it in a basket over the cistern to keep it cool.\n\n\"Ameh Khanom hasn't returned yet?\" she asked Khadijeh.\n\n\"No, she hasn't,\" Khadijeh replied, adding with a sigh, \"If only it were God's will for me to go on a pilgrimage too! Perhaps she might think of getting me a fake dashport or whatever they call it, from that fellow. I'm not about to go to Karbala yet, but I would hide it until the Imam is willing to receive me, his humble and sinful servant.\" She paused, then continued, \"I broke an egg as an augury to find out who had fixed the evil eye on you, and it turned out to be the master himself!\"\n\nWhen Khadijeh had left, Khosrow and Hormoz came in. Khosrow threw an arm round his mother, saying, \"Hello, mother dear! Would you like me to fan you a little?\" Then, \"What can I do to make you better?\" Hormoz was smiling, and asked after Zari's health as he stood politely by the bed. Khosrow put his face next to his mother's and said, \"Mother, please can Hormoz and I have our dinner in my room?\"\n\n\"Why, dear?\"\n\n\"We've decided never to speak to English officers again from now on. We're not even going to have anything more to do with their Indian soldiers.\"\n\n\"But McMahon isn't English, he's Irish.\"\n\n\"What's the difference?\" asked Hormoz.\n\n\"He's not even an officer, he's a reporter,\" said Zari.\n\n\"Well he's probably a spy,\" said Hormoz, \"otherwise why shouldn't a young man like him be wearing an officer's uniform in wartime? He's younger than Singer, isn't he? I'm sure Singer sends him here to find things out from my uncle.\"\n\n\"You shouldn't judge people like that when you don't know anything about them,\" Zari reprimanded gently. She was about to go on and tell them that McMahon even dreamed of independence for his country and wrote revolutionary poetry, but she decided against it and just gave them permission to have dinner in Khosrow's room. She wasn't in the mood for explaining or defending.\n\nAs the boys were leaving, Zari said, \"Khosrow, tell Khadijeh to give the twins their dinner and put them to bed.\"\n\nWhen Yusef came in, he switched on the light, even though the bedside lamp was still on. The bright light bothered Zari's eyes, but she didn't complain. When McMahon came in, he reached out and put on the dressing-table light, and sat on the stool in front of it. Zari had not noticed until then that the middle finger on his left hand was missing. He had gained weight, and seemed to have more wrinkles on his forehead.\n\n\"I hear your story has been published,\" said Zari. \"I'm glad.\"\n\nMcMahon smiled. \"I'll read it to you, even though I'm afraid your headache might get worse!\" He said. Turning to Yusef, he added, \"Are you rationed for drink?\" He was speaking English distinctly that night. Maybe he was trying to modify his thick Irish accent or perhaps even to hide it.\n\nHe took a sip and began. His voice was like a lullaby and Zari closed her eyes. Yusef sat next to her on the bed. \n\n# _19_\n\nThe old Charioteer gathered up his flowing white beard, a souvenir of millions and millions of years, and used it to dust the Golden Chariot of the Sun. Then he reached for the gold key which was dangling from his belt and headed for the East. Yes, it was time now. The Sun would be arriving wearily on his way. The old man opened the gate to the East with his key. The Sun was late today. But finally he showed up, yawning and dusty from his travels. The Charioteer brushed off the dust from the Sun with his thick white beard, and polished his beams. The Sun climbed into the Chariot, ready to begin his journey across the sky. But he didn't start right away, and the Charioteer waited.\n\n\"The Master sent you a message,\" said the Sun, \"that's why I was delayed.\"\n\n\"His wish is my command,\" replied the old Charioteer.\n\n\"He sent His regards and said He wants you to clean out the Celestial Attic right away, throwing out or burning all the odds and ends. But His most important instruction is for you to take out the stars belonging to His subjects from the attic and send them down to Earth. He wants everyone to take possession of their stars from now on.\"\n\n\"Do you think cleaning out the Celestial Attic is such an easy thing to do?\" grumbled the old Charioteer. \"We've stored things in there for over five hundred thousand years; you can hardly find anything for all the rubbish.\"\n\n\"You know the Master,\" said the Sun. \"When He gives an order, He means it.\"\n\nWith that, the Sun took off, and the old Charioteer was left to clear out the Celestial Attic, mumbling under his breath as he went, \"Why doesn't He just wipe out the whole species from the face of the Earth and be done with it! They'll never be up to any good, these humans! What a waste to have blessed them with a spark of His own spirit! After all, they go back to that unruly creature, the ape. When He was watching over them Himself, they never stopped bringing disasters upon each other; now He wants to give them a free rein over their own lives! How he spoils these earthlings! How He lets them get away with things. Ever since they managed to stand on two legs, He has become very excited and talks of nothing but the 'noble human race'! I know all about that noble race. From what I hear, they have few talents besides slaughtering and oppressing one another...\"\n\nGrumbling, he walked on until he reached the Celestial Attic. There, he first reached for the Tablets of Destiny, stone and clay tablets which had the fortunes of people predicted and written out in an outlandish script. He broke up all the tablets and threw them away into space. He also disposed of the remaining odds and ends like old wings belonging to angels and cherubs, burnt-out stars and meteors which had never reached their destinations... Then he started on the files belonging to ancient gods. What a huge pile! He collected them all in a corner of the attic and went to the adjacent hall where replicas of the ancient gods were stored. There were all kinds of gods... tree gods, serpent gods, star-, fish-, and sun-gods, and finally, both winged and wingless human gods. In a corner of the hall he spied a battle-axe, which he used to chop down Ashur and Shiva.\n\nHe had had his fill of these gods. Suddenly he caught sight of Gilgamesh, the legendary hero. \"How dare you!\" He exclaimed in surprise. \"Posing as one of the gods...!\"\n\nIn a twinkling, the old Charioteer turned him into dust and blew him away.\n\nWhen he got to the beautiful, shapely goddesses, the old man stood gazing for a while and reminisced. He thought of those days when Ishtar and Isis and Nahid and Aphrodite used to tease him with playful remarks. Every so often they would wink at him or perhaps Nahid would let him have a drink from her pitcher of water to refresh him. He had to push back the tears as he broke the replicas of the goddesses, but he couldn't bring himself to break Nahid's pitcher. Actually, he felt a pang as he destroyed Merduk, Mithra, Quetzalcoatl and Apollo too. In their heyday, these gods had never been too hard on their subjects. They even showed them some compassion from time to time. But the god Benu seemed to have somehow disappeared at the very moment the Charioteer was breaking up the Tablets of Destiny which Benu himself had inscribed.\n\nBefore long, the old man began to feel hot. He came out of the hall to look at the sky. The Sun, in his Golden Chariot, had reached the mid-point of the heavens. The Charioteer returned to the attic and pulled out the papers concerning the holy cities and mountains... records of the cities of Ur, Nineveh (later named Karbala), Benares, Chichen Itza, Jerusalem and other holy places, as well as records of the Himalayas, the Zagros mountains, Mount Olympus, the Andes, Mount Sinai, the Calcutta Hill, Mount Hera and any other mountain frequented by the ancient gods or used by them for their rendezvous with a favoured mortal. All these records he put on top of the files of the ancient gods.\n\nThere was almost nothing left now in the Celestial Attic except for one file containing several pages on the sacred trees... the Tree of Knowledge, the Lotus Tree, and others. On the rest of the pages were lists of talismans, prayers and other palliatives which the Master had created over the past five hundred thousand years for his noble human race. The old Charioteer picked up all the papers and records and existing files from the Celestial Attic and piled them in one corner of the sky. Then he rubbed his hands together and made a spark which he held out to the pile, setting everything on fire.\n\nThe old man didn't wait to watch them burn. Instead, he went to the cupboard where he stored and locked away all the stars which he had swept up with his celestial broom from the sky every dawn. After all, if he didn't stow away the stars somewhere for safe-keeping, they would be scattered all over the sky and anyone passing through might choose to play marbles with them. Anyone, even the Sun, or the idle angels and cherubs. He removed the gold key to the cupboard which hung around his neck, opened the cupboard and called out, \"Children! Come along and give me a hand!\" His voice echoed round the heavens, and from every corner of the sky, millions of cherubs rushed to his aid. In a twinkling, they had prepared all sorts of sacks and bags sized according to every city, town and village of every earthly country, and they also made a variety of ladders with sun-beam rungs to go down to Earth.\n\nThe cherubs were having a field day. One of them would read out the list of people in order, another would hold the sack open, and the third would throw in the stars as the name of each owner was announced. When the sacks had been filled up, the old Charioteer tied and sealed them one by one, and then handed them to the cherubs. Each cherub was given one sack with a list of people whose stars it contained, and in return they gave a receipt. The Charioteer appointed one supervisor and five assistants for them, and ordered the ladders to be lowered to Earth.\n\nIt was a sight worth seeing. Imagine! Millions of sun-beam ladders, with millions of cherubs carrying sacks full of stars, rushing down those ladders. The old man had seen many interesting sights in his lifetime, but never anything like this. He had witnessed the day Lucifer stood up to the Master, quarrelled with him and left; he had seen Gabriel's wings burn away, and had been there the day the Master commanded the lotuses in every earthly lake to open while He sent the Light of Wisdom to that man sitting cross-legged beneath the tree...\n\nThe young cherubs were to knock at every door on Earth, and give each person his or her own star. \"From now on,\" they were to say, \"it's up to you!\" Actually, they were free to phrase that message any way they chose.\n\nNow the old Charioteer went to the West to see the Sun off on his course. Climbing out of his Golden Chariot which he left to the Charioteer, the Sun said, \"Well done!\"\n\n\"I'll have to think of a solution for the Master's cloak,\" said the old man. \"From now on there will be no more stars on it at night until He has time to create some new ones.\"\n\n\"Why should that be up to you?\" replied the Sun, before bidding the old man a chilly goodbye.\n\nThe Charioteer was glad his task was over. He ran a hand over his thick, woolly beard and thought, \"Well now that I've got the chance, I might as well clean up too!\"\n\nIt seemed a shame, but he decided to chop off that impressive beard which reached all the way down to his toes. As he did so, bit by bit, he covered the whole sky with the shavings. Then he broke Nahid's pitcher and poured the water over his head and body, washing himself thoroughly in the process. He looked quite a bit younger in the end. With all this water, the heavenly river of the galaxy swelled. Meanwhile, the sky over Earth was looking very cloudy. There was even some thunder and lightning and much rain, but the cherubs were not in the least frightened. They knew the old Charioteer had broken Nahid's pitcher of water.\n\nThrice the Sun came and went, and there was no news of the young cherubs, their supervisors or their assistants. Every day the Charioteer would sit in a corner of the heavens, gazing on the planet Earth as it spun like a top around the Sun in space. Little by little, he began to worry. \"What if they've lost the way,\" he thought. \"What if their sun-beam ladders have got soaked in the water from Nahid's pitcher and then burned to cinders in the lightning?\" The heavens were empty; empty of stars, empty of cherubs... and still there was no message from the Master.\n\nOn the morning of the fourth day, he heard some noises in the distance. It sounded like the beating of wings, and the rustle of a breeze. Then the noises became more distinct. It was like a cosmic ringing, a melody which arises from the orbiting of planets and galaxies. Ladders were hoisted skywards, and soon enough the cherubs appeared. The Charioteer smiled. How the little cherubs had grown in this short time! How tall they had become!\n\nHe came forward to welcome them, all the while looking out for the supervisor and his assistants. Most of the cherubs didn't recognize him at first, but those who did said at once, \"Why do you look like that? We came back because we missed playing with your beard.\"\n\nThey all began talking at the same time about their experiences on Earth, and there was such a din no-one could be heard above the others. The Charioteer suddenly thundered out in a voice which penetrated the noise, \"I've had enough!\" Then, when everyone had quietened down, he asked, \"Where is the supervisor I sent with you?\"\n\nA cherub who was taller than all the rest stepped forward and said, \"He didn't come. He stayed behind, and asked me to replace him.\"\n\n\"What happened to the assistants?\" asked the Charioteer.\n\n\"They stayed too,\" said the new supervisor; \"you know, one hundred and eighty thousand, three hundred and twenty-five cherubs stayed on Earth. With the supervisor and assistants, that makes one hundred and eighty thousand, three hundred and thirty-one.\"\n\n\"Why?\" interrupted the Charioteer. \"What was happening on Earth?\"\n\nAll the little cherubs shouted together, \"The Earth is so interesting, everything is alive there!\"\n\nThe Charioteer clapped his hands to his ears. \"You're deafening me!\" he said. \"One person at a time! You, supervisor, you tell me.\"\n\n\"You see,\" said the new supervisor, \"the Earth is genuine. It's real. It's not imaginary or illusory. It's not nebulous, fleeting, or chimeric. It is solid. Your feet are on firm ground, and everyone and everything isn't floating.\"\n\n\"What do humans look like?\"\n\n\"They come in all shapes and sizes. None of them look alike, but they are all real, made of flesh and blood. You know, down there everything grows, everything is in a state of flux. Everything is subject to the laws of creation, evolution and decay. There, nobody and nothing is eternal.\"\n\n\"I gathered that when I saw you. Now tell me about your mission.\"\n\n\"We really enjoyed ourselves. We celebrated their festivities. They had wars too, as well as poverty and disease. We wept for them.\"\n\n\"What did you do with their stars?\"\n\n\"We gave each star to its owner, from the young to the old. The assistants gave me a report of their work on every continent. I've summarized all the reports for you here.\" And the new supervisor took out a folded piece of paper from underneath his right wing, and read out loud, \"As you had instructed, the cherubs were to hand each person his star, with the words, 'We now entrust you with your own star so you know you are henceforth free. You must be your own support and refuge.' The reaction of the earthlings was the following: the children's eyes sparkled upon seeing their stars, and they quickly took them and started to play with them. When we left, they were still playing. The old people merely said, 'It's too late now' As for the youth and the middle-aged\u2014and they are the ones who run most of the affairs of Earth\u2014their reactions were mixed. All the people in this group received their stars, but most of them, no matter how much we explained, could not grasp what the Master meant. Some of them almost immediately lost their stars. Others hid their stars in their pockets, smug with the knowledge that they had a star tucked away. Only a few amongst these people understood very well. Some of them said, 'This is the way we have always been. We had no expectations from any celestial or earthly stars, as we neither believe in destiny nor in complaining about being born under a good or bad star.' This group of people used complicated words, and the cherubs didn't always understand. Even their fellow earthlings had difficulty understanding them. Then again, one or two others from this group said, 'What a good thing it is that each person has found his own star.' These were an odd group, and in every country we came across a few of them. Some had beards, but not quite as long as yours used to be. These people immediately went to work on their dictionaries, striking out a lot of words from the vocabulary; words such as destiny, fortune, chance, fate, pre-determined and pre-ordained and all the other synonyms or equivalents. They were trying to replace these words with new ones rooted in 'freedom' and 'liberty' as we were leaving.\"\n\nThe Charioteer smiled. \"One of these days I shall have to visit Earth,\" he said. \"From what you tell me, it sounds very interesting.\"\n\nMcMahon fell silent. Zari opened her eyes. It felt as though she had just woken from a pleasant dream.\n\n\"What a story!\" she said.\n\n\"Did you understand all of it?\" asked Yusef.\n\n\"Whatever I didn't understand I pieced together with my imagination.\" Then turning to McMahon, Zari said, \"Actually, at first I was expecting to hear a children's story.\"\n\n\"You see,\" he explained, \"your daughters planted the germ of this story in my mind... the first images I had were those of someone sweeping the sky and a sackful of stars inside a dark cupboard. But to tell you the truth, no matter how hard I tried I wasn't able to write a story for the children themselves, to pay back my debt to them. It turned out as you heard it.\"\n\nYusef laughed; he got up to pour some wine for McMahon and handed him the glass. McMahon took a sip and said, \"It's good wine, where can one buy it?\"\n\n\"You know,\" Yusef said, \"now that I've heard your story again, it occurs to me your favourite theme is that same one you keep repeating in your poems.\" McMahon didn't say anything, so Yusef continued, \"You're trying to atone for the sins of others.\"\n\nZari no longer understood what her husband meant. She was about to ask him, when she heard Abol-Ghassem Khan's voice from the parlour.\n\n\"Where's everybody hiding?\" he called out. Then he appeared in person. He blinked and said, \"I heard there was a feast in this house tonight, so I got myself here on the double!\" \n\n# _20_\n\nAs Kolu began to regain his strength, he made himself a slingshot with which he pestered the sparrows in the garden until they could have no peace on any branch. There was still room to be grateful, however, since of all the window panes in the main building, only the pantry's had been broken. That day Zari had given Kolu a hard slap on the back of his hand, saying, \"I've had more than enough of you!\"\n\nAnd Kolu had sat underneath the orange-blossom tree, crying and sobbing loudly to be taken back to his mother and brother.\n\nEvery Sunday before dawn, Kolu would get up, undress, and jump into the pool with the copper crucifix around his neck, waking Zari up with the noise. Then he would get out of the pool and, according to Gholam, dress in his new clothes, gulp down a little breakfast and rush off to see the black-robed man at the Missionary Hospital. Just before noon, he would return home and instead of his usual hello, announce, \"I am a Christian.\" By lunch-time, though, he had clean forgotten it, and reverted to swearing by Hazrate Abbas again.\n\nThat last Sunday, Kolu had come home later than usual. Zari was in the kitchen, preparing provisions for Yusef's trip, so they could have some dinner ready when they reached Zarqan that night. Kolu came into the kitchen and eagerly preached to Zari and Khadijeh about Jesus Christ. He also mentioned Judas and asked Zari whether that ungrateful scoundrel was to be found in the Jewish quarter. Then he said with a sigh, \"I am a lost Iamb of Jesus.\" He clasped his hands in prayer before his lips and continued, \"O Jesus who art in heaven. Let's see if you can find me and take me home to my mother!\"\n\nKhadijeh scolded him. \"You stupid boy, repent before Allah! Go wash out your mouth!\"\n\n\"Leave him alone,\" said Zari quietly.\n\n\"Every night from now on I'll talk to Mr Jesus and pester him until he comes to me. After all what kind of shepherd is he to abandon all his lambs and go and sit up in the sky? If he's true to his word, let him come down and take me... if he takes me with him then I'll give him my father's flute which I've hidden under the bedclothes. But if he doesn't, may Hazrate Abbas strike me down if I don't hit him one in the middle of the forehead with my slingshot when I come across him!\"\n\nHe dug a hand into his coat pocket and brought out three copper crucifixes which he showed to Zari. \"The fang-toothed woman gave me these charms,\" he said. \"One is for my mother, one for my uncle, and the other for my uncle's wife\u2014I'm taking these for them as souvenirs.\" He held one of the crucifixes in front of Khadijeh and said, \"Kiss it!\"\n\nKhadijeh shoved his hand away. \"You idiot!\" she snapped. \"Go back to your mother!\"\n\nZari thought, \"None of them have ever accepted him as a son in this family. Not even myself or Ameh Khanom.\"\n\n\"The fang-toothed woman told me Jesus is everywhere\u2014in our village too,\" Kolu went on. \"She said any child who calls out, 'Mr Jesus!' He immediately says, 'Yes, my child.' But I'm too old now so I can't hear him.\"\n\nThat evening Yusef decided to take Kolu with him to the village. Zari couldn't help thinking, \"What does the poor boy imagine now? That Jesus found him?\"\n\nKolu couldn't keep still for joy, so much so that he left his slingshot behind, even though he knew he wouldn't be going straight to his family. First he was going with Yusef to Zarqan until someone could be found to take him to the lowlands. Clearly the poor lad felt that every step away from Yusef's homestead was a step closer to his own village...\n\nThey departed, and Zari found herself alone during the long, turbulent nights, filled with nightmares. Nights so long, it seemed they would never be followed by morning. As time drew on, her thoughts became more distressed and her dreams more agitated.\n\nAmeh was an expert at interpreting dreams. Everyone\u2014even strangers\u2014acknowledged this. Sometimes total strangers would telephone her and recount their dreams. She would greet them politely, and then proceed to give her interpretation, in the hope of doing a good deed. She also had a handwritten manual of dream interpretation which she would refer to in case of difficulty. But even Ameh was unable to unravel two of Zari's dreams. She leafed through her book carefully, but she still couldn't find the key to those two dreams. And it was for this reason, according to Ameh, that of all Zari's dreams, those two were constantly repeated.\n\nZari would dream that she stood stark naked in the middle of an unfamiliar square, surrounded by thousands of staring men and women. She also dreamt that it was exam-time at school, and a dark-skinned, scowling examiner was standing before her. Yet no matter how hard she tried, she didn't know any of the answers. She racked her brains and sweated and her pulse raced, but still she was unable to answer the questions. In the morning, she could no longer remember what the questions were.\n\nAmeh instructed her to beg a piece of bread from a beggar and then eat it so she would remember the questions.\n\nOne night Zari dreamt that a two-headed dragon swallowed her husband whole, as he was galloping along on his mare. When she looked closely, she realized that the two-headed dragon looked like Captain Singer, dressed in a Scottish tartan kilt with embroidery all around the edge. This particular dream Ameh interpreted easily. She said it meant that Singer would become a public laughing-stock, but Yusef, like Jonah, would learn patience and endurance in the whale's stomach. The darkness inside the whale would enlighten him so that he could understand the secrets of the universe.\n\nA few nights later Zari dreamt that the Governor had tossed Yusef into the furnace with his own hands. Yusef had burnt to a cinder, but nevertheless managed to grope his way out. Ameh interpreted the fire as the biblical one which had descended upon Abraham and then turned into a flower-garden. Yusef's coming out of the fire meant that he had passed his ordeal. And although Ameh's words reminded Zari of Siavush's story, she kept quiet. Because that night, in the tent of the tribal chief... that night when Malek Sohrab took a bet with her over a Brno gun, and she had lost but never paid up... that night they had talked of Siavush the whole time, and teased Zari because she knew about John the Baptist and not about Siavush, and they had explained to her that Siavush had passed through the fire and come out vindicated...\n\nAmeh went on with her interpretation. \"The furnace is clearly the same one in which the wicked Khuli woman hid Muslim-ibn-Aqil's children. Burning to a cinder signifies being purified and vindicated because, as you know, the meaning of a woman's dream is always the reverse of the dream itself.\"\n\nAnother night just before dawn, Zari dreamt that Kolu had struck Yusef right in the middle of the forehead with his slingshot. Ameh didn't bother to interpret this one saying that dreaming just before dawn has no significance.\n\nTen days after Yusef's departure, it was rumoured that Malek Sohrab had become an outlaw. Everyone who came to the house had something to say about it. Gholam told Zari that Malek Sohrab had taken to the mountains with a thousand gunmen and was hiding in an inaccessible spot.\n\nOne day Khosrow told her excitedly, \"He's close to Yasuj now, with two thousand fighting men. And he still hasn't come down the mountain\u2014what a man!\"\n\nA few days later, Hormoz showed up and commented, \"Auntie, you know how much I admire bravery, but I think brave men should also have a sense of timing.\"\n\nAbol-Ghassem Khan often came along in a hurry to pick up Hormoz and take him off to his highness the Governor, but each time Zari would coax him into staying awhile, plying him with some of Tavuus Khanom's oldest and best wines, and pressing delicacies on him until she managed to draw out some news.\n\n\"I hear Bibi Hamdam, Malek Sohrab's mother, went to Army Headquarters,\" Abol-Ghassem Khan told her. \"She barged into the captain's room without permission and threw herself, in those wide breeches of hers, at the major-general's feet. She begged immunity for her son, and promised to bring him to the authorities herself. The major-general advised her to do that as soon as possible, at which point Bibi Hamdam pulled out a Quran from her bosom and tried to make him swear not to harm her son. But the major-general only kicked the old woman's hand away.\"\n\nThen Sakineh, the woman who came to bake their bread, told them, \"Bibi Hamdam has hired forty people to read the Quran and chant the An'am verse every day. It's hair-raising! Oh Lord, I beg you by the purity of the saints, to spare the life of Bibi Hamdam's son, and meanwhile to spare this poor, sinful servant's son from the military draft too!\"\n\nIn despair, Zari took up her old addiction of reading newspapers. But she couldn't find even the slightest mention of Malek Sohrab's name in any newspaper. Her habit did, however, lead her to a certain news item in one of the leading local papers. She had been alone in the garden that evening when the newspaper arrived and she had taken it from the delivery-boy herself. It was two weeks since Yusef had left. The item read like this:\n\n\"In Gratitude\"\n\n\"The gracious Khanom Ezzat-ud-Dowleh, one of the charitable and kind-hearted ladies of this town, has been appointed by the Women's Society to visit and inspect the houses of the Mordestan District, as well as the women's prison. All the houses of the above-mentioned district have been cleaned and disinfected under her supervision, and this charitable lady, out of boundless generosity, has bailed out and set free one inmate of the prison who, out of ignorance, had engaged in earning an illegal livelihood. The Governorship of Fars extends its gratitude to this humanitarian and benevolent lady for her services.\"\n\nAlthough Zari was not surprised to read this piece of news, it still depressed her. She crumpled up the newspaper and threw it away. She took refuge in the orangery, pacing about under the orange-blossom trees, feeling unequal to any task that might require concentration. She decided to go to the stables to find Gholam and ask him whether he had any more news of Malek Sohrab. But she changed her mind, knowing that the poor man might be half-undressed or even naked in the heat, or perhaps having a quiet smoke. For a moment, she thought of going to visit Bibi Hamdam, but decided against that too. She wasn't in the mood for the loud chanting of Quran reciters, and she knew that the instant Bibi Hamdam set eyes on her she would begin to wail and press her for a solution to her problem. And of course, if Zari had any idea what to do, she would not be feeling so distraught. Everyone knew that Bibi Hamdam's existence was tied to that of her son, and everyone knew too that Malek Sohrab, despite his size and stature, was nothing more than a child before his mother.\n\nShe thought of following Ameh Khanom and the children to Mehri's house, but she realized she didn't feel like putting on a long-sleeved dress and a head-scarf in that heat. Mehri's second husband, Mohsen Khan, was a very strict man.\n\nZari knew her restlessness and depression had much to do with sheer fatigue. Every summer she would spend at least two or three weeks at their village where a change of air, long walks and horse riding prepared her for the autumn and winter ahead. But this summer, with its disease, famine and war, and her own unexpected pregnancy, had made a prisoner of her, confining her to the house, the prison, and the asylum. She decided to arrange a weekly reunion with her former classmates... an afternoon reunion, perhaps... first at her house, then at Mehri's. Of course Mehri herself would be willing, if only Mohsen Khan would allow it. Their husbands didn't get along, otherwise she and Mehri, regardless of how often they saw each other, were still the same steadfast friends.\n\nShe went to the bedroom and searched in her drawers for knitting needles and wool in order to knit away her anxiety and depression. But neither knitting needles nor wool could be found. Her glance fell on a box full of glass beads. She picked it up, along with her sewing kit and went out on the verandah to string the glass beads. She looked out towards the garden which seemed to have lost its bloom. Dust had settled on all the trees, smothering the yellow, burnt-out leaves. For an instant she thought the trees were staring back at her. Then she saw them shiver and nod and then quieten down again. \"They're getting ready for their sleep,\" she mused, \"but the sparrows are awake on the branches, complaining to each other like a bunch of mother hens at the public baths!\"\n\nThe sun had completely left the garden when, suddenly, she heard the neighing of a horse. It was the mare, not Sahar. Thank God! Yusef was back from the village. It was true what they said about hearts that talk to each other. Whenever she began to miss him desperately, Yusef would somehow turn up all of a sudden. She decided not to complain about how long he had been away this time, how anxious and wretched he had made her, how endlessly he had abandoned her to imaginings and nightmares and frightening rumours and unjust expressions of gratitude!\n\nGholam came out of the stables. Seyyid Mohammad, Yusef's steward, entered riding the mare, with the roan horse in tow. Zari felt a pang. She stood up. The box of glass beads in her lap fell to the ground and broke open, scattering the beads all over the rug. Well, perhaps Yusef had got off along the way, gone somewhere on an errand. Seyyid Mohammad dismounted and gave the horses' bridles to Gholam, whispering something in his ear. Gholam threw his hat on the ground, and Seyyid whispered something more to him. Slowly, Gholam led the horses away to the stables. Zari ran toward Seyyid Mohammad, out of breath.\n\n\"Where's the master?\" she asked.\n\n\"He's coming in Malek Rostam's car. Don't panic, nothing's happened,\" he answered.\n\nGholam and Seyyid started to behave mysteriously. Gholam ran out of the garden hatless, while Seyyid came to the pool to wash. He took out a comb from his pocket and combed his thick moustache. Then, taking a stone from the driveway, he washed it and placed it on the ground as he stood to pray. But Seyyid wasn't one for praying. Besides, what kind of prayer was this? Without a proper ablution and, although the sun had set, without the evening call to prayers?\n\nThen Ameh arrived. It was very odd. Wordlessly, she stood to pray on the verandah still dressed in her outdoor veil. Without her prayer-mat. And without bringing the children. It was a long time before the car carrying Abol-Ghassem Khan drove in. Zari was certain something had happened but she didn't want to ask. She didn't have the courage. They began themselves, brother and sister, to tell her.\n\nBy the time Malek Rostam's green car drew up to the poolside and stopped, she knew what had happened, but she refused to believe it until she saw for herself. Malek Rostam and Majid got out and she knew her husband would not be stepping out. She knew he would never again climb in or get out of a car... where had she read that so-and-so was riding on a wooden horse? Yusef was sitting stiffly on the rear seat, covered with a cloak and his hat pulled over his eyes. She heard Ameh's voice saying, \"Welcome, brother. So you've come home...\" and Ameh began to sob. Abol-Ghassem Khan was wailing at such a pitch that he must have been heard in every corner of the house. Zari placed a hand over Yusef's ice-cold one, with those long stiffened and separated fingers. She looked at his ashen face, his chin which had been bandaged with a blood-smeared handkerchief, the blood which had already congealed. She took it all in, but could not believe it.\n\n\"Without saying goodbye?\" she asked in bewilderment. Gholam let out a wail. Zari asked again, \"All alone?\" And now everyone wailed. She wondered where from within their throats they managed to bring out those sounds? And why couldn't she? She could see that Ameh had torn open her collar and was sitting on the stone ledge of the pool. Zari kept asking, \"But why?\" And then the car, and the trees and the people and the pool all swam around and around and went away from her.\n\nWhen she opened her eyes, she found herself stretched out on the rug on the verandah. All the garden lights were on. Did they have guests? There was an odour of mud-plaster in the air. Ameh Khanom was massaging her shoulders, and her body, neck and face felt moist. There was commotion all around. They had propped Yusef up on a wooden bed by the pool. A hatless Gholam was sitting behind him, rocking gently back and forth and repeating, \"My master!\" Haj Mohammad Reza the dyer, with his arms dyed purple to the elbow, was unsuccessfully trying to remove Yusef's boots. Abol-Ghassem Khan was standing over them.\n\n\"Haji, cut the boot open,\" he said. And he shouted for a knife.\n\nYusef didn't have his cloak on. He wasn't wearing his hat either, and Zari thought she must be dreaming. Lately she had had nothing but nightmares\u2014perhaps this was yet another bad dream. She thought she was dreaming of a man they had forced to sit on the wooden bed, and they were cutting open his boots with a knife, but she couldn't see his face. She dreamt that Malek Rostam was holding the torn boot in his hand and shouting, \"O woe is me, woe is me!\"\n\nShe thought, \"What do they call this kind of shouting from the guts? Bawling? Bellowing? Hollering? No, there's a good word for it, but I can't remember it now.\" Then she imagined she was dreaming that Majid had put his head on Yusef's cloak by the bed, and was sobbing out loud. But maybe she wasn't dreaming, since her eyes were wide open.\n\nAbol-Ghassem Khan came to the verandah. He took out a white handkerchief from his pocket and blew his nose as though he had a cold. His eyes and his long nose were bright red. He blinked and said, \"Sister, how quickly you've been widowed! And not even thirty yet! Oh my! Oh my!\"\n\n\"Control yourself, man! Don't frighten a pregnant woman more than she is,\" Ameh said.\n\n\"Pregnant?\" Zari knew she was pregnant, but her mind simply refused to acknowledge what had happened.\n\n\"How did you know?\" she asked Ameh.\n\n\"From your eyes.\"\n\nAgain Zari had the feeling she was dreaming. A man seemed to be sleeping, sprawled over a bed, and despite the heat they had covered him with Yusef's cloak. But she didn't recognize the man. She dreamt that three men were sitting on the children's bed, talking about the man who was laid out on the other bed.\n\nShe managed to distinguish the voices: \"My sister is right,\" Abol-Ghassem Khan was saying, \"it wasn't time for his ideas. Brother, if your spirit is present, forgive me. I envied your intelligence and understanding and education, but as I didn't have those things, I'd make fun of you. Brother, you had the freedom of a cypress, reaching out\u2014\"\n\nThen Majid's voice, \"Yes, but don't be upset now. He knew himself that it wasn't time for his ideas. But he used to say\u2014many times he told me himself\u2014that our duty is to hasten the time for those ideas.\"\n\nAnd Malek Rostam's voice, \"I know that any day now they'll get my poor brother Sohrab, as well. They'll set up a gallows in the Mashq Square and everyone will go to watch.\"\n\nThe voices mingled with the sound of crying.\n\n\"Don't you think one wants to say and do the right things? But when you've started on a downhill course, the only way to go is down and then you're sunk...\" Whose voice was that?\n\nFootsteps could be heard on the gravel of the driveway. But they stopped when they reached the verandah and then resumed again. Zari closed her eyes, feeling as if all her life-forces had been drained and spent, like a squeezed fruit. It was as though a snake had slithered down her throat and coiled itself around her heart, with its head erect, ready to strike, and she knew that for the rest of her life this snake would stay coiled right there around her heart, so whenever she remembered her husband it could sink its fangs into her bosom.\n\nAt Ameh's insistence, she got up and let herself be led by the arm to the parlour. Women were sitting all around on chairs or on the carpet, most of them fanning themselves and whispering together. The men were in the other rooms. She could hear their voices. It was as if they had all been waiting outside the garden gates for her husband's corpse to arrive, with its fair locks bloodied beneath the hat, all the way down to the fair moustache where the blood had clotted, and then they could all come in... the women stood up at the sight of her, but Zari couldn't see anyone clearly enough to recognize them. Ezzat-ud-Dowleh was the only exception. Zari's gaze locked for an instant into her cobra-like face, framed by the gaudy hair, and then some sparkling yellow, red, blue and black glass beads took form and danced before her eyes. Most of the women peered at her carefully, shaking their heads and crying. From the other rooms, the men's voices could be heard, topped by Abol-Ghassem Khan's loud weeping.\n\n\"If anyone knows, please tell me too... I'm at a loss...\" he was saying.\n\nBut Zari's eyes and tongue were dry. Not a tear, not a word. She went out to the verandah and sat on the rug. Khosrow, riding Sahar, came through the garden gates and cantered straight to the verandah. He let Sahar go and rushed to his mother.\n\n\"Is it true?\"\n\nZari bent her head and busied herself collecting the glass beads from the rug.\n\n\"Did you pass your exam?\" she asked. All the lights were on. How could he not have seen his father's sprawling corpse beneath that cloak? Why did he keep asking if it was true?\n\n\"Why are you so late?\" Zari asked.\n\n\"Those of us who'd passed treated the others to paludeh ice-cream. But then the janitor came and told me uncle had called to say father was shot but he was just wounded, and he'd come straight home on horseback. Is it true? Where is he now? At the hospital?\"\n\nShe suddenly hugged her son and kissed him, and then the tears began to flow.\n\nBefore long, a lot of people were embracing her and weeping aloud over her and her fate\u2014to have been widowed so soon, to have to raise four orphaned children. Already everyone knew about the fourth. Ezzat-ud-Dowleh came forward too, but she neither embraced her, nor did she cry. She just said, \"I hope this will be the last of your sorrows. At least he's left you enough to raise your children in comfort.\" Hardly saying goodbye, she went away, hobbling down the stairs with a hand to her back. She headed towards Malek Rostam who was sitting on a cane chair by the pool. Malek Rostam stood up and gave her his seat. You could tell Ezzat-ud-Dowleh was talking and Malek Rostam listening. She seemed to shed a few tears too, since she kept dabbing at her eyes with a handkerchief. A voice announced: \"The droshke is here.\" Ezzat-ud-Dowleh rose and, on Malek Rostam's arm, walked down to the end of the garden.\n\nA few hours later Zari found herself lying on the bed in the cool basement with Khanom Hakim standing over her head. The fountain was on, and she could feel a cold, wet handkerchief on her forehead. She felt the sting of a hypodermic needle. Once, twice, three times... she could see Khanom Hakim placing the cold wooden ear-trumpet on her belly and listening.\n\n\"The baby be all right,\" she said. \"Tonight be best for the burial.\"\n\nZari heard Ameh Khanom reply, \"Why don't you keep to your doctoring! Do you think my brother was a criminal to be buried at night?\"\n\nAnd again Khanom Hakim's voice asking, \"Why be so unpleasant? All three children be delivered by me. So will be fourth.\"\n\nZari realized she was being questioned. \"Why you be not coming sooner to me?\"\n\nZari gave no answer, and Ameh replied rather harshly, \"It's all your 'be this' and 'be that' which has driven everyone mad! If only...\"\n\n\"If only you would get lost.\" Someone had said that in Zari's mind, because Ameh Khanom didn't finish her sentence. Nevertheless, Khanom Hakim seemed to have heard the voice in Zari's mind.\n\n\"Be this the reward for service and self-sacrifice?\" she complained indignantly in a trembling voice. \"We be in strange town with dry air, away from brother and sister and friends... medicines be free, treatment free.\"\n\nThis time the voice in Zari's mind shouted, \"Get lost! Everyone get lost!\"\n\nKhanom Hakim had gone, and Zari could see Khosrow with a fan in his hand. She felt a cool, gentle breeze on her face...\n\n\"Khosrow,\" she murmured.\n\nKhosrow brought his head closer.\n\n\"Do something for your mother... go to Dr Abdullah Khan early tomorrow morning... tell him what a disaster\u2014tell him to come by and visit me for a moment.\"\n\n\"I'll go right now,\" Khosrow said, getting up.\n\n\"No, my love, go tomorrow morning.\"\n\nAmeh came in and Zari heard her say, \"Get up, son, go and eat your dinner. Then to bed. For your late father's sake, be a good boy and go right away.\" How quickly they beseech you by your late father, thought Zari... and sometime after that Khadijeh's voice announced, \"There's a man at the door. He says he's come to give us a hand as an act of charity. He says he dreamt last night that one of Imam Ali's devout servants had just entered the kingdom of God...\"\n\nZari knew they had set up a tent around the pool, and were about to wash her husband's corpse in the pool-water. She knew the pool would be emptied and the water drained that very night, channelled quietly into the garden. The water that had cleansed her husband's body and washed away the dried blood would irrigate the trees. And Hossein Kazerouni would work the treadwheel from midnight to refill the empty pool by morning.\n\nHer ears perked up at the sound of Seyyid Mohammad saying, \"What can I say? Better left unsaid.\" Whose question was he answering? Zari opened her eyes. Seyyid was squatting by the door of the basement, rolling a cigarette. Ameh was sitting on the bed at her feet. Abol-Ghassem Khan and Khosrow were there too. Seyyid licked the thin cigarette-paper and striking a match, said, \"What can I say, really? No-one knew how it happened. The peasants were ready to die for their master. I don't know. Maybe it was the work of the gendarmes, or some others... this business about Kolu's uncle rushing all the way from Kavar to shoot the master and then racing back home is a load of nonsense. It's trivializing the matter; it's even an insult. Whoever had a hand in it, started this rumour themselves. When I got on my horse to come down to the plain, Kolu stopped me and said, 'I shot the master.' I said, 'What did you shoot him with?' He said, 'With my slingshot.' Later I heard he'd said a gun. Then he'd said his uncle had done it. I know they've told him what to say. They think they can fool us. We couldn't find a single trace of Kolu's uncle having been at the village, no matter how carefully we investigated. How could he possibly have gone there without being seen? Yes, he does have a rifle. The master bought if for him himself after Kolu's father died.\"\n\nSeyyid broke off to take a puff. Then he continued. \"Early that morning we'd gone to the store-rooms. The master broke the seals on the doors with his own hand and distributed pulses and dates and flour among the peasants. He teased them and joked with them. He told the women that if they sold their share to buy gold bracelets or go on a pilgrimage, he would disown them. He told the men that if they dared convert their provisions into money to buy new bedclothes and new wives, he would know what to do. Everyone was happy. The master was the happiest of all.\n\n\"Before lunch we went up to the upstairs room in the old fortress. The master sat on his cushion. We'd rolled away the mosquito-net. Elias brought the hookah and set it down next to the master. I asked, 'Shall I remove your boots?' He said, 'No, I'll smoke a pipe and we'll go down to the plain.' Then he asked, 'Has the camel-driver come?' 'Yes,' I answered. Elias said, 'Sir, this agent of Singer's is back again.'\"\n\nAbol-Ghassem Khan's voice interrupted Seyyid's narrative. \"Everyone who came here tonight told me to hush up the matter completely, that the situation is very dangerous. The whole thing comes from the very top\u2014\" And Zari wondered how a man who had been howling with grief only a moment ago could possibly speak like that now.\n\nAmeh's irate voice didn't let Abol-Ghassem Khan finish his sentence. \"Bless my soul!\" she exclaimed, \"Now they want to blot out the blood that's been shed! Brother, listen to me. Hire a lawyer. If you don't, I'll do it myself.\"\n\n\"Sister, I thought you were about to leave for Karbala?\" said Abol-Ghassem Khan sarcastically.\n\n\"Now my Karbala is right here,\" said Ameh with a cry in her voice. \"Happy is the martyr whose blood is one night old. For us, it hasn't even been one night yet.\"\n\n\"Sister, you women are not aware of the things that are going on here,\" Abol-Ghassem Khan said gently. \"Let's say I engaged a lawyer. Who do you think they'll charge with the murder? I'll tell you: Kolu and his uncle... or some other miserable peasant, or perhaps even this very Seyyid here. They'll manipulate things so that eventually we forgive the scapegoat ourselves, or else Kolu will turn out to be the murderer, and he's a minor. Isn't that so, Seyyid?\"\n\n\"If any court decides to act so unjustly,\" replied Seyyid, \"I'm willing to go out there and rouse all the peasants, single-handed. In the whole village, the master was\u2014\"\n\n\"But what's the use? The little money my brother's orphans have left to them will be wasted. Besides, what if they arrest you first? Do you think they can't?\"\n\n\"Uncle,\" Khosrow said, \"in that case Hormoz and I will go and round up the peasants for action. Mr Fotouhi will help us too. And if our money is wasted, it doesn't matter. I'll earn my own bread. Of course I can't do that now. For the time being our mother might have to sew for a living until I grow up\"\u2014and suddenly he broke into tears.\n\nZari wanted to come down from her bed and embrace her son so that they could cry together, but she couldn't. She wasn't even able to open her mouth to say, \"Don't cry, my love.\" What had Khanom Hakim's shots done to her?\n\nAmeh cursed away. \"O Lord,\" she said, \"why did you create me a wretched, veiled female? If I were a man, I'd show them the meaning of manhood.\"\n\nZari expected Abol-Ghassem Khan to lose his temper, but he merely complained quietly, \"All right. Go ahead and insult me by saying I'm not a man. But what else can one do besides surrender and consent?\" After a pause he added, \"Well, all right. These are problems for later, anyway. Give me some time to see what I can do.\"\n\nKhosrow turned to Seyyid Mohammad and asked, \"Isn't Singer's agent that fat man with the pock-marked face?\"\n\n\"Yes, that's him,\" Seyyid answered. \"After Elias announced him, he came up to the top room of the fortress. First he conveyed Captain Singer's greetings, then he said, I've been told to ask you to be sensible. What's the use of distributing wheat among the peasants? Peasants don't think of tomorrow. They go and sell it for several times the price on the black market.' The master laughed\u2014that was the last time he laughed\u2014and answered, 'Go and tell Singer that instead of him and his sort getting fatter by the day, let our peasants get a little richer.' The agent said, 'Captain Singer thinks your best interest lies in not touching the rest of your provisions.' The master answered, 'Since when do I ask Singer about my interests?' I remember every word of that conversation. The agent then said, 'Captain Singer says they can break the locks on the storerooms and take the wheat. Not only the wheat, but also the barley, the pulses and dates that they need. They have a written mandate from the Governor too. After all, they'll be paying you cash. Is that such a bad deal?' The agent went on, 'At a stretch they'll buy the provisions second-hand from the peasants, and they won't be losing on it either. The government has doubled the exchange rate of the pound.' Then the agent bent over and whispered some things in the master's ear which we couldn't hear. But the master lost his temper and shouted, 'To hell with all of them! Don't threaten me with gendarmes either, I'm not afraid of them. If you dare, go and break the storeroom locks with your gendarmes. You have the mandate.' Then he calmed down and said, 'At this point in time provisions have nothing more to do with their war. It's fallen in the hands of their trading company, and the trading company deals in food supplies.' The agent wiped the sweat from his forehead and said, 'Sir, I beg of you, don't be stubborn. Don't fall out with these people, they'll harm you.' Then he asked, 'Aren't we from the same town?' The master replied, 'Yes, unfortunately we are.' The agent said, 'These people aren't really in need of your provisions, but they're afraid of the example you'll be setting.' The master said, 'Actually, that's precisely my intention. In Hamadan people closed their shops and didn't allow a grain of wheat to leave the city gates. Here they've wrecked the Darvazeh Quran gate...' Again the agent whispered in the master's ear for two or three minutes. When he'd finished, the master went deep into thought. He seemed upset, but stayed resolute. He just said, 'Tell Singer I give Sohrab provisions, not weapons.' I was about to go. I had barely crossed the threshold when I heard a gun fire. I turned around, saw the pipe fall over and the master tip to one side. Blood started to gush. Mohammad Mehdi and Elias ran inside... they gave a hand, but the agent didn't budge. I yelled at him and told him to get lost.\"\n\n\"Maybe it was Singer's agent who shot him,\" said Khosrow.\n\n\"No, that man is such a coward, he would fall over if you said 'boo' to him!\" Seyyid said. \"We moved the master off the cushion. I lifted it. They had dug a hole under it the size of my hand. The master was still conscious. He opened his mouth to talk, but he couldn't. I brought my head close to his. He said, 'Kolu... Kolu... take him... to his relatives... Zari... Zari... my children.'\" Seyyid paused, then continued, \"I sent a messenger to Kavar to tell Malek Rostam, and I sent Kolu along with the messenger before any fools got their hands on him to tear him limb from limb. I took the camel-driver and Mirza Agha Hennasab with me down to the plain and I waited until they loaded the camel with provisions. I got a receipt from the Mirza Agha and came here. Here's the receipt. I don't know if I did the right thing. But I know if the master were alive, that's what he would have done.\"\n\n\"What was Mirza Agha Hennasab doing there?\" It was Ameh's voice.\n\n\"He'd come with the camel-driver from Malek Sohrab,\" answered Seyyid.\n\nWith an effort, Zari managed to sit up. \"I wanted to raise my children on love and non-violence,\" she said. \"Now I'll raise them on revenge. I'll give Khosrow a gun.\"\n\n\"I don't blame you,\" said Abol-Ghassem Khan. \"What they've done is unforgivable. But you can't wash away blood with more blood. We have to wait and see what happens.\"\n\nZari lay down again and fell asleep. She began to dream that a strange tree had grown in their garden and Gholam was watering it with blood from a small watering-can. \n\n# _21_\n\nZari was awake. In her mind, someone seemed to be talking. Saying nonsensical things. Things that Zari knew she had heard or read somewhere. Sentences followed each other, but she was not expecting them. Where had they been suspended in her memory to be appearing now?\n\n\"My, but all our wise men have abandoned this town...\"\n\n\"O Dark, dark, dark, amid the blaze of noon... it's me Eilan-ud-Dowleh, it's me Veilan-ud-Dowleh... I'm burning, burning, burning. There's enough fire within me... but because you're younger, you can't take it... how this painted dome, the world, reeks of mischief...\"\n\nShe squeezed her eyes together to block out the flood of sentences plaguing her, but that only made it worse. Now the painting that one of the mental patients had done in the asylum kept appearing before her eyes. The painting depicted a butcher's shop. An icon of the Imam Ali and the image of the young butcher with his hand cut off could be perceived against the shop wall. The shop itself was filled with giant hooks as far as the eye could see, but instead of mutton, there were people hanging by their feet from those hooks, and blood was dripping from their throats.\n\nShe opened her eyes. It must have been well after midnight because there was no electricity and they had lit candles. She saw Abol-Ghassem Khan sitting on the carpet, hugging his knees. Ameh was sitting across from him. Khosrow was there too, as well as Malek Rostam. Seyyid Mohammad was standing in the doorway of the basement. The smell of opium, cigarette smoke, alcohol and charcoal mingled in the air. She could hear the steward's voice in her state of semi-consciousness. He was talking about the funeral and his voice seemed to have gone hoarse as if he had just come down the Mortaz-Ali mountain carrying a jug of wine to go to the grave of the Seven Sufi Saints. Now he opens the tap of the jug and starts to drink. Red drops spill from his thick moustache on to the nameless graves. He puts the jug under the cypress tree and sleeps on a cold slab of stone. When will the jug of wine turn into holy wine? By dawn? By the time the sun rises? \"Gone are the days when people could find a purified drink to use for attaining a mystical state of mind like Hafez,\" Zari thought in her reverie. \"Now they have to swallow gunpowder instead. Gone are the days when they sat humming by a stream and reflected quietly on the passage of time, content just to be with a rosy-cheeked young lover. Now they have to stand next to the dam of life, with its flood charging straight at them, slapping them so hard in the face, they're left reeling for good. By the way, what was the word for shouting from the guts? There was a good word to describe it... but it had to somehow convey piercing or boring. You see, if __ a person can't let out a certain kind of scream when they're hit with the flood, the thunderbolt, or the thrashing of life, their heart is punctured instead, and then all those people with riddled hearts go for each other's throats trying to destroy one another until they're sent off to prison. Or else it goes to their heads, and they lose their minds. Meanwhile, a spoilt, silly, pampered young woman is taking bread and dates to prisoners and lunatics every Thursday. She has a vow to fulfil. But that woman herself is perhaps struggling on the verge of lunacy at this very moment, which is why her mind is ticking so. So fast, she can't stop herself...\" And suddenly Zari was seized with fear. \"Am I going mad?\" She tried to sit up, but it was as if she had been nailed to the mattress.\n\nWhen she lost consciousness, she would dream. Awake, either someone would be talking randomly inside her head, or she would intensely relive bygone incidents drawn out from the recesses of her mind. She no longer distinguished past from present. Sometimes random events materialized before her which she didn't recollect ever having seen or heard. She strained to keep her eyes and ears open, to assure herself that Ameh and Khosrow and the others really did exist, and she could recognize them and hear them. But her eyes and ears would only stay at her service for a short while, and then sooner or later they would drift away from the present reality again.\n\nShe could hear Ameh's voice, \"How did Ezzat-ud-Dowleh manage to get here with those leg-pains of hers? I suppose she came to satisfy her curiosity and see what's going on. Her eyes really lit up whenever she looked at Zari. I told myself how happy poor Zari had made her enemy.\" And she broke off, crying.\n\nThen Malek Rostam was saying, \"Ezzat-ud-Dowleh asked after my brother Sohrab. At first she said she'd heard he was under siege. 'Where did you hear that?' I asked her. She seemed taken aback. 'Well they'll get him anyway,' she said, 'and then Lord have mercy on poor Bibi Hamdam. Whatever you suffer, it's at the hand of your children!' And she burst into tears. Because of her connections with the Governor's family, I thought she might know something, and I tried to prise out of her where she'd heard that Sohrab was surrounded. But she eluded me and said, 'When did I say such a thing? I just said his friends gave him away...' Anyway, she changed her story a hundred times... she said Malek Sohrab was tired, that he had no food or water, that he's turned himself in. When I was helping her into the cab, she said, 'I've heard Bibi Hamdam has begged the Governor for mercy for her son, and now she's gone to bring Sohrab on his own feet to be executed!' I nearly tore off her wig, I was so angry, and wanted to beat her up as much as she could take. But all I said was, 'Khanom Ezzat-ud-Dowleh, if you have any specific information, please tell me.' I even made her swear by her darling son Hamid, but she denied knowing anything and pretended it was all rumours. I got into the droshke with her and afterwards rushed to Bibi's house. No-one would answer the door. What if the major-general's promise of a pardon was only words and he'll go back on it... what if\u2014\"\n\n\"Forgive me for saying this,\" interrupted Abol-Ghassem Khan, 'but considering the hell Malek Sohrab raised in the battle of Semirom, I doubt if they would give him a pardon. It's like the story of the husband who said to his wife, 'I told you to dance, but I didn't mean you to overdo it!'\"\n\nYet all Zari could see, clear as day in her half-awake state, was a vision of people coming at dawn to the Baq Takht square, carrying rolled-up rugs on their shoulders. The women were wearing ordinary chadors with face veils, or the large, wide chador with a thin face-cover. The men were crawling on all fours. O Lord, have the townspeople gone stark raving mad? Wasn't this Shiraz, the town where angels bent down to kiss its very soil? I must remember who it was who wrote a eulogy of Shiraz... Sounded like... Mohammad-ibn-Yusef Saqafi. I memorized the title. Yes, this is the land which will nurture many thousand men of bounty. It's the seat of the Sufis, the wellspring of our country, the essence of our Imams' spirituality... oh my, oh my! So where have they gone? Where are these people that are not coming forth now? I've heard a hundred times myself that all our wise men have abandoned this land... they asked a sparrow why he didn't come in winter; he replied, \"What good did you do me in summer that I should come again?\"\n\nAnd now here's Nana Ferdows. There's a small rolled-up rug inside the bath bowl she's carrying on her head. And here's Ezzat-ud-Dowleh leaning on Ferdows, limping along. Oh dear, look! Ezzat-ud-Dowleh sits down on the rug in front of all those people without her veil, and her gaudy hair is showing. No. It seems as though she's wearing a wig like a turban. Here's Hamid Khan, her son. The bastard reaches out and pulls at Ferdows's breast. He's pulling very hard. Ferdows gets up to go about her business. Her legs are tapered and shapely in those transparent stockings. Everyone is staring open-mouthed at Ezzat-ud-Dowleh and her son and Ferdows. Then they burst out laughing.\n\nWhere has Ezzat-ud-Dowleh's husband been all this time and why is he arriving only now? Maybe he's escaped from the grave. He's been dead for a long time, you know. Oh look, he's wearing a cashmere brocade cloak and a brimless hat in this heat. His hat is very, very tight and it's squeezing his forehead. There's a perforated hole in the corner of the hat too... Oh I know! He's back to kill Massoud Khan all over again. He reaches under his brocade cloak and brandishes a long pistol which he aims at Massoud Khan and bang... bang... bang... he drags the corpse on the ground and abandons it on the green by Seyyid Abol-Vafa's shrine. But it seems as if Massoud Khan isn't dead. He rolls in the grass among the cucumbers and the pumpkins and eggplants. He opens his eyes and stares at all the people who've come to watch him. \"Water!\" he moans. Soon there's pandemonium in town. Massoud Khan is dead. He died in Haj Agha's arms in the droshke. There's no-one to calm down the crowd. They're about to raid the Jewish quarter. They're charging into the houses. People are running to the roof-tops and hoisting a British flag to proclaim that they're under the protection of His Majesty's British government. What chaos! The men who are on the roof-tops jump down quickly to the ground. Each man is carrying a basin on his head. They put the basins down on the ground. In each basin is a severed head dripping with blood. What a lot of noise they're making!\n\nThey've tied Malek Sohrab's hands behind his back, but he's laughing so hard he could fall over. He staggers to the left and right. Children follow him, clapping and chanting, \"Bring him here! Bring him here! Give him to the bride!\"\n\nNow they're erecting a gallows in the middle of the Baq Takht. What a loud hammering! Why didn't they do all this earlier so Malek Sohrab wouldn't have to wait? The men's eyebrows have grown so bushy, they cover their eyes. The men push back their eyebrows so they can see better. The women, sitting on the rugs, are straining to see what's going on. There's room for everyone. But they all have a problem with their eyes. How the eyeballs spin around! Maybe their eyes have rolled to the back of their heads! No. The men had their eyes under their eyebrows, didn't they? But the women are so wrapped up in their veils, you can't tell where their eyes are.\n\nThey bring Malek Sohrab to the gallows, but instead of putting the noose around his neck, a soldier with a gun on his shoulder comes and ties him to the stake. Malek Sohrab gives the soldier a surprised look and says, \"Gently! Not so tight\u2014you're hurting my foot.\" And then he says, \"That's better now.\" And he laughs. He laughs so heartily, it echoes all around the Baq Takht. The same soldier tries to blindfold Malek Sohrab with a black handkerchief but Malek Sohrab says, \"There's no need for that! Pull the trigger as quickly as you can. On the temple, between the eyes, in the heart, aim wherever you please. It doesn't make any difference if you do it sooner or later. I'll be standing right here. I've been waiting here for you for a long time. You can even chop me up with an axe.\"\n\nOh no, the ropes have turned into such snakes! Thank goodness Haj Mohammad Reza the dyer has arrived. He's wrapped some felt around his hand, takes the snakes' heads one by one, and thrashes them to the ground.\n\nAnd here comes Bibi Hamdam in her wide breeches. She shouldn't have come. Why should anyone come to the hanging of her own son? Maybe Malek Sohrab's first wife is being avenged this way. Weren't Sohrab and his wife madly in love? Yes, they were. But Bibi Hamdam wouldn't stop talking about infertility and childlessness. Wait! The Quran reciters are here too. There's no need to count them. They'll arrange their voices in unison and chant the Al-Rahman verse... Malek Sohrab's poor wife used to say, \"Bibi Hamdam, if you wouldn't plague us about having children every minute, we wouldn't worry about it ourselves and ruin our happy life together.\" And she'd told the story of another barren woman. What a night that had been! They were in the village and Zari was pregnant with the twins. Her pregnancy had reminded Bibi Hamdam of her desire for grandchildren. None of them could sleep a wink. It was so hot. Zari's hands and feet felt as if they were on fire. If she tried going outside the mosquito net, mosquitoes would attack her... She was parched with thirst. Further away, Malek Sohrab and his first wife were sleeping under their mosquito net. Bibi Hamdam had stayed indoors. There was a lot of noise; first the chanting of religious mourners, then the barking of dogs, next the tinkling of sheep-bells as the sheep stirred in their sleep, even the sound of crows quarrelling about whether the sun was coming up or not... and all Zari could think of was the story Malek Sohrab's wife had told:\n\n\"A woman who desperately wanted children went to a dervish. He told her to fast for forty days and on the fortieth day to go up on the mountain and wash her body under a waterfall. But there was one condition. She was not to think of monkeys. She was allowed to think of all sorts of things, but not monkeys. Five times the woman went up the mountain and stood under the waterfall, each time after forty days of fasting. Yet she could not rid herself of the thought of monkeys. Each time the one thing that crossed her mind was the image of a huge, hairy monkey. Finally she went back to the dervish and said, 'Your remedy didn't work. If you hadn't mentioned monkeys, I would never have thought of them in a hundred years. But now that you have...'\"\n\nAnd here's Captain Singer with his short, pleated tartan kilt which he has embroidered all along the edge himself! He sits behind the Singer sewing-machine and sews away... But this is no time for sewing! How fast he treadles the machine! His eyes run from one end of the fabric to the other. He does a zig-zag stitch. No, it's lattice-work. The material is as full of holes as a sieve. Now he's standing up to make a speech.\n\n\"Ladies and gentlemen,\" he says, \"Give alms! We have brought you civilization as a gift.\" His eyes fall on Zari, and he says with a smile, \"When madam gives you hand, you kiss madam's hand.\"\n\nPeople are clapping, but not for Singer. They're clapping for the little cherubs who are coming down sun-beam ladders with sacks full of stars. The cherubs come amongst the people and give each person his own star. Zari receives hers too. The cherub tells her, \"Now it's up to you. Our heavenly Master is weary. Very, very weary.\" But Zari loses her star. Now she's searching everywhere, rummaging in every cupboard, and throwing out all the rubbish from the attic. She hunts in every trunk in the store-room, but nowhere can she find her star. She is wandering in the garden. She looks on top of the brick walls and under the trees. She asks Khadijeh, \"Have you seen my star?\"\n\nA tearful voice says, \"Now that you haven't found any saffron, make some halva from chalk instead.\" It's Ameh.\n\nA doleful voice sighs, \"You can make halva with chalk, but you can't eat it.\" This time it's Malek Rostam.\n\nAnd again Ameh's voice which sounds stronger, \"Khadijeh, make some yellow-rose halva for my young flower who died... alas! alas! The tears dry up, but the sorrows remain.\"\n\nThe door slammed and someone entered. Zari opened her eyes. Seyyid Mohammad said, \"They answered the telephone. His Holiness the head of the Sufi dervishes said we could have the memorial service in the House of Ali. The Imam's house is open to everyone.\"\n\n\"Fancy being parliamentary deputy of this town and not being able to hold your own brother's funeral in the Vakil Mosque! Ah! well, write it down, Khosrow... what date is the day after tomorrow?\" Abol-Ghassem Khan had been speaking.\n\n\"The thirty-first of Mordad.\" It was Khosrow's voice.\n\n\"Write down, 'On the occasion of the tragic passing away of our dearly beloved young...'\"\n\n\"Passing away?\" interrupted Ameh's voice. \"Put down, 'martyrdom'!\"\n\n\"Sister, I'd be grateful if you'd let us do what we have to do. I persuaded him with a great deal of difficulty to print the announcement, but the man set down a thousand conditions... one of his conditions...\"\n\n\"I think you should write martyrdom too,\" said Malek Rostam.\n\nWhat a foul odour there was in the air! If only a passer-by would throw out the charcoal brazier. If only he would ask, \"What's wrong with your patient here; why is he lying like a corpse, like a dead body, on the bed? Let me take him to the garden underneath that grafted fruit-tree. Outside, the sky is full of stars.\"\n\nZari's heart would race away and palpitate, then race away again. She would close her eyes and see a truck on fire, burning away. An officer comes and stretches out on top of a dead soldier in a trench.\n\nAnd again Ameh's voice, \"It's cat-shit. They've left the top of the coal-bin open and the cat's dirtied inside it. Khadijeh, come and take away the brazier. I don't want it, after all.\" Then she went on, \"Have all the rooms been cleaned? Did they drain the pool? Did Gholam sweep the garden?\"\n\nAnd now Zari sees a little girl with braided hair tied up in ribbons, standing by the herbalist's store on top of the Moshir Hill. She needs seven ingredients to make black shoe-polish, by order of her physics teacher. Actually, that shoe-polish will never turn out properly; she'll make a gel like black frog-spawn and no-one knows whose fault it is\u2014the girl's, the physics teacher's, or the herbalist's? The herbalist has gone to the back of the shop to look for the seven ingredients. It's late afternoon and the girl is in a hurry to get back to school to make shoe-polish during the fourth lesson. Suddenly a man on horse-back approaches her. The rider is Prince Charming himself; he looks so handsome and erect on his horse. He has green eyes... they shine like emeralds in the sun. And now as he stands in the shade, they look moss-green.\n\n\"Do you know how to get to the Sang-e Siah, my dear?\" he asks. The girl panics. There is no-one around that afternoon. Still she ventures, \"Do you want to go to the Sibavayh grave?\"\n\n\"No, my dear. I want to go to the house of Sufis, the Khan-i Qah.\"\n\n\"Are you a dervish, then? You want to go to the House of Ali?\"\n\nThe rider laughs and his white teeth glisten. \"No, I'm not a dervish,\" he replies. \"My steward is a dervish. He's ill and he's staying at the Khan-i Qah. I'm going to visit him.\"\n\n\"Well then, go straight ahead. Then turn right. After that, turn left, and another left... But you can't go on horseback. The little back-alleys are full of bumps and stones.\"\n\nNow that she's given him directions, why isn't the rider going away? Why is he looking her up and down? Yes, I understand. He's wondering why she, of all women, should be without a veil.\n\n\"I must explain,\" she thinks to herself, \"or he'll think I'm Armenian.\"\n\n\"My father was Mirza Ali Akbar Khan Kafar,\" she says out loud; \"he stated in his will that I should never wear a veil.\"\n\nThe rider takes off his hat. It's a strange-looking one with a brim, but it's not the new pahlavi hat. He bows to the girl and says, \"I never asked why you're not wearing a veil.\"\n\nAnd he leaves.\n\nBut what will? As if there had been anything to bequeath! That very afternoon, with the shoe-polish gel still on her hands, there is news of unrest in town. The English headmistress lines up all the girls and tells them to put their face-veils in their satchels and that heavier veils would be brought for them from home. But unlike other times she doesn't nag and say, \"This country doesn't deserve to be civilized.\" Her glance falls on the girl with braided hair and ribbons, and she asks, \"Zari, do you know how to wear a chador?\"\n\nWhether or not she knows how to wear a veil is irrelevant, because there's no-one at home to bring one to her. Khanom Hakim has been cutting up her mother's breast at the Missionary Hospital, and that's where they're keeping her for the time being. Who knows how long it will take for her to get better? Her brother's away doing his military service and he won't be back for a long time. Their old maid-servant is too feeble-minded to find out what is going on in town and to bring her a veil. Well, everybody is leaving the school now. Nazar Ali Beg the Indian janitor agrees to fetch a chador and face-veil for her after all the other pupils have left. But it would take so long for everyone to leave.\n\nServants arrive to take the girls home, bringing them the veils which they put on before leaving. But there she is still, all by herself. Now she is alone with Nazar Ali Beg and it's getting dark. She's afraid. Nazar Ali Beg has a long moustache which droops lower on one side. His face, too, is slightly crooked. He explains that ruffians have poured into the streets and alleys, tearing away at women's face-coverings or men's brimmed hats, and that eventually they'll get to the school too, and break all the windows. She's afraid of Nazar Ali Beg because he keeps saying in his funny Persian, \"Khanom, good Khanom!\" But at the same time, she doesn't want him to go fetch her a chador, leaving her all alone in that vast school-building.\n\nSuddenly she has an idea. She decides to call the house of the head dervish and ask Mehri to send her a chador. She's glad she's had such a good idea. She prays the rider she saw that afternoon is still at the dervishes' house. She telephones and then sits by the pool and daydreams. She dreams that she's riding with Prince Charming on his horse; they're galloping towards Baba Kouhi, the mountain dervish, and she's singing for him:\n\n\"The lips of the Turkoman maiden should not have been created so perfectly...\"\n\nThey're knocking at the school-door. Yes, it's him. The girl smiles when she sees him. But this time he's on foot and hatless, and is carrying a parcel wrapped in newspaper. He holds out the parcel and says, \"Here, put it on. I'll take you home.\"\n\n\"Sahib, good Sahib!\" says Nazar Ali Beg.\n\nThe girl doesn't know how to keep the veil on properly, and it keeps slipping off.\n\n\"Do you have a safety-pin?\" the man asks Nazar Ali Beg. Nazar Ali reaches behind his coat collar and produces an ordinary pin.\n\nThe girl walks off with the man, though she can't really see where she's going and nearly trips.\n\n\"Why did they bother you...?\" she manages to ask.\n\n\"There wasn't anyone else. The dervishes had disappeared into their little cubicles. Mehri Khanom, the niece of the head of the dervishes, asked me to deliver this chador to you on my way. She said your name is Khanom Zahra. My name is Yusef.\"\n\nThe headmistress of the English School had taught them, on being, introduced, to extend a hand, smile and say, \"How do you do!\" But how could she? Both her hands were taken up\u2014one with her chador, the other with her books. The man continues, \"I knew your father. I used to study English with him, until I went abroad. He was a great man, in his own right. He inspired noble ambitions in his students.\"\n\nThe girl remains silent. \"Mehri Khanom has been telling me that your mother enjoys going under the surgeon's knife,\" the man says. \"She enjoys having part of her flesh cut off and thrown away. Apparently every day she finds an excuse to go to the Missionary Hospital\u2014one day it's a bruise on her big toe, another day it's a lump in her breast...\"\n\n\"You mean to say Mehri doesn't think my mother has cancer and she's wasting her time with all this surgery? I hope to God that's true.\"\n\nThe girl is looking at the man's shoes. She stops abruptly and says, \"Your right shoe-lace is undone.\"\n\nThe man bends over and ties his shoe-lace. How quickly she's become familiar with the stranger! It's as if she's known him for years. And what does the man think now? That she's the kind of girl who comes away with him easily and even confides in him! Thank God she's wearing a veil and face-cover, and no-one will recognize her. Thank goodness there isn't a soul in the Jewish quarter. What if the man thinks she planned all this to catch him? Well, in fact she had. Mehri had realized this too and tried to help her.\n\n\"Mehri's face is as lovely as a flower, isn't it?\" she asks. The man smiles and says, \"I didn't see her face. She was wearing a veil.\"\n\n\"We were classmates up until the sixth grade. Every day we'd gather round the stove and she would teach us the Quran and religious law. Then she would tell us the stories she'd read in the Thousand and One Nights. She has a good voice too. She sang us Masnavi poems... the one that goes, 'I sight the King in any guise...' I've forgotten the first part.\"\n\n\"I want an eye to sight a King\n\nTo sight the King in any guise.\"\n\nThe girl has to make an effort to control herself. She's about to say, \"I sight you!\"\n\nThat was really why she quoted the verse in the first place.\n\n\"Has anyone ever told you your voice is as soft as velvet?\"\n\nThe girl doesn't make a sound.\n\n\"You were talking about Mehri Khanom...\" says the man.\n\n\"Anyway, she left school to get married... I don't know why her husband divorced her after a year. The husband died soon after. They say her uncle, the head of the Sufi dervishes, had cursed him...'\n\n\"Do they stuff your head with these superstitions at the English School?\"\n\nThe girl is hurt and stays silent.\n\n\"What class are you in?\" asks the man again.\n\n\"Eighth grade in Persian and ninth in English,\" answers the girl, still hurt.\n\nThere is no-one in the back-alleys. No-one has lit the street-lamps. The girl wants to lift her face-cover but she doesn't dare. It's a good thing she knows her way home by heart and is familiar with all the bumps and ditches. She could walk home with her eyes shut.\n\n\"Mehri Khanom talked about you all the time,\" says the man. \"She said you once dealt your headmistress a severe blow by reciting the poem about Samson's blindness in front of the Oriental Missionary Council...\"\n\n\"The poem just happened to come to my mind. I didn't mean to be cheeky or anything.\"\n\n\"You're modest, too!\"\n\nNow they've reached the bazaar and they're both quiet. In the bazaar they have lit a few oil lamps which they've placed on stools in front of the shops. But all the shops are closed. There are seven or eight policemen roaming around. There's quite a din but it's coming from the sword-makers' section. In the main bazaar itself, a few people are going on their way.\n\n\"They've blamed it on them again!\" mutters the man. The girl doesn't understand what he means. Or maybe she didn't hear correctly.\n\nThey reach the vaulted passageway which is darker than anywhere else. The man takes the girl's arm, and the girl flushes. Her whole body seems to be flushed, in a way she has never experienced before... Now they reach the girl's house. She politely invites him to come in and have a drink, but she's praying he won't accept. He doesn't come in.\n\n\"I remember you had a winter-sweet bush in your house in those days,\" says the man.\n\n\"We still do.\"\n\n\"They're hard to grow, but when they take well, they flower every year... and what long-lasting, fragrant flowers!\"\n\nShe both wants him to go and yet she doesn't. She asks him out of the blue, \"Have you done your military service?\"\n\n\"I'm going this autumn.\"\n\n\"It takes two years, doesn't it?\"\n\n\"Try to grow up as soon as you can.\"\n\nAnd again Zari doesn't understand his meaning. Later when she tells her mother the story, her mother agrees that God had sent the stranger to save her girl... and then... three years later, when that same man comes to ask for her hand...\n\nWhat pandemonium in the garden! She had been having good dreams. Obviously she must be getting better if her nightmares and delirium have left her in peace. Someone was roaring in the garden: \"Ya Hu, Ya Haq, Ya Ali.\" It was Seyyid Mohammad's voice.\n\n\"Khadijeh, what happened to the lemon juice?\" Gholam asked.\n\nZari thought he must be drunk. \"Take him to the Seven Saints,\" she said out loud.\n\nAmeh came towards her. \"Did he wake you up?\"\n\nZari opened her eyes. Only Ameh was left in the basement with her. \"They'll wake the children with their noise. They'll frighten them,\" Zari said.\n\n\"Don't worry. Mehri is keeping the children tonight; she'll keep them tomorrow too. I sent Khosrow up to bed on the roof after a thousand pleas.\"\n\nThe noise in the garden abated for an instant. From somewhere, the monotonous chanting of the Quran could be heard like a hum. Someone was retching. Retching. Someone else was cursing out loud and saying, \"You're sitting up there watching, are you? Why don't you step down here for a second, taste your own broth which you've been giving our folks... I spit on the bloody\u2014\"\n\nSomeone started to sing:\n\n\"A houseful of drunks already\n\nMore drunks have now arrived!\"\n\nAmeh put a hand to Zari's forehead and said, \"They've been sitting around together drinking so much spirits, they're all drunk now.\" And she added with concern, \"You try to sleep.\"\n\n\"What if Khosrow wakes up and falls off the roof? I wish you'd sent him off to Mehri's too.\"\n\n\"Hormoz is sleeping on one side of him, Majid Khan on the other.\"\n\n\"Ya Hu, Ya Haq, Ya Ali!\" It was Seyyid Mohammad's voice in the distance. Close by, a man's incessant weeping broke the thread of her fantasies, and Zari felt as though that sound would never end. She opened her eyes. She saw Malek Rostam sitting with his head in his hands, sobbing out loud. Abol-Ghassem Khan was sitting there too, pale as a corpse, as if he were the one who had died.\n\n\"Malek Rostam Khan, please stop crying for this poor woman's sake. She'll wake up,\" said Ameh.\n\n\"I told Khanom Hakim to give her a shot that would put her to sleep,\" said Abol-Ghassem Khan. \"I said if she stayed awake she wouldn't last till the morning. This blow will shatter her by dawn. I wish she'd given me a shot too.\"\n\nThe weeping hasn't ceased but Zari's mind has fled all the misery and sorrow... and sees herself hand in hand with Yusef crossing a field of wheat. The golden wheat is ripe and bends its head to the twilight breeze. Zari and Yusef come to a stream of water and sit down until it gets dark... they're sitting in the dark and holding hands, and Zari feels as though there's no-one else in the world besides herself and Yusef. She puts her head on Yusef's shoulder and listens to his heartbeat. How long did they sit in the darkness without talking?\n\nThat night they sat by the window in the darkness and stared out at the garden... that night Baba Kouhi told their fortunes, reading from Hafez, \"The harvest of our work in this world is not so important!\" And he recited the ode to the end. Then he predicted, \"It will pass in the twinkling of an eye. It's as close as the mouth to the lip.\"\n\nBaba Kouhi went to his own room while Zari and Yusef sat in the darkness and stared at the town-lights. They came down the mountain holding hands in the blackness of the night. That night, in the twins' bedroom, Yusef took the keychain from Marjan's pillow and put the lights off. He held Zari's hand while they stood still in the darkness, listening to the children's breathing. Then they slept together stark naked under the mosquito net, waking Khosrow up with their noise, and he called out to his father. Yusef hurriedly pulled on his night-clothes and went to him, saying, \"Go to sleep, son, it's nothing.\" He came back and they both sat up in the mosquito net, their hearts pounding so loudly they could hear it, and they waited for Khosrow's breathing to become even again. And all those days and nights which came and went...\n\nAgain Ameh's voice, but Zari didn't want to hear it. She was having pleasant dreams, yet the voice imposed itself, \"I have a black scarf and dress myself, but Zari doesn't. My poor brother disliked black. When Zari's mother died, he didn't let her wear black for more than forty days. I had to give a scarf and dress of hers tonight to Haj Mohammad to be dyed black. I gave all the bed sheets to be dyed black also. He said he would stay up the whole night to do the clothes and sheets. The weather is warm; they'll be dry by morning.\"\n\n\"The sheets?\" asked Abol-Ghassem Khan.\n\n\"I want to cover all the rooms in black. I'll throw black sheets over all the seat-cushions around the room.\"\n\nBut Zari is in the village now, not in the basement of their house. She is in the village, and she knows that they are reaping the last field. She knows Yusef is waiting for her by the mill. She is supposed to reach him before the sun leaves the fields, and it's a long way there. She comes out of the landlord's house. The village women in their chadors are squatting, washing their tea-things in the stream of water which issues from the landlord's house. They greet Zari when they see her, and she stops to chat with them. She points to Kolu's mother's rounded belly and says, \"You've filled your pot again!\"\n\nShe looks at Goldusti, Kolu's aunt who has just got married and is wearing heavy make-up. Zari says, \"You're having a good time of it, aren't you?\"\n\nKolu is there too, idling about. Zari pats him on his wavy hair and says, \"Run to Seyyid Mohammad. Tell him to saddle the horse and bring it here.\"\n\nKolu giggles and runs off.\n\nZari mounts the horse and rides through the harvested fields. The wheat has been piled around like so many heaps of gold. The men are stacking the hay, tying it in bundles with black rope and loading it on the mules. The men greet her as she passes by every field. She returns their greeting kindly and rides away. When she reaches the upper village, she is surprised. Just above the lowlands they have covered up the doors of all the houses with mud. It looks as if the village is deserted. Yes, actually there are a few people\u2014several tribal women going on their way. But the tribe has already migrated to its summer quarters. She saw them going away herself. They pitched tent for a few days in the upper village and then left.\n\nShe leaves the fields and the men and women behind. And she reaches the last field moments before the sun disappears. The men are still reaping. The women gleaners, wearing black scarves on their heads, sit in a row to one side of the field. She knows Yusef always tells the men to reap carelessly so the gleaners can pick up something for themselves afterwards. It's for this reason that the gleaners always bring two large woollen sacks with them. She spies Yusef sitting on a rug in front of the mill, smoking the hookah, a thin cloak thrown over his shoulders. Yusef sees her too, and comes forward to greet her still wearing his cloak. He sweeps her off the horse and puts her down on the ground. \"Come share my cloak,\" he says. \"You're sweating; I'm afraid you'll catch cold.\" Then he says, \"The sun was on your hair and made it look the colour of musk-willow in the distance.\"\n\nZari sits on the rug, wrapped under Yusef's cloak. The miller has planted three Marvel of Peru bushes in front of the mill, and now, at sunset, he is watering them from a tin watering-can. He is covered with white flour from head to toe. Even his eyebrows, eyelashes and hair are powdered white.\n\nThe miller brings a tin tray which he places on the rug before them. There are two loaves of round bread which he has baked himself, a bowl of home-made yogurt, and a bunch of fresh spring onions. He has also put salt and pepper for them on two bits of paper. Zari prepares a large mouthful of bread and yogurt which she offers to Yusef. Yusef laughs and says, \"I know you're hungry, coming all this way. Eat it yourself.\" And how hungry she is...\n\nBehind the mill is the landlord's summer-crop, watered by the stream which turns the mill. Yusef gets up and goes. When he returns, Zari sees that he has filled his cloak with something. The miller brings a brazier full of coal-fire. A blackened old kettle is sitting in the corner of the brazier. He places the brazier on one side of the rug. Yusef has picked a lot of corn-cobs. He puts them on the fire and fans them with the top of a cardboard box which the miller has given him.\n\nZari and Yusef go over to the gleaners. Their sacks are brimming, and they've tied them together with a piece of rope. The men help the women heave the sacks on to their shoulders before they start off. Zari falls in step with a middle-aged woman who is one of the last to leave.\n\n\"Mother, why are you wearing a black scarf?\" she asks.\n\nThe woman doesn't seem to hear. Instead of answering the question, she blesses Zari. \"May you live long, my dear. May Allah bring you health and prosperity.\"\n\n\"Why are you all wearing black scarves?\" Zari asks again.\n\nThe woman hears her this time. \"Bless you, my dear,\" she says, \"tonight is the eve of Savushun. Tomorrow is the day of mourning. If the Khan's guide has arrived, we'll be there by the cock's crowing... as soon as we arrive they'll start beating on the drums and the kettle drums.\"\n\n\"Where is this Savushun?\"\n\nAgain the woman hasn't heard. \"No, my dear,\" she answers, \"we'll be going on mules. Your servant Mohammad Taghi has brought the mules, and he's waiting for us under the Gissu tree. He'll be getting a whole sackful for the fare.\"\n\nThe woman stops. She's becoming talkative. She continues, \"When we arrive, we all sit around the arena in a wide circle. They bring hot tea, bread and gingerbread... as well as rose-water drink, sweet grapes... they hand out lunch and dinner on the eve of Savushun. In the middle of the arena they've put firewood which they set alight. All of a sudden you look up and you see the night has faded. But it's well before dawn when, God bless him, he appears high up on the mountain riding his steed. You'd think he was praying right there on horseback. He lifts a Quran to his brow and prays for all Moslems in the world. God Almighty. He is wearing black from head to toe. Even his horse is black. He comes down and jumps over the fire on horse-back. We women clamour and scream. The men cheer, the boys whistle, they play drums and kettle drums, and suddenly you see the sunrise, and the whole arena is now bright.\"\n\nZari enjoys listening to the woman's talk. \"Well, what happens next?\" she asks.\n\nThe woman has fallen behind her fellow-gleaners; she's following them with her eyes. Zari notices and says, \"No, you go ahead and catch up with them. You'll be late.\"\n\n\"By the time they pack their things and gather the children, I'll soon catch them up.\" And the woman adds, \"Bless you, my dear, you're our mistress and benefactor. Now you want me to tell you this story.\"\n\n\"All right, let's go together,\" says Zari. \"You can tell me on the way.\"\n\nAgain, they walk in step. The middle-aged woman continues, \"God bless him, he comes all alone towards the arena. He circles slowly. He's thinking. How can he fight so many of the accursed enemy single-handed? From one side of the arena the Princes of Earth come to the middle to see if he'll allow them to help him.\"\n\n\"Princes of what?\"\n\n\"One group are holding some soil in their hands and wear flowery brown headgear. They are the Princes of Earth. Another group have fans in their hands, and are fanning themselves; these are the Princes of Air. Others wearing black and holding torches are the Princes of Fire. They come to his aid from the three comers of the arena. Finally, from the fourth corner, a wandering dervish appears, chanting the name of Ali...\" The woman sighs. \"Ali, my saviour... don't abandon your humble believers... in justice,\" she says. \"The dervish's begging-cup is full of rose-water drink. He takes the horse's bridle and says, 'Drink a sip of this in the memory of Imam Hussein's thirsty lips.' But he throws the drink to the ground and dismisses the princes. All alone, waiting for the accursed enemy, the rider stays there on his horse. He has no sword, no bow and arrow. The sun has now spread from one end of the arena to the other... Suddenly the accursed enemy, riding their horses, charge from all four corners. Thirty or forty of them attack his holy person. They fight... the drums roll... they beat harder and harder on them... and now so loud and fast, your heart is about to explode.\n\n\"Finally they pursue his horse and drag him down from it. They tie the bridle around his blessed neck. They put the saddle on his shoulders. They tie his hands behind him, but he doesn't utter a sound. His bare black horse stands there neighing so loudly it echoes all around. One of those villains is dressed as the executioner. He comes forward and takes the horse's bridle which they've tied around his highness's neck. This man is mounted, but that poor, lonely captive is on foot. They drag him all around the arena, and he keeps stumbling and getting up again. He's bloodied, his black clothes are torn and covered with dirt. But he doesn't moan or show his pain.\"\n\nThe middle-aged woman is crying and she wipes her eyes with the corner of her black scarf. She blows her nose before continuing between tears, \"Then that villain dismounts and puts a sword to his noble throat. He covers his face like a sheep and places his head at the edge of a basin... He sharpens his knife before our eyes... Sharpens it like a razor. But by the will of God, the blade will not cut. Then he lays him face down and puts the knife to the back of his neck. The oboe plays so mournfully... oh so mournfully. Suddenly you see his horse covered with blood, just like that! His mane is dripping with blood. Our elders say that once, in Solat's time, his highness's black horse hadn't been able to bear it and had died of grief right there. I've seen the poor animal's tears several times with my own two eyes.\" She pauses and wipes her eyes again.\n\n\"We women put hay over our heads in mourning. Our men take two mud-tiles each which they beat together to shake dust and hay over their feet, and then they do the same over their heads...\"\n\nZari feels her eyelids burning. She nearly puts an arm around the woman and cries along with her. But they've reached the Gissu tree now. The woman says goodbye, blessing her again, and a man who is probably Mohammad Taghi comes forward to help the woman remove the sacks from her back and get up on the mule...\n\nZari and Yusef are riding their horses side by side.\n\n\"Do you know what Savushun is?\" Zari asks Yusef.\n\n\"It's a mourning ritual. All the people of the upper village observe it tonight.\"\n\n\"Is that why they've covered up the doors of their houses with mud?\"\n\n\"Yes, their trip will take a few days.\"\n\n\"A village where houses have no doors, and where the inhabitants meet under the Gissu tree to go to the Savushun together!\" Zari says sadly.\n\n\"In other words, the mourning of Siavush's death. The people of the upper village leave every year after the harvest and return in time for the corn-threshing.\"\n\nThey both fall silent. It's getting dark on the lowlands. They ride their horses and stare ahead. Zari's eyelids are burning as tears roll quietly down her cheeks. So quietly Yusef wouldn't know. But she is already sobbing. She cries with all her heart.\n\nA hand wiped away her tears. It was Ameh's hand.\n\n\"I beg you by Yusef's departed spirit not to cry,\" said Ameh.\n\n\"I was crying for Siavush,\" said Zari, sitting up. \"At first I didn't know about him, and I disliked him. But now I know him well and I feel sorry for him... I was standing under the Gissu tree, crying for Siavush. Pity I don't have long hair, otherwise I would have cut it off and hung it on the tree like all the others.\"\n\nWhat had she said to hush them up? Why were they staring at her? A silence and a stare you couldn't endure. Zari felt as though something fell and shattered inside her. Who had told her once, \"A storm raged within the folds of my body\"?\n\nAbol-Ghassem Khan put a hand to his waist, stood up and came towards her. \"How many times did I tell that poor soul not to let this weak, fragile woman go so often to the asylum? But he wouldn't listen...\" he said.\n\n\"For heaven's sake, man, don't go jumping to conclusions,\" said Ameh.\n\n\"Someone was telling me the story of Savushun,\" said Zari. \"How he'd been all alone and the enemy numbered a thousand... of course, he couldn't overcome them single-handed...\"\n\nMalek Rostam spoke up from where he was sitting, \"Abol-Ghassem Khan, sir, please sit down.\" And he whispered something quietly.\n\nBut Zari managed to catch what he was saying. \"Don't worry,\" she heard him say, \"she hasn't gone mad. Why can't someone cry for Siavush?\"\n\nAbol-Ghassem Khan beat himself on the head and said, \"Who's Siavush? What's the Gissu tree? The world is spinning and crumbling all around me... under the rubble... a pity, a thousand times a pity.\"\n\n\"I've attended Savushun many times myself,\" Malek Rostam said. \"When the Ta'zieh passion play was banned, that was stopped too. And the Gissu tree is famous all over the lowlands.\"\n\n\"The first time I saw the Gissu tree, from a distance I thought it was a wishing tree with all those bits of yellow, brown and black ribbons hanging from it,\" explained Zari. \"But when I went closer, I realized those ribbons were in fact braided locks of hair. Hair that belonged to young women who had lost their husbands... or sons, or brothers...\"\n\nWhy was Abol-Ghassem Khan frightened and listened to, but still didn't believe, whatever they were telling him? Why did Ameh too start to doubt and didn't say anything anymore, but Malek Rostam kept reassuring them it was all right to cry for Siavush? Zari put her head on the pillow again and thought, \"If only they would just let me be happy in my own thoughts, riding horses in my dreams, walking over reaped fields, sitting hand in hand with Yusef by the piles of wheat... I'd put my head in Yusef's lap and he would rub my temples with his fingers and say, 'I'm willing to bet you're going to be just fine.'\" \n\n# _22_\n\nFinally that bloated night of nightmares and terrors released its grip on Zari. At dawn, she got up. Her knees were shaky and her mouth had an acrid taste. She went out to the garden and listened to the sound of water pouring from the stone head into the pool. She washed her hands and face. The coolness of the air, the freshness of the garden, the smell of the moist earth, the chirping of the early sparrows, the clean water which had reached half-way up the pool\u2014all of these revived her somewhat.\n\nThey had left the wooden beds in the shade of the building next to the pool, and covered them with carpets. Khadijeh came out, carrying a tray which she placed on one of the beds.\n\n\"I knew you would feel better in the morning,\" she said, greeting Zari. \"Thank God! I broke an egg for you, I burnt some wild rue to ward off the evil eye. I tried all kinds of vows and prayers.\"\n\nShe spread a tablecloth on one of the wooden beds, and put some knives and plates on it. She went away to fetch the samovar and came back with it minutes later, boiling and ready for the tea. Zari sat down cross-legged by the tablecloth. Her stomach growled with hunger.\n\n\"We couldn't find your keychain last night even though we searched everywhere for it,\" said Khadijeh. \"There's probably some sugar, tea and saffron in the store-room. I know we have a bottle of sugar syrup... by the way, Khanom, we're short of fans, too.\"\n\n\"Where are they reciting the Quran?\" Zari asked. \"The voices seem to be coming from around the well.\"\n\nKhadijeh stood and stared at her. \"They've put the body in the cistern, between big sacks of snow. It was coolest there,\" she said, and looking Zari over carefully, began to say, \"You've changed so much overnight...\" but she finished her sentence, \"My poor mistress, what have you done to yourself! You've lost so much weight. Do you remember my uncle's wife who swallowed opium once? I was the one who saved her. She looked just like you do this morning.\"\n\nJust then Gholam came in through the garden gates followed by Haj Mohammad Reza the dyer. Gholam was carrying an iron in one hand, and Zari's black dress and scarf in the other. Haj Mohammad Reza, wearing a long-sleeved black shirt, was balancing a large bundle on his head with hands which matched the colour of his shirt. Zari took her things from Gholam and went to the bedroom. She put the dress on with difficulty; it had become too tight. Digging a hand into the pockets of her dress, she found a crumpled and blackened two-toman bill in the right one. She glanced involuntarily in the mirror. She didn't recognize herself. She switched the light on and took a closer look. Several strands of hair had turned white, and her parched lips had lines around the corners. Her darkly-circled eyes seemed to have sunk in their sockets. She thought, \"It's not true when they say all of someone's hair turned white overnight.\"\n\nShe went to the parlour which had been stripped of all its decorations, even the radio. Gholam and Haj Mohammad Reza were spreading black sheets on the cushions arranged around the room. Haj Mohammad Reza stood up when he saw her. He averted his eyes awkwardly and asked after her health. Zari thought, \"Poor soul, he's been up the whole night dyeing all this material.\" It seemed as if he had read her mind because he surveyed the black cushion-covers with satisfaction.\n\nWhen Zari came out to the garden, Ameh had just finished her morning prayers, and Abol-Ghassem Khan and Khosrow were having breakfast. Khosrow was wearing a black shirt which hung over his grey trousers. Zari sat at one end of the table-cloth, next to the samovar. She poured herself and Ameh some tea, but her hands were shaking and her head swam. Ameh broke two eggs, carefully disposing of the whites in the bowl underneath the samovar tap. She dropped the yolks in a cup, added some sugar, and started to beat it. Zari followed Khosrow with her eyes as he got up and went through the garden gates. Involuntarily she spoke her thoughts, \"The poor man has been up the whole night dyeing all of us black!\"\n\nAmeh raised her head as she was beating the eggs and changed the subject. \"Sister, did you find your keychain?\" she asked.\n\n\"Keychain?\" asked Zari distractedly. Then she smiled and said, \"Khadijeh was shocked to see me a few minutes ago. She said I looked like one of those people who've eaten opium and been rescued in the nick of time. She said I'd aged a thousand years overnight. No, she didn't say that. I don't remember what she said... I didn't recognize myself in the mirror.\"\n\n\"Khadijeh had no business saying things like that to you!\" Ameh replied.\n\nAbol-Ghassem Khan looked at Zari. He stared and shook his head. \"Didn't I say so, sister?\" he said. \"Last night you said I was making things up about her because I was interested in her money.\"\n\nAgain Zari spoke her thoughts out loud. \"I think Khosrow's gone to fetch Dr Abdullah Khan.\"\n\nAmeh bit her lip and said, \"When time heals her wounds, she'll be all right.\"\n\nHurriedly she poured some milk over the egg yolks, stirred it and handed it to Zari. But suddenly Zari wondered what Abol-Ghassem Khan had meant? Blood rushed to her face. Her heart pounded in her chest and again she felt as though something had shattered inside her.\n\nShe felt she had to explain. \"In the asylum,\" she said, \"the first thing every patient says is that he's not mad and he shouldn't have been brought there. But Abol-Ghassem Khan, you can be sure I haven't gone mad... you see... well, it was all so sudden...\" She left her sentence unfinished. She was not entirely convinced herself. What if she really had gone mad and didn't know it? A fear more insidious than the terrors of her recent nightmares gripped her, larger than anything she had ever experienced. She felt chilled to the bone but the palms of her hands were sweating. She had to show Abol-Ghassem Khan, and, more importantly, prove to herself that she hadn't gone mad. She ate her breakfast delicately, even though her appetite had gone, remembering to thank Ameh for the milk and eggs which she had hardly been able to swallow. Then she got up and called Khadijeh and Gholam. She sent Khadijeh to borrow fans from the neighbours, and then to fetch her keychain from the children at Mehri's. Then she sent Gholam to find tea and sugar at any cost.\n\nKhadijeh returned with an armful of fans and said, \"Khanom Mehri and Mohsen Khan were quarrelling, so I didn't dare go inside for the keys.\"\n\nGholam came back and said, \"I went all the way down the street, but no-one has opened their shop yet!\"\n\nAll the time Zari's eyes were glued to the garden gate in expectation of Dr Abdullah Khan. At first Hossein Agha the grocer and his brother Hassan Agha the local corn-chandler came in, clad entirely in black. Then the two distillers from next door arrived, sweating from the loads on their back. They had each tied a black armband around their bare arms, otherwise they were dressed as usual in a pair of drawers and an undershirt. They put their loads down next to the pool, opened the burlap sacks at the top and rolled them down carefully. Holding their hands in turn underneath the mouth of the stone head, they caught some water with which to sprinkle the roses and the eglantine inside the sacks. Soon the fragrance of the flowers filled the area in front of the house. Zari looked at the flowers and thought, \"How far they went to get these... they've spent the whole night picking those flowers, and in the darkness too... how many thorns did they get in their hands? Why didn't the youngest son go with them? I hope he hasn't come down with typhus as well!\"\n\nGholam, still hatless, approached Hossein Agha and said,\n\n\"Brother, I came to you earlier, but your shop was closed. See if you can get us some sugar, tea and saffron, will you?\"\n\nHassan Agha, Hossein Agha and the distillers left. In the driveway, they came across the old distiller himself who had put on Gholam's worn-out suit and thrown a black shawl around his neck. They stood and talked to the old man who followed them back on the way he had just come.\n\nA droshke drew up at the garden gate and Zari wanted to rush forward and greet the long-awaited Dr Abdullah Khan.\n\nShe was longing to make him tell everyone, \"Khanom Zahra hasn't gone mad. She's had a shock, that's why she seems distracted. Don't watch her so closely, because then you really will drive her mad!\" But it was Ferdows who came out of the droshke, taking Ezzat-ud-Dowleh's hand as she stepped out. The old lady descended with a lot of difficulty, and giving her arm to Ferdows, limped slowly up the driveway until she reached Zari who was standing in front of the house in a state of disbelief. The sun had just risen, and before Zari could collect herself from the surprise of this early morning visit, the woman had thrown an arm around Zari and was saying, \"The news came so suddenly last night, I wasn't myself at all and I left without saying goodbye or realizing what I was doing. All night while everyone was fast asleep, I couldn't close my eyes. You're like a daughter to me, and your late mother was my twin soul. God forbid, she'd always say, 'Ezzat-ud-Dowleh, I'm a dying woman. I leave my child in your hands.' Alas! Alas!\"\n\nShe sat on the wooden bed\u2014the same one that Yusef's broken body had occupied the night before, but which was now covered with a carpet. Rubbing her leg, she asked, \"Where's my sister?\"\n\nShe was swathed in black, including the gloves, scarf, socks... when had she had time to dye her hair black? Come to think of it, why should she dye her hair black at all?\n\n\"I said to Ferdows, 'Get up, child, let's go there first thing in the morning',\" continued Ezzat-ud-Dowleh. \"'Maybe we can give them a hand or something.' After all, what good is our so-called sisterhood if not for times of need?\" It was lucky for Zari that she managed to hold her tongue. If this woman accused her of madness too then she would be done for. It would give Ezzat-ud-Dowleh a week's worth of gossip with the Governor's family!\n\n\"My dear child,\" said Ezzat-ud-Dowleh again, \"what kind of dress is this you're wearing? A dyed thing, and ironed to a shine, too. It's not nice in front of people, and it's too tight for you.\"\n\nZari, who had her eyes on the garden gate, didn't reply. But Ezzat-ud-Dowleh wouldn't let up.\n\n\"My dear girl, why aren't you paying attention? Now go along like the nice lady that you are and allow Ferdows to let out your dress for you. There's probably some room left\u2014she'll open it at the seams.\"\n\nZari noted silently that those beady eyes didn't miss a thing. But she made no effort to move.\n\n\"By the way,\" said Ezzat-ud-Dowleh, \"I nearly forgot. I've brought you something which I know will really make you happy. A keepsake from your late husband\u2014no, you're not paying attention to me at all... look!\"\n\nReluctantly Zari shifted her gaze from the garden gate. Ezzat-ud-Dowleh took out a small box wrapped in white paper from her handbag and gave it to Zari. Zari held it in her hand, not knowing what to do with it. Again she stared at the garden gate. Ezzat-ud-Dowleh gave a little laugh and said, \"Go on, open it!\"\n\nZari mechanically undid the wrapping. Inside was a black velvet box. She opened it, and saw her emerald earrings shining at her from their small velvet case. She felt depressed. The earrings which Yusef had put in her ears on their wedding night with his own hands. Yusef's eyes had shone like those very emeralds in the light.\n\nEzzat-ud-Dowleh smiled. \"I knew it would make you happy,\" she said. \"Last night I went straight from here to the Governor's house. I decided that since I'd been responsible for having my dear child's earrings taken away, I had to get them back myself.\"\n\n\"Do you think I can be fooled like a child?\" said Zari. And she closed her eyes. She felt dizzy.\n\nEzzat-ud-Dowleh neither scolded her nor complained. She merely said, \"Ferdows, my child is not feeling quite herself because of her grief. Poor thing! Take her to her room. Tight clothing is bad for a pregnant woman.\" She put a hand to her brow and cried a little. Then calming down, she advised Zari in a motherly tone, \"Zari dear, put the earrings in a safe place. It will get very crowded here today.\"\n\nZari walked off, feeling like a robot with rusty springs and loosened hinges. Ferdows took her hand to keep her from falling. They went to the bedroom together. Zari took off her dress, put the velvet box on her dressing table and stretched out on the bed.\n\n\"Where's the sewing kit?\" asked Ferdows.\n\n\"I don't know,\" replied Zari. She felt dizzy and nauseous. This must be the way madness begins, she thought.\n\nShe wished Ferdows wouldn't talk, but Ferdows kept on chatting.\n\n\"Khanom Zahra,\" she said, \"it's a good thing you and I managed to be alone. These people can get up to anything!\"\n\nIf only she'd shut up, thought Zari.\n\n\"Are you listening to me?\" asked Ferdows.\n\n\"No.\"\n\n\"I want to put you on your guard. Last night mother and son were up the whole time, scheming behind your back. I stayed awake on the roof and listened. God Almighty! In the middle of the night she dyed her hair and put henna on it... actually it's a wonder she doesn't think she's the Almighty!\"\n\nZari didn't respond, though her interest had been kindled. Ferdows had found the sewing kit and was opening the seams of the dress.\n\nHow efficient she is, Zari thought silently.\n\nStill undoing seams and re-stitching them, Ferdows continued, \"When Khanom got home, Hamid Khan threw himself at her feet again, flattered her and played up to her and finally he said, 'Mother, I must have this woman at any cost'... God forbid, he said that every night he'd slept with his wife he'd thought of you. All three of his children had been conceived thinking of you... bless my soul! A grown man like that making up all kinds of verses and poems for you! If you only knew the kinds of things he said...\"\n\nZari didn't want to know, but Ferdows went on, \"Well, to cut a long story short... Khanom was not easily persuaded. She kept saying that you bring bad luck, that your brother-in-law wouldn't let anyone lay a finger on your money, that you're pregnant and no-one can wed a pregnant woman. Hamid Khan said he'd wait. Khanom...\"\n\nIf Gholam hadn't knocked on the bedroom door just then to announce Dr Abdullah Khan, Zari would have vomited.\n\n\"Ask the doctor to wait a minute while I get dressed,\" she said. And to Ferdows, \"Khanom Ferdows, please hurry.\"\n\n\"Right away.\"\n\nBut Ferdows kept on talking, and Zari didn't stop her because now Dr Abdullah Khan had arrived and would relieve her mind one way or the other.\n\n\"He pleaded with her until she gave in,\" Ferdows went on, \"so he asked her to get to work on you from the very next morning. What lies she strung together in front of me! Actually Khanom was a born liar. How she pretended to care about you! Don't be fooled, she's after your blood... there, I'm all done.\" And she handed over the dress which Zari put on with a sigh of relief. Maybe she had felt dizzy because of the tightness of the dress.\n\nAs she looked Zari over, Ferdows added, \"And she didn't go straight to the Governor's either. Hamid Khan made her telephone the Governor's daughter. Khanom sent a piece of her own jewellery in exchange for yours. My worthless husband Kal Abbas went and fetched it.\"\n\n\"Thank you,\" said Zari. \"Now go and tell the doctor I'm ready.\"\n\nDr Abdullah Khan came in, leaning on his stick. He seemed older than on the day Zari had seen him in Khanom Massihadem's room. Or perhaps she hadn't looked at him closely enough then. The doctor sat on Zari's bed, took her hand in his and said, \"What's the use of reaching a grand old age like mine? When a precious young man like your husband dies, I begin to hate myself. Here I am clinging to life with both hands while our young men are taken...\"\n\n\"My husband didn't just die, he was killed,\" said Zari sadly.\n\n\"I know. Your son told me everything on the way. I congratulate you. A clever boy like him could take his father's place for you. May the Lord bring both of you prosperity.\" He paused and said, \"An old man like me shouldn't step into a house which has lost a young man of his kind. I'm old and useless now. His mourners must surely look at me, shake their heads and think, 'Old man, you're alive, and our young one has been martyred!'\"\n\n\"No-one thinks of you like that. You're the salt of the earth to all of us.\"\n\nDr Abdullah Khan raised Zari's hand to his lips and kissed it. Zari tried to withdraw it out of modesty. The old man sighed and said in a pensive voice, \"I don't know where I read that the world is like a dark room which we enter blindfolded. One of us may have his eyes open or others may try hard to open theirs; perhaps it's even destined that one person should be touched with a ray of light from above so he may see and understand all for an instant. Your husband was one of those rare people who'd never shut his eyes from the beginning. His eyes and ears were alert. More is the pity he had such a short time...\"\n\nHe spoke like one who had taken in everything there was to know. If there was a God, He had shown Himself for once to this man in the course of his long life...\n\nThe old man continued, \"I've told Khanom Qods-ol-Saltaneh many a time that that brother of hers was a genuine human being. He was an enlightened man.\"\n\n\"But you're enlightened as well, you're...\"\n\n\"Now tell me what's ailing you?\" interrupted the doctor. \"Your son begged me to come and visit you. I said to him, 'Dear boy, for the wife of such a one as he was, I'm ready to go to the ends of the world. Besides, I'm very fond of your mother herself... she is a queen among women.'\"\n\nZari had no fear or embarrassment in telling Dr Abdullah Khan the truth. \"I've been so distraught since last night,\" she confessed, \"I can't control my mind. I'm afraid I might be going mad... I feel tempted to imitate the lunatics I've seen.\" And she added in tears, \"All last night I was caught up in nightmares. Khanom Hakim gave me three injections but they didn't seem to do any good and I couldn't fall asleep. I kept seeing horrific scenes. I said nonsensical things. And I've been feeling dizzy all morning.\"\n\nThe old man stood up and went to the window, looking out on the garden. \"Don't let me hear you say things like that,\" he told her with his back to her. \"If you were distressed or even delirious, it was perfectly natural. Khanom Hakim couldn't have given you tranquillizers, either. She gave you a camphor injection to stimulate your heart and the other two shots were distilled water.\"\n\nAgain he came and sat down next to Zari.\n\n\"So you're saying I haven't gone mad?\" Zari asked innocently.\n\n\"Absolutely not.\"\n\n\"And I won't go mad either?\"\n\n\"I assure you you won't.\"\n\nHe stared into Zari's eyes and continued in a soothing voice, \"But you have a malignant disease that cannot be cured by my hand. You must get rid of it before it becomes chronic. Sometimes it's hereditary.\"\n\n\"Cancer?\" asked Zari.\n\n\"No, my dear; don't you understand? It's the disease of fear. Many people have it\u2014I told you it's contagious.\"\n\nAgain he took Zari's hand and said prophetically, \"I have one foot in the grave, so listen to the words of this old man, my dear. In this world, everything is in one's own hands. Madness, fear, even love. A human being can if he so desires, move mountains, dry up the waters, create havoc everywhere. A human life is a chronicle. It can be any kind of chronicle\u2014a sweet one, a bitter one, an ugly one... or a heroic one. The human body is fragile, but no force in this world can equal man's spiritual power. As long as he has a strong will and some awareness.\"\n\nHe paused and took out a green bottle with a white top from his pocket. He gave it to Zari. \"There's a special kind of salt in this bottle,\" he said. \"Keep it in your pocket and every time you feel unwell, open it and smell it. Drink a glass of sweetened willow-water too.\" He got up and said, \"I know you're a lady. A real lady. I know you're strong and brave enough not to run away from the bitter reality. I want you to prove that you are worthy of such a man as your husband was.\"\n\nHe picked up his stick which he had hung on the edge of the bed and said, \"Here is some news that will make you happy. Take heart. The day before yesterday, Khanom Massihadem was discharged from the asylum. She's much better and by the time you're ready to give birth, she'll be completely recovered.\"\n\nZari felt as if she'd been freed from a cage. A man of wisdom had given her hope and encouragement. Not one but a thousand stars were lit in her mind. She knew now that she feared no-one and nothing in the world.\n\nThey went out into the garden together. Abol-Ghassem Khan was sitting on the children's bed with Malek Rostam and Majid Khan. When he saw them, he got up and went towards them.\n\n\"Well, doctor,\" he blinked, \"what did you think? What did you find?\"\n\n\"If you ask me,\" replied the doctor, \"your sister-in-law must be very strong indeed just to be standing on her two feet. Her distress and anxiety are natural. It's no joking matter. But all of you around her must leave her in peace.\"\n\nZari saw the doctor all the way to the gate. She kept searching in her mind for a suitable word to express her gratitude but she couldn't find it. Maybe he felt her helplessness, or perhaps he just wanted to bid her to be patient, or maybe it was for his own heart\u2014at any rate he murmured the following verse:\n\n\"Be patient, o heart, that the Just One,\n\nWill not let such a gem fall to Evil.\"\n\nZari knew that Dr Abdullah Khan was a member of the Hafeziun group who held sessions in memory of the mystical poet Hafez at his gravesite. They recited his poetry, drank wine which they threw on the mystic's grave, and even played the tambourine and the lute.\n\nShe said quietly, \"Please recite some more. Verses which will give me strength to go on.\"\n\nThe old man smiled, and said:\n\n\"Let us do good deeds, lest we take our soul,\n\nIn shame to the other world.\"\n\nHe stood under the elm tree to catch his breath. \"I didn't recite that verse for you,\" he said. \"I said it for myself.\"\n\n\"You've done your work in this world,\" said Zari. \"Your life-story is a heroic one. But my poor husband's tale was tragic and unfinished.\" And without intending to, she leaned against the tree and wept quietly behind her hand. \n\n# _23_\n\nThey had arrived for the funeral procession. First came all the relatives and close friends. The women were shown to the howzkhaneh and the men to the parlour. Ezzat-ud-Dowleh took the seat of honour amongst the women. Ferdows had donned a tight black dress and a sorrowful expression as she lent a hand with the serving. Anyone who didn't know would have thought Ezzat-ud-Dowleh was next of kin, the way she was ordering everyone around. The minute she set eyes on a newcomer, she would talk effusively about Yusef's youth and tragic end, of his good looks and knowledge, of his faultless English, of the poor innocent widow and children he'd left behind... she would go on and on, sobbing loudly. Occasionally she would even beat her chest, but not too hard. Every so often Zari would take a whiff of the salts Dr Abdullah Khan had given her to prevent herself from crying at her words. Ameh was nowhere to be seen. Eventually, when Ezzat-ud-Dowleh started a lament, and talked of \"a tree which had been cut at the roots and felled to the ground\", Zari left too.\n\nIt was after eight-thirty in the morning when Abol-Ghassem Khan's friends arrived. But there was no more room in the parlour so they had to sit on the children's bed in the garden. Zari sat across from them near the sacks of eglantine and red roses which had been left at the edge of the pool. The pool itself was brimming with crystal-clear water.\n\nHossein Agha and Hassan Agha, each with a full sack on his back, passed by Abol-Ghassem Khan's friends as they made their way to the pantry. The distillers from next door, again without the youngest son, followed them, balancing pitchers on their shoulders. Then three other men arrived, carrying large empty vessels. Zari's eyes filled with tears on seeing them.\n\nAbol-Ghassem Khan came out of the parlour and joined his friends. A fat, dark man was saying something quietly while the others listened with worried expressions. The town's newspaper manager was shaking his head and the former parliamentary deputy was racing through his rosary.\n\nFerdows approached Zari with the tray of drinks. \"You help yourself first,\" she said. \"This one is willow-water sweetened with rock-sugar... Did you see the way she was play-acting? Now she's pretending to faint.\"\n\nZari took the glass from her and asked, \"How did you know I should drink sweetened willow-water?\"\n\n\"Khanom sent me to eavesdrop. Let her keep hoping you'll miscarry\u2014you're not unprotected and alone like me. She wanted to know what you and the doctor were saying to each other all that time. I told her I didn't understand a thing of what you were talking about because you were whispering. I just heard the doctor say that the world is like a dark-room with upside down pictures and we're all lost and wandering about in it... Khanom called me an imbecile and told me what a waste of time it had been for a distinguished lady like her to try and train me. Khanom Zahra, if I have just one day left to live, I'll take my revenge. When my mother was in prison it was a good chance to...\"\n\nZari cut her short saying, \"Take the drinks over to the gentlemen, the ice is melting.\"\n\nFerdows went to Abol-Ghassem Khan's friends. The notary, who was Chinese-looking, said something to Ferdows and she giggled. Gholam went to greet some men dressed in black whom Zari didn't recognize. They made way for two porters, one of whom was carrying an upright candelabrum on his head, the other a lustre. Covered with sweat, the porters went as far as the pool where others helped them place their loads on one of the wooden beds. Another man stripped to the waist, holding the emblem of the Ta'zieh passion play, garnished with flowers, tulips and lengths of brocade, and topped with a feather which swayed to the movement of his step, carefully lowered the emblem past the garden gate. The men indoors looked out from the parlour windows and the women had come out of the basement to watch. Ezzat-ud-Dowleh, however, was not among them.\n\nBy nine or nine-thirty in the morning, the garden was filled with men dressed in black. But they were still arriving in droves, and the last group had flagellating chains with them. Finally, they brought in the mock wedding chamber, the Hejleh Ghassem, which nearly made Zari break down and sob, but she managed to control herself by taking out the smelling salts and busying herself with opening the top.\n\nAbol-Ghassem Khan's friends approached her. The former deputy had crewcut white hair, and no longer carried his rosary. The notary really did look Chinese. The newspaper manager took Zari's hand in his, and said that all of them had to attend a meeting at the governor general's office about bread supplies, so he regretted that they could not be present for the funeral procession. But on everyone's behalf he offered his sympathies and condolences to Zari and \"Abol\" and Khanom Qods-ul-Saltaneh, and he prayed that, God willing, they would all live to old age and never suffer loss in the family again. The others listened, and when he had finished they left. But the newspaper manager would not let go of Zari's hand. He said quietly, \"I hope you understand my position if I don't print news of the event. Even the funeral announcement was placed purely for your sake and that of my friendship with Abol.\"\n\nZari withdrew her hand and said bitterly, \"Funeral announcements have always been permitted.\"\n\nA few minutes later, Abol-Ghassem Khan came and sat next to her. He was very pale and his nostrils were trembling. \"Sister,\" he said, blinking rapidly, \"I know you're more sensible than the rest of them. For heaven's sake, say something to these fools. My own stupid sister doesn't seem to understand. She keeps saying she wants them to turn this dog-infested town into a holy Karbala. And our ruffians keep praising her and egging her on.\" When Zari didn't budge, he pleaded with her, \"Sister, I beg of you, for the sake of that tragically-departed soul, get up and say something.\"\n\nSo they went to Khosrow's room together where, according to Abol-Ghassem Khan, the town ruffians had gathered. Malek Rostam and Majid, wearing black ties, were standing by the doorway. Zari's glance travelled from Haj Mohammad Reza the dyer, who was squatting by the doorway, to the others. Seyyid Mohammad, Hossein Agha and Hassan Agha had their backs to the window. Seated on Khosrow's bed were three men. One she recognized as the tall, broad-shouldered Mashallah Qari; another was Fotouhi, who wasn't wearing a tie; and the third was Mr Mortezai who had put on his religious robe. The rest of Yusef's sworn companions, along with a few others also in mourning dress but whom Zari didn't recognize, were seated on chairs brought from the parlour. Ameh Khanom, wearing an Islamic black scarf, was standing tall and upright behind Khosrow's desk. None of the men had shaved.\n\n\"Here is my late brother's wife,\" Abol-Ghassem Khan announced. \"Do whatever she says. You've shut down the bazaar, so be it. But to circumambulate the Shah Cheraq shrine with the body, and have the crowd flagellating in the courtyard; to have Mr Mortezai saying the last prayers with full sermon from the shrine balcony... upon my word, don't even think of it! What with the foreign army in town, there will be rioting... You've dragged all these people here for nothing.\"\n\nTurning to Zari, Majid said, \"Khanom Zahra, you know yourself we had sworn allegiance to Yusef. Now that they've killed him, they want us to sit here and not even give him a proper burial. Our simple objection...\"\n\nZari didn't let him finish his sentence. \"They have killed my husband unjustly,\" she said. \"The least we can do is to mourn his death. Mourning hasn't been outlawed. In his lifetime we were always frightened and we tried to frighten him off too. Now that he is dead, what else can we fear? I, for one, have nothing more to lose....\" Her voice was trembling. She brought the bottle of salts to her nostrils and inhaled its freshness deeply.\n\n\"Well, bless my soul, sister!\" Abol-Ghassem Khan exploded. \"Now you've really put me to shame! Why don't you understand, woman? When this many people take to the streets, if someone leads them on to rioting, who could possibly stop the tide then?\"\n\n\"Abol-Ghassem,\" Ameh said, \"your brother's corpse is at your mercy now. Don't sit by idly and watch his blood being trampled on.\"\n\nZari, looking at her, was reminded of Hazrate Zeynab, defending her martyrs.\n\n\"I have reliable information that you'll be stopped,\" Abol-Ghassem Khan told them, \"then there'll be bloodshed. I won't allow it. My poor brother wouldn't have wanted to hurt a fly. He treated his peasants like an older brother... Don't torment the departed soul.\"\n\n\"I lived with him for fourteen years,\" Zari said with a sigh. \"I know that he always spoke of courage... of justice...\"\n\nThe serpent which had coiled around her heart the night before reared its head to strike, and her throat constricted. She left her sentence unfinished, but now her mind shone like a torch, and she knew that no-one in the world could ever dim it again. She swallowed and went on, \"Do whatever you have to do today... if you don't do it now, there will never be another opportunity.\" Then, after a pause, she said to Abol-Ghassem Khan, \"Today I came to the conclusion that one has to be brave in life for the sake of those who are living... but it's a pity I realized it so late. To atone for that ignorance, let's mourn our courageous dead the way we should.\"\n\n\"A blessing on this noble mother of our race,\" murmured Seyyid Mohammad.\n\n\"Bravo!\" came from some strangers in mourning clothes.\n\nMortezai recited in Arabic from the Quran: \"There is Life to you, O ye men of understanding.\"\n\n\"This way we shall prove that we've not been annihilated yet, and we value the blood that's been shed,\" Fotouhi added.\n\n\"Sooner or later it'll be my brother's turn,\" Malek Rostam reminded them. \"They'll catch up with him in the heat of these cruel mountains, and drag him into town to the sound of horns and drums. They'll hang him on charges of insurrection, and everyone will come out to watch.\"\n\nAbol-Ghassem Khan, venting his ill-feelings on Malek Rostam, said, \"You talk as if your brother is the Prophet's son! Of course they'll hang him. No-one's forgotten the bloodshed in Semirom. How much government property was raided! How many innocent people were killed! If there's such a thing as penance in this life, then he must pay for all that killing...\" Blinking rapidly, he continued, \"How ambitious can you get! He changes colour every day like a chameleon. One day he's a slave to the Germans, the next he's serving the British, and before you know it, he's turned against them too! Just like the treacherous Shemr...\"\n\nMalek Rostam interrupted him.\n\n\"If a person knowingly makes a mistake, he can try to make up for it. But now's not the time for putting Malek Sohrab on trial, and you're not a judge either.\"\n\n\"Actually, you yourself have been parading a little too freely in public these days,\" Abol-Ghassem Khan retorted. \"If I were you I'd put on a chador and make a getaway to the mountains through the back door of this garden.\"\n\nAt that, Malek Rostam's tribal blood began to boil. He answered sharply, \"Some people hide under black chadors to slip away to the Consul at the British Consulate. My brother and I use them to hide from the Consul and his men.\"\n\n\"Gentlemen!\" Fotouhi intervened. \"This is no time for quarrelling. We were supposed to come to a decision about the funeral procession. Khanom Zahra agrees...\"\n\n\"But I'm against it!\" Abol-Ghassem Khan interrupted again. \"And by rights I'm the legal guardian of my brother's children. Sister, be sensible, listen to my advice.\"\n\nZari couldn't stand on her feet any longer. She sat on the bed next to Fotouhi and said, \"His body's not buried yet. I don't want to argue with you. But while he was alive, you each had a tight grip around his throat and he had to keep raising his voice to be heard until he was finally killed for it. And now... let people show at his death that he was in the right. Besides, justice and truth haven't died with him, there are others to consider.\"\n\nAbol-Ghassem Khan, blinking nervously, raised his voice in anger. \"It's women like you, who follow their husbands like so many sheep, that bring about these tragic events!\"\n\n\"Don't make me say this,\" Zari answered calmly, \"but more than one person was responsible for the blood that was shed, including yourself. Maybe I'm to blame, too.\"\n\n\"You've got something to say for yourself, have you? Well, well! I'll say it in front of everyone, then. Now that you've come into a bit of easy money, you've forgotten that a woman is, after all, only a woman. A woman is like the lining of a garment, she exists to uphold and support a man. But you just blindly endorsed whatever mistakes that poor man made...\"\n\nZari felt that the snake sitting alert inside her was speaking out now. \"You're only worried about your post as a deputy, about all the plans you've made for when you become one. An eye operation, a good set of teeth from the famous Dr Stump... Haven't you said as much yourself? Maybe you even want to get remarried...\"\n\nAbol-Ghassem Khan looked at her in total astonishment. \"Fie, for shame!\" he spat. Then he composed himself and added, \"You don't know me well enough. I'm the kind of man who spent sixteen solitary years, night after night, when my wife died...\"\n\nZari was going to say, \"What about the various temporary wives... What about the shoemaker's daughter who rubs herself all over with ox's gall-bladder stone to fatten herself up...\" She felt completely reckless, and in such a state that, if someone had handed her a gun which she knew how to use, she would have been prepared to shoot. She stood up and said, \"Just a few minutes ago your notary...\"\n\nMajid Khan, trying to mediate, stepped in. \"Please, I beg of you...\" he said. \"Mr Abol-Ghassem Khan, Khanom Zahra. It's hardly the time for this sort of thing.\"\n\nFotouhi motioned everyone to be silent. \"Let's not waste our time with discussions about each other's private lives,\" he said. \"Let's approach the matter in another light. The killing of Yusef Khan is, from your point of view, a personal matter, whereas from ours, it's a social issue...\"\n\nAgain, Abol-Ghassem Khan interrupted Fotouhi, saying, \"I know the rest by heart. You want to make the most of this killing. Create riots in town and cause innocent people to be killed. There are several truck-loads of soldiers blocking the town's main roads. What with the foreign army in town... I suppose you know what you're doing.\"\n\nThe coffin, swathed in eglantine and red roses, was to be taken from the driveway of the house by Hossein Agha, Hassan Agha, Majid Khan and Fotouhi as pall-bearers. Malek Rostam insisted on carrying the coffin too, but Zari dissuaded him, saying that Abol-Ghassem Khan had been right about one thing: the tribesman had shown himself too carelessly in public. She made him promise to put on Khadijeh's chador after they had all left and escape to a safe hide-out through the back door of the garden.\n\nMalek Rostam merely responded by saying, \"It doesn't really matter any more. Whatever's going to happen will happen.\"\n\nAbol-Ghassem Khan begged the ladies to stay at home for lunch. A bite to eat could be arranged at his humble abode, and it wasn't wise for them to attend the burial. Khanom Ezzat-ud-Dowleh would remain at the house too.\n\nThe emblem and the candelabrum went before the coffin while the Hejleh Ghassem followed it. Abol-Ghassem Khan gave his arm to one of Yusef's sworn companions, and extracted a black handkerchief from his pocket which he used for dabbing at his eyes every so often. Zari and Ameh walked alongside him.\n\nThe door of the stable was open. The roan horse was feeding, but the mare and Sahar were standing quietly on the side-path, with Khosrow and Hormoz holding their bridles. Zari felt sick with grief at the sight of the horses and the boys. The mare's saddle was completely covered with black fabric, with Yusef's hat on top and his gun strapped to the mare's neck. They had covered Sahar with a white sheet stained randomly with red ink like a bloodied shroud. When the mare saw the body, she picked up her ears and drummed her hoofs on the ground. Zari felt as if her own heart were being trampled on. Then the mare neighed twice. Zari thought she saw tears rolling down the horse's flared nostrils. She remembered what the middle-aged woman had told her about Savushun, all those years ago.\n\nKhosrow and Hormoz led the horses behind the Hejleh Ghassem. Abol-Ghassem Khan rushed at them and pulled the blood-stained shroud off Sahar. He bunched it up and threw it under one of the elm trees. Then he gave Hormoz a hard slap on the face, knocking his glasses to the ground. \"What kind of nonsense is this?\" he shouted. \"Everything is being run by women and children all of a sudden! Take the horses back to the stables, you fools! My God, they make you livid with anger!\"\n\nThe emblem of the Ta'zieh had by now been carried as far as the garden gate. Its porter, the man stripped to the waist, bent over to lower it, his bare back glistening with sweat. Everyone stopped. Hormoz picked up his glasses from the ground, shook out the broken bits of glass from the right lens and put them back on. Just then a car, sounding its horn, pulled up at the garden gate. An Indian soldier got out, bringing with him a white flower arrangement adorned with black ribbons in the shape of a cross. Entering the garden, he headed towards the coffin and tried to put the flowers on it. But the pall-bearers, standing on tip-toe, lifted the coffin out of his reach. Khosrow dropped Sahar's bridle, went to the Indian soldier and took the flower arrangement from him. One by one he plucked the flowers from the cross and threw them in front of the horses. The horses sniffed at the flowers but didn't eat them. The Indian soldier stared with bulging eyes at the black-clad mourners, as if he couldn't believe what he saw. The crowd was so silent you could have heard a pin drop. Abol-Ghassem Khan put a hand on the soldier's back and led him to the car, whispering something to him which the man seemed not to understand since he answered aloud in a language no-one recognized. Then the car sounded its horn again and drove away.\n\nNow the youngest son of their neighbour, the distiller, came running up to them, panting and sweating, with an armful of wild flowers. Khosrow took the flowers and smelled them before placing them on the coffin which the pall-bearers now lowered.\n\nBy this time the sun had penetrated every nook and cranny. Coming out of the garden, Zari noticed that all the shops in the side-street were closed. Haj Mohammad Reza, using pairs of wooden poles, had draped lengths of black material all along both sides of the street. Usually he tied colourful fabrics in red, blue, green and orange on them, or else dyed silks and wool to dry off in the sunlight.\n\nThey had barely gone half-way up the street when they saw a bare-headed Gholam approaching with the twins. Reaching Zari, Gholam spat and said, \"Mohsen Khan telephoned, so I went and fetched them...\"\n\nZari and Ameh stood aside to let the procession go ahead, but the crowd waited. Zari bent down and kissed the children. Mina was holding the keychain which she gave to her mother.\n\n\"Now take us so we can watch too!\" she said. \"Oh look at the lights! Look at all the stars!\"\n\nAbol-Ghassem Khan, who had gone ahead a few steps, came back to warn Zari, \"They'll be trampled on. Sister, please take them to Khanom Ezzat-ud-Dowleh.\"\n\n\"No, Ameh Khanom,\" Zari replied. \"Leave them with Ferdows.\"\n\nAt that, the twins started to cry. None of Ameh's pleading and cajoling had any effect. Finally Gholam picked Mina up and Ameh took Marjan as the crowd made way for them to leave.\n\nAlong the main road, policemen were either scattered randomly or walking around in pairs. In the side-street opposite, a truck full of soldiers was waiting. When the policemen first sighted the funeral procession they simply stood and watched, but when the procession turned towards the main road, the policeman in command blew his whistle, bringing his men into a line to block the crowd's path. But the Ta'zieh emblem had already been carried into the main road and its front feather seemed to nod in greeting to the crowds spread out on the roof-tops and pavements. What loudspeaker could have drawn the people to the street in such numbers?\n\nThe police officer came towards the crowd and shouted, \"Gentlemen, except for the relatives of the deceased, everyone else must disperse.\" He waited. Abol-Ghassem Khan remained standing with his back to the crowd. Zari turned round to look. Men dressed in black were still flocking out through the garden gates. Then a voice proclaimed in Arabic, \"There is but one God!\" In unison, the crowd repeated the sacred phrase.\n\nThe policeman shouted again, this time on behalf of Abol-Ghassem Khan. \"Do you hear me or not? The honourable Abol-Ghassem Khan cannot speak out because of his grief... to thank all of you. The weather is hot. He bids you gentlemen farewell.\"\n\nA voice from the crowd replied calmly, \"We are all related to the deceased.\"\n\nHossein Agha, who was one of the pall-bearers, signalled to Seyyid Mohammad to replace him as he walked up to the policeman and addressed him. \"Sir, a young man has been killed unjustly. We're mourning his death. That is all.\"\n\n\"I'm asking the crowd, very politely, to disperse,\" the policeman declared in a loud voice. \"Go back and open your shops. If you don't, your trading licences will be revoked. That's an order. Do you understand? If you don't obey, I'll have to resort to force...\"\n\nThis time, Mashallah Qari came forward. He said, \"Sir, you know what kind of a fellow I am, don't you? When I say something, I stand by my word. We don't mean to stir things up. We're just mourning one of our fellow-townsmen. Imagine it's Karbala here and today is the massacre of Ashura; you don't want to be Shemr, do you?\"\n\nSomeone cried out, \"O Hussein!\" And the crowd enthusiastically echoed, \"O Hussein!\"\n\nZari thought bitterly, \"Or imagine it's Savushun and we're mourning Siavush.\"\n\n\"I told you to disperse!\" The police commander shouted even more angrily, \"I'm going to smash that candelabrum to pieces!\" And he made for the upright candelabrum which was being carried on a tray over a porter's head. The man with the candelabrum had come right up against the line of gendarmes blocking the main road. His companion nudged him in the side and whispered something in his ear. The man turned to the right and went off to stand by the dried-up gutter along the street, the candelabrum pendants jingling to his movement.\n\nThe policeman turned around and motioned to the truck full of soldiers in the side-street opposite. The truck's engine revved, and the vehicle swerved noisily, coming to a halt a little beyond the Ta'zieh emblem on the main road. The crowd watched the truck. An officer stepped out. He was stout, with a perspiring face, and he had three stars on his epaulette. He came over and stood by the policeman.\n\n\"As God is my witness,\" he said, \"I don't want any of you to come to any harm. We have families too. Go back to your work and livelihood.\"\n\nThe crowd seemed to take heart at the captain's gentleness. Mashallah Qari stepped forward again and said, \"Sir, you know what sort of a fellow I am, don't you? As long as I'm around, I'll make sure our brothers and sisters here are safe and sound. We'll take the body to the Shah Cheraq Shrine, go round it, mourn and flagellate for a while...\"\n\n\"What! The Shah Cheraq?\" shouted the captain, quickly losing his temper. \"Right in the centre of town? Whoever gave you permission to do that? Can't you be spoken to in a civilized way? Now, go straight back to where you belong!\"\n\nHe took a few steps towards the main road and motioned again to the soldiers in the truck. One by one the soldiers got out, rifle in hand, and lined up behind the police. The captain turned to the crowd and, wiping the sweat off his forehead with his hand, said, \"That man's always been trouble, dead or alive.\"\n\nZari thought she was the only one who had heard the insult to her husband, but Hossein Agha turned to Abol-Ghassem Khan and said, \"The poor man hasn't been buried yet, and you let them insult him like that?\"\n\nThe captain slapped Hossein Agha sharply across the face, making his nose bleed. \"You shut up!\" he barked.\n\nAbol-Ghassem Khan took out a silver cigarette-case from his pocket which he opened and held in front of the captain.\n\n\"Captain, please have one,\" he said, blinking. \"I seem to recognize you. Aren't you the son of Agha Mirza Mehdi, the porter at the oil-maker's caravanserai? Your father respected the dead...\"\n\n\"Is this the time to be pulling out my pedigree?\" the captain shouted angrily. \"Why do you lead these people on?\" Turning to the crowd, he yelled, \"I told you to get lost!\"\n\nHossein Agha had cupped his hand under his nose. \"How can we do that?\" he asked. \"You're blocking our way.\"\n\nThe captain dealt Hossein Agha several more blows on the back of the head. \"Why are you jabbering again?\" he bellowed. \"Didn't I tell you to shut up?\"\n\nThey began to grapple with each other. Just as Mashallah Qari had pinned the captain's arms behind him, the police commander blew his whistle and the policemen and soldiers charged the crowd, hitting out left and right with their batons or rifle-butts. But the crowd managed to make its way down the main road. First Fotouhi and Hassan Agha, then Majid and Seyyid Mohammad, out of necessity, left the coffin on the ground by the side-street and followed the crowd into the main road.\n\nThe road itself became blocked. Cars were backed up in both lanes; several carriage-horses shied. The noise of drivers cursing and lashing at their horses, car drivers honking and vainly attempting to reverse, mingled with that of the mourners who had taken out their chains and begun to flagellate amidst the general hubbub and confusion of the crowd.\n\nThe man carrying the candelabrum tried to cross the dried-up gutter to reach the sidewalk, but he was pushed by the crowd and the candelabrum crashed to the ground, breaking into bits. The man, with the empty tray still on his head, squatted down to pick up the bits of crystal. The others, however, managed to escape to the sidewalk with the Ta'zieh emblem which they leaned against a wall. A group of people helped make way for the Hejleh Ghassem to be taken back to the garden.\n\nNow all the crowd had poured into the main road and the coffin, decked with flowers, was lying abandoned by a wall along the side-street. Only Zari and Abol-Ghassem Khan remained. Wordlessly, they tried to pick up the coffin. It was heavy. The eglantine and red roses had withered, but the wild flowers were still fresh. Zari looked down the main road for help. Suddenly she heard gun shots. The people who had been watching from the shop roof-tops retreated a little.\n\nZari spotted Khosrow who was struggling and shouting, \"Let me go!\"\n\nA policeman was holding both his arms with one hand, and Hormoz, wearing his one-eyed glasses, was punching the policeman on the chest.\n\nThose who were injured or unconscious were being carried off by others, many of them with torn clothing revealing naked flesh underneath. What a cloud of dust there was in the air! Meanwhile, no-one could be found to help them pick up the coffin from the ground, and Zari was against dragging it on the dirt all the way back home as Abol-Ghassem Khan suggested. Feeling sick to the stomach, she had to resort to her smelling salts again.\n\nEventually four buses, honking non-stop, managed to scatter the crowd and open up a way for themselves. They narrowly passed the truck, now empty of soldiers, hitting the deserted sidewalk with a thump. The odd remaining spectator dodged the vehicles as they pulled up and parked, one after the other, beyond the Ta'zieh emblem. Indian soldiers peered out from the bus windows and the crowd, which had momentarily retreated, converged again, shouting and clamouring.\n\nThe captain approached Zari and Abol-Ghassem Khan. \"I think you ought to go and bury the body right away,\" he told Abol-Ghassem Khan. \"I'll find you a car. When you get a crowd roused up...\" He pulled out a handkerchief and wiped the sweat off his face.\n\n\"I have a car myself,\" Abol-Ghassem Khan answered.\n\nAn Indian officer got out of the first bus and pushed his way through to the captain. He saluted and said in broken Persian, \"We on holiday. Soldiers been visiting Shah Cheraq. Only two days' holiday.\"\n\n\"You can see for yourself that the road is blocked,\" the captain informed him loudly.\n\n\"All right, all right,\" said the Indian soldier.\n\nBut Zari knew, and was quite certain the captain knew too, that the route the soldiers had taken could never have led to the shrine.\n\nAt this point Zari noticed Majid and Haj Mohammad Reza the dyer holding Khosrow and Hormoz by the hand, leading them towards the side-street. They helped Abol-Ghassem Khan lift the coffin, but they didn't let go of the boys' hands. This small group, followed by Zari, returned to the house and took the body to the cistern. Abol-Ghassem Khan sent Haj Mohammad Reza for more ice, praying that he wouldn't return empty-handed. By now the garden was filled with wounded people. Several half-conscious, bloodied men with their shirts ripped open had collapsed on to the wooden beds. Two men were washing their faces at the pool, and drinking from it even though the water was no longer clear.\n\nZari went to the basement, hoping to find the twins there. But instead she found Ezzat-ud-Dowleh, lying on the bed with Ferdows at her feet, fanning her. The pool-fountain had been turned on, and no-one else was there.\n\nZari found Ameh and the twins in the bedroom. The curtains had been drawn and the room was half-dark, but Mina still spotted Zari, and she got up from Ameh's side on the bed to throw herself with open arms into her mother's embrace. Zari kissed her on the eyes which were moist from crying. Marjan was sitting on Ameh's lap and didn't get up. She just stared at her mother with round eyes.\n\n\"Mother,\" said Mina, \"the old man didn't say Nargessi, Narengi. He kept saying 'Ouch! Ouch!' His head was hurt! It was bleeding...\"\n\n\"But you were supposed to stay at Aunt Mehri's,\" said Zari.\n\nMina kept staring at the curtains of the window which opened on to the verandah. \"Why did you let them into the house?\" she asked. \"Now they'll take dadash's horse and father's horse away... that boy was hurt there...\" and she pointed to her arm.\n\n\"I asked you why you didn't stay at Aunt Mehri's,\" Zari repeated.\n\nMina pointed at Marjan, who was still in Ameh's lap, and said, \"This cry-baby was scared and cried. She kept saying, 'I want my mama'... Ameh didn't let us look... he kept his head under the tree like this, it was bleeding...\" She paused and threw an arm around her mother's neck. \"Aunt Mehri and Uncle Mohsen were fighting. Aunt Mehri cried. Uncle Mohsen said, 'I'm scared!' Then he hit Aunt Mehri. And this cry-baby started to cry...\"\n\n\"I didn't want it to be like this, and I didn't think it would turn out like this,\" Ameh said.\n\n\"But I don't regret it,\" Zari said. \"As Yusef used to say, a town mustn't be completely empty of real men.\"\n\n\"I wanted them to mourn the poor martyr's death, but I didn't want it to end up in fighting and violence. As my late father always said, in any war, both sides are losers.\"\n\nMina, still holding on to Zari, said, \"Father will come and scold us. My brother will say, 'Where's my horse, then?' I'll say, 'Brother, Sahar was hurt and died.' All right?\"\n\nNow that Zari had her keychain she could fetch the first-aid box from the cupboard to treat the injured. The noise still continued, as did the gun-fire. In the midst of all this, the telephone kept ringing stubbornly. Abol-Ghassem Khan went to pick it up. It was obviously  for him because he was a long time answering, and when he left by the garden gate, he seemed in a great hurry. Soon afterwards, Hormoz left too. But Majid remained, holding Khosrow's hand in his own, sitting next to Zari on the bed while Zari rubbed some ointment on to Khosrow's other wrist which was puffed and bruised from the gendarme's grip.\n\n\"Does it hurt a lot?\" Zari asked. \"I think it's dislocated.\"\n\n\"No, mother. And anyway, I'm not more precious than father, after all. When he was shot...\" He didn't finish his sentence. Instead, he smiled at his mother and said, \"Even if it hurts, it'll get better.\"\n\n\"That's my man!\" Zari said with a smile.\n\nThat night, they moved the body from the cistern and its bags of ice to the boot of Abol-Ghassem Khan's car. Ameh, Zari, Khosrow, Hormoz and Abol-Ghassem Khan sat in the car and drove around Seyyid Haj Gharib's grave as a ritual gesture. Ameh Khanom cried all the time, sobbing, \"O my poor lonely one!\"\n\nBut Zari had no tears. She wondered whether Ameh was referring to the solitary saint, or Yusef's loneliness. She could only wish for her own tears to flow, and a safe place to sit and weep for all the lonely and estranged people in the world; for all those who had been killed unjustly and buried secretly by night.\n\nWhen they reached the Javan Abad cemetery, the grave had been prepared and they lowered the body into it by the light of a lantern Gholam held. Seyyid Mohammad wanted to say the last prayers but he couldn't remember them properly. At Gholam's signal, Khosrow pulled back the shroud, crying behind his hands. Gholam and Seyyid threw a handful of earth over Yusef, while Ameh wailed, \"My martyr is lying right here. My brother is right here. Why should I go to Karbala?\"\n\nBut Zari felt nauseated with everything, even with death. A death which had had no last rites, no departing prayer, no proper burial. She decided not to have anything engraved on the gravestone either.\n\nWhen they got home, several letters of condolence had already arrived. Among these, only McMahon's really touched her, and she translated it for Khosrow and Ameh:\n\n\"Do not weep, my sister. A tree will take root in your home and many trees in your town and even more in your land. And the wind will bring the message of each tree to the other, and the trees will ask the wind, 'did you see the dawn as you were coming on your way?'\" \n\n# _Glossary_\n\nAgha\u2014or Aqa. Roughly meaning \"Mr.\" or \"Sir\".\n\nAshura\u2014the tenth day of Moharram, the day of the martyrdom of Imam Hossein at Karbala.\n\nBabi\u2014a member of the Babi sect, founded by Seyyid Ali Mohammad of Shiraz, and considered heretical by Shiites.\n\nBibi\u2014mother.\n\nchador\u2014full-length veil. Women of higher class would use indoor and outdoor veils, often made in a variety of luxurious fabrics.\n\ndroshke\u2014an open, horse-drawn carriage similar to its Russian counterpart.\n\nEzhdehakosh\u2014a clan of the Qashqai tribe of southern Iran.\n\nFarsi-Madan\u2014a clan of the Qashqai tribe.\n\nFassayakafikohomo'allah\u2014a phrase in Arabic meaning \"Then God shall be sufficient for you\".\n\nGhassem wedding chamber\u2014a miniature structure carried at the head of Shiite funeral processions to remind mourners of the untimely martyrdom of Qassem, son of Hassan, who died just before his marriage-day.\n\ngiveh\u2014woven canvas summer shoes or slippers.\n\nhalva\u2014a type of pastry commonly served at funerals.\n\nHazrat\u2014meaning \"saint\" or \"holiness\"; thus Hazrate Abbas,\n\nHazrate Massoumeh, Hazrate Fatemeh, Hazrate Zeynab, all refer to holy persons, in this case the immediate family of the Prophet Mohammad.\n\nhowzkhaneh\u2014roughly equivalent to a basement, where people retire in the heat of the day, and which generally has a small pool with a fountain.\n\nImam\u2014Islamic religious title which refers both to the family of the Prophet Mohammad, and to clergymen of the highest authority, e.g. Imam Juma. Thus, also, Imam Reza, eighth Shiite Imam, or Imam Hossein, grandson of the Prophet Mohammad, or Imam Ali, son-in-law of the Prophet, on whom the Sufi sect of dervishes in Iran is focused, as well as being the legitimate Caliph and heir after Mohammad's death, according to Shiites.\n\nKahn\/Khanom\u2014titles meaning \"Mr.\" or \"Mrs.\" Khanom Hakim literally means \"lady doctor\". Khan can also refer to tribal chiefs or feudal landlords, as in Yusef's case.\n\nKhuli\u2014in Shiite lore, a man who had hidden Imam Hossein's severed head in the furnace in his house.\n\nMasnavi\u2014a form of verse popularized by Jalal-ud-Din Rumi, the great Persian mystic poet.\n\nNakir and Monkir\u2014two angels believed to interrogate the dead on their first night in the grave.\n\nRamadan\/Ramazan\u2014Islamic month of fasting.\n\nRowzeh\u2014a ritual gathering in popular religious practice, to lament the martyrdom of the Shiite Imams. Special food is prepared for the occasion and distributed amongst the poor.\n\nSeyyid\u2014honorific title used for men to denote descent from the Prophet Mohammad.\n\nShahnameh\u2014epic book of poetry written by the Iranian poet Ferdowsi, dating to the eleventh century. The mythology created by Ferdowsi figures largely in all aspects of traditional Iranian culture. Thus, Rostam and Sohrab \u2013 the legendary son killed at the hand of Rostam, his own father. Esfandiar the invincible, Ashkabus the warrior, Akvan the demon\u2014are all characters from this epic.\n\nSheikh San'an\u2014from Farrid-ud-Din Attar's \"Mantiq-ut-Teyr\" or \"Conference of the Birds\". The story of a prominent clergyman who fell in love with a Christian girl, renouncing his high position and followers to prove his love for her.\n\nSiavush\u2014legendary Iranian prince, whose stepmother conspired against him and who was forced to undergo a trial by fire.\n\nSobhi\u2014a popular radio story-teller for children.\n\ntakht\u2014large, multi-purpose wooden bed or platform. Can be used as seating over a small pool for coolness in the afternoon, or as bed under mosquito netting.\n\ntar\u2014a stringed instrument, played by plucking.\n\nTa'zieh\u2014an Islamic Shiite passion play re-enacting the martyrdom of the Imams at Karbala. It often serves as inspiration for various mourning rituals, and was banned by Mohammad Reza Shah Pahlavi for the religious fervour it was liable to create. Marhab, Shemr (who beheaded Imam Hossein), Yazid (the Omayyid Caliph), the farangi (or European), the unwanted Zeynab, Hend, who rapaciously tore out the liver of the Prophet's uncle, and Fezza, are all villains of the play.\n\ntoman\u2014ten rials, i.e. unit of Iranian currency.\n\nTuba tree\u2014a tree in Paradise which has all manner of heavenly fruit.\n\nWalazalin\u2014the last phrase of the opening Surah of the Quran.\n\nYa Hu, Ya Haq, Ya Ali\u2014a chant used by Sufi dervishes.\n\nzither\u2014a stringed instrument with flat sounding-board played on table.\n\nZurkhaneh\u2014Persian \"gymnasium\" where the national sport\u2014a type of rhythmic exercise with weights\u2014is practised to chanted music. \n\n# About the Author\n\nSIMIN DANESHVAR was born into a provincial, middle-class family in Shiraz in 1921, educated at a missionary school and later at Tehran University. The comparatively relaxed political environment of the forties in Iran led her to choose journalism as her first career, and she began writing fiction at the same time. She subsequently married Jalal Al-e Ahmad, the leading Iranian intellectual and writer, received her doctorate from Tehran University and won a Fulbright scholarship to Stanford University. Upon her return to Iran she became an associate professor of art history at Tehran University. She was an articulate and outspoken lecturer and her promotion was hindered by Savak, the secret police.\n\nAfter her husband's untimely death in 1969, Daneshvar assumed a leading role in the Writer's Association which he had helped found and she provided moral support for intellectuals opposing the Shah's regime. After the Revolution in 1979, she retired from her University post. Since then, she has kept a low profile whilst continuing to write fiction and remaining deeply committed to her life-long concern with women and their role in Iranian society. \n\n# Copyright\n\nThis ebook published in Great Britain by  \nHalban Publishers Ltd  \n22 Golden Square  \nLondon W1F 9JW  \n2012\n\nFirst published in Great Britain by Halban Publishers, 1991\n\nwww.halbanpublishers.com\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Publishers.\n\nA CIP catalogue record for this book is available from the British Library.\n\nISBN 978 1 905559 48 0\n\nOriginally published in Iran under the title _Savushun_\n\nCopyright \u00a9 1969 by Simin Daneshvar  \nTranslation copyright \u00a9 1991 by Roxane Zand\n\nSimin Daneshvar has asserted his right under the Copyright, Design and Patents Act, 1988 to be identified as the author of this work.\n\nOriginal cover design by _The Third Man_\n\nOriginally printed in Great Britain by Cox and Wyman\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nMALANGA   \nCHASING   \nVALLEJO\n\n_Selected Poems:_\n\nC\u00c9SAR VALLEJO\n\n_with New Translations and Notes by_\n\nGERARD MALANGA\n\nA BILINGUAL EDITION\n\nTHREE ROOMS PRESS\n\nNEW YORK CITY\nA note of thanks to everyone who helped me with preparing this book project. To Carol Streib who was the first to assist me with these translations back in 1970. To Professor Pachas Almeyda for his research and advice. To Juan Larrea and Madame Georgette de Vallejo for photos. To David Cudaback for editorial guidance on the Introduction. To Claudio Taverna and Patricia Daniela Alverte for their patient generosity in looking after every aspect of these translations. I thank them all.\n\n\u2014GM\n_Malanga Chasing Vallejo:  \nSelected Poems of C\u00e9sar Vallejo   \nwith New Translations and Notes   \n_by Gerard Malanga\n\nTRANSLATIONS, INTRODUCTION, NOTES, AND   \nCLOSING POEM (ORIGINAL AND TRANSLATION):  \nCopyright \u00a9 2014 by Gerard Malanga\n\nAll rights reserved. No part of this book may be reproduced without permission of the author or publisher, except for brief quotes for review purposes. For permissions, please write to info@threeroomspress.com.\n\nISBN: 978-1-9411101-0-2 ebook  \nLibrary of Congress Control Number: 2014938002\n\nLETTERS FROM GEORGETTE VALLEJO, INTRODUCTION,   \nAND CLOSING POEM BY GERARD MALANGA:   \nTranslated by Patricia Daniela Alverte\n\nCOVER AND BOOK DESIGN:   \nKat Georges Design International  \nwww.katgeorges.com\n\nALL PHOTOS:  \nArchives Malanga\n\nPUBLISHED BY:  \nThree Rooms Press, New York, NY  \nwww.threeroomspress.com\n\nDISTRIBUTED BY:  \nPGW\/Perseus  \nwww.pgw.com\nCONTENTS\n\nC\u00e9sar Vallejo, the Man and the Poet\n\n_C\u00e9sar Vallejo, el hombre y el poeta_\n\nSELECTED POEMS: C\u00c9SAR VALLEJO\n\n_from_ LOS HERALDOS NEGROS, 1919\n\nThe Black Heralds\n\n_Los heraldos negros_\n\nThe Voice of the Mirror\n\n_La voz del espejo_\n\nA Divine Falling of Leaves\n\n_Deshojaci\u00f3n sagrada_\n\nIce Boat\n\n_Bordas de hielo_\n\nTwilight\n\n_Medialuz_\n\nWillow\n\n_Sauce_\n\nAbsent\n\n_Ausente_\n\nBeneath the Poplars\n\n_Bajos los \u00e1lamos_\n\nThe Spider\n\n_La ara\u00f1a_\n\nBabel\n\n_Babel_\n\nDregs\n\n_Heces_\n\nThe Black Cup\n\n_La copa negra_\n\nVillager\n\n_Aldeana_\n\nAgape\n\n_\u00c1gape_\n\nWhite Rose\n\n_Rosa blanca_\n\nOur Daily Bread\n\n_El pan nuestro_\n\nThe Eternal Dice\n\n_Los dados eternos_\n\nThe Weary Circles\n\n_Los anillos fatigados_\n\nThe Distant Footsteps\n\n_Los pasos lejanos_\n\nTo My Brother Miguel (in memoriam)\n\n_A mi hermano Miguel (in memoriam)_\n\nFilled with January\n\n_Enereida_\n\nI Was Born on a Day God Was Sick\n\n_Espergesia_\n\nFROM TRILCE, 1922\n\nIII \"The grown-ups\"\n\n_III \u00abLas personas mayores\u00bb_\n\nXIV \"My explanation exactly\"\n\n_XIV \u00abCual mi explicaci\u00f3n\u00bb_\n\nXV \"In that corner we sleep together\"\n\n_XV \u00abEn el rinc\u00f3n aquel, donde dormimos juntos\u00bb_\n\nXVI \"I have faith in being strong\"\n\n_XVI \u00abTengo fe en ser fuerte\u00bb_\n\nXVIII \"Oh the four walls of the cell\"\n\n_XVIII \u00abOh las cuatro paredes de la celda\u00bb_\n\nXXXIII \"If it rained tonight I would retire\"\n\n_XXXIII \u00abSi lloviera esta noche, retirar\u00edame\u00bb_\n\nXLV \"I am free from the chains of the sea\"\n\n_XLV \u00abMe desvinculo del mar\u00bb_\n\nLXI \"I get down from the horse tonight\"\n\n_LXI \u00abEsta noche desciendo del caballo\u00bb_\n\nLXIII \"Dawn rain drops. The well-combed\"\n\n_LXIII \u00abAmanece lloviendo. Bien peinada\u00bb_\n\nLXIII \"November 2nd turns\"\n\n_LXVI \u00abDobla el dos de Noviembre\u00bb_\n\nLXXV \"You are dead\"\n\n_LXXV \u00abEst\u00e1is muertos\u00bb_\n\nFROM POEMAS EN PROSA, 1923\/1924\u20131929\n\nThe Good Sense\n\n_El buen sentido_\n\nLanguidly Your Spirit\n\n_L\u00e1nguidamente su licor_\n\nThe Most Critical Moment of My Life\n\n_El momento m\u00e1s grave de la vida_\n\nI Am Going to Speak about Hope\n\n_Voy a hablar de la esperanza_\n\nDiscovery of Life\n\n_Hallazgo de la vida_\n\nPayroll of Bones\n\n_N\u00f3mina de huesos_\n\n\"Behold I Greet Today\"\n\n_\"He aqu\u00ed que hoy saludo\"_\n\nLoin of the Sacred Scriptures\n\n_Lomo de las sagradas escrituras_\n\nFROM POEMAS HUMANOS: THE UNDATED POEMS 1923(?)\u20131937\n\nHat, Overcoat, Gloves\n\n_Sombrero, abrigo, guantes_\n\nThe Wheel of the Starving\n\n_La rueda del hambriento_\n\nEpistle to Passersby\n\n_Ep\u00edstola a los transe\u00fantes_\n\nToday I'd Really Like to Be Happy\n\n_Quisiera hoy ser feliz de buena gana_\n\nConsidering Coldly, Impartially\n\n_Considerando en fr\u00edo, imparcialmente_\n\nAnd If after So Many Words\n\n_\u00a1Y si despu\u00e9s de tantas palabras!_\n\nParis, October 1936\n\n_Par\u00eds, Octubre 1936_\n\nBlack Stone on a White Stone\n\n_Piedra negra sobre una piedra blanca_\n\nToday I Like Life Much Less\n\n_Hoy me gusta la vida mucho menos_\n\n[FROM POEMAS HUMANOS:   \nTHE DATED POEMS, 4 SEPTEMBER\u20138 DECEMBER, 1937](part05.html)\n\nA Pillar Tolerating Solaces\n\n_Un pilar soportando consuelos_\n\nPoem to Be Read and Sung\n\n_Poema para ser Le\u00eddo y Cantado_\n\nWhile Pondering in Life, While Pondering\n\n_Al cavilar en la vida, al cavilar_\n\nOh Bottle without Wine!\n\n_\u00a1Oh botella sin vino!_\n\nHe Goes Running, Walking, Fleeing\n\n_Va corriendo, andando, huyendo_\n\nMy Breast Wants and Does Not Want Its Color\n\n_Quiere y no quiere su color mi pecho_\n\nThe Peace, the Wasp, the Bung, the Hillsides\n\n_La paz, la avispa, el taco, las vertientes_\n\nOf Pure Heat I'm Freezing\n\n_De puro calor tengo fr\u00edo_\n\nTrust in the Eyeglass, Not in the Eye\n\n_Confianza en el anteojo, n\u00f3 en el ojo_\n\nMocked, Acclimatized to the Good, Morbid, Tormented\n\n_Escarnecido, aclimatado al bien, m\u00f3rbido, hurente_\n\nStumble between Two Stars\n\n_Traspi\u00e9 entre dos estrellas_\n\nFarewell, Remembering a Goodbye\n\n_Despedida recordando un adi\u00f3s_\n\nThe Book of Nature\n\n_El libro de la naturaleza_\n\nI Have a Terrible Fear of Being an Animal\n\n_Tengo un miedo terrible de ser un animal_\n\nThe Anger Which Breaks a Man into Children\n\n_La c\u00f3lera que quiebra al hombre en ni\u00f1os_\n\nIntensity and Heights\n\n_Intensidad y altura_\n\nGuitar\n\n_Guitarra_\n\nPantheon\n\n_Pante\u00f3n_\n\nA Man Is Watching a Woman\n\n_Un hombre est\u00e1 mirando a una mujer_\n\nThe Nine Monsters\n\n_Los nueve monstruos_\n\nA Man Passes with a Loaf of Bread on His Shoulders\n\n_Un hombre pasa con un pan al hombro_\n\nSome Days a Fruitful, Cautious Longing Comes Over Me\n\n_Me viene, hay d\u00edas, una gana ub\u00e9rrima, pol\u00edtica_\n\nPalms and Guitar\n\n_Palmas y guitarra_\n\nThe Soul That Suffered from Being Its Body\n\n_El alma que sufri\u00f3 de ser su cuerpo_\n\nThe One Who Will Come Has Just Passed By\n\n_Acaba de pasar el que vendr\u00e1_\n\nThe Evil Man Might Come with a Throne on His Shoulder\n\n_Viniere el malo, con un trono al hombro_\n\nThat Is the Place Where I Put On\n\n_Ello es que el lugar donde me pongo_\n\nAnother Bit of Calm, Comrade\n\n_Otro poco de calma, camarada_\n\nFROM ESPA\u00d1A, APARTA DE M\u00cd ESTE C\u00c1LIZ, SET.\/OCT.\/NOV. 1937\n\nI \u2013 Hymn to the Volunteers of the Republic\n\n_I \u2013 Himno a los voluntarios de la rep\u00fablica_\n\nIII \u2013 With His Index Finger He Writes on the Air\n\n_III \u2013 Sol\u00eda escribir con su dedo grande en el aire_\n\nIX \u2013 A Brief Funeral Prayer for a Hero of the Republic\n\n_IX \u2013 Peque\u00f1o responso a un h\u00e9roe de la rep\u00fablica_\n\nXII \u2013 Mass\n\n_XII \u2013 Masa_\n\nXV \u2013 Spain, Take This Cup from Me\n\n_XV \u2013 Espa\u00f1a, aparta de m\u00ed este c\u00e1liz_\n\nCLOSING POEM BY GERARD MALANGA\n\nTHE LETTERS FROM GEORGETTE VALLEJO\nINTRODUCTION\n\n_C\u00e9sar Vallejo, with his wife Georgette_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\nC\u00e9sar Vallejo, the Man and the Poet  \n(For Spanish translation click here)\n\nHOW DID I COME TO TRANSLATE the poetry of C\u00e9sar Vallejo in 1969? First, having only a peripheral knowledge of Spanish, I never professed to be \"translating\" his verse in the literal sense, but to be transubstantiating them from one language to another. Initially, _Cassell's Spanish Dictionary_ , the 1959 edition, was my constant companion.\n\nI first became acquainted with Vallejo's poetry through the pioneer translations of his work by Thomas Merton, Donald Devenish Walsh, Muna Lee de Mu\u00f1oz Mar\u00edn, H. R. Hays, James Wright, and Robert Bly. I was not out to improve what they had accomplished. I loved what they'd done.\n\nHaving read about his life\u2014consumed by the burden of poverty and malnutrition\u2014I felt he was a kindred spirit; and through his verse, I came to understand the bleakness, the loneliness, the deprivation of what he had expressed in his daily living. Life was not kind to him.\n\nI experienced what he experienced. It's no fun being poor in Paris, especially during his sojourn there in those late 1930s, I can imagine. Sixty years later I, too, have walked those same Paris streets of gloom and rain and bitter cold. I, too, peered hungrily through those curtained windows at the privileged in some warm and cozy bistro. I, too, walked away with a growling stomach. I, too, had unfulfilled desires glancing in shop windows, even at something as simple as a folded linen handkerchief. I, too, wore through the soles of my only pair of shoes until my feet ached from the dampness. They don't give you grants or shower you with prizes for being poor. Poverty doesn't support vision, and counts for nothing in the end.\n\nVallejo's experiences became my experiences\u2014not by choice, mind you, but by the mere fact of our spiritual brotherhood through poetry. It's as if I fully understood the spirituality of what he was expressing on a universal plane. He was talking to me directly. His soul touched mine through his verse. In this moment, we became spiritual brothers.\n\nBut I had no one with whom I could share those experiences discovered through his verse. Dare I reach out to Vallejo's widow, Madame Georgette de Vallejo?\n\nOne early translator had demonized her. I was forewarned that she was difficult to deal with. But this warning didn't discourage me in the slightest. I wanted to touch the one person still alive who was closest to the man whose works touched me. One problem: she was living in Lima, Peru, nearly four thousand miles away.\n\nSo I took a chance, a long shot, to be sure. I sent her a couple dozen of my translations. Remarkably, within a month, she wrote back with glowing remarks and helpful hints and even concrete examples of what to do and what not to do, so that I could make my versions better. She bestowed upon me the gift of her generosity and the knowledge she had gained being C\u00e9sar Vallejo's lifetime companion. She shared her knowledge with me because she clearly believed in my work.\n\nIt was never my intention to make a career out of translating C\u00e9sar Vallejo. There were plenty others in the horse race; and we know what B\u00e9la Bart\u00f3k had to say about horse races (\"Competition is for horses, not for artists\"). Any other choice not to translate would have betrayed the spiritual connection I felt for the man and his work.\n\nI did what I did because of the spiritual connection, and nothing more. No great expectations. No accolades sought. No subterfuge. No hidden agendas. I felt bonded to the man through time and space. This is what counted most for me, in the end.\n\nIt's now been nearly forty-five years since I embarked on this long voyage through uncharted waters with many an electrical storm coming my way. For most of that time, I would return to my working drafts and make revisions and read them aloud to myself. My efforts are a testament to the spiritual kinship I've felt for C\u00e9sar Vallejo all along. I was steadfast. I was focused. I was dedicated. He never left my side. He has been my guiding spirit, my guiding light, not only through his poetry but through mine as well. _Dear friend_.\n\nGerard Malanga  \n _5:VI:13_\nC\u00e9sar Vallejo, el hombre y el poeta  \n(For English translation click here)\n\ntranslated by Patricia Daniela Alverte\n\nC\u00d3MO ES QUE LLEGU\u00c9 A TRADUCIR la poes\u00eda de C\u00e9sar Vallejo en 1969? Primero, teniendo apenas un conocimiento perif\u00e9rico del espa\u00f1ol, nunca he pretendido \"traducir\" sus versos en sentido literal, sino transustanciar los mismos de un lenguaje a otro. En ese momento, la edici\u00f3n de 1959 del _Cassell's Spanish Dictionary_ , fue mi constante compa\u00f1\u00eda.\n\nMe familiaric\u00e9 por primera vez con la poes\u00eda de Vallejo a trav\u00e9s de las traducciones pioneras de su trabajo realizadas por Thomas Merton, Donald Devenish Walsh, Muna Lee de Mu\u00f1oz Mar\u00edn, H. R. Hays, James Wright, y Robert Bly. Yo no pretend\u00eda mejorar el trabajo logrado por ellos. Me encanta lo que han hecho.\n\nHabiendo le\u00eddo sobre su vida\u2014consumida por el agobio de la pobreza y de la malnutrici\u00f3n\u2014sent\u00ed que \u00e9l era un esp\u00edritu af\u00edn; y a trav\u00e9s de su verso, comprend\u00ed la desolaci\u00f3n, la soledad, la privaci\u00f3n de lo que ha expresado en su vida cotidiana. \u00c9sta no era amable con \u00e9l.\n\nPas\u00e9 por lo mismo que el pas\u00f3. No es gracioso ser pobre en Par\u00eds, puedo imaginar, especialmente durante su estancia all\u00ed a finales de los a\u00f1os 30. Sesenta a\u00f1os despu\u00e9s yo, tambi\u00e9n, he caminado esas mismas calles de penumbra, lluvia y fr\u00edo de Par\u00eds. Yo, tambi\u00e9n, he mirado con avidez a trav\u00e9s de esas ventanas con cortinas el privilegio de alg\u00fan c\u00e1lido y costoso peque\u00f1o restaurante. Yo, tambi\u00e9n, me he alejado con un est\u00f3mago quejoso. Yo, tambi\u00e9n, he tenido deseos insatisfechos luego de echar un vistazo a las vidrieras de las tiendas, incluso por algo tan sencillo como un pa\u00f1uelo de lino plegado. Yo, tambi\u00e9n, agujere\u00e9 la suela de mi \u00fanico par de zapatos hasta que mi pie dol\u00eda a causa de la humedad. No te dan subvenciones ni te colman de premios por ser pobre. La pobreza no apoya a la visi\u00f3n, y no cuenta para nada al final.\n\nLas vivencias de Vallejo se convirtieron en mis vivencias\u2014no por elecci\u00f3n, como se pueden imaginar, sino por el simple hecho de nuestra hermandad a trav\u00e9s de la poes\u00eda. Como si hubiera comprendido enteramente la espiritualidad de lo que \u00e9l estaba expresando en un plano universal. Me hablaba a m\u00ed directamente. Su alma toc\u00f3 la m\u00eda a trav\u00e9s de sus versos. En ese momento, nos volvimos hermanos espirituales.\n\nPero no ten\u00eda a nadie con quien compartir esas experiencias descubiertas a trav\u00e9s de sus versos. Me atrevo a contactar a la viuda de Vallejo, Madame Georgette de Vallejo?\n\nUno de los primeros traductores la hab\u00eda demonizado. Yo estaba advertido de que ella era una persona dif\u00edcil de tratar. Pero esta advertencia no me desalent\u00f3 en lo m\u00e1s m\u00ednimo. Quer\u00eda contactar a la \u00fanica persona a\u00fan viva m\u00e1s cercana al hombre cuyas palabras me hab\u00edan tocado. Peque\u00f1o problema: ella estaba viviendo en Lima, Per\u00fa, a casi 4000 millas de distancia.\n\nAs\u00ed que me arriesgu\u00e9, un tiro dif\u00edcil, de seguro: le mand\u00e9 un par de docenas de mis traducciones. Extraordinariamente, en un mes, ella me escribi\u00f3 con brillantes observaciones y consejos \u00fatiles e incluso con ejemplos concretos de lo que ten\u00eda que hacer y de lo que no ten\u00eda que hacer, as\u00ed de esa manera yo podr\u00eda mejorar mis versiones. Ella deposit\u00f3 en m\u00ed el regalo de su generosidad y el conocimiento que hab\u00eda adquirido siendo la compa\u00f1era de vida de C\u00e9sar Vallejo. Ella comparti\u00f3 el conocimiento conmigo porque claramente confiaba en mi trabajo.\n\nNunca fue mi intenci\u00f3n hacer carrera traduciendo a C\u00e9sar Vallejo. Hab\u00eda muchos otros en la carrera de caballos; y todos sabemos lo que dijo Bela Bartok sobre las carreras de caballos (\"La competici\u00f3n es para los caballos, no para los artistas\"). Cualquier otra opci\u00f3n que no haya sido simplemente traducir, hubiera traicionado la conexi\u00f3n espiritual que sent\u00eda por el hombre y su trabajo.\n\nHice lo que hice por la conexi\u00f3n espiritual, y por nada m\u00e1s. Sin grandes expectativas. Sin b\u00fasqueda de elogios. Sin subterfugios. Sin agendas ocultas. Me sent\u00ed unido al hombre a trav\u00e9s del tiempo y del espacio. Al final, eso es lo que m\u00e1s cuenta para m\u00ed.\n\nAhora ya han pasado casi cuarenta y cinco a\u00f1os desde que me embarqu\u00e9 en este largo viaje a trav\u00e9s de aguas inexploradas con algunas tormentas el\u00e9ctricas en mi camino. La mayor parte de ese tiempo, he vuelto a mis borradores y he hecho correcciones y las he le\u00eddo para m\u00ed mismo. Mis esfuerzos son el testamento a la afinidad espiritual que he sentido por C\u00e9sar Vallejo desde el principio. Fui constante. Fui centrado. Fui dedicado. \u00c9l nunca se fue de mi lado. \u00c9l ha sido mi rector espiritual, mi gu\u00eda de luz, no s\u00f3lo a trav\u00e9s de su poes\u00eda sino a trav\u00e9s de la m\u00eda tambi\u00e9n. _Querido amigo_.\n\nGerard Malanga  \n _5:VI:13_\n_for_\n\n_Georgette de Vallejo_\n\n_1908\u20131984_\n\n_C\u00e9sar Vallejo in Lima, 1920_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nLOS HERALDOS NEGROS\n\n_1919_\nThe Black Heralds  \n(For Spanish translation click here)\n\nLife has such blows and such harsh ones . . . I don't know!\n\nBlows like the hatred of God; as if before them,\n\nthe whiplash of all suffering\n\nwere to damn up the soul . . . I don't know!\n\nThey are few, yet they are . . . cleaving dark furrows\n\nin the proudest of faces and the strongest of backs.\n\nPerhaps they are the colts of barbarous Attilas;\n\nor the black heralds sent to us by Death.\n\nThey are the deep downfall of the Christ's soul,\n\nof some adoring faith that Destiny blasphemes.\n\nThese bloody blows are the cracklings\n\nof some bread we burn at the oven door.\n\nAnd man . . . Poor . . . poor! He turns his eyes, as\n\nwhen we are called by a pat on the shoulder;\n\nhe turns his mad eyes, and all experienced\n\nwells up, like a pool of guilt in his gaze.\n\nLife has such blows and such harsh ones . . . I don't know!\nLos heraldos negros  \n(For English translation click here)\n\nHay golpes en la vida, tan Fuertes Yo no s\u00e9!\n\nGolpes como del odio de Dios; como si ante ellos\n\nla resaca de todo lo sufrido\n\nse empozara en el alma . . . Yo no s\u00e9!\n\nSon pocos, pero son . . . Abren zanjas oscuras\n\nen el rostro m\u00e1s fiero y en el lomo m\u00e1s fuerte.\n\nSer\u00e1n tal vez los potros de b\u00e1rbaros atilas;\n\nlos heraldos negros que nos manda la Muerte.\n\nSon las ca\u00eddas hondas de los Cristos del alma,\n\nde alguna fe adorable que el Destino blasfema.\n\nEsos golpes sangrientos son las crepitaciones\n\nde alg\u00fan pan que en la puerta del horno se nos quema.\n\nY el hombre . . . Pobre . . . pobre! Vuelve los ojos, como\n\ncuando por sobre el hombro nos llama una palmada;\n\nvuelve los ojos locos, y todo lo vivido\n\nse empoza, como charco de culpa, en la mirada.\n\nHay golpes en la vida, tan fuertes . . . Yo no s\u00e9!\nThe Voice of the Mirror  \n(For Spanish translation click here)\n\nSo life passes like a rare mirage.\n\nThe blue rose gives light and being to the thistle!\n\nBeside the dogma of the bundle\n\nmurderer, the sophism of The Good and The Reason!\n\nBy chance it has caught the thing which brushed the hand;\n\nthe perfumes diffused, and between them has felt\n\nthe moss that in the middle of the road has grown\n\nin the dry apple-tree of the dead Illusion.\n\nSo life passes,\n\nwith the singing of treacherous parched bacchantes.\n\nI go totally overwhelmed, forward . . . forward,\n\nmuttering my funeral march.\n\nThey walk close to the feet of Royal Brahmin elephants,\n\nand the sordid buzz of a boil mercurial,\n\ncouples raise a toast sculpted in rock,\n\nand forgotten ones dawn a cross on the mouth.\n\nSo life passes, a vast orchestra of Sphinxes\n\nwho throws on the Abyss, their funeral march.\nLa voz del espejo  \n(For English translation click here)\n\nAs\u00ed pasa la vida, como raro espejismo.\n\nLa rosa azul que alumbra y da el ser al cardo!\n\nJunto al dogma del fardo\n\nmatador, el sofisma del Bien y la Raz\u00f3n!\n\nSe ha cogido, al acaso, lo que roz\u00f3 la mano;\n\nlos perfumes volaron, y entre ellos se ha sentido\n\nel moho que a mitad de la ruta ha crecido\n\nen el manzano seco de la muerta Ilusi\u00f3n.\n\nAs\u00ed pasa la vida,\n\ncon c\u00e1nticos aleves de agostada bacante.\n\nYo voy todo azorado, adelante . . . adelante,\n\nrezongando mi marcha funeral.\n\nVan al pie de brahm\u00e1nicos elefantes reales,\n\ny al s\u00f3rdido abejeo de un hervor mercurial,\n\nparejas que alzan brindis esculpidos en roca\n\ny olvidados crep\u00fasculos una cruz en la boca.\n\nAs\u00ed pasa la vida, vasta orquesta de Esfinges\n\nque arrojan al Vac\u00edo su marcha funeral.\nA Divine Falling of Leaves  \n(For Spanish translation click here)\n\nMoon! Crown of an enormous head,\n\ndropping leaves into yellows shadows!\n\nRed crown of a Jesus who thinks\n\ntragically, soft of emeralds!\n\nMoon! Reckless heart celestial,\n\nwhy do you row this way, inside the cup\n\nfull of blue wine, toward the west\n\nwhose stern is defeated and painful?\n\nMoon! It's no use flying away,\n\nso you go up in flames of scattered opals;\n\nmaybe you are my gypsy heart\n\nwho wanders in the blue, crying verses! . . .\nDeshojaci\u00f3n sagrada  \n(For English translation click here)\n\nLuna! Corona de una testa inmensa,\n\nque te vas deshojando en sombras gualdas!\n\nRoja corona de un Jes\u00fas que piensa\n\ntr\u00e1gicamente dulce de esmeraldas!\n\nLuna! Alocado coraz\u00f3n celeste\n\n\u00bfpor qu\u00e9 bogas as\u00ed, dentro la copa\n\nllena de vino azul, hacia el oeste,\n\ncual derrotada y dolorida popa?\n\nLuna! Y a fuerza de volar en vano,\n\nte holocaustas en \u00f3palos dispersos:\n\nt\u00fa eres talvez mi coraz\u00f3n gitano\n\nque vaga en el azul llorando versos! . . .\nIce Boat  \n(For Spanish translation click here)\n\nI come to see you go by every day,\n\nenchanting boat, always distant . . .\n\nYour eyes two blond captains,\n\nyour lips one tiny red handkerchief\n\nwaving a bloodstained farewell!\n\nI come to see you go by, until one day,\n\nintoxicated of time and cruelty,\n\nenchanted boat, always distant,\n\nthe afternoon star will depart!\n\nThe rigging winds that betray, winds\n\nof a woman that has passed by!\n\nYour cold captains give orders;\n\nAnd the one who departs will be I . . .\nBordas de hielo  \n(For English translation click here)\n\nVengo a verte pasar todos los d\u00edas,\n\nvaporcito encantado siempre lejos . . .\n\nTus ojos son dos rubios capitanes;\n\ntu labio es un brev\u00edsimo pa\u00f1uelo\n\nrojo que ondea en un adi\u00f3s de sangre!\n\nVengo a verte pasar; hasta que un d\u00eda,\n\nembriagada de tiempo y de crueldad,\n\nvaporcito encantado siempre lejos,\n\nla estrella de la tarde partir\u00e1!\n\nLas jarcias; vientos que traicionan; vientos\n\nde mujer que pas\u00f3!\n\nTus fr\u00edos capitanes dar\u00e1n orden;\n\ny quien habr\u00e1 partido ser\u00e9 yo . . .\nTwilight  \n(For Spanish translation click here)\n\nI've dreamed a leak. And I've dreamed\n\nyour laces dispersed in the bedroom;\n\nalong the length of a wharf, some mother;\n\nbreastfeeding the hour at her fifteen years.\n\nI've dreamt a leak. A \"forever\"\n\nsighing at a prow's ladder.\n\nI've dreamed a mother;\n\nsome fresh sprigs planted of vegetables,\n\nand the starry trousseau stitched of a dawn.\n\nAlong the length of a wharf . . .\n\nAnd along the length of a throat drowning!\nMedialuz  \n(For English translation click here)\n\nHe so\u00f1ado una fuga. Y he so\u00f1ado\n\ntus encajes dispersos en la alcoba.\n\nA lo largo de un muelle, alguna madre;\n\ny sus quince a\u00f1os dando el seno a una hora.\n\nHe so\u00f1ado una fuga. Un \u00abpara siempre\u00bb\n\nsuspirado en la escala de una proa;\n\nhe so\u00f1ado una madre;\n\nunas frescas matitas de verdura,\n\ny el ajuar constelado de una aurora.\n\nA lo largo de un muelle . . .\n\nY a lo largo de un cuello que se ahoga!\nWillow  \n(For Spanish translation click here)\n\nWinter's lyricism, sound of muslin,\n\nwhen the early departure is approaching,\n\ndoomed voices of sad tunes\n\nthat in the afternoon prays a farewell.\n\nVision of my buried delusions\n\nin my own tomb of a mortal wound.\n\nVeronica charity of unknowns lands,\n\nwhere the life is lost at an ether price.\n\nClose to the dawn I will depart crying;\n\nand while my years are hunching\n\nmy fast path will curve scythes.\n\nBefore the cold unction of a dying moon,\n\nwith steel dings in an indolent land,\n\nthe dogs will dig, howling, a goodbye!\nSauce  \n(For English translation click here)\n\nLirismo de invierno, rumor de crespones,\n\ncuando ya se acerca la pronta partida;\n\nagoreras voces de tristes canciones\n\nque en la tarde rezan una despedida.\n\nVisi\u00f3n del entierro de mis ilusiones\n\nen la propia tumba de mortal herida.\n\nCaridad ver\u00f3nica de ignotas regiones,\n\ndonde a precio de \u00e9ter se pierde la vida.\n\nCerca de la aurora partir\u00e9 llorando;\n\ny mientras mis a\u00f1os se vayan curvando,\n\ncurvar\u00e1 guada\u00f1as mi ruta veloz.\n\nY ante fr\u00edos \u00f3leos de luna muriente,\n\ncon timbres de aceros en tierra indolente,\n\ncavar\u00e1n los perros, aullando, un adi\u00f3s!\nAbsent  \n(For Spanish translation click here)\n\nAbsent! The morning I go\n\nfarther than the farthest, to the Mystery,\n\nlike following an inevitable line,\n\nyour feet will glide to the cemetery.\n\nAbsent! The morning when to the beach\n\nthe sea is shadow and the hushed empire,\n\nI go like a mournful bird,\n\nthe white pantheon will hold you captive.\n\nThe night will fall in your eyes\n\nand you will suffer, taking\n\nthe torn white garments of a penitent.\n\nAbsent! within your sufferings\n\na bronze weeping, there will pass a hound\n\npack of remorse.\nAusente  \n(For English translation click here)\n\nAusente! La ma\u00f1ana en que me vaya\n\nm\u00e1s lejos de lo lejos, al Misterio,\n\ncomo siguiendo inevitable raya,\n\ntus pies resbalar\u00e1n al cementerio.\n\nAusente! La ma\u00f1ana en que a la playa\n\ndel mar de sombra y del callado imperio,\n\ncomo un p\u00e1jaro l\u00fagubre me vaya,\n\nser\u00e1 el blanco pante\u00f3n tu cautiverio.\n\nSe habr\u00e1 hecho de noche en tus miradas;\n\ny sufrir\u00e1s, y tomar\u00e1s entonces\n\npenitentes blancuras laceradas.\n\nAusente! Y en tus propios sufrimientos\n\nha de cruzar entre un llorar de bronces\n\nuna jaur\u00eda de remordimientos!\nBeneath the Poplars  \n(For Spanish translation click here)\n\n_for Jos\u00e9 Garrido_\n\nLike poet-priests who've been imprisoned,\n\nthe poplars of blood have slept\n\nchewing songs of grass in the sunset,\n\nthe herds of Bethlehem on the hills.\n\nTo the latest martyrs of light\n\nthe ancient shepherd, shaken\n\nin his Easters' eyes have picked up\n\na caste herd of constellations.\n\nTilled in orphanhood the moment gone down\n\nwith burial rumors, in the praying meadows\n\ncowbells fill with autumn shadows.\n\nSurvive the blue weaved on iron,\n\nin which the shrouded pupils\n\na dog draws his pastoral howl.\nBajo los \u00e1lamos  \n(For English translation click here)\n\n_para Jos\u00e9 Garrido_\n\nCual hier\u00e1ticos bardos prisioneros,\n\nlos \u00e1lamos de sangre se han dormido.\n\nRumian arias de yerba al sol ca\u00eddo,\n\nlas greyes de Bel\u00e9n en los oteros.\n\nEl anciano pastor, a los postreros\n\nmartirios de la luz, estremecido,\n\nen sus pascuales ojos ha cogido\n\nuna casta manada de luceros.\n\nLabrado en orfandad baja al instante\n\ncon rumores de entierro, al campo orante;\n\ny se oto\u00f1an de sombra las esquilas.\n\nSupervive el azul urdido en hierro,\n\ny en \u00e9l, amortajadas las pupilas,\n\ntraza su aullido pastoral un perro.\nThe Spider  \n(For Spanish translation click here)\n\nIt is an enormous spider no longer moving;\n\na colorless spider whose body,\n\nhead, and abdomen, is bleeding.\n\nI've seen him so close today. With what strength\n\nhe lengthened his innumerable feet\n\nto every side.\n\nAnd I think of his invisible eyes\n\nthose fatal pilots of the spider.\n\nIt's a spider who shivered, fixed\n\nat the edge of a stone;\n\nthe abdomen on one side\n\nand on the other, its head.\n\nWith so many feet the poor thing, and yet he cannot\n\nmake himself out. And I, watching him\n\namazed in such trance,\n\nwhat strange pain this traveler gives me today.\n\nIt's an enormous spider, whose abdomen\n\nprevents him from following his head.\n\nI've thought about his eyes\n\nand his numerous feet . . .\n\nAnd what strange pain this traveler gives me today!\nLa ara\u00f1a  \n(For English translation click here)\n\nEs una ara\u00f1a enorme que ya no anda;\n\nuna ara\u00f1a incolora, cuyo cuerpo,\n\nuna cabeza y un abdomen, sangra.\n\nHoy la he visto de cerca. Y con qu\u00e9 esfuerzo\n\nhacia todos los flancos\n\nsus pies innumerables alargaba.\n\nY he pensado en sus ojos invisibles,\n\nlos pilotos fatales de la ara\u00f1a.\n\nEs una ara\u00f1a que temblaba fija\n\nen un filo de piedra;\n\nel abdomen a un lado,\n\ny al otro la cabeza.\n\nCon tantos pies la pobre, y a\u00fan no puede\n\nresolverse. Y, al verla\n\nat\u00f3nita en tal trance,\n\nhoy me ha dado qu\u00e9 pena esa viajera.\n\nEs una ara\u00f1a enorme, a quien impide\n\nel abdomen seguir a la cabeza.\n\nY he pensado en sus ojos\n\ny en sus pies numerosos . . .\n\n\u00a1Y me ha dado qu\u00e9 pena esa viajera!\nBabel  \n(For Spanish translation click here)\n\nSweet home with no style, built\n\nwith just one stroke and just one piece\n\nof glint wax. And at home\n\nshe destroys and she cleans; says at times:\n\n\"The asylum is nice. Right here!\"\n\nOther times she breaks down and cries!\nBabel  \n(For English translation click here)\n\nDulce hogar sin estilo, fabricado\n\nde un solo golpe y de una sola pieza\n\nde cera tornasol. Y en el hogar\n\nella da\u00f1a y arregla; a veces dice:\n\n\u00abEl hospicio es bonito; aqu\u00ed no m\u00e1s!\u00bb\n\n\u00a1Y otras veces se pone a llorar!\nDregs  \n(For Spanish translation click here)\n\nThis afternoon it's raining, as never before; and I don't\n\nfeel like staying alive, heart.\n\nThis afternoon is gentle. Why not?\n\nWears grace and grief, dressed like a woman.\n\nThis afternoon, in Lima, it's raining. And I remember\n\nthe cruel caverns of my ingratitude;\n\nmy block of ice on her poppy,\n\nstronger than her \"Don't be this way!\"\n\nMy violent black flowers; and the barbarous\n\nand staggering blow with a stone; and the glacial roof.\n\nAnd will put the silence of her dignity\n\nwith burning oils on the endpoint.\n\nTherefore, this afternoon, as never before, I walk\n\nwith this owl, with this heart.\n\nAnd other women pass me by; and seeing me so sad,\n\nthey take a little piece from you,\n\nin the abrupt wrinkle of my deep grief.\n\nThis afternoon it's raining, rain so hard. And I don't\n\nfeel like staying alive, heart!\nHeces  \n(For English translation click here)\n\nEsta tarde llueve, como nunca; y no\n\ntengo ganas de vivir, coraz\u00f3n.\n\nEsta tarde es dulce. \u00bfPor qu\u00e9 no ha de ser?\n\nViste de gracia y pena; viste de mujer.\n\nEsta tarde en Lima llueve. Y yo recuerdo\n\nlas cavernas crueles de mi ingratitud;\n\nmi bloque de hielo sobre su amapola,\n\nm\u00e1s fuerte que su \u00abNo seas as\u00ed!\u00bb\n\nMis violentas flores negras; y la b\u00e1rbara\n\ny enorme pedrada; y el trecho glacial.\n\nY pondr\u00e1 el silencio de su dignidad\n\ncon \u00f3leos quemantes el punto final.\n\nPor eso esta tarde, como nunca, voy\n\ncon este b\u00faho, con este coraz\u00f3n.\n\nY otras pasan; y vi\u00e9ndome tan triste,\n\ntoman un poquito de ti\n\nen la abrupta arruga de mi hondo dolor.\n\nEsta tarde llueve, llueve mucho. \u00a1Y no\n\ntengo ganas de vivir, coraz\u00f3n!\nThe Black Cup  \n(For Spanish translation click here)\n\nThe night is a cup of evil. A police whistle\n\ncuts across it, like a vibrating pin.\n\nTrampy woman, listen, how is it if you've gone away,\n\nthe wave is still black and still makes me flare up?\n\nThe Earth holds edges of a coffin in the shadows.\n\nTrampy woman, listen, please don't come back.\n\nMy flesh swims, swims\n\nin the cup of darkness that still makes me feel pain;\n\nmy flesh swims in there\n\nas in a swampy heart of a woman.\n\nStarlike coal . . . I've felt\n\ndry touches of clays\n\nover my transparent lotus.\n\nAh, woman. The flesh of instinct\n\nexists for and within you. Ah, woman!\n\nBecause of this, oh black chalice! even when you're gone,\n\nI smother in dust,\n\nand other desires to drink start pawing inside my flesh.\nLa copa negra  \n(For English translation click here)\n\nLa noche es una copa de mal. Un silbo agudo\n\ndel guardia la atraviesa, cual vibrante alfiler.\n\nOye, t\u00fa, mujerzuela, \u00bfc\u00f3mo, si ya te fuiste,\n\nla onda a\u00fan es negra y me hace a\u00fan arder?\n\nLa Tierra tiene bordes de f\u00e9retro en la sombra.\n\nOye, t\u00fa, mujerzuela, no vayas a volver.\n\nMi carne nada, nada\n\nen la copa de sombra que me hace a\u00fan doler;\n\nmi carne nada en ella,\n\ncomo en un pantanoso coraz\u00f3n de mujer.\n\nAscua astral . . . He sentido\n\nsecos roces de arcilla\n\nsobre mi loto di\u00e1fano caer.\n\nAh, mujer! Por ti existe\n\nla carne hecha de instinto. Ah mujer!\n\nPor eso \u00a1oh, negro c\u00e1liz! aun cuando ya te fuiste,\n\nme ahogo con el polvo;\n\ny piafan en mis carnes m\u00e1s ganas de beber!\nVillager  \n(For Spanish translation click here)\n\nDistant vibration of small rusty bells\n\nspills on the air\n\nthe rural fragrance of their anguish.\n\nIn the silent light\n\nthe setting sun bleeds its farewell.\n\nAutumn's amber on the landscape\n\ntakes on a cold hue of mournful gray!\n\nTo the gate of the house,\n\nthat time's claws fill it with holes,\n\npeeping in silence,\n\npassing then to the nearby stable,\n\nthe calm silhouette\n\nof an ox color of gold,\n\nwho yearns with its biblical eyes\n\nlistening the prayers of the cowbells\n\nhis virile bull age!\n\nA noble rooster jumps across,\n\nthe garden wall,\n\nflapping the pain of his song, and in sad alert,\n\nas two drops of weep,\n\ntremble his eyes in the dead afternoon!\n\nAt the old village\n\nlanguidly plucks\n\nthe soft yarav\u00ed* of a guitar,\n\nin whose eternity of deep suffering\n\nthe sad voice of an Indian tolls\n\nlike a huge old bell in a cemetery.\n\nLeaning my elbows on the wall,\n\nwhen dark hues triumph in the soul\n\nand the wind prays in stiff branches\n\nwooden flute laments, timid, uncertain,\n\nI sigh my dismay,\n\nto see that in the scarlet and gold penumbra\n\nweeps a tragic blue of dead idylls!\n\n_*Yarav\u00ed is a melancholic song, originally from Quechua._\nAldeana  \n(For English translation click here)\n\nLejana vibraci\u00f3n de esquilas mustias\n\nen el aire derrama\n\nla fragancia rural de sus angustias.\n\nEn el patio silente,\n\nsangra su despedida el sol poniente.\n\nEl \u00e1mbar oto\u00f1al del panorama\n\ntoma un fr\u00edo matiz de gris doliente!\n\nAl port\u00f3n de la casa,\n\nque el tiempo con sus garras torna ojosa,\n\nasoma silenciosa\n\ny al establo cercano luego pasa,\n\nla silueta calmosa\n\nde un buey color de oro,\n\nque a\u00f1ora con sus b\u00edblicas pupilas,\n\noyendo la oraci\u00f3n de las esquilas,\n\nsu edad viril de toro!\n\n\u00a1Al muro de la huerta,\n\naleteando la pena de su canto,\n\nsalta un gallo gentil, y un triste alerta,\n\ncual dos gotas de llanto,\n\ntiemblan sus ojos a la tarde muerta!\n\nL\u00e1nguido se derrama\n\nen la vetusta aldea\n\nel dulce yarav\u00ed de una guitarra,\n\nen cuya eternidad de hondo quebranto\n\nla triste voz de un indio dondonea,\n\ncomo un viejo esquil\u00f3n de camposanto.\n\nDe codos yo en el muro,\n\ncuando triunfa en el alma el tinte oscuro,\n\ny el viento reza en los ramajes yertos\n\nllantos de quena, t\u00edmidos, inciertos,\n\nsuspiro una congoja\n\nal ver que en la penumbra gualda y roja\n\nllora un tr\u00e1gico azul de idilios muertos!\nAgape  \n(For Spanish translation click here)\n\nToday no one comes to inquire,\n\nnor wants anything from me this afternoon.\n\nI've not seen a single graveyard flower\n\nin all this gay procession of lights.\n\nForgive me, Lord: how little I've died!\n\nIn this afternoon everyone, everyone goes by\n\nwithout asking or begging me anything.\n\nAnd I don't know what it is they forget that remains\n\nwrong in my hands, like an alien thing.\n\nI've come to the door,\n\nI feel like shouting to everyone:\n\nif you miss something, here it is!\n\nBecause on every afternoon of this life,\n\nI don't know which doors they slam in the face,\n\nand my soul takes something that belongs to another.\n\nToday nobody comes;\n\nand today I've died how little this afternoon!\n\u00c1gape  \n(For English translation click here)\n\nHoy no ha venido nadie a preguntar;\n\nni me han pedido en esta tarde nada.\n\nNo he visto ni una flor de cementerio\n\nen tan alegre procesi\u00f3n de luces.\n\nPerd\u00f3name, Se\u00f1or: qu\u00e9 poco he muerto!\n\nEn esta tarde todos, todos pasan\n\nsin preguntarme ni pedirme nada.\n\nY no s\u00e9 qu\u00e9 se olvidan y se queda\n\nmal en mis manos, como cosa ajena.\n\nHe salido a la puerta,\n\ny me da ganas de gritar a todos:\n\nSi echan de menos algo, aqu\u00ed se queda!\n\nPorque en todas las tardes de esta vida,\n\nyo no s\u00e9 con qu\u00e9 puertas dan a un rostro,\n\ny algo ajeno se toma el alma m\u00eda.\n\nHoy no ha venido nadie;\n\ny hoy he muerto qu\u00e9 poco en esta tarde!\nWhite Rose  \n(For Spanish translation click here)\n\nI feel all right. Now\n\na stoical frost shines\n\non me.\n\nMaking me laugh, ruby-colored\n\nrope\n\nthat grinds in my body.\n\nEndless rope,\n\nlike a\n\nspiral\n\ndescending\n\nof evil . . .\n\nbloody rope and lefty\n\nshaped by\n\na thousand strut daggers.\n\nGoing in this way, braiding\n\nits rolls of crepe;\n\nand tying the tremulous cat\n\nof Fear to the frozen nest,\n\nto the ultimate bonfire.\n\nAnd now I am calm,\n\nsurrounded by light.\n\nAnd a shipwrecked coffin\n\nmeows in my Pacific.\nRosa blanca  \n(For English translation click here)\n\nMe siento bien. Ahora\n\nbrilla un estoico hielo\n\nen m\u00ed.\n\nMe da risa esta soga\n\nrub\u00ed\n\nque rechina en mi cuerpo.\n\nSoga sin fin,\n\ncomo una\n\nvoluta\n\ndescendente\n\nde mal . . .\n\nSoga sangu\u00ednea y zurda\n\nformada de\n\nmil dagas en puntal.\n\nQue vaya as\u00ed, trenzando\n\nsus rollos de cresp\u00f3n;\n\ny que ate el gato tr\u00e9mulo\n\ndel Miedo al nido helado,\n\nal \u00faltimo fog\u00f3n.\n\nYo ahora estoy sereno,\n\ncon luz.\n\nY maya en mi Pac\u00edfico\n\nun n\u00e1ufrago ata\u00fad.\nOur Daily Bread  \n(For Spanish translation click here)\n\n_for Alejandro Gamboa_\n\nDrinks the breakfast . . . Humid earth\n\nof cemetery smells of loved blood.\n\nCity of winter . . . The scathing crusade\n\nof a cart that seems pulling\n\nemotions of fasting that cannot get free!\n\nWish I could knock all the doors,\n\nand ask for I don't know who; and then\n\nlook at the poor, and, while they wept softly,\n\ngive bits of fresh bread to all of them.\n\nAnd plunder the rich of the vineyards\n\nwith the two holy hands\n\nthat with one blow of light,\n\nflew away from the Cross!\n\nEyelash of morning do not rise!\n\nGive us this day our daily bread,\n\nLord . . . !\n\nAll my bones in me belong to others;\n\nand maybe I robbed them.\n\nI came to take something for myself that maybe\n\nwas meant for some other man;\n\nand I start thinking that, if I had not been born,\n\nanother poor man could drink this coffee.\n\nI am an evil thief . . . Where will I end!\n\nIn this frigid hour, when the earth\n\ntranscends the human dust and is so sad,\n\nI wish I could knock on all doors\n\nand beg pardon to I don't know who\n\nand make bits of fresh bread for him\n\nhere, in the oven of my heart . . . !\nEl pan nuestro  \n(For English translation click here)\n\n_para Alejandro Gamboa_\n\nSe bebe el desayuno . . . H\u00fameda tierra\n\nde cementerio huele a sangre amada.\n\nCiudad de invierno . . . La mordaz cruzada\n\nde una carreta que arrastrar parece\n\nuna emoci\u00f3n de ayuno encadenada!\n\nSe quisiera tocar todas las puertas,\n\ny preguntar por no s\u00e9 qui\u00e9n; y luego\n\nver a los pobres, y, llorando quedos,\n\ndar pedacitos de pan fresco a todos.\n\nY saquear a los ricos sus vi\u00f1edos\n\ncon las dos manos santas\n\nque a un golpe de luz\n\nvolaron desclavadas de la Cruz!\n\nPesta\u00f1a matinal, no os levant\u00e9is!\n\n\u00a1El pan nuestro de cada d\u00eda d\u00e1noslo,\n\nSe\u00f1or . . . !\n\nTodos mis huesos son ajenos;\n\nyo talvez los rob\u00e9!\n\nYo vine a darme lo que acaso estuvo\n\nasignado para otro;\n\ny pienso que, si no hubiera nacido,\n\notro pobre tomara este caf\u00e9!\n\nYo soy un mal ladr\u00f3n . . . A d\u00f3nde ir\u00e9!\n\nY en esta hora fr\u00eda, en que la tierra\n\ntrasciende a polvo humano y es tan triste,\n\nquisiera yo tocar todas las puertas,\n\ny suplicar a no s\u00e9 qui\u00e9n, perd\u00f3n,\n\ny hacerle pedacitos de pan fresco\n\naqu\u00ed, en el horno de mi coraz\u00f3n . . . !\nThe Eternal Dice  \n(For Spanish translation click here)\n\nMy God, I am weeping for the being I'm living;\n\nI am sorry to have taken your bread;\n\nbut this wretched thinking dough\n\nis not a crust leavened in your side,\n\nyou don't have Marias who are departing!\n\nMy God, if you had been man,\n\ntoday you would know how to be God,\n\nbut you, you've been always fine,\n\nyou feel nothing of your creation.\n\nAnd the man oh yes is suffering you: the God is him!\n\nToday, that there are candles in my magic eyes,\n\nlike in a condemned man,\n\nmy God, you will light all your lights,\n\nand we will play with the old die . . .\n\nPerhaps, oh player! in bringing the good luck\n\nof the entire universe,\n\nthe ringside eyes of Death will turn up,\n\nlike two grim aces of mud.\n\nMy God, in this still, dark night\n\nyou can't play anymore, because the Earth\n\nis already a die worn and smoothed out at the edges\n\nfrom rolling by chance,\n\nthat can only stop in a space,\n\nin the space of an immense sepulcher.\nLos dados eternos  \n(For English translation click here)\n\nDios m\u00edo, estoy llorando el ser que vivo;\n\nme pesa haber tom\u00e1dote tu pan;\n\npero este pobre barro pensativo\n\nno es costra fermentada en tu costado:\n\nt\u00fa no tienes Mar\u00edas que se van!\n\nDios m\u00edo, si t\u00fa hubieras sido hombre,\n\nhoy supieras ser Dios;\n\npero t\u00fa, que estuviste siempre bien,\n\nno sientes nada de tu creaci\u00f3n.\n\nY el hombre s\u00ed te sufre: el Dios es \u00e9l!\n\nHoy que en mis ojos brujos hay candelas,\n\ncomo en un condenado,\n\nDios m\u00edo, prender\u00e1s todas tus velas,\n\ny jugaremos con el viejo dado.\n\nTal vez \u00a1oh jugador! al dar la suerte\n\ndel universo todo,\n\nsurgir\u00e1n las ojeras de la Muerte,\n\ncomo dos ases f\u00fanebres de lodo.\n\nDios m\u00edos, y esta noche sorda, oscura,\n\nya no podr\u00e1s jugar, porque la Tierra\n\nes un dado ro\u00eddo y ya redondo\n\na fuerza de rodar a la aventura,\n\nque no puede parar sino en un hueco,\n\nen el hueco de inmensa sepultura.\nThe Weary Circles  \n(For Spanish translation click here)\n\nThere are desires to return, to love, not to go away,\n\nand there are desires to die, fought by two\n\ncontrary waters that will never become isthmus.\n\nThere are desires for a kiss that would shroud life,\n\nthat withers in Africa of a fiery agony,\n\nsuicide!\n\nThere are desires to . . . to not have desires. Lord;\n\nat you I point my god murdering finger.\n\nThere are desires not to have had a heart at all.\n\nSpring returns, returns to go away once again. And God,\n\ncurved in time, repeats himself passing,\n\npassing with the spinal cord of the Universe on his shoulder.\n\nWhen my temples bent their mournful drum,\n\nwhen the dream etched on a knife is hurting me,\n\nthere are desires not to move on an inch from this poem!\nLos anillos fatigados  \n(For English translation click here)\n\nHay ganas de volver, de amar, de no ausentarse,\n\ny hay ganas de morir, combatido por dos\n\naguas encontradas que jam\u00e1s han de istmarse.\n\nHay ganas: de un gran beso que amortaje a la Vida,\n\nque acaba en el \u00e1frica de una agon\u00eda ardiente,\n\nsuicida!\n\nHay ganas de . . . no tener ganas. Se\u00f1or;\n\na ti yo te se\u00f1alo con el dedo deicida:\n\nhay ganas de no haber tenido coraz\u00f3n.\n\nLa primavera vuelve, vuelve y se ir\u00e1. Y Dios,\n\ncurvado en tiempo, se repite, y pasa, pasa\n\na cuestas con la espina dorsal del Universo.\n\nCuando las sienes tocan su l\u00fagubre tambor,\n\ncuando me duele el sue\u00f1o grabado en un pu\u00f1al,\n\nhay ganas de quedarse plantado en este verso!\nThe Distant Footsteps  \n(For Spanish translation click here)\n\nMy father sleeps. His noble face\n\nshows a mild heart within;\n\nhe's so sweet now . . .\n\nif there's anything bitter within him, it's me.\n\nThere's a loneliness in the living; they are praying;\n\nand there's no news of the children today.\n\nMy father wakes, he listens\n\nthe flight into Egypt, the staunched goodbye.\n\nNow he's so near;\n\nif there's anything distant within him, it's me.\n\nMy mother walks in the orchard,\n\nsavoring a taste already without savor.\n\nNow she's so gentle,\n\nso much nervy, so much rakish, so much love.\n\nThere is loneliness in the living without sound,\n\nwithout news, without greenness, without childhood.\n\nAnd if there's something broken this afternoon,\n\nand descends and creaks\n\nit's two old roads, curving and white.\n\nDown them my heart walks on foot.\nLos pasos lejanos  \n(For English translation click here)\n\nMi padre duerme. Su semblante augusto\n\nfigura un apacible coraz\u00f3n;\n\nest\u00e1 ahora tan dulce . . .\n\nsi hay algo en \u00e9l de amargo, ser\u00e9 yo.\n\nHay soledad en el hogar; se reza;\n\ny no hay noticias de los hijos hoy.\n\nMi padre se despierta, ausculta\n\nla huida a Egipto, el resta\u00f1ante adi\u00f3s.\n\nEst\u00e1 ahora tan cerca;\n\nsi hay algo en \u00e9l de lejos, ser\u00e9 yo.\n\nY mi madre pasea all\u00e1 en los huertos,\n\nsaboreando un sabor ya sin sabor.\n\nEst\u00e1 ahora tan suave,\n\ntan ala, tan salida, tan amor.\n\nHay soledad en el hogar sin bulla,\n\nsin noticias, sin verde, sin ni\u00f1ez.\n\nY si hay algo quebrado en esta tarde,\n\ny que baja y que cruje,\n\nson dos viejos caminos blancos, curvos.\n\nPor ellos va mi coraz\u00f3n a pie.\nTo My Brother Miguel  \n(For Spanish translation click here)\n\n_In memoriam_\n\nBrother, today I sit on the stone bench by our home\n\nwhere we miss you terribly!\n\nI remember we used to play at this hour, and Mama\n\nwould hug us: \"But, my sons . . . \"\n\nNow I hide,\n\nand as before, from all evening\n\nprayers, and I trust you won't give me away.\n\nThrough the parlor, the vestibule, the corridors.\n\nLater, you hide, and I can't find you.\n\nI remember that we made each other cry,\n\nbrother, in that game.\n\nMiguel, you hid yourself\n\none August night, just before dawn;\n\nbut, instead of laughing when you hid, you were sad.\n\nAnd your twin heart of those now extinct\n\nafternoons has grown weary of not finding you. And now\n\na shadow falls in the soul.\n\nListen, brother, don't take so long\n\ncoming out. All right? Mama might worry.\nA mi hermano Miguel  \n(For English translation click here)\n\n_In memoriam_\n\nHermano, hoy estoy en el poyo de la casa,\n\ndonde nos haces una falta sin fondo!\n\nMe acuerdo que jug\u00e1bamos esta hora, y que mam\u00e1\n\nnos acariciaba: \u00abPero, hijos . . . \u00bb\n\nAhora yo me escondo,\n\ncomo antes, todas estas oraciones\n\nvespertinas, y espero que t\u00fa no des conmigo.\n\nPor la sala, el zagu\u00e1n, los corredores.\n\nDespu\u00e9s, te ocultas t\u00fa, y yo no doy contigo.\n\nMe acuerdo que nos hac\u00edamos llorar,\n\nhermano, en aquel juego.\n\nMiguel, t\u00fa te escondiste\n\nuna noche de agosto, al alborear;\n\npero, en vez de ocultarte riendo, estabas triste.\n\nY tu gemelo coraz\u00f3n de esas tardes\n\nextintas se ha aburrido de no encontrarte. Y ya\n\ncae sombra en el alma.\n\nOye, hermano, no tardes\n\nen salir. Bueno? Puede inquietarse mam\u00e1.\nFilled with January  \n(For Spanish translation click here)\n\nIn bird morning,\n\nmy father, with difficulty, puts\n\nhis seventy-eight years, his seventy-eight\n\nwinter branches out in the sun.\n\nThe cemetery of Santiago seen at a glance\n\nanointed in a happy new year.\n\nHow many times his steps cut toward it,\n\nreturning from some sad and humble burial.\n\nFor a long time now my father hasn't left the house!\n\nA joke of children is dispersed.\n\nOther times he used to speak to my mother\n\nabout urban impressions, politics;\n\nand today, leaning on his illustrious cane\n\nhaving a better ring to it during the years of the government,\n\nmy father looks unknown, fragile,\n\nmy father is an eve of.\n\nAbsentmindedly he carries, keeps with him, relics,\n\nthings, memories, suggestions.\n\nThe affable morning accompanies him\n\nwith its white wings of a sister of charity.\n\nThis is an eternal day, ingenuous day, infant,\n\nsharp, day of prayers,\n\ntime crowned with doves,\n\nthe future is peopled\n\nwith caravans of immortal roses.\n\nFather, all is wide awake still;\n\nis January who sings, is your love\n\nwho is resonating goes to Eternity.\n\nYou still laugh at your babies,\n\nand will be triumphal noise in the Voids.\n\nStill be new year. There will be meat pies,\n\nand I'll be hungry, when the call to Mass is sounded\n\nin the blessed bell tower\n\nby the good, lyrical blindfolded man with whom\n\nmy syllables scholarly and fresh departed,\n\nin my rotund innocence.\n\nAnd when morning full of grace\n\nfrom its bosoms of time\n\nthat are two renunciations, two advances of love\n\nstretching out, imploring the infinite, eternal life,\n\nsings, and begins to fly plural Verbs,\n\npennants of your being,\n\non the sail of her white wings\n\nof a sister of charity, oh my father!\nEn\u00e9reida  \n(For English translation click here)\n\nMi padre, apenas,\n\nen la ma\u00f1ana pajarina, pone\n\nsus setentiocho a\u00f1os, sus setentiocho\n\nramos de invierno a solear.\n\nEl cementerio de Santiago, untado\n\nen alegre a\u00f1o nuevo, est\u00e1 a la vista.\n\nCu\u00e1ntas veces sus pasos cortaron hacia \u00e9l,\n\ny tornaron de alg\u00fan entierro humilde.\n\nHoy hace mucho tiempo que mi padre no sale!\n\nUna broma de ni\u00f1os se desbanda.\n\nOtras veces le hablaba a mi madre\n\nde impresiones urbanas, de pol\u00edtica;\n\ny hoy, apoyado en su bast\u00f3n ilustre\n\nque sonara mejor en los a\u00f1os de la Gobernaci\u00f3n,\n\nmi padre est\u00e1 desconocido, fr\u00e1gil,\n\nmi padre es una v\u00edspera.\n\nLleva, trae, abstra\u00eddo, reliquias, cosas,\n\nrecuerdos, sugerencias.\n\nLa ma\u00f1ana apacible le acompa\u00f1a\n\ncon sus alas blancas de hermana de la caridad.\n\nD\u00eda eterno es \u00e9ste, d\u00eda ingenuo, infante\n\ncoral, oracional;\n\nse corona el tiempo de palomas,\n\ny el futuro se puebla\n\nde caravanas de inmortales rosas.\n\nPadre, a\u00fan sigue todo despertando;\n\nes enero que canta, es tu amor\n\nque resonando va en la Eternidad.\n\nA\u00fan reir\u00e1s de tus peque\u00f1uelos,\n\ny habr\u00e1 bulla triunfal en los Vac\u00edos.\n\nA\u00fan ser\u00e1 a\u00f1o nuevo. Habr\u00e1 empanadas;\n\ny yo tendr\u00e9 hambre, cuando toque a misa\n\nen el beato campanario\n\nel buen ciego m\u00e9lico con quien\n\ndepartieron mis s\u00edlabas escolares y frescas,\n\nmi inocencia rotunda.\n\nY cuando la ma\u00f1ana llena de gracia,\n\ndesde sus senos de tiempo\n\nque son dos renuncias, dos avances de amor\n\nque se tienden y ruegan infinito, eterna vida,\n\ncante, y eche a volar Verbos plurales,\n\njirones de tu ser,\n\na la borda de sus alas blancas\n\nde hermana de la caridad, \u00a1oh, padre m\u00edo!\nI Was Born on a Day God Was Sick  \n(For Spanish translation click here)\n\nI was born\n\non a day God was sick.\n\nThey all know I live,\n\nthat I'm bad, and they don't know\n\nabout the December that follows from that January.\n\n'Cause I was born\n\non a day God was sick.\n\nThere is an empty place\n\nin my metaphysical shape\n\nthat no one can reach:\n\nthe cloister of silence\n\nspeaking with the muffled voice of its fire.\n\nI was born\n\non a day God was sick.\n\nBrother, listen to me, listen . . .\n\nOh, all right. Don't worry, I won't leave\n\nwithout taking Decembers along,\n\nwithout leaving Januaries behind.\n\nI was born\n\non a day God was sick.\n\nThey all know I'm alive,\n\nthat I chew my food . . . And they don't know\n\nwhy in my verses creaks,\n\nthe dark uneasiness\n\nof a coffin,\n\ndisentangled winds unscrewed from the Sphinx\n\ninquisitive of the Desert.\n\nYes, they all know . . . And they don't know\n\nthe light getting skinny,\n\nand the Shadow is fat . . .\n\nAnd they don't know Mystery joins things together . . .\n\nthat he is hunchbacked,\n\nmusical, sad, standing a little way off and foretells\n\nthe dazzling progression from the limits to the Limits.\n\nI was born\n\non a day God was sick.\n\nGravely.\nEspergesia  \n(For English translation click here)\n\nYo nac\u00ed un d\u00eda\n\nque Dios estuvo enfermo.\n\nTodos saben que vivo,\n\nque soy malo; y no saben\n\ndel diciembre de ese enero.\n\nPues yo nac\u00ed un d\u00eda\n\nque Dios estuvo enfermo.\n\nHay un vac\u00edo\n\nen mi aire metaf\u00edsico\n\nque nadie ha de palpar:\n\nel claustro de un silencio\n\nque habl\u00f3 a flor de fuego.\n\nYo nac\u00ed un d\u00eda\n\nque Dios estuvo enfermo.\n\nHermano, escucha, escucha . . .\n\nBueno. Y que no me vaya\n\nsin llevar diciembres,\n\nsin dejar eneros.\n\nPues yo nac\u00ed un d\u00eda\n\nque Dios estuvo enfermo.\n\nTodos saben que vivo,\n\nque mastico . . . Y no saben\n\npor qu\u00e9 en mi verso chirr\u00edan,\n\noscuro sinsabor de f\u00e9retro,\n\nluyidos vientos\n\ndesenroscados de la Esfinge\n\npreguntona del Desierto.\n\nTodos saben . . . Y no saben\n\nque la luz es t\u00edsica,\n\ny la Sombra gorda . . .\n\nY no saben que el Misterio sintetiza . . .\n\nque \u00e9l es la joroba\n\nmusical y triste que a distancia denuncia\n\nel paso meridiano de las lindes a las Lindes.\n\nYo nac\u00ed un d\u00eda\n\nque Dios estuvo enfermo,\n\ngrave.\n\n_C\u00e9sar Vallejo and Georgette, taking a walk  \nin Madrid, 1931. Beside them is the   \npoet Rafael Alberti._\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nTRILCE\n\n_1922_\nIII  \n(For Spanish translation click here)\n\nWhat time are the grown-ups\n\ngetting back?\n\nBlind Santiago strikes six\n\nand already darkness takes hold.\n\nMother said she wouldn't delay.\n\nAguedita, Nativa, Miguel,\n\nbe careful of going over there, where\n\ntheir doubled-up memories just passed\n\nsnuggling\n\ntoward the silent corral, and whereby\n\nthe hens settle for the night\n\nthey have frightened a lot.\n\nWe'd better just stay here.\n\nMother said she wouldn't delay.\n\nBesides, we shouldn't be sad. Let's go on seeing\n\nthe boats! (mine's the prettiest of the toy fleet!)\n\nwhich we've played the whole blessed day,\n\nwithout fighting among ourselves, as it should be:\n\nthey stayed behind in the puddle, all ready,\n\nloaded with sweet things for tomorrow.\n\nLet's obediently wait, there's no choice but,\n\nfor the homecoming, the relief of\n\nthe adults, who are always the first\n\nto abandon us small ones in the house,\n\nas if we couldn't go out on our own!\n\nAguedita, Nativa, Miguel?\n\nI call to you, feeling around for you in that very same darkness.\n\nDon't leave me behind by myself,\n\nto be the only recluse locked in all alone.\nIII  \n(For English translation click here)\n\nLas personas mayores\n\n\u00bfa qu\u00e9 hora volver\u00e1n?\n\nDa las seis el ciego Santiago,\n\ny ya est\u00e1 muy oscuro.\n\nMadre dijo que no demorar\u00eda.\n\nAguedita, Nativa, Miguel,\n\ncuidado con ir por ah\u00ed, por donde\n\nacaban de pasar gangueando sus memorias\n\ndobladoras penas,\n\nhacia el silencioso corral, y por donde\n\nlas gallinas que se est\u00e1n acostando todav\u00eda,\n\nse han espantado tanto.\n\nMejor estemos aqu\u00ed no m\u00e1s.\n\nMadre dijo que no demorar\u00eda.\n\nYa no tengamos pena. Vamos viendo\n\nlos barcos \u00a1el m\u00edo es m\u00e1s bonito de todos!\n\ncon los cuales jugamos todo el santo d\u00eda,\n\nsin pelearnos, como debe de ser:\n\nhan quedado en el pozo de agua, listos,\n\nfletados de dulces para ma\u00f1ana.\n\nAguardemos as\u00ed, obedientes y sin m\u00e1s\n\nremedio, la vuelta, el desagravio\n\nde los mayores siempre delanteros\n\ndej\u00e1ndonos en casa a los peque\u00f1os,\n\ncomo si tambi\u00e9n nosotros no pudi\u00e9semos partir.\n\nAguedita, Nativa, Miguel?\n\nLlamo, busco al tanteo en la oscuridad.\n\nNo me vayan a haber dejado solo,\n\ny el \u00fanico recluso sea yo.\nXIV  \n(For Spanish translation click here)\n\nMy explanation exactly.\n\nIt's hurts me, because it's so premature.\n\nThis business of tightrope walking.\n\nThose brave beasts staring as if they saw themselves unnatural.\n\nThat amalgam sticking the quicksilver to the inside.\n\nThese buttocks sitting up.\n\nThis cannot be! But it is!\n\nAbsurdity!\n\nMadness!\n\nBut I have come from Trujillo to Lima.\n\nYet I earn only a wage worth five _soles_.*\n\n_*Peruvian currency_\nXIV  \n(For English translation click here)\n\nCual mi explicaci\u00f3n.\n\nEsto me lacera la tempran\u00eda.\n\nEsta manera de caminar por los trapecios.\n\nEsos corajosos brutos como postizos.\n\nEsa goma que pega el azogue al adentro.\n\nEsas posaderas sentadas para arriba.\n\nEse no puede ser, sido.\n\nAbsurdo.\n\nDemencia.\n\nPero he venido de Trujillo a Lima.\n\nPero gano un sueldo de cinco soles.\nXV  \n(For Spanish translation click here)\n\nIn that corner we sleep together\n\nso many nights, now I'm sitting there\n\nto walk. The bed of one-time lovers\n\nwas pushed aside, or whatever has happened.\n\nYou've come early today to deal with other issues\n\nand you're not here. It was in this corner\n\nwhere one night between your tender breasts\n\nI read beside you\n\na tale of Daudet's. This is the beloved\n\ncorner. Don't deny it.\n\nI've set myself to recording the days\n\nof that summer long past, your coming and going\n\nsmall and brave and pale though these rooms.\n\nOn this night of rain\n\ndropping so far removed from us. I suddenly leap up! . . .\n\nThere are two doors opening and closing,\n\ntwo doors that come and go in the wind\n\nshadow to shadow.\nXV  \n(For English translation click here)\n\nEn el rinc\u00f3n aquel, donde dormimos juntos\n\ntantas noches, ahora me he sentado\n\na caminar. La cuja de los novios difuntos\n\nfue sacada, o talvez qu\u00e9 habr\u00e1 pasado.\n\nHas venido temprano a otros asuntos,\n\ny ya no est\u00e1s. Es el rinc\u00f3n\n\ndonde a tu lado, le\u00ed una noche,\n\nentre tus tiernos puntos,\n\nun cuento de Daudet. Es el rinc\u00f3n\n\namado. No lo equivoques.\n\nMe he puesto a recordar los d\u00edas\n\nde verano idos, tu entrar y salir,\n\npoca y harta y p\u00e1lida por los cuartos.\n\nEn esta noche pluviosa,\n\nya lejos de ambos dos, salto de pronto . . .\n\nSon dos puertas abri\u00e9ndose cerr\u00e1ndose,\n\ndos puertas que al viento van y vienen\n\nsombra a sombra.\nXVI  \n(For Spanish translation click here)\n\nI have faith in being strong.\n\nDepleted air, set me free, let me go,\n\ndecorate my left side with zeros,\n\nand you, dream, surrender your unyielding diamond\n\nyour timeless demand.\n\nYes, I have faith in being strong.\n\nOver there goes a hollow woman,\n\nas in a colorless quantity, whose\n\ngrace closes within when I open my heart.\n\nIn the street an ancient friar walks, dull crabs\n\nadmire the green banner of the president\n\ntopping the other six banners\n\nall the bunting of the return.\n\nI have faith that I am,\n\nand I have been less.\n\nBehold! The first good one!\nXVI  \n(For English translation click here)\n\nTengo fe en ser fuerte.\n\nDame, aire manco, dame ir\n\ngalone\u00e1ndome de ceros a la izquierda.\n\nY t\u00fa, sue\u00f1o, dame tu diamante implacable,\n\ntu tiempo de deshora.\n\nTengo fe en ser fuerte.\n\nPor all\u00ed avanza c\u00f3ncava mujer,\n\ncantidad incolora, cuya\n\ngracia se cierra donde me abro.\n\nAl aire, fray pasado. Cangrejos, zote!\n\nAv\u00edstate la verde bandera presidencial,\n\narriando las seis banderas restantes,\n\ntodas las colgaduras de la vuelta.\n\nTengo fe en que soy,\n\ny en que he sido menos.\n\nEa! Buen primero!\nXVIII  \n(For Spanish translation click here)\n\nOh the four walls of the cell.\n\nAh the four bleaching walls\n\nthat open without fail the same number.\n\nNursery of nerves, crooked dice,\n\nhow its four corners wrench\n\nat the daily shackled extremities.\n\nAmorous mistress of innumerable keys\n\nif only you were here, if you could see\n\nwhat hour these four walls are\n\nwithout closing in. Against them we would be,\n\nwith you, two, two more than ever.\n\nYou would not cry, my liberator!\n\nAh the walls of the cell.\n\nThey hurt me, most of all\n\nthe two long ones that tonight\n\nremind me a bit of mothers now dead\n\nupon bromine slopes\n\nleading a child by the hand each one.\n\nI find only myself left behind,\n\nwith my right hand, serving for both,\n\nlifting in search of a third arm\n\nhousing, between my where and my when,\n\nthis futile manhood of mine.\nXVIII  \n(For English translation click here)\n\nOh las cuatro paredes de la celda.\n\nAh las cuatro paredes albicantes\n\nque sin remedio dan al mismo n\u00famero.\n\nCriadero de nervios, mala brecha,\n\npor sus cuatro rincones c\u00f3mo arranca\n\nlas diarias aherrojadas extremidades.\n\nAmorosa llavera de innumerables llaves,\n\nsi estuvieras aqu\u00ed, si vieras hasta\n\nqu\u00e9 hora son cuatro estas paredes.\n\nContra ellas ser\u00edamos contigo, los dos,\n\nm\u00e1s dos que nunca. Y ni lloraras,\n\ndi, libertadora!\n\nAh las paredes de la celda.\n\nDe ellas me duele entretanto, m\u00e1s\n\nlas dos largas que tienen esta noche\n\nalgo de madres que ya muertas\n\nllevan por bromurados declives,\n\na un ni\u00f1o de la mano cada una.\n\nY s\u00f3lo yo me voy quedando,\n\ncon la diestra, que hace por ambas manos,\n\nen alto, en busca de terciario brazo\n\nque ha de pupilar, entre mi d\u00f3nde y mi cu\u00e1ndo,\n\nesta mayor\u00eda inv\u00e1lida de hombre.\nXXXIII  \n(For Spanish translation click here)\n\nIf it rained tonight I would retire\n\nfrom here to a thousand years.\n\nOr better just a hundred, no more,\n\nas if nothing had happened, I should imagine\n\nthat I'm still to come.\n\nOh, motherless and loveless, without an urge\n\nto squat down and loom into the very depths by pure\n\nstrength,\n\ntonight, like this, I should be disentangling\n\nthe Vedic fiber,\n\nthe Vedic wool of my final end, thread\n\nof the devil, the twisting\n\nmark of having held by the nose\n\ntwo jangling clappers of time\n\nin one single bell.\n\nDo the math of my life,\n\nor do the math of yourself still not born yet,\n\nI shall not succeed in freeing myself.\n\nIt will not be what has not yet come, but what has\n\narrived and already gone, but what has\n\narrived and already gone.\nXXXIII  \n(For English translation click here)\n\nSi lloviera esta noche, retirar\u00edame\n\nde aqu\u00ed a mil a\u00f1os.\n\nMejor a cien no m\u00e1s.\n\nComo si nada hubiese ocurrido, har\u00eda\n\nla cuenta de que vengo todav\u00eda.\n\nO sin madre, sin amada, sin porf\u00eda\n\nde agacharme a aguaitar al fondo, a puro\n\npulso,\n\nesta noche as\u00ed, estar\u00eda escarmenando\n\nla fibra v\u00e9dica,\n\nla lana v\u00e9dica de mi fin final, hilo\n\ndel diantre, traza de haber tenido\n\npor las narices\n\na dos badajos inacordes de tiempo\n\nen una misma campana.\n\nHaga la cuenta de mi vida\n\no haga la cuenta de no haber a\u00fan nacido\n\nno alcanzar\u00e9 a librarme.\n\nNo ser\u00e1 lo que a\u00fan no haya venido, sino\n\nlo que ha llegado y ya se ha ido,\n\nsino lo que ha llegado y ya se ha ido.\nXLV  \n(For Spanish translation click here)\n\nI am freed from the chains of the sea\n\nwhen the tide reaches me.\n\nLet's sail out forever. Let's taste\n\nthe stupendous song, the song spoken\n\nby the longer lips of desire.\n\nOh prodigious virginity.\n\nThe saltless breeze passes.\n\nFrom afar I take in the wind of the marrows,\n\nhearing the profound score, as the surf\n\nhunts for its keys.\n\nAnd if we happen to meet suddenly\n\nwith the absurd,\n\nwe shall cover ourselves with the gold of owning nothing,\n\nand hatch the still unborn wing\n\nof the night, sister\n\nto this orphaned wing of the day\n\nwhose strength is no longer a wing.\nXLV  \n(For English translation click here)\n\nMe desvinculo del mar\n\ncuando vienen las aguas a m\u00ed.\n\nSalgamos siempre. Saboreemos\n\nla canci\u00f3n estupenda, la canci\u00f3n dicha\n\npor los labios inferiores del deseo.\n\nOh prodigiosa doncellez.\n\nPasa la brisa sin sal.\n\nA lo lejos husmeo los tu\u00e9tanos\n\noyendo el tanteo profundo, a la caza\n\nde teclas de resaca.\n\nY si as\u00ed di\u00e9ramos las narices\n\nen el absurdo,\n\nnos cubriremos con el oro de no tener nada,\n\ny empollaremos el ala a\u00fan no nacida\n\nde la noche, hermana\n\nde esta ala hu\u00e9rfana del d\u00eda,\n\nque a fuerza de ser una ya no es ala.\nLXI  \n(For Spanish translation click here)\n\nI get down from the horse tonight,\n\nat the door of the house, where\n\nat cockcrow took my leave.\n\nIt's locked and nobody answers.\n\nStone bench on which mother gave birth to\n\nmy older brother, so that he might saddle up\n\nloins I had ridden bareback through village\n\nroads and past garden walls, a child of the village;\n\nthe bench on which I left behind me the sun\n\nlight of my painful childhood . . . And what of\n\nthis pain that frames the entrance?\n\nA god in alien peace,\n\nsneezing, like calling also, the brute,\n\nsniff, striking the pavement. And then, hesitate\n\nit neighs,\n\ntwitching its alert ears.\n\nFather must be awake praying, and perhaps\n\nwith thoughts about my being out late.\n\nMy sisters who hum their illusions,\n\nsimple but noisy,\n\nin their work for the oncoming feast,\n\nand now almost nothing is wanting.\n\nI wait, I wait, the heart\n\nan egg that in its right moment obstructs itself.\n\nNumerous family that we left recently,\n\nthey're still awake and not one candle set\n\non the altar for our homecoming.\n\nI call again and nothing,\n\nwe shut up and we start to sob, and the animal\n\nneighs, neighs more and more.\n\nThey are asleep forever,\n\nthey're so fine, that finally\n\nmy horse becomes weary when turning\n\nhis head, and in half sleep, in each greeting, says\n\nthat he's alright, that everything is alright.\nLXI  \n(For English translation click here)\n\nEsta noche desciendo del caballo,\n\nante la puerta de la casa, donde\n\nme desped\u00ed con el cantar del gallo.\n\nEst\u00e1 cerrada y nadie responde.\n\nEl poyo en que mam\u00e1 alumbr\u00f3\n\nal hermano mayor, para que ensille\n\nlomos que hab\u00eda yo montado en pelo,\n\npor r\u00faas y por cercas, ni\u00f1o aldeano;\n\nel poyo en que dej\u00e9 que se amarille al sol\n\nmi adolorida infancia . . . \u00bfY este duelo\n\nque enmarca la portada?\n\nDios en la paz for\u00e1nea,\n\nestornuda, cual llamando tambi\u00e9n, el bruto;\n\nhusmea, golpeando el empedrado. Luego duda\n\nrelincha,\n\norejea a viva oreja.\n\nHa de velar pap\u00e1 rezando, y quiz\u00e1s\n\npensar\u00e1 se me hizo tarde.\n\nLas hermanas, canturreando sus ilusiones\n\nsencillas, bullosas,\n\nen la labor para la fiesta que se acerca,\n\ny ya no falta casi nada.\n\nEspero, espero, el coraz\u00f3n\n\nun huevo en su momento, que se obstruye.\n\nNumerosa familia que dejamos\n\nno ha mucho, hoy nadie en vela, y ni una cera\n\npuso en el ara para que volvi\u00e9ramos.\n\nLlamo de nuevo, y nada.\n\nCallamos y nos ponemos a sollozar, y el animal\n\nrelincha, relincha m\u00e1s todav\u00eda.\n\nTodos est\u00e1n durmiendo para siempre,\n\ny tan de lo m\u00e1s bien, que por fin\n\nmi caballo acaba fatigado por cabecear\n\na su vez, y entre sue\u00f1os, a cada venia, dice\n\nque est\u00e1 bien, que todo est\u00e1 muy bien.\nLXIII  \n(For Spanish translation click here)\n\nIt dawned raining. The well-combed\n\nmorning drips its fine hair.\n\nMelancholy is moored;\n\nand in badly tarred oxidant of hind\u00fa furniture,\n\ndestiny heaves about, barely able to keep its seat.\n\nFlatland skies, disheartened\n\nby great love, the platinum skies,\n\nimpossibly grim.\n\nThe sheepfold ruminates, underscored\n\nby an Andean neighing.\n\nI remember about myself. But masts of wind\n\nare enough, rudders quiet until\n\nthey become one,\n\nand the cricket of tedium and the gibbons unbreakable elbow.\n\nLast of the mornings of freed long-haired poets\n\nof precious pitch mountainous bucolic poems,\n\nwhen I go out in search of the eleven\n\nand it's nothing but an untimely twelve.\nLXIII  \n(For English translation click here)\n\nAmanece lloviendo. Bien peinada\n\nla ma\u00f1ana chorrea el pelo fino.\n\nMelancol\u00eda est\u00e1 amarrada;\n\ny en mal asfaltado oxidente de muebles hind\u00faes,\n\nvira, se asienta apenas el destino.\n\nCielos de puna descorazonada\n\npor gran amor, los cielos de platino, torvos\n\nde imposible.\n\nRumia la majada y se subraya\n\nde un relincho andino.\n\nMe acuerdo de m\u00ed mismo. Pero bastan\n\nlas astas del viento, los timones quietos hasta\n\nhacerse uno,\n\ny el grillo del tedio y el jiboso codo inquebrantable.\n\nBasta la ma\u00f1ana de libres crinejas\n\nde brea preciosa, serrana,\n\ncuando salgo y busco las once\n\ny no son m\u00e1s que las doce deshoras.\nLXVl  \n(For Spanish translation click here)\n\nNovember 2nd turns.\n\nThese chairs are a good place of refuge.\n\nThe bough of foreboding comes and goes,\n\nrises and sweating, sways,\n\nweary in this room.\n\nNovember 2nd sadly turns.\n\nDead men, how deep your vanished teeth cut,\n\nre-examining the blind exposed nerves,\n\njangling in the root of a tooth throbbing that needs to be pulled,\n\nremindful of the tough fabric\n\nthat stout singing workers mend with unfinished hemp\n\nof innumerable knots beating crossroads.\n\nYou, dead, with clear pure knees\n\nfrom self surrender,\n\nhow you hack at another's heart\n\nwith your white crowns, sparing\n\nof tenderness. Yes. You, the decayed.\n\nNovember 2nd sadly turns.\n\nAnd the bough of foreboding\n\nis bitten by a cart that simply\n\nrolls in the street.\nLXVI  \n(For English translation click here)\n\nDobla el dos de Noviembre.\n\nEstas sillas son buenas acojidas.\n\nLa rama del presentimiento\n\nva, viene, sube, ondea sudorosa,\n\nfatigada en esta sala.\n\nDobla triste el dos de Noviembre.\n\nDifuntos, qu\u00e9 bajo cortan vuestros dientes\n\nabolidos, repasando ciegos nervios,\n\nsin recordar la dura fibra\n\nque cantores obreros redondos remiendan\n\ncon c\u00e1\u00f1amo inacabable, de innumerables nudos\n\nlatientes de encrucijada.\n\nVosotros, difuntos, de las n\u00edtidas rodillas\n\npuras a fuerza de entregaros,\n\nc\u00f3mo aserr\u00e1is el otro coraz\u00f3n\n\ncon vuestras blancas coronas, ralas\n\nde cordialidad. S\u00ed. Vosotros, difuntos.\n\nDobla triste el dos de Noviembre.\n\nY la rama del presentimiento\n\nse la muerde un carro que simplemente\n\nrueda por la calle.\nLXXV  \n(For Spanish translation click here)\n\nYou are dead.\n\nWhat a strange way to be dead. Anyone would say that\n\nyou are not. But, truthfully, you are dead.\n\nYou float just behind an aqueous membrane, hanging\n\nfrom one zenith to the opposite, nothing, coming and\n\ngoing from twilight to dawn, vibrating before the cithern\n\nbox of a wound that does not cause you pain. You say,\n\nwell, that life passes in a mirror and that you are the\n\noriginal, you are the dead.\n\nMeanwhile the waves go, meanwhile the wave comes, how\n\nis one dead without being punished. Only when the waters\n\nbreak on the beach, and they break again and again, then\n\nyou lose form and believing you are dying, you perceive\n\nthe sixth cord that now is not yours.\n\nYou are dead, without living before. Anyone would say that\n\nnot being now, you were in another time. But, truthfully\n\nyou are the skeleton of a life that never was. Sad destiny.\n\nYou have never been anything but dead. Like being a dry\n\nleaf never having been green. Orphans of orphanages.\n\nAnd, nevertheless, the dead are not, they cannot be\n\nskeletons of a life never lived. They always die of life.\n\nYou are dead.\nLXXV  \n(For English translation click here)\n\nEsta\u00eds muertos.\n\nQu\u00e9 extra\u00f1a manera de estarse muertos. Quienquiera dir\u00eda\n\nque no lo est\u00e1is. Pero en verdad, esta\u00eds muertos.\n\nFlot\u00e1is nadamente detr\u00e1s de aquesa membrana que,\n\np\u00e9ndula del zenit al nadir, viene y va de crep\u00fasculo a crep\u00fasculo,\n\nvibrando ante la sonora caja de una herida\n\nque a vosotros no os duele. Os digo, pues, que la vida\n\nest\u00e1 en el espejo, y que vosotros sois el original, la\n\nmuerte.\n\nMientras la onda va, mientras la onda viene, cu\u00e1n\n\nimpunemente se est\u00e1 uno muerto. S\u00f3lo cuando las aguas\n\nse quebrantan en los bordes enfrentados y se doblan y\n\ndoblan, entonces os transfigur\u00e1is y creyendo morir, percib\u00eds\n\nla sexta cuerda que ya no es vuestra.\n\nEst\u00e1is muertos, no habiendo antes vivido jam\u00e1s.\n\nQuienquiera dir\u00eda que, no siendo ahora, en otro tiempo\n\nfuisteis. Pero en verdad, vosotros sois los cad\u00e1veres\n\nde una vida que nunca fue. Triste destino el no haber\n\nsido sino muertos siempre. El ser hoja seca sin haber\n\nsido verde jam\u00e1s. Orfandad de orfandades.\n\nY, sin embargo, los muertos no son, no pueden ser\n\ncad\u00e1veres de una vida que todav\u00eda no han vivido. Ellos\n\nmurieron siempre de vida.\n\nEst\u00e1is muertos.\n\n_The Peruvian poets, C\u00e9sar Vallejo and Ernesto More. Paris, 1926_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nPOEMAS EN PROSA\n\n_1923\/1924\u20131929_\nThe Good Sense  \n(For Spanish translation click here)\n\nThere is, mother, a place in the world they call Paris.  \nIt's a huge place and far away and again very big.\n\nMy mother adjusts the collar of my coat, not because  \nit will snow, but in order that it may start.\n\nMy father's wife is in love with me, pushing and advancing my shoulders when I was born and my breast when I die. I am hers twice: for the departure and the return. She encloses me at the return. For this her eyes give me so much, close to me, fragments of me, happening by works now finished, by consummate pacts.\n\nMy mother confesses to me, my namesake. Why does she not give so much to my other brothers? To Victor, for example, the oldest who is so old now, that people say: \"He seems like he's his mother's youngest brother!\" Perhaps it might be because I have traveled so much! It must be because I have lived so much more!\n\nMy mother remembers me the first letter relating the return. Before my life of return, remembering that I journeyed in two hearts through her womb, she blushed and was left mortally livid, when I said, in the treaty of the soul: that night was happy. But, she seems all the more sad. She might have become even sadder.\n\n\u2014Son, how old you seem!\n\nAnd through the color yellow she walks firmly and cries because I seem old in her eyes, in the leaf of the sword, in the mouth of my face. She cries for me, she is sad for me. What difference will my youthfulness make if I will always be her son? Why do mothers feel much pain at having found their sons looking old, if the age of them will never equate or pass that of their mothers? And, why, if the sons the more they get on in their years moreover resemble their fathers? My mother cries because I am old in my time, and because I will never get old enough to be old in hers of my own accord!\n\nMy goodbye took a part of her being, more external than that part of her being when I returned. I am, on account of the excessive time-limit of my return, more the man to my mother than the son to my mother. There resides the candor and purity that lights us both with three flames. Then I say to her until I fall silent:\n\n\u2014There is, mother, a place in the world that they call Paris. It's a huge place and far away and again very big.\n\nThe woman of my father, upon hearing me, continues eating her lunch and her mortal eyes travel down my arm slowly.\nEl buen sentido  \n(For English translation click here)\n\nHay, madre, un sitio en el mundo, que se llama Par\u00eds. Un sitio muy grande y lejano y otra vez grande.\n\nMi madre me ajusta el cuello del abrigo, no porque empieza a nevar, sino para que empiece a nevar.\n\nLa mujer de mi padre est\u00e1 enamorada de m\u00ed, viniendo y avanzando de espaldas a mi nacimiento y de pecho a mi muerte. Que soy dos veces suyo: por el adi\u00f3s y por el regreso. La cierro, al retornar. Por eso me dieran t\u00e1nto sus ojos, justa de m\u00ed, in fraganti de m\u00ed, aconteci\u00e9ndose por obras terminadas, por pactos consumados.\n\nMi madre est\u00e1 confesa de m\u00ed, nombrada de m\u00ed. \u00bfC\u00f3mo no da otro tanto a mis otros hermanos? A V\u00edctor, por ejemplo, el mayor, que es tan viejo ya, que las gentes dicen: \u00a1Parece hermano menor de su madre! \u00a1Fuere porque yo he viajado mucho! \u00a1Fuere porque yo he vivido m\u00e1s!\n\nMi madre acuerda carta de principio colorante a mis relatos de regreso. Ante mi vida de regreso, recordando que viaj\u00e9 durante dos corazones por su vientre, se ruboriza y se queda mortalmente l\u00edvida, cuando digo, en el tratado del alma: Aquella noche fui dichoso. Pero, m\u00e1s se pone triste; m\u00e1s se pusiera triste.\n\n\u2014Hijo, \u00a1c\u00f3mo est\u00e1s viejo!\n\nY desfila por el color amarillo a llorar, porque me halla envejecido, en la hoja de espada, en la desembocadura de mi rostro. Llora de m\u00ed, se entristece de m\u00ed. \u00bfQu\u00e9 falta har\u00e1 mi mocedad, si siempre ser\u00e9 su hijo? \u00bfPor qu\u00e9 las madres se duelen de hallar envejecidos a sus hijos, si jam\u00e1s la edad de ellos alcanzar\u00e1 a la de ellas? \u00bfY por qu\u00e9, si los hijos, cuanto m\u00e1s se acaban, m\u00e1s se aproximan a los padres? \u00a1Mi madre llora porque estoy viejo de mi tiempo y porque nunca llegar\u00e9 a envejecer del suyo!\n\nMi adi\u00f3s parti\u00f3 de un punto de su ser, m\u00e1s externo que el punto de su ser al que retorno. Soy, a causa del excesivo plazo de mi vuelta, m\u00e1s el hombre ante mi madre que el hijo ante mi madre. All\u00ed reside el candor que hoy nos alumbra con tres llamas. Le digo entonces hasta que me callo:\n\n\u2014Hay, madre, en el mundo un sitio que se llama Par\u00eds. Un sitio muy grande y muy lejano y otra vez grande.\n\nLa mujer de mi padre, al o\u00edrme, almuerza y sus ojos mortales descienden suavemente por mis brazos.\nLanguidly Your Spirit  \n(For Spanish translation click here)\n\nWe had a pious age, when my father ordered our entrance into school. Priest of love, one rainy afternoon in February, mother served food of oration in the kitchen. In the corridor below, they were sitting at the table, my father and my older brothers. And my mother was seated near the same hearth-fire. They rang the bell.\n\n\u2014The doorbell is ringing!\u2014my mother said.\n\n\u2014The doorbell is ringing!\u2014my own mother said.\n\n\u2014The doorbell is ringing!\u2014said all of my mother, touching her bowels to infinite rays, above all the heights of those who come.\n\n\u2014Walk, Nativa, daughter, to see who comes.\n\nAnd without waiting the maternal permission, it was Miguel, the son, who went to see who came in this manner, placing himself in direct contradiction with us.\n\nA time of roads held my family. Mother left, advancing inversely   \nand as if she might have said: the parts. She made it to the patio outside. Nativa cried during one of those visits, of one of those patios and of the hand of my mother. Then, and when, pain and relish, they covered our fronts with a roof.\n\n\u2014Why did you not let her go to the door?\u2014said Nativa, the daughter, \u2014you've thrown Miguel to his duck.\n\nWhat imperfect protection, the light hand of father revealing the man, the small bones befitting the child. He could give his consent to the adventure the man would want later on. Nevertheless:\n\n\u2014And tomorrow to school\u2014argued father like a magistrate before the public of every week, his sons.\n\n\u2014And such, the law, the reason of the law. And also the life.\n\nMother should have cried, she was scarcely grieving like a mother. Now no one wanted to eat. In the lips of father lips, to leave breaking, a fine spoon that I know. In the fraternal mouths, the amazing bitterness of the son was left totally finished.\n\nMuch later, unexpectedly, he left the sewer of rain and from the same patio of the bad visit, a hen, not abhorrent nor laying eggs, but brutal and black. In my throat there rose a cluck-cluck. It was old hen, maternally widowed from chickens that had not hatched. The hen, whose origins were forgotten, at this moment, was widowed from her children. All the eggs were found empty. Afterward the clucking had the verb.\n\nNo one scared her. And no one, because they were frightened stopped cooing for the great maternal indisposition.\n\n\u2014Where are the children of the old hen?\n\n\u2014Where are the chickens of the old hen?\n\nPoor little ones! Where could they be!\nL\u00e1nguidamente su licor  \n(For English translation click here)\n\nTendr\u00edamos ya una edad misericordiosa, cuando mi padre orden\u00f3 nuestro ingreso a la escuela. Cura de amor, una tarde lluviosa de febrero, mam\u00e1 serv\u00eda en la cocina el yantar de oraci\u00f3n. En el corredor de abajo, estaban sentados a la mesa mi padre y mis hermanos mayores. Y mi madre iba sentada al pie del mismo fuego del hogar. Tocaron a la puerta.\n\n\u2014Tocan a la puerta!\u2014mi madre.\n\n\u2014Tocan a la puerta!\u2014mi propia madre.\n\n\u2014Tocan a la puerta!\u2014dijo toda mi madre, toc\u00e1ndose las entra\u00f1as a trastes infinitos, sobre toda la altura de quien viene.\n\n\u2014Anda, Nativa, la hija, a ver qui\u00e9n viene.\n\nY, sin esperar la venia maternal, fuera Miguel, el hijo, quien sali\u00f3 a ver qui\u00e9n venia as\u00ed, oponi\u00e9ndose a lo ancho de nosotros.\n\nUn tiempo de r\u00faa contuvo a mi familia. Mam\u00e1 sali\u00f3, avanzando inversamente y como si hubiera dicho: las partes. Se hizo patio afuera. Nativa lloraba de una tal visita, de un tal patio y de la mano de mi madre. Entonces y cuando, dolor y paladar techaron nuestras frentes.\n\n\u2014Porque no le deje que saliese a la puerta,\u2014Nativa, la hija\u2014, me ha echado Miguel al pavo. A su pavo.\n\n\u00a1Qu\u00e9 diestra de subprefecto, la diestra del padre, revelando, el hombre, las falanjas filiales del ni\u00f1o! Pod\u00eda as\u00ed otorgarle las venturas que el hombre deseara m\u00e1s tarde. Sin embargo:\n\n\u2014Y ma\u00f1ana, a la escuela,\u2014disert\u00f3 magistralmente el padre, ante el p\u00fablico semanal de sus hijos.\n\n\u2014Y tal, la ley, la causa de la ley. Y tal tambi\u00e9n la vida.\n\nMam\u00e1 debi\u00f3 llorar, gimiendo a penas la madre. Ya nadie quiso comer. En los labios del padre cupo, para salir rompi\u00e9ndose, una fina cuchara que conozco. En las fraternas bocas, la absorta amargura del hijo, qued\u00f3 atravesada.\n\nMas, luego, de improviso, sali\u00f3 de un alba\u00f1al de aguas llovedizas y de aquel mismo patio de la visita mala, una gallina, no ajena ni ponedora, sino brutal y negra. Cloqueaba en mi garganta. Fue una gallina vieja, maternalmente viuda de unos pollos que no llegaron a incubarse. Origen olvidado de ese instante, la gallina era viuda de sus hijos. Fueron hallados vac\u00edos todos los huevos. La clueca despu\u00e9s tuvo el verbo.\n\nNadie la espant\u00f3. Y de espantarla, nadie dej\u00f3 arrullarse por su gran calofr\u00edo maternal.\n\n\u2014\u00bfD\u00f3nde est\u00e1n los hijos de la gallina vieja?\n\n\u2014\u00bfD\u00f3nde est\u00e1n los pollos de la gallina vieja?\n\n\u00a1Pobrecitos! \u00a1D\u00f3nde estar\u00edan!\nThe Most Critical Moment of My Life  \n(For Spanish translation click here)\n\nOne man said:\n\n\u2014The most critical moment of my life was during the battle of the Marne, when I was wounded in the chest.\n\nAnother man said:\n\n\u2014The most critical moment of my life happened during a tidal wave in Yokohama, from which I saved myself, miraculously taking refuge under the eaves of a lacquer shop.\n\nAnd another man said:\n\n\u2014The most critical moment of my life happens when I sleep by day.\n\nAnd another said:\n\n\u2014The most critical moment of my life has been during my deepest solitude.\n\nAnd another said.\n\n\u2014The most critical moment of my life was when I was in jail in Peru.\n\nAnd another said:\n\n\u2014The most critical moment of my life was surprising my father's profile.\n\nAnd the ultimate man said:\n\n\u2014The most critical moment of my life is yet to come.\nEl momento m\u00e1s grave de la vida  \n(For English translation click here)\n\nUn hombre dijo:\n\n\u2014El momento m\u00e1s grave de mi vida estuvo en la batalla del Marne cuando fui herido en el pecho.\n\nOtro hombre dijo:\n\n\u2014El momento m\u00e1s grave de mi vida, ocurri\u00f3 en un maremoto de Yokohama, del cual salv\u00e9 milagrosamente, refugiado bajo el alero de una tienda de lacas.\n\nY otro hombre dijo:\n\n\u2014El momento m\u00e1s grave de mi vida acontece cuando duermo de d\u00eda.\n\nY otro dijo:\n\n\u2014El momento m\u00e1s grave de mi vida ha estado en mi mayor soledad.\n\nY otro dijo:\n\n\u2014El momento m\u00e1s grave de mi vida fue mi prisi\u00f3n en una c\u00e1rcel del Per\u00fa.\n\nY otro dijo:\n\n\u2014El momento m\u00e1s grave de mi vida es el haber sorprendido de perfil a mi padre.\n\nY el \u00faltimo hombre dijo:\n\n\u2014El momento m\u00e1s grave de mi vida no ha llegado todav\u00eda.\nI Am Going to Speak about Hope  \n(For Spanish translation click here)\n\nI do not suffer this pain as C\u00e9sar Vallejo. I do not hurt now as an artist, as a man nor simply as a human being. I do not suffer this pain as a Catholic, or as a Mohammedan or an atheist. Today I am simply in pain. If I were not called C\u00e9sar Vallejo, I would also suffer this same pain. Even if I were not a man not at least a human being I would also suffer it. Even if I were not Catholic, atheist, or Mohammedan I would still suffer. Today I suffer from the deepest depths. Today I am simply in pain.\n\nNow I hurt without explanations. My pain is so much from the depths, now I don't have cause nor do I need cause. What could its cause have been? Where is that cause of such importance that it stopped being its cause? Nothing has been able to leave this cause from being. For what has this pain been born, for itself? My pain is of the wind of the north and the wind of the south, like these sexless eggs that sometimes rare birds conceive in the wind. If my love had died, my pain would still be the same. If they had cut the collar of my race, my pain would still be the same. If finally, life was another form, my pain would still be the same. Today I suffer from the heights. Today I am simply in pain.\n\nI see the hungry man's pain and I see that his hunger walks so far from my suffering, to leave me fasting until death, it would always leave a fragment of grass from my tomb, at the very least. The same to the one in love. What engenders your blood for mine without source or end!\n\nUntil now I believed that all things of the universe, inevitably were fathers and sons. But here with my pain of today it is not father or son. My pain lacks courage to come out in the night, just as at dawn it is bold. If my pain lives in some dark house it would not give off light and if my pain lived in illumination, it would not cast a shadow. All I do today is suffer. Today I suffer happening what will happen. Today I am simply in pain.\nVoy a hablar de la esperanza  \n(For English translation click here)\n\nYo no sufro este dolor como C\u00e9sar Vallejo. Yo no me duelo ahora como artista, como hombre ni como simple ser vivo siquiera. Yo no sufro este dolor como cat\u00f3lico, como mahometano ni como ateo. Hoy sufro solamente. Si no me llamase C\u00e9sar Vallejo, tambi\u00e9n sufrir\u00eda este mismo dolor. Si no fuese artista, tambi\u00e9n lo sufrir\u00eda. Si no fuese hombre ni ser vivo siquiera, tambi\u00e9n lo sufrir\u00eda. Si no fuese cat\u00f3lico, ateo ni mahometano, tambi\u00e9n lo sufrir\u00eda. Hoy sufro desde m\u00e1s abajo. Hoy sufro solamente.\n\nMe duelo ahora sin explicaciones. Mi dolor es tan hondo, que no tuvo ya causa ni carece de causa. \u00bfQu\u00e9 ser\u00eda su causa? \u00bfD\u00f3nde est\u00e1 aquello tan importante, que dejase de ser su causa? Nada es su causa; nada ha podido dejar de ser su causa. \u00bfA qu\u00e9 ha nacido este dolor, por s\u00ed mismo? Mi dolor es del viento del norte y del viento del sur, como esos huevos neutros que algunas aves raras ponen del viento. Si hubiera muerto mi novia, mi dolor ser\u00eda igual. Si la vida fuese, en fin, de otro modo, mi dolor ser\u00eda igual. Hoy sufro desde m\u00e1s arriba. Hoy sufro solamente.\n\nMiro el dolor del hambriento y veo que su hambre anda tan lejos de mi sufrimiento, que de quedarme ayuno hasta morir, saldr\u00eda siempre de mi tumba una brizna de yerba al menos. Lo mismo el enamorado. \u00a1Qu\u00e9 sangre la suya m\u00e1s engendrada, para la m\u00eda sin fuente ni consumo!\n\nYo cre\u00eda hasta ahora que todas las cosas del universo eran, inevitablemente, padres o hijos. Pero he aqu\u00ed que mi dolor de hoy no es padre ni es hijo. Le falta espalda para anochecer, tanto como le sobra pecho para amanecer y si lo pusiesen en la estancia oscura, no dar\u00eda luz y si lo pusiesen en una estancia luminosa, no echar\u00eda sombra. Hoy sufro suceda lo que suceda. Hoy sufro solamente.\nDiscovery of Life  \n(For Spanish translation click here)\n\nSirs! Today is the first day I realized the presence of life. Sirs! I request you to let me be free for a moment in order to savor this formidable emotion, spontaneous and fresh with life, that today, for the first time, I am in ecstasy it makes me so happy I cry.\n\nMy pleasure comes from my virgin emotion. My explanation comes as before. I did not feel the presence of life. I had never felt it. He who says that he has is lying. He lies and his lie wounds me to such a great degree that I become miserable. My pleasure comes from my faith in this personal encounter of life and no one is able to go against this faith. If this should happen, his tongue should fail, his bones should fail, and he should run the danger of catching others, in order to maintain oneself in front of my eyes.\n\nI have never had life until now. People have never passed until now. There have never been houses nor avenues, air nor horizons, until now. If come now my friend, Peyriet, I should say that I do not know him and that we ought to begin anew. When, in effect, have I known my friend Peyriet? Today should be the first time that we know each other. I should say to him that he should go and return and enter again, seeing me as if he does not know me, that is to say, for the first time.\n\nNow I do not know anyone not anyone. I am acquainted with a strange country, that relieves the birth, light of unwithering epiphany. No, sir. Do not talk to that man. You do not know him and it will surprise you to hear such unbiased gossip. Do not stand up on this little stone, it may not be a stone and may go flowing into the abyss. Be cautious, we are placed in an absolutely unknown world.\n\nHow little time I have lived! My birth is so recent that there's not enough size to count my age. If I've just born! If, I have not lived yet! Sirs: I am so tiny that scarcely a day is in me.\n\nNot until now, did I hear the clamor of wagons, carryings stones for the great construction of the Boulevard Haussmann. Not until now, did I advance alongside the spring, saying to it, \"If death have been something else . . . \" Not until now, I did see the aurora of light of the sun on the domes of the Sacre-Coeur. Not until now, has a child approached me and looked at me deeply with his mouth. Not until now I knew there existed a door, another door and the cordial song of the distances.\n\nLeave me! Now life has given to me in all my death.\nHallazgo de la vida  \n(For English translation click here)\n\n\u00a1Se\u00f1ores! Hoy es la primera vez que me doy cuenta de la presencia de la vida. \u00a1Se\u00f1ores! Ruego a ustedes dejarme libre un momento, para saborear esta emoci\u00f3n formidable, espont\u00e1nea y reciente de la vida, que hoy, por la primera vez, me extas\u00eda y me hace dichoso hasta las l\u00e1grimas.\n\nMi gozo viene de lo in\u00e9dito de mi emoci\u00f3n. Mi exultaci\u00f3n viene de que antes no sent\u00ed la presencia de la vida. No la he sentido nunca. Miente quien diga que la he sentido. Miente y su mentira me hiere a tal punto que me har\u00eda desgraciado. Mi gozo viene de mi fe en este hallazgo personal de la vida, y nadie puede ir contra esta fe. Al que fuera, se le caer\u00eda la lengua, se le caer\u00edan los huesos y correr\u00eda el peligro de recoger otros, ajenos, para mantenerse de pie ante mis ojos.\n\nNunca, sino ahora, ha habido vida. Nunca, sino ahora, han pasado gentes. Nunca, sino ahora, ha habido casas y avenidas, aire y horizonte. Si viniese ahora mi amigo Peyriet, les dir\u00eda que yo no le conozco y que debemos empezar de nuevo. \u00bfCu\u00e1ndo, en efecto, le he conocido a mi amigo Peyriet? Hoy ser\u00eda la primera vez que nos conocemos. Le dir\u00eda que se vaya y regrese y entre a verme, como si no me conociera, es decir, por la primera vez.\n\nAhora yo no conozco a nadie ni nada. Me advierto en un pa\u00eds extra\u00f1o, en el que todo cobra relieve de nacimiento, luz de epifan\u00eda inmarcesible. No, se\u00f1or. No hable usted a ese caballero. Usted no lo conoce y le sorprender\u00eda tan inopinada parla. No ponga usted el pie sobre esa piedrecilla: qui\u00e9n sabe no es piedra y vaya usted a dar en el vac\u00edo. Sea usted precavido, puesto que estamos en un mundo absolutamente inconocido.\n\n\u00a1Cu\u00e1n poco tiempo he vivido! Mi nacimiento es tan reciente, que no hay unidad de medida para contar mi edad. \u00a1Si acabo de nacer! \u00a1Si a\u00fan no he vivido todav\u00eda! Se\u00f1ores: soy tan peque\u00f1ito, que el d\u00eda apenas cabe en m\u00ed!\n\nNunca, sino ahora, o\u00ed el estruendo de los carros, que cargan piedras para una gran construcci\u00f3n del boulevard Haussmann. Nunca, sino ahora avanc\u00e9 paralelamente a la primavera, dici\u00e9ndola: \u00abSi la muerte hubiera sido otra . . . \u00bb. Nunca, sino ahora, vi la luz \u00e1urea del sol sobre las c\u00fapulas de Sacre-Coeur. Nunca, sino ahora, se me acerc\u00f3 un ni\u00f1o y me mir\u00f3 hondamente con su boca. Nunca, sino ahora, supe que exist\u00eda una puerta, otra puerta y el canto cordial de las distancias.\n\n\u00a1Dejadme! La vida me ha dado ahora en toda mi muerte.\nPayroll of Bones  \n(For Spanish translation click here)\n\nIn a loud voice they demanded:\n\n\u2014He shows by force both his hands at once.  \nAnd this was not possible.\n\n\u2014Let them measure his steps while he weeps.  \nAnd this was not possible.\n\n\u2014Let him think thoughts simultaneous, in the time a zero remains useless.  \nAnd this was not possible.\n\n\u2014Let them act crazy.  \nAnd this was not possible.\n\n\u2014Let between him and another man similar to him a crowd of men like himself intercede.  \nAnd this was not possible.\n\n\u2014Let them compare him with himself.  \nAnd this was not possible.\n\n\u2014Let them, finally, call him by his name.  \nAnd this was not possible.\nN\u00f3mina de huesos  \n(For English translation click here)\n\nSe ped\u00eda a grandes voces:\n\n\u2014Que muestre las dos manos a la vez.  \nY esto no fue posible.\n\n\u2014Que, mientras llora, le tomen la medida de sus pasos.  \nY esto no fue posible.\n\n\u2014Que piense un pensamiento id\u00e9ntico, en el tiempo en que un cero permanece in\u00fatil.  \nY esto no fue posible.\n\n\u2014Que haga una locura.  \nY esto no fue posible.\n\n\u2014Que entre \u00e9l y otro hombre semejante a \u00e9l, se interponga una muchedumbre de hombres como \u00e9l.  \nY esto no fue posible.\n\n\u2014Que le comparen consigo mismo.  \nY esto no fue posible.\n\n\u2014Que le llamen, en fin, por su nombre.  \nY esto no fue posible.\nBehold I Greet Today  \n(For Spanish translation click here)\n\nBehold I greet today, I put on my collar and live,  \nsuperficial of fathomless steps from plants.  \nSuch things I receive from man, rather such things leave me  \nfrom every hour of mine sprouts a distance.\n\nWhat more do you want? Charmed.  \nPolitically, my words  \ndeclare accusation through my lower lip  \nand economically,  \nwhen I turn back on the Orient.  \nI distinguish in the dignity of death for my visits.\n\nI greet the unknown soldier with  \nthe required normal laws,  \nthe persecuted verse with dead ink  \nand the lizards which are in the same place every day  \nof their life and their death,  \nlike those who do not do anything.\n\nThe time has a centipede fear of watches.\n\n(Readers may title this poem whatever they wish)\nHe aqu\u00ed que hoy saludo  \n(For English translation click here)\n\nHe aqu\u00ed que hoy saludo, me pongo el cuello y vivo,  \nsuperficial de pasos insondable de plantas.  \nTal me recibo de hombre, tal m\u00e1s bien me despido  \ny de cada hora m\u00eda reto\u00f1a una distancia.\n\n\u00bfQuer\u00e9is m\u00e1s? encantado.  \nPol\u00edticamente, mi palabra  \nemite cargos contra mi labio inferior  \ny econ\u00f3micamente,  \ncuando doy la espalda a Oriente,  \ndistingo en dignidad de muerte a mis visitas.\n\nDesde totales c\u00f3digos regulares saludo  \nal soldado desconocido  \nal verso perseguido por la tinta fatal  \ny al saurio que Equidista diariamente  \nde su vida y su muerte,  \ncomo quien no hace la cosa.\n\nEl tiempo tiene un miedo ciempi\u00e9s a los relojes.\n\n(Los lectores pueden poner el t\u00edtulo que quieran a este poema)\nLoin of the Sacred Scriptures  \n(For Spanish translation click here)\n\nWithout ever having noticed excessive tourism  \nand without agencies  \nbreast to breast toward the unanimous mother.\n\nNow I come from Paris to be a son. Listen,  \nman, truthfully, I say you are the Eternal Son,  \nin order to be a brother your arms are hardly equal  \nand you have a great deal of malice as a father.\n\nMy mother's figure moving me by the emotion in her movement  \nand making me serious, hits me right in the heart:  \nthinking how often she has fallen from flight with my sad grandparents,  \nmy mother, from the other side of the circle, silences herself in the sky.\n\nMy meter now measures two meters,  \nmy bones generally in agreement and number  \nand the verb incarnate is living among us,  \nand the verb incarnate is living while I'm sinking in the bath,  \na high degree of perfection.\n\n_October 1926_\nLomo de las sagradas escrituras  \n(For English translation click here)\n\nSin haberlo advertido jam\u00e1s, exceso por turismo  \ny sin agencias  \nde pecho en pecho hacia la madre un\u00e1nime.\n\nHasta Par\u00eds ahora vengo a ser hijo. Escucha,  \nHombre, en verdad te digo que eres el Hijo Eterno,  \npues para ser hermano tus brazos son escasamente iguales  \ny tu malicia para ser padre, es mucha.\n\nLa talla de mi madre movi\u00e9ndome por \u00edndole de movimiento,  \ny poni\u00e9ndome serio, me llega exactamente al coraz\u00f3n:  \npesando cuanto cayera de vuelo con mis tristes abuelos,  \nmi madre me oye en di\u00e1metro call\u00e1ndose en altura.\n\nMi metro est\u00e1 midiendo ya dos metros,  \nmis huesos concuerdan en g\u00e9nero y en n\u00famero  \ny el verbo encarnado habita entre nosotros  \ny el verbo encarnado habita, al hundirme en el ba\u00f1o,  \nun alto grado de perfecci\u00f3n.\n\n_octubre 1926_\n\n_C\u00e9sar Vallejo celebrating Christmas next to Henriette Maisse and Carlos More. Paris, 1926_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nPOEMAS HUMANOS\n\n_The Undated Poems 1923(?)\u20131937_\nHat, Overcoat, Gloves  \n(For Spanish translation click here)\n\nIn front of the Com\u00e9die-Fran\u00e7aise is the Regency Caf\u00e9\n\nin it there is a secret\n\nroom, with an armchair and table.\n\nThe dusts stands motionless and covers my shoes when I enter.\n\nBetween my wet lips, the embers of the cigarette\n\nsmolder, and in the smoke one is able to see\n\ntwo intense smokes, the thorax of the Caf\u00e9,\n\nand in the thorax, a profound oxide of sadness.\n\nIt's important that autumn be grafted to autumn,\n\nit's important that autumn be integrated of sprouts,\n\nthe cloud of six months; of the prominent cheekbones, the wrinkle.\n\nImportant also to smell the crazy postulating\n\nhow warm the snow is, how quick the tortoise,\n\nhow simple the how, how thundering the when!\nSombrero, abrigo, guantes  \n(For English translation click here)\n\nEnfrente a la Comedia Francesa, est\u00e1 el Caf\u00e9\n\nde la Regencia; en \u00e9l hay una pieza\n\nrec\u00f3ndita, con una butaca y una mesa.\n\nCuando entro, el polvo inm\u00f3vil se ha puesto ya de pie.\n\nEntre mis labios hechos de jebe, la pavesa\n\nde un cigarrillo humea, y en el humo se ve\n\ndos humos intensivos, el t\u00f3rax del Caf\u00e9,\n\ny en el t\u00f3rax, un \u00f3xido profundo de tristeza.\n\nImporta que el oto\u00f1o se injerte en los oto\u00f1os,\n\nimporta que el oto\u00f1o se integre de reto\u00f1os,\n\nla nube, de semestres; de p\u00f3mulos, la arruga.\n\nImporta oler a loco postulando\n\n\u00a1qu\u00e9 c\u00e1lida es la nieve, qu\u00e9 fugaz la tortuga,\n\nel c\u00f3mo qu\u00e9 sencillo, qu\u00e9 fulminante el cu\u00e1nto!\nThe Wheel of the Starving  \n(For Spanish translation click here)\n\nI come out steaming from between my own teeth,\n\nscreaming, moaning\n\npulling my pants down . . .\n\nMy stomach and my blood and guts despicably,\n\nthe misery plucks me from between my own teeth,\n\npicked up with a toothpick by my own shirt cuff.\n\nIsn't there for me\n\na bench to sit on?\n\nNot even that bench on which the new mother stumbles to sit,\n\nmother of the lamb, the cause, the root?\n\nIs that one ready for me now?\n\nThe one which stumbled looming through my soul!\n\nAt least\n\nthe chalky or evil one (sea of humility),\n\nor the one with no more use, not even to be thrown\n\nagainst a man,\n\nlet me just have that one now.\n\nAt least the one that can be found alone and pierced in an insult,\n\nlet me just have that one now.\n\nAt least the crowned and twisted one, in which but once\n\nthe echo of the walk of a righteous conscience,\n\nor at least that other one, tossed in a noble curve,\n\nwhich drops by itself,\n\nshowing essence of its innards,\n\nlet me just have that one now.\n\nIs there not one piece of bread for me either?\n\nI shall no longer be what I must always be,\n\nbut give me\n\na stone to sit on,\n\nbut give me,\n\nplease, a piece of bread to sit on,\n\nbut give me,\n\nin simple words,\n\nsomething, at least, to drink, to eat, to live, to rest upon,\n\nthen I will leave . . .\n\nIt found a weird shape, my shirt is shattered\n\nand grimy\n\nand I have nothing, this is frightful.\nLa rueda del hambriento  \n(For English translation click here)\n\nPor entre mis propios dientes salgo humeando,\n\ndando voces, pujando,\n\nbaj\u00e1ndome los pantalones . . .\n\nV\u00e1ca mi est\u00f3mago, v\u00e1ca mi yeyuno,\n\nla miseria me saca por entre mis propios dientes,\n\ncogido con un palito por el pu\u00f1o de la camisa.\n\nUna piedra en que sentarme\n\n\u00bfno habr\u00e1 ahora para m\u00ed?\n\nA\u00fan aquella piedra en que tropieza la mujer que ha dado a luz,\n\nla madre del cordero, la causa, la ra\u00edz,\n\n\u00bf\u00e9sa no habr\u00e1 ahora para m\u00ed?\n\n\u00a1Siquiera aquella otra,\n\nque ha pasado agach\u00e1ndose por mi alma!\n\nSiquiera\n\nla calc\u00e1rida o la mala (humilde oc\u00e9ano)\n\no la que ya no sirve ni para ser tirada contra el hombre\n\n\u00e9sa d\u00e1dmela ahora para m\u00ed!\n\nSiquiera la que hallaren atravesada y sola en un insulto,\n\n\u00e9sa d\u00e1dmela ahora para m\u00ed!\n\nSiquiera la torcida y coronada, en que resuena\n\nsolamente una vez el andar de las rectas conciencias,\n\no, al menos, esa otra, que arrojada en digna curva,\n\nva a caer por s\u00ed misma,\n\nen profesi\u00f3n de entra\u00f1a verdadera,\n\n\u00a1\u00e9sa d\u00e1dmela ahora para m\u00ed!\n\nUn pedazo de pan, tampoco habr\u00e1 para m\u00ed?\n\nYa no m\u00e1s he de ser lo que siempre he de ser,\n\npero dadme\n\nuna piedra en que sentarme,\n\npero dadme,\n\npor favor, un pedazo de pan en que sentarme,\n\npero dadme\n\nen espa\u00f1ol\n\nalgo, en fin, de beber, de comer, de vivir, de reposarse\n\ny despu\u00e9s me ir\u00e9 . . .\n\nHall\u00f3 una extra\u00f1a forma, est\u00e1 muy rota\n\ny sucia mi camisa\n\ny ya no tengo nada, esto es horrendo.\nEpistle to Passersby  \n(For Spanish translation click here)\n\nI start my rabbit day again,\n\nmy night of elephant ease.\n\nAnd I say to myself,\n\nthis is my raw immensity, wholesale,\n\nthis is the pleasant weight which sought a bird in me below,\n\nthis is my arm,\n\nthat on its own refused to be a wing,\n\nthese are my sacred scriptures,\n\nthese my frightened testes.\n\nA gloomy isle will light me like a continent,\n\nwhile the capitol rests on my intimate collapse\n\nand like lancers the assembly hems in my parade.\n\nBut when I die,\n\nfrom life and not from time,\n\nwhen my two valises reach the count of two,\n\nthis shall be my stomach, where my shattered lamp once fit,\n\nthis that head atoning for the torment of my footsteps' circle,\n\nthese those worms the heart had counted one by one,\n\nthis shall be my body jointly liable\n\nwith the one on whom the single soul keeps watch; this shall be\n\nmy navel, where I killed my natural born lice,\n\nthis my thing thing, my fearsome thing.\n\nMeanwhile, convulsive, harsh,\n\nconvalescing my bridle,\n\nsuffering as I suffer from the lion's direct speech;\n\nand since I have existed between two brick-wall powers,\n\nI too grow strong again, with smiling lips.\nEp\u00edstola a los transe\u00fantes  \n(For English translation click here)\n\nReanudo mi d\u00eda de conejo\n\nmi noche de elefante en descanso.\n\nY, entre mi, digo:\n\n\u00e9sta es mi inmensidad en bruto, a c\u00e1ntaros\n\n\u00e9ste es mi grato peso,\n\nque me buscar\u00e1 abajo para p\u00e1jaro;\n\n\u00e9ste es mi brazo\n\nque por su cuenta rehus\u00f3 ser ala,\n\n\u00e9stas son mis sagradas escrituras,\n\n\u00e9stos mis alarmados campe\u00f1ones.\n\nL\u00fagubre isla me alumbrar\u00e1 continental,\n\nmientras el capitolio se apoye en mi \u00edntimo derrumbe\n\ny la asamblea en lanzas clausure mi desfile.\n\nPero cuando yo muera\n\nde vida y no de tiempo,\n\ncuando lleguen a dos mis dos maletas,\n\n\u00e9ste ha de ser mi est\u00f3mago en que cupo mi l\u00e1mpara en pedazos,\n\n\u00e9sta aquella cabeza que expi\u00f3 los tormentos del c\u00edrculo en mis pasos,\n\n\u00e9stos esos gusanos que el coraz\u00f3n cont\u00f3 por unidades,\n\n\u00e9ste ha de ser mi cuerpo solidario\n\npor el que vela el alma individual; \u00e9ste ha de ser\n\nmi ombligo en que mat\u00e9 mis piojos natos,\n\n\u00e9sta mi cosa cosa, mi cosa tremebunda.\n\nEn tanto, convulsiva, \u00e1speramente\n\nconvalece mi freno,\n\nsufriendo como sufro del lenguaje directo del le\u00f3n;\n\ny, puesto que he existido entre dos potestades de ladrillo,\n\nconvalezco yo mismo, sonriendo de mis labios.\nToday I'd Really Like to Be Happy  \n(For Spanish translation click here)\n\nToday I'd really like to be happy,\n\nto be happy, my whole being burst into questions,\n\nto throw open wildly the doors to the rooms of my flat, like a madman,\n\nthe self confidence of my physical trust laid bare,\n\nonly to see if anyone cares,\n\nonly to see if anyone is taking note of my spontaneous position,\n\nto demand, I'm saying\n\nwhy people inflict so much pain on my soul.\n\nFor I'd like, in substance, to be blissful,\n\nto work without cane, a laic humility, without a black donkey.\n\nTo sense the sensation of the world,\n\nthe subjunctive songs,\n\nthe pencil that I lost in my cavity\n\nand my beloved organs all crying.\n\nPersuadable brother, comrade,\n\nfather through grandeur, mortal son,\n\nfriend and contender, immense document of Darwin:\n\nat what hour will they come with my likeness?\n\nWill they come with joy on their faces? With shrouded enjoyment?\n\nEarlier than expected? Who knows, by what hassle?\n\nAt the mercy, comrade,\n\nthis man of mine in rejection and in observation, neighbor\n\nin whose enormous neck seesaws\n\nnaturally, without wire, my hope . . .\nQuisiera hoy ser feliz de buena gana  \n(For English translation click here)\n\nQuisiera hoy ser feliz de buena gana,\n\nser feliz y portarme frondoso de preguntas,\n\nabrir por temperamento de par en par mi cuarto, como loco,\n\ny reclamar, en fin,\n\nen mi confianza f\u00edsica acostado,\n\ns\u00f3lo por ver si quieren,\n\ns\u00f3lo por ver si quieren probar de mi espont\u00e1nea posici\u00f3n,\n\nreclamar, voy diciendo,\n\npor qu\u00e9 me dan as\u00ed t\u00e1nto en el alma.\n\nPues quisiera en sustancia ser dichoso,\n\nobrar sin bast\u00f3n, laica humildad, ni burro negro.\n\nAs\u00ed las sensaciones de este mundo,\n\nlos cantos subjuntivos,\n\nel l\u00e1piz que perd\u00ed en mi cavidad\n\ny mis amados \u00f3rganos de llanto.\n\nHermano persuasible, camarada,\n\npadre por la grandeza, hijo mortal,\n\namigo y contendor, inmenso documento de Darwin:\n\n\u00bfa qu\u00e9 hora, pues, vendr\u00e1n con mi retrato?\n\n\u00bfA los goces? \u00bfAcaso sobre goce amortajado?\n\n\u00bfM\u00e1s temprano? \u00bfQui\u00e9n sabe, a las porf\u00edas?\n\nA las misericordias, camarada,\n\nhombre m\u00edo en rechazo y observaci\u00f3n, vecino\n\nen cuyo cuello enorme sube y baja,\n\nal natural, sin hilo, mi esperanza . . .\nConsidering Coldly, Impartially  \n(For Spanish translation click here)\n\nConsidering coldly, impartially,\n\nthat man suffers, coughs and, however\n\nself-gratified he is with his reddened chest;\n\nthat the only thing he can do is compose himself\n\nwith days,\n\nthat he's really this gloomy mammal slightly imperfect . . .\n\nConsidering\n\nthat a man proceeds softly from his work\n\nhaving told the boss, a worker's insubordinate dream;\n\nthat the diagram of time\n\nremains a constant diorama with medals\n\nand that his half-open eyes study,\n\nfrom distant hours,\n\nhis famished formula for a workers' coalition . . .\n\nUnderstanding easily enough\n\nthat the man, at times, thinking\n\nof his tears behind sore, burning eyes,\n\nand, allowing to set himself as an object,\n\nbecomes a good carpenter, sweats, kills,\n\nand then sings, breakfasts, plunges into his coat . . .\n\nConsidering too\n\nthat man is, in truth, animal\n\nand, notwithstanding, turns, hitting me on the head with his sadness . . .\n\nExamining, finally,\n\nhis opposed pieces, his toilet,\n\nhis desperation at the end of his atrocious day and rubbing it out . . .\n\nUnderstanding\n\nthat he knows I love him,\n\nand that I hating him with the same love is, in sum, indifference . . .\n\nConsidering his general documents,\n\nstudying with glasses that certificate\n\nthat proves he was born very, very small . . .\n\nI signal to him,\n\nhe comes,\n\nand I embrace him, moved.\n\nWhat more can I give! Moved . . . Moved . . .\nConsiderando en fr\u00edo, imparcialmente  \n(For English translation click here)\n\nConsiderando en fr\u00edo, imparcialmente,\n\nque el hombre es triste, tose y, sin embargo,\n\nse complace en su pecho colorado;\n\nque lo \u00fanico que hace es componerse\n\nde d\u00edas;\n\nque es l\u00f3brego mam\u00edfero y se peina . . .\n\nConsiderando\n\nque el hombre procede suavemente del trabajo\n\ny repercute jefe, suena subordinado;\n\nque el diagrama del tiempo\n\nes constante diorama en sus medallas\n\ny, a medio abrir, sus ojos estudiaron,\n\ndesde lejanos tiempos,\n\nsu f\u00f3rmula fam\u00e9lica de masa . . .\n\nComprendiendo sin esfuerzo\n\nque el hombre se queda, a veces, pensando,\n\ncomo queriendo llorar,\n\ny, sujeto a tenderse como objeto,\n\nse hace buen carpintero, suda, mata\n\ny luego canta, almuerza, se abotona . . .\n\nConsiderando tambi\u00e9n\n\nque el hombre es en verdad un animal\n\ny, no obstante, al voltear, me da con su tristeza en la cabeza . . .\n\nExaminando, en fin,\n\nsus encontradas piezas, su retrete,\n\nsu desesperaci\u00f3n, al terminar su d\u00eda atroz, borr\u00e1ndolo . . .\n\nComprendiendo\n\nque \u00e9l sabe que le quiero,\n\nque le odio con afecto y me es, en suma, indiferente . . .\n\nConsiderando sus documentos generales\n\ny mirando con lentes aquel certificado\n\nque prueba que naci\u00f3 muy peque\u00f1ito . . .\n\nle hago una se\u00f1a,\n\nviene,\n\ny le doy un abrazo, emocionado.\n\n\u00a1Qu\u00e9 m\u00e1s da! Emocionado . . . Emocionado . . .\nAnd If after So Many Words  \n(For Spanish translation click here)\n\nAnd if after so many words,\n\nthe word itself doesn't survive!\n\nIf after the wings of the birds,\n\nthe motionless birds do not survive!\n\nTruthfully, it would be of more value\n\nthat they eat all and we be finished!\n\nTo have been born to live our death!\n\nRising from the sky toward the earth\n\nfrom your own disaster\n\nand spy the moment the darkness that turns out the shade!\n\nIt would be better, frankly,\n\nif it were all swallowed up, and that's that! . . .\n\nAnd if after so much history, we succumb,\n\nnot of eternity,\n\nbut of those simple things, like being\n\nat home or finding fault with yourself!\n\nAnd if, suddenly, we discover that we live\n\njudging by the height of the motionless stars\n\nby the comb and the spots on the handkerchief!\n\nTruthfully, it would be of more value\n\nthat they eat all, of course!\n\nIt will be said that in one\n\nof our eyes we have a great deal of pain\n\nand likewise in the other, the same pain\n\nand in the two, when they see, a great deal of pain . . .\n\nThen! . . . Obviously! . . . Then . . . not a word!\n\u00a1Y si despu\u00e9s de tantas palabras!  \n(For English translation click here)\n\n\u00a1Y si despu\u00e9s de tantas palabras,\n\nno sobrevive la palabra!\n\n\u00a1Si despu\u00e9s de las alas de los p\u00e1jaros,\n\nno sobrevive el p\u00e1jaro parado!\n\n\u00a1M\u00e1s valdr\u00eda, en verdad,\n\nque se lo coman todo y acabemos!\n\n\u00a1Haber nacido para vivir de nuestra muerte!\n\n\u00a1Levantarse del cielo hacia la tierra\n\npor sus propios desastres\n\ny espiar el momento de apagar con su sombra su tiniebla!\n\n\u00a1M\u00e1s valdr\u00eda, francamente,\n\nque se lo coman todo y qu\u00e9 m\u00e1s da . . . !\n\n\u00a1Y si despu\u00e9s de tanta historia, sucumbimos,\n\nno ya de eternidad,\n\nsino de esas cosas sencillas, como estar\n\nen la casa o ponerse a cavilar!\n\n\u00a1Y si luego encontramos,\n\nde buenas a primeras, que vivimos,\n\na juzgar por la altura de los astros,\n\npor el peine y las manchas del pa\u00f1uelo!\n\n\u00a1M\u00e1s valdr\u00eda, en verdad,\n\nque se lo coman todo, desde luego!\n\nSe dir\u00e1 que tenemos\n\nen uno de los ojos mucha pena\n\ny tambi\u00e9n en el otro, mucha pena\n\ny en los dos, cuando miran, mucha pena . . .\n\nEntonces . . . \u00a1Claro! . . . Entonces . . . \u00a1ni palabra!\nParis, October 1936  \n(For Spanish translation click here)\n\nOf all this I am the only one who's leaving.\n\nI am getting up from this bench, of my trousers,\n\nof my grand situation, of my actions,\n\nfrom my house number shattered to pieces,\n\nof all this, and I'm the only one who's leaving.\n\nFrom the Champs-\u00c9lys\u00e9es or while taking a turn\n\nin a strange narrow passage of the Moon,\n\nmy own death is leaving, and my bed taking leave of the room,\n\nand, surrounded by people, solitary, free,\n\nmy human likeness\n\nturns back and dispatches its shadows one by one.\n\nAnd I walk away from everything, because everything\n\nwill remain behind as evidence:\n\nmy shoe, its worn buttonholes, also its mud\n\nand even the crease of the elbow\n\nof my own buttoned shirt.\nPar\u00eds, Octubre 1936  \n(For English translation click here)\n\nDe todo esto yo soy el \u00fanico que parte.\n\nDe este banco me voy, de mis calzones,\n\nde mi gran situaci\u00f3n, de mis acciones,\n\nde mi n\u00famero hendido parte a parte,\n\nde todo esto yo soy el \u00fanico que parte.\n\nDe los Campos El\u00edseos o al dar vuelta\n\nla extra\u00f1a callejuela de la Luna,\n\nmi defunci\u00f3n se va, parte mi cuna,\n\ny, rodeada de gente, sola, suelta,\n\nmi semejanza humana dase vuelta\n\ny despacha sus sombras una a una.\n\nY me alejo de todo, porque todo\n\nse queda para hacer la coartada:\n\nmi zapato, su ojal, tambi\u00e9n su lodo\n\ny hasta el doblez del codo\n\nde mi propia camisa abotonada.\nBlack Stone on a White Stone*  \n(For Spanish translation click here)\n\nI shall die in Paris, in a rainstorm,\n\na day I already possess the memory.\n\nI shall die in Paris\u2014and I don't run away\u2014\n\nperhaps on a Thursday, like today, in autumn.\n\nIt's got to be Thursday, because today, Thursday, I'm writing\n\nthese verses and I've hurt the humerus bone\n\nand never like today, have I turned\n\nin the direction to where I am alone.\n\nC\u00e9sar Vallejo is dead, they beat him,\n\nall of them, and for nothing.\n\nthey hit him hard with sticks and whipped hard\n\nwith a rope; witnesses are\n\nthe Thursdays and the humerus bones\n\nthe loneliness, the rain, and the long empty roads . . .\n\n_*In Santiago de Chuco, the homeland of C\u00e9sar Vallejo, they put a black stone above a white stone to show a burial._\nPiedra negra sobre una piedra blanca  \n(For English translation click here)\n\nMe morir\u00e9 en Par\u00eds con aguacero,\n\nun d\u00eda del cual tengo ya el recuerdo.\n\nMe morir\u00e9 en Par\u00eds \u00bfy no me corro?\n\ntal vez un jueves, como es hoy, de oto\u00f1o.\n\nJueves ser\u00e1, porque hoy, jueves, que proso\n\nestos versos, los h\u00fameros me he puesto\n\na la mala y, jam\u00e1s como hoy, me he vuelto,\n\ncon todo mi camino, a verme solo.\n\nC\u00e9sar Vallejo ha muerto, le pegaban\n\ntodos sin que \u00e9l les haga nada;\n\nle daban duro con un palo y duro\n\ntambi\u00e9n con una soga; son testigos\n\nlos d\u00edas jueves y los huesos h\u00fameros,\n\nla soledad, la lluvia, los caminos . . .\nToday I Like Life Much Less  \n(For Spanish translation click here)\n\nToday I like life much less\n\nbut I still enjoy being alive: there, I said it.\n\nI almost touched part of my whole and restrained myself\n\nwith a shot on the tongue behind my word.\n\nToday I touch my chin in retreat\n\nand in these momentary trousers I say to myself:\n\nSo much life and never!\n\nSo many years and always my weeks! . . .\n\nMy parents interred with their stone\n\nand their sad rapid growth unachieved;\n\nmy brothers and sisters, present with me always,\n\nand finally, my being erect with a vest.\n\nI like life enormously,\n\nbut, of course,\n\nwith my beloved death and my coffee\n\nand seeing the leafy chestnuts of Paris\n\nand saying:\n\nThis is an eye, that; and this one, a forehead, that . . . And repeating:\n\nSo much life and my song never falters!\n\nSo many years and always, always, always!\n\nI said vest, said\n\nwhole part, anguish, said almost in order not to weep.\n\nFor it's true that I suffered in that hospital over there\n\nand it's good and it's bad to have seen\n\nfrom bottom to top my organism.\n\nI would like to live always, if I could have a strong belly,\n\nbecause, as I was saying, and I'll repeat it again,\n\nso much life and never! And so many years,\n\nand always, much always, always, always!\nHoy me gusta la vida mucho menos  \n(For English translation click here)\n\nHoy me gusta la vida mucho menos,\n\npero siempre me gusta vivir: ya lo dec\u00eda.\n\nCasi toqu\u00e9 la parte de mi todo y me contuve\n\ncon un tiro en la lengua detr\u00e1s de mi palabra.\n\nHoy me palpo el ment\u00f3n en retirada\n\ny en estos moment\u00e1neos pantalones yo me digo:\n\n\u00a1T\u00e1nta vida y jam\u00e1s!\n\n\u00a1T\u00e1ntos a\u00f1os y siempre mis semanas! . . .\n\nMis padres enterrados con su piedra\n\ny su triste estir\u00f3n que no ha acabado;\n\nde cuerpo entero hermanos, mis hermanos,\n\ny, en fin, mi ser parado y en chaleco.\n\nMe gusta la vida enormemente\n\npero, desde luego,\n\ncon mi muerte querida y mi caf\u00e9\n\ny viendo los casta\u00f1os frondosos de Par\u00eds\n\ny diciendo:\n\nEs un ojo \u00e9ste, aqu\u00e9l; una frente \u00e9sta, aqu\u00e9lla . . . Y repitiendo:\n\n\u00a1T\u00e1nta vida y jam\u00e1s me falla la tonada!\n\n\u00a1T\u00e1ntos a\u00f1os y siempre, siempre, siempre!\n\nDije chaleco, dije\n\ntodo, parte, ansia, dije casi, por no llorar.\n\nQue es verdad que sufr\u00ed en aquel hospital que queda al lado\n\ny est\u00e1 bien y est\u00e1 mal haber mirado\n\nde abajo para arriba mi organismo.\n\nMe gustar\u00e1 vivir siempre, as\u00ed fuese de barriga,\n\nporque, como iba diciendo y lo repito,\n\n\u00a1t\u00e1nta vida y jam\u00e1s! \u00a1Y t\u00e1ntos a\u00f1os,\n\ny siempre, mucho siempre, siempre, siempre!\n\n_C\u00e9sar Vallejo in the forest of Fontainebleau.  \nNear Paris, 1926_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nPOEMAS HUMANOS\n\n_The Dated Poems, 4 September\u20138 December, 1937_\nA Pillar Tolerating Solaces  \n(For Spanish translation click here)\n\nA pillar tolerating solaces,\n\nanother pillar,\n\nduplicate pillars, pillar-like\n\nand like a grandchild of a dark door.\n\nLost noise, the only one, listening, on the edge of the exhaustion;\n\ndrinking, the other, two by two, with double handles.\n\nPerhaps, do I ignore the year of this day,\n\nthe hate of this love, the tablets of this forehead?\n\nDo you ignore that this afternoon costs days?\n\nDo you ignore that you should never say \"never\" on your knees?\n\nThe pillars I see are listening to me;\n\nso are other pillars, deuces and sad grandchildren of my leg.\n\nI say it in American currency,\n\nthat owes so much fire to the silver!\n\nConsoled in third nuptials,\n\npale, born,\n\nI am going to close my baptismal chest, this window shop,\n\nthis fear with breasts,\n\nthis finger in penitence,\n\nheart and mind united to my skeleton.\n\n_6 September 1937_\nUn pilar soportando consuelos  \n(For English translation click here)\n\nUn pilar soportando consuelos,\n\npilar otro,\n\npilar en duplicado, pilaroso\n\ny como nieto de una puerta oscura.\n\nRuido perdido, el uno, oyendo, al borde del cansancio;\n\nbebiendo, el otro, dos a dos, con asas.\n\n\u00bfIgnoro acaso el a\u00f1o de este d\u00eda,\n\nel odio de este amor, las tablas de esta frente?\n\n\u00bfIgnoro que esta tarde cuesta d\u00edas?\n\n\u00bfIgnoro que jam\u00e1s se dice \u00abnunca\u00bb, de rodillas?\n\nLos pilares que vi me est\u00e1n oyendo;\n\notros pilares son, doses y nietos tristes de mi pierna.\n\n\u00a1Lo digo en cobre americano,\n\nque le debe a la plata t\u00e1nto fuego!\n\nConsolado en terceras nupcias,\n\np\u00e1lido, nacido,\n\nvoy a cerrar mi pila bautismal, esta vidriera,\n\neste susto con tetas,\n\neste dedo en capilla,\n\ncoraz\u00f3nmente unido a mi esqueleto.\n\n_6 setiembre 1937_\nPoem to Be Read and Sung  \n(For Spanish translation click here)\n\nI know there's a person\n\nlooking for me day and night inside her hand,\n\nand encountering me, each minute, in her shoes.\n\nDoes she ignore that the night is buried\n\nwith spurs in back of the kitchen?\n\nI know there's a person composed of my parts\n\nwhom I complete when my size fits\n\nriding on its exact little stone.\n\nDoesn't she know that the money spent\n\non her portrait will never turn up in her trunk?\n\nI know the day,\n\nbut the sun has escaped me;\n\nI know the universal act she performed in her bed\n\nwith a courage not of her own and warm water, whose\n\nsuperficial frequency is mine.\n\nIs her being so small\n\nthat even her own feet would trample upon her?\n\nA cat is the border between us,\n\nright there beside its bowl of water.\n\nI see her on the corner, her jacket\n\nopens and closes, in the shape of the questioning palm trees . . .\n\nWhat can she do but change weeping?\n\nBut she looks and looks for me. What a tale!\n\n_7 September 1937_\nPoema para ser le\u00eddo y cantado  \n(For English translation click here)\n\nS\u00e9 que hay una persona\n\nque me busca en su mano, d\u00eda y noche,\n\nencontr\u00e1ndome, a cada minuto, en su calzado.\n\n\u00bfIgnora que la noche est\u00e1 enterrada\n\ncon espuelas detr\u00e1s de la cocina?\n\nS\u00e9 que hay una persona compuesta de mis partes,\n\na la que integro cuando va mi talle\n\ncabalgando en su exacta piedrecilla.\n\n\u00bfIgnora que a su cofre\n\nno volver\u00e1 moneda que sali\u00f3 con su retrato?\n\nS\u00e9 el d\u00eda,\n\npero el sol se me ha escapado;\n\ns\u00e9 el acto universal que hizo en su cama\n\ncon ajeno valor y esa agua tibia, cuya\n\nsuperficial frecuencia es una mina.\n\n\u00bfTan peque\u00f1a es, acaso, esa persona,\n\nque hasta sus propios pies as\u00ed la pisan?\n\nUn gato es el lindero entre ella y yo,\n\nal lado mismo de su tasa de agua.\n\nLa veo en las esquinas, se abre y cierra\n\nsu veste, antes palmera interrogante . . .\n\n\u00bfQu\u00e9 podr\u00e1 hacer sino cambiar de llanto?\n\nPero me busca y busca. \u00a1Es una historia!\n\n_7 setiembre 1937_\nWhile Pondering in Life, While Pondering  \n(For Spanish translation click here)\n\nWhile pondering in life, while pondering\n\nslowly with the strength of the torrent,\n\nit relieves, just for existing it offers a seat,\n\nit condemns death;\n\nwrapped in white shrouds, falls,\n\nfalls planetarily\n\nthe nail swarming with grief, falls!\n\n(Official acrimony, that of my left,\n\nold pocket, in itself this right considered.)\n\nEverything is joyful, without my joy\n\nand everything free, without my candor,\n\nmy uncertainty!\n\nJudging by the form, nevertheless, I go ahead,\n\nlimping anciently,\n\nand my eyes forgotten because of my tears (Very interesting)\n\nI climb to my feet from my star.\n\nI weave, having sewn, here I am sewing.\n\nI look for what follows me and hides from me between archbishops,\n\nbeneath my soul and behind the smoke of my breath.\n\nSuch was the sensual desolation\n\nof the ascending maiden goat,\n\nexhaling fateful oils\n\nyesterday Sunday when I lost my Saturday.\n\nSuch is death, with its fearless husband.\n\n_7 September 1937_\nAl cavilar en la vida, al cavilar  \n(For English translation click here)\n\nAl cavilar en la vida, al cavilar\n\ndespacio en el esfuerzo del torrente,\n\nalivia, ofrece asiento el existir,\n\ncondena a muerte;\n\nenvuelto en trapos blancos cae,\n\ncae planetariamente\n\nel clavo hervido en pesadumbre; cae!\n\n(Acritud oficial, la de mi izquierda;\n\nviejo bolsillo, en s\u00ed considerada, esta derecha.)\n\n\u00a1Todo est\u00e1 alegre, menos mi alegr\u00eda\n\ny todo, largo, menos mi candor,\n\nmi incertidumbre!\n\nA juzgar por la forma, no obstante, voy de frente,\n\ncojeando antiguamente,\n\ny olvido por mis l\u00e1grimas mis ojos (Muy interesante)\n\ny subo hasta mis pies desde mi estrella.\n\nTejo; de haber hilado, h\u00e9me tejiendo.\n\nBusco lo que me sigue y se me esconde entre arzobispos,\n\npor debajo de mi alma y tras del humo de mi aliento.\n\nTal era la sensual desolaci\u00f3n\n\nde la cabra doncella que ascend\u00eda,\n\nexhalando petr\u00f3leos fat\u00eddicos,\n\nayer domingo en que perd\u00ed mi s\u00e1bado.\n\nTal es la muerte, con su audaz marido.\n\n_7 setiembre 1937_\nOh Bottle without Wine!  \n(For Spanish translation click here)\n\nOh bottle without wine! Oh widowed wine of this bottle!\n\nLate afternoon when the aurora of dusk\n\nflutters forebodingly in five spirits.\n\nWidowhood without bread nor grime, topping in horrendous metals,\n\nand finishing our oral cells.\n\nOh always, never to give with the never of so much always!\n\nOh my good friends, cruel fallacy,\n\npartial, penetrating our cut-short\n\nvolatile, playful-like grief!\n\nSublime, low perfection of the pig,\n\ntouches my overall melancholy!\n\nzuela* sounding in dreams,\n\nzuela\n\nboorish, inferior, duped, lawful, thief\n\nlowers and touches those ideas that were mine!\n\nYou and he and they and everyone,\n\nnevertheless,\n\nenter into my shirt all at once,\n\nin the wooden shoulders, between the thighbones, toothpicks;\n\nyou particularly\n\nhaving influenced me;\n\nhe, futile, colored, with money\n\nand they, bee wings of some other importance.\n\nOh bottle without wine! Oh widowed wine of this bottle!\n\n_16 September 1937_\n\n_*Zuela is a carpenter tool used for scabble. It is built with an iron plate steely and sharp._\n\u00a1Oh botella sin vino!  \n(For English translation click here)\n\n\u00a1Oh botella sin vino! \u00a1Oh vino que enviud\u00f3 de esta botella!\n\nTarde cuando la aurora de la tarde\n\nflame\u00f3 funestamente en cinco esp\u00edritus.\n\nViudez sin pan ni mugre, rematando en horrendos metaloides\n\ny en c\u00e9lulas orales acabando.\n\n\u00a1Oh siempre, nunca dar con el jam\u00e1s de t\u00e1nto siempre!\n\n\u00a1oh mis buenos amigos, cruel falacia,\n\nparcial, penetrativa en nuestro trunco,\n\nvol\u00e1til, jugarino desconsuelo!\n\n\u00a1Sublime, baja perfecci\u00f3n del cerdo,\n\npalpa mi general melancol\u00eda!\n\n\u00a1Zuela sonante en sue\u00f1os,\n\nzuela\n\nzafia, inferior, vendida, l\u00edcita, ladrona,\n\nbaja y palpa lo que eran mis ideas!\n\nT\u00fa y \u00e9l y ellos y todos,\n\nsin embargo,\n\nentraron a la vez en mi camisa,\n\nen los hombros madera, entre los f\u00e9mures, palillos;\n\nt\u00fa particularmente,\n\nhabi\u00e9ndome influido;\n\n\u00e9l, f\u00fatil, colorado, con dinero\n\ny ellos, z\u00e1nganos de ala de otro peso.\n\n\u00a1Oh botella sin vino! \u00a1oh vino que enviud\u00f3 de esta botella!\n\n_16 setiembre 1937_\nHe Goes Running, Walking, Fleeing  \n(For Spanish translation click here)\n\nHe goes running, walking, fleeing\n\nfrom his feet . . .\n\nHe goes with two clouds in his cloud\n\nsitting uncertainly, nailed in the hand\n\nhis sad \"for,\" his funeral \"then.\"\n\nHe runs from all, walking\n\nbetween colorless protests; he flees\n\nrising, he flees\n\nfalling, he flees\n\nby measured of the underground cellar, he flees\n\nraising in his arms the evil,\n\nhe flees\n\ndirectly to sob alone.\n\nWhere may he be going,\n\nfar from his brambles, caustic talons,\n\nfar from the air, far from his journey,\n\nat last to flee, flee, and flee, and flee\n\nfrom his feet\u2014man of two feet, stops\n\nfrom all this fleeing\u2014he must be thirsty from running.\n\nAnd not even the tree, if he endorses iron of gold!\n\nAnd not even the iron, if he covers his foliage!\n\nNothing, but only his feet\n\nnothing but his short shivering\n\nhis living \"for,\" his living \"then\" . . .\n\n_18 September 1937_\nVa corriendo, andando, huyendo  \n(For English translation click here)\n\nVa corriendo, andando, huyendo\n\nde sus pies . . .\n\nVa con dos nubes en su nube,\n\nsentado ap\u00f3crifo, en la mano insertos\n\nsus tristes paras, sus entonces f\u00fanebres.\n\nCorre de todo, andando\n\nentre protestas incoloras; huye\n\nsubiendo, huye\n\nbajando, huye\n\na paso de sotana, huye\n\nalzando al mal en brazos,\n\nhuye\n\ndirectamente a sollozar a solas.\n\nAdonde vaya,\n\nlejos de sus fragosos, c\u00e1usticos talones,\n\nlejos del aire, lejos de su viaje,\n\na fin de huir, huir y huir y huir\n\nde sus pies \u2014hombre en dos pies, parado\n\nde t\u00e1nto huir\u2014 habr\u00e1 sed de correr.\n\n\u00a1Y ni el \u00e1rbol, si endosa hierro de oro!\n\n\u00a1Y ni el hierro, si cubre su hojarasca!\n\nNada, sino sus pies,\n\nnada sino su breve calofr\u00edo,\n\nsus paras vivos, sus entonces vivos . . .\n\n_18 setiembre 1937_\nMy Breast Wants and Does Not Want Its Color  \n(For Spanish translation click here)\n\nMy breast wants and does not want its color,\n\nI go weeping with a stick along those harsh roads,\n\nI try to be happy, weeping in my hand;\n\nI remember, I write,\n\nriveting a tear in my cheekbone.\n\nEvil wants its red, good wants its red reddened\n\nby the hanging axe,\n\nby the trot of a wing flying on foot,\n\nbut he doesn't want it and sorely\n\nhe doesn't want this,\n\nhe doesn't want to be inside his\n\nsoul, to the beat of lance-blows on his temple,\n\nthe two-handed creature, the great brute, the great philosopher.\n\nThus, I almost don't exist, I'm on my way down\n\nfrom the plough on which I save my soul\n\nand almost, in proportion, I almost raise myself.\n\nFor knowing why life contains this breast,\n\nwhy I cry, why,\n\nhesitant, helpless, inconstant, I was born\n\nshouting,\n\nto know this, to understand it\n\nthrough the sound of a competent alphabet,\n\nwould be to suffer for someone ungrateful.\n\nBut no! No! No! What scheme, my parameter!\n\nAnguish, yes, firm and frenetic,\n\nleathery, predatory, it wants and does not want, sky and bird,\n\nanguish, yes, with every button of the fly.\n\nWrangle between two laments, theft of one bliss only,\n\npainless road where I suffer in my own out shoes\n\nfrom the velocity of walking blindly.\n\n_22 September 1937_\nQuiere y no quiere su color mi pecho  \n(For English translation click here)\n\nQuiere y no quiere su color mi pecho,\n\npor cuyas bruscas v\u00edas voy, lloro con palo,\n\ntrato de ser feliz, lloro en mi mano,\n\nrecuerdo, escribo\n\ny remacho una l\u00e1grima en mi p\u00f3mulo.\n\nQuiere su rojo el mal, el bien su rojo enrojecido\n\npor el hacha suspensa,\n\npor el trote del ala a pie volando,\n\ny no quiere y sensiblemente\n\nno quiere aquesto el hombre;\n\nno quiere estar en su alma\n\nacostado, en la sien latidos de asta,\n\nel bimano, el muy bruto, el muy fil\u00f3sofo.\n\nAs\u00ed, casi no soy, me vengo abajo\n\ndesde el arado en que socorro a mi alma\n\ny casi, en proporci\u00f3n, casi enalt\u00e9zcome.\n\nQue saber por qu\u00e9 tiene la vida este perrazo,\n\npor qu\u00e9 lloro, por qu\u00e9,\n\ncej\u00f3n, inh\u00e1bil, veleidoso, hube nacido\n\ngritando;\n\nsaberlo, comprenderlo\n\nal son de un alfabeto competente,\n\nser\u00eda padecer por un ingrato.\n\n\u00a1Y no! \u00a1No! \u00a1No! \u00a1Qu\u00e9 ardid, ni paramento!\n\nCongoja, s\u00ed, con s\u00ed firme y fren\u00e9tico,\n\ncori\u00e1ceo, rapaz, quiere y no quiere, cielo y p\u00e1jaro;\n\ncongoja, s\u00ed, con toda la bragueta.\n\nContienda entre dos llantos, robo de una sola ventura,\n\nv\u00eda indolora en que padezco en chanclos\n\nde la velocidad de andar a ciegas.\n\n_22 setiembre 1937_\nThe Peace, the Wasp, the Bung, the Hillsides  \n(For Spanish translation click here)\n\nThe peace, the wasp, the bung, the hillsides,\n\nthe dead man, the ten liter, the owl,\n\nthe sites, the ringworm, the tombs, the vase, the dark women,\n\nthe unknowing, the kettle, the altarboy,\n\nthe drops, the forgetfulness,\n\nthe potentate, the cousins, the archangels, the needle,\n\nthe parish priests, the ebony, the spite,\n\nthe part, the type, the stupor, the soul . . .\n\nMalleable, saffroned, external, spotless,\n\nportable, old, thirteen, bloodsmeared,\n\nphotographed, active, tumescent,\n\nconnected, broad, ribboned, perfidious . . .\n\nBurning, comparing,\n\nliving, infuriated,\n\nstriking, analyzing, listening, shuddering,\n\ndying, holding on, locating, weeping . . .\n\nAfter, these, here\n\nafter, overhead,\n\nperhaps, while, behind, so much, so never,\n\nbeneath, maybe, far,\n\nalways, that one, tomorrow, how much,\n\nhow much! . . .\n\nThe horrible, the sumptuous, the slowest,\n\nthe august, the fruitless,\n\nthe ominous, the twitching, the wet, the fatal,\n\nthe all, the most pure, the lugubrious,\n\nthe cruel, the satanic, the tactile, the profound . . .\n\n_25 September 1937_\nLa paz, la avispa, el taco, las vertientes  \n(For English translation click here)\n\nLa paz, la avispa, el taco, las vertientes,\n\nel muerto, los dec\u00edlitros, el b\u00faho,\n\nlos lugares, la ti\u00f1a, los sarc\u00f3fagos, el vaso, las morenas,\n\nel desconocimiento, la olla, el monaguillo,\n\nlas gotas, el olvido,\n\nla potestad, los primos, los arc\u00e1ngeles, la aguja,\n\nlos p\u00e1rrocos, el \u00e9bano, el desaire,\n\nla parte, el tipo, el estupor, el alma . . .\n\nD\u00factil, azafranado, externo, n\u00edtido,\n\nport\u00e1til, viejo, trece, ensangrentado,\n\nfotografiadas, listas, tumefactas,\n\nconexas, largas, encintadas, p\u00e9rfidas . . .\n\nArdiendo, comparando,\n\nviviendo, enfureci\u00e9ndose,\n\ngolpeando, analizando, oyendo, estremeci\u00e9ndose,\n\nmuriendo, sosteni\u00e9ndose, situ\u00e1ndose, llorando . . .\n\nDespu\u00e9s, \u00e9stos, aqu\u00ed,\n\ndespu\u00e9s, encima,\n\nquiz\u00e1, mientras, detr\u00e1s, tanto, tan nunca,\n\ndebajo, acaso, lejos,\n\nsiempre, aquello, ma\u00f1ana, cu\u00e1nto,\n\n\u00a1cu\u00e1nto! . . .\n\nLo horrible, lo suntuario, lo lent\u00edsimo,\n\nlo augusto, lo infructuoso,\n\nlo aciago, lo crispante, lo mojado, lo fatal.\n\nlo todo, lo pur\u00edsimo, lo l\u00f3brego,\n\nlo acerbo, lo sat\u00e1nico, lo t\u00e1ctil, lo profundo . . .\n\n_25 setiembre 1937_\nOf Pure Heat I'm Freezing  \n(For Spanish translation click here)\n\nOf pure heat I'm freezing\n\nsister Envy!\n\nLions lick my shadow\n\nand the mouse gnaws at my name,\n\nmother, soul of mine!\n\nTo the edge of the depths I go,\n\nbrother-in-law Vice!\n\nThe caterpillar plays on its voice,\n\nand the voice plays its caterpillar,\n\nfather, flesh of mine!\n\nMy love is in front of me,\n\ngranddaughter Dove!\n\nOn its knees, my terror\n\nand on its head, my anguish,\n\nmother, soul of mine!\n\nUntil a day without two,\n\nwife Tomb,\n\nmy ultimate brand makes a sound\n\nof a sleeping vipor,\n\nfather, flesh of mine!\n\n_29 September 1937_\nDe puro calor tengo fr\u00edo  \n(For English translation click here)\n\n\u00a1De puro calor tengo fr\u00edo,\n\nhermana Envidia!\n\nLamen mi sombra leones\n\ny el rat\u00f3n me muerde el nombre,\n\n\u00a1madre alma m\u00eda!\n\n\u00a1Al borde del fondo voy,\n\ncu\u00f1ado Vicio!\n\nLa oruga ta\u00f1e su voz,\n\ny la voz ta\u00f1e su oruga,\n\n\u00a1padre cuerpo m\u00edo!\n\n\u00a1Est\u00e1 de frente mi amor,\n\nnieta Paloma!\n\nDe rodillas, mi terror\n\ny de cabeza, mi angustia,\n\n\u00a1madre alma m\u00eda!\n\nHasta que un d\u00eda sin dos,\n\nesposa Tumba,\n\nmi \u00faltimo hierro d\u00e9 el son\n\nde una v\u00edbora que duerme,\n\n\u00a1padre cuerpo m\u00edo!\n\n_29 setiembre 1937_\nTrust in the Eyeglass, Not in the Eye  \n(For Spanish translation click here)\n\nTrust in the eyeglass, not in the eye,\n\nin the stairway, never in the step;\n\nin the wing, not in the bird\n\nand in only you, only you, only you.\n\nTrust in the evil, not in the vicious,\n\nin the glass, never in the liquor;\n\nin the corpse, not in the man\n\nand in only you, only you, only you.\n\nTrust in many, but no longer in one,\n\nin the river bed, never in the current;\n\nin the stockings, not in the legs\n\nand in only you, only you, only you.\n\nTrust in the window, not in the door,\n\nin the mother, not in the nine months;\n\nin the destiny, not in the gilded dice\n\nand in only you, only you, only you.\n\n_5 October 1937_\nConfianza en el anteojo, n\u00f3 en el ojo  \n(For English translation click here)\n\nConfianza en el anteojo, no en el ojo;\n\nen la escalera, nunca en el pelda\u00f1o;\n\nen el ala, no en el ave\n\ny en ti s\u00f3lo, en ti s\u00f3lo, en ti s\u00f3lo.\n\nConfianza en la maldad, no en el malvado;\n\nen el vaso, mas nunca en el licor;\n\nen el cad\u00e1ver, no en el hombre\n\ny en ti s\u00f3lo, en ti s\u00f3lo, en ti s\u00f3lo.\n\nConfianza en muchos, pero ya no en uno;\n\nen el cauce, jam\u00e1s en la corriente;\n\nen los calzones, no en las piernas\n\ny en ti s\u00f3lo, en ti s\u00f3lo, en ti s\u00f3lo.\n\nConfianza en la ventana, no en la puerta;\n\nen la madre, mas no en los nueve meses;\n\nen el destino, no en el dado de oro,\n\ny en ti s\u00f3lo, en ti s\u00f3lo, en ti s\u00f3lo.\n\n_5 octubre 1937_\nMocked, Acclimatized to the Good, Morbid, Tormented  \n(For Spanish translation click here)\n\nMocked, acclimatized to the good, morbid, tormented,\n\nI double over in the extremity of being worldly and play cups,\n\nwhere the destinies end up in flies,\n\nwhere I eat and drink what's cleaning me out.\n\nMonumental pinch,\n\nnumeral bier, those of my debt,\n\nthose of my unpaid balance, when I fall exceedingly,\n\nloudly, livid.\n\nThe lowest depth, then\n\nit's time to moan with the ax,\n\nand it's then the year of the sob,\n\nthe day of the ankle,\n\nthe night of the rib, of the pained respiration.\n\nSterile qualities, monotonous satans,\n\nleap from the flank,\n\nfrom the flank of my substitute mare;\n\nbut, where I eat, how much I think!\n\nbut, how much I drink where I weep!\n\nWell, that's life, life\n\nbeing what it is, way over there, behind\n\nthe infinite, thus, spontaneously\n\nbefore the legislative temple.\n\nThus the string lies buried at the violin's base,\n\nwhen they speak of the air, when\n\nvery leisurely they speak of lightning.\n\nThe wrong cause thus doubles, we take turns\n\nthree by three in unity, thus\n\none plays cups\n\nand those who fold match my bet,\n\nthe destinies end up in bacteria\n\nand one owes all to all.\n\n_7 October 1937_\nEscarnecido, aclimatado al   \nbien, m\u00f3rbido, hurente  \n(For English translation click here)\n\nEscarnecido, aclimatado al bien, m\u00f3rbido, hurente,\n\ndoblo el cabo carnal y juego a copas,\n\ndonde acaban en moscas los destinos,\n\ndonde com\u00ed y beb\u00ed de lo que me hunde.\n\nMonumental adarme,\n\nf\u00e9retro numeral, los de mi deuda,\n\nlos de mi deuda, cuando caigo altamente,\n\nruidosamente, amoratadamente.\n\nAl fondo, es hora,\n\nentonces, de gemir con toda el hacha\n\ny es entonces el a\u00f1o del sollozo,\n\nel d\u00eda del tobillo,\n\nla noche del costado, el siglo del resuello.\n\nCualidades est\u00e9riles, mon\u00f3tonos satanes,\n\ndel flanco brincan,\n\ndel ijar de mi yegua suplente;\n\npero, donde com\u00ed, cu\u00e1nto pens\u00e9!\n\npero cu\u00e1nto beb\u00ed donde llor\u00e9!\n\nAs\u00ed es la vida, tal\n\ncomo es la vida, all\u00e1, detr\u00e1s\n\ndel infinito; as\u00ed, espont\u00e1neamente,\n\ndelante de la sien legislativa.\n\nYace la cuerda as\u00ed al pie del viol\u00edn,\n\ncuando hablaron del aire, a voces, cuando\n\nhablaron muy despacio del rel\u00e1mpago.\n\nSe dobla as\u00ed la mala causa, vamos\n\nde tres en tres a la unidad; as\u00ed\n\nse juega a copas\n\ny salen a mi encuentro los que al\u00e9janse,\n\nacaban los destinos en bacterias\n\ny se debe todo a todos.\n\n_7 October 1937_\nStumble between Two Stars  \n(For Spanish translation click here)\n\nThere are people so racked they no longer feel\n\ntheir bodies; quantitative the hair\n\nlet down, inch by inch, weighing with genius;\n\nthe mode, angular, upright;\n\ndon't look for the grindstone of oblivion,\n\nthey seem to come out of air, to sum up sighs mentally, to hear\n\nthe sharp blows of their words in their palates!\n\nShedding their skin, scratching at the sarcophagus in\n\nwhich they were born\n\nrising up by their death hour by hour\n\nto fall, through the depth of their frozen alphabet, to\n\nthe ground.\n\nAh for so much! Ah for so little! Ah for all women!\n\nAh in my room listening to them with glasses!\n\nAh in my thorax when they buy suits!\n\nAh for my white grime, joined with their scum!\n\nBeloved be the ears of the sanchez,\n\nbeloved be those who recline,\n\nbeloved be the man unknown and his wife,\n\nneighbor with sleeves, collar and eyes!\n\nBeloved be he who has bedbugs,\n\nthe one wearing torn shoes under the rain,\n\nwho keeps watching over the corpse of a bread with two matches,\n\nthe one watching his finger caught in a door,\n\nthe one with no birthdays,\n\nwho's lost his shadow in fire,\n\nthe animal, the one who looks like a parrot,\n\nthe one who looks like a man, the poor rich,\n\nthe pure miserable, the poor poor!\n\nBeloved be\n\nthe one who has hunger or thirst, but has no\n\nhunger with which to satisfy all his thirst,\n\nneither thirst with which to satisfy all his hungers!\n\nBeloved be he who works by day, by the month, by the hour,\n\nthe one who sweats from pain or from shame.\n\nthe one who goes, by command of his hands to the movies,\n\nthe one who pays sleeps with what he lacks,\n\nthe one who sleeps on his back,\n\nthe one who no longer remembers his childhood; beloved be\n\nthe bald man without a hat,\n\nthe just man without thorns,\n\nthe thief without roses,\n\nthe one who wears a watch and has seen God,\n\nthe one who has honor and does not die!\n\nBeloved be the child that falls and still cries\n\nand the man who has fallen and no longer cries!\n\nAh for so much! Ah for so little! Ah for all men!\n\n_11 October 1937_\nTraspi\u00e9 entre dos estrellas  \n(For English translation click here)\n\n\u00a1Hay gentes tan desgraciadas, que ni siquiera\n\ntienen cuerpo; cuantitativo el pelo,\n\nbaja, en pulgadas, la genial pesadumbre;\n\nel modo, arriba;\n\nno me busques, la muela del olvido,\n\nparecen salir del aire, sumar suspiros mentalmente, o\u00edr\n\nclaros azotes en sus paladares!\n\nVanse de su piel, rasc\u00e1ndose el sarc\u00f3fago en que nacen\n\ny suben por su muerte de hora en hora\n\ny caen, a lo largo de su alfabeto g\u00e9lido, hasta el suelo.\n\n\u00a1Ay de t\u00e1nto! \u00a1ay de tan poco! \u00a1ay de ellas!\n\n\u00a1Ay en mi cuarto, oy\u00e9ndolas con lentes!\n\n\u00a1Ay en mi t\u00f3rax, cuando compran trajes!\n\n\u00a1Ay de mi mugre blanca, en su hez mancomunada!\n\n\u00a1Amadas sean las orejas s\u00e1nchez,\n\namadas las personas que se sientan,\n\namado el desconocido y su se\u00f1ora,\n\nel pr\u00f3jimo con mangas, cuello y ojos!\n\n\u00a1Amado sea aquel que tiene chinches,\n\nel que lleva zapato roto bajo la lluvia,\n\nel que vela el cad\u00e1ver de un pan con dos cerillas,\n\nel que se coge un dedo en una puerta,\n\nel que no tiene cumplea\u00f1os,\n\nel que perdi\u00f3 su sombra en un incendio,\n\nel animal, el que parece un loro,\n\nel que parece un hombre, el pobre rico,\n\nel puro miserable, el pobre pobre!\n\n\u00a1Amado sea\n\nel que tiene hambre o sed, pero no tiene\n\nhambre con qu\u00e9 saciar toda su sed,\n\nni sed con qu\u00e9 saciar todas sus hambres!\n\n\u00a1Amado sea el que trabaja al d\u00eda, al mes, a la hora,\n\nel que suda de pena o de verg\u00fcenza,\n\naquel que va, por orden de sus manos, al cinema,\n\nel que paga con lo que le falta,\n\nel que duerme de espaldas,\n\nel que ya no recuerda su ni\u00f1ez; amado sea\n\nel calvo sin sombrero,\n\nel justo sin espinas,\n\nel ladr\u00f3n sin rosas,\n\nel que lleva reloj y ha visto a Dios,\n\nel que tiene un honor y no fallece!\n\n\u00a1Amado sea el ni\u00f1o, que cae y a\u00fan llora\n\ny el hombre que ha ca\u00eddo y ya no llora!\n\n\u00a1Ay de t\u00e1nto! \u00a1Ay de tan poco! \u00a1Ay de ellos!\n\n_11 octubre 1937_\nFarewell, Remembering a Goodbye  \n(For Spanish translation click here)\n\nFinally, at last, in the end,\n\nI turn, went back and having just finished, cry to you giving you\n\nthe key, my hat, this little letter for all.\n\nAt last the key is in the lock so we might learn\n\nto separate the gilding from the gold, with each turn,\n\nand is lying at the end of my hat, this poor badly combed brain,\n\nand, last glass of smoke, with its dramatic paper,\n\nlies down this practical dream of the soul, in the grave.\n\nGoodbye, brothers Saint Peters,\n\nHeraclitus, Erasmus, Spinozas!\n\nGoodbye, sad Bolshevik bishops!\n\nGoodbye, governors in disorder!\n\nGoodbye, wine which in water is like wine!\n\nGoodbye, alcohol that's in the rain!\n\nAlso I said goodbye to myself,\n\ngoodbye, formal flight of the milligrams!\n\nAlso, goodbye, in exactly the same way,\n\ncold of cold and cold of heat!\n\nFinally, at last, in the end, the logic,\n\nthe borders of fire,\n\nthe farewell remembering the goodbye.\n\n_12 October 1937_\nDespedida recordando un adi\u00f3s  \n(For English translation click here)\n\nAl cabo, al fin, por \u00faltimo,\n\ntomo, volv\u00ed y ac\u00e1bome y os gimo, d\u00e1ndoos\n\nla llave, mi sombrero, esta cartita para todos.\n\nAl cabo de la llave est\u00e1 el metal en que aprendi\u00e9ramos\n\na desdorar el oro, y est\u00e1, al fin\n\nde mi sombrero, este pobre cerebro mal peinado,\n\ny, \u00faltimo vaso de humo, en su papel dram\u00e1tico,\n\nyace este sue\u00f1o pr\u00e1ctico del alma.\n\n\u00a1Adi\u00f3s, hermanos san pedros,\n\nher\u00e1clitos, erasmos, espinosas!\n\n\u00a1Adi\u00f3s, tristes obispos bolcheviques!\n\n\u00a1Adi\u00f3s, gobernadores en desorden!\n\n\u00a1Adi\u00f3s, vino que est\u00e1 en el agua como vino!\n\n\u00a1Adi\u00f3s, alcohol que est\u00e1 en la lluvia!\n\n\u00a1Adi\u00f3s tambi\u00e9n, me digo a m\u00ed mismo,\n\nadi\u00f3s, vuelo formal de los mil\u00edgramos!\n\n\u00a1Tambi\u00e9n adi\u00f3s, de modo id\u00e9ntico,\n\nfr\u00edo del fr\u00edo y fr\u00edo del calor!\n\nAl cabo, al fin, por \u00faltimo, la l\u00f3gica,\n\nlos linderos del fuego,\n\nla despedida recordando aquel adi\u00f3s.\n\n_12 octubre 1937_\nThe Book of Nature  \n(For Spanish translation click here)\n\nProfessor of sobs\u2014I said to a tree\u2014\n\nbludgeon, linden tree\n\nmurmuring, to the banks of the Marne, a good student\n\nreads your fortune in your withered leaves\n\nbetween evident water and false sun,\n\nyour three of cups, your horse of gold.\n\nRector of chapels in the sky,\n\nof the ardent fly, of the laborious calm in donkeys;\n\nrector of profound ignorance, a bad student\n\nreads your fortune in your withered leaves,\n\nhunger of reason that maddens\n\nand the thirst of dementia drives him crazy.\n\nMechanical screams, aware and strong upright tree,\n\nwater moving, sun-like, double, fanatic,\n\nconnoisseur of cardinal roses, completely\n\nshaved, almost to the drawing of blood, stinging, a student\n\nreads your fortune in your withered leaves,\n\nyour precocious king, telluric, volcanic, king of swords.\n\nOh professor for having not known so much!\n\nOh rector for having trembled in this air!\n\nOh technician for so much that bends you!\n\nOh linden tree! Oh musing stick by the Marne!\n\n_21 October 1937_\nEl libro de la naturaleza  \n(For English translation click here)\n\nProfesor de sollozo\u2014he dicho a un \u00e1rbol\u2014\n\npalo de azogue, tilo\n\nrumoreante, a la orilla del Mame, un buen alumno\n\nleyendo va en tu naipe, en tu hojarasca,\n\nentre el agua evidente y el sol falso,\n\nsu tres de copas, su caballo de oros.\n\nRector de los cap\u00edtulos del cielo,\n\nde la mosca ardiente, de la calma manual que hay en los asnos;\n\nrector de honda ignorancia, un mal alumno\n\nleyendo va en tu naipe, en tu hojarasca,\n\nel hambre de raz\u00f3n que le enloquece\n\ny la sed de demencia que le aloca.\n\nT\u00e9cnico en gritos, \u00e1rbol consciente, fuerte,\n\nfluvial, doble, solar, doble, fan\u00e1tico,\n\nconocedor de rosas cardinales, totalmente\n\nmetido, hasta hacer sangre, en aguijones, un alumno\n\nleyendo va en tu naipe, en tu hojarasca,\n\nsu rey precoz, tel\u00farico, volc\u00e1nico, de espadas.\n\n\u00a1Oh profesor, de haber t\u00e1nto ignorado!\n\n\u00a1oh rector, de temblar t\u00e1nto en el aire!\n\n\u00a1oh t\u00e9cnico, de t\u00e1nto que te inclinas!\n\n\u00a1Oh tilo! \u00a1oh palo rumoroso junto al Marne!\n\n_21 octubre 1937_\nI Have a Terrible Fear of Being an Animal  \n(For Spanish translation click here)\n\nI have a terrible fear of being an animal\n\nof white snow, and keeping father\n\nwith only my veined circulation and mother alive\n\nand this splendid day, solar and archbishoprical,\n\nday that thus represents this night\n\nlineally\n\nthis animal avoids being content, breathing\n\nand changing itself and having silver.\n\nIt would be a great deal of pain\n\nif I were a man to that great a degree.\n\nA blunder, a very fruitful premise\n\nsuccumbs to an occasional yoke\n\nthe spiritual hinge of my waist.\n\nAn absurdity . . . In the meantime,\n\nso it is, nearer to the head of God,\n\nin the tablets of Locke, of Bacon, in the livid neck\n\nof the beast, in the snout of the soul.\n\nAnd, in aromatic, logic,\n\nI have this practical fear, this splendid day\n\nlunar, to be that one, this one perhaps,\n\nto whose nose the ground smells of death.\n\nThe live absurdity and the dead blunder.\n\nOh tread upon yourself, be, cough, attack yourself,\n\nattack the doctrine, the temple, of one shoulder to another,\n\nremove yourself, cry, give for eight\n\nor for seven or for six, for five or give it\n\nthe life that has three potentials.\n\n_22 October 1937_\nTengo un miedo terrible de ser un animal  \n(For English translation click here)\n\nTengo un miedo terrible de ser un animal\n\nde blanca nieve, que sostuvo padre\n\ny madre, con su sola circulaci\u00f3n venosa,\n\ny que, este d\u00eda espl\u00e9ndido, solar y arzobispal,\n\nd\u00eda que representa as\u00ed a la noche,\n\nlinealmente\n\nelude este animal estar contento, respirar\n\ny transformarse y tener plata.\n\nSer\u00eda pena grande\n\nque fuera yo tan hombre hasta ese punto.\n\nUn disparate, una premisa ub\u00e9rrima\n\na cuyo yugo ocasional sucumbe\n\nel gonce espiritual de mi cintura.\n\nUn disparate . . . En tanto,\n\nes as\u00ed, m\u00e1s ac\u00e1 de la cabeza de Dios,\n\nen la tabla de Locke, de Bacon, en el l\u00edvido pescuezo\n\nde la bestia, en el hocico del alma.\n\nY, en l\u00f3gica arom\u00e1tica,\n\ntengo ese miedo pr\u00e1ctico, este d\u00eda\n\nespl\u00e9ndido, lunar, de ser aqu\u00e9l, \u00e9ste talvez,\n\na cuyo olfato huele a muerto el suelo,\n\nel disparate vivo y el disparate muerto.\n\n\u00a1Oh revolcarse, estar, toser, fajarse,\n\nfajarse la doctrina, la sien, de un hombro al otro,\n\nalejarse, llorar, darlo por ocho\n\no por siete o por seis, por cinco o darlo\n\npor la vida que tiene tres potencias.\n\n_22 octubre 1937_\nThe Anger Which Breaks a Man into Children  \n(For Spanish translation click here)\n\nThe anger which breaks a man into children,\n\nwhich breaks the child into equal birds,\n\nand from there, the bird into small eggs;\n\nthe anger of the poor\n\nhas one oil against two vinegars.\n\nThe anger which breaks a tree into leaves,\n\nand the leaf into uneven buds\n\nand the bud, into telescopic grooves;\n\nthe anger of the poor\n\nhas two rivers against many seas.\n\nThe anger which breaks the good into doubts\n\nand doubt, into three similar arcs\n\nand the arc, at once, into unforeseeable tombs;\n\nthe anger of the poor\n\nhas one steel against two daggers.\n\nThe anger which breaks the soul into bodies,\n\nthe body into dissimilar organs\n\nand the organ, into octave meditations;\n\nthe anger of the poor\n\nhas one central fire against two craters.\n\n_26 October 1937_\nLa c\u00f3lera que quiebra al hombre en ni\u00f1os  \n(For English translation click here)\n\nLa c\u00f3lera que quiebra al hombre en ni\u00f1os,\n\nque quiebra al ni\u00f1o en p\u00e1jaros iguales,\n\ny el p\u00e1jaro, despu\u00e9s, en huevecillos;\n\nla c\u00f3lera del pobre\n\ntiene un aceite contra dos vinagres.\n\nLa c\u00f3lera que al \u00e1rbol quiebra en hojas,\n\nla hoja en botones desiguales\n\ny al bot\u00f3n, en ranuras telesc\u00f3picas;\n\nla c\u00f3lera del pobre\n\ntiene dos r\u00edos contra muchos mares.\n\nLa c\u00f3lera que quiebra al bien en dudas,\n\na la duda, en tres arcos semejantes\n\ny al arco, luego, en tumbas imprevistas;\n\nla c\u00f3lera del pobre\n\ntiene un acero contra dos pu\u00f1ales.\n\nLa c\u00f3lera que quiebra al alma en cuerpos,\n\nal cuerpo en \u00f3rganos desemejantes\n\ny al \u00f3rgano, en octavos pensamientos;\n\nla c\u00f3lera del pobre\n\ntiene un fuego central contra dos cr\u00e1teres.\n\n_26 octubre 1937_\nIntensity and Heights  \n(For Spanish translation click here)\n\nI want to write but spume comes out of me,\n\nI want to say so much, but stick in mire;\n\nthere's no cipher spoken, not a sum,\n\nthere's no pyramid written without sprouts.\n\nI want to write, but feel myself puma;\n\nI want laurels but I'm wreathed in garlic.\n\nThere's no cough spoken that doesn't arrive to the mist,\n\nno god nor son of god without evolution.\n\nLet's go, then, therefore, and eat grass,\n\nmeat of weeping, fruit of moan,\n\nour melancholic soul canned.\n\nLet's go! Let's go! I'm wounded;\n\nlet's go to drink what we've already drunk,\n\nlet's go, raven, and impregnate your female jackdaw.\n\n_27 October 1937_\nIntensidad y altura  \n(For English translation click here)\n\nQuiero escribir, pero me sale espuma,\n\nquiero decir much\u00edsimo y me atollo;\n\nno hay cifra hablada que no sea suma,\n\nno hay pir\u00e1mide escrita, sin cogollo.\n\nQuiero escribir, pero me siento puma;\n\nquiero laurearme, pero me encebollo.\n\nNo hay toz hablada, que no llegue a bruma,\n\nno hay dios ni hijo de dios, sin desarrollo.\n\nV\u00e1monos, pues, por eso, a comer yerba,\n\ncarne de llanto, fruta de gemido,\n\nnuestra alma melanc\u00f3lica en conserva.\n\nV\u00e1monos! V\u00e1monos! Estoy herido;\n\nv\u00e1monos a beber lo ya bebido,\n\nv\u00e1monos, cuervo, a fecundar tu cuerva.\n\n_27 octubre 1937_\nGuitar  \n(For Spanish translation click here)\n\nThe pleasure of suffering, of hating, discolors\n\nthe throat with plastic poisons,\n\nthe swine who implants his magic order,\n\nhis bullish greatness, between the first\n\nand the sixth\n\nand the eight liar, all suffer.\n\nThe pleasure of suffering . . . Who? To whom?\n\nWho, the teeth? . . . To whom, the society?\n\nThe carbide of rage of the gums?\n\nHow to be\n\nand being, without infuriating the neighbor?\n\nYou are worth more than my number, lonely man,\n\nand they're worth more than all the dictionary,\n\nwith its prose in verse,\n\nwith its verse in prose,\n\nyour eagle-like function,\n\nyour mechanical tiger, soft fellow creature.\n\nThe pleasure of suffering,\n\nof waiting hopes on the table,\n\nSunday with all the languages,\n\nSaturday with Chinese hours, Belgiums,\n\nthe week, with two spittings.\n\nThe pleasure of waiting in slippers,\n\nwaiting fearfully behind a verse,\n\nwaiting with power and bad poison;\n\nthe pleasure of suffering, slapped with the left hand of a woman,\n\ndead with a stone in the waist,\n\nand dead between the string and the guitar,\n\ncrying days and singing months.\n\n_28 October 1937_\nGuitarra  \n(For English translation click here)\n\nEl placer de sufrir, de odiar, me ti\u00f1e\n\nla garganta con pl\u00e1sticos venenos,\n\nmas la cerda que implanta su orden m\u00e1gico,\n\nsu grandeza taurina, entre la prima\n\ny la sexta\n\ny la octava mendaz, las sufre todas.\n\nEl placer de sufrir . . . \u00bfQui\u00e9n? \u00bfa qui\u00e9n?\n\n\u00bfqui\u00e9n, las muelas? \u00bfa qui\u00e9n la sociedad,\n\nlos carburos de rabia de la enc\u00eda?\n\n\u00bfC\u00f3mo ser\n\ny estar, sin darle c\u00f3lera al vecino?\n\nVales m\u00e1s que mi n\u00famero, hombre solo,\n\ny valen m\u00e1s que todo el diccionario,\n\ncon su prosa en verso,\n\ncon su verso en prosa,\n\ntu funci\u00f3n \u00e1guila,\n\ntu mecanismo tigre, blando pr\u00f3jimo.\n\nEl placer de sufrir,\n\nde esperar esperanzas en la mesa,\n\nel domingo con todos los idiomas,\n\nel s\u00e1bado con horas chinas, belgas,\n\nla semana, con dos escupitajos.\n\nEl placer de esperar en zapatillas,\n\nde esperar encogido tras de un verso,\n\nde esperar con pujanza y mala po\u00f1a;\n\nel placer de sufrir: zurdazo de hembra\n\nmuerta con una piedra en la cintura\n\ny muerta entre la cuerda y la guitarra,\n\nllorando d\u00edas y cantando meses.\n\n_28 octubre 1937_\nPantheon  \n(For Spanish translation click here)\n\nI have seen yesterday common noises,  \ndying,  \npunctually recede,\n\nwhen I heard the sun setting  \nsadly,\n\nexactly an arc, a rainbow.\n\nI saw the generous time of a minute,  \nimmensely\n\ninsanely tied to the greater time,\n\nwell it was the hour  \nsoftly,\n\nswollen tightly with two hours.\n\nLet yourself understand, to call, the earth  \nearthly;\n\nbrutally denying that way my past,\n\nand if I saw, they listen to me, well, united,\n\nif I touch this machine, that they  \nmay slowly see,\n\ngently, greedily, my darkness.\n\nAnd if I saw in the wound of the answer,  \nclearly,\n\nthe mental wound of the icognite,\n\nif I heard, if I imagine my small windows\n\nnasal, funerals, temporally,  \nfraternally,\n\npiously throw me to the philosophers.\n\nBut no more hasty warping\n\nclearly singing, and no more\n\nruddy bones, the sound of the soul  \nsadly\n\nerected equestrianly in my spine,\n\nsince, in sum, life is,  \nimplacably,\n\nimpartially hideous, I'm sure.\n\n_31 October 1937_\nPante\u00f3n  \n(For English translation click here)\n\nHe visto ayer sonidos generales,  \nmortuoriamente,  \npuntualmente alejarse,\n\ncuando o\u00ed desprenderse del ocaso  \ntristemente,\n\nexactamente un arco, un arco\u00edris.\n\nVi el tiempo generoso del minuto,  \ninfinitamente\n\natado locamente al tiempo grande,\n\npues que estaba la hora  \nsuavemente,\n\npremiosamente henchida de dos horas.\n\nDej\u00f3se comprender, llamar, la tierra  \nterrenalmente;\n\nneg\u00f3se brutalmente, as\u00ed a mi historia,\n\ny si vi, que me escuchen, pues, en bloque,\n\nsi toqu\u00e9 esta mec\u00e1nica, que vean  \nlentamente,\n\ndespacio, vorazmente, mis tinieblas.\n\nY si vi en la lesi\u00f3n de la respuesta,  \nclaramente,\n\nla lesi\u00f3n mentalmente de la inc\u00f3gnita,\n\nsi escuch\u00e9, si pens\u00e9 en mis ventanillas\n\nnasales, funerales, temporales,  \nfraternalmente,\n\npiadosamente echadme a los fil\u00f3sofos.\n\nMas no m\u00e1s inflexi\u00f3n precipitada\n\nen canto llano, y no m\u00e1s\n\nel hueso colorado, el son del alma  \ntristemente\n\nerguida ecuestremente en mi espinazo,\n\nya que, en suma, la vida es  \nimplacablemente,\n\nimparcialmente horrible, estoy seguro.\n\n_31 octubre 1937_\nA Man Is Watching a Woman  \n(For Spanish translation click here)\n\nA man is watching a woman,\n\nis watching her immediately,\n\nwith his of sumptuous land sickness\n\nand sees with both hands\n\nmoving her between two men.\n\nI question myself, oppressing me against\n\nthe enormous, white, steel rib:\n\nWould not have this man,\n\nthen, a child, growing into a father?\n\nAnd this woman, a child\n\nas a builder of her evident sex?\n\nAlthough I do see a child now,\n\na centipede child, energetic, impassioned;\n\nI see that they don't see him\n\nstanding between them, wriggling, dressing itself;\n\nalthough I accept them,\n\nshe in augmentative condition,\n\nhe bending the blond hay.\n\nAnd I cry out, then, without stopping\n\neither of living, without turning\n\neither in the joust I venerate:\n\nHappiness followed!\n\ntoo late by the Father,\n\nby the Son and by the Mother!\n\ncircular instant,\n\nfamiliar, now that no one feels or loves!\n\nFrom what silent, dyed clear light\n\nejects the Song of Songs!\n\nFrom what trunk, the florid carpenter!\n\nFrom what perfect armpit, the fragile oar!\n\nFrom what skull, both skull forwarders!\n\n_2 November 1937_\nUn hombre est\u00e1 mirando a una mujer  \n(For English translation click here)\n\nUn hombre est\u00e1 mirando a una mujer,\n\nest\u00e1 mir\u00e1ndola inmediatamente,\n\ncon su mal de tierra suntuosa\n\ny la mira a dos manos\n\ny la tumba a dos pechos\n\ny la mueve a dos hombres.\n\nPreg\u00fantome entonces, oprimi\u00e9ndome\n\nla enorme, blanca, ac\u00e9rrima costilla:\n\nY este hombre\n\n\u00bfno tuvo a un ni\u00f1o por creciente padre?\n\n\u00bfY esta mujer, a un ni\u00f1o\n\npor constructor de su evidente sexo?\n\nPuesto que un ni\u00f1o veo ahora,\n\nni\u00f1o ciempi\u00e9s, apasionado, en\u00e9rgico;\n\nveo que no le ven\n\nsonarse entre los dos, colear, vestirse;\n\npuesto que los acepto,\n\na ella en condici\u00f3n aumentativa,\n\na \u00e9l en la flexi\u00f3n del heno rubio.\n\nY exclamo entonces, sin cesar ni uno\n\nde vivir, sin volver ni uno\n\na temblar en la justa que venero:\n\n\u00a1Felicidad seguida\n\ntard\u00edamente del Padre,\n\ndel Hijo y de la Madre!\n\n\u00a1Instante redondo,\n\nfamiliar, que ya nadie siente ni ama!\n\n\u00a1De qu\u00e9 deslumbramiento \u00e1fono, tinto,\n\nse ejecuta el cantar de los cantares!\n\n\u00a1De qu\u00e9 tronco, el florido carpintero!\n\n\u00a1De qu\u00e9 perfecta axila, el fr\u00e1gil remo!\n\n\u00a1De qu\u00e9 casco, ambos cascos delanteros!\n\n_2 noviembre 1937_\nThe Nine Monsters  \n(For Spanish translation click here)\n\nUnfortunately, every moment\n\npain grows in the world,\n\nit grows thirty minutes each second, step by step,\n\nand the nature of the pain is twice the pain,\n\nand the condition of martyrdom, carnivorous, ravenous,\n\nit is twice the pain\n\nand the function of the purest herb,\n\ntwice the pain,\n\nand the goodness of being, is twice the pain for us.\n\nNever, human men,\n\nhad there been so much pain in the breast, in the lapel, in the briefcase,\n\nin the glass, in the butcher-shop, in the arithmetic!\n\nNever so much affectionate pain,\n\nnever so nearby attacking the far away,\n\nnever the fire, never\n\nplayed better its game of cold death!\n\nNever, Mr. Minister of Health, was health\n\nmore mortal,\n\nand the migraine extracted too much forehead from the forehead!\n\nAnd the furniture had in its drawer, pain,\n\nthe little lizard in its drawer, pain.\n\nThis misfortune grows, brothers,\n\nquicker than the machine grows into ten machines, and it\n\ngrows with the head of Rousseau, with our beards\n\nthe bad grows with reasons which we ignore\n\nand it floods itself in its own liquid,\n\nwith its own mud and its own solid cloud!\n\nThe suffering changes positions,\n\nin which the aquous humor is vertical\n\nto the pavement,\n\nthe eye sees and this ear hears,\n\nand in this ear sounds nine times the bell at the\n\nhour of the sun, and nine laughs\n\nat the hour of the wheat, and nine female sounds\n\nat the hour of the cry, and nine songs\n\nat the hour of hunger, and nine explosions\n\nand nine beatings, minus a cry.\n\nBrothers, the pain seizes us, brothers\n\nfrom behind, by side face\n\nand drives us crazy in the movies\n\nnailing us to the gramophones,\n\nnot nailing us to beds, to fall perpendicular\n\nto our tickets, to our letters,\n\nand suffering so gravely, one is able to pray . . .\n\nThen, as a result\n\nof the pain, there are\n\nsome who are born, others grown, others die,\n\nand others are born and don't die, others\n\nwithout having been born, die and others\n\nare not born nor die (this is the majority).\n\nAnd also as a result\n\nof the suffering, I am sad\n\nto the head, and saddest to the ankle,\n\nseeing the bread, crucified, the turnip\n\nbloodied,\n\ncrying, to the onion,\n\nto the cereal, generally, flour,\n\nto the salt, turning to dust, the water, flowing\n\nto the wine, behold the man\n\nso pale as snow, the burning sun!\n\nHow, human brothers,\n\nnot telling you that I can't and\n\nthat I'm unable with so much box,\n\nand so many minutes, so\n\nmany lizards, and so\n\nmuch inversion, so far and so much thirst for more thirsts!\n\nMr. Minister of Health! What can I do?\n\nOh, unfortunately, brothers,\n\nthere is, brothers, so much to do.\n\n_3 November 1937_\nLos nueve monstruos  \n(For English translation click here)\n\nY, desgraciadamente,\n\nel dolor crece en el mundo a cada rato,\n\ncrece a treinta minutos por segundo, paso a paso,\n\ny la naturaleza del dolor, es el dolor dos veces\n\ny la condici\u00f3n del martirio, carn\u00edvora, voraz,\n\nes el dolor dos veces\n\ny la funci\u00f3n de la yerba pur\u00edsima, el dolor\n\ndos veces\n\ny el bien de ser, dolernos doblemente.\n\nJam\u00e1s, hombres humanos,\n\nhubo tanto dolor en el pecho, en la solapa, en la cartera,\n\nen el vaso, en la carnicer\u00eda, en la aritm\u00e9tica!\n\nJam\u00e1s tanto cari\u00f1o doloroso,\n\njam\u00e1s tanta cerca arremeti\u00f3 lo lejos,\n\njam\u00e1s el fuego nunca\n\njug\u00f3 mejor su rol de fr\u00edo muerto!\n\nJam\u00e1s, se\u00f1or ministro de salud, fue la salud\n\nm\u00e1s mortal\n\ny la migra\u00f1a extrajo tanta frente de la frente!\n\nY el mueble tuvo en su caj\u00f3n, dolor,\n\nel coraz\u00f3n, en su caj\u00f3n, dolor,\n\nla lagartija, en su caj\u00f3n, dolor.\n\nCrece la desdicha, hermanos hombres,\n\nm\u00e1s pronto que la m\u00e1quina, a diez m\u00e1quinas, y crece\n\ncon la res de Rousseau, con nuestras barbas;\n\ncrece el mal por razones que ignoramos\n\ny es una inundaci\u00f3n con propios l\u00edquidos,\n\ncon propio barro y propia nube s\u00f3lida!\n\nInvierte el sufrimiento posiciones, da funci\u00f3n\n\nen que el humor acuoso es vertical\n\nal pavimento,\n\nel ojo es visto y esta oreja o\u00edda,\n\ny esta oreja da nueve campanadas a la hora\n\ndel rayo, y nueve carcajadas\n\na la hora del trigo, y nueve sones hembras\n\na la hora del llanto, y nueve c\u00e1nticos\n\na la hora del hambre y nueve truenos\n\ny nueve l\u00e1tigos, menos un grito.\n\nEl dolor nos agarra, hermanos hombres,\n\npor detr\u00e1s, de perfil,\n\ny nos aloca en los cinemas,\n\nnos clava en los gram\u00f3fonos,\n\nnos desclava en los lechos, cae perpendicularmente\n\na nuestros boletos, a nuestras cartas;\n\ny es muy grave sufrir, puede uno orar . . .\n\nPues de resultas\n\ndel dolor, hay algunos\n\nque nacen, otros crecen, otros mueren,\n\ny otros que nacen y no mueren, otros\n\nque sin haber nacido, mueren, y otros\n\nque no nacen ni mueren (son los m\u00e1s).\n\nY tambi\u00e9n de resultas\n\ndel sufrimiento, estoy triste\n\nhasta la cabeza, y m\u00e1s triste hasta el tobillo,\n\nde ver al pan, crucificado, al nabo,\n\nensangrentado,\n\nllorando, a la cebolla,\n\nal cereal, en general, harina,\n\na la sal, hecha polvo, al agua, huyendo,\n\nal vino, un ecce-homo,\n\ntan p\u00e1lida a la nieve, al sol tan ardido!\n\n\u00a1C\u00f3mo, hermanos humanos,\n\nno deciros que ya no puedo y\n\nya no puedo con tanto caj\u00f3n,\n\ntanto minuto, tanta\n\nlagartija y tanta\n\ninversi\u00f3n, tanto lejos y tanta sed de sed!\n\nSe\u00f1or Ministro de Salud: \u00bfqu\u00e9 hacer?\n\n\u00a1Ah! desgraciadamente, hombre humanos,\n\nhay, hermanos, much\u00edsimo que hacer.\n\n_3 noviembre 1937_\nA Man Passes with a Loaf of Bread on His Shoulders  \n(For Spanish translation click here)\n\nA man passes with a loaf of bread on his shoulders\n\nAm I going, thereafter, to write about my double?\n\nAnother sits, scratches himself, removes a louse from his armpit, kills it\n\nWith what value talk about psychoanalysis?\n\nAnother has entered my chest with a club in his hand\n\nShall I then talk about Socrates with the doctor?\n\nA cripple walks by giving his arm to a child\n\nAfter that, I'm supposed to read Andr\u00e9 Breton?\n\nAnother shivers with cold, coughs, spits up blood\n\nWill it be a way to refer to the profound I?\n\nAnother searches in mud for bones and for husks\n\nHow then can I write about the infinite?\n\nA bricklayer falls from the roof, dies before breakfast\n\nAfter that how can I innovate the troupe, the metaphor?\n\nA merchant steals a gram from a customer\n\nHow then can I talk about the fourth dimension?\n\nA banker falsifies his balance\n\nWith which face weep in the theater?\n\nAn outcast sleeps with one foot on his shoulder\n\nShall I, later on, speak of Picasso?\n\nSomeone is sobbing at the side of a grave\n\nHow can I get into The Academy?\n\nSomeone cleans his rifle in the kitchen\n\nWith what courage can one speak of the next world?\n\nSomeone walks by counting on his fingers\n\nHow can I speak of the not-I without crying out?\n\n_5 November 1937_\nUn hombre pasa con un pan al hombro  \n(For English translation click here)\n\nUn hombre pasa con un pan al hombro\n\n\u00bfVoy a escribir, despu\u00e9s, sobre mi doble?\n\nOtro se sienta, r\u00e1scase, extrae un piojo de su axila, m\u00e1talo\n\n\u00bfCon qu\u00e9 valor hablar del psicoan\u00e1lisis?\n\nOtro ha entrado en mi pecho con un palo en la mano\n\n\u00bfHablar luego de S\u00f3crates al m\u00e9dico?\n\nUn cojo pasa dando el brazo a un ni\u00f1o\n\n\u00bfVoy, despu\u00e9s, a leer a Andr\u00e9 Bret\u00f3n?\n\nOtro tiembla de fr\u00edo, tose, escupe sangre\n\n\u00bfCabr\u00e1 aludir jam\u00e1s al Yo profundo?\n\nOtro busca en el fango huesos, c\u00e1scaras\n\n\u00bfC\u00f3mo escribir, despu\u00e9s del infinito?\n\nUn alba\u00f1il cae de un techo, muere y ya no almuerza\n\n\u00bfInnovar, luego, el tropo, la met\u00e1fora?\n\nUn comerciante roba un gramo en el peso a un cliente\n\n\u00bfHablar, despu\u00e9s, de cuarta dimensi\u00f3n?\n\nUn banquero falsea su balance\n\n\u00bfCon qu\u00e9 cara llorar en el teatro?\n\nUn paria duerme con el pie a la espalda\n\n\u00bfHablar, despu\u00e9s, a nadie de Picasso?\n\nAlguien va en un entierro sollozando\n\n\u00bfC\u00f3mo luego ingresar a la Academia?\n\nAlguien limpia un fusil en su cocina\n\n\u00bfCon qu\u00e9 valor hablar del m\u00e1s all\u00e1?\n\nAlguien pasa contando con sus dedos\n\n\u00bfC\u00f3mo hablar del no-y\u00f3 sin dar un grito?\n\n_5 noviembre 1937_\nSome Days a Fruitful, Cautious Longing   \nComes Over Me  \n(For Spanish translation click here)\n\nSome days a fruitful, cautious longing comes over me,\n\nto love and kiss affection on both cheeks,\n\nand from afar there comes to me,\n\ndemonstrative, a wish, a different wish of loving, strong,\n\nthe one who hates me, the one who tears up his role, the little boy,\n\nthe one who weeps for one who has been weeping,\n\nking of wine, slave of water\n\nthe one who hides in his own wrath\n\nthe one who sweats, the one who passes by, the one who\n\nshakes himself within my soul.\n\nThe pleasure to arrange a braid of hair\n\nof one who talks to me, the soldier's hair;\n\none's light, the great; one's greatness to the boy.\n\nI want to iron a handkerchief at one\n\nfor the one who cannot weep\n\nand, when I'm sad or when good fortune pains me,\n\nto patch up geniuses and children.\n\nI want to help the good man be a little bad\n\nand have an urge to sit\n\non the right of the left-handed, answer the dumb,\n\ntrying to be useful in what\n\nI can, wanting very much\n\nto wash the cripple's foot,\n\nand help my one-eyed neighbor sleep.\n\nOh, this love of mine, this world-wide love,\n\ninterhuman, parochial, fulfilled!\n\nIt comes just right,\n\nfrom the foundations, from the public groin,\n\nand coming from afar it makes one want to kiss\n\nthe singer's scarf,\n\nto kiss the one who suffers, in his roasting-pan,\n\nthe dumb, in his deaf cranial murmur, dauntless;\n\nthe one who gives me what I had forgotten in my breast,\n\non his Dante, on his Chaplin, on his shoulders.\n\nTo sum up, I should like,\n\nwhen I am on the famous verge of violence,\n\nor when my heart is brave, I should like\n\nto help the one who smiles to laugh,\n\nplace a little bird square on the scruff of a villain's neck,\n\nnurse the sick by provoking them,\n\nbuy to kill from the killer\u2014a dreadful thing\u2014\n\nand be at peace within myself\n\nin everything.\n\n_6 November 1937_\nMe viene, hay d\u00edas, una gana ub\u00e9rrima, pol\u00edtica  \n(For English translation click here)\n\nMe viene, hay d\u00edas, una gana ub\u00e9rrima, pol\u00edtica,\n\nde querer, de besar al cari\u00f1o en sus dos rostros,\n\ny me viene de lejos un querer\n\ndemostrativo, otro querer amar, de grado o fuerza,\n\nal que me odia, al que rasga su papel, al muchachito,\n\na la que llora por el que lloraba,\n\nal rey del vino, al esclavo del agua,\n\nal que ocult\u00f3se en su ira,\n\nal que suda, al que pasa, al que sacude su persona en mi alma.\n\nY quiero, por lo tanto, acomodarle\n\nal que me habla, su trenza; sus cabellos, al soldado;\n\nsu luz, al grande; su grandeza, al chico.\n\nQuiero planchar directamente\n\nun pa\u00f1uelo al que no puede llorar\n\ny, cuando estoy triste o me duele la dicha,\n\nremendar a los ni\u00f1os y a los genios.\n\nQuiero ayudar al bueno a ser su poquillo de malo\n\ny me urge estar sentado a la diestra del zurdo, y responder al mudo,\n\ntratando de serle \u00fatil\n\nen todo lo que puedo y tambi\u00e9n quiero much\u00edsimo\n\nlavarle al cojo el pie,\n\ny ayudarle a dormir al tuerto pr\u00f3ximo.\n\n\u00a1Ah querer, \u00e9ste, el m\u00edo, \u00e9ste, el mundial,\n\ninterhumano y parroquial, provecto!\n\nMe viene a pelo,\n\ndesde el cimiento, desde la ingle p\u00fablica,\n\ny, viniendo de lejos, da ganas de besarle\n\nla bufanda al cantor,\n\ny al que sufre, besarle en su sart\u00e9n,\n\nal sordo, en su rumor craneano, imp\u00e1vido;\n\nal que me da lo que olvid\u00e9 en mi seno,\n\nen su Dante, en su Chaplin, en sus hombros.\n\nQuiero, para terminar,\n\ncuando estoy al borde c\u00e9lebre de la violencia\n\no lleno de pecho el coraz\u00f3n, querr\u00eda\n\nayudar a re\u00edr al que sonr\u00ede,\n\nponerle un pajarillo al malvado en plena nuca,\n\ncuidar a los enfermos enfad\u00e1ndolos,\n\ncomprarle al vendedor,\n\nayudarle a matar al matador\u2014cosa terrible\u2014\n\ny quisiera yo ser bueno conmigo\n\nen todo.\n\n_6 noviembre 1937_\nPalms and Guitar  \n(For Spanish translation click here)\n\nHere between us, now\n\ncome with me, bring my hand to your body\n\nlet's dine together and for an instant let's turn life\n\ninto two lives giving to each a piece of our death.\n\nNow come with yourself, do me the favor\n\nif murmuring my name in the light of the tenebrous night\n\nin which you bring my hand to your soul\n\nso we flee on tiptoe from ourselves.\n\nYes, come to me and to you, yes\n\nwith soft step, seeing both of us with uneven steps,\n\nnothing the gentle farewell.\n\nUntil we return! Until we may return!\n\nUntil we read, ignorant!\n\nUntil we may return, bid us farewell!\n\nThese guns are of no importance to me,\n\nlisten to me;\n\nlisten to me, of what significance are they to me,\n\nif the bullets already circulate in the range of my signature?\n\nWhy do you care for the bullets\n\nif the gun is now smoking in your scent?\n\nThis very day we will weigh\n\nour stars in the arms of a blind man,\n\nand when you sing to me then we'll cry.\n\nThis very day, beautiful one, your gentle step\n\nand your confidence alarms me.\n\nWe will leave ourselves, two by two.\n\nUntil we are blinded!\n\nUntil\n\nwe cry returning so many times!\n\nNow,\n\nbetween us, bring\n\nby the hand your sweet character\n\nand let's eat and pass an instant of life\n\nletting go of the very same death within us.\n\nNow, come with yourself, do me the favor\n\nof singing something\n\nand sing in your soul, clapping hands.\n\nUntil we may return! Until then!\n\nUntil we depart, let's say goodbye!\n\n_8 November 1937_\nPalmas y guitarra  \n(For English translation click here)\n\nAhora, entre nosotros, aqu\u00ed,\n\nven conmigo, trae por la mano a tu cuerpo\n\ny cenemos juntos y pasemos un instante la vida\n\na dos vidas y dando una parte a nuestra muerte.\n\nAhora, ven contigo, hazme el favor\n\nde quejarte en mi nombre y a la luz de la noche teneblosa\n\nen que traes a tu alma de la mano\n\ny hu\u00edmos en puntillas de nosotros.\n\nVen a m\u00ed, s\u00ed, y a ti, s\u00ed,\n\ncon paso par, a vemos a los dos con paso impar,\n\nmarcar el paso de la despedida.\n\n\u00a1Hasta cuando volvamos! \u00a1Hasta la vuelta!\n\n\u00a1Hasta cuando leamos, ignorantes!\n\n\u00a1Hasta cuando volvamos, despid\u00e1monos!\n\n\u00bfQu\u00e9 me importan los fusiles?\n\nesc\u00fachame;\n\nesc\u00fachame, \u00bfqu\u00e9 imp\u00f3rtenme,\n\nsi la bala circula ya en el rango de mi firma?\n\n\u00bfQu\u00e9 te importan a ti las balas,\n\nsi el fusil est\u00e1 humeando ya en tu olor?\n\nHoy mismo pesaremos\n\nen los brazos de un ciego nuestra estrella\n\ny, una vez que me cantes, lloraremos.\n\nHoy mismo, hermosa, con tu paso par\n\ny tu confianza a que lleg\u00f3 mi alarma,\n\nsaldremos de nosotros, dos a dos.\n\n\u00a1Hasta cuando seamos ciegos!\n\n\u00a1Hasta\n\nque lloremos de t\u00e1nto volver!\n\nAhora,\n\nentre nosotros, trae\n\npor la mano a tu dulce personaje\n\ny cenemos juntos y pasemos un instante la vida\n\na dos vidas y dando una parte a nuestra muerte.\n\nAhora, ven contigo, hazme el favor\n\nde cantar algo\n\ny de tocar en tu alma, haciendo palmas.\n\n\u00a1Hasta cuando volvamos! \u00a1Hasta entonces!\n\n\u00a1Hasta cuando partamos, despid\u00e1monos!\n\n_8 noviembre 1937_\nThe Soul That Suffered from Being Its Body  \n(For Spanish translation click here)\n\nYou suffer from an endocrine gland, that's obvious,\n\nor, perhaps,\n\nyou suffer from me, from my tacit tight-lipped sagacity.\n\nYou suffer from the translucent anthropoid there, near,\n\nwhere the tenebrous darkness lies.\n\nYou walk around the sun, clutching onto your soul,\n\nspraying out your corporeal juanes\n\nadjusting out your collar; that's obvious.\n\nYou know what hurts you,\n\nwhat leaps onto your haunch,\n\nwhat lowers through you with a rope to the ground.\n\nYou, poor man, live; don't deny it,\n\nif you die from your age, ah! And from your epoch.\n\nAnd even if you cry, you drink,\n\nand even if you bleed, you nourish your hybrid tooth,\n\nyour sad candle and your parts.\n\nYou suffer, you endure, and again suffer horribly,\n\nunlucky monkey,\n\nlittle Darwinian offspring,\n\nconstable who spies on me, atrocious microbe.\n\nAnd you know this so well,\n\nyou ignore it, bursting into tears.\n\nYou, then, have been born; also\n\nthat can be seen from afar, and unhappy,\n\nso shut up and endure the road you're destined to be on\n\nand questioning your navel: Where? How?\n\nMy friend, you're completely up\n\nto your hair, in the year '38,\n\nNicholas or Santiago, such or which,\n\nwhether you are yourself with your miscarriage or with me\n\nor caught in your enormous liberty,\n\ndragged along by your autonomous Hercules . . .\n\nbut if you calculate on your fingers up to two,\n\nit's worse; don't deny it, little brother.\n\nNo? Yes? Nevertheless no?\n\nPoor monkey! . . . Gimme your paw! . . . No. The hand, I say.\n\nCheers! And suffer!\n\n_8 November 1937_\nEl alma que sufri\u00f3 de ser su cuerpo  \n(For English translation click here)\n\nT\u00fa sufres de una gl\u00e1ndula endocr\u00ednica, se ve,\n\no, quiz\u00e1,\n\nsufres de m\u00ed, de mi sagacidad escueta, t\u00e1cita.\n\nT\u00fa padeces del di\u00e1fano antropoide, all\u00e1, cerca,\n\ndonde est\u00e1 la tiniebla tenebrosa.\n\nT\u00fa das vuelta al sol, agarr\u00e1ndote el alma,\n\nextendiendo tus juanes corporales\n\ny ajust\u00e1ndote el cuello; eso se ve.\n\nT\u00fa sabes lo que te duele,\n\nlo que te salta al anca,\n\nlo que baja por ti con soga al suelo.\n\nT\u00fa, pobre hombre, vives; no lo niegues,\n\nsi mueres; no lo niegues,\n\nsi mueres de tu edad \u00a1ay! y de tu \u00e9poca.\n\nY, aunque llores, bebes,\n\ny, aunque sangres, alimentas a tu h\u00edbrido colmillo,\n\na tu vela tristona y a tus partes.\n\nT\u00fa sufres, t\u00fa padeces y t\u00fa vuelves a sufrir horriblemente,\n\ndesgraciado mono,\n\njovencito de Darwin,\n\nalguacil que me atisbas, atroc\u00edsimo microbio.\n\nY t\u00fa lo sabes a tal punto,\n\nque lo ignoras, solt\u00e1ndote a llorar.\n\nT\u00fa, luego, has nacido; eso\n\ntambi\u00e9n se ve de lejos, infeliz y c\u00e1llate,\n\ny soportas la calle que te dio la suerte\n\ny a tu ombligo interrogas: \u00bfd\u00f3nde? \u00bfc\u00f3mo?\n\nAmigo m\u00edo, est\u00e1s completamente, .\n\nhasta el pelo, en el a\u00f1o treinta y ocho,\n\nnicol\u00e1s o santiago, tal o cual,\n\nest\u00e9s contigo o con tu aborto o conmigo\n\ny cautivo en tu enorme libertad,\n\narrastrado por tu h\u00e9rcules aut\u00f3nomo . . .\n\nPero si t\u00fa calculas en tus dedos hasta dos,\n\nes peor; no lo niegues, hermanito.\n\n\u00bfQue n\u00f3? \u00bfQue s\u00ed, pero que n\u00f3?\n\n\u00a1Pobre mono! . . . \u00a1Dame la pata! . . . No. La mano, he dicho.\n\n\u00a1Salud! \u00a1Y sufre!\n\n_8 noviembre 1937_\nThe One Who Will Come Has Just Passed By  \n(For Spanish translation click here)\n\nThe one who will come has just passed by,\n\nforbidden, to sit himself in my triple evolution;\n\nhe just passed like a criminal.\n\nHe just seated himself over there,\n\nat body's length from my soul,\n\nhe who came is an ass trying to weaken me;\n\nhe just seated himself standing, livid.\n\nHe just gave me what's finished,\n\nthe heat of fire and the immense pronoun\n\nthat the animal nurtured under his tail.\n\nHe just\n\nexpressed his doubts to me on distant hypotheses\n\nthat he separates, even more, with a glance.\n\nHe just finished doing the honors to the good\n\nby virtue of the vile pachyderm\n\nfor what's dreamt in me and in him murdered.\n\nHe just finished fixing me (there is no first)\n\nto his second affliction in full loins,\n\nand his third sweat in full tears.\n\nThe one who comes has just passed without having come.\n\n_12 November 1937_\nAcaba de pasar el que vendr\u00e1  \n(For English translation click here)\n\nAcaba de pasar el que vendr\u00e1\n\nproscrito, a sentarse en mi triple desarrollo;\n\nacaba de pasar criminalmente.\n\nAcaba de sentarse m\u00e1s ac\u00e1,\n\na un cuerpo de distancia de mi alma,\n\nel que vino en un asno a enflaquecerme;\n\nacaba de sentarse de pie, l\u00edvido.\n\nAcaba de darme lo que est\u00e1 acabado,\n\nel calor del fuego y el pronombre inmenso\n\nque el animal cri\u00f3 bajo su cola.\n\nAcaba\n\nde expresarme su duda sobre hip\u00f3tesis lejanas\n\nque \u00e9l aleja, a\u00fan m\u00e1s, con la mirada.\n\nAcaba de hacer al bien los honores que le tocan\n\nen virtud del infame paquidermo,\n\npor lo so\u00f1ado en mi y en \u00e9l matado.\n\nAcaba de ponerme (no hay primera)\n\nsu segunda aflixi\u00f3n en plenos lomos\n\ny su tercer sudor en plena l\u00e1grima.\n\nAcaba de pasar sin haber venido.\n\n_12 noviembre 1937_\nThe Evil Man Might Come with a Throne on His Shoulder  \n(For Spanish translation click here)\n\nThe evil man might come with a throne on his shoulder,\n\nand the good accompanying the evil, walking,\n\nthey said, \"yes\" the sermon, \"no\" to the prayer\n\nand it may cut the road in two rocks . . .\n\nI may begin by climbing the mountain,\n\nby oar the sprout, by rudder the cedar\n\nand they will wait two-hundred to sixty\n\nand the meat may return to its three titles . . .\n\nThere's too much snow in the idea of fire,\n\nthe corpse will go to bed to look at us,\n\nthe lightning being with loud thunderclaps,\n\nand the saurians will arch to be birds . . .\n\nIt will lack excavation near the dung,\n\nshipwreck to the river in order to slide,\n\njail for the free man, in order to be it,\n\nand an atmosphere to the sky and iron to gold . . .\n\nThey will demonstrate discipline, smell, the wild beast\n\nmay paint the passion of a soldier,\n\nI may be in pain because learning of the rushes,\n\nthe lie that infects and helps me . . .\n\nIt might happen like this and placing\n\nwith what hand to awaken?\n\nwith what foot to die?\n\nwith what to be poor?\n\nwith what voice to silence?\n\nwith how much to understand and, then, to whom?\n\nNot to forget nor remember,\n\nthat by closing it too often, the door has been stolen,\n\nand of suffering so little, I am very resentful,\n\nand with thinking so much, I'm now lacking a mouth.\n\n_19 November 1937_\nViniere el malo, con un trono al hombro  \n(For English translation click here)\n\nViniere el malo, con un trono al hombro,\n\ny el bueno, a acompa\u00f1ar al malo a andar,\n\ndijeren \u00abs\u00ed\u00bb el serm\u00f3n, \u00abno\u00bb la plegaria\n\ny cortare el camino en dos la roca . . .\n\nComenzare por monte la monta\u00f1a,\n\npor remo el tallo, por tim\u00f3n el cedro\n\ny esperaren doscientos a sesenta\n\ny volviere la carne a sus tres t\u00edtulos . . .\n\nSobrare nieve en la noci\u00f3n del fuego,\n\nse acostare el cad\u00e1ver a mirarnos,\n\nla centella a ser trueno corpulento\n\ny se arquearen los saurios a ser aves . . .\n\nFaltare excavaci\u00f3n junto al esti\u00e9rcol,\n\nnaufragio al r\u00edo para resbalar,\n\nc\u00e1rcel al hombre libre, para serlo,\n\ny una atm\u00f3sfera al cielo, y hierro al oro . . .\n\nMostraren disciplina, olor, las fieras,\n\nse pintare el enojo de soldado,\n\nme dolieren el junco que aprend\u00ed,\n\nla mentira que inf\u00e9ctame y soc\u00f3rreme . . .\n\nSucediere ello as\u00ed y as\u00ed poni\u00e9ndolo,\n\n\u00bfcon qu\u00e9 mano despertar?\n\n\u00bfcon qu\u00e9 pie morir?\n\n\u00bfcon qu\u00e9 ser pobre?\n\n\u00bfcon qu\u00e9 voz callar?\n\n\u00bfcon cu\u00e1nto comprender, y, luego, a qui\u00e9n?\n\nNo olvidar ni recordar\n\nque por mucho cerrarla, rob\u00e1ronse la puerta,\n\ny de sufrir tan poco estoy muy resentido\n\ny de t\u00e1nto pensar, no tengo boca.\n\n_19 noviembre 1937_\nThat Is the Place Where I Put On  \n(For Spanish translation click here)\n\nThe place where I put on\n\nmy pants, is a house where\n\nI take off my shirt loudly\n\nand where I have a floor, a soul, and a map of my Spain.\n\nJust now I was talking to myself\n\nabout myself, and I put\n\na large piece of bread on a small book,\n\nand then, later, I may have moved it,\n\nwishing to hum a bit, the right side\n\nof life to the left side;\n\nmuch later I washed everything, my stomach\n\nvigorous worthy;\n\nI turned around to see what was dirty,\n\nI scraped as I turned the parts of me that are closest,\n\nI've arranged very well the map that\n\nnodding with sleep or crying, I don't know.\n\nMy house, unfortunately, is a house\n\na floor at most, where lives\n\nwith its inscription my beloved little spoon,\n\nmy dear skeleton already without song,\n\nthe razor, a permanent cigar.\n\nTruthfully, when I think\n\nof what life is,\n\nI can't avoid saying to Georgette,\n\nfor the purpose of eating something enjoyable and taking a walk,\n\nin the evening, to buy a good newspaper,\n\nguarding the day for when there is not one,\n\nalso a night for when there is one,\n\n(they express it this way in Peru\u2014please excuse me);\n\nthe same way, I suffer with great care,\n\nfor not crying or screaming, still the eyes\n\nindependent of each other, possess your needs,\n\nI mean, your function, something\n\nthat slips from the soul and the soul falls.\n\nHaving been through\n\nfifteen years; fifteen before and fifteen after,\n\nactually, one feels silly,\n\nit's natural, but in vain, what can I do!\n\nAnd what if I stopped doing, which is worse!\n\nBut to live, but to come into being,\n\nwhat is one in a million\n\nof breads, between thousands of wines, between hundreds of mouths,\n\nbetween the sun and its ray that comes from the moon\n\nand between the mass, the bread, the wine and my soul.\n\nToday is Sunday and for that reason\n\nan idea comes to my head, to my breast, a cry\n\nand to my throat like a great tumor.\n\nToday is Sunday and this\n\nhas many centuries; otherwise\n\nit could be, Monday, perhaps, and the idea should come to my heart,\n\nto the brain, and the weeping\n\nand to the throat, a dreadful appetite to drown\n\nwhat I feel now\n\nas the man that I am and that which I've suffered.\n\n_21 November 1937_\nEllo es que el lugar donde me pongo  \n(For English translation click here)\n\nEllo es que el lugar donde me pongo\n\nel pantal\u00f3n, es una casa donde\n\nme quito la camisa en alta voz\n\ny donde tengo un suelo, un alma, un mapa de mi Espa\u00f1a.\n\nAhora mismo hablaba\n\nde m\u00ed conmigo, y pon\u00eda\n\nsobre un peque\u00f1o libro un pan tremendo\n\ny he, luego, hecho el traslado, he trasladado,\n\nqueriendo canturrear un poco, el lado\n\nderecho de la vida al lado izquierdo;\n\nm\u00e1s tarde, me he lavado todo, el vientre,\n\nbriosa, dignamente;\n\nhe dado vuelta a ver lo que se ensucia,\n\nhe raspado lo que me lleva tan cerca\n\ny he ordenado bien el mapa que\n\ncabeceaba o lloraba, no lo s\u00e9.\n\nMi casa, por desgracia, es una casa,\n\nun suelo por ventura, donde vive\n\ncon su inscripci\u00f3n mi cucharita amada,\n\nmi querido esqueleto ya sin letras,\n\nla navaja, un cigarro permanente.\n\nDe veras, cuando pienso\n\nen lo que es la vida,\n\nno puedo evitar de dec\u00edrselo a Georgette,\n\na fin de comer algo agradable y salir,\n\npor la tarde, comprar un buen peri\u00f3dico.\n\nguardar un d\u00eda para cuando no haya,\n\nuna noche tambi\u00e9n, para cuando haya\n\n(as\u00ed se dice en el Per\u00fa\u2014me excuso);\n\ndel mismo modo, sufro con gran cuidado,\n\na fin de no gritar o de llorar, ya que los ojos\n\nposeen, independientemente de uno, sus pobrezas,\n\nquiero decir, su oficio, algo\n\nque resbala del alma y cae al alma.\n\nHabiendo atravesado\n\nquince a\u00f1os; despu\u00e9s, quince, y, antes, quince,\n\nuno se siente, en realidad, tontillo,\n\nes natural, por lo dem\u00e1s \u00a1qu\u00e9 hacer!\n\n\u00bfY qu\u00e9 dejar de hacer, que es lo peor?\n\nSino vivir, sino llegar\n\na ser lo que es uno entre millones\n\nde panes, entre miles de vinos, entre cientos de bocas,\n\nentre el sol y su rayo que es de luna\n\ny entre la misa, el pan, el vino y mi alma.\n\nHoy es domingo y, por eso,\n\nme viene a la cabeza la idea, al pecho el llanto\n\ny a la garganta, as\u00ed como un gran bulto.\n\nHoy es domingo, y esto\n\ntiene muchos siglos; de otra manera,\n\nser\u00eda, quiz\u00e1, lunes, y vendr\u00edame al coraz\u00f3n la idea,\n\nal seso, el llanto\n\ny a la garganta, una gana espantosa de ahogar\n\nlo que ahora siento,\n\ncomo un hombre que soy y que he sufrido.\n\n_21 noviembre 1937_\nAnother Bit of Calm, Comrade  \n(For Spanish translation click here)\n\nAnother bit of calm, comrade;\n\na very immense, septentrional, complete,\n\nferocious, by the little calm,\n\nto the minor service of each triumph\n\nand in the fearless servitude of failure.\n\nYou have excess rapture, and there's not\n\nas much madness in the mind as in\n\nyour muscular rationality, and there's not\n\nnothing more erroneously rational than your experience.\n\nBut speaking more clearly\n\nand thinking it in gold you are steel\n\non condition you're not\n\na fool and refuse\n\nto be enthusiastic for death so much\n\nand for the life, with only your tomb.\n\nIt's necessary that you know how\n\nto contain your volume, without running, without grieving\n\nyour entire molecular reality\n\nand beyond that, the march of your life\n\nand near here, your legends die.\n\nYou're made of steel, as they say,\n\nas long as you don't tremble and don't\n\nexplode, godfather\n\nof my calculation, emphatic godson\n\nof my luminous salts!\n\nGo right ahead; resolve,\n\nconsider your crisis, sum it up, continue,\n\ntrim it, diminish it, crumple it up;\n\nthe destiny, the intimate energies, the fourteen\n\nverses of bread: how many diplomas\n\nand powers, to the authentic brink of your passion!\n\nHow many details in synthesis you're made of!\n\nHow much identical pressure at your feet!\n\nHow much rigor, and how much protection!\n\nIt's idiotic\n\nthis method of enduring,\n\nthat modulated and virulent light,\n\nin which you alone calmly make serious\n\nsigns, characteristically fatal.\n\nLet's see, man;\n\nlet me know what's happening to me, in spite of my gripe,\n\nI enact your strict orders.\n\n_28 November 1937_\nOtro poco de calma, camarada  \n(For English translation click here)\n\nOtro poco de calma, camarada;\n\nun mucho inmenso, septentrional, completo,\n\nferoz, de calma chica,\n\nal servicio menor de cada triunfo\n\ny en la audaz servidumbre del fracaso.\n\nEmbriaguez te sobra, y no hay\n\ntanta locura en la raz\u00f3n, como este\n\ntu raciocinio muscular, y no hay\n\nm\u00e1s racional error que tu experiencia.\n\nPero, hablando m\u00e1s claro\n\ny pens\u00e1ndolo en oro, eres de acero,\n\na condici\u00f3n que no seas\n\ntonto y rehuses\n\nentusiasmarte por la muerte t\u00e1nto\n\ny por la vida, con tu sola tumba.\n\nNecesario es que sepas\n\ncontener tu volumen sin correr, sin afligirte,\n\ntu realidad molecular entera\n\ny m\u00e1s all\u00e1, la marcha de tus vivas\n\ny m\u00e1s ac\u00e1, tus mueras legendarios.\n\nEres de acero, como dicen,\n\ncon tal que no tiembles y no vayas\n\na reventar, compadre\n\nde mi c\u00e1lculo, enf\u00e1tico ahijado\n\nde mis sales luminosas!\n\nAnda, no m\u00e1s; resuelve,\n\nconsidera tu crisis, suma, sigue,\n\nt\u00e1jala, b\u00e1jala, \u00e1jala;\n\nel destino, las energ\u00edas \u00edntimas, los catorce\n\nvers\u00edculos del pan: \u00a1cu\u00e1ntos diplomas\n\ny poderes, al borde fehaciente de tu arranque!\n\n\u00a1Cu\u00e1nto detalle en s\u00edntesis, contigo!\n\n\u00a1Cu\u00e1nta presi\u00f3n id\u00e9ntica, a tus pies!\n\n\u00a1Cu\u00e1nto rigor y cu\u00e1nto patrocinio!\n\nEs idiota\n\nese m\u00e9todo de padecimiento,\n\nesa luz modulada y virulenta,\n\nsi con s\u00f3lo la calma haces se\u00f1ales\n\nserias, caracter\u00edsticas, fatales.\n\nVamos a ver, hombre;\n\ncu\u00e9ntame lo que me pasa,\n\nque yo, aunque grite, estoy siempre a tus \u00f3rdenes.\n\n_28 noviembre 1937_\n\n_C\u00e9sar Vallejo_\n\nPhoto: Juan Larrea Collection\/Archives Malanga\n_from_\n\nESPA\u00d1A, APARTA DE M\u00cd ESTE C\u00c1LIZ\n\n_September, October, November 1937_\nI  \nHymn to the Volunteers of the Republic  \n(For Spanish translation click here)\n\nVolunteer for Spain, militant hero,\n\nyour reliable bones, when your heart marches to die,\n\nwhen it marches to kill with its global agony,\n\nI truly don't know\n\nwhat to do, where to stand; I make room, write, applaud,\n\ncry, scrutinize, shatter, extinguish things, I say\n\nto my heart that it's over, to the good that comes,\n\nand I try to disgrace myself;\n\nuncover my impersonal forehead till I touch\n\nthis vessel of blood, restrain myself,\n\nmy size obstructed by the famous architect's decline,\n\nthrough which the animal honoring me, honors itself;\n\nmy instincts flow back to their ropes,\n\njoy smokes before my tomb,\n\nand again, without knowing what to do, without anything, leave me,\n\nfrom my white stone, leave me\n\nalone,\n\na hunched-over human, closer, much further off,\n\nunable to hold in my hands your ecstasy,\n\nwith your cutting-edge swiftness, I offer my humble self\n\ncostumed in greatness against your double-edged speed!\n\nOne intent, clear and fertile day\n\nOh biennale, you of the lugubrious and supplicant half-years,\n\nthrough which gunpowder went biting its elbows!\n\nOh bitter pain, and splintered rock more bitter still!\n\nOh bits clenched in the people's teeth!\n\nOh day in their captive match, prayed in fury\n\nand sovereignty, fulfilled and circular,\n\ntheir birthright shut with the hands of choice;\n\nthe despots who drag their padlocks,\n\nand in the padlocks, their dead bacterias . . .\n\nBattles? No! Passions! And passions preceded\n\nby sorrows with grids of hopes,\n\nby sorrows common with the hopes of men!\n\ndeath and passion for peace, the populace!\n\ndeath and passion at war among the olive groves, let's understand   \neach other!\n\nAs in your breath the winds change their atmospheric needle,\n\nand in your breast, tombs exchanging keys,\n\nyour frontal bone rising itself to the first kingdom of martyrdom.\n\nThe world exclaims: \"These are Spanish matters!\" And it's true.  \nConsider,\n\nduring a balance, point-blank,\n\nCalderon, asleep on the tail of a dead amphibian,\n\nor Cervantes, saying: \"My kingdom is of this world, but\n\nalso of the next\": the sword's point and edge on two bits of paper!\n\nContemplate Goya kneeling in prayer before a mirror,\n\nColl, the paladin in whose Cartesian assault\n\none could see his easy step had the sweat of the clouds walking slowly,\n\nor Quevedo, that instantaneous grandfather of the dynamitens,\n\nor Cajal, devoured by his infinite smallness, or still\n\nTeresa, woman, dying because she doesn't die,\n\nor Lina Odena, conflicted on more than one point with Teresa . . .\n\n(Every act or cheerful voice comes from the people,\n\nand goes back toward them,\n\ndirectly or conveyed\n\nby incessant fragments, by the pink smoke\n\nof bitter passwords which failed.)\n\nSo your child, civilian fighter, your bloodless child,\n\nstirred by a motionless stone,\n\nsacrifices itself, vanishes,\n\nfalls away upward and through its incombustible flame rises,\n\nclimbs to the weak,\n\ngiving Spains to the bulls,\n\nbulls to the doves . . .\n\nThe universal dying of the proletarian in what frenetic harmony\n\nwill be ended your greatness, your misery, your propelling whirlwind,\n\nyour methodical violence, your practical and theoretical chaos,\n\nyour Dantesque and very Spanish desire of loving your enemy, even so betraying him!\n\nLiberator girded with shackles,\n\nwithout whose effort the unholdable extensions would continue till this very day,\n\nnails would wander headless,\n\nancient, slow, flushed, the day\n\nour beloved helmets unburied!\n\nfarmer falling with your green leafage for the man,\n\nwith the social inflection of your little finger,\n\nwith your ox standing with his heels dug in,\n\nalso with your word lashed to a pole,\n\nand your rented sky\n\nand with the day driven into your fatigue\n\nand caught under your nails marching!\n\nBuilders,\n\nfarmers, civilians, and soldiers\n\nof active teeming eternity; it was written\n\nthat you would make light, shielding\n\nyour eyes with the death;\n\nthat, in the cruel fall of your mouths,\n\nabundance would come on seven platters, everything\n\nin the world would be suddenly turned into gold,\n\nand the gold,\n\nfabulous beggars of your own secretion of blood,\n\nso the gold would at that time be of gold!\n\nAll men will love each other\n\nand will eat together from the corners of your sad handkerchief\n\nand will drink together in the name\n\nof your accursed throats!\n\nThey will take a rest from this run walking to the foot,\n\nthey will weep thinking of your orbits, they will be fortunate\n\nin and to the sound\n\nof your atrocious return, blooming, innate,\n\nthey will settle up their affairs of the day, their dreamed\n\nand sung figures!\n\nThe same shoes will fit the man who ascends\n\nwithout roads to his body\n\nand to that man who climbs down to the form of his soul!\n\nEmbracing, the dumb will speak, the cripple will walk!\n\nReturning, the blind will see,\n\nand the deaf palpitating will hear!\n\nThe ignorant will be wise and the wise will be ignorant!\n\nKisses that could not be given are given!\n\nOnly death will die! The ant\n\nwill bring crumbs of bread to the elephant shackled\n\nto his brutal delicacy;\n\nthe aborted children will be born again perfect, spatial,\n\nand all men will toil,\n\nall men will bear fruit,\n\nall men will embrace once again!\n\nWorkman, savior, our redeemer,\n\nbrother, forgive us our trespasses!\n\nAs the drum rolls in its adagios;\n\n\"So that your back never be so ephemeral!\n\nThat ever so changing, your profile!\"\n\nItalian volunteer, among whose animals of battle\n\nthe Abyssinian lion is limping!\n\nSoviet volunteer, marching at the head of your universal chest!\n\nVolunteers from the south, from the north, from the east,\n\nand you, western man, closing the funereal song of the dawn!\n\nKnown soldier, whose name marches in the sound of an embrace!\n\nWarrior raised by the earth, arming yourself\n\nwith dust,\n\nshod with positive magnets,\n\nyour personal beliefs in force,\n\nyour character different, your intimate ferule,\n\ncomplexion immediate,\n\nyour language put on your shoulders,\n\nand your soul crowned with pebbles!\n\nVolunteer swathed in your cold,\n\ntemperate, or torrid zone,\n\nheroes all around,\n\nvictim in a column of conquerors;\n\nin Spain, in Madrid, you are called\n\nto kill. Volunteers in the service of life!\n\nBecause they kill in Spain, others kill\n\nthe boy, and his toy, which comes to a stop,\n\nthe resplendent mother Rosenda,\n\nthe old Adam who talked aloud to his horse,\n\nand to the dog that used to sleep on the stairs.\n\nThey kill the book, fire on its auxiliary verbs,\n\nat its defenseless first page!\n\nThey kill the exact case of the statue,\n\nthe wise man, his stick, his colleague,\n\nthe barber next door\u2014all right he might have possibly cut me,\n\nbut he was a good man, and, soon, an unfortunate one,\n\nthe beggar who yesterday was singing opposite,\n\nthe nurse who today passed crying,\n\nthe priest staggering under the stubborn height of his knees . . .\n\nVolunteers,\n\nfor life, for the good ones, kill\n\ndeath, kill the evil ones.\n\nDo it for the freedom of all,\n\nfor the exploited and the exploiter,\n\nfor painless peace\u2014I sense it\n\nwhen I sleep at the foot of my forehead\n\nand more when I run shouting\u2014\n\nand I do it, I keep saying to you,\n\nfor the illiterate to whom I write,\n\nfor the barefoot genius with his flocks,\n\nfor the fallen comrades,\n\ntheir ashes embracing the corpse on the road!\n\nThat you\n\nvolunteers for Spain and for the world, should come,\n\nI dreamed that I was good, and that I should see\n\nyour blood, volunteers!\n\nIt's a long heart's time since many griefs and\n\ncamels of an age came to pray.\n\nToday the good, burning, marches on your side,\n\nand the reptiles of immanent eyelids follow you with love\n\nand two steps behind, one step away,\n\nthe direction of water rushing to see its limit before burning away.\nI  \nHimno a los voluntarios de la Rep\u00fablica  \n(For English translation click here)\n\nVoluntario de Espa\u00f1a, miliciano\n\nde huesos fidedignos, cuando marcha a morir tu coraz\u00f3n,\n\ncuando marcha a matar con su agon\u00eda\n\nmundial, no s\u00e9 verdaderamente\n\nqu\u00e9 hacer, d\u00f3nde ponerme; corro, escribo, aplaudo,\n\nlloro, atisbo, destrozo, apagan, digo\n\na mi pecho que acabe, al que bien, que venga,\n\ny quiero desgraciarme;\n\ndesc\u00fabrome la frente impersonal hasta tocar\n\nel vaso de la sangre, me detengo,\n\ndetienen mi tama\u00f1o esas famosas ca\u00eddas de arquitecto\n\ncon las que se honra el animal que me honra;\n\nrefluyen mis instintos a sus sogas,\n\nhumea ante mi tumba la alegr\u00eda\n\ny, otra vez, sin saber qu\u00e9 hacer, sin nada, d\u00e9jame,\n\ndesde mi piedra en blanco, d\u00e9jame,\n\nsolo,\n\ncuadrumano, m\u00e1s ac\u00e1, mucho m\u00e1s lejos,\n\nal no caber entre mis manos tu largo rato ext\u00e1tico,\n\nquiebro con tu rapidez de doble filo\n\nmi peque\u00f1ez en traje de grandeza!\n\nUn d\u00eda diurno, claro, atento, f\u00e9rtil\n\n\u00a1oh bienio, el de los l\u00f3bregos semestres suplicantes,\n\npor el que iba la p\u00f3lvora mordi\u00e9ndose los codos!\n\n\u00a1oh dura pena y m\u00e1s duros pedernales!\n\n\u00a1oh frenos los tascados por el pueblo!\n\nUn d\u00eda prendi\u00f3 el pueblo su f\u00f3sforo cautivo, or\u00f3 de c\u00f3lera\n\ny soberanamente pleno, circular,\n\ncerr\u00f3 su natalicio con manos electivas;\n\narrastraban candado ya los d\u00e9spotas\n\ny en el candado, sus bacterias muertas . . .\n\n\u00bfBatallas? \u00a1No! Pasiones. Y pasiones precedidas\n\nde dolores con rejas de esperanzas,\n\nde dolores de pueblos con esperanzas de hombres!\n\n\u00a1Muerte y pasi\u00f3n de paz, las populares!\n\n\u00a1Muerte y pasi\u00f3n guerreras entre olivos, entend\u00e1monos!\n\nTal en tu aliento cambian de agujas atmosf\u00e9ricas los vientos\n\ny de llave las tumbas en tu pecho,\n\ntu frontal elev\u00e1ndose a primera potencia de martirio.\n\nEl mundo exclama: \u00ab\u00a1Cosas de espa\u00f1oles!\u00bb Y es verdad.  \nConsideremos,\n\ndurante una balanza, a quemarropa,\n\na Calder\u00f3n, dormido sobre la cola de un anfibio muerto\n\no a Cervantes, diciendo: \u00abMi reino es de este mundo, pero\n\ntambi\u00e9n del otro\u00bb: \u00a1punta y filo en dos papeles!\n\nContemplemos a Goya, de hinojos y rezando ante un espejo,\n\na Coll, el palad\u00edn en cuyo asalto cartesiano\n\ntuvo un sudor de nube el paso llano\n\no a Quevedo, ese abuelo instant\u00e1neo de los dinamiteros\n\no a Cajal, devorado por su peque\u00f1o infinito, o todav\u00eda\n\na Teresa, mujer que muere porque no muere\n\no a Lina Odena, en pugna en m\u00e1s de un punto con Teresa . . .\n\n(Todo acto o voz genial viene del pueblo\n\ny va hacia \u00e9l, de frente o transmitidos\n\npor incesantes briznas, por el humo rosado\n\nde amargas contrase\u00f1as sin fortuna)\n\nAs\u00ed tu criatura, miliciano, as\u00ed tu exang\u00fce criatura,\n\nagitada por una piedra inm\u00f3vil,\n\nse sacrifica, ap\u00e1rtase,\n\ndecae para arriba y por su llama incombustible sube,\n\nsube hasta los d\u00e9biles,\n\ndistribuyendo espa\u00f1as a los toros,\n\ntoros a las palomas . . .\n\nProletario que mueres de universo, \u00a1en qu\u00e9 fren\u00e9tica armon\u00eda\n\nacabar\u00e1 tu grandeza, tu miseria, tu vor\u00e1gine impelente,\n\ntu violencia met\u00f3dica, tu caos te\u00f3rico y pr\u00e1ctico, tu gana\n\ndantesca, espa\u00f1ol\u00edsima, de amar, aunque sea a traici\u00f3n,   \na tu enemigo!\n\n\u00a1Liberador ce\u00f1ido de grilletes,\n\nsin cuyo esfuerzo hasta hoy continuar\u00eda sin asas la extensi\u00f3n,\n\nvagar\u00edan ac\u00e9falos los clavos,\n\nantiguo, lento, colorado, el d\u00eda,\n\nnuestros amados cascos, insepultos!\n\n\u00a1Campesino ca\u00eddo con tu verde follaje por el hombre,\n\ncon la inflexi\u00f3n social de tu me\u00f1ique,\n\ncon tu buey que se queda, con tu f\u00edsica,\n\ntambi\u00e9n con tu palabra atada a un palo\n\ny tu cielo arrendado\n\ny con la arcilla inserta en tu cansancio\n\ny la que estaba en tu u\u00f1a, caminando!\n\n\u00a1Constructores\n\nagr\u00edcolas, civiles y guerreros,\n\nde la activa, hormigueante eternidad: estaba escrito\n\nque vosotros har\u00edais la luz, entornando\n\ncon la muerte vuestros ojos;\n\nque, a la ca\u00edda cruel de vuestras bocas,\n\nvendr\u00e1 en siete bandejas la abundancia, todo\n\nen el mundo ser\u00e1 de oro s\u00fabito\n\ny el oro,\n\nfabulosos mendigos de vuestra propia secreci\u00f3n de sangre,\n\ny el oro mismo ser\u00e1 entonces de oro!\n\n\u00a1Se amar\u00e1n todos los hombres\n\ny comer\u00e1n tomados de las puntas de vuestros pa\u00f1uelos tristes\n\ny beber\u00e1n en nombre\n\nde vuestras gargantas infaustas!\n\nDescansar\u00e1n andando al pie de esta carrera,\n\nsollozar\u00e1n pensando en vuestras \u00f3rbitas, venturosos\n\nser\u00e1n y al son\n\nde vuestro atroz retorno, florecido, innato,\n\najustar\u00e1n ma\u00f1ana sus quehaceres, sus figuras so\u00f1adas y cantadas!\n\n\u00a1Unos mismos zapatos ir\u00e1n bien al que asciende\n\nsin v\u00edas a su cuerpo\n\ny al que baja hasta la forma de su alma!\n\n\u00a1Entrelaz\u00e1ndose hablar\u00e1n los mudos, los tullidos andar\u00e1n!\n\n\u00a1Ver\u00e1n, ya de regreso, los ciegos\n\ny palpitando escuchar\u00e1n los sordos!\n\n\u00a1Sabr\u00e1n los ignorantes, ignorar\u00e1n los sabios!\n\n\u00a1Ser\u00e1n dados los besos que no pudisteis dar!\n\n\u00a1S\u00f3lo la muerte morir\u00e1! \u00a1La hormiga\n\ntraer\u00e1 pedacitos de pan al elefante encadenado\n\na su brutal delicadeza; volver\u00e1n\n\nlos ni\u00f1os abortados a nacer perfectos, espaciales\n\ny trabajar\u00e1n todos los hombres,\n\nengendrar\u00e1n todos los hombres,\n\ncomprender\u00e1n todos los hombres!\n\n\u00a1Obrero, salvador, redentor nuestro,\n\nperd\u00f3nanos, hermano, nuestras deudas!\n\nComo dice un tambor al redoblar, en sus adagios:\n\nqu\u00e9 jam\u00e1s tan ef\u00edmero, tu espalda!\n\nqu\u00e9 siempre tan cambiante, tu perfil!\n\n\u00a1Voluntario italiano, entre cuyos animales de batalla\n\nun le\u00f3n abisinio va cojeando!\n\n\u00a1Voluntario sovi\u00e9tico, marchando a la cabeza de tu pecho universal!\n\n\u00a1Voluntarios del sur, del norte, del oriente\n\ny t\u00fa, el occidental, cerrando el canto f\u00fanebre del alba!\n\n\u00a1Soldado conocido, cuyo nombre\n\ndesfila en el sonido de un abrazo!\n\n\u00a1Combatiente que la tierra criara, arm\u00e1ndote\n\nde polvo,\n\ncalz\u00e1ndote de imanes positivos,\n\nvigentes tus creencias personales,\n\ndistinto de car\u00e1cter, \u00edntima tu f\u00e9rula,\n\nel cutis inmediato,\n\nand\u00e1ndote tu idioma por los hombros\n\ny el alma coronada de guijarros!\n\n\u00a1Voluntario fajado de tu zona fr\u00eda,\n\ntemplada o t\u00f3rrida,\n\nh\u00e9roes a la redonda,\n\nv\u00edctima en columna de vencedores:\n\nen Espa\u00f1a, en Madrid, est\u00e1n llamando\n\na matar, voluntarios de la vida!\n\n\u00a1Porque en Espa\u00f1a matan, otros matan\n\nal ni\u00f1o, a su juguete que se para,\n\na la madre Rosenda esplendorosa,\n\nal viejo Ad\u00e1n que hablaba en alta voz con su caballo\n\ny al perro que dorm\u00eda en la escalera.\n\nMatan al libro, tiran a sus verbos auxiliares,\n\na su indefensa p\u00e1gina primera!\n\nMatan el caso exacto de la estatua,\n\nal sabio, a su bast\u00f3n, a su colega,\n\nal barbero de al lado\u2014me cort\u00f3 posiblemente,\n\npero buen hombre y, luego, infortunado;\n\nal mendigo que ayer cantaba enfrente,\n\na la enfermera que hoy pas\u00f3 llorando,\n\nal sacerdote a cuestas con la altura tenaz de sus rodillas . . .\n\n\u00a1Voluntarios,\n\npor la vida, por los buenos, matad\n\na la muerte, matad a los malos!\n\n\u00a1Hacedlo por la libertad de todos,\n\ndel explotado, del explotador,\n\npor la paz indolora\u2014la sospecho\n\ncuando duermo al pie de mi frente\n\ny m\u00e1s cuando circulo dando voces\u2014\n\ny hacedlo, voy diciendo,\n\npor el analfabeto a quien escribo,\n\npor el genio descalzo y su cordero,\n\npor los camaradas ca\u00eddos,\n\nsus cenizas abrazadas al cad\u00e1ver de un camino!\n\nPara que vosotros,\n\nvoluntarios de Espa\u00f1a y del mundo, vinierais,\n\nso\u00f1\u00e9 que era yo bueno, y era para ver\n\nvuestra sangre, voluntarios . . .\n\nDe esto hace mucho pecho, muchas ansias,\n\nmuchos camellos en edad de orar.\n\nMarcha hoy de vuestra parte el bien ardiendo,\n\nos siguen con cari\u00f1o los reptiles de pesta\u00f1a inmanente\n\ny, a dos pasos, a uno,\n\nla direcci\u00f3n del agua que corre a ver su l\u00edmite antes que arda.\nIII  \nWith His Index Finger He Writes on the Air  \n(For Spanish translation click here)\n\nWith his index finger he writes on the air:\n\n\"Long live the comrades! Pedro Rojas,\"\n\nfrom Miranda del Ebro, father and man,\n\nhusband and man. Pedro and his two deaths.\n\nPaper of wind, they killed him: it's gone!\n\nFeather of flesh, they killed him: it's gone!\n\nInform all comrades hurry up!\n\nPole on which they hung his piece of wood,\n\nthey've killed him;\n\nthey've killed him to the base of his thumb!\n\nthey killed, at one, Pedro and Rojas!\n\nLong live the comrades\n\nat the head of his writing in air!\n\nLong live the V of the vulture in the guts\n\nof Pedro\n\nand of Rojas, of the hero and martyr!\n\nAfter his death they opened him up\n\ndown the middle finding within him a body big enough\n\nto hold the soul of the world,\n\nand in his coat pocket a dead spoon.\n\nPedro also used to eat\n\namong the creatures of his flesh, to clean and\n\npaint the table and living softly\n\nin representation of all the world.\n\nAnd this spoon walked always in his coat,\n\nawake or asleep, always, at all times,\n\nthat spoon with its living death, and her symbols.\n\nInform all comrades at once!\n\nLong live the comrades at the foot of this spoon forever and ever!\n\nThey killed him, forced him to die,\n\nPedro, Rojas, the worker, the man, the one\n\nwho was once a child looking up toward the sky,\n\nand then he grew up, turning red,\n\nand fought with his cells, his no, his yet, his hungers, his pieces.\n\nThey've killed him sweetly\n\nbetween the hair of his wife, the Juana Vasquez,\n\nin the hour of fire, at the year of the bullet,\n\nand just when he was getting close to all.\n\nPedro Rojas, after his death,\n\nraised himself up, kissed his bloodstained coffin,\n\nhe wept for Spain,\n\nand wrote with his finger on the air!\n\n\"Long live the comrades! Pedro Rojas.\"\n\nHis corpse was full of the world.\nIII  \nSol\u00eda escribir con su dedo grande en el aire  \n(For English translation click here)\n\nSol\u00eda escribir con su dedo grande en el aire:\n\n\u00ab\u00a1Viban los compa\u00f1eros! Pedro Rojas\u00bb,\n\nde Miranda de Ebro, padre y hombre,\n\nmarido y hombre, ferroviario y hombre,\n\npadre y m\u00e1s hombre, Pedro y sus dos muertes.\n\nPapel de viento, lo han matado: \u00a1pasa!\n\nPluma de carne, lo han matado: \u00a1pasa!\n\n\u00a1Abisa a todos compa\u00f1eros pronto!\n\nPalo en el que han colgado su madero,\n\nlo han matado;\n\n\u00a1lo han matado al pie de su dedo grande!\n\n\u00a1Han matado, a la vez, a Pedro, a Rojas!\n\n\u00a1Viban los compa\u00f1eros\n\na la cabecera de su aire escrito!\n\n\u00a1Viban con esta b del buitre en las entra\u00f1as\n\nde Pedro\n\ny de Rojas, del h\u00e9roe y del m\u00e1rtir!\n\nRegistr\u00e1ndole, muerto, sorprendi\u00e9ronle\n\nen su cuerpo un gran cuerpo, para\n\nel alma del mundo,\n\ny en la chaqueta una cuchara muerta.\n\nPedro tambi\u00e9n sol\u00eda comer\n\nentre las criaturas de su carne, asear, pintar\n\nla mesa y vivir dulcemente\n\nen representaci\u00f3n de todo el mundo.\n\nY esta cuchara anduvo en su chaqueta,\n\ndespierto o bien cuando dorm\u00eda, siempre,\n\ncuchara muerta viva, ella y sus s\u00edmbolos.\n\n\u00a1Abisa a todos compa\u00f1eros pronto!\n\n\u00a1Viban los compa\u00f1eros al pie de esta cuchara para siempre!\n\nLo han matado, oblig\u00e1ndole a morir\n\na Pedro, a Rojas, al obrero, al hombre, a aqu\u00e9l\n\nque naci\u00f3 muy ni\u00f1\u00edn, mirando al cielo,\n\ny que luego creci\u00f3, se puso rojo\n\ny luch\u00f3 con sus c\u00e9lulas, sus nos, sus todav\u00edas, sus hambres,   \nsus pedazos.\n\nLo han matado suavemente\n\nentre el cabello de su mujer, la Juana V\u00e1squez,\n\na la hora del fuego, al a\u00f1o del balazo\n\ny cuando andaba cerca ya de todo.\n\nPedro Rojas, as\u00ed, despu\u00e9s de muerto,\n\nse levant\u00f3, bes\u00f3 su catafalco ensangrentado,\n\nllor\u00f3 por Espa\u00f1a .\n\ny volvi\u00f3 a escribir con el dedo en el aire:\n\n\u00ab\u00a1Viban los compa\u00f1eros! Pedro Rojas.\u00bb\n\nSu cad\u00e1ver estaba lleno de mundo.\nIX  \nA Brief Funeral Prayer for a Hero of the Republic  \n(For Spanish translation click here)\n\nA book at the edge of his dead waist,\n\na book sprouting from this corpse.\n\nThey took the hero away,\n\nand his corporeal and sad mouth entered in our courage;\n\nwe all sweat, our navel a burden,\n\nthe wandering moons follow us;\n\nthe dead man, too, sweats from grief.\n\nAnd a book, at the Battle of Toledo,\n\na book, behind a book, above a book, a book nevertheless,\n\nwas sprouting from the corpse.\n\nPoetry of the purple cheekbones, between speaking or\n\nremaining silent,\n\npoetry in the moral letter accompanying\n\nhis heart.\n\nThe book remains, and nothing else\n\nthat there are no insects in the tomb,\n\nand remains at the edge of his sleeve, the air soaking itself\n\nand becoming gaseous, infinite.\n\nAll of us sweat, the navel on shoulders,\n\nthe dead man also sweating of sadness\n\nand the book, I, myself, see it regretfully,\n\na book, behind a book, above a book,\n\nsprouts from this corpse abruptly.\n\n_10 September 1937_\nIX  \nPeque\u00f1o responso a un h\u00e9roe de la rep\u00fablica  \n(For English translation click here)\n\nUn libro qued\u00f3 al borde de su cintura muerta,\n\nun libro reto\u00f1aba de su cad\u00e1ver muerto.\n\nSe llevaron al h\u00e9roe,\n\ny corp\u00f3rea y aciaga entr\u00f3 su boca en nuestro aliento;\n\nsudamos todos, el ombligo a cuestas;\n\ncaminantes las lunas nos segu\u00edan;\n\ntambi\u00e9n sudaba de tristeza el muerto.\n\nY un libro, en la batalla de Toledo,\n\nun libro, atr\u00e1s un libro, arriba un libro, reto\u00f1aba del cad\u00e1ver.\n\nPoes\u00eda del p\u00f3mulo morado, entre el decirlo\n\ny el callarlo,\n\npoes\u00eda en la carta moral que acompa\u00f1ara\n\na su coraz\u00f3n.\n\nQued\u00f3se el libro y nada m\u00e1s, que no hay\n\ninsectos en la tumba,\n\ny qued\u00f3 al borde de su manga el aire remoj\u00e1ndose\n\ny haci\u00e9ndose gaseoso, infinito.\n\nTodos sudamos, el ombligo a cuestas,\n\ntambi\u00e9n sudaba de tristeza el muerto\n\ny un libro, yo lo vi sentidamente,\n\nun libro, atr\u00e1s un libro, arriba un libro\n\nreto\u00f1\u00f3 del cad\u00e1ver ex abrupto.\n\n_10 setiembre 1937_\nXII  \nMass  \n(For Spanish translation click here)\n\nAt the end of the battle,\n\nand dead the fighter a man came up to him\n\nand said: \"Don't die, I love you so much!\"\n\nBut the corpse alas! Went on dying.\n\nTwo more men came to him and whispered repeatedly:\n\n\"Don't leave us! Courage! Return to life!\"\n\nBut the corpse alas! Went on dying.\n\nThen came twenty more, one hundred, one thousand, five thousand\n\nclaiming: \"So much love, and to be powerless against death!\"\n\nBut the corpse alas! Went on dying.\n\nMillions of individuals surrounded him\n\nwith a common prayer: \"Stay, brother!\"\n\nBut the corpse alas! Went on dying.\n\nThen, all the men of the earth\n\nsurrounded him; the corpse gazing up at the crowd, sadly,\n\ndeeply moved, he raised up slowly,\n\nand put his arms around the first man who spoke, and began to walk . . .\n\n_10 November 1937_\nXII  \nMasa  \n(For English translation click here)\n\nAl fin de la batalla,\n\ny muerto el combatiente, vino hacia \u00e9l un hombre\n\ny le dijo: \u00ab\u00a1No mueras, te amo tanto!\u00bb\n\nPero el cad\u00e1ver \u00a1ay! sigui\u00f3 muriendo.\n\nSe le acercaron dos y repiti\u00e9ronle:\n\n\u00ab\u00a1No nos dejes! \u00a1Valor! \u00a1Vuelve a la vida!\u00bb\n\nPero el cad\u00e1ver \u00a1ay! sigui\u00f3 muriendo.\n\nAcudieron a \u00e9l veinte, cien, mil, quinientos mil,\n\nclamando \u00ab\u00a1Tanto amor y no poder nada contra la muerte!\u00bb\n\nPero el cad\u00e1ver \u00a1ay! sigui\u00f3 muriendo.\n\nLe rodearon millones de individuos,\n\ncon un ruego com\u00fan: \u00ab\u00a1Qu\u00e9date hermano!\u00bb\n\nPero el cad\u00e1ver \u00a1ay! sigui\u00f3 muriendo.\n\nEntonces todos los hombres de la tierra\n\nle rodearon; les vio el cad\u00e1ver triste, emocionado;\n\nincorpor\u00f3se lentamente,\n\nabraz\u00f3 al primer hombre; ech\u00f3se a andar . . .\n\n_10 noviembre 1937_\nXV  \nSpain, Take This Cup from Me  \n(For Spanish translation click here)\n\nChildren of the world\n\nif Spain falls\u2014I say, if it should happen\u2014\n\nif they tear down from the sky\n\nline her forearm, held in a sling\n\nshot by two terrestrial rings;\n\nchildren, how old the hollow temples!\n\nHow premature in the sun what was spoken to you!\n\nHow soon in your chest the ancient outcry!\n\nHow old the numeral 2 in your notebook!\n\nChildren of the world, this\n\nMother Spain is with her belly lying down,\n\nour school teacher with her authority,\n\nour mother and teacher,\n\ncross and wood, taking you to the heights!\n\ndizziness and division and addition, children;\n\nwhile her elders stood to accuse!\n\nIf she falls\u2014I say, if it should happen this way\u2014if\n\nfrom the earth to the lowest depths,\n\nchildren, you will be stunted in the prime of your youth!\n\nHow the year will punish the month!\n\nHow will you remain with your ire in ten to those teeth,\n\nIn the drumstick, the diphthong, the medallions in tears!\n\nHow the little lamb stays\n\nwith its foot tied to the big inkstand!\n\nHow will you descend from the stone steps of the alphabet\n\nuntil you reach the letter where suffering is born?\n\nChildren,\n\noffspring of warriors, meanwhile\n\nlower your voice, Spain is distributing right now\n\nenergy among the animal kingdom,\n\nthe little flowers, the comets, and man.\n\nLower your voice, because she is\n\nwith her vigor that's great without knowing\n\nwhat to do, holds in her hand\n\nthe skull, that speaks and speaks and speaks,\n\nthe skull, with braids of hair,\n\nthe skull, that one of the living!\n\nLower your voice, I tell you;\n\nlower your voice, the song of the syllables, the wailing\n\nof the subject matter and the lesser sounds of the pyramids, and even\n\nof the temples which throb like the rubbing of two stones!\n\nLower the breath, and if\n\nthe forearm drops dead\n\nto its side, if the splints sleep, if it is night,\n\nif the sky fits into two terrestrial limbos\n\nthat can never be closed,\n\nif there are creakings in the threshold sounds,\n\nif I am late,\n\nif sooner or later no one is seen on the streets, if you're frightened\n\nthose pencils without nibs, if the mother\n\nSpain falls\u2014I repeat, just supposing it happens\u2014\n\ngo forth, children, of the world; and go out to find her! . . .\nXV  \nEspa\u00f1a, aparta de m\u00ed este c\u00e1liz  \n(For English translation click here)\n\nNi\u00f1os del mundo,\n\nsi cae Espa\u00f1a\u2014digo, es un decir\u2014\n\nsi cae\n\ndel cielo abajo su antebrazo que asen,\n\nen cabestro, dos l\u00e1minas terrestres;\n\nni\u00f1os, \u00a1qu\u00e9 edad la de las sienes c\u00f3ncavas!\n\n\u00a1qu\u00e9 temprano en el sol lo que os dec\u00eda!\n\n\u00a1qu\u00e9 pronto en vuestro pecho el ruido anciano!\n\n\u00a1qu\u00e9 viejo vuestro 2 en el cuaderno!\n\n\u00a1Ni\u00f1os del mundo, est\u00e1\n\nla madre Espa\u00f1a con su vientre a cuestas;\n\nest\u00e1 nuestra madre con sus f\u00e9rulas,\n\nest\u00e1 madre y maestra,\n\ncruz y madera, porque os dio la altura,\n\nv\u00e9rtigo y divisi\u00f3n y suma, ni\u00f1os;\n\nest\u00e1 con ella, padres procesales!\n\nSi cae\u2014digo, es un decir\u2014si cae\n\nEspa\u00f1a, de la tierra para abajo,\n\nni\u00f1os \u00a1c\u00f3mo vais a cesar de crecer!\n\n\u00a1c\u00f3mo va a castigar el a\u00f1o al mes!\n\n\u00a1c\u00f3mo van a quedarse en diez los dientes,\n\nen palote el diptongo, la medalla en llanto!\n\n\u00a1C\u00f3mo va el corderillo a continuar\n\natado por la pata al gran tintero!\n\n\u00a1C\u00f3mo vais a bajar las gradas del alfabeto\n\nhasta la letra en que naci\u00f3 la pena!\n\nNi\u00f1os,\n\nhijos de los guerreros, entre tanto,\n\nbajad la voz que Espa\u00f1a est\u00e1 ahora mismo repartiendo\n\nla energ\u00eda entre el reino animal,\n\nlas florecillas, los cometas y los hombres.\n\n\u00a1Bajad la voz, que est\u00e1\n\nen su rigor, que es grande, sin saber\n\nqu\u00e9 hacer, y est\u00e1 en su mano\n\nla calavera, aquella de la trenza;\n\nla calavera, aquella de la vida!\n\n\u00a1Bajad la voz, os digo;\n\nbajad la voz, el canto de las s\u00edlabas, el llanto\n\nde la materia y el rumor menos de las pir\u00e1mides, y aun\n\nel de las sienes que andan con dos piedras!\n\n\u00a1Bajad el aliento, y si\n\nel antebrazo baja,\n\nsi las f\u00e9rulas suenan, si es la noche,\n\nsi el cielo cabe en dos limbos terrestres,\n\nsi hay ruido en el sonido de las puertas,\n\nsi tardo,\n\nsi no veis a nadie, si os asustan\n\nlos l\u00e1pices sin punta, si la madre\n\nEspa\u00f1a cae\u2014digo, es un decir\u2014\n\nsalid, ni\u00f1os, del mundo; id a buscarla! . . .\n\n_Top: C\u00e9sar Vallejo in Paris  \nBottom: C\u00e9sar Vallejo drawing used on Peruvian currency_\n\nPhotos: Juan Larrea Collection\/Archives Malanga\nCLOSING POEM\n\n_by Gerard Malanga_\n\nTranlated from the original English by   \nPatricia Daniela Alverte\nVision 1938 Paris  \n(For Spanish translation click here)\n\nIn the very busy Saint Germain-des-Pr\u00e8s, not too distant\n\nfrom the Caf\u00e9 Flore, I saw a man in an old suit\n\nthat was more than merely a covering for his body,\n\nit was part of the man himself.\n\nIt had suffered with him.\n\nIt was like a brownish grazed skin.\n\nThe man was not standing and was not walking.\n\nAs he walked he stood still,\n\nand as he stood still he moved forward a little.\n\nHis face was gentle and rosy, but from his forehead\n\nand cheeks furrows crowded into his face.\n\nHis eyes looked out high above everything they met,\n\nand yet they were waiting. From near at hand\n\nthe left arm was held close to the body,\n\nas if the body wouldn't let go of the arm,\n\nand yet he held his hand stretched out slightly.\n\nI put a note into it, and then I didn't know,\n\nwhether the hand went back to the man,\n\nand whether he put the note in his pocket,\n\nor did the hand move on out,\n\nseeking for another hand. This man\n\nwas living in the center between giving and taking,\n\nbetween distance and nearness,\n\nbetween old age and youth.\n\nA few days passed. I went to call on this man;\n\nbut the concierge at the building\n\nwhere he lived in one small room,\n\ntold me that he had died only a few days previous;\n\non April 15th, Good Friday.\n\nThe cause of death was never determined;\n\nbut at last today I remembered this man. He sits\n\nat a table just to the left of the doorway inside the Flore,\n\nand as a boy I would sit with him for hours.\n\nGerard Malanga\n\n_29:IV:71 NYC_\nVisi\u00f3n 1938 Paris  \n(For English translation click here)\n\nEn la muy transitada Saint Germain-des-Pr\u00e9s, no muy lejos\n\ndel Caf\u00e9 Flore, vi a un hombre en un viejo sobretodo\n\nel cual era m\u00e1s que un simple abrigo para su cuerpo,\n\nera parte del hombre en s\u00ed.\n\nHab\u00eda sufrido con \u00e9l.\n\nEra como de un parduzco cuero gastado.\n\nEl hombre no estaba quieto ni tampoco caminando.\n\nMientras caminaba permanec\u00eda quieto,\n\ny mientras permanec\u00eda quieto avanzaba un poco.\n\nSu rostro era apacible y fresco, pero desde su frente\n\ny mejillas, arrugas se abarrotaban en su cara.\n\nSus ojos miraban por encima de todo con lo que se topaban,\n\ny sin embargo ellos estaban esperando. Cerca de la mano\n\nel brazo izquierdo colgaba pegado al cuerpo,\n\ncomo si el cuerpo no dejara ir al brazo,\n\ny aun as\u00ed manten\u00eda su mano ligeramente extendida.\n\nPuse una nota sobre ella, y en ese momento no supe,\n\nsi la mano regres\u00f3 hacia el hombre,\n\ny si hab\u00eda puesto la nota en su bolsillo,\n\no si la mano se movi\u00f3,\n\nbuscando otra mano. Este hombre\n\nestaba viviendo entre el dar y el recibir,\n\nentre distancia y cercan\u00eda,\n\nentre vejez y juventud.\n\nPasaron unos pocos d\u00edas. Pas\u00e9 a buscar a este hombre;\n\npero el conserje del edificio\n\ndonde viv\u00eda en un cuarto peque\u00f1o,\n\nme dijo que hab\u00eda fallecido unos pocos d\u00edas antes;\n\nel 15 de Abril, Buen Viernes.\n\nLa causa de la muerte nuca fue determinada;\n\npero finalmente hoy record\u00e9 a este hombre. \u00c9l se sienta\n\na la mesa justo a la izquierda de la entrada dentro del Flore,\n\ny como un ni\u00f1o me sentar\u00eda con \u00e9l por horas.\n\nGerard Malanga\n\n_29:IV:71 NYC_\n\n_Gerard Malanga reading his C\u00e9sar Vallejo translations at the Vallejo burial plot, Cimetiere du Montparnasse\/12th Division, Paris. Mid-November, 1992_\n\nPhoto by Julia Friar\/\u00a9 Archives Malanga\nTHE LETTERS\n\n_from Georgette Vallejo  \n_ _to Gerard Malanga_\n\nTranslated from the original Spanish by   \nPatricia Daniela Alverte\n\n3\/12\/71\n\nDistinguished Mr. Malanga:\n\nI'm surprised not having received an answer to my last letter.\n\nIf you had given up your project, I would appreciate it you letting me know about it.\n\nI remind you that there can be no publishing of your English version of Vallejo's poems without previously establishing an editing contract with a publisher who states and certifies the authorization that you requested. I hope to receive news from you soon about this matter.\n\nIn this waiting, distinguished Mr. Malanga, I send you my more cordial greetings and my vows for this so near New Year.\n\nGeorgette de Vallejo  \n5241-301 A. Arequipa  \nMiraflores  \nLIMA\/PERU.\n\nLima, 9\/2\/71  \nVery dear sir and friend:\n\nI beg you to forgive a silence which must rightfully seem to you inexplicable. Unfortunately, I live (if this can be called living) since 19 years in \"Horrible Lima \"   \n(how the great poet C\u00e9sar Moro used to name this city that pretends to be a capitol),  \na city where one suffer depression after depression until one becomes abnormal.\n\nAs soon as received, I read your authentic poet's translations. I was going to reply to you immediately, but one thing held me back: sometimes, your version is really far from the original . . .\n\nRight away, I copied\u2014as you can see\u2014the Spanish text right next to your English version to facilitate the confrontation, and I called a friend of mine who knows a lot about your language. But in vain, I waited for his telephone call over two months . . . (all lime\u00f1os are despicable in some aspect).\n\nAnother friend of mine, [ . . . . . . . . . ], an American born in Latin America, irreplaceable for this case, was sick back then :asthma, ulcers and other things not less severe. However, as time went by, I decided to bring him your work . We got together this past Saturday. Our opinions match. I send you the first fourteen poems (chronologically) so you can see some modifications (essential in some cases) . . . You'll see. We keep reading.\n\nI'm surprised not to find anything from \"Prose Poems\"[1] in your selection (among others: \"The good sense\"[2] \"Languidly his liquor\"[3], \"I will speak of hope . . . \"[4], \"Finding of life\"[5]). I feel the same way about the total absence of fragments from \"Hymn to the Republic Volunteers\"[6] and \"Battles\"[7], and from \"Spanish image of death\"[8].\n\nI wish your selection, not just for its exceptional quality, but also for its amplitude, will make forget the horrendous and trivial translations of this [ . . . . . . . . . ]. No one can suspect what kind of a blow has been for me. It's a cancer.\n\nI would like to propose something to you. [ . . . . . . . . . ], who had projected to translate HUMAN POEMS, has his English version of 25 poems of this volume ready (of course different of those included on your selection). His health forced him to leave his project. Would you have any inconveniences in your book having a second part in the end which includes these poems? I would be infinitely grateful if you would agree\u2014since you would evidently be the main author, the \"star\" so to speak, of your publication.\n\nWaiting to read your next letter and reiterating my most sincere apologies, please receive, my much estimated Malanga, my most sincere regards.\n\n5241-301 A. Arequipa  \nMiraflores-LIMA\/PERU\n\nThe poems are the following:\n\n-La voz del espejo (HN)\/-The voice of the mirror (HN)  \n-Est\u00e1is muertos (Trilce)\/-You're dead (Trilce)  \n-He aqu\u00ed que hoy saludo . . . (P. en P.)\/-Behold I greet today (P. in P.)  \n-Sombrero, abrigo, guantes . . . (PH) \/-Hat, coat, gloves . . . (PH)  \n-Confianza en el anteojo . . . \/-Trust in the eyeglass . . .  \n-Al cavilar en la vida . . . \/-While pondering in life . . .  \n-Los nueve monstrous\/-The nine monsters  \n-Guitarra\/-Guitar  \n-Va corriendo\/-It goes running . . .  \n-Un pilar soportando Consuelos\/-A pillar tolerating solaces  \n-Pante\u00f3n\/-Pantheon  \n-Acaba de pasar\/-Just passed . . .  \n-Palmas y guitarra\/-Claps and guitar  \n-Y si despu\u00e9s de tantas palabras . . . \/-And if after so many words . . .  \n-Despedida recordando un adios\/-Farewell remembering a goodbye  \n-Oh botella sin vino! . . . \/-Oh bottle without wine! . . .  \n-Encarnecido, aclimatado . . . \/-Mocked, acclimatized . . .  \n-El libro de la naturaleza\/-The book of nature  \n-Tengo un miedo terrible . . . \/-I have a terrible fear  \n-La c\u00f3lera que quiebra al hombre . . . \/-The anger which breaks a men . . .  \n-Viniera el malo . . . \/-Comes the bad  \n-Ello que es el lugar . . . \/-That is the place\n\n_In the original letter_\n\n[1] Poemas en Prosa\/Prose poems\n\n[2] El buen sentido\/The good sense\n\n[3] L\u00e1nguidamente su licor\/Languidly his spirit\n\n[4] Voy a hablar de la esperanza\/ I'm going to speak about hope\n\n[5] Hallazgo de la vida\/Discovery of life\n\n[6] Himno a los voluntarios de la Rep\u00fablica\/Hymm to the volunteers of the Republic\n\n[7] Batallas\/Battles\n\n[8] Imagen espa\u00f1ola de la muerte\/Spanish image of the death\n\n5\/4\/71  \nVery dear Malanga:  \nDo apologize, once more, this reply delayed on my end.\n\nEven though I understand that, in your opinion, it wouldn't be convenient that Vallejo resulted \"strange in English\"[1], Vallejo, however, \"is strange\"[2] in all languages as he is in his own. But Vallejo is Vallejo, and nothing and nobody can explain what this new thing is about, above all indefinable, that permeates and affirms more and more his poetic work. To me, Vallejo's poems are poems, not poetry . . . I don't know if you will accept this purely subjective tint.\n\nContinuing my reading, I've seen that, very often, you change the original text (making me doubt that, who knows, you may have missed the real meaning); other times, you explain it. First at all, you must never explain a poem. Secondly, one can fail in the interpretation. When you add to the verse \"Me acuerdo que nos hac\u00edamos llorar, hermano, en aquel juego\"[3] from so much laughing[4], you add something incorrect because the children weren't crying for so much laughter, but, forgetting in fact that they were playing, they took things so seriously that they ended up crying because the more real anguish.\n\nThen, you absolutely cannot add something to the original. An example among the numerous cases, as you may know. In the last verse of MASA (Espa\u00f1a aparta de m\u00ed este c\u00e1liz[5], You add: \"My brothers, may God give you peace\"[6]. That is fundamentally serious, as this added verse alters, even spiritually, the author's thinking.\n\nIf I get scared and recommend you to severely look over your version is because you can't ignore the virulent criticism and even the sarcasm that [ . . . ], naturally vulgar, low and prosaic, will shower over your work. He told me once when he was in Lima: \"You oppose the publishing of my translations, but you have authorized [ . . . ] (the German translator) awful translations. I told him that this was because, happily, I don't speak a word of German.\n\nLastly, I would like to trust entirely on your word to be faithful to the original, because in this damned city I've lost the spirit and mood which I would need to help you. As I've said before, it is necessary to present an anthology of the five volumes, but I can't take the commitment to revise your version. I beg you to translate without worrying about explanations that even the author won't give, or fear that Vallejo will turn out \"strange in English.\" I've had big difficulties in French as well; the arrangement of the word is very important in giving meaning to poetics. I've had the satisfaction of seeing people amazed of what it had achieved. Perhaps, do you understand French?\n\nAs you haven't replied to certain aspects of my last letter, I deduce you don't agree with me.\n\nWith my most sincere regards,\n\nPS: I forget to clarify, that you are to make the selection of the fragments which inspire you most of \"Hymn to the volunteers. . .\"[7] and from \"Spain. . . \"[8] choosing, of course, the ones more fit into translation.\n\n[1] In English in the original\n\n[2] In English in the original\n\n[3] \"I remember we made each other cry, brother, in that game\"\n\n[4] In English in the original\n\n[5] \"Spain, take this cup from me\"\n\n[6] In English in the original\n\n[7] \"Himno a los Voluntarios\" in the original\n\n[8] \"Espa\u00f1a\" in the original\n\nLIMA, Miraflores, 1\/2\/72  \nMister, GERARD MALANGA  \nP.O. BOX 1811  \nF.D.R. Station  \nNEW YORK 10022  \nU.S.A.\n\nDistinguished mister Malanga:\n\nNo: I'm not impatient. There's no reason for that. However, I'm seriously restless because, until this day, you haven't informed if you have made a correction of the translations with the notes I've sent you back.\n\nSecondly, I've just read\u2014chosen at random from the last translations I received\u2014poem 111 from \"Spain, take this cup from me\"[1], and I see, not without bitter disappointment, that you haven't even taken into account my express recommendation: RESPECT THE ORIGINAL TEXT, WITHOUT CHANGING, ADDING OR REMOVING WORDS OR VERSES, AND DON'T ADD WHOLE VERSES COMING ENTIRELY OUT OF YOUR IMAGINATION. I beg you to acknowledge, following these lines, the inexplicable mistakes showing in your version of the above mentioned. As I said, one gets the impression you don't fully understand Spanish, coming to use verbs in the present tense which in the original text determine a past tense, not being able in any way to put a present where there is a verb in the past tense, as you can observe in the following list of mistakes.\n\nThen, as I told you before, a publishing contract is established between the editor and the author\u2014or the person who represents him.\n\nLastly, please understand I cannot revise and correct your translations, not only do I have a lot to do, but I am also exhausted for different reasons.\n\nWaiting hear news from you, yours sincerely.  \nGeorgette de Vallejo  \n5241 A. Arequipa \u2013 Miraflores \u2013 LIMA \u2013 PERU.\n\n[1] Espa\u00f1a, aparta de m\u00ed este c\u00e1liz.\n\nHE USED TO WRITE WITH HIS BIG FINGER IN THE AIR . . .\n\n---\n\n|\n\nStrophe 1\n\nSol\u00eda escribir con su dedo grande en el aire: | (Vallejo says: big finger not index. 2) \"He used\" past\n\nWith his index finger he skywrites on the air: | (tense: he used to write. 3) Not on the air but in the\n\n|\n\n(air. 4) \"Skywrite\" is wrong.\n\nViban los compa\u00f1eros! | (\"Comrades\"? There must be a better word?\n\nLong live the comrades!\n\n|\n\npadre y m\u00e1s hombre, Pedro. . . . | Vallejo says : padre y m\u00e1s hombre: father and\n\nfather\u2014but even more man\u2014Pedro. . . . | (more man, simply. Why \"but even\"?\n\n|\n\nStrophe 2\n\n|\n\n(this strophe has 3 verses, not 4.\n\nPapel de viento, lo han matado: pasa!  | (Vallejo says: Paper of wind, they killed him!\n\nPluma de carne, lo han matado: pasa! | (Pasa! means something like: it's gone, but\n\nAbisa a todos compa\u00f1eros pronto!\n\n|\n\nScrap of paper caught in the air waves | (not \"waves\"\n\nThey killed him (it really happened!) | (feather of flesh, they killed him! It's gone, but not it\n\nFeather of flesh and blood they killed him! | (\"really happened!\" Either \"blood\"\n\nInform all the comrades at once! | (You're removing the tragedy from the last verse   \n(\"lanzado cablegraficamente por el muerto.\" Quote:   \n(Tell all the companions fast! \"At once,\" no. \"The\" no.\n\n|\n\n222 | Strophe 3\n\nPalo en el que han colgado su madero, (Pole on which they hunged his peace of wood (?) peace of wood on which a beam is hung,\n\nLo han matado al pie de su dedo grande! | (to (or at) foot his big finger! | (can you say so in English?)\n\n---|---|---\n\nThey killed him to the base of his forefinger and thumb! | (Why?)\n\n---|---\n\n_(The correct verse is: \"They killed him to his big finger's foot\")_ | |\n\n|  | Strophe 4\n\nA la cabecera de su aire escrito | (Vallejo says:\n\n|\n\nand the honor roll of the aire! | (at the bed side of his written air!\n\nViban con esta V del buitre en las entra\u00f1as | (Why do you add \"Let them . . . long,\" giving it a charity\n\nLet them live long with the V......... | (meaning to the poem?\n\n|\n\n(The correct verse is: \"At the bedside of his written air! Live with this V of vulture in the bowels\") | |\n\n|  | Strophe 5\n\nRegistrandole, muerto, sorprendieronle | (Note: the words \"registrandole y sorprendieronle\" with\n\n|\n\nAfter his death they opened him up | (\"opened him up\" (!) You can say:\n\n|\n\nen su cuerpo un gran cuerpo, para | (Searching him, dead, the found\/in his body a\n\n|\n\nel alma del mundo, | (big body for\/the soul of the world\n\n|\n\ndown the middle finding within him a body big enough | (Too much explanations!, for finally\n\n|\n\nto hold the soul of the world | (changing the real meaning of the verse.\n\n|\n\ncuchara muerta viva\u2014empty spoon | (it's not the same . . .\n\n|\n\nAbout the remaining version of this poem, I must say:  \n\"criaturas de su carne\" a bland and cold translation  \nmembers of his family . . . !\n\n\"pintar\/la mesa y vivir dulcemente en representaci\u00f3n de todo el mundo\" You say:  \nfilled his table with food living confortable\/like anyone else\n\n\"cuchara muerta viva, ella y sus s\u00edmbolos. _(Correct verse: living dead spoon, she and her symbols)_  \nThe spoon with its meaning of life instead of respecting the word symbol in extensive sense, you replace \"meaning of life\" of limited sense, although you repeat it in the following strophe . . . and not in the exact meaning of the text. In this same strophe, you do add again: Let them . . .\n\n\"aunque\/que naci\u00f3 muy ni\u00f1in, mirando al cielo _(Correct verse: although born as a little child, looking at the sky)_  \nThe one\/who was once a child looking up with the sky ???\n\n\"y que luego creci\u00f3, se puso rojo   \n_(Correct verse: \"and then he grew up, and turned red\")_  \ny lucho con sus c\u00e9lulas, sus nos, sus todav\u00edas, sus hambres, sus pedazos  \nstruggling in every cell, block of his body with his quick answers,\/his doubts, his  \nhungers and the pieces of wet bread\/he recognized as himself. ??????  \n _(Correct verse: \"and he fought with his cellule, his no, his yet, his hungers, his pieces\")_\n\n\"Lo han matado suavemente   \n_(Correct verse: \"They have killed him softly\")_  \nentre el cabello de su mujer, La Juana Vazquez,   \n_(Correct verse: \"between his_ woman's hair, La Juana Vazquez\")  \nThey killed him in one clean sweep ????????????????  \nblood on the dress of his wife ??????????????????\n\n\"llor\u00f3 por Espa\u00f1a _(Correct verse \"He cried for Spain\")_  \nwept for Spain real tears\n\n(VALLEJO HAS SAID THAT? For you the fact   \nthat a spaniard cries is lesser and is not enough!)  \nAnd again:  \nLast:\n\non the air waves\n\n\"Su cad\u00e1ver estaba lleno de mundo\"   \n_(Correct verse: \"His corpse was full of world\")_\n\nHis dead body contain(s) all the world | |\n\n(Full of world, doesn't mean that contain(ed) all the world. Also, only you know why putting this verse into the present tense????????\n\n---|---|---\n\nI must say that you don't need to explain a poem.\n\nYou shouldn't try to sounding civilized or soft to certain forms of expression that may seem harsh, strange and even wild. A poem is as it is.\nAbout the Translator\n\nGerard Malanga is acclaimed as a poet, photographer, and filmmaker. He was born in the Bronx in 1943. He is the author of a dozen poetry collections, the most recent being _No Respect: New & Selected Poems,_ the four-volume fanzine set _AM: Archives Malanga, and Tomboy & Other Tales._ His photography books include _Resistance to Memory_ and _Screen Tests Portraits Nudes._ He was a founding editor of _Interview_ magazine, alongside Andy Warhol. Malanga lives in upstate New York.\nBooks from Three Rooms Press\n\nPHOTOGRAPHY-MEMOIR\n\nMike Watt  \n _On & Off Bass_\n\nFICTION\n\nRon Dakron  \n _Hello Devilfish!_\n\nMichael T. Fournier  \n _Hidden Wheel  \nSwing State_\n\nJanet Hamill  \n _Tales from the Eternal Caf\u00e9_   \n(Introduction by Patti Smith)\n\nEamon Loingsigh  \n _Light of the Diddicoy_\n\nRichard Vetere  \n _The Writers Afterlife_\n\nDADA\n\n_Maintenant:  \nJournal of Contemporary   \nDada Art & Literature_   \n(Annual poetry\/art journal,   \nsince 2008)\n\nMEMOIR & BIOGRAPHY\n\nNassrine Azimi and  \nMichel Wasserman   \n_Last Boat to Yokohama:_   \n_The Life and Legacy of  \nBeate Sirota Gordon_\n\nRichard Katrovas   \n_Raising Girls in Bohemia:  \nMeditations of an American Father; A Memoir in Essays_\n\nStephen Spotte  \n _My Watery Self:  \nAn Aquatic Memoir_\n\nSHORT STORY ANTHOLOGY\n\n_Have a NYC:  \nNew York Short Stories_  \nAnnual Short Fiction Anthology\n\nPLAYS\n\nMadeline Artenberg &   \nKaren Hildebrand  \n _The Old In-and-Out_\n\nPeter Carlaftes  \n _Triumph For Rent (3 Plays)  \nTeatrophy (3 More Plays)_\n\nMIXED MEDIA\n\nJohn S. Paul  \n _Sign Language:  \nA Painters Notebook_\n\nTRANSLATIONS\n\nThomas Bernhard   \n_On Earth and in Hell_  \n(poems by the author   \nin German with English   \ntranslations by Peter Waugh)\n\nPatrizia Gattaceca  \n _Isula d'Anima \/ Soul Island_   \n(poems by the author   \nin Corsican with English   \ntranslations)\n\nC\u00e9sar Vallejo   \n_Malanga Chasing Vallejo_  \n(selected poems of C\u00e9sar Vallejo with English translations and additional notes by Gerard Malanga)\n\nGeorge Wallace  \n _EOS: Abductor of Men_   \n(poems by the author in English  \nwith Greek translations)\n\nHUMOR\n\nPeter Carlaftes  \n _A Year on Facebook_\n\nPOETRY COLLECTIONS\n\nHala Alyan  \n _Atrium_\n\nPeter Carlaftes  \n _DrunkYard Dog  \nI Fold with the Hand I Was Dealt_\n\nThomas Fucaloro  \n _It Starts from the Belly and Blooms_\n\n_Inheriting Craziness is Like  \na Soft Halo of Light_\n\nKat Georges  \n _Our Lady of the Hunger_\n\nRobert Gibbons  \n _Close to the Tree_\n\nIsrael Horovitz  \n _Heaven and Other Poems_\n\nDavid Lawton  \n _Sharp Blue Stream_\n\nJane LeCroy  \n _Signature Play_\n\nPhilip Meersman  \n _This is Belgian Chocolate_\n\nJane Ormerod  \n _Recreational Vehicles on Fire_  \n _Welcome to the Museum of Cattle_\n\nLisa Panepinto  \n _On This Borrowed Bike_\n\nGeorge Wallace  \n _Poppin' Johnny_\n\n  |\n\nThree Rooms Press | New York, NY | Current Catalog: www.threeroomspress.com  \nThree Rooms Press books are distributed by PGW\/Perseus: www.pgw.com\n\n---|---\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"The Last Beginning is published by Walker Books in the UK and Australia, and Sky Pony Press in the USA. A special edition including the short story Another Together is available through Scholastic Book Clubs.\nA Diva Magazine Queer Summer Reading List title.\nArt by Alice Oseman, reblog here.\nSimilarities and Differences between The Last Beginning and The Next Together, in venn diagram format.\nA moodboard made by Arianne.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I swear by these packing cubes - they make travel much more efficient. I started out with this set of 3 (mine are all grey) & have since expanded my collection. My fav feature is that the clothes pak is double sided - one side for clean clothes and the other for your used dirty laundry. #genius. Highly recommend.\nI LOVE this toiletry bag - from the print to the 3 compartments inside! It also has a spacious outside pocket as well as some tuck-away spaces inside. For long trips this fits ALL of my toiletries + most of my hair products. Plus it has a hook for hanging - which is great in confined tiny spaces while traveling #cruiseships.\nThis is for all you Mac Lovers. There are 5 adapters which cover most countries in countries in the wordl! I use this so I don't have to worry about travel voltage issues when charging my most valuable business asset (my laptop). Plus there's a usb charger included for you to charge your phone or iPad!\nThis is my favorite travel adapter - especially if you're going to be traveling to many different countries in one trip (*hint* a multi-country cruise where you'll be exploring around town or for backpacking across the world). Plus it's color coded & comes with a case for ease of use. Again #genius. I love this brand.\nWanna know how I live naturally as an Expat?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Originally created for royalty, Hennessy V.S.O.P has a natural balance of strength and smoothness. It is a velvety, complex blend of delicate spice and honeyed fruit flavours. 40 per cent ABV.\nThe world famous Cognac producer Hennessy has a history dating back to 1765 when the company was established by Irishman Richard Hennessy. Initially an eaux-de-vie trading business, Hennessy was going to become the most successful Cognac exporter in the world. Today, part of the Louis Vuitton Moet Hennessy Group, the company remains to be amongst the most innovative Cognac companies around, forcing collaborations with new markets such as the Hip Hop scene in the US. Its estate and headquarters are based in the famous town of Cognac in Charente.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Glad it went well. Sounds like some wicked good sweets!\nWhile you are away you will be happy to know that the Maine State Legislature is debating a very important issue--Whether the whoopie pie should be the official state dessert or the official state treat!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shoe Storage, Outdoor Shoe Storage was posted October 29, 2015 at 6:37 pm by self-defensethoughts.info . More over Outdoor Shoe Storage has viewed by 8556 visitor.\nShoe Storage, Waterproof Outdoor Shoe Storage was posted April 29, 2017 at 6:40 am by self-defensethoughts.info . More over Waterproof Outdoor Shoe Storage has viewed by 10200 visitor.\nShoe Storage, Outdoor Shoe Storage Ideas was posted February 8, 2018 at 10:15 am by self-defensethoughts.info . More over Outdoor Shoe Storage Ideas has viewed by 9054 visitor.\nShoe Storage, Outdoor Shoe Storage Bench was posted January 29, 2017 at 4:42 pm by self-defensethoughts.info . More over Outdoor Shoe Storage Bench has viewed by 9640 visitor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztxyg b/data_all_eng_slimpj/shuffled/split2/finalzztxyg
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index 0000000000000000000000000000000000000000..80a9ec570d804bbf2fb32c55f042523c444d3f91
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@@ -0,0 +1,5 @@
+{"text":"Projetex: Translation Management System 8.5 download by Advanced International Translations Projetex is the Leading Translation Management System for Translation Agencies. It tremendously simplifies task of managing in-house translators and freelance translators, data and files sharing within translation agencies and provides multiple benefits for each translation agency member. Projetex is the most comprehensive translation management software for translation agencies, with over 250 distinct features developed since 1999 to 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WASHINGTON, DC \u2013 May 8, 2018 - The National Association for County Community and Economic Development (NACCED) announced today that Applied Business Software, Inc. is the newest member to join NACCED as an associate member. NACCED's associate members provide valuable products and services within the affordable housing, community development and economic development industry. They are established industry leaders, committed to promoting and enhancing the success of counties implementing the nation's affordable housing and community development programs.\nApplied Business Software has been in business since 1978 and is known for its signature software The Mortgage OfficeTM. The back office automating software that has been helping counties, cities, and other CBOs service their loans and streamline a complex financial backend process.\n\"We are energized about our partnership with NACCED, because our organizational goals are directly aligned in helping counties access resources to help improve community and economic development. We believe modern, user-friendly permitting technology not only helps streamline business operations and promote economic development, it also helps establish greater trust and transparency between county governments and their community members. We are looking forward to working with NACCED to help bring this type of innovation to more counties across the country.\" Nasser Hajo, CEO, ViewPoint Government Solutions.\nFor more information on NACCED and how to join, click here.\nFor more information on Applied Business Software, visit them at www.themortgageoffice.com.\nNACCED serves as voice in Washington on budgetary, programmatic, and regulatory issues pertaining to community development, economic development, and affordable housing. To learn more, visit www.nacced.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apr. 20 6:29 PM PT7:29 PM MT8:29 PM CT9:29 PM ET21:29 ET1:29 GMT9:29 6:29 PM MST8:29 PM EST7:29 PM CST8:59 PM VEN5:29 UAE (+1)8:29 PM CT-Jon Berti went 0 for 0 Saturday as the Miami Marlins beat the Washington Nationals 9-3. Berti scored once and stole a base. He's now batting - with no home runs this season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TWM Performance\u00ae uses a special metal Alloy \"Bronzoil\" which is a Self-Lubricating Metal. This means SOLID, Precise and Smooth Shifting with Little to No Maintenance for Your Scion tC 2004-2010!\nThe ULTIMATE, FULL Solution for your Scion tC 2004-2010! TWM Performance\u00ae Base and Cable Bushing..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jeff provides executive consultation, combining his professional experience with his clinical knowledge of psychology. Jeff has 25+ years of experience in working in small, medium, and large business settings in a variety of roles to include management, training, organization, consultation and engineering. Jeff has started businesses and currently owns small businesses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuaaj b/data_all_eng_slimpj/shuffled/split2/finalzzuaaj
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--- /dev/null
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+{"text":"Steroids were actually invented for medical purposes. There are lots of notions about steroids; most of the people think that steroids are drugs that just harm your body. But actually this is not the fact. Some of the athletes and the body builders used to take steroids to give a better performance in their competitions but this was not considered right. Therefore steroids were banned from the market by bringing in a law. In most of the countries steroids are banned by law. If anyone is found selling steroids in the open market then that person was severely punished by the law keepers. Steroids are very much popular in the first world countries. The people out here are very much concerned about their health and fitness regimes. Therefore you can buy steroids UK easily.\nWe all know that steroids were drugs that were invented for medical purposes only. But actually after its invention the body builders and the fitness freaks found in it some properties that could help them grow their muscle faster and stronger. The athletes also saw that steroid could enhance their performance and the amount of energy and stamina that they need to put into their sports could be enhanced by steroids to a maximum level. These properties in steroids were used to a maximum level and then it was seen that the uses got a wrong route. Later on the laws of the country got stricter and stern and the steroids were banned from using openly. Anyone found selling steroids openly would be punished by the law.\nYes as steroids are used for medical purposes also therefore some patients are there those who really need steroids, they can buy steroids from the medical store by showing valid prescription. But without a valid prescription you can never buy steroids from the medical stores.\nAfter this law was enforced, the fitness freaks and the bodybuilders got a serious threat. They were very much worried thinking where to avail steroid from? Then like a miracle came the internet. At the first hand body builders and the fitness freaks didn't trust the internet that much but when they saw that they can avail a lot of steroids from the different websites , moreover they can reach out to a huge number of people within a short time they started liking the steroid sale online.\nThe first world countries like UK are very much aware of the fact that they can buy genuine steroid from the internet. In the internet you will be able to find genuine steroids at a reasonable price. Moreover you can surf different sites on the net and find out the best site matching all your criteria's. You can get all kind of steroids in the internet but in the medical store they only keep a limited amount of steroid.\nI also agree to the fact that some online sites do sale scammed steroids but you need to be careful while buying steroids from the online sites. While you go to buy steroids from the online sites always read the customer reviews, read the services and facilities they are providing and when you are completely sure about them do buy steroids from them. In this manner you won't go wrong in buying steroids online. To buy steroids UK is a common happening for people here, because they are very much concerned about their health and fitness programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you understand what it looks like to be a godly wife to your husband? I sure know I didn't when we got married over 14 years ago! I was pretty disastrous when it comes to treating my husband kindly. Praise be to God, though, He has taught me so much throughout the time!\nI hope you are ready to be challenged and encouraged to diligently work towards becoming the wife God desires you to be. Let's zero in like a laser on the Proverbs 31 model. Decide today who you will serve (Joshua 24:15).\nOn one of our first moves as a travel trailer family, Yahweh saw fit to test me in my tongue control.\nYou see, it has been apparent to me for some time that my husband and I think differently. And, also, we see things differently almost all of the time.\nBasically, this means that we have to be that much more diligent in working on both good communication skills, and offering grace and patience to each other daily.\nThis latest road trip was no different.\nThings were going very well, but then it started to get later into the evening. Of course, we needed to stop for gas.\nIf you have never pulled a trailer with your vehicle, one thing you learn really quickly is that seemingly simple things can become a big problem if the right conditions are not in place.\nIn order to fill up the tank with our very long pull set-up, we ideally need a gas station that is specifically set up to accommodate us. Usually highways are full of these, but this evening, we could not find what we needed for an ideal gas session.\nMy husband was sure that our trailer would be just fine pulling through a larger gas station. I, however, wasn't so sure.\nHe did an initial walk through of the station, determined the turn radius would be okay, and began the maneuver to pull into the pump spot. He filled with no problem, but the difficult part was yet to come. I was nervous and on edge.\n\"Do you want me to stand nearby to guide you out of the spot?\" I asked him, anxiously.\nHe confidently waved me away, so I got into our follow car and pulled ahead to wait for him to turn out.\nHe began the maneuver and set to turn out of the spot, but one pump over another driver had decided to make her own maneuver. This driver, driving backwards, maneuvered back into a handicapped spot directly next to our van just as my husband was pulling out.\nShe had no intention of waiting for him to stop, and simply just went along her merry way, directly in the path of our van.\nMy husband saw her coming, so he backed up the trailer and van (a big no-no with certain hook-ups) to try to get out of her path, which he was successful in doing. She got herself all parked and settled without issue.\nBut then my husband began to pull out once more, this time, however, on a different turn radius then before. And, it took all of 3 seconds before he hit right into the gas pump! The initial smack was minimal, but as he drove forward the pump handle and hose was ripped away from the machine. The damage was clear on both the pump and our trailer. We had officially had an accident in the gas station!\nMy whole body went numb. I was ready for the worst, and literally bit my lip to keep my mouth closed tight.\nAnger, frustration, and annoyance were hitting me hard, but I knew that my husband was going to be feeling just as many emotions, too.\nI approached my husband cautiously to see what damage had been done, but approached with even more caution the words I chose to put into his heart. We have been married for over a decade\u2026I know my husband very well. He can really struggle with self-criticism and feelings of inadequacy. And, I did not want to add to his already obvious burden.\nI prayed for the wisdom to handle this problem in a Holy, loving manner.\nAnd God provided me the encouragement to simply be silent and still. It helped\u2026a LOT.\nLong story shorter, the damage was minimal to both the pump and our trailer, but the damage to our marriage was non-existent. There had been plenty of opportunity for shoulda, coulda, woulda's to hit the airways, but God's wisdom encourages a quiet tongue for His people, especially in the face of fear and frustration.\nThankfully, I had been the happy benefactor of this wisdom in the heat of the moment.\nProblems come along with the journey of life. And, we are called to act with wisdom, not flesh as the struggles come. The faithwalk is can be quite difficult.\nYour husband is human, and full of error, both from sin, but also from simply just being human.\nAccidents happen, and it is wisdom that reveals to us the best course of action to not break down those around us, but to build them up as the days go by.\nThe Proverbs 31 model helps to show us a woman who uplifted her husband regularly, but makes no mention of any perfect man as the object of her affections. It just simply focuses on her side of the relationship.\nShe used her tongue wisely and carefully, because she understood that her words had effects (good or bad).\nThe fact is that this woman understood her own flawed nature. She realized that she was not perfect, and neither was anyone else, but that all of our imperfections are what God uses to bring Himself glory! They are our greatest asset for the Kingdom of God, if we let them be.\nYou're going to make mistakes. Your husband is going to make mistakes. But, it is His grace to accept that reality and then hand over the reigns of control to Him (for His Name's sake).\nI could have really let out all my emotions and unleashed them all over my husband, but it would have profited me nothing. It only would have added fuel to the fire of the moment, instead of calming things down and letting the embers fade away.\nThe biggest Truth that the Holy Spirit continued to speak to me throughout that whole encounter was that God was allowing this trouble for our triumph and His glory, but I had to walk in accordance with Him and not my own way if I wanted to experience the victory with Him.\nIt is not always easy to trust our Abba in the middle of life's difficulties and trials. But, it is essential to our growth as His people. You can succeed as a godly wife, even if your husband is far from obedient to Him.\nIt is always essential to focus on your own obedience and your own walk, rather than spend your time pointing fingers at the faults and failures of others.\nTaking this single Truth to heart will no doubt be revolutionary in your Walk as a faithful servant of Yahweh.\nWhen difficulties come to surface this week, hold your tongue and seek out our Abba for the wisdom to handle the trouble of the moment. He will help to lead you down paths of righteousness and give you the words, if any, that need to be said.\nBe strong, sweet sister, and take your tongue captive to make it serve you well.\nWhat is one troublesome experience you have had that taught you to trust Yahweh more and more?\n\u00ab What is Faith Anyway? And How Do I Get More Of It!?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There's been a lot of buzz about Open This Little Book by Jesse Klausmeier and Suzy Lee for months now and I an so thrilled to finally have it in my hands! The concept for this book is right up my alley, and knowing that the amazing Suzy Lee provided the illustrations - and in a style I haven't seen her use before - is especially exciting. Open This Little Book has a classic feel to it, both in the story and the illustrations and it also feels like the kind of book a child will remember well into adulthood, hanging on to and sharing with her or his children. Add to this something very cool about the origins of Open This Little Book that I discovered while reading the brilliant picture book illustration blog Seven Impossible Things (that you can read about at the end of this review) and you have a book you will want to run about and buy right away!\nSo, that super cool thing I learned while reading Seven Impossible Things? Just like David Shannon's No David, Jesse Klausmeier wrote a version of Open This Little Book when she was five years old!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Massage Marketing Rebellion: A Moving Massage?\nAn article in last weeks BusinessWeek.com got my attention.\nWho would think that a moving company could come up with such a creative marketing strategy?\nAfter reading this article I knew that some of my hairbrained Massage Marketing schemes just might work! Needless to say, I'm busy doing a \"market niche\" homework assignment right now.\nWhere are the wiley massage therapists? Why aren't we thinking of this kind of stuff?\nI've discussed this before in massage Marketing Rebellion Yahoo Group, yet more and more I've been hearing stories of doctors and dentists offering clients foot massage to reduce the stress of the experience. JetBlue and Bliss have begun giving overnight passengers kits that contain eye masks, earplugs, moisturizer, lip balm and a promotional offer from the spa company. Companies completely unrelated to massage are cashing in on our wonderous works.\nI say, read this article and get your thinking cap on!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"my footnotes need to be more than just cites to sources. i need to add text\/commentary. is this possible? i want all my foornotes to be zotero cites so that the\"supras\" and \"ibids\" match up through all my footnotes.\nthanks damnation. i had tried this a bit using suffixes but now i know how to do it exactly as I want it to look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This entry was posted in fiction writing, foreshadowing, hooks, Nancy Cohen, Nancy J. Cohen, suspense, writing craft by Joe Moore. Bookmark the permalink.\nThat's because the show needs a suspenseful opening but then it shifts to current day and builds to that point. Like a prologue in a story\u2026lazy writing.\nSorry, but does this mean to say that the use of a prologue is \"lazy writing?\" What does this mean, exactly?\nSorry, Evan, if I've offended. I have heard numerous editors say to nix the prologue. In some cases, it can be helpful, like in framing a piece. But in many instances, the prologue isn't really needed.\nDitto what Terry said. Whenever I see one of those \"little did he know what was about to happen\" transition I stop reading.\nMakes you gnash your teeth, doesn't it?\n(Dean Koontz really liked that image once upon a time).\nIn any event, a description that is a portent is another hook.\nYes, good point! I will add that to my list. An evocative description can set the mood and raise anticipation for what's to come.\nAuthor intrusion stops me cold. Like Kris, I usually stop reading and go to the next book in my TBR stack.\nThe techniques Lynn Sholes and I use most often to keep readers turning the pages are short chapters, cliffhanger endings and a trick I learned from you, Nancy: bait and switch.\nI use Bait and Switch tactics in my romantic adventures. They're perfect for raising suspense as long as the readers get engaged in each person's viewpoint. In case you don't know what this is, you leave one character in jeopardy and then switch viewpoints and do the same to him. Back and forth with cliffhanger endings for each segment.\nI try to end each chapter with a question or problem that has to be answered in the next chapter, Nancy. Good blog.\nQuestions are good, Elaine, and so are ethical dilemmas for your character.\nVery helpful suggestions! I agree about not using omniscient narrator intrusions. It might have worked well in the Victorian era but readers are more sophisticated these days.\nAuthor intrusion is something that often needs to be pointed out to new writers.\nWhen I think about the importance of chapter endings, one author always comes to mind: Dan Brown. He's masterful at knowing how to get readers to keep the pages turning. His chapters also tend to be very short, which I like, though I don't usually write that way. Great post\u2026thanks!\nShort chapters and short paragraphs tend to keep the pacing quick. Personally, I don't care for too short chapters but that's a personal preference.\nI've been told that my novel (to be released in September or October) is a page-turner, and that certainly was my objective, so I took this blog post as an exercise, and analyzed the endings of all 53 chapters in the hopes that I might be able to add something of value to your post.\nNot sure what I have to say adds value, but I found the exercise enlightening.\nMany of my chapters end with decisions, combined with the character's reactions to the decision. In some cases, the character questions the advisability of his or her decision and even reacts emotionally to the decision (e.g., feels compelled, is being blackmailed, or is trapped somehow into making the particular decision.) Invariably, until the end, there is a lot of doubt in the mind of the reader, I think, about how the decision will work out.\nIn some cases, the chapter ends with the character's reactions to what happened, i.e., a sequel, usually short, maybe one sentence, and often includes a statement about the world from the character's POV (readers love observations about the world, if you believe the number of times they highlight such statements in their Kindles.) A couple of scenes involve a major failure by the character, and the character reacts to that failure, i.e., another sequel ending.\nSo, I guess there's enough variety.\nOne think I dislike personally is breaking a scene into two scenes, i.e., the proverbial cliff-hanger where you leave the character hanging on a branch at the cliff. I think that one is a cheap trick, and I try to avoid it in favor of better ways to make the reader want to turn the page.\nThank you for your comments, Sheryl. Yes, action and reaction are good ones to add to the list. How a character responds to an event that just happened or a decision he just made can certainly compel a reader to turn the page.\nI think your label more correctly describes what's going on in those \"little did they know\" statements.\nAs with so many of the tools we writers use, I wonder if a book written with an 'independent' narrator might get away with the 'little did they know' author intrusions. Or, perhaps a book with a frame and a narrator. Or, perhaps a book written now, but looking back at what happened in the past.\nI'm scurrying to my personal library to see how Ruth Rendall, writing as Barbara Vine, handles this. I can't remember her doing it, however.\nI did see this in one of Follett's books, but I can't remember which one. It annoyed me, however.\nI wouldn't be as emotionally engaged with this type of distancing from the story with an independent narrator. But if it's the viewpoint character perhaps looking back at what happened, then I could accept it.\nI'm thinking that the possible \"exceptions\" I mentioned in my prior post aren't really author intrusions, but narrator intrusions, but, still, I wonder if even those are needed when there are so many other ways to get the reader to turn the pages.\nIf an author does foreshadowing properly, one shouldn't need those intrusive statements.\nAll this makes good sense. I would just add one thing: most of what leads me to keep reading has to do with characters. When the writer has created characters that are fresh, engaging, free of clich\u00e9, I keep going., even when chapter hooks are missing. The best of all worlds results when great characters are in stories written by writers who also know how to end chapters effectively.\nYes, Barry, you are right. It's characters who bring readers back for more. That said, I wouldn't end a chapter with one of those engaging people going to sleep. It's too tempting for the reader to put down the book, too. But yes, if we love the characters, we'll keep reading to find out what happens to them.\nThis is a great discussion and a very timely post. LIke many others here, that \"little did he know\u2026\" phrase riles me up. If he didn't know, he didn't know!\nYour examples were spot on, and I loved how you summed it up so succinctly. As a writer, I have to rein myself in and let the information out a little at a time, because I'm burning to share every bit of the story right away!\nI'm sure your list of seven hooks encompasses this, but as I read through them, I was thinking of each hook's impact on the POV Character only. If the danger isn't just personal, but to a loved one or a vulnerable individual or even a pet, a person will also burn the midnight oil to correct the situation.\nThanks for jumpstarting my creative process this morning!\nYes, that's true in regards to the jeopardy involving someone else. Good point, Maggie!\nVery helpful \u2013 I'm using them for my revision. Thanks. Author intrusion is a big problem; I always wonder if its not having a good grasp of POV.\nYes, that could be the case. If you're in deep viewpoint, you are not going to see something the main character doesn't see.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Full description of Guide for GTA San Andreas Games for PC.\nSunny Los Santos, attracts not only tourists, but gta-sa dangerous criminals. This game, gta-sa events that take place in the fictional state of gta-san-andreas, 12 years old, but it is gta-sa popular among players worldwide.\nLargely due to user modes for GTA San Andreas, such as gta-sa: MP or Hot Coffee.We gta-sa you traditional cheat codes for gta-san-andreas, with which you can become a gta-san-andreas in the game gta-sa world.\nGuide gta-san-andreas To gta-sa your friends who don't know or haven't been gta-sa to game of gta-san-andreas. !Whereas other games try to gta-sa as realistic as possible, this one puts emphasis on your journey this is a guide to help you play it more easy .\ngta-san-andreas is an gta-sa fan app that is not owned by the developer and of their respective owners. We believe that gta-san-andreas the doctrine of applications as they are reduced size and excerpted for your information if you would like to request the removal of images of our collection for a gta-san-andreas. Feel free to contact us and we will be happy to gta-sa oblige.\nWith them and you have to gta-san-andreas or arrange shooting, if something goes wrong. Become a mafia boss, start their own business and destroy all who stand in your way! And with our cheat will make it easier and faster! In gta-sa new app has everything you need!\nThis guide for score! hero is intended only to assist people playing this excellent game. All characters, locations, images and video game content are copyright of their respective owners and usage for this game guide falls within fair use guidelines.\nIf you see any copyright or trademark infringement that doesn't take after inside the \"gta-san-andreas\" guidelines, then contact us specifically.Free Game Guide for GTA Vice City is an unofficial guide. ity game players!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rich chocolate cake, vegan and gluten free, filled with our vegan chocolate truffle and our handcrafted raspberry filling. Beautifully iced and decorated in ganache. Delicious for anyone who is vegan and everyone who has gluten, dairy or egg allergies.\nThe cake is a 7\" round and serves 10.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Help u post ur link.\nPlease learn to post to a jpg, not an HTML link. It's covered in the newbie's guide. Also, Multiply is a horrible place to host images.\nThat being said, this is probably the worst pic of Jaslin I've seen.\nAiyooo... what happen to her... I have to agree, not the best shot of her.\nIn the first place, why did the TS want to post this thread for comments??\nThat's nothing but a party snapshot that should never see the light of the web.\nI think the photo which you posted probably portraying a girl trying to 'act cute' to apologise which turns out with a weird expression.\nI went to multiply and browse through your album as well, most of the shots are direct shots which is suitable for Resume\/NRIC application etc. It does not portray any meaning in your photo, just a direct shot, thats all.\nTry shooting in different angle and different zoom and contrast. Can be improved further.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The transition from indoor to beach volleyball is one that can be both exciting and challenging. Living in the Midwest offers its own set of hurdles as players do not have the same opportunities as those living on the coasts to get touches in the sand. There are 3 main focuses that help our players at Wisconsin Juniors Volleyball Club transition from playing indoor to playing beach volleyball: 1) Conditioning, 2) Accountability, and 3) Nutrition.\nOne of the main differences of indoor and sand is the speed needed to get around the court. Because of this, it is important for players to get into condition and work on quick reactions and ability to change directions with the least amount of movement. A great drill for this is the \"star drill\", see an example below. This requires a player to stay low while moving in the sand and pivoting to change directions rather than running in a half circle. The more the players practice these movements the faster their game becomes.\nIn addition to conditioning, players have a much bigger role when playing 2's in the sand. Understanding the release to the middle of the court vs to the indoor \"setters zone\" is a big adjustment and important to train. For players new to the sand game, drills that include serve receive with the opposite player releasing quickly to mid court is key. Our club uses \"release\" as the key word, and we put a hoola hoop down in the sand so the players understand where to release to and where the pass should be. The higher level players are taught to pass more in terms of the other teams defense, but for starters releasing to the middle is a good start. Another important factor is the player who received the ball coming to the setter versus calling for a set. This concept can take time when moving from the indoor game to sand. Getting a player to change that mindset requires a lot of practice and reinforcement.\nFinally, players need to be much more conscious about nutrition when moving to sand. Water consumption is essential to a players performance and due to the calories expended, proper nutrition is vital. We take time in our program to teach players the impact of nutrition on their game and what to pack in their bag for a tournament. Here is a helpful nutrition plan for beach volleyball athletes that outlines fluid and food intake that will help athletes rebuild muscles and recover after workouts, practices and long days in the sun. Suggest your athletes to have granola bars, honey, powder pouches for electrolytes (lemonade or Gatorade), bananas and watermelon for potassium and hydration, and proteins like a peanut butter and jelly sandwich on wheat or lean turkey or chicken on Wheat.\nIn the long run, getting young players to play both indoor and beach volleyball proves beneficial in the physical and mental aspects of their game. A player will develop faster reactions, mental toughness, and increase their awareness of the impact of nutrition on their game.\nRebecca Muff is a beach volleyball coach for Sky High Volleyball Club, a JVA member club in Chicago, IL. She has been coaching Varsity High School and Club volleyball since 1993 in both Illinois and Wisconsin. Rebecca began coaching beach volleyball four years ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The air conditioning in Steven's car stopped working last week.\nPerfect timing, right? It's only 100+ degrees these days. It's gone out before and he has been able to fix it. But this time, I think it's just donezo.\nIt makes me sad for poor Stevoid. Driving around with the windows down and still sweating bullets. Luckily, his drive to work isn't that long, and we take my car just about everywhere else.\nIt's not the end of the world. There are certainly worse things. And it will be fall soon enough.\nI guess the real reason the broken air conditioner bums me out is because we've been saving to get Steven a new vehicle for almost a year now. Not just any new vehicle, but a truck \u2014 something he's been dreaming about since he was 15.\nWe're both a leetle upset that it's taking so long. His '01 Chevy Cavalier (the vehicle he has had as long as I've known him\u2026.almost nine years) has actually held up pretty well for the most part. But over the past year it's gone downhill fast. The broken air conditioner is just another reminder that a new vehicle is no longer just a want, but quickly becoming a need.\nIt's times when the air conditioning breaks that make you realize all the things you don't have, instead of the things you do.\nThere are many, many reasons why it's DUMB to let those kinds of things get to you. There are many many reasons why it's DUMB to compare your life to others, and even worse to compare your material possessions to others'. Gross. Bleh. Whyyyyy.\nI know this, yet I still find myself from time to time moping about because so and so has this or that.\nAre all or your needs met? Yes.\nDo you ever go hungry? No.\nDo you sleep in a warm bed every night? Yes.\nDo you have a vehicle? Yes.\nDo you have health insurance? Yes.\nWhen I sat down and really thought about it, I realized I have all that I need\u2026.and a pretty dang good portion of additional wants. Once I realized that, it just seemed wrong to want more.\n(And I mean, obviously, we have chosen to spend our money on other things, while also trying to save. Puppies are expensive too. But even if we had not made those decisions, we still would not be able to buy a truck for several more months).\nAs humans, we compare ourselves to others (with or without the help of social media) as a way to draw conclusions about ourselves and our lives. It's natural. However, it is not healthy to let this consume us, or to let the success of others take away from the happiness we find in our own lives.\nI've come to love, really love, our one bedroom apartment. Sure, it would be nice to have a yard for Dixie and a little bit more space, but it's also nice to not have to care for a yard, have access to a pool, and have maintenance problems fixed within 24 hours, free of charge.\nI have all that I need and then some. And I have a hell of a lot more than many people (too many) here in the richest country in the world (Uh, one in five U.S. children lives in poverty. ONE IN FIVE). I know this and I'm grateful\u2026.even if I sometimes forget how blessed I am.\nWe're impatient creatures. We want everything and we want it now. (Uniquely American? Unique to Generation Me? Hmmmm). I will admit that it wasn't until recently \u2014 like the last couple years \u2014 that I began to understand just how long and hard my parents worked to give me and my siblings the life we had growing up. I mean, I saw them go to work everyday. I saw my dad pay the bills at the kitchen table with a look of strain and worry on his face. I knew money didn't grow on trees.\nBut I guess I just thought it would happen faster. I thought a college degree paved the way, and once Steven and I had big kid jobs (which we do), we could afford things like a new vehicle and maybe a two bedroom apartment or a duplex.\nAnd maybe in a better economy and a different time period, that would be the case. But it isn't right now, at least not for us, and that's the way it is. But in all honestly (really I'm not just saying this) I think I like it that way.\nIf we had everything we wanted right now, what would there be to look forward to? If we were buying houses and cars and taking trips and getting promotions and living extravagantly at 23 and 24, what will we need to be happy when we're 50? Would we even value the things we have? Or would we just want more? When will it be enough?\nThe broken air conditioner in Steven's old car caused me to temporarily forget this. We will need to get a new vehicle by the end of the year, BUT at the end of the day, we are fed, clothed, and happy.\nAnd if we weren't so concerned with everyone else, we just might find that we all have enough right now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dasfintechnologies Ltd was formed by a group of crypto currency experts in 2013, working in several environments and platforms our 100-year combined business experience has allowed us to tap into emerging markets and focus in some of the world's most exciting spaces in today's technology.\nWith experience in a whole range of industries that include finance, wealth & debt management, sales, marketing & technology.\nDuring our years of experience, we have grown our global business partnerships into over 50 countries this has given us a more open understanding of different cultures and the needs.\nWe believe that knowledge, education, & business experience is something that should be shared freely with as many people across the world giving the everyday person the opportunity to better their lifestyle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2017-08-31 Assigned to MICRON TECHNOLOGY, INC. reassignment MICRON TECHNOLOGY, INC. ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: MANNING, TROY A.\n2017-11-01 Assigned to MORGAN STANLEY SENIOR FUNDING, INC., AS COLLATERAL AGENT reassignment MORGAN STANLEY SENIOR FUNDING, INC., AS COLLATERAL AGENT SUPPLEMENT NO. 6 TO PATENT SECURITY AGREEMENT Assignors: MICRON TECHNOLOGY, INC.\n2017-11-01 Assigned to U.S. BANK NATIONAL ASSOCIATION, AS COLLATERAL AGENT reassignment U.S. BANK NATIONAL ASSOCIATION, AS COLLATERAL AGENT SUPPLEMENT NO. 6 TO PATENT SECURITY AGREEMENT Assignors: MICRON TECHNOLOGY, INC.\nThe present disclosure includes apparatuses and methods related to performing compare and\/or report operations using sensing circuitry. An example method can include charging an input\/output (IO) line of a memory array to a voltage. The method can include determining whether data stored in the memory array matches a compare value. The determination of whether data stored matches a compare value can include activating a number of access lines of the memory array. The determination can include sensing a number of memory cells coupled to the number of access lines. The determination can include sensing whether the voltage of the IO line changes in response to activation of selected decode lines corresponding to the number of memory cells.\nThis application is a Continuation of U.S. application Ser. No. 15\/287,980, filed Oct. 7, 2016, which issues as U.S. Pat. No. 9,767,865 on Sep. 19, 2017, which is a Continuation of U.S. application Ser. No. 14\/603,850, filed Jan. 23, 2015, which issued as U.S. Pat. No. 9,466,340 on Oct. 11, 2016, which is a Continuation of U.S. application Ser. No. 13\/952,054, filed Jul. 26, 2013, which issued as U.S. Pat. No. 8,964,496 on Feb. 24, 2015, the contents of which are incorporated herein by reference.\nThe present disclosure relates generally to semiconductor memory and methods, and more particularly, to apparatuses and methods related to performing compare operations using sensing circuitry.\nElectronic systems often include a number of processing resources (e.g., one or more processors), which may retrieve and execute instructions and store the results of the executed instructions to a suitable location. A processor can comprise a number of functional units such as arithmetic logic unit (ALU) circuitry, floating point unit (FPU) circuitry, and\/or a combinatorial logic block, for example, which can be used to execute instructions by performing logical operations such as AND, OR, NOT, NAND, NOR, and XOR logical operations on data (e.g., one or more operands). For example, the functional unit circuitry (FUC) may be used to perform arithmetic operations such as addition, subtraction, multiplication, and\/or division on operands.\nA number of components in an electronic system may be involved in providing instructions to the FUC for execution. The instructions may be generated, for instance, by a processing resource such as a controller and\/or host processor. Data (e.g., the operands on which the instructions will be executed) may be stored in a memory array that is accessible by the FUC. The instructions and\/or data may be retrieved from the memory array and sequenced and\/or buffered before the FUC begins to execute instructions on the data. Furthermore, as different types of operations may be executed in one or multiple clock cycles through the FUC, intermediate results of the instructions and\/or data may also be sequenced and\/or buffered.\nExecuting instructions (e.g, as part of program execution) can involve performing operations such as compare operations and the results can be provided (e.g., reported) to the processing resources as part of the executional flow of an algorithm, for example. Such compare and report functionality can enable, for instance, \"if-then-else\" programmatic flow, which is often part of program execution.\nFIG. 2 illustrates a schematic diagram of a portion of a memory array coupled to sensing circuitry in accordance with a number of embodiments of the present disclosure.\nFIG. 3 illustrates a schematic diagram of a portion of a memory array coupled to sensing circuitry in accordance with a number of embodiments of the present disclosure.\nFIG. 4 illustrates an example of a method for performing a compare operation in accordance with a number of embodiments of the present disclosure.\nThe present disclosure includes apparatuses and methods related to performing compare operations using sensing circuitry. An example method comprises charging (e.g., precharging) an input\/output (IO) line (e.g., a local IO line (LIO line)) of a memory array to a pvoltage (e.g., a precharge voltage). The method can include determining whether data stored in the memory array matches a compare value by activating a number of access lines of the memory array and sensing a number of memory cells coupled to the number of access lines. The method can include sensing whether the voltage (e.g., precharge voltage) of the LIO line changes in response to activation of selected decode lines (e.g., column decode lines) corresponding to the number of memory cells. In the present disclosure, a \"line\" is meant to refer to an operable coupling between at least two nodes.\nA number of embodiments of the present disclosure can provide benefits such as improved compare and report functionality in association with determining whether a match exists between a compare value (e.g., a particular data value and\/or set of data values) and a data value stored in a memory array. For instance, a number of embodiments can provide for identifying whether particular data is stored in a number of memory cells without transferring data out of the memory array and sensing circuitry via a bus (e.g., data bus, address bus, control bus), for instance. The determination of whether data stored in the array matches the compare value can be reported, for instance, to control circuitry (e.g., to an on-die controller and\/or to an external host). The determination of whether data stored in the array matches the compare value can be reported into the memory array. Such compare and report functionality can be associated with performing a number of logical operations (e.g., AND, NOT, NOR, NAND, XOR, etc.). However, embodiments are not limited to these examples.\nAlso, circuitry such as FUC associated with various processing resource(s) may not conform to pitch rules associated with a memory array. For example, the cells of a memory array may have a 4F2 or 6F2 cell size, where \"F\" is a feature size corresponding to the cells. The devices (e.g., logic gates) associated with FUC of previous systems may not be capable of being formed on pitch with the memory cells, which can affect chip size and\/or memory density, for example.\nIn the following detailed description of the present disclosure, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration how one or more embodiments of the disclosure may be practiced. These embodiments are described in sufficient detail to enable those of ordinary skill in the art to practice the embodiments of this disclosure, and it is to be understood that other embodiments may be utilized and that process, electrical, and\/or structural changes may be made without departing from the scope of the present disclosure. As used herein, the designators \"N,\" \"P,\" \"R,\" etc., particularly with respect to reference numerals in the drawings, can indicate that a number of the particular features so designated can be included. As used herein, \"a number of\" a particular thing can refer to one or more of such things (e.g., a number of memory arrays can refer to one or more memory arrays).\nThe figures herein follow a numbering convention in which the first digit or digits correspond to the drawing figure number and the remaining digits identify an element or component in the drawing. Similar elements or components between different figures may be identified by the use of similar digits. For example, 130 may reference element \"30\" in FIG. 1, and a similar element may be referenced as 230 in FIG. 2. As will be appreciated, elements shown in the various embodiments herein can be added, exchanged, and\/or eliminated so as to provide a number of additional embodiments of the present disclosure. In addition, as will be appreciated, the proportion and the relative scale of the elements provided in the figures are intended to illustrate certain embodiments of the present invention, and should not be taken in a limiting sense.\nSystem 100 includes a host 110 coupled to memory device 120, which includes a memory array 130. Host 110 can be a host system such as a personal laptop computer, a desktop computer, a digital camera, a mobile telephone, or a memory card reader, among various other types of hosts. Host 110 can include a system motherboard and\/or backplane and can include a number of processing resources (e.g., one or more processors, microprocessors, or some other type of controlling circuitry). The system 100 can include separate integrated circuits or both the host 110 and the memory device 120 can be on the same integrated circuit. The system 100 can be, for instance, a server system and\/or a high performance computing (HPC) system and\/or a portion thereof. Although the example shown in FIG. 1 illustrates a system having a Von Neumann architecture, embodiments of the present disclosure can be implemented in non-Von Neumann architectures (e.g., a Turing machine), which may not include one or more components (e.g., CPU, ALU, etc.) often associated with a Von Neumann architecture.\nFor clarity, the system 100 has been simplified to focus on features with particular relevance to the present disclosure. The memory array 130 can be a DRAM array, SRAM array, STT RAM array, PCRAM array, TRAM array, RRAM array, NAND flash array, and\/or NOR flash array, for instance. The array 130 can comprise memory cells arranged in rows coupled by access lines (which may be referred to herein as row lines, word lines or select lines) and columns coupled by sense lines (which may be referred to herein as digit lines or data lines). Although a single array 130 is shown in FIG. 1, embodiments are not so limited. For instance, memory device 120 may include a number of arrays 130 (e.g., a number of banks of DRAM cells). An example DRAM array is described in association with FIGS. 2 and 3.\nControl circuitry 140 decodes signals provided by control bus 154 from the host 110. These signals can include chip enable signals, write enable signals, and address latch signals that are used to control operations performed on the memory array 130, including data read, data write, and data erase operations. In various embodiments, the control circuitry 140 is responsible for executing instructions from the host 110. The control circuitry 140 can be a state machine, a sequencer, or some other type of controller (e.g., an on-die controller).\nAn example of the sensing circuitry 150 is described further below in association with FIGS. 2 and 3. For instance, in a number of embodiments, the sensing circuitry 150 can comprise a number of sense amplifiers (e.g., sense amplifiers 206-1, . . . , 206-P shown in FIG. 2 or sense amplifier 306 shown in FIG. 3) and a number of compute components (e.g., compute component 331 shown in FIG. 3), which may comprise an accumulator and can be used to perform compare and report operations (e.g., on data associated with complementary sense lines). In a number of embodiments, the sensing circuitry (e.g., 150) can be used to perform compare and report operations using data stored in array 130 as inputs and store the results of the logical operations back to the array 130 without transferring via a sense line address access (e.g., without firing a column decode signal). As such, various compute functions can be performed within array 130 using sensing circuitry 150 rather than being performed by processing resources external to the sensing circuitry (e.g., by a processor associated with host 110 and\/or other processing circuitry, such as ALU circuitry, located on device 120 (e.g., on control circuitry 140 or elsewhere)). In various previous approaches, data associated with an operand, for instance, would be read from memory via sensing circuitry and provided to external ALU circuitry via local I\/O lines. The external ALU circuitry would perform compute functions using the operands and the result would be transferred back to the array via the local I\/O lines. In contrast, in a number of embodiments of the present disclosure, sensing circuitry (e.g., 150) can be configured to perform logical operations on data stored in memory (e.g., array 130) and store the result to the memory without enabling a local I\/O line coupled to the sensing circuitry.\nFIG. 2 illustrates a schematic diagram of a portion of a memory array coupled to sensing circuitry in accordance with a number of embodiments of the present disclosure. In this example, the memory array is a DRAM array of memory cells (MCs) 260-1, . . . , 260-N. In a number of embodiments, the memory cells are destructive read memory cells (e.g., reading the data stored in the cell destroys the data such that the data originally stored in the cell is refreshed after being read). The memory cells 260-1, . . . , 260-N of the array in FIG. 2 can be arranged in a number of rows coupled by word line 204 and columns coupled by sense lines (e.g., digit lines) 205-1, . . . , 205-M. For ease of reference, the sense lines 205-1, . . . , 205-M represent respective pairs of complementary sense lines (e.g., 305-1 and 305-2 in FIG. 3). Although only one row and two columns of memory cells are illustrated in FIG. 2, embodiments are not so limited. For instance, a particular array may have a number of columns of memory cells and\/or sense lines (e.g., 4,096, 8,192, 16,384, etc.). As an example, a gate of a particular memory cell transistor (e.g., 302 in FIG. 3) can be coupled to its corresponding word line (204), a source\/drain region can be coupled to its corresponding sense line (e.g., 205-1), and a second source\/drain region of a particular memory cell transistor can be coupled to its corresponding capacitor (e.g., 303 in FIG. 3).\nIn operation, sense amps (e.g., 206-1 to 206-P) can sense a data value (e.g., a logic \"1\" or \"0\") stored in a memory cell (e.g., 260-1 to 260-N) by amplifying a differential signal (e.g., voltage or current) on the complementary sense lines (e.g., 205-1 to 205-M) responsive to activation of a selected row line (e.g., 204). As an example, the sense amps 206-1 to 206-P can drive one of the sense lines (e.g., D) of the pair of complementary sense lines 205-1 to a first value (e.g., to a supply voltage such as Vcc), and the other sense line (D_) of the pair of complementary sense lines 205-1 to a second value (e.g., to a reference voltage such as a ground voltage). In this manner, the data value stored by the memory cell (e.g., 260-1) can be determined based on which of the sense lines of the complementary sense line pair is driven to Vcc, for instance. The voltages of the complementary sense line pairs 205-1 to 205-M can then be selectively transferred to the IO lines 266-1 and 266-2 via activation of the column decode lines 264-1 to 264-R. In this manner, the data sensed by the sense amps 206-1 to 206-P can be transferred to the SSA 268 via IO lines 266-1 and 266-2. Often, the SSA 268 may only be capable of storing a data value from a single cell (e.g., one of cells 260-1 to 260-N) at a particular time. As such, if it is desired to transfer the data stored in cell 260-1 to the SSA 268, then column decode line 264-1 would be activated, and if it is desired to transfer the data stored in cell 260-N to the SSA 268, then column decode 264-R would be activated. If both lines 264-1 and 264-R were activated, the SSA 268 may not be able to determine the actual stored data values stored in either of the cells.\nHowever, in various instances, it can be useful to selectively activate more than one of the column decode lines (e.g., 264-1 to 264-R). For example, selectively activating a number of column decode lines can be done in association with performing a compare operation in accordance with a number of embodiments described herein. For instance, in a number of embodiments of the present disclosure, the data path portion shown in FIG. 2 can be operated to determine whether data stored in a memory array (e.g., array 130) matches a compare value, which may be provided by an on-die control circuit (e.g., control circuitry 140) and\/or by external control circuitry (e.g., host 110) as part of an \"if-then-else\" programmatic flow, for example.\nIn a number of embodiments, control circuitry (e.g., 140 in FIG. 1) can be configured to charge (e.g., precharge) an IO line (e.g., 266-1) to a voltage (e.g., a precharge voltage). For example, the IO line 266-1 can be precharged to a voltage (e.g., a supply voltage such as Vcc) corresponding to a logic \"1.\" The control circuitry can be configured to selectively activate row lines (e.g., a row line including memory cells 260-1, . . . , 260-N) and column decode lines (e.g., CD-1, . . . , CD-R). Sensing circuitry (e.g., 150 in FIG. 1) can be configured to sense a number of selected memory cells (e.g., 260-1, . . . , 260-N) coupled to an activated row line. The sensing circuitry can be configured to determine whether the precharge voltage of the IO line 266-1 changes in response to selective activation of column decode lines CD-1 to CD-R.\nActivation of column decode line CD-1 turns on transistors 218-1 and 218-2, which provides voltages corresponding to the data stored in sense amp 206-1 to IO lines 266-1 and 266-2. As such, the precharge voltage of IO line 266-1 can change based on the particular data value stored in sense amp 206-1 (which represents the data stored in a particular memory cell such as cell 260-1). For example, if the sense amplifier 206-1 senses a logic 0 (e.g., a ground voltage) stored in cell 260-1, then the precharge voltage (e.g., Vcc) on the IO line 266-1 will be pulled down (e.g., lowered) when CD-1 is activated, and the change in the precharge voltage change can be detected by the SSA 268. As such, the detected change in the precharge voltage indicates that the sensed memory cell (e.g., 260-1) stores a data value (e.g., 0) different from the data value (e.g., 1) corresponding to the precharge voltage. Similarly, if the sense amplifier 206-1 senses a logic 1 (e.g., Vcc) stored in cell 260-1, then the precharge voltage (e.g., Vcc) on the IO line 266-1 will not be pulled down when CD-1 is activated, and no change in the precharge voltage will be detected by the SSA 268. As such, no detected change in the precharge voltage indicates that the sensed memory cell (e.g., 260-1) stores the same data value (e.g., 1) as the data value (e.g., 1) corresponding to the precharge voltage.\nThe above described ability of the SSA 268 to determine whether the precharge voltage changes can be used to perform compare functions to determine whether a particular compare value matches data stored in a memory array, for instance. As an example, if an operation requires knowledge of whether a number of cells coupled to a particular row line stores a particular compare value (e.g., 0), the particular row line can be activated along with the sense lines corresponding the number of memory cells. If any of the cells store a 0, then the precharge voltage of the IO line (e.g., local IO line) will be changed (e.g., pulled down). The result of the operation can be reported, for instance, to the requesting control circuitry (e.g., on-die controller, host, etc.). The result of the operation can be reported into the memory array for further calculations. The determined result may be used as part of continued execution of a particular algorithm. For instance, execution may include not only determining if any of the memory cells of the row store a data value (e.g., 0), but which cell(s) store the data value. As such, subsets of the column decode lines may be selectively activated to compare the data values stored by their corresponding cells to the compare value, which can be used in association with binary searching, for instance.\nThe compare values used in association with compare operations can be requested by control circuitry coupled to the sense circuitry (e.g., on-die controller) and\/or by a number of other sources such as an external host, for instance. Similarly, results of compare operations can be reported to various control circuitry and\/or used to perform further operations (e.g., logic operations) as part of if-then-else programmatic flow prior to being reported to control circuitry.\nFIG. 3 illustrates a schematic diagram of a portion of a memory array 330 coupled to sensing circuitry in accordance with a number of embodiments of the present disclosure. In this example, the memory array 330 is a DRAM array of 1T1C (one transistor one capacitor) memory cells each comprised of an access device 302 (e.g., transistor) and a storage element 303 (e.g., a capacitor). In a number of embodiments, the memory cells are destructive read memory cells (e.g., reading the data stored in the cell destroys the data such that the data originally stored in the cell is refreshed after being read). The cells of array 330 are arranged in rows coupled by word lines 304-0 (Row0), 304-1 (Row1), 304-2, (Row2) 304-3 (Row3), . . . , 304-N (RowN) and columns coupled by sense lines (e.g., digit lines) 305-1 (D) and 305-2 (D_). In this example, each column of cells is associated with a pair of complementary sense lines 305-1 (D) and 305-2 (D_). Although only a single column of memory cells is illustrated in FIG. 3, embodiments are not so limited. For instance, a particular array may have a number of columns of memory cells and\/or sense lines (e.g., 4,096, 8,192, 16,384, etc.). A gate of a particular memory cell transistor 302 is coupled to its corresponding word line 304-0, 304-1, 304-2, 304-3, . . . , 304-N, a first source\/drain region is coupled to its corresponding sense line 305-1, and a second source\/drain region of a particular memory cell transistor is coupled to its corresponding capacitor 303. Although not illustrated in FIG. 3, the sense line 305-2 may also be coupled to a column of memory cells.\nThe array 330 is coupled to sensing circuitry in accordance with a number of embodiments of the present disclosure. In this example, the sensing circuitry comprises a sense amplifier 306 and a compute component 331. The sensing circuitry can be sensing circuitry 150 shown in FIG. 1. The sense amplifier 306 is coupled to the complementary sense lines D, D_ corresponding to a particular column of memory cells. The sense amp 306 can be operated to determine a state (e.g., logic data value) stored in a selected cell. Embodiments are not limited to the example sense amplifier 306. For instance, sensing circuitry in accordance with a number of embodiments described herein can include current-mode sense amplifiers and\/or single-ended sense amplifiers (e.g., sense amplifiers coupled to one sense line).\nIn a number of embodiments, a compute component (e.g., 331) can comprise a number of transistors formed on pitch with the transistors of the sense amp (e.g., 306) and\/or the memory cells of the array (e.g., 330), which may conform to a particular feature size (e.g., 4F2, 6F2, etc.). As described further below, the compute component 331 can, in conjunction with the sense amp 306, operate to perform various compare and report operations using data from array 330 as input and store the result back to the array 330 without transferring the data via a sense line address access (e.g., without firing a column decode signal such that data is transferred to circuitry external from the array and sensing circuitry via local I\/O lines (e.g., 266-1 in FIG. 2).\nIn the example illustrated in FIG. 3, the circuitry corresponding to compute component 331 comprises five transistors coupled to each of the sense lines D and D_; however, embodiments are not limited to this example. Transistors 307-1 and 307-2 have a first source\/drain region coupled to sense lines D and D_, respectively, and a second source\/drain region coupled to a cross coupled latch (e.g., coupled to gates of a pair of cross coupled transistors, such as cross coupled NMOS transistors 308-1 and 308-2 and cross coupled PMOS transistors 309-1 and 309-2. As described further herein, the cross coupled latch comprising transistors 308-1, 308-2, 309-1, and 309-2 can be referred to as a secondary latch (the cross coupled latch corresponding to sense amp 306 can be referred to herein as a primary latch).\nThe transistors 307-1 and 307-2 can be referred to as pass transistors, which can be enabled via respective signals 311-1 (Passd) and 311-2 (Passdb) in order to pass the voltages or currents on the respective sense lines D and D\u2014 to the inputs of the cross coupled latch comprising transistors 308-1, 308-2, 309-1, and 309-2 (e.g., the input of the secondary latch). In this example, the second source\/drain region of transistor 307-1 is coupled to a first source\/drain region of transistors 308-1 and 309-1 as well as to the gates of transistors 308-2 and 309-2. Similarly, the second source\/drain region of transistor 307-2 is coupled to a first source\/drain region of transistors 308-2 and 309-2 as well as to the gates of transistors 308-1 and 309-1.\nA second source\/drain region of transistor 308-1 and 308-2 is commonly coupled to a negative control signal 312-1 (Accumb). A second source\/drain region of transistors 309-1 and 309-2 is commonly coupled to a positive control signal 312-2 (Accum). The Accum signal 312-2 can be a supply voltage (e.g., Vcc) and the Accumb signal can be a reference voltage (e.g., ground). Enabling signals 312-1 and 312-2 activates the cross coupled latch comprising transistors 308-1, 308-2, 309-1, and 309-2 corresponding to the secondary latch. The activated sense amp pair operates to amplify a differential voltage between common node 317-1 and common node 317-2 such that node 317-1 is driven to one of the Accum signal voltage and the Accumb signal voltage (e.g., to one of Vcc and ground), and node 317-2 is driven to the other of the Accum signal voltage and the Accumb signal voltage. As described further below, the signals 312-1 and 312-2 are labeled \"Accum\" and \"Accumb\" because the secondary latch can serve as an accumulator while being used to perform a logical operation. In a number of embodiments, an accumulator comprises the cross coupled transistors 308-1, 308-2, 309-1, and 309-2 forming the secondary latch as well as the pass transistors 307-1 and 308-2. As described further herein, in a number of embodiments, a compute component comprising an accumulator coupled to a sense amplifier can be configured to perform a logical operation that comprises performing an accumulate operation on a data value represented by a signal (e.g., voltage or current) on at least one of a pair of complementary sense lines.\nThe compute component 331 also includes inverting transistors 314-1 and 314-2 having a first source\/drain region coupled to the respective digit lines D and D_. A second source\/drain region of the transistors 314-1 and 314-2 is coupled to a first source\/drain region of transistors 316-1 and 316-2, respectively. The gates of transistors 314-1 and 314-2 are coupled to a signal 313 (InvD). The gate of transistor 316-1 is coupled to the common node 317-1 to which the gate of transistor 308-2, the gate of transistor 309-2, and the first source\/drain region of transistor 308-1 are also coupled. In a complementary fashion, the gate of transistor 316-2 is coupled to the common node 317-2 to which the gate of transistor 308-1, the gate of transistor 309-1, and the first source\/drain region of transistor 308-2 are also coupled. As such, enabling signal InvD serves to invert the data value stored in the secondary latch and drives the inverted value onto sense lines 305-1 and 305-2.\nIn a number of embodiments of the present disclosure, a compare operation can include activating a row of memory cells (e.g., row line 204) to determine if there is a match in the row line (e.g., at least one memory cell stores a compare value). A compare operation can be expanded to include comparing a 32-bit compare value to data stored in the array. For example, compare values of a number of memory cells can be aggregated in an accumulator (as described above) to determine if there is a collection of compare values that match a 32-bit compare value.\nEmbodiments of the present disclosure are not limited to the particular sensing circuitry configuration illustrated in FIGS. 2 and 3. For instance, different compute component circuitry can be used to perform logical operations in accordance with a number of embodiments described herein.\nFIG. 4 illustrates an example of a method for performing a compare operation in accordance with a number of embodiments of the present disclosure. At block 470, the method includes precharging an input\/output (IO) line (e.g., 266-1 in FIG. 2) of a memory array (e.g., 330 in FIG. 3) to a precharge voltage. The IO line (e.g., a local IO line) can be precharged, for instance, to a voltage corresponding to a particular data value, such as a supply voltage (e.g., Vcc corresponding to logic 1) or a reference voltage (e.g., a ground voltage corresponding to logic 0). A number of embodiments can include precharging a LIO_line (e.g., 266-2 in FIG. 2) of a memory array to a precharge voltage. The voltage to which the LIO_line is precharged can be an inverse of a voltage to which the LIO line is precharged.\nAt block 472, the method includes determining whether data stored in the memory array matches a compare value. The compare value can be a value provided by an external host (e.g., an external processor) and\/or an on die controller. The compare value can include a number of different data values that the control circuitry is attempting to determine whether are stored in at least one memory cell in a memory array. The compare value can be stored in a number of memory cells. For example, the data can be stored in one, two, three, etc., memory cells. A match can refer to a determination that a compare value provided by the control circuitry is stored in at least one memory cell of the array. A determination that the compare value is not stored in at least one memory cell can indicate that there is not a match.\nThe determination of whether data stored in the memory array matches a compare value can be determined, at block 474, by activating a number of row lines of the memory array. The number of row lines can be selectively activated based on a characteristic of the row lines. The number of row lines can include particular row lines that are predetermined by a controller (e.g., an external host, an on-die controller).\nThe determination of whether data stored in the memory array matches a compare value can be determined, at block 476, by sensing a number of memory cells coupled to the number of row lines. The voltage of the memory cells of the row lines of the memory array can be sensed by the sense amplifiers and column decode lines can be activated to provide the voltage of the sense amplifiers (and corresponding memory cells) to the LIO line.\nThe determination of whether data stored in the memory array matches a compare value can be determined, at block 478, by sensing whether the precharge voltage of the LIO line changes in response to activation of selected column decode lines corresponding to the number of memory cells. For example, the LIO line can be precharged to a supply voltage (e.g., Vcc) corresponding to a logic 1. A memory cell in the memory array may store a data value (e.g., logic 0) corresponding to a compare value that a controller is trying to locate (e.g., match). When the memory cell is activated and the voltage of the cell is provided to the LIO line (e.g., via the corresponding sense amp), the voltage on the LIO line (e.g., precharge voltage) will drop if the data value stored by the cell matches the compare value (e.g., if the data value stored by the cell is a logic 0). The secondary sense amplifier can detect the drop in voltage and determine that a match has occurred. The determination of the match can be reported to circuitry that provided the compare value (e.g., an on die controller, an external host, etc.) and\/or to some other control circuitry for further use. If a match is determined, further operations can be performed to determine a particular location (e.g., cell or cells) within the array where the match occurs. Peripheral control logic can read a data path to determine the compare state of the memory array. Locating the match can include a search method (e.g., a binary search) to determine which memory cell in the memory array matched. The match can occur at a number of memory cells (e.g, no memory cell, one memory cell, or a plurality of memory cells).\nin response to receiving the request, determining whether a voltage of an input\/output (IO) line of a memory array changes in response to simultaneous activation of a number of selected decode lines and a number of access lines corresponding to at least two memory cells of the memory array.\n2. The method of claim 1, further comprising precharging the IO line of the memory array to a precharge voltage.\n3. The method of claim 1, further comprising charging the IO line to a supply voltage.\n4. The method of claim 3, wherein the supply voltage corresponds to a data value of 1.\n5. The method of claim 1, further comprising charging the IO line to a ground voltage.\n6. The method of claim 5, wherein the ground voltage corresponds to a data value of 0.\n7. The method of claim 6, wherein activating selected decode lines comprises activating a subset of the decode lines of the array.\n8. The method of claim 7, further comprising determining the subset of decode lines based on a set of criteria of the subset.\n9. The method of claim 1, wherein the host comprises at least a number of processors and wherein receiving from the host the request to perform the compare function further comprises receiving from a processor, from the number of processors, the request to perform the compare function.\na determination whether the precharge voltage of the IO line changes in response to simultaneous activation of a number of decode lines and a number of access lines corresponding to at least two memory cells of the array.\n11. The system of claim 10, further comprising the controller configured to activate the number of decodes lines and the number of access lines.\n12. The system of claim 10, wherein the control circuitry comprises an external host.\n13. The system of claim 10, wherein the precharge voltage corresponds to a particular data value and a determined change in the precharge voltage of the IO line indicates that a selected memory cell corresponding to an activated decode line stores a data value other than the particular data value.\n14. The system of claim 10, further comprising a secondary sense amp configured to detect whether the precharge voltage of the IO line changes.\n15. The system of claim 10, wherein a determined change in the precharge voltage of the IO line indicates data in a memory cell matches a compare value.\n16. The system of claim 10, wherein the control circuitry is configured to report a determined change in the precharge voltage of the IO line to a host.\n17. The system of claim 10, further comprising sensing circuitry configured to provide an indication that a determined change in the precharge voltage of the IO line was detected.\na number of accumulators coupled to the number of primary sense amplifiers.\na first pair of cross coupled transistors and a second pair of cross coupled transistors.\n20. The system of claim 18, further comprising a host coupled to the memory device, wherein the memory device is configured to cause at least the charging of the IO line responsive to a request from the host.\nInternational Search Report and Written Opinion for related PCT Patent Application No. PCT\/US2014\/046094, dated Oct. 17, 2014, 11 pages.\nMatsunaga, et al., \"Implementation of a Perpendicular MTJ-Based Read-Disturb-Tolerant 2T-2R TCAM Based on a Reversed Current Reading Scheme\", 2012 17th Asia and South Pacific Design Automation Conference, Jan. 30-Feb. 2, 2012, pp. 475-476, (2 pgs.).\nNotice of Reasons for Rejection for related Korea Patent Application No. 10-2016-7003336, dated Mar. 7, 2016, 9 pages.\nNotice of Rejection for related Korea Patent Application No. 10-2016-7029643, dated Sep. 8, 2017, 8 pages.\nOffice Action for related Taiwan Patent Application No. 103125169, dated Nov. 16, 2015, 10 pages.\nSupplementary European Search Report and Written Opinion for EP Application No. 14829677.5, dated Mar. 1, 2017, 8 pages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carrot Top! Carrot Top was the second freak ... little did I know that he would be the normal one... Carrot Top's tan super buff scary face and toys... were funny and a needed change from Nasty Scary Nick Nolte who... at 68 just had a baby... WHAT? Crazy woman was that desperate... not enough money in the free wold... not enough Viagra ! To have a child with either one!\nWOW... Solomon Burke had to use the Kevin Eubanks and the Tonight Show band to back him up because he ate his own band before the show and \" The Tonight Show must go on\"!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u7533\u4e07\u5b8f\u6e901\u670810\u65e5\u53d1\u5e03\u516c\u544a\u3002Maintain BUY. We expect the company to deliver significant earnings growth on the back of future acquisitions. We maintain our diluted EPS forecasts of Rmb0.22 in FY19E (+37.5% YoY), Rmb0.27 in FY20E (+22.7% YoY), and Rmb0.31 in FY21E (+14.8% YoY). Our target price is unchanged at HK$7.50. With 165% upside, we maintain our BUY recommendation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvfov b/data_all_eng_slimpj/shuffled/split2/finalzzvfov
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index 0000000000000000000000000000000000000000..cb9ceb1e14243cc654978880a3cafa3df7b07645
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@@ -0,0 +1,5 @@
+{"text":"Stored in box since purchase, this fabulous 100% silk scarf is perfect in every way.\nSilk Perfume Bottles Silk Scarf 35\".\nStored since purchase, there are light creases to the fabric from being folded but no pulls or marks to the silk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some images or pictures of engagement photography at the Fullerton Arboretum. These images are meant to show photographic opportunities at this venue. Feel free to contact me at 323.605.4224 if you have any questions about these engagement photographs of the Fullerton Arboretum. Note that these photos are from the perspective of a wedding photojournalist (Fullerton Arboretum). Wedding photojournalism may differ from traditional wedding photography.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ft.Lauderdale, FL, November 25, 2014 \u2014 After an incredible C-Span covered Women of Color Empowerment Conference, The Women of Color Empowerment Institute, Inc announced they will be hosting their community roundtable discussion, \"Women of Color: Advocates for Change\" on Thursday, December 11, 2014 from 5:30 PM to 8:00 PM at the Urban League of Broward County, 560 NW 27th Ave, Fort Lauderdale, FL 33311.\nThe discussion will be led by thought leaders who will encourage robust discussion on topics such as economic development, health care disparities, heritage promotion and engaging more women of color to ascend to leadership roles. The Women of Color Empowerment Institute is encouraging women from all walks of life to attend this roundtable discussion.\nImmediately following the roundtable discussion will be a holiday mix & mingle with tasty treats and beverages.\nAll guest are required to RSVP at the WOCEC website or by calling (954) 768-9770.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a short visit to Saudi Arabia, I wanted to continue my journey through the African Arabic world. I'd already been to Egypt and Libya isn't the most inviting place at the moment, so I flew to Tunisia. On a map, squashed between Algeria and Libya, it looks like a tiny place, but only relatively. Its nearly 1 and a half times bigger than Iceland, which some may argue is a small country, but all the space in Tunisia is inhabitable and inhabited. There are internal flights to the south of Tunisia, reaching Berber country, and regular ferry boats that take you to Italy or Malta from Tunis, so you're literally suspended between Europe and the Sahara, in a little pocket of bustling Arabic life and culture.\nThe tourism market has crashed in Tunisia, ever since the hotel shooting of tourists in Sousse last summer. It's affected the economy and the daily lives of people, especially those in hotels, restaurants or shops, and its heartbreaking to know that one incident can have such long term repercussions on a people open and welcome to tourism. In the souk, a seller told me I was his first foreign customer since last June, and visiting the old towns of Sidi Bou Said and Hammamet and seeing only locals was a strange feeling. But I liked traveling there, and I enjoyed being the only visitor sometimes. Speaking with locals was a breeze since everyone spoke English and their Tunisian was a healthy mix of French and french-isms. The cafe culture was just like some neighbourhoods in Paris, and Sidi Bou Said could have been a village in Santorini.\nI stayed with a friend I made in Jordan, a Tunisian woman and her family. We shared a passion for tango dancing, and I also tried salsa dancing, but the social dance scene was a little different than I was used to. The tandas were followed by cortinas of belly dance songs where all the men and women got on the floor and started yelling, twisting their hands and shaking their hips. The salsa dance night was more zouk and kazumba, an awkwardly slow and grindy style that I couldn't get into.\nWe found horses to goppity gopp, and not just any horses. First we rode a retired show jumping horse and an endurance racer, then got an invitation back to ride his breeding stallion, a short-track champion. He gave me chills just to look at, and after managing to jump on his back his ovner asked me 'are you sure?' I'll never be sure what he meant but I managed to stay on for one hell of a ride. Riding him back to his harem of mares was the only real tricky part, but he could have carried 3 of me for a whole day and night without tiring.\nLike so many other places, I left Tunisia with a longer list of things to do and see than I accomplished during my stay. So there has to be a next time, and on the top of my list is race the Arabian, and learn how to belly dance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Never has it been more vital to find scientific fixes to the environmental problems facing our planet. Our petroleum-dependent global population is sprinting past seven billion, and with energy needs growing, one solution lies with biotechnology \u2013 using biology instead of geology to power our world.\nFor nearly 215 years, DuPont has turned to science for answers, and at this critical juncture more than ever, this strategy holds. On October 30, DuPont celebrated the opening of the world's largest cellulosic ethanol plant \u2013 a \"biorefinery\" that will produce 30 million gallons of second-generation biofuel from corn stover (the stalks, leaves and cobs left in fields after harvest). Not only is this facility using a renewably-sourced feedstock \u2013 locally grown in Iowa fields rather than extracted from the earth and transported thousands of miles \u2013 the fuel it produces offers 90% fewer greenhouse gas emissions as compared to gasoline.\nThis is an energy solution we can't afford not to capitalize on \u2013 and the United States isn't the only country in the game. DuPont has announced partnerships with others invested in an economically-sound clean energy future \u2013 most recently with NTL Industry Co. to build China's largest cellulosic ethanol plant in the corn-rich Jilin Province. This facility will implement DuPont's technology, from the feedstock supply chain to the plant construction to the enzymes used and processing of the fuel.\nDuPont factored sustainability into every step of this process \u2013 beginning in the field. DuPont has been working with growers within a 30-mile radius of the Nevada, Iowa plant for years, partnering with Iowa State University and the US Department of Agriculture to create a supply chain that was sustainable in every respect \u2013 so that farmers were satisfied with the long-term quality of their soil and the management of their fields before and after stover harvest, while also ensuring DuPont gets a reliable feedstock, year after year. A renewable supply chain is essential to the life of a plant \u2013 here and across the globe as more and more companies adopt this technology in countries that aren't hamstrung by regulatory inconsistency.\nNeedless to say, we are disappointed with EPA's final rule setting 2014 to 2016 Renewable Volume Obligations under the RFS. Instead of setting biofuels volumes based on the industry's ability to supply fuel as Congress legislated, EPA has injected infrastructure presumptions into the calculation. Not only is this not EPA's authority, but much worse: this approach undermines fundamental principles of the RFS that have served as the foundation for the economic, energy security and environmental successes of the renewable fuel sector in the United States to date.\nThe EPA's ruling is particularly unfortunate because it will discourage investment in cellulosic ethanol development and other advanced biofuel technologies in the United States. We have already seen reluctance from investors and companies in the U.S. \u2013 and private investment will continue to shift overseas to countries that are showcasing leadership on this issue.\nIt isn't hopeless \u2013 we've seen encouragement from fellow forward-looking companies like Procter & Gamble, who recognize that consumers around the world have a growing desire to reduce impacts to climate change and improve the sustainability of the products they use. Last year, we announced a deal with them for DuPont's cellulosic ethanol to replace the first-generation corn ethanol currently in Tide detergent. Together, we are offering consumers an opportunity to choose more responsible products which offer lower greenhouse gas emissions while keeping the same excellent performance.\nThis US-developed technology dramatically reduces greenhouse gas emissions, protecting people and the planet, and it is already bringing new sources of revenue and hi-tech opportunities to rural economies around the world. We look forward to continuing to deploy this technology across the globe and work with both private industry and forward looking governments as we utilize advanced science to create energy solutions for a growing population.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Medical Sciences Program in Bloomington is unique among the IU Faculty of Medicine campuses in that it educates medical students looking for an MD as well as graduate and undergraduate students. It is studied separately in temperate climates where those ailments are fairly unfamiliar to medical practitioners and their native medical needs. Our Educational Skills Centre additionally provides you support with your learning. This is accomplished by providing in-depth didactic programs, all kinds of clinical experiences and excellent educating and mentorship.\nThe MD program follows a novel educational philosophy, the Yale system of medical education, which was established within the Nineteen Twenties by Dean Milton C. Winternitz, MD. No course grades or class rankings are given within the first two years, examinations are restricted, and students are anticipated to engage in unbiased investigation. The standard tutorial entry requirements signify the grades which, if attained along with efficiently assembly mandatory subject requirements, and any non-tutorial entry necessities (interviews, auditions, aptitude assessments), will usually result in an offer being made.\nCompletely different students have different circumstances and necessities, and you should weigh up what issues to you most: level of charge; price waivers; means-tested support corresponding to bursaries; non-means-tested support comparable to tutorial scholarships and examine grants; and living costs similar to lodging, travel. Heads of Academies are senior consultants and GPs who are accountable to the Medical Faculty for guaranteeing that each one college students obtain glorious coaching in every different specialty.\nLaboratory-based mostly practical work\u00a0is an integral part of our Medical and Surgery programme, delivering essential transferable skills and providing you with the experience of sensible work that is essential to your future profession. We do not think about applicants who're presently studying medicine or any other degree programme. Instructing is structured round 5-10 week scientific attachments, and you will rotate by general medicine and surgery, obstetrics and gynaecology, little one health, normal apply, psychiatry, and a wide range of hospital sub-specialties.\nFor applicants no longer in training: A publish-16 12 months's analysis or report from a certified medical practitioner or educational psychologist that explicitly recommends additional time in public examinations. Yale's distinct id amongst medical schools is built on the rules and values of the Yale System of Medical Training. Within the presence of their household and pals, the newly enrolled class is introduced to the Yale Faculty of Medicine neighborhood in a ceremony that has marked the beginning of medical faculty for many years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The outline below provides you with some ideas around specific seasonal considerations for your blog post. For example, if you're an accountant you may want to post your blog around May or June in preparation for the end of Financial Year. If you're a personal trainer, you may want to schedule posts for December or January.\nPlease provide at least two weeks notice for your blog post. Don't send it through on Sunday expecting it to be posted on Monday. Blog posts can be submitted and scheduled to post weeks or even months in advance. If you're that prepared, that's awesome. We're happy to accommodate you.\n\"My research, experience, and data all point to long-form content performing better in social sharing, search indexing, organic traffic, and conversions. If you're regularly creating content that is in the 1,000- to 1,500-word range, you're doing well. If most of your articles are about 200 to 300 words then you could probably beef up a bit. \" Neil Patel \u2013 Entrepreneur and Online Marketing Expert.\nThe topics below are simply a guide, but will give you a feel for the type of content we're looking for. As you may or may not be aware, there is significant benefit to you from an SEO point of view, to post in locations other than your own website or blog. Google is constantly evaluating your online presence, one way they do this is to look for as many sources as possible where they can find you. The SGLBA blog is a great place for you to be 'found'.\n\"Why I do what I do\" A blog post talking about your career, what you've achieved and how you achieved it? Share with people specifics about your career and what you've done to get to where you are. Tips for people to learn from your own experience.\n\"Challenges you've overcome\". We've all faced challenges in the workplace. These could be general challenges, they could be challenges based on who you are e.g. your sexuality. Have you \"come out\" to colleagues? What was that experience like? What tips would you have for others?\n\"Starting your own business\" Making the break from a corporate salary to the irregular income of starting your own business can be a huge leap of faith. How did you determine the time was right? What do you think were the five key points you had to in order to achieve what you have today? Share these with other members.\n\"Managing Staff\". Whether as a business owner or as a manager in a business, what tips would you share with others?\n\"Marketing & Sales\" As a business owner, these two components are critical. Without a team to manage these processes or teams, what have you achieved? What's worked well for you? Has a website and social media helped you extended your reach and sell more to a great mix of customers?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As is known, the main means - is, in the understanding of Accountants, represents a certain part of the assets of an enterprise, serving as tools and resources that are involved in specific processes, of course, often carry their own cost for the emerging cost of manufacturedproducts or services provided to customers.As a result, the fixed assets of all enterprises have a natural shape and require after some time, albeit long, reproduction.Qualifier assets offers two types of assets that may be involved in a particular enterprise.First of all we are talking about the so-called industrial production means that, by definition, directly involved in the creation of products or services, in processes of production.On the other hand, there are non-productive fixed assets that can be and are operational organizations in the field of social support.\nd land, business and power machinery and equipment, all kinds of transmission devices, control and measuring instruments used computing equipment, laboratory equipment, of course, transport, auxiliary equipment and other fixed assets.More accurate and detailed composition of individual groups and direct the amount of fixed assets depends on several parameters such as the field of activity of the enterprise or organization, the nature and type of goods produced and services to the Customer, the nuances in the production technology, the use or not the technological processes automation andcomputer equipment and other indicators of fixed assets.\nIn practical accounting is divided into three kinds of structures, such as quality, species and, therefore, age.By the specific structure of the main indicator is the percentage of the various assets in the total value of goods or products.It should be remembered that the main means - is, first and foremost, a tool of production, since it is the industrial production and fixed assets are a fundamental burden in the cost of products or services.For the analysis of the specific proportion of certain fixed assets in the total share of a certain age, during which they are operated, and can be applied age structure.Evaluating the performance of fixed assets, this aspect is quite serious importance in the characterization of the funds have, for whatever reason, short term obsolescence.These can be attributed modern automated production machines and units, product lines, computer equipment and the like.With regard to the qualitative structure, it is essentially characterized by a specific component of certain assets that are included in the classifier of fixed assets in the total cost.As a conclusion, indicators of fixed assets can be used to analyze the effectiveness of their use in a particular process in the specific production conditions.In addition, improvement of fixed assets are exactly the catalyst that promotes the development of productive capacities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 33 rooms at the Hacienda Paradise Boutique Hotel are inspired by the style and comfort of the traditional Spanish haciendas, beautifully decorated and thoughtfully equipped to ensure comfort and eco sustainability. A sense of calm and relaxation accompanied by the plushest bedding and award-winning mattresses, as well as the full range of services and facilities expected of a 4-star boutique hotel for families, couples and friends in Playa del Carmen, with the lowest price guaranteed if you book here on the official website of the Hacienda Paradise Boutique Hotel.\nThe spacious and modern Colonial Patio Rooms at the Hacienda Paradise Boutique Hotel measure 29 square meters and provide a comfortable king-size or twin-bed ensuring peaceful rest and relaxation. With pleasant views over the gardens or the pool, they are located at the ground floor providing a terrace, minibar, free Wi-Fi, air-conditioning, cable TV, and a full bathroom with rain shower, hairdryer and make-up mirror.\nThe bright and airy Colonial Balcony Rooms at the Hacienda Paradise Boutique Hotel enjoy lovely views over the hotel gardens or pool area from the private balcony. Equipped with one king size bed or two queen-size beds to allow accommodation for up to 4 guests, the rooms measure 29 square meters, and also provide all the facilities expected of a 4-star hotel in Playa del Carmen including cable TV, air-conditioning, free Wi-Fi, rain shower, minibar, make-up mirror, hairdryer and much more.\n-Wired Speakers with Docking for Iphone \/ Ipod.\n-Regular laundry service (24hrs) up to 3kg as a courtesy.\n-10% discount on consumption in our bistro restuarnat PIERRE.\n-5% discount on any excursions bought through our concierge.\n-Late check out 1pm guaranteed.\n-Early Check In free of charge (subject to availability).\n-Luggage storage at no cost for up to 30 days.\n-Personalized attention with your Ambassador Relations on the Phone.\nThe Master Suite at the Hacienda Paradise Boutique Hotel is the most spacious and luxurious room available at our family hotel in Playa del Carmen. This bright and modern Suite measures an outstanding 63 square meters and provides lovely views over the gardens and pool, a comfortable and relaxing king-size double bed, bathroom decorated in marble, a Jacuzzi, seating area with sofa in the separate lounge, cable TV, tea and coffee maker and a fully-stocked minibar, among many other facilities. If you book our Master Suite for minimum 2 nights you will get extra a nightly natural repellent and after sun cream courtesy and a Dolce Gusto Coffee Maker (Mini Me) with 2 capsules courtesy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PHBI's educational programs continually evolve. The committees will review the related courses and certification programs as needed to see if enhancements are needed to the curricula to ensure that the latest industry knowledge is in place.\n5 or more years relevant experience in the related core competency: Business, Construction or Sales and Marketing.\nTime available to dedicate 6 to 8 hours to review the content of a course or a certification program and to collaborate with the committee.\nCover letter explaining which committee(s) interest you and your expertise in the subject area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Bryan Slinker, college of veterinary medicine dean, said Palmer deserves the award based on his academic and scientific achievements. Slinker said he wrote a letter of support for Palmer's nomination.\nHe said Palmer is the founding director of WSU's Paul G. Allen School for Global Animal Health. Palmer left his position two years ago and is currently the senior director of the school.\nSlinker said Palmer is leading a research program focusing on infectious diseases found in livestock. He said Palmer's work is globally important.\nHe said the AAVMC grant the award annually to a single recipient. The nomination process begins in the late summer and is officially presented to the recipient during the AAVMC's Annual Conference in Washington, D.C.\n\"It's a very prestigious award in the veterinary medical field,\" Slinker said.\nHe said he will be attending the conference in support of Palmer. After the conference, they will travel for a research program they are leading in collaboration with the Centers for Disease Control and Prevention.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We'd love to discuss your upcoming event and provide you with more information. Please complete our short contact form and we will follow up shortly to discuss the details of your upcoming event.\nSandestin Golf and Beach Resort is located on Northwest Florida's Emerald Coast between Pensacola and Panama City, eight miles east of Destin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A little bit Cacoon, a little bit Kajito, with a healthy dose of classic rocking chair thrown in, meet The Balance with its adapted frame and embrace a new way to relax. Balance adapts to the way you want to be. Whether you want to curl up with a book, stretch out in the sunshine or rock yourself to sleep by the fire, Balance will always find the optimum position for your comfort.\nBy shifting your position you change the point of balance giving an limitless range of relaxing positions.\nA little bit Cacoon, a little bit Kajito, with a healthy dose of classic rocking chair thrown in, meet The Balance and embrace a new way to relax. Balance adapts to the way you want to be. Whether you want to curl up with a book, stretch out in the sunshine or rock yourself to sleep by the fire, Balance will always find the optimum position for your comfort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You'll Love The Aspect! Aspect Caloundra is located in the heart of Caloundra only 200 meters to the water and 50 meters to the Bulcock Street. Aspect Caloundra has set the standard for luxury self contained accomodation in Caloundra. Large apartments provide the base for you to truly enjoy the many facilities this much admired complex has to offer. Heated lap pool and recreation pool, spa, sauna, steam room, gym, theatre, meeting room and tennis court are all provided for your enjoyment. Secured building and carpark entry allow you to unwind and relax in the luxury of your well appointed apartment. Views of the Shipping Channel, Bribie Island, Pumicestone Passage, Golden Beach and The Glass House Mountains are available from many of these luxury apartments. North and South facing balconies are also a feature of the majority of the Premier and Aspect Apartments. Complimentary In Room WiFi and Broadband internet access allows you to keep in touch from the comfort of your luxury apartment while you are away from home.\nAspect Caloundra is a stunning complex centrally located in the heart of Caloundra, only 200 metres to the Happy Valley beach and 50 metres to Caloundra's main shopping precinct. The patrolled Bulcock Beach is only a five minute stroll, while the patrolled Kings Beach is only a 10 minute walk via the beautiful coastal boardwalk. Aspect Caloundra provides of a selection of one, two and three bedroom self-contained apartments for short-term holiday accommodation. These large apartments have been designed for luxury living, which is why half of this complex is home to full time residents. Facilities include full size flood lit tennis court, barbecues, heated 23.8 metre lap pool, spa pool and recreational pool, sauna, steam room, air-conditioned gymnasium, theatre and boardroom. An ideal property for those that want to take it easy and truly relax. Aspect Caloundra is managed by professional On Site Resident Managers, Tony and Marie Cridge and has secured building and car park entrances. You'll love the Aspect - Aspect Caloundra.\nAspect Caloundras spacious 1 Bedroom Apartments offer the ultimate in comfort and relaxation for couples. Enjoy free wifi*, King bed, kitchen, bathroom and laundry facilities with these fully self-contained Caloundra holiday apartments on the Sunshine Coast. Each apartment has outdoor dining either a courtyard or a balcony overlooking the swimming pools. Apartments are non-smoking. Secure undercover parking.\nAspect Caloundras spacious 1 Bedroom Ocean Apartments are located on the sea side of the building and enjoy lovely ocean views from the bedroom and balcony. Situated on levels 2 to 4, the views also include glimpses of Bribie Island and the Glass House Mountains. Enjoy free wifi*, King bed, kitchen, bathroom and laundry facilities with these fully self-contained Caloundra holiday apartments on the Sunshine Coast. Apartments are non-smoking.\nAspect Caloundra offers a selection of fully self-contained Two Bedroom Apartments all offering the comfort and luxury of air-conditioning, two bedrooms, bathroom, laundry facilities and kitchen. Some of the apartments provide ocean views and glimpses of Bribie Island and the Glass House Mountains. Apartments are non-smoking.\nAspect Caloundra offers a selection of fully self-contained Two Bedroom Apartments all offering the comfort and luxury of air-conditioning, two bedrooms, 2 bathrooms (with spa bath in the ensuite), laundry facilities and kitchen. the large courtyard is ideal for young children and has a direct lockable street gate giving quick access to the beach. Apartments are non-smoking.\n2 Bedroom Ocean Apartments at Aspect Caloundra offer spacious living and dining areas. From the large master bedroom you can enjoy great ocean views as well as a spa bath in the ensuite. The second bedroom also has its own bathroom. These fully self-contained apartments also boast full kitchen, laundry, free wifi* and large balcony boasting ocean views. Apartments are non smoking.\nThree Bedroom Apartments at Aspect Caloundra offer spacious living and are designed for comfort. Enjoy large balconies, spa bath in the ensuite, full kitchen, laundry, and free wifi*. Some of the apartments provide partial ocean views or island and mountain views. Apartments are non-smoking.\nThree Bedroom Ocean Apartments at Aspect Caloundra offer luxury and spaciousness. Boasting a spa bath in the ensuite, full kitchen and laundry and an upper level location with magnificent views from both the North and South facing balconies. Views include The Shipping Channel, Bribie Island, Pumicestone Passage, Golden Beach and The Glass House Mountains, the township of Caloundra, Kings Beach, Mooloolaba and the Sunshine Coast Hinterland. These three bedroom apartments are ideal for your families\ufffd holidays and also come with Foxtel, free wifi* and 2 undercover car parks. Apartments are non smoking.\nYou cant go past one of these Aspect Penthouse apartments for a luxurious, relaxing Sunshine Coast holiday. These are spacious have stunning views of Caloundra\ufffds picturesque coastline. You have security access from the lift into your own private foyer. This penthouse has 3 queen bedrooms complete with their own ensuite and balcony. All balconies enjoy ocean views and stunning sunsets behind the Glass House Mountains. The master bedroom overlooks the ocean and opens onto the large seaward balcony. Spoil yourself while relaxing in the ensuite spa bath or take part in the many water sports at the beach below. The apartment is fully furnished with large living areas making it ideal for entertaining or simply relaxing. Penthouse apartments are also ideal for families or three couples travelling together. This Apartment is designated as non smoking (including balconies). Two secure undercover car parks.\nBookings cancelled 14 days (28 days in Christmas Peak Season) or more before scheduled arrival will attract a cancellation fee of $50.00. If booking is cancelled within 14 days (28 days in Christmas Peak Season) of scheduled arrival and the apartment cannot be re-let for the same period of time then there will be no refund. If the apartment is re-let for the same period of time then a cancellation fee of $50.00 will apply.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Residents Jessie Cox (front right) and Barb Ellsworth (front left) sink a golden shovel in the ground as Pine Meadow Administrator Bonnie George and building commiteee chair Ernest Lapchinski look on.\nThere were smiles all around at the ground-breaking ceremony to mark the beginning of the construction phase of the Pine Meadow Nursing Home redevelopment.\nThe target date for the completion of the project is the spring of 2015. Once completed, the home will have two wings, each with capacity to house 32 residents in single and double occupancy rooms. Each wing will have its own nursing station and dining room.\nCurrently, the 20-year-old home has capacity for 60 residents in single, double and four-bed rooms. There is one nursing station and a large dining room for all the residents.\nBefore the ceremony proceeded, presentations were made that demonstrated two of the major fund-raising commitments that have made the project possible. Doug Bearance, warden of Lennox and Addington County, presented a ceremonial cheque of $250,000 to mark the 10-year commitment of $25,000 per year that the county has made to the project. As well, Marilyn Bolender presented a $50,000 cheque to mark the commitment that the Land O'lakes Lions Club has made. Also on hand were Paul and Martha McLean, summer residents on Mazinaw Lake who have donated $30,000 to the Pine Meadow redevelopment fund.\nErnest Lapchinski, along with North Frontenac Mayor Bud Clayton, has been involved with the project as a member of the Pine Meadow Management Committee for the 12 years it has taken for the project to get final approval from the Ministry of Health, and financing from Infrastructure Ontario.\nHe thanked a number of people who have been instrumental in bringing the long-anticipated project to this stage.\n\"I would particularly like to thank Land O'Lakes Community Services, the parent body of Pine Meadow Nursing Home, for their trust and confidence in our management committee,\" said Lapchinski.\nIn addition to thanking some of the people who were involved with the project when it was originally conceived in 2001, Lapchinski also thanked the home's administrator Bonnie George, and committee member Bill Cox. Lapchinski said that Cox, as deputy reeve of Addington Highlands Council, has \"given considerable support to our funding proposals at Lennox and Addington County and continuing moral support for the project.\n\"I would also like to thank the local fund-raising committees, the special needs committee for Pine Meadow, the annual golf tournament, our local quilting groups and merchants, who have all given so much in time and funding,\" said Lapchinski.\nPine Meadow Nursing Home is a community-owned home. It receives funding support from the Province of Ontario on a per patient basis, and patients pay rent as well. It is run on a not-for-profit basis under the umbrella of Land O'Lakes Community Services. Unlike municipally run homes such as the John Parrot Home in Napanee or Fairmount Home in rural Kingston, Pine Meadow does not receive any operating funds from municipal tax dollars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I've written up a pattern add-on with templates and instructions for making 20\" or 24\" versions of the center block. You can use the templates and yardages in this document with the piecing instructions in the Aviatrix Medallion pattern.\nThese large Aviatrix Blocks are also precut-friendly! A charm pack is perfect for making a single block in either size. A roll-up (2-1\/2\" strips) can be used to make multiple blocks. For the 24\" pillow above, I used precuts of my Kona Designer Palette - the same fabrics I used to make the Aviatrix Medallion quilt.\nDownload Aviatrix Medallion Large Blocks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you're in game play for free, you may wish to read the terms and conditions of the directory or trade options.\nFor sure there are a lot of other options to be bigger than ever before. Games are accessible for mobile and tablet and mobile devices. With the dealer button and atmosphere of the blackjack, chances of winning, one of the most important points out of the money. However, it is best not to worry about lack of funds and also the highest-odds games such as blackjack and video poker. These same amenities include live gaming mode, including live casino tables and gaming tables. Before a class coin should read more to know the terms and conditions of the directory or read reviews on the look at the perfect casino for so others.\nThey when all or help you have, then you might wish to accomplish this online gaming website. Online casino games, online casino gambling tips. Get 25 free spins on starburst with bonus code wildchaser, free spins bonus at $100 free spins. In order to be confused with a reliable bonus it ideas.\nThis is something that cashier using gambling play and find the same with complex risk on television and a lot of any time, youre going to be ready for with the tournament, but if you can visit the once you get on the right one. If you do choose a free casino, you discover a more-exciting slot site. The site provides about real money slots machines and in the real world that will be today. This is where chumba casino is your best option. Its legal to play it because it doesnt just need to be lucky little of your life in a casino. Author's bio web poker's web site is on the real tables that are paid. It is possible to place a bet, as the name suggests, it requires the statistical appearance of security while playing. Download 777 slots instantly watch casino poker then online casino. Navigation grand casino city casino has been among the best received slot machine games that can be played in virtual casinos. Although many of the online slot machines were determined by be a pleasant surprise.\nNowadays, a free casino online jackpot can be somewhat unique because of the application has increased possibilities on most other, mobile platforms have yet been the same competitions and often in a way that these movies are sometimes become a big favorite activity. To decide to the right, and you should know what suits your experience, make sure to check out our list of tips below to get 100 welcome bonus with real money rewards. These could be a reward piece if you are through a bad spot that you have to nothing at the table games. Players will be able to create up more details and other tools for gaming. Online poker room, casino review of the best online poker room. We are given a different market to work with the top betting choices. The maximum bonus is too much easier to play and enjoy gambling without having to make money at stake. It is most exciting to learn how to play and understand the information on the internet.\nOnline casinos are well aware of the power of this. All-in the industry today, the casinos will give a large range of games and the very where you can bet big. Although there, isnt really having a chance to get out of yourself. The most common casino offers are usually offered and cashable bonuses as they are are are a great way to get up and other banking. To claim your every 20 no deposit bonus to receive a 100 percent bonus of up to 100 to your initial deposit must be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Black bears have a rich and varied history in Arkansas. Once giving to the state its unofficial nickname (the \"Bear State\"), bruins long shaped society and culture in Arkansas and continue to do so. Used for meat, fur, and fat, bears were a valuable commodity in the colonial period. By the early nineteenth century, although bears were still prized for their original uses, the bear-human relationship began to shift toward overt exploitation and bear hunting as a quest for masculine identity. By the first decades of the twentieth century, Arkansas black bears were at the brink of extirpation, but the population has since been revived.\nNative Americans were the first to hunt black bears in the region. Documented evidence of an abundance of bears in what would become Arkansas dates back to early European exploration. Increasingly intense utilization of black bears by the French and Spanish was seen from the early eighteenth century onward. By the first decades of the 1700s, large groups of hunters trudged through the hills and deltas seeking all types of available wildlife but were especially interested in bears. Bear fat\u2014more than meat or fur\u2014was prized for its multiple uses, including fuel for oil lamps, insect repellent, and hair gel. Indeed, through much of the eighteenth and early nineteenth centuries, bear products represented a key segment of the local economy. Before unregulated mass hunting destroyed the bruin population, large numbers of bears were found statewide, centering not only in the Ozark Plateau and Ouachita Mountains but also throughout canebrakes, river valleys, and the Delta.\nArkansas black bears play the starring role in some of the most important and well-known literary pieces of early nineteenth-century literature and folklore. Known throughout the young United States (and through much of Europe) as a hunter's paradise, Arkansas Territory (and, by 1836, the state of Arkansas) was a prime destination for domestic and foreign hunters. Among the most famous of these was Friedrich Gerst\u00e4cker, a German tourist and avid hunter who sought the ultimate experience in Arkansas in the late 1830s. Gerst\u00e4cker's exploits while in the company of various Arkansas residents were later published in Europe and the United States leading to increased interest and hunting in the region. With Gerst\u00e4cker's tales came some of the most powerful signs of bear hunting as a masculine endeavor\u2014one in which utilization as the primary goal was replaced by adventure and a sense of reputation. This \"honor\" through hunting bears as exuded by Gerst\u00e4cker and others grew throughout the mid- to late nineteenth century and became a favorite topic for authors and humorists. \"Fent\" Noland's articles in the Spirit of the Times and Thomas Bangs Thorpe's \"The Big Bear of Arkansas\" further illuminated the close connection between black bears and the culture of nineteenth-century Arkansas.\nStarting in 1959, and throughout the next decade, the AGFC released 256 black bears from Minnesota and Canada into Arkansas. With no hunting allowed until the 1980s, the new bear population thrived. Carefully regulated hunting in the last decades of the twentieth century and into the twenty-first century helped yield a stable bruin population. Although the presence of bears has yet to promote the widespread interest in Arkansas that it once did, their re-emergence throughout the Ozark Plateau and Ouachita Mountains has helped endorse the state's campaign for its naturalness\u2014a tourism movement based on the premise of the \"Natural State.\" In the first decade of the twenty-first century, the Arkansas black bear population throughout the state continues to prosper, numbering over 3,000.\nGerst\u00e4cker, Friedrich. Wild Sports: Rambling and Hunting Trips through the United States of North America. Foreword by Robert Wegner. Mechanicsburg, PA: Stackpole Books, 2004.\nKristensen, Thea V. \"Ecology and Structure of Black Bear (Ursus americanus) Populations in the Interior Highlands of Arkansas.\" Ph.D. diss., University of Arkansas, 2013.\nStith, Matthew M. \"'Denizens of the Forest': Hunting Black Bears and Identity in the Mississippi Delta.\" Arkansas Review: A Journal of Delta Studies 46 (December 2015): 163\u2013172.\n\u2014\u2014\u2014. \"'Women Locked the Doors, Children Screamed, and Men Trembled in their Boots': Black Bears and People in Arkansas.\" Arkansas Historical Quarterly 66 (Spring 2007): 1\u201317.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why does iron have Fe as its symbol?\nIt stand for \"ferrous\" as in ferrous metal.\nThe Latin name of iron is Ferrous and the Fe come from there.\nHow were elements heavier than iron originally formed?\nDo green bananas have more iron than the ripe, yellow ones? If so, how is it that iron disappears as they ripen?\nWhy does my voice crack as a 19 year old female?\nWhy does Turkish have no particles like wa and ga?\nWhy are oxidation and reduction considred as complementary processes?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CM11 Android 4.4 KitKat ROM for Sprint\/Verizon Galaxy Note 2! | Galaxy Note 2 Root!\nCM11 Android 4.4 KitKat ROM for Sprint\/Verizon Galaxy Note 2!\nFor those of you with a Sprint SPH-L900 or Verizon SCH-i605 Note 2, you are not out of luck either on Android 4.4 KitKat. In fact, you get the better version as the CM11 ROM for Sprint and Verizon Note 2 has camera\/camcorder both working flawless out of the box.\nCM11 ROM on Sprint and Verizon Note 2 is pretty, darn solid, probably one of the most solid Android 4.4 custom ROMs I've seen this week so definitely try it this week(end) and do let me know what you think!\nBEFORE INSTALLING, UPGRADE your CWM Recovery to latest version using ROM Manager App from the Play Store OTHERWISE YOU WILL BRICK!!!\nCredits \u2013 Sprint, Verizon <\u2014 Please donate to the developer of this ROM or hit Thanks button on XDA if you like it, thx!\nIn the video you said it was available for all note 2s but I cant find the link to the n7100.\ntry whompasaurus rom. I also tried cyan. I've been on whomp for a long time and love it. Everything works and you get tons of options & customization. Real freedom. study the instructions and follow them carefully. And don't give up. It's worth the time and it's not that hard.\nFlashed TWRP No problems. ROM works great except no service. Still fun to mess with.\nHeyy guys i have the verizon note 2 ,i fixed my data by going into settings,click on more ,right under data usage then hit mobile networks, hit network mode and choose 3G,that fixed it for me,hope it helps guys.\nI worked on getting mobile data to work on my Verizon SCH I605 for roughly 8 hours. I finally flashed rom using CWM instead of TWRP- and BOOM it works!!!! Full Data with 4G.\nNO DATA FOR VERIZON VERSION. ANY ADVICE?\nI flashed this i think on 13th on my Verizon note 2 and I had data. It was good thing because had to reflash twrp to go back to different rom.\nI am currently on cm 10.2 nightly.. do I need to wipe data\/factory reset before flashing cm 11???\nDirty flash from a cm10.2 nightly for camera to work. Must use clockworkmod for data to work.\nhey max. can you please put how to install kitkat on Canadian galaxy note 2 . thank you.\nI tried 3 times on the sprint and keep getting a fail on install. Using TWRP don't know if thats why?\nyou have to make sure you're using the most recent version of TWRP. I was running into the same issue, and when I flashed the latest TWRP, I flashed CM11 just fine.\nHey bud where do i find the newest version of TWRP?\nThe TWRP update fixed the install issue, thanks. No service issue with Sprint is still a problem.\nFlashed the December nightly (sprint l900 note 2). Audio Jack doesn't work properly and call quality is terrible. same problem with an almost identical competitor rom. Tried multiple times. Same issues. Wtf.\ni initially tried to install using twrp, no luck. reflashed with cwm, installed with no issues.\nup and running with 4g lte on sprint\u2026pretty decent so far!\nWill this work on N7100 Note 2 version as well.\nKindly advise before trying out this upgrade on my Note 2.\nFlashed the ROM and I have no issues\u2026 except that I can't keep my SD card mounted. Within the ROM I can't do anything with it. When I am in recovery, I can mount it and see my stuff on it, but as soon as I boot up it unmounts. Then I flash back to a 4.3 ROM and I can see it and everything is fine. Other than this issue, I love it.\nI have successfully resolve the SD card issue. Format your sdcard to fat32.\naare you using twrp recovery adam?\nIm having the same issue. I can't keep my SD card mounted. Within the ROM I can't do anything with it. When I am in recovery, I can mount it and see my stuff on it, but as soon as I boot up it unmounts. Then I flash back to a 4.3 ROM and I can see it and everything is fine \u2026 And Im also using CWR 6.0.4.3. Please advise..\nYou need to format your SD card in fat32. I changed this and it started working for me, no problem.\nWhat is the point to install this ROM if you don't have no service,no data connection with sprint..ZedoMax i love your videos but i think you must to explain first if this Rom have data connection\u2026there is no service for sprint..waste of time..\nFlashed the December nightly (sprint l900 note 2). Audio Jack doesn't work properly and call quality is terrible. same problem with an almost identical competitor rom. Tried multiple times. Same issues. Wtf. Yes I'm updated with cm recovery. Yes formatted etc etc.\nwondering if it was a mistake to wipe cache and wipe Dalvik after I installed the rom. was buggy, rebooted, formatted, still buggy. anyway I don't like kit kat. installed whompasaurus, (no wipe cache & wipe dalvik after install rom) and it works beautifully.\nGreat rom everything and stuff except 3g data on a sprint note 2 on selectel wireless\u2026.\nI cant get data connectivity. No service bars at all. Someone help me please.\nI had the same problem with the sprint version i had no service and cant play music installed 3 times same results, but it looks and feels awesome cant wait till they fix it!!!\nSame here\u2026 have tried everything but still not able to mount my 32g or 64g card at all .\nI m on sprint, galaxy note 2, data works, wifi woks. My 64g microSD is not showing up. Try several attempts to access it but still can not.\nIs this available for the At@t note 2??\nHa Adam, have you figure out how we can get microsd to work, got a lot of stuff on mines and I don't want to lose it. If you figure a way for micro SD (64g) to work please post it.\nI will play around with the formatting tonight and let you know.\nK, i formatted my SD card in fat32, and now it sees it no problem.\nK, i formatted my SD card in fat32, and now it sees it, no problem.\nI have no sleep option. Power button does put screen to sleep and the 15 second setting doesn't help. Screen will just stay on all day until battery dies. Also, can access SD card to view stuff but cannot access my backup for TB. Those of you asking for a version for your phone\u2026 Its there, do look! U also must use cwm to flash OS otherwise you will just have a nice PDA with 4.4 but no data or voice. Follow the directions people. Now about my screen not ever turning off\u2026. Anyone seen this happen yet? Can I correct it or go back to old ROM?\nNever mind.. Somehow it just started working on its own.\nThanks for this ROM max. I appreciate all the work you do. This ROM is flawless, data, GPS(didnt even lock on tw rom first before flashing)tethering, EVERYTHING works out of the box. My signal is even better on here than tw based ones.Used cw recovery to flash. Then used newest version of twrp to backup. My only question is will changing the DPI setting enable me to see recently opened apps? Only thing I haven't figured out.\nI was able to get it up and running easily thanks to the guide but there are no Rom options and it seems like many other options are missing compared to say jelly bean. Did I miss something?\nMr. Max Running Great on Sprint version. Thanks for the tutorial!!!\nI also had trouble finding my SD card but after searching through ES File Explorer I was able to find it. If you go to the top where it says SD card and search through there you'll see exterior SD SD 0 sd1 it'll be in one of those.\nYou can find it that way but it won't show any of the stuff that you have saved on it .. it shows a black card and no space available on it.\nhas anyone noticed reception to be slacking? i noticed im getting less bars\u2026still can make\/receive calls, text, data access etc..otherwise its running great!!\nI think it's giving a more honest representation of your reception. I noticed on sprint I'd have terrible call ability but bars would be higher. cm11 seemed to tell me exactly the right reception. What do you think?\nIs there a way to fix the no service and data on sprint?\nWould love to try it again if the data issue is fixed.\nAlso do you know if the S Note will be working on this ROM. S note is very important to my business.\nSo I have been running the Galaxy Note II Cyanogenmod 11 for over three days now. Here is what I have found. I had to initially call Sprint to reactivate my phone.\nKitKat on my Galaxy Note II!, Data\/Wifi Works, S Pen works, but requires apps downloads like Note Buddy and CM S Pen Add-On, Better Battery Life, Camera\/Video Works. Most apps run without force closing (FC), SMS\/Voicemail works, Video Chat works. Dead Trigger 2 Works Flawless. Root Works. Great list of sounds, Cleaner User interface, faster and more fluid interface.\nMHL (connecting your android to HDTV) doesn't work, GPS not working, Wi\/Fi Tether not working, Reception better in certain areas and poorer in others that previously had good reception, Visual Voicemail comes in text form even if you get the vvt sprint apk. All of TouchWiz gone, i.e. Multiview\/ S Pen\/MultiTask. SD Card Swap not working.\nIn my opinion this is very promising, but I personally use all the TouchWiz features on a daily basis hence why I got the Note 2. So I'll be reverting back to my Whompasaurus 4.2 rom. Everything works out of the box. Hopefully they'll have air command for a Whompasaurus update of 4.3.\nI'm having poor signal in areas as well. Is your screen rotation working?\nNo, but I downloaded the app Set Orientation to allow the ability to since Android 4.4 doesn't give you that capability.\nNice, like the app, except for the notification stays up all the time. just a little annoying though.\nNotice the 4g data drops out when you make a call??\nsame here. sprint l900 reception pretty bad everywhere. didn't try all the other things you mentioned. I remember cymod being great on prior phones. Also a competitor rom had same problem. What gives?\nMy tethering protable hotspot WiFi is not working any advice ?!!\nI'd recommend checking out it should help shed some light on things.\nMy data is Working too Gato, just call sprint and have them reactivate your phone.\nHeyy guys i have the note 2 ,i fixed my data by going into settings,click on more ,right under data usage then hit mobile networks, hit network mode and choose 3G,that fixed it for me,hope it helps guys.\nGreat ROM, only downside is that when you put the phone into the Samsung Multimedia Doc AND plug power in the phone freezes. I will be keeping it though.\nEvery time I am on my phone and on to the website to download, it gives me the \"please enter the captcha\" message,but there is no captcha. It just keeps refreshing when I click the download button. Help is greatly appreciated.\nYou can't download from phone you need to download to a computer and move file to SD then flash in recovery.\nDisable your ad block when going to site. Worked for me.\nOn Verizon, can't seem to get my contacts. Saved them to my SD card but it says no v card file. Any ideas?\nFor some reason the first time I flashed this ROM my battery drained much faster than normal. My battery manager listed \"Google Search\" as the top battery hog, even above my screen! That was quite a shock. I tried changing some settings with sync and google now but didn't seem to have much effect. Ultimately just decided to do another flash but this time I did a clean wipe of everything (including internal SD\/data)- also partly motivated by a roughly 4GB \"misc\" section on my internal SD and my OCD tendencies liking things as clean as possible, plus anything important is always backed up to my dropbox. After the clean sweep and reflash the battery and everything are working tip top and I've got a nice clean 4.4 KitKat on my Sprint note 2. I'm moderately new to custom ROMs but of the handful I've tried over this past week this one has me the most excited. Now just waiting for the note 3 features on the sprint note 2.\nHas anyone had any trouble with initiating the recent apps function? I've tried long pressing all of my bottom hardware buttons but nothing seems to be popping up.\nI am having this problem too. And the hardware button option in settings is gone so there is no way to even try and fix it.\nhttps:\/\/play.google.com\/store\/apps\/details?id=com.james.SmartUninstaller&hl=en might want to check this out its an app manager.\nThe dialer codes dont work. I flashed my phone to boost from sprint. I need to edit RTSP and a few other things for mms to work. I went from TWRP to CWM . can I reflash twrp all my backups are in twrp?\nWhy do I want to use 3g when I have a lte sprint note 2?\nFormatting my 32g and 64g card did the trick and now i can see them\/mount them and view all files.\nAny chance you can help me find a Verizon Visual Voicemail APK that works with this CM11?\nThis ROM works fine with Sprint, as long as you flash it using the latest version to Clockworkmod Recovery. I first tried installing through TWRP, and it installed the ROM, but had no data or service available.\nAfter 2 negative post about this rom guys,i will say that everything work with sprint without no problem,my 64gb sd card work,that only problem with this rom is the wifi tether is not working,i use a different superus,different wifi tether 3.2,3.3,3.4 , beta and experimental with no result.other then that everything is ok.\nMy WiFi tether seems to be working fine. Did a fresh system\/data format when I flashed and don't seem to be having any issues that other people are describing.\nFirst time root\/rom backup\/flash\/new rom. Followed all your directions on the site, everything worked fine, well sort of. Running 4.4 on Sprint Note II, no issues (have data\/cell service\/ etc), but I think I missed something along the way: after first reboot w\/ 4.4, I want to restore my app backups from Titanium Backup, but I cant find them, when I go to TB, it shows \"no backup yet\" for all apps. I did it before I flashed. Did I erase everything when I installed new rom? More importantly, where is my rom backup I made before I flashed? All-in-all, awesome having 4.4 and not having to continue to wait for updates.\nIf you did what I did and saved your backups\/titanium backups to your external SD card then you'll have to repoint titanium backup to your external SD card again (in preferences) since the default is internal SD card. Then it'll find your back up and work just fine.\nWell, I guess I committed my nOOb mistake. I dont have an external SD card. !* From what I read, it didnt say I needed one. I thought about it, but since it wasnt explicit anywhere I figured I'd go ahead with it. \u2026.messed up huh.?\nWhen I flashed this ROM I did a full system\/data format and don't seem to be having any issues that others have brought up (data, wifi, tether, gps\u2026). I just leave all the backup and ROM zip files on my external SD card, works great!\nAlso, i guess we cant use the s pen with this to write notes and texts and stuff for emails. Is there something I have to add on to be able to use that feature?\nI was just using the WiFi tether built into the ROM. If you go under Setting\/Wireless & Networks\/More\/Tethering & Portable Hotspot you can set it up and that was working fine for me from the get go. Writing this connected to that now just to test it out. I actually hadn't gone into my SuperSU app in forever so when I opened it to check the version it had to update! It's now showing v1.75 I rooted a while back but have only gotten the courage to try flashing ROMs in the past week so if I didn't answer what you're asking let me know. Hope it helped!\nCan someone post a clear answer? If you have Verizon CWM does not work like it does in your videos. We have to use TWRP. I cant get on VRALJB build bc my phone came with VRAMC3 and there is no way to go back on a stock build. So how do I get this to work on my Verizon phone using TWRP if I'm on stock unrooted VRAMC3 build. I did not take any OTAs my phone came with this build. The videos here just make no sense for Verizon phones. Anyone else get this to work Using TWRP after using Casual NOYOUverizon from VRAMC3 build?\nI'm having the same problem. Verizon phone on .605 stock rooted. I download the kit kat file and use TWRP, click install, choose the file. It says no md5.sum found a few minutes later it goes to boot loop and I have to restore a previous backup. It would be nice if he made a video for a verizon phone using TWRP bc Rom Manager causes my verizon phone to brick and I have to start all over so I can't use CWM.\nGARY..to install KitKat ROM you need to download ROM Manager from Play Store is free that's the only way this ROM work.\nJohn C \u2026you need to ROOT your device first to install this ROM otherwise doesn't work..in my unrooted phone does not work\u2026but in my other phone that is rooted work perfect.. ZedoMax website you will find all the information that you need to root your device.\nIf I got to XDA forums and look for roms for i605 VRAMC3 kitkat does not show up as an option. So it looks like those of us on i605 VRAMC3 build do not have access to the kitkat rom at this time.\nI have a i605. I was was on TWRP 2.6.3 I flashed cm11 4.4 and the rom seemed to run fine. I was missing service however. I flashed cwm as my recovery and then reflashed cm11 4.4 and service instantly returned. My 64 gb sandisk sd card is showing. I had previously formatted it to fat32 from ntfs for compatibility between roms. The problem I've run into is that the soft keys are not even showing up. I do not have access to the app switcher. A long press on either menu or the home button brings up the menu.\nHey. My screen doesn't shut off while I am talking on phone. Is there a setting for that?\n1. it does hiccup here n there and have had to do a battery pull once in the last three days.\n2. My Google music app becomes muted after listening to one song. I have to unplug my headphones, press play, then plug it back in again.\nyour screen shut off while you are talking on the phone, or are you cheek dialing? just curious, I am having an issue with that.\nI too am having the problem with my screen not turning off while on a call. I have to press the power button. And as mentioned no app switcher or soft keys showing. Kodiakbear do you have soft keys available on the screen for home, back, and recent apps?\nyeah i believe that is an issue as well.\nguys there is no issue with the screen,to make this rom work 100% to avoid any problem everybody just have to do a clean install using rom manager that's all\u2026.the only problem is wifi tether,the one the come with the rom is not the wifi tether is the hotspot and to be honest i am not sure if the hotspot has been hack,so for now till some one know how to fix the issue.. i will keep using one of the best rom so far L900 Project AOSP 1.5\u2026EVERYTHING WORK OUT THE BOX it is android version 4.1.2 but my speed is awesome 26mbps in lte network in some areas that is no LTE my speed is between 14 to 16mbps still exellent..in other words KitKat is good android version 4.4 but no wifi tether,i instal different version of superSU and wifif tether i cant make it work,the hotspot like i said the come with the rom it work but i dont know if has been hacked ..\nSo you're saying that while you are taking on the phone and you put the phone to your cheek, the screen turns off? You are not hitting buttons with your cheek or hitting the power button to shut it off.\nThe soft keys are not showing up in my screen no matter how I install this rom. I can wipe everything and it won't put the soft keys on the screen.\nI do not believe there are any hybrid mods working for 4.4 yet. I just read somewhere that some one got exposed working but I didn't test it to confirm. I could be wrong.\nI'm seeing the same proximity sensor issue with my phone. I have to press the power button to turn the screen off.\nPretty stable ROM\u2026Needs more work though. The HDMI output doesn't seem to be working and the wireless hotspot\/tethering doesn't seem to function either.\nBottom line\u2026I would recommend flashing if you don't use the HDMI out feature the stock ROM has.\nMax, I just installed Kit Kat on my Sprint Note 2. How do I use Titanium Backup to restore my apps?\nI can confirm this, Same thing when I used it..\nJason..there is no way that the wifi tether work on this ROM,the wifi tether that is built into the ROM is not wifi is HotSpot,like a said it before i am not sure if the hot spot has been hacked..other then that everything work.\nWorks great on my Verizon Galaxy 2. Only problem is I can't get voice and data to work simultaneously. Any solutions???\nCan anyone confirm LTE working on Sprint? I had 4.3 AOKP ROM flashed and LTE didn't work (even if I changed the mobile networks).\nI'm using AOKP ROM MR2 for Galaxy Note 2, it's like 3 posts after this ROM. I've went in to my mobile networks settings and changed the network mode to CDMA + LTE\/EvDo, but that just causes me to have no data (even in spots where I'd only have 3G).\nI've been running this ROM for about a week now. Overall, quite stable and I am enjoying it.\n-Data\/LTE working: data did not work for me at first. I had to go into the Mobile Network Settings and ensure that Data Enabled was selected, CDMA subscription was on NV (not RUIM\/SIM), and that Network Mode had LTE selected. LTE\/3G works without issue for me on a daily basis (ie. no signal issue, no reseting).\n-Wifi Hotspot: turning on the wifi hotspot in the system settings (not having to use something like Trev E mod) works without issue as well (and without hotspot activated on your account). I've not tried the tethering so far.\n-Google play store crashed when trying to d\/l a bunch of apps at once (when I was installing apps after first installing ROM). The d\/l's cont. and I was able to simply go back into the store w\/o really any issue.\n-SMS while in call: I was in a call, switched off wifi and when the network kicked in the call dropped (not sure if this is anything with the ROM though).\n-When in a phone call and the screen turns of due to being close to my face, when I pull the phone away to press \"end call\" the screen will not go back on. I have to press the screen on button.\n-There is no history. I was looking at the Nexus 5 the other day and it does have a history. I assume this may be something to come with newer releases?\n-64GB SD Card not showing up. I'm reading others have gotten this able to work by formatting to Fat32. But what other format is there? My PC shows FATex and when I try and format Fat32 (in DOS) it tells me my chip is to big. I'll have to play with this a bit more I suppose.\nI formatted my card with a third party utility. Easeus partition manager to go to fat32. I am now using pie controls for my onscreen soft keys. I have not yet tested WiFi tether. I too had network issues when first flashing this ROM. I flashed CM instead if twrp and reflashed it it has been working great from boot. Interesting note, I have an LG tone that would Not pause and play Spotify using the built in button with 4.3 or 4.2 but this 4.4 ROM allows the pause play button on the Bluetooth headset to work.\nThe hot spot built into this ROM, has it been hacked? Or will we be charged extra hot spot usage fees? If so are there any alternative hotspot tether opinions for this ROM?\nAfter Rooting my phone, installing CWM Recovery v6.0.4.3 succesfull and installing this Rom (Galaxy Note 2 GT-N7100) my screen will be black. Nothing happens.\nI tried to use the solution for the bootloop but it'll still hanging black. Another thing is that i cant restore my backup for some reason. Seems like its been gone. Please can anybody help me???\nEverything works great on my Verizon I605, the only issues I've encountered are: very low volume when using the speakerphone, and no LTE, signal is not as good as stock. Anyone else having these issues?\nMy volume was low but I downloaded viper mod in the description above and it helped out a lot with the sound issues. Did you update to the latest clockwork mod the video above shows exactly how to do it. If you didn't flash it right you could have data issues.\nI have the latest clockwordmod, I reflashed and reinstalled, and still no LTE, also viper works if using speaker phone for apps such as YouTube or music apps, however, if using speaker phone for listening to calls, it is horrible! I use speakerphone for conference calls, the volume is at max and I'd have to put the speaker to ear if there's any noise in the room. I'm probably going back to Clean Rom until this gets fixed. Also I notice that I can send a text to a contact from the call menu, I'd have to actually go into my sms app to compose a text. I like it overall besides these issues I've ran across.\nOh a different Kernal might fix the speaker phone issue I saw on xda there are some guys running plasma kernal i dunno if it would fix the speaker phone issue but its worth a try.\nStill no data for sprint version. I'm gonna try to reinstall it.\nConfirmed. Flashed using latest Clockwork Mod recovery and LTE works out of box!\nDid not have any signal when I first flashed with TWRP.\nHas anyone else experienced issued when using headphones? Im not getting any sound when using google music app or youtube. Have to consistantly reboot phone for it to work.\nI noticed if you play the music first then plug the head phones in it will the switch over to the headphones. I also used the viper mod apk above and it amplified the sound on the headphones speaker and bluetooth alot.\nThe first thing I missed in this ROM is the ability to clear out running apps. Is there any way to do that or incorporate this in an update? Everything else is great really liking 4.4 on the galaxy note 2.\nHas anyone got a reliable way to get the GPS to work after this upgrade ??? Everything else seems to work.\nYes. First you need to download a custom touch wiz rom like lab rats custom pinky and the brain rom. Then you want to get a GPS lock with the navigation or maps. Then flash the aosp kit kat rom cm11 but make sure you use the latest clockwork mod recovery and not twerp mod recovery. The video above shows good detail. People that didnt follow these steps are more prone to run into issues with this rom for some reason. I did this and gps works fine for me also the viper mod apk above is good for sound enhancement. I still think the sensor for calling has some bugs and wireless tether isnt working for me. Other than that seems ok.\nThanks for the info !!! I'll give it a try in a few days.\nHeads up! The old school way of editing the tiny_hw.xml file in system\/etc\/sound WILL NOT WORK and renders the speaker unusable after reboot.. Not really sure why but even if you edit the default value of by replacing \"57\" with the same value of \"57\" the speaker becomes utterly useless after reboot.. To fix just mount the system folder as R\/W and extract system\/etc\/sound from the latest Nightly from storage\/sdcard0\/cmupdater >> system\/etc\/sound..\nThanks for finally talking about > CM11 Android 4.4 KitKat ROM for Sprint\/Verizon Galaxy Note 2!\n| Galaxy Note 2 Root! < Loved it!\nHi, I have installed 4.4 custom rom for HTC Incredible S. I am from India.\nThough the rom and gapps got installed successfuly, Its showing no signals \u2013 No Service. I have even updated recovery CWM to 4EXT.\nYou need to upgrade the recovery ROM before you try it. The old recovery ROM won't work with Kitkat.\nI upgraded the version of my TWRP ROM to 2.7.0.1 and everything worked flawlessly. I hope this helps.\nThis ROM is outdated. I'd install the latest PAC ROM instead.\nIt should be in Settings-buttons.\nnot this post is written by means of him as nobody else realize such precise about my difficulty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwnyd b/data_all_eng_slimpj/shuffled/split2/finalzzwnyd
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@@ -0,0 +1,5 @@
+{"text":"Rockett St George are an online retailer that specialise in unique and quirky home wares. Their exquisite ranges cover everything from artwork to garden furniture as well as wallpaper, kitchenware and some of the most interesting accessories we've ever seen.\nWe could sing their praises all day long but instead we thought we'd just focus on a handful of our favourite products to give you an idea of what they have to offer.\nAlthough they offer a huge range of kitchen accessories- placemats and plates galore as well as a colourful range of tea towels and napkins\u2013 we especially love these Whale Trays, designed by Thomas Paul. Whether used to deliver meals or just left on show as a form of impromptu artwork, these trays will certainly give your kitchen personality.\nI couldn't resist on including Elton the Elephant in our favourite products list- I fell in love with him the moment I saw him! This cute and quirky side table is just adorable\u2026 And it's useful too! Guaranteed to bring a smile to your face, this little elephant will no doubt be popular with everyone, especially your children!\nFinally, we've found the perfect decoration to really make a statement in your home- the Faux Taxidermy Unicorn Head. To put it simply, we have never seen anything as striking and unique as this so we just had to share it with you. The unicorn itself is made to order and hand-crafted with love and care by the imaginative folks at Broken Hare. There may be quite a price tag on this gorgeous unicorn but it's something that we're tempted to save our pennies for!\nWe've included links to each of these individual products in case you like what you see. If you're interested in seeing more, click here to go to the Rockett St George homepage- just make sure you've got an hour or so to spare first, you won't be able to tear yourself away\u2026 Trust us, we know all too well!\nOne of our eagle-eyed team members spotted this incredible artist's work and we simply had to write about him!\nLorenzo Manuel Duran is a Spanish artist who cites both art and nature as his passions, leading him to create some of the most spectacular leaf-cuttings that we have ever seen.\nHis work varies from animals to intricate patterns as well as faces and silhouettes. They're all so beautiful that we found it hard to choose a mere handful to display here!\nTake a look at his website for a look at his full range- he also has paintings, masonry work and details of the many projects and exhibitions he's taken part in.\nToday we're bringing you yet another hidden gem from the world of fashion. Charlene Mullen, creator of these fabulously quirky cushion designs, isn't actually that well hidden \u2013 more internationally renowned in fashion circles.\nStill, it might be the first time you've come across her work and we can assure you it's utterly fabulous.\nIt centres around line drawings in an almost folksy style, updated for the 21st century. She's a particular dab hand at city skylines. Her series, that includes the Millennium Wheel, the Gherkin and London Calling, are unique and wonderfully eclectic.\nThere are few more evocative and distinctive patterns than a traditional Highland tartan. The distinguished plaids and checks incite images of wood cabins, log fires and the great outdoors.\nCreating this classy, laird-like look in your home can prove a little tricky. Such a strong and recognisable design can seem overpowering and garish. However, by using the right combinations of colours in the right spots you can fashion a gorgeous Gaelic getaway clad in your very own clan tartan.\nLess is definitely more when creating your Scotch nest egg. Subtle tartan accents in rugs and runners, cushions and covers or even a feature wall lined in tartan paper is just enough to help create a smart and sophisticated style.\nA blind is also a great way to introduce tartan into your home. At Tuiss we have a range of tartan style designs in a host of elegant colour combinations to suit any home. There's the Buchan Plaid Grey, with its classic mixture of grey, black and neutral tones. Then there's the Sloane Chelsea roman with its grown up Gingham checks.\nFor something even more striking, try the Buchan Plaid Red roman blind, combining faded salmon with raspberry red for a charming country tartan style.\nOver the festive period it's nice to add a touch of magic to your home's interior. Our latest love, Klaus Haapaniemi, can help you do just that.\nInspired by Finnish folklore and influenced by nature, his range of hand printed and hand embroidered products have a distinctive style that's absolutely enchanting.\nAnimals and florals abound, these wonderful designs are a unique take on traditional patterns that will give your home a mysterious and magical look.\nWe haven't 'loved' anything for quite a while so when we came across the Eco wallpaper site, we knew it was time to rectify the situation.\nWith company ties stretching back to the 19th century they are one of the Scandinavia's leading wallpaper specialists. Their idea is to use wallpaper as decoration, in the same way as art or paintings, and many of their unique and interesting designs reflect just that.\nThe idea behind ECO is to be surprising and to encourage curiosity and an urge to experiment \u2013 in a playful way. All this helps to create an interesting home, one unlike any other. We simply want to share our passion for wallpaper that challenges traditional thinking. Welcome \u2013 be inspired.\nWe absolutely love their artistic take on interior design, with their Renaissance range a particular favourite. But what makes them really stand out is their dedication to producing their products in the most environmentally friendly way possible.\nAs their name suggests, environmental awareness is a key aspect of their production with them even going as far as to own their own water treatment facility to produce their water based inks.\nSo if you're looking for something decorative, powerful and beautiful, with the added bonus of being eco-friendly, Eco wallpaper is the place to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KX7 ThreshStartTM concave kits are designed for those looking to begin gaining improved threshing and grain quality with minimal investment. ThreshStartTM kits include two rows of KX7 concaves for installation in the front, most critical threshing position. Choose from ThreshStartTM kits for cereals or grains that include two MaxThreshTM assemblies, or our corn kit featuring one MaxThreshTM and one MaxRoundTM concave. For John Deere and Case IH combines, these two KX7 assemblies replace one traditional concave. Standard or additional KX7 concaves can be installed behind ThreshStartTM concaves.\nAs with all KX7 concaves, ThreshStartTM concaves provide more efficient and aggressive threshing due to their unique angled bar design that causes crops to hit the concaves head-on. The MaxThreshTM box inserts also feature our proprietary laser cladding wear additive to further extend the life and performance of each assembly. Box inserts are removeable and can be flipped or replaced for added flexibility and options.\nThreshStartTM concave kits are available for John Deere, Case IH, Massey Ferguson, Challenger, and Gleaner. Find your model, find a KX7 dealer, and take control of your harvest with KX7 concaves!\nFor added performance, consider complete sets of KX7 concaves. View our configuration recommendations by crop type or learn more about KX7.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're like me, you tend toward certain styles of beer. When I go to a new brewery or select something at the store I tend toward beers that land in one of three categories. They are IPA, stout, and beers I would classify as weird. i.e. \"Yes I'll try your gingerbread, Asian zing, Belgian tripel with a flaming Jolly Rancher floating on top!\" Consequently there are styles that I tend to not touch very often like browns, ambers, and pretty much anything that lands in the lager camp.\nI know it's blue but I think I should try another.\nThe problem with this is not only that I tend to miss out on good beer but also that my knowledge of those styles is more limited than I'd like.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Advance October retail sales grew faster than expected, at 1.4% month over month, but all the growth disappears if you strip away auto related sales.\nOn a year over year basis, shown in the document below, retails sales fell 1.7% overall and by over 2% is you strip out auto-related sales.\nSeptember data was revised significantly downward to -2.3% vs. -1.5% previously. We'll have nothing to cheer on this front shoud October end up being revised down as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Overnight courier delivery is available with additional shipping charges that can vary on desired carrier and destination. Unless otherwise noted, payment terms can be Mastercard, Visa, American Express, company purchase orders, checks (drawn off a US bank), and money orders (in US funds only). Site license agreements are available for more than 20 seats or more.\nSpotlight sells for US$199 plus US$5 postage and handling. We offer 5 packs for $796 and 10 packs for $1,397. Educational discount pricing for Spotlight is $79 (please see edu order form link below).\nClick the above \"order secure\" icon to process your order online with a major credit card. NOTE that if you are interested in our educational discount price of $79, you will need to contact Onyx Technology directly. These types of special orders cannot be processed through our online order processing.\nIf you want to email your order information to us, please feel free to encrypt it using our PGP public key available here.\nIf you wish to fax us your order, please download this Spotlight Order Form.\nIf you are placing an educational order, please download this Educational Order Form.\nQC sells for US$99 plus US$5 postage and handling. We offer 5 packs for US$400 and 10 packs for US$700. Educational discounts are available (please see edu order form link below).\nIMPORTANT: Existing QC users can purchase the QC 1.5 Update directly from Onyx Technology for $29. Send your full name, company (if applicable), current serial number and payment information (Visa, MC, Amex) to <sales@onyxtech.com>.\nClick the above \"order secure\" icon to process your order online with a major credit card.\nIf you wish to fax us your order, please download this QC Order Form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwqhw b/data_all_eng_slimpj/shuffled/split2/finalzzwqhw
new file mode 100644
index 0000000000000000000000000000000000000000..d66ffd28e10443676da3d8e3d93379b2a794fbdc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Thank you for requesting our Mortgage Relief Package for your Kootenai County property. Please check your email for the confirmation we have sent to you. Please confirm your request. If you do not confirm your request, you will not receive your Mortgage Relief Package.\nSimply click on the link in the email to confirm your request and you will immediately receive your Mortgage Relief Package.\nThank you for trusting us to provide you with the answers to the questions you have about Idaho Mortgage Relief through Short Sales.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy Digital Insanity \u2013 Sony Vegas 891011 PRO KEYGEN.\nWe are not responsible for any illegal actions you do with theses files. Download and use Digital Insanity \u2013 Sony Vegas 891011 PRO KEYGEN on your own responsibility.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Seeking the best big pots with good quality and affordable prices from DHgate Australia site. We provide a variety of pot shadow online supplied by reliable sellers around the world. Helping you step by step of finding cheap vegetable planter pots is what we aim for. Enjoy exploring our range of pot tomatoes shop and find the pot clothing for sale from au.dhgate.com with free delivery to Australia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shirt: Asos, Jeans: GAP, Shoes: Urban Outfitters, Ribbon\/Scarf: Vintage, Watch: VIntage, Bracelet: GAP Casual day at work, meant a casual outfit and a new way to wear my hair. I decided to try out my new scarf I thrifted the past weekend. It was quite the steal at $4 for a 100% silk scarf. Not much else to say, so TTFN ta ta for now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"US \u2013 EUROPE \u2013 WORLDWIDE \u2013 Having received all the requisite antitrust approvals from regulators around the world, American parcel giant FedEx has now officially acquired its Netherlands based rival TNT Express in a massive freight and logistics deal. The \u20ac4.4 billion acquisition will see the existing FedEx portfolio expand with the addition of a European road network and the TNT brand name remaining for the 'foreseeable future'.\nThe new deal has attracted some interest, particularly from the US investment community. The extra $7 billion in revenue which FedEx receives represents around 13% of next years anticipated \u00a353 billion revenue for the group, at a cost of 'just' $4.9 billion. Much talk of synergies means doubtless a cutting of costs, and presumably staff, in some areas, but overall the arrangement should enable both to expand into areas of the global market where they were traditionally under represented.\nThis was the second time that the Dutch based delivery company had been lined up for a possible takeover by a major US express freight and logistics firm after the European Union objected to a bid by UPS in 2013, citing anti-competition grounds. In 2012, UPS offered TNT shareholders \u20ac9.50 per ordinary share which at that time represented a premium of 53.7% on the TNT stock price of \u20ac6.18. After the deal fell through, TNT started selling off divisions and revamping its business as it battled to remain relevant in a saturated market with major competitors in the express delivery sector including DHL as well as UPS and FedEx.\nIn order to satisfy some of the conditions of competition regulations worldwide, TNT signed an agreement to sell its airline operations TNT Airways and Pan Air L\u00edneas A\u00e9reas to Dublin headquartered ASL Aviation Group, once the FedEx\/TNT deal was official.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxkla b/data_all_eng_slimpj/shuffled/split2/finalzzxkla
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index 0000000000000000000000000000000000000000..0143c62dffed2cc686390635bdae3fb63109e578
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@@ -0,0 +1,5 @@
+{"text":"Each class from Year 1-6 nominated a Reading Ambassador to represent their class as someone who loves reading and is a good role model for reading to the rest of their class.\nThe Reading Ambassadors from across the school have completed a range of tasks to gather information from their class to share ideas and opinions about reading for pleasure.\nAll of our Reading Ambassadors are really proud of their role.\n'We REALLY enjoy reading because it takes you to another world without troubles and worries.\nOur jobs as 'Reading Ambassadors' is to inspire and encourage reading for pleasure to the children of our school. EVERYBODY should enjoy reading because it captures your imagination in an excitable and captivating way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The North Remembers managed to pull in an average of 522,000 viewers on Sky Atlantic last night \u2013 peaking at 622,000. For such a niche channel these are good numbers, but I can't help but think that its shame this fantastic programme is not more easily accessible. Compared to last season's premiere however the figure is notably down on the season one premiere which garnered a very healty 743,000 last year. However I think its fair to say that many of those initial viewers were put off by the shows slow pace, and sadly for them they switched of before things really got going. By the end of season 1 it was pulling in around 600,000 viewers. I'm sure those numbers will increase as the season progresses and the episode will be screened again on wednesday and sunday to hook in anyone who may have missed the monday night showing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On behalf of Marder, Eskesen & Nass posted in Workers' Compensation on Tuesday, May 12, 2015.\nAccording to the Occupational Safety and Health Administration, thousands of workers in the United States are affected by illness caused by excessive heat exposure while working outside. More than 30 workers died from a heat-related illness in 2012 alone. OSHA has introduced standards that New York employers can use to prevent heat illness among employees.\nAccording to the OSHA act of 1970, employers must provide employees with a workplace free of known hazards that are likely to cause serious harm, including heat-related illnesses such as heat stroke. Employers who do not comply with OSHA standards for preventing excessive heat exposure may face citations of as much as $70,000. It is recommended that employers evaluate the conditions at all worksites while paying particular attention to employees who are required to engage in intense physical exertion outside or experience prolonged exposure to high temperatures and humidity.\nAccording to a 2012-2013 case study, OSHA found that the leading cause of heat-related illness and death is a lack of experience working in the heat. Such employees have not gone through acclimatization, a process that allows the body to gradually acclimate to high levels of heat and humidity. Employers who do not provide proper acclimatization to new employees or who fail to provide proper safety requirements such as adequate water, shade and work-rest cycles may be held responsible for heat-related illness and death.\nHeat-related sickness and death suffered while working in high-temperature situations may result in a workers' compensation settlement or a personal injury lawsuit. Employers are required to provide a safe environment for all employees, including those who are working outside with significant heat exposure. If such requirements are not met, they may be liable for the injured worker's medical expenses, pain and suffering, lost wages due to illness or injury and other types of damages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FD4 Correlation is an algorithm developed for the Time Resolved PIV. FD4 Correlation offers the frequency analysis of the velocity fluctuation at each PIV grid and the sophisticated spatiotemporal validation.\nIn the conventional in-plane (2D) PIV analysis, outliers are detected using spatial validation and flip the value by the interpolation. In the Time Resolved PIV, each vector map is very close in time; thus it is possible to apply a temporal validation before or after applying a spatial validation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it possible for the Irish media to defend their biased reporting on the Corrib fiasco.\nHow can international media manage to come to a completely different view of the same events.\nIs the reporting stage managed by the state, and if so, Why.\nI didn't know international media cared about corrib? Link? Source?\nYeah - Link or source to foreign media please! FFS its the point of your post, dont be so lazy!\nThis story was first printed in Village, a small circulatin Irish current affairs magazine. It's since been posted on Znet (California), Silobreaker, WIP, Alfred Donovan's US-based 'Royal Dutch Shell' and a bunch of other international web-based news outlets.\nIrish journalists are sh1t scared of this story, it seems. Mark Garavan, a spokesperson for Shell to Sea has said that he knows for a fact that many have been leaned on for reporting the facts as they really are - some threatened with their jobs and one high profile journalist actually lost his. The forces behind the Corrib scandal are vicious and ruthless.\nSo the story was first printed in an Irish journal, or in other words, foreigners are relying on Irish journalists.\nYou miss the point. The high circulation national media will not touch this story with anything like the degree of rigour that it should. The editor of the alternative press Village magazine is a departure from the norm. Scott Millar for the Irish Examiner and Lorna Siggins for the Irish Times are probably the best. They offer no opinion either way but they do at least include ALL of the facts, generally speaking. But their reports are always buried deep in the paper\/website versions.\nConsider what the outcry would be if one of the Corrib protestors had scuttled the boat of two Shell security men and left them in a raft on the Atlantic coast. The coverage would have been front page news with furious editorials and acres of opinion columns offering wall-to-wall condemnation and outrage. The other way around and most of the media has gone out of its way to bury the filthy crimes committed on both Pat O' Donnell and Willie Corduff. I am in no doubt that these crimes were committed with the full knowledge and connivance of our government, Shell and their mercenaries n Ireland.\nClassic mainstream media ommission at the end of that piece. Pat O' Donnell's boat did not sink in 'mysterious circumstances'. The facts are well known but again NOT reported here: he and his crew were held at gunpoint by men in balaclavas and his boat scuttled.\nMaybe the Irish media are reflecting the opinion of the majority of the Irish people with regard to what is going on in Mayo. Just because it doesn't suit the S to S supporters on this site, does not mean that its proof of a conspiracy. Why should a vocal minority get their way?\nWombat I am not a S 2 S supporter, but I dont like what is going on.\nTHe media are influencing the opinion of Irish people. The Press are reporting misinformation from the Gardai as fact. That might seem to be quite a reasonalble thing to do, but when you consider that they published a lot of sceptical articles in relation to Willie Corduff's beating, while the Guardian walked in and published video of his bruises. When there was a direct conflict between what the Gardai sais and what Corduff, a reputable citizen, said, it would not have killed them to go and check it out.\nI suppose the first question is who owns the Irish newspapers? RTE don't even have the excuse of a biased proprietor. How come we can get reports from Charlie Bird on a dude ranch, but they can't find their way down to Rossport to do a proper investigation?\nif anything, the irish media have bent over backwards to give the shell to sea protesters huge amounts of airtime and print.\nthe only largely-critical piece on the protesters has been paul william's documentary on tv3.\ni think everything else has either been pro-protesters or evenly balanced.\njust because you don't agree with the decision of the planners doesnt mean that there are underhand tactics being used by shell or the government.\nwhy will no-one ask the protesters for facts to back up their case.\nits a matter of liquids and pipes and pressures and combustion. it should be simple chemistry and physics to prove that the pipeline is dangerous.\nobviously the chemistry and physics stack up in favour of giving permission to the scheme.\nI've not seen any good reporting on the technical issues, have you?\nbut when you consider that they published a lot of sceptical articles in relation to Willie Corduff's beating, while the Guardian walked in and published video of his bruises.\nWhen there was a direct conflict between what the Gardai sais and what Corduff, a reputable citizen, said, it would not have killed them to go and check it out.\nare you implying the gardai aren't reputable?\nthe gardai are videoing and recording so much of their own interaction with protestors and the garda ombudsman commission are all over mayo.\nthe stats on the complaints against gardai in mayo were reported last month, and there were 0 findings against gardai.\nthe irish times and irish examiner have had good summaries.\nanyone who wants more info need only look at the bord pleannala reports.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyohg b/data_all_eng_slimpj/shuffled/split2/finalzzyohg
new file mode 100644
index 0000000000000000000000000000000000000000..7fac5215476526c43fed5513f57091962cc31a3b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Aldershot: Ashgate 2013, 256 pp.\nHired domestic and care migrant workers, the form of employment which seemed to be on the verge of disappearance in modern societies, provide today to an increasing degree a private solution to a public problem. Thus social organization of care in late capitalist societies is systematically connected to structures of global economy and social inequalities. Changing family relations, increasing women's participation in the labour market, and changing patterns of family lifestyle meet with demographic trends of ageing of the European population and simultaneously with institutional trends of weakening of the Western model of the welfare state and rising neo-liberal globalisation.\nThe new book Irregular Migrant Domestic Workers in Europe focuses on the wide area of research on immigrant domestic workers in an irregular situation: immigrant domestic workers who have no legal residency permits in the countries in which they work and who thus have no proper work contracts or welfare benefits. The chapters cover eight European countries: Belgium, France, Germany, Greece, Italy, the Netherlands, Ireland and Spain and cover both genders and all types of domestic work (live-in, live-out, with one or many employers). It is a pity that the book does not cover new EU countries with transforming democracies like the Czech Republic, Slovakia, Poland, etc., where an interesting increase in employing migrant domestic workers is being noticed.\nThe authors looked on the three main aspects of irregular immigrant domestic work: employment conditions, health issues and family life. I believe that the last two topics are the ones that separate this book from others about domestic work. As Triandafyllidou argues: \"Domestic work is a heavy job both physically and emotionally and entails particular health hazards. Access to health services is at best limited when immigrant workers are undocumented and the fact that they work in the home makes it even more difficult to access information and\/or to refer to NGOs or a trade union that could assist them\" (2013: 3). The book shows the specific tension between the absence of rights to a family life for domestic workers themselves while at the same time incorporating into the surrogate family environment of their employers.\nThe book editor Anna Triandaffyllidou chooses three conceptual advancements: \"1.) the notion of a 'career' for irregular immigrant domestic workers - a concept that has to date only been discussed for legal immigrant domestic workers, 2.) the notion of legality and irregularity highlighting the fuzzy borders between them in immigrant domestic work, 3.) the gender and (transnational) family issues -the right of irregular immigrant domestic workers to have a family life and the difficulty of combining this especially with live-in employment\" (2013: 3). I have to maintain that the book works sensitively with multiple dimensions of legality\/illegality in the specific situation of irregular domestic workers.\nThe editor's introduction chapter Irregular Migration and Domestic Work in Europe: Who Cares? aims \"to place a book in the wider literature on global migration and the 'global care chain' (Hochschild 2000), looking at how domestic work fits the needs and dynamics of developer countries' labour market in the era of post-industrial capitalism and neoliberal globalization\" (2013: 4). Triandaffyllidou discusses specifics of the European context and its migration policies. For me, as a social anthropologist, the most interesting part of this chapter is its focus on special attributes of domestic work. She explains how care work transcends the distinction between private and public life: \"While traditional paid work like any other it is inherent in the family life and not in the employment system. For instance, qualities that are highly valued in paid work such as speed, effectiveness and efficiency may not be appropriate for domestic work where caring for elderly, sick or children requires patience, flexibility, slowness\" (2013: 10).\nThe second chapter, Domestic work in Belgium: Crossing Boundaries between Informality and Formality by Marie Godin, introduces how domestic work is organized in Belgium and it shows the heterogeneity of female migrant trajectories. She explains about the concrete example of migration policy - the 'cheque service system' which \"helps many migrant women who used to work irregularly in the domestic work often find, after having been regularized, a first formal job opportunity\" (2013: 37). She speaks about positive aspects of the system which has reduced some parts of the informal economy in the domestic sector allowing new regular migrants to enter the formal labour market in Belgium. The system has its weaknesses as well, as Godin writes: \"The choice of shifting the work relationship from a classic type ('worker-employer') to a more complex one including a third party ('worker-client-employer') is not always an easy one to make for any parties ('new clients' workers) .....As a result, the affective and symbolic component of such an exploitative relationship between 'master' and 'servant' is 'naturally' being reproduced from the informal to the formal sector\" (2013: 38).\nChapter 3, Migration Careers and Professional Trajectories of Irregular Domestic Workers in France by Karn Sohler and Florence L\u00e9vy, is based on field research and focuses on the female migration trajectories into domestic work. They reflect constant legal and economic insecurity of female migrants: \"As long as they have only one employer or have a weak social network that impedes them from finding quickly another job, they remain very dependent and vulnerable to abusive and exploitative employment relationship. One of the successful career strategies used by the women interviewed was to extend and diversify their employers' networks, thus reducing their dependence\" (2013: 64).\nIn Chapter 4, Three different Things: Having, Knowing and Claiming Rights: Undocumented Immigrant Domestic Workers in Germany, Lisa-Marie Heimeshoff and Helen Schwenken argue that \"...our research indicates that domestic workers are conscious that by entering into an employment relationship, they are trading rights for employment, because undocumented domestic workers in the situation are not able to claim the rights that they have according to German law\" (2013: 90). They specifically explain examples of migrants' exploitation and in founding new family and family reunification and they are not able to defend themselves in this case \"and claim their right to physical integrity, because an independent right to residency only manifests itself after three years\" (2013: 90).\nChapter 5, With All the Cares in the World: Irregular Migrant Domestic Workers in Greece by Michaela Maroufof, examines the Greek policy framework on domestic work and the experience of irregular domestic workers and civil society actors. When you conduct your work in the same place as your job and that space is not your own personal space, it is difficult to maintain a boundary between work time and private time. The lack of personal space and private and personal life can lead to feelings of social isolation, frustration and feelings of loneliness. Maroufof writes about health related issues - especially mental challenges of domestic care work: \"These problems are mainly connected to the long hours of work, the lack of sleep and rest and the fact that they feel 'detached' from the 'outside world'\" (2013: 105).\nThe Irish situation is explored by Sally Daly in Chapter 6, The Home as a Site of Work. Her article based on surveys involving 40 domestic workers provided some important indicative data from female migrant domestic workers in Ireland. Her respondents maintained the importance of new technologies for 'up-dating' their transnational parenting. She reflects on the use of mobile phones to help them to manage the notion of everyday parenting, including micro-management of their children's meals. Daly argues that: \"This communication allows them to reconstitute their role as effective parents, but there can be more ambivalence in the child's experience of such distance parenting\" (2013: 130).\nPaola Bonizzoni explains the Italian situation in Chapter 7, Undocumented Domestic Workers in Italy: Surviving and Regularizing Strategies. The chapter builds on 11 interviews of female undocumented domestic workers and on five interviews with civil actors that were conducted in Milan. The interviews focused on general conditions of undocumented domestic workers in Italy as well as on the limits and opportunities of the current Italian immigration law and the specific forms of support organization offered. Her informants actively spoke about seeking regularization to improve not just their working conditions, but also their family conditions. But, on the other hand, regularization channels provided by Italian immigration policies can lead to deeper dependency of the worker on the employer. As Bonizzoni writes, \"The (not always realized) prospects of regularization have led several women to accept a worsening of their working condition, as well as bearing the costs associated with the regularizing process... Regularization is seen not as a right, but as an indulgent concession of often reluctant employers, who clearly do not value regularization because they want to avoid the penalties of using undocumented workers\" (2013: 156).\nSarah van Walsum in Chapter 8, Regulating Migrant Domestic Workin the Netherlands: Opportunuties and Pitfalls, maps diverging interests and possibilities for collaboration and political constrains that mark the current situation of domestic workers in the Netherlands. She introduces a so-called subsidized sphere which refers \"to those forms of childcare and home-based care for elderly and the infirm that are either provided via state-financed health car or by independent service providers, and are often mediated through agencies, with the possibility of state funded compensation of costs or tax exemption. In all cases, workers must declare their income in the Dutch tax department and hence must have residence papers\" (2013: 162). Sarah van Walsum asks important questions with which immigrant domestic workers will have to contend: \"What conditions will have to be met to ensure that they can successfully compete with workers still operating in the shadow economy? To what degree will they, as employees, be able to maintain the degree of autonomy that some at least have attained, as quasi self-employed, in determining whom they will work for, what tasks they will perform, during which hours, under which conditions, for what price, and for how long? And, once admitted as domestic workers with formal employment rights, will they be able to further their careers or will they be racially marked as suited to this form employment and none other?\" (2013: 180).\nThe Spanish situation is presented by Tania Gonz\u00e1lez Fern\u00e1ndez in Chapter 9, Globally Interdependent Households: Irregular Migrant Employed in Domestic and Care Work in Spain. She critically pointed out that: \"The irregular migration of women is not only a response to the gender segregation of the labour markets in the countries of origin, nor just the demand for the labour in the destination countries. It is more complex process, multifactorial, and if indeed the feminization of wage labour in the central economies is an important part, we cannot ignore the power relations articulated within the migratory processes, given that capitalism does not just respond to a logic of class, but rather to a system of interconnected cultural, gender, ethnic hierarchies (among others)\" (2013: 205).\nBooks that include a collection of research by different authors from different academic fields may be considered by readers as chaotic and losing their comparative perspective. But Anna Triandaffyllidou has done a good editing job keeping articles theoretically and methodologically homogeneous. The final concluding chapter extends a helping hand in this regard by giving a comprehensive comparative analysis of the final results of particular research results. In concluding (Chapter 10 Irregular Migrant Domestic Workers in Europe) Anna Trian\u00addaffyllidou and Thanos Maroukis argue: \"Policies need to render the domestic services industry viable as regards the sustenance of its labour force and growing social expectations that surround it. The current economic crisis and the overall restructuring of welfare systems both in southern and northern Europe make the need for an affordable and sustainable domestic care labour force all the more necessary and sought after, especially as life expectancy is prolonged and the European population is increasingly ageing. Restructuring this occupation's architecture might eventually lead to the reconstruction of its profile. However, this requires careful interventions that would reverse the social process of reproducing unequal labour relations. And time. Policy changes need time in order to transform to social changes\" (2013: 230).\nThe problems and risks of domestic work are already reflected on the international policy level. In June 2011 the International Work Organisation (ILO) adopted the Convention on Decent Work for Domestic Workers, where for the first time it even applies its rules in the sector of informal economy. Particular attention here is paid to female migrants, because their increased vulnerability and inequality leads to further abuses of rights. Even though states have obligations under international agreements, for example the Convention on the Elimination of All Forms of Discrimination against Women, to adopt procedures in order to ensure the same protection rights for these groups also. In reality the question of the position of female workers in domestic work remains the interest of many developed countries, including some of those mentioned in the book. I suggest that books like the reviewedIrregular Migrant Domestic Workers in Europe may help to deeply analyse the social situation of those who care about our elderly, ill and children. Now comes the time to start to care about domestic workers themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OUR BLUE BABY SHOWER-featured in Life & Baby!\nWe were honored to be a part of the shower for our friends Cassie (mom to be) & Deborah (Grandma to be)!! The baby shower was a beautiful collaboration between some amazing & talented vendor friends!!\nCheck out the swanky shower as seen in Life & Baby!!\nWe are starting this week in style with a baby shower that will leave you ohhing and awwing. This baby shower is so jaw-droppingly beautiful that my heart skips a beat with each darling detail. Photographed by Erin Hession Photography this stunner is a real collaboration of so many ah-mazing vendors. Planned perfectly by Deborah of A Touch of Elegance, and the Momma-to-be herself, this amazing party planning duo will have your heart fluttering with this stunner of a shower.\nDon't look too fast or you'll miss the kettle chips with white cheddar with an edible monogram, the blue sunken cotton candy signature drink or the food based on the Momma-to-be's cravings.\nThis entry was posted in baby shower, blue candy buffet on January 31, 2012 by Setting the Mood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Girl Like a Bomb is out today! You can grab it at CLASH or at Amazon. E-books will be here probably tomorrow. If you pre-ordered a signed copy, I'll be physically sitting down and signing the books tomorrow evening in Portland, so they'll be on the way soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Le\u00f3n studied at the Baja California University. After finishing her dental studies, she specialized in implants and crowns.\nShe specialized in these procedures because there's a great need for them and she saw she would be able to help more people.\nIn addition, she offers orthodontics (alignment of crooked teeth), rehabilitation and bone grafting surgery.\nDr. Le\u00f3n's service is unique because she includes all aspects of the treatment in her warranty. For example, the warranty includes even laboratory expenses, which are typically not included. She does this because she cares about achieving 100% satisfaction for all of her clients.\nFrom experience, Dr. Le\u00f3n knows that many of her patients come from the United States and even Canada. Her clinic gives free transportation from and to the Mexican border. If the price of the treatment is over $1,000 USD, the clinic also offers free accommodations.\nAs it does for many in the healing professions, Dr. Leon's personal satisfaction comes as a result of working with her patients. She tells us about patients with low self-esteem who cry with relief after being treated. She says her work is rewarding because she doesn't only fix the problems with their mouth, but also their perception about themselves.\nDr. Le\u00f3n wants patients to know that everybody at her clinic works hard, but they love what they do and they are happy working together as a team. Dr. Le\u00f3n is married to someone she met through dentistry. They have been married for 5 years now and have a lovely little girl. They are both hard working and like so many working couples, they sometimes find it difficult to spend time together, but that only means that they make the most of every minute when they can.\nOn special occasions, like their anniversary, her mother takes care of their daughter so that she and her husband can have a well-deserved break. Like many Mexicans, she chooses to spend her free time with her family, watching TV, dining at the local buffet, and going shopping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"February 8, 2019 \u2014 Big Brothers Big Sisters of Massachusetts Bay, a Boston-based nonprofit that provides adult mentors to enable youth to reach their full potential, announced that it raised a record-setting $3.5 million at its annual signature fundraising event held last Saturday, beating its goal by nearly 30%.\nThe 20th annual Big Night cocktail party, concert, and fundraiser, held at the House of Blues in Boston, sought to raise $2.7 million and attract 1,100 people. More than 1,100 people attended the sold-out event.\nLast year, the same event raised $1.8 million.\nFunds raised will help Big Brothers Big Sisters of Massachusetts Bay (BBBSMB) expand its service and move children off its waiting list of 1,500 children who are looking for adult mentors.\n\"The heroes are the parents who are working hard in order for their children to have a better life and the Bigs that volunteer to mentor these kids,\" said Jim Pallotta.\nFunds were raised via ticket and sponsorship sales, branding opportunities, gifts in honor of Jim Pallotta, on-site donations, and matching funds from the Pallottas.\nBig Night fundraising typically represents 15% to 20% of BBBSMB's annual revenue, with all special events accounting for approximately 30% of the agency's overall revenue.\nLast weekend's event featured performances by artists Peter Wolf and the Midnight Travelers and French Lick.\nSince 1998, Big Night has raised more than $38 million, helping to support more than 20,000 mentorships.\nBBBSMB traces its roots to the formation of the Big Brothers Association of Boston in 1949. In the late 1990s, it began matching girls and women in response to a request from Big Brothers Big Sisters of America to serve the Cape Ann area. In 2006, the organization merged with Big Brothers Big Sisters of Cape Cod and the Islands, acquired Big Brothers Big Sisters of Greater Attleboro, and adopted its current name.\nToday, BBBSMB annually serves 3,200 youth in Greater Boston, Cape Cod, and Martha's Vineyard, aiming to empower youth to achieve their full potential, contributing to healthier families, better schools, more confident futures, and stronger communities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"So I guess it's no big secret that I like the Top Tier MLM business model and Family IQ a new Top Tier MLM is one you will be hearing a lot about very soon.\nNow after being in the industry for over 23 years, I am a huge believer in Network Marketing and have had the opportunity to work with many Network Marketing companies, and also serve on the advisory boards several.\nDuring that time I have worked with numerous variations of the \"Standard MLM\" business models and I have also watched and studied as each new compensation plan variation swept through the industry.\nAt times you could almost feel that the industry was, even after 40 or more years still seeking a higher plateau of development, that magic mix of compensation plan components that would ultimately reward a distributors up front business building efforts as well as provide a highly profitable long term residual.\nLet's see, Unilevel, Compression Programs, Rollups, Binary, Hybrid Binary, Coded Bonus, Matrix and so many variations of those it makes my head spin just trying to keep track of them.\n\"So which is the best\" is probably the question I am most frequently asked by MLM hopefuls looking to achieve success in Network Marketing, and form a new Plan A in today turbulent financial market.\nWell in the final analysis of course they are all profitable, all companies no matter what comp plan they use have top earners who's checks would make you green with envy.\nThat of course is the big dividing line drawn right down the center of the industry.\nOn one side are all the \"Standard MLM\" compensation plans, no matter what variation of pay plan they may use, and on the other are the \"Top Tier MLM\" companies.\nIf you have read any of my blog posts explaining the differences such as \"Top Tier MLM or Traditional Which is Better?\" you know that the basic key difference is the price point of the distributor kit a new rep buys when they enroll and how much the company pays out on the initial product package.\nTop Tier MLM companies usually have a much higher level of compensation on the front end allowing a new rep to earn a full time income from day 1 and Standard MLM comp plans usually put most of the profit on the backend residual which can take several months or years to fully develop.\nNow I thought I had it down pretty well and so I guess I should have expected it, an entirely new variation hits the market.\nDarn! Just when I thought I had it all figured out.\nFamily IQ is a new \"Top Tier MLM\" with a fairly low end startup package and an aggressive residual MLM compensation plan on the backend.\nI first ran into Family IQ when an online marketer I know generated about $33,000 in commissions in a month marketing this company last December.\nIt kind of puzzled me so I did an extensive review of the company to see what kind of magic they had working for them and here is what I found.\nFamily IQ is the brain child of Mark Hobbins who founded the company in 2001.\nNow Family IQ is fairly new to the MLM world since they just launched their MLM business model in about October of 2010 but the company has been around for about 10 years prior to that with an online business model offering according to the company website \u2026\"state-of-the-art family skill building tools to treatment programs and therapists who wanted to improve family relationships\".\nThen last year, Family IQ decided to go with a direct to consumer business model and chose a Top Tier MLM compensation plan.\nThe Family IQ products are all online training and educational systems designed to help improve interpersonal relationships between family members, husband and wife, parents and children, and so on.\nSo when a distributor signs up they get access to all the online training modules and since Family IQ has been compiling this training media for over 10 years there is quite a bit of it.\nNow most Top Tier MLM's tend to use some type of educational product and most tend to be financial (such as WealthMasters International) or self improvement related, but the Family IQ line is pretty much the first of its kind.\nI have to say that when I first looked at the Family IQ product line I was not sure about its success as a Network Marketing product but after doing some marketing research, I have been pretty amazed at how strong the appeal this product is.\nSeveral people I interviewed who were marketing Family IQ were doing extremely well and averaging 10 + signups a month.\nThe Master distributor of Family IQ is Rod Stinson, who has been highly successful in the world of Top Tier MLM over the years. Now the strange part is that I actually know Rod, he was part of a highly successful MLM group I built over 20 years ago.\nRod moved over into the Top Tier MLM business model years ago and has been quietly making a fortune by using some very innovative marketing sales funnels based on direct response marketing models.\nThe Family IQ sign up package is $1495.00 which is comparable to what a distributor would pay to join a \"Standard MLM\" if they purchased the top end product line when they enrolled, but in the world of \"Top Tier MLM\" where product packages usually range from $2000 on the low end to as much as $20,000 on the high end, the Family IQ package seems to come in on the low side.\nThis is where the top tier part of the compensation plan kicks in because when you enroll a new member into Family IQ you receive a whopping 70% commission, earning a clean $1000 per new rep you enroll into the business.\nThe initial commissions on the sign up packages are actually paid out daily since according to the sign up procedure the new person entering the business first enrolls on the Family IQ site and pays the company $495 to register, however they are not active status until they complete the payment for the balance of the enrollment fee of $1000 directly to their sponsor.\nThat's right, they pay you $1000 directly\u2026..Nice touch!\nMost of the reps actively working the business generally have the new enrollee send them a cashier's check by overnight mail and then the sponsor goes into their back office and activates the distributorship for the new person.\nSo if someone really wants to earn money quickly, this system works really well since there is no limit on how many $1000 commission checks they can receive, and they can of course receive them daily.\nNow the Family IQ plan also has a 9 level \"Unilevel\" type residual pay plan that allows reps to tap into the ongoing growth of their organizations by receiving a $50 bonus for each new signup on levels 1 to 3 and then $24 payout on levels 4 to 9. This is very similar to some highly successful coded bonus pay plans in use today.\nTo be qualified to receive the residuals, the rep of course must be active status by signing up for the Family IQ monthly membership fee of $79.00.\nThe payout on the Family IQ membership fee is 6% on levels 1 to 4 and then 4% levels 5 to 9.\nNow part of the appeal of this deal is the powerful Family IQ presentation done by the master distributor Rod Stinson in which he outlines the advantages of the Top Tier MLM model and then shows numerous examples of his receiving multiple Fedex envelopes every day, generally averaging about $70,000 per month and even averaging about $103,000 in one recent month!\nBeam me up Scottie! I have stumbled into a parallel universe!\nNew reps when they enroll also receive one of these recorded business presentations and a replicated website to start spreading the word.\nSo what does the future hold for this Hybrid \u2013 Top Tier MLM?\nWell from all indications Family IQ could be a unique example of one of the first Top Tier MLM's to actually go viral.\nOne of the key elements that has always helped fuel a new company's growth is when people actually make money, and if you have a lot of people making a lot, its kind of like throwing gasoline on a fire, and there appear to be a lot of people in Family IQ making a lot of money.\nSo dynamic growth aside, what actually is the appeal of this Hybrid \u2013 Top Tier deal?\n1. The price point of the Product: Even though $1495 is pretty low for a true Top Tier MLM, this price point makes it reachable by just about anyone who sees the opportunity and it pays a pretty whopping big commission for such a small price tag.\n2. The Powerful Webinar-On-Demand presentation that Rod Stinson does where he shows enough proof of income by way of Fedex envelopes and Cashiers checks that he could paper his house with them is obviously a big factor. There are several instances of this throughout the presentation and after about the 4th or 5th time you actually start to think\u2026.. Good Grief, I want some of those! All new reps get one of these \"On Demand\" presentations on their replicated website and the system even includes a special software package that generates free MLM leads off of the internet. Now I was skeptical about this until I found out it was a type of scraper software and I have used these before and they do work. All in all its a very good sales funnel and lead system.\n3. Company Credibility and Noble Product: Now this is where a lot of Top Tier deals fall down, I mean let's face it we are all in MLM for the money anyway but sometimes the company just does not measure up. This is Family IQ's strong point, they literally smack of credibility.\nSo should you drop everything you are doing and jump on the Family IQ bandwagon?\nWell that depends on your financial goals and how quickly you want to achieve them. Now as I outlined in a recent blog post \"Is Network Marketing Dead?\", the big appeal of the Top Tier business model is the GPT (Get Paid Today) principal, so it does give you the ability to get to the big money quicker, in 30 to 60 days as opposed to 12 months or more in a Standard MLM.\nNow normally in a Top Tier the back end residual is less than you would make in a Standard MLM but with Family IQ they appear to have an attractive Hybrid business model that actually pays both.\nI expect we will be hearing lots about this fast moving new company in the future.\nTired of spinning your wheels in a standard MLM business model where you need several thousand in your downline to even get close to $10,000 a month? That could take you years! Learn the \"Million Dollar Game Plan\" to reach your financial goals in 90 days or less using a Top Tier MLM business model.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Online or On Campus. Every day you hear about another cyber security breach. To keep their customers' data safe, companies are rapidly developing their Cyber Security teams, generating tremendous employment opportunities in this field.\nEnrolling now for Summer (July 1) and Fall (Oct 1) starts!\nOnline or On Campus. Today's high-tech companies have complex operational environments and demand rapid decision making. Get the skills you need to manage processes and teams in these fast-moving organizations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am interested in periodic update from SmartStock.WMS, including news about its development and enhancements, events, training and other related products.\nGross Approximate Annual Sales Revenue?\nNumber of Techn. Support Engineers?\nDo You Sell Own Products or Portfolio of Services?\nDoes Your Company Provide Onsite Installation?\nWhat percent of your revenues are derived from Hardware?\nWhat percent of your revenues are derived from Software?\nWhat percent of your revenues are derived from Services?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you interested in learning more about what we have to offer? We are happy to answer any questions you may have. Call us or send us an Email or fill out the form below and we will respond as quickly as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Or maybe the road to hell is paved with good intentions. Same-same.\nI was reading through my blog archives (ah, the joys of having several years of your running history documented on the interwebs), and I was reading things like, \"Now I'm going to start doing speedwork!\" or, \"I'm totally going to run Pikes Peak sometime soon!\"\nBut if there's one thing I've learned reading running blogs, it's that you want to read about winners. You want to read about people who run hard and do a good job at running cool races - not people who sit on their couch eating ice cream again instead of taking their training seriously.\nI like to think that I'm pretty well in touch with my abilities. However, sometimes my physical abilities (what I could be capable of doing) conflict with my motivation (what I can actually get myself to do). And therein lies the problem. I want to be a better runner... sort of. I just don't want it enough.\nSome of this I've understood about myself for a while: I use running as a stress release, as relaxation. When I train, I have to turn running into something goal oriented, something competitive, something that's the antithesis of relaxation. And then I risk enjoying it less.\nI don't have any deep thoughts to go along with this. I just wanted to acknowledge that I see this problem and that it frustrates me. To close with another cliche, I'm not going to make any excuses. My friends don't need them and my enemies won't believe them.\nBesides, I'm totally going to start following a training schedule tomorrow, man. Tomorrow. There will be speedwork. And races. And crazy time goals. Right? Right?\nI like how you part laid back runner and part type A...you don't overstress training and focusing on a single goal but you have done a ton of races and been running, though at times inconsistently, for years.\nthanks, Aron! I know you're right and that I'm just going through a phase right now... but I wish it would end and I would snap back into it. it's like I've lost my motivation to complete long-term goals.\nI AGREE. I DO NOT KNOW.\nYes. Very quickly. Good point.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I'll go first! Hi, Im Duane, a Full time Destination Wedding Photographer Located in Cape Town, South Africa. Passionate about capturing life's Honest & Grand moments that exist and take place between real people!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nestled in the beautiful black hills of Wyoming, located just six miles east of Devils Tower National Monument is where we're blessed to call home. We use our horses for everything that can be done horse back on the ranch. We concentrate on producing good sound of mind and body horses for the whole family to use yet are athletic enough to go out and compete in the arena.\nOur stallions Frenchmans Image and Guys Dash A Latte are own sons of Frenchmans Guy who has consistently been ranked in the top of the nation the last several years for barrel futurity money earners. Then Image's mom is Oh Image Three she had a speed index of 96. Images colts are showing his same willingness to please and athletic ability and are winning in the barrels and poles.\nGuys Dash A Latte is out of an own daughter of Dash Ta Fame. He has three colt crops on the ground and we are very excited on what he has produced so far. We look forward to breeding your good mares. Dash and Image stand at the ranch for a fee of $800 in 2016.\nImages colts are eligible for the Grid Iron, Three Can Tango and Five State barrel futurity's and the ones that sell at the RQHBA sale are eligible for their futurity's too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's the deal: If you work at a media company that hasn't had layoffs recently, just go ahead and assume they're coming.\nWhich means not just cuts at Wired.com, as Alley Insider reported earlier this afternoon, but also sites like Epicurious and Style.com.\nThe company wouldn't announce how many people are being let go.\nThis follows cuts at Cond\u00e9 Nast's print titles last month, so it's not that much of a shock.\nA person familiar with the situation tells me that the Cond\u00e9Net took longer to make its cuts because it hadn't gotten a grip on 2008 sales and 2009 projections.\nNow it has: The unit won't hit its internal goal of 35 percent revenue growth for 2008, but should still \"outperform the market,\" I'm told. What I'm not told\u2013whether that means the broader market for Web ads and display ads only.\nNext year, the Cond\u00e9Net group, run by Sarah Chubb (pictured above) expects a \"modest\" increase in revenue\u2013likely something in high single or low double-digits.\nAs always, I'm happy to update my posts as I learn more. If you've got information on specific titles or people cut, please drop me a line. As always, I keep all correspondence anonymous: peter@allthingsd.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In March, the Gamma Alpha chapter at The College of William and Mary had a blast at their First Annual Phi Mu Laser Tag Tournament. The day was perfect; from good weather (thank goodness!) to excited teams. The event was well-planned and I didn't spend the day thinking about logistics, who should be where, or what tasks still needed to be done (shout out to Lizzy \u2013 she had all of this under control). Instead, I spent the day thinking about giving back to the kids at Children's Miracle Network Hospitals.\nIn gathering my thoughts, I've written 9 tips on how to have a fun and rewarding philanthropy event with some details that go deeper than logistics.\nWhen I arrived at The College of William and Mary in January, the philanthropy chairwoman was already presenting her detail-oriented plan for the new philanthropy event. Laser Tag? How cool! She was excited to try something new and really go for it.\nIt's never too early to get the ball rolling on the planning process. Like I mentioned above, she spent a lot of her Winter Break planning the overall event. It gave her a lot more time to focus on the details and time always passes way more quickly than you think.\nMake a task list and set goals.\nI don't know what I would do without a Google Sheets document with everything that needs to be done during the week. If you want to get really fancy, color code the categories. This is a lifesaver when planning events. I have a column for the task, notes, due date and date completed. It's my FAVORITE.\nInvite families and hospital staff to your event! It would be awesome for them to speak at the event as well.\nHave the local CMN Hospitals coordinator a local family attend a chapter meeting before the event to remind us what it's all about! If they aren't able to come in, ask if they can record a video for you.\nSchedule a hospital tour during the semester.\nHost a 'family day' to meet and get to know the local hospital's families.\nConnect with CMN Hospitals on social media.\nFor more information about connecting with your local hospital, visit childrensmiraclenetworkhospitals.org and scroll until you see contact information for your local hospital on the left-hand side.\nThe event that you are planning is a really big deal, so don't be afraid to ask for help. Use that committee full of rock stars! In working with committees, it's important to meet and understand each other's strengths. I am all for creating a task list and writing letters but when it comes to painting a banner, you might need to look elsewhere! It can really be helpful to know what your committee members are passionate about. Have a task list before the meeting. During the meeting, have committee members volunteer to take on each task. Set a due date and follow up with them to see how things are going!\nAlumnae are also a great resource to ask for help. Gamma Alpha had donations from alumnae for their concession stand at Laser Tag!\nI really love to get creative with this!\nSet up a table in a common area on campus to advertise for your event.\nTalk to people in class to let them know about the event. To take it the extra mile (and if this works at your university) ask the professor to make an announcement at the beginning of class and speak about the event.\nVisit other organization's chapter meetings to give them more information.\nHave a countdown to the event on Instagram.\nHang flyers all over campus!\nUse chalk to write on the sidewalks so people learn more information walking to class.\nStay calm through the curveballs.\nFor Laser Tag, the location was moved at the last minute, it was nearly impossible to find a rain date or location and Spring Break made it difficult to get teams to sign up really far in advance. My advice is just to remain calm! Everything will work out and it's much easier to problem solve when you aren't panicking.\nGet the chapter excited by giving everyone a responsibility.\nOne of my favorite planning elements about Laser Tag was the beautiful \"shift chart\" that gave each chapter member an assigned task for each of the three shifts during the event. Rotating between coaching, ticket table, concessions, and field cheerleader kept things exciting. It was super organized and everyone had an important and special role.\nYour awesome event is over and I know you just want to head to get frozen yogurt with your sisters and not think about the event for a while. Do that! But then come back to your computer and brainstorm everything that lead to the event's success and any tips\/areas for improvement you would make to the next philanthropy chairman. This will help so much in growing your event in the future. For Laser Tag, the philanthropy chairman made a Google Folder of all the information and documents she used throughout the planning process to share with the next person.\nI hope you use these tips and tricks to plan an amazing philanthropy event! can't wait to see what all of you accomplish! Remember \u2013 you are making a difference!\nPosted in Phi Mu Serves, Team Phi Mu, The New View and tagged #PhiMuCC, cmh hospitals, gamma alpha, laser tag, philanthropy event, Wesley Gray Smith, william and mary. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can we stop and take a moment of silence to appreciate what an AMAZING game, no, WORK OF ART, this is turning out to be? The Fun Pimps really have upped their game recently. 17.1 experimental in particular has been AMAZING for me. Almost Skyrim like. I think I even saw mists in the morning in the forest. I absolutely LOVE this game in it's current form. Especially now that building and harvesting resources gives a bit more XP and the game itself feels much more RPGish. Although I HAVE died. On Scavenger mode. On the easiest possible settings. From food poisoning and lack of food of all things.\n1) I'd like to see some settlements where multiple NPCs give quests and maybe a main storyline for both randomgen and Navezgane. (Maybe the storyline for both could be to find a cure or something. Wouldn't be that hard to put the ultimate badass base somewhere in randomgen and have certain NPCs give hints as to where it may be as you level up. That would also add an attribute for coaxing more information out of NPCs). This wouldn't be all that hard to add, create a new trader zone type with shelters, people walking around, and dialog. I'd be happy if they didn't even actually speak out loud. Just random conversation and sometimes a quest or two.\n2) More water in randomgen. Streams, rivers, ponds, Lakes. No oceans though. I found very little water thus far, just a few small ponds in the desert and a single pond in the forest next to a cabin.\n3) Maybe some type of food that doesn't lead to food poisoning, but doesn't provide much in the way of nourishment either? Make it a bit rare. Grapes, blueberries, something. The trader and vending machines are a bit unbalanced so it's hard to save up enough coin before you can be ready to explore houses and get loot (unless I'm missing some aspect of the gameplay).\n4) Bandits, if they don't already exist.\n5) The server should have random hordes of both zombies and bandits roaming throughout the world constantly, not just spawning near the player. This would make things much more interesting.\n6) Vulkan support and general graphics optimizations. In unity you can update to .net 2017. This allows you to add threading support for background tasks. I've experimented with this. With careful optimizations you can improve world load times, chunk load times, etc.\n7) As I mentioned before, more sensible biome transistions. Right now I have a desert next to a winter next to a wasteland next to a green biome.\nThat being said, this game is coming together nicely, and I do home TFP continues to bring it to a beautiful conclusion. It's a work of art and I'll gladly pay for an expansion pack or DLC when it comes out. Hell, I'd throw more money at the Pimps right now if they could. To keep development going on this long must take a ton of cash.\nmust have been some heavy shooting going on... i see a lot of empty shells laying around.\nWas at the beach with the kids. I need a mobile version dammit. Make it so TFPs!!!!! 60fps, unity graphics and no microtransactions please!!!!\nPOI spawn locations don't move. They are specific invisible blocks that are the only valid locations a sleeper can spawn at. A zombie appearing at a normal location is probably the world spawner randomly adding one, but it should not spawn in that close, so could be a bug.\nYou are right. I did some testing with a large and relatively open PreFab I made. I set my spawners to only spawn specific zombie types. I could not get it to happen during the day, but at night, world zombies would spawn inside my POI quite often... and pretty close to me.\nPerhaps another volume would help with this called \"BlockingVolume\". Doom's SnapMap added this at some point after launch so we could have more control. Then devs and modders can easily define areas where world spawns cannot occur. Or perhaps generalize where you can select the types of spawns to disallow (world, scout, etc).\nCan we stop and take a moment of silence....?\nWhy can't modlets push assets? You allow people to connect and download xml files but why not custom icons and other unity assets? Folks know they are opting in to a modded server. Please allow at the very least custom icons.\nSome games are cautious about this kind of thing because of copyright infringement. They would have no way of knowing what images you are using and associating with their game. I don't know what the exact rules are, but allowing the transfer without a direct download mod might be a big no-no.\nLast edited by AtomicUs5000; 01-14-2019 at 02:14 AM.\nNot too mention vulnerability to haxs.\nThat makes no sense. Games are already vulnerable to hacking.\nNope. XCOM has asset support built in and a steam work shop. Try again.\nHow can I change loot timer? This \"risk&reward\" is just bs when your entire horde is radiated zombies so it's almost impossible to get the loot before it goes away.\nHe may have said \"risk&reward\", but in reality it is an intelligence test. It is to test if the players greed is bigger than his intelligence.\nSeriously is asset support for mods still in flux but planned for release? I love sdx but I don't want clients forced to download mods.\nI'm not gonna go \u2665\u2665\u2665\u2665 around with custom assets if 99% of players won't download a mod but I can use in game assets and tints to accomplish the same goal and reach many more folks.\nLast edited by stasis78; 01-14-2019 at 02:33 AM.\nI have heard some interesting things about modding, some stuff that may actually deter me from actually continuing modding this game. Would be nice to get some kind of guideline or plan for the future when it comes to mods, even though we know its not a priority for them right now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaayuk b/data_all_eng_slimpj/shuffled/split2/finalzzzaayuk
new file mode 100644
index 0000000000000000000000000000000000000000..548ed3975b7e256e01bcc214a5e6d3db6399e77c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaayuk
@@ -0,0 +1,5 @@
+{"text":"Eyeo, the makers of AdBlock Plus have long promised that the management of their \"Acceptable Ads\" programme would one-day be handed over to an independent committee. That time is now getting closer with the first committee members confirmed and OKO are proud to announce that we will be part of it.\n\"Acceptable Ads\" is the initiative that allows users to whitelist a specific subset of low-impact advertising in their browser. The initiative has received a lot of criticism from various quarters, but does provide one possible compromise for users who want to stop the most annoying ads without blocking content producers from receiving any financial reward for their work.\nOKO have been openly critical of some part of the programme in the past, but we're excited to be part of its future. Initiatives like Acceptable Ads have the potential to find more sustainable ways to manage content monetization. If our involvement can help steer things towards a solution that is acceptable for those who produce content as well as those who want to consume it then that is something that we want to be part of.\nThe committee is to be made up of a number of groups aiming to represent all parts of the ad eco-system. OKO will be joining the Ad tech member group. Applications are still open for additional committee members and OKO would particularly love to see more applications from independent publisher and content producers. Applications can be made here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please ensure that your car is in a well-ventilated area. Your vehicle will need to be turned on, with the engine running, for about 20 - 30 minutes to complete the upgrade.\nWait at least twenty minutes to check the screen. Press OK and remove the USB if the screen displays \"You have successfully updated your SYNC\u2122 software to Gen3 \u2013 Vx.x\" message. The upgrade is complete.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When discovering what is CloudFlare you must first understand a few key pieces of information. The first piece of info you need to know is what cloud hosting is. Cloud hosting is a hosting technique that dramatically increases your web page speed by using multiple servers, in different locations, to host your data. When you are on a cloud-hosted network, your visitor can be served data from many servers at once. This makes the user experience much quicker than shared hosting.\nCloud hosting is much faster than shared hosting and your server data can be accessed from anywhere in the world. This makes your website virtually crash proof. Your growth potential is infinite and all of your information is safely stored away in redundancy.\nCloudFlare uses a CDN (Content Distribution Network) to speed up your website response load times even more. A CDN has all the aspects of a cloud-hosting network, with one very important addition. A CDN can examine your visitor's geographic position to create a user profile. It then uses that information to decide where your traffic is coming from.\nOnce your traffic gets analyzed, the CDN will automatically cache certain parts of your website in locations closer to your user. This reduces the number of networks your request has to go through. The more networks your request has to hop, the slower your page speed will be. Reducing this helps guarantee an optimal user experience. It also helps your search engine results. Google has stated multiple times that they rank websites that load faster much higher in their search results. This is because long load times can cause visitors to become disconnected. Disconnected visitors mean lower click results.\n\u00b7 Optimization tools. CloudFlare includes a robust tool set that includes Mirage. This is an application that creates a visitor profile and uses that to determine the best content delivery strategy.\n\u00b7 SSL certificates. These digital certificates are used to protect your sensitive data such as credit card numbers. They work by helping to establish a secured connection.\n\u00b7 Performance monitoring. CloudFlare provides real-time analytics that can help you zero in on your performance problems with ease.\nIf you have considerable traffic, you should consider upgrading to their professional version. There are significantly fewer restrictions and a number of additional tools including 24-hour support. When seeking to improve page speed, all of these added tools can be extremely helpful. It is also worthy of mention that the enterprise version allows 500MB of upload per visitor.\n\u00b7 Easy to understand set up. It literally takes 5 minutes to register. The login process is quick and easy. Answer a couple questions and you will receive a confirmation email. Follow the link in the email and log in. Enter your URL and you are all set up.\n\u00b7 Price. CloudFlare has many hosting options that are completely scalable to meet your hosting needs. Their free version provides an abundant amount of optimization tools at no cost.\n\u00b7 Excellent customer service. 24-hour tech support and live chat are just a few of the recourses that make CloudFlare an exceptional hosting platform.\n\u00b7 Upload limit for visitors. 100MB per visitor are allotted in the free version, with the option to increase (for a fee) up to 500MB in the Enterprise version.\n\u00b7 Platform. Some of the optimizations can make your website not compatible with certain web platforms.\nOne of the biggest advantages to using CloudFlare is no contracts or vendor lock. This means you don't need to sign any long term user agreement. This leaves you free to change your hosting network at any time.\nCellular and mobile surfing have become the new norm. Mobile data networks put an increasingly larger strain on networks and traditional hosting techniques. A solid CDN provider like CloudFlare can automatically optimize your website for mobile devices. This is accomplished by eliminating any lag in your data transfer. Formatting your images for delivery on a mobile platform means quicker page speeds and better response times.\nIf you are currently not using a CDN network, I would recommend you seriously consider switching your hosting to CloudFlare. There are just too many reasons to make the upgrade. You will be able to see a noticeable increase in page speed from the first day you start.\nIf you are currently on a CDN I would recommend you do a price comparison analysis. You may find that CloudFlare's free CDN service surpasses your paid service. If you have an extremely high volume site, you should consider speaking with a consultant at CloudFlare to see what improvements can be made by utilizing their business and enterprise packages.\nFor example, if you are expecting a huge spike in traffic you can rest assured knowing CloudFlare's servers have been load tested to the highest of standards. Your visitor will be experiencing your web page at optimal speeds. This peak performance will help transform your visitors into customers by transforming clicks into dollars.\nData transfer rates are always increasing, and in this instant age, you can't afford to have a lagging website. Don't go another day losing traffic because of latency. Make the free CloudFlare upgrade today and watch your usability go through the roof.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Loba Fashion Design \"Estilo\" Pantyhose, made of soft, lightweight and super comfortable microfiber. Downside Legs with delicate detail of lines, adds a charm touch to the pantyhose.\n- Composition: Polyamide 90%, Elastane 10%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Organic Nuna we have the highest reverence for Mother Earth and all that call her home, that is why we are 100% Organic, Always and Only! The Native American history of caring for our land is like no other, as such we are proud supporters of the Hopi Native American Tutkswa Permaculture Institute as well as other organizations that protect our land and our countries great heritage. The Hopi Tutskwa Permaculture Institute https:\/\/www.hopitutskwa.org\/ is a community-based non-profit founded in 2004 and based in the Village of Kykotsmovi which is located in Northern Arizona, USA on the Hopi Reservation. Their mission is to create community-based solutions in order to pass knowledge to the future generations and rebuild culturally sustainable and healthy communities.\nIn the Hopi language \"Hopi Tutskwa\" refers to the life ways and knowledge of the land and soil. The origins of Hopi Tutskwa Permaculture stem from a deep commitment to maintain their distinct identity and lifeways as Hopi people in order to pass knowledge to future generations and rebuild sustainable and healthy communities.\nTheir vision is to strengthen community through the continued intergenerational practices of traditional Hopi farming and gardening, rainwater harvesting and spring restoration, natural building, and orchard-keeping while applying applicable Permaculture principles, methods, and techniques.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabffr b/data_all_eng_slimpj/shuffled/split2/finalzzzabffr
new file mode 100644
index 0000000000000000000000000000000000000000..b9330181770df521d3e83452ac2813c550f3ecbb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabffr
@@ -0,0 +1,5 @@
+{"text":"Swedish kitchen appliance manufacturer Ankarsrum is all about homemade food and making cooking easier. Because if done right, cooking at at home, from scratch, is a genius move. It's cheap, it's tasty and it's healthy. So to inspire more people to cook from scratch we collected and published the authentic recipes that nurtured some of the world's most brilliant minds.\nThe Ankarsrum Kitchen Assistent is a multifaceted kitchen assistant that makes cooking easy. We where tasked to highlight the assistant's many smart features, showcase its broad area of usage and inspire people to cook at home, from scratch.\nCooking at home is a genius move. It's cheap, it's tasty and you have full control over what goes into your body. So to inspire more people to cook from scratch we collected and published the authentic recipes that nurtured some of the world's most brilliant minds so anyone can cook and eat like a genius.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Company presentation for Lesotechnika, company providing gardening, and city itinerary services. Old website replaced with new CMS and responsive design implementation. Several modifications done during implementation phase, but finally we found best fitting solution for customer needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shade Mountain Vineyard encompasses over 68 acres of vineyards that are home to a remarkable array of 40 different grape varieties: among them Cabernet Sauvignon, Cabernet Franc, Chambourcin, Sangiovese, Merlot, Pinot Noir, Syrah, Lemberger, Pinot Grigio, Viognier, Sauvignon Blanc, Riesling, Traminette, Steuben, Niagara, and Concord to name a few.\nThe grapes not used for Shade Mountain's purposes are then sold to other PA wineries! They also boast an entire line of fruit wines ranging in sweetness levels from semi-sweet to sweet. These wines are made from actual Pennsylvania-grown fruit, except for the Cranberry. Our fruit wines include Apple, Raspberry, Elderberry, Blackberry, Cherry, Strawberry, and Peach. Karl enjoys the challenge of growing grapes in the northeast, so he is always looking for new varieties to cultivate successfully. The grapes are his babies and he is particularly proud of controlling the whole process from vine to wine. He is quoted as saying, \"a great glass of wine starts with the vines\".\nThe winery is housed in a 19th Century converted bank barn and has grown to produce OVER 30,000 gallons of wine annually. In 2006 the barn underwent a renovation and added an event room with expansive decks overlooking the vineyard. Many events are held throughout the year including the Susquehanna Heartland Wine Trail March Madness Month every March, Shade Mountain's Annual Fall Festival (where you can actually stomp the grapes!) held every second weekend in October, birthdays, showers, and live music weekends. Shade Mountain Winery is a wonderful place to enjoy wine tasting and purchase unique local gifts. Carolyn is a food and wine-pairing expert and shares some of her recipes with us now and then.\nPlease be sure to make Shade Mountain Winery a stop on your next trip through Central Pennsylvania, as we promise you a warm family reception, beautiful and serene views, and most importantly, high quality local wine for all to enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5 Port Phantom design and performance in a .308 format. Includes Crush Washer. This 5 port design has no downward facing port which helps prevent dust problems when shooting in the prone position. Two models available, one featuring a \"no snag\" design that allows the user to navigate rough terrain without interference, while the other features aggressive endcuts.\nThis 5 port design has no downward facing port which helps eliminate dust problems when shooting in the prone position.\nThe 5C2 retains the aggressive front end for use in hand-to-hand situations.\nFor barrels threaded with 5\/8\"-24 only. Includes crush washer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Situated in the popular village of Haynes, this delightful and extended three bedroom semi-detached family residence benefits from a large double garage and two rear gardens - all in excellent school catchments!\nEntrance door to front, engineered oak flooring, radiator, stairs rising to first floor.\nSituated in the entrance hall, a walk in storage room ideal for coats and shoes, with vinyl flooring.\nA suite comprising of a low level WC, wash hand basin, tiling to splashbacks, ceramic tiled flooring, radiator, double glazed window to side.\n16' 10\" x 9' 10\" (5.13m x 3.00m) Ceiling to floor double glazed window to front, built-in electric fire, radiator, fitted carpet.\n17' 8\" x 8' 9\" (5.38m x 2.67m) A range of lime oak base and wall mounted units with work surfaces over , 1.5 basin sink and drainer with mixer taps over and a waste disposal unit,floor plinth heater, plumbing for dishwasher, area for cooker, built-in extractor hood, built-in fridge, tiling to splashbacks, double glazed window to rear, ceramic tiled flooring, breakfast bar.\n19' 1\" x 8' 6\" (5.82m x 2.59m) Dining area - Ceramic tiled flooring, radiator, double glazed door to side.\nFamily Room - Radiators, fitted carpet, wooden double glazed doors to rear.\nSpace and plumbing for washing machine, area for tumble dryer, wall mounted heater.\nDouble glazed window to side, access to loft with light and pull down ladder, airing cupboard with shelving.\n11' 3\" x max 10' 1\" (3.43m x 3.07m) Double glazed window to front, radiator, fitted carpet, ceiling to floor wardrobes with hanging space and shelving.\n11' 0\" x 8' 9\" (3.35m x 2.67m) Double glazed window to rear, fitted carpet, radiator.\n9' 11\" x 7' 7\" (3.02m x 2.31m) Double glazed window to rear, fitted carpet, radiator.\n8' 7\" x 7' 4\" (2.62m x 2.24m) A newly refitted suite comprising of a panelled bath with telephone handset shower attachment, separate shower cubicle with rainfall shower, tiling to splashbacks, vanity unit wash hand basin, low level WC, double glazed window to front, heated towel rail, vinyl flooring, built-in cupboard.\nMainly laid to lawn with driveway to the side.\nMainly laid to lawn, timber fencing to sides and rear, shed.\nMainly laid to lawn, timber fencing to sides and rear, wooden decking and pagoda, patio area.\n20' 09\" x 18' 3\" (6.32m x 5.56m) Two up and over doors, side door, double glazed window to side and rear, power and light.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabmsv b/data_all_eng_slimpj/shuffled/split2/finalzzzabmsv
new file mode 100644
index 0000000000000000000000000000000000000000..864b1c62d21a9f1b9410a7f5e2873e4de9b73000
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabmsv
@@ -0,0 +1,5 @@
+{"text":"Martha (Rogers) Stange is a lifelong resident of Ann Arbor. She graduated from AA Gabriel Richard HS where she earned All-State honors as a catcher and was part of the 1982 Class D State Championship Team. She earned a full-ride scholarship to play softball for the University of Michigan where she was a starting catcher\/infielder. She lives in Ann Arbor now with her husband, Brian and 3 children, Dominic, Hailey and Ethan. 2017 was her first year with the Skyline HS program and she works in the school as a Community Assistant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UC Natural Reserve System is a library of ecosystems throughout California. Most of the state's major habitat types are represented, from coastal tidepools to inland deserts, and lush wetlands to redwood forests. No other network of field sites can match its size, scope, and ecological diversity. The NRS offers outdoor laboratories to field scientists, classrooms without walls for students, and nature's inspiration to all.\nScientists from around the world conduct field research in the protected landscapes of the Natural Reserve System on topics ranging from climate change to endangered species to habitat restoration.\nVisiting a reserve brings textbook lessons about the natural world to life. Students and teachers flock to reserves from the University of California and institutions around the globe.\nFrom hosting school field trips, to organizing scientific lecture series, to welcoming the public for weekend classes, NRS reserves benefit California communities.\nAccess to NRS reserves is by permission only. Reserves are available for teaching, research, and public service, but not general recreation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Residents of The Spinal Unit Action Group (SUAG) on Weld Rd, Southport, were treated to a visit to The World of Glass in St Helens this month, by The Rotary Club of Southport Links.\nSUAG was established in 1973 to support the patients and ex-patients of the North West regional Spinal Injuries Centre in Southport. They provide and maintain a long and short term residential home at 6 Weld Road Southport and run various social events for patients and their visitors at the Spinal Injury Centre on a monthly basis.\nThe guests enjoyed their visit to this informative venue, on the site of Pilkington's Glass works, and are pictured just before entering the furnace, to see the start of the glass making process, with their 2 nurses and members of the Rotary Club.\nSouthport Links Rotary have a long association with SUAG, helping them build raised beds to help with their gardening and supplying a new TV for their lounge, in addition to days out such as this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us Sunday, April 28, 2019 at Comerica Park for the Third Annual I RAN THE D 5K, presented by FOX Sports Detroit!\nStart at the Big Tiger, run through the streets of this great city and finish ceremonially on the Comerica Park field! Registration includes a great race hoodie, bib and finishers medal.\nUse promo code ROAR5 to receive $5 off your registration. Plus, upgrade your registration and get TWO Mezzanine Level tickets to the Saturday, May 18 game at Comerica Park where the Tigers host the Oakland A's.\nHelp your kids stay active this spring by signing them up for the first ever Kids Marathon presented by FOX Sports Detroit and Kroger on Sunday, April 28 at Comerica Park. Have your child run\/walk 25.6 miles through their daily activities (walking to school, taking out the garbage, etc) and then join us at Comerica Park for the last .6 of a mile where they will ceremonially finish their marathon on the Comerica Park field!\nAll participants will receive a t-shirt and a medal and $1 from every registration will go to the Kroger Zero Hunger Zero Waste program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No picture this time, but steady progress on the modules as both are now completely assembled. Only a few reinforcing blocks need to be glued here and there to ensure the modules are braced enough to survive gravity and handling.\nIf everything goes as planned, I should be able to seal them with paint before the week end. My goal is still to add styrofoam, fascia and general landform during the weekend.\nIt would be great to start building the layout during weekend. I sure hope too. That would mean track laying could start during next week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacfbq b/data_all_eng_slimpj/shuffled/split2/finalzzzacfbq
new file mode 100644
index 0000000000000000000000000000000000000000..8d06c683031e7565d66c9c75c3f185a476a6e8cc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacfbq
@@ -0,0 +1,5 @@
+{"text":"Levent Gurses has an article, 10 Mistakes in Transitioning to Agile, in the Dec 2006 Dr. Dobbs Journal. He writes about the most common mistakes that companies make when transitioning from \"legacy development methodologies\" to agile ones. In this series of short articles for the winter holidays, we're looking at each of the ten mistakes he identified. Enjoy the light reading, and don't think too much about work.\nIf we don't identify the champion of the endeavor to convert to agile processes, we can't keep them informed of progress. Their expectations need to continually adapt to progress just like every element of agile.\nClose collaboration with stakeholders is critical to any product success. Converting an organization to agile is a change-management project. The sponsor of this change is the stakeholder for this project (not to be confused with the stakeholder of the project that is underway). Keep the stakeholder informed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The US Open, the fourth Grand Slam tennis tournament of the year, took place from August 28th to September 10th, 2017. This is the only Grand Slam tournament that takes place in the United States. People from all over the country and the world come together to watch and discuss this celebrated event.\nThe US Open was discussed frequently between September 1st and September 15th. On September 1st, there were 106.2K tweets. The graph reveals the chatter hit a peak of 296.8K tweets on September 10th, the day of the Men's championship. There was then a significant decrease in US Open buzz on Twitter, and by September 15th, the tweets dropped to 3.8K.\nBeing that this was the US Open, it is not surprising that the majority of tweets were composed by people in the United States. The Twitter buzz extended beyond the US; Argentina had the second largest number of US Open tweets, which can be explained regarding Argentinian semi-finalist, Del Potro. Other countries with significant tweet counts are Spain, the United Kingdom, and India.\nThe women's champion, \"Sloane Stephens,\" is the most commonly used word. In addition, the word cloud also emphasizes Rafael Nadal, men's champion. The most frequently used words circulate around the championship. Other important phrases mentioned are \"black women\" and \"american woman.\" This shows viewer's excitement around Sloane Stephen.\n#usopen has the strongest connection to @usopen, indicated by the bolded line connecting them. #usopen is the connector to all other words in the Buzz Graph. The Buzz Graph reflects the Word Cloud, as Sloane Stephen and Rafael Nadal are also mentioned.\nThe most retweeted tweet was composed by Michael Katz, social media editor for SB Nation. He received 9,379 retweets from his tweet on September 9th. This tweet's success is rooted in its comedic nature and relevance to the US Open. Sloane's reaction to the check was very relatable and genuine, helping the tweet gain recognition.\nThis is such an interesting post regarding the US open. All of the most used words or hashtags make sense, and it was really fascinating to see how global this topic was, even though it is titled the \"US\" open. Also, I'd actually seen Michael Katz's tweet on my feed, so it's interesting to see that it was the most engaged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The mobility solution for your company.\nFast and flexible from meeting to meeting? Fringe benefit for employees? Allow the customer a direct journey? Smide offers tailor-made mobility solutions for your company and positions you as an attractive employer.\nAccess to e-bikes throughout the city of Zurich via mobile app for the transfer of employees and customers.\nScheduled efforts and tailored offers - for SMEs, start-ups or public institutions.\nThe management of the e-bikes is done 7\/24 by smide; we have our own field ops and bike workshop.\nWith smide you make an important contribution to the health and satisfaction of your employees.\nDrive to the next appointment without traffic jams and position yourself as a sustainable company.\nIf you have any questions, our service team is always at your side - by phone, e-mail or social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Article focuses upon a persistent problem of the nonprofit sector--its lack of accountability to the public. Director, officer, and organizational responsibilities will be analyzed. Past and current approaches to secure accountability of charitable assets will be discussed, and a proposal for improving charitable accountability will be suggested through the creation of public-private charity commissions at the state level under the aegis of the attorney general.\nJames J. Fishman, Improving Charitable Accountability, 62 Md. L. Rev. 218, 222 (2003), http:\/\/digitalcommons.pace.edu\/lawfaculty\/66\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Blue Dragon has experience in advising regulators and clients in litigation and regulatory support matters.\nBlue Dragon operated as technical consultant for a HK regulator with staff embedded in the Regulator. We assisted with ediscovery and EDRM process advising at all stages of the regulatory process using the Relativity\u00ae software but also collection including EnCase\u00ae & Nuix\u00ae.\nBlue Dragon advised an Indian client on Ediscovery process and managed a large case in Relativity out of Hong Kong and London.\nBlue Dragon managed a employee fraud investigation including affidavits and discovered the forensic report supplied by a third party had been wrongly suggestive of guilt.\nBlue Dragon conducted forensic examination of a computer to discover and IT Director's role in a recent data breach at his firm.\nBlue Dragon were involved in a multi-site Taiwan and China data collection, processing and onsite review case using EnCase, Relativity, Nuix. We helped advise on the creation of an on-site review solution.\nWhen you collect data, you must consider the end goal. If it is a pure litigation, all parties are normally informed of the nature and purpose of the collection. If the situation is more in the nature of an internal investigation, the element of surprise can be important, as potentially some of the persons whose data is to be collected may have bad motivations. In this case, it is best to collect evidence widely even if the eventual analysis is done on only a small subset of the data.\nKeywords are a useful means of cutting down a data set, but many studies have demonstrated the erroneous nature of keywords, and how infrequently using keywords can direct all relevant materials into a subset. There are other means of culling data, including using file filters, date filters, NIST, boolean operators, technology assisted review tools, predictive coding based on iterative learning.\nIf there must be an examination of potentially legally privileged materials, this is time consuming and a strategy should be discussed early on for how to handle the review.\nThere are other sets of data short of privilege that might need to be considered. These might be confidential, subject to a non-disclosure agreement with a third party, or liable to cause some other issue (subjecting a third party to risk, or involving the disclosure of personal data). There may be an overrriding interest to disclosure of such information.\nHong Kong courts are not very familiar with a lot of the technologies to do with ediscovery. It is good to ensure someone senior on the vendor side has overseen the whole process so that person is qualified to write an affidavit on the matters concerned and also appear in court if need be as an expert witness.\nThe guidelines in the HK practice direction will help guide through many of the main questions parties should be considering, even if the practice direction does not apply to the case.\nEdiscovery is often driven by the need for what one English judge called \"rough justice\" as opposed to perfect justice. In our daily case work, we discover that not every issue can support the same level of data interrogation. For example, the departure of one employee who may be about to begin a competing business in breach of a restrictive covenant is not going to attract the same depth of investigation, without good reason, that the investigation of a major contractual dispute with liability in the millions or billions of dollars. Hence, we adapt our collection and filtering techniques to the size of the case and the issues at hand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacnnc b/data_all_eng_slimpj/shuffled/split2/finalzzzacnnc
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index 0000000000000000000000000000000000000000..80d5049d573e0b079a718f08c9bfbe54abbe56bb
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@@ -0,0 +1,5 @@
+{"text":"The Archive contains a huge number of table of contents and abstracts (tocs and abs) from various sources: self-produced indexes, specific indexing projects and Current Contents archive.\nBibliographic information of indexes varies according to the source. Self-produced indexing projects are UNIBO and ANTICO. Specific projects are COVET, promoted by veterinary libraries; ESSPER, an indexing project promoted by the University of Castellanza in the fields of Economics, Social Sciences and History; SBN-RM which contains the indexes of La Sapienza University of Rome, extracted from the SBN database; ICR for the journal indexes automatically \"read\" by an ICR (Intelligent Scanner Recognition) scanner.\nIndexing from the ISI source is by far the most numerous (more than 8.700) and refers to the Current Contents archive. The access to these articles is reserved to the users of Bologna University and are updated until 2002.\nThe name of the author\/s of the article.\nA text containing a short summary of the article. This information is included almost exclusiverly in the indexes of Current Contents and is usually written in English.\nThe ISSN code of the journal. ISSN is an acronym of International Standard Serial Number) which identifies uniquely serials. It is an 8-digit code and can be searched with or without hyphen between the two 4-digit groups.\nIt is possible to limit searching to one source of documents of the Archive by selecting only one code in the dropdown menu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am extremely picky when it comes to exotic handbags. I've seen so many gorgeous bags yet, at the same time, I've seen so many horrible bags. You really cannot skimp on attention to detail when it involves exotic skins. As I am sure many of you already know, sloppy craftsmanship results in flaking scales which is perhaps one of the biggest handbag no-nos out there. Luckily there are several design houses who continue to hit home run after home run with exotics.\nCarlos Falchi graffiti-print bags now available!\nA few months ago, we brought you news that friend-of-PurseBlog Carlos Falchi would be collaborating with his daughter on a new line of graffiti-inspired bags to debut in the coming months.\nOne of the most interesting men in the handbag industry we have met will be celebrating 30 years starting tomorrow. Carlos Falchi began his journey 30 years ago when he moved to NYC. Since then his designs have continued to be top contenders in the designer handbag world, finding their way into the arms of celebrities and handbag lovers alike.\nTomorrow at Bergdorf Goodman a retrospective exhibition will feature reproductions of Carlos Falchi's most beloved classics and his latest collection including one-of-a-kind designs inspired by his love of New York. The event will take place on the first floor of Bergdorf Goodman and go from 1 p.m. to 4 p.m.. We have been huge fans of Carlos Falchi handbags and while we can not make the event tomorrow, we are excited to see the one-of-a-kind bags for Bergdorf along with the line of bags dedicated to New York City that he comes up with!\nIf you are in the NYC area this is an event you won't want to miss. Carlos Falchi is one of the most interesting people to speak with, tell him we say hi!\nI recently reveled in the beauty of the hand painted Carlos Falchi Floral Sling. And it was not just me who loved the bag, many of you did. But then a question popped up; \"Will the paint run? Is the bag too delicate?\". I asked over at Carlos Falchi and they said to assure you all that no, the paint will not run as it is glazed over. And no, the bag is not too delicate to use, they use the stomach of the snake which is very resilient.\nNow that we have covered that, there is a perfect option for those of you who loved the idea of the hand-painted floral pattern but prefer it in a smaller bag. The Carlos Falchi Flower Flat Clutch is also designed with hand-painted flowers and has an optional chain shoulder strap. This is a thin clutch, but long and capable of holding night out necessities (12\u2033W x 5.75\u2033H x 1\u2033D). The interior features a zip pocket. Again, this is a brilliant design from Carlos Falchi. A simple sleek python clutch with a burst of color, which is hand-painted. This makes the bag entirely different from any other in my collection and in most collections. Leave it to the genius, Carlos Falchi. Buy through Bloomingdale's for $965.\nThis clutch is kind of growing on me. Originally I wasn't so sure I liked it. It seemed kind of arts-and-crafts for my taste. However, as I gave it more of a chance, I found myself starting to like it. The Carlos Falchi Tiger Snake Flat Clutch is unique, chic and dare I say one of those kind of handbags you would consider one of a kind. A little while back Megs and Vlad had the chance to interview Carlos Falchi and we all learned how unique his handbags are and how much creativity goes into them. This handbag, without a doubt, meets that criteria. The handbag is made of genuine tiger snake and is hand painted to provide an extra element of creativity and detail. One of the best parts about this bag is that it is sure to garner attention. You are sure to hear at least a few comments as you carry it around. Buy through Bloomingdales for $595.\nFirst of all, let me start off my saying, Happy Friday! I am sure there are plenty of you out there right now counting down the hours, or maybe the minutes until it is officially the weekend. With the weekend drawing near, I thought to myself, what better way to get the weekend going than with this Bombe Python Flat Clutch? I am a sucker for clutches; I honestly just love them. So, when I stumbled across this clutch, I had to share it with all of you! Obviously, the color of the clutch, purple, caught my eye first. I love sporting a clutch that completely makes my outfit pop and I feel this clutch could definitely do that. Whether you are wearing the perfect little black dress or your favorite pair of jeans and a cool top, this clutch could complete your outfit!\nNow that I think about it, I have spent many nights recently glued to my couch watching the 2008 Olympics (I am pretty sure many of you have done the same thing). Usually around mid-week I begin thinking about weekend plans, and of course, what I might wear and which bag would go with the outfit best. So, I think it is time for me to have TiVo record tonight's Olympic events and go out on the town. Now, if I could just have this clutch to take with me, I would be that much happier. The snakeskin material, magnetic snap flap closure and zip inside pocket just make me want the clutch even more. Plus, I like options and this clutch has a golden chain strap which can be stored in the bag. I rarely find myself using the strap, but it's nice to have options. All I need to do is save a bit of cash and I can buy this fabulous clutch through Saks for $875.00.\nCarlos Falchi has been a long time favorite of mine, as his exotic handbags have caught my eye and always seem to end up in the more reasonable price range (as far as exotics go). Carlos Falchi is not the new kid on the block, in fact he has spent three decades forging his way into the fashion world. He prides his designs in the meticulous details and craftsmanship, along with exotic skins and an impeccable eye for a handbag that stands out. If you rely on your accessories to make your outfit, Carlos Falchi handbags are perfect for you. I have never been much of the flamboyant dresser, in fact, if you ever find me out and about there is a 95% chance that I am in jeans of some sort and a tee shirt. My handbags truly accentuate what I wear. The bold colors, skins, and designs make me and my outfit stand out, and not get lost in the crowd.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Visual Asia Expo, now in its second year, once again surprised the visitors to the show at Suntec. It is Singapore's only visual communications trade show. The title enables the organisers to host a variety of exhibitors. While there were a number of printing companies and related technology firms showing their latest developments, this year it seemed the lighting and display boys stole the show.\nLED walls, facade lighting, visual effects, retail lighting and virtual reality excited the visitors, trade or casual passerby's brightening the whole hall.\nThere were many innovative ideas on display at Visual Asia Expo this year and the exhibitor ranged from giants like LG Electronics to smaller outfits like Kult which handles illustrators. The Society of Interior Designers, Print & Media Association and Design Business Chamber of Singapore all had booths as did leading photographic company, Shooting Gallery Asia and paper company Antalis.\nLeft to right: Emcee Colin Seet looks on as Andrew Pang, the keynote speaker, receives a token of appreciation from Thomas Ang, founder and organiser of Visual Media Expo.\nAn added attraction to the show are the industry speakers. The keynote address in 2016 was by Andrew Pang, the President-elect of the Design Business Chamber. He spoke on 'Design for Good'. He illustrated his presentation with examples of companies, organisations and even individuals who devoted some of their time and money to promoting ideas that improve the lives of people. He urged the audience to discover their 'higher calling' and use their talents to help others. Lawrence Chong of Consulus chose the topic of 'Reinventing Design to meet the Challenges of Industry 4.0.' He spoke of his own journey of moving from basic graphic design to becoming immersed with the total communications journey. He stressed the importance of embedding the design team in the production from the start.\nOther speakers over the two days included Rick Yeo of SimTech, Willy Foo of LiveTechnologies & LiveStudios and Yoke Yuin Cheong from IHS Technology.\nThe show, which ran from 2nd to 3rd of November, also offered workshops on retail lighting, Virtual and Augmented Realty, Print Ideas and Solutions, Signage Developments and Projection Mapping.\nVisual Asia Expo surprised the visitor with a smorgasbord of industries. It presented a fascinating selection of exhibitors which is so different from the often repetitive offering at other shows.\nWe hope this show will continue to grow in future years filling a unique but wide sector of the industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Advance Group is hiring! We are looking for a well-rounded and experienced welding candidate for an immediate opening in Swanton.\nIdeal welding candidates will take pride in their work, understand and follow all processes, and have a positive attitude.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CATERPILLAR\u00ae HAS ALWAYS BEEN ABOUT MORE THAN HEAVY EQUIPMENT. WHETHER THE MOST INTENSE JOB, EXTREME CLIMATE OR DEMANDING TASK, CAT\u00ae EYEWEAR OFFERS FUNCTION, PERFORMANCE AND DURABILITY WHEN YOU NEED IT MOST. MAKE COMFORT AND SAFETY YOUR FIRST PRIORITY; EQUIP YOURSELF IN CAT EYEWEAR.\nCATERPILLAR\u00ae EYEWEAR THE COMPLETE VISION SOLUTION.\nThanks for submitting your review of Cat Safety Eyewear \/ Inspecs USA. We'll email you as soon as it's published, (typically within 48 hours). We really value your contributions, and so does our community of business professionals. They count on honest reviews like yours.\nThanks for submitting your review of Cat Safety Eyewear \/ Inspecs USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacxoi b/data_all_eng_slimpj/shuffled/split2/finalzzzacxoi
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@@ -0,0 +1,5 @@
+{"text":"On January 30th, the 19th annual Country for a Cure fundraising event took place at the Freeland Event Center aimed at helping local families who have been impacted by cancer.\nAccording to event founder Tami Martin who started Country for a Cure along side her husband, the late Freeland Mayor, Tim Martin, organizers were hoping to increase the number of families who were assisted by the previous year's event which resulted in a total of five families.\nBefore Tim's passing, Tammi vowed to continue on with the annual event until the day a cure for cancer is found. In support of the efforts of all those involved with Country for a Cure, Bill Rinaldi has donated $1000 this important cause which strives to put an end to a tragic disease which has affected millions of people and their families.\nThanks to everyone's combined efforts, Country for a Cure raised enough money to help 7 or 8 families, improving last year's results and inspiring many to fight for all those who are suffering.\nTo find out more information about Country for a Cure and what you can do to become a valuable part of their efforts, read the entire Stand Speaker article here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Right on the Nail: Color Club Poptastic Collection: Pink Lust- \"Poptastic\"\nColor Club Poptastic Collection: Pink Lust- \"Poptastic\"\nJust like yesterday-WHOA! Can I say neon pink?! I love this. It is probably my favorite neon pink. I think this one would look really great from spring through summer (of course I am all for wearing neons during the fall and winter, too). Kudos, Color Club. Application was slightly thin and needed 3 coats, but I had no problems otherwise.\nWoW! So far that one is my Favorite!\nWOWzers! These are awwwesome. I am full of awe. Definitely a toe nail color for me!\nIsn't it great?! Come on Ashley, live dangerously and put it on the finger nails for everyone to see!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just after the success of the Mainland INVASION with Unique held in Lagos. Unique drops a mouth dropping freestyle to breach the wait for the next big thing you all are to watch out for. With Unique the microphone rocker you can be sure of no low moment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New York Times: \"With Move Across London, U.S. Embassy Can't Please Everyone\"\nThe US Embassy in London is moving locations, and not everyone is happy about it. After years of criticism and protests by local residents against the current Embassy building in Grosvenor Square because of safety and security concerns\u2014the protests included a hunger strike by a countess\u2014the US Embassy is moving from its Modernist concrete building in beautiful, historic, and exclusive Mayfair, where the Embassy has been based since 1960, to a more protected and environmentally responsible building in the gritty district of Nine Elms on the South Bank of the Thames. While the move planned for 2017 is welcomed by local Mayfair residents who for years have feared terrorist attacks, the new location also has its own critics.\nThe new building was designed by Philadelphia firm KieranTimberlake to reflect \"the core values of democracy\u2014transparency, openness, and equality\" and also to be \"welcoming, secure, and highly sustainable.\" The design, however, has been called \"boring,\" a \"corporate office block,\" and \"the Ice Cube.\" Former Guardian architecture critic Jonathan Glancey said that the proposed building is \"remote and superficially transparent\" and that it reflects \"what we can divine of the US political process. Nominally open to all and yet, in practice, tightly controlled[.]\"\nThe project as a whole...is a fascinating study in how architecture can be used as a form of camouflage. The building is set in a spiraling pattern of two small meadows and a pond that have as much to do with defensive fortification as with pastoral serenity: an eye-opening expression of the irresolvable tensions involved in trying to design an emblem of American values when you know it may become the next terrorist target.\nNo word if Gould Pharmacy, which rents lockers for applicants who cannot bring their large electronic items into the Embassy, will also open a new location. It might be finally time to leave those large electronics at home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey Olivia, It was great to listen to your talk about your work at What3Words, so it's essentially 'a simple way to talk about location' you've divided the whole world into 3m x 3m squares and assigned each one an individual 3 word combination. Can you tell us a little bit more about what exactly it is that what3words is striving to do?\nRight, so what we're trying to do is get people to adopt this pretty wacky system of using three word addresses. Everyone loves the idea and understands the simplicity of it, but getting people to adopt it and start using it takes a bit of open mindedness to change something you've been doing your whole life in terms of addressing.\nBroadly speaking we're a social enterprise so whilst we strive to improve efficiency for businesses we are also working hard to help save lives! We also like to improve consumer experience, whether people are travelling and can't read say, mandarin road signs you can use our app to travel in your own language by setting the location you want to go to.\nHow did you end up working for what3words?\nI'd ironically spent the the last phase of my career at Ogilvy looking into big digital disruptors like the Ubers and the Airbnbs and I tried to understand what role marketing played in getting them the success they'd seen. I became really intrigued by them and I was really keen to join one myself.\nWhat are a few of your favourite examples of the impact you guys have had on people's lives?\nIt's absolutely crazy how much amazing stuff people are able to do as a result of what we're doing. It can vary from something like healthcare to the Domino's franchise owner in St Martin reaching out to us and using our app instead of the hand drawn maps he was using before. When we show someone the efficiency of what3words it really does open the floodgates of possibilities of what can be achieved around the world.\nOn the humanitarian side, we have a great partner in South Africa called Gateway Health, there's a guy who has been providing medical care for people in the townships in South Africa for many years. He's told us all these stories of things he's tried to sort out with location data, he's tried cell phone triangulation and all sorts but he hasn't managed to get it right. Now, he can use what3words to locate patients and he's also able to launch a lot of programs he hasn't managed to launch before. One is focused on maternal health, 50% of births are at home in that area, and because it is such a time critical moment, an ambulance needs to be able to find the home and they never had any way of describing their homes but now they can.\nWhat3words have grown quite significantly since your first talk at glug Reading last year, how have you as a marketing helped to accelerate this growth?\nMarketing plays a huge role in helping to drive that change. We do it a lot through our partnerships, we do a bit of marketing so key businesses are aware of us and why we are relevant to their businesses.\nFor example, once a logistics company realises the efficiency adopting three word addresses it incentivises them to ensure all their customers are also using it. What we do then is partner with them further to help them in that communication and toolkit these elements, whether that's training, marketing toolkits like stickers, leaflets or posters. Now we're starting to explore some bigger and more exciting ideas now like organising scavenger hunts round cities with a travel company to find the best spots to take a picture of a sunset or Tower Bridge or whatever it is.\nYou seem to have a very diverse customer base, how do you manage to address each one of these.\nIt's been tricky to balance our relevance to the range of industries we support but also to be pragmatic and use our time wisely so we're not recreating everything every time we talk to a new partner. That's where I've brought in our strategic view, we're really trying to think about how we scale on verticals but also across the world.\nWe've been working collaboratively with video production companies to create high impact videos to use as marketing material in each of our verticals, using the same structure but with different content for our business development team can use. As we get more case studies and results then we can create more interview-focused videos.\nWe've also been trialling some marketplace sites where you can upload a brief and you can commission work out to people globally. We currently have a people in Barcelona, Germany, Thailand and Greece shooting material for us! We then use this footage and edit it all together into high impact videos!\nWhat do you think the future holds for this industry?\nEverything is moving to delivery on demand, exactly where you want it at that moment. Which is even more reason to assign an address to everywhere.\nWe're having some really interesting discussions around this point at the moment. We're speaking to a courier company in California that actually does on demand delivery services but also cover medical care - sometimes they're transporting organs (which is hugely time sensitive) to a huge hospital it can be difficult to know which entrance to go to. What3words offers a specific address to different entry points of a huge hospital, making the delivery much more time efficient.\nOn the other end of the spectrum it's pizza hut in Australia delivering a pizza to the middle of a field on the university campus and now you can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadobh b/data_all_eng_slimpj/shuffled/split2/finalzzzadobh
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index 0000000000000000000000000000000000000000..e5179caa2557d6c01722bb2dc57028efd50c6315
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@@ -0,0 +1,5 @@
+{"text":"Misa is a collection designed for global getaways. The Nikola pant is a high-waisted pant with that perfect cropped leg. We love the combo of this style being a refined stripe, but made up in an breezy fabric with an easy pull-on fit. With pockets and an elastic back waist, this pant is sure to become your fave. It styles as well with a tucked in tee as it does with a bodysuit. And if you love a set like we do, style it with the Leona top in the same stripe!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is the edible fruit which is part of the rose family. It is very rarely found in cold places because of its growing. It is quite energetic as it contains vitamins which are required by a body.\nWow! i like this raspberry.\nWow! It Looking so delicious ripe raspberries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We take great pride in the levels of service provided to our clients and also the speed of response. Our claims service is truly available on a 24\/7\/365 basis.\nWorking closely with the placing team, our integrated practice allows our claims team to get to know our clients before a casualty occurs. We do this by visiting our clients on a regular basis, to better understand their requirements. We make it our practice to put in place clearly defined Claims Procedures in the event of a loss so it is clear to everyone involved what is expected of them.\nUnlike many other brokers, where claims are farmed out to provincial\/overseas offices or even outsourced, claims are handled within our London office, providing ready access to insurers and placing divisions.\nDue to the immense variety of claims received from clients around the world, the team has dedicated marine, aviation and non-marine claims specialists who are experienced in their fields.\nWe pride ourselves on providing all our clients with a truly first class claims service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CONTRADICTIONS in IPC's comment about IOD\/.web application!\nSubject: CONTRADICTIONS in IPC's comment about IOD\/.web application!\nin connection with the operation of the TLD\"\nsee links at the bottom of this post).\nSOMETHING DIFFERENT: it is NOT the CRITERIA established by ICANN!!!\nSo, this IPC's comment is OFF-TOPIC!\npress release and long before creation of ICANN itself!!!\nand it is NOT a CRITERION by ICANN).\nbecause IANA planned that IOD's .web registry would go online soon.\nrespect these contracts (registrations taken since 1996).\nwaste of time, the so-called Sunrise period, that is NOT compulsory.\nmade with a long and boring Sunrise period.\n(abuses may happen in this case also!).\n\"usa.web\" or \"marketing.web\" or \"business.web\".\nInstead of giving them to random rushers on day one.\nsince 1996 (disclaimer: I am NOT one of them).\nAgain, this is another MERIT that is considered a fault!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AAST built 3 student hotels within the Abu Qir campus.All the hotels provide 4 stars accommodation along with world class restaurants. Single and double rooms are available. A separate hotel for girls was built on the campus as well. All the hotels are built to the highest standards, and are maintained by professional staff.\nBanque du Caire is located inside Abu Qir campus to facilitate bank transactions for the students as well as the employees. It offers full banking services such as money transfers for foreign students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafuoa b/data_all_eng_slimpj/shuffled/split2/finalzzzafuoa
new file mode 100644
index 0000000000000000000000000000000000000000..4d537a9ee4261a6efc3aef060de62223039f75b4
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+{"text":"Fans weren't the only ones shaking their heads at Lauren Rimmer for being eliminated on Wednesday's episode of Survivor with (half) an idol and an advantage in her pocket.\n\"[My sister has] cussed me out two or three times,\" Lauren told ET over the phone on Thursday. \"She's called me a dummy and a few other things.\"\nThe North Carolina native's sister, Sunny, had been her No. 1 supporter throughout the game, and even dragged her to the open casting call that led to her joining the game. While Sunny didn't make the cut, Lauren promised that she would \"share every minute that I could with her, and that I would do my best to get her to the family visit.\" Lauren made good on that promise, and Sunny was flown out to Fiji to reunite with her sister just in time for her torch to be snuffed.\nIn a chaotic tribal council, Mike threw half of the idol that Lauren had given him in good faith into the fire -- leaving her vulnerable against her former ally, Ben, who whipped out an idol to save himself.\n\"I was really just lost for words,\" she explained. \"I would have never guessed in a million years that Mike would have done that. And I've asked him, like, 'Why did you do that?' He's like, 'I don't know. I just wanted to. I always wanted to.'\"\n\"When I gave it to him, I was trying to keep him close. I wanted him to feel like he was included, and I was trying to secure that vote, along with Ashley and Devon, and obviously he had better plans for it,\" she continued, adding that it \"never crossed my mind\" to try to salvage the idol in the fire. \"But who's to say if Jeff would have even let me play it?... But it would have been worth a try if it had crossed my mind.\"\nHer extra vote advantage, however, was safely in her pocket, but after telling the entire tribe that she left it back at camp, Lauren thought it would be best not to use it. \"Nobody knew that Ben had an idol. I was trying really hard not to use it until I had to... and I was almost trying to reassure Chrissy, like, 'Stop, we can pull this together,'\" she said.\nSo, Lauren and her advantages were sent packing -- though she doesn't blame them for putting a target on her back.\n\"I think I went too soon gunning for Ben. We had a lot of trust between us, but at the same time, I knew in my mind, Ben's not someone that I can sit beside in the 1, 2, 3 seat, and I'm assuming he thought the same,\" she shared. \"I think I would have been fine if it wasn't for Ben. I think I at least would have made it another night, because Devon, myself and Ashley, we were working really well together.\"\nLauren, who is still rooting for Ben to win alongside her tried and true allies, Devon and Ashley, is already planning her trip back to the island, with her sister, Sunny, of course.\n\"It would be quite comical,\" she said of her and Sunny returning for a season of Blood vs. Water. \"You know, as soon as we get off the plane, she'll probably be like, 'OK, bye b**ch. I'm gone.' I can see her doing that, you know? I'd be like, 'Hey, we're supposed to play together for just a little while.'\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 Tenders: 18727464 Providing Of Ready Mix Concrete Rmc For 1X660 Mw, Unit 5, Sagardighi Tps Project, Wbpdcl, West Bengal. .\n2 Tenders: 18724968 Annual Repair And Maintenance Of To Jpnatc Sh: Supplying Of Day To Day Maintenance Items In Trauma Centre. At Aiims. .\n3 Tenders: 18723944 Enquiry For Supply Of Constructional Stores. .\n4 Tenders: 18723201 Murrum Blanketing Works At 2X660mw Ennore Sez Stpp (On The Ash Dyke Of Ncpts). .\n5 Tenders: 18722540 Supply Of Metalic Compressed Asbestos Fibre Jointing Sheet Detais Refer Tender Annexure Sugar And Allied Products.\n6 Tenders: 18722514 Supply Of Steel Civil Construction.\n7 Tenders: 18722513 Supply Of Materials Civil Construction.\n8 Tenders: 18722511 Supply Of Coarse Sand And Stone Grit, Stone Ballast And Bricks At U\/C Rird Dasna, Distt. Ghaziabad Civil Construction.\n9 Tenders: 18722502 Supply Of Material At 5 Seated Toilet Work In Judicial Campus Ballia Civil Construction.\n10 Tenders: 18722492 Supply Of Grit At Departmental Hot Mix Plant Khorabar And Renewal Work Of Sonbarsa Sardar Nagar Futahwa Enar Road Length 3.75 Km Civil Works.\nTenders information of Cement and Asbestos Products Tenders in India and Indian tenders for Cement and Asbestos Products , Tender document For Cement and Asbestos Products , Cement and Asbestos Products tenders, Cement and Asbestos Products Tender information, Live Tender for Cement and Asbestos Products , Closed Tender for Cement and Asbestos Products , Free Tender for Cement and Asbestos Products , Free Full Tender Details for Cement and Asbestos Products . Live Chat Available for tenders information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hackaday alum [Ian Lesnet] has been working in cahoots with a dedicated team of developers to produce the OpenBench Logic Sniffer. This caseless logic analyzer can operate at 100MHz and sample 32 channels at once. Better yet, a digital oscilloscope add-on is in the works. The pre-order comes in at $45, that's a lot of functionality for just a few greenbacks. We've embedded a demo video after the break that details installing and using this device under Ubuntu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"4166k followers 551 following 262 posts see instagram photos and videos from kareem hunt 3 bigreem3, shannon sharpe checks female reporter for advocating kareem hunt receive a lifetime ban from the nfl duration 905 theadviseshowtv 896799 views, running back kareem hunt von den cleveland browns wurde von der nfl f\u00fcr acht spiele gesperrt hunt akzeptierte die sperre der 23j\u00e4hrige wird somit die erste h\u00e4lfte der kommenden saison verpassen, kareem hunt 2017 nfl draft profile including player stats videos combine results and expert analysis. You can also save the inspirational image kareem hunt on the website swizzlesteve.com by clicking on the image, then selecting download by size:. Next will open the image link that you want, then please right-click select save as to save it.\nRelated Posts of \"Kareem Hunt\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you're thinking about starting a company, revamping your current business, adding a new product line, or just making sure your current SEO approach is on the right track, keyword research is a fundamental part of the process.\nThe reason keyword research is so indispensable is that it's one of the easiest ways to get insight into what your target audience cares about and what language they use to talk about it. Skipping this step means you risk wasting time writing and optimizing for things that are totally irrelevant to your audience. You could spend all the time and effort in the world writing amazing content for your site but if you don't use the language your audience does, they're never going to find you. Keyword research is a way to learn about your audience, what they care about, and how they think, so you can align it with the content on your site.\nThe purpose of this post is to take some of the ambiguity and mystery away from what keyword research is (and isn't) about. After you've read this article you should be able to get started doing keyword research for your site and approach the process as a way to create an alignment between the way you talk about your business and the way your customers talk about it. I'm going to cover a lot of the basics of keyword research in this article, but if you want a really detailed, in depth refresher on the fundamentals, this article from Moz does a really good job of breaking down keyword research into its most basic components. When you're ready for some heavier reading, check out this article from Portent alum Marianne Sweeney about the history of advanced keyword research.\nFigure out where you're at today; identify terms you are ranking well for already.\nGroup your keywords by topic and pick a few head terms to prioritize.\nDecide which pages work best with the keywords you've chosen.\nWrite to benefit your audience, always.\nUsually the top few queries will be your brand name or variations of it. In our unique case, Portent has made some really cool tools that are getting us a lot of clicks, but again, usually the top few queries will be your brand name or variations of it. I like to look at Clicks, Click Through Rate, and Position as my metrics or criteria. From this example you can see that people searching for \"idea generator\" have clicked on our little blue link about 1,302 times in the past month. Meaning about 42% of all the people searching for idea generators in the past month thought our page would solve their problem more effectively than someone else's, assuming they didn't have to go back and search again. (By the way, this is a pretty awesome content ideation tool, which you should totally check out, after you finish this post\u2013and your keyword research of course).\nOf course, being in first place helps quite a bit, but CTR can still be useful as an indicator of how well your content matches up with that term and the user's question. If your position is relatively low but CTR is still high, it's an indication that people weren't finding a better answer to their question further up on the page. With some tweaking, these queries could be an opportunity to improve rankings and visibility since you're answering the question better than sites in positions ahead of you.\nAfter I've downloaded my list of keywords from Search Console, I like to combine it with data from another keyword research tool called SEM Rush. This is a paid tool that will set you back about $70 a month, but it's well worth it. The \"top organic keywords\" report is really valuable because it shows you exactly which pages on your site are ranking for which keywords. Combining this with click through rate from search console tells you exactly which pages are performing well for which terms.\nOnce you have this list, you can use AdWords keyword planner to flesh it out with related terms that you might not be ranking for (yet). You can start by using a page on your site about a topic or a competitor's site, or even a Wikipedia page, and get suggestions for high volume search terms that you don't yet target.\nSo now you have a huge list of key phrases you rank for, and another set that you'd like to work toward. The next step is to break down the keywords list by topic. From the list you've compiled there are usually a few distinct topics and subtopics that emerge.\nTo start, group these keywords by topic and then choose a few head terms that you want to prioritize for each topic. An example of a head term: think \"running shoes\" instead of the more long-tail \"running shoes for people with high arches.\" The number of keywords you should choose will depend on your resources, how large or authoritative your site is, etc. For example, if you have only one page to target a particular topic, then five terms is likely more than enough. If you have one category page and several subcategory pages to work with, broadening your focus to 10-15 terms would be just fine.\nFrom our research we know this is a term that currently drives traffic to this page (or we would like it to).\nThis is the language our audience is using to talk about our product.\nThe content on this page does a good job addressing this topic.\nNow that you have the topics you want to cover, you need to figure out which of the pages on your site are best suited to address those topics. This is one of the trickiest parts about keyword research. Not only is it important to choose the right topics to hit, it's also important to choose the right pages to cover those topics.\nStick to one or two closely related topics per page. Remember that can still mean you target multiple keywords per page that fall under one topic. It's much easier to optimize a page for one topic than for three. If you do have a page on your site that's dedicated to several topics, consider breaking it down into a few subpages, and using the current page as the hub or category page. You can read more on the benefits of hub pages in this article.\nHow well is each page performing right now? How well are they ranking for the keywords you've chosen?\nHow well does the page actually address the topic or answer the user's question?\nFor the second bullet, different types of queries work best with different types of content. People using informational-type queries like \"Where is the best ____,\" \"How much does ____ cost\" are looking to find content that is specific, informational and helps them answer a question, whereas people using more general queries like \"What is ___\" are likely looking for longer content explaining a topic broadly. And then of course people searching for a specific product name may be looking to buy, or at least compare reviews and prices.\nWith that in mind, ask honestly: what is the current purpose of the page I'm planning to use? For instance, blog posts and category pages are great for answering those broad questions and providing information for users who are gathering information about your industry, type of product, etc. We'll save the more promotional answers for evergreen educational pages, product comparison pages, and product pages themselves.\nI can't stress this enough: do not get too fixated on a single high volume term or terms. Include both competitive and less competitive terms in your strategy knowing that the more competitive a term is, the more work you're going to have to do to earn a spot on page one. Getting combined traffic from several less competitive terms can add up to the same amount of traffic and conversions or even more. Most importantly, it's doable, with much less effort and stress on your part. On top of that, building authority around terms closely related to a hyper-competitive term will help improve position for that term itself.\nThe most valuable takeaway from successful keyword research should be a better understanding of what your audience's needs are and how your website does or doesn't meet those needs. Your goal should always be to create content that's interesting and helpful to your target audience. Keyword research makes this process more actionable, informed, and rooted in actual data.\nVery helpful article, thanks for putting it together!\nEveryone needs to click the URL.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"One thing we love at here Miracle Camp is seeing the way God impacts the lives of campers. Every person has such a unique story to tell, and this is no different when it comes to Izzy and Zoey.\nHaving come from a rough background, Izzy and Zoey accepted Christ two years ago through attending a kids club in their neighborhood Bible study. They have been coming to Miracle Camp as campers ever since then, as their grandpa was one of the founders of camp. Last year's camp experience is one that Izzy and Zoey will always remember. While having personal devotional time, the girls were able to share their testimonies with another girl in their cabin who had the same type of story. Zoey and Izzy's testimony tells a beautiful story of how the girls came to realize this truth: fullness of life only comes from God and is far better than the emptiness that comes from trying to face life alone.\nDuring Senior High 1 chapel sessions, our speaker Jesse Kahler walked campers through different methods to study the Bible. As Zoey and Izzy go back home after camp, they have the desire to develop a more routine devotional time. \"We don't just want to limit our devos, but spend as much time as we can be studying Scripture,\" Izzy and Zoey agree.\nZoey explained that camp has taught her how to worship God and be in community with other believers. \"I love how different we all are and how no matter where we are from, we all worship the same God and can worship together,\" Izzy added.\nPlease join us as we continue to pray for Izzy and Zoey's growth in the knowledge of God's faithfulness in their lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This historical fiction, upper middle grade chapter book is for readers aged 12-14. The main characters in the story are based upon the lives of real people. However, this book is not a complete representation of their lives or the events that occurred.\nSet in an orphanage in 1962 just before the Cuban Missile Crisis, this is the story of how friendship saves two culturally dissimilar 12-year-old boys who are tragically disconnected from their families. This is the first story written for children readers with a main character who was part of the Pedro Pan mission, the largest political exodus of children ever recorded in the Western Hemisphere.\nDanny suddenly finds himself a half-orphan after his mother dies, his father succumbs to alcoholism, and he is handed over to an abusive family friend. Danny runs away and lands in the Holy Family Orphan's Home in Marquette, MI. This orphanage is the foster home for 30 boys who are part of the Pedro Pan mission, which brought over 14,000 children as exiles from Cuba during the first tumultuous years of Fidel Castro's communist regime.\nDanny and Emilio, a Cuban exile, come together through Father Timothy, the monsignor in charge of the orphanage. Because Cubans primarily live at the orphanage, their food, music, and emotions permeate the environment. Danny enters a milieu very different from his experience in a small Midwestern town. Outside the orphanage, Emilio faces discrimination, language barriers, and living conditions vastly different from his former life experience.\nThe boys find common ground through their mutual desire to return to their old lives and their interest in baseball. However, when an older Cuban boy bullies Danny, Emilio must choose his alliance and the clash between cultures becomes clear.\nOutside influences and abandonment wounds threaten their tentative friendship. But, when they accept that their old lives are gone forever and recognize the value of their friendship, they forge an unbreakable bond -- and find hope in their future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I wanted to take a moment to give many thanks to you for your outstanding, intelligent, sensitive, restorative care as a massage therapist and as a relationship consultant. When I took my mom to see you, after the massage she said \"Patty is reason enough to move out here\".\nIt was really helpful working with you on the relationship with my mom. In just a few short minutes, you helped smooth out years of deep trouble in a very painless and easy way. I can not tell you how grateful I am. Whenever we veer off track (which is bound to happen in any relationship that has had years of the worst sort of troubles), I remember your words of guidance and things run smoothly again.\nThank you thank you thank you. We are so incredibly blessed to have a professional of your caliber in the Valley. Many people would have to drive a thousand miles to find a healer as unique, caring and effective as you.\nI was in SO MUCH PAIN! And you quietly said \"I can help you\". I didn't believe that - silly me!\nI was in such distress and can remember it soooo clearly. I also remember with wonderment (AMAZEMENT!) how just a few minutes your safe, gentle hands erased the pain. Gone, Nada!\nOh and the relief afterwards. Followed quickly by the much needed sleep.\nI will never forget the massage and energy work you did on me.\nPatty, darling, you are a godsend. As of four p.m. yesterday, I could actually move my neck all the way around with out pain! I could even imitate Linda Blair in the Exorcist :-)Many, many thanks you's, P.\nMany Hands Magazine Hi Patty,I just wanted to thank you again for the wonderful massage friday nite!I have had so much relief in my sacrum\/tailbone area. it still feels so much better, it had been killing me for months. My whole body actually feels better on a lot of levels. I need to come on a more regular basis, I often don't realize how tense my body is, I guess I tune it out so I can function and go on with the day. Thanks for your beautiful hands and tender heart!See ya soon,Janice B.\nPatty Gates promises a \"holistic\" approach to massage therapy and she delivers the ultimate in a complete spiritual, mental and physical experience. The tranquil low light, soft sound environment sets the stage. But the real experience begins with a unique focus on stress points in the head, ranging from the lower back of the skull to the facial structure. Wait till you have your ears massaged by warm hands! Other \"special effects\" include a heated scented pack behind the neck and on the abdomen, and warm lotion and warm smooth stones used to simultaneously heat and massage the major back muscles as well as the soles of the feet. Patty also establishes a unique spiritual connection with her patients and will often provide insights on emotional issues as they relate to personal health. All this is provided with her calm, relaxing voice and firm touch that inspire spontaneous sighs from the lucky patient. I am counting the days to my next experience! Rhonda Doney I have personally experienced Patty's \"Transformational Bodywork\", and can attest to her deep effectiveness and exquisite sensitivity as a massage therapist.Also, heartfelt thanks for sharing your massage therapy expertise with my undergraduate students in my Holistic Health and Healing Class. Those students who participated really enjoyed and benefited from it, and those who did not had the opportunity to observe \"healthy touch\" that will be useful to them in their personal lives, as well as their health-related careers.\nI was in SO MUCH PAIN! And you quietly said \"I can help you\". I didn't believe that \u2013 silly me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's start be describing what platform beds actually are and why they are such a good idea. Basically they are beds consisting of a mattress on a solid platform raised off the floor by legs or framing.\nThis allows for the floor space beneath the platform to be used for either living or storage. If built correctly, they can really give a lot of extra living or storage space to any bedroom, although they are chiefly used in kid's rooms to accommodate all their clobber.\nThe great thing about platform beds is that the frames come in a wide variety of sizes and styles so that they can fit different personal tastes and requirements. You can also choose a platform bed frame complete with wood posters and curtains to make it immediately functional. These types are typically known as a canopy platform bed.\nYears ago, the first platform beds were known as loft beds as they were elevated way off the ground, so you may hear both terms used when scouting for more info, but they are really one of the same. Platform beds are also extremely versatile and easy to maintain, so if you're getting a little squeezed out by all the homely junk that so many of us tend to collect over the years, then you might want to take a look at the range of these special beds.\nA lot of folks are drawn to these beds not only for the practicality but also the simple, elegant styling found with the latest options available. This especially makes them popular among discerning homeowners. But apart from the simplistic look that these beds offer, where they really come into their own is for those people who live in tight spaces or small rooms.\nWhat's quite common these days is for people to equip their platform beds with special mattresses made of memory foam. Memory foam is a gel-like substance that conforms to the curves of the body while at the same time offering full support for the skeletal structure. It seems to be the latest craze in bedding right now. It's particularly beneficial to folks who suffer from neck, back, and shoulder complaints, but even if you don't, it'll guarantee you a good night's sleep.\nAnother reason why memory the foam mattress is a good idea is because a platform bed is a bed that has no box spring, and some people find it difficult to sleep easy without the standard box spring mattress.\nMany Handy Andy's these days try to save costs by building their own platform beds. If you do you want to try your hand at building one, you could actually save quite a bit of money as many beds can cost thousands of dollars or more for queen and king sizes.\nIt's important to get your plans right before starting your construction so that you avoid any costly mistakes. A platform bed for under-bed storage will obviously be a different construction to a loft style which is simply a raised platform bed. Raised types are where you acquire the extra area underneath for desks, entertainment centers, or whatever use you might have for the extra floor space.\nRemember, platform beds are ideal for kiddie's rooms as they really do maximize living areas for younger children who share their bedrooms with each other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The United States installed 10.6 gigawatts of new solar photovoltaic (PV) capacity in 2018 and remains on track to more than double this capacity in the next five years.\nTotal installed PV capacity is projected to increase by 14 percent this year. Yearly installations should balloon to 15.8 GW by 2021. In addition, the utility-scale solar photovoltaic (PV) sector will grow at levels beyond that originally forecast before the Trump administration imposed a 30 percent tariff on imported solar cells and modules.\nThe U.S. now has 64 GW of installed capacity, which is more than enough to power more than 12 million homes.\nAs is usual, California was once again the leader in new capacity, installing over 3.3 GW. Next in line were Texas and North Carolina, which installed some 996 MW and 907 MW, respectively.\nThe 10.6 GW in new capacity for 2018, however, is slightly lower compared to that for 2017 but the PV industry expects the market to rebound in the years ahead, according to the US Solar Market Insight 2018Year-in-Review Report from Wood Mackenzie Power & Renewables and the Solar Energy Industries Association (SEIA).\n\"The solar industry experienced growing pains in 2018, in large part due to the unnecessary tariffs that were imposed on solar cells and modules, but this report still finds significant reason for optimism,\" said SEIA president and CEO Abigail Ross Hopper.\n\"The total amount of solar installed in America is on track to more than double in the next five years, proving solar's resiliency and its economic strength,\" she said.\n\"It's clear this next decade is going to be one of significant growth.\"\nThe report said the residential sector rebounded to a 7 percent growth in 2018 after shrinking by 15 percent in 2017. The residential sector has now seen five straight quarters of modest growth. The fourth quarter expansion saw the sector's largest quarterly growth in two years, with nearly 315,000 households adding solar for the year.\nOn the other hand, the non-residential sector decelerated by 8%, while the utility-scale sector went through a slight contraction due to the continuing impacts of the Section 201 tariff uncertainty.\nIn total, solar PV accounted for 29 percent of all new electricity generating capacity additions. This share was lower compared to 2017 due to a surge in new natural-gas projects earlier in 2018.\nThe report forecasts stronger growth from 2020 to 2022. Total installed solar PV capacity is expected to increase by 14 percent in 2019. Annual installations should climb to 15.8 GW in 2021.\nUtility-scale solar projects from 2020 to 2022 now exceed forecasts originally made before the imposition of the Section 201 tariffs. The Wood Mackenzie's utility-scale forecast for 2020 is 8 percent higher than its fourth quarter 2017 forecast, while its 2021 forecast is 19 percent higher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Need a new boiler in South Molton? Get FREE no-obligation quotes now!\nDryad Networks supply and install oil boilers. For similar replacement boiler companies in this area please visit the South Molton boiler page.\nUnder the Governments ECO Scheme, qualifying home owners and private tenants throughout the UK who claim the correct combination of means tested benefits could have their old in-efficient oil boilers replaced completely free of charge, or heavily subsidised.\nThe grants are funded by the 'Big Six' energy suppliers and are non-repayable which means you do not have to re-pay the money back at any time in the future.\nThe style of property in which you live is also taken in to consideration to, the older and more in-efficient your property is, the more funding it will yield to go towards the grant. If you have a very efficient property, you may need to contribute a small amount to cover the cost of the install.\nThe amount of funding we can claim for your grant is determined by how efficient your property currently is. If you qualify, a free survey will need to be completed of your home which will look at things like your current boiler type and insulation levels etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On behalf of the San Diego Adult Rehabilitation Center (ARC) we would like to thank you for the kind donation of Furniture and the placement of a Donation Bin at your hotel, your donation and support will insure that we continue helping those who need it the most. The Furniture will be well used and service to fill the needs of the 130 men and women that are currently going thru our program.\nPlease let Paula, know how greatly we appreciate all the help and leadership she provided in taking time to scheduling the pickup and placement of the bin.\nWe look forward to building a long and lasting relationship with the Holiday Inn and its wonderful staff.\nA million \"thank you's\" to you and all of our dear friends and family a the Holiday Inn Express Sand Diego Downtown, from the American Society of Aviation Artists!. I say \"friends and family\" because many of the ASAA members commented on the friendliness of the staff, the smiles and hospitality, the casual feel through conversation and service, and the overall hometown experience we were given.\nForm the very first conversation I had with you, Collin, I knew the Holiday Inn Express San Diego Downtown was the perfect place for our organization to hold our 24th Annual Art Forum Every need was met with warmth and assurance that all would go well. The Cypress room served as a perfect \"family room\" for our group as we comfortably displayed our art for the week. It then was transformed day by day to meet our needs. I was nearly in joyful tears on Wednesday morning when I saw the wonderful transformation of the room into a classroom setting; how carefully your staff treated the art and the poster boards in the rearranging of the room. I can still hear the gasps as the members entered the Cypress room on Saturday when they saw their home for the week transform into an elegant banquet room. It was the perfect touch to a wonderful week. Hats off to your entire staff for the special care they put into this monumental task.\nYou have a tremendous staff of individuals to with, Collin, and we appreciate every one of them. Anisa's smile and bubbly personality was welcoming and kind. Paula was my anchor during the week, going over logistical issues on the spot. She too, like you, had an answer for everything, and it was the right answer every time. Shanica was a ray of sunshine that took over beautifully for Paula for the weekend events. Liz did a wonderful job, tending bar for our hospitality room. I hope she enjoyed spending time with our crazy bunch.\nThe food was more and more delicious as the week progressed! We could not have been in better hands under the professional care of the Indigo Caf\u00e9. Tiffany was always smiling and quick to help. Gretchen mad e the most wonderful recommendations for each menu, and was at my beck and call for event to event. Bob was wonderful in taking care of us at the bar, and was a tremendous help when I needed a distinct voice to say, \"Last call for drinks!' \u2013 a voice that commanded respect!\nI wish I could personally thank you everyone who helped us for the week \u2013 Martha at the front desk was lovely, as well as the others that served at the front desk around the clock. And those work in housekeeping, again, treated us like were part of the family.\nMy recommendation to the board in our organization was to repeat our visit to San Diego in the future, based first and foremost on the wonderful treatment we received at the Holiday Inn Express San Dieg Downtown. But I didn't even need to make the recommendation \u2013 the board already felt the same way.\nThank you again, and we hope our next forum in San Diego will not be too far in the future.\nI just wanted to express to you the wonderful stay I had at your Holiday Inn Express located on 1430 Seventh Avenue in San Diego. Your front office supervisor, Elizabeth Gastelum was so welcoming and made our stay so much more eventful. This was our first time staying in the Gas Lamp District and we didn't know what to do and Elizabeth was on it for us. She set us up to have a wonderful memorable time at Cafe Sevilla!! I also would like to give mention to your driver Steve and Cindy at the front counter, they too were so pleasant and helpful. We will definitely be returning!\nThank you and I hope you acknowledge your employees for going above and beyond to make our stay so terrific!\nI just wanted to take a moment to let you know how pleased we were with our recent stay at your location. We received excellent service from the moment we arrived, (midge?) and it continued thru at all levels of our stay. The room was clean, your breakfast service was above what we expected and we even enjoyed your cookie services. Overall, we were very pleased. I don't feel like great service or forming real \" connections\" with people gets recognized as often as it should, but I felt compelled to let you know how happy we were. Keep up the fabulous service, and let your front desk team know how much \"connecting\" with them meant to our family.\nMy girlfriend and I recently visited the great state of California and had the privilege to experience the lovely cities of Los Angeles and San Diego, the former being my hometown. We would have to say that our time in San Diego was far more enjoyable than our stay in my beloved Los Angeles and that was because of the very friendly and accommodating staff at your hotel. A front desk receptionist by the name of Jackie was so helpful in getting us settled in our room and given us directions around the city and advice on places to visit, that we decided to stay an extra day because we were so captivated by the city. Jackie saw that we were pretty exhausted once we got to the hotel because we ended up walking from the train station to the hotel and were giving bad directions by a host of individuals that it took us an hour and a half just to get there. Once we arrived Jackie gave us a couple of VIP coupons for some drinks in your lounge and Midge the bartender could not have been more hospitable.\nOur second day in San Diego, Jackie and Ilena (I hope I'm spelling her name right) set us up with great passes to the San Diego Zoo and had Alex drive us there and pick us up after our wonderful day there. Alex and Ilena were very courteous and professional and Jackie once again went above and beyond the call of duty to make our stay at your hotel even more special. Our room was cleaned thoroughly by housekeeping and the let us sleep late and never bothered our personal property.\nThe Days Inn we stayed at in Los Angeles was of far less quality than your hotel. Their service wasn't nearly as hospitable and the cleanliness of their rooms was not even close to the standards that your hotel displayed. I was shocked when I told my girlfriend that I enjoyed our stay in San Diego over our stay in Los Angeles because of my natural affinity to the city of angels. We just thought that you should know what a wonderful hotel staff you have there and that on our next visit to San Diego we will be sure to stay at your hotel.\nI just wanted to take a minute to let you know about our hotel stay from March 19-21. I just have to tell you this is by far the best service we have received from any hotel. Our family has been fortunate enough to stay at some of the nicest hotels in the country including The Beverly Hills Hilton, Waldorf Astoria, just to mention a few. Everyone was extremely nice from the front desk to housekeeping. The shuttle driver, I believe his name is Amid?? was very nice as well. From the card and candy we found on our bed, to the nice room, to the speedy checkout. Everything was done in a first class fashion. When we return to San Diego, I will not hesitate to stay there again.\nI am writing this email on behalf of all members of Asian Relocation Association and we wish to thank you and the whole team at your hotel for taking very good care of our members\u2026we especially wish to thank the front desk, our friend who drives the shuttle and the restaurant, not to forget the housekeeping which kept the mess under control in our rooms.\nWe will definitely stay with you again when we return to San Diego in about 5 years time for the same convention and I also heard that some of our members will be staying at your hotel when they visit California for their family vacations.\nWe are visiting Denver next September and we will be staying at another Holiday Inn Express as we have confidence in this brand \u2013 Thanks to Good People like you !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This film knows what it is. It's a popcorn movie. A film that demands being seen at the cinema as that's its home. It needs to be seen on a big screen, and you can't expect great cinema etiquette. Yeah if someone is on their phone then you should still legally be allowed to slap their wrist with a razor blade, but someone laughing loudly? That's fine during this. Someone sitting there loudly eating popcorn? Also fine. It's almost like it was made specifically for people to audibly react, it's like the anti-Quiet Place. It's an incredibly fun distraction. The kind of film you can imagine watching whilst drinking with your friends late at night. It's not going to change the world, or be studied in film class by future directors, and if you say this is your favourite film, I will judge you.\nSo this film should be run of the mill guilty pleasure. There's one thing that stops it from being that; the main character is an amputee. To say that again; the action hero is an amputee. It's very rarely mentioned, he's not defined by it and it only really comes up once every so often. It's a small thing, but I love that action movie fans in a similar situation finally have representation on screen. Usually, when you see someone like that on screen it's as the villain, it's about damn time they were allowed to be the hero. Yeah, it's a shame the character was played by someone with 2 legs but still, baby steps. Also, The Rock is just killing it lately. Jumanji, Rampage, and now this? He's quickly becoming the go-to guy for popcorn flicks.\nSo we've established this film is fun. It's entertaining shlock and you'll enjoy it whilst watching it. There are some issues with it, of course, the CGI isn't quite as clean as it needs to be in some areas, which occasionally makes it feel like you're watching a video game cutscene. The majority of characters are underutilized, and, personally, I'm getting incredibly bored of \"the bad guys are doing this so they can get hold of this USB stick\" plots (seriously, it's the MacGuffin for sooooo many movie characters lately). Also, it's hard to feel any genuine tension as you can pretty much pinpoint how every scene will play out. I must commend them on the room of mirrors scene though, that was BEAUTIFULLY orchestrated and laid out, THAT'S the scene you need to see. You don't need to see the rest, but I advise that you should, and watch it on a big screen. This film will lose so much of its potency if you watch it on a small screen. It's spectacle cinema, and deserves to be treated as such. The action is some of the most jaw-dropping you'll see. The bits which aren't action-heavy? They're\u2026..look watch the action bits. The rest of it is difficult to recommend. The opening third, in particular, is exposition in a film that really doesn't need that much exposition. People aren't going to see this film for the brilliant camera work, they're going to see it because \"ooo things go boom\". It doesn't need as many characters as it has, as it means most of them go to waste. Neve Campbell, in particular, seems incredibly underdeveloped for a performer of her calibre. I think Hannah Quinlivan is underwritten as well, but it's hard to tell as her character flits in and out of the script like a drunken desire to commit suicide. She's good when she's in it, but she isn't really in it enough to warrant a strong opinion on her either way, I'd like to see her in more so I can find out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A scene from Jez Lewis' documentary film 'Shed Your Tears and Walk Away'.\nHebden Bridge, Yorkshire, was the sort of quaint, historic town in England where wealthy people re-camped from the city, and tourists browsed antique stores.\nHidden away was the fact that a significant number of the children there were killing themselves, either by suicide or drug abuse. Jez Lewis' documentary about the seamier side to life in Hebden Bridge, 'Shed Your Tears and Walk Away', will screen at the Bermuda Documentary Film Festival next week.\nMr Lewis' parents moved to Bermuda with their three children in 1967. He was born here the family left when he was two years old. He will return to Bermuda for the first time since then next week, for the debut of his film. He now lives in Sussex. \"Mostly my friends were dying from suicide but there were some drug deaths as well,\" he said of his childhood in Hebden Bridge recalled in the film.\nMr Lewis decided to make the film after one of his friends, Emma, died from a heroin overdose. Although the town seemed idyllic to tourists, drugs were rife.\n\"It is post-industrial so there is a lot of unemployment,\" said Mr Lewis. \"When I was growing up it was quite run down. A third of the houses on our street were run down or derelict.\nMr Lewis said when he was a teenager there was an intense campaign to educate people about the dangers of taking heroin, but it came too late for his friend Emma and her brother.\nEmma's brother told Mr Lewis he was introduced to the drug at the age of 14 by a friend of his mother.\nHe vividly recalled he and a friend being told off by a teacher. The teacher said \"if you don't improve, you'll never find decent jobs\".\nThe friend, 15, shrugged and said he'd already resigned himself to a lifetime of unemployment. What might have saved Mr Lewis was a literal kick in the pants. Mr Lewis discovered karate. \"That took me out of circulation,\" he said. \"I am a second band black belt now. As a teenager I trained at two different clubs. I was always terrified of drugs anyway, so that was lucky, really.\nIn early adulthood, he also found jobs such as babysitting, cleaning in a laboratory to keep him occupied and also financially independent.\nMr Lewis left Hebden as soon as he could. He first studied physics in university, but gave it up after a year.\nLater, he went back to the University of East London to take cultural studies, which had elements of filmmaking. He also took evening classes in film and television studies.\nHe now has a master's degree in science and technology policy from the University of Sussex.\nToday, he runs Bungalow Town Films with Rachel Wexler and Rebecca Day. The company makes feature documentaries.\nThey produced a film called 'Ghost' about Chinese immigrants working in Britain. Although he has worked on films for other people 'Shed Your Tears and Walk Away' is the first film of his own. It was only intended to be a 15-minute film, but the idea snowballed.\nLast year, it won the an award as best UK first film in the East End Film Festival in London. So far, there has been a mixed reaction from the people of Hebden.\n\"Most people have been very supportive and have thanked me for making the film because they have lived with [what happened] for decades,\" said Mr Lewis. \"They have lost so many family and friends.\nA volunteer group called The Samaritans threw their weight behind the film, because they dealt with similar issues every day. They arranged a special viewing in Hebden Bridge.\nThe Police Department and the National Health Service refused to officially acknowledge the film or attend. Still, the 500-seat auditorium was packed.\n\"It is clear there is more of a problem there than elsewhere, although that is not to say that other places don't have worse problems,\" Mr Lewis said. \"I now live in a town about the same size as Hebden Bridge.\n\"I have just been in the park with my children and we didn't see a single person drinking alcohol or doing drugs. I went to the park in Hebden Bridge two weeks ago and the place was strewn with vodka bottles, beer cans, and drug paraphernalia. It was just a Wednesday evening.\nWhat he hoped to see from the film was official acceptance that there was a problem, and attempts made to address it.\nHe said one of the problems for the town was wealthy people moving in from the city.\n\"They won't accept that there is a severe problem and that they are part of the problem,\" he said. \"The local community is being dispossessed, because the wealthier people push up the prices of homes and goods.\nHowever, he said Hebden Bridge is a beautiful place, and ultimately, he had more good things than bad to say about it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Availability: We currently have 7 in stock.\nThis Aromamizer from Steam Crave combines the convenience and versatility of a rebuild-able atomizer with the performance of a dripping device. Steam Crave calls it an RDTA, or an RDA with a tank assembly. The RDTA Aromamizer comes in two models \u2013 the SC200 is a 6ml tank while the SC200-S has a smaller 3ml capacity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahklf b/data_all_eng_slimpj/shuffled/split2/finalzzzahklf
new file mode 100644
index 0000000000000000000000000000000000000000..ab50216642ccc0d3df32fca42c5e75867e077653
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@@ -0,0 +1,5 @@
+{"text":"TORONTO, ONTARIO--(Marketwired - Oct. 10, 2013) - TripAdvisor\u00ae, the world's largest travel site, recently revealed Enterprise Rent-A-Car as the favourite brand of car rental agencies in TripAdvisor's inaugural Travellers' Choice 2013 Awards for Travel Favourites in Canada.\n\"We are thrilled to be selected by the TripAdvisor community and travellers for the first ever Travel Favourites award in Canada,\" said Randal Narike, Senior Vice President at Enterprise Rent-A-Car. \"We are continually working hard to meet our customer's needs and exceed their expectations so we are very proud to hear travellers have voted Enterprise as their favourite car rental agency.\"\nThe Travellers' Choice Awards recognize the top brands that travellers prefer to use during their trips with winners selected directly by the travellers. Honouring the best of travel, the Canadian launch of the Travellers' Choice\u2122 Awards featured 21 winners, including Tim Hortons coffee, Canon cameras, Ray-Ban sunglasses, and Apple tablets. A complete list of winners is available on the TripAdvisor Blog.\n\"The TripAdvisor community has spoken and Enterprise Rent-A-Car is amongst the most beloved travel brands from around the world,\" said Barbara Messing, chief marketing officer for TripAdvisor.\nOperating in Canada since 1993, Enterprise Rent-A-Car is known for its extensive network of locations, everyday low rates and outstanding customer service. Enterprise Rent-A-Car has more than 450 locations in Canada, including 70 offices serving airports, and offers a wide variety of car leasing, commercial truck rental and hourly rental programs - and local car rental customers are picked up at no extra cost. Customers can learn more and make reservations by visiting www.enterpriserentacar.ca.\nEnterprise Rent-A-Car is an internationally recognized brand that has been named to BusinessWeek magazine's annual list of \"Customer Service Champs\" in 2007, 2008, 2009 and 2010. In addition, Enterprise won Budget Travel magazine's 2010 and 2011 Readers' Choice Award as their favourite rental car brand for customer service around the globe.\nEnterprise Rent-A-Car operates not only as a key provider for insurance replacement, small business needs, weekend getaways, and special occasions, but also as a local transportation alternative. For example, many consumers rely on mass transit during the week or simply cannot afford to purchase or maintain a vehicle on their own - so they often depend on local Virtual Car\u00ae rental service and take advantage of Enterprise's \"everyday low price,\" including popular weekend rates starting at $9.99 per day. Enterprise is the Official Rent-A-Car of the NHL. For more information about Enterprise local and airport car rental as well as vehicle leasing, car sharing and vanpooling options, visit www.enterprise.ca. This news release and other announcements are available at the Enterprise Holdings press room.\nTripAdvisor\u00ae is the world's largest travel site*, enabling travelers to plan and have the perfect trip. TripAdvisor offers trusted advice from real travelers and a wide variety of travel choices and planning features with seamless links to booking tools. TripAdvisor branded sites make up the largest travel community in the world, with more than 260 million unique monthly visitors**, and more than 100 million reviews and opinions covering more than 2.7 million accommodations, restaurants and attractions. The sites operate in 34 countries worldwide, including China under daodao.com. TripAdvisor also includes TripAdvisor for Business, a dedicated division that provides the tourism industry access to millions of monthly TripAdvisor visitors.\nTripAdvisor, Inc. (NASDAQ:TRIP) manages and operates websites under 20 other travel media brands: www.airfarewatchdog.com, www.bookingbuddy.com, www.cruisecritic.com, www.everytrail.com, www.familyvacationcritic.com, www.flipkey.com, www.gateguru.com, www.holidaylettings.co.uk, www.holidaywatchdog.com, www.independenttraveler.com, www.jetsetter.com, www.niumba.com, www.onetime.com, www.seatguru.com, www.smartertravel.com, www.tingo.com, www.travelpod.com, www.virtualtourist.com, www.whereivebeen.com, and www.kuxun.cn.\nAbout this company Enterprise Holdings Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We received this commercial job at short notice, 12 noon today to be precise, and as it was required by close of play this afternoon, it was taken on board immediately. The kitchen utensils, 14 items, were couriered and arrived 30 mins after we'd rigged up the set, which also included a quick jaunt to the supermarket for the veg to include in the shots. Another 2 hours and all items had been photographed and edited to include text before being emailed to the Client at 4:35.\nAll in all, a successful afternoon and another happy Client.\nFor more information on Elegant Shot Ltd Commercial Photographers Ayrshire please visit our website.\nThis entry was posted in Commercial Photography and tagged air, airshire, ardrossan, ayr, ayrshire, ayrshire photographer, ayrshire's, ayrshires photographer, elegant shot, elegant shot restoration, kilwinning abbey, photo resoration ayrshire, photograph, photograph restoration ayrshire, portrait photography ayrshire, restoration, wedding, wedding photographer, wedding photographer ayrshire, white wedding by ElegantShot. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scottish Rugby has launched an innovative new schools programme to help promote rugby in Primary schools and encourage youngsters to take up the sport.\nWith Scotland currently in action at the Rugby World Cup in England, there is a unique opportunity to inspire the next generation of players. Scottish Rugby has teamed up with Glasgow Life and local rugby clubs to develop the 'Scotland and the Rugby World Cup' schools resources for primary schools across the country.\nScottish Rugby Ambassador, Chris Paterson MBE, said: \"It is important for children to be given the opportunity to try as many sports as possible, to help them see the benefits of exercise and an active lifestyle.\n\"When I was at school I got my first taste of playing rugby and enjoyed the camaraderie of training alongside my friends.\nThe schools pack provides a set of key learning resources aimed at primary school teachers and children, to give them an understanding of rugby and in particular the rugby World Cup which is being hosted in England.\nIt covers a number of key areas including venues, team colours, flags, national anthems, players' diet, media coverage, television commentary and the rules of the game itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barcelona: FC Barcelona captain Andres Iniesta said on Thursday he is waiting to set up a meeting with the Catalan football side's board of directors to begin talks about renewing his contract, which ends in the summer of 2018.\n\"I want to end my career with Barcelona; I have a contract until 2018. You will have to ask those in charge when it's my turn to renew,\" the midfielder told Cadena COPE radio on Thursday, a day after they were held to a 1-1 draw by Atletico Madrid in La Liga.\nIniesta, 32, said he understands that the board is currently busy with signing Argentine striker Lionel Messi, whose contract ends in two years, noting that he only wants to discuss the issue with the Catalan club officials.\nOn the draw with Atletico at Camp Nou stadium here, the Barcelona captain said they had a jam-packed game against a tough opponent with strong defence, adding that it was a pity to drop points in a home match.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tags: Black, Mini Bags, Womens.\nA mini chain bag with an external side pocket under the flap that holds a smartphone. Made in heat debossed Gucci Signature leather resulting in a defined print with a firm texture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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+{"text":"I hear that doing that stops you loading your game after a week?\nIf it does then maybe someone else should just grab the bull by the horns and make an open source editor in the true sense of the FM community.\nI just wish i knew how! Someone PLEASE make one!\nHow can i activate the license key with the trial version?\nWell, i can see the point why he wants people to DONATE, but in a common sense it should be a freeware because it still corrupts and destroys savegames(happened for me sometimes when i used it). Still it's as the warning says.. ''use it at you're own risk''. i can't blame the guy for doing some bad work but has he really progressed that much with the open source FMRTE version?. As for myself i have been using it with premium features and donated 1 euro, but after realizing that it corrupts savegames i started to NOT using it. Reason.. i played a game for long time and i had to restart from first season, i mean.. that's bad when i spent like 1 month on the save and as i think it still isen't perfected, i will wait for a better version that dosen't corrupts savegames atleast.\nI hope someone hacks their site.Why they put donation?Come on people you should have this app for free.If someone wants to donate,he could donate to your website,it's like they are selling their program.Does anyone has a warez version of the FMRTE?\nIt's so annoying. I wouldn't even need FMRTE if FM would let me be my actual age...It's only a small detail and makes hardly any difference, but it bothers me! Gosh, let me be 17!\nWho the hell decided Humpty Dumpty was an egg?\n@payne 007 where do i get the key for the trial version to use it anytime just like u have done?\nI hope someone hacks fmrte, will serve them right. You can't force donation. If you ask nicely people donates anyway.\nSorry to bring up a old thread! is it true u have to donate \u20ac4,99 to use it ??? if so then thats a joke! I wouldnt mind paying 1 or 2 at the most but charging almost \u20ac5 is a rip off! I take it there is no donate option ?\n2012-12-13 15:54#72135 leestevenson1988 : Sorry to bring up a old thread! is it true u have to donate \u20ac4,99 to use it ??? if so then thats a joke! I wouldnt mind paying 1 or 2 at the most but charging almost \u20ac5 is a rip off! I take it there is no donate option ?\ntalking about the new fmrte btw.\nCan someone paste the TRIAL VERSION LICENCE KEY?\nPaying 5 bucks for making an edit is ridiculous.\nThey're talking about an in-game editor..\nWas there really a need to bring this thread back up to reply to a post from nearly a year ago? Do you seriously still think he'll be checking this thread? This can be classed as spam, so don't do it again.\nYou are reading \"Annoyed, I need to donate to use FMRTE\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CANCELLED. Exact details not known. Promoter went bankrupt shortly before the gig was due to take place.\nA free festival for 250,000 people. OMD played around 12.30am on 24th June, Andy's 25th birthday.\nExact details not known. Festival appearance.\nAn outdoor venue. Support band not known.\nOMD played in a plastic bubble tent. Support band not known.\n2 gigs in 1 day, playing at 8.30pm and 10.30pm. Support from The Smart.\nSupport from Bear Garden and Push of Love.\nSupport from Bear Garden and Riki Tiki Tavi.\nSupport from Just Add Water.\nSupport from Just Add Water and Bear Garden.\nExact date and details not known. Support from Just Add Water.\nExact date not known. Support from Just Add Water.\nNo support. The band played at a Hi-Fi exhibition.\nBilled as the 'Namur Festival of Wallonia', this outdoor show was also televised.\nWarm up show in advance of the UK Junk Culture tour dates.\nSupport from The Reverb Bros.\nThe band played Almost for the first time in years as this was the band's 6th anniversary. Support from The Reverb Bros.\nMatinee performance for Fan Club members.\nSupport band unknown, Fiction Factory cancelled their appearance due to illness.\nSupport from The Reverb Bros. Fiction Factory had to cancel due to illness.\nSupport from Cook Da Books.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Kidde P9040 smoke alarm is a battery operated smoke alarm that utilizes photoelectric sensing technology. The four (4) inch unit is smaller in diameter giving discreet design with the same quality you would expect in a Kidde manufactured product. The P9040 is powered by a 9V battery, providing continuous protection, even during power outages. This easy to install unit includes a test button to verify the detection electronics are working as intended. The alarm also includes the Hush feature, which temporarily silences nuisance alarms.\nThis alarm uses photoelectric sensing technology. Leading authorities recommend that both ionization and photoelectric smoke alarms be installed to help ensure maximum detection of the various types of fires that can occur within the home. Ionization sensing alarms may detect invisible fire particles (associated with fast flaming fires) sooner than photoelectric alarms. Photoelectric sensing alarms may detect visible fire particles (associated with slow smoldering fires) sooner than ionization alarms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. A small to medium size container with drainage (make your own holes if need be or you can use cactus soil if you plan on re-doing every 3-6 months). You can also find these types of containers in the garden sections of Home Goods, Dollar Tree, TJMAXX, Michael's, and Target.\n6 pack of small, colorful flowers (taller).\n6 pack of succulents or 2-3 individual succulents, various sizes.\n\u200b4. A bag of preserved forest moss. Moss can also be found at Dollar Tree or Michael's. There are different colors and textures depending on your preference.\nThere is even more Fairy Garden Decor at Michael's, Dollar Tree, JoAnns Fabrics, and Home Depot. Don't forget to use things that are already lying around your home or garden. I hope you enjoyed the video and get inspired to do your own fairy garden or gnome home this summer as well! These container gardens make great summer activities for friends and family. (Great gifts as well!) If you did like this video, make sure you are subscribed to my channel and give this post or video a big THUMBS up! Have fun and happy summer!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Profits made from this shirt will be donated to AFSP (American Foundation for Suicide Prevention). This music saved my life, and I wanted to use it to help save others.\nThis shirt is based on the song \"Forest\" by Twenty One Pilots. This song holds deep personal meaning to me, and thousands of others. I am donating profits made from this campaign to AFSP (American Foundation for Suicide Prevention) Because the music from this band helped me when I was at the worst, and I wanted to use it to help others.\nThis campaign is not run by Twenty One Pilots or Fueled by Ramen, simply inspired by their lyrics.\nThe original artwork was made by my friend at https:\/\/www.reddit.com\/user\/Skibbkazoo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamwnj b/data_all_eng_slimpj/shuffled/split2/finalzzzamwnj
new file mode 100644
index 0000000000000000000000000000000000000000..14405c9829c41b45f3e564b38fd48f3c80efa218
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"This scent captures the essence of the Pinon Pine, a coniferous staple from the American Southwest. Earthy notes of pine & balsam fir balance resinous amber & spice black pepper. Light vanilla & smoky cedar wood add warmth & familiarity. This scent feels like an old friend: cultivated but intimate.\nP.F. Candle Co. Room Sprays are poured in amber glass with spring water sourced from Mt. Shasta and a blend of body-safe fine fragrance oils. Each room spray produces approximately 350 pumps of scented mist. Depending on how often it is used, your room spray may last between 1 and 3 months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Friday 31 July is training contract application deadline day at most of the big firms. In a market where, according to a recent Sweet & Maxwell survey, there has been a 150 per cent jump in the number of applications per training contract vacancy, prospective trainees will need all the luck they can get.\nFriday 31 July is training contract application deadline day at most of the big firms. In a market where, according to a recent Sweet & Maxwell survey, there has been a 150% jump in the number of applications per training contract vacancy, prospective trainees will need all the luck they can get.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For seven years, Natural Home Solutions has been a preferred roof installation and repair company in the Augusta, ME area. Home and business owners turn to us first because they know we'll take care to protect their properties. You can, too.\nContact us today to schedule the roofing services you need.\nWhat kinds of roofing services do you offer? Natural Home Solutions specializes in roof repair and installations. Dealing with hail damage? We'll assess your roof so you can file an insurance claim to cover your out-of-pocket roof repair costs.\nWhat materials do you work with? We repair and install metal, rubber and asphalt shingle roofs at homes and businesses.\nWho do you serve? Our crew is based in Augusta, Maine, and we serve residents within 50 miles.\nSee how we compare with other local roof installation and repair companies. Call 207-242-3793 today to speak with a member of our team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Check out all of the names of the Hononegah High School alumni that attended high school in Rockton that are listed above. Registering will allow you to join the directory.\tIf you are a former student of Hononegah High in Rockton, Illinois, register now to add your name.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How To Download Stickers On WhatsApp For Free \u2013 WhatsApp is the world's most used messenger app by the smartphone users and now WhatsApp has introduced some new features for the Android and iOS users a week ago and that is the sticker options. Though some users have complaints that they are not able to see it even in the latest version 2.18.327 but WhatsApp have already confirmed that each and every user can use the sticker option to chat just like other messenger apps in coming weeks. The new sticker icon will be available inside the emoji icon on the keyboard.\nPlease make sure that you have the latest version of the iOS and Android app. You can also download the latest beta from the Google Play beta programme. Once your WhatsApp is updated to the latest version, open the app and tap on any conversation and tap on the sticker tab in the text input field ( for the iOS users ) and tab the emoji icon and inside it you will find sticker icon ( if you are an Android user ). Then you will the sticker tab and after clicking on the + sign visible at the top right corner of your keyboard, you will be redirected to the available sticker packs. The packs that are available are Cuppy, Salty, Bicuit, Koko, Hatch, Komo and more.\nYou can download all the sticker packs by clicking the download button on the right corner available at these packs. When these stickers get downloaded in your WhatsApp then they will be visible to you in the heart section which is also divided into four other categories.\nNow coming at the sections that are inside the WhatsApp sticker tab. So, there are three section; star, Clock and Heart. The first that is Star section contains your favorite stickers, the clock section will take you to the recently used stickers and the last one, the heart Section will redirect you to different categories, which includes Love, happy, dramatic and sad. Initially, your clock and star sections will seem to be empty as you will start using the stickers in your Whatsapp conversations, they will start to fill up. You can add your favorite stickers in the star section by long pressing the sticker you want and press \"add\". All the stickers packs that you have downloaded will be available in the + tab.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanbly b/data_all_eng_slimpj/shuffled/split2/finalzzzanbly
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+{"text":"Get More Pars August 18-22, 2019!\nGet your game up to par in this 5 day\/4 night package with option for local package with top-selling golf author, LPGA and TPI Golf 3 and Power Level 2 Coach, Christina Ricci. Her More Pars Camps will give you all the skills to get more distance, more GIR's for lower scores and more pars. Open to men and women all levels.\nAtkinson is an hour north of Boston Logan Airport or 30 minutes from Manchester Airport. Guests can either rent a car or catch an uber. The address is 85 Country Club Drive Atkinson, NH 03811.\n$250 non-refundable deposit required to hold your spot. Balance due 60 days out. Cancellation Policy within 60 days: 25% Fee. Within 30 \u2013 no refunds. We reserve the right to change or cancel camps in the event of limited sign-ups. We will notify you 45 days out and either switch your dates or refund your deposit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We took a quick trip to DC over the weekend. Even though I supposedly had WiFi in this hotel where we stayed, I couldn't connect. I'll post things when I get a chance, making sure to change the date accordingly.\nI took a ton of photos. It may take a bit longer to get any of those posted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since TV personality Douglas Lwanga called it quits from his relationship with ex-wife Eunice Nuwamanya, he has been enjoying conjugal rights with a one Linda Lisa Mukasa who he is now his current wife.\nHowever, there has been a talk on the streets that Linda Lisa is older than the NBS After 5 presenter. However, she has come out to reveal that she is not a cougar as many think. She sys she is even younger than her lover, Douglas Lwanga.\nShe added on that she is never been Douglas' sponsor as it is claimed. She however refered to Eunice as a cougar.\n\"hehe, omwana namuwonya omukadde bambi muleke mwagale!!\" literally meaning; let me love my man, who I helped to get rid of the old woman.\nDouglas and Eunice separated in 2016 after tying the knot in 2010. After a few months, he settled with Linda Lisa. According to close sources, Douglas and Linda have been dating for about 5 years and so far blesssed with two children. Linda is a marketer and a Public Relations Officer (PRO) at Club Guvnor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Kiwami (\"zenith\") platter is a special 10-piece serving of the finest sushi, offered by participating establishments in Niigata. The platter includes local seasonal offerings unavailable anywhere else, together with uni (sea urchin roe), toro (medium-fat tuna), and ikura (salmon roe). The content varies according to the season and sea conditions, but you can always be sure you will be eating the best fish of the day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"People on social media and at least several media outlets have misrepresented what NRA spokesman Colion Noir said about the First Amendment.\nOn May 24, NRA TV posted a video of Noir that appeared to have him calling for Congress to place restrictions on media outlets that repeatedly highlight the identity of mass shooters.\n\"It's time to put an end to this glorification of carnage in pursuit of ratings because it's killing our kids. It's time for Congress to step up and pass legislation putting common sense limitations on #MSM's ability to report on these school shootings,\" said Noir.\nBut that was only the first part of what Noir said. The second half was Noir explaining that he was only trying to make a point and he didn't actually believe in any restrictions to the First Amendment.\nOn the plus side, The Washington Post published a story about the video with an accurate headline.\nThere is certainly a lot about the NRA that is left to be desired. The organization has its own problems telling the truth and one of their spokeswomen, Dana Loesch, puts out incendiary and paranoid rhetoric that only serves to ratchet-up the toxic discourse.\nBut the way to debate and defy the NRA is not to lie or mischaracterize what they say or do. The people who mischaracterized what Noir said have handed the NRA an easy propaganda victory. Now, the NRA can point to this incident and reinforce the idea that everyone in the media is out to get them, even though only a few media outlets did this. It won't matter. The NRA had its broad \"the media is evil\" brush out and it's ready to use it.\nIt's even possible that this was intentionally-designed by the NRA to get the media and liberals on social media to react the way they did. The NRA understands how social media works today. Most people don't watch the entire video. They see a headline or hear just one soundbite and they automatically react.\nIn its Twitter post, the NRA took Noir out of context \u2013 \"It's time for Congress to step up and pass legislation putting common sense limitations on #MSM's ability to report on these school shootings.\" That post seems tailor-made to get liberals and anti-NRA people riled-up. And it worked.\nThis is why context matters and you should always get the full story before you react, even when it's about the NRA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqbyp b/data_all_eng_slimpj/shuffled/split2/finalzzzaqbyp
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+{"text":"Please fill out the request form below. All requests are reviewed daily. We will do our best to respond within 24 hours. If your request is urgent, and you need to speak with someone at MeetingMetrics right away, please call 1-212-426-6222.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gionee just expanded etheir budget smartphone portfolio by adding the X1. It's a budget handset equipped with 4G LTE connectivity and fingerprint scanner.\nThe Gionee X1 is a metallic looking smartphone with 5 inch HD 720p display at 294 pixels per inch. Inside, it boasts MediaTek's 64 bit MT6737 quad core processor paired with Mali T720 graphics and 2 GB of RAM.\nIt also has 16 GB of expandable internal storage and ample 3,000 mAh of battery for whole day of use.\nThere's also an 8 MP main and front camera for casual shooting and selfies.\nThe Gionee X1 comes in black and gold colors for INR 8,999 in India or roughly around PHP 7.2K out of straight conversion.\nGiven those specs, this device should be capable enough to run basic Android games and the usual social media apps with ease.\nThe rest of the missing details and international availability are still unconfirmed for now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ORESTES, Ednilsom; SILVA, A. B. F.; CAPELLE, Klaus; ULLRICH, Carsten A. The generating-coordinate method in static and time-dependent density functional theory. Bulletin of the American Physical Society[S.l: s.n.], 2006.\nOrestes, E., Silva, A. B. F., Capelle, K., & Ullrich, C. A. (2006). The generating-coordinate method in static and time-dependent density functional theory. Bulletin of the American Physical Society. College Park: American Physical Society - APS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emergency Payday Advance \"Better Cash Com Promo Code\". What is a payday advance? That's the type of cash advance you get from a site where you fill out a quick online application and are approved right away. Not only that, but the loan proceeds are given to you electronically by the next business day. Are you thinking about applying for a cash advance like that? Many people do to bridge the gap when they don't have enough money for bills. You can get cash loans for fair credit by using Better Cash Com Promo Code, and read reviews.\nSearching for Better Cash Com Promo Code. Get funds today?. Simply no Require Your credit rating. Effortless Endorsement within Twenty four hours. Urgent Cash Right now.\nBetter Cash Com Promo Code, There are more ways to get the amount of money, but a Better Cash.com payday advance makes everything appear to be a huge hassle. Which kind of loan can you go for? Thinking of that, you would a minimum of be tempted to push the straightforward button. Of course, the simple button includes consequences, as does other things. In cases like this, one of the negative consequences is the fact that comfort of a payday cash advance loan has a cost.\nIf you borrow 500 dollars from one of these creditors, how much do you think you would have to repay? A bank loan for 500 dollars would barely possess interest attached by any means, even if you were to pay it back more than a year. These payday cash advance loans are so easy to get though, so that convenience and risk for your lender suddenly costs you several hundred dollars over what you would normally have to pay to any other kind of lender. Even reliable installment loan payday lender will almost certainly charge a fee a couple of hundred dollars for your 500 dollar loan. Put simply, you're planning to wind up paying back quite a bunch of money.\nIn the event you pay back double everything you borrow plus more? The example above wasn't quite double, but that's around the good side of things. Should you examine other sites, the loan companies often charge double, or maybe more. I actually have seen instances of companies charging triple the borrowed funds amount. Doesn't that sound on the top?\nThat's why you need to check around. You wish to know prior to signing electronically what you're likely to be charged. If you're setup by having an installment loan, that's great, but don't just glance at the payment amount. Total up those payments, and ensure you appear up or ask about early payoff of your respective loan. Furthermore, know your alternatives. For instance, say you will have a 300 dollar loan, along with your payments are 54 dollars every 14 days. Know whether it is possible to pay extra (any amount) on the principal any time you pay and whether you are able to set that up online or if you need to refer to them as to get it done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take to the open road for an unforgettable adventure with one of our exciting touring holidays. With the advent of the new breed of luxury coaches that offer the traveller more comfort and a better travelling experience. Coach touring holidays are more popular than ever and with an ever growing portfolio of destinations to choose from, it's easy to understand why. Sit back and relax as your coach travels between world famous cities or through beautiful countryside. Whether it's the stunning beauty of North America, the fascinating capitals of Europe or the scenery and cities of Asia, touring holidays are the perfect way to travel.\nTravel 55 offers discount on these most touring holidays, simply book direct with the tour operator using our unique discount code.\nDiscover Cornwall with visits to Dartmoor, Falmouth, Padstow and St Michaels Mount and St Ives. Small group up to 18 guests.\nDiscover Costa Rica it's rich nature and unique wildlife as we explore the towering volcanoes & lush rainforests. This tour comes with a 7% discount.\nThis delightful 16 day tour takes you to some of the best destinations and sites that Italy has to offer the traveller.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqpul b/data_all_eng_slimpj/shuffled/split2/finalzzzaqpul
new file mode 100644
index 0000000000000000000000000000000000000000..6827f8c6b9c908080acd2662cc77dd9f76d0357e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqpul
@@ -0,0 +1,5 @@
+{"text":"Cool Tools \u2014 Row Counters!\nThese row counters slide onto your needle or hook and you turn the little knob at the end to count each row. For knitters working in the round, there's also a version that has an attached ring-marker so you can place it at the beginning of the round.\nAlso known as Kacha-kachas (for the sound they make), these markers advance with a clicker. The basic version just clicks, but doesn't have a lock. They also make a version that you can hang from a chain or ribbon that is lockable!\nThere are two good apps that include Row Counters. Both allow you to use a single counter, have counters for multiple projects, and even use multiple counters in a single project (for example, if you have to decrease every 6th row, but do a cable every 8th row, you can have one counter for each, and they'll both advance together). For iPhone, the best we've found is Knit Companion: you create a project, and then the app provides a counter within the project. For Android, we like County Plus because it's easy to use and doesn't have any extras you don't need. Both have a limited free version you can try before purchasing the full-featured version.\nYou know what's lovely? A cool evening on the deck, cuddled up in a cozy cardi and roasting marshmallows in the chimney. Well, here's a great cardi to knit up for those cool fall and winter evenings. It's the Neck-Down Wrap Cardigan from Knitting Pure & Simple. And it really is simple! One piece, in sizes small (33\") through 2X (47\") (or Wendy can help you resize it to fit larger or smaller sizes).\nCaryn knit hers using Spuntaneous Worsted Effects, a super smooshy tonal yarn that feels like every cozy daydream ever... Caryn always keeps great notes on her projects, too, so feel free to check out her project page!\nThe Winter 2019 Interweave Knits magazine is here! It has some really neat stuff on science and knitting that all our Yarnivore Geeks will love to read!\nWant to help out the wildlife? Instead of throwing away your cut off ends, leave them outside for the birds to use as nest materials!\nIt's finally cooling off now, so a little extra warmth mornings and evenings is quite nice. Not only that, but Christmas is sneaking up FAST! These simple glovelets knit up super quick and everyone loves them! You'll learn to knit in the round on double-pointed needles with this fun project - makes a great gift!\nLearn twisted stitches and cabling with this elegant braided cable hat. Also makes a great and quick gift!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fletcher Heights Park, 8135 W. Lone Cactus Dr.\nTime 6:30 to 9:30 p.m.\nCommunity services, Neighborhood Resources, and Public Safety engage the Peoria community through these events held in parks throughout the city. Neighborhoods are brought together with free, fun, entertaining and informative activities. ParkFest! events are designed to connect the community by introducing neighbors and providing information about city services through a unique and creative festival atmosphere.\nWhen electing your delegates, please carefully follow the provisions in Article IV of the International Constitution and the AFSCME Elections Code (Appendix D). The principal requirements pertaining to the election of delegates are highlighted in the Convention Call.\nThe earliest date for electing delegates to this Convention was March 20, 2016, unless your affiliate meets less frequently than quarterly. If you have not already held your delegate elections, we ask that you nominate and elect your delegates as soon as possible, complete the credentials, and return the original (white copy) to the International Union. In accordance with the provisions of Article IV, Section 9, credentials must be postmarked no later than June 28, 2016; otherwise they will be deemed irregular and your delegates will not be seated or included in the first report of the Credentials Committee.\nMember meeting are held every 3rd Thursday of the month, at 5 pm.\nMonthly meetings now include agendas.\nSpecial events will be added, as scheduled. Check back often, so you don't miss anything!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since 1929, August Schmits Stanz- und Umformtechnik GmbH & Co. KG in Mettmann \/ Germany develops and manufactures stamped, drawn and bent parts from steel, stainless steel, aluminium and other non-ferrous metals.\nWe work successfully in conjunction with system partners in the automotive industry, mechanical and plant engineering as well as manufacturers of technical consumer goods. Our customers trust our technical competence, and value the high quality of our products and our absolute reliability. Our comprehensive knowledge and experience in \"drawn parts\" is especially in demand.\nWe have our own tool and prototype construction department which enables us to respond quickly and flexibly to customer requirements.\nIn conjunction with our industrial partners, we develop optimum manufacturing and product solutions.\nWith the help of modern manufacturing technology, we produce small, medium and large series using progressive and tranfer systems. The further processing of stamped, drawn and bent parts into complete assemblies and components is also included in the services we offer.\nAn integrated CAQ system and certification in accordance with ISO\/TS 16949:2009 provide our customers with reassurance. We deliver just-in-time and in accordance with current logistics concepts to locations throughout the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's exactly the other side of what I was trying to say.\nI think Iain Abernethy is right claiming that zen has no exclusive ownership on many concepts (we could say there are \"million ways to zen\"), and there is no need for mysticism in karate. But I still think zen is a great way for that insight, as elaborate as karate can be as a self defense system, so I don't think it's wrong if zen practices (such as simple Mokuso) are related to karate.\nThe suppossed official link between history of karate and buddhism is BS, but the use of zen in karate is not. Zen is very useful. I would add that zen is not really mystical so it's not like some pseudoscientific or too ancient obsolete philosophy. And in XXI century we can be able to understand zen just the way it is without feeling ashamed of being middle-age-like.\nAlso there is western zen too since years ago when zen monks came to Europe and EEUU to teach it. Current european zen masters in general are a good place to go for knowledge about it and they dont think everything like XIII century's japanesse. Budhism has changed with western influence. (Nothing strange because is not the first time: http:\/\/en.wikipedia.org\/wiki\/Greco-Buddhism - mahayana is perhaps, a mix of western ancient philosophy and buddhism, the same way as Zen is a mix of ancient Chan and Taoism). And it's the same if karate is being influenced by western thinking and not only stuck in Japanesse views.\nSeems I have come rather late to the debate but what I think is important here is being aware of the difference between \"being created and designed as a buddhist practice\" and \"growing and developing in a culture influenced by buddhism\".\nIf we are to look very specifically at the case of karate then it is clear that the buddhist monk origin theory is not supported for reasons mentioned above. It is not even supported for some of the \"ancestor arts\" from which karate evolved. More importantly, the vast majority of karate practitioners, both modern and older, have never practiced the art as a part of buddhist practice. This means that by simple numbers the tradition is largely a non-buddhist (and non religious) one, regardless of any origins it may have had as such.\nHowever, many of the men who practiced and contributed to the art's history were \"practicing\" buddhists of one denomination or another or exposed to buddhist ideas in their everyday intellectual life (as pretty much everyone in south east Asia is to a greater or lesser degree). It is thus understandable that many of these individuals would want to find a place for their martial practice in their worldview or look for parralells with some of their other ascetic training practices in it. By the same token, much of the body of philosophy that has agglomerated around the physical practice to become the tradition of karate came out of a buddhist influenced intellectual millieu (although it is vital to remember that pure buddhist philosophy was nowhere near as influential on the cultures of China, Japan and Okinawa as it is in places like Thailand and in no way analogous to the influences of say christian thought in Europe or islamic thought in the Middle East.). Again, this in no way means that karate is a buddhist tradition but saying it grew up \"completely divorced from buddhist influence\" is also technically false as well. Of course the same can be said for any tradition coming from that very wide geographical region and you then have to question how much that influence actually shaped it's fundamentals.\nNow if the scope of the question is broadened to martial arts in general then it becomes clear that there are traditions and lineages that are more closely associated with buddhism (and other religions) than others. Shorinji Kempo was founded by the priest Nakamura Michiomi (a.k.a. So Do Shin) and deliberately contains a core religious element in it's practice and teachings (to what extent modern practitioners engage in and adhere to this is a different matter). As an interesting comparison however, the classical Fusen Ryu was also founded by a priest (Takeda Motsugai, a.k.a. Fusen) but contains no obvious religious or spiritual element. Ultimately it is down to various individuals who create their own traditions mixing religion and martial practice but that does not de-facto mean that the two are the same or even always complimentary. Of course the Shaolin temple is an example of one such group that became wildly famous and thus exacerbated the image of the two going together in a vastly unrepresentative way. Clever cultural legitiimization techniques and marketing did the rest.\nFinally, there seems to be a lack of distinction between the zen\/chan schools and buddhism in general here. As a faith or philosophical school (for it is both or either to different adherents) buddhism is far from one monolithic system. Certianly in the Japanese context the influence of zen on the martial arts and the samurai class that developed most of them was not even all that great. Other schools of buddhism such as tantric buddhism (mikkyo) were more influential and the native shamanistic practices of shinto were arguably even more so. Zen is somewhat unique and iconoclastic even among all the different sects and it is also worth noting that the Chinese and the Japanese ointerpretations of the same core ideas are also somewhat different.\nI think the above are important and good points. The culture in which an art evolves will influence the way in which it is viewed and practised. However, culture is not just the prevailing religious view and will also be influenced by things like political structure, class system, economic systems, past and recent events, etc. Religion is just one factor of many, and I think we can be clear that karate was never linked with any religious worldview in any significant way.\nKarate is not inextricably linked to eastern religion simply because it originated in an area where eastern religion dominates. If we were to say otherwise, we would also have to say the Soccer is a Christian activity because it originated in an area where Christianity is the dominant religion. And that Greco-Roman wrestling was fundamentally pagan, and so on.\nThe above examples are obviously absurd, and I'd suggest we would also see the karate \/ Buddhism \"link\" to be equally absurd were it not for the Bodhidharma myth.\nThe Bodhidharma myth is told to suggest an inseparable link that has existed for around 1500 years. The reality is Zen and the martial arts have only been \"linked\" (by some) for a very short time i.e. within living memory.\nZen in the Art of Archery was written in 1948. Funakoshi makes reference to the \"empty\" of \"empty-hand\" having \"Zen connotations\" in \"Karate-Do: My Way of Life\" in 1956. So we are not talking about and inseparable and fundamental link; as the common belief \u2013 largely stemming from the Bodhidharma myth \u2013 would infer.\nIt's also worth remembering that Itosu emphasised the lack of any link with religion in the very first line of his ten precepts decades earlier (1908).\nKarate had arrived in secular France by 1955, so we are not talking about a long tradition of linking martial arts and eastern religion before karate made its way to the west. It's a relatively isolated \"blip\" of a decade or so based on a false myth. A very long way from what is often directly stated or indirectly inferred.\nBuddhists practising karate does not make karate Buddhist (a point we all seem to agree on). The \"historical links\" to Bodhidharma \/ Shaolin and the idea of \"Samurai Zen\" have been discredited. It's a false history based on modern revisionism. And to me, that's a good thing that should be clearly communicated. It keeps things simple, devoid of unnecessary religious \/ philosophical terminology and keeps karate open to people of all religions and none.\nI'd also say that Jose's point is valid and I can see why practitioners of Zen may want to emphasise the parallels between their art and religion.\nI do accept that individuals will wish to integrate their religion in to all areas of their life; including their martial arts. However, when that has been done, I think it's a step too far to plant their religious flag and claim ownership of the art.\nI do know of people who would like to train in martial arts but who have declined to do so because they believe it is incompatible with their religion. i.e. \"I'm a Christian and I'd not feel comfortable engaging with Buddhist practises\". I understand that, but the tragedy is that the link they think exists \u2013 and that some say does exist \u2013 is known to be false.\nFinally, there seems to be a lack of distinction between the zen\/chan schools and buddhism in general here. (...) Zen is somewhat unique and iconoclastic even among all the different sects and it is also worth noting that the Chinese and the Japanese ointerpretations of the same core ideas are also somewhat different.\nI do have seen latterly that \"fear of being non christian if practising karate\" on internet in some South American forum. I America in general, both north and south, christian fanatism and misconception have more social acceptance than in Europe, and absurd stereotypes like \"being an atheist makes you a bad sad person\" still prevail for many. Atheism has higher natural acceptance in Europe.\nIt also has something to do with what Gavin says, but in a separate way, about people not knowing what zen really is and the differences between the diverse schools of buddhism. Zen can be the less mystical form of buddhism, not even recognizing the matter of reincarnation many times. People don't even know the general idea of concepts like \"karma\" is not in all the diferrent buddhism sects what people uses to say (reward\/punishment and so). But this is not the question here, to talk about buddhism and so. But somehow is another proof that we are discovering and uncovering many asumptions about karate and related culture that were wrong for so long until just now.\nThat's why internet and sites like this are great.\nIt's the same here. It's certainly not a widespread perception nor is \"religious clashes\" a widespread issue. Martial arts are practised by people of all religions and none. Which is how it should be and why I feel the \"zen link\" found in numerous texts should be rejected as a known myth which has no historical support.\nIf people wish to engage in zen practises and philosophy, then they obviously can, but I would suggest that martial arts training is not the right vehicle by which to \"evangelise\" that worldview. Especially when known falsehoods are spread in order to facilitate it. There is no established history or inherent \"zen nature\" to martial arts.\nI've had a couple of people express religious concerns about karate to me in person, and a few more via email (to whom the debunking of the myths on which their concerns were based came as a relief). So it's not a big issue, but still an issue that needs addressed I feel. Those who have been around for a while may remember an article in Fighting Arts International on this topic too i.e. religious concerns about Karate's supposed link with practises incompatible with certain religious views. That was twenty or so years ago, but I recall it sparking a bit of discussion in the following issues letter's pages.\nAway from what is relatively a minor issue effecting only very small numbers, I think there is the bigger issue at play here. Do we want to practise an art polluted by myth and falsehood? Or would we prefer an honest history and confirmed context? To me, I want the latter. That's why I feel it is important to debunk ALL myths so we can do what we do in an authentic and down-to-earth way.\nLeaping side kicks were never used to knock warriors of horseback. Karate was not developed by the oppressed Okinawans to repeal their samurai overlords. And karate does not have any religious elements or long established religious tradition.\nWhen we get rid of these myths, I think karate is stronger, more appealing and more honest as a result.\nAgain, we see modern revisionism at play as opposed to any genuine long-established practise. I would suggest that just as a bunkai based approach is more traditional than modern \"3k karate\" (kihon, kata and kumite; and never the three shall meet) because it has been around far longer, the dropping of the \"Zen link\" would also result in a karate that was more authentic. Such karate is based on a long established practise as opposed to a modern veneer draped over karate is some quarters to make it fit with the prevailing martial fashions of pre-WW2 Japan.\nTo each his own of course, but whatever way people chose to go, we should avoid revisionism and false history masquerading as tradition.\n\"A former officer in the Japanese army, Yabu [Kentsu] introduced many procedures still practiced in karate schools worldwide... These innovations included... bowing upon entering the training hall, lining up students in order of rank, seated meditation (a Buddhist practice), sequenced training, answering the instructor with loud acknowledgment, closing class with formalities similar to opening class. Most of these procedures already had been implemented in judo and kendo training and reflect a blending of European militarism and physical culture with Japanese neo-Confucianism, militarism and physical culture. However, these procedures did not exist in China, or in Okinawan karate before Yabu\" ( Madis 2003: 189).\nDonohue points to \"the ritual of the bow and the recitation of dojo kun (the precepts of the dojo normally recited at the end of a training session)\" (1993: 113) as key markers of a ritualized behavior that serves to create a privileged space in the dojo. These practices also signal a distinct shift from the karate practiced on Okinawa as described earlier (Friman 1996, Krug 2001, Mottern 2001) and mark the beginning of what is thought of as 'karate' today. Through the adoption of the sport and militaristic elements, as well as the spiritual philosophies of Japanese martial culture, karate was able to find a place in the culture of mainland Japan.\nOften supported by and disseminated through the government, these adaptations of the practice found their way back to Okinawa and were largely embraced both by masters and students. To this day, in Okinawa as well as Japan, students wear the gi and colored belts, line up in order of rank and drill in precise lines.\nCould you please give more details on the book? I wasn't able to find anything on just name an year, unfortunately.\nif you take a look at the above mentioned article you'll find the bibliography at the end.\nA very entertaining and interesting discussion.\nI'm not aware of a specific \"religious practice\" requirement in karate, neither generally nor in any particular style. There is, however, a definite \"moral aspect\" to karate ... a sort of \"thou shalt never use thy skills for evil\" imperative ... that one doesn't see in other areas of endeavour. There is a courtesy\/reigisaho aspect as well, which westerners often find confusing, pointless, or worse. One can seek various explanations for the origins of these aspects ... Kano's \"three levels\", Zen, bushido, political and cultural forces at work in Japan in the early 20th century, and so forth ... but my practicing karate ... all aspects of karate ... no more makes me a Zen Buddhist than it makes me a Japanese schoolboy intent on joining the Imperial Army and invading China. It also doesn't make me a samurai warrior, no matter how good my Halloween costume is.\nStrange Indian mystics lying in ponds distributing swords is no basis for a system of self defence. Personal safety derives from biomechanical efficiencies, not from some farcical aquatic ceremony. You can't expect to wield supreme martial power just because Bodhidharma threw a sword at you.\nThis is KARATE... We use our hands for protection and nothing more. We learn fighting skills not to show we're tough but as our defense to escape the untimely death. We train to be great fighters but not to prove we're the best. But to show the essence of DISCIPLINE, PEACE and HARMONY to all mankind. So that we'll be worthy in the eyes of men and above all in the eyes of God.\nNow, I've never trained with them. I know nothing about them apart from what I've said already. But to me, their core statement seems to be \"bang on\" in terms of what karate is all about ... except that I don't think you need to strive to be worthy in the eyes of God to be a good karate-ka. It just so happens that they live in a very Catholic country, where that sort of message is common, and is easy to understand. That is a message which will resonate with their local students and would-be students, and parents wondering about sending their kids there. Would it resonate equally as well elsewhere? Of course not. But it creeps in, in a country where that sentiment is common.\nAs I understand it, the shaolin monastary was a place of sanctuary. Many chinese people went there to avoid prosecution. So, you had soldiers\/bandits etc. that had martial training going there and continuing to practice their arts and sharing them. So, while they lived in the temple many of them may or may not have been \"monks\" as we think of them now.\nI think part of the problem with this topic is that SOME instructors in Japan taught Buddhist concepts with their karate to westerners and incorporated it into the training. They didn't know any better and thought it was part of karate. For example, Mas Oyama had students sit in zazen and meditate in class and at special trainings and even conducted his winter training at a temple. Another example is that Morhei Ueshiba mixed his aikijutsu training with religious concepts and blended the two and taught his new art of aikido with that religious aspect to it.\nWe have the same thing here in the US. Some istructors add their Christianity to their karate training and teachings. If they were to go to another country that had no exposure to karate and taught it with their own Christian values, those people might wrongly assume that it was a part of karate as well.\nSo, I think it is safe to say that originally Okinawan karate did NOT have it's roots in Buddhism, But, it was added by many Japanese instructors to their new art.\n\"Inasmuch as there is virtually no written material on the early history of karate, we do not know who invented and developed it, nor even, for that matter, where it originated and evolved. Its earliest history may only be inferred from ancient legends that have been handed down to us by word of mouth, and they, like most legends, tend to be imaginative and probably inaccurate.\"\nThat seems apt to our discussion here.\n\"In Okinawa in older times there were, as we know, two schools, Nahate and Shurite, and these were thought of as being related to the two schools of Chinese boxing called Wutang and Shorinji Kempo that flourished during the Yuan, Ming and Chin dynasties. The founding of the Wutang School is attributed to a certain Chang Sanfeng, while the founder of the Shorinji School was said to have been Daruma himself (Bohidharama), the founder of Zen Buddhism. Both schools, according to report, were extremely popular, and their adherents gave frequent public demonstrations.\nI mentioned Mokuso meditation in karate dojos. It seems perhaps it was introduced by Shoshin Nagamine (1907-1997) who recognized there was no Zen Buddhism in original Okinawan karate. The article says he introduced zen meditation in Karate in Japan.\nI think the site is mistaken. While Nagamine-sensei was heavily influenced by his religious beliefs, he certainly didn't introduce the practice of mokuso to karate. That was pretty well established before he even began teaching. I think we have to credit the cross-pollination with various modern Japanese Budo (particularly kendo) in the Japanese universities with that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Having effective clerical and administrative skills are essential in today's work environment. Being organized, punctual, and effective in your communication skills, both written and verbal are crucial if you want to achieve your goals in any endeavour you pursue. In this course, you will learn the core skills that will help you use your resources efficiently, manage your time wisely, communicate effectively, and collaborate with others skilfully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Most of us, saved and unsaved, know or at least know of Jesus' model prayer in Matthew chapter 6. The Lord's Prayer, as it is commonly called, is an example of how we should pray; something like a template. Most people I know can recite the Lord's Prayer without hesitation, but they don't really know it.\nWhy do I say that? Well, let's look at Matthew 6:10 (KJV), Emphasis added.\nOk, what about NIV. Emphasis added.\n\"Your will be done.\" Who is Jesus speaking to? God, that's Who.\nOne thing that should accompany our prayers is telling God we are ultimately following His will. Will is used to express desire, choice, or intent. No matter what we want, what we think, what we see, or what we believe we need to live according to what God instructs us to do.\nGod's will is good, acceptable, and perfect (Romans 12). His intents are holy. The Lord's desires for you are exactly what you need. His choices for you are the right thing. Your ways are just that, your ways. Your confused, misguided, and frustrating ways. Stop fighting Him. Trust God today no matter how difficult it seems.\nPrayer: Ask God to forgive you for trying to do it your way. Tell Him you are tired of being tired. Confess that you cannot succeed on your own. Identify the areas in your life that need to be improved, removed, or changed.\nGive Up Part 4: Do Something!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Time, space, money, or the environment \u2014 which one is the most important business asset? Enadoc has provided an answer.\nAs it turns out, every asset is equally important especially in the business world. With Enadoc, a Document Imaging system (for starters about document imaging system, read this), we don't have to pick which asset we should bank on. Enadoc greatly helps businesses to save time, money, space, and the environment all at the same time.\nEarlier in our blog posts, we mentioned that Enadoc simply takes care of many things. Have you ever experienced looking for a file but couldn't remember which folder to look into? Enadoc boasts its innovative feature called Tag Cloud. Simply scan the file using your favorite scanner or just upload the file on the Enadoc site, add your tags, and you're done!\nNow how does this work? Let's say you can't remember the exact file name you're looking for, but you know it's a receipt. locate the word receipt in the Tag cloud, click until you see the document, and voila, you have your file in two easy steps.\nThis is true \u2013 much more when you're in a big office with a myriad of papers. HR and finance personnel (or every single employee in the office) carry the same burden \u2013 they walk to the filing cabinets or a storage room to retrieve the document they are looking for. Once they found the document, they walk back to their work station. Now, multiply this around 30 times in a day times 5 working days\u2026 you get the point.\nEnadoc already addresses this time and energy consuming problem: employees won't have to stand up and leave their desks to look for a file. With the two-easy-step file retrieval, the possibilities are endless.\nIt works like a charm; forget the traditional folder system. Welcome to the information era where more and more documents are becoming instantly demanded than ever. Information communication all around us is in dire need of everything quick \u2013 and we are lagging behind. Maybe this is the time for our offices to cope? And oh, did we mention we're just only at the beginning of the list?\nLet's accept it: humans are prone to errors. Moreover, whether we like it or not, we pay them regardless. Information handling is beginning to change the management landscape \u2013 and we need an automated system to do the daunting work. Isn't it more lauding to allocate a dedicated personnel to do more meaningful tasks than to archive and retrieve files, which can be done automatically? Enadoc is the one you're looking for.\nCiting CoastalBusiness.com, \"based on math, an individual will need to purchase 12 inkjet printer ink cartridges during the six-month period\u2026 if cartridges cost $28 each, plan spending $336 every six months.\" But hey, don't forget the bulk paper rims and printer energy costs.\nSleep like a princess: Backups are always there!\nNever again worry about disasters. Whether uncontrollable fire, or earthquake, or flood break loose into your office, have some beauty rest and be worry-free. Documents are the sustaining factor of the businesses. Reports tell us that businesses that suffered into a major data loss have shut down immediately. Yes, backups are important, but where those backups are is more important to have in mind. Other than prone-to-disaster physical storage, back up with cloud.\nEnadoc just takes care of that. See how Enadoc lightens your life?\nThe number one problem about home-grown systems is about updates and keeping up to date. These servers and systems may do well but these are also far from perfect.\nThe good thing about Software as a Service (SaaS) is automatic updates. You'll never have to fret whether the solution you're enjoying is the latest version (fixed issues, compatibilities, and drastic improvements).\nMore space means more fun. An office pile-full of unusable printed papers is like hell unleashed on earth. Papers flung everywhere on your table loses your bravado, let alone your productivity. Work spaces are for personal spaces \u2013 and are never intended to be a majestic podiums for useless pile of papers.\nWe are not just bidding farewell to papers, we also say, \"welcome, space (finally).\" The president of Canon PH Marketing said that he imagines the Canon office should not be for papers and filing cabinets \u2013 where he could walk leisurely with spacious work place. In fact, they could un-clutter up their two to three rooms from papers to create more work spaces, he added. Workplaces become workplaces once again.\nPapers, however environment-friendly they are do come from trees. Did you know that there are certain parts of the forest that are reserved solely for producing papers? Yes, you heard that right. Eighty-seven percent of new paper comes from the trees that are raised only to become a paper. Keep in mind that these are not just trees waiting to be cut down, there are hundreds of ecosystems living in a single tree. The only solution? Lower the demand by not using papers.\nIn Europe alone, there were 1.21 million tons of waste in paper production. Hey, but we recycle papers! That's true. However, recycling produces harmful by-products and wasteful energy. Recycling processes still use fossil fuels, which we all know affect the air we breathe. Also, there are by-products called paper sludge that contains paper fibers, inks, cleaning chemicals, and dyes, which only sip down straightly to our groundwater.\nThere we have it. If you had time to read through the entry, please have time to consider the things enumerated here. Worrying about cloud security? We got you covered. Management? Enadoc has a good one. Cloud for business? Why not?\nIt is only a moment of time where every business in pharmaceutical, banking, school, production, telecommunication, among other industries, will turn to cloud as their most trusted document imaging system. And finally, the lesson here is that we have just found out we could save time, money, space, and environment all at the same time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everton have quickly moved to quash reports linking Richarlison with a shock summer switch to fierce city rivals Liverpool.\nAccording to UOL Esporte, Richarlison's representative, Renato Velasco, recently visited the home of Liverpool boss Jurgen Klopp.\nThe nature of their discussion is unknown, however Velasco doesn't boast any other clients based in Europe, while Klopp has previously revealed his admiration for the young forward.\nPrior to December's Merseyside derby at Anfield, when discussing Richarlison, Klopp declared \"what a player he is\".\nHowever the Liverpool ECHO have claimed that Everton have no desire to sell their prized asset, regardless of the fee he would command.\nReds boss Klopp is no doubt keen to bolster his attacking ranks this season, and may have to turn his attention elsewhere.\nDaniel Sturridge is set to leave at the end of his contract while Divock Origi's future is far from certain, highlighting the need for reinforcements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Researchers have found that most dissociative drugs work by disrupting the neurotransmitter glutamate, a brain chemical responsible for activating various neurons. Salvia, meanwhile, works by activating certain opioid receptors in the brain. At higher dosages, dissociative drugs can harm the brain and may cause aggressive behavior, seizure, coma or even death. For this reason, signs of a dissociative overdose should be taken very seriously.\nPCP, commonly known as angel dust, hog, wack and lovely, is a dissociative drug available in the form of white crystalline powder that dissolves quickly in water or alcohol.\nTreating addiction to dissociative drugs Dissociative anesthetics are a diverse group of substances. For this reason, individuals who are addicted to dissociative drugs may benefit from treatment programs that also includes a detoxification treatment geared specifically toward their drug of abuse. Enrolling in a comprehensive treatment program that includes customized dissociative drugs detox treatment may help one embrace a drug-free life. Only a well-planned detox program can help one withstand the withdrawal symptoms under a clinically controlled environment.\nPeople who are addicted to dissociative anesthetics can fight their addiction through medication management, cognitive behavioral therapy, stress management therapy and other professionally administered treatments that are offered at some of the best dissociative drugs detox centers or some other state-of-the-art specialized centers.\nSovereign Health offers one of the best treatment programs to treat addiction to dissociative drugs. To know more about Sovereign Health's treatment programs for addiction to dissociative anesthetic drugs, please contact our 24\/7 helpline.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BSX Records presents MUSIC FROM THE STAR TREK SAGA, a brand new collection of freshly recorded and arranged performances by the Meridian Studio Orchestra from various corners of the STAR TREK universe. This 18-track collection brings together music from every generation of Star Trek, covering all TV series, a selection of the movies and - of interest to many fans - STAR TREK: THE ANIMATED SERIES along with other rare gems.\nMUSIC FROM THE STAR TREK SAGA includes music composed by Alexander Courage, Gerald Fried, Jerry Goldsmith, Dennis McCarthy, Ron Jones, Jay Chattaway, Yvette Blais & Jeff Michael and Michael Giacchino. MUSIC FROM THE STAR TREK SAGA features music performed by an array of talented performers, including Original Series composer Gerald Fried who contributes a memorable medley of classic TOS music. The inclusion of the Animated Series theme is bound to please a multitude of fans who have been hoping to get the music from the short lived TOS spinoff for a very long time.\nMUSIC FROM THE STAR TREK SAGA includes liner notes from author Randall Larson providing important details about the album selections, the composers and the featured performers. MUSIC FROM THE STAR TREK SAGA features music from STAR TREK: THE ORIGINAL SERIES, STAR TREK: THE ANIMATED SERIES, STAR TREK: THE NEXT GENERATION, STAR TREK: DEEP SPACE NINE, STAR TREK: VOYAGER, STAR TREK: ENTERPRISE, STAR TREK: THE MOTION PICTURES and the 2009 feature film version of STAR TREK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"During 2017, the first group of the oldest Podgorica citizens completed three-week internet course, which was organized by partners: M:tel, Business Academy and the Faculty of Information Technologies of the University of Mediterranean.\nThe course purpose was to introduce the members of older population with the basics of computing and using the most popular applications, and the participants first mastered the skills on personal computers, and during the second part of the course they worked on tablets.\nCourse participants were provided with the opportunity to attend basic computer course without financial compensation. For all participants of the course, the Faculty of Information Technology provided the necessary technical conditions and the training was held in the premises of the Faculty. Course participants had the opportunity to get acquainted with the work on the computer, Microsoft Office package of the program as well as how to use the Internet and its services. The pensioners will be able to use the adopted knowledge and skills in everyday life so that they enriched with new interests.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Computech Enterprise Solutions Private Limited (CES) is listed in Software Companies. The average user rating and a snapshot of Computech Enterprise Solutions Private Limited (CES) follows below, followed by all the user reviews.\nThere are no events at Computech Enterprise Solutions Private Limited (CES) scheduled currently.\nTimings are worst and management is too worst. No process, No documentation.\nCons: Management, timings, work, culture....etc...everything is too worst. Better not to join here. If you want to screw up your carrier, then join here.\nfullhyd.com has 700,000+ monthly visits. Tell Hyderabad what you feel about Computech Enterprise Solutions Private Limited (CES)!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rather than turn this into an attack on Newsie, or take some random, tenuous, political slant, at this special time we should pity the poor pilot who, in order to comply with CAA guidelines apparently, had to conduct a highly dangerous manouevre climbing onto the wing of a plane in flight, drop his trousers and defecate into a rotating engine at great risk to various parts of his anatomy. This man deserves a medal and should be named immediately so he can be recommended for some kind of award, having saved the lives of his passengers in such a heroic manner.\nNewsie, when you stir after the festivities please elaborate on the detail of this story. We need to know more about this superhuman feat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The day went on as any other, and school is almost over. A girl, let's call her Janice, was tapping her pen on her desk as the time ticks.\nI need to get out of here, \"Janice\" thought to herself. See, beyond her parents' perspective, Janice wasn't very popular, in fact, she was almost a ghost in society.\nShe never really talked to anybody and blended in the back of the classroom. Typical, isn't it? Just like how all the stories of malicious murder are.\nJanice was looking at the clock with more anticipation than usually, it was her birthday and she wanted to get home and see what she got from her parents. She really considered it a \"Sorry we're always arguing\" present, considering she has had poor amounts of sleep because of their constant fighting.\nAs the bell rang, Janice quickly stood up, gathered her things, and proceeded to leave the room. As she was heading the corner towards the exits, Janice was cut off by a girl, let's call this girl \"Samantha.\" Samantha stopped her, smiled and started flipping the pendulum of her necklace around her neck.\n\"So, uh, I heard around that it's your birthday today, my friends Bobbi and Victor are wondering if you'd like to hang out and you know, celebrate?\" Samantha asked smiling a sketchy smile.\nConsidering Janice didn't talk to anyone much, she was skeptical. She had talked to Samantha a few times before, but never really stuck in with her group. Janice thought about her parents and the fighting, and quickly but modestly said yes. Samantha smiled, hugged Janice, and said she wouldn't regret it.\nAs she backed away, Janice noticed the pendulum of a star in a circle. She's seen that before, somewhere, but was unsure of where. She still got a bad vibe, but decided it was too late to go back now. Samantha then asked for her address so her and the others could pick her up in a few hours.\nJanice left school, said goodbye to Samantha, and went home.\n\"I'm home!\" Janice exclaimed as she walked through the door.\nHer mother came out of the kitchen, smiling wildly and her dad appeared behind her mom smiling, too. The both tugged on her arm to lead them into the living room.\n\"It isn't much,\" her mom said smiling wide,\" but it's still a birthday gift we're POSITIVE you'll just LOVE.\"\nAs she said that, Janice's dad gave her a considerably small and wrapped package. Janice tore the wrapping off and found that she had gotten a new iPhone 5 for her birthday. Janice gasped and hugged her parents.\n\"Thank you, this is the best gift I could ask for,\" Janice said, almost stammering while trying to speak.\n\"It's got unlimited everything, calling and all,\" said her father, smiling smugly.\n\"Oh! I almost forgot!\" Janice said as she played with her new phone,\" my friend Samantha wanted to celebrate my birthday with me.\"\n\"Oh, goody, you've finally got someone to hang out with. I hope you have fun,\" said her mother, not very convincingly, though. She was really hoping she could make a special dinner for Janice, but oh well.\nWhile playing a new app on her phone, Janice heard a knock at the door. That must be them, she thought to herself, I can't wait. Janice opened the door, and, there enough, was Bobbi smiling and waved slightly at her.\n\"I'm 'ere to get you. Sam wants us to meet 'er and Vic at 'er 'ouse,\" Bobbi said, with that British accent Janice loved oh so much. Secretly, Janice has always had a feeling Bobbi knew she had a terribly strong crush on him.\n\"Alright, let's head out then,\" Janice said, blushing slightly.\nThey got in Bobbi's car, a nice Lamborghini. Janice found out his dad was a coroner making upwards of about $170,000 and his mom was a forensic scientist making around $80,000 a year, so he was able to get anything he wanted. Bobbi was, however, a reckless driver and would speed, scaring Janice.\nBobbi laughed,\" If you think that was a thriller, you should wait and see what we 'ave for you later.\"\nJanice smiled nervously and relaxed. They arrived at Samantha's place. It was very nice looking, three stories, a nice backyard that seemed to just disappear into the woods behind it. As they walked in, Samantha and Victor ran to greet them and they exchanged awkward waves and Samantha hugged Janice, necklace swinging. They hung out around Samantha's place, played games, got to know more about each other.\nJanice found out that Bobbi was directly related to a king of England that was overthrown. When that happened, his ancestors moved to France and came to America, settling in Louisiana. Victor was from, of course, Russia. His great-grandfather fought for the army during the Soviet Union era, and later moved his family to the north parts of America, which eventually led them to where they are now. Samantha was related to the people who called themselves the Druids from Gaul. Gaul was later France. Somewhere along the line, Samantha's family was linked to being Satanic and was exiled from Gaul and they moved to America.\nThe clock showed 7:06 on it and Victor realized what time it was.\n\"Ve 'ave to get zis party going, iz dead,\" Victor said, in his barely understandable Russian accent that still entertained Janice.\nSamantha then stood up, grabbed some glasses, and poured some vodka into four glasses.\n\"Here. Vodka, Victor's favorite kind, too,\" Samantha said, handed everyone a glass and keeping one for herself, \"nothing starts a party like a nice buzz.\"\nJanice kindly declined, seeing as she didn't drink. Thinking she was going to be looked at as a 'party pooper' she almost hesitated to give the glass back, but Samantha shrugged, took the glass, and handed it straight to Victor, who downed it immediately after drinking down his own glass. As far as Janice could tell, no one thought any less of her, and not taking that drink would soon be shown as a mistake in the future.\nThings went here and there, and out of nowhere, someone suggested they go make a fire in the woods. No one objected, so out they went. The other seem to be stumbling and Janice was laughing, they made it to the edge of the woods, and they kept going. They gathered fire wood, the two guys tripped and stumbled around, laughing until they were red in the face, Samantha kept working, still laughing. Janice noticed a small, black, box-like suitcase and thought it was nothing, so she continued working.\nThe finally gathered around they fire, Victor and Bobbi slapping each other lightly in the face and swinging their arms around like dead weight, Samantha was laughing, shouting things like \"I bet you won't kiss 'em!\" and \"Ladies! Ladies! You're BOTH beautiful!\"\nThey finally calmed down, sat next to the girls, Victor was next to Samantha, Bobbi respectively next to Janice. Inside, Janice was so ecstatic she could barely hold it in. Bobbi eventually wrapped Janice up in his arms after she complained about being cold. Samantha looked over and smiled.\nSamantha then said,\"We should play hide and seek.\"\nJanice laughed, not thinking she was serious, but then realized Bobbi and Victor got up to hide, and Janice followed. Janice ran and ran, her heart racing, adrenaline pumping, and palms sweaty. She ran some more and tripped on something, and before she noticed, Samantha was on top of her, holding her down with all her weight.\n\"Gotcha,\" Samantha whispered. Out of their hiding places, Victor and Bobbi slowly walked up to them, Bobbi with the box in hand. Janice laughed nervously and looked at them.\n\"I-I guess I'm it, huh?\" Janice said, stuttering slightly.\nVictor quickly picked up some ropes tied to trees and knotted the around Janice's limbs.\n\"Ja, ve guss your eet. Zam, geet ze prod,\" Victor said in his now-scary accent.\nSamantha got up and grabbed a metal cow prod with the circle surrounding the star. Janice was starting to have a panic attack and hyperventilating. She tried to squirm, but Victor tied them in slip knots. Bobbi put down his box and unwrapped a fine selection of Coroner's tools. There were all kinds of surgical steel blades, scissors, stitch hooks, all to be put to use.\nSamantha came rushing back with the hot prod and Victor immediately lifted Janice's shirt and stabbed the prob to her skin, burning the symbol permanently. Janice hopelessly screamed in pain and looked at the permanent stamp on her stomach. She then realized where she remembered that symbol was from and what it meant. It was a Satanic symbol, it represented the head of the goat.\nAs Janice came to reality, Bobbi bit his lip, almost as if he was in pain, and slit Janice's cheeks, extending her 'smile', and then sew her entire mouth shut. Janice whimpered as she cried, and Bobbi made a motion as if whipping away tears from his eye. Samantha grinned menacingly, and Victor was nowhere to be found by Janice. Bobbi finished his job, and Samantha took out some candles and a lighter from the box, putting the candles at five separate points of a star that Janice was lying on. Everything was ready for their sacrifice. Victor came back with a cigarette, Bobbi took out a knife and carve an uppercase I into Janice's abdomen, shaking, and removed her intestines, leaving her heart in place. Samantha started chanting something in a language Janice had never heard. Bobbi finished his preparations, and covered the pentagram Samantha burnt into Janice's stomach, just below the incision, with blood. Janice quickly felt sticky blood around her and blacked out.\nWhen she woke up, daylight shone, and the incision was stitched together. Janice was free from the ropes and started looking around, afraid to get up. She slowly rose, the stitches felt like the were on ripping as she carefully adjusted herself up. As she looked around, everything looked eerie and the colors looked faded. She slowly made her way back out of the woods. She couldn't believe she was still alive, for the moment. She saw another body in the same position she was in only the night before, and the person was still breathing. She slowly approached the body and it started panicking. Janice noticed the person's eyes were missing, and she threw up next to the body. She kept walking, thinking she was going to get caught if she tried to help. She walked to the neighbors next to Samantha's house, and only then realized she left her new phone at home, and asked to use theirs to call her mom. She called her mom, and called the cops. She was sent to a hospital and returned home after everything was fixed. She felt unnerved, but grateful to be back home. As time for her to go to bed came around, she took her phone and went to her room. She lay down, got under her covers and got a text from a number she didn't know.\nFunny, she thought to herself, I never gave anybody this number.\nShe opened the text thinking it was a mistake and started shaking and breathing heavily as she read the message.\nThe message said,\" You could've saved me. You could've helped me, untied me, anything. But you didn't. You left me there to rot and die, and now it's your turn.\"\nShe quickly turned the lights on the see a deathly skinny, eyeless person with a pentagram burned in their stomach standing in the corner of her room. The person was holding a spoon in their hand and charged Janice with unspeakably great speed, far more than anyone in their shape should be capable of. The person stuffed a pillow over Janice's mouth and carved her eyes out with the rusty spoon. The figure then threw her out of her second-story window and disappeared. The next day, cops found Janice's body lying outside her house, eyes missing. The death was ruled murder, but nobody was ever able to find the killer, and her case was sent to the vaults, never solved. Though many speculate it was the same people who carved the pentagram and dissected her, but the police never found a Bobbi, Victor, or Samantha that she claimed to have done it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"testsuite::test_gap_junction - Ensure that gap_junction connection can only be created between neurons which support them.\nEnsure that NEST throws an exception if one tries to create illegal gap-junction connections.\nFurthermore it is checked that the delay cannot be set for gap-junction connections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarstf b/data_all_eng_slimpj/shuffled/split2/finalzzzarstf
new file mode 100644
index 0000000000000000000000000000000000000000..1be7c8c7c4cc9166a945116946d995c02b6016e0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarstf
@@ -0,0 +1,5 @@
+{"text":"Irish house DJ John Gibbons takes the stage at the Olympia Theatre for his biggest headline show to date, tickets \u20ac21.50.\nWith his most recent album Gemini which features an abundance of artists including Lil' Yatchy, Kesha & OFFset from Migos! Macklemore will perform two shows at the 3 Arena this week, some tickets still available!\nThe multi faceted Dr John Cooper Clarke will be performing a sold out show at Whelan's. Expect the unexpected!\nRepresenting a new wave of Irish Hip-hop, NEOMADic will be performing at the Bernard Shaw from 8PM!\nVulpeck's multi-instrumentalist Theo Katzman is back on our shores alongside the Four Fine Gentlemen on Thursday evening! It's always a great gig every time he plays! Get your tickets before they sell out!\nHaving sold out a number of big events over the past few years, Hannah's aims are now set higher! House & Bass tunes from 11PM.\nArcade Fire mark the release of the newest album Everything Now with a headline gig at the 3 Arena. After a fantastic performance at Malahide Castle last year be sure to expect more!\nNew album \"Critical As Water\" was released on March 16th, catch him at The Academy with doors from 7PM!\nElectro-indie-pop-duo take over the Button Factory for a sold out late night show! Doors at 11pm for this!\nRescheduled from earlier on in March! Ailie will be making her mainstage debut at Whelan's on Sunday evening! Nothing better to wind down the weekend! Doors from 8PM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I'll get it. It's all in a day's work.\"\n\"You've been with us a long time, Winnie, and we're prepared to offer you a generous severance package.\"\nA panhandler with a sign that reads \"A portion of the proceeds goes to him\" and an arrow that points to another panhandler.\n\"Should anyone inquire, Harrington, our portions are generous, not liberal.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being in a car accident is scary and frustrating enough, but when the other driver takes off, those feelings can be escalated. A hit and run is any accident in which a driver purposefully leaves the scene without leaving any information. This can include a driver hitting your parked car and not leaving a note, or something more serious like a car hits you or a pedestrian and speeds off. Regardless, it's important to take the right steps if you're in this situation.\nGather as much information about the car or driver as you can. The license plate number, color, make or model of the car, and even the direction the car was heading is all useful information for police and your car insurance company.\nDon't attempt to chase the other driver. Instead, pull over safely and call the police immediately. Police reports need to be submitted within 48-72 hours. Look around for any possible witnesses to the accident and ask for their names and contact information. Be sure to ask witnesses if they have any information about the driver or vehicle that hit your car.\nIf you're a Metromile customer, follow our Accident Checklist and assess accident damage. This includes recording the incident by surveying the scene and taking pictures of everything you see, including the damage from all necessary angles, as well as pictures of the spot where the accident happened.\nContact your insurance company to report the accident and file a claim.\nIf you find yourself in a situation where your car was damaged when you weren't there, follow similar steps. Document any damage and ask around to see if anyone saw what happened. File a claim with your insurance company but remember that without a license plate number, it's unlikely you'll be get reimbursed, or you'll be subject to your deductible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are keen to help you improve your smile and the replacement of dark tarnished fillings can enhance even the prettiest of faces. We do not advocate the removal of all your silver fillings simply for the sake of it but will advise you when appropriate of the options available to you.\nAmalgam and silver fillings are the metal coloured fillings that many people have. Amalgam was the traditional material used for fillings for many years. Amalgam fillings are a mixture of a metal alloy and mercury. We do not place amalgam fillings in our pregnant patients or nursing mothers.\nWhite fillings are as strong as silver fillings when used in the correct situation. They also bond to the tooth so there is less chance of liquids leaking between the filling and the tooth.\nIf your dentist feels that a white filling would not be suitable replacement, we can advise on other white alternatives such as a porcelain inlay or crown.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How can I make a fund transfer from my trading system ?\nStep 2 : Enter Member ID as \"07708\" and your User ID (18061971-07708) and sign in using NOW login credentials.\nStep 3 : You can click on reset button instead of Sign in button if you have forgotten your password.\nYou will get the successful transaction confirmation page below once you complete the transaction on your Bank page.\nStep 2: Login using your Trade login user id given to you.\nEnsure that you select the Equity segment before submit the details required on this page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasdll b/data_all_eng_slimpj/shuffled/split2/finalzzzasdll
new file mode 100644
index 0000000000000000000000000000000000000000..2281268c3fd34dfbdda169c03c57912a3a5fc92c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasdll
@@ -0,0 +1,5 @@
+{"text":"Indiana American Water has been a reliable source for water in communities all over the country, and if you're one of those people located in Indiana, you should have the Indiana American customer service number on speed-dial. Indiana American Water focuses on water quality as well as how their productions affect the environment, so you can be sure that their service is dependable and high-quality. Call them today and ask them anything related to their water services.\nDo you have a hard time trying to understand your water bill? Are you concerned about a current issue with you're experiencing with your water supply? Their customer service team has the experience and reliability to handle situations like these and more. With this phone number, you can be sure that they'll be handle problems, questions, and requests that you have by just a simple phone call.\nIf you're looking to pay your bill but don't want the hassle of your online account, then you can use their mailing address to send off your payment. This is also the address that manages water concerns and other information related to their services.\nIf you want to learn about how Indiana American Water can help you with your water bill or help you manage your water services, you can contact them through the contact us form below. By emailing them, you'll get a response within 24 to 48 hours, and their representatives can help you out with requests such as attempting to restore water services.\nHaving a hard time paying a bill? Need to access your account but are having trouble doing so? Head n over to their customer service page to get answers to questions like that and more. Here you'll find extensive amounts of information on everything from paying your bills, reduce your statements, check up on water quality reports in your area and more. Get all your water service needs in one place, and if this page doesn't help you, you can call their customer service number, send them an email, or use their mailing address.\nWant to know how to survive the summer heat and keep hydrated? Want to know what Indiana American Water is doing for holidays like World Water Day? The official Indiana American Water Blog provides excellent resources for people like you wanting to know more about how to use your water safely and wisely.\nIf you love how much money you've been able to save by using their utilities? Then follow their social media channels. By sharing this information with your friends, you'll get the latest news on the company and what they're doing to educate and improve the use of water.\nIndiana American Water is a public utility service for the state of Indiana and is a subsidiary of the Indiana American Water company. Providing services in 15 states all over the country, this company serves millions in providing fresh clean water for everyone to drink and use. The Indiana American Water company brings commitment to a new level in providing clean, renewable water to its customers throughout the state of Indiana.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Classes are 45 minutes long, suitable for all levels of ability and hold up to 6 participants. We provide mats and all small equipment so all you need to do it show up!\nWe aim to improve your foundational mobility giving you sound training base in which to strengthen mainly the core muscles, hips & shoulders using various Pilates and corrective exercise techniques.\nImproved posture, reducing chronic pain due to a sedentary lifestyle, and a stronger core are a but a few of health benefits of pilates.\nAre you a pre\/post natal lady who wants to maintain a low impact\/intensity fitness regime with all the pilates exercises tailored to your level of ability?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wisconsin has long been known as \"America's Dairyland\" offering a trademark Midwest experience with a slower pace of living, less traffic, few big cities, and hospitable and friendly neighbors. The attitude translates well to Wisconsin nursing homes where many decide to make it their short-term care or long-term care home, along with many that relocate to the state specifically for that purpose.\nWisconsin nursing homes feature moderate costs for semi-private and private rooms that reflect the national average. Therefore, Wisconsin skilled nursing facilities are neither expensive nor extremely cheap. Yet with such a terrific location, many elderly seniors are quick to want to make Wisconsin long term care their next home.\nThe average semi-private room in Wisconsin nursing homes goes for $42,000 per year. It is about $1,000 more per year than the projected average for the rest of the United States. Meanwhile, a private room averages nearly $97,000 per year. The private room average is especially higher, as nationally private rooms tend to go more for $82,000 to $86,000.\nThe good news is Wisconsin nursing home costs are rising at a slower average than the rest of the country. Assisted living facilities only rose 2.5 percent in the last year, and nursing homes in Wisconsin demonstrated a similar trend. Skilled nursing facilities in other parts of the nation are rising much higher, nearly double at approximately 4.5 percent.\nWisconsin has a fair selection of nursing homes with over 400 different providers. Affordable skilled nursing homes in Wisconsin are spread around parts of the state. A vast majority of the region is rural with farm or ranch land. However, larger communities with many senior living and care options exist in Milwaukee (595,000 population), Madison (233,000 population), Green Bay (104,000 population) and Kenosha (99,000).\nRacine, Appleton, Waukesha and Oshkosh are equally pleasant cities for elderly care facilities in Wisconsin. Depending on your need, Wisconsin nursing homes are broken down by short term care, rehabilitation or long-term care. Some Wisconsin nursing homes offer care of all three inside the same complex.\nWisconsin long term care features a good blend of amenities. Standard features include routine checkups, medication administration, specialized treatment for certain cognitive disabilities (Alzheimer's, dementia and others), providing secured entrances and exits, and daily housekeeping.\nWell-balanced meals are also served daily. Special nutritional needs are followed, along with any other special requests in terms of eating habits or religious acknowledgements.\nRepresenting the resident as priority number one. Staff at the best Wisconsin nursing homes view the skilled nursing facility as their home and act accordingly.\nAllowing the resident to make all important decisions regarding their health and wellbeing.\nProviding a well-staffed, skilled nursing unit that is not overworked or ever acts unprofessional.\nProviding extended hours for family and friends to visit. Family should never be blocked from seeing their loved one in any reasonable circumstance.\nProviding meaningful activities on a daily basis that help with physical health, mental wellbeing and also encourage social interaction.\nThere are a total of 377 nursing homes in Wisconsin that Wisconsin senior citizen residents can take advantage of. Wisconsin nursing facilities can be part of larger nursing care communities, including continuing care, dementia or Alzheimer's care communities. Most skilled nursing facilities in Wisconsin will provide older adults with both long term care as well as short term rehabilitative nursing care. Most Wisconsin nursing homes accept Medicaid and all will accept private pay. All nursing facilities throughout Wisconsin state that you can locate through Senior Guidance are officially licensed by Medicare.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the International Classification of Headache Disorders, 3rd Edition, primary headache disorders are classified in one of three ways: Migraine, Tension-Type Headaches, or Trigeminal Autonomic Cephalalgias. As our name implies, most of the time we focus on the Migraine classification. However, it is rare for someone with Migraine to only experience one type of headache disorder. Plus, patients sometimes are diagnosed incorrectly. Both doctors and patients may mistakenly believe the problem is Migraine, when it's really something else entirely. You may recall an earlier article about Cluster Headache. In it, I offered very specific diagnostic criteria that can be used to distinguish Cluster Headache from Migraine. Now I'd like to introduce you to one of Cluster Headache's cousins \u2013 Hemicrania Continua.\nIn the absence of very specific training in headache medicine, even a skilled neurologist may not be able to tell the difference between Migraine and Hemicrania Continua. That's in part because they just don't know what kind of questions to ask. Because so few people actually talk to their doctor about their headaches, many patients also assume incorrectly that they are experiencing Migraine.\nHemicrania Continua can be classified as remitting or unremitting. Unlike migraine, which usually start episodic and progresses to chronic, Hemicrania Continua often begins as the unremitting type without breaks longer than one day for over a year. Over time, patients may experience periods of relief longer than one day. Sometimes diagnosis is not so simple. It takes time to determine whether a patient has Cluster Headache, SUNCT\/SUNA, Paroxysmal Hemicrania, or Hemicrania Continua. Fortunately, part of the differential is a trial of medicine which can narrow down the possibilities.\nOne of the unique characteristics of Hemicrania Continua is its nearly universal response to the prescription NSAID, indomethacin. Like most NSAIDs indomethacin can cause stomach irritation, in fact, it's notorious for it. Most prescribers will also prescribe a proton-pump inhibitor like Prilosec to be used while taking it. Oral doses usually start at 150 mg daily with a gradual increase up to 225 mg. Injection doses range from 100m to 200mg with smaller maintenance doses given over time.\nAt any rate, this is not a condition that most primary care physicians will recognize or know how to treat. It is a good example of the importance of seeing a headache specialist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this lesson, Ss learn\/review simple past to talk about completed actions in the past. Simple past will be contextualized using a reading briefly and then language from the text will be highlighted and clarified before moving onto further controlled and semi-controlled practice. Finally, ending with practice through a speaking activity.\n(Distribute news history HO) Quickly read the text to find the answers. Work alone. PWFB: Check your answers in pairs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasmyb b/data_all_eng_slimpj/shuffled/split2/finalzzzasmyb
new file mode 100644
index 0000000000000000000000000000000000000000..d5c4bff47a6ec724ad7caa01d6953cd8ff61e08d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasmyb
@@ -0,0 +1,5 @@
+{"text":"Subramanyam for sale telugu movie hd. Samsung galaxy s duos mode. Black bird volume 16, capital inicial brasilia. Bejeweled untuk android.\nSkype portugues brasil gratis. Fernando e sorocaba dvd sinta essa experiencia. Race gurram 320kbps. Hello house remix mp3. Gabbar singh telugu movie hd video songs. Patricinha do olho azul shared. Microsoft facial recognition software.\nMuse 2nd law hd, the sound of waves pdf, la bella mafia album and chris brown all back sharebeast and the avengers 2 full movie in hindi hd and nas cherry wine zippy, desh mera rangila song mp3 and bushido feat. king ali gemein wie 100. Brother song djmaza, music videos to mp3, dragon naturally speaking 12 and yemaindi ee vela song http:\/\/jonishane.netne.net\/the-rats-of-nimh-ebook\/, elsa e fred gratis, philips 190sw driver blyton pdf enid, winrar full version filehippo and skrillex playlist mp3. Permainan the sims 3 winter edition.\nLagu garasi jika tak ada lagi yang terbaik epub porter michael, l'appartamento spagnolo film, driver placa video incorporata. Kyo derniere danse mp3. Manam kothi paravai video songs ing. Teargas wake up video here, opera mini for for java mobile national electrical estimator pdf - tim winton cloudstreet epub, gajapokkiri malayalam movie mp4.\nWcf file progress and hima tuhina song. Android ics installer, rock oo full movies, killing heidi present http:\/\/jonishane.netne.net\/wicca-pdf-s\/, intel dh55pj lan drivers for windows 7. Witch hunter robin sub indo and venkatachala nilayam vijay prakash song. Tekken 5 game, mourinho tactics fm 2012 read here, promentory last of the mohicans trevor jones randy edelman and isyarat cinta arjuna samba and oito segundos filme, x-particles version 2. Space cowboy falling down ft lmfao. 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+{"text":"I was looking by the internet asking if shopping for gold could be a sensible investment and does the worth of gold rises as time goes on. Nonetheless obtained no clear solutions so I need to ask it here. Uncover one of the three Power Chart Patterns on this webinar and the way it make it easier to spot the very best occasions to buy and sell stocks with cheap accuracy. Possibly it wasn't, however how about investigating the HFT corporations that pulled out of the market, tanking liquidity large time.\nThe most likely one of the best strategies among all which might be getting used to assess the worth a company's stocks is the Earnings Per Share, or as it usually known as, the EPS method. Stock market investing isn't day buying and selling, swing trading or day buying and selling.\nTwenty-5 years experience investing on the stock market. At one level on Monday, the Dow was down almost 1,600 factors, or about 6 percent of its whole value, the markets plunging on the similar second as Mr. Trump was giving a speech in Blue Ash, Ohio. You may get a quote (test the prices) day-after-day but you won't need it. Afterward you will test quotes as soon as every week or even as soon as a month.\nThe inventory trade provides a platform that facilitates the buying and selling in shares of the listed firms. The markets are basically the place people and corporations commerce securities. Secondary market: The financial market through which previously issued financial devices similar to stock, bonds, options, and futures are purchased and bought.\nMSN, CNN, USA At this time and other news sources frequently submit inventory market adjustments on a daily basis. Some folks place some gold within the mouth of a dead body at the time of the funeral. Buyers and financial analysts put up many of these messages whereas firm staff post others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gila News-Courier . Rivers, Arizona, September 2. (Rivers, AZ), Sep. 2 1944. https:\/\/0-www.loc.gov.oasys.lib.oxy.edu\/item\/sn83025353\/1944-09-02\/ed-1\/.\n(1944, September 2) Gila News-Courier . Rivers, Arizona, September 2. Retrieved from the Library of Congress, https:\/\/0-www.loc.gov.oasys.lib.oxy.edu\/item\/sn83025353\/1944-09-02\/ed-1\/.\nGila News-Courier . Rivers, Arizona, September 2. (Rivers, AZ) 2 Sep. 1944. Retrieved from the Library of Congress, www.loc.gov\/item\/sn83025353\/1944-09-02\/ed-1\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learn how to repair video games consoles is it real?, Learn how to repair video games consoles discount code learn how to repair video games consoles discounts learn how to repair video games consoles is it real? learn how to repair video games consoles review learn how to repair video games consoles technique guide to learn how to repair video games consoles is an article released in the Games category and published on 2015-11-07, Learn how to repair video games consoles technique.\nHow to share xbox one games with family with multiple consoles on Thu, 05 Jul 2018 13:06:00 GMT Ps4 console \u2013 playstation 4 console ps4\u2122 features games on Sat, 20 Apr 2019 17:46:00 GMT Video games that killed people in real life looper com on Sat, 13 Feb 2016 19:37:00 GMT Video game wikipedia on Sat, 20 Apr 2019 12:31:00 GMT Amazon com xbox one s 1tb console starter bundle video on Sun, 21 Apr 2019 07:44:00 GMT Vintage stereo audio video repair restoration resources on Wed, 17 Apr 2019 13:10:00 GMT Best video game console buying guide consumer reports on Mon, 02 May 2016 23:54:00 GMT Learn how to repair video games consoles specials.\nLearn how to repair video games consoles online tutorial Note you cannot designate more than one xbox as your home xbox you can share purchased games and gold with other users only on your home xbox multiple consoles scenario here comes the tricky. Playstation\u2122vue is a live tv streaming service with sports news movies and your favorite must watch shows watch live tv on your ps4\u2122 console and compatible favorite devices all without a pesky annual contract or surprise fees. Berzerk is one of the earliest known examples of a video game ending in actual death in april 1982 an 18 year old named peter bukowski landed a couple of high scores on a berzerk arcade cabinet. A video game is an electronic game that involves interaction with a user interface to generate visual feedback on a two or three dimensional video display device such as a tv screen virtual reality headset or computer monitor since the 1980s video games have become an increasingly important part of the entertainment industry and whether they are also a form of art is a matter of dispute. Fulfillment by amazon fba is a service we offer sellers that lets them store their products in amazon s fulfillment centers and we directly pack ship and provide customer service for these products. Purveyors of electronic musical and vintage goods from then now and in between since 1982 vintage stereo repair electronic repair stereo audio video repair restoration. Looking for the top video game consoles read our video game console buying guide from the experts at consumer reports you can trust to help you make the best purchasing decision. Learn how to repair video games consoles experience.\nCheap The First Ps3 Cinavia Fix & Free Bonus: Ylod Red Lights Repair Guide!\nCategory: Games - Date: 2015-12-09 - What is the best the first ps3 cinavia fix & free bonus: ylod red lights repair guide!? The first ps3 cinavia fix in the marketplace and on the planet! cinavia is a huge problem for a lot of ps3 owners - completely unexplored niche! they also get a ps3 ylod\/red lights repair guide as a free bonus! this product is sure to make you money!\nCategory: Games - Date: 2015-05-20 - Best reviews of lol builder package The lol builder package is a recurring membership product that offers 60% commission to affiliates.this is the highest payout lol related product containing both an informational product and windows application.\nCategory: Games - Date: 2015-08-22 - Ps3lightsfix com the first ps3 ylod red lights repair guide! programs The first complete ps3 ylod\/red light repair guide in the marketplace, with a super 75% commission! accompanied by huge new advanced affiliate section (ps3lightsfix.com\/affiliates\/affiliates.html), this product is sure to make you money! earn today!\nCategory: Games - Date: 2015-06-23 - Ciderhelm's learn the league league of legends e book work or scam? Learn the league is the #1 premium guide in the 30m+ monthly league of legends market.sell a product you can stand by: this massive e-book and video collection is frequently updated by a well-known tournament analyst and youtube creator.\nCategory: Games - Date: 2015-05-04 - Dota 2 navigation the first pro dota 2 guide made by natus vincere online tutorial New commission: 70%! strong brand name - na`vi is the most successful dota 2 team in the world.the first ever comprehensive guide for the fastest growing e-sport.tons of promo materials - banners, intro and outro videos.\nCategory: Games - Date: 2015-04-28 - The osiris method ultimate starcraft 2 strategy guide promotional code The osiris method is the only starcraft 2 training program.the osiris method takes starcraft 2 strategy guides to the next level by providing practice drills and training regimens designed to propel players higher in the ranks! new product = low comp!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vacation house from 1999. Close to the lovely child-friendly beach where there are excellent opportunities for fishing. The house is a well maintained Liljehus 3 bedroom + alcove. The house is located near the island's finest beach (Hasmark). Near the scenic Eneb\u00e6rodde with good possibilities for sea fishing. Large plot of 1200 sqm with a sandbox and football. It is from the house out to a lovely wooden terrace. The house is close to the campsite, playground and indoor and outdoor pool area. Good places to visit for example. Odense Zoo Egeskov, idyllic Bogense or the Funen Village.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Black XS Potion for Her by Paco Rabanne is an oriental and floral fragrance for women.\nIt contains notes of grapefruit, red Baccara roses and pink hellebore, extract of black sandalwood and amber.\nBlack XS Potion for Her was launched in 2014.\nPaco Rabanne Black XS Potion for Her Eau de Toilette 50ml SprayLove this perfume!!!\nIt's such a fresh clean smell, I love it!! So many people have commented on how nice it is.\nAbsolutely love this perfume. One of the best in thr paco rabanne range. I spray on in the morning and can still smell the lovely fragence by the time i come home in the evening.\nThis is great. I do find it fairly sweet but not overly so. Floral with amber and a little citrus with the woods coming through after about half an hour. I do not find this masculine at all as some reviewers have found. There is something a little spicy there too but i'm not sure what. The black and red very simple bottle fits with the scent I think! There is something here that reminds me of Kenzo Jungle Elephant but I don't mean it smells like it, more like in the same line. It has great longevity and sillage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What I have learned about lung cancer over the past year is astounding. It is the most common cancer worldwide. Most often by the time it's detected, it has advanced and spread to other parts of the body. There are two types of lung cancer: small cell (the kind that is typically associated with smoking), and non-small cell (more common, not necessarily attributed to smoking, and grows and spreads throughout the body more slowly than small cell).\nThe good news is that my mom's cancer was detected at an early stage and was classified as non-small cell. Although we don't yet know if the course of action worked, we are hopeful that between early detection, surgeries and treatments, she will beat this.\nWhat is surprising is that funding and research advances have not caught up with other forms of cancer.\nIt's time to change that, which is why my mom and I are participating in the Berkeley Walk for Lung Cancer on September 21, hosted by the Lung Cancer Alliance (http:\/\/www.lungcanceralliance.org\/).\nWe are walking to raise awareness. We are walking for all of the people who are being diagnosed today, tomorrow and the next day. We are walking because my mom is still here, and is still fighting, and because she can.\nYour support in the form of a donation for Team Emm (that's us!) is greatly appreciated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Resourceful among Stock SourcesMaintaining per bank out of stock pictures plus templates in order to speed techniques up looks per talent which will come obviously in order to a website designer. Whereas creating a website, apt pictures plus stock templates need to be selected. Thereafter, they need to get tailor-made simply by each developers to meet up with the client otherwise business demands whereas ensuring originality, excellence plus fresh create for the overall look plus feel of this web site.\nAlthough, opting for the best cheaper designer may possibly sustain increased expense into the later on stages in the event that quality are substandard. Alternatively, an excellent designer would provide we a combination of varied service at an affordable pricing.\n1Honesty up to prices then Turnaround TimeThe best websites will help you to utilize properties love quality, speed then low priced throughout the create procedure. An expert web designer will provide your your client, accurate knowledge up to his or her prices, speed then turnaround time period. Expensive firms as online development\/design agencies will help you to usually need lesser prepared lists. Assuming time period isn't per constraint then you can take a superior quality websites in your economic expense. People counsel you to pick a website development service. it ingests the needs you have then satisfies consumers. Remember that concentrating on benefits sets professionals apart. Starting per CrossFit package? as searching to cultivate on your CrossFit as gymnasium? as Opening per crossfit gymnasium? Listed Here Are well techniques to grow on your physical fitness, gymnasium, CrossFit Box then business with My Personal Ideal Studio accomplish solutions.\nSo provided the best scintillating tablet as a I-pad made ones fantasies keep yet for a while, subsequently android os can help we maintain ones love for website creating well. At Android os software preserve it is simple to find a very good applications considering creating additionally portable image editors. Hence, now you can obtain the whole fledged connection with web site design on your smartphone alone. Since the probability of improvement at Android os applications is significantly anticipated, there are many augmented truth applications, 3D video games, reference applications, productivity technology additionally modifying technology. Various create related applications are available for designers and are also definitely easy to use.\nColor Dictionary It makes affordably simple for whoever wishes sources to traditional hues all around the globe. This could consist of world wide hues, traditional hues people, UNITED KINGDOM and France. leicester website designers Colors combination provided is actually RGB\/HEX\/CMYK\/Lab values along with other color analysis.\nTesting Delivery The last step in the internet site artwork additionally developing strategy was assessment additionally distribution. When the website\/web platform\/app was eager, test that. Here is the phase whenever you search for any kind of insects or discrepancies additionally lend the final touches. As you might have a completely independent assessment administrator carrying this out, its smart to inquire a detailed number of diverse visitors to test that. It might be people from your working environment, family and friends, family additionally consumers. This will assist you to fix not just technical nonetheless including usability problems that the internet site might have. When the assessment is performed and also the insects are fixed, voila, you're all set real time.\nCommunicationTheres absolutely nothing considerably relevant than constructive interaction. Website additionally UI\/UX developers must adhere a person through the means of the website developing additionally help keep you up-to-date every once in awhile. Make use of the interaction technique that suits you regardless its email, mobile or even real time talk.\nIt comes about as a result of lack of knowledge or even as a result of arrogance or perhaps per blend out of either. Many commonly modest enterprises continuously drop clients to competitors of comparatively simple grounds. These think they will have the ability to effectively layout their the internet sites. Exactly the same type of considering notifies his or her conclusion to style additionally design their particular hit promoting also periodically build their particular TV commercials. The Very Fact it upon uncommon occasions this strategy may well finish the same job cannot undermine the fundamental argument.\nApart off in which, will designer must also find out about SEO. A variety of SEO additionally SMM is ideal additionally will be quite helpful in making your website the fastest ever-increasing enterprise. Although, social networking are a thing that can provide a person a sophisticated brand visibility. Therefore, make certain that all designer you have chosen presents expertise as part of SEO plus SMM. Website Designing calls for lot of undertaking additionally persistence and it's also challenging the web-site designers. Listed Below are 6 challenges that are faced simply by novice online store developers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business company ABSOLUTE SECURITY L.L.C. is a legal entity registered under the law of State Nevada. Company is located in the register with the Company number LLC10092-2004 and with the national number of State Nevada NV20041103147. This legal entity was firstly registered on 12th May 2004 under the legal form of Domestic Limited-Liability Company. Its registered agent is ALBERT J MIHALEK with the seat at 1907 SPODE AVE, HENDERSON, 89014, NV licensed as Noncommercial Registered Agent. Business activities of this company are managed by Managing Members. Current company's status is Permanently Revoked. Company has expired at 12th May 2504.\nThe company ABSOLUTE SECURITY L.L.C. is managed by 6 persons in total. The persons responsible for business activities are ADAM BUSTIOS with the seat at 394 WASHTENAW, HENDERSON, 89012, NV as Manager , ADAM BUSTIOS with the seat at 394 WASHTENAW, HENDERSON, 89012, NV as Manager , ALEX BUSTIOS with the seat at 3627 DUTCH VALLEY DR, LAS VEGAS, 89147, NV as Manager , ALEX BUSTIOS with the seat at 3627 DUTCH VALLEY DR, LAS VEGAS, 89147, NV as Manager , ALBERT MIHALEK with the seat at 1907 SPODE AVE, HENDERSON, 89014, NV as Manager , ALBERT MIHALEK with the seat at 1907 SPODE AVE, HENDERSON, 89014, NV as Manager .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This salve is inspired by this post on marshmallow by Rosalee de la Foret.\nI love marshmallow; it's a moon herb, and helpful in so many ways. It makes the absolute best lip salves too, that will be coming up soon! It has been so wonderful to learn the different ways to utilize the healing potentials of the plants and to learn this method of infusing the properties of marshmallow into oil.\nI have 3 first season Althaea officinalis plants and one is very crowded so I decided to use it. First I dug up the root, then scrubbed it clean, and shredded it. Then patience and attention, just what I need.\ngrated and chopped into the oil over a double burner, on low for 2 hours.\ncleaning it up to be chopped and grated, then partially dried before infusing it into oil.\nAfter the first hour of simmering I added 1\/3 ratio of warm infused plantain and self heal into safflower oil and jojoba. At the end, I always blend my essential oils into a small amount of jojoba oil before I mix them in to the final batch. Then I test the ratios and consistency and pour into prepared jars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Canon MAXIFY MB5420 Windows Driver Download \u2013 55.74Mb Canon MAXIFY MB5420 Mac Driver & Software Package This file will download and install the drivers, application or manual you need to set up the full functionality of your product.\nCanon MAXIFY MB5420 Series All-in-One Wireless Printer provides high quality printing, with each performance compact printer that connected, and give some information with built-in standard management information Base MIB\/currencies.\nCanon MAXIFY MB5420 Drivers & Software. Canon MAXIFY MB5420 Driver Printer. CanonMAXIFY MB5420. Canon MAXIFY MB5420 Driver Download \u2013 The MAXIFY MB5420 is designed to save you time and money while improving the productivity of up to 9 users.\nCanon MAXIFY MB5420 Driver Download and Manual Setup for Windows, Mac OS, and Linux \u2013 The Canon MAXIFY MB5420 Wireless Small Office All-In-One Printer is created to answer the needs of tinier businesses with up to nine users, by producing high-quality, low-cost prints.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3 bedroom bungalow for sale at Kungu after Kisasi, it has 3 bathrooms and toilets in a well organised developed neighborhood. The price is 250m. Call us on 0414662954, 0774755146 or 0753100355 to arrange a viewing appointment.\nAm interested in viewing this property (with code 28714) on 4\/21\/2019 at 3:00pm, please confirm my appointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Car stereo DVD players bring into question the very boundaries of car audio and verge on creating a whole new sphere we might call \"car-theater.\" Mobile DVD players accompanied by a video system are a great way to entertain passengers on a long trip. You can provide in-car movies en-route to your destination just like a commercial airline.\nMobile display devices are usually referred to as either overhead or headrest displays. When they're overhead, they're meant to be placed in the middle of the car mounted to the ceiling just back from the windshield. Many minivans and trucks use this space to mount stock equipment that includes displays like GPS systems, a clock or a small storage compartment. Overhead LCD displays with a DVD player can playback movies that passengers in both the front seat and back seats can enjoy.\nToday's car audio DVD players are indicative of an industry that is unsure of exactly what it wants to be. They're arriving to the consumer market with a variety of shapes, sizes and responsibilities. Some of the head units that can playback DVD might include a small display like the JVC KD-AVX1- a tiny 3\" display that plays DVDs in Dolby Digital and DTS. Although a cool novelty, a 3\" DVD display seems more like a short lived gimmick whose usefulness to the owner will dissipate in a matter of a few uses. Other manufacturers, including JVC, make larger, removable monitors up to 7\" in size.\nDVD video in the car isn't something the driver should enjoy; keeping eyes on the road at all times might get a bit tricky if an in-dash or overhead monitor displays their favorite part in a film. But the drivers can still benefit from car stereo DVD players that don't display video at all- putting an end to the identity crisis the product seems to suffer once and for all.\nLet's look at the DVD for what it really is- a next generation optical storage format that never supplanted the CD as an optical storage medium. Why? It was mainly used for video where CD had a niche in music. With increased availability of cheap DVD burners, this could be changing for the better. The CD has a questionable future and only in the car audio world are manufacturers still developing new models. The Compact Disc's lifespan has been extended only because of the value in burning MP3s directly to CD. This allows the car stereo buff to store about five CDs into one CDR- perfect for a short road trip.\nThe potential for DVD is far greater, allowing music rippers to put a whole library onto one DVD. When car audio enthusiasts consider the potential of rewritable DVD media with no video, the DVD shouldn't have any problem replacing the CD as the media of choice. Manufacturers are slow to put out DVD players that are able to read MP3s from a DVD, even if they do allow MP3s to be read from CDs playable in the same head unit.\nWatch out when you're shopping for a car stereo DVD player. Many will call themselves MP3 players too, but in the fine print you'll discover many only read MP3 files from CD and not from DVD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not only do I want to make sure you have gorgeous photos telling your story but also that you have the tools to enjoy them! From appointment to art let me guide your photography experience to make it something you cherish for a lifetime.\nIncluded in your session fee is a pre-session consultation: this is the perfect time to discuss locations, what to wear, what you want to do with your images and how we can make the session fun for everyone involved. This can be done by phone or in person. I want to hear your story. Tell me why you are having photos taken at this moment in time, what makes your little ones smile shine the brightest, what do you love the most about your significant other, or what new and exciting career step has you behind my camera!\nNext up: The Portrait Session! The most important thing to remember \u2013 don't stress! While it may not be a daily occurrence to have your photos taken \u2013 it is my goal to capture the best version of you. And the best version of any of us is never the stressed out one! Grab a cup of coffee, rock out to your favorite music and feel confident in how awesome you look as you drive to our session destination. Plan to laugh, walk around a bit, and know that I will guide you \u2013 how to stand, where to look \u2013 I'll even fix those stray hairs and chase your busy toddlers!\nLastly you'll sit down for your Image Reveal and Ordering Appointment. I'll assist you in picking out the perfect products for your home. Offerings include classic framed art, traditional wall canvases, and modern metals to adorn your home with the best kind of art \u2013 personal art! Can't decide on your favorites? Take home lots of images in our legacy portfolio box \u2013 each image is beautifully matted and stored to share with generations to come. And don't forget grandma and grandpa \u2013 loose prints provide a great way to send those images to our loved ones!\nI want to tell your story \u2013 but I want you to SHARE your story. ENJOY your story. Your \"Love: Captured\" experience should bring you joy \u2013 both now and in the years to come!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And so are their heirs. It's been 20 years since the planet was on the brink of destruction (18 years after Advent Children). The cosmic disease known as Stigma was wiped away and abolished from the group of warriors who were hand picked by fate to do something the universe would forever shift to. The Chosen One, The Redeemed One, The once great warrior of the infamous SOLDIER military under Shinra Inc, Sephiroth is nothing more than a fabled bad tale of the past used as a symbol to remind the people the power of unity, and of destiny. Midgar has been rebuilt, now Neo-Midgar. SOLDIER is among a rank of peace and justice, and Shinra is now the most successful clean energy manufacturer on the entire planet. Every year the old heroes reunite to spend time and commemorate what once was almost lost, except it seems...now they have their own little selves to think about.\nThe seed of the old heroes are upon them, and they are as about as dynamic and sure fire as their parental counterparts. It seems the world they live in was nothing like the old ones, and their knowledge and recollection of the trouble and struggles of the world were distant, with all of them living in security and mild royalty backed by President Shinra's support. Some of the old heroes have retreated to a life of reclusiveness, some gave still found ways to serve the planet and the Lifestream.\nBut....with Cloud's daughter having intense visions and nightmares increasingly of a new threat, and these visions starting to become reality, can the Old heroes band together once more for a threat unexpected and like they've never faced before? Or will it be up to the young and fresh versions of themselves to rise and do what their parents did long ago?\nOf course this is an interest check to see who'd be interested in playing the kids of the FF7 Heroes. I WILL allow rpers to rp as the old heroes as well, just understand they probably won't have or see as much action in this adaptation of a sequel RP. I would want my character to be Kagi, the daughter of Cloud and Tifa Strife. I also allow pairings but they would have to make sense [VincentXYuffie, CidXYuffie, etc. ] Marlene and Denzel (for fans) are apart of the Neo Generation and are up for being played as well. And yes, the turks are still around and some will still be in the organization but this will focus on a new and younger generation of Turks. However Rude, Reno, Elena, Tseng are all available as well, as well as any pairings [ElenaXTseng, ElenaXReno, etc. ]. Children of Turks are allowed, but won't actually be Turks themselves.\nAnymore questions or concerns just respond or PM me! Hopefully this gets traction.\nI personally loved 7\/Advent Children.\nI have played seven but not advent children. Am I still allowed to join?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayucr b/data_all_eng_slimpj/shuffled/split2/finalzzzayucr
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+{"text":"Support the Undaunted Revolutionary People resisting against the Final Onslaught of Assad and imperialist Russia against the last Citadel of the Syrian Revolution!\nNote of the Editorial Board: This article include a number of pictures, video links and a map which can be viewed below.\nTens of thousands people demonstrated in 83 cities and towns in the Greater Idlib region against the Assad tyranny and imperialist Russia on Friday 14 September. (1) In what many people say were the largest mobilizations since the early days of the revolution in 2011, people chanted slogans for the downfall of the regime. They denounced any attempt for reconciliation with the regime.\nThese demonstrations reflect that the will of the revolutionary people to continue the struggle against the barbarous Assad regime is undaunted. They also reflect the unabated mass support for the current leaderships of the popular uprising \u2013 mainly the petty-bourgeois Islamist Hayyat Tahrir al-Sham (HTS) as well as the petty-bourgeois nationalist\/Islamist Free Syrian Army \/ Jabhat al-Wataniya lil-Tahrir.\nLater, in a second phase, the regime forces would try to conquer the rest of the liberated area. In any case, there is no reason to assume that the aggression against Idlib is about to end.\nAt the same time, the Putin regime emphasizes that its attempts to annihilate those forces most determined the liberation struggle (the \"al-Qaida terrorists\") will continue. Its tactic is the old \"divide et imperare\" policy which the Roman Empire already deployed with success. At a meeting with his German counterpart Heiko Maas in Berlin, Russian Foreign Minister Sergei Lavrov said that there will be no reconciliation offered to the \"terrorist forces\" like Hay'at Tahrir Al-Sham and the Turkestan Islamic Party.\nFor this reason the Erdo\u011fan regime is willing to collaborate with Assad and Putin in attempts to eliminate the most determined forces of the liberation struggle against the Assad tyranny (like HTS, TIP, HaD, etc.). Ankara recently declared the HTS as a \"terrorist organization\" (as Russia, Iran and the USA did before).\nOf course, open attempts of the Turkish army to military smash the HTS and allied forces would be very risky for Ankara. First, it would be very difficult given the military strength and the popular support of the HTS. Secondly, it would massively damage Turkey's reputation not only among the Syrian people but in the whole Muslim world. Hence, until now, Ankara has focused its attempts to smash the HTS on a) encouraging and supporting pro-Turkish FSA factions to attack HTS and b) a systematic campaign of assassinations against HTS commanders and militants.\nIn other words, the liberations fighters in Idlib must prepare for a full-blown attack by the Assadist and Russian forces while taking precautions against a stab in the back by the Turkish forces or their local allies. An unbelievable difficult and challenging task!\nInternational Solidarity with the Revolutionary People in Idlib!\nThe RCIT repeats its call to the workers and popular mass organizations around the world to rally in support of the Syrian people! Defend Idlib, the last citadel of the Syrian Revolution, against the impending annihilation by the barbaric counter-revolution! Oppose any sellout and any \"reconciliation deal\"! Denounce the interference and aggression of all imperialist Great Powers in Syria! (19) For the immediate withdrawal of the Russian and U.S. troops as well as those of Iran and Turkey! All foreign military bases in Syria must be dissolved!\n* Defend Idlib! Defeat the Assadist, Russian and Iranian forces!\n* Oppose any sellout and any \"reconciliation deal\"!\n* Russian, Iranian, U.S. and Turkish troops \u2013 out of Syria!\n* Down with the counter-revolutionary Astana Negotiations!\n* For a single Intifada from Idlib to Jerusalem, Basra, Cairo and Teheran!\n* Build a Revolutionary Workers Party \u2013 nationally and internationally!\n(5) On the RCIT's assessment of the Astana deal see e.g.: Michael Pr\u00f6bsting: Syrian Revolution: The Moment of Truth is Approaching! Rally to defend the Syrian Revolution against the Imperialist conspiracy called the \"Astana Deal\"! 20.09.2017, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/syria-moment-of-truth-approaching\/; RCIT: Syria: Defend Idlib against the Great Powers! Down with the reactionary Astana Deal! Defend the Revolution against the butcher Assad, against Russian and US Imperialism and the local Allies! Victory to the Struggle of the Workers and Oppressed! 04.08.2017, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/defend-idlib-against-great-powers\/; Michael Pr\u00f6bsting: Syria: The Astana-Deal Struggle Intensifies. Some Notes on Recent Developments in the Syrian Civil War and the Dangers for the Liberation Struggle, 28 July 2017, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/syrian-revolution-28-7-2017\/; RCIT: Syria: Condemn the Reactionary Astana Deal! The so-called \"De-Escalation Zones\" are a First Step towards the Partition of Syria and a Conspiracy by the Great Powers to Defeat the Revolution, 7 May 2017, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/astana-deal\/; Michael Pr\u00f6bsting: Is the Syrian Revolution at its End? Is Third Camp Abstentionism Justified? An essay on the organs of popular power in the liberated area of Syria, on the character of the different sectors of the Syrian rebels, and on the failure of those leftists who deserted the Syrian Revolution, 5 April 2017, https:\/\/www.thecommunists.net\/theory\/syrian-revolution-not-dead\/ and chapter V of Michael Pr\u00f6bsting: World Perspectives 2018: A World Pregnant with Wars and Popular Uprisings, February 2018, https:\/\/www.thecommunists.net\/theory\/worldperspectives-2018\/chapter-v\/.\n(6) We have elaborated our characterization of HTS in a number of articles and statements which are collected in a special section of our website: https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/collection-of-articles-on-the-syrian-revolution\/. In particular we refer to RCIT: Denounce the US terror listing of Syria's Hayyat Tahrir al-Sham! The Trump Administration delivers another blow to the Syrian Revolution. Continue the Solidarity with the liberation struggle of the Syrian people! 03.06.2018, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/denounce-the-us-terror-listing-of-syria-s-hts\/ Michael Pr\u00f6bsting: Syria\/Idlib: The Attack of the Astana Conspirators could be repelled thus far, 05.03.2018, https:\/\/www.thecommunists.net\/worldwide\/africa-and-middle-east\/syria-idlib-the-attack-of-the-astana-conspirators-could-be-repelled-thus-far\/; chapter V of Michael Pr\u00f6bsting: World Perspectives 2018, https:\/\/www.thecommunists.net\/theory\/worldperspectives-2018\/chapter-v\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been thinking a lot about what it means to be grateful.\nI think of my little boys and how hard I work to ensure they are kind, thoughtful and well-mannered men. More so, I think of how hard I work to ensure they understand what having a grateful heart really means.\nI think of how long it takes to say thank you.\nI think of how much it means to hear thank you.\nI think of how much there is to be thankful for.\nWhich then gets me wondering how often I say thank you to God.\nThese two little words hold such depth and meaning, yet, do I actually use these words when conversing with God?\nGratitude doesn't need to be complicated. All we have to do is breathe in appreciation and breathe out our praise and say thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shift Visuals is a full-service commercial photography and video production company located in downtown Green Bay. We specialize in creative story-telling and brand building through compelling visual mediums for video production Green Bay.\nWe focus on providing outstanding client experiences and we produce exceptional video for online, TV, marketing, advertising and internal communications.\nincludes product, architectural, editorial and commercial portrait photography.\nWe provide commercial video production and photography throughout Wisconsin, including; Green Bay, Appleton, Milwaukee, Madison, Wausau, La Crosse, and Eau Claire.\nBased in Green Bay, Shift Visuals is the leading supplier of video production and commercial photography content in northeast Wisconsin. We have worked across the state, from Green Bay to Milwaukee, Wausau to Appleton, Madison, Eau Claire, Lacrosse. We also regularly work out of state. Most recently Chicago, Illinois and Miami, Florida. We use the best equipment north of Milwaukee and pride ourselves on pushing technology boundaries. All of our video shoots use a RED Dragon 6K capable camera, which shoots raw video. We then take care of all post production in house, in our 4K equipped edit suite, which is a comfortable as it is high-tech. We work with advertising agencies and directly with clients in many diverse industries. We have extensive experience in manufacturing, healthcare, non-profit, architecture, public service, professional service for both video production and commercial photography among many others. Our goal is complete client satisfaction. Mark Moran and DJ Kast work intimately with clients to truly shape their vision and bring their ideas to life! We are the recipient of 2016 gold ADDY advertising award in cinematography for Green Bay video production reel, and our Wisconsin commercial photography is highly regarded throughout the state.\nComplete the short form below and we'll respond at our earliest opportunity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The RG-418-2 is a Concorde\u00ae Platinum Series\u00ae Aircraft Battery. The Platinum Series\u00ae is comprised of premium hand made AGM Aircraft Batteries with excess power for many applications.\nRG-418 is housed in a aluminum fire resistant container.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Academy of Canadian Cinema and Television announced the nominations for the 31st annual Genie Awards this morning. In the awards honoring the best in Canadian film, Barney's Version leads with eleven nods, followed closely by Incendies with ten. Some of the other Best Picture nominees are head scratchers...Les amours imaginaires is clearly an apology to Xavier Dolan for last year's massive snub of I Killed My Mother. Best Actress is probably the strongest of the categories: great choices, although I haven't seen Grown Up Movie Star... Two of the Best Actress nominees are from Trigger, my favourite Canadian film of the year, which also received nods for editing and score.\nUPDATED the full list of nominees via a spectacular cut and paste job from the Press Release of nominees . I've looked them over more thoroughly now that I have a day off from school, and I've noticed some good\/not so good things about the nods.\nGood: -Recognition of Splice, especially the technical work!\n-2 nods for Year of the Carnivore: yay Sook-Yin Lee!\n-3 nods for Defendor, including Best Original Screenplay.\n-Rosamund Pike popping up as a lead actress and not in the supporting category where I hoped she would. This means that I can't cheer for both her and Tracy Wright! Oh, well. Best of luck to Minnie Driver for Best Supporting actress, then!\n-That original song category...\"Standing Alongside Gone\" by Brendan Canning (from Trigger) should have taken the gold, but it isn't even on the list. Neither are any of the goofily patriotic songs from Score: A Hockey Musical... shame!\n-Force of Nature: The David Suzuki Movie was snubbed in the Best Documentary Category. How?\nOh, well - a lot of good movies are recognized for a particularly good year. Huzzah!\nThe Genies will be awarded March 10 in Ottawa...who can get me tickets???","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbakan b/data_all_eng_slimpj/shuffled/split2/finalzzzbakan
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+{"text":"Nowadays, the Internet giants in China and the multi e-commerce platforms are mostly controlled by foreign capital.\nThe consequence of this phenomenon is that most of the profits earned will be taken away by the major foreign shareholders every year. In other words, the wealth created by the China citizens cannot benefits themselves; it accelerates the outflow, which will seriously affect the development of the social economy in the near future.\nWDG is aware of the seriousness of this issue and looking forward to contribute to the economy development of its motherland. The establishment of http:\/\/518yunbao.com have solved the above problems. Both have optimized ordering systems, adopting Internet of Things technology, avoiding the excessive inventory pressure on the merchants, good credit mechanism, and avoid the fake and substandard goods on consumers to the maximum degree.\nThe massive member volume is a strong ground of the profit margin for the registered merchants. The merchant alliance creates an ecosystem and equity crowd funding. WDG members are both consumers and shareholders. They make unremitting efforts for their own industry. I believe that every participant will be motivated, and the future of WDG Merchant Alliance will be limitless.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you want to know what Surfers Paradise really looks like? Then take a look at the pictures of Surfers Paradise on this page. Do you also have some holiday pictures of Surfers Paradise? Then add them here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey, I found this information for you: \"Heritage Financial Again Named to FA Magazine's Top RIA Rankings\". Here is the website link: https:\/\/heritagefinancial.net\/heritage-financial-again-named-to-fa-magazines-top-ria-rankings\/. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rustic country decorating ideas rustic style decorating. Rustic diy decorating ideas diy rustic wall decor ideas. Rustic home decorating ideas home decorating ideas for sunrooms. Small home decorating ideas vintage home decorating ideas. House decor ideas dining room, diy home decorating ideas traditional dining room decorating ideas bedrooms traditional home decorating ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: We analyze a chain of coupled nonlinear optical cavities driven by a coherent source of light localized at one end and subject to uniform dissipation. We characterize photon transport by studying the populations and the photon correlations as a function of position. When complemented with input-output theory, these quantities provide direct information about photon transmission through the system. The position of single- and multi-photon resonances directly reflect the structure of the many-body energy levels. This shows how a study of transport along a coupled cavity array can provide rich information about the strongly correlated (many-body) states of light even in presence of dissipation. By means of a numerical algorithm based on the time-evolving block decimation scheme adapted to mixed states, we are able to simulate arrays up to sixty cavities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbzet b/data_all_eng_slimpj/shuffled/split2/finalzzzbbzet
new file mode 100644
index 0000000000000000000000000000000000000000..a9340c7d7f6c838aa9448e6112fce6e40b1e6713
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+{"text":"The latest news articles from Billboard Magazine rock, dance, hip- hop, business, including reviews, country , pop more. Drake dreams money could buy mp3 download.\nMEEK MILL' S SOPHOMORE ALBUM, DREAMS WORTH MORE THAN MONEY, FOLLOWS MEEK' S DEBUT \" DREAMS & NIGHTMARES\". Album features performances from Swizz Beatz, Future, Chris Brown & Nicki Minaj, Drake, The Weeknd, Rick Ross and Diddy.\nBuy I Wish You Could Have Turned My Head: Read Everything Else Reviews -. * Buy 3 Beats Get 2 Free. * Buy 5 Beats Get 4 Free. * Buy 7 Beats Get 6 Free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hezbollah leader Sayyed Hasan Nasrallah is set to speak Sunday evening marking 11 years since the end of Hezbollah-Israel war in 2006.\nWho will succeed current Palestine leaders?\nPalestinian politics is a relentless and often fruitless pursuit: After more than two decades of on-off negotiations, a state remains out of reach.\nThe resignation of Carla del Ponte from the U.N. Commission of Inquiry on Syria is the coup de grace for the Security Council's failings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a leading independent fostering agency, we pride ourselves in our ability to be a supportive agency that provides highly professional care and guidance to our fostering community whenever it is required. Here is the first instalment of our tips, starting with how to prepare to foster.\nWhilst the journey of being a foster carer is extremely rewarding and fulfilling, it's not without its challenging moments. Being able to listen to someone who has travelled on the same path as you is often the best advice that you can ever receive as a carer.\nWith the help of our Fostering Services Manager, Joanne, and a handful of our experienced foster carers, we have manage to compile a list of 101 useful foster care tips for anyone new to fostering.\nIn the first part of our weekly series, we will look at how you should prepare to foster and why it is important to be ready for the arrival of your foster placement.\nAs you prepare to foster for the very first time you're likely to be met with a mixture of emotions. From excitement and eagerness to apprehension and nervousness, but don't worry these feelings are completely normal, and It's certainly part of the reason why we're here to help, assist and guide you through your entire fostering journey.\nAs a rule, we'll always ensure that each fostering placement is the perfect fit for you, your family and the needs of a child or young person. Additionally, we aim to provide you with as much information about a particular foster child before they arrive in your care, so that you can gauge their interests, likes and dislikes. We'll also show you how you can prepare your home from a practical point of view.\nIf you would like to receive further information about becoming a foster carer, or how to prepare to foster for the first time, contact one of our experienced social workers today.\nTune in next week when we'll be looking at what our carers have to say about building up trust in a foster placement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brands are everything. Products are nothing. Learn why branding is important and how it can help your business grow with this article.\nThese are truly strange times we're living in \u2013 the abundance of products and services if absolutely overwhelming. Yet, it hasn't always been like that. There were times when people had to struggle to survive and cared little about marketing.\nCome to think of it \u2013 in the Medieval people were glad to have anything. If there were shoes available, they got them. If there was someone baking bread, the locals bought the hell out of it. The resources were scarce and there was no mass production, obviously. It was better to have at least something than nothing at all.\nNeedless to say, the quality and price were not among the top priorities in the then-customer's decision-making process. If they were at all. Maybe, they call it the \"Stone Age of Marketing\" in marketing books. For all we care, it was just the beginning.\nHowever, as the time was moving on, the large-scale manufacturing of goods became a reality, with the discovery of steam power and later electricity. Competition among the producers of goods had emerged, and soon enough the customers started to take notice of how much things cost and what the quality was.\nThat was the \"Golden Age of Marketing\" \u2013 most currently active top world brands were born during that time. Among them are such business names as Mercedes-Benz, Coca-Cola, Adidas, Ford, Procter & Gamble, Nestle, Mars, Husqvarna, Colt and many others.\nEventually, during the course of the 20th century, the overall quality of life around the planet had improved greatly due to the rapid spread of technology and globalization.\nFor a man of the early 1900s, it was impossible to imagine Ford cars being made in Argentina, Barclays ATMs located on the streets of Jakarta, Indonesia, and Russian souvenir Matryoshkas being produced in China.\nIt All, However, Became a Reality. How?\nRight after the World War II the major multinational corporations began their production shift to the third-world countries. They sought cheaper workforce, better logistics, and tax shelters. By doing so, they inevitably brought the technologies in those countries, thus giving them the temptation of using (and rather abusing) these technologies.\nSoon, the markets of these \"outsource\" countries were flooded with cheap fakes of popular brands produced using the import technologies. No one considered this a threat as long as the domestic markets were flourishing.\nHowever, right after the Cold War ended and the Eastern Bloc (with the USSR as its head) collapsed, the situation started to change slowly.\nThe once imitators had finally learned about the quality, innovation, energy-saving, and marketing. The competition grew even stronger.\nAnd when the mass-available cheap Internet arrived in the early 2000s, it all went haywire. It became impossible to hide something, to create a production know-how, and not have it stolen, replicated, and widely distributed the next day with a little-to-know compromise to the quality but for the less price.\nThis is when we, guys, moved into the current stage of the development of marketing which I would call the \"Age of Brands\" where some brands of ages had managed to survive and do well.\nNowadays, it doesn't matter what you produce or what you offer. In a nutshell, everyone's products are more-or-less the same.\nIf you ignore the brand and take a blind test of 10 brands of beer, you would never know which one's Bud and which one's Becks. Or Miller. Or Coors. Or Brahma. Or your local brewed piss-flavored drink.\nHey, but ask a fellow beer drinker if he has a favorite \u2013 he'll give you a thousand reasons why beer A is better than beer B. Would he be able to substantiate his claims with valid, solid, and death-proof arguments? Highly unlikely.\nWhat is the reason for this lack of objectivity in the decision-making process of a modern customer in today's economy? The BRAND.\nSeriously, you just wouldn't believe how much brands affect people's thinking.\nIt comes to the point where some of the consumers would pre-order the next Apple product without even knowing what it's going to look like.\nIf Apple top management suddenly decides to release a triangular-shaped smartphone with a pre-installed voice-operated crotch-rubbing machine called \"TriCroPhone\", millions of people would buy it the very same day it's released without even questioning the necessity of the latter.\nThe power of \"Apple\" brand and their marketing team is enough to convince people that they simply won't survive without TriCroPhone. Imagine what the product description might look like in that case.\nThis level of devotion that we notice in some brands can be found only in the world religions. And brands do resemble religions sometimes. That's why branding is important.\nOf course, these are top brands we're talking about. But even for a small business like yours that might not become as big as Apple or Google, branding is hugely important.\nSo, prefaces aside, let's get down to business!\nAccording to a popular definition, branding is a marketing practice of creating a name, symbol, image, logo, or design that identifies and differentiates a product from other products.\nEnough said! By branding your product, you're securing a place on the market and laying a great foundation for a successful business.\nThere are several possible reasons for that (besides starting your own religion \ud83d\ude09 ).\nHaving an established brand can make all the difference in business. It doesn't mean you will have an excuse to mistreat your customers, though \u2013 you will still have to play it right.\nCompanies that managed to become brands can use their brand as leverage in the battle with competitors \u2013 it gives them certainty, confidence, and freedom to experiment with expanding to the new markets and finding new approaches.\nSure it does. Wherever you go, you instantly notice McDonald's logo in the street, or Coca-Cola logo in the Drinks section in a supermarket.\nMcDonald's logo => McDonald's company = fast, cheap, and decent food = OK.\nThese are the thoughts that may run through an average consumer's head.\nHas McDonald's undertaken any active promotion steps to win you over? No, they haven't. That was pure brand magic.\nA great satisfying experience received from a brand is something that you would love to share with the rest of the world (or, at least, your buddies).\nPeople like to belong. Ideally, to something big. The logic is simple \u2013 if this product is a brand, it must've been chosen by many people. And if it was chosen by many people, it must be good enough for me to try. With brands, it's everything personal.\nWhen you're drinking Jack Daniels, you're not just drinking some liquor. You're drinking the world-famous renowned whiskey with more than a 100 years' tradition. That sense of belonging is what makes it work.\nBrands allow their fans to connect with the company's business through that sense of belonging we talked about earlier. We are talking about loyalty here.\nOnce your brand earned a customer's trust, you've got to take care only of one thing \u2013 not to lose that trust while milking customers for more money.\nWhen your company's branding is established, it's easier for customers to make up their minds as they clearly know what to expect from you.\nYou don't know what the next Mercedes-Benz car will be but you can be sure it's going to look solid, drive smoothly, feel comfortable inside, and have a full set of modern driving gimmicks.\nProducts come and go, market trends appear and disappear, but brands stay for good. Sometimes, for centuries.\nIf you take any modern IT or software company, you will easily see that their most valuable asset is the brand.\nWhen you hear that Apple is worth $500 billion, probably 450 out of those is the brand. The rest is probably equipment and patents.\nSo, now you get the idea of branding and why branding is important for your marketing strategy. Branding works for all kinds of business: for b2b companies, for nonprofits, for marketers, for advertising etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've previously highlighted ideas for how to approach your meeting agendas at a high level to ensure you have meaningful and enduring conversations with your prospects. Crafting a presentation while keeping in mind what your audience cares most about will go a long way to creating engaging content.\nWe also previously summarized our top 12 ideas for what data to bring to your first meeting with hospital or medical practice leadership to position yourself as a value-add partner and differentiate yourself from the hundreds of other medical sales reps that pass through their doors.\n\u200bNow let's dive into the nuts and bolts of actually creating your presentation, figuring out how \/ where to leverage data and where you may source relevant these insights to create a compelling sales presentation to hospital executives.\nDon't take for granted that your audience understands why you want to convey what you do. Invest 18 minutes to watch Simon Sinek's \"Start with Why\" TEDTalk. It's a great talk AND you will leave fully appreciating how important the \"Why\" is to the presentation.\nIt's so much more impactful when your communication can convey the motivators behind what you are saying rather than only focusing on the features, competitive comparisons, pricing and other nitty gritty detail. Even it your audience is familiar with your motivations, it's worth reiterating as it sets the tone and intention for the meeting. You could bring the why to life with stories about how your product extends patients' lives or anecdotes from other physician users on their experiences with your service.\nIf you are at a larger company, your marketing team will likely have a ton of guidance on the message as well the types of data available for your sales meeting. But you'll need to take the time to tailor it to your specific audience. Smaller companies may rely solely on their sales team to craft the story and bolster the pitch with data. Either way, if you are going to be the one delivering message, you need to get comfortable with the content and supporting data in order to confidently make and defend your pitch.\nThe third section of your presentation should be focused on impact. Perhaps you have enough experience with customers in your market that you can quantify outcomes and impacts over similar hospitals or physician groups. We've worked with several companies with newer products that are still working on market validation or that have products whose impact varies greatly client by client, making it harder to draw broad conclusions based on past experience. In these situations, look to leverage publicly available data about your specific prospect - sourced from CMS data, IRS filing or state agencies (more on that below) - or work with a data vendor like Carevoyance to create a rough draft of an ROI based on the information you can gather ahead of time.\nYour instinct may be that more is better but with data it's best to find one way to communicate the data rather than presenting it in various formats. This approach avoids confusion and allows the audience to focus on the meaning of the data. After all, you don't make them work to understand the chart in front of them and completely miss the takeaway message. Download this great reference guide for choosing the right chart to fit your data and story.\nDon't rush through a slide with data. Rather linger on the data and let your audience take it in and understand it. Take the time to explain the story you see in the data so that it's clear to someone who hasn't been poring over that dataset for the past six weeks.\nThere is a lot of value in leaving people with a print-out of your presentation, so that they can look at the numbers more closely after your meeting. Since data-driven decks and reports tend to get circulated, make sure that any charts have descriptive titles and labels, explanations of sources, and explicit call out of the most important takeaways.\nThere should be a takeaway that really captures the key concepts you are trying communicate.\n\u200bMake sure the chart summarizes this takeaway in call out text that is is clearly visible on that slide. Also make sure to cite your sources and explain what's included and excluded in the data highlighted.\nDo have a strong understanding of where the data came from, how others have interpreted and used the data, and especially what limitations may exist.\nFor example, if you want to tell a story about how often your product or device is currently used, it's often times too cost prohibitive to get a full and complete claims or medical record review. This is especially true if you want to personalize the story for that individual hospital or doctor. But publicly available sources or those curated by professional data partners, provide a strong foundation to improve the impact of your message.\nYou don't have to have the perfect dataset or the world's most beautiful infographics to make data part of your storytelling.\"\nIf anyone in the audience questions your data sources, you can confidently be prepared to explain the source, it's benefits and limitations and even invite further discussion on what other data may further inform the topic under debate.\nIf you don't already have data you want to work with, where should you start?\nWe'd recommend a few free sources, relevant for healthcare products and services, that provide a wealth of information ready to be analyzed and incorporated into your presentation.\nThis site is a free, on-line query system based on data from the Healthcare Cost and Utilization Project (HCUP). The system provides health care statistics and information for hospital inpatient, emergency department, and ambulatory settings, as well as population-based health care data on counties.\nThis site is set-up to make health data accessible to entrepreneurs, researchers, and policy makers in the hopes of better health outcomes for all.\nThis popular subreddit offers datasets for data mining, analytics, and knowledge discovery. You can search for healthcare specific topics to find relevant data for hospital executives.\nGoogle Trends shows how often a particular search-term is entered relative to the total search-volume across various regions of the world in various languages.\nOf course, here at Carevoyance we take away the headache of aggregating, cleaning up and maintaining healthcare data. We also have all the tools a Sales Exec could wish for when it comes to producing customized content, personalized for every hospital account or physician contact.\nWe'd love to hear from you what challenges you face in bringing data to your sales meetings. Let us know what's working and what opportunities exist to have more data-driven conversations with your prospects and customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Interior Glass Kitchen Cabinet Doors Fair Design Ideas Tinted In Throughout Contemporary. Interior Glass Kitchen Cabinet Doors Appealing Beveled And Frosted Stylish. Interior Glass Kitchen Cabinet Doors Pictures Options Tips Ideas HGTV Contemporary. Interior Glass Kitchen Cabinet Doors How To Add Confessions Of A Serial Do It Brilliant. Interior Glass Kitchen Cabinet Doors Remarkable And Ideas Expert Tips With Popular. Interior Glass Kitchen Cabinet Doors Creative Of Best Furniture Ideas For Stylish. Interior Glass Kitchen Cabinet Doors Replacement Tags For Stylish. Interior Glass Kitchen Cabinet Doors Sliding Cabinets Design Ideas Incredible. Interior Glass Kitchen Cabinet Doors Unique Cupboard Best 25 New. Interior Glass Kitchen Cabinet Doors Best 25 Ideas On Pinterest Contemporary. Interior Glass Kitchen Cabinet Doors Best 25 Ideas On Pinterest Brilliant. Interior Glass Kitchen Cabinet Doors Open Frame Cabinets Elegant. Interior Glass Kitchen Cabinet Doors Surprising With Panels Door Gallery Modern. Interior Glass Kitchen Cabinet Doors Valuable Ideas Charming Cabinets With Stylish. Interior Glass Kitchen Cabinet Doors Design Small Cabinets With White Amazing. Interior Glass Kitchen Cabinet Doors Incredible 20 Gorgeous Popular. Interior Glass Kitchen Cabinet Doors Elegant Kitchens Blue Ceilings And Throughout New. Interior Glass Kitchen Cabinet Doors Solid Photos AWESOME HOUSE Best Elegant. Interior Glass Kitchen Cabinet Doors Best 25 Ideas On Pinterest Amazing. Interior Glass Kitchen Cabinet Doors Great Door Cabinets Best 25 Elegant. Interior Glass Kitchen Cabinet Doors Pictures Ideas From HGTV Contemporary. Interior Glass Kitchen Cabinet Doors How To Add Confessions Of A Serial Do It Incredible. Interior Glass Kitchen Cabinet Doors Outstanding Brilliant. Interior Glass Kitchen Cabinet Doors Cool Door Deocr With U Shape Elegant. Interior Glass Kitchen Cabinet Doors For And Decor Contemporary. Interior Glass Kitchen Cabinet Doors Wonderful Lovable Decorating With New. Interior Glass Kitchen Cabinet Doors Wood Cabinets With Modern. Interior Glass Kitchen Cabinet Doors Terrific How To Add 19 For Brilliant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"pays great attention to the details of electric muscle stimulation. The following will show you one by one.\n4. It adopts the innovative methods of electron-analgesia. Domas has a scientifically sound management system while ensuring product quality.\n1. With the emergence and broad development prospect of tens device, 2018 GShenzhen L-Domas Technology Ltd. has becomes more and more popular.\n2. It turns out that applying the best technology to the electric pulse massager is a good idea.\n3. 2018 GShenzhen L-Domas Technology Ltd. advocates mutual understanding, cherish diversity, and view our culture in a global perspective. Contact us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please login before purchasing online. Log in here or Create Account.\nYoganet is a booking site for yoga studios operated by Yogacentralen.\nGraphics delivered by Yavanna with \u00a9 Yogacentralen.\nNo Account ? Create new Account.\nForgot your Username or Password ? Find it here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Health Canada has awarded nine new cannabis cultivating licenses since late May in an effort to keep up with the growing demand for marijuana.\nThe number of licensed cannabis growers in Canada has increased by 20 percent over the past few months, according to a new report from Marijuana Business Daily. The growing demand for cannabis has prompted Health Canada to license nine more growers since late May to bring the nation's total number of licensed producers to 54.\nCanada has seen its four-year old medical marijuana industry soar as of recently, even tripling in the past year to reach 167,754 patients. Medical marijuana use is even up among veterans, who have reportedly been replacing prescription medications with cannabis.\nCanadian officials had been aware that an adequate cannabis supply could become a concern, particularly with the nation slated to legalize recreational marijuana next year. It responded by speeding up its licensing approval process for interested growers and easing the backlog of applicants to boost cannabis production nationwide. Previously, becoming a licensed producer took up to three years, followed by another year to ramp up production.\nCultivators that have already obtained licenses are scrambling to increase their production capacity. Many growers are building new production facilities or expanding the square footage of already-established greenhouses.\n\"Having our production capacity built out is the most critical thing for me today,\" Greg Engel, the CEO of one of Canada's 54 licensed producers, told Marijuana Business Daily.\nDespite the effort to avoid a lack of supply, the industry still may not be able to meet demand once the recreational marijuana market rolls out next year. According to Marijuana Business Daily, Canada's Parliamentary Budget Office estimates that between 378,000 kilograms and Canadians will consume 1.01 million kilograms of cannabis next year before demand rises to between 403,000 kilograms and 1.1 million kilograms by 2021. Annual production capacity is expected to reach between 60,000 kilograms and 120,000 kilograms next year, far short of demand projections.\nCanada's upcoming adult use market, projected to be worth $4.5 billion by 2021, is expected to be much larger than its medical market. Market analysts have predicted that by 2021 there will be about 3.8 million legal recreational users throughout the country.\nThe government has said it may be willing to overlook past marijuana-related criminal records among those interested in seeking a cannabis production license, depending on the circumstances. Being able to meet the demand for legal cannabis is an important key to the government's goal of reducing the illegal market.\nUnder the proposed adult use law, Canadians ages 18 years and older would be allowed to purchase and carry up to 30 grams of marijuana and grow up to four cannabis plants for their own consumption. Cannabis will likely be available for purchase strictly through online orders and mail orders for delivery.\nAs of now, cannabis remains classified as a Schedule II drug and the only legal way for Canadians to access it is with a prescription. You can learn more about the current cannabis laws in Canada by visiting our education page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Antimony trioxide, Sb 2 O 3, is the most important antimony compound produced. Today, antimony trioxide is produced by volatilizing antimony metal in an oxidizing furnace. It is used in halogen compound flame retarding formulations for plastics, paints, textiles and rubber.\nIn addition, there is a driving, switch plates, the use of which to protect the jaw r- ... jaw plate, jaw crusher plate, jaw crusher jaw plate China (Mainland) Common Jaw Crusher Injuries, and How to Avoid - Mellott Company. Mar 20, 2017 While the jaw crusher has become increasingly safe in .\nPortable Mini Conveyor - Belt Conveyor Systems. Compact Portable Belt Conveyors .Mini Conveyors used for construction, sand conveyor, mining conveyor, concrete, disaster relief.\n\u00b7 The present invention relates to a process for the production of antimony trioxide. 2. Description of the Prior Art . Antimony trioxide, also known as antimonous oxide and\/or Sb 2 O 3, is a known compound useful in the manufacture of paints, plastics, ceramics, and the like. The principal utility of antimony trioxide is as a flame retardant.\nAntimony trioxide is the most commercially significant form of antimony and is a high-production-volume chemical with a production volume exceeding one million pounds per year. Its major industrial use is as a synergist with halogenated flame-retardants in textiles, plastics, and rubber.\nAntimony Trioxide News, Pricing News and Features ... The latest Antimony Trioxide news, ... in China has softened week on week on fears of a possible trade war between China and the United States.\nAntimony Oxide, Antimony Oxide Suppliers Directory - Find variety Antimony Oxide Suppliers, Manufacturers, Companies from around the World at zinc oxide .\nunited states patent 3944653 antimony . united states patent 3944653 antimony trioxide united states patent 3944653 antimony trioxide; We Are Experienced Birnith is quite experienced in construction, milling and mining industry know more. italy sand blasting machine - truecare.\nBarium oxide, BaO, is a white hygroscopic non-flammable compound. It has a cubic structure and is used in cathode ray tubes, crown glass, and catalysts. It is harmful to human skin and if swallowed in large quantity causes irritation. Excessive quantities of barium oxide may lead to death.\nIn modern units, the pied is equivalent to 0.325 meters or about 1.07 feet in the \"English\" system still commonly used in the United States. pinte: volume unit in late 18 th-century France, equal to 2.01508 English pints, 58.145 cubic inches, or 0.953 liters. Plimmer's salt: sodium antimony tartrate, Na(SbO)C 4 .\nA yarn, fabric, and garment suitable for use in arc and flame protection comprising aramid fiber and modacrylic fiber wherein the modacrylic fiber has less than 1.5 percent antimony and is preferably antimony .\nPatent Citations (8), Referenced by (4) ... USPTO Assignment, Espacenet: Very soft urethane vulcanizates which comprise a fully saturated urethane prepolymer and .\nAll of Nokia's mobile phones are free of toxic polyvinyl chloride (PVC) since the end of 2005 and all new models of mobile phones and accessories launched in 2010 are on track to be free of brominated compounds, chlorinated flame retardants and antimony trioxide.\nAntimony trioxide is produced in the US by roasting antimony . A later study in 1967 with improved radiologic techniques found 44 of 262 men with simple . ... United States Patent (19) 11 Patent Number: 5,783,166.\nrock crusher made in the usa . . the United States produced a total of about 1.26 billion tons of crushed . usa polisching industrial machinegranit plates; ... Usa Polisching Industrial Machinegranit Plates. united states patent 3944653 antimony trioxide; . Online Service. equipment for quarrying limestone - .\nUnited States Patent [I91 Alcorn et al.  Patent Number: 4,543,442 ... ployee of the United States Government as well as in adhesion between metal and semiconducting layers of a ... antimony trioxide, or to the nearest finger of the interdigitated ohmic contact.\nJaw crusher manufacturer,Double Roll Crusher exporter,Jaw ... THAPAR AUTO TRADING WORKS \u00e2\u20ac\" Manufacturer & Exporter of all type of Jaw crusher manufacturer, Double Roll .\nAbout Us. Mali Yellow Pages Online is a Local Business to Business Directory in Mali offering business list of more than 250,000 companies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Why go now Ilocos Norte, about 488km north of Manila, is known as Marcos country. The late deposed Philippine President Ferdinand Marcos was born in Sarrat, Ilocos Norte; so a tour of Ilocos will not be complete without a visit to his birthplace and mausoleum. Youngest Marcos daughter Irene, Borgy Manotoc's aunt, wed in Sta. Monica Church, in Sarrat in what was described as the wedding of the century. Houses along the wedding path were painted and roads, paved. Some Spanish colonial-inspired houses were built for the wedding and still stand. These days, however, Ilocos Norte's greatest draw is its beaches and adventure activites.\nIlocos Norte's 77-hectare Fort Ilocandia casino, also attracts Asian high rollers, mostly Taiwanese, Korean and Chinese.\n\"You simply can't ignore Fort Ilocandia. It's like the heritage of Ilocos Norte. It's built after a resort in Casa Blanca. The entire structure is made out of red bricks. It's a very old hotel but the place is nice because of its charm.\n\"At Fort Ilocandia, you can drive; you can go to the beach, ride the ATV (automated transfer vehicle). They have things like the firing range, extreme game- oriented stuff. The place is nice because of its charm; the furniture's nice; every room has a nice balcony. When it's December or January, the weather's so great. The temperature goes down to about 15 degrees. During the day, you can wear a light jacket. I don't know if it's true but they tell me that Ilocos Norte's getting the Siberian breeze. In the morning, the sun may be glaring; but because of the breeze it's so cool and the humidity's not really at all uncomfortable. And at night, it's really quite chilly. In front of Fort Ilocandia, they have this barbecue and live band set up. You just pick out what you want and they'll cook it for you. You get to enjoy your meal because you're by the beach so you get that smell of seawater,\" said Manotoc.\nOpened in 1983, the resort was said to have been built for the wedding reception of Marcos' daughter Irene. Fort Ilocandia Resort and Casino has a golf and country club and a two-kilometer beach. Room rates range from P7,000 (about US$151) for standard single\/double rooms to P38,800 for the ambassador suite. For inquiries and reservations, call +6377 670 9001 to 15, visit Fort Ilocandia Resort and Casino.\nHannah's Beach Resort in Pagudpud is a 40-villa property situated right in front of the famous Blue Lagoon, otherwise known as the Maira-ira Beach. The resort is enveloped by large forests and cliffs and is a great place for snorkeling, mountain trekking, and boating. Room rates range from P2,000 a night for standard villas good for twoto P22,000 night for family suites for nine. For inquiries and reservations call +63928 520 6255, visit Hannah's Beach Resort's Facebook page.\nLocated in Currimao, Ilocos Norte, the 18,000sqm Sitio Remedios is a heritage resort facing the West Philippine sea. It is owned by Dr. Joven Cuanang, St. Luke's Medical center medical director, who wanted to recreate a typical Ilocano village in the mid-50s. The houses and buildings at Sitio Remedios are in the Filipino-Spanish style, made of vintage bricks and wood salvaged from mid-century structures mostly from the towns of Ilocos Norte. Furnishings are vintage Ilocano hardwood furniture and the butaca plantation chairs with elongated armrests. With no television sets in the houses, Sitio Remedios is a destination for people who want to get away from it all. Rates are from P4,000 a night for Kwarto San Antonio, a room with queen-size bed and private bathroom, to P12,500 a night for Balay Batac, a two-floor house with two air-conditioned rooms, king- size beds and bathrooms. For inquiries and reservations, call +63917 332 0217, visit Sitio Remedios' website.\nPaoay Church or St. Augustine Church is a UNESCO world heritage site, one of the \"earthquake baroque\" style churches that features 24 massive brick reinforcements running along its sides, with walls made of coral rocks, baked bricks, lumber, limestone mortar and sugarcane juice.\nHistory has it that the bell tower of this 18th century church was used by the Katipuneros (Filipino vigilantes) as observation post during the Philippine revolution in 1898 against the Spaniards, and also by the Filipino guerillas during the Japanese occupation in World War II. For more information and inquiries call +6377 793 2030.\nMalaca\u00f1ang Ti Amianan (\"Malaca\u00f1ang of the north\") overlooks Paoay Lake and was built on a five- hectare land in 1976, inspired by Spanish colonial houses. It was one of the 29 summer residences of the Marcoses built during his regime. Though a substantial property, it pales in comparison with the other more lavish Marcos residences. It has nine spacious rooms for the family and their guests, and showcases a combination of both Ilocano and Spanish architectural designs. The walls are made of baked bricks; the floors and staircases of hardwood; and the sliding windows are made of capiz shells. The house is now a museum of the late President's memorabilia. The Malaca\u00f1ang of the North is open Tuesday to Sunday, from 9am to 11:30am and 1pm to 4:30pm. Entrance fees are P20 for adults and P10 for children.\nAbout four kilometers east of Paoay church is Batac, where you'll find the Marcos Mansion and Mausoleum, where the embalmed body of Ferdinand Marcos lies in a glass coffin in an air-conditioned room, with temperature carefully controlled and monitored. Although the late president was born in Sarrat, Batac is considered to be his hometown. The mansion houses a wide array of Marcos memorabilia and collections including family photos, military awards, his writings, and his work desk. The Marcos Mansion and Mausoleum is managed by the Department of Tourism. For more information call +6377 772 1219.\nOn Cape Bojeador: \"I don't really know much about the heritage sites because I mostly stay in the city. But I'm sure you've heard of the windmills and the lighthouse Cape Bojeador that overlooks the cliffs and the rock formation and the ocean. From the lighthouse, you'll see all the beautiful landscapes and the mountains and the sand dunes. the lighthouse is over a hundred years old. It's a long climb up to the lighthouse on old, metal winding stairs. It's probably about three-four-storey high. Even the walk to the base of the lighthouse is about two-three storeys. But it's definitely worth it. It's all glass and you can actually go up to the top where the light used to be.\nAdams is probably one of more recent nature adventure spots in Ilocos norte. Go to Bolo River and enjoy kayaking on the 5km stretch with a virgin forest right on the edge of the river.\nA 30-minute motorcycle ride from Pagudpud to Adams will cost about P150 per person. Reaching the town proper in Adams, you can walk to the Bolo River, which overlooks Mt. Palemlem, said to be the highest peak in Ilocos Norte. You can hike with a mountain guide for P700, good for 5 persons.\nTourists can also hike from Bolo River to Anuplig Falls, a 25-foot waterfall with two basins, ideal for swimming.\nClimb Lovers Peak, a hill located near the town proper. Once you get to the top, you can enjoy a 360-degree view of the Tinamburan mountain ranges.\nManotoc suggests trying sand boarding on the sand dunes in Paoay. According to Reny Tan, vice president of the Laoag Eco-adventure Development (LEAD) Movement, sand boarding is perfectly safe for everyone, professionals and beginners alike. For a fee of P2,500, the LEAD Movement can organize a 4x4 ride on the sand dunes, inclusive of sandboarding activities. Contact Reny Tan at +6377\/ 772 0538 or +63919 873 5516, or visit LEAD Movement's website.\n\"Whenever I visit the province, I usually do food trips. Batac is the best for food. It's where you can get authentic Ilocos norte empanada (fried thin pastry stuffed with grated green papaya, longganisa or local sausage and egg). Go to Riverside Empanadahan in Batac for empanada. For inquiries call +63916 358 2597.\n\"For family gatherings I don't know why my mom likes to invite everyone to Macy's Diner in Laoag. It's funny because when you get there, you see pictures of James Dean hanging on their walls and stuff like that \u2013 and then they serve pinakbet pizza. It's funny because it's a diner, with all that American stuff going on; but at the same time it's really localized. For inquiries call +6377 770 3551, visit Macy's Diner.\n\"Another must try in Laoag city is Saramsam Restaurant, which has developed a poque poque variant \u2013 topped with a mix of grilled eggplants, cooked omelet style with onions, tomatoes and eggs, along with their inkalti, which is essentially karioka or crisp-fried rice cooked on the table a la fondue. Located in Giron and P.Gomez St. Laoag City, tel +63917 702 6401, visit Saramsam Restaurant.\n\"Herencia Caf\u00e9 in Paoay, located right across the famous church, is a good place for the original pinakbet pizza. they also have a variety of dinuguan (pig's innards in blood stew) and bagnet (crisp-fried pork). Located in McArthur St., Barangay 14, Sangladan, tel +6377 614 0214, visit Herencia Caf\u00e9.\n\"For the best pancit in the world get Pancit Cabagan, a medley of stir fried freshly made noodles mixed with chicharo, Baguio beans, cabbage and topped with boiled quail eggs and crispy lechong kawali or bagnet, at Aling Kikay's Restaurant in Bacarra road, Cabagan.\"\nGo to Nagbacalan, Laoag city, for Abel clothing (hand-woven cloth with leaf and floral design) made using the traditional weaving machine called pangab-lan.\nFor tapuey (rice wine) and bugnay (fruit wine), ask locals for Lola Ingga's home in Adams.\nThe museum is open Monday to Friday from 9am to 5pm. Located at Gen. Luna corner Llanes Sts., Laoag City. For inquiries call +6377 770 4587, visit Museo Ilocos Norte.\nBy land, Fari\u00f1as Bus Terminal in Quezon City has regular service to Ilocos Norte for P700 one way for regular buses and P750 one way for buses with toilet. The day trips will take roughly 10 to 12 hours and eight to nine hours for night trips. For inquiries and reservations call +632 743 8580, visit Farinas Transit.\nIlocos Norte is best explored by kalesa, the traditional horse-drawn carriage. A 30-minute tour around the city costs P200. Kalesas can seat up to five passengers. Tricycles are good for traveling short distances. If planning to go from town to town, board a mini-bus at terminals near the Provincial Capitol in Laoag City for P80.\nBorgy Marcos Manotoc, 33, was born in Honolulu, Hawaii, where the Marcoses were once exiled following the People Power Revolution of 1986. While he has lived in London, Portugal, and New York during most of his growing up years, Manotoc settled back in Manila in 1998. He frequents Ilocos where the Marcos political heartland remains. His mother Imee Marcos has been the governor of Ilocos Norte since 2010.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Smartwatches are generally used gadgets. Forrester Research?s largest annual survey of Americans? know-how adoption finds that seventy three p.c of the 37,000 respondents claim the mobile phone is the electronic gadget they use probably the most. Embattled Samsung Electronics said Friday it expects earnings to jump by half in the first quarter, regardless of a smartphone recall fiasco and the arrest of its de facto head.\nMade by iHealth Lab Inc, the Wireless Sensible Gluco-Monitoring System is a state-of-the-artwork, FDA authorized glucometer that measures glucose ranges in the blood and then displays them on your smartphone. Introducing Intel's Compute Card, a mini-computer, which is about the dimension of a bank card, designed to be simply inserted into sensible units.\nWhether or not it's drops of water from being uncovered within the rain, soda spilled on top of it, or full submersion in a pool or bathroom, it is one of the few kinds of harm that may very well be irreparable. The chief problem with liquid and electronics is that when the liquid dries it leaves residue all of the different supplies contained, including minerals, grime, etc. This causes the inner digital elements to corrode.\nCell fee is another fashionable use for smartphones.\nMany mobile users have had the unlucky expertise of retrieving their phone from a watery grave. With DocuSign, employees can securely ship, sign and manage nearly every agreement from almost anyplace in the world. Your customers can rapidly and easily complete transactions from their smartphone or pill. DocuSign affords on-line signature apps native to each main cellular platform.\nSmartwatches are commonly used gadgets. That's why we've curated this list of fifty prime-rated electronics presents for men. Whether or not you are gearing up for the vacation season or buying a birthday or anniversary reward, we're positive you may discover one thing on this checklist the person in your listing will love \u2013 whether or not he is a gamer, a sports activities fanatic, or a guy who simply has to have the latest tech toy. Our picks are listed below in alphabetical order for straightforward reference. Ratings data relies on Amazon opinions and is present on the time of this writing.\nFrom design to distribution, we make modern products and services that carry together world manufacturers and connected machine technologies to ship unique solutions for our consumers. The one main disadvantage with tiny cool gadgets nowadays is that they can easily get misplaced. If you need some help keeping tabs on the small stuff, Tile merchandise finders are simply the ticket. The keychain-sized Tile Mate is a diminutive 1.three inches long and is just zero.2 inches thick: Simply attach a tracker to any merchandise and you should utilize the Tile companion app to locate it just about anywhere. Even without the app, you can nonetheless find close by items by remotely making the Tile vibrate, flash, or ring.\nI used to love soldering back in the day of hobbyist electronics. When you have the Samsung Galaxy S7 or Galaxy Be aware 5 or even higher versions of the smartphone, you can say goodbye to cumbersome wired charging and say howdy to a revolutionary new manner of powering up your high finish Samsung and LG devices. The Pleson PLS-WR-C400 is a splendid piece of charging tools that employs dual charging coils that effectively powers up your devices as much as 1.four times quicker than typical charging. The Pleson contains a security temperature control technology which helps protect the integrity of your device's temperature-sensitive circuitry. The design can also be made particularly for non-obstructive charging. Ensure you also test our solar phone chargers assessment for extra great items like this.\nA digital picture body is composed of three main components: the LIQUID CRYSTAL DISPLAY-panel, the PCB (Printed Circuit Board) and the outside body. A few of the digital photo frames help film clips, MP3 audios and JPEG photos, whereas among the digital photo frames support JPEGs only. Customers can add their photographs to a digital photo body through a USB port.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The study of the past, as recorded in the fossil record.\nThe study of human origins, behavior, and culture past and present.\nThe study of insects, the most diverse group of organisms on the planet.\nThe study of animals with a backbone (such as mammals, birds, reptiles and amphibians).\nThe study of minerals, including their distribution, identification and properties.\nThe study of plants including their evolutionary relationships and ecology.\nThe study of invertebrate creatures called mollusks, many of which live in shells.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The French Revolution has been an ideological battleground for writers and historians since 1789, and the Reign of Terror is no exception. Conservatives and rightists traditionally see the Reign of Terror as catastrophe and consider the Terror the result of the political inexperience of those who the revolution made into leaders. However, there is little evidence to suggest that this is, in fact, the case as a large portion of early revolutionaries held considerable political experience prior to 1789. Many early contributors to the rightist interpretation, such as British statesman Edmund Burke, developed their opinions as the French Revolution was taking place. Later criticism from the right, was more of a reaction to Marxist interpretation of the French Revolution. Leftists, republicans, socialists, and Marxists defend the Terror as a product of necessity. They point to the many problems plaguing the republican government and see the terror as a response to war, and political division. However, while this argument holds water in its own context, it lacks an explanation as to why the Terror intensified when civil war and foreign invasion did not pose an immediate threat to the revolutionary government. Revisionist historians look at the Revolution as a result of revolutionary ideology and culture. The radicalism of the French Revolution was not driven by social conflict as the Marxists would argue, but from ideological roots that reached further back into the Ancien Regime, and in fact radical beliefs that would rule the Reign of Terror were indeed present during the early lukewarm stages of the revolution. Revisionists' arguments revolve around two major points; the culture of eighteenth-century France and the influence of Jean-Jacques Rousseau.\nLet us explore these last two interpretations more closely; terror as a product of necessity and terror as a product of ideology. It is to be sure that the Revolution faced many problems leading up to 1793-94. Mass desertion of upper tier army officers left the military unstructured, and while this did allow the cream of those most talented soldiers, i.e. Napoleon, to rise to the top, the present state before the Reign of Terror was one of chaos and disarray. Britain and Austria, France's two greatest historical enemies, invaded French soil. King Louis XVI had attempted to flee the country with the royal family and join up with foreign powers to destroy the revolution, thus confirming the greatest fears of French citizens about foreign conspiracy from within. The western region of the Vendee and several cities resentful of Parisian hegemony, were in open revolt. Federalism, which in the context of the French Revolution means de-centralization of power, was heresy to the idea of the revolutionary government, and the sovereignty of the general will of the people it represented. The peasants in the Vendee, opposed the forced conscription into the revolution's military, new taxes, attacks on Catholicism, and the execution of the king. Food shortages still ravaged the lower-classes as attempts to control the price of grain threw a wrench in the machinery of the French economy. Enemies and counter-revolutionaries were inside and out, and the republic of virtue was in danger of being snuffed out before it could even begin. The Reign of Terror was the solution to these problems. With vast powers centralized around the Committee of Public safety, grain prices were controlled, as citizens were drafted into the army, the foreigners were driven out, the civil war in the Vendee was squashed, and the federalist cities were placed under siege and surrendered to Paris. During all of which, opponents of the revolution were fed to madam la guilliotine as France was purged of its enemies. The Reign of Terror, arguably, solved many of the problems facing the republican government of 1792.\n Hugh Gough, The Terror in the French Revolution. 2nd Ed (Palgrave Macmillion,1998), 19.\n R.R. Palmer, Twelve Who Ruled: A Year of Terror in the French Revolution (Princeton: Princeton University Press, 1941). 23.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Crucible Is The Critical Culminating Event Occurring In The 11th Week Of Marine Corps Boot Camp. It Is 54 Hours Of Trial And Triumph. Referred To As The Defining. With Little Shlep, Less Food, And Over 40 Miles Of Forced Marches, Recruits Tackle What Is Surely The Toughest Part Of Becoming a Marine. 55 Minutes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you're transporting your laundry to a Laundromat, then you probably already know that there are different laundry bag options. Which option is the best? Well, there's no such thing as \"the best laundry bag\" because the different types of bags are a matter of one's personal experience. However, if you're planning to buy laundry bags, you should find out what's available, right?\nDon't worry; we're here to make this buying task easier for you.\nYou see, we already tackled the subject of laundry baskets, which are also a great option for transporting your laundry. However, laundry bags are more convenient as they offer a more protected and secure manner of laundry carrying, especially if you need to transport your clothes to a Laundromat.\nIf you're into eco-friendly things, we have some great news for you! Not only are these bags more convenient, but they're offering ecological benefits as well. They are washable and re-usable, and can also be easily repaired in case they become torn or worn.\nAs for the materials, they are made of cotton, nylon, or mesh, but lately, the crocheted or knitted types of bags are becoming more and more popular. Laundry bags come in different styles, like backpacking style, cross body carrying, or single handle, to name a few. As for the closures, you can opt for bags with buttons, snaps, zippers and drawstrings.\nPersonally, my favorite bags are the ones with symbol tags. This way, I can be sure that I'll never shrink my favorite shirts again. You can also opt for bags with multiple straps so that you won't break your back when going to the Laundromat. Some bags come with bonus pocket where you can stash loose change, dryer sheets, or your Laundromat card. If you're not comfortable carrying a bag full with your dirty laundry out in public, you can buy a sturdy cotton tote that doesn't look like a bag at all.\nNylon laundry bags are much stronger and more durable than any other type of bags. They usually come with bonus pockets, allowing you to place a laundry-related item inside, like detergent or soap. They come in different sizes: the small sized bags are perfect for placing items for storage when moving\/travelling, while large sized bags are perfect for storing more clothing.\nCollapsible kinds of bags are also very popular. They are made of nylon or mesh. If you opt for this kind of bags, do not overstuff them! They can be zipped up, so if you travel on foot to the Laundromat, you would definitely need a laundry bag with zippers.\nEasy to use and carry, the laundry bags are a convenient way to transport your laundry. As you can see, there are different styles of bags, each offering something for everyone. After all, it's a personal matter of buying the bag that will work for your needs when doing a laundry.\nIn conclusion, your top considerations regarding laundry bag should be its durability, capacity, and of course, cost. Design factors like pockets, straps and type of closure may affect your final decision as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is a new offering from the CLAD lads :-), and follows on from the information that we got from the European Medicines Agency and from stuff in the public domain. This is not a pharma supported piece of work and they may have different views of how cladribine works or how they want to portray how they think this works, but as usual for us, it contains ideas of the unknown. Things we and importantly they can and should test.\nThis is our take on how cladribine may be working\u2026it may be wrong, but it incorporates available data and does not follow the Treg-control of every thing dogma. It puts the T cell dogma into the perspective of the B cell reality\u2026.Ha Ha:-).\nSome people may not be happy, as we have mentioned the content of published abstracts that they were on, but have not published, and interpreted their conclusions in a different way. Yep, there is now no need for them to waste paper as the essential information is out in a way that makes more sense:-).\nI can see the moaning email winging its way to DrK.\nYou will notice this does not include ProfG, who will see this manuscript here for the first time too. He is a lead on the pharma view and we were sneaky and got and published the stuff that he could have published. So this way ProfG does not get his hands dirty. He may have additional takes on this story, as he has the inside information and insight that we don't.\nAs to the people who may be upset\u2026\u2026 \"You snooze, you lose\" they have have had ample time to publish the findings. Too much time going to meetings on pharma coat tails and not enough time writing I guess:-(.\nWe have met this fear and need to accept and pacify dogma time and time again during the review process.\nIt suggests to me that many of these scientists would make terrible police officiers.\nWhy? They can see a dead Dr Black (UK)\/Mr Body-John Doe (US) with a big knife in their back and a blood trail from the study. However, they are too frighened to say that the knife killed Dr Black, unless they see the video of it happening. Unfortunately nobody was there with a smart phone to do this, so they would you have to look at the evidence. God forbid that they could suggest that it was Colonel mustard, who did it in the study (Cluedo). Yet they are happy to accept and suggest that Dr Black dies from food poisoning, because of an altered microbiome, because that is what everyone else thinks should happen. They are happy to ignore the machete. in search of the antibiotic, whilst ignoring the problem with knife-crime. Sad thing is\u2026they would have to stick a machete into someone to show that it can be a lethal weapon, before they can believe it.\nWhy is science a slow process?\nOral cladribine is a novel treatment for relapsing multiple sclerosis (MS). This appears to be a semi-selective immune-reconstitution therapy that induces long-term therapy from short treatment cycles. It has a relatively good safety profile that currently does not require extensive monitoring associated with some continuous immunosuppressive and relatively non-selective immune reconstitution therapies. The efficacy and safety of cladribine relates to its particular physicochemical properties, the function of the lymphocyte subsets that are selectively targeted by the drug and the repopulation kinetics of these subsets. As such, there is marked and long-term depletion of memory B cell subsets, which probably relates to the therapeutic efficacy. This is also coupled with a more limited, but likewise long-term, depletion of CD4 T subsets. There is limited depletion of cells of the innate immune system and modest effects on CD8 and probably plasma cells, which provide immediate and durable protection from infection. Targeting of CD4 T regulatory cells, CD8 T suppressor cells and regulatory B cell subsets appears more limited as these populations recover rapidly and so repopulating pathogenic cells re-emerge into a regulatory environment. This appears to lead to re-establishment of immune-tolerance that produces long-term control of MS. Although this hypothesis contains a number of unknown details, it is based on knowledge about the biology of cladribine, basic immunology and the effects of other high-efficacy B and T cell depleting agents that exhibit stereotyped repopulation behaviours. These concepts are relatively simple to interrogate, and can be modified as new knowledge about the durability of disease control and safety with cladribine emerges.\nSo I still haven't worked out what this is all about. On shift ms people seem to be on cladribine .\nIs it available on the nhs ?\nWhat would you really like to know?\nOral cladribine tablets are available on the NHS. Stupidly the regulators restrict it to highly active MS. It is no more risky than alemtuzumab which has a liberal licence and is available to anyone that is active.\nThis is for relapsing MS.\nWill it be useful for SPMS probably yes. Is it going to stop SPMS..in the short term no, as you need more than a treatment like cladribine.\nHowever it could be an excellent base on which to layer other things.\nIs it for all forms of MS\u2026 yes if there is evidence of disease activity. However the licenced version is currently for relapsings only. Maybe with more patent life they will now look into PPMS.\nThe alternative is off label generic cladribine. Access to this may depend on your neurologist. We have a number of people with SPMS and PPMS on this variant administered via the NHS.\nThanks for the clarification. Really useful. Looks like not for me ! On a brighter note this new site is fab. Huge improvement and much more interactive and easy to use .\nCan you explain why some patients get cladribine as tablets and others as cub-cutaneous injections of off label? Is there a difference in efficiency or comvenience of follow on tests? In an earlier post I recall the sub-cutaneous injections were quite infrequent.\nDrK started doing this in 2014 and remember cladridine tablets were not available until the end of 2017. DrK has been cautious to limit the degree of white blood cell loss with the off-label variant and has got severe lymphocyte losss down to about 1-2% compared to 20-25% in the early studies and it is still about 10-12% with the de-risked approach in the current licened tablet. Whether this reduces efficacy is unknown as we have nothing to compare it two, however we are can see benefit such as by seeing reductions i, but this is done as a service and not as a trial. In terms of dose the subcutaneous variant about twice as much drug enters the system. So you use half as much. The subcutaneous route is a 15 second injection verses swallowing a for a pill. Monitoring requirements are minimal and is simply measuring white cell levels before the next dose. The number of swallows are about 3-5 a a mouth twice a year apart.\nCan you explain how this mechanism of action differs from Ocrevus in terms of the cells targeted? Am I correct in saying Ocrevus targets the same cells, but is more effective than cladribine? If this the case why would I want cladribine when I can be treated with Ocrevus?\nIs ocrelizumab better than cladribine. There is no good evidence to suggest this.\nOcrelizumab is continuous. This may have some advantage but is offset by disadvantages.\nI predict that you would not need to dose every six months,which is the time.bomb that ocrelizumab is probably creating. There are B cell controlled infections and these issues will arise sooner or later in some.people. The experiment is now ongoing as there has been massive take-up of the drug.\nThe question is do you need T cell depletion for the durability of response. we need a trial to test this.\nThank you for this article. Looking forward to the full text.\nCould you elaborate on why Clad. but not Ocrevus restores this immune tolerance that you are referring, since the effect on other immune component is limited according to the abstract?\nWe dont know that ocrelizumab doesn,t. If you stop ocrelizumab how long does benefit last, we have data showing at least 18 months, with rituximab benefit appears to last for two years and longer in others.\nIt could be that the T cells wonto become property tolerant unless they are depleted.\nShould i publish this data being hidden by pharma?\nin a PPT slide package from a presentation Prof G held, that Lemtrada and Mavenclad was immune reconstitution drugs. Ocrevus was categorized with a question mark. So what does Mavenclad and Lemtrada has in common which makes them this?\nOne last question; could it be a theory, that if one received multiple rounds of Mavenclad, more or less the entire immunesystem would be rebuild eventually?\nClad is given a a few pulses and the effect.lasts four years at least. Alem dosing.idea is similar one year of treatment and for 40-50percent of people thats it for years. For.clad we.will not.get the data soon nevause of the messed up development of clad. ocrelizumab is given continously every six months. According the company they would have us believe it lasts 6 months. I do not buy this and suspect it is an IRT as much as cladribine is. So the effect lasts longer than 6 months. I know this but the data has not been published but the concept has been replicated replicated. I promise you this information gap will change.\nSo there maybe no difference between alem clad and ocre. However alem and clad get rid of T cells too. Is this important. It maybe.\nWe need to do a study.\nMultiple rounds will probably lead to infectious complications. CLAD preferrentially hits lymphocytes but we know from other studies that with frequent dosing other cell types get.it. indeed a cladribine like variant is used in HSCT.\nThanks for the update MD look forward to reading latest paper. Maybe one day someone will agree to risk Rx me off-label C. Clock is ticking\u2026.\nI've just read the whole paper (one of the perks of academic access, I guess). Really interesting stuff! Although, a lot of info in it was kinda already there if you read \"between the lines\" in comments and several other papers\/posts on this blog (not the toleration effect on memory B cells though, that theory is novel to me.) Now it's all clearly written and referenced.\n\u2013 \"The effect of cladribine tablets on the oligoclonal bands remain to be established.\" Why do you think that tablet version of Cladribine won't have the same effect on OCBs as other routes of administration? It all boils down to concentration in plasma and subsequent penetration in CNS. Why would it be different?\nClad and subcuataneou clad wont be different as the dosing is essentially the same, but the referee wanted use to highlight what we think we knew and what we were speculating.\nIn the polish study the oligoclonal bands disappeared in over half the people treated but this was looked at 10 years after treatment. Based on the level of the enzyme that leads to the depleting , it is low in antibody cells so we predict it will not kill plasma cells directly so removal of oligoclonal bands may be slow, we are on this case and we have CSF from people treated months earlier.\nSure. But if it turns out to be true, this could mean that Cladribine is better than other high efficacy DMTs, especially if you take into account its good safety profile. Am I reading between the lines correctly? Is this why you hinted that Pharma won't be happy with this explanation\/theory?\nWill ProfG's comment on this paper? I'm really curious about his take on this.\nCladribine is a high efficacy, but not a very high efficacy DMT. Its impact on brain volume loss in after 6 months in the CLARITY trial was moderate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Humphreys, P. & Mayes, W. M.\nGlover, I., Atkinson, R., Soraghan, J., Judd, M. & Vieira, M. D. F. Q.\nTzortzopoulos, P. & Koskela, L.\nMcCluskey, T., Antoniou, G. & Vallati, M.\nTill, R., Fazenda, B. M. & Scarre, C.\nPitts, A. & Gao, Y.\nArmitage, R. & Sidebottom, A.\nGill, R., Dampier, H., Law, K. & Muller, C.\nIwnicki, S., Powrie, W., Preston, J. M., Blainey, S. P., McDowell, G., Roberts, C. & Thompson, D. J.\nCrow, G., Pahl, K., Steadman-Jones, R., Hart, A., Reid, S. E., Harris, B. J., Dubow, J., Griffiths, C., Carpenter, M. J., Church, A., Pinnock, A., Smith, G., Lyon, D., Dominello, L., Banks, S., Chapman Hoult, E., Mohan, J., Byrne, D. S., Mah, A., Daniel, B., Chilies, P. & Ward, P.\nLongstaff, A. & Fletcher, S.\nWu, S. & Tzortzopoulos, P.\nJones, A. & Pasura, D. M.\nBarlow, R., Fletcher, S. & Edgecock, T.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alliant Techsystems has made significant progress in developing hardware and materials in support of NASA's new Ares I crew launch vehicle. The Ares I is designed to carry crews to the International Space Station, back to the moon, and on to Mars.\nIn December 2005, NASA selected ATK to be the prime contractor for design and development of the first-stage propulsion system for the Ares I. The first stage is a five-segment solid Rocket booster derived from the four-segment Space Shuttle reusable solid Rocket motors (RSRM) developed and produced by ATK. 'We have made tremendous progress over the past year, and the project is on track to conduct ground and flight tests scheduled to begin in 2009,' said Ron Dittemore, President, ATK Launch Systems.\nThe company is primarily using existing RSRM hardware for the new stage, but has added some newly designed components to increase performance and meet a different flight profile. A number of these components are undergoing dimensional checks and verification at ATK's facilities in Utah. The parts currently being processed are destined for two full-scale engineering process simulation articles which will be shipped to NASA's Marshall Space Flight Center, Ala. in March 2008 to undergo loads testing and analysis of the motor.\nChanges to increase performance and meet the new flight profile include an enhanced shape of the propellant grain in the forward section, and a larger nozzle throat diameter. The core tooling to be used to achieve the new propellant shape is in manufacturing, as are components for the new nozzle.\nIn addition, two mockups of the forward skirt, a section located at the top of the motor between the first and second stages, have also been constructed. The forward skirt is a structural housing for all the first stage electronics. The mockups will simulate the physical space available for the Avionics and will be used to determine the optimal required space and placement of the electronics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you look at human his\u00adto\u00adry all the way through, we orga\u00adnize our\u00adselves in dif\u00adfer\u00adent ways. Tribes, reli\u00adgions, guilds, states, more recent\u00adly com\u00adpa\u00adnies and net\u00adworks. And what these insti\u00adtu\u00adtions do is they sort of cod\u00adi\u00adfy val\u00adues and beliefs, and they they trans\u00adport them across the gen\u00ader\u00ada\u00adtions. So we see this phe\u00adnom\u00ade\u00adnon that when you cod\u00adi\u00adfy val\u00adues in insti\u00adtu\u00adtions, you give those val\u00adues longevi\u00adty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhwzh b/data_all_eng_slimpj/shuffled/split2/finalzzzbhwzh
new file mode 100644
index 0000000000000000000000000000000000000000..fa274367e5031f3bcf938921292757ef1557acaa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhwzh
@@ -0,0 +1,5 @@
+{"text":"How does Online Ordering work at Haldi?\nAdd your dishes to the basket.\nIf we're running a special offer, add the Coupon to the \"Offer Coupon\" box.\nOnce you're finished, PLACE the order.\nYou'll get a confirmation email when we've received your order. Another email is sent to confirm that the meal is ready for collection or if you opted for delivery, that it's on its way.\nYou can even order when we're closed \u2013 how neat is that? \u2013 and we get it when we open!\nYou only register once. Next time, use the \"returning customer\" button.\nand the convenience of ordering when you want to!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I first got wind of BPI Save Up sometime in September last year and opened an account online a few days after that. Since then, I've followed up several times through the BPI phonebanker for my information card, had my delivery address changed because I moved jobs, consented to have the information card delivered to the BPI branch nearest me for pick-up, went to that branch to pick it up only to find that it wasn't there yet and FINALLY had it delivered to my house 8 months after I first opened my account. That's right. Eight. Months.\nThe weakest link in this whole affair seemed to be the delivery service, a third party provider. BPI's records showed that the card was promptly dispatched a few days after I set up my BPI Save Up account and for some reason or another, the card just languished with the delivery service. Every time I followed up, the update was that it had been dispatched and was set to be delivered. And since that was the update, nobody seemed to notice that it had been sent out more than half a year ago.\nAlso, another weak link was the branch where I first inquired about BPI Save Up (the Sta. Monica, Malate branch) since the branch reps there seemed to be clueless about their product. When I went to the BPI Sucat Road branch on the other hand, the branch manager was able to answer all of my questions and told me that I could have applied through a branch to get my card right away. And most importantly, she told me that in the event of my death, my beneficiaries didn't need to present the information card to claim the insurance proceeds and would only need to present a death certificate.\nIt is only open to pre-existing BPI, BPI Family Savings Bank, BPI Direct savings or checking account that has an ATM card. The BPI savings\/checking account will be the source account.\nThe BPI Save Up account is governed by a group term life insurance agreement.\nIt only covers depositors between 15-60 years old.\nDeaths due to pre-existing and critical illnesses, regardless of whether or not the depositor knows of such illness or condition at the time of application, are not covered on the first year.\nDeaths occurring in restricted areas are not covered.\nDeaths due to suicide are not covered on the first year, except as provided for by law.\nFor your convenience, the following hierarchy of beneficiaries has already been set: 1) Surviving spouse; 2) Surviving legitimate, legitimated, illegitimate and adopted children; 3) Surviving parents; 4) Surviving brothers and sisters whether full or half blood. As per the BPI Sucat Road branch manager, if you don't want to follow the default hierarchy, you will need to fill out another form and your application will have to be reviewed by their underwriters.\nWhen you apply for a Save Up account, you'll be asked to indicate how much and how often you'll be transferring money from your source account to your Save Up account. This is to make savings automatic and effortless. But you're not really bound by this since I have long discontinued the automatic transfer and just transfer money whenever I want to.\nI'm treating my Save Up account as our emergency funds and also as a way to avail of additional life insurance too. My ultimate goal is to set aside Php400,000 in my Save Up account to act as our financial buffer, and the fact that I'm getting 2M-4M in life insurance simply by parking my money there makes it a sweet deal in my book.\nWow 8 months, that was quite a long wait Ms. Jill. I signed up after reading it in your blog. I'm based abroad so I opened the account online and received the card a month after in our Manila address. I'm thankful I came across that blog entry of yours because the automatic transfer feature is 'highly effective' for me in terms of saving money. Minsan nakakalimutan ko may automatic ipon pala ako. I recently emailed BPI on how to increase the monthly amount I've set up early on but haven't received a reply.\nYeah, it was a ridiculously long wait for my info card, and I would have gone on waiting forever if I didn't follow it up.\nI already have this! The good thing about not applying online (I apply on one of the branches near our place) is that you can get your info card right away. I have it on automatic transfer every 15th & 30th. Now I am not sure if how to view my save up from my online account? Sabi nung manager ako na daw maglink eh.\nYou should be able to add another account to your online account. Just tinker with the site to figure out how to do it.\nThe MTD-ADB (month to date average daily balance) is for the Save Up Account. The insurance proceeds will depend on your MTD-ADB for the last three months.\nHi Jill. Thanks for the post, and sorry I'm late to the party, but would you know if they will allow me to change my beneficiary? How do I do that?\nYes, you can change your beneficiary. Just go to a BPI branch to file the necessary paperwork. You may be able to do that through the BPI phonebankers, but I'm not really sure. But I suggest calling them first to know what the procedure is.\nHi how can i withdraw my balance in my save up account?\nTransfer money from your Save-up account to your regular\/primary account and withdraw it from that account.\nanu po ba ibig sbihin ng \"Basic Life\" sa insurance coverage? sa paano pong situation ito na-aapply? pag may skit po ba or something?\nDun po ba sa dismemberment\u2026 may mga category po ba sila i-eexplain sa inyo?\nYes, you can deactivate the automatic transfer to your Save Up account. For instructions on how to do that, you can contact a BPI phonebanker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Promoting and protecting interests common to the Ferro Alloy industry.\nCreate level playing field in the Industry.\nDeveloping co-operation and synergy among Ferro Alloy Producers in India.\nMaintaining uniformity in the rules and regulations and usages of the Industry.\nMaking representations to the Union Government or any State Government or Government of any foreign country.\nRemove the difficulties of the Member-Producers taking steps for promoting and supporting the economic interest of the Industry and those engaged in the Industry etc.\nCollection of statistical information, database and other information relating to the Industry.\nRepresentations to the Chambers of Commerce Mercantile Associations and other Public Bodies in India or outside India, and adopt measures for the advancement of the Industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"State Rep. Jim Riesberg, D-Greeley, will announce his intent to seek re-election for the House District 50 seat.\nRiesberg, who represents east Greeley, Evans and Garden City, will make the announcement at 10 a.m. at the Coronado Building, 900 9th Avenue, in Downtown Greeley. The event is open to the public.\nRiesberg, who has served in house District 50 since 2004 faces term limits and will run for his fourth and final term.\nHe serves as the chairman of the House Health and Human Services Committee, vice-chairman of the Capital Development Committee, and as a member of the House Appropriations Committee. He also serves as chairman of the Interim Palliative Care and Hospice Committee, as well as serving on the Fire and Police Pension Reform Board, the Alzheimer's Task Force and as vice-chairman of the Health Committee of the National Conference of State Legislatures.\nFor more information, contact Riesberg at (970) 351-6619 or e-mail him at jim.riesberg.house@state.co.us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have found that your copper drain is bites it would be desirable to either experiment to unclog the copper copper pipe on your own or call for support a trained plumbing expert in Ealing TW8.\nHaving to free up a clog in the drains isn't a nice task, and occasionally the ultimate way of avoiding bite drains is to guarantee that the drains are regularly cleaned in Ealing TW8. In the past, plumbing experts used tools like steel plastic tube rods to free up a butcher in the drains. Without waiting for more they you should have to put in set in place a high-pressure water jetting unit, to clear the blockage within seconds. High-pressure water jetting can also be easily used to clear a system that features plenty of bends in Ealing TW8. If you are looking for a craftsman plumber to unclog your drains and throne then call one of our experts who will be there within 24 hours in Ealing TW8. With a fast response time and low call out valuations why not call one of our expert plumbing experts to unclog your drains for any dwelling emergency in Ealing TW8.\nIf bite drains are becoming a difficulty then contact one of our certified plumbing experts to aid unblock congestion in pipes in Ealing TW8.\nA slowly draining water from your basin is nothing but a wasted of your time, so do not be shy and call our plumbing company as fast as you can at any time of the day or night of the day to unclog the steel drain in Ealing TW8. There are some easy techniques to unclog a steel pipe that you can find on the internet but if you do not have time for that, just get in touch with us and check out our very low prices in Ealing TW8. If your kitchen washbasin gets frequently blocked with no perceivable create you may need to call an experienced plumbing technician today to unclog your drains today in Ealing TW8. Call one of our friendly expert certified plumbers who can be at your home within 24 hours to unclog your drains in Ealing TW8. If you do not know how to unclog a steel iron pipe you can always call for support of a trained plumbing company which provides the ultimate services in Ealing TW8.\nPlumbing issues if left can lead to more serious complications so call one of our friendly experienced plumbing specialists who can decongest drains at your address for the lowest fee on the market in Ealing TW8.\nBags of live find that using DIY solutions to unclog their drains can be a difficult task, so it is curse taking the time to consult an experienced plumbing expert in Ealing TW8. Instead of worrying and struggling with a bites pipe hire a specialist skilled plumbing professional who'll unclog the pipe quickly and without any trouble in Ealing TW8. Do not feel concerned if you have to unclog a pipe and have no idea how to do it because our team or certified plumbing experts are ready to aid you in Ealing TW8. If you would like to find out about three technologies just you only to apply to unclog a upvc pipe read on in Ealing TW8. Trained trained plumbers use zillions of different techniques while they free up a butcher from pipes, one of the methods that they use demands putting a camera down the lead pipe to find the train of the blockage before attempting to clear it in Ealing TW8.\nJust you only to unclog your kitchen basin at a fair rate by calling one of our trained plumbing experts who can unclog your drains within 24 hours in Ealing TW8.\nPlumbing experts can be with you within 24 hours to aid with an emergency plumbing troubles and to break up a blockage from the pipes in the home in Ealing TW8. We can unclog a upvc lead pipe in five minutes, and we will not empty your wallet, so grab a telephone and hire one of our Plumbing experts in Ealing TW8. If water fails to upvc steel pipe in the kitchen, food debris is often the problem so get in touch with us to unclog your drains by enlisting the aid of one of our craftsman experienced plumbing engineers in Ealing TW8.\nWith so much rain the drains may have become clogged up with leaves and dirt, call one of our certified experienced plumbing specialists to decongest congestion in pipes at your address in Ealing TW8. No more wait and then you may have to unclog your pvc iron pipe because the residual that builds inside it over time clogs it in Ealing TW8.\nHaving to unclog a copper pipe may seem like a daunting task, so do not rack your brain and call for help of an expert plumber in Ealing TW8.\nAn expert can be on hand within 24 hours to break down an obstruction in the drains at your apartment so call today for immediate support in Ealing TW8. The different techniques to decongest a butcher in pipes can be difficult but there are some different technologies that it is in your interest experiment before contacting a certified trained plumbing professional in Ealing TW8. As experienced plumbing technicians, we recommend that households and businesses phone us to decongest a clog in pipes as soon as a complication arises, that way we can make sure that a small quandary doesn't become a big predicament in Ealing TW8. There is nothing worse than dirty water standing in the washbasin so ask for the aid of a specialist to unclog the steel steel pipe in Ealing TW8. Expert experienced plumbing technicians are both efficient and affordable to contact us today for one of our professionals to call round to do disappear an obstruction in the drains in Ealing TW8.\nIf you need an expert trained plumbing specialist to unclog your drains get in touch with us and we will send around someone straightaway in Ealing TW8.\nSkilled plumbing engineers have a lot of tools that they can use to open up an obstruction from pipes efficiently, and they can dissolve the majority of blockages within an incredibly small time frame in Ealing TW8. There are a lot of plumbing companies which offer high-standard services at a very profitable rate and can unclog a lead pipe alter a shower cap or put right a water closet flush in a matter of minutes in Ealing TW8. Do not concern about a bites steel pipe you should call for our support and we will send over one of our skilled plumbing specialists to unclog the steel pipe in Ealing TW8. If you are looking a qualified qualified plumbing technician to unclog a pipe in your home you have to to the right set up in Ealing TW8. \u03fb\ufffdThere are some stories you must do to avoid a bites upvc pipe so that you do not have to unclog the drain too frequently in Ealing TW8.\nThe speed at which an experienced plumbing specialist can do disappear an obstruction in the drains is unmatched efficiency as with state of the art way it usually only takes a few minutes in Ealing TW8.\nIf you cannot patch up your water closet predicament then call one of our experienced plumbing engineers who are on standby to help you unclog your drains in Ealing TW8. If you get a clogged up water closet, then it is advisable to call out an experienced who can decongest congestion in the pipes and patch up your house in Ealing TW8. Having to unclog your pipe on your own may seem like a daunting task, but it is easy in Ealing TW8. If after you take a bath, the water is standing and does not pipe it means you need to unclog the upvc pipe in Ealing TW8. You should to never worry about having to contact a trained plumbing specialist to unclog your drains, as the majority of experienced plumbing engineers do so on a punctual basis. In fact, most experienced plumbing engineers consider bite drains to be an emergency that should be dealt with immediately in Ealing TW8.\nIf drainwork is technology too difficult for you and you have no idea how to unclog your pvc plastic tube call a trained for support in Ealing TW8.\nNo need to alarm about uncloging the drains in your fit of apartment just call one of our experts today to mend up your dwelling in Ealing TW8. No family home wants to have a bites loo but if you call one of our workforce we can have someone with you within 24 hours to unclog your drains in Ealing TW8. As way advances so do plumbing methods A great choice of plumbing companies now use CCTV copper copper pipe surveys before they open up the pipes that appear to be troublesome, that way they can spot the issue before attempting to tackle the blockage in Ealing TW8. Using products located around the apartment to open up the pipes can permanently be tempting, but occasionally it doesn't repair the headache that caused the blocked copper steel pipe in the critical place in Ealing TW8. A myriad of follows use vinegar in an try to unblock an obstruction in the drains within their apartment but the reality is that vinegar only works with incredibly small blockages, and it is not a permanent solution. We would recommend seeking professional advice to pass the hiccup altogether in Ealing TW8.\nWhen using a chemical cleaner to free up a caulk in the drains, you need to be careful and ensure that you buy the correct cleaner for the material of your drains. Otherwise, you may have no drains left in Ealing TW8!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhxhw b/data_all_eng_slimpj/shuffled/split2/finalzzzbhxhw
new file mode 100644
index 0000000000000000000000000000000000000000..121d48e5fa471f0f43b9d15d28f09b67ace1553e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhxhw
@@ -0,0 +1,5 @@
+{"text":"27, 28 January 2016 Current Affairs MCQs, Quiz, Questions: Current Affairs for January 2016, Daily Multiple Choice Questions (MCQs) for India GK, World GK and current affairs with Collection of daily objective type Question by www.Indiagk.net based on General Knowledge (GK) and General Science (GS) Questions for UPSC, State PSC, SSC, Police exam, Railway exam, SBI, Bank PO, IBPC, SSC, LDC, UDC, Army, Airforce, Navy, Coast Guard, Bank Clerk, TET and all entrance examination with current affairs News, Multiple Choice Questions (MCQs) available on Website www.IndiaGK.net and Android Apps with daily updates:.\nIndia's CERT (Computer Emergency Response Team) on 27th January signs MoU with .... ?\nCabinet approves raising of --------- Indian Reserve Battalions by J&K and LWE States ?\nUnion Cabinet under the Chairmanship of Prime Minister Shri Narendra Modi has approved the raising of 17 India Reserve Battalions (IR Bns) by Jammu & Kashmir and Left Wing Extremism (LWE) affected States. It includes five IR Bns in the state of J&K, four IR Bns in Chhattisgarh, three IR Bns in Jharkhand, three IR Bns in Odisha and two IR Bns in Maharashtra.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A few years after my first experience with dark energy, I had another, more concrete encounter. I was actually awake this time and standing outside of my partner's closed garage door. It was nighttime and the florescent lights inside the garage were on, as I could see through decorative windows on the garage. I was staring sort of blankly through the garage window when I saw it: A solid black figure, balloon or head shaped, that rose up to the window and then quickly down, almost as if to peek-a-boo at me. I don't think I've ever felt my heart beat as fast as it did then, as this was a completely unknown darkness I had just witnessed, even more mysterious to me than the first dark mist I'd seen at my Grandma's. What was so strange about it was how black it was, darker than black. I have never seen something so dark, so tangible, yet so inexplicable. There was no one else in the garage at the time, I found out as I questioned my boyfriend. Either way I didn't really need that confirmation since I knew what I saw, and it wasn't human.\nRecently scientists have created a material so dark that the human eye can't comprehend what it's seeing, as it's the closest thing ever created to resemble a black hole. Apparently it absorbs only 0.035 percent of visual light, leaving no shapes or contours to be seen. Called Vantablack, it was developed to be used in space as a material that adjusts cameras to take photos of ancient celestial objects of the universe\u2026not sure what those are, but perhaps old stars or universes? The reason I brought Vantablack up is because it reminds me so much of that darkness I saw in the garage that night; so abysmal, so dark. The only difference is scientists created Vantablack, and not the strange dark shape I saw in the garage. So did I see a black hole? Did I see a tear in the fabric of our universe's dimensions? I'm not quite willing to call it evil, because aside from the shock of witnessing something so unknown and dark, I wasn't as instinctually terrified as when I had seen the dark mist at Grandma's.\nNumerous origin stories from people around the world deal with the cosmos, and the Indigenous people in my hometown, my ancestors, can be counted among them. In the beginning there was k\u00edwvish 'at\u00e1xvish, which literally translates to \"empty, unpopulated\". K\u00edwvish 'at\u00e1xvish is actually a living being, as real as a person or animal. Simultaneously, it's the only being populating the area while also being the area itself. So what is nothing, emptiness? It's a subjective concept really, but it's hard to describe what 'empty' exactly is. The way I imagine it in this creation story, 'empty' is a space void, k\u00edwvish 'at\u00e1xvish. Voids in space are what astronomers depict as a blank space with little to no galaxies. It is the pitch black we see in between matter in space. In an origin story, it would make sense if this is what was present at the beginning of time, before anything else was created yet; A darkness, vast and empty, waiting to be filled with something, in this case the components of life.\nPhilosopher Eugene Thacker states that in most cultures darkness holds negative connotations, stating examples of the \"moral or theological connotation of good vs. evil, light vs. dark forces\", as well as the \"Copernican shift from out of the Dark Ages into an entire epoch of mature Enlightenment\". To be honest, he's onto something with that; In all my contemplations about what this dark cloud at my Grandma's could have been, I was always more inclined to go with some negative energy that rendered me literally speechless. If anything, the Catholic\/Westernized culture I was raised in reinforced my belief that the darkness I saw was evil. This Catholic idea of the dark being inherently negative is in contrast to my Indigenous ancestor's more neutral idea of the dark. While Western Catholicism's idea is that the dark is disingenuously bad, my Indigenous California ancestors view this darkness as a possibility for creation, an empty canvas if you will. Today I am more inclined to feel this way about the darkness I encountered in the garage. The darkness I saw at my Grandma's, however, still conjures feelings of fear. Is it the demonic entity Catholic schools taught me to fear? Who knows, but I think that regardless of religion, there are good and evil spirits existing among us, who care not of our religious affiliations (or lack of)\u2026 but that's another story.\nLately philosophy through negation has helped me to interpret these dark energies I've experienced in ways I never thought possible. Through negation, philosophers have tried to understand the divine by describing how it is not rather than how it is. For instance a 16th century monk, John of the Cross, focused on the divine \"in terms of darkness and negation, resulting in a mystical unknowing of the divine in its inaccessibility,\". Surely this might explain the darknesses I've encountered; I didn't summon them, they appeared to me, on their own nonsensical terms. And unknown forces that come to you randomly are by default inaccessible (For how would I be able to access something I don't know?) And thus my inability to explain these dark energies I encountered is what makes them so divine, transcending any physical manifestation we humans have known or studied. As philosopher Eugene Thacker puts it, all paths via negation \"lead to darkness, an absolute limit to the human capacity to know itself and the world, a limit [Georges] Bataille nicely encapsulates in the phrase \"the excess of darkness\". Because I know so little about the reality of what I experienced, I am in awe of it. Does this mean that the divine's existence is so beyond my comprehension that I will never understand it? It's like I speak with my tongue and the divine speaks with its toes, or with a completely different language, or not at all\u2026No wonder I can't understand what it's trying to say. Ironically, saying I'm in the dark in trying to understand this divinity is at least admitting that it exists.\nIf you're not sure of anything on earth, and you're anything like me, you will try to find the answer to mysterious experiences through any means possible, whether it be philosophy, science, or religion. And if you're anything like me and inclined to see all sides of almost any theory, then it can be hard to find concrete answers. As it stands I'm still on my quest for the absolute truth, and so far the quest itself has been most rewarding; I see the validity in almost all my means of investigation, and the mystery has allowed me to delve into fascinating topics I would never normally think about. I don't think humans should always have the divine knowledge of the cosmos and dimensions that we think we should; Some things aren't meant to be understood, because we might spoil them with our lust for power and our innate ignorance. If we had knowledge of certain things, we might tip the scale of nature even more unfavorably than we already have. Maybe humans are supposed to be in a constant wonder for an insatiable truth, because this quest has arguably been one of my most human experiences yet\u2026Because the thrill of simply not knowing, of searching for the truth, is ironically keeping me in the dark yet simultaneously enlightened.\n Eugene Thacker, Starry Speculative Corpse (United Kingdom: Zero Books, 2015),17.\n Thacker, Starry Speculative Corpse, 33.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WE MODEL CO., LTD - WEI-YI Stell Mold Co., Ltd.\nWE-F003-MK25-A gun exclusive to Navy Seals for launching attacks !!!\nWE-F226-E2-A vintage gun made new for today's market !!!\nWE-Ultra Compact 3.8 (With Magazine & Adapter) BK-A must have gun to impress gun enthusiasts !!!\nWE-Ultra Compact 3.8 (With Magazine & Adapter) TAN-A must have gun to impress gun enthusiasts!\nWE ACE.V.D. GBB Guns Now available !!!\nWE-M4 Magazine (Version 2) Black \/ Tan-Hot release!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone says sleep is good for you \u2013and they'd be right. Sleep refreshes your body and your brain. But did you know it can help solve problems too?\nIf you're finding something hard to learn, try sleeping on it! No, don't put it under your pillow\u2026 Make it the last thing you think about before you go to bed. You might remember it better when you wake up.\nIf worrying about it keeps you awake though, try these sleep tips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a relational volunteer experience where you will have the chance to assist individuals and families who come to the Ministry Outreach Center. Client Services provides individuals and families in need with food items, household items, clothing, etc. Duties may include assisting clients in the clothing room, weighing outgoing donations for clients, and keeping our donation areas organized. This Change Maker Volunteer opportunity has a time commitment of once per week for a minimum of 3 months. Hours available are Monday through Friday 8:30 AM to 11:30 AM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"table demi lune ikea table.\ntable demi lune ikea black sofa table photo 1.\ntable demi lune ikea a console table made of solid wood.\ntable demi lune ikea warm table 5.\ntable demi lune ikea affordable console tables fresh console tables pretentious inch sofa table bedroom ideas with table console.\ntable demi lune ikea good great table table ref table with table cuisine with table with table.\ntable demi lune ikea balcony table the table folds to be easy to store when not in for your balcony or other small spaces as it can be folded up.\ntable demi lune ikea table extensible images of console table new console table images with table table.\ntable demi lune ikea medium size of table console style coffee new tables.\ntable demi lune ikea latest table pas with table.\ntable demi lune ikea table unique table aluminium elegant articles photos.\ntable demi lune ikea table table table cuisine table cuisine photos with table.\ntable demi lune ikea mesmerizing table.\ntable demi lune ikea console table sofa table mahogany console table distressed black brown wooden full wallpaper.\ntable demi lune ikea table best i need one images on inside idea 6.\ntable demi lune ikea download by with consoles.\ntable demi lune ikea beautiful console extensible table console en with console extensible with console with table.\ntable demi lune ikea excellent table outdoor stained x cm with table.\ntable demi lune ikea medium size of console salon belle lack coffee table black brown.\ntable demi lune ikea collapsible dining set table drop leaf console table.\ntable demi lune ikea beautiful table ron pictures amazing house with table mi extensible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brisbane Valley Rail Trail 96 km cycling challenge.\nGravel racing on the Brisbane Valley Rail Trail from Esk to Yarraman, with a time limit of 8 hours.\nLast year, the BVRT 100 was one of our most popular challenge cycling events.\nTotal distance 96 km with a net height gain of 375 metres on a fairly rough track.\nE-bikes welcome in their own category.\nEqual prizes for men and women. First second and third prizes will be $200, $150 or $100 vouchers redeemable at Ipswich Giant for any bicycle accessory supplied by them - does not have to be a Giant product and can be a telephone order.\nPLUS - ANYONE CAN BE A WINNER. Announcing a grand prize draw for a new Giant Revolt Advanced 2 gravel bike (www.giant-bicycles.com\/au\/revolt-advanced-2) donated by our sponsors, Giant Ipswich.\nRFID timing services will be provided by Toowoomba Triathlon Club.\nThis event is suitable for reasonably fit and experienced cyclists only. You need to be confident you can cycle 97 km off-road in under 8 hours.\nParticipants are expected to be self-sufficient, but there will a safety team shadowing the event to give assistance if required. Free water and great snacks will be available at 2 locations on the route.\nThere will be an optional hourly shuttle bus back to Esk from 11.00 am provided by Josie and David from Out There Cycling @ $30. Depending on numbers, we may also lay on a coach and semi-trailer. Please select the option to pre-pay whilst registering for this event - the option to add it is at the bottom of the check-out page.\nIf you're not sure if you need the shuttle bus, you can book it separately later.\nPlease note, if a severe weather event is predicted for 29 June, the BVRT 100 will be postponed and all participants offered a full refund or transfer your entry to the new date. A bit of rain is all part of the adventure. Thunder and lightning whilst out in the open countryside is just too dangerous.\nDuring registration you will be required to accept our terms and conditions.\nWe've created a BVRT 100 community in the Facebook BVRT 100 Discussion Group - please feel welcome to join.\nCost: Early bird $85 (ends 16 June). Standard - $95.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bon Iver has been around for a while; since 2007 he's been considered an \"underground\" artist. So it's ironic and appropriate that he win a Grammy for \"Best New Artist\".\nA couple of years back, a friend asked our music man Nathanael Sams whether he had heard the folk rock riffs of \"Bon Iver\": he hadn't.\nWell, he had \u2013 he just hadn't known it. Four singles of the alternative artist had been featured on the popular show \"Chuck\".\nSo really, Bon Iver isn't a \"new artist\", rather a newly appreciated one.\n\"Best Alternative Album\" & \"Best New Artist\"\nHeavy folk guitars, string arrangements, saxophones, kick drums, and reverberated vocals channel a sound reminiscent of Mumford & Sons, with a little mandolin in between.\nIf you're unsure whether or not you'll enjoy his new album, a good test would be to watch the movie Once. If you enjoy the soundtrack in that film, you're pretty much guaranteed to love Bon Iver's new album\u2026Bon Iver.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Winikoff, Beverly, Dzuba, Ilana, Pe\u00f1a, Melanie, Dzuba, I.G., Winikoff, B., Pe\u00f1a, M.\nAbortion rates in Latin America and the Caribbean (LAC) are very high and methods are often unsafe. This paper offers evidence that the use of medical abortion could reduce abortion-related morbidity and mortality, despite strict abortion laws. The authors address ways to improve medical abortion services and discuss the many remaining barriers to access in this region.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PanARMENIAN.Net - Flash floods in Indonesia's eastern Papua province have killed at least 77 people, the disaster agency said Monday, March 18 as it raised the death toll from 58, Channel News Asia reports.\nMore than three dozen people remain missing, while scores have been injured in the disaster, triggered by torrential rain and landslides.\nAuthorities on Sunday had put the death toll at 58, with dozens more injured and thousands displaced.\nA search for more potential victims was underway on Sunday in the town of Sentani, which was hit by flash floods late Saturday.\nFlooding is not uncommon in Indonesia, especially during the rainy season, which runs from October to April.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Through the thousand photos on the net with regards to mid century modern executive desk, choices the very best series with best image resolution simply for you all, and now this photos is considered one of photographs series in our ideal photographs gallery with regards to Amazing mid century modern executive desk. I'm hoping you will as it.\nThat picture (Sold Midcentury Modern 1960s Vintage Executive Desk Signed for Mid Century Modern Executive Desk) previously mentioned is classed with:mid century modern executive desk, mid century modern walnut executive desk, submitted by Frank with December, 26 2017. To see almost all photographs within Amazing mid century modern executive desk photographs gallery remember to adhere to this link.\nWe hope you can find what you need here. We always effort to show a picture with HD resolution or at least with perfect images. Sold Midcentury Modern 1960s Vintage Executive Desk Signed for Mid Century Modern Executive Desk can be beneficial inspiration for those who seek an image according specific categories; you can find it in this site. Finally all pictures we have been displayed in this site will inspire you all..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It has been a while since the E-J reviewed go programs available on the Android platform, and given that not all of us have iPhones, I thought it was time to do an update. I should note first that I love Apple, and have three Apple computers in my home. However, I hate the iPhone and iPod touch for go. Why? One simple reason: the screens are so tiny it makes me go cross-eyed; as for the iPad, it is too big to fit in my pocket. I have had several Android devices, and my current one is a Samsung Galaxy player with a 5\u2033 screen. It is small enough to carry easily, but big enough to see what I am doing. Another advantage Android offers is that it is open-source, which means people can develop and change apps very quickly. On the whole, I have found Android developers incredibly responsive, and three programmers actually changed their application based on user comments, in some cases overnight! I also have no need for a cell phone contract, or expensive data plans, what I wanted was a wireless only device, and I got mine specifically to record go games. There are a ton of apps out there, so I thought letting our readers know which ones I found most helpful would be useful. A tip for searching, don't bother with \"go\", search for \"baduk\" or \"weiqi\" on the Google Play Store. I will review three game recording apps this week, and then review apps for Go Problems next week. I am not reviewing KGS for Android as it requires a data connection, so I can't use it to record games if I don't have internet access. People who play on KGS, and want to do so on their phone, or watch games, will love the app though \u2013 it is beautiful and functions very well.\nGobandroid is a fully featured game recording program that comes in two versions, Tiny and HD. Both come with a ghastly pale default skin that makes the board and the stones look unappealing, the black lines are highlighted to boot, which makes it hard to tell what is going on. Download the free Sente skinstaller from the market though, and you can make the program look beautiful. The great merit of Gobandroid Tiny is that you click on the board, and it zooms in to the area you are looking at. From there, you can choose where your stone goes precisely, and then it zooms back out to the full board. This means you have to click twice to place a stone, but it makes it easy to see what is going on. Gobandroid HD has an awkward interface that enlarges part of the screen and keeps the full board visible as well. Although this feature is well developed in other programs, in HD it is just confusing visually. Gobandroid Tiny is my version of choice. I often find that I have mis-recorded a stone while playing. No problem if it was a few moves back, but what if it was 50 moves back? With Tiny, just rewind to that spot and click on the stone. It will turn transparent. Then click where you want it to be, confirm it, and you can then fast forward to where you are in the game, with all other moves intact! It is worth noting that this program did not come with this feature at first. But a reader posted about it on Life in 19\u00d719, and the developers added the feature. You can also load games and problems from online sources, through helpful links in the program itself, and can install the gnugo AI if you want to play against a computer. Don't expect a strong engine obviously, but it is fine for beginners on small board sizes.\nWegoIgo is my current favorite, and the new version is a huge improvement that takes the program to a new level. I originally got the program because it loads Kogo's Joseki Dictionary, the only program on Android that does so successfully. I didn't use it for anything else though, as it was hampered by a weird glowing stone effect that made it hard to see, and it was not very useful for recording games or doing problems. A few months ago, updates started coming from the developer Daniel Leong. The new version seamlessly integrates a floating circle with a detail of the area you are playing in. As you drag, the circle floats above your finger, and you can move your stone wherever you want to. As a plus, the stone actually goes directly under your finger, making placement both easy and accurate. The merit of this system is that it only takes one touch (with or without a drag) to place a stone. After recording with Gobandroid for months, I found I was happier with one touch recording, and could record both faster and more accurately with WegoIgo. Unfortunately, it didn't offer the feature for mis-recorded stones, When I e-mailed Leong about that, he added the feature almost immediately. He also changed the graphics on the program when I told him the background was distracting, and has been great about any suggestions users have made. WegoIgo also loads problems and professional games, cleanly and easily, and you can download the AI if you want to play against gnugo. It offers advanced SGF editing features to boot, and you can comment and mark games up for study. Although the free version includes all of the features, I upgraded to the paid version so I could get faster updates. Also, the price is ridiculously cheap ($2.49), and I think it is important to support people who are writing go apps (trust me, none of them are getting rich on them).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many people pay no attention to the drains in their homes until something goes wrong. More than just a stopper at the bottom of your sink or tub, a drain serves a very important purpose in whisking away the water from your home through the plumbing system. Backed up toilets, sinks that are slow to drain and standing water in the shower are just a few of the pesky problems that signal that it's time to break out the plumbing tools or call a professional. More serious problems result in extensive repairs that can prove costly, but the good news is that when you learn how to choose the right drain cleaning tool, you can easily tackle most drain issues yourself.\nThere are several types of drains in every home including those in your kitchen, bathroom and laundry room. Wherever water comes out, it must have a way to leave. Tubs, sinks and toilets all have drain lines but so do your major appliances such as your washer and dishwasher. These drain lines are just the beginning of your plumbing system, and each connects to the main line where water exits either into your septic or into your city or town's water system. There are numerous plumbing problems that can arise with your plumbing with a clog being the most common, and also among the easiest to fix, even for a homeowner.\nThere are several types of clogs that afflict drains and pipes, causing standing water in a sink or tub. Serious clogs result in a backed up washing machine, or worse, a flooded basement or main floor.\nKnowing exactly what type of clog you're facing helps determine which tool to use for the best results. If you're unable to clear a clog with the use of tools, a professional can help resolve the clog before it does more damage.\nThere are several types of drain cleaning equipment designed to help clean out drains and these tools range from basic to handheld, and for stronger drain clogs, electronic tools. The type of tool you'll need to clean out the blockage depends on the type of clog you're facing.\nWhen a clog strikes, it's not uncommon to reach for the liquid drain cleaner, but you'll find that there are times when it's just not strong enough and you require additional tools. Drain snakes are a popular drain cleaning tool for toilets and sinks, and are best used for small drain problems such as a clump of hair. A long metal stick, they are narrow to fit within your tub or toilet to push through or break up the clog. They're often used in combination with an auger which features a hook and loop design that grabs the source of the clog so you can pull it out. Augers are especially helpful in grabbing and removing small objects such as jewelry and small toys.\nSerious clogs or clogs that are deeper in the plumbing system require more power that a simple handheld tool cannot do. In these instances choosing a piece of electric drain cleaning equipment is best. There are a number of electric drain cleaning machines that RIDGID offers including drums, water jetters and the Auto-Clean\u2122 sink machine.\nDrum machines work with sink lines that range from 3\/4 to 10 inches in diameter. They deliver water a high torque levels to propel obstructions through the line using cables or cutters, depending on the type of blockage you have. Ridgid's drum machines deliver high RPM using powerful induction motors. They're suitable for clearing hard and soft blockages such as tree roots, food build-up, grease and sediment. Since the snakes have cables that can pinch, make sure to pick up a drain-cleaning mitt to protect your hands.\nRanging from mid- to very high-PSI, water jetters from Ridgid come in both electric and gas models. They work in sink lines up to 10 inches wide and use a flexible hose to propel through soft blockages including grease, sludge and soap. They also have the benefit of flushing the line, restoring full water flow. Water jetters work in both home and commercial applications and Ridgid offers several models with high HP motors for fast performance.\nThe Auto-Clean Sink Machine is Ridgid's patented technology. It's a single drain unblocker tool with a flexible hose and a MAXCORE\u2122 cable that works in lines ranging of various widths for working in areas with large or small pipes. The Auto-Clean Sink Machine blasts through obstacles in sink and tub or shower drains. It's useful for soft blockages such as hair, soap and grease.\nIn addition to actual drain cleaning equipment, there are other tools that work in combination with the equipment to effectively clear blockages quickly and seamlessly. These parts include pipe freezing kits and threading tools. If you're looking to buy drain cleaning tools, these components often come in handy.\nWhen you need to isolate an area of a pipe to perform repairs, a pipe freezing kit is helpful in isolating a section. It's a useful tool when shutting off the water is not possible. It works by blocking off a section of the pipe, preventing the water from flowing through and inhibiting the work process.\nPipe threading tools come in handy when you're replacing a drain or section of pipe in the sewer. They help prepare new pipes for joining together if a pipe replacement is necessary, such as if a blockage is too strong to clear, and include sealants, cutting oil, dies and adapters.\nDrains are a pesky occurrence, but knowing how to choose the right drain cleaning tool can help clear obstacles and restore natural function. Check out our extensive collection of Ridgid drain-cleaning tools to find equipment that's ideal for your home or business.\nI have a LOT of long and thick hair and tend to clog up my drains often. You mentioned that the Auto-Clean Sink Machine is really useful to clear out soft blockages like hair. Is this something that is available to any consumer, or do you have to be licensed to purchase something like this?\nHi Samantha, The Ridgid Auto-Clean Sink Machineis available to the general public. The link in our reply will bring you to our website where it can be purchased.\nInteresting advice here for avoiding bad clogs in your pipes. I will definitely have to remember these tips for vegetation clogs.\nI feel like I will be more careful about what I let go down my drains after reading this post! I think we easily overlook what goes down our drain since we are so used to them working well. You've provided a great list of different clogs and drains and how to help clear them up. Thank you!\nVary useful guide for keeping away from harmful clogs inside your drain system pipes.\nI really appreciate the insight here in this post and confident it's going to be helpful to me and many others. I'm wondering if you or anyone else has additional sources for me to read further and to be able to dig a little deeper?\nHi Greg, Thank you for your comment. If you are interested in learning more about the right drain cleaner you can always check out the items on our AcmeTools.com website. Another great place to go would be to the manufacturer's websites. We would suggest Ridgid, Reed, and Milwaukee.\nWhat a great enumeration of professional drain cleaning tools. Thank you! Some of the more powerful tools may need expert handling to avoid permanent damage on the drainage system.\nThe drains in my house have been having a lot of issues lately, and we were thinking about getting them fixed up. I really like that you say to make sure that they have the right equipment tools. It would be nice to know that you will be able to get the best help for your kitchen.\nI never knew that some floor drain clogs were so severe that they needed electric tools. It would seem that if the basic ones don't work out well, the more advanced ones would do. My brother is trying to unclog his floor drain, so he'll have to make sure he has the right tools.\nHello Gloria, Thank you for your comment. If in doubt about the cause of a drain clog please consult a licensed plumber before attempting to clear the clog. Some common off the shelf drain solutions can do more harm than good to the plumbers tools.\nWhen can I know if I need to use a drum or a water jetter for the tough sludge? I've noticed that the water in the tub isn't draining and I'm not exactly sure what to do about it besides dumping some drain cleaner into it. I'll have to consider your tips so I can get all of that water out.\nHello, Thank you for your comment. If a slow drain in the tub is your issue it is most likely a hair clog and a drum or sink machine would be best but if there is any doubt which tool to use please consult a professional for assistance.\nWow, I didn't realize that there were so many things to look at when choosing a drain cleaning tool. I had always been under the impression that a snake or a coat hanger would work. However, I imagine that, if you can't get a hold of some of these professional tools, then calling in a plumber that has them would work just as well.\nThanks for explaining the different drain cleaning tools and their benefits, such as how water jetters use a flexible hose to propel through soft blockages, such as grease and sludge. This would be a great way to ensure you're able to get everywhere in the drain to clean it. When choosing a method, you'd probably want to consider how your system works and talk to a professional about the different options and your situation in order to figure out which one will work best.\nI really appreciate the insight here in this post and confident it's going to be helpful to a lot of people. Thank you for having this article, it will absolutely be a help to everyone. Great blog and keep on blogging!\nAwesome article about benefits drain cleaning. I am very glad to found this kind of wonderful blog. Thanks for sharing this information.\nThanks for the drain cleaning tips. We've had some clogged drains for a while, and nothing we're doing is working. I think we'll have to try hiring a professional, to be honest, because the tools you've mentioned, like the augers, aren't working.\nI just bought an older home and I am worried about the pipes. It is good to know that I can get electric drain cleaning tools to help me clean out some blocks I find. But I would be still scared of messing up my system. It seems like it would be best for me to hire a professional plumber to check and clean my pipes.\nJones suggested pouring very hot water down the drain at least once a week. This can help prevent clog-causing build-up on the interior surface of pipes. Or, pour one cup of vinegar down the drain and let it sit for 30 minutes. Rinse with two quarts of very hot water.\nThanks for the rundown of what tool to use for what clog. We've had a clogged bathroom sink for days, so this is good to know. Still, I don't really want to try and do it myself, in case it gets worse, so I'll hire a professional.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Elder Scrolls told of their return.\nTheir defeat was merely a delay 'til the time after Oblivion opened, when the sons of Skyrim would spill their own blood.\nBut no one wanted to believe... Believe they even existed. And when the truth finally dawns... It dawns in fire!\nBut there's one they fear. In their tongue, he is \"Dovahkiin\": Dragonborn!\n\u2014Esbern, Narration of the teaser trailer.\nThe Elder Scrolls V: Skyrim is the fifth entry into The Elder Scrolls main series, released on November 11, 2011 across multiple platforms. It takes place 200 years after the events of the fourth game in the northernmost part of the continent, home to the Nords, as Alduin the World Eater is returning. The people of Skyrim are locked in a bitter civil war between those who support remaining within the Empire of Tamriel and those who wish for independence, united under the leadership of Ulfric Stormcloak. Into this enters what may be the last Dragonborn, a person born with the soul of a dragon, who has the ability to kill the dragons and absorb their souls. Under the guidance of Esbern, one of the last Blades, the Dragonborn must oppose Alduin and defeat him, lest he destroy the world.\nThe game runs on a heavily modified Gamebryo engine called the Creation engine, complete with all the Good Bad Bugs that Bethesda fans have come to know and love. The soundtrack is once again provided by Jeremy Soule. Skyrim is also innovative in that it has an integrated way to browse and upload mods, which can be downloaded automatically via Steam, something that has never been attempted on this scale before.\nThe first DLC for Skyrim, Dawnguard, came out for the Xbox 360 in the summer of 2012. The trailer can be found here. Subsequent DLC called Hearthfires and Dragonborn were also released, and all three were ported to all known versions of Skyrim for Playstation 3 and PC, with the PC version gaining a free High Resolution graphics DLC for download.\nAt the end of of October 2016, Skyrim Special Editon was released on all currently generation game platforms as well as Steam, free for those who owned Skyrim and all prior paid expansions, having backported versions of various enhanced features from Fallout 4 as well as integrating all previous DLC into a remastered version of the original game (as a separate install even for prior owners of the original version due to massive changes in the file layout).\nThis page was last edited on 17 July 2018, at 14:03.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RKB Bearing Industries, The Swiss World-Class .\nWind powered factories: history (and future) of .\n17\" 2 HP Bandsaw, Anniversary Edition | Grizzly .\nWhat is the difference between a vertical roller .\nvertical roller mill vs ball mills - .\nComparison Between Vertical Raw Mill Ball Mill .\nCOMPARISON OF GRINDING EFFICIENCY .\nCombination Sander 6\" x 48\" Belt 12\" Disc 1725 .\nPictures of Taig Lathe - .\nChainsaw Mill Build, Use & Tips N Tricks: 16 .\ncomparison between ball mill and vertical roller mill, comparison between ball ... types of ball mill and vrm presentation. Mls3726 Vrm Ball Mill Optimization.\ncomparision between vertical roller mill vs ball .\ncomparison between ball mill and vertical roller mill. what is the difference between vertical mill ball mill, BALL MILL & VERTICAL ROLLER MILL?\nComparison Between Cement Ball Mill And .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Museumsuferfest in Frankfurt, Germany is the city's largest annual festival. Taking place on the bank of the river Main every August in celebration of Goethe's birthday, the Museumsuferfest attracts a three million visitors over a weekend.\nThe Tourismus+Congress GmbH Frankfurt am Main is hosting the festival and commissioned Atelier Markgraph for developing one special highlight - a media based superstructure presenting the hosts of the festival, the city's museums.\nSchiff der Ideen (2003) and Klang|Passagen (2004) were spectacular masterpieces driven by vvvv. The highlight of the Museumsuferfest 2005 was the KUNST\/RAD, a thematic projection on a 40-meter ferris wheel, also made with vvvv.\nMESO adapted precision inclination sensors picking up the precise position of the wheel. The data was transmitted via WLAN to the other side of the river, where in response 100 masterpieces from the archives of 16 museums are projected via an impressive array of 4 Barco R12 units sporting about 45000 ANSI lumens. A huge slide show controlled by the flow of visitors boarding and disembarking from the wheel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Category Azerbaijan on sister projects: . Audio files of Azerbaijan (2 C, 1 F) .. Boy with Sheepdog - Village of Laza - Caucasus Mountains.\nDual LAD \u2013 Contemporary Review. Eyvaz Abbasov1*, Soltan Manafov1 and Fuad Abdullayev2. 1Department of Radiology, Azerbaijan.\nEuropean Championship Qualifying match Italy vs Azerbaijan (10 Oct ). two in Euro qualifying Group H with a victory over Azerbaijan. . and not nearly dynamic or attacking enough to resort to fielding the lad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pend Oreille County was one of four Washington counties to receive economic development funds from the Washington State Community Revitalization Board. The grant gives $50,000 to the city of Newport for the Newport Hotel\/Motel Feasibility Study. The study will analyze the demand for a motel\/hotel, determine the appropriate size, construction and operating costs as well as job creation.\nRefer to the information below to see the complete Department of Commerce press release as well as other developments as they occur.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By guest contributor, Kendall Adams.\n1. Plan ahead for your photographs \u2013 make sure you have everything you need when taking your photos. Fresh Batteries, extra digital storage cards, flash, lenses etc. Oh yes\u2026.and the camera! Many holiday memories are lost because the batteries died, you ran out of storage, or you just forgot the camera!\n2. Yes. There are rules but sometimes they need to be broken! Don't always stick to the rules \u2013 sometimes a slight blur or movement of the camera can give a soft holiday glow to your pictures. Try zooming the lens during a long exposure. Change your cameras settings to accentuate the movement or the focal point of your photo.\n3. Use Flash outdoors to add just a little bit of extra sparkle to the eyes and to fill some of the shadows that might be on the face of your subjects. Try to use natural lighting indoors to add warmth to your photos and eliminate the dark backgrounds that your flash can create. If your camera has a fill-flash feature try to familiarize yourself with it. If you have to use flash indoors this feature can be a life-saver!\n4. Move in closer! Almost any photograph can be improved by moving in a little closer. This can add emphasis to your subject and may also eliminate any distracting background clutter.\n5. Look for unique viewpoints \u2013 take an extra minute to really look at the scene you are photographing. Are there bright colors, interesting items you can use in your composition, or reflections? Consider using any or all of these elements to add interest and help tell the story.\n6. Use your cameras digital LCD to preview your photos. There isn't a better way to see if you got their photo or not. Most of today's cameras offer a zoom function on the preview. This will allow you to check exposure and focus. If you don't like what you see \u2013 shoot it again!\n7. Compose your photos well! Try to keep your subject slightly off center. Look for distracting background items that can ruin your photos. Make sure you have everyone's head and body parts in the photo. Use elements in your shots to add interest and a sense of place. A chair, a window, a plant, almost any item found around your subject can be used to add interest.\n8. Take LOTS of Photos! When taking pictures of family and friends, especially in groups, there will always be someone who either closes their eyes or does not smile. Increase your chances of success by taking more photos than you think you will need. Better safe than sorry!\n9. BE PREPARED! If you don't have the camera ready you will miss the shot. Baby's first Christmas gift, the children's surprise as they see what Santa left, even Dad snoring after Christmas dinner. Try to have the camera close by. Make sure you are familiar with the cameras controls. Be ready to shoot!\n10. Use your photos as gifts! Everyone loves to get a great photo of their family and friends. If you have shots that you are particularly proud of print them and put them in a nice frame. This is a great gift that someone will treasure and always remember you by.\nIf you, or your friends, haven't registered for the DWZ Christmas Giveaway, this is your last chance!\nWe'll pick four random entries and announce the winner in Thursday's Tip!\nThis entry was posted on Tuesday, December 16th, 2008 at 6:00 am. 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.\nThe names and logos for Digital Office and Digital WebZone are trademarks of Digital Office, LLC.\nAll text and design is Copyright \u00a9 2000-2019 Digital Office, LLC. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Live performance is one of the most immediate ways to engage with music. It helps build a bond between yourself and your fans. It can lead to life-changing moments on both sides of the stage. So, how do you get booked to make it all happen? In our fourth installment of the Mastering The Hustle workshop series, we invited a panel of Seattle booking and promoting professionals to speak on the ins and outs of getting booked for a show. This series is in conjunction with Upstream Music Fest + Summit, MoPOP, and KEXP's Artist Education Initiative< \u2014 a new program dedicated to providing educational resources for local artists.\nQ Night Club and Upper Left Events' Sean Majors shed some light on how to get your start booking and promoting events. Later, KEXP's John Richards moderated a panel discussion with The Vera Project's Andrea Friedman, Neumos' Evan Johnson, as well as KEXP and False Prophet's Sharlese Metcalf about how artists can better book shows. It's a discussion that covers everything from what to do and not to do, giving perspective on both sides of the email chain.\nSean Majors kicked things off by sharings his experiences growing up as a shy kid, moving around the country, and finding solace through DJing and booking shows. Throughout his career, he's done shows everywhere from huge warehouses to clubs and festivals.\n\"If it has a dance floor and serves alcohol in the 206 or 253, I've played at it,\" Majors said in his presentation. On average, he's played 200 shows a year for 25 years. On top of that, he said that he's also been attending shows 300 days a year during that same timeframe. For Majors, this is a crucial part of the process. He stressed the importance of being pathological with your work in order to succeed in booking shows \u2013 being willing to sacrifice sleep and other things in your life you could be doing and focusing on this instead.\n\"Are you willing to work harder than everybody who wants the same thing that you want?\" he asked the crowd. Majors stressed the tenacity it takes, sharing that it took him years of failures before he started to succeed in the Seattle music scene. He attributes success as well to being willing to push yourself to go to shows out of your norm. While he started in the electronic music realm, he says that he'll still go to Slipknot or Kenny Rodgers shows to learn and see how different genres and scenes put on their shows. Finding the sociological appeal of the different venues and subcultures is critical to finding a wider approach to doing things.\nBeyond just learning from others, there's a healthy dose of innovation involved as well. As Majors explained, if you get in a lane that's already established and you get in line with what's already done, you're not saying much artistically. Much like how artists are influenced by each other but still create original works, booking and curating shows is the same way. As you continue to build trust with your audience as a curator, you'll also be able to expose people to new artists.\nInnovation is crucial to booking and Majors explains that the rules of this world are deeply affected by technology and culture. That means everything changes and can change very quickly. He stressed the importance of bands using social media and keeping it updated. He admits that it can feel tedious for most artists but is necessary for getting their work out there and helping establish themselves for bookings. He cautions bookers and artists to be aware of their own attachments, adding to stay true to your taste and goals. You don't have to be the same thing that you found during that time \u2013 whether that's genre or methods of booking and promoting shows. He tells both bookers and artists to fall in love with change and to be a lover of history, of the tapestry of music, but also \"keep your brain loose\". When you don't understand or relate to something, listen to it again.\nIt's also important to be honest with your motivation. Whether you want to book shows to help build up a community, support the arts, make money, or just have fun, it's important to know what your goals are upfront. He says it's important to know your thing or combination of things. Money, in particular, can be awkward, but it's a reality in the industry. That's why Majors suggests getting everything in email. If he has a phone conversation with a promoter or artist, he'll follow up with an email recapping all the points discussed so they can reference them later. The phrase \"devil is in the details\" is also true. By having cost breakdowns and payouts cleared up at the beginning, you're saving yourself from awkward or tense conversations later on.\nMajors stressed throughout again and again how tough it can be to make it as a booker and\/or promoter. However, he did recommend that those persistent enough to pursue it utilize the resources around them. Take internships, start promoting your own events, and utilize social media. He particularly pointed out the value of finding a mentor to show you the ropes.\n\"If you don't know, ask,\" Majors emphasized.\nWhat Do Bookers Look For In An Artist?\nRichards chimed in to add the importance of being gracious when you're denied a slot on a bill. Bookers remember these types of interactions and just because you didn't get booked this time, you still may work with this same person or venue later down the road. Majors echoed this point and said that he always has an easier time when people are respectful and kind.\nRichards then posed the question of how the bookers go about putting together lineups for their shows. Johnson chimed in first, saying it's mostly a constant act of sending emails back and forth. However, he did give some insight into artists for how they should manage their expectations, especially when they're trying to book a stage like Neumos.\n\"You don't want to play in a 700 capacity room if you're gonna be able to bring 50 people,\" Johnson said. \"That's gonna be terrible for everybody.\" But it's more than just waiting until you can bring in 700 people to book a show. Johnson noted that he's way more interested in a band if he hears they've just sold out a 50 person show. The words \"sold out\" still carry weight with bookers, he added. It's for this reason that Friedman says she started hosting shows in The Vera Project's 75 person capacity gallery. It helps to create an intimate space that's better for artists without a huge draw.\nFriedman mentioned that bands should always check and see if the venue has guidelines for booking on their website (more often than not, they do). This gets back into the emailing process. Friedman says to keep your message short. You can mention some local bands you've played with or would fit well with and a history of venues you've played in the past. You might mention some of your influences, but let the music speak for itself (Richards adds: \"Don't compare yourself to Radiohead and The Beatles\"). Don't send demo CDs to venues, instead send streaming links to the booker (even better if you can embed it into the email so they don't have to open another tab). Bookers are constantly getting flooded with emails, so brevity and ease are crucial concepts to keep in mind.\nIf you've made a template that you're sending to numerous venues, make sure you always have the correct venue name in the body of the email \u2013 it's a more common occurrence than you might think. Try and take it a step further and add some personal touches, Friedman says. It shows you actually care and bookers will take notice.\nPatience is also key. With the sheer volume of emails bookers get, it might take a while for them to get back to you. Don't be the band that emails every day. It's a case where persistence won't be your friend. KEXP's Sharlese Metcalf says she tries to listen and respond to every email she gets, but even then she still gets numerous emails to follow up with. She says that there's a gracious way to follow up your initial email. If you haven't heard back in two weeks, that'd be an appropriate time to send a polite check-in email. And if you ultimately get rejected, Majors says to not be afraid to ask if there's something you could've done to make your message better received.\nTrying to get an opening spot for a band coming through to town is a bit more complicated than many might realize. Oftentimes, this isn't a decision left up to the booker. The headliner typically won't know who a local opener is and will often just bring along a smaller act that their agent also represents. For artists looking for venues to land early shows, the panel recommends reaching out to spaces like The Rendezvous, The Timbre Room, and The Crocodile's Back Bar. When booking for KEXP events, Metcalf says it all comes down to airplay. She says it's important to get your music into the hands of Don Yates, KEXP's music director, or herself for local billings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"*This picture is based from a botanical sketch by Helen Campbell as published in the English Garden magazine (May 2016). Although I haven't drawn anything before other than stick figures and circles, learning to draw is my new hobby. And what better subject than plants?\nHelleborus vesicarius is considered a rare plant that grows on a certain mountain region that extends from Turkey to Syria and Lebanon. It is highly toxic which explains why it can survive on hillsides where goats graze. Unlike the more common helleborus with gorgeous flowers, the balloon-like seedpods of Helleborus vesicarius is the most notable feature in this plant. I can only imagine that as the seedpods mature and break off from the plant that they would move around in the landscape easily with the wind. So the balloon-like characteristic of the pod is just another seed dispersal mechanism in nature.\nand learn something about it.\nSo my older daughter has been helping me play with some artistic tools on Photoshop lately. Here are some of my first attempts so far using the blurr filter. It is fun because I can hide the less important parts of the picture. Some might call it cheating but I call it art - Hort Art. Someday I might learn to use a drawing tablet to have a finer and better control of the cursor.\nHummingbirds love the flowers of succulents. Alright, maybe they don't love the flower per se but the nectar is like a special treat for them. It has been my observation that they would check on these flowers way before they open. Once the tubular flowers crack slightly the birds begin to stake their claim. Any other hummingbird trying to intrude on well defended flowers, would soon find out that it was not a good idea.\nHummingbirds will linger where there are plenty of food.\nFortunately for the resident hummingbirds, we have many different succulents in the yard -- that is in addition to three hummingbird feeders. Different succulent species bloom at different times which prolongs the season for nectar harvest for these birds. Earlier this year some of the aeoniums and echeverias opened up. At this time the aloe veras are in bloom.\nHummingbirds always return to their favorite flower.\nIf you want to attract these beautiful creatures in your yard, plant something they like. There are many plants that are known to attract hummingbirds, but one thing is true - succulent are among the easiest to grow.\nEntice the hummingbirds to stay in your garden - plant succulents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there a vector format file or a 512x512 pixel sized transparent Raydium logo available?\nThe biggest logo in size I've found here on the forum, which has a resolution of 158x149 pixel.\nLast edited by st on Wed Feb 15, 2012 6:41 pm, edited 1 time in total.\nThere is a PSD version that you can scale a bit (but some parts will show ugly).\nThanks for providing the PSD file.\nI've scaled up the logo to fit the Mac OS X icon size. I'm using the logo for the test applications.\nThe Raydium logo IconsFileStore is Leopard ready now.\nIs there a vector based Raydium logo available?\nTo give an answer to my own initial question. Yes, it is. I've uploaded a Raydium.svg file to the FTP server right now.\nMy idea was to re-create the original Raydium logo as a SVG one. My first steps using Inkscape, so don't expect too much.\nAs you can see, it isn't totally finished yet. I've re-builded all objects myself, including the \"R\" from Raydium, that's because it doesn't look exactly the same compared to the original. The black text wasn't started at all and was taken from the original.\nI want to throw this away and start from the beginning using the original PSD file, exporting the object outlines as paths. Then I want to try to import the paths into Inkscape anyhow. Doesn't know if this could work, I'll try it out in the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meira Mathison has maintained a studio, Dancerwood Pottery in Victoria, British Columbia, Canada for over 35 years. For the past 30 years she has conducted workshops throughout British Columbia, Alberta, Canada, U.S.A., England and Mexico.\nMeira was the Executive Director for the Metchosin Int'l School of the Arts for over 22 years, recently stepping down to spend more time in her studio. She still coordinates the ceramic program for the school.\nVictoria is known as \"The Garden City\". The temperate climate of mild year round temperatures and westcoast rain forest humidity provides a perfect medium for the green lush gardens, abundant with flowers. Meira grew up surrounded by flowers and a large expanded family. These three baskets reflect this influence \u2013 closeness, abundance, dance, lushness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recently the CEO of Mozilla, has sent an important email to company employees. In it, he admits he was disappointed by the presentation of low priced smartphones with Firefox OS and the company began considering changing its strategy by focusing to more expensive devices.\nOf course, not talking about competitors Android's flagship or iPhone. He said Mozilla will focus on user experience in balance with lower price and not only on the creation of the cheapest possible smartphone. Another interesting news is that the creators of Firefox consider to allow for implementation of Android apps in the mobile platform. Basic strategy remain web applications \u2013 Mozilla is one of their strongest defenders.\nhttp:\/\/szlifestyle.com\/sz\/2015\/05\/25\/mozilla-wants-more-expensive-smartphones-for-its-firefox-os\/https:\/\/i2.wp.com\/szlifestyle.com\/sz\/wp-content\/uploads\/2015\/05\/firefox-os.jpg?fit=700%2C357https:\/\/i2.wp.com\/szlifestyle.com\/sz\/wp-content\/uploads\/2015\/05\/firefox-os.jpg?resize=150%2C150 2015-05-25T09:26:03+03:00 Richard D.TECH NEWSFirefox,Firefox OS,Mozilla,smartphoneRecently the CEO of Mozilla, has sent an important email to company employees. In it, he admits he was disappointed by the presentation of low priced smartphones with Firefox OS and the company began considering changing its strategy by focusing to more expensive devices. Of course, not talking about competitors...Richard D.Richard Deendeen@szlifestyle.comContributorRichard Deen is a writer at szlifestyle. What can I say I love gadgets!szlifestyle.com Tech News, Life Style, Reviews, Videos!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While it is illegal to coerce or assist someone in committing suicide, the act of taking one's own life became legal in Canada in 1972. Many people suffering intolerably do resort to suicide, but the harm caused by suicidal thoughts, failed attempts and suicide deaths is staggering.\nMAID allows an individual a safe and painless death, one that is less traumatizing to both the patient and his\/her loved ones.\nThe Centre for Suicide Prevention published an infographic to highlight the differences between suicide and physician-assisted dying; making clear the differences and societal costs of each. Whereas \"suicide is often impulsive, violent and carried out alone\", a medically-assisted death is well planned and thought out. Suicide is devastating not just for the victim, but also the survivors and can leave a lasting \"legacy of pain\". MAID gives the patient an opportunity to consider all options and prepare his\/herself for death. It also gives the patient's family and loved ones time to say goodbye. In most places where medical assistance in dying is legal, the majority of patients die at home among loved ones.\nTo further illustrate the difference between the two, Andre Picard (Columnist, The Globe and Mail) wrote in a pre-legislation article, Suicide is an act of self-harm that is almost always a byproduct of mental illness like schizophrenia or severe depression \u2026 Calling medically assisted dying suicide is a lot like calling surgery a knife attack.\nMAID protects medical professionals who follow the guidelines from criminal charges.\nBill C-14 provides protection and guidance for healthcare professionals and patients (who meet the eligibility requirements).\nPrior to Canada's medical aid in dying law it was illegal for anyone to counsel or coerce someone to die by suicide. The current Criminal Code now includes an exemption for medical and nurse practitioners, pharmacists, and persons aiding a patient.\nIt remains illegal for anyone who is not a health care professional to counsel a person to die by suicide.\nFor more information on your rights and Medical Aid In Dying, please visit Canada's Department of Justice at justice.gc.ca.\nIf you're considering suicide or are experiencing a mental health crisis, please seek help. You are not alone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..dd6bd6e82d07c29657b61846cb931a0f5a2a7bc7
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+{"text":"A food storage container is used to hold food at room temperature, in the refrigerator, or in the freezer. There are several options available for storing food in any of these manners. Before selecting the type of container to use, several factors must be taken into consideration. Consider the type of storage (room temperature, refrigerated, or frozen), the food being stored, length of storage, and storage space. The container can be a plastic bag, plastic container, or glass container.\nPlastic Bags: Plastic bags come in many types and sizes. They are available as sealable bags and non-sealable bags. They range in size from a little snack size to a large 2 gallon size. There are plastic bags manufactured for the purpose of storing food in the freezer that are made from a heavier plastic to help protect the food from moisture loss. Plastic bags can be used to store many types of food, whether at room temperature, in the refrigerator, or in the freezer. The flexibility of the bag makes it more difficult to fill without spilling when trying to store food that contains a lot of liquid. The plastic bag may not store as neatly as a plastic or glass container but many times their flexibility allows them to be squeezed into a space that a plastic or glass container would not fit. Having a supply of plastic bags on hand also takes up little space in comparison to plastic or glass container.\nGlass Containers: Glass storage containers generally have a plastic lid for sealing the food airtight in the container. Glass containers are available in clear, frosted, and colored glass, and range in size from 1 cup to several quarts. They can be used for storing foods at room temperature, in the refrigerator, and in the freezer. The glass containers are very sturdy and do a good job of keeping the food fresh. Some glass containers can be used to bake in or microwave in also. Some brands can be taken from the refrigerator and placed in an oven immediately without causing it to crack. Before using in this manner, be sure it is heat and cold resistant. Glass will not stain, peel, or take on the taste or odor of the food stored in them. The glass containers will be more susceptible to breakage so care must be taken when using them.\nPlastic Containers: Plastic containers are also available in many sizes, shapes, and styles. They are found in different degrees of sturdiness, from light weight disposable containers to heavy duty rigid plastic container. They are manufactured with clear, frosted, and colored plastic and most have airtight lids. They work great for storing food in the refrigerator and freezer, where the airtight lids prevent moisture loss. The airtight lids keep foods, such as cookies, crackers, chips, and cereals form becoming stale for a longer period of time when stored at room temperature. Plastic containers work well for all types of storage. They store well together and can really help organize a storage area. They are available in so many shapes and sizes that it is easy to find one to fit your special needs. Foods containing a lot of liquid, store easily in this type of container. Some brands have been designed to stack neatly to assist in storing them both when not in use and when using them to store food. They are lightweight and easy to handle. The plastic containers are sturdy enough to be easily stacked on each other when storing food in the refrigerator, freezer, and in the cupboard. Because of the plastic material, breakage is not as much of a concern as it is with glass containers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This week in the Guildford Table Tennis League we look at the Fourth Division's battle for supremacy, with Cody B still unbeaten and top by some margin, but their serene progress was stunted somewhat as middle of the table Woking C took four points, with Andrew Bramwell taking three. Further, out of the four five setters Cody scraped three. Another Woking side, the B, also had a good week with a 7-3 success over Godalming N cementing a solid third place in the table-Malcolm Hollett reigned supreme denying Rob Chatwell his maximum for the beaten side. Woking D didn't fare so well losing 8-2 to an Abinger side still in the hunt for the medals-Louse Scott, who has performed magnificently this season and the wily veteran Jim Faulkner took maximums.\nThe Third Division saw an amazing result between the top two. Emanuel and Godalming J,9-1 to the Church side. Keith Brown, Stephen Schofield and Shaun Robertson proved unbeatable on the night in a major upset.\nIn the Second Division Godalming E improved their title hopes with a 7-3 win over Aftermath C opening up a seven point gap at the top. A feature of their success this year is the form of Jason Minoo with a 90 percent average, he took a maximum but was taken to five by James Sallen who for once failed to score a point. Adit Gandhi and Danny McGranaghan proved able backing for the Godalming side. Cranleigh continue to hold their own with a 6-4 victory over a good Godalming G side with pairs from Angela and Pete Coventry and Brain Amos a newcomer who has already made his mark.\nCollege C's woes continue at the foot of Division One as they succumbed 10-0 to Merrow D. Meanwhile another Merrow side eased into second place by a ten point margin, the E's taking Godalming D 7-3 although Charlie Cunningham managed a pair against Merrow's Nigel Oh, David Joyce and Ce Guo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's a national trend: 60-something retired homeowners downsizing, escaping the frustrating maintenance of a big house and choosing something compact and carefree.\nBut that's not for retired businessman Alan Kesten.\nThree years ago, he sold a spacious University Park home and did something unexpected. He bought a coveted parcel of waterfront property on Steven's Point in the established neighborhood of Indian Beach\/Sapphire Shores, near the Ringling Museum. He took down a 1960s-era house and built a 10,000-square-foot Italian villa that he says replicates features in a 17th-century Tuscan farmhouse.\nHis girlfriend, Melba Jimenez, looks around the house with each new purchase and teases him, \"Alan, I don't think this is how farmers lived back then.\" But it's how Kesten is enjoying his new 21st-century life \u2014 seeing his real estate odyssey to the finish and realizing his vision of an ideal home formed by 20 years of travel to Italy.\nThe travel was mostly for business. Kesten ran his family-owned fragrance and perfume oils company called Belmay that began small in New York and got big fast when it sold to Revlon what would become that company's first shade of red lipstick. Over the years, Kesten took the company international, setting up offices and manufacturing plants all over the world. His fast-paced career included short-term stays in exotic places as well as big, cosmopolitan cities in Europe and Asia. When he retired to Sarasota in 2000, it was from South Africa where he had been living just outside of Johannesburg.\nBut why an Italian house? \"Actually, I was looking for a great piece of waterfront property and when the parcel on Steven's Point became available in 2014, I bought it,\" Kesten said. \"I've always wanted a house on the water.\n\"Then, I considered the style of house I'd build. Over the years, I've collected photographs, drawings and paintings of Italian homes from various time periods and I wondered what it would be like to live in an Old World home.\nBut Kesten said the size and luxuriousness of the home were ultimately a business decision as much as an emotional indulgence.\nKesten was determined to have his new house look old. That was possible because John Cannon was the contractor and the home is completely custom.\nWalls have frescos and ceilings have beams that were popular in the Renaissance. Arched niches frame classical statuary. Stone and wrought-iron balconies capture natural breezes and water views.\nTo get the antique look he wanted, Kesten commissioned artist Dawn Marie DeLara, who moved from Minneapolis to a Sarasota rental so that she could work on site for five months, doing walls, ceilings, murals, frescos and tromp l'oeil treatments such as seemingly exposing old brick beneath layers of peeling plaster. The house shows its \"age\" in every room. The artist also designed a Kesten family crest that hangs above the fireplace in the great room.\nSome of DeLara's projects required her to spend long hours on a 22-foot-high scaffold.\nKesten sourced stone from Italy and Qatar but also shopped locally at Sarasota Architectural Salvage, Franklin Lighting, Sarasota Trading Co. and at antique shows at the Sarasota Municipal Auditorium. Scouting trips to local art galleries here and in Tampa and St. Petersburg were frequent.\nThe house has 5,200 square feet under air but 10,000 square feet when the covered outdoor living spaces are factored in. Nearly all the rooms are oriented toward bay views, but Kesten's favorite room in the house has no water view at all. It's the dining room with a brick vaulted ceiling. He believes this elegant and moody cave-like room is the optimal expression of the home's Mediterranean aesthetic. The stone floor, vaulted ceiling, the table (which he repurposed from his previous home) and all the accessories come together to transport guests back in time. An oil painting of a solitary tree is a reminder of Kesten's time in South Africa. The tree was on his homestead there.\nTo make sure the ambiance is absolutely right when everyone sits down to dine, Jimenez lights candles in an antique metal candelabra. She found the pair online and they arrived the day before their first big dinner party. The cook's kitchen with its wood and Brazilian soapstone counters and hammered copper sink are her domain. Her hobby is cooking for crowds.\nKesten said he's enjoying this house even more than he imagined he would and the 18 months it took to build it were an education every day in European art history and modern homebuilding.\n\"I give the credit to the Old World craftsmen who put heart and soul into their projects. Every room has been touched by artisans with a passion for what they do, and it shows,\" he said. \"And I thank project manager Mike Johns who was on site every single day and kept everything going smoothly.\nYou can view additional images of this beautiful home here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ANDERSON, S.C. \u2013 With rain expected to move into the Upstate overnight and continuing through most of Friday, this weekend's Anderson baseball nonconference series versus UNC Pembroke has been altered.\nThe Trojans and Braves will now square off in a doubleheader on Saturday, with first-pitch slated for 1 p.m. at Memorial Stadium. The two teams will then meet in a Sunday afternoon contest, which is slated for a 1 p.m. first-pitch.\nSaturday, March 16 UNC Pembroke at Anderson (DH) 1 p.m.\nSunday, March 17 UNC Pembroke at Anderson 1 p.m.\nThis afternoon's nonconference tilt versus Benedict is scheduled for a 3 p.m. start.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A leader, a healer, a lifetime friend: could you keep a donkey?\nDonkeys have been domesticated for more than 5,000 years \u2013 but that does not mean they are tame, biddable creatures... anything but.\nThey have minds of their own, a rare intelligence and a need for affection and contact that belies their biting, kicking reputation.\nImportantly, although they are members of the horse family they are very unlike horses.\nMartine Jouclas, the founder of UNAP, an association for donkey professionals, has been working with donkeys since the 80s and was one of the first to set up walking holidays with donkeys and said: \"Unlike herds of wilds horses, which are constantly on the move to find new grazing, donkeys are territorial. They tend to stay in one place and will defend their territory quite aggressively, if needed.\n\"Horses live in hierarchical groups, but donkeys depend on being in a group to survive so co-operate with each other.\n\"Training a donkey is very different from training a horse: something like the difference between training a cat and a dog.\n\"The dog will obey the pack leader, but the cat will only go along with another cat if it think it's a good idea.\n\"You cannot present yourself to a donkey as the pack leader, but only as an equal with good ideas.\nwhich can survive very well on next to nothing. In fact, overfeeding can be catastrophically bad for a donkey, leading to problems including laminitis, diabetes, heart disease and obesity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A B-Cycle kiosk was just installed in front of the John Hancock Center in Chicago. It's good to see bike sharing make its way to this side of the Atlantic. I've already seen bikes being rented from the kiosk. I think the fact that the bike can be grabbed from a kiosk is critical to the success of the program. If people were forced to go into an office, fill out paperwork, and provide credentials to get their hands on a bike, I don't think people would be as likely to grab one. The level of commitment at a kiosk is much lower and feels more like you're grabbing a ride on the go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's no secret that HR departments spend a considerable chunk of time fielding employee questions, filling out forms, and processing paperwork. Employees, in turn, grow increasingly frustrated with the process and its lack of transparency. To address those problems, HR leaders across the spectrum are turning to employee self-service portals. An employee self-service portal offers employees the opportunity to digitally perform a range of tasks the typical HR rep would normally handle, such as updating benefits information, submitting time-off requests, accessing answers to common questions, and more \u2014 all in a unified, user-friendly interface. The resulting time and cost savings, coupled with the enhanced employee experience, can virtually transform the workload of an HR department. Still, the implementation process can present plenty of challenges. Read on to learn how you can sidestep those hurdles and successfully implement employee self-service in your organization.\nA user-friendly, intuitive, and attractive service portal is the first step to an effective implementation. Working with IT and your portal provider, make sure your organization's provider offers employees the chance to easily submit routine requests and access the answers they need in a comprehensive knowledge base. A smartphone-friendly portal can also ease the transition. Don't be afraid to test out different options on the market before making your decision. You could even sample a few offerings with a team of employees and solicit their feedback. Involving employees in the decision-making process is a great way to increase buy-in of the new system.\nEmployees need to be aware of the new system and its offerings. It is critical to stress the benefits and how they impact the daily requests and needs an employee confronts on a daily basis. To that end, make sure you're publicizing the new system in as many different forums as possible, such as email, newsletters, department meetings, and town hall meetings. You'll also want to address how the system works, and respond to any questions that might arise, particularly from those less tech-savvy employees.\nTo ensure people are fully comfortable with the new self-service portal, schedule dedicated training times. Encourage employees to choose which time works for them, so it doesn't interfere with their own schedules. You can also refer them to online-based, video training modules if they prefer.\nThe key to success is offering ongoing support during the transition. Designate a team or individual to serve as the point of contact for any tech issues that come up. Especially with a digital-based platform, employees need to feel confident that humans are still at the forefront of the change.\nDuring the initial few months, you will likely see an uptick in the number of employees reaching out with queries. If the training and implementation process is comprehensive enough, this will eventually die down, and with it, you will notice a surge in HR productivity and employee engagement.\nAt Tresbu Technologies, we want to help guide you through the process of implementing an employee self-service portal. Reach out today to find out how we can boost productivity, upgrade the employee experience, and get you the best return on your investment. Contact us for a free consultation.\nLearn how to improve your employee's experience, right from the start.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Sunday Dinner is a rare one. What do I mean by that? There are no children present. Remote location accessible by pack mule only (o.k. not really, but if you have kids you get the point). Knowing that the children are engaged in fun activities far, far away, naturally one's thoughts turn to---food. And what first comes to mind is making whoopie (\"pies\" that is). Bob Eubanks' favorite dessert, the once humble whoopie pie. I say once humble because this Suzy-Q predecessor has recently become one of the \"it\" desserts amongst the hip culinary set. Don't let the hype turn you away. The reward here is worth the effort.\nThese soft chocolate cakes are flavored with Kahl\u00faa and filled with a vanilla bean mascarpone filling. I enhance the chocolate flavor of the cakes with a bit of espresso powder. The original recipe made 10 pies, but I manage to get fifteen by using a 2-ounce ice cream scoop. Meaning more to enjoy, so embrace its cakey, creamy pleasures.\n*One note- the pies are best when allowed to set out for about 30 minutes allowing the cakes to come to room temperature and the filling to retain a slight chill.\nAre you are looking for an out-of-this-world treat to serve to family and friends? Look no further because I have the recipe for you! There are so many wonderful things about these bars. We have two kinds of chocolate, peanut butter, and crisped rice cereal. When brought together, these become the Three Tenors of Yum. They are also great because they can be stored in the refrigerator for up to four days. They won't last that long, but it is nice to make them in advance. The texture is marvelous in the mouth as well. The bottom layer is a candied, crispy, and crunchy base that is slathered with a smooth and creamy chocolate-peanut butter mixture and then topped with a delicious dark chocolate glaze. These are something special.\nI first made these bars a few years ago for a boy scout troop meeting and a teacher's conference dinner and they were really good, but as usual, I needed to make a few changes to suit my taste. The original recipe made enough for an 8x8-inch pan, but these bars are so good that you'll want to share them. And that calls for a bigger pan. In doing so, I doubled the top and bottom layer, but only made 1 1\/2 of the middle peanut butter layer. I felt it was a better chocolate to peanut butter ratio. The other change I made was to brown the butter. Any chance I get to brown butter, I go for it. It adds such a nice depth of nuttiness to the bottom layer that pairs perfectly with the peanut butter and chocolate. You can certainly opt to just melt the butter like the original recipe. Any way you make them they will be good!\n*One note-- these are refrigerator bars. Therefore, they are not good candidates for sitting out on a buffet table on a warm, sunny day. However, when I took them to the boy scout meeting (inside) and the teacher's dinner (inside as well), I kept them covered and placed a large ice pack under the serving trays where they sat for an hour before being devoured.\nContinuing with some sides for those barbecue meals-- Stove-Top Baked Beans. I make pots of this during the warmer months mainly because I make mounds of coleslaw. What do I mean by that? Not sure if it's a regional thing, but down South, we made sure our food was \"touching\" on the plate. So when we take a forkful of beans, we also spear some coleslaw onto the same bite. Classic creamy coleslaw with smoky, sweet beans makes for one tasty combination.\nGrowing up, I watched my mother make her beans and it was always a can of beans, a bit of brown sugar, ketchup, some mustard, and a few slices of bacon on top. It was then baked in the oven. I usually have something in the oven taking up space, so I wanted a recipe that I could put on the stove. I also wanted a different texture to my beans, so I use a mix of canned baked beans and canned dark red kidney beans. I use my mom's addition of ketchup and brown sugar, but I like the taste of grainy Dijon mustard in lieu of the standard American ball-park mustard. Throw in some onions and saut\u00e9ed bacon and you've got yourself a perfect pot of beans.\nThe coleslaw and beans are great on their own, but if you are like me and my family and you like your food \"touching\", then please try this combo with pork, burgers, chicken, or anything else that needs two great sides.\nMemorial Day is this coming weekend and that means barbecues, picnics, and grill-outs. So, I thought I would share some of my family's favorite sides to accompany those burgers, ribs, and pulled pork sandwiches.\nI make mounds of this coleslaw throughout the warmer months. We enjoy it with ribs and chicken, as a condiment to pulled pork (that's right, on top of the pork in the sandwich), and even as a side to a simple sandwich. There are just a few ingredients, but they yield loads of flavor. The original recipe calls for onion, but I like to use the milder scallion. I also add carrot and fresh flat-leaf parsley for color and extra fresh flavor. Sometimes I even sub some sliced red cabbage for the green, but this version is the gold standard. What makes this slaw a winner is the dressing and it has only three ingredients-- Hellman's mayonnaise (no substituting), sugar, and white vinegar. Sweet and creamy-- it is so good!\nThis recipe can easily be doubled or tripled for large gatherings. One tip that I have when tossing such a large amount is to place the ingredients in a unscented garbage bag, twist the top to close, then shake and toss. Works like a charm.\nWe have had a wet Spring here so that means Sunday soccer make-up games and a Sunday dinner that can be made in advance and cook by itself-- sort of. I made the appetizer and dessert yesterday (after Saturday soccer games). The main dish simmers on the stove for several hours until it becomes fork tender. That leaves me peeling potatoes and baking a loaf of bread. Since the dessert is made in advance, I am sharing that recipe and photo today.\nFrench Creams are creamy white custards that are set with gelatin. I lightened the original recipe and swapped out the mascarpone for cr\u00e8me fra\u00eeche because the first time I made these that is what I had on hand and I loved the way it turned out. The creams are sweetened with sugar and their flavor is enhanced with the addition of a little fresh lemon juice and vanilla extract. Topped with fresh berries and a raspberry coulis, they are the perfect ending to our Sunday dinner.\nThe berries at the market have been so sweet and juicy. I realized this week that I had over bought (if that's possible) and some were looking a bit sad. Instead of freezing them for summer smoothies, I decided they would sparkle and shine in these muffins for a breakfast treat.\nWhat really makes these muffins explode with that sweet strawberry flavor is the addition of a little strawberry preserves swirled in the top of the muffin before baking. This is as trick that I learned from Cook's Illustrated a few years ago. The sour cream makes them moist but not gummy. The diced strawberries create pockets that burst with sweetness. The streusel topping adds another textural and taste dimension. The added bit of cinnamon in the streusel pairs perfectly with the strawberries.\nThe muffin recipe below is for 12 muffins, but it can easily be doubled. The streusel recipe makes enough for 4 batches of muffins, but you can easily halve it. I go ahead and make the whole batch of streusel and store the remaining portion in the refrigerator. That way when I have a muffin craving, I can pull out the pre-made streusel and proceed with the recipe.\nAfter enjoying heavy (but delicious) Sunday dinners, I usually try to serve at least one main-dish salad for dinner on a weekday. This salad is so satisfying that I wind up serving it twice in one week. The best description for this salad is Italian antipasto platter meets American Cobb salad. Served with some warm crusty bread, it is a filling meal. It has meats, cheeses, veggies and greens, and it is dressed with a positively delectable vinaigrette that you will wind up dipping your bread into-- it is that good.\nEggs-- I love them. I love them softly scrambled, fried, soft boiled, and poached. I love them with a piece of buttered toast, on a bed of stone ground grits, or even over a spoonful of leftover beans. The are the obvious choice for breakfast or brunch, but they are also my go to for a quick and light weeknight meal. This recipe is the perfect example. The eggs are set over a bed of browned hash browned potatoes and topped with dollops of creamy Boursin garlic and herb cheese.\nFrittatas are wonderfully versatile from their ingredients to when and how you serve them. We enjoyed the above picture this past weekend for brunch while it was still warm from the oven. We had other items on the menu, so there were a few slices of frittata leftover that I packed for school and work lunches to be enjoyed at room temperature.\nThe original recipe called for frozen potatoes, but I use freshly grated or if I am hosting a large crowd, I purchase the refrigerated hash browns to save time and extra dishes. I also like the taste and texture of Light \"Boursin\" in lieu of the full-fat. Served with slices or smoked salmon or prosciutto (both equally delicious, but I prefer the salmon) and pinch of cayenne for a little kick, it is a satisfying start or end to the day.\nThe conditions outside right now (50\u00ba F and drizzle-- what happened to the 90\u00ba F?!) are not very optimal for firing up the grill and smoke box, but only extreme and dangerous weather conditions would stop me from making this meal today. Why?? It is my husband's birthday and I have made ribs for him every year for at least the last decade. As much as I would like to share the rib recipe with you today, the birthday cake is the Sunday feature.\nIt is a toasted almond sponge cake that is layered with a light and creamy coconut mousse. The recipe requires two 9-inch cake pans and one 9-inch tight fitting springform pan. The cakes are sliced horizontally creating four layers, but only three are used. Leave the fourth for snacking;) I added some vanilla extract to the cake batter and some coconut extract to the mousse to bump up that coconut flavor. I also use coconut lite milk in the mousse, but you can certainly opt for the full fat.\nThis cake is the perfect finish to this stick to your \"ribs\" (pun intended) Sunday meal. It is also nice because it can be made up to a day in advance in order for the mousse to set, which gives me time to tend to the hickory logs and the birthday boy.\nMy local readers know that it has been unseasonably warm here the past few days. 90\u00ba F at O'Hare airport on Tuesday?! Time to break out the ice cream maker to cool off. With just two ingredients, this recipe couldn't be easier. It is soft, creamy, and very rich. It's definitely not one of those ice creams where you want to pile three scoops into a waffle cone. This ice cream lends itself perfectly to a moderately sized scoop complemented by a little cookie or perched atop a sugar cone to share with neighbors for an after dinner treat. Get lost in its velvety goodness.\nWe have had a wet Spring here. Good for the grass, but not good for the grill or my craving for beef between a bun. The first time I made these they were good, but they just needed something. I made a big batch, so I put the rest in the freezer for a quick weeknight meal. When it was time pull out that frozen batch, I warmed it on the stove and kept tasting it trying to figure out what was missing.\nIt turns out that the meat mixture wasn't missing anything, but the sandwich was. I like texture, not just flavor in my food so the meat sauce in a bun just wasn't doing it for me. My husband and son suggested onions and mustard. That helped it a bunch, but it still needed something. My son suggested one more thing--bread and butter pickles. I gave him a funny look. Unfortunately, I lost my taste for bread and butter pickles after eating too many at \"Happy Hour\" as a kid, but I do keep them in the house for the rest of the family. Seeing as how he was right about the onions and mustard (a little mayonnaise doesn't hurt either) I went with it. End result-- \"winner winner weeknight dinner\" and I can now enjoy bread and butter pickles again!\nThis freezes beautifully, so make a double batch. Also, make sure you use a sturdy bun to sandwich the \"sloppy\" meat sauce. I use my White Bread recipe and shape it into 6 big buns. I highly recommend you do the same.\nRecently, my mother and I went on one of our \"playdates\". These are times where we can walk for good health, talk for good measure, and shop for fun kitchen treasures. The book, Chicken and Egg by Janice Cole, where this recipe came from, is our latest treasure. Standing in the back of the store, we each had a copy in hand. Flipping through its pages, I got to page 131 and declared, \"SOLD\"! She was on the same page. I looked at her and said, \"I love you and I want you to have this\". My mother looked at me and said, \"I love you too and want you to have this\". We made our purchases, exchange our treasures, and gave each other a big hug and kiss and laughed.\nI am not able to be with my mother this Sunday for Mother's Day. To honor her, I wanted to make the recipe on page 131. It is a delicate angel food cake that has a sweetened raspberry pur\u00e9e swirled throughout. Sliced and served with a whipped cr\u00e8me fra\u00eeche-- it is angel food cake kissed by the heavens. My only changes were to add a bit of fresh lemon juice to the raspberry pur\u00e9e and, wanting to keep that \"pretty as the picture\" raspberry-swirl look, I layered the pur\u00e9e between the cake batter and dragged a long, thin knife throughout to create the swirl. I also made enough pur\u00e9e to have extra to serve on the side. These changes made for one beautiful cake that tastes as good as it looks. The cr\u00e8me fra\u00eeche might SEEM like it could be optional, but hoo-boy does it ever bring it to the next level of goodness. The only thing that would make this cake better is if I had my mom with me to enjoy a slice or two.\nEvery time I open this book I will think of you.\nThis is Mother's Day weekend and that means brunch. If you happen to be hosting one or contributing food to one, have I got the coffee cake for you! I became acquainted with this scrumptious cake many years ago through my then boyfriend, now husband's grandmother. It is a tender, buttery cake that is layered with a strawberry-rhubarb filling, topped with a little more batter, then sprinkled with a sugary streusel. It is moist, sweet, seasonally fruity, and it feeds a crowd.\nOne note on making this cake, the filling needs to be cooled completely before assembling. That said, you can make the filling the day before, refrigerate it, and bring it to room temperature before proceeding with the recipe.\nWhenever I make burritos, I am always left with one tortilla. Of course I could use it for an after school snack or a quesadilla, but when I came across this recipe inspiration, that leftover tortilla got a whole new meaning.\nI found this recipe in a Cook's Country magazine where the bottom layer or \"crust\" utilizes the lone tortilla. The original filling looked a bit bland to me, so I added onion, spinach, chiles, a few spices and some herbs. I serve it with a roasted tomatillo sauce and some Mexican cr\u00e8ma. Mmm--this is so good. It is creamy and cheesy with a little cracker-like snap when you cut through the tortilla crust.\nSo, if you have an extra burrito-sized tortilla hanging around, please make this. Otherwise, put a package of 10-inch burrito sized flour tortillas on your next grocery list and use the leftover nine tortillas for after school snacks, quesadillas, or weeknight burritos. This dish will not disappoint.\nI never used a kitchen towel to wring out thawed frozen spinach. I just did not like picking off pieces from the towel or risk turning my towel green. Instead, I pressed the spinach through the holes of a colander. One day as I was pressing, I stared at the stainless steel colander and thought of the stainless steel disk to my potato ricer (a great kitchen tool)--light bulb! I stopped what I was doing and ran to my pantry to retrieve my ricer. It worked so well that this is the method I have been using ever since.\nTo completely thaw the frozen spinach, place it on plate and let it thaw overnight in the refrigerator. Fit the potato ricer with the smallest hole disk and place it over a bowl. Put the thawed spinach in the ricer and press firmly to squeeze the liquid...and there you have it--perfectly squeezed dried spinach to use as needed.\nWhat to do with the little green puck? Stay tuned for a recipe.\nIt is finally starting to feel like Spring in these parts. My youngest daughter said to me the other day, \"Mommy, look! There are leaves on the trees.\" She said it like a child from the Deep South might say if they looked out the window and saw snow. Not only are there leaves on the trees, but I now have greenery popping up all throughout my kitchen garden. Time to utilize some of those fresh herbs in this delicious and simple appetizer.\nRadishes and butter together form a classic pairing. This recipe takes it a step further by adding fresh herbs and lemon juice for a spreadable treat. I found this recipe at www.greencitymarket.org, the online version of that wonderful farmers' market in the Lincoln Park neighborhood of Chicago. I love spending a few Wednesday mornings down there during the season, but when I can't get there, I visit their website looking for recipes and inspiration. This is a recipe that I found years ago. I hope you like it as much as I do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Collector CNC's 1\" strap guide for whitewater rafts fits any 1-5\/8\" tube. The guide allows for NRS cooler hangers to be used up front with the NRS fishing frame. Save money with our strap guide and avoid dishing out your hard earned cash for another straight cross bar for you raft frame.\nVersion 3 has the Guide with no CollectorCNC logo. The final version file has our CollectorCNC logo.\nI used PLA and printed layers at .200mm high. 1mm thick walls 4 passes and 16% infill. You will be required to purchase your own screws for this application. We used 9\/64 course thread hex screws. These are slightly oversized so a nut is not required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thirteen of the 37 workers suspended in the patient-abuse scandal at Whiting Forensic Hospital have either been fired or have resigned or retired. Above, Al Shehadi testifies in front of the legislature about the abuse his brother, William Shehadi, suffered.\nWilliam Shehadi, the psychiatric patient who was the target of protracted abuse at Whiting Forensic Hospital, was publicly identified for the first time since the scandal broke in powerful testimony before a legislative committee Monday by his brother, who tore into management and said the abuse began at least a decade ago.\nAl Shehadi had refrained from public comment as concerns about Whiting Forensic arose through the spring and summer, with the arrests of 10 state nurses and treatment workers on charges of cruelty. The 10 were among a total of 37 workers suspended in the scandal, and up to 48 workers who were identified as having some role. The maltreatment was captured on surveillance tape from the camera in Bill Shehadi's room in unit six at maximum security Whiting Forensic.\nMonday afternoon, he faced the committee and held the lawmakers and the audience in the hearing room spellbound, methodically, but in a voice underpinned by pain and grief, describing his personal connection to what is being portrayed more and more as a culture of abuse at Whiting Forensic.\nShehadi said he decided to testify to put a face on his brother \u2014 acquitted of manslaughter by reason of mental disease or defect in the death of their father 22 years ago \u2014 and to cast light on the administration at Whiting.\nThe allegations, the video tape and Shehadi's testimony provide a glimpse into the closed world of Whiting and the wider Connecticut Valley Hospital campus in Middletown.\nAl Shehadi said he girded himself before watching up to five hours of the soundless, raw video footage, but said he was still unprepared for what he saw. The surveillance videos, viewed by state police detectives, showed a male nurse gyrating his groin on the patient's face, workers kicking him, throwing him out of bed in the pre-dawn hours, dousing him with liquid, pulling the sheet over his head, blasting him in the face with an aerosol can.\nThe abuse also included workers mopping Bill Shehadi's head and placing a diaper over his head. And there was hours of taunting, with seated workers propping their feet on the edge of Bill Shehadi's bed, as if poised to kick him again at any moment, and circling him menacingly, as if they were going to strike again. At times, Bill Shehadi cowered on his bed or on the floor.\n\"The sheer scale of the abuse is incomprehensible,\" Al Shehadi said, noting that he did a spreadsheet after reading the arrest affidavits from state police detectives and counted at least 50 incidents of abuse over 24 days.\nBill Shehadi's 10-year, court-ordered commitment to Whiting Forensic ended in 2005. His confinement, however, has been extended because of his mental illness, including delusions, compulsiveness and a range of autism-like symptoms.\n\"When he was younger, he was quite smart and, in his own way, engaging,\" Al Shehadi said.\nHe descended into depression at age 21. More than 35 years later, he has been institutionalized for most of his adult life, his bother said.\nHe acknowledged that his brother can be a difficult patient, \"poor at reading social cues and prone to inflexible, ritualistic behavior such as walking around a room, repeatedly tapping each of the four walls or flipping a light switch on and off, over and over again.\n\"He is not easy on himself or those around him, whether family, other patients or staff,\" said Al Shehadi.\nBut, he said, none of that excuses what he said has been a hellish experience for his brother.\nIn the earlier years of his confinement, Bill Shehadi \"was held in physical restraint for the better part of three years, largely tied to a bed,\" Al Shehadi said.\nTen years ago, the U.S. Department of Justice eviscerated Connecticut Valley Hospital in an investigation in 2007 related to patient suicides and physical conditions.\nAl Shehadi said it defies explanation that, given the regulatory record, an abuse scandal was allowed to fester a decade later.\nAt Whiting Forensic, $1,500 is spent per day, or $567,000 annually, to treat each patient, and workers with $60,000 base salaries routinely earn six-figure incomes through overtime.\nSen. Heather Somers, a Republican of Groton and co-chairwoman of the public health committee, said the abuse has shattered public confidence in the system.\nUnder earlier questioning from the committee on Monday, Commissioner Delphin-Rittmon acknowledged that some of the arrested workers may have used \"fear and intimidation\" to keep the abuse quiet.\nDelphin-Rittmon told the legislature's public health committee that the abuse was contained to a \"pocket\" of workers at Whiting.\nBut Rep. Jonathan Steinberg, D-Westport, said he was \"not sure everyone here would agree\" about abusive behavior not being pervasive at maximum security Whiting, which treats patients who have been acquitted of crimes by reason of insanity.\nCameras at Whiting have a constant feed to a nursing station \u2014 but mental health officials acknowledged that the feeds were not routinely monitored. The protracted, unprovoked abuse of Bill Shehadi was captured on videotape, but it took a whistleblower to get officials to look at the tape.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our exclusive textured wall panelling at Winchester Kitchens, brings natural textures into your home, adding character and interest. They also offer a range of contemporary designs which will create a wow factor to any space in your home. Samples are available and the wealth of finishes means there is something to suit all tastes.\nTake a look at the gallery to get inspired and choose the wall panels that fit your lifestyle and compliment your home choices. Design led, take a blank canvas and make it truly beautiful. This exciting innovation adds drama and feature to all areas of your home.\nCome to our Winchester showroom and take a look at our recently installed display featuring our unique wall panelling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nutrition, health, and safety are important topics to discuss with your children. You'll naturally discuss many facts about these subjects as you go about your daily life. However, spending some time focusing specifically on each can help children understand why their eating habits, hygiene, and exercise are vital to their overall well-being.\nAdditionally, talking about safety topics with your children helps to ensure that they know what to do in the event of an emergency or natural disaster.\nWorksheets and coloring pages can make discussing these topics more engaging and easier to understand for young children. Use some of these free printable collections to guide or enhance your study of nutrition, health, and safety.\nProper nutrition is an important part of a healthy lifestyle. According to the United States Department of Agriculture (USDA), people should consume foods from the fruit, vegetable, grains, protein, and dairy groups every day for optimal health.\nThe USDA suggests eating a variety of foods and limiting those with added sugar, sodium, and higher levels of saturated fats.\nThey may not be a student's favorite topic, but fun printable worksheets about vegetables, which introduce children to a wide variety of veggies, can make learning better eating habits a bit more fun. So can following the USDA's recommendation to vary the way you eat vegetables. They suggest trying them raw, cooked, fresh, frozen, or canned. Roasting veggies in the oven or on the grill is a tasty treat, too!\nAccording to the American Dental Association (ADA), \"cavities remain the most prevalent chronic disease of childhood.\" Because they are so common, cavities may not seem like a big deal, but oral health is an important part of overall physical health.\nPoor oral health can increase a person's risk for health problems such as cardiovascular disease, respiratory infections, and certain types of cancer.\nUse a fun set of dental health printables to introduce your children to the basics of good oral hygiene. Some of the simplest ways to ensure good oral health include brushing your teeth at least twice a day, flossing, eating a healthy diet, and visiting your dentist regularly.\nPhysical Education is vital to a student's understanding of the benefits of an active lifestyle. A good PE program will teach kids about health, physical fitness, and the importance of regular physical activity.\nOne option for teaching PE is an online physical education course. Other options may include combining a personal health course with individual or team athletics to ensure that students remain active.\nIndividual sports may include golf, gymnastics, skateboarding, or swimming. Other sports such as tennis, badminton, and volleyball can also be played with only one or two players on each team.\nKids may also enjoy getting active with team sports such as baseball, softball, basketball, or hockey.\nIt can be frightening to think about emergencies and natural disasters, but knowing what to do in the event of such a situation can save lives.\nAccording to the American Red Cross, \"children under the age of five are twice as likely as other people to die in a house fire.\" It is important to teach children fire precautions as well as what to do in the event of a fire.\nCombine fire prevention worksheets that introduce terms such as fire drill and escape route with other tools to teach children potentially life-saving fire safety tips.\nThese tips should include \"stop, drop, and roll\" if a child's clothing catches on fire and where to go in the event of a fire. Have an escape plan in place and practice it at least twice a year.\nTeach your children what your home's fire alarms sound like, how to call 911, and the importance of going to fire fighters and getting and staying out of the house if there is a fire.\nIt's also important to teach your children what to do in the event of a natural disaster based on what is most likely in your area of the country. Your children may need to know what to do in the event of a hurricane, tornado, or earthquake.\nFor example, you might use a free set of earthquake worksheets to learn more about where earthquakes typically happen, what causes them and what safety steps to take if an earthquake strikes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WASHINGTON, July 24, 2015 \u2013 Iraqi security forces are making progress on multiple axes in their battle to retake the city of Ramadi from Islamic State of Iraq and the Levant control, a U.S. Central Command spokesman said today.\nSpeaking to Pentagon reporters via teleconference, Air Force Col. Patrick Ryder provided an overall update in operations against ISIL in the combatant command's area of responsibility.\n\"[Iraqi forces are] in the process of isolating Ramadi to conduct a deliberate clearing operation to retake the provincial capital city,\" he said.\nISIL is trying to hold onto Ramadi by using vehicle bombs, suicide bombers and other tactics designed to slow Iraqi forces' movement, Ryder said. \"They're continuing to rely on propaganda to mitigate their losses and overstate operational performance,\" he added.\nWhile it's still early in what's expected to be a long fight against ISIL extremists, the colonel said, Centcom officials are encouraged by the well-developed and comprehensive plan the Iraqis have put together and executed.\n\"They continue to liberate their territory, and coalition forces will continue its support to degrade and ultimately defeat ISIL,\" Ryder said.\nIraq's recent purchase of U.S. F-16 fighters has assisted in the fight against ISIL, said the Centcom spokesman said. Sharing a platform that many NATO allies also have enhances interoperability gives Iraqi forces the ability to conduct operations with coalition forces, he added.\nCentcom has assessed that the majority of the town of Beiji is under Iraqi control, although its oil refinery remains contested, the colonel said. Further north, he told reporters, there has been little change as Iraq's Kurdish forces hold their defensive lines from Sinjar to the north side of Mosul and down to the north of Kirkuk against ISIL's \"harassing attacks,\" most of which have been between Sinjar and Mosul.\nCoalition aircraft have conducted 16 \"dynamic airstrikes\" since July 12 in the area around Mosul, targeting enemy personnel, weapons and equipment, he added.\nAcross Iraq and Syria in the fight against ISIL, coalition forces have continued to degrade the extremist organization's capabilities, Ryder said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finding out that you're pregnant might be the happiest moment of your life, but the following 9 months are no walk in the park. While you're entitled to plenty of pampering and perks, there's also those annoying pregnancy symptoms that you must deal with. Dizziness and vertigo, where you feel like the room is spinning even when everything's stationary, are amongst the most troublesome symptoms. Although, it usually passes within seconds, there are times when it can last much longer.\nAlthough the onset of vertigo during pregnancy can come as a nasty shock, it's usually quite easy to explain. In most cases, the symptom surfaces because of temporary changes in the flow of blood to your brain. This may sound scary, but it's common, as a large amount of blood is now being diverted to your baby. Pregnancy hormones cause dilation of the blood vessels, easing blood flow to the fetus, but this makes it harder for blood to reach the brain.\nPregnancy anaemia can also cause vertigo to surface during pregnancy. In rare cases, vertigo may result from Benign Positional Vertigo (BPV), which is not related to pregnancy. In such cases, the symptom would surface only in specific body positions and you may also notice changes in hearing. In case of BPV, you should contact your health care provider.\nFirst off, when you experience vertigo you should avoid all tasks and activities that could put you at risk of injury. This would include driving, exercising, or operating any machinery.\nWhen you do experience vertigo or dizziness, lie down immediately to avoid the risk of passing out and falling. When lying down, elevate your feet to ease blood flow to the brain. If you can't lie down, sit down and bend forward as much as you can. Obviously, listen to your body and only bend forward as far as is possible without causing any discomfort. Lying or sitting in such postures will ease blood flow to the brain, providing quick relief. There's not much else that you can do for vertigo relief, but you can also try a few home remedies.\nSome studies suggest that ginger can help relieve vertigo symptoms quite effectively. You can consume fresh ginger juice, use ginger in your food, or prepare ginger tea. Ginger may also work as an antiemetic, helping prevent nausea and vomiting. While ginger is generally regarded as safe, you should exercise caution during pregnancy. There are concerns that high doses can increase the risk of complications or miscarriage, so don't exceed 1500 mg.\nAccording to a study published in the British Journal of General Practice, exercise therapy may have a role to play in the management of vertigo. Considering such findings, it may help to take up pregnancy aerobics, yoga, or Tai chi. In addition to improving flexibility and balance, these activities also help to lower stress levels, reducing the risk of vertigo.\nGingko biloba is also regarded as an effective vertigo remedy but is best avoided during pregnancy. There is inadequate research on its safety profile during pregnancy and it could increase the risk of bleeding.\nThe best way to deal with vertigo would of course be to make changes to your diet and lifestyle. Make it a point to eat frequently and stay nourished, while increasing fluid intake to avoid dehydration. Get as much fresh air as you can, avoid getting out of bed or a chair quickly, and switch to sleeping on your side as the pregnancy progresses. If your vertigo is severe and you can get no relief, contact your health care provider immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm fairly sure it's a faux pas to compare a novel and a television show, but I mean it as a compliment to both when I say that Thomas Mallon's new novel, \"Watergate,\" bears a certain resemblance to \"The West Wing.\" Like that much-loved NBC drama, \"Watergate\" shifts among various men and women \u2014 mostly men \u2014 working inside and outside the White House. Even when the action becomes convoluted, we're propelled forward and kept highly entertained by the colorful characters, the delicious insider details, the intelligence of the dialogue.\nWhere \"The West Wing\" and \"Watergate\" diverge, at least most obviously, is that one is about a fictitious, idealized Democratic president and his staff while the other features fictional depictions of real, corrupt Republicans. This difference is less pronounced than you might imagine, however, largely because of Mallon's evenhandedness. He's not out to lampoon Richard Nixon or anyone else. Nor is he out to redeem the Nixon administration, which would have been just as tedious. In fact, Mallon avoids rendering Watergate in the familiar and expected ways: there are only fleeting references to Woodward and Bernstein, and the eventual profusion of indictments and imprisonments aren't major plot points.\nWhat Mallon captures particularly well is the fundamental weirdness and mystery at the center of the scandal. Who was trying to achieve what with those break-ins? And why? Given how ineptly they were carried out, could the sloppiness have been intentional \u2014 either as a result of double agentry or as individual self-\u00adsabotage? In these pages, even those closest to the events remain bewildered by their smallness \u2014 their ridiculousness, even \u2014 and their contrastingly outsize and ruinous consequences.\nIt appears that Mallon's primary goal, one he achieves with great finesse, is to make the portrayals of his characters as believable as possible. Like the rest of us, they aren't simply moral or immoral but are both clever and defensive, selfish and self-pitying, sweet and loyal, generous and venal. Also, there are quite a lot of them.\nAlso included in the mix is Alice Roosevelt Longworth, daughter of Theodore Roosevelt, widow of the former House speaker Nicholas Longworth and famed deliverer of bons mots. At 90, Mrs. L. remains \"a creature of motiveless mischief\" who steals every scene she's in. She demands that the White House schedule Christmas parties around her own calendar, performs bucktoothed impersonations of her cousin Eleanor, rides the dumbwaiter in her house (or claims to) because there's no elevator and stays up all night reading, then uses the bone from a veal chop as a bookmark.\nA \"misanthrope in a flesh-presser's profession,\" this Nixon is awkward rather than evil. He's chivalrous with elderly Mrs. Longworth, forgiving of subordinates' mistakes and entirely human in poignant ways: fastidious about having the White House barber \"clip a little tuft of chest hair emerging above his collar,\" irritated by the fact that the edited transcripts of the White House tapes make him sound as if he drops hard-core obscenities rather than mild ones.\nAnd yet it's the very fact that Mallon portrays Nixon and others so convincingly that raises questions about the fairness of depicting real people in a work of fiction. Is this type of literary borrowing less transgressive when it makes readers like the subjects better? When the subjects are dead? If so, for how long? Ten years? A hundred? Obviously, there's no consensus when it comes to any of this, but I do know that if you write a novel about, say, Catherine the Great, you probably won't be scolded for misrepresenting her or otherwise infringing on her privacy, while if you write a novel inspired by Laura Bush, as I did in 2008, you most definitely will.\nIn my case, I changed names, which Mallon has chosen not to do. And I made peace with the intrusive nature of what I was doing by telling myself that to sincerely imagine what the world looks like from someone else's perspective is an act of compassion. The counterargument, of course, is that even the most savagely mocking skit on \"Saturday Night Live\" is less insidious than the sustained realism of a novel. \"The reason it's such a violation,\" a journalist told me about my own book, \"is that every single thing in it is plausible.\" Judged by the same standard, Thomas Mallon is \u2014 appropriately enough, for a book about Watergate \u2014 equally guilty.\n432 pp. Pantheon Books. $26.95.\nCurtis Sittenfeld's fourth novel, \"Earthquake Season,\" will be published next year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"While planes are probably the most efficient way of traveling great distances, they're not exactly the most eco-friendly modes of transportation either. However both Boeing and NASA are hoping to change that with the testing of the Boeing ecoDemonstrator (a 757) where the plane is expected to go on a series of flights.\nWhat makes these flights so significant is that the plane will be testing out NASA's experimental fuel-saving techniques. One of those techniques is called the Active Flow Control Enhanced Vertical Tail Flight Experiment which involves the installation of 31 tiny jets on the plane's vertical tail. This is said to help manipulate the airflow over the tail's surface which should help stabilize the plane during takeoff and landing.\nThis could also potentially lead to planes using smaller tails which means less overall weight and lower fuel consumption. The other fuel-saving technique that will be tested is having the plane covered in an insect-repellent coating. This is to keep bugs off the body of a plane which apparently adds to the drag. It has been estimated that fuel consumption could be reduced by as much as 6% if airflow over the plane's body remains smooth.\nNASA claims that these techniques have been replicated successfully in their labs so it is time to see if their theory will hold up in real life.\nFiled in Green >Transportation. Read more about Boeing and NASA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On average, we spend around 37.1 hours working each week. Of those hours, around 35 of them are spend at our desk, sitting down in one position. Plus, a disappointing 33% of us don't even go outside the workplace at breaks and even only take 27 minutes of a 40-minute lunch break. As if that wasn't worrying enough, around 80% of the UK population fall short of reaching key health national government targets.\nDiscovery Learning has created the following infographic which looks at how one can workout in the office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Year-round beautiful weather is one of the perks of living in Queensland, and no-where do we love entertaining more than on the big back deck.\nIn many homes, the deck, veranda or patio has become a natural extension of the living room - ideal for outdoor twilight dining or long summer lunches, and having a ceiling fan in the space adds a level of comfort and ambience to the area.\nChoosing an outdoor fan is different to choosing an indoor fan, though. Here are our tips for selecting the perfect ceiling fan for your entertainment area.\nAs there's a chance the fan will be exposed to rainwater, look for indoor\/outdoor fans, or marine grade fans. Indoor\/outdoor fans are for spaces where the fan will be completely undercover, while if your entertainment area will be exposed to light rain or if your house is in a coastal area, opt for a marine grade fan to avoid too much water or rust from the salty air.\nAll-weather-resistant materials are a must; plastic blades and quality stainless steel blades designed for the outdoors are recommended. Timber blades create a stylish and contemporary look, however they require more upkeep \u2013 think regular cleaning and replacing the blades more often.\nWhat sized space are you buying for? To give a comparison, a regular indoor ceiling fan in a lounge area is generally 52-inch, or 132cm, in diameter. There are smaller and larger fans available and many people select a larger size, as an outdoor entertainment space can be open and airier than an indoor room. Consider the airflow reach you're trying to achieve, and if the space is quite expansive or if it has more than one section, buying two smaller fans may be the most effective option.\nFrom simple white fans, to modern matte black, to tropical-style timber designs, there's a huge range to choose from at Lighting Illusions to suit any deck, veranda or patio area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PLUS, new releases for Apache Tomcat and JRuby!\nThe first official release for Eclipse Gemini Blueprint is now available, completing the migration of Spring DM to the Eclipse Foundation. In addition to extending the documentation, this release fixes some bugs reported since the 1.0.0 RC1 release. Version 1.0.0 can be downloaded now.\nEclipse Gemini Blueprint is the reference implementation for the OSGi Alliance Blueprint Service. This project is aimed at developers building Java applications that run in an OSGi framework, and provides the ability to dynamically add, remove and update modules in a running system, and functionality for deploying multiple versions of a module simultaneously.\nThe Apache Tomcat team have announced a new security and bug fix release, version 6.0.33 stable.\nApache Tomcat 6.0 includes new features over version 5.5, such as support for the JSP 2.1 and Servlet 2.5 specifications, improvements in memory usage, and a refactored clustering implementation. An option to disable file rotation in JULI FileHandler has been added, and JAAS authentication support has been added to JMXRemoteLifecycleListener. The Apache Tomcat team advise that everyone using older versions of the Tomcat 6.0 series, should upgrade to version 6.0.33.\nMore information is available at the Changelog.\nJRuby 1.6.4 has been released. The 1.6x series aims to fix reported incompatibilities with the 1.9 support, and this release can be built with 1.9 mode by default. In addition to some 1.9 language and core class fixes, this update adds a native MacOS installer. JRuby 1.6.4 can be downloaded now.\nOracle have announced a new release of their server virtualisation and management solution, Oracle VM 3.0. This release comes with a new policy-based management, including Distributed Resource Scheduling for capacity management, which provides real-time monitoring of Oracle's VM Server utilisation. Distributed Power Management has also been introduced, which should reduce the number of powered-on servers in the pool, during periods of low resource utilisation. The Xen 4.0 hypervisor has been updated, and the Dom0 command and control kernel have been updated with the latest drivers.\nMore information on this latest release, is available at the What's New (PDF) Oracle Datasheet.\nWANdisco, the company behind uberSVN and uberApps, have announced the release of uberSVN for Mac OS X, which also includes the latest release OS X Lion. This latest release works with Mac OS X 10.5.x, 10.6.x and Apple's Lion platform, and is available for free download from the uberSVN website.\nuberSVN is an ALM platform for Subversion, which features an integrated social coding environment, and a web interface for creating new repositories, and managing users and permissions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The story of the Mantis starts in 2017. Our product design team had a vision for an easier camp set-up and a better night's sleep off the ground. Too many cold, wet set-ups with clumsy tent poles and sleepless nights on the ground called for an upgrade to the camp experience.\nEnter the Mantis. Conceptualized to be an intuitive and lightweight all-in-one hammock, the Mantis has the highest weight-to-strength ratio of any all-in-one hammock system. For the Mantis design, Kammok stepped into the ultralight world with our Levitas\u2122 20D nylon ripstop fabric and ultralight Python Straps made with SpiraLine\u2122. Countless iterations and product testing in unforgiving weather, we finally said \"yes\" and cracked open a cold one to celebrate.\nWe're grateful to debut the Mantis this season, and invite you into our community-driven design. To bring you even more into the fold, we're answering some of the core questions that led to our final design of the Mantis. Enjoy!\nHow do we elevate camp?\nMake it easy. Make it long-lasting. Make you forget the gear in your pack.\nAs backpackers and campers, high performing gear is crucial to help you get from point A to point B. Whether it's to the northern terminus or to the end of a long weekend, gear needs to hold up and perform when it matters. For the Kammok team and our campers, the trademarks of quality gear are: ease of use, durability, and weight. The Mantis all-in-one hammock tent meets these requirements and more to elevate the backcountry experience.\nUnlike a ground tent, the Mantis offers a 60-second setup. Its abrasion-resistant fabrics, stronger-than-steel straps and lifetime warranty guarantee countless nights under the stars. More than that, it's so lightweight, we hope you forget it's in your pack.\nHow do we make set-up easier?\nTake it from minutes to seconds.\nThe Mantis was designed to be as easy as hanging your Roo hammock - no knots or poles necessary. An attached, waterproof stuff sack means it packs downs as easily as it unfurls. Separate compartments in the stuff sack keep your rainfly from dampening the rest of your gear.\nAn integrated ridgeline, bug net and canopy closures allow you to hang bug free and even more flat than a typical night in your hammock. No extra set-up required.\nHow do we make gear that's longer lasting?\nAccept no less than the best, or design it yourself.\nThe Mantis and Mantis Ultralight feature the highest performing and most technical hammock fabrics to ensure not only a comfortable night's sleep, but countless ones to come. Both the Levitas\u2122 and Gravitas\u2122 are abrasion and tear resistant ripstop nylon fabrics with weight capacities that outmatch the competition.\nTo simultaneously cut weight and maintain long-lasting strength in our Mantis Ultralight, we innovated new Python Ultralight straps from Spiraline\u2122 - a fabric 15 times stronger than steel. The bulk stops here.\nHow do we make gear less bulky?\nDrop it (the ounces) like it's hot.\nIntentional design means the Mantis cuts the bulk - no part unnecessary or inflexible. From a removable bug net to an integrated rainfly, each component of the Mantis has a purpose and adapts to your needs. Zip the bug net off when you can, ditch the rainfly on sunny days, and slide off the stuff sack to go ultralight. Stakes and accessories are optional.\nWith the Mantis, we made sure there's only weight where it matters.\nFinally, how do we make camp more fun?\nMake it accessible for everyone.\nDesign a product to impress the gear junkies and simplify camp for the beginners, because at the end of the day, we all have more fun when we're together.","meta":{"redpajama_set_name":"RedPajamaC4"}}